0. Then, (ρ c , z c ) is found to satisfy\n\nθ(ρ c , z c ) = -ϕ/2, (ρ c -ρ 0 ) 2 + z 2 c = r 0 , ( 33\n\n)\n\ni.e., the trap center lies on the surface of the \"resonance toroid\" at ρ = ρ 0 with a radius r 0 as shown in Fig. 1 . The relative angle of the trap center with respect to the center of the toroid cross-section is given by -ϕ/2. On this \"resonance toroid,\" the rf-field is resonant with the static field, i.e., B eff z vanishes. As a result, the \"effective B-field\" lies again in the x-y plane on the \"resonance toroid,\" which gives cos β eff (ρ c , z c ) = 0 and leads to the result A (φ) n (ρ c , z c ) = γ n = 0 as shown in the trap center for the storage ring considered before in Ref. [18] .\n\nFrom the expression (29) of the static field and the definition of the angle θ(ρ, z), we find a simple relationship β s (ρ, z) = π/2 + θ(ρ, z), with which the gauge potential A (φ) n (ρ, φ) in (28) can be further simplified as\n\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z) sin θ(ρ, z) 7 ≈ n ρ cos β eff (ρ, z) sin θ(ρ c , z c ), ( 34\n\n)\n\nnear the trap center. Thus, the spatial fluctuation of the gauge potential A (φ) n (ρ, z) in the region around the trap center is closely related to the angle θ(ρ c , z c ) of the trap center, or the parameter ϕ of the oscillating field B o . When ϕ = 0, the atom is trapped in the region with θ ≈ 0 or π, where the fluctuation of A (φ) n (ρ, z) is suppressed significantly due to the small value of sin θ. On the other hand, if the angle ϕ is set to π with the trap center located in the region with θ ≈ ±π/2, the fluctuation of the gauge potential becomes amplified.\n\nIn Fig. 2 , we illustrate numerical results for the distribution of the geometric phase γ 1 (ρ, z) = 2πρA φ 1 (ρ, z) in the region near the trap center at ϕ = 0, π/2, π. We see clearly decreased fluctuations of γ 1 when the absolute value of sin θ(ρ c , z c ) = -sin(ϕ/2) is decreased.\n\nNext we turn to the storage ring of Ref. [19] constructed from a quadrupole static B-field B s ( r) = B ′ (x, y, -2z) and an r-independent rf field B o = B rf cos(ωt)ê z along the z direction. The resulting ARFP provides a 2D ring shaped trap in the x-y plane. In addition, a 1D optical potential along the z direction is employed to confine atoms in the transverse plane at z = 0 [19]. The \"effective B-field\" takes the form\n\nB eff ( r) = B ′ (ρ -ρ 0 )ê z - 1 2 B rf êρ , ( 35\n\n)\n\nin the plane at z = 0, with\n\nρ 0 = ω/|µ B g F B ′ |.\n\nBecause the strength of B eff is near minimum at the ring ρ = ρ 0 , the trap center for this storage ring is located at ρ c = ρ 0 and z c = 0. At the trap center, the \"effective Bfield\" is along the direction of êρ . Thus, according to Eq. ( 28 ), the geometric phase γ (c) n at the trap center again vanishes.\n\nIn Fig. 3 , we show the distribution of the geometric phase γ 1 in the region near the trap center for B rf = 0.05|B ′ |ρ 0 and B rf = 0.15|B ′ |ρ 0 . We see that the fluctuation is relatively small when the strength of the rf-field is large. This can be explained by Eq. (28) , which shows that A (φ) n is proportional to cos β eff and can be approximated as 2B eff z /B rf near the trap center. When B rf is large, the gauge potential becomes a relatively slow varying function of ρ and z. In this case, the presence of a 1D optical potential allows for the possibility of tuning the trap center position to a nonzero value of z, with the storage ring remaining in the x-y plane. Then cos β s is assumed to a nonzero value, leading to increased fluctuations for the geometric phase.\n\nFinally, we discuss the geometric phase in the \"time averaged\" ARFP storage ring proposed in Ref. [20] . Unlike previously considered ARFP based storage rings, the time dependence now exists in both the \"static B-field\" and the frequency of the rf field given by Color online) The distribution of the geometric phase γ1 near the trap center (ρc, zc) of the storage ring proposed in Ref. [18] at (a) ϕ = 0, (b) ϕ = π/2, and (c) ϕ = π, clearly displaying the sin(ϕ/2) dependence.\n\nB s ( r, t) = B ′ ρê ρ -2B ′ zê z + B m sin(ω m t)ê z , -0.1 0 0.1 -0.1 0 0.1 -1 0 1 (z-z c )/ρ 0 γ 1 (a) (ρ-ρ c )/ρ 0 -0.1 0 0.1 -0.1 0 0.1 -5 0 5 (z-z c )/ρ 0 (ρ-ρ c )/ρ 0 γ 1 (b) -0.1 0 0.1 -0.1 0 0.1 -10 0 10 (z-z c )/ρ 0 γ 1 (c) (ρ-ρ c )/ρ 0 FIG. 2: (\n\nB o (t) = B rf sin[ω(t)t]ê z , ω(t) = ω 0 1 + (B m /B ′ ρ 0 ) 2 sin 2 (ω m t) . (36)\n\nThe frequency ω m is assumed to be much smaller than ω 0 but much larger than the trap frequency. The radius ρ 0 is now defined as\n\nρ 0 = ω 0 /|µ B g F B ′ |,\n\nand the \"effective 8 -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 (a) -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 (b) z/ρ 0 (ρ-ρ 0 )/ρ 0 γ 1 FIG. 3: (Color online) The geometric phase γ1 for the storage ring of Ref. [19] with (a) B rf = 0.15B ′ ρ0 and (b) B rf = 0.05B ′ ρ0.\n\nB-field\" takes the form\n\nB eff ( r, t) = ∆( r, t)ê z - B ′ ρ |2 B s ( r, t)| B rf êφ . ( 37\n\n)\n\nThe operating principle for the time averaged storage ring of Ref. [20] is similar to the well-known TOP [2] and TORT traps [3, 4] . The effective trap potential experienced by the atom is proportional to the time averaged value of the \"effective B-field\"\n\n2π/ωm 0 | B eff ( r, t)|dt.\n\nWhen B rf and B m are much smaller than B ′ ρ 0 , the center of the storage ring is located approximately at ρ c = ρ 0 , z c = 0. Using the earlier result [33] , we find that in the time averaged storage ring, the effective gauge potential Ã(φ) n (ρ, z) is reduced simply to the time averaged instantaneous gauge potential 28) . The geometric phase then is given approximately by γ n (ρ, z) = 2π Ã(φ) n (ρ, z). In this case, we find that the geometric phase always vanishes at -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 z/ρ 0 (ρ-ρ 0 )/ρ 0 (a) γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 Color online) The geometric phase γ1 for the storage ring of Ref. [20] at (a) B rf = 0.3B ′ ρ0 and (b) B rf = 0.1B ′ ρ0. Bm = 0.05B ′ ρ0.\n\nÃ(φ) n (ρ, z) = ω m 2π 2π/ωm 0 A (φ) n (ρ, z, t)dt, ( 38\n\n) with A (φ) n (ρ, z, t) given in (\n\n(b) (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 FIG. 4: (\n\nthe trap center (ρ c , z c ). Figure 4 illustrates the distribution of the geometric phase in the region near the trap center for two different values of the rf-field amplitude B rf . Similar to the storage ring of Ref. [19] , the fluctuation of the geometric phase is suppressed in this case for large B rf .\n\nB. The storage ring proposals of Refs. [21, 22] Next we consider the ARFP based storage ring proposed in Refs. [21, 22] . In this case, the static B-field is that of a Ioffe-Pritchard trap on an atom chip. In the Cartesian coordinate (x, y, z), it takes the form\n\nB s = B ′ xê x -B ′ yê y + B ′ Lê z , ( 39\n\n)\n\nwhere B ′ is the B-field gradient and the bias field along the z-direction is denoted as B ′ L. The amplitudes B\n\n(a) rf and B (b) rf (z) of the rf field are B\n\n(a) rf = [B rf (z)/ √ 2]ê x and B (b) rf = [B rf (z)/ √ 2]ê y with B rf (z) = B ( 0\n\n) rf + B ′′ z 2 . ( 40\n\n)\n\n9 In the schemes of Ref. [21, 22] considered earlier, the phase η of the rf field is assumed to be κπ/2. The xand y-components of the \"effective B-field\" B eff ( r) then become\n\nB eff x ( r) = B rf (z) 2 √ 2 (1 + cos β s (ρ, z)), B eff y ( r) = 0, ( 41\n\n)\n\naccording to Eq. (10). Then the strength of the \"effective B-field\" B eff has its minimum along a circle with a nonzero radius ρ c , provided a positive detuning ∆ exists at the origin (0, 0, 0) [21, 22] . The \"effective B-field\" B eff is easily shown to lie in the x-z plane along the trap bottom mapped out by the atomic center of mass motion. This gives rise to a vanishing γ (I) F . With a proper choice for the local phase of |n( r) eff , the gauge potential A (φ) n takes the form\n\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z) (1 -cos β s (ρ, z)) . ( 42\n\n)\n\nFigure 5 displays the geometric phase along a closed path for a spin-1 atom as a function of ρ c for the ARFP storage ring proposed in Refs. [21, 22] . The parameter λ is defined as\n\nλ = √ 2 ∆[ r = 0] |g F µ B B (0) rf | . ( 43\n\n)\n\nTo assure the validity of the rotating wave approximation, we find that the maximal values of ∆[ r = 0]/(|g F |µ B ) and B (0) rf / √ 2 must be restricted to the region of λ ∈ [0, 0.15].\n\nAs shown in Fig. 1 , the geometric phase remains much smaller than 2π in this situation. This fact can be appreciated easily if we look at the distribution of the \"effective B-field\" B eff . According to Eq. (41), the component B eff x has a nonzero minimal value B rf /(2 √ 2), while |B eff z | can become arbitrarily small, although not necessarily zero in general. Therefore, at the trap center where | B eff | is a minimum, the value of cos β eff = B eff z /| B eff | can become very small, leading to small geometric phases. Yet, despite the relatively small geometric phase found here, our result remains important because it could represent a systematic error if not properly included in a Sagnac interference experiment.\n\nIn Fig. 6 , we show the spatial distribution of the geometric phase γ 1 around the trap center with λ = 1/3 and λ = 3. The fluctuation is found to be relatively small when λ is small or when the rf-field amplitude B rf is large.\n\nAlthough not discussed in Refs. [21, 22] , a ring shaped trap also can be realized if we take η = -κπ/2. The \"effective B-field\" B eff still lies in the x-y plane\n\nB eff x ( r) = - B rf (z) 2 √ 2 cos(2φ)(1 -cos β s (ρ, z)), B eff y ( r) = B rf (z) 2 √ 2 sin(2φ)(1 -cos β s (ρ, z)), ( 44\n\n) 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 -0.08 -0.06 -0.04 -0.02 0 λ=1/3 λ=3 λ=1 ρ c /L γ (c) 1\n\nFIG. 5: (Color online) The geometric phase γ1 is plotted against the radius ρc for the ARFP storage ring of Ref. [21, 22] with η = κπ/2 at λ = 3, λ = 1, and λ = 1/3. To assure the validity of the rotating wave approximation, in the solid lines, the maximal value of ∆[ r = 0]/|gF µBB ′ ρ0| or B rf /( √ 2B ′ ρ0) are restricted to be smaller than 0.15. The extending dashed line is beyond the rotating wave approximation for λ = 1/3 and B rf /( √ 2B ′ ρ0) ∈ [0.15, 0.3].\n\nclearly giving rise to a non-zero solid angle with respect to a closed path along the storage ring. Therefore, the term γ (eff) F is non-zero in this case. We choose the eigenstates |n( r\n\n) s and |n( r) eff as |n( r) s = exp[-i F • ns ( r)β s (ρ, z)]|n z , |n( r) eff = exp[-i F • n eff ⊥ ( r)β eff (ρ, z)]|n z . ( 45\n\n)\n\nwith the unit vector ns ⊥ ( r) in the x-y plane orthogonal to B s ( r). In this case, the gauge potential A (φ) n can be expressed as\n\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z)[1 + cos β s (ρ, z)]. ( 46\n\n)\n\nIn Figure 7 , we show the fluctuation of the geometric phase γ 1 for a closed path with a new parameter\n\nλ ′ = 6 √ 2 ∆[ r = 0] |g F µ B B (0) rf | , ( 47\n\n)\n\nequal to 3 and 1/3. The fluctuation for γ 1 is found to be much larger than the case of η = κπ/2, which can be explained by the transverse components B eff x,y of the \"effective B-field.\" Because cos β s is always close to unity.\n\nIn the case of η = -κπ/2, B eff x,y can take only small positive values. Therefore, at the minimum of the ARFP\n\ntrap ρ = ρ 0 of | B eff |, both B eff z\n\nand B eff x,y have to be close to zero. In this case the value for cos β eff becomes a rapidly changing function of ρ in the region near ρ c .\n\nOur above calculations have obtained analytical expressions of the geometric phases in an ARFP based storage ring for η = ±κπ/2. We have further investigated the 10 -0.05 0 0.05 -0.05 0 0.05 -0.06 -0.02 0.02 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.6 0 0.6 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 6: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ = 1/3 and (b) λ = 3. η = κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.\n\nfluctuations of the geometric phase for the two cases of η = ±κπ/2. It seems one benefits from implementing a Sagnac interferometer in the discussed ARFP storage ring with η = κπ/2 and operating at a relatively large λ.\n\nBefore proceeding onto the concluding section, we will discuss the geometric phase in an ARFP based beam splitter created via a double potential [14, 21] . In such an implementation, the static field B s is created from a Ioffe-Pritchard trap, while the oscillating rf field components are B (a) rf = B rf [z]ê x and B (b) rf = 0. By spatially tuning the amplitude of B rf from zero to a significant value, in the x-y plane, an ARFP can be tuned from a single well centered near the origin to a double well with two minimal points at the point with nonzero radius ρ 0 and φ = 0, π. Therefore, a Y-shaped atom beam splitter can be accomplished when the B rf [z] initially is increased along the z-axis to a large value, and then decreased to zero. In such an arrangement, the atom beam moving along the z direction can be separated into two beams that move along the z-axis at φ = 0, π for a while, and then can be recombined again into a single beam.\n\n-0.05 0 0.05 -0.05 0 0.05 -10 0 10 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -15 0 15 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 7: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ ′ = 3 and (b)λ ′ = 1/3. η = -κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.\n\nIn the atom interferometer considered above, both the static field B s and the \"effective B-field\" B eff are limited to the x-z plane. Therefore, for motion along the closed path of the trap bottom, the solid angle enclosed by the trajectory of B eff is zero. Thus, the geometric phase in (16) can be expressed as\n\nγ n (t) = -i t 0 l | eff n( r)|l z | 2 s l( r)|∇|l( r) s • vdt ′ . ( 48\n\n)\n\nWe can show that the product r) s is a function of ρ c and is independent of z. Thus, the geometric phase can be expressed as an integral of this function with respect to ρ c , from zero to a large value and then back to zero. Therefore, the value of the geometric phase would be zero in the end.\n\neff n( r)|l z | 2 s l( r)|∇|l(\n\n11 V. CONCLUSION\n\nIn this study, we develop theoretical formalisms for the calculation of the atomic geometric phase inside an ARFP. We show that, due to the complexity of the ARFP, the geometric phase depends on the spatial variation of both the static field and an \"effective B-field\" B eff . We provide general expressions for the geometric phase and the corresponding adiabatic gauge potential in Eqs. (16) and (23), respectively.\n\nTo shed light on actual applications of the atomic geometric phase, we investigate the distribution of atomic geometric phases for several proposed or ongoing experiments with ARFP based storage rings and atom beam splitters. We prove rigorously that the geometric phase in the center of the storage rings proposed in Refs. [18, 19] is always zero. In addition, we find that in the storage ring of Ref. [18] , the spatial fluctuation of the geometric phase sensitively depends on the position of the trap center on the \"resonance toroid.\" In the proposals of Refs. [19, 20, 21, 22] , the fluctuation for the geometric phase becomes significantly suppressed when the amplitude B rf of the rf-field is large. In the proposals of [21, 22] , the fluctuations of the geometric phase also is suppressed if the angle η is set to be κ2π. In the beam splitter realized with the double well potential ARFP [14, 21], the geometric phase is shown to be zero.\n\nOur work helps to clarify the working principle of trapping neutral atoms in an ARFP and the validity conditions for the various approximations involved. We hope our results will shine new light on the proposed inertial sensing experiments based on trapped atoms in ARFP."},{"section_type":"OTHER","section_title":"Acknowledgments","text":"We thank Dr. T. Uzer and Dr. B. Sun for helpful discussions. This work is supported by NASA, NSF, CNSF, and the 863 and 973 programs of the MOST of China.\n\n[1] D. E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983). [2] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A.\n\nCornell, Phys. Rev. Lett. 74, 3352 (1995). [3] A.S. Arnold and E. Riis, J. Mod. Opt. 49, 959 (2002); C. S. Garvie, E. Riis, and A. S. Arnold, Laser Spectroscopy XVI, edited by P. Hannaford et al. 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For instance, the widely used quadrupole trap and the Ioffe-Pritchard trap [1] are usually created with static B-fields, while the time averaged orbiting potential (TOP) [2] and time orbiting ring trap (TORT) [3, 4] are created using oscillating B-fields with frequencies larger than the effective trap frequencies. Atom chips [5] have brought further developments to magnetic trap technology, as they can provide larger B-fields and gradients at reduced power-consumptions or electric currents using micro-fabricated coils. Today, magnetic trapping is a versatile tool used in many laboratories around the world for controlling atomic spatial motion in regions of different scales and geometric shapes, e.g., 3D or 2D traps, double well traps, and storage ring traps [1, 2, 3, 4, 5, 6, 7] .\\n\\nRecently, magnetic traps based on adiabatic microwave [8, 9] and adiabatic radio frequency (rf) potentials (ARFP) [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] have attracted considerable attention. An ARFP is typically created with the combination of a static B-field and an rf field. The idea for an ARFP has been around for some time [10, 11, 12] , and experimental demonstrations recently have been carried out for confining both thermal [13] and Bose condensed atoms [14, 15, 16, 17] . Further development of improved ARFP with atom chip technology likely will assist in practical applications of atom interferometry. For instance, a double well potential was constructed recently using low order multi-poles capable of atomic beam splitting while maintaining tight spatial confinement [21] . Several interesting recent proposals outline the construction of small storage rings with radii of the order 1µm [18, 19, 20, 21, 22] , which could become useful if implemented for atom Sagnac interferometry [27] setups on atom chips.\\n\\nWhen a neutral atom is confined in a magnetic potential, its hyperfine spin is assumed to follow adiabatically the spatial variation of the B-field direction during its spatial translational motion. As a result of this adiabatic approximation, the center of mass motion for the atom experiences an induced gauge field [28] , giving rise to a geometric phase (or Berry's phase) to the atomic internal spin state [29, 30] . The effect of this geometric phase is widely known, and is first addressed carefully in a meaningful way for atomic quantum gases by an explicit calculation of the resulting geometric phase in a static or a time averaged magnetic trap in Ref. [31] . Several important consequences are predicted to occur for a magnetically trapped atomic condensate in a quadrupole trap, a Ioffe-Pritchard trap [31, 32] , or a TORT based storage ring [33] . To our knowledge, this geometric phase effect has not been investigated in any detail for an atom inside an ARFP.\\n\\nIn a recent paper, we show that this geometric phase causes an effective Aharonov-Bohm-type [34] phase shift in a magnetic storage ring based atom interferometer [33] . In addition, our studies imply that the spatial fluctuation of the geometric phase can lead to a reduction of the visibility of the interference contrast. In view of this, we decided to carry out this study as reported here for the atomic geometric phase in an ARFP in order to shed light on the proposed high precision atom Sagnac interference experiment [27] . Analytical derivations for this study at some places become rather tedious and complicated. We therefore first will summarize our major results here for readers who may not be interested in the intricate details. We find that the geometric phase in an ARFP generally takes a more complicated form in comparison to the case of a static trap or a time averaged trap. In an ARFP, this phase factor is found to be determined by the trajectory of the time independent component of the trap field as well as an \\\"effective B-field\\\" that depends on the total B-field. In contrast to the earlier result found for a static trap or a time averaged trap [31, 33] , the final result turns out to be not expressible as a functional of the trajectory 2 for the direction of the total B-field in the parameter space. This paper is organized as follows. In sec. II, we generalize the semi-classical approach as outlined in Ref. [21] for the operating principle of an ARFP to a form more convenient for discussing the geometric phase. Section III parallels that of sec. II by reformulating a full quantum theory for discussing the geometric phase inside an ARFP [22] . The explicit expression for the geometric phase inside an ARFP is given in a readily adaptable form for specifical calculations. In sec. IV, we discuss the effect of the geometric phase in several types of ARFP recently proposed for atomic splitters and storage rings [18, 19, 20, 21, 22] and beam splitters [14, 21] . Finally, concluding remarks are given in sec. V.\"\n },\n {\n \"section_type\": \"METHOD\",\n \"section_title\": \"II. A SEMI-CLASSICAL APPROACH\",\n \"text\": \"In this section, we provide a semi-classical formulation for calculating the atomic geometric phase inside an ARFP. The semi-classical working principle for an ARFP is described in Ref. [21] , although only for the special case when the atomic center of mass is assumed at a fixed location. In order to calculate the geometric phase, our formulation allows for the explicit consideration of atomic center of mass motion classically. In our approach, the geometric phase is obtained naturally, and the validity conditions for both the adiabatic and the rotating wave approximations are clearly shown for an ARFP.\\n\\nInside an ARFP [21] , the total B-field B( r, t) is the sum of a static field component B s ( r) and an oscillatory rf field B o ( r, t), which conveniently is expressed as B o ( r, t) = B (a) rf ( r, t) cos(ωt) + B (b) rf ( r, t) cos(ωt + η). (1)\\n\\nwhere r is the spatial position vector of the atom, ω is frequency of the rf field, and η is a relative phase factor.\\n\\nIn this section, we will assume that the atomic spatial motion is pre-determined, i.e., r(t) is given (as a slowly varying function of time t). For weak B-fields, the system Hamiltonian is simply the linear Zemman interaction\\n\\nH(t) = g F µ B F • B[ r(t), t], ( 2\\n\\n)\\n\\nwhere g F is the corresponding Lande g-factor and µ B denotes the Bohr magneton. = 1 is assumed. For a static or a time averaged magnetic trap, the Hamiltonian (2) varies slowly over time scales of the Larmor precession of the atomic spin in the total B-field.\\n\\nDuring the effectively slow trapped motion, the atomic hyperfine spin is assumed to be fixed at the instantaneous eigenstate of the Hamiltonian (2). The geometric phase then can be calculated straightforwardly from the variation of the B-field direction in the parameter space [31, 33] .\\n\\nIn an ARFP, the situation is more complicated. Although the variation of B s [ r(t)] remains much slower than the Larmor precession, the rf frequency ω usually is assumed to be nearly resonant with the precession frequency. Thus, the Hamiltonian (2) contains both fast and slow time varying components, making the direct calculation of the geometric phase a more involved task.\\n\\nIn the following, we will proceed step by step, clarifying the various approximations adopted along the way.\\n\\nTo understand the working principle for an ARFP, we first decompose the Hamiltonian H(t) (2) into the following form\\n\\nH(t) = H s [ r(t)] + H + [ r(t)]e -iωt + H -[ r(t)]e iωt , ( 3\\n\\n)\\n\\nwhere H s and H ± are all slow varying functions of time and are given by\\n\\nH s [ r(t)] = g F µ B F • B s [ r(t)], H + [ r(t)] = 1 2 g F µ B F • B (a) rf [ r(t)] + e -iη B (b) rf [ r(t)] , H -[ r(t)] = H † + [ r(t)]. ( 4\\n\\n)\\n\\nH s is diagonal in the spin angular momentum basis defined along the local direction of the static Bfield B s [ r(t)]. The eigenstate takes the familiar form |m F [ r(t)] s , quantized along the direction of B s [ r(t)], with the eigenvalue\\n\\nm F |B s [ r(t)]| for B s [ r(t)] • F and m F ∈ [-F, F ], in analogy with the usual case of the z- quantized representation result of F z |m F z = m F |m F z .\\n\\nNext we introduce a unitary transformation\\n\\nU (t) = F mF =-F |m F z s m F [ r(t)]|e imF κωt , ( 5\\n\\n)\\n\\nwith κ = sign(g F ) for the rotating wave approximation. The quantum state in the interaction picture |Ψ(t) I = U (t)|Ψ(t) defined by U (t) is governed by the Schroedinger equation\\n\\ni∂ t |Ψ(t) I = H I (t)|Ψ(t) I , with\\n\\nthe Hamiltonian in the interaction picture given by\\n\\nH I (t) = U HU † + i(∂ t U )U † = F m=-F mκ∆[ r(t)]|m z z m| -i F m,n=-F |m z s m[ r(t)]| d dt |n[ r(t)] s z n|e i(m-n)κωt\\n\\n3 + F m=-F +1 h (+) m [ r(t)]|m z z m -1| + h (-) m [ r(t)]|m z z m -1|e 2iκωt + h.c.\\n\\n+ F m=-F h m [ r(t)]|m z z m|e iκωt + h.c. , ( 6\\n\\n)\\n\\nwhere the time dependent parameters are defined as\\n\\n∆[ r(t)] = µ B |g F B[ r(t)]| -ω, h (±) m [ r(t)] = s m[ r(t)]|H ± [ r(t)]|(m -1)[ r(t)] s , and (7) h m [ r(t)] = s m[ r(t)]|H ± [ r(t)]|m[ r(t)] s .\\n\\nThe above result is obtained easily if we note that the matrix element s m[ r(t)|H ± (t)|m ′ [ r(t) s is non-zero only when m -m ′ = 0, ±1. So far, we have always assumed that |m[ r(t)] s is a single valued function of the atomic position r. A careful examination shows that the eigenstate |m[ r(t)] s cannot be determined uniquely because of the presence of the U (1) gauge freedom for selecting a local phase factor exp{iφ[ r(t)]}, which consequently affects the resulting expressions for h\\n\\n(±) m (t) and s m[ r(t)]|d/dt|m ′ [ r(t)] s .\\n\\nThe rotating wave approximation neglects of the oscillating terms proportional to e imωt (m = 0) in the Hamiltonian H I (6) . The error for this approximation is estimated easily from a time dependent perturbation calculation. The sufficient condition for its validity requires that all factors such as\\n\\nt 0 dt ′ h m (t ′ ) exp[iκωt ′ ], t 0 dt ′ h (-) m (t ′ )ξ m,m-1 (t ′ ) exp[iκ(2ω + ∆)t ′ ], and t 0 dt ′ m[ r(t ′ )]|d/dt ′ |n[ r(t ′ )] ξ mn (t ′ ) exp[i(m -n)κ(ω + ∆)t ′ ] are negligible, where ξ mn (t) = exp t 0 dt ′ s m[ r(t ′ )]| d dt ′ |m[ r(t ′ )] s × exp - t 0 dt ′ s n[ r(t ′ )]| d dt ′ |n[ r(t ′ )] s . ( 8\\n\\n)\\n\\nThus, the gauge independent factors h (-)\\n\\nm ξ m,m-1 , m s |d/dt|n s ξ mn , h m ,\\n\\nand ∆ should all vary slowly with time and with the modulus of their amplitudes much less than ω.\\n\\nThe effective Hamiltonian in the interaction picture under the rotating wave approximation then becomes\\n\\nH (I) eff (t) = µ B g F F • B eff [ r(t)] -i F m=-F |m z s m[ r(t)]| d dt |m[ r(t)] s z m|, ( 9\\n\\n)\\n\\nwhere the first term resembles a coupling between the atomic spin and an \\\"effective B-field\\\" B eff ( r), whose components in real space are given by\\n\\nB eff x ( r) = Re 2 s m( r)|H ± ( r)|(m -1)( r) s µ B g F (F + m)(F -m -1) , B eff y ( r) = -Im 2 s m( r)|H ± ( r)|(m -1)( r) s µ B g F (F + m)(F -m -1) , and\\n\\nB eff z ( r) = B s ( r) - ω µ B |g F | . ( 10\\n\\n)\\n\\nClearly, the x-and y-components of the effective field B eff ( r) depend on the explicit form of the eigenstate |m( r) s . In fact, it easily can be seen that different choices of the local phase factor for the |m( r) s actually lead to different values of B eff ( r) related to each other through r-dependent rotations in the x-y plane. In practice, the eigenstate |n( r) s and the effective field B eff can sometimes be constructed more simply, as in Ref. [21] . For any spatial position r, we first choose a rotation R[ m( r), χ( r)] along the axis m( r) with an angle χ( r\\n\\n) that satisfies R[ n( r), χ( r)] B s ( r) = | B s ( r)|ê z . It is then easy to show that the eigenstate |n[ r(t)] s can be chosen as |n( r) s = exp i F • m( r)χ( r) |n z . ( 11\\n\\n)\\n\\nUnfortunately, the choice for R is not unique in a given static field B s ( r), an analogous result to the U (1) gauge freedom for the the egienstate |n( r) s . Corresponding to the choice (11) given above for |n( r) s , the unitary transformation U defined in (5) would become\\n\\nU (t) = exp(-iF z ωt) • exp -i F • m( r)χ( r) , ( 12\\n\\n)\\n\\nand the transverse components of the \\\"effective B-field\\\" given by\\n\\nB eff x,y ( r) = B x,y ( r)/2 [21] with B( r) = R[ m( r), χ( r)] B (a) rf ( r) +R[ê z , -κη]R[ m( r), χ( r)] B (b) rf ( r). (13)\\n\\nIn earlier discussions of an ARFP [21, 22] , the atomic internal state is assumed uniformly to remain adiabatically in a certain eigenstate of the first term of H (I) eff (t). To fully appreciate this adiabatic approximation and to calculate the geometric phase, we expand |Ψ(t) I into the instantaneous eigenstate basis |n[ r(t)] eff quantized along the direction of the effective B-field\\n\\nB eff accord- ing to |Ψ(t) I = n C n (t)|n[ r(t)] eff .\\n\\nThe first term of H (I) eff (t) is simply the effective Zemman interaction between the atomic hyperfine spin and the effective B-field. The corresponding Schroedinger equation for the Hamiltonian H (I) eff (t) of (9) then becomes\\n\\ni d dt C n (t) = [ǫ (n) I (t) + ν nn (t)]C n (t) + m =n ν nm (t)C m (t), 4 with ǫ (n) I (t) = nµ B g F | B eff (t)|, ν pq (t) = -i l eff p[ r(t)]|l z s l[ r(t)]| d dt |l[ r(t)] s z l|q[ r(t)] eff -i eff p[ r(t)]| d dt |q[ r(t)] eff . ( 14\\n\\n)\\n\\nUnder the adiabatic approximation, the atomic internal state remains in a given eigenstate |n[ r(t)] eff with transitions to states |m[ r(t)] eff (m = n) being negligibly small. Thus, the transition probability, as estimated from the first order perturbation theory,\\n\\nt 0 dt ′ ν nm (t ′ )e i R t ′ 0 dt ′′ [ǫ (n) I (t ′′ )+νnn(t ′′ )-ǫ m I (t ′′ )-νmm(t ′′ )]\\n\\nshould be much less than one. As before, we find the sufficient condition for the validity of the adiabatic approximation is given by\\n\\n|ν mn (t)| |ǫ m I (t ′ ) -ǫ (n) I (t ′ )| ≪ 1, ( 15\\n\\n) provided that ν mn (t) exp[i t ′ 0 [ν nn (t ′′ ) -ν mm (t ′′\\n\\n)]dt ′ ], which is independent of the local phase factor for |n[ r(t)] eff and |n[ r(t)] s , remains a slowly varying function of time.\\n\\nA straight forward calculation from the effective Hamiltonian (9) then gives the general expression for the geometric phase in an ARFP\\n\\nγ n (t) = t 0 ν nn (t ′ )dt ′ = -i t 0 l | eff n[ r(t ′ )|l z | 2 × s l[ r(t ′ )]| d dt ′ |l[ r(t ′ )] s dt ′ +γ (I) n (t), ( 16\\n\\n) and γ (I) n (t) = -i t 0 dt ′ eff n[ r(t ′ )|d/dt ′ |n[ r(t ′ ) eff .\\n\\nDuring the adiabatic motion in a given internal state, the time evolution of the coefficient C n (t) takes the form (16) is the central result of this work. The geometric phase of an atom inside an ARFP is shown to contain two parts. The second part, γ 9) , with its value determined by the trajectory of the direction for the \\\"effective B-field\\\" B eff . The first part arises from the second term of H (I) eff (9). It is determined by the trajectories of both the static field B s and the effective B-field B eff . The expression for γ n in an ARFP is complicated because the internal quantum state in an ARFP is assumed to be adiabatically kept in an eigenstate of µ B g F F • B eff , rather than an eigenstate of the total interaction Hamiltonian H (I) eff (t). In section IV, we will perform explicit calculations for several examples of ARFP proposed for various applications: e.g., as atomic storage rings or atomic beam splitters. Most often we find that only the first part of Eq. (16) contributes a non-zero value to the geometric phase.\\n\\nC n (t) = C n (0)e -i R t 0 ǫ (n) I (t ′ )dt ′ e -iγn(t) . ( 17\\n\\n) Equation\\n\\n(I) n (t), is clearly due to the interaction term µ B g F F • B eff in H (I) eff (\\n\\nBefore proceeding to the next section for a quantal treatment of the geometric phase, we find the time evolution of the atomic spin state in the Schroedinger picture\\n\\n|Ψ(t) = ml C l (0) z m|l[ r(t)] eff × e -i R t 0 ǫ l I (t ′ )dt ′ e -iγ l (t) e -imωt |m[ r(t)] s , ( 18\\n\\n) obtained directly from |Ψ(t) = U † (t)|Ψ(t) I after the\\n\\napplications of the rotating wave and adiabatic approximations. When the atom is prepared initially in a specific adiabatic state |n[ r(t)] eff of the interaction picture, we arrive at the simple case of C l (0) = δ ln .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. A QUANTUM MECHANICAL TREATMENT\",\n \"text\": \"In the previous section, we provided the result for the geometric phase γ n (t) in an ARFP based on a semiclassical approach, where the atomic center of mass motion is described classically. A clear physical picture exists in this case for the appearance of the geometric phase in a certain parameter space. The validity conditions for the rotating wave and the adiabatic approximations as obtained above are all formulated in terms of gauge independent forms. However, if the influence of the geometric phase on the atomic spatial motion is to be included, e.g., as in the Aharonov-Bohm-type, phase shift, interference arrangement in an atomic Sagnac interferometer discussed earlier [33] , we would need an improved description where both the atomic spin and its center of mass motion are treated quantum mechanically.\\n\\nIn a full quantum treatment of the atomic motion, the quantum state of an atom can be expressed as |Φ(t) = φ l ( r, t)|l z , where φ l ( r, t) is the atomic spatial wave function for the internal state |l z of F z . The state |Φ(t) then satisfies the Schroedinger equation governed by the Hamiltonian\\n\\nH = P 2 2M + g F µ B F • B( r, t), ( 19\\n\\n)\\n\\nwith P being the kinetic momentum and M the atomic mass. The rotating wave and adiabatic approximations can be introduced now by defining the interaction picture 5 with the unitary transformation\\n\\nU(t) = F m=-F |m z eff m( r)| × F n=-F |n z s n( r)|e inκωt . ( 20\\n\\n)\\n\\nThe state in the interaction picture |Φ(t) I = U(t)|Φ(t) now is governed by the Schroedinger equation with the Hamiltonian H eff = UHU † . Under the rotating wave and adiabatic approximations, we neglect transitions between states |m z and |n z (m = n) as well as the rapidly oscillating terms. We then obtain\\n\\nH eff ≈ n |n z z n|H eff |n z z n| ≈ n H (n) ad |n z z n|, ( 21\\n\\n)\\n\\nwhere the adiabatic Hamiltonian H (n) ad for the n-th adiabatic branch is defined as\\n\\nH (n) ad = P -A n 2 2M + ǫ (n) I ( r), ( 22\\n\\n)\\n\\nwith the effective gauge potential\\n\\nA n ( r) = -i l | eff n( r)|l z | 2 s l( r)|∇|l( r) s -i eff n( r)|∇|n( r) eff . ( 23\\n\\n)\\n\\nIn this form, it is well known that the geometric phase γ n can be expressed as the integral of the gauge potential A n along the spatial trajectory for the atomic center of mass in an ARFP, i.e., one would expect generally that γ n = A n • d r. Similar to the result of the semiclassical approach, the gauge potential A n ( r) can be expressed as the sum of two parts. The first part in Eq. (23) is the weighted sum of the atomic gauge potential -i s l( r)|∇|l( r) s from the static field B s , while the second term is the atomic gauge potential from the \\\"effective B-field\\\" B eff .\\n\\nA full quantum treatment for atomic motion in an ARFP has been attempted earlier [22] . In fact, many of our formulations are identical to the results of Ref. [22] . For instance, it is easy to show that the unitary transformations U S , U R , and U F in [22]\\n\\nare related directly to ours as U † F U † R U † S = U.\\n\\nThe only difference concerns the gauge potential A n that was neglected in Ref. [22] . Thus, they did not give the expression for the gauge potential, and the result for the geometric phase was not obtained either [22] . Our study shows that the neglect of the adiabatic gauge potential potentially can give rise to a final result, dependent on the choice of the local phase factors for the internal eigenstate.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. GEOMETRIC PHASES IN ARFP BASED APPLICATIONS\",\n \"text\": \"In the above two sections, we obtain the expression for the atomic geometric phase in an ARFP. This section is devoted to the calculations of the geometric phases for several proposed applications of ARFP, such as storage rings or beam splitters for neutral atoms [18, 19, 20, 21, 22] .\\n\\nBefore presenting our results for the more specific cases, we provide some general discussions of the geometric phases in several ARFP based storage rings. As was pointed out earlier, the geometric phase γ n is given by the line integral of the gauge potential A n along the trajectory for the atomic center of mass motion. For a closed path in the storage ring at a fixed ρ = ρ c and z = z c , this can be further reduced to\\n\\nγ n = q 2π 0 A (φ) n (ρ, φ, z)ρdφ, ( 24\\n\\n)\\n\\nwhere the integer q is the winding number of the path and A (φ) n is the component of A n along the azimuthal direction êφ of the familiar cylindrical coordinate system (ρ, φ, z). Without loss of generality, we take q = 1 in this paper. For the storage rings proposed in Refs. [18, 19, 20, 21, 22] , the gauge potentials A (φ) n (ρ, φ, z) are actually independent of the angle φ. Therefore, the geometric phase is simply given by\\n\\nγ (c) n = 2πρ c A (φ) n (ρ c , z c ), ( 25\\n\\n)\\n\\ngiven out in explicit forms for different storage ring schemes [18, 19, 20, 21, 22] .\\n\\nIn reality, because of thermal motion or when the atomic transverse motional state is considered, the center of mass for an atom can deviate from (ρ c , z c ) even for a closed trajectory. This uncertainty in the exact shape of the closed trajectory gives rise to a fluctuating geometric phase and is usually difficult to study. Assuming a simple closed path at fixed ρ and z, we have found previously that the subsequently fluctuations could decrease the visibility of the interference pattern [33] . Quantum mechanically, such destructive interference can be explained as resulting from entanglement between the freedoms for φ and (ρ, z) because of the dependence of the gauge potential A φ n on ρ and z. Therefore, it is important to investigate this dependence near the trap center.\\n\\nFor simplicity, our discussions below will focus on the closed loops where ρ and z are φ-independent constants. In this case, the geometric phase can be expressed as ρ, z) . We will show numerically the distributions for γ n (ρ, z) obtained this way near the central region of (ρ c , z c ). If needed, a more rigorous approach can be developed to investigate the fluctuations of the resulting geometric phase from the gauge potential A (φ) n (ρ, z).\\n\\nγ n (ρ, z) = 2πρA (φ) n (\\n\\nA. The storage ring proposals of Refs. [18, 19, 20] This subsection is devoted to a detailed calculation of the geometric phases for the ARFP storage ring proposals of Refs. [18, 19, 20] . We will derive the analytical expressions for the azimuthal component A (φ) n of the gauge potential that arises in both cases from cylindrically symmetric static B-field and rf fields. Because of the cylindrical symmetry, the angle β s (ρ, z) between the local static B-field and the z-axis is required to be analytical in the region near the storage ring. Therefore, the eigenstate |n( r) s of F • B s can be chosen as\\n\\n|n( r) s = exp{-i[ F • êφ β s (ρ, z) + nφ]}|n z . ( 26\\n\\n)\\n\\nConsequently, B eff ( r) is also cylindrically symmetric, which leads to the eigenstate |n( r\\n\\n) eff of F • B eff as |n( r) eff = exp{-i[ F • n eff ⊥ ( r)β eff (ρ, z) + nφ]}|n z , ( 27\\n\\n)\\n\\nwith the unit vector n eff ⊥ ( r) in the x-y plane orthogonal to B eff ( r) and β eff (ρ, z) denoting the angle between B eff ( r) and the z-axis. We note that the unit vector field n eff ⊥ ( r) also possesses cylindrical symmetry, i.e., remains invariant under rotation around the z-axis. The expressions of (26) and (27) allow us to obtain the simple expression of the gauge potential\\n\\nA (φ) n (ρ, z) = - n ρ cos β eff (ρ, z) cos β s (ρ, z), ( 28\\n\\n)\\n\\nafter straightforward calculations. In the scheme of Ref. [18] , the static B-field is a \\\"ringshaped quadrupole field\\\" that vanishes along a circle of a radius ρ 0 in the x-y plane. Near ρ = ρ 0 , the B-field is given approximately by\\n\\nB s ( r) = B ′ (ρ -ρ 0 )ê ρ -B ′ zê z , ( 29\\n\\n) like a quadrupole field, while the rf-field takes a complicated form\\n\\nB o ( r, t) = a √ 2 cos(ωt) + b √ 2 cos(ωt + ϕ) êρ + - a √ 2 sin(ωt) + b √ 2 sin(ωt + ϕ) êz , ( 30\\n\\n)\\n\\nwith constants a and b independent of r. From the expression of (26) for the eigenstate |n( r) s , the \\\"effective B-field\\\" B eff becomes\\n\\nB eff ( r) = B ′ [ (ρ -ρ 0 ) 2 + z 2 -r 0 ]ê z - b √ 2 cos(θ + ϕ) + a √ 2 cos θ êρ + b √ 2 sin(θ + ϕ) - a √ 2 sin θ êφ , ( 31\\n\\n)\\n\\nFIG. 1: (Color online) A cross-sectional view for the storage ring of Ref. [18] . The static field is zero in the ring at the fixed radius ρ0. The addition of rf-fields creates an ARFP centered at a ring through (ρc, zc). The distance from the trap center to the ring with radius ρ0 in the plane z = 0 is r0.\\n\\nwhere r 0 and θ are given by\\n\\nr 0 = ω |µ B g F B ′ | , cos θ(ρ, z) = ρ -ρ 0 (ρ -ρ 0 ) 2 + z 2 , sin θ(ρ, z) = z (ρ -ρ 0 ) 2 + z 2 . ( 32\\n\\n)\\n\\nIn an ARFP, as discussed here, the trap center at (ρ c , z c ) is determined by minimizing both the zcomponent and the transverse component of B eff . Without loss of generality, we will assume a, b > 0. Then, (ρ c , z c ) is found to satisfy\\n\\nθ(ρ c , z c ) = -ϕ/2, (ρ c -ρ 0 ) 2 + z 2 c = r 0 , ( 33\\n\\n)\\n\\ni.e., the trap center lies on the surface of the \\\"resonance toroid\\\" at ρ = ρ 0 with a radius r 0 as shown in Fig. 1 . The relative angle of the trap center with respect to the center of the toroid cross-section is given by -ϕ/2. On this \\\"resonance toroid,\\\" the rf-field is resonant with the static field, i.e., B eff z vanishes. As a result, the \\\"effective B-field\\\" lies again in the x-y plane on the \\\"resonance toroid,\\\" which gives cos β eff (ρ c , z c ) = 0 and leads to the result A (φ) n (ρ c , z c ) = γ n = 0 as shown in the trap center for the storage ring considered before in Ref. [18] .\\n\\nFrom the expression (29) of the static field and the definition of the angle θ(ρ, z), we find a simple relationship β s (ρ, z) = π/2 + θ(ρ, z), with which the gauge potential A (φ) n (ρ, φ) in (28) can be further simplified as\\n\\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z) sin θ(ρ, z) 7 ≈ n ρ cos β eff (ρ, z) sin θ(ρ c , z c ), ( 34\\n\\n)\\n\\nnear the trap center. Thus, the spatial fluctuation of the gauge potential A (φ) n (ρ, z) in the region around the trap center is closely related to the angle θ(ρ c , z c ) of the trap center, or the parameter ϕ of the oscillating field B o . When ϕ = 0, the atom is trapped in the region with θ ≈ 0 or π, where the fluctuation of A (φ) n (ρ, z) is suppressed significantly due to the small value of sin θ. On the other hand, if the angle ϕ is set to π with the trap center located in the region with θ ≈ ±π/2, the fluctuation of the gauge potential becomes amplified.\\n\\nIn Fig. 2 , we illustrate numerical results for the distribution of the geometric phase γ 1 (ρ, z) = 2πρA φ 1 (ρ, z) in the region near the trap center at ϕ = 0, π/2, π. We see clearly decreased fluctuations of γ 1 when the absolute value of sin θ(ρ c , z c ) = -sin(ϕ/2) is decreased.\\n\\nNext we turn to the storage ring of Ref. [19] constructed from a quadrupole static B-field B s ( r) = B ′ (x, y, -2z) and an r-independent rf field B o = B rf cos(ωt)ê z along the z direction. The resulting ARFP provides a 2D ring shaped trap in the x-y plane. In addition, a 1D optical potential along the z direction is employed to confine atoms in the transverse plane at z = 0 [19]. The \\\"effective B-field\\\" takes the form\\n\\nB eff ( r) = B ′ (ρ -ρ 0 )ê z - 1 2 B rf êρ , ( 35\\n\\n)\\n\\nin the plane at z = 0, with\\n\\nρ 0 = ω/|µ B g F B ′ |.\\n\\nBecause the strength of B eff is near minimum at the ring ρ = ρ 0 , the trap center for this storage ring is located at ρ c = ρ 0 and z c = 0. At the trap center, the \\\"effective Bfield\\\" is along the direction of êρ . Thus, according to Eq. ( 28 ), the geometric phase γ (c) n at the trap center again vanishes.\\n\\nIn Fig. 3 , we show the distribution of the geometric phase γ 1 in the region near the trap center for B rf = 0.05|B ′ |ρ 0 and B rf = 0.15|B ′ |ρ 0 . We see that the fluctuation is relatively small when the strength of the rf-field is large. This can be explained by Eq. (28) , which shows that A (φ) n is proportional to cos β eff and can be approximated as 2B eff z /B rf near the trap center. When B rf is large, the gauge potential becomes a relatively slow varying function of ρ and z. In this case, the presence of a 1D optical potential allows for the possibility of tuning the trap center position to a nonzero value of z, with the storage ring remaining in the x-y plane. Then cos β s is assumed to a nonzero value, leading to increased fluctuations for the geometric phase.\\n\\nFinally, we discuss the geometric phase in the \\\"time averaged\\\" ARFP storage ring proposed in Ref. [20] . Unlike previously considered ARFP based storage rings, the time dependence now exists in both the \\\"static B-field\\\" and the frequency of the rf field given by Color online) The distribution of the geometric phase γ1 near the trap center (ρc, zc) of the storage ring proposed in Ref. [18] at (a) ϕ = 0, (b) ϕ = π/2, and (c) ϕ = π, clearly displaying the sin(ϕ/2) dependence.\\n\\nB s ( r, t) = B ′ ρê ρ -2B ′ zê z + B m sin(ω m t)ê z , -0.1 0 0.1 -0.1 0 0.1 -1 0 1 (z-z c )/ρ 0 γ 1 (a) (ρ-ρ c )/ρ 0 -0.1 0 0.1 -0.1 0 0.1 -5 0 5 (z-z c )/ρ 0 (ρ-ρ c )/ρ 0 γ 1 (b) -0.1 0 0.1 -0.1 0 0.1 -10 0 10 (z-z c )/ρ 0 γ 1 (c) (ρ-ρ c )/ρ 0 FIG. 2: (\\n\\nB o (t) = B rf sin[ω(t)t]ê z , ω(t) = ω 0 1 + (B m /B ′ ρ 0 ) 2 sin 2 (ω m t) . (36)\\n\\nThe frequency ω m is assumed to be much smaller than ω 0 but much larger than the trap frequency. The radius ρ 0 is now defined as\\n\\nρ 0 = ω 0 /|µ B g F B ′ |,\\n\\nand the \\\"effective 8 -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 (a) -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 (b) z/ρ 0 (ρ-ρ 0 )/ρ 0 γ 1 FIG. 3: (Color online) The geometric phase γ1 for the storage ring of Ref. [19] with (a) B rf = 0.15B ′ ρ0 and (b) B rf = 0.05B ′ ρ0.\\n\\nB-field\\\" takes the form\\n\\nB eff ( r, t) = ∆( r, t)ê z - B ′ ρ |2 B s ( r, t)| B rf êφ . ( 37\\n\\n)\\n\\nThe operating principle for the time averaged storage ring of Ref. [20] is similar to the well-known TOP [2] and TORT traps [3, 4] . The effective trap potential experienced by the atom is proportional to the time averaged value of the \\\"effective B-field\\\"\\n\\n2π/ωm 0 | B eff ( r, t)|dt.\\n\\nWhen B rf and B m are much smaller than B ′ ρ 0 , the center of the storage ring is located approximately at ρ c = ρ 0 , z c = 0. Using the earlier result [33] , we find that in the time averaged storage ring, the effective gauge potential Ã(φ) n (ρ, z) is reduced simply to the time averaged instantaneous gauge potential 28) . The geometric phase then is given approximately by γ n (ρ, z) = 2π Ã(φ) n (ρ, z). In this case, we find that the geometric phase always vanishes at -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 z/ρ 0 (ρ-ρ 0 )/ρ 0 (a) γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 Color online) The geometric phase γ1 for the storage ring of Ref. [20] at (a) B rf = 0.3B ′ ρ0 and (b) B rf = 0.1B ′ ρ0. Bm = 0.05B ′ ρ0.\\n\\nÃ(φ) n (ρ, z) = ω m 2π 2π/ωm 0 A (φ) n (ρ, z, t)dt, ( 38\\n\\n) with A (φ) n (ρ, z, t) given in (\\n\\n(b) (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 FIG. 4: (\\n\\nthe trap center (ρ c , z c ). Figure 4 illustrates the distribution of the geometric phase in the region near the trap center for two different values of the rf-field amplitude B rf . Similar to the storage ring of Ref. [19] , the fluctuation of the geometric phase is suppressed in this case for large B rf .\\n\\nB. The storage ring proposals of Refs. [21, 22] Next we consider the ARFP based storage ring proposed in Refs. [21, 22] . In this case, the static B-field is that of a Ioffe-Pritchard trap on an atom chip. In the Cartesian coordinate (x, y, z), it takes the form\\n\\nB s = B ′ xê x -B ′ yê y + B ′ Lê z , ( 39\\n\\n)\\n\\nwhere B ′ is the B-field gradient and the bias field along the z-direction is denoted as B ′ L. The amplitudes B\\n\\n(a) rf and B (b) rf (z) of the rf field are B\\n\\n(a) rf = [B rf (z)/ √ 2]ê x and B (b) rf = [B rf (z)/ √ 2]ê y with B rf (z) = B ( 0\\n\\n) rf + B ′′ z 2 . ( 40\\n\\n)\\n\\n9 In the schemes of Ref. [21, 22] considered earlier, the phase η of the rf field is assumed to be κπ/2. The xand y-components of the \\\"effective B-field\\\" B eff ( r) then become\\n\\nB eff x ( r) = B rf (z) 2 √ 2 (1 + cos β s (ρ, z)), B eff y ( r) = 0, ( 41\\n\\n)\\n\\naccording to Eq. (10). Then the strength of the \\\"effective B-field\\\" B eff has its minimum along a circle with a nonzero radius ρ c , provided a positive detuning ∆ exists at the origin (0, 0, 0) [21, 22] . The \\\"effective B-field\\\" B eff is easily shown to lie in the x-z plane along the trap bottom mapped out by the atomic center of mass motion. This gives rise to a vanishing γ (I) F . With a proper choice for the local phase of |n( r) eff , the gauge potential A (φ) n takes the form\\n\\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z) (1 -cos β s (ρ, z)) . ( 42\\n\\n)\\n\\nFigure 5 displays the geometric phase along a closed path for a spin-1 atom as a function of ρ c for the ARFP storage ring proposed in Refs. [21, 22] . The parameter λ is defined as\\n\\nλ = √ 2 ∆[ r = 0] |g F µ B B (0) rf | . ( 43\\n\\n)\\n\\nTo assure the validity of the rotating wave approximation, we find that the maximal values of ∆[ r = 0]/(|g F |µ B ) and B (0) rf / √ 2 must be restricted to the region of λ ∈ [0, 0.15].\\n\\nAs shown in Fig. 1 , the geometric phase remains much smaller than 2π in this situation. This fact can be appreciated easily if we look at the distribution of the \\\"effective B-field\\\" B eff . According to Eq. (41), the component B eff x has a nonzero minimal value B rf /(2 √ 2), while |B eff z | can become arbitrarily small, although not necessarily zero in general. Therefore, at the trap center where | B eff | is a minimum, the value of cos β eff = B eff z /| B eff | can become very small, leading to small geometric phases. Yet, despite the relatively small geometric phase found here, our result remains important because it could represent a systematic error if not properly included in a Sagnac interference experiment.\\n\\nIn Fig. 6 , we show the spatial distribution of the geometric phase γ 1 around the trap center with λ = 1/3 and λ = 3. The fluctuation is found to be relatively small when λ is small or when the rf-field amplitude B rf is large.\\n\\nAlthough not discussed in Refs. [21, 22] , a ring shaped trap also can be realized if we take η = -κπ/2. The \\\"effective B-field\\\" B eff still lies in the x-y plane\\n\\nB eff x ( r) = - B rf (z) 2 √ 2 cos(2φ)(1 -cos β s (ρ, z)), B eff y ( r) = B rf (z) 2 √ 2 sin(2φ)(1 -cos β s (ρ, z)), ( 44\\n\\n) 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 -0.08 -0.06 -0.04 -0.02 0 λ=1/3 λ=3 λ=1 ρ c /L γ (c) 1\\n\\nFIG. 5: (Color online) The geometric phase γ1 is plotted against the radius ρc for the ARFP storage ring of Ref. [21, 22] with η = κπ/2 at λ = 3, λ = 1, and λ = 1/3. To assure the validity of the rotating wave approximation, in the solid lines, the maximal value of ∆[ r = 0]/|gF µBB ′ ρ0| or B rf /( √ 2B ′ ρ0) are restricted to be smaller than 0.15. The extending dashed line is beyond the rotating wave approximation for λ = 1/3 and B rf /( √ 2B ′ ρ0) ∈ [0.15, 0.3].\\n\\nclearly giving rise to a non-zero solid angle with respect to a closed path along the storage ring. Therefore, the term γ (eff) F is non-zero in this case. We choose the eigenstates |n( r\\n\\n) s and |n( r) eff as |n( r) s = exp[-i F • ns ( r)β s (ρ, z)]|n z , |n( r) eff = exp[-i F • n eff ⊥ ( r)β eff (ρ, z)]|n z . ( 45\\n\\n)\\n\\nwith the unit vector ns ⊥ ( r) in the x-y plane orthogonal to B s ( r). In this case, the gauge potential A (φ) n can be expressed as\\n\\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z)[1 + cos β s (ρ, z)]. ( 46\\n\\n)\\n\\nIn Figure 7 , we show the fluctuation of the geometric phase γ 1 for a closed path with a new parameter\\n\\nλ ′ = 6 √ 2 ∆[ r = 0] |g F µ B B (0) rf | , ( 47\\n\\n)\\n\\nequal to 3 and 1/3. The fluctuation for γ 1 is found to be much larger than the case of η = κπ/2, which can be explained by the transverse components B eff x,y of the \\\"effective B-field.\\\" Because cos β s is always close to unity.\\n\\nIn the case of η = -κπ/2, B eff x,y can take only small positive values. Therefore, at the minimum of the ARFP\\n\\ntrap ρ = ρ 0 of | B eff |, both B eff z\\n\\nand B eff x,y have to be close to zero. In this case the value for cos β eff becomes a rapidly changing function of ρ in the region near ρ c .\\n\\nOur above calculations have obtained analytical expressions of the geometric phases in an ARFP based storage ring for η = ±κπ/2. We have further investigated the 10 -0.05 0 0.05 -0.05 0 0.05 -0.06 -0.02 0.02 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.6 0 0.6 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 6: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ = 1/3 and (b) λ = 3. η = κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.\\n\\nfluctuations of the geometric phase for the two cases of η = ±κπ/2. It seems one benefits from implementing a Sagnac interferometer in the discussed ARFP storage ring with η = κπ/2 and operating at a relatively large λ.\\n\\nBefore proceeding onto the concluding section, we will discuss the geometric phase in an ARFP based beam splitter created via a double potential [14, 21] . In such an implementation, the static field B s is created from a Ioffe-Pritchard trap, while the oscillating rf field components are B (a) rf = B rf [z]ê x and B (b) rf = 0. By spatially tuning the amplitude of B rf from zero to a significant value, in the x-y plane, an ARFP can be tuned from a single well centered near the origin to a double well with two minimal points at the point with nonzero radius ρ 0 and φ = 0, π. Therefore, a Y-shaped atom beam splitter can be accomplished when the B rf [z] initially is increased along the z-axis to a large value, and then decreased to zero. In such an arrangement, the atom beam moving along the z direction can be separated into two beams that move along the z-axis at φ = 0, π for a while, and then can be recombined again into a single beam.\\n\\n-0.05 0 0.05 -0.05 0 0.05 -10 0 10 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -15 0 15 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 7: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ ′ = 3 and (b)λ ′ = 1/3. η = -κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.\\n\\nIn the atom interferometer considered above, both the static field B s and the \\\"effective B-field\\\" B eff are limited to the x-z plane. Therefore, for motion along the closed path of the trap bottom, the solid angle enclosed by the trajectory of B eff is zero. Thus, the geometric phase in (16) can be expressed as\\n\\nγ n (t) = -i t 0 l | eff n( r)|l z | 2 s l( r)|∇|l( r) s • vdt ′ . ( 48\\n\\n)\\n\\nWe can show that the product r) s is a function of ρ c and is independent of z. Thus, the geometric phase can be expressed as an integral of this function with respect to ρ c , from zero to a large value and then back to zero. Therefore, the value of the geometric phase would be zero in the end.\\n\\neff n( r)|l z | 2 s l( r)|∇|l(\\n\\n11 V. CONCLUSION\\n\\nIn this study, we develop theoretical formalisms for the calculation of the atomic geometric phase inside an ARFP. We show that, due to the complexity of the ARFP, the geometric phase depends on the spatial variation of both the static field and an \\\"effective B-field\\\" B eff . We provide general expressions for the geometric phase and the corresponding adiabatic gauge potential in Eqs. (16) and (23), respectively.\\n\\nTo shed light on actual applications of the atomic geometric phase, we investigate the distribution of atomic geometric phases for several proposed or ongoing experiments with ARFP based storage rings and atom beam splitters. We prove rigorously that the geometric phase in the center of the storage rings proposed in Refs. [18, 19] is always zero. In addition, we find that in the storage ring of Ref. [18] , the spatial fluctuation of the geometric phase sensitively depends on the position of the trap center on the \\\"resonance toroid.\\\" In the proposals of Refs. [19, 20, 21, 22] , the fluctuation for the geometric phase becomes significantly suppressed when the amplitude B rf of the rf-field is large. In the proposals of [21, 22] , the fluctuations of the geometric phase also is suppressed if the angle η is set to be κ2π. In the beam splitter realized with the double well potential ARFP [14, 21], the geometric phase is shown to be zero.\\n\\nOur work helps to clarify the working principle of trapping neutral atoms in an ARFP and the validity conditions for the various approximations involved. We hope our results will shine new light on the proposed inertial sensing experiments based on trapped atoms in ARFP.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments\",\n \"text\": \"We thank Dr. T. Uzer and Dr. B. Sun for helpful discussions. This work is supported by NASA, NSF, CNSF, and the 863 and 973 programs of the MOST of China.\\n\\n[1] D. E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983). [2] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A.\\n\\nCornell, Phys. Rev. Lett. 74, 3352 (1995). [3] A.S. Arnold and E. Riis, J. Mod. Opt. 49, 959 (2002); C. S. Garvie, E. Riis, and A. S. Arnold, Laser Spectroscopy XVI, edited by P. Hannaford et al. 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Rev. 115, 485 (1959).\"\n }\n]"}}},{"rowIdx":85,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0482"},"arxiv_id":{"kind":"string","value":"0704.0482"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevA.76.062311"},"content_hash":{"kind":"string","value":"4caec5185f281f351422ba4711e16fc0b298b4789732d3ff316689474088ac07"},"title":{"kind":"string","value":"Implementation of holonomic quantum computation through engineering and\n manipulating environment"},"abstract":{"kind":"string","value":"We consider an atom-field coupled system, in which two pairs of four-level\natoms are respectively driven by laser fields and trapped in two distant\ncavities that are connected by an optical fiber. First, we show that an\neffective squeezing reservoir can be engineered under appropriate conditions.\nThen, we show that a two-qubit geometric CPHASE gate between the atoms in the\ntwo cavities can be implemented through adiabatically manipulating the\nengineered reservoir along a closed loop. This scheme that combines engineering\nenvironment with decoherence-free space and geometric phase quantum computation\ntogether has the remarkable feature: a CPHASE gate with arbitrary phase shift\nis implemented by simply changing the strength and relative phase of the\ndriving fields."},"authors":{"kind":"list like","value":["Zhang-qi Yin","Fu-li Li","Peng Peng"],"string":"[\n \"Zhang-qi Yin\",\n \"Fu-li Li\",\n \"Peng Peng\"\n]"},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]"},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"Quantum computation, attracting much current interest since Shor's algorithm [1] was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevitably coupled to their environment so that entanglement is too fragile to be retained. This makes the realization of quantum computation extremely difficult in the real world. In order to overcome this difficulty, one proposed the decoherence-free space concept [2, 3] . It is found that when qubits involved in quantum computation collectively interact with a same environment there exists a \"protected\" subspace in the entire Hilbert space, in which the qubits are immune to the decoherence effects induced by the environment. This subspace is called decoherence-free space (DFS). To perform quantum computation in a DFS, one has to design the specific Hamiltonian containing controlling parameters, which eigenspace is spanned by DFS states and the state-unitary manipulation related to quantum computation goal is implemented by changing the controlling parameters [4] .\n\nAs well known, instantaneous eigenstates of a quantum system with the time-dependent Hamiltonian may acquire a geometric phase when the time-dependent parameters adiabatically undergo a closed loop in the parameter space [5] . The phase depends only on the swept solid angle by the parameter vector in the parameter space. This feature can be utilized to implement geometric quantum computation (GQC) which is resilient to stochastic control errors [6, 7, 8] . On combining the DFS approach with the GQC scheme, one may build quantum gates which may be immune to both the environment-induced decoherence effects and the control-led errors [9] . In the scheme, quantum logical bits are represented by degenerate eigenstates of the parameterized Hamiltonian. These states have the features: they belong to DFS, and unitarily evolve in time and acquire a geometric phase when the controlling parameters adiabatically vary and undergo a closed loop.\n\nIn the recent paper [10], Carollo and coworkers showed that a cascade three-level atom interacting with a broadband squeezed vacuum bosonic bath can be prepared in a state which is decoupled to the environment. This state depends on the reservoir parameters such as squeezing degree and phase angle. As the squeezing parameters smoothly vary, the atomic state can unitarily evolve in time and always be in the manifold of the DFS. Moreover, after a cyclic evolution of the squeezing parameters, the state acquires a geometric phase. This investigation has been generalized to cases where both quantum systems and manipulated reservoir under consideration are not restricted to cascade three-level atoms and squeezed vacuum [11] . These results strongly inspire us that instead of engineering Hamiltonian one may implement the decoherence-free GQC by engineering and manipulating reservoir.\n\nIn this paper, we propose a scheme in which the quantum-reservoir engineering [12, 13, 14] is combined with DFS and Berry phase together to realize a twoqubit CPHASE gate [15] . We show that atomic states can unitarily evolve in time in a DFS if the change rate of reservoir parameters is much smaller than the characteristic relaxation time of an atom-reservoir coupled system. Moreover, we find that as the reservoir parameters adiabatically change in time along an appropriate closed loop, the atomic state in the DFS acquires a Berry phase and a CPHASE gate with arbitrary phase shift can be realized. To our knowledge, it is the first proposal for the realization of quantum gates by engineering and steering the environment.\n\nThis paper is organized as follows. In Sec. II, we introduce a cavity-atom coupling model in which two pairs of four-level atoms are respectively trapped in two distant cavities that are connected by an optical fiber. In the model, each of pairs of the atoms are simultaneously driven by laser fields and coupled to the local cavity 2 modes through the double Raman transition configuration. Under large detuning and bad cavity limits, we investigate to engineer an effective broadband squeezing reservoir for the atoms. In Sec. III, we analyze how to realize controlling gates between the atoms trapped in the two cavities by steering the squeezing reservoir. Section IV contains conclusions of our investigations.\n\nFIG. 1. Atom-field coupling scheme.\n\n|g jn |r jn |s jn ∆ r jn g s jn ∆ s jn δ jn |e jn g r jn Ω r jn Ω s jn\n\nFIG. 2. Atomic level configuration for atom j in cavity n.\n\nOur scheme is shown in Fig. 1 . A pair of four-level atoms are trapped in each of two distant cavities, respectively, which are connected through an optical fiber. In the short fiber limit [16, 17, 18] , only one fiber mode b is excited and coupled to cavity modes a 1 and a 2 with strength ν [19] . We assume that the cavity modes and the fiber mode have the same frequency ω. The level scheme of atoms is shown in Fig. 2 . Atom j in cavity n is labeled by the index jn with j, n = 1, 2. The distance between the atoms in the same cavity is assumed to be large enough that there is no direct interaction between the atoms. The levels |g jn and |e jn of atom j in cavity n, with j, n = 1, 2 are stable with a long life time. The energy of the level |g jn is taken to be zero as the energy reference point. The lower lying level |e jn , and upper levels |r jn and |s jn have the energy δ jn , and ω r jn and ω s jn , respectively, in the unit with = 1. Transitions |g jn ↔ |s jn and |e jn ↔ |r jn are driven by laser fields of frequencies ω Ls jn and ω Lr jn with Rabi frequencies Ω s jn and Ω r jn and relative phase ϕ, respectively. Transitions |g jn ↔ |r jn and |e jn ↔ |s jn are coupled to the cavity mode a n with the strengths g r jn and g s jn , respectively. Here, we set ∆ r jn = ω r jn -ω = ω r jn -ω Lr jn -δ jn , and ∆ s jn = ω s jn -ω -δ jn = ω s jn -ω Ls jn . Under the Markovian approximation, the master equation of the density matrix for the whole system under consideration can be written as [14]\n\nρT = -i[H, ρ T ] + L cav1 ρ T + L cav2 ρ T + L f iber ρ T , (1) where H = H 0 + H d + H ac + H cf with H 0 = 2 j,n=1 ω r jn |r jn r jn | + ω s jn |s jn s jn | + δ jn |e jn e jn |) + ω( 2 n=1 a † n a n + b † b), H d = 2 j,n=1 Ω s jn 2 e -iω Ls jn t |s jn g jn | + Ω r jn 2 e -i(ω Lr jn t+ϕ) |r jn e jn | + H.c. , H ac = 2 j,n=1 (g r jn |r jn g jn |a n + g s jn |s jn e jn |a n + H.c.), H cf =ν b(a † 1 + a † 2 ) + H.c. . (2)\n\nHere, H 0 is the free energy of atoms and cavity fields, H d is the interaction energy between the atoms and laser fields, H ac is the interaction energy between the atoms and the cavity fields, and H cf describes the interaction between the cavity modes and the fiber mode. The last three terms in (1) describe the relaxation processes of the cavity and fibre modes in the usual vacuum reservoir, taking the forms\n\nL cavn ρ T =κ n (2a n ρ T a † n -a † n a n ρ T -ρ T a † n a n ), L f iber ρ T =κ f (2bρ T b † -b † bρ T -ρ T b † b), ( 3\n\n)\n\nwhere κ n is the leakage rate of photons from cavity n, and κ f is the decay rate of the fiber mode.\n\nLet's introduce collective basis:\n\n|a n = (|g 1n |e 2n - |e 1n |g 2n )/ √ 2, | -1 n = |g 1n |g 2n , |0 n = (|g 1n |e 2n + |e 1n |g 2n )/ √ 2, |1 n = |e 1n |e 2n .\n\nThe states |a n and | -1 n are taken as a qubit n for quantum computation. In the large detuning limit, adiabatically eliminating the excited states and setting 2 ), we obtain the effective interaction Hamiltonian\n\nΩ 1n g r 1n 2∆ r 1n = Ω r 2n g r 2n 2∆ r 2n = β r n and Ω s 1n g s 1n 2∆ s 1n = Ω s 2n g s 2n 2∆ s 2n = β s n , from (\n\nH ef f = n √ 2 a n (β r n e iϕ S + n + β s n S n ) + H.c. + H cf , ( 4\n\n) where S + n = |0 nn -1|+|1 nn 0|.\n\nIn the derivation of (4), we have assumed the resonant condition\n\ng s jn 2 ∆ s jn a † n a n + 3 Ω r jn 2 4∆ r jn = Ω s jn 2 4∆ s jn + g r jn 2 ∆ r jn a † n a n + δ ′ jn .\n\nIn order to satisfy the condition with the flexible choice of Ω r jn , Ω s jn , ∆ r jn and ∆ s jn , we have introduced additional ac-Stark shifts δ ′ jn to states |g jn , which can be generated by using a laser field to couple the level |g jn to an ancillary level.\n\nWe now introduce three normal modes c and c ± with frequencies ω and ω ± √ 2ν by use of the unitary transfor-\n\nmation a 1 = 1 2 (c + + c -+ √ 2c), a 2 = 1 2 (c + + c -- √ 2c), b = 1 √ 2 (c + -c -) [17, 18]. In the limit ν ≫ |β r j |, |β s j |,\n\nneglecting the far off-resonant modes c ± and setting β p 1 = -β p 2 = β p with p = r, s, we can approximately write the effective Hamiltonian (4) as\n\nH ef f = (β r e iϕ S + + β s S)c + H.c., ( 5\n\n)\n\nwhere\n\nS + = S + 1 + S + 2 .\n\nSince the modes c ± are nearly not excited and decoupled with the resonant mode c, the fiber mode b is mostly in the vacuum state, therefore, L f iber ρ T can be neglected, and L cav1 ρ T + L cav2 ρ T can be approximated as\n\nL cav ρ T = κ(2cρ T c † -c † cρ T -ρ T c † c), ( 6\n\n)\n\nwhere κ = (κ 1 + κ 2 )/2. In the bad cavity limit, κ ≫ β, adiabatically eliminating the mode c [12, 14] , from Eq. ( 1 ) with the replacement of the Hamiltonian (2) and the relaxation terms (3) by the effective Hamiltonian (5) and the relaxation term (6), respectively, we can obtain the master equation for the density matrix of the atoms\n\nρ = - Γ 2 (R + Rρ + ρR + R -2RρR + ), ( 7\n\n)\n\nwhere ρ = Tr f (ρ T ), R = S cosh r + e iϕ S † sinh r, r = cosh -1 (β r / β r 2 -β s 2\n\n) and Γ = 2(β r 2 -β s2 )/κ. Eq. ( 7 ) describes the collective interaction of two cascade threelevel atoms with the effective squeezed vacuum reservoir [10] . The parameters β r , β s and ϕ are easily changed and controlled at will by varying the strength and phase of the driving lasers [8] . We will show that a geometric phase gate can be realized through changing these parameters.\n\nThe DFS of the atomic system is spanned by the states which satisfy the equation R(r, ϕ\n\n)|ψ DF (r, ϕ) = 0 [10]. In terms of basis states |e 1 = |a 1 |a 2 , |e 2 = |a 1 | -1 2 , |e 3 = | -1 1 |a 2 , |e 4 = |a 1 |0 2 , |e 5 = |0 1 |a 2 , |e 6 = |a 1 |1 2 , |e 7 = |1 1 |a 2 , |e 8 = |1 1 |1 2 , |e 9 = 1 √ 2 (|1 1 |0 2 + |0 1 |1 2 ), |e 10 = | - 1 1 | -1 2 , |e 11 = 1 √ 2 (|0 1 | -1 2 + | -1 1 |0 2 ), |e 12 = 1 √ 6 (|1 1 | -1 2 + | -1 1 |1 2 ) + 2 √ 6 |0 1 |0 2 )\n\n, the DFS states can be written as\n\n|ψ DF (r, ϕ) 1 =|e 1 , |ψ DF (r, ϕ) j = cosh r √ cosh 2r |e j -e iϕ sinh r √ cosh 2r |e j+4 , j = 2, 3, |ψ DF (r, ϕ) 4 = e 2iϕ (tanh r) 2 |e 8 -2 3 e iϕ tanh r|e 12 + |e 10 (tanh r) 4 + 2 3 (tanh r) 2 + 1 . ( 8\n\n)\n\nLet's introduce a unitary transformation O(r, ϕ) 1, 2, 3, 4. For the transformed density matrix ρ = O † ρO, we have\n\nby |φ i = 12 i=1 O ij (r, ϕ)|e j , where |φ i = |ψ DF i for i =\n\ndρ dt = i[G, ρ] + O † dρ dt O, ( 9\n\n)\n\nwhere G(r, ϕ) = iO † dO dt = iO † [ ṙ dO dr + φ dO dϕ ]. To solve Eq. (9) in the DFS, let's define the time-independent projector Π(0 9) , we obtain the equation of motion for ρDF = Π(0)ρΠ(0)\n\n) = O † Π(r, ϕ)O = 4 i=1 O † |φ i φ i |O = 3 j=1 |e j e j | + |e 10 e 10 | onto the DFS. From (\n\ndρ DF dt =i[G DF , ρDF ] + iΠ(0)GΠ ⊥ (0)ρΠ( 0\n\n)\n\n-iΠ(0)ρΠ ⊥ (0)GΠ(0) + Π(0)O † dρ dt OΠ(0), (10) where Π ⊥ (0) = 1 -Π(0) and G DF = Π(0)GΠ(0). In the limit of ṙ, φ ≪ Γ, the last three terms in Eq. (10) can be neglected [11] . In this way, Eq. (10) is reduced to\n\ndρ DF dt = i[G DF , ρDF ]. ( 11\n\n)\n\nTherefore, in the frame dragged adiabatically by the reservoir, the state of the atoms in the DFS unitarily evolves in time.\n\nIn this section, we investigate how to realize a CPHASE gate through manipulating the engineered reservoir. Suppose that at the initial time the laser field driving the transition |g ↔ |s is switched off but the laser field driving the transition |r ↔ |e is switched on and the atoms are in the DFS state |Ψ(0\n\n) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ) = 4 j=1 |ψ DF (0, 0) j /2.\n\nTo generate a geometric phase for the atomic state, we smoothly change the parameters of the engineered reservoir along a closed loop, which is divided into the following three steps: (1) From time 0 to T 1 , hold on ϕ = 0, and adiabatically increase the parameter r from 0 to r 0 ; (2) From time T 1 to T 2 , hold on r = r 0 , and adiabatically change the phase ϕ from 0 to ϕ 0 ; (3) From time T 2 to T 3 , hold on ϕ = ϕ 0 , and adiabatically decrease r from r 0 to 0. When the cyclic evolution ends, the atomic state becomes\n\n|Ψ(T 3 ) a = 1 2 (|e 1 + e iχ1 |e 2 + e iχ1 |e 3 + e iχ12 |e 10 ), (12) where geometric phases χ 1 = -ν 1 ϕ 0 , χ 12 = -ν 12 ϕ 0 with ν 1 = sinh 2 r0 sinh 2 r0+cosh 2 r0 , ν 12 = 2 tanh 4 r0+ 2 3 tanh 2 r0 tanh 4 r0+ 2 3 tanh 2 r0+1 . By per- forming local transformations U 1 = e -iχ1 |-1 11 -1| and 4 U 2 = e -iχ1 | -1 22 -1|,\n\nthe state (12) can be written as\n\n|Ψ ′ (T 3 ) a = U 1 U 2 |Ψ(T 3 ) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + e i∆ | -1 1 | -1 2 ), where ∆ = χ 12 -2χ 1 = (2ν 1 -ν 12 )ϕ 0 .\n\nThus, the CPHASE gate with the phase shift ∆ is realized. If both the atoms in cavity 1 and the atoms in cavity 2 \"see\" different environments, |χ 12 | must be equal to |2χ 1 | and ∆ = 0. Therefore, the phase shift ∆ results from the collective coupling of the atoms in both cavities with the same engineered environment.\n\nIf r 0 = atanh( 4/3 -1) ≃ 0.4157, |ν 12 | = |ν 1 |. Under this condition with ϕ 0 = π/ν 1 ,\n\nthe state of the atoms at the time T\n\n3 is |Ψ ′′ (T 3 ) a = -1 2 (-|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ).\n\nIn this case, the Controlled-Z gate between the two qubits is realized without local transformations. 0 0.2 0.4 0.6 0.8 0.9965 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 r 0 F r T=100/Γ T=200/Γ T=400/Γ (a)\n\nFIG. 3. Fidelity F r of the atomic state. 0 0.2 0.4 0.6 0.8 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 r 0 F p T = 200/Γ T = 400/Γ T = 1000/Γ (b) FIG. 4. Fidelity F p of the atomic state.\n\nThe above results depend on the adiabatical approximation. To check the adiabatical condition, we numerically simulate the following two examples. In the first example, we suppose that at the initial time the atoms are in the state |Ψ 1 a = (|a 1 |a 2 + |ψ DF (0, 0) 2 )/ √ 2 and the laser field driving the transition |e ↔ |r are turned on. Then, by slowly switching the laser field driving the transition |g ↔ |s , we increase the parameter r from 0 to r 0 according to the linear function r(t) = r 0 t/T . In the adiabatical limit (T ≫ Γ -1 ), the atomic state becomes\n\n|Ψ ′ 1 a = (|a 1 |a 2 + |ψ DF (r 0 , 0) 2 )/ √ 2\n\nat the time T . On the other hand, in the Hilbert space spanned by the basis states {|e i } for i = 1, 2, • • • , 12 , we can numerically solve Eq. ( 7 ) and obtain the density matrix ρ 1 (T ) of the atoms. Let's define\n\nF r = a Ψ ′ 1 |ρ 1 (T )|Ψ ′ 1 a as\n\nthe fidelity for this process. As shown in Fig. 3 , if T > 100/Γ, F r is always bigger than 0.997 if r ∈ (0, 0.8), corresponding to the almost perfect evolution.\n\nIn the second example, we suppose that the atoms are initially in the state\n\n|Ψ 2 a = (|a 1 |a 2 + |ψ DF (r, 0) 4 )/\n\n√ 2 and all the driving fields are turned on to hold the parameters r = r 0 and ϕ = 0. By adiabatically changing the phase ϕ from 0 to 2π at the rate φ = 2π/T , the atomic state at the time\n\nT becomes |Ψ ′ 2 a = (|a 1 |a 2 + e iχ12 |ψ DF (r, 2π) 4 )/ √ 2.\n\nLet's define the fidelity for this example as\n\nF p = a Ψ ′ 2 |ρ(T )|Ψ ′ a ,\n\nwhere ρ(T ) is the numerical solution of Eq. ( 7 ). As shown in Fig. 4 , F p increases as T increases but decreases as the parameter r 0 increases. If T > 1000/Γ, F p is larger than 0.992 for 0 < r 0 < 0.8. From these two examples, we find that to fulfill the adiabatical condition the time used in the step 2 should be much longer than in the steps 1 and 3.\n\nA controlled-Z gate has been numerically simulated by directly solving Eq. ( 7 ) with r 0 = 0.5, and ϕ\n\n0 = π/|2ν 1 -ν 12 |.\n\nIn the simulation, we set ṙ = r 0 /T 1 in the steps 1 and 3, and φ = ϕ 0 /(T 2 -T 1 ) in the step 2 with T 1 = 0.05T 3 and T 2 -T 1 = 0.90T 3 . If T 3 > 1100/Γ, we find that the fidelity F = a Ψ(T 3 )|ρ(T 3 )|Ψ(T 3 ) a is larger than 0.95. For an almost perfect controlled-Z gate with F > 0.99, we find that T 3 must be longer than 6000/Γ. Now let's briefly discuss the effects of the atomic spontaneous emission, the fiber mode decay and cavity photon leakage. For simplicity but without the loss of generality, we suppose that atomic spontaneous emission rates of the excited levels are equal to γ. In the large detunig limit, the characteristic spontaneous emission rate of the atoms is γ ef f = γ(Ω 2 /2∆ 2 ) [14, 16] and the effective decay rate of the fiber mode is\n\nκ ef f = κ f Ω 2 g 2 /(4∆ 2 ν 2 ). If κ f ≤ γ and g 2 ≪ ν 2 , κ\n\nef f be much smaller than γ ef f . Under this condition, the present scheme is feasible if Γ ≫ γ ef f . In the current cavity quantum dynamic (CQED) experiment, the parameters (g, κ, γ) = (2000, 10, 10) MHz could be available [20] . If setting Ω/(2∆) = 1 √ 2 × 10 -3 , we have Γ ≃ 4 × 10 4 γ ef f . The condition is held. In the present scheme, the large cavity decay rate is required to ensure that the cavity modes are in a broadband squeezed vacuum reservoir and then the atoms always \"see\" the broadband squeezed vacuum reservoir during the dynamic evolution. For an arbitrary small but nonzero value of the squeezing degree of the reservoir, a CPHASE gate with arbitrary high fidelity can always be realized in the represent scheme. The cavity decay does 5 not directly affect the fidelity of the realized CPHASE gates. However, the larger the decay rate is, the longer the operation time of the CPHASE gates is. Thus, we have the condition κ >> β, γ ef f for realizing the reliable CPHASE gates. Based on the parameters quoted above, this condition can be well satisfied. With the parameters of the current CQED experiment, we find that the operation time of the controlled-Z gate, with fidelity larger than 0.95, is about 2.8 ms. It is much shorter than both 1/γ ef f and the single-atom trapping time in cavity [21] . On the other hand, the present scheme needs a strong coupling between the cavity and the fiber. This could be realized at the current experiment [22] . Therefore, the requirement for the realization of the present scheme can be satisfied with the current technology.\n\nWe propose a cavity-atom coupled scheme for the realization of quantum controlling gates, in which each of two pairs of four-level atoms in two distant cavities connected by a short optical fibre are simultaneously driven by laser fields and coupled to the local cavity modes through the double Raman transition configuration. We show that an effective squeezing reservoir coupled to the multilevel atoms can be engineered under appropriate driving condition and bad cavity limit. We find that in the scheme a CPHASE gate with arbitrary phase shift can be implemented through adiabatically changing the strength and phase of driving fields along a closed loop. It is also noticed that the larger the effective coupling strength between the environment and the atoms is, the more reliable the realized CPHASE gate is.\n\nWe thank Yun-feng Xiao and Wen-ping He for valuable discussions and suggestions. This work was supported by the Natural Science Foundation of China (Grant Nos. 10674106, 60778021 and 05-06-01).\n\n[1] P. W. Shor, in Proceeings, 35 th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Press, Los Almitos, CA, 1994), p. 124. [2] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997). [3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). [4] D. A. Lidar and et al., Phys. Rev. Lett. 81, 2594 (1998). [5] M. V. Berry, Proc. R. Soc. A 392, 45 (1984). [6] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [7] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301 (2005). [8] L.-M. Duan and et al., Science 292, 1695 (2001). [9] L.-A. Wu and et al., Phys. Rev. Lett. 95, 130501 (2005). [10] Angelo Carollo et al., Phys. Rev. Lett. 96, 150403 (2006). [11] Angelo Carollo et al., Phys. Rev. Lett. 96, 020403 (2006). [12] J. I. Cirac, Phys. Rev. A 46, 4354 (1992). [13] C. J. Myatt and et al., Nature (London) 403, 269 (2000). [14] S. G. Clark and A. S. Parkins, Phys. Rev. Lett. 90, 047905 (2003). [15] S. Lloyd, Phys. Rev. Lett. 75, 346 (1995). [16] T. Pellizzari, Phys. Rev. Lett. 79, 5242 (1997). [17] A. Serafini and et al., Phys. Rev. Lett. 96, 010503 (2006) . [18] Zhang-qi Yin and Fu-li Li, Phys. Rev. A 75, 012324 (2007). [19] S. J. van Enk and et al., Phys. Rev. A 59, 2659 (1999). [20] S. M. Spillane et al., Phys. Rev. A 71, 013817 (2005). [21] Stefan Nuβmann and et al., Nature Phys. 1, 122 (2005). [22] S. M. Spillane et al., Phys. Rev. Lett. 91, 043902 (2003)."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"Quantum computation, attracting much current interest since Shor's algorithm [1] was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevitably coupled to their environment so that entanglement is too fragile to be retained. This makes the realization of quantum computation extremely difficult in the real world. In order to overcome this difficulty, one proposed the decoherence-free space concept [2, 3] . It is found that when qubits involved in quantum computation collectively interact with a same environment there exists a \"protected\" subspace in the entire Hilbert space, in which the qubits are immune to the decoherence effects induced by the environment. This subspace is called decoherence-free space (DFS). To perform quantum computation in a DFS, one has to design the specific Hamiltonian containing controlling parameters, which eigenspace is spanned by DFS states and the state-unitary manipulation related to quantum computation goal is implemented by changing the controlling parameters [4] .\n\nAs well known, instantaneous eigenstates of a quantum system with the time-dependent Hamiltonian may acquire a geometric phase when the time-dependent parameters adiabatically undergo a closed loop in the parameter space [5] . The phase depends only on the swept solid angle by the parameter vector in the parameter space. This feature can be utilized to implement geometric quantum computation (GQC) which is resilient to stochastic control errors [6, 7, 8] . On combining the DFS approach with the GQC scheme, one may build quantum gates which may be immune to both the environment-induced decoherence effects and the control-led errors [9] . In the scheme, quantum logical bits are represented by degenerate eigenstates of the parameterized Hamiltonian. These states have the features: they belong to DFS, and unitarily evolve in time and acquire a geometric phase when the controlling parameters adiabatically vary and undergo a closed loop.\n\nIn the recent paper [10], Carollo and coworkers showed that a cascade three-level atom interacting with a broadband squeezed vacuum bosonic bath can be prepared in a state which is decoupled to the environment. This state depends on the reservoir parameters such as squeezing degree and phase angle. As the squeezing parameters smoothly vary, the atomic state can unitarily evolve in time and always be in the manifold of the DFS. Moreover, after a cyclic evolution of the squeezing parameters, the state acquires a geometric phase. This investigation has been generalized to cases where both quantum systems and manipulated reservoir under consideration are not restricted to cascade three-level atoms and squeezed vacuum [11] . These results strongly inspire us that instead of engineering Hamiltonian one may implement the decoherence-free GQC by engineering and manipulating reservoir.\n\nIn this paper, we propose a scheme in which the quantum-reservoir engineering [12, 13, 14] is combined with DFS and Berry phase together to realize a twoqubit CPHASE gate [15] . We show that atomic states can unitarily evolve in time in a DFS if the change rate of reservoir parameters is much smaller than the characteristic relaxation time of an atom-reservoir coupled system. Moreover, we find that as the reservoir parameters adiabatically change in time along an appropriate closed loop, the atomic state in the DFS acquires a Berry phase and a CPHASE gate with arbitrary phase shift can be realized. To our knowledge, it is the first proposal for the realization of quantum gates by engineering and steering the environment.\n\nThis paper is organized as follows. In Sec. II, we introduce a cavity-atom coupling model in which two pairs of four-level atoms are respectively trapped in two distant cavities that are connected by an optical fiber. In the model, each of pairs of the atoms are simultaneously driven by laser fields and coupled to the local cavity 2 modes through the double Raman transition configuration. Under large detuning and bad cavity limits, we investigate to engineer an effective broadband squeezing reservoir for the atoms. In Sec. III, we analyze how to realize controlling gates between the atoms trapped in the two cavities by steering the squeezing reservoir. Section IV contains conclusions of our investigations."},{"section_type":"OTHER","section_title":"laser laser fibre","text":"FIG. 1. Atom-field coupling scheme.\n\n|g jn |r jn |s jn ∆ r jn g s jn ∆ s jn δ jn |e jn g r jn Ω r jn Ω s jn\n\nFIG. 2. Atomic level configuration for atom j in cavity n."},{"section_type":"OTHER","section_title":"II. ENGINEERING A SQUEEZING ENVIRONMENT AND GENERATING A DECOHERENCE-FREE SUBSPACE","text":"Our scheme is shown in Fig. 1 . A pair of four-level atoms are trapped in each of two distant cavities, respectively, which are connected through an optical fiber. In the short fiber limit [16, 17, 18] , only one fiber mode b is excited and coupled to cavity modes a 1 and a 2 with strength ν [19] . We assume that the cavity modes and the fiber mode have the same frequency ω. The level scheme of atoms is shown in Fig. 2 . Atom j in cavity n is labeled by the index jn with j, n = 1, 2. The distance between the atoms in the same cavity is assumed to be large enough that there is no direct interaction between the atoms. The levels |g jn and |e jn of atom j in cavity n, with j, n = 1, 2 are stable with a long life time. The energy of the level |g jn is taken to be zero as the energy reference point. The lower lying level |e jn , and upper levels |r jn and |s jn have the energy δ jn , and ω r jn and ω s jn , respectively, in the unit with = 1. Transitions |g jn ↔ |s jn and |e jn ↔ |r jn are driven by laser fields of frequencies ω Ls jn and ω Lr jn with Rabi frequencies Ω s jn and Ω r jn and relative phase ϕ, respectively. Transitions |g jn ↔ |r jn and |e jn ↔ |s jn are coupled to the cavity mode a n with the strengths g r jn and g s jn , respectively. Here, we set ∆ r jn = ω r jn -ω = ω r jn -ω Lr jn -δ jn , and ∆ s jn = ω s jn -ω -δ jn = ω s jn -ω Ls jn . Under the Markovian approximation, the master equation of the density matrix for the whole system under consideration can be written as [14]\n\nρT = -i[H, ρ T ] + L cav1 ρ T + L cav2 ρ T + L f iber ρ T , (1) where H = H 0 + H d + H ac + H cf with H 0 = 2 j,n=1 ω r jn |r jn r jn | + ω s jn |s jn s jn | + δ jn |e jn e jn |) + ω( 2 n=1 a † n a n + b † b), H d = 2 j,n=1 Ω s jn 2 e -iω Ls jn t |s jn g jn | + Ω r jn 2 e -i(ω Lr jn t+ϕ) |r jn e jn | + H.c. , H ac = 2 j,n=1 (g r jn |r jn g jn |a n + g s jn |s jn e jn |a n + H.c.), H cf =ν b(a † 1 + a † 2 ) + H.c. . (2)\n\nHere, H 0 is the free energy of atoms and cavity fields, H d is the interaction energy between the atoms and laser fields, H ac is the interaction energy between the atoms and the cavity fields, and H cf describes the interaction between the cavity modes and the fiber mode. The last three terms in (1) describe the relaxation processes of the cavity and fibre modes in the usual vacuum reservoir, taking the forms\n\nL cavn ρ T =κ n (2a n ρ T a † n -a † n a n ρ T -ρ T a † n a n ), L f iber ρ T =κ f (2bρ T b † -b † bρ T -ρ T b † b), ( 3\n\n)\n\nwhere κ n is the leakage rate of photons from cavity n, and κ f is the decay rate of the fiber mode.\n\nLet's introduce collective basis:\n\n|a n = (|g 1n |e 2n - |e 1n |g 2n )/ √ 2, | -1 n = |g 1n |g 2n , |0 n = (|g 1n |e 2n + |e 1n |g 2n )/ √ 2, |1 n = |e 1n |e 2n .\n\nThe states |a n and | -1 n are taken as a qubit n for quantum computation. In the large detuning limit, adiabatically eliminating the excited states and setting 2 ), we obtain the effective interaction Hamiltonian\n\nΩ 1n g r 1n 2∆ r 1n = Ω r 2n g r 2n 2∆ r 2n = β r n and Ω s 1n g s 1n 2∆ s 1n = Ω s 2n g s 2n 2∆ s 2n = β s n , from (\n\nH ef f = n √ 2 a n (β r n e iϕ S + n + β s n S n ) + H.c. + H cf , ( 4\n\n) where S + n = |0 nn -1|+|1 nn 0|.\n\nIn the derivation of (4), we have assumed the resonant condition\n\ng s jn 2 ∆ s jn a † n a n + 3 Ω r jn 2 4∆ r jn = Ω s jn 2 4∆ s jn + g r jn 2 ∆ r jn a † n a n + δ ′ jn .\n\nIn order to satisfy the condition with the flexible choice of Ω r jn , Ω s jn , ∆ r jn and ∆ s jn , we have introduced additional ac-Stark shifts δ ′ jn to states |g jn , which can be generated by using a laser field to couple the level |g jn to an ancillary level.\n\nWe now introduce three normal modes c and c ± with frequencies ω and ω ± √ 2ν by use of the unitary transfor-\n\nmation a 1 = 1 2 (c + + c -+ √ 2c), a 2 = 1 2 (c + + c -- √ 2c), b = 1 √ 2 (c + -c -) [17, 18]. In the limit ν ≫ |β r j |, |β s j |,\n\nneglecting the far off-resonant modes c ± and setting β p 1 = -β p 2 = β p with p = r, s, we can approximately write the effective Hamiltonian (4) as\n\nH ef f = (β r e iϕ S + + β s S)c + H.c., ( 5\n\n)\n\nwhere\n\nS + = S + 1 + S + 2 .\n\nSince the modes c ± are nearly not excited and decoupled with the resonant mode c, the fiber mode b is mostly in the vacuum state, therefore, L f iber ρ T can be neglected, and L cav1 ρ T + L cav2 ρ T can be approximated as\n\nL cav ρ T = κ(2cρ T c † -c † cρ T -ρ T c † c), ( 6\n\n)\n\nwhere κ = (κ 1 + κ 2 )/2. In the bad cavity limit, κ ≫ β, adiabatically eliminating the mode c [12, 14] , from Eq. ( 1 ) with the replacement of the Hamiltonian (2) and the relaxation terms (3) by the effective Hamiltonian (5) and the relaxation term (6), respectively, we can obtain the master equation for the density matrix of the atoms\n\nρ = - Γ 2 (R + Rρ + ρR + R -2RρR + ), ( 7\n\n)\n\nwhere ρ = Tr f (ρ T ), R = S cosh r + e iϕ S † sinh r, r = cosh -1 (β r / β r 2 -β s 2\n\n) and Γ = 2(β r 2 -β s2 )/κ. Eq. ( 7 ) describes the collective interaction of two cascade threelevel atoms with the effective squeezed vacuum reservoir [10] . The parameters β r , β s and ϕ are easily changed and controlled at will by varying the strength and phase of the driving lasers [8] . We will show that a geometric phase gate can be realized through changing these parameters.\n\nThe DFS of the atomic system is spanned by the states which satisfy the equation R(r, ϕ\n\n)|ψ DF (r, ϕ) = 0 [10]. In terms of basis states |e 1 = |a 1 |a 2 , |e 2 = |a 1 | -1 2 , |e 3 = | -1 1 |a 2 , |e 4 = |a 1 |0 2 , |e 5 = |0 1 |a 2 , |e 6 = |a 1 |1 2 , |e 7 = |1 1 |a 2 , |e 8 = |1 1 |1 2 , |e 9 = 1 √ 2 (|1 1 |0 2 + |0 1 |1 2 ), |e 10 = | - 1 1 | -1 2 , |e 11 = 1 √ 2 (|0 1 | -1 2 + | -1 1 |0 2 ), |e 12 = 1 √ 6 (|1 1 | -1 2 + | -1 1 |1 2 ) + 2 √ 6 |0 1 |0 2 )\n\n, the DFS states can be written as\n\n|ψ DF (r, ϕ) 1 =|e 1 , |ψ DF (r, ϕ) j = cosh r √ cosh 2r |e j -e iϕ sinh r √ cosh 2r |e j+4 , j = 2, 3, |ψ DF (r, ϕ) 4 = e 2iϕ (tanh r) 2 |e 8 -2 3 e iϕ tanh r|e 12 + |e 10 (tanh r) 4 + 2 3 (tanh r) 2 + 1 . ( 8\n\n)\n\nLet's introduce a unitary transformation O(r, ϕ) 1, 2, 3, 4. For the transformed density matrix ρ = O † ρO, we have\n\nby |φ i = 12 i=1 O ij (r, ϕ)|e j , where |φ i = |ψ DF i for i =\n\ndρ dt = i[G, ρ] + O † dρ dt O, ( 9\n\n)\n\nwhere G(r, ϕ) = iO † dO dt = iO † [ ṙ dO dr + φ dO dϕ ]. To solve Eq. (9) in the DFS, let's define the time-independent projector Π(0 9) , we obtain the equation of motion for ρDF = Π(0)ρΠ(0)\n\n) = O † Π(r, ϕ)O = 4 i=1 O † |φ i φ i |O = 3 j=1 |e j e j | + |e 10 e 10 | onto the DFS. From (\n\ndρ DF dt =i[G DF , ρDF ] + iΠ(0)GΠ ⊥ (0)ρΠ( 0\n\n)\n\n-iΠ(0)ρΠ ⊥ (0)GΠ(0) + Π(0)O † dρ dt OΠ(0), (10) where Π ⊥ (0) = 1 -Π(0) and G DF = Π(0)GΠ(0). In the limit of ṙ, φ ≪ Γ, the last three terms in Eq. (10) can be neglected [11] . In this way, Eq. (10) is reduced to\n\ndρ DF dt = i[G DF , ρDF ]. ( 11\n\n)\n\nTherefore, in the frame dragged adiabatically by the reservoir, the state of the atoms in the DFS unitarily evolves in time."},{"section_type":"OTHER","section_title":"III. REALIZING CONTROLLING PHASE GATES THROUGH MANIPULATING THE SQUEEZING ENVIRONMENT","text":"In this section, we investigate how to realize a CPHASE gate through manipulating the engineered reservoir. Suppose that at the initial time the laser field driving the transition |g ↔ |s is switched off but the laser field driving the transition |r ↔ |e is switched on and the atoms are in the DFS state |Ψ(0\n\n) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ) = 4 j=1 |ψ DF (0, 0) j /2.\n\nTo generate a geometric phase for the atomic state, we smoothly change the parameters of the engineered reservoir along a closed loop, which is divided into the following three steps: (1) From time 0 to T 1 , hold on ϕ = 0, and adiabatically increase the parameter r from 0 to r 0 ; (2) From time T 1 to T 2 , hold on r = r 0 , and adiabatically change the phase ϕ from 0 to ϕ 0 ; (3) From time T 2 to T 3 , hold on ϕ = ϕ 0 , and adiabatically decrease r from r 0 to 0. When the cyclic evolution ends, the atomic state becomes\n\n|Ψ(T 3 ) a = 1 2 (|e 1 + e iχ1 |e 2 + e iχ1 |e 3 + e iχ12 |e 10 ), (12) where geometric phases χ 1 = -ν 1 ϕ 0 , χ 12 = -ν 12 ϕ 0 with ν 1 = sinh 2 r0 sinh 2 r0+cosh 2 r0 , ν 12 = 2 tanh 4 r0+ 2 3 tanh 2 r0 tanh 4 r0+ 2 3 tanh 2 r0+1 . By per- forming local transformations U 1 = e -iχ1 |-1 11 -1| and 4 U 2 = e -iχ1 | -1 22 -1|,\n\nthe state (12) can be written as\n\n|Ψ ′ (T 3 ) a = U 1 U 2 |Ψ(T 3 ) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + e i∆ | -1 1 | -1 2 ), where ∆ = χ 12 -2χ 1 = (2ν 1 -ν 12 )ϕ 0 .\n\nThus, the CPHASE gate with the phase shift ∆ is realized. If both the atoms in cavity 1 and the atoms in cavity 2 \"see\" different environments, |χ 12 | must be equal to |2χ 1 | and ∆ = 0. Therefore, the phase shift ∆ results from the collective coupling of the atoms in both cavities with the same engineered environment.\n\nIf r 0 = atanh( 4/3 -1) ≃ 0.4157, |ν 12 | = |ν 1 |. Under this condition with ϕ 0 = π/ν 1 ,\n\nthe state of the atoms at the time T\n\n3 is |Ψ ′′ (T 3 ) a = -1 2 (-|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ).\n\nIn this case, the Controlled-Z gate between the two qubits is realized without local transformations. 0 0.2 0.4 0.6 0.8 0.9965 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 r 0 F r T=100/Γ T=200/Γ T=400/Γ (a)\n\nFIG. 3. Fidelity F r of the atomic state. 0 0.2 0.4 0.6 0.8 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 r 0 F p T = 200/Γ T = 400/Γ T = 1000/Γ (b) FIG. 4. Fidelity F p of the atomic state.\n\nThe above results depend on the adiabatical approximation. To check the adiabatical condition, we numerically simulate the following two examples. In the first example, we suppose that at the initial time the atoms are in the state |Ψ 1 a = (|a 1 |a 2 + |ψ DF (0, 0) 2 )/ √ 2 and the laser field driving the transition |e ↔ |r are turned on. Then, by slowly switching the laser field driving the transition |g ↔ |s , we increase the parameter r from 0 to r 0 according to the linear function r(t) = r 0 t/T . In the adiabatical limit (T ≫ Γ -1 ), the atomic state becomes\n\n|Ψ ′ 1 a = (|a 1 |a 2 + |ψ DF (r 0 , 0) 2 )/ √ 2\n\nat the time T . On the other hand, in the Hilbert space spanned by the basis states {|e i } for i = 1, 2, • • • , 12 , we can numerically solve Eq. ( 7 ) and obtain the density matrix ρ 1 (T ) of the atoms. Let's define\n\nF r = a Ψ ′ 1 |ρ 1 (T )|Ψ ′ 1 a as\n\nthe fidelity for this process. As shown in Fig. 3 , if T > 100/Γ, F r is always bigger than 0.997 if r ∈ (0, 0.8), corresponding to the almost perfect evolution.\n\nIn the second example, we suppose that the atoms are initially in the state\n\n|Ψ 2 a = (|a 1 |a 2 + |ψ DF (r, 0) 4 )/\n\n√ 2 and all the driving fields are turned on to hold the parameters r = r 0 and ϕ = 0. By adiabatically changing the phase ϕ from 0 to 2π at the rate φ = 2π/T , the atomic state at the time\n\nT becomes |Ψ ′ 2 a = (|a 1 |a 2 + e iχ12 |ψ DF (r, 2π) 4 )/ √ 2.\n\nLet's define the fidelity for this example as\n\nF p = a Ψ ′ 2 |ρ(T )|Ψ ′ a ,\n\nwhere ρ(T ) is the numerical solution of Eq. ( 7 ). As shown in Fig. 4 , F p increases as T increases but decreases as the parameter r 0 increases. If T > 1000/Γ, F p is larger than 0.992 for 0 < r 0 < 0.8. From these two examples, we find that to fulfill the adiabatical condition the time used in the step 2 should be much longer than in the steps 1 and 3.\n\nA controlled-Z gate has been numerically simulated by directly solving Eq. ( 7 ) with r 0 = 0.5, and ϕ\n\n0 = π/|2ν 1 -ν 12 |.\n\nIn the simulation, we set ṙ = r 0 /T 1 in the steps 1 and 3, and φ = ϕ 0 /(T 2 -T 1 ) in the step 2 with T 1 = 0.05T 3 and T 2 -T 1 = 0.90T 3 . If T 3 > 1100/Γ, we find that the fidelity F = a Ψ(T 3 )|ρ(T 3 )|Ψ(T 3 ) a is larger than 0.95. For an almost perfect controlled-Z gate with F > 0.99, we find that T 3 must be longer than 6000/Γ. Now let's briefly discuss the effects of the atomic spontaneous emission, the fiber mode decay and cavity photon leakage. For simplicity but without the loss of generality, we suppose that atomic spontaneous emission rates of the excited levels are equal to γ. In the large detunig limit, the characteristic spontaneous emission rate of the atoms is γ ef f = γ(Ω 2 /2∆ 2 ) [14, 16] and the effective decay rate of the fiber mode is\n\nκ ef f = κ f Ω 2 g 2 /(4∆ 2 ν 2 ). If κ f ≤ γ and g 2 ≪ ν 2 , κ\n\nef f be much smaller than γ ef f . Under this condition, the present scheme is feasible if Γ ≫ γ ef f . In the current cavity quantum dynamic (CQED) experiment, the parameters (g, κ, γ) = (2000, 10, 10) MHz could be available [20] . If setting Ω/(2∆) = 1 √ 2 × 10 -3 , we have Γ ≃ 4 × 10 4 γ ef f . The condition is held. In the present scheme, the large cavity decay rate is required to ensure that the cavity modes are in a broadband squeezed vacuum reservoir and then the atoms always \"see\" the broadband squeezed vacuum reservoir during the dynamic evolution. For an arbitrary small but nonzero value of the squeezing degree of the reservoir, a CPHASE gate with arbitrary high fidelity can always be realized in the represent scheme. The cavity decay does 5 not directly affect the fidelity of the realized CPHASE gates. However, the larger the decay rate is, the longer the operation time of the CPHASE gates is. Thus, we have the condition κ >> β, γ ef f for realizing the reliable CPHASE gates. Based on the parameters quoted above, this condition can be well satisfied. With the parameters of the current CQED experiment, we find that the operation time of the controlled-Z gate, with fidelity larger than 0.95, is about 2.8 ms. It is much shorter than both 1/γ ef f and the single-atom trapping time in cavity [21] . On the other hand, the present scheme needs a strong coupling between the cavity and the fiber. This could be realized at the current experiment [22] . Therefore, the requirement for the realization of the present scheme can be satisfied with the current technology."},{"section_type":"CONCLUSION","section_title":"IV. CONCLUSIONS","text":"We propose a cavity-atom coupled scheme for the realization of quantum controlling gates, in which each of two pairs of four-level atoms in two distant cavities connected by a short optical fibre are simultaneously driven by laser fields and coupled to the local cavity modes through the double Raman transition configuration. We show that an effective squeezing reservoir coupled to the multilevel atoms can be engineered under appropriate driving condition and bad cavity limit. We find that in the scheme a CPHASE gate with arbitrary phase shift can be implemented through adiabatically changing the strength and phase of driving fields along a closed loop. It is also noticed that the larger the effective coupling strength between the environment and the atoms is, the more reliable the realized CPHASE gate is."},{"section_type":"OTHER","section_title":"Acknowledgments","text":"We thank Yun-feng Xiao and Wen-ping He for valuable discussions and suggestions. This work was supported by the Natural Science Foundation of China (Grant Nos. 10674106, 60778021 and 05-06-01).\n\n[1] P. W. Shor, in Proceeings, 35 th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Press, Los Almitos, CA, 1994), p. 124. [2] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997). [3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). [4] D. A. Lidar and et al., Phys. Rev. Lett. 81, 2594 (1998). [5] M. V. Berry, Proc. R. Soc. A 392, 45 (1984). [6] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [7] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301 (2005). [8] L.-M. Duan and et al., Science 292, 1695 (2001). [9] L.-A. Wu and et al., Phys. Rev. Lett. 95, 130501 (2005). [10] Angelo Carollo et al., Phys. Rev. Lett. 96, 150403 (2006). [11] Angelo Carollo et al., Phys. Rev. Lett. 96, 020403 (2006). [12] J. I. Cirac, Phys. Rev. A 46, 4354 (1992). [13] C. J. Myatt and et al., Nature (London) 403, 269 (2000). [14] S. G. Clark and A. S. Parkins, Phys. Rev. Lett. 90, 047905 (2003). [15] S. Lloyd, Phys. Rev. Lett. 75, 346 (1995). [16] T. Pellizzari, Phys. Rev. Lett. 79, 5242 (1997). [17] A. Serafini and et al., Phys. Rev. Lett. 96, 010503 (2006) . [18] Zhang-qi Yin and Fu-li Li, Phys. Rev. A 75, 012324 (2007). [19] S. J. van Enk and et al., Phys. Rev. A 59, 2659 (1999). [20] S. M. Spillane et al., Phys. Rev. A 71, 013817 (2005). [21] Stefan Nuβmann and et al., Nature Phys. 1, 122 (2005). [22] S. M. Spillane et al., Phys. Rev. Lett. 91, 043902 (2003)."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"Quantum computation, attracting much current interest since Shor's algorithm [1] was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevitably coupled to their environment so that entanglement is too fragile to be retained. This makes the realization of quantum computation extremely difficult in the real world. In order to overcome this difficulty, one proposed the decoherence-free space concept [2, 3] . It is found that when qubits involved in quantum computation collectively interact with a same environment there exists a \\\"protected\\\" subspace in the entire Hilbert space, in which the qubits are immune to the decoherence effects induced by the environment. This subspace is called decoherence-free space (DFS). To perform quantum computation in a DFS, one has to design the specific Hamiltonian containing controlling parameters, which eigenspace is spanned by DFS states and the state-unitary manipulation related to quantum computation goal is implemented by changing the controlling parameters [4] .\\n\\nAs well known, instantaneous eigenstates of a quantum system with the time-dependent Hamiltonian may acquire a geometric phase when the time-dependent parameters adiabatically undergo a closed loop in the parameter space [5] . The phase depends only on the swept solid angle by the parameter vector in the parameter space. This feature can be utilized to implement geometric quantum computation (GQC) which is resilient to stochastic control errors [6, 7, 8] . On combining the DFS approach with the GQC scheme, one may build quantum gates which may be immune to both the environment-induced decoherence effects and the control-led errors [9] . In the scheme, quantum logical bits are represented by degenerate eigenstates of the parameterized Hamiltonian. These states have the features: they belong to DFS, and unitarily evolve in time and acquire a geometric phase when the controlling parameters adiabatically vary and undergo a closed loop.\\n\\nIn the recent paper [10], Carollo and coworkers showed that a cascade three-level atom interacting with a broadband squeezed vacuum bosonic bath can be prepared in a state which is decoupled to the environment. This state depends on the reservoir parameters such as squeezing degree and phase angle. As the squeezing parameters smoothly vary, the atomic state can unitarily evolve in time and always be in the manifold of the DFS. Moreover, after a cyclic evolution of the squeezing parameters, the state acquires a geometric phase. This investigation has been generalized to cases where both quantum systems and manipulated reservoir under consideration are not restricted to cascade three-level atoms and squeezed vacuum [11] . These results strongly inspire us that instead of engineering Hamiltonian one may implement the decoherence-free GQC by engineering and manipulating reservoir.\\n\\nIn this paper, we propose a scheme in which the quantum-reservoir engineering [12, 13, 14] is combined with DFS and Berry phase together to realize a twoqubit CPHASE gate [15] . We show that atomic states can unitarily evolve in time in a DFS if the change rate of reservoir parameters is much smaller than the characteristic relaxation time of an atom-reservoir coupled system. Moreover, we find that as the reservoir parameters adiabatically change in time along an appropriate closed loop, the atomic state in the DFS acquires a Berry phase and a CPHASE gate with arbitrary phase shift can be realized. To our knowledge, it is the first proposal for the realization of quantum gates by engineering and steering the environment.\\n\\nThis paper is organized as follows. In Sec. II, we introduce a cavity-atom coupling model in which two pairs of four-level atoms are respectively trapped in two distant cavities that are connected by an optical fiber. In the model, each of pairs of the atoms are simultaneously driven by laser fields and coupled to the local cavity 2 modes through the double Raman transition configuration. Under large detuning and bad cavity limits, we investigate to engineer an effective broadband squeezing reservoir for the atoms. In Sec. III, we analyze how to realize controlling gates between the atoms trapped in the two cavities by steering the squeezing reservoir. Section IV contains conclusions of our investigations.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"laser laser fibre\",\n \"text\": \"FIG. 1. Atom-field coupling scheme.\\n\\n|g jn |r jn |s jn ∆ r jn g s jn ∆ s jn δ jn |e jn g r jn Ω r jn Ω s jn\\n\\nFIG. 2. Atomic level configuration for atom j in cavity n.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. ENGINEERING A SQUEEZING ENVIRONMENT AND GENERATING A DECOHERENCE-FREE SUBSPACE\",\n \"text\": \"Our scheme is shown in Fig. 1 . A pair of four-level atoms are trapped in each of two distant cavities, respectively, which are connected through an optical fiber. In the short fiber limit [16, 17, 18] , only one fiber mode b is excited and coupled to cavity modes a 1 and a 2 with strength ν [19] . We assume that the cavity modes and the fiber mode have the same frequency ω. The level scheme of atoms is shown in Fig. 2 . Atom j in cavity n is labeled by the index jn with j, n = 1, 2. The distance between the atoms in the same cavity is assumed to be large enough that there is no direct interaction between the atoms. The levels |g jn and |e jn of atom j in cavity n, with j, n = 1, 2 are stable with a long life time. The energy of the level |g jn is taken to be zero as the energy reference point. The lower lying level |e jn , and upper levels |r jn and |s jn have the energy δ jn , and ω r jn and ω s jn , respectively, in the unit with = 1. Transitions |g jn ↔ |s jn and |e jn ↔ |r jn are driven by laser fields of frequencies ω Ls jn and ω Lr jn with Rabi frequencies Ω s jn and Ω r jn and relative phase ϕ, respectively. Transitions |g jn ↔ |r jn and |e jn ↔ |s jn are coupled to the cavity mode a n with the strengths g r jn and g s jn , respectively. Here, we set ∆ r jn = ω r jn -ω = ω r jn -ω Lr jn -δ jn , and ∆ s jn = ω s jn -ω -δ jn = ω s jn -ω Ls jn . Under the Markovian approximation, the master equation of the density matrix for the whole system under consideration can be written as [14]\\n\\nρT = -i[H, ρ T ] + L cav1 ρ T + L cav2 ρ T + L f iber ρ T , (1) where H = H 0 + H d + H ac + H cf with H 0 = 2 j,n=1 ω r jn |r jn r jn | + ω s jn |s jn s jn | + δ jn |e jn e jn |) + ω( 2 n=1 a † n a n + b † b), H d = 2 j,n=1 Ω s jn 2 e -iω Ls jn t |s jn g jn | + Ω r jn 2 e -i(ω Lr jn t+ϕ) |r jn e jn | + H.c. , H ac = 2 j,n=1 (g r jn |r jn g jn |a n + g s jn |s jn e jn |a n + H.c.), H cf =ν b(a † 1 + a † 2 ) + H.c. . (2)\\n\\nHere, H 0 is the free energy of atoms and cavity fields, H d is the interaction energy between the atoms and laser fields, H ac is the interaction energy between the atoms and the cavity fields, and H cf describes the interaction between the cavity modes and the fiber mode. The last three terms in (1) describe the relaxation processes of the cavity and fibre modes in the usual vacuum reservoir, taking the forms\\n\\nL cavn ρ T =κ n (2a n ρ T a † n -a † n a n ρ T -ρ T a † n a n ), L f iber ρ T =κ f (2bρ T b † -b † bρ T -ρ T b † b), ( 3\\n\\n)\\n\\nwhere κ n is the leakage rate of photons from cavity n, and κ f is the decay rate of the fiber mode.\\n\\nLet's introduce collective basis:\\n\\n|a n = (|g 1n |e 2n - |e 1n |g 2n )/ √ 2, | -1 n = |g 1n |g 2n , |0 n = (|g 1n |e 2n + |e 1n |g 2n )/ √ 2, |1 n = |e 1n |e 2n .\\n\\nThe states |a n and | -1 n are taken as a qubit n for quantum computation. In the large detuning limit, adiabatically eliminating the excited states and setting 2 ), we obtain the effective interaction Hamiltonian\\n\\nΩ 1n g r 1n 2∆ r 1n = Ω r 2n g r 2n 2∆ r 2n = β r n and Ω s 1n g s 1n 2∆ s 1n = Ω s 2n g s 2n 2∆ s 2n = β s n , from (\\n\\nH ef f = n √ 2 a n (β r n e iϕ S + n + β s n S n ) + H.c. + H cf , ( 4\\n\\n) where S + n = |0 nn -1|+|1 nn 0|.\\n\\nIn the derivation of (4), we have assumed the resonant condition\\n\\ng s jn 2 ∆ s jn a † n a n + 3 Ω r jn 2 4∆ r jn = Ω s jn 2 4∆ s jn + g r jn 2 ∆ r jn a † n a n + δ ′ jn .\\n\\nIn order to satisfy the condition with the flexible choice of Ω r jn , Ω s jn , ∆ r jn and ∆ s jn , we have introduced additional ac-Stark shifts δ ′ jn to states |g jn , which can be generated by using a laser field to couple the level |g jn to an ancillary level.\\n\\nWe now introduce three normal modes c and c ± with frequencies ω and ω ± √ 2ν by use of the unitary transfor-\\n\\nmation a 1 = 1 2 (c + + c -+ √ 2c), a 2 = 1 2 (c + + c -- √ 2c), b = 1 √ 2 (c + -c -) [17, 18]. In the limit ν ≫ |β r j |, |β s j |,\\n\\nneglecting the far off-resonant modes c ± and setting β p 1 = -β p 2 = β p with p = r, s, we can approximately write the effective Hamiltonian (4) as\\n\\nH ef f = (β r e iϕ S + + β s S)c + H.c., ( 5\\n\\n)\\n\\nwhere\\n\\nS + = S + 1 + S + 2 .\\n\\nSince the modes c ± are nearly not excited and decoupled with the resonant mode c, the fiber mode b is mostly in the vacuum state, therefore, L f iber ρ T can be neglected, and L cav1 ρ T + L cav2 ρ T can be approximated as\\n\\nL cav ρ T = κ(2cρ T c † -c † cρ T -ρ T c † c), ( 6\\n\\n)\\n\\nwhere κ = (κ 1 + κ 2 )/2. In the bad cavity limit, κ ≫ β, adiabatically eliminating the mode c [12, 14] , from Eq. ( 1 ) with the replacement of the Hamiltonian (2) and the relaxation terms (3) by the effective Hamiltonian (5) and the relaxation term (6), respectively, we can obtain the master equation for the density matrix of the atoms\\n\\nρ = - Γ 2 (R + Rρ + ρR + R -2RρR + ), ( 7\\n\\n)\\n\\nwhere ρ = Tr f (ρ T ), R = S cosh r + e iϕ S † sinh r, r = cosh -1 (β r / β r 2 -β s 2\\n\\n) and Γ = 2(β r 2 -β s2 )/κ. Eq. ( 7 ) describes the collective interaction of two cascade threelevel atoms with the effective squeezed vacuum reservoir [10] . The parameters β r , β s and ϕ are easily changed and controlled at will by varying the strength and phase of the driving lasers [8] . We will show that a geometric phase gate can be realized through changing these parameters.\\n\\nThe DFS of the atomic system is spanned by the states which satisfy the equation R(r, ϕ\\n\\n)|ψ DF (r, ϕ) = 0 [10]. In terms of basis states |e 1 = |a 1 |a 2 , |e 2 = |a 1 | -1 2 , |e 3 = | -1 1 |a 2 , |e 4 = |a 1 |0 2 , |e 5 = |0 1 |a 2 , |e 6 = |a 1 |1 2 , |e 7 = |1 1 |a 2 , |e 8 = |1 1 |1 2 , |e 9 = 1 √ 2 (|1 1 |0 2 + |0 1 |1 2 ), |e 10 = | - 1 1 | -1 2 , |e 11 = 1 √ 2 (|0 1 | -1 2 + | -1 1 |0 2 ), |e 12 = 1 √ 6 (|1 1 | -1 2 + | -1 1 |1 2 ) + 2 √ 6 |0 1 |0 2 )\\n\\n, the DFS states can be written as\\n\\n|ψ DF (r, ϕ) 1 =|e 1 , |ψ DF (r, ϕ) j = cosh r √ cosh 2r |e j -e iϕ sinh r √ cosh 2r |e j+4 , j = 2, 3, |ψ DF (r, ϕ) 4 = e 2iϕ (tanh r) 2 |e 8 -2 3 e iϕ tanh r|e 12 + |e 10 (tanh r) 4 + 2 3 (tanh r) 2 + 1 . ( 8\\n\\n)\\n\\nLet's introduce a unitary transformation O(r, ϕ) 1, 2, 3, 4. For the transformed density matrix ρ = O † ρO, we have\\n\\nby |φ i = 12 i=1 O ij (r, ϕ)|e j , where |φ i = |ψ DF i for i =\\n\\ndρ dt = i[G, ρ] + O † dρ dt O, ( 9\\n\\n)\\n\\nwhere G(r, ϕ) = iO † dO dt = iO † [ ṙ dO dr + φ dO dϕ ]. To solve Eq. (9) in the DFS, let's define the time-independent projector Π(0 9) , we obtain the equation of motion for ρDF = Π(0)ρΠ(0)\\n\\n) = O † Π(r, ϕ)O = 4 i=1 O † |φ i φ i |O = 3 j=1 |e j e j | + |e 10 e 10 | onto the DFS. From (\\n\\ndρ DF dt =i[G DF , ρDF ] + iΠ(0)GΠ ⊥ (0)ρΠ( 0\\n\\n)\\n\\n-iΠ(0)ρΠ ⊥ (0)GΠ(0) + Π(0)O † dρ dt OΠ(0), (10) where Π ⊥ (0) = 1 -Π(0) and G DF = Π(0)GΠ(0). In the limit of ṙ, φ ≪ Γ, the last three terms in Eq. (10) can be neglected [11] . In this way, Eq. (10) is reduced to\\n\\ndρ DF dt = i[G DF , ρDF ]. ( 11\\n\\n)\\n\\nTherefore, in the frame dragged adiabatically by the reservoir, the state of the atoms in the DFS unitarily evolves in time.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. REALIZING CONTROLLING PHASE GATES THROUGH MANIPULATING THE SQUEEZING ENVIRONMENT\",\n \"text\": \"In this section, we investigate how to realize a CPHASE gate through manipulating the engineered reservoir. Suppose that at the initial time the laser field driving the transition |g ↔ |s is switched off but the laser field driving the transition |r ↔ |e is switched on and the atoms are in the DFS state |Ψ(0\\n\\n) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ) = 4 j=1 |ψ DF (0, 0) j /2.\\n\\nTo generate a geometric phase for the atomic state, we smoothly change the parameters of the engineered reservoir along a closed loop, which is divided into the following three steps: (1) From time 0 to T 1 , hold on ϕ = 0, and adiabatically increase the parameter r from 0 to r 0 ; (2) From time T 1 to T 2 , hold on r = r 0 , and adiabatically change the phase ϕ from 0 to ϕ 0 ; (3) From time T 2 to T 3 , hold on ϕ = ϕ 0 , and adiabatically decrease r from r 0 to 0. When the cyclic evolution ends, the atomic state becomes\\n\\n|Ψ(T 3 ) a = 1 2 (|e 1 + e iχ1 |e 2 + e iχ1 |e 3 + e iχ12 |e 10 ), (12) where geometric phases χ 1 = -ν 1 ϕ 0 , χ 12 = -ν 12 ϕ 0 with ν 1 = sinh 2 r0 sinh 2 r0+cosh 2 r0 , ν 12 = 2 tanh 4 r0+ 2 3 tanh 2 r0 tanh 4 r0+ 2 3 tanh 2 r0+1 . By per- forming local transformations U 1 = e -iχ1 |-1 11 -1| and 4 U 2 = e -iχ1 | -1 22 -1|,\\n\\nthe state (12) can be written as\\n\\n|Ψ ′ (T 3 ) a = U 1 U 2 |Ψ(T 3 ) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + e i∆ | -1 1 | -1 2 ), where ∆ = χ 12 -2χ 1 = (2ν 1 -ν 12 )ϕ 0 .\\n\\nThus, the CPHASE gate with the phase shift ∆ is realized. If both the atoms in cavity 1 and the atoms in cavity 2 \\\"see\\\" different environments, |χ 12 | must be equal to |2χ 1 | and ∆ = 0. Therefore, the phase shift ∆ results from the collective coupling of the atoms in both cavities with the same engineered environment.\\n\\nIf r 0 = atanh( 4/3 -1) ≃ 0.4157, |ν 12 | = |ν 1 |. Under this condition with ϕ 0 = π/ν 1 ,\\n\\nthe state of the atoms at the time T\\n\\n3 is |Ψ ′′ (T 3 ) a = -1 2 (-|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ).\\n\\nIn this case, the Controlled-Z gate between the two qubits is realized without local transformations. 0 0.2 0.4 0.6 0.8 0.9965 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 r 0 F r T=100/Γ T=200/Γ T=400/Γ (a)\\n\\nFIG. 3. Fidelity F r of the atomic state. 0 0.2 0.4 0.6 0.8 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 r 0 F p T = 200/Γ T = 400/Γ T = 1000/Γ (b) FIG. 4. Fidelity F p of the atomic state.\\n\\nThe above results depend on the adiabatical approximation. To check the adiabatical condition, we numerically simulate the following two examples. In the first example, we suppose that at the initial time the atoms are in the state |Ψ 1 a = (|a 1 |a 2 + |ψ DF (0, 0) 2 )/ √ 2 and the laser field driving the transition |e ↔ |r are turned on. Then, by slowly switching the laser field driving the transition |g ↔ |s , we increase the parameter r from 0 to r 0 according to the linear function r(t) = r 0 t/T . In the adiabatical limit (T ≫ Γ -1 ), the atomic state becomes\\n\\n|Ψ ′ 1 a = (|a 1 |a 2 + |ψ DF (r 0 , 0) 2 )/ √ 2\\n\\nat the time T . On the other hand, in the Hilbert space spanned by the basis states {|e i } for i = 1, 2, • • • , 12 , we can numerically solve Eq. ( 7 ) and obtain the density matrix ρ 1 (T ) of the atoms. Let's define\\n\\nF r = a Ψ ′ 1 |ρ 1 (T )|Ψ ′ 1 a as\\n\\nthe fidelity for this process. As shown in Fig. 3 , if T > 100/Γ, F r is always bigger than 0.997 if r ∈ (0, 0.8), corresponding to the almost perfect evolution.\\n\\nIn the second example, we suppose that the atoms are initially in the state\\n\\n|Ψ 2 a = (|a 1 |a 2 + |ψ DF (r, 0) 4 )/\\n\\n√ 2 and all the driving fields are turned on to hold the parameters r = r 0 and ϕ = 0. By adiabatically changing the phase ϕ from 0 to 2π at the rate φ = 2π/T , the atomic state at the time\\n\\nT becomes |Ψ ′ 2 a = (|a 1 |a 2 + e iχ12 |ψ DF (r, 2π) 4 )/ √ 2.\\n\\nLet's define the fidelity for this example as\\n\\nF p = a Ψ ′ 2 |ρ(T )|Ψ ′ a ,\\n\\nwhere ρ(T ) is the numerical solution of Eq. ( 7 ). As shown in Fig. 4 , F p increases as T increases but decreases as the parameter r 0 increases. If T > 1000/Γ, F p is larger than 0.992 for 0 < r 0 < 0.8. From these two examples, we find that to fulfill the adiabatical condition the time used in the step 2 should be much longer than in the steps 1 and 3.\\n\\nA controlled-Z gate has been numerically simulated by directly solving Eq. ( 7 ) with r 0 = 0.5, and ϕ\\n\\n0 = π/|2ν 1 -ν 12 |.\\n\\nIn the simulation, we set ṙ = r 0 /T 1 in the steps 1 and 3, and φ = ϕ 0 /(T 2 -T 1 ) in the step 2 with T 1 = 0.05T 3 and T 2 -T 1 = 0.90T 3 . If T 3 > 1100/Γ, we find that the fidelity F = a Ψ(T 3 )|ρ(T 3 )|Ψ(T 3 ) a is larger than 0.95. For an almost perfect controlled-Z gate with F > 0.99, we find that T 3 must be longer than 6000/Γ. Now let's briefly discuss the effects of the atomic spontaneous emission, the fiber mode decay and cavity photon leakage. For simplicity but without the loss of generality, we suppose that atomic spontaneous emission rates of the excited levels are equal to γ. In the large detunig limit, the characteristic spontaneous emission rate of the atoms is γ ef f = γ(Ω 2 /2∆ 2 ) [14, 16] and the effective decay rate of the fiber mode is\\n\\nκ ef f = κ f Ω 2 g 2 /(4∆ 2 ν 2 ). If κ f ≤ γ and g 2 ≪ ν 2 , κ\\n\\nef f be much smaller than γ ef f . Under this condition, the present scheme is feasible if Γ ≫ γ ef f . In the current cavity quantum dynamic (CQED) experiment, the parameters (g, κ, γ) = (2000, 10, 10) MHz could be available [20] . If setting Ω/(2∆) = 1 √ 2 × 10 -3 , we have Γ ≃ 4 × 10 4 γ ef f . The condition is held. In the present scheme, the large cavity decay rate is required to ensure that the cavity modes are in a broadband squeezed vacuum reservoir and then the atoms always \\\"see\\\" the broadband squeezed vacuum reservoir during the dynamic evolution. For an arbitrary small but nonzero value of the squeezing degree of the reservoir, a CPHASE gate with arbitrary high fidelity can always be realized in the represent scheme. The cavity decay does 5 not directly affect the fidelity of the realized CPHASE gates. However, the larger the decay rate is, the longer the operation time of the CPHASE gates is. Thus, we have the condition κ >> β, γ ef f for realizing the reliable CPHASE gates. Based on the parameters quoted above, this condition can be well satisfied. With the parameters of the current CQED experiment, we find that the operation time of the controlled-Z gate, with fidelity larger than 0.95, is about 2.8 ms. It is much shorter than both 1/γ ef f and the single-atom trapping time in cavity [21] . On the other hand, the present scheme needs a strong coupling between the cavity and the fiber. This could be realized at the current experiment [22] . Therefore, the requirement for the realization of the present scheme can be satisfied with the current technology.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"IV. CONCLUSIONS\",\n \"text\": \"We propose a cavity-atom coupled scheme for the realization of quantum controlling gates, in which each of two pairs of four-level atoms in two distant cavities connected by a short optical fibre are simultaneously driven by laser fields and coupled to the local cavity modes through the double Raman transition configuration. We show that an effective squeezing reservoir coupled to the multilevel atoms can be engineered under appropriate driving condition and bad cavity limit. We find that in the scheme a CPHASE gate with arbitrary phase shift can be implemented through adiabatically changing the strength and phase of driving fields along a closed loop. It is also noticed that the larger the effective coupling strength between the environment and the atoms is, the more reliable the realized CPHASE gate is.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments\",\n \"text\": \"We thank Yun-feng Xiao and Wen-ping He for valuable discussions and suggestions. This work was supported by the Natural Science Foundation of China (Grant Nos. 10674106, 60778021 and 05-06-01).\\n\\n[1] P. W. Shor, in Proceeings, 35 th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Press, Los Almitos, CA, 1994), p. 124. [2] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997). [3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). [4] D. A. Lidar and et al., Phys. Rev. Lett. 81, 2594 (1998). [5] M. V. Berry, Proc. R. Soc. A 392, 45 (1984). [6] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [7] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301 (2005). [8] L.-M. Duan and et al., Science 292, 1695 (2001). [9] L.-A. Wu and et al., Phys. Rev. Lett. 95, 130501 (2005). [10] Angelo Carollo et al., Phys. Rev. Lett. 96, 150403 (2006). [11] Angelo Carollo et al., Phys. Rev. Lett. 96, 020403 (2006). [12] J. I. Cirac, Phys. Rev. A 46, 4354 (1992). [13] C. J. Myatt and et al., Nature (London) 403, 269 (2000). [14] S. G. Clark and A. S. Parkins, Phys. Rev. Lett. 90, 047905 (2003). [15] S. Lloyd, Phys. Rev. Lett. 75, 346 (1995). [16] T. Pellizzari, Phys. Rev. Lett. 79, 5242 (1997). [17] A. Serafini and et al., Phys. Rev. Lett. 96, 010503 (2006) . [18] Zhang-qi Yin and Fu-li Li, Phys. Rev. A 75, 012324 (2007). [19] S. J. van Enk and et al., Phys. Rev. A 59, 2659 (1999). [20] S. M. Spillane et al., Phys. Rev. A 71, 013817 (2005). [21] Stefan Nuβmann and et al., Nature Phys. 1, 122 (2005). [22] S. M. Spillane et al., Phys. Rev. Lett. 91, 043902 (2003).\"\n }\n]"}}},{"rowIdx":86,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0488"},"arxiv_id":{"kind":"string","value":"0704.0488"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1007/s10714-007-0431-5"},"content_hash":{"kind":"string","value":"5164ad34ca4de631f8a4c37f1aba38127c537f07d91eec4dfc4322512c2ad2cd"},"title":{"kind":"string","value":"Teleparallel Version of the Stationary Axisymmetric Solutions and their\n Energy Contents"},"abstract":{"kind":"string","value":"This work contains the teleparallel version of the stationary axisymmetric\nsolutions. We obtain the tetrad and the torsion fields representing these\nsolutions. The tensor, vector and axial-vector parts of the torsion tensor are\nevaluated. It is found that the axial-vector has component only along $\\rho$\nand $z$ directions. The three possibilities of the axial vector depending on\nthe metric function $B$ are discussed. The vector related with spin has also\nbeen evaluated and the corresponding extra Hamiltonian is furnished. Further,\nwe use the teleparallel version of M$\\ddot{o}$ller prescription to find the\nenergy-momentum distribution of the solutions. It is interesting to note that\n(for $\\lambda=1$) energy and momentum densities in teleparallel theory are\nequal to the corresponding quantities in GR plus an additional quantity in\neach, which may become equal under certain conditions. Finally, we discuss the\ntwo special cases of the stationary axisymmetric solutions."},"authors":{"kind":"list like","value":["M. Sharif and M. Jamil Amir"],"string":"[\n \"M. Sharif and M. Jamil Amir\"\n]"},"arxiv_categories":{"kind":"list like","value":["gr-qc"],"string":"[\n \"gr-qc\"\n]"},"primary_category":{"kind":"string","value":"gr-qc"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along ρ and z directions. The three possibilities of the axial vector depending on the metric function B are discussed. The vector related with spin has also been evaluated and the corresponding extra Hamiltonian is furnished. Further, we use the teleparallel version of Möller prescription to find the energy-momentum distribution of the solutions. It is interesting to note that (for λ = 1) energy and momentum densities in teleparallel theory are equal to the corresponding quantities in GR plus an additional quantity in each, which may become equal under certain conditions. Finally, we discuss the two special cases of the stationary axisymmetric solutions.\n\nThe attempts made by Einstein and his followers to unify gravitation with other interactions led to the investigation of structures of gravitation other than the metric tensor. These structures yield the metric tensor as a by product. Tetrad field is one of these structures which leads to the theory of teleparallel gravity (TPG) [1, 2] . TPG is an alternative theory of gravity which corresponds to a gauge theory of translation group [3, 4] based on Weitzenböck geometry [5] . This theory is characterized by the vanishing of curvature identically while the torsion is taken to be non-zero. In TPG, gravitation is attributed to torsion [4] which plays a role of force [6] . In General Relativity (GR), gravitation geometrizes the underlying spacetime. The translational gauge potentials appear as a non-trivial part of the tetrad field and induce a teleparallel (TP) structure on spacetime which is directly related to the presence of a gravitational field. In some other theories [3] [4] [5] [6] [7] [8] , torsion is only relevant when spins are important [9] . This point of view indicates that torsion might represent additional degrees of freedom as compared to curvature. As a result, some new physics may be associated with it. Teleparallelism is naturally formulated by gauging external (spacetime) translations which are closely related to the group of general coordinate transformations underlying GR. Thus the energy-momentum tensor represents the matter source in the field equations of tetradic theories of gravity like in GR.\n\nThere is a large literature available [10] about the study of TP versions of the exact solutions of GR. Recently, Pereira, et al. [11] obtained the TP versions of the Schwarzschild and the stationary axisymmetric Kerr solutions of GR. They proved that the axial-vector torsion plays the role of the gravitomagnetic component of the gravitational field in the case of slow rotation and weak field approximations. In a previous paper [12] , we have found the TP versions of the Friedmann models and of the Lewis-Papapetrou spacetimes, and also discussed their axial-vectors.\n\nThere has been a longstanding, controversial and still unresolved problem of the localization of energy (i.e., to express it as a unique tensor quantity) in GR [13] . Einstein [14] introduced the energy-momentum pseudo-tensor and then Landau-Lifshitz [15] , Papapetrou [16] , Bergmann [17] , Tolman [18] and Weinberg [19] proposed their own prescriptions to resolve this issue. All these prescriptions can provide meaningful results only in Cartesian coordinates. But Möller [20] introduced a coordinate-independent prescription. The idea of coordinate-independent quasi-local mass was introduced [21] by associ-ating a Hamiltonian term to each gravitational energy-momentum pseudotensor. Later, a Hamiltonian approach in the frame of Schwinger condition [22] was developed, followed by the construction of a Lagrangian density of TP equivalent to GR [4, 6, 23, 24] . Many authors explored several examples in the framework of GR and found that different energy-momentum complexes can give either the same [25] or different [26] results for a given spacetime.\n\nMikhail et al. [27] defined the superpotential in the Moller's tetrad theory which has been used to find the energy in TPG. Vargas [28] defined the TP version of Bergman, Einstein and Landau-Lifshitz prescriptions and found that the total energy of the closed Friedman-Robinson-Walker universe is zero by using the last two prescriptions. This agrees with the results of GR available in literature [29] . Later, many authors [30] used TP version of these prescriptions and showed that energy may be localized in TPG. Keeping these points in mind, this paper is addressed to the following two problems: We obtain TP version of the stationary axisymmetric solutions and then calculate the axial-vector part of the torsion tensor. The energymomentum distribution of the solutions is explored by using the TP version of Möller prescription.\n\nThe scheme adopted in this paper is as follows. In section 2, we shall review the basic concepts of TP theory. Section 3 contains the TP version of the stationary axisymmetric solutions and the tensor, vector and axial-vector parts of the torsion tensor. Section 4 is devoted to evaluate the energymomentum distribution for this family of solutions using the TP version of Möller prescription. In section 5, we present two special solutions for this class of metrics and investigate the corresponding quantities. The last section contains a summary and a discussion of the results obtained.\n\nIn teleparallel theory, the connection is a Weitzenböck connection given as [31] Γ\n\nθ µν = h a θ ∂ ν h a µ , (1)\n\nwhere h a ν is a non-trivial tetrad. Its inverse field is denoted by h a µ and satisfy the relations\n\nh a µ h a ν = δ µ ν ; h a µ h b µ = δ a b . (2)\n\nIn this paper, the Latin alphabet (a, b, c, ... = 0, 1, 2, 3) will be used to denote tangent space indices and the Greek alphabet (µ, ν, ρ, ... = 0, 1, 2, 3) to denote spacetime indices. The Riemannian metric in TP theory arises as a by product [4] of the tetrad field given by\n\ng µν = η ab h a µ h b ν , (3)\n\nwhere η ab is the Minkowski metric η ab = diag(+1, -1, -1, -1). For the Weitzenböck spacetime, the torsion is defined as [2]\n\nT θ µν = Γ θ νµ -Γ θ µν (4)\n\nwhich is antisymmetric w.r.t. its last two indices. Due to the requirement of absolute parallelism, the curvature of the Weitzenböck connection vanishes identically. The Weitzenböck connection also satisfies the relation\n\nΓ 0 θ µν = Γ θ µν -K θ µν , (5)\n\nwhere\n\nK θ µν = 1 2 [T µ θ ν + T ν θ µ -T θ µν ] (6)\n\nis the contortion tensor and Γ 0 θ µν are the Christoffel symbols in GR. The torsion tensor of the Weitzenböck connection can be decomposed into three irreducible parts under the group of global Lorentz transformations [4] : the tensor part\n\nt λµν = 1 2 (T λµν + T µλν ) + 1 6 (g νλ V µ + g νµ V λ ) - 1 3 g λµ V ν , (7)\n\nthe vector part\n\nV µ = T ν νµ ( 8\n\n)\n\nand the axial-vector part\n\nA µ = 1 6 ǫ µνρσ T νρσ . (9)\n\nThe torsion tensor can now be expressed in terms of these irreducible components as follows\n\nT λµν = 1 2 (t λµν -t λνµ ) + 1 3 (g λµ V ν -g λν V µ ) + ǫ λµνρ A ρ , (10)\n\nwhere\n\nǫ λµνρ = 1 √ -g δ λµνρ . (11)\n\nHere δ = {δ λµνρ } and δ * = {δ λµνρ } are completely skew symmetric tensor densities of weight -1 and +1 respectively [4] . TP theory provides an alternate description of the Einstein's field equations which is given by the teleparallel equivalent of GR [24, 31] . Mikhail et al. [27] defined the super-potential (which is antisymmetric in its last two indices) of the Möller tetrad theory as\n\nU µ νβ = √ -g 2κ P τ νβ χρσ [V ρ g σχ g µτ -λg τ µ K χρσ -(1 -2λ)g τ µ K σρχ ], (12)\n\nwhere\n\nP τ νβ χρσ = δ χ τ g νβ ρσ + δ ρ τ g νβ σχ -δ σ τ g νβ χρ ( 13\n\n)\n\nand g νβ ρσ is a tensor quantity defined by\n\ng νβ ρσ = δ ρ ν δ σ β -δ σ ν δ ρ β . ( 14\n\n)\n\nK σρχ is the contortion tensor given by Eq.( 6 ), g is the determinant of the metric tensor g µν , λ is the free dimensionless coupling constant of TPG, κ is the Einstein constant and V µ is the basic vector field given by Eq.( 8 ). The energy-momentum density is defined as\n\nΞ ν µ = U νρ µ , ρ , (15)\n\nwhere comma means ordinary differentiation. The momentum 4-vector of Möller prescription can be expressed as\n\nP µ = Σ Ξ 0 µ dxdydz, (16)\n\nwhere P 0 gives the energy and P 1 , P 2 and P 3 are the momentum components while the integration is taken over the hypersurface element Σ described by x 0 = t = constant. The energy may be given in the form of surface integral [20] as\n\nE = lim r→∞ r=constant U 0 0ρ u ρ dS, (17)\n\nwhere u ρ is the unit three-vector normal to the surface element dS.\n\nTupper [32] found five classes of non-null electromagnetic field plus perfect fluid solutions in which the electromagnetic field does not inherit one of the symmetries of the spacetime. The metric representing the stationary axisymmetric solutions is given by [32 ]\n\nds 2 = dt 2 -e 2K dρ 2 -(F 2 -B 2 )dφ 2 -e 2K dz 2 + 2Bdtdφ, (18)\n\nwhere B = B(ρ, z), K = K(ρ, z) and F = F (ρ) are such functions which satisfy the following relations . The metric given by Eq.( 18 ) represents five classes of non-null electromagnetic field and perfect fluid solutions which possesses a metric symmetry not inherited by the electromagnetic field and admits a homothetic vector field. Two of these classes contain electrovac solutions as special cases, while the other three necessarily contain some fluid. The generalization of this metric is given in [34] .\n\nḂ = F W ′ , B ′ = - 1 4 aF ( Ẇ 2 -W ′2 ), K ′ = - 1 2 aF Ẇ W ′ , Ẅ + Ḟ F -1 Ẇ + W ′′ = 0, (19)\n\nUsing the procedure adopted in the papers [11, 12] , the tetrad components of the above metric can be written as\n\nh a µ = 1 0 B 0 0 e K cos φ -F sin φ 0 0 e K sin φ F cos φ 0 0 0 0 e K (20)\n\nwith its inverse\n\nh a µ = 1 0 0 0 B F sin φ e -K cos φ -1 F sin φ 0 -B F cos φ e -K sin φ 1 F cos φ 0 0 0 0 e -K . ( 21\n\n)\n\nThe non-vanishing components of the torsion tensor are\n\nT 0 12 = Ḃ + B F (e K -Ḟ ), T 0 32 = B ′ , T 1 13 = -K ′ , T 2 12 = 1 F ( Ḟ -e K ), T 3 31 = -K. ( 22\n\n)\n\nUsing these expressions in Eqs.( 7 )-( 9 ), we obtain the following non-zero components of the tensor part\n\nt 001 = 1 3 [ K + 1 F ( Ḟ -e K )], t 003 = 1 3 K ′ , t 010 = 1 6 { 1 F (e K -Ḟ ) -K} = t 100 , t 030 = - 1 6 K ′ = t 300 , t 012 = 1 2 Ḃ + B 6 { 1 F (e K -Ḟ ) -K} = t 102 , t 021 = - 1 2 Ḃ - B 3 { 1 F (e K -Ḟ ) -K} = t 201 , t 023 = - 1 2 B ′ + 1 3 BK ′ = t 203 , t 032 = 1 2 B ′ - 1 6 BK ′ = t 302 , t 122 = 1 2 {F (e K -Ḟ ) + B Ḃ} + 1 6 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K} = t 212 , t 120 = B 6 { 1 F (e K -Ḟ ) -K} = t 210 , t 113 = 2K ′ 3 e 2K , t 131 = - K ′ 3 e 2K = t 311 , t 133 = - e 2K 6 { 1 F (e K -Ḟ ) + 2 K} = t 313 , t 221 = -F (e K -Ḟ ) -B Ḃ - 1 3 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K}, t 223 = -BB ′ + K ′ 3 (B 2 -F 2 ), t 331 = e 2K 3 { 1 F (e K -Ḟ ) + 2 K}, t 322 = 1 2 BB ′ - K ′ 6 (B 2 -F 2 ) = t 232 , t 320 = - 1 6 BK ′ = t 232 , (23)\n\nthe vector part\n\nV 1 = - 1 F ( Ḟ -e K ) -K, (24)\n\nV 3 = -K ′ , (25)\n\nand the axial-vector part\n\nA 1 = B ′ 3F e -2K , (26)\n\nA 3 = Ḃ 3F e -2K , (27)\n\nrespectively. The axial-vector component along the φ-direction vanishes and hence the spacelike axial-vector can be written as\n\nA = √ -g 11 A 1 êρ + √ -g 33 A 3 êz , (28)\n\nwhere êρ and êz are unit vectors along the radial and z-directions respectively. Substituting A 1 , A 3 , g 11 and g 33 in Eq.( 28 ), it follows that\n\nA = e -K 3F (B ′ êρ + Ḃê z ). ( 29\n\n)\n\nThis shows that the axial-vector lies along radial direction if B = B(z), along z-direction if B = B(ρ) and vanishes identically if B is constant. As the axialvector torsion represents the deviation of axial symmetry from cylindrical symmetry, the symmetry of the underlying spacetime will not be affected even for B constant. Also, the torsion plays the role of the gravitational force in TP theory, hence a spinless particle will obey the force equation [11, 24]\n\ndu ρ ds -Γ µρν u µ u ν = T µρν u µ u ν . ( 30\n\n)\n\nThe left hand side of this equation is the Weitzenböck covariant derivative of u ρ along the particle world-line. The appearance of the torsion tensor on its right hand side indicates that the torsion plays the role of an external force in TPG. It has been shown, both in GR and TP theories, by many authors [4, 35] that the spin precession of a Dirac particle in torsion gravity is related to the torsion axial-vector by\n\ndS dt = -b × S, ( 31\n\n)\n\nwhere S is the spin vector of a Dirac particle and b = 3 2 A, with A the spacelike part of the torsion axial-vector. Thus\n\nb = e -K 2F {B ′ êρ + Ḃê z }. (32)\n\nThe corresponding extra Hamiltonian [36] is given by\n\nδH = -b.σ, ( 33\n\n)\n\nwhere σ is the spin of the particle [35] . Using Eq.( 32 ), this takes the form\n\nδH = - e -K 2F (B ′ êρ + Ḃê z ).σ. ( 34\n\n)\n\n4 Teleparallel Energy of the Stationary Axisymmetric Solutions\n\nIn this section we evaluate the component of energy-momentum densities by using the teleparallel version of Möller prescription. Multiplying Eqs.( 24 ) and ( 25 ) by g 11 and g 33 respectively, it follows that\n\nV 1 = Ke -2K + e -2K F ( Ḟ -e K ), (35)\n\nV 3 = K ′ e -2K . (36)\n\nIn view of Eqs.( 6 ) and ( 22 ), the non-vanishing components of the contorsion tensor are\n\nK 100 = -e -2K { B 2 F 3 (e K -Ḟ ) + B Ḃ F 2 } = -K 010 , K 300 = - BB ′ F 2 e -2K = -K 030 , K 122 = - e -2K F 3 (e K -Ḟ ) = -K 212 , K 133 = Ke -4K = -K 313 , K 311 = K ′ e -4K = -K 131 , K 102 = K 120 = e -2K { B F 3 (e K -Ḟ ) + Ḃ 2F 2 } = -K 012 = -K 210 , K 302 = K 320 = K 023 = B ′ 2F 2 e -2K = -K 032 = -K 230 = -K 203 , K 021 = Ḃ 2F 2 e -2K = -K 201 . ( 37\n\n)\n\nIt should be mentioned here that the contorsion tensor is antisymmetric w.r.t. its first two indices. Making use of Eqs.( 35 )- (37) in Eq.( 12 ), we obtain the required independent non-vanishing components of the supperpotential in Möller's tetrad theory as\n\nU 01 0 = 1 κ [e K -Ḟ -F K + 1 2 (1 + λ) B Ḃ F ] = -U 10 0 , U 03 0 = 1 κ [-F K ′ + 1 2 (1 + λ) BB ′ F ] = -U 30 0 , U 21 0 = - 1 2κ (1 + λ) Ḃ F = -U 12 0 , U 23 0 = - 1 2κ (1 + λ) B ′ F = -U 32 0 , U 01 2 = 1 κ [B(e K -Ḟ ) + 1 2 (1 + λ) B 2 Ḃ F + 1 2 (1 -λ) ḂF ] = -U 10 2 , U 03 2 = 1 κ [ 1 2 (1 + λ) B 2 B ′ F + 1 2 (1 -λ)B ′ F ] = -U 30 2 , U 02 1 = 1 2κF (λ -1) Ḃe 2K = -U 20 1 , U 02 3 = 1 2κF (λ -1)B ′ e 2K = -U 30 1 . (38)\n\nIt is worth mentioning here that the supperpotential is skew symmetric w.r.t. its last two indices. When we make use of Eqs.( 15 ), ( 37 ), (38) and take λ = 1, the energy density turns out to be\n\nΞ 0 0 = 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ ) + 1 F 2 {BF ( B + E ′′ ) + ( Ḃ2 + B ′ 2 )F -B Ḃ Ḟ }]. ( 39\n\n)\n\nThis implies that\n\nE d T P T = E d GR + 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ )], (40)\n\nwhere E d stands for energy density. The only non-zero component of momentum density is along φ-direction and (for λ = 1) it takes the form\n\nΞ 0 2 = 1 κF 2 {F 3 ( B + B ′′ ) + B 2 F ( B + B ′′ ) + 2BF ( Ḃ2 + B ′ 2 ) -Ḃ Ḟ (B 2 + F 2 )} + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (41)\n\nthat is,\n\nM d T P T = M d GR + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (42)\n\nwhere M d stands for momentum density.\n\nIn this section, we evaluate the above quantities for some special cases of the non-null Einstein Maxwell solutions.\n\nA special case of the non-null Einstein-Maxwell solutions can be obtained by choosing\n\nB = m n e nρ , F = e nρ , K = 0. ( 43\n\n)\n\nThis is known as electromagnetic generalization of the Gödel solution [32] .\n\nWhen we make use of Eq.( 43 ) in Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42), the corresponding results reduce to\n\nt 001 = 1 3 (n -e -nρ ), t 010 = 1 6 (e -nρ -n) = t 100 , t 012 = m 6n (1 + 2ne nρ ) = t 102 , t 021 = - m 3n (1 + 2ne nρ ) = t 201 , t 122 = e nρ 6n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ } = t 212 , t 120 = m 6n (1 -ne nρ ) = t 210 , t 133 = 1 6 (n -e -nρ ) = t 313 , t 221 = - e nρ 3n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ },\n\nt 331 = - 1 3 (n -e -nρ ), (44)\n\nV 1 = e -nρ -n, V 3 = 0, (45)\n\nA 1 = 0, A 3 = m 3 , (46)\n\nA = m 3 êz , b = m 2 êz , (47)\n\nδH = m 2 êz .σ, (48)\n\nΞ 0 0 = 1 κ (m 2 -n 2 )e nρ , (49)\n\nE d T P T = E d GR - n 2 κ e nρ (50)\n\nΞ 0 2 = 1 κ ( 2m 3 n ) + m κ (1 -2ne nρ )e nρ , (51)\n\nM d T P T = M d GR + m κ (1 -2ne nρ )e nρ . (52)\n\nThe metric (43) reduces to the usual perfect fluid solution when m = √ 2n [32] , i.e., B = √ 2e nρ . The corresponding energy and momentum densities take the form as\n\nE d T P T = E d GR - n 2 κ e nρ (53)\n\nM d T P T = M d GR + √ 2n κ (1 -2ne nρ )e nρ . (54)\n\nWhen we choose B = e aρ , F = e aρ √ 2 and K = 0, the metric given by Eq.( 18 ) reduces to the Gödel metric [32] . The results corresponding to Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42) take the following form\n\nt 001 = 1 3 (a - √ 2e -aρ\n\n), t 010 = -1 6 (a -√ 2e -aρ ) = t 100 , t 012 = 1 6 ( √ 2 + 2ae aρ ) = t 102 , t 021 = -1 6 (2 √ 2 + ae aρ ) = t 102 , t 122 = e aρ 6 (2 √ 2 + ae aρ ) = t 212 , t 120 = 1 6 ( √ 2ae aρ ) = t 210 , t 133 = 1 6 (a -√ 2e -aρ ) = t 313 , t 221 = -e aρ 3 (2 √ 2 + ae aρ ), (55)\n\nt 331 = 1 3 (a - √ 2e -aρ ), ( 56\n\n) V 1 = √ 2e -aρ -a, V 3 = 0, (57)\n\nA 1 = 0, A 3 = √ 2a 3 , (58)\n\nA = √ 2a 3 êz , b = a √ 2 êz , (59)\n\nδH = a √ 2 êz .σ, (60)\n\nΞ 0 0 = √ 2 κ a 2 e aρ - a 2 κ √ 2 e aρ , (61)\n\nE d T P T = E d GR - a 2 κ √ 2 e aρ (62) Ξ 0 2 = a 2 κ √ 2 e 2aρ + a κ (1 - √ 2ae aρ )e aρ , (63)\n\nM d T P T = M d GR + a κ (1 - √ 2ae aρ )e aρ . ( 64\n\n)\n\nThe purpose of this paper is twofold: Firstly, we have found the TP version of the non-null Einstein Maxwell solutions. This provides some interesting features about the axial vector and the corresponding quantities. Secondly, we have used the TP version of Möller prescription to evaluate the energymomentum distribution of the solutions. The axial-vector torsion of these solutions has been evaluated. The only non-vanishing components of the vector part are along the radial and the z-directions due to the cross term dx 0 dx 2 involving in the metric. This corresponds to the case of Kerr metric [11] , which involves the cross term dx 0 dx 3 . We also find the vector b which is related to the spin vector [4] as given by Eq.( 32 ). The axial-vector torsion lies in the ρz-plane, as its component along the φ-direction vanishes everywhere.\n\nThe non-inertial force on the Dirac particle can be represented as a rotation induced torsion of spacetime. There arise three possibilities for the axial-vector, depending upon the metric function B(ρ, z). When B is a function of z only, the axial-vector lies only along the radial direction. When B is a function of ρ only, the axialvector will lie along z-direction. The axial-vector vanishes identically for B to be constant. As the axial-vector represents the deviation from the symmetry of the underlying spacetime corresponding to an inertial field with respect to the Dirac particle, the symmetry of the spacetime will not be affected in the third possibility. Consequently there exists no inertial field with respect to the Dirac particle and the spin vector of the Dirac particle becomes constant. The corresponding extra Hamiltonian is expressed in terms of the vector b which vanishes when the metric function B is constant, i.e., when the axialvector becomes zero.\n\nThe energy-momentum distribution of the non-null Einstein-Maxwell solutions has been explored by using the TP version of Möller prescription. It is found that energy in the TP theory is equal to the energy in GR (as found by Sharif and Fatima [37] ) plus some additional part. If, for a particular case, we have K = 0 and K ′ , Ḟ = constant (or if Ḟ , K = 0 and K\n\n′ = constant), then E d T P G = E d GR . (65)\n\nOn the other hand, the only non-vanishing component of the momentum density lies along φ-direction, similar to the case of Kerr metric [11] , due to the cross term appearing in both the metrics. When we choose λ = 1, it becomes equal to be the momentum in GR [37] plus an additional quantity. If F , Ḃ, B ′′ , K all vanish, then\n\nM d T P G = M d GR . (66)\n\nBy taking particular values of E, F and K, we obtain the electromagnetic generalization of the Gödel solution and the Gödel metric as two special cases. The corresponding results for both the special cases are obtained. It is shown that, for the electromagnetic generalization of the Gödel solution, Eq.(65) does not hold, while Eq.(66) holds when m = 0. However, for the perfect fluid case, i.e., when m = √ 2n, both Eqs.( 65 ) and (66) hold by taking n = 0. In the case of the Gödel metric, these equations hold if we choose the arbitrary constant a = 0. For the special solutions, the vector part lies along the radial direction while the axial-vector part along z-direction.\n\nWe would like to re-iterate here that the tetrad formalism itself has some advantages which comes mainly from its independence from the equivalence principle and consequent suitability to the discussion of quantum issues. In TPG, an energy-momentum gauge current j i µ for the gravitational field can be defined. This is covariant under a spacetime general coordinate transformation and transforms covariantly under a global tangent space Lorentz transformation [38] . It, then, follows that j i µ is a true spacetime tensor but not a tangent space tensor. When we re-write the gauge field equations in a purely spacetime form, they lead to the Einstein field equations and the gauge current j i µ reduces to the canonical energy-momentum pseudo-tensor of the gravitational field. Thus TPG seems to provide a more appropriate environment to deal with the energy problem.\n\nFinally, it is pointed out that we are not claiming that this paper has resolved the problems of GR using the TPG. This is an attempt to touch some issues in TPG with the hope that this alternative may provide more feasible results. Also, it is always an interesting and enriching to look at things from another point of view. This endeavor is in itself commendable."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along ρ and z directions. The three possibilities of the axial vector depending on the metric function B are discussed. The vector related with spin has also been evaluated and the corresponding extra Hamiltonian is furnished. Further, we use the teleparallel version of Möller prescription to find the energy-momentum distribution of the solutions. It is interesting to note that (for λ = 1) energy and momentum densities in teleparallel theory are equal to the corresponding quantities in GR plus an additional quantity in each, which may become equal under certain conditions. Finally, we discuss the two special cases of the stationary axisymmetric solutions."},{"section_type":"BACKGROUND","section_title":"Introduction","text":"The attempts made by Einstein and his followers to unify gravitation with other interactions led to the investigation of structures of gravitation other than the metric tensor. These structures yield the metric tensor as a by product. Tetrad field is one of these structures which leads to the theory of teleparallel gravity (TPG) [1, 2] . TPG is an alternative theory of gravity which corresponds to a gauge theory of translation group [3, 4] based on Weitzenböck geometry [5] . This theory is characterized by the vanishing of curvature identically while the torsion is taken to be non-zero. In TPG, gravitation is attributed to torsion [4] which plays a role of force [6] . In General Relativity (GR), gravitation geometrizes the underlying spacetime. The translational gauge potentials appear as a non-trivial part of the tetrad field and induce a teleparallel (TP) structure on spacetime which is directly related to the presence of a gravitational field. In some other theories [3] [4] [5] [6] [7] [8] , torsion is only relevant when spins are important [9] . This point of view indicates that torsion might represent additional degrees of freedom as compared to curvature. As a result, some new physics may be associated with it. Teleparallelism is naturally formulated by gauging external (spacetime) translations which are closely related to the group of general coordinate transformations underlying GR. Thus the energy-momentum tensor represents the matter source in the field equations of tetradic theories of gravity like in GR.\n\nThere is a large literature available [10] about the study of TP versions of the exact solutions of GR. Recently, Pereira, et al. [11] obtained the TP versions of the Schwarzschild and the stationary axisymmetric Kerr solutions of GR. They proved that the axial-vector torsion plays the role of the gravitomagnetic component of the gravitational field in the case of slow rotation and weak field approximations. In a previous paper [12] , we have found the TP versions of the Friedmann models and of the Lewis-Papapetrou spacetimes, and also discussed their axial-vectors.\n\nThere has been a longstanding, controversial and still unresolved problem of the localization of energy (i.e., to express it as a unique tensor quantity) in GR [13] . Einstein [14] introduced the energy-momentum pseudo-tensor and then Landau-Lifshitz [15] , Papapetrou [16] , Bergmann [17] , Tolman [18] and Weinberg [19] proposed their own prescriptions to resolve this issue. All these prescriptions can provide meaningful results only in Cartesian coordinates. But Möller [20] introduced a coordinate-independent prescription. The idea of coordinate-independent quasi-local mass was introduced [21] by associ-ating a Hamiltonian term to each gravitational energy-momentum pseudotensor. Later, a Hamiltonian approach in the frame of Schwinger condition [22] was developed, followed by the construction of a Lagrangian density of TP equivalent to GR [4, 6, 23, 24] . Many authors explored several examples in the framework of GR and found that different energy-momentum complexes can give either the same [25] or different [26] results for a given spacetime.\n\nMikhail et al. [27] defined the superpotential in the Moller's tetrad theory which has been used to find the energy in TPG. Vargas [28] defined the TP version of Bergman, Einstein and Landau-Lifshitz prescriptions and found that the total energy of the closed Friedman-Robinson-Walker universe is zero by using the last two prescriptions. This agrees with the results of GR available in literature [29] . Later, many authors [30] used TP version of these prescriptions and showed that energy may be localized in TPG. Keeping these points in mind, this paper is addressed to the following two problems: We obtain TP version of the stationary axisymmetric solutions and then calculate the axial-vector part of the torsion tensor. The energymomentum distribution of the solutions is explored by using the TP version of Möller prescription.\n\nThe scheme adopted in this paper is as follows. In section 2, we shall review the basic concepts of TP theory. Section 3 contains the TP version of the stationary axisymmetric solutions and the tensor, vector and axial-vector parts of the torsion tensor. Section 4 is devoted to evaluate the energymomentum distribution for this family of solutions using the TP version of Möller prescription. In section 5, we present two special solutions for this class of metrics and investigate the corresponding quantities. The last section contains a summary and a discussion of the results obtained."},{"section_type":"OTHER","section_title":"An Overview of the Teleparallel Theory","text":"In teleparallel theory, the connection is a Weitzenböck connection given as [31] Γ\n\nθ µν = h a θ ∂ ν h a µ , (1)\n\nwhere h a ν is a non-trivial tetrad. Its inverse field is denoted by h a µ and satisfy the relations\n\nh a µ h a ν = δ µ ν ; h a µ h b µ = δ a b . (2)\n\nIn this paper, the Latin alphabet (a, b, c, ... = 0, 1, 2, 3) will be used to denote tangent space indices and the Greek alphabet (µ, ν, ρ, ... = 0, 1, 2, 3) to denote spacetime indices. The Riemannian metric in TP theory arises as a by product [4] of the tetrad field given by\n\ng µν = η ab h a µ h b ν , (3)\n\nwhere η ab is the Minkowski metric η ab = diag(+1, -1, -1, -1). For the Weitzenböck spacetime, the torsion is defined as [2]\n\nT θ µν = Γ θ νµ -Γ θ µν (4)\n\nwhich is antisymmetric w.r.t. its last two indices. Due to the requirement of absolute parallelism, the curvature of the Weitzenböck connection vanishes identically. The Weitzenböck connection also satisfies the relation\n\nΓ 0 θ µν = Γ θ µν -K θ µν , (5)\n\nwhere\n\nK θ µν = 1 2 [T µ θ ν + T ν θ µ -T θ µν ] (6)\n\nis the contortion tensor and Γ 0 θ µν are the Christoffel symbols in GR. The torsion tensor of the Weitzenböck connection can be decomposed into three irreducible parts under the group of global Lorentz transformations [4] : the tensor part\n\nt λµν = 1 2 (T λµν + T µλν ) + 1 6 (g νλ V µ + g νµ V λ ) - 1 3 g λµ V ν , (7)\n\nthe vector part\n\nV µ = T ν νµ ( 8\n\n)\n\nand the axial-vector part\n\nA µ = 1 6 ǫ µνρσ T νρσ . (9)\n\nThe torsion tensor can now be expressed in terms of these irreducible components as follows\n\nT λµν = 1 2 (t λµν -t λνµ ) + 1 3 (g λµ V ν -g λν V µ ) + ǫ λµνρ A ρ , (10)\n\nwhere\n\nǫ λµνρ = 1 √ -g δ λµνρ . (11)\n\nHere δ = {δ λµνρ } and δ * = {δ λµνρ } are completely skew symmetric tensor densities of weight -1 and +1 respectively [4] . TP theory provides an alternate description of the Einstein's field equations which is given by the teleparallel equivalent of GR [24, 31] . Mikhail et al. [27] defined the super-potential (which is antisymmetric in its last two indices) of the Möller tetrad theory as\n\nU µ νβ = √ -g 2κ P τ νβ χρσ [V ρ g σχ g µτ -λg τ µ K χρσ -(1 -2λ)g τ µ K σρχ ], (12)\n\nwhere\n\nP τ νβ χρσ = δ χ τ g νβ ρσ + δ ρ τ g νβ σχ -δ σ τ g νβ χρ ( 13\n\n)\n\nand g νβ ρσ is a tensor quantity defined by\n\ng νβ ρσ = δ ρ ν δ σ β -δ σ ν δ ρ β . ( 14\n\n)\n\nK σρχ is the contortion tensor given by Eq.( 6 ), g is the determinant of the metric tensor g µν , λ is the free dimensionless coupling constant of TPG, κ is the Einstein constant and V µ is the basic vector field given by Eq.( 8 ). The energy-momentum density is defined as\n\nΞ ν µ = U νρ µ , ρ , (15)\n\nwhere comma means ordinary differentiation. The momentum 4-vector of Möller prescription can be expressed as\n\nP µ = Σ Ξ 0 µ dxdydz, (16)\n\nwhere P 0 gives the energy and P 1 , P 2 and P 3 are the momentum components while the integration is taken over the hypersurface element Σ described by x 0 = t = constant. The energy may be given in the form of surface integral [20] as\n\nE = lim r→∞ r=constant U 0 0ρ u ρ dS, (17)\n\nwhere u ρ is the unit three-vector normal to the surface element dS."},{"section_type":"OTHER","section_title":"Teleparallel Version of the Stationary Axisymmetric Solutions","text":"Tupper [32] found five classes of non-null electromagnetic field plus perfect fluid solutions in which the electromagnetic field does not inherit one of the symmetries of the spacetime. The metric representing the stationary axisymmetric solutions is given by [32 ]\n\nds 2 = dt 2 -e 2K dρ 2 -(F 2 -B 2 )dφ 2 -e 2K dz 2 + 2Bdtdφ, (18)\n\nwhere B = B(ρ, z), K = K(ρ, z) and F = F (ρ) are such functions which satisfy the following relations . The metric given by Eq.( 18 ) represents five classes of non-null electromagnetic field and perfect fluid solutions which possesses a metric symmetry not inherited by the electromagnetic field and admits a homothetic vector field. Two of these classes contain electrovac solutions as special cases, while the other three necessarily contain some fluid. The generalization of this metric is given in [34] .\n\nḂ = F W ′ , B ′ = - 1 4 aF ( Ẇ 2 -W ′2 ), K ′ = - 1 2 aF Ẇ W ′ , Ẅ + Ḟ F -1 Ẇ + W ′′ = 0, (19)\n\nUsing the procedure adopted in the papers [11, 12] , the tetrad components of the above metric can be written as\n\nh a µ = 1 0 B 0 0 e K cos φ -F sin φ 0 0 e K sin φ F cos φ 0 0 0 0 e K (20)\n\nwith its inverse\n\nh a µ = 1 0 0 0 B F sin φ e -K cos φ -1 F sin φ 0 -B F cos φ e -K sin φ 1 F cos φ 0 0 0 0 e -K . ( 21\n\n)\n\nThe non-vanishing components of the torsion tensor are\n\nT 0 12 = Ḃ + B F (e K -Ḟ ), T 0 32 = B ′ , T 1 13 = -K ′ , T 2 12 = 1 F ( Ḟ -e K ), T 3 31 = -K. ( 22\n\n)\n\nUsing these expressions in Eqs.( 7 )-( 9 ), we obtain the following non-zero components of the tensor part\n\nt 001 = 1 3 [ K + 1 F ( Ḟ -e K )], t 003 = 1 3 K ′ , t 010 = 1 6 { 1 F (e K -Ḟ ) -K} = t 100 , t 030 = - 1 6 K ′ = t 300 , t 012 = 1 2 Ḃ + B 6 { 1 F (e K -Ḟ ) -K} = t 102 , t 021 = - 1 2 Ḃ - B 3 { 1 F (e K -Ḟ ) -K} = t 201 , t 023 = - 1 2 B ′ + 1 3 BK ′ = t 203 , t 032 = 1 2 B ′ - 1 6 BK ′ = t 302 , t 122 = 1 2 {F (e K -Ḟ ) + B Ḃ} + 1 6 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K} = t 212 , t 120 = B 6 { 1 F (e K -Ḟ ) -K} = t 210 , t 113 = 2K ′ 3 e 2K , t 131 = - K ′ 3 e 2K = t 311 , t 133 = - e 2K 6 { 1 F (e K -Ḟ ) + 2 K} = t 313 , t 221 = -F (e K -Ḟ ) -B Ḃ - 1 3 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K}, t 223 = -BB ′ + K ′ 3 (B 2 -F 2 ), t 331 = e 2K 3 { 1 F (e K -Ḟ ) + 2 K}, t 322 = 1 2 BB ′ - K ′ 6 (B 2 -F 2 ) = t 232 , t 320 = - 1 6 BK ′ = t 232 , (23)\n\nthe vector part\n\nV 1 = - 1 F ( Ḟ -e K ) -K, (24)\n\nV 3 = -K ′ , (25)\n\nand the axial-vector part\n\nA 1 = B ′ 3F e -2K , (26)\n\nA 3 = Ḃ 3F e -2K , (27)\n\nrespectively. The axial-vector component along the φ-direction vanishes and hence the spacelike axial-vector can be written as\n\nA = √ -g 11 A 1 êρ + √ -g 33 A 3 êz , (28)\n\nwhere êρ and êz are unit vectors along the radial and z-directions respectively. Substituting A 1 , A 3 , g 11 and g 33 in Eq.( 28 ), it follows that\n\nA = e -K 3F (B ′ êρ + Ḃê z ). ( 29\n\n)\n\nThis shows that the axial-vector lies along radial direction if B = B(z), along z-direction if B = B(ρ) and vanishes identically if B is constant. As the axialvector torsion represents the deviation of axial symmetry from cylindrical symmetry, the symmetry of the underlying spacetime will not be affected even for B constant. Also, the torsion plays the role of the gravitational force in TP theory, hence a spinless particle will obey the force equation [11, 24]\n\ndu ρ ds -Γ µρν u µ u ν = T µρν u µ u ν . ( 30\n\n)\n\nThe left hand side of this equation is the Weitzenböck covariant derivative of u ρ along the particle world-line. The appearance of the torsion tensor on its right hand side indicates that the torsion plays the role of an external force in TPG. It has been shown, both in GR and TP theories, by many authors [4, 35] that the spin precession of a Dirac particle in torsion gravity is related to the torsion axial-vector by\n\ndS dt = -b × S, ( 31\n\n)\n\nwhere S is the spin vector of a Dirac particle and b = 3 2 A, with A the spacelike part of the torsion axial-vector. Thus\n\nb = e -K 2F {B ′ êρ + Ḃê z }. (32)\n\nThe corresponding extra Hamiltonian [36] is given by\n\nδH = -b.σ, ( 33\n\n)\n\nwhere σ is the spin of the particle [35] . Using Eq.( 32 ), this takes the form\n\nδH = - e -K 2F (B ′ êρ + Ḃê z ).σ. ( 34\n\n)\n\n4 Teleparallel Energy of the Stationary Axisymmetric Solutions\n\nIn this section we evaluate the component of energy-momentum densities by using the teleparallel version of Möller prescription. Multiplying Eqs.( 24 ) and ( 25 ) by g 11 and g 33 respectively, it follows that\n\nV 1 = Ke -2K + e -2K F ( Ḟ -e K ), (35)\n\nV 3 = K ′ e -2K . (36)\n\nIn view of Eqs.( 6 ) and ( 22 ), the non-vanishing components of the contorsion tensor are\n\nK 100 = -e -2K { B 2 F 3 (e K -Ḟ ) + B Ḃ F 2 } = -K 010 , K 300 = - BB ′ F 2 e -2K = -K 030 , K 122 = - e -2K F 3 (e K -Ḟ ) = -K 212 , K 133 = Ke -4K = -K 313 , K 311 = K ′ e -4K = -K 131 , K 102 = K 120 = e -2K { B F 3 (e K -Ḟ ) + Ḃ 2F 2 } = -K 012 = -K 210 , K 302 = K 320 = K 023 = B ′ 2F 2 e -2K = -K 032 = -K 230 = -K 203 , K 021 = Ḃ 2F 2 e -2K = -K 201 . ( 37\n\n)\n\nIt should be mentioned here that the contorsion tensor is antisymmetric w.r.t. its first two indices. Making use of Eqs.( 35 )- (37) in Eq.( 12 ), we obtain the required independent non-vanishing components of the supperpotential in Möller's tetrad theory as\n\nU 01 0 = 1 κ [e K -Ḟ -F K + 1 2 (1 + λ) B Ḃ F ] = -U 10 0 , U 03 0 = 1 κ [-F K ′ + 1 2 (1 + λ) BB ′ F ] = -U 30 0 , U 21 0 = - 1 2κ (1 + λ) Ḃ F = -U 12 0 , U 23 0 = - 1 2κ (1 + λ) B ′ F = -U 32 0 , U 01 2 = 1 κ [B(e K -Ḟ ) + 1 2 (1 + λ) B 2 Ḃ F + 1 2 (1 -λ) ḂF ] = -U 10 2 , U 03 2 = 1 κ [ 1 2 (1 + λ) B 2 B ′ F + 1 2 (1 -λ)B ′ F ] = -U 30 2 , U 02 1 = 1 2κF (λ -1) Ḃe 2K = -U 20 1 , U 02 3 = 1 2κF (λ -1)B ′ e 2K = -U 30 1 . (38)\n\nIt is worth mentioning here that the supperpotential is skew symmetric w.r.t. its last two indices. When we make use of Eqs.( 15 ), ( 37 ), (38) and take λ = 1, the energy density turns out to be\n\nΞ 0 0 = 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ ) + 1 F 2 {BF ( B + E ′′ ) + ( Ḃ2 + B ′ 2 )F -B Ḃ Ḟ }]. ( 39\n\n)\n\nThis implies that\n\nE d T P T = E d GR + 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ )], (40)\n\nwhere E d stands for energy density. The only non-zero component of momentum density is along φ-direction and (for λ = 1) it takes the form\n\nΞ 0 2 = 1 κF 2 {F 3 ( B + B ′′ ) + B 2 F ( B + B ′′ ) + 2BF ( Ḃ2 + B ′ 2 ) -Ḃ Ḟ (B 2 + F 2 )} + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (41)\n\nthat is,\n\nM d T P T = M d GR + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (42)\n\nwhere M d stands for momentum density."},{"section_type":"OTHER","section_title":"Special Solutions of the Non-Null Einstein Maxwell Solutions","text":"In this section, we evaluate the above quantities for some special cases of the non-null Einstein Maxwell solutions."},{"section_type":"OTHER","section_title":"Electromagnetic Generalization of the Gödel Solution","text":"A special case of the non-null Einstein-Maxwell solutions can be obtained by choosing\n\nB = m n e nρ , F = e nρ , K = 0. ( 43\n\n)\n\nThis is known as electromagnetic generalization of the Gödel solution [32] .\n\nWhen we make use of Eq.( 43 ) in Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42), the corresponding results reduce to\n\nt 001 = 1 3 (n -e -nρ ), t 010 = 1 6 (e -nρ -n) = t 100 , t 012 = m 6n (1 + 2ne nρ ) = t 102 , t 021 = - m 3n (1 + 2ne nρ ) = t 201 , t 122 = e nρ 6n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ } = t 212 , t 120 = m 6n (1 -ne nρ ) = t 210 , t 133 = 1 6 (n -e -nρ ) = t 313 , t 221 = - e nρ 3n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ },\n\nt 331 = - 1 3 (n -e -nρ ), (44)\n\nV 1 = e -nρ -n, V 3 = 0, (45)\n\nA 1 = 0, A 3 = m 3 , (46)\n\nA = m 3 êz , b = m 2 êz , (47)\n\nδH = m 2 êz .σ, (48)\n\nΞ 0 0 = 1 κ (m 2 -n 2 )e nρ , (49)\n\nE d T P T = E d GR - n 2 κ e nρ (50)\n\nΞ 0 2 = 1 κ ( 2m 3 n ) + m κ (1 -2ne nρ )e nρ , (51)\n\nM d T P T = M d GR + m κ (1 -2ne nρ )e nρ . (52)\n\nThe metric (43) reduces to the usual perfect fluid solution when m = √ 2n [32] , i.e., B = √ 2e nρ . The corresponding energy and momentum densities take the form as\n\nE d T P T = E d GR - n 2 κ e nρ (53)\n\nM d T P T = M d GR + √ 2n κ (1 -2ne nρ )e nρ . (54)"},{"section_type":"OTHER","section_title":"The Gödel Metric","text":"When we choose B = e aρ , F = e aρ √ 2 and K = 0, the metric given by Eq.( 18 ) reduces to the Gödel metric [32] . The results corresponding to Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42) take the following form\n\nt 001 = 1 3 (a - √ 2e -aρ\n\n), t 010 = -1 6 (a -√ 2e -aρ ) = t 100 , t 012 = 1 6 ( √ 2 + 2ae aρ ) = t 102 , t 021 = -1 6 (2 √ 2 + ae aρ ) = t 102 , t 122 = e aρ 6 (2 √ 2 + ae aρ ) = t 212 , t 120 = 1 6 ( √ 2ae aρ ) = t 210 , t 133 = 1 6 (a -√ 2e -aρ ) = t 313 , t 221 = -e aρ 3 (2 √ 2 + ae aρ ), (55)\n\nt 331 = 1 3 (a - √ 2e -aρ ), ( 56\n\n) V 1 = √ 2e -aρ -a, V 3 = 0, (57)\n\nA 1 = 0, A 3 = √ 2a 3 , (58)\n\nA = √ 2a 3 êz , b = a √ 2 êz , (59)\n\nδH = a √ 2 êz .σ, (60)\n\nΞ 0 0 = √ 2 κ a 2 e aρ - a 2 κ √ 2 e aρ , (61)\n\nE d T P T = E d GR - a 2 κ √ 2 e aρ (62) Ξ 0 2 = a 2 κ √ 2 e 2aρ + a κ (1 - √ 2ae aρ )e aρ , (63)\n\nM d T P T = M d GR + a κ (1 - √ 2ae aρ )e aρ . ( 64\n\n)"},{"section_type":"DISCUSSION","section_title":"Summary and Discussion","text":"The purpose of this paper is twofold: Firstly, we have found the TP version of the non-null Einstein Maxwell solutions. This provides some interesting features about the axial vector and the corresponding quantities. Secondly, we have used the TP version of Möller prescription to evaluate the energymomentum distribution of the solutions. The axial-vector torsion of these solutions has been evaluated. The only non-vanishing components of the vector part are along the radial and the z-directions due to the cross term dx 0 dx 2 involving in the metric. This corresponds to the case of Kerr metric [11] , which involves the cross term dx 0 dx 3 . We also find the vector b which is related to the spin vector [4] as given by Eq.( 32 ). The axial-vector torsion lies in the ρz-plane, as its component along the φ-direction vanishes everywhere.\n\nThe non-inertial force on the Dirac particle can be represented as a rotation induced torsion of spacetime. There arise three possibilities for the axial-vector, depending upon the metric function B(ρ, z). When B is a function of z only, the axial-vector lies only along the radial direction. When B is a function of ρ only, the axialvector will lie along z-direction. The axial-vector vanishes identically for B to be constant. As the axial-vector represents the deviation from the symmetry of the underlying spacetime corresponding to an inertial field with respect to the Dirac particle, the symmetry of the spacetime will not be affected in the third possibility. Consequently there exists no inertial field with respect to the Dirac particle and the spin vector of the Dirac particle becomes constant. The corresponding extra Hamiltonian is expressed in terms of the vector b which vanishes when the metric function B is constant, i.e., when the axialvector becomes zero.\n\nThe energy-momentum distribution of the non-null Einstein-Maxwell solutions has been explored by using the TP version of Möller prescription. It is found that energy in the TP theory is equal to the energy in GR (as found by Sharif and Fatima [37] ) plus some additional part. If, for a particular case, we have K = 0 and K ′ , Ḟ = constant (or if Ḟ , K = 0 and K\n\n′ = constant), then E d T P G = E d GR . (65)\n\nOn the other hand, the only non-vanishing component of the momentum density lies along φ-direction, similar to the case of Kerr metric [11] , due to the cross term appearing in both the metrics. When we choose λ = 1, it becomes equal to be the momentum in GR [37] plus an additional quantity. If F , Ḃ, B ′′ , K all vanish, then\n\nM d T P G = M d GR . (66)\n\nBy taking particular values of E, F and K, we obtain the electromagnetic generalization of the Gödel solution and the Gödel metric as two special cases. The corresponding results for both the special cases are obtained. It is shown that, for the electromagnetic generalization of the Gödel solution, Eq.(65) does not hold, while Eq.(66) holds when m = 0. However, for the perfect fluid case, i.e., when m = √ 2n, both Eqs.( 65 ) and (66) hold by taking n = 0. In the case of the Gödel metric, these equations hold if we choose the arbitrary constant a = 0. For the special solutions, the vector part lies along the radial direction while the axial-vector part along z-direction.\n\nWe would like to re-iterate here that the tetrad formalism itself has some advantages which comes mainly from its independence from the equivalence principle and consequent suitability to the discussion of quantum issues. In TPG, an energy-momentum gauge current j i µ for the gravitational field can be defined. This is covariant under a spacetime general coordinate transformation and transforms covariantly under a global tangent space Lorentz transformation [38] . It, then, follows that j i µ is a true spacetime tensor but not a tangent space tensor. When we re-write the gauge field equations in a purely spacetime form, they lead to the Einstein field equations and the gauge current j i µ reduces to the canonical energy-momentum pseudo-tensor of the gravitational field. Thus TPG seems to provide a more appropriate environment to deal with the energy problem.\n\nFinally, it is pointed out that we are not claiming that this paper has resolved the problems of GR using the TPG. This is an attempt to touch some issues in TPG with the hope that this alternative may provide more feasible results. Also, it is always an interesting and enriching to look at things from another point of view. This endeavor is in itself commendable."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along ρ and z directions. The three possibilities of the axial vector depending on the metric function B are discussed. The vector related with spin has also been evaluated and the corresponding extra Hamiltonian is furnished. Further, we use the teleparallel version of Möller prescription to find the energy-momentum distribution of the solutions. It is interesting to note that (for λ = 1) energy and momentum densities in teleparallel theory are equal to the corresponding quantities in GR plus an additional quantity in each, which may become equal under certain conditions. Finally, we discuss the two special cases of the stationary axisymmetric solutions.\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"The attempts made by Einstein and his followers to unify gravitation with other interactions led to the investigation of structures of gravitation other than the metric tensor. These structures yield the metric tensor as a by product. Tetrad field is one of these structures which leads to the theory of teleparallel gravity (TPG) [1, 2] . TPG is an alternative theory of gravity which corresponds to a gauge theory of translation group [3, 4] based on Weitzenböck geometry [5] . This theory is characterized by the vanishing of curvature identically while the torsion is taken to be non-zero. In TPG, gravitation is attributed to torsion [4] which plays a role of force [6] . In General Relativity (GR), gravitation geometrizes the underlying spacetime. The translational gauge potentials appear as a non-trivial part of the tetrad field and induce a teleparallel (TP) structure on spacetime which is directly related to the presence of a gravitational field. In some other theories [3] [4] [5] [6] [7] [8] , torsion is only relevant when spins are important [9] . This point of view indicates that torsion might represent additional degrees of freedom as compared to curvature. As a result, some new physics may be associated with it. Teleparallelism is naturally formulated by gauging external (spacetime) translations which are closely related to the group of general coordinate transformations underlying GR. Thus the energy-momentum tensor represents the matter source in the field equations of tetradic theories of gravity like in GR.\\n\\nThere is a large literature available [10] about the study of TP versions of the exact solutions of GR. Recently, Pereira, et al. [11] obtained the TP versions of the Schwarzschild and the stationary axisymmetric Kerr solutions of GR. They proved that the axial-vector torsion plays the role of the gravitomagnetic component of the gravitational field in the case of slow rotation and weak field approximations. In a previous paper [12] , we have found the TP versions of the Friedmann models and of the Lewis-Papapetrou spacetimes, and also discussed their axial-vectors.\\n\\nThere has been a longstanding, controversial and still unresolved problem of the localization of energy (i.e., to express it as a unique tensor quantity) in GR [13] . Einstein [14] introduced the energy-momentum pseudo-tensor and then Landau-Lifshitz [15] , Papapetrou [16] , Bergmann [17] , Tolman [18] and Weinberg [19] proposed their own prescriptions to resolve this issue. All these prescriptions can provide meaningful results only in Cartesian coordinates. But Möller [20] introduced a coordinate-independent prescription. The idea of coordinate-independent quasi-local mass was introduced [21] by associ-ating a Hamiltonian term to each gravitational energy-momentum pseudotensor. Later, a Hamiltonian approach in the frame of Schwinger condition [22] was developed, followed by the construction of a Lagrangian density of TP equivalent to GR [4, 6, 23, 24] . Many authors explored several examples in the framework of GR and found that different energy-momentum complexes can give either the same [25] or different [26] results for a given spacetime.\\n\\nMikhail et al. [27] defined the superpotential in the Moller's tetrad theory which has been used to find the energy in TPG. Vargas [28] defined the TP version of Bergman, Einstein and Landau-Lifshitz prescriptions and found that the total energy of the closed Friedman-Robinson-Walker universe is zero by using the last two prescriptions. This agrees with the results of GR available in literature [29] . Later, many authors [30] used TP version of these prescriptions and showed that energy may be localized in TPG. Keeping these points in mind, this paper is addressed to the following two problems: We obtain TP version of the stationary axisymmetric solutions and then calculate the axial-vector part of the torsion tensor. The energymomentum distribution of the solutions is explored by using the TP version of Möller prescription.\\n\\nThe scheme adopted in this paper is as follows. In section 2, we shall review the basic concepts of TP theory. Section 3 contains the TP version of the stationary axisymmetric solutions and the tensor, vector and axial-vector parts of the torsion tensor. Section 4 is devoted to evaluate the energymomentum distribution for this family of solutions using the TP version of Möller prescription. In section 5, we present two special solutions for this class of metrics and investigate the corresponding quantities. The last section contains a summary and a discussion of the results obtained.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"An Overview of the Teleparallel Theory\",\n \"text\": \"In teleparallel theory, the connection is a Weitzenböck connection given as [31] Γ\\n\\nθ µν = h a θ ∂ ν h a µ , (1)\\n\\nwhere h a ν is a non-trivial tetrad. Its inverse field is denoted by h a µ and satisfy the relations\\n\\nh a µ h a ν = δ µ ν ; h a µ h b µ = δ a b . (2)\\n\\nIn this paper, the Latin alphabet (a, b, c, ... = 0, 1, 2, 3) will be used to denote tangent space indices and the Greek alphabet (µ, ν, ρ, ... = 0, 1, 2, 3) to denote spacetime indices. The Riemannian metric in TP theory arises as a by product [4] of the tetrad field given by\\n\\ng µν = η ab h a µ h b ν , (3)\\n\\nwhere η ab is the Minkowski metric η ab = diag(+1, -1, -1, -1). For the Weitzenböck spacetime, the torsion is defined as [2]\\n\\nT θ µν = Γ θ νµ -Γ θ µν (4)\\n\\nwhich is antisymmetric w.r.t. its last two indices. Due to the requirement of absolute parallelism, the curvature of the Weitzenböck connection vanishes identically. The Weitzenböck connection also satisfies the relation\\n\\nΓ 0 θ µν = Γ θ µν -K θ µν , (5)\\n\\nwhere\\n\\nK θ µν = 1 2 [T µ θ ν + T ν θ µ -T θ µν ] (6)\\n\\nis the contortion tensor and Γ 0 θ µν are the Christoffel symbols in GR. The torsion tensor of the Weitzenböck connection can be decomposed into three irreducible parts under the group of global Lorentz transformations [4] : the tensor part\\n\\nt λµν = 1 2 (T λµν + T µλν ) + 1 6 (g νλ V µ + g νµ V λ ) - 1 3 g λµ V ν , (7)\\n\\nthe vector part\\n\\nV µ = T ν νµ ( 8\\n\\n)\\n\\nand the axial-vector part\\n\\nA µ = 1 6 ǫ µνρσ T νρσ . (9)\\n\\nThe torsion tensor can now be expressed in terms of these irreducible components as follows\\n\\nT λµν = 1 2 (t λµν -t λνµ ) + 1 3 (g λµ V ν -g λν V µ ) + ǫ λµνρ A ρ , (10)\\n\\nwhere\\n\\nǫ λµνρ = 1 √ -g δ λµνρ . (11)\\n\\nHere δ = {δ λµνρ } and δ * = {δ λµνρ } are completely skew symmetric tensor densities of weight -1 and +1 respectively [4] . TP theory provides an alternate description of the Einstein's field equations which is given by the teleparallel equivalent of GR [24, 31] . Mikhail et al. [27] defined the super-potential (which is antisymmetric in its last two indices) of the Möller tetrad theory as\\n\\nU µ νβ = √ -g 2κ P τ νβ χρσ [V ρ g σχ g µτ -λg τ µ K χρσ -(1 -2λ)g τ µ K σρχ ], (12)\\n\\nwhere\\n\\nP τ νβ χρσ = δ χ τ g νβ ρσ + δ ρ τ g νβ σχ -δ σ τ g νβ χρ ( 13\\n\\n)\\n\\nand g νβ ρσ is a tensor quantity defined by\\n\\ng νβ ρσ = δ ρ ν δ σ β -δ σ ν δ ρ β . ( 14\\n\\n)\\n\\nK σρχ is the contortion tensor given by Eq.( 6 ), g is the determinant of the metric tensor g µν , λ is the free dimensionless coupling constant of TPG, κ is the Einstein constant and V µ is the basic vector field given by Eq.( 8 ). The energy-momentum density is defined as\\n\\nΞ ν µ = U νρ µ , ρ , (15)\\n\\nwhere comma means ordinary differentiation. The momentum 4-vector of Möller prescription can be expressed as\\n\\nP µ = Σ Ξ 0 µ dxdydz, (16)\\n\\nwhere P 0 gives the energy and P 1 , P 2 and P 3 are the momentum components while the integration is taken over the hypersurface element Σ described by x 0 = t = constant. The energy may be given in the form of surface integral [20] as\\n\\nE = lim r→∞ r=constant U 0 0ρ u ρ dS, (17)\\n\\nwhere u ρ is the unit three-vector normal to the surface element dS.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Teleparallel Version of the Stationary Axisymmetric Solutions\",\n \"text\": \"Tupper [32] found five classes of non-null electromagnetic field plus perfect fluid solutions in which the electromagnetic field does not inherit one of the symmetries of the spacetime. The metric representing the stationary axisymmetric solutions is given by [32 ]\\n\\nds 2 = dt 2 -e 2K dρ 2 -(F 2 -B 2 )dφ 2 -e 2K dz 2 + 2Bdtdφ, (18)\\n\\nwhere B = B(ρ, z), K = K(ρ, z) and F = F (ρ) are such functions which satisfy the following relations . The metric given by Eq.( 18 ) represents five classes of non-null electromagnetic field and perfect fluid solutions which possesses a metric symmetry not inherited by the electromagnetic field and admits a homothetic vector field. Two of these classes contain electrovac solutions as special cases, while the other three necessarily contain some fluid. The generalization of this metric is given in [34] .\\n\\nḂ = F W ′ , B ′ = - 1 4 aF ( Ẇ 2 -W ′2 ), K ′ = - 1 2 aF Ẇ W ′ , Ẅ + Ḟ F -1 Ẇ + W ′′ = 0, (19)\\n\\nUsing the procedure adopted in the papers [11, 12] , the tetrad components of the above metric can be written as\\n\\nh a µ = 1 0 B 0 0 e K cos φ -F sin φ 0 0 e K sin φ F cos φ 0 0 0 0 e K (20)\\n\\nwith its inverse\\n\\nh a µ = 1 0 0 0 B F sin φ e -K cos φ -1 F sin φ 0 -B F cos φ e -K sin φ 1 F cos φ 0 0 0 0 e -K . ( 21\\n\\n)\\n\\nThe non-vanishing components of the torsion tensor are\\n\\nT 0 12 = Ḃ + B F (e K -Ḟ ), T 0 32 = B ′ , T 1 13 = -K ′ , T 2 12 = 1 F ( Ḟ -e K ), T 3 31 = -K. ( 22\\n\\n)\\n\\nUsing these expressions in Eqs.( 7 )-( 9 ), we obtain the following non-zero components of the tensor part\\n\\nt 001 = 1 3 [ K + 1 F ( Ḟ -e K )], t 003 = 1 3 K ′ , t 010 = 1 6 { 1 F (e K -Ḟ ) -K} = t 100 , t 030 = - 1 6 K ′ = t 300 , t 012 = 1 2 Ḃ + B 6 { 1 F (e K -Ḟ ) -K} = t 102 , t 021 = - 1 2 Ḃ - B 3 { 1 F (e K -Ḟ ) -K} = t 201 , t 023 = - 1 2 B ′ + 1 3 BK ′ = t 203 , t 032 = 1 2 B ′ - 1 6 BK ′ = t 302 , t 122 = 1 2 {F (e K -Ḟ ) + B Ḃ} + 1 6 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K} = t 212 , t 120 = B 6 { 1 F (e K -Ḟ ) -K} = t 210 , t 113 = 2K ′ 3 e 2K , t 131 = - K ′ 3 e 2K = t 311 , t 133 = - e 2K 6 { 1 F (e K -Ḟ ) + 2 K} = t 313 , t 221 = -F (e K -Ḟ ) -B Ḃ - 1 3 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K}, t 223 = -BB ′ + K ′ 3 (B 2 -F 2 ), t 331 = e 2K 3 { 1 F (e K -Ḟ ) + 2 K}, t 322 = 1 2 BB ′ - K ′ 6 (B 2 -F 2 ) = t 232 , t 320 = - 1 6 BK ′ = t 232 , (23)\\n\\nthe vector part\\n\\nV 1 = - 1 F ( Ḟ -e K ) -K, (24)\\n\\nV 3 = -K ′ , (25)\\n\\nand the axial-vector part\\n\\nA 1 = B ′ 3F e -2K , (26)\\n\\nA 3 = Ḃ 3F e -2K , (27)\\n\\nrespectively. The axial-vector component along the φ-direction vanishes and hence the spacelike axial-vector can be written as\\n\\nA = √ -g 11 A 1 êρ + √ -g 33 A 3 êz , (28)\\n\\nwhere êρ and êz are unit vectors along the radial and z-directions respectively. Substituting A 1 , A 3 , g 11 and g 33 in Eq.( 28 ), it follows that\\n\\nA = e -K 3F (B ′ êρ + Ḃê z ). ( 29\\n\\n)\\n\\nThis shows that the axial-vector lies along radial direction if B = B(z), along z-direction if B = B(ρ) and vanishes identically if B is constant. As the axialvector torsion represents the deviation of axial symmetry from cylindrical symmetry, the symmetry of the underlying spacetime will not be affected even for B constant. Also, the torsion plays the role of the gravitational force in TP theory, hence a spinless particle will obey the force equation [11, 24]\\n\\ndu ρ ds -Γ µρν u µ u ν = T µρν u µ u ν . ( 30\\n\\n)\\n\\nThe left hand side of this equation is the Weitzenböck covariant derivative of u ρ along the particle world-line. The appearance of the torsion tensor on its right hand side indicates that the torsion plays the role of an external force in TPG. It has been shown, both in GR and TP theories, by many authors [4, 35] that the spin precession of a Dirac particle in torsion gravity is related to the torsion axial-vector by\\n\\ndS dt = -b × S, ( 31\\n\\n)\\n\\nwhere S is the spin vector of a Dirac particle and b = 3 2 A, with A the spacelike part of the torsion axial-vector. Thus\\n\\nb = e -K 2F {B ′ êρ + Ḃê z }. (32)\\n\\nThe corresponding extra Hamiltonian [36] is given by\\n\\nδH = -b.σ, ( 33\\n\\n)\\n\\nwhere σ is the spin of the particle [35] . Using Eq.( 32 ), this takes the form\\n\\nδH = - e -K 2F (B ′ êρ + Ḃê z ).σ. ( 34\\n\\n)\\n\\n4 Teleparallel Energy of the Stationary Axisymmetric Solutions\\n\\nIn this section we evaluate the component of energy-momentum densities by using the teleparallel version of Möller prescription. Multiplying Eqs.( 24 ) and ( 25 ) by g 11 and g 33 respectively, it follows that\\n\\nV 1 = Ke -2K + e -2K F ( Ḟ -e K ), (35)\\n\\nV 3 = K ′ e -2K . (36)\\n\\nIn view of Eqs.( 6 ) and ( 22 ), the non-vanishing components of the contorsion tensor are\\n\\nK 100 = -e -2K { B 2 F 3 (e K -Ḟ ) + B Ḃ F 2 } = -K 010 , K 300 = - BB ′ F 2 e -2K = -K 030 , K 122 = - e -2K F 3 (e K -Ḟ ) = -K 212 , K 133 = Ke -4K = -K 313 , K 311 = K ′ e -4K = -K 131 , K 102 = K 120 = e -2K { B F 3 (e K -Ḟ ) + Ḃ 2F 2 } = -K 012 = -K 210 , K 302 = K 320 = K 023 = B ′ 2F 2 e -2K = -K 032 = -K 230 = -K 203 , K 021 = Ḃ 2F 2 e -2K = -K 201 . ( 37\\n\\n)\\n\\nIt should be mentioned here that the contorsion tensor is antisymmetric w.r.t. its first two indices. Making use of Eqs.( 35 )- (37) in Eq.( 12 ), we obtain the required independent non-vanishing components of the supperpotential in Möller's tetrad theory as\\n\\nU 01 0 = 1 κ [e K -Ḟ -F K + 1 2 (1 + λ) B Ḃ F ] = -U 10 0 , U 03 0 = 1 κ [-F K ′ + 1 2 (1 + λ) BB ′ F ] = -U 30 0 , U 21 0 = - 1 2κ (1 + λ) Ḃ F = -U 12 0 , U 23 0 = - 1 2κ (1 + λ) B ′ F = -U 32 0 , U 01 2 = 1 κ [B(e K -Ḟ ) + 1 2 (1 + λ) B 2 Ḃ F + 1 2 (1 -λ) ḂF ] = -U 10 2 , U 03 2 = 1 κ [ 1 2 (1 + λ) B 2 B ′ F + 1 2 (1 -λ)B ′ F ] = -U 30 2 , U 02 1 = 1 2κF (λ -1) Ḃe 2K = -U 20 1 , U 02 3 = 1 2κF (λ -1)B ′ e 2K = -U 30 1 . (38)\\n\\nIt is worth mentioning here that the supperpotential is skew symmetric w.r.t. its last two indices. When we make use of Eqs.( 15 ), ( 37 ), (38) and take λ = 1, the energy density turns out to be\\n\\nΞ 0 0 = 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ ) + 1 F 2 {BF ( B + E ′′ ) + ( Ḃ2 + B ′ 2 )F -B Ḃ Ḟ }]. ( 39\\n\\n)\\n\\nThis implies that\\n\\nE d T P T = E d GR + 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ )], (40)\\n\\nwhere E d stands for energy density. The only non-zero component of momentum density is along φ-direction and (for λ = 1) it takes the form\\n\\nΞ 0 2 = 1 κF 2 {F 3 ( B + B ′′ ) + B 2 F ( B + B ′′ ) + 2BF ( Ḃ2 + B ′ 2 ) -Ḃ Ḟ (B 2 + F 2 )} + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (41)\\n\\nthat is,\\n\\nM d T P T = M d GR + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (42)\\n\\nwhere M d stands for momentum density.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Special Solutions of the Non-Null Einstein Maxwell Solutions\",\n \"text\": \"In this section, we evaluate the above quantities for some special cases of the non-null Einstein Maxwell solutions.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Electromagnetic Generalization of the Gödel Solution\",\n \"text\": \"A special case of the non-null Einstein-Maxwell solutions can be obtained by choosing\\n\\nB = m n e nρ , F = e nρ , K = 0. ( 43\\n\\n)\\n\\nThis is known as electromagnetic generalization of the Gödel solution [32] .\\n\\nWhen we make use of Eq.( 43 ) in Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42), the corresponding results reduce to\\n\\nt 001 = 1 3 (n -e -nρ ), t 010 = 1 6 (e -nρ -n) = t 100 , t 012 = m 6n (1 + 2ne nρ ) = t 102 , t 021 = - m 3n (1 + 2ne nρ ) = t 201 , t 122 = e nρ 6n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ } = t 212 , t 120 = m 6n (1 -ne nρ ) = t 210 , t 133 = 1 6 (n -e -nρ ) = t 313 , t 221 = - e nρ 3n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ },\\n\\nt 331 = - 1 3 (n -e -nρ ), (44)\\n\\nV 1 = e -nρ -n, V 3 = 0, (45)\\n\\nA 1 = 0, A 3 = m 3 , (46)\\n\\nA = m 3 êz , b = m 2 êz , (47)\\n\\nδH = m 2 êz .σ, (48)\\n\\nΞ 0 0 = 1 κ (m 2 -n 2 )e nρ , (49)\\n\\nE d T P T = E d GR - n 2 κ e nρ (50)\\n\\nΞ 0 2 = 1 κ ( 2m 3 n ) + m κ (1 -2ne nρ )e nρ , (51)\\n\\nM d T P T = M d GR + m κ (1 -2ne nρ )e nρ . (52)\\n\\nThe metric (43) reduces to the usual perfect fluid solution when m = √ 2n [32] , i.e., B = √ 2e nρ . The corresponding energy and momentum densities take the form as\\n\\nE d T P T = E d GR - n 2 κ e nρ (53)\\n\\nM d T P T = M d GR + √ 2n κ (1 -2ne nρ )e nρ . (54)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Gödel Metric\",\n \"text\": \"When we choose B = e aρ , F = e aρ √ 2 and K = 0, the metric given by Eq.( 18 ) reduces to the Gödel metric [32] . The results corresponding to Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42) take the following form\\n\\nt 001 = 1 3 (a - √ 2e -aρ\\n\\n), t 010 = -1 6 (a -√ 2e -aρ ) = t 100 , t 012 = 1 6 ( √ 2 + 2ae aρ ) = t 102 , t 021 = -1 6 (2 √ 2 + ae aρ ) = t 102 , t 122 = e aρ 6 (2 √ 2 + ae aρ ) = t 212 , t 120 = 1 6 ( √ 2ae aρ ) = t 210 , t 133 = 1 6 (a -√ 2e -aρ ) = t 313 , t 221 = -e aρ 3 (2 √ 2 + ae aρ ), (55)\\n\\nt 331 = 1 3 (a - √ 2e -aρ ), ( 56\\n\\n) V 1 = √ 2e -aρ -a, V 3 = 0, (57)\\n\\nA 1 = 0, A 3 = √ 2a 3 , (58)\\n\\nA = √ 2a 3 êz , b = a √ 2 êz , (59)\\n\\nδH = a √ 2 êz .σ, (60)\\n\\nΞ 0 0 = √ 2 κ a 2 e aρ - a 2 κ √ 2 e aρ , (61)\\n\\nE d T P T = E d GR - a 2 κ √ 2 e aρ (62) Ξ 0 2 = a 2 κ √ 2 e 2aρ + a κ (1 - √ 2ae aρ )e aρ , (63)\\n\\nM d T P T = M d GR + a κ (1 - √ 2ae aρ )e aρ . ( 64\\n\\n)\"\n },\n {\n \"section_type\": \"DISCUSSION\",\n \"section_title\": \"Summary and Discussion\",\n \"text\": \"The purpose of this paper is twofold: Firstly, we have found the TP version of the non-null Einstein Maxwell solutions. This provides some interesting features about the axial vector and the corresponding quantities. Secondly, we have used the TP version of Möller prescription to evaluate the energymomentum distribution of the solutions. The axial-vector torsion of these solutions has been evaluated. The only non-vanishing components of the vector part are along the radial and the z-directions due to the cross term dx 0 dx 2 involving in the metric. This corresponds to the case of Kerr metric [11] , which involves the cross term dx 0 dx 3 . We also find the vector b which is related to the spin vector [4] as given by Eq.( 32 ). The axial-vector torsion lies in the ρz-plane, as its component along the φ-direction vanishes everywhere.\\n\\nThe non-inertial force on the Dirac particle can be represented as a rotation induced torsion of spacetime. There arise three possibilities for the axial-vector, depending upon the metric function B(ρ, z). When B is a function of z only, the axial-vector lies only along the radial direction. When B is a function of ρ only, the axialvector will lie along z-direction. The axial-vector vanishes identically for B to be constant. As the axial-vector represents the deviation from the symmetry of the underlying spacetime corresponding to an inertial field with respect to the Dirac particle, the symmetry of the spacetime will not be affected in the third possibility. Consequently there exists no inertial field with respect to the Dirac particle and the spin vector of the Dirac particle becomes constant. The corresponding extra Hamiltonian is expressed in terms of the vector b which vanishes when the metric function B is constant, i.e., when the axialvector becomes zero.\\n\\nThe energy-momentum distribution of the non-null Einstein-Maxwell solutions has been explored by using the TP version of Möller prescription. It is found that energy in the TP theory is equal to the energy in GR (as found by Sharif and Fatima [37] ) plus some additional part. If, for a particular case, we have K = 0 and K ′ , Ḟ = constant (or if Ḟ , K = 0 and K\\n\\n′ = constant), then E d T P G = E d GR . (65)\\n\\nOn the other hand, the only non-vanishing component of the momentum density lies along φ-direction, similar to the case of Kerr metric [11] , due to the cross term appearing in both the metrics. When we choose λ = 1, it becomes equal to be the momentum in GR [37] plus an additional quantity. If F , Ḃ, B ′′ , K all vanish, then\\n\\nM d T P G = M d GR . (66)\\n\\nBy taking particular values of E, F and K, we obtain the electromagnetic generalization of the Gödel solution and the Gödel metric as two special cases. The corresponding results for both the special cases are obtained. It is shown that, for the electromagnetic generalization of the Gödel solution, Eq.(65) does not hold, while Eq.(66) holds when m = 0. However, for the perfect fluid case, i.e., when m = √ 2n, both Eqs.( 65 ) and (66) hold by taking n = 0. In the case of the Gödel metric, these equations hold if we choose the arbitrary constant a = 0. For the special solutions, the vector part lies along the radial direction while the axial-vector part along z-direction.\\n\\nWe would like to re-iterate here that the tetrad formalism itself has some advantages which comes mainly from its independence from the equivalence principle and consequent suitability to the discussion of quantum issues. In TPG, an energy-momentum gauge current j i µ for the gravitational field can be defined. This is covariant under a spacetime general coordinate transformation and transforms covariantly under a global tangent space Lorentz transformation [38] . It, then, follows that j i µ is a true spacetime tensor but not a tangent space tensor. When we re-write the gauge field equations in a purely spacetime form, they lead to the Einstein field equations and the gauge current j i µ reduces to the canonical energy-momentum pseudo-tensor of the gravitational field. Thus TPG seems to provide a more appropriate environment to deal with the energy problem.\\n\\nFinally, it is pointed out that we are not claiming that this paper has resolved the problems of GR using the TPG. This is an attempt to touch some issues in TPG with the hope that this alternative may provide more feasible results. Also, it is always an interesting and enriching to look at things from another point of view. This endeavor is in itself commendable.\"\n }\n]"}}},{"rowIdx":87,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0489"},"arxiv_id":{"kind":"string","value":"0704.0489"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.2478/s11534-008-0018-0"},"content_hash":{"kind":"string","value":"05a07c71e5ae114221137d0f5aa9063589e159ec0dabb4c42451345ca0534803"},"title":{"kind":"string","value":"Relativistic treatment in D-dimensions to a spin-zero particle with\n noncentral equal scalar and vector ring-shaped Kratzer potential"},"abstract":{"kind":"string","value":"The Klein-Gordon equation in D-dimensions for a recently proposed Kratzer\npotential plus ring-shaped potential is solved analytically by means of the\nconventional Nikiforov-Uvarov method. The exact energy bound-states and the\ncorresponding wave functions of the Klein-Gordon are obtained in the presence\nof the noncentral equal scalar and vector potentials. The results obtained in\nthis work are more general and can be reduced to the standard forms in\nthree-dimensions given by other works."},"authors":{"kind":"list like","value":["Sameer M. Ikhdair","Ramazan Sever"],"string":"[\n \"Sameer M. Ikhdair\",\n \"Ramazan Sever\"\n]"},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]"},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics. The Klein-Gordon equation has also been used to understand the motion of a spin-0 particle in large class of potentials. In recent years, much efforts have been paid to solve these relativistic wave equations for various potentials by using different methods. These relativistic equations contain two objects: the four-vector linear momentum operator and the scalar rest mass. They allow us to introduce two types of potential coupling, which are the four-vector potential (V) and the space-time scalar potential (S).\n\nRecently, many authors have worked on solving these equations with physical potentials including Morse potential [3], Hulthen potential [4] , Woods-Saxon potential [5] , Pösch-Teller potential [6] , reflectionless-type potential [7] , pseudoharmonic oscillator [8] , ring-shaped harmonic oscillator [9], V 0 tanh 2 (r/r 0 ) potential [10] , five-parameter exponential potential [11] , Rosen-Morse potential [12] , and generalized symmetrical double-well potential [13] , etc. It is remarkable that in most works in this area, the scalar and vector potentials are almost taken to be equal (i.e., S = V ) [2, 14] . However, in some few other cases, it is considered the case where the scalar potential is greater than the vector potential (in order to guarantee the existence of Klein-Gordon bound states) (i.e., S > V ) [15] [16] [17] [18] [19] . Nonetheless, such physical potentials are very few. The bound-state solutions for the last case is obtained for the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is greater than the vector potential [15] .\n\nThe study of exact solutions of the nonrelativistic equation for a class of non-central potentials with a vector potential and a non-central scalar potential is of considerable interest in quantum chemistry [20] [21] [22] . In recent years, numerous studies [23] have been made in 2 analyzing the bound states of an electron in a Coulomb field with simultaneous presence of Aharanov-Bohm (AB) [24] field, and/or a magnetic Dirac monopole [25] , and Aharanov-Bohm plus oscillator (ABO) systems. In most of these works, the eigenvalues and eigenfunctions are obtained by means of seperation of variables in spherical or other orthogonal curvilinear coordinate systems. The path integral for particles moving in non-central potentials is evaluated to derive the energy spectrum of this system analytically [26] . In addition, the idea of SUSY and shape invariance is also used to obtain exact solutions of such noncentral but seperable potentials [27, 28] . Very recently, the conventional Nikiforov-Uvarov (NU) method [29] has been used to give a clear recipe of how to obtain an explicit exact bound-states solutions for the energy eigenvalues and their corresponding wave functions in terms of orthogonal polynomials for a class of non-central potentials [30] .\n\nAnother type of noncentral potentials is the ring-shaped Kratzer potential, which is a combination of a Coulomb potential plus an inverse square potential plus a noncentral angular part [31, 32] . The Kratzer potential has been used to describe the vibrational-rotational motion of isolated diatomic molecules [33] and has a mixed-energy spectrum containing both bound and scattering states with bound-states have been widely used in molecular spectroscopy [34] . The ring-shaped Kratzer potential consists of radial and angular-dependent potentials and is useful in studying ring-shaped molecules [22] . In taking the relativistic effects into account for spin-0 particle in the presence of a class of noncentral potentials, Yasuk et al [35] applied the NU method to solve the Klein-Gordon equation for the noncentral Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [36] also used the same method to solve the Klein-Gordon equation for the Kratzer-type potential.\n\nRecently, Chen and Dong [37] proposed a new ring-shaped potential and obtained the exact solution of the Schrödinger equation for the Coulomb potential plus this new ringshaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. This type of potential used by Chen and Dong [37] appears to be very similar to the potential used by Yasuk et al [35] . Moreover, Cheng and Dai [38] , proposed a new potential consisting from the modified Kratzer's potential [33] plus the 3 new proposed ring-shaped potential in [37] . They have presented the energy eigenvalues for this proposed exactly-solvable non-central potential in three dimensional (i.e., D = 3)-Schrödinger equation by means of the NU method. The two quantum systems solved by Refs [37, 38] are closely relevant to each other as they deal with a Coulombic field interaction except for a slight change in the angular momentum barrier acts as a repulsive core which is for any arbitrary angular momentum ℓ prevents collapse of the system in any dimensional space due to the slight perturbation to the original angular momentum barrier. Very recently, we have also applied the NU method to solve the Schrödinger equation in any arbitrary Ddimension to this new modified Kratzer-type potential [39, 40] .\n\nThe aim of the present paper is to consider the relativistic effects for the spin-0 particle in our recent works [39, 40] . So we want to present a systematic recipe to solving the D-dimensional Klein-Gordon equation for the Kratzer plus the new ring-shaped potential proposed in [38] using the simple NU method. This method is based on solving the Klein-Gordon equation by reducing it to a generalized hypergeometric equation.\n\nThis work is organized as follows: in section II, we shall present the Klein-Gordon equation in spherical coordinates for spin-0 particle in the presence of equal scalar and vector noncentral Kratzer plus the new ring-shaped potential and we also separate it into radial and angular parts. Section III is devoted to a brief description of the NU method.\n\nIn section IV, we present the exact solutions to the radial and angular parts of the Klein-Gordon equation in D-dimensions. Finally, the relevant conclusions are given in section V.\n\nIn relativistic quantum mechanics, we usually use the Klein-Gordon equation for describing a scalar particle, i.e., the spin-0 particle dynamics. The discussion of the relativistic behavior of spin-zero particles requires understanding the single particle spectrum and the 4 exact solutions to the Klein Gordon equation which are constructed by using the four-vector potential A λ (λ = 0, 1, 2, 3) and the scalar potential (S). In order to simplify the solution of the Klein-Gordon equation, the four-vector potential can be written as A λ = (A 0 , 0, 0, 0).\n\nThe first component of the four-vector potential is represented by a vector potential (V ), i.e., A 0 = V. In this case, the motion of a relativistic spin-0 particle in a potential is described by the Klein-Gordon equation with the potentials V and S [1]. For the case S ≥ V, there exist bound-state (real) solutions for a relativistic spin-zero particle [15] [16] [17] [18] [19] . On the other hand, for S = V, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby the bound-state solutions are easily obtained by using the well-known methods developed in nonrelativistic quantum mechanics [2] .\n\nThe Klein-Gordon equation describing a scalar particle (spin-0 particle) with scalar S(r, θ, ϕ) and vector V (r, θ, ϕ) potentials is given by [2,14]\n\nP 2 -[E R -V (r, θ, ϕ)/2] 2 + [µ + S(r, θ, ϕ)/2] 2 ψ(r, θ, ϕ) = 0, ( 1\n\n)\n\nwhere E R , P and µ are the relativistic energy, momentum operator and rest mass of the particle, respectively. The potential terms are scaled in (1) by Alhaidari et al [14] so that in the nonrelativistic limit the interaction potential becomes V.\n\nIn this work, we consider the equal scalar and vector potentials case, that is, S(r, θ, ϕ) = V (r, θ, ϕ) with the recently proposed general non-central potential taken in the form of the Kratzer plus ring-shaped potential [38-40]:\n\nV (r, θ, ϕ) = V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ , ( 2\n\n) V 1 (r) = - A r + B r 2 , V 2 (θ) = Cctg 2 θ, V 3 (ϕ) = 0, ( 3\n\n)\n\nwhere A = 2a 0 r 0 , B = a 0 r 2 0 and C is positive real constant with a 0 is the dissociation energy and r 0 is the equilibrium internuclear distance [33] . The potentials in Eq. ( 3 ) introduced by Cheng-Dai [38] reduce to the Kratzer potential in the limiting case of C = 0 [33]. In fact the energy spectrum for this potential can be obtained directly by considering it as special case of the general non-central seperable potentials [30] .\n\nIn the relativistic atomic units (h = c = 1), the D-dimensional Klein-Gordon equation in (1) becomes [41]\n\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 -(E R + µ) V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ + E 2 R -µ 2 ψ(r, θ, ϕ) = 0. ( 4\n\n)\n\nwith ψ(r, θ, ϕ) being the spherical total wave function separated as follows\n\nψ njm (r, θ, ϕ) = R(r)Y m j (θ, ϕ), R(r) = r -(D-1)/2 g(r), Y m j (θ, ϕ) = H(θ)Φ(ϕ). ( 5\n\n)\n\nInserting Eqs (3) and (5) into Eq. ( 4 ) and using the method of separation of variables, the following differential equations are obtained:\n\n1 r D-1 d dr r D-1 dR(r) dr - j(j + D -2) r 2 + α 2 2 α 2 1 - A r + B r 2 R(r) = 0, ( 6\n\n) 1 sin θ d dθ sin θ d dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ + j(j + D -2) H(θ) = 0, ( 7\n\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 8\n\n)\n\nwhere α 2 1 = µ -E R , α 2 2 = µ + E R , m\n\nand j are constants and with m 2 and λ j = j(j + D -2) are the separation constants.\n\nFor a nonrelativistic treatment with the same potential, the Schrödinger equation in spherical coordinates is\n\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 + 2µ E N R -V 1 (r) - V 2 (θ) r 2 - V 3 (ϕ) r 2 sin 2 θ ψ(r, θ, ϕ) = 0. ( 9\n\n)\n\nwhere µ and E N R are the reduced mass and the nonrelativistic energy, respectively. Besides, the spherical total wave function appearing in Eq. (9) has the same representation as in Eq. 6 (5) but with the transformation j → ℓ. Inserting Eq. (5) into Eq. (9) leads to the following differential equations [39,40]:\n\n1 r D-1 d dr r D-1 dR(r) dr - λ D r 2 -2µ E N R + A r - B r 2 R(r) = 0, ( 10\n\n) 1 sin θ d dθ sin θ d dθ - m 2 + 2µC cos 2 θ sin 2 θ + ℓ(ℓ + D -2) H(θ) = 0, ( 11\n\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 12\n\n)\n\nwhere m 2 and λ ℓ = ℓ(ℓ + D -2) are the separation constants. Equations (6)-(8) have the same functional form as Eqs (10)-(12). Therefore, the solution of the Klein-Gordon equation can be reduced to the solution of the Schrödinger equation with the appropriate choice of parameters: j → ℓ, α 2 1 → -E N R and α 2 2 → 2µ.\n\nThe solution of Eq. ( 8 ) is well-known periodic and must satisfy the period boundary condition Φ(ϕ + 2π) = Φ(ϕ) which is the azimuthal angle solution:\n\nΦ m (ϕ) = 1 √ 2π exp(±imϕ), m = 0, 1, 2, ..... ( 13\n\n)\n\nAdditionally, Eqs (6) and (7) are radial and polar angle equations and they will be solved by using the Nikiforov-Uvarov (NU) method [29] which is given briefly in the following section.\n\nThe NU method is based on reducing the second-order differential equation to a generalized equation of hypergeometric type [29] . In this sense, the Schrödinger equation, after employing an appropriate coordinate transformation s = s(r), transforms to the following form:\n\nψ ′′ n (s) + τ (s) σ(s) ψ ′ n (s) + σ(s) σ 2 (s) ψ n (s) = 0, ( 14\n\n) 7\n\nwhere σ(s) and σ(s) are polynomials, at most of second-degree, and τ (s) is a first-degree polynomial. Using a wave function, ψ n (s), of the simple ansatz:\n\nψ n (s) = φ n (s)y n (s), ( 15\n\n)\n\nreduces (14) into an equation of a hypergeometric type\n\nσ(s)y ′′ n (s) + τ (s)y ′ n (s) + λy n (s) = 0, ( 16\n\n)\n\nwhere\n\nσ(s) = π(s) φ(s) φ ′ (s) , ( 17\n\n)\n\nτ (s) = τ (s) + 2π(s), τ ′ (s) < 0, ( 18\n\n)\n\nand λ is a parameter defined as\n\nλ = λ n = -nτ ′ (s) - n (n -1) 2 σ ′′ (s), n = 0, 1, 2, .... ( 19\n\n)\n\nThe polynomial τ (s) with the parameter s and prime factors show the differentials at first degree be negative. It is worthwhile to note that λ or λ n are obtained from a particular solution of the form y(s) = y n (s) which is a polynomial of degree n. Further, the other part y n (s) of the wave function (14) is the hypergeometric-type function whose polynomial solutions are given by Rodrigues relation\n\ny n (s) = B n ρ(s) d n ds n [σ n (s)ρ(s)] , ( 20\n\n)\n\nwhere B n is the normalization constant and the weight function ρ(s) must satisfy the condition [29]\n\nd ds w(s) = τ (s) σ(s) w(s), w(s) = σ(s)ρ(s). ( 21\n\n)\n\nThe function π and the parameter λ are defined as\n\nπ(s) = σ ′ (s) -τ (s) 2 ± σ ′ (s) -τ (s) 2 2 -σ(s) + kσ(s), ( 22\n\n) 8 λ = k + π ′ (s). ( 23\n\n)\n\nIn principle, since π(s) has to be a polynomial of degree at most one, the expression under the square root sign in (22) can be arranged to be the square of a polynomial of first degree [29] . This is possible only if its discriminant is zero. In this case, an equation for k is obtained. After solving this equation, the obtained values of k are substituted in (22) . In addition, by comparing equations (19) and (23), we obtain the energy eigenvalues.\n\nWe seek to solving the radial and angular parts of the Klein-Gordon equation given by Eqs (6) and (7). Equation (6) involving the radial part can be written simply in the following form [39-41]:\n\nd 2 g(r) dr 2 - (M -1)(M -3) 4r 2 -α 2 2 A r - B r 2 + α 2 1 α 2 2 g(r) = 0, ( 24\n\n)\n\nwhere\n\nM = D + 2j. ( 25\n\n)\n\nOn the other hand, Eq. ( 7 ) involving the angular part of Klein-Gordon equation retakes the simple form\n\nd 2 H(θ) dθ 2 + ctg(θ) dH(θ) dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ -j(j + D -2) H(θ) = 0. ( 26\n\n)\n\nThus, Eqs (24) and (26) have to be solved latter through the NU method in the following subsections. 9 B. Eigenvalues and eigenfunctions of the angle-dependent equation\n\nIn order to apply NU method [29, 30, 33, 35, 36, [38] [39] [40] [42] [43] [44] , we use a suitable transformation variable s = cos θ. The polar angle part of the Klein Gordon equation in (26) can be written in the following universal associated-Legendre differential equation form [38-40]\n\nd 2 H(s) ds 2 - 2s 1 -s 2 dH(s) ds + 1 (1 -s 2 ) 2 j(j + D -2)(1 -s 2 ) -m 2 -Cα 2 2 s 2 H(θ) = 0. ( 27\n\n)\n\nEquation (27) has already been solved for the three-dimensional Schrödinger equation through the NU method in [38] . However, the aim in this subsection is to solve with different parameters resulting from the D-space-dimensions of Klein-Gordon equation. Further, Eq.\n\n(27) is compared with (14) and the following identifications are obtained\n\nτ (s) = -2s, σ(s) = 1 -s 2 , σ(s) = -m ′2 + (1 -s 2 )ν ′ , ( 28\n\n)\n\nwhere\n\nν ′ = j ′ (j ′ + D -2) = j(j + D -2) + Cα 2 2 , m ′2 = m 2 + Cα 2 2 . ( 29\n\n)\n\nInserting the above expressions into equation (22), one obtains the following function:\n\nπ(s) = ± (ν ′ -k)s 2 + k -ν ′ + m ′2 , ( 30\n\n)\n\nFollowing the method, the polynomial π(s) is found in the following possible values\n\nπ(s) = m ′ s for k 1 = ν ′ -m ′2 , -m ′ s for k 1 = ν ′ -m ′2 , m ′ for k 2 = ν ′ , -m ′ for k 2 = ν ′ . ( 31\n\n)\n\nImposing the condition τ ′ (s) < 0, for equation (18), one selects\n\nk 1 = ν ′ -m ′2 and π(s) = -m ′ s, ( 32\n\n) which yields 10 τ (s) = -2(1 + m ′ )s. ( 33\n\n)\n\nUsing equations (19) and (23), the following expressions for λ are obtained, respectively,\n\nλ = λ n = 2 n(1 + m ′ ) + n( n -1), ( 34\n\n) λ = ν ′ -m ′ (1 + m ′ ). ( 35\n\n)\n\nWe compare equations (34) and (35), the new angular momentum j values are obtained as\n\nj = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -4Cα 2 2 -1, ( 36\n\n) or j ′ = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -1. ( 37\n\n)\n\nUsing Eqs (15)-(17) and (20)-(21), the polynomial solution of y n is expressed in terms of Jacobi polynomials [39,40] which are one of the orthogonal polynomials:\n\nH n (θ) = N n sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ), ( 38\n\n) where N n = 1 2 m ′ ( n+m ′ )! (2 n+2m ′ +1)( n+2m ′ )! n! 2\n\nis the normalization constant determined by +1 -1 [H n (s)] 2 ds = 1 and using the orthogonality relation of Jacobi polynomials [35, 45, 46] .\n\nBesides n = - (1 + 2m ′ ) 2 + 1 2 (2j + 1) 2 + 4j(D -3) + 4Cα 2 2 , ( 39\n\n)\n\nwhere m ′ is defined by equation (29).\n\nThe solution of the radial part of Klein-Gordon equation, Eq. ( 24 ), for the Kratzer's potential has already been solved by means of NU-method in [39] . Very recently, using the same method, the problem for the non-central potential in (2) has been solved in three 11 dimensions (3D) by Cheng and Dai [36] . However, the aim of this subsection is to solve the problem with a different radial separation function g(r) in any arbitrary dimensions. In what follows, we present the exact bound-states (real) solution of Eq. (24). Letting\n\nε 2 = α 2 1 α 2 2 , 4γ 2 = (M -1)(M -3) + 4Bα 2 2 , β 2 = Aα 2 2 , ( 40\n\n)\n\nand substituting these expressions in equation (24), one gets\n\nd 2 g(r) dr 2 + -ε 2 r 2 + β 2 r -γ 2 r 2 g(r) = 0. ( 41\n\n)\n\nTo apply the conventional NU-method, equation (41) is compared with (14), resulting in the following expressions:\n\nτ (r) = 0, σ(r) = r, σ(r) = -ε 2 r 2 + β 2 r -γ 2 . ( 42\n\n)\n\nSubstituting the above expressions into equation (22) gives\n\nπ(r) = 1 2 ± 1 2 4ε 2 r 2 + 4(k -β 2 )r + 4γ 2 + 1. ( 43\n\n)\n\nTherefore, we can determine the constant k by using the condition that the discriminant of the square root is zero, that is\n\nk = β 2 ± ε 4γ 2 + 1, 4γ 2 + 1 = (D + 2j -2) 2 + 4Bα 2 2 . ( 44\n\n)\n\nIn view of that, we arrive at the following four possible functions of π(r) :\n\nπ(r) = 1 2 + εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 -εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 + εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1, 1 2 -εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1. ( 45\n\n)\n\nThe correct value of π(r) is chosen such that the function τ (r) given by Eq. (18) will have negative derivative [29] . So we can select the physical values to be\n\nk = β 2 -ε 4γ 2 + 1 and π(r) = 1 2 -εr - 1 2 4γ 2 + 1 , ( 46\n\n)\n\n12 which yield τ (r) = -2εr + (1 + 4γ 2 + 1), τ ′ (r) = -2ε < 0. (47) Using Eqs (19) and (23), the following expressions for λ are obtained, respectively,\n\nλ = λ n = 2nε, n = 0, 1, 2, ..., ( 48\n\n) λ = δ 2 -ε(1 + 4γ 2 + 1). ( 49\n\n)\n\nSo we can obtain the Klein Gordon energy eigenvalues from the following relation:\n\n1 + 2n + (D + 2j -2) 2 + 4(µ + E R )B µ -E R = A µ + E R , ( 50\n\n)\n\nand hence for the Kratzer plus the new ring-shaped potential, it becomes\n\n1 + 2n + (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 51\n\n)\n\nwith j defined in (36) . Although Eq. (51) is exactly solvable for E R but it looks to be little complicated. Further, it is interesting to investigate the solution for the Coulomb potential.\n\nTherefore, applying the following transformations: A = Ze 2 , B = 0, and j = ℓ, the central part of the potential in (3) turns into the Coulomb potential with Klein Gordon solution for the energy spectra given by\n\nE R = µ 1 - 2q 2 e 2 q 2 e 2 + (2n + 2ℓ + D -1) 2 , n, ℓ = 0, 1, 2, ..., ( 52\n\n)\n\nwhere q = Ze is the charge of the nucleus. Further, Eq. ( 52 ) can be expanded as a series in the nucleus charge as\n\nE R = µ - 2µq 2 e 2 (2n + 2ℓ + D -1) 2 + 2µq 4 e 4 (2n + 2ℓ + D -1) 4 -O(qe) 6 , ( 53\n\n)\n\nThe physical meaning of each term in the last equation was given in detail by Ref. [36] .\n\nBesides, the difference from the conventional nonrelativistic form is because of the choice of the vector V (r, θ, ϕ) and scalar S(r, θ, ϕ) parts of the potential in Eq. ( 1 ). 13 Overmore, if the value of j obtained by Eq.(36) is inserted into the eigenvalues of the radial part of the Klein Gordon equation with the noncentral potential given by Eq. (51), we finally find the energy eigenvalues for a bound electron in the presence of a noncentral potential by Eq. (2) as\n\n1 + 2n + (2j ′ + D -2) 2 + 4(a 0 r 2 0 -C)(µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 54\n\n)\n\nwhere m ′ = m 2 + C(µ + E R ) and n is given by Eq. (39). On the other hand, the solution of the Schrödinger equation, Eq. ( 9 ), for this potential has already been obtained by using the same method in Ref. [39] and it is in the Coulombic-like form:\n\nE N R = - 8µa 2 0 r 2 0 2n + 1 + (2ℓ ′ + D -2) 2 + 8µ(a 0 r 2 0 -C) 2 , n = 0, 1, 2, ... ( 55\n\n) 2ℓ ′ + D -2 = (D -2) 2 + (2 n + 2m ′ + 1) 2 -1, ( 56\n\n)\n\nwhere m ′ = √ m 2 + 2µC. Also, applying the following appropriate transformation: 54 ) provides exactly the nonrelativistic limit given by Eq. (55).\n\nµ+E R → 2µ, µ -E R → -E N R , j → ℓ to Eq. (\n\nIn what follows, let us now turn attention to find the radial wavefunctions for this potential. Substituting the values of σ(r), π(r) and τ (r) in Eqs (42), (45) and (47) into Eqs.\n\n(17) and (21), we find φ(r) = r (ζ+1)\n\n/2 e -εr , ( 57\n\n) ρ(r) = r ζ e -2εr , ( 58\n\n)\n\nwhere ζ = √ 4γ 2 + 1. Then from equation (20), we obtain\n\ny nj (r) = B nj r -ζ e 2εr d n dr n r n+ζ e -2εr , ( 59\n\n)\n\nand the wave function g(r) can be written in the form of the generalized Laguerre polynomials as 14 g(ρ) = C nj ρ 2ε\n\n(1+ζ)/2\n\ne -ρ/2 L ζ n (ρ), ( 60\n\n)\n\nwhere for Kratzer's potential we have\n\nζ = (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ), ρ = 2εr. ( 61\n\n)\n\nFinally, the radial wave functions of the Klein-Gordon equation are obtained R(ρ) = C nj ρ 2ε\n\n(ζ+2-D)/2\n\ne -ρ/2 L ζ n (ρ), ( 62\n\n)\n\nwhere C nj is the normalization constant to be determined below. Using the normalization condition, ∞ 0 R 2 (r)r D-1 dr = 1, and the orthogonality relation of the generalized Laguerre polynomials, (2n+η+1)(n+η)! n! , we have\n\n∞ 0 z η+1 e -z [L η n (z)] 2 dz =\n\nC nj = 2 µ 2 -E 2 R 1+ ζ 2 n! (2n + ζ + 1) (n + ζ)! . ( 63\n\n)\n\nFinally, we may express the normalized total wave functions as\n\nψ(r, θ, ϕ) = 2 µ 2 -E 2 R 1+ ζ 2 2 m ′ ( n + m ′ )! (2 n + 2m ′ + 1)( n + 2m ′ )! n!n! 2π (2n + ζ + 1) (n + ζ)! ×r (ζ+2-D) 2 exp(-µ 2 -E 2 R r)L ζ n (2 µ 2 -E 2 R r) sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ) exp(±imϕ). ( 64\n\n)\n\nwhere ζ is defined in Eq. ( 61 ) and m ′ is given after the Eq. (54).\n\nThe relativistic spin-0 particle D-dimensional Klein-Gordon equation has been solved easily for its exact bound-states with equal scalar and vector ring-shaped Kratzer potential through the conventional NU method. The analytical expressions for the total energy levels and eigenfunctions of this system can be reduced to their conventional three-dimensional space form upon setting D = 3. Further, the noncentral potentials treated in [30] can be introduced as perturbation to the Kratzer's potential by adjusting the strength of the 15 coupling constant C in terms of a 0 , which is the coupling constant of the Kratzer's potential.\n\nAdditionally, the radial and polar angle wave functions of Klein-Gordon equation are found in terms of Laguerre and Jacobi polynomials, respectively. The method presented in this paper is general and worth extending to the solution of other interaction problems. This method is very simple and useful in solving other complicated systems analytically without given a restiction conditions on the solution of some quantum systems as the case in the other models. We have seen that for the nonrelativistic model, the exact energy spectra can be obtained either by solving the Schrödinger equation in (9) (cf. Ref. [39] or Eq. ( 55 )) or by applying appropriate transformation to the relativistic solution given by Eq. ( 54 ). 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[38] Y.F. Cheng and T.Q. Dai, Phys. Scr. 75 (2007) 274. [39] S.M. Ikhdair and R. Sever, preprint quant-ph/0703008; quant-ph/0703042. [40] S.M. Ikhdair and R. Sever, preprint quant-ph/0703131. [41] S.M. Ikhdair and R. Sever, Z. Phys. C 56 (1992) 155; C 58 (1993) 153; D 28 (1993) 1; Hadronic J. 15 (1992) 389; Int. J. Mod. Phys. A 18 (2003) 4215; A 19 (2004) 1771; A 20 (2005) 4035; A 20 (2005) 6509; A 21 (2006) 2191; A 21 (2006) 3989; A 21 (2006) 6699; Int. J. Mod. Phys. E (in press) (preprint hep-ph/0504176); S. Ikhdair et al, Tr. J. Phys. 16 (1992) 510; 17 (1993) 474. [42] S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. (DOI 10.1007/s10773-006-9317-7; J.\n\nMath. Chem. (DOI 10.1007/s10910-006-9115-8). [43] S.M. Ikhdair and R. Sever, Ann. Phys. (Leipzig) 16 (3) (2007) 218. [44] S.M. Ikhdair and R. Sever, to appear in the Int. J. Mod. Phys. E (preprint quant-19 ph/0611065). [45] G. Sezgo, Orthogonal Polynomials (American Mathematical Society, New York, 1939). [46] N.N. Lebedev, Special Functions and Their Applications (Prentice-Hall, Englewood Cliffs, NJ, 1965). 20"},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics. The Klein-Gordon equation has also been used to understand the motion of a spin-0 particle in large class of potentials. In recent years, much efforts have been paid to solve these relativistic wave equations for various potentials by using different methods. These relativistic equations contain two objects: the four-vector linear momentum operator and the scalar rest mass. They allow us to introduce two types of potential coupling, which are the four-vector potential (V) and the space-time scalar potential (S).\n\nRecently, many authors have worked on solving these equations with physical potentials including Morse potential [3], Hulthen potential [4] , Woods-Saxon potential [5] , Pösch-Teller potential [6] , reflectionless-type potential [7] , pseudoharmonic oscillator [8] , ring-shaped harmonic oscillator [9], V 0 tanh 2 (r/r 0 ) potential [10] , five-parameter exponential potential [11] , Rosen-Morse potential [12] , and generalized symmetrical double-well potential [13] , etc. It is remarkable that in most works in this area, the scalar and vector potentials are almost taken to be equal (i.e., S = V ) [2, 14] . However, in some few other cases, it is considered the case where the scalar potential is greater than the vector potential (in order to guarantee the existence of Klein-Gordon bound states) (i.e., S > V ) [15] [16] [17] [18] [19] . Nonetheless, such physical potentials are very few. The bound-state solutions for the last case is obtained for the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is greater than the vector potential [15] .\n\nThe study of exact solutions of the nonrelativistic equation for a class of non-central potentials with a vector potential and a non-central scalar potential is of considerable interest in quantum chemistry [20] [21] [22] . In recent years, numerous studies [23] have been made in 2 analyzing the bound states of an electron in a Coulomb field with simultaneous presence of Aharanov-Bohm (AB) [24] field, and/or a magnetic Dirac monopole [25] , and Aharanov-Bohm plus oscillator (ABO) systems. In most of these works, the eigenvalues and eigenfunctions are obtained by means of seperation of variables in spherical or other orthogonal curvilinear coordinate systems. The path integral for particles moving in non-central potentials is evaluated to derive the energy spectrum of this system analytically [26] . In addition, the idea of SUSY and shape invariance is also used to obtain exact solutions of such noncentral but seperable potentials [27, 28] . Very recently, the conventional Nikiforov-Uvarov (NU) method [29] has been used to give a clear recipe of how to obtain an explicit exact bound-states solutions for the energy eigenvalues and their corresponding wave functions in terms of orthogonal polynomials for a class of non-central potentials [30] .\n\nAnother type of noncentral potentials is the ring-shaped Kratzer potential, which is a combination of a Coulomb potential plus an inverse square potential plus a noncentral angular part [31, 32] . The Kratzer potential has been used to describe the vibrational-rotational motion of isolated diatomic molecules [33] and has a mixed-energy spectrum containing both bound and scattering states with bound-states have been widely used in molecular spectroscopy [34] . The ring-shaped Kratzer potential consists of radial and angular-dependent potentials and is useful in studying ring-shaped molecules [22] . In taking the relativistic effects into account for spin-0 particle in the presence of a class of noncentral potentials, Yasuk et al [35] applied the NU method to solve the Klein-Gordon equation for the noncentral Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [36] also used the same method to solve the Klein-Gordon equation for the Kratzer-type potential.\n\nRecently, Chen and Dong [37] proposed a new ring-shaped potential and obtained the exact solution of the Schrödinger equation for the Coulomb potential plus this new ringshaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. This type of potential used by Chen and Dong [37] appears to be very similar to the potential used by Yasuk et al [35] . Moreover, Cheng and Dai [38] , proposed a new potential consisting from the modified Kratzer's potential [33] plus the 3 new proposed ring-shaped potential in [37] . They have presented the energy eigenvalues for this proposed exactly-solvable non-central potential in three dimensional (i.e., D = 3)-Schrödinger equation by means of the NU method. The two quantum systems solved by Refs [37, 38] are closely relevant to each other as they deal with a Coulombic field interaction except for a slight change in the angular momentum barrier acts as a repulsive core which is for any arbitrary angular momentum ℓ prevents collapse of the system in any dimensional space due to the slight perturbation to the original angular momentum barrier. Very recently, we have also applied the NU method to solve the Schrödinger equation in any arbitrary Ddimension to this new modified Kratzer-type potential [39, 40] .\n\nThe aim of the present paper is to consider the relativistic effects for the spin-0 particle in our recent works [39, 40] . So we want to present a systematic recipe to solving the D-dimensional Klein-Gordon equation for the Kratzer plus the new ring-shaped potential proposed in [38] using the simple NU method. This method is based on solving the Klein-Gordon equation by reducing it to a generalized hypergeometric equation.\n\nThis work is organized as follows: in section II, we shall present the Klein-Gordon equation in spherical coordinates for spin-0 particle in the presence of equal scalar and vector noncentral Kratzer plus the new ring-shaped potential and we also separate it into radial and angular parts. Section III is devoted to a brief description of the NU method.\n\nIn section IV, we present the exact solutions to the radial and angular parts of the Klein-Gordon equation in D-dimensions. Finally, the relevant conclusions are given in section V."},{"section_type":"OTHER","section_title":"II. THE KLEIN-GORDON EQUATION WITH EQUAL SCALAR AND VECTOR POTENTIALS","text":"In relativistic quantum mechanics, we usually use the Klein-Gordon equation for describing a scalar particle, i.e., the spin-0 particle dynamics. The discussion of the relativistic behavior of spin-zero particles requires understanding the single particle spectrum and the 4 exact solutions to the Klein Gordon equation which are constructed by using the four-vector potential A λ (λ = 0, 1, 2, 3) and the scalar potential (S). In order to simplify the solution of the Klein-Gordon equation, the four-vector potential can be written as A λ = (A 0 , 0, 0, 0).\n\nThe first component of the four-vector potential is represented by a vector potential (V ), i.e., A 0 = V. In this case, the motion of a relativistic spin-0 particle in a potential is described by the Klein-Gordon equation with the potentials V and S [1]. For the case S ≥ V, there exist bound-state (real) solutions for a relativistic spin-zero particle [15] [16] [17] [18] [19] . On the other hand, for S = V, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby the bound-state solutions are easily obtained by using the well-known methods developed in nonrelativistic quantum mechanics [2] .\n\nThe Klein-Gordon equation describing a scalar particle (spin-0 particle) with scalar S(r, θ, ϕ) and vector V (r, θ, ϕ) potentials is given by [2,14]\n\nP 2 -[E R -V (r, θ, ϕ)/2] 2 + [µ + S(r, θ, ϕ)/2] 2 ψ(r, θ, ϕ) = 0, ( 1\n\n)\n\nwhere E R , P and µ are the relativistic energy, momentum operator and rest mass of the particle, respectively. The potential terms are scaled in (1) by Alhaidari et al [14] so that in the nonrelativistic limit the interaction potential becomes V.\n\nIn this work, we consider the equal scalar and vector potentials case, that is, S(r, θ, ϕ) = V (r, θ, ϕ) with the recently proposed general non-central potential taken in the form of the Kratzer plus ring-shaped potential [38-40]:\n\nV (r, θ, ϕ) = V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ , ( 2\n\n) V 1 (r) = - A r + B r 2 , V 2 (θ) = Cctg 2 θ, V 3 (ϕ) = 0, ( 3\n\n)\n\nwhere A = 2a 0 r 0 , B = a 0 r 2 0 and C is positive real constant with a 0 is the dissociation energy and r 0 is the equilibrium internuclear distance [33] . The potentials in Eq. ( 3 ) introduced by Cheng-Dai [38] reduce to the Kratzer potential in the limiting case of C = 0 [33]. In fact the energy spectrum for this potential can be obtained directly by considering it as special case of the general non-central seperable potentials [30] .\n\nIn the relativistic atomic units (h = c = 1), the D-dimensional Klein-Gordon equation in (1) becomes [41]\n\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 -(E R + µ) V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ + E 2 R -µ 2 ψ(r, θ, ϕ) = 0. ( 4\n\n)\n\nwith ψ(r, θ, ϕ) being the spherical total wave function separated as follows\n\nψ njm (r, θ, ϕ) = R(r)Y m j (θ, ϕ), R(r) = r -(D-1)/2 g(r), Y m j (θ, ϕ) = H(θ)Φ(ϕ). ( 5\n\n)\n\nInserting Eqs (3) and (5) into Eq. ( 4 ) and using the method of separation of variables, the following differential equations are obtained:\n\n1 r D-1 d dr r D-1 dR(r) dr - j(j + D -2) r 2 + α 2 2 α 2 1 - A r + B r 2 R(r) = 0, ( 6\n\n) 1 sin θ d dθ sin θ d dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ + j(j + D -2) H(θ) = 0, ( 7\n\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 8\n\n)\n\nwhere α 2 1 = µ -E R , α 2 2 = µ + E R , m\n\nand j are constants and with m 2 and λ j = j(j + D -2) are the separation constants.\n\nFor a nonrelativistic treatment with the same potential, the Schrödinger equation in spherical coordinates is\n\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 + 2µ E N R -V 1 (r) - V 2 (θ) r 2 - V 3 (ϕ) r 2 sin 2 θ ψ(r, θ, ϕ) = 0. ( 9\n\n)\n\nwhere µ and E N R are the reduced mass and the nonrelativistic energy, respectively. Besides, the spherical total wave function appearing in Eq. (9) has the same representation as in Eq. 6 (5) but with the transformation j → ℓ. Inserting Eq. (5) into Eq. (9) leads to the following differential equations [39,40]:\n\n1 r D-1 d dr r D-1 dR(r) dr - λ D r 2 -2µ E N R + A r - B r 2 R(r) = 0, ( 10\n\n) 1 sin θ d dθ sin θ d dθ - m 2 + 2µC cos 2 θ sin 2 θ + ℓ(ℓ + D -2) H(θ) = 0, ( 11\n\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 12\n\n)\n\nwhere m 2 and λ ℓ = ℓ(ℓ + D -2) are the separation constants. Equations (6)-(8) have the same functional form as Eqs (10)-(12). Therefore, the solution of the Klein-Gordon equation can be reduced to the solution of the Schrödinger equation with the appropriate choice of parameters: j → ℓ, α 2 1 → -E N R and α 2 2 → 2µ.\n\nThe solution of Eq. ( 8 ) is well-known periodic and must satisfy the period boundary condition Φ(ϕ + 2π) = Φ(ϕ) which is the azimuthal angle solution:\n\nΦ m (ϕ) = 1 √ 2π exp(±imϕ), m = 0, 1, 2, ..... ( 13\n\n)\n\nAdditionally, Eqs (6) and (7) are radial and polar angle equations and they will be solved by using the Nikiforov-Uvarov (NU) method [29] which is given briefly in the following section."},{"section_type":"METHOD","section_title":"III. NIKIFOROV-UVAROV METHOD","text":"The NU method is based on reducing the second-order differential equation to a generalized equation of hypergeometric type [29] . In this sense, the Schrödinger equation, after employing an appropriate coordinate transformation s = s(r), transforms to the following form:\n\nψ ′′ n (s) + τ (s) σ(s) ψ ′ n (s) + σ(s) σ 2 (s) ψ n (s) = 0, ( 14\n\n) 7\n\nwhere σ(s) and σ(s) are polynomials, at most of second-degree, and τ (s) is a first-degree polynomial. Using a wave function, ψ n (s), of the simple ansatz:\n\nψ n (s) = φ n (s)y n (s), ( 15\n\n)\n\nreduces (14) into an equation of a hypergeometric type\n\nσ(s)y ′′ n (s) + τ (s)y ′ n (s) + λy n (s) = 0, ( 16\n\n)\n\nwhere\n\nσ(s) = π(s) φ(s) φ ′ (s) , ( 17\n\n)\n\nτ (s) = τ (s) + 2π(s), τ ′ (s) < 0, ( 18\n\n)\n\nand λ is a parameter defined as\n\nλ = λ n = -nτ ′ (s) - n (n -1) 2 σ ′′ (s), n = 0, 1, 2, .... ( 19\n\n)\n\nThe polynomial τ (s) with the parameter s and prime factors show the differentials at first degree be negative. It is worthwhile to note that λ or λ n are obtained from a particular solution of the form y(s) = y n (s) which is a polynomial of degree n. Further, the other part y n (s) of the wave function (14) is the hypergeometric-type function whose polynomial solutions are given by Rodrigues relation\n\ny n (s) = B n ρ(s) d n ds n [σ n (s)ρ(s)] , ( 20\n\n)\n\nwhere B n is the normalization constant and the weight function ρ(s) must satisfy the condition [29]\n\nd ds w(s) = τ (s) σ(s) w(s), w(s) = σ(s)ρ(s). ( 21\n\n)\n\nThe function π and the parameter λ are defined as\n\nπ(s) = σ ′ (s) -τ (s) 2 ± σ ′ (s) -τ (s) 2 2 -σ(s) + kσ(s), ( 22\n\n) 8 λ = k + π ′ (s). ( 23\n\n)\n\nIn principle, since π(s) has to be a polynomial of degree at most one, the expression under the square root sign in (22) can be arranged to be the square of a polynomial of first degree [29] . This is possible only if its discriminant is zero. In this case, an equation for k is obtained. After solving this equation, the obtained values of k are substituted in (22) . In addition, by comparing equations (19) and (23), we obtain the energy eigenvalues."},{"section_type":"OTHER","section_title":"IV. EXACT SOLUTIONS OF THE RADIAL AND ANGLE-DEPENDENT EQUATIONS A. Separating variables of the Klein-Gordon equation","text":"We seek to solving the radial and angular parts of the Klein-Gordon equation given by Eqs (6) and (7). Equation (6) involving the radial part can be written simply in the following form [39-41]:\n\nd 2 g(r) dr 2 - (M -1)(M -3) 4r 2 -α 2 2 A r - B r 2 + α 2 1 α 2 2 g(r) = 0, ( 24\n\n)\n\nwhere\n\nM = D + 2j. ( 25\n\n)\n\nOn the other hand, Eq. ( 7 ) involving the angular part of Klein-Gordon equation retakes the simple form\n\nd 2 H(θ) dθ 2 + ctg(θ) dH(θ) dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ -j(j + D -2) H(θ) = 0. ( 26\n\n)\n\nThus, Eqs (24) and (26) have to be solved latter through the NU method in the following subsections. 9 B. Eigenvalues and eigenfunctions of the angle-dependent equation\n\nIn order to apply NU method [29, 30, 33, 35, 36, [38] [39] [40] [42] [43] [44] , we use a suitable transformation variable s = cos θ. The polar angle part of the Klein Gordon equation in (26) can be written in the following universal associated-Legendre differential equation form [38-40]\n\nd 2 H(s) ds 2 - 2s 1 -s 2 dH(s) ds + 1 (1 -s 2 ) 2 j(j + D -2)(1 -s 2 ) -m 2 -Cα 2 2 s 2 H(θ) = 0. ( 27\n\n)\n\nEquation (27) has already been solved for the three-dimensional Schrödinger equation through the NU method in [38] . However, the aim in this subsection is to solve with different parameters resulting from the D-space-dimensions of Klein-Gordon equation. Further, Eq.\n\n(27) is compared with (14) and the following identifications are obtained\n\nτ (s) = -2s, σ(s) = 1 -s 2 , σ(s) = -m ′2 + (1 -s 2 )ν ′ , ( 28\n\n)\n\nwhere\n\nν ′ = j ′ (j ′ + D -2) = j(j + D -2) + Cα 2 2 , m ′2 = m 2 + Cα 2 2 . ( 29\n\n)\n\nInserting the above expressions into equation (22), one obtains the following function:\n\nπ(s) = ± (ν ′ -k)s 2 + k -ν ′ + m ′2 , ( 30\n\n)\n\nFollowing the method, the polynomial π(s) is found in the following possible values\n\nπ(s) = m ′ s for k 1 = ν ′ -m ′2 , -m ′ s for k 1 = ν ′ -m ′2 , m ′ for k 2 = ν ′ , -m ′ for k 2 = ν ′ . ( 31\n\n)\n\nImposing the condition τ ′ (s) < 0, for equation (18), one selects\n\nk 1 = ν ′ -m ′2 and π(s) = -m ′ s, ( 32\n\n) which yields 10 τ (s) = -2(1 + m ′ )s. ( 33\n\n)\n\nUsing equations (19) and (23), the following expressions for λ are obtained, respectively,\n\nλ = λ n = 2 n(1 + m ′ ) + n( n -1), ( 34\n\n) λ = ν ′ -m ′ (1 + m ′ ). ( 35\n\n)\n\nWe compare equations (34) and (35), the new angular momentum j values are obtained as\n\nj = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -4Cα 2 2 -1, ( 36\n\n) or j ′ = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -1. ( 37\n\n)\n\nUsing Eqs (15)-(17) and (20)-(21), the polynomial solution of y n is expressed in terms of Jacobi polynomials [39,40] which are one of the orthogonal polynomials:\n\nH n (θ) = N n sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ), ( 38\n\n) where N n = 1 2 m ′ ( n+m ′ )! (2 n+2m ′ +1)( n+2m ′ )! n! 2\n\nis the normalization constant determined by +1 -1 [H n (s)] 2 ds = 1 and using the orthogonality relation of Jacobi polynomials [35, 45, 46] .\n\nBesides n = - (1 + 2m ′ ) 2 + 1 2 (2j + 1) 2 + 4j(D -3) + 4Cα 2 2 , ( 39\n\n)\n\nwhere m ′ is defined by equation (29)."},{"section_type":"OTHER","section_title":"C. Eigensolutions of the radial equation","text":"The solution of the radial part of Klein-Gordon equation, Eq. ( 24 ), for the Kratzer's potential has already been solved by means of NU-method in [39] . Very recently, using the same method, the problem for the non-central potential in (2) has been solved in three 11 dimensions (3D) by Cheng and Dai [36] . However, the aim of this subsection is to solve the problem with a different radial separation function g(r) in any arbitrary dimensions. In what follows, we present the exact bound-states (real) solution of Eq. (24). Letting\n\nε 2 = α 2 1 α 2 2 , 4γ 2 = (M -1)(M -3) + 4Bα 2 2 , β 2 = Aα 2 2 , ( 40\n\n)\n\nand substituting these expressions in equation (24), one gets\n\nd 2 g(r) dr 2 + -ε 2 r 2 + β 2 r -γ 2 r 2 g(r) = 0. ( 41\n\n)\n\nTo apply the conventional NU-method, equation (41) is compared with (14), resulting in the following expressions:\n\nτ (r) = 0, σ(r) = r, σ(r) = -ε 2 r 2 + β 2 r -γ 2 . ( 42\n\n)\n\nSubstituting the above expressions into equation (22) gives\n\nπ(r) = 1 2 ± 1 2 4ε 2 r 2 + 4(k -β 2 )r + 4γ 2 + 1. ( 43\n\n)\n\nTherefore, we can determine the constant k by using the condition that the discriminant of the square root is zero, that is\n\nk = β 2 ± ε 4γ 2 + 1, 4γ 2 + 1 = (D + 2j -2) 2 + 4Bα 2 2 . ( 44\n\n)\n\nIn view of that, we arrive at the following four possible functions of π(r) :\n\nπ(r) = 1 2 + εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 -εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 + εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1, 1 2 -εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1. ( 45\n\n)\n\nThe correct value of π(r) is chosen such that the function τ (r) given by Eq. (18) will have negative derivative [29] . So we can select the physical values to be\n\nk = β 2 -ε 4γ 2 + 1 and π(r) = 1 2 -εr - 1 2 4γ 2 + 1 , ( 46\n\n)\n\n12 which yield τ (r) = -2εr + (1 + 4γ 2 + 1), τ ′ (r) = -2ε < 0. (47) Using Eqs (19) and (23), the following expressions for λ are obtained, respectively,\n\nλ = λ n = 2nε, n = 0, 1, 2, ..., ( 48\n\n) λ = δ 2 -ε(1 + 4γ 2 + 1). ( 49\n\n)\n\nSo we can obtain the Klein Gordon energy eigenvalues from the following relation:\n\n1 + 2n + (D + 2j -2) 2 + 4(µ + E R )B µ -E R = A µ + E R , ( 50\n\n)\n\nand hence for the Kratzer plus the new ring-shaped potential, it becomes\n\n1 + 2n + (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 51\n\n)\n\nwith j defined in (36) . Although Eq. (51) is exactly solvable for E R but it looks to be little complicated. Further, it is interesting to investigate the solution for the Coulomb potential.\n\nTherefore, applying the following transformations: A = Ze 2 , B = 0, and j = ℓ, the central part of the potential in (3) turns into the Coulomb potential with Klein Gordon solution for the energy spectra given by\n\nE R = µ 1 - 2q 2 e 2 q 2 e 2 + (2n + 2ℓ + D -1) 2 , n, ℓ = 0, 1, 2, ..., ( 52\n\n)\n\nwhere q = Ze is the charge of the nucleus. Further, Eq. ( 52 ) can be expanded as a series in the nucleus charge as\n\nE R = µ - 2µq 2 e 2 (2n + 2ℓ + D -1) 2 + 2µq 4 e 4 (2n + 2ℓ + D -1) 4 -O(qe) 6 , ( 53\n\n)\n\nThe physical meaning of each term in the last equation was given in detail by Ref. [36] .\n\nBesides, the difference from the conventional nonrelativistic form is because of the choice of the vector V (r, θ, ϕ) and scalar S(r, θ, ϕ) parts of the potential in Eq. ( 1 ). 13 Overmore, if the value of j obtained by Eq.(36) is inserted into the eigenvalues of the radial part of the Klein Gordon equation with the noncentral potential given by Eq. (51), we finally find the energy eigenvalues for a bound electron in the presence of a noncentral potential by Eq. (2) as\n\n1 + 2n + (2j ′ + D -2) 2 + 4(a 0 r 2 0 -C)(µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 54\n\n)\n\nwhere m ′ = m 2 + C(µ + E R ) and n is given by Eq. (39). On the other hand, the solution of the Schrödinger equation, Eq. ( 9 ), for this potential has already been obtained by using the same method in Ref. [39] and it is in the Coulombic-like form:\n\nE N R = - 8µa 2 0 r 2 0 2n + 1 + (2ℓ ′ + D -2) 2 + 8µ(a 0 r 2 0 -C) 2 , n = 0, 1, 2, ... ( 55\n\n) 2ℓ ′ + D -2 = (D -2) 2 + (2 n + 2m ′ + 1) 2 -1, ( 56\n\n)\n\nwhere m ′ = √ m 2 + 2µC. Also, applying the following appropriate transformation: 54 ) provides exactly the nonrelativistic limit given by Eq. (55).\n\nµ+E R → 2µ, µ -E R → -E N R , j → ℓ to Eq. (\n\nIn what follows, let us now turn attention to find the radial wavefunctions for this potential. Substituting the values of σ(r), π(r) and τ (r) in Eqs (42), (45) and (47) into Eqs.\n\n(17) and (21), we find φ(r) = r (ζ+1)\n\n/2 e -εr , ( 57\n\n) ρ(r) = r ζ e -2εr , ( 58\n\n)\n\nwhere ζ = √ 4γ 2 + 1. Then from equation (20), we obtain\n\ny nj (r) = B nj r -ζ e 2εr d n dr n r n+ζ e -2εr , ( 59\n\n)\n\nand the wave function g(r) can be written in the form of the generalized Laguerre polynomials as 14 g(ρ) = C nj ρ 2ε\n\n(1+ζ)/2\n\ne -ρ/2 L ζ n (ρ), ( 60\n\n)\n\nwhere for Kratzer's potential we have\n\nζ = (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ), ρ = 2εr. ( 61\n\n)\n\nFinally, the radial wave functions of the Klein-Gordon equation are obtained R(ρ) = C nj ρ 2ε\n\n(ζ+2-D)/2\n\ne -ρ/2 L ζ n (ρ), ( 62\n\n)\n\nwhere C nj is the normalization constant to be determined below. Using the normalization condition, ∞ 0 R 2 (r)r D-1 dr = 1, and the orthogonality relation of the generalized Laguerre polynomials, (2n+η+1)(n+η)! n! , we have\n\n∞ 0 z η+1 e -z [L η n (z)] 2 dz =\n\nC nj = 2 µ 2 -E 2 R 1+ ζ 2 n! (2n + ζ + 1) (n + ζ)! . ( 63\n\n)\n\nFinally, we may express the normalized total wave functions as\n\nψ(r, θ, ϕ) = 2 µ 2 -E 2 R 1+ ζ 2 2 m ′ ( n + m ′ )! (2 n + 2m ′ + 1)( n + 2m ′ )! n!n! 2π (2n + ζ + 1) (n + ζ)! ×r (ζ+2-D) 2 exp(-µ 2 -E 2 R r)L ζ n (2 µ 2 -E 2 R r) sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ) exp(±imϕ). ( 64\n\n)\n\nwhere ζ is defined in Eq. ( 61 ) and m ′ is given after the Eq. (54)."},{"section_type":"CONCLUSION","section_title":"V. CONCLUSIONS","text":"The relativistic spin-0 particle D-dimensional Klein-Gordon equation has been solved easily for its exact bound-states with equal scalar and vector ring-shaped Kratzer potential through the conventional NU method. The analytical expressions for the total energy levels and eigenfunctions of this system can be reduced to their conventional three-dimensional space form upon setting D = 3. Further, the noncentral potentials treated in [30] can be introduced as perturbation to the Kratzer's potential by adjusting the strength of the 15 coupling constant C in terms of a 0 , which is the coupling constant of the Kratzer's potential.\n\nAdditionally, the radial and polar angle wave functions of Klein-Gordon equation are found in terms of Laguerre and Jacobi polynomials, respectively. The method presented in this paper is general and worth extending to the solution of other interaction problems. This method is very simple and useful in solving other complicated systems analytically without given a restiction conditions on the solution of some quantum systems as the case in the other models. We have seen that for the nonrelativistic model, the exact energy spectra can be obtained either by solving the Schrödinger equation in (9) (cf. Ref. [39] or Eq. ( 55 )) or by applying appropriate transformation to the relativistic solution given by Eq. ( 54 ). Finally, we point out that these exact results obtained for this new proposed form of the potential (2) may have some interesting applications in the study of different quantum mechanical systems, atomic and molecular physics."},{"section_type":"OTHER","section_title":"ACKNOWLEDGMENTS","text":"This research was partially supported by the Scientific and Technological Research Council of Turkey. S.M. Ikhdair wishes to dedicate this work to his family for their love and assistance. 16 REFERENCES [1] T.Y. Wu and W.Y. Pauchy Hwang, Relativistic Quantum Mechanics and Quantum Fields (World Scientific, Singapore, 1991). [2] W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd edn (springer, Berlin, 2000). [3] A.D. Alhaidari, Phys. Rev. Lett. 87 (2001) 210405; 88 (2002) 189901.\n\n[4] G. Chen, Mod. Phys. Lett. A 19 (2004) 2009; J.-Y. Guo, J. Meng and F.-X. Xu, Chin.\n\nPhys. Lett. 20 (2003) 602; A.D. Alhaidari, J. Phys. A: Math. Gen. 34 (2001) 9827; 35 (2002) 6207; M. 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J. Phys. 75 (2007) 81. [37] C.Y. Chen and S.H. Dong, Phys. Lett. A 335 (2005) 374. [38] Y.F. Cheng and T.Q. Dai, Phys. Scr. 75 (2007) 274. [39] S.M. Ikhdair and R. Sever, preprint quant-ph/0703008; quant-ph/0703042. [40] S.M. Ikhdair and R. Sever, preprint quant-ph/0703131. [41] S.M. Ikhdair and R. Sever, Z. Phys. C 56 (1992) 155; C 58 (1993) 153; D 28 (1993) 1; Hadronic J. 15 (1992) 389; Int. J. Mod. Phys. A 18 (2003) 4215; A 19 (2004) 1771; A 20 (2005) 4035; A 20 (2005) 6509; A 21 (2006) 2191; A 21 (2006) 3989; A 21 (2006) 6699; Int. J. Mod. Phys. E (in press) (preprint hep-ph/0504176); S. Ikhdair et al, Tr. J. Phys. 16 (1992) 510; 17 (1993) 474. [42] S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. (DOI 10.1007/s10773-006-9317-7; J.\n\nMath. Chem. (DOI 10.1007/s10910-006-9115-8). [43] S.M. Ikhdair and R. Sever, Ann. Phys. (Leipzig) 16 (3) (2007) 218. [44] S.M. Ikhdair and R. Sever, to appear in the Int. J. Mod. Phys. E (preprint quant-19 ph/0611065). [45] G. Sezgo, Orthogonal Polynomials (American Mathematical Society, New York, 1939). [46] N.N. Lebedev, Special Functions and Their Applications (Prentice-Hall, Englewood Cliffs, NJ, 1965). 20"}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics. The Klein-Gordon equation has also been used to understand the motion of a spin-0 particle in large class of potentials. In recent years, much efforts have been paid to solve these relativistic wave equations for various potentials by using different methods. These relativistic equations contain two objects: the four-vector linear momentum operator and the scalar rest mass. They allow us to introduce two types of potential coupling, which are the four-vector potential (V) and the space-time scalar potential (S).\\n\\nRecently, many authors have worked on solving these equations with physical potentials including Morse potential [3], Hulthen potential [4] , Woods-Saxon potential [5] , Pösch-Teller potential [6] , reflectionless-type potential [7] , pseudoharmonic oscillator [8] , ring-shaped harmonic oscillator [9], V 0 tanh 2 (r/r 0 ) potential [10] , five-parameter exponential potential [11] , Rosen-Morse potential [12] , and generalized symmetrical double-well potential [13] , etc. It is remarkable that in most works in this area, the scalar and vector potentials are almost taken to be equal (i.e., S = V ) [2, 14] . However, in some few other cases, it is considered the case where the scalar potential is greater than the vector potential (in order to guarantee the existence of Klein-Gordon bound states) (i.e., S > V ) [15] [16] [17] [18] [19] . Nonetheless, such physical potentials are very few. The bound-state solutions for the last case is obtained for the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is greater than the vector potential [15] .\\n\\nThe study of exact solutions of the nonrelativistic equation for a class of non-central potentials with a vector potential and a non-central scalar potential is of considerable interest in quantum chemistry [20] [21] [22] . In recent years, numerous studies [23] have been made in 2 analyzing the bound states of an electron in a Coulomb field with simultaneous presence of Aharanov-Bohm (AB) [24] field, and/or a magnetic Dirac monopole [25] , and Aharanov-Bohm plus oscillator (ABO) systems. In most of these works, the eigenvalues and eigenfunctions are obtained by means of seperation of variables in spherical or other orthogonal curvilinear coordinate systems. The path integral for particles moving in non-central potentials is evaluated to derive the energy spectrum of this system analytically [26] . In addition, the idea of SUSY and shape invariance is also used to obtain exact solutions of such noncentral but seperable potentials [27, 28] . Very recently, the conventional Nikiforov-Uvarov (NU) method [29] has been used to give a clear recipe of how to obtain an explicit exact bound-states solutions for the energy eigenvalues and their corresponding wave functions in terms of orthogonal polynomials for a class of non-central potentials [30] .\\n\\nAnother type of noncentral potentials is the ring-shaped Kratzer potential, which is a combination of a Coulomb potential plus an inverse square potential plus a noncentral angular part [31, 32] . The Kratzer potential has been used to describe the vibrational-rotational motion of isolated diatomic molecules [33] and has a mixed-energy spectrum containing both bound and scattering states with bound-states have been widely used in molecular spectroscopy [34] . The ring-shaped Kratzer potential consists of radial and angular-dependent potentials and is useful in studying ring-shaped molecules [22] . In taking the relativistic effects into account for spin-0 particle in the presence of a class of noncentral potentials, Yasuk et al [35] applied the NU method to solve the Klein-Gordon equation for the noncentral Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [36] also used the same method to solve the Klein-Gordon equation for the Kratzer-type potential.\\n\\nRecently, Chen and Dong [37] proposed a new ring-shaped potential and obtained the exact solution of the Schrödinger equation for the Coulomb potential plus this new ringshaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. This type of potential used by Chen and Dong [37] appears to be very similar to the potential used by Yasuk et al [35] . Moreover, Cheng and Dai [38] , proposed a new potential consisting from the modified Kratzer's potential [33] plus the 3 new proposed ring-shaped potential in [37] . They have presented the energy eigenvalues for this proposed exactly-solvable non-central potential in three dimensional (i.e., D = 3)-Schrödinger equation by means of the NU method. The two quantum systems solved by Refs [37, 38] are closely relevant to each other as they deal with a Coulombic field interaction except for a slight change in the angular momentum barrier acts as a repulsive core which is for any arbitrary angular momentum ℓ prevents collapse of the system in any dimensional space due to the slight perturbation to the original angular momentum barrier. Very recently, we have also applied the NU method to solve the Schrödinger equation in any arbitrary Ddimension to this new modified Kratzer-type potential [39, 40] .\\n\\nThe aim of the present paper is to consider the relativistic effects for the spin-0 particle in our recent works [39, 40] . So we want to present a systematic recipe to solving the D-dimensional Klein-Gordon equation for the Kratzer plus the new ring-shaped potential proposed in [38] using the simple NU method. This method is based on solving the Klein-Gordon equation by reducing it to a generalized hypergeometric equation.\\n\\nThis work is organized as follows: in section II, we shall present the Klein-Gordon equation in spherical coordinates for spin-0 particle in the presence of equal scalar and vector noncentral Kratzer plus the new ring-shaped potential and we also separate it into radial and angular parts. Section III is devoted to a brief description of the NU method.\\n\\nIn section IV, we present the exact solutions to the radial and angular parts of the Klein-Gordon equation in D-dimensions. Finally, the relevant conclusions are given in section V.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. THE KLEIN-GORDON EQUATION WITH EQUAL SCALAR AND VECTOR POTENTIALS\",\n \"text\": \"In relativistic quantum mechanics, we usually use the Klein-Gordon equation for describing a scalar particle, i.e., the spin-0 particle dynamics. The discussion of the relativistic behavior of spin-zero particles requires understanding the single particle spectrum and the 4 exact solutions to the Klein Gordon equation which are constructed by using the four-vector potential A λ (λ = 0, 1, 2, 3) and the scalar potential (S). In order to simplify the solution of the Klein-Gordon equation, the four-vector potential can be written as A λ = (A 0 , 0, 0, 0).\\n\\nThe first component of the four-vector potential is represented by a vector potential (V ), i.e., A 0 = V. In this case, the motion of a relativistic spin-0 particle in a potential is described by the Klein-Gordon equation with the potentials V and S [1]. For the case S ≥ V, there exist bound-state (real) solutions for a relativistic spin-zero particle [15] [16] [17] [18] [19] . On the other hand, for S = V, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby the bound-state solutions are easily obtained by using the well-known methods developed in nonrelativistic quantum mechanics [2] .\\n\\nThe Klein-Gordon equation describing a scalar particle (spin-0 particle) with scalar S(r, θ, ϕ) and vector V (r, θ, ϕ) potentials is given by [2,14]\\n\\nP 2 -[E R -V (r, θ, ϕ)/2] 2 + [µ + S(r, θ, ϕ)/2] 2 ψ(r, θ, ϕ) = 0, ( 1\\n\\n)\\n\\nwhere E R , P and µ are the relativistic energy, momentum operator and rest mass of the particle, respectively. The potential terms are scaled in (1) by Alhaidari et al [14] so that in the nonrelativistic limit the interaction potential becomes V.\\n\\nIn this work, we consider the equal scalar and vector potentials case, that is, S(r, θ, ϕ) = V (r, θ, ϕ) with the recently proposed general non-central potential taken in the form of the Kratzer plus ring-shaped potential [38-40]:\\n\\nV (r, θ, ϕ) = V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ , ( 2\\n\\n) V 1 (r) = - A r + B r 2 , V 2 (θ) = Cctg 2 θ, V 3 (ϕ) = 0, ( 3\\n\\n)\\n\\nwhere A = 2a 0 r 0 , B = a 0 r 2 0 and C is positive real constant with a 0 is the dissociation energy and r 0 is the equilibrium internuclear distance [33] . The potentials in Eq. ( 3 ) introduced by Cheng-Dai [38] reduce to the Kratzer potential in the limiting case of C = 0 [33]. In fact the energy spectrum for this potential can be obtained directly by considering it as special case of the general non-central seperable potentials [30] .\\n\\nIn the relativistic atomic units (h = c = 1), the D-dimensional Klein-Gordon equation in (1) becomes [41]\\n\\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 -(E R + µ) V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ + E 2 R -µ 2 ψ(r, θ, ϕ) = 0. ( 4\\n\\n)\\n\\nwith ψ(r, θ, ϕ) being the spherical total wave function separated as follows\\n\\nψ njm (r, θ, ϕ) = R(r)Y m j (θ, ϕ), R(r) = r -(D-1)/2 g(r), Y m j (θ, ϕ) = H(θ)Φ(ϕ). ( 5\\n\\n)\\n\\nInserting Eqs (3) and (5) into Eq. ( 4 ) and using the method of separation of variables, the following differential equations are obtained:\\n\\n1 r D-1 d dr r D-1 dR(r) dr - j(j + D -2) r 2 + α 2 2 α 2 1 - A r + B r 2 R(r) = 0, ( 6\\n\\n) 1 sin θ d dθ sin θ d dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ + j(j + D -2) H(θ) = 0, ( 7\\n\\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 8\\n\\n)\\n\\nwhere α 2 1 = µ -E R , α 2 2 = µ + E R , m\\n\\nand j are constants and with m 2 and λ j = j(j + D -2) are the separation constants.\\n\\nFor a nonrelativistic treatment with the same potential, the Schrödinger equation in spherical coordinates is\\n\\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 + 2µ E N R -V 1 (r) - V 2 (θ) r 2 - V 3 (ϕ) r 2 sin 2 θ ψ(r, θ, ϕ) = 0. ( 9\\n\\n)\\n\\nwhere µ and E N R are the reduced mass and the nonrelativistic energy, respectively. Besides, the spherical total wave function appearing in Eq. (9) has the same representation as in Eq. 6 (5) but with the transformation j → ℓ. Inserting Eq. (5) into Eq. (9) leads to the following differential equations [39,40]:\\n\\n1 r D-1 d dr r D-1 dR(r) dr - λ D r 2 -2µ E N R + A r - B r 2 R(r) = 0, ( 10\\n\\n) 1 sin θ d dθ sin θ d dθ - m 2 + 2µC cos 2 θ sin 2 θ + ℓ(ℓ + D -2) H(θ) = 0, ( 11\\n\\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 12\\n\\n)\\n\\nwhere m 2 and λ ℓ = ℓ(ℓ + D -2) are the separation constants. Equations (6)-(8) have the same functional form as Eqs (10)-(12). Therefore, the solution of the Klein-Gordon equation can be reduced to the solution of the Schrödinger equation with the appropriate choice of parameters: j → ℓ, α 2 1 → -E N R and α 2 2 → 2µ.\\n\\nThe solution of Eq. ( 8 ) is well-known periodic and must satisfy the period boundary condition Φ(ϕ + 2π) = Φ(ϕ) which is the azimuthal angle solution:\\n\\nΦ m (ϕ) = 1 √ 2π exp(±imϕ), m = 0, 1, 2, ..... ( 13\\n\\n)\\n\\nAdditionally, Eqs (6) and (7) are radial and polar angle equations and they will be solved by using the Nikiforov-Uvarov (NU) method [29] which is given briefly in the following section.\"\n },\n {\n \"section_type\": \"METHOD\",\n \"section_title\": \"III. NIKIFOROV-UVAROV METHOD\",\n \"text\": \"The NU method is based on reducing the second-order differential equation to a generalized equation of hypergeometric type [29] . In this sense, the Schrödinger equation, after employing an appropriate coordinate transformation s = s(r), transforms to the following form:\\n\\nψ ′′ n (s) + τ (s) σ(s) ψ ′ n (s) + σ(s) σ 2 (s) ψ n (s) = 0, ( 14\\n\\n) 7\\n\\nwhere σ(s) and σ(s) are polynomials, at most of second-degree, and τ (s) is a first-degree polynomial. Using a wave function, ψ n (s), of the simple ansatz:\\n\\nψ n (s) = φ n (s)y n (s), ( 15\\n\\n)\\n\\nreduces (14) into an equation of a hypergeometric type\\n\\nσ(s)y ′′ n (s) + τ (s)y ′ n (s) + λy n (s) = 0, ( 16\\n\\n)\\n\\nwhere\\n\\nσ(s) = π(s) φ(s) φ ′ (s) , ( 17\\n\\n)\\n\\nτ (s) = τ (s) + 2π(s), τ ′ (s) < 0, ( 18\\n\\n)\\n\\nand λ is a parameter defined as\\n\\nλ = λ n = -nτ ′ (s) - n (n -1) 2 σ ′′ (s), n = 0, 1, 2, .... ( 19\\n\\n)\\n\\nThe polynomial τ (s) with the parameter s and prime factors show the differentials at first degree be negative. It is worthwhile to note that λ or λ n are obtained from a particular solution of the form y(s) = y n (s) which is a polynomial of degree n. Further, the other part y n (s) of the wave function (14) is the hypergeometric-type function whose polynomial solutions are given by Rodrigues relation\\n\\ny n (s) = B n ρ(s) d n ds n [σ n (s)ρ(s)] , ( 20\\n\\n)\\n\\nwhere B n is the normalization constant and the weight function ρ(s) must satisfy the condition [29]\\n\\nd ds w(s) = τ (s) σ(s) w(s), w(s) = σ(s)ρ(s). ( 21\\n\\n)\\n\\nThe function π and the parameter λ are defined as\\n\\nπ(s) = σ ′ (s) -τ (s) 2 ± σ ′ (s) -τ (s) 2 2 -σ(s) + kσ(s), ( 22\\n\\n) 8 λ = k + π ′ (s). ( 23\\n\\n)\\n\\nIn principle, since π(s) has to be a polynomial of degree at most one, the expression under the square root sign in (22) can be arranged to be the square of a polynomial of first degree [29] . This is possible only if its discriminant is zero. In this case, an equation for k is obtained. After solving this equation, the obtained values of k are substituted in (22) . In addition, by comparing equations (19) and (23), we obtain the energy eigenvalues.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. EXACT SOLUTIONS OF THE RADIAL AND ANGLE-DEPENDENT EQUATIONS A. Separating variables of the Klein-Gordon equation\",\n \"text\": \"We seek to solving the radial and angular parts of the Klein-Gordon equation given by Eqs (6) and (7). Equation (6) involving the radial part can be written simply in the following form [39-41]:\\n\\nd 2 g(r) dr 2 - (M -1)(M -3) 4r 2 -α 2 2 A r - B r 2 + α 2 1 α 2 2 g(r) = 0, ( 24\\n\\n)\\n\\nwhere\\n\\nM = D + 2j. ( 25\\n\\n)\\n\\nOn the other hand, Eq. ( 7 ) involving the angular part of Klein-Gordon equation retakes the simple form\\n\\nd 2 H(θ) dθ 2 + ctg(θ) dH(θ) dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ -j(j + D -2) H(θ) = 0. ( 26\\n\\n)\\n\\nThus, Eqs (24) and (26) have to be solved latter through the NU method in the following subsections. 9 B. Eigenvalues and eigenfunctions of the angle-dependent equation\\n\\nIn order to apply NU method [29, 30, 33, 35, 36, [38] [39] [40] [42] [43] [44] , we use a suitable transformation variable s = cos θ. The polar angle part of the Klein Gordon equation in (26) can be written in the following universal associated-Legendre differential equation form [38-40]\\n\\nd 2 H(s) ds 2 - 2s 1 -s 2 dH(s) ds + 1 (1 -s 2 ) 2 j(j + D -2)(1 -s 2 ) -m 2 -Cα 2 2 s 2 H(θ) = 0. ( 27\\n\\n)\\n\\nEquation (27) has already been solved for the three-dimensional Schrödinger equation through the NU method in [38] . However, the aim in this subsection is to solve with different parameters resulting from the D-space-dimensions of Klein-Gordon equation. Further, Eq.\\n\\n(27) is compared with (14) and the following identifications are obtained\\n\\nτ (s) = -2s, σ(s) = 1 -s 2 , σ(s) = -m ′2 + (1 -s 2 )ν ′ , ( 28\\n\\n)\\n\\nwhere\\n\\nν ′ = j ′ (j ′ + D -2) = j(j + D -2) + Cα 2 2 , m ′2 = m 2 + Cα 2 2 . ( 29\\n\\n)\\n\\nInserting the above expressions into equation (22), one obtains the following function:\\n\\nπ(s) = ± (ν ′ -k)s 2 + k -ν ′ + m ′2 , ( 30\\n\\n)\\n\\nFollowing the method, the polynomial π(s) is found in the following possible values\\n\\nπ(s) = m ′ s for k 1 = ν ′ -m ′2 , -m ′ s for k 1 = ν ′ -m ′2 , m ′ for k 2 = ν ′ , -m ′ for k 2 = ν ′ . ( 31\\n\\n)\\n\\nImposing the condition τ ′ (s) < 0, for equation (18), one selects\\n\\nk 1 = ν ′ -m ′2 and π(s) = -m ′ s, ( 32\\n\\n) which yields 10 τ (s) = -2(1 + m ′ )s. ( 33\\n\\n)\\n\\nUsing equations (19) and (23), the following expressions for λ are obtained, respectively,\\n\\nλ = λ n = 2 n(1 + m ′ ) + n( n -1), ( 34\\n\\n) λ = ν ′ -m ′ (1 + m ′ ). ( 35\\n\\n)\\n\\nWe compare equations (34) and (35), the new angular momentum j values are obtained as\\n\\nj = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -4Cα 2 2 -1, ( 36\\n\\n) or j ′ = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -1. ( 37\\n\\n)\\n\\nUsing Eqs (15)-(17) and (20)-(21), the polynomial solution of y n is expressed in terms of Jacobi polynomials [39,40] which are one of the orthogonal polynomials:\\n\\nH n (θ) = N n sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ), ( 38\\n\\n) where N n = 1 2 m ′ ( n+m ′ )! (2 n+2m ′ +1)( n+2m ′ )! n! 2\\n\\nis the normalization constant determined by +1 -1 [H n (s)] 2 ds = 1 and using the orthogonality relation of Jacobi polynomials [35, 45, 46] .\\n\\nBesides n = - (1 + 2m ′ ) 2 + 1 2 (2j + 1) 2 + 4j(D -3) + 4Cα 2 2 , ( 39\\n\\n)\\n\\nwhere m ′ is defined by equation (29).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"C. Eigensolutions of the radial equation\",\n \"text\": \"The solution of the radial part of Klein-Gordon equation, Eq. ( 24 ), for the Kratzer's potential has already been solved by means of NU-method in [39] . Very recently, using the same method, the problem for the non-central potential in (2) has been solved in three 11 dimensions (3D) by Cheng and Dai [36] . However, the aim of this subsection is to solve the problem with a different radial separation function g(r) in any arbitrary dimensions. In what follows, we present the exact bound-states (real) solution of Eq. (24). Letting\\n\\nε 2 = α 2 1 α 2 2 , 4γ 2 = (M -1)(M -3) + 4Bα 2 2 , β 2 = Aα 2 2 , ( 40\\n\\n)\\n\\nand substituting these expressions in equation (24), one gets\\n\\nd 2 g(r) dr 2 + -ε 2 r 2 + β 2 r -γ 2 r 2 g(r) = 0. ( 41\\n\\n)\\n\\nTo apply the conventional NU-method, equation (41) is compared with (14), resulting in the following expressions:\\n\\nτ (r) = 0, σ(r) = r, σ(r) = -ε 2 r 2 + β 2 r -γ 2 . ( 42\\n\\n)\\n\\nSubstituting the above expressions into equation (22) gives\\n\\nπ(r) = 1 2 ± 1 2 4ε 2 r 2 + 4(k -β 2 )r + 4γ 2 + 1. ( 43\\n\\n)\\n\\nTherefore, we can determine the constant k by using the condition that the discriminant of the square root is zero, that is\\n\\nk = β 2 ± ε 4γ 2 + 1, 4γ 2 + 1 = (D + 2j -2) 2 + 4Bα 2 2 . ( 44\\n\\n)\\n\\nIn view of that, we arrive at the following four possible functions of π(r) :\\n\\nπ(r) = 1 2 + εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 -εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 + εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1, 1 2 -εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1. ( 45\\n\\n)\\n\\nThe correct value of π(r) is chosen such that the function τ (r) given by Eq. (18) will have negative derivative [29] . So we can select the physical values to be\\n\\nk = β 2 -ε 4γ 2 + 1 and π(r) = 1 2 -εr - 1 2 4γ 2 + 1 , ( 46\\n\\n)\\n\\n12 which yield τ (r) = -2εr + (1 + 4γ 2 + 1), τ ′ (r) = -2ε < 0. (47) Using Eqs (19) and (23), the following expressions for λ are obtained, respectively,\\n\\nλ = λ n = 2nε, n = 0, 1, 2, ..., ( 48\\n\\n) λ = δ 2 -ε(1 + 4γ 2 + 1). ( 49\\n\\n)\\n\\nSo we can obtain the Klein Gordon energy eigenvalues from the following relation:\\n\\n1 + 2n + (D + 2j -2) 2 + 4(µ + E R )B µ -E R = A µ + E R , ( 50\\n\\n)\\n\\nand hence for the Kratzer plus the new ring-shaped potential, it becomes\\n\\n1 + 2n + (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 51\\n\\n)\\n\\nwith j defined in (36) . Although Eq. (51) is exactly solvable for E R but it looks to be little complicated. Further, it is interesting to investigate the solution for the Coulomb potential.\\n\\nTherefore, applying the following transformations: A = Ze 2 , B = 0, and j = ℓ, the central part of the potential in (3) turns into the Coulomb potential with Klein Gordon solution for the energy spectra given by\\n\\nE R = µ 1 - 2q 2 e 2 q 2 e 2 + (2n + 2ℓ + D -1) 2 , n, ℓ = 0, 1, 2, ..., ( 52\\n\\n)\\n\\nwhere q = Ze is the charge of the nucleus. Further, Eq. ( 52 ) can be expanded as a series in the nucleus charge as\\n\\nE R = µ - 2µq 2 e 2 (2n + 2ℓ + D -1) 2 + 2µq 4 e 4 (2n + 2ℓ + D -1) 4 -O(qe) 6 , ( 53\\n\\n)\\n\\nThe physical meaning of each term in the last equation was given in detail by Ref. [36] .\\n\\nBesides, the difference from the conventional nonrelativistic form is because of the choice of the vector V (r, θ, ϕ) and scalar S(r, θ, ϕ) parts of the potential in Eq. ( 1 ). 13 Overmore, if the value of j obtained by Eq.(36) is inserted into the eigenvalues of the radial part of the Klein Gordon equation with the noncentral potential given by Eq. (51), we finally find the energy eigenvalues for a bound electron in the presence of a noncentral potential by Eq. (2) as\\n\\n1 + 2n + (2j ′ + D -2) 2 + 4(a 0 r 2 0 -C)(µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 54\\n\\n)\\n\\nwhere m ′ = m 2 + C(µ + E R ) and n is given by Eq. (39). On the other hand, the solution of the Schrödinger equation, Eq. ( 9 ), for this potential has already been obtained by using the same method in Ref. [39] and it is in the Coulombic-like form:\\n\\nE N R = - 8µa 2 0 r 2 0 2n + 1 + (2ℓ ′ + D -2) 2 + 8µ(a 0 r 2 0 -C) 2 , n = 0, 1, 2, ... ( 55\\n\\n) 2ℓ ′ + D -2 = (D -2) 2 + (2 n + 2m ′ + 1) 2 -1, ( 56\\n\\n)\\n\\nwhere m ′ = √ m 2 + 2µC. Also, applying the following appropriate transformation: 54 ) provides exactly the nonrelativistic limit given by Eq. (55).\\n\\nµ+E R → 2µ, µ -E R → -E N R , j → ℓ to Eq. (\\n\\nIn what follows, let us now turn attention to find the radial wavefunctions for this potential. Substituting the values of σ(r), π(r) and τ (r) in Eqs (42), (45) and (47) into Eqs.\\n\\n(17) and (21), we find φ(r) = r (ζ+1)\\n\\n/2 e -εr , ( 57\\n\\n) ρ(r) = r ζ e -2εr , ( 58\\n\\n)\\n\\nwhere ζ = √ 4γ 2 + 1. Then from equation (20), we obtain\\n\\ny nj (r) = B nj r -ζ e 2εr d n dr n r n+ζ e -2εr , ( 59\\n\\n)\\n\\nand the wave function g(r) can be written in the form of the generalized Laguerre polynomials as 14 g(ρ) = C nj ρ 2ε\\n\\n(1+ζ)/2\\n\\ne -ρ/2 L ζ n (ρ), ( 60\\n\\n)\\n\\nwhere for Kratzer's potential we have\\n\\nζ = (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ), ρ = 2εr. ( 61\\n\\n)\\n\\nFinally, the radial wave functions of the Klein-Gordon equation are obtained R(ρ) = C nj ρ 2ε\\n\\n(ζ+2-D)/2\\n\\ne -ρ/2 L ζ n (ρ), ( 62\\n\\n)\\n\\nwhere C nj is the normalization constant to be determined below. Using the normalization condition, ∞ 0 R 2 (r)r D-1 dr = 1, and the orthogonality relation of the generalized Laguerre polynomials, (2n+η+1)(n+η)! n! , we have\\n\\n∞ 0 z η+1 e -z [L η n (z)] 2 dz =\\n\\nC nj = 2 µ 2 -E 2 R 1+ ζ 2 n! (2n + ζ + 1) (n + ζ)! . ( 63\\n\\n)\\n\\nFinally, we may express the normalized total wave functions as\\n\\nψ(r, θ, ϕ) = 2 µ 2 -E 2 R 1+ ζ 2 2 m ′ ( n + m ′ )! (2 n + 2m ′ + 1)( n + 2m ′ )! n!n! 2π (2n + ζ + 1) (n + ζ)! ×r (ζ+2-D) 2 exp(-µ 2 -E 2 R r)L ζ n (2 µ 2 -E 2 R r) sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ) exp(±imϕ). ( 64\\n\\n)\\n\\nwhere ζ is defined in Eq. ( 61 ) and m ′ is given after the Eq. (54).\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"V. CONCLUSIONS\",\n \"text\": \"The relativistic spin-0 particle D-dimensional Klein-Gordon equation has been solved easily for its exact bound-states with equal scalar and vector ring-shaped Kratzer potential through the conventional NU method. The analytical expressions for the total energy levels and eigenfunctions of this system can be reduced to their conventional three-dimensional space form upon setting D = 3. Further, the noncentral potentials treated in [30] can be introduced as perturbation to the Kratzer's potential by adjusting the strength of the 15 coupling constant C in terms of a 0 , which is the coupling constant of the Kratzer's potential.\\n\\nAdditionally, the radial and polar angle wave functions of Klein-Gordon equation are found in terms of Laguerre and Jacobi polynomials, respectively. The method presented in this paper is general and worth extending to the solution of other interaction problems. This method is very simple and useful in solving other complicated systems analytically without given a restiction conditions on the solution of some quantum systems as the case in the other models. We have seen that for the nonrelativistic model, the exact energy spectra can be obtained either by solving the Schrödinger equation in (9) (cf. Ref. [39] or Eq. ( 55 )) or by applying appropriate transformation to the relativistic solution given by Eq. ( 54 ). Finally, we point out that these exact results obtained for this new proposed form of the potential (2) may have some interesting applications in the study of different quantum mechanical systems, atomic and molecular physics.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"ACKNOWLEDGMENTS\",\n \"text\": \"This research was partially supported by the Scientific and Technological Research Council of Turkey. S.M. Ikhdair wishes to dedicate this work to his family for their love and assistance. 16 REFERENCES [1] T.Y. Wu and W.Y. Pauchy Hwang, Relativistic Quantum Mechanics and Quantum Fields (World Scientific, Singapore, 1991). [2] W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd edn (springer, Berlin, 2000). [3] A.D. Alhaidari, Phys. Rev. Lett. 87 (2001) 210405; 88 (2002) 189901.\\n\\n[4] G. Chen, Mod. Phys. Lett. A 19 (2004) 2009; J.-Y. Guo, J. Meng and F.-X. Xu, Chin.\\n\\nPhys. Lett. 20 (2003) 602; A.D. Alhaidari, J. Phys. A: Math. Gen. 34 (2001) 9827; 35 (2002) 6207; M. S ¸imşek and H. Egrifes, J. Phys. A: Math. Gen. 37 (2004) 4379.\\n\\n[5] J.-Y. Guo, X.-Z. Fang and F.-X. Xu, Phys. Rev. A 66 (2002) 062105; C. Berkdemir, A.\\n\\nBerkdemir and R. Sever, J. Phys. A: Math. Gen. 39 (2006) 13455. [6] G. Chen, Acta Phys. Sinica 50 (2001) 1651; Ö. Yeşiltaş, Phys. Scr. 75 (2007) 41.\\n\\n[7] G. Chen and Z.M. Lou, Acta Phys. Sinica 52 (2003) 1071. [8] S.M. Ikhdair and R. Sever, J. Mol. Structure:THEOCHEM 806 (2007) 155; G. Chen, Z.D. Chen and Z.M. Lou, Chin. Phys. 13 (2004) 279. [9] W.C. Qiang, Chin. Phys. 12 (2003) 136.\\n\\n[10] W.C. Qiang, Chin. Phys. 13 (2004) 571. [11] G. Chen, Phys. Lett. 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Zhedabov, Phys. Lett. A 153 (1991) 177; C. Quesne, J. Phys. A 21 (1988) 3093.\\n\\n[21] M. Kibler and P. Winternitz, J. Phys. A 20 (1987) 4097.\\n\\n[22] H. Hartmann and D. Schuch, Int. J. Quantum Chem. 18 (1980) 125. [23] M. Kibler and T. Negadi, Phys. Lett. A 124 (1987) 42; A. Guha and S. Mukherjee, J.\\n\\nMath. Phys. 28 (1989) 840; G. E. Draganescu, C. Campiogotto and M. Kibler, Phys.\\n\\nLett. A 170 (1992) 339; M. Kibler and C. Campiogotto, Phys. Lett. A 181 (1993) 1; V. M. Villalba, Phys. Lett. A 193 (1994) 218. [24] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [25] P. A. M. Dirac, Proc. R. Soc. London Ser. A 133 (1931) 60. [26] B.P. Mandal, Int. J. Mod. Phys. A 15 (2000) 1225.\\n\\n[27] R. Dutt, A. Gangopadhyaya and U.P. Sukhatme, Am. J. Phys. 65 (5) (1997) 400. [28] B. Gönül and İ. Zorba, Phys. Lett. A 269 (2000) 83. [29] A.F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics 18 (Birkhauser, Bassel, 1988). [30] S.M. Ikhdair and R. Sever, to appear in the Int. J. Theor. Phys. (preprint quantph//0702186). [31] H.S. Valk, Am. J. Phys. 54 (1986) 921. [32] Q.W. Chao, Chin. Phys. 13 (5) (2004) 575.\\n\\n[33] C. Berkdemir, A. Berkdemir and J.G. Han, Chem. Phys. Lett. 417 (2006) 326. [34] A. Bastida et al, J. Chem. Phys. 93 (1990) 3408. [35] F. Yasuk, A. Durmuş and I. Boztosun, J. Math. Phys. 47 (2006) 082302. [36] C. Berkdemir, Am. J. Phys. 75 (2007) 81. [37] C.Y. Chen and S.H. Dong, Phys. Lett. A 335 (2005) 374. [38] Y.F. Cheng and T.Q. Dai, Phys. Scr. 75 (2007) 274. [39] S.M. Ikhdair and R. Sever, preprint quant-ph/0703008; quant-ph/0703042. [40] S.M. Ikhdair and R. Sever, preprint quant-ph/0703131. [41] S.M. Ikhdair and R. Sever, Z. Phys. C 56 (1992) 155; C 58 (1993) 153; D 28 (1993) 1; Hadronic J. 15 (1992) 389; Int. J. Mod. Phys. A 18 (2003) 4215; A 19 (2004) 1771; A 20 (2005) 4035; A 20 (2005) 6509; A 21 (2006) 2191; A 21 (2006) 3989; A 21 (2006) 6699; Int. J. Mod. Phys. E (in press) (preprint hep-ph/0504176); S. Ikhdair et al, Tr. J. Phys. 16 (1992) 510; 17 (1993) 474. [42] S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. (DOI 10.1007/s10773-006-9317-7; J.\\n\\nMath. Chem. (DOI 10.1007/s10910-006-9115-8). [43] S.M. Ikhdair and R. Sever, Ann. Phys. (Leipzig) 16 (3) (2007) 218. [44] S.M. Ikhdair and R. Sever, to appear in the Int. J. Mod. Phys. E (preprint quant-19 ph/0611065). [45] G. Sezgo, Orthogonal Polynomials (American Mathematical Society, New York, 1939). [46] N.N. Lebedev, Special Functions and Their Applications (Prentice-Hall, Englewood Cliffs, NJ, 1965). 20\"\n }\n]"}}},{"rowIdx":88,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0490"},"arxiv_id":{"kind":"string","value":"0704.0490"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.76.111701"},"content_hash":{"kind":"string","value":"33a1d1b81936bb4d490d360820f2123e47c0f5994f09751e7382ad683a799b76"},"title":{"kind":"string","value":"Long Distance Signaling Using Axion-like Particles"},"abstract":{"kind":"string","value":"The possible existence of axion-like particles could lead to a new type of\nlong distance communication. In this work, basic antenna concepts are defined\nand a Friis-like equation is derived to facilitate long-distance link\ncalculations. An example calculation is presented showing that communication\nover distances of 1000 km or more may be possible for $m_{a}< 3.5$ meV and\n$g_{a\\gamma \\gamma} > 5 \\times 10^{- 8} {\\text{GeV}}^{- 1}$."},"authors":{"kind":"list like","value":["Daniel D. Stancil"],"string":"[\n \"Daniel D. Stancil\"\n]"},"arxiv_categories":{"kind":"list like","value":["hep-ph","astro-ph","hep-th"],"string":"[\n \"hep-ph\",\n \"astro-ph\",\n \"hep-th\"\n]"},"primary_category":{"kind":"string","value":"hep-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The possible existence of axion-like particles could lead to a new type of long distance communication. In this work, basic antenna concepts are defined and a Friis-like equation is derived to facilitate long-distance link calculations. An example calculation is presented showing that communication over distances of 1000 km or more may be possible for 3.5\n\nsuggested coupling between the axion and electromagnetic fields that was much larger than thought possible based on solar axion observations by the CAST collaboration [5] .\n\nAlthough mechanisms have been proposed to reconcile the reports [6] , the PVLAS collaboration recently retracted the results [7] and an independent group has reported a negative result from a photon regeneration experiment that excludes the PVLAS result [8] .\n\nThere does not now appear to be any experimental evidence of a coupling strength inconsistent with CAST observations. However, since recent work has suggested mechanisms whereby such strong coupling may be possible, I believe it remains interesting to consider the implications of stronger-than-expected axion-photon coupling.\n\nIn particular, I would like to call attention to the observation that a new type of longdistance signaling and communication may be possible. It may be possible to construct a communication system that cannot be blocked-even communicating directly through the diameter of the earth. This would make reliable worldwide signaling possible without the use of either satellites or the ionosphere, and would enable communication to locations previously inaccessible, such as submarines at the bottom of the sea and mines deep beneath the earth. The signal would also be very difficult to intercept since the axion beam would be essentially as narrow as a laser beam used to create it, and most of the path would be underground. With advances in power and sensitivity, it may also be possible to use axion signaling in space communications. For example, using such a system, communication with points on the far side of the moon may be possible without the use of lunar satellites.\n\nCommunication systems using neutrinos have also been proposed [9], and would have many of the same characteristics as the proposed axion system. However, the generation and detection of neutrinos requires massive particle accelerators and scintillation detectors [10] . Also, full deflection over 4π steradians would not be practical, though limited beam steering could be achieved using a magnetic field to deflect the precursor pion beam. Finally, it would be difficult to consider modulation techniques more sophisticated than simple amplitude modulation. In contrast, using axion mass and coupling values not yet experimentally explored, it appears that worldwide communication would be possible with a fully steerable axion system about the size of a medium-size telescope. Further, since the signals at the input and output would be electromagnetic waves, any existing modulation technology could be used. However, as with neutrinos, the lack of strong interactions with matter presents challenges with respect to the generation and detection of axions. Sikivie proposed an experimental approach for detecting axions via their coupling to the electromagnetic field [11] . The coupling was obtained by considering the Lagrangian density\n\n2 2 1 1 1 1 4 4 2 2 a a L F F g aF F a a m a μν μν μ μν γγ μν μ = - - + ∂ ∂ - , (1)\n\nwhere\n\nF A A μν μ ν ν μ = ∂ -∂ is the electromagnetic field tensor, ( ) , A V A α =\n\nwhere V is the electric scalar potential and A is the magnetic vector potential, a is the axion field, a g γγ is the coupling constant between the electromagnetic and axion fields, and the electromagnetic dual tensor is given by 1 2 F F γδ αβ αβγδ ε = . In these equations we have\n\ntaken 1 h c = = .\n\nAs an example, consider the coupling between plane waves propagating along the z direction caused by a strong static magnetic field parallel to the polarization of the incident electromagnetic wave. If the time dependence is exp(\n\n) i t ω -\n\n, where ω is the frequency of the incident linearly polarized electromagnetic wave, the equations of motion obtained from (1) reduce to the coupled equations ( )\n\n2 2 2 2 2 0 0 2 2 , . a a a a A m a i g B A A i g B a z z γγ γγ ω ω ω ω ∂ ∂ + - =- + = ∂ ∂ (2)\n\nThus the static magnetic field B 0 couples the photon and axion fields. An apparatus for the generation and detection of axions based on this coupling is shown in Figure 1(a) . This is sometimes referred to as an \"invisible light through walls\" experiment [4, 12, 13 ].\n\nAn electromagnetic wave with amplitude (0) A enters a region of magnetic field of strength B OT extending over a distance L T . If the coupling is sufficiently weak that the change in A over the transmit conversion region and the change in a over the receive conversion region are negligible, then the conversion loss through the system will be [12] 0 / in R T\n\nP P p p = , (3)\n\nwhere in P is the optical input power, 0 P is the optical output power, T p , R p are the probabilities of photon-axion (and axion-photon) conversion in the transmitter and receiver, respectively, and the conversion probability is [11, 12, 14] 2\n\n0 1 sin /2 2 /2 a a qL p g BL k q L γγ ω ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ . (4)\n\nHere a q k k γ =indicates the phase mismatch between the photon and axion fields.\n\nThe efficiency of generating axions and regenerating photons can be greatly increased by adding electromagnetic resonators, as shown in Figure 1 (b) [4, 15] . In this light to pass through the magnetic field multiple times, increasing the conversion probability by the factor 2 / T F π , where F T is the finesse of the resonator in the axion generator (transmitter). This factor can be interpreted as the effective number of photon passes in the resonator. Axions are also emitted in the backward direction owing to the counter-propagating light in the resonator, resulting in half the particles traveling in an unwanted direction. Consequently, the probability of conversion in a given direction is increased by the factor / T F π . As also shown in Figure 1(b), a resonator on the photon regenerator (receiver) likewise increases the axion-photon conversion probability [15]. Since regenerated photons will be emitted in both directions, detectors are placed on both ends of the receiving optical resonator, enhancing the photon regeneration probability by the factor 2 / R F π , where F R is the finesse of the receiving resonator. (If the power from a single end is collected, the factor would be / R F π , as with the transmitter.) Finally, to turn this into a communication system, we add appropriate electromagnetic wave modulators and detectors as shown in Figure 1(b).\n\nThe conversion loss equation ( 3 ) is valid when the transmitter and receiver are sufficiently close together that beam diffraction can be neglected, and when the transmitter and receiver have equal cross-sectional areas. For signaling over long distances, neither assumption will be valid in general. To treat the long-distance case, we first calculate the radiated axion field, then calculate the regenerated photons resulting when the radiated field reaches the receiver.\n\nThe general solution to Eq. ( 2 ) is given by [12] 3 0\n\n( ) ( ) ( ) 4 a ik r r a V e a r i g d r A r B r r r γγ ω π ′ - ′ ′ ′ = ′ - ∫ i . ( 5\n\n)\n\nIf the observation point r is very far away from all points in the source volume V, then we obtain the far-field approximation for the axion field\n\nT A x F A ik z γ π =\n\n, where 0 A is the amplitude of the incident electromagnetic wave, and the cylinder is contained in a resonant cavity with finesse T F .\n\nUsing Eq. ( 6 ), the far-field potential is found to be\n\n( ) ( ) ( ) 1 0 0 sin / 2 ( ) 2 / 2 a ik r az T a T T a T T T a T az T k k L J k R F e a r i g A B L s r k R k k L γ ρ γγ ρ γ ω π π ⎡ ⎤⎡ ⎤ - ⎣ ⎦ ≈ ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ , (7)\n\nwhere sin , cos\n\n1 0 lim / 1/ 2 a a T a T k J k R k R ρ ρ ρ → ⎡ ⎤ = ⎣ ⎦ .\n\nConsequently, the axion field on axis is\n\n[ ] 0 0 sin / 2 ( ) 4 / 2 a ik r T T a T T T T qL F e a r i g A B L s r q L γγ ω π π ≈ . ( 8\n\n)\n\nThe time averaged transmitted power density is ( )\n\n2 2 2 1 ( ) ( ) 2 2 a T a a T Ti n k F S r k a r s p P r ω π π = = , (9)\n\nwhere (0)\n\nin T P S s γ = , and\n\n( ) 2 1 (0) 0 2 S k A γ γ ω = .\n\nIf this axion power density is incident upon a photon regenerator at distance r that is perfectly aligned with the transmitter, then the received power is\n\n0 (2 / ) ( ) R R a R P F p S r s π = . ( 10\n\n)\n\nSubstituting Eq. (9) for the power flux ( )\n\na S r gives 2 0 2 2 4 in a R T R R T T P k F F P ps ps r π π π π = . ( 11\n\n)\n\nThis expression can be understood in terms of antenna theory for electromagnetic waves.\n\nIn this context, we refer to the apparatus consisting of the resonator and the structure creating the magnetic field as an axion antenna. In analogy with conventional antenna theory, we define the directivity as 2 ( ) /( 4 )\n\na rad S r D P r π = , ( 12\n\n)\n\nwhere P rad is the total power radiated by the transmitting antenna. To obtain the total power radiated, we could integrate the power flux (9) over a sphere enclosing the antenna. However, it is easier to do the calculation in the near field using the axion field at the aperture of the transmitting antenna. Using ( ) ( / ) (0)\n\na T T T S L F p S γ π = , we have 2 ( ) (2 / ) rad a T T T T in P S L s F p P π = = . (13)\n\nSubstituting ( 13 ) for the total radiated power and (9) for the power flux, the directivity simplifies to\n\n2 2 (2 / ) (4 / )( /2) T a T a T D s s π λ π λ = = . ( 14\n\n)\n\nThe relation between the directivity and physical area ( 14 ) is ½ that found in conventional antenna theory, or equivalently, the area appears to be half the physical area. This results from the bi-directional radiation properties of the resonator. Defining an efficiency as /\n\nrad in P P η = , we also define the antenna gain as 2 (2 / ) (2 / ) T T T T T a T G D F p s η π π λ = = , (15) where (2 / ) T T T F p η π = .\n\nNext suppose that at some distant location this transmitted field is incident upon a receive antenna with length L R , radius R R , and finesse F R . From (10) and assuming the photons emitted from both ends of the receive antenna are collected, the total power collected will be 0 As with electromagnetic antennas, the ratio of effective area to gain is found to be independent of the details of the antenna, other than whether or not photons are collected from both ends of the antenna when used to receive:\n\n2 , , / / / (4 ) . e R R e T T c a s G s G n λ π = = ( 17\n\n)\n\nIf photons were collected only from one end of the receive antenna ( 1 c n = ), then Eq. ( 17 ) would be identical to conventional antenna theory.\n\nWith these definitions, the expression for the received power ( 11 ) can be interpreted as a Friis-like equation:\n\n( ) ( )( )\n\n2 2 2 0 , , , / / 4 / ( ) / in c T R a e T c T a e R P P n G G r s n s r λ π λ = = . (18) Here , c T n is the value used to compute , e T s according to Eq. (16). The ratio , , / e T c T s n is independent of the choice of , c T\n\nn , as it should be since the number of detectors that might be used on receive is independent of the transmit properties of the antenna.\n\nIt is also useful to note that if the magnetic field is uniform (i.e., wigglers, or quasi-phase matching, are not used [12] ), then there is an optimum length for the conversion region of an antenna. This occurs when ( )\n\nsin / 2 1 qL = , or qL π = . The optimum length is found to be 2 ( / ) opt a L m γ λ ω = . ( 19\n\n)\n\nIn obtaining this expression, we have used 2 /(2 ) a q m ω ≈ , which is valid for a m ω .\n\nThe diffraction-limited power pattern of the radiated axion field is determined by the aperture size in wavelengths through the Bessel function term in ( 7 ):\n\n( ) ( )\n\n2 1 4 sin /( sin ) d a a P J k R k R θ θ θ = ⎡ ⎤ ⎣ ⎦ . ( 20\n\n)\n\nThe diffraction beam width between first nulls is determined by the first zero of the Airy disc, and for small angles is given by the well-known expression\n\n1.22 / d FWFN T R γ θ λ ≈ . (21)\n\nSimilarly, the diffraction beam width at half maximum is determined by the roots of ( )\n\n) / 0.514 / d FWHM a T T R R γ θ πλ λ = ≈ . 1/ 2 d P θ = , or (1.616 /\n\nIn contrast, the conversion-limited power pattern is given by ( ) ( )\n\n2 sin cos / 2 ( ) 2 cos / 2 a c a k k L P k k L γ γ θ π θ θ ⎡ ⎤ ⎡ ⎤ - ⎣ ⎦ ⎢ ⎥ = ⎢ ⎥ - ⎣ ⎦ , (23)\n\nand depends on both the length in wavelengths and the velocity mismatch. For the optimum length L given by ( 19 ), the conversion beam widths are approximately given by 2( / ), 1.06( / )\n\nc c FWFN a FWHM a m m θ ω θ ω ≈ ≈ . ( 24\n\n)\n\nFor quantum-limited detection, the channel capacity is [16] log [17], and c is the velocity of light. Equation (25) can be combined with (11) or (18) to find the axion parameters that would permit a particular channel capacity at a given distance for a particular experimental apparatus.\n\n(1 / ) d R C N ν η ν = Δ + Δ , (25)\n\nAs an example, consider transmitters and receivers with 1064 nm (1.17 eV)\n\nγ λ = , 10 W in P = , 2 0.01 m s = , 0 10 T B = , 3 L = m, 5 3.1 10 F = × , 0.5 d η =\n\n, and a minimum information capacity of 1 bps. Figure 3 shows the inverse coupling strength curves\n\n( ) 1/ a a M m g γγ =\n\nfor distances of 1000 km and the diameter of the earth. Also shown is the curve for communication between the earth and the far side of the moon using the \"4+4\" experimental apparatus proposed in [15] . The shaded regions are excluded by the results from the BFRT collaboration [13] and Robilliard et al. [8] . The dots represent extensions of the curves if quasi-phase matching (QPM) is used by periodically reversing the magnetic field [12] . The minimum period of the reversal is taken to be twice the beam diameter for the 3 m system example, and 28.6 m for the 4+4 system. From the figure, communication over distances in excess of 1000 km should be possible for 3.5\n\na m < meV and 7 2 10 GeV M < × ( 8 1 5 10 GeV a g γγ -- Note that the receive resonator must be tuned to the same frequency as the transmit resonator to within a fraction of the resonator line width which is 161 Hz in this example. This will present a significant challenge, especially since the resonators are remote from one another. A possible approach would be to use atomic clocks to stabilize both the transmit frequency and a local reference frequency at the receiver. The receive resonator would then be locked to the reference to get as close as possible to the correct frequency, then slowly tuned until the signal is located. A feedback loop could then be closed to lock the receive resonator to the signal.\n\nIn summary, for 3.5 > × ), it may be possible to realize a new type of wireless signaling that cannot be blocked or shielded. An example calculation shows that communication between points located diametrically opposite on the earth should be possible. This could enable world-wide communication without the use of satellites or the ionosphere. However, with present knowledge, the signaling will be limited to low data rates, perhaps on the order of a few bits per second for terrestrial links. This estimate assumes 3 m long generation/regeneration regions to allow fully steerable instruments. Though not easily steerable, the apparatus in the \"4+4\" experiment proposed by Sikivie et al., [15] may enable communication to the far side of the moon for 0.3 a m < meV and 6 6 10 GeV M < × . I would like to acknowledge helpful discussions with Jim Lesh, Rich Holman, Jeff Peterson, Pierre Sikivie, and David Tanner during the development of these ideas. Electronic address: stancil@cmu.edu 17. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6 th Edition (Oxford Univ. Press, New York, 2007), pp. 171. 18. C-C. Chen, op cit., p. 106. B 0R L T L R Mod Laser Demod Data In Data Out B 0T Optical Detectors B 0T B 0R Photon input Photon output Axions penetrate barrier opaque to photons L T L R (a) Photons Axions Cavity mirrors (b) Figure 1, Stancil, Phys. Rev. D. x y z r θ -L T /2 L T /2"},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"The possible existence of axion-like particles could lead to a new type of long distance communication. In this work, basic antenna concepts are defined and a Friis-like equation is derived to facilitate long-distance link calculations. An example calculation is presented showing that communication over distances of 1000 km or more may be possible for 3.5"},{"section_type":"OTHER","section_title":"Untitled Section","text":"suggested coupling between the axion and electromagnetic fields that was much larger than thought possible based on solar axion observations by the CAST collaboration [5] .\n\nAlthough mechanisms have been proposed to reconcile the reports [6] , the PVLAS collaboration recently retracted the results [7] and an independent group has reported a negative result from a photon regeneration experiment that excludes the PVLAS result [8] .\n\nThere does not now appear to be any experimental evidence of a coupling strength inconsistent with CAST observations. However, since recent work has suggested mechanisms whereby such strong coupling may be possible, I believe it remains interesting to consider the implications of stronger-than-expected axion-photon coupling.\n\nIn particular, I would like to call attention to the observation that a new type of longdistance signaling and communication may be possible. It may be possible to construct a communication system that cannot be blocked-even communicating directly through the diameter of the earth. This would make reliable worldwide signaling possible without the use of either satellites or the ionosphere, and would enable communication to locations previously inaccessible, such as submarines at the bottom of the sea and mines deep beneath the earth. The signal would also be very difficult to intercept since the axion beam would be essentially as narrow as a laser beam used to create it, and most of the path would be underground. With advances in power and sensitivity, it may also be possible to use axion signaling in space communications. For example, using such a system, communication with points on the far side of the moon may be possible without the use of lunar satellites.\n\nCommunication systems using neutrinos have also been proposed [9], and would have many of the same characteristics as the proposed axion system. However, the generation and detection of neutrinos requires massive particle accelerators and scintillation detectors [10] . Also, full deflection over 4π steradians would not be practical, though limited beam steering could be achieved using a magnetic field to deflect the precursor pion beam. Finally, it would be difficult to consider modulation techniques more sophisticated than simple amplitude modulation. In contrast, using axion mass and coupling values not yet experimentally explored, it appears that worldwide communication would be possible with a fully steerable axion system about the size of a medium-size telescope. Further, since the signals at the input and output would be electromagnetic waves, any existing modulation technology could be used. However, as with neutrinos, the lack of strong interactions with matter presents challenges with respect to the generation and detection of axions. Sikivie proposed an experimental approach for detecting axions via their coupling to the electromagnetic field [11] . The coupling was obtained by considering the Lagrangian density\n\n2 2 1 1 1 1 4 4 2 2 a a L F F g aF F a a m a μν μν μ μν γγ μν μ = - - + ∂ ∂ - , (1)\n\nwhere\n\nF A A μν μ ν ν μ = ∂ -∂ is the electromagnetic field tensor, ( ) , A V A α =\n\nwhere V is the electric scalar potential and A is the magnetic vector potential, a is the axion field, a g γγ is the coupling constant between the electromagnetic and axion fields, and the electromagnetic dual tensor is given by 1 2 F F γδ αβ αβγδ ε = . In these equations we have\n\ntaken 1 h c = = .\n\nAs an example, consider the coupling between plane waves propagating along the z direction caused by a strong static magnetic field parallel to the polarization of the incident electromagnetic wave. If the time dependence is exp(\n\n) i t ω -\n\n, where ω is the frequency of the incident linearly polarized electromagnetic wave, the equations of motion obtained from (1) reduce to the coupled equations ( )\n\n2 2 2 2 2 0 0 2 2 , . a a a a A m a i g B A A i g B a z z γγ γγ ω ω ω ω ∂ ∂ + - =- + = ∂ ∂ (2)\n\nThus the static magnetic field B 0 couples the photon and axion fields. An apparatus for the generation and detection of axions based on this coupling is shown in Figure 1(a) . This is sometimes referred to as an \"invisible light through walls\" experiment [4, 12, 13 ].\n\nAn electromagnetic wave with amplitude (0) A enters a region of magnetic field of strength B OT extending over a distance L T . If the coupling is sufficiently weak that the change in A over the transmit conversion region and the change in a over the receive conversion region are negligible, then the conversion loss through the system will be [12] 0 / in R T\n\nP P p p = , (3)\n\nwhere in P is the optical input power, 0 P is the optical output power, T p , R p are the probabilities of photon-axion (and axion-photon) conversion in the transmitter and receiver, respectively, and the conversion probability is [11, 12, 14] 2\n\n0 1 sin /2 2 /2 a a qL p g BL k q L γγ ω ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ . (4)\n\nHere a q k k γ =indicates the phase mismatch between the photon and axion fields.\n\nThe efficiency of generating axions and regenerating photons can be greatly increased by adding electromagnetic resonators, as shown in Figure 1 (b) [4, 15] . In this light to pass through the magnetic field multiple times, increasing the conversion probability by the factor 2 / T F π , where F T is the finesse of the resonator in the axion generator (transmitter). This factor can be interpreted as the effective number of photon passes in the resonator. Axions are also emitted in the backward direction owing to the counter-propagating light in the resonator, resulting in half the particles traveling in an unwanted direction. Consequently, the probability of conversion in a given direction is increased by the factor / T F π . As also shown in Figure 1(b), a resonator on the photon regenerator (receiver) likewise increases the axion-photon conversion probability [15]. Since regenerated photons will be emitted in both directions, detectors are placed on both ends of the receiving optical resonator, enhancing the photon regeneration probability by the factor 2 / R F π , where F R is the finesse of the receiving resonator. (If the power from a single end is collected, the factor would be / R F π , as with the transmitter.) Finally, to turn this into a communication system, we add appropriate electromagnetic wave modulators and detectors as shown in Figure 1(b).\n\nThe conversion loss equation ( 3 ) is valid when the transmitter and receiver are sufficiently close together that beam diffraction can be neglected, and when the transmitter and receiver have equal cross-sectional areas. For signaling over long distances, neither assumption will be valid in general. To treat the long-distance case, we first calculate the radiated axion field, then calculate the regenerated photons resulting when the radiated field reaches the receiver.\n\nThe general solution to Eq. ( 2 ) is given by [12] 3 0\n\n( ) ( ) ( ) 4 a ik r r a V e a r i g d r A r B r r r γγ ω π ′ - ′ ′ ′ = ′ - ∫ i . ( 5\n\n)\n\nIf the observation point r is very far away from all points in the source volume V, then we obtain the far-field approximation for the axion field\n\nT A x F A ik z γ π =\n\n, where 0 A is the amplitude of the incident electromagnetic wave, and the cylinder is contained in a resonant cavity with finesse T F .\n\nUsing Eq. ( 6 ), the far-field potential is found to be\n\n( ) ( ) ( ) 1 0 0 sin / 2 ( ) 2 / 2 a ik r az T a T T a T T T a T az T k k L J k R F e a r i g A B L s r k R k k L γ ρ γγ ρ γ ω π π ⎡ ⎤⎡ ⎤ - ⎣ ⎦ ≈ ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ , (7)\n\nwhere sin , cos\n\n1 0 lim / 1/ 2 a a T a T k J k R k R ρ ρ ρ → ⎡ ⎤ = ⎣ ⎦ .\n\nConsequently, the axion field on axis is\n\n[ ] 0 0 sin / 2 ( ) 4 / 2 a ik r T T a T T T T qL F e a r i g A B L s r q L γγ ω π π ≈ . ( 8\n\n)\n\nThe time averaged transmitted power density is ( )\n\n2 2 2 1 ( ) ( ) 2 2 a T a a T Ti n k F S r k a r s p P r ω π π = = , (9)\n\nwhere (0)\n\nin T P S s γ = , and\n\n( ) 2 1 (0) 0 2 S k A γ γ ω = .\n\nIf this axion power density is incident upon a photon regenerator at distance r that is perfectly aligned with the transmitter, then the received power is\n\n0 (2 / ) ( ) R R a R P F p S r s π = . ( 10\n\n)\n\nSubstituting Eq. (9) for the power flux ( )\n\na S r gives 2 0 2 2 4 in a R T R R T T P k F F P ps ps r π π π π = . ( 11\n\n)\n\nThis expression can be understood in terms of antenna theory for electromagnetic waves.\n\nIn this context, we refer to the apparatus consisting of the resonator and the structure creating the magnetic field as an axion antenna. In analogy with conventional antenna theory, we define the directivity as 2 ( ) /( 4 )\n\na rad S r D P r π = , ( 12\n\n)\n\nwhere P rad is the total power radiated by the transmitting antenna. To obtain the total power radiated, we could integrate the power flux (9) over a sphere enclosing the antenna. However, it is easier to do the calculation in the near field using the axion field at the aperture of the transmitting antenna. Using ( ) ( / ) (0)\n\na T T T S L F p S γ π = , we have 2 ( ) (2 / ) rad a T T T T in P S L s F p P π = = . (13)\n\nSubstituting ( 13 ) for the total radiated power and (9) for the power flux, the directivity simplifies to\n\n2 2 (2 / ) (4 / )( /2) T a T a T D s s π λ π λ = = . ( 14\n\n)\n\nThe relation between the directivity and physical area ( 14 ) is ½ that found in conventional antenna theory, or equivalently, the area appears to be half the physical area. This results from the bi-directional radiation properties of the resonator. Defining an efficiency as /\n\nrad in P P η = , we also define the antenna gain as 2 (2 / ) (2 / ) T T T T T a T G D F p s η π π λ = = , (15) where (2 / ) T T T F p η π = .\n\nNext suppose that at some distant location this transmitted field is incident upon a receive antenna with length L R , radius R R , and finesse F R . From (10) and assuming the photons emitted from both ends of the receive antenna are collected, the total power collected will be 0 As with electromagnetic antennas, the ratio of effective area to gain is found to be independent of the details of the antenna, other than whether or not photons are collected from both ends of the antenna when used to receive:\n\n2 , , / / / (4 ) . e R R e T T c a s G s G n λ π = = ( 17\n\n)\n\nIf photons were collected only from one end of the receive antenna ( 1 c n = ), then Eq. ( 17 ) would be identical to conventional antenna theory.\n\nWith these definitions, the expression for the received power ( 11 ) can be interpreted as a Friis-like equation:\n\n( ) ( )( )\n\n2 2 2 0 , , , / / 4 / ( ) / in c T R a e T c T a e R P P n G G r s n s r λ π λ = = . (18) Here , c T n is the value used to compute , e T s according to Eq. (16). The ratio , , / e T c T s n is independent of the choice of , c T\n\nn , as it should be since the number of detectors that might be used on receive is independent of the transmit properties of the antenna.\n\nIt is also useful to note that if the magnetic field is uniform (i.e., wigglers, or quasi-phase matching, are not used [12] ), then there is an optimum length for the conversion region of an antenna. This occurs when ( )\n\nsin / 2 1 qL = , or qL π = . The optimum length is found to be 2 ( / ) opt a L m γ λ ω = . ( 19\n\n)\n\nIn obtaining this expression, we have used 2 /(2 ) a q m ω ≈ , which is valid for a m ω .\n\nThe diffraction-limited power pattern of the radiated axion field is determined by the aperture size in wavelengths through the Bessel function term in ( 7 ):\n\n( ) ( )\n\n2 1 4 sin /( sin ) d a a P J k R k R θ θ θ = ⎡ ⎤ ⎣ ⎦ . ( 20\n\n)\n\nThe diffraction beam width between first nulls is determined by the first zero of the Airy disc, and for small angles is given by the well-known expression\n\n1.22 / d FWFN T R γ θ λ ≈ . (21)\n\nSimilarly, the diffraction beam width at half maximum is determined by the roots of ( )\n\n) / 0.514 / d FWHM a T T R R γ θ πλ λ = ≈ . 1/ 2 d P θ = , or (1.616 /\n\nIn contrast, the conversion-limited power pattern is given by ( ) ( )\n\n2 sin cos / 2 ( ) 2 cos / 2 a c a k k L P k k L γ γ θ π θ θ ⎡ ⎤ ⎡ ⎤ - ⎣ ⎦ ⎢ ⎥ = ⎢ ⎥ - ⎣ ⎦ , (23)\n\nand depends on both the length in wavelengths and the velocity mismatch. For the optimum length L given by ( 19 ), the conversion beam widths are approximately given by 2( / ), 1.06( / )\n\nc c FWFN a FWHM a m m θ ω θ ω ≈ ≈ . ( 24\n\n)\n\nFor quantum-limited detection, the channel capacity is [16] log [17], and c is the velocity of light. Equation (25) can be combined with (11) or (18) to find the axion parameters that would permit a particular channel capacity at a given distance for a particular experimental apparatus.\n\n(1 / ) d R C N ν η ν = Δ + Δ , (25)\n\nAs an example, consider transmitters and receivers with 1064 nm (1.17 eV)\n\nγ λ = , 10 W in P = , 2 0.01 m s = , 0 10 T B = , 3 L = m, 5 3.1 10 F = × , 0.5 d η =\n\n, and a minimum information capacity of 1 bps. Figure 3 shows the inverse coupling strength curves\n\n( ) 1/ a a M m g γγ =\n\nfor distances of 1000 km and the diameter of the earth. Also shown is the curve for communication between the earth and the far side of the moon using the \"4+4\" experimental apparatus proposed in [15] . The shaded regions are excluded by the results from the BFRT collaboration [13] and Robilliard et al. [8] . The dots represent extensions of the curves if quasi-phase matching (QPM) is used by periodically reversing the magnetic field [12] . The minimum period of the reversal is taken to be twice the beam diameter for the 3 m system example, and 28.6 m for the 4+4 system. From the figure, communication over distances in excess of 1000 km should be possible for 3.5\n\na m < meV and 7 2 10 GeV M < × ( 8 1 5 10 GeV a g γγ -- Note that the receive resonator must be tuned to the same frequency as the transmit resonator to within a fraction of the resonator line width which is 161 Hz in this example. This will present a significant challenge, especially since the resonators are remote from one another. A possible approach would be to use atomic clocks to stabilize both the transmit frequency and a local reference frequency at the receiver. The receive resonator would then be locked to the reference to get as close as possible to the correct frequency, then slowly tuned until the signal is located. A feedback loop could then be closed to lock the receive resonator to the signal.\n\nIn summary, for 3.5 > × ), it may be possible to realize a new type of wireless signaling that cannot be blocked or shielded. An example calculation shows that communication between points located diametrically opposite on the earth should be possible. This could enable world-wide communication without the use of satellites or the ionosphere. However, with present knowledge, the signaling will be limited to low data rates, perhaps on the order of a few bits per second for terrestrial links. This estimate assumes 3 m long generation/regeneration regions to allow fully steerable instruments. Though not easily steerable, the apparatus in the \"4+4\" experiment proposed by Sikivie et al., [15] may enable communication to the far side of the moon for 0.3 a m < meV and 6 6 10 GeV M < × . I would like to acknowledge helpful discussions with Jim Lesh, Rich Holman, Jeff Peterson, Pierre Sikivie, and David Tanner during the development of these ideas. Electronic address: stancil@cmu.edu 17. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6 th Edition (Oxford Univ. Press, New York, 2007), pp. 171. 18. C-C. Chen, op cit., p. 106. B 0R L T L R Mod Laser Demod Data In Data Out B 0T Optical Detectors B 0T B 0R Photon input Photon output Axions penetrate barrier opaque to photons L T L R (a) Photons Axions Cavity mirrors (b) Figure 1, Stancil, Phys. Rev. D. x y z r θ -L T /2 L T /2"}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"The possible existence of axion-like particles could lead to a new type of long distance communication. In this work, basic antenna concepts are defined and a Friis-like equation is derived to facilitate long-distance link calculations. An example calculation is presented showing that communication over distances of 1000 km or more may be possible for 3.5\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"suggested coupling between the axion and electromagnetic fields that was much larger than thought possible based on solar axion observations by the CAST collaboration [5] .\\n\\nAlthough mechanisms have been proposed to reconcile the reports [6] , the PVLAS collaboration recently retracted the results [7] and an independent group has reported a negative result from a photon regeneration experiment that excludes the PVLAS result [8] .\\n\\nThere does not now appear to be any experimental evidence of a coupling strength inconsistent with CAST observations. However, since recent work has suggested mechanisms whereby such strong coupling may be possible, I believe it remains interesting to consider the implications of stronger-than-expected axion-photon coupling.\\n\\nIn particular, I would like to call attention to the observation that a new type of longdistance signaling and communication may be possible. It may be possible to construct a communication system that cannot be blocked-even communicating directly through the diameter of the earth. This would make reliable worldwide signaling possible without the use of either satellites or the ionosphere, and would enable communication to locations previously inaccessible, such as submarines at the bottom of the sea and mines deep beneath the earth. The signal would also be very difficult to intercept since the axion beam would be essentially as narrow as a laser beam used to create it, and most of the path would be underground. With advances in power and sensitivity, it may also be possible to use axion signaling in space communications. For example, using such a system, communication with points on the far side of the moon may be possible without the use of lunar satellites.\\n\\nCommunication systems using neutrinos have also been proposed [9], and would have many of the same characteristics as the proposed axion system. However, the generation and detection of neutrinos requires massive particle accelerators and scintillation detectors [10] . Also, full deflection over 4π steradians would not be practical, though limited beam steering could be achieved using a magnetic field to deflect the precursor pion beam. Finally, it would be difficult to consider modulation techniques more sophisticated than simple amplitude modulation. In contrast, using axion mass and coupling values not yet experimentally explored, it appears that worldwide communication would be possible with a fully steerable axion system about the size of a medium-size telescope. Further, since the signals at the input and output would be electromagnetic waves, any existing modulation technology could be used. However, as with neutrinos, the lack of strong interactions with matter presents challenges with respect to the generation and detection of axions. Sikivie proposed an experimental approach for detecting axions via their coupling to the electromagnetic field [11] . The coupling was obtained by considering the Lagrangian density\\n\\n2 2 1 1 1 1 4 4 2 2 a a L F F g aF F a a m a μν μν μ μν γγ μν μ = - - + ∂ ∂ - , (1)\\n\\nwhere\\n\\nF A A μν μ ν ν μ = ∂ -∂ is the electromagnetic field tensor, ( ) , A V A α =\\n\\nwhere V is the electric scalar potential and A is the magnetic vector potential, a is the axion field, a g γγ is the coupling constant between the electromagnetic and axion fields, and the electromagnetic dual tensor is given by 1 2 F F γδ αβ αβγδ ε = . In these equations we have\\n\\ntaken 1 h c = = .\\n\\nAs an example, consider the coupling between plane waves propagating along the z direction caused by a strong static magnetic field parallel to the polarization of the incident electromagnetic wave. If the time dependence is exp(\\n\\n) i t ω -\\n\\n, where ω is the frequency of the incident linearly polarized electromagnetic wave, the equations of motion obtained from (1) reduce to the coupled equations ( )\\n\\n2 2 2 2 2 0 0 2 2 , . a a a a A m a i g B A A i g B a z z γγ γγ ω ω ω ω ∂ ∂ + - =- + = ∂ ∂ (2)\\n\\nThus the static magnetic field B 0 couples the photon and axion fields. An apparatus for the generation and detection of axions based on this coupling is shown in Figure 1(a) . This is sometimes referred to as an \\\"invisible light through walls\\\" experiment [4, 12, 13 ].\\n\\nAn electromagnetic wave with amplitude (0) A enters a region of magnetic field of strength B OT extending over a distance L T . If the coupling is sufficiently weak that the change in A over the transmit conversion region and the change in a over the receive conversion region are negligible, then the conversion loss through the system will be [12] 0 / in R T\\n\\nP P p p = , (3)\\n\\nwhere in P is the optical input power, 0 P is the optical output power, T p , R p are the probabilities of photon-axion (and axion-photon) conversion in the transmitter and receiver, respectively, and the conversion probability is [11, 12, 14] 2\\n\\n0 1 sin /2 2 /2 a a qL p g BL k q L γγ ω ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ . (4)\\n\\nHere a q k k γ =indicates the phase mismatch between the photon and axion fields.\\n\\nThe efficiency of generating axions and regenerating photons can be greatly increased by adding electromagnetic resonators, as shown in Figure 1 (b) [4, 15] . In this light to pass through the magnetic field multiple times, increasing the conversion probability by the factor 2 / T F π , where F T is the finesse of the resonator in the axion generator (transmitter). This factor can be interpreted as the effective number of photon passes in the resonator. Axions are also emitted in the backward direction owing to the counter-propagating light in the resonator, resulting in half the particles traveling in an unwanted direction. Consequently, the probability of conversion in a given direction is increased by the factor / T F π . As also shown in Figure 1(b), a resonator on the photon regenerator (receiver) likewise increases the axion-photon conversion probability [15]. Since regenerated photons will be emitted in both directions, detectors are placed on both ends of the receiving optical resonator, enhancing the photon regeneration probability by the factor 2 / R F π , where F R is the finesse of the receiving resonator. (If the power from a single end is collected, the factor would be / R F π , as with the transmitter.) Finally, to turn this into a communication system, we add appropriate electromagnetic wave modulators and detectors as shown in Figure 1(b).\\n\\nThe conversion loss equation ( 3 ) is valid when the transmitter and receiver are sufficiently close together that beam diffraction can be neglected, and when the transmitter and receiver have equal cross-sectional areas. For signaling over long distances, neither assumption will be valid in general. To treat the long-distance case, we first calculate the radiated axion field, then calculate the regenerated photons resulting when the radiated field reaches the receiver.\\n\\nThe general solution to Eq. ( 2 ) is given by [12] 3 0\\n\\n( ) ( ) ( ) 4 a ik r r a V e a r i g d r A r B r r r γγ ω π ′ - ′ ′ ′ = ′ - ∫ i . ( 5\\n\\n)\\n\\nIf the observation point r is very far away from all points in the source volume V, then we obtain the far-field approximation for the axion field\\n\\nT A x F A ik z γ π =\\n\\n, where 0 A is the amplitude of the incident electromagnetic wave, and the cylinder is contained in a resonant cavity with finesse T F .\\n\\nUsing Eq. ( 6 ), the far-field potential is found to be\\n\\n( ) ( ) ( ) 1 0 0 sin / 2 ( ) 2 / 2 a ik r az T a T T a T T T a T az T k k L J k R F e a r i g A B L s r k R k k L γ ρ γγ ρ γ ω π π ⎡ ⎤⎡ ⎤ - ⎣ ⎦ ≈ ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ , (7)\\n\\nwhere sin , cos\\n\\n1 0 lim / 1/ 2 a a T a T k J k R k R ρ ρ ρ → ⎡ ⎤ = ⎣ ⎦ .\\n\\nConsequently, the axion field on axis is\\n\\n[ ] 0 0 sin / 2 ( ) 4 / 2 a ik r T T a T T T T qL F e a r i g A B L s r q L γγ ω π π ≈ . ( 8\\n\\n)\\n\\nThe time averaged transmitted power density is ( )\\n\\n2 2 2 1 ( ) ( ) 2 2 a T a a T Ti n k F S r k a r s p P r ω π π = = , (9)\\n\\nwhere (0)\\n\\nin T P S s γ = , and\\n\\n( ) 2 1 (0) 0 2 S k A γ γ ω = .\\n\\nIf this axion power density is incident upon a photon regenerator at distance r that is perfectly aligned with the transmitter, then the received power is\\n\\n0 (2 / ) ( ) R R a R P F p S r s π = . ( 10\\n\\n)\\n\\nSubstituting Eq. (9) for the power flux ( )\\n\\na S r gives 2 0 2 2 4 in a R T R R T T P k F F P ps ps r π π π π = . ( 11\\n\\n)\\n\\nThis expression can be understood in terms of antenna theory for electromagnetic waves.\\n\\nIn this context, we refer to the apparatus consisting of the resonator and the structure creating the magnetic field as an axion antenna. In analogy with conventional antenna theory, we define the directivity as 2 ( ) /( 4 )\\n\\na rad S r D P r π = , ( 12\\n\\n)\\n\\nwhere P rad is the total power radiated by the transmitting antenna. To obtain the total power radiated, we could integrate the power flux (9) over a sphere enclosing the antenna. However, it is easier to do the calculation in the near field using the axion field at the aperture of the transmitting antenna. Using ( ) ( / ) (0)\\n\\na T T T S L F p S γ π = , we have 2 ( ) (2 / ) rad a T T T T in P S L s F p P π = = . (13)\\n\\nSubstituting ( 13 ) for the total radiated power and (9) for the power flux, the directivity simplifies to\\n\\n2 2 (2 / ) (4 / )( /2) T a T a T D s s π λ π λ = = . ( 14\\n\\n)\\n\\nThe relation between the directivity and physical area ( 14 ) is ½ that found in conventional antenna theory, or equivalently, the area appears to be half the physical area. This results from the bi-directional radiation properties of the resonator. Defining an efficiency as /\\n\\nrad in P P η = , we also define the antenna gain as 2 (2 / ) (2 / ) T T T T T a T G D F p s η π π λ = = , (15) where (2 / ) T T T F p η π = .\\n\\nNext suppose that at some distant location this transmitted field is incident upon a receive antenna with length L R , radius R R , and finesse F R . From (10) and assuming the photons emitted from both ends of the receive antenna are collected, the total power collected will be 0 As with electromagnetic antennas, the ratio of effective area to gain is found to be independent of the details of the antenna, other than whether or not photons are collected from both ends of the antenna when used to receive:\\n\\n2 , , / / / (4 ) . e R R e T T c a s G s G n λ π = = ( 17\\n\\n)\\n\\nIf photons were collected only from one end of the receive antenna ( 1 c n = ), then Eq. ( 17 ) would be identical to conventional antenna theory.\\n\\nWith these definitions, the expression for the received power ( 11 ) can be interpreted as a Friis-like equation:\\n\\n( ) ( )( )\\n\\n2 2 2 0 , , , / / 4 / ( ) / in c T R a e T c T a e R P P n G G r s n s r λ π λ = = . (18) Here , c T n is the value used to compute , e T s according to Eq. (16). The ratio , , / e T c T s n is independent of the choice of , c T\\n\\nn , as it should be since the number of detectors that might be used on receive is independent of the transmit properties of the antenna.\\n\\nIt is also useful to note that if the magnetic field is uniform (i.e., wigglers, or quasi-phase matching, are not used [12] ), then there is an optimum length for the conversion region of an antenna. This occurs when ( )\\n\\nsin / 2 1 qL = , or qL π = . The optimum length is found to be 2 ( / ) opt a L m γ λ ω = . ( 19\\n\\n)\\n\\nIn obtaining this expression, we have used 2 /(2 ) a q m ω ≈ , which is valid for a m ω .\\n\\nThe diffraction-limited power pattern of the radiated axion field is determined by the aperture size in wavelengths through the Bessel function term in ( 7 ):\\n\\n( ) ( )\\n\\n2 1 4 sin /( sin ) d a a P J k R k R θ θ θ = ⎡ ⎤ ⎣ ⎦ . ( 20\\n\\n)\\n\\nThe diffraction beam width between first nulls is determined by the first zero of the Airy disc, and for small angles is given by the well-known expression\\n\\n1.22 / d FWFN T R γ θ λ ≈ . (21)\\n\\nSimilarly, the diffraction beam width at half maximum is determined by the roots of ( )\\n\\n) / 0.514 / d FWHM a T T R R γ θ πλ λ = ≈ . 1/ 2 d P θ = , or (1.616 /\\n\\nIn contrast, the conversion-limited power pattern is given by ( ) ( )\\n\\n2 sin cos / 2 ( ) 2 cos / 2 a c a k k L P k k L γ γ θ π θ θ ⎡ ⎤ ⎡ ⎤ - ⎣ ⎦ ⎢ ⎥ = ⎢ ⎥ - ⎣ ⎦ , (23)\\n\\nand depends on both the length in wavelengths and the velocity mismatch. For the optimum length L given by ( 19 ), the conversion beam widths are approximately given by 2( / ), 1.06( / )\\n\\nc c FWFN a FWHM a m m θ ω θ ω ≈ ≈ . ( 24\\n\\n)\\n\\nFor quantum-limited detection, the channel capacity is [16] log [17], and c is the velocity of light. Equation (25) can be combined with (11) or (18) to find the axion parameters that would permit a particular channel capacity at a given distance for a particular experimental apparatus.\\n\\n(1 / ) d R C N ν η ν = Δ + Δ , (25)\\n\\nAs an example, consider transmitters and receivers with 1064 nm (1.17 eV)\\n\\nγ λ = , 10 W in P = , 2 0.01 m s = , 0 10 T B = , 3 L = m, 5 3.1 10 F = × , 0.5 d η =\\n\\n, and a minimum information capacity of 1 bps. Figure 3 shows the inverse coupling strength curves\\n\\n( ) 1/ a a M m g γγ =\\n\\nfor distances of 1000 km and the diameter of the earth. Also shown is the curve for communication between the earth and the far side of the moon using the \\\"4+4\\\" experimental apparatus proposed in [15] . The shaded regions are excluded by the results from the BFRT collaboration [13] and Robilliard et al. [8] . The dots represent extensions of the curves if quasi-phase matching (QPM) is used by periodically reversing the magnetic field [12] . The minimum period of the reversal is taken to be twice the beam diameter for the 3 m system example, and 28.6 m for the 4+4 system. From the figure, communication over distances in excess of 1000 km should be possible for 3.5\\n\\na m < meV and 7 2 10 GeV M < × ( 8 1 5 10 GeV a g γγ -- Note that the receive resonator must be tuned to the same frequency as the transmit resonator to within a fraction of the resonator line width which is 161 Hz in this example. This will present a significant challenge, especially since the resonators are remote from one another. A possible approach would be to use atomic clocks to stabilize both the transmit frequency and a local reference frequency at the receiver. The receive resonator would then be locked to the reference to get as close as possible to the correct frequency, then slowly tuned until the signal is located. A feedback loop could then be closed to lock the receive resonator to the signal.\\n\\nIn summary, for 3.5 > × ), it may be possible to realize a new type of wireless signaling that cannot be blocked or shielded. An example calculation shows that communication between points located diametrically opposite on the earth should be possible. This could enable world-wide communication without the use of satellites or the ionosphere. However, with present knowledge, the signaling will be limited to low data rates, perhaps on the order of a few bits per second for terrestrial links. This estimate assumes 3 m long generation/regeneration regions to allow fully steerable instruments. Though not easily steerable, the apparatus in the \\\"4+4\\\" experiment proposed by Sikivie et al., [15] may enable communication to the far side of the moon for 0.3 a m < meV and 6 6 10 GeV M < × . I would like to acknowledge helpful discussions with Jim Lesh, Rich Holman, Jeff Peterson, Pierre Sikivie, and David Tanner during the development of these ideas. Electronic address: stancil@cmu.edu 17. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6 th Edition (Oxford Univ. Press, New York, 2007), pp. 171. 18. C-C. Chen, op cit., p. 106. B 0R L T L R Mod Laser Demod Data In Data Out B 0T Optical Detectors B 0T B 0R Photon input Photon output Axions penetrate barrier opaque to photons L T L R (a) Photons Axions Cavity mirrors (b) Figure 1, Stancil, Phys. Rev. D. x y z r θ -L T /2 L T /2\"\n }\n]"}}},{"rowIdx":89,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0495"},"arxiv_id":{"kind":"string","value":"0704.0495"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.3842/SIGMA.2007.075"},"content_hash":{"kind":"string","value":"1901032a0ef23b90804048b2aef0d82a6e8e98bc6f56ec72e49cd08936aac7b0"},"title":{"kind":"string","value":"The Veldkamp Space of Two-Qubits"},"abstract":{"kind":"string","value":"Given a remarkable representation of the generalized Pauli operators of\ntwo-qubits in terms of the points of the generalized quadrangle of order two,\nW(2), it is shown that specific subsets of these operators can also be\nassociated with the points and lines of the four-dimensional projective space\nover the Galois field with two elements - the so-called Veldkamp space of W(2).\nAn intriguing novelty is the recognition of (uni- and tri-centric) triads and\nspecific pentads of the Pauli operators in addition to the \"classical\" subsets\nanswering to geometric hyperplanes of W(2)."},"authors":{"kind":"list like","value":["Metod Saniga (ASTRINSTSAV)","Michel Planat (FEMTO-ST)","Petr Pracna\n (JH-Inst)","Hans Havlicek (TUW)"],"string":"[\n \"Metod Saniga (ASTRINSTSAV)\",\n \"Michel Planat (FEMTO-ST)\",\n \"Petr Pracna\\n (JH-Inst)\",\n \"Hans Havlicek (TUW)\"\n]"},"arxiv_categories":{"kind":"list like","value":["quant-ph","math-ph","math.MP"],"string":"[\n \"quant-ph\",\n \"math-ph\",\n \"math.MP\"\n]"},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2 N -dimensional Hilbert spaces is embodied in the geometry of the symplectic polar space of rank N and order two [1, 2, 3] . The case of two-qubit operator space, N = 2, was scrutinized in very detail [1, 3] by explicitly demonstrating, in different ways, the correspondence between various subsets of the generalized Pauli operators/matrices and the fundamental subgeometries of the associated rank-two polar space -the (unique) generalized quadrangle of order two. In this paper we will reveal another interesting geometry hidden behind the Pauli operators of two-qubits, namely that of the Veldkamp space defined on this generalized quadrangle.\n\nIn this section we will briefly highlight the basics of the theory of finite generalized quadrangles [4] and introduce the concept of the Veldkamp space of a point-line incidence geometry [5] to be employed in what follows. 2 M. Saniga, M. Planat, P. Pracna and H. Havlicek A finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), is an incidence structure S = (P, B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [4]: (i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique pair (y, M ) ∈ P × B for which xIM IyIL; from these axioms it readily follows that |P | = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). It is obvious that there exists a point-line duality with respect to which each of the axioms is self-dual. Interchanging points and lines in S thus yields a generalized quadrangle S D of order (t, s), called the dual of S. If s = t, S is said to have order s. The generalized quadrangle of order (s, 1) is called a grid and that of order (1, t) a dual grid. A generalized quadrangle with both s > 1 and t > 1 is called thick.\n\nGiven two points x and y of S one writes x ∼ y and says that x and y are collinear if there exists a line L of S incident with both. For any x ∈ P denote x ⊥ = {y ∈ P |y ∼ x} and note that x ∈ x ⊥ ; obviously, x ⊥ = 1 + s + st. Given an arbitrary subset A of P , the perp(-set) of A, A ⊥ , is defined as A ⊥ = {x ⊥ |x ∈ A} and A ⊥⊥ := (A ⊥ ) ⊥ . A triple of pairwise non-collinear points of S is called a triad; given any triad T , a point of T ⊥ is called its center and we say that T is acentric, centric or unicentric according as |T ⊥ | is, respectively, zero, non-zero or one. An ovoid of a generalized quadrangle S is a set of points of S such that each line of S is incident with exactly one point of the set; hence, each ovoid contains st + 1 points.\n\nThe concept of crucial importance is a geometric hyperplane H of a point-line geometry Γ(P, B), which is a proper subset of P such that each line of Γ meets H in one or all points [6] . For Γ = GQ(s, t), it is well known that H is one of the following three kinds: (i) the perp-set of a point x, x ⊥ ; (ii) a (full) subquadrangle of order (s, t ′ ), t ′ < t; and (iii) an ovoid.\n\nFinally, we need to introduce the notion of the Veldkamp space of a point-line incidence geometry Γ(P, B), V(Γ) [5] . V(Γ) is the space in which (i) a point is a geometric hyperplane of Γ and (ii) a line is the collection i = 1, 2) , where H 1 and H 2 are distinct points of V(Γ). 1 If Γ = S, from the preceding paragraph we learn that the points of V(S) are, in general, of three different types.\n\nH 1 H 2 of all geometric hyperplanes H of Γ such that H 1 H 2 = H 1 H = H 2 H or H = H i (\n\nThe smallest thick GQ is obviously the one with s = t = 2, dubbed the \"doily.\" This quadrangle has a number of interesting representations of which we mention the most important two [4]. One, frequently denoted as W 3 (2) or simply W (2), is in terms of the points of P G(3, 2) (i.e., the three-dimensional projective space over the Galois field with two elements) together with the totally isotropic lines with respect to a symplectic polarity. The other, usually denoted as Q(4, 2), is in terms of points and lines of a parabolic quadric in P G (4, 2) . By abuse of notation, any GQ isomorphic to W (2) will also be denoted by this symbol. From the preceding section we readily get that W (2) is endowed with 15 points/lines, each line contains three points and, dually, each point is on three lines; moreover, it is a self-dual object, i.e., isomorphic to its dual. W (2) features all the three kinds of hyperplanes, of the following cardinalities [5]: 15 perp-sets, x ⊥ , seven points each; 10 grids (of order (2, 1)), nine points each; and six ovoids, five points each -as depicted in Fig. 1 . The quadrangle exhibits two distinct kinds of triads, viz. unicentric and tricentric. A point of W (2) is the center of four distinct unicentric triads (Fig. 2 , left ); hence, 1 It is important to mention here that the definition of Veldkamp space given by Shult in [7] is more restrictive than that of Buekenhout and Cohen [5] adopted in this paper.\n\nThe Veldkamp Space of Two-Qubits 3 Figure 1. The three kinds of geometric hyperplanes of W (2). The points of the quadrangle are represented by small circles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the quadrangle. The upper panel shows the points' perp-sets (yellow bullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouring will become clear later. Each pictureexcept that in the bottom right-hand corner -stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon.\n\nFigure 2. Left: -The four distinct unicentric triads (grey bullets) and their common center (black bullet); note that the triads intersect pairwise in a single point and their union covers fully the center's perp-set. Right: -A grid (red bullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); the two triads are also seen to comprise a dual grid (of order (1, 2)). 4 M. Saniga, M. Planat, P. Pracna and H. Havlicek Figure 3. The five different kinds of the lines of V(W (2)), each being uniquely determined by the properties of its core-set (black bullets). Note that the \"yellow\" hyperplanes (i.e., perp-sets) occur in each type, and yellow is also the colour of two homogeneous (i.e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also worth mentioning that the cardinality of core-sets is an odd number not exceeding five. The three hyperplanes of any line are always in such relation to each other that their union comprises all the points of W (2).\n\nThe Veldkamp Space of Two-Qubits 5 the number of such triads is 4 × 15 = 60. Tricentric triads always come in \"complementary\" pairs, one representing the centers of the other, and each such pair is the complement of a grid of W (2) (Fig. 2, right); hence, the number of such triads is 2 × 10 = 20. A unicentric triad is always a subset of an ovoid, which is never the case for a tricentric triad; the latter, in graphcombinatorial terms, representing a complete bipartite graph on six vertices. Now, we have enough background information at hand to reveal the structure of the Veldkamp space of our \"doily\", V(W (2)). 2\n\nFrom the definition given in Section 2, we easily see that V(W (2)) consists of 31 points of which 15 are represented/generated by single-point perp-sets, 10 by grids and six by ovoids. The lines of V(W (2)) feature three points each and are of five distinct types, as illustrated in Fig. 3. These types differ from each other in the cardinality and structure of \"core-sets\", i.e., the sets of points of W (2) shared by all the three hyperplanes forming a given line. As it is obvious from Fig. 3, the lines of the first three types (the first three rows of the figure) have the core-sets of the same cardinality, three, differing from each other only in the structure of these sets as being unicentric triads, tricentric triads and triples of collinear points, respectively. The lines of the fourth type have as core-sets pentads of points, each being a quadruple of points collinear with a given point of W (2), whereas core-sets of the last type's lines feature just a single point. A much more interesting issue is the composition of the lines. Just a brief look at Fig. 3 reveals that geometric hyperplanes of only one kind, namely perp-sets, are present on each line of V(W (2)); grids and ovoids occur only on two kinds of the lines. We also see that the purely homogeneous types are those whose core-sets feature collinear triples and tricentric triads, the most heterogeneous type -the one exhibiting all the three kinds of hyperplanes -being that characterized by unicentric triads. We also notice that there are no lines comprising solely grids and/or solely ovoids, nor the lines featuring only grids and ovoids, which seems to be connected with the fact that the cardinality of a core-set is an odd number. From the properties of W (2) and its triads as discussed above it readily follows that the number of the lines of type one to five is 60, 20, 15, 45 and 15, respectively, totalling 155. All these observations and facts are gathered in Table 1 . We conclude this section with the observation that V(W (2)) has the same number of points (31) and lines (155) as P G(4, 2), the four-dimensional projective space over the Galois field of two elements [8] ; this is not a coincidence, as the two spaces are, in fact, isomorphic to each other [5].\n\n4 Pauli operators of two-qubits in light of V(W (2))\n\nAs discovered in [1] (see also [3]), the fifteen generalized Pauli operators/matrices associated with the Hilbert space of two-qubits (see, e.g., [9]) can be put into a one-to-one correspondence with the fifteen points of the generalized quadrangle W (2) in such a way that their commutation algebra is completely and uniquely reproduced by the geometry of W (2) in which the concept commuting/non-commuting translates into that of collinear/non-collinear. Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [1, 3] for more details). Yet, V(W (2)) puts this bijection in a different light, in which other three subsets of the Pauli operators come into play, namely those represented by the two types of a triad and by the specific pentads occurring as the core-sets of the lines of V(W (2)) (Table 1 ). As already mentioned, the role of tricentric triads of the operators has 2 As this paper is primarily aimed at physicists rather than mathematicians, in what follows we opt for an elementary and self-contained exposition of the Veldkamp space of W (2); this explanation is based only upon some very simple properties of W (2) readily to be grasped from its depiction as \"the doily\", and does not presuppose/require any further background from the reader. 6 M. Saniga, M. Planat, P. Pracna and H. Havlicek Table 1. A succinct summary of the properties of the five different types of the lines of V(W (2)) in terms of the core-sets and the types of geometric hyperplanes featured by a generic line of a given type. The last column gives the total number of lines per the corresponding type.\n\nType of Core-Set Perp-Sets Grids Ovoids # Single Point 1 0 2 15 Collinear Triple 3 0 0 15 Unicentric Triad 1 1 1 60 Tricentric Triad 3 0 0 20 Pentad 1 2 0 45 Table 2. Three kinds of the distinguished subsets of the generalized Pauli operators of two-qubits (PO) viewed as the geometric hyperplanes in the generalized quadrangle of order two (GQ) [1, 3].\n\nPO set of five mutually set of six operators nine operators of non-commuting operators commuting with a given one a Mermin's square GQ ovoid perp-set\\{reference point} grid been recognized in disguise of complete bipartite graphs on six vertices [3]. A true novelty here is obviously unicentric triads and pentads of the generalized Pauli operators as these are all intimately connected with single-point perp-sets; given a point of W (2) (i.e., a generalized Pauli operator of two-qubits), its perp-set fully encompasses four unicentric triads (Fig. 2 , left) and three pentads (Fig. 3 , 4th row) of points/operators. This feature has also a very interesting aspect in connection with the conjecture relating the existence of mutually unbiased bases and finite projective planes raised in [10] , because with each point x of W (2) there is associated a projective plane of order two (the Fano plane) whose points are the elements of x ⊥ and whose lines are the spans {u,\n\nv} ⊥⊥ , where u, v ∈ x ⊥ with u = v [4].\n\nIdentifying the Pauli operators of a two-qubit system with the points of the generalized quadrangle of order two led to the discovery of three distinguished subsets of the operators in terms of geometric hyperplanes of the quadrangle. Here we go one level higher, and identifying these subsets with the points of the associated Veldkamp space leads to recognition of nother remarkable subsets of the Pauli operators, viz. unicentric triads and pentads. It is really intriguing to see that these are the core-sets of the two kinds of lines that both feature grids alias Mermin squares. As it is well known, Mermin squares, which reveal certain important aspects of the entanglement of the system, play a crucial role in the proof of the Kochen-Specker theorem in dimension four and our approach gives a novel geometrical meaning to this [3, 11] . At the Veldkamp space level it turns out of particular importance to study relations between eigenvectors of the above-mentioned unicentric triads and pentad of operators in order to reveal finer, hitherto unnoticed traits of the structure of Mermin squares. These seem to be intimately connected with the existence of outer automorphisms of the symmetric group on six letters, which is the full group of automorphisms of our quadrangle; as this group is the only symmetric group possessing (non-trivial) outer automorphisms, this implies that two-qubits have a rather special footing among multiple qubit systems. All these aspects deserve special attention and will therefore be dealt with in a separate paper.\n\nConcerning three-qubits, our preliminary study indicates that the corresponding finite geometry differs fundamentally from that of W (2) in the sense that it contains multi-lines, i.e., two or more lines passing through two distinct points [12] . As we do not have a full picture at hand yet, we cannot see if it admits hyperplanes and so lends itself to constructing the corresponding Veldkamp space. If the latter does exist, it is likely to differ substantially from that of twoqubits, which would imply the expected difference between entanglement properties of the two\n\nThe Veldkamp Space of Two-Qubits 7 kinds of systems; if not, this will only further strengthen the above-mentioned uniqueness of two-qubits.\n\nBy employing the concept of the Veldkamp space of the generalized quadrangle of order two, we were able to recognize other, on top of those examined in [1, 2, 3] , distinguished subsets of generalized Pauli operators of two-level quantum systems, namely unicentric triads and pentads of them. It may well be that these two kinds of subsets of the two-qubit Pauli operators hold an important key for getting deeper insights into the nature of finite geometries underlying multiple higher-level quantum systems [12, 13] , in particular when the dimension of Hilbert space is not a power of a prime [14].\n\nThis work was partially supported by the Science and Technology Assistance Agency under the contract # APVT-51-012704, the VEGA grant agency projects # 2/6070/26 and # 7012 (all from Slovak Republic), the trans-national ECO-NET project # 12651NJ \"Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases\" (France), the CNRS-SAV Project # 20246 \"Projective and Related Geometries for Quantum Information\" (France/Slovakia) and by the Action Austria-Slovakia project # 58s2 \"Finite Geometries Behind Hilbert Spaces\"."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"Introduction","text":"A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2 N -dimensional Hilbert spaces is embodied in the geometry of the symplectic polar space of rank N and order two [1, 2, 3] . The case of two-qubit operator space, N = 2, was scrutinized in very detail [1, 3] by explicitly demonstrating, in different ways, the correspondence between various subsets of the generalized Pauli operators/matrices and the fundamental subgeometries of the associated rank-two polar space -the (unique) generalized quadrangle of order two. In this paper we will reveal another interesting geometry hidden behind the Pauli operators of two-qubits, namely that of the Veldkamp space defined on this generalized quadrangle."},{"section_type":"OTHER","section_title":"Finite generalized quadrangles and Veldkamp spaces","text":"In this section we will briefly highlight the basics of the theory of finite generalized quadrangles [4] and introduce the concept of the Veldkamp space of a point-line incidence geometry [5] to be employed in what follows. 2 M. Saniga, M. Planat, P. Pracna and H. Havlicek A finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), is an incidence structure S = (P, B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [4]: (i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique pair (y, M ) ∈ P × B for which xIM IyIL; from these axioms it readily follows that |P | = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). It is obvious that there exists a point-line duality with respect to which each of the axioms is self-dual. Interchanging points and lines in S thus yields a generalized quadrangle S D of order (t, s), called the dual of S. If s = t, S is said to have order s. The generalized quadrangle of order (s, 1) is called a grid and that of order (1, t) a dual grid. A generalized quadrangle with both s > 1 and t > 1 is called thick.\n\nGiven two points x and y of S one writes x ∼ y and says that x and y are collinear if there exists a line L of S incident with both. For any x ∈ P denote x ⊥ = {y ∈ P |y ∼ x} and note that x ∈ x ⊥ ; obviously, x ⊥ = 1 + s + st. Given an arbitrary subset A of P , the perp(-set) of A, A ⊥ , is defined as A ⊥ = {x ⊥ |x ∈ A} and A ⊥⊥ := (A ⊥ ) ⊥ . A triple of pairwise non-collinear points of S is called a triad; given any triad T , a point of T ⊥ is called its center and we say that T is acentric, centric or unicentric according as |T ⊥ | is, respectively, zero, non-zero or one. An ovoid of a generalized quadrangle S is a set of points of S such that each line of S is incident with exactly one point of the set; hence, each ovoid contains st + 1 points.\n\nThe concept of crucial importance is a geometric hyperplane H of a point-line geometry Γ(P, B), which is a proper subset of P such that each line of Γ meets H in one or all points [6] . For Γ = GQ(s, t), it is well known that H is one of the following three kinds: (i) the perp-set of a point x, x ⊥ ; (ii) a (full) subquadrangle of order (s, t ′ ), t ′ < t; and (iii) an ovoid.\n\nFinally, we need to introduce the notion of the Veldkamp space of a point-line incidence geometry Γ(P, B), V(Γ) [5] . V(Γ) is the space in which (i) a point is a geometric hyperplane of Γ and (ii) a line is the collection i = 1, 2) , where H 1 and H 2 are distinct points of V(Γ). 1 If Γ = S, from the preceding paragraph we learn that the points of V(S) are, in general, of three different types.\n\nH 1 H 2 of all geometric hyperplanes H of Γ such that H 1 H 2 = H 1 H = H 2 H or H = H i ("},{"section_type":"OTHER","section_title":"The smallest thick GQ and its Veldkamp space","text":"The smallest thick GQ is obviously the one with s = t = 2, dubbed the \"doily.\" This quadrangle has a number of interesting representations of which we mention the most important two [4]. One, frequently denoted as W 3 (2) or simply W (2), is in terms of the points of P G(3, 2) (i.e., the three-dimensional projective space over the Galois field with two elements) together with the totally isotropic lines with respect to a symplectic polarity. The other, usually denoted as Q(4, 2), is in terms of points and lines of a parabolic quadric in P G (4, 2) . By abuse of notation, any GQ isomorphic to W (2) will also be denoted by this symbol. From the preceding section we readily get that W (2) is endowed with 15 points/lines, each line contains three points and, dually, each point is on three lines; moreover, it is a self-dual object, i.e., isomorphic to its dual. W (2) features all the three kinds of hyperplanes, of the following cardinalities [5]: 15 perp-sets, x ⊥ , seven points each; 10 grids (of order (2, 1)), nine points each; and six ovoids, five points each -as depicted in Fig. 1 . The quadrangle exhibits two distinct kinds of triads, viz. unicentric and tricentric. A point of W (2) is the center of four distinct unicentric triads (Fig. 2 , left ); hence, 1 It is important to mention here that the definition of Veldkamp space given by Shult in [7] is more restrictive than that of Buekenhout and Cohen [5] adopted in this paper.\n\nThe Veldkamp Space of Two-Qubits 3 Figure 1. The three kinds of geometric hyperplanes of W (2). The points of the quadrangle are represented by small circles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the quadrangle. The upper panel shows the points' perp-sets (yellow bullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouring will become clear later. Each pictureexcept that in the bottom right-hand corner -stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon.\n\nFigure 2. Left: -The four distinct unicentric triads (grey bullets) and their common center (black bullet); note that the triads intersect pairwise in a single point and their union covers fully the center's perp-set. Right: -A grid (red bullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); the two triads are also seen to comprise a dual grid (of order (1, 2)). 4 M. Saniga, M. Planat, P. Pracna and H. Havlicek Figure 3. The five different kinds of the lines of V(W (2)), each being uniquely determined by the properties of its core-set (black bullets). Note that the \"yellow\" hyperplanes (i.e., perp-sets) occur in each type, and yellow is also the colour of two homogeneous (i.e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also worth mentioning that the cardinality of core-sets is an odd number not exceeding five. The three hyperplanes of any line are always in such relation to each other that their union comprises all the points of W (2).\n\nThe Veldkamp Space of Two-Qubits 5 the number of such triads is 4 × 15 = 60. Tricentric triads always come in \"complementary\" pairs, one representing the centers of the other, and each such pair is the complement of a grid of W (2) (Fig. 2, right); hence, the number of such triads is 2 × 10 = 20. A unicentric triad is always a subset of an ovoid, which is never the case for a tricentric triad; the latter, in graphcombinatorial terms, representing a complete bipartite graph on six vertices. Now, we have enough background information at hand to reveal the structure of the Veldkamp space of our \"doily\", V(W (2)). 2\n\nFrom the definition given in Section 2, we easily see that V(W (2)) consists of 31 points of which 15 are represented/generated by single-point perp-sets, 10 by grids and six by ovoids. The lines of V(W (2)) feature three points each and are of five distinct types, as illustrated in Fig. 3. These types differ from each other in the cardinality and structure of \"core-sets\", i.e., the sets of points of W (2) shared by all the three hyperplanes forming a given line. As it is obvious from Fig. 3, the lines of the first three types (the first three rows of the figure) have the core-sets of the same cardinality, three, differing from each other only in the structure of these sets as being unicentric triads, tricentric triads and triples of collinear points, respectively. The lines of the fourth type have as core-sets pentads of points, each being a quadruple of points collinear with a given point of W (2), whereas core-sets of the last type's lines feature just a single point. A much more interesting issue is the composition of the lines. Just a brief look at Fig. 3 reveals that geometric hyperplanes of only one kind, namely perp-sets, are present on each line of V(W (2)); grids and ovoids occur only on two kinds of the lines. We also see that the purely homogeneous types are those whose core-sets feature collinear triples and tricentric triads, the most heterogeneous type -the one exhibiting all the three kinds of hyperplanes -being that characterized by unicentric triads. We also notice that there are no lines comprising solely grids and/or solely ovoids, nor the lines featuring only grids and ovoids, which seems to be connected with the fact that the cardinality of a core-set is an odd number. From the properties of W (2) and its triads as discussed above it readily follows that the number of the lines of type one to five is 60, 20, 15, 45 and 15, respectively, totalling 155. All these observations and facts are gathered in Table 1 . We conclude this section with the observation that V(W (2)) has the same number of points (31) and lines (155) as P G(4, 2), the four-dimensional projective space over the Galois field of two elements [8] ; this is not a coincidence, as the two spaces are, in fact, isomorphic to each other [5].\n\n4 Pauli operators of two-qubits in light of V(W (2))\n\nAs discovered in [1] (see also [3]), the fifteen generalized Pauli operators/matrices associated with the Hilbert space of two-qubits (see, e.g., [9]) can be put into a one-to-one correspondence with the fifteen points of the generalized quadrangle W (2) in such a way that their commutation algebra is completely and uniquely reproduced by the geometry of W (2) in which the concept commuting/non-commuting translates into that of collinear/non-collinear. Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [1, 3] for more details). Yet, V(W (2)) puts this bijection in a different light, in which other three subsets of the Pauli operators come into play, namely those represented by the two types of a triad and by the specific pentads occurring as the core-sets of the lines of V(W (2)) (Table 1 ). As already mentioned, the role of tricentric triads of the operators has 2 As this paper is primarily aimed at physicists rather than mathematicians, in what follows we opt for an elementary and self-contained exposition of the Veldkamp space of W (2); this explanation is based only upon some very simple properties of W (2) readily to be grasped from its depiction as \"the doily\", and does not presuppose/require any further background from the reader. 6 M. Saniga, M. Planat, P. Pracna and H. Havlicek Table 1. A succinct summary of the properties of the five different types of the lines of V(W (2)) in terms of the core-sets and the types of geometric hyperplanes featured by a generic line of a given type. The last column gives the total number of lines per the corresponding type.\n\nType of Core-Set Perp-Sets Grids Ovoids # Single Point 1 0 2 15 Collinear Triple 3 0 0 15 Unicentric Triad 1 1 1 60 Tricentric Triad 3 0 0 20 Pentad 1 2 0 45 Table 2. Three kinds of the distinguished subsets of the generalized Pauli operators of two-qubits (PO) viewed as the geometric hyperplanes in the generalized quadrangle of order two (GQ) [1, 3].\n\nPO set of five mutually set of six operators nine operators of non-commuting operators commuting with a given one a Mermin's square GQ ovoid perp-set\\{reference point} grid been recognized in disguise of complete bipartite graphs on six vertices [3]. A true novelty here is obviously unicentric triads and pentads of the generalized Pauli operators as these are all intimately connected with single-point perp-sets; given a point of W (2) (i.e., a generalized Pauli operator of two-qubits), its perp-set fully encompasses four unicentric triads (Fig. 2 , left) and three pentads (Fig. 3 , 4th row) of points/operators. This feature has also a very interesting aspect in connection with the conjecture relating the existence of mutually unbiased bases and finite projective planes raised in [10] , because with each point x of W (2) there is associated a projective plane of order two (the Fano plane) whose points are the elements of x ⊥ and whose lines are the spans {u,\n\nv} ⊥⊥ , where u, v ∈ x ⊥ with u = v [4].\n\nIdentifying the Pauli operators of a two-qubit system with the points of the generalized quadrangle of order two led to the discovery of three distinguished subsets of the operators in terms of geometric hyperplanes of the quadrangle. Here we go one level higher, and identifying these subsets with the points of the associated Veldkamp space leads to recognition of nother remarkable subsets of the Pauli operators, viz. unicentric triads and pentads. It is really intriguing to see that these are the core-sets of the two kinds of lines that both feature grids alias Mermin squares. As it is well known, Mermin squares, which reveal certain important aspects of the entanglement of the system, play a crucial role in the proof of the Kochen-Specker theorem in dimension four and our approach gives a novel geometrical meaning to this [3, 11] . At the Veldkamp space level it turns out of particular importance to study relations between eigenvectors of the above-mentioned unicentric triads and pentad of operators in order to reveal finer, hitherto unnoticed traits of the structure of Mermin squares. These seem to be intimately connected with the existence of outer automorphisms of the symmetric group on six letters, which is the full group of automorphisms of our quadrangle; as this group is the only symmetric group possessing (non-trivial) outer automorphisms, this implies that two-qubits have a rather special footing among multiple qubit systems. All these aspects deserve special attention and will therefore be dealt with in a separate paper.\n\nConcerning three-qubits, our preliminary study indicates that the corresponding finite geometry differs fundamentally from that of W (2) in the sense that it contains multi-lines, i.e., two or more lines passing through two distinct points [12] . As we do not have a full picture at hand yet, we cannot see if it admits hyperplanes and so lends itself to constructing the corresponding Veldkamp space. If the latter does exist, it is likely to differ substantially from that of twoqubits, which would imply the expected difference between entanglement properties of the two\n\nThe Veldkamp Space of Two-Qubits 7 kinds of systems; if not, this will only further strengthen the above-mentioned uniqueness of two-qubits."},{"section_type":"CONCLUSION","section_title":"Conclusion","text":"By employing the concept of the Veldkamp space of the generalized quadrangle of order two, we were able to recognize other, on top of those examined in [1, 2, 3] , distinguished subsets of generalized Pauli operators of two-level quantum systems, namely unicentric triads and pentads of them. It may well be that these two kinds of subsets of the two-qubit Pauli operators hold an important key for getting deeper insights into the nature of finite geometries underlying multiple higher-level quantum systems [12, 13] , in particular when the dimension of Hilbert space is not a power of a prime [14]."},{"section_type":"OTHER","section_title":"Acknowledgements","text":"This work was partially supported by the Science and Technology Assistance Agency under the contract # APVT-51-012704, the VEGA grant agency projects # 2/6070/26 and # 7012 (all from Slovak Republic), the trans-national ECO-NET project # 12651NJ \"Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases\" (France), the CNRS-SAV Project # 20246 \"Projective and Related Geometries for Quantum Information\" (France/Slovakia) and by the Action Austria-Slovakia project # 58s2 \"Finite Geometries Behind Hilbert Spaces\"."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2 N -dimensional Hilbert spaces is embodied in the geometry of the symplectic polar space of rank N and order two [1, 2, 3] . The case of two-qubit operator space, N = 2, was scrutinized in very detail [1, 3] by explicitly demonstrating, in different ways, the correspondence between various subsets of the generalized Pauli operators/matrices and the fundamental subgeometries of the associated rank-two polar space -the (unique) generalized quadrangle of order two. In this paper we will reveal another interesting geometry hidden behind the Pauli operators of two-qubits, namely that of the Veldkamp space defined on this generalized quadrangle.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Finite generalized quadrangles and Veldkamp spaces\",\n \"text\": \"In this section we will briefly highlight the basics of the theory of finite generalized quadrangles [4] and introduce the concept of the Veldkamp space of a point-line incidence geometry [5] to be employed in what follows. 2 M. Saniga, M. Planat, P. Pracna and H. Havlicek A finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), is an incidence structure S = (P, B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [4]: (i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique pair (y, M ) ∈ P × B for which xIM IyIL; from these axioms it readily follows that |P | = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). It is obvious that there exists a point-line duality with respect to which each of the axioms is self-dual. Interchanging points and lines in S thus yields a generalized quadrangle S D of order (t, s), called the dual of S. If s = t, S is said to have order s. The generalized quadrangle of order (s, 1) is called a grid and that of order (1, t) a dual grid. A generalized quadrangle with both s > 1 and t > 1 is called thick.\\n\\nGiven two points x and y of S one writes x ∼ y and says that x and y are collinear if there exists a line L of S incident with both. For any x ∈ P denote x ⊥ = {y ∈ P |y ∼ x} and note that x ∈ x ⊥ ; obviously, x ⊥ = 1 + s + st. Given an arbitrary subset A of P , the perp(-set) of A, A ⊥ , is defined as A ⊥ = {x ⊥ |x ∈ A} and A ⊥⊥ := (A ⊥ ) ⊥ . A triple of pairwise non-collinear points of S is called a triad; given any triad T , a point of T ⊥ is called its center and we say that T is acentric, centric or unicentric according as |T ⊥ | is, respectively, zero, non-zero or one. An ovoid of a generalized quadrangle S is a set of points of S such that each line of S is incident with exactly one point of the set; hence, each ovoid contains st + 1 points.\\n\\nThe concept of crucial importance is a geometric hyperplane H of a point-line geometry Γ(P, B), which is a proper subset of P such that each line of Γ meets H in one or all points [6] . For Γ = GQ(s, t), it is well known that H is one of the following three kinds: (i) the perp-set of a point x, x ⊥ ; (ii) a (full) subquadrangle of order (s, t ′ ), t ′ < t; and (iii) an ovoid.\\n\\nFinally, we need to introduce the notion of the Veldkamp space of a point-line incidence geometry Γ(P, B), V(Γ) [5] . V(Γ) is the space in which (i) a point is a geometric hyperplane of Γ and (ii) a line is the collection i = 1, 2) , where H 1 and H 2 are distinct points of V(Γ). 1 If Γ = S, from the preceding paragraph we learn that the points of V(S) are, in general, of three different types.\\n\\nH 1 H 2 of all geometric hyperplanes H of Γ such that H 1 H 2 = H 1 H = H 2 H or H = H i (\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The smallest thick GQ and its Veldkamp space\",\n \"text\": \"The smallest thick GQ is obviously the one with s = t = 2, dubbed the \\\"doily.\\\" This quadrangle has a number of interesting representations of which we mention the most important two [4]. One, frequently denoted as W 3 (2) or simply W (2), is in terms of the points of P G(3, 2) (i.e., the three-dimensional projective space over the Galois field with two elements) together with the totally isotropic lines with respect to a symplectic polarity. The other, usually denoted as Q(4, 2), is in terms of points and lines of a parabolic quadric in P G (4, 2) . By abuse of notation, any GQ isomorphic to W (2) will also be denoted by this symbol. From the preceding section we readily get that W (2) is endowed with 15 points/lines, each line contains three points and, dually, each point is on three lines; moreover, it is a self-dual object, i.e., isomorphic to its dual. W (2) features all the three kinds of hyperplanes, of the following cardinalities [5]: 15 perp-sets, x ⊥ , seven points each; 10 grids (of order (2, 1)), nine points each; and six ovoids, five points each -as depicted in Fig. 1 . The quadrangle exhibits two distinct kinds of triads, viz. unicentric and tricentric. A point of W (2) is the center of four distinct unicentric triads (Fig. 2 , left ); hence, 1 It is important to mention here that the definition of Veldkamp space given by Shult in [7] is more restrictive than that of Buekenhout and Cohen [5] adopted in this paper.\\n\\nThe Veldkamp Space of Two-Qubits 3 Figure 1. The three kinds of geometric hyperplanes of W (2). The points of the quadrangle are represented by small circles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the quadrangle. The upper panel shows the points' perp-sets (yellow bullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouring will become clear later. Each pictureexcept that in the bottom right-hand corner -stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon.\\n\\nFigure 2. Left: -The four distinct unicentric triads (grey bullets) and their common center (black bullet); note that the triads intersect pairwise in a single point and their union covers fully the center's perp-set. Right: -A grid (red bullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); the two triads are also seen to comprise a dual grid (of order (1, 2)). 4 M. Saniga, M. Planat, P. Pracna and H. Havlicek Figure 3. The five different kinds of the lines of V(W (2)), each being uniquely determined by the properties of its core-set (black bullets). Note that the \\\"yellow\\\" hyperplanes (i.e., perp-sets) occur in each type, and yellow is also the colour of two homogeneous (i.e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also worth mentioning that the cardinality of core-sets is an odd number not exceeding five. The three hyperplanes of any line are always in such relation to each other that their union comprises all the points of W (2).\\n\\nThe Veldkamp Space of Two-Qubits 5 the number of such triads is 4 × 15 = 60. Tricentric triads always come in \\\"complementary\\\" pairs, one representing the centers of the other, and each such pair is the complement of a grid of W (2) (Fig. 2, right); hence, the number of such triads is 2 × 10 = 20. A unicentric triad is always a subset of an ovoid, which is never the case for a tricentric triad; the latter, in graphcombinatorial terms, representing a complete bipartite graph on six vertices. Now, we have enough background information at hand to reveal the structure of the Veldkamp space of our \\\"doily\\\", V(W (2)). 2\\n\\nFrom the definition given in Section 2, we easily see that V(W (2)) consists of 31 points of which 15 are represented/generated by single-point perp-sets, 10 by grids and six by ovoids. The lines of V(W (2)) feature three points each and are of five distinct types, as illustrated in Fig. 3. These types differ from each other in the cardinality and structure of \\\"core-sets\\\", i.e., the sets of points of W (2) shared by all the three hyperplanes forming a given line. As it is obvious from Fig. 3, the lines of the first three types (the first three rows of the figure) have the core-sets of the same cardinality, three, differing from each other only in the structure of these sets as being unicentric triads, tricentric triads and triples of collinear points, respectively. The lines of the fourth type have as core-sets pentads of points, each being a quadruple of points collinear with a given point of W (2), whereas core-sets of the last type's lines feature just a single point. A much more interesting issue is the composition of the lines. Just a brief look at Fig. 3 reveals that geometric hyperplanes of only one kind, namely perp-sets, are present on each line of V(W (2)); grids and ovoids occur only on two kinds of the lines. We also see that the purely homogeneous types are those whose core-sets feature collinear triples and tricentric triads, the most heterogeneous type -the one exhibiting all the three kinds of hyperplanes -being that characterized by unicentric triads. We also notice that there are no lines comprising solely grids and/or solely ovoids, nor the lines featuring only grids and ovoids, which seems to be connected with the fact that the cardinality of a core-set is an odd number. From the properties of W (2) and its triads as discussed above it readily follows that the number of the lines of type one to five is 60, 20, 15, 45 and 15, respectively, totalling 155. All these observations and facts are gathered in Table 1 . We conclude this section with the observation that V(W (2)) has the same number of points (31) and lines (155) as P G(4, 2), the four-dimensional projective space over the Galois field of two elements [8] ; this is not a coincidence, as the two spaces are, in fact, isomorphic to each other [5].\\n\\n4 Pauli operators of two-qubits in light of V(W (2))\\n\\nAs discovered in [1] (see also [3]), the fifteen generalized Pauli operators/matrices associated with the Hilbert space of two-qubits (see, e.g., [9]) can be put into a one-to-one correspondence with the fifteen points of the generalized quadrangle W (2) in such a way that their commutation algebra is completely and uniquely reproduced by the geometry of W (2) in which the concept commuting/non-commuting translates into that of collinear/non-collinear. Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [1, 3] for more details). Yet, V(W (2)) puts this bijection in a different light, in which other three subsets of the Pauli operators come into play, namely those represented by the two types of a triad and by the specific pentads occurring as the core-sets of the lines of V(W (2)) (Table 1 ). As already mentioned, the role of tricentric triads of the operators has 2 As this paper is primarily aimed at physicists rather than mathematicians, in what follows we opt for an elementary and self-contained exposition of the Veldkamp space of W (2); this explanation is based only upon some very simple properties of W (2) readily to be grasped from its depiction as \\\"the doily\\\", and does not presuppose/require any further background from the reader. 6 M. Saniga, M. Planat, P. Pracna and H. Havlicek Table 1. A succinct summary of the properties of the five different types of the lines of V(W (2)) in terms of the core-sets and the types of geometric hyperplanes featured by a generic line of a given type. The last column gives the total number of lines per the corresponding type.\\n\\nType of Core-Set Perp-Sets Grids Ovoids # Single Point 1 0 2 15 Collinear Triple 3 0 0 15 Unicentric Triad 1 1 1 60 Tricentric Triad 3 0 0 20 Pentad 1 2 0 45 Table 2. Three kinds of the distinguished subsets of the generalized Pauli operators of two-qubits (PO) viewed as the geometric hyperplanes in the generalized quadrangle of order two (GQ) [1, 3].\\n\\nPO set of five mutually set of six operators nine operators of non-commuting operators commuting with a given one a Mermin's square GQ ovoid perp-set\\\\{reference point} grid been recognized in disguise of complete bipartite graphs on six vertices [3]. A true novelty here is obviously unicentric triads and pentads of the generalized Pauli operators as these are all intimately connected with single-point perp-sets; given a point of W (2) (i.e., a generalized Pauli operator of two-qubits), its perp-set fully encompasses four unicentric triads (Fig. 2 , left) and three pentads (Fig. 3 , 4th row) of points/operators. This feature has also a very interesting aspect in connection with the conjecture relating the existence of mutually unbiased bases and finite projective planes raised in [10] , because with each point x of W (2) there is associated a projective plane of order two (the Fano plane) whose points are the elements of x ⊥ and whose lines are the spans {u,\\n\\nv} ⊥⊥ , where u, v ∈ x ⊥ with u = v [4].\\n\\nIdentifying the Pauli operators of a two-qubit system with the points of the generalized quadrangle of order two led to the discovery of three distinguished subsets of the operators in terms of geometric hyperplanes of the quadrangle. Here we go one level higher, and identifying these subsets with the points of the associated Veldkamp space leads to recognition of nother remarkable subsets of the Pauli operators, viz. unicentric triads and pentads. It is really intriguing to see that these are the core-sets of the two kinds of lines that both feature grids alias Mermin squares. As it is well known, Mermin squares, which reveal certain important aspects of the entanglement of the system, play a crucial role in the proof of the Kochen-Specker theorem in dimension four and our approach gives a novel geometrical meaning to this [3, 11] . At the Veldkamp space level it turns out of particular importance to study relations between eigenvectors of the above-mentioned unicentric triads and pentad of operators in order to reveal finer, hitherto unnoticed traits of the structure of Mermin squares. These seem to be intimately connected with the existence of outer automorphisms of the symmetric group on six letters, which is the full group of automorphisms of our quadrangle; as this group is the only symmetric group possessing (non-trivial) outer automorphisms, this implies that two-qubits have a rather special footing among multiple qubit systems. All these aspects deserve special attention and will therefore be dealt with in a separate paper.\\n\\nConcerning three-qubits, our preliminary study indicates that the corresponding finite geometry differs fundamentally from that of W (2) in the sense that it contains multi-lines, i.e., two or more lines passing through two distinct points [12] . As we do not have a full picture at hand yet, we cannot see if it admits hyperplanes and so lends itself to constructing the corresponding Veldkamp space. If the latter does exist, it is likely to differ substantially from that of twoqubits, which would imply the expected difference between entanglement properties of the two\\n\\nThe Veldkamp Space of Two-Qubits 7 kinds of systems; if not, this will only further strengthen the above-mentioned uniqueness of two-qubits.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusion\",\n \"text\": \"By employing the concept of the Veldkamp space of the generalized quadrangle of order two, we were able to recognize other, on top of those examined in [1, 2, 3] , distinguished subsets of generalized Pauli operators of two-level quantum systems, namely unicentric triads and pentads of them. It may well be that these two kinds of subsets of the two-qubit Pauli operators hold an important key for getting deeper insights into the nature of finite geometries underlying multiple higher-level quantum systems [12, 13] , in particular when the dimension of Hilbert space is not a power of a prime [14].\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgements\",\n \"text\": \"This work was partially supported by the Science and Technology Assistance Agency under the contract # APVT-51-012704, the VEGA grant agency projects # 2/6070/26 and # 7012 (all from Slovak Republic), the trans-national ECO-NET project # 12651NJ \\\"Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases\\\" (France), the CNRS-SAV Project # 20246 \\\"Projective and Related Geometries for Quantum Information\\\" (France/Slovakia) and by the Action Austria-Slovakia project # 58s2 \\\"Finite Geometries Behind Hilbert Spaces\\\".\"\n }\n]"}}},{"rowIdx":90,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0505"},"arxiv_id":{"kind":"string","value":"0704.0505"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.76.085001"},"content_hash":{"kind":"string","value":"aeda426d1071e75e895afdc97caa1fa9e76af03cbfa583d2180a72529e4ba2a9"},"title":{"kind":"string","value":"Exact Solutions of Einstein-Yang-Mills Theory with Higher-Derivative\n Coupling"},"abstract":{"kind":"string","value":"We construct a classical solution of an Einstein-Yang-Mills system with a\nfourth order term with respect to the field strength of the Yang-Mills field.\nThe solution provides a spontaneous compactification proposed by Cremmer and\nScherk; ten-dimensional space-time with a cosmological constant is compactified\nto the four-dimensional Minkowski space with a six-dimensional sphere S^6 on\nwhich an instanton solution exists. The radius of the sphere is not a modulus\nbut is determined by the gauge coupling and the four-derivative coupling\nconstants and the Newton's constant. We also construct a solution of\nten-dimensional theory without a cosmological constant compactified to AdS_4 x\nS^6."},"authors":{"kind":"list like","value":["Hironobu Kihara","Muneto Nitta"],"string":"[\n \"Hironobu Kihara\",\n \"Muneto Nitta\"\n]"},"arxiv_categories":{"kind":"list like","value":["hep-th","gr-qc"],"string":"[\n \"hep-th\",\n \"gr-qc\"\n]"},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term with respect to the field strength of the Yang-Mills field. The solution provides a compactification proposed by Cremmer and Scherk; ten-dimensional space-time with a cosmological constant is compactified to the four-dimensional Minkowski space with a six-dimensional sphere S 6 on which an instanton solution exists. The radius of the sphere is not a modulus but is determined by the gauge coupling and the four-derivative coupling constants and the Newton's constant. We also construct a solution of ten-dimensional theory without a cosmological constant compactified to AdS 4 × S 6 .\n\nUnification of fundamental forces with space-time and matter often requires higherdimensional space-time rather than our four-dimensional Universe. The early Kaluza-Klein theory unifies gravity and the electro-magnetic interaction by considering five-dimensional space-time with one direction compactified into a circle S 1 [1] . This old idea has been revisited several times. After supergravity was discovered many people tried to unify all forces and matter in higher-dimensional space-time compactified on various internal manifolds [2] .\n\nString theory was proposed as the most attractive candidate of unification, but it is defined only in ten-dimensional space-time. In order to realize four-dimensional Universe one has to find a suitable six-dimensional internal space. So many candidates of such spaces were proposed; Calabi-Yau manifolds and orbifold models. Internal manifolds can be deformed with satisfying the Einstein equation, and these degrees of freedom are called the moduli.\n\nThe moduli introduce unwanted massless particles in four-dimensional world. Recently a new mechanism has been suggested to fix these moduli by turning on the Ramond-Ramond flux on the internal space [3, 4] . This flux compactification has been extensively studied in these years.\n\nWe would like to revise the compactification scenario with fixed moduli proposed by Cremmer and Scherk long time ago [5] (see also [6, 7] ) in a theory with a cosmological constant. By placing solitons on a compact internal space they showed decompactifying limit with large radius of the internal space is disfavored and the radius is fixed to a certain value determined by coupling constants. They considered the 't Hooft-Polyakov monopole [8] on S 2 and the Yang-Mills instanton [9] on S 4 , both of which can satisfy, with proper coupling constants, the first order (self-dual) equations rather than the second order equations of motion, but their solutions on higher dimensional sphere are not the case. Since string theory is defined in ten dimensions, it is natural to consider this scenario with stable BPS solitons on a six-dimensional internal space like S 6 .\n\nHigher dimensional generalization of self-dual equations was suggested by Tchrakian some years ago [10] . Eight dimensional case is known as octonionic instantons [11] . Though several works have been done for generalized self-dual equations [12, 13] , a six-dimensional case was not discussed because of the lack of conformal property. Recently we have found a new solution to the generalized self-dual equations in an SO (6) pure Yang-Mills theory with a fourth order term with respect to the field strength of the Yang-Mills field (a four-derivative term) on a six-dimensional sphere S 6 [14] .\n\nIn this letter we propose to use this solution in the context of a compactification of the Cremmer-Scherk type. In our model ten-dimensional space-time with (without) a cosmological constant is compactified to a four-dimensional Minkowski space M 4 (anti de Sitter space AdS 4 ) with a six-dimensional sphere S 6 , where dimensionality of the internal space, six, is required by the four derivative term. Unlike the case of the absence of gravity [14] the four-derivative coupling constant α can differ from the constant β in the generalized self-dual equations. When the relation α = β holds the generalized self-dual equations become the Bogomol'nyi equations and solutions are BPS. We find for both M 4 × S 6 and AdS 4 × S 6 that certain relations exist between the radius of S 6 , the gauge coupling, the four-derivative coupling α and the gravitational coupling constants. When the four-derivative coupling constant α vanishes in the case of M 4 × S 6 , these relations reduce to those of the original work by Cremmer and Scherk. The advantage of our model to the Cremmer-Scherk model is that the Yang-Mills soliton in our model satisfies the self-dual equations (the Bogomol'nyi equations for α = β) rather than usual equations of motion in the case of the Cremmer-Scherk model. This ensures the stability of configuration at least for the sector of Yang-Mills fields.\n\nLet us consider that space-time is a ten-dimensional manifold. We consider an Einstein-Yang-Mills theory. Our action contains as dynamical variables the Yang-Mills (gauge) fields\n\nA [ab] μ\n\nand a graviton field or the metric ĝμν . Indices with a hat \"ˆ\" will refer to a tendimensional space-time (X 0 , X 1 , • • • , X 9 ). Latin indices (a, b, • • • ) run from 1 to 6 and refer to an internal space. The Clifford algebra associated with the orthogonal group SO( 6 ) is useful and we represent generators of the Lie algebra so(6) as their elements. The Clifford algebra is defined by gamma matrices {Γ a } which satisfy the following anti-commutation relations, {Γ a , Γ b } = 2δ ab . These matrices can be realized as 8 × 8 matrices with complex coefficients. The generators of so( 6 ) are represented by\n\nΓ ab = 1 2 [Γ a , Γ b ].\n\nWe often abbreviate the Yang-Mills fields as\n\nA μ = 1 2 A [ab]\n\nμ Γ ab and we also use notations with differential forms. Thus the gauge fields are expressed as A = A μdX μ. In this notation, the corresponding field strength F is written as F = dA + eA ∧ A, where e is a gauge coupling. Covariant derivative\n\nD μ on an adjoint representation Y = 1 2 Y [ab] Γ ab is defined as D μY = ∂ μY + e(A μY -Y A μ),\n\nwhere Y is a scalar multiplet. The action S total consists of two parts. One is the Einstein-Hilbert action S E and the other S Y M T is a Yang-Mills action with a term which is the fourth power of the field strength F . Such a quartic term has been studied by Tchrakian [10] and so we call it the Tchrakian term. The total action is:\n\nS total = S E + S Y M T , S E = 1 16πG dvR , S Y M T = 1 16 Tr -F ∧ * F + α 2 (F ∧ F ) ∧ * (F ∧ F ) -V 0 dv . (1)\n\nHere the 10-form dv is an invariant volume form with respect to the metric ĝ and is written as dv = √ -ĝd 10 X in a local patch. The scalar curvature is denoted by R. The asterisk \" * \" denotes the Hodge dual operator. This operator defines an inner product over differential forms, and for a given form ω, ω ∧ * ω is proportional to the invariant volume form dv. 1 The parameters of this action are the Newton's gravitational constant G, the gauge coupling e, the four-derivative coupling α and the cosmological constant V 0 .\n\nWe show the explicit form of the Yang-Mills part with components of A and F ,\n\nS Y M T = -dv 1 8 F [ab] μν F μν,[ab] + α 2 8 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 16 S μν ρσ S μν ρσ + 1 2 V 0 , (2)\n\nF = 1 4 F [ab] μν dX μ ∧ dX ν Γ ab , S μν ρσ = F [ab] μν F [ab] ρσ + F [ab] μρ F [ab] σ ν + F [ab] μσ F [ab] ν ρ , (3)\n\nT [ab][cd] μν ρσ = F [ab] μν F [cd] ρσ + F [ab] μρ F [cd] σ ν + F [ab] μσ F [cd] ν ρ , T [abcd] μν ρσ = 1 6 T [ab][cd]\n\nμν ρσ + T\n\n[ac][db] μν ρσ + T [ad][bc] μν ρσ + T [cd][ab] μν ρσ + T [db][ac] μν ρσ + T [bc][ad] μν ρσ .\n\nThe Euler-Lagrange equations from these actions read the usual Einstein equation and the equations for the Yang-Mills fields:\n\nR μν - 1 2 ĝμν R = 8πGT μν , D μ √ -gF μν -2α 2 √ -gF [μν F ρσ] F ρσ = 0 . (5)\n\nHere the energy-momentum tensor T μν is obtained by the variation of the Yang-Mills part with respect to the metric:\n\nT μν = 1 2 F μ ρ,[ab] F [ab] ν ρ + α 2 T [abcd] μρσ τ T ν ρστ ,[ab][cd] + α 2 3 • 2 S μρσ τ S ν ρσ τ - 1 2 g μν χ χ = 1 4 F [ab] μν F μν,[ab] + α 2 4 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 8 S μν ρσ S μν ρσ + V 0 . (6)\n\nTo solve these equations, we make an ansatz which is the same as that of Cremmer-Scherk.\n\nOur ansatz for the metric is the following:\n\nds 2 = η µν dx µ dx ν + δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J = ĝμν dX μdX ν , y 2 = 6 a=1 (y I ) 2 , ( 7\n\n)\n\n1 The Hodge dual operator acting on a differential form on a space with Minkowski signature satisfies the following relation:\n\n(F µν dx µν ) ∧ * (F ρσ dx ρσ ) = -F µν F µν dv.\n\nwhere the coordinates X are the total space-time coordinates. The metric η µν = diag(-+ ++) is the Lorentz metric on the four-dimensional Minkowski space. Greek indices without a hat \"ˆ\", for instance µ will refer to the first four variables. Capital indices (I, J, • • • ) run from one to six and refer to the compact space. The six-dimensional space is taken as a sphere with a radius R 0 . The Riemann tensor, Ricci tensor and scalar curvature are\n\nR I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ , R = 30 R 2 0 . (8)\n\nThe rest components of the curvature tensor vanish. In this space, the Einstein equations in (5) reduce to simple equations,\n\n- 1 2 η µν 30 R 2 0 = 8πGT µν , 0 = T µI , - 1 2 20 R 2 0 g IJ = 8πGT IJ . (9)\n\nWe now make ansatzes for the gauge fields. We assume that the fields A do not depend on the four-dimensional directions, ∂ µ A = 0, and they have no four-dimensional components A µ = 0. This implies that the field strengths are two forms on the six-dimensional sphere:\n\nA = A I (y)dy I , F = 1 2\n\nF IJ dy I ∧ dy J . With these ansatzes, the four-dimensional part of the energy-momentum tensor becomes -1 2 η µν χ, and the equation reduces to 30/R 2 0 = 8πGχ. This equation requires that the χ is a constant. Suppose that the field strength fulfils the generalized self-dual condition\n\nF = iβγ 7 * 6 (F ∧ F ), ( 10\n\n)\n\nwhere β is a real parameter. Here \" * 6 \" means the Hodge dual on the six-dimensional sphere.\n\nThen the second part of the equations of motion ( 5 ) is fulfilled automatically by the relation DF = 0, where the exterior covariant derivative is defined as DF = dF +e (A ∧ F -F ∧ A).\n\nIn fact we have an explicit solution to the self-dual equation:\n\nA = 1 4eR 2 0 y a e b Γ ab , F = 1 4eR 2 0 e a ∧ e b Γ ab , β = eR 2 0 3 . ( 11\n\n)\n\nHere we identify the internal space index and the sphere index. The energy-momentum tensor of this configuration becomes\n\nζ ≡ α 2 β 2 , χ = (1 + ζ) 15 4e 2 R 4 0 + V 0 , T IJ = - 1 2 (1 -ζ) 5 4e 2 R 4 0 + V 0 g IJ . (12)\n\nWith these ansatzes we obtain algebraic equations from the Einstein equations:\n\n30 R 2 0 = 8πG 1 2 (1 + ζ) 15 2e 2 R 4 0 + V 0 , 10 R 2 0 = 8πG (1 -ζ) 5 8e 2 R 4 0 + V 0 2 , (13)\n\nFrom these we finally obtain\n\n1 πG = 1 e 2 R 2 0 (2 + 4ζ) , V 0 = 15 4e 2 R 4 0 (1 + 3ζ) . (14)\n\nWhen the four-derivative coupling vanishes, α = 0 and therefore ζ = 0, these relations reduce to those of the Cremmer and Scherk [5] . 2 When the relation α = β holds (ζ = 1) our solution saturates the Bogomol'nyi bound and becomes a BPS state. The energy density is given by an integral over S 6 as follows:\n\nE S 6 Y M T = 1 16 S 6 Tr -F ∧ * 6 F + α 2 (F ∧ F ) ∧ * 6 (F ∧ F ) = 1 16 S 6 Tr (iF ∓ αγ 7 * 6 (F ∧ F )) ∧ * 6 (iF ∓ αγ 7 * 6 (F ∧ F )) ± i 8 α S 6 Trγ 7 F ∧ F ∧ F ≥ ± i 8 α S 6 Trγ 7 F ∧ F ∧ F = ∓ 1 2 3 S 6 ǫ abcdef F [ab] ∧ F [cd] ∧ F [ef ] ≡ ±Q , (15)\n\nwhere the field strength F has only components along S 6 . If the coupling α is equal to β, the solution of eq. ( 10 ) satisfies the Bogomol'nyi equation and the energy attains the local minimum. We can also consider a system coupled with scalar fields. Suppose that scalar fields Q m transform as a representation of SO (6) . The index m labels the representation space. Let us add an action S Q of the scalar fields Q with a Higgs potential\n\nS Q = 1 2 dvD μQ m D μQ m + V (Q 2 ) , D μQ m = ∂ μQ m - 1 2 ieA [ab] μ R(Γ ab ) mm ′ Q m ′ (16)\n\nto S total . The equations of motion are modified. In general, our solution mentioned above does not satisfy the modified equations any more. However, for the scalars which fulfil the covariantly constant condition D μQ m = 0 and attain the absolute minimum V (Q) = 0, the configurations of A and g in equations ( 7 ), (11) are still solutions for the modified equations. Here the constant value of the minimum is shifted to 0. Thus we can argue the Higgs mechanism around our solutions.\n\nNext we suppose that the four-dimensional part is an anti-de Sitter space AdS 4 of radius R A . Our ansatz for the metric is the following:\n\nds 2 = η µν (x)dx µ dx ν + g IJ (y)dy I dy J = ĝμν dX μdX ν , (17)\n\ng IJ (y)dy\n\nI dy J = δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J y 2 = 6 a=1 (y I ) 2 , η µν (x)dx µ dx ν = R 2 A cos 2 θ -dτ 2 + dθ 2 + sin 2 θdΩ 2 , dΩ 2 = |dz| 2 (1 + |z| 2 /4) 2 , (18)\n\nwhere z parametrizes a whole complex plane. The metric η µν (x) is a maximally symmetric metric on the four-dimensional anti-de Sitter space. The Riemann tensor and the Ricci tensor are\n\nR µ νρσ = - 1 R 2 A δ µ ρ η νσ -δ µ σ η νρ , R µν = - 3 R 2 A η µν , R I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ . (19)\n\nThe total scalar curvature is obtained by a summation of those of two parts: R = -\n\n12 R 2 A + 30 R 2 0 .\n\nIn this space, the Einstein equations are\n\nR µν - 1 2 η µν R = 8πGT µν , R IJ - 1 2 g IJ R = 8πGT IJ . (20)\n\nThe ansatz for the gauge fields is the same as previous one and the energy momentum tensor does not change. With these ansatzes, we obtain algebraic equations from the Einstein\n\nequations as 3 R 2 A - 15 R 2 0 = -4πG (1 + ζ) 15 4e 2 R 4 0 + V 0 , 6 R 2 A - 10 R 2 0 = -4πG (1 -ζ) 5 4e 2 R 4 0 + V 0 . (21)\n\nWe are interested in a possible relation to string theory and therefore we consider the case with the vanishing cosmological constant, V 0 = 0. In this case, the radii (R A , R 0 ) are written by the couplings,\n\nR 2 0 = (5 + 7ζ) πG 4e 2 , R 2 A = 5 + 7ζ 5 + 15ζ R 2 0 . (22)\n\nThus the additional higher derivative coupling term of the Tchrakian type does not affect critically to the equations of motion. When ζ = 1 our solution becomes a solution of the Bogomol'nyi equation again.\n\nOur solutions introduced in this letter are new solutions of the system with a Tchrakian term. The origin of this term has not been clear so far but it seems rather universal in order to construct solitons with codimensions higher than four: for instance it has played a crucial role to construct a finite energy monopole (with codimension five) in a six-dimensional spacetime [13] . Though the parameter ζ(= α 2 /β 2 ) is a free parameter, we expect that the system goes to ζ = 1 because it becomes BPS. There are several discussions on the (in)stability of higher-dimensional Yang-Mills theories [15] . To compute the mass spectra of the fluctuations around our solutions is a future work. When the scalar fields Q m are non-trivially coupled, the system may allow BPS composite solitons which are made of solitons with different codimensions, as in the case of usual self-dual Yang-Mills equations coupled to Higgs fields [16] .\n\nFinally our solution of AdS 4 × S 6 may have a relation with D2-branes, and we hope that there exists some impact on AdS/CFT duality [17] ."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term with respect to the field strength of the Yang-Mills field. The solution provides a compactification proposed by Cremmer and Scherk; ten-dimensional space-time with a cosmological constant is compactified to the four-dimensional Minkowski space with a six-dimensional sphere S 6 on which an instanton solution exists. The radius of the sphere is not a modulus but is determined by the gauge coupling and the four-derivative coupling constants and the Newton's constant. We also construct a solution of ten-dimensional theory without a cosmological constant compactified to AdS 4 × S 6 ."},{"section_type":"OTHER","section_title":"Untitled Section","text":"Unification of fundamental forces with space-time and matter often requires higherdimensional space-time rather than our four-dimensional Universe. The early Kaluza-Klein theory unifies gravity and the electro-magnetic interaction by considering five-dimensional space-time with one direction compactified into a circle S 1 [1] . This old idea has been revisited several times. After supergravity was discovered many people tried to unify all forces and matter in higher-dimensional space-time compactified on various internal manifolds [2] .\n\nString theory was proposed as the most attractive candidate of unification, but it is defined only in ten-dimensional space-time. In order to realize four-dimensional Universe one has to find a suitable six-dimensional internal space. So many candidates of such spaces were proposed; Calabi-Yau manifolds and orbifold models. Internal manifolds can be deformed with satisfying the Einstein equation, and these degrees of freedom are called the moduli.\n\nThe moduli introduce unwanted massless particles in four-dimensional world. Recently a new mechanism has been suggested to fix these moduli by turning on the Ramond-Ramond flux on the internal space [3, 4] . This flux compactification has been extensively studied in these years.\n\nWe would like to revise the compactification scenario with fixed moduli proposed by Cremmer and Scherk long time ago [5] (see also [6, 7] ) in a theory with a cosmological constant. By placing solitons on a compact internal space they showed decompactifying limit with large radius of the internal space is disfavored and the radius is fixed to a certain value determined by coupling constants. They considered the 't Hooft-Polyakov monopole [8] on S 2 and the Yang-Mills instanton [9] on S 4 , both of which can satisfy, with proper coupling constants, the first order (self-dual) equations rather than the second order equations of motion, but their solutions on higher dimensional sphere are not the case. Since string theory is defined in ten dimensions, it is natural to consider this scenario with stable BPS solitons on a six-dimensional internal space like S 6 .\n\nHigher dimensional generalization of self-dual equations was suggested by Tchrakian some years ago [10] . Eight dimensional case is known as octonionic instantons [11] . Though several works have been done for generalized self-dual equations [12, 13] , a six-dimensional case was not discussed because of the lack of conformal property. Recently we have found a new solution to the generalized self-dual equations in an SO (6) pure Yang-Mills theory with a fourth order term with respect to the field strength of the Yang-Mills field (a four-derivative term) on a six-dimensional sphere S 6 [14] .\n\nIn this letter we propose to use this solution in the context of a compactification of the Cremmer-Scherk type. In our model ten-dimensional space-time with (without) a cosmological constant is compactified to a four-dimensional Minkowski space M 4 (anti de Sitter space AdS 4 ) with a six-dimensional sphere S 6 , where dimensionality of the internal space, six, is required by the four derivative term. Unlike the case of the absence of gravity [14] the four-derivative coupling constant α can differ from the constant β in the generalized self-dual equations. When the relation α = β holds the generalized self-dual equations become the Bogomol'nyi equations and solutions are BPS. We find for both M 4 × S 6 and AdS 4 × S 6 that certain relations exist between the radius of S 6 , the gauge coupling, the four-derivative coupling α and the gravitational coupling constants. When the four-derivative coupling constant α vanishes in the case of M 4 × S 6 , these relations reduce to those of the original work by Cremmer and Scherk. The advantage of our model to the Cremmer-Scherk model is that the Yang-Mills soliton in our model satisfies the self-dual equations (the Bogomol'nyi equations for α = β) rather than usual equations of motion in the case of the Cremmer-Scherk model. This ensures the stability of configuration at least for the sector of Yang-Mills fields.\n\nLet us consider that space-time is a ten-dimensional manifold. We consider an Einstein-Yang-Mills theory. Our action contains as dynamical variables the Yang-Mills (gauge) fields\n\nA [ab] μ\n\nand a graviton field or the metric ĝμν . Indices with a hat \"ˆ\" will refer to a tendimensional space-time (X 0 , X 1 , • • • , X 9 ). Latin indices (a, b, • • • ) run from 1 to 6 and refer to an internal space. The Clifford algebra associated with the orthogonal group SO( 6 ) is useful and we represent generators of the Lie algebra so(6) as their elements. The Clifford algebra is defined by gamma matrices {Γ a } which satisfy the following anti-commutation relations, {Γ a , Γ b } = 2δ ab . These matrices can be realized as 8 × 8 matrices with complex coefficients. The generators of so( 6 ) are represented by\n\nΓ ab = 1 2 [Γ a , Γ b ].\n\nWe often abbreviate the Yang-Mills fields as\n\nA μ = 1 2 A [ab]\n\nμ Γ ab and we also use notations with differential forms. Thus the gauge fields are expressed as A = A μdX μ. In this notation, the corresponding field strength F is written as F = dA + eA ∧ A, where e is a gauge coupling. Covariant derivative\n\nD μ on an adjoint representation Y = 1 2 Y [ab] Γ ab is defined as D μY = ∂ μY + e(A μY -Y A μ),\n\nwhere Y is a scalar multiplet. The action S total consists of two parts. One is the Einstein-Hilbert action S E and the other S Y M T is a Yang-Mills action with a term which is the fourth power of the field strength F . Such a quartic term has been studied by Tchrakian [10] and so we call it the Tchrakian term. The total action is:\n\nS total = S E + S Y M T , S E = 1 16πG dvR , S Y M T = 1 16 Tr -F ∧ * F + α 2 (F ∧ F ) ∧ * (F ∧ F ) -V 0 dv . (1)\n\nHere the 10-form dv is an invariant volume form with respect to the metric ĝ and is written as dv = √ -ĝd 10 X in a local patch. The scalar curvature is denoted by R. The asterisk \" * \" denotes the Hodge dual operator. This operator defines an inner product over differential forms, and for a given form ω, ω ∧ * ω is proportional to the invariant volume form dv. 1 The parameters of this action are the Newton's gravitational constant G, the gauge coupling e, the four-derivative coupling α and the cosmological constant V 0 .\n\nWe show the explicit form of the Yang-Mills part with components of A and F ,\n\nS Y M T = -dv 1 8 F [ab] μν F μν,[ab] + α 2 8 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 16 S μν ρσ S μν ρσ + 1 2 V 0 , (2)\n\nF = 1 4 F [ab] μν dX μ ∧ dX ν Γ ab , S μν ρσ = F [ab] μν F [ab] ρσ + F [ab] μρ F [ab] σ ν + F [ab] μσ F [ab] ν ρ , (3)\n\nT [ab][cd] μν ρσ = F [ab] μν F [cd] ρσ + F [ab] μρ F [cd] σ ν + F [ab] μσ F [cd] ν ρ , T [abcd] μν ρσ = 1 6 T [ab][cd]\n\nμν ρσ + T\n\n[ac][db] μν ρσ + T [ad][bc] μν ρσ + T [cd][ab] μν ρσ + T [db][ac] μν ρσ + T [bc][ad] μν ρσ .\n\nThe Euler-Lagrange equations from these actions read the usual Einstein equation and the equations for the Yang-Mills fields:\n\nR μν - 1 2 ĝμν R = 8πGT μν , D μ √ -gF μν -2α 2 √ -gF [μν F ρσ] F ρσ = 0 . (5)\n\nHere the energy-momentum tensor T μν is obtained by the variation of the Yang-Mills part with respect to the metric:\n\nT μν = 1 2 F μ ρ,[ab] F [ab] ν ρ + α 2 T [abcd] μρσ τ T ν ρστ ,[ab][cd] + α 2 3 • 2 S μρσ τ S ν ρσ τ - 1 2 g μν χ χ = 1 4 F [ab] μν F μν,[ab] + α 2 4 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 8 S μν ρσ S μν ρσ + V 0 . (6)\n\nTo solve these equations, we make an ansatz which is the same as that of Cremmer-Scherk.\n\nOur ansatz for the metric is the following:\n\nds 2 = η µν dx µ dx ν + δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J = ĝμν dX μdX ν , y 2 = 6 a=1 (y I ) 2 , ( 7\n\n)\n\n1 The Hodge dual operator acting on a differential form on a space with Minkowski signature satisfies the following relation:\n\n(F µν dx µν ) ∧ * (F ρσ dx ρσ ) = -F µν F µν dv.\n\nwhere the coordinates X are the total space-time coordinates. The metric η µν = diag(-+ ++) is the Lorentz metric on the four-dimensional Minkowski space. Greek indices without a hat \"ˆ\", for instance µ will refer to the first four variables. Capital indices (I, J, • • • ) run from one to six and refer to the compact space. The six-dimensional space is taken as a sphere with a radius R 0 . The Riemann tensor, Ricci tensor and scalar curvature are\n\nR I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ , R = 30 R 2 0 . (8)\n\nThe rest components of the curvature tensor vanish. In this space, the Einstein equations in (5) reduce to simple equations,\n\n- 1 2 η µν 30 R 2 0 = 8πGT µν , 0 = T µI , - 1 2 20 R 2 0 g IJ = 8πGT IJ . (9)\n\nWe now make ansatzes for the gauge fields. We assume that the fields A do not depend on the four-dimensional directions, ∂ µ A = 0, and they have no four-dimensional components A µ = 0. This implies that the field strengths are two forms on the six-dimensional sphere:\n\nA = A I (y)dy I , F = 1 2\n\nF IJ dy I ∧ dy J . With these ansatzes, the four-dimensional part of the energy-momentum tensor becomes -1 2 η µν χ, and the equation reduces to 30/R 2 0 = 8πGχ. This equation requires that the χ is a constant. Suppose that the field strength fulfils the generalized self-dual condition\n\nF = iβγ 7 * 6 (F ∧ F ), ( 10\n\n)\n\nwhere β is a real parameter. Here \" * 6 \" means the Hodge dual on the six-dimensional sphere.\n\nThen the second part of the equations of motion ( 5 ) is fulfilled automatically by the relation DF = 0, where the exterior covariant derivative is defined as DF = dF +e (A ∧ F -F ∧ A).\n\nIn fact we have an explicit solution to the self-dual equation:\n\nA = 1 4eR 2 0 y a e b Γ ab , F = 1 4eR 2 0 e a ∧ e b Γ ab , β = eR 2 0 3 . ( 11\n\n)\n\nHere we identify the internal space index and the sphere index. The energy-momentum tensor of this configuration becomes\n\nζ ≡ α 2 β 2 , χ = (1 + ζ) 15 4e 2 R 4 0 + V 0 , T IJ = - 1 2 (1 -ζ) 5 4e 2 R 4 0 + V 0 g IJ . (12)\n\nWith these ansatzes we obtain algebraic equations from the Einstein equations:\n\n30 R 2 0 = 8πG 1 2 (1 + ζ) 15 2e 2 R 4 0 + V 0 , 10 R 2 0 = 8πG (1 -ζ) 5 8e 2 R 4 0 + V 0 2 , (13)\n\nFrom these we finally obtain\n\n1 πG = 1 e 2 R 2 0 (2 + 4ζ) , V 0 = 15 4e 2 R 4 0 (1 + 3ζ) . (14)\n\nWhen the four-derivative coupling vanishes, α = 0 and therefore ζ = 0, these relations reduce to those of the Cremmer and Scherk [5] . 2 When the relation α = β holds (ζ = 1) our solution saturates the Bogomol'nyi bound and becomes a BPS state. The energy density is given by an integral over S 6 as follows:\n\nE S 6 Y M T = 1 16 S 6 Tr -F ∧ * 6 F + α 2 (F ∧ F ) ∧ * 6 (F ∧ F ) = 1 16 S 6 Tr (iF ∓ αγ 7 * 6 (F ∧ F )) ∧ * 6 (iF ∓ αγ 7 * 6 (F ∧ F )) ± i 8 α S 6 Trγ 7 F ∧ F ∧ F ≥ ± i 8 α S 6 Trγ 7 F ∧ F ∧ F = ∓ 1 2 3 S 6 ǫ abcdef F [ab] ∧ F [cd] ∧ F [ef ] ≡ ±Q , (15)\n\nwhere the field strength F has only components along S 6 . If the coupling α is equal to β, the solution of eq. ( 10 ) satisfies the Bogomol'nyi equation and the energy attains the local minimum. We can also consider a system coupled with scalar fields. Suppose that scalar fields Q m transform as a representation of SO (6) . The index m labels the representation space. Let us add an action S Q of the scalar fields Q with a Higgs potential\n\nS Q = 1 2 dvD μQ m D μQ m + V (Q 2 ) , D μQ m = ∂ μQ m - 1 2 ieA [ab] μ R(Γ ab ) mm ′ Q m ′ (16)\n\nto S total . The equations of motion are modified. In general, our solution mentioned above does not satisfy the modified equations any more. However, for the scalars which fulfil the covariantly constant condition D μQ m = 0 and attain the absolute minimum V (Q) = 0, the configurations of A and g in equations ( 7 ), (11) are still solutions for the modified equations. Here the constant value of the minimum is shifted to 0. Thus we can argue the Higgs mechanism around our solutions.\n\nNext we suppose that the four-dimensional part is an anti-de Sitter space AdS 4 of radius R A . Our ansatz for the metric is the following:\n\nds 2 = η µν (x)dx µ dx ν + g IJ (y)dy I dy J = ĝμν dX μdX ν , (17)\n\ng IJ (y)dy\n\nI dy J = δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J y 2 = 6 a=1 (y I ) 2 , η µν (x)dx µ dx ν = R 2 A cos 2 θ -dτ 2 + dθ 2 + sin 2 θdΩ 2 , dΩ 2 = |dz| 2 (1 + |z| 2 /4) 2 , (18)\n\nwhere z parametrizes a whole complex plane. The metric η µν (x) is a maximally symmetric metric on the four-dimensional anti-de Sitter space. The Riemann tensor and the Ricci tensor are\n\nR µ νρσ = - 1 R 2 A δ µ ρ η νσ -δ µ σ η νρ , R µν = - 3 R 2 A η µν , R I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ . (19)\n\nThe total scalar curvature is obtained by a summation of those of two parts: R = -\n\n12 R 2 A + 30 R 2 0 .\n\nIn this space, the Einstein equations are\n\nR µν - 1 2 η µν R = 8πGT µν , R IJ - 1 2 g IJ R = 8πGT IJ . (20)\n\nThe ansatz for the gauge fields is the same as previous one and the energy momentum tensor does not change. With these ansatzes, we obtain algebraic equations from the Einstein\n\nequations as 3 R 2 A - 15 R 2 0 = -4πG (1 + ζ) 15 4e 2 R 4 0 + V 0 , 6 R 2 A - 10 R 2 0 = -4πG (1 -ζ) 5 4e 2 R 4 0 + V 0 . (21)\n\nWe are interested in a possible relation to string theory and therefore we consider the case with the vanishing cosmological constant, V 0 = 0. In this case, the radii (R A , R 0 ) are written by the couplings,\n\nR 2 0 = (5 + 7ζ) πG 4e 2 , R 2 A = 5 + 7ζ 5 + 15ζ R 2 0 . (22)\n\nThus the additional higher derivative coupling term of the Tchrakian type does not affect critically to the equations of motion. When ζ = 1 our solution becomes a solution of the Bogomol'nyi equation again.\n\nOur solutions introduced in this letter are new solutions of the system with a Tchrakian term. The origin of this term has not been clear so far but it seems rather universal in order to construct solitons with codimensions higher than four: for instance it has played a crucial role to construct a finite energy monopole (with codimension five) in a six-dimensional spacetime [13] . Though the parameter ζ(= α 2 /β 2 ) is a free parameter, we expect that the system goes to ζ = 1 because it becomes BPS. There are several discussions on the (in)stability of higher-dimensional Yang-Mills theories [15] . To compute the mass spectra of the fluctuations around our solutions is a future work. When the scalar fields Q m are non-trivially coupled, the system may allow BPS composite solitons which are made of solitons with different codimensions, as in the case of usual self-dual Yang-Mills equations coupled to Higgs fields [16] .\n\nFinally our solution of AdS 4 × S 6 may have a relation with D2-branes, and we hope that there exists some impact on AdS/CFT duality [17] ."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term with respect to the field strength of the Yang-Mills field. The solution provides a compactification proposed by Cremmer and Scherk; ten-dimensional space-time with a cosmological constant is compactified to the four-dimensional Minkowski space with a six-dimensional sphere S 6 on which an instanton solution exists. The radius of the sphere is not a modulus but is determined by the gauge coupling and the four-derivative coupling constants and the Newton's constant. We also construct a solution of ten-dimensional theory without a cosmological constant compactified to AdS 4 × S 6 .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"Unification of fundamental forces with space-time and matter often requires higherdimensional space-time rather than our four-dimensional Universe. The early Kaluza-Klein theory unifies gravity and the electro-magnetic interaction by considering five-dimensional space-time with one direction compactified into a circle S 1 [1] . This old idea has been revisited several times. After supergravity was discovered many people tried to unify all forces and matter in higher-dimensional space-time compactified on various internal manifolds [2] .\\n\\nString theory was proposed as the most attractive candidate of unification, but it is defined only in ten-dimensional space-time. In order to realize four-dimensional Universe one has to find a suitable six-dimensional internal space. So many candidates of such spaces were proposed; Calabi-Yau manifolds and orbifold models. Internal manifolds can be deformed with satisfying the Einstein equation, and these degrees of freedom are called the moduli.\\n\\nThe moduli introduce unwanted massless particles in four-dimensional world. Recently a new mechanism has been suggested to fix these moduli by turning on the Ramond-Ramond flux on the internal space [3, 4] . This flux compactification has been extensively studied in these years.\\n\\nWe would like to revise the compactification scenario with fixed moduli proposed by Cremmer and Scherk long time ago [5] (see also [6, 7] ) in a theory with a cosmological constant. By placing solitons on a compact internal space they showed decompactifying limit with large radius of the internal space is disfavored and the radius is fixed to a certain value determined by coupling constants. They considered the 't Hooft-Polyakov monopole [8] on S 2 and the Yang-Mills instanton [9] on S 4 , both of which can satisfy, with proper coupling constants, the first order (self-dual) equations rather than the second order equations of motion, but their solutions on higher dimensional sphere are not the case. Since string theory is defined in ten dimensions, it is natural to consider this scenario with stable BPS solitons on a six-dimensional internal space like S 6 .\\n\\nHigher dimensional generalization of self-dual equations was suggested by Tchrakian some years ago [10] . Eight dimensional case is known as octonionic instantons [11] . Though several works have been done for generalized self-dual equations [12, 13] , a six-dimensional case was not discussed because of the lack of conformal property. Recently we have found a new solution to the generalized self-dual equations in an SO (6) pure Yang-Mills theory with a fourth order term with respect to the field strength of the Yang-Mills field (a four-derivative term) on a six-dimensional sphere S 6 [14] .\\n\\nIn this letter we propose to use this solution in the context of a compactification of the Cremmer-Scherk type. In our model ten-dimensional space-time with (without) a cosmological constant is compactified to a four-dimensional Minkowski space M 4 (anti de Sitter space AdS 4 ) with a six-dimensional sphere S 6 , where dimensionality of the internal space, six, is required by the four derivative term. Unlike the case of the absence of gravity [14] the four-derivative coupling constant α can differ from the constant β in the generalized self-dual equations. When the relation α = β holds the generalized self-dual equations become the Bogomol'nyi equations and solutions are BPS. We find for both M 4 × S 6 and AdS 4 × S 6 that certain relations exist between the radius of S 6 , the gauge coupling, the four-derivative coupling α and the gravitational coupling constants. When the four-derivative coupling constant α vanishes in the case of M 4 × S 6 , these relations reduce to those of the original work by Cremmer and Scherk. The advantage of our model to the Cremmer-Scherk model is that the Yang-Mills soliton in our model satisfies the self-dual equations (the Bogomol'nyi equations for α = β) rather than usual equations of motion in the case of the Cremmer-Scherk model. This ensures the stability of configuration at least for the sector of Yang-Mills fields.\\n\\nLet us consider that space-time is a ten-dimensional manifold. We consider an Einstein-Yang-Mills theory. Our action contains as dynamical variables the Yang-Mills (gauge) fields\\n\\nA [ab] μ\\n\\nand a graviton field or the metric ĝμν . Indices with a hat \\\"ˆ\\\" will refer to a tendimensional space-time (X 0 , X 1 , • • • , X 9 ). Latin indices (a, b, • • • ) run from 1 to 6 and refer to an internal space. The Clifford algebra associated with the orthogonal group SO( 6 ) is useful and we represent generators of the Lie algebra so(6) as their elements. The Clifford algebra is defined by gamma matrices {Γ a } which satisfy the following anti-commutation relations, {Γ a , Γ b } = 2δ ab . These matrices can be realized as 8 × 8 matrices with complex coefficients. The generators of so( 6 ) are represented by\\n\\nΓ ab = 1 2 [Γ a , Γ b ].\\n\\nWe often abbreviate the Yang-Mills fields as\\n\\nA μ = 1 2 A [ab]\\n\\nμ Γ ab and we also use notations with differential forms. Thus the gauge fields are expressed as A = A μdX μ. In this notation, the corresponding field strength F is written as F = dA + eA ∧ A, where e is a gauge coupling. Covariant derivative\\n\\nD μ on an adjoint representation Y = 1 2 Y [ab] Γ ab is defined as D μY = ∂ μY + e(A μY -Y A μ),\\n\\nwhere Y is a scalar multiplet. The action S total consists of two parts. One is the Einstein-Hilbert action S E and the other S Y M T is a Yang-Mills action with a term which is the fourth power of the field strength F . Such a quartic term has been studied by Tchrakian [10] and so we call it the Tchrakian term. The total action is:\\n\\nS total = S E + S Y M T , S E = 1 16πG dvR , S Y M T = 1 16 Tr -F ∧ * F + α 2 (F ∧ F ) ∧ * (F ∧ F ) -V 0 dv . (1)\\n\\nHere the 10-form dv is an invariant volume form with respect to the metric ĝ and is written as dv = √ -ĝd 10 X in a local patch. The scalar curvature is denoted by R. The asterisk \\\" * \\\" denotes the Hodge dual operator. This operator defines an inner product over differential forms, and for a given form ω, ω ∧ * ω is proportional to the invariant volume form dv. 1 The parameters of this action are the Newton's gravitational constant G, the gauge coupling e, the four-derivative coupling α and the cosmological constant V 0 .\\n\\nWe show the explicit form of the Yang-Mills part with components of A and F ,\\n\\nS Y M T = -dv 1 8 F [ab] μν F μν,[ab] + α 2 8 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 16 S μν ρσ S μν ρσ + 1 2 V 0 , (2)\\n\\nF = 1 4 F [ab] μν dX μ ∧ dX ν Γ ab , S μν ρσ = F [ab] μν F [ab] ρσ + F [ab] μρ F [ab] σ ν + F [ab] μσ F [ab] ν ρ , (3)\\n\\nT [ab][cd] μν ρσ = F [ab] μν F [cd] ρσ + F [ab] μρ F [cd] σ ν + F [ab] μσ F [cd] ν ρ , T [abcd] μν ρσ = 1 6 T [ab][cd]\\n\\nμν ρσ + T\\n\\n[ac][db] μν ρσ + T [ad][bc] μν ρσ + T [cd][ab] μν ρσ + T [db][ac] μν ρσ + T [bc][ad] μν ρσ .\\n\\nThe Euler-Lagrange equations from these actions read the usual Einstein equation and the equations for the Yang-Mills fields:\\n\\nR μν - 1 2 ĝμν R = 8πGT μν , D μ √ -gF μν -2α 2 √ -gF [μν F ρσ] F ρσ = 0 . (5)\\n\\nHere the energy-momentum tensor T μν is obtained by the variation of the Yang-Mills part with respect to the metric:\\n\\nT μν = 1 2 F μ ρ,[ab] F [ab] ν ρ + α 2 T [abcd] μρσ τ T ν ρστ ,[ab][cd] + α 2 3 • 2 S μρσ τ S ν ρσ τ - 1 2 g μν χ χ = 1 4 F [ab] μν F μν,[ab] + α 2 4 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 8 S μν ρσ S μν ρσ + V 0 . (6)\\n\\nTo solve these equations, we make an ansatz which is the same as that of Cremmer-Scherk.\\n\\nOur ansatz for the metric is the following:\\n\\nds 2 = η µν dx µ dx ν + δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J = ĝμν dX μdX ν , y 2 = 6 a=1 (y I ) 2 , ( 7\\n\\n)\\n\\n1 The Hodge dual operator acting on a differential form on a space with Minkowski signature satisfies the following relation:\\n\\n(F µν dx µν ) ∧ * (F ρσ dx ρσ ) = -F µν F µν dv.\\n\\nwhere the coordinates X are the total space-time coordinates. The metric η µν = diag(-+ ++) is the Lorentz metric on the four-dimensional Minkowski space. Greek indices without a hat \\\"ˆ\\\", for instance µ will refer to the first four variables. Capital indices (I, J, • • • ) run from one to six and refer to the compact space. The six-dimensional space is taken as a sphere with a radius R 0 . The Riemann tensor, Ricci tensor and scalar curvature are\\n\\nR I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ , R = 30 R 2 0 . (8)\\n\\nThe rest components of the curvature tensor vanish. In this space, the Einstein equations in (5) reduce to simple equations,\\n\\n- 1 2 η µν 30 R 2 0 = 8πGT µν , 0 = T µI , - 1 2 20 R 2 0 g IJ = 8πGT IJ . (9)\\n\\nWe now make ansatzes for the gauge fields. We assume that the fields A do not depend on the four-dimensional directions, ∂ µ A = 0, and they have no four-dimensional components A µ = 0. This implies that the field strengths are two forms on the six-dimensional sphere:\\n\\nA = A I (y)dy I , F = 1 2\\n\\nF IJ dy I ∧ dy J . With these ansatzes, the four-dimensional part of the energy-momentum tensor becomes -1 2 η µν χ, and the equation reduces to 30/R 2 0 = 8πGχ. This equation requires that the χ is a constant. Suppose that the field strength fulfils the generalized self-dual condition\\n\\nF = iβγ 7 * 6 (F ∧ F ), ( 10\\n\\n)\\n\\nwhere β is a real parameter. Here \\\" * 6 \\\" means the Hodge dual on the six-dimensional sphere.\\n\\nThen the second part of the equations of motion ( 5 ) is fulfilled automatically by the relation DF = 0, where the exterior covariant derivative is defined as DF = dF +e (A ∧ F -F ∧ A).\\n\\nIn fact we have an explicit solution to the self-dual equation:\\n\\nA = 1 4eR 2 0 y a e b Γ ab , F = 1 4eR 2 0 e a ∧ e b Γ ab , β = eR 2 0 3 . ( 11\\n\\n)\\n\\nHere we identify the internal space index and the sphere index. The energy-momentum tensor of this configuration becomes\\n\\nζ ≡ α 2 β 2 , χ = (1 + ζ) 15 4e 2 R 4 0 + V 0 , T IJ = - 1 2 (1 -ζ) 5 4e 2 R 4 0 + V 0 g IJ . (12)\\n\\nWith these ansatzes we obtain algebraic equations from the Einstein equations:\\n\\n30 R 2 0 = 8πG 1 2 (1 + ζ) 15 2e 2 R 4 0 + V 0 , 10 R 2 0 = 8πG (1 -ζ) 5 8e 2 R 4 0 + V 0 2 , (13)\\n\\nFrom these we finally obtain\\n\\n1 πG = 1 e 2 R 2 0 (2 + 4ζ) , V 0 = 15 4e 2 R 4 0 (1 + 3ζ) . (14)\\n\\nWhen the four-derivative coupling vanishes, α = 0 and therefore ζ = 0, these relations reduce to those of the Cremmer and Scherk [5] . 2 When the relation α = β holds (ζ = 1) our solution saturates the Bogomol'nyi bound and becomes a BPS state. The energy density is given by an integral over S 6 as follows:\\n\\nE S 6 Y M T = 1 16 S 6 Tr -F ∧ * 6 F + α 2 (F ∧ F ) ∧ * 6 (F ∧ F ) = 1 16 S 6 Tr (iF ∓ αγ 7 * 6 (F ∧ F )) ∧ * 6 (iF ∓ αγ 7 * 6 (F ∧ F )) ± i 8 α S 6 Trγ 7 F ∧ F ∧ F ≥ ± i 8 α S 6 Trγ 7 F ∧ F ∧ F = ∓ 1 2 3 S 6 ǫ abcdef F [ab] ∧ F [cd] ∧ F [ef ] ≡ ±Q , (15)\\n\\nwhere the field strength F has only components along S 6 . If the coupling α is equal to β, the solution of eq. ( 10 ) satisfies the Bogomol'nyi equation and the energy attains the local minimum. We can also consider a system coupled with scalar fields. Suppose that scalar fields Q m transform as a representation of SO (6) . The index m labels the representation space. Let us add an action S Q of the scalar fields Q with a Higgs potential\\n\\nS Q = 1 2 dvD μQ m D μQ m + V (Q 2 ) , D μQ m = ∂ μQ m - 1 2 ieA [ab] μ R(Γ ab ) mm ′ Q m ′ (16)\\n\\nto S total . The equations of motion are modified. In general, our solution mentioned above does not satisfy the modified equations any more. However, for the scalars which fulfil the covariantly constant condition D μQ m = 0 and attain the absolute minimum V (Q) = 0, the configurations of A and g in equations ( 7 ), (11) are still solutions for the modified equations. Here the constant value of the minimum is shifted to 0. Thus we can argue the Higgs mechanism around our solutions.\\n\\nNext we suppose that the four-dimensional part is an anti-de Sitter space AdS 4 of radius R A . Our ansatz for the metric is the following:\\n\\nds 2 = η µν (x)dx µ dx ν + g IJ (y)dy I dy J = ĝμν dX μdX ν , (17)\\n\\ng IJ (y)dy\\n\\nI dy J = δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J y 2 = 6 a=1 (y I ) 2 , η µν (x)dx µ dx ν = R 2 A cos 2 θ -dτ 2 + dθ 2 + sin 2 θdΩ 2 , dΩ 2 = |dz| 2 (1 + |z| 2 /4) 2 , (18)\\n\\nwhere z parametrizes a whole complex plane. The metric η µν (x) is a maximally symmetric metric on the four-dimensional anti-de Sitter space. The Riemann tensor and the Ricci tensor are\\n\\nR µ νρσ = - 1 R 2 A δ µ ρ η νσ -δ µ σ η νρ , R µν = - 3 R 2 A η µν , R I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ . (19)\\n\\nThe total scalar curvature is obtained by a summation of those of two parts: R = -\\n\\n12 R 2 A + 30 R 2 0 .\\n\\nIn this space, the Einstein equations are\\n\\nR µν - 1 2 η µν R = 8πGT µν , R IJ - 1 2 g IJ R = 8πGT IJ . (20)\\n\\nThe ansatz for the gauge fields is the same as previous one and the energy momentum tensor does not change. With these ansatzes, we obtain algebraic equations from the Einstein\\n\\nequations as 3 R 2 A - 15 R 2 0 = -4πG (1 + ζ) 15 4e 2 R 4 0 + V 0 , 6 R 2 A - 10 R 2 0 = -4πG (1 -ζ) 5 4e 2 R 4 0 + V 0 . (21)\\n\\nWe are interested in a possible relation to string theory and therefore we consider the case with the vanishing cosmological constant, V 0 = 0. In this case, the radii (R A , R 0 ) are written by the couplings,\\n\\nR 2 0 = (5 + 7ζ) πG 4e 2 , R 2 A = 5 + 7ζ 5 + 15ζ R 2 0 . (22)\\n\\nThus the additional higher derivative coupling term of the Tchrakian type does not affect critically to the equations of motion. When ζ = 1 our solution becomes a solution of the Bogomol'nyi equation again.\\n\\nOur solutions introduced in this letter are new solutions of the system with a Tchrakian term. The origin of this term has not been clear so far but it seems rather universal in order to construct solitons with codimensions higher than four: for instance it has played a crucial role to construct a finite energy monopole (with codimension five) in a six-dimensional spacetime [13] . Though the parameter ζ(= α 2 /β 2 ) is a free parameter, we expect that the system goes to ζ = 1 because it becomes BPS. There are several discussions on the (in)stability of higher-dimensional Yang-Mills theories [15] . To compute the mass spectra of the fluctuations around our solutions is a future work. When the scalar fields Q m are non-trivially coupled, the system may allow BPS composite solitons which are made of solitons with different codimensions, as in the case of usual self-dual Yang-Mills equations coupled to Higgs fields [16] .\\n\\nFinally our solution of AdS 4 × S 6 may have a relation with D2-branes, and we hope that there exists some impact on AdS/CFT duality [17] .\"\n }\n]"}}},{"rowIdx":91,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0507"},"arxiv_id":{"kind":"string","value":"0704.0507"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.76.124023"},"content_hash":{"kind":"string","value":"22f44bb72e223a65c9213c7d3684c85f57dc3b9bd321cac32466088966d0a47e"},"title":{"kind":"string","value":"E_6 and the bipartite entanglement of three qutrits"},"abstract":{"kind":"string","value":"Recent investigations have established an analogy between the entropy of\nfour-dimensional supersymmetric black holes in string theory and entanglement\nin quantum information theory. Examples include: (1) N=2 STU black holes and\nthe tripartite entanglement of three qubits (2-state systems), where the common\nsymmetry is [SL(2)]^3 and (2) N=8 black holes and the tripartite entanglement\nof seven qubits where the common symmetry is E_7 which contains [SL(2)]^7. Here\nwe present another example: N=8 black holes (or black strings) in five\ndimensions and the bipartite entanglement of three qutrits (3-state systems),\nwhere the common symmetry is E_6 which contains [SL(3)]^3. Both the black hole\n(or black string) entropy and the entanglement measure are provided by the\nCartan cubic E_6 invariant. Similar analogies exist for ``magic'' N=2\nsupergravity black holes in both four and five dimensions."},"authors":{"kind":"list like","value":["M. J. Duff and S. Ferrara"],"string":"[\n \"M. J. Duff and S. Ferrara\"\n]"},"arxiv_categories":{"kind":"list like","value":["hep-th","quant-ph"],"string":"[\n \"hep-th\",\n \"quant-ph\"\n]"},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"1 m.duff@imperial.ac.uk 2 Sergio.Ferrara@cern.ch Contents 1 D = 4 black holes and qubits 1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 1.2 N = 2 black holes and the bipartite entanglement of two qubits . . . . . . . 1.3 N = 8 black holes and the tripartite entanglement of seven qubits . . . . . . 1.4 Magic supergravities in D = 4 . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n2 Five-dimensional supergravity 3 D = 5 black holes and qutrits 3.1 N = 2 black holes and the bipartite entanglement of two qutrits . . . . . . . 3.2 N = 8 black holes and the bipartite entanglement of three qutrits . . . . . . 3.3 Magic supergravities in D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n4 Conclusions 5 Acknowledgements 2 1 D = 4 black holes and qubits\n\nIt sometimes happens that two very different areas of theoretical physics share the same mathematics. This may eventually lead to the realisation that they are, in fact, dual descriptions of the same physical phenomena, or it may not. Either way, it frequently leads to new insights in both areas. Recent papers [1, 2, 3, 4, 5, 6] have established an analogy between the entropy of certain four-dimensional supersymmetric black holes in string theory and entanglement measures in quantum information theory. In this paper we extend the analogy from four dimensions to five which also involves going from two-state systems (qubits) to three-state systems (qutrits). We begin by recalling the four-dimensional examples:\n\n1.1 N = 2 black holes and the tripartite entanglement of three qubits\n\nThe three qubit system (Alice, Bob, Charlie) is described by the state\n\n|Ψ = a ABC |ABC (1.1)\n\nwhere A = 0, 1, so the Hilbert space has dimension 2 3 = 8. The complex numbers a ABC transforms as a (2, 2, 2) under SL(2, C) A × SL(2, C) B × SL(2, C) C . The tripartite entanglement is measured by the 3-tangle [7, 8]\n\nτ 3 (ABC) = 4|Det a ABC |. (1.2)\n\nwhere Det a ABC is Cayley's hyperdeterminant [9] .\n\nDet a = - 1 2 ǫ A 1 A 2 ǫ B 1 B 2 ǫ A 3 A 4 ǫ B 3 B 4 ǫ C 1 C 4 ǫ C 2 C 3 a A 1 B 1 C 1 a A 2 B 2 C 2 a A 3 B 3 C 3 a A 4 B 4 C 4 (1.\n\n3) The hyperdeterminant is invariant under SL(2) A × SL(2) B × SL(2) C and under a triality that interchanges A, B and C. In the context of stringy black holes the 8 a ABC are the 4 electric and 4 magnetic charges of the N = 2 STU black hole [10] and hence take on real (integer) values. The ST U model corresponds to N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . The Bekenstein-Hawking entropy of the black hole, S, was first calculated in [11] . The connection to quantum information theory arises by noting [1] that it can also be expressed in terms of Cayley's hyperdeterminant\n\nS = π |Det a ABC |.\n\n(1.4) One can then establish a dictionary between the classification of various entangled states (separable A-B-C; bipartite entangled A-BC, B-CA, C-AB; tripartite entangled W; tripartite entangled GHZ) and the classfication of various \"small\" and \"large\" BPS and non-BPS black holes [1, 2, 3, 4, 5, 6] . For example, the GHZ state [12] |Ψ ∼ |111 + |000 (1.5) 3 with Det a ABC ≥ 0 corresponds to a large non-BPS 2-charge black hole; the W-state |Ψ ∼ |100 + |010 + |001 (1.6) with Det a ABC = 0 corresponds to a small-BPS 3-charge black hole; the GHZ-state\n\n|Ψ = -|000 + |011 + |101 + |110 (1.7)\n\ncorresponds to a large BPS 4-charge black hole.\n\n1.2 N = 2 black holes and the bipartite entanglement of two qubits\n\nAn even simpler example [2] is provided by the two qubit system (Alice and Bob) described by the state |Ψ = a AB |AB (1.8) where A = 0, 1, and the Hilbert space has dimension 2 2 = 4. The a AB transforms as a (2, 2) under SL(2, C) A × SL(2, C) B . The entanglement is measured by the 2-tangle τ 2 (AB) = C 2 (AB) (1.9) where C(AB) = 2 |det a AB | (1.10) is the concurrence. The determinant is invariant under SL(2, C) A × SL(2, C) B and under a duality that interchanges A and B. Here it is sufficient to look at N = 2 supergravity coupled to just one vector multiplet and the 4 a AB are the 2 electric and 2 magnetic charges of the axion-dilaton black hole with entropy S = π|det a AB | (1.11) For example, the Bell state |Ψ ∼ |11 + |00 (1.12) with det a AB ≥ 0 corresponds to a large non-BPS 2-charge black hole.\n\nWe recall that in the case of D = 4, N = 8 supergravity, the the 28 electric and 28 magnetic charges belong to the 56 of E 7 (7) . The black hole entropy is [15, 18] (1.13) where J 4 is Cartan's quartic E 7 invariant [13, 14] . It may be written\n\nS = π |J 4 |\n\nJ 4 = P ij Q jk P kl Q li -1 4 P ij Q ij P kl Q kl + 1 96 ǫ ijklmnop Q ij Q kl Q mn Q op + ǫ ijklmnop P ij P kl P mn P op . (1.14) 4\n\nwhere P ij and Q jk are 8 × 8 antisymmetric matrices. The qubit interpretation [4] relies on the decomposition E 7 (C) ⊃ [SL(2, C)] 7 (1.15) under which 56 → 2, 2, 1, 2, 1, 1, 1) + (1, 2, 2, 1, 2, 1, 1 1, 1, 2, 2, 1, 2, 1) +(1, 1, 1, 2, 2, 1, 2) +(2, 1, 1, 1, 2, 2, 1) + (1, 2, 1, 1, 1, 2, 2 2, 1, 2, 1, 1, 1, 2) (1.16) suggesting the tripartite entanglement of seven qubits (Alice, Bob, Charlie, Daisy, Emma, Fred and George) described by the state.\n\n(\n\n) +(\n\n) +(\n\n|Ψ = a ABD |ABD +b BCE |BCE +c CDF |CDF +d DEG |DEG +e EF A |EF A +f F GB |F GB +g GAC |GAC (1.17)\n\nwhere A = 0, 1, so the Hilbert space has dimension 7.2 3 = 56. The a, b, c, d, e, f, g transform as a 56 of E 7 (C). The entanglement may be represented by a heptagon where the vertices A,B,C,D,E,F,G represent the seven qubits and the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC represent the tripartite entanglement. See Figure 1 . Alternatively, we can use the Fano plane. See Figure 2 . The Fano plane also corresponds to the multiplication table of the octonions 3 The measure of the tripartite entanglement of the seven qubits is provided by the 3-tangle\n\nτ 3 (ABCDEF G) = 4|J 4 | (1.18) with J 4 ∼ a 4 + b 4 + c 4 + d 4 + e 4 + f 4 + g 4 +\n\n3 Not the \"split\" octonions as was incorrectly stated in the published version of [4].\n\nA B C D E F G Figure 1: The E 7 entanglement diagram. Each of the seven vertices A,B,C,D,E,F,G represents a qubit and each of the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC describes a tripartite entanglement.\n\n2[a 2 b 2 + b 2 c 2 + c 2 d 2 + d 2 e 2 + e 2 f 2 + f 2 g 2 + g 2 a 2 + a 2 c 2 + b 2 d 2 + c 2 e 2 + d 2 f 2 + e 2 g 2 + f 2 a 2 + g 2 b 2 + a 2 d 2 + b 2 e 2 + c 2 f 2 + d 2 g 2 + e 2 a 2 + f 2 b 2 + g 2 c 2 ] +8[bcdf + cdeg + def a + ef gb + f gac + gabd + abce] (1.19)\n\nwhere products like a 4 = (ABD)(ABD)(ABD)(ABD)\n\n= ǫ A 1 A 2 ǫ B 1 B 2 ǫ D 1 D 4 ǫ A 3 A 4 ǫ B 3 B 4 ǫ D 2 D 3 a A 1 B 1 D 1 a A 2 B 2 D 2 a A 3 B 3 D 3 a A 4 B 4 D 4 (1.20)\n\nexclude four individuals (here Charlie, Emma, Fred and George), products like\n\na 2 b 2 = (ABD)(ABD)(F GB)(F GB) = ǫ A 1 A 2 ǫ B 1 B 3 ǫ D 1 D 2 ǫ F 3 F 4 ǫ G 3 G 4 ǫ B 2 B 4 a A 1 B 1 D 1 a A 2 B 2 D 2 b F 3 G 3 B 3 b F 4 G 4 B 4 (1.21)\n\nexclude two individuals (here Charlie and Emma), and products like\n\nabce = (ABD)(BCE)(CDF )(EF A) = ǫ A 1 A 4 ǫ B 1 B 2 ǫ C 2 C 3 ǫ D 1 D 3 ǫ E 2 E 4 ǫ F 3 F 4 a A 1 B 1 D 1 b B 2 C 2 E 2 c C 3 D 3 F 3 e E 4 F 4 A 4 (1.22)\n\nexclude one individual (here George) 4 . Once again large non-BPS, small BPS and large BPS black holes correspond to states with J 4 > 0, J 4 = 0 and J 4 < 0, respectively.\n\n4 This corrects the corresponding equation in the published version of [4] which had the wrong index contraction.\n\nF E B G C A D\n\nFigure 2: The Fano plane has seven points, representing the seven qubits, and seven lines (the circle counts as a line) with three points on every line, representing the tripartite entanglement, and three lines through every point.\n\nThe black holes described by Cayley's hyperdeterminant are those of N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . In [4] the following four-dimensional generalizations were considered: 1) N = 2 supergravity coupled to l vector multiplets where the symmetry is SL(2, Z) × SO(l -1, 2, Z) and the black holes carry charges belonging to the (2, l + 1) representation (l + 1 electric plus l + 1 magnetic).\n\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SL(2, Z) × SO(6, m, Z) where the black holes carry charges belonging to the (2, 6 + m) representation (m + 6 electric plus m + 6 magnetic).\n\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 7(7) (Z) and the black holes carry charges belonging to the fundamental 56-dimensional representation (28 electric plus 28 magnetic).\n\nIn all three case there exist quartic invariants akin to Cayley's hyperdeterminant whose square root yields the corresponding black hole entropy. In [4] we succeeded in giving a quantum theoretic interpretation in the N = 8 case together with its truncations to N = 4 (with m = 6) and N = 2 (with l = 3, the case we already knew [1]).\n\nHowever, as suggested by Levay [5], one might also consider the \"magic\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 4 supergravity coupled to 6, 9, 15 and 27 vector multiplets with symmetries Sp(6, Z), SU(3, 3), SO * (12) and E 7(-25) , respectively. Once again, as has been shown just recently [20] , in all cases there are quartic invariants whose square root yields the corresponding black hole entropy.\n\nHere we demonstrate that the black-hole/qubit correspondence does indeed continue to 7 hold for magic supergravities. The crucial observation is that, although the black hole charges a ABC are real (integer) numbers and the entropy (1.13) is invariant under E 7 (7)(Z), the coefficients a ABC that appear in the qutrit state (1.17) are complex. So the three tangle (1.18) is invariant under E 7 (C) which contains both E 7(7) (Z) and E 7(-25) (Z) as subgroups.\n\nTo find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\n\nE 7(7) (Z) ⊃ [SL(2)(Z)] 7 (1.23) but E 7(-25) (Z) ⊃ [SL(2)(Z)] 7 (1.24)\n\nis irrelevant. All that matters is that\n\nE 7 (C) ⊃ [SL(2)(C)] 7 (1.25)\n\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic) . Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) above. When we looked at the seven qubit subsector E 7 (C) ⊃ SL(2, C) × SO(12, C), we gave an N = 4 supergravity interpretationwith symmetry SL(2, R)×SO(6, 6) [4], but we could equally have given an interpretation in terms of N = 2 supergravity coupled to 11 vector multiplets with symmetry SL(2, R) × SO(10, 2).\n\nMoreover, SO(l -1, 2) is contained in SO(l + 1, C) and SO(6, m) is contained in SO(12 + m, C) so we can give a qubit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(4n, C) which contains [SL(2, C)] 2n .\n\nIn five dimensions we might consider: 1) N = 2 supergravity coupled to l+1 vector multiplets where the symmetry is SO(1, 1, Z)× SO(l, 1, Z) and the black holes carry charges belonging to the (l + 1) representation (all electric) .\n\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SO(1, 1, Z)× SO(m, 5, Z) where the black holes carry charges belonging to the (m + 5) representation (all electric).\n\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 6(6) (Z) and the black holes carry charges belonging to the fundamental 27-dimensional representation (all electric).\n\nThe electrically charged objects are point-like and the magnetic duals are one-dimensional, or string-like, transforming according to the contrgredient representation. In all three cases above there exist cubic invariants akin to the determinant which yield the corresponding black hole or black string entropy.\n\nIn this section we briefly describe the salient properties of maximal N = 8 case, following [16] . We have 27 abelian gauge fields which transform in the fundamental representation of 8 6(6) . The first invariant of E 6(6) is the cubic invariant [13, 17, 16, 18, 19]\n\nE\n\nJ 3 = q ij Ω jl q lm Ω mn q np Ω pi (2.1)\n\nwhere q ij is the charge vector transforming as a 27 which can be represented as traceless Sp(8) matrix. The entropy of a black hole with charges q ij is then given by\n\nS = π |J 3 | (2.2)\n\nWe will see that a configuration with J 3 = 0 preserves 1/8 of the supersymmetries. If J 3 = 0 and ∂J 3 ∂q i = 0 then it preserves 1/4 of the supersymmetries, and finally if ∂J 3 ∂q i = 0 (and the charge vector q i is non-zero), the configuration preserves 1/2 of the supersymmetries. We will show this by choosing a particular basis for the charges, the general result following by U-duality.\n\nIn five dimensions the compact group H is USp(8). We choose our conventions so that USp(2) = SU(2). In the commutator of the supersymmetry generators we have a central charge matrix Z ab which can be brought to a normal form by a USp(8) transformation. In the normal form the central charge matrix can be written as\n\ne ab = s 1 + s 2 -s 3 0 0 0 0 s 1 + s 3 -s 2 0 0 0 0 s 2 + s 3 -s 1 0 0 0 0 -(s 1 + s 2 + s 3 ) × 0 1 -1 0 (2.3) we can order s i so that s 1 ≥ s 2 ≥ |s 3 |.\n\nThe cubic invariant, in this basis, becomes\n\nJ 3 = s 1 s 2 s 3 (2.4)\n\nEven though the eigenvalues s i might depend on the moduli, the invariant (2.4) only depends on the quantized values of the charges. We can write a generic charge configuration as UeU t , where e is the normal frame as above, and the invariant will then be (2.4). There are three distinct possibilities\n\nJ 3 = 0 s 1 , s 2 , s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 , s 2 = 0, s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 = 0, s 2 , s 3 = 0 (2.5)\n\nTaking the case of type II on T 5 we can choose the rotation in such a way that, for example, s 1 corresponds to solitonic five-brane charge, s 2 to fundamental string winding charge along some direction and s 3 to Kaluza-Klein momentum along the same direction. We can see that in this specific example the three possibilities in (2.5) break 1/8, 1/4 and 1/2 supersymmetries. The respective orbits are\n\nE 6 (6) F 4(4) E 6 (6) SO(5, 4) ×T 16 9 E 6 (6) SO(5, 5) ×T 16 (2.6)\n\nThis also shows that one can generically choose a basis for the charges so that all others are related by U-duality. The basis chosen here is the S-dual of the D-brane basis usually chosen for describing black holes in type II B on T 5 . All others are related by U-duality to this particular choice. Note that, in contrast to the four-dimensional case where flipping the sign of J 4 (1.14) interchanges BPS and non-BPS black holes, the sign of the J 3 (2.4) is not important since it changes under a CPT transformation. There is no non-BPS orbit in five dimensions.\n\nIn five dimensions there are also string-like configurations which are the magnetic duals of the configurations considered here. They transform in the contragredient 27 ′ representation and the solutions preserving 1/2, 1/4, 1/8 supersymmetries are characterized in an analogous way. We could also have configurations where we have both point-like and string-like ch the point-like charge is uniformly distributed along the string, it is more natural to consider this configuration as a point-like object in D = 4 by dimensional reduction.\n\nIt is useful to decompose the U-duality group into the T-duality group and the S-duality group. The decomposition reads E 6 → SO(5, 5) × SO(1, 1), leading to\n\n27 → 16 1 + 10 -2 + 1 4 (2.7)\n\nThe last term in (2.7) corresponds to the NS five-brane charge. The 16 correspond to the D-brane charges and the 10 correspond to the 5 directions of KK momentum and the 5 directions of fundamental string winding, which are the charges that explicitly appear in string perturbation theory. The cubic invariant has the decomposition (27) 3 → 10 -2 10 -2 1 4 + 16 1 16 1 10 -2 (2.8) This is saying that in order to have a non-zero area black hole we must have three NS charges (more precisely some \"perturbative\" charges and a solitonic five-brane); or we can have two D-brane charges and one NS charge. In particular, it is not possible to have a black hole with a non-zero horizon area with purely D-brane charges. Notice that the non-compact nature of the groups is crucial in this classification.\n\nSo far, all the quantum information analogies involve four-dimensional black holes and qubits. In order to find an analogy with five-dimensional black holes we invoke three state systems called qutrits.\n\nThe two qutrit system (Alice and Bob) is described by the state\n\n|Ψ = a AB |AB 10\n\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3 2 = 9. The a AB transforms as a (3, 3) under SL(3) A × SL(3) B . The bipartite entanglement is measured by the concurrence [21]\n\nC(AB) = 3 3/2 |det a AB |. (3.1)\n\nThe determinant is invariant under SL(3, C) A × SL(3, C) B and under a duality that interchanges A and B. The black hole interpretation is provided by N = 2 supergravity coupled to 8 vector multiplets with symmetry SL(3, C) where the black hole charges transform as a 9. The entropy is given by\n\nS = π|det a AB | (3.2)\n\n3.2 N = 8 black holes and the bipartite entanglement of three qutrits\n\nAs we have seen in section (2) in the case of D = 5, N = 8 supergravity, the black hole charges belong to the 27 of E 6(6) and the entropy is given by (2.2).\n\nThe qutrit interpretation now relies on the decomposition\n\nE 6 (C) ⊃ SL(3, C) A × SL(3, C) B × SL(3, C) C (3.3) under which 27 → (3, 3, 1) + (3 ′ , 1, 3) + (1, 3 ′ , 3 ′ ) (3.4)\n\nsuggesting the bipartite entanglement of three qutrits (Alice, Bob, Charlie). However, the larger symmetry requires that they undergo at most bipartite entanglement of a very specific kind, where each person has bipartite entanglement with the other two:\n\n|Ψ = a AB |AB + b B C |BC + c CA |CA (3.5)\n\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3.3 2 = 27. The three states transforms as a pair of triplets under two of the SL(3)'s and singlets under the remaining one. Individually, therefore, the bipartite entanglement of each of the three states is given by the determinant (3.1). Taken together however, we see from (3.4) that they transform as a complex 27 of E 6 (C). The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements. See Fig. 3. The N=2 truncation of section 3.1 is represented by just the line AB with endpoints A and B. Note that: 1) Any pair of states has an individual in common 2) Each individual is excluded from one out of the three states The entanglement measure will be given by the concurrence\n\nC(ABC) = 3 3/2 |J 3 | (3.6)\n\nJ 3 being the singlet in 27 × 27 × 27:\n\nJ 3 ∼ a 3 + b 3 + c 3 + 6abc (3.7) 11 A B C\n\nFigure 3: The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements.\n\nwhere the products\n\na 3 = ǫ A 1 A 2 A 3 ǫ B 1 B 2 B 3 a A 1 B 1 a A 2 B 2 a A 3 B 3 (3.8) b 3 = ǫ B 1 B 2 B 3 ǫ C 1 C 2 C 3 b B 1 C 1 b B 2 C 2 b B 3 C 3 (3.9) c 3 = ǫ C 1 C 2 C 3 ǫ A 1 A 2 A 3 c C 1 A 1 c C 2 A 2 c C 3 A 3 (3.10)\n\nexclude one individual (Charlie, Alice, and Bob respectively), and the product abc = a AB b B C c CA (3.11) excludes none.\n\nJust as in four dimensions, one might also consider the \"magic\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 5 supergravity coupled to 5, 8, 14 and 26 vector multiplets with symmetries SL(3, R), SL(3, C), SU * (6) and E 6(-26) respectively. Once again, in all cases there are cubic invariants whose square root yields the corresponding black hole entropy [20] . Here we demonstrate that the black-hole/qubit correspondence continue to hold for these D = 5 magic supergravities, as well as D = 4 . Once again, the crucial observation is that, although the black hole charges a AB are real (integer) numbers and the entropy (2.2) is invariant under E 6(6) (Z), the coefficients a AB that appear in the wave function (3.5) are complex. So the 2-tangle (3.6) is invariant under E 6 (C) which contains both E 6(6) (Z) and E 6(-26) (Z) as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\n\nE 6(6) (Z) ⊃ [SL(3)(Z)] 3 (3.12) 12 but E 6(-26) (Z) ⊃ [SL(3)(Z)] 3 ( 3\n\n.13) is irrelevant. All that matters is that\n\nE 6 (C) ⊃ [SL(3)(C)] 3 (3.14)\n\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic). In fact, the example of section 3.1 corresponds to the complex case. Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) of section (2). SO(l, 1) is contained in SO(l + 1, C) and SO(m, 5) is contained in SO(5 + m, C), so we can give a qutrit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(6n, C) which contains [SL(3, C)] n .\n\nWe note that the 27-dimensional Hilbert space given in (3.4) and (3.5) is not a subspace of the 3 3 -dimensional three qutrit Hilbert space given by (3, 3, 3) , but rather a direct sum of three 3 2 -dimensional Hilbert spaces. It is, however, a subspace of the 7 3 -dimensional three 7-dit Hilbert space given by (7, 7, 7) . Consider the decomposition 3, 3, 1) + (3, 1, 3) + (1, 3, 3) 1, 1, 1) This contains the subspace that describes the bipartite entanglement of three qutrits, namely\n\nSL(7) A × SL(7) B × SL(7) C → SL(3) A × SL(3) B × SL(3) C under which (7, 7, 7) → (3 ′ , 3 ′ , 3 ′ ) + (3 ′ , 3 ′ , 3) + (3 ′ , 3, 3 ′ ) + (3, 3 ′ , 3 ′ ) + (3 ′ , 3, 3) + (3, 3 ′ , 3) + (3, 3, 3 ′ ) + (3, 3, 3) +(3 ′ , 3 ′ , 1) + (3 ′ , 1, 3 ′ ) + (1, 3 ′ , 3 ′ ) + (3 ′ , 1, 3) + (3 ′ , 3, 1) + (1, 3, 3 ′ ) +(\n\n+ (3, 1, 3 ′ ) + (3, 3 ′ , 1) + (1, 3 ′ , 3) +(3 ′ , 1, 1) + (1, 3 ′ , 1) + (1, 1, 3 ′ ) + (3, 1, 1) + (1, 3, 1) + (1, 1, 3) +(\n\n(3 ′ , 3, 1) + (3, 1, 3) + (1, 3 ′ , 3 ′ )\n\nSo the triangle entanglement we have described fits within conventional quantum information theory. Our analogy between black holes and quantum information remains, for the moment, just that. We know of no physics connecting them.\n\nNevertheless, just as the exceptional group E 7 describes the tripartite entanglement of seven qubits [4, 5] , we have seen is this paper that the exceptional group E 6 describes the bipartite entanglement of three qutrits. In the E 7 case, the quartic Cartan invariant provides both the measure of entanglement and the entropy of the four-dimensional N = 8 black hole, 13 whereas in the E 6 case, the cubic Cartan invariant provides both the measure of entanglement and the entropy of the five-dimensional N = 8 black hole.\n\nMoreover, we have seen that similar analogies exist not only for the N = 4 and N = 2 truncations, but also for the magic N = 2 supergravities in both four and five dimensions (In the four-dimensional case, this had previously been conjectured by Levay[4, 5]). Murat Gunaydin has suggested (private communication) that the appearance of octonions implies a connection to quaternionic and/or octonionic quantum mechanics. This was not apparent (at least to us) in the four-dimensional N = 8 case [4], but the appearance in the five dimensional magic N = 2 case of SL(3, R), SL(3, C), SL(3, H) and SL(3, O) is more suggestive.\n\nMJD has enjoyed useful conversations with Leron Borsten, Hajar Ebrahim, Chris Hull, Martin Plenio and Tony Sudbery. This work was supported in part by the National Science Foundation under grant number PHY-0245337 and PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The work of S.F. has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 Constituents, fundamental forces and symmetries of the universe, in association with INFN Frascati National Laboratories and by the D.O.E grant DE-FG03-91ER40662, Task C. The work of MJD is supported in part by PPARC under rolling grant PPA/G/O/2002/00474, PP/D50744X/1."},"sections":{"kind":"list like","value":[{"section_type":"OTHER","section_title":"Untitled Section","text":"1 m.duff@imperial.ac.uk 2 Sergio.Ferrara@cern.ch Contents 1 D = 4 black holes and qubits 1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 1.2 N = 2 black holes and the bipartite entanglement of two qubits . . . . . . . 1.3 N = 8 black holes and the tripartite entanglement of seven qubits . . . . . . 1.4 Magic supergravities in D = 4 . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n2 Five-dimensional supergravity 3 D = 5 black holes and qutrits 3.1 N = 2 black holes and the bipartite entanglement of two qutrits . . . . . . . 3.2 N = 8 black holes and the bipartite entanglement of three qutrits . . . . . . 3.3 Magic supergravities in D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n4 Conclusions 5 Acknowledgements 2 1 D = 4 black holes and qubits\n\nIt sometimes happens that two very different areas of theoretical physics share the same mathematics. This may eventually lead to the realisation that they are, in fact, dual descriptions of the same physical phenomena, or it may not. Either way, it frequently leads to new insights in both areas. Recent papers [1, 2, 3, 4, 5, 6] have established an analogy between the entropy of certain four-dimensional supersymmetric black holes in string theory and entanglement measures in quantum information theory. In this paper we extend the analogy from four dimensions to five which also involves going from two-state systems (qubits) to three-state systems (qutrits). We begin by recalling the four-dimensional examples:\n\n1.1 N = 2 black holes and the tripartite entanglement of three qubits\n\nThe three qubit system (Alice, Bob, Charlie) is described by the state\n\n|Ψ = a ABC |ABC (1.1)\n\nwhere A = 0, 1, so the Hilbert space has dimension 2 3 = 8. The complex numbers a ABC transforms as a (2, 2, 2) under SL(2, C) A × SL(2, C) B × SL(2, C) C . The tripartite entanglement is measured by the 3-tangle [7, 8]\n\nτ 3 (ABC) = 4|Det a ABC |. (1.2)\n\nwhere Det a ABC is Cayley's hyperdeterminant [9] .\n\nDet a = - 1 2 ǫ A 1 A 2 ǫ B 1 B 2 ǫ A 3 A 4 ǫ B 3 B 4 ǫ C 1 C 4 ǫ C 2 C 3 a A 1 B 1 C 1 a A 2 B 2 C 2 a A 3 B 3 C 3 a A 4 B 4 C 4 (1.\n\n3) The hyperdeterminant is invariant under SL(2) A × SL(2) B × SL(2) C and under a triality that interchanges A, B and C. In the context of stringy black holes the 8 a ABC are the 4 electric and 4 magnetic charges of the N = 2 STU black hole [10] and hence take on real (integer) values. The ST U model corresponds to N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . The Bekenstein-Hawking entropy of the black hole, S, was first calculated in [11] . The connection to quantum information theory arises by noting [1] that it can also be expressed in terms of Cayley's hyperdeterminant\n\nS = π |Det a ABC |.\n\n(1.4) One can then establish a dictionary between the classification of various entangled states (separable A-B-C; bipartite entangled A-BC, B-CA, C-AB; tripartite entangled W; tripartite entangled GHZ) and the classfication of various \"small\" and \"large\" BPS and non-BPS black holes [1, 2, 3, 4, 5, 6] . For example, the GHZ state [12] |Ψ ∼ |111 + |000 (1.5) 3 with Det a ABC ≥ 0 corresponds to a large non-BPS 2-charge black hole; the W-state |Ψ ∼ |100 + |010 + |001 (1.6) with Det a ABC = 0 corresponds to a small-BPS 3-charge black hole; the GHZ-state\n\n|Ψ = -|000 + |011 + |101 + |110 (1.7)\n\ncorresponds to a large BPS 4-charge black hole.\n\n1.2 N = 2 black holes and the bipartite entanglement of two qubits\n\nAn even simpler example [2] is provided by the two qubit system (Alice and Bob) described by the state |Ψ = a AB |AB (1.8) where A = 0, 1, and the Hilbert space has dimension 2 2 = 4. The a AB transforms as a (2, 2) under SL(2, C) A × SL(2, C) B . The entanglement is measured by the 2-tangle τ 2 (AB) = C 2 (AB) (1.9) where C(AB) = 2 |det a AB | (1.10) is the concurrence. The determinant is invariant under SL(2, C) A × SL(2, C) B and under a duality that interchanges A and B. Here it is sufficient to look at N = 2 supergravity coupled to just one vector multiplet and the 4 a AB are the 2 electric and 2 magnetic charges of the axion-dilaton black hole with entropy S = π|det a AB | (1.11) For example, the Bell state |Ψ ∼ |11 + |00 (1.12) with det a AB ≥ 0 corresponds to a large non-BPS 2-charge black hole."},{"section_type":"OTHER","section_title":"N = 8 black holes and the tripartite entanglement of seven qubits","text":"We recall that in the case of D = 4, N = 8 supergravity, the the 28 electric and 28 magnetic charges belong to the 56 of E 7 (7) . The black hole entropy is [15, 18] (1.13) where J 4 is Cartan's quartic E 7 invariant [13, 14] . It may be written\n\nS = π |J 4 |\n\nJ 4 = P ij Q jk P kl Q li -1 4 P ij Q ij P kl Q kl + 1 96 ǫ ijklmnop Q ij Q kl Q mn Q op + ǫ ijklmnop P ij P kl P mn P op . (1.14) 4\n\nwhere P ij and Q jk are 8 × 8 antisymmetric matrices. The qubit interpretation [4] relies on the decomposition E 7 (C) ⊃ [SL(2, C)] 7 (1.15) under which 56 → 2, 2, 1, 2, 1, 1, 1) + (1, 2, 2, 1, 2, 1, 1 1, 1, 2, 2, 1, 2, 1) +(1, 1, 1, 2, 2, 1, 2) +(2, 1, 1, 1, 2, 2, 1) + (1, 2, 1, 1, 1, 2, 2 2, 1, 2, 1, 1, 1, 2) (1.16) suggesting the tripartite entanglement of seven qubits (Alice, Bob, Charlie, Daisy, Emma, Fred and George) described by the state.\n\n(\n\n) +(\n\n) +(\n\n|Ψ = a ABD |ABD +b BCE |BCE +c CDF |CDF +d DEG |DEG +e EF A |EF A +f F GB |F GB +g GAC |GAC (1.17)\n\nwhere A = 0, 1, so the Hilbert space has dimension 7.2 3 = 56. The a, b, c, d, e, f, g transform as a 56 of E 7 (C). The entanglement may be represented by a heptagon where the vertices A,B,C,D,E,F,G represent the seven qubits and the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC represent the tripartite entanglement. See Figure 1 . Alternatively, we can use the Fano plane. See Figure 2 . The Fano plane also corresponds to the multiplication table of the octonions 3 The measure of the tripartite entanglement of the seven qubits is provided by the 3-tangle\n\nτ 3 (ABCDEF G) = 4|J 4 | (1.18) with J 4 ∼ a 4 + b 4 + c 4 + d 4 + e 4 + f 4 + g 4 +\n\n3 Not the \"split\" octonions as was incorrectly stated in the published version of [4].\n\nA B C D E F G Figure 1: The E 7 entanglement diagram. Each of the seven vertices A,B,C,D,E,F,G represents a qubit and each of the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC describes a tripartite entanglement.\n\n2[a 2 b 2 + b 2 c 2 + c 2 d 2 + d 2 e 2 + e 2 f 2 + f 2 g 2 + g 2 a 2 + a 2 c 2 + b 2 d 2 + c 2 e 2 + d 2 f 2 + e 2 g 2 + f 2 a 2 + g 2 b 2 + a 2 d 2 + b 2 e 2 + c 2 f 2 + d 2 g 2 + e 2 a 2 + f 2 b 2 + g 2 c 2 ] +8[bcdf + cdeg + def a + ef gb + f gac + gabd + abce] (1.19)\n\nwhere products like a 4 = (ABD)(ABD)(ABD)(ABD)\n\n= ǫ A 1 A 2 ǫ B 1 B 2 ǫ D 1 D 4 ǫ A 3 A 4 ǫ B 3 B 4 ǫ D 2 D 3 a A 1 B 1 D 1 a A 2 B 2 D 2 a A 3 B 3 D 3 a A 4 B 4 D 4 (1.20)\n\nexclude four individuals (here Charlie, Emma, Fred and George), products like\n\na 2 b 2 = (ABD)(ABD)(F GB)(F GB) = ǫ A 1 A 2 ǫ B 1 B 3 ǫ D 1 D 2 ǫ F 3 F 4 ǫ G 3 G 4 ǫ B 2 B 4 a A 1 B 1 D 1 a A 2 B 2 D 2 b F 3 G 3 B 3 b F 4 G 4 B 4 (1.21)\n\nexclude two individuals (here Charlie and Emma), and products like\n\nabce = (ABD)(BCE)(CDF )(EF A) = ǫ A 1 A 4 ǫ B 1 B 2 ǫ C 2 C 3 ǫ D 1 D 3 ǫ E 2 E 4 ǫ F 3 F 4 a A 1 B 1 D 1 b B 2 C 2 E 2 c C 3 D 3 F 3 e E 4 F 4 A 4 (1.22)\n\nexclude one individual (here George) 4 . Once again large non-BPS, small BPS and large BPS black holes correspond to states with J 4 > 0, J 4 = 0 and J 4 < 0, respectively.\n\n4 This corrects the corresponding equation in the published version of [4] which had the wrong index contraction.\n\nF E B G C A D\n\nFigure 2: The Fano plane has seven points, representing the seven qubits, and seven lines (the circle counts as a line) with three points on every line, representing the tripartite entanglement, and three lines through every point."},{"section_type":"OTHER","section_title":"Magic supergravities in D = 4","text":"The black holes described by Cayley's hyperdeterminant are those of N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . In [4] the following four-dimensional generalizations were considered: 1) N = 2 supergravity coupled to l vector multiplets where the symmetry is SL(2, Z) × SO(l -1, 2, Z) and the black holes carry charges belonging to the (2, l + 1) representation (l + 1 electric plus l + 1 magnetic).\n\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SL(2, Z) × SO(6, m, Z) where the black holes carry charges belonging to the (2, 6 + m) representation (m + 6 electric plus m + 6 magnetic).\n\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 7(7) (Z) and the black holes carry charges belonging to the fundamental 56-dimensional representation (28 electric plus 28 magnetic).\n\nIn all three case there exist quartic invariants akin to Cayley's hyperdeterminant whose square root yields the corresponding black hole entropy. In [4] we succeeded in giving a quantum theoretic interpretation in the N = 8 case together with its truncations to N = 4 (with m = 6) and N = 2 (with l = 3, the case we already knew [1]).\n\nHowever, as suggested by Levay [5], one might also consider the \"magic\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 4 supergravity coupled to 6, 9, 15 and 27 vector multiplets with symmetries Sp(6, Z), SU(3, 3), SO * (12) and E 7(-25) , respectively. Once again, as has been shown just recently [20] , in all cases there are quartic invariants whose square root yields the corresponding black hole entropy.\n\nHere we demonstrate that the black-hole/qubit correspondence does indeed continue to 7 hold for magic supergravities. The crucial observation is that, although the black hole charges a ABC are real (integer) numbers and the entropy (1.13) is invariant under E 7 (7)(Z), the coefficients a ABC that appear in the qutrit state (1.17) are complex. So the three tangle (1.18) is invariant under E 7 (C) which contains both E 7(7) (Z) and E 7(-25) (Z) as subgroups.\n\nTo find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\n\nE 7(7) (Z) ⊃ [SL(2)(Z)] 7 (1.23) but E 7(-25) (Z) ⊃ [SL(2)(Z)] 7 (1.24)\n\nis irrelevant. All that matters is that\n\nE 7 (C) ⊃ [SL(2)(C)] 7 (1.25)\n\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic) . Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) above. When we looked at the seven qubit subsector E 7 (C) ⊃ SL(2, C) × SO(12, C), we gave an N = 4 supergravity interpretationwith symmetry SL(2, R)×SO(6, 6) [4], but we could equally have given an interpretation in terms of N = 2 supergravity coupled to 11 vector multiplets with symmetry SL(2, R) × SO(10, 2).\n\nMoreover, SO(l -1, 2) is contained in SO(l + 1, C) and SO(6, m) is contained in SO(12 + m, C) so we can give a qubit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(4n, C) which contains [SL(2, C)] 2n ."},{"section_type":"OTHER","section_title":"Five-dimensional supergravity","text":"In five dimensions we might consider: 1) N = 2 supergravity coupled to l+1 vector multiplets where the symmetry is SO(1, 1, Z)× SO(l, 1, Z) and the black holes carry charges belonging to the (l + 1) representation (all electric) .\n\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SO(1, 1, Z)× SO(m, 5, Z) where the black holes carry charges belonging to the (m + 5) representation (all electric).\n\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 6(6) (Z) and the black holes carry charges belonging to the fundamental 27-dimensional representation (all electric).\n\nThe electrically charged objects are point-like and the magnetic duals are one-dimensional, or string-like, transforming according to the contrgredient representation. In all three cases above there exist cubic invariants akin to the determinant which yield the corresponding black hole or black string entropy.\n\nIn this section we briefly describe the salient properties of maximal N = 8 case, following [16] . We have 27 abelian gauge fields which transform in the fundamental representation of 8 6(6) . The first invariant of E 6(6) is the cubic invariant [13, 17, 16, 18, 19]\n\nE\n\nJ 3 = q ij Ω jl q lm Ω mn q np Ω pi (2.1)\n\nwhere q ij is the charge vector transforming as a 27 which can be represented as traceless Sp(8) matrix. The entropy of a black hole with charges q ij is then given by\n\nS = π |J 3 | (2.2)\n\nWe will see that a configuration with J 3 = 0 preserves 1/8 of the supersymmetries. If J 3 = 0 and ∂J 3 ∂q i = 0 then it preserves 1/4 of the supersymmetries, and finally if ∂J 3 ∂q i = 0 (and the charge vector q i is non-zero), the configuration preserves 1/2 of the supersymmetries. We will show this by choosing a particular basis for the charges, the general result following by U-duality.\n\nIn five dimensions the compact group H is USp(8). We choose our conventions so that USp(2) = SU(2). In the commutator of the supersymmetry generators we have a central charge matrix Z ab which can be brought to a normal form by a USp(8) transformation. In the normal form the central charge matrix can be written as\n\ne ab = s 1 + s 2 -s 3 0 0 0 0 s 1 + s 3 -s 2 0 0 0 0 s 2 + s 3 -s 1 0 0 0 0 -(s 1 + s 2 + s 3 ) × 0 1 -1 0 (2.3) we can order s i so that s 1 ≥ s 2 ≥ |s 3 |.\n\nThe cubic invariant, in this basis, becomes\n\nJ 3 = s 1 s 2 s 3 (2.4)\n\nEven though the eigenvalues s i might depend on the moduli, the invariant (2.4) only depends on the quantized values of the charges. We can write a generic charge configuration as UeU t , where e is the normal frame as above, and the invariant will then be (2.4). There are three distinct possibilities\n\nJ 3 = 0 s 1 , s 2 , s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 , s 2 = 0, s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 = 0, s 2 , s 3 = 0 (2.5)\n\nTaking the case of type II on T 5 we can choose the rotation in such a way that, for example, s 1 corresponds to solitonic five-brane charge, s 2 to fundamental string winding charge along some direction and s 3 to Kaluza-Klein momentum along the same direction. We can see that in this specific example the three possibilities in (2.5) break 1/8, 1/4 and 1/2 supersymmetries. The respective orbits are\n\nE 6 (6) F 4(4) E 6 (6) SO(5, 4) ×T 16 9 E 6 (6) SO(5, 5) ×T 16 (2.6)\n\nThis also shows that one can generically choose a basis for the charges so that all others are related by U-duality. The basis chosen here is the S-dual of the D-brane basis usually chosen for describing black holes in type II B on T 5 . All others are related by U-duality to this particular choice. Note that, in contrast to the four-dimensional case where flipping the sign of J 4 (1.14) interchanges BPS and non-BPS black holes, the sign of the J 3 (2.4) is not important since it changes under a CPT transformation. There is no non-BPS orbit in five dimensions.\n\nIn five dimensions there are also string-like configurations which are the magnetic duals of the configurations considered here. They transform in the contragredient 27 ′ representation and the solutions preserving 1/2, 1/4, 1/8 supersymmetries are characterized in an analogous way. We could also have configurations where we have both point-like and string-like ch the point-like charge is uniformly distributed along the string, it is more natural to consider this configuration as a point-like object in D = 4 by dimensional reduction.\n\nIt is useful to decompose the U-duality group into the T-duality group and the S-duality group. The decomposition reads E 6 → SO(5, 5) × SO(1, 1), leading to\n\n27 → 16 1 + 10 -2 + 1 4 (2.7)\n\nThe last term in (2.7) corresponds to the NS five-brane charge. The 16 correspond to the D-brane charges and the 10 correspond to the 5 directions of KK momentum and the 5 directions of fundamental string winding, which are the charges that explicitly appear in string perturbation theory. The cubic invariant has the decomposition (27) 3 → 10 -2 10 -2 1 4 + 16 1 16 1 10 -2 (2.8) This is saying that in order to have a non-zero area black hole we must have three NS charges (more precisely some \"perturbative\" charges and a solitonic five-brane); or we can have two D-brane charges and one NS charge. In particular, it is not possible to have a black hole with a non-zero horizon area with purely D-brane charges. Notice that the non-compact nature of the groups is crucial in this classification."},{"section_type":"OTHER","section_title":"D = 5 black holes and qutrits","text":"So far, all the quantum information analogies involve four-dimensional black holes and qubits. In order to find an analogy with five-dimensional black holes we invoke three state systems called qutrits."},{"section_type":"OTHER","section_title":"N = 2 black holes and the bipartite entanglement of two qutrits","text":"The two qutrit system (Alice and Bob) is described by the state\n\n|Ψ = a AB |AB 10\n\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3 2 = 9. The a AB transforms as a (3, 3) under SL(3) A × SL(3) B . The bipartite entanglement is measured by the concurrence [21]\n\nC(AB) = 3 3/2 |det a AB |. (3.1)\n\nThe determinant is invariant under SL(3, C) A × SL(3, C) B and under a duality that interchanges A and B. The black hole interpretation is provided by N = 2 supergravity coupled to 8 vector multiplets with symmetry SL(3, C) where the black hole charges transform as a 9. The entropy is given by\n\nS = π|det a AB | (3.2)\n\n3.2 N = 8 black holes and the bipartite entanglement of three qutrits\n\nAs we have seen in section (2) in the case of D = 5, N = 8 supergravity, the black hole charges belong to the 27 of E 6(6) and the entropy is given by (2.2).\n\nThe qutrit interpretation now relies on the decomposition\n\nE 6 (C) ⊃ SL(3, C) A × SL(3, C) B × SL(3, C) C (3.3) under which 27 → (3, 3, 1) + (3 ′ , 1, 3) + (1, 3 ′ , 3 ′ ) (3.4)\n\nsuggesting the bipartite entanglement of three qutrits (Alice, Bob, Charlie). However, the larger symmetry requires that they undergo at most bipartite entanglement of a very specific kind, where each person has bipartite entanglement with the other two:\n\n|Ψ = a AB |AB + b B C |BC + c CA |CA (3.5)\n\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3.3 2 = 27. The three states transforms as a pair of triplets under two of the SL(3)'s and singlets under the remaining one. Individually, therefore, the bipartite entanglement of each of the three states is given by the determinant (3.1). Taken together however, we see from (3.4) that they transform as a complex 27 of E 6 (C). The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements. See Fig. 3. The N=2 truncation of section 3.1 is represented by just the line AB with endpoints A and B. Note that: 1) Any pair of states has an individual in common 2) Each individual is excluded from one out of the three states The entanglement measure will be given by the concurrence\n\nC(ABC) = 3 3/2 |J 3 | (3.6)\n\nJ 3 being the singlet in 27 × 27 × 27:\n\nJ 3 ∼ a 3 + b 3 + c 3 + 6abc (3.7) 11 A B C\n\nFigure 3: The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements.\n\nwhere the products\n\na 3 = ǫ A 1 A 2 A 3 ǫ B 1 B 2 B 3 a A 1 B 1 a A 2 B 2 a A 3 B 3 (3.8) b 3 = ǫ B 1 B 2 B 3 ǫ C 1 C 2 C 3 b B 1 C 1 b B 2 C 2 b B 3 C 3 (3.9) c 3 = ǫ C 1 C 2 C 3 ǫ A 1 A 2 A 3 c C 1 A 1 c C 2 A 2 c C 3 A 3 (3.10)\n\nexclude one individual (Charlie, Alice, and Bob respectively), and the product abc = a AB b B C c CA (3.11) excludes none."},{"section_type":"OTHER","section_title":"Magic supergravities in D = 5","text":"Just as in four dimensions, one might also consider the \"magic\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 5 supergravity coupled to 5, 8, 14 and 26 vector multiplets with symmetries SL(3, R), SL(3, C), SU * (6) and E 6(-26) respectively. Once again, in all cases there are cubic invariants whose square root yields the corresponding black hole entropy [20] . Here we demonstrate that the black-hole/qubit correspondence continue to hold for these D = 5 magic supergravities, as well as D = 4 . Once again, the crucial observation is that, although the black hole charges a AB are real (integer) numbers and the entropy (2.2) is invariant under E 6(6) (Z), the coefficients a AB that appear in the wave function (3.5) are complex. So the 2-tangle (3.6) is invariant under E 6 (C) which contains both E 6(6) (Z) and E 6(-26) (Z) as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\n\nE 6(6) (Z) ⊃ [SL(3)(Z)] 3 (3.12) 12 but E 6(-26) (Z) ⊃ [SL(3)(Z)] 3 ( 3\n\n.13) is irrelevant. All that matters is that\n\nE 6 (C) ⊃ [SL(3)(C)] 3 (3.14)\n\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic). In fact, the example of section 3.1 corresponds to the complex case. Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) of section (2). SO(l, 1) is contained in SO(l + 1, C) and SO(m, 5) is contained in SO(5 + m, C), so we can give a qutrit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(6n, C) which contains [SL(3, C)] n ."},{"section_type":"CONCLUSION","section_title":"Conclusions","text":"We note that the 27-dimensional Hilbert space given in (3.4) and (3.5) is not a subspace of the 3 3 -dimensional three qutrit Hilbert space given by (3, 3, 3) , but rather a direct sum of three 3 2 -dimensional Hilbert spaces. It is, however, a subspace of the 7 3 -dimensional three 7-dit Hilbert space given by (7, 7, 7) . Consider the decomposition 3, 3, 1) + (3, 1, 3) + (1, 3, 3) 1, 1, 1) This contains the subspace that describes the bipartite entanglement of three qutrits, namely\n\nSL(7) A × SL(7) B × SL(7) C → SL(3) A × SL(3) B × SL(3) C under which (7, 7, 7) → (3 ′ , 3 ′ , 3 ′ ) + (3 ′ , 3 ′ , 3) + (3 ′ , 3, 3 ′ ) + (3, 3 ′ , 3 ′ ) + (3 ′ , 3, 3) + (3, 3 ′ , 3) + (3, 3, 3 ′ ) + (3, 3, 3) +(3 ′ , 3 ′ , 1) + (3 ′ , 1, 3 ′ ) + (1, 3 ′ , 3 ′ ) + (3 ′ , 1, 3) + (3 ′ , 3, 1) + (1, 3, 3 ′ ) +(\n\n+ (3, 1, 3 ′ ) + (3, 3 ′ , 1) + (1, 3 ′ , 3) +(3 ′ , 1, 1) + (1, 3 ′ , 1) + (1, 1, 3 ′ ) + (3, 1, 1) + (1, 3, 1) + (1, 1, 3) +(\n\n(3 ′ , 3, 1) + (3, 1, 3) + (1, 3 ′ , 3 ′ )\n\nSo the triangle entanglement we have described fits within conventional quantum information theory. Our analogy between black holes and quantum information remains, for the moment, just that. We know of no physics connecting them.\n\nNevertheless, just as the exceptional group E 7 describes the tripartite entanglement of seven qubits [4, 5] , we have seen is this paper that the exceptional group E 6 describes the bipartite entanglement of three qutrits. In the E 7 case, the quartic Cartan invariant provides both the measure of entanglement and the entropy of the four-dimensional N = 8 black hole, 13 whereas in the E 6 case, the cubic Cartan invariant provides both the measure of entanglement and the entropy of the five-dimensional N = 8 black hole.\n\nMoreover, we have seen that similar analogies exist not only for the N = 4 and N = 2 truncations, but also for the magic N = 2 supergravities in both four and five dimensions (In the four-dimensional case, this had previously been conjectured by Levay[4, 5]). Murat Gunaydin has suggested (private communication) that the appearance of octonions implies a connection to quaternionic and/or octonionic quantum mechanics. This was not apparent (at least to us) in the four-dimensional N = 8 case [4], but the appearance in the five dimensional magic N = 2 case of SL(3, R), SL(3, C), SL(3, H) and SL(3, O) is more suggestive."},{"section_type":"OTHER","section_title":"Acknowledgements","text":"MJD has enjoyed useful conversations with Leron Borsten, Hajar Ebrahim, Chris Hull, Martin Plenio and Tony Sudbery. This work was supported in part by the National Science Foundation under grant number PHY-0245337 and PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The work of S.F. has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 Constituents, fundamental forces and symmetries of the universe, in association with INFN Frascati National Laboratories and by the D.O.E grant DE-FG03-91ER40662, Task C. The work of MJD is supported in part by PPARC under rolling grant PPA/G/O/2002/00474, PP/D50744X/1."}],"string":"[\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"1 m.duff@imperial.ac.uk 2 Sergio.Ferrara@cern.ch Contents 1 D = 4 black holes and qubits 1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 1.2 N = 2 black holes and the bipartite entanglement of two qubits . . . . . . . 1.3 N = 8 black holes and the tripartite entanglement of seven qubits . . . . . . 1.4 Magic supergravities in D = 4 . . . . . . . . . . . . . . . . . . . . . . . . . .\\n\\n2 Five-dimensional supergravity 3 D = 5 black holes and qutrits 3.1 N = 2 black holes and the bipartite entanglement of two qutrits . . . . . . . 3.2 N = 8 black holes and the bipartite entanglement of three qutrits . . . . . . 3.3 Magic supergravities in D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . .\\n\\n4 Conclusions 5 Acknowledgements 2 1 D = 4 black holes and qubits\\n\\nIt sometimes happens that two very different areas of theoretical physics share the same mathematics. This may eventually lead to the realisation that they are, in fact, dual descriptions of the same physical phenomena, or it may not. Either way, it frequently leads to new insights in both areas. Recent papers [1, 2, 3, 4, 5, 6] have established an analogy between the entropy of certain four-dimensional supersymmetric black holes in string theory and entanglement measures in quantum information theory. In this paper we extend the analogy from four dimensions to five which also involves going from two-state systems (qubits) to three-state systems (qutrits). We begin by recalling the four-dimensional examples:\\n\\n1.1 N = 2 black holes and the tripartite entanglement of three qubits\\n\\nThe three qubit system (Alice, Bob, Charlie) is described by the state\\n\\n|Ψ = a ABC |ABC (1.1)\\n\\nwhere A = 0, 1, so the Hilbert space has dimension 2 3 = 8. The complex numbers a ABC transforms as a (2, 2, 2) under SL(2, C) A × SL(2, C) B × SL(2, C) C . The tripartite entanglement is measured by the 3-tangle [7, 8]\\n\\nτ 3 (ABC) = 4|Det a ABC |. (1.2)\\n\\nwhere Det a ABC is Cayley's hyperdeterminant [9] .\\n\\nDet a = - 1 2 ǫ A 1 A 2 ǫ B 1 B 2 ǫ A 3 A 4 ǫ B 3 B 4 ǫ C 1 C 4 ǫ C 2 C 3 a A 1 B 1 C 1 a A 2 B 2 C 2 a A 3 B 3 C 3 a A 4 B 4 C 4 (1.\\n\\n3) The hyperdeterminant is invariant under SL(2) A × SL(2) B × SL(2) C and under a triality that interchanges A, B and C. In the context of stringy black holes the 8 a ABC are the 4 electric and 4 magnetic charges of the N = 2 STU black hole [10] and hence take on real (integer) values. The ST U model corresponds to N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . The Bekenstein-Hawking entropy of the black hole, S, was first calculated in [11] . The connection to quantum information theory arises by noting [1] that it can also be expressed in terms of Cayley's hyperdeterminant\\n\\nS = π |Det a ABC |.\\n\\n(1.4) One can then establish a dictionary between the classification of various entangled states (separable A-B-C; bipartite entangled A-BC, B-CA, C-AB; tripartite entangled W; tripartite entangled GHZ) and the classfication of various \\\"small\\\" and \\\"large\\\" BPS and non-BPS black holes [1, 2, 3, 4, 5, 6] . For example, the GHZ state [12] |Ψ ∼ |111 + |000 (1.5) 3 with Det a ABC ≥ 0 corresponds to a large non-BPS 2-charge black hole; the W-state |Ψ ∼ |100 + |010 + |001 (1.6) with Det a ABC = 0 corresponds to a small-BPS 3-charge black hole; the GHZ-state\\n\\n|Ψ = -|000 + |011 + |101 + |110 (1.7)\\n\\ncorresponds to a large BPS 4-charge black hole.\\n\\n1.2 N = 2 black holes and the bipartite entanglement of two qubits\\n\\nAn even simpler example [2] is provided by the two qubit system (Alice and Bob) described by the state |Ψ = a AB |AB (1.8) where A = 0, 1, and the Hilbert space has dimension 2 2 = 4. The a AB transforms as a (2, 2) under SL(2, C) A × SL(2, C) B . The entanglement is measured by the 2-tangle τ 2 (AB) = C 2 (AB) (1.9) where C(AB) = 2 |det a AB | (1.10) is the concurrence. The determinant is invariant under SL(2, C) A × SL(2, C) B and under a duality that interchanges A and B. Here it is sufficient to look at N = 2 supergravity coupled to just one vector multiplet and the 4 a AB are the 2 electric and 2 magnetic charges of the axion-dilaton black hole with entropy S = π|det a AB | (1.11) For example, the Bell state |Ψ ∼ |11 + |00 (1.12) with det a AB ≥ 0 corresponds to a large non-BPS 2-charge black hole.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"N = 8 black holes and the tripartite entanglement of seven qubits\",\n \"text\": \"We recall that in the case of D = 4, N = 8 supergravity, the the 28 electric and 28 magnetic charges belong to the 56 of E 7 (7) . The black hole entropy is [15, 18] (1.13) where J 4 is Cartan's quartic E 7 invariant [13, 14] . It may be written\\n\\nS = π |J 4 |\\n\\nJ 4 = P ij Q jk P kl Q li -1 4 P ij Q ij P kl Q kl + 1 96 ǫ ijklmnop Q ij Q kl Q mn Q op + ǫ ijklmnop P ij P kl P mn P op . (1.14) 4\\n\\nwhere P ij and Q jk are 8 × 8 antisymmetric matrices. The qubit interpretation [4] relies on the decomposition E 7 (C) ⊃ [SL(2, C)] 7 (1.15) under which 56 → 2, 2, 1, 2, 1, 1, 1) + (1, 2, 2, 1, 2, 1, 1 1, 1, 2, 2, 1, 2, 1) +(1, 1, 1, 2, 2, 1, 2) +(2, 1, 1, 1, 2, 2, 1) + (1, 2, 1, 1, 1, 2, 2 2, 1, 2, 1, 1, 1, 2) (1.16) suggesting the tripartite entanglement of seven qubits (Alice, Bob, Charlie, Daisy, Emma, Fred and George) described by the state.\\n\\n(\\n\\n) +(\\n\\n) +(\\n\\n|Ψ = a ABD |ABD +b BCE |BCE +c CDF |CDF +d DEG |DEG +e EF A |EF A +f F GB |F GB +g GAC |GAC (1.17)\\n\\nwhere A = 0, 1, so the Hilbert space has dimension 7.2 3 = 56. The a, b, c, d, e, f, g transform as a 56 of E 7 (C). The entanglement may be represented by a heptagon where the vertices A,B,C,D,E,F,G represent the seven qubits and the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC represent the tripartite entanglement. See Figure 1 . Alternatively, we can use the Fano plane. See Figure 2 . The Fano plane also corresponds to the multiplication table of the octonions 3 The measure of the tripartite entanglement of the seven qubits is provided by the 3-tangle\\n\\nτ 3 (ABCDEF G) = 4|J 4 | (1.18) with J 4 ∼ a 4 + b 4 + c 4 + d 4 + e 4 + f 4 + g 4 +\\n\\n3 Not the \\\"split\\\" octonions as was incorrectly stated in the published version of [4].\\n\\nA B C D E F G Figure 1: The E 7 entanglement diagram. Each of the seven vertices A,B,C,D,E,F,G represents a qubit and each of the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC describes a tripartite entanglement.\\n\\n2[a 2 b 2 + b 2 c 2 + c 2 d 2 + d 2 e 2 + e 2 f 2 + f 2 g 2 + g 2 a 2 + a 2 c 2 + b 2 d 2 + c 2 e 2 + d 2 f 2 + e 2 g 2 + f 2 a 2 + g 2 b 2 + a 2 d 2 + b 2 e 2 + c 2 f 2 + d 2 g 2 + e 2 a 2 + f 2 b 2 + g 2 c 2 ] +8[bcdf + cdeg + def a + ef gb + f gac + gabd + abce] (1.19)\\n\\nwhere products like a 4 = (ABD)(ABD)(ABD)(ABD)\\n\\n= ǫ A 1 A 2 ǫ B 1 B 2 ǫ D 1 D 4 ǫ A 3 A 4 ǫ B 3 B 4 ǫ D 2 D 3 a A 1 B 1 D 1 a A 2 B 2 D 2 a A 3 B 3 D 3 a A 4 B 4 D 4 (1.20)\\n\\nexclude four individuals (here Charlie, Emma, Fred and George), products like\\n\\na 2 b 2 = (ABD)(ABD)(F GB)(F GB) = ǫ A 1 A 2 ǫ B 1 B 3 ǫ D 1 D 2 ǫ F 3 F 4 ǫ G 3 G 4 ǫ B 2 B 4 a A 1 B 1 D 1 a A 2 B 2 D 2 b F 3 G 3 B 3 b F 4 G 4 B 4 (1.21)\\n\\nexclude two individuals (here Charlie and Emma), and products like\\n\\nabce = (ABD)(BCE)(CDF )(EF A) = ǫ A 1 A 4 ǫ B 1 B 2 ǫ C 2 C 3 ǫ D 1 D 3 ǫ E 2 E 4 ǫ F 3 F 4 a A 1 B 1 D 1 b B 2 C 2 E 2 c C 3 D 3 F 3 e E 4 F 4 A 4 (1.22)\\n\\nexclude one individual (here George) 4 . Once again large non-BPS, small BPS and large BPS black holes correspond to states with J 4 > 0, J 4 = 0 and J 4 < 0, respectively.\\n\\n4 This corrects the corresponding equation in the published version of [4] which had the wrong index contraction.\\n\\nF E B G C A D\\n\\nFigure 2: The Fano plane has seven points, representing the seven qubits, and seven lines (the circle counts as a line) with three points on every line, representing the tripartite entanglement, and three lines through every point.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Magic supergravities in D = 4\",\n \"text\": \"The black holes described by Cayley's hyperdeterminant are those of N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . In [4] the following four-dimensional generalizations were considered: 1) N = 2 supergravity coupled to l vector multiplets where the symmetry is SL(2, Z) × SO(l -1, 2, Z) and the black holes carry charges belonging to the (2, l + 1) representation (l + 1 electric plus l + 1 magnetic).\\n\\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SL(2, Z) × SO(6, m, Z) where the black holes carry charges belonging to the (2, 6 + m) representation (m + 6 electric plus m + 6 magnetic).\\n\\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 7(7) (Z) and the black holes carry charges belonging to the fundamental 56-dimensional representation (28 electric plus 28 magnetic).\\n\\nIn all three case there exist quartic invariants akin to Cayley's hyperdeterminant whose square root yields the corresponding black hole entropy. In [4] we succeeded in giving a quantum theoretic interpretation in the N = 8 case together with its truncations to N = 4 (with m = 6) and N = 2 (with l = 3, the case we already knew [1]).\\n\\nHowever, as suggested by Levay [5], one might also consider the \\\"magic\\\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 4 supergravity coupled to 6, 9, 15 and 27 vector multiplets with symmetries Sp(6, Z), SU(3, 3), SO * (12) and E 7(-25) , respectively. Once again, as has been shown just recently [20] , in all cases there are quartic invariants whose square root yields the corresponding black hole entropy.\\n\\nHere we demonstrate that the black-hole/qubit correspondence does indeed continue to 7 hold for magic supergravities. The crucial observation is that, although the black hole charges a ABC are real (integer) numbers and the entropy (1.13) is invariant under E 7 (7)(Z), the coefficients a ABC that appear in the qutrit state (1.17) are complex. So the three tangle (1.18) is invariant under E 7 (C) which contains both E 7(7) (Z) and E 7(-25) (Z) as subgroups.\\n\\nTo find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\\n\\nE 7(7) (Z) ⊃ [SL(2)(Z)] 7 (1.23) but E 7(-25) (Z) ⊃ [SL(2)(Z)] 7 (1.24)\\n\\nis irrelevant. All that matters is that\\n\\nE 7 (C) ⊃ [SL(2)(C)] 7 (1.25)\\n\\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic) . Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) above. When we looked at the seven qubit subsector E 7 (C) ⊃ SL(2, C) × SO(12, C), we gave an N = 4 supergravity interpretationwith symmetry SL(2, R)×SO(6, 6) [4], but we could equally have given an interpretation in terms of N = 2 supergravity coupled to 11 vector multiplets with symmetry SL(2, R) × SO(10, 2).\\n\\nMoreover, SO(l -1, 2) is contained in SO(l + 1, C) and SO(6, m) is contained in SO(12 + m, C) so we can give a qubit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(4n, C) which contains [SL(2, C)] 2n .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Five-dimensional supergravity\",\n \"text\": \"In five dimensions we might consider: 1) N = 2 supergravity coupled to l+1 vector multiplets where the symmetry is SO(1, 1, Z)× SO(l, 1, Z) and the black holes carry charges belonging to the (l + 1) representation (all electric) .\\n\\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SO(1, 1, Z)× SO(m, 5, Z) where the black holes carry charges belonging to the (m + 5) representation (all electric).\\n\\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 6(6) (Z) and the black holes carry charges belonging to the fundamental 27-dimensional representation (all electric).\\n\\nThe electrically charged objects are point-like and the magnetic duals are one-dimensional, or string-like, transforming according to the contrgredient representation. In all three cases above there exist cubic invariants akin to the determinant which yield the corresponding black hole or black string entropy.\\n\\nIn this section we briefly describe the salient properties of maximal N = 8 case, following [16] . We have 27 abelian gauge fields which transform in the fundamental representation of 8 6(6) . The first invariant of E 6(6) is the cubic invariant [13, 17, 16, 18, 19]\\n\\nE\\n\\nJ 3 = q ij Ω jl q lm Ω mn q np Ω pi (2.1)\\n\\nwhere q ij is the charge vector transforming as a 27 which can be represented as traceless Sp(8) matrix. The entropy of a black hole with charges q ij is then given by\\n\\nS = π |J 3 | (2.2)\\n\\nWe will see that a configuration with J 3 = 0 preserves 1/8 of the supersymmetries. If J 3 = 0 and ∂J 3 ∂q i = 0 then it preserves 1/4 of the supersymmetries, and finally if ∂J 3 ∂q i = 0 (and the charge vector q i is non-zero), the configuration preserves 1/2 of the supersymmetries. We will show this by choosing a particular basis for the charges, the general result following by U-duality.\\n\\nIn five dimensions the compact group H is USp(8). We choose our conventions so that USp(2) = SU(2). In the commutator of the supersymmetry generators we have a central charge matrix Z ab which can be brought to a normal form by a USp(8) transformation. In the normal form the central charge matrix can be written as\\n\\ne ab = s 1 + s 2 -s 3 0 0 0 0 s 1 + s 3 -s 2 0 0 0 0 s 2 + s 3 -s 1 0 0 0 0 -(s 1 + s 2 + s 3 ) × 0 1 -1 0 (2.3) we can order s i so that s 1 ≥ s 2 ≥ |s 3 |.\\n\\nThe cubic invariant, in this basis, becomes\\n\\nJ 3 = s 1 s 2 s 3 (2.4)\\n\\nEven though the eigenvalues s i might depend on the moduli, the invariant (2.4) only depends on the quantized values of the charges. We can write a generic charge configuration as UeU t , where e is the normal frame as above, and the invariant will then be (2.4). There are three distinct possibilities\\n\\nJ 3 = 0 s 1 , s 2 , s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 , s 2 = 0, s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 = 0, s 2 , s 3 = 0 (2.5)\\n\\nTaking the case of type II on T 5 we can choose the rotation in such a way that, for example, s 1 corresponds to solitonic five-brane charge, s 2 to fundamental string winding charge along some direction and s 3 to Kaluza-Klein momentum along the same direction. We can see that in this specific example the three possibilities in (2.5) break 1/8, 1/4 and 1/2 supersymmetries. The respective orbits are\\n\\nE 6 (6) F 4(4) E 6 (6) SO(5, 4) ×T 16 9 E 6 (6) SO(5, 5) ×T 16 (2.6)\\n\\nThis also shows that one can generically choose a basis for the charges so that all others are related by U-duality. The basis chosen here is the S-dual of the D-brane basis usually chosen for describing black holes in type II B on T 5 . All others are related by U-duality to this particular choice. Note that, in contrast to the four-dimensional case where flipping the sign of J 4 (1.14) interchanges BPS and non-BPS black holes, the sign of the J 3 (2.4) is not important since it changes under a CPT transformation. There is no non-BPS orbit in five dimensions.\\n\\nIn five dimensions there are also string-like configurations which are the magnetic duals of the configurations considered here. They transform in the contragredient 27 ′ representation and the solutions preserving 1/2, 1/4, 1/8 supersymmetries are characterized in an analogous way. We could also have configurations where we have both point-like and string-like ch the point-like charge is uniformly distributed along the string, it is more natural to consider this configuration as a point-like object in D = 4 by dimensional reduction.\\n\\nIt is useful to decompose the U-duality group into the T-duality group and the S-duality group. The decomposition reads E 6 → SO(5, 5) × SO(1, 1), leading to\\n\\n27 → 16 1 + 10 -2 + 1 4 (2.7)\\n\\nThe last term in (2.7) corresponds to the NS five-brane charge. The 16 correspond to the D-brane charges and the 10 correspond to the 5 directions of KK momentum and the 5 directions of fundamental string winding, which are the charges that explicitly appear in string perturbation theory. The cubic invariant has the decomposition (27) 3 → 10 -2 10 -2 1 4 + 16 1 16 1 10 -2 (2.8) This is saying that in order to have a non-zero area black hole we must have three NS charges (more precisely some \\\"perturbative\\\" charges and a solitonic five-brane); or we can have two D-brane charges and one NS charge. In particular, it is not possible to have a black hole with a non-zero horizon area with purely D-brane charges. Notice that the non-compact nature of the groups is crucial in this classification.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"D = 5 black holes and qutrits\",\n \"text\": \"So far, all the quantum information analogies involve four-dimensional black holes and qubits. In order to find an analogy with five-dimensional black holes we invoke three state systems called qutrits.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"N = 2 black holes and the bipartite entanglement of two qutrits\",\n \"text\": \"The two qutrit system (Alice and Bob) is described by the state\\n\\n|Ψ = a AB |AB 10\\n\\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3 2 = 9. The a AB transforms as a (3, 3) under SL(3) A × SL(3) B . The bipartite entanglement is measured by the concurrence [21]\\n\\nC(AB) = 3 3/2 |det a AB |. (3.1)\\n\\nThe determinant is invariant under SL(3, C) A × SL(3, C) B and under a duality that interchanges A and B. The black hole interpretation is provided by N = 2 supergravity coupled to 8 vector multiplets with symmetry SL(3, C) where the black hole charges transform as a 9. The entropy is given by\\n\\nS = π|det a AB | (3.2)\\n\\n3.2 N = 8 black holes and the bipartite entanglement of three qutrits\\n\\nAs we have seen in section (2) in the case of D = 5, N = 8 supergravity, the black hole charges belong to the 27 of E 6(6) and the entropy is given by (2.2).\\n\\nThe qutrit interpretation now relies on the decomposition\\n\\nE 6 (C) ⊃ SL(3, C) A × SL(3, C) B × SL(3, C) C (3.3) under which 27 → (3, 3, 1) + (3 ′ , 1, 3) + (1, 3 ′ , 3 ′ ) (3.4)\\n\\nsuggesting the bipartite entanglement of three qutrits (Alice, Bob, Charlie). However, the larger symmetry requires that they undergo at most bipartite entanglement of a very specific kind, where each person has bipartite entanglement with the other two:\\n\\n|Ψ = a AB |AB + b B C |BC + c CA |CA (3.5)\\n\\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3.3 2 = 27. The three states transforms as a pair of triplets under two of the SL(3)'s and singlets under the remaining one. Individually, therefore, the bipartite entanglement of each of the three states is given by the determinant (3.1). Taken together however, we see from (3.4) that they transform as a complex 27 of E 6 (C). The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements. See Fig. 3. The N=2 truncation of section 3.1 is represented by just the line AB with endpoints A and B. Note that: 1) Any pair of states has an individual in common 2) Each individual is excluded from one out of the three states The entanglement measure will be given by the concurrence\\n\\nC(ABC) = 3 3/2 |J 3 | (3.6)\\n\\nJ 3 being the singlet in 27 × 27 × 27:\\n\\nJ 3 ∼ a 3 + b 3 + c 3 + 6abc (3.7) 11 A B C\\n\\nFigure 3: The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements.\\n\\nwhere the products\\n\\na 3 = ǫ A 1 A 2 A 3 ǫ B 1 B 2 B 3 a A 1 B 1 a A 2 B 2 a A 3 B 3 (3.8) b 3 = ǫ B 1 B 2 B 3 ǫ C 1 C 2 C 3 b B 1 C 1 b B 2 C 2 b B 3 C 3 (3.9) c 3 = ǫ C 1 C 2 C 3 ǫ A 1 A 2 A 3 c C 1 A 1 c C 2 A 2 c C 3 A 3 (3.10)\\n\\nexclude one individual (Charlie, Alice, and Bob respectively), and the product abc = a AB b B C c CA (3.11) excludes none.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Magic supergravities in D = 5\",\n \"text\": \"Just as in four dimensions, one might also consider the \\\"magic\\\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 5 supergravity coupled to 5, 8, 14 and 26 vector multiplets with symmetries SL(3, R), SL(3, C), SU * (6) and E 6(-26) respectively. Once again, in all cases there are cubic invariants whose square root yields the corresponding black hole entropy [20] . Here we demonstrate that the black-hole/qubit correspondence continue to hold for these D = 5 magic supergravities, as well as D = 4 . Once again, the crucial observation is that, although the black hole charges a AB are real (integer) numbers and the entropy (2.2) is invariant under E 6(6) (Z), the coefficients a AB that appear in the wave function (3.5) are complex. So the 2-tangle (3.6) is invariant under E 6 (C) which contains both E 6(6) (Z) and E 6(-26) (Z) as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\\n\\nE 6(6) (Z) ⊃ [SL(3)(Z)] 3 (3.12) 12 but E 6(-26) (Z) ⊃ [SL(3)(Z)] 3 ( 3\\n\\n.13) is irrelevant. All that matters is that\\n\\nE 6 (C) ⊃ [SL(3)(C)] 3 (3.14)\\n\\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic). In fact, the example of section 3.1 corresponds to the complex case. Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) of section (2). SO(l, 1) is contained in SO(l + 1, C) and SO(m, 5) is contained in SO(5 + m, C), so we can give a qutrit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(6n, C) which contains [SL(3, C)] n .\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusions\",\n \"text\": \"We note that the 27-dimensional Hilbert space given in (3.4) and (3.5) is not a subspace of the 3 3 -dimensional three qutrit Hilbert space given by (3, 3, 3) , but rather a direct sum of three 3 2 -dimensional Hilbert spaces. It is, however, a subspace of the 7 3 -dimensional three 7-dit Hilbert space given by (7, 7, 7) . Consider the decomposition 3, 3, 1) + (3, 1, 3) + (1, 3, 3) 1, 1, 1) This contains the subspace that describes the bipartite entanglement of three qutrits, namely\\n\\nSL(7) A × SL(7) B × SL(7) C → SL(3) A × SL(3) B × SL(3) C under which (7, 7, 7) → (3 ′ , 3 ′ , 3 ′ ) + (3 ′ , 3 ′ , 3) + (3 ′ , 3, 3 ′ ) + (3, 3 ′ , 3 ′ ) + (3 ′ , 3, 3) + (3, 3 ′ , 3) + (3, 3, 3 ′ ) + (3, 3, 3) +(3 ′ , 3 ′ , 1) + (3 ′ , 1, 3 ′ ) + (1, 3 ′ , 3 ′ ) + (3 ′ , 1, 3) + (3 ′ , 3, 1) + (1, 3, 3 ′ ) +(\\n\\n+ (3, 1, 3 ′ ) + (3, 3 ′ , 1) + (1, 3 ′ , 3) +(3 ′ , 1, 1) + (1, 3 ′ , 1) + (1, 1, 3 ′ ) + (3, 1, 1) + (1, 3, 1) + (1, 1, 3) +(\\n\\n(3 ′ , 3, 1) + (3, 1, 3) + (1, 3 ′ , 3 ′ )\\n\\nSo the triangle entanglement we have described fits within conventional quantum information theory. Our analogy between black holes and quantum information remains, for the moment, just that. We know of no physics connecting them.\\n\\nNevertheless, just as the exceptional group E 7 describes the tripartite entanglement of seven qubits [4, 5] , we have seen is this paper that the exceptional group E 6 describes the bipartite entanglement of three qutrits. In the E 7 case, the quartic Cartan invariant provides both the measure of entanglement and the entropy of the four-dimensional N = 8 black hole, 13 whereas in the E 6 case, the cubic Cartan invariant provides both the measure of entanglement and the entropy of the five-dimensional N = 8 black hole.\\n\\nMoreover, we have seen that similar analogies exist not only for the N = 4 and N = 2 truncations, but also for the magic N = 2 supergravities in both four and five dimensions (In the four-dimensional case, this had previously been conjectured by Levay[4, 5]). Murat Gunaydin has suggested (private communication) that the appearance of octonions implies a connection to quaternionic and/or octonionic quantum mechanics. This was not apparent (at least to us) in the four-dimensional N = 8 case [4], but the appearance in the five dimensional magic N = 2 case of SL(3, R), SL(3, C), SL(3, H) and SL(3, O) is more suggestive.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgements\",\n \"text\": \"MJD has enjoyed useful conversations with Leron Borsten, Hajar Ebrahim, Chris Hull, Martin Plenio and Tony Sudbery. This work was supported in part by the National Science Foundation under grant number PHY-0245337 and PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The work of S.F. has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 Constituents, fundamental forces and symmetries of the universe, in association with INFN Frascati National Laboratories and by the D.O.E grant DE-FG03-91ER40662, Task C. The work of MJD is supported in part by PPARC under rolling grant PPA/G/O/2002/00474, PP/D50744X/1.\"\n }\n]"}}},{"rowIdx":92,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0511"},"arxiv_id":{"kind":"string","value":"0704.0511"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1007/s10946-007-0032-5"},"content_hash":{"kind":"string","value":"be892cb8194712d11f25f536c9a91df2f945e518862e27506bb42e0662579c98"},"title":{"kind":"string","value":"A unified approach to SIC-POVMs and MUBs"},"abstract":{"kind":"string","value":"A unified approach to (symmetric informationally complete) positive operator\nvalued measures and mutually unbiased bases is developed in this article. The\napproach is based on the use of operator equivalents expanded in the enveloping\nalgebra of SU(2). Emphasis is put on similarities and differences between\nSIC-POVMs and MUBs."},"authors":{"kind":"list like","value":["O. Albouy (IPNL)","M. R. Kibler (IPNL)"],"string":"[\n \"O. Albouy (IPNL)\",\n \"M. R. Kibler (IPNL)\"\n]"},"arxiv_categories":{"kind":"list like","value":["quant-ph","math-ph","math.MP"],"string":"[\n \"quant-ph\",\n \"math-ph\",\n \"math.MP\"\n]"},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The importance of finite-dimensional spaces for quantum mechanics is well recognized (see for instance [1]-[3]). In particular, such spaces play a major role in quantum information theory, especially for quantum cryptography and quantum state tomography [4]- [27] .\n\nAlong this vein, a symmetric informationally complete (SIC) positive operator valued measure (POVM) is a set of operators acting on a finite Hilbert space [4]-[14] (see also [3] for an infinite Hilbert space) and mutually unbiased bases (MUBs) are specific bases for such a space [15]- [27] .\n\nThe introduction of POVMs goes back to the seventies [4]- [7] . The most general quantum measurement is represented by a POVM. In the present work, we will be interested 1 in SIC-POVMs, for which the statistics of the measurement allows the reconstruction of the quantum state. Moreover, those POVMs are endowed with an extra symmetry condition (see definition in Sec. 2). The notion of MUBs (see definition in Sec. 3), implicit or explicit in the seminal works of [15] - [18] , has been the object of numerous mathematical and physical investigations during the last two decades in connection with the so-called complementary observables. Unfortunately, the question to know, for a given Hilbert space of finite dimension d, whether there exist SIC-POVMs and how many MUBs there exist has remained an open one.\n\nThe aim of this note is to develop a unified approach to SIC-POVMs and MUBs based on a complex vector space of higher dimension, viz. d 2 instead of d. We then give a specific example of this approach grounded on the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) recently used for a study of entanglement of rotationally invariant spin systems [28] and for an angular momentum study of MUBs [26, 27] .\n\nMost of the notations in this work are standard. Let us simply mention that I is the identity operator, the bar indicates complex conjugation, A † denotes the adjoint of the operator A, δ a,b stands for the Kronecker symbol for a and b, and ∆(a, b, c) is 1 or 0 according as a, b and c satisfy or not the triangular inequality.\n\nLet C d be the standard Hilbert space of dimension d endowed with its usual inner product denoted by | . As is usual, we will identify a POVM with a nonorthogonal decomposition of the identity. Thus, a discrete SIC-POVM is a set {P x : x = 1, 2, • • • , d 2 } of d 2 nonnegative operators P x acting on C d , such that:\n\n• they satisfy the trace or symmetry condition\n\nTr (P x P y ) = 1 d + 1 , x = y; ( 1\n\n)\n\nmoreover, we will assume the operators P x are normalized, thus completing this 2 condition with\n\nTr P 2 x = 1; ( 2\n\n)\n\n• they form a decomposition of the identity\n\n1 d d 2 x=1 P x = I; ( 3\n\n)\n\n• they satisfy a completeness condition: the knowledge of the probabilities p x defined by p x = Tr(P x ρ) is sufficient to reconstruct the density matrix ρ.\n\nNow, let us develop each of the operators P x on an orthonormal (with respect to the Hilbert-Schmidt product) basis {u i : i = 1, 2, • • • , d 2 } of the space of linear operators on\n\nC d P x = d 2 i=1 v i (x)u i , ( 4\n\n)\n\nwhere the operators u i satisfy Tr(u † i u j ) = δ i,j . The operators P x are thus considered as vectors\n\nv(x) = (v 1 (x), v 2 (x), • • • , v d 2 (x)) ( 5\n\n)\n\nin the Hilbert space C d 2 of dimension d 2 and the determination of the operators P x is equivalent to the determination of the components v i (x) of v(x). In this language, the trace property (1) together with the normalization condition (2) give y) is the usual Hermitian product in C d 2 . In order to compare Eq. ( 6 ) with what usually happens in the search for SIC-POVMs, we suppose from now on that the operators P x are rank-one operators. Therefore, by putting\n\nv(x) • v(y) = 1 d + 1 (dδ x,y + 1) , ( 6\n\n) where v(x) • v(y) = d 2 i=1 v i (x)v i (\n\nP x = |Φ x Φ x | ( 7\n\n) 3 with |φ x ∈ C d , the trace property (1, 2) reads | Φ x |Φ y | 2 = 1 d + 1 (dδ x,y + 1) . ( 8\n\n)\n\nFrom this point of view, to find d 2 operators P x is equivalent to finding d 2 vectors |φ x in C d satisfying Eq. ( 8 ). At the price of an increase in the number of components from d 3 (for d 2 vectors in C d ) to d 4 (for d 2 vectors in C d 2 ), we have got rid of the square modulus to result in a single scalar product (compare Eqs. (6) and (8)), what may prove to be suitable for another way to search for SIC-POVMs. Moreover, our relation (6) is independent of any hypothesis on the rank of the operators P x . In fact, there exists a lot of relations among these d 4 coefficients that decrease the effective number of coefficients to be found and give structural constraints on them. Those relations are highly sensitive to the choice of the basis {u i : i = 1, 2, • • • , d 2 } and we are going to exhibit an example of such a set of relations by choosing the basis to consist of Racah unit tensors.\n\nThe cornerstone of this approach is to identify C d with a subspace ε(j) of constant angular momentum j = (d -1)/2. Such a subspace is spanned by the set {|j, m : m = -j, -j + 1, • • • , j}, where |j, m is an eigenvector of the square and the z-component of a generalized angular momentum operator. Let u (k) be the Racah unit tensor [29]\n\nof order k (with k = 0, 1, • • • , 2j) defined by its 2k + 1 components u (k) q (where q = -k, -k + 1, • • • , k) through u (k) q = j m=-j j m ′ =-j (-1) j-m j k j -m q m ′ |j, m j, m ′ |, ( 9\n\n)\n\nwhere (• • •) denotes a 3-jm Wigner symbol. For fixed j, the (2j + 1) 2 operators u\n\n(k) q (with k = 0, 1, • • • , 2j and q = -k, -k + 1, • • • , k) act on ε(j) ∼ C d and form a basis\n\nof the Hilbert space C N of dimension N = (2j + 1) 2 , the inner product in C N being the Hilbert-Schmidt product. The formulas (involving unit tensors, 3-jm and 6-j symbols) relevant for this work are given in Appendix (see also [29] to [31] ). We must remember that those Racah operators are not normalized to unity (see relation (46)). So this will generate an extra factor when defining v i (x).\n\n4 Each operator P x can be developed as a linear combination of the operators u (k) q .\n\nHence, we have\n\nP x = 2j k=0 k q=-k c kq (x)u (k) q , ( 10\n\n)\n\nwhere the unknown expansion coefficients c kq (x) are a priori complex numbers. The determination of the operators P x is thus equivalent to the determination of the coefficients c kq (x), which are formally given by\n\nc kq (x) = (2k + 1) Φ x |u (k) q |Φ x , ( 11\n\n)\n\nas can be seen by multiplying each member of Eq. (10) by the adjoint of u (ℓ) p and then using Eq. (46) of Appendix.\n\nBy defining the vector\n\nv(x) = (v 1 (x), v 2 (x), • • • , v N (x)), N = (2j + 1) 2 ( 12\n\n) via v i (x) = 1 √ 2k + 1 c kq (x), i = k 2 + k + q + 1, ( 13\n\n)\n\nthe following properties and relations are obtained.\n\n• The first component v 1 (x) of v(x) does not depend on x since c 00 (x) = 1 √ 2j + 1 ( 14\n\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }.\n\nProof: Take the trace of Eq. (10) and use Eq. (48) of Appendix.\n\n• The components v i (x) of v(x) satisfy the complex conjugation property described by c kq (x) = (-1) q c k-q (x) ( 15\n\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\n\nProof: Use the Hermitian property of P x and Eq. (43) of Appendix.\n\n• In terms of c kq , Eq. (6) reads\n\n2j k=0 1 2k + 1 k q=-k c kq (x)c kq (y) = 1 2(j + 1) [(2j + 1)δ x,y + 1] ( 16\n\n) for all x, y ∈ {1, 2, • • • , (2j + 1) 2 },\n\nwhere the sum over q is SO(3) rotationally invariant.\n\nProof: The proof is trivial.\n\n• The coefficients c kq (x) are solutions of the nonlinear system given by\n\n1 2K + 1 c KQ (x) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ k ℓ K -q -p Q × k ℓ K j j j c kq (x)c ℓp (x) ( 17\n\n) for all x ∈ {1, 2, • • • , (2j+1) 2 }, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K+1, • • • , K}.\n\nProof: Consider P 2 x = P x and use the coupling relation (51) of Appendix involving a 3-jm and a 6-j Wigner symbols.\n\nAs a corollary of the latter property, by taking K = 0 and using Eqs. (47) and (50) of Appendix, we get again the normalization relation v(x\n\n) 2 = v(x) • v(x) = 1.\n\n• All coefficients c kq (x) are connected through the sum rule\n\n(2j+1) 2 x=1 2j k=0 k q=-k c kq (x) j k j -m q m ′ = (-1) j-m (2j + 1)δ m,m ′ , ( 18\n\n)\n\nwhich turns out to be useful for global checking purposes.\n\nProof: Take the jm-jm ′ matrix element of the resolution of the identity in terms of the operators P x /(2j + 1).\n\nA complete set of MUBs in the Hilbert space\n\nC d is a set of d(d + 1) vectors |aα ∈ C d such that | aα|bβ | 2 = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 19\n\n) 6 where a = 0, 1, • • • , d and α = 0, 1, • • • , d -1.\n\nThe indices of type a refer to the bases and, for fixed a, the index α refers to one of the d vectors of the basis corresponding to a.\n\nWe know that such a complete set exists if d is a prime or the power of a prime (e.g., see [16]-[24]).\n\nThe approach developed in Sec. 2 for SIC-POVMs can be applied to MUBs too. Let us suppose that it is possible to\n\nfind d + 1 sets S a (with a = 0, 1, • • • , d) of vectors in C d , each set S a = {|aα : α = 0, 1, • • • , d -1} containing d vectors |aα such that Eq. ( 19\n\n)\n\nbe satisfied. This amounts to finding d(d + 1) projection operators\n\nΠ aα = |aα aα| ( 20\n\n)\n\nsatisfying the trace condition\n\nTr (Π aα Π bβ ) = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 21\n\n)\n\nwhere the trace is taken on C d . Therefore, they also form a nonorthogonal decomposition of the identity\n\n1 d + 1 d a=0 d-1 α=0 Π aα = I. ( 22\n\n)\n\nAs in Sec. 2, we develop each operator Π aα on an orthonormal basis with expansion coefficients w i (aα). Thus we get vectors w(aα) in\n\nC d 2 w(aα) = (w 1 (aα), w 2 (aα), • • • , w d 2 (aα)) ( 23\n\n) such that w(aα) • w(bβ) = δ α,β δ a,b + 1 d (1 -δ a,b ) ( 24\n\n) for all a, b ∈ {0, 1, • • • , d} and α, β ∈ {0, 1, • • • , d -1}.\n\nNow we draw the same relations as for POVMs by choosing the Racah operators to be our basis in C d 2 . We assume once again that the Hilbert space C d is realized by ε(j) with j = (d -1)/2. Then, each operator Π aα can be developed on the basis of the (2j + 1) 2 7 operators u (k) q as\n\nΠ aα = 2j k=0 k q=-k d kq (aα)u (k) q , ( 25\n\n)\n\nto be compared with Eq. (10). The expansion coefficients are d kq (aα) = (2k + 1) aα|u\n\n(k) q |aα ( 26\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1,\n\n• • • , 2j} and q ∈ {-k, -k + 1, • • • , k}. For a and α fixed, the complex coefficients d kq (aα) define a vector\n\nw(aα) = (w 1 (aα), w 2 (aα), • • • , w N (aα)) , N = (2j + 1) 2 ( 27\n\n)\n\nin the Hilbert space C N , the components of which are given by\n\nw i (aα) = 1 √ 2k + 1 d kq (aα), i = k 2 + k + q + 1. ( 28\n\n)\n\nWe are thus led to the following properties and relations. The proofs are similar to those in Sec. 2.\n\n• First component w 1 (aα) of w(aα):\n\nd 00 (aα) = 1 √ 2j + 1 ( 29\n\n) for all a ∈ {0, 1, • • • , 2j + 1} and α ∈ {0, 1, • • • , 2j}.\n\n• Complex conjugation property:\n\nd kq (aα) = (-1) q d k-q (aα) ( 30\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\n\n• Rotational invariance:\n\n2j k=0 1 2k + 1 k q=-k d kq (aα)d kq (bβ) = δ α,β δ a,b + 1 2j + 1 (1 -δ a,b ) ( 31\n\n) for all a, b ∈ {0, 1, • • • , 2j + 1} and α, β ∈ {0, 1, • • • , 2j}.\n\n• Tensor product formula: 1 2K + 1 d KQ (aα) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ\n\nk ℓ K -q -p Q × k ℓ K j j j d kq (aα)d ℓp (aα) ( 32\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K + 1, • • • , K}.\n\n• Sum rule:\n\n2j+1 a=0 2j α=0 2j k=0 k q=-k d kq (aα) j k j -m q m ′ = (-1) j-m 2(2j + 1)δ m,m ′ ( 33\n\n)\n\nwhich involves all coefficients d kq (aα).\n\nAlthough the structure of the relations in Sec. 1 on the one hand and Sec. 2 on the other hand is very similar, there are deep differences between the two sets of results. The similarities are reminiscent of the fact that both MUBs and SIC-POVMs can be linked to finite affine planes [12, 13, 22, 23, 25] and to complex projective 2-designs [8, 10, 19, 24] .\n\nOn the other side, there are two arguments in favor of the differences between relations (6) and (24). First, the problem of constructing SIC-POVMs in dimension d is not equivalent to the existence of an affine plane of order d [12, 13] . Second, there is a consensus around the conjecture according to which there exists a complete set of MUBs in dimension d if and only if there exists an affine plane of order d [22].\n\nIn dimension d, to find d 2 operators P x of a SIC-POVM acting on the Hilbert space\n\nC d amounts to find d 2 vectors v(x) in the Hilbert space C N with N = d 2 satisfying v x = 1, v(x) • v(y) = 1 d + 1 for x = y ( 34\n\n)\n\n(the norm v(x) of each vector v(x) is 1 and the angle ω xy of any pair of vectors v(x) and v(y) is ω xy = cos -1 [1/(d + 1)] for x = y).\n\nIn a similar way, to find d + 1 MUBs of C d is equivalent to find d + 1 sets S a (with a = 0, 1, • • • , d) of d vectors, i.e., d(d + 1) vectors in all, w(aα) in C N with N = d 2 satisfying w(aα) • w(aβ) = δ α,β , w(aα) • w(bβ) = 1 d for a = b (35) (each set S a consists of d orthonormalized vectors and the angle ω aαbβ of any vector w(aα) of a set S a with any vector w(bβ) of a set S b is ω aαbβ = cos -1 (1/d) for a = b).\n\nAccording to a well accepted conjecture [8, 10], SIC-POVMs should exist in any\n\ndimension. The present study shows that in order to prove this conjecture it is sufficient to prove that Eq. (34) admits solutions for any value of d.\n\nThe situation is different for MUBs. In dimension d, it is known that there exist d + 1 sets of d vectors of type |aα in C d satisfying Eq. (19) when d is a prime or the power of a prime. This shows that Eq. ( 35 ) can be solved for d prime or power of a prime. For d prime, it is possible to find an explicit solution of Eq. ( 19 ). In fact, we have [26, 27] |aα = 1 √ 2j + 1 j m=-j ω (j+m)(j-m+1)a/2+(j+m)\n\nα |j, m , ( 36\n\n) ω = exp i 2π 2j + 1 , j = 1 2 (d -1) ( 37\n\n) for a, α ∈ {0, 1, • • • , 2j} while |aα = |j, m ( 38\n\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j. Then, Eq. (26) yields d kq (aα) = 2k + 1 2j + 1 j m=-j j m ′ =-j ω θ(m,m ′ ) (-1) j-m j k j -m q m ′ , ( 39\n\n) θ(m, m ′ ) = (m -m ′ ) 1 2 (1 -m -m ′ )a + α ( 40\n\n) for a, α ∈ {0, 1, • • • , 2j} while d kq (aα) = δ q,0 (2k + 1)(-1) j-m j k j -m 0 m ( 41\n\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j.\n\nIt can be shown that Eqs. (40) and (41) are in agreement with the results of Sec. 3. We thus have a solution of the equations for 10 the results of Sec. 3 when d is prime. As an open problem, it would be worthwhile to find an explicit solution for the coefficients d kq (aα) when d = 2j + 1 is any positive power of a prime. Finally, note that to prove (or disprove) the conjecture according to which a complete set of MUBs in dimension d exists only if d is a prime or the power of a prime is equivalent to prove (or disprove) that Eq. ( 35 ) has a solution only if d is a prime or the power of a prime.\n\nAPPENDIX: WIGNER-RACAH ALGEBRA OF SU(2) ⊃ U(1)\n\nWe limit ourselves to those basic formulas for the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) which are necessary to derive the results of this paper. The summations in this appendix have to be extended to the allowed values for the involved magnetic and angular momentum quantum numbers.\n\nThe definition (9) of the components u (k) q of the Racah unit tensor u (k) yields\n\nj, m|u (k) q |j, m ′ = (-1) j-m j k j -m q m ′ , ( 42\n\n)\n\nfrom which we easily obtain the Hermitian conjugation property\n\nu (k) q † = (-1) q u (k) -q . ( 43\n\n)\n\nThe 3-jm Wigner symbol in Eq. (42) satisfies the orthogonality relations\n\nmm ′ j j ′ k m m ′ q j j ′ ℓ m m ′ p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j ′ , k) ( 44\n\n) and kq (2k + 1) j j ′ k m m ′ q j j ′ k M M ′ q = δ m,M δ m ′ ,M ′ . ( 45\n\n)\n\nThe trace relation on the space ε(j)\n\nTr u (k) q † u (ℓ) p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j, k) ( 46\n\n)\n\n11 easily follows by combining Eqs. (42) and (44). Furthermore, by introducing\n\nj j ′ 0 m -m ′ 0 = δ j,j ′ δ m,m ′ (-1) j-m 1 √ 2j + 1 ( 47\n\n)\n\nin Eq. ( 44 ), we obtain the sum rule\n\nm (-1) j-m j k j -m q m = 2j + 1δ k,0 δ q,0 ∆(j, k, j), ( 48\n\n)\n\nknown in spectroscopy as the barycenter theorem.\n\nThere are several relations involving 3-jm and 6-j symbols. In particular, we have\n\nmm ′ M (-1) j-M j j -m q M j ℓ j -M p m ′ j K j -m Q m ′ = (-1) 2j-Q k ℓ K -q -p Q k ℓ K j j j , ( 49\n\n)\n\nwhere {• • •} denotes a 6-j Wigner symbol (or W Racah coefficient). Note that the introduction of k ℓ 0 j j J = δ k,ℓ (-1) j+k+J 1 (2k + 1)(2j + 1) (50) in Eq. (49) gives back Eq. ( 44 ). Equation (49) is central in the derivation of the coupling relation\n\nu (k) q u (ℓ) p = KQ (-1) 2j-Q (2K + 1) k ℓ K -q -p Q k ℓ K j j j u (K) Q . ( 51\n\n)\n\nEquation (51) makes it possible to calculate the commutator [u\n\n(k) q , u (ℓ)\n\np ] which shows that the set {u (k) q\n\n: k = 0, 1, • • • , 2j; q = -k, -k + 1, • • •\n\n, k} can be used to span the Lie algebra of the unitary group U(2j + 1). The latter result is at the root of the expansions (17) and (32).\n\nAfter the submission of the present paper for publication in Journal of Russian Laser Research, a pre-print dealing with the existence of SIC-POVMs was posted on arXiv [32] .\n\nThe main result in [32] is that SIC-POVMs exist in all dimensions. As a corollary of this result, Eq. (34) admits solutions in any dimension.\n\n12\n\nThis work was presented at the International Conference on Squeezed States and Uncertainty Relations, University of Bradford, England (ICSSUR'07). The authors wish to thank the organizer A. Vourdas and are grateful to D. M. Appleby, V. I. Man'ko and M.\n\nPlanat for interesting comments. 13 References [1] A. 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B 20, 1942 (2006) [15] J. Schwinger, Proc. Nat. Acad. Sci. USA 46, 570 (1960) [16] P. Delsarte, J. M. Goethals and J. J. Seidel, Philips Res. Repts. 30, 91 (1975) 14 [17] I. D. Ivanović, J. Phys. A: Math. Gen. 14, 3241 (1981) [18] W. K. Wootters, Ann. Phys. (N.Y.) 176, 1 (1987) [19] H. Barnum, Preprint quant-ph/0205155 (2002) [20] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, Algorithmica 34, 512 (2002) [21] A. O. Pittenger and M. H. Rubin, Linear Alg. Appl. 390, 255 (2004) [22] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum Semiclassical Opt. 6, L19 (2004) [23] I. Bengtsson and Å. Ericsson, Open Syst. Inf. Dyn. 12, 107 (2005) [24] A. Klappenecker and M. Rötteler, Preprint quant-ph/0502031 (2005) [25] W. K. Wootters, Found. Phys. 36, 112 (2006) [26] M. R. Kibler and M. Planat, Int. J. Mod. Phys. B 20, 1802 (2006) [27] O. Albouy and M. R. Kibler, SIGMA 3, article 076 (2007) [28] H.-P. Breuer, J. Phys. A: Math. Gen. 38, 9019 (2005) [29] G. Racah, Phys. Rev. 62, 438 (1942) [30] U. Fano and G. Racah, \"Irreducible Tensorial Sets\", New York: Academic (1959) [31] M. Kibler and G. Grenet, J. Math. Phys. 21, 422 (1980) [32] J.L. Hall and A. Rao, Preprint quant-ph/0707.3002v1 (20 July 2007) 15"},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"INTRODUCTION","text":"The importance of finite-dimensional spaces for quantum mechanics is well recognized (see for instance [1]-[3]). In particular, such spaces play a major role in quantum information theory, especially for quantum cryptography and quantum state tomography [4]- [27] .\n\nAlong this vein, a symmetric informationally complete (SIC) positive operator valued measure (POVM) is a set of operators acting on a finite Hilbert space [4]-[14] (see also [3] for an infinite Hilbert space) and mutually unbiased bases (MUBs) are specific bases for such a space [15]- [27] .\n\nThe introduction of POVMs goes back to the seventies [4]- [7] . The most general quantum measurement is represented by a POVM. In the present work, we will be interested 1 in SIC-POVMs, for which the statistics of the measurement allows the reconstruction of the quantum state. Moreover, those POVMs are endowed with an extra symmetry condition (see definition in Sec. 2). The notion of MUBs (see definition in Sec. 3), implicit or explicit in the seminal works of [15] - [18] , has been the object of numerous mathematical and physical investigations during the last two decades in connection with the so-called complementary observables. Unfortunately, the question to know, for a given Hilbert space of finite dimension d, whether there exist SIC-POVMs and how many MUBs there exist has remained an open one.\n\nThe aim of this note is to develop a unified approach to SIC-POVMs and MUBs based on a complex vector space of higher dimension, viz. d 2 instead of d. We then give a specific example of this approach grounded on the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) recently used for a study of entanglement of rotationally invariant spin systems [28] and for an angular momentum study of MUBs [26, 27] .\n\nMost of the notations in this work are standard. Let us simply mention that I is the identity operator, the bar indicates complex conjugation, A † denotes the adjoint of the operator A, δ a,b stands for the Kronecker symbol for a and b, and ∆(a, b, c) is 1 or 0 according as a, b and c satisfy or not the triangular inequality."},{"section_type":"OTHER","section_title":"SIC-POVMs","text":"Let C d be the standard Hilbert space of dimension d endowed with its usual inner product denoted by | . As is usual, we will identify a POVM with a nonorthogonal decomposition of the identity. Thus, a discrete SIC-POVM is a set {P x : x = 1, 2, • • • , d 2 } of d 2 nonnegative operators P x acting on C d , such that:\n\n• they satisfy the trace or symmetry condition\n\nTr (P x P y ) = 1 d + 1 , x = y; ( 1\n\n)\n\nmoreover, we will assume the operators P x are normalized, thus completing this 2 condition with\n\nTr P 2 x = 1; ( 2\n\n)\n\n• they form a decomposition of the identity\n\n1 d d 2 x=1 P x = I; ( 3\n\n)\n\n• they satisfy a completeness condition: the knowledge of the probabilities p x defined by p x = Tr(P x ρ) is sufficient to reconstruct the density matrix ρ.\n\nNow, let us develop each of the operators P x on an orthonormal (with respect to the Hilbert-Schmidt product) basis {u i : i = 1, 2, • • • , d 2 } of the space of linear operators on\n\nC d P x = d 2 i=1 v i (x)u i , ( 4\n\n)\n\nwhere the operators u i satisfy Tr(u † i u j ) = δ i,j . The operators P x are thus considered as vectors\n\nv(x) = (v 1 (x), v 2 (x), • • • , v d 2 (x)) ( 5\n\n)\n\nin the Hilbert space C d 2 of dimension d 2 and the determination of the operators P x is equivalent to the determination of the components v i (x) of v(x). In this language, the trace property (1) together with the normalization condition (2) give y) is the usual Hermitian product in C d 2 . In order to compare Eq. ( 6 ) with what usually happens in the search for SIC-POVMs, we suppose from now on that the operators P x are rank-one operators. Therefore, by putting\n\nv(x) • v(y) = 1 d + 1 (dδ x,y + 1) , ( 6\n\n) where v(x) • v(y) = d 2 i=1 v i (x)v i (\n\nP x = |Φ x Φ x | ( 7\n\n) 3 with |φ x ∈ C d , the trace property (1, 2) reads | Φ x |Φ y | 2 = 1 d + 1 (dδ x,y + 1) . ( 8\n\n)\n\nFrom this point of view, to find d 2 operators P x is equivalent to finding d 2 vectors |φ x in C d satisfying Eq. ( 8 ). At the price of an increase in the number of components from d 3 (for d 2 vectors in C d ) to d 4 (for d 2 vectors in C d 2 ), we have got rid of the square modulus to result in a single scalar product (compare Eqs. (6) and (8)), what may prove to be suitable for another way to search for SIC-POVMs. Moreover, our relation (6) is independent of any hypothesis on the rank of the operators P x . In fact, there exists a lot of relations among these d 4 coefficients that decrease the effective number of coefficients to be found and give structural constraints on them. Those relations are highly sensitive to the choice of the basis {u i : i = 1, 2, • • • , d 2 } and we are going to exhibit an example of such a set of relations by choosing the basis to consist of Racah unit tensors.\n\nThe cornerstone of this approach is to identify C d with a subspace ε(j) of constant angular momentum j = (d -1)/2. Such a subspace is spanned by the set {|j, m : m = -j, -j + 1, • • • , j}, where |j, m is an eigenvector of the square and the z-component of a generalized angular momentum operator. Let u (k) be the Racah unit tensor [29]\n\nof order k (with k = 0, 1, • • • , 2j) defined by its 2k + 1 components u (k) q (where q = -k, -k + 1, • • • , k) through u (k) q = j m=-j j m ′ =-j (-1) j-m j k j -m q m ′ |j, m j, m ′ |, ( 9\n\n)\n\nwhere (• • •) denotes a 3-jm Wigner symbol. For fixed j, the (2j + 1) 2 operators u\n\n(k) q (with k = 0, 1, • • • , 2j and q = -k, -k + 1, • • • , k) act on ε(j) ∼ C d and form a basis\n\nof the Hilbert space C N of dimension N = (2j + 1) 2 , the inner product in C N being the Hilbert-Schmidt product. The formulas (involving unit tensors, 3-jm and 6-j symbols) relevant for this work are given in Appendix (see also [29] to [31] ). We must remember that those Racah operators are not normalized to unity (see relation (46)). So this will generate an extra factor when defining v i (x).\n\n4 Each operator P x can be developed as a linear combination of the operators u (k) q .\n\nHence, we have\n\nP x = 2j k=0 k q=-k c kq (x)u (k) q , ( 10\n\n)\n\nwhere the unknown expansion coefficients c kq (x) are a priori complex numbers. The determination of the operators P x is thus equivalent to the determination of the coefficients c kq (x), which are formally given by\n\nc kq (x) = (2k + 1) Φ x |u (k) q |Φ x , ( 11\n\n)\n\nas can be seen by multiplying each member of Eq. (10) by the adjoint of u (ℓ) p and then using Eq. (46) of Appendix.\n\nBy defining the vector\n\nv(x) = (v 1 (x), v 2 (x), • • • , v N (x)), N = (2j + 1) 2 ( 12\n\n) via v i (x) = 1 √ 2k + 1 c kq (x), i = k 2 + k + q + 1, ( 13\n\n)\n\nthe following properties and relations are obtained.\n\n• The first component v 1 (x) of v(x) does not depend on x since c 00 (x) = 1 √ 2j + 1 ( 14\n\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }.\n\nProof: Take the trace of Eq. (10) and use Eq. (48) of Appendix.\n\n• The components v i (x) of v(x) satisfy the complex conjugation property described by c kq (x) = (-1) q c k-q (x) ( 15\n\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\n\nProof: Use the Hermitian property of P x and Eq. (43) of Appendix.\n\n• In terms of c kq , Eq. (6) reads\n\n2j k=0 1 2k + 1 k q=-k c kq (x)c kq (y) = 1 2(j + 1) [(2j + 1)δ x,y + 1] ( 16\n\n) for all x, y ∈ {1, 2, • • • , (2j + 1) 2 },\n\nwhere the sum over q is SO(3) rotationally invariant.\n\nProof: The proof is trivial.\n\n• The coefficients c kq (x) are solutions of the nonlinear system given by\n\n1 2K + 1 c KQ (x) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ k ℓ K -q -p Q × k ℓ K j j j c kq (x)c ℓp (x) ( 17\n\n) for all x ∈ {1, 2, • • • , (2j+1) 2 }, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K+1, • • • , K}.\n\nProof: Consider P 2 x = P x and use the coupling relation (51) of Appendix involving a 3-jm and a 6-j Wigner symbols.\n\nAs a corollary of the latter property, by taking K = 0 and using Eqs. (47) and (50) of Appendix, we get again the normalization relation v(x\n\n) 2 = v(x) • v(x) = 1.\n\n• All coefficients c kq (x) are connected through the sum rule\n\n(2j+1) 2 x=1 2j k=0 k q=-k c kq (x) j k j -m q m ′ = (-1) j-m (2j + 1)δ m,m ′ , ( 18\n\n)\n\nwhich turns out to be useful for global checking purposes.\n\nProof: Take the jm-jm ′ matrix element of the resolution of the identity in terms of the operators P x /(2j + 1).\n\nA complete set of MUBs in the Hilbert space\n\nC d is a set of d(d + 1) vectors |aα ∈ C d such that | aα|bβ | 2 = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 19\n\n) 6 where a = 0, 1, • • • , d and α = 0, 1, • • • , d -1.\n\nThe indices of type a refer to the bases and, for fixed a, the index α refers to one of the d vectors of the basis corresponding to a.\n\nWe know that such a complete set exists if d is a prime or the power of a prime (e.g., see [16]-[24]).\n\nThe approach developed in Sec. 2 for SIC-POVMs can be applied to MUBs too. Let us suppose that it is possible to\n\nfind d + 1 sets S a (with a = 0, 1, • • • , d) of vectors in C d , each set S a = {|aα : α = 0, 1, • • • , d -1} containing d vectors |aα such that Eq. ( 19\n\n)\n\nbe satisfied. This amounts to finding d(d + 1) projection operators\n\nΠ aα = |aα aα| ( 20\n\n)\n\nsatisfying the trace condition\n\nTr (Π aα Π bβ ) = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 21\n\n)\n\nwhere the trace is taken on C d . Therefore, they also form a nonorthogonal decomposition of the identity\n\n1 d + 1 d a=0 d-1 α=0 Π aα = I. ( 22\n\n)\n\nAs in Sec. 2, we develop each operator Π aα on an orthonormal basis with expansion coefficients w i (aα). Thus we get vectors w(aα) in\n\nC d 2 w(aα) = (w 1 (aα), w 2 (aα), • • • , w d 2 (aα)) ( 23\n\n) such that w(aα) • w(bβ) = δ α,β δ a,b + 1 d (1 -δ a,b ) ( 24\n\n) for all a, b ∈ {0, 1, • • • , d} and α, β ∈ {0, 1, • • • , d -1}.\n\nNow we draw the same relations as for POVMs by choosing the Racah operators to be our basis in C d 2 . We assume once again that the Hilbert space C d is realized by ε(j) with j = (d -1)/2. Then, each operator Π aα can be developed on the basis of the (2j + 1) 2 7 operators u (k) q as\n\nΠ aα = 2j k=0 k q=-k d kq (aα)u (k) q , ( 25\n\n)\n\nto be compared with Eq. (10). The expansion coefficients are d kq (aα) = (2k + 1) aα|u\n\n(k) q |aα ( 26\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1,\n\n• • • , 2j} and q ∈ {-k, -k + 1, • • • , k}. For a and α fixed, the complex coefficients d kq (aα) define a vector\n\nw(aα) = (w 1 (aα), w 2 (aα), • • • , w N (aα)) , N = (2j + 1) 2 ( 27\n\n)\n\nin the Hilbert space C N , the components of which are given by\n\nw i (aα) = 1 √ 2k + 1 d kq (aα), i = k 2 + k + q + 1. ( 28\n\n)\n\nWe are thus led to the following properties and relations. The proofs are similar to those in Sec. 2.\n\n• First component w 1 (aα) of w(aα):\n\nd 00 (aα) = 1 √ 2j + 1 ( 29\n\n) for all a ∈ {0, 1, • • • , 2j + 1} and α ∈ {0, 1, • • • , 2j}.\n\n• Complex conjugation property:\n\nd kq (aα) = (-1) q d k-q (aα) ( 30\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\n\n• Rotational invariance:\n\n2j k=0 1 2k + 1 k q=-k d kq (aα)d kq (bβ) = δ α,β δ a,b + 1 2j + 1 (1 -δ a,b ) ( 31\n\n) for all a, b ∈ {0, 1, • • • , 2j + 1} and α, β ∈ {0, 1, • • • , 2j}.\n\n• Tensor product formula: 1 2K + 1 d KQ (aα) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ\n\nk ℓ K -q -p Q × k ℓ K j j j d kq (aα)d ℓp (aα) ( 32\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K + 1, • • • , K}.\n\n• Sum rule:\n\n2j+1 a=0 2j α=0 2j k=0 k q=-k d kq (aα) j k j -m q m ′ = (-1) j-m 2(2j + 1)δ m,m ′ ( 33\n\n)\n\nwhich involves all coefficients d kq (aα)."},{"section_type":"CONCLUSION","section_title":"CONCLUSIONS","text":"Although the structure of the relations in Sec. 1 on the one hand and Sec. 2 on the other hand is very similar, there are deep differences between the two sets of results. The similarities are reminiscent of the fact that both MUBs and SIC-POVMs can be linked to finite affine planes [12, 13, 22, 23, 25] and to complex projective 2-designs [8, 10, 19, 24] .\n\nOn the other side, there are two arguments in favor of the differences between relations (6) and (24). First, the problem of constructing SIC-POVMs in dimension d is not equivalent to the existence of an affine plane of order d [12, 13] . Second, there is a consensus around the conjecture according to which there exists a complete set of MUBs in dimension d if and only if there exists an affine plane of order d [22].\n\nIn dimension d, to find d 2 operators P x of a SIC-POVM acting on the Hilbert space\n\nC d amounts to find d 2 vectors v(x) in the Hilbert space C N with N = d 2 satisfying v x = 1, v(x) • v(y) = 1 d + 1 for x = y ( 34\n\n)\n\n(the norm v(x) of each vector v(x) is 1 and the angle ω xy of any pair of vectors v(x) and v(y) is ω xy = cos -1 [1/(d + 1)] for x = y).\n\nIn a similar way, to find d + 1 MUBs of C d is equivalent to find d + 1 sets S a (with a = 0, 1, • • • , d) of d vectors, i.e., d(d + 1) vectors in all, w(aα) in C N with N = d 2 satisfying w(aα) • w(aβ) = δ α,β , w(aα) • w(bβ) = 1 d for a = b (35) (each set S a consists of d orthonormalized vectors and the angle ω aαbβ of any vector w(aα) of a set S a with any vector w(bβ) of a set S b is ω aαbβ = cos -1 (1/d) for a = b).\n\nAccording to a well accepted conjecture [8, 10], SIC-POVMs should exist in any\n\ndimension. The present study shows that in order to prove this conjecture it is sufficient to prove that Eq. (34) admits solutions for any value of d.\n\nThe situation is different for MUBs. In dimension d, it is known that there exist d + 1 sets of d vectors of type |aα in C d satisfying Eq. (19) when d is a prime or the power of a prime. This shows that Eq. ( 35 ) can be solved for d prime or power of a prime. For d prime, it is possible to find an explicit solution of Eq. ( 19 ). In fact, we have [26, 27] |aα = 1 √ 2j + 1 j m=-j ω (j+m)(j-m+1)a/2+(j+m)\n\nα |j, m , ( 36\n\n) ω = exp i 2π 2j + 1 , j = 1 2 (d -1) ( 37\n\n) for a, α ∈ {0, 1, • • • , 2j} while |aα = |j, m ( 38\n\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j. Then, Eq. (26) yields d kq (aα) = 2k + 1 2j + 1 j m=-j j m ′ =-j ω θ(m,m ′ ) (-1) j-m j k j -m q m ′ , ( 39\n\n) θ(m, m ′ ) = (m -m ′ ) 1 2 (1 -m -m ′ )a + α ( 40\n\n) for a, α ∈ {0, 1, • • • , 2j} while d kq (aα) = δ q,0 (2k + 1)(-1) j-m j k j -m 0 m ( 41\n\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j.\n\nIt can be shown that Eqs. (40) and (41) are in agreement with the results of Sec. 3. We thus have a solution of the equations for 10 the results of Sec. 3 when d is prime. As an open problem, it would be worthwhile to find an explicit solution for the coefficients d kq (aα) when d = 2j + 1 is any positive power of a prime. Finally, note that to prove (or disprove) the conjecture according to which a complete set of MUBs in dimension d exists only if d is a prime or the power of a prime is equivalent to prove (or disprove) that Eq. ( 35 ) has a solution only if d is a prime or the power of a prime.\n\nAPPENDIX: WIGNER-RACAH ALGEBRA OF SU(2) ⊃ U(1)\n\nWe limit ourselves to those basic formulas for the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) which are necessary to derive the results of this paper. The summations in this appendix have to be extended to the allowed values for the involved magnetic and angular momentum quantum numbers.\n\nThe definition (9) of the components u (k) q of the Racah unit tensor u (k) yields\n\nj, m|u (k) q |j, m ′ = (-1) j-m j k j -m q m ′ , ( 42\n\n)\n\nfrom which we easily obtain the Hermitian conjugation property\n\nu (k) q † = (-1) q u (k) -q . ( 43\n\n)\n\nThe 3-jm Wigner symbol in Eq. (42) satisfies the orthogonality relations\n\nmm ′ j j ′ k m m ′ q j j ′ ℓ m m ′ p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j ′ , k) ( 44\n\n) and kq (2k + 1) j j ′ k m m ′ q j j ′ k M M ′ q = δ m,M δ m ′ ,M ′ . ( 45\n\n)\n\nThe trace relation on the space ε(j)\n\nTr u (k) q † u (ℓ) p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j, k) ( 46\n\n)\n\n11 easily follows by combining Eqs. (42) and (44). Furthermore, by introducing\n\nj j ′ 0 m -m ′ 0 = δ j,j ′ δ m,m ′ (-1) j-m 1 √ 2j + 1 ( 47\n\n)\n\nin Eq. ( 44 ), we obtain the sum rule\n\nm (-1) j-m j k j -m q m = 2j + 1δ k,0 δ q,0 ∆(j, k, j), ( 48\n\n)\n\nknown in spectroscopy as the barycenter theorem.\n\nThere are several relations involving 3-jm and 6-j symbols. In particular, we have\n\nmm ′ M (-1) j-M j j -m q M j ℓ j -M p m ′ j K j -m Q m ′ = (-1) 2j-Q k ℓ K -q -p Q k ℓ K j j j , ( 49\n\n)\n\nwhere {• • •} denotes a 6-j Wigner symbol (or W Racah coefficient). Note that the introduction of k ℓ 0 j j J = δ k,ℓ (-1) j+k+J 1 (2k + 1)(2j + 1) (50) in Eq. (49) gives back Eq. ( 44 ). Equation (49) is central in the derivation of the coupling relation\n\nu (k) q u (ℓ) p = KQ (-1) 2j-Q (2K + 1) k ℓ K -q -p Q k ℓ K j j j u (K) Q . ( 51\n\n)\n\nEquation (51) makes it possible to calculate the commutator [u\n\n(k) q , u (ℓ)\n\np ] which shows that the set {u (k) q\n\n: k = 0, 1, • • • , 2j; q = -k, -k + 1, • • •\n\n, k} can be used to span the Lie algebra of the unitary group U(2j + 1). The latter result is at the root of the expansions (17) and (32)."},{"section_type":"OTHER","section_title":"Note added in version 3","text":"After the submission of the present paper for publication in Journal of Russian Laser Research, a pre-print dealing with the existence of SIC-POVMs was posted on arXiv [32] .\n\nThe main result in [32] is that SIC-POVMs exist in all dimensions. As a corollary of this result, Eq. (34) admits solutions in any dimension.\n\n12"},{"section_type":"OTHER","section_title":"Acknowledgements","text":"This work was presented at the International Conference on Squeezed States and Uncertainty Relations, University of Bradford, England (ICSSUR'07). The authors wish to thank the organizer A. Vourdas and are grateful to D. M. Appleby, V. I. Man'ko and M.\n\nPlanat for interesting comments. 13 References [1] A. Peres, \"Quantum Theory: Concepts and Methods\", Dordrecht: Kluwer (1995) [2] A. Vourdas, J. Phys. A: Math. Gen. 38, 8453 (2005) [3] W. M. de Muynck, \"Foundations of Quantum Mechanics, an Empiricist Approach\", Dordrecht: Kluwer (2002) [4] J. M. Jauch and C. Piron, Helv. Phys. Acta 40, 559 (1967) [5] E. B. Davies and J. T. Levis, Comm. Math. Phys. 17, 239 (1970) [6] E. B. Davies, IEEE Trans. Inform. Theory IT-24, 596 (1978) [7] K. Kraus, \"States, Effects, and Operations\", Lect. Notes Phys. 190 (1983) [8] G. Zauner, Diploma Thesis, University of Wien (1999) [9] C. M. Caves, C. A. Fuchs and R. Schack, J. Math. Phys. 43, 4537 (2002) [10] J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, J. Math. Phys. 45, 2171 (2004) [11] D. M. Appleby, J. Math. Phys. 46, 052107 (2005) [12] M. Grassl, Proc. ERATO Conf. Quant. Inf. Science (EQIS 2004) ed. J. Gruska, Tokyo (2005) [13] M. Grassl, Elec. Notes Discrete Math. 20, 151 (2005) [14] S. Weigert, Int. J. Mod. Phys. B 20, 1942 (2006) [15] J. Schwinger, Proc. Nat. Acad. Sci. USA 46, 570 (1960) [16] P. Delsarte, J. M. Goethals and J. J. Seidel, Philips Res. Repts. 30, 91 (1975) 14 [17] I. D. Ivanović, J. Phys. A: Math. Gen. 14, 3241 (1981) [18] W. K. Wootters, Ann. Phys. (N.Y.) 176, 1 (1987) [19] H. Barnum, Preprint quant-ph/0205155 (2002) [20] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, Algorithmica 34, 512 (2002) [21] A. O. Pittenger and M. H. Rubin, Linear Alg. Appl. 390, 255 (2004) [22] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum Semiclassical Opt. 6, L19 (2004) [23] I. Bengtsson and Å. Ericsson, Open Syst. Inf. Dyn. 12, 107 (2005) [24] A. Klappenecker and M. Rötteler, Preprint quant-ph/0502031 (2005) [25] W. K. Wootters, Found. Phys. 36, 112 (2006) [26] M. R. Kibler and M. Planat, Int. J. Mod. Phys. B 20, 1802 (2006) [27] O. Albouy and M. R. Kibler, SIGMA 3, article 076 (2007) [28] H.-P. Breuer, J. Phys. A: Math. Gen. 38, 9019 (2005) [29] G. Racah, Phys. Rev. 62, 438 (1942) [30] U. Fano and G. Racah, \"Irreducible Tensorial Sets\", New York: Academic (1959) [31] M. Kibler and G. Grenet, J. Math. Phys. 21, 422 (1980) [32] J.L. Hall and A. Rao, Preprint quant-ph/0707.3002v1 (20 July 2007) 15"}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"INTRODUCTION\",\n \"text\": \"The importance of finite-dimensional spaces for quantum mechanics is well recognized (see for instance [1]-[3]). In particular, such spaces play a major role in quantum information theory, especially for quantum cryptography and quantum state tomography [4]- [27] .\\n\\nAlong this vein, a symmetric informationally complete (SIC) positive operator valued measure (POVM) is a set of operators acting on a finite Hilbert space [4]-[14] (see also [3] for an infinite Hilbert space) and mutually unbiased bases (MUBs) are specific bases for such a space [15]- [27] .\\n\\nThe introduction of POVMs goes back to the seventies [4]- [7] . The most general quantum measurement is represented by a POVM. In the present work, we will be interested 1 in SIC-POVMs, for which the statistics of the measurement allows the reconstruction of the quantum state. Moreover, those POVMs are endowed with an extra symmetry condition (see definition in Sec. 2). The notion of MUBs (see definition in Sec. 3), implicit or explicit in the seminal works of [15] - [18] , has been the object of numerous mathematical and physical investigations during the last two decades in connection with the so-called complementary observables. Unfortunately, the question to know, for a given Hilbert space of finite dimension d, whether there exist SIC-POVMs and how many MUBs there exist has remained an open one.\\n\\nThe aim of this note is to develop a unified approach to SIC-POVMs and MUBs based on a complex vector space of higher dimension, viz. d 2 instead of d. We then give a specific example of this approach grounded on the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) recently used for a study of entanglement of rotationally invariant spin systems [28] and for an angular momentum study of MUBs [26, 27] .\\n\\nMost of the notations in this work are standard. Let us simply mention that I is the identity operator, the bar indicates complex conjugation, A † denotes the adjoint of the operator A, δ a,b stands for the Kronecker symbol for a and b, and ∆(a, b, c) is 1 or 0 according as a, b and c satisfy or not the triangular inequality.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"SIC-POVMs\",\n \"text\": \"Let C d be the standard Hilbert space of dimension d endowed with its usual inner product denoted by | . As is usual, we will identify a POVM with a nonorthogonal decomposition of the identity. Thus, a discrete SIC-POVM is a set {P x : x = 1, 2, • • • , d 2 } of d 2 nonnegative operators P x acting on C d , such that:\\n\\n• they satisfy the trace or symmetry condition\\n\\nTr (P x P y ) = 1 d + 1 , x = y; ( 1\\n\\n)\\n\\nmoreover, we will assume the operators P x are normalized, thus completing this 2 condition with\\n\\nTr P 2 x = 1; ( 2\\n\\n)\\n\\n• they form a decomposition of the identity\\n\\n1 d d 2 x=1 P x = I; ( 3\\n\\n)\\n\\n• they satisfy a completeness condition: the knowledge of the probabilities p x defined by p x = Tr(P x ρ) is sufficient to reconstruct the density matrix ρ.\\n\\nNow, let us develop each of the operators P x on an orthonormal (with respect to the Hilbert-Schmidt product) basis {u i : i = 1, 2, • • • , d 2 } of the space of linear operators on\\n\\nC d P x = d 2 i=1 v i (x)u i , ( 4\\n\\n)\\n\\nwhere the operators u i satisfy Tr(u † i u j ) = δ i,j . The operators P x are thus considered as vectors\\n\\nv(x) = (v 1 (x), v 2 (x), • • • , v d 2 (x)) ( 5\\n\\n)\\n\\nin the Hilbert space C d 2 of dimension d 2 and the determination of the operators P x is equivalent to the determination of the components v i (x) of v(x). In this language, the trace property (1) together with the normalization condition (2) give y) is the usual Hermitian product in C d 2 . In order to compare Eq. ( 6 ) with what usually happens in the search for SIC-POVMs, we suppose from now on that the operators P x are rank-one operators. Therefore, by putting\\n\\nv(x) • v(y) = 1 d + 1 (dδ x,y + 1) , ( 6\\n\\n) where v(x) • v(y) = d 2 i=1 v i (x)v i (\\n\\nP x = |Φ x Φ x | ( 7\\n\\n) 3 with |φ x ∈ C d , the trace property (1, 2) reads | Φ x |Φ y | 2 = 1 d + 1 (dδ x,y + 1) . ( 8\\n\\n)\\n\\nFrom this point of view, to find d 2 operators P x is equivalent to finding d 2 vectors |φ x in C d satisfying Eq. ( 8 ). At the price of an increase in the number of components from d 3 (for d 2 vectors in C d ) to d 4 (for d 2 vectors in C d 2 ), we have got rid of the square modulus to result in a single scalar product (compare Eqs. (6) and (8)), what may prove to be suitable for another way to search for SIC-POVMs. Moreover, our relation (6) is independent of any hypothesis on the rank of the operators P x . In fact, there exists a lot of relations among these d 4 coefficients that decrease the effective number of coefficients to be found and give structural constraints on them. Those relations are highly sensitive to the choice of the basis {u i : i = 1, 2, • • • , d 2 } and we are going to exhibit an example of such a set of relations by choosing the basis to consist of Racah unit tensors.\\n\\nThe cornerstone of this approach is to identify C d with a subspace ε(j) of constant angular momentum j = (d -1)/2. Such a subspace is spanned by the set {|j, m : m = -j, -j + 1, • • • , j}, where |j, m is an eigenvector of the square and the z-component of a generalized angular momentum operator. Let u (k) be the Racah unit tensor [29]\\n\\nof order k (with k = 0, 1, • • • , 2j) defined by its 2k + 1 components u (k) q (where q = -k, -k + 1, • • • , k) through u (k) q = j m=-j j m ′ =-j (-1) j-m j k j -m q m ′ |j, m j, m ′ |, ( 9\\n\\n)\\n\\nwhere (• • •) denotes a 3-jm Wigner symbol. For fixed j, the (2j + 1) 2 operators u\\n\\n(k) q (with k = 0, 1, • • • , 2j and q = -k, -k + 1, • • • , k) act on ε(j) ∼ C d and form a basis\\n\\nof the Hilbert space C N of dimension N = (2j + 1) 2 , the inner product in C N being the Hilbert-Schmidt product. The formulas (involving unit tensors, 3-jm and 6-j symbols) relevant for this work are given in Appendix (see also [29] to [31] ). We must remember that those Racah operators are not normalized to unity (see relation (46)). So this will generate an extra factor when defining v i (x).\\n\\n4 Each operator P x can be developed as a linear combination of the operators u (k) q .\\n\\nHence, we have\\n\\nP x = 2j k=0 k q=-k c kq (x)u (k) q , ( 10\\n\\n)\\n\\nwhere the unknown expansion coefficients c kq (x) are a priori complex numbers. The determination of the operators P x is thus equivalent to the determination of the coefficients c kq (x), which are formally given by\\n\\nc kq (x) = (2k + 1) Φ x |u (k) q |Φ x , ( 11\\n\\n)\\n\\nas can be seen by multiplying each member of Eq. (10) by the adjoint of u (ℓ) p and then using Eq. (46) of Appendix.\\n\\nBy defining the vector\\n\\nv(x) = (v 1 (x), v 2 (x), • • • , v N (x)), N = (2j + 1) 2 ( 12\\n\\n) via v i (x) = 1 √ 2k + 1 c kq (x), i = k 2 + k + q + 1, ( 13\\n\\n)\\n\\nthe following properties and relations are obtained.\\n\\n• The first component v 1 (x) of v(x) does not depend on x since c 00 (x) = 1 √ 2j + 1 ( 14\\n\\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }.\\n\\nProof: Take the trace of Eq. (10) and use Eq. (48) of Appendix.\\n\\n• The components v i (x) of v(x) satisfy the complex conjugation property described by c kq (x) = (-1) q c k-q (x) ( 15\\n\\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\\n\\nProof: Use the Hermitian property of P x and Eq. (43) of Appendix.\\n\\n• In terms of c kq , Eq. (6) reads\\n\\n2j k=0 1 2k + 1 k q=-k c kq (x)c kq (y) = 1 2(j + 1) [(2j + 1)δ x,y + 1] ( 16\\n\\n) for all x, y ∈ {1, 2, • • • , (2j + 1) 2 },\\n\\nwhere the sum over q is SO(3) rotationally invariant.\\n\\nProof: The proof is trivial.\\n\\n• The coefficients c kq (x) are solutions of the nonlinear system given by\\n\\n1 2K + 1 c KQ (x) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ k ℓ K -q -p Q × k ℓ K j j j c kq (x)c ℓp (x) ( 17\\n\\n) for all x ∈ {1, 2, • • • , (2j+1) 2 }, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K+1, • • • , K}.\\n\\nProof: Consider P 2 x = P x and use the coupling relation (51) of Appendix involving a 3-jm and a 6-j Wigner symbols.\\n\\nAs a corollary of the latter property, by taking K = 0 and using Eqs. (47) and (50) of Appendix, we get again the normalization relation v(x\\n\\n) 2 = v(x) • v(x) = 1.\\n\\n• All coefficients c kq (x) are connected through the sum rule\\n\\n(2j+1) 2 x=1 2j k=0 k q=-k c kq (x) j k j -m q m ′ = (-1) j-m (2j + 1)δ m,m ′ , ( 18\\n\\n)\\n\\nwhich turns out to be useful for global checking purposes.\\n\\nProof: Take the jm-jm ′ matrix element of the resolution of the identity in terms of the operators P x /(2j + 1).\\n\\nA complete set of MUBs in the Hilbert space\\n\\nC d is a set of d(d + 1) vectors |aα ∈ C d such that | aα|bβ | 2 = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 19\\n\\n) 6 where a = 0, 1, • • • , d and α = 0, 1, • • • , d -1.\\n\\nThe indices of type a refer to the bases and, for fixed a, the index α refers to one of the d vectors of the basis corresponding to a.\\n\\nWe know that such a complete set exists if d is a prime or the power of a prime (e.g., see [16]-[24]).\\n\\nThe approach developed in Sec. 2 for SIC-POVMs can be applied to MUBs too. Let us suppose that it is possible to\\n\\nfind d + 1 sets S a (with a = 0, 1, • • • , d) of vectors in C d , each set S a = {|aα : α = 0, 1, • • • , d -1} containing d vectors |aα such that Eq. ( 19\\n\\n)\\n\\nbe satisfied. This amounts to finding d(d + 1) projection operators\\n\\nΠ aα = |aα aα| ( 20\\n\\n)\\n\\nsatisfying the trace condition\\n\\nTr (Π aα Π bβ ) = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 21\\n\\n)\\n\\nwhere the trace is taken on C d . Therefore, they also form a nonorthogonal decomposition of the identity\\n\\n1 d + 1 d a=0 d-1 α=0 Π aα = I. ( 22\\n\\n)\\n\\nAs in Sec. 2, we develop each operator Π aα on an orthonormal basis with expansion coefficients w i (aα). Thus we get vectors w(aα) in\\n\\nC d 2 w(aα) = (w 1 (aα), w 2 (aα), • • • , w d 2 (aα)) ( 23\\n\\n) such that w(aα) • w(bβ) = δ α,β δ a,b + 1 d (1 -δ a,b ) ( 24\\n\\n) for all a, b ∈ {0, 1, • • • , d} and α, β ∈ {0, 1, • • • , d -1}.\\n\\nNow we draw the same relations as for POVMs by choosing the Racah operators to be our basis in C d 2 . We assume once again that the Hilbert space C d is realized by ε(j) with j = (d -1)/2. Then, each operator Π aα can be developed on the basis of the (2j + 1) 2 7 operators u (k) q as\\n\\nΠ aα = 2j k=0 k q=-k d kq (aα)u (k) q , ( 25\\n\\n)\\n\\nto be compared with Eq. (10). The expansion coefficients are d kq (aα) = (2k + 1) aα|u\\n\\n(k) q |aα ( 26\\n\\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1,\\n\\n• • • , 2j} and q ∈ {-k, -k + 1, • • • , k}. For a and α fixed, the complex coefficients d kq (aα) define a vector\\n\\nw(aα) = (w 1 (aα), w 2 (aα), • • • , w N (aα)) , N = (2j + 1) 2 ( 27\\n\\n)\\n\\nin the Hilbert space C N , the components of which are given by\\n\\nw i (aα) = 1 √ 2k + 1 d kq (aα), i = k 2 + k + q + 1. ( 28\\n\\n)\\n\\nWe are thus led to the following properties and relations. The proofs are similar to those in Sec. 2.\\n\\n• First component w 1 (aα) of w(aα):\\n\\nd 00 (aα) = 1 √ 2j + 1 ( 29\\n\\n) for all a ∈ {0, 1, • • • , 2j + 1} and α ∈ {0, 1, • • • , 2j}.\\n\\n• Complex conjugation property:\\n\\nd kq (aα) = (-1) q d k-q (aα) ( 30\\n\\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\\n\\n• Rotational invariance:\\n\\n2j k=0 1 2k + 1 k q=-k d kq (aα)d kq (bβ) = δ α,β δ a,b + 1 2j + 1 (1 -δ a,b ) ( 31\\n\\n) for all a, b ∈ {0, 1, • • • , 2j + 1} and α, β ∈ {0, 1, • • • , 2j}.\\n\\n• Tensor product formula: 1 2K + 1 d KQ (aα) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ\\n\\nk ℓ K -q -p Q × k ℓ K j j j d kq (aα)d ℓp (aα) ( 32\\n\\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K + 1, • • • , K}.\\n\\n• Sum rule:\\n\\n2j+1 a=0 2j α=0 2j k=0 k q=-k d kq (aα) j k j -m q m ′ = (-1) j-m 2(2j + 1)δ m,m ′ ( 33\\n\\n)\\n\\nwhich involves all coefficients d kq (aα).\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"CONCLUSIONS\",\n \"text\": \"Although the structure of the relations in Sec. 1 on the one hand and Sec. 2 on the other hand is very similar, there are deep differences between the two sets of results. The similarities are reminiscent of the fact that both MUBs and SIC-POVMs can be linked to finite affine planes [12, 13, 22, 23, 25] and to complex projective 2-designs [8, 10, 19, 24] .\\n\\nOn the other side, there are two arguments in favor of the differences between relations (6) and (24). First, the problem of constructing SIC-POVMs in dimension d is not equivalent to the existence of an affine plane of order d [12, 13] . Second, there is a consensus around the conjecture according to which there exists a complete set of MUBs in dimension d if and only if there exists an affine plane of order d [22].\\n\\nIn dimension d, to find d 2 operators P x of a SIC-POVM acting on the Hilbert space\\n\\nC d amounts to find d 2 vectors v(x) in the Hilbert space C N with N = d 2 satisfying v x = 1, v(x) • v(y) = 1 d + 1 for x = y ( 34\\n\\n)\\n\\n(the norm v(x) of each vector v(x) is 1 and the angle ω xy of any pair of vectors v(x) and v(y) is ω xy = cos -1 [1/(d + 1)] for x = y).\\n\\nIn a similar way, to find d + 1 MUBs of C d is equivalent to find d + 1 sets S a (with a = 0, 1, • • • , d) of d vectors, i.e., d(d + 1) vectors in all, w(aα) in C N with N = d 2 satisfying w(aα) • w(aβ) = δ α,β , w(aα) • w(bβ) = 1 d for a = b (35) (each set S a consists of d orthonormalized vectors and the angle ω aαbβ of any vector w(aα) of a set S a with any vector w(bβ) of a set S b is ω aαbβ = cos -1 (1/d) for a = b).\\n\\nAccording to a well accepted conjecture [8, 10], SIC-POVMs should exist in any\\n\\ndimension. The present study shows that in order to prove this conjecture it is sufficient to prove that Eq. (34) admits solutions for any value of d.\\n\\nThe situation is different for MUBs. In dimension d, it is known that there exist d + 1 sets of d vectors of type |aα in C d satisfying Eq. (19) when d is a prime or the power of a prime. This shows that Eq. ( 35 ) can be solved for d prime or power of a prime. For d prime, it is possible to find an explicit solution of Eq. ( 19 ). In fact, we have [26, 27] |aα = 1 √ 2j + 1 j m=-j ω (j+m)(j-m+1)a/2+(j+m)\\n\\nα |j, m , ( 36\\n\\n) ω = exp i 2π 2j + 1 , j = 1 2 (d -1) ( 37\\n\\n) for a, α ∈ {0, 1, • • • , 2j} while |aα = |j, m ( 38\\n\\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j. Then, Eq. (26) yields d kq (aα) = 2k + 1 2j + 1 j m=-j j m ′ =-j ω θ(m,m ′ ) (-1) j-m j k j -m q m ′ , ( 39\\n\\n) θ(m, m ′ ) = (m -m ′ ) 1 2 (1 -m -m ′ )a + α ( 40\\n\\n) for a, α ∈ {0, 1, • • • , 2j} while d kq (aα) = δ q,0 (2k + 1)(-1) j-m j k j -m 0 m ( 41\\n\\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j.\\n\\nIt can be shown that Eqs. (40) and (41) are in agreement with the results of Sec. 3. We thus have a solution of the equations for 10 the results of Sec. 3 when d is prime. As an open problem, it would be worthwhile to find an explicit solution for the coefficients d kq (aα) when d = 2j + 1 is any positive power of a prime. Finally, note that to prove (or disprove) the conjecture according to which a complete set of MUBs in dimension d exists only if d is a prime or the power of a prime is equivalent to prove (or disprove) that Eq. ( 35 ) has a solution only if d is a prime or the power of a prime.\\n\\nAPPENDIX: WIGNER-RACAH ALGEBRA OF SU(2) ⊃ U(1)\\n\\nWe limit ourselves to those basic formulas for the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) which are necessary to derive the results of this paper. The summations in this appendix have to be extended to the allowed values for the involved magnetic and angular momentum quantum numbers.\\n\\nThe definition (9) of the components u (k) q of the Racah unit tensor u (k) yields\\n\\nj, m|u (k) q |j, m ′ = (-1) j-m j k j -m q m ′ , ( 42\\n\\n)\\n\\nfrom which we easily obtain the Hermitian conjugation property\\n\\nu (k) q † = (-1) q u (k) -q . ( 43\\n\\n)\\n\\nThe 3-jm Wigner symbol in Eq. (42) satisfies the orthogonality relations\\n\\nmm ′ j j ′ k m m ′ q j j ′ ℓ m m ′ p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j ′ , k) ( 44\\n\\n) and kq (2k + 1) j j ′ k m m ′ q j j ′ k M M ′ q = δ m,M δ m ′ ,M ′ . ( 45\\n\\n)\\n\\nThe trace relation on the space ε(j)\\n\\nTr u (k) q † u (ℓ) p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j, k) ( 46\\n\\n)\\n\\n11 easily follows by combining Eqs. (42) and (44). Furthermore, by introducing\\n\\nj j ′ 0 m -m ′ 0 = δ j,j ′ δ m,m ′ (-1) j-m 1 √ 2j + 1 ( 47\\n\\n)\\n\\nin Eq. ( 44 ), we obtain the sum rule\\n\\nm (-1) j-m j k j -m q m = 2j + 1δ k,0 δ q,0 ∆(j, k, j), ( 48\\n\\n)\\n\\nknown in spectroscopy as the barycenter theorem.\\n\\nThere are several relations involving 3-jm and 6-j symbols. In particular, we have\\n\\nmm ′ M (-1) j-M j j -m q M j ℓ j -M p m ′ j K j -m Q m ′ = (-1) 2j-Q k ℓ K -q -p Q k ℓ K j j j , ( 49\\n\\n)\\n\\nwhere {• • •} denotes a 6-j Wigner symbol (or W Racah coefficient). Note that the introduction of k ℓ 0 j j J = δ k,ℓ (-1) j+k+J 1 (2k + 1)(2j + 1) (50) in Eq. (49) gives back Eq. ( 44 ). Equation (49) is central in the derivation of the coupling relation\\n\\nu (k) q u (ℓ) p = KQ (-1) 2j-Q (2K + 1) k ℓ K -q -p Q k ℓ K j j j u (K) Q . ( 51\\n\\n)\\n\\nEquation (51) makes it possible to calculate the commutator [u\\n\\n(k) q , u (ℓ)\\n\\np ] which shows that the set {u (k) q\\n\\n: k = 0, 1, • • • , 2j; q = -k, -k + 1, • • •\\n\\n, k} can be used to span the Lie algebra of the unitary group U(2j + 1). The latter result is at the root of the expansions (17) and (32).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Note added in version 3\",\n \"text\": \"After the submission of the present paper for publication in Journal of Russian Laser Research, a pre-print dealing with the existence of SIC-POVMs was posted on arXiv [32] .\\n\\nThe main result in [32] is that SIC-POVMs exist in all dimensions. As a corollary of this result, Eq. (34) admits solutions in any dimension.\\n\\n12\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgements\",\n \"text\": \"This work was presented at the International Conference on Squeezed States and Uncertainty Relations, University of Bradford, England (ICSSUR'07). The authors wish to thank the organizer A. Vourdas and are grateful to D. M. Appleby, V. I. Man'ko and M.\\n\\nPlanat for interesting comments. 13 References [1] A. Peres, \\\"Quantum Theory: Concepts and Methods\\\", Dordrecht: Kluwer (1995) [2] A. Vourdas, J. Phys. A: Math. Gen. 38, 8453 (2005) [3] W. M. de Muynck, \\\"Foundations of Quantum Mechanics, an Empiricist Approach\\\", Dordrecht: Kluwer (2002) [4] J. M. Jauch and C. Piron, Helv. Phys. Acta 40, 559 (1967) [5] E. B. Davies and J. T. Levis, Comm. Math. Phys. 17, 239 (1970) [6] E. B. Davies, IEEE Trans. Inform. Theory IT-24, 596 (1978) [7] K. Kraus, \\\"States, Effects, and Operations\\\", Lect. Notes Phys. 190 (1983) [8] G. Zauner, Diploma Thesis, University of Wien (1999) [9] C. M. Caves, C. A. Fuchs and R. Schack, J. Math. Phys. 43, 4537 (2002) [10] J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, J. Math. Phys. 45, 2171 (2004) [11] D. M. Appleby, J. Math. Phys. 46, 052107 (2005) [12] M. Grassl, Proc. ERATO Conf. Quant. Inf. Science (EQIS 2004) ed. J. Gruska, Tokyo (2005) [13] M. Grassl, Elec. Notes Discrete Math. 20, 151 (2005) [14] S. Weigert, Int. J. Mod. Phys. B 20, 1942 (2006) [15] J. Schwinger, Proc. Nat. Acad. Sci. USA 46, 570 (1960) [16] P. Delsarte, J. M. Goethals and J. J. Seidel, Philips Res. Repts. 30, 91 (1975) 14 [17] I. D. Ivanović, J. Phys. A: Math. Gen. 14, 3241 (1981) [18] W. K. Wootters, Ann. Phys. (N.Y.) 176, 1 (1987) [19] H. Barnum, Preprint quant-ph/0205155 (2002) [20] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, Algorithmica 34, 512 (2002) [21] A. O. Pittenger and M. H. Rubin, Linear Alg. Appl. 390, 255 (2004) [22] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum Semiclassical Opt. 6, L19 (2004) [23] I. Bengtsson and Å. Ericsson, Open Syst. Inf. Dyn. 12, 107 (2005) [24] A. Klappenecker and M. Rötteler, Preprint quant-ph/0502031 (2005) [25] W. K. Wootters, Found. Phys. 36, 112 (2006) [26] M. R. Kibler and M. Planat, Int. J. Mod. Phys. B 20, 1802 (2006) [27] O. Albouy and M. R. Kibler, SIGMA 3, article 076 (2007) [28] H.-P. Breuer, J. Phys. A: Math. Gen. 38, 9019 (2005) [29] G. Racah, Phys. Rev. 62, 438 (1942) [30] U. Fano and G. Racah, \\\"Irreducible Tensorial Sets\\\", New York: Academic (1959) [31] M. Kibler and G. Grenet, J. Math. Phys. 21, 422 (1980) [32] J.L. Hall and A. Rao, Preprint quant-ph/0707.3002v1 (20 July 2007) 15\"\n }\n]"}}},{"rowIdx":93,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0516"},"arxiv_id":{"kind":"string","value":"0704.0516"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":""},"content_hash":{"kind":"string","value":"4a119e65b80170869ee3687296117b16f01ab82d1c97a18278fec0312c5f9da7"},"title":{"kind":"string","value":"Effects of Imperfect Gate Operations in Shor's Prime Factorization\n Algorithm"},"abstract":{"kind":"string","value":"The effects of imperfect gate operations in implementation of Shor's prime\nfactorization algorithm are investigated. The gate imperfections may be\nclassified into three categories: the systematic error, the random error, and\nthe one with combined errors. It is found that Shor's algorithm is robust\nagainst the systematic errors but is vulnerable to the random errors. Error\nthreshold is given to the algorithm for a given number $N$ to be factorized."},"authors":{"kind":"list like","value":["Hao Guo","Gui Lu Long and Yang Sun"],"string":"[\n \"Hao Guo\",\n \"Gui Lu Long and Yang Sun\"\n]"},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]"},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"Shor's factorization algorithm [1] is a very important quantum algorithm, through which one has demonstrated the power of quantum computers. It has greatly promoted the worldwide research in quantum computing over the past few years. In practice, however, quantum systems are subject to influence of environment, and in addition, quantum gate operations are often imperfect [2, 3] . Environment influence on the system can cause decoherence of quantum states, and gate imperfection leads to errors in quantum computing. Thanks to Shor's another important work, in which he showed that quantum error correlation can be corrected [4] . With quantum error correction scheme, errors arising from both decoherence and imperfection can be corrected.\n\nThere have been several works on the effects of decoherence on Shor's algorithm. Sun et al. discussed the effect of decoherence on the algorithm by modeling the environment [5] . Palma studied the effects of both decoherence and gate imperfection in ion trap quantum computers [6] . There have also been many other studies on the quantum algorithm [7, 8, 9, 10] .\n\nThe error correction scheme uses available resources. Thus it is important to study the robustness of the algorithm itself so that one can strike a balance between the amount of quantum error correction and the amount of qubits available. In this paper, we investigate the effects of gate imperfection on the efficiency of Shor's factorization algorithm. The results may guide us in practice to suppress deliberately those errors that influence the algorithm most sensitively. For those errors that do not affect the algorithm very much, we may ignore them as a good approximation. In addition, study of the robustness of algorithm to errors is important where one can not apply the quantum error correction at all, for instance, in cases that there are not enough qubits available.\n\nThe paper is organized as follows. Section II is devoted to an outline of Shor's algorithm and different error's modes. In Section III, we present the results. Finally, a short summary is given in Section IV.\n\nShor's algorithm consists of the following steps: 1) preparing a superposition of evenly distributed states |ψ = 1 √ q q-1 a=0 |a |0 ,\n\nwhere q = 2 L and N 2 ≤ q ≤ 2N 2 with N being the number to be factorized; 2) implementing y a modN and putting the results into the 2nd register\n\n|ψ 1 = 1 √ q q-1 a=0 |a |y a modN ;\n\n3) making a measument on the 2nd register; The state of the register is then\n\n|φ 2 = 1 √ A + 1 A j=0 |jr + l |z = y l = y jr+l modN\n\nwhere j ≤ q-l r = A.\n\n4) performing discrete Fourier transformation (DFT) on the first\n\nregister |φ 3 = c f (c) |c |z , where f (c) = √ r q q r -1 j=0 exp 2πi(jr + l) q = √ r q e 2πilc q q r -1 j=0 exp 2πijrc q .\n\n2 This term is nonzero only when c = k q r , with k = 0, 1, 2...r -1, which correspond to the peaks of the distribution in the measured results, and thus this term becomes f (c) = 1 √ r e 2πilc q . The Fourier transformation is important because it makes the state in the first register the same for all possible values in the 2nd register. The DFT is constructed by two basic gate operations: the single bit gate operation A j = 1 √ 2 1 1 1 -1 , which is also called the Walsh-Hadmard transformation, and the 2-bits controlled rotation\n\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e iθ jk with θ jk = π 2 k-j .\n\nThe gate sequence for implementing DFT is\n\n(A q-1 )(B q-2q-1 A q-2 ) . . . (B 0q-1 B 0q-2 . . . B 01 A 0 ). Errors can occur in both A j and B jk . A j is actually a rotation about y-axis through π 2 A j (θ) = e i h Syθ = I cos( θ 2 )-i sin( θ 2 )σ y = cos( θ 2 ) -sin( θ 2 ) sin( θ 2 ) cos( θ 2 )\n\n.\n\nIf the gate operation is not perfect, the rotation is not exactly π 2 . In this case, A j is a rotation of π 2 + 2δ A j (δ) = 1 √ 2 cos(δ) -sin(δ) -(sin(δ) + cos(δ)) sin(δ) + cos(δ) cos(δ) -sin(δ) .\n\nIf δ is very small, we have:\n\nA j (θ) = 1 √ 2 1 -δ -(1 + δ) 1 + δ+ 1 -δ .\n\nSimilarly, errors in B jk can be written as\n\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e i(θ jk +δ) .\n\nWith these errors, the DFT becomes\n\n|a → 1 √ q q-1 c=0 e i( 2π q/c +δc)a (1 + δ ′ c )|c = 1 √ q q-1 c=0 e i( 2πc q +δc)a (1 + δ ′ c )|c ,( 1\n\n)\n\nwhere δ c and δ ′ c denote the error of A j and B jk , respectively.\n\nLet us assume the following error modes: 1) systematic errors, where δ c or δ ′ c in (1) can only have systematic errors (EM 1 ); 2) random errors (EM 2 ), for which we assume that δ c or δ ′ c can only be random errors of the Gaussian or the uniform type; 3) coexistence of both systematic and random errors (EM 3 ). In the next section, we shall present the results of numerical simulations and discuss the effects of imperfect gate operation on the DFT algorithm, and thus on the Shor's algorithm.\n\nWe first discuss the influence of imperfect gate operations in the initial preparation\n\nA l-1 A l-2 ...A 0 |0...0 = 1 √ 2 (|0 + |1 + δ 1 (|0 -|1 )) ⊗ (|0 + |1 + δ 2 . = 1 √ 2 l 1 i1i2...in=0 |i 1 i 2 ...i n + 1 √ 2 l n R=1 δ n 1 i\n\nIf the errors are systematic, for instance, caused by the inaccurate calibration of the rotations, then δ 1 = δ 2 = . . . = δ n = δ. In this case, we can write the 2nd term as\n\n|ψ = 1 √ 2 l δ 1 i1i2...in=0 (2s -n)|i 1 i 2 ...i n ,\n\nwhere s stands for the number of 1's, and 2sn = s -(ns) is the difference in the number of 1's and 0's. Thus the results after the first procedure is\n\n1 √ 2 l 2 L -1 a=0 (|a + δ(2s -n)|a ) = 1 √ 2 l a=0 (1 + δ a )|a . (2)\n\nThis implies that after the procedure, the amplitude of each state is no longer equal, but have slight difference. Combining the effect in the initialization and in the DFT, we have\n\n(1 + δ a )(1 + δ c )e i( 2πc q +δ ′ c )a . = (1 + δ ′′ )e i( 2πc q +δ ′ c )a , where δ ′′ c = δ c + δ a . In the DFT, we have |ψ ⇒ √ r q q-1 c=0 q r -1 j=0 (1 + δ j )e i( 2πc q +δ ′ j )(jr+l) |c ,\n\nwhere we have rewrite δ ′′ as δ j here. Let P c denote the probability of getting the state |c after we perform a measurement, we have 3 ), we find that after the last measurement, each state can be extracted with a probability which is nonzero, and the offset l can't be eliminated. Eq. ( 3 ) is very complicated, so we will make some predigestions to discuss different error modes for convenience. Generally speaking, the influence of exponential error δ j is more remarkable than δ j , so we can omit the error δ j , thus\n\nP c = r q 2 q r -1 m=0 q r -1 k=0 (1 + δ m )(1 + δ k )e i( 2πc q +δ ′ m )(mr+l) ×e -i( 2πc q +δ ′ k )(kr+l) = r q 2 m k (1 + δ m )(1 + δ k ) cos[ 2πc q r(m -k) + (mr + l)δ ′ m -( From Eq. (\n\nDFT q |φ = √ r q q-1 c=0 q r j=0 e i( 2πc q +δ ′ j )(jr+l) |c . 3\n\nA. Case 1 If only systematic errors (EM 1 ) are considered, namely, all the δ j 's are equal, then f (c) can be given analytically f (c) = √ r q q r -1 j=0 e i( 2πc q +δ)(jr+l) = √ r q e il( 2πc q +δ) 1e i( 2πc q +δ)q\n\n1 -e i( 2πc q +δ)r ( 4\n\n)\n\nThe relative probability of finding c is\n\nP c = f (c) 2 = r q 2 sin 2 ( δq 2 ) sin 2 ( πcr q + δr 2 ) , and if c = k q r , then P c = r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) .\n\nIt can be easily seen that lim δ→0 P c = 1 r , which is just the case that no error is considered.\n\nWhen δ takes certain values, say, δ = 2 r (k -r q )π where k is an integer, then the summation in Eq. (4) is on longer valid. In our simulation, δ does not take these values. Here we consider the case where q = 2 7 = 128 and r = 4. For comparisons, we have drawn the relative probability for obtaining state c in Fig.1. for this given example. We have found the following results: (i) When δ is small, the errors do hardly influence the final result, for instance when c = k q r , then\n\nlim δ→0 P c = lim δ→0 r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) = 1 r .\n\nThe probability distribution is almost identical to those without errors.\n\n(ii) Let us increase δ gradually, from Fig. 2 , we see that a gradual change in the probability distribution takes place. (Here, we again consider the relative probabilities) When δ is increased to certain values, the positions of peaks change greatly. For instance at δ = 0.05, there appears a peak at c=127, whereas it is P c = 0 when no systematic errors are present. In general, the influence of systematic errors on the algorithm is a shift of the peak positions. This influences the final results directly.\n\nWhen both random errors and systematic errors are present, we add random errors to the simulation. To see the effect of different mode of random errors, we use two random number generators. One is the Gaussian mode and the other is the uniform mode. In this case, the error has the form δ = δ 0 + s, where δ 0 is the systematic error. s has a probability distribution with respect to c, depending on the uniform or the Gaussian distribution. When δ 0 = 0, we have only random errors which is our error mode 2. When δ 0 = 0, we have error mode 3. For the uniform distribution, s ∼ ±s max × u(0, 1) where u(0, 1) is evenly distributed in [0, 1] . s max indicates the maximum deviation from δ 0 . For Gaussian distribution, s ∼ N (0, σ 0 ). Through the figure, we see the following: (1) When only random errors are present (δ 0 = 0), the peak positions are not affected by these random errors. However, different random error modes cause similar results. The results for uniform random error mode are shown in Fig. 3 . For the uniform distribution error mode, with increasing δ max , the final probability distribution of the final results become irregular. In particular, when δ max is very large, all the patterns are destroyed and is hardly recognizable. Many unexpected small peaks appear. For the Gaussian distribution error mode, as shown in Fig. 4 , the influence of the error is more serious. This is because in Gaussian distribution, there is no cut-off of errors. Large errors can occur although their probability is small. The influence of σ 0 on the final results is also sensitive, because it determines the shape of the distribution. When σ 0 increases, the final probability distribution becomes very messy. A small change in σ 0 can cause a big change in the final results. (2) When δ 0 = 0, which corresponds to error mode 3, the effect is seen as to shift the positions of the peaks in addition to the influences of the random errors.\n\nTo summarize, we have analyzed the errors in Shor's factorization algorithm. It has been seen that the effect of the systematic errors is to shift the positions of the peaks, whereas the random errors change the shape of the probability distribution. For systematic errors, the shape of the distribution of the final results is hardly destroyed, though displaced. We can still use the result with several trial guesses to obtain the right results because the peak positions are shifted only slightly. However, the random errors are detrimental to the algorithm and should be reduced as much as possible. It is different from the case with Grover's algorithm where systematic errors are disastrous while random errors are less harmful [10].\n\n[1] P.W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S.\n\nGoldwasser (IEEE Computer Society Press, Los Alamitos, CA, 1994) p.124.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 p c FIG. 1: Relative probability for finding state c in the absence of errors.\n\n[2] A. Ekert and R. Jozsa, Rev. Mod. Phys. 68 (1996) 733. [3] W.G. Unruh, Phys. Rev A51 (1995) 992. [4] I. Chuang and R. laflamme, \"Quantum error correction by codding\" (1995) quant-ph/9511003. [5] C.P. Sun, H. Zhan and X.F. Liu, Phys. Rew. A58 (1998) 1810. [6] G.M. Palma, K.A. Suominen and A.K. Ekert, Proc. R. Soc. London, A 452 (1996) 567. [7] R.P. Feynman, Int. J. Theo. Phys., 21 (1982) 467. [8] D. Deutsch, Proc. R. Soc. Land. A 400 (1985) 97. [9] L.K. Grover, Phys. Rev, Lett, 79 (1997) 325. [10] G.L. Long, Y.S. Li, W.L. Zhang, C.C. Tu, Phys. Rev. A 61 (2000) 042305. [11] L.K. Grover, Phys. Rev. Lett, 80 (1998) 4329.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (4) (3) c P (2) -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 c P -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c FIG. 2: The same as Fig.1. with systematic errors. In subfigures (1), (2), (3), (4), δ are 0.02, 0.03, 0.05 respectively. In sub-figure (4), the curve with solid circles(with higher peaks) is the result with δ = 0.1, and the one without solid circles(with lower peaks) denotes the result with δ = 0.33.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (3) c P (1) -20 0 20 40 60 80 100 120 140 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (4) c P (2)\n\nFIG. 3: The same as Fig.1. with uniform random errors. In sub-figures (1), (2), (3), (4), smax are set to 0.01, 0.03, 0.05, 0.1 respectively.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 (2) P c -20 0 20 40 60 80 100 120 140 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 c P (3) -20 0 20 40 60 80 100 120 140 0.00 0.02 0.04 0.06 0.08 c P (4)\n\nFIG. 4: The same as Fig.1. with Gaussian random errors and systematic errors. In sub-figures (1), (2), and (3) τ are set to 0.01, 0.03 and 0.05 respectively, and δ0 = 0(without systematic errors). In sub-figure (4), both systematic and random Gaussian errors exist, where δ0 = 0.33, τ = 0.02."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"Shor's factorization algorithm [1] is a very important quantum algorithm, through which one has demonstrated the power of quantum computers. It has greatly promoted the worldwide research in quantum computing over the past few years. In practice, however, quantum systems are subject to influence of environment, and in addition, quantum gate operations are often imperfect [2, 3] . Environment influence on the system can cause decoherence of quantum states, and gate imperfection leads to errors in quantum computing. Thanks to Shor's another important work, in which he showed that quantum error correlation can be corrected [4] . With quantum error correction scheme, errors arising from both decoherence and imperfection can be corrected.\n\nThere have been several works on the effects of decoherence on Shor's algorithm. Sun et al. discussed the effect of decoherence on the algorithm by modeling the environment [5] . Palma studied the effects of both decoherence and gate imperfection in ion trap quantum computers [6] . There have also been many other studies on the quantum algorithm [7, 8, 9, 10] .\n\nThe error correction scheme uses available resources. Thus it is important to study the robustness of the algorithm itself so that one can strike a balance between the amount of quantum error correction and the amount of qubits available. In this paper, we investigate the effects of gate imperfection on the efficiency of Shor's factorization algorithm. The results may guide us in practice to suppress deliberately those errors that influence the algorithm most sensitively. For those errors that do not affect the algorithm very much, we may ignore them as a good approximation. In addition, study of the robustness of algorithm to errors is important where one can not apply the quantum error correction at all, for instance, in cases that there are not enough qubits available.\n\nThe paper is organized as follows. Section II is devoted to an outline of Shor's algorithm and different error's modes. In Section III, we present the results. Finally, a short summary is given in Section IV."},{"section_type":"OTHER","section_title":"II. SHOR'S ALGORITHM AND ERROR'S MODES","text":"Shor's algorithm consists of the following steps: 1) preparing a superposition of evenly distributed states |ψ = 1 √ q q-1 a=0 |a |0 ,\n\nwhere q = 2 L and N 2 ≤ q ≤ 2N 2 with N being the number to be factorized; 2) implementing y a modN and putting the results into the 2nd register\n\n|ψ 1 = 1 √ q q-1 a=0 |a |y a modN ;\n\n3) making a measument on the 2nd register; The state of the register is then\n\n|φ 2 = 1 √ A + 1 A j=0 |jr + l |z = y l = y jr+l modN\n\nwhere j ≤ q-l r = A.\n\n4) performing discrete Fourier transformation (DFT) on the first\n\nregister |φ 3 = c f (c) |c |z , where f (c) = √ r q q r -1 j=0 exp 2πi(jr + l) q = √ r q e 2πilc q q r -1 j=0 exp 2πijrc q .\n\n2 This term is nonzero only when c = k q r , with k = 0, 1, 2...r -1, which correspond to the peaks of the distribution in the measured results, and thus this term becomes f (c) = 1 √ r e 2πilc q . The Fourier transformation is important because it makes the state in the first register the same for all possible values in the 2nd register. The DFT is constructed by two basic gate operations: the single bit gate operation A j = 1 √ 2 1 1 1 -1 , which is also called the Walsh-Hadmard transformation, and the 2-bits controlled rotation\n\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e iθ jk with θ jk = π 2 k-j .\n\nThe gate sequence for implementing DFT is\n\n(A q-1 )(B q-2q-1 A q-2 ) . . . (B 0q-1 B 0q-2 . . . B 01 A 0 ). Errors can occur in both A j and B jk . A j is actually a rotation about y-axis through π 2 A j (θ) = e i h Syθ = I cos( θ 2 )-i sin( θ 2 )σ y = cos( θ 2 ) -sin( θ 2 ) sin( θ 2 ) cos( θ 2 )\n\n.\n\nIf the gate operation is not perfect, the rotation is not exactly π 2 . In this case, A j is a rotation of π 2 + 2δ A j (δ) = 1 √ 2 cos(δ) -sin(δ) -(sin(δ) + cos(δ)) sin(δ) + cos(δ) cos(δ) -sin(δ) .\n\nIf δ is very small, we have:\n\nA j (θ) = 1 √ 2 1 -δ -(1 + δ) 1 + δ+ 1 -δ .\n\nSimilarly, errors in B jk can be written as\n\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e i(θ jk +δ) .\n\nWith these errors, the DFT becomes\n\n|a → 1 √ q q-1 c=0 e i( 2π q/c +δc)a (1 + δ ′ c )|c = 1 √ q q-1 c=0 e i( 2πc q +δc)a (1 + δ ′ c )|c ,( 1\n\n)\n\nwhere δ c and δ ′ c denote the error of A j and B jk , respectively.\n\nLet us assume the following error modes: 1) systematic errors, where δ c or δ ′ c in (1) can only have systematic errors (EM 1 ); 2) random errors (EM 2 ), for which we assume that δ c or δ ′ c can only be random errors of the Gaussian or the uniform type; 3) coexistence of both systematic and random errors (EM 3 ). In the next section, we shall present the results of numerical simulations and discuss the effects of imperfect gate operation on the DFT algorithm, and thus on the Shor's algorithm."},{"section_type":"OTHER","section_title":"III. INFLUENCE OF IMPERFECT GATE OPERATIONS","text":"We first discuss the influence of imperfect gate operations in the initial preparation\n\nA l-1 A l-2 ...A 0 |0...0 = 1 √ 2 (|0 + |1 + δ 1 (|0 -|1 )) ⊗ (|0 + |1 + δ 2 . = 1 √ 2 l 1 i1i2...in=0 |i 1 i 2 ...i n + 1 √ 2 l n R=1 δ n 1 i\n\nIf the errors are systematic, for instance, caused by the inaccurate calibration of the rotations, then δ 1 = δ 2 = . . . = δ n = δ. In this case, we can write the 2nd term as\n\n|ψ = 1 √ 2 l δ 1 i1i2...in=0 (2s -n)|i 1 i 2 ...i n ,\n\nwhere s stands for the number of 1's, and 2sn = s -(ns) is the difference in the number of 1's and 0's. Thus the results after the first procedure is\n\n1 √ 2 l 2 L -1 a=0 (|a + δ(2s -n)|a ) = 1 √ 2 l a=0 (1 + δ a )|a . (2)\n\nThis implies that after the procedure, the amplitude of each state is no longer equal, but have slight difference. Combining the effect in the initialization and in the DFT, we have\n\n(1 + δ a )(1 + δ c )e i( 2πc q +δ ′ c )a . = (1 + δ ′′ )e i( 2πc q +δ ′ c )a , where δ ′′ c = δ c + δ a . In the DFT, we have |ψ ⇒ √ r q q-1 c=0 q r -1 j=0 (1 + δ j )e i( 2πc q +δ ′ j )(jr+l) |c ,\n\nwhere we have rewrite δ ′′ as δ j here. Let P c denote the probability of getting the state |c after we perform a measurement, we have 3 ), we find that after the last measurement, each state can be extracted with a probability which is nonzero, and the offset l can't be eliminated. Eq. ( 3 ) is very complicated, so we will make some predigestions to discuss different error modes for convenience. Generally speaking, the influence of exponential error δ j is more remarkable than δ j , so we can omit the error δ j , thus\n\nP c = r q 2 q r -1 m=0 q r -1 k=0 (1 + δ m )(1 + δ k )e i( 2πc q +δ ′ m )(mr+l) ×e -i( 2πc q +δ ′ k )(kr+l) = r q 2 m k (1 + δ m )(1 + δ k ) cos[ 2πc q r(m -k) + (mr + l)δ ′ m -( From Eq. (\n\nDFT q |φ = √ r q q-1 c=0 q r j=0 e i( 2πc q +δ ′ j )(jr+l) |c . 3\n\nA. Case 1 If only systematic errors (EM 1 ) are considered, namely, all the δ j 's are equal, then f (c) can be given analytically f (c) = √ r q q r -1 j=0 e i( 2πc q +δ)(jr+l) = √ r q e il( 2πc q +δ) 1e i( 2πc q +δ)q\n\n1 -e i( 2πc q +δ)r ( 4\n\n)\n\nThe relative probability of finding c is\n\nP c = f (c) 2 = r q 2 sin 2 ( δq 2 ) sin 2 ( πcr q + δr 2 ) , and if c = k q r , then P c = r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) .\n\nIt can be easily seen that lim δ→0 P c = 1 r , which is just the case that no error is considered.\n\nWhen δ takes certain values, say, δ = 2 r (k -r q )π where k is an integer, then the summation in Eq. (4) is on longer valid. In our simulation, δ does not take these values. Here we consider the case where q = 2 7 = 128 and r = 4. For comparisons, we have drawn the relative probability for obtaining state c in Fig.1. for this given example. We have found the following results: (i) When δ is small, the errors do hardly influence the final result, for instance when c = k q r , then\n\nlim δ→0 P c = lim δ→0 r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) = 1 r .\n\nThe probability distribution is almost identical to those without errors.\n\n(ii) Let us increase δ gradually, from Fig. 2 , we see that a gradual change in the probability distribution takes place. (Here, we again consider the relative probabilities) When δ is increased to certain values, the positions of peaks change greatly. For instance at δ = 0.05, there appears a peak at c=127, whereas it is P c = 0 when no systematic errors are present. In general, the influence of systematic errors on the algorithm is a shift of the peak positions. This influences the final results directly."},{"section_type":"OTHER","section_title":"B. Case 2","text":"When both random errors and systematic errors are present, we add random errors to the simulation. To see the effect of different mode of random errors, we use two random number generators. One is the Gaussian mode and the other is the uniform mode. In this case, the error has the form δ = δ 0 + s, where δ 0 is the systematic error. s has a probability distribution with respect to c, depending on the uniform or the Gaussian distribution. When δ 0 = 0, we have only random errors which is our error mode 2. When δ 0 = 0, we have error mode 3. For the uniform distribution, s ∼ ±s max × u(0, 1) where u(0, 1) is evenly distributed in [0, 1] . s max indicates the maximum deviation from δ 0 . For Gaussian distribution, s ∼ N (0, σ 0 ). Through the figure, we see the following: (1) When only random errors are present (δ 0 = 0), the peak positions are not affected by these random errors. However, different random error modes cause similar results. The results for uniform random error mode are shown in Fig. 3 . For the uniform distribution error mode, with increasing δ max , the final probability distribution of the final results become irregular. In particular, when δ max is very large, all the patterns are destroyed and is hardly recognizable. Many unexpected small peaks appear. For the Gaussian distribution error mode, as shown in Fig. 4 , the influence of the error is more serious. This is because in Gaussian distribution, there is no cut-off of errors. Large errors can occur although their probability is small. The influence of σ 0 on the final results is also sensitive, because it determines the shape of the distribution. When σ 0 increases, the final probability distribution becomes very messy. A small change in σ 0 can cause a big change in the final results. (2) When δ 0 = 0, which corresponds to error mode 3, the effect is seen as to shift the positions of the peaks in addition to the influences of the random errors."},{"section_type":"OTHER","section_title":"IV. SUMMARY","text":"To summarize, we have analyzed the errors in Shor's factorization algorithm. It has been seen that the effect of the systematic errors is to shift the positions of the peaks, whereas the random errors change the shape of the probability distribution. For systematic errors, the shape of the distribution of the final results is hardly destroyed, though displaced. We can still use the result with several trial guesses to obtain the right results because the peak positions are shifted only slightly. However, the random errors are detrimental to the algorithm and should be reduced as much as possible. It is different from the case with Grover's algorithm where systematic errors are disastrous while random errors are less harmful [10].\n\n[1] P.W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S.\n\nGoldwasser (IEEE Computer Society Press, Los Alamitos, CA, 1994) p.124.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 p c FIG. 1: Relative probability for finding state c in the absence of errors.\n\n[2] A. Ekert and R. Jozsa, Rev. Mod. Phys. 68 (1996) 733. [3] W.G. Unruh, Phys. Rev A51 (1995) 992. [4] I. Chuang and R. laflamme, \"Quantum error correction by codding\" (1995) quant-ph/9511003. [5] C.P. Sun, H. Zhan and X.F. Liu, Phys. Rew. A58 (1998) 1810. [6] G.M. Palma, K.A. Suominen and A.K. Ekert, Proc. R. Soc. London, A 452 (1996) 567. [7] R.P. Feynman, Int. J. Theo. Phys., 21 (1982) 467. [8] D. Deutsch, Proc. R. Soc. Land. A 400 (1985) 97. [9] L.K. Grover, Phys. Rev, Lett, 79 (1997) 325. [10] G.L. Long, Y.S. Li, W.L. Zhang, C.C. Tu, Phys. Rev. A 61 (2000) 042305. [11] L.K. Grover, Phys. Rev. Lett, 80 (1998) 4329.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (4) (3) c P (2) -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 c P -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c FIG. 2: The same as Fig.1. with systematic errors. In subfigures (1), (2), (3), (4), δ are 0.02, 0.03, 0.05 respectively. In sub-figure (4), the curve with solid circles(with higher peaks) is the result with δ = 0.1, and the one without solid circles(with lower peaks) denotes the result with δ = 0.33.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (3) c P (1) -20 0 20 40 60 80 100 120 140 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (4) c P (2)\n\nFIG. 3: The same as Fig.1. with uniform random errors. In sub-figures (1), (2), (3), (4), smax are set to 0.01, 0.03, 0.05, 0.1 respectively.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 (2) P c -20 0 20 40 60 80 100 120 140 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 c P (3) -20 0 20 40 60 80 100 120 140 0.00 0.02 0.04 0.06 0.08 c P (4)\n\nFIG. 4: The same as Fig.1. with Gaussian random errors and systematic errors. In sub-figures (1), (2), and (3) τ are set to 0.01, 0.03 and 0.05 respectively, and δ0 = 0(without systematic errors). In sub-figure (4), both systematic and random Gaussian errors exist, where δ0 = 0.33, τ = 0.02."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"Shor's factorization algorithm [1] is a very important quantum algorithm, through which one has demonstrated the power of quantum computers. It has greatly promoted the worldwide research in quantum computing over the past few years. In practice, however, quantum systems are subject to influence of environment, and in addition, quantum gate operations are often imperfect [2, 3] . Environment influence on the system can cause decoherence of quantum states, and gate imperfection leads to errors in quantum computing. Thanks to Shor's another important work, in which he showed that quantum error correlation can be corrected [4] . With quantum error correction scheme, errors arising from both decoherence and imperfection can be corrected.\\n\\nThere have been several works on the effects of decoherence on Shor's algorithm. Sun et al. discussed the effect of decoherence on the algorithm by modeling the environment [5] . Palma studied the effects of both decoherence and gate imperfection in ion trap quantum computers [6] . There have also been many other studies on the quantum algorithm [7, 8, 9, 10] .\\n\\nThe error correction scheme uses available resources. Thus it is important to study the robustness of the algorithm itself so that one can strike a balance between the amount of quantum error correction and the amount of qubits available. In this paper, we investigate the effects of gate imperfection on the efficiency of Shor's factorization algorithm. The results may guide us in practice to suppress deliberately those errors that influence the algorithm most sensitively. For those errors that do not affect the algorithm very much, we may ignore them as a good approximation. In addition, study of the robustness of algorithm to errors is important where one can not apply the quantum error correction at all, for instance, in cases that there are not enough qubits available.\\n\\nThe paper is organized as follows. Section II is devoted to an outline of Shor's algorithm and different error's modes. In Section III, we present the results. Finally, a short summary is given in Section IV.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. SHOR'S ALGORITHM AND ERROR'S MODES\",\n \"text\": \"Shor's algorithm consists of the following steps: 1) preparing a superposition of evenly distributed states |ψ = 1 √ q q-1 a=0 |a |0 ,\\n\\nwhere q = 2 L and N 2 ≤ q ≤ 2N 2 with N being the number to be factorized; 2) implementing y a modN and putting the results into the 2nd register\\n\\n|ψ 1 = 1 √ q q-1 a=0 |a |y a modN ;\\n\\n3) making a measument on the 2nd register; The state of the register is then\\n\\n|φ 2 = 1 √ A + 1 A j=0 |jr + l |z = y l = y jr+l modN\\n\\nwhere j ≤ q-l r = A.\\n\\n4) performing discrete Fourier transformation (DFT) on the first\\n\\nregister |φ 3 = c f (c) |c |z , where f (c) = √ r q q r -1 j=0 exp 2πi(jr + l) q = √ r q e 2πilc q q r -1 j=0 exp 2πijrc q .\\n\\n2 This term is nonzero only when c = k q r , with k = 0, 1, 2...r -1, which correspond to the peaks of the distribution in the measured results, and thus this term becomes f (c) = 1 √ r e 2πilc q . The Fourier transformation is important because it makes the state in the first register the same for all possible values in the 2nd register. The DFT is constructed by two basic gate operations: the single bit gate operation A j = 1 √ 2 1 1 1 -1 , which is also called the Walsh-Hadmard transformation, and the 2-bits controlled rotation\\n\\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e iθ jk with θ jk = π 2 k-j .\\n\\nThe gate sequence for implementing DFT is\\n\\n(A q-1 )(B q-2q-1 A q-2 ) . . . (B 0q-1 B 0q-2 . . . B 01 A 0 ). Errors can occur in both A j and B jk . A j is actually a rotation about y-axis through π 2 A j (θ) = e i h Syθ = I cos( θ 2 )-i sin( θ 2 )σ y = cos( θ 2 ) -sin( θ 2 ) sin( θ 2 ) cos( θ 2 )\\n\\n.\\n\\nIf the gate operation is not perfect, the rotation is not exactly π 2 . In this case, A j is a rotation of π 2 + 2δ A j (δ) = 1 √ 2 cos(δ) -sin(δ) -(sin(δ) + cos(δ)) sin(δ) + cos(δ) cos(δ) -sin(δ) .\\n\\nIf δ is very small, we have:\\n\\nA j (θ) = 1 √ 2 1 -δ -(1 + δ) 1 + δ+ 1 -δ .\\n\\nSimilarly, errors in B jk can be written as\\n\\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e i(θ jk +δ) .\\n\\nWith these errors, the DFT becomes\\n\\n|a → 1 √ q q-1 c=0 e i( 2π q/c +δc)a (1 + δ ′ c )|c = 1 √ q q-1 c=0 e i( 2πc q +δc)a (1 + δ ′ c )|c ,( 1\\n\\n)\\n\\nwhere δ c and δ ′ c denote the error of A j and B jk , respectively.\\n\\nLet us assume the following error modes: 1) systematic errors, where δ c or δ ′ c in (1) can only have systematic errors (EM 1 ); 2) random errors (EM 2 ), for which we assume that δ c or δ ′ c can only be random errors of the Gaussian or the uniform type; 3) coexistence of both systematic and random errors (EM 3 ). In the next section, we shall present the results of numerical simulations and discuss the effects of imperfect gate operation on the DFT algorithm, and thus on the Shor's algorithm.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. INFLUENCE OF IMPERFECT GATE OPERATIONS\",\n \"text\": \"We first discuss the influence of imperfect gate operations in the initial preparation\\n\\nA l-1 A l-2 ...A 0 |0...0 = 1 √ 2 (|0 + |1 + δ 1 (|0 -|1 )) ⊗ (|0 + |1 + δ 2 . = 1 √ 2 l 1 i1i2...in=0 |i 1 i 2 ...i n + 1 √ 2 l n R=1 δ n 1 i\\n\\nIf the errors are systematic, for instance, caused by the inaccurate calibration of the rotations, then δ 1 = δ 2 = . . . = δ n = δ. In this case, we can write the 2nd term as\\n\\n|ψ = 1 √ 2 l δ 1 i1i2...in=0 (2s -n)|i 1 i 2 ...i n ,\\n\\nwhere s stands for the number of 1's, and 2sn = s -(ns) is the difference in the number of 1's and 0's. Thus the results after the first procedure is\\n\\n1 √ 2 l 2 L -1 a=0 (|a + δ(2s -n)|a ) = 1 √ 2 l a=0 (1 + δ a )|a . (2)\\n\\nThis implies that after the procedure, the amplitude of each state is no longer equal, but have slight difference. Combining the effect in the initialization and in the DFT, we have\\n\\n(1 + δ a )(1 + δ c )e i( 2πc q +δ ′ c )a . = (1 + δ ′′ )e i( 2πc q +δ ′ c )a , where δ ′′ c = δ c + δ a . In the DFT, we have |ψ ⇒ √ r q q-1 c=0 q r -1 j=0 (1 + δ j )e i( 2πc q +δ ′ j )(jr+l) |c ,\\n\\nwhere we have rewrite δ ′′ as δ j here. Let P c denote the probability of getting the state |c after we perform a measurement, we have 3 ), we find that after the last measurement, each state can be extracted with a probability which is nonzero, and the offset l can't be eliminated. Eq. ( 3 ) is very complicated, so we will make some predigestions to discuss different error modes for convenience. Generally speaking, the influence of exponential error δ j is more remarkable than δ j , so we can omit the error δ j , thus\\n\\nP c = r q 2 q r -1 m=0 q r -1 k=0 (1 + δ m )(1 + δ k )e i( 2πc q +δ ′ m )(mr+l) ×e -i( 2πc q +δ ′ k )(kr+l) = r q 2 m k (1 + δ m )(1 + δ k ) cos[ 2πc q r(m -k) + (mr + l)δ ′ m -( From Eq. (\\n\\nDFT q |φ = √ r q q-1 c=0 q r j=0 e i( 2πc q +δ ′ j )(jr+l) |c . 3\\n\\nA. Case 1 If only systematic errors (EM 1 ) are considered, namely, all the δ j 's are equal, then f (c) can be given analytically f (c) = √ r q q r -1 j=0 e i( 2πc q +δ)(jr+l) = √ r q e il( 2πc q +δ) 1e i( 2πc q +δ)q\\n\\n1 -e i( 2πc q +δ)r ( 4\\n\\n)\\n\\nThe relative probability of finding c is\\n\\nP c = f (c) 2 = r q 2 sin 2 ( δq 2 ) sin 2 ( πcr q + δr 2 ) , and if c = k q r , then P c = r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) .\\n\\nIt can be easily seen that lim δ→0 P c = 1 r , which is just the case that no error is considered.\\n\\nWhen δ takes certain values, say, δ = 2 r (k -r q )π where k is an integer, then the summation in Eq. (4) is on longer valid. In our simulation, δ does not take these values. Here we consider the case where q = 2 7 = 128 and r = 4. For comparisons, we have drawn the relative probability for obtaining state c in Fig.1. for this given example. We have found the following results: (i) When δ is small, the errors do hardly influence the final result, for instance when c = k q r , then\\n\\nlim δ→0 P c = lim δ→0 r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) = 1 r .\\n\\nThe probability distribution is almost identical to those without errors.\\n\\n(ii) Let us increase δ gradually, from Fig. 2 , we see that a gradual change in the probability distribution takes place. (Here, we again consider the relative probabilities) When δ is increased to certain values, the positions of peaks change greatly. For instance at δ = 0.05, there appears a peak at c=127, whereas it is P c = 0 when no systematic errors are present. In general, the influence of systematic errors on the algorithm is a shift of the peak positions. This influences the final results directly.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Case 2\",\n \"text\": \"When both random errors and systematic errors are present, we add random errors to the simulation. To see the effect of different mode of random errors, we use two random number generators. One is the Gaussian mode and the other is the uniform mode. In this case, the error has the form δ = δ 0 + s, where δ 0 is the systematic error. s has a probability distribution with respect to c, depending on the uniform or the Gaussian distribution. When δ 0 = 0, we have only random errors which is our error mode 2. When δ 0 = 0, we have error mode 3. For the uniform distribution, s ∼ ±s max × u(0, 1) where u(0, 1) is evenly distributed in [0, 1] . s max indicates the maximum deviation from δ 0 . For Gaussian distribution, s ∼ N (0, σ 0 ). Through the figure, we see the following: (1) When only random errors are present (δ 0 = 0), the peak positions are not affected by these random errors. However, different random error modes cause similar results. The results for uniform random error mode are shown in Fig. 3 . For the uniform distribution error mode, with increasing δ max , the final probability distribution of the final results become irregular. In particular, when δ max is very large, all the patterns are destroyed and is hardly recognizable. Many unexpected small peaks appear. For the Gaussian distribution error mode, as shown in Fig. 4 , the influence of the error is more serious. This is because in Gaussian distribution, there is no cut-off of errors. Large errors can occur although their probability is small. The influence of σ 0 on the final results is also sensitive, because it determines the shape of the distribution. When σ 0 increases, the final probability distribution becomes very messy. A small change in σ 0 can cause a big change in the final results. (2) When δ 0 = 0, which corresponds to error mode 3, the effect is seen as to shift the positions of the peaks in addition to the influences of the random errors.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. SUMMARY\",\n \"text\": \"To summarize, we have analyzed the errors in Shor's factorization algorithm. It has been seen that the effect of the systematic errors is to shift the positions of the peaks, whereas the random errors change the shape of the probability distribution. For systematic errors, the shape of the distribution of the final results is hardly destroyed, though displaced. We can still use the result with several trial guesses to obtain the right results because the peak positions are shifted only slightly. However, the random errors are detrimental to the algorithm and should be reduced as much as possible. It is different from the case with Grover's algorithm where systematic errors are disastrous while random errors are less harmful [10].\\n\\n[1] P.W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S.\\n\\nGoldwasser (IEEE Computer Society Press, Los Alamitos, CA, 1994) p.124.\\n\\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 p c FIG. 1: Relative probability for finding state c in the absence of errors.\\n\\n[2] A. Ekert and R. Jozsa, Rev. Mod. Phys. 68 (1996) 733. [3] W.G. Unruh, Phys. Rev A51 (1995) 992. [4] I. Chuang and R. laflamme, \\\"Quantum error correction by codding\\\" (1995) quant-ph/9511003. [5] C.P. Sun, H. Zhan and X.F. Liu, Phys. Rew. A58 (1998) 1810. [6] G.M. Palma, K.A. Suominen and A.K. Ekert, Proc. R. Soc. London, A 452 (1996) 567. [7] R.P. Feynman, Int. J. Theo. Phys., 21 (1982) 467. [8] D. Deutsch, Proc. R. Soc. Land. A 400 (1985) 97. [9] L.K. Grover, Phys. Rev, Lett, 79 (1997) 325. [10] G.L. Long, Y.S. Li, W.L. Zhang, C.C. Tu, Phys. Rev. A 61 (2000) 042305. [11] L.K. Grover, Phys. Rev. Lett, 80 (1998) 4329.\\n\\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (4) (3) c P (2) -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 c P -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c FIG. 2: The same as Fig.1. with systematic errors. In subfigures (1), (2), (3), (4), δ are 0.02, 0.03, 0.05 respectively. In sub-figure (4), the curve with solid circles(with higher peaks) is the result with δ = 0.1, and the one without solid circles(with lower peaks) denotes the result with δ = 0.33.\\n\\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (3) c P (1) -20 0 20 40 60 80 100 120 140 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (4) c P (2)\\n\\nFIG. 3: The same as Fig.1. with uniform random errors. In sub-figures (1), (2), (3), (4), smax are set to 0.01, 0.03, 0.05, 0.1 respectively.\\n\\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 (2) P c -20 0 20 40 60 80 100 120 140 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 c P (3) -20 0 20 40 60 80 100 120 140 0.00 0.02 0.04 0.06 0.08 c P (4)\\n\\nFIG. 4: The same as Fig.1. with Gaussian random errors and systematic errors. In sub-figures (1), (2), and (3) τ are set to 0.01, 0.03 and 0.05 respectively, and δ0 = 0(without systematic errors). In sub-figure (4), both systematic and random Gaussian errors exist, where δ0 = 0.33, τ = 0.02.\"\n }\n]"}}},{"rowIdx":94,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0520"},"arxiv_id":{"kind":"string","value":"0704.0520"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":""},"content_hash":{"kind":"string","value":"e2ed075f4ade80918e070091647f48ec5d73e5a1179cb737dc87b8224a6ec3ab"},"title":{"kind":"string","value":"A critical theory of quantum entanglement for the Hydrogen molecule"},"abstract":{"kind":"string","value":"In this paper we investigate some entanglement properties for the Hydrogen\nmolecule considered as a two interacting spin 1/2 (qubit) model. The\nentanglement related to the $H_{2}$ molecule is evaluated both using the von\nNeumann entropy and the Concurrence and it is compared with the corresponding\nquantities for the two interacting spin system. Many aspects of these functions\nare examinated employing in part analytical and, essentially, numerical\ntechniques. We have compared analogous results obtained by Huang and Kais a few\nyears ago. In this respect, some possible controversial situations are\npresented and discussed."},"authors":{"kind":"list like","value":["Tina A.C. Maiolo","Luigi Martina","Giulio Soliani"],"string":"[\n \"Tina A.C. Maiolo\",\n \"Luigi Martina\",\n \"Giulio Soliani\"\n]"},"arxiv_categories":{"kind":"list like","value":["quant-ph","cond-mat.other","physics.atom-ph"],"string":"[\n \"quant-ph\",\n \"cond-mat.other\",\n \"physics.atom-ph\"\n]"},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"Entanglement is a physical observable measured by the von Neumann entropy or, alternatively, by the Concurrence of the system under consideration.\n\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The electron correlation is not directly observable, since it is defined as the difference between the exact ground state energy of the many electrons Schrödinger equation and the Hartree-Fock energy.\n\nIn this paper we discuss the Hamiltonian which describes the Hydrogen molecule regarded as a two interacting spin 1 2 (qubit) model. In [1] it was argued that the entanglement (a quantum observable) can be used in analyzing the so-called correlation energy which is not directly observable. From our point of view, the Hydrogen molecule is dealt with a bipartite system governed by the Hamiltonian\n\nH H2 = - J 2 (1 + g)σ 1 ⊗ σ 1 - J 2 (1 -g)σ 2 ⊗ σ 2 -B(σ 3 ⊗ σ 3 + σ 0 ⊗ σ 3 ), ( 1\n\n) 1\n\nwhere σ i stand for the Pauli matrices (σ 0 = I). Actually, this model was considered in [1] in order to illustrate their method. However, here we will make some interpretative changes. Indeed, from our point of view, the states of an isolated atom are strongly reduced to a system with two energy levels related to the intensity of the magnetic field B. Relatively to this scale, the exchange interaction constant J is usually smaller than B, in order to represent the residual interatomic interactions. From the point of view of quantum chemistry, one may interpret the discrete spectrum as provided by the Hartree-Fock calculations, while the interaction coupling J models the residual multielectronic effects, not taken into account by the mean field approximation. For simplicity we limit ourselves to the ferromagnetic phase with J > 0. The parameter g, such that 0 ≤ g ≤ 1, describes the degree of anisotropy corresponding for g = 0 to the completely isotropic XY spin model. Conversely, g = 1 provides the anisotropic XY spin model, the so-called Ising model.\n\nWe notice that when the atoms are far apart, their interaction is quite weak. This corresponds to a vanishing value of J. In this situation the state of the system is completely factorized in the product state of the ground states of the indipendent spins. The corresponding total energy, in unit of B, is just the sum of the two fundamental levels, E 0 = -2, which we may consider as the Hartree-Fock approximated fundamental level in molecular structure calculations.\n\nWhen J = 0, the fundamental energy eigenvalue is E= -4 + g 2 λ 2 in Region I defined by 0 < λ ≤ 2 √ 1-g 2 , otherwise E = -λ (λ means the coupling constant) in Region II, which is the complement of I which respect to positive real axis. The corresponding (non normalized) eigenstates are |Ψ I = √ g 2 λ 2 +4+2 gλ , 0, 0, 1 and |Ψ II = 0, 1, 1, 0 , respectively. In both cases the state is entangled.\n\nSince we are dealing with pure states, the von Neumann entropy [2]\n\nS vN = -Tr ρ 1 log 2 ρ 1 ( 2\n\n)\n\nis chosen to be a measurement of the entanglement, where ρ 1 is the 1-particle reduced density matrix. However, for general mixed states other entanglement estimators (for instance, the Concurrence [4]) have to be used. In the considered case, one has S vN,I = -g 2 g 2 λ 2 + 4 log 1 2 -1 g 2 λ 2 + 4 λ 2 -g 2 g 2 λ 2 + 4 + 4 λ 2 + 8 g 2 λ 2 + 4 + 2 log 1 2 + 1 g 2 λ 2 + 4 1 g 2 λ 2 + 4 g 2 λ 2 + 2 g 2 λ 2 + 4 + 4 log(4) (3) S vN,II = 1. (4) 2 Scrutinizing Eq. (3) and Eq. ( 4 ) it emerges that the entropy is an increasing function of the coupling constant λ in Region I, but the state is maximally entangled in Region II independently from the anisotropy parameter g. One sees that, as it arises graphycally, for g = 1 the entanglement is a monotonic increasing function of the interaction coupling λ. Moreover for weak (< 1) coupling values it is always less than the 30%. Of course, for large coupling constants the entropy approaches 1, meaning that all levels are equiprobably visited by the considered spin.\n\nLimiting all further considerations to the case of weak interaction, we observe that at the boundary point λ b = 2 √ 1-g 2 a discontinuity occurs, signaling a crossing of the lowest eigenvalues and, in a more general context, a quantum phase transition [5] .\n\nAs it was pointed out in [6] , for quantifying the entanglement we can resort to the reduced density matrix. Furthermore, in [7] , Wootters has shown that for a pair of binary qubits one can use the concept of Concurrence C to measure the entanglement.\n\nThe Concurrence reads\n\nC(ρ) = max(0, ν 1 -ν 2 -ν 3 -ν 4 ), ( 5\n\n)\n\nwhere the ν i 's are the eigenvalues of the Hermitian matrix\n\nR = √ ρρ √ ρ 1 2 ,\n\nwhere ρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ), ρ * being the complex conjugate of ρ taken in the standard basis [7] . Some interesting results on the simple model (1) of the Hydrogen molecule can be achieved by realizing a comparative study of the von Neumann entropy and the Concurrence.\n\nTo this aim, we compute the Concurrence C I and C II , i. e.\n\nC I = gλ 1 g 2 λ 2 + 4 , C II = 1. ( 6\n\n)\n\nwhere I and II refer to Regions I and II, where 0 ≤ λ ≤ 2 1-g 2 , and E = -λ, respectively.\n\nIn Figure 1 a comparison between the Concurrence and the von Neumann entropy for two spins system as a function of the coupling λ for g = 1 is presented. Sec. 2 contains a comparison between the entanglement and the correlation energy. In Sec. 3 the Configuration Interaction method is introduced to compare entanglement and correlation energy. In Sec. 4 some differences between the Configuration Interaction approach and the two spin Ising model are presented. Finally, our main results are summarized in Sec. 5. 3 1 2 3 4 5 Λ 0.2 0.4 0.6 0.8 1 S vN Conc.\n\nFigure 1: Comparison between the Concurrence and the von Neumann entropy for the two spins system as a function of the coupling constant λ for g = 1.\n\n2 A comparison between the entanglement and the correlation energy\n\nNow we look for a comparison between the entanglement with the energy correlation, which as we have already recalled, it is understood as the difference of the fundamental energy level compared with respect to the corresponding value at vanishing coupling constant λ. For g = 1 and in unities of B it is given by\n\nE corr = |E 0 | -2 = 4 + λ 2 -2. ( 7\n\n)\n\nWe observe that the entanglement measure is always bounded, while E corr is a divergent function of λ. So it does not make much sense to look for simple relations valid on the entire λ-axes. Consequently, limiting ourselves to weak couplings, for 0 ≤ λ ≤ 1, we minimize the mean squared deviation\n\nI α = 1 0 ∆S 2 α dλ, with ∆S α = E corr -α S vN . ( 8\n\n)\n\nThus the minimizing parameter α min will be given by\n\nα min = 1 0 E corr S vN dλ 1 0 S 2 vN dλ ≈ -0.691217. ( 9\n\n)\n\nA formula analogous to (9) can be obtained by using the Concurrence as a measure of entanglement. In this case, by minimizing the mean squared deviation we have\n\nI C α ′ = 1 0 ∆C 2 α ′ dλ, with ∆C α ′ = E corr -α ′ C. ( 10\n\n)\n\nNow, in order to estimate the relative deviation of\n\nS vN with respect to E corr , let us report |∆S αmin |/S vN and |∆S αmin /E corr | as\n\nfunctions of λ at the optimal value α min . The graphs of these functions are shown in Figure 2. 4 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min S vN 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min E corr Figure 2: The relative quadratic deviation between the von Neumann entropy and the correlation energy with respect to the former and the latter, respectively, at the optimal value α min as a function of the coupling constant λ for g = 1.\n\nIn Figure 3 , the relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, at the optimal values α ′ min , is represented.\n\n0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min C 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min E corr Figure 3: The relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, respectively, at the optimal value α ′ min as a function of the coupling constant λ for g = 1.\n\nFrom these graphs, one can argue that the agreement between the two functions S vN and E corr is only qualitatively good, in fact, for very small λ, it is not good at all. However, in an intermediate range of values, i. e., 0.6 ≤ λ ≤ 1 the two functions are almost proportional within the 10%. Analogously, the same is true between energy and Concurrence. Even, the agreement becomes worst comparing the relative deviation of the Concurrence with respect to the correlation energy, since the range in which the relative deviations become smaller than 10% are narrower. Then, the question is whether the above results are i) sufficient to justify the conjecture advanced in [1] , i.e., entanglement can be considered as an estimation of correlation energy; ii) if such a relation has a 5 more concrete physical meaning, in particular whether the minimizing parameter α min and the vanishing point of ∆S αmin does possess any physical meaning (or α ′ min and the vanishing point of ∆C α ′ min ). Notice that in the case of the comparison for the Concurrence simpler analytical expressions appear. For instance one finds ∆C α ′ min = 0.383249 λ\n\n√ λ 2 +4 - √ λ 2 + 4 + 2 2 . Remark 2\n\nWe note that in an interval of values around α min , the deviation function (8) possesses a minimum in the interval of interest 0 ≤ λ ≤ 1, otherwise the minimum is achieved at larger value of λ, or the function is monotonically increasing (see Figure 4 ).\n\n0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 S Α 0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 d S Α dΛ\n\nFigure 4: The deviation ∆S α and its derivative with respect to λ are computed for values of -1.29(red) ≤ α ≤ -0.091(violet), for steps of 0.06. The curve drawn thicker corresponds to α min This behavior suggests to consider the function ∆S αmin as a sort of \"free energy\" , where α min mimics the \"temperature\" specific of the system. If, for some reason, we allow λ to change, then we expect that spontaneously the interaction coupling adjusts itself to the minimum of ∆S αmin . Similar considerations can be made looking at the graphs drawn for the function ∆C α ′ min and its derivative with respect to λ (see Figure 5 ). The function ∆S αmin or, alternatively, the minimum of ∆C α ′ min can be obtained algebraically. Such a minimum is at the value of the coupling constant λ SvN min ≈ 0.485 and λ C min ≈ 0.371, respectively. The authors in [1] studied numerically the von Neumann entropy and the correlation function for a Hydrogen molecule, using an old result by Herring and Flicker [8], going back to an oldest idea by Heitler and London [9], which consists in substituting the molecular binding with a position dependent exchange coupling:\n\nJ(r) ≈ 1.641 r 5 2 e -2 r Ry, ( 11\n\n)\n\nwhere r is given in Bohr radius, see Figure 6 . The maximum value taken by this function is at the point r max = 1.25. Assuming B = 0.5 Ry, i.e. 1 2 of the fundamental level of the Hydrogen atom, the maximum value λ ′ max = J(r max )/B ≈ 6 0.2 0.4 0.6 0.8 1 Λ -0.2 -0.1 0.1 0.2 0.3 C Α' 0.2 0.4 0.6 0.8 1 Λ -0.4 -0.2 0.2 0.4 d C Α' dΛ 5: The deviation ∆C α ′ and its derivative with respect to λ are computed for values of -0.98(red) ≤ α ′ ≤ 0.22(violet), for steps of 0.06. The curve drawn thicker corresponds to α ′ min 0.5 1 1.5 2 2.5 3 r 0.05 0.1 0.15 0.2 J r Ry\n\nFigure 6: The effective interaction Hydrogen-Hydrogen atom 0.470628 < λ SvN min , i.e. the value of the effective interaction value is less than the minimum for the deviation function ∆S αmin . Then, the equilibrium balance between entanglement (as von Neumann entropy) and correlation energy predicts a length of the molecule equal to r max (see the first panel of Figure 7 ). On the other hand, if we consider the energy gap 2B = 3/4 Ry, i.e. the energy step to the first excited state, one obtains the new value λ ′′ max ≈ 0.628, which goes beyond λ min , even if it is always less than 1. Now, the deviation function ∆S αmin has two minima as seen in the second panel of Figure 7 , one of which is at r ′′ -≈ 0.76 , the other one being at r ′′ + ≈ 1.91. These results should be compared with the experimental equilibrium length of the Hydrogen molecule, which is r exp ≈ 2.0. We point out that although the spin-model described by the Hamiltonian (1) is characterized by features which are essentially rough, however we are induced to answer positively to the quest for a physical meaning of the deviation function ∆S αmin . Indeed, the results elucidated in Figure 7 encourage, on one part, improvement of the computation of r in order to make more accurate the comparison with the experimental value r exp . 7 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .5 Ry 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .375 Ry r'' r''\n\nFigure 7: The von Neumann entropy for the 2-spin model for B = .5 Ry (left panel) and for B = .375 Ry (right panel) and the position depending interaction given by ( 11 ).\n\nThe first question to answer is whether this draft works also for the Concurrence. A statement about it is not obvious, since the von Neumann entropy is a nonlinear function of the Concurrence in the 2-qubits case. However, from Figure 8 one can see that the minimized deviation of the Concurrence takes one minimum for relatively large intensity of the magnetic field ( say B ≥ 0.6 Ry), while for weak fields two minima appear, corresponding to the situation depicted nearby. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 r Bohr B Ry 0.2 0.4 0.6 0.8 1 0 1 2 3 4 B(Ry) r(Bohr)\n\nFigure 8: Two contour plots of the minimized deviation of the Concurrence as a function of the magnetic field B (Ry) and of the internuclear distance r, as given by (11) . The range of values divided by the contour lines is [-0.038, 0, 04] for the left panel and [-0.03705, -0, 03000] for the right one that approximatively corresponding to the black area in the left panel.\n\nIn correspondence of the same values considered above, for B = 0.5 Ry the function ∆C α ′ min (r) has two minima at r = 0.79 and r = 1.88, while for B = 0.375 Ry they are located at r = 0.60 and r = 2.25. So one sees that the resulting equilibrium configurations are not much very close to the experimental one. The equilibrium configuration more closest to the experimental one is the minimum occurring at r = 1.88 (B = 1 2 Ry) for the function ∆C α ′ min (r).\n\n8 One sees that one of the resulting equilibrium configurations is only roughly close to the experimental one.\n\nIn other words, to conclude monitoring numerically B the equilibrium configuration more closest to experimental one in the minimum occurring at r = 1.88 for B = 1 2 Ry and at r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r).\n\nIn this Section we represent the results produced in [1], where the electron entanglement in the Hydrogen molecule, calculated by the von Neumann entropy of the reduced density matrix ρ 1 , is obtained starting by the excitation coefficients of the wave function expanded by a configuration interaction method:\n\nS ρ CISD 1 = -T r ρ CISD 1 log 2 ρ CISD 1 = = - m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 log 2 m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 + -|c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 log 2 |c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 , ( 12\n\n)\n\nwhere c 1 is the coefficient for a single excitation, and c 1,2 is the double excitation (in Appendix A of [10] more details are shown).\n\nIn this framework, entanglement (S) and correlation energy (E corr ), as functions of nucleus -nucleus separation are those in Figure 9 0 0.01 0.02 0.03 0.04 0.05 E c [ a. u. ] E c 0 1 2 3 4 5 R ( Å ) 0 0.2 0.4 0.6 0.8 1 Entanglement (S) S ( ρ 1 CISD ) H 2 Figure 9: Comparison between the entanglement, calculated by the von Neumann entropy of the reduced density matrix, and the electron correlation energy in the Hydrogen molecule.\n\nBy the results given by this model, we want to discuss and to suggest some answers to the questions i) and ii) presented in Remark 1. Even if, in order 9 to represent correlation energy and entanglement, we use two different scales, in Figure 9 we can see that entanglement has a small value in the united atom limit after it is growing for small distances till it arrives at a maximum value then it decrease till it assumes zero value at the separated atom limit and it is exactly the progress of the correlation curve.\n\nIn order to compare the entropy S with the electron correlation energy E corr , we rescale S with the parameter α min calculated with some procedure illustrated in Eq. ( 8 ) and Eq. (9) replacing the integration variable λ with R; in this way we extract\n\nα = E corr S vN dR S vN dR ≈ 0.009. ( 13\n\n)\n\nThe corresponding ∆S αmin = E corr -αS allows us to answer to the question ii); in fact, as it is shown in Figure 10 , the vanishing point of ∆S αmin is, according to the two -spin Ising model, nearby R ≈ 2 Å that corresponds to the equilibrium configuration of the Hydrogen molecule.\n\n0 1 2 3 4 5 R(Å) -0.01 0 0.01 0.02 0.03 0.04 Ec-αS\n\nFigure 10: ∆S αmin for the H 2 molecule as a function of nucleus-nucleus distance.\n\nThe model proposed in Sec. 1 provides us with a measurement of entanglement: indeed, Eq. (3) describes the von Neumann entropy as a function of coupling constant λ, for small λ. By using Eq. ( 7 ), we can express λ in terms of correlation energy and substituting it in Eq. ( 3 ) we can obtain the variation of S vN in terms of E corr .\n\nS vN = - E corr Log Ecorr 2(Ecorr +2) + (E corr + 4)Log Ecorr +4 2(Ecorr +2) (E corr + 2)Log4 . ( 14\n\n)\n\n10 In order to calculated the coefficient of proportionality among S vN and E corr we make an expansion of S vN for E corr → 0 (or equivalently for λ → 0) at the first order, obtaining a straight line characterized by an angular coefficient given by m SvN (Ecorr) = ( 1 4 )(1 + 1 Log2 ). Since this behavior is uncorrect to represent the logatithmic singularity of S vN in the origin, we make an expansion of Eq. ( 14 ), preserving the logarithmic deviation, and we obtain an expression of the form\n\nS vN = AE corr + BE corr Log(E corr ), ( 15\n\n)\n\nwhere A = 1/2 and B = -1/(4Log2).\n\n0.02 0.04 0.06 0.08 0.1 0.025 0.05 0.075 0.1 0.125 0.15 Ecorr S Linear AE+BELogE Figure 11: A comparison among the behavior of Eq. ( 14 ) and its linear approximation and the logarithmic one, for the Ising model.\n\nIn order to compare the behavior of S vN in Eq. ( 14 ), we have organized the numerical data, calculated with the method proposed in [1] , by making a correspondence between each value of E corr and its respective value of S vN , obtaining the plot in Figure 12 0 0.01 0.02 0.03 0.04 0.05 E c 0 0.2 0.4 0.6 0.8 1 S Figure 12: A correspondence of E corr and S vN by the numerical procedure suggested by [1] Of particular significance is the fact that, in the range where S is monotonically increasing, the correlation energy has its maximum, consequently S seems to be not a function. Moreover, it is important to note that E corr begins to decrease for R > 1 Å, region where the states become mixed, i. e. ,T rρ = T rρ 2 ; as depicted in Figure 13. 11 0 1 2 3 4 R(Å ) 0.4 0.6 0.8 1 1.2 1.4 Trρ 2\n\nFigure 13: The increasing of the degree of mixing in the two electron state: in black we depict the trace of ρ, in red the trace of ρ 2 .\n\nProbably, for this reason, the procedure adopted in [1] seems to be not correct: the density matrix, in fact, is calculated starting by the excitation coefficient of a wave function obtained developping with the Configuration Interaction Single Double method a pure two electrons state.\n\nHowever, even if we consider only the first branch of the plot in Figure 12 , i.e. , the numerical values of S vN corresponding with increasing values of E corr , and we fit the values around E corr → 0 with a F = AE corr + BE corr Log(E corr ) we draw out numerical values of the coefficient different from the ones used in Eq. (15) . This result is shown in Figure 14. 0.01 0.02 0.03 0.04 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 S Ecorr A=17.1 B=3.3 Figure 14: A fit of S vN as a function of E corr , around the origin, with a function of the form F = AE corr + BE corr Log(E corr ) whose coefficients A and B assume the numerical values in Figure.\n\nIn particular the arithmetic sign of the coefficient B in the two models are opposite and this implies the opposite concavity of the curve.\n\nThis fact, clearly demonstrates a not satisfactory agreement between the Ising model and the one proposed in [1] .\n\nWe have explored the role of entanglement in the model of two qubits describing the Hydrogen molecule (1), considered as a bipartite system. In our discussion we have limited to the ferromagnetic case governed by the interaction coupling parameter J > 0.\n\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The entanglement can be measured by using the von Neumann entropy or, alternatively, the notion of Concurrence [7] . To compute the entanglement it is convenient to consider two Regions, say I and II, which provide two different reduced density matrices. The entropy turns out to be an increasing function of the coupling constant λ in Region I, but the state under consideration is maximally entangled in Region II indipendently from the anisotropy parameter g.\n\nAn interesting result is that for large coupling constants the entropy approach 1, meaning that all levels are equiprobably visited by the considered spin.\n\nFor weak interactions, at the boundary point λ b = 2 √ 1-g 2 the von Neumann entropy admits a discontinuity, indicating a crossing of the lowest eigenvalues and, in a more general constext, a quantum phase transition [5] .\n\nIn Sec. 2 a comparison between the entanglement and the correlation energy is performed.\n\nTo quantifying the entanglement we resort to the reduced density matrix. The entanglement can also be measured by exploiting the concept of Concurrence.\n\nThe entanglement measure is always bounded, while the energy correlation,\n\nE corr = |E 0 | -2 =\n\n√ 4 + λ 2 -2, is a divergent function of λ. This fact tells us that to look for simple relations valid on the whole λ-axes has no sense.\n\nThus, by limiting ourselves to weak couplings, we have minimized the mean square deviation given by Eq. ( 8 ). This procedure leads to the value α min ≈ -0.691217 for the minimizing parameter (see Eq. (9)). Sec. 1 contains a comparison between the von Neumann entropy and the Concurrence. Such a comparison is illustrated in Figure 1 , for two spin system as a function of the coupling λ for g = 1. Some important points are commented in Remark 1 and Remark 2 .\n\nIn Figure 4 the deviation ∆S α and its derivatives with respect to λ are computed and α min is evaluated for α ranging in the interval -1.29 ≤ α ≤ -0.091.\n\nIn Figure 5 the minimized Concurrence deviation ∆C α (i) α ′\n\nfor the four eigenstates of the 2-spin model is shown.\n\nWe point out the existence of a perfect symmetry among the Concurrence deviations for pairs of eigenstates of opposite eigenvalues.\n\nFormula (11), due to Heitler-London [9], is reported, where the position dependent exchange coupling J(r) is expressed in term of the length r of the 13 nucleus-nucleus separation in the Hydrogen molecule.\n\nTo conclude, the magnetic field B has been monitored such that the equilibrium configuration more closest to the experimental one, r ≈ 2.00, is the minimum occurring at r = 1.88 for B = 1 2 Ry and r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r). We observe also that in the intermediate range of values, i. e., for 0.6 ≤ λ ≤ 1, the two functions SvN and the correlation energy are almost proportional within the 10%.\n\nHowever, when we organized the pairs of points (E corr , S vN ) calculated by following the procedure described by [1], it is clear that the von Neumann entropy cannot be considered a function of correlation energy. The principle cause is that the function E corr presents a maximum in the region where S vN is monotonically increasing.\n\nThe reversing behavior of correlation energy occurs in correspondence with an increase of the mixing degree of the two electrons state. The function E corr in terms of the nucleus -nucleus distance R, increases till the state is pure, on the contrary, when T r(ρ 2 ) becomes discordant from T r(ρ), the function E corr decreases.\n\nThis fact suggests us that the numerical model based on the calculation of S vN starting by the excitation coefficients c i , isn't completley correct because the density matrix is obtained as a product of two electron pure states. However, even if we consider only a branch of the plot in Figure 12 , the function obtained by the two spin Ising model, i. e., Eq. ( 14 ), is unsuitable for fitting these numerical data.\n\nOn the basis of our results, essentially grounded on numerical considerations, in the near feature we would explore more complicated systems of molecules, such as for example the ethylene or other hydrocarbons, and compare these studies with the goals obtained for the Hydrogen molecule.\n\nThe authors acknowledge the Italian Ministry of Scientific Researches (MIUR) for partial support of the present work under the project SINTESI 2004/06 and the INFN for partial support under the project Iniziativa Specifica LE41."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"Introduction and the model","text":"Entanglement is a physical observable measured by the von Neumann entropy or, alternatively, by the Concurrence of the system under consideration.\n\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The electron correlation is not directly observable, since it is defined as the difference between the exact ground state energy of the many electrons Schrödinger equation and the Hartree-Fock energy.\n\nIn this paper we discuss the Hamiltonian which describes the Hydrogen molecule regarded as a two interacting spin 1 2 (qubit) model. In [1] it was argued that the entanglement (a quantum observable) can be used in analyzing the so-called correlation energy which is not directly observable. From our point of view, the Hydrogen molecule is dealt with a bipartite system governed by the Hamiltonian\n\nH H2 = - J 2 (1 + g)σ 1 ⊗ σ 1 - J 2 (1 -g)σ 2 ⊗ σ 2 -B(σ 3 ⊗ σ 3 + σ 0 ⊗ σ 3 ), ( 1\n\n) 1\n\nwhere σ i stand for the Pauli matrices (σ 0 = I). Actually, this model was considered in [1] in order to illustrate their method. However, here we will make some interpretative changes. Indeed, from our point of view, the states of an isolated atom are strongly reduced to a system with two energy levels related to the intensity of the magnetic field B. Relatively to this scale, the exchange interaction constant J is usually smaller than B, in order to represent the residual interatomic interactions. From the point of view of quantum chemistry, one may interpret the discrete spectrum as provided by the Hartree-Fock calculations, while the interaction coupling J models the residual multielectronic effects, not taken into account by the mean field approximation. For simplicity we limit ourselves to the ferromagnetic phase with J > 0. The parameter g, such that 0 ≤ g ≤ 1, describes the degree of anisotropy corresponding for g = 0 to the completely isotropic XY spin model. Conversely, g = 1 provides the anisotropic XY spin model, the so-called Ising model.\n\nWe notice that when the atoms are far apart, their interaction is quite weak. This corresponds to a vanishing value of J. In this situation the state of the system is completely factorized in the product state of the ground states of the indipendent spins. The corresponding total energy, in unit of B, is just the sum of the two fundamental levels, E 0 = -2, which we may consider as the Hartree-Fock approximated fundamental level in molecular structure calculations.\n\nWhen J = 0, the fundamental energy eigenvalue is E= -4 + g 2 λ 2 in Region I defined by 0 < λ ≤ 2 √ 1-g 2 , otherwise E = -λ (λ means the coupling constant) in Region II, which is the complement of I which respect to positive real axis. The corresponding (non normalized) eigenstates are |Ψ I = √ g 2 λ 2 +4+2 gλ , 0, 0, 1 and |Ψ II = 0, 1, 1, 0 , respectively. In both cases the state is entangled.\n\nSince we are dealing with pure states, the von Neumann entropy [2]\n\nS vN = -Tr ρ 1 log 2 ρ 1 ( 2\n\n)\n\nis chosen to be a measurement of the entanglement, where ρ 1 is the 1-particle reduced density matrix. However, for general mixed states other entanglement estimators (for instance, the Concurrence [4]) have to be used. In the considered case, one has S vN,I = -g 2 g 2 λ 2 + 4 log 1 2 -1 g 2 λ 2 + 4 λ 2 -g 2 g 2 λ 2 + 4 + 4 λ 2 + 8 g 2 λ 2 + 4 + 2 log 1 2 + 1 g 2 λ 2 + 4 1 g 2 λ 2 + 4 g 2 λ 2 + 2 g 2 λ 2 + 4 + 4 log(4) (3) S vN,II = 1. (4) 2 Scrutinizing Eq. (3) and Eq. ( 4 ) it emerges that the entropy is an increasing function of the coupling constant λ in Region I, but the state is maximally entangled in Region II independently from the anisotropy parameter g. One sees that, as it arises graphycally, for g = 1 the entanglement is a monotonic increasing function of the interaction coupling λ. Moreover for weak (< 1) coupling values it is always less than the 30%. Of course, for large coupling constants the entropy approaches 1, meaning that all levels are equiprobably visited by the considered spin.\n\nLimiting all further considerations to the case of weak interaction, we observe that at the boundary point λ b = 2 √ 1-g 2 a discontinuity occurs, signaling a crossing of the lowest eigenvalues and, in a more general context, a quantum phase transition [5] .\n\nAs it was pointed out in [6] , for quantifying the entanglement we can resort to the reduced density matrix. Furthermore, in [7] , Wootters has shown that for a pair of binary qubits one can use the concept of Concurrence C to measure the entanglement.\n\nThe Concurrence reads\n\nC(ρ) = max(0, ν 1 -ν 2 -ν 3 -ν 4 ), ( 5\n\n)\n\nwhere the ν i 's are the eigenvalues of the Hermitian matrix\n\nR = √ ρρ √ ρ 1 2 ,\n\nwhere ρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ), ρ * being the complex conjugate of ρ taken in the standard basis [7] . Some interesting results on the simple model (1) of the Hydrogen molecule can be achieved by realizing a comparative study of the von Neumann entropy and the Concurrence.\n\nTo this aim, we compute the Concurrence C I and C II , i. e.\n\nC I = gλ 1 g 2 λ 2 + 4 , C II = 1. ( 6\n\n)\n\nwhere I and II refer to Regions I and II, where 0 ≤ λ ≤ 2 1-g 2 , and E = -λ, respectively.\n\nIn Figure 1 a comparison between the Concurrence and the von Neumann entropy for two spins system as a function of the coupling λ for g = 1 is presented. Sec. 2 contains a comparison between the entanglement and the correlation energy. In Sec. 3 the Configuration Interaction method is introduced to compare entanglement and correlation energy. In Sec. 4 some differences between the Configuration Interaction approach and the two spin Ising model are presented. Finally, our main results are summarized in Sec. 5. 3 1 2 3 4 5 Λ 0.2 0.4 0.6 0.8 1 S vN Conc.\n\nFigure 1: Comparison between the Concurrence and the von Neumann entropy for the two spins system as a function of the coupling constant λ for g = 1.\n\n2 A comparison between the entanglement and the correlation energy\n\nNow we look for a comparison between the entanglement with the energy correlation, which as we have already recalled, it is understood as the difference of the fundamental energy level compared with respect to the corresponding value at vanishing coupling constant λ. For g = 1 and in unities of B it is given by\n\nE corr = |E 0 | -2 = 4 + λ 2 -2. ( 7\n\n)\n\nWe observe that the entanglement measure is always bounded, while E corr is a divergent function of λ. So it does not make much sense to look for simple relations valid on the entire λ-axes. Consequently, limiting ourselves to weak couplings, for 0 ≤ λ ≤ 1, we minimize the mean squared deviation\n\nI α = 1 0 ∆S 2 α dλ, with ∆S α = E corr -α S vN . ( 8\n\n)\n\nThus the minimizing parameter α min will be given by\n\nα min = 1 0 E corr S vN dλ 1 0 S 2 vN dλ ≈ -0.691217. ( 9\n\n)\n\nA formula analogous to (9) can be obtained by using the Concurrence as a measure of entanglement. In this case, by minimizing the mean squared deviation we have\n\nI C α ′ = 1 0 ∆C 2 α ′ dλ, with ∆C α ′ = E corr -α ′ C. ( 10\n\n)\n\nNow, in order to estimate the relative deviation of\n\nS vN with respect to E corr , let us report |∆S αmin |/S vN and |∆S αmin /E corr | as\n\nfunctions of λ at the optimal value α min . The graphs of these functions are shown in Figure 2. 4 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min S vN 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min E corr Figure 2: The relative quadratic deviation between the von Neumann entropy and the correlation energy with respect to the former and the latter, respectively, at the optimal value α min as a function of the coupling constant λ for g = 1.\n\nIn Figure 3 , the relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, at the optimal values α ′ min , is represented.\n\n0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min C 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min E corr Figure 3: The relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, respectively, at the optimal value α ′ min as a function of the coupling constant λ for g = 1."},{"section_type":"OTHER","section_title":"Remark 1","text":"From these graphs, one can argue that the agreement between the two functions S vN and E corr is only qualitatively good, in fact, for very small λ, it is not good at all. However, in an intermediate range of values, i. e., 0.6 ≤ λ ≤ 1 the two functions are almost proportional within the 10%. Analogously, the same is true between energy and Concurrence. Even, the agreement becomes worst comparing the relative deviation of the Concurrence with respect to the correlation energy, since the range in which the relative deviations become smaller than 10% are narrower. Then, the question is whether the above results are i) sufficient to justify the conjecture advanced in [1] , i.e., entanglement can be considered as an estimation of correlation energy; ii) if such a relation has a 5 more concrete physical meaning, in particular whether the minimizing parameter α min and the vanishing point of ∆S αmin does possess any physical meaning (or α ′ min and the vanishing point of ∆C α ′ min ). Notice that in the case of the comparison for the Concurrence simpler analytical expressions appear. For instance one finds ∆C α ′ min = 0.383249 λ\n\n√ λ 2 +4 - √ λ 2 + 4 + 2 2 . Remark 2\n\nWe note that in an interval of values around α min , the deviation function (8) possesses a minimum in the interval of interest 0 ≤ λ ≤ 1, otherwise the minimum is achieved at larger value of λ, or the function is monotonically increasing (see Figure 4 ).\n\n0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 S Α 0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 d S Α dΛ\n\nFigure 4: The deviation ∆S α and its derivative with respect to λ are computed for values of -1.29(red) ≤ α ≤ -0.091(violet), for steps of 0.06. The curve drawn thicker corresponds to α min This behavior suggests to consider the function ∆S αmin as a sort of \"free energy\" , where α min mimics the \"temperature\" specific of the system. If, for some reason, we allow λ to change, then we expect that spontaneously the interaction coupling adjusts itself to the minimum of ∆S αmin . Similar considerations can be made looking at the graphs drawn for the function ∆C α ′ min and its derivative with respect to λ (see Figure 5 ). The function ∆S αmin or, alternatively, the minimum of ∆C α ′ min can be obtained algebraically. Such a minimum is at the value of the coupling constant λ SvN min ≈ 0.485 and λ C min ≈ 0.371, respectively. The authors in [1] studied numerically the von Neumann entropy and the correlation function for a Hydrogen molecule, using an old result by Herring and Flicker [8], going back to an oldest idea by Heitler and London [9], which consists in substituting the molecular binding with a position dependent exchange coupling:\n\nJ(r) ≈ 1.641 r 5 2 e -2 r Ry, ( 11\n\n)\n\nwhere r is given in Bohr radius, see Figure 6 . The maximum value taken by this function is at the point r max = 1.25. Assuming B = 0.5 Ry, i.e. 1 2 of the fundamental level of the Hydrogen atom, the maximum value λ ′ max = J(r max )/B ≈ 6 0.2 0.4 0.6 0.8 1 Λ -0.2 -0.1 0.1 0.2 0.3 C Α' 0.2 0.4 0.6 0.8 1 Λ -0.4 -0.2 0.2 0.4 d C Α' dΛ 5: The deviation ∆C α ′ and its derivative with respect to λ are computed for values of -0.98(red) ≤ α ′ ≤ 0.22(violet), for steps of 0.06. The curve drawn thicker corresponds to α ′ min 0.5 1 1.5 2 2.5 3 r 0.05 0.1 0.15 0.2 J r Ry\n\nFigure 6: The effective interaction Hydrogen-Hydrogen atom 0.470628 < λ SvN min , i.e. the value of the effective interaction value is less than the minimum for the deviation function ∆S αmin . Then, the equilibrium balance between entanglement (as von Neumann entropy) and correlation energy predicts a length of the molecule equal to r max (see the first panel of Figure 7 ). On the other hand, if we consider the energy gap 2B = 3/4 Ry, i.e. the energy step to the first excited state, one obtains the new value λ ′′ max ≈ 0.628, which goes beyond λ min , even if it is always less than 1. Now, the deviation function ∆S αmin has two minima as seen in the second panel of Figure 7 , one of which is at r ′′ -≈ 0.76 , the other one being at r ′′ + ≈ 1.91. These results should be compared with the experimental equilibrium length of the Hydrogen molecule, which is r exp ≈ 2.0. We point out that although the spin-model described by the Hamiltonian (1) is characterized by features which are essentially rough, however we are induced to answer positively to the quest for a physical meaning of the deviation function ∆S αmin . Indeed, the results elucidated in Figure 7 encourage, on one part, improvement of the computation of r in order to make more accurate the comparison with the experimental value r exp . 7 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .5 Ry 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .375 Ry r'' r''\n\nFigure 7: The von Neumann entropy for the 2-spin model for B = .5 Ry (left panel) and for B = .375 Ry (right panel) and the position depending interaction given by ( 11 ).\n\nThe first question to answer is whether this draft works also for the Concurrence. A statement about it is not obvious, since the von Neumann entropy is a nonlinear function of the Concurrence in the 2-qubits case. However, from Figure 8 one can see that the minimized deviation of the Concurrence takes one minimum for relatively large intensity of the magnetic field ( say B ≥ 0.6 Ry), while for weak fields two minima appear, corresponding to the situation depicted nearby. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 r Bohr B Ry 0.2 0.4 0.6 0.8 1 0 1 2 3 4 B(Ry) r(Bohr)\n\nFigure 8: Two contour plots of the minimized deviation of the Concurrence as a function of the magnetic field B (Ry) and of the internuclear distance r, as given by (11) . The range of values divided by the contour lines is [-0.038, 0, 04] for the left panel and [-0.03705, -0, 03000] for the right one that approximatively corresponding to the black area in the left panel.\n\nIn correspondence of the same values considered above, for B = 0.5 Ry the function ∆C α ′ min (r) has two minima at r = 0.79 and r = 1.88, while for B = 0.375 Ry they are located at r = 0.60 and r = 2.25. So one sees that the resulting equilibrium configurations are not much very close to the experimental one. The equilibrium configuration more closest to the experimental one is the minimum occurring at r = 1.88 (B = 1 2 Ry) for the function ∆C α ′ min (r).\n\n8 One sees that one of the resulting equilibrium configurations is only roughly close to the experimental one.\n\nIn other words, to conclude monitoring numerically B the equilibrium configuration more closest to experimental one in the minimum occurring at r = 1.88 for B = 1 2 Ry and at r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r)."},{"section_type":"OTHER","section_title":"A quantum chemical framework to compare entanglement and correlation energy","text":"In this Section we represent the results produced in [1], where the electron entanglement in the Hydrogen molecule, calculated by the von Neumann entropy of the reduced density matrix ρ 1 , is obtained starting by the excitation coefficients of the wave function expanded by a configuration interaction method:\n\nS ρ CISD 1 = -T r ρ CISD 1 log 2 ρ CISD 1 = = - m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 log 2 m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 + -|c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 log 2 |c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 , ( 12\n\n)\n\nwhere c 1 is the coefficient for a single excitation, and c 1,2 is the double excitation (in Appendix A of [10] more details are shown).\n\nIn this framework, entanglement (S) and correlation energy (E corr ), as functions of nucleus -nucleus separation are those in Figure 9 0 0.01 0.02 0.03 0.04 0.05 E c [ a. u. ] E c 0 1 2 3 4 5 R ( Å ) 0 0.2 0.4 0.6 0.8 1 Entanglement (S) S ( ρ 1 CISD ) H 2 Figure 9: Comparison between the entanglement, calculated by the von Neumann entropy of the reduced density matrix, and the electron correlation energy in the Hydrogen molecule.\n\nBy the results given by this model, we want to discuss and to suggest some answers to the questions i) and ii) presented in Remark 1. Even if, in order 9 to represent correlation energy and entanglement, we use two different scales, in Figure 9 we can see that entanglement has a small value in the united atom limit after it is growing for small distances till it arrives at a maximum value then it decrease till it assumes zero value at the separated atom limit and it is exactly the progress of the correlation curve.\n\nIn order to compare the entropy S with the electron correlation energy E corr , we rescale S with the parameter α min calculated with some procedure illustrated in Eq. ( 8 ) and Eq. (9) replacing the integration variable λ with R; in this way we extract\n\nα = E corr S vN dR S vN dR ≈ 0.009. ( 13\n\n)\n\nThe corresponding ∆S αmin = E corr -αS allows us to answer to the question ii); in fact, as it is shown in Figure 10 , the vanishing point of ∆S αmin is, according to the two -spin Ising model, nearby R ≈ 2 Å that corresponds to the equilibrium configuration of the Hydrogen molecule.\n\n0 1 2 3 4 5 R(Å) -0.01 0 0.01 0.02 0.03 0.04 Ec-αS\n\nFigure 10: ∆S αmin for the H 2 molecule as a function of nucleus-nucleus distance."},{"section_type":"METHOD","section_title":"Differences between the Configuration Interaction approach and the two-spin Ising model","text":"The model proposed in Sec. 1 provides us with a measurement of entanglement: indeed, Eq. (3) describes the von Neumann entropy as a function of coupling constant λ, for small λ. By using Eq. ( 7 ), we can express λ in terms of correlation energy and substituting it in Eq. ( 3 ) we can obtain the variation of S vN in terms of E corr .\n\nS vN = - E corr Log Ecorr 2(Ecorr +2) + (E corr + 4)Log Ecorr +4 2(Ecorr +2) (E corr + 2)Log4 . ( 14\n\n)\n\n10 In order to calculated the coefficient of proportionality among S vN and E corr we make an expansion of S vN for E corr → 0 (or equivalently for λ → 0) at the first order, obtaining a straight line characterized by an angular coefficient given by m SvN (Ecorr) = ( 1 4 )(1 + 1 Log2 ). Since this behavior is uncorrect to represent the logatithmic singularity of S vN in the origin, we make an expansion of Eq. ( 14 ), preserving the logarithmic deviation, and we obtain an expression of the form\n\nS vN = AE corr + BE corr Log(E corr ), ( 15\n\n)\n\nwhere A = 1/2 and B = -1/(4Log2).\n\n0.02 0.04 0.06 0.08 0.1 0.025 0.05 0.075 0.1 0.125 0.15 Ecorr S Linear AE+BELogE Figure 11: A comparison among the behavior of Eq. ( 14 ) and its linear approximation and the logarithmic one, for the Ising model.\n\nIn order to compare the behavior of S vN in Eq. ( 14 ), we have organized the numerical data, calculated with the method proposed in [1] , by making a correspondence between each value of E corr and its respective value of S vN , obtaining the plot in Figure 12 0 0.01 0.02 0.03 0.04 0.05 E c 0 0.2 0.4 0.6 0.8 1 S Figure 12: A correspondence of E corr and S vN by the numerical procedure suggested by [1] Of particular significance is the fact that, in the range where S is monotonically increasing, the correlation energy has its maximum, consequently S seems to be not a function. Moreover, it is important to note that E corr begins to decrease for R > 1 Å, region where the states become mixed, i. e. ,T rρ = T rρ 2 ; as depicted in Figure 13. 11 0 1 2 3 4 R(Å ) 0.4 0.6 0.8 1 1.2 1.4 Trρ 2\n\nFigure 13: The increasing of the degree of mixing in the two electron state: in black we depict the trace of ρ, in red the trace of ρ 2 .\n\nProbably, for this reason, the procedure adopted in [1] seems to be not correct: the density matrix, in fact, is calculated starting by the excitation coefficient of a wave function obtained developping with the Configuration Interaction Single Double method a pure two electrons state.\n\nHowever, even if we consider only the first branch of the plot in Figure 12 , i.e. , the numerical values of S vN corresponding with increasing values of E corr , and we fit the values around E corr → 0 with a F = AE corr + BE corr Log(E corr ) we draw out numerical values of the coefficient different from the ones used in Eq. (15) . This result is shown in Figure 14. 0.01 0.02 0.03 0.04 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 S Ecorr A=17.1 B=3.3 Figure 14: A fit of S vN as a function of E corr , around the origin, with a function of the form F = AE corr + BE corr Log(E corr ) whose coefficients A and B assume the numerical values in Figure.\n\nIn particular the arithmetic sign of the coefficient B in the two models are opposite and this implies the opposite concavity of the curve.\n\nThis fact, clearly demonstrates a not satisfactory agreement between the Ising model and the one proposed in [1] ."},{"section_type":"OTHER","section_title":"5 Concluding remarks","text":"We have explored the role of entanglement in the model of two qubits describing the Hydrogen molecule (1), considered as a bipartite system. In our discussion we have limited to the ferromagnetic case governed by the interaction coupling parameter J > 0.\n\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The entanglement can be measured by using the von Neumann entropy or, alternatively, the notion of Concurrence [7] . To compute the entanglement it is convenient to consider two Regions, say I and II, which provide two different reduced density matrices. The entropy turns out to be an increasing function of the coupling constant λ in Region I, but the state under consideration is maximally entangled in Region II indipendently from the anisotropy parameter g.\n\nAn interesting result is that for large coupling constants the entropy approach 1, meaning that all levels are equiprobably visited by the considered spin.\n\nFor weak interactions, at the boundary point λ b = 2 √ 1-g 2 the von Neumann entropy admits a discontinuity, indicating a crossing of the lowest eigenvalues and, in a more general constext, a quantum phase transition [5] .\n\nIn Sec. 2 a comparison between the entanglement and the correlation energy is performed.\n\nTo quantifying the entanglement we resort to the reduced density matrix. The entanglement can also be measured by exploiting the concept of Concurrence.\n\nThe entanglement measure is always bounded, while the energy correlation,\n\nE corr = |E 0 | -2 =\n\n√ 4 + λ 2 -2, is a divergent function of λ. This fact tells us that to look for simple relations valid on the whole λ-axes has no sense.\n\nThus, by limiting ourselves to weak couplings, we have minimized the mean square deviation given by Eq. ( 8 ). This procedure leads to the value α min ≈ -0.691217 for the minimizing parameter (see Eq. (9)). Sec. 1 contains a comparison between the von Neumann entropy and the Concurrence. Such a comparison is illustrated in Figure 1 , for two spin system as a function of the coupling λ for g = 1. Some important points are commented in Remark 1 and Remark 2 .\n\nIn Figure 4 the deviation ∆S α and its derivatives with respect to λ are computed and α min is evaluated for α ranging in the interval -1.29 ≤ α ≤ -0.091.\n\nIn Figure 5 the minimized Concurrence deviation ∆C α (i) α ′\n\nfor the four eigenstates of the 2-spin model is shown.\n\nWe point out the existence of a perfect symmetry among the Concurrence deviations for pairs of eigenstates of opposite eigenvalues.\n\nFormula (11), due to Heitler-London [9], is reported, where the position dependent exchange coupling J(r) is expressed in term of the length r of the 13 nucleus-nucleus separation in the Hydrogen molecule.\n\nTo conclude, the magnetic field B has been monitored such that the equilibrium configuration more closest to the experimental one, r ≈ 2.00, is the minimum occurring at r = 1.88 for B = 1 2 Ry and r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r). We observe also that in the intermediate range of values, i. e., for 0.6 ≤ λ ≤ 1, the two functions SvN and the correlation energy are almost proportional within the 10%.\n\nHowever, when we organized the pairs of points (E corr , S vN ) calculated by following the procedure described by [1], it is clear that the von Neumann entropy cannot be considered a function of correlation energy. The principle cause is that the function E corr presents a maximum in the region where S vN is monotonically increasing.\n\nThe reversing behavior of correlation energy occurs in correspondence with an increase of the mixing degree of the two electrons state. The function E corr in terms of the nucleus -nucleus distance R, increases till the state is pure, on the contrary, when T r(ρ 2 ) becomes discordant from T r(ρ), the function E corr decreases.\n\nThis fact suggests us that the numerical model based on the calculation of S vN starting by the excitation coefficients c i , isn't completley correct because the density matrix is obtained as a product of two electron pure states. However, even if we consider only a branch of the plot in Figure 12 , the function obtained by the two spin Ising model, i. e., Eq. ( 14 ), is unsuitable for fitting these numerical data.\n\nOn the basis of our results, essentially grounded on numerical considerations, in the near feature we would explore more complicated systems of molecules, such as for example the ethylene or other hydrocarbons, and compare these studies with the goals obtained for the Hydrogen molecule."},{"section_type":"OTHER","section_title":"Acknowledgments","text":"The authors acknowledge the Italian Ministry of Scientific Researches (MIUR) for partial support of the present work under the project SINTESI 2004/06 and the INFN for partial support under the project Iniziativa Specifica LE41."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction and the model\",\n \"text\": \"Entanglement is a physical observable measured by the von Neumann entropy or, alternatively, by the Concurrence of the system under consideration.\\n\\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The electron correlation is not directly observable, since it is defined as the difference between the exact ground state energy of the many electrons Schrödinger equation and the Hartree-Fock energy.\\n\\nIn this paper we discuss the Hamiltonian which describes the Hydrogen molecule regarded as a two interacting spin 1 2 (qubit) model. In [1] it was argued that the entanglement (a quantum observable) can be used in analyzing the so-called correlation energy which is not directly observable. From our point of view, the Hydrogen molecule is dealt with a bipartite system governed by the Hamiltonian\\n\\nH H2 = - J 2 (1 + g)σ 1 ⊗ σ 1 - J 2 (1 -g)σ 2 ⊗ σ 2 -B(σ 3 ⊗ σ 3 + σ 0 ⊗ σ 3 ), ( 1\\n\\n) 1\\n\\nwhere σ i stand for the Pauli matrices (σ 0 = I). Actually, this model was considered in [1] in order to illustrate their method. However, here we will make some interpretative changes. Indeed, from our point of view, the states of an isolated atom are strongly reduced to a system with two energy levels related to the intensity of the magnetic field B. Relatively to this scale, the exchange interaction constant J is usually smaller than B, in order to represent the residual interatomic interactions. From the point of view of quantum chemistry, one may interpret the discrete spectrum as provided by the Hartree-Fock calculations, while the interaction coupling J models the residual multielectronic effects, not taken into account by the mean field approximation. For simplicity we limit ourselves to the ferromagnetic phase with J > 0. The parameter g, such that 0 ≤ g ≤ 1, describes the degree of anisotropy corresponding for g = 0 to the completely isotropic XY spin model. Conversely, g = 1 provides the anisotropic XY spin model, the so-called Ising model.\\n\\nWe notice that when the atoms are far apart, their interaction is quite weak. This corresponds to a vanishing value of J. In this situation the state of the system is completely factorized in the product state of the ground states of the indipendent spins. The corresponding total energy, in unit of B, is just the sum of the two fundamental levels, E 0 = -2, which we may consider as the Hartree-Fock approximated fundamental level in molecular structure calculations.\\n\\nWhen J = 0, the fundamental energy eigenvalue is E= -4 + g 2 λ 2 in Region I defined by 0 < λ ≤ 2 √ 1-g 2 , otherwise E = -λ (λ means the coupling constant) in Region II, which is the complement of I which respect to positive real axis. The corresponding (non normalized) eigenstates are |Ψ I = √ g 2 λ 2 +4+2 gλ , 0, 0, 1 and |Ψ II = 0, 1, 1, 0 , respectively. In both cases the state is entangled.\\n\\nSince we are dealing with pure states, the von Neumann entropy [2]\\n\\nS vN = -Tr ρ 1 log 2 ρ 1 ( 2\\n\\n)\\n\\nis chosen to be a measurement of the entanglement, where ρ 1 is the 1-particle reduced density matrix. However, for general mixed states other entanglement estimators (for instance, the Concurrence [4]) have to be used. In the considered case, one has S vN,I = -g 2 g 2 λ 2 + 4 log 1 2 -1 g 2 λ 2 + 4 λ 2 -g 2 g 2 λ 2 + 4 + 4 λ 2 + 8 g 2 λ 2 + 4 + 2 log 1 2 + 1 g 2 λ 2 + 4 1 g 2 λ 2 + 4 g 2 λ 2 + 2 g 2 λ 2 + 4 + 4 log(4) (3) S vN,II = 1. (4) 2 Scrutinizing Eq. (3) and Eq. ( 4 ) it emerges that the entropy is an increasing function of the coupling constant λ in Region I, but the state is maximally entangled in Region II independently from the anisotropy parameter g. One sees that, as it arises graphycally, for g = 1 the entanglement is a monotonic increasing function of the interaction coupling λ. Moreover for weak (< 1) coupling values it is always less than the 30%. Of course, for large coupling constants the entropy approaches 1, meaning that all levels are equiprobably visited by the considered spin.\\n\\nLimiting all further considerations to the case of weak interaction, we observe that at the boundary point λ b = 2 √ 1-g 2 a discontinuity occurs, signaling a crossing of the lowest eigenvalues and, in a more general context, a quantum phase transition [5] .\\n\\nAs it was pointed out in [6] , for quantifying the entanglement we can resort to the reduced density matrix. Furthermore, in [7] , Wootters has shown that for a pair of binary qubits one can use the concept of Concurrence C to measure the entanglement.\\n\\nThe Concurrence reads\\n\\nC(ρ) = max(0, ν 1 -ν 2 -ν 3 -ν 4 ), ( 5\\n\\n)\\n\\nwhere the ν i 's are the eigenvalues of the Hermitian matrix\\n\\nR = √ ρρ √ ρ 1 2 ,\\n\\nwhere ρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ), ρ * being the complex conjugate of ρ taken in the standard basis [7] . Some interesting results on the simple model (1) of the Hydrogen molecule can be achieved by realizing a comparative study of the von Neumann entropy and the Concurrence.\\n\\nTo this aim, we compute the Concurrence C I and C II , i. e.\\n\\nC I = gλ 1 g 2 λ 2 + 4 , C II = 1. ( 6\\n\\n)\\n\\nwhere I and II refer to Regions I and II, where 0 ≤ λ ≤ 2 1-g 2 , and E = -λ, respectively.\\n\\nIn Figure 1 a comparison between the Concurrence and the von Neumann entropy for two spins system as a function of the coupling λ for g = 1 is presented. Sec. 2 contains a comparison between the entanglement and the correlation energy. In Sec. 3 the Configuration Interaction method is introduced to compare entanglement and correlation energy. In Sec. 4 some differences between the Configuration Interaction approach and the two spin Ising model are presented. Finally, our main results are summarized in Sec. 5. 3 1 2 3 4 5 Λ 0.2 0.4 0.6 0.8 1 S vN Conc.\\n\\nFigure 1: Comparison between the Concurrence and the von Neumann entropy for the two spins system as a function of the coupling constant λ for g = 1.\\n\\n2 A comparison between the entanglement and the correlation energy\\n\\nNow we look for a comparison between the entanglement with the energy correlation, which as we have already recalled, it is understood as the difference of the fundamental energy level compared with respect to the corresponding value at vanishing coupling constant λ. For g = 1 and in unities of B it is given by\\n\\nE corr = |E 0 | -2 = 4 + λ 2 -2. ( 7\\n\\n)\\n\\nWe observe that the entanglement measure is always bounded, while E corr is a divergent function of λ. So it does not make much sense to look for simple relations valid on the entire λ-axes. Consequently, limiting ourselves to weak couplings, for 0 ≤ λ ≤ 1, we minimize the mean squared deviation\\n\\nI α = 1 0 ∆S 2 α dλ, with ∆S α = E corr -α S vN . ( 8\\n\\n)\\n\\nThus the minimizing parameter α min will be given by\\n\\nα min = 1 0 E corr S vN dλ 1 0 S 2 vN dλ ≈ -0.691217. ( 9\\n\\n)\\n\\nA formula analogous to (9) can be obtained by using the Concurrence as a measure of entanglement. In this case, by minimizing the mean squared deviation we have\\n\\nI C α ′ = 1 0 ∆C 2 α ′ dλ, with ∆C α ′ = E corr -α ′ C. ( 10\\n\\n)\\n\\nNow, in order to estimate the relative deviation of\\n\\nS vN with respect to E corr , let us report |∆S αmin |/S vN and |∆S αmin /E corr | as\\n\\nfunctions of λ at the optimal value α min . The graphs of these functions are shown in Figure 2. 4 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min S vN 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min E corr Figure 2: The relative quadratic deviation between the von Neumann entropy and the correlation energy with respect to the former and the latter, respectively, at the optimal value α min as a function of the coupling constant λ for g = 1.\\n\\nIn Figure 3 , the relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, at the optimal values α ′ min , is represented.\\n\\n0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min C 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min E corr Figure 3: The relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, respectively, at the optimal value α ′ min as a function of the coupling constant λ for g = 1.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Remark 1\",\n \"text\": \"From these graphs, one can argue that the agreement between the two functions S vN and E corr is only qualitatively good, in fact, for very small λ, it is not good at all. However, in an intermediate range of values, i. e., 0.6 ≤ λ ≤ 1 the two functions are almost proportional within the 10%. Analogously, the same is true between energy and Concurrence. Even, the agreement becomes worst comparing the relative deviation of the Concurrence with respect to the correlation energy, since the range in which the relative deviations become smaller than 10% are narrower. Then, the question is whether the above results are i) sufficient to justify the conjecture advanced in [1] , i.e., entanglement can be considered as an estimation of correlation energy; ii) if such a relation has a 5 more concrete physical meaning, in particular whether the minimizing parameter α min and the vanishing point of ∆S αmin does possess any physical meaning (or α ′ min and the vanishing point of ∆C α ′ min ). Notice that in the case of the comparison for the Concurrence simpler analytical expressions appear. For instance one finds ∆C α ′ min = 0.383249 λ\\n\\n√ λ 2 +4 - √ λ 2 + 4 + 2 2 . Remark 2\\n\\nWe note that in an interval of values around α min , the deviation function (8) possesses a minimum in the interval of interest 0 ≤ λ ≤ 1, otherwise the minimum is achieved at larger value of λ, or the function is monotonically increasing (see Figure 4 ).\\n\\n0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 S Α 0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 d S Α dΛ\\n\\nFigure 4: The deviation ∆S α and its derivative with respect to λ are computed for values of -1.29(red) ≤ α ≤ -0.091(violet), for steps of 0.06. The curve drawn thicker corresponds to α min This behavior suggests to consider the function ∆S αmin as a sort of \\\"free energy\\\" , where α min mimics the \\\"temperature\\\" specific of the system. If, for some reason, we allow λ to change, then we expect that spontaneously the interaction coupling adjusts itself to the minimum of ∆S αmin . Similar considerations can be made looking at the graphs drawn for the function ∆C α ′ min and its derivative with respect to λ (see Figure 5 ). The function ∆S αmin or, alternatively, the minimum of ∆C α ′ min can be obtained algebraically. Such a minimum is at the value of the coupling constant λ SvN min ≈ 0.485 and λ C min ≈ 0.371, respectively. The authors in [1] studied numerically the von Neumann entropy and the correlation function for a Hydrogen molecule, using an old result by Herring and Flicker [8], going back to an oldest idea by Heitler and London [9], which consists in substituting the molecular binding with a position dependent exchange coupling:\\n\\nJ(r) ≈ 1.641 r 5 2 e -2 r Ry, ( 11\\n\\n)\\n\\nwhere r is given in Bohr radius, see Figure 6 . The maximum value taken by this function is at the point r max = 1.25. Assuming B = 0.5 Ry, i.e. 1 2 of the fundamental level of the Hydrogen atom, the maximum value λ ′ max = J(r max )/B ≈ 6 0.2 0.4 0.6 0.8 1 Λ -0.2 -0.1 0.1 0.2 0.3 C Α' 0.2 0.4 0.6 0.8 1 Λ -0.4 -0.2 0.2 0.4 d C Α' dΛ 5: The deviation ∆C α ′ and its derivative with respect to λ are computed for values of -0.98(red) ≤ α ′ ≤ 0.22(violet), for steps of 0.06. The curve drawn thicker corresponds to α ′ min 0.5 1 1.5 2 2.5 3 r 0.05 0.1 0.15 0.2 J r Ry\\n\\nFigure 6: The effective interaction Hydrogen-Hydrogen atom 0.470628 < λ SvN min , i.e. the value of the effective interaction value is less than the minimum for the deviation function ∆S αmin . Then, the equilibrium balance between entanglement (as von Neumann entropy) and correlation energy predicts a length of the molecule equal to r max (see the first panel of Figure 7 ). On the other hand, if we consider the energy gap 2B = 3/4 Ry, i.e. the energy step to the first excited state, one obtains the new value λ ′′ max ≈ 0.628, which goes beyond λ min , even if it is always less than 1. Now, the deviation function ∆S αmin has two minima as seen in the second panel of Figure 7 , one of which is at r ′′ -≈ 0.76 , the other one being at r ′′ + ≈ 1.91. These results should be compared with the experimental equilibrium length of the Hydrogen molecule, which is r exp ≈ 2.0. We point out that although the spin-model described by the Hamiltonian (1) is characterized by features which are essentially rough, however we are induced to answer positively to the quest for a physical meaning of the deviation function ∆S αmin . Indeed, the results elucidated in Figure 7 encourage, on one part, improvement of the computation of r in order to make more accurate the comparison with the experimental value r exp . 7 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .5 Ry 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .375 Ry r'' r''\\n\\nFigure 7: The von Neumann entropy for the 2-spin model for B = .5 Ry (left panel) and for B = .375 Ry (right panel) and the position depending interaction given by ( 11 ).\\n\\nThe first question to answer is whether this draft works also for the Concurrence. A statement about it is not obvious, since the von Neumann entropy is a nonlinear function of the Concurrence in the 2-qubits case. However, from Figure 8 one can see that the minimized deviation of the Concurrence takes one minimum for relatively large intensity of the magnetic field ( say B ≥ 0.6 Ry), while for weak fields two minima appear, corresponding to the situation depicted nearby. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 r Bohr B Ry 0.2 0.4 0.6 0.8 1 0 1 2 3 4 B(Ry) r(Bohr)\\n\\nFigure 8: Two contour plots of the minimized deviation of the Concurrence as a function of the magnetic field B (Ry) and of the internuclear distance r, as given by (11) . The range of values divided by the contour lines is [-0.038, 0, 04] for the left panel and [-0.03705, -0, 03000] for the right one that approximatively corresponding to the black area in the left panel.\\n\\nIn correspondence of the same values considered above, for B = 0.5 Ry the function ∆C α ′ min (r) has two minima at r = 0.79 and r = 1.88, while for B = 0.375 Ry they are located at r = 0.60 and r = 2.25. So one sees that the resulting equilibrium configurations are not much very close to the experimental one. The equilibrium configuration more closest to the experimental one is the minimum occurring at r = 1.88 (B = 1 2 Ry) for the function ∆C α ′ min (r).\\n\\n8 One sees that one of the resulting equilibrium configurations is only roughly close to the experimental one.\\n\\nIn other words, to conclude monitoring numerically B the equilibrium configuration more closest to experimental one in the minimum occurring at r = 1.88 for B = 1 2 Ry and at r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A quantum chemical framework to compare entanglement and correlation energy\",\n \"text\": \"In this Section we represent the results produced in [1], where the electron entanglement in the Hydrogen molecule, calculated by the von Neumann entropy of the reduced density matrix ρ 1 , is obtained starting by the excitation coefficients of the wave function expanded by a configuration interaction method:\\n\\nS ρ CISD 1 = -T r ρ CISD 1 log 2 ρ CISD 1 = = - m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 log 2 m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 + -|c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 log 2 |c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 , ( 12\\n\\n)\\n\\nwhere c 1 is the coefficient for a single excitation, and c 1,2 is the double excitation (in Appendix A of [10] more details are shown).\\n\\nIn this framework, entanglement (S) and correlation energy (E corr ), as functions of nucleus -nucleus separation are those in Figure 9 0 0.01 0.02 0.03 0.04 0.05 E c [ a. u. ] E c 0 1 2 3 4 5 R ( Å ) 0 0.2 0.4 0.6 0.8 1 Entanglement (S) S ( ρ 1 CISD ) H 2 Figure 9: Comparison between the entanglement, calculated by the von Neumann entropy of the reduced density matrix, and the electron correlation energy in the Hydrogen molecule.\\n\\nBy the results given by this model, we want to discuss and to suggest some answers to the questions i) and ii) presented in Remark 1. Even if, in order 9 to represent correlation energy and entanglement, we use two different scales, in Figure 9 we can see that entanglement has a small value in the united atom limit after it is growing for small distances till it arrives at a maximum value then it decrease till it assumes zero value at the separated atom limit and it is exactly the progress of the correlation curve.\\n\\nIn order to compare the entropy S with the electron correlation energy E corr , we rescale S with the parameter α min calculated with some procedure illustrated in Eq. ( 8 ) and Eq. (9) replacing the integration variable λ with R; in this way we extract\\n\\nα = E corr S vN dR S vN dR ≈ 0.009. ( 13\\n\\n)\\n\\nThe corresponding ∆S αmin = E corr -αS allows us to answer to the question ii); in fact, as it is shown in Figure 10 , the vanishing point of ∆S αmin is, according to the two -spin Ising model, nearby R ≈ 2 Å that corresponds to the equilibrium configuration of the Hydrogen molecule.\\n\\n0 1 2 3 4 5 R(Å) -0.01 0 0.01 0.02 0.03 0.04 Ec-αS\\n\\nFigure 10: ∆S αmin for the H 2 molecule as a function of nucleus-nucleus distance.\"\n },\n {\n \"section_type\": \"METHOD\",\n \"section_title\": \"Differences between the Configuration Interaction approach and the two-spin Ising model\",\n \"text\": \"The model proposed in Sec. 1 provides us with a measurement of entanglement: indeed, Eq. (3) describes the von Neumann entropy as a function of coupling constant λ, for small λ. By using Eq. ( 7 ), we can express λ in terms of correlation energy and substituting it in Eq. ( 3 ) we can obtain the variation of S vN in terms of E corr .\\n\\nS vN = - E corr Log Ecorr 2(Ecorr +2) + (E corr + 4)Log Ecorr +4 2(Ecorr +2) (E corr + 2)Log4 . ( 14\\n\\n)\\n\\n10 In order to calculated the coefficient of proportionality among S vN and E corr we make an expansion of S vN for E corr → 0 (or equivalently for λ → 0) at the first order, obtaining a straight line characterized by an angular coefficient given by m SvN (Ecorr) = ( 1 4 )(1 + 1 Log2 ). Since this behavior is uncorrect to represent the logatithmic singularity of S vN in the origin, we make an expansion of Eq. ( 14 ), preserving the logarithmic deviation, and we obtain an expression of the form\\n\\nS vN = AE corr + BE corr Log(E corr ), ( 15\\n\\n)\\n\\nwhere A = 1/2 and B = -1/(4Log2).\\n\\n0.02 0.04 0.06 0.08 0.1 0.025 0.05 0.075 0.1 0.125 0.15 Ecorr S Linear AE+BELogE Figure 11: A comparison among the behavior of Eq. ( 14 ) and its linear approximation and the logarithmic one, for the Ising model.\\n\\nIn order to compare the behavior of S vN in Eq. ( 14 ), we have organized the numerical data, calculated with the method proposed in [1] , by making a correspondence between each value of E corr and its respective value of S vN , obtaining the plot in Figure 12 0 0.01 0.02 0.03 0.04 0.05 E c 0 0.2 0.4 0.6 0.8 1 S Figure 12: A correspondence of E corr and S vN by the numerical procedure suggested by [1] Of particular significance is the fact that, in the range where S is monotonically increasing, the correlation energy has its maximum, consequently S seems to be not a function. Moreover, it is important to note that E corr begins to decrease for R > 1 Å, region where the states become mixed, i. e. ,T rρ = T rρ 2 ; as depicted in Figure 13. 11 0 1 2 3 4 R(Å ) 0.4 0.6 0.8 1 1.2 1.4 Trρ 2\\n\\nFigure 13: The increasing of the degree of mixing in the two electron state: in black we depict the trace of ρ, in red the trace of ρ 2 .\\n\\nProbably, for this reason, the procedure adopted in [1] seems to be not correct: the density matrix, in fact, is calculated starting by the excitation coefficient of a wave function obtained developping with the Configuration Interaction Single Double method a pure two electrons state.\\n\\nHowever, even if we consider only the first branch of the plot in Figure 12 , i.e. , the numerical values of S vN corresponding with increasing values of E corr , and we fit the values around E corr → 0 with a F = AE corr + BE corr Log(E corr ) we draw out numerical values of the coefficient different from the ones used in Eq. (15) . This result is shown in Figure 14. 0.01 0.02 0.03 0.04 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 S Ecorr A=17.1 B=3.3 Figure 14: A fit of S vN as a function of E corr , around the origin, with a function of the form F = AE corr + BE corr Log(E corr ) whose coefficients A and B assume the numerical values in Figure.\\n\\nIn particular the arithmetic sign of the coefficient B in the two models are opposite and this implies the opposite concavity of the curve.\\n\\nThis fact, clearly demonstrates a not satisfactory agreement between the Ising model and the one proposed in [1] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"5 Concluding remarks\",\n \"text\": \"We have explored the role of entanglement in the model of two qubits describing the Hydrogen molecule (1), considered as a bipartite system. In our discussion we have limited to the ferromagnetic case governed by the interaction coupling parameter J > 0.\\n\\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The entanglement can be measured by using the von Neumann entropy or, alternatively, the notion of Concurrence [7] . To compute the entanglement it is convenient to consider two Regions, say I and II, which provide two different reduced density matrices. The entropy turns out to be an increasing function of the coupling constant λ in Region I, but the state under consideration is maximally entangled in Region II indipendently from the anisotropy parameter g.\\n\\nAn interesting result is that for large coupling constants the entropy approach 1, meaning that all levels are equiprobably visited by the considered spin.\\n\\nFor weak interactions, at the boundary point λ b = 2 √ 1-g 2 the von Neumann entropy admits a discontinuity, indicating a crossing of the lowest eigenvalues and, in a more general constext, a quantum phase transition [5] .\\n\\nIn Sec. 2 a comparison between the entanglement and the correlation energy is performed.\\n\\nTo quantifying the entanglement we resort to the reduced density matrix. The entanglement can also be measured by exploiting the concept of Concurrence.\\n\\nThe entanglement measure is always bounded, while the energy correlation,\\n\\nE corr = |E 0 | -2 =\\n\\n√ 4 + λ 2 -2, is a divergent function of λ. This fact tells us that to look for simple relations valid on the whole λ-axes has no sense.\\n\\nThus, by limiting ourselves to weak couplings, we have minimized the mean square deviation given by Eq. ( 8 ). This procedure leads to the value α min ≈ -0.691217 for the minimizing parameter (see Eq. (9)). Sec. 1 contains a comparison between the von Neumann entropy and the Concurrence. Such a comparison is illustrated in Figure 1 , for two spin system as a function of the coupling λ for g = 1. Some important points are commented in Remark 1 and Remark 2 .\\n\\nIn Figure 4 the deviation ∆S α and its derivatives with respect to λ are computed and α min is evaluated for α ranging in the interval -1.29 ≤ α ≤ -0.091.\\n\\nIn Figure 5 the minimized Concurrence deviation ∆C α (i) α ′\\n\\nfor the four eigenstates of the 2-spin model is shown.\\n\\nWe point out the existence of a perfect symmetry among the Concurrence deviations for pairs of eigenstates of opposite eigenvalues.\\n\\nFormula (11), due to Heitler-London [9], is reported, where the position dependent exchange coupling J(r) is expressed in term of the length r of the 13 nucleus-nucleus separation in the Hydrogen molecule.\\n\\nTo conclude, the magnetic field B has been monitored such that the equilibrium configuration more closest to the experimental one, r ≈ 2.00, is the minimum occurring at r = 1.88 for B = 1 2 Ry and r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r). We observe also that in the intermediate range of values, i. e., for 0.6 ≤ λ ≤ 1, the two functions SvN and the correlation energy are almost proportional within the 10%.\\n\\nHowever, when we organized the pairs of points (E corr , S vN ) calculated by following the procedure described by [1], it is clear that the von Neumann entropy cannot be considered a function of correlation energy. The principle cause is that the function E corr presents a maximum in the region where S vN is monotonically increasing.\\n\\nThe reversing behavior of correlation energy occurs in correspondence with an increase of the mixing degree of the two electrons state. The function E corr in terms of the nucleus -nucleus distance R, increases till the state is pure, on the contrary, when T r(ρ 2 ) becomes discordant from T r(ρ), the function E corr decreases.\\n\\nThis fact suggests us that the numerical model based on the calculation of S vN starting by the excitation coefficients c i , isn't completley correct because the density matrix is obtained as a product of two electron pure states. However, even if we consider only a branch of the plot in Figure 12 , the function obtained by the two spin Ising model, i. e., Eq. ( 14 ), is unsuitable for fitting these numerical data.\\n\\nOn the basis of our results, essentially grounded on numerical considerations, in the near feature we would explore more complicated systems of molecules, such as for example the ethylene or other hydrocarbons, and compare these studies with the goals obtained for the Hydrogen molecule.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments\",\n \"text\": \"The authors acknowledge the Italian Ministry of Scientific Researches (MIUR) for partial support of the present work under the project SINTESI 2004/06 and the INFN for partial support under the project Iniziativa Specifica LE41.\"\n }\n]"}}},{"rowIdx":95,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0523"},"arxiv_id":{"kind":"string","value":"0704.0523"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevA.76.042103"},"content_hash":{"kind":"string","value":"1ca28034a1181cbdbddecf9d042ad8b097ea11a943736a666791eb33df4d9ea3"},"title":{"kind":"string","value":"Quantum superpositions and entanglement of thermal states at high\n temperatures and their applications to quantum information processing"},"abstract":{"kind":"string","value":"We study characteristics of superpositions and entanglement of thermal states\nat high temperatures and discuss their applications to quantum information\nprocessing. We introduce thermal-state qubits and thermal-Bell states, which\nare a generalization of pure-state qubits and Bell states to thermal mixtures.\nA scheme is then presented to discriminate between the four thermal-Bell states\nwithout photon number resolving detection but with Kerr nonlinear interactions\nand two single-photon detectors. This enables one to perform quantum\nteleportation and gate operations for quantum computation with thermal-state\nqubits."},"authors":{"kind":"list like","value":["Hyunseok Jeong and Timothy C. Ralph"],"string":"[\n \"Hyunseok Jeong and Timothy C. Ralph\"\n]"},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]"},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states without photon number resolving detection but with Kerr nonlinear interactions and two singlephoton detectors. This enables one to perform quantum teleportation and gate operations for quantum computation with thermal-state qubits.\n\nIn many problems considered within the framework of quantum physics, physical systems are treated as pure states that can be represented by state vectors, or equivalently, by wave functions. Even though such an approach is simple and useful to address certain problems, it could often be quite different from real conditions of physical systems. This may be particularly true when one deals with macroscopic physical systems in terms of quantum physics. A macroscopic object is a complex open system which cannot avoid continuous interactions with the environment. Such a physical system is generally in a significantly mixed state and cannot be represented by a state vector. In general, mixed states are subtle objects whose properties are significantly more difficult to characterize than pure states.\n\nSchrödinger's famous cat paradox is a typical example where a massive classical object was assumed to be a pure state. It describes a counter-intuitive feature of quantum physics which dramatically appears when the principle of quantum superposition is applied to macroscopic objects. In the original paradox and its various explanations, the initial cat isolated in the steel chamber is considered a pure state that can be represented by a state vector such as |alive (or a wave function such as ψ alive ). The cat isolated from the environment is then assumed to interact with a microscopic superposition state, (|g +|e )/ √ 2, where |g and |e are the ground and excited states of a two-level atom. The cat will be dead if the atom is found in the excited state, |e , while it will remain alive if otherwise. Thus in Schrödinger's gedanken experiment the cat is entangled with the atom as (|g |alive + |e |dead ) √ 2, where the alive and dead statuses of the cat are described by the state vectors |alive and |dead . If one measures out the atomic system on the superposed basis, (|g ± |e )/ √ 2, the cat will be in a superposition of alive and dead states such as (|alive ± |dead )/ √ 2. It is often argued that such superposed states and entangled states can theoretically exist but are virtually impossible to observe because one cannot perfectly isolate a macroscopic object such as the cat from its environment [4] .\n\nHowever, this explanation is not fully satisfactory because the cat, a macroscopic object, is a complex open system which cannot be represented by a state vector. One may argue that the cat could be assumed to be in an unknown pure state such that the cat was certainly alive but the exact state of the cat was unknown. However, the interactions between the cat and its environment can cause the cat to become entangled with the environment [5] . In such a case, even though one can perfectly isolate the cat in the steel chamber from the enviroment, the cat will remain entangled with the environment due to its pre-interactions with the environment. Therefore, strictly speaking, even to assume a cat as an unknown pure state in the steel chamber is not legitimate. Thus a key point here is that it is unsatisfactory to describe the cat by a pure state such as |alive and |dead . We may need a more realistic assumption that the \"cat\" in Schrödinger's paradox was in a significantly mixed classical state. An intriguing question is then whether the quantum properties of the resulting state would still remain or diminish under such an assumption.\n\nRecently, such an analogy of Schrödinger's cat paradox, where the state corresponding to the virtual cat is a significantly mixed thermal state, was investigated [6] . A thermal state with a high temperature is considered a classical state in quantum optics. As the temperature of the thermal state increases, the degree of mixedness, which can be quantified by linear entropy, rapidly approaches the maximum value. When the temperature approaches infinity, the thermal state does not show any quantum properties. As a comparison, coherent states with large amplitudes are known as the most classical pure states [7] , and their superposition is often regarded as a superposition of classical states [8] . However, coherent states are still pure states which may not well represent truly classical systems, and they display some nonclassical features [9] . In Ref. [6] , it was shown that prominent quantum properties can actually be transferred from a microscopic superposition to a significantly mixed thermal state (i.e. a thermal state of which the degree of mixedness is close to the maximum value) at a high temperature through an experimentally feasible process. This result clarifies that unavoidable initial mixedness of the cat does not preclude strong quantum phenomena.\n\nOne of the results in Ref. [6] is that quantum entanglement can be produced between thermal states with nearly the maximum Bell-inequality violation when the temperatures of both modes goes to infinity. In previous related results, Bose et al. showed that entanglement can arise when two systems interact if one of the system are pure even when the other system is extremely mixed [10] . There is an interesting previous example shown by Filip et al. for the maximum violation of Bell's inequality when one of the modes is an extremely mixed thermal state [11] . Very recently, Ferreira et al. showed that entanglement can be generated at any finite temperature between high Q cavity mode field and a movable mirror thermal state [12] . However, in these example [10, 11, 12] only one of the modes is considered a large thermal state [10, 11, 12] and entanglement vanishes in the infinite temperature limit [10, 12] , which is obviously in contrast to the result presented in Ref. [6] . Entanglement for both of the modes at the thermal limit of the infinitely high temperature has not been found before. Remarkably, the violation of Bell's inequality in our examples reaches up to Cirel'son's bound [13] even in this infinite-temperature limit for both modes. As Vedral [14] and Ferreira et al. [12] pointed out it is believed that high temperatures reduce entanglement and all entanglement vanishes if the temperature is high enough, which is obviously not the case in Ref. [6] .\n\nThe purpose of this paper is twofold. Firstly, we review and further investigate various properties of superpositions and entanglement of thermal states at high temperatures [6] . In particular, we investigate two classes of highly mixed symmetric states in the phase space. Both the classes of these states do not show typical interference patterns in the phase space while they manifest strong singular behaviors. Interestingly, the first class of states has neither squeezing properties nor negative values in their Wigner functions, however, they are found to be highly nonclassical states. The second class of states has the maximum negativity in the Wigner function. Further, we discuss the possibility of quantum information processing with thermal-state qubits. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure Bell states. We show that four thermal-Bell states can be well discriminated by nonlinear interactions without photon number resolving measurements. Quantum teleportation and gate operations for thermal-state qubits can be realized using the Bell measurement scheme.\n\nThis paper is organized as follows. In Sec. II, we review the generation process of superpositions of thermal states and study their characteristics. In Sec. III, we study en-tanglement of thermal states, i.e., Bell inequality violations. In Sec. IV, we discuss the possibility of quantum information processing using thermal states. We first define the thermal-state qubit and the Bell-basis states using thermal-state entanglement. We then show that the four Bell states can be well discriminated by homodyne detection and two Kerr nonlinearities. It follows that quantum teleportation and quantum gate operations can be realized with thermal-state qubits. We conclude with final remarks in Sec. V.\n\nLet us first consider a two-mode harmonic oscillator system. A displaced thermal state can be defined as\n\nρ th (V, d) = d 2 αP th (V, d)|α α| ( 1\n\n)\n\nwhere |α is a coherent state of amplitude α and\n\nP th α (V, d) = 2 π(V -1) exp[- 2|α -d| 2 V -1 ] (2)\n\nwith variance V and displacement d in the phase space.\n\nThe thermal temperature τ increases as V increases as e ν/τ = (V + 1)/(V -1), where is Planck's constant and ν is the frequency [15] . Suppose that a microscopic superposition state\n\n|ψ a = 1 √ 2 (|0 a + |1 a ), (3)\n\nwhere |0 and |1 are the ground and first excited states of the harmonic oscillator, interacts with a thermal state ρ th b (V, d) and the interaction Hamiltonian is\n\nH K = λâ † âb † b (4)\n\nwhich corresponds to the cross Kerr nonlinear interaction. The resulting state is then\n\nρ ent ab = 1 2 d 2 αP th (V, d) |0 0| ⊗ |α α| + |1 0| ⊗ |αe iϕ α| + |0 1| ⊗ |α αe iϕ | + |1 1| ⊗ |αe iϕ αe iϕ | ( 5\n\n)\n\nand ϕ is determined by the strength of the nonlinearity λ and the interaction time. The Wigner representation of ρ ent ab is\n\nW ent ab (α, β) = 1 π e -2|α| 2 W th (β; d) + 2αV c (β; d) + 2[αV c (β; d)] * + (4|α| 2 -1)W th (β; de iϕ ) (6)\n\nwhere α and β are complex numbers parametrizing the phase spaces of the microscopic and macroscopic systems respectively and\n\nW th (α; d) = 2 πV exp[- 2|α -d| 2 V ], (7)\n\nV c (α; d) = 2 πJK exp[- 2 K (1 -e iϕ )d 2 - 1 J (α - 2e iϕ d K )(α * - 2d K )], (8)\n\nK = 2 + (V -1)(1 -e iϕ\n\n), J = (sin ϕ/2 + iV cos ϕ/2)/(2V sin ϕ/2 + 2i cos ϕ/2), and d has been assumed real without loss of generality. If one traces ρ ent ab over mode a, the remaining state will be simply in a classical mixture of two thermal states and its Wigner function will be positive everywhere. However, if one measures out the \"microscopic part\" on the superposed basis, i.e., (|0 a ± |1 a )/ √ 2, the \"macroscopic part\" for mode b may not lose its nonclassical characteristics. Such a measurement on the the superposed basis will reduce the remaining state to\n\nρ sup(±) = N ± s d 2 αP th (V, d) |α α| ± |αe iϕ α| ± |α αe iϕ | + |αe iϕ αe iϕ | , (9)\n\nwhere N ± s are the normalization factors, and its Wigner function is\n\nW sup(±) (α) = N ± s {W th (α; d) ± V c (α; d) ± {V c (α; d)} * + W th (α; de iϕ )}. ( 10\n\n)\n\nThe ± signs in Eqs. ( 8 ) and (9) correspond to the two possible results from the measurement of the microscopic system. The state in Eq. ( 10 ) is a superposition of two thermal states. A feasible experimental setup to generate superpositions of thermal states is atom-field interactions in cavities, where a π/2 pulse can be used to prepare the atom in a superposed state. This type of experiment has already been performed to produce a superposition of coherent states [16] . In our cases, simply thermal states can be used instead of coherent states. Another possible setup is an all-optical scheme with free-traveling fields and a cross-Kerr medium, where a standard singlephoton qubit could be used as the microscopic superposition. Recently, there have been theoretical and experimental efforts to produce and observe giant Kerr nonlinearities using electromagnetically induced transparency [17] . Furthermore, it was shown that a weak Kerr nonlinearity can still be useful if a initially strong field is employed in this type of experiment [18] . We shall further explain this with examples in Sec. III.\n\nThe negativity of the Wigner function is known as an indicator of non-classicality of quantum states. In order to observe negativity of the Wigner function in a real experiment, its absolute minimum negativity should be large enough. The minimum negativity of the Wigner function in Eq. ( 6 ) for V = 1 is -0.144 for d = 0 and -0.246 for d → ∞. Now suppose the initial state can be considered a classical thermal state by letting V ≫ 1. One might expect that the negativity would be washed out as the initial state becomes mixed, but this is not the case. The minimum negativity actually increases as V gets larger. If V → ∞, the minimum negativity of the Wigner function ( 6 ) is -0.246 regardless of d: no matter how mixed the initial thermal state was, the minimum negativity of Wigner function is found to be a large value. The point in the phase space which gives the minimum negativity when V ≫ 1 or d ≫ 0 is (-1 2 , 0) and has negativity\n\nW neg ≡ W ent ab (- 1 2 , 0) = 2(-2 + 1 V exp[-2d 2 V ]) π 2 √ e . (11)\n\nIt can be shown that\n\nW neg approaches -4/(π 2 √ e) ≈ -0.246 when either d → ∞ or V → ∞.\n\nThis effect is obviously due to the interaction between the microscopic superposition and the macroscopic thermal state. If the initial microscopic state is not superposed, e.g., |ψ a = |1 a , the resulting state will be a simple direct product, (|1 1|) a ⊗ ρ th b (V, -d). Whilst for V = 1 this state will exhibit negativity, this is washed out and tends to zero as V → ∞. Needless to say, if it was |0 a instead of |1 a , the resulting Wigner function will be a direct product of two Gaussian states whose Wigner fucntion can never be negative. The superpositon state (3) plays the crucial role in making the minimum negativity of the resulting Wigner function always saturate to a certain negative value no matter how mixed and classical the initial state of the other mode becomes. The Wigner functions of the single-mode states, W sup(±) (α), in Eq. (10) show large negative values. The minimum negativity of the Wigner function W sup(-) (α) is W sup(-) (0) = 2/π regardless of the values of V and d. On the other hand, the minimum negativity of the Wigner function W sup(+) (α) approaches 2/π for d → ∞ and disappears when d = 0.\n\nWhen ϕ = π, the state (9) becomes (12) where\n\nρ ± = N (ρ th (V, d)±σ(V, d)±σ(V, -d)+ρ th (V, -d)),\n\nσ(V, d) = d 2 αP th (V, d)| -α α| and N = 2 1 ± exp[-2d 2 V ] V 2 . ( 13\n\n)\n\nIf the initial state for mode b is a pure coherent state, i.e., V = 1, the measurement on the superposed basis for mode a will produce a superposition of two pure coherent states as\n\n| Ψ ± = 1 √ 1 ± e -2|α| 2 (|α ± | -α ), (14)\n\nwhere α = d. The probability P ± to obtain the state ρ ± is obtained as [19 ] where\n\nP ± = ψ ± |Tr b [ρ ent ab ]|ψ ± = 1 2 (1 ± exp[-2d 2 V ] V ), ( 15\n\n)\n\n|ψ ± = (|0 ±|1 )/ √ 2.\n\nThe probability approaches P ± = 1/2 when either d or V becomes large.\n\nAs an analogy of Schrödinger's cat paradox, the variance V corresponds to the size the initial \"cat\", and the distance d between the two thermal component states corresponds to distinguishability between the \"alive cat\" and the \"dead cat\". Suppose that both V and d are very large for the initial thermal state. The two thermal states ρ th (V, ±d) become macroscopically distinguishable when d ≫ √ V , and our example may become a more realistic analogy of the cat paradox in this limit. Both the states ρ ± in this case show probability distributions with two Gaussian peaks and interference fringes [6] . Figure 1 presents the probability distributions of x (≡ Re[α]) and p (≡ Im[α]) for ρ -(a) when V = 100 and d = 100 and (b) when V = 1000 and d = 300. The probability distribution of x (p) for ρ ± can be obtained by integrating the Wigner function of ρ ± over p (x). The two Gaussian peaks along the x axis and interference fringes along the p axis shown in Fig. 1 are a typical signature of a quantum superposition between macroscopically distinguishable states. The visibility v of the interference fringes is defined as [15]\n\nv = I max -I min I max + I min , (16)\n\nwhere I = dxW sup(-) (α) and the maximum should be taken over p. It can be simply shown that the visibility v is always 1 regardless of the value of V . Note that d should increase proportionally to √ V to maintain the condition of classical distingushability between the two component thermal states ρ th (V, ±d). The interference fringes with high visibility are incompatible with classical physics and evidence of quantum coherence. The fringe spacing (the distance between the fringes) does not depend on V but only on d, i.e., a pure superposition of coherent states shows the same fringe spacing for a given d. We emphasize that the states shown in Fig. 1 are \"superpositions\" of severely mixed thermal states.\n\nAn experimental realization of a nonlinear effect corresponding to ϕ = π is very demanding particularly in the presence of decoherence. Here we point out that the method using a weak nonlinear effect (ϕ ≪ π) combined with a strong field (d ≫ 1) [18] can be useful to generate a thermal-state superposition with prominent interference patterns. In Fig. 2 , we have used experimentally accessible values, V = 5, d = 2000 and ϕ = π/1000, but the fringe visibility is still 1. In this case, decoherence during the nonlinear interaction would be significantly reduced because of the decrease of the interaction time [18] . Note also that, if required, the state in Fig. 2 can be moved to the center of the phase space, for example, using a biased beam splitter (BS) and a strong coherent field [18] .\n\nLet us assume that d = 0, i.e., the initial state is the thermal state, ρ th (V, 0), at the origin of the phase space. In this case, the thermal-state superpositions, ρ ± , are produced with probabilities, P ± = (1/2){1 ± (1/V )}, respectively. Figure 3 shows the Wigner functions of ρ + dependent on the interaction time between the macro-scopic thermal state and the microscopic superposition in a cross Kerr medium. The state is always symmetric in the phase space regardless of the interaction time as shown in Fig. 3 . In this figure, the initial state is a thermal state of V = 100 (Fig. 3(a) ). In a relatively short time (θ = π/32 and θ = π/16), the state shows some interference patterns. When θ = π, the evolved state looks very localized around the origin as shown in Fig 3 . The generated state at θ = π does not show negativity of the Wigner function nor squeezing properties. On the other hand, a well defined P function does not exist for this state.\n\nIn the case of ρ -, with the same assumption d = 0, the Wigner function at ϕ = π has the minimum negativity (-2/π) at the origin regardless of V . As a result of the interaction with the microscopic superposition, a deep hole to the negative direction below zero has been formed around the origin for ρ -as shown in Fig. 4 .\n\nEntanglement between macroscopic objects and its Bell-type inequality tests are an important issue. In this section, we shall show that entanglement can be generated between high-temperature thermal states even when the temperature of each mode goes to infinity. 2 200 1 400 600 800 1000 V 2.1 2.2 2.3 B (a) 50 100 150 d 2.2 2 0 2.4 2.6 2.8 B (b) FIG. 5: (a) The optimized violation, B ≡ |B + |max, of Bell-\n\nCHSH inequality for the \"thermal-state entenglement\", ρ+, of V = 1000 (solid curve) and V = 100 (dashed curve). The Bell-violation of a pure entangled coherent state, i.e., V = 1, has been plotted for comparison (dotted curve). The Bell-violation B approaches its maximum bound, 2 √ 2, when d ≫ √ V regardless of the level of the mixedness V . (b) The optimized Bell-violation B against d for the different type of thermal-state entanglement generated using a 50:50 beam splitter from ρ + . V = 1000 (solid curve), V = 100 (dashed curve) and V = 1 (dotted curve).\n\nIf the microscopic superposition interacts with two thermal states, ρ th b (V, d) and ρ th c (V, d), and the micro-scopic particle is measured out on the superposed basis, the resulting state will be\n\nρ tm(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (17)\n\nwhere\n\nN t = 2 1 ± exp[-4d 2 V ] V 2 . ( 18\n\n)\n\nSuch two-mode thermal-state entanglement can be generated using two cavities and an atomic state detector [20] .\n\nExtending the two cavities to N cavities, entanglement of N -mode thermal states can also be generated. Such a state is an analogy of the N -mode pure GHZ state [21] but each mode is extremely mixed. Here we shall consider the Bell-CHSH inequality [22, 23] with photon number parity measurements [20, 24] . The parity measurements can be performed in a high-Q cavity using a far-off-resonant interaction between a two-level atom and the field [25] . The Bell-CHSH inequality can be represented in terms of the Winger function as [24 ]\n\n|B (±) | = π 2 4 |W tm(±) (α, β) + W tm(±) (α, β ′ ) + W tm(±) (α ′ , β) -W tm(±) (α ′ , β ′ )| ≤ 2, (19)\n\nwhere W tm(±) (α, β) is the Wigner function of ρ tm(±) in Eq. ( 17 ). As shown in Fig. 5 , the Bell-violation approaches the maximum bound for a bipartite measurement, 2 √ 2 [13] , when d ≫ √\n\nV regardless of the level of the mixedness V , i.e., the temperatures of the thermal states. Note that it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(a) . This implies that entanglement of nearly 1 ebit has been produced between the two significantly mixed thermal states for d ≫ √ V , and such \"thermal-state entanglement\" cannot be described by a local theory.\n\nA different type of macroscopic entanglement can be generated by applying the beam splitter operation\n\nexp[θ/2(e iφ â † s âd -e -iφ â † d âs )], (20)\n\non the \"thermal-state superpositions\" in Eq. ( 9 ). The state after passing through a 50:50 beam splitter can be represented as\n\nN d 2 αP th α (V, d) | α √ 2 , - α √ 2 ± | - α √ 2 , α √ 2 α √ 2 , - α √ 2 | ± - α √ 2 , α √ 2 | , (21\n\n) 200 400 600 800 1000 V 2.1 2 0 2.2 2.3 B FIG. 6: The optimized Bell-violation B against V for the slightly different type of thermal-state entanglement generated using a 50:50 beam splitter using ρ + when d = 0. 3.13 3.14 3.15 3.16 ϕ 1.6 2.2 2 2.4 2.6 2.8 B (a) 3.13 3.14 3.15 3.16 ϕ 1.6 2 2.2 2.4 2.6 2.8 B (b) where N is defined in Eq. ( 13 ). When d is large, this state violates the Bell-CHSH inequality to the maximum bound 2 √ 2 regardless of the level of mixedness V as shown in Fig. 5(b ). Again, it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(b ). Furthermore, these states severely violate Bell's inequality even when d = 0 as V increases as shown in Fig. 6 . We have found that the optimized Bell violation of these states approaches 2.32449 for V → ∞. Interestingly, this value is exactly the same as the optimized Bell-CHSH violation for a pure two-mode squeezed state in the infinite squeezing limit [26] . Note that multilmode entangled states can be generated using multiple beam splitters.\n\nIt should be noted that the Bell violations are more sensitive to the interaction time when either V or d is larger. Figure 7 clearly shows this tendency. Therefore, in order to observe the Bell violations using the mixed state of V (and d) large, the interaction time in the Kerr medium should be more accurate.\n\nIn this section, we discuss the possibility of quantum information processing with thermal-state qubits and thermal-state entanglement.\n\nWe introduce a thermal-state qubit\n\nρ ψ = |a| 2 ρ th (V, d) ± ab * σ(V, d) ± a * bσ(V, -d) + |b| 2 ρ th (V, -d), (22)\n\nwhere a and b are arbitrary complex numbers. The basis states, ρ th (V, d) and ρ th (V, -d), can be well discriminated by a homodyne measurement when d is larger than V . The thermal state qubit (22) can be re-written as\n\nρ ψ = d 2 αP th α (V, d) a|α + b| -α a * α| + b * -α| , (23)\n\nwhich can be understood as a generalization of the coherent state qubit, a|d + b|d , where |d is a coherent state of amplitude d. The thermal-state qubit (23) becomes identical to the coherent-state qubit when V = 1.\n\nWe also define four thermal-Bell states as\n\nρ Φ(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (24) ρ Ψ(±) = N t ρ th (V, d) ⊗ ρ th (V, -d) ± σ(V, d) ⊗ σ(V, -d) ± σ(V, -d) ⊗ σ(V, d) + ρ th (V, -d) ⊗ ρ th (V, d) (25)\n\nwhere N t was defined in Eq. ( 18 ). The thermal-Bell states can be written as For quantum information processing applications, it is an important task to discriminate between the four Bell states. Here we discuss two possible ways to discriminate between the thermal-Bell states (25) . We shall only briefly describe the first scheme using photon number resolving measurements and focus on the second scheme using nonlinear interactions. The first method is to simply use a 50-50 beam splitter and two photon number resolving detectors as shown in Fig. 8(a) . This scheme is basically the same as the Bellstate measurement scheme with pure entangled coherent states [27, 28] . Let us suppose that the amplitude, d, is large enough, i.e., d ≫ √ V . If the incident state was ρ Φ(+) or ρ Φ(-) , most of the photons are detected on detector A in in Fig. 8(a) . Meanwhile, most of the photons are detected on detector B when the incident state was ρ Ψ(+) or ρ Ψ(-) . The average photon numbers between the \"many-photon case\" and the \"few-photon case\" are compared in Fig. 9 . Furthermore, the states ρ Ψ(+) and ρ Φ(+) contain only even numbers of photons while ρ Ψ(-) and ρ Φ(-) contain only odd numbers of photons. Therefore, all the four Bell states can be well discriminated by analyzing numbers of photons detected at detectors A and B. For example, if detector A detects many photons while detector B detects few and the total photon number detected by the two detectors are even, this means that state ρ Φ(+) was measured by the thermal-Bell measurement. The nonzero failure probability can be made arbitrarily small by increasing d.\n\nρ Φ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, β ± | -α, -β α, β| ± -α, -β| , ( 26\n\n)\n\nρ Ψ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, -β ± | -α, β α, -β| ± -α, β| . (27)\n\nHowever, the average photon numbers of the thermal-Bell states are high when V ≫ 1 and d ≫ 1. In this case, it would be unrealistic to use photon number resolving detectors. It would be an interesting question whether\n\n0.5 1 1.5 2 2.5 3 d 10 0 0 20 30 N (a) 0.5 1 1.5 2 2.5 3 d 0 0 10 20 30 N (b)\n\nFIG. 9: The average photon number N for the \"many-photon case\" (solid line) and the \"few-photon case\" (dashed line) for V = 10 against d (a) when the input state is either ρ Φ(+) or ρ Ψ(+) and (b) when the input state is either ρ Φ(-) or ρ Ψ(-) .\n\nthese four thermal-Bell states can be distinguished by classical measurements, such as homodyne detection, instead of photon number resolving detection. Our alternative scheme employs cross-Kerr nonlinearities and single photon detectors as shown in Fig. 8(b ). Let us first suppose that the input field was ρ Φ(+) . The incident twomode state passes through a 50-50 beam splitter, BS1. The state after passing through the 50:50 beam splitter, BS1, is\n\nρ B = N t d 2 αd 2 βP th α (V, d)P th β (V, d) |η, -ξ η, -ξ| + |η, -ξ -η, ξ| + | -η, ξ η, -ξ| + | -η, ξ -η, ξ| (28)\n\nwhere η = (α+β)/ √ 2 and ξ = (α-β)/ √ 2. Two dual-rail single photon qubits, |ψ + ee ′ and |ψ + f f ′ , where\n\n|ψ + = 1 √ 2 (|0 |1 + |1 |0 ), (29)\n\nare prepared using two single photons and 50:50 beam splitters, BS2 and BS3, as shown in Fig. 8(b ). Then, traveling fields at modes c and d interacts with those of modes e and f , respectively, in cross-Kerr nonlinear media. We suppose that the interaction time is t = π/λ, and the resulting state is then\n\nρ B ′ = U ce U df ρ B cd ρ q ee ′ ρ q f f ′ U † ce U † df (30)\n\nwhere U ce = exp[iπH K ce /λ ] and ρ q = |ψ q ψ q |. An explicit form of Eq. ( 30 ) can then be simply obtained using the identity\n\nU ce |α c |0 e = |α c |0 e , U ce |α c |1 e = | -α c |1 e ( 31\n\n)\n\nwhere |α is a coherent state. However, we omit such an explicit expression in this paper for it is too lengthy. After the nonlinear interactions, the qubit parts, modes e, e ′ , f and f ′ , should be measured with the measurement basis\n\n{| + + , | + -, | -+ , | --} (32)\n\nwhere\n\n| + + = |ψ + ee ′ |ψ + f f ′ , | + -= |ψ + ee ′ |ψ -f f ′ , | -+ = |ψ -ee ′ |ψ + f f ′ , | --= |ψ -ee ′ |ψ -f f ′ , and |ψ -= (|0 |1 -|1 |0 )/ √ 2.\n\nThis measurement can be performed using two 50:50 beam splitters, BS4 and BS5, and four detectors, A1, A2, B1 and B2, as shown in Fig. 8(b ). If detector A1 and B1 click, i.e., the measurement result is | + + , the resulting state at modes c and d is\n\nρ ++ = N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η + | -η )( η| + -η|) c ⊗ (|ξ + | -ξ )( ξ| + -ξ|) d (33)\n\nNote that state ρ ++ is not normalized, which implies that the probability of obtaining the corresponding measurement result is not unity. The probability of obtaining this result is\n\nP ++ = (V + 1)(V + e -4d 2 V ) 2(V 2 + e -4d 2 V ) . (34)\n\nWhen the result is either | +or | -+ , the result is\n\nψ 2 |ρ B ′ |ψ 2 = ψ 3 |ρ B ′ |ψ 3 = 0, (35)\n\nwhich obviously means that the probability of the obtaining this result is zero. When the result is | --, i.e., detector A2 and B2 click,\n\nρ --= N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η -| -η )( η| --η|) c ⊗ (|ξ -| -ξ )( ξ| --ξ|) d , (36)\n\nwhich is not normalized. The probability of obtaining this result is\n\nP --= (V -1)(V -e -4d 2 V ) 2(V 2 + e -4d 2 V ) , (37)\n\nand it can be simply verified that P ++ +P --= 1. Therefore, only the measurement results | + + and | -can be obtained in the case of the input state ρ Φ(+) . This is exactly the same for the case of ρ Ψ(+) . In the same way, it can be shown that if either the input state was ρ Φ(-) or ρ Ψ(-) , only the measurement results | +and | -+ can be obtained. In other words, the parity of the toincoming state is perfectly well discriminated by the measurements on single-photon qubits.\n\nSubsequently, a homodyne measurement is performed for mode c by homodyne detector C as shown in Fig. 8(b) . We assume that ideal homodyne measurements are performed, i.e., when a homodyne measurement is performed the state is projected onto eigenstate |x of operator X with eigenvalue x, where\n\nX = 1 √ 2 (a + a † ). ( 38\n\n)\n\nLet us first consider the case when the measurement result for the single photon qubits is | + + . In this case, the remaining state is ρ ++ in Eq. ( 33 ). The probability distribution P ++ Φ (+) for the homodyne measurement at detector C is\n\nP ++ Φ (+) = x|Tr d [ρ ++ ]|x = V 1 2 (e -V x 2 + e -x 2 V ) π 1 2 (V + 1)\n\n. (39) Note that the superscript, ++, denotes that the qubit measurement result was | + + , and the subscript, Φ (+) , denotes that the input state was ρ Φ (+) . These notations will be used also for the other cases in this section. The same analysis can be performed for the other possible measurement outcome | --:\n\nP -- Φ (+) = x|Tr d [ρ --]|x = V 1 2 (e -V x 2 -e -x 2 V ) π 1 2 (V -1)\n\n. (40)\n\nIn the same way, for another input state, ρ Φ(-) , it is straightforward to show:\n\nP +- Φ (-) = P ++ Φ (+) , P -+ Φ (-) = P -- Φ (+) , (41)\n\nand P ++ Φ (-) = P -- Φ (-) = 0. On the other hand, if the input state was ρ Ψ(+) , the probability distributions P ++ Ψ (+) and P -- Ψ (+) at detector C are It is straightforward to show for the other input state ρ Ψ(-) :\n\nP ++ Ψ (+) = x|Tr c [ρ ++ ]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V + 2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) + 1 , (42)\n\nP -- Ψ (+) = x|Tr c [ρ --]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V -2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) -1 . ( 43\n\nP +- Ψ (-) = P -- Ψ (+) , P -+ Ψ (-) = P ++ Ψ (+) . (44)\n\nThe probability distributions P ++ Φ (±) and P ++ Ψ (±) are plotted in Fig. 10 . Figure 10 shows that when the input state was ρ Φ(+) or ρ Φ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Φ (+) and P -- Φ (+) , is located around the origin. However, when the input state was ρ Ψ(+) or ρ Ψ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Ψ (+) and P -- Ψ (+) , is located far from the origin. Therefore, two of the Bell states, ρ Φ(+) or ρ Φ(-) , can be well distinguished from the other two by the homodyne detector C for the case of the measurement outcome | + + . Finally, by combining the homodyne measurement result and the qubit measurement result, all four Bell states can be effectively distinguished. For example, let us assume that the measurement outcome of the single photon detectors was | + + and the homodyne detection outcome was around the origin, i.e., x ≈ 0. Then, one can say that state ρ Ψ(-) has been measured for the result of the thermal-Bell measurement.\n\nAs implied in Fig. 10 , the overlaps between the probability distributions around the origin, P ++ Φ (+) and P -- Φ (+) , and the other distributions, P ++ Ψ (+) and P -- Ψ (+) , are extremely small for a sufficiently large d. In other words, the distinguishability by the homodyne detection rapidly approaches 1 as d increases. As an example, we can calculate the distinguishability between the states ρ Ψ(+) and ρ Φ(+) by the homodyne measurement by detector C. The distinguishability by homodyne detection is Note also that the second scheme using homodyne detection is robust to detection inefficiency compared with the first scheme using photon number resolving measurements. In the first scheme, even if a detector misses only one photon, it will result in a completely wrong measurement outcome. In the second scheme, however, the measurement outcome will not be affected in that way. If a single photon detector misses a photon, it will be immediately recognized. Such a case can simply be discarded so that it will only degrade the success probability of the Bell measurement. The homodyne detection inefficiency will not significantly affect the result when the distributions around the origin and the distributions far from the origin are well separated, i.e., when d ≫ √ V , as shown in Fig. 10 . On the other hand, loss in the Kerr medium will have a detrimental affect.\n\nP s = 1 2 |x|<d dxP ++ c (x) + |x|≥d dxP ++ d (x) (45\n\nQuantum teleportation of a thermal-state qubit can be performed using one of the Bell states as the quantum channel. Let us assume that Alice needs to teleport a thermal-state qubit, ρ ψ , to Bob using a thermal-state entanglement, ρ Ψ(-) , shared by the two parties. The total state can be represented as\n\nAlice first needs to perform the thermal-Bell measurement described in the previous subsection. To complete the teleportation process, Bob should perform an appropriate unitary transformation on his part of the quantum channel according to the measurement result sent from Alice via a classical channel. It is straightforward to show that the required transformations are exactly the same to those for the coherent-state qubit [27] . When the measurement outcome is ρ Ψ(-) , Bob obtains a perfect replica of the original unknown qubit without any operation.\n\nWhen the measurement outcome is ρ Φ(-) , Bob should perform |α ↔ |α on his qubit in Eq. ( 23 ). Such a phase shift by π can be done using a phase shifter whose action is described by P (ϕ) = e iϕa † a , where a and a † are the annihilation and creation operators. When the outcome is ρ Ψ(+) , the transformation should be performed as |α → |α and |α → -|α . It is known that the displacement operator is a good approximation of this transformation for d ≫ 1 [29] . This transformation can also be achieved by teleporting the state again locally and repeating until the required phase shift is obtained [30] . When the outcome is ρ Φ(+) , σ x and σ z should be successively applied.\n\nIn this paper, we have studied characteristics of superpositions and entanglement of thermal states at high temperatures and discussed their applications to quantum information processing. The superpositions and entanglement of thermal states show various nonclassical properties such as interference patterns, negativity of the Wigner functions, and violations of the Bell-CHSH inequality. The Bell violations are more sensitive to the interaction time during the generation process when the thermal temperature (i.e. mixedness) of the thermalstate entanglement is larger. Therefore, in order to observe the Bell violations using the mixed state at a high temperature, the interaction time in the Kerr medium should be accurate. We have pointed out that certain superpositions of high-temperature thermal states, symmetric in the phase space, can also be generated. Some of these states have neither squeezing properties nor negative values in their Wigner functions but they are found to be highly nonclassical.\n\nWe have introduced the thermal-state qubit and thermal-Bell states for applications to quantum information processing. We have presented two possible methods for the Bell-state measurement. The Bell-state measurement enables one to perform quantum teleportation and gate operations for quantum computation with thermalstate qubits. The first scheme uses two photon number resolving detectors and a 50-50 beam splitter to discriminate the thermal-Bell states. Using the second scheme, it is possible to effectively discriminate the thermal-Bell states without photon number resolving detection. The required resources for the second scheme are two Kerr nonlinear interactions, two single photon detectors, two 50:50 beam splitters and one homodyne detector. The second scheme is more robust to inefficiency of the detectors: the inefficiency of the single photon detectors only degrades the success probability of the Bell measurement."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states without photon number resolving detection but with Kerr nonlinear interactions and two singlephoton detectors. This enables one to perform quantum teleportation and gate operations for quantum computation with thermal-state qubits."},{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"In many problems considered within the framework of quantum physics, physical systems are treated as pure states that can be represented by state vectors, or equivalently, by wave functions. Even though such an approach is simple and useful to address certain problems, it could often be quite different from real conditions of physical systems. This may be particularly true when one deals with macroscopic physical systems in terms of quantum physics. A macroscopic object is a complex open system which cannot avoid continuous interactions with the environment. Such a physical system is generally in a significantly mixed state and cannot be represented by a state vector. In general, mixed states are subtle objects whose properties are significantly more difficult to characterize than pure states.\n\nSchrödinger's famous cat paradox is a typical example where a massive classical object was assumed to be a pure state. It describes a counter-intuitive feature of quantum physics which dramatically appears when the principle of quantum superposition is applied to macroscopic objects. In the original paradox and its various explanations, the initial cat isolated in the steel chamber is considered a pure state that can be represented by a state vector such as |alive (or a wave function such as ψ alive ). The cat isolated from the environment is then assumed to interact with a microscopic superposition state, (|g +|e )/ √ 2, where |g and |e are the ground and excited states of a two-level atom. The cat will be dead if the atom is found in the excited state, |e , while it will remain alive if otherwise. Thus in Schrödinger's gedanken experiment the cat is entangled with the atom as (|g |alive + |e |dead ) √ 2, where the alive and dead statuses of the cat are described by the state vectors |alive and |dead . If one measures out the atomic system on the superposed basis, (|g ± |e )/ √ 2, the cat will be in a superposition of alive and dead states such as (|alive ± |dead )/ √ 2. It is often argued that such superposed states and entangled states can theoretically exist but are virtually impossible to observe because one cannot perfectly isolate a macroscopic object such as the cat from its environment [4] .\n\nHowever, this explanation is not fully satisfactory because the cat, a macroscopic object, is a complex open system which cannot be represented by a state vector. One may argue that the cat could be assumed to be in an unknown pure state such that the cat was certainly alive but the exact state of the cat was unknown. However, the interactions between the cat and its environment can cause the cat to become entangled with the environment [5] . In such a case, even though one can perfectly isolate the cat in the steel chamber from the enviroment, the cat will remain entangled with the environment due to its pre-interactions with the environment. Therefore, strictly speaking, even to assume a cat as an unknown pure state in the steel chamber is not legitimate. Thus a key point here is that it is unsatisfactory to describe the cat by a pure state such as |alive and |dead . We may need a more realistic assumption that the \"cat\" in Schrödinger's paradox was in a significantly mixed classical state. An intriguing question is then whether the quantum properties of the resulting state would still remain or diminish under such an assumption.\n\nRecently, such an analogy of Schrödinger's cat paradox, where the state corresponding to the virtual cat is a significantly mixed thermal state, was investigated [6] . A thermal state with a high temperature is considered a classical state in quantum optics. As the temperature of the thermal state increases, the degree of mixedness, which can be quantified by linear entropy, rapidly approaches the maximum value. When the temperature approaches infinity, the thermal state does not show any quantum properties. As a comparison, coherent states with large amplitudes are known as the most classical pure states [7] , and their superposition is often regarded as a superposition of classical states [8] . However, coherent states are still pure states which may not well represent truly classical systems, and they display some nonclassical features [9] . In Ref. [6] , it was shown that prominent quantum properties can actually be transferred from a microscopic superposition to a significantly mixed thermal state (i.e. a thermal state of which the degree of mixedness is close to the maximum value) at a high temperature through an experimentally feasible process. This result clarifies that unavoidable initial mixedness of the cat does not preclude strong quantum phenomena.\n\nOne of the results in Ref. [6] is that quantum entanglement can be produced between thermal states with nearly the maximum Bell-inequality violation when the temperatures of both modes goes to infinity. In previous related results, Bose et al. showed that entanglement can arise when two systems interact if one of the system are pure even when the other system is extremely mixed [10] . There is an interesting previous example shown by Filip et al. for the maximum violation of Bell's inequality when one of the modes is an extremely mixed thermal state [11] . Very recently, Ferreira et al. showed that entanglement can be generated at any finite temperature between high Q cavity mode field and a movable mirror thermal state [12] . However, in these example [10, 11, 12] only one of the modes is considered a large thermal state [10, 11, 12] and entanglement vanishes in the infinite temperature limit [10, 12] , which is obviously in contrast to the result presented in Ref. [6] . Entanglement for both of the modes at the thermal limit of the infinitely high temperature has not been found before. Remarkably, the violation of Bell's inequality in our examples reaches up to Cirel'son's bound [13] even in this infinite-temperature limit for both modes. As Vedral [14] and Ferreira et al. [12] pointed out it is believed that high temperatures reduce entanglement and all entanglement vanishes if the temperature is high enough, which is obviously not the case in Ref. [6] .\n\nThe purpose of this paper is twofold. Firstly, we review and further investigate various properties of superpositions and entanglement of thermal states at high temperatures [6] . In particular, we investigate two classes of highly mixed symmetric states in the phase space. Both the classes of these states do not show typical interference patterns in the phase space while they manifest strong singular behaviors. Interestingly, the first class of states has neither squeezing properties nor negative values in their Wigner functions, however, they are found to be highly nonclassical states. The second class of states has the maximum negativity in the Wigner function. Further, we discuss the possibility of quantum information processing with thermal-state qubits. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure Bell states. We show that four thermal-Bell states can be well discriminated by nonlinear interactions without photon number resolving measurements. Quantum teleportation and gate operations for thermal-state qubits can be realized using the Bell measurement scheme.\n\nThis paper is organized as follows. In Sec. II, we review the generation process of superpositions of thermal states and study their characteristics. In Sec. III, we study en-tanglement of thermal states, i.e., Bell inequality violations. In Sec. IV, we discuss the possibility of quantum information processing using thermal states. We first define the thermal-state qubit and the Bell-basis states using thermal-state entanglement. We then show that the four Bell states can be well discriminated by homodyne detection and two Kerr nonlinearities. It follows that quantum teleportation and quantum gate operations can be realized with thermal-state qubits. We conclude with final remarks in Sec. V."},{"section_type":"OTHER","section_title":"II. SUPERPOSITIONS OF THERMAL STATES A. Generation of thermal-state superpositions","text":"Let us first consider a two-mode harmonic oscillator system. A displaced thermal state can be defined as\n\nρ th (V, d) = d 2 αP th (V, d)|α α| ( 1\n\n)\n\nwhere |α is a coherent state of amplitude α and\n\nP th α (V, d) = 2 π(V -1) exp[- 2|α -d| 2 V -1 ] (2)\n\nwith variance V and displacement d in the phase space.\n\nThe thermal temperature τ increases as V increases as e ν/τ = (V + 1)/(V -1), where is Planck's constant and ν is the frequency [15] . Suppose that a microscopic superposition state\n\n|ψ a = 1 √ 2 (|0 a + |1 a ), (3)\n\nwhere |0 and |1 are the ground and first excited states of the harmonic oscillator, interacts with a thermal state ρ th b (V, d) and the interaction Hamiltonian is\n\nH K = λâ † âb † b (4)\n\nwhich corresponds to the cross Kerr nonlinear interaction. The resulting state is then\n\nρ ent ab = 1 2 d 2 αP th (V, d) |0 0| ⊗ |α α| + |1 0| ⊗ |αe iϕ α| + |0 1| ⊗ |α αe iϕ | + |1 1| ⊗ |αe iϕ αe iϕ | ( 5\n\n)\n\nand ϕ is determined by the strength of the nonlinearity λ and the interaction time. The Wigner representation of ρ ent ab is\n\nW ent ab (α, β) = 1 π e -2|α| 2 W th (β; d) + 2αV c (β; d) + 2[αV c (β; d)] * + (4|α| 2 -1)W th (β; de iϕ ) (6)\n\nwhere α and β are complex numbers parametrizing the phase spaces of the microscopic and macroscopic systems respectively and\n\nW th (α; d) = 2 πV exp[- 2|α -d| 2 V ], (7)\n\nV c (α; d) = 2 πJK exp[- 2 K (1 -e iϕ )d 2 - 1 J (α - 2e iϕ d K )(α * - 2d K )], (8)\n\nK = 2 + (V -1)(1 -e iϕ\n\n), J = (sin ϕ/2 + iV cos ϕ/2)/(2V sin ϕ/2 + 2i cos ϕ/2), and d has been assumed real without loss of generality. If one traces ρ ent ab over mode a, the remaining state will be simply in a classical mixture of two thermal states and its Wigner function will be positive everywhere. However, if one measures out the \"microscopic part\" on the superposed basis, i.e., (|0 a ± |1 a )/ √ 2, the \"macroscopic part\" for mode b may not lose its nonclassical characteristics. Such a measurement on the the superposed basis will reduce the remaining state to\n\nρ sup(±) = N ± s d 2 αP th (V, d) |α α| ± |αe iϕ α| ± |α αe iϕ | + |αe iϕ αe iϕ | , (9)\n\nwhere N ± s are the normalization factors, and its Wigner function is\n\nW sup(±) (α) = N ± s {W th (α; d) ± V c (α; d) ± {V c (α; d)} * + W th (α; de iϕ )}. ( 10\n\n)\n\nThe ± signs in Eqs. ( 8 ) and (9) correspond to the two possible results from the measurement of the microscopic system. The state in Eq. ( 10 ) is a superposition of two thermal states. A feasible experimental setup to generate superpositions of thermal states is atom-field interactions in cavities, where a π/2 pulse can be used to prepare the atom in a superposed state. This type of experiment has already been performed to produce a superposition of coherent states [16] . In our cases, simply thermal states can be used instead of coherent states. Another possible setup is an all-optical scheme with free-traveling fields and a cross-Kerr medium, where a standard singlephoton qubit could be used as the microscopic superposition. Recently, there have been theoretical and experimental efforts to produce and observe giant Kerr nonlinearities using electromagnetically induced transparency [17] . Furthermore, it was shown that a weak Kerr nonlinearity can still be useful if a initially strong field is employed in this type of experiment [18] . We shall further explain this with examples in Sec. III."},{"section_type":"OTHER","section_title":"B. Negativity of the Wigner function","text":"The negativity of the Wigner function is known as an indicator of non-classicality of quantum states. In order to observe negativity of the Wigner function in a real experiment, its absolute minimum negativity should be large enough. The minimum negativity of the Wigner function in Eq. ( 6 ) for V = 1 is -0.144 for d = 0 and -0.246 for d → ∞. Now suppose the initial state can be considered a classical thermal state by letting V ≫ 1. One might expect that the negativity would be washed out as the initial state becomes mixed, but this is not the case. The minimum negativity actually increases as V gets larger. If V → ∞, the minimum negativity of the Wigner function ( 6 ) is -0.246 regardless of d: no matter how mixed the initial thermal state was, the minimum negativity of Wigner function is found to be a large value. The point in the phase space which gives the minimum negativity when V ≫ 1 or d ≫ 0 is (-1 2 , 0) and has negativity\n\nW neg ≡ W ent ab (- 1 2 , 0) = 2(-2 + 1 V exp[-2d 2 V ]) π 2 √ e . (11)\n\nIt can be shown that\n\nW neg approaches -4/(π 2 √ e) ≈ -0.246 when either d → ∞ or V → ∞.\n\nThis effect is obviously due to the interaction between the microscopic superposition and the macroscopic thermal state. If the initial microscopic state is not superposed, e.g., |ψ a = |1 a , the resulting state will be a simple direct product, (|1 1|) a ⊗ ρ th b (V, -d). Whilst for V = 1 this state will exhibit negativity, this is washed out and tends to zero as V → ∞. Needless to say, if it was |0 a instead of |1 a , the resulting Wigner function will be a direct product of two Gaussian states whose Wigner fucntion can never be negative. The superpositon state (3) plays the crucial role in making the minimum negativity of the resulting Wigner function always saturate to a certain negative value no matter how mixed and classical the initial state of the other mode becomes. The Wigner functions of the single-mode states, W sup(±) (α), in Eq. (10) show large negative values. The minimum negativity of the Wigner function W sup(-) (α) is W sup(-) (0) = 2/π regardless of the values of V and d. On the other hand, the minimum negativity of the Wigner function W sup(+) (α) approaches 2/π for d → ∞ and disappears when d = 0."},{"section_type":"OTHER","section_title":"C. Quantum interference in the phase space","text":"When ϕ = π, the state (9) becomes (12) where\n\nρ ± = N (ρ th (V, d)±σ(V, d)±σ(V, -d)+ρ th (V, -d)),\n\nσ(V, d) = d 2 αP th (V, d)| -α α| and N = 2 1 ± exp[-2d 2 V ] V 2 . ( 13\n\n)\n\nIf the initial state for mode b is a pure coherent state, i.e., V = 1, the measurement on the superposed basis for mode a will produce a superposition of two pure coherent states as\n\n| Ψ ± = 1 √ 1 ± e -2|α| 2 (|α ± | -α ), (14)\n\nwhere α = d. The probability P ± to obtain the state ρ ± is obtained as [19 ] where\n\nP ± = ψ ± |Tr b [ρ ent ab ]|ψ ± = 1 2 (1 ± exp[-2d 2 V ] V ), ( 15\n\n)\n\n|ψ ± = (|0 ±|1 )/ √ 2.\n\nThe probability approaches P ± = 1/2 when either d or V becomes large.\n\nAs an analogy of Schrödinger's cat paradox, the variance V corresponds to the size the initial \"cat\", and the distance d between the two thermal component states corresponds to distinguishability between the \"alive cat\" and the \"dead cat\". Suppose that both V and d are very large for the initial thermal state. The two thermal states ρ th (V, ±d) become macroscopically distinguishable when d ≫ √ V , and our example may become a more realistic analogy of the cat paradox in this limit. Both the states ρ ± in this case show probability distributions with two Gaussian peaks and interference fringes [6] . Figure 1 presents the probability distributions of x (≡ Re[α]) and p (≡ Im[α]) for ρ -(a) when V = 100 and d = 100 and (b) when V = 1000 and d = 300. The probability distribution of x (p) for ρ ± can be obtained by integrating the Wigner function of ρ ± over p (x). The two Gaussian peaks along the x axis and interference fringes along the p axis shown in Fig. 1 are a typical signature of a quantum superposition between macroscopically distinguishable states. The visibility v of the interference fringes is defined as [15]\n\nv = I max -I min I max + I min , (16)\n\nwhere I = dxW sup(-) (α) and the maximum should be taken over p. It can be simply shown that the visibility v is always 1 regardless of the value of V . Note that d should increase proportionally to √ V to maintain the condition of classical distingushability between the two component thermal states ρ th (V, ±d). The interference fringes with high visibility are incompatible with classical physics and evidence of quantum coherence. The fringe spacing (the distance between the fringes) does not depend on V but only on d, i.e., a pure superposition of coherent states shows the same fringe spacing for a given d. We emphasize that the states shown in Fig. 1 are \"superpositions\" of severely mixed thermal states.\n\nAn experimental realization of a nonlinear effect corresponding to ϕ = π is very demanding particularly in the presence of decoherence. Here we point out that the method using a weak nonlinear effect (ϕ ≪ π) combined with a strong field (d ≫ 1) [18] can be useful to generate a thermal-state superposition with prominent interference patterns. In Fig. 2 , we have used experimentally accessible values, V = 5, d = 2000 and ϕ = π/1000, but the fringe visibility is still 1. In this case, decoherence during the nonlinear interaction would be significantly reduced because of the decrease of the interaction time [18] . Note also that, if required, the state in Fig. 2 can be moved to the center of the phase space, for example, using a biased beam splitter (BS) and a strong coherent field [18] ."},{"section_type":"OTHER","section_title":"D. Symmetric macroscopic quantum states","text":"Let us assume that d = 0, i.e., the initial state is the thermal state, ρ th (V, 0), at the origin of the phase space. In this case, the thermal-state superpositions, ρ ± , are produced with probabilities, P ± = (1/2){1 ± (1/V )}, respectively. Figure 3 shows the Wigner functions of ρ + dependent on the interaction time between the macro-scopic thermal state and the microscopic superposition in a cross Kerr medium. The state is always symmetric in the phase space regardless of the interaction time as shown in Fig. 3 . In this figure, the initial state is a thermal state of V = 100 (Fig. 3(a) ). In a relatively short time (θ = π/32 and θ = π/16), the state shows some interference patterns. When θ = π, the evolved state looks very localized around the origin as shown in Fig 3 . The generated state at θ = π does not show negativity of the Wigner function nor squeezing properties. On the other hand, a well defined P function does not exist for this state.\n\nIn the case of ρ -, with the same assumption d = 0, the Wigner function at ϕ = π has the minimum negativity (-2/π) at the origin regardless of V . As a result of the interaction with the microscopic superposition, a deep hole to the negative direction below zero has been formed around the origin for ρ -as shown in Fig. 4 ."},{"section_type":"OTHER","section_title":"III. ENTANGLEMENT BETWEEN THERMAL STATES","text":"Entanglement between macroscopic objects and its Bell-type inequality tests are an important issue. In this section, we shall show that entanglement can be generated between high-temperature thermal states even when the temperature of each mode goes to infinity. 2 200 1 400 600 800 1000 V 2.1 2.2 2.3 B (a) 50 100 150 d 2.2 2 0 2.4 2.6 2.8 B (b) FIG. 5: (a) The optimized violation, B ≡ |B + |max, of Bell-\n\nCHSH inequality for the \"thermal-state entenglement\", ρ+, of V = 1000 (solid curve) and V = 100 (dashed curve). The Bell-violation of a pure entangled coherent state, i.e., V = 1, has been plotted for comparison (dotted curve). The Bell-violation B approaches its maximum bound, 2 √ 2, when d ≫ √ V regardless of the level of the mixedness V . (b) The optimized Bell-violation B against d for the different type of thermal-state entanglement generated using a 50:50 beam splitter from ρ + . V = 1000 (solid curve), V = 100 (dashed curve) and V = 1 (dotted curve)."},{"section_type":"OTHER","section_title":"A. Entanglement using two initial thermal states","text":"If the microscopic superposition interacts with two thermal states, ρ th b (V, d) and ρ th c (V, d), and the micro-scopic particle is measured out on the superposed basis, the resulting state will be\n\nρ tm(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (17)\n\nwhere\n\nN t = 2 1 ± exp[-4d 2 V ] V 2 . ( 18\n\n)\n\nSuch two-mode thermal-state entanglement can be generated using two cavities and an atomic state detector [20] .\n\nExtending the two cavities to N cavities, entanglement of N -mode thermal states can also be generated. Such a state is an analogy of the N -mode pure GHZ state [21] but each mode is extremely mixed. Here we shall consider the Bell-CHSH inequality [22, 23] with photon number parity measurements [20, 24] . The parity measurements can be performed in a high-Q cavity using a far-off-resonant interaction between a two-level atom and the field [25] . The Bell-CHSH inequality can be represented in terms of the Winger function as [24 ]\n\n|B (±) | = π 2 4 |W tm(±) (α, β) + W tm(±) (α, β ′ ) + W tm(±) (α ′ , β) -W tm(±) (α ′ , β ′ )| ≤ 2, (19)\n\nwhere W tm(±) (α, β) is the Wigner function of ρ tm(±) in Eq. ( 17 ). As shown in Fig. 5 , the Bell-violation approaches the maximum bound for a bipartite measurement, 2 √ 2 [13] , when d ≫ √\n\nV regardless of the level of the mixedness V , i.e., the temperatures of the thermal states. Note that it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(a) . This implies that entanglement of nearly 1 ebit has been produced between the two significantly mixed thermal states for d ≫ √ V , and such \"thermal-state entanglement\" cannot be described by a local theory."},{"section_type":"OTHER","section_title":"B. Entanglement using a beam splitter","text":"A different type of macroscopic entanglement can be generated by applying the beam splitter operation\n\nexp[θ/2(e iφ â † s âd -e -iφ â † d âs )], (20)\n\non the \"thermal-state superpositions\" in Eq. ( 9 ). The state after passing through a 50:50 beam splitter can be represented as\n\nN d 2 αP th α (V, d) | α √ 2 , - α √ 2 ± | - α √ 2 , α √ 2 α √ 2 , - α √ 2 | ± - α √ 2 , α √ 2 | , (21\n\n) 200 400 600 800 1000 V 2.1 2 0 2.2 2.3 B FIG. 6: The optimized Bell-violation B against V for the slightly different type of thermal-state entanglement generated using a 50:50 beam splitter using ρ + when d = 0. 3.13 3.14 3.15 3.16 ϕ 1.6 2.2 2 2.4 2.6 2.8 B (a) 3.13 3.14 3.15 3.16 ϕ 1.6 2 2.2 2.4 2.6 2.8 B (b) where N is defined in Eq. ( 13 ). When d is large, this state violates the Bell-CHSH inequality to the maximum bound 2 √ 2 regardless of the level of mixedness V as shown in Fig. 5(b ). Again, it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(b ). Furthermore, these states severely violate Bell's inequality even when d = 0 as V increases as shown in Fig. 6 . We have found that the optimized Bell violation of these states approaches 2.32449 for V → ∞. Interestingly, this value is exactly the same as the optimized Bell-CHSH violation for a pure two-mode squeezed state in the infinite squeezing limit [26] . Note that multilmode entangled states can be generated using multiple beam splitters.\n\nIt should be noted that the Bell violations are more sensitive to the interaction time when either V or d is larger. Figure 7 clearly shows this tendency. Therefore, in order to observe the Bell violations using the mixed state of V (and d) large, the interaction time in the Kerr medium should be more accurate."},{"section_type":"OTHER","section_title":"IV. QUANTUM INFORMATION PROCESSING WITH THERMAL-STATE QUBITS","text":"In this section, we discuss the possibility of quantum information processing with thermal-state qubits and thermal-state entanglement."},{"section_type":"OTHER","section_title":"A. Qubits and Bell-state measurements","text":"We introduce a thermal-state qubit\n\nρ ψ = |a| 2 ρ th (V, d) ± ab * σ(V, d) ± a * bσ(V, -d) + |b| 2 ρ th (V, -d), (22)\n\nwhere a and b are arbitrary complex numbers. The basis states, ρ th (V, d) and ρ th (V, -d), can be well discriminated by a homodyne measurement when d is larger than V . The thermal state qubit (22) can be re-written as\n\nρ ψ = d 2 αP th α (V, d) a|α + b| -α a * α| + b * -α| , (23)\n\nwhich can be understood as a generalization of the coherent state qubit, a|d + b|d , where |d is a coherent state of amplitude d. The thermal-state qubit (23) becomes identical to the coherent-state qubit when V = 1.\n\nWe also define four thermal-Bell states as\n\nρ Φ(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (24) ρ Ψ(±) = N t ρ th (V, d) ⊗ ρ th (V, -d) ± σ(V, d) ⊗ σ(V, -d) ± σ(V, -d) ⊗ σ(V, d) + ρ th (V, -d) ⊗ ρ th (V, d) (25)\n\nwhere N t was defined in Eq. ( 18 ). The thermal-Bell states can be written as For quantum information processing applications, it is an important task to discriminate between the four Bell states. Here we discuss two possible ways to discriminate between the thermal-Bell states (25) . We shall only briefly describe the first scheme using photon number resolving measurements and focus on the second scheme using nonlinear interactions. The first method is to simply use a 50-50 beam splitter and two photon number resolving detectors as shown in Fig. 8(a) . This scheme is basically the same as the Bellstate measurement scheme with pure entangled coherent states [27, 28] . Let us suppose that the amplitude, d, is large enough, i.e., d ≫ √ V . If the incident state was ρ Φ(+) or ρ Φ(-) , most of the photons are detected on detector A in in Fig. 8(a) . Meanwhile, most of the photons are detected on detector B when the incident state was ρ Ψ(+) or ρ Ψ(-) . The average photon numbers between the \"many-photon case\" and the \"few-photon case\" are compared in Fig. 9 . Furthermore, the states ρ Ψ(+) and ρ Φ(+) contain only even numbers of photons while ρ Ψ(-) and ρ Φ(-) contain only odd numbers of photons. Therefore, all the four Bell states can be well discriminated by analyzing numbers of photons detected at detectors A and B. For example, if detector A detects many photons while detector B detects few and the total photon number detected by the two detectors are even, this means that state ρ Φ(+) was measured by the thermal-Bell measurement. The nonzero failure probability can be made arbitrarily small by increasing d.\n\nρ Φ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, β ± | -α, -β α, β| ± -α, -β| , ( 26\n\n)\n\nρ Ψ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, -β ± | -α, β α, -β| ± -α, β| . (27)\n\nHowever, the average photon numbers of the thermal-Bell states are high when V ≫ 1 and d ≫ 1. In this case, it would be unrealistic to use photon number resolving detectors. It would be an interesting question whether\n\n0.5 1 1.5 2 2.5 3 d 10 0 0 20 30 N (a) 0.5 1 1.5 2 2.5 3 d 0 0 10 20 30 N (b)\n\nFIG. 9: The average photon number N for the \"many-photon case\" (solid line) and the \"few-photon case\" (dashed line) for V = 10 against d (a) when the input state is either ρ Φ(+) or ρ Ψ(+) and (b) when the input state is either ρ Φ(-) or ρ Ψ(-) .\n\nthese four thermal-Bell states can be distinguished by classical measurements, such as homodyne detection, instead of photon number resolving detection. Our alternative scheme employs cross-Kerr nonlinearities and single photon detectors as shown in Fig. 8(b ). Let us first suppose that the input field was ρ Φ(+) . The incident twomode state passes through a 50-50 beam splitter, BS1. The state after passing through the 50:50 beam splitter, BS1, is\n\nρ B = N t d 2 αd 2 βP th α (V, d)P th β (V, d) |η, -ξ η, -ξ| + |η, -ξ -η, ξ| + | -η, ξ η, -ξ| + | -η, ξ -η, ξ| (28)\n\nwhere η = (α+β)/ √ 2 and ξ = (α-β)/ √ 2. Two dual-rail single photon qubits, |ψ + ee ′ and |ψ + f f ′ , where\n\n|ψ + = 1 √ 2 (|0 |1 + |1 |0 ), (29)\n\nare prepared using two single photons and 50:50 beam splitters, BS2 and BS3, as shown in Fig. 8(b ). Then, traveling fields at modes c and d interacts with those of modes e and f , respectively, in cross-Kerr nonlinear media. We suppose that the interaction time is t = π/λ, and the resulting state is then\n\nρ B ′ = U ce U df ρ B cd ρ q ee ′ ρ q f f ′ U † ce U † df (30)\n\nwhere U ce = exp[iπH K ce /λ ] and ρ q = |ψ q ψ q |. An explicit form of Eq. ( 30 ) can then be simply obtained using the identity\n\nU ce |α c |0 e = |α c |0 e , U ce |α c |1 e = | -α c |1 e ( 31\n\n)\n\nwhere |α is a coherent state. However, we omit such an explicit expression in this paper for it is too lengthy. After the nonlinear interactions, the qubit parts, modes e, e ′ , f and f ′ , should be measured with the measurement basis\n\n{| + + , | + -, | -+ , | --} (32)\n\nwhere\n\n| + + = |ψ + ee ′ |ψ + f f ′ , | + -= |ψ + ee ′ |ψ -f f ′ , | -+ = |ψ -ee ′ |ψ + f f ′ , | --= |ψ -ee ′ |ψ -f f ′ , and |ψ -= (|0 |1 -|1 |0 )/ √ 2.\n\nThis measurement can be performed using two 50:50 beam splitters, BS4 and BS5, and four detectors, A1, A2, B1 and B2, as shown in Fig. 8(b ). If detector A1 and B1 click, i.e., the measurement result is | + + , the resulting state at modes c and d is\n\nρ ++ = N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η + | -η )( η| + -η|) c ⊗ (|ξ + | -ξ )( ξ| + -ξ|) d (33)\n\nNote that state ρ ++ is not normalized, which implies that the probability of obtaining the corresponding measurement result is not unity. The probability of obtaining this result is\n\nP ++ = (V + 1)(V + e -4d 2 V ) 2(V 2 + e -4d 2 V ) . (34)\n\nWhen the result is either | +or | -+ , the result is\n\nψ 2 |ρ B ′ |ψ 2 = ψ 3 |ρ B ′ |ψ 3 = 0, (35)\n\nwhich obviously means that the probability of the obtaining this result is zero. When the result is | --, i.e., detector A2 and B2 click,\n\nρ --= N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η -| -η )( η| --η|) c ⊗ (|ξ -| -ξ )( ξ| --ξ|) d , (36)\n\nwhich is not normalized. The probability of obtaining this result is\n\nP --= (V -1)(V -e -4d 2 V ) 2(V 2 + e -4d 2 V ) , (37)\n\nand it can be simply verified that P ++ +P --= 1. Therefore, only the measurement results | + + and | -can be obtained in the case of the input state ρ Φ(+) . This is exactly the same for the case of ρ Ψ(+) . In the same way, it can be shown that if either the input state was ρ Φ(-) or ρ Ψ(-) , only the measurement results | +and | -+ can be obtained. In other words, the parity of the toincoming state is perfectly well discriminated by the measurements on single-photon qubits.\n\nSubsequently, a homodyne measurement is performed for mode c by homodyne detector C as shown in Fig. 8(b) . We assume that ideal homodyne measurements are performed, i.e., when a homodyne measurement is performed the state is projected onto eigenstate |x of operator X with eigenvalue x, where\n\nX = 1 √ 2 (a + a † ). ( 38\n\n)\n\nLet us first consider the case when the measurement result for the single photon qubits is | + + . In this case, the remaining state is ρ ++ in Eq. ( 33 ). The probability distribution P ++ Φ (+) for the homodyne measurement at detector C is\n\nP ++ Φ (+) = x|Tr d [ρ ++ ]|x = V 1 2 (e -V x 2 + e -x 2 V ) π 1 2 (V + 1)\n\n. (39) Note that the superscript, ++, denotes that the qubit measurement result was | + + , and the subscript, Φ (+) , denotes that the input state was ρ Φ (+) . These notations will be used also for the other cases in this section. The same analysis can be performed for the other possible measurement outcome | --:\n\nP -- Φ (+) = x|Tr d [ρ --]|x = V 1 2 (e -V x 2 -e -x 2 V ) π 1 2 (V -1)\n\n. (40)\n\nIn the same way, for another input state, ρ Φ(-) , it is straightforward to show:\n\nP +- Φ (-) = P ++ Φ (+) , P -+ Φ (-) = P -- Φ (+) , (41)\n\nand P ++ Φ (-) = P -- Φ (-) = 0. On the other hand, if the input state was ρ Ψ(+) , the probability distributions P ++ Ψ (+) and P -- Ψ (+) at detector C are It is straightforward to show for the other input state ρ Ψ(-) :\n\nP ++ Ψ (+) = x|Tr c [ρ ++ ]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V + 2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) + 1 , (42)\n\nP -- Ψ (+) = x|Tr c [ρ --]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V -2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) -1 . ( 43\n\nP +- Ψ (-) = P -- Ψ (+) , P -+ Ψ (-) = P ++ Ψ (+) . (44)\n\nThe probability distributions P ++ Φ (±) and P ++ Ψ (±) are plotted in Fig. 10 . Figure 10 shows that when the input state was ρ Φ(+) or ρ Φ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Φ (+) and P -- Φ (+) , is located around the origin. However, when the input state was ρ Ψ(+) or ρ Ψ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Ψ (+) and P -- Ψ (+) , is located far from the origin. Therefore, two of the Bell states, ρ Φ(+) or ρ Φ(-) , can be well distinguished from the other two by the homodyne detector C for the case of the measurement outcome | + + . Finally, by combining the homodyne measurement result and the qubit measurement result, all four Bell states can be effectively distinguished. For example, let us assume that the measurement outcome of the single photon detectors was | + + and the homodyne detection outcome was around the origin, i.e., x ≈ 0. Then, one can say that state ρ Ψ(-) has been measured for the result of the thermal-Bell measurement.\n\nAs implied in Fig. 10 , the overlaps between the probability distributions around the origin, P ++ Φ (+) and P -- Φ (+) , and the other distributions, P ++ Ψ (+) and P -- Ψ (+) , are extremely small for a sufficiently large d. In other words, the distinguishability by the homodyne detection rapidly approaches 1 as d increases. As an example, we can calculate the distinguishability between the states ρ Ψ(+) and ρ Φ(+) by the homodyne measurement by detector C. The distinguishability by homodyne detection is Note also that the second scheme using homodyne detection is robust to detection inefficiency compared with the first scheme using photon number resolving measurements. In the first scheme, even if a detector misses only one photon, it will result in a completely wrong measurement outcome. In the second scheme, however, the measurement outcome will not be affected in that way. If a single photon detector misses a photon, it will be immediately recognized. Such a case can simply be discarded so that it will only degrade the success probability of the Bell measurement. The homodyne detection inefficiency will not significantly affect the result when the distributions around the origin and the distributions far from the origin are well separated, i.e., when d ≫ √ V , as shown in Fig. 10 . On the other hand, loss in the Kerr medium will have a detrimental affect.\n\nP s = 1 2 |x|<d dxP ++ c (x) + |x|≥d dxP ++ d (x) (45"},{"section_type":"OTHER","section_title":"B. Quantum teleportation and computation","text":"Quantum teleportation of a thermal-state qubit can be performed using one of the Bell states as the quantum channel. Let us assume that Alice needs to teleport a thermal-state qubit, ρ ψ , to Bob using a thermal-state entanglement, ρ Ψ(-) , shared by the two parties. The total state can be represented as\n\nAlice first needs to perform the thermal-Bell measurement described in the previous subsection. To complete the teleportation process, Bob should perform an appropriate unitary transformation on his part of the quantum channel according to the measurement result sent from Alice via a classical channel. It is straightforward to show that the required transformations are exactly the same to those for the coherent-state qubit [27] . When the measurement outcome is ρ Ψ(-) , Bob obtains a perfect replica of the original unknown qubit without any operation.\n\nWhen the measurement outcome is ρ Φ(-) , Bob should perform |α ↔ |α on his qubit in Eq. ( 23 ). Such a phase shift by π can be done using a phase shifter whose action is described by P (ϕ) = e iϕa † a , where a and a † are the annihilation and creation operators. When the outcome is ρ Ψ(+) , the transformation should be performed as |α → |α and |α → -|α . It is known that the displacement operator is a good approximation of this transformation for d ≫ 1 [29] . This transformation can also be achieved by teleporting the state again locally and repeating until the required phase shift is obtained [30] . When the outcome is ρ Φ(+) , σ x and σ z should be successively applied."},{"section_type":"CONCLUSION","section_title":"V. CONCLUSION","text":"In this paper, we have studied characteristics of superpositions and entanglement of thermal states at high temperatures and discussed their applications to quantum information processing. The superpositions and entanglement of thermal states show various nonclassical properties such as interference patterns, negativity of the Wigner functions, and violations of the Bell-CHSH inequality. The Bell violations are more sensitive to the interaction time during the generation process when the thermal temperature (i.e. mixedness) of the thermalstate entanglement is larger. Therefore, in order to observe the Bell violations using the mixed state at a high temperature, the interaction time in the Kerr medium should be accurate. We have pointed out that certain superpositions of high-temperature thermal states, symmetric in the phase space, can also be generated. Some of these states have neither squeezing properties nor negative values in their Wigner functions but they are found to be highly nonclassical.\n\nWe have introduced the thermal-state qubit and thermal-Bell states for applications to quantum information processing. We have presented two possible methods for the Bell-state measurement. The Bell-state measurement enables one to perform quantum teleportation and gate operations for quantum computation with thermalstate qubits. The first scheme uses two photon number resolving detectors and a 50-50 beam splitter to discriminate the thermal-Bell states. Using the second scheme, it is possible to effectively discriminate the thermal-Bell states without photon number resolving detection. The required resources for the second scheme are two Kerr nonlinear interactions, two single photon detectors, two 50:50 beam splitters and one homodyne detector. The second scheme is more robust to inefficiency of the detectors: the inefficiency of the single photon detectors only degrades the success probability of the Bell measurement."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states without photon number resolving detection but with Kerr nonlinear interactions and two singlephoton detectors. This enables one to perform quantum teleportation and gate operations for quantum computation with thermal-state qubits.\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"In many problems considered within the framework of quantum physics, physical systems are treated as pure states that can be represented by state vectors, or equivalently, by wave functions. Even though such an approach is simple and useful to address certain problems, it could often be quite different from real conditions of physical systems. This may be particularly true when one deals with macroscopic physical systems in terms of quantum physics. A macroscopic object is a complex open system which cannot avoid continuous interactions with the environment. Such a physical system is generally in a significantly mixed state and cannot be represented by a state vector. In general, mixed states are subtle objects whose properties are significantly more difficult to characterize than pure states.\\n\\nSchrödinger's famous cat paradox is a typical example where a massive classical object was assumed to be a pure state. It describes a counter-intuitive feature of quantum physics which dramatically appears when the principle of quantum superposition is applied to macroscopic objects. In the original paradox and its various explanations, the initial cat isolated in the steel chamber is considered a pure state that can be represented by a state vector such as |alive (or a wave function such as ψ alive ). The cat isolated from the environment is then assumed to interact with a microscopic superposition state, (|g +|e )/ √ 2, where |g and |e are the ground and excited states of a two-level atom. The cat will be dead if the atom is found in the excited state, |e , while it will remain alive if otherwise. Thus in Schrödinger's gedanken experiment the cat is entangled with the atom as (|g |alive + |e |dead ) √ 2, where the alive and dead statuses of the cat are described by the state vectors |alive and |dead . If one measures out the atomic system on the superposed basis, (|g ± |e )/ √ 2, the cat will be in a superposition of alive and dead states such as (|alive ± |dead )/ √ 2. It is often argued that such superposed states and entangled states can theoretically exist but are virtually impossible to observe because one cannot perfectly isolate a macroscopic object such as the cat from its environment [4] .\\n\\nHowever, this explanation is not fully satisfactory because the cat, a macroscopic object, is a complex open system which cannot be represented by a state vector. One may argue that the cat could be assumed to be in an unknown pure state such that the cat was certainly alive but the exact state of the cat was unknown. However, the interactions between the cat and its environment can cause the cat to become entangled with the environment [5] . In such a case, even though one can perfectly isolate the cat in the steel chamber from the enviroment, the cat will remain entangled with the environment due to its pre-interactions with the environment. Therefore, strictly speaking, even to assume a cat as an unknown pure state in the steel chamber is not legitimate. Thus a key point here is that it is unsatisfactory to describe the cat by a pure state such as |alive and |dead . We may need a more realistic assumption that the \\\"cat\\\" in Schrödinger's paradox was in a significantly mixed classical state. An intriguing question is then whether the quantum properties of the resulting state would still remain or diminish under such an assumption.\\n\\nRecently, such an analogy of Schrödinger's cat paradox, where the state corresponding to the virtual cat is a significantly mixed thermal state, was investigated [6] . A thermal state with a high temperature is considered a classical state in quantum optics. As the temperature of the thermal state increases, the degree of mixedness, which can be quantified by linear entropy, rapidly approaches the maximum value. When the temperature approaches infinity, the thermal state does not show any quantum properties. As a comparison, coherent states with large amplitudes are known as the most classical pure states [7] , and their superposition is often regarded as a superposition of classical states [8] . However, coherent states are still pure states which may not well represent truly classical systems, and they display some nonclassical features [9] . In Ref. [6] , it was shown that prominent quantum properties can actually be transferred from a microscopic superposition to a significantly mixed thermal state (i.e. a thermal state of which the degree of mixedness is close to the maximum value) at a high temperature through an experimentally feasible process. This result clarifies that unavoidable initial mixedness of the cat does not preclude strong quantum phenomena.\\n\\nOne of the results in Ref. [6] is that quantum entanglement can be produced between thermal states with nearly the maximum Bell-inequality violation when the temperatures of both modes goes to infinity. In previous related results, Bose et al. showed that entanglement can arise when two systems interact if one of the system are pure even when the other system is extremely mixed [10] . There is an interesting previous example shown by Filip et al. for the maximum violation of Bell's inequality when one of the modes is an extremely mixed thermal state [11] . Very recently, Ferreira et al. showed that entanglement can be generated at any finite temperature between high Q cavity mode field and a movable mirror thermal state [12] . However, in these example [10, 11, 12] only one of the modes is considered a large thermal state [10, 11, 12] and entanglement vanishes in the infinite temperature limit [10, 12] , which is obviously in contrast to the result presented in Ref. [6] . Entanglement for both of the modes at the thermal limit of the infinitely high temperature has not been found before. Remarkably, the violation of Bell's inequality in our examples reaches up to Cirel'son's bound [13] even in this infinite-temperature limit for both modes. As Vedral [14] and Ferreira et al. [12] pointed out it is believed that high temperatures reduce entanglement and all entanglement vanishes if the temperature is high enough, which is obviously not the case in Ref. [6] .\\n\\nThe purpose of this paper is twofold. Firstly, we review and further investigate various properties of superpositions and entanglement of thermal states at high temperatures [6] . In particular, we investigate two classes of highly mixed symmetric states in the phase space. Both the classes of these states do not show typical interference patterns in the phase space while they manifest strong singular behaviors. Interestingly, the first class of states has neither squeezing properties nor negative values in their Wigner functions, however, they are found to be highly nonclassical states. The second class of states has the maximum negativity in the Wigner function. Further, we discuss the possibility of quantum information processing with thermal-state qubits. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure Bell states. We show that four thermal-Bell states can be well discriminated by nonlinear interactions without photon number resolving measurements. Quantum teleportation and gate operations for thermal-state qubits can be realized using the Bell measurement scheme.\\n\\nThis paper is organized as follows. In Sec. II, we review the generation process of superpositions of thermal states and study their characteristics. In Sec. III, we study en-tanglement of thermal states, i.e., Bell inequality violations. In Sec. IV, we discuss the possibility of quantum information processing using thermal states. We first define the thermal-state qubit and the Bell-basis states using thermal-state entanglement. We then show that the four Bell states can be well discriminated by homodyne detection and two Kerr nonlinearities. It follows that quantum teleportation and quantum gate operations can be realized with thermal-state qubits. We conclude with final remarks in Sec. V.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. SUPERPOSITIONS OF THERMAL STATES A. Generation of thermal-state superpositions\",\n \"text\": \"Let us first consider a two-mode harmonic oscillator system. A displaced thermal state can be defined as\\n\\nρ th (V, d) = d 2 αP th (V, d)|α α| ( 1\\n\\n)\\n\\nwhere |α is a coherent state of amplitude α and\\n\\nP th α (V, d) = 2 π(V -1) exp[- 2|α -d| 2 V -1 ] (2)\\n\\nwith variance V and displacement d in the phase space.\\n\\nThe thermal temperature τ increases as V increases as e ν/τ = (V + 1)/(V -1), where is Planck's constant and ν is the frequency [15] . Suppose that a microscopic superposition state\\n\\n|ψ a = 1 √ 2 (|0 a + |1 a ), (3)\\n\\nwhere |0 and |1 are the ground and first excited states of the harmonic oscillator, interacts with a thermal state ρ th b (V, d) and the interaction Hamiltonian is\\n\\nH K = λâ † âb † b (4)\\n\\nwhich corresponds to the cross Kerr nonlinear interaction. The resulting state is then\\n\\nρ ent ab = 1 2 d 2 αP th (V, d) |0 0| ⊗ |α α| + |1 0| ⊗ |αe iϕ α| + |0 1| ⊗ |α αe iϕ | + |1 1| ⊗ |αe iϕ αe iϕ | ( 5\\n\\n)\\n\\nand ϕ is determined by the strength of the nonlinearity λ and the interaction time. The Wigner representation of ρ ent ab is\\n\\nW ent ab (α, β) = 1 π e -2|α| 2 W th (β; d) + 2αV c (β; d) + 2[αV c (β; d)] * + (4|α| 2 -1)W th (β; de iϕ ) (6)\\n\\nwhere α and β are complex numbers parametrizing the phase spaces of the microscopic and macroscopic systems respectively and\\n\\nW th (α; d) = 2 πV exp[- 2|α -d| 2 V ], (7)\\n\\nV c (α; d) = 2 πJK exp[- 2 K (1 -e iϕ )d 2 - 1 J (α - 2e iϕ d K )(α * - 2d K )], (8)\\n\\nK = 2 + (V -1)(1 -e iϕ\\n\\n), J = (sin ϕ/2 + iV cos ϕ/2)/(2V sin ϕ/2 + 2i cos ϕ/2), and d has been assumed real without loss of generality. If one traces ρ ent ab over mode a, the remaining state will be simply in a classical mixture of two thermal states and its Wigner function will be positive everywhere. However, if one measures out the \\\"microscopic part\\\" on the superposed basis, i.e., (|0 a ± |1 a )/ √ 2, the \\\"macroscopic part\\\" for mode b may not lose its nonclassical characteristics. Such a measurement on the the superposed basis will reduce the remaining state to\\n\\nρ sup(±) = N ± s d 2 αP th (V, d) |α α| ± |αe iϕ α| ± |α αe iϕ | + |αe iϕ αe iϕ | , (9)\\n\\nwhere N ± s are the normalization factors, and its Wigner function is\\n\\nW sup(±) (α) = N ± s {W th (α; d) ± V c (α; d) ± {V c (α; d)} * + W th (α; de iϕ )}. ( 10\\n\\n)\\n\\nThe ± signs in Eqs. ( 8 ) and (9) correspond to the two possible results from the measurement of the microscopic system. The state in Eq. ( 10 ) is a superposition of two thermal states. A feasible experimental setup to generate superpositions of thermal states is atom-field interactions in cavities, where a π/2 pulse can be used to prepare the atom in a superposed state. This type of experiment has already been performed to produce a superposition of coherent states [16] . In our cases, simply thermal states can be used instead of coherent states. Another possible setup is an all-optical scheme with free-traveling fields and a cross-Kerr medium, where a standard singlephoton qubit could be used as the microscopic superposition. Recently, there have been theoretical and experimental efforts to produce and observe giant Kerr nonlinearities using electromagnetically induced transparency [17] . Furthermore, it was shown that a weak Kerr nonlinearity can still be useful if a initially strong field is employed in this type of experiment [18] . We shall further explain this with examples in Sec. III.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Negativity of the Wigner function\",\n \"text\": \"The negativity of the Wigner function is known as an indicator of non-classicality of quantum states. In order to observe negativity of the Wigner function in a real experiment, its absolute minimum negativity should be large enough. The minimum negativity of the Wigner function in Eq. ( 6 ) for V = 1 is -0.144 for d = 0 and -0.246 for d → ∞. Now suppose the initial state can be considered a classical thermal state by letting V ≫ 1. One might expect that the negativity would be washed out as the initial state becomes mixed, but this is not the case. The minimum negativity actually increases as V gets larger. If V → ∞, the minimum negativity of the Wigner function ( 6 ) is -0.246 regardless of d: no matter how mixed the initial thermal state was, the minimum negativity of Wigner function is found to be a large value. The point in the phase space which gives the minimum negativity when V ≫ 1 or d ≫ 0 is (-1 2 , 0) and has negativity\\n\\nW neg ≡ W ent ab (- 1 2 , 0) = 2(-2 + 1 V exp[-2d 2 V ]) π 2 √ e . (11)\\n\\nIt can be shown that\\n\\nW neg approaches -4/(π 2 √ e) ≈ -0.246 when either d → ∞ or V → ∞.\\n\\nThis effect is obviously due to the interaction between the microscopic superposition and the macroscopic thermal state. If the initial microscopic state is not superposed, e.g., |ψ a = |1 a , the resulting state will be a simple direct product, (|1 1|) a ⊗ ρ th b (V, -d). Whilst for V = 1 this state will exhibit negativity, this is washed out and tends to zero as V → ∞. Needless to say, if it was |0 a instead of |1 a , the resulting Wigner function will be a direct product of two Gaussian states whose Wigner fucntion can never be negative. The superpositon state (3) plays the crucial role in making the minimum negativity of the resulting Wigner function always saturate to a certain negative value no matter how mixed and classical the initial state of the other mode becomes. The Wigner functions of the single-mode states, W sup(±) (α), in Eq. (10) show large negative values. The minimum negativity of the Wigner function W sup(-) (α) is W sup(-) (0) = 2/π regardless of the values of V and d. On the other hand, the minimum negativity of the Wigner function W sup(+) (α) approaches 2/π for d → ∞ and disappears when d = 0.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"C. Quantum interference in the phase space\",\n \"text\": \"When ϕ = π, the state (9) becomes (12) where\\n\\nρ ± = N (ρ th (V, d)±σ(V, d)±σ(V, -d)+ρ th (V, -d)),\\n\\nσ(V, d) = d 2 αP th (V, d)| -α α| and N = 2 1 ± exp[-2d 2 V ] V 2 . ( 13\\n\\n)\\n\\nIf the initial state for mode b is a pure coherent state, i.e., V = 1, the measurement on the superposed basis for mode a will produce a superposition of two pure coherent states as\\n\\n| Ψ ± = 1 √ 1 ± e -2|α| 2 (|α ± | -α ), (14)\\n\\nwhere α = d. The probability P ± to obtain the state ρ ± is obtained as [19 ] where\\n\\nP ± = ψ ± |Tr b [ρ ent ab ]|ψ ± = 1 2 (1 ± exp[-2d 2 V ] V ), ( 15\\n\\n)\\n\\n|ψ ± = (|0 ±|1 )/ √ 2.\\n\\nThe probability approaches P ± = 1/2 when either d or V becomes large.\\n\\nAs an analogy of Schrödinger's cat paradox, the variance V corresponds to the size the initial \\\"cat\\\", and the distance d between the two thermal component states corresponds to distinguishability between the \\\"alive cat\\\" and the \\\"dead cat\\\". Suppose that both V and d are very large for the initial thermal state. The two thermal states ρ th (V, ±d) become macroscopically distinguishable when d ≫ √ V , and our example may become a more realistic analogy of the cat paradox in this limit. Both the states ρ ± in this case show probability distributions with two Gaussian peaks and interference fringes [6] . Figure 1 presents the probability distributions of x (≡ Re[α]) and p (≡ Im[α]) for ρ -(a) when V = 100 and d = 100 and (b) when V = 1000 and d = 300. The probability distribution of x (p) for ρ ± can be obtained by integrating the Wigner function of ρ ± over p (x). The two Gaussian peaks along the x axis and interference fringes along the p axis shown in Fig. 1 are a typical signature of a quantum superposition between macroscopically distinguishable states. The visibility v of the interference fringes is defined as [15]\\n\\nv = I max -I min I max + I min , (16)\\n\\nwhere I = dxW sup(-) (α) and the maximum should be taken over p. It can be simply shown that the visibility v is always 1 regardless of the value of V . Note that d should increase proportionally to √ V to maintain the condition of classical distingushability between the two component thermal states ρ th (V, ±d). The interference fringes with high visibility are incompatible with classical physics and evidence of quantum coherence. The fringe spacing (the distance between the fringes) does not depend on V but only on d, i.e., a pure superposition of coherent states shows the same fringe spacing for a given d. We emphasize that the states shown in Fig. 1 are \\\"superpositions\\\" of severely mixed thermal states.\\n\\nAn experimental realization of a nonlinear effect corresponding to ϕ = π is very demanding particularly in the presence of decoherence. Here we point out that the method using a weak nonlinear effect (ϕ ≪ π) combined with a strong field (d ≫ 1) [18] can be useful to generate a thermal-state superposition with prominent interference patterns. In Fig. 2 , we have used experimentally accessible values, V = 5, d = 2000 and ϕ = π/1000, but the fringe visibility is still 1. In this case, decoherence during the nonlinear interaction would be significantly reduced because of the decrease of the interaction time [18] . Note also that, if required, the state in Fig. 2 can be moved to the center of the phase space, for example, using a biased beam splitter (BS) and a strong coherent field [18] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"D. Symmetric macroscopic quantum states\",\n \"text\": \"Let us assume that d = 0, i.e., the initial state is the thermal state, ρ th (V, 0), at the origin of the phase space. In this case, the thermal-state superpositions, ρ ± , are produced with probabilities, P ± = (1/2){1 ± (1/V )}, respectively. Figure 3 shows the Wigner functions of ρ + dependent on the interaction time between the macro-scopic thermal state and the microscopic superposition in a cross Kerr medium. The state is always symmetric in the phase space regardless of the interaction time as shown in Fig. 3 . In this figure, the initial state is a thermal state of V = 100 (Fig. 3(a) ). In a relatively short time (θ = π/32 and θ = π/16), the state shows some interference patterns. When θ = π, the evolved state looks very localized around the origin as shown in Fig 3 . The generated state at θ = π does not show negativity of the Wigner function nor squeezing properties. On the other hand, a well defined P function does not exist for this state.\\n\\nIn the case of ρ -, with the same assumption d = 0, the Wigner function at ϕ = π has the minimum negativity (-2/π) at the origin regardless of V . As a result of the interaction with the microscopic superposition, a deep hole to the negative direction below zero has been formed around the origin for ρ -as shown in Fig. 4 .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. ENTANGLEMENT BETWEEN THERMAL STATES\",\n \"text\": \"Entanglement between macroscopic objects and its Bell-type inequality tests are an important issue. In this section, we shall show that entanglement can be generated between high-temperature thermal states even when the temperature of each mode goes to infinity. 2 200 1 400 600 800 1000 V 2.1 2.2 2.3 B (a) 50 100 150 d 2.2 2 0 2.4 2.6 2.8 B (b) FIG. 5: (a) The optimized violation, B ≡ |B + |max, of Bell-\\n\\nCHSH inequality for the \\\"thermal-state entenglement\\\", ρ+, of V = 1000 (solid curve) and V = 100 (dashed curve). The Bell-violation of a pure entangled coherent state, i.e., V = 1, has been plotted for comparison (dotted curve). The Bell-violation B approaches its maximum bound, 2 √ 2, when d ≫ √ V regardless of the level of the mixedness V . (b) The optimized Bell-violation B against d for the different type of thermal-state entanglement generated using a 50:50 beam splitter from ρ + . V = 1000 (solid curve), V = 100 (dashed curve) and V = 1 (dotted curve).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. Entanglement using two initial thermal states\",\n \"text\": \"If the microscopic superposition interacts with two thermal states, ρ th b (V, d) and ρ th c (V, d), and the micro-scopic particle is measured out on the superposed basis, the resulting state will be\\n\\nρ tm(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (17)\\n\\nwhere\\n\\nN t = 2 1 ± exp[-4d 2 V ] V 2 . ( 18\\n\\n)\\n\\nSuch two-mode thermal-state entanglement can be generated using two cavities and an atomic state detector [20] .\\n\\nExtending the two cavities to N cavities, entanglement of N -mode thermal states can also be generated. Such a state is an analogy of the N -mode pure GHZ state [21] but each mode is extremely mixed. Here we shall consider the Bell-CHSH inequality [22, 23] with photon number parity measurements [20, 24] . The parity measurements can be performed in a high-Q cavity using a far-off-resonant interaction between a two-level atom and the field [25] . The Bell-CHSH inequality can be represented in terms of the Winger function as [24 ]\\n\\n|B (±) | = π 2 4 |W tm(±) (α, β) + W tm(±) (α, β ′ ) + W tm(±) (α ′ , β) -W tm(±) (α ′ , β ′ )| ≤ 2, (19)\\n\\nwhere W tm(±) (α, β) is the Wigner function of ρ tm(±) in Eq. ( 17 ). As shown in Fig. 5 , the Bell-violation approaches the maximum bound for a bipartite measurement, 2 √ 2 [13] , when d ≫ √\\n\\nV regardless of the level of the mixedness V , i.e., the temperatures of the thermal states. Note that it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(a) . This implies that entanglement of nearly 1 ebit has been produced between the two significantly mixed thermal states for d ≫ √ V , and such \\\"thermal-state entanglement\\\" cannot be described by a local theory.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Entanglement using a beam splitter\",\n \"text\": \"A different type of macroscopic entanglement can be generated by applying the beam splitter operation\\n\\nexp[θ/2(e iφ â † s âd -e -iφ â † d âs )], (20)\\n\\non the \\\"thermal-state superpositions\\\" in Eq. ( 9 ). The state after passing through a 50:50 beam splitter can be represented as\\n\\nN d 2 αP th α (V, d) | α √ 2 , - α √ 2 ± | - α √ 2 , α √ 2 α √ 2 , - α √ 2 | ± - α √ 2 , α √ 2 | , (21\\n\\n) 200 400 600 800 1000 V 2.1 2 0 2.2 2.3 B FIG. 6: The optimized Bell-violation B against V for the slightly different type of thermal-state entanglement generated using a 50:50 beam splitter using ρ + when d = 0. 3.13 3.14 3.15 3.16 ϕ 1.6 2.2 2 2.4 2.6 2.8 B (a) 3.13 3.14 3.15 3.16 ϕ 1.6 2 2.2 2.4 2.6 2.8 B (b) where N is defined in Eq. ( 13 ). When d is large, this state violates the Bell-CHSH inequality to the maximum bound 2 √ 2 regardless of the level of mixedness V as shown in Fig. 5(b ). Again, it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(b ). Furthermore, these states severely violate Bell's inequality even when d = 0 as V increases as shown in Fig. 6 . We have found that the optimized Bell violation of these states approaches 2.32449 for V → ∞. Interestingly, this value is exactly the same as the optimized Bell-CHSH violation for a pure two-mode squeezed state in the infinite squeezing limit [26] . Note that multilmode entangled states can be generated using multiple beam splitters.\\n\\nIt should be noted that the Bell violations are more sensitive to the interaction time when either V or d is larger. Figure 7 clearly shows this tendency. Therefore, in order to observe the Bell violations using the mixed state of V (and d) large, the interaction time in the Kerr medium should be more accurate.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. QUANTUM INFORMATION PROCESSING WITH THERMAL-STATE QUBITS\",\n \"text\": \"In this section, we discuss the possibility of quantum information processing with thermal-state qubits and thermal-state entanglement.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. Qubits and Bell-state measurements\",\n \"text\": \"We introduce a thermal-state qubit\\n\\nρ ψ = |a| 2 ρ th (V, d) ± ab * σ(V, d) ± a * bσ(V, -d) + |b| 2 ρ th (V, -d), (22)\\n\\nwhere a and b are arbitrary complex numbers. The basis states, ρ th (V, d) and ρ th (V, -d), can be well discriminated by a homodyne measurement when d is larger than V . The thermal state qubit (22) can be re-written as\\n\\nρ ψ = d 2 αP th α (V, d) a|α + b| -α a * α| + b * -α| , (23)\\n\\nwhich can be understood as a generalization of the coherent state qubit, a|d + b|d , where |d is a coherent state of amplitude d. The thermal-state qubit (23) becomes identical to the coherent-state qubit when V = 1.\\n\\nWe also define four thermal-Bell states as\\n\\nρ Φ(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (24) ρ Ψ(±) = N t ρ th (V, d) ⊗ ρ th (V, -d) ± σ(V, d) ⊗ σ(V, -d) ± σ(V, -d) ⊗ σ(V, d) + ρ th (V, -d) ⊗ ρ th (V, d) (25)\\n\\nwhere N t was defined in Eq. ( 18 ). The thermal-Bell states can be written as For quantum information processing applications, it is an important task to discriminate between the four Bell states. Here we discuss two possible ways to discriminate between the thermal-Bell states (25) . We shall only briefly describe the first scheme using photon number resolving measurements and focus on the second scheme using nonlinear interactions. The first method is to simply use a 50-50 beam splitter and two photon number resolving detectors as shown in Fig. 8(a) . This scheme is basically the same as the Bellstate measurement scheme with pure entangled coherent states [27, 28] . Let us suppose that the amplitude, d, is large enough, i.e., d ≫ √ V . If the incident state was ρ Φ(+) or ρ Φ(-) , most of the photons are detected on detector A in in Fig. 8(a) . Meanwhile, most of the photons are detected on detector B when the incident state was ρ Ψ(+) or ρ Ψ(-) . The average photon numbers between the \\\"many-photon case\\\" and the \\\"few-photon case\\\" are compared in Fig. 9 . Furthermore, the states ρ Ψ(+) and ρ Φ(+) contain only even numbers of photons while ρ Ψ(-) and ρ Φ(-) contain only odd numbers of photons. Therefore, all the four Bell states can be well discriminated by analyzing numbers of photons detected at detectors A and B. For example, if detector A detects many photons while detector B detects few and the total photon number detected by the two detectors are even, this means that state ρ Φ(+) was measured by the thermal-Bell measurement. The nonzero failure probability can be made arbitrarily small by increasing d.\\n\\nρ Φ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, β ± | -α, -β α, β| ± -α, -β| , ( 26\\n\\n)\\n\\nρ Ψ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, -β ± | -α, β α, -β| ± -α, β| . (27)\\n\\nHowever, the average photon numbers of the thermal-Bell states are high when V ≫ 1 and d ≫ 1. In this case, it would be unrealistic to use photon number resolving detectors. It would be an interesting question whether\\n\\n0.5 1 1.5 2 2.5 3 d 10 0 0 20 30 N (a) 0.5 1 1.5 2 2.5 3 d 0 0 10 20 30 N (b)\\n\\nFIG. 9: The average photon number N for the \\\"many-photon case\\\" (solid line) and the \\\"few-photon case\\\" (dashed line) for V = 10 against d (a) when the input state is either ρ Φ(+) or ρ Ψ(+) and (b) when the input state is either ρ Φ(-) or ρ Ψ(-) .\\n\\nthese four thermal-Bell states can be distinguished by classical measurements, such as homodyne detection, instead of photon number resolving detection. Our alternative scheme employs cross-Kerr nonlinearities and single photon detectors as shown in Fig. 8(b ). Let us first suppose that the input field was ρ Φ(+) . The incident twomode state passes through a 50-50 beam splitter, BS1. The state after passing through the 50:50 beam splitter, BS1, is\\n\\nρ B = N t d 2 αd 2 βP th α (V, d)P th β (V, d) |η, -ξ η, -ξ| + |η, -ξ -η, ξ| + | -η, ξ η, -ξ| + | -η, ξ -η, ξ| (28)\\n\\nwhere η = (α+β)/ √ 2 and ξ = (α-β)/ √ 2. Two dual-rail single photon qubits, |ψ + ee ′ and |ψ + f f ′ , where\\n\\n|ψ + = 1 √ 2 (|0 |1 + |1 |0 ), (29)\\n\\nare prepared using two single photons and 50:50 beam splitters, BS2 and BS3, as shown in Fig. 8(b ). Then, traveling fields at modes c and d interacts with those of modes e and f , respectively, in cross-Kerr nonlinear media. We suppose that the interaction time is t = π/λ, and the resulting state is then\\n\\nρ B ′ = U ce U df ρ B cd ρ q ee ′ ρ q f f ′ U † ce U † df (30)\\n\\nwhere U ce = exp[iπH K ce /λ ] and ρ q = |ψ q ψ q |. An explicit form of Eq. ( 30 ) can then be simply obtained using the identity\\n\\nU ce |α c |0 e = |α c |0 e , U ce |α c |1 e = | -α c |1 e ( 31\\n\\n)\\n\\nwhere |α is a coherent state. However, we omit such an explicit expression in this paper for it is too lengthy. After the nonlinear interactions, the qubit parts, modes e, e ′ , f and f ′ , should be measured with the measurement basis\\n\\n{| + + , | + -, | -+ , | --} (32)\\n\\nwhere\\n\\n| + + = |ψ + ee ′ |ψ + f f ′ , | + -= |ψ + ee ′ |ψ -f f ′ , | -+ = |ψ -ee ′ |ψ + f f ′ , | --= |ψ -ee ′ |ψ -f f ′ , and |ψ -= (|0 |1 -|1 |0 )/ √ 2.\\n\\nThis measurement can be performed using two 50:50 beam splitters, BS4 and BS5, and four detectors, A1, A2, B1 and B2, as shown in Fig. 8(b ). If detector A1 and B1 click, i.e., the measurement result is | + + , the resulting state at modes c and d is\\n\\nρ ++ = N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η + | -η )( η| + -η|) c ⊗ (|ξ + | -ξ )( ξ| + -ξ|) d (33)\\n\\nNote that state ρ ++ is not normalized, which implies that the probability of obtaining the corresponding measurement result is not unity. The probability of obtaining this result is\\n\\nP ++ = (V + 1)(V + e -4d 2 V ) 2(V 2 + e -4d 2 V ) . (34)\\n\\nWhen the result is either | +or | -+ , the result is\\n\\nψ 2 |ρ B ′ |ψ 2 = ψ 3 |ρ B ′ |ψ 3 = 0, (35)\\n\\nwhich obviously means that the probability of the obtaining this result is zero. When the result is | --, i.e., detector A2 and B2 click,\\n\\nρ --= N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η -| -η )( η| --η|) c ⊗ (|ξ -| -ξ )( ξ| --ξ|) d , (36)\\n\\nwhich is not normalized. The probability of obtaining this result is\\n\\nP --= (V -1)(V -e -4d 2 V ) 2(V 2 + e -4d 2 V ) , (37)\\n\\nand it can be simply verified that P ++ +P --= 1. Therefore, only the measurement results | + + and | -can be obtained in the case of the input state ρ Φ(+) . This is exactly the same for the case of ρ Ψ(+) . In the same way, it can be shown that if either the input state was ρ Φ(-) or ρ Ψ(-) , only the measurement results | +and | -+ can be obtained. In other words, the parity of the toincoming state is perfectly well discriminated by the measurements on single-photon qubits.\\n\\nSubsequently, a homodyne measurement is performed for mode c by homodyne detector C as shown in Fig. 8(b) . We assume that ideal homodyne measurements are performed, i.e., when a homodyne measurement is performed the state is projected onto eigenstate |x of operator X with eigenvalue x, where\\n\\nX = 1 √ 2 (a + a † ). ( 38\\n\\n)\\n\\nLet us first consider the case when the measurement result for the single photon qubits is | + + . In this case, the remaining state is ρ ++ in Eq. ( 33 ). The probability distribution P ++ Φ (+) for the homodyne measurement at detector C is\\n\\nP ++ Φ (+) = x|Tr d [ρ ++ ]|x = V 1 2 (e -V x 2 + e -x 2 V ) π 1 2 (V + 1)\\n\\n. (39) Note that the superscript, ++, denotes that the qubit measurement result was | + + , and the subscript, Φ (+) , denotes that the input state was ρ Φ (+) . These notations will be used also for the other cases in this section. The same analysis can be performed for the other possible measurement outcome | --:\\n\\nP -- Φ (+) = x|Tr d [ρ --]|x = V 1 2 (e -V x 2 -e -x 2 V ) π 1 2 (V -1)\\n\\n. (40)\\n\\nIn the same way, for another input state, ρ Φ(-) , it is straightforward to show:\\n\\nP +- Φ (-) = P ++ Φ (+) , P -+ Φ (-) = P -- Φ (+) , (41)\\n\\nand P ++ Φ (-) = P -- Φ (-) = 0. On the other hand, if the input state was ρ Ψ(+) , the probability distributions P ++ Ψ (+) and P -- Ψ (+) at detector C are It is straightforward to show for the other input state ρ Ψ(-) :\\n\\nP ++ Ψ (+) = x|Tr c [ρ ++ ]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V + 2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) + 1 , (42)\\n\\nP -- Ψ (+) = x|Tr c [ρ --]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V -2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) -1 . ( 43\\n\\nP +- Ψ (-) = P -- Ψ (+) , P -+ Ψ (-) = P ++ Ψ (+) . (44)\\n\\nThe probability distributions P ++ Φ (±) and P ++ Ψ (±) are plotted in Fig. 10 . Figure 10 shows that when the input state was ρ Φ(+) or ρ Φ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Φ (+) and P -- Φ (+) , is located around the origin. However, when the input state was ρ Ψ(+) or ρ Ψ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Ψ (+) and P -- Ψ (+) , is located far from the origin. Therefore, two of the Bell states, ρ Φ(+) or ρ Φ(-) , can be well distinguished from the other two by the homodyne detector C for the case of the measurement outcome | + + . Finally, by combining the homodyne measurement result and the qubit measurement result, all four Bell states can be effectively distinguished. For example, let us assume that the measurement outcome of the single photon detectors was | + + and the homodyne detection outcome was around the origin, i.e., x ≈ 0. Then, one can say that state ρ Ψ(-) has been measured for the result of the thermal-Bell measurement.\\n\\nAs implied in Fig. 10 , the overlaps between the probability distributions around the origin, P ++ Φ (+) and P -- Φ (+) , and the other distributions, P ++ Ψ (+) and P -- Ψ (+) , are extremely small for a sufficiently large d. In other words, the distinguishability by the homodyne detection rapidly approaches 1 as d increases. As an example, we can calculate the distinguishability between the states ρ Ψ(+) and ρ Φ(+) by the homodyne measurement by detector C. The distinguishability by homodyne detection is Note also that the second scheme using homodyne detection is robust to detection inefficiency compared with the first scheme using photon number resolving measurements. In the first scheme, even if a detector misses only one photon, it will result in a completely wrong measurement outcome. In the second scheme, however, the measurement outcome will not be affected in that way. If a single photon detector misses a photon, it will be immediately recognized. Such a case can simply be discarded so that it will only degrade the success probability of the Bell measurement. The homodyne detection inefficiency will not significantly affect the result when the distributions around the origin and the distributions far from the origin are well separated, i.e., when d ≫ √ V , as shown in Fig. 10 . On the other hand, loss in the Kerr medium will have a detrimental affect.\\n\\nP s = 1 2 |x|<d dxP ++ c (x) + |x|≥d dxP ++ d (x) (45\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Quantum teleportation and computation\",\n \"text\": \"Quantum teleportation of a thermal-state qubit can be performed using one of the Bell states as the quantum channel. Let us assume that Alice needs to teleport a thermal-state qubit, ρ ψ , to Bob using a thermal-state entanglement, ρ Ψ(-) , shared by the two parties. The total state can be represented as\\n\\nAlice first needs to perform the thermal-Bell measurement described in the previous subsection. To complete the teleportation process, Bob should perform an appropriate unitary transformation on his part of the quantum channel according to the measurement result sent from Alice via a classical channel. It is straightforward to show that the required transformations are exactly the same to those for the coherent-state qubit [27] . When the measurement outcome is ρ Ψ(-) , Bob obtains a perfect replica of the original unknown qubit without any operation.\\n\\nWhen the measurement outcome is ρ Φ(-) , Bob should perform |α ↔ |α on his qubit in Eq. ( 23 ). Such a phase shift by π can be done using a phase shifter whose action is described by P (ϕ) = e iϕa † a , where a and a † are the annihilation and creation operators. When the outcome is ρ Ψ(+) , the transformation should be performed as |α → |α and |α → -|α . It is known that the displacement operator is a good approximation of this transformation for d ≫ 1 [29] . This transformation can also be achieved by teleporting the state again locally and repeating until the required phase shift is obtained [30] . When the outcome is ρ Φ(+) , σ x and σ z should be successively applied.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"V. CONCLUSION\",\n \"text\": \"In this paper, we have studied characteristics of superpositions and entanglement of thermal states at high temperatures and discussed their applications to quantum information processing. The superpositions and entanglement of thermal states show various nonclassical properties such as interference patterns, negativity of the Wigner functions, and violations of the Bell-CHSH inequality. The Bell violations are more sensitive to the interaction time during the generation process when the thermal temperature (i.e. mixedness) of the thermalstate entanglement is larger. Therefore, in order to observe the Bell violations using the mixed state at a high temperature, the interaction time in the Kerr medium should be accurate. We have pointed out that certain superpositions of high-temperature thermal states, symmetric in the phase space, can also be generated. Some of these states have neither squeezing properties nor negative values in their Wigner functions but they are found to be highly nonclassical.\\n\\nWe have introduced the thermal-state qubit and thermal-Bell states for applications to quantum information processing. We have presented two possible methods for the Bell-state measurement. The Bell-state measurement enables one to perform quantum teleportation and gate operations for quantum computation with thermalstate qubits. The first scheme uses two photon number resolving detectors and a 50-50 beam splitter to discriminate the thermal-Bell states. Using the second scheme, it is possible to effectively discriminate the thermal-Bell states without photon number resolving detection. The required resources for the second scheme are two Kerr nonlinear interactions, two single photon detectors, two 50:50 beam splitters and one homodyne detector. The second scheme is more robust to inefficiency of the detectors: the inefficiency of the single photon detectors only degrades the success probability of the Bell measurement.\"\n }\n]"}}},{"rowIdx":96,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0526"},"arxiv_id":{"kind":"string","value":"0704.0526"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":""},"content_hash":{"kind":"string","value":"a1a26ce33595a638cf1c21c15b33b4f6450ff8e123099f4f85a016d59c12c65c"},"title":{"kind":"string","value":"Fractional WKB Approximation"},"abstract":{"kind":"string","value":"Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is\ninvestigated in this paper using the fractional calculus. In the fractional\ncase the wave function is constructed such that the phase factor is the same as\nthe Hamilton's principle function \"S\". To demonstrate our proposed approach two\nexamples are investigated in details."},"authors":{"kind":"list like","value":["Eqab M. Rabei","Ibrahim M. A. Altarazi","Sami I. Muslih","Dumitru Baleanu"],"string":"[\n \"Eqab M. Rabei\",\n \"Ibrahim M. A. Altarazi\",\n \"Sami I. Muslih\",\n \"Dumitru Baleanu\"\n]"},"arxiv_categories":{"kind":"list like","value":["math-ph","hep-th","math.MP"],"string":"[\n \"math-ph\",\n \"hep-th\",\n \"math.MP\"\n]"},"primary_category":{"kind":"string","value":"math-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function \"S\". To demonstrate our proposed approach two examples are investigated in details.\n\nFractional calculus is a branch of mathematics that deal with a generalization of well-known operations of differentiations and integrations to arbitrary non-integer order, which can be real non-integer or even imaginary number.\n\nNowadays physicists have used this powerful tool to deal with some problems which were not solvable in the classical sense. Therefore, the fractional calculus became one of the most powerful and widely useful tools in describing and explaining some physical complex systems.\n\nRecently, the Euler-Lagrange equations has been presented for unconstrained and constrained fractional variational problems [1 and other references] . This technique enable us to solve some problems including describing the behavior of non-conservative systems developed by Riewe [2] , where he used the fractional derivative to construct the Lagrangian and Hamiltonian for non-conservative systems.\n\nFrom these reasons in [3] was developed a general formula for the potential of any arbitrary force conservative or not conservative, which leads directly to the consideration of dissipative effect in Lagrangian and Hamiltonian formulation. Also, the canonical quantization of non-conservative systems has been carried out in [4] .\n\nStarting from a Lagrangian containing a fractional derivative, the fractional Hamiltonian is achieved in [5] . In addition, the passage from Hamiltonian containing fractional derivatives to the fractional Hamilton-Jaccobi is achieved by Rabei et.al [6] . The equations of motion are obtained in a similar manner to the usual mechanics.\n\nAll these outstanding results using the fractional derivative make us concentrate on another branch of quntam physics. WKB approximation [7, 8, 9, 10, 14] . In this paper we are mainly interested to construct the solution of Schroödinger equation in an exponential form (Griffith 1995) starting from fractional Hamilton-Jaccobi equation and how it leads naturally to this semi-classical approximation namely fractional WKB.\n\nThe purpose of this paper is to find the solution of Schrödinger equation for some systems that have a fractional behavior in their Lagrangians and obey the WKB approximation assumptions.\n\nThe plan of this paper is as follows: In section II the derivation of generalized Hamilton-Jaccobi partial differential equation which given in [6] is briefly reviewed. In section III the fractional WKB approximation is derived. In Section IV some examples with the fractional WKB technique is reported. Section V is dedicated to conclusions.\n\nThe left and right Reimann-Loville fractional derivative are defined as follows [3] The left Riemann-Liouville fractional derivative is given by\n\n( ) τ τ τ α α α d f x dx d n x f D x a n n x a ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - Γ = ) ( ) ( 1 ) ( 1 (1)\n\nThe right Riemann-Liouville fractional derivative has the form\n\n( ) τ τ τ α β β d f x dx d n x f D b x n n b x ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - Γ = ) ( ) ( 1 ) ( 1 (2)\n\nHere α, β are the order of derivation such that n-1≤α <n, n-1≤β<n, and they are not zero.\n\nIf α is an integer, these derivatives are defined in usual sense as\n\n) ( ) ( x f dx d x f D x a α α ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = (3-a) ) ( ) ( x f dx d x f D b x β β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = (3-b)\n\nHamilton formalism with fractional derivative was proposed in [5]\n\nnamely ) , , , ( ) , , , ( t q D q D q L q D p q D p t p p q H b t t a b t t a β α β β α α β α - + = , ( 4\n\n) where L represents the fractional Lagrangian obtained by replacing the classical derivatives with the corresponding fractional ones [5] .\n\nHamilton's equations of motion are obtained as follows [5]\n\nt L t H ∂ ∂ - = ∂ ∂ ; ; q D p H t a α α = ∂ ∂ ; q D p H b t β β = ∂ ∂ α α β β p D p D q H b t t a + = ∂ ∂ (5)\n\nIn [6] based on the sequential derivatives the fractional Hamilton-Jacobi partial differential equation is obtained. The Hamilton-Jacobi function in configuration space is written in a similar manner to the usual mechanics by using the Reimann-Loville fractional derivative. In [6] the following generating function is used, where α and β are bigger or equal to 1. Thus, the new Hamiltonian is expressed as\n\nS t P P q D q D F F b t t a = = - - ) , , , , ( 1 1 2 β α β α ) , , , ( ' ) , , , ( t Q D Q D Q L Q D P Q D P t P P Q K b t t a b t t a β α β β α α β α - + = (7)\n\nIt is concluded that, the following relation relates the two Hamiltonians\n\ndt dF K Q D P Q D P H q D p q D p b t t a b t t a + - + = - + β β α α β β α α (8)\n\nAccording to reference [6] the function F is proposed as ) , , , , (\n\n1 1 t P P q D q D S F b t t a β α β α - - = Q D P Q D P B t t a 1 1 - - - - β β α α , (9)\n\nThe function S is called Hamilton's principle function.\n\nTherefore, requiring that the transformed Hamiltonian K shall be zero the Hamilton-Jacobi equation is satisfied. In other words Q, P α , P β are constants.\n\n0 = ∂ ∂ + t S H (10)\n\nSince Q, P α , P β are constants, The Hamilton's principle function is written as (11) where\n\n) , , , , ( 2 1 1 1 t E E q D q D S S b t t a - - = β α\n\n1 E P = α 2 E P = β\n\nIf the Hamiltonian is explicitly independent of time, then S can be written as follows (12) ) , , ( ) , ( ) , (\n\n2 1 2 1 2 1 1 1 t E E f E q D W E q D W S b t t a + + = - - β α\n\nW represents the Hamilton's characteristic function; therefore, the following equations of motion are obtained in [6] as:\n\nq D W P t a 1 1 - ∂ ∂ = α α q D W P b t 1 1 - ∂ ∂ = β β (13) 1 1 1 1 λ α = ∂ ∂ = - E W Q D t a 2 2 2 1 λ β = ∂ ∂ = - E W Q D b t (14)\n\nHere λ 1 ,λ 2 are constants.\n\nThe outstanding result regarding the meaning of the state function ψ and its relationship to Hamilton's principle function S enables us to write the exponential solution of Schrödinger equation [13] .\n\n⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = h ) , ( exp ) , ( t q S i t q ψ (15)\n\nThe phase of state function obeys the same mathematical equation, as does Hamilton's principle function S. The physical significance of S in classical mechanics is that it represents the generator of trajectories [12] for fractional systems; the fractional Hamilton's principle function is become the phase of the state function ψ. One can write the solution of Schrödinger equation under the postulated constrains by the WKB approximation and using the fractional Hamilton's principle function eq (12) . Thus we propose the fractional state function as:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t q D q D S i t q D q D b t t a b t t a , , exp ) , , ( 1 1 1 1 β α β α ψ h (16)\n\nFrom the quantization using WKB approximation [7, 8, 9, 10, 14] a general solution of Schrödinger equation is obtained using the expansion for S and then using the transformation to the N-dimensional system as:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ∏ = h t q S i q i N i i o i , exp ) ( 1 ψ ψ (17)\n\nwhere ( )\n\n) ( 1 i i io q p q = ψ (18)\n\nIn our case, S behaves like a 2-dimensional problem with two distinct momenta. Thus,\n\n(19) q D q t a 1 1 - ≡ α α P P 1 ≡ (20) q D q b t 1 2 - ≡ β β P P 2 ≡\n\nAnd the momenta are defined as operators. Therefore, we can propose the wave function ψ of the fractional system in the following form\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t E E q D q D S i P P t q D q D b t t a b t t a , , , , exp 1 ) , , ( 2 1 1 1 1 1 β α β α β α ψ h (21)\n\nand the momenta operators in the form\n\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 , ˆβ β α α h h (22)\n\nWe conclude that (21) is the solution of Schrödinger equation for any given fractional systems. If α and β both are equal to unity, then we will return to the usual classical solution of Schrödinger equation, also we can notice how the probability is inversely proportional to the momentum . ) (\n\n1 2 q p ≅ ψ IV. Examples\n\nAs a first model let us consider the following fractional Lagrangian,\n\n( ) ( ) 2 1 2 0 2 1 2 1 q D q D L t t β α + = (23)\n\nThe fractional Hamilton-Jacobi equation for this fractional Lagrangian can be calculated as:\n\n( ) ( ) . 0 2 1 2 1 2 2 = ∂ ∂ + + t S P P β α (24) where q D L P t α α 0 ∂ ∂ = ; q D L P t β β 1 ∂ ∂ =\n\nMaking use of equation ( 13 ), the fractional Hamilton-Jacobi equation (24) becomes:\n\n0 2 1 2 1 2 1 1 2 2 1 0 1 = ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - t S q D W q D W t t β α (25) Taking into account t S H ∂ ∂ - = (26)\n\nIf we apply (26) on a wave function it gives:\n\n) ( 2 1 E E E t S + - ≡ - = ∂ ∂ (27)\n\nBy using the fact that E is the total energy of the system and taking into account (27) we obtain 0 2\n\n1 2 1 2 2 1 1 2 1 2 1 0 1 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - E q D W E q D W t t β α (28)\n\nThus, both sides of (28) should be zero, and we obtain\n\nq D E W q D E W t t 1 1 2 2 1 0 1 1 2 , 2 - - = = β α\n\n(29) By using ( 12 ) and ( 21 ) we obtain It's the same as the classical solution. Also, when applying the energy operator it gives the energy eigenvalues:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + = - - - - t E q D E q D E i P P t q D q D t t t t 1 1 2 1 0 1 1 1 1 0 2 2 exp 1 ) , , (\n\n( ) ( )ψ ψ ψ β α 2 2 2 1 2 1 P P H + = ( ) ψ 2 1 E E + = (34)\n\nas in the classical case.\n\nAs a second example let us consider the following fractional Lagrangian ( ) ( )\n\n2 1 0 2 1 2 0 2 1 2 1 2 1 q q D q D q D q D L t t t t + + + + = β α β α (35)\n\nThe corresponding fractional Hamilton is calculated as follows\n\n( ) ( ) 2 2 2 2 1 1 2 1 1 2 1 q P P H - - + - = β α (36)\n\nThus, the fractional Hamilton-Jacobi equation becomes ( ) ( )\n\n0 2 1 1 2 1 1 2 1 2 2 2 = ∂ ∂ + - - + - t S q P P β α (37)\n\nThe fractional Hamilton's principle function is calculated as,\n\n) ( )\n\nt E E q D E q D E q S t t ) ( 1 2 1 2 2 1 1 1 2 1 0 1 2 + - + + + + = - - β α (38)\n\nAs a result the wave function can be written in the form ( )\n\n( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + + + + = - - - - t E q D E q D E q i P P t q D q D t t t t 1 1 2 1 0 1 2 1 1 1 0 1 2 1 2 exp 1 ) , , ( β α β α β α ψ h (39)\n\nTo identify the influence of the operators let us test the effect of the momenta\n\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 ˆβ β α α h h (40)\n\nUsing the characteristic equations, it can be shown that\n\n1 2 , 1 2 ˆ2 1 2 + = + + = E P E q P β α (41) The result shown in (41 ) is the same classical solution. When applying the energy operator it will give the energy eigenvalues ( ) ( ) ψ ψ ψ ψ β α 2 2 2 2 1 1 2 1 1 2 1 q P P H --+ -= (42) ( ) ( ) ψ ψ ψ ψ β α β α 2 2 2 2 1 2 1 q P P P P -+ + -+ = ( ) ( ) ψ } 2 1 1 2 2 2 2 2 2 2 2 2 2 1 { 2 2 1 2 2 1 2 2 q E E q E E q E q -+ + + + -+ + + + + = Then we get\n\nψ ψ E H = (43)\n\nwhich is exactly the total energy as the case for the classical systems.\n\nWe use the generating function \"S\" of the Hamilton-Jaccobi equation in its fractional form to be the phase factor of the wave function describing some potentials valid for the assumptions suggested by the WKB approximation The proof of our results arises from the new proposed concepts of the momentum and energy operators, that they give the same eigenvalues producing the ordinary results achieved by the classical approach.\n\nGiving the same eigenvalues that means this form of fractional operator also eigen, valid, and useful in effecting on a state functions."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function \"S\". To demonstrate our proposed approach two examples are investigated in details."},{"section_type":"BACKGROUND","section_title":"I. Introduction","text":"Fractional calculus is a branch of mathematics that deal with a generalization of well-known operations of differentiations and integrations to arbitrary non-integer order, which can be real non-integer or even imaginary number.\n\nNowadays physicists have used this powerful tool to deal with some problems which were not solvable in the classical sense. Therefore, the fractional calculus became one of the most powerful and widely useful tools in describing and explaining some physical complex systems.\n\nRecently, the Euler-Lagrange equations has been presented for unconstrained and constrained fractional variational problems [1 and other references] . This technique enable us to solve some problems including describing the behavior of non-conservative systems developed by Riewe [2] , where he used the fractional derivative to construct the Lagrangian and Hamiltonian for non-conservative systems.\n\nFrom these reasons in [3] was developed a general formula for the potential of any arbitrary force conservative or not conservative, which leads directly to the consideration of dissipative effect in Lagrangian and Hamiltonian formulation. Also, the canonical quantization of non-conservative systems has been carried out in [4] .\n\nStarting from a Lagrangian containing a fractional derivative, the fractional Hamiltonian is achieved in [5] . In addition, the passage from Hamiltonian containing fractional derivatives to the fractional Hamilton-Jaccobi is achieved by Rabei et.al [6] . The equations of motion are obtained in a similar manner to the usual mechanics.\n\nAll these outstanding results using the fractional derivative make us concentrate on another branch of quntam physics. WKB approximation [7, 8, 9, 10, 14] . In this paper we are mainly interested to construct the solution of Schroödinger equation in an exponential form (Griffith 1995) starting from fractional Hamilton-Jaccobi equation and how it leads naturally to this semi-classical approximation namely fractional WKB.\n\nThe purpose of this paper is to find the solution of Schrödinger equation for some systems that have a fractional behavior in their Lagrangians and obey the WKB approximation assumptions.\n\nThe plan of this paper is as follows: In section II the derivation of generalized Hamilton-Jaccobi partial differential equation which given in [6] is briefly reviewed. In section III the fractional WKB approximation is derived. In Section IV some examples with the fractional WKB technique is reported. Section V is dedicated to conclusions."},{"section_type":"OTHER","section_title":"II. Basic Tools","text":"The left and right Reimann-Loville fractional derivative are defined as follows [3] The left Riemann-Liouville fractional derivative is given by\n\n( ) τ τ τ α α α d f x dx d n x f D x a n n x a ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - Γ = ) ( ) ( 1 ) ( 1 (1)\n\nThe right Riemann-Liouville fractional derivative has the form\n\n( ) τ τ τ α β β d f x dx d n x f D b x n n b x ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - Γ = ) ( ) ( 1 ) ( 1 (2)\n\nHere α, β are the order of derivation such that n-1≤α <n, n-1≤β<n, and they are not zero.\n\nIf α is an integer, these derivatives are defined in usual sense as\n\n) ( ) ( x f dx d x f D x a α α ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = (3-a) ) ( ) ( x f dx d x f D b x β β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = (3-b)\n\nHamilton formalism with fractional derivative was proposed in [5]\n\nnamely ) , , , ( ) , , , ( t q D q D q L q D p q D p t p p q H b t t a b t t a β α β β α α β α - + = , ( 4\n\n) where L represents the fractional Lagrangian obtained by replacing the classical derivatives with the corresponding fractional ones [5] .\n\nHamilton's equations of motion are obtained as follows [5]\n\nt L t H ∂ ∂ - = ∂ ∂ ; ; q D p H t a α α = ∂ ∂ ; q D p H b t β β = ∂ ∂ α α β β p D p D q H b t t a + = ∂ ∂ (5)\n\nIn [6] based on the sequential derivatives the fractional Hamilton-Jacobi partial differential equation is obtained. The Hamilton-Jacobi function in configuration space is written in a similar manner to the usual mechanics by using the Reimann-Loville fractional derivative. In [6] the following generating function is used, where α and β are bigger or equal to 1. Thus, the new Hamiltonian is expressed as\n\nS t P P q D q D F F b t t a = = - - ) , , , , ( 1 1 2 β α β α ) , , , ( ' ) , , , ( t Q D Q D Q L Q D P Q D P t P P Q K b t t a b t t a β α β β α α β α - + = (7)\n\nIt is concluded that, the following relation relates the two Hamiltonians\n\ndt dF K Q D P Q D P H q D p q D p b t t a b t t a + - + = - + β β α α β β α α (8)\n\nAccording to reference [6] the function F is proposed as ) , , , , (\n\n1 1 t P P q D q D S F b t t a β α β α - - = Q D P Q D P B t t a 1 1 - - - - β β α α , (9)\n\nThe function S is called Hamilton's principle function.\n\nTherefore, requiring that the transformed Hamiltonian K shall be zero the Hamilton-Jacobi equation is satisfied. In other words Q, P α , P β are constants.\n\n0 = ∂ ∂ + t S H (10)\n\nSince Q, P α , P β are constants, The Hamilton's principle function is written as (11) where\n\n) , , , , ( 2 1 1 1 t E E q D q D S S b t t a - - = β α\n\n1 E P = α 2 E P = β\n\nIf the Hamiltonian is explicitly independent of time, then S can be written as follows (12) ) , , ( ) , ( ) , (\n\n2 1 2 1 2 1 1 1 t E E f E q D W E q D W S b t t a + + = - - β α\n\nW represents the Hamilton's characteristic function; therefore, the following equations of motion are obtained in [6] as:\n\nq D W P t a 1 1 - ∂ ∂ = α α q D W P b t 1 1 - ∂ ∂ = β β (13) 1 1 1 1 λ α = ∂ ∂ = - E W Q D t a 2 2 2 1 λ β = ∂ ∂ = - E W Q D b t (14)\n\nHere λ 1 ,λ 2 are constants."},{"section_type":"OTHER","section_title":"III. Fractional WKB approximation","text":"The outstanding result regarding the meaning of the state function ψ and its relationship to Hamilton's principle function S enables us to write the exponential solution of Schrödinger equation [13] .\n\n⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = h ) , ( exp ) , ( t q S i t q ψ (15)\n\nThe phase of state function obeys the same mathematical equation, as does Hamilton's principle function S. The physical significance of S in classical mechanics is that it represents the generator of trajectories [12] for fractional systems; the fractional Hamilton's principle function is become the phase of the state function ψ. One can write the solution of Schrödinger equation under the postulated constrains by the WKB approximation and using the fractional Hamilton's principle function eq (12) . Thus we propose the fractional state function as:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t q D q D S i t q D q D b t t a b t t a , , exp ) , , ( 1 1 1 1 β α β α ψ h (16)\n\nFrom the quantization using WKB approximation [7, 8, 9, 10, 14] a general solution of Schrödinger equation is obtained using the expansion for S and then using the transformation to the N-dimensional system as:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ∏ = h t q S i q i N i i o i , exp ) ( 1 ψ ψ (17)\n\nwhere ( )\n\n) ( 1 i i io q p q = ψ (18)\n\nIn our case, S behaves like a 2-dimensional problem with two distinct momenta. Thus,\n\n(19) q D q t a 1 1 - ≡ α α P P 1 ≡ (20) q D q b t 1 2 - ≡ β β P P 2 ≡\n\nAnd the momenta are defined as operators. Therefore, we can propose the wave function ψ of the fractional system in the following form\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t E E q D q D S i P P t q D q D b t t a b t t a , , , , exp 1 ) , , ( 2 1 1 1 1 1 β α β α β α ψ h (21)\n\nand the momenta operators in the form\n\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 , ˆβ β α α h h (22)\n\nWe conclude that (21) is the solution of Schrödinger equation for any given fractional systems. If α and β both are equal to unity, then we will return to the usual classical solution of Schrödinger equation, also we can notice how the probability is inversely proportional to the momentum . ) (\n\n1 2 q p ≅ ψ IV. Examples"},{"section_type":"OTHER","section_title":"IV. a) Example 1:","text":"As a first model let us consider the following fractional Lagrangian,\n\n( ) ( ) 2 1 2 0 2 1 2 1 q D q D L t t β α + = (23)\n\nThe fractional Hamilton-Jacobi equation for this fractional Lagrangian can be calculated as:\n\n( ) ( ) . 0 2 1 2 1 2 2 = ∂ ∂ + + t S P P β α (24) where q D L P t α α 0 ∂ ∂ = ; q D L P t β β 1 ∂ ∂ =\n\nMaking use of equation ( 13 ), the fractional Hamilton-Jacobi equation (24) becomes:\n\n0 2 1 2 1 2 1 1 2 2 1 0 1 = ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - t S q D W q D W t t β α (25) Taking into account t S H ∂ ∂ - = (26)\n\nIf we apply (26) on a wave function it gives:\n\n) ( 2 1 E E E t S + - ≡ - = ∂ ∂ (27)\n\nBy using the fact that E is the total energy of the system and taking into account (27) we obtain 0 2\n\n1 2 1 2 2 1 1 2 1 2 1 0 1 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - E q D W E q D W t t β α (28)\n\nThus, both sides of (28) should be zero, and we obtain\n\nq D E W q D E W t t 1 1 2 2 1 0 1 1 2 , 2 - - = = β α\n\n(29) By using ( 12 ) and ( 21 ) we obtain It's the same as the classical solution. Also, when applying the energy operator it gives the energy eigenvalues:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + = - - - - t E q D E q D E i P P t q D q D t t t t 1 1 2 1 0 1 1 1 1 0 2 2 exp 1 ) , , (\n\n( ) ( )ψ ψ ψ β α 2 2 2 1 2 1 P P H + = ( ) ψ 2 1 E E + = (34)\n\nas in the classical case."},{"section_type":"OTHER","section_title":"IVb.)Example 2:","text":"As a second example let us consider the following fractional Lagrangian ( ) ( )\n\n2 1 0 2 1 2 0 2 1 2 1 2 1 q q D q D q D q D L t t t t + + + + = β α β α (35)\n\nThe corresponding fractional Hamilton is calculated as follows\n\n( ) ( ) 2 2 2 2 1 1 2 1 1 2 1 q P P H - - + - = β α (36)\n\nThus, the fractional Hamilton-Jacobi equation becomes ( ) ( )\n\n0 2 1 1 2 1 1 2 1 2 2 2 = ∂ ∂ + - - + - t S q P P β α (37)\n\nThe fractional Hamilton's principle function is calculated as,\n\n) ( )\n\nt E E q D E q D E q S t t ) ( 1 2 1 2 2 1 1 1 2 1 0 1 2 + - + + + + = - - β α (38)\n\nAs a result the wave function can be written in the form ( )\n\n( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + + + + = - - - - t E q D E q D E q i P P t q D q D t t t t 1 1 2 1 0 1 2 1 1 1 0 1 2 1 2 exp 1 ) , , ( β α β α β α ψ h (39)\n\nTo identify the influence of the operators let us test the effect of the momenta\n\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 ˆβ β α α h h (40)\n\nUsing the characteristic equations, it can be shown that\n\n1 2 , 1 2 ˆ2 1 2 + = + + = E P E q P β α (41) The result shown in (41 ) is the same classical solution. When applying the energy operator it will give the energy eigenvalues ( ) ( ) ψ ψ ψ ψ β α 2 2 2 2 1 1 2 1 1 2 1 q P P H --+ -= (42) ( ) ( ) ψ ψ ψ ψ β α β α 2 2 2 2 1 2 1 q P P P P -+ + -+ = ( ) ( ) ψ } 2 1 1 2 2 2 2 2 2 2 2 2 2 1 { 2 2 1 2 2 1 2 2 q E E q E E q E q -+ + + + -+ + + + + = Then we get\n\nψ ψ E H = (43)\n\nwhich is exactly the total energy as the case for the classical systems."},{"section_type":"CONCLUSION","section_title":"V. Conclusions","text":"We use the generating function \"S\" of the Hamilton-Jaccobi equation in its fractional form to be the phase factor of the wave function describing some potentials valid for the assumptions suggested by the WKB approximation The proof of our results arises from the new proposed concepts of the momentum and energy operators, that they give the same eigenvalues producing the ordinary results achieved by the classical approach.\n\nGiving the same eigenvalues that means this form of fractional operator also eigen, valid, and useful in effecting on a state functions."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function \\\"S\\\". To demonstrate our proposed approach two examples are investigated in details.\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. Introduction\",\n \"text\": \"Fractional calculus is a branch of mathematics that deal with a generalization of well-known operations of differentiations and integrations to arbitrary non-integer order, which can be real non-integer or even imaginary number.\\n\\nNowadays physicists have used this powerful tool to deal with some problems which were not solvable in the classical sense. Therefore, the fractional calculus became one of the most powerful and widely useful tools in describing and explaining some physical complex systems.\\n\\nRecently, the Euler-Lagrange equations has been presented for unconstrained and constrained fractional variational problems [1 and other references] . This technique enable us to solve some problems including describing the behavior of non-conservative systems developed by Riewe [2] , where he used the fractional derivative to construct the Lagrangian and Hamiltonian for non-conservative systems.\\n\\nFrom these reasons in [3] was developed a general formula for the potential of any arbitrary force conservative or not conservative, which leads directly to the consideration of dissipative effect in Lagrangian and Hamiltonian formulation. Also, the canonical quantization of non-conservative systems has been carried out in [4] .\\n\\nStarting from a Lagrangian containing a fractional derivative, the fractional Hamiltonian is achieved in [5] . In addition, the passage from Hamiltonian containing fractional derivatives to the fractional Hamilton-Jaccobi is achieved by Rabei et.al [6] . The equations of motion are obtained in a similar manner to the usual mechanics.\\n\\nAll these outstanding results using the fractional derivative make us concentrate on another branch of quntam physics. WKB approximation [7, 8, 9, 10, 14] . In this paper we are mainly interested to construct the solution of Schroödinger equation in an exponential form (Griffith 1995) starting from fractional Hamilton-Jaccobi equation and how it leads naturally to this semi-classical approximation namely fractional WKB.\\n\\nThe purpose of this paper is to find the solution of Schrödinger equation for some systems that have a fractional behavior in their Lagrangians and obey the WKB approximation assumptions.\\n\\nThe plan of this paper is as follows: In section II the derivation of generalized Hamilton-Jaccobi partial differential equation which given in [6] is briefly reviewed. In section III the fractional WKB approximation is derived. In Section IV some examples with the fractional WKB technique is reported. Section V is dedicated to conclusions.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. Basic Tools\",\n \"text\": \"The left and right Reimann-Loville fractional derivative are defined as follows [3] The left Riemann-Liouville fractional derivative is given by\\n\\n( ) τ τ τ α α α d f x dx d n x f D x a n n x a ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - Γ = ) ( ) ( 1 ) ( 1 (1)\\n\\nThe right Riemann-Liouville fractional derivative has the form\\n\\n( ) τ τ τ α β β d f x dx d n x f D b x n n b x ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - Γ = ) ( ) ( 1 ) ( 1 (2)\\n\\nHere α, β are the order of derivation such that n-1≤α <n, n-1≤β<n, and they are not zero.\\n\\nIf α is an integer, these derivatives are defined in usual sense as\\n\\n) ( ) ( x f dx d x f D x a α α ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = (3-a) ) ( ) ( x f dx d x f D b x β β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = (3-b)\\n\\nHamilton formalism with fractional derivative was proposed in [5]\\n\\nnamely ) , , , ( ) , , , ( t q D q D q L q D p q D p t p p q H b t t a b t t a β α β β α α β α - + = , ( 4\\n\\n) where L represents the fractional Lagrangian obtained by replacing the classical derivatives with the corresponding fractional ones [5] .\\n\\nHamilton's equations of motion are obtained as follows [5]\\n\\nt L t H ∂ ∂ - = ∂ ∂ ; ; q D p H t a α α = ∂ ∂ ; q D p H b t β β = ∂ ∂ α α β β p D p D q H b t t a + = ∂ ∂ (5)\\n\\nIn [6] based on the sequential derivatives the fractional Hamilton-Jacobi partial differential equation is obtained. The Hamilton-Jacobi function in configuration space is written in a similar manner to the usual mechanics by using the Reimann-Loville fractional derivative. In [6] the following generating function is used, where α and β are bigger or equal to 1. Thus, the new Hamiltonian is expressed as\\n\\nS t P P q D q D F F b t t a = = - - ) , , , , ( 1 1 2 β α β α ) , , , ( ' ) , , , ( t Q D Q D Q L Q D P Q D P t P P Q K b t t a b t t a β α β β α α β α - + = (7)\\n\\nIt is concluded that, the following relation relates the two Hamiltonians\\n\\ndt dF K Q D P Q D P H q D p q D p b t t a b t t a + - + = - + β β α α β β α α (8)\\n\\nAccording to reference [6] the function F is proposed as ) , , , , (\\n\\n1 1 t P P q D q D S F b t t a β α β α - - = Q D P Q D P B t t a 1 1 - - - - β β α α , (9)\\n\\nThe function S is called Hamilton's principle function.\\n\\nTherefore, requiring that the transformed Hamiltonian K shall be zero the Hamilton-Jacobi equation is satisfied. In other words Q, P α , P β are constants.\\n\\n0 = ∂ ∂ + t S H (10)\\n\\nSince Q, P α , P β are constants, The Hamilton's principle function is written as (11) where\\n\\n) , , , , ( 2 1 1 1 t E E q D q D S S b t t a - - = β α\\n\\n1 E P = α 2 E P = β\\n\\nIf the Hamiltonian is explicitly independent of time, then S can be written as follows (12) ) , , ( ) , ( ) , (\\n\\n2 1 2 1 2 1 1 1 t E E f E q D W E q D W S b t t a + + = - - β α\\n\\nW represents the Hamilton's characteristic function; therefore, the following equations of motion are obtained in [6] as:\\n\\nq D W P t a 1 1 - ∂ ∂ = α α q D W P b t 1 1 - ∂ ∂ = β β (13) 1 1 1 1 λ α = ∂ ∂ = - E W Q D t a 2 2 2 1 λ β = ∂ ∂ = - E W Q D b t (14)\\n\\nHere λ 1 ,λ 2 are constants.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. Fractional WKB approximation\",\n \"text\": \"The outstanding result regarding the meaning of the state function ψ and its relationship to Hamilton's principle function S enables us to write the exponential solution of Schrödinger equation [13] .\\n\\n⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = h ) , ( exp ) , ( t q S i t q ψ (15)\\n\\nThe phase of state function obeys the same mathematical equation, as does Hamilton's principle function S. The physical significance of S in classical mechanics is that it represents the generator of trajectories [12] for fractional systems; the fractional Hamilton's principle function is become the phase of the state function ψ. One can write the solution of Schrödinger equation under the postulated constrains by the WKB approximation and using the fractional Hamilton's principle function eq (12) . Thus we propose the fractional state function as:\\n\\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t q D q D S i t q D q D b t t a b t t a , , exp ) , , ( 1 1 1 1 β α β α ψ h (16)\\n\\nFrom the quantization using WKB approximation [7, 8, 9, 10, 14] a general solution of Schrödinger equation is obtained using the expansion for S and then using the transformation to the N-dimensional system as:\\n\\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ∏ = h t q S i q i N i i o i , exp ) ( 1 ψ ψ (17)\\n\\nwhere ( )\\n\\n) ( 1 i i io q p q = ψ (18)\\n\\nIn our case, S behaves like a 2-dimensional problem with two distinct momenta. Thus,\\n\\n(19) q D q t a 1 1 - ≡ α α P P 1 ≡ (20) q D q b t 1 2 - ≡ β β P P 2 ≡\\n\\nAnd the momenta are defined as operators. Therefore, we can propose the wave function ψ of the fractional system in the following form\\n\\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t E E q D q D S i P P t q D q D b t t a b t t a , , , , exp 1 ) , , ( 2 1 1 1 1 1 β α β α β α ψ h (21)\\n\\nand the momenta operators in the form\\n\\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 , ˆβ β α α h h (22)\\n\\nWe conclude that (21) is the solution of Schrödinger equation for any given fractional systems. If α and β both are equal to unity, then we will return to the usual classical solution of Schrödinger equation, also we can notice how the probability is inversely proportional to the momentum . ) (\\n\\n1 2 q p ≅ ψ IV. Examples\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. a) Example 1:\",\n \"text\": \"As a first model let us consider the following fractional Lagrangian,\\n\\n( ) ( ) 2 1 2 0 2 1 2 1 q D q D L t t β α + = (23)\\n\\nThe fractional Hamilton-Jacobi equation for this fractional Lagrangian can be calculated as:\\n\\n( ) ( ) . 0 2 1 2 1 2 2 = ∂ ∂ + + t S P P β α (24) where q D L P t α α 0 ∂ ∂ = ; q D L P t β β 1 ∂ ∂ =\\n\\nMaking use of equation ( 13 ), the fractional Hamilton-Jacobi equation (24) becomes:\\n\\n0 2 1 2 1 2 1 1 2 2 1 0 1 = ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - t S q D W q D W t t β α (25) Taking into account t S H ∂ ∂ - = (26)\\n\\nIf we apply (26) on a wave function it gives:\\n\\n) ( 2 1 E E E t S + - ≡ - = ∂ ∂ (27)\\n\\nBy using the fact that E is the total energy of the system and taking into account (27) we obtain 0 2\\n\\n1 2 1 2 2 1 1 2 1 2 1 0 1 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - E q D W E q D W t t β α (28)\\n\\nThus, both sides of (28) should be zero, and we obtain\\n\\nq D E W q D E W t t 1 1 2 2 1 0 1 1 2 , 2 - - = = β α\\n\\n(29) By using ( 12 ) and ( 21 ) we obtain It's the same as the classical solution. Also, when applying the energy operator it gives the energy eigenvalues:\\n\\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + = - - - - t E q D E q D E i P P t q D q D t t t t 1 1 2 1 0 1 1 1 1 0 2 2 exp 1 ) , , (\\n\\n( ) ( )ψ ψ ψ β α 2 2 2 1 2 1 P P H + = ( ) ψ 2 1 E E + = (34)\\n\\nas in the classical case.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IVb.)Example 2:\",\n \"text\": \"As a second example let us consider the following fractional Lagrangian ( ) ( )\\n\\n2 1 0 2 1 2 0 2 1 2 1 2 1 q q D q D q D q D L t t t t + + + + = β α β α (35)\\n\\nThe corresponding fractional Hamilton is calculated as follows\\n\\n( ) ( ) 2 2 2 2 1 1 2 1 1 2 1 q P P H - - + - = β α (36)\\n\\nThus, the fractional Hamilton-Jacobi equation becomes ( ) ( )\\n\\n0 2 1 1 2 1 1 2 1 2 2 2 = ∂ ∂ + - - + - t S q P P β α (37)\\n\\nThe fractional Hamilton's principle function is calculated as,\\n\\n) ( )\\n\\nt E E q D E q D E q S t t ) ( 1 2 1 2 2 1 1 1 2 1 0 1 2 + - + + + + = - - β α (38)\\n\\nAs a result the wave function can be written in the form ( )\\n\\n( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + + + + = - - - - t E q D E q D E q i P P t q D q D t t t t 1 1 2 1 0 1 2 1 1 1 0 1 2 1 2 exp 1 ) , , ( β α β α β α ψ h (39)\\n\\nTo identify the influence of the operators let us test the effect of the momenta\\n\\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 ˆβ β α α h h (40)\\n\\nUsing the characteristic equations, it can be shown that\\n\\n1 2 , 1 2 ˆ2 1 2 + = + + = E P E q P β α (41) The result shown in (41 ) is the same classical solution. When applying the energy operator it will give the energy eigenvalues ( ) ( ) ψ ψ ψ ψ β α 2 2 2 2 1 1 2 1 1 2 1 q P P H --+ -= (42) ( ) ( ) ψ ψ ψ ψ β α β α 2 2 2 2 1 2 1 q P P P P -+ + -+ = ( ) ( ) ψ } 2 1 1 2 2 2 2 2 2 2 2 2 2 1 { 2 2 1 2 2 1 2 2 q E E q E E q E q -+ + + + -+ + + + + = Then we get\\n\\nψ ψ E H = (43)\\n\\nwhich is exactly the total energy as the case for the classical systems.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"V. Conclusions\",\n \"text\": \"We use the generating function \\\"S\\\" of the Hamilton-Jaccobi equation in its fractional form to be the phase factor of the wave function describing some potentials valid for the assumptions suggested by the WKB approximation The proof of our results arises from the new proposed concepts of the momentum and energy operators, that they give the same eigenvalues producing the ordinary results achieved by the classical approach.\\n\\nGiving the same eigenvalues that means this form of fractional operator also eigen, valid, and useful in effecting on a state functions.\"\n }\n]"}}},{"rowIdx":97,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0527"},"arxiv_id":{"kind":"string","value":"0704.0527"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.75.125023"},"content_hash":{"kind":"string","value":"edd179bddd702d53493d85c4e4f40b62a69e7e5d63dd63196e931a133fab844c"},"title":{"kind":"string","value":"Towards Skyrmion Stars: Large Baryon Configurations in the\n Einstein-Skyrme Model"},"abstract":{"kind":"string","value":"We investigate the large baryon number sector of the Einstein-Skyrme model as\na possible model for baryon stars. Gravitating hedgehog skyrmions have been\ninvestigated previously and the existence of stable solitonic stars excluded\ndue to energy considerations. However, in this paper we demonstrate that by\ngenerating gravitating skyrmions using rational maps, we can achieve\nmulti-baryon bound states whilst recovering spherical symmetry in the limit\nwhere B becomes large."},"authors":{"kind":"list like","value":["B.M.A.G. Piette and G.I. Probert"],"string":"[\n \"B.M.A.G. Piette and G.I. Probert\"\n]"},"arxiv_categories":{"kind":"list like","value":["hep-th"],"string":"[\n \"hep-th\"\n]"},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in a series of papers as a non-linear field theory of pions [2] , [3] . Skyrme's initial idea was to think of baryons (in particular the nucleons) as secondary structures arising from a more fundamental mesonic fluid. The key property of the model was that the baryons arose as solitons in a topological manner and thus possessed a conserved topological charge identified with the baryon number.\n\nThe lowest energy stable solutions of the model are termed Skyrmions and can be thought of as baryonic solitons. The Skyrme model has been very successful in modelling the structures of various nuclei and has been shown by Witten et al. [4] to possess the general features of a low energy effective field theory for QCD. Some studies of the Skyrme model coupled to gravity have previously been undertaken [1], [5] , [6] , mainly with the motivation of a comparison of its features with those of other nonlinear field theories coupled to gravity. Of particular note is the Einstein-Yang-Mills theory, in which gravitationally bound configurations of non-abelian gauge fields are produced.\n\nOther reasons for studying the Einstein-Skyrme model are cosmological and astrophysical ones. Various authors have studied black hole formation in the model, with the conclusion that the so-called no-hair conjecture may not hold [7] , [8] .\n\nThe purpose of this paper is to study large baryon number Skyrmions or configurations of Skyrmions in the Einstein-Skyrme model. In particular, we wish to investigate if stable solitonic stars could exist within the model and to compare their properties to those of neutron stars. Preliminary studies of Skyrmion stars have predicted instability to single particle decay [1] . However this was done using the hedgehog ansatz for baryon number larger than 1 which is known to lead to unstable solutions even for the usual Skyrme model. Since then, it has been shown that the Skyrme model has stable shell-like solutions [9] which can be well approximated by the so called rational map ansatz [10] .\n\nIn this paper we use the rational map ansatz and its extension to multiple shells to construct configurations in the Gravitating Skyrme model that have a very large number of baryon. We show that those configurations, contrary to the hedgehog ansatz are bound even for very large baryon numbers.\n\nTo construct configurations that have a baryon number comparable to that of neutron star, we have to introduce a further approximation, which we call the ramp ansatz. We show 2 that this anstaz introduces further errors of only a few percent and we use it to compute very large Skyrmion configurations.\n\nThe paper is organised as follow: first we outline the Einstein-Skyrme model and discuss the main features of the results on static gravitating SU (2) hedgehogs obtained by Bizon and Chmaj [1] . We then use the rational map ansatz to construct shell like gravitating multibaryon configurations and show that for a fixed value of the coupling constant, the configurations exist only when the baryon number is below a certain critical value. Finally we introduce a ramp profile approximation to construct solutions with extremely high baryon numbers. We show how accurate it is and use it to construct Skyrmion stars configuration.\n\nThe action for gravitating Skyrmions is formed from the standard Skyrme action for the matter field and the Einstein-Hilbert action for the gravitational field.\n\nS = Z M √ -g \" L Sk - R 16πG « d 4 x. ( 1\n\n)\n\nHere L Sk is the Lagrangian density for the Skyrme model defined on the manifold M :\n\nL Sk = F 2 π 16\n\nT r(∇µU ∇ µ U -1 ) + 1 32e 2 T r[(∇µU )U -1 , (∇νU )U -1 ] 2 , (\n\n) 2\n\nwhere U belongs to SU (2). As we eventually wish to study baryon stars, we take a spherically symmetric metric, such as associated with the line element ds 2 = -A 2 (r) \" 1 -2m(r) r « dt 2 + \" 1 -2m(r) r\n\n« -1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), ( 3\n\n)\n\nwhere A(r) and m(r) are two profile functions that must be determined by solving the Einstein equations for the model. Our choice of ansatz is motivated by the fact that although in some cases we will be studying non-spherical Skyrmion configurations, the regime we are primarily interested in (i.e. Skyrmions of extremely high baryon number) will be shown to admit quasi spherical solutions. Also, for realistic values of the couplings, the gravitational interaction is small compared to the Skyrme interaction and thus the use of a spherical metric even with non-spherical configurations, is not a great problem.\n\nFrom (3), it can be shown that the Ricci scalar is\n\nR = -2 Ar 2 `-A ′′ r 2 -2A ′ r + 2A ′′ rm + A ′ m + 3A ′ rm ′ + Arm ′′ + 2Am ′ ´( 4\n\n)\n\nwhich, after integrating various terms by parts and noting that asymptotic flatness requires both A(r) and m(r) to take a constant value at spatial infinity, reduces the gravitational part of the action to Sgr = Z A(r) \" -m ′ (r) G « dr + m(∞) G . (5) For what follows, it will be convenient to scale to dimensionless variables by defining x = eFπr and µ(x) = eFπm(r)/2, resulting in one dimensionless coupling parameter for the model, α = πF 2 π G. We note that taking Fπ = 186M ev and G = 6.72 × 10 -45 M ev -2 , then the physical value of the coupling is α = 7.3 × 10 -40 .\n\nAs the Skyrme field is an SU (2) valued scalar field, at any given time one can think of it as a map from R 3 to the SU (2) manifold. Finite energy considerations impose that the field at spatial infinity should map to the same point on SU (2), say the identity. Thus, one can simply think of the Skyrme field as a map between three-spheres. All such maps fall into disjoint homotopy classes characterised by their winding number. This winding number is a conserved topological charge because no continuous deformation of the field and thus no time evolution, can allow transitions between homotopy classes. It is this topological charge that is interpreted as the baryon number.\n\nGravitating Skyrmions were first studied by Bizon and Chmaj[1] who analysed the properties of static spherically symmetric gravitating SU (2) skyrmions. Taking the Hedgehog Ansatz for the Skyrme field\n\nU = exp(i -→ σ .rF (r)) ( 6\n\n)\n\nsubject to the boundary conditions\n\nF (r = 0) = Bπ ( 7\n\n) F (r = ∞) = 0 ( 8\n\n)\n\nwhere B is the Baryon number associated with the Skyrmion configuration, they derived the Euler-Lagrange equation for the profiles F (r), (A(r) and m(r) and found that the model admit two branches of global solitonic solutions at each given baryon number, which annihilate at a critical value of the coupling parameter. Above αcrit no further solutions were found. In particular the value of the critical coupling decreased quite considerably with increasing baryon number as αcrit ≈ 0.040378/B 2 . It appears that the existence of a 4 critical coupling does not signal the collapse of a Skyrmion to form a black hole. In fact the metric factor S(x) = (1 -2µ(x) x ) is non-zero at αcrit; there simply ceases to be any stationary points of the action above the critical coupling.\n\nThe major problem with the ansatz (7) is that it leads to unstable solutions, i.e. for any given value of α, MADM (B = N ) > N MADM (B = 1). This is actually the case for the pure Skyrme model as well where the hedgehog anstaz (7) with B > 1 does not correspond to the lowest energy solution for the model. The solutions of the pure Skyrme model when B > 1 are known not to be spherically symmetric [11] but are stable i.e. E(B = N ) < N * E(B = 1).\n\nIt was actually shown by Houghton et al [10] , [12] that the multi-baryon solutions of the pure Skyrme model can be well approximated by the so called rational maps ansatz which is a generalisation of the hedgehog ansatz. While not radially symmetric, the ansatz separates its radial and angular dependence through a profile function and a rational map respectively.\n\nIn the following sections we will generalise the construction of Houghton et al to approximate the solution of the Einstein-Skyrme model.\n\nThe rational map ansatz introduced by Houghton et al.[10] works by decomposing the field into angular and radial parts. Using the polar coordinates in R 3 and defining the stereographic coordinates z = tan(θ/2) exp iφ the ansatz reads [10]\n\nU = exp (i σ • nRF (r, t)) ( 9\n\n) where nR = 1 1 + |R| 2 `2ℜ(R), 2ℑ(R), 1 -|R| 2 ´( 10\n\n)\n\nis a unit vector where R is a rational function of z.\n\nIt can be shown that the baryon number for Skyrmions constructed in this way, is equal to the degree of the rational map providing we take the boundary conditions\n\nF (r = 0) = π F (r = ∞) = 0. ( 11\n\n)\n\nSubstituting the ansatz (9) into the action for the model and scaling to dimensionless\n\n5\n\nvariables as earlier, we obtain the following reduced Hamiltonian\n\nH = 16πFπ e »Z ∞ 0 A(x) \" 1 2 S(x)F (x) ′2 x 2 + Bsin 2 F (x)(1 + S(x)F (x) ′2 ) + Isin 4 F (x) 2x 2 - µ ′ (x) α « dx + µ(∞) α - ( 12\n\n)\n\nwhere\n\nS(x) = 1 - 2µ(x) x ( 13\n\n)\n\nFrom which one obtains the following field equations\n\nµ(x) ′ = α \" 1 2 S(x)x 2 F (x) ′2 + B sin 2 F (x) + S(x)BF (x) ′2 sin 2 F (x) + I sin 4 F (x) 2x 2 « ( 14\n\n) F (x) ′′ = 1 S(x)V (x) » sin 2F (x) \" B + S(x)BF (x) ′2 + I sin 2 F (x) x 2 « - αS(x)F (x) ′3 V (x) 2 x -S(x) ′ F (x) ′ V (x) -S(x)F (x) ′ V (x) ′ ˜( 15\n\n) and A(x) ′ = αA(x)F (x) ′2 \" x + 2B sin 2 F (x) x « ( 16\n\n)\n\nwhere, for convenience, we have defined V (x) as\n\nV (x) = x 2 + 2B sin 2 F (x). ( 17\n\n)\n\nB is the baryon number and\n\nI = 1 4π Z \" 1 + |z| 2 1 + |R| 2 ˛dR dz ˛«4 2idzdz (1 + |z| 2 ) 2 ( 18\n\n)\n\nIts value depends on the chosen rational map R. To compute low energy configurations for a given baryon charge B one must find the rational map R or degree B that minimize I.\n\nThis has been done in [10] and [11] for several values of B. Moreover when b is large, one can use the approximation[11] I ≈ 1.28B 2 . The value of I so obtained is then used as a parameter and one can solve equations (14) -(16) for the radial profiles F (x), A(x) and µ(x).\n\nWe should point out here that for the pure Skyrme model the rational map ansatz produce very good approximation to the multi skyrmion solutions [10]: the energies are only 3 or 4 percent higher and the energy densities exhibit the same symmetries and differ by very little. All the solutions computed by Battye and Sutcliffe [11] , when B is not too small, have somehow the shape of a hollow shell. The baryon density is very small everywhere outside the shell, while on the shell itself, it forms a lattice of hexagons and pentagons.\n\nUsing the rational map ansatz, we will now solves the field equations (14) -(16) to compute some low action configurations. These solutions will correspond, initially, to a hollow shell of Skyrmions similar to the configuration obtained with the rational map anstaz for the pure Skyrme model. In the following sections we will show how our ansatz can be generalised to allow for more realistic configuration made out of embedded shells.\n\nThe first thing to note about our solutions is that we again obtain two branches of solutions at each baryon number (Fig. 1 ). Obtaining this same qualitative behaviour is not surprising when one considers that the B = 1 rational map Skyrmion reproduces the usual B = 1 hedgehog. However, the behaviour of the critical coupling itself is drastically altered for the rational map generated configurations. Namely, we observe that it decreases as approximately 0.040378/B 1 2 (Fig. 2 ). In particular this means that for a given value of the coupling, the rational map generated skyrmions can possess a much higher topological charge than their hedgehog counterparts, before there ceases to be any solutions. Quantitatively if B hedgehog is the maximum baryon number for which hedgehog solutions can be found at a given value of the coupling, then the highest baryon number rational map solution found at the same value of α will be approximately B 4 hedgehog . Again we observe that the metric function S(x) is non-zero at the critical coupling for all the solutions we have found and as such a horizon has not formed.\n\nIn Table 1 we present the radius, ADM mass per baryon and minimum value of the metric function, S(x), for configurations up to the maximum baryon number allowed at α = 1 × 10 -6 . These values were obtained by direct numerical solution of equations (14) -(16), where we have used the boundary data as specified in (11) .\n\nWe didn't didn't use the physical value of α (7.3×10 -40 ) because for this value, the ratio between the width of the shell and its radius is so small when we reach the maximum value of B that it becomes very difficult to solve the equation reliably. The value α = 1 × 10 -6 is small enough to allow for a shell with a large baryon number to exist but large enough to make it possible to compute these solution nears the critical value of B for a single shell configuration. 7 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 M ADM α Figure 1: Plot of the two branches of solutions found for B=2 configurations generated with the rational map ansatz.\n\nThe major difference between these configuration and the solutions of Bizon and Chmadj is that the rational map ansatz configurations become more bound when the baryon number increases This suggests the possibility that giant gravitating Skyrmions can be bound and consequently, that the Skyrme model can be used to study baryon stars.\n\nAnother interesting feature of the data is the observed change in the radius of the solutions with increasing baryon number. We note that the radius grows as approximately B 1 2 .\n\nHowever there are two main deviations from this. Firstly, the constant of proportionality relating the radius to the square root of the baryon number decreases slightly but persistently as we increase the baryon number, indicating the gravitational interaction becoming more important as the number of baryons increases.\n\nAs we approach the maximum baryon charge that can exist at α = 1 × 10 -6 , we also notice that the radius of the skyrmion actually decreases as we add more baryons. This shows that the gravitation pull plays a crucial role near the critical value of the skyrmion.\n\nThis is a tantalising property when one considers that generally a neutron star's radius must decrease for an increase in mass in order to achieve sufficient degenerate neutron pressure 8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 2 4 6 8 10 12 14 16 18 20 α crit B Figure 2: Plot of the decrease in α crit with increasing baryon number, for configurations generated with the rational map ansatz. +: α c r for the minimum value of I; curve: α c r = 0.0404B -1/2 .\n\nto support the star.\n\nTo motivate the further approximation that we will introduce in the next section, we now look at the profiles of the configuration that we have computed. First of all, we observe that the profile function F (x) stays approximately at its boundary value, π, for a finite radial distance before decreasing monotonically over some small region and finally attaining its second boundary value, 0. A similar behaviour is seen for both the mass field µ(x) and the metric field A(x) (see Fig. 3 ). Furthermore, as we increase the baryon number the structure becomes more pronounced, with the distance before the fields change (shell radius) increasing significantly, whilst the distance over which the fields change (shell width) settles to a constant size. We conclude that at large baryon numbers, those configurations correspond to hollow shells where the baryons are distributed on a tight lattice over the shell. As such the, structures are nearly spherical, validating our choice of radial metric.\n\nSuch structures immediately pose an interesting question. Can the gravitating Skyrmions exist as shells with more than one layer? To investigate this we note that it is possible to 9 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 1 0.8763 1.2315 1.0000 4 1.7728 1.1365 1.0000 8 2.5065 1.1180 1.0000 100 8.6829 1.0845 0.9999 500 19.3994 1.0827 0.9998 1 × 10 3 27.4314 1.0825 0.9997 1 × 10 4 86.7192 1.0821 0.9989 1 × 10 5 274.0397 1.0814 0.9963 1 × 10 6 864.6968 1.0792 0.9883 1 × 10 7 2715.0729 1.0722 0.9628 1 × 10 8 8377.4601 1.0500 0.88192 1 × 10 9 23585.5315 0.9743 0.6107 1.5 × 10 9 26860.2040 0.9463 0.5020 1.8 × 10 9 27470.2449 0.9302 0.4256 1.81 × 10 9 27456.5804 0.9296 0.4225 1.85 × 10 9 27357.9201 0.9274 0.4090 1.9 × 10 9 27078.6014 0.9246 0.3886 1.95 × 10 9 26126.5508 0.9217 0.3517 1.951 × 10 9 26050.7695 0.9217 0.3495 1.952 × 10 9 25937.4210 0.9216 0.3463 Table 1: Properties of the one shell low energy configuration for α = 1 × 10-6 10 0 0.5 1 1.5 2 2.5 3 3.5 4 1210 1215 1220 1225 1230 1235 F(x) x 0 2 4 6 8 10 1210 1215 1220 1225 1230 1235 µ(x) x 0.99 0.995 1 1.005 1.01 1210 1215 1220 1225 1230 1235 A(x) x Figure 3: Numerical solutions for the profiles F (x), µ(x) and A(x) when B = 2 × 10 6 and α = 1 × 10 -6 .\n\nmodify the boundary condition (11) to read\n\nF (r = 0) = N π ( 19\n\n) F (r = ∞) = 0 ( 20\n\n)\n\nwhilst still ensuring that the Skyrme field is well defined at the origin. This idea was first used in [12] to construct two shell configurations for the pure Skyrme model.\n\nThe baryon charge is now N times the degree of the rational map. Fig. 4 shows the structure of the solutions we find in this case when N = 2. They suggest that the Skyrmion now exists as a N -layered structure. This is exhibited in the form of the profile, mass and metric functions which interpolate between the boundary values in N distinct steps of equal 11 0 1 2 3 4 5 6 7 1200 1205 1210 1215 1220 1225 1230 F(x) x 0 5 10 15 20 1200 1205 1210 1215 1220 1225 1230 µ(x) x 0.995 1 1.005 1.01 1.015 1.02 1200 1205 1210 1215 1220 1225 1230 A(x) x Figure 4: Numerical solutions for for the profiles F (x), µ(x) and A(x) for 2 layers configurations (F (0) = 2π) when B = 2 × 10 6 and α = 1 × 10 -6 .\n\nsize stacked next to each other.\n\nWe can therefore think of this as a naive way of constructing a gravitating Skyrmion.\n\nInstead of using the boundary conditions as in (11) and a rational map of degree B we consider constructing the B-Skyrmion using a rational map of degree B/N (with the associated value of I) and the boundary condition (20). This is a crude construction as we are effectively considering N adjacent shells of baryons, all with the same baryon number. We might realistically expect that the baryon number per shell and distribution of shells may vary significantly for the minimum energy configuration. Nevertheless we shall study the properties of such structures. In fact, in the case where the baryon number is large and the number of shells is small, we expect this crude construction to be quite valid. That is, we do not expect the baryon number to change significantly over the few shells at large radius.\n\n12 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 4 1.2898 1.6179 1.0000 8 1.7858 1.4072 1.0000 100 6.1754 1.1363 0.9999 1 × 10 3 19.4157 1.0913 0.9996 1 × 10 4 61.3207 1.0833 0.9985 1 × 10 5 193.7006 1.0812 0.9949 1 × 10 6 610.6271 1.0779 0.9835 1 × 10 7 1911.3704 1.0680 0.9475 1 × 10 8 5825.2626 1.0362 0.8325 9.0 × 10 8 13736.9982 0.9302 0.4258 9.7 × 10 8 13263.0853 0.9224 0.3644 9.76 × 10 8 12998.2817 0.9217 0.3480 9.764 × 10 8 12931.5189 0.9216 0.3444 9.7647 × 10 8 12895.6984 0.9216 0.3425 9.76472 × 10 8 12891.4247 0.9216 0.3423 9.764724 × 10 8 12889.8645 0.9216 0.3422 Table 2: Table of properties of double layer solutions obtained numerically at α = 1 × 10-6\n\nFor the remainder of this section we will restrict ourselves to the case where N = 2.\n\nTable 2 summarises the properties of double layered gravitating skyrmions up to the maximum baryon charge allowed at α = 1 × 10 -6 . Briefly, we note the main features.\n\nFirstly, for all baryon numbers, the radius of the double layered solutions is significantly less than their single layered counterparts. This is not surprising as the baryon charge exists over a thicker region and so the mean radius can decrease with the baryon density remaining the same. Secondly, when B is large enough, i.e. when the double layer starts to make sense, the double layer solutions are energetically favourable when one compares the ADM mass with the single layer solutions. Finally we note that the maximum baryon number allowed (at the given coupling) is almost twice as much in the case of the single layer skyrmions.\n\nOf course the results of this section are not really the main regime of interest. We clearly 13 need to study configurations of extremely high baryon number (of order 10 58 ) relevant for baryon stars.\n\nWe will now discuss this high baryon number regime.\n\nUnfortunately, at very high baryon numbers, eqns. ((14) -(16)) become difficult to handle numerically. This is largely because the radius of the solutions becomes much larger than the distance over which the fields change. That is, we need to integrate over a region which is much less than 10 -16 radius, and so even double precision data types have insufficient precision.\n\nMoreover, single shell configurations are not physically relevant and multiple shells will only yield configuration that looks like a star if the number of layers is very large, typically well over 10 17 . With such a large number of layers we won't be able to solve the equation numerically as we will need at least 10 times as many sampling points for the profile functions.\n\nWe must thus resort to another level of approximation: approximate the profile functions by profiles that are piecewise linear. This is inspired by the work of Kopeliovich [13] [14] except that our ansatz has to be piecewise linear to be able to generate configurations with a huge number of layers. After defining the ansatz for an arbitrary number of layers, we will show that for a single layer configuration the ansatz produces configurations that are in good agreements with the rational map ansatz configuration. Then we will use the new ansatz to construct configurations that are made out of a very large no of layers.\n\nWe have shown, in the previous section, that one can construct shell like structures with very large Baryon numbers. At large baryon numbers, the Skyrmions resemble shell like structures. That is, the fields are constant nearly everywhere except in a small region corresponding to the shell. In that region, the profile look like linear functions smoothly linked to the constant parts at the edges (cfr. Fig 3 ). Motivated by this we approximate the fields by the ramp-functions\n\nF (x) = N π 2 -(x -x0) π W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 21\n\n) µ(x) = M 2 + (x -x0) M N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 22\n\n) A(x) = (1 + A0) 2 + (x -x0) (1 -A0) N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 23\n\n)\n\nIn the above there are four free parameters, namely the central radius x0 of the shell\n\nover which the fields change, the width of the shell W , the mass field at spatial infinity M and the value of the metric field at the origin A0 such that limx→∞ = 0. N is the number of layers we wish to study and, as such, is treated as an input parameter.\n\nThe picture is of a gravitating skyrmion with very high baryon number existing as N thin layers or shell of small thickness.\n\nThe above ansatz, allow us to find an approximation to the integrated energy. To do this we use the fact that the shell width is much smaller than the radius at large baryon numbers. In particular to evaluate the action integral we can approximate expressions of the type R G(x) sin p F (x) for any function G(x) that varies very little over the width of the shell by R G(x0) sin p F (x). We then use the fact that\n\nZ x 0 +NW/2 x 0 -NW/2 sin p F (x) = N π W Z π 0 sin p y dy. ( 24\n\n)\n\nThis leads to the following expression for the energy:\n\nE = - 16πFπ e » M α \" 1 + A0 2 « - π 2 W » \" 1 + A0 2 « \" x 2 0 -M x0 + W 2 12 - M W 6 « + \" 1 -A0 W « \" W 2 x0 6 - M W 2 12 - M W x0 6 « - + B \" 1 + A0 2 « 1 2 \" M π 2 W x0 -W - π 2 W « - 3IW 16x 2 0 \" 1 + A0 2 « - M α - ( 25\n\n)\n\nTo find the configurations which minimize this energy we first minimised it with respect to A0 and M algebraically in order to find an expression for the energy as a function of the width and radius only. Then we minimised this numerically using Mathematica. We will now discuss the features of these configurations.\n\nFirst of all, we must compare the results obtained with the ramp-profile when N = 1 and compare them to the result obtained with the full profile. Tables. 3 and 4 show the properties of solutions we obtained using the ramp-profile approximation, again at α = 1 × 10 -6 . All the general features of the full numerical solutions are reproduced. In particular, the approximate B 1 2 scaling and then decrease of the radius, the decreasing ADM mass and the differences between the double and single layer solutions are all exhibited by the data obtained using the ramp-profile approximation.\n\nQuantitatively though, there are some differences. The approximation allows a significant increase in the maximum allowed baryon charge. Also, the radius of configurations obtained using the approximation, tend to be smaller than those obtained numerically. If we concentrate on the baryon numbers greater than 10 5 so as to ensure our approximation, 15 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 8.3063 3.1286 1.1023 0.9999 500 18.6031 3.1386 1.1160 0.9997 1 × 10 3 26.313 3.1397 1.1195 0.9996 1 × 10 4 83.206 3.1396 1.1254 0.9987 1 × 10 5 262.94 3.1357 1.1266 0.9960 1 × 10 6 829.60 3.1230 1.1251 0.9872 1 × 10 7 2604.2 3.0825 1.1186 0.9595 1 × 10 8 8032.8 2.9512 1.0972 0.8713 1 × 10 9 22899 2.4837 1.0272 0.5772 2 × 10 9 29121 2.1092 0.9818 0.3645 2.8 × 10 9 29098 1.6623 0.9505 0.1380 2.83 × 10 9 28514 1.6066 0.9495 0.1119 2.839 × 10 9 28024 1.5671 0.94922 0.09373 2.8397 × 10 9 27869.3 1.5556 0.94924 0.08845 2.83975 × 10 9 27869.8 1.5524 0.94925 0.08699 2.839752 × 10 9 27822 1.5521 0.94925 0.08687 Table 3: Table of properties of the single layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6 16 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 5.7924 3.0428 1.0692 0.9999 1 × 10 3 18.5788 3.1305 1.1047 0.9995 1 × 10 4 58.8202 3.1380 1.1201 0.9983 1 × 10 5 185.8420 3.1332 1.1246 0.9944 1 × 10 6 585.7950 3.1153 1.1233 0.9820 1 × 10 7 1833.0500 3.0578 1.1143 0.9428 1 × 10 8 5587.3600 2.8688 1.0840 0.8172 9 × 10 8 14147.1782 2.1859 0.9900 0.4065 9.764724 × 10 8 14472.3851 2.1276 0.9837 0.3746 1 × 10 9 14560.5000 2.1092 0.9818 0.3646 1.4 × 10 9 14549.0000 1.6623 0.9505 0.1381 1.41963 × 10 9 13994.0523 1.5644 0.9492 0.0926 1.419635134 × 10 9 13993.2000 1.5643 0.9492 0.0925 Table 4: Table of properties of the double layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6\n\nthat the width is much smaller than the radius, is valid, then at worst we find a discrepancy in the ADM mass of 11% and in the radius of 7%.\n\nIn general then, the data seems to confirm the reliability the ramp-profile approximation. In fact the approach will be even more reliable at the extremely high values of the baryon number that we are interested in. This is because the radius of solutions is of orders of magnitudes greater than the width in such a regime, consistent with the approximations we have made.\n\nMoreover, whilst searching for minima of the energy does not allow us to probe both branches of solutions, it does allow us to locate the value of αcrit. We again obtain the approximate trend αcrit ∝ B -1 2 , for large B. Now in order to say anything about the possibility of baryon stars in the Skyrme model we need to be able to verify that the decrease in the ADM mass per baryon we observed at low and moderate baryon numbers, extends to baryon numbers of order 10 58 for α = 7.3 × 10 -40 .\n\nTable. 5 summarizes our solutions in such a regime. Firstly we consider constructing a single layer self-gravitating Skyrmion with these values. We do indeed see that the configuration is bound. This is verified by checking that the ADM mass is lower (even at this significantly lower value of α) than for the B = 1 hedgehog. So the possibility of baryon stars in the Einstein-Skyrme model cannot be ruled out on the grounds of energy.\n\nThe Skyrmion exists as a giant thin shell, and the large baryon charge is distributed as a tight lattice over this. However a hollow shell is clearly not a realistic construction for a neutron star. This fact manifests itself in the extremely high radius of the configuration.\n\nTransferring to standard units, the single layer B = 10 58 gravitating Skyrmion has a radius of 2.42 × 10 10 km ! To address this issue, we can use a large number of layered Skyrmions as discussed earlier. This has several benefits. Firstly, as we are distributing the baryon number through a larger volume, then at a given baryon density the necessary radius can decrease. Similar to what we see in the double layer results. On top of this, we expect the radius to decrease further due to extra gravitational compression, as the outer layers of the Skyrmions feel the attraction of inner layers. Finally, the many layer approach is also a more realistic construction of a solid baryon star.\n\nThe results for using more and more layers in the construction (for fixed B and α), are also presented in Fig. 5 . We note that not only does the radius decrease significantly, but\n\nthe added gravitational binding further improves the energies of the configurations, reflected in the low ADM masses obtained. There appears to be a critical number of layers that can be used before there ceases to be any solutions and although the value of Smin is close zero at this point, the star still has not collapsed to form a black hole. Finally, we note that the radius of the Skyrmion at the critical number of layers is approximately 20.91km. This is comparable to a real neutron star, with a typical radius of 10km.\n\nWe reemphasise here that our approach to embedding shells of baryons is quite crude.\n\nFor few shells and large baryon number, we might reasonably believe that baryon number does not chance significantly from one shell to the next. However, when we embed many shells we should really consider that the baryon number of the inner most shells would likely be significantly less than the that of the outer shells. Nevertheless, our naive embedding has produced some interesting properties. In a future work we hope to improve our multi-layer construction to obtain a more realistic description of a baryon star. 19 N Shell R( 2 Fπe ) W M ADM /(6π 2 B) S min 1 × 10 2 8.3236 × 10 27 3.1416 1.1285 1.0000 1 × 10 3 2.6321 × 10 27 3.1416 1.1285 1.0000 1 × 10 4 × 10 26 3.1416 1.1285 1.0000 1 × 10 5 2.6321 × 10 26 3.1416 1.1285 1.0000 1 × 10 6 8.3236 × 10 25 3.1416 1.1285 1.0000 1 × 10 7 2.6321 × 10 25 3.1416 1.1285 1.0000 1 × 10 8 8.3236 × 10 24 3.1416 1.1285 1.0000 1 × 10 9 2.6321 × 10 24 3.1415 1.1285 1.0000 1 × 10 10 8.3234 × 10 23 3.1415 1.1285 0.9999 1 × 10 11 2.6319 × 10 23 3.1412 1.1285 0.9997 1 × 10 12 8.3216 × 10 22 3.1402 1.1283 0.9991 1 × 10 13 2.6301 × 10 22 3.1373 1.1278 0.9971 1 × 10 14 8.3034 × 10 21 3.1280 1.1263 0.9907 1 × 10 15 2.6118 × 10 21 3.0986 1.1213 0.9705 1 × 10 16 8.1147 × 10 20 3.0037 1.1057 0.9063 1 × 10 17 2.4001 × 10 20 2.6810 1.0552 0.6977 5 × 10 17 8.2066 × 10 19 1.7888 0.9552 0.2036 5.3 × 10 17 7.4172 × 10 19 1.6227 0.9491 0.1247 5.33 × 10 17 7.1871 × 10 19 1.5625 0.94866 0.0971 5.3306 × 10 17 7.1597 × 10 19 1.5549 0.94868 0.0936 5.33065 × 10 17 7.1525 × 10 19 1.5528 0.948694 0.0927 5.330657 × 10 17 7.1506 × 10 19 1.5523 0.948692 0.0924 Table 5: Table of properties of the step ansatz configurations for varying the number of embedded shells at fixed B = 10 58 and α = 7.3 × 10 -40 .\n\nPrevious work on the Einstein-Skyrme model highlighted a considerable problem with using the Skyrmions as a model for baryon stars. Namely, multibaryon hedgehog Skyrmions were simply not energetically favourable states. We have shown that this is simply a consequence of a poor ansatz for the true Skyrmion and, having used the more appropriate rational map ansatz, we have generated energetically favourable configurations of multibaryons.\n\nWe also observe the interesting property that near the critical coupling, the Skyrmions can decrease in radius as we add more baryons. This hints towards the similar behaviour exhibited by real neutron stars.\n\nAlthough the rational map ansatz does not have an exact radial symmetry, at large scale it does. The anisotropy only appears at the nucleon scale.\n\nFinally, since we started with the motivation of studying baryon stars within the Skyrme model, it is interesting to compare the features of our configurations with those of neutron stars. For realistic values, B = 10 58 and α = 7.3 × 10 -40 we find a minimal energy single layer configuration with radius=2.42 × 10 10 km. This is clearly too large for a neutron star (which is of order 10km. in radius). This is to be expected however due to the shell model we have taken. Firstly, as we are distributing the baryons over the surface area rather than throughout the volume of the star we naturally must require a much larger star for a given baryon number. This effect is two-fold in that if we were distributing the baryons throughout the volume, outer layers would feel the attraction of inner layers and enhanced radial compression would occur. The loss of such an effect is pronounced when we are considering realistically small values of the coupling.\n\nIt seems therefore that the way to construct baryon stars in the Skyrme model is to consider embedding shells of baryons within shells. This gives rise to more appropriate specifications for the star and is also more realistic. We do indeed observe such improvements for a many layered configuration. In fact the radius of B = 10 58 gravitating Skyrmion (at realistic α), can be decreased in this manner to approximately 20.91km.\n\nWe note however that this approach to shell embedding has only be done naively thus far.\n\nWe have only considered the case where the baryon number is equal for each shell. We really should allow the baryon number(and hence the rational map quantities) to vary over the shells. One approach towards this would be to assume that the baryon density is a constant over the shells. An even better approach would be to allow this to be a smoothly varying function that must be determined by minimising the energy. This will give a more realistic 21 description of baryon stars within the Einstein-Skyrme model, as traditional descriptions of neutron stars also involve many strata, of differing neutron density. We are currently investigating such configurations.\n\n[13] V. B. Kopeliovich \"The Bubbles of Matter from MultiSkyrmions\" JETP Lett. 73 (2001), 587-591; Pisma Zh.Eksp.Teor.Fiz. 73 (2001), 667-671 [14] V. B. Kopeliovich \"MultiSkyrmions and Baryonic Bags\" J.Phys. G28 (2002), 103-120 23\n\nGIP is supported by a PPARC studentship."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"Introduction","text":"The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in a series of papers as a non-linear field theory of pions [2] , [3] . Skyrme's initial idea was to think of baryons (in particular the nucleons) as secondary structures arising from a more fundamental mesonic fluid. The key property of the model was that the baryons arose as solitons in a topological manner and thus possessed a conserved topological charge identified with the baryon number.\n\nThe lowest energy stable solutions of the model are termed Skyrmions and can be thought of as baryonic solitons. The Skyrme model has been very successful in modelling the structures of various nuclei and has been shown by Witten et al. [4] to possess the general features of a low energy effective field theory for QCD. Some studies of the Skyrme model coupled to gravity have previously been undertaken [1], [5] , [6] , mainly with the motivation of a comparison of its features with those of other nonlinear field theories coupled to gravity. Of particular note is the Einstein-Yang-Mills theory, in which gravitationally bound configurations of non-abelian gauge fields are produced.\n\nOther reasons for studying the Einstein-Skyrme model are cosmological and astrophysical ones. Various authors have studied black hole formation in the model, with the conclusion that the so-called no-hair conjecture may not hold [7] , [8] .\n\nThe purpose of this paper is to study large baryon number Skyrmions or configurations of Skyrmions in the Einstein-Skyrme model. In particular, we wish to investigate if stable solitonic stars could exist within the model and to compare their properties to those of neutron stars. Preliminary studies of Skyrmion stars have predicted instability to single particle decay [1] . However this was done using the hedgehog ansatz for baryon number larger than 1 which is known to lead to unstable solutions even for the usual Skyrme model. Since then, it has been shown that the Skyrme model has stable shell-like solutions [9] which can be well approximated by the so called rational map ansatz [10] .\n\nIn this paper we use the rational map ansatz and its extension to multiple shells to construct configurations in the Gravitating Skyrme model that have a very large number of baryon. We show that those configurations, contrary to the hedgehog ansatz are bound even for very large baryon numbers.\n\nTo construct configurations that have a baryon number comparable to that of neutron star, we have to introduce a further approximation, which we call the ramp ansatz. We show 2 that this anstaz introduces further errors of only a few percent and we use it to compute very large Skyrmion configurations.\n\nThe paper is organised as follow: first we outline the Einstein-Skyrme model and discuss the main features of the results on static gravitating SU (2) hedgehogs obtained by Bizon and Chmaj [1] . We then use the rational map ansatz to construct shell like gravitating multibaryon configurations and show that for a fixed value of the coupling constant, the configurations exist only when the baryon number is below a certain critical value. Finally we introduce a ramp profile approximation to construct solutions with extremely high baryon numbers. We show how accurate it is and use it to construct Skyrmion stars configuration."},{"section_type":"OTHER","section_title":"The Einstein-Skyrme Model","text":"The action for gravitating Skyrmions is formed from the standard Skyrme action for the matter field and the Einstein-Hilbert action for the gravitational field.\n\nS = Z M √ -g \" L Sk - R 16πG « d 4 x. ( 1\n\n)\n\nHere L Sk is the Lagrangian density for the Skyrme model defined on the manifold M :\n\nL Sk = F 2 π 16\n\nT r(∇µU ∇ µ U -1 ) + 1 32e 2 T r[(∇µU )U -1 , (∇νU )U -1 ] 2 , (\n\n) 2\n\nwhere U belongs to SU (2). As we eventually wish to study baryon stars, we take a spherically symmetric metric, such as associated with the line element ds 2 = -A 2 (r) \" 1 -2m(r) r « dt 2 + \" 1 -2m(r) r\n\n« -1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), ( 3\n\n)\n\nwhere A(r) and m(r) are two profile functions that must be determined by solving the Einstein equations for the model. Our choice of ansatz is motivated by the fact that although in some cases we will be studying non-spherical Skyrmion configurations, the regime we are primarily interested in (i.e. Skyrmions of extremely high baryon number) will be shown to admit quasi spherical solutions. Also, for realistic values of the couplings, the gravitational interaction is small compared to the Skyrme interaction and thus the use of a spherical metric even with non-spherical configurations, is not a great problem.\n\nFrom (3), it can be shown that the Ricci scalar is\n\nR = -2 Ar 2 `-A ′′ r 2 -2A ′ r + 2A ′′ rm + A ′ m + 3A ′ rm ′ + Arm ′′ + 2Am ′ ´( 4\n\n)\n\nwhich, after integrating various terms by parts and noting that asymptotic flatness requires both A(r) and m(r) to take a constant value at spatial infinity, reduces the gravitational part of the action to Sgr = Z A(r) \" -m ′ (r) G « dr + m(∞) G . (5) For what follows, it will be convenient to scale to dimensionless variables by defining x = eFπr and µ(x) = eFπm(r)/2, resulting in one dimensionless coupling parameter for the model, α = πF 2 π G. We note that taking Fπ = 186M ev and G = 6.72 × 10 -45 M ev -2 , then the physical value of the coupling is α = 7.3 × 10 -40 .\n\nAs the Skyrme field is an SU (2) valued scalar field, at any given time one can think of it as a map from R 3 to the SU (2) manifold. Finite energy considerations impose that the field at spatial infinity should map to the same point on SU (2), say the identity. Thus, one can simply think of the Skyrme field as a map between three-spheres. All such maps fall into disjoint homotopy classes characterised by their winding number. This winding number is a conserved topological charge because no continuous deformation of the field and thus no time evolution, can allow transitions between homotopy classes. It is this topological charge that is interpreted as the baryon number."},{"section_type":"OTHER","section_title":"Gravitating Hedgehog Skyrmions","text":"Gravitating Skyrmions were first studied by Bizon and Chmaj[1] who analysed the properties of static spherically symmetric gravitating SU (2) skyrmions. Taking the Hedgehog Ansatz for the Skyrme field\n\nU = exp(i -→ σ .rF (r)) ( 6\n\n)\n\nsubject to the boundary conditions\n\nF (r = 0) = Bπ ( 7\n\n) F (r = ∞) = 0 ( 8\n\n)\n\nwhere B is the Baryon number associated with the Skyrmion configuration, they derived the Euler-Lagrange equation for the profiles F (r), (A(r) and m(r) and found that the model admit two branches of global solitonic solutions at each given baryon number, which annihilate at a critical value of the coupling parameter. Above αcrit no further solutions were found. In particular the value of the critical coupling decreased quite considerably with increasing baryon number as αcrit ≈ 0.040378/B 2 . It appears that the existence of a 4 critical coupling does not signal the collapse of a Skyrmion to form a black hole. In fact the metric factor S(x) = (1 -2µ(x) x ) is non-zero at αcrit; there simply ceases to be any stationary points of the action above the critical coupling.\n\nThe major problem with the ansatz (7) is that it leads to unstable solutions, i.e. for any given value of α, MADM (B = N ) > N MADM (B = 1). This is actually the case for the pure Skyrme model as well where the hedgehog anstaz (7) with B > 1 does not correspond to the lowest energy solution for the model. The solutions of the pure Skyrme model when B > 1 are known not to be spherically symmetric [11] but are stable i.e. E(B = N ) < N * E(B = 1).\n\nIt was actually shown by Houghton et al [10] , [12] that the multi-baryon solutions of the pure Skyrme model can be well approximated by the so called rational maps ansatz which is a generalisation of the hedgehog ansatz. While not radially symmetric, the ansatz separates its radial and angular dependence through a profile function and a rational map respectively.\n\nIn the following sections we will generalise the construction of Houghton et al to approximate the solution of the Einstein-Skyrme model."},{"section_type":"OTHER","section_title":"The Rational Map Ansatz","text":"The rational map ansatz introduced by Houghton et al.[10] works by decomposing the field into angular and radial parts. Using the polar coordinates in R 3 and defining the stereographic coordinates z = tan(θ/2) exp iφ the ansatz reads [10]\n\nU = exp (i σ • nRF (r, t)) ( 9\n\n) where nR = 1 1 + |R| 2 `2ℜ(R), 2ℑ(R), 1 -|R| 2 ´( 10\n\n)\n\nis a unit vector where R is a rational function of z.\n\nIt can be shown that the baryon number for Skyrmions constructed in this way, is equal to the degree of the rational map providing we take the boundary conditions\n\nF (r = 0) = π F (r = ∞) = 0. ( 11\n\n)\n\nSubstituting the ansatz (9) into the action for the model and scaling to dimensionless\n\n5\n\nvariables as earlier, we obtain the following reduced Hamiltonian\n\nH = 16πFπ e »Z ∞ 0 A(x) \" 1 2 S(x)F (x) ′2 x 2 + Bsin 2 F (x)(1 + S(x)F (x) ′2 ) + Isin 4 F (x) 2x 2 - µ ′ (x) α « dx + µ(∞) α - ( 12\n\n)\n\nwhere\n\nS(x) = 1 - 2µ(x) x ( 13\n\n)\n\nFrom which one obtains the following field equations\n\nµ(x) ′ = α \" 1 2 S(x)x 2 F (x) ′2 + B sin 2 F (x) + S(x)BF (x) ′2 sin 2 F (x) + I sin 4 F (x) 2x 2 « ( 14\n\n) F (x) ′′ = 1 S(x)V (x) » sin 2F (x) \" B + S(x)BF (x) ′2 + I sin 2 F (x) x 2 « - αS(x)F (x) ′3 V (x) 2 x -S(x) ′ F (x) ′ V (x) -S(x)F (x) ′ V (x) ′ ˜( 15\n\n) and A(x) ′ = αA(x)F (x) ′2 \" x + 2B sin 2 F (x) x « ( 16\n\n)\n\nwhere, for convenience, we have defined V (x) as\n\nV (x) = x 2 + 2B sin 2 F (x). ( 17\n\n)\n\nB is the baryon number and\n\nI = 1 4π Z \" 1 + |z| 2 1 + |R| 2 ˛dR dz ˛«4 2idzdz (1 + |z| 2 ) 2 ( 18\n\n)\n\nIts value depends on the chosen rational map R. To compute low energy configurations for a given baryon charge B one must find the rational map R or degree B that minimize I.\n\nThis has been done in [10] and [11] for several values of B. Moreover when b is large, one can use the approximation[11] I ≈ 1.28B 2 . The value of I so obtained is then used as a parameter and one can solve equations (14) -(16) for the radial profiles F (x), A(x) and µ(x).\n\nWe should point out here that for the pure Skyrme model the rational map ansatz produce very good approximation to the multi skyrmion solutions [10]: the energies are only 3 or 4 percent higher and the energy densities exhibit the same symmetries and differ by very little. All the solutions computed by Battye and Sutcliffe [11] , when B is not too small, have somehow the shape of a hollow shell. The baryon density is very small everywhere outside the shell, while on the shell itself, it forms a lattice of hexagons and pentagons."},{"section_type":"OTHER","section_title":"5 Hollow Skyrmion Shells","text":"Using the rational map ansatz, we will now solves the field equations (14) -(16) to compute some low action configurations. These solutions will correspond, initially, to a hollow shell of Skyrmions similar to the configuration obtained with the rational map anstaz for the pure Skyrme model. In the following sections we will show how our ansatz can be generalised to allow for more realistic configuration made out of embedded shells.\n\nThe first thing to note about our solutions is that we again obtain two branches of solutions at each baryon number (Fig. 1 ). Obtaining this same qualitative behaviour is not surprising when one considers that the B = 1 rational map Skyrmion reproduces the usual B = 1 hedgehog. However, the behaviour of the critical coupling itself is drastically altered for the rational map generated configurations. Namely, we observe that it decreases as approximately 0.040378/B 1 2 (Fig. 2 ). In particular this means that for a given value of the coupling, the rational map generated skyrmions can possess a much higher topological charge than their hedgehog counterparts, before there ceases to be any solutions. Quantitatively if B hedgehog is the maximum baryon number for which hedgehog solutions can be found at a given value of the coupling, then the highest baryon number rational map solution found at the same value of α will be approximately B 4 hedgehog . Again we observe that the metric function S(x) is non-zero at the critical coupling for all the solutions we have found and as such a horizon has not formed.\n\nIn Table 1 we present the radius, ADM mass per baryon and minimum value of the metric function, S(x), for configurations up to the maximum baryon number allowed at α = 1 × 10 -6 . These values were obtained by direct numerical solution of equations (14) -(16), where we have used the boundary data as specified in (11) .\n\nWe didn't didn't use the physical value of α (7.3×10 -40 ) because for this value, the ratio between the width of the shell and its radius is so small when we reach the maximum value of B that it becomes very difficult to solve the equation reliably. The value α = 1 × 10 -6 is small enough to allow for a shell with a large baryon number to exist but large enough to make it possible to compute these solution nears the critical value of B for a single shell configuration. 7 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 M ADM α Figure 1: Plot of the two branches of solutions found for B=2 configurations generated with the rational map ansatz.\n\nThe major difference between these configuration and the solutions of Bizon and Chmadj is that the rational map ansatz configurations become more bound when the baryon number increases This suggests the possibility that giant gravitating Skyrmions can be bound and consequently, that the Skyrme model can be used to study baryon stars.\n\nAnother interesting feature of the data is the observed change in the radius of the solutions with increasing baryon number. We note that the radius grows as approximately B 1 2 .\n\nHowever there are two main deviations from this. Firstly, the constant of proportionality relating the radius to the square root of the baryon number decreases slightly but persistently as we increase the baryon number, indicating the gravitational interaction becoming more important as the number of baryons increases.\n\nAs we approach the maximum baryon charge that can exist at α = 1 × 10 -6 , we also notice that the radius of the skyrmion actually decreases as we add more baryons. This shows that the gravitation pull plays a crucial role near the critical value of the skyrmion.\n\nThis is a tantalising property when one considers that generally a neutron star's radius must decrease for an increase in mass in order to achieve sufficient degenerate neutron pressure 8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 2 4 6 8 10 12 14 16 18 20 α crit B Figure 2: Plot of the decrease in α crit with increasing baryon number, for configurations generated with the rational map ansatz. +: α c r for the minimum value of I; curve: α c r = 0.0404B -1/2 .\n\nto support the star.\n\nTo motivate the further approximation that we will introduce in the next section, we now look at the profiles of the configuration that we have computed. First of all, we observe that the profile function F (x) stays approximately at its boundary value, π, for a finite radial distance before decreasing monotonically over some small region and finally attaining its second boundary value, 0. A similar behaviour is seen for both the mass field µ(x) and the metric field A(x) (see Fig. 3 ). Furthermore, as we increase the baryon number the structure becomes more pronounced, with the distance before the fields change (shell radius) increasing significantly, whilst the distance over which the fields change (shell width) settles to a constant size. We conclude that at large baryon numbers, those configurations correspond to hollow shells where the baryons are distributed on a tight lattice over the shell. As such the, structures are nearly spherical, validating our choice of radial metric.\n\nSuch structures immediately pose an interesting question. Can the gravitating Skyrmions exist as shells with more than one layer? To investigate this we note that it is possible to 9 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 1 0.8763 1.2315 1.0000 4 1.7728 1.1365 1.0000 8 2.5065 1.1180 1.0000 100 8.6829 1.0845 0.9999 500 19.3994 1.0827 0.9998 1 × 10 3 27.4314 1.0825 0.9997 1 × 10 4 86.7192 1.0821 0.9989 1 × 10 5 274.0397 1.0814 0.9963 1 × 10 6 864.6968 1.0792 0.9883 1 × 10 7 2715.0729 1.0722 0.9628 1 × 10 8 8377.4601 1.0500 0.88192 1 × 10 9 23585.5315 0.9743 0.6107 1.5 × 10 9 26860.2040 0.9463 0.5020 1.8 × 10 9 27470.2449 0.9302 0.4256 1.81 × 10 9 27456.5804 0.9296 0.4225 1.85 × 10 9 27357.9201 0.9274 0.4090 1.9 × 10 9 27078.6014 0.9246 0.3886 1.95 × 10 9 26126.5508 0.9217 0.3517 1.951 × 10 9 26050.7695 0.9217 0.3495 1.952 × 10 9 25937.4210 0.9216 0.3463 Table 1: Properties of the one shell low energy configuration for α = 1 × 10-6 10 0 0.5 1 1.5 2 2.5 3 3.5 4 1210 1215 1220 1225 1230 1235 F(x) x 0 2 4 6 8 10 1210 1215 1220 1225 1230 1235 µ(x) x 0.99 0.995 1 1.005 1.01 1210 1215 1220 1225 1230 1235 A(x) x Figure 3: Numerical solutions for the profiles F (x), µ(x) and A(x) when B = 2 × 10 6 and α = 1 × 10 -6 .\n\nmodify the boundary condition (11) to read\n\nF (r = 0) = N π ( 19\n\n) F (r = ∞) = 0 ( 20\n\n)\n\nwhilst still ensuring that the Skyrme field is well defined at the origin. This idea was first used in [12] to construct two shell configurations for the pure Skyrme model.\n\nThe baryon charge is now N times the degree of the rational map. Fig. 4 shows the structure of the solutions we find in this case when N = 2. They suggest that the Skyrmion now exists as a N -layered structure. This is exhibited in the form of the profile, mass and metric functions which interpolate between the boundary values in N distinct steps of equal 11 0 1 2 3 4 5 6 7 1200 1205 1210 1215 1220 1225 1230 F(x) x 0 5 10 15 20 1200 1205 1210 1215 1220 1225 1230 µ(x) x 0.995 1 1.005 1.01 1.015 1.02 1200 1205 1210 1215 1220 1225 1230 A(x) x Figure 4: Numerical solutions for for the profiles F (x), µ(x) and A(x) for 2 layers configurations (F (0) = 2π) when B = 2 × 10 6 and α = 1 × 10 -6 .\n\nsize stacked next to each other.\n\nWe can therefore think of this as a naive way of constructing a gravitating Skyrmion.\n\nInstead of using the boundary conditions as in (11) and a rational map of degree B we consider constructing the B-Skyrmion using a rational map of degree B/N (with the associated value of I) and the boundary condition (20). This is a crude construction as we are effectively considering N adjacent shells of baryons, all with the same baryon number. We might realistically expect that the baryon number per shell and distribution of shells may vary significantly for the minimum energy configuration. Nevertheless we shall study the properties of such structures. In fact, in the case where the baryon number is large and the number of shells is small, we expect this crude construction to be quite valid. That is, we do not expect the baryon number to change significantly over the few shells at large radius.\n\n12 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 4 1.2898 1.6179 1.0000 8 1.7858 1.4072 1.0000 100 6.1754 1.1363 0.9999 1 × 10 3 19.4157 1.0913 0.9996 1 × 10 4 61.3207 1.0833 0.9985 1 × 10 5 193.7006 1.0812 0.9949 1 × 10 6 610.6271 1.0779 0.9835 1 × 10 7 1911.3704 1.0680 0.9475 1 × 10 8 5825.2626 1.0362 0.8325 9.0 × 10 8 13736.9982 0.9302 0.4258 9.7 × 10 8 13263.0853 0.9224 0.3644 9.76 × 10 8 12998.2817 0.9217 0.3480 9.764 × 10 8 12931.5189 0.9216 0.3444 9.7647 × 10 8 12895.6984 0.9216 0.3425 9.76472 × 10 8 12891.4247 0.9216 0.3423 9.764724 × 10 8 12889.8645 0.9216 0.3422 Table 2: Table of properties of double layer solutions obtained numerically at α = 1 × 10-6\n\nFor the remainder of this section we will restrict ourselves to the case where N = 2.\n\nTable 2 summarises the properties of double layered gravitating skyrmions up to the maximum baryon charge allowed at α = 1 × 10 -6 . Briefly, we note the main features.\n\nFirstly, for all baryon numbers, the radius of the double layered solutions is significantly less than their single layered counterparts. This is not surprising as the baryon charge exists over a thicker region and so the mean radius can decrease with the baryon density remaining the same. Secondly, when B is large enough, i.e. when the double layer starts to make sense, the double layer solutions are energetically favourable when one compares the ADM mass with the single layer solutions. Finally we note that the maximum baryon number allowed (at the given coupling) is almost twice as much in the case of the single layer skyrmions.\n\nOf course the results of this section are not really the main regime of interest. We clearly 13 need to study configurations of extremely high baryon number (of order 10 58 ) relevant for baryon stars.\n\nWe will now discuss this high baryon number regime."},{"section_type":"OTHER","section_title":"The Ramp-profile Approximation","text":"Unfortunately, at very high baryon numbers, eqns. ((14) -(16)) become difficult to handle numerically. This is largely because the radius of the solutions becomes much larger than the distance over which the fields change. That is, we need to integrate over a region which is much less than 10 -16 radius, and so even double precision data types have insufficient precision.\n\nMoreover, single shell configurations are not physically relevant and multiple shells will only yield configuration that looks like a star if the number of layers is very large, typically well over 10 17 . With such a large number of layers we won't be able to solve the equation numerically as we will need at least 10 times as many sampling points for the profile functions.\n\nWe must thus resort to another level of approximation: approximate the profile functions by profiles that are piecewise linear. This is inspired by the work of Kopeliovich [13] [14] except that our ansatz has to be piecewise linear to be able to generate configurations with a huge number of layers. After defining the ansatz for an arbitrary number of layers, we will show that for a single layer configuration the ansatz produces configurations that are in good agreements with the rational map ansatz configuration. Then we will use the new ansatz to construct configurations that are made out of a very large no of layers.\n\nWe have shown, in the previous section, that one can construct shell like structures with very large Baryon numbers. At large baryon numbers, the Skyrmions resemble shell like structures. That is, the fields are constant nearly everywhere except in a small region corresponding to the shell. In that region, the profile look like linear functions smoothly linked to the constant parts at the edges (cfr. Fig 3 ). Motivated by this we approximate the fields by the ramp-functions\n\nF (x) = N π 2 -(x -x0) π W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 21\n\n) µ(x) = M 2 + (x -x0) M N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 22\n\n) A(x) = (1 + A0) 2 + (x -x0) (1 -A0) N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 23\n\n)\n\nIn the above there are four free parameters, namely the central radius x0 of the shell\n\nover which the fields change, the width of the shell W , the mass field at spatial infinity M and the value of the metric field at the origin A0 such that limx→∞ = 0. N is the number of layers we wish to study and, as such, is treated as an input parameter.\n\nThe picture is of a gravitating skyrmion with very high baryon number existing as N thin layers or shell of small thickness.\n\nThe above ansatz, allow us to find an approximation to the integrated energy. To do this we use the fact that the shell width is much smaller than the radius at large baryon numbers. In particular to evaluate the action integral we can approximate expressions of the type R G(x) sin p F (x) for any function G(x) that varies very little over the width of the shell by R G(x0) sin p F (x). We then use the fact that\n\nZ x 0 +NW/2 x 0 -NW/2 sin p F (x) = N π W Z π 0 sin p y dy. ( 24\n\n)\n\nThis leads to the following expression for the energy:\n\nE = - 16πFπ e » M α \" 1 + A0 2 « - π 2 W » \" 1 + A0 2 « \" x 2 0 -M x0 + W 2 12 - M W 6 « + \" 1 -A0 W « \" W 2 x0 6 - M W 2 12 - M W x0 6 « - + B \" 1 + A0 2 « 1 2 \" M π 2 W x0 -W - π 2 W « - 3IW 16x 2 0 \" 1 + A0 2 « - M α - ( 25\n\n)\n\nTo find the configurations which minimize this energy we first minimised it with respect to A0 and M algebraically in order to find an expression for the energy as a function of the width and radius only. Then we minimised this numerically using Mathematica. We will now discuss the features of these configurations.\n\nFirst of all, we must compare the results obtained with the ramp-profile when N = 1 and compare them to the result obtained with the full profile. Tables. 3 and 4 show the properties of solutions we obtained using the ramp-profile approximation, again at α = 1 × 10 -6 . All the general features of the full numerical solutions are reproduced. In particular, the approximate B 1 2 scaling and then decrease of the radius, the decreasing ADM mass and the differences between the double and single layer solutions are all exhibited by the data obtained using the ramp-profile approximation.\n\nQuantitatively though, there are some differences. The approximation allows a significant increase in the maximum allowed baryon charge. Also, the radius of configurations obtained using the approximation, tend to be smaller than those obtained numerically. If we concentrate on the baryon numbers greater than 10 5 so as to ensure our approximation, 15 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 8.3063 3.1286 1.1023 0.9999 500 18.6031 3.1386 1.1160 0.9997 1 × 10 3 26.313 3.1397 1.1195 0.9996 1 × 10 4 83.206 3.1396 1.1254 0.9987 1 × 10 5 262.94 3.1357 1.1266 0.9960 1 × 10 6 829.60 3.1230 1.1251 0.9872 1 × 10 7 2604.2 3.0825 1.1186 0.9595 1 × 10 8 8032.8 2.9512 1.0972 0.8713 1 × 10 9 22899 2.4837 1.0272 0.5772 2 × 10 9 29121 2.1092 0.9818 0.3645 2.8 × 10 9 29098 1.6623 0.9505 0.1380 2.83 × 10 9 28514 1.6066 0.9495 0.1119 2.839 × 10 9 28024 1.5671 0.94922 0.09373 2.8397 × 10 9 27869.3 1.5556 0.94924 0.08845 2.83975 × 10 9 27869.8 1.5524 0.94925 0.08699 2.839752 × 10 9 27822 1.5521 0.94925 0.08687 Table 3: Table of properties of the single layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6 16 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 5.7924 3.0428 1.0692 0.9999 1 × 10 3 18.5788 3.1305 1.1047 0.9995 1 × 10 4 58.8202 3.1380 1.1201 0.9983 1 × 10 5 185.8420 3.1332 1.1246 0.9944 1 × 10 6 585.7950 3.1153 1.1233 0.9820 1 × 10 7 1833.0500 3.0578 1.1143 0.9428 1 × 10 8 5587.3600 2.8688 1.0840 0.8172 9 × 10 8 14147.1782 2.1859 0.9900 0.4065 9.764724 × 10 8 14472.3851 2.1276 0.9837 0.3746 1 × 10 9 14560.5000 2.1092 0.9818 0.3646 1.4 × 10 9 14549.0000 1.6623 0.9505 0.1381 1.41963 × 10 9 13994.0523 1.5644 0.9492 0.0926 1.419635134 × 10 9 13993.2000 1.5643 0.9492 0.0925 Table 4: Table of properties of the double layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6\n\nthat the width is much smaller than the radius, is valid, then at worst we find a discrepancy in the ADM mass of 11% and in the radius of 7%.\n\nIn general then, the data seems to confirm the reliability the ramp-profile approximation. In fact the approach will be even more reliable at the extremely high values of the baryon number that we are interested in. This is because the radius of solutions is of orders of magnitudes greater than the width in such a regime, consistent with the approximations we have made.\n\nMoreover, whilst searching for minima of the energy does not allow us to probe both branches of solutions, it does allow us to locate the value of αcrit. We again obtain the approximate trend αcrit ∝ B -1 2 , for large B. Now in order to say anything about the possibility of baryon stars in the Skyrme model we need to be able to verify that the decrease in the ADM mass per baryon we observed at low and moderate baryon numbers, extends to baryon numbers of order 10 58 for α = 7.3 × 10 -40 .\n\nTable. 5 summarizes our solutions in such a regime. Firstly we consider constructing a single layer self-gravitating Skyrmion with these values. We do indeed see that the configuration is bound. This is verified by checking that the ADM mass is lower (even at this significantly lower value of α) than for the B = 1 hedgehog. So the possibility of baryon stars in the Einstein-Skyrme model cannot be ruled out on the grounds of energy.\n\nThe Skyrmion exists as a giant thin shell, and the large baryon charge is distributed as a tight lattice over this. However a hollow shell is clearly not a realistic construction for a neutron star. This fact manifests itself in the extremely high radius of the configuration.\n\nTransferring to standard units, the single layer B = 10 58 gravitating Skyrmion has a radius of 2.42 × 10 10 km ! To address this issue, we can use a large number of layered Skyrmions as discussed earlier. This has several benefits. Firstly, as we are distributing the baryon number through a larger volume, then at a given baryon density the necessary radius can decrease. Similar to what we see in the double layer results. On top of this, we expect the radius to decrease further due to extra gravitational compression, as the outer layers of the Skyrmions feel the attraction of inner layers. Finally, the many layer approach is also a more realistic construction of a solid baryon star.\n\nThe results for using more and more layers in the construction (for fixed B and α), are also presented in Fig. 5 . We note that not only does the radius decrease significantly, but\n\nthe added gravitational binding further improves the energies of the configurations, reflected in the low ADM masses obtained. There appears to be a critical number of layers that can be used before there ceases to be any solutions and although the value of Smin is close zero at this point, the star still has not collapsed to form a black hole. Finally, we note that the radius of the Skyrmion at the critical number of layers is approximately 20.91km. This is comparable to a real neutron star, with a typical radius of 10km.\n\nWe reemphasise here that our approach to embedding shells of baryons is quite crude.\n\nFor few shells and large baryon number, we might reasonably believe that baryon number does not chance significantly from one shell to the next. However, when we embed many shells we should really consider that the baryon number of the inner most shells would likely be significantly less than the that of the outer shells. Nevertheless, our naive embedding has produced some interesting properties. In a future work we hope to improve our multi-layer construction to obtain a more realistic description of a baryon star. 19 N Shell R( 2 Fπe ) W M ADM /(6π 2 B) S min 1 × 10 2 8.3236 × 10 27 3.1416 1.1285 1.0000 1 × 10 3 2.6321 × 10 27 3.1416 1.1285 1.0000 1 × 10 4 × 10 26 3.1416 1.1285 1.0000 1 × 10 5 2.6321 × 10 26 3.1416 1.1285 1.0000 1 × 10 6 8.3236 × 10 25 3.1416 1.1285 1.0000 1 × 10 7 2.6321 × 10 25 3.1416 1.1285 1.0000 1 × 10 8 8.3236 × 10 24 3.1416 1.1285 1.0000 1 × 10 9 2.6321 × 10 24 3.1415 1.1285 1.0000 1 × 10 10 8.3234 × 10 23 3.1415 1.1285 0.9999 1 × 10 11 2.6319 × 10 23 3.1412 1.1285 0.9997 1 × 10 12 8.3216 × 10 22 3.1402 1.1283 0.9991 1 × 10 13 2.6301 × 10 22 3.1373 1.1278 0.9971 1 × 10 14 8.3034 × 10 21 3.1280 1.1263 0.9907 1 × 10 15 2.6118 × 10 21 3.0986 1.1213 0.9705 1 × 10 16 8.1147 × 10 20 3.0037 1.1057 0.9063 1 × 10 17 2.4001 × 10 20 2.6810 1.0552 0.6977 5 × 10 17 8.2066 × 10 19 1.7888 0.9552 0.2036 5.3 × 10 17 7.4172 × 10 19 1.6227 0.9491 0.1247 5.33 × 10 17 7.1871 × 10 19 1.5625 0.94866 0.0971 5.3306 × 10 17 7.1597 × 10 19 1.5549 0.94868 0.0936 5.33065 × 10 17 7.1525 × 10 19 1.5528 0.948694 0.0927 5.330657 × 10 17 7.1506 × 10 19 1.5523 0.948692 0.0924 Table 5: Table of properties of the step ansatz configurations for varying the number of embedded shells at fixed B = 10 58 and α = 7.3 × 10 -40 ."},{"section_type":"CONCLUSION","section_title":"7 Conclusions - Acknowledgements","text":"Previous work on the Einstein-Skyrme model highlighted a considerable problem with using the Skyrmions as a model for baryon stars. Namely, multibaryon hedgehog Skyrmions were simply not energetically favourable states. We have shown that this is simply a consequence of a poor ansatz for the true Skyrmion and, having used the more appropriate rational map ansatz, we have generated energetically favourable configurations of multibaryons.\n\nWe also observe the interesting property that near the critical coupling, the Skyrmions can decrease in radius as we add more baryons. This hints towards the similar behaviour exhibited by real neutron stars.\n\nAlthough the rational map ansatz does not have an exact radial symmetry, at large scale it does. The anisotropy only appears at the nucleon scale.\n\nFinally, since we started with the motivation of studying baryon stars within the Skyrme model, it is interesting to compare the features of our configurations with those of neutron stars. For realistic values, B = 10 58 and α = 7.3 × 10 -40 we find a minimal energy single layer configuration with radius=2.42 × 10 10 km. This is clearly too large for a neutron star (which is of order 10km. in radius). This is to be expected however due to the shell model we have taken. Firstly, as we are distributing the baryons over the surface area rather than throughout the volume of the star we naturally must require a much larger star for a given baryon number. This effect is two-fold in that if we were distributing the baryons throughout the volume, outer layers would feel the attraction of inner layers and enhanced radial compression would occur. The loss of such an effect is pronounced when we are considering realistically small values of the coupling.\n\nIt seems therefore that the way to construct baryon stars in the Skyrme model is to consider embedding shells of baryons within shells. This gives rise to more appropriate specifications for the star and is also more realistic. We do indeed observe such improvements for a many layered configuration. In fact the radius of B = 10 58 gravitating Skyrmion (at realistic α), can be decreased in this manner to approximately 20.91km.\n\nWe note however that this approach to shell embedding has only be done naively thus far.\n\nWe have only considered the case where the baryon number is equal for each shell. We really should allow the baryon number(and hence the rational map quantities) to vary over the shells. One approach towards this would be to assume that the baryon density is a constant over the shells. An even better approach would be to allow this to be a smoothly varying function that must be determined by minimising the energy. This will give a more realistic 21 description of baryon stars within the Einstein-Skyrme model, as traditional descriptions of neutron stars also involve many strata, of differing neutron density. We are currently investigating such configurations.\n\n[13] V. B. Kopeliovich \"The Bubbles of Matter from MultiSkyrmions\" JETP Lett. 73 (2001), 587-591; Pisma Zh.Eksp.Teor.Fiz. 73 (2001), 667-671 [14] V. B. Kopeliovich \"MultiSkyrmions and Baryonic Bags\" J.Phys. G28 (2002), 103-120 23\n\nGIP is supported by a PPARC studentship."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in a series of papers as a non-linear field theory of pions [2] , [3] . Skyrme's initial idea was to think of baryons (in particular the nucleons) as secondary structures arising from a more fundamental mesonic fluid. The key property of the model was that the baryons arose as solitons in a topological manner and thus possessed a conserved topological charge identified with the baryon number.\\n\\nThe lowest energy stable solutions of the model are termed Skyrmions and can be thought of as baryonic solitons. The Skyrme model has been very successful in modelling the structures of various nuclei and has been shown by Witten et al. [4] to possess the general features of a low energy effective field theory for QCD. Some studies of the Skyrme model coupled to gravity have previously been undertaken [1], [5] , [6] , mainly with the motivation of a comparison of its features with those of other nonlinear field theories coupled to gravity. Of particular note is the Einstein-Yang-Mills theory, in which gravitationally bound configurations of non-abelian gauge fields are produced.\\n\\nOther reasons for studying the Einstein-Skyrme model are cosmological and astrophysical ones. Various authors have studied black hole formation in the model, with the conclusion that the so-called no-hair conjecture may not hold [7] , [8] .\\n\\nThe purpose of this paper is to study large baryon number Skyrmions or configurations of Skyrmions in the Einstein-Skyrme model. In particular, we wish to investigate if stable solitonic stars could exist within the model and to compare their properties to those of neutron stars. Preliminary studies of Skyrmion stars have predicted instability to single particle decay [1] . However this was done using the hedgehog ansatz for baryon number larger than 1 which is known to lead to unstable solutions even for the usual Skyrme model. Since then, it has been shown that the Skyrme model has stable shell-like solutions [9] which can be well approximated by the so called rational map ansatz [10] .\\n\\nIn this paper we use the rational map ansatz and its extension to multiple shells to construct configurations in the Gravitating Skyrme model that have a very large number of baryon. We show that those configurations, contrary to the hedgehog ansatz are bound even for very large baryon numbers.\\n\\nTo construct configurations that have a baryon number comparable to that of neutron star, we have to introduce a further approximation, which we call the ramp ansatz. We show 2 that this anstaz introduces further errors of only a few percent and we use it to compute very large Skyrmion configurations.\\n\\nThe paper is organised as follow: first we outline the Einstein-Skyrme model and discuss the main features of the results on static gravitating SU (2) hedgehogs obtained by Bizon and Chmaj [1] . We then use the rational map ansatz to construct shell like gravitating multibaryon configurations and show that for a fixed value of the coupling constant, the configurations exist only when the baryon number is below a certain critical value. Finally we introduce a ramp profile approximation to construct solutions with extremely high baryon numbers. We show how accurate it is and use it to construct Skyrmion stars configuration.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Einstein-Skyrme Model\",\n \"text\": \"The action for gravitating Skyrmions is formed from the standard Skyrme action for the matter field and the Einstein-Hilbert action for the gravitational field.\\n\\nS = Z M √ -g \\\" L Sk - R 16πG « d 4 x. ( 1\\n\\n)\\n\\nHere L Sk is the Lagrangian density for the Skyrme model defined on the manifold M :\\n\\nL Sk = F 2 π 16\\n\\nT r(∇µU ∇ µ U -1 ) + 1 32e 2 T r[(∇µU )U -1 , (∇νU )U -1 ] 2 , (\\n\\n) 2\\n\\nwhere U belongs to SU (2). As we eventually wish to study baryon stars, we take a spherically symmetric metric, such as associated with the line element ds 2 = -A 2 (r) \\\" 1 -2m(r) r « dt 2 + \\\" 1 -2m(r) r\\n\\n« -1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), ( 3\\n\\n)\\n\\nwhere A(r) and m(r) are two profile functions that must be determined by solving the Einstein equations for the model. Our choice of ansatz is motivated by the fact that although in some cases we will be studying non-spherical Skyrmion configurations, the regime we are primarily interested in (i.e. Skyrmions of extremely high baryon number) will be shown to admit quasi spherical solutions. Also, for realistic values of the couplings, the gravitational interaction is small compared to the Skyrme interaction and thus the use of a spherical metric even with non-spherical configurations, is not a great problem.\\n\\nFrom (3), it can be shown that the Ricci scalar is\\n\\nR = -2 Ar 2 `-A ′′ r 2 -2A ′ r + 2A ′′ rm + A ′ m + 3A ′ rm ′ + Arm ′′ + 2Am ′ ´( 4\\n\\n)\\n\\nwhich, after integrating various terms by parts and noting that asymptotic flatness requires both A(r) and m(r) to take a constant value at spatial infinity, reduces the gravitational part of the action to Sgr = Z A(r) \\\" -m ′ (r) G « dr + m(∞) G . (5) For what follows, it will be convenient to scale to dimensionless variables by defining x = eFπr and µ(x) = eFπm(r)/2, resulting in one dimensionless coupling parameter for the model, α = πF 2 π G. We note that taking Fπ = 186M ev and G = 6.72 × 10 -45 M ev -2 , then the physical value of the coupling is α = 7.3 × 10 -40 .\\n\\nAs the Skyrme field is an SU (2) valued scalar field, at any given time one can think of it as a map from R 3 to the SU (2) manifold. Finite energy considerations impose that the field at spatial infinity should map to the same point on SU (2), say the identity. Thus, one can simply think of the Skyrme field as a map between three-spheres. All such maps fall into disjoint homotopy classes characterised by their winding number. This winding number is a conserved topological charge because no continuous deformation of the field and thus no time evolution, can allow transitions between homotopy classes. It is this topological charge that is interpreted as the baryon number.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Gravitating Hedgehog Skyrmions\",\n \"text\": \"Gravitating Skyrmions were first studied by Bizon and Chmaj[1] who analysed the properties of static spherically symmetric gravitating SU (2) skyrmions. Taking the Hedgehog Ansatz for the Skyrme field\\n\\nU = exp(i -→ σ .rF (r)) ( 6\\n\\n)\\n\\nsubject to the boundary conditions\\n\\nF (r = 0) = Bπ ( 7\\n\\n) F (r = ∞) = 0 ( 8\\n\\n)\\n\\nwhere B is the Baryon number associated with the Skyrmion configuration, they derived the Euler-Lagrange equation for the profiles F (r), (A(r) and m(r) and found that the model admit two branches of global solitonic solutions at each given baryon number, which annihilate at a critical value of the coupling parameter. Above αcrit no further solutions were found. In particular the value of the critical coupling decreased quite considerably with increasing baryon number as αcrit ≈ 0.040378/B 2 . It appears that the existence of a 4 critical coupling does not signal the collapse of a Skyrmion to form a black hole. In fact the metric factor S(x) = (1 -2µ(x) x ) is non-zero at αcrit; there simply ceases to be any stationary points of the action above the critical coupling.\\n\\nThe major problem with the ansatz (7) is that it leads to unstable solutions, i.e. for any given value of α, MADM (B = N ) > N MADM (B = 1). This is actually the case for the pure Skyrme model as well where the hedgehog anstaz (7) with B > 1 does not correspond to the lowest energy solution for the model. The solutions of the pure Skyrme model when B > 1 are known not to be spherically symmetric [11] but are stable i.e. E(B = N ) < N * E(B = 1).\\n\\nIt was actually shown by Houghton et al [10] , [12] that the multi-baryon solutions of the pure Skyrme model can be well approximated by the so called rational maps ansatz which is a generalisation of the hedgehog ansatz. While not radially symmetric, the ansatz separates its radial and angular dependence through a profile function and a rational map respectively.\\n\\nIn the following sections we will generalise the construction of Houghton et al to approximate the solution of the Einstein-Skyrme model.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Rational Map Ansatz\",\n \"text\": \"The rational map ansatz introduced by Houghton et al.[10] works by decomposing the field into angular and radial parts. Using the polar coordinates in R 3 and defining the stereographic coordinates z = tan(θ/2) exp iφ the ansatz reads [10]\\n\\nU = exp (i σ • nRF (r, t)) ( 9\\n\\n) where nR = 1 1 + |R| 2 `2ℜ(R), 2ℑ(R), 1 -|R| 2 ´( 10\\n\\n)\\n\\nis a unit vector where R is a rational function of z.\\n\\nIt can be shown that the baryon number for Skyrmions constructed in this way, is equal to the degree of the rational map providing we take the boundary conditions\\n\\nF (r = 0) = π F (r = ∞) = 0. ( 11\\n\\n)\\n\\nSubstituting the ansatz (9) into the action for the model and scaling to dimensionless\\n\\n5\\n\\nvariables as earlier, we obtain the following reduced Hamiltonian\\n\\nH = 16πFπ e »Z ∞ 0 A(x) \\\" 1 2 S(x)F (x) ′2 x 2 + Bsin 2 F (x)(1 + S(x)F (x) ′2 ) + Isin 4 F (x) 2x 2 - µ ′ (x) α « dx + µ(∞) α - ( 12\\n\\n)\\n\\nwhere\\n\\nS(x) = 1 - 2µ(x) x ( 13\\n\\n)\\n\\nFrom which one obtains the following field equations\\n\\nµ(x) ′ = α \\\" 1 2 S(x)x 2 F (x) ′2 + B sin 2 F (x) + S(x)BF (x) ′2 sin 2 F (x) + I sin 4 F (x) 2x 2 « ( 14\\n\\n) F (x) ′′ = 1 S(x)V (x) » sin 2F (x) \\\" B + S(x)BF (x) ′2 + I sin 2 F (x) x 2 « - αS(x)F (x) ′3 V (x) 2 x -S(x) ′ F (x) ′ V (x) -S(x)F (x) ′ V (x) ′ ˜( 15\\n\\n) and A(x) ′ = αA(x)F (x) ′2 \\\" x + 2B sin 2 F (x) x « ( 16\\n\\n)\\n\\nwhere, for convenience, we have defined V (x) as\\n\\nV (x) = x 2 + 2B sin 2 F (x). ( 17\\n\\n)\\n\\nB is the baryon number and\\n\\nI = 1 4π Z \\\" 1 + |z| 2 1 + |R| 2 ˛dR dz ˛«4 2idzdz (1 + |z| 2 ) 2 ( 18\\n\\n)\\n\\nIts value depends on the chosen rational map R. To compute low energy configurations for a given baryon charge B one must find the rational map R or degree B that minimize I.\\n\\nThis has been done in [10] and [11] for several values of B. Moreover when b is large, one can use the approximation[11] I ≈ 1.28B 2 . The value of I so obtained is then used as a parameter and one can solve equations (14) -(16) for the radial profiles F (x), A(x) and µ(x).\\n\\nWe should point out here that for the pure Skyrme model the rational map ansatz produce very good approximation to the multi skyrmion solutions [10]: the energies are only 3 or 4 percent higher and the energy densities exhibit the same symmetries and differ by very little. All the solutions computed by Battye and Sutcliffe [11] , when B is not too small, have somehow the shape of a hollow shell. The baryon density is very small everywhere outside the shell, while on the shell itself, it forms a lattice of hexagons and pentagons.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"5 Hollow Skyrmion Shells\",\n \"text\": \"Using the rational map ansatz, we will now solves the field equations (14) -(16) to compute some low action configurations. These solutions will correspond, initially, to a hollow shell of Skyrmions similar to the configuration obtained with the rational map anstaz for the pure Skyrme model. In the following sections we will show how our ansatz can be generalised to allow for more realistic configuration made out of embedded shells.\\n\\nThe first thing to note about our solutions is that we again obtain two branches of solutions at each baryon number (Fig. 1 ). Obtaining this same qualitative behaviour is not surprising when one considers that the B = 1 rational map Skyrmion reproduces the usual B = 1 hedgehog. However, the behaviour of the critical coupling itself is drastically altered for the rational map generated configurations. Namely, we observe that it decreases as approximately 0.040378/B 1 2 (Fig. 2 ). In particular this means that for a given value of the coupling, the rational map generated skyrmions can possess a much higher topological charge than their hedgehog counterparts, before there ceases to be any solutions. Quantitatively if B hedgehog is the maximum baryon number for which hedgehog solutions can be found at a given value of the coupling, then the highest baryon number rational map solution found at the same value of α will be approximately B 4 hedgehog . Again we observe that the metric function S(x) is non-zero at the critical coupling for all the solutions we have found and as such a horizon has not formed.\\n\\nIn Table 1 we present the radius, ADM mass per baryon and minimum value of the metric function, S(x), for configurations up to the maximum baryon number allowed at α = 1 × 10 -6 . These values were obtained by direct numerical solution of equations (14) -(16), where we have used the boundary data as specified in (11) .\\n\\nWe didn't didn't use the physical value of α (7.3×10 -40 ) because for this value, the ratio between the width of the shell and its radius is so small when we reach the maximum value of B that it becomes very difficult to solve the equation reliably. The value α = 1 × 10 -6 is small enough to allow for a shell with a large baryon number to exist but large enough to make it possible to compute these solution nears the critical value of B for a single shell configuration. 7 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 M ADM α Figure 1: Plot of the two branches of solutions found for B=2 configurations generated with the rational map ansatz.\\n\\nThe major difference between these configuration and the solutions of Bizon and Chmadj is that the rational map ansatz configurations become more bound when the baryon number increases This suggests the possibility that giant gravitating Skyrmions can be bound and consequently, that the Skyrme model can be used to study baryon stars.\\n\\nAnother interesting feature of the data is the observed change in the radius of the solutions with increasing baryon number. We note that the radius grows as approximately B 1 2 .\\n\\nHowever there are two main deviations from this. Firstly, the constant of proportionality relating the radius to the square root of the baryon number decreases slightly but persistently as we increase the baryon number, indicating the gravitational interaction becoming more important as the number of baryons increases.\\n\\nAs we approach the maximum baryon charge that can exist at α = 1 × 10 -6 , we also notice that the radius of the skyrmion actually decreases as we add more baryons. This shows that the gravitation pull plays a crucial role near the critical value of the skyrmion.\\n\\nThis is a tantalising property when one considers that generally a neutron star's radius must decrease for an increase in mass in order to achieve sufficient degenerate neutron pressure 8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 2 4 6 8 10 12 14 16 18 20 α crit B Figure 2: Plot of the decrease in α crit with increasing baryon number, for configurations generated with the rational map ansatz. +: α c r for the minimum value of I; curve: α c r = 0.0404B -1/2 .\\n\\nto support the star.\\n\\nTo motivate the further approximation that we will introduce in the next section, we now look at the profiles of the configuration that we have computed. First of all, we observe that the profile function F (x) stays approximately at its boundary value, π, for a finite radial distance before decreasing monotonically over some small region and finally attaining its second boundary value, 0. A similar behaviour is seen for both the mass field µ(x) and the metric field A(x) (see Fig. 3 ). Furthermore, as we increase the baryon number the structure becomes more pronounced, with the distance before the fields change (shell radius) increasing significantly, whilst the distance over which the fields change (shell width) settles to a constant size. We conclude that at large baryon numbers, those configurations correspond to hollow shells where the baryons are distributed on a tight lattice over the shell. As such the, structures are nearly spherical, validating our choice of radial metric.\\n\\nSuch structures immediately pose an interesting question. Can the gravitating Skyrmions exist as shells with more than one layer? To investigate this we note that it is possible to 9 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 1 0.8763 1.2315 1.0000 4 1.7728 1.1365 1.0000 8 2.5065 1.1180 1.0000 100 8.6829 1.0845 0.9999 500 19.3994 1.0827 0.9998 1 × 10 3 27.4314 1.0825 0.9997 1 × 10 4 86.7192 1.0821 0.9989 1 × 10 5 274.0397 1.0814 0.9963 1 × 10 6 864.6968 1.0792 0.9883 1 × 10 7 2715.0729 1.0722 0.9628 1 × 10 8 8377.4601 1.0500 0.88192 1 × 10 9 23585.5315 0.9743 0.6107 1.5 × 10 9 26860.2040 0.9463 0.5020 1.8 × 10 9 27470.2449 0.9302 0.4256 1.81 × 10 9 27456.5804 0.9296 0.4225 1.85 × 10 9 27357.9201 0.9274 0.4090 1.9 × 10 9 27078.6014 0.9246 0.3886 1.95 × 10 9 26126.5508 0.9217 0.3517 1.951 × 10 9 26050.7695 0.9217 0.3495 1.952 × 10 9 25937.4210 0.9216 0.3463 Table 1: Properties of the one shell low energy configuration for α = 1 × 10-6 10 0 0.5 1 1.5 2 2.5 3 3.5 4 1210 1215 1220 1225 1230 1235 F(x) x 0 2 4 6 8 10 1210 1215 1220 1225 1230 1235 µ(x) x 0.99 0.995 1 1.005 1.01 1210 1215 1220 1225 1230 1235 A(x) x Figure 3: Numerical solutions for the profiles F (x), µ(x) and A(x) when B = 2 × 10 6 and α = 1 × 10 -6 .\\n\\nmodify the boundary condition (11) to read\\n\\nF (r = 0) = N π ( 19\\n\\n) F (r = ∞) = 0 ( 20\\n\\n)\\n\\nwhilst still ensuring that the Skyrme field is well defined at the origin. This idea was first used in [12] to construct two shell configurations for the pure Skyrme model.\\n\\nThe baryon charge is now N times the degree of the rational map. Fig. 4 shows the structure of the solutions we find in this case when N = 2. They suggest that the Skyrmion now exists as a N -layered structure. This is exhibited in the form of the profile, mass and metric functions which interpolate between the boundary values in N distinct steps of equal 11 0 1 2 3 4 5 6 7 1200 1205 1210 1215 1220 1225 1230 F(x) x 0 5 10 15 20 1200 1205 1210 1215 1220 1225 1230 µ(x) x 0.995 1 1.005 1.01 1.015 1.02 1200 1205 1210 1215 1220 1225 1230 A(x) x Figure 4: Numerical solutions for for the profiles F (x), µ(x) and A(x) for 2 layers configurations (F (0) = 2π) when B = 2 × 10 6 and α = 1 × 10 -6 .\\n\\nsize stacked next to each other.\\n\\nWe can therefore think of this as a naive way of constructing a gravitating Skyrmion.\\n\\nInstead of using the boundary conditions as in (11) and a rational map of degree B we consider constructing the B-Skyrmion using a rational map of degree B/N (with the associated value of I) and the boundary condition (20). This is a crude construction as we are effectively considering N adjacent shells of baryons, all with the same baryon number. We might realistically expect that the baryon number per shell and distribution of shells may vary significantly for the minimum energy configuration. Nevertheless we shall study the properties of such structures. In fact, in the case where the baryon number is large and the number of shells is small, we expect this crude construction to be quite valid. That is, we do not expect the baryon number to change significantly over the few shells at large radius.\\n\\n12 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 4 1.2898 1.6179 1.0000 8 1.7858 1.4072 1.0000 100 6.1754 1.1363 0.9999 1 × 10 3 19.4157 1.0913 0.9996 1 × 10 4 61.3207 1.0833 0.9985 1 × 10 5 193.7006 1.0812 0.9949 1 × 10 6 610.6271 1.0779 0.9835 1 × 10 7 1911.3704 1.0680 0.9475 1 × 10 8 5825.2626 1.0362 0.8325 9.0 × 10 8 13736.9982 0.9302 0.4258 9.7 × 10 8 13263.0853 0.9224 0.3644 9.76 × 10 8 12998.2817 0.9217 0.3480 9.764 × 10 8 12931.5189 0.9216 0.3444 9.7647 × 10 8 12895.6984 0.9216 0.3425 9.76472 × 10 8 12891.4247 0.9216 0.3423 9.764724 × 10 8 12889.8645 0.9216 0.3422 Table 2: Table of properties of double layer solutions obtained numerically at α = 1 × 10-6\\n\\nFor the remainder of this section we will restrict ourselves to the case where N = 2.\\n\\nTable 2 summarises the properties of double layered gravitating skyrmions up to the maximum baryon charge allowed at α = 1 × 10 -6 . Briefly, we note the main features.\\n\\nFirstly, for all baryon numbers, the radius of the double layered solutions is significantly less than their single layered counterparts. This is not surprising as the baryon charge exists over a thicker region and so the mean radius can decrease with the baryon density remaining the same. Secondly, when B is large enough, i.e. when the double layer starts to make sense, the double layer solutions are energetically favourable when one compares the ADM mass with the single layer solutions. Finally we note that the maximum baryon number allowed (at the given coupling) is almost twice as much in the case of the single layer skyrmions.\\n\\nOf course the results of this section are not really the main regime of interest. We clearly 13 need to study configurations of extremely high baryon number (of order 10 58 ) relevant for baryon stars.\\n\\nWe will now discuss this high baryon number regime.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Ramp-profile Approximation\",\n \"text\": \"Unfortunately, at very high baryon numbers, eqns. ((14) -(16)) become difficult to handle numerically. This is largely because the radius of the solutions becomes much larger than the distance over which the fields change. That is, we need to integrate over a region which is much less than 10 -16 radius, and so even double precision data types have insufficient precision.\\n\\nMoreover, single shell configurations are not physically relevant and multiple shells will only yield configuration that looks like a star if the number of layers is very large, typically well over 10 17 . With such a large number of layers we won't be able to solve the equation numerically as we will need at least 10 times as many sampling points for the profile functions.\\n\\nWe must thus resort to another level of approximation: approximate the profile functions by profiles that are piecewise linear. This is inspired by the work of Kopeliovich [13] [14] except that our ansatz has to be piecewise linear to be able to generate configurations with a huge number of layers. After defining the ansatz for an arbitrary number of layers, we will show that for a single layer configuration the ansatz produces configurations that are in good agreements with the rational map ansatz configuration. Then we will use the new ansatz to construct configurations that are made out of a very large no of layers.\\n\\nWe have shown, in the previous section, that one can construct shell like structures with very large Baryon numbers. At large baryon numbers, the Skyrmions resemble shell like structures. That is, the fields are constant nearly everywhere except in a small region corresponding to the shell. In that region, the profile look like linear functions smoothly linked to the constant parts at the edges (cfr. Fig 3 ). Motivated by this we approximate the fields by the ramp-functions\\n\\nF (x) = N π 2 -(x -x0) π W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 21\\n\\n) µ(x) = M 2 + (x -x0) M N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 22\\n\\n) A(x) = (1 + A0) 2 + (x -x0) (1 -A0) N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 23\\n\\n)\\n\\nIn the above there are four free parameters, namely the central radius x0 of the shell\\n\\nover which the fields change, the width of the shell W , the mass field at spatial infinity M and the value of the metric field at the origin A0 such that limx→∞ = 0. N is the number of layers we wish to study and, as such, is treated as an input parameter.\\n\\nThe picture is of a gravitating skyrmion with very high baryon number existing as N thin layers or shell of small thickness.\\n\\nThe above ansatz, allow us to find an approximation to the integrated energy. To do this we use the fact that the shell width is much smaller than the radius at large baryon numbers. In particular to evaluate the action integral we can approximate expressions of the type R G(x) sin p F (x) for any function G(x) that varies very little over the width of the shell by R G(x0) sin p F (x). We then use the fact that\\n\\nZ x 0 +NW/2 x 0 -NW/2 sin p F (x) = N π W Z π 0 sin p y dy. ( 24\\n\\n)\\n\\nThis leads to the following expression for the energy:\\n\\nE = - 16πFπ e » M α \\\" 1 + A0 2 « - π 2 W » \\\" 1 + A0 2 « \\\" x 2 0 -M x0 + W 2 12 - M W 6 « + \\\" 1 -A0 W « \\\" W 2 x0 6 - M W 2 12 - M W x0 6 « - + B \\\" 1 + A0 2 « 1 2 \\\" M π 2 W x0 -W - π 2 W « - 3IW 16x 2 0 \\\" 1 + A0 2 « - M α - ( 25\\n\\n)\\n\\nTo find the configurations which minimize this energy we first minimised it with respect to A0 and M algebraically in order to find an expression for the energy as a function of the width and radius only. Then we minimised this numerically using Mathematica. We will now discuss the features of these configurations.\\n\\nFirst of all, we must compare the results obtained with the ramp-profile when N = 1 and compare them to the result obtained with the full profile. Tables. 3 and 4 show the properties of solutions we obtained using the ramp-profile approximation, again at α = 1 × 10 -6 . All the general features of the full numerical solutions are reproduced. In particular, the approximate B 1 2 scaling and then decrease of the radius, the decreasing ADM mass and the differences between the double and single layer solutions are all exhibited by the data obtained using the ramp-profile approximation.\\n\\nQuantitatively though, there are some differences. The approximation allows a significant increase in the maximum allowed baryon charge. Also, the radius of configurations obtained using the approximation, tend to be smaller than those obtained numerically. If we concentrate on the baryon numbers greater than 10 5 so as to ensure our approximation, 15 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 8.3063 3.1286 1.1023 0.9999 500 18.6031 3.1386 1.1160 0.9997 1 × 10 3 26.313 3.1397 1.1195 0.9996 1 × 10 4 83.206 3.1396 1.1254 0.9987 1 × 10 5 262.94 3.1357 1.1266 0.9960 1 × 10 6 829.60 3.1230 1.1251 0.9872 1 × 10 7 2604.2 3.0825 1.1186 0.9595 1 × 10 8 8032.8 2.9512 1.0972 0.8713 1 × 10 9 22899 2.4837 1.0272 0.5772 2 × 10 9 29121 2.1092 0.9818 0.3645 2.8 × 10 9 29098 1.6623 0.9505 0.1380 2.83 × 10 9 28514 1.6066 0.9495 0.1119 2.839 × 10 9 28024 1.5671 0.94922 0.09373 2.8397 × 10 9 27869.3 1.5556 0.94924 0.08845 2.83975 × 10 9 27869.8 1.5524 0.94925 0.08699 2.839752 × 10 9 27822 1.5521 0.94925 0.08687 Table 3: Table of properties of the single layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6 16 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 5.7924 3.0428 1.0692 0.9999 1 × 10 3 18.5788 3.1305 1.1047 0.9995 1 × 10 4 58.8202 3.1380 1.1201 0.9983 1 × 10 5 185.8420 3.1332 1.1246 0.9944 1 × 10 6 585.7950 3.1153 1.1233 0.9820 1 × 10 7 1833.0500 3.0578 1.1143 0.9428 1 × 10 8 5587.3600 2.8688 1.0840 0.8172 9 × 10 8 14147.1782 2.1859 0.9900 0.4065 9.764724 × 10 8 14472.3851 2.1276 0.9837 0.3746 1 × 10 9 14560.5000 2.1092 0.9818 0.3646 1.4 × 10 9 14549.0000 1.6623 0.9505 0.1381 1.41963 × 10 9 13994.0523 1.5644 0.9492 0.0926 1.419635134 × 10 9 13993.2000 1.5643 0.9492 0.0925 Table 4: Table of properties of the double layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6\\n\\nthat the width is much smaller than the radius, is valid, then at worst we find a discrepancy in the ADM mass of 11% and in the radius of 7%.\\n\\nIn general then, the data seems to confirm the reliability the ramp-profile approximation. In fact the approach will be even more reliable at the extremely high values of the baryon number that we are interested in. This is because the radius of solutions is of orders of magnitudes greater than the width in such a regime, consistent with the approximations we have made.\\n\\nMoreover, whilst searching for minima of the energy does not allow us to probe both branches of solutions, it does allow us to locate the value of αcrit. We again obtain the approximate trend αcrit ∝ B -1 2 , for large B. Now in order to say anything about the possibility of baryon stars in the Skyrme model we need to be able to verify that the decrease in the ADM mass per baryon we observed at low and moderate baryon numbers, extends to baryon numbers of order 10 58 for α = 7.3 × 10 -40 .\\n\\nTable. 5 summarizes our solutions in such a regime. Firstly we consider constructing a single layer self-gravitating Skyrmion with these values. We do indeed see that the configuration is bound. This is verified by checking that the ADM mass is lower (even at this significantly lower value of α) than for the B = 1 hedgehog. So the possibility of baryon stars in the Einstein-Skyrme model cannot be ruled out on the grounds of energy.\\n\\nThe Skyrmion exists as a giant thin shell, and the large baryon charge is distributed as a tight lattice over this. However a hollow shell is clearly not a realistic construction for a neutron star. This fact manifests itself in the extremely high radius of the configuration.\\n\\nTransferring to standard units, the single layer B = 10 58 gravitating Skyrmion has a radius of 2.42 × 10 10 km ! To address this issue, we can use a large number of layered Skyrmions as discussed earlier. This has several benefits. Firstly, as we are distributing the baryon number through a larger volume, then at a given baryon density the necessary radius can decrease. Similar to what we see in the double layer results. On top of this, we expect the radius to decrease further due to extra gravitational compression, as the outer layers of the Skyrmions feel the attraction of inner layers. Finally, the many layer approach is also a more realistic construction of a solid baryon star.\\n\\nThe results for using more and more layers in the construction (for fixed B and α), are also presented in Fig. 5 . We note that not only does the radius decrease significantly, but\\n\\nthe added gravitational binding further improves the energies of the configurations, reflected in the low ADM masses obtained. There appears to be a critical number of layers that can be used before there ceases to be any solutions and although the value of Smin is close zero at this point, the star still has not collapsed to form a black hole. Finally, we note that the radius of the Skyrmion at the critical number of layers is approximately 20.91km. This is comparable to a real neutron star, with a typical radius of 10km.\\n\\nWe reemphasise here that our approach to embedding shells of baryons is quite crude.\\n\\nFor few shells and large baryon number, we might reasonably believe that baryon number does not chance significantly from one shell to the next. However, when we embed many shells we should really consider that the baryon number of the inner most shells would likely be significantly less than the that of the outer shells. Nevertheless, our naive embedding has produced some interesting properties. In a future work we hope to improve our multi-layer construction to obtain a more realistic description of a baryon star. 19 N Shell R( 2 Fπe ) W M ADM /(6π 2 B) S min 1 × 10 2 8.3236 × 10 27 3.1416 1.1285 1.0000 1 × 10 3 2.6321 × 10 27 3.1416 1.1285 1.0000 1 × 10 4 × 10 26 3.1416 1.1285 1.0000 1 × 10 5 2.6321 × 10 26 3.1416 1.1285 1.0000 1 × 10 6 8.3236 × 10 25 3.1416 1.1285 1.0000 1 × 10 7 2.6321 × 10 25 3.1416 1.1285 1.0000 1 × 10 8 8.3236 × 10 24 3.1416 1.1285 1.0000 1 × 10 9 2.6321 × 10 24 3.1415 1.1285 1.0000 1 × 10 10 8.3234 × 10 23 3.1415 1.1285 0.9999 1 × 10 11 2.6319 × 10 23 3.1412 1.1285 0.9997 1 × 10 12 8.3216 × 10 22 3.1402 1.1283 0.9991 1 × 10 13 2.6301 × 10 22 3.1373 1.1278 0.9971 1 × 10 14 8.3034 × 10 21 3.1280 1.1263 0.9907 1 × 10 15 2.6118 × 10 21 3.0986 1.1213 0.9705 1 × 10 16 8.1147 × 10 20 3.0037 1.1057 0.9063 1 × 10 17 2.4001 × 10 20 2.6810 1.0552 0.6977 5 × 10 17 8.2066 × 10 19 1.7888 0.9552 0.2036 5.3 × 10 17 7.4172 × 10 19 1.6227 0.9491 0.1247 5.33 × 10 17 7.1871 × 10 19 1.5625 0.94866 0.0971 5.3306 × 10 17 7.1597 × 10 19 1.5549 0.94868 0.0936 5.33065 × 10 17 7.1525 × 10 19 1.5528 0.948694 0.0927 5.330657 × 10 17 7.1506 × 10 19 1.5523 0.948692 0.0924 Table 5: Table of properties of the step ansatz configurations for varying the number of embedded shells at fixed B = 10 58 and α = 7.3 × 10 -40 .\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"7 Conclusions - Acknowledgements\",\n \"text\": \"Previous work on the Einstein-Skyrme model highlighted a considerable problem with using the Skyrmions as a model for baryon stars. Namely, multibaryon hedgehog Skyrmions were simply not energetically favourable states. We have shown that this is simply a consequence of a poor ansatz for the true Skyrmion and, having used the more appropriate rational map ansatz, we have generated energetically favourable configurations of multibaryons.\\n\\nWe also observe the interesting property that near the critical coupling, the Skyrmions can decrease in radius as we add more baryons. This hints towards the similar behaviour exhibited by real neutron stars.\\n\\nAlthough the rational map ansatz does not have an exact radial symmetry, at large scale it does. The anisotropy only appears at the nucleon scale.\\n\\nFinally, since we started with the motivation of studying baryon stars within the Skyrme model, it is interesting to compare the features of our configurations with those of neutron stars. For realistic values, B = 10 58 and α = 7.3 × 10 -40 we find a minimal energy single layer configuration with radius=2.42 × 10 10 km. This is clearly too large for a neutron star (which is of order 10km. in radius). This is to be expected however due to the shell model we have taken. Firstly, as we are distributing the baryons over the surface area rather than throughout the volume of the star we naturally must require a much larger star for a given baryon number. This effect is two-fold in that if we were distributing the baryons throughout the volume, outer layers would feel the attraction of inner layers and enhanced radial compression would occur. The loss of such an effect is pronounced when we are considering realistically small values of the coupling.\\n\\nIt seems therefore that the way to construct baryon stars in the Skyrme model is to consider embedding shells of baryons within shells. This gives rise to more appropriate specifications for the star and is also more realistic. We do indeed observe such improvements for a many layered configuration. In fact the radius of B = 10 58 gravitating Skyrmion (at realistic α), can be decreased in this manner to approximately 20.91km.\\n\\nWe note however that this approach to shell embedding has only be done naively thus far.\\n\\nWe have only considered the case where the baryon number is equal for each shell. We really should allow the baryon number(and hence the rational map quantities) to vary over the shells. One approach towards this would be to assume that the baryon density is a constant over the shells. An even better approach would be to allow this to be a smoothly varying function that must be determined by minimising the energy. This will give a more realistic 21 description of baryon stars within the Einstein-Skyrme model, as traditional descriptions of neutron stars also involve many strata, of differing neutron density. We are currently investigating such configurations.\\n\\n[13] V. B. Kopeliovich \\\"The Bubbles of Matter from MultiSkyrmions\\\" JETP Lett. 73 (2001), 587-591; Pisma Zh.Eksp.Teor.Fiz. 73 (2001), 667-671 [14] V. B. Kopeliovich \\\"MultiSkyrmions and Baryonic Bags\\\" J.Phys. G28 (2002), 103-120 23\\n\\nGIP is supported by a PPARC studentship.\"\n }\n]"}}},{"rowIdx":98,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0530"},"arxiv_id":{"kind":"string","value":"0704.0530"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1088/1126-6708/2007/06/065"},"content_hash":{"kind":"string","value":"e09094677ad52079a512d808ae2511bbf4a65c00c4c0f9afba29ba8ee99b8a1e"},"title":{"kind":"string","value":"Noncommutative Solitons in a Supersymmetric Chiral Model in 2+1\n Dimensions"},"abstract":{"kind":"string","value":"We consider a supersymmetric Bogomolny-type model in 2+1 dimensions\noriginating from twistor string theory. By a gauge fixing this model is reduced\nto a modified U(n) chiral model with N<=8 supersymmetries in 2+1 dimensions.\nAfter a Moyal-type deformation of the model, we employ the dressing method to\nexplicitly construct multi-soliton configurations on noncommutative R^{2,1} and\nanalyze some of their properties."},"authors":{"kind":"list like","value":["Olaf Lechtenfeld","Alexander D. Popov"],"string":"[\n \"Olaf Lechtenfeld\",\n \"Alexander D. Popov\"\n]"},"arxiv_categories":{"kind":"list like","value":["hep-th"],"string":"[\n \"hep-th\"\n]"},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model, we employ the dressing method to explicitly construct multi-soliton configurations on noncommutative R 2,1 and analyze some of their properties.\n\nIn the low-energy limit string theory with D-branes gives rise to noncommutative field theory on the branes when the string propagates in a nontrivial NS-NS two-form (B-field) background [1, 2, 3, 4] . In particular, if the open string has N =2 worldsheet supersymmetry, the tree-level target space dynamics is described by a noncommutative self-dual Yang-Mills (SDYM) theory in 2+2 dimensions [5] . Furthermore, open N =2 strings in a B-field background induce on the worldvolume of n coincident D2-branes a noncommutative Yang-Mills-Higgs Bogomolny-type system in 2+1 dimensions which is equivalent to a noncommutative generalization [6] of the modified U(n) chiral model known as the Ward model [7] . The topological nature of N =2 strings and the integrability of their tree-level dynamics [8] render this noncommutative sigma model integrable. 1 Being integrable, the commutative U(n≥2) Ward model features a plethora of exact scattering and no-scattering multi-soliton and wave solutions, i.e. time-dependent stable configurations on R foot_1 . These are not only a rich testing ground for physical properties such as adiabatic dynamics or quantization, but also descend to more standard multi-solitons of various integrable systems in 2+0 and 1+1 dimensions, such as sine-Gordon, upon dimensional and algebraic reduction. There is a price to pay however: Nonlinear sigma models in 2+1 dimensions may be Lorentz-invariant or integrable but not both [7, 11] . In fact, Derrick's theorem prohibits the existence of stable solitons in Lorentz-invariant scalar field theories above 1+1 dimensions. A Moyal deformation, however, overcomes this hurdle, but of course replaces Lorentz invariance by a Drinfeld-twisted version. There is another gain: The deformed Ward model possesses not only deformed versions of the just-mentioned multi-solitons, but in addition allows for a whole new class of genuinely noncommutative (multi-)solitons, in particular for the U(1) group [12, 13] ! Moreover, this class is related to the generic but perturbatively constructed noncommutative scalar-field solitons [14, 15] by an infinite-stiffness limit of the potential [16] .\n\nIn [12, 13] and [17] - [20] families of multi-solitons as well as their reduction to solitons of the noncommutative sine-Gordon equations were described and studied. In the nonabelian case both scattering and nonscattering configurations were obtained. For static configurations the issue of their stability was analyzed [21] . The full moduli space metric for the abelian model was computed and its adiabatic two-soliton dynamics was discussed [16] .\n\nRecall that the critical N =2 string theory has a four-dimensional target space, and its open string effective field theory is self-dual Yang-Mills [8] , which gets deformed noncommutatively in the presence of a B-field [5] . Conversely, the noncommutative SDYM equations are contained [19] in the equations of motion of N =2 string field theory (SFT) [22] in a B-field background. This SFT formulation is based on the N =4 topological string description [23] . It is well known that the SDYM model can be described in terms of holomorphic bundles over (an open subset of) the twistor space 2 [26] CP 3 and the topological N =4 string theory contains twistors from the outset. The Lax pair, integrability and the solutions to the equations of motion by twistor and dressing methods were incorporated into the N =2 open SFT in [27, 28] . However, this theory reproduces only bosonic SDYM theory, its symmetries (see e.g. [29, 30, 31] ) and integrability properties. It is natural to ask: What string theory can describe supersymmetric SDYM theory [32, 33] in four dimensions?\n\nThere are some proposals [33, 34, 35, 36] for extending N =2 open string theory (and its SFT) to be space-time supersymmetric. Moreover, it was shown by Witten [37] that N =4 supersymmetric SDYM theory appears in twistor string theory, which is a B-type open topological string with the supertwistor space CP 3|4 as a target space. 3 Note that N <4 SDYM theory forms a BPS subsector of N -extended super Yang-Mills theory, and N =4 SDYM can be considered as a truncation of the full N =4 super Yang-Mills theory [37] . It is believed [43, 39] that twistor string theory is related with the previous proposals [33, 34, 35, 36] for a Lorentz-invariant supersymmetric extension of N =2 (and topological N =4) string theory which also leads to the N =4 SDYM model.\n\nA dimensional reduction of the above relations between twistor strings and N =4 super Yang-Mills and SDYM models was considered in [44, 45, 46, 47] . The corresponding twistor string theory after this reduction is the topological B-model on the mini-supertwistor space P 2|4 . In [47] it was shown that the 2N =8 supersymmetric extension of the Bogomolny-type model in 2+1 dimensions is equivalent to an 2N =8 supersymmetric modified U(n) chiral model on R 2,1 . The subject of the current paper is an 2N ≤8 version of the above supersymmetric Bogomolny-type Yang-Mills-Higgs model in signature (-+ +), its relation with an N -extended supersymmetric modified integrable U(n) chiral model (to be defined) in 2+1 dimensions and the Moyal-type noncommutative deformation of this chiral model. We go on to explicitly construct multi-soliton configurations on noncommutative R 2,1 for the corresponding supersymmetric sigma model field equations. By studying the scattering properties of the constructed configurations, we prove their asymptotic factorization without scattering for large times. We also briefly discuss a D-brane interpretation of these soliton configurations from the viewpoint of twistor string theory.\n\nSpace R 2,2 . Let us consider the four-dimensional space R 2,2 = (R 4 , g) with the metric\n\nds 2 = g µν dx µ dx ν = det(dx α α) = dx 1 1dx 2 2 -dx 2 1dx 1 2 (2.1)\n\nwith (g µν ) = diag(-1, +1, +1, -1), where µ, ν, . . . = 1, . . . , 4 are space-time indices and α = 1, 2, α = 1, 2 are spinor indices. We choose the coordinates 4\n\n(x µ ) = (x a , t) = (t, x, y, t) with a, b, . . . = 1, 2, 3 , (2.2)\n\nand the signature (-+ + -) allows us to introduce real isotropic coordinates (cf. [19, 6] )\n\nx 1 1 = 1 2 (t -y) , x 1 2 = 1 2 (x + t) , x 2 1 = 1 2 (x -t) , x 2 2 = 1 2 (t + y) . (2.3)\n\nSDYM. Recall that the SDYM equations for a field strength tensor\n\nF µν on R 2,2 read 1 2 ε µνρσ F ρσ = F µν , (2.4)\n\nwhere ε µνρσ is a completely antisymmetric tensor on R 2,2 and ε 1234 = 1. In the coordinates (2.3) we have the decomposition\n\nF α α,β β = ∂ α αA β β -∂ β β A α α + [A α α, A β β ] = ε αβ F α β + ε α β F αβ (2.5) with F α β := -1 2 ε αβ F α α,β β and F αβ := -1 2 ε α β F α α,β β , (2.6)\n\nwhere ε αβ is antisymmetric, ε αβ ε βγ = δ γ α , and similar for ε α β , with ε 12 = ε 1 2 = 1. The gauge potential (A α α) will appear in the covariant derivative\n\nD α β = ∂ α β + [A α β , • ] . (2.7)\n\nIn spinor notation, (2.4) is equivalently written as\n\nF α β = 0 ⇔ F α α,β β = ε α β F αβ . (2.8)\n\nSolutions {A α α} to these equations form a subset (a BPS sector) of the solution space of Yang-Mills theory on R 2,2 .\n\nN -extended SDYM in component fields. The field content of N -extended super SDYM is foot_4\n\nN = 0 A α α (2.9a) N = 1 A α α, χ i α with i = 1 (2.9b) N = 2 A α α, χ i α , φ [ij] with i, j = 1, 2 (2.9c)\n\nN = 3 A α α, χ i α , φ [ij] , χ[ijk] α with i, j, k = 1, 2, 3 (2.9d) N = 4 A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl] α β\n\nwith i, j, k, l = 1, 2, 3, 4 .\n\n(2.9e)\n\nHere (A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl]\n\nα β ) are fields of helicities (+1, + 1 2 , 0, -1 2 , -1). These fields obey the field equations of the N = 4 SDYM model, namely [33, 37]\n\nF α β = 0 , (2.10a)\n\nD α αχ iα = 0 , (2.10b)\n\nD α αD α αφ ij + 2{χ iα , χ j α } = 0 , (2.10c)\n\nD α α χ α[ijk] -6[χ [i α , φ jk] ] = 0 , (2.10d)\n\nD γ α G [ijkl] γ β + 12{χ [i α , χjkl] β } -18[φ [ij , D α β φ kl] ] = 0 . (2.10e)\n\nNote that the N < 4 SDYM field equations are governed by the first N +1 equations of (2.10), where F α β = 0 is counted as one equation and so on.\n\nN -extended SDYM Superspace R 4|4N .\n\nRecall that in the space R 2,2 = (R 4 , g) with the metric g given in (2.1) one may introduce purely real Majorana-Weyl spinors foot_5 θ α and η α of helicities + 1 2 and -1 2 as anticommuting (Grassmann-algebra) objects. Using 2N such spinors with components θ iα and η α i for i = 1, . . . , N , one can define the N -extended superspace R 4|4N and the N -extended supersymmetry algebra generated by the supertranslation operators\n\nP α α = ∂ α α , Q iα = ∂ iα -η α i ∂ α α and Q i α = ∂ i α -θ iα ∂ α α , (2.11)\n\nwhere\n\n∂ α α := ∂ ∂x α α , ∂ iα := ∂ ∂θ iα and ∂ i α := ∂ ∂η α i .\n\n(2.12)\n\nThe commutation relations for the generators (2.11) read\n\n{Q iα , Q j α} = -2δ j i P α α , [P α α, Q iβ ] = 0 and [P α α, Q i β ] = 0 . (2.13)\n\nTo rewrite equations of motion in terms of R 4|4N superfields one uses the additional operators\n\nD iα = ∂ iα + η α i ∂ α α and D i α = ∂ i α + θ iα ∂ α α , (2.14)\n\nwhich (anti)commute with the operators (2.11) and satisfy\n\n{D iα , D j β } = 2δ j i P α β , [P α α, D iβ ] = 0 and [P α α, D j β ] = 0 . (2.15)\n\nAntichiral superspace R 4|2N . On the superspace R 4|4N one may introduce tensor fields depending on bosonic and fermionic coordinates (superfields), differential forms, Lie derivatives L X etc.. Furthermore, on any such superfield A one can impose the constraint equations L D iα A = 0, which for a scalar superfield f reduce to the so-called antichirality conditions\n\nD iα f = 0 . (2.16)\n\nThese are easily solved by using a coordinate transformation on R 4|4N ,\n\n(x α α, η α i , θ iα ) → (x α α = x α α-θ iα η α i , η α i , θ iα ) , (2.17)\n\nunder which ∂ α α, D iα and D i α transform to the operators\n\n∂α α = ∂ α α , Diα = ∂ iα and Di α = ∂ i α + 2θ iα ∂ α α . ( 2\n\n.18) Then (2.16) simply means that f is defined on a sub-superspace R 4|2N ⊂ R 4|4N with coordinates xα α and η α i . (2.19)\n\nThis space is called antichiral superspace. In the following we will usually omit the tildes when working on the antichiral superspace.\n\nN -extended SDYM in superfields. The N -extended SDYM equations can be rewritten in terms of superfields on the antichiral superspace R 4|2N [33, 48] . Namely, for any given 0 ≤ N ≤ 4, fields of a proper multiplet from (2.9) can be combined into superfields A α α and A i α depending on x α α, η α i ∈ R 4|2N and giving rise to covariant derivatives\n\n∇ α α := ∂ α α + A α α and ∇ i α := ∂ i α + A i α .\n\n(2.20)\n\nIn such terms the N -extended SDYM equations (2.10) read\n\n[∇ α α, ∇ β β ] + [∇ α β , ∇ β α] = 0 , [∇ i α, ∇ β β ] + [∇ i β , ∇ β α] = 0 , {∇ i α, ∇ j β } + {∇ i β , ∇ j α} = 0 , (2.21)\n\nwhich is equivalent to\n\n[∇ α α, ∇ β β ] = ε α β F αβ , [∇ i α, ∇ β β ] = ε α β F i β and {∇ i α, ∇ j β } = ε α β F ij , (2.22)\n\nwhere F ij is antisymmetric and F αβ is symmetric in their indices.\n\nThe above gauge potential superfields (A α α, A i α) as well as the gauge strength superfields (F αβ , F i α , F ij ) contain all physical component fields of the N -extended SDYM model. For instance, the lowest component of the triple (F αβ , F i α , F ij ) in an η-expansion is (F αβ , χ i α , φ ij ), with zeros in case N is too small. By employing Bianchi identities for the gauge strength superfields, one successively obtains [48] the superfield expansions and the field equations (2.10) for all component fields.\n\nIt is instructive to extend the antichiral combination in (2.18) to potentials and covariant derivatives, Di\n\nα = ∂ i α + 2 θ iα ∂ α α + + + Ãi α := A i α + 2 θ iα A α α ∇i α := ∇ i α + 2 θ iα ∇ α α ( 2\n\n.23) where ∇ α α, ∇ i α and Di α are given by (2.20) and (2.18), while A i α and A α α depend on x α α and η α i only. With the antichiral covariant derivatives, one may condense (2.21) or (2.22) into the single set { ∇i α, ∇j β\n\n} + { ∇i β , ∇j α} = 0 ⇔ { ∇i α, ∇j β } = ε α β Fij , (2.24)\n\nwith Fij =\n\nF ij + 4 θ [iα F j] α + 4 θ iα θ jβ F αβ .\n\nThe concise form (2.24) of the N -extended SDYM equations is quite convenient, and we will use it interchangeable with (2.21).\n\nLinear system for N -extended SDYM. It is well known that the superfield SDYM equations (2.21) can be seen as the compatibility conditions for the linear system of differential equations\n\nζ α(∂ α α + A α α) ψ = 0 and ζ α(∂ i α + A i α) ψ = 0 , (2.25)\n\nwhere\n\n(ζ β ) = 1 ζ and ζ α = ε α β ζ β . The extra (spectral) parameter 7 ζ lies in the extended complex plane C ∪ ∞ = CP 1 .\n\nHere ψ is a matrix-valued function depending not only on x α α and η α i but also (meromorphically) on ζ ∈ CP 1 . We subject the n×n matrix ψ to the following reality condition:\n\nψ(x α α, η α i , ζ) ψ(x α α, η α i , ζ) † = 1l , (2.26)\n\nwhere \" †\" denotes hermitian conjugation and ζ is complex conjugate to ζ. This condition guarantees that all physical fields of the N -extended SDYM model will take values in the adjoint representation of the algebra u(n). In the concise form the linear system (2.25) is written as\n\nζ α(∇ i α + 2θ iα ∇ α α) ψ = 0 ⇔ ζ α( Di α + Ãi α) ψ = 0 ⇔ ζ α ∇i α ψ = 0 . (2.27) 2.3 Reduction of N -extended SDYM to 2+1 dimensions\n\nThe supersymmetric Bogomolny-type Yang-Mills-Higgs equations in 2+1 dimensions are obtained from the described N -extended super SDYM equations by a dimensional reduction R 2,2 → R 2,1 .\n\nIn particular, for the N =0 sector we demand the components A µ of a gauge potential to be independent of x 4 and put A 4 =: ϕ. Here, ϕ is a Lie-algebra valued scalar field in three dimensions (the Higgs field) which enters into the Bogomolny-type equations. Similarly, for N ≥ 1 one can reduce the N -extended SDYM equations on R 2,2 by imposing the ∂ 4 -invariance condition on all the fields (A α α,\n\nχ i α , φ [ij] , χ[ijk] α , G [ijkl]\n\nα β ) from the N =4 supermultiplet or its truncation to N <4 and obtain supersymmetric Bogomolny-type equations on R 2,1 .\n\nSpinors in R 2,1 .\n\nRecall that on R 2,2 both N =4 SDYM theory and full N =4 super Yang-Mills theory have an SL(4, R) ∼ = Spin(3,3) R-symmetry group [33] . A dimensional reduction to R 2,1 enlarges the supersymmetry and R-symmetry to 2N =8 and Spin(4,4), respectively, for both theories (cf. [49] for Minkowski signature). More generally, any number N of supersymmetries gets doubled to 2N in the reduction. Since dimensional reduction collapses the rotation group Spin(2,2)\n\n∼ = Spin(2,1) L ×Spin(2,1) R of R 2,2\n\nto its diagonal subgroup Spin(2,1) D as the local rotation group of R 2,1 , the distinction between undotted and dotted indices disappears. We shall use undotted indices henceforth.\n\nCoordinates and derivatives in R 2,1 . The above discussion implies that one can relabel the bosonic coordinates x α β from (2.3) by x αβ and split them as\n\nx αβ = 1 2 (x αβ + x βα ) + 1 2 (x αβ -x βα ) = x (αβ) + x [αβ]\n\n(2.28) into antisymmetric and symmetric parts,\n\nx [αβ] = 1 2 ε αβ x 4 = 1 2 ε αβ t and x (αβ) =: y αβ , (2.29)\n\nrespectively, with\n\ny 11 = x 11 = 1 2 (t -y) , y 12 = 1 2 (x 12 + x 21 ) = 1 2 x , y 22 = x 22 = 1 2 (t + y) . (2.30)\n\nWe also have θ iα → θ iα and η α i → η α i for the fermionic coordinates on R 4|4N reduced to R 3|4N . Bosonic coordinate derivatives reduce in 2+1 dimensions to the operators\n\n∂ (αβ) = 1 2 (∂ αβ + ∂ βα ) (2.31)\n\nwhich read explicitly as\n\n∂ (11) = ∂ ∂y 11 = ∂ t -∂ y , ∂ (12) = ∂ (21) = 1 2 ∂ ∂y 12 = ∂ x , ∂ (22) = ∂ ∂y 22 = ∂ t + ∂ y . (2.32)\n\nWe thus have\n\n∂ ∂x αβ = ∂ (αβ) -ε αβ ∂ 4 = ∂ (αβ) -ε αβ ∂ t , (2.33)\n\nwhere\n\nε 12 = -ε 21 = -1, ∂ 4 = ∂/∂x 4 and ∂ t = ∂/∂ t.\n\nThe operators D iα and D i α acting on t-independent superfields reduce to\n\nD iα = ∂ iα + η β i ∂ (αβ) and D i α = ∂ i α + θ iβ ∂ (αβ) , (2.34)\n\nwhere\n\n∂ iα = ∂/∂θ iα and ∂ i α = ∂/∂η α i .\n\nSimilarly, the antichiral operators Diα and Di α in (2.18) become Diα = ∂ iα and Di α = ∂ i α + 2θ iβ ∂ (αβ) . (2.35) Supersymmetric Bogomolny-type equations in component fields. According to (2.33), the components A α β of a gauge potential in four dimensions split into the components A (αβ) of a gauge potential in three dimensions and a Higgs field\n\nA [αβ] = -ε αβ ϕ, i.e. A αβ = A (αβ) + A [αβ] = A (αβ) -ε αβ ϕ . (2.36)\n\nThen the covariant derivatives D α β reduced to three dimensions become the differential operators\n\nD αβ -ε αβ ϕ = ∂ (αβ) + [A (αβ) , • ] -ε αβ [ϕ, • ] , (2.37)\n\nand the Yang-Mills field strength on R 2,1 decomposes as\n\nF αβ, γδ = [D αβ , D γδ ] = ε αγ f βδ + ε βδ f αγ with f αβ = f βα . (2.38)\n\nSubstituting (2.36) and (2.37) into (2.10), i.e. demanding that all fields in (2.10) are independent of x 4 = t, we obtain the following supersymmetric Bogomolny-type equations on R 2,1 :\n\nf αβ + D αβ ϕ = 0 , (2.39a)\n\nD αβ χ iβ + ε αβ [ϕ, χ iβ ] = 0 , (2.39b)\n\nD αβ D αβ φ ij + 2[ϕ, [ϕ, φ ij ]] + 2{χ iα , χ j α } = 0 , (2.39c)\n\nD αβ χβ[ijk] -ε αβ [ϕ, χβ[ijk] ] -6[χ [i α , φ jk] ] = 0 , (2.39d)\n\nD γ α G [ijkl] γβ + [ϕ, G [ijkl] αβ ] + 12{χ [i α , χjkl] β } -18[φ [ij , D αβ φ kl] ] -18ε αβ [φ [ij , [φ kl] , ϕ]] = 0 .(2.\n\nSupersymmetric Bogomolny-type equations in terms of superfields. Translations generated by the vector field ∂ 4 = ∂ t are isometries of superspaces R 4|4N and R 4|2N . By taking the quotient with respect to the action of the abelian group G generated by ∂ 4 , we obtain the reduced full superspace R 3|4N ∼ = R 4|4N /G and the reduced antichiral superspace R 3|2N ∼ = R 4|2N /G. In the following, we shall work on R 3|2N and R 3|2N × CP 1 , since the reduced ψ-function from (2.25) and (2.27) is defined on the latter space.\n\nThe linear system stays in the center of the superfield approach to the N -extended SDYM equations. After imposing t-independence on all fields in the linear system (2.27), we arrive at the linear equations\n\nζ α ∇i α ψ ≡ ζ α ( Di α + Âi α ) ψ = 0 (2.40)\n\nof the same form but with\n\nDi α = ∂ i α + 2θ iβ ∂ (αβ) and Âi α = A i α + 2θ iβ (A (αβ) -ε αβ Ξ) , (2.41)\n\nwhere A i α , A (αβ) and Ξ are superfields depending on y αβ and η α i only. These linear equations expand again to the pair (cf. (2.25))\n\nζ β (∂ (αβ) + A (αβ) -ε αβ Ξ) ψ = 0 and ζ α (∂ i α + A i α ) ψ = 0 . (2.42)\n\nThe compatibility conditions for the linear system (2.40) read\n\n{ ∇i α , ∇j β } + { ∇i β , ∇j α } = 0 ⇔ { ∇i α , ∇j β } = ε αβ Fij (2.43)\n\nand present a condensed form of (2.39) rewritten in terms of R 3|2N superfields. Similarly, these equations can also be written in more expanded forms analogously to (2.21) or using the superfield analog of (2.37). However, we will not do this since all these sets of equations are equivalent.\n\nAs has been known for some time, nonlinear sigma models in 2 + 1 dimensions may be Lorentzinvariant or integrable but not both [7, 11] . We will show that the super Bogomolny-type model discussed in Section 2 after a gauge fixing is equivalent to a super extension of the modified U(n) chiral model (so as to be integrable) first formulated by Ward [7] . Since integrability is compatible with noncommutative deformation (if introduced properly, see e.g. [9] - [20] ) we choose from the beginning to formulate our super extension of this chiral model on Moyal-deformed R 2,1 with noncommutativity parameter θ ≥ 0. Ordinary space-time R 2,1 can always be restored by taking the commutative limit θ → 0.\n\nStar-product formulation. Classical field theory on noncommutative spaces may be realized in a star-product formulation or in an operator formalism 8 . The first approach is closer to the commutative field theory: it is obtained by simply deforming the ordinary product of classical fields (or their components) to the noncommutative star product\n\n(f ⋆ g)(x) = f (x) exp{ i 2 ← - ∂ a θ ab -→ ∂ b } g(x) ⇒ x a ⋆ x b -x b ⋆ x a = iθ ab (3.1)\n\nwith a constant antisymmetric tensor θ ab . Specializing to R 2,1 , we use real coordinates (x a ) = (t, x, y) in which the Minkowski metric g on R 3 reads (g ab ) = diag(-1, +1, +1) with a, b, . . . = 1, 2, 3 (cf. Section 2). It is straightforward to generalize the Moyal deformation (3.1) to the superspaces introduced in the previous section, allowing in particular for non-anticommuting Grassmann-odd coordinates. Deferring general superspace deformations and their consequences to future work, we here content ourselves with the simple embedding of the \"bosonic\" Moyal deformation into superspace, meaning that (3.1) is also valid for superfields f and g depending on Grassmann variables θ iα and η α i . For later use we consider not only isotropic coordinates and vector fields\n\nu := 1 2 (t+y) = y 22 , v := 1 2 (t-y) = y 11 , ∂ u = ∂ t + ∂ y = ∂ (22) , ∂ v = ∂ t -∂ y = ∂ (11) (3.2)\n\nintroduced in Section 2, but also the complex combinations\n\nz := x + iy , z := x -iy , ∂ z = 1 2 (∂ x -i∂ y ) , ∂ z = 1 2 (∂ x + i∂ y ) . (3.3)\n\nSince the time coordinate t remains commutative, the only nonvanishing component of the noncommutativity tensor θ ab is Operator formalism. The nonlocality of the star products renders explicit computation cumbersome. We therefore pass to the operator formalism, which trades the star product for operatorvalued spatial coordinates (x, ŷ) or their complex combinations (ẑ, ẑ), subject to\n\nθ xy = -θ yx =: θ > 0 ⇒ θ z z = -θ zz = -2i θ . ( 3\n\n[t, x] = [t, ŷ] = 0 but [x, ŷ] = iθ ⇒ [ẑ, ẑ] = 2 θ . (3.6)\n\nThe latter equation suggests the introduction of annihilation and creation operators,\n\na = 1 √ 2θ ẑ and a † = 1 √ 2θ ẑ with [a , a † ] = 1 , (3.7)\n\nwhich act on a harmonic-oscillator Fock space H with an orthonormal basis { |ℓ , ℓ = 0, 1, 2, . . .} such that\n\na |ℓ = √ ℓ |ℓ-1 and a † |ℓ = √ ℓ+1 |ℓ+1 . (3.8)\n\nAny superfield f (t, z, z, η α i ) on R 3|2N can be related to an operator-valued superfield f (t, η α i ) ≡ F (t, a, a † , η α i ) on R 1|2N acting in H, with the help of the Moyal-Weyl map\n\nf (t, z, z, η α i ) → f (t, η α i ) = Weyl-ordered f t, √ 2θa, √ 2θa † , η α i . (3.9)\n\nThe inverse transformation recovers the ordinary superfield,\n\nf (t, η α i ) ≡ F (t, a, a † , η α i ) → f (t, z, z, η α i ) = F ⋆ t, z √ 2θ , z √ 2θ , η α i , (3.10)\n\nwhere F ⋆ is obtained from F by replacing ordinary with star products. Under the Moyal-Weyl map, we have\n\nf ⋆ g → f ĝ and dx dy f = 2π θ Tr f = 2π θ ℓ≥0 ℓ| f |ℓ , (3.11)\n\nand the spatial derivatives are mapped into commutators,\n\n∂ z f → ∂z f = -1 √ 2θ [a † , f ] and ∂ z f → ∂z f = 1 √ 2θ [a , f ] . (3.12)\n\nFor notational simplicity we will from now on omit the hats over the operators except when confusion may arise.\n\nGauge fixing for ψ. Note that the linear system (2.40) and the compatibility conditions (2.43) are invariant under a gauge transformation\n\nψ → ψ ′ = g -1 ψ , (3.13a)\n\nA → A ′ = g -1 A g + g -1 ∂ g (with appropriate indices) , (3.13b)\n\nΞ → Ξ ′ = g -1 Ξ g , (3.13c)\n\nwhere g = g(x a , η α i ) is a U(n)-valued superfield globally defined on the deformed superspace R 3|2N θ × CP 1 . Using a gauge transformation of the form (3.13), we can choose ψ such that it will satisfy the standard asymptotic conditions (see e.g. [51] )\n\nψ = Φ -1 + O(ζ) for ζ → 0 , (3.14a\n\n)\n\nψ = 1l + ζ -1 Υ + O(ζ -2 ) for ζ → ∞ , (3.14b)\n\nwhere the U(n)-valued function Φ and u(n)-valued function Υ depend on x a and η α i . This \"unitary\" gauge is compatible with the reality condition for ψ,\n\nψ(x a , η α i , ζ) ψ(x a , η α i , ζ) † = 1l , (3.15)\n\nobtained by reduction from (2.26).\n\nA i 1 = 0 and A i 2 = Φ -1 ∂ i 2 Φ = ∂ i 1 Υ , (3.17)\n\nA (11) = 0 and\n\nA (12) + Ξ = Φ -1 ∂ (12) Φ = ∂ (11) Υ , (3.18)\n\nA (21) -Ξ = 0 and\n\nA (22) = Φ -1 ∂ (22) Φ = ∂ (12) Υ . (3.19)\n\nUsing (2.32), we can rewrite the nonzero components as\n\nA := Φ -1 ∂ u Φ = ∂ x Υ , B := Φ -1 ∂ x Φ = ∂ v Υ , C i := Φ -1 ∂ i 2 Φ = ∂ i 1 Υ . (3.20)\n\nRecall that the superfields Φ and Υ depend on x a and η α i .\n\nLinear system. In the above-introduced unitary gauge the linear system (2.42) reads\n\n(ζ∂ x -∂ u -A) ψ = 0 , (ζ∂ v -∂ x -B) ψ = 0 , (ζ∂ i 1 -∂ i 2 -C i ) ψ = 0 , (3.21)\n\nwhich adds the last equation to the linear system of the Ward model [7] and generalizes it to superfields A(x a , η α j ), B(x a , η α j ) and C i (x a , η α j ). The concise form of (3.21) reads\n\nζ Di 1 -Di 2 -Âi 2 ψ = 0 (3.22)\n\nor, in more explicit form,\n\nζ ∂ i 1 + 2θ i1 ∂ v + 2θ i2 ∂ x -∂ i 2 + C i + 2θ i1 (∂ x + B) + 2θ i2 (∂ u + A) ψ = 0 . (3.23)\n\nN -extended sigma model.\n\nThe compatibility conditions of this linear system are the Nextended noncommutative sigma model equations Di\n\n1 (Φ -1 Dj 2 Φ) + Dj 1 (Φ -1 Di 2 Φ) = 0 (3.24)\n\nwhich in expanded form reads\n\n(g ab + v c ε cab ) ∂ a (Φ -1 ∂ b Φ) = 0 ⇔ ∂ x (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ u Φ) = 0 , (3.25a) ∂ i 1 (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ i 2 Φ) = 0 , ∂ i 1 (Φ -1 ∂ u Φ) -∂ x (Φ -1 ∂ i 2 Φ) = 0 , (3.25b)\n\n∂ i 1 (Φ -1 ∂ j 2 Φ) + ∂ j 1 (Φ -1 ∂ i 2 Φ) = 0 . (3.25c)\n\nHere, the first line contains the Wess-Zumino-Witten term with a constant vector (v c ) = (0, 1, 0) which spoils the standard Lorentz invariance but yields an integrable chiral model in 2+1 dimensions. Recall that Φ is a U(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . As discussed in Section 2, the compatibility conditions of the linear equations (3.22) (or (3.21)) are equivalent to the N -extended Bogomolnytype equations (2.39) for the component (physical) fields. Thus, chiral model field equations (3.25) are equivalent to a gauge fixed form of equations (2.39). Υ-formulation. Instead of Φ-parametrization of (A, B, C i ) given in (3.17)-(3.20) we may use the equivalent Υ-parametrization also given there. In this case, the compatibility conditions for the linear system (3.21) reduce to\n\n(∂ 2 x -∂ u ∂ v )Υ + [∂ v Υ , ∂ x Υ] = 0 , (3.26a)\n\n(∂ i 2 ∂ v -∂ i 1 ∂ x )Υ + [∂ i 1 Υ , ∂ v Υ] = 0 , (∂ i 2 ∂ x -∂ i 1 ∂ u )Υ + [∂ i 1 Υ , ∂ x Υ] = 0 , (3.26b) (∂ i 2 ∂ j 1 + ∂ j 2 ∂ i 1 )Υ + {∂ i 1 Υ , ∂ j 1 Υ} = 0 , (3.26c)\n\nwhich in concise form read\n\n( Di 2 Dj 1 + Dj 2 Di 1 ) Υ + { Di 1 Υ , Dj 1 Υ} = 0 . (3.27)\n\nRecall that Υ is a u(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . For N =4, the commutative limit of (3.27) can be considered as Siegel's equation [33] reduced to 2+1 dimensions. According to Siegel, one can extract the multiplet of physical fields appearing in (2.39) from the prepotential Υ via\n\n∂ i 1 Υ = A i 2 , ∂ i 1 ∂ j 1 Υ = φ ij , ∂ i 1 ∂ j 1 ∂ k 1 Υ = χ[ijk] 2 , ∂ i 1 ∂ j 1 ∂ k 1 ∂ l 1 Υ = G [ijkl] 22 , (3.28a)\n\n∂ (α1) Υ = A (α2) -ε α2 ϕ , ∂ (α1) ∂ i 1 Υ = χ i α , ∂ (α1) ∂ (β1) Υ = f αβ , (3.28b)\n\nwhere one takes Υ and its derivatives at η 2 i = 0. The other components of the physical fields, i.e. χ[ijk]\n\n1 , G [ijkl] 11 , G [ijkl]\n\n21 , A (11) and A (21) -ϕ, vanish in this light-cone gauge.\n\nSupersymmetry transformations. The 4N supercharges given in (2.11) reduce in 2+1 dimensions to the form\n\nQ iα = ∂ iα -η β i ∂ (αβ) and Q i α = ∂ i α -θ iβ ∂ (αβ) . (3.29)\n\nTheir antichiral version, matching to Diα and Dj β of (2.35), reads\n\nQiα = ∂ iα -2η β i ∂ (αβ) and Qj β = ∂ j β , (3.30)\n\nso that { Qiα , Qj β } = -2 δ j i ∂ (αβ) . (3.31)\n\nOn a (scalar) R 3|2N superfield Σ these supersymmetry transformations act as\n\nδ Σ := ε iα Qiα Σ + ε α i Qi α Σ (3.32)\n\nand are induced by the coordinate shifts\n\nδ y αβ = -2ε i(α η β) i and δ η α i = ε α i , (3.33)\n\nwhere ε iα and ε α i are 4N real Grassmann parameters. It is easy to see that our equations (3.24) and (3.27) are invariant under the supersymmetry transformations (3.32) (applied to Φ or Υ). This is simply because the operators Diα and Dj β anticommute with the supersymmetry generators Qiα and Qj β . Therefore, the equations of motion (3.25) of the modified N -extended chiral model in 2+1 dimensions as well as their reductions to 2+0 and 1+1 dimensions carry 2N supersymmetries and are genuine supersymmetric extensions of the corresponding bosonic equations. Note that this type of extension is not the standard one since the R-symmetry groups are Spin(N , N ) in 2+1 and Spin(N , N )× Spin(N , N ) in 1+1 dimensions, which differ from the compact unitary R-symmetry groups of standard sigma models. Contrary to the standard case of two-dimensional sigma models the above \"noncompact\" 2N supersymmetries do not impose any constraints on the geometry of the target space, e.g. they do not demand it to be Kähler [52] or hyper-Kähler [53] . This may be of interest and deserves further study.\n\nAction functionals. In either formulation of the N -extended supersymmetric SDYM model on R 2,2 there are difficulties with finding a proper action functional generalizing the one [54, 55] for the purely bosonic case. These difficulties persist after the reduction to 2+1 dimensions, i.e. for the equations (3.25) and (3.26) describing our supersymmetric modified U(n) chiral model. It is the price to be paid for overcoming the no-go barrier N ≤ 4 and the absence of geometric target-space constraints. On a more formal level, the problem is related to the chiral character of (3.24) as well as (3.27) , where only the operators Di α but not Diα appear. Note however, that for N = 4 one can write an action functional in component fields producing the equations (2.39), which are equivalent to the superspace equations (3.24) when i, j = 1, . . . , 4 (see e.g. [47] ).\n\nOne proposal for an action functional stems from Siegel's idea [33] for the Υ-formulation of the N -extended SDYM equations. Namely, one sees that ∂ i 2 Υ enters only linearly into the last two lines in (3.26) . Therefore, if we introduce\n\nΥ (1) := Υ| η 2 i =0 (3.34)\n\nthen it must satisfy the first equation from (3.26) , and the remaining equations iteratively define the dependence of Υ on η 2 i starting from Υ (1) . Hence, all information is contained in Υ (1) , as can also be seen from (3.28) . In other words, the dependence of Υ on η 2 i is not 'dynamical'. For an action one can then take (cf. [33] )\n\nS = d 3 x d N η 1 Υ (1) ∂ (αβ) ∂ (αβ) Υ (1) + 2 3 Υ (1) ε αβ ∂ (α1) Υ (1) ∂ (β1) Υ (1) . (3.35)\n\nExtremizing this functional yields the first line of (3.26) at η 2 i = 0. Except for the Grassmann integration, this action has the same form as the purely bosonic one [55] . One may apply the same logic to the Φ-formulation where the action for the purely bosonic case is also known [54, 56] .\n\nThe existence of the linear system (3.22) (equivalent to (3.21)) encoding solutions of the N -extended U(n) chiral model in an auxiliary matrix ψ allows for powerful methods to systematically construct explicit solutions for ψ and hence for Φ † = ψ| ζ=0 and Υ = lim ζ→∞ ζ (ψ-1l). For our purposes the so-called dressing method [57, 51] proves to be the most practical [12] - [20] , and so we shall use it here for our linear system, i.e. already in the N -extended noncommutative case.\n\nMulti-pole ansatz for ψ. The dressing method is a recursive procedure for generating a new solution from an old one. More concretely, we rewrite the linear system (3.21) in the form\n\nψ(∂ u -ζ∂ x )ψ † = A , ψ(∂ x -ζ∂ v )ψ † = B , ψ(∂ i 2 -ζ∂ i 1 )ψ † = C i . (4.1)\n\nRecall that ψ † := (ψ(x a , η α i , ζ)) † and (A, B, C i ) depend only on x a and η α i . The central idea is to demand analyticity in the spectral parameter ζ, which strongly restricts the possible form of ψ. One way to exploit this constraint starts from the observation that the left hand sides of (4.1) as well as of the reality condition (3.15) do not depend on ζ while ψ is expected to be a nontrivial function of ζ globally defined on CP 1 . Therefore, it must be a meromorphic function on CP 1 possessing some poles which we choose to lie at finite points with constant coordinates µ k ∈ CP 1 .\n\nHere we will build a (multi-soliton) solution ψ m featuring m simple poles at positions µ 1 , . . . , µ m with foot_8 Im µ k < 0 by left-multiplying an (m-1)-pole solution ψ m-1 with a single-pole factor of the form 1l\n\n+ µ m -μm ζ -µ m P m (x a , η α i ) , (4.2)\n\nwhere the n×n matrix function P m is yet to be determined. Starting from the trivial (vacuum) solution ψ 0 = 1l, the iteration ψ 0 → ψ 1 → . . . → ψ m yields a multiplicative ansatz for ψ m ,\n\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ , (4.3)\n\nwhich, via partial fraction decomposition, may be rewritten in the additive form\n\nψ m = 1l + m k=1 Λ mk S † k ζ -µ k , (4.4)\n\nwhere Λ mk and S k are some n×r k matrices depending on x a and η α i , with r k ≤ n.\n\nEquations for S k . Let us first consider the additive parametrization (4.4) of ψ m . This ansatz must satisfy the reality condition (3.15) as well as our linear equations in the form (4.1). In particular, the poles at ζ = μk on the left hand sides of these equations have to be removable since the right hand sides are independent of ζ. Inserting the ansatz (4.4) and putting to zero the corresponding residues, we learn from (3.15) that\n\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ S k = 0 , (4.5)\n\nwhile from (4.1) we obtain the differential equations\n\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ LA,B,i k S k = 0 , (4.6)\n\nwhere LA,B,i\n\nk stands for either LA k = ∂ u -μk ∂ x , LB k = µ k (∂ x -μk ∂ v ) or Li k = ∂ i 2 -μk ∂ i 1 . (4.7)\n\nNote that we consider a recursive procedure starting from m=1, and operators (4.7) will appear with k = 1, . . . , m if we consider poles at ζ = μk .\n\nBecause the LA,B,i k for k = 1, . . . , m are linear differential operators, it is easy to write down the general solution for (4.6) at any given k, by passing from the coordinates (u, v, x; η 1 i , η 2 i ) to \"co-moving coordinates\" (w k , wk , s k ; η i k , ηi k ). The precise relation for k = 1, . . . , m is [12, 58]\n\nw k := x + μk u + μ-1 k v = x + 1 2 (μ k -μ -1 k )y + 1 2 (μ k +μ -1 k )t and η i k := η 1 i + μk η 2 i , (4.8)\n\nwith wk and ηi k obtained by complex conjugation and the co-moving time s k being inessential because by definition nothing will depend on it. The kth moving frame travels with a constant velocity\n\n(v x , v y ) k = - µ k + μk µ k μk + 1 , µ k μk -1 µ k μk + 1 , (4.9)\n\nso that the static case w k =z is recovered for µ k = -i. On functions of (w k , η i k , wk , ηi k ) alone the operators (4.7) act as\n\nLA k = LB k = (µ k -μ k ) ∂ ∂ wk =: Lk and Li k = (µ k -μ k ) ∂ ∂ ηi k . (4.10)\n\nBy induction in k = 1, . . . , m we learn that, due to (4.5), a necessary and sufficient condition for a solution of (4.6) is Lk\n\nS k = S k Zk and Li k S k = S k Zi k (4.11)\n\nwith some r k ×r k matrices Zk and Zi k depending on (w k , wk , η j k , ηj k ). Passing to the noncommutative bosonic coordinates we obtain ŵk , ŵk = 2θ ν k νk with\n\nν k νk = 4i µ k -μ k -µ -1 k +μ -1 k . (4.12)\n\nThus, we can introduce annihilation and creation operators\n\nc k = 1 √ 2θ ŵk ν k and c † k = 1 √ 2θ ŵk νk so that [c k , c † k ] = 1 (4.13)\n\nfor k = 1, . . . , m. Naturally, this Heisenberg algebra is realized on a \"co-moving\" Fock space H k , with basis states |ℓ k and a \"co-moving\" vacuum |0 k subject to c k |0 k = 0. Each co-moving vacuum |0 k (annihilated by c k ) is related to the static vacuum |0 (annihilated by a) through an ISU(1,1) squeezing transformation (cf. [12] ) which is time-dependent. The fermionic coordinates η i k and ηi k remain spectators in the deformation. Coordinate derivatives are represented in the standard fashion as\n\nν k √ 2θ ∂ ∂w k → -[c † k , • ] and νk √ 2θ ∂ ∂ wk → [c k , • ] . (4.14)\n\nAfter the Moyal deformation, the n×r k matrices S k have become operator-valued, but are still functions of the Grassmann coordinates η i k and ηi k . The noncommutative version of the BPS conditions (4.11) naturally reads\n\nc k S k = S k Z k and ∂ ∂ ηi k S k = S k Z i k (4.15)\n\nwhere Z k and Z i k are some operator-valued r k ×r k matrix functions of η j k and ηj k .\n\nNonabelian solutions for S k . For general data Z k and Z i k it is difficult to solve (4.15), but it is also unnecessary because the final expression ψ m turns out not to depend on them. Therefore, we conveniently choose\n\nZ k = c k ⊗ 1l r k ×r k and Z i k = 0 ⇒ S k = R k (c k , η i k ) , (4.16)\n\nwhere R k is an arbitrary n×r k matrix function independent of c † k and ηi k . 10 It is known that nonabelian (multi-) solitons arise for algebraic functions R k (cf. e.g. [7] for the commutative and [12] for the noncommutative N =0 case). Their common feature is a smooth commutative limit. The only novelty of the supersymmetric extension is the η i k dependence, i.e.\n\nR k = R k,0 + η i k R k,i + η i k η j k R k,ij + η i k η j k η p k R k,ijp + η i k η j k η p k η q k R k,ijpq . (4.17)\n\nAbelian solutions for S k . It is useful to view S k as a map from C r k ⊗H k to C n ⊗H k (momentarily suppressing the η dependence). The noncommutative setup now allows us to generalize the domain of this map to any subspace of C n ⊗ H k . In particular, we may choose it to be finite-dimensional, say C q k , and represent the map by an n×q k array |S k of kets in H. In this situation, Z k and Z i k in (4.15) are just number -valued q k ×q k matrix functions of η j k and ηj k . In case they do not depend on ηj k , we can write down the most general solution as\n\n|S k = R k (c k , η j k ) |Z k exp i Z i k (η j k ) ηi k with |Z k := exp Z k (η j k ) c † k |0 k . (4.18)\n\nAs before, we may put Z i k = 0 without loss of generality, but now the choice of Z k does matter. For any given k generically there exists a q k -dimensional basis change which diagonalizes the ket-valued matrix\n\n|Z k → diag e α 1 k c † , e α 2 k c † , . . . , e α q k k c † |0 k = diag |α 1 k , |α 2 k , . . . , |α q k k , (4.19)\n\nwhere we defined coherent states\n\n|α l k := e α l k c † |0 k so that c k |α l k = α l k |α l k for l = 1, . . . , q k and α l k ∈ C . (4.20)\n\nNote that not only the entries of R k but also the α l k are holomorphic functions of the co-moving Grassmann parameters η j k and thus can be expanded like in (4.17). In the U(1) model, we must use ket-valued 1×q k matrices |S k for all k, yielding rows\n\n|S k = R 1 k |α 1 k , R 2 k |α 2 k , . . . , R q k k |α q k k for k = 1, . . . , m , (4.21)\n\nwith functions α l k (η j k ). Here, the R l k only affect the states' normalization and can be collected in a diagonal matrix to the right, hence will drop out later and thus may all be put to one. Formally, we have recovered the known abelian (multi-) soliton solutions, but the supersymmetric extension has generalized |S k → |S k (η j k ) . Explicit form of P k . Let us now consider the multiplicative parametrization (4.3) of ψ m which also allows us to solve (4.5) . First of all, note that the reality condition (3.15) is satisfied if\n\nP k = P † k = P 2 k ⇔ P k = T k (T † k T k ) -1 T † k for k = 1, . . . , m , (4.22)\n\nmeaning that P k is an operator-valued hermitian projector (of group-space rank r k ≤ n) built from an n×r k matrix function T k (the abelian case of n=1 is included). The reality condition follows just because The r k columns of T k span the image of P k and obey\n\n1l + µ k -μk ζ -µ k P k 1l + μk -µ k ζ -μk P k =\n\nP k T k = T k ⇔ (1l-P k ) T k = 0 . (4.24)\n\nFurthermore, the equation (4.5) with m = k (induction) rewritten in the form\n\n(1l-P k ) k-1 ℓ=1 1l + µ k-ℓ -μk-ℓ μk -µ k-ℓ P k-ℓ S k = 0 (4.25)\n\nreveals that (cf. (4.24))\n\nT 1 = S 1 and T k = k-1 ℓ=1 1l - µ k-ℓ -μk-ℓ µ k-ℓ -μk P k-ℓ S k for k ≥ 2 , (4.26)\n\nwhere the explicit form of S k for k = 1, . . . , m is given in (4.16) or (4.18) . The final result reads\n\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ = 1l + m k=1 Λ mk S † k ζ -µ k (4.27)\n\nwith hermitian projectors P k given by (4.22), T k given by (4.26) and S k given by (4.16) or (4.18) . The explicit form of Λ mk (which we do not need) can be found in [12] . The corresponding superfields Φ and Υ are\n\nΦ m = ψ † m | ζ=0 = m k=1 (1l -ρ k P k ) with ρ k = 1 - µ k μk , (4.28a)\n\nΥ m = lim ζ→∞ ζ (ψ m -1l) = m k=1 (µ k -μ k ) P k . (4.28b)\n\nFrom (4.22) it is obvious that P k is invariant under a similarity transformation\n\nT k → T k Λ k ⇔ S k → S k Λ k (4.29)\n\nfor an invertible operator-valued r k ×r k matrix Λ k . This justifies putting Z i k = 0 from the beginning and also the restriction to Z k = c k ⊗ 1l r k ×r k in the nonabelian case, both without loss of generality. Hence, the nonabelian solution space constructed here is parametrized by the set {R k } m 1 of matrixvalued functions of c k and η i k and the pole positions µ k . The abelian moduli space, however, is larger by the set {Z k } m 1 of matrix-values functions of η i k which generically contain the coherentstate parameter functions {α l k (η i k )}. Restricting to η i k =0 reproduces the soliton configurations of the bosonic model [12] .\n\nStatic solutions. Let us consider the reduction to 2+0 dimensions, i.e. the static case. Recall that static solutions correspond to the choice m = 1 and µ 1 ≡ µ = -i implying w 1 = z, so we drop the index k. Specializing (4.27), we have\n\nψ = 1l - 2 i ζ + i P so that Φ = Φ † = 1l -2P , (4.30)\n\nwhere a hermitian projector P of group-space rank r satisfies the BPS equations (1l-P ) a P = 0 ⇒ (1l-P ) a T = 0 ,\n\n(1l-P ) ∂ ∂ ηi P = 0 ⇒ (1l-P ) ∂ ∂ ηi T = 0 , (4.31a)\n\nwith P = T (T † T ) -1 T † and η i = η 1 i + iη 2 i . In this case T = S, and for a nonabelian r=1 projector P we get T = T (a, η i ) as an n×1 column. For the simplest case of N =1 we just have (cf. [59] )\n\nT = T e (a) + η T o (a) with η = η 1 + iη 2 , (4.32)\n\nwhere T e (a) and T o (a) are rational functions of a (e.g. polynomials) taking values in the even and odd parts of the Grassmann algebra. Similarly, an abelian N =1 projector (for n=1) is built from\n\n|T = |α 1 , |α 2 , . . . , |α q . (4.33)\n\nAt θ=0, the static solution (4.32) of our supersymmetric U(n) sigma model is also a solution of the standard N =1 supersymmetric CP n-1 sigma model in two dimensions (see e.g. [59] ). 11 For this reason, one can overcome the previously mentioned difficulty with constructing an action (or energy from the viewpoint of 2+1 dimensions) for static configurations. Moreover, on solutions obeying the BPS conditions (4.31) the topological charge\n\nQ = 2πθ dη 1 dη 2 Tr tr Φ D + Φ , D -Φ (4.34)\n\nis proportional to the action (BPS bound)\n\nS = 2πθ dη 1 dη 2 Tr tr D + Φ , D -Φ (4.35)\n\nand is finite for algebraic functions T e and T o . Here, the standard superderivatives D ± are defined as\n\nD + = ∂ ∂η + iη ∂ z and D -= ∂ ∂ η + iη ∂ z . ( 4\n\nOne-soliton configuration. For one moving soliton, from (4.27) and (4.28) we obtain\n\nψ 1 = 1l + µ - μ ζ -µ P with P = T (T † T ) -1 T † (4.37) and Φ = 1l -ρ P with ρ = 1 - µ μ . (4.38)\n\nNow our n×r matrix T must satisfy (putting\n\nZ i = 0 and Z = c ⊗ 1l r×r ) [c , T ] = 0 and ∂ ∂ ηi T = 0 with η i = η 1 i + μ η 2 i , (4.39)\n\nwhere c is the moving-frame annihilation operator given by (4.13) for k=1.\n\nRecall that the operators c and c † and therefore the matrix T and the projector P can be expressed in terms of the corresponding static objects by a unitary squeezing transformation (see e.g. (4.8) and (4.13)). For simplicity we again consider the case N =1 and a nonabelian projector with r=1. Then (4.39) tells us that T is a holomorphic function of c and η, i.e.\n\nT = T e (c) + η T o (c) = T 1 e (c) + η T 1 o (c)\n\n. . .\n\nT n e (c) + η T n o (c) (4.40)\n\nwith polynomials T a e and T a o of order q, say, analogously to the static case (4.32). Note that, for T a o to be Grassmann-odd and nonzero, some extraneous Grassmann parameter must appear. Similarly, abelian projectors for a moving one-soliton obtain by subjecting (4.33) to a squeezing transformation.\n\nFor N =1 the moving frame was defined in (4.8) (dropping the index k) via\n\nw = x + 1 2 (μ-μ -1 )y + 1 2 (μ+μ -1 )t and η = η 1 + μη 2 hence ∂ t η = 0 . (4.41)\n\nConsider the moving frame with the coordinates (w, w, s; η, η) with the choice s = t and the related change of the derivatives (see [12, 58] )\n\n∂ x = ∂ w + ∂ w , (4.42a)\n\n∂ y = 1 2 (μ-μ -1 ) ∂ w + 1 2 (µ-µ -1 ) ∂ w , (4.42b)\n\n∂ t = 1 2 (μ+μ -1 ) ∂ w + 1 2 (µ+µ -1 ) ∂ w + ∂ s , (4.42c)\n\n∂ η 1 = ∂ η + ∂ η , (4.42d)\n\n∂ η 2 = μ ∂ η + µ ∂ η . (4.42e)\n\nIn the moving frame our solution (4.38) is static, i.e. ∂ s Φ = 0, and the projector P has the same form as in the static case. The only difference is the coefficient ρ instead of 2 in (4.38). Therefore, by computing the action (4.35) in (w, w; η 1 , η 2 ) coordinates, we obtain for algebraic functions T in (4.40) a finite answer, which differs from the static one by a kinematical prefactor depending on µ (cf. [12] for the bosonic case).\n\nLarge-time asymptotics. Note that in the distinguished (z, z, t) coordinate frame (4.41) implies that at large times w → κ t with κ = 1 2 (μ+μ -1 ). As a consequence, the t q term in each polynomial in (4.40) will dominate, i.e.\n\nT → t q a 1 + η b 1 . . .\n\nan + η bn =: t q Γ , (4.43)\n\nwhere Γ is a fixed vector in C n . It is easy to see that in the distinguished frame the large-time limit of Φ given by (4. being the projector on the constant vector Γ.\n\nConsider now the m-soliton configuration (4.28). By induction of the above argument one easily arrives at the m-soliton generalization of (4.44). Namely, in the frame moving with the ℓth lump we have lim t→±∞ Φ m = (1lρ 1 Π 1 ) . . . (1lρ ℓ-1 Π ℓ-1 )(1lρ ℓ P ℓ )(1lρ ℓ+1 Π ℓ+1 ) . . . (1lρ m Π m ) , (4.45) where the Π m are constant projectors. This large-time factorization of multi-soliton solutions provides a proof of the no-scattering property because the asymptotic configurations are identical for large negative and large positive times.\n\nIn this paper we introduced a generalization of the modified integrable U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions and considered a Moyal deformation of this model. It was shown that this N -extended chiral model is equivalent to a gauge-fixed BPS subsector of an N -extended super Yang-Mills model in 2+1 dimensions originating from twistor string theory. The dressing method was applied to generate a wide class of multi-soliton configurations, which are time-dependent finite-energy solutions to the equations of motion. Compared to the N =0 model, the supersymmetric extension was seen to promote the configurations' building blocks to holomorphic functions of suitable Grassmann coordinates. By considering the large-time asymptotic factorization into a product of single soliton solutions we have shown that no scattering occurs within the dressing ansatz chosen here.\n\nThe considered model does not stand alone but is motivated by twistor string theory [37] with a target space reduced to the mini-supertwistor space [44, 45, 47] . In this context, the obtained multi-soliton solutions are to be regarded as D(0|2N )-branes moving inside D(2|2N )-branes [60] . Here 2N appears due to fermionic worldvolume directions of our branes in the superspace description [60] . Switching on a constant B-field simply deforms the sigma model and D-brane worldvolumes noncommutatively, thereby admitting also regular supersymmetric noncommutative abelian solutions.\n\nRestricting to static configurations, the models can be specialized to Grassmannian supersymmetric sigma models, where the superfield Φ takes values in Gr(r, n), and the field equations are invariant under 2N supersymmetry transformations with 0 ≤ N ≤ 4. This differs from the results for standard 2D sigma models [52, 53] where the target spaces have to be Kähler or hyper-Kähler for admitting two or four supersymmetries, respectively. This difference will be discussed in more details elsewhere.\n\nWe derived the supersymmetric chiral model in 2+1 dimensions through dimensional reduction and gauge fixing of the N -extended supersymmetric SDYM equations in 2+2 dimensions. Recall that for the purely bosonic case most (if not all) integrable equations in three and fewer dimensions can be obtained from the SDYM equations (or their hierarchy [25] ) by suitable dimensional reductions (see e.g. [61] - [65] and references therein). Moreover, this Ward conjecture [61] was extended to the noncommutative case (see e.g. [66, 67] ). It will be interesting to consider similar reductions of the N -extended supersymmetric SDYM equations (and their hierarchy [68] ) to supersymmetric integrable equations in three and two dimensions generalizing earlier results [69] ."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model, we employ the dressing method to explicitly construct multi-soliton configurations on noncommutative R 2,1 and analyze some of their properties."},{"section_type":"BACKGROUND","section_title":"Introduction","text":"In the low-energy limit string theory with D-branes gives rise to noncommutative field theory on the branes when the string propagates in a nontrivial NS-NS two-form (B-field) background [1, 2, 3, 4] . In particular, if the open string has N =2 worldsheet supersymmetry, the tree-level target space dynamics is described by a noncommutative self-dual Yang-Mills (SDYM) theory in 2+2 dimensions [5] . Furthermore, open N =2 strings in a B-field background induce on the worldvolume of n coincident D2-branes a noncommutative Yang-Mills-Higgs Bogomolny-type system in 2+1 dimensions which is equivalent to a noncommutative generalization [6] of the modified U(n) chiral model known as the Ward model [7] . The topological nature of N =2 strings and the integrability of their tree-level dynamics [8] render this noncommutative sigma model integrable. 1 Being integrable, the commutative U(n≥2) Ward model features a plethora of exact scattering and no-scattering multi-soliton and wave solutions, i.e. time-dependent stable configurations on R foot_1 . These are not only a rich testing ground for physical properties such as adiabatic dynamics or quantization, but also descend to more standard multi-solitons of various integrable systems in 2+0 and 1+1 dimensions, such as sine-Gordon, upon dimensional and algebraic reduction. There is a price to pay however: Nonlinear sigma models in 2+1 dimensions may be Lorentz-invariant or integrable but not both [7, 11] . In fact, Derrick's theorem prohibits the existence of stable solitons in Lorentz-invariant scalar field theories above 1+1 dimensions. A Moyal deformation, however, overcomes this hurdle, but of course replaces Lorentz invariance by a Drinfeld-twisted version. There is another gain: The deformed Ward model possesses not only deformed versions of the just-mentioned multi-solitons, but in addition allows for a whole new class of genuinely noncommutative (multi-)solitons, in particular for the U(1) group [12, 13] ! Moreover, this class is related to the generic but perturbatively constructed noncommutative scalar-field solitons [14, 15] by an infinite-stiffness limit of the potential [16] .\n\nIn [12, 13] and [17] - [20] families of multi-solitons as well as their reduction to solitons of the noncommutative sine-Gordon equations were described and studied. In the nonabelian case both scattering and nonscattering configurations were obtained. For static configurations the issue of their stability was analyzed [21] . The full moduli space metric for the abelian model was computed and its adiabatic two-soliton dynamics was discussed [16] .\n\nRecall that the critical N =2 string theory has a four-dimensional target space, and its open string effective field theory is self-dual Yang-Mills [8] , which gets deformed noncommutatively in the presence of a B-field [5] . Conversely, the noncommutative SDYM equations are contained [19] in the equations of motion of N =2 string field theory (SFT) [22] in a B-field background. This SFT formulation is based on the N =4 topological string description [23] . It is well known that the SDYM model can be described in terms of holomorphic bundles over (an open subset of) the twistor space 2 [26] CP 3 and the topological N =4 string theory contains twistors from the outset. The Lax pair, integrability and the solutions to the equations of motion by twistor and dressing methods were incorporated into the N =2 open SFT in [27, 28] . However, this theory reproduces only bosonic SDYM theory, its symmetries (see e.g. [29, 30, 31] ) and integrability properties. It is natural to ask: What string theory can describe supersymmetric SDYM theory [32, 33] in four dimensions?\n\nThere are some proposals [33, 34, 35, 36] for extending N =2 open string theory (and its SFT) to be space-time supersymmetric. Moreover, it was shown by Witten [37] that N =4 supersymmetric SDYM theory appears in twistor string theory, which is a B-type open topological string with the supertwistor space CP 3|4 as a target space. 3 Note that N <4 SDYM theory forms a BPS subsector of N -extended super Yang-Mills theory, and N =4 SDYM can be considered as a truncation of the full N =4 super Yang-Mills theory [37] . It is believed [43, 39] that twistor string theory is related with the previous proposals [33, 34, 35, 36] for a Lorentz-invariant supersymmetric extension of N =2 (and topological N =4) string theory which also leads to the N =4 SDYM model.\n\nA dimensional reduction of the above relations between twistor strings and N =4 super Yang-Mills and SDYM models was considered in [44, 45, 46, 47] . The corresponding twistor string theory after this reduction is the topological B-model on the mini-supertwistor space P 2|4 . In [47] it was shown that the 2N =8 supersymmetric extension of the Bogomolny-type model in 2+1 dimensions is equivalent to an 2N =8 supersymmetric modified U(n) chiral model on R 2,1 . The subject of the current paper is an 2N ≤8 version of the above supersymmetric Bogomolny-type Yang-Mills-Higgs model in signature (-+ +), its relation with an N -extended supersymmetric modified integrable U(n) chiral model (to be defined) in 2+1 dimensions and the Moyal-type noncommutative deformation of this chiral model. We go on to explicitly construct multi-soliton configurations on noncommutative R 2,1 for the corresponding supersymmetric sigma model field equations. By studying the scattering properties of the constructed configurations, we prove their asymptotic factorization without scattering for large times. We also briefly discuss a D-brane interpretation of these soliton configurations from the viewpoint of twistor string theory."},{"section_type":"OTHER","section_title":"N -extended SDYM equations in 2+2 dimensions","text":"Space R 2,2 . Let us consider the four-dimensional space R 2,2 = (R 4 , g) with the metric\n\nds 2 = g µν dx µ dx ν = det(dx α α) = dx 1 1dx 2 2 -dx 2 1dx 1 2 (2.1)\n\nwith (g µν ) = diag(-1, +1, +1, -1), where µ, ν, . . . = 1, . . . , 4 are space-time indices and α = 1, 2, α = 1, 2 are spinor indices. We choose the coordinates 4\n\n(x µ ) = (x a , t) = (t, x, y, t) with a, b, . . . = 1, 2, 3 , (2.2)\n\nand the signature (-+ + -) allows us to introduce real isotropic coordinates (cf. [19, 6] )\n\nx 1 1 = 1 2 (t -y) , x 1 2 = 1 2 (x + t) , x 2 1 = 1 2 (x -t) , x 2 2 = 1 2 (t + y) . (2.3)\n\nSDYM. Recall that the SDYM equations for a field strength tensor\n\nF µν on R 2,2 read 1 2 ε µνρσ F ρσ = F µν , (2.4)\n\nwhere ε µνρσ is a completely antisymmetric tensor on R 2,2 and ε 1234 = 1. In the coordinates (2.3) we have the decomposition\n\nF α α,β β = ∂ α αA β β -∂ β β A α α + [A α α, A β β ] = ε αβ F α β + ε α β F αβ (2.5) with F α β := -1 2 ε αβ F α α,β β and F αβ := -1 2 ε α β F α α,β β , (2.6)\n\nwhere ε αβ is antisymmetric, ε αβ ε βγ = δ γ α , and similar for ε α β , with ε 12 = ε 1 2 = 1. The gauge potential (A α α) will appear in the covariant derivative\n\nD α β = ∂ α β + [A α β , • ] . (2.7)\n\nIn spinor notation, (2.4) is equivalently written as\n\nF α β = 0 ⇔ F α α,β β = ε α β F αβ . (2.8)\n\nSolutions {A α α} to these equations form a subset (a BPS sector) of the solution space of Yang-Mills theory on R 2,2 .\n\nN -extended SDYM in component fields. The field content of N -extended super SDYM is foot_4\n\nN = 0 A α α (2.9a) N = 1 A α α, χ i α with i = 1 (2.9b) N = 2 A α α, χ i α , φ [ij] with i, j = 1, 2 (2.9c)\n\nN = 3 A α α, χ i α , φ [ij] , χ[ijk] α with i, j, k = 1, 2, 3 (2.9d) N = 4 A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl] α β\n\nwith i, j, k, l = 1, 2, 3, 4 .\n\n(2.9e)\n\nHere (A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl]\n\nα β ) are fields of helicities (+1, + 1 2 , 0, -1 2 , -1). These fields obey the field equations of the N = 4 SDYM model, namely [33, 37]\n\nF α β = 0 , (2.10a)\n\nD α αχ iα = 0 , (2.10b)\n\nD α αD α αφ ij + 2{χ iα , χ j α } = 0 , (2.10c)\n\nD α α χ α[ijk] -6[χ [i α , φ jk] ] = 0 , (2.10d)\n\nD γ α G [ijkl] γ β + 12{χ [i α , χjkl] β } -18[φ [ij , D α β φ kl] ] = 0 . (2.10e)\n\nNote that the N < 4 SDYM field equations are governed by the first N +1 equations of (2.10), where F α β = 0 is counted as one equation and so on."},{"section_type":"OTHER","section_title":"Superfield formulation of","text":"N -extended SDYM Superspace R 4|4N .\n\nRecall that in the space R 2,2 = (R 4 , g) with the metric g given in (2.1) one may introduce purely real Majorana-Weyl spinors foot_5 θ α and η α of helicities + 1 2 and -1 2 as anticommuting (Grassmann-algebra) objects. Using 2N such spinors with components θ iα and η α i for i = 1, . . . , N , one can define the N -extended superspace R 4|4N and the N -extended supersymmetry algebra generated by the supertranslation operators\n\nP α α = ∂ α α , Q iα = ∂ iα -η α i ∂ α α and Q i α = ∂ i α -θ iα ∂ α α , (2.11)\n\nwhere\n\n∂ α α := ∂ ∂x α α , ∂ iα := ∂ ∂θ iα and ∂ i α := ∂ ∂η α i .\n\n(2.12)\n\nThe commutation relations for the generators (2.11) read\n\n{Q iα , Q j α} = -2δ j i P α α , [P α α, Q iβ ] = 0 and [P α α, Q i β ] = 0 . (2.13)\n\nTo rewrite equations of motion in terms of R 4|4N superfields one uses the additional operators\n\nD iα = ∂ iα + η α i ∂ α α and D i α = ∂ i α + θ iα ∂ α α , (2.14)\n\nwhich (anti)commute with the operators (2.11) and satisfy\n\n{D iα , D j β } = 2δ j i P α β , [P α α, D iβ ] = 0 and [P α α, D j β ] = 0 . (2.15)\n\nAntichiral superspace R 4|2N . On the superspace R 4|4N one may introduce tensor fields depending on bosonic and fermionic coordinates (superfields), differential forms, Lie derivatives L X etc.. Furthermore, on any such superfield A one can impose the constraint equations L D iα A = 0, which for a scalar superfield f reduce to the so-called antichirality conditions\n\nD iα f = 0 . (2.16)\n\nThese are easily solved by using a coordinate transformation on R 4|4N ,\n\n(x α α, η α i , θ iα ) → (x α α = x α α-θ iα η α i , η α i , θ iα ) , (2.17)\n\nunder which ∂ α α, D iα and D i α transform to the operators\n\n∂α α = ∂ α α , Diα = ∂ iα and Di α = ∂ i α + 2θ iα ∂ α α . ( 2\n\n.18) Then (2.16) simply means that f is defined on a sub-superspace R 4|2N ⊂ R 4|4N with coordinates xα α and η α i . (2.19)\n\nThis space is called antichiral superspace. In the following we will usually omit the tildes when working on the antichiral superspace.\n\nN -extended SDYM in superfields. The N -extended SDYM equations can be rewritten in terms of superfields on the antichiral superspace R 4|2N [33, 48] . Namely, for any given 0 ≤ N ≤ 4, fields of a proper multiplet from (2.9) can be combined into superfields A α α and A i α depending on x α α, η α i ∈ R 4|2N and giving rise to covariant derivatives\n\n∇ α α := ∂ α α + A α α and ∇ i α := ∂ i α + A i α .\n\n(2.20)\n\nIn such terms the N -extended SDYM equations (2.10) read\n\n[∇ α α, ∇ β β ] + [∇ α β , ∇ β α] = 0 , [∇ i α, ∇ β β ] + [∇ i β , ∇ β α] = 0 , {∇ i α, ∇ j β } + {∇ i β , ∇ j α} = 0 , (2.21)\n\nwhich is equivalent to\n\n[∇ α α, ∇ β β ] = ε α β F αβ , [∇ i α, ∇ β β ] = ε α β F i β and {∇ i α, ∇ j β } = ε α β F ij , (2.22)\n\nwhere F ij is antisymmetric and F αβ is symmetric in their indices.\n\nThe above gauge potential superfields (A α α, A i α) as well as the gauge strength superfields (F αβ , F i α , F ij ) contain all physical component fields of the N -extended SDYM model. For instance, the lowest component of the triple (F αβ , F i α , F ij ) in an η-expansion is (F αβ , χ i α , φ ij ), with zeros in case N is too small. By employing Bianchi identities for the gauge strength superfields, one successively obtains [48] the superfield expansions and the field equations (2.10) for all component fields.\n\nIt is instructive to extend the antichiral combination in (2.18) to potentials and covariant derivatives, Di\n\nα = ∂ i α + 2 θ iα ∂ α α + + + Ãi α := A i α + 2 θ iα A α α ∇i α := ∇ i α + 2 θ iα ∇ α α ( 2\n\n.23) where ∇ α α, ∇ i α and Di α are given by (2.20) and (2.18), while A i α and A α α depend on x α α and η α i only. With the antichiral covariant derivatives, one may condense (2.21) or (2.22) into the single set { ∇i α, ∇j β\n\n} + { ∇i β , ∇j α} = 0 ⇔ { ∇i α, ∇j β } = ε α β Fij , (2.24)\n\nwith Fij =\n\nF ij + 4 θ [iα F j] α + 4 θ iα θ jβ F αβ .\n\nThe concise form (2.24) of the N -extended SDYM equations is quite convenient, and we will use it interchangeable with (2.21).\n\nLinear system for N -extended SDYM. It is well known that the superfield SDYM equations (2.21) can be seen as the compatibility conditions for the linear system of differential equations\n\nζ α(∂ α α + A α α) ψ = 0 and ζ α(∂ i α + A i α) ψ = 0 , (2.25)\n\nwhere\n\n(ζ β ) = 1 ζ and ζ α = ε α β ζ β . The extra (spectral) parameter 7 ζ lies in the extended complex plane C ∪ ∞ = CP 1 .\n\nHere ψ is a matrix-valued function depending not only on x α α and η α i but also (meromorphically) on ζ ∈ CP 1 . We subject the n×n matrix ψ to the following reality condition:\n\nψ(x α α, η α i , ζ) ψ(x α α, η α i , ζ) † = 1l , (2.26)\n\nwhere \" †\" denotes hermitian conjugation and ζ is complex conjugate to ζ. This condition guarantees that all physical fields of the N -extended SDYM model will take values in the adjoint representation of the algebra u(n). In the concise form the linear system (2.25) is written as\n\nζ α(∇ i α + 2θ iα ∇ α α) ψ = 0 ⇔ ζ α( Di α + Ãi α) ψ = 0 ⇔ ζ α ∇i α ψ = 0 . (2.27) 2.3 Reduction of N -extended SDYM to 2+1 dimensions\n\nThe supersymmetric Bogomolny-type Yang-Mills-Higgs equations in 2+1 dimensions are obtained from the described N -extended super SDYM equations by a dimensional reduction R 2,2 → R 2,1 .\n\nIn particular, for the N =0 sector we demand the components A µ of a gauge potential to be independent of x 4 and put A 4 =: ϕ. Here, ϕ is a Lie-algebra valued scalar field in three dimensions (the Higgs field) which enters into the Bogomolny-type equations. Similarly, for N ≥ 1 one can reduce the N -extended SDYM equations on R 2,2 by imposing the ∂ 4 -invariance condition on all the fields (A α α,\n\nχ i α , φ [ij] , χ[ijk] α , G [ijkl]\n\nα β ) from the N =4 supermultiplet or its truncation to N <4 and obtain supersymmetric Bogomolny-type equations on R 2,1 .\n\nSpinors in R 2,1 .\n\nRecall that on R 2,2 both N =4 SDYM theory and full N =4 super Yang-Mills theory have an SL(4, R) ∼ = Spin(3,3) R-symmetry group [33] . A dimensional reduction to R 2,1 enlarges the supersymmetry and R-symmetry to 2N =8 and Spin(4,4), respectively, for both theories (cf. [49] for Minkowski signature). More generally, any number N of supersymmetries gets doubled to 2N in the reduction. Since dimensional reduction collapses the rotation group Spin(2,2)\n\n∼ = Spin(2,1) L ×Spin(2,1) R of R 2,2\n\nto its diagonal subgroup Spin(2,1) D as the local rotation group of R 2,1 , the distinction between undotted and dotted indices disappears. We shall use undotted indices henceforth.\n\nCoordinates and derivatives in R 2,1 . The above discussion implies that one can relabel the bosonic coordinates x α β from (2.3) by x αβ and split them as\n\nx αβ = 1 2 (x αβ + x βα ) + 1 2 (x αβ -x βα ) = x (αβ) + x [αβ]\n\n(2.28) into antisymmetric and symmetric parts,\n\nx [αβ] = 1 2 ε αβ x 4 = 1 2 ε αβ t and x (αβ) =: y αβ , (2.29)\n\nrespectively, with\n\ny 11 = x 11 = 1 2 (t -y) , y 12 = 1 2 (x 12 + x 21 ) = 1 2 x , y 22 = x 22 = 1 2 (t + y) . (2.30)\n\nWe also have θ iα → θ iα and η α i → η α i for the fermionic coordinates on R 4|4N reduced to R 3|4N . Bosonic coordinate derivatives reduce in 2+1 dimensions to the operators\n\n∂ (αβ) = 1 2 (∂ αβ + ∂ βα ) (2.31)\n\nwhich read explicitly as\n\n∂ (11) = ∂ ∂y 11 = ∂ t -∂ y , ∂ (12) = ∂ (21) = 1 2 ∂ ∂y 12 = ∂ x , ∂ (22) = ∂ ∂y 22 = ∂ t + ∂ y . (2.32)\n\nWe thus have\n\n∂ ∂x αβ = ∂ (αβ) -ε αβ ∂ 4 = ∂ (αβ) -ε αβ ∂ t , (2.33)\n\nwhere\n\nε 12 = -ε 21 = -1, ∂ 4 = ∂/∂x 4 and ∂ t = ∂/∂ t.\n\nThe operators D iα and D i α acting on t-independent superfields reduce to\n\nD iα = ∂ iα + η β i ∂ (αβ) and D i α = ∂ i α + θ iβ ∂ (αβ) , (2.34)\n\nwhere\n\n∂ iα = ∂/∂θ iα and ∂ i α = ∂/∂η α i .\n\nSimilarly, the antichiral operators Diα and Di α in (2.18) become Diα = ∂ iα and Di α = ∂ i α + 2θ iβ ∂ (αβ) . (2.35) Supersymmetric Bogomolny-type equations in component fields. According to (2.33), the components A α β of a gauge potential in four dimensions split into the components A (αβ) of a gauge potential in three dimensions and a Higgs field\n\nA [αβ] = -ε αβ ϕ, i.e. A αβ = A (αβ) + A [αβ] = A (αβ) -ε αβ ϕ . (2.36)\n\nThen the covariant derivatives D α β reduced to three dimensions become the differential operators\n\nD αβ -ε αβ ϕ = ∂ (αβ) + [A (αβ) , • ] -ε αβ [ϕ, • ] , (2.37)\n\nand the Yang-Mills field strength on R 2,1 decomposes as\n\nF αβ, γδ = [D αβ , D γδ ] = ε αγ f βδ + ε βδ f αγ with f αβ = f βα . (2.38)\n\nSubstituting (2.36) and (2.37) into (2.10), i.e. demanding that all fields in (2.10) are independent of x 4 = t, we obtain the following supersymmetric Bogomolny-type equations on R 2,1 :\n\nf αβ + D αβ ϕ = 0 , (2.39a)\n\nD αβ χ iβ + ε αβ [ϕ, χ iβ ] = 0 , (2.39b)\n\nD αβ D αβ φ ij + 2[ϕ, [ϕ, φ ij ]] + 2{χ iα , χ j α } = 0 , (2.39c)\n\nD αβ χβ[ijk] -ε αβ [ϕ, χβ[ijk] ] -6[χ [i α , φ jk] ] = 0 , (2.39d)\n\nD γ α G [ijkl] γβ + [ϕ, G [ijkl] αβ ] + 12{χ [i α , χjkl] β } -18[φ [ij , D αβ φ kl] ] -18ε αβ [φ [ij , [φ kl] , ϕ]] = 0 .(2.\n\nSupersymmetric Bogomolny-type equations in terms of superfields. Translations generated by the vector field ∂ 4 = ∂ t are isometries of superspaces R 4|4N and R 4|2N . By taking the quotient with respect to the action of the abelian group G generated by ∂ 4 , we obtain the reduced full superspace R 3|4N ∼ = R 4|4N /G and the reduced antichiral superspace R 3|2N ∼ = R 4|2N /G. In the following, we shall work on R 3|2N and R 3|2N × CP 1 , since the reduced ψ-function from (2.25) and (2.27) is defined on the latter space.\n\nThe linear system stays in the center of the superfield approach to the N -extended SDYM equations. After imposing t-independence on all fields in the linear system (2.27), we arrive at the linear equations\n\nζ α ∇i α ψ ≡ ζ α ( Di α + Âi α ) ψ = 0 (2.40)\n\nof the same form but with\n\nDi α = ∂ i α + 2θ iβ ∂ (αβ) and Âi α = A i α + 2θ iβ (A (αβ) -ε αβ Ξ) , (2.41)\n\nwhere A i α , A (αβ) and Ξ are superfields depending on y αβ and η α i only. These linear equations expand again to the pair (cf. (2.25))\n\nζ β (∂ (αβ) + A (αβ) -ε αβ Ξ) ψ = 0 and ζ α (∂ i α + A i α ) ψ = 0 . (2.42)\n\nThe compatibility conditions for the linear system (2.40) read\n\n{ ∇i α , ∇j β } + { ∇i β , ∇j α } = 0 ⇔ { ∇i α , ∇j β } = ε αβ Fij (2.43)\n\nand present a condensed form of (2.39) rewritten in terms of R 3|2N superfields. Similarly, these equations can also be written in more expanded forms analogously to (2.21) or using the superfield analog of (2.37). However, we will not do this since all these sets of equations are equivalent."},{"section_type":"OTHER","section_title":"Noncommutative N -extended U(n) chiral model in 2+1 dimensions","text":"As has been known for some time, nonlinear sigma models in 2 + 1 dimensions may be Lorentzinvariant or integrable but not both [7, 11] . We will show that the super Bogomolny-type model discussed in Section 2 after a gauge fixing is equivalent to a super extension of the modified U(n) chiral model (so as to be integrable) first formulated by Ward [7] . Since integrability is compatible with noncommutative deformation (if introduced properly, see e.g. [9] - [20] ) we choose from the beginning to formulate our super extension of this chiral model on Moyal-deformed R 2,1 with noncommutativity parameter θ ≥ 0. Ordinary space-time R 2,1 can always be restored by taking the commutative limit θ → 0.\n\nStar-product formulation. Classical field theory on noncommutative spaces may be realized in a star-product formulation or in an operator formalism 8 . The first approach is closer to the commutative field theory: it is obtained by simply deforming the ordinary product of classical fields (or their components) to the noncommutative star product\n\n(f ⋆ g)(x) = f (x) exp{ i 2 ← - ∂ a θ ab -→ ∂ b } g(x) ⇒ x a ⋆ x b -x b ⋆ x a = iθ ab (3.1)\n\nwith a constant antisymmetric tensor θ ab . Specializing to R 2,1 , we use real coordinates (x a ) = (t, x, y) in which the Minkowski metric g on R 3 reads (g ab ) = diag(-1, +1, +1) with a, b, . . . = 1, 2, 3 (cf. Section 2). It is straightforward to generalize the Moyal deformation (3.1) to the superspaces introduced in the previous section, allowing in particular for non-anticommuting Grassmann-odd coordinates. Deferring general superspace deformations and their consequences to future work, we here content ourselves with the simple embedding of the \"bosonic\" Moyal deformation into superspace, meaning that (3.1) is also valid for superfields f and g depending on Grassmann variables θ iα and η α i . For later use we consider not only isotropic coordinates and vector fields\n\nu := 1 2 (t+y) = y 22 , v := 1 2 (t-y) = y 11 , ∂ u = ∂ t + ∂ y = ∂ (22) , ∂ v = ∂ t -∂ y = ∂ (11) (3.2)\n\nintroduced in Section 2, but also the complex combinations\n\nz := x + iy , z := x -iy , ∂ z = 1 2 (∂ x -i∂ y ) , ∂ z = 1 2 (∂ x + i∂ y ) . (3.3)\n\nSince the time coordinate t remains commutative, the only nonvanishing component of the noncommutativity tensor θ ab is Operator formalism. The nonlocality of the star products renders explicit computation cumbersome. We therefore pass to the operator formalism, which trades the star product for operatorvalued spatial coordinates (x, ŷ) or their complex combinations (ẑ, ẑ), subject to\n\nθ xy = -θ yx =: θ > 0 ⇒ θ z z = -θ zz = -2i θ . ( 3\n\n[t, x] = [t, ŷ] = 0 but [x, ŷ] = iθ ⇒ [ẑ, ẑ] = 2 θ . (3.6)\n\nThe latter equation suggests the introduction of annihilation and creation operators,\n\na = 1 √ 2θ ẑ and a † = 1 √ 2θ ẑ with [a , a † ] = 1 , (3.7)\n\nwhich act on a harmonic-oscillator Fock space H with an orthonormal basis { |ℓ , ℓ = 0, 1, 2, . . .} such that\n\na |ℓ = √ ℓ |ℓ-1 and a † |ℓ = √ ℓ+1 |ℓ+1 . (3.8)\n\nAny superfield f (t, z, z, η α i ) on R 3|2N can be related to an operator-valued superfield f (t, η α i ) ≡ F (t, a, a † , η α i ) on R 1|2N acting in H, with the help of the Moyal-Weyl map\n\nf (t, z, z, η α i ) → f (t, η α i ) = Weyl-ordered f t, √ 2θa, √ 2θa † , η α i . (3.9)\n\nThe inverse transformation recovers the ordinary superfield,\n\nf (t, η α i ) ≡ F (t, a, a † , η α i ) → f (t, z, z, η α i ) = F ⋆ t, z √ 2θ , z √ 2θ , η α i , (3.10)\n\nwhere F ⋆ is obtained from F by replacing ordinary with star products. Under the Moyal-Weyl map, we have\n\nf ⋆ g → f ĝ and dx dy f = 2π θ Tr f = 2π θ ℓ≥0 ℓ| f |ℓ , (3.11)\n\nand the spatial derivatives are mapped into commutators,\n\n∂ z f → ∂z f = -1 √ 2θ [a † , f ] and ∂ z f → ∂z f = 1 √ 2θ [a , f ] . (3.12)\n\nFor notational simplicity we will from now on omit the hats over the operators except when confusion may arise.\n\nGauge fixing for ψ. Note that the linear system (2.40) and the compatibility conditions (2.43) are invariant under a gauge transformation\n\nψ → ψ ′ = g -1 ψ , (3.13a)\n\nA → A ′ = g -1 A g + g -1 ∂ g (with appropriate indices) , (3.13b)\n\nΞ → Ξ ′ = g -1 Ξ g , (3.13c)\n\nwhere g = g(x a , η α i ) is a U(n)-valued superfield globally defined on the deformed superspace R 3|2N θ × CP 1 . Using a gauge transformation of the form (3.13), we can choose ψ such that it will satisfy the standard asymptotic conditions (see e.g. [51] )\n\nψ = Φ -1 + O(ζ) for ζ → 0 , (3.14a\n\n)\n\nψ = 1l + ζ -1 Υ + O(ζ -2 ) for ζ → ∞ , (3.14b)\n\nwhere the U(n)-valued function Φ and u(n)-valued function Υ depend on x a and η α i . This \"unitary\" gauge is compatible with the reality condition for ψ,\n\nψ(x a , η α i , ζ) ψ(x a , η α i , ζ) † = 1l , (3.15)\n\nobtained by reduction from (2.26)."},{"section_type":"OTHER","section_title":"Gauge fixing for","text":"A i 1 = 0 and A i 2 = Φ -1 ∂ i 2 Φ = ∂ i 1 Υ , (3.17)\n\nA (11) = 0 and\n\nA (12) + Ξ = Φ -1 ∂ (12) Φ = ∂ (11) Υ , (3.18)\n\nA (21) -Ξ = 0 and\n\nA (22) = Φ -1 ∂ (22) Φ = ∂ (12) Υ . (3.19)\n\nUsing (2.32), we can rewrite the nonzero components as\n\nA := Φ -1 ∂ u Φ = ∂ x Υ , B := Φ -1 ∂ x Φ = ∂ v Υ , C i := Φ -1 ∂ i 2 Φ = ∂ i 1 Υ . (3.20)\n\nRecall that the superfields Φ and Υ depend on x a and η α i .\n\nLinear system. In the above-introduced unitary gauge the linear system (2.42) reads\n\n(ζ∂ x -∂ u -A) ψ = 0 , (ζ∂ v -∂ x -B) ψ = 0 , (ζ∂ i 1 -∂ i 2 -C i ) ψ = 0 , (3.21)\n\nwhich adds the last equation to the linear system of the Ward model [7] and generalizes it to superfields A(x a , η α j ), B(x a , η α j ) and C i (x a , η α j ). The concise form of (3.21) reads\n\nζ Di 1 -Di 2 -Âi 2 ψ = 0 (3.22)\n\nor, in more explicit form,\n\nζ ∂ i 1 + 2θ i1 ∂ v + 2θ i2 ∂ x -∂ i 2 + C i + 2θ i1 (∂ x + B) + 2θ i2 (∂ u + A) ψ = 0 . (3.23)\n\nN -extended sigma model.\n\nThe compatibility conditions of this linear system are the Nextended noncommutative sigma model equations Di\n\n1 (Φ -1 Dj 2 Φ) + Dj 1 (Φ -1 Di 2 Φ) = 0 (3.24)\n\nwhich in expanded form reads\n\n(g ab + v c ε cab ) ∂ a (Φ -1 ∂ b Φ) = 0 ⇔ ∂ x (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ u Φ) = 0 , (3.25a) ∂ i 1 (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ i 2 Φ) = 0 , ∂ i 1 (Φ -1 ∂ u Φ) -∂ x (Φ -1 ∂ i 2 Φ) = 0 , (3.25b)\n\n∂ i 1 (Φ -1 ∂ j 2 Φ) + ∂ j 1 (Φ -1 ∂ i 2 Φ) = 0 . (3.25c)\n\nHere, the first line contains the Wess-Zumino-Witten term with a constant vector (v c ) = (0, 1, 0) which spoils the standard Lorentz invariance but yields an integrable chiral model in 2+1 dimensions. Recall that Φ is a U(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . As discussed in Section 2, the compatibility conditions of the linear equations (3.22) (or (3.21)) are equivalent to the N -extended Bogomolnytype equations (2.39) for the component (physical) fields. Thus, chiral model field equations (3.25) are equivalent to a gauge fixed form of equations (2.39). Υ-formulation. Instead of Φ-parametrization of (A, B, C i ) given in (3.17)-(3.20) we may use the equivalent Υ-parametrization also given there. In this case, the compatibility conditions for the linear system (3.21) reduce to\n\n(∂ 2 x -∂ u ∂ v )Υ + [∂ v Υ , ∂ x Υ] = 0 , (3.26a)\n\n(∂ i 2 ∂ v -∂ i 1 ∂ x )Υ + [∂ i 1 Υ , ∂ v Υ] = 0 , (∂ i 2 ∂ x -∂ i 1 ∂ u )Υ + [∂ i 1 Υ , ∂ x Υ] = 0 , (3.26b) (∂ i 2 ∂ j 1 + ∂ j 2 ∂ i 1 )Υ + {∂ i 1 Υ , ∂ j 1 Υ} = 0 , (3.26c)\n\nwhich in concise form read\n\n( Di 2 Dj 1 + Dj 2 Di 1 ) Υ + { Di 1 Υ , Dj 1 Υ} = 0 . (3.27)\n\nRecall that Υ is a u(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . For N =4, the commutative limit of (3.27) can be considered as Siegel's equation [33] reduced to 2+1 dimensions. According to Siegel, one can extract the multiplet of physical fields appearing in (2.39) from the prepotential Υ via\n\n∂ i 1 Υ = A i 2 , ∂ i 1 ∂ j 1 Υ = φ ij , ∂ i 1 ∂ j 1 ∂ k 1 Υ = χ[ijk] 2 , ∂ i 1 ∂ j 1 ∂ k 1 ∂ l 1 Υ = G [ijkl] 22 , (3.28a)\n\n∂ (α1) Υ = A (α2) -ε α2 ϕ , ∂ (α1) ∂ i 1 Υ = χ i α , ∂ (α1) ∂ (β1) Υ = f αβ , (3.28b)\n\nwhere one takes Υ and its derivatives at η 2 i = 0. The other components of the physical fields, i.e. χ[ijk]\n\n1 , G [ijkl] 11 , G [ijkl]\n\n21 , A (11) and A (21) -ϕ, vanish in this light-cone gauge.\n\nSupersymmetry transformations. The 4N supercharges given in (2.11) reduce in 2+1 dimensions to the form\n\nQ iα = ∂ iα -η β i ∂ (αβ) and Q i α = ∂ i α -θ iβ ∂ (αβ) . (3.29)\n\nTheir antichiral version, matching to Diα and Dj β of (2.35), reads\n\nQiα = ∂ iα -2η β i ∂ (αβ) and Qj β = ∂ j β , (3.30)\n\nso that { Qiα , Qj β } = -2 δ j i ∂ (αβ) . (3.31)\n\nOn a (scalar) R 3|2N superfield Σ these supersymmetry transformations act as\n\nδ Σ := ε iα Qiα Σ + ε α i Qi α Σ (3.32)\n\nand are induced by the coordinate shifts\n\nδ y αβ = -2ε i(α η β) i and δ η α i = ε α i , (3.33)\n\nwhere ε iα and ε α i are 4N real Grassmann parameters. It is easy to see that our equations (3.24) and (3.27) are invariant under the supersymmetry transformations (3.32) (applied to Φ or Υ). This is simply because the operators Diα and Dj β anticommute with the supersymmetry generators Qiα and Qj β . Therefore, the equations of motion (3.25) of the modified N -extended chiral model in 2+1 dimensions as well as their reductions to 2+0 and 1+1 dimensions carry 2N supersymmetries and are genuine supersymmetric extensions of the corresponding bosonic equations. Note that this type of extension is not the standard one since the R-symmetry groups are Spin(N , N ) in 2+1 and Spin(N , N )× Spin(N , N ) in 1+1 dimensions, which differ from the compact unitary R-symmetry groups of standard sigma models. Contrary to the standard case of two-dimensional sigma models the above \"noncompact\" 2N supersymmetries do not impose any constraints on the geometry of the target space, e.g. they do not demand it to be Kähler [52] or hyper-Kähler [53] . This may be of interest and deserves further study.\n\nAction functionals. In either formulation of the N -extended supersymmetric SDYM model on R 2,2 there are difficulties with finding a proper action functional generalizing the one [54, 55] for the purely bosonic case. These difficulties persist after the reduction to 2+1 dimensions, i.e. for the equations (3.25) and (3.26) describing our supersymmetric modified U(n) chiral model. It is the price to be paid for overcoming the no-go barrier N ≤ 4 and the absence of geometric target-space constraints. On a more formal level, the problem is related to the chiral character of (3.24) as well as (3.27) , where only the operators Di α but not Diα appear. Note however, that for N = 4 one can write an action functional in component fields producing the equations (2.39), which are equivalent to the superspace equations (3.24) when i, j = 1, . . . , 4 (see e.g. [47] ).\n\nOne proposal for an action functional stems from Siegel's idea [33] for the Υ-formulation of the N -extended SDYM equations. Namely, one sees that ∂ i 2 Υ enters only linearly into the last two lines in (3.26) . Therefore, if we introduce\n\nΥ (1) := Υ| η 2 i =0 (3.34)\n\nthen it must satisfy the first equation from (3.26) , and the remaining equations iteratively define the dependence of Υ on η 2 i starting from Υ (1) . Hence, all information is contained in Υ (1) , as can also be seen from (3.28) . In other words, the dependence of Υ on η 2 i is not 'dynamical'. For an action one can then take (cf. [33] )\n\nS = d 3 x d N η 1 Υ (1) ∂ (αβ) ∂ (αβ) Υ (1) + 2 3 Υ (1) ε αβ ∂ (α1) Υ (1) ∂ (β1) Υ (1) . (3.35)\n\nExtremizing this functional yields the first line of (3.26) at η 2 i = 0. Except for the Grassmann integration, this action has the same form as the purely bosonic one [55] . One may apply the same logic to the Φ-formulation where the action for the purely bosonic case is also known [54, 56] ."},{"section_type":"OTHER","section_title":"N -extended multi-soliton configurations via dressing","text":"The existence of the linear system (3.22) (equivalent to (3.21)) encoding solutions of the N -extended U(n) chiral model in an auxiliary matrix ψ allows for powerful methods to systematically construct explicit solutions for ψ and hence for Φ † = ψ| ζ=0 and Υ = lim ζ→∞ ζ (ψ-1l). For our purposes the so-called dressing method [57, 51] proves to be the most practical [12] - [20] , and so we shall use it here for our linear system, i.e. already in the N -extended noncommutative case.\n\nMulti-pole ansatz for ψ. The dressing method is a recursive procedure for generating a new solution from an old one. More concretely, we rewrite the linear system (3.21) in the form\n\nψ(∂ u -ζ∂ x )ψ † = A , ψ(∂ x -ζ∂ v )ψ † = B , ψ(∂ i 2 -ζ∂ i 1 )ψ † = C i . (4.1)\n\nRecall that ψ † := (ψ(x a , η α i , ζ)) † and (A, B, C i ) depend only on x a and η α i . The central idea is to demand analyticity in the spectral parameter ζ, which strongly restricts the possible form of ψ. One way to exploit this constraint starts from the observation that the left hand sides of (4.1) as well as of the reality condition (3.15) do not depend on ζ while ψ is expected to be a nontrivial function of ζ globally defined on CP 1 . Therefore, it must be a meromorphic function on CP 1 possessing some poles which we choose to lie at finite points with constant coordinates µ k ∈ CP 1 .\n\nHere we will build a (multi-soliton) solution ψ m featuring m simple poles at positions µ 1 , . . . , µ m with foot_8 Im µ k < 0 by left-multiplying an (m-1)-pole solution ψ m-1 with a single-pole factor of the form 1l\n\n+ µ m -μm ζ -µ m P m (x a , η α i ) , (4.2)\n\nwhere the n×n matrix function P m is yet to be determined. Starting from the trivial (vacuum) solution ψ 0 = 1l, the iteration ψ 0 → ψ 1 → . . . → ψ m yields a multiplicative ansatz for ψ m ,\n\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ , (4.3)\n\nwhich, via partial fraction decomposition, may be rewritten in the additive form\n\nψ m = 1l + m k=1 Λ mk S † k ζ -µ k , (4.4)\n\nwhere Λ mk and S k are some n×r k matrices depending on x a and η α i , with r k ≤ n.\n\nEquations for S k . Let us first consider the additive parametrization (4.4) of ψ m . This ansatz must satisfy the reality condition (3.15) as well as our linear equations in the form (4.1). In particular, the poles at ζ = μk on the left hand sides of these equations have to be removable since the right hand sides are independent of ζ. Inserting the ansatz (4.4) and putting to zero the corresponding residues, we learn from (3.15) that\n\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ S k = 0 , (4.5)\n\nwhile from (4.1) we obtain the differential equations\n\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ LA,B,i k S k = 0 , (4.6)\n\nwhere LA,B,i\n\nk stands for either LA k = ∂ u -μk ∂ x , LB k = µ k (∂ x -μk ∂ v ) or Li k = ∂ i 2 -μk ∂ i 1 . (4.7)\n\nNote that we consider a recursive procedure starting from m=1, and operators (4.7) will appear with k = 1, . . . , m if we consider poles at ζ = μk .\n\nBecause the LA,B,i k for k = 1, . . . , m are linear differential operators, it is easy to write down the general solution for (4.6) at any given k, by passing from the coordinates (u, v, x; η 1 i , η 2 i ) to \"co-moving coordinates\" (w k , wk , s k ; η i k , ηi k ). The precise relation for k = 1, . . . , m is [12, 58]\n\nw k := x + μk u + μ-1 k v = x + 1 2 (μ k -μ -1 k )y + 1 2 (μ k +μ -1 k )t and η i k := η 1 i + μk η 2 i , (4.8)\n\nwith wk and ηi k obtained by complex conjugation and the co-moving time s k being inessential because by definition nothing will depend on it. The kth moving frame travels with a constant velocity\n\n(v x , v y ) k = - µ k + μk µ k μk + 1 , µ k μk -1 µ k μk + 1 , (4.9)\n\nso that the static case w k =z is recovered for µ k = -i. On functions of (w k , η i k , wk , ηi k ) alone the operators (4.7) act as\n\nLA k = LB k = (µ k -μ k ) ∂ ∂ wk =: Lk and Li k = (µ k -μ k ) ∂ ∂ ηi k . (4.10)\n\nBy induction in k = 1, . . . , m we learn that, due to (4.5), a necessary and sufficient condition for a solution of (4.6) is Lk\n\nS k = S k Zk and Li k S k = S k Zi k (4.11)\n\nwith some r k ×r k matrices Zk and Zi k depending on (w k , wk , η j k , ηj k ). Passing to the noncommutative bosonic coordinates we obtain ŵk , ŵk = 2θ ν k νk with\n\nν k νk = 4i µ k -μ k -µ -1 k +μ -1 k . (4.12)\n\nThus, we can introduce annihilation and creation operators\n\nc k = 1 √ 2θ ŵk ν k and c † k = 1 √ 2θ ŵk νk so that [c k , c † k ] = 1 (4.13)\n\nfor k = 1, . . . , m. Naturally, this Heisenberg algebra is realized on a \"co-moving\" Fock space H k , with basis states |ℓ k and a \"co-moving\" vacuum |0 k subject to c k |0 k = 0. Each co-moving vacuum |0 k (annihilated by c k ) is related to the static vacuum |0 (annihilated by a) through an ISU(1,1) squeezing transformation (cf. [12] ) which is time-dependent. The fermionic coordinates η i k and ηi k remain spectators in the deformation. Coordinate derivatives are represented in the standard fashion as\n\nν k √ 2θ ∂ ∂w k → -[c † k , • ] and νk √ 2θ ∂ ∂ wk → [c k , • ] . (4.14)\n\nAfter the Moyal deformation, the n×r k matrices S k have become operator-valued, but are still functions of the Grassmann coordinates η i k and ηi k . The noncommutative version of the BPS conditions (4.11) naturally reads\n\nc k S k = S k Z k and ∂ ∂ ηi k S k = S k Z i k (4.15)\n\nwhere Z k and Z i k are some operator-valued r k ×r k matrix functions of η j k and ηj k .\n\nNonabelian solutions for S k . For general data Z k and Z i k it is difficult to solve (4.15), but it is also unnecessary because the final expression ψ m turns out not to depend on them. Therefore, we conveniently choose\n\nZ k = c k ⊗ 1l r k ×r k and Z i k = 0 ⇒ S k = R k (c k , η i k ) , (4.16)\n\nwhere R k is an arbitrary n×r k matrix function independent of c † k and ηi k . 10 It is known that nonabelian (multi-) solitons arise for algebraic functions R k (cf. e.g. [7] for the commutative and [12] for the noncommutative N =0 case). Their common feature is a smooth commutative limit. The only novelty of the supersymmetric extension is the η i k dependence, i.e.\n\nR k = R k,0 + η i k R k,i + η i k η j k R k,ij + η i k η j k η p k R k,ijp + η i k η j k η p k η q k R k,ijpq . (4.17)\n\nAbelian solutions for S k . It is useful to view S k as a map from C r k ⊗H k to C n ⊗H k (momentarily suppressing the η dependence). The noncommutative setup now allows us to generalize the domain of this map to any subspace of C n ⊗ H k . In particular, we may choose it to be finite-dimensional, say C q k , and represent the map by an n×q k array |S k of kets in H. In this situation, Z k and Z i k in (4.15) are just number -valued q k ×q k matrix functions of η j k and ηj k . In case they do not depend on ηj k , we can write down the most general solution as\n\n|S k = R k (c k , η j k ) |Z k exp i Z i k (η j k ) ηi k with |Z k := exp Z k (η j k ) c † k |0 k . (4.18)\n\nAs before, we may put Z i k = 0 without loss of generality, but now the choice of Z k does matter. For any given k generically there exists a q k -dimensional basis change which diagonalizes the ket-valued matrix\n\n|Z k → diag e α 1 k c † , e α 2 k c † , . . . , e α q k k c † |0 k = diag |α 1 k , |α 2 k , . . . , |α q k k , (4.19)\n\nwhere we defined coherent states\n\n|α l k := e α l k c † |0 k so that c k |α l k = α l k |α l k for l = 1, . . . , q k and α l k ∈ C . (4.20)\n\nNote that not only the entries of R k but also the α l k are holomorphic functions of the co-moving Grassmann parameters η j k and thus can be expanded like in (4.17). In the U(1) model, we must use ket-valued 1×q k matrices |S k for all k, yielding rows\n\n|S k = R 1 k |α 1 k , R 2 k |α 2 k , . . . , R q k k |α q k k for k = 1, . . . , m , (4.21)\n\nwith functions α l k (η j k ). Here, the R l k only affect the states' normalization and can be collected in a diagonal matrix to the right, hence will drop out later and thus may all be put to one. Formally, we have recovered the known abelian (multi-) soliton solutions, but the supersymmetric extension has generalized |S k → |S k (η j k ) . Explicit form of P k . Let us now consider the multiplicative parametrization (4.3) of ψ m which also allows us to solve (4.5) . First of all, note that the reality condition (3.15) is satisfied if\n\nP k = P † k = P 2 k ⇔ P k = T k (T † k T k ) -1 T † k for k = 1, . . . , m , (4.22)\n\nmeaning that P k is an operator-valued hermitian projector (of group-space rank r k ≤ n) built from an n×r k matrix function T k (the abelian case of n=1 is included). The reality condition follows just because The r k columns of T k span the image of P k and obey\n\n1l + µ k -μk ζ -µ k P k 1l + μk -µ k ζ -μk P k =\n\nP k T k = T k ⇔ (1l-P k ) T k = 0 . (4.24)\n\nFurthermore, the equation (4.5) with m = k (induction) rewritten in the form\n\n(1l-P k ) k-1 ℓ=1 1l + µ k-ℓ -μk-ℓ μk -µ k-ℓ P k-ℓ S k = 0 (4.25)\n\nreveals that (cf. (4.24))\n\nT 1 = S 1 and T k = k-1 ℓ=1 1l - µ k-ℓ -μk-ℓ µ k-ℓ -μk P k-ℓ S k for k ≥ 2 , (4.26)\n\nwhere the explicit form of S k for k = 1, . . . , m is given in (4.16) or (4.18) . The final result reads\n\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ = 1l + m k=1 Λ mk S † k ζ -µ k (4.27)\n\nwith hermitian projectors P k given by (4.22), T k given by (4.26) and S k given by (4.16) or (4.18) . The explicit form of Λ mk (which we do not need) can be found in [12] . The corresponding superfields Φ and Υ are\n\nΦ m = ψ † m | ζ=0 = m k=1 (1l -ρ k P k ) with ρ k = 1 - µ k μk , (4.28a)\n\nΥ m = lim ζ→∞ ζ (ψ m -1l) = m k=1 (µ k -μ k ) P k . (4.28b)\n\nFrom (4.22) it is obvious that P k is invariant under a similarity transformation\n\nT k → T k Λ k ⇔ S k → S k Λ k (4.29)\n\nfor an invertible operator-valued r k ×r k matrix Λ k . This justifies putting Z i k = 0 from the beginning and also the restriction to Z k = c k ⊗ 1l r k ×r k in the nonabelian case, both without loss of generality. Hence, the nonabelian solution space constructed here is parametrized by the set {R k } m 1 of matrixvalued functions of c k and η i k and the pole positions µ k . The abelian moduli space, however, is larger by the set {Z k } m 1 of matrix-values functions of η i k which generically contain the coherentstate parameter functions {α l k (η i k )}. Restricting to η i k =0 reproduces the soliton configurations of the bosonic model [12] .\n\nStatic solutions. Let us consider the reduction to 2+0 dimensions, i.e. the static case. Recall that static solutions correspond to the choice m = 1 and µ 1 ≡ µ = -i implying w 1 = z, so we drop the index k. Specializing (4.27), we have\n\nψ = 1l - 2 i ζ + i P so that Φ = Φ † = 1l -2P , (4.30)\n\nwhere a hermitian projector P of group-space rank r satisfies the BPS equations (1l-P ) a P = 0 ⇒ (1l-P ) a T = 0 ,\n\n(1l-P ) ∂ ∂ ηi P = 0 ⇒ (1l-P ) ∂ ∂ ηi T = 0 , (4.31a)\n\nwith P = T (T † T ) -1 T † and η i = η 1 i + iη 2 i . In this case T = S, and for a nonabelian r=1 projector P we get T = T (a, η i ) as an n×1 column. For the simplest case of N =1 we just have (cf. [59] )\n\nT = T e (a) + η T o (a) with η = η 1 + iη 2 , (4.32)\n\nwhere T e (a) and T o (a) are rational functions of a (e.g. polynomials) taking values in the even and odd parts of the Grassmann algebra. Similarly, an abelian N =1 projector (for n=1) is built from\n\n|T = |α 1 , |α 2 , . . . , |α q . (4.33)\n\nAt θ=0, the static solution (4.32) of our supersymmetric U(n) sigma model is also a solution of the standard N =1 supersymmetric CP n-1 sigma model in two dimensions (see e.g. [59] ). 11 For this reason, one can overcome the previously mentioned difficulty with constructing an action (or energy from the viewpoint of 2+1 dimensions) for static configurations. Moreover, on solutions obeying the BPS conditions (4.31) the topological charge\n\nQ = 2πθ dη 1 dη 2 Tr tr Φ D + Φ , D -Φ (4.34)\n\nis proportional to the action (BPS bound)\n\nS = 2πθ dη 1 dη 2 Tr tr D + Φ , D -Φ (4.35)\n\nand is finite for algebraic functions T e and T o . Here, the standard superderivatives D ± are defined as\n\nD + = ∂ ∂η + iη ∂ z and D -= ∂ ∂ η + iη ∂ z . ( 4\n\nOne-soliton configuration. For one moving soliton, from (4.27) and (4.28) we obtain\n\nψ 1 = 1l + µ - μ ζ -µ P with P = T (T † T ) -1 T † (4.37) and Φ = 1l -ρ P with ρ = 1 - µ μ . (4.38)\n\nNow our n×r matrix T must satisfy (putting\n\nZ i = 0 and Z = c ⊗ 1l r×r ) [c , T ] = 0 and ∂ ∂ ηi T = 0 with η i = η 1 i + μ η 2 i , (4.39)\n\nwhere c is the moving-frame annihilation operator given by (4.13) for k=1.\n\nRecall that the operators c and c † and therefore the matrix T and the projector P can be expressed in terms of the corresponding static objects by a unitary squeezing transformation (see e.g. (4.8) and (4.13)). For simplicity we again consider the case N =1 and a nonabelian projector with r=1. Then (4.39) tells us that T is a holomorphic function of c and η, i.e.\n\nT = T e (c) + η T o (c) = T 1 e (c) + η T 1 o (c)\n\n. . .\n\nT n e (c) + η T n o (c) (4.40)\n\nwith polynomials T a e and T a o of order q, say, analogously to the static case (4.32). Note that, for T a o to be Grassmann-odd and nonzero, some extraneous Grassmann parameter must appear. Similarly, abelian projectors for a moving one-soliton obtain by subjecting (4.33) to a squeezing transformation.\n\nFor N =1 the moving frame was defined in (4.8) (dropping the index k) via\n\nw = x + 1 2 (μ-μ -1 )y + 1 2 (μ+μ -1 )t and η = η 1 + μη 2 hence ∂ t η = 0 . (4.41)\n\nConsider the moving frame with the coordinates (w, w, s; η, η) with the choice s = t and the related change of the derivatives (see [12, 58] )\n\n∂ x = ∂ w + ∂ w , (4.42a)\n\n∂ y = 1 2 (μ-μ -1 ) ∂ w + 1 2 (µ-µ -1 ) ∂ w , (4.42b)\n\n∂ t = 1 2 (μ+μ -1 ) ∂ w + 1 2 (µ+µ -1 ) ∂ w + ∂ s , (4.42c)\n\n∂ η 1 = ∂ η + ∂ η , (4.42d)\n\n∂ η 2 = μ ∂ η + µ ∂ η . (4.42e)\n\nIn the moving frame our solution (4.38) is static, i.e. ∂ s Φ = 0, and the projector P has the same form as in the static case. The only difference is the coefficient ρ instead of 2 in (4.38). Therefore, by computing the action (4.35) in (w, w; η 1 , η 2 ) coordinates, we obtain for algebraic functions T in (4.40) a finite answer, which differs from the static one by a kinematical prefactor depending on µ (cf. [12] for the bosonic case).\n\nLarge-time asymptotics. Note that in the distinguished (z, z, t) coordinate frame (4.41) implies that at large times w → κ t with κ = 1 2 (μ+μ -1 ). As a consequence, the t q term in each polynomial in (4.40) will dominate, i.e.\n\nT → t q a 1 + η b 1 . . .\n\nan + η bn =: t q Γ , (4.43)\n\nwhere Γ is a fixed vector in C n . It is easy to see that in the distinguished frame the large-time limit of Φ given by (4. being the projector on the constant vector Γ.\n\nConsider now the m-soliton configuration (4.28). By induction of the above argument one easily arrives at the m-soliton generalization of (4.44). Namely, in the frame moving with the ℓth lump we have lim t→±∞ Φ m = (1lρ 1 Π 1 ) . . . (1lρ ℓ-1 Π ℓ-1 )(1lρ ℓ P ℓ )(1lρ ℓ+1 Π ℓ+1 ) . . . (1lρ m Π m ) , (4.45) where the Π m are constant projectors. This large-time factorization of multi-soliton solutions provides a proof of the no-scattering property because the asymptotic configurations are identical for large negative and large positive times."},{"section_type":"CONCLUSION","section_title":"Conclusions","text":"In this paper we introduced a generalization of the modified integrable U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions and considered a Moyal deformation of this model. It was shown that this N -extended chiral model is equivalent to a gauge-fixed BPS subsector of an N -extended super Yang-Mills model in 2+1 dimensions originating from twistor string theory. The dressing method was applied to generate a wide class of multi-soliton configurations, which are time-dependent finite-energy solutions to the equations of motion. Compared to the N =0 model, the supersymmetric extension was seen to promote the configurations' building blocks to holomorphic functions of suitable Grassmann coordinates. By considering the large-time asymptotic factorization into a product of single soliton solutions we have shown that no scattering occurs within the dressing ansatz chosen here.\n\nThe considered model does not stand alone but is motivated by twistor string theory [37] with a target space reduced to the mini-supertwistor space [44, 45, 47] . In this context, the obtained multi-soliton solutions are to be regarded as D(0|2N )-branes moving inside D(2|2N )-branes [60] . Here 2N appears due to fermionic worldvolume directions of our branes in the superspace description [60] . Switching on a constant B-field simply deforms the sigma model and D-brane worldvolumes noncommutatively, thereby admitting also regular supersymmetric noncommutative abelian solutions.\n\nRestricting to static configurations, the models can be specialized to Grassmannian supersymmetric sigma models, where the superfield Φ takes values in Gr(r, n), and the field equations are invariant under 2N supersymmetry transformations with 0 ≤ N ≤ 4. This differs from the results for standard 2D sigma models [52, 53] where the target spaces have to be Kähler or hyper-Kähler for admitting two or four supersymmetries, respectively. This difference will be discussed in more details elsewhere.\n\nWe derived the supersymmetric chiral model in 2+1 dimensions through dimensional reduction and gauge fixing of the N -extended supersymmetric SDYM equations in 2+2 dimensions. Recall that for the purely bosonic case most (if not all) integrable equations in three and fewer dimensions can be obtained from the SDYM equations (or their hierarchy [25] ) by suitable dimensional reductions (see e.g. [61] - [65] and references therein). Moreover, this Ward conjecture [61] was extended to the noncommutative case (see e.g. [66, 67] ). It will be interesting to consider similar reductions of the N -extended supersymmetric SDYM equations (and their hierarchy [68] ) to supersymmetric integrable equations in three and two dimensions generalizing earlier results [69] ."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model, we employ the dressing method to explicitly construct multi-soliton configurations on noncommutative R 2,1 and analyze some of their properties.\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"In the low-energy limit string theory with D-branes gives rise to noncommutative field theory on the branes when the string propagates in a nontrivial NS-NS two-form (B-field) background [1, 2, 3, 4] . In particular, if the open string has N =2 worldsheet supersymmetry, the tree-level target space dynamics is described by a noncommutative self-dual Yang-Mills (SDYM) theory in 2+2 dimensions [5] . Furthermore, open N =2 strings in a B-field background induce on the worldvolume of n coincident D2-branes a noncommutative Yang-Mills-Higgs Bogomolny-type system in 2+1 dimensions which is equivalent to a noncommutative generalization [6] of the modified U(n) chiral model known as the Ward model [7] . The topological nature of N =2 strings and the integrability of their tree-level dynamics [8] render this noncommutative sigma model integrable. 1 Being integrable, the commutative U(n≥2) Ward model features a plethora of exact scattering and no-scattering multi-soliton and wave solutions, i.e. time-dependent stable configurations on R foot_1 . These are not only a rich testing ground for physical properties such as adiabatic dynamics or quantization, but also descend to more standard multi-solitons of various integrable systems in 2+0 and 1+1 dimensions, such as sine-Gordon, upon dimensional and algebraic reduction. There is a price to pay however: Nonlinear sigma models in 2+1 dimensions may be Lorentz-invariant or integrable but not both [7, 11] . In fact, Derrick's theorem prohibits the existence of stable solitons in Lorentz-invariant scalar field theories above 1+1 dimensions. A Moyal deformation, however, overcomes this hurdle, but of course replaces Lorentz invariance by a Drinfeld-twisted version. There is another gain: The deformed Ward model possesses not only deformed versions of the just-mentioned multi-solitons, but in addition allows for a whole new class of genuinely noncommutative (multi-)solitons, in particular for the U(1) group [12, 13] ! Moreover, this class is related to the generic but perturbatively constructed noncommutative scalar-field solitons [14, 15] by an infinite-stiffness limit of the potential [16] .\\n\\nIn [12, 13] and [17] - [20] families of multi-solitons as well as their reduction to solitons of the noncommutative sine-Gordon equations were described and studied. In the nonabelian case both scattering and nonscattering configurations were obtained. For static configurations the issue of their stability was analyzed [21] . The full moduli space metric for the abelian model was computed and its adiabatic two-soliton dynamics was discussed [16] .\\n\\nRecall that the critical N =2 string theory has a four-dimensional target space, and its open string effective field theory is self-dual Yang-Mills [8] , which gets deformed noncommutatively in the presence of a B-field [5] . Conversely, the noncommutative SDYM equations are contained [19] in the equations of motion of N =2 string field theory (SFT) [22] in a B-field background. This SFT formulation is based on the N =4 topological string description [23] . It is well known that the SDYM model can be described in terms of holomorphic bundles over (an open subset of) the twistor space 2 [26] CP 3 and the topological N =4 string theory contains twistors from the outset. The Lax pair, integrability and the solutions to the equations of motion by twistor and dressing methods were incorporated into the N =2 open SFT in [27, 28] . However, this theory reproduces only bosonic SDYM theory, its symmetries (see e.g. [29, 30, 31] ) and integrability properties. It is natural to ask: What string theory can describe supersymmetric SDYM theory [32, 33] in four dimensions?\\n\\nThere are some proposals [33, 34, 35, 36] for extending N =2 open string theory (and its SFT) to be space-time supersymmetric. Moreover, it was shown by Witten [37] that N =4 supersymmetric SDYM theory appears in twistor string theory, which is a B-type open topological string with the supertwistor space CP 3|4 as a target space. 3 Note that N <4 SDYM theory forms a BPS subsector of N -extended super Yang-Mills theory, and N =4 SDYM can be considered as a truncation of the full N =4 super Yang-Mills theory [37] . It is believed [43, 39] that twistor string theory is related with the previous proposals [33, 34, 35, 36] for a Lorentz-invariant supersymmetric extension of N =2 (and topological N =4) string theory which also leads to the N =4 SDYM model.\\n\\nA dimensional reduction of the above relations between twistor strings and N =4 super Yang-Mills and SDYM models was considered in [44, 45, 46, 47] . The corresponding twistor string theory after this reduction is the topological B-model on the mini-supertwistor space P 2|4 . In [47] it was shown that the 2N =8 supersymmetric extension of the Bogomolny-type model in 2+1 dimensions is equivalent to an 2N =8 supersymmetric modified U(n) chiral model on R 2,1 . The subject of the current paper is an 2N ≤8 version of the above supersymmetric Bogomolny-type Yang-Mills-Higgs model in signature (-+ +), its relation with an N -extended supersymmetric modified integrable U(n) chiral model (to be defined) in 2+1 dimensions and the Moyal-type noncommutative deformation of this chiral model. We go on to explicitly construct multi-soliton configurations on noncommutative R 2,1 for the corresponding supersymmetric sigma model field equations. By studying the scattering properties of the constructed configurations, we prove their asymptotic factorization without scattering for large times. We also briefly discuss a D-brane interpretation of these soliton configurations from the viewpoint of twistor string theory.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"N -extended SDYM equations in 2+2 dimensions\",\n \"text\": \"Space R 2,2 . Let us consider the four-dimensional space R 2,2 = (R 4 , g) with the metric\\n\\nds 2 = g µν dx µ dx ν = det(dx α α) = dx 1 1dx 2 2 -dx 2 1dx 1 2 (2.1)\\n\\nwith (g µν ) = diag(-1, +1, +1, -1), where µ, ν, . . . = 1, . . . , 4 are space-time indices and α = 1, 2, α = 1, 2 are spinor indices. We choose the coordinates 4\\n\\n(x µ ) = (x a , t) = (t, x, y, t) with a, b, . . . = 1, 2, 3 , (2.2)\\n\\nand the signature (-+ + -) allows us to introduce real isotropic coordinates (cf. [19, 6] )\\n\\nx 1 1 = 1 2 (t -y) , x 1 2 = 1 2 (x + t) , x 2 1 = 1 2 (x -t) , x 2 2 = 1 2 (t + y) . (2.3)\\n\\nSDYM. Recall that the SDYM equations for a field strength tensor\\n\\nF µν on R 2,2 read 1 2 ε µνρσ F ρσ = F µν , (2.4)\\n\\nwhere ε µνρσ is a completely antisymmetric tensor on R 2,2 and ε 1234 = 1. In the coordinates (2.3) we have the decomposition\\n\\nF α α,β β = ∂ α αA β β -∂ β β A α α + [A α α, A β β ] = ε αβ F α β + ε α β F αβ (2.5) with F α β := -1 2 ε αβ F α α,β β and F αβ := -1 2 ε α β F α α,β β , (2.6)\\n\\nwhere ε αβ is antisymmetric, ε αβ ε βγ = δ γ α , and similar for ε α β , with ε 12 = ε 1 2 = 1. The gauge potential (A α α) will appear in the covariant derivative\\n\\nD α β = ∂ α β + [A α β , • ] . (2.7)\\n\\nIn spinor notation, (2.4) is equivalently written as\\n\\nF α β = 0 ⇔ F α α,β β = ε α β F αβ . (2.8)\\n\\nSolutions {A α α} to these equations form a subset (a BPS sector) of the solution space of Yang-Mills theory on R 2,2 .\\n\\nN -extended SDYM in component fields. The field content of N -extended super SDYM is foot_4\\n\\nN = 0 A α α (2.9a) N = 1 A α α, χ i α with i = 1 (2.9b) N = 2 A α α, χ i α , φ [ij] with i, j = 1, 2 (2.9c)\\n\\nN = 3 A α α, χ i α , φ [ij] , χ[ijk] α with i, j, k = 1, 2, 3 (2.9d) N = 4 A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl] α β\\n\\nwith i, j, k, l = 1, 2, 3, 4 .\\n\\n(2.9e)\\n\\nHere (A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl]\\n\\nα β ) are fields of helicities (+1, + 1 2 , 0, -1 2 , -1). These fields obey the field equations of the N = 4 SDYM model, namely [33, 37]\\n\\nF α β = 0 , (2.10a)\\n\\nD α αχ iα = 0 , (2.10b)\\n\\nD α αD α αφ ij + 2{χ iα , χ j α } = 0 , (2.10c)\\n\\nD α α χ α[ijk] -6[χ [i α , φ jk] ] = 0 , (2.10d)\\n\\nD γ α G [ijkl] γ β + 12{χ [i α , χjkl] β } -18[φ [ij , D α β φ kl] ] = 0 . (2.10e)\\n\\nNote that the N < 4 SDYM field equations are governed by the first N +1 equations of (2.10), where F α β = 0 is counted as one equation and so on.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Superfield formulation of\",\n \"text\": \"N -extended SDYM Superspace R 4|4N .\\n\\nRecall that in the space R 2,2 = (R 4 , g) with the metric g given in (2.1) one may introduce purely real Majorana-Weyl spinors foot_5 θ α and η α of helicities + 1 2 and -1 2 as anticommuting (Grassmann-algebra) objects. Using 2N such spinors with components θ iα and η α i for i = 1, . . . , N , one can define the N -extended superspace R 4|4N and the N -extended supersymmetry algebra generated by the supertranslation operators\\n\\nP α α = ∂ α α , Q iα = ∂ iα -η α i ∂ α α and Q i α = ∂ i α -θ iα ∂ α α , (2.11)\\n\\nwhere\\n\\n∂ α α := ∂ ∂x α α , ∂ iα := ∂ ∂θ iα and ∂ i α := ∂ ∂η α i .\\n\\n(2.12)\\n\\nThe commutation relations for the generators (2.11) read\\n\\n{Q iα , Q j α} = -2δ j i P α α , [P α α, Q iβ ] = 0 and [P α α, Q i β ] = 0 . (2.13)\\n\\nTo rewrite equations of motion in terms of R 4|4N superfields one uses the additional operators\\n\\nD iα = ∂ iα + η α i ∂ α α and D i α = ∂ i α + θ iα ∂ α α , (2.14)\\n\\nwhich (anti)commute with the operators (2.11) and satisfy\\n\\n{D iα , D j β } = 2δ j i P α β , [P α α, D iβ ] = 0 and [P α α, D j β ] = 0 . (2.15)\\n\\nAntichiral superspace R 4|2N . On the superspace R 4|4N one may introduce tensor fields depending on bosonic and fermionic coordinates (superfields), differential forms, Lie derivatives L X etc.. Furthermore, on any such superfield A one can impose the constraint equations L D iα A = 0, which for a scalar superfield f reduce to the so-called antichirality conditions\\n\\nD iα f = 0 . (2.16)\\n\\nThese are easily solved by using a coordinate transformation on R 4|4N ,\\n\\n(x α α, η α i , θ iα ) → (x α α = x α α-θ iα η α i , η α i , θ iα ) , (2.17)\\n\\nunder which ∂ α α, D iα and D i α transform to the operators\\n\\n∂α α = ∂ α α , Diα = ∂ iα and Di α = ∂ i α + 2θ iα ∂ α α . ( 2\\n\\n.18) Then (2.16) simply means that f is defined on a sub-superspace R 4|2N ⊂ R 4|4N with coordinates xα α and η α i . (2.19)\\n\\nThis space is called antichiral superspace. In the following we will usually omit the tildes when working on the antichiral superspace.\\n\\nN -extended SDYM in superfields. The N -extended SDYM equations can be rewritten in terms of superfields on the antichiral superspace R 4|2N [33, 48] . Namely, for any given 0 ≤ N ≤ 4, fields of a proper multiplet from (2.9) can be combined into superfields A α α and A i α depending on x α α, η α i ∈ R 4|2N and giving rise to covariant derivatives\\n\\n∇ α α := ∂ α α + A α α and ∇ i α := ∂ i α + A i α .\\n\\n(2.20)\\n\\nIn such terms the N -extended SDYM equations (2.10) read\\n\\n[∇ α α, ∇ β β ] + [∇ α β , ∇ β α] = 0 , [∇ i α, ∇ β β ] + [∇ i β , ∇ β α] = 0 , {∇ i α, ∇ j β } + {∇ i β , ∇ j α} = 0 , (2.21)\\n\\nwhich is equivalent to\\n\\n[∇ α α, ∇ β β ] = ε α β F αβ , [∇ i α, ∇ β β ] = ε α β F i β and {∇ i α, ∇ j β } = ε α β F ij , (2.22)\\n\\nwhere F ij is antisymmetric and F αβ is symmetric in their indices.\\n\\nThe above gauge potential superfields (A α α, A i α) as well as the gauge strength superfields (F αβ , F i α , F ij ) contain all physical component fields of the N -extended SDYM model. For instance, the lowest component of the triple (F αβ , F i α , F ij ) in an η-expansion is (F αβ , χ i α , φ ij ), with zeros in case N is too small. By employing Bianchi identities for the gauge strength superfields, one successively obtains [48] the superfield expansions and the field equations (2.10) for all component fields.\\n\\nIt is instructive to extend the antichiral combination in (2.18) to potentials and covariant derivatives, Di\\n\\nα = ∂ i α + 2 θ iα ∂ α α + + + Ãi α := A i α + 2 θ iα A α α ∇i α := ∇ i α + 2 θ iα ∇ α α ( 2\\n\\n.23) where ∇ α α, ∇ i α and Di α are given by (2.20) and (2.18), while A i α and A α α depend on x α α and η α i only. With the antichiral covariant derivatives, one may condense (2.21) or (2.22) into the single set { ∇i α, ∇j β\\n\\n} + { ∇i β , ∇j α} = 0 ⇔ { ∇i α, ∇j β } = ε α β Fij , (2.24)\\n\\nwith Fij =\\n\\nF ij + 4 θ [iα F j] α + 4 θ iα θ jβ F αβ .\\n\\nThe concise form (2.24) of the N -extended SDYM equations is quite convenient, and we will use it interchangeable with (2.21).\\n\\nLinear system for N -extended SDYM. It is well known that the superfield SDYM equations (2.21) can be seen as the compatibility conditions for the linear system of differential equations\\n\\nζ α(∂ α α + A α α) ψ = 0 and ζ α(∂ i α + A i α) ψ = 0 , (2.25)\\n\\nwhere\\n\\n(ζ β ) = 1 ζ and ζ α = ε α β ζ β . The extra (spectral) parameter 7 ζ lies in the extended complex plane C ∪ ∞ = CP 1 .\\n\\nHere ψ is a matrix-valued function depending not only on x α α and η α i but also (meromorphically) on ζ ∈ CP 1 . We subject the n×n matrix ψ to the following reality condition:\\n\\nψ(x α α, η α i , ζ) ψ(x α α, η α i , ζ) † = 1l , (2.26)\\n\\nwhere \\\" †\\\" denotes hermitian conjugation and ζ is complex conjugate to ζ. This condition guarantees that all physical fields of the N -extended SDYM model will take values in the adjoint representation of the algebra u(n). In the concise form the linear system (2.25) is written as\\n\\nζ α(∇ i α + 2θ iα ∇ α α) ψ = 0 ⇔ ζ α( Di α + Ãi α) ψ = 0 ⇔ ζ α ∇i α ψ = 0 . (2.27) 2.3 Reduction of N -extended SDYM to 2+1 dimensions\\n\\nThe supersymmetric Bogomolny-type Yang-Mills-Higgs equations in 2+1 dimensions are obtained from the described N -extended super SDYM equations by a dimensional reduction R 2,2 → R 2,1 .\\n\\nIn particular, for the N =0 sector we demand the components A µ of a gauge potential to be independent of x 4 and put A 4 =: ϕ. Here, ϕ is a Lie-algebra valued scalar field in three dimensions (the Higgs field) which enters into the Bogomolny-type equations. Similarly, for N ≥ 1 one can reduce the N -extended SDYM equations on R 2,2 by imposing the ∂ 4 -invariance condition on all the fields (A α α,\\n\\nχ i α , φ [ij] , χ[ijk] α , G [ijkl]\\n\\nα β ) from the N =4 supermultiplet or its truncation to N <4 and obtain supersymmetric Bogomolny-type equations on R 2,1 .\\n\\nSpinors in R 2,1 .\\n\\nRecall that on R 2,2 both N =4 SDYM theory and full N =4 super Yang-Mills theory have an SL(4, R) ∼ = Spin(3,3) R-symmetry group [33] . A dimensional reduction to R 2,1 enlarges the supersymmetry and R-symmetry to 2N =8 and Spin(4,4), respectively, for both theories (cf. [49] for Minkowski signature). More generally, any number N of supersymmetries gets doubled to 2N in the reduction. Since dimensional reduction collapses the rotation group Spin(2,2)\\n\\n∼ = Spin(2,1) L ×Spin(2,1) R of R 2,2\\n\\nto its diagonal subgroup Spin(2,1) D as the local rotation group of R 2,1 , the distinction between undotted and dotted indices disappears. We shall use undotted indices henceforth.\\n\\nCoordinates and derivatives in R 2,1 . The above discussion implies that one can relabel the bosonic coordinates x α β from (2.3) by x αβ and split them as\\n\\nx αβ = 1 2 (x αβ + x βα ) + 1 2 (x αβ -x βα ) = x (αβ) + x [αβ]\\n\\n(2.28) into antisymmetric and symmetric parts,\\n\\nx [αβ] = 1 2 ε αβ x 4 = 1 2 ε αβ t and x (αβ) =: y αβ , (2.29)\\n\\nrespectively, with\\n\\ny 11 = x 11 = 1 2 (t -y) , y 12 = 1 2 (x 12 + x 21 ) = 1 2 x , y 22 = x 22 = 1 2 (t + y) . (2.30)\\n\\nWe also have θ iα → θ iα and η α i → η α i for the fermionic coordinates on R 4|4N reduced to R 3|4N . Bosonic coordinate derivatives reduce in 2+1 dimensions to the operators\\n\\n∂ (αβ) = 1 2 (∂ αβ + ∂ βα ) (2.31)\\n\\nwhich read explicitly as\\n\\n∂ (11) = ∂ ∂y 11 = ∂ t -∂ y , ∂ (12) = ∂ (21) = 1 2 ∂ ∂y 12 = ∂ x , ∂ (22) = ∂ ∂y 22 = ∂ t + ∂ y . (2.32)\\n\\nWe thus have\\n\\n∂ ∂x αβ = ∂ (αβ) -ε αβ ∂ 4 = ∂ (αβ) -ε αβ ∂ t , (2.33)\\n\\nwhere\\n\\nε 12 = -ε 21 = -1, ∂ 4 = ∂/∂x 4 and ∂ t = ∂/∂ t.\\n\\nThe operators D iα and D i α acting on t-independent superfields reduce to\\n\\nD iα = ∂ iα + η β i ∂ (αβ) and D i α = ∂ i α + θ iβ ∂ (αβ) , (2.34)\\n\\nwhere\\n\\n∂ iα = ∂/∂θ iα and ∂ i α = ∂/∂η α i .\\n\\nSimilarly, the antichiral operators Diα and Di α in (2.18) become Diα = ∂ iα and Di α = ∂ i α + 2θ iβ ∂ (αβ) . (2.35) Supersymmetric Bogomolny-type equations in component fields. According to (2.33), the components A α β of a gauge potential in four dimensions split into the components A (αβ) of a gauge potential in three dimensions and a Higgs field\\n\\nA [αβ] = -ε αβ ϕ, i.e. A αβ = A (αβ) + A [αβ] = A (αβ) -ε αβ ϕ . (2.36)\\n\\nThen the covariant derivatives D α β reduced to three dimensions become the differential operators\\n\\nD αβ -ε αβ ϕ = ∂ (αβ) + [A (αβ) , • ] -ε αβ [ϕ, • ] , (2.37)\\n\\nand the Yang-Mills field strength on R 2,1 decomposes as\\n\\nF αβ, γδ = [D αβ , D γδ ] = ε αγ f βδ + ε βδ f αγ with f αβ = f βα . (2.38)\\n\\nSubstituting (2.36) and (2.37) into (2.10), i.e. demanding that all fields in (2.10) are independent of x 4 = t, we obtain the following supersymmetric Bogomolny-type equations on R 2,1 :\\n\\nf αβ + D αβ ϕ = 0 , (2.39a)\\n\\nD αβ χ iβ + ε αβ [ϕ, χ iβ ] = 0 , (2.39b)\\n\\nD αβ D αβ φ ij + 2[ϕ, [ϕ, φ ij ]] + 2{χ iα , χ j α } = 0 , (2.39c)\\n\\nD αβ χβ[ijk] -ε αβ [ϕ, χβ[ijk] ] -6[χ [i α , φ jk] ] = 0 , (2.39d)\\n\\nD γ α G [ijkl] γβ + [ϕ, G [ijkl] αβ ] + 12{χ [i α , χjkl] β } -18[φ [ij , D αβ φ kl] ] -18ε αβ [φ [ij , [φ kl] , ϕ]] = 0 .(2.\\n\\nSupersymmetric Bogomolny-type equations in terms of superfields. Translations generated by the vector field ∂ 4 = ∂ t are isometries of superspaces R 4|4N and R 4|2N . By taking the quotient with respect to the action of the abelian group G generated by ∂ 4 , we obtain the reduced full superspace R 3|4N ∼ = R 4|4N /G and the reduced antichiral superspace R 3|2N ∼ = R 4|2N /G. In the following, we shall work on R 3|2N and R 3|2N × CP 1 , since the reduced ψ-function from (2.25) and (2.27) is defined on the latter space.\\n\\nThe linear system stays in the center of the superfield approach to the N -extended SDYM equations. After imposing t-independence on all fields in the linear system (2.27), we arrive at the linear equations\\n\\nζ α ∇i α ψ ≡ ζ α ( Di α + Âi α ) ψ = 0 (2.40)\\n\\nof the same form but with\\n\\nDi α = ∂ i α + 2θ iβ ∂ (αβ) and Âi α = A i α + 2θ iβ (A (αβ) -ε αβ Ξ) , (2.41)\\n\\nwhere A i α , A (αβ) and Ξ are superfields depending on y αβ and η α i only. These linear equations expand again to the pair (cf. (2.25))\\n\\nζ β (∂ (αβ) + A (αβ) -ε αβ Ξ) ψ = 0 and ζ α (∂ i α + A i α ) ψ = 0 . (2.42)\\n\\nThe compatibility conditions for the linear system (2.40) read\\n\\n{ ∇i α , ∇j β } + { ∇i β , ∇j α } = 0 ⇔ { ∇i α , ∇j β } = ε αβ Fij (2.43)\\n\\nand present a condensed form of (2.39) rewritten in terms of R 3|2N superfields. Similarly, these equations can also be written in more expanded forms analogously to (2.21) or using the superfield analog of (2.37). However, we will not do this since all these sets of equations are equivalent.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Noncommutative N -extended U(n) chiral model in 2+1 dimensions\",\n \"text\": \"As has been known for some time, nonlinear sigma models in 2 + 1 dimensions may be Lorentzinvariant or integrable but not both [7, 11] . We will show that the super Bogomolny-type model discussed in Section 2 after a gauge fixing is equivalent to a super extension of the modified U(n) chiral model (so as to be integrable) first formulated by Ward [7] . Since integrability is compatible with noncommutative deformation (if introduced properly, see e.g. [9] - [20] ) we choose from the beginning to formulate our super extension of this chiral model on Moyal-deformed R 2,1 with noncommutativity parameter θ ≥ 0. Ordinary space-time R 2,1 can always be restored by taking the commutative limit θ → 0.\\n\\nStar-product formulation. Classical field theory on noncommutative spaces may be realized in a star-product formulation or in an operator formalism 8 . The first approach is closer to the commutative field theory: it is obtained by simply deforming the ordinary product of classical fields (or their components) to the noncommutative star product\\n\\n(f ⋆ g)(x) = f (x) exp{ i 2 ← - ∂ a θ ab -→ ∂ b } g(x) ⇒ x a ⋆ x b -x b ⋆ x a = iθ ab (3.1)\\n\\nwith a constant antisymmetric tensor θ ab . Specializing to R 2,1 , we use real coordinates (x a ) = (t, x, y) in which the Minkowski metric g on R 3 reads (g ab ) = diag(-1, +1, +1) with a, b, . . . = 1, 2, 3 (cf. Section 2). It is straightforward to generalize the Moyal deformation (3.1) to the superspaces introduced in the previous section, allowing in particular for non-anticommuting Grassmann-odd coordinates. Deferring general superspace deformations and their consequences to future work, we here content ourselves with the simple embedding of the \\\"bosonic\\\" Moyal deformation into superspace, meaning that (3.1) is also valid for superfields f and g depending on Grassmann variables θ iα and η α i . For later use we consider not only isotropic coordinates and vector fields\\n\\nu := 1 2 (t+y) = y 22 , v := 1 2 (t-y) = y 11 , ∂ u = ∂ t + ∂ y = ∂ (22) , ∂ v = ∂ t -∂ y = ∂ (11) (3.2)\\n\\nintroduced in Section 2, but also the complex combinations\\n\\nz := x + iy , z := x -iy , ∂ z = 1 2 (∂ x -i∂ y ) , ∂ z = 1 2 (∂ x + i∂ y ) . (3.3)\\n\\nSince the time coordinate t remains commutative, the only nonvanishing component of the noncommutativity tensor θ ab is Operator formalism. The nonlocality of the star products renders explicit computation cumbersome. We therefore pass to the operator formalism, which trades the star product for operatorvalued spatial coordinates (x, ŷ) or their complex combinations (ẑ, ẑ), subject to\\n\\nθ xy = -θ yx =: θ > 0 ⇒ θ z z = -θ zz = -2i θ . ( 3\\n\\n[t, x] = [t, ŷ] = 0 but [x, ŷ] = iθ ⇒ [ẑ, ẑ] = 2 θ . (3.6)\\n\\nThe latter equation suggests the introduction of annihilation and creation operators,\\n\\na = 1 √ 2θ ẑ and a † = 1 √ 2θ ẑ with [a , a † ] = 1 , (3.7)\\n\\nwhich act on a harmonic-oscillator Fock space H with an orthonormal basis { |ℓ , ℓ = 0, 1, 2, . . .} such that\\n\\na |ℓ = √ ℓ |ℓ-1 and a † |ℓ = √ ℓ+1 |ℓ+1 . (3.8)\\n\\nAny superfield f (t, z, z, η α i ) on R 3|2N can be related to an operator-valued superfield f (t, η α i ) ≡ F (t, a, a † , η α i ) on R 1|2N acting in H, with the help of the Moyal-Weyl map\\n\\nf (t, z, z, η α i ) → f (t, η α i ) = Weyl-ordered f t, √ 2θa, √ 2θa † , η α i . (3.9)\\n\\nThe inverse transformation recovers the ordinary superfield,\\n\\nf (t, η α i ) ≡ F (t, a, a † , η α i ) → f (t, z, z, η α i ) = F ⋆ t, z √ 2θ , z √ 2θ , η α i , (3.10)\\n\\nwhere F ⋆ is obtained from F by replacing ordinary with star products. Under the Moyal-Weyl map, we have\\n\\nf ⋆ g → f ĝ and dx dy f = 2π θ Tr f = 2π θ ℓ≥0 ℓ| f |ℓ , (3.11)\\n\\nand the spatial derivatives are mapped into commutators,\\n\\n∂ z f → ∂z f = -1 √ 2θ [a † , f ] and ∂ z f → ∂z f = 1 √ 2θ [a , f ] . (3.12)\\n\\nFor notational simplicity we will from now on omit the hats over the operators except when confusion may arise.\\n\\nGauge fixing for ψ. Note that the linear system (2.40) and the compatibility conditions (2.43) are invariant under a gauge transformation\\n\\nψ → ψ ′ = g -1 ψ , (3.13a)\\n\\nA → A ′ = g -1 A g + g -1 ∂ g (with appropriate indices) , (3.13b)\\n\\nΞ → Ξ ′ = g -1 Ξ g , (3.13c)\\n\\nwhere g = g(x a , η α i ) is a U(n)-valued superfield globally defined on the deformed superspace R 3|2N θ × CP 1 . Using a gauge transformation of the form (3.13), we can choose ψ such that it will satisfy the standard asymptotic conditions (see e.g. [51] )\\n\\nψ = Φ -1 + O(ζ) for ζ → 0 , (3.14a\\n\\n)\\n\\nψ = 1l + ζ -1 Υ + O(ζ -2 ) for ζ → ∞ , (3.14b)\\n\\nwhere the U(n)-valued function Φ and u(n)-valued function Υ depend on x a and η α i . This \\\"unitary\\\" gauge is compatible with the reality condition for ψ,\\n\\nψ(x a , η α i , ζ) ψ(x a , η α i , ζ) † = 1l , (3.15)\\n\\nobtained by reduction from (2.26).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Gauge fixing for\",\n \"text\": \"A i 1 = 0 and A i 2 = Φ -1 ∂ i 2 Φ = ∂ i 1 Υ , (3.17)\\n\\nA (11) = 0 and\\n\\nA (12) + Ξ = Φ -1 ∂ (12) Φ = ∂ (11) Υ , (3.18)\\n\\nA (21) -Ξ = 0 and\\n\\nA (22) = Φ -1 ∂ (22) Φ = ∂ (12) Υ . (3.19)\\n\\nUsing (2.32), we can rewrite the nonzero components as\\n\\nA := Φ -1 ∂ u Φ = ∂ x Υ , B := Φ -1 ∂ x Φ = ∂ v Υ , C i := Φ -1 ∂ i 2 Φ = ∂ i 1 Υ . (3.20)\\n\\nRecall that the superfields Φ and Υ depend on x a and η α i .\\n\\nLinear system. In the above-introduced unitary gauge the linear system (2.42) reads\\n\\n(ζ∂ x -∂ u -A) ψ = 0 , (ζ∂ v -∂ x -B) ψ = 0 , (ζ∂ i 1 -∂ i 2 -C i ) ψ = 0 , (3.21)\\n\\nwhich adds the last equation to the linear system of the Ward model [7] and generalizes it to superfields A(x a , η α j ), B(x a , η α j ) and C i (x a , η α j ). The concise form of (3.21) reads\\n\\nζ Di 1 -Di 2 -Âi 2 ψ = 0 (3.22)\\n\\nor, in more explicit form,\\n\\nζ ∂ i 1 + 2θ i1 ∂ v + 2θ i2 ∂ x -∂ i 2 + C i + 2θ i1 (∂ x + B) + 2θ i2 (∂ u + A) ψ = 0 . (3.23)\\n\\nN -extended sigma model.\\n\\nThe compatibility conditions of this linear system are the Nextended noncommutative sigma model equations Di\\n\\n1 (Φ -1 Dj 2 Φ) + Dj 1 (Φ -1 Di 2 Φ) = 0 (3.24)\\n\\nwhich in expanded form reads\\n\\n(g ab + v c ε cab ) ∂ a (Φ -1 ∂ b Φ) = 0 ⇔ ∂ x (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ u Φ) = 0 , (3.25a) ∂ i 1 (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ i 2 Φ) = 0 , ∂ i 1 (Φ -1 ∂ u Φ) -∂ x (Φ -1 ∂ i 2 Φ) = 0 , (3.25b)\\n\\n∂ i 1 (Φ -1 ∂ j 2 Φ) + ∂ j 1 (Φ -1 ∂ i 2 Φ) = 0 . (3.25c)\\n\\nHere, the first line contains the Wess-Zumino-Witten term with a constant vector (v c ) = (0, 1, 0) which spoils the standard Lorentz invariance but yields an integrable chiral model in 2+1 dimensions. Recall that Φ is a U(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . As discussed in Section 2, the compatibility conditions of the linear equations (3.22) (or (3.21)) are equivalent to the N -extended Bogomolnytype equations (2.39) for the component (physical) fields. Thus, chiral model field equations (3.25) are equivalent to a gauge fixed form of equations (2.39). Υ-formulation. Instead of Φ-parametrization of (A, B, C i ) given in (3.17)-(3.20) we may use the equivalent Υ-parametrization also given there. In this case, the compatibility conditions for the linear system (3.21) reduce to\\n\\n(∂ 2 x -∂ u ∂ v )Υ + [∂ v Υ , ∂ x Υ] = 0 , (3.26a)\\n\\n(∂ i 2 ∂ v -∂ i 1 ∂ x )Υ + [∂ i 1 Υ , ∂ v Υ] = 0 , (∂ i 2 ∂ x -∂ i 1 ∂ u )Υ + [∂ i 1 Υ , ∂ x Υ] = 0 , (3.26b) (∂ i 2 ∂ j 1 + ∂ j 2 ∂ i 1 )Υ + {∂ i 1 Υ , ∂ j 1 Υ} = 0 , (3.26c)\\n\\nwhich in concise form read\\n\\n( Di 2 Dj 1 + Dj 2 Di 1 ) Υ + { Di 1 Υ , Dj 1 Υ} = 0 . (3.27)\\n\\nRecall that Υ is a u(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . For N =4, the commutative limit of (3.27) can be considered as Siegel's equation [33] reduced to 2+1 dimensions. According to Siegel, one can extract the multiplet of physical fields appearing in (2.39) from the prepotential Υ via\\n\\n∂ i 1 Υ = A i 2 , ∂ i 1 ∂ j 1 Υ = φ ij , ∂ i 1 ∂ j 1 ∂ k 1 Υ = χ[ijk] 2 , ∂ i 1 ∂ j 1 ∂ k 1 ∂ l 1 Υ = G [ijkl] 22 , (3.28a)\\n\\n∂ (α1) Υ = A (α2) -ε α2 ϕ , ∂ (α1) ∂ i 1 Υ = χ i α , ∂ (α1) ∂ (β1) Υ = f αβ , (3.28b)\\n\\nwhere one takes Υ and its derivatives at η 2 i = 0. The other components of the physical fields, i.e. χ[ijk]\\n\\n1 , G [ijkl] 11 , G [ijkl]\\n\\n21 , A (11) and A (21) -ϕ, vanish in this light-cone gauge.\\n\\nSupersymmetry transformations. The 4N supercharges given in (2.11) reduce in 2+1 dimensions to the form\\n\\nQ iα = ∂ iα -η β i ∂ (αβ) and Q i α = ∂ i α -θ iβ ∂ (αβ) . (3.29)\\n\\nTheir antichiral version, matching to Diα and Dj β of (2.35), reads\\n\\nQiα = ∂ iα -2η β i ∂ (αβ) and Qj β = ∂ j β , (3.30)\\n\\nso that { Qiα , Qj β } = -2 δ j i ∂ (αβ) . (3.31)\\n\\nOn a (scalar) R 3|2N superfield Σ these supersymmetry transformations act as\\n\\nδ Σ := ε iα Qiα Σ + ε α i Qi α Σ (3.32)\\n\\nand are induced by the coordinate shifts\\n\\nδ y αβ = -2ε i(α η β) i and δ η α i = ε α i , (3.33)\\n\\nwhere ε iα and ε α i are 4N real Grassmann parameters. It is easy to see that our equations (3.24) and (3.27) are invariant under the supersymmetry transformations (3.32) (applied to Φ or Υ). This is simply because the operators Diα and Dj β anticommute with the supersymmetry generators Qiα and Qj β . Therefore, the equations of motion (3.25) of the modified N -extended chiral model in 2+1 dimensions as well as their reductions to 2+0 and 1+1 dimensions carry 2N supersymmetries and are genuine supersymmetric extensions of the corresponding bosonic equations. Note that this type of extension is not the standard one since the R-symmetry groups are Spin(N , N ) in 2+1 and Spin(N , N )× Spin(N , N ) in 1+1 dimensions, which differ from the compact unitary R-symmetry groups of standard sigma models. Contrary to the standard case of two-dimensional sigma models the above \\\"noncompact\\\" 2N supersymmetries do not impose any constraints on the geometry of the target space, e.g. they do not demand it to be Kähler [52] or hyper-Kähler [53] . This may be of interest and deserves further study.\\n\\nAction functionals. In either formulation of the N -extended supersymmetric SDYM model on R 2,2 there are difficulties with finding a proper action functional generalizing the one [54, 55] for the purely bosonic case. These difficulties persist after the reduction to 2+1 dimensions, i.e. for the equations (3.25) and (3.26) describing our supersymmetric modified U(n) chiral model. It is the price to be paid for overcoming the no-go barrier N ≤ 4 and the absence of geometric target-space constraints. On a more formal level, the problem is related to the chiral character of (3.24) as well as (3.27) , where only the operators Di α but not Diα appear. Note however, that for N = 4 one can write an action functional in component fields producing the equations (2.39), which are equivalent to the superspace equations (3.24) when i, j = 1, . . . , 4 (see e.g. [47] ).\\n\\nOne proposal for an action functional stems from Siegel's idea [33] for the Υ-formulation of the N -extended SDYM equations. Namely, one sees that ∂ i 2 Υ enters only linearly into the last two lines in (3.26) . Therefore, if we introduce\\n\\nΥ (1) := Υ| η 2 i =0 (3.34)\\n\\nthen it must satisfy the first equation from (3.26) , and the remaining equations iteratively define the dependence of Υ on η 2 i starting from Υ (1) . Hence, all information is contained in Υ (1) , as can also be seen from (3.28) . In other words, the dependence of Υ on η 2 i is not 'dynamical'. For an action one can then take (cf. [33] )\\n\\nS = d 3 x d N η 1 Υ (1) ∂ (αβ) ∂ (αβ) Υ (1) + 2 3 Υ (1) ε αβ ∂ (α1) Υ (1) ∂ (β1) Υ (1) . (3.35)\\n\\nExtremizing this functional yields the first line of (3.26) at η 2 i = 0. Except for the Grassmann integration, this action has the same form as the purely bosonic one [55] . One may apply the same logic to the Φ-formulation where the action for the purely bosonic case is also known [54, 56] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"N -extended multi-soliton configurations via dressing\",\n \"text\": \"The existence of the linear system (3.22) (equivalent to (3.21)) encoding solutions of the N -extended U(n) chiral model in an auxiliary matrix ψ allows for powerful methods to systematically construct explicit solutions for ψ and hence for Φ † = ψ| ζ=0 and Υ = lim ζ→∞ ζ (ψ-1l). For our purposes the so-called dressing method [57, 51] proves to be the most practical [12] - [20] , and so we shall use it here for our linear system, i.e. already in the N -extended noncommutative case.\\n\\nMulti-pole ansatz for ψ. The dressing method is a recursive procedure for generating a new solution from an old one. More concretely, we rewrite the linear system (3.21) in the form\\n\\nψ(∂ u -ζ∂ x )ψ † = A , ψ(∂ x -ζ∂ v )ψ † = B , ψ(∂ i 2 -ζ∂ i 1 )ψ † = C i . (4.1)\\n\\nRecall that ψ † := (ψ(x a , η α i , ζ)) † and (A, B, C i ) depend only on x a and η α i . The central idea is to demand analyticity in the spectral parameter ζ, which strongly restricts the possible form of ψ. One way to exploit this constraint starts from the observation that the left hand sides of (4.1) as well as of the reality condition (3.15) do not depend on ζ while ψ is expected to be a nontrivial function of ζ globally defined on CP 1 . Therefore, it must be a meromorphic function on CP 1 possessing some poles which we choose to lie at finite points with constant coordinates µ k ∈ CP 1 .\\n\\nHere we will build a (multi-soliton) solution ψ m featuring m simple poles at positions µ 1 , . . . , µ m with foot_8 Im µ k < 0 by left-multiplying an (m-1)-pole solution ψ m-1 with a single-pole factor of the form 1l\\n\\n+ µ m -μm ζ -µ m P m (x a , η α i ) , (4.2)\\n\\nwhere the n×n matrix function P m is yet to be determined. Starting from the trivial (vacuum) solution ψ 0 = 1l, the iteration ψ 0 → ψ 1 → . . . → ψ m yields a multiplicative ansatz for ψ m ,\\n\\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ , (4.3)\\n\\nwhich, via partial fraction decomposition, may be rewritten in the additive form\\n\\nψ m = 1l + m k=1 Λ mk S † k ζ -µ k , (4.4)\\n\\nwhere Λ mk and S k are some n×r k matrices depending on x a and η α i , with r k ≤ n.\\n\\nEquations for S k . Let us first consider the additive parametrization (4.4) of ψ m . This ansatz must satisfy the reality condition (3.15) as well as our linear equations in the form (4.1). In particular, the poles at ζ = μk on the left hand sides of these equations have to be removable since the right hand sides are independent of ζ. Inserting the ansatz (4.4) and putting to zero the corresponding residues, we learn from (3.15) that\\n\\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ S k = 0 , (4.5)\\n\\nwhile from (4.1) we obtain the differential equations\\n\\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ LA,B,i k S k = 0 , (4.6)\\n\\nwhere LA,B,i\\n\\nk stands for either LA k = ∂ u -μk ∂ x , LB k = µ k (∂ x -μk ∂ v ) or Li k = ∂ i 2 -μk ∂ i 1 . (4.7)\\n\\nNote that we consider a recursive procedure starting from m=1, and operators (4.7) will appear with k = 1, . . . , m if we consider poles at ζ = μk .\\n\\nBecause the LA,B,i k for k = 1, . . . , m are linear differential operators, it is easy to write down the general solution for (4.6) at any given k, by passing from the coordinates (u, v, x; η 1 i , η 2 i ) to \\\"co-moving coordinates\\\" (w k , wk , s k ; η i k , ηi k ). The precise relation for k = 1, . . . , m is [12, 58]\\n\\nw k := x + μk u + μ-1 k v = x + 1 2 (μ k -μ -1 k )y + 1 2 (μ k +μ -1 k )t and η i k := η 1 i + μk η 2 i , (4.8)\\n\\nwith wk and ηi k obtained by complex conjugation and the co-moving time s k being inessential because by definition nothing will depend on it. The kth moving frame travels with a constant velocity\\n\\n(v x , v y ) k = - µ k + μk µ k μk + 1 , µ k μk -1 µ k μk + 1 , (4.9)\\n\\nso that the static case w k =z is recovered for µ k = -i. On functions of (w k , η i k , wk , ηi k ) alone the operators (4.7) act as\\n\\nLA k = LB k = (µ k -μ k ) ∂ ∂ wk =: Lk and Li k = (µ k -μ k ) ∂ ∂ ηi k . (4.10)\\n\\nBy induction in k = 1, . . . , m we learn that, due to (4.5), a necessary and sufficient condition for a solution of (4.6) is Lk\\n\\nS k = S k Zk and Li k S k = S k Zi k (4.11)\\n\\nwith some r k ×r k matrices Zk and Zi k depending on (w k , wk , η j k , ηj k ). Passing to the noncommutative bosonic coordinates we obtain ŵk , ŵk = 2θ ν k νk with\\n\\nν k νk = 4i µ k -μ k -µ -1 k +μ -1 k . (4.12)\\n\\nThus, we can introduce annihilation and creation operators\\n\\nc k = 1 √ 2θ ŵk ν k and c † k = 1 √ 2θ ŵk νk so that [c k , c † k ] = 1 (4.13)\\n\\nfor k = 1, . . . , m. Naturally, this Heisenberg algebra is realized on a \\\"co-moving\\\" Fock space H k , with basis states |ℓ k and a \\\"co-moving\\\" vacuum |0 k subject to c k |0 k = 0. Each co-moving vacuum |0 k (annihilated by c k ) is related to the static vacuum |0 (annihilated by a) through an ISU(1,1) squeezing transformation (cf. [12] ) which is time-dependent. The fermionic coordinates η i k and ηi k remain spectators in the deformation. Coordinate derivatives are represented in the standard fashion as\\n\\nν k √ 2θ ∂ ∂w k → -[c † k , • ] and νk √ 2θ ∂ ∂ wk → [c k , • ] . (4.14)\\n\\nAfter the Moyal deformation, the n×r k matrices S k have become operator-valued, but are still functions of the Grassmann coordinates η i k and ηi k . The noncommutative version of the BPS conditions (4.11) naturally reads\\n\\nc k S k = S k Z k and ∂ ∂ ηi k S k = S k Z i k (4.15)\\n\\nwhere Z k and Z i k are some operator-valued r k ×r k matrix functions of η j k and ηj k .\\n\\nNonabelian solutions for S k . For general data Z k and Z i k it is difficult to solve (4.15), but it is also unnecessary because the final expression ψ m turns out not to depend on them. Therefore, we conveniently choose\\n\\nZ k = c k ⊗ 1l r k ×r k and Z i k = 0 ⇒ S k = R k (c k , η i k ) , (4.16)\\n\\nwhere R k is an arbitrary n×r k matrix function independent of c † k and ηi k . 10 It is known that nonabelian (multi-) solitons arise for algebraic functions R k (cf. e.g. [7] for the commutative and [12] for the noncommutative N =0 case). Their common feature is a smooth commutative limit. The only novelty of the supersymmetric extension is the η i k dependence, i.e.\\n\\nR k = R k,0 + η i k R k,i + η i k η j k R k,ij + η i k η j k η p k R k,ijp + η i k η j k η p k η q k R k,ijpq . (4.17)\\n\\nAbelian solutions for S k . It is useful to view S k as a map from C r k ⊗H k to C n ⊗H k (momentarily suppressing the η dependence). The noncommutative setup now allows us to generalize the domain of this map to any subspace of C n ⊗ H k . In particular, we may choose it to be finite-dimensional, say C q k , and represent the map by an n×q k array |S k of kets in H. In this situation, Z k and Z i k in (4.15) are just number -valued q k ×q k matrix functions of η j k and ηj k . In case they do not depend on ηj k , we can write down the most general solution as\\n\\n|S k = R k (c k , η j k ) |Z k exp i Z i k (η j k ) ηi k with |Z k := exp Z k (η j k ) c † k |0 k . (4.18)\\n\\nAs before, we may put Z i k = 0 without loss of generality, but now the choice of Z k does matter. For any given k generically there exists a q k -dimensional basis change which diagonalizes the ket-valued matrix\\n\\n|Z k → diag e α 1 k c † , e α 2 k c † , . . . , e α q k k c † |0 k = diag |α 1 k , |α 2 k , . . . , |α q k k , (4.19)\\n\\nwhere we defined coherent states\\n\\n|α l k := e α l k c † |0 k so that c k |α l k = α l k |α l k for l = 1, . . . , q k and α l k ∈ C . (4.20)\\n\\nNote that not only the entries of R k but also the α l k are holomorphic functions of the co-moving Grassmann parameters η j k and thus can be expanded like in (4.17). In the U(1) model, we must use ket-valued 1×q k matrices |S k for all k, yielding rows\\n\\n|S k = R 1 k |α 1 k , R 2 k |α 2 k , . . . , R q k k |α q k k for k = 1, . . . , m , (4.21)\\n\\nwith functions α l k (η j k ). Here, the R l k only affect the states' normalization and can be collected in a diagonal matrix to the right, hence will drop out later and thus may all be put to one. Formally, we have recovered the known abelian (multi-) soliton solutions, but the supersymmetric extension has generalized |S k → |S k (η j k ) . Explicit form of P k . Let us now consider the multiplicative parametrization (4.3) of ψ m which also allows us to solve (4.5) . First of all, note that the reality condition (3.15) is satisfied if\\n\\nP k = P † k = P 2 k ⇔ P k = T k (T † k T k ) -1 T † k for k = 1, . . . , m , (4.22)\\n\\nmeaning that P k is an operator-valued hermitian projector (of group-space rank r k ≤ n) built from an n×r k matrix function T k (the abelian case of n=1 is included). The reality condition follows just because The r k columns of T k span the image of P k and obey\\n\\n1l + µ k -μk ζ -µ k P k 1l + μk -µ k ζ -μk P k =\\n\\nP k T k = T k ⇔ (1l-P k ) T k = 0 . (4.24)\\n\\nFurthermore, the equation (4.5) with m = k (induction) rewritten in the form\\n\\n(1l-P k ) k-1 ℓ=1 1l + µ k-ℓ -μk-ℓ μk -µ k-ℓ P k-ℓ S k = 0 (4.25)\\n\\nreveals that (cf. (4.24))\\n\\nT 1 = S 1 and T k = k-1 ℓ=1 1l - µ k-ℓ -μk-ℓ µ k-ℓ -μk P k-ℓ S k for k ≥ 2 , (4.26)\\n\\nwhere the explicit form of S k for k = 1, . . . , m is given in (4.16) or (4.18) . The final result reads\\n\\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ = 1l + m k=1 Λ mk S † k ζ -µ k (4.27)\\n\\nwith hermitian projectors P k given by (4.22), T k given by (4.26) and S k given by (4.16) or (4.18) . The explicit form of Λ mk (which we do not need) can be found in [12] . The corresponding superfields Φ and Υ are\\n\\nΦ m = ψ † m | ζ=0 = m k=1 (1l -ρ k P k ) with ρ k = 1 - µ k μk , (4.28a)\\n\\nΥ m = lim ζ→∞ ζ (ψ m -1l) = m k=1 (µ k -μ k ) P k . (4.28b)\\n\\nFrom (4.22) it is obvious that P k is invariant under a similarity transformation\\n\\nT k → T k Λ k ⇔ S k → S k Λ k (4.29)\\n\\nfor an invertible operator-valued r k ×r k matrix Λ k . This justifies putting Z i k = 0 from the beginning and also the restriction to Z k = c k ⊗ 1l r k ×r k in the nonabelian case, both without loss of generality. Hence, the nonabelian solution space constructed here is parametrized by the set {R k } m 1 of matrixvalued functions of c k and η i k and the pole positions µ k . The abelian moduli space, however, is larger by the set {Z k } m 1 of matrix-values functions of η i k which generically contain the coherentstate parameter functions {α l k (η i k )}. Restricting to η i k =0 reproduces the soliton configurations of the bosonic model [12] .\\n\\nStatic solutions. Let us consider the reduction to 2+0 dimensions, i.e. the static case. Recall that static solutions correspond to the choice m = 1 and µ 1 ≡ µ = -i implying w 1 = z, so we drop the index k. Specializing (4.27), we have\\n\\nψ = 1l - 2 i ζ + i P so that Φ = Φ † = 1l -2P , (4.30)\\n\\nwhere a hermitian projector P of group-space rank r satisfies the BPS equations (1l-P ) a P = 0 ⇒ (1l-P ) a T = 0 ,\\n\\n(1l-P ) ∂ ∂ ηi P = 0 ⇒ (1l-P ) ∂ ∂ ηi T = 0 , (4.31a)\\n\\nwith P = T (T † T ) -1 T † and η i = η 1 i + iη 2 i . In this case T = S, and for a nonabelian r=1 projector P we get T = T (a, η i ) as an n×1 column. For the simplest case of N =1 we just have (cf. [59] )\\n\\nT = T e (a) + η T o (a) with η = η 1 + iη 2 , (4.32)\\n\\nwhere T e (a) and T o (a) are rational functions of a (e.g. polynomials) taking values in the even and odd parts of the Grassmann algebra. Similarly, an abelian N =1 projector (for n=1) is built from\\n\\n|T = |α 1 , |α 2 , . . . , |α q . (4.33)\\n\\nAt θ=0, the static solution (4.32) of our supersymmetric U(n) sigma model is also a solution of the standard N =1 supersymmetric CP n-1 sigma model in two dimensions (see e.g. [59] ). 11 For this reason, one can overcome the previously mentioned difficulty with constructing an action (or energy from the viewpoint of 2+1 dimensions) for static configurations. Moreover, on solutions obeying the BPS conditions (4.31) the topological charge\\n\\nQ = 2πθ dη 1 dη 2 Tr tr Φ D + Φ , D -Φ (4.34)\\n\\nis proportional to the action (BPS bound)\\n\\nS = 2πθ dη 1 dη 2 Tr tr D + Φ , D -Φ (4.35)\\n\\nand is finite for algebraic functions T e and T o . Here, the standard superderivatives D ± are defined as\\n\\nD + = ∂ ∂η + iη ∂ z and D -= ∂ ∂ η + iη ∂ z . ( 4\\n\\nOne-soliton configuration. For one moving soliton, from (4.27) and (4.28) we obtain\\n\\nψ 1 = 1l + µ - μ ζ -µ P with P = T (T † T ) -1 T † (4.37) and Φ = 1l -ρ P with ρ = 1 - µ μ . (4.38)\\n\\nNow our n×r matrix T must satisfy (putting\\n\\nZ i = 0 and Z = c ⊗ 1l r×r ) [c , T ] = 0 and ∂ ∂ ηi T = 0 with η i = η 1 i + μ η 2 i , (4.39)\\n\\nwhere c is the moving-frame annihilation operator given by (4.13) for k=1.\\n\\nRecall that the operators c and c † and therefore the matrix T and the projector P can be expressed in terms of the corresponding static objects by a unitary squeezing transformation (see e.g. (4.8) and (4.13)). For simplicity we again consider the case N =1 and a nonabelian projector with r=1. Then (4.39) tells us that T is a holomorphic function of c and η, i.e.\\n\\nT = T e (c) + η T o (c) = T 1 e (c) + η T 1 o (c)\\n\\n. . .\\n\\nT n e (c) + η T n o (c) (4.40)\\n\\nwith polynomials T a e and T a o of order q, say, analogously to the static case (4.32). Note that, for T a o to be Grassmann-odd and nonzero, some extraneous Grassmann parameter must appear. Similarly, abelian projectors for a moving one-soliton obtain by subjecting (4.33) to a squeezing transformation.\\n\\nFor N =1 the moving frame was defined in (4.8) (dropping the index k) via\\n\\nw = x + 1 2 (μ-μ -1 )y + 1 2 (μ+μ -1 )t and η = η 1 + μη 2 hence ∂ t η = 0 . (4.41)\\n\\nConsider the moving frame with the coordinates (w, w, s; η, η) with the choice s = t and the related change of the derivatives (see [12, 58] )\\n\\n∂ x = ∂ w + ∂ w , (4.42a)\\n\\n∂ y = 1 2 (μ-μ -1 ) ∂ w + 1 2 (µ-µ -1 ) ∂ w , (4.42b)\\n\\n∂ t = 1 2 (μ+μ -1 ) ∂ w + 1 2 (µ+µ -1 ) ∂ w + ∂ s , (4.42c)\\n\\n∂ η 1 = ∂ η + ∂ η , (4.42d)\\n\\n∂ η 2 = μ ∂ η + µ ∂ η . (4.42e)\\n\\nIn the moving frame our solution (4.38) is static, i.e. ∂ s Φ = 0, and the projector P has the same form as in the static case. The only difference is the coefficient ρ instead of 2 in (4.38). Therefore, by computing the action (4.35) in (w, w; η 1 , η 2 ) coordinates, we obtain for algebraic functions T in (4.40) a finite answer, which differs from the static one by a kinematical prefactor depending on µ (cf. [12] for the bosonic case).\\n\\nLarge-time asymptotics. Note that in the distinguished (z, z, t) coordinate frame (4.41) implies that at large times w → κ t with κ = 1 2 (μ+μ -1 ). As a consequence, the t q term in each polynomial in (4.40) will dominate, i.e.\\n\\nT → t q a 1 + η b 1 . . .\\n\\nan + η bn =: t q Γ , (4.43)\\n\\nwhere Γ is a fixed vector in C n . It is easy to see that in the distinguished frame the large-time limit of Φ given by (4. being the projector on the constant vector Γ.\\n\\nConsider now the m-soliton configuration (4.28). By induction of the above argument one easily arrives at the m-soliton generalization of (4.44). Namely, in the frame moving with the ℓth lump we have lim t→±∞ Φ m = (1lρ 1 Π 1 ) . . . (1lρ ℓ-1 Π ℓ-1 )(1lρ ℓ P ℓ )(1lρ ℓ+1 Π ℓ+1 ) . . . (1lρ m Π m ) , (4.45) where the Π m are constant projectors. This large-time factorization of multi-soliton solutions provides a proof of the no-scattering property because the asymptotic configurations are identical for large negative and large positive times.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusions\",\n \"text\": \"In this paper we introduced a generalization of the modified integrable U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions and considered a Moyal deformation of this model. It was shown that this N -extended chiral model is equivalent to a gauge-fixed BPS subsector of an N -extended super Yang-Mills model in 2+1 dimensions originating from twistor string theory. The dressing method was applied to generate a wide class of multi-soliton configurations, which are time-dependent finite-energy solutions to the equations of motion. Compared to the N =0 model, the supersymmetric extension was seen to promote the configurations' building blocks to holomorphic functions of suitable Grassmann coordinates. By considering the large-time asymptotic factorization into a product of single soliton solutions we have shown that no scattering occurs within the dressing ansatz chosen here.\\n\\nThe considered model does not stand alone but is motivated by twistor string theory [37] with a target space reduced to the mini-supertwistor space [44, 45, 47] . In this context, the obtained multi-soliton solutions are to be regarded as D(0|2N )-branes moving inside D(2|2N )-branes [60] . Here 2N appears due to fermionic worldvolume directions of our branes in the superspace description [60] . Switching on a constant B-field simply deforms the sigma model and D-brane worldvolumes noncommutatively, thereby admitting also regular supersymmetric noncommutative abelian solutions.\\n\\nRestricting to static configurations, the models can be specialized to Grassmannian supersymmetric sigma models, where the superfield Φ takes values in Gr(r, n), and the field equations are invariant under 2N supersymmetry transformations with 0 ≤ N ≤ 4. This differs from the results for standard 2D sigma models [52, 53] where the target spaces have to be Kähler or hyper-Kähler for admitting two or four supersymmetries, respectively. This difference will be discussed in more details elsewhere.\\n\\nWe derived the supersymmetric chiral model in 2+1 dimensions through dimensional reduction and gauge fixing of the N -extended supersymmetric SDYM equations in 2+2 dimensions. Recall that for the purely bosonic case most (if not all) integrable equations in three and fewer dimensions can be obtained from the SDYM equations (or their hierarchy [25] ) by suitable dimensional reductions (see e.g. [61] - [65] and references therein). Moreover, this Ward conjecture [61] was extended to the noncommutative case (see e.g. [66, 67] ). It will be interesting to consider similar reductions of the N -extended supersymmetric SDYM equations (and their hierarchy [68] ) to supersymmetric integrable equations in three and two dimensions generalizing earlier results [69] .\"\n }\n]"}}},{"rowIdx":99,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0531"},"arxiv_id":{"kind":"string","value":"0704.0531"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1088/1126-6708/2007/11/079"},"content_hash":{"kind":"string","value":"b0e695f3b0dd96c805ccd0cc547d6cced9bf6b91d8fab221fb029b978f81b498"},"title":{"kind":"string","value":"Gravitational Duality Transformations on (A)dS4"},"abstract":{"kind":"string","value":"We discuss the implementation of electric-magnetic duality transformations in\nfour-dimensional gravity linearized around Minkowski or (A)dS4 backgrounds. In\nthe presence of a cosmological constant duality generically modifies the\nHamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular\nattention to the boundary terms generated by the duality transformations and\ndiscuss their implications for holography."},"authors":{"kind":"list like","value":["Robert G. Leigh","Anastasios C. Petkou"],"string":"[\n \"Robert G. Leigh\",\n \"Anastasios C. Petkou\"\n]"},"arxiv_categories":{"kind":"list like","value":["hep-th"],"string":"[\n \"hep-th\"\n]"},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]"},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"We discuss the implementation of electric-magnetic duality transformations in fourdimensional gravity linearized around Minkowski or (A)dS 4 backgrounds. In the presence of a cosmological constant duality generically modifies the Hamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular attention to the boundary terms generated by the duality transformations and discuss their implications for holography. 1 For reviews of higher-spin theories see e.g. [1]. 2 See [4] and [5] for more recent works.\n\nContents 1. Introduction and Summary 2. Action and Hamiltonian 2.1 The 3 + 1 split 2.2 Shifted Variables 2.3 Linearization 3. Linearized Gravitational Duality and Holography 3.1 Duality and Holography 3.2 Linearized gravitational duality 3.3 Linearized Constraints and Bianchi Identities 3.4 Connection with other known dualities 4. The Effect on the Boundary Theory 5. Conclusions and Outlook 6. Appendix: other duality mappings\n\nDuality has played an important role in our understanding of Yang-Mill theories and it is believed that it will play an important role also in gravity and in higher-spin gauge theories. Indeed, although it is less clear what could be the implications of duality for theories whose quantum versions are still unknown, gravity and higher-spin gauge theories 1 are intimately connected to a quantum string theory where certainly duality plays a crucial role. The recent advent of holography raises some intriguing questions for duality. For example one may wonder what is the holographic image of a duality invariant spectrum, a duality transformation or a possible quantization condition that usually duality implies for charges. Some of these issues were raised by Witten in [2] where it was argued that the standard electric-magnetic duality of a U(1) gauge theory on AdS 4 is responsible for a \"natural\" SL(2, Z) action on current two-point functions in three-dimensional CFTs. 2 Shortly afterwards it was shown in [3] that such an SL(2, Z) action is intimately related to certain \"double-trace\" deformations in the boundary, assuming suitable large-N limits and existence of non-trivial fixed points. The latter assumptions are strengthened by the fact that there exist models (e.g., see [6] and references therein) which exhibit the required behavior. In particular, it was shown in [3] that certain \"double-trace\" deformations induce an SL(2, Z) action on two-point functions of higher-spin (i.e. spin s ≥ 2) currents. This has led to the Duality Conjecture of [3] : linearized higher-spin theories on AdS 4 spaces possess a generalization of electric-magnetic duality whose holographic image is the natural SL(2, Z) action on boundary two-point functions.\n\nSurprisingly, even the duality for linearized spin-2 gauge fields (linearized gravity) was not widely known by the time of this conjecture. 3 Second order linearized gravitational duality was discussed among other in [8, 9, 10, 11, 12] . More recently, the duality properties of linearized gravity around flat space were studied in [13] and were further discussed in [14] . The duality of linearized gravity around dS 4 was later studied in [15] .\n\nIn this note we present our calculations regarding the duality properties of gravity in the presence of a cosmological constant. Having in mind applications to higher-spin gauge theories we use forms and work in the first order formalism where duality is also manifested at the level of the action [16] . Moreover, the first order formalism is relevant for applications of duality to holography, since the correlation functions of the boundary theory are essentially determined by the bulk canonical momenta (see e.g. [17] ).\n\nOur aim in this work is to formulate linearized first order gravity using suitable \"electric\" and \"magnetic\" variables, in close analogy with electromagnetism. We find that this is possible only when the background geometry is Minkowski or (A)dS 4 . Then we implement the standard electric-magnetic duality rotations. We find that, up to \"boundary\" terms, the linearized Hamiltonian changes by terms that do not alter the bulk dynamics i.e. do not alter the second order bulk equations of motion. Moreover, the duality rotation interchanges the (linearized) constraints with the (linearized) Bianchi identitites. The \"boundary\" terms have important holographic consequences since they correspond to marginal \"double-trace\" deformations [3] that induce the boundary SL(2, Z) action. In the Appendix we exhibit a modified duality rotation that leaves the bulk Hamiltonian invariant and induces \"boundary\" terms that correspond to relevant deformations as in [3] .\n\nHaving in mind the extension of our results to higher-spin gauge theorieswe start from the MacDowell-Mansouri form [18] of the gravitational action foot_1\n\nI M M = 1 2Λ M ǫ abcd R ab ∧ R cd + 2Λe a ∧ e b ∧ R cd + Λ 2 e a ∧ e b ∧ e c ∧ e d , (2.1)\n\nwhere a, b, ... are Lorentz indices. In this formalism, the vierbein e a and the spin connection ω a b are initially thought of as independent variables. The curvature 2-form is\n\nR a b = dω a b + ω a c ∧ ω c b = 1 2 R a bcd e c ∧ e d .\n\nVarying the action with respect to e a and ω a b , we find\n\nR ab + Λe a ∧ e b = 0 , (2.2)\n\nT a = de a + ω a b ∧ e b = 0 . (2.\n\n3)\n\nThe relation to gravity is established via the vanishing torsion equation (2.3), which relates e and ω in the familiar way. The above equations are equivalent to the Einstein equation in metric variables R µν -\n\n1 2 Rg µν = +3Λg µν . (2.4)\n\nand the scalar curvature is R = -d(d -1)Λ = -12Λ. Note that our Λ is related to the cosmological constant in its usual definition via Λ cosm = -6Λ. Λ > 0 corresponds to AdS. Note that this is actually SO(3, 2) covariant, as we can combine ω, e into a super-connection. Note that Λ has units (Length) -2 . In the SO(3, 2)-invariant formalism, I M M arises from\n\nI M M = 1 2Λ ǫ ABCDE V E R AB ∧ R CD , (2.5)\n\nwhere V E is a non-dynamical 0-form field (that we take to have value V -1 = 1 to gauge back to the SO(3, 1) formalism) and R A B is the curvature of Ω A B ≡ {e a , ω a b }. There are also quasitopological terms of the form\n\nI top = θ 2Λ R A B ∧ R B A + α Λ R A B ∧ R AC V B V C (2.6)\n\nthat we could add to the action. In the stated gauge, this reduces to\n\nI top = θ 2Λ P 2 + (θ + α)C N Y + α R ab ∧ e a ∧ e b (2.7)\n\nwhere\n\nP 2 = R a b ∧ R b a is the Pontryagin class, C N Y = (T a ∧ T a -R ab ∧ e a ∧ e b\n\n) is the Nieh-Yan class and we also note the Euler class E 2 = ǫ abcd R ab ∧ R cd . Note that in the presence of torsion, the action (2.7) contains the non-topological term R ab ∧ e a ∧ e b with \"Immirzi parameter\" γ = -2/α. In the absence of torsion, this term is a total derivative.\n\nThe Hilbert-Palatini action is\n\nI HP = I M M - 1 2Λ E 2 .\n\n(2.8)\n\nIt differs from I M M by a boundary term, is smooth as Λ → 0 but is not manifestly SO(3, 2)invariant.\n\nNext, we carefully consider the 3 + 1 split. Although much of the discussion here is familiar from the ADM formalism, we feel it is important to set notation carefully, as we will introduce some new ingredients. To accommodate both AdS and dS signatures simultaneously, we will introduce a 'time' function t and a foliation of space-time Σ t ֒→ M. In dS, t is time-like, and this corresponds to the usual Hamiltonian foliation; in AdS on the other hand, we will take t to be the (space-like) radial coordinate. We will keep track of the resulting signs by a parameter σ ⊥ , equal to ±1 in dS(AdS).\n\nProceeding as usual then, we get a vector field t that satisfies ∇ t t = 1 ≡ t(t) (so t = ∂ ∂t ) and a 1-form dt. Given a 4-metric, we can introduce the normal 1-form n as\n\nn = σ ⊥ Ndt , (2.9)\n\nwhich is normalized as (n, n) = σ ⊥ . The dual vector field n can be expanded as\n\nn = 1 N t - 1 N N , (2.10)\n\nwhere the shift N satisfies (N, n) = 0, and thus (t, n) = σ ⊥ N.\n\nNext, we will locally choose a basis of 1-forms\n\ne 0 = σ ⊥ n = Ndt , (2.11)\n\ne α = ẽα + N α dt .\n\n(2.12)\n\nThe ẽα span T * Σ t , and correspond to a 3-metric h ij = ẽα i ẽβ j η αβ . The quantities N α are the components of N: N α = e α i N i . These basis 1-forms are dual to {e 0 = n, e α = ẽα }, with e a (e b ) = δ b a . We expand the spin connection in the same basis 5\n\nω a b = q a b dt + ωa b , (2.13)\n\nwhich leads to\n\nR a b = Ra b + dt ∧ r a b , (2.14)\n\nwhere R is formed from ω and d only, and\n\nr a b = ωa b -dq a b -ωa c q c b + q a c ωc b . (2.15)\n\nNote that these quantities are merely decompositions along T * Σ t in the 4-geometry; we will introduce the intrinsically defined objects shortly.\n\nWe then find\n\nI HP = 2ǫ αβγ dt ∧ N( Rαβ + Λẽ α ∧ ẽβ ) ∧ ẽγ -2N α ( R0β ) ∧ ẽγ + r 0α ∧ ẽβ ∧ ẽγ .\n\n(2.16) 5 We have\n\nq a b = N ω 0 a b + N α ω α a b and ωa b ≡ ω α a b ẽα .\n\nAs is familiar, the lapse and shift appear as Lagrange multipliers. The constraints that they multiply are of course zero in any background (i.e. vacuum solution), such as (A)dS 4 . The final term in the action contains the real dynamics -r 0α depends on the components R 0α 0β of the Riemann tensor.\n\nNote though that the tensors used here are 4-dimensional. Let us define the \"electric field\"\n\nK α = σ ⊥ ω0 α = K βα ẽβ .\n\n(2.17)\n\nIn the case that ω is the torsion-free Levi-Civita connection, this agrees with the standard definition for extrinsic curvature, regarded as a vector-valued one-form. We then find\n\nRα β = (3) R α β -σ ⊥ K α ∧ K β , (2.18)\n\nand\n\nR0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ ( DK) α . (2.19)\n\nThese equations amount to the Gauss-Codazzi relations. Furthermore, r 0α contains time derivatives of ω0α as well as terms linear in components of q. We find\n\n2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = 2ǫ αβγ σ ⊥ Kα -( Dq) 0α ∧ ẽβ ∧ ẽγ , (2.20)\n\n= 2σ ⊥ ǫ αβγ Kα + q αδ K δ ∧ ẽβ ∧ ẽγ + 4q 0α ǫ αβγ T β ∧ ẽγ up to a total 3-derivative. We have defined the intrinsic 3-torsion T α = dẽ α + ωα β ∧ ẽβ . Since we wish to regard the ẽ as coordinate variables, 6 we integrate the first term by parts to obtain (up to the total time-derivative\n\n∂ ∂t 2σ ⊥ K α ∧ ẽβ ∧ ẽγ ǫ αβγ ) 2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = Π α ∧ ėα + 4q 0α ǫ αβγ T β ∧ ẽγ + 2σ ⊥ q αδ ǫ αβγ K δ ∧ ẽβ ∧ ẽγ . (2.21)\n\nwhere we have defined the momentum 2-form\n\nΠ α = -4σ ⊥ ǫ αβγ K β ∧ ẽγ . (2.22)\n\nThe q ab appear as Lagrange multipliers. In particular, the q αβ constraint precisely sets the antisymmetric (torsional) part of the extrinsic curvature tensor K [αβ] to zero. Next, we define the \"magnetic field\"\n\nB α = 1 2 σ ⊥ ǫ αβγ ωβγ , ω αβ = -ǫ αβγ B γ . (2.23)\n\nand we find that the q 0α constraint\n\nǫ αβγ T β ∧ ẽγ = ǫ αβγ dẽ β ∧ ẽγ -σ ⊥ B β ∧ ẽβ ∧ ẽα = 0 , (2.24)\n\ninvolves only the antisymmetric part B [α,β] of the magnetic field B α = B αβ ẽβ . The antisymmetric part of B α spoils the gauge covariance of the constraint (2.24) under an SO(3) rotation of the dreibein ẽα , hence it represents degrees of freedom that can be gauged fixed to zero by an SO(3) rotation. On the other hand, an algebraic equation of motion connects the symmetric part of B αβ to derivatives of ẽα as dẽ α + ǫ αβγ B β ∧ ẽγ = 0 (2.25)\n\nAt the end, one is left with the canonically conjugate variables ẽα and Π α . These results are familiar from the metric formalism.\n\nDropping the torsional terms, we then arrive at the action\n\nI HP = dt ∧ ėα ∧ Π α + 2Nǫ αβγ ( (3) R αβ -σ ⊥ K α ∧ K β + Λẽ α ∧ ẽβ ) ∧ ẽγ -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ . (2.26) Furthermore, using * 3 ẽα ∧ ẽβ = 1 2 η γδ ǫ αβγ ẽδ , we have Πα = * 3 Π α = -2(K αβ -η αβ trK)ẽ β , (2.27)\n\nwhere trK = η αβ K αβ . We can solve the above equation to get\n\nK α = - 1 2 ( Παβ - 1 2 η αβ tr Π)ẽ β .\n\n(2.28)\n\nAs stated above, K αβ (and Παβ ) is symmetric when the torsion vanishes. Finally, with the definition (2.23) we find foot_3\n\nǫ αβγ (3) R αβ ∧ ẽγ = ǫ αβγ dω αβ + ωα δ ∧ ωδβ ∧ ẽγ = σ ⊥ 2 dB γ + ǫ αβγ B α ∧ B β ∧ ẽγ . (2.29)\n\nIntroducing B α is an unusual thing to do but it will play a role in duality: in this form, the Hamiltonian contains terms which are reminiscent of those of the Maxwell theory. The full HP action is of the form\n\nI HP = dt ∧ ėα ∧ Π α -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ +2σ ⊥ N(2 dB γ + ǫ αβγ B α ∧ B β -ǫ αβγ K α ∧ K β + σ ⊥ Λǫ αβγ ẽα ∧ ẽβ ) ∧ ẽγ . (2.30)\n\nNote that the entire contribution of the cosmological constant appears in the last term of the Hamiltonian constraint.\n\nIt is possible to make a transformation of the canonical variables in order to absorb the cosmological constant term in (2.30 ). This can be achieved by introducing the new variables\n\nKα = K α -ρẽ α , (2.31)\n\nand requiring that\n\nρ 2 = σ ⊥ Λ . (2.32)\n\nThis is positive only when σ ⊥ and Λ are simultaneously positive or negative, as it is the case for both AdS 4 (Λ > 0) and dS 4 (Λ < 0). We will often write Λ = σ ⊥ /L 2 where L is a length scale. Under (2.31) the momentum 2-form becomes\n\nΠ α → P α -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ . (2.33)\n\nThe last term in (2.33) contributes a total time derivative to the action (of the form of a boundary cosmological term). We have introduced a new momentum variable\n\nP α = -4σ ⊥ ǫ αβγ Kβ ∧ ẽγ .\n\nThen, we get the action\n\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K + ρ T ) β ∧ ẽγ - 4 3 σ ⊥ ρǫ αβγ ∂ ∂t (ẽ α ∧ ẽβ ∧ ẽγ ) +2σ ⊥ N 2 d(B α ∧ ẽα ) + 2B γ ∧ T γ -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ .(2.34)\n\nNote that the shift constraint is still written in terms of the ordinary covariant derivative, and thus involves a non-linear term coupling B to K. Consistent with our previous discussion, we drop the terms involving the torsion T , and disregard the boundary term to obtain\n\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K) β ∧ ẽγ +2σ ⊥ N 2 d(B α ∧ ẽα ) -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ . (2.35)\n\nWe note that the parameter ρ can be of either sign (although, this sign does not appear in the second order equations of motion).\n\nNext, we linearize the above action around an appropriate fixed background. We expand as\n\nẽα = ẽα + E α , N = 1 + n, N α = n α , B α = B α + b α , Kα = Kα + k α . (2.36)\n\nThe background values should satisfy the constraints. The simplest choice is the background where Kα = 0 = B α .\n\n(2.37)\n\nIn fact, reaching this simple form was a motivation for the shift (2.31). Then, to quadratic order in the fluctuating fields the Hamiltonian gives\n\nI HP = dt ∧ Ėα ∧ p α -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + 4σ ⊥ n d(b α ∧ ẽα ) -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β + 2ρk α ∧ E β ∧ ẽγ , (2.38)\n\nwhere\n\np α = -4σ ⊥ ǫ αβγ k β ∧ ẽγ (2.39)\n\nare the linearized momentum variables conjugate to E α . In order to reach the form (2.38) the linear terms in the fluctuations must vanish. For this to happen we find the relationships ėα + ρẽ α = 0 .\n\n(2.40)\n\nNotice that we can also write the linearized action in the form\n\nI HP = dt ∧ ( Ėα + ρE α ) ∧ p α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + n 4σ ⊥ db γ + ρp γ ∧ ẽγ . (2.41)\n\nThe form of the first term, involving the momentum, makes clear that longitudinal fluctuations are non-dynamical. The natural time dependence of E α is of the form e -ρt (correspondingly, the natural time dependence of p α is e +ρt ). Other than that, we see that in comparing to the flat space action, in these variables, the only change is that the Hamiltonian constraint is modified. The solutions of (2.40) and (2.37) are components of (A)dS 4 spacetimes. We can solve (2.40) to obtain e 0 = dt, e α = e -ρt dx α .\n\n(2.42)\n\nWith these we construct the usual Poincaré metric on (A)dS which, however, covers only half of the space even though the parameter t runs from -∞ to +∞. The conformal boundary in these coordinates is at t = +∞. Then we derive\n\nω α 0 = -ρe -ρt dx α = -ρe α , (2.43)\n\nand so\n\nR α β = -σ ⊥ L 2 e α ∧ e β R α 0 = -1 L 2 e α ∧ e 0 R a b = - σ ⊥ L 2 e a ∧ e b . ( 2\n\nHence Ric ab = -3σ ⊥ L 2 η ab and R = -12σ ⊥ /L 2 = -12Λ. We also evaluate Π α = -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ , Πα = 4ρẽ α , tr Π = 12ρ (2.45)\n\nB α = 0, K α = ρẽ α ⇒ Kα = 0 (2.46)\n\nNote that in this gauge, ( DK) α = 1 L T α = 0, which solves the shift constraint, while the Hamiltonian constraint is satisfied through a cancellation between the K 2 term and the cosmological term.\n\nLet us summarize what we have obtained so far. In the presence of a cosmological constant we have defined variables such that the action resembles most closely the action without the cosmological constant. This was done in order to look for a suitable background around which linear fluctuations are as simple as possible. Requiring that K (the \"electric field\") and B (the \"magnetic field\") vanish in such a background -as they do around flat space -we found that the background should be (A)dS 4 . Quite satisfactorily, both sign choices for ρ in the change of variables (2.31) lead to (A)dS 4 spacetimes.\n\nThis is the appropriate point to recall some salient features of duality rotations. In simple Hamiltonian systems the effect of the canonical transformation p → q and q → -p to the action is (see e.g. [19] )\n\nI = t 2 t 1 dt[p q -H(p, q)] → I D = t 2 t 1 dt[-q ṗ -H(q, -p)] . (3.1)\n\nNotice that I D involves the dual variables, for which we have however kept the same notation for simplicity. The transformed Hamiltonian H(q, -p) is in general not related to H(p, q). However, if H(q, -p) = H(p, q) we call the above transformation a duality. It then holds\n\nI D = I -qp t 2 t 1 . (3.2)\n\nThe dual action describes exactly the same dynamics as the initial one, up to a modification of the boundary conditions. For example, if I is stationary on the e.o.m for fixed q in the boundary, I D is stationary on the same e.o.m. for fixed p in the boundary. This simple example illustrates the role of duality in holography; a bulk duality transformation corresponds to a particular modification of the boundary conditions. This property of duality transformations is behind the remarkable holographic properties of electormagnetism in (A)dS 4 [2, 3] .\n\nClearly, the crucial properties of a duality transformation are to be canonical and to leave the Hamiltonian unchanged. However, consider a slight generalization\n\nS = t 2 t 1 dt[p q - 1 2 (p 2 + q 2 + 2λpq)] (3.3)\n\nwhere λ is an arbitrary parameter. The Hamiltonian now is not invariant under the canonical transformation p → q and q → -p -the pq term changes sign. Consequently, the first order form of the equations of motion are also not duality invariant. Nevertheless, the second order equation of motion is invariant. We will find that gravity in the presence of a cosmological constant follows precisely this model. Of course, gravity is a much more complicated constrained system, but as we will show, the constraints and Bianchi identities transform appropriately.\n\nWe also note that the canonical transformation (implemented by a generating functional of the first kind) p → q + 2λp , q → -p . (3.4) is of interest here. The above does not change the Hamiltonian and the transformed action differs from the initial one by total time derivative terms foot_4 S → S D = S -pq\n\nt 2 t 1 -λp 2 t 2 t 1 . (3.5)\n\nAs a preamble to gravity we recall the duality properties of Maxwell theory\n\nI M ax = 1 2g 2 dt ∧ Ȧ ∧ * 3 E - 1 2 (E ∧ * 3 E + B ∧ * 3 B) -A 0 d * 3 E , (3.6)\n\nUnder the duality E → - * 3 B, B → * 3 E, Ã → ÃD , we find\n\nI M ax → I M ax,D = 1 2g 2 dt ∧ -ȦD ∧ B - 1 2 (E ∧ * 3 E + B ∧ * 3 B) + A 0 dB . (3.7)\n\nE and B in (3.7) should be expressed through ÃD . We observe that the kinetic term has changed sign, while the Hamiltonian remains invariant. In addition, the (Gauss) constraint is dualized to the trivial 'Bianchi' identity dB = 0 for the dual magnetic field.\n\nNext we try to apply a Maxwell-type duality map in gravity. We consider the following transformation around the fixed background (2.40)\n\nk α → -b α , b α → k α . (3.8)\n\nTo implement the map (3.8) we need to specify the mapping of E α to a 'dual 3-bein' E α . We do that using the linearized form of (2.25) as\n\nǫ αβγ b β ∧ ẽγ + dE α = 0 → ǫ αβγ k β ∧ ẽγ + dE α = 0 = dE α - 1 4σ ⊥ p α (3.9) Since p α = 4σ ⊥ dE α , it is natural to define p D,α = 4σ ⊥ dE α = -4σ ⊥ ǫ αβγ b β ∧ ẽγ , (3.10)\n\nand thus the mapping (3.8) is supplemented by\n\nE → E , E → -E , p → -p D , p D → p (3.11)\n\nNow, let us see the effects of the above duality mapping. The action transforms to\n\nI HP → I HP,D = dt ∧ -Ėα ∧ p D,α -ρE α ∧ p D,α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ (3.12) +4σ ⊥ n α ǫ αβγ db β ∧ ẽγ + n 4σ ⊥ dk γ + ρp D,α ∧ ẽγ\n\nwhere now k α and b α should be expressed in terms of the dual variables E α and p D,α via (3.9) and (3.10). We notice that the 'kinetic' part Ė ∧ p of the action changes sign under the duality map, in direct analogy with the Maxwell case. However, the Hamiltonian is not invariant due to the change of sign of the second term in the first line of (3.12). We will discuss this further in a later section. For now, we note that this sign change would not show up in the equations of motion, written in second order form. It is important to also note that the constraints are transformed into quantities which in the next subsection we will recognize as the linearized Bianchi identities. This is to be expected since the duality transformations are canonical. We also note that it may be possible to choose an alternative canonical transformation, designed to leave the Hamiltonian invariant. The latter is presumably related to the work of Julia et. al. [15] and is considered in the Appendix.\n\nBy virtue of the discussion above we may now demonstrate that under the duality mapping (3.8) the linearized constraints transform to the linearized Bianchi identities as\n\nC α ≡ ǫ αβγ dk β ∧ ẽγ → -ǫ αβγ db β ∧ ẽγ (3.13) C 0 ≡ -σ ⊥ db γ -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ → -σ ⊥ dk γ + ρǫ αβγ b α ∧ ẽβ ∧ ẽγ (3.14)\n\nTo identify the right hand sides, we first note that the Bianchi identities are\n\nB R a b = dR a b -R a c ∧ ω c b + ω a c ∧ R c b = 0 (3.15) B a T = dT a -R a b ∧ e b + ω a b ∧ T b = 0 (3.16)\n\nwhich are obtained from the definitions of R a b and T a by exterior differentiation. The first equation is satisfied identically. Since the torsion vanishes, the second equation tells us only that R a b ∧ e b = 0. If we do the 3+1 split, we find two equations. The first is\n\nB T α = -( (3) R α β -σ ⊥ K α ∧ K β ) ∧ ẽβ = 0 (3.17)\n\nwhich upon using the symmetry of K α linearizes to\n\nB T α = -ǫ αβγ db β ∧ ẽγ + . . . (3.18)\n\nNote that this is the image under duality of the shift constraint as in (3.13) .\n\nThe second identity is\n\nB T 0 = -R0 α ∧ ẽα = -σ ⊥ ( DK) α = -σ ⊥ dk α + ρǫ αβγ b β ∧ ẽγ ∧ ẽα = 0 (3.19)\n\nwhere to arrive in the second line we used (2.46) . This is the image of the Hamiltonian constraint as in (3.14) . Summarizing, the duality transformations between linearized constraints and Bianchi identities are\n\nC α → B T,α C 0 → B 0 T (3.20) B T,α → -C α B 0 T → -C 0 (3.21)\n\nThe Maxwell-type duality operation (3. We begin with the spatial 2-forms when we have\n\nRαβ = -ǫ αβγ dB γ + σ ⊥ (B α ∧ B β -K α ∧ K β ) (3.23) R0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ (\n\nDK) α (3.24) and S0 γ = 1 2 σ ⊥ ǫ αβγ Rαβ (3.25) Sαβ = ǫ αβγ R0γ (3.26) If we linearize these expressions, we find under the duality transformation (3.8) Rab → -σ ⊥ Sab (3.27) Because the expressions (3.24) involve derivatives of B and K, the duality (3.8) is an 'integrated form' of the usual Riemann tensor duality, but implies it. Similarly, if we investigate the spatial 1-forms, we find r ab → -σ ⊥ s ab (3.28)\n\nTo arrive at this result we have set to zero the Lagrange multiplier field q.\n\nIt is well known that AdS is holographic. We may well ask, in the context of AdS/CFT, how the duality transformation that we have defined here acts in the boundary. We are instructed to consider the on-shell bulk action as a function of bulk fields. So, we evaluate the action on a solution to the equation of motion, resulting in a pure boundary term which is of the form\n\nS bdy = ∂M p α ∧ E α (4.1)\n\nApplying the duality transformation to the bulk theory, although the bulk action is not invariant as we have discussed above, nevertheless it may be easily shown that it induces a simple transformation on the (linearized) boundary term: it simply changes its sign.\n\nS dual bdy = - ∂M p D,α ∧ E α (4.2)\n\nThis transformation is exactly analogous to what happens in the Maxwell case: it amounts to the result [21] .\n\n10 G 2 G dual 2 = -1 . (4.3)\n\nMotivated by possible application in holography and in higher-spin gauge theory we have studied the duality properties of gravity in the Hamiltonian formulation. We have presented the gravity action in terms of suitable variables that closely resemble the electric and magnetic fields in Maxwell theory. We have found suitable \"electric\" and \"magnetic\" field variables, such that at the linearized level first order gravity most closely resembles electromagnetism. This can be done only around Minkowksi and (A)dS 4 backgrounds. We have implemented duality transformations in the linearized gravity fluctuations around these backgrounds. In the presence of a cosmological constant, the Hamiltonian changes, nevertheless the bulk dynamics remains unaltered, while the linearized lapse and shift constraints are mapped into the linearized Bianchi identities. Moreover, the duality transformations induce boundary terms whose relevance in holography we have briefly discussed. Finally, we have exhibited a modified duality rotation that leaves the bulk Hamiltonian invariant, while it induces boundary terms corresponding to relevant deformations.\n\nThe main implication of our results is that certain properties of correlations functions in three-dimensional CFTs mimic the duality of gravity. It would be interesting to extend our results to black-hole backgrounds and also when topological terms are present in the bulk. We also expect that one can analyze the duality of higher-spin gauge theories based on our first-order approach.\n\nThe work of A. C. P. was partially supported by the research program \"PYTHAGORAS II\" of the Greek Ministry of Education. RGL was supported in part by the U.S. Department of Energy under contract DE-FG02-91ER40709.\n\nIt is possible to find a transformation that leaves the Hamiltonian unchanged. Consider the following transformation in the fixed background (2.40)\n\nk α → -b α -2ρE α , b α → k α . (6.1)\n\nThe mapping to the dual dreibein is still specified by (3.9). A straightforward calculation reveals that the action transforms as The transformations (6.1) leaves unchanged the Hamiltonian and changes the action by the total \"time\" derivative terms shown in the first line of (6.2). Moreover, the linearized constraints transform into the linearized Bianchi identities. Let us see that in some detail. The second term in the shift constraint is zero since k α is a symmetric one form k α = k αβ ẽβ with k αβ = k βα ; see (2.21) . The term proportional to Λ in the lapse constraint is also zero. This is slightly more involved to see and it is based on the possibility of solving (3.9) for E α after gauge fixing. 11 One way to see this is in components. Write E α = E α β ẽβ and (3.9) becomes\n\nI\n\n∂ α E β γ -∂ γ E β α = ǫ β δα k δ γ -ǫ β δγ k δ α (6.3)\n\nIn the \"Lorentz gauge\" where ∂ α E β α = 0 = ∂ α k β α the above can be inverted as\n\nE α β = 1 ∂ 2 ǫ α δγ ∂ γ k δ β (6.4)\n\nUsing (6.4) one verifies that the last term in the lapse constraint vanishes. This modified duality transformation is probably related to the one considered by Julia et. al. in [15] ."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"We discuss the implementation of electric-magnetic duality transformations in fourdimensional gravity linearized around Minkowski or (A)dS 4 backgrounds. In the presence of a cosmological constant duality generically modifies the Hamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular attention to the boundary terms generated by the duality transformations and discuss their implications for holography. 1 For reviews of higher-spin theories see e.g. [1]. 2 See [4] and [5] for more recent works."},{"section_type":"OTHER","section_title":"Untitled Section","text":"Contents 1. Introduction and Summary 2. Action and Hamiltonian 2.1 The 3 + 1 split 2.2 Shifted Variables 2.3 Linearization 3. Linearized Gravitational Duality and Holography 3.1 Duality and Holography 3.2 Linearized gravitational duality 3.3 Linearized Constraints and Bianchi Identities 3.4 Connection with other known dualities 4. The Effect on the Boundary Theory 5. Conclusions and Outlook 6. Appendix: other duality mappings"},{"section_type":"BACKGROUND","section_title":"Introduction and Summary","text":"Duality has played an important role in our understanding of Yang-Mill theories and it is believed that it will play an important role also in gravity and in higher-spin gauge theories. Indeed, although it is less clear what could be the implications of duality for theories whose quantum versions are still unknown, gravity and higher-spin gauge theories 1 are intimately connected to a quantum string theory where certainly duality plays a crucial role. The recent advent of holography raises some intriguing questions for duality. For example one may wonder what is the holographic image of a duality invariant spectrum, a duality transformation or a possible quantization condition that usually duality implies for charges. Some of these issues were raised by Witten in [2] where it was argued that the standard electric-magnetic duality of a U(1) gauge theory on AdS 4 is responsible for a \"natural\" SL(2, Z) action on current two-point functions in three-dimensional CFTs. 2 Shortly afterwards it was shown in [3] that such an SL(2, Z) action is intimately related to certain \"double-trace\" deformations in the boundary, assuming suitable large-N limits and existence of non-trivial fixed points. The latter assumptions are strengthened by the fact that there exist models (e.g., see [6] and references therein) which exhibit the required behavior. In particular, it was shown in [3] that certain \"double-trace\" deformations induce an SL(2, Z) action on two-point functions of higher-spin (i.e. spin s ≥ 2) currents. This has led to the Duality Conjecture of [3] : linearized higher-spin theories on AdS 4 spaces possess a generalization of electric-magnetic duality whose holographic image is the natural SL(2, Z) action on boundary two-point functions.\n\nSurprisingly, even the duality for linearized spin-2 gauge fields (linearized gravity) was not widely known by the time of this conjecture. 3 Second order linearized gravitational duality was discussed among other in [8, 9, 10, 11, 12] . More recently, the duality properties of linearized gravity around flat space were studied in [13] and were further discussed in [14] . The duality of linearized gravity around dS 4 was later studied in [15] .\n\nIn this note we present our calculations regarding the duality properties of gravity in the presence of a cosmological constant. Having in mind applications to higher-spin gauge theories we use forms and work in the first order formalism where duality is also manifested at the level of the action [16] . Moreover, the first order formalism is relevant for applications of duality to holography, since the correlation functions of the boundary theory are essentially determined by the bulk canonical momenta (see e.g. [17] ).\n\nOur aim in this work is to formulate linearized first order gravity using suitable \"electric\" and \"magnetic\" variables, in close analogy with electromagnetism. We find that this is possible only when the background geometry is Minkowski or (A)dS 4 . Then we implement the standard electric-magnetic duality rotations. We find that, up to \"boundary\" terms, the linearized Hamiltonian changes by terms that do not alter the bulk dynamics i.e. do not alter the second order bulk equations of motion. Moreover, the duality rotation interchanges the (linearized) constraints with the (linearized) Bianchi identitites. The \"boundary\" terms have important holographic consequences since they correspond to marginal \"double-trace\" deformations [3] that induce the boundary SL(2, Z) action. In the Appendix we exhibit a modified duality rotation that leaves the bulk Hamiltonian invariant and induces \"boundary\" terms that correspond to relevant deformations as in [3] ."},{"section_type":"OTHER","section_title":"Action and Hamiltonian","text":"Having in mind the extension of our results to higher-spin gauge theorieswe start from the MacDowell-Mansouri form [18] of the gravitational action foot_1\n\nI M M = 1 2Λ M ǫ abcd R ab ∧ R cd + 2Λe a ∧ e b ∧ R cd + Λ 2 e a ∧ e b ∧ e c ∧ e d , (2.1)\n\nwhere a, b, ... are Lorentz indices. In this formalism, the vierbein e a and the spin connection ω a b are initially thought of as independent variables. The curvature 2-form is\n\nR a b = dω a b + ω a c ∧ ω c b = 1 2 R a bcd e c ∧ e d .\n\nVarying the action with respect to e a and ω a b , we find\n\nR ab + Λe a ∧ e b = 0 , (2.2)\n\nT a = de a + ω a b ∧ e b = 0 . (2.\n\n3)\n\nThe relation to gravity is established via the vanishing torsion equation (2.3), which relates e and ω in the familiar way. The above equations are equivalent to the Einstein equation in metric variables R µν -\n\n1 2 Rg µν = +3Λg µν . (2.4)\n\nand the scalar curvature is R = -d(d -1)Λ = -12Λ. Note that our Λ is related to the cosmological constant in its usual definition via Λ cosm = -6Λ. Λ > 0 corresponds to AdS. Note that this is actually SO(3, 2) covariant, as we can combine ω, e into a super-connection. Note that Λ has units (Length) -2 . In the SO(3, 2)-invariant formalism, I M M arises from\n\nI M M = 1 2Λ ǫ ABCDE V E R AB ∧ R CD , (2.5)\n\nwhere V E is a non-dynamical 0-form field (that we take to have value V -1 = 1 to gauge back to the SO(3, 1) formalism) and R A B is the curvature of Ω A B ≡ {e a , ω a b }. There are also quasitopological terms of the form\n\nI top = θ 2Λ R A B ∧ R B A + α Λ R A B ∧ R AC V B V C (2.6)\n\nthat we could add to the action. In the stated gauge, this reduces to\n\nI top = θ 2Λ P 2 + (θ + α)C N Y + α R ab ∧ e a ∧ e b (2.7)\n\nwhere\n\nP 2 = R a b ∧ R b a is the Pontryagin class, C N Y = (T a ∧ T a -R ab ∧ e a ∧ e b\n\n) is the Nieh-Yan class and we also note the Euler class E 2 = ǫ abcd R ab ∧ R cd . Note that in the presence of torsion, the action (2.7) contains the non-topological term R ab ∧ e a ∧ e b with \"Immirzi parameter\" γ = -2/α. In the absence of torsion, this term is a total derivative.\n\nThe Hilbert-Palatini action is\n\nI HP = I M M - 1 2Λ E 2 .\n\n(2.8)\n\nIt differs from I M M by a boundary term, is smooth as Λ → 0 but is not manifestly SO(3, 2)invariant."},{"section_type":"OTHER","section_title":"The 3 + 1 split","text":"Next, we carefully consider the 3 + 1 split. Although much of the discussion here is familiar from the ADM formalism, we feel it is important to set notation carefully, as we will introduce some new ingredients. To accommodate both AdS and dS signatures simultaneously, we will introduce a 'time' function t and a foliation of space-time Σ t ֒→ M. In dS, t is time-like, and this corresponds to the usual Hamiltonian foliation; in AdS on the other hand, we will take t to be the (space-like) radial coordinate. We will keep track of the resulting signs by a parameter σ ⊥ , equal to ±1 in dS(AdS).\n\nProceeding as usual then, we get a vector field t that satisfies ∇ t t = 1 ≡ t(t) (so t = ∂ ∂t ) and a 1-form dt. Given a 4-metric, we can introduce the normal 1-form n as\n\nn = σ ⊥ Ndt , (2.9)\n\nwhich is normalized as (n, n) = σ ⊥ . The dual vector field n can be expanded as\n\nn = 1 N t - 1 N N , (2.10)\n\nwhere the shift N satisfies (N, n) = 0, and thus (t, n) = σ ⊥ N.\n\nNext, we will locally choose a basis of 1-forms\n\ne 0 = σ ⊥ n = Ndt , (2.11)\n\ne α = ẽα + N α dt .\n\n(2.12)\n\nThe ẽα span T * Σ t , and correspond to a 3-metric h ij = ẽα i ẽβ j η αβ . The quantities N α are the components of N: N α = e α i N i . These basis 1-forms are dual to {e 0 = n, e α = ẽα }, with e a (e b ) = δ b a . We expand the spin connection in the same basis 5\n\nω a b = q a b dt + ωa b , (2.13)\n\nwhich leads to\n\nR a b = Ra b + dt ∧ r a b , (2.14)\n\nwhere R is formed from ω and d only, and\n\nr a b = ωa b -dq a b -ωa c q c b + q a c ωc b . (2.15)\n\nNote that these quantities are merely decompositions along T * Σ t in the 4-geometry; we will introduce the intrinsically defined objects shortly.\n\nWe then find\n\nI HP = 2ǫ αβγ dt ∧ N( Rαβ + Λẽ α ∧ ẽβ ) ∧ ẽγ -2N α ( R0β ) ∧ ẽγ + r 0α ∧ ẽβ ∧ ẽγ .\n\n(2.16) 5 We have\n\nq a b = N ω 0 a b + N α ω α a b and ωa b ≡ ω α a b ẽα .\n\nAs is familiar, the lapse and shift appear as Lagrange multipliers. The constraints that they multiply are of course zero in any background (i.e. vacuum solution), such as (A)dS 4 . The final term in the action contains the real dynamics -r 0α depends on the components R 0α 0β of the Riemann tensor.\n\nNote though that the tensors used here are 4-dimensional. Let us define the \"electric field\"\n\nK α = σ ⊥ ω0 α = K βα ẽβ .\n\n(2.17)\n\nIn the case that ω is the torsion-free Levi-Civita connection, this agrees with the standard definition for extrinsic curvature, regarded as a vector-valued one-form. We then find\n\nRα β = (3) R α β -σ ⊥ K α ∧ K β , (2.18)\n\nand\n\nR0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ ( DK) α . (2.19)\n\nThese equations amount to the Gauss-Codazzi relations. Furthermore, r 0α contains time derivatives of ω0α as well as terms linear in components of q. We find\n\n2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = 2ǫ αβγ σ ⊥ Kα -( Dq) 0α ∧ ẽβ ∧ ẽγ , (2.20)\n\n= 2σ ⊥ ǫ αβγ Kα + q αδ K δ ∧ ẽβ ∧ ẽγ + 4q 0α ǫ αβγ T β ∧ ẽγ up to a total 3-derivative. We have defined the intrinsic 3-torsion T α = dẽ α + ωα β ∧ ẽβ . Since we wish to regard the ẽ as coordinate variables, 6 we integrate the first term by parts to obtain (up to the total time-derivative\n\n∂ ∂t 2σ ⊥ K α ∧ ẽβ ∧ ẽγ ǫ αβγ ) 2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = Π α ∧ ėα + 4q 0α ǫ αβγ T β ∧ ẽγ + 2σ ⊥ q αδ ǫ αβγ K δ ∧ ẽβ ∧ ẽγ . (2.21)\n\nwhere we have defined the momentum 2-form\n\nΠ α = -4σ ⊥ ǫ αβγ K β ∧ ẽγ . (2.22)\n\nThe q ab appear as Lagrange multipliers. In particular, the q αβ constraint precisely sets the antisymmetric (torsional) part of the extrinsic curvature tensor K [αβ] to zero. Next, we define the \"magnetic field\"\n\nB α = 1 2 σ ⊥ ǫ αβγ ωβγ , ω αβ = -ǫ αβγ B γ . (2.23)\n\nand we find that the q 0α constraint\n\nǫ αβγ T β ∧ ẽγ = ǫ αβγ dẽ β ∧ ẽγ -σ ⊥ B β ∧ ẽβ ∧ ẽα = 0 , (2.24)\n\ninvolves only the antisymmetric part B [α,β] of the magnetic field B α = B αβ ẽβ . The antisymmetric part of B α spoils the gauge covariance of the constraint (2.24) under an SO(3) rotation of the dreibein ẽα , hence it represents degrees of freedom that can be gauged fixed to zero by an SO(3) rotation. On the other hand, an algebraic equation of motion connects the symmetric part of B αβ to derivatives of ẽα as dẽ α + ǫ αβγ B β ∧ ẽγ = 0 (2.25)\n\nAt the end, one is left with the canonically conjugate variables ẽα and Π α . These results are familiar from the metric formalism.\n\nDropping the torsional terms, we then arrive at the action\n\nI HP = dt ∧ ėα ∧ Π α + 2Nǫ αβγ ( (3) R αβ -σ ⊥ K α ∧ K β + Λẽ α ∧ ẽβ ) ∧ ẽγ -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ . (2.26) Furthermore, using * 3 ẽα ∧ ẽβ = 1 2 η γδ ǫ αβγ ẽδ , we have Πα = * 3 Π α = -2(K αβ -η αβ trK)ẽ β , (2.27)\n\nwhere trK = η αβ K αβ . We can solve the above equation to get\n\nK α = - 1 2 ( Παβ - 1 2 η αβ tr Π)ẽ β .\n\n(2.28)\n\nAs stated above, K αβ (and Παβ ) is symmetric when the torsion vanishes. Finally, with the definition (2.23) we find foot_3\n\nǫ αβγ (3) R αβ ∧ ẽγ = ǫ αβγ dω αβ + ωα δ ∧ ωδβ ∧ ẽγ = σ ⊥ 2 dB γ + ǫ αβγ B α ∧ B β ∧ ẽγ . (2.29)\n\nIntroducing B α is an unusual thing to do but it will play a role in duality: in this form, the Hamiltonian contains terms which are reminiscent of those of the Maxwell theory. The full HP action is of the form\n\nI HP = dt ∧ ėα ∧ Π α -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ +2σ ⊥ N(2 dB γ + ǫ αβγ B α ∧ B β -ǫ αβγ K α ∧ K β + σ ⊥ Λǫ αβγ ẽα ∧ ẽβ ) ∧ ẽγ . (2.30)\n\nNote that the entire contribution of the cosmological constant appears in the last term of the Hamiltonian constraint."},{"section_type":"OTHER","section_title":"Shifted Variables","text":"It is possible to make a transformation of the canonical variables in order to absorb the cosmological constant term in (2.30 ). This can be achieved by introducing the new variables\n\nKα = K α -ρẽ α , (2.31)\n\nand requiring that\n\nρ 2 = σ ⊥ Λ . (2.32)\n\nThis is positive only when σ ⊥ and Λ are simultaneously positive or negative, as it is the case for both AdS 4 (Λ > 0) and dS 4 (Λ < 0). We will often write Λ = σ ⊥ /L 2 where L is a length scale. Under (2.31) the momentum 2-form becomes\n\nΠ α → P α -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ . (2.33)\n\nThe last term in (2.33) contributes a total time derivative to the action (of the form of a boundary cosmological term). We have introduced a new momentum variable\n\nP α = -4σ ⊥ ǫ αβγ Kβ ∧ ẽγ .\n\nThen, we get the action\n\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K + ρ T ) β ∧ ẽγ - 4 3 σ ⊥ ρǫ αβγ ∂ ∂t (ẽ α ∧ ẽβ ∧ ẽγ ) +2σ ⊥ N 2 d(B α ∧ ẽα ) + 2B γ ∧ T γ -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ .(2.34)\n\nNote that the shift constraint is still written in terms of the ordinary covariant derivative, and thus involves a non-linear term coupling B to K. Consistent with our previous discussion, we drop the terms involving the torsion T , and disregard the boundary term to obtain\n\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K) β ∧ ẽγ +2σ ⊥ N 2 d(B α ∧ ẽα ) -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ . (2.35)\n\nWe note that the parameter ρ can be of either sign (although, this sign does not appear in the second order equations of motion)."},{"section_type":"OTHER","section_title":"Linearization","text":"Next, we linearize the above action around an appropriate fixed background. We expand as\n\nẽα = ẽα + E α , N = 1 + n, N α = n α , B α = B α + b α , Kα = Kα + k α . (2.36)\n\nThe background values should satisfy the constraints. The simplest choice is the background where Kα = 0 = B α .\n\n(2.37)\n\nIn fact, reaching this simple form was a motivation for the shift (2.31). Then, to quadratic order in the fluctuating fields the Hamiltonian gives\n\nI HP = dt ∧ Ėα ∧ p α -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + 4σ ⊥ n d(b α ∧ ẽα ) -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β + 2ρk α ∧ E β ∧ ẽγ , (2.38)\n\nwhere\n\np α = -4σ ⊥ ǫ αβγ k β ∧ ẽγ (2.39)\n\nare the linearized momentum variables conjugate to E α . In order to reach the form (2.38) the linear terms in the fluctuations must vanish. For this to happen we find the relationships ėα + ρẽ α = 0 .\n\n(2.40)\n\nNotice that we can also write the linearized action in the form\n\nI HP = dt ∧ ( Ėα + ρE α ) ∧ p α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + n 4σ ⊥ db γ + ρp γ ∧ ẽγ . (2.41)\n\nThe form of the first term, involving the momentum, makes clear that longitudinal fluctuations are non-dynamical. The natural time dependence of E α is of the form e -ρt (correspondingly, the natural time dependence of p α is e +ρt ). Other than that, we see that in comparing to the flat space action, in these variables, the only change is that the Hamiltonian constraint is modified. The solutions of (2.40) and (2.37) are components of (A)dS 4 spacetimes. We can solve (2.40) to obtain e 0 = dt, e α = e -ρt dx α .\n\n(2.42)\n\nWith these we construct the usual Poincaré metric on (A)dS which, however, covers only half of the space even though the parameter t runs from -∞ to +∞. The conformal boundary in these coordinates is at t = +∞. Then we derive\n\nω α 0 = -ρe -ρt dx α = -ρe α , (2.43)\n\nand so\n\nR α β = -σ ⊥ L 2 e α ∧ e β R α 0 = -1 L 2 e α ∧ e 0 R a b = - σ ⊥ L 2 e a ∧ e b . ( 2\n\nHence Ric ab = -3σ ⊥ L 2 η ab and R = -12σ ⊥ /L 2 = -12Λ. We also evaluate Π α = -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ , Πα = 4ρẽ α , tr Π = 12ρ (2.45)\n\nB α = 0, K α = ρẽ α ⇒ Kα = 0 (2.46)\n\nNote that in this gauge, ( DK) α = 1 L T α = 0, which solves the shift constraint, while the Hamiltonian constraint is satisfied through a cancellation between the K 2 term and the cosmological term."},{"section_type":"OTHER","section_title":"Linearized Gravitational Duality and Holography","text":"Let us summarize what we have obtained so far. In the presence of a cosmological constant we have defined variables such that the action resembles most closely the action without the cosmological constant. This was done in order to look for a suitable background around which linear fluctuations are as simple as possible. Requiring that K (the \"electric field\") and B (the \"magnetic field\") vanish in such a background -as they do around flat space -we found that the background should be (A)dS 4 . Quite satisfactorily, both sign choices for ρ in the change of variables (2.31) lead to (A)dS 4 spacetimes."},{"section_type":"OTHER","section_title":"Duality and Holography","text":"This is the appropriate point to recall some salient features of duality rotations. In simple Hamiltonian systems the effect of the canonical transformation p → q and q → -p to the action is (see e.g. [19] )\n\nI = t 2 t 1 dt[p q -H(p, q)] → I D = t 2 t 1 dt[-q ṗ -H(q, -p)] . (3.1)\n\nNotice that I D involves the dual variables, for which we have however kept the same notation for simplicity. The transformed Hamiltonian H(q, -p) is in general not related to H(p, q). However, if H(q, -p) = H(p, q) we call the above transformation a duality. It then holds\n\nI D = I -qp t 2 t 1 . (3.2)\n\nThe dual action describes exactly the same dynamics as the initial one, up to a modification of the boundary conditions. For example, if I is stationary on the e.o.m for fixed q in the boundary, I D is stationary on the same e.o.m. for fixed p in the boundary. This simple example illustrates the role of duality in holography; a bulk duality transformation corresponds to a particular modification of the boundary conditions. This property of duality transformations is behind the remarkable holographic properties of electormagnetism in (A)dS 4 [2, 3] .\n\nClearly, the crucial properties of a duality transformation are to be canonical and to leave the Hamiltonian unchanged. However, consider a slight generalization\n\nS = t 2 t 1 dt[p q - 1 2 (p 2 + q 2 + 2λpq)] (3.3)\n\nwhere λ is an arbitrary parameter. The Hamiltonian now is not invariant under the canonical transformation p → q and q → -p -the pq term changes sign. Consequently, the first order form of the equations of motion are also not duality invariant. Nevertheless, the second order equation of motion is invariant. We will find that gravity in the presence of a cosmological constant follows precisely this model. Of course, gravity is a much more complicated constrained system, but as we will show, the constraints and Bianchi identities transform appropriately.\n\nWe also note that the canonical transformation (implemented by a generating functional of the first kind) p → q + 2λp , q → -p . (3.4) is of interest here. The above does not change the Hamiltonian and the transformed action differs from the initial one by total time derivative terms foot_4 S → S D = S -pq\n\nt 2 t 1 -λp 2 t 2 t 1 . (3.5)"},{"section_type":"OTHER","section_title":"Linearized gravitational duality","text":"As a preamble to gravity we recall the duality properties of Maxwell theory\n\nI M ax = 1 2g 2 dt ∧ Ȧ ∧ * 3 E - 1 2 (E ∧ * 3 E + B ∧ * 3 B) -A 0 d * 3 E , (3.6)\n\nUnder the duality E → - * 3 B, B → * 3 E, Ã → ÃD , we find\n\nI M ax → I M ax,D = 1 2g 2 dt ∧ -ȦD ∧ B - 1 2 (E ∧ * 3 E + B ∧ * 3 B) + A 0 dB . (3.7)\n\nE and B in (3.7) should be expressed through ÃD . We observe that the kinetic term has changed sign, while the Hamiltonian remains invariant. In addition, the (Gauss) constraint is dualized to the trivial 'Bianchi' identity dB = 0 for the dual magnetic field.\n\nNext we try to apply a Maxwell-type duality map in gravity. We consider the following transformation around the fixed background (2.40)\n\nk α → -b α , b α → k α . (3.8)\n\nTo implement the map (3.8) we need to specify the mapping of E α to a 'dual 3-bein' E α . We do that using the linearized form of (2.25) as\n\nǫ αβγ b β ∧ ẽγ + dE α = 0 → ǫ αβγ k β ∧ ẽγ + dE α = 0 = dE α - 1 4σ ⊥ p α (3.9) Since p α = 4σ ⊥ dE α , it is natural to define p D,α = 4σ ⊥ dE α = -4σ ⊥ ǫ αβγ b β ∧ ẽγ , (3.10)\n\nand thus the mapping (3.8) is supplemented by\n\nE → E , E → -E , p → -p D , p D → p (3.11)\n\nNow, let us see the effects of the above duality mapping. The action transforms to\n\nI HP → I HP,D = dt ∧ -Ėα ∧ p D,α -ρE α ∧ p D,α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ (3.12) +4σ ⊥ n α ǫ αβγ db β ∧ ẽγ + n 4σ ⊥ dk γ + ρp D,α ∧ ẽγ\n\nwhere now k α and b α should be expressed in terms of the dual variables E α and p D,α via (3.9) and (3.10). We notice that the 'kinetic' part Ė ∧ p of the action changes sign under the duality map, in direct analogy with the Maxwell case. However, the Hamiltonian is not invariant due to the change of sign of the second term in the first line of (3.12). We will discuss this further in a later section. For now, we note that this sign change would not show up in the equations of motion, written in second order form. It is important to also note that the constraints are transformed into quantities which in the next subsection we will recognize as the linearized Bianchi identities. This is to be expected since the duality transformations are canonical. We also note that it may be possible to choose an alternative canonical transformation, designed to leave the Hamiltonian invariant. The latter is presumably related to the work of Julia et. al. [15] and is considered in the Appendix."},{"section_type":"OTHER","section_title":"Linearized Constraints and Bianchi Identities","text":"By virtue of the discussion above we may now demonstrate that under the duality mapping (3.8) the linearized constraints transform to the linearized Bianchi identities as\n\nC α ≡ ǫ αβγ dk β ∧ ẽγ → -ǫ αβγ db β ∧ ẽγ (3.13) C 0 ≡ -σ ⊥ db γ -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ → -σ ⊥ dk γ + ρǫ αβγ b α ∧ ẽβ ∧ ẽγ (3.14)\n\nTo identify the right hand sides, we first note that the Bianchi identities are\n\nB R a b = dR a b -R a c ∧ ω c b + ω a c ∧ R c b = 0 (3.15) B a T = dT a -R a b ∧ e b + ω a b ∧ T b = 0 (3.16)\n\nwhich are obtained from the definitions of R a b and T a by exterior differentiation. The first equation is satisfied identically. Since the torsion vanishes, the second equation tells us only that R a b ∧ e b = 0. If we do the 3+1 split, we find two equations. The first is\n\nB T α = -( (3) R α β -σ ⊥ K α ∧ K β ) ∧ ẽβ = 0 (3.17)\n\nwhich upon using the symmetry of K α linearizes to\n\nB T α = -ǫ αβγ db β ∧ ẽγ + . . . (3.18)\n\nNote that this is the image under duality of the shift constraint as in (3.13) .\n\nThe second identity is\n\nB T 0 = -R0 α ∧ ẽα = -σ ⊥ ( DK) α = -σ ⊥ dk α + ρǫ αβγ b β ∧ ẽγ ∧ ẽα = 0 (3.19)\n\nwhere to arrive in the second line we used (2.46) . This is the image of the Hamiltonian constraint as in (3.14) . Summarizing, the duality transformations between linearized constraints and Bianchi identities are\n\nC α → B T,α C 0 → B 0 T (3.20) B T,α → -C α B 0 T → -C 0 (3.21)"},{"section_type":"OTHER","section_title":"Connection with other known dualities","text":"The Maxwell-type duality operation (3. We begin with the spatial 2-forms when we have\n\nRαβ = -ǫ αβγ dB γ + σ ⊥ (B α ∧ B β -K α ∧ K β ) (3.23) R0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ (\n\nDK) α (3.24) and S0 γ = 1 2 σ ⊥ ǫ αβγ Rαβ (3.25) Sαβ = ǫ αβγ R0γ (3.26) If we linearize these expressions, we find under the duality transformation (3.8) Rab → -σ ⊥ Sab (3.27) Because the expressions (3.24) involve derivatives of B and K, the duality (3.8) is an 'integrated form' of the usual Riemann tensor duality, but implies it. Similarly, if we investigate the spatial 1-forms, we find r ab → -σ ⊥ s ab (3.28)\n\nTo arrive at this result we have set to zero the Lagrange multiplier field q."},{"section_type":"OTHER","section_title":"The Effect on the Boundary Theory","text":"It is well known that AdS is holographic. We may well ask, in the context of AdS/CFT, how the duality transformation that we have defined here acts in the boundary. We are instructed to consider the on-shell bulk action as a function of bulk fields. So, we evaluate the action on a solution to the equation of motion, resulting in a pure boundary term which is of the form\n\nS bdy = ∂M p α ∧ E α (4.1)\n\nApplying the duality transformation to the bulk theory, although the bulk action is not invariant as we have discussed above, nevertheless it may be easily shown that it induces a simple transformation on the (linearized) boundary term: it simply changes its sign.\n\nS dual bdy = - ∂M p D,α ∧ E α (4.2)\n\nThis transformation is exactly analogous to what happens in the Maxwell case: it amounts to the result [21] .\n\n10 G 2 G dual 2 = -1 . (4.3)"},{"section_type":"CONCLUSION","section_title":"Conclusions and Outlook","text":"Motivated by possible application in holography and in higher-spin gauge theory we have studied the duality properties of gravity in the Hamiltonian formulation. We have presented the gravity action in terms of suitable variables that closely resemble the electric and magnetic fields in Maxwell theory. We have found suitable \"electric\" and \"magnetic\" field variables, such that at the linearized level first order gravity most closely resembles electromagnetism. This can be done only around Minkowksi and (A)dS 4 backgrounds. We have implemented duality transformations in the linearized gravity fluctuations around these backgrounds. In the presence of a cosmological constant, the Hamiltonian changes, nevertheless the bulk dynamics remains unaltered, while the linearized lapse and shift constraints are mapped into the linearized Bianchi identities. Moreover, the duality transformations induce boundary terms whose relevance in holography we have briefly discussed. Finally, we have exhibited a modified duality rotation that leaves the bulk Hamiltonian invariant, while it induces boundary terms corresponding to relevant deformations.\n\nThe main implication of our results is that certain properties of correlations functions in three-dimensional CFTs mimic the duality of gravity. It would be interesting to extend our results to black-hole backgrounds and also when topological terms are present in the bulk. We also expect that one can analyze the duality of higher-spin gauge theories based on our first-order approach.\n\nThe work of A. C. P. was partially supported by the research program \"PYTHAGORAS II\" of the Greek Ministry of Education. RGL was supported in part by the U.S. Department of Energy under contract DE-FG02-91ER40709."},{"section_type":"OTHER","section_title":"Appendix: other duality mappings","text":"It is possible to find a transformation that leaves the Hamiltonian unchanged. Consider the following transformation in the fixed background (2.40)\n\nk α → -b α -2ρE α , b α → k α . (6.1)\n\nThe mapping to the dual dreibein is still specified by (3.9). A straightforward calculation reveals that the action transforms as The transformations (6.1) leaves unchanged the Hamiltonian and changes the action by the total \"time\" derivative terms shown in the first line of (6.2). Moreover, the linearized constraints transform into the linearized Bianchi identities. Let us see that in some detail. The second term in the shift constraint is zero since k α is a symmetric one form k α = k αβ ẽβ with k αβ = k βα ; see (2.21) . The term proportional to Λ in the lapse constraint is also zero. This is slightly more involved to see and it is based on the possibility of solving (3.9) for E α after gauge fixing. 11 One way to see this is in components. Write E α = E α β ẽβ and (3.9) becomes\n\nI\n\n∂ α E β γ -∂ γ E β α = ǫ β δα k δ γ -ǫ β δγ k δ α (6.3)\n\nIn the \"Lorentz gauge\" where ∂ α E β α = 0 = ∂ α k β α the above can be inverted as\n\nE α β = 1 ∂ 2 ǫ α δγ ∂ γ k δ β (6.4)\n\nUsing (6.4) one verifies that the last term in the lapse constraint vanishes. This modified duality transformation is probably related to the one considered by Julia et. al. in [15] ."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"We discuss the implementation of electric-magnetic duality transformations in fourdimensional gravity linearized around Minkowski or (A)dS 4 backgrounds. In the presence of a cosmological constant duality generically modifies the Hamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular attention to the boundary terms generated by the duality transformations and discuss their implications for holography. 1 For reviews of higher-spin theories see e.g. [1]. 2 See [4] and [5] for more recent works.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"Contents 1. Introduction and Summary 2. Action and Hamiltonian 2.1 The 3 + 1 split 2.2 Shifted Variables 2.3 Linearization 3. Linearized Gravitational Duality and Holography 3.1 Duality and Holography 3.2 Linearized gravitational duality 3.3 Linearized Constraints and Bianchi Identities 3.4 Connection with other known dualities 4. The Effect on the Boundary Theory 5. Conclusions and Outlook 6. Appendix: other duality mappings\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction and Summary\",\n \"text\": \"Duality has played an important role in our understanding of Yang-Mill theories and it is believed that it will play an important role also in gravity and in higher-spin gauge theories. Indeed, although it is less clear what could be the implications of duality for theories whose quantum versions are still unknown, gravity and higher-spin gauge theories 1 are intimately connected to a quantum string theory where certainly duality plays a crucial role. The recent advent of holography raises some intriguing questions for duality. For example one may wonder what is the holographic image of a duality invariant spectrum, a duality transformation or a possible quantization condition that usually duality implies for charges. Some of these issues were raised by Witten in [2] where it was argued that the standard electric-magnetic duality of a U(1) gauge theory on AdS 4 is responsible for a \\\"natural\\\" SL(2, Z) action on current two-point functions in three-dimensional CFTs. 2 Shortly afterwards it was shown in [3] that such an SL(2, Z) action is intimately related to certain \\\"double-trace\\\" deformations in the boundary, assuming suitable large-N limits and existence of non-trivial fixed points. The latter assumptions are strengthened by the fact that there exist models (e.g., see [6] and references therein) which exhibit the required behavior. In particular, it was shown in [3] that certain \\\"double-trace\\\" deformations induce an SL(2, Z) action on two-point functions of higher-spin (i.e. spin s ≥ 2) currents. This has led to the Duality Conjecture of [3] : linearized higher-spin theories on AdS 4 spaces possess a generalization of electric-magnetic duality whose holographic image is the natural SL(2, Z) action on boundary two-point functions.\\n\\nSurprisingly, even the duality for linearized spin-2 gauge fields (linearized gravity) was not widely known by the time of this conjecture. 3 Second order linearized gravitational duality was discussed among other in [8, 9, 10, 11, 12] . More recently, the duality properties of linearized gravity around flat space were studied in [13] and were further discussed in [14] . The duality of linearized gravity around dS 4 was later studied in [15] .\\n\\nIn this note we present our calculations regarding the duality properties of gravity in the presence of a cosmological constant. Having in mind applications to higher-spin gauge theories we use forms and work in the first order formalism where duality is also manifested at the level of the action [16] . Moreover, the first order formalism is relevant for applications of duality to holography, since the correlation functions of the boundary theory are essentially determined by the bulk canonical momenta (see e.g. [17] ).\\n\\nOur aim in this work is to formulate linearized first order gravity using suitable \\\"electric\\\" and \\\"magnetic\\\" variables, in close analogy with electromagnetism. We find that this is possible only when the background geometry is Minkowski or (A)dS 4 . Then we implement the standard electric-magnetic duality rotations. We find that, up to \\\"boundary\\\" terms, the linearized Hamiltonian changes by terms that do not alter the bulk dynamics i.e. do not alter the second order bulk equations of motion. Moreover, the duality rotation interchanges the (linearized) constraints with the (linearized) Bianchi identitites. The \\\"boundary\\\" terms have important holographic consequences since they correspond to marginal \\\"double-trace\\\" deformations [3] that induce the boundary SL(2, Z) action. In the Appendix we exhibit a modified duality rotation that leaves the bulk Hamiltonian invariant and induces \\\"boundary\\\" terms that correspond to relevant deformations as in [3] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Action and Hamiltonian\",\n \"text\": \"Having in mind the extension of our results to higher-spin gauge theorieswe start from the MacDowell-Mansouri form [18] of the gravitational action foot_1\\n\\nI M M = 1 2Λ M ǫ abcd R ab ∧ R cd + 2Λe a ∧ e b ∧ R cd + Λ 2 e a ∧ e b ∧ e c ∧ e d , (2.1)\\n\\nwhere a, b, ... are Lorentz indices. In this formalism, the vierbein e a and the spin connection ω a b are initially thought of as independent variables. The curvature 2-form is\\n\\nR a b = dω a b + ω a c ∧ ω c b = 1 2 R a bcd e c ∧ e d .\\n\\nVarying the action with respect to e a and ω a b , we find\\n\\nR ab + Λe a ∧ e b = 0 , (2.2)\\n\\nT a = de a + ω a b ∧ e b = 0 . (2.\\n\\n3)\\n\\nThe relation to gravity is established via the vanishing torsion equation (2.3), which relates e and ω in the familiar way. The above equations are equivalent to the Einstein equation in metric variables R µν -\\n\\n1 2 Rg µν = +3Λg µν . (2.4)\\n\\nand the scalar curvature is R = -d(d -1)Λ = -12Λ. Note that our Λ is related to the cosmological constant in its usual definition via Λ cosm = -6Λ. Λ > 0 corresponds to AdS. Note that this is actually SO(3, 2) covariant, as we can combine ω, e into a super-connection. Note that Λ has units (Length) -2 . In the SO(3, 2)-invariant formalism, I M M arises from\\n\\nI M M = 1 2Λ ǫ ABCDE V E R AB ∧ R CD , (2.5)\\n\\nwhere V E is a non-dynamical 0-form field (that we take to have value V -1 = 1 to gauge back to the SO(3, 1) formalism) and R A B is the curvature of Ω A B ≡ {e a , ω a b }. There are also quasitopological terms of the form\\n\\nI top = θ 2Λ R A B ∧ R B A + α Λ R A B ∧ R AC V B V C (2.6)\\n\\nthat we could add to the action. In the stated gauge, this reduces to\\n\\nI top = θ 2Λ P 2 + (θ + α)C N Y + α R ab ∧ e a ∧ e b (2.7)\\n\\nwhere\\n\\nP 2 = R a b ∧ R b a is the Pontryagin class, C N Y = (T a ∧ T a -R ab ∧ e a ∧ e b\\n\\n) is the Nieh-Yan class and we also note the Euler class E 2 = ǫ abcd R ab ∧ R cd . Note that in the presence of torsion, the action (2.7) contains the non-topological term R ab ∧ e a ∧ e b with \\\"Immirzi parameter\\\" γ = -2/α. In the absence of torsion, this term is a total derivative.\\n\\nThe Hilbert-Palatini action is\\n\\nI HP = I M M - 1 2Λ E 2 .\\n\\n(2.8)\\n\\nIt differs from I M M by a boundary term, is smooth as Λ → 0 but is not manifestly SO(3, 2)invariant.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The 3 + 1 split\",\n \"text\": \"Next, we carefully consider the 3 + 1 split. Although much of the discussion here is familiar from the ADM formalism, we feel it is important to set notation carefully, as we will introduce some new ingredients. To accommodate both AdS and dS signatures simultaneously, we will introduce a 'time' function t and a foliation of space-time Σ t ֒→ M. In dS, t is time-like, and this corresponds to the usual Hamiltonian foliation; in AdS on the other hand, we will take t to be the (space-like) radial coordinate. We will keep track of the resulting signs by a parameter σ ⊥ , equal to ±1 in dS(AdS).\\n\\nProceeding as usual then, we get a vector field t that satisfies ∇ t t = 1 ≡ t(t) (so t = ∂ ∂t ) and a 1-form dt. Given a 4-metric, we can introduce the normal 1-form n as\\n\\nn = σ ⊥ Ndt , (2.9)\\n\\nwhich is normalized as (n, n) = σ ⊥ . The dual vector field n can be expanded as\\n\\nn = 1 N t - 1 N N , (2.10)\\n\\nwhere the shift N satisfies (N, n) = 0, and thus (t, n) = σ ⊥ N.\\n\\nNext, we will locally choose a basis of 1-forms\\n\\ne 0 = σ ⊥ n = Ndt , (2.11)\\n\\ne α = ẽα + N α dt .\\n\\n(2.12)\\n\\nThe ẽα span T * Σ t , and correspond to a 3-metric h ij = ẽα i ẽβ j η αβ . The quantities N α are the components of N: N α = e α i N i . These basis 1-forms are dual to {e 0 = n, e α = ẽα }, with e a (e b ) = δ b a . We expand the spin connection in the same basis 5\\n\\nω a b = q a b dt + ωa b , (2.13)\\n\\nwhich leads to\\n\\nR a b = Ra b + dt ∧ r a b , (2.14)\\n\\nwhere R is formed from ω and d only, and\\n\\nr a b = ωa b -dq a b -ωa c q c b + q a c ωc b . (2.15)\\n\\nNote that these quantities are merely decompositions along T * Σ t in the 4-geometry; we will introduce the intrinsically defined objects shortly.\\n\\nWe then find\\n\\nI HP = 2ǫ αβγ dt ∧ N( Rαβ + Λẽ α ∧ ẽβ ) ∧ ẽγ -2N α ( R0β ) ∧ ẽγ + r 0α ∧ ẽβ ∧ ẽγ .\\n\\n(2.16) 5 We have\\n\\nq a b = N ω 0 a b + N α ω α a b and ωa b ≡ ω α a b ẽα .\\n\\nAs is familiar, the lapse and shift appear as Lagrange multipliers. The constraints that they multiply are of course zero in any background (i.e. vacuum solution), such as (A)dS 4 . The final term in the action contains the real dynamics -r 0α depends on the components R 0α 0β of the Riemann tensor.\\n\\nNote though that the tensors used here are 4-dimensional. Let us define the \\\"electric field\\\"\\n\\nK α = σ ⊥ ω0 α = K βα ẽβ .\\n\\n(2.17)\\n\\nIn the case that ω is the torsion-free Levi-Civita connection, this agrees with the standard definition for extrinsic curvature, regarded as a vector-valued one-form. We then find\\n\\nRα β = (3) R α β -σ ⊥ K α ∧ K β , (2.18)\\n\\nand\\n\\nR0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ ( DK) α . (2.19)\\n\\nThese equations amount to the Gauss-Codazzi relations. Furthermore, r 0α contains time derivatives of ω0α as well as terms linear in components of q. We find\\n\\n2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = 2ǫ αβγ σ ⊥ Kα -( Dq) 0α ∧ ẽβ ∧ ẽγ , (2.20)\\n\\n= 2σ ⊥ ǫ αβγ Kα + q αδ K δ ∧ ẽβ ∧ ẽγ + 4q 0α ǫ αβγ T β ∧ ẽγ up to a total 3-derivative. We have defined the intrinsic 3-torsion T α = dẽ α + ωα β ∧ ẽβ . Since we wish to regard the ẽ as coordinate variables, 6 we integrate the first term by parts to obtain (up to the total time-derivative\\n\\n∂ ∂t 2σ ⊥ K α ∧ ẽβ ∧ ẽγ ǫ αβγ ) 2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = Π α ∧ ėα + 4q 0α ǫ αβγ T β ∧ ẽγ + 2σ ⊥ q αδ ǫ αβγ K δ ∧ ẽβ ∧ ẽγ . (2.21)\\n\\nwhere we have defined the momentum 2-form\\n\\nΠ α = -4σ ⊥ ǫ αβγ K β ∧ ẽγ . (2.22)\\n\\nThe q ab appear as Lagrange multipliers. In particular, the q αβ constraint precisely sets the antisymmetric (torsional) part of the extrinsic curvature tensor K [αβ] to zero. Next, we define the \\\"magnetic field\\\"\\n\\nB α = 1 2 σ ⊥ ǫ αβγ ωβγ , ω αβ = -ǫ αβγ B γ . (2.23)\\n\\nand we find that the q 0α constraint\\n\\nǫ αβγ T β ∧ ẽγ = ǫ αβγ dẽ β ∧ ẽγ -σ ⊥ B β ∧ ẽβ ∧ ẽα = 0 , (2.24)\\n\\ninvolves only the antisymmetric part B [α,β] of the magnetic field B α = B αβ ẽβ . The antisymmetric part of B α spoils the gauge covariance of the constraint (2.24) under an SO(3) rotation of the dreibein ẽα , hence it represents degrees of freedom that can be gauged fixed to zero by an SO(3) rotation. On the other hand, an algebraic equation of motion connects the symmetric part of B αβ to derivatives of ẽα as dẽ α + ǫ αβγ B β ∧ ẽγ = 0 (2.25)\\n\\nAt the end, one is left with the canonically conjugate variables ẽα and Π α . These results are familiar from the metric formalism.\\n\\nDropping the torsional terms, we then arrive at the action\\n\\nI HP = dt ∧ ėα ∧ Π α + 2Nǫ αβγ ( (3) R αβ -σ ⊥ K α ∧ K β + Λẽ α ∧ ẽβ ) ∧ ẽγ -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ . (2.26) Furthermore, using * 3 ẽα ∧ ẽβ = 1 2 η γδ ǫ αβγ ẽδ , we have Πα = * 3 Π α = -2(K αβ -η αβ trK)ẽ β , (2.27)\\n\\nwhere trK = η αβ K αβ . We can solve the above equation to get\\n\\nK α = - 1 2 ( Παβ - 1 2 η αβ tr Π)ẽ β .\\n\\n(2.28)\\n\\nAs stated above, K αβ (and Παβ ) is symmetric when the torsion vanishes. Finally, with the definition (2.23) we find foot_3\\n\\nǫ αβγ (3) R αβ ∧ ẽγ = ǫ αβγ dω αβ + ωα δ ∧ ωδβ ∧ ẽγ = σ ⊥ 2 dB γ + ǫ αβγ B α ∧ B β ∧ ẽγ . (2.29)\\n\\nIntroducing B α is an unusual thing to do but it will play a role in duality: in this form, the Hamiltonian contains terms which are reminiscent of those of the Maxwell theory. The full HP action is of the form\\n\\nI HP = dt ∧ ėα ∧ Π α -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ +2σ ⊥ N(2 dB γ + ǫ αβγ B α ∧ B β -ǫ αβγ K α ∧ K β + σ ⊥ Λǫ αβγ ẽα ∧ ẽβ ) ∧ ẽγ . (2.30)\\n\\nNote that the entire contribution of the cosmological constant appears in the last term of the Hamiltonian constraint.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Shifted Variables\",\n \"text\": \"It is possible to make a transformation of the canonical variables in order to absorb the cosmological constant term in (2.30 ). This can be achieved by introducing the new variables\\n\\nKα = K α -ρẽ α , (2.31)\\n\\nand requiring that\\n\\nρ 2 = σ ⊥ Λ . (2.32)\\n\\nThis is positive only when σ ⊥ and Λ are simultaneously positive or negative, as it is the case for both AdS 4 (Λ > 0) and dS 4 (Λ < 0). We will often write Λ = σ ⊥ /L 2 where L is a length scale. Under (2.31) the momentum 2-form becomes\\n\\nΠ α → P α -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ . (2.33)\\n\\nThe last term in (2.33) contributes a total time derivative to the action (of the form of a boundary cosmological term). We have introduced a new momentum variable\\n\\nP α = -4σ ⊥ ǫ αβγ Kβ ∧ ẽγ .\\n\\nThen, we get the action\\n\\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K + ρ T ) β ∧ ẽγ - 4 3 σ ⊥ ρǫ αβγ ∂ ∂t (ẽ α ∧ ẽβ ∧ ẽγ ) +2σ ⊥ N 2 d(B α ∧ ẽα ) + 2B γ ∧ T γ -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ .(2.34)\\n\\nNote that the shift constraint is still written in terms of the ordinary covariant derivative, and thus involves a non-linear term coupling B to K. Consistent with our previous discussion, we drop the terms involving the torsion T , and disregard the boundary term to obtain\\n\\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K) β ∧ ẽγ +2σ ⊥ N 2 d(B α ∧ ẽα ) -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ . (2.35)\\n\\nWe note that the parameter ρ can be of either sign (although, this sign does not appear in the second order equations of motion).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Linearization\",\n \"text\": \"Next, we linearize the above action around an appropriate fixed background. We expand as\\n\\nẽα = ẽα + E α , N = 1 + n, N α = n α , B α = B α + b α , Kα = Kα + k α . (2.36)\\n\\nThe background values should satisfy the constraints. The simplest choice is the background where Kα = 0 = B α .\\n\\n(2.37)\\n\\nIn fact, reaching this simple form was a motivation for the shift (2.31). Then, to quadratic order in the fluctuating fields the Hamiltonian gives\\n\\nI HP = dt ∧ Ėα ∧ p α -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + 4σ ⊥ n d(b α ∧ ẽα ) -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β + 2ρk α ∧ E β ∧ ẽγ , (2.38)\\n\\nwhere\\n\\np α = -4σ ⊥ ǫ αβγ k β ∧ ẽγ (2.39)\\n\\nare the linearized momentum variables conjugate to E α . In order to reach the form (2.38) the linear terms in the fluctuations must vanish. For this to happen we find the relationships ėα + ρẽ α = 0 .\\n\\n(2.40)\\n\\nNotice that we can also write the linearized action in the form\\n\\nI HP = dt ∧ ( Ėα + ρE α ) ∧ p α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + n 4σ ⊥ db γ + ρp γ ∧ ẽγ . (2.41)\\n\\nThe form of the first term, involving the momentum, makes clear that longitudinal fluctuations are non-dynamical. The natural time dependence of E α is of the form e -ρt (correspondingly, the natural time dependence of p α is e +ρt ). Other than that, we see that in comparing to the flat space action, in these variables, the only change is that the Hamiltonian constraint is modified. The solutions of (2.40) and (2.37) are components of (A)dS 4 spacetimes. We can solve (2.40) to obtain e 0 = dt, e α = e -ρt dx α .\\n\\n(2.42)\\n\\nWith these we construct the usual Poincaré metric on (A)dS which, however, covers only half of the space even though the parameter t runs from -∞ to +∞. The conformal boundary in these coordinates is at t = +∞. Then we derive\\n\\nω α 0 = -ρe -ρt dx α = -ρe α , (2.43)\\n\\nand so\\n\\nR α β = -σ ⊥ L 2 e α ∧ e β R α 0 = -1 L 2 e α ∧ e 0 R a b = - σ ⊥ L 2 e a ∧ e b . ( 2\\n\\nHence Ric ab = -3σ ⊥ L 2 η ab and R = -12σ ⊥ /L 2 = -12Λ. We also evaluate Π α = -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ , Πα = 4ρẽ α , tr Π = 12ρ (2.45)\\n\\nB α = 0, K α = ρẽ α ⇒ Kα = 0 (2.46)\\n\\nNote that in this gauge, ( DK) α = 1 L T α = 0, which solves the shift constraint, while the Hamiltonian constraint is satisfied through a cancellation between the K 2 term and the cosmological term.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Linearized Gravitational Duality and Holography\",\n \"text\": \"Let us summarize what we have obtained so far. In the presence of a cosmological constant we have defined variables such that the action resembles most closely the action without the cosmological constant. This was done in order to look for a suitable background around which linear fluctuations are as simple as possible. Requiring that K (the \\\"electric field\\\") and B (the \\\"magnetic field\\\") vanish in such a background -as they do around flat space -we found that the background should be (A)dS 4 . Quite satisfactorily, both sign choices for ρ in the change of variables (2.31) lead to (A)dS 4 spacetimes.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Duality and Holography\",\n \"text\": \"This is the appropriate point to recall some salient features of duality rotations. In simple Hamiltonian systems the effect of the canonical transformation p → q and q → -p to the action is (see e.g. [19] )\\n\\nI = t 2 t 1 dt[p q -H(p, q)] → I D = t 2 t 1 dt[-q ṗ -H(q, -p)] . (3.1)\\n\\nNotice that I D involves the dual variables, for which we have however kept the same notation for simplicity. The transformed Hamiltonian H(q, -p) is in general not related to H(p, q). However, if H(q, -p) = H(p, q) we call the above transformation a duality. It then holds\\n\\nI D = I -qp t 2 t 1 . (3.2)\\n\\nThe dual action describes exactly the same dynamics as the initial one, up to a modification of the boundary conditions. For example, if I is stationary on the e.o.m for fixed q in the boundary, I D is stationary on the same e.o.m. for fixed p in the boundary. This simple example illustrates the role of duality in holography; a bulk duality transformation corresponds to a particular modification of the boundary conditions. This property of duality transformations is behind the remarkable holographic properties of electormagnetism in (A)dS 4 [2, 3] .\\n\\nClearly, the crucial properties of a duality transformation are to be canonical and to leave the Hamiltonian unchanged. However, consider a slight generalization\\n\\nS = t 2 t 1 dt[p q - 1 2 (p 2 + q 2 + 2λpq)] (3.3)\\n\\nwhere λ is an arbitrary parameter. The Hamiltonian now is not invariant under the canonical transformation p → q and q → -p -the pq term changes sign. Consequently, the first order form of the equations of motion are also not duality invariant. Nevertheless, the second order equation of motion is invariant. We will find that gravity in the presence of a cosmological constant follows precisely this model. Of course, gravity is a much more complicated constrained system, but as we will show, the constraints and Bianchi identities transform appropriately.\\n\\nWe also note that the canonical transformation (implemented by a generating functional of the first kind) p → q + 2λp , q → -p . (3.4) is of interest here. The above does not change the Hamiltonian and the transformed action differs from the initial one by total time derivative terms foot_4 S → S D = S -pq\\n\\nt 2 t 1 -λp 2 t 2 t 1 . (3.5)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Linearized gravitational duality\",\n \"text\": \"As a preamble to gravity we recall the duality properties of Maxwell theory\\n\\nI M ax = 1 2g 2 dt ∧ Ȧ ∧ * 3 E - 1 2 (E ∧ * 3 E + B ∧ * 3 B) -A 0 d * 3 E , (3.6)\\n\\nUnder the duality E → - * 3 B, B → * 3 E, Ã → ÃD , we find\\n\\nI M ax → I M ax,D = 1 2g 2 dt ∧ -ȦD ∧ B - 1 2 (E ∧ * 3 E + B ∧ * 3 B) + A 0 dB . (3.7)\\n\\nE and B in (3.7) should be expressed through ÃD . We observe that the kinetic term has changed sign, while the Hamiltonian remains invariant. In addition, the (Gauss) constraint is dualized to the trivial 'Bianchi' identity dB = 0 for the dual magnetic field.\\n\\nNext we try to apply a Maxwell-type duality map in gravity. We consider the following transformation around the fixed background (2.40)\\n\\nk α → -b α , b α → k α . (3.8)\\n\\nTo implement the map (3.8) we need to specify the mapping of E α to a 'dual 3-bein' E α . We do that using the linearized form of (2.25) as\\n\\nǫ αβγ b β ∧ ẽγ + dE α = 0 → ǫ αβγ k β ∧ ẽγ + dE α = 0 = dE α - 1 4σ ⊥ p α (3.9) Since p α = 4σ ⊥ dE α , it is natural to define p D,α = 4σ ⊥ dE α = -4σ ⊥ ǫ αβγ b β ∧ ẽγ , (3.10)\\n\\nand thus the mapping (3.8) is supplemented by\\n\\nE → E , E → -E , p → -p D , p D → p (3.11)\\n\\nNow, let us see the effects of the above duality mapping. The action transforms to\\n\\nI HP → I HP,D = dt ∧ -Ėα ∧ p D,α -ρE α ∧ p D,α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ (3.12) +4σ ⊥ n α ǫ αβγ db β ∧ ẽγ + n 4σ ⊥ dk γ + ρp D,α ∧ ẽγ\\n\\nwhere now k α and b α should be expressed in terms of the dual variables E α and p D,α via (3.9) and (3.10). We notice that the 'kinetic' part Ė ∧ p of the action changes sign under the duality map, in direct analogy with the Maxwell case. However, the Hamiltonian is not invariant due to the change of sign of the second term in the first line of (3.12). We will discuss this further in a later section. For now, we note that this sign change would not show up in the equations of motion, written in second order form. It is important to also note that the constraints are transformed into quantities which in the next subsection we will recognize as the linearized Bianchi identities. This is to be expected since the duality transformations are canonical. We also note that it may be possible to choose an alternative canonical transformation, designed to leave the Hamiltonian invariant. The latter is presumably related to the work of Julia et. al. [15] and is considered in the Appendix.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Linearized Constraints and Bianchi Identities\",\n \"text\": \"By virtue of the discussion above we may now demonstrate that under the duality mapping (3.8) the linearized constraints transform to the linearized Bianchi identities as\\n\\nC α ≡ ǫ αβγ dk β ∧ ẽγ → -ǫ αβγ db β ∧ ẽγ (3.13) C 0 ≡ -σ ⊥ db γ -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ → -σ ⊥ dk γ + ρǫ αβγ b α ∧ ẽβ ∧ ẽγ (3.14)\\n\\nTo identify the right hand sides, we first note that the Bianchi identities are\\n\\nB R a b = dR a b -R a c ∧ ω c b + ω a c ∧ R c b = 0 (3.15) B a T = dT a -R a b ∧ e b + ω a b ∧ T b = 0 (3.16)\\n\\nwhich are obtained from the definitions of R a b and T a by exterior differentiation. The first equation is satisfied identically. Since the torsion vanishes, the second equation tells us only that R a b ∧ e b = 0. If we do the 3+1 split, we find two equations. The first is\\n\\nB T α = -( (3) R α β -σ ⊥ K α ∧ K β ) ∧ ẽβ = 0 (3.17)\\n\\nwhich upon using the symmetry of K α linearizes to\\n\\nB T α = -ǫ αβγ db β ∧ ẽγ + . . . (3.18)\\n\\nNote that this is the image under duality of the shift constraint as in (3.13) .\\n\\nThe second identity is\\n\\nB T 0 = -R0 α ∧ ẽα = -σ ⊥ ( DK) α = -σ ⊥ dk α + ρǫ αβγ b β ∧ ẽγ ∧ ẽα = 0 (3.19)\\n\\nwhere to arrive in the second line we used (2.46) . This is the image of the Hamiltonian constraint as in (3.14) . Summarizing, the duality transformations between linearized constraints and Bianchi identities are\\n\\nC α → B T,α C 0 → B 0 T (3.20) B T,α → -C α B 0 T → -C 0 (3.21)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Connection with other known dualities\",\n \"text\": \"The Maxwell-type duality operation (3. We begin with the spatial 2-forms when we have\\n\\nRαβ = -ǫ αβγ dB γ + σ ⊥ (B α ∧ B β -K α ∧ K β ) (3.23) R0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ (\\n\\nDK) α (3.24) and S0 γ = 1 2 σ ⊥ ǫ αβγ Rαβ (3.25) Sαβ = ǫ αβγ R0γ (3.26) If we linearize these expressions, we find under the duality transformation (3.8) Rab → -σ ⊥ Sab (3.27) Because the expressions (3.24) involve derivatives of B and K, the duality (3.8) is an 'integrated form' of the usual Riemann tensor duality, but implies it. Similarly, if we investigate the spatial 1-forms, we find r ab → -σ ⊥ s ab (3.28)\\n\\nTo arrive at this result we have set to zero the Lagrange multiplier field q.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Effect on the Boundary Theory\",\n \"text\": \"It is well known that AdS is holographic. We may well ask, in the context of AdS/CFT, how the duality transformation that we have defined here acts in the boundary. We are instructed to consider the on-shell bulk action as a function of bulk fields. So, we evaluate the action on a solution to the equation of motion, resulting in a pure boundary term which is of the form\\n\\nS bdy = ∂M p α ∧ E α (4.1)\\n\\nApplying the duality transformation to the bulk theory, although the bulk action is not invariant as we have discussed above, nevertheless it may be easily shown that it induces a simple transformation on the (linearized) boundary term: it simply changes its sign.\\n\\nS dual bdy = - ∂M p D,α ∧ E α (4.2)\\n\\nThis transformation is exactly analogous to what happens in the Maxwell case: it amounts to the result [21] .\\n\\n10 G 2 G dual 2 = -1 . (4.3)\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusions and Outlook\",\n \"text\": \"Motivated by possible application in holography and in higher-spin gauge theory we have studied the duality properties of gravity in the Hamiltonian formulation. We have presented the gravity action in terms of suitable variables that closely resemble the electric and magnetic fields in Maxwell theory. We have found suitable \\\"electric\\\" and \\\"magnetic\\\" field variables, such that at the linearized level first order gravity most closely resembles electromagnetism. This can be done only around Minkowksi and (A)dS 4 backgrounds. We have implemented duality transformations in the linearized gravity fluctuations around these backgrounds. In the presence of a cosmological constant, the Hamiltonian changes, nevertheless the bulk dynamics remains unaltered, while the linearized lapse and shift constraints are mapped into the linearized Bianchi identities. Moreover, the duality transformations induce boundary terms whose relevance in holography we have briefly discussed. Finally, we have exhibited a modified duality rotation that leaves the bulk Hamiltonian invariant, while it induces boundary terms corresponding to relevant deformations.\\n\\nThe main implication of our results is that certain properties of correlations functions in three-dimensional CFTs mimic the duality of gravity. It would be interesting to extend our results to black-hole backgrounds and also when topological terms are present in the bulk. We also expect that one can analyze the duality of higher-spin gauge theories based on our first-order approach.\\n\\nThe work of A. C. P. was partially supported by the research program \\\"PYTHAGORAS II\\\" of the Greek Ministry of Education. RGL was supported in part by the U.S. Department of Energy under contract DE-FG02-91ER40709.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Appendix: other duality mappings\",\n \"text\": \"It is possible to find a transformation that leaves the Hamiltonian unchanged. Consider the following transformation in the fixed background (2.40)\\n\\nk α → -b α -2ρE α , b α → k α . (6.1)\\n\\nThe mapping to the dual dreibein is still specified by (3.9). A straightforward calculation reveals that the action transforms as The transformations (6.1) leaves unchanged the Hamiltonian and changes the action by the total \\\"time\\\" derivative terms shown in the first line of (6.2). Moreover, the linearized constraints transform into the linearized Bianchi identities. Let us see that in some detail. The second term in the shift constraint is zero since k α is a symmetric one form k α = k αβ ẽβ with k αβ = k βα ; see (2.21) . The term proportional to Λ in the lapse constraint is also zero. This is slightly more involved to see and it is based on the possibility of solving (3.9) for E α after gauge fixing. 11 One way to see this is in components. Write E α = E α β ẽβ and (3.9) becomes\\n\\nI\\n\\n∂ α E β γ -∂ γ E β α = ǫ β δα k δ γ -ǫ β δγ k δ α (6.3)\\n\\nIn the \\\"Lorentz gauge\\\" where ∂ α E β α = 0 = ∂ α k β α the above can be inverted as\\n\\nE α β = 1 ∂ 2 ǫ α δγ ∂ γ k δ β (6.4)\\n\\nUsing (6.4) one verifies that the last term in the lapse constraint vanishes. This modified duality transformation is probably related to the one considered by Julia et. al. in [15] .\"\n }\n]"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":0,"numItemsPerPage":100,"numTotalItems":343056,"offset":0,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3MjM5NjQ1Mywic3ViIjoiL2RhdGFzZXRzL2tvbnNtYW4vcGh5c2ljcy1jb3JwdXMiLCJleHAiOjE3NzI0MDAwNTMsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.0Z6RrbUOasHRJebG4p5SeWaCBt4aAr2nTidGKHTG58OAYkm54PmYeuBtSjf7IfXq6hydfpSW2fD2mdyFnrB_Ag","displayUrls":true,"splitSizeSummaries":[{"config":"quantum-physics","split":"train","numRows":343056,"numBytesParquet":15067827363}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
paper_id stringlengths 15 15 | arxiv_id stringlengths 9 9 | arxiv_version int32 1 1 | doi stringlengths 0 94 | content_hash stringlengths 64 64 | title stringlengths 5 243 | abstract stringlengths 13 2.11k | authors listlengths 1 816 | arxiv_categories listlengths 1 13 | primary_category stringclasses 116 values | journal stringclasses 1 value | keywords listlengths 0 0 | submission_date stringdate 2007-03-31 00:00:00 2012-12-06 00:00:00 | processed_date stringdate 2026-02-26 00:00:00 2026-02-26 00:00:00 | full_text stringlengths 29 1.72M | sections listlengths 1 507 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
arxiv:0704.0007 | 0704.0007 | 1 | 10.1103/PhysRevD.76.044016 | 8ae0e0f35013725cd790673ceb8a5ca5fa7caa17c3f9774fc6aa4adfcfb62edd | Polymer Quantum Mechanics and its Continuum Limit | A rather non-standard quantum representation of the canonical commutation
relations of quantum mechanics systems, known as the polymer representation has
gained some attention in recent years, due to its possible relation with Planck
scale physics. In particular, this approach has been followed in a symmetric
sector of loop quantum gravity known as loop quantum cosmology. Here we explore
different aspects of the relation between the ordinary Schroedinger theory and
the polymer description. The paper has two parts. In the first one, we derive
the polymer quantum mechanics starting from the ordinary Schroedinger theory
and show that the polymer description arises as an appropriate limit. In the
second part we consider the continuum limit of this theory, namely, the reverse
process in which one starts from the discrete theory and tries to recover back
the ordinary Schroedinger quantum mechanics. We consider several examples of
interest, including the harmonic oscillator, the free particle and a simple
cosmological model. | [
"Alejandro Corichi",
"Tatjana Vukasinac and Jose A. Zapata"
] | [
"gr-qc"
] | gr-qc | [] | 2007-03-31 | 2026-02-26 | The so-called polymer quantum mechanics, a nonregular and somewhat 'exotic' representation of the canonical commutation relations (CCR) [1], has been used to explore both mathematical and physical issues in background independent theories such as quantum gravity [2, 3] . A notable example of this type of quantization, when applied to minisuperspace models has given way to what is known as loop quantum cosmology [4, 5] . As in any toy model situation, one hopes to learn about the subtle technical and conceptual issues that are present in full quantum gravity by means of simple, finite dimensional examples. This formalism is not an exception in this regard. Apart from this motivation coming from physics at the Planck scale, one can independently ask for the relation between the standard continuous representations and their polymer cousins at the level of mathematical physics. A deeper understanding of this relation becomes important on its own.
The polymer quantization is made of several steps. The first one is to build a representation of the Heisenberg-Weyl algebra on a Kinematical Hilbert space that is "background independent", and that is sometimes referred to as the polymeric Hilbert space H poly . The second and most important part, the implementation of dynamics, deals with the definition of a Hamiltonian (or Hamiltonian constraint) on this space. In the examples studied so far, the first part is fairly well understood, yielding the kinematical Hilbert space H poly that is, however, non-separable. For the second step, a natural implementation of the dynamics has proved to be a bit more difficult, given that a direct definition of the Hamiltonian Ĥ of, say, a particle on a potential on the space H poly is not possible since one of the main features of this representation is that the operators q and p cannot be both simultaneously defined (nor their analogues in theories involving more elaborate variables). Thus, any operator that involves (powers of) the not defined variable has to be regulated by a well defined operator which normally involves introducing some extra structure on the configuration (or momentum) space, namely a lattice. However, this new structure that plays the role of a regulator can not be removed when working in H poly and one is left with the ambiguity that is present in any regularization. The freedom in choosing it can be sometimes associated with a length scale (the lattice spacing). For ordinary quantum systems such as a simple harmonic oscillator, that has been studied in detail from the polymer viewpoint, it has been argued that if this length scale is taken to be 'sufficiently small', one can arbitrarily approximate standard Schrödinger quantum mechanics [2, 3] . In the case of loop quantum cosmology, the minimum area gap A 0 of the full quantum gravity theory imposes such a scale, that is then taken to be fundamental [4] .
A natural question is to ask what happens when we change this scale and go to even smaller 'distances', that is, when we refine the lattice on which the dynamics of the theory is defined. Can we define consistency conditions between these scales? Or even better, can we take the limit and find thus a continuum limit? As it has been shown recently in detail, the answer to both questions is in the affirmative [6] . There, an appropriate notion of scale was defined in such a way that one could define refinements of the theory and pose in a precise fashion the question of the continuum limit of the theory. These results could also be seen as handing a procedure to remove the regulator when working on the appropriate space. The purpose of this paper is to further explore different aspects of the relation between the continuum and the polymer representation. In particular in the first part we put forward a novel way of deriving the polymer representation from the ordinary Schrödinger representation as an appropriate limit. In Sec. II we derive two versions of the polymer representation as different limits of the Schrödinger theory. In Sec. III we show that these two versions can be seen as different polarizations of the 'abstract' polymer representation. These results, to the best of our knowledge, are new and have not been reported elsewhere. In Sec. IV we pose the problem of implementing the dynamics on the polymer representation. In Sec. V we motivate further the question of the continuum limit (i.e. the proper removal of the regulator) and recall the basic constructions of [6] . Several examples are considered in Sec. VI. In particular a simple harmonic oscillator, the polymer free particle and a simple quantum cosmology model are considered. The free particle and the cosmological model represent a generalization of the results obtained in [6] where only systems with a discrete and non-degenerate spectrum where considered. We end the paper with a discussion in Sec. VII. In order to make the paper self-contained, we will keep the level of rigor in the presentation to that found in the standard theoretical physics literature.
In this section we derive the so called polymer representation of quantum mechanics starting from a specific reformulation of the ordinary Schrödinger representation. Our starting point will be the simplest of all possible phase spaces, namely Γ = R 2 corresponding to a particle living on the real line R. Let us choose coordinates (q, p) thereon. As a first step we shall consider the quantization of this system that leads to the standard quantum theory in the Schrödinger description. A convenient route is to introduce the necessary structure to define the Fock representation of such system. From this perspective, the passage to the polymeric case becomes clearest. Roughly speaking by a quantization one means a passage from the classical algebraic bracket, the Poisson bracket,
{q, p} = 1 ( 1
)
to a quantum bracket given by the commutator of the corresponding operators,
[ q, p] = i 1 ( 2
)
These relations, known as the canonical commutation relation (CCR) become the most common corner stone of the (kinematics of the) quantum theory; they should be satisfied by the quantum system, when represented on a Hilbert space H.
There are alternative points of departure for quantum kinematics. Here we consider the algebra generated by the exponentiated versions of q and p that are denoted by, U (α) = e i(α q)/ ; V (β) = e i(β p)/
where α and β have dimensions of momentum and length, respectively. The CCR now become U (α) • V (β) = e (-iα β)/ V (β) • U (α) (3) and the rest of the product is
U (α 1 )•U (α 2 ) = U (α 1 +α 2 ) ; V (β 1 )•V (β 2 ) = V (β 1 +β 2 )
The Weyl algebra W is generated by taking finite linear combinations of the generators U (α i ) and V (β i ) where the product (3) is extended by linearity,
i (A i U (α i ) + B i V (β i ))
From this perspective, quantization means finding an unitary representation of the Weyl algebra W on a Hilbert space H ′ (that could be different from the ordinary Schrödinger representation). At first it might look weird to attempt this approach given that we know how to quantize such a simple system; what do we need such a complicated object as W for? It is infinite dimensional, whereas the set S = { 1, q, p}, the starting point of the ordinary Dirac quantization, is rather simple. It is in the quantization of field systems that the advantages of the Weyl approach can be fully appreciated, but it is also useful for introducing the polymer quantization and comparing it to the standard quantization. This is the strategy that we follow. A question that one can ask is whether there is any freedom in quantizing the system to obtain the ordinary Schrödinger representation. On a first sight it might seem that there is none given the Stone-Von Neumann uniqueness theorem. Let us review what would be the argument for the standard construction. Let us ask that the representation we want to build up is of the Schrödinger type, namely, where states are wave functions of configuration space ψ(q). There are two ingredients to the construction of the representation, namely the specification of how the basic operators (q, p) will act, and the nature of the space of functions that ψ belongs to, that is normally fixed by the choice of inner product on H, or measure µ on R.
The standard choice is to select the Hilbert space to be, H = L 2 (R, dq) 2 the space of square-integrable functions with respect to the Lebesgue measure dq (invariant under constant translations) on R. The operators are then represented as, q • ψ(q) = (q ψ)(q) and p • ψ(q) = -i ∂ ∂q ψ(q) (4) Is it possible to find other representations? In order to appreciate this freedom we go to the Weyl algebra and build the quantum theory thereon. The representation of the Weyl algebra that can be called of the 'Fock type' involves the definition of an extra structure on the phase space Γ: a complex structure J. That is, a linear mapping from Γ to itself such that J 2 = -1. In 2 dimensions, all the freedom in the choice of J is contained in the choice of a parameter d with dimensions of length. It is also convenient to define: k = p/ that has dimensions of 1/L. We have then, J d : (q, k) → (-d 2 k, q/d 2 ) This object together with the symplectic structure: Ω((q, p); (q ′ , p ′ )) = q p ′ -p q ′ define an inner product on Γ by the formula g d (• ; •) = Ω(• ; J d •) such that:
g d ((q, p); (q ′ , p ′ )) = 1 d 2 q q ′ + d 2 2 p p ′
which is dimension-less and positive definite. Note that with this quantities one can define complex coordinates (ζ, ζ) as usual:
ζ = 1 d q + i d p ; ζ = 1 d q -i d p from which one can build the standard Fock representation. Thus, one can alternatively view the introduction of the length parameter d as the quantity needed to define (dimensionless) complex coordinates on the phase space. But what is the relevance of this object (J or d)? The definition of complex coordinates is useful for the construction of the Fock space since from them one can define, in a natural way, creation and annihilation operators. But for the Schrödinger representation we are interested here, it is a bit more subtle. The subtlety is that within this approach one uses the algebraic properties of W to construct the Hilbert space via what is known as the Gel'fand-Naimark-Segal (GNS) construction. This implies that the measure in the Schrödinger representation becomes non trivial and thus the momentum operator acquires an extra term in order to render the operator self-adjoint. The representation of the Weyl algebra is then, when acting on functions φ(q) [7]: Û (α) • φ(q) := (e iα q/ φ)(q) and, V (β) • φ(q) := e β d 2 (q-β/2) φ(q -β)
The Hilbert space structure is introduced by the definition of an algebraic state (a positive linear functional) ω d : W → C, that must coincide with the expectation value in the Hilbert space taken on a special state refered to as the vacuum: ω d (a) = â vac , for all a ∈ W.
In our case this specification of J induces such a unique state ω d that yields, Û (α) vac = e -1 4 d 2 α 2 2 (5) and V (β) vac = e -1 4 β 2 d 2 (6) Note that the exponents in the vacuum expectation values correspond to the metric constructed out of J:
d 2 α 2 2 = g d ((0, α); (0, α)) and β 2 d 2 = g d ((β, 0); (β, 0)). Wave functions belong to the space L 2 (R, dµ d ), where the measure that dictates the inner product in this representation is given by,
dµ d = 1 d √ π e -q 2 d 2 dq
In this representation, the vacuum is given by the identity function φ 0 (q) = 1 that is, just as any plane wave, normalized. Note that for each value of d > 0, the representation is well defined and continuous in α and β. Note also that there is an equivalence between the qrepresentation defined by d and the k-representation defined by 1/d. How can we recover then the standard representation in which the measure is given by the Lebesgue measure and the operators are represented as in (4) ? It is easy to see that there is an isometric isomorphism K that maps the d-representation in H d to the standard Schrödinger representation in H schr by: ψ(q) = K • φ(q) = e -q 2 2 d 2 d 1/2 π 1/4 φ(q) ∈ H schr = L 2 (R, dq) Thus we see that all d-representations are unitarily equivalent. This was to be expected in view of the Stone-Von Neumann uniqueness result. Note also that the vacuum now becomes ψ 0 (q) = 1 d 1/2 π 1/4 e -q 2 2 d 2 , so even when there is no information about the parameter d in the representation itself, it is contained in the vacuum state. This procedure for constructing the GNS-Schrödinger representation for quantum mechanics has also been generalized to scalar fields on arbitrary curved space in [8] . Note, however that so far the treatment has all been kinematical, without any knowledge of a Hamiltonian. For the Simple Harmonic Oscillator of mass m and frequency ω, there is a natural choice compatible with the dynamics given by d = m ω , in which some calculations simplify (for instance for coherent states), but in principle one can use any value of d.
Our study will be simplified by focusing on the fundamental entities in the Hilbert Space H d , namely those states generated by acting with Û(α) on the vacuum φ 0 (q) = 1. Let us denote those states by,
φ α (q) = Û (α) • φ 0 (q) = e i 1 α q
The inner product between two such states is given by
φ α , φ λ d = dµ d e -iαq e iλq = e -(λ-α) 2 d 2 4 2 ( 7
)
Note incidentally that, contrary to some common belief, the 'plane waves' in this GNS Hilbert space are indeed normalizable.
Let us now consider the polymer representation. For that, it is important to note that there are two possible limiting cases for the parameter d: i) The limit 1/d → 0 and ii) The case d → 0. In both cases, we have expressions that become ill defined in the representation or measure, so one needs to be careful.
A. The 1/d → 0 case.
The first observation is that from the expressions (5) and (6) for the algebraic state ω d , we see that the limiting cases are indeed well defined. In our case we get, ω A := lim 1/d→0 ω d such that,
ω A ( Û (α)) = δ α,0 and ω A ( V (β)) = 1 ( 8
)
From this, we can indeed construct the representation by means of the GNS construction. In order to do that and to show how this is obtained we shall consider several expressions. One has to be careful though, since the limit has to be taken with care. Let us consider the measure on the representation that behaves as:
dµ d = 1 d √ π e -q 2 d 2 dq → 1 d √ π dq
so the measures tends to an homogeneous measure but whose 'normalization constant' goes to zero, so the limit becomes somewhat subtle. We shall return to this point later. Let us now see what happens to the inner product between the fundamental entities in the Hilbert Space H d given by (7) . It is immediate to see that in the 1/d → 0 limit the inner product becomes,
φ α , φ λ d → δ α,λ ( 9
)
with δ α,λ being Kronecker's delta. We see then that the plane waves φ α (q) become an orthonormal basis for the new Hilbert space. Therefore, there is a delicate interplay between the two terms that contribute to the measure in order to maintain the normalizability of these functions; we need the measure to become damped (by 1/d) in order to avoid that the plane waves acquire an infinite norm (as happens with the standard Lebesgue measure), but on the other hand the measure, that for any finite value of d is a Gaussian, becomes more and more spread. It is important to note that, in this limit, the operators Û (α) become discontinuous with respect to α, given that for any given α 1 and α 2 (different), its action on a given basis vector ψ λ (q) yields orthogonal vectors. Since the continuity of these operators is one of the hypotesis of the Stone-Von Neumann theorem, the uniqueness result does not apply here. The representation is inequivalent to the standard one.
Let us now analyze the other operator, namely the action of the operator V (β) on the basis φ α (q): V (β) • φ α (q) = e -β 2 2d 2 -i αβ e (β/d 2 +iα/ )q which in the limit 1/d → 0 goes to,
V (β) • φ α (q) → e i αβ φ α (q)
that is continuous on β. Thus, in the limit, the operator p = -i ∂ q is well defined. Also, note that in this limit the operator p has φ α (q) as its eigenstate with eigenvalue given by α:
p • φ α (q) → α φ α (q)
To summarize, the resulting theory obtained by taking the limit 1/d → 0 of the ordinary Schrödinger description, that we shall call the 'polymer representation of type A', has the following features: the operators U (α) are well defined but not continuous in α, so there is no generator (no operator associated to q). The basis vectors φ α are orthonormal (for α taking values on a continuous set) and are eigenvectors of the operator p that is well defined. The resulting Hilbert space H A will be the (A-version of the) polymer representation. Let us now consider the other case, namely, the limit when d → 0.
B. The d → 0 case
Let us now explore the other limiting case of the Schrödinger/Fock representations labelled by the parameter d. Just as in the previous case, the limiting algebraic state becomes, ω B := lim d→0 ω d such that,
ω B ( Û (α)) = 1 and ω B ( V (β)) = δ β,0 ( 10
)
From this positive linear function, one can indeed construct the representation using the GNS construction. First let us note that the measure, even when the limit has to be taken with due care, behaves as:
dµ d = 1 d √ π e -q 2 d 2 dq → δ(q) dq
That is, as Dirac's delta distribution. It is immediate to see that, in the d → 0 limit, the inner product between the fundamental states φ α (q) becomes,
φ α , φ λ d → 1 ( 11
) 4
This in fact means that the vector ξ = φ α -φ λ belongs to the Kernel of the limiting inner product, so one has to mod out by these (and all) zero norm states in order to get the Hilbert space.
Let us now analyze the other operator, namely the action of the operator V (β) on the vacuum φ 0 (q) = 1, which for arbitrary d has the form,
φβ := V (β) • φ 0 (q) = e β d 2 (q-β/2)
The inner product between two such states is given by φα , φβ d = e -1 4d 2 (α-β) 2
In the limit d → 0, φα , φβ d → δ α,β . We can see then that it is these functions that become the orthonormal, 'discrete basis' in the theory. However, the function φβ (q) in this limit becomes ill defined. For example, for β > 0, it grows unboundedly for q > β/2, is equal to one if q = β/2 and zero otherwise. In order to overcome these difficulties and make more transparent the resulting theory, we shall consider the other form of the representation in which the measure is incorporated into the states (and the resulting Hilbert space is L 2 (R, dq)). Thus the new state
ψ β (q) := K • ( V (β) • φ 0 (q)) = = 1 (d √ π) 1 2 e -1 2d 2 (q-β) 2 ( 12
)
We can now take the limit and what we get is lim d →0 ψ β (q) := δ 1/2 (q, β)
where by δ 1/2 (q, β) we mean something like 'the square root of the Dirac distribution'. What we really mean is an object that satisfies the following property:
δ 1/2 (q, β) • δ 1/2 (q, α) = δ(q, β) δ β,α
That is, if α = β then it is just the ordinary delta, otherwise it is zero. In a sense these object can be regarded as half-densities that can not be integrated by themselves, but whose product can. We conclude then that the inner product is,
ψ β , ψ α = R dq ψ β (q) ψ α (q) = R dq δ(q, α) δ β,α = δ β,α
(13) which is just what we expected. Note that in this representation, the vacuum state becomes ψ 0 (q) := δ 1/2 (q, 0), namely, the half-delta with support in the origin. It is important to note that we are arriving in a natural way to states as half-densities, whose squares can be integrated without the need of a nontrivial measure on the configuration space. Diffeomorphism invariance arises then in a natural but subtle manner.
Note that as the end result we recover the Kronecker delta inner product for the new fundamental states:
χ β (q) := δ 1/2 (q, β).
Thus, in this new B-polymer representation, the Hilbert space H B is the completion with respect to the inner product (13) of the states generated by taking (finite) linear combinations of basis elements of the form χ β :
Ψ(q) = i b i χ βi (q) ( 14
)
Let us now introduce an equivalent description of this Hilbert space. Instead of having the basis elements be half-deltas as elements of the Hilbert space where the inner product is given by the ordinary Lebesgue measure dq, we redefine both the basis and the measure. We could consider, instead of a half-delta with support β, a Kronecker delta or characteristic function with support on β: χ ′ β (q) := δ q,β These functions have a similar behavior with respect to the product as the half-deltas, namely:
χ ′ β (q) • χ ′ α (q) = δ β,α .
The main difference is that neither χ ′ nor their squares are integrable with respect to the Lebesgue measure (having zero norm). In order to fix that problem we have to change the measure so that we recover the basic inner product (13) with our new basis. The needed measure turns out to be the discrete counting measure on R. Thus any state in the 'half density basis' can be written (using the same expression) in terms of the 'Kronecker basis'. For more details and further motivation see the next section.
Note that in this B-polymer representation, both Û and V have their roles interchanged with that of the A-polymer representation: while U (α) is discontinuous and thus q is not defined in the A-representation, we have that it is V (β) in the B-representation that has this property. In this case, it is the operator p that can not be defined. We see then that given a physical system for which the configuration space has a well defined physical meaning, within the possible representation in which wave-functions are functions of the configuration variable q, the A and B polymer representations are radically different and inequivalent.
Having said this, it is also true that the A and B representations are equivalent in a different sense, by means of the duality between q and p representations and the d ↔ 1/d duality: The A-polymer representation in the "q-representation" is equivalent to the B-polymer representation in the "p-representation", and conversely. When studying a problem, it is important to decide from the beginning which polymer representation (if any) one should be using (for instance in the q-polarization). This has as a consequence an implication on which variable is naturally "quantized" (even if continuous): p for A and q for B. There could be for instance a physical criteria for this choice. For example a fundamental symmetry could suggest that one representation is more natural than another one. This indeed has been recently noted by Chiou in [10] , where the Galileo group is investigated and where it is shown that the B representation is better behaved.
In the other polarization, namely for wavefunctions of p, the picture gets reversed: q is discrete for the Arepresentation, while p is for the B-case. Let us end this section by noting that the procedure of obtaining the polymer quantization by means of an appropriate limit of Fock-Schrödinger representations might prove useful in more general settings in field theory or quantum gravity.
In previous sections we have derived what we have called the A and B polymer representations (in the qpolarization) as limiting cases of ordinary Fock representations. In this section, we shall describe, without any reference to the Schrödinger representation, the 'abstract' polymer representation and then make contact with its two possible realizations, closely related to the A and B cases studied before. What we will see is that one of them (the A case) will correspond to the p-polarization while the other one corresponds to the q-representation, when a choice is made about the physical significance of the variables.
We can start by defining abstract kets |µ labelled by a real number µ. These shall belong to the Hilbert space H poly . From these states, we define a generic 'cylinder states' that correspond to a choice of a finite collection of numbers µ i ∈ R with i = 1, 2, . . . , N . Associated to this choice, there are N vectors |µ i , so we can take a linear combination of them
|ψ = N i=1 a i |µ i ( 15
)
The polymer inner product between the fundamental kets is given by,
ν|µ = δ ν,µ ( 16
)
That is, the kets are orthogonal to each other (when ν = µ) and they are normalized ( µ|µ = 1). Immediately, this implies that, given any two vectors |φ =
M j=1 b j |ν j and |ψ = N i=1 a i |µ i , the inner product between them is given by, φ|ψ = i j bj a i ν j |µ i = k bk a k
where the sum is over k that labels the intersection points between the set of labels {ν j } and {µ i }. The Hilbert space H poly is the Cauchy completion of finite linear combination of the form (15) with respect to the inner product (16) . H poly is non-separable. There are two basic operators on this Hilbert space: the 'label operator' ε:
ε |µ := µ |µ
and the displacement operator ŝ (λ), ŝ (λ
) |µ := |µ + λ
The operator ε is symmetric and the operator(s) ŝ(λ) defines a one-parameter family of unitary operators on H poly , where its adjoint is given by ŝ † (λ) = ŝ (-λ). This action is however, discontinuous with respect to λ given that |µ and |µ + λ are always orthogonal, no matter how small is λ. Thus, there is no (Hermitian) operator that could generate ŝ (λ) by exponentiation. So far we have given the abstract characterization of the Hilbert space, but one would like to make contact with concrete realizations as wave functions, or by identifying the abstract operators ε and ŝ with physical operators.
Suppose we have a system with a configuration space with coordinate given by q, and p denotes its canonical conjugate momenta. Suppose also that for physical reasons we decide that the configuration coordinate q will have some "discrete character" (for instance, if it is to be identified with position, one could say that there is an underlying discreteness in position at a small scale). How can we implement such requirements by means of the polymer representation? There are two possibilities, depending on the choice of 'polarizations' for the wavefunctions, namely whether they will be functions of configuration q or momenta p. Let us the divide the discussion into two parts.
In this polarization, states will be denoted by,
ψ(p) = p|ψ where ψ µ (p) = p|µ = e i µp
How are then the operators ε and ŝ represented? Note that if we associate the multiplicative operator
V (λ) • ψ µ (p) = e i λ p e i µ p = e i (µ+λ) p = ψ (µ+λ) (p)
we see then that the operator V (λ) corresponds precisely to the shift operator ŝ (λ). Thus we can also conclude that the operator p does not exist. It is now easy to identify the operator q with:
q • ψ µ (p) = -i ∂ ∂p ψ µ (p) = µ e i µ p = µ ψ µ (p)
namely, with the abstract operator ε. The reason we say that q is discrete is because this operator has as its eigenvalue the label µ of the elementary state ψ µ (p), and this label, even when it can take value in a continuum of possible values, is to be understood as a discrete set, given that the states are orthonormal for all values of µ. Given that states are now functions of p, the inner product (16) should be defined by a measure µ on the space on which the wave-functions are defined. In order 6 to know what these two objects are, namely, the quantum "configuration" space C and the measure thereon 1 , we have to make use of the tools available to us from the theory of C * -algebras. If we consider the operators V (λ), together with their natural product and * -relation given by V * (λ) = V (-λ), they have the structure of an Abelian C * -algebra (with unit) A. We know from the representation theory of such objects that A is isomorphic to the space of continuous functions C 0 (∆) on a compact space ∆, the spectrum of A. Any representation of A on a Hilbert space as multiplication operator will be on spaces of the form L 2 (∆, dµ). That is, our quantum configuration space is the spectrum of the algebra, which in our case corresponds to the Bohr compactification R b of the real line [11] . This space is a compact group and there is a natural probability measure defined on it, the Haar measure µ H . Thus, our Hilbert space H poly will be isomorphic to the space,
H poly,p = L 2 (R b , dµ H ) ( 17
)
In terms of 'quasi periodic functions' generated by ψ µ (p), the inner product takes the form
ψ µ |ψ λ := R b dµ H ψ µ (p) ψ λ (p) := = lim L →∞ 1 2L L -L dp ψ µ (p) ψ λ (p) = δ µ,λ ( 18
)
note that in the p-polarization, this characterization corresponds to the 'A-version' of the polymer representation of Sec. II (where p and q are interchanged).
Let us now consider the other polarization in which wave functions will depend on the configuration coordinate q:
ψ(q) = q|ψ
The basic functions, that now will be called ψµ (q), should be, in a sense, the dual of the functions ψ µ (p) of the previous subsection. We can try to define them via a 'Fourier transform':
ψµ (q) := q|µ = q| R b dµ H |p p|µ which is given by ψµ (q) := R b dµ H q|p ψ µ (p) = = R b dµ H e -i p q e i µ p = δ q,µ ( 19
)
1 here we use the standard terminology of 'configuration space' to denote the domain of the wave function even when, in this case, it corresponds to the physical momenta p.
That is, the basic objects in this representation are Kronecker deltas. This is precisely what we had found in Sec. II for the B-type representation. How are now the basic operators represented and what is the form of the inner product? Regarding the operators, we expect that they are represented in the opposite manner as in the previous p-polarization case, but that they preserve the same features: p does not exist (the derivative of the Kronecker delta is ill defined), but its exponentiated version V (λ) does:
V (λ) • ψ(q) = ψ(q + λ)
and the operator q that now acts as multiplication has as its eigenstates, the functions ψν (q) = δ ν,q : q • ψµ (q) := µ ψµ (q) What is now the nature of the quantum configurations space Q? And what is the measure thereon dµ q ? that defines the inner product we should have: ψµ (q), ψλ (q) = δ µ,λ
The answer comes from one of the characterizations of the Bohr compactification: we know that it is, in a precise sense, dual to the real line but when equipped with the discrete topology R d . Furthermore, the measure on R d will be the 'counting measure'. In this way we recover the same properties we had for the previous characterization of the polymer Hilbert space. We can thus write:
H poly,x := L 2 (R d , dµ c ) ( 20
)
This completes a precise construction of the B-type polymer representation sketched in the previous section. Note that if we had chosen the opposite physical situation, namely that q, the configuration observable, be the quantity that does not have a corresponding operator, then we would have had the opposite realization: In the qpolarization we would have had the type-A polymer representation and the type-B for the p-polarization. As we shall see both scenarios have been considered in the literature. Up to now we have only focused our discussion on the kinematical aspects of the quantization process. Let us now consider in the following section the issue of dynamics and recall the approach that had been adopted in the literature, before the issue of the removal of the regulator was reexamined in [6] .
As we have seen the construction of the polymer representation is rather natural and leads to a quantum theory with different properties than the usual Schrödinger counterpart such as its non-separability, the 7 non-existence of certain operators and the existence of normalized eigen-vectors that yield a precise value for one of the phase space coordinates. This has been done without any regard for a Hamiltonian that endows the system with a dynamics, energy and so on.
First let us consider the simplest case of a particle of mass m in a potential V (q), in which the Hamiltonian H takes the form,
H = 1 2m p 2 + V (q)
Suppose furthermore that the potential is given by a nonperiodic function, such as a polynomial or a rational function. We can immediately see that a direct implementation of the Hamiltonian is out of our reach, for the simple reason that, as we have seen, in the polymer representation we can either represent q or p, but not both! What has been done so far in the literature? The simplest thing possible: approximate the non-existing term by a well defined function that can be quantized and hope for the best. As we shall see in next sections, there is indeed more that one can do. At this point there is also an important decision to be made: which variable q or p should be regarded as "discrete"? Once this choice is made, then it implies that the other variable will not exist: if q is regarded as discrete, then p will not exist and we need to approximate the kinetic term p 2 /2m by something else; if p is to be the discrete quantity, then q will not be defined and then we need to approximate the potential V (q). What happens with a periodic potential? In this case one would be modelling, for instance, a particle on a regular lattice such as a phonon living on a crystal, and then the natural choice is to have q not well defined. Furthermore, the potential will be well defined and there is no approximation needed.
In the literature both scenarios have been considered. For instance, when considering a quantum mechanical system in [2] , the position was chosen to be discrete, so p does not exist, and one is then in the A type for the momentum polarization (or the type B for the qpolarization). With this choice, it is the kinetic term the one that has to be approximated, so once one has done this, then it is immediate to consider any potential that will thus be well defined. On the other hand, when considering loop quantum cosmology (LQC), the standard choice is that the configuration variable is not defined [4] . This choice is made given that LQC is regarded as the symmetric sector of full loop quantum gravity where the connection (that is regarded as the configuration variable) can not be promoted to an operator and one can only define its exponentiated version, namely, the holonomy. In that case, the canonically conjugate variable, closely related to the volume, becomes 'discrete', just as in the full theory. This case is however, different from the particle in a potential example. First we could mention that the functional form of the Hamiltonian constraint that implements dynamics has a different structure, but the more important difference lies in that the system is constrained.
Let us return to the case of the particle in a potential and for definiteness, let us start with the auxiliary kinematical framework in which: q is discrete, p can not be promoted and thus we have to approximate the kinetic term p2 /2m. How is this done? The standard prescription is to define, on the configuration space C, a regular 'graph' γ µ0 . This consists of a numerable set of points, equidistant, and characterized by a parameter µ 0 that is the (constant) separation between points. The simplest example would be to consider the set
γ µ0 = {q ∈ R | q = n µ 0 , ∀ n ∈ Z}.
This means that the basic kets that will be considered |µ n will correspond precisely to labels µ n belonging to the graph γ µ0 , that is, µ n = n µ 0 . Thus, we shall only consider states of the form,
|ψ = n b n |µ n . ( 21
)
This 'small' Hilbert space H γµ 0 , the graph Hilbert space, is a subspace of the 'large' polymer Hilbert space H poly but it is separable. The condition for a state of the form (21) to belong to the Hilbert space H γµ 0 is that the coefficients b n satisfy:
n |b n | 2 < ∞.
Let us now consider the kinetic term p2 /2m. We have to approximate it by means of trigonometric functions, that can be built out of the functions of the form e iλ p/ . As we have seen in previous sections, these functions can indeed be promoted to operators and act as translation operators on the kets |µ . If we want to remain in the graph γ, and not create 'new points', then one is constrained to considering operators that displace the kets by just the right amount. That is, we want the basic shift operator V (λ) to be such that it maps the ket with label |µ n to the next ket, namely |µ n+1 . This can indeed achieved by fixing, once and for all, the value of the allowed parameter λ to be λ = µ 0 . We have then,
V (µ 0 ) • |µ n = |µ n + µ 0 = |µ n+1
which is what we wanted. This basic 'shift operator' will be the building block for approximating any (polynomial) function of p. In order to do that we notice that the function p can be approximated by,
p ≈ µ 0 sin µ 0 p = 2iµ 0 e i µ 0 p -e -i µ 0 p
where the approximation is good for p << /µ 0 . Thus, one can define a regulated operator pµ0 that depends on the 'scale' µ 0 as:
pµ0 • |µ n := 2iµ 0 [V (µ 0 ) -V (-µ 0 )] • |µ n = = i 2µ 0 (|µ n+1 -|µ n-1 ) ( 22
)
In order to regulate the operator p2 , there are (at least) two possibilities, namely to compose the operator pµ0 8 with itself or to define a new approximation. The operator pµ0 • pµ0 has the feature that shifts the states two steps in the graph to both sides. There is however another operator that only involves shifting once:
p2 µ0 • |ν n := 2 µ 2 0 [2 -V (µ 0 ) -V (-µ 0 )] • |ν n = = 2 µ 2 0 (2|ν n -|ν n+1 -|ν n-1 ) ( 23
)
which corresponds to the approximation p 2 ≈ 2 2 µ 2 0 (1cos(µ 0 p/ )), valid also in the regime p << /µ 0 . With these considerations, one can define the operator Ĥµ0 , the Hamiltonian at scale µ 0 , that in practice 'lives' on the space H γµ 0 as,
Ĥµ0 := 1 2m p2 µ0 + V (q) , ( 24
)
that is a well defined, symmetric operator on H γµ 0 . Notice that the operator is also defined on H poly , but there its physical interpretation is problematic. For example, it turns out that the expectation value of the kinetic term calculated on most states (states which are not tailored to the exact value of the parameter µ 0 ) is zero. Even if one takes a state that gives "reasonable" expectation values of the µ 0 -kinetic term and uses it to calculate the expectation value of the kinetic term corresponding to a slight perturbation of the parameter µ 0 one would get zero. This problem, and others that arise when working on H poly , forces one to assign a physical interpretation to the Hamiltonian Ĥµ0 only when its action is restricted to the subspace H γµ 0 . Let us now explore the form that the Hamiltonian takes in the two possible polarizations. In the q-polarization, the basis, labelled by n is given by the functions χ n (q) = δ q,µn . That is, the wave functions will only have support on the set γ µ0 . Alternatively, one can think of a state as completely characterized by the 'Fourier coefficients' a n : ψ(q) ↔ a n , which is the value that the wave function ψ(q) takes at the point q = µ n = n µ 0 . Thus, the Hamiltonian takes the form of a difference equation when acting on a general state ψ(q). Solving the time independent Schrödinger equation Ĥ • ψ = E ψ amounts to solving the difference equation for the coefficients a n .
The momentum polarization has a different structure. In this case, the operator p2 µ0 acts as a multiplication operator,
p2 µ0 • ψ(p) = 2 2 µ 2 0 1 -cos µ 0 p ψ(p) ( 25
)
The operator corresponding to q will be represented as a derivative operator q • ψ(p) := i ∂ p ψ(p).
For a generic potential V (q), it has to be defined by means of spectral theory defined now on a circle. Why on a circle? For the simple reason that by restricting ourselves to a regular graph γ µ0 , the functions of p that preserve it (when acting as shift operators) are of the form e (i m µ0 p/ ) for m integer. That is, what we have are Fourier modes, labelled by m, of period 2π /µ 0 in p.
Can we pretend then that the phase space variable p is now compactified? The answer is in the affirmative. The inner product on periodic functions ψ µ0 (p) of p coming from the full Hilbert space H poly and given by
φ(p)|ψ(p) poly = lim L →∞ 1 2L L -L dp φ(p) ψ(p)
is precisely equivalent to the inner product on the circle given by the uniform measure φ(p)|ψ(p) µ0 = µ 0 2π π /µ0 -π /µ0 dp φ(p) ψ(p) with p ∈ (-π /µ 0 , π /µ 0 ). As long as one restricts attention to the graph γ µ0 , one can work in this separable Hilbert space H γµ 0 of square integrable functions on S 1 . Immediately, one can see the limitations of this description. If the mechanical system to be quantized is such that its orbits have values of the momenta p that are not small compared with π /µ 0 then the approximation taken will be very poor, and we don't expect neither the effective classical description nor its quantization to be close to the standard one. If, on the other hand, one is always within the region in which the approximation can be regarded as reliable, then both classical and quantum descriptions should approximate the standard description. What does 'close to the standard description' exactly mean needs, of course, some further clarification. In particular one is assuming the existence of the usual Schrödinger representation in which the system has a behavior that is also consistent with observations. If this is the case, the natural question is: How can we approximate such description from the polymer picture? Is there a fine enough graph γ µ0 that will approximate the system in such a way that all observations are indistinguishable? Or even better, can we define a procedure, that involves a refinement of the graph γ µ0 such that one recovers the standard picture? It could also happen that a continuum limit can be defined but does not coincide with the 'expected one'. But there might be also physical systems for which there is no standard description, or it just does not make sense. Can in those cases the polymer representation, if it exists, provide the correct physical description of the system under consideration? For instance, if there exists a physical limitation to the minimum scale set by µ 0 , as could be the case for a quantum theory of gravity, then the polymer description would provide a true physical bound on the value of certain quantities, such as p in our example. This could be the case for loop quantum cosmology, where there is a minimum value for physical volume (coming from the full theory), and phase space points near the 'singularity' lie at the region where the 9 approximation induced by the scale µ 0 departs from the standard classical description. If in that case the polymer quantum system is regarded as more fundamental than the classical system (or its standard Wheeler-De Witt quantization), then one would interpret this discrepancies in the behavior as a signal of the breakdown of classical description (or its 'naive' quantization).
In the next section we present a method to remove the regulator µ 0 which was introduced as an intermediate step to construct the dynamics. More precisely, we shall consider the construction of a continuum limit of the polymer description by means of a renormalization procedure.
This section has two parts. In the first one we motivate the need for a precise notion of the continuum limit of the polymeric representation, explaining why the most direct, and naive approach does not work. In the second part, we shall present the main ideas and results of the paper [6], where the Hamiltonian and the physical Hilbert space in polymer quantum mechanics are constructed as a continuum limit of effective theories, following Wilson's renormalization group ideas. The resulting physical Hilbert space turns out to be unitarily isomorphic to the ordinary H s = L 2 (R, dq) of the Schrödinger theory.
Before describing the results of [6] we should discuss the precise meaning of reaching a theory in the continuum. Let us for concreteness consider the B-type representation in the q-polarization. That is, states are functions of q and the orthonormal basis χ µ (q) is given by characteristic functions with support on q = µ. Let us now suppose we have a Schrödinger state Ψ(q) ∈ H s = L 2 (R, dq). What is the relation between Ψ(q) and a state in H poly,x ? We are also interested in the opposite question, that is, we would like to know if there is a preferred state in H s that is approximated by an arbitrary state ψ(q) in H poly,x . The first obvious observation is that a Schödinger state Ψ(q) does not belong to H poly,x since it would have an infinite norm. To see that note that even when the would-be state can be formally expanded in the χ µ basis as,
Ψ(q) = µ Ψ(µ) χ µ (q)
where the sum is over the parameter µ ∈ R. Its associated norm in H poly,x would be:
|Ψ(q)| 2 poly = µ |Ψ(µ)| 2 → ∞
which blows up. Note that in order to define a mapping P : H s → H poly,x , there is a huge ambiguity since the values of the function Ψ(q) are needed in order to expand the polymer wave function. Thus we can only define a mapping in a dense subset D of H s where the values of the functions are well defined (recall that in H s the value of functions at a given point has no meaning since states are equivalence classes of functions). We could for instance ask that the mapping be defined for representatives of the equivalence classes in H s that are piecewise continuous. From now on, when we refer to an element of the space H s we shall be refereeing to one of those representatives. Notice then that an element of H s does define an element of Cyl * γ , the dual to the space Cyl γ , that is, the space of cylinder functions with support on the (finite) lattice γ = {µ 1 , µ 2 , . . . , µ N }, in the following way:
Ψ(q) : Cyl γ -→ C such that Ψ(q)[ψ(q)] = (Ψ|ψ := µ Ψ(µ) χ µ | N i=1 ψ i χ µi poly γ = N i=1 Ψ(µ i ) ψ i < ∞ ( 26
)
Note that this mapping could be seen as consisting of two parts: First, a projection
P γ : Cyl * → Cyl γ such that P γ (Ψ) = Ψ γ (q) := i Ψ(µ i ) χ µi (q) ∈ Cyl γ .
The state Ψ γ is sometimes refereed to as the 'shadow of Ψ(q) on the lattice γ'. The second step is then to take the inner product between the shadow Ψ γ (q) and the state ψ(q) with respect to the polymer inner product Ψ γ |ψ poly γ . Now this inner product is well defined. Notice that for any given lattice γ the corresponding projector P γ can be intuitively interpreted as some kind of 'coarse graining map' from the continuum to the lattice γ. In terms of functions of q the projection is replacing a continuous function defined on R with a function over the lattice γ ⊂ R which is a discrete set simply by restricting Ψ to γ. The finer the lattice the more points that we have on the curve. As we shall see in the second part of this section, there is indeed a precise notion of coarse graining that implements this intuitive idea in a concrete fashion.
In particular, we shall need to replace the lattice γ with a decomposition of the real line in intervals (having the lattice points as end points). Let us now consider a system in the polymer representation in which a particular lattice γ 0 was chosen, say with points of the form
{q k ∈ R |q k = ka 0 , ∀ k ∈ Z},
namely a uniform lattice with spacing equal to a 0 . In this case, any Schrödinger wave function (of the type that we consider) will have a unique shadow on the lattice γ 0 . If we refine the lattice γ → γ n by dividing each interval in 2 n new intervals of length a n = a 0 /2 n we have new shadows that have more and more points on the curve. Intuitively, by refining infinitely the graph we would recover the original function Ψ(q). Even when at each finite step the corresponding shadow has a finite norm in the polymer Hilbert space, the norm grows unboundedly and the 10 limit can not be taken, precisely because we can not embed H s into H poly . Suppose now that we are interested in the reverse process, namely starting from a polymer theory on a lattice and asking for the 'continuum wave function' that is best approximated by a wave function over a graph. Suppose furthermore that we want to consider the limit of the graph becoming finer. In order to give precise answers to these (and other) questions we need to introduce some new technology that will allow us to overcome these apparent difficulties. In the remaining of this section we shall recall these constructions for the benefit of the reader. Details can be found in [6] (which is an application of the general formalism discussed in [9] ).
The starting point in this construction is the concept of a scale C, which allows us to define the effective theories and the concept of continuum limit. In our case a scale is a decomposition of the real line in the union of closed-open intervals, that cover the whole line and do not intersect. Intuitively, we are shifting the emphasis from the lattice points to the intervals defined by the same points with the objective of approximating continuous functions defined on R with functions that are constant on the intervals defined by the lattice. To be precise, we define an embedding, for each scale
C n from H poly to H s by means of a step function: m Ψ(ma n ) χ man (q) → m Ψ(ma n ) χ αm (q) ∈ H s with χ αn (q) a characteristic function on the interval α m = [ma n , (m + 1)a n ).
Thus, the shadows (living on the lattice) were just an intermediate step in the construction of the approximating function; this function is piece-wise constant and can be written as a linear combination of step functions with the coefficients provided by the shadows.
The challenge now is to define in an appropriate sense how one can approximate all the aspects of the theory by means of this constant by pieces functions. Then the strategy is that, for any given scale, one can define an effective theory by approximating the kinetic operator by a combination of the translation operators that shift between the vertices of the given decomposition, in other words by a periodic function in p. As a result one has a set of effective theories at given scales which are mutually related by coarse graining maps. This framework was developed in [6] . For the convenience of the reader we briefly recall part of that framework.
Let us denote the kinematic polymer Hilbert space at the scale C n as H Cn , and its basis elements as e αi,Cn , where α i = [ia n , (i + 1)a n ) ∈ C n . By construction this basis is orthonormal. The basis elements in the dual Hilbert space H * Cn are denoted by ω αi,Cn ; they are also orthonormal. The states ω αi,Cn have a simple action on Cyl, ω αi,Cn (δ x0,q ) = χ αi,Cn (x 0 ). That is, if x 0 is in the interval α i of C n the result is one and it is zero if it is not there.
Given any m ≤ n, we define d * m,n : H * Cn → H * Cm as the 'coarse graining' map between the dual Hilbert spaces, that sends the part of the elements of the dual basis to zero while keeping the information of the rest: d * m,n (ω αi,Cn ) = ω βj ,Cm if i = j2 n-m , in the opposite case d * m,n (ω αi,Cn ) = 0. At every scale the corresponding effective theory is given by the hamiltonian H n . These Hamiltonians will be treated as quadratic forms,
h n : H Cn → R, given by h n (ψ) = λ 2 Cn (ψ, H n ψ) , ( 27
)
where λ 2 Cn is a normalizaton factor. We will see later that this rescaling of the inner product is necessary in order to guarantee the convergence of the renormalized theory. The completely renormalized theory at this scale is obtained as
h ren m := lim Cn→R d ⋆ m,n h n . ( 28
)
and the renormalized Hamiltonians are compatible with each other, in the sense that
d ⋆ m,n h ren n = h ren m .
In order to analyze the conditions for the convergence in (28) let us express the Hamiltonian in terms of its eigen-covectors end eigenvalues. We will work with effective Hamiltonians that have a purely discrete spectrum
(labelled by ν) H n • Ψ ν,Cn = E ν,Cn Ψ ν,Cn .
We shall also introduce, as an intermediate step, a cut-off in the energy levels. The origin of this cut-off is in the approximation of the Hamiltonian of our system at a given scale with a Hamiltonian of a periodic system in a regime of small energies, as we explained earlier. Thus, we can write
h ν cut-off m = ν cut-off ν=0 E ν,Cm Ψ ν,Cm ⊗ Ψ ν,Cm , ( 29
)
where the eigen covectors Ψ ν,Cm are normalized according to the inner product rescaled by 1 λ 2 Cn , and the cutoff can vary up to a scale dependent bound, ν cut-off ≤ ν max (C m ). The Hilbert space of covectors together with such inner product will be called H ⋆ren Cm . In the presence of a cut-off, the convergence of the microscopically corrected Hamiltonians, equation (28) is equivalent to the existence of the following two limits. The first one is the convergence of the energy levels,
lim Cn→R E ν,Cn = E ren ν . ( 30
)
Second is the existence of the completely renormalized eigen covectors,
lim Cn→R d ⋆ m,n Ψ ν,Cn = Ψ ren ν,Cm ∈ H ⋆ren Cm ⊂ Cyl ⋆ . ( 31
)
We clarify that the existence of the above limit means that Ψ ren ν,Cm (δ x0,q ) is well defined for any δ x0,q ∈ Cyl. Notice that this point-wise convergence, if it can take place 11 at all, will require the tuning of the normalization factors λ 2 Cn . Now we turn to the question of the continuum limit of the renormalized covectors. First we can ask for the existence of the limit lim Cn→R
Ψ ren ν,Cn (δ x0,q ) ( 32
)
for any δ x0,q ∈ Cyl. When this limits exists there is a natural action of the eigen covectors in the continuum limit. Below we consider another notion of the continuum limit of the renormalized eigen covectors. When the completely renormalized eigen covectors exist, they form a collection that is d ⋆ -compatible, d ⋆ m,n Ψ ren ν,Cn = Ψ ren ν,Cm . A sequence of d ⋆ -compatible normalizable covectors define an element of ←-H ⋆ren R , which is the projective limit of the renormalized spaces of covectors ←-
H ⋆ren R := ←- lim Cn→R H ⋆ren Cn . ( 33
)
The inner product in this space is defined by
({Ψ Cn }, {Φ Cn }) ren R := lim Cn→R (Ψ Cn , Φ Cn ) ren Cn .
The natural inclusion of
C ∞ 0 in ←- H ⋆ren R is by an antilinear map which assigns to any Ψ ∈ C ∞ 0 the d ⋆ -compatible collection Ψ shad Cn := αi ω αi Ψ(L(α i )) ∈ H ⋆ren Cn ⊂ Cyl ⋆ ; Ψ shad
Cn will be called the shadow of Ψ at scale C n and acts in Cyl as a piecewise constant function. Clearly other types of test functions like Schwartz functions are also naturally included in ←-H ⋆ren R . In this context a shadow is a state of the effective theory that approximates a state in the continuum theory.
Since the inner product in ←-H ⋆ren R is degenerate, the physical Hilbert space is defined as
H ⋆ phys := ←- H ⋆ren R / ker(•, •) ren R H phys := H ⋆⋆ phys
The nature of the physical Hilbert space, whether it is isomorphic to the Schrödinger Hilber space, H s , or not, is determined by the normalization factors λ 2 Cn which can be obtained from the conditions asking for compatibility of the dynamics of the effective theories at different scales. The dynamics of the system under consideration selects the continuum limit.
Let us now return to the definition of the Hamiltonian in the continuum limit. First consider the continuum limit of the Hamiltonian (with cut-off) in the sense of its point-wise convergence as a quadratic form. It turns out that if the limit of equation (32) exists for all the eigencovectors allowed by the cut-off, we have h
ν cut-off ren R : H poly,x → R defined by h ν cut-off ren R (δ x0,q ) := lim Cn→R h ν cut-off ren n ([δ x0,q ] Cn ). (34)
This Hamiltonian quadratic form in the continuum can be coarse grained to any scale and, as can be expected, it yields the completely renormalized Hamiltonian quadratic forms at that scale. However, this is not a completely satisfactory continuum limit because we can not remove the auxiliary cut-off ν cut-off . If we tried, as we include more and more eigencovectors in the Hamiltonian the calculations done at a given scale would diverge and doing them in the continuum is just as divergent. Below we explore a more successful path.
We can use the renormalized inner product to induce an action of the cut-off Hamiltonians on ←-
H ⋆ren R h ν cut-off ren R ({Ψ Cn }) := lim Cn→R h ν cut-off ren n ((Ψ Cn , •) ren Cn ),
where we have used the fact that (Ψ Cn , •) ren Cn ∈ H Cn . The existence of this limit is trivial because the renormalized Hamiltonians are finite sums and the limit exists term by term.
These cut-off Hamiltonians descend to the physical Hilbert space
h ν cut-off ren R ([{Ψ Cn }]) := h ν cut-off ren R ({Ψ Cn }) for any representative {Ψ Cn } ∈ [{Ψ Cn }] ∈ H ⋆ phys .
Finally we can address the issue of removal of the cutoff. The Hamiltonian h ren R : ←-H ⋆ren R → R is defined by the limit h ren R := lim ν cut-off →∞ h ν cut-off ren R when the limit exists. Its corresponding Hermitian form in H phys is defined whenever the above limit exists. This concludes our presentation of the main results of [6] . Let us now consider several examples of systems for which the continuum limit can be investigated.
In this section we shall develop several examples of systems that have been treated with the polymer quantization. These examples are simple quantum mechanical systems, such as the simple harmonic oscillator and the free particle, as well as a quantum cosmological model known as loop quantum cosmology.
In this part, let us consider the example of a Simple Harmonic Oscillator (SHO) with parameters m and ω, classically described by the following Hamiltonian
H = 1 2m p 2 + 1 2 m ω 2 x 2 .
Recall that from these parameters one can define a length scale D = /mω. In the standard treatment one uses 12 this scale to define a complex structure J D (and an inner product from it), as we have described in detail that uniquely selects the standard Schrödinger representation. At scale C n we have an effective Hamiltonian for the Simple Harmonic Oscillator (SHO) given by
H Cn = 2 ma 2 n 1 -cos a n p + 1 2 m ω 2 x 2 . ( 35
)
If we interchange position and momentum, this Hamiltonian is exactly that of a pendulum of mass m, length l and subject to a constant gravitational field g:
ĤCn = - 2 2ml 2 d 2 dθ 2 + mgl(1 -cos θ)
where those quantities are related to our system by,
l = m ω a n , g = ω m a n , θ = p a n
That is, we are approximating, for each scale C n the SHO by a pendulum. There is, however, an important difference. From our knowledge of the pendulum system, we know that the quantum system will have a spectrum for the energy that has two different asymptotic behaviors, the SHO for low energies and the planar rotor in the higher end, corresponding to oscillating and rotating solutions respectively 2 . As we refine our scale and both the length of the pendulum and the height of the periodic potential increase, we expect to have an increasing number of oscillating states (for a given pendulum system, there is only a finite number of such states). Thus, it is justified to consider the cut-off in the energy eigenvalues, as discussed in the last section, given that we only expect a finite number of states of the pendulum to approximate SHO eigenstates. With these consideration in mind, the relevant question is whether the conditions for the continuum limit to exist are satisfied. This question has been answered in the affirmative in [6] . What was shown there was that the eigen-values and eigen functions of the discrete systems, which represent a discrete and non-degenerate set, approximate those of the continuum, namely, of the standard harmonic oscillator when the inner product is renormalized by a factor λ 2 Cn = 1/2 n . This convergence implies that the continuum limit exists as we understand it. Let us now consider the simplest possible system, a free particle, that has nevertheless the particular feature that the spectrum of the energy is continuous.
2 Note that both types of solutions are, in the phase space, closed.
This is the reason behind the purely discrete spectrum. The distinction we are making is between those solutions inside the separatrix, that we call oscillating, and those that are above it that we call rotating.
In the limit ω → 0, the Hamiltonian of the Simple Harmonic oscillator (35) goes to the Hamiltonian of a free particle and the corresponding time independent Schrödinger equation, in the p-polarization, is given by
2 ma 2 n (1 -cos a n p ) -E Cn ψ(p) = 0
where we now have that p ∈ S 1 , with p ∈ (-π an , π an ). Thus, we have
E Cn = 2 ma 2 n 1 -cos a n p ≤ E Cn,max ≡ 2 2 ma 2 n . ( 36
)
At each scale the energy of the particle we can describe is bounded from above and the bound depends on the scale. Note that in this case the spectrum is continuous, which implies that the ordinary eigenfunctions of the Hilbert are not normalizable. This imposes an upper bound in the value that the energy of the particle can have, in addition to the bound in the momentum due to its "compactification". Let us first look for eigen-solutions to the time independent Schrödinger equation, that is, for energy eigenstates. In the case of the ordinary free particle, these correspond to constant momentum plane waves of the form e ±( ipx ) and such that the ordinary dispersion relation p 2 /2m = E is satisfied. These plane waves are not square integrable and do not belong to the ordinary Hilbert space of the Schrödinger theory but they are still useful for extracting information about the system. For the polymer free particle we have,
ψCn (p) = c 1 δ(p -P Cn ) + c 2 δ(p + P Cn )
where P Cn is a solution of the previous equation considering a fixed value of E Cn . That is,
P Cn = P (E Cn ) = a n arccos 1 - ma 2 n 2 E Cn
The inverse Fourier transform yields, in the 'x representation',
ψ Cn (x j ) = 1 √ 2π π /an -π /an ψ(p) e i an p j dp = = √ 2π a n c 1 e ixj PC n / + c 2 e -ixj PC n / . ( 37
)
with x j = a n j for j ∈ Z. Note that the eigenfunctions are still delta functions (in the p representation) and thus not (square) normalizable with respect to the polymer inner product, that in the p polarization is just given by the ordinary Haar measure on S 1 , and there is no quantization of the momentum (its spectrum is still truly continuous).
Let us now consider the time dependent Schrödinger equation, i ∂ t Ψ(p, t) = Ĥ • Ψ(p, t).
Which now takes the form,
∂ ∂t Ψ(p, t) = -i m a n (1 -cos (a n p/ )) Ψ(p, t)
that has as its solution, Ψ(p, t) = e -i m an (1-cos (an p/ )) t ψ(p) = e (-iEC n / ) t ψ(p) for any initial function ψ(p), where E Cn satisfy the dispersion relation (36) . The wave function Ψ(x j , t), the x j -representation of the wave function, can be obtained for any given time t by Fourier transforming with (37) the wave function Ψ(p, t).
In order to check out the convergence of the microscopically corrected Hamiltonians we should analyze the convergence of the energy levels and of the proper covectors. In the limit n → ∞, E Cn → E = p 2 /2m so we can be certain that the eigen-values for the energy converge (when fixing the value of p). Let us write the proper covector as Ψ Cn = (ψ Cn , •) ren Cn ∈ H ⋆ren Cn . Then we can bring microscopic corrections to scale C m and look for convergence of such corrections
Ψ ren Cm . = lim n→∞ d ⋆ m,n Ψ Cn .
It is easy to see that given any basis vector e αi ∈ H Cm the following limit Ψ ren Cm (e αi,Cm ) = lim Cn→∞ Ψ Cn (d n,m (e αi,Cm )) exists and is equal to
Ψ shad Cm (e αi,Cm ) = [d ⋆ Ψ Schr ](e αi,Cm ) = Ψ Schr (ia m )
where Ψ shad Cm is calculated using the free particle Hamiltonian in the Schrödinger representation. This expression defines the completely renormalized proper covector at the scale C m .
In this section we shall present a version of quantum cosmology that we call polymer quantum cosmology. The idea behind this name is that the main input in the quantization of the corresponding mini-superspace model is the use of a polymer representation as here understood. Another important input is the choice of fundamental variables to be used and the definition of the Hamiltonian constraint. Different research groups have made different choices. We shall take here a simple model that has received much attention recently, namely an isotropic, homogeneous FRW cosmology with k = 0 and coupled to a massless scalar field ϕ. As we shall see, a proper treatment of the continuum limit of this system requires new tools under development that are beyond the scope of this work. We will thus restrict ourselves to the introduction of the system and the problems that need to be solved.
The system to be quantized corresponds to the phase space of cosmological spacetimes that are homogeneous and isotropic and for which the homogeneous spatial slices have a flat intrinsic geometry (k = 0 condition). The only matter content is a mass-less scalar field ϕ. In this case the spacetime geometry is given by metrics of the form:
ds 2 = -dt 2 + a 2 (t) (dx 2 + dy 2 + dz 2 )
where the function a(t) carries all the information and degrees of freedom of the gravity part. In terms of the coordinates (a, p a , ϕ, p ϕ ) for the phase space Γ of the theory, all the dynamics is captured in the Hamiltonian constraint C := -3 8
p 2 a |a| + 8πG p 2 ϕ 2|a| 3 ≈ 0
The first step is to define the constraint on the kinematical Hilbert space to find physical states and then a physical inner product to construct the physical Hilbert space. First note that one can rewrite the equation as:
3 8 p 2 a a 2 = 8πG p 2 ϕ 2
If, as is normally done, one chooses ϕ to act as an internal time, the right hand side would be promoted, in the quantum theory, to a second derivative. The left hand side is, furthermore, symmetric in a and p a . At this point we have the freedom in choosing the variable that will be quantized and the variable that will not be well defined in the polymer representation. The standard choice is that p a is not well defined and thus, a and any geometrical quantity derived from it, is quantized. Furthermore, we have the choice of polarization on the wave function. In this respect the standard choice is to select the a-polarization, in which a acts as multiplication and the approximation of p a , namely sin(λ p a )/λ acts as a difference operator on wave functions of a. For details of this particular choice see [5] . Here we shall adopt the opposite polarization, that is, we shall have wave functions Ψ(p a , ϕ). Just as we did in the previous cases, in order to gain intuition about the behavior of the polymer quantized theory, it is convenient to look at the equivalent problem in the classical theory, namely the classical system we would get be approximating the non-well defined observable (p a in our present case) by a well defined object (made of trigonometric functions). Let us for simplicity choose to replace p a → sin(λ p a )/λ. With this choice we get an effective classical Hamiltonian constraint that 14 depends on λ:
C λ := - 3 8 sin(λ p a ) 2 λ 2 |a| + 8πG p 2 ϕ 2|a| 3 ≈ 0
We can now compute effective equations of motion by means of the equations: Ḟ := {F, C λ }, for any observable F ∈ C ∞ (Γ), and where we are using the effective (first order) action:
S λ = dτ (p a ȧ + p ϕ φ -N C λ )
with the choice N = 1. The first thing to notice is that the quantity p ϕ is a constant of the motion, given that the variable ϕ is cyclic. The second observation is that φ = 8π G pϕ |a| 3 has the same sign as p ϕ and never vanishes. Thus ϕ can be used as a (n internal) time variable. The next observation is that the equation for ȧ a 2 , namely the effective Friedman equation, will have a zero for a non-zero value of a given by
a * = 32πG 3 λ 2 p 2 ϕ .
This is the value at which there will be bounce if the trajectory started with a large value of a and was contracting. Note that the 'size' of the universe when the bounce occurs depends on both the constant p ϕ (that dictates the matter density) and the value of the lattice size λ. Here it is important to stress that for any value of p ϕ (that uniquely fixes the trajectory in the (a, p a ) plane), there will be a bounce. In the original description in terms of Einstein's equations (without the approximation that depends on λ), there in no such bounce. If ȧ < 0 initially, it will remain negative and the universe collapses, reaching the singularity in a finite proper time. What happens within the effective description if we refine the lattice and go from λ to λ n := λ/2 n ? The only thing that changes, for the same classical orbit labelled by p ϕ , is that the bounce occurs at a 'later time' and for a smaller value of a * but the qualitative picture remains the same. This is the main difference with the systems considered before. In those cases, one could have classical trajectories that remained, for a given choice of parameter λ, within the region where sin(λp)/λ is a good approximation to p. Of course there were also classical trajectories that were outside this region but we could then refine the lattice and find a new value λ ′ for which the new classical trajectory is well approximated. In the case of the polymer cosmology, this is never the case: Every classical trajectory will pass from a region where the approximation is good to a region where it is not; this is precisely where the 'quantum corrections' kick in and the universes bounces.
Given that in the classical description, the 'original' and the 'corrected' descriptions are so different we expect that, upon quantization, the corresponding quantum theories, namely the polymeric and the Wheeler-DeWitt will be related in a non-trivial way (if at all).
In this case, with the choice of polarization and for a particular factor ordering we have,
1 λ sin(λp a ) ∂ ∂p a 2 + 32π 3 ℓ 2 p ∂ 2 ∂ϕ 2 • Ψ(p a , ϕ) = 0
as the Polymer Wheeler-DeWitt equation.
In order to approach the problem of the continuum limit of this quantum theory, we have to realize that the task is now somewhat different than before. This is so given that the system is now a constrained system with a constraint operator rather than a regular non-singular system with an ordinary Hamiltonian evolution. Fortunately for the system under consideration, the fact that the variable ϕ can be regarded as an internal time allows us to interpret the quantum constraint as a generalized Klein-Gordon equation of the form
∂ 2 ∂ϕ 2 Ψ = Θ λ • Ψ
where the operator Θ λ is 'time independent'. This allows us to split the space of solutions into 'positive and negative frequency', introduce a physical inner product on the positive frequency solutions of this equation and a set of physical observables in terms of which to describe the system. That is, one reduces in practice the system to one very similar to the Schrödinger case by taking the positive square root of the previous equation:
∂ ∂ϕ Ψ = √ Θ λ • Ψ.
The question we are interested is whether the continuum limit of these theories (labelled by λ) exists and whether it corresponds to the Wheeler-DeWitt theory. A complete treatment of this problem lies, unfortunately, outside the scope of this work and will be reported elsewhere [12] .
Let us summarize our results. In the first part of the article we showed that the polymer representation of the canonical commutation relations can be obtained as the limiting case of the ordinary Fock-Schrödinger representation in terms of the algebraic state that defines the representation. These limiting cases can also be interpreted in terms of the naturally defined coherent states associated to each representation labelled by the parameter d, when they become infinitely 'squeezed'. The two possible limits of squeezing lead to two different polymer descriptions that can nevertheless be identified, as we have also shown, with the two possible polarizations for an abstract polymer representation. This resulting theory has, however, very different behavior as the standard one: The Hilbert space is non-separable, the representation is unitarily inequivalent to the Schrödinger one, and natural operators such as p are no longer well defined. This particular limiting construction of the polymer theory can shed some light for more complicated systems such as field theories and gravity.
In the regular treatments of dynamics within the polymer representation, one needs to introduce some extra structure, such as a lattice on configuration space, to construct a Hamiltonian and implement the dynamics for the system via a regularization procedure. How does this resulting theory compare to the original continuum theory one had from the beginning? Can one hope to remove the regulator in the polymer description? As they stand there is no direct relation or mapping from the polymer to a continuum theory (in case there is one defined). As we have shown, one can indeed construct in a systematic fashion such relation by means of some appropriate notions related to the definition of a scale, closely related to the lattice one had to introduce in the regularization. With this important shift in perspective, and an appropriate renormalization of the polymer inner product at each scale one can, subject to some consistency conditions, define a procedure to remove the regulator, and arrive to a Hamiltonian and a Hilbert space.
As we have seen, for some simple examples such as a free particle and the harmonic oscillator one indeed recovers the Schrödinger description back. For other systems, such as quantum cosmological models, the answer is not as clear, since the structure of the space of classical solutions is such that the 'effective description' intro-duced by the polymer regularization at different scales is qualitatively different from the original dynamics. A proper treatment of these class of systems is underway and will be reported elsewhere [12] .
Perhaps the most important lesson that we have learned here is that there indeed exists a rich interplay between the polymer description and the ordinary Schrödinger representation. The full structure of such relation still needs to be unravelled. We can only hope that a full understanding of these issues will shed some light in the ultimate goal of treating the quantum dynamics of background independent field systems such as general relativity.
We thank A. Ashtekar, G. Hossain, T. Pawlowski and P. Singh for discussions. This work was in part supported by CONACyT U47857-F and 40035-F grants, by NSF PHY04-56913, by the Eberly Research Funds of Penn State, by the AMC-FUMEC exchange program and by funds of the CIC-Universidad Michoacana de San Nicolás de Hidalgo.
[1] R. Beaume, J. Manuceau, A. Pellet and M. Sirugue, "Translation Invariant States In Quantum Mechanics," Commun. Math. Phys. 38, 29 (1974); W. E. Thirring and H. Narnhofer, "Covariant QED without indefinite metric," Rev. Math. Phys. 4, 197 (1992); F. Acerbi, G. Morchio and F. Strocchi, "Infrared singular fields and nonregular representations of canonical commutation relation algebras", J. Math. Phys. 34, 899 (1993); F. Cavallaro, G. Morchio and F. Strocchi, "A generalization of the Stone-von Neumann theorem to non-regular representations of the CCR-algebra", Lett. Math. Phys. 47 307 (1999); H. Halvorson, "Complementarity of Representations in quantum mechanics", Studies in History and Philosophy of Modern Physics 35 45 (2004). [2] A. Ashtekar, S. Fairhurst and J.L. Willis, "Quantum gravity, shadow states, and quantum mechanics", Class. Quant. Grav. 20 1031 (2003) [arXiv:gr-qc/0207106]. [3] K. Fredenhagen and F. Reszewski, "Polymer state approximations of Schrödinger wave functions", Class. Quant. Grav. 23 6577 (2006) [arXiv:gr-qc/0606090]. [4] M. Bojowald, "Loop quantum cosmology", Living Rev. Rel. 8, 11 (2005) [arXiv:gr-qc/0601085]; A. Ashtekar, M. Bojowald and J. Lewandowski, "Mathematical structure of loop quantum cosmology", Adv. Theor. Math. Phys. 7 233 (2003) [arXiv:gr-qc/0304074]; A. Ashtekar, T. Pawlowski and P. Singh, "Quantum nature of the big bang: Improved dynamics" Phys. Rev. D 74 084003 (2006) [arXiv:gr-qc/0607039] [5] V. Husain and O. Winkler, "Semiclassical states for quantum cosmology" Phys. Rev. D 75 024014 (2007) [arXiv:gr-qc/0607097]; V. Husain V and O. Winkler, "On singularity resolution in quantum gravity", Phys. Rev. D 69 084016 (2004). [arXiv:gr-qc/0312094]. [6] A. Corichi, T. Vukasinac and J.A. Zapata. "Hamiltonian and physical Hilbert space in polymer quantum mechanics", Class. Quant. Grav. 24 1495 (2007) [arXiv:gr-qc/0610072] [7] A. Corichi and J. Cortez, "Canonical quantization from an algebraic perspective" (preprint) [8] A. Corichi, J. Cortez and H. Quevedo, "Schrödinger and Fock Representations for a Field Theory on Curved Spacetime", Annals Phys. (NY) 313 446 (2004) [arXiv:hep-th/0202070]. [9] E. Manrique, R. Oeckl, A. Weber and J.A. Zapata, "Loop quantization as a continuum limit" Class. Quant. Grav. 23 3393 (2006) [arXiv:hep-th/0511222]; E. Manrique, R. Oeckl, A. Weber and J.A. Zapata, "Effective theories and continuum limit for canonical loop quantization" (preprint) [10] D.W. Chiou, "Galileo symmetries in polymer particle representation", Class. Quant. Grav. 24, 2603 (2007) [arXiv:gr-qc/0612155]. [11] W. Rudin, Fourier analysis on groups, (Interscience, New York, 1962) [12] A. Ashtekar, A. Corichi, P. Singh, "Contrasting LQC and WDW using an exactly soluble model" (preprint); A. Corichi, T. Vukasinac, and J.A. Zapata, "Continuum limit for quantum constrained system" (preprint).
16 | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The so-called polymer quantum mechanics, a nonregular and somewhat 'exotic' representation of the canonical commutation relations (CCR) [1], has been used to explore both mathematical and physical issues in background independent theories such as quantum gravity [2, 3] . A notable example of this type of quantization, when applied to minisuperspace models has given way to what is known as loop quantum cosmology [4, 5] . As in any toy model situation, one hopes to learn about the subtle technical and conceptual issues that are present in full quantum gravity by means of simple, finite dimensional examples. This formalism is not an exception in this regard. Apart from this motivation coming from physics at the Planck scale, one can independently ask for the relation between the standard continuous representations and their polymer cousins at the level of mathematical physics. A deeper understanding of this relation becomes important on its own.\n\nThe polymer quantization is made of several steps. The first one is to build a representation of the Heisenberg-Weyl algebra on a Kinematical Hilbert space that is \"background independent\", and that is sometimes referred to as the polymeric Hilbert space H poly . The second and most important part, the implementation of dynamics, deals with the definition of a Hamiltonian (or Hamiltonian constraint) on this space. In the examples studied so far, the first part is fairly well understood, yielding the kinematical Hilbert space H poly that is, however, non-separable. For the second step, a natural implementation of the dynamics has proved to be a bit more difficult, given that a direct definition of the Hamiltonian Ĥ of, say, a particle on a potential on the space H poly is not possible since one of the main features of this representation is that the operators q and p cannot be both simultaneously defined (nor their analogues in theories involving more elaborate variables). Thus, any operator that involves (powers of) the not defined variable has to be regulated by a well defined operator which normally involves introducing some extra structure on the configuration (or momentum) space, namely a lattice. However, this new structure that plays the role of a regulator can not be removed when working in H poly and one is left with the ambiguity that is present in any regularization. The freedom in choosing it can be sometimes associated with a length scale (the lattice spacing). For ordinary quantum systems such as a simple harmonic oscillator, that has been studied in detail from the polymer viewpoint, it has been argued that if this length scale is taken to be 'sufficiently small', one can arbitrarily approximate standard Schrödinger quantum mechanics [2, 3] . In the case of loop quantum cosmology, the minimum area gap A 0 of the full quantum gravity theory imposes such a scale, that is then taken to be fundamental [4] .\n\nA natural question is to ask what happens when we change this scale and go to even smaller 'distances', that is, when we refine the lattice on which the dynamics of the theory is defined. Can we define consistency conditions between these scales? Or even better, can we take the limit and find thus a continuum limit? As it has been shown recently in detail, the answer to both questions is in the affirmative [6] . There, an appropriate notion of scale was defined in such a way that one could define refinements of the theory and pose in a precise fashion the question of the continuum limit of the theory. These results could also be seen as handing a procedure to remove the regulator when working on the appropriate space. The purpose of this paper is to further explore different aspects of the relation between the continuum and the polymer representation. In particular in the first part we put forward a novel way of deriving the polymer representation from the ordinary Schrödinger representation as an appropriate limit. In Sec. II we derive two versions of the polymer representation as different limits of the Schrödinger theory. In Sec. III we show that these two versions can be seen as different polarizations of the 'abstract' polymer representation. These results, to the best of our knowledge, are new and have not been reported elsewhere. In Sec. IV we pose the problem of implementing the dynamics on the polymer representation. In Sec. V we motivate further the question of the continuum limit (i.e. the proper removal of the regulator) and recall the basic constructions of [6] . Several examples are considered in Sec. VI. In particular a simple harmonic oscillator, the polymer free particle and a simple quantum cosmology model are considered. The free particle and the cosmological model represent a generalization of the results obtained in [6] where only systems with a discrete and non-degenerate spectrum where considered. We end the paper with a discussion in Sec. VII. In order to make the paper self-contained, we will keep the level of rigor in the presentation to that found in the standard theoretical physics literature."
},
{
"section_type": "OTHER",
"section_title": "II. QUANTIZATION AND POLYMER REPRESENTATION",
"text": "In this section we derive the so called polymer representation of quantum mechanics starting from a specific reformulation of the ordinary Schrödinger representation. Our starting point will be the simplest of all possible phase spaces, namely Γ = R 2 corresponding to a particle living on the real line R. Let us choose coordinates (q, p) thereon. As a first step we shall consider the quantization of this system that leads to the standard quantum theory in the Schrödinger description. A convenient route is to introduce the necessary structure to define the Fock representation of such system. From this perspective, the passage to the polymeric case becomes clearest. Roughly speaking by a quantization one means a passage from the classical algebraic bracket, the Poisson bracket,\n\n{q, p} = 1 ( 1\n\n)\n\nto a quantum bracket given by the commutator of the corresponding operators,\n\n[ q, p] = i 1 ( 2\n\n)\n\nThese relations, known as the canonical commutation relation (CCR) become the most common corner stone of the (kinematics of the) quantum theory; they should be satisfied by the quantum system, when represented on a Hilbert space H.\n\nThere are alternative points of departure for quantum kinematics. Here we consider the algebra generated by the exponentiated versions of q and p that are denoted by, U (α) = e i(α q)/ ; V (β) = e i(β p)/\n\nwhere α and β have dimensions of momentum and length, respectively. The CCR now become U (α) • V (β) = e (-iα β)/ V (β) • U (α) (3) and the rest of the product is\n\nU (α 1 )•U (α 2 ) = U (α 1 +α 2 ) ; V (β 1 )•V (β 2 ) = V (β 1 +β 2 )\n\nThe Weyl algebra W is generated by taking finite linear combinations of the generators U (α i ) and V (β i ) where the product (3) is extended by linearity,\n\ni (A i U (α i ) + B i V (β i ))\n\nFrom this perspective, quantization means finding an unitary representation of the Weyl algebra W on a Hilbert space H ′ (that could be different from the ordinary Schrödinger representation). At first it might look weird to attempt this approach given that we know how to quantize such a simple system; what do we need such a complicated object as W for? It is infinite dimensional, whereas the set S = { 1, q, p}, the starting point of the ordinary Dirac quantization, is rather simple. It is in the quantization of field systems that the advantages of the Weyl approach can be fully appreciated, but it is also useful for introducing the polymer quantization and comparing it to the standard quantization. This is the strategy that we follow. A question that one can ask is whether there is any freedom in quantizing the system to obtain the ordinary Schrödinger representation. On a first sight it might seem that there is none given the Stone-Von Neumann uniqueness theorem. Let us review what would be the argument for the standard construction. Let us ask that the representation we want to build up is of the Schrödinger type, namely, where states are wave functions of configuration space ψ(q). There are two ingredients to the construction of the representation, namely the specification of how the basic operators (q, p) will act, and the nature of the space of functions that ψ belongs to, that is normally fixed by the choice of inner product on H, or measure µ on R.\n\nThe standard choice is to select the Hilbert space to be, H = L 2 (R, dq) 2 the space of square-integrable functions with respect to the Lebesgue measure dq (invariant under constant translations) on R. The operators are then represented as, q • ψ(q) = (q ψ)(q) and p • ψ(q) = -i ∂ ∂q ψ(q) (4) Is it possible to find other representations? In order to appreciate this freedom we go to the Weyl algebra and build the quantum theory thereon. The representation of the Weyl algebra that can be called of the 'Fock type' involves the definition of an extra structure on the phase space Γ: a complex structure J. That is, a linear mapping from Γ to itself such that J 2 = -1. In 2 dimensions, all the freedom in the choice of J is contained in the choice of a parameter d with dimensions of length. It is also convenient to define: k = p/ that has dimensions of 1/L. We have then, J d : (q, k) → (-d 2 k, q/d 2 ) This object together with the symplectic structure: Ω((q, p); (q ′ , p ′ )) = q p ′ -p q ′ define an inner product on Γ by the formula g d (• ; •) = Ω(• ; J d •) such that:\n\ng d ((q, p); (q ′ , p ′ )) = 1 d 2 q q ′ + d 2 2 p p ′\n\nwhich is dimension-less and positive definite. Note that with this quantities one can define complex coordinates (ζ, ζ) as usual:\n\nζ = 1 d q + i d p ; ζ = 1 d q -i d p from which one can build the standard Fock representation. Thus, one can alternatively view the introduction of the length parameter d as the quantity needed to define (dimensionless) complex coordinates on the phase space. But what is the relevance of this object (J or d)? The definition of complex coordinates is useful for the construction of the Fock space since from them one can define, in a natural way, creation and annihilation operators. But for the Schrödinger representation we are interested here, it is a bit more subtle. The subtlety is that within this approach one uses the algebraic properties of W to construct the Hilbert space via what is known as the Gel'fand-Naimark-Segal (GNS) construction. This implies that the measure in the Schrödinger representation becomes non trivial and thus the momentum operator acquires an extra term in order to render the operator self-adjoint. The representation of the Weyl algebra is then, when acting on functions φ(q) [7]: Û (α) • φ(q) := (e iα q/ φ)(q) and, V (β) • φ(q) := e β d 2 (q-β/2) φ(q -β)\n\nThe Hilbert space structure is introduced by the definition of an algebraic state (a positive linear functional) ω d : W → C, that must coincide with the expectation value in the Hilbert space taken on a special state refered to as the vacuum: ω d (a) = â vac , for all a ∈ W.\n\nIn our case this specification of J induces such a unique state ω d that yields, Û (α) vac = e -1 4 d 2 α 2 2 (5) and V (β) vac = e -1 4 β 2 d 2 (6) Note that the exponents in the vacuum expectation values correspond to the metric constructed out of J:\n\nd 2 α 2 2 = g d ((0, α); (0, α)) and β 2 d 2 = g d ((β, 0); (β, 0)). Wave functions belong to the space L 2 (R, dµ d ), where the measure that dictates the inner product in this representation is given by,\n\ndµ d = 1 d √ π e -q 2 d 2 dq\n\nIn this representation, the vacuum is given by the identity function φ 0 (q) = 1 that is, just as any plane wave, normalized. Note that for each value of d > 0, the representation is well defined and continuous in α and β. Note also that there is an equivalence between the qrepresentation defined by d and the k-representation defined by 1/d. How can we recover then the standard representation in which the measure is given by the Lebesgue measure and the operators are represented as in (4) ? It is easy to see that there is an isometric isomorphism K that maps the d-representation in H d to the standard Schrödinger representation in H schr by: ψ(q) = K • φ(q) = e -q 2 2 d 2 d 1/2 π 1/4 φ(q) ∈ H schr = L 2 (R, dq) Thus we see that all d-representations are unitarily equivalent. This was to be expected in view of the Stone-Von Neumann uniqueness result. Note also that the vacuum now becomes ψ 0 (q) = 1 d 1/2 π 1/4 e -q 2 2 d 2 , so even when there is no information about the parameter d in the representation itself, it is contained in the vacuum state. This procedure for constructing the GNS-Schrödinger representation for quantum mechanics has also been generalized to scalar fields on arbitrary curved space in [8] . Note, however that so far the treatment has all been kinematical, without any knowledge of a Hamiltonian. For the Simple Harmonic Oscillator of mass m and frequency ω, there is a natural choice compatible with the dynamics given by d = m ω , in which some calculations simplify (for instance for coherent states), but in principle one can use any value of d.\n\nOur study will be simplified by focusing on the fundamental entities in the Hilbert Space H d , namely those states generated by acting with Û(α) on the vacuum φ 0 (q) = 1. Let us denote those states by,\n\nφ α (q) = Û (α) • φ 0 (q) = e i 1 α q\n\nThe inner product between two such states is given by\n\nφ α , φ λ d = dµ d e -iαq e iλq = e -(λ-α) 2 d 2 4 2 ( 7\n\n)\n\nNote incidentally that, contrary to some common belief, the 'plane waves' in this GNS Hilbert space are indeed normalizable.\n\nLet us now consider the polymer representation. For that, it is important to note that there are two possible limiting cases for the parameter d: i) The limit 1/d → 0 and ii) The case d → 0. In both cases, we have expressions that become ill defined in the representation or measure, so one needs to be careful.\n\nA. The 1/d → 0 case.\n\nThe first observation is that from the expressions (5) and (6) for the algebraic state ω d , we see that the limiting cases are indeed well defined. In our case we get, ω A := lim 1/d→0 ω d such that,\n\nω A ( Û (α)) = δ α,0 and ω A ( V (β)) = 1 ( 8\n\n)\n\nFrom this, we can indeed construct the representation by means of the GNS construction. In order to do that and to show how this is obtained we shall consider several expressions. One has to be careful though, since the limit has to be taken with care. Let us consider the measure on the representation that behaves as:\n\ndµ d = 1 d √ π e -q 2 d 2 dq → 1 d √ π dq\n\nso the measures tends to an homogeneous measure but whose 'normalization constant' goes to zero, so the limit becomes somewhat subtle. We shall return to this point later. Let us now see what happens to the inner product between the fundamental entities in the Hilbert Space H d given by (7) . It is immediate to see that in the 1/d → 0 limit the inner product becomes,\n\nφ α , φ λ d → δ α,λ ( 9\n\n)\n\nwith δ α,λ being Kronecker's delta. We see then that the plane waves φ α (q) become an orthonormal basis for the new Hilbert space. Therefore, there is a delicate interplay between the two terms that contribute to the measure in order to maintain the normalizability of these functions; we need the measure to become damped (by 1/d) in order to avoid that the plane waves acquire an infinite norm (as happens with the standard Lebesgue measure), but on the other hand the measure, that for any finite value of d is a Gaussian, becomes more and more spread. It is important to note that, in this limit, the operators Û (α) become discontinuous with respect to α, given that for any given α 1 and α 2 (different), its action on a given basis vector ψ λ (q) yields orthogonal vectors. Since the continuity of these operators is one of the hypotesis of the Stone-Von Neumann theorem, the uniqueness result does not apply here. The representation is inequivalent to the standard one.\n\nLet us now analyze the other operator, namely the action of the operator V (β) on the basis φ α (q): V (β) • φ α (q) = e -β 2 2d 2 -i αβ e (β/d 2 +iα/ )q which in the limit 1/d → 0 goes to,\n\nV (β) • φ α (q) → e i αβ φ α (q)\n\nthat is continuous on β. Thus, in the limit, the operator p = -i ∂ q is well defined. Also, note that in this limit the operator p has φ α (q) as its eigenstate with eigenvalue given by α:\n\np • φ α (q) → α φ α (q)\n\nTo summarize, the resulting theory obtained by taking the limit 1/d → 0 of the ordinary Schrödinger description, that we shall call the 'polymer representation of type A', has the following features: the operators U (α) are well defined but not continuous in α, so there is no generator (no operator associated to q). The basis vectors φ α are orthonormal (for α taking values on a continuous set) and are eigenvectors of the operator p that is well defined. The resulting Hilbert space H A will be the (A-version of the) polymer representation. Let us now consider the other case, namely, the limit when d → 0.\n\nB. The d → 0 case\n\nLet us now explore the other limiting case of the Schrödinger/Fock representations labelled by the parameter d. Just as in the previous case, the limiting algebraic state becomes, ω B := lim d→0 ω d such that,\n\nω B ( Û (α)) = 1 and ω B ( V (β)) = δ β,0 ( 10\n\n)\n\nFrom this positive linear function, one can indeed construct the representation using the GNS construction. First let us note that the measure, even when the limit has to be taken with due care, behaves as:\n\ndµ d = 1 d √ π e -q 2 d 2 dq → δ(q) dq\n\nThat is, as Dirac's delta distribution. It is immediate to see that, in the d → 0 limit, the inner product between the fundamental states φ α (q) becomes,\n\nφ α , φ λ d → 1 ( 11\n\n) 4\n\nThis in fact means that the vector ξ = φ α -φ λ belongs to the Kernel of the limiting inner product, so one has to mod out by these (and all) zero norm states in order to get the Hilbert space.\n\nLet us now analyze the other operator, namely the action of the operator V (β) on the vacuum φ 0 (q) = 1, which for arbitrary d has the form,\n\nφβ := V (β) • φ 0 (q) = e β d 2 (q-β/2)\n\nThe inner product between two such states is given by φα , φβ d = e -1 4d 2 (α-β) 2\n\nIn the limit d → 0, φα , φβ d → δ α,β . We can see then that it is these functions that become the orthonormal, 'discrete basis' in the theory. However, the function φβ (q) in this limit becomes ill defined. For example, for β > 0, it grows unboundedly for q > β/2, is equal to one if q = β/2 and zero otherwise. In order to overcome these difficulties and make more transparent the resulting theory, we shall consider the other form of the representation in which the measure is incorporated into the states (and the resulting Hilbert space is L 2 (R, dq)). Thus the new state\n\nψ β (q) := K • ( V (β) • φ 0 (q)) = = 1 (d √ π) 1 2 e -1 2d 2 (q-β) 2 ( 12\n\n)\n\nWe can now take the limit and what we get is lim d →0 ψ β (q) := δ 1/2 (q, β)\n\nwhere by δ 1/2 (q, β) we mean something like 'the square root of the Dirac distribution'. What we really mean is an object that satisfies the following property:\n\nδ 1/2 (q, β) • δ 1/2 (q, α) = δ(q, β) δ β,α\n\nThat is, if α = β then it is just the ordinary delta, otherwise it is zero. In a sense these object can be regarded as half-densities that can not be integrated by themselves, but whose product can. We conclude then that the inner product is,\n\nψ β , ψ α = R dq ψ β (q) ψ α (q) = R dq δ(q, α) δ β,α = δ β,α\n\n(13) which is just what we expected. Note that in this representation, the vacuum state becomes ψ 0 (q) := δ 1/2 (q, 0), namely, the half-delta with support in the origin. It is important to note that we are arriving in a natural way to states as half-densities, whose squares can be integrated without the need of a nontrivial measure on the configuration space. Diffeomorphism invariance arises then in a natural but subtle manner.\n\nNote that as the end result we recover the Kronecker delta inner product for the new fundamental states:\n\nχ β (q) := δ 1/2 (q, β).\n\nThus, in this new B-polymer representation, the Hilbert space H B is the completion with respect to the inner product (13) of the states generated by taking (finite) linear combinations of basis elements of the form χ β :\n\nΨ(q) = i b i χ βi (q) ( 14\n\n)\n\nLet us now introduce an equivalent description of this Hilbert space. Instead of having the basis elements be half-deltas as elements of the Hilbert space where the inner product is given by the ordinary Lebesgue measure dq, we redefine both the basis and the measure. We could consider, instead of a half-delta with support β, a Kronecker delta or characteristic function with support on β: χ ′ β (q) := δ q,β These functions have a similar behavior with respect to the product as the half-deltas, namely:\n\nχ ′ β (q) • χ ′ α (q) = δ β,α .\n\nThe main difference is that neither χ ′ nor their squares are integrable with respect to the Lebesgue measure (having zero norm). In order to fix that problem we have to change the measure so that we recover the basic inner product (13) with our new basis. The needed measure turns out to be the discrete counting measure on R. Thus any state in the 'half density basis' can be written (using the same expression) in terms of the 'Kronecker basis'. For more details and further motivation see the next section.\n\nNote that in this B-polymer representation, both Û and V have their roles interchanged with that of the A-polymer representation: while U (α) is discontinuous and thus q is not defined in the A-representation, we have that it is V (β) in the B-representation that has this property. In this case, it is the operator p that can not be defined. We see then that given a physical system for which the configuration space has a well defined physical meaning, within the possible representation in which wave-functions are functions of the configuration variable q, the A and B polymer representations are radically different and inequivalent.\n\nHaving said this, it is also true that the A and B representations are equivalent in a different sense, by means of the duality between q and p representations and the d ↔ 1/d duality: The A-polymer representation in the \"q-representation\" is equivalent to the B-polymer representation in the \"p-representation\", and conversely. When studying a problem, it is important to decide from the beginning which polymer representation (if any) one should be using (for instance in the q-polarization). This has as a consequence an implication on which variable is naturally \"quantized\" (even if continuous): p for A and q for B. There could be for instance a physical criteria for this choice. For example a fundamental symmetry could suggest that one representation is more natural than another one. This indeed has been recently noted by Chiou in [10] , where the Galileo group is investigated and where it is shown that the B representation is better behaved.\n\nIn the other polarization, namely for wavefunctions of p, the picture gets reversed: q is discrete for the Arepresentation, while p is for the B-case. Let us end this section by noting that the procedure of obtaining the polymer quantization by means of an appropriate limit of Fock-Schrödinger representations might prove useful in more general settings in field theory or quantum gravity."
},
{
"section_type": "OTHER",
"section_title": "III. POLYMER QUANTUM MECHANICS: KINEMATICS",
"text": "In previous sections we have derived what we have called the A and B polymer representations (in the qpolarization) as limiting cases of ordinary Fock representations. In this section, we shall describe, without any reference to the Schrödinger representation, the 'abstract' polymer representation and then make contact with its two possible realizations, closely related to the A and B cases studied before. What we will see is that one of them (the A case) will correspond to the p-polarization while the other one corresponds to the q-representation, when a choice is made about the physical significance of the variables.\n\nWe can start by defining abstract kets |µ labelled by a real number µ. These shall belong to the Hilbert space H poly . From these states, we define a generic 'cylinder states' that correspond to a choice of a finite collection of numbers µ i ∈ R with i = 1, 2, . . . , N . Associated to this choice, there are N vectors |µ i , so we can take a linear combination of them\n\n|ψ = N i=1 a i |µ i ( 15\n\n)\n\nThe polymer inner product between the fundamental kets is given by,\n\nν|µ = δ ν,µ ( 16\n\n)\n\nThat is, the kets are orthogonal to each other (when ν = µ) and they are normalized ( µ|µ = 1). Immediately, this implies that, given any two vectors |φ =\n\nM j=1 b j |ν j and |ψ = N i=1 a i |µ i , the inner product between them is given by, φ|ψ = i j bj a i ν j |µ i = k bk a k\n\nwhere the sum is over k that labels the intersection points between the set of labels {ν j } and {µ i }. The Hilbert space H poly is the Cauchy completion of finite linear combination of the form (15) with respect to the inner product (16) . H poly is non-separable. There are two basic operators on this Hilbert space: the 'label operator' ε:\n\nε |µ := µ |µ\n\nand the displacement operator ŝ (λ), ŝ (λ\n\n) |µ := |µ + λ\n\nThe operator ε is symmetric and the operator(s) ŝ(λ) defines a one-parameter family of unitary operators on H poly , where its adjoint is given by ŝ † (λ) = ŝ (-λ). This action is however, discontinuous with respect to λ given that |µ and |µ + λ are always orthogonal, no matter how small is λ. Thus, there is no (Hermitian) operator that could generate ŝ (λ) by exponentiation. So far we have given the abstract characterization of the Hilbert space, but one would like to make contact with concrete realizations as wave functions, or by identifying the abstract operators ε and ŝ with physical operators.\n\nSuppose we have a system with a configuration space with coordinate given by q, and p denotes its canonical conjugate momenta. Suppose also that for physical reasons we decide that the configuration coordinate q will have some \"discrete character\" (for instance, if it is to be identified with position, one could say that there is an underlying discreteness in position at a small scale). How can we implement such requirements by means of the polymer representation? There are two possibilities, depending on the choice of 'polarizations' for the wavefunctions, namely whether they will be functions of configuration q or momenta p. Let us the divide the discussion into two parts."
},
{
"section_type": "OTHER",
"section_title": "A. Momentum polarization",
"text": "In this polarization, states will be denoted by,\n\nψ(p) = p|ψ where ψ µ (p) = p|µ = e i µp\n\nHow are then the operators ε and ŝ represented? Note that if we associate the multiplicative operator\n\nV (λ) • ψ µ (p) = e i λ p e i µ p = e i (µ+λ) p = ψ (µ+λ) (p)\n\nwe see then that the operator V (λ) corresponds precisely to the shift operator ŝ (λ). Thus we can also conclude that the operator p does not exist. It is now easy to identify the operator q with:\n\nq • ψ µ (p) = -i ∂ ∂p ψ µ (p) = µ e i µ p = µ ψ µ (p)\n\nnamely, with the abstract operator ε. The reason we say that q is discrete is because this operator has as its eigenvalue the label µ of the elementary state ψ µ (p), and this label, even when it can take value in a continuum of possible values, is to be understood as a discrete set, given that the states are orthonormal for all values of µ. Given that states are now functions of p, the inner product (16) should be defined by a measure µ on the space on which the wave-functions are defined. In order 6 to know what these two objects are, namely, the quantum \"configuration\" space C and the measure thereon 1 , we have to make use of the tools available to us from the theory of C * -algebras. If we consider the operators V (λ), together with their natural product and * -relation given by V * (λ) = V (-λ), they have the structure of an Abelian C * -algebra (with unit) A. We know from the representation theory of such objects that A is isomorphic to the space of continuous functions C 0 (∆) on a compact space ∆, the spectrum of A. Any representation of A on a Hilbert space as multiplication operator will be on spaces of the form L 2 (∆, dµ). That is, our quantum configuration space is the spectrum of the algebra, which in our case corresponds to the Bohr compactification R b of the real line [11] . This space is a compact group and there is a natural probability measure defined on it, the Haar measure µ H . Thus, our Hilbert space H poly will be isomorphic to the space,\n\nH poly,p = L 2 (R b , dµ H ) ( 17\n\n)\n\nIn terms of 'quasi periodic functions' generated by ψ µ (p), the inner product takes the form\n\nψ µ |ψ λ := R b dµ H ψ µ (p) ψ λ (p) := = lim L →∞ 1 2L L -L dp ψ µ (p) ψ λ (p) = δ µ,λ ( 18\n\n)\n\nnote that in the p-polarization, this characterization corresponds to the 'A-version' of the polymer representation of Sec. II (where p and q are interchanged)."
},
{
"section_type": "OTHER",
"section_title": "B. q-polarization",
"text": "Let us now consider the other polarization in which wave functions will depend on the configuration coordinate q:\n\nψ(q) = q|ψ\n\nThe basic functions, that now will be called ψµ (q), should be, in a sense, the dual of the functions ψ µ (p) of the previous subsection. We can try to define them via a 'Fourier transform':\n\nψµ (q) := q|µ = q| R b dµ H |p p|µ which is given by ψµ (q) := R b dµ H q|p ψ µ (p) = = R b dµ H e -i p q e i µ p = δ q,µ ( 19\n\n)\n\n1 here we use the standard terminology of 'configuration space' to denote the domain of the wave function even when, in this case, it corresponds to the physical momenta p.\n\nThat is, the basic objects in this representation are Kronecker deltas. This is precisely what we had found in Sec. II for the B-type representation. How are now the basic operators represented and what is the form of the inner product? Regarding the operators, we expect that they are represented in the opposite manner as in the previous p-polarization case, but that they preserve the same features: p does not exist (the derivative of the Kronecker delta is ill defined), but its exponentiated version V (λ) does:\n\nV (λ) • ψ(q) = ψ(q + λ)\n\nand the operator q that now acts as multiplication has as its eigenstates, the functions ψν (q) = δ ν,q : q • ψµ (q) := µ ψµ (q) What is now the nature of the quantum configurations space Q? And what is the measure thereon dµ q ? that defines the inner product we should have: ψµ (q), ψλ (q) = δ µ,λ\n\nThe answer comes from one of the characterizations of the Bohr compactification: we know that it is, in a precise sense, dual to the real line but when equipped with the discrete topology R d . Furthermore, the measure on R d will be the 'counting measure'. In this way we recover the same properties we had for the previous characterization of the polymer Hilbert space. We can thus write:\n\nH poly,x := L 2 (R d , dµ c ) ( 20\n\n)\n\nThis completes a precise construction of the B-type polymer representation sketched in the previous section. Note that if we had chosen the opposite physical situation, namely that q, the configuration observable, be the quantity that does not have a corresponding operator, then we would have had the opposite realization: In the qpolarization we would have had the type-A polymer representation and the type-B for the p-polarization. As we shall see both scenarios have been considered in the literature. Up to now we have only focused our discussion on the kinematical aspects of the quantization process. Let us now consider in the following section the issue of dynamics and recall the approach that had been adopted in the literature, before the issue of the removal of the regulator was reexamined in [6] ."
},
{
"section_type": "OTHER",
"section_title": "IV. POLYMER QUANTUM MECHANICS: DYNAMICS",
"text": "As we have seen the construction of the polymer representation is rather natural and leads to a quantum theory with different properties than the usual Schrödinger counterpart such as its non-separability, the 7 non-existence of certain operators and the existence of normalized eigen-vectors that yield a precise value for one of the phase space coordinates. This has been done without any regard for a Hamiltonian that endows the system with a dynamics, energy and so on.\n\nFirst let us consider the simplest case of a particle of mass m in a potential V (q), in which the Hamiltonian H takes the form,\n\nH = 1 2m p 2 + V (q)\n\nSuppose furthermore that the potential is given by a nonperiodic function, such as a polynomial or a rational function. We can immediately see that a direct implementation of the Hamiltonian is out of our reach, for the simple reason that, as we have seen, in the polymer representation we can either represent q or p, but not both! What has been done so far in the literature? The simplest thing possible: approximate the non-existing term by a well defined function that can be quantized and hope for the best. As we shall see in next sections, there is indeed more that one can do. At this point there is also an important decision to be made: which variable q or p should be regarded as \"discrete\"? Once this choice is made, then it implies that the other variable will not exist: if q is regarded as discrete, then p will not exist and we need to approximate the kinetic term p 2 /2m by something else; if p is to be the discrete quantity, then q will not be defined and then we need to approximate the potential V (q). What happens with a periodic potential? In this case one would be modelling, for instance, a particle on a regular lattice such as a phonon living on a crystal, and then the natural choice is to have q not well defined. Furthermore, the potential will be well defined and there is no approximation needed.\n\nIn the literature both scenarios have been considered. For instance, when considering a quantum mechanical system in [2] , the position was chosen to be discrete, so p does not exist, and one is then in the A type for the momentum polarization (or the type B for the qpolarization). With this choice, it is the kinetic term the one that has to be approximated, so once one has done this, then it is immediate to consider any potential that will thus be well defined. On the other hand, when considering loop quantum cosmology (LQC), the standard choice is that the configuration variable is not defined [4] . This choice is made given that LQC is regarded as the symmetric sector of full loop quantum gravity where the connection (that is regarded as the configuration variable) can not be promoted to an operator and one can only define its exponentiated version, namely, the holonomy. In that case, the canonically conjugate variable, closely related to the volume, becomes 'discrete', just as in the full theory. This case is however, different from the particle in a potential example. First we could mention that the functional form of the Hamiltonian constraint that implements dynamics has a different structure, but the more important difference lies in that the system is constrained.\n\nLet us return to the case of the particle in a potential and for definiteness, let us start with the auxiliary kinematical framework in which: q is discrete, p can not be promoted and thus we have to approximate the kinetic term p2 /2m. How is this done? The standard prescription is to define, on the configuration space C, a regular 'graph' γ µ0 . This consists of a numerable set of points, equidistant, and characterized by a parameter µ 0 that is the (constant) separation between points. The simplest example would be to consider the set\n\nγ µ0 = {q ∈ R | q = n µ 0 , ∀ n ∈ Z}.\n\nThis means that the basic kets that will be considered |µ n will correspond precisely to labels µ n belonging to the graph γ µ0 , that is, µ n = n µ 0 . Thus, we shall only consider states of the form,\n\n|ψ = n b n |µ n . ( 21\n\n)\n\nThis 'small' Hilbert space H γµ 0 , the graph Hilbert space, is a subspace of the 'large' polymer Hilbert space H poly but it is separable. The condition for a state of the form (21) to belong to the Hilbert space H γµ 0 is that the coefficients b n satisfy:\n\nn |b n | 2 < ∞.\n\nLet us now consider the kinetic term p2 /2m. We have to approximate it by means of trigonometric functions, that can be built out of the functions of the form e iλ p/ . As we have seen in previous sections, these functions can indeed be promoted to operators and act as translation operators on the kets |µ . If we want to remain in the graph γ, and not create 'new points', then one is constrained to considering operators that displace the kets by just the right amount. That is, we want the basic shift operator V (λ) to be such that it maps the ket with label |µ n to the next ket, namely |µ n+1 . This can indeed achieved by fixing, once and for all, the value of the allowed parameter λ to be λ = µ 0 . We have then,\n\nV (µ 0 ) • |µ n = |µ n + µ 0 = |µ n+1\n\nwhich is what we wanted. This basic 'shift operator' will be the building block for approximating any (polynomial) function of p. In order to do that we notice that the function p can be approximated by,\n\np ≈ µ 0 sin µ 0 p = 2iµ 0 e i µ 0 p -e -i µ 0 p\n\nwhere the approximation is good for p << /µ 0 . Thus, one can define a regulated operator pµ0 that depends on the 'scale' µ 0 as:\n\npµ0 • |µ n := 2iµ 0 [V (µ 0 ) -V (-µ 0 )] • |µ n = = i 2µ 0 (|µ n+1 -|µ n-1 ) ( 22\n\n)\n\nIn order to regulate the operator p2 , there are (at least) two possibilities, namely to compose the operator pµ0 8 with itself or to define a new approximation. The operator pµ0 • pµ0 has the feature that shifts the states two steps in the graph to both sides. There is however another operator that only involves shifting once:\n\np2 µ0 • |ν n := 2 µ 2 0 [2 -V (µ 0 ) -V (-µ 0 )] • |ν n = = 2 µ 2 0 (2|ν n -|ν n+1 -|ν n-1 ) ( 23\n\n)\n\nwhich corresponds to the approximation p 2 ≈ 2 2 µ 2 0 (1cos(µ 0 p/ )), valid also in the regime p << /µ 0 . With these considerations, one can define the operator Ĥµ0 , the Hamiltonian at scale µ 0 , that in practice 'lives' on the space H γµ 0 as,\n\nĤµ0 := 1 2m p2 µ0 + V (q) , ( 24\n\n)\n\nthat is a well defined, symmetric operator on H γµ 0 . Notice that the operator is also defined on H poly , but there its physical interpretation is problematic. For example, it turns out that the expectation value of the kinetic term calculated on most states (states which are not tailored to the exact value of the parameter µ 0 ) is zero. Even if one takes a state that gives \"reasonable\" expectation values of the µ 0 -kinetic term and uses it to calculate the expectation value of the kinetic term corresponding to a slight perturbation of the parameter µ 0 one would get zero. This problem, and others that arise when working on H poly , forces one to assign a physical interpretation to the Hamiltonian Ĥµ0 only when its action is restricted to the subspace H γµ 0 . Let us now explore the form that the Hamiltonian takes in the two possible polarizations. In the q-polarization, the basis, labelled by n is given by the functions χ n (q) = δ q,µn . That is, the wave functions will only have support on the set γ µ0 . Alternatively, one can think of a state as completely characterized by the 'Fourier coefficients' a n : ψ(q) ↔ a n , which is the value that the wave function ψ(q) takes at the point q = µ n = n µ 0 . Thus, the Hamiltonian takes the form of a difference equation when acting on a general state ψ(q). Solving the time independent Schrödinger equation Ĥ • ψ = E ψ amounts to solving the difference equation for the coefficients a n .\n\nThe momentum polarization has a different structure. In this case, the operator p2 µ0 acts as a multiplication operator,\n\np2 µ0 • ψ(p) = 2 2 µ 2 0 1 -cos µ 0 p ψ(p) ( 25\n\n)\n\nThe operator corresponding to q will be represented as a derivative operator q • ψ(p) := i ∂ p ψ(p).\n\nFor a generic potential V (q), it has to be defined by means of spectral theory defined now on a circle. Why on a circle? For the simple reason that by restricting ourselves to a regular graph γ µ0 , the functions of p that preserve it (when acting as shift operators) are of the form e (i m µ0 p/ ) for m integer. That is, what we have are Fourier modes, labelled by m, of period 2π /µ 0 in p.\n\nCan we pretend then that the phase space variable p is now compactified? The answer is in the affirmative. The inner product on periodic functions ψ µ0 (p) of p coming from the full Hilbert space H poly and given by\n\nφ(p)|ψ(p) poly = lim L →∞ 1 2L L -L dp φ(p) ψ(p)\n\nis precisely equivalent to the inner product on the circle given by the uniform measure φ(p)|ψ(p) µ0 = µ 0 2π π /µ0 -π /µ0 dp φ(p) ψ(p) with p ∈ (-π /µ 0 , π /µ 0 ). As long as one restricts attention to the graph γ µ0 , one can work in this separable Hilbert space H γµ 0 of square integrable functions on S 1 . Immediately, one can see the limitations of this description. If the mechanical system to be quantized is such that its orbits have values of the momenta p that are not small compared with π /µ 0 then the approximation taken will be very poor, and we don't expect neither the effective classical description nor its quantization to be close to the standard one. If, on the other hand, one is always within the region in which the approximation can be regarded as reliable, then both classical and quantum descriptions should approximate the standard description. What does 'close to the standard description' exactly mean needs, of course, some further clarification. In particular one is assuming the existence of the usual Schrödinger representation in which the system has a behavior that is also consistent with observations. If this is the case, the natural question is: How can we approximate such description from the polymer picture? Is there a fine enough graph γ µ0 that will approximate the system in such a way that all observations are indistinguishable? Or even better, can we define a procedure, that involves a refinement of the graph γ µ0 such that one recovers the standard picture? It could also happen that a continuum limit can be defined but does not coincide with the 'expected one'. But there might be also physical systems for which there is no standard description, or it just does not make sense. Can in those cases the polymer representation, if it exists, provide the correct physical description of the system under consideration? For instance, if there exists a physical limitation to the minimum scale set by µ 0 , as could be the case for a quantum theory of gravity, then the polymer description would provide a true physical bound on the value of certain quantities, such as p in our example. This could be the case for loop quantum cosmology, where there is a minimum value for physical volume (coming from the full theory), and phase space points near the 'singularity' lie at the region where the 9 approximation induced by the scale µ 0 departs from the standard classical description. If in that case the polymer quantum system is regarded as more fundamental than the classical system (or its standard Wheeler-De Witt quantization), then one would interpret this discrepancies in the behavior as a signal of the breakdown of classical description (or its 'naive' quantization).\n\nIn the next section we present a method to remove the regulator µ 0 which was introduced as an intermediate step to construct the dynamics. More precisely, we shall consider the construction of a continuum limit of the polymer description by means of a renormalization procedure."
},
{
"section_type": "OTHER",
"section_title": "V. THE CONTINUUM LIMIT",
"text": "This section has two parts. In the first one we motivate the need for a precise notion of the continuum limit of the polymeric representation, explaining why the most direct, and naive approach does not work. In the second part, we shall present the main ideas and results of the paper [6], where the Hamiltonian and the physical Hilbert space in polymer quantum mechanics are constructed as a continuum limit of effective theories, following Wilson's renormalization group ideas. The resulting physical Hilbert space turns out to be unitarily isomorphic to the ordinary H s = L 2 (R, dq) of the Schrödinger theory.\n\nBefore describing the results of [6] we should discuss the precise meaning of reaching a theory in the continuum. Let us for concreteness consider the B-type representation in the q-polarization. That is, states are functions of q and the orthonormal basis χ µ (q) is given by characteristic functions with support on q = µ. Let us now suppose we have a Schrödinger state Ψ(q) ∈ H s = L 2 (R, dq). What is the relation between Ψ(q) and a state in H poly,x ? We are also interested in the opposite question, that is, we would like to know if there is a preferred state in H s that is approximated by an arbitrary state ψ(q) in H poly,x . The first obvious observation is that a Schödinger state Ψ(q) does not belong to H poly,x since it would have an infinite norm. To see that note that even when the would-be state can be formally expanded in the χ µ basis as,\n\nΨ(q) = µ Ψ(µ) χ µ (q)\n\nwhere the sum is over the parameter µ ∈ R. Its associated norm in H poly,x would be:\n\n|Ψ(q)| 2 poly = µ |Ψ(µ)| 2 → ∞\n\nwhich blows up. Note that in order to define a mapping P : H s → H poly,x , there is a huge ambiguity since the values of the function Ψ(q) are needed in order to expand the polymer wave function. Thus we can only define a mapping in a dense subset D of H s where the values of the functions are well defined (recall that in H s the value of functions at a given point has no meaning since states are equivalence classes of functions). We could for instance ask that the mapping be defined for representatives of the equivalence classes in H s that are piecewise continuous. From now on, when we refer to an element of the space H s we shall be refereeing to one of those representatives. Notice then that an element of H s does define an element of Cyl * γ , the dual to the space Cyl γ , that is, the space of cylinder functions with support on the (finite) lattice γ = {µ 1 , µ 2 , . . . , µ N }, in the following way:\n\nΨ(q) : Cyl γ -→ C such that Ψ(q)[ψ(q)] = (Ψ|ψ := µ Ψ(µ) χ µ | N i=1 ψ i χ µi poly γ = N i=1 Ψ(µ i ) ψ i < ∞ ( 26\n\n)\n\nNote that this mapping could be seen as consisting of two parts: First, a projection\n\nP γ : Cyl * → Cyl γ such that P γ (Ψ) = Ψ γ (q) := i Ψ(µ i ) χ µi (q) ∈ Cyl γ .\n\nThe state Ψ γ is sometimes refereed to as the 'shadow of Ψ(q) on the lattice γ'. The second step is then to take the inner product between the shadow Ψ γ (q) and the state ψ(q) with respect to the polymer inner product Ψ γ |ψ poly γ . Now this inner product is well defined. Notice that for any given lattice γ the corresponding projector P γ can be intuitively interpreted as some kind of 'coarse graining map' from the continuum to the lattice γ. In terms of functions of q the projection is replacing a continuous function defined on R with a function over the lattice γ ⊂ R which is a discrete set simply by restricting Ψ to γ. The finer the lattice the more points that we have on the curve. As we shall see in the second part of this section, there is indeed a precise notion of coarse graining that implements this intuitive idea in a concrete fashion.\n\nIn particular, we shall need to replace the lattice γ with a decomposition of the real line in intervals (having the lattice points as end points). Let us now consider a system in the polymer representation in which a particular lattice γ 0 was chosen, say with points of the form\n\n{q k ∈ R |q k = ka 0 , ∀ k ∈ Z},\n\nnamely a uniform lattice with spacing equal to a 0 . In this case, any Schrödinger wave function (of the type that we consider) will have a unique shadow on the lattice γ 0 . If we refine the lattice γ → γ n by dividing each interval in 2 n new intervals of length a n = a 0 /2 n we have new shadows that have more and more points on the curve. Intuitively, by refining infinitely the graph we would recover the original function Ψ(q). Even when at each finite step the corresponding shadow has a finite norm in the polymer Hilbert space, the norm grows unboundedly and the 10 limit can not be taken, precisely because we can not embed H s into H poly . Suppose now that we are interested in the reverse process, namely starting from a polymer theory on a lattice and asking for the 'continuum wave function' that is best approximated by a wave function over a graph. Suppose furthermore that we want to consider the limit of the graph becoming finer. In order to give precise answers to these (and other) questions we need to introduce some new technology that will allow us to overcome these apparent difficulties. In the remaining of this section we shall recall these constructions for the benefit of the reader. Details can be found in [6] (which is an application of the general formalism discussed in [9] ).\n\nThe starting point in this construction is the concept of a scale C, which allows us to define the effective theories and the concept of continuum limit. In our case a scale is a decomposition of the real line in the union of closed-open intervals, that cover the whole line and do not intersect. Intuitively, we are shifting the emphasis from the lattice points to the intervals defined by the same points with the objective of approximating continuous functions defined on R with functions that are constant on the intervals defined by the lattice. To be precise, we define an embedding, for each scale\n\nC n from H poly to H s by means of a step function: m Ψ(ma n ) χ man (q) → m Ψ(ma n ) χ αm (q) ∈ H s with χ αn (q) a characteristic function on the interval α m = [ma n , (m + 1)a n ).\n\nThus, the shadows (living on the lattice) were just an intermediate step in the construction of the approximating function; this function is piece-wise constant and can be written as a linear combination of step functions with the coefficients provided by the shadows.\n\nThe challenge now is to define in an appropriate sense how one can approximate all the aspects of the theory by means of this constant by pieces functions. Then the strategy is that, for any given scale, one can define an effective theory by approximating the kinetic operator by a combination of the translation operators that shift between the vertices of the given decomposition, in other words by a periodic function in p. As a result one has a set of effective theories at given scales which are mutually related by coarse graining maps. This framework was developed in [6] . For the convenience of the reader we briefly recall part of that framework.\n\nLet us denote the kinematic polymer Hilbert space at the scale C n as H Cn , and its basis elements as e αi,Cn , where α i = [ia n , (i + 1)a n ) ∈ C n . By construction this basis is orthonormal. The basis elements in the dual Hilbert space H * Cn are denoted by ω αi,Cn ; they are also orthonormal. The states ω αi,Cn have a simple action on Cyl, ω αi,Cn (δ x0,q ) = χ αi,Cn (x 0 ). That is, if x 0 is in the interval α i of C n the result is one and it is zero if it is not there.\n\nGiven any m ≤ n, we define d * m,n : H * Cn → H * Cm as the 'coarse graining' map between the dual Hilbert spaces, that sends the part of the elements of the dual basis to zero while keeping the information of the rest: d * m,n (ω αi,Cn ) = ω βj ,Cm if i = j2 n-m , in the opposite case d * m,n (ω αi,Cn ) = 0. At every scale the corresponding effective theory is given by the hamiltonian H n . These Hamiltonians will be treated as quadratic forms,\n\nh n : H Cn → R, given by h n (ψ) = λ 2 Cn (ψ, H n ψ) , ( 27\n\n)\n\nwhere λ 2 Cn is a normalizaton factor. We will see later that this rescaling of the inner product is necessary in order to guarantee the convergence of the renormalized theory. The completely renormalized theory at this scale is obtained as\n\nh ren m := lim Cn→R d ⋆ m,n h n . ( 28\n\n)\n\nand the renormalized Hamiltonians are compatible with each other, in the sense that\n\nd ⋆ m,n h ren n = h ren m .\n\nIn order to analyze the conditions for the convergence in (28) let us express the Hamiltonian in terms of its eigen-covectors end eigenvalues. We will work with effective Hamiltonians that have a purely discrete spectrum\n\n(labelled by ν) H n • Ψ ν,Cn = E ν,Cn Ψ ν,Cn .\n\nWe shall also introduce, as an intermediate step, a cut-off in the energy levels. The origin of this cut-off is in the approximation of the Hamiltonian of our system at a given scale with a Hamiltonian of a periodic system in a regime of small energies, as we explained earlier. Thus, we can write\n\nh ν cut-off m = ν cut-off ν=0 E ν,Cm Ψ ν,Cm ⊗ Ψ ν,Cm , ( 29\n\n)\n\nwhere the eigen covectors Ψ ν,Cm are normalized according to the inner product rescaled by 1 λ 2 Cn , and the cutoff can vary up to a scale dependent bound, ν cut-off ≤ ν max (C m ). The Hilbert space of covectors together with such inner product will be called H ⋆ren Cm . In the presence of a cut-off, the convergence of the microscopically corrected Hamiltonians, equation (28) is equivalent to the existence of the following two limits. The first one is the convergence of the energy levels,\n\nlim Cn→R E ν,Cn = E ren ν . ( 30\n\n)\n\nSecond is the existence of the completely renormalized eigen covectors,\n\nlim Cn→R d ⋆ m,n Ψ ν,Cn = Ψ ren ν,Cm ∈ H ⋆ren Cm ⊂ Cyl ⋆ . ( 31\n\n)\n\nWe clarify that the existence of the above limit means that Ψ ren ν,Cm (δ x0,q ) is well defined for any δ x0,q ∈ Cyl. Notice that this point-wise convergence, if it can take place 11 at all, will require the tuning of the normalization factors λ 2 Cn . Now we turn to the question of the continuum limit of the renormalized covectors. First we can ask for the existence of the limit lim Cn→R\n\nΨ ren ν,Cn (δ x0,q ) ( 32\n\n)\n\nfor any δ x0,q ∈ Cyl. When this limits exists there is a natural action of the eigen covectors in the continuum limit. Below we consider another notion of the continuum limit of the renormalized eigen covectors. When the completely renormalized eigen covectors exist, they form a collection that is d ⋆ -compatible, d ⋆ m,n Ψ ren ν,Cn = Ψ ren ν,Cm . A sequence of d ⋆ -compatible normalizable covectors define an element of ←-H ⋆ren R , which is the projective limit of the renormalized spaces of covectors ←-\n\nH ⋆ren R := ←- lim Cn→R H ⋆ren Cn . ( 33\n\n)\n\nThe inner product in this space is defined by\n\n({Ψ Cn }, {Φ Cn }) ren R := lim Cn→R (Ψ Cn , Φ Cn ) ren Cn .\n\nThe natural inclusion of\n\nC ∞ 0 in ←- H ⋆ren R is by an antilinear map which assigns to any Ψ ∈ C ∞ 0 the d ⋆ -compatible collection Ψ shad Cn := αi ω αi Ψ(L(α i )) ∈ H ⋆ren Cn ⊂ Cyl ⋆ ; Ψ shad\n\nCn will be called the shadow of Ψ at scale C n and acts in Cyl as a piecewise constant function. Clearly other types of test functions like Schwartz functions are also naturally included in ←-H ⋆ren R . In this context a shadow is a state of the effective theory that approximates a state in the continuum theory.\n\nSince the inner product in ←-H ⋆ren R is degenerate, the physical Hilbert space is defined as\n\nH ⋆ phys := ←- H ⋆ren R / ker(•, •) ren R H phys := H ⋆⋆ phys\n\nThe nature of the physical Hilbert space, whether it is isomorphic to the Schrödinger Hilber space, H s , or not, is determined by the normalization factors λ 2 Cn which can be obtained from the conditions asking for compatibility of the dynamics of the effective theories at different scales. The dynamics of the system under consideration selects the continuum limit.\n\nLet us now return to the definition of the Hamiltonian in the continuum limit. First consider the continuum limit of the Hamiltonian (with cut-off) in the sense of its point-wise convergence as a quadratic form. It turns out that if the limit of equation (32) exists for all the eigencovectors allowed by the cut-off, we have h\n\nν cut-off ren R : H poly,x → R defined by h ν cut-off ren R (δ x0,q ) := lim Cn→R h ν cut-off ren n ([δ x0,q ] Cn ). (34)\n\nThis Hamiltonian quadratic form in the continuum can be coarse grained to any scale and, as can be expected, it yields the completely renormalized Hamiltonian quadratic forms at that scale. However, this is not a completely satisfactory continuum limit because we can not remove the auxiliary cut-off ν cut-off . If we tried, as we include more and more eigencovectors in the Hamiltonian the calculations done at a given scale would diverge and doing them in the continuum is just as divergent. Below we explore a more successful path.\n\nWe can use the renormalized inner product to induce an action of the cut-off Hamiltonians on ←-\n\nH ⋆ren R h ν cut-off ren R ({Ψ Cn }) := lim Cn→R h ν cut-off ren n ((Ψ Cn , •) ren Cn ),\n\nwhere we have used the fact that (Ψ Cn , •) ren Cn ∈ H Cn . The existence of this limit is trivial because the renormalized Hamiltonians are finite sums and the limit exists term by term.\n\nThese cut-off Hamiltonians descend to the physical Hilbert space\n\nh ν cut-off ren R ([{Ψ Cn }]) := h ν cut-off ren R ({Ψ Cn }) for any representative {Ψ Cn } ∈ [{Ψ Cn }] ∈ H ⋆ phys .\n\nFinally we can address the issue of removal of the cutoff. The Hamiltonian h ren R : ←-H ⋆ren R → R is defined by the limit h ren R := lim ν cut-off →∞ h ν cut-off ren R when the limit exists. Its corresponding Hermitian form in H phys is defined whenever the above limit exists. This concludes our presentation of the main results of [6] . Let us now consider several examples of systems for which the continuum limit can be investigated."
},
{
"section_type": "OTHER",
"section_title": "VI. EXAMPLES",
"text": "In this section we shall develop several examples of systems that have been treated with the polymer quantization. These examples are simple quantum mechanical systems, such as the simple harmonic oscillator and the free particle, as well as a quantum cosmological model known as loop quantum cosmology."
},
{
"section_type": "OTHER",
"section_title": "A. The Simple Harmonic Oscillator",
"text": "In this part, let us consider the example of a Simple Harmonic Oscillator (SHO) with parameters m and ω, classically described by the following Hamiltonian\n\nH = 1 2m p 2 + 1 2 m ω 2 x 2 .\n\nRecall that from these parameters one can define a length scale D = /mω. In the standard treatment one uses 12 this scale to define a complex structure J D (and an inner product from it), as we have described in detail that uniquely selects the standard Schrödinger representation. At scale C n we have an effective Hamiltonian for the Simple Harmonic Oscillator (SHO) given by\n\nH Cn = 2 ma 2 n 1 -cos a n p + 1 2 m ω 2 x 2 . ( 35\n\n)\n\nIf we interchange position and momentum, this Hamiltonian is exactly that of a pendulum of mass m, length l and subject to a constant gravitational field g:\n\nĤCn = - 2 2ml 2 d 2 dθ 2 + mgl(1 -cos θ)\n\nwhere those quantities are related to our system by,\n\nl = m ω a n , g = ω m a n , θ = p a n\n\nThat is, we are approximating, for each scale C n the SHO by a pendulum. There is, however, an important difference. From our knowledge of the pendulum system, we know that the quantum system will have a spectrum for the energy that has two different asymptotic behaviors, the SHO for low energies and the planar rotor in the higher end, corresponding to oscillating and rotating solutions respectively 2 . As we refine our scale and both the length of the pendulum and the height of the periodic potential increase, we expect to have an increasing number of oscillating states (for a given pendulum system, there is only a finite number of such states). Thus, it is justified to consider the cut-off in the energy eigenvalues, as discussed in the last section, given that we only expect a finite number of states of the pendulum to approximate SHO eigenstates. With these consideration in mind, the relevant question is whether the conditions for the continuum limit to exist are satisfied. This question has been answered in the affirmative in [6] . What was shown there was that the eigen-values and eigen functions of the discrete systems, which represent a discrete and non-degenerate set, approximate those of the continuum, namely, of the standard harmonic oscillator when the inner product is renormalized by a factor λ 2 Cn = 1/2 n . This convergence implies that the continuum limit exists as we understand it. Let us now consider the simplest possible system, a free particle, that has nevertheless the particular feature that the spectrum of the energy is continuous.\n\n2 Note that both types of solutions are, in the phase space, closed.\n\nThis is the reason behind the purely discrete spectrum. The distinction we are making is between those solutions inside the separatrix, that we call oscillating, and those that are above it that we call rotating."
},
{
"section_type": "OTHER",
"section_title": "B. Free Polymer Particle",
"text": "In the limit ω → 0, the Hamiltonian of the Simple Harmonic oscillator (35) goes to the Hamiltonian of a free particle and the corresponding time independent Schrödinger equation, in the p-polarization, is given by\n\n2 ma 2 n (1 -cos a n p ) -E Cn ψ(p) = 0\n\nwhere we now have that p ∈ S 1 , with p ∈ (-π an , π an ). Thus, we have\n\nE Cn = 2 ma 2 n 1 -cos a n p ≤ E Cn,max ≡ 2 2 ma 2 n . ( 36\n\n)\n\nAt each scale the energy of the particle we can describe is bounded from above and the bound depends on the scale. Note that in this case the spectrum is continuous, which implies that the ordinary eigenfunctions of the Hilbert are not normalizable. This imposes an upper bound in the value that the energy of the particle can have, in addition to the bound in the momentum due to its \"compactification\". Let us first look for eigen-solutions to the time independent Schrödinger equation, that is, for energy eigenstates. In the case of the ordinary free particle, these correspond to constant momentum plane waves of the form e ±( ipx ) and such that the ordinary dispersion relation p 2 /2m = E is satisfied. These plane waves are not square integrable and do not belong to the ordinary Hilbert space of the Schrödinger theory but they are still useful for extracting information about the system. For the polymer free particle we have,\n\nψCn (p) = c 1 δ(p -P Cn ) + c 2 δ(p + P Cn )\n\nwhere P Cn is a solution of the previous equation considering a fixed value of E Cn . That is,\n\nP Cn = P (E Cn ) = a n arccos 1 - ma 2 n 2 E Cn\n\nThe inverse Fourier transform yields, in the 'x representation',\n\nψ Cn (x j ) = 1 √ 2π π /an -π /an ψ(p) e i an p j dp = = √ 2π a n c 1 e ixj PC n / + c 2 e -ixj PC n / . ( 37\n\n)\n\nwith x j = a n j for j ∈ Z. Note that the eigenfunctions are still delta functions (in the p representation) and thus not (square) normalizable with respect to the polymer inner product, that in the p polarization is just given by the ordinary Haar measure on S 1 , and there is no quantization of the momentum (its spectrum is still truly continuous).\n\nLet us now consider the time dependent Schrödinger equation, i ∂ t Ψ(p, t) = Ĥ • Ψ(p, t).\n\nWhich now takes the form,\n\n∂ ∂t Ψ(p, t) = -i m a n (1 -cos (a n p/ )) Ψ(p, t)\n\nthat has as its solution, Ψ(p, t) = e -i m an (1-cos (an p/ )) t ψ(p) = e (-iEC n / ) t ψ(p) for any initial function ψ(p), where E Cn satisfy the dispersion relation (36) . The wave function Ψ(x j , t), the x j -representation of the wave function, can be obtained for any given time t by Fourier transforming with (37) the wave function Ψ(p, t).\n\nIn order to check out the convergence of the microscopically corrected Hamiltonians we should analyze the convergence of the energy levels and of the proper covectors. In the limit n → ∞, E Cn → E = p 2 /2m so we can be certain that the eigen-values for the energy converge (when fixing the value of p). Let us write the proper covector as Ψ Cn = (ψ Cn , •) ren Cn ∈ H ⋆ren Cn . Then we can bring microscopic corrections to scale C m and look for convergence of such corrections\n\nΨ ren Cm . = lim n→∞ d ⋆ m,n Ψ Cn .\n\nIt is easy to see that given any basis vector e αi ∈ H Cm the following limit Ψ ren Cm (e αi,Cm ) = lim Cn→∞ Ψ Cn (d n,m (e αi,Cm )) exists and is equal to\n\nΨ shad Cm (e αi,Cm ) = [d ⋆ Ψ Schr ](e αi,Cm ) = Ψ Schr (ia m )\n\nwhere Ψ shad Cm is calculated using the free particle Hamiltonian in the Schrödinger representation. This expression defines the completely renormalized proper covector at the scale C m ."
},
{
"section_type": "OTHER",
"section_title": "C. Polymer Quantum Cosmology",
"text": "In this section we shall present a version of quantum cosmology that we call polymer quantum cosmology. The idea behind this name is that the main input in the quantization of the corresponding mini-superspace model is the use of a polymer representation as here understood. Another important input is the choice of fundamental variables to be used and the definition of the Hamiltonian constraint. Different research groups have made different choices. We shall take here a simple model that has received much attention recently, namely an isotropic, homogeneous FRW cosmology with k = 0 and coupled to a massless scalar field ϕ. As we shall see, a proper treatment of the continuum limit of this system requires new tools under development that are beyond the scope of this work. We will thus restrict ourselves to the introduction of the system and the problems that need to be solved.\n\nThe system to be quantized corresponds to the phase space of cosmological spacetimes that are homogeneous and isotropic and for which the homogeneous spatial slices have a flat intrinsic geometry (k = 0 condition). The only matter content is a mass-less scalar field ϕ. In this case the spacetime geometry is given by metrics of the form:\n\nds 2 = -dt 2 + a 2 (t) (dx 2 + dy 2 + dz 2 )\n\nwhere the function a(t) carries all the information and degrees of freedom of the gravity part. In terms of the coordinates (a, p a , ϕ, p ϕ ) for the phase space Γ of the theory, all the dynamics is captured in the Hamiltonian constraint C := -3 8\n\np 2 a |a| + 8πG p 2 ϕ 2|a| 3 ≈ 0\n\nThe first step is to define the constraint on the kinematical Hilbert space to find physical states and then a physical inner product to construct the physical Hilbert space. First note that one can rewrite the equation as:\n\n3 8 p 2 a a 2 = 8πG p 2 ϕ 2\n\nIf, as is normally done, one chooses ϕ to act as an internal time, the right hand side would be promoted, in the quantum theory, to a second derivative. The left hand side is, furthermore, symmetric in a and p a . At this point we have the freedom in choosing the variable that will be quantized and the variable that will not be well defined in the polymer representation. The standard choice is that p a is not well defined and thus, a and any geometrical quantity derived from it, is quantized. Furthermore, we have the choice of polarization on the wave function. In this respect the standard choice is to select the a-polarization, in which a acts as multiplication and the approximation of p a , namely sin(λ p a )/λ acts as a difference operator on wave functions of a. For details of this particular choice see [5] . Here we shall adopt the opposite polarization, that is, we shall have wave functions Ψ(p a , ϕ). Just as we did in the previous cases, in order to gain intuition about the behavior of the polymer quantized theory, it is convenient to look at the equivalent problem in the classical theory, namely the classical system we would get be approximating the non-well defined observable (p a in our present case) by a well defined object (made of trigonometric functions). Let us for simplicity choose to replace p a → sin(λ p a )/λ. With this choice we get an effective classical Hamiltonian constraint that 14 depends on λ:\n\nC λ := - 3 8 sin(λ p a ) 2 λ 2 |a| + 8πG p 2 ϕ 2|a| 3 ≈ 0\n\nWe can now compute effective equations of motion by means of the equations: Ḟ := {F, C λ }, for any observable F ∈ C ∞ (Γ), and where we are using the effective (first order) action:\n\nS λ = dτ (p a ȧ + p ϕ φ -N C λ )\n\nwith the choice N = 1. The first thing to notice is that the quantity p ϕ is a constant of the motion, given that the variable ϕ is cyclic. The second observation is that φ = 8π G pϕ |a| 3 has the same sign as p ϕ and never vanishes. Thus ϕ can be used as a (n internal) time variable. The next observation is that the equation for ȧ a 2 , namely the effective Friedman equation, will have a zero for a non-zero value of a given by\n\na * = 32πG 3 λ 2 p 2 ϕ .\n\nThis is the value at which there will be bounce if the trajectory started with a large value of a and was contracting. Note that the 'size' of the universe when the bounce occurs depends on both the constant p ϕ (that dictates the matter density) and the value of the lattice size λ. Here it is important to stress that for any value of p ϕ (that uniquely fixes the trajectory in the (a, p a ) plane), there will be a bounce. In the original description in terms of Einstein's equations (without the approximation that depends on λ), there in no such bounce. If ȧ < 0 initially, it will remain negative and the universe collapses, reaching the singularity in a finite proper time. What happens within the effective description if we refine the lattice and go from λ to λ n := λ/2 n ? The only thing that changes, for the same classical orbit labelled by p ϕ , is that the bounce occurs at a 'later time' and for a smaller value of a * but the qualitative picture remains the same. This is the main difference with the systems considered before. In those cases, one could have classical trajectories that remained, for a given choice of parameter λ, within the region where sin(λp)/λ is a good approximation to p. Of course there were also classical trajectories that were outside this region but we could then refine the lattice and find a new value λ ′ for which the new classical trajectory is well approximated. In the case of the polymer cosmology, this is never the case: Every classical trajectory will pass from a region where the approximation is good to a region where it is not; this is precisely where the 'quantum corrections' kick in and the universes bounces.\n\nGiven that in the classical description, the 'original' and the 'corrected' descriptions are so different we expect that, upon quantization, the corresponding quantum theories, namely the polymeric and the Wheeler-DeWitt will be related in a non-trivial way (if at all).\n\nIn this case, with the choice of polarization and for a particular factor ordering we have,\n\n1 λ sin(λp a ) ∂ ∂p a 2 + 32π 3 ℓ 2 p ∂ 2 ∂ϕ 2 • Ψ(p a , ϕ) = 0\n\nas the Polymer Wheeler-DeWitt equation.\n\nIn order to approach the problem of the continuum limit of this quantum theory, we have to realize that the task is now somewhat different than before. This is so given that the system is now a constrained system with a constraint operator rather than a regular non-singular system with an ordinary Hamiltonian evolution. Fortunately for the system under consideration, the fact that the variable ϕ can be regarded as an internal time allows us to interpret the quantum constraint as a generalized Klein-Gordon equation of the form\n\n∂ 2 ∂ϕ 2 Ψ = Θ λ • Ψ\n\nwhere the operator Θ λ is 'time independent'. This allows us to split the space of solutions into 'positive and negative frequency', introduce a physical inner product on the positive frequency solutions of this equation and a set of physical observables in terms of which to describe the system. That is, one reduces in practice the system to one very similar to the Schrödinger case by taking the positive square root of the previous equation:\n\n∂ ∂ϕ Ψ = √ Θ λ • Ψ.\n\nThe question we are interested is whether the continuum limit of these theories (labelled by λ) exists and whether it corresponds to the Wheeler-DeWitt theory. A complete treatment of this problem lies, unfortunately, outside the scope of this work and will be reported elsewhere [12] ."
},
{
"section_type": "DISCUSSION",
"section_title": "VII. DISCUSSION",
"text": "Let us summarize our results. In the first part of the article we showed that the polymer representation of the canonical commutation relations can be obtained as the limiting case of the ordinary Fock-Schrödinger representation in terms of the algebraic state that defines the representation. These limiting cases can also be interpreted in terms of the naturally defined coherent states associated to each representation labelled by the parameter d, when they become infinitely 'squeezed'. The two possible limits of squeezing lead to two different polymer descriptions that can nevertheless be identified, as we have also shown, with the two possible polarizations for an abstract polymer representation. This resulting theory has, however, very different behavior as the standard one: The Hilbert space is non-separable, the representation is unitarily inequivalent to the Schrödinger one, and natural operators such as p are no longer well defined. This particular limiting construction of the polymer theory can shed some light for more complicated systems such as field theories and gravity.\n\nIn the regular treatments of dynamics within the polymer representation, one needs to introduce some extra structure, such as a lattice on configuration space, to construct a Hamiltonian and implement the dynamics for the system via a regularization procedure. How does this resulting theory compare to the original continuum theory one had from the beginning? Can one hope to remove the regulator in the polymer description? As they stand there is no direct relation or mapping from the polymer to a continuum theory (in case there is one defined). As we have shown, one can indeed construct in a systematic fashion such relation by means of some appropriate notions related to the definition of a scale, closely related to the lattice one had to introduce in the regularization. With this important shift in perspective, and an appropriate renormalization of the polymer inner product at each scale one can, subject to some consistency conditions, define a procedure to remove the regulator, and arrive to a Hamiltonian and a Hilbert space.\n\nAs we have seen, for some simple examples such as a free particle and the harmonic oscillator one indeed recovers the Schrödinger description back. For other systems, such as quantum cosmological models, the answer is not as clear, since the structure of the space of classical solutions is such that the 'effective description' intro-duced by the polymer regularization at different scales is qualitatively different from the original dynamics. A proper treatment of these class of systems is underway and will be reported elsewhere [12] .\n\nPerhaps the most important lesson that we have learned here is that there indeed exists a rich interplay between the polymer description and the ordinary Schrödinger representation. The full structure of such relation still needs to be unravelled. We can only hope that a full understanding of these issues will shed some light in the ultimate goal of treating the quantum dynamics of background independent field systems such as general relativity."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "We thank A. Ashtekar, G. Hossain, T. Pawlowski and P. Singh for discussions. This work was in part supported by CONACyT U47857-F and 40035-F grants, by NSF PHY04-56913, by the Eberly Research Funds of Penn State, by the AMC-FUMEC exchange program and by funds of the CIC-Universidad Michoacana de San Nicolás de Hidalgo.\n\n[1] R. Beaume, J. Manuceau, A. Pellet and M. Sirugue, \"Translation Invariant States In Quantum Mechanics,\" Commun. Math. Phys. 38, 29 (1974); W. E. Thirring and H. Narnhofer, \"Covariant QED without indefinite metric,\" Rev. Math. Phys. 4, 197 (1992); F. Acerbi, G. Morchio and F. Strocchi, \"Infrared singular fields and nonregular representations of canonical commutation relation algebras\", J. Math. Phys. 34, 899 (1993); F. Cavallaro, G. Morchio and F. Strocchi, \"A generalization of the Stone-von Neumann theorem to non-regular representations of the CCR-algebra\", Lett. Math. Phys. 47 307 (1999); H. Halvorson, \"Complementarity of Representations in quantum mechanics\", Studies in History and Philosophy of Modern Physics 35 45 (2004). [2] A. Ashtekar, S. Fairhurst and J.L. Willis, \"Quantum gravity, shadow states, and quantum mechanics\", Class. Quant. Grav. 20 1031 (2003) [arXiv:gr-qc/0207106]. [3] K. Fredenhagen and F. Reszewski, \"Polymer state approximations of Schrödinger wave functions\", Class. Quant. Grav. 23 6577 (2006) [arXiv:gr-qc/0606090]. [4] M. Bojowald, \"Loop quantum cosmology\", Living Rev. Rel. 8, 11 (2005) [arXiv:gr-qc/0601085]; A. Ashtekar, M. Bojowald and J. Lewandowski, \"Mathematical structure of loop quantum cosmology\", Adv. Theor. Math. Phys. 7 233 (2003) [arXiv:gr-qc/0304074]; A. Ashtekar, T. Pawlowski and P. Singh, \"Quantum nature of the big bang: Improved dynamics\" Phys. Rev. D 74 084003 (2006) [arXiv:gr-qc/0607039] [5] V. Husain and O. Winkler, \"Semiclassical states for quantum cosmology\" Phys. Rev. D 75 024014 (2007) [arXiv:gr-qc/0607097]; V. Husain V and O. Winkler, \"On singularity resolution in quantum gravity\", Phys. Rev. D 69 084016 (2004). [arXiv:gr-qc/0312094]. [6] A. Corichi, T. Vukasinac and J.A. Zapata. \"Hamiltonian and physical Hilbert space in polymer quantum mechanics\", Class. Quant. Grav. 24 1495 (2007) [arXiv:gr-qc/0610072] [7] A. Corichi and J. Cortez, \"Canonical quantization from an algebraic perspective\" (preprint) [8] A. Corichi, J. Cortez and H. Quevedo, \"Schrödinger and Fock Representations for a Field Theory on Curved Spacetime\", Annals Phys. (NY) 313 446 (2004) [arXiv:hep-th/0202070]. [9] E. Manrique, R. Oeckl, A. Weber and J.A. Zapata, \"Loop quantization as a continuum limit\" Class. Quant. Grav. 23 3393 (2006) [arXiv:hep-th/0511222]; E. Manrique, R. Oeckl, A. Weber and J.A. Zapata, \"Effective theories and continuum limit for canonical loop quantization\" (preprint) [10] D.W. Chiou, \"Galileo symmetries in polymer particle representation\", Class. Quant. Grav. 24, 2603 (2007) [arXiv:gr-qc/0612155]. [11] W. Rudin, Fourier analysis on groups, (Interscience, New York, 1962) [12] A. Ashtekar, A. Corichi, P. Singh, \"Contrasting LQC and WDW using an exactly soluble model\" (preprint); A. Corichi, T. Vukasinac, and J.A. Zapata, \"Continuum limit for quantum constrained system\" (preprint).\n\n16"
}
] | |
arxiv:0704.0015 | 0704.0015 | 1 | 10.1088/1126-6708/2007/05/034 | 736f71b146882dce29dd074d1d2d87f98333dd269e5d47f089d31b0d7812e8e5 | Fermionic superstring loop amplitudes in the pure spinor formalism | The pure spinor formulation of the ten-dimensional superstring leads to
manifestly supersymmetric loop amplitudes, expressed as integrals in pure
spinor superspace. This paper explores different methods to evaluate these
integrals and then uses them to calculate the kinematic factors of the one-loop
and two-loop massless four-point amplitudes involving two and four Ramond
states. | [
"Christian Stahn"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | The pure spinor formulation of the ten-dimensional superstring leads to manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor superspace. This paper explores different methods to evaluate these integrals and then uses them to calculate the kinematic factors of the one-loop and two-loop massless four-point amplitudes involving two and four Ramond states.
Contents 1. Introduction 1 2. Zero mode integration 2 2.1 Symmetry considerations and tensorial formulae 3 2.2 A spinorial formula 5 2.3 Component-based approach 7 3. One-loop amplitudes 7 3.1 Review: four bosons 8 3.2 Four fermions 10 3.3 Two bosons, two fermions 10 4. Two-loop amplitudes 12 4.1 Review: four bosons 13 4.2 Four fermions 14 4.3 Two bosons, two fermions 15
The quantisation of the ten-dimensional superstring using pure spinors as world-sheet ghosts [1] has overcome many difficulties encountered in the Green-Schwarz (GS) and Ramond-Neveu-Schwarz (RNS) formalisms. Most notably, by maintaining manifest spacetime supersymmetry, the pure spinor formalism has yielded super-Poincaré covariant multiloop amplitudes, leading to new insights into perturbative finiteness of superstring theory [2, 3] .
Counting fermionic zero modes is a powerful technique in the computation of loop amplitudes in the pure spinor formalism and has for example been used to show that at least four external states are needed for a non-vanishing massless loop amplitude [2] . Furthermore, the structure of massless four-point amplitudes is relatively simple because all fermionic worldsheet variables contribute only through their zero modes. In the expressions derived for the one-loop [2] and two-loop [4] amplitudes, supersymmetry was kept manifest by expressing the kinematic factors as integrals over pure spinor superspace [5] involving three pure spinors λ and five fermionic superspace coordinates θ, K 1-loop = (λA)(λγ m W )(λγ n W )F mn , K 2-loop = (λγ mnpqr λ)(λγ s W )F mn F pq F rs ,
where the pure spinor superspace integration is denoted by . . . , and A α (x, θ), W α (x, θ) and F mn (x, θ) are the superfields of ten-dimensional Yang-Mills theory.
The kinematic factors in (1.1) have been explicitly evaluated for Neveu-Schwarz states at two loops [6] and one loop [7] , and were found to match the amplitudes derived in the RNS formalism [8] . This provided important consistency checks in establishing the validity of the pure spinor amplitude prescriptions. (Related one-loop calculations had been reported in [9] .)
In this paper, it will be shown how to compute the kinematic factors in (1.1) when the superfields are allowed to contribute fermionic fields, as is relevant for the scattering of fermionic closed string states as well as Ramond/Ramond bosons. It turns out that the calculation of fermionic amplitudes presents no additional difficulties, making (1.1) a good practical starting point for the computation of four-point loop amplitudes in a unified fashion. This practical aspect of the supersymmetric pure spinor amplitudes was also emphasised in [10] , where the tree-level amplitudes were used to construct the fermion and Ramond/Ramond form contributions to the four-point effective action of the type II theories.
This paper is organised as follows. In section 2, different methods to compute pure spinor superspace integrals are explored. These methods are then applied to the explicit evaluation of the kinematic factors of massless four-point amplitudes at the one-loop level in section 3, and at the two-loop level in section 4. In both these sections, the bosonic calculations are briefly reviewed before separately considering the cases of two and four Ramond states. Particular attention will be paid to the constraints imposed by simple exchange symmetries. An appendix contains algorithms which were used to reduce intermediate expressions encountered in the amplitude calculations to a canonical form.
The calculation of scattering amplitudes in the pure spinor formalism leads to integrals over zero modes of the fermionic worldsheet variables λ and θ. Both θ and λ are 16-component Weyl spinors, the λ are commuting and the θ anticommuting, and λ is subject to the pure spinor constraint (λγ m λ) = 0. The amplitude prescriptions [1, 2] require three zero modes of λ and five zero modes of θ to be present, and a Lorentz covariant object T αβγ,δ 1 ...δ 5 ≡ λ α λ β λ γ θ δ 1 . . . θ δ 5 = T (αβγ),[δ 1 ...δ 5 ]
(2.1) was constructed such that the Yang-Mills antighost vertex operator V = (λγ m θ)(λγ n θ)(λγ p θ)(θγ mnp θ) has V = 1 .
(2.2)
In this section, different methods of computing such "pure superspace integrals" are explored. As an example, a typical correlator encountered in the two-loop calculations of section 4 is considered:
F (k i , u i ) = k 2 a k 2 m k 3 p k 4 r (λγ mnpq[r λ)(λγ s] u 1 )(θγ n ab θ)(θγ b u 2 )(θγ q u 3 )(θγ s u 4 ) (2.3)
Here, k i and u i are the momenta and spinor wavefunctions of the four external particles.
One systematic approach to evaluate the zero mode integrals is to find expressions for all tensors that can be formed from (2.1). By Fierz transformations, one can always write the product of two θ spinors as (θγ [3] θ), where γ [k] denotes the antisymmetrised product of k gamma matrices. Due to the pure spinor constraint, the only bilinear in λ is (λγ [5] λ), and it is thus sufficient to consider the three cases (λγ [5] λ)(λ{γ [1] or γ [3] or γ [5] }θ)(θγ [3] θ)(θγ [3] θ) .
(2.4)
Lorentz invariance then implies that it must be possible to express these tensors as sums of suitably symmetrised products of metric tensors, resulting in a parity-even expression, plus a parity-odd part made up from terms which in addition contain an epsilon tensor. The parity-even parts may be constructed [6] starting from the most general ansatz compatible with the symmetries of the correlator and then using spinor identities along with the normalisation (2.2) to determine all coefficients in the ansatz. Duality properties of the spinor bilinears can be used to determine the parity-odd part [7] . An extensive (and almost exhaustive) list of correlators is found in [11] , including the (λγ [1] θ) and (λγ [3] θ) cases of the above list:
(λγ mnpqr λ)(λγ u θ)(θγ f gh θ)(θγ jkl θ) = -4 35 δ mnpqr mnpqr + 1 5! ε mnpqr mnpqr δ mn f g δ pq jk (δ r l δ h u + δ r h δ l u -δ r u δ h l ) [f gh][jkl]
(2.5)
(λγ mnpqr λ)(λγ stu θ)(θγ f gh θ)(θγ jkl θ) = -24 35 δ mnpqr mnpqr + 1 5! ε mnpqr mnpqr δ m j δ np f g δ q s δ t l (δ r h δ k u -δ k h δ r u ) [f gh][jkl](f gh↔jkl) (2.6)
(Here, the brackets (f gh ↔ jkl) denote symmetrisation under simultaneous interchange of f gh with ijk, with weight one.) The remaining correlator with the (λγ [5] θ) factor can be derived in the same way, using an ansatz consisting of six parity-even structures. Taking a trace between the two γ [5] factors and noting that η ar (λγ mnpqr λ)(λγ abcde θ) . . . = -4 (λγ mnpq[b λ)(λγ cde] θ) . . . , one finds a relation to (2.6) . This is sufficient to determine all coefficients in the ansatz, and the result is
(λγ mnpqr λ)(λγ abcde θ)(θγ f gh θ)(θγ jkl θ) = 16 7 δ mnpqr mnpqr + 1 5! ε mnpqr mnpqr × δ mnp abc δ f j δ d g δ q k (-δ e h δ r l + 2δ e l δ r h ) + δ mn ab δ cd f g δ pq jk (δ e h δ r l -3δ e l δ r h ) [abcde][f gh][jkl](f gh↔jkl) (2.7)
One may find it surprising that the derivation of these tensorial expressions only made use of properties of (pure) spinors, and of the normalisation condition (2.2). However, it can be seen from representation theory that the correlator (2.1) is uniquely characterised, up to normalisation, by its symmetry. To see this, note that [12] the spinor products λ 3 and θ 5 transform in
λ (α λ β λ γ) : Sym 3 S + = [00003] ⊕ [10001] θ [δ 1 . . . θ δ 5 ] : Alt 5 S + = [00030] ⊕ [11010] . (2.8)
(Here, λ and θ are taken to be in the S + irrep of SO (1, 9) , with Dynkin label [00001] .) The tensor product of these contains only one copy of the trivial representation. This applies to any spinors λ, which means that the pure spinor property cannot be essential to the derivation of the tensorial identities. The use of the pure spinor constraint merely allows for simpler derivations of the same identities.
As an illustration of this approach, consider the correlator of eq. ( 2 .3). Leaving the momenta aside for the moment by setting
F = k 2 a k 2 m k 3 p k 4 r F , the task is to compute F = (λγ mnpq[r λ)(λγ s] u 1 )(θγ n ab θ)(θγ b u 2 )(θγ q u 3 )(θγ s u 4 ) .
After applying two Fierz transformations,
F = 1 16 (λγ mnpq[r| λ)(λγ c θ)(θγ n ab θ)(θγ jkl θ) (u 1 γ |s] γ c γ b u 2 ) + 1 3!•16 (λγ mnpq[r| λ)(λγ cde θ)(θγ n ab θ)(θγ jkl θ) (u 1 γ |s] γ cde γ b u 2 ) + 1 2•5!•16 (λγ mnpq[r| λ)(λγ cdef g θ)(θγ n ab θ)(θγ jkl θ) (u 1 γ |s] γ cdef g γ b u 2 )
× 1 3!•16 (u 3 γ q γ jkl γ s u 4 ) , one obtains a combination of the fundamental correlators listed in (2.5), (2.6) and (2.7). A reliable evaluation of the numerous index symmetrisations is made possible by the use of a computer algebra program. In doing these calculations with Mathematica, an essential tool is the GAMMA package [13] , expanding the products of gamma matrices in a γ [k] basis. The result consists of two parts, F = F (δ) + F (ε) , with
F (δ) = 1 560 (u 1 γ mpr u 2 )(u 3 γ a u 4 ) + 7 720 δ a p δ m r (u 1 γ i u 2 )(u 3 γ i u 4 ) + . . . -1 1680 (u 1 γ ai 1 i 2 u 2 )(u 3 γ mpr i 1 i 2 u 4 ) (92 terms) (2.9) F (ε) = -1 1209600 ε i 1 ...i 7 mpr (u 1 γ i 1 ...i 7 u 2 )(u 3 γ a u 4 ) + . . . -1 604800 ε ampr i 1 ...i 6 (u 1 γ i 3 ...i 9 u 2 )(u 3 γ i 1 i 2 i 7 i 8 i 9 u 4 ) (34 terms) (2.10)
The epsilon tensors in the second part can be eliminated using the fact that the u i are chiral spinors: If all the indices on γ [k] u i are contracted into an epsilon tensor, one uses
ε i 1 ...i k ′ j 1 ...j k γ j 1 ...j k γ 11 = (-) 1 2 k(k+1) k! γ i 1 ...i k ′ , (2.11)
where γ 11 = 1 10! ε i 0 ...i 9 γ i 0 ...i 9 . More generally, if all but r indices of γ [k] u i are contracted,
ε i 1 ...i k ′ j 1 ...j k γ p 1 ...prj 1 ...j k γ 11 = (-) 1 2 k(k+1) k! k ′ ! (k ′ -r)! δ pr...p 1 [i 1 ...ir γ i r+1 ...i ′ k ] .
(2.12)
The result of these manipulations is
F (ε) = -1 560 (u 1 γ mpr u 2 )(u 3 γ a u 4 ) -1 280 δ p r (u 1 γ ami u 2 )(u 3 γ i u 4 ) + . . . + 9 11200 (u 1 γ i 1 i 2 i 3 u 2 )(u 3 γ ampr i 1 i 2 i 3 u 4 ) (53 terms) (2.13)
(Note that while the epsilon terms in the basic correlator formulae were easily obtained from the delta terms by using Poincaré duality, this cannot be done here in any obvious way.) The last step in the evaluation of (2.3) is to contract with the momenta, F = k 2 a k 2 m k 3 p k 4 r F , and to simplify the expressions using the on-shell identities i k i = 0, k 2 i = 0, / k i u i = 0. It is shown in appendix A.2 that there are only ten independent scalars, denoted by B 1 . . . B 10 , that can be formed from four momenta and the four spinors u 1 . . . u 4 . With respect to this basis, the result is
F (δ) = 1 48•10080 695s 12 (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) + • • • + 233s 2 13 (u 1 γ a u 2 )(u 3 γ a u 4 ) ( 7 terms
) = 1 48•10080 (695, 775, 0, -80, 356, 356, 0, 233, 233, 0) B 1 ...B 10 , F (ε) = 1 48•10080 (-23, -7, 0, -16, 28, 28, 0, 7, 7, 0) B 1 ...B 10 , F = 1 10080 (14, 16, 0, -2, 8, 8, 0, 5, 5, 0) B 1 ...B 10 , (2.14)
where
s ij = k i • k j .
While the derivation of tensorial identities for correlators of the form (2.4) is relatively straightforward and elegant, it may be a tedious task to transform the expressions encountered in amplitude calculations to match this pattern. As seen in the example calculated above, this is particularly true if additional spinors are involved, making it necessary to apply Fierz transformations. It is therefore desirable to use a covariant correlator expression with open spinor indices. Such an expression was given in [1, 2] :
T αβγ,δ 1 ...δ 5 = N -1 (γ m ) αδ 1 (γ n ) βδ 2 (γ p ) γδ 3 (γ mnp ) δ 4 δ 5 (αβγ)[δ 1 ...δ 5 ] , (2.15)
where N is a normalisation constant and the brackets ()[] denote (anti-)symmetrisation with weight one. (Note that the right hand side is automatically gamma-matrix traceless: any gamma-trace
(γ r ) αβ × (γ m ) α[δ 1 | (γ n ) β|δ 2 | (γ p ) γ|δ 3 (γ mnp ) δ 4 δ 5 ] = -(γ mnr ) [δ 1 δ 2 (γ mnp ) δ 3 δ 4 (γ p ) δ 5 ]γ = 0
vanishes due to the double-trace identity (γ ab θ) α (θγ abc θ) = 0, which follows from the fact that the tensor product (Alt 3 S + ) ⊗ S -does not contain a vector representation and therefore the vector (ψγ ab θ)(θγ abc θ) has to vanish for all spinors ψ, and can also be shown by applying a Fierz transformation.) This prescription was originally motivated [2] by the fermionic expansion of the Yang-Mills antighost vertex operator V ,
V = T αβγ,δ 1 ...δ 5 λ α λ β λ γ θ δ 1 . . . θ δ 5 (2.16) with T αβγ,δ 1 ...δ 5 = (γ m ) αδ 1 (γ n ) βδ 2 (γ p ) γδ 3 (γ mnp ) δ 4 δ 5 (αβγ)[δ 1 ...δ 5 ] ,
where T is related to T by a parity transformation, up to the overall constant N . (Since T is uniquely determined by its symmetries, any covariant expression will be proportional to T , after symmetrisation of the spinor indices, and this is merely the simplest choice.) Equation (2.15) immediately yields an algorithm to convert any correlator into traces of gamma matrices or, if additional spinors are involved, bilinears in those spinors. It is, however, already very tiresome to determine the normalisation constant N by hand. The main advantage of this approach is that it clearly lends itself to implementation on a computer algebra system, which can easily carry out the spinor index symmetrisations, simplify the gamma products (again using the GAMMA package), and compute the traces. For example,
N V = (γ m ) αδ 1 (γ n ) βδ 2 (γ p ) γδ 3 (γ mnp ) δ 4 δ 5 (αβγ)[δ 1 ...δ 5 ] (γ x ) αδ 1 (γ y ) βδ 2 (γ z ) γδ 3 (γ xyz ) δ 4 δ 5 = -1 60 Tr(γ x γ m ) Tr(γ y γ n ) Tr(γ z γ p ) Tr(γ xyz γ pnm ) + . . . -1 60 Tr(γ z γ pnm γ zyx γ n γ x γ m γ y γ p ) (60 terms) = 5160960 .
The correct normalisation is therefore obtained by setting N = 5160960.
Returning to the example correlator (2.3), one finds that the calculation is by far simpler than with the previous method. After carrying out the symmetrisations (αβγ)[δ i ], one obtains
N F = 1 60 Tr(γ x γ ab n γ y γ mnpq[r| )(u 3 γ q γ xyz γ s u 4 )(u 1 γ |s] γ z γ b u 2 ) + . . . -1 30 (u 2 γ b γ xyz γ q u 3 )(u 1 γ s γ y γ ab n γ x γ mnpq[r γ z γ s] u 4 ) , ( 24 terms)
where elementary index re-sorting has reduced the number of terms from 60 to 24. Expanding the gamma products leads to
N F = 476 5 δ p r (u 1 γ m u 4 )(u 2 γ a u 3 ) + • • • + 8 15 (u 1 γ ai 1 i 2 i 3 i 4 u 2 )(u 3 γ mpr i 1 i 2 i 3 i 4 u 4 ) , ( 294 terms)
which, in contrast to (2.10), contains no epsilon terms as there are not enough free indices present. Note that this intermediate result contains terms with with u 1 paired with u 3 or u 4 , so it is not possible to directly compare to eqs. (2.9) and (2.13). However, after contracting with the momenta k foot_0 a k 2 m k 3 p k 4 r and decomposing the result in the basis B 1 . . . B 10 , one again obtains
F = 1 10080 (14, 16, 0, -2, 8, 8, 0, 5, 5, 0) B 1 ...B 10 , (2.17)
in agreement with (2.14) .
The algorithm just outlined will be the method of choice for all correlator calculations in the later sections of this paper and can easily be applied to a wider range of problems. The only limitation is that the larger the number of gamma matrices and open indices of the correlator, the slower the computer evaluation will be. For example, the correlator considered in eq. (5.2) of [11] ,
t mnm 1 n 1 ...m 4 n 4 10 ≡ (λγ p γ m 1 n 1 θ)(λγ q γ m 2 n 2 θ)(λγ r γ m 3 n 3 θ)(θγ m γ n γ pqr γ m 4 n 4 θ) = -2 45 η mn t m 1 n 1 ...m 4 n 4
A third method to evaluate the zero mode integrals consists of choosing a gamma matrix representation, expanding the integrand as a polynomial in spinor components, and then applying (2.15) to the individual monomials. This procedure seems particularly appealing if at some stage of the calculation one works with a matrix representation anyhow, in order to reduce the results to a canonical form (e.g. as outlined in appendix A). An efficient decomposition algorithm (of k 4 u 1 u 2 u 3 u 4 scalars, say) only needs a few non-zero momentum and spinor wavefunction components to distinguish all independent scalars, and therefore k and u can be replaced by sparse vectors. Furthermore, a trivial observation allows for a much quicker numeric evaluation of correlator components than a naive use of (2.15): In view of (2.16), one can equivalently compute the components of the paritytransformed expression V = ( λγ m θ)( λγ n θ)( λγ p θ)( θγ mnp θ), where λ and θ are spinors of chirality opposite to that of λ, θ. In the representation given in appendix B, V coincides with V | λ→ λ,θ→ θ, and
V = 192 λ 9 λ 9 λ 9 θ 1 θ 2 θ 3 θ 4 θ 9 + • • • + 480 λ 1 λ 2 λ 3 θ 1 θ 9 θ 10 θ 13 θ 15 + . . . ( 100352 terms)
The monomials in the fermionic expansion of V then correspond to the arguments of non-zero correlators, and the coefficients of those monomials are, up to normalisation and symmetry factors, the correlator values.
Unfortunately, it turns out that the complexity of typical correlators (e.g. the one given in (2.3)) makes it difficult to carry out the expansion in fermionic components in any straightforward way and limits this method to special applications. For example, the coefficients in (2.18) can be checked relatively easily by choosing particular index values, such as (λγ p γ 12 θ)(λγ q γ 21 θ)(λγ r γ 34 θ)(θγ 0 γ 0 γ pqr γ 43 θ)
= 12 λ 1 λ 1 λ 1 θ 1 θ 9 θ 10 θ 11 θ 12 + • • • + 12 λ 16 λ 16 λ 16 θ 5 θ 6 θ 7 θ 8 θ 16 = 1 45 .
(For fixed values of pqr, one gets no more than about 10 5 monomials of the form λ 3 θ 5 ). This approach may thus still be helpful in situations where the result has been narrowed down to a simple ansatz.
The amplitude for the scattering of four massless states of the type IIB superstring was computed [2] in the pure spinor formalism as
A = K K d 2 τ (Im τ ) 5 d 2 z 2 d 2 z 3 d 2 z 4 i<j G(z i , z j ) k i •k j , (3.1)
where G(z i , z j ) is the scalar Green's function, and the kinematic factor is given by the product K K of left-and right-moving open superstring expressions,
K 1-loop = (λA 1 )(λγ m W 2 )(λγ n W 3 )F 4,mn + cycl(234) . (3.2)
Here the indices 1 . . . 4 label the external states and "• • •+ cycl(234) " denotes the addition of two other terms obtained by cyclic permutation of the indices 234. The spinor superfield A α and its supercovariant derivatives, the vector gauge superfield A m = foot_1 8 γ αβ m D α A β as well as the spinor and vector field strengths The superfields A α and W α as well as the gaugino field ûα are anticommuting. 1 To facilitate computer calculations involving polynomials in the spinor components, and for easier comparison with the literature, it will be more convenient to work with commuting fermion wavefunctions u α . Fortunately, as the kinematic factors with fermionic external states are multilinear functions of the distinctly labelled spinors ûi , it is straightforward to translate between the two conventions: Any monomial expression in û1 . . . û4 (and possibly fermionic coordinates θ) corresponds to the same expression in u 1 . . . u 4 , multiplied by the signature of the permutation sorting the ûi (and any θ variables) into some fixed order, such as (θ
W α = 1 10 (γ m ) αβ (D β A m -∂ m A β ) and F mn = 1 8 (γ mn ) α β D α W β = 2∂ [m A n] ,
• • • θ)û α 1 1 ûα 2 2 ûα 3 3 ûα 4
4 . Choosing a gauge where θ α A α = 0, the on-shell identities
2D (α A β) = γ m αβ A m , D α W β = 1 4 (γ mn ) α β F mn
have been used to derive recursive relations [10, 14, 15] for the fermionic expansion
A (n) α = 1 n+1 (γ m θ) α A (n-1) m , A (n) m = 1 n (θγ m W (n-1) ) , W α(n) = -1 2n (γ mn θ) α ∂ m A (n-1) n ,
where
f (n) = 1 n! θ αn • • • θ α 1 (D α 1 • • • D αn f )|.
These recursion relations were explicitly solved in [10] , reducing the fermionic expansion to a simple repeated application of the derivative operator O m q = 1 2 (θγ m qp θ)∂ p :
A (2k) m = 1 (2k)! [O k ] m q ζ q , A (2k+1) m = 1 (2k+1)! [O k ] m q (θγ q û) . (3.3)
With this solution at hand, one has all ingredients to evaluate the kinematic factor (3.2) for the three cases of zero, two, or four fermionic states.
The kinematic factor involving four bosons was considered in [7] and this calculation will now be reviewed briefly. First, note that the outcome is not fixed by symmetry: The result must be gauge invariant [2] and therefore expressible in terms of the field strengths F 1 . . . F 4 .
The cyclic symmetrisation in (3.2) yields expressions symmetric in F 2 , F 3 , F 4 , and acting on scalars constructed from the F i only, the (234) symmetrisation is equivalent to complete symmetrisation in all labels (1234). Thus the result must be a linear combination of the two gauge invariant symmetric F 4 scalars, namely the single trace Tr(F (1 F 2 F 3 F 4) ) and double trace Tr(F (1 F 2 ) Tr(F 3 F 4) ), leaving one relative coefficient to be determined. Since all four states are of the same kind, one may first evaluate the correlator for one labelling and then carry out the cyclic symmetrisation:
K (4B) 1-loop = (λA 1 )(λγ m W 2 )(λγ n W 3 )F 4,mn 4B + cycl (234) .
The different ways to saturate θ 5 result in a sum of terms of the form
X ABCD = (λA (A) 1 )(λγ m W (B) 2 )(λγ n W (C) 3 )F (D) 4,mn
(λA 1 )(λγ m W 2 )(λγ n W 3 )F 4,mn 4B = X 3110 + X 1310 + X 1130 + X 1112 .
Note that X 1310 and X 1130 are related by exchange of the labels 2 and 3. This exchange can be carried out after computing the correlator, an operation which will in the following be denoted by π 23 . Using (3.3) for the superfield expansions and replacing ∂ m → ik m , one obtains
X 3110 = -1 512 F 1 mn F 2 pq F 3 rs F 4 tu X3110 , X3110 = (λγ [t| γ pq θ)(λγ |u] γ rs θ)(λγ a θ)(θγ amn θ) , X 1112 = -1 128 ik 4 m ζ 1 n F 2 pq F 3 rs F 4 tu X1112 , X1112 = (λγ [m| γ pq θ)(λγ |a] γ rs θ)(λγ n θ)(θγ a tu θ) , X 1310 = -1 384 ik 3 m ζ 1 n F 2 pq F 3 rs F 4 tu X1310 , X1310 = (λγ [t| γ ma θ)(λγ |u] γ rs θ)(λγ n θ)(θγ a pq θ) .
The method outlined in section 2.2 is readily applicable to these correlators. For example, for X 3111 , the trace evaluation yields
X3110 = N -1 1 60 Tr(γ a γ z ) Tr(γ xyz γ anm ) Tr(γ x γ qp γ [t| ) Tr(γ y γ sr γ |u] ) + • • • • • • + 1 60 Tr(γ [u| γ rs γ zyx γ qp γ |t] γ x γ a γ y γ mna γ z ) ( 60
terms) = 2 35 δ mp rs δ nq tu -1 315 δ mn tu δ pq rs -1 45 δ mn rs δ pq tu + 26 315 δ mn pr δ qs tu [mn][pq][rs][tu](pq↔rs)
Upon contracting with the field strengths, momenta and polarisations, and symmetrising over the cyclic permutations (234) (with weight 3), one finds that all three contributions are separately gauge invariant:
X 3110 + cycl(234) = -11 13440 Tr(F (1 F 2 F 3 F 4) ) + 1 6720 Tr(F (1 F 2 ) Tr(F 3 F 4) ) X 1112 + cycl(234) = -19 53760 Tr(F (1 F 2 F 3 F 4) ) + 31 215040 Tr(F (1 F 2 ) Tr(F 3 F 4) ) (1 + π 23 )X 1310 + cycl(234) = -1 10240 4 Tr(F (1 F 2 F 3 F 4) ) -Tr(F (1 F 2 ) Tr(F 3 F 4) )
The sum X 3110 + X 1112 has the right ratio of single-and double-trace terms to be proportional to the well-known result t 8 F 4 , and the last line exhibits the right ratio by itself. The overall kinematic factor is therefore
K 4B 1-loop = -1 2560 4 Tr(F (1 F 2 F 3 F 4) ) -Tr(F (1 F 2 ) Tr(F 3 F 4) ) = -1 15360 t 8 F 4 , (3.5)
in agreement with the expressions derived in the RNS [16] and Green-Schwarz [17] formalisms.
The four-fermion kinematic factor could be evaluated in the same way as in the four-boson case by summing up all terms X ABCD , A + B + C + D = 5, now with A, B, C even and D odd. Note however that this time, the outcome is fixed by symmetry: The cyclic symmetrisation in (3.2) leads to a completely symmetric dependence on û2 , û3 , û4 , and therefore to a completely antisymmetric dependence on u 2 , u 3 , u 4 . Acting on scalars of the form k 2 u 1 u 2 u 3 u 4 , antisymmetrising over [234] is equivalent to antisymmetrising over [1234], and there is only one completely antisymmetric k 2 u 1 u 2 u 3 u 4 scalar. Without further calculation, one can infer that the kinematic factor is proportional to that scalar,
K 4F 1-loop = const • (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) -(u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) + (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) ,
which of course agrees with the RNS amplitude (see e.g. [16] , eq. (3.67)).
In evaluating (3.2) for two bosons and two fermions, the cyclic symmetrisations affect whether the W and F superfields contribute bosons or fermions. Only the label of the A α superfield stays unaffected, and one has to choose whether it should contribute a boson or a fermion. Since its fermionic expansion starts with the bosonic polarisation vector, A 1,α ∼ (/ ζ 1 θ) α , the calculation can be simplified by choosing a labelling where particle 1 is a fermion. (Of course, the final result must be independent of this choice.) The assignment of the other three labels is then irrelevant and will be chosen as f 1 f 2 b 3 b 4 . Writing out the cyclic permutations, two of the three terms are essentially the same because they are related by interchange of the labels 3 and 4. The kinematic factor is then
K 2B2F 1-loop (f 1 f 2 b 3 b 4 ) = (1 + π 34 ) (λA (even) 1
)(λγ m W (even) 2
)(λγ n W (odd) 3
)F
(even) 4,mn
+ (λA (even) 1
)(λγ m W (odd) 3
)(λγ n W (odd) 4
)F
(odd) 2,mn .
Unlike in the four-fermion calculation, the result is not fixed by symmetry. There are five independent ku 1 u 2 F 3 F 4 scalars (see appendix A, eq. (A.6)), denoted by C 1 . . . C 5 , and there are two independent combinations of these scalars with the required [12] (34) symmetry.
Expanding the superfields and collecting terms with θ 5 , the first line yields a combination of terms X ABCD with A, B, D odd and C even. There is only one θ 5 combination coming from the second line, which will be denoted by X ′ 2111 ≡ (-π 24 )X 2111 :
K 2B2F 1-loop = (1 + π 34 ) (X 4010 + X 2210 + X 2030 + X 2012 ) + X ′ 2111 ,
with the correlators
X 4010 = i 60 k 1 q k 3 b ζ 3 c k 4 m ζ 4 n X4010 , X4010 = (λγ a θ)(θγ a pq θ)(θγ p u 1 )(λγ [m u 2 )(λγ n] γ bc θ) X 2210 = -i 12 k 2 b k 3 d ζ 3 e k 4 m ζ 4 n X2210 , X2210 = (λγ a θ)(θγ a u 1 )(λγ [m| γ bc θ)(θγ c u 2 )(λγ |n] γ de θ) X 2030 = -i 36 k 3 b k 3 d ζ 3 e k 4 m ζ 4 n X2030 , X2030 = (λγ a θ)(θγ a u 1 )(λγ [m u 2 )(λγ n] γ bc θ)(θγ c de θ) X 2012 = -i 12 k 3 b ζ 3 c k 4 m k 4 d ζ 4 e X2012 , X2012 = (λγ a θ)(θγ a u 1 )(λγ [m u 2 )(λγ n] γ bc θ)(θγ n de θ) X ′ 2111 = i 6 k 3 b ζ 3 c k 4 d ζ 4 e k 2 m X′ 2111 , X′ 2111 = (λγ a θ)(θγ a u 1 )(λγ [m| γ bc θ)(λγ |n] γ de θ)(θγ n u 2 )
(The numerical coefficient in X ′ 2111 includes a sign coming from the θ, û ordering: there is an odd number of θs between u 1 and u 2 .) Evaluating these expressions as outlined in section 2.2, the spinor wavefunctions u i present no complication. The last part takes the simplest form: One finds
(λγ a θ)(θγ a u 1 )(λγ m γ bc θ)(λγ n γ de θ)(θγ n u 2 ) = -1 240 (2δ bc m[d (u 1 γ e] u 2 ) + δ [b m (u 1 γ c]de u 2 ))
and therefore
X′ 2111 = -1 480 δ [b m (u 1 γ c] γ de u 2 ) + δ [d m (u 1 γ e] γ bc u 2 ) .
The result for X4010 is
X4010 = -1 360 δ bq mn (u 1 γ c u 2 ) -1 90 δ bc mq (u 1 γ n u 2 ) + 1 720 δ bc mn (u 1 γ q u 2 ) -1 2520 δ m q (u 1 γ bcn u 2 ) -1 720 δ b q (u 1 γ cmn u 2 ) + 1 1260 δ b m (u 1 γ cnq u 2 ) + 1 3360 (u 1 γ bcmnq u 2 ) [bc][mn]
.
For the evaluation of X2210 , it is useful to consider the more general correlator
(λγ a θ)(θγ a u 1 )(λγ [m| γ bc θ)(λγ |n] γ de θ)(θγ x u 2 ) = -13 5040 δ d x δ be mn (u 1 γ c u 2 ) + . . . + 11 201600 δ m x (u 1 γ bcden u 2 ) + • • • -11 403200 (u 1 γ bcdemnx u 2 ) [mn][bc][de] ( 27 terms)
+ 1 9676800 ε bcdemni 1 i 2 i 3 i 4 (u 1 γ i 1 i 2 i 3 i 4 x u 2 ) - 1 2419200 ε bcdemnxi 1 i 2 i 3 (u 1 γ i 1 i 2 i 3 u 2 )
. This time, even using the method of section 2.2, there are sufficiently many open indices and long enough traces for epsilon tensors to appear. Using eqs. (2.11) and (2.12), they can be re-written into γ [5,7] terms:
(λγ a θ)(θγ a u 1 )(λγ [m| γ bc θ)(λγ |n] γ de θ)(θγ x u 2 ) = -13 5040 δ d x δ be mn (u 1 γ c u 2 ) + . . .
+ 1 16800 δ m x (u 1 γ bcden u 2 ) + • • • -1 33600 (u 1 γ bcdemnx u 2 ) [mn][bc][de] ( 27 terms)
A good check on the sign of the epsilon contributions is that X′ 2111 is recovered when contracting with η nx , involving a cancellation of all γ [5] terms. To obtain X2210 , one multiplies by -η cx :
X2210 = 1 720 δ de mn (u 1 γ b u 2 ) + 29 2880 δ bd mn (u 1 γ e u 2 ) + 11 2880 δ bm de (u 1 γ n u 2 ) + 1 20160 δ d m (u 1 γ ben u 2 ) + 1 2880 δ b m (u 1 γ den u 2 ) + 11 20160 δ b d (u 1 γ emn u 2 ) + 1 4480 (u 1 γ bdemn u 2 ) [de][mn]
For the calculation of X 2030 and X 2012 , one may first evaluate a more general correlator (λγ a θ)(θγ a u 1 )(λγ [m u 2 )(λγ n] γ bc θ)(θγ x γ de θ) and then contract with η cx and η nx , respectively. The results are X2030 = -1 720 δ de mn (u
1 γ b u 2 ) + 1 288 δ bd mn (u 1 γ e u 2 ) -1 1440 δ bm de (u 1 γ n u 2 ) -17 10080 δ d m (u 1 γ ben u 2 ) -23 10080 δ b m (u 1 γ den u 2 ) -1 1440 δ b d (u 1 γ emn u 2 ) + 1 6720 (u 1 γ bdemn u 2 ) [mn][de] , X2012 = 1 288 δ de bm (u 1 γ c u 2 ) + 1 288 δ bc dm (u 1 γ e u 2 ) -1 1440 δ bc de (u 1 γ m u 2 ) + 1 2016 δ d m (u 1 γ bce u 2 ) -11 10080 δ b m (u 1 γ cde u 2 ) + 17 10080 δ b d (u 1 γ cem u 2 ) -1 3360 (u 1 γ bcdem u 2 ) [bc][de]
.
After multiplication with the momenta and polarisations, all individual contributions are gauge invariant and can be expanded in the basis C 1 . . . C 5 listed in (A.6):
(1 + π 34 )X 4010 = i 483840 (-6, -16, -40, 6, 0) C 1 ...C 5 (1 + π 34 )X 2210 = i 483840 (-18, -104, -176, 18, 0) C 1 ...C 5 (1 + π 34 )X 2030 = i 483840 (-21, 42, -42, 21, 0) C 1 ...C 5 (1 + π 34 )X 2012 = i 483840 (-39, 78, -78, 39, 0) C 1
...C 5 X ′ 2111 = -i 11520 (1, 0, 4, -1, 0) C 1 ...C 5
The sum can be written as
K 2B2F 1-loop = X ′ 2111 = -i 3840 (1, 0, 4, -1, 0) C 1 ...C 5 = -i 1920 s 13 (u 2 / ζ 3 (/ k 2 + / k 3 )/ ζ 4 u 1 ) + s 23 (u 2 / ζ 4 (/ k 2 + / k 4 )/ ζ 3 u 1 ) (3.6)
and again agrees with the amplitude computed in the RNS result, see [16] eq. (3.37).
The pure spinor formalism was used in [4, 2] to compute the two-loop type-IIB amplitude involving four massless states,
A = d 2 Ω 11 d 2 Ω 12 d 2 Ω 22 4 i=1 d 2 z i exp -i,j k i • k j G(z i , z j ) (det Im Ω) 5 K 2-loop (k i , z i ) ,
where Ω is the genus-two period matrix, and the integration over fermionic zero modes is encapsulated in
K 2-loop = ∆ 12 ∆ 34 (λγ mnpqr λ)(λγ s W 1 )F
2,mn F 3,pq F 4,rs + perm(1234) (4.1) ≡ ∆ 12 ∆ 34 K 12 + ∆ 13 ∆ 24 K 13 + ∆ 14 ∆ 23 K 14 . (4.2)
The kinematic factors K 12 , K 13 , K 14 are accompanied by the basic antisymmetric biholomorphic 1-form ∆, which is related to a canonical basis
ω 1 , ω 2 of holomorphic differentials via ∆ ij = ∆(z i , z j ) = ω 1 (z i )ω 2 (z j ) -ω 2 (z i )ω 1 (z j ).
The superfields W α i and F i,mn are the spinor and vector field strengths of the i-th external state, as in section 3. One encounters superspace integrals of the form
Y (abcd) = (λγ mnpqr λ)(λγ s W a )F b,mn F c,pq F d,rs . (4.
3)
The symmetries of the λ 3 combination [4] in this correlator include the obvious symmetry under mn ↔ pq, and also (λγ [mnpqr λ)(λγ s] ) α = 0 (this holds for pure spinors λ and can be seen by dualising, and holds for unconstrained spinors λ as part of a λ 3 θ 5 scalar, as seen from the representation content (2.8)), and allow one to shuffle the F factors:
Y (abcd) = Y (acbd) , Y (abcd) + Y (acdb) + Y (adbc) = 0 . (4.4)
The case of four Neveu-Schwarz states was considered in [6] and will be briefly reviewed here. As all three kinematic factors K 12 , K 13 and K 14 are equivalent, it is sufficient to consider K 12 in detail. With all external states being identical, the symmetrisations of (4.1) can be carried out at the end of the calculation:
K 4B 12 = 4 W [1 F 2] F [3 F 4] 4B + 4 W [3 F 4] F [1 F 2] 4B = (1 -π 12 )(1 -π 34 )(1 + π 13 π 24 ) W 1 F 2 F 3 F 4 4B
Expanding the superfields and adopting the notation
Y ABCD (abcd) = (λγ mnpqr λ)(λγ s W (A) a )F (B) b,mn F (C) c,pq F (D) d,rs ,
the Neveu-Schwarz states come from terms of the form Y ABCD ≡ Y ABCD (1234) with A odd and B, C, D even. Using the shuffling identities (4.4) to simplify, one obtains
W 1 F 2 F 3 F 4 4B = Y 5000 + Y 1400 + Y 1040 + Y 1004 + Y 3200 + Y 3020 + Y 3002 + Y 1220 + Y 1202 + Y 1022 = (1 + π 23 )(1 -π 24 ) 1 3 Y 5000 + Y 1400 + Y 3200 + Y 1022 ,
and therefore K 4B 12 can be written as the image of a symmetrisation operator S 4B :
K 4B 12 = S 4B 1 3 Y 5000 + Y 1400 + Y 3200 + Y 1022 S 4B = (1 -π 12 )(1 -π 34 )(1 + π 13 π 24 )(1 + π 23 )(1 -π 24 )
It is worth noting at this point that, on the sixteen-dimensional space of Lorentz scalars built from the four field strengths F i and two momenta, the symmetriser S 4B has rank four. The correlators were computed in [6] , using the method outlined in section 2.1. Two are zero, Y 5000 = Y 1400 = 0, and the remaining ones are
Y 3200 = 1 192 k 1 a F 1 cd k 2 m F 2 ef F 3 pq F 4 rs (λγ mnpqr λ)(λγ s γ ab θ)(θγ b cd θ)(θγ n ef θ) , Y 1022 = 1 64 F 1 ab F 2 mn k 3 p F 3 cd k 4 r F 4 ef (λγ mnpq[r λ)(λγ s] γ ab θ)(θγ q cd θ)(θγ s ef θ) .
In reducing those two contributions to a set of independent scalars, one finds that they both are not just sums of (k • k)F 4 terms but also contain terms of the form k • F terms. The latter are projected out by the symmetriser S 4B , and the result is
K 4B 12 = S 4B (Y 3200 + Y 1022 ) = 1 120 (s 13 -s 23 ) 4 Tr(F (1 F 2 F 3 F 4) ) -Tr(F (1 F 2 ) Tr(F 3 F 4) ) , = 1 720 (s 13 -s 23 )t 8 F 4 .
By trivial index exchange, one obtains K 13 and K 14 , and the total is
K 4B 2-loop =
1 720 (s 13 -s 23 )∆ 12 ∆ 34 + (s 12 -s 23 )∆ 13 ∆ 24 + (s 12 -s 13 )∆ 14 ∆ 23 t 8 F 4 , (4.5)
a product of the completely symmetric one-loop kinematic factor t 8 F 4 and a completely symmetric combination of the momenta and the ∆ ij .
The calculation involving four Ramond states is very similar to the bosonic one. Focussing on the K 12 part, the symmetrisations in (4.1) can again be rewritten as action of symmetrisation operators on the correlator of superfields with one particular labelling:
K 4F 12 (û i ) = (1 -π 12 )(1 -π 34 )(1 + π 13 π 24 ) W 1 F 2 F 3 F 4 û1 û2 û3 û4 = 4(1 -π 12 ) W 1 F 2 F 3 F 4 û1 û2 û3 û4
The last step follows from the fact that all scalars of the form k 4 u 4 (see appendix A.2), and therefore all k 4 û4 scalars, are invariant under π 13 π 24 and have π 12 = π 34 . This time, on expanding the superfields, one collects the terms Y ABCD with A even and B, C, D odd. After using (4.4) to simplify,
W 1 F 2 F 3 F 4 û1 û2 û3 û4 = Y 2111 + Y 0311 + Y 0131 + Y 0113 = (1 + π 23 )(1 -π 24 ) 1 3 Y 2111 + Y 0311
, and after translating to commuting wavefunctions u i , which multiplies every permutation operator with its signature, one obtains
K 4F 12 (u i ) = S 4F 1 3 Y 2111 (u i ) + Y 0311 (u i ) , S 4F = 4(1 + π 12 )(1 -π 23 )(1 + π 24 ) .
This symmetriser has rank three, and the result is again not determined by symmetry. Two correlators have to be computed:
Y 2111 (u i ) = (-2)k 1 a k 2 m k 3 p k 4 r (λγ mnpq[r λ)(λγ s] γ ab θ)(θγ b u 1 )(θγ n u 2 )(θγ q u 3 )(θγ s u 4 ) Y 0311 (u i ) = (-2 3 )k 2 a k 2 m k 3 p k 4 r (λγ mnpq[r λ)(λγ s] u 1 )(θγ n ab θ)(θγ b u 2 )(θγ q u 3 )(θγ s u 4 )
With four fermions present, the method of section 2.2 is preferred as it does not involve rearranging the fermions using Fierz identities. The first correlator was covered as an example in that section, and the second one can be evaluated in the same fashion. Expressed in the basis listed in (A.5), the results are
Y 2111 (u i ) = 1 5040 (-19, -21, 21, 19, -17, -17, 0, 0, 0, 0) B 1 ...B 10 , Y 0311 (u i ) = 1
15120 (-14, -16, 0, 2, -8, -8, 0, -5, -5, 0) B 1 ...B 10 . After acting with the symmetriser S 4F , one obtains the same u 4 scalar encountered in the one-loop amplitude,
K 4F 12 (u i ) = S 4F ( 1 3 Y 2111 (u i ) + Y 0311 (u i )) = 1 45 (-1, -2, 1, 2, -1, -2, 0, 0, 0, 0) B 1 ...B 10 = 1 45 (s 23 -s 13 ) (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) -(u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) + (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) .
The K 13 and K 14 parts again follow by index exchange, and the total result
K 4F 2-loop (u i ) = 1 45 (s 23 -s 13 )∆ 12 ∆ 34 + (s 23 -s 12 )∆ 13 ∆ 24 + (s 13 -s 12 )∆ 14 ∆ 23 × (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) -(u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) + (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) (4.6)
is again a simple product of the one-loop kinematic factor and a combination of the ∆ ij and momenta.
As in the one-loop calculation of section 3.3, in the mixed case one has to pay some attention to the permutations in (4.1) since they affect which superfields contribute fermionic fields.
The complete symmetrisation makes it irrelevant which labels are assigned to the two fermions, and the convention f 1 f 2 b 3 b 4 will be used here. The kinematic factor K 2B2F
is then distinguished from the other two, K 2B2F 13 and K 2B2F
. Carrying out the symmetrisations in (4.1) and using the identities (4.4), one finds
K 12 (û 1 , û2 , ζ 3 , ζ 4 ) = (1 -π 12 )(1 -π 34 ) K , K 13 (û 1 , û2 , ζ 3 , ζ 4 ) = (2 • 1 + π 12 + π 34 + 2π 12 π 34 ) K , K 14 (û 1 , û2 , ζ 3 , ζ 4 ) = (1 + 2π 12 + 2π 34 + π 12 π 34 ) K , where, schematically, K = (λ 3 W (even) 1 )F (odd) 2 F (even) 3 F (even) 4 + (λ 3 W (odd) 3 )F (even) 4 F (odd) 1 F (odd) 2 . ( 4.7)
In translating to commuting variables u 1 and u 2 , the permutation operator π 12 changes sign, and therefore foot_2
K 12 (u 1 , u 2 , ζ 3 , ζ 4 ) = (1 + π 12 )(1 -π 34 ) K , K 13 (u 1 , u 2 , ζ 3 , ζ 4 ) = (2 • 1 -π 12 + π 34 -2π 12 π 34 ) K , K 14 (u 1 , u 2 , ζ 3 , ζ 4 ) = (1 -2π 12 + 2π 34 -π 12 π 34 ) K .
Expanding the superfields, the contributions to K are:
Y 4100 = -i 48 k 1 a k 1 d k 2 m F 3 pq F 4 rs (λγ mnpqr λ)(λγ s γ ab θ)(θγ b γ cd θ)(θγ c u 1 )(θγ n u 2 ) Y 0500 = i 240 k 2 m k 2 a k 2 c F 3 pq F 4 rs (λγ mnpqr λ)(λγ s u 1 )(θγ n ab θ)(θγ b cd θ)(θγ d u 2 ) Y 0140 = i 48 k 2 m k 3 p k 3 a F 3 cd F 4 rs (λγ mnpqr λ)(λγ s u 1 )(θγ n u 2 )(θγ q ab θ)(θγ b cd θ) Y 0104 = i 48 k 2 m F 3 pq k 4 a F 4 cd k 4 [r| (λγ mnpqr λ)(λγ s u 1 )(θγ n u 2 )(θγ |s] ab θ)(θγ b cd θ) Y 2300 = i 24 k 1 a k 2 m k 2 c F 3 pq F 4 rs (λγ mnpqr λ)(λγ s γ ab θ)(θγ b u 1 )(θγ n cd θ)(θγ e u 2 ) Y 2120 = i 8 k 1 a k 2 m k 3 p F 3 cd F 4 rs (λγ mnpqr λ)(λγ s γ ab θ)(θγ b u 1 )(θγ n u 2 )(θγ q cd θ) Y 2102 = i 8 k 1 a k 2 m F 3 pq F 4 cd k 4 [r| (λγ mnpqr λ)(λγ s γ ab θ)(θγ b u 1 )(θγ n u 2 )(θγ |s] cd θ) Y 0320 = i 24 k 2 m k 2 a k 3 p F 3 cd F 4 rs (λγ mnpqr λ)(λγ s u 1 )(θγ n ab θ)(θγ b u 2 )(θγ q cd θ) Y 0302 = i 24 k 2 m k 2 a F 3 pq F 4 cd k 4 [r| (λγ mnpqr λ)(λγ s u 1 )(θγ n ab θ)(θγ b u 2 )(θγ |s] cd θ) Y 0122 = i 4 k 2 m k 3 p F 3 ab F 4 cd k 4 [r| (λγ mnpqr λ)(λγ s u 1 )(θγ n u 2 )(θγ q ab θ)(θγ s] cd θ) Y 3011 = i 12 k 3 a F 3 cd F 4 mn k 1 p k 2 [r| (λγ mnpqr λ)(λγ s γ ab θ)(θγ b cd θ)(θγ c u 1 )(θγ n u 2 ) Y 1211 = i 2 F 3 ab k 4 m F 4 cd k 1 p k 2 [r| (λγ mnpqr λ)(λγ s γ ab θ)(θγ n cd θ)(θγ q u 1 )(θγ |s] u 2 ) Y 1031 = i 12 F 3 ab F 4 mn k 1 p k 1 c k 2 [r| (λγ mnpqr λ)(λγ s γ ab θ)(θγ q cd θ)(θγ d u 1 )(θγ |s] u 2 ) Y 1013 = i 12 F 3 ab F 4 mn k 1 p k 2 c k 2 [r| (λγ mnpqr λ)(λγ s γ ab θ)(θγ q u 1 )(θγ |s] cd θ)(θγ d u 2 )
These correlators can be evaluated exactly as described in section 3.3. One finds that Y 0500 = Y 0140 = Y 0104 = 0, and the sum of the remaining terms reduces to
K = Y 4100 + Y 2300 + Y 2120 + Y 2102 + Y 0320 + Y 0302 + Y 0122 + Y 3011 + Y 1211 + Y 1031 + Y 1013 = i 360 (s 12 + s 13 ) × (1, 0, 4, -1, 0) C 1 ...C 5 .
After applying the symmetrisation operators,
(1 + π 12 )(1 -π 34 ) K = i 180 (s 12 + 2s 13 ) × (1, 0, 4, -1, 0) C 1 ...C 5 , (2 • 1 -π 12 + π 34 -2π 12 π 34 ) K = i 180 (2s 12 + s 13 ) × (1, 0, 4, -1, 0) C 1 ...C 5 , (1 -2π 12 + 2π 34 -π 12 π 34 ) K = i 180 (s 12 -s 13 ) × (1, 0, 4, -1, 0) C 1 ...C 5 ,
the total kinematic factor is seen to be
K 2-loop (u 1 , u 2 , ζ 3 , ζ 4 ) = -i
180 (s 23 -s 13 )∆ 12 ∆ 34 + (s 23 -s 12 )∆ 12 ∆ 34 + (s 13 -s 12 )∆ 12 ∆ 34 × (1, 0, 4, -1, 0) C 1 ...C 5 (4.8)
and displays the same simple product form as in the four-boson and four-fermion case.
In this paper, different methods were discussed to efficiently evaluate the superspace integrals appearing in multiloop amplitudes derived in the pure spinor formalism. Extending previous calculations [6, 7] restricted to Neveu-Schwarz states, it was then shown how the treatment of Ramond states poses no additional difficulties. While the bosonic calculations of [6, 7] have, in conjunction with supersymmetry, already established the equivalence of the massless four-point amplitudes derived in the pure spinor and RNS formalisms, it would be interesting to make contact between the results of sections 4.2 / 4.3 and two-loop amplitudes involving Ramond states as computed in the RNS formalism (see for example [19] ).
The assistance of a computer algebra system seems indispensible in explicitly evaluating pure spinor superspace integrals. To avoid excessive use of custom-made algorithms, it would be desirable to implement these calculations in a wider computational framework particular adapted to field theory calculations [20] .
The methods outlined in this paper should be easily applicable to future higher-loop amplitude expressions derived from the pure spinor formalism, and, it is hoped, to other superspace integrals.
In calculating scattering amplitudes one encounters kinematic factors which are Lorentz invariant polynomials in the momenta, polarisations and/or spinor wavefunctions of the scattered particles. It can be a non-trivial task to simplify such expressions, taking into account the on-shell identities i k i = 0, k 2 i = 0, k i • ζ i = 0, / k i u i = 0, and, in the case of fermions, re-arrangements stemming from Fierz identities.
More generally, one would like to know how many independent combinations of some given fields (subject to on-shell identitites) there are, and how to reduce an arbitrary expression with respect to some chosen basis. This appendix outlines methods to address these problems, with an emphasis on algorithms which can easily be transferred to a computer algebra system. These methods are not limited to dealing with pure spinor calculations but the scope will be restricted to amplitudes of four massless vector or spinor particles in ten dimensions.
It is not difficult to reduce polynomials in the momenta and polarisations to a canonical form. The momentum conservation constraint i k i = 0 is solved by eliminating one momentum (for example k 4 ), all k 2 i are set to zero, and one of the two remaining quadratic combinations of momenta is eliminated (for example s 23 → -s 12 -s 13 , where
s ij ≡ k i • k j ).
Then all products k i • ζ i are set to zero, and one extra k • ζ product is replaced (when eliminating k 4 , the replacement is
k 3 • ζ 4 → (-k 1 -k 2 ) • ζ 4
). The remaining monomials are then independent. (This is at least the case with the low powers of momenta encountered in the calculations of sections 3 and 4, where there are enough spatial directions for all momenta/polarisations to be linearly independent.)
The implementation of these reduction rules on a computer is straightforward. The easiest way to obtain scalars which are also invariant under the gauge symmetry k i → ζ i is to start with expressions constructed from the field strengths
F ab i = 2∂ [a ζ b]
i . For the one-loop calculations of section 3.1, the relevant basis consists of gauge invariant scalars containing only the four field strengths F 1 . . . F 4 . One finds six independent combinations,
Tr(F 1 F 2 F 3 F 4 ) Tr(F 1 F 2 F 4 F 3 ) Tr(F 1 F 3 F 2 F 4 ) Tr(F 1 F 2 ) Tr(F 3 F 4 ) Tr(F 1 F 3 ) Tr(F 2 F 4 ) Tr(F 1 F 4 ) Tr(F 2 F 3 )
In the two-loop calculations of section 4.1, all monomials have two more momenta. There are sixteen independent gauge invariant scalars of the form kkF 1 F 2 F 3 F 4 , and twelve of them may be constructed from the previous basis by multiplication with s 12 and s 13 : A 1 = s 12 Tr(F 1 F 2 F 3 F 4 ), A 2 = s 13 Tr(F 1 F 2 F 3 F 4 ), etc. One choice for the additional four is
A 13 = k 3 • F 1 • F 2 • k 3 Tr(F 3 F 4 ) A 15 = k 3 • F 1 • F 4 • k 2 Tr(F 2 F 3 ) A 14 = k 4 • F 1 • F 3 • k 2 Tr(F 2 F 4 ) A 16 = k 4 • F 2 • F 3 • k 4 Tr(F 1 F 4 ) .
As an example application of the computer algorithms, one may check that the symmetrisation operator of section 4.1,
S 4B = (1 -π 12 )(1 -π 34 )(1 + π 13 π 24 )(1 + π 23 )(1 -π 24 ) ,
acts as
S 4B A 1 = 8A 1 + 4A 2 -4A 3 + 4A 4 + 8A 5 + 16A 6 . . . S 4B A 16 = -6A 1 + 6A 3 -6A 5 -12A 6 + 3 2 A 7 + 3A 8 + 3 2 A 9 + 3A 10 + 3 2 A 11 + 3A 12
and has rank four.
In dealing with the spinor wavefunctions u i one has to face two issues: Fierz identities, and the Dirac equation. Fierz identities not only allow one to change the order of the spinors but also give rise to relations between different expressions in one spinor order. The Dirac equation often simplifies terms with momenta contracted into (u i γ [n] u j ) bilinears.
In this section it is shown how to construct bases for terms of the form (k 2 or k 4 ) × u 1 u 2 u 3 u 4 . A significant simplification comes from noting that the Dirac equation allows one to rewrite (u i γ [n] u j ) bilinears into terms with lower n if more than one momentum is contracted into the γ [n] . A good first step is therefore to disregard the momenta temporarily and find all independent scalars and two-index tensors built from u 1 , . . . , u 4 . From the SO(10) representation content, (S + ) ⊗4 = 2 • 1 + 6 • + 3 • + (tensors with rank > 2) , one expects two scalars and nine 2-tensors. The scalars are easily found by considering, as in [21] ,
T 1 (1234) = (u 1 γ a u 2 )(u 3 γ a u 4 ) , T 3 (1234) = (u 1 γ abc u 2 )(u 3 γ abc u 4 ) .
and similarly for the other two inequivalent orders of the four spinors. (Note there is no T 5 because of self-duality of the γ [5] .) From Fierz transformations, one learns that all T 3 terms can be reduced to T 1 by T 3 (1234) = -12T 1 (1234) -24T 1 (1324) and permutations, and the identity (γ a ) (αβ (γ a ) γ)δ = 0 implies that T 1 (1234) + T 1 (1324) + T 1 (1423) = 0, leaving for example T 1 (1234) and T 1 (1324) as independent scalars.
Generalising this approach to two-index tensors, it turns out that it is sufficient to start with
T 11 (1234) = (u 1 γ m u 2 )(u 3 γ n u 4 ) , T 31 (1234) = (u 1 γ a γ m γ n u 2 )(u 3 γ a u 4 ) , T 33 (1234) = (u 1 γ ab γ m u 2 )(u 3 γ ab γ n u 4 ) ,
and permutations of the spinor labels. It would be very tiresome to systematically apply a variety of Fierz transformations by hand and to find an independent set. Fortunately, by choosing a gamma matrix representation (such as the one listed in appendix B) and reducing all expressions to polynomials in the independent spinor components u 1 i , . . . , u 16 i , this problem can be solved with computer help. As expected, one finds that the T ij (abcd) span a nine-dimensional space, and a basis can be chosen as Having solved the first step, it is now easy to include the two or four momenta, taking the Dirac equation into account. Consider first the case of two momenta. Starting from the two-tensors in (A.1), one gets the three independent scalars
(u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) , (u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) , (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) .
In addition, there are four products of the two independent scalars T 1 (1234) and T 1 (1324) with the two independent momentum invariants s 12 and s 13 . By contracting (A.2) with momenta, one can show that
s 12 T 1 (1324) -s 13 T 1 (1234) = -(u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) + (u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) -(u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) , (A.3)
and this relation can be used to eliminate s 12 T 1 (1324). (It will become clear later that there are no independent other relations like this one.) There are thus six independent
k 2 u 1 • • • u 4 scalars: (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) s 12 T 1 (1234) (u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) s 13 T 1 (1234) (A.4) (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) s 13 T 1 (1324)
Note that there is only one completely antisymmetric combination of those, given by the right hand side of (A.3). Similarly, in the case of four momenta, one finds ten independent k 4 u 1
• • • u 4 scalars: B 1 = s 12 (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) B 2 = s 13 (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) B 3 = s 12 (u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) B 4 = s 13 (u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) B 5 = s 12 (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) B 6 = s 13 (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) (A.5) B 7 = s 2 12 T 1 (1234) B 8 = s 12 s 13 T 1 (1234) B 9 = s 2 13 T 1 (1234) B 10 = s 2 13 T 1 (1324)
Working in a gamma matrix representation, it is again simple to construct a computer algorithm which reduces any given k 2 u 1 • • • u 4 or k 4 u 1 • • • u 4 scalar into polynomials of the spinor and momentum components. The Dirac equation can then be solved by breaking up the sixteen-component spinors u i into eight-dimensional chiral spinors u s i and u c i , as in eq. (B.1). One obtains polynomials in the momentum components k a i and the independent spinor components (u c i ) 1...8 . However, a great disadvantage of this procedure is that it breaks manifest Lorentz invariance. For example, one encounters expressions which contain subsets of terms proportional to the square of a single momentum and are therefore equal to zero, but it is difficult to recognise this with a simple algorithm. The easiest solution is to choose several sets of particular vectors k i satisfying k 2 i = 0 and i k i = 0 and to evaluate all expressions on these vectors. (By choosing integer arithmetic, one easily avoids issues of numerical accuracy.) Substituting these sets of momentum vectors in the bases (A.4) and (A.5) gives full rank six and ten respectively, showing they are indeed linearly independent.
Equipped with a computer algorithm for these basis decompositions, one finds, for example, that the symmetriser S 4F of section 4. and has rank three.
The combined methods of the last two sections can easily be extended to the mixed case of two bosons and two fermions. In the one-loop calculation of section 3.3, one encounters scalars of the form ku 1 u 2 F 3 F 4 . A basis of such objects is given by There are two combinations antisymmetric in [12] and symmetric in (34):
C 1 = (u 1 γ a u 2 )k 3 a F 3 bc F 4
-C 1 + 4C 2 + C 4 and C 2 + C 3 .
Finally, there are ten independent scalars of the form k 3 u 1 u 2 F 3 F 4 (relevant to the two-loop calculation of section 4.3), and they can all be obtained by multiplication of C 1 . . . C 5 with the two momentum invariants s 12 and s 13 .
A convenient representation of the SO(1,9) gamma matrices is given by the 32×32 matrices (with eight-dimensional dot products). These can be solved for u s in terms of u c :
u s = -i √ 2k + (σ • ∂)u c = 1 √ 2k + (σ • k)u c , (B.1)
where k + = -i∂ + = -i √ 2 (∂ 0 + ∂ 9 ). | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The pure spinor formulation of the ten-dimensional superstring leads to manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor superspace. This paper explores different methods to evaluate these integrals and then uses them to calculate the kinematic factors of the one-loop and two-loop massless four-point amplitudes involving two and four Ramond states."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Contents 1. Introduction 1 2. Zero mode integration 2 2.1 Symmetry considerations and tensorial formulae 3 2.2 A spinorial formula 5 2.3 Component-based approach 7 3. One-loop amplitudes 7 3.1 Review: four bosons 8 3.2 Four fermions 10 3.3 Two bosons, two fermions 10 4. Two-loop amplitudes 12 4.1 Review: four bosons 13 4.2 Four fermions 14 4.3 Two bosons, two fermions 15"
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The quantisation of the ten-dimensional superstring using pure spinors as world-sheet ghosts [1] has overcome many difficulties encountered in the Green-Schwarz (GS) and Ramond-Neveu-Schwarz (RNS) formalisms. Most notably, by maintaining manifest spacetime supersymmetry, the pure spinor formalism has yielded super-Poincaré covariant multiloop amplitudes, leading to new insights into perturbative finiteness of superstring theory [2, 3] .\n\nCounting fermionic zero modes is a powerful technique in the computation of loop amplitudes in the pure spinor formalism and has for example been used to show that at least four external states are needed for a non-vanishing massless loop amplitude [2] . Furthermore, the structure of massless four-point amplitudes is relatively simple because all fermionic worldsheet variables contribute only through their zero modes. In the expressions derived for the one-loop [2] and two-loop [4] amplitudes, supersymmetry was kept manifest by expressing the kinematic factors as integrals over pure spinor superspace [5] involving three pure spinors λ and five fermionic superspace coordinates θ, K 1-loop = (λA)(λγ m W )(λγ n W )F mn , K 2-loop = (λγ mnpqr λ)(λγ s W )F mn F pq F rs ,\n\nwhere the pure spinor superspace integration is denoted by . . . , and A α (x, θ), W α (x, θ) and F mn (x, θ) are the superfields of ten-dimensional Yang-Mills theory.\n\nThe kinematic factors in (1.1) have been explicitly evaluated for Neveu-Schwarz states at two loops [6] and one loop [7] , and were found to match the amplitudes derived in the RNS formalism [8] . This provided important consistency checks in establishing the validity of the pure spinor amplitude prescriptions. (Related one-loop calculations had been reported in [9] .)\n\nIn this paper, it will be shown how to compute the kinematic factors in (1.1) when the superfields are allowed to contribute fermionic fields, as is relevant for the scattering of fermionic closed string states as well as Ramond/Ramond bosons. It turns out that the calculation of fermionic amplitudes presents no additional difficulties, making (1.1) a good practical starting point for the computation of four-point loop amplitudes in a unified fashion. This practical aspect of the supersymmetric pure spinor amplitudes was also emphasised in [10] , where the tree-level amplitudes were used to construct the fermion and Ramond/Ramond form contributions to the four-point effective action of the type II theories.\n\nThis paper is organised as follows. In section 2, different methods to compute pure spinor superspace integrals are explored. These methods are then applied to the explicit evaluation of the kinematic factors of massless four-point amplitudes at the one-loop level in section 3, and at the two-loop level in section 4. In both these sections, the bosonic calculations are briefly reviewed before separately considering the cases of two and four Ramond states. Particular attention will be paid to the constraints imposed by simple exchange symmetries. An appendix contains algorithms which were used to reduce intermediate expressions encountered in the amplitude calculations to a canonical form."
},
{
"section_type": "OTHER",
"section_title": "Zero mode integration",
"text": "The calculation of scattering amplitudes in the pure spinor formalism leads to integrals over zero modes of the fermionic worldsheet variables λ and θ. Both θ and λ are 16-component Weyl spinors, the λ are commuting and the θ anticommuting, and λ is subject to the pure spinor constraint (λγ m λ) = 0. The amplitude prescriptions [1, 2] require three zero modes of λ and five zero modes of θ to be present, and a Lorentz covariant object T αβγ,δ 1 ...δ 5 ≡ λ α λ β λ γ θ δ 1 . . . θ δ 5 = T (αβγ),[δ 1 ...δ 5 ]\n\n(2.1) was constructed such that the Yang-Mills antighost vertex operator V = (λγ m θ)(λγ n θ)(λγ p θ)(θγ mnp θ) has V = 1 .\n\n(2.2)\n\nIn this section, different methods of computing such \"pure superspace integrals\" are explored. As an example, a typical correlator encountered in the two-loop calculations of section 4 is considered:\n\nF (k i , u i ) = k 2 a k 2 m k 3 p k 4 r (λγ mnpq[r λ)(λγ s] u 1 )(θγ n ab θ)(θγ b u 2 )(θγ q u 3 )(θγ s u 4 ) (2.3)\n\nHere, k i and u i are the momenta and spinor wavefunctions of the four external particles."
},
{
"section_type": "OTHER",
"section_title": "Symmetry considerations and tensorial formulae",
"text": "One systematic approach to evaluate the zero mode integrals is to find expressions for all tensors that can be formed from (2.1). By Fierz transformations, one can always write the product of two θ spinors as (θγ [3] θ), where γ [k] denotes the antisymmetrised product of k gamma matrices. Due to the pure spinor constraint, the only bilinear in λ is (λγ [5] λ), and it is thus sufficient to consider the three cases (λγ [5] λ)(λ{γ [1] or γ [3] or γ [5] }θ)(θγ [3] θ)(θγ [3] θ) .\n\n(2.4)\n\nLorentz invariance then implies that it must be possible to express these tensors as sums of suitably symmetrised products of metric tensors, resulting in a parity-even expression, plus a parity-odd part made up from terms which in addition contain an epsilon tensor. The parity-even parts may be constructed [6] starting from the most general ansatz compatible with the symmetries of the correlator and then using spinor identities along with the normalisation (2.2) to determine all coefficients in the ansatz. Duality properties of the spinor bilinears can be used to determine the parity-odd part [7] . An extensive (and almost exhaustive) list of correlators is found in [11] , including the (λγ [1] θ) and (λγ [3] θ) cases of the above list:\n\n(λγ mnpqr λ)(λγ u θ)(θγ f gh θ)(θγ jkl θ) = -4 35 δ mnpqr mnpqr + 1 5! ε mnpqr mnpqr δ mn f g δ pq jk (δ r l δ h u + δ r h δ l u -δ r u δ h l ) [f gh][jkl]\n\n(2.5)\n\n(λγ mnpqr λ)(λγ stu θ)(θγ f gh θ)(θγ jkl θ) = -24 35 δ mnpqr mnpqr + 1 5! ε mnpqr mnpqr δ m j δ np f g δ q s δ t l (δ r h δ k u -δ k h δ r u ) [f gh][jkl](f gh↔jkl) (2.6)\n\n(Here, the brackets (f gh ↔ jkl) denote symmetrisation under simultaneous interchange of f gh with ijk, with weight one.) The remaining correlator with the (λγ [5] θ) factor can be derived in the same way, using an ansatz consisting of six parity-even structures. Taking a trace between the two γ [5] factors and noting that η ar (λγ mnpqr λ)(λγ abcde θ) . . . = -4 (λγ mnpq[b λ)(λγ cde] θ) . . . , one finds a relation to (2.6) . This is sufficient to determine all coefficients in the ansatz, and the result is\n\n(λγ mnpqr λ)(λγ abcde θ)(θγ f gh θ)(θγ jkl θ) = 16 7 δ mnpqr mnpqr + 1 5! ε mnpqr mnpqr × δ mnp abc δ f j δ d g δ q k (-δ e h δ r l + 2δ e l δ r h ) + δ mn ab δ cd f g δ pq jk (δ e h δ r l -3δ e l δ r h ) [abcde][f gh][jkl](f gh↔jkl) (2.7)\n\nOne may find it surprising that the derivation of these tensorial expressions only made use of properties of (pure) spinors, and of the normalisation condition (2.2). However, it can be seen from representation theory that the correlator (2.1) is uniquely characterised, up to normalisation, by its symmetry. To see this, note that [12] the spinor products λ 3 and θ 5 transform in\n\nλ (α λ β λ γ) : Sym 3 S + = [00003] ⊕ [10001] θ [δ 1 . . . θ δ 5 ] : Alt 5 S + = [00030] ⊕ [11010] . (2.8)\n\n(Here, λ and θ are taken to be in the S + irrep of SO (1, 9) , with Dynkin label [00001] .) The tensor product of these contains only one copy of the trivial representation. This applies to any spinors λ, which means that the pure spinor property cannot be essential to the derivation of the tensorial identities. The use of the pure spinor constraint merely allows for simpler derivations of the same identities.\n\nAs an illustration of this approach, consider the correlator of eq. ( 2 .3). Leaving the momenta aside for the moment by setting\n\nF = k 2 a k 2 m k 3 p k 4 r F , the task is to compute F = (λγ mnpq[r λ)(λγ s] u 1 )(θγ n ab θ)(θγ b u 2 )(θγ q u 3 )(θγ s u 4 ) .\n\nAfter applying two Fierz transformations,\n\nF = 1 16 (λγ mnpq[r| λ)(λγ c θ)(θγ n ab θ)(θγ jkl θ) (u 1 γ |s] γ c γ b u 2 ) + 1 3!•16 (λγ mnpq[r| λ)(λγ cde θ)(θγ n ab θ)(θγ jkl θ) (u 1 γ |s] γ cde γ b u 2 ) + 1 2•5!•16 (λγ mnpq[r| λ)(λγ cdef g θ)(θγ n ab θ)(θγ jkl θ) (u 1 γ |s] γ cdef g γ b u 2 )\n\n× 1 3!•16 (u 3 γ q γ jkl γ s u 4 ) , one obtains a combination of the fundamental correlators listed in (2.5), (2.6) and (2.7). A reliable evaluation of the numerous index symmetrisations is made possible by the use of a computer algebra program. In doing these calculations with Mathematica, an essential tool is the GAMMA package [13] , expanding the products of gamma matrices in a γ [k] basis. The result consists of two parts, F = F (δ) + F (ε) , with\n\nF (δ) = 1 560 (u 1 γ mpr u 2 )(u 3 γ a u 4 ) + 7 720 δ a p δ m r (u 1 γ i u 2 )(u 3 γ i u 4 ) + . . . -1 1680 (u 1 γ ai 1 i 2 u 2 )(u 3 γ mpr i 1 i 2 u 4 ) (92 terms) (2.9) F (ε) = -1 1209600 ε i 1 ...i 7 mpr (u 1 γ i 1 ...i 7 u 2 )(u 3 γ a u 4 ) + . . . -1 604800 ε ampr i 1 ...i 6 (u 1 γ i 3 ...i 9 u 2 )(u 3 γ i 1 i 2 i 7 i 8 i 9 u 4 ) (34 terms) (2.10)\n\nThe epsilon tensors in the second part can be eliminated using the fact that the u i are chiral spinors: If all the indices on γ [k] u i are contracted into an epsilon tensor, one uses\n\nε i 1 ...i k ′ j 1 ...j k γ j 1 ...j k γ 11 = (-) 1 2 k(k+1) k! γ i 1 ...i k ′ , (2.11)\n\nwhere γ 11 = 1 10! ε i 0 ...i 9 γ i 0 ...i 9 . More generally, if all but r indices of γ [k] u i are contracted,\n\nε i 1 ...i k ′ j 1 ...j k γ p 1 ...prj 1 ...j k γ 11 = (-) 1 2 k(k+1) k! k ′ ! (k ′ -r)! δ pr...p 1 [i 1 ...ir γ i r+1 ...i ′ k ] .\n\n(2.12)\n\nThe result of these manipulations is\n\nF (ε) = -1 560 (u 1 γ mpr u 2 )(u 3 γ a u 4 ) -1 280 δ p r (u 1 γ ami u 2 )(u 3 γ i u 4 ) + . . . + 9 11200 (u 1 γ i 1 i 2 i 3 u 2 )(u 3 γ ampr i 1 i 2 i 3 u 4 ) (53 terms) (2.13)\n\n(Note that while the epsilon terms in the basic correlator formulae were easily obtained from the delta terms by using Poincaré duality, this cannot be done here in any obvious way.) The last step in the evaluation of (2.3) is to contract with the momenta, F = k 2 a k 2 m k 3 p k 4 r F , and to simplify the expressions using the on-shell identities i k i = 0, k 2 i = 0, / k i u i = 0. It is shown in appendix A.2 that there are only ten independent scalars, denoted by B 1 . . . B 10 , that can be formed from four momenta and the four spinors u 1 . . . u 4 . With respect to this basis, the result is\n\nF (δ) = 1 48•10080 695s 12 (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) + • • • + 233s 2 13 (u 1 γ a u 2 )(u 3 γ a u 4 ) ( 7 terms\n\n) = 1 48•10080 (695, 775, 0, -80, 356, 356, 0, 233, 233, 0) B 1 ...B 10 , F (ε) = 1 48•10080 (-23, -7, 0, -16, 28, 28, 0, 7, 7, 0) B 1 ...B 10 , F = 1 10080 (14, 16, 0, -2, 8, 8, 0, 5, 5, 0) B 1 ...B 10 , (2.14)\n\nwhere\n\ns ij = k i • k j ."
},
{
"section_type": "OTHER",
"section_title": "A spinorial formula",
"text": "While the derivation of tensorial identities for correlators of the form (2.4) is relatively straightforward and elegant, it may be a tedious task to transform the expressions encountered in amplitude calculations to match this pattern. As seen in the example calculated above, this is particularly true if additional spinors are involved, making it necessary to apply Fierz transformations. It is therefore desirable to use a covariant correlator expression with open spinor indices. Such an expression was given in [1, 2] :\n\nT αβγ,δ 1 ...δ 5 = N -1 (γ m ) αδ 1 (γ n ) βδ 2 (γ p ) γδ 3 (γ mnp ) δ 4 δ 5 (αβγ)[δ 1 ...δ 5 ] , (2.15)\n\nwhere N is a normalisation constant and the brackets ()[] denote (anti-)symmetrisation with weight one. (Note that the right hand side is automatically gamma-matrix traceless: any gamma-trace\n\n(γ r ) αβ × (γ m ) α[δ 1 | (γ n ) β|δ 2 | (γ p ) γ|δ 3 (γ mnp ) δ 4 δ 5 ] = -(γ mnr ) [δ 1 δ 2 (γ mnp ) δ 3 δ 4 (γ p ) δ 5 ]γ = 0\n\nvanishes due to the double-trace identity (γ ab θ) α (θγ abc θ) = 0, which follows from the fact that the tensor product (Alt 3 S + ) ⊗ S -does not contain a vector representation and therefore the vector (ψγ ab θ)(θγ abc θ) has to vanish for all spinors ψ, and can also be shown by applying a Fierz transformation.) This prescription was originally motivated [2] by the fermionic expansion of the Yang-Mills antighost vertex operator V ,\n\nV = T αβγ,δ 1 ...δ 5 λ α λ β λ γ θ δ 1 . . . θ δ 5 (2.16) with T αβγ,δ 1 ...δ 5 = (γ m ) αδ 1 (γ n ) βδ 2 (γ p ) γδ 3 (γ mnp ) δ 4 δ 5 (αβγ)[δ 1 ...δ 5 ] ,\n\nwhere T is related to T by a parity transformation, up to the overall constant N . (Since T is uniquely determined by its symmetries, any covariant expression will be proportional to T , after symmetrisation of the spinor indices, and this is merely the simplest choice.) Equation (2.15) immediately yields an algorithm to convert any correlator into traces of gamma matrices or, if additional spinors are involved, bilinears in those spinors. It is, however, already very tiresome to determine the normalisation constant N by hand. The main advantage of this approach is that it clearly lends itself to implementation on a computer algebra system, which can easily carry out the spinor index symmetrisations, simplify the gamma products (again using the GAMMA package), and compute the traces. For example,\n\nN V = (γ m ) αδ 1 (γ n ) βδ 2 (γ p ) γδ 3 (γ mnp ) δ 4 δ 5 (αβγ)[δ 1 ...δ 5 ] (γ x ) αδ 1 (γ y ) βδ 2 (γ z ) γδ 3 (γ xyz ) δ 4 δ 5 = -1 60 Tr(γ x γ m ) Tr(γ y γ n ) Tr(γ z γ p ) Tr(γ xyz γ pnm ) + . . . -1 60 Tr(γ z γ pnm γ zyx γ n γ x γ m γ y γ p ) (60 terms) = 5160960 .\n\nThe correct normalisation is therefore obtained by setting N = 5160960.\n\nReturning to the example correlator (2.3), one finds that the calculation is by far simpler than with the previous method. After carrying out the symmetrisations (αβγ)[δ i ], one obtains\n\nN F = 1 60 Tr(γ x γ ab n γ y γ mnpq[r| )(u 3 γ q γ xyz γ s u 4 )(u 1 γ |s] γ z γ b u 2 ) + . . . -1 30 (u 2 γ b γ xyz γ q u 3 )(u 1 γ s γ y γ ab n γ x γ mnpq[r γ z γ s] u 4 ) , ( 24 terms)\n\nwhere elementary index re-sorting has reduced the number of terms from 60 to 24. Expanding the gamma products leads to\n\nN F = 476 5 δ p r (u 1 γ m u 4 )(u 2 γ a u 3 ) + • • • + 8 15 (u 1 γ ai 1 i 2 i 3 i 4 u 2 )(u 3 γ mpr i 1 i 2 i 3 i 4 u 4 ) , ( 294 terms)\n\nwhich, in contrast to (2.10), contains no epsilon terms as there are not enough free indices present. Note that this intermediate result contains terms with with u 1 paired with u 3 or u 4 , so it is not possible to directly compare to eqs. (2.9) and (2.13). However, after contracting with the momenta k foot_0 a k 2 m k 3 p k 4 r and decomposing the result in the basis B 1 . . . B 10 , one again obtains\n\nF = 1 10080 (14, 16, 0, -2, 8, 8, 0, 5, 5, 0) B 1 ...B 10 , (2.17)\n\nin agreement with (2.14) .\n\nThe algorithm just outlined will be the method of choice for all correlator calculations in the later sections of this paper and can easily be applied to a wider range of problems. The only limitation is that the larger the number of gamma matrices and open indices of the correlator, the slower the computer evaluation will be. For example, the correlator considered in eq. (5.2) of [11] ,\n\nt mnm 1 n 1 ...m 4 n 4 10 ≡ (λγ p γ m 1 n 1 θ)(λγ q γ m 2 n 2 θ)(λγ r γ m 3 n 3 θ)(θγ m γ n γ pqr γ m 4 n 4 θ) = -2 45 η mn t m 1 n 1 ...m 4 n 4"
},
{
"section_type": "METHOD",
"section_title": "Component-based approach",
"text": "A third method to evaluate the zero mode integrals consists of choosing a gamma matrix representation, expanding the integrand as a polynomial in spinor components, and then applying (2.15) to the individual monomials. This procedure seems particularly appealing if at some stage of the calculation one works with a matrix representation anyhow, in order to reduce the results to a canonical form (e.g. as outlined in appendix A). An efficient decomposition algorithm (of k 4 u 1 u 2 u 3 u 4 scalars, say) only needs a few non-zero momentum and spinor wavefunction components to distinguish all independent scalars, and therefore k and u can be replaced by sparse vectors. Furthermore, a trivial observation allows for a much quicker numeric evaluation of correlator components than a naive use of (2.15): In view of (2.16), one can equivalently compute the components of the paritytransformed expression V = ( λγ m θ)( λγ n θ)( λγ p θ)( θγ mnp θ), where λ and θ are spinors of chirality opposite to that of λ, θ. In the representation given in appendix B, V coincides with V | λ→ λ,θ→ θ, and\n\nV = 192 λ 9 λ 9 λ 9 θ 1 θ 2 θ 3 θ 4 θ 9 + • • • + 480 λ 1 λ 2 λ 3 θ 1 θ 9 θ 10 θ 13 θ 15 + . . . ( 100352 terms)\n\nThe monomials in the fermionic expansion of V then correspond to the arguments of non-zero correlators, and the coefficients of those monomials are, up to normalisation and symmetry factors, the correlator values.\n\nUnfortunately, it turns out that the complexity of typical correlators (e.g. the one given in (2.3)) makes it difficult to carry out the expansion in fermionic components in any straightforward way and limits this method to special applications. For example, the coefficients in (2.18) can be checked relatively easily by choosing particular index values, such as (λγ p γ 12 θ)(λγ q γ 21 θ)(λγ r γ 34 θ)(θγ 0 γ 0 γ pqr γ 43 θ)\n\n= 12 λ 1 λ 1 λ 1 θ 1 θ 9 θ 10 θ 11 θ 12 + • • • + 12 λ 16 λ 16 λ 16 θ 5 θ 6 θ 7 θ 8 θ 16 = 1 45 .\n\n(For fixed values of pqr, one gets no more than about 10 5 monomials of the form λ 3 θ 5 ). This approach may thus still be helpful in situations where the result has been narrowed down to a simple ansatz."
},
{
"section_type": "OTHER",
"section_title": "One-loop amplitudes",
"text": "The amplitude for the scattering of four massless states of the type IIB superstring was computed [2] in the pure spinor formalism as\n\nA = K K d 2 τ (Im τ ) 5 d 2 z 2 d 2 z 3 d 2 z 4 i<j G(z i , z j ) k i •k j , (3.1)\n\nwhere G(z i , z j ) is the scalar Green's function, and the kinematic factor is given by the product K K of left-and right-moving open superstring expressions,\n\nK 1-loop = (λA 1 )(λγ m W 2 )(λγ n W 3 )F 4,mn + cycl(234) . (3.2)\n\nHere the indices 1 . . . 4 label the external states and \"• • •+ cycl(234) \" denotes the addition of two other terms obtained by cyclic permutation of the indices 234. The spinor superfield A α and its supercovariant derivatives, the vector gauge superfield A m = foot_1 8 γ αβ m D α A β as well as the spinor and vector field strengths The superfields A α and W α as well as the gaugino field ûα are anticommuting. 1 To facilitate computer calculations involving polynomials in the spinor components, and for easier comparison with the literature, it will be more convenient to work with commuting fermion wavefunctions u α . Fortunately, as the kinematic factors with fermionic external states are multilinear functions of the distinctly labelled spinors ûi , it is straightforward to translate between the two conventions: Any monomial expression in û1 . . . û4 (and possibly fermionic coordinates θ) corresponds to the same expression in u 1 . . . u 4 , multiplied by the signature of the permutation sorting the ûi (and any θ variables) into some fixed order, such as (θ\n\nW α = 1 10 (γ m ) αβ (D β A m -∂ m A β ) and F mn = 1 8 (γ mn ) α β D α W β = 2∂ [m A n] ,\n\n• • • θ)û α 1 1 ûα 2 2 ûα 3 3 ûα 4\n\n4 . Choosing a gauge where θ α A α = 0, the on-shell identities\n\n2D (α A β) = γ m αβ A m , D α W β = 1 4 (γ mn ) α β F mn\n\nhave been used to derive recursive relations [10, 14, 15] for the fermionic expansion\n\nA (n) α = 1 n+1 (γ m θ) α A (n-1) m , A (n) m = 1 n (θγ m W (n-1) ) , W α(n) = -1 2n (γ mn θ) α ∂ m A (n-1) n ,\n\nwhere\n\nf (n) = 1 n! θ αn • • • θ α 1 (D α 1 • • • D αn f )|.\n\nThese recursion relations were explicitly solved in [10] , reducing the fermionic expansion to a simple repeated application of the derivative operator O m q = 1 2 (θγ m qp θ)∂ p :\n\nA (2k) m = 1 (2k)! [O k ] m q ζ q , A (2k+1) m = 1 (2k+1)! [O k ] m q (θγ q û) . (3.3)\n\nWith this solution at hand, one has all ingredients to evaluate the kinematic factor (3.2) for the three cases of zero, two, or four fermionic states."
},
{
"section_type": "OTHER",
"section_title": "Review: four bosons",
"text": "The kinematic factor involving four bosons was considered in [7] and this calculation will now be reviewed briefly. First, note that the outcome is not fixed by symmetry: The result must be gauge invariant [2] and therefore expressible in terms of the field strengths F 1 . . . F 4 .\n\nThe cyclic symmetrisation in (3.2) yields expressions symmetric in F 2 , F 3 , F 4 , and acting on scalars constructed from the F i only, the (234) symmetrisation is equivalent to complete symmetrisation in all labels (1234). Thus the result must be a linear combination of the two gauge invariant symmetric F 4 scalars, namely the single trace Tr(F (1 F 2 F 3 F 4) ) and double trace Tr(F (1 F 2 ) Tr(F 3 F 4) ), leaving one relative coefficient to be determined. Since all four states are of the same kind, one may first evaluate the correlator for one labelling and then carry out the cyclic symmetrisation:\n\nK (4B) 1-loop = (λA 1 )(λγ m W 2 )(λγ n W 3 )F 4,mn 4B + cycl (234) .\n\nThe different ways to saturate θ 5 result in a sum of terms of the form\n\nX ABCD = (λA (A) 1 )(λγ m W (B) 2 )(λγ n W (C) 3 )F (D) 4,mn\n\n(λA 1 )(λγ m W 2 )(λγ n W 3 )F 4,mn 4B = X 3110 + X 1310 + X 1130 + X 1112 .\n\nNote that X 1310 and X 1130 are related by exchange of the labels 2 and 3. This exchange can be carried out after computing the correlator, an operation which will in the following be denoted by π 23 . Using (3.3) for the superfield expansions and replacing ∂ m → ik m , one obtains\n\nX 3110 = -1 512 F 1 mn F 2 pq F 3 rs F 4 tu X3110 , X3110 = (λγ [t| γ pq θ)(λγ |u] γ rs θ)(λγ a θ)(θγ amn θ) , X 1112 = -1 128 ik 4 m ζ 1 n F 2 pq F 3 rs F 4 tu X1112 , X1112 = (λγ [m| γ pq θ)(λγ |a] γ rs θ)(λγ n θ)(θγ a tu θ) , X 1310 = -1 384 ik 3 m ζ 1 n F 2 pq F 3 rs F 4 tu X1310 , X1310 = (λγ [t| γ ma θ)(λγ |u] γ rs θ)(λγ n θ)(θγ a pq θ) .\n\nThe method outlined in section 2.2 is readily applicable to these correlators. For example, for X 3111 , the trace evaluation yields\n\nX3110 = N -1 1 60 Tr(γ a γ z ) Tr(γ xyz γ anm ) Tr(γ x γ qp γ [t| ) Tr(γ y γ sr γ |u] ) + • • • • • • + 1 60 Tr(γ [u| γ rs γ zyx γ qp γ |t] γ x γ a γ y γ mna γ z ) ( 60\n\nterms) = 2 35 δ mp rs δ nq tu -1 315 δ mn tu δ pq rs -1 45 δ mn rs δ pq tu + 26 315 δ mn pr δ qs tu [mn][pq][rs][tu](pq↔rs)\n\nUpon contracting with the field strengths, momenta and polarisations, and symmetrising over the cyclic permutations (234) (with weight 3), one finds that all three contributions are separately gauge invariant:\n\nX 3110 + cycl(234) = -11 13440 Tr(F (1 F 2 F 3 F 4) ) + 1 6720 Tr(F (1 F 2 ) Tr(F 3 F 4) ) X 1112 + cycl(234) = -19 53760 Tr(F (1 F 2 F 3 F 4) ) + 31 215040 Tr(F (1 F 2 ) Tr(F 3 F 4) ) (1 + π 23 )X 1310 + cycl(234) = -1 10240 4 Tr(F (1 F 2 F 3 F 4) ) -Tr(F (1 F 2 ) Tr(F 3 F 4) )\n\nThe sum X 3110 + X 1112 has the right ratio of single-and double-trace terms to be proportional to the well-known result t 8 F 4 , and the last line exhibits the right ratio by itself. The overall kinematic factor is therefore\n\nK 4B 1-loop = -1 2560 4 Tr(F (1 F 2 F 3 F 4) ) -Tr(F (1 F 2 ) Tr(F 3 F 4) ) = -1 15360 t 8 F 4 , (3.5)\n\nin agreement with the expressions derived in the RNS [16] and Green-Schwarz [17] formalisms."
},
{
"section_type": "OTHER",
"section_title": "Four fermions",
"text": "The four-fermion kinematic factor could be evaluated in the same way as in the four-boson case by summing up all terms X ABCD , A + B + C + D = 5, now with A, B, C even and D odd. Note however that this time, the outcome is fixed by symmetry: The cyclic symmetrisation in (3.2) leads to a completely symmetric dependence on û2 , û3 , û4 , and therefore to a completely antisymmetric dependence on u 2 , u 3 , u 4 . Acting on scalars of the form k 2 u 1 u 2 u 3 u 4 , antisymmetrising over [234] is equivalent to antisymmetrising over [1234], and there is only one completely antisymmetric k 2 u 1 u 2 u 3 u 4 scalar. Without further calculation, one can infer that the kinematic factor is proportional to that scalar,\n\nK 4F 1-loop = const • (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) -(u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) + (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) ,\n\nwhich of course agrees with the RNS amplitude (see e.g. [16] , eq. (3.67))."
},
{
"section_type": "OTHER",
"section_title": "Two bosons, two fermions",
"text": "In evaluating (3.2) for two bosons and two fermions, the cyclic symmetrisations affect whether the W and F superfields contribute bosons or fermions. Only the label of the A α superfield stays unaffected, and one has to choose whether it should contribute a boson or a fermion. Since its fermionic expansion starts with the bosonic polarisation vector, A 1,α ∼ (/ ζ 1 θ) α , the calculation can be simplified by choosing a labelling where particle 1 is a fermion. (Of course, the final result must be independent of this choice.) The assignment of the other three labels is then irrelevant and will be chosen as f 1 f 2 b 3 b 4 . Writing out the cyclic permutations, two of the three terms are essentially the same because they are related by interchange of the labels 3 and 4. The kinematic factor is then\n\nK 2B2F 1-loop (f 1 f 2 b 3 b 4 ) = (1 + π 34 ) (λA (even) 1\n\n)(λγ m W (even) 2\n\n)(λγ n W (odd) 3\n\n)F\n\n(even) 4,mn\n\n+ (λA (even) 1\n\n)(λγ m W (odd) 3\n\n)(λγ n W (odd) 4\n\n)F\n\n(odd) 2,mn .\n\nUnlike in the four-fermion calculation, the result is not fixed by symmetry. There are five independent ku 1 u 2 F 3 F 4 scalars (see appendix A, eq. (A.6)), denoted by C 1 . . . C 5 , and there are two independent combinations of these scalars with the required [12] (34) symmetry.\n\nExpanding the superfields and collecting terms with θ 5 , the first line yields a combination of terms X ABCD with A, B, D odd and C even. There is only one θ 5 combination coming from the second line, which will be denoted by X ′ 2111 ≡ (-π 24 )X 2111 :\n\nK 2B2F 1-loop = (1 + π 34 ) (X 4010 + X 2210 + X 2030 + X 2012 ) + X ′ 2111 ,\n\nwith the correlators\n\nX 4010 = i 60 k 1 q k 3 b ζ 3 c k 4 m ζ 4 n X4010 , X4010 = (λγ a θ)(θγ a pq θ)(θγ p u 1 )(λγ [m u 2 )(λγ n] γ bc θ) X 2210 = -i 12 k 2 b k 3 d ζ 3 e k 4 m ζ 4 n X2210 , X2210 = (λγ a θ)(θγ a u 1 )(λγ [m| γ bc θ)(θγ c u 2 )(λγ |n] γ de θ) X 2030 = -i 36 k 3 b k 3 d ζ 3 e k 4 m ζ 4 n X2030 , X2030 = (λγ a θ)(θγ a u 1 )(λγ [m u 2 )(λγ n] γ bc θ)(θγ c de θ) X 2012 = -i 12 k 3 b ζ 3 c k 4 m k 4 d ζ 4 e X2012 , X2012 = (λγ a θ)(θγ a u 1 )(λγ [m u 2 )(λγ n] γ bc θ)(θγ n de θ) X ′ 2111 = i 6 k 3 b ζ 3 c k 4 d ζ 4 e k 2 m X′ 2111 , X′ 2111 = (λγ a θ)(θγ a u 1 )(λγ [m| γ bc θ)(λγ |n] γ de θ)(θγ n u 2 )\n\n(The numerical coefficient in X ′ 2111 includes a sign coming from the θ, û ordering: there is an odd number of θs between u 1 and u 2 .) Evaluating these expressions as outlined in section 2.2, the spinor wavefunctions u i present no complication. The last part takes the simplest form: One finds\n\n(λγ a θ)(θγ a u 1 )(λγ m γ bc θ)(λγ n γ de θ)(θγ n u 2 ) = -1 240 (2δ bc m[d (u 1 γ e] u 2 ) + δ [b m (u 1 γ c]de u 2 ))\n\nand therefore\n\nX′ 2111 = -1 480 δ [b m (u 1 γ c] γ de u 2 ) + δ [d m (u 1 γ e] γ bc u 2 ) .\n\nThe result for X4010 is\n\nX4010 = -1 360 δ bq mn (u 1 γ c u 2 ) -1 90 δ bc mq (u 1 γ n u 2 ) + 1 720 δ bc mn (u 1 γ q u 2 ) -1 2520 δ m q (u 1 γ bcn u 2 ) -1 720 δ b q (u 1 γ cmn u 2 ) + 1 1260 δ b m (u 1 γ cnq u 2 ) + 1 3360 (u 1 γ bcmnq u 2 ) [bc][mn]\n\n.\n\nFor the evaluation of X2210 , it is useful to consider the more general correlator\n\n(λγ a θ)(θγ a u 1 )(λγ [m| γ bc θ)(λγ |n] γ de θ)(θγ x u 2 ) = -13 5040 δ d x δ be mn (u 1 γ c u 2 ) + . . . + 11 201600 δ m x (u 1 γ bcden u 2 ) + • • • -11 403200 (u 1 γ bcdemnx u 2 ) [mn][bc][de] ( 27 terms)\n\n+ 1 9676800 ε bcdemni 1 i 2 i 3 i 4 (u 1 γ i 1 i 2 i 3 i 4 x u 2 ) - 1 2419200 ε bcdemnxi 1 i 2 i 3 (u 1 γ i 1 i 2 i 3 u 2 )\n\n. This time, even using the method of section 2.2, there are sufficiently many open indices and long enough traces for epsilon tensors to appear. Using eqs. (2.11) and (2.12), they can be re-written into γ [5,7] terms:\n\n(λγ a θ)(θγ a u 1 )(λγ [m| γ bc θ)(λγ |n] γ de θ)(θγ x u 2 ) = -13 5040 δ d x δ be mn (u 1 γ c u 2 ) + . . .\n\n+ 1 16800 δ m x (u 1 γ bcden u 2 ) + • • • -1 33600 (u 1 γ bcdemnx u 2 ) [mn][bc][de] ( 27 terms)\n\nA good check on the sign of the epsilon contributions is that X′ 2111 is recovered when contracting with η nx , involving a cancellation of all γ [5] terms. To obtain X2210 , one multiplies by -η cx :\n\nX2210 = 1 720 δ de mn (u 1 γ b u 2 ) + 29 2880 δ bd mn (u 1 γ e u 2 ) + 11 2880 δ bm de (u 1 γ n u 2 ) + 1 20160 δ d m (u 1 γ ben u 2 ) + 1 2880 δ b m (u 1 γ den u 2 ) + 11 20160 δ b d (u 1 γ emn u 2 ) + 1 4480 (u 1 γ bdemn u 2 ) [de][mn]\n\nFor the calculation of X 2030 and X 2012 , one may first evaluate a more general correlator (λγ a θ)(θγ a u 1 )(λγ [m u 2 )(λγ n] γ bc θ)(θγ x γ de θ) and then contract with η cx and η nx , respectively. The results are X2030 = -1 720 δ de mn (u\n\n1 γ b u 2 ) + 1 288 δ bd mn (u 1 γ e u 2 ) -1 1440 δ bm de (u 1 γ n u 2 ) -17 10080 δ d m (u 1 γ ben u 2 ) -23 10080 δ b m (u 1 γ den u 2 ) -1 1440 δ b d (u 1 γ emn u 2 ) + 1 6720 (u 1 γ bdemn u 2 ) [mn][de] , X2012 = 1 288 δ de bm (u 1 γ c u 2 ) + 1 288 δ bc dm (u 1 γ e u 2 ) -1 1440 δ bc de (u 1 γ m u 2 ) + 1 2016 δ d m (u 1 γ bce u 2 ) -11 10080 δ b m (u 1 γ cde u 2 ) + 17 10080 δ b d (u 1 γ cem u 2 ) -1 3360 (u 1 γ bcdem u 2 ) [bc][de]\n\n.\n\nAfter multiplication with the momenta and polarisations, all individual contributions are gauge invariant and can be expanded in the basis C 1 . . . C 5 listed in (A.6):\n\n(1 + π 34 )X 4010 = i 483840 (-6, -16, -40, 6, 0) C 1 ...C 5 (1 + π 34 )X 2210 = i 483840 (-18, -104, -176, 18, 0) C 1 ...C 5 (1 + π 34 )X 2030 = i 483840 (-21, 42, -42, 21, 0) C 1 ...C 5 (1 + π 34 )X 2012 = i 483840 (-39, 78, -78, 39, 0) C 1\n\n...C 5 X ′ 2111 = -i 11520 (1, 0, 4, -1, 0) C 1 ...C 5\n\nThe sum can be written as\n\nK 2B2F 1-loop = X ′ 2111 = -i 3840 (1, 0, 4, -1, 0) C 1 ...C 5 = -i 1920 s 13 (u 2 / ζ 3 (/ k 2 + / k 3 )/ ζ 4 u 1 ) + s 23 (u 2 / ζ 4 (/ k 2 + / k 4 )/ ζ 3 u 1 ) (3.6)\n\nand again agrees with the amplitude computed in the RNS result, see [16] eq. (3.37)."
},
{
"section_type": "OTHER",
"section_title": "Two-loop amplitudes",
"text": "The pure spinor formalism was used in [4, 2] to compute the two-loop type-IIB amplitude involving four massless states,\n\nA = d 2 Ω 11 d 2 Ω 12 d 2 Ω 22 4 i=1 d 2 z i exp -i,j k i • k j G(z i , z j ) (det Im Ω) 5 K 2-loop (k i , z i ) ,\n\nwhere Ω is the genus-two period matrix, and the integration over fermionic zero modes is encapsulated in\n\nK 2-loop = ∆ 12 ∆ 34 (λγ mnpqr λ)(λγ s W 1 )F\n\n2,mn F 3,pq F 4,rs + perm(1234) (4.1) ≡ ∆ 12 ∆ 34 K 12 + ∆ 13 ∆ 24 K 13 + ∆ 14 ∆ 23 K 14 . (4.2)\n\nThe kinematic factors K 12 , K 13 , K 14 are accompanied by the basic antisymmetric biholomorphic 1-form ∆, which is related to a canonical basis\n\nω 1 , ω 2 of holomorphic differentials via ∆ ij = ∆(z i , z j ) = ω 1 (z i )ω 2 (z j ) -ω 2 (z i )ω 1 (z j ).\n\nThe superfields W α i and F i,mn are the spinor and vector field strengths of the i-th external state, as in section 3. One encounters superspace integrals of the form\n\nY (abcd) = (λγ mnpqr λ)(λγ s W a )F b,mn F c,pq F d,rs . (4.\n\n3)\n\nThe symmetries of the λ 3 combination [4] in this correlator include the obvious symmetry under mn ↔ pq, and also (λγ [mnpqr λ)(λγ s] ) α = 0 (this holds for pure spinors λ and can be seen by dualising, and holds for unconstrained spinors λ as part of a λ 3 θ 5 scalar, as seen from the representation content (2.8)), and allow one to shuffle the F factors:\n\nY (abcd) = Y (acbd) , Y (abcd) + Y (acdb) + Y (adbc) = 0 . (4.4)"
},
{
"section_type": "OTHER",
"section_title": "Review: four bosons",
"text": "The case of four Neveu-Schwarz states was considered in [6] and will be briefly reviewed here. As all three kinematic factors K 12 , K 13 and K 14 are equivalent, it is sufficient to consider K 12 in detail. With all external states being identical, the symmetrisations of (4.1) can be carried out at the end of the calculation:\n\nK 4B 12 = 4 W [1 F 2] F [3 F 4] 4B + 4 W [3 F 4] F [1 F 2] 4B = (1 -π 12 )(1 -π 34 )(1 + π 13 π 24 ) W 1 F 2 F 3 F 4 4B\n\nExpanding the superfields and adopting the notation\n\nY ABCD (abcd) = (λγ mnpqr λ)(λγ s W (A) a )F (B) b,mn F (C) c,pq F (D) d,rs ,\n\nthe Neveu-Schwarz states come from terms of the form Y ABCD ≡ Y ABCD (1234) with A odd and B, C, D even. Using the shuffling identities (4.4) to simplify, one obtains\n\nW 1 F 2 F 3 F 4 4B = Y 5000 + Y 1400 + Y 1040 + Y 1004 + Y 3200 + Y 3020 + Y 3002 + Y 1220 + Y 1202 + Y 1022 = (1 + π 23 )(1 -π 24 ) 1 3 Y 5000 + Y 1400 + Y 3200 + Y 1022 ,\n\nand therefore K 4B 12 can be written as the image of a symmetrisation operator S 4B :\n\nK 4B 12 = S 4B 1 3 Y 5000 + Y 1400 + Y 3200 + Y 1022 S 4B = (1 -π 12 )(1 -π 34 )(1 + π 13 π 24 )(1 + π 23 )(1 -π 24 )\n\nIt is worth noting at this point that, on the sixteen-dimensional space of Lorentz scalars built from the four field strengths F i and two momenta, the symmetriser S 4B has rank four. The correlators were computed in [6] , using the method outlined in section 2.1. Two are zero, Y 5000 = Y 1400 = 0, and the remaining ones are\n\nY 3200 = 1 192 k 1 a F 1 cd k 2 m F 2 ef F 3 pq F 4 rs (λγ mnpqr λ)(λγ s γ ab θ)(θγ b cd θ)(θγ n ef θ) , Y 1022 = 1 64 F 1 ab F 2 mn k 3 p F 3 cd k 4 r F 4 ef (λγ mnpq[r λ)(λγ s] γ ab θ)(θγ q cd θ)(θγ s ef θ) .\n\nIn reducing those two contributions to a set of independent scalars, one finds that they both are not just sums of (k • k)F 4 terms but also contain terms of the form k • F terms. The latter are projected out by the symmetriser S 4B , and the result is\n\nK 4B 12 = S 4B (Y 3200 + Y 1022 ) = 1 120 (s 13 -s 23 ) 4 Tr(F (1 F 2 F 3 F 4) ) -Tr(F (1 F 2 ) Tr(F 3 F 4) ) , = 1 720 (s 13 -s 23 )t 8 F 4 .\n\nBy trivial index exchange, one obtains K 13 and K 14 , and the total is\n\nK 4B 2-loop =\n\n1 720 (s 13 -s 23 )∆ 12 ∆ 34 + (s 12 -s 23 )∆ 13 ∆ 24 + (s 12 -s 13 )∆ 14 ∆ 23 t 8 F 4 , (4.5)\n\na product of the completely symmetric one-loop kinematic factor t 8 F 4 and a completely symmetric combination of the momenta and the ∆ ij ."
},
{
"section_type": "OTHER",
"section_title": "Four fermions",
"text": "The calculation involving four Ramond states is very similar to the bosonic one. Focussing on the K 12 part, the symmetrisations in (4.1) can again be rewritten as action of symmetrisation operators on the correlator of superfields with one particular labelling:\n\nK 4F 12 (û i ) = (1 -π 12 )(1 -π 34 )(1 + π 13 π 24 ) W 1 F 2 F 3 F 4 û1 û2 û3 û4 = 4(1 -π 12 ) W 1 F 2 F 3 F 4 û1 û2 û3 û4\n\nThe last step follows from the fact that all scalars of the form k 4 u 4 (see appendix A.2), and therefore all k 4 û4 scalars, are invariant under π 13 π 24 and have π 12 = π 34 . This time, on expanding the superfields, one collects the terms Y ABCD with A even and B, C, D odd. After using (4.4) to simplify,\n\nW 1 F 2 F 3 F 4 û1 û2 û3 û4 = Y 2111 + Y 0311 + Y 0131 + Y 0113 = (1 + π 23 )(1 -π 24 ) 1 3 Y 2111 + Y 0311\n\n, and after translating to commuting wavefunctions u i , which multiplies every permutation operator with its signature, one obtains\n\nK 4F 12 (u i ) = S 4F 1 3 Y 2111 (u i ) + Y 0311 (u i ) , S 4F = 4(1 + π 12 )(1 -π 23 )(1 + π 24 ) .\n\nThis symmetriser has rank three, and the result is again not determined by symmetry. Two correlators have to be computed:\n\nY 2111 (u i ) = (-2)k 1 a k 2 m k 3 p k 4 r (λγ mnpq[r λ)(λγ s] γ ab θ)(θγ b u 1 )(θγ n u 2 )(θγ q u 3 )(θγ s u 4 ) Y 0311 (u i ) = (-2 3 )k 2 a k 2 m k 3 p k 4 r (λγ mnpq[r λ)(λγ s] u 1 )(θγ n ab θ)(θγ b u 2 )(θγ q u 3 )(θγ s u 4 )\n\nWith four fermions present, the method of section 2.2 is preferred as it does not involve rearranging the fermions using Fierz identities. The first correlator was covered as an example in that section, and the second one can be evaluated in the same fashion. Expressed in the basis listed in (A.5), the results are\n\nY 2111 (u i ) = 1 5040 (-19, -21, 21, 19, -17, -17, 0, 0, 0, 0) B 1 ...B 10 , Y 0311 (u i ) = 1\n\n15120 (-14, -16, 0, 2, -8, -8, 0, -5, -5, 0) B 1 ...B 10 . After acting with the symmetriser S 4F , one obtains the same u 4 scalar encountered in the one-loop amplitude,\n\nK 4F 12 (u i ) = S 4F ( 1 3 Y 2111 (u i ) + Y 0311 (u i )) = 1 45 (-1, -2, 1, 2, -1, -2, 0, 0, 0, 0) B 1 ...B 10 = 1 45 (s 23 -s 13 ) (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) -(u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) + (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) .\n\nThe K 13 and K 14 parts again follow by index exchange, and the total result\n\nK 4F 2-loop (u i ) = 1 45 (s 23 -s 13 )∆ 12 ∆ 34 + (s 23 -s 12 )∆ 13 ∆ 24 + (s 13 -s 12 )∆ 14 ∆ 23 × (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) -(u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) + (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) (4.6)\n\nis again a simple product of the one-loop kinematic factor and a combination of the ∆ ij and momenta."
},
{
"section_type": "OTHER",
"section_title": "Two bosons, two fermions",
"text": "As in the one-loop calculation of section 3.3, in the mixed case one has to pay some attention to the permutations in (4.1) since they affect which superfields contribute fermionic fields.\n\nThe complete symmetrisation makes it irrelevant which labels are assigned to the two fermions, and the convention f 1 f 2 b 3 b 4 will be used here. The kinematic factor K 2B2F\n\nis then distinguished from the other two, K 2B2F 13 and K 2B2F\n\n. Carrying out the symmetrisations in (4.1) and using the identities (4.4), one finds\n\nK 12 (û 1 , û2 , ζ 3 , ζ 4 ) = (1 -π 12 )(1 -π 34 ) K , K 13 (û 1 , û2 , ζ 3 , ζ 4 ) = (2 • 1 + π 12 + π 34 + 2π 12 π 34 ) K , K 14 (û 1 , û2 , ζ 3 , ζ 4 ) = (1 + 2π 12 + 2π 34 + π 12 π 34 ) K , where, schematically, K = (λ 3 W (even) 1 )F (odd) 2 F (even) 3 F (even) 4 + (λ 3 W (odd) 3 )F (even) 4 F (odd) 1 F (odd) 2 . ( 4.7)\n\nIn translating to commuting variables u 1 and u 2 , the permutation operator π 12 changes sign, and therefore foot_2\n\nK 12 (u 1 , u 2 , ζ 3 , ζ 4 ) = (1 + π 12 )(1 -π 34 ) K , K 13 (u 1 , u 2 , ζ 3 , ζ 4 ) = (2 • 1 -π 12 + π 34 -2π 12 π 34 ) K , K 14 (u 1 , u 2 , ζ 3 , ζ 4 ) = (1 -2π 12 + 2π 34 -π 12 π 34 ) K .\n\nExpanding the superfields, the contributions to K are:\n\nY 4100 = -i 48 k 1 a k 1 d k 2 m F 3 pq F 4 rs (λγ mnpqr λ)(λγ s γ ab θ)(θγ b γ cd θ)(θγ c u 1 )(θγ n u 2 ) Y 0500 = i 240 k 2 m k 2 a k 2 c F 3 pq F 4 rs (λγ mnpqr λ)(λγ s u 1 )(θγ n ab θ)(θγ b cd θ)(θγ d u 2 ) Y 0140 = i 48 k 2 m k 3 p k 3 a F 3 cd F 4 rs (λγ mnpqr λ)(λγ s u 1 )(θγ n u 2 )(θγ q ab θ)(θγ b cd θ) Y 0104 = i 48 k 2 m F 3 pq k 4 a F 4 cd k 4 [r| (λγ mnpqr λ)(λγ s u 1 )(θγ n u 2 )(θγ |s] ab θ)(θγ b cd θ) Y 2300 = i 24 k 1 a k 2 m k 2 c F 3 pq F 4 rs (λγ mnpqr λ)(λγ s γ ab θ)(θγ b u 1 )(θγ n cd θ)(θγ e u 2 ) Y 2120 = i 8 k 1 a k 2 m k 3 p F 3 cd F 4 rs (λγ mnpqr λ)(λγ s γ ab θ)(θγ b u 1 )(θγ n u 2 )(θγ q cd θ) Y 2102 = i 8 k 1 a k 2 m F 3 pq F 4 cd k 4 [r| (λγ mnpqr λ)(λγ s γ ab θ)(θγ b u 1 )(θγ n u 2 )(θγ |s] cd θ) Y 0320 = i 24 k 2 m k 2 a k 3 p F 3 cd F 4 rs (λγ mnpqr λ)(λγ s u 1 )(θγ n ab θ)(θγ b u 2 )(θγ q cd θ) Y 0302 = i 24 k 2 m k 2 a F 3 pq F 4 cd k 4 [r| (λγ mnpqr λ)(λγ s u 1 )(θγ n ab θ)(θγ b u 2 )(θγ |s] cd θ) Y 0122 = i 4 k 2 m k 3 p F 3 ab F 4 cd k 4 [r| (λγ mnpqr λ)(λγ s u 1 )(θγ n u 2 )(θγ q ab θ)(θγ s] cd θ) Y 3011 = i 12 k 3 a F 3 cd F 4 mn k 1 p k 2 [r| (λγ mnpqr λ)(λγ s γ ab θ)(θγ b cd θ)(θγ c u 1 )(θγ n u 2 ) Y 1211 = i 2 F 3 ab k 4 m F 4 cd k 1 p k 2 [r| (λγ mnpqr λ)(λγ s γ ab θ)(θγ n cd θ)(θγ q u 1 )(θγ |s] u 2 ) Y 1031 = i 12 F 3 ab F 4 mn k 1 p k 1 c k 2 [r| (λγ mnpqr λ)(λγ s γ ab θ)(θγ q cd θ)(θγ d u 1 )(θγ |s] u 2 ) Y 1013 = i 12 F 3 ab F 4 mn k 1 p k 2 c k 2 [r| (λγ mnpqr λ)(λγ s γ ab θ)(θγ q u 1 )(θγ |s] cd θ)(θγ d u 2 )\n\nThese correlators can be evaluated exactly as described in section 3.3. One finds that Y 0500 = Y 0140 = Y 0104 = 0, and the sum of the remaining terms reduces to\n\nK = Y 4100 + Y 2300 + Y 2120 + Y 2102 + Y 0320 + Y 0302 + Y 0122 + Y 3011 + Y 1211 + Y 1031 + Y 1013 = i 360 (s 12 + s 13 ) × (1, 0, 4, -1, 0) C 1 ...C 5 .\n\nAfter applying the symmetrisation operators,\n\n(1 + π 12 )(1 -π 34 ) K = i 180 (s 12 + 2s 13 ) × (1, 0, 4, -1, 0) C 1 ...C 5 , (2 • 1 -π 12 + π 34 -2π 12 π 34 ) K = i 180 (2s 12 + s 13 ) × (1, 0, 4, -1, 0) C 1 ...C 5 , (1 -2π 12 + 2π 34 -π 12 π 34 ) K = i 180 (s 12 -s 13 ) × (1, 0, 4, -1, 0) C 1 ...C 5 ,\n\nthe total kinematic factor is seen to be\n\nK 2-loop (u 1 , u 2 , ζ 3 , ζ 4 ) = -i\n\n180 (s 23 -s 13 )∆ 12 ∆ 34 + (s 23 -s 12 )∆ 12 ∆ 34 + (s 13 -s 12 )∆ 12 ∆ 34 × (1, 0, 4, -1, 0) C 1 ...C 5 (4.8)\n\nand displays the same simple product form as in the four-boson and four-fermion case."
},
{
"section_type": "DISCUSSION",
"section_title": "Discussion",
"text": "In this paper, different methods were discussed to efficiently evaluate the superspace integrals appearing in multiloop amplitudes derived in the pure spinor formalism. Extending previous calculations [6, 7] restricted to Neveu-Schwarz states, it was then shown how the treatment of Ramond states poses no additional difficulties. While the bosonic calculations of [6, 7] have, in conjunction with supersymmetry, already established the equivalence of the massless four-point amplitudes derived in the pure spinor and RNS formalisms, it would be interesting to make contact between the results of sections 4.2 / 4.3 and two-loop amplitudes involving Ramond states as computed in the RNS formalism (see for example [19] ).\n\nThe assistance of a computer algebra system seems indispensible in explicitly evaluating pure spinor superspace integrals. To avoid excessive use of custom-made algorithms, it would be desirable to implement these calculations in a wider computational framework particular adapted to field theory calculations [20] .\n\nThe methods outlined in this paper should be easily applicable to future higher-loop amplitude expressions derived from the pure spinor formalism, and, it is hoped, to other superspace integrals."
},
{
"section_type": "OTHER",
"section_title": "A. Reduction to kinematic bases",
"text": "In calculating scattering amplitudes one encounters kinematic factors which are Lorentz invariant polynomials in the momenta, polarisations and/or spinor wavefunctions of the scattered particles. It can be a non-trivial task to simplify such expressions, taking into account the on-shell identities i k i = 0, k 2 i = 0, k i • ζ i = 0, / k i u i = 0, and, in the case of fermions, re-arrangements stemming from Fierz identities.\n\nMore generally, one would like to know how many independent combinations of some given fields (subject to on-shell identitites) there are, and how to reduce an arbitrary expression with respect to some chosen basis. This appendix outlines methods to address these problems, with an emphasis on algorithms which can easily be transferred to a computer algebra system. These methods are not limited to dealing with pure spinor calculations but the scope will be restricted to amplitudes of four massless vector or spinor particles in ten dimensions."
},
{
"section_type": "OTHER",
"section_title": "A.1 Four bosons",
"text": "It is not difficult to reduce polynomials in the momenta and polarisations to a canonical form. The momentum conservation constraint i k i = 0 is solved by eliminating one momentum (for example k 4 ), all k 2 i are set to zero, and one of the two remaining quadratic combinations of momenta is eliminated (for example s 23 → -s 12 -s 13 , where\n\ns ij ≡ k i • k j ).\n\nThen all products k i • ζ i are set to zero, and one extra k • ζ product is replaced (when eliminating k 4 , the replacement is\n\nk 3 • ζ 4 → (-k 1 -k 2 ) • ζ 4\n\n). The remaining monomials are then independent. (This is at least the case with the low powers of momenta encountered in the calculations of sections 3 and 4, where there are enough spatial directions for all momenta/polarisations to be linearly independent.)\n\nThe implementation of these reduction rules on a computer is straightforward. The easiest way to obtain scalars which are also invariant under the gauge symmetry k i → ζ i is to start with expressions constructed from the field strengths\n\nF ab i = 2∂ [a ζ b]\n\ni . For the one-loop calculations of section 3.1, the relevant basis consists of gauge invariant scalars containing only the four field strengths F 1 . . . F 4 . One finds six independent combinations,\n\nTr(F 1 F 2 F 3 F 4 ) Tr(F 1 F 2 F 4 F 3 ) Tr(F 1 F 3 F 2 F 4 ) Tr(F 1 F 2 ) Tr(F 3 F 4 ) Tr(F 1 F 3 ) Tr(F 2 F 4 ) Tr(F 1 F 4 ) Tr(F 2 F 3 )\n\nIn the two-loop calculations of section 4.1, all monomials have two more momenta. There are sixteen independent gauge invariant scalars of the form kkF 1 F 2 F 3 F 4 , and twelve of them may be constructed from the previous basis by multiplication with s 12 and s 13 : A 1 = s 12 Tr(F 1 F 2 F 3 F 4 ), A 2 = s 13 Tr(F 1 F 2 F 3 F 4 ), etc. One choice for the additional four is\n\nA 13 = k 3 • F 1 • F 2 • k 3 Tr(F 3 F 4 ) A 15 = k 3 • F 1 • F 4 • k 2 Tr(F 2 F 3 ) A 14 = k 4 • F 1 • F 3 • k 2 Tr(F 2 F 4 ) A 16 = k 4 • F 2 • F 3 • k 4 Tr(F 1 F 4 ) .\n\nAs an example application of the computer algorithms, one may check that the symmetrisation operator of section 4.1,\n\nS 4B = (1 -π 12 )(1 -π 34 )(1 + π 13 π 24 )(1 + π 23 )(1 -π 24 ) ,\n\nacts as\n\nS 4B A 1 = 8A 1 + 4A 2 -4A 3 + 4A 4 + 8A 5 + 16A 6 . . . S 4B A 16 = -6A 1 + 6A 3 -6A 5 -12A 6 + 3 2 A 7 + 3A 8 + 3 2 A 9 + 3A 10 + 3 2 A 11 + 3A 12\n\nand has rank four."
},
{
"section_type": "OTHER",
"section_title": "A.2 Four fermions",
"text": "In dealing with the spinor wavefunctions u i one has to face two issues: Fierz identities, and the Dirac equation. Fierz identities not only allow one to change the order of the spinors but also give rise to relations between different expressions in one spinor order. The Dirac equation often simplifies terms with momenta contracted into (u i γ [n] u j ) bilinears.\n\nIn this section it is shown how to construct bases for terms of the form (k 2 or k 4 ) × u 1 u 2 u 3 u 4 . A significant simplification comes from noting that the Dirac equation allows one to rewrite (u i γ [n] u j ) bilinears into terms with lower n if more than one momentum is contracted into the γ [n] . A good first step is therefore to disregard the momenta temporarily and find all independent scalars and two-index tensors built from u 1 , . . . , u 4 . From the SO(10) representation content, (S + ) ⊗4 = 2 • 1 + 6 • + 3 • + (tensors with rank > 2) , one expects two scalars and nine 2-tensors. The scalars are easily found by considering, as in [21] ,\n\nT 1 (1234) = (u 1 γ a u 2 )(u 3 γ a u 4 ) , T 3 (1234) = (u 1 γ abc u 2 )(u 3 γ abc u 4 ) .\n\nand similarly for the other two inequivalent orders of the four spinors. (Note there is no T 5 because of self-duality of the γ [5] .) From Fierz transformations, one learns that all T 3 terms can be reduced to T 1 by T 3 (1234) = -12T 1 (1234) -24T 1 (1324) and permutations, and the identity (γ a ) (αβ (γ a ) γ)δ = 0 implies that T 1 (1234) + T 1 (1324) + T 1 (1423) = 0, leaving for example T 1 (1234) and T 1 (1324) as independent scalars.\n\nGeneralising this approach to two-index tensors, it turns out that it is sufficient to start with\n\nT 11 (1234) = (u 1 γ m u 2 )(u 3 γ n u 4 ) , T 31 (1234) = (u 1 γ a γ m γ n u 2 )(u 3 γ a u 4 ) , T 33 (1234) = (u 1 γ ab γ m u 2 )(u 3 γ ab γ n u 4 ) ,\n\nand permutations of the spinor labels. It would be very tiresome to systematically apply a variety of Fierz transformations by hand and to find an independent set. Fortunately, by choosing a gamma matrix representation (such as the one listed in appendix B) and reducing all expressions to polynomials in the independent spinor components u 1 i , . . . , u 16 i , this problem can be solved with computer help. As expected, one finds that the T ij (abcd) span a nine-dimensional space, and a basis can be chosen as Having solved the first step, it is now easy to include the two or four momenta, taking the Dirac equation into account. Consider first the case of two momenta. Starting from the two-tensors in (A.1), one gets the three independent scalars\n\n(u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) , (u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) , (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) .\n\nIn addition, there are four products of the two independent scalars T 1 (1234) and T 1 (1324) with the two independent momentum invariants s 12 and s 13 . By contracting (A.2) with momenta, one can show that\n\ns 12 T 1 (1324) -s 13 T 1 (1234) = -(u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) + (u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) -(u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) , (A.3)\n\nand this relation can be used to eliminate s 12 T 1 (1324). (It will become clear later that there are no independent other relations like this one.) There are thus six independent\n\nk 2 u 1 • • • u 4 scalars: (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) s 12 T 1 (1234) (u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) s 13 T 1 (1234) (A.4) (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) s 13 T 1 (1324)\n\nNote that there is only one completely antisymmetric combination of those, given by the right hand side of (A.3). Similarly, in the case of four momenta, one finds ten independent k 4 u 1\n\n• • • u 4 scalars: B 1 = s 12 (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) B 2 = s 13 (u 1 / k 3 u 2 )(u 3 / k 1 u 4 ) B 3 = s 12 (u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) B 4 = s 13 (u 1 / k 2 u 3 )(u 2 / k 1 u 4 ) B 5 = s 12 (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) B 6 = s 13 (u 1 / k 2 u 4 )(u 2 / k 1 u 3 ) (A.5) B 7 = s 2 12 T 1 (1234) B 8 = s 12 s 13 T 1 (1234) B 9 = s 2 13 T 1 (1234) B 10 = s 2 13 T 1 (1324)\n\nWorking in a gamma matrix representation, it is again simple to construct a computer algorithm which reduces any given k 2 u 1 • • • u 4 or k 4 u 1 • • • u 4 scalar into polynomials of the spinor and momentum components. The Dirac equation can then be solved by breaking up the sixteen-component spinors u i into eight-dimensional chiral spinors u s i and u c i , as in eq. (B.1). One obtains polynomials in the momentum components k a i and the independent spinor components (u c i ) 1...8 . However, a great disadvantage of this procedure is that it breaks manifest Lorentz invariance. For example, one encounters expressions which contain subsets of terms proportional to the square of a single momentum and are therefore equal to zero, but it is difficult to recognise this with a simple algorithm. The easiest solution is to choose several sets of particular vectors k i satisfying k 2 i = 0 and i k i = 0 and to evaluate all expressions on these vectors. (By choosing integer arithmetic, one easily avoids issues of numerical accuracy.) Substituting these sets of momentum vectors in the bases (A.4) and (A.5) gives full rank six and ten respectively, showing they are indeed linearly independent.\n\nEquipped with a computer algorithm for these basis decompositions, one finds, for example, that the symmetriser S 4F of section 4. and has rank three."
},
{
"section_type": "OTHER",
"section_title": "A.3 Two bosons, two fermions",
"text": "The combined methods of the last two sections can easily be extended to the mixed case of two bosons and two fermions. In the one-loop calculation of section 3.3, one encounters scalars of the form ku 1 u 2 F 3 F 4 . A basis of such objects is given by There are two combinations antisymmetric in [12] and symmetric in (34):\n\nC 1 = (u 1 γ a u 2 )k 3 a F 3 bc F 4\n\n-C 1 + 4C 2 + C 4 and C 2 + C 3 .\n\nFinally, there are ten independent scalars of the form k 3 u 1 u 2 F 3 F 4 (relevant to the two-loop calculation of section 4.3), and they can all be obtained by multiplication of C 1 . . . C 5 with the two momentum invariants s 12 and s 13 ."
},
{
"section_type": "OTHER",
"section_title": "B. A gamma matrix representation",
"text": "A convenient representation of the SO(1,9) gamma matrices is given by the 32×32 matrices (with eight-dimensional dot products). These can be solved for u s in terms of u c :\n\nu s = -i √ 2k + (σ • ∂)u c = 1 √ 2k + (σ • k)u c , (B.1)\n\nwhere k + = -i∂ + = -i √ 2 (∂ 0 + ∂ 9 )."
}
] | |
arxiv:0704.0018 | 0704.0018 | 1 | ee51631f7ef4614c82d5dfcb38c0d10d13a798d2b8a12e1d77f585492414ebce | In quest of a generalized Callias index theorem | We give a prescription for how to compute the Callias index, using as
regulator an exponential function. We find agreement with old results in all
odd dimensions. We show that the problem of computing the dimension of the
moduli space of self-dual strings can be formulated as an index problem in
even-dimensional (loop-)space. We think that the regulator used in this Letter
can be applied to this index problem. | [
"Andreas Gustavsson"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | We give a prescription for how to compute the Callias index, using as regulator an exponential function. We find agreement with old results in all odd dimensions. We show that the problem of computing the dimension of the moduli space of self-dual strings can be formulated as an index problem in even-dimensional (loop-)space. We think that the regulator used in this Letter can be applied to this index problem.
We do not know what six-dimensional (2, 0) theory really is. It is believed that it can sustain solitonic self-dual strings [1] , although no one today knows what a (non-Abelian) self-dual string really is. But if we break the gauge group maximally to U (1) r , then we should be able to define the charges of these mysterious self-dual strings by the asymptotic behaviour of the U (1) gauge fields. One should expect these asymptotic U (1) fields to be (at least isomorphic with) a copy of the familiar abelian two-form gauge potentials (with self-dual field strengths).
It now seems to make sense to ask a question like, what is the dimension of the moduli space of self-dual strings of a given charge?
If the gauge group is SU (2) and is broken to U (1) by the Higgs vacuum expectation value (that should also determine the tension of the string), then the intuitive answer to this question is 4N where N is the U (1) charge in a suitable normalization, such that N = 1 corresponds to one self-dual string. One may argue that half the supersymmetry is broken by the string. Therefore one string should sustain 4 fermionic zero modes. Since some (half) of the supersymmety is unbroken there should also be 4 corresponding bosonic zero modes. These are naturally identified with the translational zero modes associated with the four transverse directions to the string. Furthermore, the strings being BPS, should be possible to separate at no cost of energy (thus staying in the moduli space approximation). If we take them far from each other, one may suspect that we can just add 4 bosonic zero modes from each string, to get 4N bosonic zero modes in total in a configuration of N strings [2] .
It would of course be nice to have a proof of this conjecture. Could it be proven if one had some index theorem? We will not provide a full solution to this problem in this Letter. But we will make it plausible that the problem can indeed be solved by computing the index of a certain Dirac operator in loop space.
To address our index problem, we think that one can lend the methods that Callias [3] used to prove his index theorem in odd-dimensional spaces. In our case we have an even number of dimensions (namely the four transverse direction) so it is apparent that we would have to construct a new type of index. This we do in section 3.
In section 2 we recall the Callias method [3] to address index problems in open spaces, though we will modify Callias' regularization, using the more convergent exponential function to obtain the index, as the limit
lim s→∞ Tr γe -sD 2 , (1)
(here D 2 > 0 and γ =diag (1, -1)) rather than lim
M→0 Tr γ M 2 D 2 + M 2 , (2)
which is the regularization that Callias used. We think that using the more convergent regularization of an exponential function is interesting in itself, as it could possibly extend the Callias index theorem to a wider class of index problems. Therefore we will devote the first part of this Letter on this subject.
But let us at once say that our regulator probably has no advantages when attacking these old problems. It does not provide us with a solution for how to count the number of zero modes in a multimonopole configuration with a nonmaximally broken gauge group, where the index can not be reliable computed due to a contribution from the continuum portion of the spectrum. What we hope though, is that our regulatization can be useful when attacking our new index problem associated with the moduli space of self-dual strings.
In section 2 we obtain the index in one and three dimensions. In three dimensions we apply this on the multimonopole moduli space and re-derive the result in [4] . A recent review article on monopoles and supersymmetry is [5] . The one and three-dimensional index problems have also been studied in [6] .
We then indicate how our method manages to reproduce the correct results in any odd dimensions. In section 3 we show how one at least in principle should be able to compute the dimension of the moduli space of N self-dual strings by computing a certain index.
For Dirac operators on open n -1-dimensional space where n -1 is odd, there is an index theorem by Callias [3] . This applies to Dirac equations of the form
Dψ = 0 (3)
where the Dirac operator D is of the form
D = γ i iD i + γ n φ. (4)
Here i = 1, ..., n -1 and γ µ ≡ (γ i , γ n ) denote the Dirac gamma matrices,
{γ µ , γ ν } = 2δ µν . (5)
We define the gauge covariant derivative as iD is = i∂ is + A is and all our fields are hermitian. If n -1 is odd, the gamma matrices can be represented as
γ i = 0 γ i γ i 0 , γ n = 0 i -i 0 (6)
One may use the n-dimesional notation A µ = (A i , φ), D = γ µ iD µ , but one must then remember that space is really n -1 dimensional.
If n -1 is even there is no Weyl representation of the gamma matrices (because of the inclusion of the 'gamma-five'), and no index theorem of this form exists.
We define the 'gamma-five' for even n as
γ ≡ -i -n 2 γ 1•••n (7)
which then is hermitian, and we define the projectors
P ± = 1 2 (1 ∓ γ) . (8)
In odd dimensions n -1, the Dirac operator splits into two Weyl operators
D ≡ P + DP - D † ≡ P -DP + (9)
Because P ± and D are all hermitian, it follows that D † is the hermitian conjugate of D. Also, because D is already of an off-block diagonal form, it suffices to include just one of the projectors, so we can just as well write this as
D = P + D = DP - D † = P -D = DP + (10)
The index can now be defined as
dim ker D -dim ker D † (11)
Since ker D = ker D † D and ker D † = ker DD † we can express this as
2 dim ker D † D -dim ker DD † = dim ker γD 2 . ( 12
)
where we have noted that γ = P --P + .
Callias, Weinberg and others used the regulator
I(M 2 ) = Tr γ M 2 D 2 + M 2 (13)
to obtain the index as the limit M 2 → 0. In this Letter we will be slightly more general. We define
J i (x, y) ≡ tr x |γγ i f (D)| y , (14)
for any function f (and of course D is not dimensionless, so D has to be accompanied by M in a suitable way). Then we notice that
W (x, y) ≡ (iγ i ∂ x i + γ µ A µ (x) + M ) x |f (D)| y = x |f (D)| y -iγ i ∂ y i + γ µ A µ (y) + M (15)
where (manifestly)
W (x, y) = x |(D + M )f (D)| y . ( 16
)
From this, we obtain the following identity
i ∂ x i + ∂ y i J i (x, y) = 2tr x |γDf (D)| y +tr (A µ (y) -A µ (x)) x |γγ µ f (D)| y (17)
In odd dimensions, the second term in the right hand side vanishes as x approaches y. This can be seen as being equivalent to the statement that there is no chiral anomaly in odd dimensions (by using point-splitting and inserting a Wilson line). So we get
i∂ i J i (x, x) = 2tr x |γDf (D)| x ( 18
)
2 To see this that ker D = ker D † D we apply the definition of hermitian conjugate with respect to the inner product (ψ, χ) = R dxψ † χ and the property of the norm, to 0 = (ψ, D † Dψ) = (Dψ, Dψ).
If we wish to compute the index as in Eq (13), then we can take
f (D) = 1 D M 2 D 2 + M 2 (19)
(however there is no unique choice of J i ). We then get
J i (x, y) = tr x γγ i 1 D M 2 D 2 + M 2 y = tr x γγ i 1 D -D 2 + D 2 + M 2 1 D 2 + M 2 y = -tr x γγ i D 1 D 2 + M 2 y . ( 20
) provided tr x γγ i 1 D y = 0 (21)
We will see in the next few paragraphs how one can achieve this by using a principal value prescription. The virtue of expressing Eq (13) as a total divergence, is that we then can compute the index as a boundary integral over an (n -2)-sphere at infinity as
I(M 2 ) = i 2 S n-2 ∞ dΩ n-2 r n-2 xi J i (x, x). ( 22
)
where r is the radius of the sphere and dΩ n-2 denotes the volume element of the unit sphere.
If instead we wish to compute the index as the limit of
I(s) = Tr γe -sD 2 . ( 23
)
as s → ∞, then we get
J i (x, y) = tr x γγ i 1 D e -sD 2 y . (24)
It might seem confusing that we can have a plus sign here, when we have a minus sign in Eq (20). These peculiar signs seem to be correct though. Why we can have opposite signs should be a reflection of the fact that these expressions can not be continuously connected with each other, at least not in any obvious way (like taking M to zero and s to zero. In fact s should be taken to plus infinity as M goes to zero). We will now illustrate how one can use this J i to compute the index in odd dimensions.
We choose our gamma matrices as
γ 1 = 0 1 1 0 , γ 2 = 0 i -i 0 (25)
and we have
γ = iγ 1 γ 2 = 1 0 0 -1 . ( 26
)
The Dirac operator reads
D = iγ 1 ∂ + γ 2 φ (27)
We need the square of the Dirac operator,
D 2 = -∂ 2 + φ 2 + γ∂φ. ( 28
)
We make the choice
J 1 (x, y) = -tr x γγ 1 D 1 D 2 + M 2 y (29)
We assume that φ(x) converges towards some constant values at x = -∞ and x = +∞. That means that we may ignore ∂φ(x) for sufficiently large |x|, where we then get
J 1 (x, x) = -tr (γγ 1 γ 2 ) ∞ -∞ dk 2π φ k 2 + φ 2 + M 2 = i φ φ 2 + M 2 (30)
The index is now given by lim
M→0 i 2 (J 1 (+∞) -J 1 (-∞)) = ±1 (31)
if φ flips the sign an odd number of times when going from -∞ to +∞, and 0 otherwise.
If instead we choose
J(x, y) = tr x γγ 1 D 1 D 2 e -sD 2 y (32)
then we get
J(x, x) = tr (γγ 1 γ 2 ) dk 2π φ k 2 + φ 2 e -s(k 2 +φ 2 ) (33)
If we compute the integral over k in the most natural way, then we get a result that vanishes in the limit s → ∞. Could there be another way of defining this integral, such that we do not get zero as the result? We notice that the integral
A(s) ≡ dk e -s(k 2 +1) k 2 + 1 (34)
for s > 0 is convergent only if we integrate k along a line in the complex plane which is such that it asymptotically is such that -π 2 < θ < π 2 where k = |k|e iθ . Integrating along any such line in the complex plane, we get the same value of this integral. If on the other hand we integrate over a line that asymptotically lies outside this cone, then we get a divergent integral for s > 0. But we get a convergent integral for s < 0. We then define the value of the integral for s > 0 as the analytic continuation of the same integral for s < 0. It remains to compute this convergent integral. Replacing k by ik and s by -s, we get the integral
A(-s) = -i ∞ -∞ dk e -s(k 2 -1) k 2 -1 (35)
We can compute its derivative
A ′ (-s) = -i ∞ -∞ dke -s(k 2 -1) = -i π s e s (36)
The right-hand side can obviously be analytically continued to -s, and that is how we will define A(s) where the integral representation does not converge. We can then integrate up A ′ (s),
A(∞) = A(0) - ∞ o ds π s e -s = A(0) - √ πΓ 1 2 = A(0) -π (37)
and we then need to compute
A(0) = i ∞ -∞ dk 1 k 2 -1 (38)
We define this as the principal value. This is ad hoc -we have no argument why one should define it like this. But if we accept this, then we get A(0) = 0. We conclude that we could just as well define the integral that we had, as
lim s→∞ dk e -s(k 2 +1) k 2 + 1 = -π. ( 39
)
But this requires us to perform the integration of k in the cone where it diverges for s > 0, and then define this integral by analytic continuation. This seem to be rather ad hoc. We have three rather week arguments why one should Wick rotate. First, if we keep xy as a small number, then we get the factor e ik(x-y) and this can act as a convergence factor only if we Wick rotate. (We illustrate this in the Appendix where we compute the corresponding integral in any complex number of dimensions.) Second, it seems to be the only way that we could produce a non-trivial answer. Third, with this prescription we will manage to reproduce the right answer in any odd number of dimensions, where we can check our result against the safer regularization used by Callias.
If we compute the integral by this prescription, then we get
J(x, x) = tr (γγ 1 γ 2 ) lim s→∞ dk 2π φ k 2 + φ 2 e -s(k 2 +φ 2 ) = i φ φ 2 (40)
and we see that we indeed get the right answer.
The physics problem that we will consider in three dimensions, is to compute number of zero modes of the Bogomolnyi equation
F ij = ǫ ijk D k φ (41)
We choose the convention that our fields are hermitian. It is convenient to group the fields into 'gauge potential'
A µ = (A i , φ) (42)
We define D µ = (D i , φ) such that iD µ = i∂ µ + A µ and we let G µν = i[D µ , D ν ] be the associated 'field strength'. Then the Bogomolnyi equation reads
G µν = 1 2 ǫ µνρσ G ρσ . ( 43
)
Linearizing this, we get
D µ δA ν = 1 2 ǫ µνρσ D ρ δA σ (44)
Contracting with γ µν , we get
(1 + γ)γ µν D µ δA ν = 0 (45)
and if we impose the background gauge condition
D µ δA µ = 0 (46)
which is to say that zero modes are orthogonal to gauge variations with respect to the moduli space metric, then we can write this linearized equation as a Dirac equation
Dψ ≡ γ µ D µ ψ = 0 ( 47
)
where
ψ := (1 + γ)γ µ δA µ . (48)
We compute
D 2 = -D 2 i + φ 2 + 1 2 iγ µν G µν (49)
Inserting the Bogomolnyi configuration we can write this, thus using the fact that G µν is selfdual,
D 2 = -D 2 i + φ 2 + 1 4 (1 + γ)iγ µν G µν . ( 50
)
and get a vanishing theorem. Namely, dim ker DD † = 0 as DD † > 0 is strictly postive. Hence we can compute the dimension of the moduli space dim ker D ≡ dim ker D † D just by computing the index of D. To compute the index, we now wish to compute
J i (x, x) = tr x γγ i γ k D k 1 D 2 e -sD 2 x (51)
We assume that asymptotically φ approaches a constant value at infinity. This corresponds to a gauge choice where we have a Dirac string singularity. Some further examination reveals that we get a non-negligible contribution to J i , for a sufficiently large two-sphere, only from the term
J i (x, x) = tr γγ i γ 4 φ d 3 k (2π) 3 1 k 2 + φ 2 + 1 2 iγ µν G µν e -s(k 2 +φ 2 + 1 2 iγµν Gµν ) (52)
We thus need to perform an integral of the form
A(s) = dk k 2 k 2 + 1 e -s(k 2 +1) (53)
If we choose the same prescription as we did in one dimension, then we get the result
A(+∞) = π. ( 54
)
For details of such a computation we refer to appendix A.
If we apply this result to the integral that we had, we get
J i (x, x) = 1 2π tr γγ i γ 4 φ φ 2 + 1 2 iγ µν G µν (55)
We expand the square root,
φ 2 + 1 2 iγ µν G µν = φ + 1 4 φ 2 iγ µν G µν + ... (56)
In the far distance, in a charge Q monopole configuration, we find that
γ µν G µν = 2γ k γ 4 (1 -γ) xk r 2 Q (57)
and so when we trace over the gamma matrices, we get
J i (x, x) = ix i 2πr 2 tr φQ φ 2 . ( 58
)
If we now for instance assume SU (2) gauge group, broken to U (1), then if we integrate i 2 J i over S 2 , we get the index 2Q. The number of bosonic zero modes is twice the index, i.e. -4Q in our conventions [4, 5] .
In 2m + 1 dimensions we get the integral
A(µ) ≡ lim s→∞ dk k 2m k 2 + µ 2 e -s(k 2 +µ 2 )
(59) if we use our regulator. Here
µ 2 ≡ v 2 + G (60)
(and G is an abbreviation for 1 2 iγ µν G µν .) This should be compared to the integral
B(µ) ≡ -lim M→0 (-1) m dk k 2m (k 2 + v 2 + M 2 ) m+1 G m (61)
that we get using the Callias regulator. 3 In order to compare these integrals, we rewrite them as
A(µ) = µ 2m-1 a B(µ) = v -1 bG m (63)
where
a = lim s→∞ dξ ξ 2m ξ 2 + 1 e -s(ξ 2 +1) b = -lim M→0 (-1) m dξ ξ 2m ξ 2 + 1 + M 2 m+1 (64)
We compute a according the prescription introduced above in one and three dimensions, that is by Wick rotating ξ and continue analytically in s. (Details are in appendix A.) We can compute b using residue calculus (introducing a regulator so that we can close the contour on a semi-circle at infinity). The result is
a = -(-1) m π b = (-1) m 1 2 π Γ m -1 2 Γ 1 2 (65)
We next expand
vA(µ) = v v 2 + G m-1 2 a = v 2m a + ... + Γ m -1 2 Γ -1 2 aG m + ... vB(µ) = bG m (66)
and we find that the coefficient of G m becomes equal to
-(-1) m Γ m -1 2 Γ -1 2 π ( 67
)
if one uses our regularization, and equal to
(-1) m 1 2 Γ m -1 2 Γ 1 2 π ( 68
)
3 This integral comes from expanding
1 k 2 + v 2 + G + M 2 = 1 k 2 + v 2 + M 2 + ... ( 62
)
in powers of G as a geometric series [4] .
if one uses the Callias regularization. We see that the two expressions coincide for all m.
We have now showed that if we use our prescription of Wick rotating k to compute the integrals over the exponential, then we get the right answer for all cases that can be safely computed using a regulator that is less convergent. We are inclined to think that our prescription for how to compute the integral, will also work for index problems where the Callias regulator diverges. But we have no proof. It is perhaps not so obvious that more general index problems can be formulated. In the next section we will give one example of a more general type of index problem.
To introduce the notation, we first consider the free Abelian tensor multiplet theory in 1 + 5 dimensions. The on-shell field content is a two-form gauge potential B µν , five scalar fields φ A and corresponding Weyl fermions ψ. The field strength
H µνρ = ∂ µ B νρ + ∂ ρ B µν + ∂ ν B ρµ is selfdual. The supersymmetry variation of the Weyl fermions is δψ = 1 12 Γ µνρ H µνρ + Γ µ Γ A ∂ µ φ A ǫ (69)
where we use eleven-dimensional gamma matrices splitted into SO(1, 5)×SO( 5 ), so that in particular
{Γ µ , Γ A } = 0. ( 70
)
In a static and x 5 independent field configuration, in which only φ 5 =: φ is non-zero, we find the SUSY variation
δψ = Γ 0i5 H 0i5 + Γ i Γ A=5 ∂ i φ ǫ (71)
If we assume that the classical bosonic field configuration is such that
∂ i φ = H 0i5 (72)
then the SUSY variation reduces to
δψ = ∂ i φΓ i Γ 05 + Γ A=5 ǫ (73)
and we find the condition for unbroken SUSY as
1 + Γ 05 Γ A=5 ǫ = 0 (74) If we use the Weyl condition Γǫ = -ǫ (75)
of the (2, 0) supersymmetry parameter ǫ, then we can also write this as
1 + Γ 1234 Γ A=5 ǫ = 0. ( 76
)
We may represent the gamma matrices as
Γ µ = (Γ 0 , Γ i , Γ 5 ) = 1 ⊗ iσ 2 ⊗ 1, γ i ⊗ σ 1 ⊗ 1, γ ⊗ σ 1 ⊗ 1 Γ A = 1 ⊗ iσ 2 ⊗ σ A ( 77
)
where σ 1,2,3 are the Pauli sigma matrices, γ = γ 1234 . Then the condition for unbroken SUSY is
(1 + γ ⊗ σ) ǫ = 0 ( 78
)
where σ = σ 1234 = σ A=5 .
We have found that if
H ijk = ǫ ijkl ∂ l φ (79)
then half SUSY is unbroken. This equation is the Bogomolnyi equation for selfdual strings [1] . We are interested in finding the number of parameters needed to describe solutions of this equation. We can linearize it and get the equation
γ i ∂ i χ = 0 (80)
for the bosonic zero modes, that we have gathered into a matrix
χ ≡ γ ij δB ij + γδφ. (81)
For this to work we must also assume the background gauge condition
∂ i B ij = 0. ( 82
)
Now this linearized equation Eq (80) does not make any reference to the gauge field. So there is no way that we could count the number of parameters of a multi-string configuration just using this equation. This should of course not be a surprise. The strings that we have in the Abelian theory are not solutions of the field equations. They have to be inserted by hand, that is we need to insert delta function sources by hand, in the same spirit as for Dirac monopoles.
To be able to count the number of zero modes, we must consider some interacting theory which (at the classical level) has solitonic string solutions.
To pass to non-Abelian theory we begin by rewriting the Abelian theory in loop space. Loop space consists of parametrized loops C: s → C µ (s). We introduce the Abelian 'loop fields' [7] A µs = B µν (C(s)) Ċν (s)
φ µs = φ(C(s)) Ċµ (s) ψ µs = ψ(C(s)) Ċµ (s) ( 83
)
With these definitions, a short computation reveals that A µs transforms as a vector and φ µs a contra-variant vector under diffeomorphisms in loop space induced by diffeomorphisms in space-time. One may then extend these transformation properties to any diffeomorphism in loop space. Space-time diffeomorphism and reparametrizations of the loops then get unified and are both diffemorphisms in loop space. The only thing to remember is what is kept fixed under the variation. If it is the parameter of the loop, or the loop itself.
The field strength becomes
F µs,νt = H µνρ (C(s)) Ċρ (s)δ(s -t) (84)
In terms of these fields, the Bogomolnyi equation will read foot_0
F is,jt = ǫ ijkl ∂ k(s φ lt) . (85)
We pass to the non-Abelian theory by letting these loop fields become non-Abelian, in the sense that A µs = A a µs λ a (s) where λ a (s) are generators of a loop algebra associated to the gauge group [7] . We introduce a covariant derivative
D µs = ∂ µs + A µs . (86)
Local gauge transformations act as
δ Λ A µs = D µs Λ δ Λ φ µs = [φ µs , Λ]. (87)
Given a loop C, we automatically get a tangent vector Ċµ (s) that makes no reference to space-time. We can therefore impose the loop space constraints Ċµ (s)A µs = 0 (88)
for each s, and also
φ µs = Ċµ (s)φ(s; C) (89)
for some subtle field φ(s; C) on loop space. As a consequence, we find that
A µs φ µs = 0. ( 90
)
These constraints are covariant under diffeomorphisms of space-time and reparametrizations of loops. They are invariant also under local gauge transformations, provided that the gauge parameter is subject to the condition
Ċµ (s)∂ µs Λ = 0 (91)
which is the condition of reparametrization invariance. With the assumption made that λ a (s) are generators of a loop algebra, we find that the constraint can also be written as
[A µs , φ µt ] = 0 (92)
A local gauge variation of this constraint is
[D µs Λ, φ µt ] + [A µs , [φ µt , Λ]] = [∂ µs Λ, φ µt ] + [[A µs , Λ], φ µt ] + [A µs , [φ µt , Λ]] = [∂ µs Λ, φ µt ] + [Λ, [φ µt , A µs ]] (93)
The last term vanishes by the constraint. The first term gives us the constraint Eq (92) that we must impose on the gauge parameter
Λ = dsΛ a (s, C)λ a (s). (94)
We have now introduced non-local non-Abelian fields with infinitely many components. It is also likely that consisteny of the theory requires an infinite set of constraints on these fields. Maybe then, it could be that we may in the end descend to a finite degrees of freedom. But this is just a speculation. The problem appears to be difficult and ill-defined -How should one define a degree of freedom in a strongly coupled non-local theory?
The non-Abelian generalization of the Bogomolnyi equation should be given by [7]
F is,jt = ±ǫ ijkl D k(s φ lt) . ( 95
)
This equation is gauge invariant and invariant under the residual SO(4) Lorentz group that is preserved by the strings. We can not think of any reasonable modification of this equation that would preserve these symmetries, so on this grounds alone one could suspect this equation to be correct. Of course this is not the only requirement that the BPS condition imposes. We also get conditions on the 0s and the 5s components. But these BPS equations will be of no interest to us right now. We will show below that the linearized Bogomolnyi equation can be written as
γ i D i(s + σφ i(s χ t) = 0 (96)
We will also see below that we (presumably) can actually drop the symmetrization in s and t in this equation. The fields transform in the adjoint representation of the loop algebra, by which we mean that φ is χ t = [φ is , χ t ]. We define the Dirac operator
D s = γ i (D is + σφ is ) (97)
and the projectors
P ± ≡ 1 2 (1 ∓ γσ) , (98)
We can now formulate an index problem, in an even-dimensional (loop-)space.
The even-dimensional space in this case is given by the 4-dimensional transverse space to the strings, and the index is given by
dim ker D s -dim ker D † s ( 99
)
where
D s = P + D s = D s P - D † s = P -D s = D s P + . (100)
Since D s and P ± are hermitian, it is manifest that D † s defined this way will be the hermitian conjugate of D s , thus justifying the notation.
Computing the index alone is not sufficient in order to obtain the dimension of the moduli space of self-dual strings. We also need a vanishing theorem that says that dim ker D † s = 0. Linearizing the Bogomolnyi equation, we get
2D [is δA jt] = ±ǫ ijkl (D ks δφ lt + φ ks δA lt ) (101)
Contracting by γ ij , we get
γ ij Dis χ jt = 0 ( 102
)
where we have defined
Dis ≡ D is ∓ γφ is χ is ≡ δA is ∓ γδφ is ( 103
)
To see that the linearized BPS equation can be written like this, one must use the constraint
γ ij φ is δφ jt = 0. ( 104
)
We can avoid having explicit ± signs by introducing the other chiraly matrix at our disposal, namely σ that lives in a different vector space than γ. We can then hide the ± signs in the tensor product
γ ⊗ σ = ±1 (105)
which amounts to
Dis ≡ D is + σφ is χ is ≡ δA is + σδφ is (106)
without any ±. 5 If we define
χ s ≡ γ i χ is (108)
then we can write the zero mode equation as
γ i Dis χ t + Di s χ it = 0. ( 109
)
Let us analyze the second term in this equation. It is given by
D i s δA it + φ i s δφ it -γ φ i s δA it + D i s δφ it (110)
We should not count variations that are gauge variations as bosonic zero modes. We can insure this by demanding the zero modes to be orthogonal to gauge variations, with respect to the metric on the moduli space,
(δ Λ A is , δA it ) + (δ Λ φ is , δφ jt ) = 0 (111)
This leads to the background gauge condition
D i s δA it + φ i s δφ it = 0.
(112) 5 To really understand what is going on, one should apply (1 ± γσ) on everything, on ψs and on Ds. Then one notices that
∓γ (1 ∓ γσ) = σ (1 ∓ γσ) . (107)
That is, we can trade ∓γ for σ, once we apply (1 ± γσ) on everything. This is what we really should do, but to keep the notation simple, we do not spell this out.
This condition implies that the gauge variation of the zero modes vanishes,
δ Λ δA is = 0 = δ Λ δφ is (113)
To see this, we make a gauge variation δ Λ δA is = D is Λ, δ Λ φ is = φ is Λ, and ask which gauge parameters Λ will respect the background gauge condition. Inserting this gauge variation into the background gauge condition, we get
D i s D it + φ i s φ it Λ = 0. ( 114
)
For this to work nicely, it seems that we must constrain the non-locality of our loop field such that ∂ i (s ∂ it) < 0. Then the only solution to this equation is Λ = 0. In other words all gauge variations of the zero modes have to vanish.
Furthermore we want the variation to preserve the orthogonality between A is and φ is ,
(A is , δφ it ) + (δA is , φ it ) = 0 (115)
If we make a gauge variation of this, then we get the condition
(δ Λ A is , δφ it ) + (δA is , δ Λ φ it ) = 0 (116)
which amounts to
φ i s δA it + D i s δφ it = 0. ( 117
)
We conclude that the zero mode equation can be written as
D s χ t = 0 (118)
where
D s = γ i (D is + σφ is ) (119)
We are interested in counting the number of such modes in a background of k BPS strings. We compute
D 2 = (D is ) 2 + (φ is ) 2 + 1 2 γ ij (F is,js + γσǫ ijkl D ks φ ls ) (120)
(Here D 2 ≡ D s D s ≡ ds 2π D s D s , and analogously for the other fields or operators.) In a BPS configuration, we get is
D 2 = (D is ) 2 + (φ is ) 2 + 1 2 γ ij (1 + γσ) F is,js (121)
Furthermore, in the subspace where 1 + γσ = 0, we find that
D 2 = (D is ) 2 + (φ is ) 2 (122)
is a strictly negative operator, hence has no zero modes. This means that we have a vanishing theorem, dim ker D † = 0.
The zero mode equation was really
D (s χ t) = 0 ( 123
)
where we should symmetrize in s and t. That means that we should rather consider
D s D (s χ t) = 1 2 (D s D s χ t + D s D t χ s ) = 1 2 (D s D s χ t + D t D s χ s + [D s , D t ]χ s ) . (124)
If now D [s D t] = 0 and D s χ s = 0, then we get
D s D s χ t = 0 (125)
The latter condition, D s χ s = 0 is of course a consequence of D (s χ t) = 0 with s = t. The former condition reads
0 = D [s D t] = D i[s D it] + φ i[s φ it] + σD i[s φ it] (126)
which we would like to impose as a constraint. Restricting to the abelian case this is condition is of course true as 0 ≡ ∂ i[s ∂ |i|t] . If we can impose this as a constraint on the non-abelian fields, then we have now seen that the zero mode equation Eq (123) implies that dsD † s D s χ t = 0 (127) because D s is anti-self-adjoint with respect to the inner product (ψ s , χ t ) = DCtr ψ † s (C)χ t (C) (128) on loop space. We can also go in the opposite direction. Assuming that Eq (127) holds, we get 0 = χ t , D † s D s χ t = (D s χ t , D s χ t ) (129) and we conclude that (123) implies D s χ t = 0 (130) with no symmetrization in s, t.
We should now be able to compute an index associated to self-dual strings, as the limit
I(s) = Tr γσe sD 2 (131)
when s → ∞. We define the quantity
J is (C, C ′ ) = tr C γσγ i γ k (D ks + σφ ks ) 1 D 2 e sD 2 C ′ (132)
(it should be clear that the two s's involved in this formula are totally unrelated) and find that
I(s) = ds 2π DC∂ is J is (C, C) (133)
We can separate the functional integral over parametrized loops C into several pieces. We can keep a point on the loops C(s) = x fixed, and separate it as
DC = d 4 x D x C (134)
Then we can write I(s) as an integral over a large three-sphere at spatial infinity,
ds 2π d 4 x D x C ∂J is (C) ∂C i (s) = ds 2π S 3 dΩ 3 xi D x CJ is (C, C) (135)
where thus x = C(s).
If we assume that the gauge group is maximally broken to a product of U (1)'s by the Higgs vacuum expectation values, then we should have U (1) loop fields at spatial infinity.
If we assume that the gauge group is SU (2) and that it is broken to U (1), then we need only the asymptotic form of the U (1) fields at spatial infinity, could certainly arise somewhere (in odd dimensions a corresponding term vanished since there is no chiral anomaly in odd dimensions). In our case this term vanishes identically by the Bogomolnyi equation and the constraint foot_1
F is,jt D is φ jt = 0. ( 139
)
Then there can be a term
ǫ ijkl D x Ctr F is,jt φ ks v (140)
that should arise in a very similar way as the corresponding term arose for monopoles. If we insert the asymptotic U (1) fields, this term becomes proportional to
ǫ ijkl H ijk (x) (141)
That means that the index should be given by some numerical constant, times the magnetic charge
S 3 ∞ H. (142) | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We give a prescription for how to compute the Callias index, using as regulator an exponential function. We find agreement with old results in all odd dimensions. We show that the problem of computing the dimension of the moduli space of self-dual strings can be formulated as an index problem in even-dimensional (loop-)space. We think that the regulator used in this Letter can be applied to this index problem."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "We do not know what six-dimensional (2, 0) theory really is. It is believed that it can sustain solitonic self-dual strings [1] , although no one today knows what a (non-Abelian) self-dual string really is. But if we break the gauge group maximally to U (1) r , then we should be able to define the charges of these mysterious self-dual strings by the asymptotic behaviour of the U (1) gauge fields. One should expect these asymptotic U (1) fields to be (at least isomorphic with) a copy of the familiar abelian two-form gauge potentials (with self-dual field strengths).\n\nIt now seems to make sense to ask a question like, what is the dimension of the moduli space of self-dual strings of a given charge?\n\nIf the gauge group is SU (2) and is broken to U (1) by the Higgs vacuum expectation value (that should also determine the tension of the string), then the intuitive answer to this question is 4N where N is the U (1) charge in a suitable normalization, such that N = 1 corresponds to one self-dual string. One may argue that half the supersymmetry is broken by the string. Therefore one string should sustain 4 fermionic zero modes. Since some (half) of the supersymmety is unbroken there should also be 4 corresponding bosonic zero modes. These are naturally identified with the translational zero modes associated with the four transverse directions to the string. Furthermore, the strings being BPS, should be possible to separate at no cost of energy (thus staying in the moduli space approximation). If we take them far from each other, one may suspect that we can just add 4 bosonic zero modes from each string, to get 4N bosonic zero modes in total in a configuration of N strings [2] .\n\nIt would of course be nice to have a proof of this conjecture. Could it be proven if one had some index theorem? We will not provide a full solution to this problem in this Letter. But we will make it plausible that the problem can indeed be solved by computing the index of a certain Dirac operator in loop space.\n\nTo address our index problem, we think that one can lend the methods that Callias [3] used to prove his index theorem in odd-dimensional spaces. In our case we have an even number of dimensions (namely the four transverse direction) so it is apparent that we would have to construct a new type of index. This we do in section 3.\n\nIn section 2 we recall the Callias method [3] to address index problems in open spaces, though we will modify Callias' regularization, using the more convergent exponential function to obtain the index, as the limit\n\nlim s→∞ Tr γe -sD 2 , (1)\n\n(here D 2 > 0 and γ =diag (1, -1)) rather than lim\n\nM→0 Tr γ M 2 D 2 + M 2 , (2)\n\nwhich is the regularization that Callias used. We think that using the more convergent regularization of an exponential function is interesting in itself, as it could possibly extend the Callias index theorem to a wider class of index problems. Therefore we will devote the first part of this Letter on this subject.\n\nBut let us at once say that our regulator probably has no advantages when attacking these old problems. It does not provide us with a solution for how to count the number of zero modes in a multimonopole configuration with a nonmaximally broken gauge group, where the index can not be reliable computed due to a contribution from the continuum portion of the spectrum. What we hope though, is that our regulatization can be useful when attacking our new index problem associated with the moduli space of self-dual strings.\n\nIn section 2 we obtain the index in one and three dimensions. In three dimensions we apply this on the multimonopole moduli space and re-derive the result in [4] . A recent review article on monopoles and supersymmetry is [5] . The one and three-dimensional index problems have also been studied in [6] .\n\nWe then indicate how our method manages to reproduce the correct results in any odd dimensions. In section 3 we show how one at least in principle should be able to compute the dimension of the moduli space of N self-dual strings by computing a certain index."
},
{
"section_type": "OTHER",
"section_title": "Computing the Callias index in odd-dimensional spaces",
"text": "For Dirac operators on open n -1-dimensional space where n -1 is odd, there is an index theorem by Callias [3] . This applies to Dirac equations of the form\n\nDψ = 0 (3)\n\nwhere the Dirac operator D is of the form\n\nD = γ i iD i + γ n φ. (4)\n\nHere i = 1, ..., n -1 and γ µ ≡ (γ i , γ n ) denote the Dirac gamma matrices,\n\n{γ µ , γ ν } = 2δ µν . (5)\n\nWe define the gauge covariant derivative as iD is = i∂ is + A is and all our fields are hermitian. If n -1 is odd, the gamma matrices can be represented as\n\nγ i = 0 γ i γ i 0 , γ n = 0 i -i 0 (6)\n\nOne may use the n-dimesional notation A µ = (A i , φ), D = γ µ iD µ , but one must then remember that space is really n -1 dimensional.\n\nIf n -1 is even there is no Weyl representation of the gamma matrices (because of the inclusion of the 'gamma-five'), and no index theorem of this form exists.\n\nWe define the 'gamma-five' for even n as\n\nγ ≡ -i -n 2 γ 1•••n (7)\n\nwhich then is hermitian, and we define the projectors\n\nP ± = 1 2 (1 ∓ γ) . (8)\n\nIn odd dimensions n -1, the Dirac operator splits into two Weyl operators\n\nD ≡ P + DP - D † ≡ P -DP + (9)\n\nBecause P ± and D are all hermitian, it follows that D † is the hermitian conjugate of D. Also, because D is already of an off-block diagonal form, it suffices to include just one of the projectors, so we can just as well write this as\n\nD = P + D = DP - D † = P -D = DP + (10)\n\nThe index can now be defined as\n\ndim ker D -dim ker D † (11)\n\nSince ker D = ker D † D and ker D † = ker DD † we can express this as\n\n2 dim ker D † D -dim ker DD † = dim ker γD 2 . ( 12\n\n)\n\nwhere we have noted that γ = P --P + .\n\nCallias, Weinberg and others used the regulator\n\nI(M 2 ) = Tr γ M 2 D 2 + M 2 (13)\n\nto obtain the index as the limit M 2 → 0. In this Letter we will be slightly more general. We define\n\nJ i (x, y) ≡ tr x |γγ i f (D)| y , (14)\n\nfor any function f (and of course D is not dimensionless, so D has to be accompanied by M in a suitable way). Then we notice that\n\nW (x, y) ≡ (iγ i ∂ x i + γ µ A µ (x) + M ) x |f (D)| y = x |f (D)| y -iγ i ∂ y i + γ µ A µ (y) + M (15)\n\nwhere (manifestly)\n\nW (x, y) = x |(D + M )f (D)| y . ( 16\n\n)\n\nFrom this, we obtain the following identity\n\ni ∂ x i + ∂ y i J i (x, y) = 2tr x |γDf (D)| y +tr (A µ (y) -A µ (x)) x |γγ µ f (D)| y (17)\n\nIn odd dimensions, the second term in the right hand side vanishes as x approaches y. This can be seen as being equivalent to the statement that there is no chiral anomaly in odd dimensions (by using point-splitting and inserting a Wilson line). So we get\n\ni∂ i J i (x, x) = 2tr x |γDf (D)| x ( 18\n\n)\n\n2 To see this that ker D = ker D † D we apply the definition of hermitian conjugate with respect to the inner product (ψ, χ) = R dxψ † χ and the property of the norm, to 0 = (ψ, D † Dψ) = (Dψ, Dψ).\n\nIf we wish to compute the index as in Eq (13), then we can take\n\nf (D) = 1 D M 2 D 2 + M 2 (19)\n\n(however there is no unique choice of J i ). We then get\n\nJ i (x, y) = tr x γγ i 1 D M 2 D 2 + M 2 y = tr x γγ i 1 D -D 2 + D 2 + M 2 1 D 2 + M 2 y = -tr x γγ i D 1 D 2 + M 2 y . ( 20\n\n) provided tr x γγ i 1 D y = 0 (21)\n\nWe will see in the next few paragraphs how one can achieve this by using a principal value prescription. The virtue of expressing Eq (13) as a total divergence, is that we then can compute the index as a boundary integral over an (n -2)-sphere at infinity as\n\nI(M 2 ) = i 2 S n-2 ∞ dΩ n-2 r n-2 xi J i (x, x). ( 22\n\n)\n\nwhere r is the radius of the sphere and dΩ n-2 denotes the volume element of the unit sphere.\n\nIf instead we wish to compute the index as the limit of\n\nI(s) = Tr γe -sD 2 . ( 23\n\n)\n\nas s → ∞, then we get\n\nJ i (x, y) = tr x γγ i 1 D e -sD 2 y . (24)\n\nIt might seem confusing that we can have a plus sign here, when we have a minus sign in Eq (20). These peculiar signs seem to be correct though. Why we can have opposite signs should be a reflection of the fact that these expressions can not be continuously connected with each other, at least not in any obvious way (like taking M to zero and s to zero. In fact s should be taken to plus infinity as M goes to zero). We will now illustrate how one can use this J i to compute the index in odd dimensions."
},
{
"section_type": "OTHER",
"section_title": "One dimension",
"text": "We choose our gamma matrices as\n\nγ 1 = 0 1 1 0 , γ 2 = 0 i -i 0 (25)\n\nand we have\n\nγ = iγ 1 γ 2 = 1 0 0 -1 . ( 26\n\n)\n\nThe Dirac operator reads\n\nD = iγ 1 ∂ + γ 2 φ (27)\n\nWe need the square of the Dirac operator,\n\nD 2 = -∂ 2 + φ 2 + γ∂φ. ( 28\n\n)\n\nWe make the choice\n\nJ 1 (x, y) = -tr x γγ 1 D 1 D 2 + M 2 y (29)\n\nWe assume that φ(x) converges towards some constant values at x = -∞ and x = +∞. That means that we may ignore ∂φ(x) for sufficiently large |x|, where we then get\n\nJ 1 (x, x) = -tr (γγ 1 γ 2 ) ∞ -∞ dk 2π φ k 2 + φ 2 + M 2 = i φ φ 2 + M 2 (30)\n\nThe index is now given by lim\n\nM→0 i 2 (J 1 (+∞) -J 1 (-∞)) = ±1 (31)\n\nif φ flips the sign an odd number of times when going from -∞ to +∞, and 0 otherwise.\n\nIf instead we choose\n\nJ(x, y) = tr x γγ 1 D 1 D 2 e -sD 2 y (32)\n\nthen we get\n\nJ(x, x) = tr (γγ 1 γ 2 ) dk 2π φ k 2 + φ 2 e -s(k 2 +φ 2 ) (33)\n\nIf we compute the integral over k in the most natural way, then we get a result that vanishes in the limit s → ∞. Could there be another way of defining this integral, such that we do not get zero as the result? We notice that the integral\n\nA(s) ≡ dk e -s(k 2 +1) k 2 + 1 (34)\n\nfor s > 0 is convergent only if we integrate k along a line in the complex plane which is such that it asymptotically is such that -π 2 < θ < π 2 where k = |k|e iθ . Integrating along any such line in the complex plane, we get the same value of this integral. If on the other hand we integrate over a line that asymptotically lies outside this cone, then we get a divergent integral for s > 0. But we get a convergent integral for s < 0. We then define the value of the integral for s > 0 as the analytic continuation of the same integral for s < 0. It remains to compute this convergent integral. Replacing k by ik and s by -s, we get the integral\n\nA(-s) = -i ∞ -∞ dk e -s(k 2 -1) k 2 -1 (35)\n\nWe can compute its derivative\n\nA ′ (-s) = -i ∞ -∞ dke -s(k 2 -1) = -i π s e s (36)\n\nThe right-hand side can obviously be analytically continued to -s, and that is how we will define A(s) where the integral representation does not converge. We can then integrate up A ′ (s),\n\nA(∞) = A(0) - ∞ o ds π s e -s = A(0) - √ πΓ 1 2 = A(0) -π (37)\n\nand we then need to compute\n\nA(0) = i ∞ -∞ dk 1 k 2 -1 (38)\n\nWe define this as the principal value. This is ad hoc -we have no argument why one should define it like this. But if we accept this, then we get A(0) = 0. We conclude that we could just as well define the integral that we had, as\n\nlim s→∞ dk e -s(k 2 +1) k 2 + 1 = -π. ( 39\n\n)\n\nBut this requires us to perform the integration of k in the cone where it diverges for s > 0, and then define this integral by analytic continuation. This seem to be rather ad hoc. We have three rather week arguments why one should Wick rotate. First, if we keep xy as a small number, then we get the factor e ik(x-y) and this can act as a convergence factor only if we Wick rotate. (We illustrate this in the Appendix where we compute the corresponding integral in any complex number of dimensions.) Second, it seems to be the only way that we could produce a non-trivial answer. Third, with this prescription we will manage to reproduce the right answer in any odd number of dimensions, where we can check our result against the safer regularization used by Callias.\n\nIf we compute the integral by this prescription, then we get\n\nJ(x, x) = tr (γγ 1 γ 2 ) lim s→∞ dk 2π φ k 2 + φ 2 e -s(k 2 +φ 2 ) = i φ φ 2 (40)\n\nand we see that we indeed get the right answer."
},
{
"section_type": "OTHER",
"section_title": "Three dimensions and magnetic monopoles",
"text": "The physics problem that we will consider in three dimensions, is to compute number of zero modes of the Bogomolnyi equation\n\nF ij = ǫ ijk D k φ (41)\n\nWe choose the convention that our fields are hermitian. It is convenient to group the fields into 'gauge potential'\n\nA µ = (A i , φ) (42)\n\nWe define D µ = (D i , φ) such that iD µ = i∂ µ + A µ and we let G µν = i[D µ , D ν ] be the associated 'field strength'. Then the Bogomolnyi equation reads\n\nG µν = 1 2 ǫ µνρσ G ρσ . ( 43\n\n)\n\nLinearizing this, we get\n\nD µ δA ν = 1 2 ǫ µνρσ D ρ δA σ (44)\n\nContracting with γ µν , we get\n\n(1 + γ)γ µν D µ δA ν = 0 (45)\n\nand if we impose the background gauge condition\n\nD µ δA µ = 0 (46)\n\nwhich is to say that zero modes are orthogonal to gauge variations with respect to the moduli space metric, then we can write this linearized equation as a Dirac equation\n\nDψ ≡ γ µ D µ ψ = 0 ( 47\n\n)\n\nwhere\n\nψ := (1 + γ)γ µ δA µ . (48)\n\nWe compute\n\nD 2 = -D 2 i + φ 2 + 1 2 iγ µν G µν (49)\n\nInserting the Bogomolnyi configuration we can write this, thus using the fact that G µν is selfdual,\n\nD 2 = -D 2 i + φ 2 + 1 4 (1 + γ)iγ µν G µν . ( 50\n\n)\n\nand get a vanishing theorem. Namely, dim ker DD † = 0 as DD † > 0 is strictly postive. Hence we can compute the dimension of the moduli space dim ker D ≡ dim ker D † D just by computing the index of D. To compute the index, we now wish to compute\n\nJ i (x, x) = tr x γγ i γ k D k 1 D 2 e -sD 2 x (51)\n\nWe assume that asymptotically φ approaches a constant value at infinity. This corresponds to a gauge choice where we have a Dirac string singularity. Some further examination reveals that we get a non-negligible contribution to J i , for a sufficiently large two-sphere, only from the term\n\nJ i (x, x) = tr γγ i γ 4 φ d 3 k (2π) 3 1 k 2 + φ 2 + 1 2 iγ µν G µν e -s(k 2 +φ 2 + 1 2 iγµν Gµν ) (52)\n\nWe thus need to perform an integral of the form\n\nA(s) = dk k 2 k 2 + 1 e -s(k 2 +1) (53)\n\nIf we choose the same prescription as we did in one dimension, then we get the result\n\nA(+∞) = π. ( 54\n\n)\n\nFor details of such a computation we refer to appendix A.\n\nIf we apply this result to the integral that we had, we get\n\nJ i (x, x) = 1 2π tr γγ i γ 4 φ φ 2 + 1 2 iγ µν G µν (55)\n\nWe expand the square root,\n\nφ 2 + 1 2 iγ µν G µν = φ + 1 4 φ 2 iγ µν G µν + ... (56)\n\nIn the far distance, in a charge Q monopole configuration, we find that\n\nγ µν G µν = 2γ k γ 4 (1 -γ) xk r 2 Q (57)\n\nand so when we trace over the gamma matrices, we get\n\nJ i (x, x) = ix i 2πr 2 tr φQ φ 2 . ( 58\n\n)\n\nIf we now for instance assume SU (2) gauge group, broken to U (1), then if we integrate i 2 J i over S 2 , we get the index 2Q. The number of bosonic zero modes is twice the index, i.e. -4Q in our conventions [4, 5] ."
},
{
"section_type": "OTHER",
"section_title": "(2m + 1) dimensions",
"text": "In 2m + 1 dimensions we get the integral\n\nA(µ) ≡ lim s→∞ dk k 2m k 2 + µ 2 e -s(k 2 +µ 2 )\n\n(59) if we use our regulator. Here\n\nµ 2 ≡ v 2 + G (60)\n\n(and G is an abbreviation for 1 2 iγ µν G µν .) This should be compared to the integral\n\nB(µ) ≡ -lim M→0 (-1) m dk k 2m (k 2 + v 2 + M 2 ) m+1 G m (61)\n\nthat we get using the Callias regulator. 3 In order to compare these integrals, we rewrite them as\n\nA(µ) = µ 2m-1 a B(µ) = v -1 bG m (63)\n\nwhere\n\na = lim s→∞ dξ ξ 2m ξ 2 + 1 e -s(ξ 2 +1) b = -lim M→0 (-1) m dξ ξ 2m ξ 2 + 1 + M 2 m+1 (64)\n\nWe compute a according the prescription introduced above in one and three dimensions, that is by Wick rotating ξ and continue analytically in s. (Details are in appendix A.) We can compute b using residue calculus (introducing a regulator so that we can close the contour on a semi-circle at infinity). The result is\n\na = -(-1) m π b = (-1) m 1 2 π Γ m -1 2 Γ 1 2 (65)\n\nWe next expand\n\nvA(µ) = v v 2 + G m-1 2 a = v 2m a + ... + Γ m -1 2 Γ -1 2 aG m + ... vB(µ) = bG m (66)\n\nand we find that the coefficient of G m becomes equal to\n\n-(-1) m Γ m -1 2 Γ -1 2 π ( 67\n\n)\n\nif one uses our regularization, and equal to\n\n(-1) m 1 2 Γ m -1 2 Γ 1 2 π ( 68\n\n)\n\n3 This integral comes from expanding\n\n1 k 2 + v 2 + G + M 2 = 1 k 2 + v 2 + M 2 + ... ( 62\n\n)\n\nin powers of G as a geometric series [4] .\n\nif one uses the Callias regularization. We see that the two expressions coincide for all m.\n\nWe have now showed that if we use our prescription of Wick rotating k to compute the integrals over the exponential, then we get the right answer for all cases that can be safely computed using a regulator that is less convergent. We are inclined to think that our prescription for how to compute the integral, will also work for index problems where the Callias regulator diverges. But we have no proof. It is perhaps not so obvious that more general index problems can be formulated. In the next section we will give one example of a more general type of index problem."
},
{
"section_type": "OTHER",
"section_title": "Four dimensions and self-dual strings",
"text": "To introduce the notation, we first consider the free Abelian tensor multiplet theory in 1 + 5 dimensions. The on-shell field content is a two-form gauge potential B µν , five scalar fields φ A and corresponding Weyl fermions ψ. The field strength\n\nH µνρ = ∂ µ B νρ + ∂ ρ B µν + ∂ ν B ρµ is selfdual. The supersymmetry variation of the Weyl fermions is δψ = 1 12 Γ µνρ H µνρ + Γ µ Γ A ∂ µ φ A ǫ (69)\n\nwhere we use eleven-dimensional gamma matrices splitted into SO(1, 5)×SO( 5 ), so that in particular\n\n{Γ µ , Γ A } = 0. ( 70\n\n)\n\nIn a static and x 5 independent field configuration, in which only φ 5 =: φ is non-zero, we find the SUSY variation\n\nδψ = Γ 0i5 H 0i5 + Γ i Γ A=5 ∂ i φ ǫ (71)\n\nIf we assume that the classical bosonic field configuration is such that\n\n∂ i φ = H 0i5 (72)\n\nthen the SUSY variation reduces to\n\nδψ = ∂ i φΓ i Γ 05 + Γ A=5 ǫ (73)\n\nand we find the condition for unbroken SUSY as\n\n1 + Γ 05 Γ A=5 ǫ = 0 (74) If we use the Weyl condition Γǫ = -ǫ (75)\n\nof the (2, 0) supersymmetry parameter ǫ, then we can also write this as\n\n1 + Γ 1234 Γ A=5 ǫ = 0. ( 76\n\n)\n\nWe may represent the gamma matrices as\n\nΓ µ = (Γ 0 , Γ i , Γ 5 ) = 1 ⊗ iσ 2 ⊗ 1, γ i ⊗ σ 1 ⊗ 1, γ ⊗ σ 1 ⊗ 1 Γ A = 1 ⊗ iσ 2 ⊗ σ A ( 77\n\n)\n\nwhere σ 1,2,3 are the Pauli sigma matrices, γ = γ 1234 . Then the condition for unbroken SUSY is\n\n(1 + γ ⊗ σ) ǫ = 0 ( 78\n\n)\n\nwhere σ = σ 1234 = σ A=5 .\n\nWe have found that if\n\nH ijk = ǫ ijkl ∂ l φ (79)\n\nthen half SUSY is unbroken. This equation is the Bogomolnyi equation for selfdual strings [1] . We are interested in finding the number of parameters needed to describe solutions of this equation. We can linearize it and get the equation\n\nγ i ∂ i χ = 0 (80)\n\nfor the bosonic zero modes, that we have gathered into a matrix\n\nχ ≡ γ ij δB ij + γδφ. (81)\n\nFor this to work we must also assume the background gauge condition\n\n∂ i B ij = 0. ( 82\n\n)\n\nNow this linearized equation Eq (80) does not make any reference to the gauge field. So there is no way that we could count the number of parameters of a multi-string configuration just using this equation. This should of course not be a surprise. The strings that we have in the Abelian theory are not solutions of the field equations. They have to be inserted by hand, that is we need to insert delta function sources by hand, in the same spirit as for Dirac monopoles.\n\nTo be able to count the number of zero modes, we must consider some interacting theory which (at the classical level) has solitonic string solutions.\n\nTo pass to non-Abelian theory we begin by rewriting the Abelian theory in loop space. Loop space consists of parametrized loops C: s → C µ (s). We introduce the Abelian 'loop fields' [7] A µs = B µν (C(s)) Ċν (s)\n\nφ µs = φ(C(s)) Ċµ (s) ψ µs = ψ(C(s)) Ċµ (s) ( 83\n\n)\n\nWith these definitions, a short computation reveals that A µs transforms as a vector and φ µs a contra-variant vector under diffeomorphisms in loop space induced by diffeomorphisms in space-time. One may then extend these transformation properties to any diffeomorphism in loop space. Space-time diffeomorphism and reparametrizations of the loops then get unified and are both diffemorphisms in loop space. The only thing to remember is what is kept fixed under the variation. If it is the parameter of the loop, or the loop itself.\n\nThe field strength becomes\n\nF µs,νt = H µνρ (C(s)) Ċρ (s)δ(s -t) (84)\n\nIn terms of these fields, the Bogomolnyi equation will read foot_0\n\nF is,jt = ǫ ijkl ∂ k(s φ lt) . (85)\n\nWe pass to the non-Abelian theory by letting these loop fields become non-Abelian, in the sense that A µs = A a µs λ a (s) where λ a (s) are generators of a loop algebra associated to the gauge group [7] . We introduce a covariant derivative\n\nD µs = ∂ µs + A µs . (86)\n\nLocal gauge transformations act as\n\nδ Λ A µs = D µs Λ δ Λ φ µs = [φ µs , Λ]. (87)\n\nGiven a loop C, we automatically get a tangent vector Ċµ (s) that makes no reference to space-time. We can therefore impose the loop space constraints Ċµ (s)A µs = 0 (88)\n\nfor each s, and also\n\nφ µs = Ċµ (s)φ(s; C) (89)\n\nfor some subtle field φ(s; C) on loop space. As a consequence, we find that\n\nA µs φ µs = 0. ( 90\n\n)\n\nThese constraints are covariant under diffeomorphisms of space-time and reparametrizations of loops. They are invariant also under local gauge transformations, provided that the gauge parameter is subject to the condition\n\nĊµ (s)∂ µs Λ = 0 (91)\n\nwhich is the condition of reparametrization invariance. With the assumption made that λ a (s) are generators of a loop algebra, we find that the constraint can also be written as\n\n[A µs , φ µt ] = 0 (92)\n\nA local gauge variation of this constraint is\n\n[D µs Λ, φ µt ] + [A µs , [φ µt , Λ]] = [∂ µs Λ, φ µt ] + [[A µs , Λ], φ µt ] + [A µs , [φ µt , Λ]] = [∂ µs Λ, φ µt ] + [Λ, [φ µt , A µs ]] (93)\n\nThe last term vanishes by the constraint. The first term gives us the constraint Eq (92) that we must impose on the gauge parameter\n\nΛ = dsΛ a (s, C)λ a (s). (94)\n\nWe have now introduced non-local non-Abelian fields with infinitely many components. It is also likely that consisteny of the theory requires an infinite set of constraints on these fields. Maybe then, it could be that we may in the end descend to a finite degrees of freedom. But this is just a speculation. The problem appears to be difficult and ill-defined -How should one define a degree of freedom in a strongly coupled non-local theory?\n\nThe non-Abelian generalization of the Bogomolnyi equation should be given by [7]\n\nF is,jt = ±ǫ ijkl D k(s φ lt) . ( 95\n\n)\n\nThis equation is gauge invariant and invariant under the residual SO(4) Lorentz group that is preserved by the strings. We can not think of any reasonable modification of this equation that would preserve these symmetries, so on this grounds alone one could suspect this equation to be correct. Of course this is not the only requirement that the BPS condition imposes. We also get conditions on the 0s and the 5s components. But these BPS equations will be of no interest to us right now. We will show below that the linearized Bogomolnyi equation can be written as\n\nγ i D i(s + σφ i(s χ t) = 0 (96)\n\nWe will also see below that we (presumably) can actually drop the symmetrization in s and t in this equation. The fields transform in the adjoint representation of the loop algebra, by which we mean that φ is χ t = [φ is , χ t ]. We define the Dirac operator\n\nD s = γ i (D is + σφ is ) (97)\n\nand the projectors\n\nP ± ≡ 1 2 (1 ∓ γσ) , (98)\n\nWe can now formulate an index problem, in an even-dimensional (loop-)space.\n\nThe even-dimensional space in this case is given by the 4-dimensional transverse space to the strings, and the index is given by\n\ndim ker D s -dim ker D † s ( 99\n\n)\n\nwhere\n\nD s = P + D s = D s P - D † s = P -D s = D s P + . (100)\n\nSince D s and P ± are hermitian, it is manifest that D † s defined this way will be the hermitian conjugate of D s , thus justifying the notation.\n\nComputing the index alone is not sufficient in order to obtain the dimension of the moduli space of self-dual strings. We also need a vanishing theorem that says that dim ker D † s = 0. Linearizing the Bogomolnyi equation, we get\n\n2D [is δA jt] = ±ǫ ijkl (D ks δφ lt + φ ks δA lt ) (101)\n\nContracting by γ ij , we get\n\nγ ij Dis χ jt = 0 ( 102\n\n)\n\nwhere we have defined\n\nDis ≡ D is ∓ γφ is χ is ≡ δA is ∓ γδφ is ( 103\n\n)\n\nTo see that the linearized BPS equation can be written like this, one must use the constraint\n\nγ ij φ is δφ jt = 0. ( 104\n\n)\n\nWe can avoid having explicit ± signs by introducing the other chiraly matrix at our disposal, namely σ that lives in a different vector space than γ. We can then hide the ± signs in the tensor product\n\nγ ⊗ σ = ±1 (105)\n\nwhich amounts to\n\nDis ≡ D is + σφ is χ is ≡ δA is + σδφ is (106)\n\nwithout any ±. 5 If we define\n\nχ s ≡ γ i χ is (108)\n\nthen we can write the zero mode equation as\n\nγ i Dis χ t + Di s χ it = 0. ( 109\n\n)\n\nLet us analyze the second term in this equation. It is given by\n\nD i s δA it + φ i s δφ it -γ φ i s δA it + D i s δφ it (110)\n\nWe should not count variations that are gauge variations as bosonic zero modes. We can insure this by demanding the zero modes to be orthogonal to gauge variations, with respect to the metric on the moduli space,\n\n(δ Λ A is , δA it ) + (δ Λ φ is , δφ jt ) = 0 (111)\n\nThis leads to the background gauge condition\n\nD i s δA it + φ i s δφ it = 0.\n\n(112) 5 To really understand what is going on, one should apply (1 ± γσ) on everything, on ψs and on Ds. Then one notices that\n\n∓γ (1 ∓ γσ) = σ (1 ∓ γσ) . (107)\n\nThat is, we can trade ∓γ for σ, once we apply (1 ± γσ) on everything. This is what we really should do, but to keep the notation simple, we do not spell this out.\n\nThis condition implies that the gauge variation of the zero modes vanishes,\n\nδ Λ δA is = 0 = δ Λ δφ is (113)\n\nTo see this, we make a gauge variation δ Λ δA is = D is Λ, δ Λ φ is = φ is Λ, and ask which gauge parameters Λ will respect the background gauge condition. Inserting this gauge variation into the background gauge condition, we get\n\nD i s D it + φ i s φ it Λ = 0. ( 114\n\n)\n\nFor this to work nicely, it seems that we must constrain the non-locality of our loop field such that ∂ i (s ∂ it) < 0. Then the only solution to this equation is Λ = 0. In other words all gauge variations of the zero modes have to vanish.\n\nFurthermore we want the variation to preserve the orthogonality between A is and φ is ,\n\n(A is , δφ it ) + (δA is , φ it ) = 0 (115)\n\nIf we make a gauge variation of this, then we get the condition\n\n(δ Λ A is , δφ it ) + (δA is , δ Λ φ it ) = 0 (116)\n\nwhich amounts to\n\nφ i s δA it + D i s δφ it = 0. ( 117\n\n)\n\nWe conclude that the zero mode equation can be written as\n\nD s χ t = 0 (118)\n\nwhere\n\nD s = γ i (D is + σφ is ) (119)\n\nWe are interested in counting the number of such modes in a background of k BPS strings. We compute\n\nD 2 = (D is ) 2 + (φ is ) 2 + 1 2 γ ij (F is,js + γσǫ ijkl D ks φ ls ) (120)\n\n(Here D 2 ≡ D s D s ≡ ds 2π D s D s , and analogously for the other fields or operators.) In a BPS configuration, we get is\n\nD 2 = (D is ) 2 + (φ is ) 2 + 1 2 γ ij (1 + γσ) F is,js (121)\n\nFurthermore, in the subspace where 1 + γσ = 0, we find that\n\nD 2 = (D is ) 2 + (φ is ) 2 (122)\n\nis a strictly negative operator, hence has no zero modes. This means that we have a vanishing theorem, dim ker D † = 0."
},
{
"section_type": "OTHER",
"section_title": "A small comment",
"text": "The zero mode equation was really\n\nD (s χ t) = 0 ( 123\n\n)\n\nwhere we should symmetrize in s and t. That means that we should rather consider\n\nD s D (s χ t) = 1 2 (D s D s χ t + D s D t χ s ) = 1 2 (D s D s χ t + D t D s χ s + [D s , D t ]χ s ) . (124)\n\nIf now D [s D t] = 0 and D s χ s = 0, then we get\n\nD s D s χ t = 0 (125)\n\nThe latter condition, D s χ s = 0 is of course a consequence of D (s χ t) = 0 with s = t. The former condition reads\n\n0 = D [s D t] = D i[s D it] + φ i[s φ it] + σD i[s φ it] (126)\n\nwhich we would like to impose as a constraint. Restricting to the abelian case this is condition is of course true as 0 ≡ ∂ i[s ∂ |i|t] . If we can impose this as a constraint on the non-abelian fields, then we have now seen that the zero mode equation Eq (123) implies that dsD † s D s χ t = 0 (127) because D s is anti-self-adjoint with respect to the inner product (ψ s , χ t ) = DCtr ψ † s (C)χ t (C) (128) on loop space. We can also go in the opposite direction. Assuming that Eq (127) holds, we get 0 = χ t , D † s D s χ t = (D s χ t , D s χ t ) (129) and we conclude that (123) implies D s χ t = 0 (130) with no symmetrization in s, t."
},
{
"section_type": "OTHER",
"section_title": "How to compute the index",
"text": "We should now be able to compute an index associated to self-dual strings, as the limit\n\nI(s) = Tr γσe sD 2 (131)\n\nwhen s → ∞. We define the quantity\n\nJ is (C, C ′ ) = tr C γσγ i γ k (D ks + σφ ks ) 1 D 2 e sD 2 C ′ (132)\n\n(it should be clear that the two s's involved in this formula are totally unrelated) and find that\n\nI(s) = ds 2π DC∂ is J is (C, C) (133)\n\nWe can separate the functional integral over parametrized loops C into several pieces. We can keep a point on the loops C(s) = x fixed, and separate it as\n\nDC = d 4 x D x C (134)\n\nThen we can write I(s) as an integral over a large three-sphere at spatial infinity,\n\nds 2π d 4 x D x C ∂J is (C) ∂C i (s) = ds 2π S 3 dΩ 3 xi D x CJ is (C, C) (135)\n\nwhere thus x = C(s).\n\nIf we assume that the gauge group is maximally broken to a product of U (1)'s by the Higgs vacuum expectation values, then we should have U (1) loop fields at spatial infinity.\n\nIf we assume that the gauge group is SU (2) and that it is broken to U (1), then we need only the asymptotic form of the U (1) fields at spatial infinity, could certainly arise somewhere (in odd dimensions a corresponding term vanished since there is no chiral anomaly in odd dimensions). In our case this term vanishes identically by the Bogomolnyi equation and the constraint foot_1\n\nF is,jt D is φ jt = 0. ( 139\n\n)\n\nThen there can be a term\n\nǫ ijkl D x Ctr F is,jt φ ks v (140)\n\nthat should arise in a very similar way as the corresponding term arose for monopoles. If we insert the asymptotic U (1) fields, this term becomes proportional to\n\nǫ ijkl H ijk (x) (141)\n\nThat means that the index should be given by some numerical constant, times the magnetic charge\n\nS 3 ∞ H. (142)"
}
] | ||
arxiv:0704.0034 | 0704.0034 | 1 | a05703051fdf4cf4df48610e330d72278db2f90f7aef16c061572996b8a9c0c0 | Origin of adaptive mutants: a quantum measurement? | This is a supplement to the paper arXiv:q-bio/0701050, containing the text of
correspondence sent to Nature in 1990. | [
"Vasily Ogryzko"
] | [
"q-bio.PE",
"q-bio.CB",
"quant-ph"
] | q-bio.PE | [] | 2007-03-31 | 2026-02-26 | This is a supplement to the paper arXiv:q-bio/0701050, containing the text of correspondence sent to Nature in 1990. Origin of adaptive mutants: a quantum measurement? Sir, -Several recent works described non-random induction of adaptive mutations by environmental stimuli 1-3 . The most obvious explanation of this striking phenomenon would be that activation of gene expression leads to the enhancement of its mutation rate 4 . However, this does not work with the lacZ mutations described by Cairns and co-workers as the true inducer of the lac-operon is not lactose as such, but allolactose, a by-product of the β-galactosidase reaction 5 . So, in lacZ mutants the operon is not induced by lactose 6 . Besides, induction of respective genes would not explain the high fraction, among the revertants, of suppressor mutations in tRNA genes 1,7 Other explanations suggest some special mechanisms for the "acceleration of adaptive evolution", like selection of "useful" protein coupled to specific reverse transcription 1 . However, any mechanism of this type also should have emerged in evolution. I propose that, to explain the adaptive mutation phenomenon, there is no need for any new ad hoc mechanism. The only thing that is necessary is to return to the old discussion of the role of quantum concepts in our understanding of life. This alone will allow the explanation of this manifestly Lamarckian phenomenon by Darwinian selection, occurring not in a population of organisms as usual, but in a "population" of virtual, in the direct quantum theory sense, states of each distinct cell. Thus, this hypothesis may be called "selection of virtual mutations". Detailed substantiation of this concept will be presented in a special publication; below I briefly show how this explanation might work. It has been shown by the Cairns group that the mutations ensuring cell growth begin to accumulate not immediately after plating, but only after conditions are created under which such mutations become "useful", as if the mutations are induced by these conditions 1 . I suggest that, to explain this phenomenon, we should change our ideas about what a cell is, and consider not actual but virtual mutations. An important distinction of virtual mutations is that they do not accumulate with time in stationary cell, whereas the number of actual mutants would grow linearly from the moment of plating, and this would yield drastically different results in experiments like those shown in Fig. 3 of Ref. 1. Virtual mutations produce "delocalization" of the cell among different states, similarly to the delocalization of electron in physical space. However, for a virtual mutation to become an actual one, certain conditions are necessary, namely the possibility to grow, leading the system away irreversibly from the initial state. Such conditions arise when, for example, lactose is added to a plate with lacZ bacteria. Briefly, this is the essence of the proposed explanation. What is a virtual mutation? The main cause of usual spontaneous mutations is the wellknown base tautomerization 8 (having the in vitro frequency of about 10 -4 ). Thus could we reduce 'virtual mutation' to such tautomerization? I believe that this view is not consistent with experiments, as it implies that the same rare tautomeric form should work both in transcription and in replication. If these processes are considered independent, we logically arrive to the leaky mutant, which was refuted by Cairns and coworkers 1 . Thus we need to postulate a correlation
between the recognition of the tautomeric forms in transcription and in replication, making us to define "virtual mutation" as a certain state of the cell as a whole. Analogous reasoning is applicable to the "adaptive transpositions" discussed by Cairns. In other words, we consider the whole cell as a quantum system, with non-negligible nonlocality inherent in such systems. Most of all it resembles the systems of "generalized rigidity" 9 , such as superfluid or superconducing states of matter, whose behavior is linked to quantum correlations; and I believe, similar correlations take place in the cell too.
I would like to emphasize that the proposed approach does not require detalization of molecular processes in the cell. Its main focus is the behavior of the cell as a whole. Similarly, to explain gyroscopic precession there is no need to consider interactions between elements inside the gyroscope; it's enough to know some motion invariants, defined by space-time symmetries.
Starting from this general view, one may express the above ideas using the operator formalism, and considering experiments conducted by Cairns as measurement of the cells' capability to propagate under given conditions. I suggest that the trait "ability to reproduce on lactose" (as an example) can be represented by an operator which one may designate "Lac". Importantly, this new operator will act on the state Ψ of the whole cell because the ability to reproduce is a property of the cell as a whole, and not of any part of it. Generally, "Lac " will decompose this Ψ into a superposition of some eigenfunctions. The components of this superposition are those functions that do not change upon the action of this operator, but are only multiplied by a constant. It reflects the essence of operator formalism in quantum theory, which chooses states compatible with given experimental conditions. There are three such eigenfunctions (I intentionally simplify the situation): ψ 1 corresponds to cell death, ψ 2 to the stationary state, and ψ 3 to the self-reproduction (that is the virtual mutation, in our case). Each function will enter the decomposition of Ψ with a coefficient c i related to the probability of this or that outcome, i.e.:
Ψ = c 1 ψ 1 + c 2 ψ 2 + c 3 ψ 3 , where Σ| c i | 2 = 1
By plating the cells on lactose agar we, in fact, begin to measure their ability to grow under these particular conditions. The rate of accumulation of lac revertants, i.e. the probability to obtain a cell in the mutant state, will correspond to |c 3 | 2 , being a small, but finite quantity, appearing, for example, due to base tautomerization. Here, the role of cell growth is dual: on the one hand, it is a factor of irreversibility amplifying the "quantum fluctuation", and on the other hand, it is a selection criterion, as each kind of virtual mutants capable of growth under these conditions can lead to colony formation. Another situation, i.e. glucose/valine agar, will be represented by another operator (Val r ), which will decompose the same Ψ function according to another basis, and Val r mutants will be obtained with certain rate. In fact, this is the essence of adaptive mutation phenomenon, where a particular condition induces emergence of respective mutants.
Thus, the proposed change of our view on the cell suggests that, in accord with quantum concepts, we are not dealing with the probability for a cell to mutate by itself, independent of experimental conditions. Rather, we are dealing with the probability to observe the cell in the mutant state by plating it on lactose. We are certainly simplifying situation, as spontaneous mutations that accumulate during cell growth before plating, make our ensemble 'mixed'. However, this complication does not change the essence of the explanation, according to which adaptive mutations emerge by measurement of 'pure' state. This resembles the passage of a polarized photon through a polarizer turned under some angle to the photon polarization. It will be incorrect to say that the polarization of the photon could turn by the necessary angle by chance, prior to interaction. It is the specific experimental situation that makes us to decompose the state vector according to the respective basis states, and to evaluate the fraction of the component that will pass through polarizer. On the other hand, one may speak about "adaptation" of photon polarization by selection of "fit" eigenstate, and consider this case as the model for our phenomenon.
How are all these ideas applicable to the living bacterial cell? Discussion of the possible role of quantum concepts in biology has a rather long history, initiated by Niels Bohr ('the complementarity principle'). Briefly, one might reduce the essence of this discussion to the principal impossibility to predict precisely the fate of an individual cell. For example, any attempt to determine, whether it is able to reproduce under certain conditions, will lead to irreversible change of the state of the cell, even to its death. This is reminiscent of the two-slit diffraction experiment, where an attempt to determine through which of the two slits the electron actually passes will lead to disappearance of the interference. The two trajectories of the electron can be made physically discernable only by the cost of changing the experimental situation. Similarly, the notorious phenomenon of the "wholeness" of the living organism can be formally expressed according to the Feynman rules of calculating probabilities: different indiscernible (in the given experimental conditions) variants should be included in the pure state (i.e. their amplitudes, and not probabilities, should be summed, leading to interference and other quantum effects). Thus, as long as a whole cell exists and is alive, we are obligated to treat its different indiscernible states in this way. Such consideration of operational limitations allows us to explain the adaptive mutation phenomenon (and hopefully other adaptations too) as the consequence of unavoidable quantum scatter in measurement of the cell's capability to propagate under given conditions.
In spite of its apparent formal character, this hypothesis allows us to make some predictions of applied (in particular, medical) interest. It predicts that in processes involving somatic mutations (e.g. oncogenesis, or specific antibody generation) the mutations may be induced by conditions allowing the cell that happened to be in the state of virtual mutation to proliferate irreversibly. I believe, this possibility can be tested experimentally. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "This is a supplement to the paper arXiv:q-bio/0701050, containing the text of correspondence sent to Nature in 1990. Origin of adaptive mutants: a quantum measurement? Sir, -Several recent works described non-random induction of adaptive mutations by environmental stimuli 1-3 . The most obvious explanation of this striking phenomenon would be that activation of gene expression leads to the enhancement of its mutation rate 4 . However, this does not work with the lacZ mutations described by Cairns and co-workers as the true inducer of the lac-operon is not lactose as such, but allolactose, a by-product of the β-galactosidase reaction 5 . So, in lacZ mutants the operon is not induced by lactose 6 . Besides, induction of respective genes would not explain the high fraction, among the revertants, of suppressor mutations in tRNA genes 1,7 Other explanations suggest some special mechanisms for the \"acceleration of adaptive evolution\", like selection of \"useful\" protein coupled to specific reverse transcription 1 . However, any mechanism of this type also should have emerged in evolution. I propose that, to explain the adaptive mutation phenomenon, there is no need for any new ad hoc mechanism. The only thing that is necessary is to return to the old discussion of the role of quantum concepts in our understanding of life. This alone will allow the explanation of this manifestly Lamarckian phenomenon by Darwinian selection, occurring not in a population of organisms as usual, but in a \"population\" of virtual, in the direct quantum theory sense, states of each distinct cell. Thus, this hypothesis may be called \"selection of virtual mutations\". Detailed substantiation of this concept will be presented in a special publication; below I briefly show how this explanation might work. It has been shown by the Cairns group that the mutations ensuring cell growth begin to accumulate not immediately after plating, but only after conditions are created under which such mutations become \"useful\", as if the mutations are induced by these conditions 1 . I suggest that, to explain this phenomenon, we should change our ideas about what a cell is, and consider not actual but virtual mutations. An important distinction of virtual mutations is that they do not accumulate with time in stationary cell, whereas the number of actual mutants would grow linearly from the moment of plating, and this would yield drastically different results in experiments like those shown in Fig. 3 of Ref. 1. Virtual mutations produce \"delocalization\" of the cell among different states, similarly to the delocalization of electron in physical space. However, for a virtual mutation to become an actual one, certain conditions are necessary, namely the possibility to grow, leading the system away irreversibly from the initial state. Such conditions arise when, for example, lactose is added to a plate with lacZ bacteria. Briefly, this is the essence of the proposed explanation. What is a virtual mutation? The main cause of usual spontaneous mutations is the wellknown base tautomerization 8 (having the in vitro frequency of about 10 -4 ). Thus could we reduce 'virtual mutation' to such tautomerization? I believe that this view is not consistent with experiments, as it implies that the same rare tautomeric form should work both in transcription and in replication. If these processes are considered independent, we logically arrive to the leaky mutant, which was refuted by Cairns and coworkers 1 . Thus we need to postulate a correlation"
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "between the recognition of the tautomeric forms in transcription and in replication, making us to define \"virtual mutation\" as a certain state of the cell as a whole. Analogous reasoning is applicable to the \"adaptive transpositions\" discussed by Cairns. In other words, we consider the whole cell as a quantum system, with non-negligible nonlocality inherent in such systems. Most of all it resembles the systems of \"generalized rigidity\" 9 , such as superfluid or superconducing states of matter, whose behavior is linked to quantum correlations; and I believe, similar correlations take place in the cell too.\n\nI would like to emphasize that the proposed approach does not require detalization of molecular processes in the cell. Its main focus is the behavior of the cell as a whole. Similarly, to explain gyroscopic precession there is no need to consider interactions between elements inside the gyroscope; it's enough to know some motion invariants, defined by space-time symmetries.\n\nStarting from this general view, one may express the above ideas using the operator formalism, and considering experiments conducted by Cairns as measurement of the cells' capability to propagate under given conditions. I suggest that the trait \"ability to reproduce on lactose\" (as an example) can be represented by an operator which one may designate \"Lac\". Importantly, this new operator will act on the state Ψ of the whole cell because the ability to reproduce is a property of the cell as a whole, and not of any part of it. Generally, \"Lac \" will decompose this Ψ into a superposition of some eigenfunctions. The components of this superposition are those functions that do not change upon the action of this operator, but are only multiplied by a constant. It reflects the essence of operator formalism in quantum theory, which chooses states compatible with given experimental conditions. There are three such eigenfunctions (I intentionally simplify the situation): ψ 1 corresponds to cell death, ψ 2 to the stationary state, and ψ 3 to the self-reproduction (that is the virtual mutation, in our case). Each function will enter the decomposition of Ψ with a coefficient c i related to the probability of this or that outcome, i.e.:\n\nΨ = c 1 ψ 1 + c 2 ψ 2 + c 3 ψ 3 , where Σ| c i | 2 = 1\n\nBy plating the cells on lactose agar we, in fact, begin to measure their ability to grow under these particular conditions. The rate of accumulation of lac revertants, i.e. the probability to obtain a cell in the mutant state, will correspond to |c 3 | 2 , being a small, but finite quantity, appearing, for example, due to base tautomerization. Here, the role of cell growth is dual: on the one hand, it is a factor of irreversibility amplifying the \"quantum fluctuation\", and on the other hand, it is a selection criterion, as each kind of virtual mutants capable of growth under these conditions can lead to colony formation. Another situation, i.e. glucose/valine agar, will be represented by another operator (Val r ), which will decompose the same Ψ function according to another basis, and Val r mutants will be obtained with certain rate. In fact, this is the essence of adaptive mutation phenomenon, where a particular condition induces emergence of respective mutants.\n\nThus, the proposed change of our view on the cell suggests that, in accord with quantum concepts, we are not dealing with the probability for a cell to mutate by itself, independent of experimental conditions. Rather, we are dealing with the probability to observe the cell in the mutant state by plating it on lactose. We are certainly simplifying situation, as spontaneous mutations that accumulate during cell growth before plating, make our ensemble 'mixed'. However, this complication does not change the essence of the explanation, according to which adaptive mutations emerge by measurement of 'pure' state. This resembles the passage of a polarized photon through a polarizer turned under some angle to the photon polarization. It will be incorrect to say that the polarization of the photon could turn by the necessary angle by chance, prior to interaction. It is the specific experimental situation that makes us to decompose the state vector according to the respective basis states, and to evaluate the fraction of the component that will pass through polarizer. On the other hand, one may speak about \"adaptation\" of photon polarization by selection of \"fit\" eigenstate, and consider this case as the model for our phenomenon.\n\nHow are all these ideas applicable to the living bacterial cell? Discussion of the possible role of quantum concepts in biology has a rather long history, initiated by Niels Bohr ('the complementarity principle'). Briefly, one might reduce the essence of this discussion to the principal impossibility to predict precisely the fate of an individual cell. For example, any attempt to determine, whether it is able to reproduce under certain conditions, will lead to irreversible change of the state of the cell, even to its death. This is reminiscent of the two-slit diffraction experiment, where an attempt to determine through which of the two slits the electron actually passes will lead to disappearance of the interference. The two trajectories of the electron can be made physically discernable only by the cost of changing the experimental situation. Similarly, the notorious phenomenon of the \"wholeness\" of the living organism can be formally expressed according to the Feynman rules of calculating probabilities: different indiscernible (in the given experimental conditions) variants should be included in the pure state (i.e. their amplitudes, and not probabilities, should be summed, leading to interference and other quantum effects). Thus, as long as a whole cell exists and is alive, we are obligated to treat its different indiscernible states in this way. Such consideration of operational limitations allows us to explain the adaptive mutation phenomenon (and hopefully other adaptations too) as the consequence of unavoidable quantum scatter in measurement of the cell's capability to propagate under given conditions.\n\nIn spite of its apparent formal character, this hypothesis allows us to make some predictions of applied (in particular, medical) interest. It predicts that in processes involving somatic mutations (e.g. oncogenesis, or specific antibody generation) the mutations may be induced by conditions allowing the cell that happened to be in the state of virtual mutation to proliferate irreversibly. I believe, this possibility can be tested experimentally."
}
] | ||
arxiv:0704.0042 | 0704.0042 | 1 | ff68e0616e33788a1cc39c55250b83169d82f127c9ade78202d572a62a8993c2 | General System theory, Like-Quantum Semantics and Fuzzy Sets | It is outlined the possibility to extend the quantum formalism in relation to
the requirements of the general systems theory. It can be done by using a
quantum semantics arising from the deep logical structure of quantum theory. It
is so possible taking into account the logical openness relationship between
observer and system. We are going to show how considering the truth-values of
quantum propositions within the context of the fuzzy sets is here more useful
for systemics . In conclusion we propose an example of formal quantum
coherence. | [
"Ignazio Licata"
] | [
"physics.gen-ph",
"quant-ph"
] | physics.gen-ph | [] | 2007-03-31 | 2026-02-26 | It is outlined the possibility to extend the quantum formalism in relation to the requirements of the general systems theory. It can be done by using a quantum semantics arising from the deep logical structure of quantum theory. It is so possible taking into account the logical openness relationship between observer and system. We are going to show how considering the truth-values of quantum propositions within the context of the fuzzy sets is here more useful for systemics . In conclusion we propose an example of formal quantum coherence.
relationships. The GST main goal is delineating a formal epistemology to study the scientific knowledge formation, a science able to speak about science. Succeeding to outline such panorama will make possible analysing those inter-disciplinary processes which are more and more important in studying complex systems and they will be guaranteed the "transportability" conditions of a modellistic set from a field to another one. For instance, during a theory developing, syntax gets more and more structured by putting univocal constraints on semantics according to the operative requirements of the problem. Sometimes it can be useful generalising a syntactic tool in a new semantic domain so as to formulate new problems. Such work, a typically transdisciplinary one, can only be done by the tools of a GST able to discuss new relations between syntactics (formal model) and semantics ( model usage). It is here useful to consider again the omologic perspective, which not only identifies analogies and isomorphisms in pre-defined structures, but aims to find out a structural and dynamical relation among theories to an higher level of analysis, so providing new use possibilities (Rossi-Landi, 1985) . Which thing is particularly useful in studying complex systems, where the very essence of the problem itself makes a dynamic use of models necessary to describe the emergent features of the system (Minati & Brahms, 2002; Collen, 2002) .
We want here to briefly discuss such GST acceptation, and then showing the possibility of modifying the semantics of Quantum Mechanics (QM) so to get a conceptual tool fit for the systemic requirements.
What we look at is not Nature in itself, but Nature unveiling to our questioning methods.
W. Heisenberg, 1958 A very important and interesting question in system theory can be stated as follows: given a set of measurement systems M and of theories T related to a system S, is it always possible to order them, such that T i-1 T i , where the partial order symbol is used to denote the relationship "physically weaker than" ? We shall point out that, in this case, the i th theory of the chain contains more information than the preceding ones. This consequently leads to a second key question: can an unique final theory T f describe exhaustively each and every aspect of system S ? From the informational and metrical side, this is equivalent to state that all of the information contained in a system S can be extracted, by means of adequate measurement processes.
The fundamental proposition for reductionism is, in fact, the idea that such a theory chain will be sufficient to give a coherent and complete description for a system S. Reductionism, in the light of our definitions, coincides therefore with the highest degree of semantic space "compression"; each object D ∈ T i in S has a definition in a theory T i belonging to the theory chain, and the latter is -on its turn -related to the fundamental explanatory level of the "final" theory T f . This implies that each aspect in a system S is unambiguously determined by the syntax described in T f . Each system S can be described at a fundamental level, but also with many phenomenological descriptions, each of these descriptions can be considered an approximation of the "final" theory.
Anyway, most of the "interesting" systems we deal with cannot be included in this chainedtheory syntax compatibility program: we have to consider this important aspect for a correct epistemic definition of systems "complexity". Let us illustrate this point with a simple reasoning, based upon the concepts of logical openness and intrinsic emergence (Minati, Pessa, Penna, 1998; Licata, 2003b) .
Each measurement operation can be theoretically coded on a Turing machine. If a coherent and complete fundamental description T f exists, then there will also exist a finite set -or, at most, countably infinite -of measurement operations M which can extract each and every single information that describes the system S. We shall call such a measurement set Turing-observer.
We can easily imagine Turing-observer as a robot that executes a series of measurements on a system. The robot is guided by a program built upon rules belonging to the theory T. It can be proved, though, that this is only possible for logically closed systems, or at most for systems with a very low degree of logical openness. When dealing with highly logically open systems, no recursive formal criterion exists that can be as selective as requested (i.e., automatically choose which information is relevant to describe and characterize the system, and which one is not), simply because it is not possible to isolate the system from the environment. This implies that the Turingobserver hypothesis does not hold for fundamental reasons, strongly related to Zermelo-Fraenkel's choice axiom and to classical Godel's decision problems. In other words, our robot executes the measurements always following the same syntactics, whereas the scenario showing intrinsic emergence is semantically modified. So it is impossible thinking to codify any possible measurement in a logically open system!
The observer therefore plays a key rule, unavoidable as a semantic ambiguity solver: only the observer can and will single out intrinsic-observational emergence properties ( Bass & Emmeche,1997; Cariani, 1991) , and subsequently plan adequate measurement processes to describe what -as a matter of fact-have turned in new systems. System complexity is structurally bound to logical openness and is, at the same time, both an expression of highly organized system behaviours (long-range correlations, hierarchical structure, and so on) and an observer's request for new explanatory models.
So, a GST has to allow -in the very same theoretical context -to deal with the observer as an emergence surveyor in a logical open system. In particular, it is clear that the observer itself is a logical open system.
Moreover, it has to be pointed out that the co-existence of many description levels -compatible but not each other deductible -leads to intrinsic uncertainty situations, linked to the different frameworks by which a system property can be defined.
I'm not happy with all the analyses that go with just the classical theory, because nature isn't classical, damm it, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy. Thank you.
When we modify and/or amplify a theory so as to being able to speak about different systems from the ones they were fitted for, it could be better to look at the theory deep structural features so as to get an abstract perspective able to fulfil the omologic approach requirements, aiming to point out a non-banal conceptual convergence.
As everybody knows, the logic of classical physics is a dichotomic language (tertium non datur), relatively orthocomplemented and able to fulfil the weak distributivity relations by the logical connectives AND/OR. Such features are the core of the Boolean commutative elements of this logic because disjunctions and conjunctions are symmetrical and associative operations. We shall here dwell on the systemic consequences of these properties. A system S can get or not to get a given property P. Once we fix the P truth-value it is possible to keep on our research over a new preposition P subordinated to the previous one's truth-value. Going ahead, we add a new piece of information to our knowledge about the system. So the relative orthocomplementation axiom grants that we keep on following a successions of steps, each one making our uncertainty about the system to diminish or, in case of a finite amount of steps, to let us defining the state of the system by determining all its properties. Each system's property can be described by a countable infinity of atomic propositions. So, such axiom plays the role of a describable axiom for classical systems.
The unconstrained use of such kind of axiom tends to hide the conceptual problems spreading up from the fact that every description implies a context, as we have seen in the case of Turingobserver analysis, and it seems to imply that systemic properties are independent of the observer, it surely is a non-valid statement when we deal with open logical systems. In particular, the Boolean features point out that it is always possible carrying out exhaustively a synchronic description of the properties of a systems. In other words, every question about the system is not depending on the order we ask it and it is liable to a fixed answer we will indicate as 0-false / 1true. It can be suddenly noticed that the emergent features otherwise get a diachronic nature and can easily make such characteristics not taken for granted. By using Venn diagrams it is possible providing a representation of the complete descriptiveness of a system ruled by classical logics. If the system's state is represented by a point and a property of its by a set of points, then it is always possible a complete "blanketing" of the universal set I, which means the always universally true proposition. (see fig. 1 ).
The quantum logics shows deep differences which could be extremely useful for our goals (Birkhoff & von Neumann, 1936; Piron, 1964) . At the beginning it was born to clarify some QM's counter-intuitive sides , later it has developed as an autonomous field greatly independent from the matters which gave birth to it. We will abridge here the formal references to an essential survey, focusing on some points of general interest in systemics.
The quantum language is a non-Boolean orthomodular structure, which is to say it is relatively orthocomplemented but non-commutative, for the crack down of the distributivity axiom. Such thing comes naturally from the Heisenberg Indetermination Principle and binds the truth-value of an assertion to the context and the order by which it has been investigated (Griffiths, 1995) . A wellknown example is the one of a particle's spin measurement along a given direction. In this case we deal with semantically well defined possibilities and yet intrinsically uncertain. Let put x Ψ the spin measurement along the direction x. For the indetermination principle the value y Ψ will be totally uncertain, yet the proposition y Ψ =0 ∨ y Ψ =1 is necessarily true. In general, if P is a proposition , (-P ) its negation and Q the property which does not commute with P, then we will get a situation that can be represented by a "patchy" blanketing of the set I (see fig. 2 ). Such configuration finds its essential meaning just in its relation with the observer. So we can state that when a situation can be described by a quantum logics, a system is never completely defined a priori. The measurement process by which the observer's action takes place is a choice fixing some system's characteristics and letting other ones undefined. It happens just for the nature itself of the observer-system inter-relationship. Each observation act gives birth to new descriptive possibilities. The proposition Q -in the above example -describes properties that cannot be defined by any implicational chain of propositions P. Since the intrinsic emergence cannot be regarded as a system property independent of the observer action-as in naïve classical emergentism -, Q can be formally considered the expression of an emergent property. Now we are strongly tempted to define as emergent the undefined proposition of quantum-like anticommutative language. In particular, it can be showed that a non-Boolean and irreducible orthomodular language arises infinite propositions. It means that for each couple of propositions P 1 and P 2 such that non of them imply the other , there exists infinite propositions Q which imply P 1 ∨ P 2 without necessarily implying the two of them separately: tertium datur. In a sense, the disjunction of the two propositions gets more information than their mere set-sum, that is the entirely opposite of what happens in the Boolean case. It is now easy to comprehend the deep relation binding the anti-commutativity, indetermination principles and system's holistic global structure. A system describable by a Boolean structure can be completely "solved" by analysing the sub-systems defined by a fit decomposition process ( Heylighen, 1990; Abram, 2002) . On the contrary, in the anti-commutative case studying any sub-system modifies the entire system in an irreversible and structural way and produces uncertainty correlated to the gained information, which think makes absolutely natural extending the indetermination principles to a big deal of spheres of strong interest for systemics (Volkenshtein , 1988) .
A particularly key-matter is how to conceptually managing the infinite cardinality of emergent propositions in a lik-quantum semantics. As everybody knows traditional QM refers to the frequentistic probability worked out within the Copenhagen Interpretation (CIQM). It is essentially a sub specie probabilitatis Boolean logics extension. The values between [ ] 1 , 0 -i.e. between the completely and always true proposition I and the always false one O -are meant as expectation values, or the probabilities associated to any measurable property. Without dwelling on the complex -and as for many questions still open -debate on QM interpretation, we can here ask if the probabilistic acception of truth-values is the fittest for system theory. As it usually happens when we deal with trans-disciplinary feels, it will bring us to add a new, and of remarkable interest for the "ordinary" QM too, step to our search.
A slight variation in the founding axioms of a theory can give way to huge changings on the frontier.
S. Gudder, 1988 The study of the structural and logical facets of quantum semanics does not provide any necessary indications about the most suitable algebraic space to implement its own ideas. One of the thing which made a big merit of such researches has been to put under discussion the key role of Hilbert space. In our approach we have kept the QM "internal" problems and its extension to systemic questions well separated. Anyway, the last ones suggest an interpretative possibility bounded to fuzzy logic, which thing can considerably affect the traditional QM too. The fuzzy set theory is , in its essence, a formal tool created to deal with information characterized with vagueness and indeterminacy. The by-now classical paper of Lotfi Zadeh (Zadeh, 1965) brings to a conclusion an old tradition of logics, which counts Charles S. Peirce, Jan C. Smuts, Bertrand Russell, Max Black and Ian Lukasiewicz among its forerunners. At the core of the fuzzy theory lies the idea that an element can belong to a set to a variable degree of membership; the same goes for a proposition and its variable relation to the true and false logical constants. We underline here two aspects of particular interest for our aims. The fuzziness' definition concerns single elements and properties, but not a statistical ensemble, so it has to be considered a completely different concept from the probability one, it should -by now-be widely clarified (Mamdani, 1977; Kosko, 1990) . A further essential -even maybe less evident -point is that fuzzy theory calls up a nonalgorithmic "oracle", an observator (i.e. a logical open system and a semantic ambiguity solver) to make a choice as for the membership degree. In fact, the most part of the theory in its structure is free-model; no equation and no numerical value create constraints to the quantitative evaluation, being the last one the model builder's task. There consequently exists a deep bound between systemics and fuzziness successfully expressed by the Zadeh's incompatibility principle (Zadeh, 1972) which satisfies our requirement for a generalized indeterminacy principle. It states that by increasing the system complexity (i.e. its logical openness degree), it will decrease our ability to make exact statements and proved predictions about its behaviour. There already exists many examples of crossing between fuzzy theory and QM (Dalla Chiara, Giuntini, 1995; Cattaneo, Dalla Chiara, Giuntini 1993) . We want here to delineate the utility of fuzzy polyvalence for systemic interpretation of quantum semantics.
Let us consider a complex system, such as a social group, a mind and a biological organism.
Each of these cases show typical emergent features owed both to the interaction among its components and the inter-relations with the environment. An act of the observer will fix some properties and will let some others undetermined according to a non-Boolean logic. The recording of such properties will depend on the succession of the measurement acts and their very nature.
The kind of complexity into play, on the other hand, prevents us by stating what the system state is so as to associate to the measurement of a property an expectation probabilistic value. In fact, just the above-mentioned examples are related to macroscopic systems for which the probabilistic interpretation of QM is patently not valid. Moreover, the traditional application of the probability concept implies the notion of "possible cases", and so it also implies a pre-defined knowledge of systems' properties. However, the non-commutative logical structure here outlined does not provide any cogent indication on probability usage.
Therefore, it would be proper to look at a fuzzy approach so to describe the measurement acts.
We can state that given a generic system endowed with high logical openness and an indefinite set of properties able of describing it, each of them will belong to the system in a variable degree.
Such viewpoint expressing the famous theorem of fuzzy "subsetness" -also known as "the whole into the part" principle -could seem to be too strong , indeed it is nothing else than the most natural expression of the actual scientific praxis facing intrinsic emergent systems. At the beginning, we have at our disposal indefinite information progressively structuring thanks to the feedback between models and measurements. It can be shown that any logically open model of degree n -where n is an integer -will let a wide range of properties and propositions indeterminate (the Qs in fig. 2 ).The above-mentioned model is a "static" approximation of a process showing aspects of variable closeness and openness. The latter ones varies in time, intensity, different levels and context. It is remarkable pointing out how such systems are "flexible" and context-sensitive, change the rules and make use of "contradictions" . This point has to be stressed to understand the link between fuzzy logic and quantum languages. By increasing the logical openness and the unsharp properties of a system, it will be less and less fit to be described by a Boolean logic. It brings as a consequence that for a complex system the intersection between a set (properties, propositions) and its complement is not equal to the empty set, but it includes they both in a fuzzy sense. So we get a polyvalent semantic situation which is well fitted for being described by a quantum language. As for our systemic goal it is the probabilistic interpretation to be useless, so we are going to build a fuzzy acception of the semantics of the formalism. In our case, given a system S and a property Q,, let Ψ be a function which associates Q to S, the expression
( ) [ ] 1 , 0 ∈ Ψ Q S
has not to be meant as a probability value, but as a degree of membership. Such union between the non-commutative sides of quantum languages and fuzzy polyvalence appears to be the most suitable and fecund for systemics.
Let us consider the traditional expression of quantum coherence (the property expressing the QM global and non-local characteristics, i.e. superposition principle, uncertainty, interference of probabilities),
2 2 1 1 Ψ + Ψ = Ψ a a
. In the fuzzy interpretation, it means that the properties 1 Ψ e 2 Ψ belong to Ψ with degrees of membership 1 a e 2 a respectively. In other words, for complex systems the Schrödinger's cat can be simultaneously both alive and dead ! Indeed the recent experiments with SQUIDs and the other ones investigating the so-called macroscopic quantum states suggest a form of macro-realism quite close to our fuzzy acception (Leggett, 1980; Chiatti, Cini, Serva, 1995) . It can provide in nuce an hint which could show up to be interesting for the QM old-questioned interpretative problems.
In general, let x be a position coordinate of a quantum object and Ψ its wave function,
( ) dV x 2 Ψ
is usually meant as the probability of finding the particle in a region dV of space. On the contrary, in the fuzzy interpretation we will be compelled to look at the Ψ square modulus as the degree of membership of the particle to the region dV of space. How unusual it may seem, such idea has not to be regarded thoughtlessly at. As a matter of fact, in Quantum Field Theory and in other more advanced quantum scenarios, a particle is not only a localized object in the space, but rather an event emerging from the non-local networks elementary quantum transition (Licata, 2003a) . Thus, the measurement is a "defuzzification" process which, according to the stated, reduces the system ambiguity by limiting the semantic space and by defining a fixed information quantity.
If we agree with such interpretation we will easily and immediately realize that we will able to observate quantum coherence behaviours in non-quantum and quite far from the range of Plank's h constant situations. We reconsider here a situation owed to Yuri Orlov (Orlov, 1997) .
Let us consider a Riemann's sphere (Dirac, 1947 ) -see fig. In conclusion, the generalized using of quantum semantics associated to new interpretative possibilities gives to systemics a very powerful tool to describe the observator-environment relation and to convey the several, partial attempts -till now undertaken -of applying the quantum formalism to the study of complex systems into a comprehensive conceptual root. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "It is outlined the possibility to extend the quantum formalism in relation to the requirements of the general systems theory. It can be done by using a quantum semantics arising from the deep logical structure of quantum theory. It is so possible taking into account the logical openness relationship between observer and system. We are going to show how considering the truth-values of quantum propositions within the context of the fuzzy sets is here more useful for systemics . In conclusion we propose an example of formal quantum coherence."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "relationships. The GST main goal is delineating a formal epistemology to study the scientific knowledge formation, a science able to speak about science. Succeeding to outline such panorama will make possible analysing those inter-disciplinary processes which are more and more important in studying complex systems and they will be guaranteed the \"transportability\" conditions of a modellistic set from a field to another one. For instance, during a theory developing, syntax gets more and more structured by putting univocal constraints on semantics according to the operative requirements of the problem. Sometimes it can be useful generalising a syntactic tool in a new semantic domain so as to formulate new problems. Such work, a typically transdisciplinary one, can only be done by the tools of a GST able to discuss new relations between syntactics (formal model) and semantics ( model usage). It is here useful to consider again the omologic perspective, which not only identifies analogies and isomorphisms in pre-defined structures, but aims to find out a structural and dynamical relation among theories to an higher level of analysis, so providing new use possibilities (Rossi-Landi, 1985) . Which thing is particularly useful in studying complex systems, where the very essence of the problem itself makes a dynamic use of models necessary to describe the emergent features of the system (Minati & Brahms, 2002; Collen, 2002) .\n\nWe want here to briefly discuss such GST acceptation, and then showing the possibility of modifying the semantics of Quantum Mechanics (QM) so to get a conceptual tool fit for the systemic requirements."
},
{
"section_type": "OTHER",
"section_title": "Observer as emergence surveyor and semantic ambiguity solver",
"text": "What we look at is not Nature in itself, but Nature unveiling to our questioning methods.\n\nW. Heisenberg, 1958 A very important and interesting question in system theory can be stated as follows: given a set of measurement systems M and of theories T related to a system S, is it always possible to order them, such that T i-1 T i , where the partial order symbol is used to denote the relationship \"physically weaker than\" ? We shall point out that, in this case, the i th theory of the chain contains more information than the preceding ones. This consequently leads to a second key question: can an unique final theory T f describe exhaustively each and every aspect of system S ? From the informational and metrical side, this is equivalent to state that all of the information contained in a system S can be extracted, by means of adequate measurement processes.\n\nThe fundamental proposition for reductionism is, in fact, the idea that such a theory chain will be sufficient to give a coherent and complete description for a system S. Reductionism, in the light of our definitions, coincides therefore with the highest degree of semantic space \"compression\"; each object D ∈ T i in S has a definition in a theory T i belonging to the theory chain, and the latter is -on its turn -related to the fundamental explanatory level of the \"final\" theory T f . This implies that each aspect in a system S is unambiguously determined by the syntax described in T f . Each system S can be described at a fundamental level, but also with many phenomenological descriptions, each of these descriptions can be considered an approximation of the \"final\" theory.\n\nAnyway, most of the \"interesting\" systems we deal with cannot be included in this chainedtheory syntax compatibility program: we have to consider this important aspect for a correct epistemic definition of systems \"complexity\". Let us illustrate this point with a simple reasoning, based upon the concepts of logical openness and intrinsic emergence (Minati, Pessa, Penna, 1998; Licata, 2003b) .\n\nEach measurement operation can be theoretically coded on a Turing machine. If a coherent and complete fundamental description T f exists, then there will also exist a finite set -or, at most, countably infinite -of measurement operations M which can extract each and every single information that describes the system S. We shall call such a measurement set Turing-observer.\n\nWe can easily imagine Turing-observer as a robot that executes a series of measurements on a system. The robot is guided by a program built upon rules belonging to the theory T. It can be proved, though, that this is only possible for logically closed systems, or at most for systems with a very low degree of logical openness. When dealing with highly logically open systems, no recursive formal criterion exists that can be as selective as requested (i.e., automatically choose which information is relevant to describe and characterize the system, and which one is not), simply because it is not possible to isolate the system from the environment. This implies that the Turingobserver hypothesis does not hold for fundamental reasons, strongly related to Zermelo-Fraenkel's choice axiom and to classical Godel's decision problems. In other words, our robot executes the measurements always following the same syntactics, whereas the scenario showing intrinsic emergence is semantically modified. So it is impossible thinking to codify any possible measurement in a logically open system!\n\nThe observer therefore plays a key rule, unavoidable as a semantic ambiguity solver: only the observer can and will single out intrinsic-observational emergence properties ( Bass & Emmeche,1997; Cariani, 1991) , and subsequently plan adequate measurement processes to describe what -as a matter of fact-have turned in new systems. System complexity is structurally bound to logical openness and is, at the same time, both an expression of highly organized system behaviours (long-range correlations, hierarchical structure, and so on) and an observer's request for new explanatory models.\n\nSo, a GST has to allow -in the very same theoretical context -to deal with the observer as an emergence surveyor in a logical open system. In particular, it is clear that the observer itself is a logical open system.\n\nMoreover, it has to be pointed out that the co-existence of many description levels -compatible but not each other deductible -leads to intrinsic uncertainty situations, linked to the different frameworks by which a system property can be defined."
},
{
"section_type": "OTHER",
"section_title": "Like-quantum semantics",
"text": "I'm not happy with all the analyses that go with just the classical theory, because nature isn't classical, damm it, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy. Thank you."
},
{
"section_type": "OTHER",
"section_title": "R. P. Feyman, 1981",
"text": "When we modify and/or amplify a theory so as to being able to speak about different systems from the ones they were fitted for, it could be better to look at the theory deep structural features so as to get an abstract perspective able to fulfil the omologic approach requirements, aiming to point out a non-banal conceptual convergence.\n\nAs everybody knows, the logic of classical physics is a dichotomic language (tertium non datur), relatively orthocomplemented and able to fulfil the weak distributivity relations by the logical connectives AND/OR. Such features are the core of the Boolean commutative elements of this logic because disjunctions and conjunctions are symmetrical and associative operations. We shall here dwell on the systemic consequences of these properties. A system S can get or not to get a given property P. Once we fix the P truth-value it is possible to keep on our research over a new preposition P subordinated to the previous one's truth-value. Going ahead, we add a new piece of information to our knowledge about the system. So the relative orthocomplementation axiom grants that we keep on following a successions of steps, each one making our uncertainty about the system to diminish or, in case of a finite amount of steps, to let us defining the state of the system by determining all its properties. Each system's property can be described by a countable infinity of atomic propositions. So, such axiom plays the role of a describable axiom for classical systems.\n\nThe unconstrained use of such kind of axiom tends to hide the conceptual problems spreading up from the fact that every description implies a context, as we have seen in the case of Turingobserver analysis, and it seems to imply that systemic properties are independent of the observer, it surely is a non-valid statement when we deal with open logical systems. In particular, the Boolean features point out that it is always possible carrying out exhaustively a synchronic description of the properties of a systems. In other words, every question about the system is not depending on the order we ask it and it is liable to a fixed answer we will indicate as 0-false / 1true. It can be suddenly noticed that the emergent features otherwise get a diachronic nature and can easily make such characteristics not taken for granted. By using Venn diagrams it is possible providing a representation of the complete descriptiveness of a system ruled by classical logics. If the system's state is represented by a point and a property of its by a set of points, then it is always possible a complete \"blanketing\" of the universal set I, which means the always universally true proposition. (see fig. 1 ).\n\nThe quantum logics shows deep differences which could be extremely useful for our goals (Birkhoff & von Neumann, 1936; Piron, 1964) . At the beginning it was born to clarify some QM's counter-intuitive sides , later it has developed as an autonomous field greatly independent from the matters which gave birth to it. We will abridge here the formal references to an essential survey, focusing on some points of general interest in systemics.\n\nThe quantum language is a non-Boolean orthomodular structure, which is to say it is relatively orthocomplemented but non-commutative, for the crack down of the distributivity axiom. Such thing comes naturally from the Heisenberg Indetermination Principle and binds the truth-value of an assertion to the context and the order by which it has been investigated (Griffiths, 1995) . A wellknown example is the one of a particle's spin measurement along a given direction. In this case we deal with semantically well defined possibilities and yet intrinsically uncertain. Let put x Ψ the spin measurement along the direction x. For the indetermination principle the value y Ψ will be totally uncertain, yet the proposition y Ψ =0 ∨ y Ψ =1 is necessarily true. In general, if P is a proposition , (-P ) its negation and Q the property which does not commute with P, then we will get a situation that can be represented by a \"patchy\" blanketing of the set I (see fig. 2 ). Such configuration finds its essential meaning just in its relation with the observer. So we can state that when a situation can be described by a quantum logics, a system is never completely defined a priori. The measurement process by which the observer's action takes place is a choice fixing some system's characteristics and letting other ones undefined. It happens just for the nature itself of the observer-system inter-relationship. Each observation act gives birth to new descriptive possibilities. The proposition Q -in the above example -describes properties that cannot be defined by any implicational chain of propositions P. Since the intrinsic emergence cannot be regarded as a system property independent of the observer action-as in naïve classical emergentism -, Q can be formally considered the expression of an emergent property. Now we are strongly tempted to define as emergent the undefined proposition of quantum-like anticommutative language. In particular, it can be showed that a non-Boolean and irreducible orthomodular language arises infinite propositions. It means that for each couple of propositions P 1 and P 2 such that non of them imply the other , there exists infinite propositions Q which imply P 1 ∨ P 2 without necessarily implying the two of them separately: tertium datur. In a sense, the disjunction of the two propositions gets more information than their mere set-sum, that is the entirely opposite of what happens in the Boolean case. It is now easy to comprehend the deep relation binding the anti-commutativity, indetermination principles and system's holistic global structure. A system describable by a Boolean structure can be completely \"solved\" by analysing the sub-systems defined by a fit decomposition process ( Heylighen, 1990; Abram, 2002) . On the contrary, in the anti-commutative case studying any sub-system modifies the entire system in an irreversible and structural way and produces uncertainty correlated to the gained information, which think makes absolutely natural extending the indetermination principles to a big deal of spheres of strong interest for systemics (Volkenshtein , 1988) .\n\nA particularly key-matter is how to conceptually managing the infinite cardinality of emergent propositions in a lik-quantum semantics. As everybody knows traditional QM refers to the frequentistic probability worked out within the Copenhagen Interpretation (CIQM). It is essentially a sub specie probabilitatis Boolean logics extension. The values between [ ] 1 , 0 -i.e. between the completely and always true proposition I and the always false one O -are meant as expectation values, or the probabilities associated to any measurable property. Without dwelling on the complex -and as for many questions still open -debate on QM interpretation, we can here ask if the probabilistic acception of truth-values is the fittest for system theory. As it usually happens when we deal with trans-disciplinary feels, it will bring us to add a new, and of remarkable interest for the \"ordinary\" QM too, step to our search."
},
{
"section_type": "OTHER",
"section_title": "A Fuzzy Interpretation of Quantum Languages",
"text": "A slight variation in the founding axioms of a theory can give way to huge changings on the frontier.\n\nS. Gudder, 1988 The study of the structural and logical facets of quantum semanics does not provide any necessary indications about the most suitable algebraic space to implement its own ideas. One of the thing which made a big merit of such researches has been to put under discussion the key role of Hilbert space. In our approach we have kept the QM \"internal\" problems and its extension to systemic questions well separated. Anyway, the last ones suggest an interpretative possibility bounded to fuzzy logic, which thing can considerably affect the traditional QM too. The fuzzy set theory is , in its essence, a formal tool created to deal with information characterized with vagueness and indeterminacy. The by-now classical paper of Lotfi Zadeh (Zadeh, 1965) brings to a conclusion an old tradition of logics, which counts Charles S. Peirce, Jan C. Smuts, Bertrand Russell, Max Black and Ian Lukasiewicz among its forerunners. At the core of the fuzzy theory lies the idea that an element can belong to a set to a variable degree of membership; the same goes for a proposition and its variable relation to the true and false logical constants. We underline here two aspects of particular interest for our aims. The fuzziness' definition concerns single elements and properties, but not a statistical ensemble, so it has to be considered a completely different concept from the probability one, it should -by now-be widely clarified (Mamdani, 1977; Kosko, 1990) . A further essential -even maybe less evident -point is that fuzzy theory calls up a nonalgorithmic \"oracle\", an observator (i.e. a logical open system and a semantic ambiguity solver) to make a choice as for the membership degree. In fact, the most part of the theory in its structure is free-model; no equation and no numerical value create constraints to the quantitative evaluation, being the last one the model builder's task. There consequently exists a deep bound between systemics and fuzziness successfully expressed by the Zadeh's incompatibility principle (Zadeh, 1972) which satisfies our requirement for a generalized indeterminacy principle. It states that by increasing the system complexity (i.e. its logical openness degree), it will decrease our ability to make exact statements and proved predictions about its behaviour. There already exists many examples of crossing between fuzzy theory and QM (Dalla Chiara, Giuntini, 1995; Cattaneo, Dalla Chiara, Giuntini 1993) . We want here to delineate the utility of fuzzy polyvalence for systemic interpretation of quantum semantics.\n\nLet us consider a complex system, such as a social group, a mind and a biological organism.\n\nEach of these cases show typical emergent features owed both to the interaction among its components and the inter-relations with the environment. An act of the observer will fix some properties and will let some others undetermined according to a non-Boolean logic. The recording of such properties will depend on the succession of the measurement acts and their very nature.\n\nThe kind of complexity into play, on the other hand, prevents us by stating what the system state is so as to associate to the measurement of a property an expectation probabilistic value. In fact, just the above-mentioned examples are related to macroscopic systems for which the probabilistic interpretation of QM is patently not valid. Moreover, the traditional application of the probability concept implies the notion of \"possible cases\", and so it also implies a pre-defined knowledge of systems' properties. However, the non-commutative logical structure here outlined does not provide any cogent indication on probability usage.\n\nTherefore, it would be proper to look at a fuzzy approach so to describe the measurement acts.\n\nWe can state that given a generic system endowed with high logical openness and an indefinite set of properties able of describing it, each of them will belong to the system in a variable degree.\n\nSuch viewpoint expressing the famous theorem of fuzzy \"subsetness\" -also known as \"the whole into the part\" principle -could seem to be too strong , indeed it is nothing else than the most natural expression of the actual scientific praxis facing intrinsic emergent systems. At the beginning, we have at our disposal indefinite information progressively structuring thanks to the feedback between models and measurements. It can be shown that any logically open model of degree n -where n is an integer -will let a wide range of properties and propositions indeterminate (the Qs in fig. 2 ).The above-mentioned model is a \"static\" approximation of a process showing aspects of variable closeness and openness. The latter ones varies in time, intensity, different levels and context. It is remarkable pointing out how such systems are \"flexible\" and context-sensitive, change the rules and make use of \"contradictions\" . This point has to be stressed to understand the link between fuzzy logic and quantum languages. By increasing the logical openness and the unsharp properties of a system, it will be less and less fit to be described by a Boolean logic. It brings as a consequence that for a complex system the intersection between a set (properties, propositions) and its complement is not equal to the empty set, but it includes they both in a fuzzy sense. So we get a polyvalent semantic situation which is well fitted for being described by a quantum language. As for our systemic goal it is the probabilistic interpretation to be useless, so we are going to build a fuzzy acception of the semantics of the formalism. In our case, given a system S and a property Q,, let Ψ be a function which associates Q to S, the expression\n\n( ) [ ] 1 , 0 ∈ Ψ Q S\n\nhas not to be meant as a probability value, but as a degree of membership. Such union between the non-commutative sides of quantum languages and fuzzy polyvalence appears to be the most suitable and fecund for systemics.\n\nLet us consider the traditional expression of quantum coherence (the property expressing the QM global and non-local characteristics, i.e. superposition principle, uncertainty, interference of probabilities),\n\n2 2 1 1 Ψ + Ψ = Ψ a a\n\n. In the fuzzy interpretation, it means that the properties 1 Ψ e 2 Ψ belong to Ψ with degrees of membership 1 a e 2 a respectively. In other words, for complex systems the Schrödinger's cat can be simultaneously both alive and dead ! Indeed the recent experiments with SQUIDs and the other ones investigating the so-called macroscopic quantum states suggest a form of macro-realism quite close to our fuzzy acception (Leggett, 1980; Chiatti, Cini, Serva, 1995) . It can provide in nuce an hint which could show up to be interesting for the QM old-questioned interpretative problems.\n\nIn general, let x be a position coordinate of a quantum object and Ψ its wave function,\n\n( ) dV x 2 Ψ\n\nis usually meant as the probability of finding the particle in a region dV of space. On the contrary, in the fuzzy interpretation we will be compelled to look at the Ψ square modulus as the degree of membership of the particle to the region dV of space. How unusual it may seem, such idea has not to be regarded thoughtlessly at. As a matter of fact, in Quantum Field Theory and in other more advanced quantum scenarios, a particle is not only a localized object in the space, but rather an event emerging from the non-local networks elementary quantum transition (Licata, 2003a) . Thus, the measurement is a \"defuzzification\" process which, according to the stated, reduces the system ambiguity by limiting the semantic space and by defining a fixed information quantity.\n\nIf we agree with such interpretation we will easily and immediately realize that we will able to observate quantum coherence behaviours in non-quantum and quite far from the range of Plank's h constant situations. We reconsider here a situation owed to Yuri Orlov (Orlov, 1997) .\n\nLet us consider a Riemann's sphere (Dirac, 1947 ) -see fig. In conclusion, the generalized using of quantum semantics associated to new interpretative possibilities gives to systemics a very powerful tool to describe the observator-environment relation and to convey the several, partial attempts -till now undertaken -of applying the quantum formalism to the study of complex systems into a comprehensive conceptual root."
}
] | ||
arxiv:0704.0046 | 0704.0046 | 1 | 10.1063/1.2779138 | f01371fa48c30b75baa9584d06be7c51746e1c39eccfa73e742b5f5b675b7ccb | A limit relation for entropy and channel capacity per unit cost | In a quantum mechanical model, Diosi, Feldmann and Kosloff arrived at a
conjecture stating that the limit of the entropy of certain mixtures is the
relative entropy as system size goes to infinity. The conjecture is proven in
this paper for density matrices. The first proof is analytic and uses the
quantum law of large numbers. The second one clarifies the relation to channel
capacity per unit cost for classical-quantum channels. Both proofs lead to
generalization of the conjecture. | [
"I. Csiszar",
"F. Hiai and D. Petz"
] | [
"quant-ph",
"cs.IT",
"math.IT"
] | quant-ph | [] | 2007-04-01 | 2026-02-26 | In a quantum mechanical model, Diósi, Feldmann and Kosloff arrived at a conjecture stating that the limit of the entropy of certain mixtures is the relative entropy as system size goes to infinity. The conjecture is proven in this paper for density matrices. The first proof is analytic and uses the quantum law of large numbers. The second one clarifies the relation to channel capacity per unit cost for classical-quantum channels. Both proofs lead to generalizations of the conjecture.
It was conjectured by Diósi, Feldmann and Kosloff in [4] , based on thermodynamical considerations, that the von Neumann entropy of a quantum state equal to a mixture
R n := 1 n σ ⊗ ρ ⊗(n-1) + ρ ⊗ σ ⊗ ρ ⊗(n-2) + • • • + ρ ⊗(n-1) ⊗ σ
exceeds the entropy of a component asymptotically by the Umegaki relative entropy S(σ ρ), that is,
S(R n ) -(n -1)S(ρ) -S(σ) → S(σ ρ) (1)
as n → ∞. Here ρ and σ are density matrices acting on a finite dimensional Hilbert space. Recall that S(σ) = -Tr σ log σ and
S(σ ρ) = Tr σ(log σ -log ρ) if supp σ ≤ supp ρ +∞ otherwise.
Concerning the background of quantum entropy quantities, we refer to [10, 12] .
Apparently no exact proof of (1) has been published even for the classical case, although for that case a heuristic proof is offered in [4] .
In the paper first an analytic proof of ( 1 ) is given for the case supp σ ≤ supp ρ, using an inequality between the Umegaki and the Belavkin-Staszewski relative entropies, and the weak law of large numbers in the quantum case. In the second part of the paper, it is clarified that the problem is related to the theory of classical-quantum channels. The essential observation is the fact that S(R n ) -(n -1)S(ρ) -S(σ) in the conjecture is a Holevo quantity (classical-quantum mutual information) for a certain channel for which the relative entropy emerges as the capacity per unit cost.
The two different proofs lead to two different generalizations of the conjecture.
In this section we assume that supp σ ≤ supp ρ for the support projections of σ and ρ. One can simply compute:
S(R n ρ ⊗n ) = Tr(R n log R n -R n log ρ ⊗n ) = -S(R n ) -(n -1)Tr ρ log ρ -Tr σ log ρ.
Hence the identity
S(R n ρ ⊗n ) = -S(R n ) + (n -1)S(ρ) + S(σ ρ) + S(σ)
holds. It follows that the conjecture (1) is equivalent to the statement
S(R n ρ ⊗n ) → 0 as n → ∞ when supp σ ≤ supp ρ.
Recall the Belavkin-Staszewski relative entropy
S BS (ω ρ) = Tr(ω log(ω 1/2 ρ -1 ω 1/2 )) = -Tr(ρ η(ρ -1/2 ωρ -1/2 ))
if supp ω ≤ supp ρ, where η(t) := -t log t, see [1, 10] . It was proved by Hiai and Petz that S(ω ρ) ≤ S BS (ω ρ),
see [6] , or Proposition 7.11 in [10] .
Theorem 1. If supp σ ≤ supp ρ, then S(R n ) -(n -1)S(ρ) -S(σ) → S(σ ρ) as n → ∞.
Proof: We want to use the quantum law of large numbers, see Proposition 1.17 in [10] . Assume that ρ and σ are d × d density matrices and we may suppose that ρ is invertible. Due to the GNS-construction with respect to the limit ϕ ∞ of the product states ϕ n (A) = Tr ρ ⊗n A on the n-fold tensor product M d (C) ⊗n , n ∈ N, all finite tensor products M d (C) ⊗n are embedded into a von Neumann algebra M acting on a Hilbert space H. If γ denotes the right shift and X := ρ -1/2 σρ -1/2 , then R n is written as
R n = (ρ 1/2 ) ⊗n 1 n n-1 i=0 γ i (X) (ρ 1/2 ) ⊗n .
By inequality (2), we get
0 ≤ S(R n ρ ⊗n ) ≤ S BS (R n ρ ⊗n ) = -Tr ρ ⊗n η (ρ -1/2 ) ⊗n R n (ρ -1/2 ) ⊗n = Ω, η 1 n n-1 i=0 γ i (X) Ω , (3)
where Ω is the cyclic vector in the GNS-construction.
The law of large numbers gives
1 n n-1 i=0 γ i (X) → I
in the strong operator topology in B(H), since ϕ(X) = Tr ρρ -1/2 σρ -1/2 = 1.
Since the continuous functional calculus preserves the strong convergence (simply due to approximation by polynomials on a compact set), we obtain
η 1 n n-1 i=0 γ i (X) → η(I) = 0 strongly.
This shows that the upper bound (3) converges to 0 and the proof is complete.
By the same proof one can obtain that for
R m,n := 1 n σ ⊗m ⊗ ρ ⊗(n-1) + ρ ⊗ σ ⊗m ⊗ ρ ⊗(n-2) + • • • + ρ ⊗(n-1) ⊗ σ ⊗m ,
the limit relation
S(R m,n ) -(n -1)S(ρ) -mS(σ) → mS(σ ρ) (4)
holds as n → ∞ when m is fixed.
In the next theorem we treat the probabilistic case in a matrix language. The proof includes the case when supp σ ≤ supp ρ is not true. Those readers who are not familiar with the quantum setting of the previous theorem are suggested to follow the arguments below.
Theorem 2. Assume that ρ and σ are commuting density matrices. Then S(R n ) -(n -1)S(ρ) -S(σ) → S(σ ρ) as n → ∞.
Proof: We may assume that ρ = Diag(µ 1 , . . . , µ ℓ , 0, . . . , 0) and σ = Diag(λ 1 , . . . , λ d ) are d × d diagonal matrices, µ 1 , . . . , µ ℓ > 0 and ℓ < d. (We may consider ρ, σ in a matrix algebra of bigger size if
ρ is invertible.) If supp σ ≤ supp ρ, then λ ℓ+1 = • • • = λ d = 0;
this will be called the regular case. When supp σ ≤ supp ρ is not true, we may assume that λ d > 0 and we refer to the singular case.
The eigenvalues of R n correspond to elements (i 1 , . . . , i n ) of {1, . . . , d} n :
1 n (λ i 1 µ i 2 • • • µ in + µ i 1 λ i 2 µ i 3 • • • µ in + • • • + µ i 1 • • • µ i n-1 λ in ). ( 5
)
We divide the eigenvalues in three different groups as follows:
(a) A corresponds to (i 1 , . . . , i n ) ∈ {1, . . . , d} n with 1 ≤ i 1 , . . . , i n ≤ ℓ, (b) B corresponds to (i 1 , . . . , i n ) ∈ {1, . . . , d} n which contains exactly one d, (c) C is the rest of the eigenvalues.
If the eigenvalue ( 5 ) is in group A, then it is
(λ i 1 /µ i 1 ) + • • • + (λ in /µ in ) n µ i 1 µ i 2 • • • µ in . First we compute κ∈A η(κ) = i 1 ,...,in η (λ i 1 /µ i 1 ) + • • • + (λ in /µ in ) n µ i 1 • • • µ in .
Below the summations are over 1 ≤ i 1 , . . . , i n ≤ ℓ:
i 1 ,...,in η (λ i 1 /µ i 1 ) + • • • + (λ in /µ in ) n µ i 1 • • • µ in = - i 1 ,...,in ( (λ i 1 /µ i 1 ) + • • • + (λ in /µ in ) n µ i 1 • • • µ in log(µ i 1 • • • µ in ) + Q n = - 1 n n k=1 i 1 ,...,in λ i 1 µ i 2 • • • µ in log µ i k + i 1 ,...,in λ i 1 µ i 2 • • • µ in log µ i k + • • • + i 1 ,...,in λ i 1 µ i 2 • • • µ in log µ i k + Q n = - 1 n n k=1 (n -1) i k µ i k log µ i k + i k λ i k log µ i k + Q n = (n -1)S(ρ) - ℓ i=1 λ i log µ i + Q n ,
where
Q n := i 1 ,...,in (µ i 1 • • • µ in )η (λ i 1 /µ i 1 ) + • • • + (λ in /µ in ) n .
Consider a probability space (Ω, P) := {1, . . . , ℓ} N , (µ 1 , . . . , µ ℓ ) N , where (µ 1 , . . . , µ ℓ ) N is the product of the measure on {1, . . . , ℓ} with the distribution (µ 1 , . . . , µ ℓ ). For each n ∈ N let X n be a random variable on Ω depending on the nth {1, . . . , ℓ} so that the value of X n at i ∈ {1, . . . , ℓ} is λ i /µ i . Then X 1 , X 2 , . . . are identically distributed independent random variables and Q n is the expectation value of
η X 1 + • • • + X n n .
The strong law of large numbers says that
X 1 + • • • + X n n → E(X 1 ) = ℓ i=1 λ i µ i µ i = ℓ i=1 λ i almost surely. Since η((X 1 + • • • + X n )/n) is uniformly bounded, the Lebesgue bounded convergence theorem implies that Q n → η ℓ i=1 λ i as n → ∞.
In the regular case ℓ i=1 λ i = 1, Q n → 0 and all non-zero eigenvalues are in group A. Hence we have
S(R n ) -(n -1)S(ρ) -S(σ) = - ℓ i=1 λ i log µ i + ℓ i=1 λ i log λ i + Q n = S(σ ρ) + Q n
and the statement is clear.
Next we consider the singular case, when we have
κ∈A η(κ) = (n -1)S(ρ) + O(1),
and we turn to eigenvalues in B. If the eigenvalue corresponding to (i 1 , . . . , i n ) ∈ {1, . . . , d} n is in group B and i 1 = d, then the eigenvalue is
1 n λ d µ i 2 . . . µ in .
It follows that
- i 2 ,...,in λ d µ i 2 • • • µ in n log λ d µ i 2 • • • µ in n = - λ d n i 2 ,...,in (µ i 2 • • • µ in ) log(µ i 2 • • • µ in ) - λ d n log λ d n = λ d n (n -1)S(ρ) - λ d n log λ d n .
When i 2 = d, . . . , i n = d, we get the same quantity, so this should be multiplied with n:
κ∈B η(κ) = λ d (n -1)S(ρ) -λ d log λ d n .
We make a lower estimate to the entropy of R n in such a way that we compute κ η(κ) when κ runs over A and B. It is clear now that
S(R n ) -(n -1)S(ρ) -S(σ) ≥ κ∈A η(κ) + κ∈B η(κ) -(n -1)S(ρ) -S(σ) ≥ λ d (n -1)S(ρ) + λ d log n + O(1) → +∞
as n → ∞.
A classical-quantum channel with classical input alphabet X transfers the input x ∈ X into the output W (x) ≡ ρ x which is a density matrix acting on a Hilbert space K. We restrict ourselves to the case when X is finite and K is finite dimensional.
If a classical random variable X is chosen to be the input, with probability distribution P = {p(x) : x ∈ X }, then the corresponding output is the quantum state ρ X := x∈X p(x)ρ x . When a measurement is performed on the output quantum system, it gives rise to an output random variable Y which is jointly distributed with the input X. If a partition of unity {F y : y ∈ X } in B(K) describes the measurement, then
Prob(Y = y | X = x) = Tr ρ x F y (x, y ∈ X ). (6)
According to the Holevo bound, we have
I(X ∧ Y ) := H(Y ) -H(Y |X) ≤ I(X, W ) := S(ρ X ) - x∈X p(x)S(ρ x ), (7)
which is actually a simple consequence of the monotonicity of the relative entropy under state transformation [7] , see also [11] . I(X, W ) is the so-called Holevo quantity or classical-quantum mutual information, and it satisfies the identity
x∈X p(x)S(ρ x ρ) = I(X, W ) + S(ρ X ρ), (8)
where ρ is an arbitrary density.
The channel is used to transfer sequences from the classical alphabet; x = (x 1 , x 2 , . . . , x n ) ∈ X n is transferred into the quantum state W ⊗n (x) = ρ x := ρ x 1 ⊗ρ x 2 ⊗. . .⊗ρ xn . A code for the channel W ⊗n is defined by a subset A n ⊂ X n , which is called a codeword set. The decoder is a measurement {F y : y ∈ X n }. The probability of error is Prob(X = Y ), where X is the input random variable uniformly distributed on A n and the output random variable is determined by (6) , where x and y are replaced by x and y.
The essential observation is the fact that S(R n ) -(n -1)S(ρ) -S(σ) in the conjecture is a Holevo quantity in case of a channel with input sequences (x 1 , x 2 , . . . , x n ) ∈ {0, 1} n and outputs ρ x 1 ⊗ ρ x 2 ⊗ . . . ⊗ ρ xn , where ρ 0 = σ, ρ 1 = ρ and the codewords are all sequences containing exactly one 0. More generally, we shall consider Holevo quantities
I(A, ρ 0 , ρ 1 ) := S 1 |A| x∈A ρ x - 1 |A| x∈A S(ρ x ).
defined for any set A ⊂ {0, 1} n of binary sequences of length n.
The concept related to the conjecture we study is the channel capacity per unit cost which is defined next for simplicity only in the case where X = {0, 1}, the cost of a character 0 ∈ X is 1, while the cost of 1 ∈ X is 0.
For a memoryless channel with a binary input alphabet X = {0, 1} and an ε > 0, a number R > 0 is called an ε-achievable rate per unit cost if for every δ > 0 and for any sufficiently large T , there exists a code of length n > T with at least e T (R-δ) codewords such that each of the codewords contains at most T 0's and the error probability is at most ε. The largest R which is an ε-achievable per unit cost for every ε > 0 is the channel capacity per unit cost. for I(A, ρ 0 , ρ 1 ).
Lemma 2. If A ⊂ {0, 1} n is a code of the channel W ⊗n , whose probability of error (for some decoding scheme) does not exceed a given 0 < ε < 1, then
(1 -ε) log |A| -log 2 ≤ I(A, ρ 0 , ρ 1 )
.
Proof: The right-hand side is a bound for the classical mutual information
I(X ∧Y ) = H(Y ) -H(Y |X)
, where Y is the channel output, see (7) . Since the error probability Prob(X = Y ) is smaller than ε, application of the Fano inequality (see [3] ) gives
H(X|Y ) ≤ ε log |A| + log 2. Therefore I(X ∧ Y ) = H(X) -H(X|Y ) ≥ (1 -ε) log |A| -log 2,
and the proof is complete.
The above two lemmas shows that the relative entropy S(ρ 0 ρ 1 ) is an upper bound for the channel capacity per unit cost of the channel W (0) = ρ 0 and W (1) = ρ 1 with a binary input alphabet. In fact, assume that R > 0 is an ε-achievable rate. For every δ > 0 and T > 0 there is a code A ⊂ {0, 1} n for which we get by Lemmas 1 and 2
T S(ρ 0 ρ 1 ) ≥ c(A)S(ρ 0 ρ 1 ) ≥ I(A, ρ 0 , ρ 1 ) ≥ (1 -ε) log |A| -log 2 ≥ (1 -ε)T (R -δ) -log 2.
Since T is arbitrarily large and ε, δ are arbitrarily small, R ≤ S(ρ 0 ρ 1 ) follows. That S(ρ 0 ρ 1 ) equals the channel capacity per unit cost will be verified below.
Theorem 3. Let the classical-quantum channel W : X = {0, 1} → B(K) be defined as
W (0) = ρ 0 ≡ σ and W (1) = ρ 1 ≡ ρ. Assume that A n ⊂ {0, 1} n is chosen such that (a) each element x = (x 1 , x 2 , . . . , x n ) ∈ A n contains at most ℓ copies of 0, (b) log |A n |/ log n → c as n → ∞, (c)
c(A n ) := 1 |A n | x∈An |{i : x i = 0}| → c as n → ∞
for some real number c > 0 and for some natural number ℓ. If the random variable X n has a uniform distribution on A n , then
lim n→∞ S(ρ Xn ) - 1 |A n | x∈An S(ρ x ) = cS(σ ρ).
The proof of the theorem is divided into lemmas. We need the direct part of the so-called quantum Stein lemma obtained in [6] , see also [2, 5, 9, 12] . Lemma 3. Let ρ 0 and ρ 1 be density matrices. For every η > 0 and 0 < R < S(ρ 0 ρ 1 ), if N is sufficiently large, then there is a projection E ∈ B(K ⊗N ) such that
α N [E] := Tr ρ ⊗N 0 (I -E) < η and for β N [E] := Tr ρ ⊗N 1 E the estimate 1 N log β N [E] < -R holds.
Note that α N is called the error of the first kind, while β N is the error of the second kind.
Lemma 4. Assume that ε > 0, 0 < R < S(ρ 0 ρ 1 ), ℓ is a positive integer and the sequences x in A n ⊂ {0, 1} n contain at most ℓ copies of 0. Let the codewords be the N-fold repetitions x N = (x, x, . . . , x) of the sequences x ∈ A n . If N is the integer part of 1 R log 2n ε and n is large enough, then there is a decoding scheme such that the error probability is smaller than ε.
Proof: We follow the probabilistic construction in [13] . Let the codewords be the Nfold repetitions x N = (x, x, . . . , x) of the sequences x ∈ A n . The corresponding output density matrices act on the Hilbert space K ⊗N n ≡ (K ⊗n ) ⊗N . We decompose this Hilbert space into an N-fold product in a different way. For each 1 ≤ i ≤ n, let K i be the tensor product of the factors i, i + n, i + 2n, . . . , i + (N -1)n. So K is identified with
K 1 ⊗ K 2 ⊗ . . . ⊗ K n .
For each 1 ≤ i ≤ n we perform a hypothesis testing on the Hilbert space K i . The 0-hypothesis is that the ith component of the actually chosen x ∈ A n is 0. Based on the channel outputs at time instances i, i + n, . . . , i + (N -1)n, the 0-hypothesis is tested against the alternative hypothesis that the ith component of x is 1. According to the quantum Stein lemma (Lemma 3), given any η > 0 and 0 < R < S(σ ρ), for N sufficiently large, there exists a test E i such that the probability of error of the first kind is smaller than η, while the probability of error of the second kind is smaller than e -N R . The projections E i and I -E i form a partition of unity in the Hilbert space K i , and the n-fold tensor product of these commuting projection will give a partition of unity in K ⊗N n . Let y ∈ {0, 1} n and set F y := ⊗ n i=1 F y i , where
F y i = E i if y i = 0 and F y i = I -E i if y i = 1.
Therefore, the result of decoding can be an arbitrary 0-1 sequence in {0, 1} n .
The decoding scheme gives y ∈ {0, 1} n in such a way that y i = 0 if the tests accepted the 0-hypothesis for i and y i = 1 if the alternative was accepted. The error probability should be estimated:
Prob(Y = X|X = x) = y:y =x Tr ρ ⊗N x F y = y:y =x n i=1 Tr ρ ⊗N x i F y i ≤ n i=1 y:y i =x i n j=1 Tr ρ ⊗N x j F y j ≤ n i=1 Tr ρ ⊗N x i (I -F x i ).
If x i = 0, then Tr ρ ⊗N x i (I -F x i ) = Tr ρ ⊗N 0 (I -E i ) ≤ η, because it is an error of the first kind. When x i = 1, Tr ρ ⊗N
x i (I -
F x i ) = Tr ρ ⊗N 1 E i ≤ e -RN
from the error of the second kind. It follows that ℓη + ne -N R is a bound for the error probability. The first term will be small if η is small. The second term will be small if N is large enough. If both terms are majorized by ε/2, then the statement of the lemma holds. We can choose n so large that N defined by the statement should be large enough.
Proof of Theorem 3: Since Lemma 1 gives an upper bound, that is, Lemma 4 is about the N-times repeated input X N and describes a decoding scheme with error probability at most ε. According to Lemma 2 we have
(1 -ε) log |A n | -1 ≤ S(ρ X N ) - 1 |A| x∈An S(ρ x N ).
From the subadditivity of the entropy we have S(ρ X N ) ≤ NS(ρ X ) and S(ρ x N ) = NS(ρ x ) holds due to the additivity for product. It follows that
(1 -ε) log |A n | N - 1 N ≤ S(ρ X ) - 1 |A n | x∈An S(ρ x ).
From the choice of N in Lemma 4 we have
R log |A n | log n log n log n + log 2 -log ε ≤ log |A n | N
and the lower bound is arbitrarily close to cR. Since R < S(ρ 0 ρ 1 ) was arbitrary, the proof is complete. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "In a quantum mechanical model, Diósi, Feldmann and Kosloff arrived at a conjecture stating that the limit of the entropy of certain mixtures is the relative entropy as system size goes to infinity. The conjecture is proven in this paper for density matrices. The first proof is analytic and uses the quantum law of large numbers. The second one clarifies the relation to channel capacity per unit cost for classical-quantum channels. Both proofs lead to generalizations of the conjecture."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "It was conjectured by Diósi, Feldmann and Kosloff in [4] , based on thermodynamical considerations, that the von Neumann entropy of a quantum state equal to a mixture\n\nR n := 1 n σ ⊗ ρ ⊗(n-1) + ρ ⊗ σ ⊗ ρ ⊗(n-2) + • • • + ρ ⊗(n-1) ⊗ σ\n\nexceeds the entropy of a component asymptotically by the Umegaki relative entropy S(σ ρ), that is,\n\nS(R n ) -(n -1)S(ρ) -S(σ) → S(σ ρ) (1)\n\nas n → ∞. Here ρ and σ are density matrices acting on a finite dimensional Hilbert space. Recall that S(σ) = -Tr σ log σ and\n\nS(σ ρ) = Tr σ(log σ -log ρ) if supp σ ≤ supp ρ +∞ otherwise.\n\nConcerning the background of quantum entropy quantities, we refer to [10, 12] .\n\nApparently no exact proof of (1) has been published even for the classical case, although for that case a heuristic proof is offered in [4] .\n\nIn the paper first an analytic proof of ( 1 ) is given for the case supp σ ≤ supp ρ, using an inequality between the Umegaki and the Belavkin-Staszewski relative entropies, and the weak law of large numbers in the quantum case. In the second part of the paper, it is clarified that the problem is related to the theory of classical-quantum channels. The essential observation is the fact that S(R n ) -(n -1)S(ρ) -S(σ) in the conjecture is a Holevo quantity (classical-quantum mutual information) for a certain channel for which the relative entropy emerges as the capacity per unit cost.\n\nThe two different proofs lead to two different generalizations of the conjecture."
},
{
"section_type": "OTHER",
"section_title": "An analytic proof of the conjecture",
"text": "In this section we assume that supp σ ≤ supp ρ for the support projections of σ and ρ. One can simply compute:\n\nS(R n ρ ⊗n ) = Tr(R n log R n -R n log ρ ⊗n ) = -S(R n ) -(n -1)Tr ρ log ρ -Tr σ log ρ.\n\nHence the identity\n\nS(R n ρ ⊗n ) = -S(R n ) + (n -1)S(ρ) + S(σ ρ) + S(σ)\n\nholds. It follows that the conjecture (1) is equivalent to the statement\n\nS(R n ρ ⊗n ) → 0 as n → ∞ when supp σ ≤ supp ρ.\n\nRecall the Belavkin-Staszewski relative entropy\n\nS BS (ω ρ) = Tr(ω log(ω 1/2 ρ -1 ω 1/2 )) = -Tr(ρ η(ρ -1/2 ωρ -1/2 ))\n\nif supp ω ≤ supp ρ, where η(t) := -t log t, see [1, 10] . It was proved by Hiai and Petz that S(ω ρ) ≤ S BS (ω ρ),\n\nsee [6] , or Proposition 7.11 in [10] .\n\nTheorem 1. If supp σ ≤ supp ρ, then S(R n ) -(n -1)S(ρ) -S(σ) → S(σ ρ) as n → ∞.\n\nProof: We want to use the quantum law of large numbers, see Proposition 1.17 in [10] . Assume that ρ and σ are d × d density matrices and we may suppose that ρ is invertible. Due to the GNS-construction with respect to the limit ϕ ∞ of the product states ϕ n (A) = Tr ρ ⊗n A on the n-fold tensor product M d (C) ⊗n , n ∈ N, all finite tensor products M d (C) ⊗n are embedded into a von Neumann algebra M acting on a Hilbert space H. If γ denotes the right shift and X := ρ -1/2 σρ -1/2 , then R n is written as\n\nR n = (ρ 1/2 ) ⊗n 1 n n-1 i=0 γ i (X) (ρ 1/2 ) ⊗n .\n\nBy inequality (2), we get\n\n0 ≤ S(R n ρ ⊗n ) ≤ S BS (R n ρ ⊗n ) = -Tr ρ ⊗n η (ρ -1/2 ) ⊗n R n (ρ -1/2 ) ⊗n = Ω, η 1 n n-1 i=0 γ i (X) Ω , (3)\n\nwhere Ω is the cyclic vector in the GNS-construction.\n\nThe law of large numbers gives\n\n1 n n-1 i=0 γ i (X) → I\n\nin the strong operator topology in B(H), since ϕ(X) = Tr ρρ -1/2 σρ -1/2 = 1.\n\nSince the continuous functional calculus preserves the strong convergence (simply due to approximation by polynomials on a compact set), we obtain\n\nη 1 n n-1 i=0 γ i (X) → η(I) = 0 strongly.\n\nThis shows that the upper bound (3) converges to 0 and the proof is complete.\n\nBy the same proof one can obtain that for\n\nR m,n := 1 n σ ⊗m ⊗ ρ ⊗(n-1) + ρ ⊗ σ ⊗m ⊗ ρ ⊗(n-2) + • • • + ρ ⊗(n-1) ⊗ σ ⊗m ,\n\nthe limit relation\n\nS(R m,n ) -(n -1)S(ρ) -mS(σ) → mS(σ ρ) (4)\n\nholds as n → ∞ when m is fixed.\n\nIn the next theorem we treat the probabilistic case in a matrix language. The proof includes the case when supp σ ≤ supp ρ is not true. Those readers who are not familiar with the quantum setting of the previous theorem are suggested to follow the arguments below.\n\nTheorem 2. Assume that ρ and σ are commuting density matrices. Then S(R n ) -(n -1)S(ρ) -S(σ) → S(σ ρ) as n → ∞.\n\nProof: We may assume that ρ = Diag(µ 1 , . . . , µ ℓ , 0, . . . , 0) and σ = Diag(λ 1 , . . . , λ d ) are d × d diagonal matrices, µ 1 , . . . , µ ℓ > 0 and ℓ < d. (We may consider ρ, σ in a matrix algebra of bigger size if\n\nρ is invertible.) If supp σ ≤ supp ρ, then λ ℓ+1 = • • • = λ d = 0;\n\nthis will be called the regular case. When supp σ ≤ supp ρ is not true, we may assume that λ d > 0 and we refer to the singular case.\n\nThe eigenvalues of R n correspond to elements (i 1 , . . . , i n ) of {1, . . . , d} n :\n\n1 n (λ i 1 µ i 2 • • • µ in + µ i 1 λ i 2 µ i 3 • • • µ in + • • • + µ i 1 • • • µ i n-1 λ in ). ( 5\n\n)\n\nWe divide the eigenvalues in three different groups as follows:\n\n(a) A corresponds to (i 1 , . . . , i n ) ∈ {1, . . . , d} n with 1 ≤ i 1 , . . . , i n ≤ ℓ, (b) B corresponds to (i 1 , . . . , i n ) ∈ {1, . . . , d} n which contains exactly one d, (c) C is the rest of the eigenvalues.\n\nIf the eigenvalue ( 5 ) is in group A, then it is\n\n(λ i 1 /µ i 1 ) + • • • + (λ in /µ in ) n µ i 1 µ i 2 • • • µ in . First we compute κ∈A η(κ) = i 1 ,...,in η (λ i 1 /µ i 1 ) + • • • + (λ in /µ in ) n µ i 1 • • • µ in .\n\nBelow the summations are over 1 ≤ i 1 , . . . , i n ≤ ℓ:\n\ni 1 ,...,in η (λ i 1 /µ i 1 ) + • • • + (λ in /µ in ) n µ i 1 • • • µ in = - i 1 ,...,in ( (λ i 1 /µ i 1 ) + • • • + (λ in /µ in ) n µ i 1 • • • µ in log(µ i 1 • • • µ in ) + Q n = - 1 n n k=1 i 1 ,...,in λ i 1 µ i 2 • • • µ in log µ i k + i 1 ,...,in λ i 1 µ i 2 • • • µ in log µ i k + • • • + i 1 ,...,in λ i 1 µ i 2 • • • µ in log µ i k + Q n = - 1 n n k=1 (n -1) i k µ i k log µ i k + i k λ i k log µ i k + Q n = (n -1)S(ρ) - ℓ i=1 λ i log µ i + Q n ,\n\nwhere\n\nQ n := i 1 ,...,in (µ i 1 • • • µ in )η (λ i 1 /µ i 1 ) + • • • + (λ in /µ in ) n .\n\nConsider a probability space (Ω, P) := {1, . . . , ℓ} N , (µ 1 , . . . , µ ℓ ) N , where (µ 1 , . . . , µ ℓ ) N is the product of the measure on {1, . . . , ℓ} with the distribution (µ 1 , . . . , µ ℓ ). For each n ∈ N let X n be a random variable on Ω depending on the nth {1, . . . , ℓ} so that the value of X n at i ∈ {1, . . . , ℓ} is λ i /µ i . Then X 1 , X 2 , . . . are identically distributed independent random variables and Q n is the expectation value of\n\nη X 1 + • • • + X n n .\n\nThe strong law of large numbers says that\n\nX 1 + • • • + X n n → E(X 1 ) = ℓ i=1 λ i µ i µ i = ℓ i=1 λ i almost surely. Since η((X 1 + • • • + X n )/n) is uniformly bounded, the Lebesgue bounded convergence theorem implies that Q n → η ℓ i=1 λ i as n → ∞.\n\nIn the regular case ℓ i=1 λ i = 1, Q n → 0 and all non-zero eigenvalues are in group A. Hence we have\n\nS(R n ) -(n -1)S(ρ) -S(σ) = - ℓ i=1 λ i log µ i + ℓ i=1 λ i log λ i + Q n = S(σ ρ) + Q n\n\nand the statement is clear.\n\nNext we consider the singular case, when we have\n\nκ∈A η(κ) = (n -1)S(ρ) + O(1),\n\nand we turn to eigenvalues in B. If the eigenvalue corresponding to (i 1 , . . . , i n ) ∈ {1, . . . , d} n is in group B and i 1 = d, then the eigenvalue is\n\n1 n λ d µ i 2 . . . µ in .\n\nIt follows that\n\n- i 2 ,...,in λ d µ i 2 • • • µ in n log λ d µ i 2 • • • µ in n = - λ d n i 2 ,...,in (µ i 2 • • • µ in ) log(µ i 2 • • • µ in ) - λ d n log λ d n = λ d n (n -1)S(ρ) - λ d n log λ d n .\n\nWhen i 2 = d, . . . , i n = d, we get the same quantity, so this should be multiplied with n:\n\nκ∈B η(κ) = λ d (n -1)S(ρ) -λ d log λ d n .\n\nWe make a lower estimate to the entropy of R n in such a way that we compute κ η(κ) when κ runs over A and B. It is clear now that\n\nS(R n ) -(n -1)S(ρ) -S(σ) ≥ κ∈A η(κ) + κ∈B η(κ) -(n -1)S(ρ) -S(σ) ≥ λ d (n -1)S(ρ) + λ d log n + O(1) → +∞\n\nas n → ∞."
},
{
"section_type": "OTHER",
"section_title": "Interpretation as capacity",
"text": "A classical-quantum channel with classical input alphabet X transfers the input x ∈ X into the output W (x) ≡ ρ x which is a density matrix acting on a Hilbert space K. We restrict ourselves to the case when X is finite and K is finite dimensional.\n\nIf a classical random variable X is chosen to be the input, with probability distribution P = {p(x) : x ∈ X }, then the corresponding output is the quantum state ρ X := x∈X p(x)ρ x . When a measurement is performed on the output quantum system, it gives rise to an output random variable Y which is jointly distributed with the input X. If a partition of unity {F y : y ∈ X } in B(K) describes the measurement, then\n\nProb(Y = y | X = x) = Tr ρ x F y (x, y ∈ X ). (6)\n\nAccording to the Holevo bound, we have\n\nI(X ∧ Y ) := H(Y ) -H(Y |X) ≤ I(X, W ) := S(ρ X ) - x∈X p(x)S(ρ x ), (7)\n\nwhich is actually a simple consequence of the monotonicity of the relative entropy under state transformation [7] , see also [11] . I(X, W ) is the so-called Holevo quantity or classical-quantum mutual information, and it satisfies the identity\n\nx∈X p(x)S(ρ x ρ) = I(X, W ) + S(ρ X ρ), (8)\n\nwhere ρ is an arbitrary density.\n\nThe channel is used to transfer sequences from the classical alphabet; x = (x 1 , x 2 , . . . , x n ) ∈ X n is transferred into the quantum state W ⊗n (x) = ρ x := ρ x 1 ⊗ρ x 2 ⊗. . .⊗ρ xn . A code for the channel W ⊗n is defined by a subset A n ⊂ X n , which is called a codeword set. The decoder is a measurement {F y : y ∈ X n }. The probability of error is Prob(X = Y ), where X is the input random variable uniformly distributed on A n and the output random variable is determined by (6) , where x and y are replaced by x and y.\n\nThe essential observation is the fact that S(R n ) -(n -1)S(ρ) -S(σ) in the conjecture is a Holevo quantity in case of a channel with input sequences (x 1 , x 2 , . . . , x n ) ∈ {0, 1} n and outputs ρ x 1 ⊗ ρ x 2 ⊗ . . . ⊗ ρ xn , where ρ 0 = σ, ρ 1 = ρ and the codewords are all sequences containing exactly one 0. More generally, we shall consider Holevo quantities\n\nI(A, ρ 0 , ρ 1 ) := S 1 |A| x∈A ρ x - 1 |A| x∈A S(ρ x ).\n\ndefined for any set A ⊂ {0, 1} n of binary sequences of length n.\n\nThe concept related to the conjecture we study is the channel capacity per unit cost which is defined next for simplicity only in the case where X = {0, 1}, the cost of a character 0 ∈ X is 1, while the cost of 1 ∈ X is 0.\n\nFor a memoryless channel with a binary input alphabet X = {0, 1} and an ε > 0, a number R > 0 is called an ε-achievable rate per unit cost if for every δ > 0 and for any sufficiently large T , there exists a code of length n > T with at least e T (R-δ) codewords such that each of the codewords contains at most T 0's and the error probability is at most ε. The largest R which is an ε-achievable per unit cost for every ε > 0 is the channel capacity per unit cost. for I(A, ρ 0 , ρ 1 ).\n\nLemma 2. If A ⊂ {0, 1} n is a code of the channel W ⊗n , whose probability of error (for some decoding scheme) does not exceed a given 0 < ε < 1, then\n\n(1 -ε) log |A| -log 2 ≤ I(A, ρ 0 , ρ 1 )\n\n.\n\nProof: The right-hand side is a bound for the classical mutual information\n\nI(X ∧Y ) = H(Y ) -H(Y |X)\n\n, where Y is the channel output, see (7) . Since the error probability Prob(X = Y ) is smaller than ε, application of the Fano inequality (see [3] ) gives\n\nH(X|Y ) ≤ ε log |A| + log 2. Therefore I(X ∧ Y ) = H(X) -H(X|Y ) ≥ (1 -ε) log |A| -log 2,\n\nand the proof is complete.\n\nThe above two lemmas shows that the relative entropy S(ρ 0 ρ 1 ) is an upper bound for the channel capacity per unit cost of the channel W (0) = ρ 0 and W (1) = ρ 1 with a binary input alphabet. In fact, assume that R > 0 is an ε-achievable rate. For every δ > 0 and T > 0 there is a code A ⊂ {0, 1} n for which we get by Lemmas 1 and 2\n\nT S(ρ 0 ρ 1 ) ≥ c(A)S(ρ 0 ρ 1 ) ≥ I(A, ρ 0 , ρ 1 ) ≥ (1 -ε) log |A| -log 2 ≥ (1 -ε)T (R -δ) -log 2.\n\nSince T is arbitrarily large and ε, δ are arbitrarily small, R ≤ S(ρ 0 ρ 1 ) follows. That S(ρ 0 ρ 1 ) equals the channel capacity per unit cost will be verified below.\n\nTheorem 3. Let the classical-quantum channel W : X = {0, 1} → B(K) be defined as\n\nW (0) = ρ 0 ≡ σ and W (1) = ρ 1 ≡ ρ. Assume that A n ⊂ {0, 1} n is chosen such that (a) each element x = (x 1 , x 2 , . . . , x n ) ∈ A n contains at most ℓ copies of 0, (b) log |A n |/ log n → c as n → ∞, (c)\n\nc(A n ) := 1 |A n | x∈An |{i : x i = 0}| → c as n → ∞\n\nfor some real number c > 0 and for some natural number ℓ. If the random variable X n has a uniform distribution on A n , then\n\nlim n→∞ S(ρ Xn ) - 1 |A n | x∈An S(ρ x ) = cS(σ ρ).\n\nThe proof of the theorem is divided into lemmas. We need the direct part of the so-called quantum Stein lemma obtained in [6] , see also [2, 5, 9, 12] . Lemma 3. Let ρ 0 and ρ 1 be density matrices. For every η > 0 and 0 < R < S(ρ 0 ρ 1 ), if N is sufficiently large, then there is a projection E ∈ B(K ⊗N ) such that\n\nα N [E] := Tr ρ ⊗N 0 (I -E) < η and for β N [E] := Tr ρ ⊗N 1 E the estimate 1 N log β N [E] < -R holds.\n\nNote that α N is called the error of the first kind, while β N is the error of the second kind.\n\nLemma 4. Assume that ε > 0, 0 < R < S(ρ 0 ρ 1 ), ℓ is a positive integer and the sequences x in A n ⊂ {0, 1} n contain at most ℓ copies of 0. Let the codewords be the N-fold repetitions x N = (x, x, . . . , x) of the sequences x ∈ A n . If N is the integer part of 1 R log 2n ε and n is large enough, then there is a decoding scheme such that the error probability is smaller than ε.\n\nProof: We follow the probabilistic construction in [13] . Let the codewords be the Nfold repetitions x N = (x, x, . . . , x) of the sequences x ∈ A n . The corresponding output density matrices act on the Hilbert space K ⊗N n ≡ (K ⊗n ) ⊗N . We decompose this Hilbert space into an N-fold product in a different way. For each 1 ≤ i ≤ n, let K i be the tensor product of the factors i, i + n, i + 2n, . . . , i + (N -1)n. So K is identified with\n\nK 1 ⊗ K 2 ⊗ . . . ⊗ K n .\n\nFor each 1 ≤ i ≤ n we perform a hypothesis testing on the Hilbert space K i . The 0-hypothesis is that the ith component of the actually chosen x ∈ A n is 0. Based on the channel outputs at time instances i, i + n, . . . , i + (N -1)n, the 0-hypothesis is tested against the alternative hypothesis that the ith component of x is 1. According to the quantum Stein lemma (Lemma 3), given any η > 0 and 0 < R < S(σ ρ), for N sufficiently large, there exists a test E i such that the probability of error of the first kind is smaller than η, while the probability of error of the second kind is smaller than e -N R . The projections E i and I -E i form a partition of unity in the Hilbert space K i , and the n-fold tensor product of these commuting projection will give a partition of unity in K ⊗N n . Let y ∈ {0, 1} n and set F y := ⊗ n i=1 F y i , where\n\nF y i = E i if y i = 0 and F y i = I -E i if y i = 1.\n\nTherefore, the result of decoding can be an arbitrary 0-1 sequence in {0, 1} n .\n\nThe decoding scheme gives y ∈ {0, 1} n in such a way that y i = 0 if the tests accepted the 0-hypothesis for i and y i = 1 if the alternative was accepted. The error probability should be estimated:\n\nProb(Y = X|X = x) = y:y =x Tr ρ ⊗N x F y = y:y =x n i=1 Tr ρ ⊗N x i F y i ≤ n i=1 y:y i =x i n j=1 Tr ρ ⊗N x j F y j ≤ n i=1 Tr ρ ⊗N x i (I -F x i ).\n\nIf x i = 0, then Tr ρ ⊗N x i (I -F x i ) = Tr ρ ⊗N 0 (I -E i ) ≤ η, because it is an error of the first kind. When x i = 1, Tr ρ ⊗N\n\nx i (I -\n\nF x i ) = Tr ρ ⊗N 1 E i ≤ e -RN\n\nfrom the error of the second kind. It follows that ℓη + ne -N R is a bound for the error probability. The first term will be small if η is small. The second term will be small if N is large enough. If both terms are majorized by ε/2, then the statement of the lemma holds. We can choose n so large that N defined by the statement should be large enough.\n\nProof of Theorem 3: Since Lemma 1 gives an upper bound, that is, Lemma 4 is about the N-times repeated input X N and describes a decoding scheme with error probability at most ε. According to Lemma 2 we have\n\n(1 -ε) log |A n | -1 ≤ S(ρ X N ) - 1 |A| x∈An S(ρ x N ).\n\nFrom the subadditivity of the entropy we have S(ρ X N ) ≤ NS(ρ X ) and S(ρ x N ) = NS(ρ x ) holds due to the additivity for product. It follows that\n\n(1 -ε) log |A n | N - 1 N ≤ S(ρ X ) - 1 |A n | x∈An S(ρ x ).\n\nFrom the choice of N in Lemma 4 we have\n\nR log |A n | log n log n log n + log 2 -log ε ≤ log |A n | N\n\nand the lower bound is arbitrarily close to cR. Since R < S(ρ 0 ρ 1 ) was arbitrary, the proof is complete."
}
] | |
arxiv:0704.0048 | 0704.0048 | 1 | 10.1088/0264-9381/24/19/S17 | 5831c84409a92a3da91bce6c44bb910afdfbe6594b6b49eb409254e409cc6c92 | Inference on white dwarf binary systems using the first round Mock LISA
Data Challenges data sets | We report on the analysis of selected single source data sets from the first
round of the Mock LISA Data Challenges (MLDC) for white dwarf binaries. We
implemented an end-to-end pipeline consisting of a grid-based coherent
pre-processing unit for signal detection, and an automatic Markov Chain Monte
Carlo post-processing unit for signal evaluation. We demonstrate that signal
detection with our coherent approach is secure and accurate, and is increased
in accuracy and supplemented with additional information on the signal
parameters by our Markov Chain Monte Carlo approach. We also demonstrate that
the Markov Chain Monte Carlo routine is additionally able to determine
accurately the noise level in the frequency window of interest. | [
"Alexander Stroeer",
"John Veitch",
"Christian Roever",
"Ed Bloomer",
"James\n Clark",
"Nelson Christensen",
"Martin Hendry",
"Chris Messenger",
"Renate Meyer",
"Matthew Pitkin",
"Jennifer Toher",
"Richard Umstaetter",
"Alberto Vecchio and\n Graham Woan"
] | [
"gr-qc",
"astro-ph"
] | gr-qc | [] | 2007-03-31 | 2026-02-26 | The data obtained from LISA [1] will contain a large number of white dwarf binary systems across the whole observational window [2] . At frequencies below ∼ 3 mHz the sources are so abundant that they produce a stochastic foreground whose intensity dominates the instrumental noise [3] . The closer (and louder) sources will still be sufficiently bright to be individually resolvable. Above ∼ 3 mHz the sources become sufficiently sparse in parameter space (and in particular in the frequency domain) that the detectable sources become individually resolvable. The identification of white dwarfs in the LISA data set represents one of the most interesting analysis problems posed by the mission: the total number of signals in the data set is unknown, the effective noise WD MLDC1 2 level affecting the measurements is not easily estimated from the data streams, and there is a large number of overlapping sources to the limit of confusion.
Bayesian inference provides a clear framework to tackle such a problem [4, 5, 6] . Some of us have carried out exploratory studies and "proof of concept" analyses on simplified problems that have demonstrated that Bayesian techniques do indeed show good potential for LISA applications [11, 10, 12] . Similarly other authors have successfully implemented techniques using Bayesian inference [18, 17, 16] . In this paper we present the first results of an end-to-end analysis pipeline developed in the context of the Mock LISA Data Challenges that has evolved from our earlier work. This pipeline is applied to the simplest single-source challenge data sets 1.1.1a and 1.1.1b and all the results presented here are obtained after the release of the key files. In a companion paper [19] , we present results that we have obtained for the analysis of the data sets containing gravitational radiation from a massive-black-hole binary inspiral. Our group submitted an entry for the MLDC analysing the blind data set 1.1.1c [13, 14]: however that result suffered from the fact that the pipeline was not complete, the analysis code was inefficient and we encountered hardware problems with the Beowulf cluster used to perform the analysis.
The results that we present here are obtained with a two-stage end-to-end analysis pipeline: (i) we first process the data set with a grid-based coherent algorithm to identify candidate signals; (ii) we then follow up the candidate signals with a Markov Chain Monte Carlo code to obtain probability density function on the model parameters. Our method differs from other MCMC methods that have been proposed and applied to the MLDC data in the context of white dwarf binaries [18, 17, 16] : the MCMC is not used to search, but only in the final stage of the analysis to produce posterior density functions of the model parameters. The noise spectral level is included as one of the unknown parameters and is estimated together with the parameters of the gravitational wave source(s).
In this section we describe the two stage approach that we have adopted for the analysis. The signal produced by a white dwarf binary system is modelled as monochromatic in the source reference frame, following the conventions adopted in the first MLDC [7, 8, 9] . It is described by 7 parameters: ecliptic latitude ϑ e and longitude ϕ e , inclination ι and polarisation angle Ψ, frequency at a reference time f 0 and corresponding overall phase Φ 0 and amplitude A.
The data distributed for the MLDC are the three TDI v1.5 Michelson observables X, Y and Z ‡. From those we construct the two orthogonal TDI outputs
A = (2X -Y -Z)/3 ( 1
) E = (Z -Y )/ √ 3 ( 2
)
‡ In our MCMC analysis we use the data set produced using the LISA Simulator.
WD MLDC1 3 by diagonalizing the noise covariance matrix following the procedure presented in [23] . The noise affecting the channels A and E is uncorrelated and described by the one-sided noise spectral density S n (f ). We model the LISA response function in the low frequency limit in order to improve the computational efficiency of our analysis.
2.1. First stage: Grid based search
The first stage of the pipeline consists of a fast search of the data for the best matched filter based on the well-known F -statistic algorithm, first developed for triaxial pulsar signals in the context of ground-based observations [20] . This exploits the Fast Fourier Transform to perform matching in the frequency domain to templates which are generated at an array of fixed points in the parameter space. The data from an individual detector in the frequency domain d(f ) is supposed to contain a signal plus Gaussian noise, d(f ) = h(f ) + ñ(f ). We define the logarithmic likelihood as a measure of match, as given by log L ≈ ( d| h) -1 2 ( h| h) with (•|•) denoting the scalar product as defined in [20] . A single signal in the F -statistic algorithm is re-parameterised as a linear function of four orthogonal variables, and the frequency f 0 . The detection statistic is based on four parameters A F , B F , C F and D F , found by integrating over the response functions a(t) and b(t) to the two polarisation states of the gravitational wave signal [20] ,
A F = 2 T obs T obs 0 a(t) 2 dt ( 3
) B F = 2 T obs T obs 0 b(t) 2 dt ( 4
) C F = 2 T obs T obs 0 a(t)b(t)dt, ( 5
) D F = A F B F -C 2 F ( 6
)
T obs denotes the total observed time for the data set being analysed. The optimal detection statistic 2F , which is pre-maximised over the nuisance parameters h 0 , ι, φ 0 and ψ is
2F = 8 S n (f )T obs B F |F a | 2 + A F |F b | 2 -2C F × R(F a F b ) D F . ( 7
)
F a and F b are the demodulated Fourier transforms of the data, iΦ(t) dt; F b = T obs 0 d(t)b(t)e -iΦ(t) dt, (8) Φ(t) is the phase of the gravitational wave signal, as is described in [22] .
F a = T obs 0 d(t)a(t)e -
As the LISA array moves in space, the frequency f 0 is affected by Doppler modulations. This modulation changes with differing position of the source in the sky, implying the need to recalculate the modulations and thus a(t) and b(t) for each sky position that is tested -a significant factor in the performance of this approach. The differing modulation structure however also allows us to estimate the location of the WD MLDC1 4 source in the sky by maximising the 2F value. The resolution possible on the sky with this method is not as good as from a full Bayesian posterior probability calculation as performed in the parameter estimation stage, as shown in an example for Challenge 1.1.1a in figure 1 . Nevertheless, since this statistic can be computed fairly quickly it serves as a useful way of finding initial values to feed into the MCMC routine, as adopted within the pipeline. The resolution achievable on the sky increases with frequency, which implies that the mismatch between filter and signal falls off more rapidly at higher frequencies, requiring a greater number of templates to cover the sky. Therefore for challenge 1.1.1b at f ≈ 3 mHz a sky grid of size 5,752 points was used, in comparison with 765 points for challenge 1.1.1a at f ≈ 1 mHz.
The F -statistic search was implemented using the LIGO "Lalapps" suite of software [24] , in which the pulsar search code was modified by Reinhard Prix and John Whelan to use the LISA response function for the TDI variables X, Y , and Z [21] . These input data streams were given in the form of Short Fourier Transforms, each of length one day, created from the MLDC1 challenge data. For each challenge the full specified range of frequencies was searched for the signal as it would be in a blind search. The code was run on a single CPU and executed in a few hours, with the run-time increasing at higher frequency due to the higher resolution of sky and frequency grid that had to be used. The candidate chosen to pass to the MCMC stage was simply that which triggered the highest value of 2F .
According to Bayes' theorem, the posterior probability, p( m| d) of a model m given the data d depends on the prior distribution p( m), containing the information known before the analysis, the likelihood L( d| m) of the model and a normalisation factor p( d)
p( m| d) = L( d| m)p( m) p( d) ( 9
)
The posterior probability density function shows the joint probability density of given values of parameters of the model m, conditional on the data d. We implemented Bayes' theorem using data in the form of TDI variables A and E and modelled our template according to the Long Wavelength Approximation directly in the Fourier domain [25] to gain computational speed. The logarithmic likelihood L( d| m) in this stage explicitly included its dependence on the one-sided noise spectral density S n (f )
log L( d| m) = const. - 1 2 log S n (f ) -( d -h| d -h), ( 10
)
shown here for either A or E, with the combined likelihood as sum of the individual likelihoods. We restricted our analysis to a sufficiently narrow frequency window in order to be able to approximate the noise spectral density as constant, S n (f ) = S 0 . This window was set as the interval in frequency that contains at least 98% of the power of our WD MLDC1 5 Ecliptic Longitude Ecliptic Latitude 2F as a function of sky position, at a frequency 0.001063 Hz 1 2 3 4 5 6 -1.5 -1 -0.5 0 0.5 1 1.5 2F 500 1000 1500 2000 2500 Figure 1. The variation of 2F values for the search for unknown signal 1.1.1a, as a function of sky position, parameterised by ecliptic latitude β and longitude λ. The distribution is multi-modal and non-Gaussian, and has a poor resolution in comparison with that can be achieved with the MCMC and a Bayesian likelihood, but by finding the maximum it serves well as a starting point for the more refined parameter estimation below.
model m, with the interval's upper and lower limits given by f ±(2/year)(5+2πf 0 AU/c) [25]. S 0 is therefore an additional parameter to be inferred within the model m in Eq. 10.
We implemented an automatic Random Walk Metropolis sampler (Stroeer & Vecchio 2007, in. prep.) to sample from the posterior probability density function in form of a Markov chain. Metropolis sampling eliminates the need to explicitly calculate the normalisation constant in Bayes' theorem, and the evolving Markov chain gives easy access to joint as well as marginalised posterior density distribution. The sampler was started from the parameter set which triggered the highest value of 2F in our grid based coherent run of the analysis (see former section). The automated function of the Metropolis sampling is achieved by controlling the sampling step-size with adaptive acceptance probability techniques [26] . The sampler therefore does not depend on assumptions about the signal in the data set in order to perform successfully and reliably; it develops a suitable algorithm and approach by itself based on the properties of the likelihood as found on the fly, in the initial steps of the sampler. The length of our Markov chain was pre-set to 10 6 , with the initial 10 4 chain states discarded as the "burn-in" phase of our sampler. The runtime for one data analysis run is 5 hours on a single 2 GHz CPU on the Tsunami cluster of the University of Birmingham. WD MLDC1 6 Figure 2. The marginalised posterior probability density functions of the eight unknown parameters -the seven parameters that describe the signal and the noise spectral density S 0 -for the the challenge data set 1.1.1a. The vertical black solid line denotes the true value of the parameter (for the polarisation angle the true value modulo π/2), and the grey dashed line the initial value for the MCMC analysis as determined by the template of the first-stage that produces the maximum value of the F -statistic. In the case of the noise spectral density the first stage of the analysis does not provide an estimate; the true value of this parameter is taken to be the value of the instrumental noise spectrum used to generate the data set and provided in [9]. WD MLDC1 7 Figure 3. The marginalised posterior probability density functions of the eight unknown parameters for the the challenge data set 1.1.1b. Labels are as in Figure 2.
We found that the most promising candidate signal from the F -statistic search already matched the true embedded signal to high accuracy, particularly in frequency and sky location. Our MCMC sampler, as a post-processing unit, thus only needed 1000 iterations to burn in and to establish a reliable sampling from the posterior. The marginalised posteriors are shown in Figs. 2 and 3. We found, as seen in latter figures, that the MCMC sampler further refined the initial guesses from the F -statistic, as measured by the absolute difference between the true value of a given parameter and the median of the marginalised posterior recovered for that parameter. Table 1
WD MLDC1 8 Table 1. Details about the results from Challenge 1.1.1a and Challenge 1.1.1b. S 0 , the constant one-sided noise spectral density within our narrow frequency window, is compared to the true one sided noise spectral density at the true frequency of the signal, Ψ is given modulo π/2. Int 90 denotes the minimum interval to include 90% of MCMC states for given parameter, ∆mode denotes the absolute difference between the true value of a signal parameter and the mode of its recovered posterior; ∆median and ∆mean denote the equivalent absolute difference for median and mean of the posterior respectively; σ denotes the sampled standard deviation of the posterior as derived from the median. We further quote the signal-to-noise ratio (SNR) for a template using the true values of the source and the recovered values of the data analysis run, as derived from the median of the individual posterior distributions, and the correlation C between these two templates. Int 90 ∆mode ∆median ∆mean σ Challenge 1.1.1a S 0 10 -41 Hz -1 (3.53257, 4.72639) -0.42084 -0.440278 -0.452456 0.36704 ϑ e / rad (0.958409, 1.03165) -0.0147383 -0.0149381 -0.0148725 0.0222861 ϕ e / rad (5.05376, 5.13528) -0.00550139 -0.00569547 -0.00579889 0.0247886 Ψ/ rad (1.32475, 0.500553) 0.1768 0.1823 0.1902 0.1908 ι/ rad (0.097761, 1.0008) -0.0459747 0.190001 0.23459 0.295211 A/10 -22 (1.61976, 2.67967) 0.664371 0.358844 0.298978 0.368524 f 0 / mHz (1.06273, 1.06273) -1.19664e-06 -1.22207e-06 -1.22259e-06 1.04422e-06 Φ 0 / rad (3.10668, 5.808) -0.164989 0.00998525 0.229659 0.829146 SNR true = 51.024497 recovered = 50.648600 C true vs. recovered = 0.99689 Challenge 1.1.1b S 0 10 -41 Hz -1 (0.876833, 1.38959) -0.0679571 -0.0906557 -0.0996144 0.16017 ϑ e / rad (-0.121611, 0.0116916) -0.0343353 -0.151185 -0.150328 0.0406552 ϕ e / rad (4.60969, 4.63537) 0.00265723 0.00305564 0.00302203 0.00779893 Ψ/ rad (0.246328, 0.362409) 0.0301541 0.0311747 0.0311268 0.0353938 ι/ rad (1.22036, 1.33338) -0.0430412 -0.040458 -0.0394818 0.0348383 A/10 -22 (0.45001, 0.542454) -0.016442 -0.0151921 -0.0149907 0.0281154 f 0 / mHz (3.00036, 3.00036) 3.1221e-07 2.49289e-07 2.42807e-07 8.18111e-07 Φ 0 / rad (5.83869, 6.19411) 0.137219 0.119301 0.119921 0.502384 SNR true = 36.587444 recovered = 37.368806 C true vs. recovered = 0.97897
shows details of the statistics of recovered posterior distributions. We highlight that the majority of the true values of the parameters are within one standard deviation of the median of the posterior, with a small percentage within two sampled standard deviations. In addition, every true value of a parameter of the signal is within the minimum interval of the posterior to cover 90% of all MCMC state values. Recovered signal-to-noise ratios are measured as SNR = (s|h)/ (h|h), and the match C = (h true |h med )/ (h true |h true ) (h med |h med ) between a template constructed from the true values and a template from the median values of the individual posterior distributions, yielding a correlation that is always higher than 0.97. Noise levels are determined accurately and within 1 to 1.5 sampled standard deviations. Nevertheless we note that WD MLDC1 9 our run on Challenge 1.1.1a shows a lower match and higher differences between true value and recovered value of parameters as compared to the run on Challenge 1.1.1b. It also exhibits tailing posterior distributions in inclination and amplitude, although the SNR of Challenge 1.1.1a is twice the value of Challenge 1.1.1b.
We have presented a new approach to LISA data analysis in the form of an end-to-end pipeline. We first detected and identified candidate signals in the LISA data stream with a grid-based coherent algorithm, and then post-processed the most promising candidate signals with an automatic Markov Chain Monte Carlo code to obtain probability densities for the model's parameters. We demonstrated successful identification and post-processing of the signals from the double white dwarf single source MLDC data sets 1.1.1a and 1.1.1b. Furthermore, the automatic Markov Chain Monte Carlo code successfully identified the noise level within a small frequency window of interest in these data sets. We note that a parallel approach to the data analysis of binary inspiral signals is being developed by Röver et al, with a Markov Chain Monte Carlo method that can successfully post-process a candidate signal generated from the true parameters of the signal. Signal detection in a pre-processing stage is currently being tested within parallel tempered MCMC methods and/or time-frequency analyses [19] . We identify two prominent and promising features of our pipeline: its ability to determine good initial conditions for the MCMC and its ability to run the MCMC automatically. As we have demonstrated in this paper, the width of the marginalised posterior density for the frequency parameter is extremely narrow. It is therefore vital that the initial estimate of the frequency is within this region, as the almost flat structure of the posterior PDF outside this region gives little to no information on the location of the peak. The chances of finding the mode through a random sampling are decreased further still with a larger prior range for the parameter. Adding an F -statistic search as the first stage in the pipeline solves this problem, since the frequency and position in the sky are recovered very accurately, to within the limits of the posterior probability region of interest, before the MCMC performs post-processing and parameter estimation. The automatic feature of the MCMC ensures a successful post-processing for the other astrophysical parameters that may have been located outside the posterior region of interest by the F -statistic approach, as in the case for the amplitude of Challenge 1.1.1a. Convergence is aided by the ability of our code to increase or decrease sampling step-sizes according to its experience of the sampling quality of the posterior during the burn-in phase.
We are working on an extension of the pipeline as shown in this document to successfully tackle multi-source data sets, required for the second round of the MLDC. Current work includes the exploration of our grid-based coherent search on such data streams in order to automatically identify the most promising individual candidate signals, and the implementation of an automatic Reversible Jump Markov Chain Monte WD MLDC1 10 Carlo routine (e.g. as already demonstrated in [10] ) to find the trans-dimensional probability density functions of the parameters of an unknown total number of signals. We highlight that the noise level determination presented here already serves as a key ingredient to round 2, where the simulation of a galactic white dwarf binary population introduces additional confusion noise levels from unresolvable sources.
Nelson Christensen's work was supported by the National Science Foundation grant PHY-0553422 and the Fulbright Scholar Program. Alberto Vecchio's work was partially supported by the Packard Foundation and the National Science Foundation. The University of Auckland group was supported by the Royal Society of New Zealand Marsden Fund Grant UOA-204.
References [1] Bender B L et al 1998 LISA Pre-Phase A Report; Second Edition, MPQ 233 [2] Nelemans G, Yungelson L R and Portegies Zwart S F 2001 Astron. and Astrophys. 375 890 [3] Farmer A J and Phinney E S 2003 Mon. Not. R. Astron. Soc 346 1197 [4] Jaynes E T Probability theory: The logic of science 2003 Cambridge University Press [5] Gregory P C Bayesian logical data analysis for the physical sciences 2005 Cambridge University Press [6] Gelman A, Carlin J B, Stern H, and Rubin D B Bayesian data analysis 1997 Chapman & Hall CRC Boca Raton [7] Arnaud K A et al 2006 AIP Conf. Proc. 873 619 Preprint gr-qc/0609105 [8] Arnaud K A et al 2006 AIP Conf. Proc. 873 625 Preprint gr-qc/0609106 [9] Mock LISA Data Challenge Task Force, "Document for Challenge 1," svn.sourceforge.net/viewvc/lisatools/Docs/challenge1.pdf. [10] Stroeer A, Gair J and Vecchio A 2006 Automatic Bayesian inference for LISA data analysis strategies Preprint gr-qc/0609010 [11] Umstätter R, Christensen N, Hendry M, Meyer R, Simha V, Veitch J, Vigeland S and Woan G 2005 Phys Rev D 72 022001 [12] Wickham E D L, Stroeer A and Vecchio A 2006 Class Quantum Grav 23 819 [13] Bloomer E et al Report on MLDC1 available at http://astrogravs.nasa.gov/docs/mldc/round1/entries.html [14] Arnaud K A et al 2007 Preprint gr-qc/0701139 [15] Arnaud K A et al 2007 Preprint gr-qc/0701170 [16] Crowder, J., and Cornish, N. J. 2007 Phys. Rev. D 75 043008 [17] Crowder J, Cornish N J and Reddinger J L 2006 Phys. Rev. D 73 063011 [18] Cornish N J and Crowder J 2005 Phys. Rev. D 72 043005 [19] Röver C et al in this volume [20] Jaranowski P, Królak A and Schutz B F 1998 Phys. Rev. D 58 063001 [21] Prix R and Whelan J 2006 Technical note [22] Brady P R, Creighton T, Cutler C and Schutz B F 1997 Phys. Rev. D 57 2101 [23] Prince T A, Tinto M, Larson S L and Armstrong J W 2002 Phys. Rev. D 66 122002 [24] LAL Home Page: http://www.lsc-group.phys.uwm.edu/daswg/projects/lal.html [25] Cornish N J, Larson S L 2003 Phys. rev. D 67 103001 [26] Atchade Y F, Rosenthal J S 2005 Bernoulli 11 815-828 | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The data obtained from LISA [1] will contain a large number of white dwarf binary systems across the whole observational window [2] . At frequencies below ∼ 3 mHz the sources are so abundant that they produce a stochastic foreground whose intensity dominates the instrumental noise [3] . The closer (and louder) sources will still be sufficiently bright to be individually resolvable. Above ∼ 3 mHz the sources become sufficiently sparse in parameter space (and in particular in the frequency domain) that the detectable sources become individually resolvable. The identification of white dwarfs in the LISA data set represents one of the most interesting analysis problems posed by the mission: the total number of signals in the data set is unknown, the effective noise WD MLDC1 2 level affecting the measurements is not easily estimated from the data streams, and there is a large number of overlapping sources to the limit of confusion.\n\nBayesian inference provides a clear framework to tackle such a problem [4, 5, 6] . Some of us have carried out exploratory studies and \"proof of concept\" analyses on simplified problems that have demonstrated that Bayesian techniques do indeed show good potential for LISA applications [11, 10, 12] . Similarly other authors have successfully implemented techniques using Bayesian inference [18, 17, 16] . In this paper we present the first results of an end-to-end analysis pipeline developed in the context of the Mock LISA Data Challenges that has evolved from our earlier work. This pipeline is applied to the simplest single-source challenge data sets 1.1.1a and 1.1.1b and all the results presented here are obtained after the release of the key files. In a companion paper [19] , we present results that we have obtained for the analysis of the data sets containing gravitational radiation from a massive-black-hole binary inspiral. Our group submitted an entry for the MLDC analysing the blind data set 1.1.1c [13, 14]: however that result suffered from the fact that the pipeline was not complete, the analysis code was inefficient and we encountered hardware problems with the Beowulf cluster used to perform the analysis.\n\nThe results that we present here are obtained with a two-stage end-to-end analysis pipeline: (i) we first process the data set with a grid-based coherent algorithm to identify candidate signals; (ii) we then follow up the candidate signals with a Markov Chain Monte Carlo code to obtain probability density function on the model parameters. Our method differs from other MCMC methods that have been proposed and applied to the MLDC data in the context of white dwarf binaries [18, 17, 16] : the MCMC is not used to search, but only in the final stage of the analysis to produce posterior density functions of the model parameters. The noise spectral level is included as one of the unknown parameters and is estimated together with the parameters of the gravitational wave source(s)."
},
{
"section_type": "METHOD",
"section_title": "Analysis method",
"text": "In this section we describe the two stage approach that we have adopted for the analysis. The signal produced by a white dwarf binary system is modelled as monochromatic in the source reference frame, following the conventions adopted in the first MLDC [7, 8, 9] . It is described by 7 parameters: ecliptic latitude ϑ e and longitude ϕ e , inclination ι and polarisation angle Ψ, frequency at a reference time f 0 and corresponding overall phase Φ 0 and amplitude A.\n\nThe data distributed for the MLDC are the three TDI v1.5 Michelson observables X, Y and Z ‡. From those we construct the two orthogonal TDI outputs\n\nA = (2X -Y -Z)/3 ( 1\n\n) E = (Z -Y )/ √ 3 ( 2\n\n)\n\n‡ In our MCMC analysis we use the data set produced using the LISA Simulator.\n\nWD MLDC1 3 by diagonalizing the noise covariance matrix following the procedure presented in [23] . The noise affecting the channels A and E is uncorrelated and described by the one-sided noise spectral density S n (f ). We model the LISA response function in the low frequency limit in order to improve the computational efficiency of our analysis.\n\n2.1. First stage: Grid based search\n\nThe first stage of the pipeline consists of a fast search of the data for the best matched filter based on the well-known F -statistic algorithm, first developed for triaxial pulsar signals in the context of ground-based observations [20] . This exploits the Fast Fourier Transform to perform matching in the frequency domain to templates which are generated at an array of fixed points in the parameter space. The data from an individual detector in the frequency domain d(f ) is supposed to contain a signal plus Gaussian noise, d(f ) = h(f ) + ñ(f ). We define the logarithmic likelihood as a measure of match, as given by log L ≈ ( d| h) -1 2 ( h| h) with (•|•) denoting the scalar product as defined in [20] . A single signal in the F -statistic algorithm is re-parameterised as a linear function of four orthogonal variables, and the frequency f 0 . The detection statistic is based on four parameters A F , B F , C F and D F , found by integrating over the response functions a(t) and b(t) to the two polarisation states of the gravitational wave signal [20] ,\n\nA F = 2 T obs T obs 0 a(t) 2 dt ( 3\n\n) B F = 2 T obs T obs 0 b(t) 2 dt ( 4\n\n) C F = 2 T obs T obs 0 a(t)b(t)dt, ( 5\n\n) D F = A F B F -C 2 F ( 6\n\n)\n\nT obs denotes the total observed time for the data set being analysed. The optimal detection statistic 2F , which is pre-maximised over the nuisance parameters h 0 , ι, φ 0 and ψ is\n\n2F = 8 S n (f )T obs B F |F a | 2 + A F |F b | 2 -2C F × R(F a F b ) D F . ( 7\n\n)\n\nF a and F b are the demodulated Fourier transforms of the data, iΦ(t) dt; F b = T obs 0 d(t)b(t)e -iΦ(t) dt, (8) Φ(t) is the phase of the gravitational wave signal, as is described in [22] .\n\nF a = T obs 0 d(t)a(t)e -\n\nAs the LISA array moves in space, the frequency f 0 is affected by Doppler modulations. This modulation changes with differing position of the source in the sky, implying the need to recalculate the modulations and thus a(t) and b(t) for each sky position that is tested -a significant factor in the performance of this approach. The differing modulation structure however also allows us to estimate the location of the WD MLDC1 4 source in the sky by maximising the 2F value. The resolution possible on the sky with this method is not as good as from a full Bayesian posterior probability calculation as performed in the parameter estimation stage, as shown in an example for Challenge 1.1.1a in figure 1 . Nevertheless, since this statistic can be computed fairly quickly it serves as a useful way of finding initial values to feed into the MCMC routine, as adopted within the pipeline. The resolution achievable on the sky increases with frequency, which implies that the mismatch between filter and signal falls off more rapidly at higher frequencies, requiring a greater number of templates to cover the sky. Therefore for challenge 1.1.1b at f ≈ 3 mHz a sky grid of size 5,752 points was used, in comparison with 765 points for challenge 1.1.1a at f ≈ 1 mHz.\n\nThe F -statistic search was implemented using the LIGO \"Lalapps\" suite of software [24] , in which the pulsar search code was modified by Reinhard Prix and John Whelan to use the LISA response function for the TDI variables X, Y , and Z [21] . These input data streams were given in the form of Short Fourier Transforms, each of length one day, created from the MLDC1 challenge data. For each challenge the full specified range of frequencies was searched for the signal as it would be in a blind search. The code was run on a single CPU and executed in a few hours, with the run-time increasing at higher frequency due to the higher resolution of sky and frequency grid that had to be used. The candidate chosen to pass to the MCMC stage was simply that which triggered the highest value of 2F ."
},
{
"section_type": "OTHER",
"section_title": "Second stage: Markov Chain Monte Carlo follow-up",
"text": "According to Bayes' theorem, the posterior probability, p( m| d) of a model m given the data d depends on the prior distribution p( m), containing the information known before the analysis, the likelihood L( d| m) of the model and a normalisation factor p( d)\n\np( m| d) = L( d| m)p( m) p( d) ( 9\n\n)\n\nThe posterior probability density function shows the joint probability density of given values of parameters of the model m, conditional on the data d. We implemented Bayes' theorem using data in the form of TDI variables A and E and modelled our template according to the Long Wavelength Approximation directly in the Fourier domain [25] to gain computational speed. The logarithmic likelihood L( d| m) in this stage explicitly included its dependence on the one-sided noise spectral density S n (f )\n\nlog L( d| m) = const. - 1 2 log S n (f ) -( d -h| d -h), ( 10\n\n)\n\nshown here for either A or E, with the combined likelihood as sum of the individual likelihoods. We restricted our analysis to a sufficiently narrow frequency window in order to be able to approximate the noise spectral density as constant, S n (f ) = S 0 . This window was set as the interval in frequency that contains at least 98% of the power of our WD MLDC1 5 Ecliptic Longitude Ecliptic Latitude 2F as a function of sky position, at a frequency 0.001063 Hz 1 2 3 4 5 6 -1.5 -1 -0.5 0 0.5 1 1.5 2F 500 1000 1500 2000 2500 Figure 1. The variation of 2F values for the search for unknown signal 1.1.1a, as a function of sky position, parameterised by ecliptic latitude β and longitude λ. The distribution is multi-modal and non-Gaussian, and has a poor resolution in comparison with that can be achieved with the MCMC and a Bayesian likelihood, but by finding the maximum it serves well as a starting point for the more refined parameter estimation below.\n\nmodel m, with the interval's upper and lower limits given by f ±(2/year)(5+2πf 0 AU/c) [25]. S 0 is therefore an additional parameter to be inferred within the model m in Eq. 10.\n\nWe implemented an automatic Random Walk Metropolis sampler (Stroeer & Vecchio 2007, in. prep.) to sample from the posterior probability density function in form of a Markov chain. Metropolis sampling eliminates the need to explicitly calculate the normalisation constant in Bayes' theorem, and the evolving Markov chain gives easy access to joint as well as marginalised posterior density distribution. The sampler was started from the parameter set which triggered the highest value of 2F in our grid based coherent run of the analysis (see former section). The automated function of the Metropolis sampling is achieved by controlling the sampling step-size with adaptive acceptance probability techniques [26] . The sampler therefore does not depend on assumptions about the signal in the data set in order to perform successfully and reliably; it develops a suitable algorithm and approach by itself based on the properties of the likelihood as found on the fly, in the initial steps of the sampler. The length of our Markov chain was pre-set to 10 6 , with the initial 10 4 chain states discarded as the \"burn-in\" phase of our sampler. The runtime for one data analysis run is 5 hours on a single 2 GHz CPU on the Tsunami cluster of the University of Birmingham. WD MLDC1 6 Figure 2. The marginalised posterior probability density functions of the eight unknown parameters -the seven parameters that describe the signal and the noise spectral density S 0 -for the the challenge data set 1.1.1a. The vertical black solid line denotes the true value of the parameter (for the polarisation angle the true value modulo π/2), and the grey dashed line the initial value for the MCMC analysis as determined by the template of the first-stage that produces the maximum value of the F -statistic. In the case of the noise spectral density the first stage of the analysis does not provide an estimate; the true value of this parameter is taken to be the value of the instrumental noise spectrum used to generate the data set and provided in [9]. WD MLDC1 7 Figure 3. The marginalised posterior probability density functions of the eight unknown parameters for the the challenge data set 1.1.1b. Labels are as in Figure 2."
},
{
"section_type": "RESULTS",
"section_title": "Results",
"text": "We found that the most promising candidate signal from the F -statistic search already matched the true embedded signal to high accuracy, particularly in frequency and sky location. Our MCMC sampler, as a post-processing unit, thus only needed 1000 iterations to burn in and to establish a reliable sampling from the posterior. The marginalised posteriors are shown in Figs. 2 and 3. We found, as seen in latter figures, that the MCMC sampler further refined the initial guesses from the F -statistic, as measured by the absolute difference between the true value of a given parameter and the median of the marginalised posterior recovered for that parameter. Table 1\n\nWD MLDC1 8 Table 1. Details about the results from Challenge 1.1.1a and Challenge 1.1.1b. S 0 , the constant one-sided noise spectral density within our narrow frequency window, is compared to the true one sided noise spectral density at the true frequency of the signal, Ψ is given modulo π/2. Int 90 denotes the minimum interval to include 90% of MCMC states for given parameter, ∆mode denotes the absolute difference between the true value of a signal parameter and the mode of its recovered posterior; ∆median and ∆mean denote the equivalent absolute difference for median and mean of the posterior respectively; σ denotes the sampled standard deviation of the posterior as derived from the median. We further quote the signal-to-noise ratio (SNR) for a template using the true values of the source and the recovered values of the data analysis run, as derived from the median of the individual posterior distributions, and the correlation C between these two templates. Int 90 ∆mode ∆median ∆mean σ Challenge 1.1.1a S 0 10 -41 Hz -1 (3.53257, 4.72639) -0.42084 -0.440278 -0.452456 0.36704 ϑ e / rad (0.958409, 1.03165) -0.0147383 -0.0149381 -0.0148725 0.0222861 ϕ e / rad (5.05376, 5.13528) -0.00550139 -0.00569547 -0.00579889 0.0247886 Ψ/ rad (1.32475, 0.500553) 0.1768 0.1823 0.1902 0.1908 ι/ rad (0.097761, 1.0008) -0.0459747 0.190001 0.23459 0.295211 A/10 -22 (1.61976, 2.67967) 0.664371 0.358844 0.298978 0.368524 f 0 / mHz (1.06273, 1.06273) -1.19664e-06 -1.22207e-06 -1.22259e-06 1.04422e-06 Φ 0 / rad (3.10668, 5.808) -0.164989 0.00998525 0.229659 0.829146 SNR true = 51.024497 recovered = 50.648600 C true vs. recovered = 0.99689 Challenge 1.1.1b S 0 10 -41 Hz -1 (0.876833, 1.38959) -0.0679571 -0.0906557 -0.0996144 0.16017 ϑ e / rad (-0.121611, 0.0116916) -0.0343353 -0.151185 -0.150328 0.0406552 ϕ e / rad (4.60969, 4.63537) 0.00265723 0.00305564 0.00302203 0.00779893 Ψ/ rad (0.246328, 0.362409) 0.0301541 0.0311747 0.0311268 0.0353938 ι/ rad (1.22036, 1.33338) -0.0430412 -0.040458 -0.0394818 0.0348383 A/10 -22 (0.45001, 0.542454) -0.016442 -0.0151921 -0.0149907 0.0281154 f 0 / mHz (3.00036, 3.00036) 3.1221e-07 2.49289e-07 2.42807e-07 8.18111e-07 Φ 0 / rad (5.83869, 6.19411) 0.137219 0.119301 0.119921 0.502384 SNR true = 36.587444 recovered = 37.368806 C true vs. recovered = 0.97897\n\nshows details of the statistics of recovered posterior distributions. We highlight that the majority of the true values of the parameters are within one standard deviation of the median of the posterior, with a small percentage within two sampled standard deviations. In addition, every true value of a parameter of the signal is within the minimum interval of the posterior to cover 90% of all MCMC state values. Recovered signal-to-noise ratios are measured as SNR = (s|h)/ (h|h), and the match C = (h true |h med )/ (h true |h true ) (h med |h med ) between a template constructed from the true values and a template from the median values of the individual posterior distributions, yielding a correlation that is always higher than 0.97. Noise levels are determined accurately and within 1 to 1.5 sampled standard deviations. Nevertheless we note that WD MLDC1 9 our run on Challenge 1.1.1a shows a lower match and higher differences between true value and recovered value of parameters as compared to the run on Challenge 1.1.1b. It also exhibits tailing posterior distributions in inclination and amplitude, although the SNR of Challenge 1.1.1a is twice the value of Challenge 1.1.1b."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusions",
"text": "We have presented a new approach to LISA data analysis in the form of an end-to-end pipeline. We first detected and identified candidate signals in the LISA data stream with a grid-based coherent algorithm, and then post-processed the most promising candidate signals with an automatic Markov Chain Monte Carlo code to obtain probability densities for the model's parameters. We demonstrated successful identification and post-processing of the signals from the double white dwarf single source MLDC data sets 1.1.1a and 1.1.1b. Furthermore, the automatic Markov Chain Monte Carlo code successfully identified the noise level within a small frequency window of interest in these data sets. We note that a parallel approach to the data analysis of binary inspiral signals is being developed by Röver et al, with a Markov Chain Monte Carlo method that can successfully post-process a candidate signal generated from the true parameters of the signal. Signal detection in a pre-processing stage is currently being tested within parallel tempered MCMC methods and/or time-frequency analyses [19] . We identify two prominent and promising features of our pipeline: its ability to determine good initial conditions for the MCMC and its ability to run the MCMC automatically. As we have demonstrated in this paper, the width of the marginalised posterior density for the frequency parameter is extremely narrow. It is therefore vital that the initial estimate of the frequency is within this region, as the almost flat structure of the posterior PDF outside this region gives little to no information on the location of the peak. The chances of finding the mode through a random sampling are decreased further still with a larger prior range for the parameter. Adding an F -statistic search as the first stage in the pipeline solves this problem, since the frequency and position in the sky are recovered very accurately, to within the limits of the posterior probability region of interest, before the MCMC performs post-processing and parameter estimation. The automatic feature of the MCMC ensures a successful post-processing for the other astrophysical parameters that may have been located outside the posterior region of interest by the F -statistic approach, as in the case for the amplitude of Challenge 1.1.1a. Convergence is aided by the ability of our code to increase or decrease sampling step-sizes according to its experience of the sampling quality of the posterior during the burn-in phase.\n\nWe are working on an extension of the pipeline as shown in this document to successfully tackle multi-source data sets, required for the second round of the MLDC. Current work includes the exploration of our grid-based coherent search on such data streams in order to automatically identify the most promising individual candidate signals, and the implementation of an automatic Reversible Jump Markov Chain Monte WD MLDC1 10 Carlo routine (e.g. as already demonstrated in [10] ) to find the trans-dimensional probability density functions of the parameters of an unknown total number of signals. We highlight that the noise level determination presented here already serves as a key ingredient to round 2, where the simulation of a galactic white dwarf binary population introduces additional confusion noise levels from unresolvable sources."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgements",
"text": "Nelson Christensen's work was supported by the National Science Foundation grant PHY-0553422 and the Fulbright Scholar Program. Alberto Vecchio's work was partially supported by the Packard Foundation and the National Science Foundation. The University of Auckland group was supported by the Royal Society of New Zealand Marsden Fund Grant UOA-204.\n\nReferences [1] Bender B L et al 1998 LISA Pre-Phase A Report; Second Edition, MPQ 233 [2] Nelemans G, Yungelson L R and Portegies Zwart S F 2001 Astron. and Astrophys. 375 890 [3] Farmer A J and Phinney E S 2003 Mon. Not. R. Astron. Soc 346 1197 [4] Jaynes E T Probability theory: The logic of science 2003 Cambridge University Press [5] Gregory P C Bayesian logical data analysis for the physical sciences 2005 Cambridge University Press [6] Gelman A, Carlin J B, Stern H, and Rubin D B Bayesian data analysis 1997 Chapman & Hall CRC Boca Raton [7] Arnaud K A et al 2006 AIP Conf. Proc. 873 619 Preprint gr-qc/0609105 [8] Arnaud K A et al 2006 AIP Conf. Proc. 873 625 Preprint gr-qc/0609106 [9] Mock LISA Data Challenge Task Force, \"Document for Challenge 1,\" svn.sourceforge.net/viewvc/lisatools/Docs/challenge1.pdf. [10] Stroeer A, Gair J and Vecchio A 2006 Automatic Bayesian inference for LISA data analysis strategies Preprint gr-qc/0609010 [11] Umstätter R, Christensen N, Hendry M, Meyer R, Simha V, Veitch J, Vigeland S and Woan G 2005 Phys Rev D 72 022001 [12] Wickham E D L, Stroeer A and Vecchio A 2006 Class Quantum Grav 23 819 [13] Bloomer E et al Report on MLDC1 available at http://astrogravs.nasa.gov/docs/mldc/round1/entries.html [14] Arnaud K A et al 2007 Preprint gr-qc/0701139 [15] Arnaud K A et al 2007 Preprint gr-qc/0701170 [16] Crowder, J., and Cornish, N. J. 2007 Phys. Rev. D 75 043008 [17] Crowder J, Cornish N J and Reddinger J L 2006 Phys. Rev. D 73 063011 [18] Cornish N J and Crowder J 2005 Phys. Rev. D 72 043005 [19] Röver C et al in this volume [20] Jaranowski P, Królak A and Schutz B F 1998 Phys. Rev. D 58 063001 [21] Prix R and Whelan J 2006 Technical note [22] Brady P R, Creighton T, Cutler C and Schutz B F 1997 Phys. Rev. D 57 2101 [23] Prince T A, Tinto M, Larson S L and Armstrong J W 2002 Phys. Rev. D 66 122002 [24] LAL Home Page: http://www.lsc-group.phys.uwm.edu/daswg/projects/lal.html [25] Cornish N J, Larson S L 2003 Phys. rev. D 67 103001 [26] Atchade Y F, Rosenthal J S 2005 Bernoulli 11 815-828"
}
] | |
arxiv:0704.0051 | 0704.0051 | 1 | 89bfe7bf5b6e413c61845825ba2507e7c5d1bbea46c828b96f384fd5323b3afc | Visualizing Teleportation | A novel way of picturing the processing of quantum information is described,
allowing a direct visualization of teleportation of quantum states and
providing a simple and intuitive understanding of this fascinating phenomenon.
The discussion is aimed at providing physicists a method of explaining
teleportation to non-scientists. The basic ideas of quantum physics are first
explained in lay terms, after which these ideas are used with a graphical
description, out of which teleportation arises naturally. | [
"Scott M. Cohen"
] | [
"physics.ed-ph",
"quant-ph"
] | physics.ed-ph | [] | 2007-04-02 | 2026-02-26 | One of the most exciting and fastest-growing fields of physics today is quantum information. Especially since the discovery by Shor [1, 2] that there exist calculations for which a quantum computer is apparently far more efficient than a classical computer, interest in understanding quantum information has increased at an impressive rate. One widely publicized discovery that has emerged from work in this field is teleportation [3] . While not precisely equivalent to the process enjoying widespread fame amongst fans of Star Trek ("Beam me up, Scotty"), the phenomenon referred to here is nonetheless fascinating, and perhaps even astonishing. The reason for the widespread publicity of this rigorously proven (and experimentally tested [4, 5, 6, 7, 8] , though not yet unambiguously demonstrated) scientific prediction is almost certainly in large part due to the fact that it shares the same name as the just-mentioned, intriguing idea from science-fiction.
The usual way of describing teleportation is through mathematical equations, and this mathematics is relatively straightforward, as has been amply demonstrated elsewhere [3, 9] . Hence, an understanding of this phenomenon is accessible to physicists, other scientists, and those possessing a reasonably strong level of mathematical skill. There does, on the other hand, seem to be a good deal of misunderstanding of teleportation amongst non-scientists, with the notion floating around that the amazing phenomenon shown regularly in episodes of Star Trek -that is, of material objects being teleported from one place to another -has actually turned out to be possible in real life. Nothing could be further from the truth, of course, so we are left wondering how to rectify this unfortunate state of affairs. The question I address here is the following: can the true (scientific) phenomenon of teleportation be understood by others, those without much skill in mathematics? The usual explanations will certainly fail in this regard, even if carefully presented by a competent physicist, because mathematics has a well-known tendency to scare people away, and in any case, the mathematics of teleportation is not all that simple. The paper is addressed to physicists possessing a solid understanding of quantum physics (including graduate students), with the aim to provide a method by which such a physicist can explain teleportation to someone who is not mathematically inclined. Thus, the objective is ultimately, though indirectly, to educate the general public about teleportation, and by extension, quantum mechanics itself. The approach involves only the most basic ideas about quantum physics, and while it does not entirely avoid mathematical expressions, it uses only the simplest mathematics (one only needs to accept that certain objects are either 0 or 1) and relies almost entirely on "pictures", allowing the layperson to visualize -and thus, understand -what is happening.
In the following sections, I will describe my method of directly visualizing teleportation. These sections are written as if addressed to the layperson. The next section explains the probabilistic nature of quantum physics by considering "quantum coins", which are examples of two-level systems. This section describes how one should think about measurements, what is meant by probabilities for classical systems, and then how these ideas can be used to describe quantum systems. Then, in Section III, I present my graphical approach to understanding the dynamics of quantum information processing, which is then used in Section III B to explain in pictures how teleportation of quantum states is possible. One of the crucial observations will be that a shared entangled state on, say, systems a and b, provides the parties with multiple "images" of the state of an additional system A. The ability to manipulate these images -independently by each party, and differently from one image to the next -is what allows teleportation to be accomplished. More generally, these ideas provide important insights into why entanglement is a valuable resource, as I have described in detail elsewhere, and they have been useful in understanding other aspects of quantum information 2 processing [10, 11] .
Perhaps the most fundamental aspect of quantum theory is that it can only make predictions in terms of probabilities. In general even if one has a complete description of the state of a quantum system, one will not know ahead of time what the outcome of a given measurement will be. This is in direct contradiction with our everyday experience, which we refer to as "classical". For example, a flipped classical coin which lands heads ("heads" is then a complete description of the state of this coin), is known with certainty to be heads, and also with certainty to not be tails. That is, if we know the state of a classical coin (in this case "heads"), we can predict with certainty the answer to any reasonable question we choose to ask (or "measure") about that coin (for example, "Is it tails?"). We therefore need to understand what is meant by the "state" of a quantum system and how this state relates to probabilities and outcomes of measurements. The following definition of a measurement will be adequate for our purposes.
Definition: A measurement is a procedure that provides answers to a collection of yes-no questions, which is both mutually exclusive (when the answer to one of the questions is "yes", the answer to all the others is "no") and complete (all possibilities are included; that is, one of the questions will always be answered in the affirmative). The single question that receives the "yes" answer is referred to as the outcome of the measurement.
For example, since a classical coin is either heads or tails, and these two possibilities are mutually exclusive, a measurement on a classical coin is a procedure that answers the two questions "Is it heads?" and "Is it tails?" Since the coin will always be one or the other, there will always be a "yes" answer to one of these questions, and then the other question is always answered "no". Hence these two questions do indeed constitute a measurement according to the above definition. If "Is it heads?" is answered affirmatively, then "heads" is the outcome of the measurement.
It turns out that these two questions also constitute a measurement on quantum coins. However, in contrast to the classical case in which this is the only possible measurement, there is a vast array of possible measurements on quantum coins. This will become clearer from the discussion in the following sections, where we introduce a compact way of describing these things, a way commonly used in quantum mechanics.
Consider again a flipped classical coin. The coin lands either heads or tails. It will be useful to use a somewhat abbreviated notation: |H for heads and |T for tails. The statement that "if it is heads, it is not tails" (that is, has zero probability of being tails) will be represented as
T |H = 0.
The left-facing bracket |H represents the known initial state ("It is heads.") and the right-facing bracket T | represents the question ("Is it tails?"). The number (0) appearing on the right-hand side of the equal sign then gives the probability that with this initial state, the answer to this question will be yes. For the above example, we have that the probability is 0, which is as expected since when the coin is H it will never be T . Note that it is useful to use the left-and right-facing brackets, so that we can easily read off what is the initial state and what is the question being asked about it. Simply writing T H = 0 in the above equation would lead to confusion when we discuss two coins (see below), which might have an initial state where one is tails, the other heads, represented by |T H .
Perhaps an even more trivial statement "if it is heads, then it is heads" (with certainty, or with probability one), will similarly be represented as H|H = 1.
Again, the right-facing bracket contains the question H|, or "Is it heads?", and the fact that the expression is equal to 1 indicates that the answer to this question will always be "yes" when the initial state is |H . These statements are trivial because if we know the state of a classical coin, we can predict with certainty whether it will be heads or tails when we look at it. Although the remaining equations will look a bit more involved, the only mathematics the reader need understand is contained in the above two equations, along with two others that are almost exactly the same. The discussion in the remainder of this paper will follow from the four simple statements, H|H = 1, T |T = 1, T |H = 0, H|T = 0.
Next let us consider two coins. In this case, a complete list of mutually exclusive possibilities is HH, HT, T H, T T . We can make statements in exactly the same way we did above, for example "if they are HH, then they are not HT ", 3 which in our notation is written
H 1 T 2 |H 1 H 2 = H 1 |H 1 × T 2 |H 2 = (1) × (0) = 0,
where the subscripts (1, 2) have been inserted for clarity to indicate which coin is which. Note that in this equation, we have equated the expression H 1 T 2 |H 1 H 2 with the product of two expressions, H 1 |H 1 and T 2 |H 2 . This is because any question about the two coins jointly is the same as two questions, one about each of the coins separately.
It is obviously also true that "if they are HH, then they are HH", so
H 1 H 2 |H 1 H 2 = H 1 |H 1 × H 2 |H 2 = (1) × (1) = 1.
For three coins, there are eight possibilities (HHH, HHT, HT H, T HH, HT T, T HT, T T H, T T T ) and the same notation will readily account for this case, as well. We will not need to consider more than three coins here, though it is in principle straightforward to do so.
Quantum coins behave very differently as compared to their classical counterparts, and quantum probabilities must be understood in very different ways. We still have heads and tails, |H and |T , as possible states of a quantum coin. We refer to these two states as being "orthogonal" to each other, by which we simply mean that they are mutually exclusive: if the quantum coin is H, it is definitely (with certainty) not T , and vice-versa. We note that the four equations appearing in the previous section are equally true for both quantum and classical coins. However, there now exist some very strange possibilities. If I were to suggest that a classical coin can be both H and T at one and the same time, you would be completely justified in thinking I'd gone slightly crazy. I am going to tell you, though, that at least in a certain (though very real) sense, this is the case for quantum coins (though you may still wonder a bit about my sanity). The point is that, in the quantum case, it makes complete sense to ask questions such as: "If the coin is H, is it half H and half T ?"; or we can turn this around and ask "If the coin is half H and half T , is it H?" Neither of these questions makes any sense whatsoever when referred to a classical coin. On the other hand, for a quantum coin these are not only legitimate questions, but they are in fact very important ones (we do not consider the negligible possibility of a classical coin landing on its edge, and in any case this bears no relationship to what we mean by a quantum coin being half H and half T ).
To represent these questions, we can write the state (Q) of a quantum coin that is half H and half T as
|Q = 1 2 |H + 1 2 |T .
Then the answer to the question, "If the coin is half H and half T , is it H?" is answered by the equation,
H|Q = H| 1 2 |H + 1 2 |T = 1 2 H|H + 1 2 H|T = 1 2 (1) + 1 2 (0) = 1 2 ,
which should be interpreted as meaning "yes, with probability 1/2", implying also "no, with probability 1-1/2 = 1/2" [In quantum mechanics, it is actually the square of the object on the left-hand side of the foregoing equation that represents the probability, rather than that object itself, which is known as the "probability amplitude"; however, although the difference between probabilities and probability amplitudes is crucial to the understanding of quantum mechanics, I have chosen in the present discussion to overlook this distinction for the benefit of the layperson to whom these ideas are aimed, as they would only serve to complicate matters, causing unnecessary confusion amongst the intended audience]. The left-facing bracket |Q represents the known initial state, and the right-facing bracket H| represents the question ("Is it heads?"). The number 1/2 appearing on the right-hand side of the last line then gives the probability that with this initial state, the answer to this question will be yes. The point to understand here is that even though we have a complete description (Q) of the state of the quantum coin, we do not generally know in advance whether the coin will be H or T when we look at it. We can only predict in terms of probabilities: if we perform this experiment many times, half the time the answer will be yes and the other half of the time it will be no. Furthermore, there are many more questions we can ask in the quantum, as compared to the classical, case. We are 4 no longer restricted to asking "is the coin H?" or "is it T ?", but we can ask other questions, such as the reverse of the question we just answered,
Q|H = 1 2 H| + 1 2 T | |H = 1 2 H|H + 1 2 T |H = 1 2 (1) + 1 2 (0) = 1 2 .
We see that the question "If the coin is H, is it half H and half T ?" has the same answer as the previous question: "yes, with probability 1/2; and no, with probability 1/2." We note that in the remainder of the paper, instead of phrasing questions as "is the coin half H and half T ?", we instead ask whether it is "equal parts" H and T . While there is no real difference between these two questions, this rephrasing allows us to simplify the notation by dispensing with the factors of 1/2 that have appeared in the above discussion. In doing so, the equations will not yield the same numbers as probabilities for the various questions, but this will not hamper the presentation since the numerical values of the probabilities are not crucial to the ideas we wish to convey: we just need to remember that certain objects are equal to 1 and others are equal to 0.
What exactly do we mean by teleportation in the context of quantum information? It is not a material object that is being teleported, but rather the state of a quantum system. We will assume that the system is a quantum coin, with a complete set of mutually exclusive (orthogonal) states being "heads" and "tails", which we may denote as |H and |T . Suppose Alice and Bob are physicists in locations widely separated from each other. They each have a quantum coin -labeled a and b, respectively -and these two coins are in the state
|B 0 ab = |H a H b + |T a T b ,
where the subscripts used here refer to system a (b) in Alice's (Bob's) possession. This state of two quantum coins has a very strange property, which is known as entanglement, and the state itself is an example of a maximally entangled state. Entanglement is a rather strange sort of correlation between quantum systems, which manifests itself in the state B 0 by the fact that neither system a nor b can be considered to have a definite "state of its own" independent of the other system: whatever is the state of coin a, coin b will have the same state, but one cannot say anything about the state of either coin independent of the other one. It is this property of entanglement that is credited with enabling Alice and Bob to accomplish teleportation.
Alice is given another coin (system A), prepared in a state
|S A = c H |H A + c T |T A
with arbitrary coefficients c H and c T that are completely unknown to her and to Bob. If c H = 1/2 and c T = 1/2, we have the case discussed in the previous section, where the coin is equally likely to be found to be H or T . For other values of these coefficients, the two possibilities will in general not be equally likely. Alice's task is to perform operations on the systems in her possession (a and A) in such a way that Bob will end up with his system (b) in precisely the state |S b , which is the same state as |S A , but now on the distant system b. It turns out that this task can be accomplished if Alice communicates information to Bob (perhaps via a telephone) about what she ended up doing to her systems, after which Bob performs a rather simple quantum operation, dependent on the information obtained from Alice, on system b. An important point to understand in what follows is that nothing either of them does in this process provides even the slightest information about the coefficients c H and c T , so the state (S) that has been teleported remains completely unknown to the parties. This aspect of teleportation becomes even more amazing if one considers the amount of information that is conveyed: the information contained in a quantum state is far greater than the amount actually transmitted from Alice to Bob via the telephone (as we will see below, the amount transmitted via the telephone is two classical bits, enough to convey which one of four possibilities has been chosen). True, the classical information one can encode in a two-level quantum system cannot exceed one bit (one bit is the amount of information needed to choose between two possibilities, such as |H and |T ). But if Alice wanted to tell Bob how to create the state in his own lab by communicating with him over a phone line, this would require an infinite amount of classical information; that is, enough information to completely describe the arbitrary numbers, c H and c T (it is infinite because 5 one of these numbers might well be an irrational number such as π, having a decimal expansion that is unending, never repeating itself). Of course, Alice and Bob are both completely ignorant of what these numbers are, so even if it were possible to transmit an infinite amount of information, they don't even know what information they would need to send! Nonetheless, when they share entanglement, it is possible for the two of them, by working together, to create the unknown state on Bob's coin b with the communication of only two classical bits.
Let us now introduce the pictorial method which will be used to visualize teleportation. The simple diagrams we will use to depict states of multiple quantum coins, held by two different parties, are familiar to many researchers working in quantum information. We will now illustrate how these diagrams are used to represent quantum states, and then how they can be used to follow what happens to these coins when measurements are performed by one of the parties. Then, we will be ready to use them for visualizing teleportation.
To depict the state of a single quantum coin labeled A (standing for Alice; she will also have the other coin labeled a, while Bob's single coin is labeled b), we may use a simple box diagram,
|S A = c H |H A + c T |T A = |T A |H A c T c H .
The coefficients c H and c T appearing in the boxes indicate "how much" is in that part of the state S A of coin A. The next example illustrates the case where there are two coins (A and b) held by two different parties. Then, the state of these two coins might be
|S A H b = |H b |T b |T A |H A c T c H ,
with S A as given above. The empty squares on the right-hand side of this diagram represent the fact that system b is "not T " (has zero probability of being tails); the c H in the upper-left corner represents the probability the coins are both heads; and the c T in the lower-left, the probability Bob's coin is heads and Alice's is tails. If there are three parties involved, a three-dimensional cube could be used to represent this situation. However, it will serve our present purposes to represent both of Alice's systems along the vertical dimension of the diagram. We might have coins A and b as in the previous example, and coin a being heads, the overall state of these three coins represented as
|S A H a H b = |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H .
6 If instead the a,b systems are both T , this picture is
|S A T a T b = |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H .
Now consider what happens if we add the previous two equations together. Then our two coins a,b are "equal parts in HH and in T T ", which is what we previously referred to as the "maximally entangled state" |B 0 ab = |H a H b +|T a T b . The corresponding diagram looks like
|S A (|H a H b + |T a T b ) = |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H c T c H = |H b |T b |T a |H a S A S A .
Notice how there are now two images of the state |S A . This observation turns out to be rather useful in understanding entanglement [10, 11], but we will not need to discuss such issues here. Let us now look at how to represent measurements by use of these diagrams.
Suppose Alice and Bob share three quantum coins in the state represented in the last equation of the previous section, and Alice wants to know something about her coins. If she measures coin a and discovers it is H, then we have
H a | × |H b |T b |T a |H a S A S A = |H b |T b S A .
Recall that when the right-facing bracket H a | is attached to the left-facing one |H a on the left of this equation, we get H a |H a = 1, which "preserves" the upper row, whereas H a |T a = 0, indicating that the bottom row is annihilated (multiplied by 0), which is why it no longer appears on the far right of this equation. The interpretation is as follows: when the question "Is coin a heads?" is answered in the affirmative the other coins are left in the state |S A H b . We see how this measurement acts on both of the images simultaneously, rather than on the two independently. The upper-left image has been preserved intact, but the other image was annihilated, disappearing altogether. On the other hand, if the outcome of Alice's measurement had been that coin a was T , this would be represented as
T a | × |H b |T b |T a |H a S A S A = |H b |T b S A .
In this case, the upper-left image has disappeared and the one in the lower-right has been preserved intact. In each of these cases, the state of coin A is unchanged, but that of coin b is left in a state that corresponds directly to the outcome of Alice's measurement on a. If she discovers that coin a was H (or T ), then coin b ends up H (or T ). 7 Alternatively, she could do a measurement that includes the question "Is coin a equal parts H and T ?" If the answer to this question is yes, then
( H a | + T a |) × |H b |T b |T a |H a S A S A = |H b |T b S A S A = |S A (|H b + |T b ) ,
which is just a sum of the previous two equations (notice how after each of the three measurement outcomes we have just considered, the two images have been collapsed into a single row). Once again we see that the state of coin b ends up corresponding to the outcome of Alice's measurement on coin a. This illustrates some of the strangeness that resides in entangled states of quantum systems: no matter what measurement Alice makes on coin a and no matter what outcome she obtains from that measurement, the resulting state of coin b will correspond directly to that outcome. The way the images of S A appear in the diagram is crucial. The fact that the two start out in different rows and in different columns will be important in what is to come. If entanglement between systems a,b was absent, for example if they were in the (unentangled)
state (|H a + |T a ) |H b , then this would be represented by (recall that |S A = c H |H A + c T |T A ) |S A (|H a + |T a ) |H b = |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H c T c H = |H b |T b |T a |H a S A S A .
Under these circumstances, Bob's view of the lower image of S A is "obstructed" by the presence of the upper image; the two images effectively appear as one to him. As will become clear in the following section, the presence of entanglement between the a,b coins will be necessary for them to accomplish teleportation. We will see that it is Bob's (and Alice's) ability to "see" the two images separately, and the consequent ability for each of them to act differently on one of the images as compared to the other, that is crucial to their success.
In the next section, we turn to the task of teleporting the state S A onto Bob's coin b. To begin this process, Alice will perform a measurement that asks "joint" questions; that is, questions about both coins in her possession simultaneously. As an example, she could ask if they are both H. That is,
H A H a | × |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H c T c H = |H b |T b c H .
The c H appearing in the box on the right corresponds to the probability that the answer to this question will be "yes". More important for our purposes is to recognize that when this is the outcome of the measurement, coin b ends up H, once again a consequence of the initial entanglement between coins a and b. Now let us see how teleportation is possible.
Teleportation is accomplished with the aid of the extra systems a, b in the entangled state |B 0 ab . System A starts in state |S A , discussed above, and this is the state they will teleport. Alice will ask a set of joint questions, which 8 together constitute a measurement, about the state of the two coins in her possession, a and A. The first question she asks is whether these two coins are equal parts HH and T T . When the answer is yes, we have
( H A H a | + T A T a |) × |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H c T c H = |H b |T b c H c T .
Notice how the middle two rows are annihilated by this outcome (because these rows correspond to a situation where the two coins are different -one H and one T -whereas we are asking if they are the same), and the remaining rows are collapsed into a single row. Now, if we look carefully (or perhaps, not even so carefully) at the final diagram in this picture, we will arrive at a rather startling conclusion. We see that the state of Bob's system b
is now |S b = c H |H b + c T |T b .
That is, the unknown state |S A , originally on system A, is now on Bob's system b. Furthermore, the question asked by Alice had nothing whatsoever to do with the coefficients c H and c T , which determine what the original state of coin A was. Hence, the parties remain completely ignorant of the state S, yet that state has been successfully teleported! We are not quite finished, however, since we would like for Alice and Bob to be able to teleport no matter which joint question ends up being the outcome of Alice's measurement. Because of the probabilistic nature of the quantum world, she cannot choose the outcome of her measurement. Instead, Alice effectively asks all of the questions in her chosen measurement and then must wait for Nature to decide which question she (Nature, that is) will choose as the outcome. The nice thing about Nature is that she will tell Alice which question was chosen.
There must be four questions in a complete set of questions making up a joint measurement on coins A, a. Let me illustrate with one other question how Alice and Bob can succeed with teleportation, and then the reader is asked to believe that they can also succeed with either of the remaining two questions (these can be treated in a very similar way to the one shown here [12] ). The second question is: Are coins A, a equal parts T H and HT ? The corresponding diagram is
( T A H a | + H A T a |) × |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H c T c H = |H b |T b c T c H .
Here, the first and last rows are annihilated by this outcome, and the middle two are collapsed into a single row. Looking at the final diagram, we see that coin b is left in the state c T |H b + c H |T b , which has the coefficients c H and c T exchanged in comparison to the state S that we want it to be in. However, Alice knows that this is the question to which Nature answered yes, and she can call Bob on the telephone and inform him of this fact. Once he knows this is the question that was answered affirmatively, all he needs to do is "flip his quantum coin". Recall that this is a quantum coin, which he cannot simply pick up and turn over. Instead what we mean by this is that he exchanges H for T and vice-versa. In turns out that this is a legitimate action that can be performed on a quantum coin, and it will leave coin b in the desired state:
c T |H b + c H |T b → c T |T b + c H |H b = |S b .
Notice also that they again remain completely ignorant of the coefficients c H and c T -nothing that has happened has provided them with any such information, nor have they needed it. It turns out that no matter which of the four outcomes of Alice's measurement she obtains, once she informs Bob of that outcome, he will be able to perform a legitimate action on his quantum coin that will leave his coin in the state |S b . All Bob needs to know, in order to choose which action to perform, is the outcome of Alice's measurement: Alice needs only to send him two bits of information, enough to choose between one of the four possible outcomes. Furthermore, none of the four outcomes provides either party with any information about the coefficients, c H and c T , so they both remain completely ignorant of the original state they have just successfully teleported. 9 The diagrams provide a great deal of insight into what is going on. The crucial observation is the presence of two images of the state S, resulting from the entanglement between coins a, b. Alice does a measurement that, while not acting independently on these two images, does act differently on them, as we alluded to above. This measurement picks out different parts of S from the different images in a way such that all of S is preserved and none of it is repeated. For example, in the previous example, the H part is preserved from the lower-right image, and the T part from the upper-left one (and vice-versa in the example before that). This suggests (and it is indeed the case) that for coins that have more than two sides (a six-sided quantum die, for example), the parties can teleport the state of such an n-sided coin by sharing a maximally-entangled state that is "large enough" to provide them with n images of the unknown state S. Then Alice can design her measurement such that for each outcome: (1) a different part of S is extracted from each image; and (2) the whole state is preserved across the n images. Afterward, Bob can recover the state simply by rearranging the various parts, which he will be able to do once Alice informs him of the outcome of her measurement. Alice's measurement does not provide them with any information about the original state they are attempting to teleport, nor does Bob's rearrangement require that they know anything about it. In all cases, they remain ignorant of the state they are teleporting. The reader is encouraged to draw a diagram (perhaps for coins with n = 3 sides each; see the following paragraph) and follow through the argument to be sure it is clear how this is done. The diagram will have n × n = n 2 horizontal rows (representing Alice's two n-sided coins A, a, each row corresponding to one of the combinations of sides of these two coins: HH, HT, HU, HV, • • •, where H, T, U, V, • • • label the various sides), and n vertical columns (representing Bob's coin b).
A complete measurement for the n = 3 case will include n 2 = 9 outcomes, but an essentially complete understanding can be gained if the reader considers only the three outcomes corresponding to and (3)
(1) H A H a | + T A T a | + U A U a |; (2) H A T a | + T A U a | + U A H a |;
H A U a | + T A H a | + U A T a |,
where U is the third side of these coins (the other six outcomes involve additional complications that I have not explained here, but these outcomes are not crucial for the general kind of understanding we are aiming for here). In this case the appropriate generalization of B 0 is the state |H a H b + |T a T b + |U a U b , and the three terms in this expression yield the three images needed for teleportation.
In this section, I consider teleportation of classical coins, which turns out to be possible using a method that bears a striking resemblance to the method used for quantum coins. Imagine that Chloe prepares a classical coin (labeled A) as either H or T , and gives it to Alice, who is not allowed to look at the coin. Chloe also prepares classical coins a and b such that they are either HH (both H) or T T (both T ). She then gives coin a to Alice and coin b to Bob, but again does not allow these parties to look at their coins. Alice now asks Chloe the following two questions: Are coins A and a the same? Or are they different? This pair of "yes-no" questions represents a measurement, as defined earlier, on this pair of coins. If Chloe informs her they are the same, then Alice knows that coin b, which is guaranteed to be the same as a, is also the same as A; if, on the other hand, Chloe says coins A and a are different, then coin b is also different from A. Alice now calls Bob on the telephone and tells him to "flip" or "don't flip". In the first case (A same as a) she tells him not to flip, while when A and a are different, she tells him to flip. After he follows her instruction, Bob's coin b will with certainty match coin A. The state of coin A has been teleported onto coin b.
It is instructive to look at why the quantum case is astonishing while the classical one is rather mundane. There are three important differences between classical and quantum teleportation. The first difference has to do with the information that Alice would need to transmit to Bob in order to inform him of the state of coin A, if she happened to know that state. For a classical coin, there are only two possibilities, H or T , so she would need to transmit only one bit to Bob. This is the same amount of information that is actually transmitted when she tells him "flip" or "don't flip" -again, two possibilities. In contrast, as was discussed at the beginning of Section III for the case of quantum coins, it would require an infinite amount of information for Alice to inform Bob of the state of coin A, whereas she only actually transmits two bits of information when informing him which of her four questions was the outcome of her measurement. We see that the two cases, classical vs. quantum, are dramatically different in terms of the amounts of information involved.
The second difference between these two cases is a bit more subtle. In the classical case, if Alice were to cheat and actually look at coin A, she would automatically know what state that coin is in and be able to tell Bob what to do with his coin -turn it H or turn it T ; another one-bit message encompassing these two possibilities. This absolutely will not work for a quantum coin, which Alice cannot simply "look" at to discover its state. The reason is the following: To begin with, in contrast to a classical coin, when Alice looks at her quantum coin, she invariably disturbs it in the process. That is, no matter what state the coin was in before she looked at it, the state after she looks at it is with certainty given by the outcome of her measurement. For example, even if the state is "equal parts H and T " before she asks if it is H or T , if the answer is H (T ), then the coin is now H (T ). Or if it is H to begin with and she does a measurement that answers yes to the question "Is it equal parts H and T ?", then the state of 10 the coin will now be equal parts H and T . Hence, when she looks at it, she will get one of two answers (those being the two possible outcomes of her measurement) as to the state of the coin, but if she looks at it wrong, that answer will not tell her what the state was beforehand, but only what it is now. Furthermore, since she has now disturbed the state, there is no way to go back and try again, since the coin is now in a completely different state than the one she is trying to discover. The moral of this story is twofold: With quantum coins (1) don't bother trying to cheat; and (2) there's no point in asking for a "do-over".
The third difference between the quantum and classical cases is even more subtle and is related to entanglement, for which there is no counterpart with classical coins. For classical teleportation, Chloe must tell Alice whether or not coins a, A are the same or different. When these coins are classical, and since Chloe is the one that prepared them, she is certainly able to do so. However, in the quantum case coins a, b are entangled, which means that neither one has a definite state of its own. Since coin a does not have a definite state, the question whether coins a, A are the same (have the same state) has no answer ! Even if we assume that Chloe prepared coins a, b in their entangled state, there is nothing whatsoever that she (or anyone else) can say about the state of coin a, except that it is entangled with b and has no definite state of its own. It is worth noting that in both the quantum and classical cases, coins a, b are correlated with each other in ways that at first glance appear to be very similar -when one is measured and found to be H (T ), the other one will also be H (T ). Nonetheless, the correlations present in the entangled state B 0 of quantum coins a, b have no analog in the case of classical coins. One reason is precisely what we have just discussed: that the quantum coins can be correlated in this way even though neither of the individual coins has a definite state of its own (a classical coin always has a definite state of its own).
I have described a novel way to visualize the processing of quantum information, and used this picture to give a simple way to "see" how teleportation is possible. The picture turns out to be useful beyond just providing an understanding of previously known phenomena (teleportation), however. Indeed, it has given us a deeper understanding of the process of deterministically implementing non-local unitaries by local operations and classical communication (when shared entanglement is available as a resource), allowing us to construct new protocols [10] that go far beyond what was previously known to be possible [13] . We have also used this picture to study the question of what entanglement resources are required to locally implement other non-local operations, such as measurement protocols for the purpose of distinguishing sets of quantum states that are indistinguishable without the extra entangled resource [11] .
This work has been supported in part by the National Science Foundation through Grant No. PHY-0456951. I am very grateful for numerous discussions with Bob Griffiths and others in his research group.
[1] P. W. Shor, in Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, 1994). [2] E. Gerjuoy, Am. J. Phys. 73, 521 (2005). [3] C. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993). [4] M. Barrett et al., Nature 429, 737 (2004). [5] M. Riebe et al., Nature 429, 734 (2004). [6] D. Bouwmeester et al., Nature 390, 575 (1997). [7] A. Furusawa et al., Science 282, 706 (1998). [8] J. F. Sherson et al., Nature 443, 557 (2006) . [9] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000). [10] L. Yu, R. B. Griffiths, and S.M. Cohen, to be published. [11] S. M. Cohen, Phys. Rev. A 77, 012304: 1 (2008). [12] The other two questions involve minus signs and are represented as (3) HAHa| -TATa| and (4) HATa| -TAHa|, respectively. Multiplying one or the other of the states H and T by a minus sign is another valid quantum operation, which Bob can perform on his coin. By doing so, he effectively turns each of these cases into one of the two already shown explicitly in the paper. When question (3) is answered affirmatively, multiplication of T b by the minus sign is all that is needed to complete the teleportation; and for question (4), the minus sign operation need only be followed by Bob flipping his quantum coin.
[13] B. Reznik, Y. Aharonov, and B. Groisman, Phys. Rev. A 65, 032312: 1 (2002). | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "One of the most exciting and fastest-growing fields of physics today is quantum information. Especially since the discovery by Shor [1, 2] that there exist calculations for which a quantum computer is apparently far more efficient than a classical computer, interest in understanding quantum information has increased at an impressive rate. One widely publicized discovery that has emerged from work in this field is teleportation [3] . While not precisely equivalent to the process enjoying widespread fame amongst fans of Star Trek (\"Beam me up, Scotty\"), the phenomenon referred to here is nonetheless fascinating, and perhaps even astonishing. The reason for the widespread publicity of this rigorously proven (and experimentally tested [4, 5, 6, 7, 8] , though not yet unambiguously demonstrated) scientific prediction is almost certainly in large part due to the fact that it shares the same name as the just-mentioned, intriguing idea from science-fiction.\n\nThe usual way of describing teleportation is through mathematical equations, and this mathematics is relatively straightforward, as has been amply demonstrated elsewhere [3, 9] . Hence, an understanding of this phenomenon is accessible to physicists, other scientists, and those possessing a reasonably strong level of mathematical skill. There does, on the other hand, seem to be a good deal of misunderstanding of teleportation amongst non-scientists, with the notion floating around that the amazing phenomenon shown regularly in episodes of Star Trek -that is, of material objects being teleported from one place to another -has actually turned out to be possible in real life. Nothing could be further from the truth, of course, so we are left wondering how to rectify this unfortunate state of affairs. The question I address here is the following: can the true (scientific) phenomenon of teleportation be understood by others, those without much skill in mathematics? The usual explanations will certainly fail in this regard, even if carefully presented by a competent physicist, because mathematics has a well-known tendency to scare people away, and in any case, the mathematics of teleportation is not all that simple. The paper is addressed to physicists possessing a solid understanding of quantum physics (including graduate students), with the aim to provide a method by which such a physicist can explain teleportation to someone who is not mathematically inclined. Thus, the objective is ultimately, though indirectly, to educate the general public about teleportation, and by extension, quantum mechanics itself. The approach involves only the most basic ideas about quantum physics, and while it does not entirely avoid mathematical expressions, it uses only the simplest mathematics (one only needs to accept that certain objects are either 0 or 1) and relies almost entirely on \"pictures\", allowing the layperson to visualize -and thus, understand -what is happening.\n\nIn the following sections, I will describe my method of directly visualizing teleportation. These sections are written as if addressed to the layperson. The next section explains the probabilistic nature of quantum physics by considering \"quantum coins\", which are examples of two-level systems. This section describes how one should think about measurements, what is meant by probabilities for classical systems, and then how these ideas can be used to describe quantum systems. Then, in Section III, I present my graphical approach to understanding the dynamics of quantum information processing, which is then used in Section III B to explain in pictures how teleportation of quantum states is possible. One of the crucial observations will be that a shared entangled state on, say, systems a and b, provides the parties with multiple \"images\" of the state of an additional system A. The ability to manipulate these images -independently by each party, and differently from one image to the next -is what allows teleportation to be accomplished. More generally, these ideas provide important insights into why entanglement is a valuable resource, as I have described in detail elsewhere, and they have been useful in understanding other aspects of quantum information 2 processing [10, 11] ."
},
{
"section_type": "OTHER",
"section_title": "II. PROBABILITIES",
"text": "Perhaps the most fundamental aspect of quantum theory is that it can only make predictions in terms of probabilities. In general even if one has a complete description of the state of a quantum system, one will not know ahead of time what the outcome of a given measurement will be. This is in direct contradiction with our everyday experience, which we refer to as \"classical\". For example, a flipped classical coin which lands heads (\"heads\" is then a complete description of the state of this coin), is known with certainty to be heads, and also with certainty to not be tails. That is, if we know the state of a classical coin (in this case \"heads\"), we can predict with certainty the answer to any reasonable question we choose to ask (or \"measure\") about that coin (for example, \"Is it tails?\"). We therefore need to understand what is meant by the \"state\" of a quantum system and how this state relates to probabilities and outcomes of measurements. The following definition of a measurement will be adequate for our purposes.\n\nDefinition: A measurement is a procedure that provides answers to a collection of yes-no questions, which is both mutually exclusive (when the answer to one of the questions is \"yes\", the answer to all the others is \"no\") and complete (all possibilities are included; that is, one of the questions will always be answered in the affirmative). The single question that receives the \"yes\" answer is referred to as the outcome of the measurement.\n\nFor example, since a classical coin is either heads or tails, and these two possibilities are mutually exclusive, a measurement on a classical coin is a procedure that answers the two questions \"Is it heads?\" and \"Is it tails?\" Since the coin will always be one or the other, there will always be a \"yes\" answer to one of these questions, and then the other question is always answered \"no\". Hence these two questions do indeed constitute a measurement according to the above definition. If \"Is it heads?\" is answered affirmatively, then \"heads\" is the outcome of the measurement.\n\nIt turns out that these two questions also constitute a measurement on quantum coins. However, in contrast to the classical case in which this is the only possible measurement, there is a vast array of possible measurements on quantum coins. This will become clearer from the discussion in the following sections, where we introduce a compact way of describing these things, a way commonly used in quantum mechanics."
},
{
"section_type": "OTHER",
"section_title": "A. Classical coins and classical probabilities",
"text": "Consider again a flipped classical coin. The coin lands either heads or tails. It will be useful to use a somewhat abbreviated notation: |H for heads and |T for tails. The statement that \"if it is heads, it is not tails\" (that is, has zero probability of being tails) will be represented as\n\nT |H = 0.\n\nThe left-facing bracket |H represents the known initial state (\"It is heads.\") and the right-facing bracket T | represents the question (\"Is it tails?\"). The number (0) appearing on the right-hand side of the equal sign then gives the probability that with this initial state, the answer to this question will be yes. For the above example, we have that the probability is 0, which is as expected since when the coin is H it will never be T . Note that it is useful to use the left-and right-facing brackets, so that we can easily read off what is the initial state and what is the question being asked about it. Simply writing T H = 0 in the above equation would lead to confusion when we discuss two coins (see below), which might have an initial state where one is tails, the other heads, represented by |T H .\n\nPerhaps an even more trivial statement \"if it is heads, then it is heads\" (with certainty, or with probability one), will similarly be represented as H|H = 1.\n\nAgain, the right-facing bracket contains the question H|, or \"Is it heads?\", and the fact that the expression is equal to 1 indicates that the answer to this question will always be \"yes\" when the initial state is |H . These statements are trivial because if we know the state of a classical coin, we can predict with certainty whether it will be heads or tails when we look at it. Although the remaining equations will look a bit more involved, the only mathematics the reader need understand is contained in the above two equations, along with two others that are almost exactly the same. The discussion in the remainder of this paper will follow from the four simple statements, H|H = 1, T |T = 1, T |H = 0, H|T = 0.\n\nNext let us consider two coins. In this case, a complete list of mutually exclusive possibilities is HH, HT, T H, T T . We can make statements in exactly the same way we did above, for example \"if they are HH, then they are not HT \", 3 which in our notation is written\n\nH 1 T 2 |H 1 H 2 = H 1 |H 1 × T 2 |H 2 = (1) × (0) = 0,\n\nwhere the subscripts (1, 2) have been inserted for clarity to indicate which coin is which. Note that in this equation, we have equated the expression H 1 T 2 |H 1 H 2 with the product of two expressions, H 1 |H 1 and T 2 |H 2 . This is because any question about the two coins jointly is the same as two questions, one about each of the coins separately.\n\nIt is obviously also true that \"if they are HH, then they are HH\", so\n\nH 1 H 2 |H 1 H 2 = H 1 |H 1 × H 2 |H 2 = (1) × (1) = 1.\n\nFor three coins, there are eight possibilities (HHH, HHT, HT H, T HH, HT T, T HT, T T H, T T T ) and the same notation will readily account for this case, as well. We will not need to consider more than three coins here, though it is in principle straightforward to do so."
},
{
"section_type": "OTHER",
"section_title": "B. Quantum coins and quantum probabilities",
"text": "Quantum coins behave very differently as compared to their classical counterparts, and quantum probabilities must be understood in very different ways. We still have heads and tails, |H and |T , as possible states of a quantum coin. We refer to these two states as being \"orthogonal\" to each other, by which we simply mean that they are mutually exclusive: if the quantum coin is H, it is definitely (with certainty) not T , and vice-versa. We note that the four equations appearing in the previous section are equally true for both quantum and classical coins. However, there now exist some very strange possibilities. If I were to suggest that a classical coin can be both H and T at one and the same time, you would be completely justified in thinking I'd gone slightly crazy. I am going to tell you, though, that at least in a certain (though very real) sense, this is the case for quantum coins (though you may still wonder a bit about my sanity). The point is that, in the quantum case, it makes complete sense to ask questions such as: \"If the coin is H, is it half H and half T ?\"; or we can turn this around and ask \"If the coin is half H and half T , is it H?\" Neither of these questions makes any sense whatsoever when referred to a classical coin. On the other hand, for a quantum coin these are not only legitimate questions, but they are in fact very important ones (we do not consider the negligible possibility of a classical coin landing on its edge, and in any case this bears no relationship to what we mean by a quantum coin being half H and half T ).\n\nTo represent these questions, we can write the state (Q) of a quantum coin that is half H and half T as\n\n|Q = 1 2 |H + 1 2 |T .\n\nThen the answer to the question, \"If the coin is half H and half T , is it H?\" is answered by the equation,\n\nH|Q = H| 1 2 |H + 1 2 |T = 1 2 H|H + 1 2 H|T = 1 2 (1) + 1 2 (0) = 1 2 ,\n\nwhich should be interpreted as meaning \"yes, with probability 1/2\", implying also \"no, with probability 1-1/2 = 1/2\" [In quantum mechanics, it is actually the square of the object on the left-hand side of the foregoing equation that represents the probability, rather than that object itself, which is known as the \"probability amplitude\"; however, although the difference between probabilities and probability amplitudes is crucial to the understanding of quantum mechanics, I have chosen in the present discussion to overlook this distinction for the benefit of the layperson to whom these ideas are aimed, as they would only serve to complicate matters, causing unnecessary confusion amongst the intended audience]. The left-facing bracket |Q represents the known initial state, and the right-facing bracket H| represents the question (\"Is it heads?\"). The number 1/2 appearing on the right-hand side of the last line then gives the probability that with this initial state, the answer to this question will be yes. The point to understand here is that even though we have a complete description (Q) of the state of the quantum coin, we do not generally know in advance whether the coin will be H or T when we look at it. We can only predict in terms of probabilities: if we perform this experiment many times, half the time the answer will be yes and the other half of the time it will be no. Furthermore, there are many more questions we can ask in the quantum, as compared to the classical, case. We are 4 no longer restricted to asking \"is the coin H?\" or \"is it T ?\", but we can ask other questions, such as the reverse of the question we just answered,\n\nQ|H = 1 2 H| + 1 2 T | |H = 1 2 H|H + 1 2 T |H = 1 2 (1) + 1 2 (0) = 1 2 .\n\nWe see that the question \"If the coin is H, is it half H and half T ?\" has the same answer as the previous question: \"yes, with probability 1/2; and no, with probability 1/2.\" We note that in the remainder of the paper, instead of phrasing questions as \"is the coin half H and half T ?\", we instead ask whether it is \"equal parts\" H and T . While there is no real difference between these two questions, this rephrasing allows us to simplify the notation by dispensing with the factors of 1/2 that have appeared in the above discussion. In doing so, the equations will not yield the same numbers as probabilities for the various questions, but this will not hamper the presentation since the numerical values of the probabilities are not crucial to the ideas we wish to convey: we just need to remember that certain objects are equal to 1 and others are equal to 0."
},
{
"section_type": "OTHER",
"section_title": "III. TELEPORTATION",
"text": "What exactly do we mean by teleportation in the context of quantum information? It is not a material object that is being teleported, but rather the state of a quantum system. We will assume that the system is a quantum coin, with a complete set of mutually exclusive (orthogonal) states being \"heads\" and \"tails\", which we may denote as |H and |T . Suppose Alice and Bob are physicists in locations widely separated from each other. They each have a quantum coin -labeled a and b, respectively -and these two coins are in the state\n\n|B 0 ab = |H a H b + |T a T b ,\n\nwhere the subscripts used here refer to system a (b) in Alice's (Bob's) possession. This state of two quantum coins has a very strange property, which is known as entanglement, and the state itself is an example of a maximally entangled state. Entanglement is a rather strange sort of correlation between quantum systems, which manifests itself in the state B 0 by the fact that neither system a nor b can be considered to have a definite \"state of its own\" independent of the other system: whatever is the state of coin a, coin b will have the same state, but one cannot say anything about the state of either coin independent of the other one. It is this property of entanglement that is credited with enabling Alice and Bob to accomplish teleportation.\n\nAlice is given another coin (system A), prepared in a state\n\n|S A = c H |H A + c T |T A\n\nwith arbitrary coefficients c H and c T that are completely unknown to her and to Bob. If c H = 1/2 and c T = 1/2, we have the case discussed in the previous section, where the coin is equally likely to be found to be H or T . For other values of these coefficients, the two possibilities will in general not be equally likely. Alice's task is to perform operations on the systems in her possession (a and A) in such a way that Bob will end up with his system (b) in precisely the state |S b , which is the same state as |S A , but now on the distant system b. It turns out that this task can be accomplished if Alice communicates information to Bob (perhaps via a telephone) about what she ended up doing to her systems, after which Bob performs a rather simple quantum operation, dependent on the information obtained from Alice, on system b. An important point to understand in what follows is that nothing either of them does in this process provides even the slightest information about the coefficients c H and c T , so the state (S) that has been teleported remains completely unknown to the parties. This aspect of teleportation becomes even more amazing if one considers the amount of information that is conveyed: the information contained in a quantum state is far greater than the amount actually transmitted from Alice to Bob via the telephone (as we will see below, the amount transmitted via the telephone is two classical bits, enough to convey which one of four possibilities has been chosen). True, the classical information one can encode in a two-level quantum system cannot exceed one bit (one bit is the amount of information needed to choose between two possibilities, such as |H and |T ). But if Alice wanted to tell Bob how to create the state in his own lab by communicating with him over a phone line, this would require an infinite amount of classical information; that is, enough information to completely describe the arbitrary numbers, c H and c T (it is infinite because 5 one of these numbers might well be an irrational number such as π, having a decimal expansion that is unending, never repeating itself). Of course, Alice and Bob are both completely ignorant of what these numbers are, so even if it were possible to transmit an infinite amount of information, they don't even know what information they would need to send! Nonetheless, when they share entanglement, it is possible for the two of them, by working together, to create the unknown state on Bob's coin b with the communication of only two classical bits."
},
{
"section_type": "OTHER",
"section_title": "A. Visualizing quantum information processing",
"text": "Let us now introduce the pictorial method which will be used to visualize teleportation. The simple diagrams we will use to depict states of multiple quantum coins, held by two different parties, are familiar to many researchers working in quantum information. We will now illustrate how these diagrams are used to represent quantum states, and then how they can be used to follow what happens to these coins when measurements are performed by one of the parties. Then, we will be ready to use them for visualizing teleportation."
},
{
"section_type": "OTHER",
"section_title": "States of quantum coins",
"text": "To depict the state of a single quantum coin labeled A (standing for Alice; she will also have the other coin labeled a, while Bob's single coin is labeled b), we may use a simple box diagram,\n\n|S A = c H |H A + c T |T A = |T A |H A c T c H .\n\nThe coefficients c H and c T appearing in the boxes indicate \"how much\" is in that part of the state S A of coin A. The next example illustrates the case where there are two coins (A and b) held by two different parties. Then, the state of these two coins might be\n\n|S A H b = |H b |T b |T A |H A c T c H ,\n\nwith S A as given above. The empty squares on the right-hand side of this diagram represent the fact that system b is \"not T \" (has zero probability of being tails); the c H in the upper-left corner represents the probability the coins are both heads; and the c T in the lower-left, the probability Bob's coin is heads and Alice's is tails. If there are three parties involved, a three-dimensional cube could be used to represent this situation. However, it will serve our present purposes to represent both of Alice's systems along the vertical dimension of the diagram. We might have coins A and b as in the previous example, and coin a being heads, the overall state of these three coins represented as\n\n|S A H a H b = |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H .\n\n6 If instead the a,b systems are both T , this picture is\n\n|S A T a T b = |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H .\n\nNow consider what happens if we add the previous two equations together. Then our two coins a,b are \"equal parts in HH and in T T \", which is what we previously referred to as the \"maximally entangled state\" |B 0 ab = |H a H b +|T a T b . The corresponding diagram looks like\n\n|S A (|H a H b + |T a T b ) = |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H c T c H = |H b |T b |T a |H a S A S A .\n\nNotice how there are now two images of the state |S A . This observation turns out to be rather useful in understanding entanglement [10, 11], but we will not need to discuss such issues here. Let us now look at how to represent measurements by use of these diagrams."
},
{
"section_type": "OTHER",
"section_title": "Measurements on quantum coins",
"text": "Suppose Alice and Bob share three quantum coins in the state represented in the last equation of the previous section, and Alice wants to know something about her coins. If she measures coin a and discovers it is H, then we have\n\nH a | × |H b |T b |T a |H a S A S A = |H b |T b S A .\n\nRecall that when the right-facing bracket H a | is attached to the left-facing one |H a on the left of this equation, we get H a |H a = 1, which \"preserves\" the upper row, whereas H a |T a = 0, indicating that the bottom row is annihilated (multiplied by 0), which is why it no longer appears on the far right of this equation. The interpretation is as follows: when the question \"Is coin a heads?\" is answered in the affirmative the other coins are left in the state |S A H b . We see how this measurement acts on both of the images simultaneously, rather than on the two independently. The upper-left image has been preserved intact, but the other image was annihilated, disappearing altogether. On the other hand, if the outcome of Alice's measurement had been that coin a was T , this would be represented as\n\nT a | × |H b |T b |T a |H a S A S A = |H b |T b S A .\n\nIn this case, the upper-left image has disappeared and the one in the lower-right has been preserved intact. In each of these cases, the state of coin A is unchanged, but that of coin b is left in a state that corresponds directly to the outcome of Alice's measurement on a. If she discovers that coin a was H (or T ), then coin b ends up H (or T ). 7 Alternatively, she could do a measurement that includes the question \"Is coin a equal parts H and T ?\" If the answer to this question is yes, then\n\n( H a | + T a |) × |H b |T b |T a |H a S A S A = |H b |T b S A S A = |S A (|H b + |T b ) ,\n\nwhich is just a sum of the previous two equations (notice how after each of the three measurement outcomes we have just considered, the two images have been collapsed into a single row). Once again we see that the state of coin b ends up corresponding to the outcome of Alice's measurement on coin a. This illustrates some of the strangeness that resides in entangled states of quantum systems: no matter what measurement Alice makes on coin a and no matter what outcome she obtains from that measurement, the resulting state of coin b will correspond directly to that outcome. The way the images of S A appear in the diagram is crucial. The fact that the two start out in different rows and in different columns will be important in what is to come. If entanglement between systems a,b was absent, for example if they were in the (unentangled)\n\nstate (|H a + |T a ) |H b , then this would be represented by (recall that |S A = c H |H A + c T |T A ) |S A (|H a + |T a ) |H b = |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H c T c H = |H b |T b |T a |H a S A S A .\n\nUnder these circumstances, Bob's view of the lower image of S A is \"obstructed\" by the presence of the upper image; the two images effectively appear as one to him. As will become clear in the following section, the presence of entanglement between the a,b coins will be necessary for them to accomplish teleportation. We will see that it is Bob's (and Alice's) ability to \"see\" the two images separately, and the consequent ability for each of them to act differently on one of the images as compared to the other, that is crucial to their success.\n\nIn the next section, we turn to the task of teleporting the state S A onto Bob's coin b. To begin this process, Alice will perform a measurement that asks \"joint\" questions; that is, questions about both coins in her possession simultaneously. As an example, she could ask if they are both H. That is,\n\nH A H a | × |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H c T c H = |H b |T b c H .\n\nThe c H appearing in the box on the right corresponds to the probability that the answer to this question will be \"yes\". More important for our purposes is to recognize that when this is the outcome of the measurement, coin b ends up H, once again a consequence of the initial entanglement between coins a and b. Now let us see how teleportation is possible."
},
{
"section_type": "OTHER",
"section_title": "B. Visualizing teleportation",
"text": "Teleportation is accomplished with the aid of the extra systems a, b in the entangled state |B 0 ab . System A starts in state |S A , discussed above, and this is the state they will teleport. Alice will ask a set of joint questions, which 8 together constitute a measurement, about the state of the two coins in her possession, a and A. The first question she asks is whether these two coins are equal parts HH and T T . When the answer is yes, we have\n\n( H A H a | + T A T a |) × |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H c T c H = |H b |T b c H c T .\n\nNotice how the middle two rows are annihilated by this outcome (because these rows correspond to a situation where the two coins are different -one H and one T -whereas we are asking if they are the same), and the remaining rows are collapsed into a single row. Now, if we look carefully (or perhaps, not even so carefully) at the final diagram in this picture, we will arrive at a rather startling conclusion. We see that the state of Bob's system b\n\nis now |S b = c H |H b + c T |T b .\n\nThat is, the unknown state |S A , originally on system A, is now on Bob's system b. Furthermore, the question asked by Alice had nothing whatsoever to do with the coefficients c H and c T , which determine what the original state of coin A was. Hence, the parties remain completely ignorant of the state S, yet that state has been successfully teleported! We are not quite finished, however, since we would like for Alice and Bob to be able to teleport no matter which joint question ends up being the outcome of Alice's measurement. Because of the probabilistic nature of the quantum world, she cannot choose the outcome of her measurement. Instead, Alice effectively asks all of the questions in her chosen measurement and then must wait for Nature to decide which question she (Nature, that is) will choose as the outcome. The nice thing about Nature is that she will tell Alice which question was chosen.\n\nThere must be four questions in a complete set of questions making up a joint measurement on coins A, a. Let me illustrate with one other question how Alice and Bob can succeed with teleportation, and then the reader is asked to believe that they can also succeed with either of the remaining two questions (these can be treated in a very similar way to the one shown here [12] ). The second question is: Are coins A, a equal parts T H and HT ? The corresponding diagram is\n\n( T A H a | + H A T a |) × |H b |T b |T A H a |H A H a |T A T a |H A T a c T c H c T c H = |H b |T b c T c H .\n\nHere, the first and last rows are annihilated by this outcome, and the middle two are collapsed into a single row. Looking at the final diagram, we see that coin b is left in the state c T |H b + c H |T b , which has the coefficients c H and c T exchanged in comparison to the state S that we want it to be in. However, Alice knows that this is the question to which Nature answered yes, and she can call Bob on the telephone and inform him of this fact. Once he knows this is the question that was answered affirmatively, all he needs to do is \"flip his quantum coin\". Recall that this is a quantum coin, which he cannot simply pick up and turn over. Instead what we mean by this is that he exchanges H for T and vice-versa. In turns out that this is a legitimate action that can be performed on a quantum coin, and it will leave coin b in the desired state:\n\nc T |H b + c H |T b → c T |T b + c H |H b = |S b .\n\nNotice also that they again remain completely ignorant of the coefficients c H and c T -nothing that has happened has provided them with any such information, nor have they needed it. It turns out that no matter which of the four outcomes of Alice's measurement she obtains, once she informs Bob of that outcome, he will be able to perform a legitimate action on his quantum coin that will leave his coin in the state |S b . All Bob needs to know, in order to choose which action to perform, is the outcome of Alice's measurement: Alice needs only to send him two bits of information, enough to choose between one of the four possible outcomes. Furthermore, none of the four outcomes provides either party with any information about the coefficients, c H and c T , so they both remain completely ignorant of the original state they have just successfully teleported. 9 The diagrams provide a great deal of insight into what is going on. The crucial observation is the presence of two images of the state S, resulting from the entanglement between coins a, b. Alice does a measurement that, while not acting independently on these two images, does act differently on them, as we alluded to above. This measurement picks out different parts of S from the different images in a way such that all of S is preserved and none of it is repeated. For example, in the previous example, the H part is preserved from the lower-right image, and the T part from the upper-left one (and vice-versa in the example before that). This suggests (and it is indeed the case) that for coins that have more than two sides (a six-sided quantum die, for example), the parties can teleport the state of such an n-sided coin by sharing a maximally-entangled state that is \"large enough\" to provide them with n images of the unknown state S. Then Alice can design her measurement such that for each outcome: (1) a different part of S is extracted from each image; and (2) the whole state is preserved across the n images. Afterward, Bob can recover the state simply by rearranging the various parts, which he will be able to do once Alice informs him of the outcome of her measurement. Alice's measurement does not provide them with any information about the original state they are attempting to teleport, nor does Bob's rearrangement require that they know anything about it. In all cases, they remain ignorant of the state they are teleporting. The reader is encouraged to draw a diagram (perhaps for coins with n = 3 sides each; see the following paragraph) and follow through the argument to be sure it is clear how this is done. The diagram will have n × n = n 2 horizontal rows (representing Alice's two n-sided coins A, a, each row corresponding to one of the combinations of sides of these two coins: HH, HT, HU, HV, • • •, where H, T, U, V, • • • label the various sides), and n vertical columns (representing Bob's coin b).\n\nA complete measurement for the n = 3 case will include n 2 = 9 outcomes, but an essentially complete understanding can be gained if the reader considers only the three outcomes corresponding to and (3)\n\n(1) H A H a | + T A T a | + U A U a |; (2) H A T a | + T A U a | + U A H a |;\n\nH A U a | + T A H a | + U A T a |,\n\nwhere U is the third side of these coins (the other six outcomes involve additional complications that I have not explained here, but these outcomes are not crucial for the general kind of understanding we are aiming for here). In this case the appropriate generalization of B 0 is the state |H a H b + |T a T b + |U a U b , and the three terms in this expression yield the three images needed for teleportation."
},
{
"section_type": "OTHER",
"section_title": "IV. TELEPORTING CLASSICAL COINS",
"text": "In this section, I consider teleportation of classical coins, which turns out to be possible using a method that bears a striking resemblance to the method used for quantum coins. Imagine that Chloe prepares a classical coin (labeled A) as either H or T , and gives it to Alice, who is not allowed to look at the coin. Chloe also prepares classical coins a and b such that they are either HH (both H) or T T (both T ). She then gives coin a to Alice and coin b to Bob, but again does not allow these parties to look at their coins. Alice now asks Chloe the following two questions: Are coins A and a the same? Or are they different? This pair of \"yes-no\" questions represents a measurement, as defined earlier, on this pair of coins. If Chloe informs her they are the same, then Alice knows that coin b, which is guaranteed to be the same as a, is also the same as A; if, on the other hand, Chloe says coins A and a are different, then coin b is also different from A. Alice now calls Bob on the telephone and tells him to \"flip\" or \"don't flip\". In the first case (A same as a) she tells him not to flip, while when A and a are different, she tells him to flip. After he follows her instruction, Bob's coin b will with certainty match coin A. The state of coin A has been teleported onto coin b.\n\nIt is instructive to look at why the quantum case is astonishing while the classical one is rather mundane. There are three important differences between classical and quantum teleportation. The first difference has to do with the information that Alice would need to transmit to Bob in order to inform him of the state of coin A, if she happened to know that state. For a classical coin, there are only two possibilities, H or T , so she would need to transmit only one bit to Bob. This is the same amount of information that is actually transmitted when she tells him \"flip\" or \"don't flip\" -again, two possibilities. In contrast, as was discussed at the beginning of Section III for the case of quantum coins, it would require an infinite amount of information for Alice to inform Bob of the state of coin A, whereas she only actually transmits two bits of information when informing him which of her four questions was the outcome of her measurement. We see that the two cases, classical vs. quantum, are dramatically different in terms of the amounts of information involved.\n\nThe second difference between these two cases is a bit more subtle. In the classical case, if Alice were to cheat and actually look at coin A, she would automatically know what state that coin is in and be able to tell Bob what to do with his coin -turn it H or turn it T ; another one-bit message encompassing these two possibilities. This absolutely will not work for a quantum coin, which Alice cannot simply \"look\" at to discover its state. The reason is the following: To begin with, in contrast to a classical coin, when Alice looks at her quantum coin, she invariably disturbs it in the process. That is, no matter what state the coin was in before she looked at it, the state after she looks at it is with certainty given by the outcome of her measurement. For example, even if the state is \"equal parts H and T \" before she asks if it is H or T , if the answer is H (T ), then the coin is now H (T ). Or if it is H to begin with and she does a measurement that answers yes to the question \"Is it equal parts H and T ?\", then the state of 10 the coin will now be equal parts H and T . Hence, when she looks at it, she will get one of two answers (those being the two possible outcomes of her measurement) as to the state of the coin, but if she looks at it wrong, that answer will not tell her what the state was beforehand, but only what it is now. Furthermore, since she has now disturbed the state, there is no way to go back and try again, since the coin is now in a completely different state than the one she is trying to discover. The moral of this story is twofold: With quantum coins (1) don't bother trying to cheat; and (2) there's no point in asking for a \"do-over\".\n\nThe third difference between the quantum and classical cases is even more subtle and is related to entanglement, for which there is no counterpart with classical coins. For classical teleportation, Chloe must tell Alice whether or not coins a, A are the same or different. When these coins are classical, and since Chloe is the one that prepared them, she is certainly able to do so. However, in the quantum case coins a, b are entangled, which means that neither one has a definite state of its own. Since coin a does not have a definite state, the question whether coins a, A are the same (have the same state) has no answer ! Even if we assume that Chloe prepared coins a, b in their entangled state, there is nothing whatsoever that she (or anyone else) can say about the state of coin a, except that it is entangled with b and has no definite state of its own. It is worth noting that in both the quantum and classical cases, coins a, b are correlated with each other in ways that at first glance appear to be very similar -when one is measured and found to be H (T ), the other one will also be H (T ). Nonetheless, the correlations present in the entangled state B 0 of quantum coins a, b have no analog in the case of classical coins. One reason is precisely what we have just discussed: that the quantum coins can be correlated in this way even though neither of the individual coins has a definite state of its own (a classical coin always has a definite state of its own)."
},
{
"section_type": "CONCLUSION",
"section_title": "V. CONCLUSION",
"text": "I have described a novel way to visualize the processing of quantum information, and used this picture to give a simple way to \"see\" how teleportation is possible. The picture turns out to be useful beyond just providing an understanding of previously known phenomena (teleportation), however. Indeed, it has given us a deeper understanding of the process of deterministically implementing non-local unitaries by local operations and classical communication (when shared entanglement is available as a resource), allowing us to construct new protocols [10] that go far beyond what was previously known to be possible [13] . We have also used this picture to study the question of what entanglement resources are required to locally implement other non-local operations, such as measurement protocols for the purpose of distinguishing sets of quantum states that are indistinguishable without the extra entangled resource [11] ."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "This work has been supported in part by the National Science Foundation through Grant No. PHY-0456951. I am very grateful for numerous discussions with Bob Griffiths and others in his research group.\n\n[1] P. W. Shor, in Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, 1994). [2] E. Gerjuoy, Am. J. Phys. 73, 521 (2005). [3] C. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993). [4] M. Barrett et al., Nature 429, 737 (2004). [5] M. Riebe et al., Nature 429, 734 (2004). [6] D. Bouwmeester et al., Nature 390, 575 (1997). [7] A. Furusawa et al., Science 282, 706 (1998). [8] J. F. Sherson et al., Nature 443, 557 (2006) . [9] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000). [10] L. Yu, R. B. Griffiths, and S.M. Cohen, to be published. [11] S. M. Cohen, Phys. Rev. A 77, 012304: 1 (2008). [12] The other two questions involve minus signs and are represented as (3) HAHa| -TATa| and (4) HATa| -TAHa|, respectively. Multiplying one or the other of the states H and T by a minus sign is another valid quantum operation, which Bob can perform on his coin. By doing so, he effectively turns each of these cases into one of the two already shown explicitly in the paper. When question (3) is answered affirmatively, multiplication of T b by the minus sign is all that is needed to complete the teleportation; and for question (4), the minus sign operation need only be followed by Bob flipping his quantum coin.\n\n[13] B. Reznik, Y. Aharonov, and B. Groisman, Phys. Rev. A 65, 032312: 1 (2002)."
}
] | ||
arxiv:0704.0052 | 0704.0052 | 1 | 67ed34caf79c65f4a9d124daf92524fdfa9e55ddc4435a33fc69e1973c007809 | Quantum Field Theory on Curved Backgrounds. II. Spacetime Symmetries | We study space-time symmetries in scalar quantum field theory (including
interacting theories) on static space-times. We first consider Euclidean
quantum field theory on a static Riemannian manifold, and show that the
isometry group is generated by one-parameter subgroups which have either
self-adjoint or unitary quantizations. We analytically continue the
self-adjoint semigroups to one-parameter unitary groups, and thus construct a
unitary representation of the isometry group of the associated Lorentzian
manifold. The method is illustrated for the example of hyperbolic space, whose
Lorentzian continuation is Anti-de Sitter space. | [
"Arthur Jaffe (1) and Gordon Ritter (1) ((1) Harvard University)"
] | [
"hep-th"
] | hep-th | [] | 2007-03-31 | 2026-02-26 | The extension of quantum field theory to curved space-times has led to the discovery of many qualitatively new phenomena which do not occur in the simpler theory on Minkowski space, such as Hawking radiation; for background and historical references, see [2, 6, 18] . The reconstruction of quantum field theory on a Lorentz-signature spacetime from the corresponding Euclidean quantum field theory makes use of Osterwalder-Schrader (OS) positivity [15, 16] and analytic continuation. On a curved background, there may be no proper definition of time-translation and no Hamiltonian; thus, the mathematical framework of Euclidean quantum field theory may break down. However, on static space-times there is a Hamiltonian and it makes sense to define Euclidean QFT. This approach was recently taken by the authors [11] , in which the fundamental properties of Osterwalder-Schrader quantization and some of the fundamental estimates of constructive quantum field theory 1 were generalized to static space-times.
The previous work [11] , however, did not address the analytic continuation which leads from a Euclidean theory to a real-time theory. In the present article, we initiate a treatment of the analytic continuation by constructing unitary operators which form a representation of the isometry group of the Lorentz-signature space-time associated to a static Riemannian space-time.
Our approach is similar in spirit to that of Fröhlich [4] and of Klein and 1 2 ARTHUR JAFFE AND GORDON RITTER Landau [13] , who showed how to go from the Euclidean group to the Poincaré group without using the field operators on flat space-time.
This work also has applications to representation theory, as it provides a natural (functorial) quantization procedure which constructs nontrivial unitary representations of those Lie groups which arise as isometry groups of static, Lorentz-signature space-times. These groups are typically noncompact. For example, when applied to AdS d+1 , our procedure gives a unitary representation of the identity component of SO(d, 2). Moreover, our procedure makes use of the Cartan decomposition, a standard tool in representation theory.
2. Classical Space-Time 2.1. Structure of Static Space-Times. Definition 2.1. A quantizable static space-time is a complete, connected orientable Riemannian manifold (M, g ab ) with a globally-defined (smooth) Killing field ξ which is orthogonal to a codimension-one hypersurface Σ ⊂ M , such that the orbits of ξ are complete and each orbit intersects Σ exactly once.
Throughout this paper, we assume that M is a quantizable static spacetime. Definition 2.1 implies that there is a global time function t defined up to a constant by the requirement that ξ = ∂/∂t. Thus M is foliated by time-slices M t , and M = Ω -∪ Σ ∪ Ω +
where the unions are disjoint, Σ = M 0 , and Ω ± are open sets corresponding to t > 0 and t < 0 respectively. We infer existence of an isometry θ which reverses the sign of t,
θ : Ω ± → Ω ∓ such that θ 2 = 1, θ| Σ = id.
Fix a self-adjoint extension of the Laplacian, and let C = (-∆ + m 2 ) -1 be the resolvent of the Laplacian (also called the free covariance), where m 2 > 0. Then C is a bounded self-adjoint operator on L 2 (M ). For each s ∈ R, the Sobolev space H s (M ) is a real Hilbert space, defined as completion of C ∞ c (M ) in the norm (2.1) f 2 s = f, C -s f . The inclusion H s ֒→ H s+k for k > 0 is Hilbert-Schmidt. Define S := s<0 H s (M ) and S ′ := s>0 H s (M ). Then S ⊂ H -1 (M ) ⊂ S ′ form a Gelfand triple, and S is a nuclear space. Recall that S ′ has a natural σ-algebra of measurable sets (see for instance [7, 8, 17] ). There is a unique Gaussian probability measure µ with mean zero and covariance C defined on the cylinder sets in S ′ (see [7]).
More generally, one may consider a non-Gaussian, countably-additive measure µ on S ′ and the space E := L 2 (S ′ , µ).
We are interested in the case that the monomials of the form A(Φ) = Φ(f 1 ) . . . Φ(f n ) for f i ∈ S are all elements of E , and for which their span is dense in E . This is of course true if µ is the Gaussian measure with covariance C.
For an open set Ω ⊂ M , let E Ω denote the closure in E of the set of monomials A(Φ) = i Φ(f i ) where supp(f i ) ⊂ Ω for all i. Of particular importance for Euclidean quantum field theory is the positive-time subspace E + := E Ω + .
2.2. The Operator Induced by an Isometry. Isometries of the underlying space-time manifold act on a Hilbert space of classical fields arising in the study of a classical field theory. For f ∈ C ∞ (M ) and ψ : M → M an isometry, define f ψ ≡ (ψ -1 ) * f = f • ψ -1 .
Since det(dψ) = 1, the operation f → f ψ extends to a bounded operator on H ±1 (M ) or on L 2 (M ). A treatment of isometries for static space-times appears in [11] . Definition 2.2. Let ψ be an isometry, and A(Φ) = Φ(f 1 ) . . . Φ(f n ) ∈ E a monomial. Define the induced operator (2.2) Γ(ψ)A ≡ Φ(f 1 ψ ) . . . Φ(f n ψ ) , and extend Γ(ψ) by linearity to the domain of polynomials in the fields, which is dense in E .
3. Osterwalder-Schrader Quantization 3.1. Quantization of Vectors (The Hilbert Space H of Quantum Theory). In this section we define the quantization map E + → H , where H is the Hilbert space of quantum theory. The existence of the quantization map relies on a condition known as Osterwalder-Schrader (or reflection) positivity. A probability measure µ on S ′ is said to be reflection positive if (3.1) Γ(θ)F F dµ ≥ 0 for all F in the positive-time subspace E + ⊂ E . Let Θ = Γ(θ) be the reflection on E induced by θ. Define the sesquilinear form (A, B) on E + × E + as (A, B) = ΘA, B E , so (3.1) states that (F, F ) ≥ 0. Assumption 1 (O-S Positivity). Any measure dµ that we consider is reflection positive with respect to the time-reflection Θ.
4 ARTHUR JAFFE AND GORDON RITTER Definition 3.1 (OS-Quantization). Given a reflection-positive measure dµ, the Hilbert space H of quantum theory is the completion of E + /N with respect to the inner product given by the sesquilinear form (A, B). Denote the quantization map Π for vectors E + → H by Π(A) = Â, and write (3.2) Â, B H = (A, B) = ΘA, B E for A, B ∈ E + .
3.2. Quantization of Operators. The basic quantization theorem gives a sufficient condition to map a (possibly unbounded) linear operator T on E to its quantization, a linear operator T on H . Consider a densely-defined operator T on E , the unitary time-reflection Θ, and the adjoint T + = ΘT * Θ. A preliminary version of the following was also given in [10] .
Definition 3.2 (Quantization Condition I). The operator T satisfies QC-I if: i. The operator T has a domain D(T ) dense in E . ii. There is a subdomain
D 0 ⊂ E + ∩ D(T ) ∩ D(T + ), for which D 0 ⊂ H is dense.
iii. The transformations T and T + both map D 0 into E + .
Theorem 3.3 (Quantization I). If T satisfies QC-I, then i. The operators T ↾D 0 and T + ↾D 0 have quantizations T and T + with domain D0 . ii. The operators T * = T ↾ D0 * and T + agree on D0 .
iii. The operator T ↾D 0 has a closure, namely T * * .
Proof. We wish to define the quantization T with the putative domain D0 by (3.3) T Â = T A .
For any vector A ∈ D 0 and for any B ∈ (D 0 ∩ N ), it is the case that  = A + B. The transformation T is defined by (3.3) iff T A = T (A + B) = T A + T B. Hence one needs to verify that T : D 0 ∩ N → N , which we now do.
The assumption D 0 ⊂ D(T + ), along with the fact that Θ is unitary, ensures that ΘD 0 ⊂ D(T * ). Therefore for any
F ∈ D 0 , (3.4) ΘF, T B E = T * ΘF, B E = Θ (ΘT * ΘF ) , B E = ΘT + F, B E = T + F , B H .
In the last step we use the fact assumed in part (iii) of QC-I that T + : D 0 → E + , yielding the inner product of two vectors in H . We infer from the Schwarz inequality in H that
| ΘF, T B E | ≤ T + F H B H = 0 . As ΘF, T B E = F , T B H , this means that T B ⊥ D0 .
As D0 is dense in H by QC-I.ii, we infer T B = 0. In other words, T B ∈ N as required to define T .
In order show that D0 ⊂ D( T * ), perform a similar calculation to (3.4) with arbitrary A ∈ D 0 replacing B, namely (3.5)
F , T Â H = ΘF, T A E = Θ (ΘT * ΘF ) , A E = ΘT + F , A E = T + F , Â H .
The right side is continuous in  ∈ H , and therefore F ∈ D(T * ). Furthermore T * F = T + F . This identity shows that if F ∈ N , then T + F = 0. Hence T + ↾D 0 has a quantization T + , and we may write (3.5) as (3.6) T * F = T + F , for all F ∈ D 0 .
In particular T * is densely defined so T has a closure. This completes the proof. Definition 3.4 (Quantization Condition II). The operator T satisfies QC-II if i. Both the operator T and its adjoint T * have dense domains D(T ), D(T * ) ⊂ E . ii. There is a domain D 0 ⊂ E + in the common domain of T , T + , T + T , and T T + . iii. Each operator T , T + , T + T , and T T + maps D 0 into E + .
Theorem 3.5 (Quantization II). If T satisfies QC-II, then i. The operators T ↾D 0 and T + ↾D 0 have quantizations T and T + with domain D0 . ii. If A, B ∈ D 0 , one has B, T Â H = T + B, Â H .
i. In Theorem 3.5 we drop the assumption that the domain D0 is dense, obtaining quantizations T and T + whose domains are not necessarily dense. In order to compensate for this, we assume more properties concerning the domain and the range of T + on E . ii. As D0 need not be dense in H , the adjoint of T need not be defined.
Nevertheless, one calls the operator T symmetric in case one has
(3.7) B, T Â H = T B, Â H , for all A, B ∈ D 0 .
iii. If Ŝ ⊃ T is a densely-defined extension of T , then Ŝ * = T + on the domain D0 .
Proof. We define the quantization T with the putative domain D0 . As in the proof of Theorem 3.3, this quantization T is well-defined iff it is the case that T :
D 0 ∩ N → N . For any F ∈ D 0 ∩ N , by definition F H = 0. Also T F, T F H = (T F, T F ) = ΘT F, T F E = F, T * ΘT F E ,
where one uses the fact that
D 0 ⊂ D(T + T ). Thus T F, T F H = ΘF, T + T F E = F, T + T F H .
Here we use the fact that T + T maps D 0 to E + . Thus one can use the Schwarz inequality on H to obtain
T F, T F H ≤ F H T + T F H = 0 . Hence T : D 0 ∩ N → N , and T has a quantization T with domain D0 .
In order verify that T + ↾D 0 has a quantization, one needs to show that T + : D 0 ∩ N ⊂ N . Repeat the argument above with T + replacing T . The assumption T T + :
D 0 → E + yields for F ∈ D 0 ∩ N , T + F, T + F H = T * ΘF, T + F E = ΘF, T T + F E = F , T T + F H .
Use the Schwarz inequality in H to obtain the desired result that
T + F, T + F H ≤ F H T T + F H = 0 . Hence T + has a quantization T + with domain D0 , and for B ∈ D 0 one has T + B = T + B. In order to establish (ii), assume that A, B ∈ D 0 . Then B, T Â H = ΘB, T A E = Θ (ΘT * ΘB) , A E = ΘT + B, A E = T + B, Â H = T + B, Â H . (3.8)
This completes the proof.
For the remainder of this paper we assume the following, which is clearly true in the Gaussian case as the Laplacian commutes with the isometry group G. (A further explanation was given in [11] .) Assumption 2. The isometry groups G that we consider leave the measure dµ invariant, in the sense that Γ, defined above, is a unitary representation of G on E .
The Representation of g on E . Lemma 4.1. Let G i be an analytic group with Lie algebra g i (i = 1, 2), and let λ : g 1 → g 2 be a homomorphism. There cannot exist more than one analytic homomorphism π : G 1 → G 2 for which dπ = λ. If G 1 is simply connected then there is always one such π.
Let D = d/dt denote the canonical unit vector field on R. Let G be a real Lie group with algebra g, and let X ∈ g. The map tD → tX(t ∈ R) is a homomorphism of Lie(R) → g, so by the Lemma there is a unique analytic homomorphism
ξ X : R → G such that dξ X (D) = X. Conversely, if η is an analytic homomorphism of R → G, and if we let X = dη(D), it is obvious that η = ξ X . Thus X → ξ X is a bijection of g onto the set of analytic homomorphisms R → G.
The exponential map is defined by 1) . For complex Lie groups, the same argument applies, replacing R with C throughout. Since g is connected, so is exp(g). Hence exp(g) ⊆ G 0 , where G 0 denotes the connected component of the identity in G. It need not be the case for a general Lie group that exp(g) = G 0 , but for a large class of examples (the so-called exponential groups) this does hold. For any Lie group, exp(g) contains an open neighborhood of the identity, so the subgroup generated by exp(g) always coincides with G 0 .
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 7 exp(X) := ξ X (
We will apply the above results with G = Iso(M ), the isometry group of M , and g = Lie(G) the algebra of global Killing fields. Thus we have a bijective correspondence between Killing fields and 1-parameter groups of isometries. This correspondence has a geometric realization: the 1-parameter group of isometries
φ s = ξ X (s) = exp(sX)
corresponding to X ∈ g is the flow generated by X.
Consider the two different 1-parameter groups of unitary operators: (1) the unitary group φ * s on L 2 (M ), and (2) the unitary group Γ(φ s ) on E .
Stone's theorem applies to both of these unitary groups to yield denselydefined self-adjoint operators on the respective Hilbert spaces.
In the first case, the relevant self-adjoint operator is simply an extension of -iX, viewed as a differential operator on C ∞ c (M ). This is because for f ∈ C ∞ c (M ) and p ∈ M , we have:
X p f = (L X f )(p) = d ds f (φ s (p))| s=0 .
Thus -iX is a densely-defined symmetric operator on L 2 (M ), and Stone's theorem implies that -iX has self-adjoint extensions. In the second case, the unitary group Γ(φ s ) on E also has a self-adjoint generator Γ(X), which can be calculated explicitly. By definition,
e -isΓ(X) n i=1 Φ(f i ) = n i=1 Φ(f i • φ -s ).
Now replace s → -s and calculate d/ds| s=0 applied to both sides of the last equation to see that
Γ(X) n i=1 Φ(f i ) = n j=1 Φ(f 1 ) . . . Φ(-iXf j )Φ(f j+1 ) . . . Φ(f n ) .
One may check that Γ is a Lie algebra representation of g,
i.e. Γ([X, Y ]) = [Γ(X), Γ(Y )].
4.2. The Cartan Decomposition of g. For each ξ ∈ g, there exists some dense domain in E on which Γ(ξ) is self-adjoint, as discussed previously.
8 ARTHUR JAFFE AND GORDON RITTER However, the quantizations Γ(ξ) acting on H may be hermitian, antihermitian, or neither depending on whether there holds a relation of the form (4.1) Γ(ξ)Θ = ±ΘΓ(ξ), with one of the two possible signs, or whether no such relation holds. Even if (4.1) holds, to complete the construction of a unitary representation one must prove that there exists a dense domain in H on which Γ(ξ) is self-adjoint or skew-adjoint. This nontrivial problem will be dealt with in a later section using Theorems 3.3 and 3.5 and the theory of symmetric local semigroups [12, 4] . Presently we determine which elements within g satisfy relations of the form (4.1).
Let ϑ := θ * as an operator on C ∞ (M ), and consider a Killing field X ∈ g also as an operator on C ∞ (M ). Define T : g → g by (4.2) T (X) := ϑXϑ.
From (4.2) it is not obvious that the range of T is contained in g. To prove this, we recall some geometric constructions.
Let M, N be manifolds, let ψ : M → N be a diffeomorphism, and X ∈ Vect(M ). Then
(4.3) ψ -1 * Xψ * = X(• • ψ) • ψ -1 .
defines an operator on C ∞ (N ). One may check that this operator is a derivation, thus (4.3) defines a vector field on N . The vector field (4.3) is usually denoted ψ * X = dψ(X ψ -1 (p) ) and referred to as the push-forward of X.
We now wish to show that g = g + ⊕ g -, where g ± are the ±1-eigenspaces of T . This is proven by introducing an inner product (X, Y ) g on g with respect to which T is self-adjoint.
Theorem 4.2. Consider g as a Hilbert space with inner product (X, Y ) g . The operator T : g → g is self-adjoint with T 2 = I; hence (4.4) g = g + ⊕ g - as an orthogonal direct sum of Hilbert spaces, where g ± are the ±1-eigenspaces of T . Further, ∂ t ∈ g -hence dim(g -) ≥ 1. Elements of g -have hermitian quantizations, while elements of g + have anti-hermitian quantizations. 2 Proof. Write (4.2) as (4.5) T (X) = θ -1 * Xθ * = θ * X .
Thus T is the operator of push-forward by θ. The push-forward of a Killing field by an isometry is another Killing field, hence the range of T is contained 2 It is not the case that g-consists only of ∂t. In particular, dim(g-) = 2 for M = H 2.
It can occur that dim g+ = 0.
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 9 in g. Also, T must have a trivial kernel since T 2 = I, and this implies that T is surjective. It follows from (4.5) that T is a Hermitian operator on g. Hence T is diagonalizable and has real eigenvalues which are square roots of 1. This establishes the decomposition (4.4). That elements of g -have hermitian quantizations, while elements of g + have anti-hermitian quantizations follows from Theorem 3.3.
A Cartan involution is a Lie algebra homomorphism g → g which squares to the identity. It follows from (4.2) that T is a Lie algebra homomorphism; thus, Theorem 4.2 implies that T is a Cartan involution of g. This implies that the eigenspaces (g + , g -) form a Cartan pair, meaning that
(4.6) [g + , g + ] ⊂ g + , [g + , g -] ⊂ g -, and [g -, g -] ⊂ g + .
Clearly g + is a subalgebra while g -is not, and any subalgebra contained in g -is abelian.
Let G = Iso(M ) denote the isometry group of M , as above. Then G has a Z 2 subgroup containing {1, θ}. This subgroup acts on G by conjugation, which is just the action ψ → ψ θ := θψθ. Conjugation is an (inner) automorphism of the group, so
(ψφ) θ = ψ θ φ θ , (ψ θ ) -1 = (ψ -1 ) θ .
Definition 5.1. We say that ψ ∈ G is reflection-invariant if
ψ θ = ψ,
and that ψ is reflected if ψ θ = ψ -1 .
Let G RI denote the subgroup of G consisting of reflection-invariant elements, and let G R denote the subset of reflected elements.
Note that G RI is the stabilizer of the Z 2 action, hence a subgroup. An alternate proof of this proceeds using G RI = exp(g + ). Although G R is closed under the taking of inverses and does contain the identity, the product of two reflected isometries is no longer reflected unless they commute. Generally, the product of an element of G R with an element of G RI is neither an element of G R nor of G RI . The only isometry that is both reflection-invariant and reflected is θ itself. Thus we have:
G R ∩ G RI = {1, θ} ⊂ G R ∪ G RI G.
Theorem 5.2. Let G 0 denote the connected component of the identity in G. Then G 0 is generated by G R ∪ G RI . (This is a form of the Cartan decomposition for G.)
10 ARTHUR JAFFE AND GORDON RITTER Proof. Since g = g + ⊕ g -as a direct sum of vector spaces (though not of Lie algebras), we have
G 0 = exp(g) = exp(g + ) ∪ exp(g -) .
Choose bases {ξ ±,i } i=1,...,n ± for g ± respectively. Then we have:
G 0 = {exp(sξ +,i ) : 1 ≤ i ≤ n + , s ∈ R} ∪ {exp(sξ -,j ) : 1 ≤ j ≤ n -, s ∈ R} .
Furthermore, exp(sξ -,i ) is reflected, while exp(sξ +,i ) is reflection-invariant, completing the proof.
Corollary 5.3. The Lie algebra of the subgroup G RI is g + .
To summarize, the isometry group of a static space-time can always be generated by a collection of n (= dim g) one-parameter subgroups, each of which consists either of reflected isometries, or reflection-invariant isometries.
6.1. Self-adjointness of Semigroups. In this section, we recall several known results on self-adjointness of semigroups. Roughly speaking, these results imply that if a one-parameter family S α of unbounded symmetric operators satisfies a semigroup condition of the form S α S β = S α+β , then under suitable conditions one may conclude essential self-adjointness.
A theorem of this type appeared in a 1970 paper of Nussbaum [14], who assumed that the semigroup operators have a common dense domain. The result was rediscovered independently by Fröhlich, who applied it to quantum field theory in several important papers [5, 3] . For our intended application to quantum field theory, it turns out to be very convenient to drop the assumption that ∃ a such that the S α all have a common dense domain for |α| < a, in favor of the weaker assumption that α>0 D(S α ) is dense.
A generalization of Nussbaum's theorem which allows the domains of the semigroup operators to vary with the parameter, and which only requires the union of the domains to be dense, was later formulated and two independent proofs were given: one by Fröhlich [4], and another by Klein and Landau [12] . The latter also used this theorem in their construction of representations of the Euclidean group and the corresponding analytic continuation to the Lorentz group [13] .
In order to keep the present article self-contained, we first define symmetric local semigroups and then recall the refined self-adjointness theorem of Fröhlich, and Klein and Landau. Definition 6.1. Let H be a Hilbert space, let T > 0 and for each α ∈ [0, T ], let S α be a symmetric linear operator on the domain
D α ⊂ H , such that: (i) D α ⊃ D β if α ≤ β and D := 0<α≤T D α is dense in H , (ii) α → S α is weakly continuous, (iii) S 0 = I, S β (D α ) ⊂ D α-β for 0 ≤ β ≤ α ≤ T , and
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 11
(iv) S α S β = S α+β on D α+β for α, β, α + β ∈ [0, T ]. In this situation, we say that (S α , D α , T ) is a symmetric local semigroup.
It is important that D α is not required to be dense in H for each α; the only density requirement is (i).
Theorem 6.2 ([12, 4]). For each symmetric local semigroup (S α , D α , T ), there exists a unique self-adjoint operator A such that 3
D α ⊂ D(e -αA ) and S α = e -αA | Dα for all α ∈ [0, T ].
Also, A ≥ -c if and only if S α f ≤ e cα f for all f ∈ D α and 0 < α < T .
6.2. Reflection-Invariant Isometries. Lemma 6.3. Let ψ be a reflection-invariant isometry and assume ∃ p ∈ Ω + such that ψ(p) ∈ Ω + . Then ψ preserves the positive-time subspace, i.e. ψ(Ω + ) ⊆ Ω + .
Proof. We first prove that ψ(Σ) ⊆ Σ. Suppose not; then ∃ p ∈ Σ with ψ(p) ∈ Σ. Assume ψ(p) ∈ Ω + (without loss of generality: we could repeat the same argument with ψ(p) ∈ Ω -). Then Ω + contains (θψθ)(p) = θψ(p) ∈ Ω -, a contradiction since Ω -∩Ω + = ∅. We used the fact that θ| Σ = id so θ(p) = p. Hence ψ restricts to an isometry of Σ. It follows that the restriction of ψ to M ′ = M \ Σ is also an isometry. However, M ′ = Ω -⊔ Ω + , where ⊔ denotes the disjoint union. Therefore ψ(Ω + ) is wholly contained in either Ω + or Ω -, since ψ is a homeomorphism and so ψ(Ω + ) is connected. The possibility that ψ(Ω + ) ⊆ Ω -is ruled out by our assumption, so ψ(Ω + ) ⊆ Ω + . Lemma 6.3 has the immediate consequence that if ξ ∈ g + then the oneparameter group associated to ξ is positive-time-invariant. This result plays a key role in the proof of Theorem 6.4.
The rest of this section is devoted to proving that the theory of symmetric local semigroups can be applied to the quantized operators on H corresponding to each of a set of 1-parameter subgroups of G = Iso(M ). The proof relies upon Lemma 6.3, and Theorems 3.3, 3.5 and 6.2. Theorem 6.4. Let (M, g ab ) be a quantizable static space-time. Let ξ be a Killing field which lies in g + or g -, with associated one-parameter group of isometries {φ α } α∈R . Then there exists a densely-defined self-adjoint operator A ξ on H such that
Γ(φ α ) = e -αA ξ , if ξ ∈ g - e iαA ξ if ξ ∈ g + .
3 The authors of [4, 12] also showed that
b D := [ 0<α≤S h [ 0<β<α S β (Dα) i , where 0 < S ≤ T,
is a core for A, i.e. (A, b D) is essentially self-adjoint.
12 ARTHUR JAFFE AND GORDON RITTER
Proof. First suppose that ξ ∈ g -, which implies that the isometries φ α are reflected, and so Γ(φ α ) + = Γ(φ α ). Define Ω ξ,α := φ -1 α (Ω + ). For all α in some neighborhood of zero, Ω ξ,α is a nonempty open subset of Ω + , and moreover, as α → 0 + , Ω ξ,α increases to fill Ω + with Ω ξ,0 = Ω + . These statements follow immediately from the fact that, for each p ∈ Ω + , φ α (p) is continuous with respect to α, and φ 0 is the identity map. Since φ α (Ω ξ,α ) ⊆ Ω + , we infer that Γ(φ α )E Ω ξ,α ⊆ E + . By Theorem 3.5, Γ(φ α ) has a quantization which is a symmetric operator on the domain
D ξ,α := Π(E Ω ξ,α ).
Note that D ξ,α is not necessarily dense in H . 4 We now show that Theorem 6.2 can be applied.
Fix some positive constant a with Ω ξ,a nonempty. Note that 0<α≤a
Ω ξ,α = Ω + ⇒ 0<α≤a E Ω ξ,α = E + . It follows that D ξ := 0<α≤a D ξ,α
is dense in H . This establishes condition (i) of Definition 6.1, and the other conditions are routine verifications. Theorem 6.2 implies existence of a densely-defined self-adjoint operator A ξ on H , such that Γ(φ α ) = exp(-αA ξ ) for all α ∈ [0, a] .
This proves the theorem in case ξ ∈ g -. Now suppose that ξ ∈ g + , implying that the isometries φ α are reflectioninvariant, and Γ(φ α
) + = Γ(φ α ) -1 = Γ(φ -α ) on E . Lemma 6.3 implies that Γ(φ α )E + ⊆ E + . By Theorem 3.3, Γ(φ α ) has a quantization Γ(φ α ) which is defined and satisfies Γ(φ α ) * = Γ(φ α ) -1 on the domain Π(E + ), which is dense in H by definition.
In this case we do not need Theorem 6.2; for each α, Γ(φ α ) extends by continuity to a oneparameter unitary group defined on all of H (not only for a dense subspace). By Stone's theorem, Γ(φ α ) = exp(iαA ξ ) for A ξ self-adjoint and for all α ∈ R. The proof is complete.
4 Density of D ξ,α would be implied by a Reeh-Schlieder theorem, which we do not prove except in the free case. Theorem 6.2 removes the need for a Reeh-Schlieder theorem in this argument.
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 13
Each Riemannian static space-time (M, g ab ) has a Lorentzian continuation M lor , which we construct as follows. In adapted coordinates, the metric g ab on M takes the form (7.1)
ds 2 = F (x)dt 2 + G µν (x)dx µ dx ν .
The analytic continuation t → -it of (7.1) is standard and gives a metric of Lorentz signature, ds 2 lor = -F dt 2 + G dx 2 , by which we define the Lorentzian space-time M lor . Einstein's equation Ric g = k g is preserved by the analytic continuation, but we do not use this fact anywhere in the present paper. Let {ξ
(±) i : 1 ≤ i ≤ n ± } be bases of g ± , respectively. Let A (±) i = A ξ (±) i be the densely-defined self-adjoint operators on H , constructed by Theorem 6.4. Let (7.2) U (±) i (α) = exp(iαA (±) i ) , for 1 ≤ i ≤ n ± be the associated one-parameter unitary groups on H .
We claim that the group generated by the n = n + + n -one-parameter unitary groups (7.2) is isomorphic to the identity component of
G lor := Iso(M lor ),
the group of Lorentzian isometries. Since locally, the group structure is determined by its Lie algebra, it suffices to check that the generators satisfy the defining relations of g lor := Lie(G lor ).
Since quantization of operators preserves multiplication, we have
(7.3) X, Y, Z ∈ g, [X, Y ] = Z ⇒ [ Γ(X), Γ(Y )] = Γ(Z).
In what follows, we will use the notation
g ± for { Γ(X) : X ∈ g ± }.
Quantization converts the elements of g -from skew operators into Hermitian operators; i.e. elements of g -are Hermitian on H and hence, elements of i g -are skew-symmetric on H . Thus g + ⊕ i g -is a Lie algebra represented by skew-symmetric operators on H .
Theorem 7.1. We have an isomorphism of Lie algebras:
(7.4) g lor ∼ = g + ⊕ i g -.
Proof. Let M C be the manifold obtained by allowing the t coordinate to take values in C. Define ψ :
M C → M C by t → -it. Then g lor is generated by {ξ (+) i } 1≤i≤n + ∪ {η j } 1≤j≤n -,
where η j := iψ * ξ (-) j .
It is possible to define a set of real structure constants f ijk such that (7.5) [ξ
(-) i , ξ (-) j ] = n + k=1 f ijk ξ (+) k .
14 ARTHUR JAFFE AND GORDON RITTER Applying ψ * to both sides of (7.5), the commutation relations of g lor are seen to be
(7.6) [η i , η j ] = -f ijk ξ
(+) k , together with the same relations for g + as before. Now (7.3) implies that (7.6) are the precisely the commutation relations of g + ⊕ i g -, completing the proof of (7.4).
Corollary 7.2. Let (M, g ab ) be a quantizable static space-time. The unitary groups (7.2) determine a unitary representation of G 0 lor on H .
7.1. Conclusions. We have obtained the following conclusions. There is a unitary representation of the group G 0 lor on the physical Hilbert space H of quantum field theory on the static space-time M . This representation maps the time-translation subgroup into the unitary group exp(itH), where the energy H ≥ 0 is a positive, densely-defined self-adjoint operator corresponding to the Hamiltonian of the theory. The Hilbert space H contains a ground state Ψ 0 = 1 which is such that HΨ 0 = 0 and Ψ 0 is invariant under the action of all spacetime symmetries. We obtain these results via analytic continuation from the Euclidean path integral, under mild assumptions on the measure which should include all physically interesting examples. This is done without introducing the field operators; nonetheless, Theorems 3.3 and 3.5 do suffice to construct field operators. In the special case M = R d with G = SO(4), we obtain a unitary representation of the proper orthochronous Lorentz group, G 0 lor = L ↑ + = SO 0 (3, 1).
Consider Euclidean quantum field theory on M = H d . The metric is
ds 2 = r -2 d i=1 dx 2 i ,
where we define r = x d for convenience. The Laplacian is
(8.1) ∆ H d = (2 -d)r ∂ ∂r + r 2 ∆ R d .
The d -1 coordinate vector fields {∂/∂x i : i = d} are all static Killing fields, and any one of the coordinates x i (i = d) is a satisfactory representation of time in this space-time. It is convenient to define t = x 1 as before, and to identify t with time. The time-zero slice is M 0 = H d-1 . From d, 1) and the orientation-preserving isometry group is SO + (d, 1).
H d = {v ∈ R d,1 | v, v = -1, v 0 > 0} it follows that Isom(H d ) = O + (
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 15
Figure 1. Flow lines of the Killing field ζ = (t 2r 2
)∂ t + 2tr ∂ r on H d .
For constant curvature spaces, one may solve Killing's equation L K g = 0 explicitly. Let us illustrate the solutions and their quantizations for d = 2. The three Killing fields
(8.2) ξ = ∂ t , η = t∂ t + r∂ r , ζ = (t 2 -r 2 )∂ t + 2tr ∂ r
are a convenient basis for g. Any d-dimensional manifold satisfies dim g ≤ d(d + 1)/2, manifolds saturating the bound are said to be maximally symmetric, and H d is maximally symmetric. Now, ∂ t f (-t) = -f ′ (-t) so ∂ t Θ = -Θ∂ t , hence ∂ t ∈ g -. Similar calculations show [Θ, η] = 0 and Θζ = -ζΘ. Thus η spans g + , while ∂ t , ζ span g -. The commutation relations 5 for g are:
[η, ζ] = ζ, [η, ∂ t ] = -∂ t , [ζ, ∂ t ] = -2η.
These calculations verify that (g + , g -) forms a Cartan pair, as defined in (4.6) .
The flows associated to (8.2) are easily visualized: ξ is a right-translation, and η flow-lines are radially outward from the Euclidean origin. The flows of ζ are Euclidean circles, indicated by the darker lines in Figure 1 . Hence the flows of η are defined on all of E + , while the flows of ζ are analogous to space-time rotations in R 2 , and hence, must be defined on a wedge of the form W α = {(t, r) : t, r > 0, tan -1 (r/t) < α}.
The simple geometric idea of Section 6.2 is nicely confirmed in this case: the flows of η (the generator of g + ) preserve the t = 0 plane, and are separately isometries of Ω + and Ω -. Corollary 7.2 implies that the procedure outlined above defines a unitary representation of the identity component of Iso(AdS 2 ) on the physical Hilbert space H for quantum field theory on this background, including theories with interactions that preserve the symmetry. Since Iso(AdS d+1 ) = 5 Note that quite generally [g-, g-] ⊆ g+ so it's automatic that [ζ, ∂t] is proportional to η.
16 ARTHUR JAFFE AND GORDON RITTER SO(d, 2), we have a unitary representation of SO 0 (1, 2). The latter is a noncompact, semisimple real Lie group, and thus it has no finite-dimensional unitary representations, but a host of interesting infinite-dimensional ones.
We prove the Euclidean Reeh-Schlieder property for free theories on curved backgrounds. It is reasonable to expect this property to extend to interacting theories on curved backgrounds, but it would have to be established for each such model since it depends explicitly on the two-point function.
The Reeh-Schlieder theorem guarantees the existence of a dense quantization domain based on any open subset of Ω + . For this reason, one could use the Reeh-Schlieder (RS) theorem with Nussbaum's theorem [14] to construct a second proof of Theorem 6.4 under the additional assumption that M is real-analytic.
Fortunately, our proof of Theorem 6.4 is completely independent of the Reeh-Schlieder property. This has two advantages: we do not have to assume M is a real-analytic manifold and, more importantly, our proof of Theorem 6.4 generalizes immediately and transparently to interacting theories as long as the Hilbert space H is not modified by the interaction.
We state and prove this using the one-particle space; however, the result clearly extends to the quantum-field Hilbert space.
Theorem A.1. Let M be a quantizable static space-time endowed with a real-analytic structure, and assume that g ab is real-analytic. Let O ⊂ Ω + and
D = C ∞ (O) ⊂ L 2 (Ω + ). Then D ⊥ = {0}. Proof. Let f ∈ L 2 (Ω + ) with f ⊥ D. For x ∈ Ω + , define η(x) := f , δx H = Θf, Cδ x L 2 .
Real-analyticity of η(x) follows from the real-analyticity of (the integral kernel of) C, which in turn follows from the elliptic regularity theorem in the real-analytic category (see for instance [1, Sec. II.1.3]). Now by assumption, for any g ∈ C ∞ c (O), we have 0 = ĝ, f H = Θf, Cg L 2 (M ) .
Let g → δ x for x ∈ O. Then 0 = Θf, Cδ x L 2 ≡ η(x). Since η| O = 0, by real-analyticity we infer the vanishing of η on Ω + , completing the proof.
Acknowledgements. We are grateful to Hanno Gottschalk and Alexander Strohmaier for helpful discussions, and G.R. is grateful to the Universität Bonn for their hospitality during February 2007.
References [1] Lipman Bers, Fritz John, and Martin Schechter. Partial differential equations. American Mathematical Society, Providence, R.I., 1979. Lectures in Applied Mathematics 3.
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 17 [2] N. D. Birrell and P. C. W. Davies. Quantum fields in curved space, volume 7 of Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1982. [3] W. Driessler and J. Fröhlich. The reconstruction of local observable algebras from the Euclidean Green's functions of relativistic quantum field theory. Ann. Inst. H. Poincaré Sect. A (N.S.), 27(3):221-236, 1977. [4] J. Fröhlich. Unbounded, symmetric semigroups on a separable Hilbert space are essentially selfadjoint. Adv. in Appl. Math., 1(3):237-256, 1980. [5] Jürg Fröhlich. The pure phases, the irreducible quantum fields, and dynamical symmetry breaking in Symanzik-Nelson positive quantum field theories. Ann. Physics, 97(1):1-54, 1976. [6] Stephen A. Fulling. Aspects of quantum field theory in curved spacetime, volume 17 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1989. [7] I. M. Gel ′ fand and N. Ya. Vilenkin. Generalized functions. Vol. 4. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977]. Applications of harmonic analysis, Translated from the Russian by Amiel Feinstein. [8] James Glimm and Arthur Jaffe. Quantum physics. Springer-Verlag, New York, second edition, 1987. A functional integral point of view. [9] Arthur Jaffe. Constructive quantum field theory. In Mathematical physics 2000, pages 111-127. Imp. Coll. Press, London, 2000. [10] Arthur Jaffe. Introduction to Quantum Field Theory. 2005. Lecture notes from Harvard Physics 289r, available in part online at http://www.arthurjaffe.com/Assets/pdf/IntroQFT.pdf .
[11] Arthur Jaffe and Gordon Ritter. Quantum field theory on curved backgrounds. i. the euclidean functional integral. Comm. Math. Phys., 270(2):545-572, 2007. [12] Abel Klein and Lawrence J. Landau. Construction of a unique selfadjoint generator for a symmetric local semigroup. J. Funct. Anal., 44(2):121-137, 1981. [13] Abel Klein and Lawrence J. Landau. From the Euclidean group to the Poincaré group via Osterwalder-Schrader positivity. Comm. Math. Phys., 87(4):469-484, 1983. [14] A. E. Nussbaum. Spectral representation of certain one-parametric families of symmetric operators in Hilbert space. Trans. Amer. Math. Soc., 152:419-429, 1970. [15] Konrad Osterwalder and Robert Schrader. Axioms for Euclidean Green's functions. Comm. Math. Phys., 31:83-112, 1973. [16] Konrad Osterwalder and Robert Schrader. Axioms for Euclidean Green's functions. II. Comm. Math. Phys., 42:281-305, 1975. With an appendix by Stephen Summers.
18 ARTHUR JAFFE AND GORDON RITTER [17] Barry Simon. The P (φ) 2 Euclidean (quantum) field theory. Princeton University Press, Princeton, N.J., 1974. Princeton Series in Physics. [18] Robert M. Wald. Quantum field theory in curved space-time. In Gravitation et quantifications (Les Houches, 1992), pages 63-167. North-Holland, Amsterdam, 1995. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The extension of quantum field theory to curved space-times has led to the discovery of many qualitatively new phenomena which do not occur in the simpler theory on Minkowski space, such as Hawking radiation; for background and historical references, see [2, 6, 18] . The reconstruction of quantum field theory on a Lorentz-signature spacetime from the corresponding Euclidean quantum field theory makes use of Osterwalder-Schrader (OS) positivity [15, 16] and analytic continuation. On a curved background, there may be no proper definition of time-translation and no Hamiltonian; thus, the mathematical framework of Euclidean quantum field theory may break down. However, on static space-times there is a Hamiltonian and it makes sense to define Euclidean QFT. This approach was recently taken by the authors [11] , in which the fundamental properties of Osterwalder-Schrader quantization and some of the fundamental estimates of constructive quantum field theory 1 were generalized to static space-times.\n\nThe previous work [11] , however, did not address the analytic continuation which leads from a Euclidean theory to a real-time theory. In the present article, we initiate a treatment of the analytic continuation by constructing unitary operators which form a representation of the isometry group of the Lorentz-signature space-time associated to a static Riemannian space-time.\n\nOur approach is similar in spirit to that of Fröhlich [4] and of Klein and 1 2 ARTHUR JAFFE AND GORDON RITTER Landau [13] , who showed how to go from the Euclidean group to the Poincaré group without using the field operators on flat space-time.\n\nThis work also has applications to representation theory, as it provides a natural (functorial) quantization procedure which constructs nontrivial unitary representations of those Lie groups which arise as isometry groups of static, Lorentz-signature space-times. These groups are typically noncompact. For example, when applied to AdS d+1 , our procedure gives a unitary representation of the identity component of SO(d, 2). Moreover, our procedure makes use of the Cartan decomposition, a standard tool in representation theory.\n\n2. Classical Space-Time 2.1. Structure of Static Space-Times. Definition 2.1. A quantizable static space-time is a complete, connected orientable Riemannian manifold (M, g ab ) with a globally-defined (smooth) Killing field ξ which is orthogonal to a codimension-one hypersurface Σ ⊂ M , such that the orbits of ξ are complete and each orbit intersects Σ exactly once.\n\nThroughout this paper, we assume that M is a quantizable static spacetime. Definition 2.1 implies that there is a global time function t defined up to a constant by the requirement that ξ = ∂/∂t. Thus M is foliated by time-slices M t , and M = Ω -∪ Σ ∪ Ω +\n\nwhere the unions are disjoint, Σ = M 0 , and Ω ± are open sets corresponding to t > 0 and t < 0 respectively. We infer existence of an isometry θ which reverses the sign of t,\n\nθ : Ω ± → Ω ∓ such that θ 2 = 1, θ| Σ = id.\n\nFix a self-adjoint extension of the Laplacian, and let C = (-∆ + m 2 ) -1 be the resolvent of the Laplacian (also called the free covariance), where m 2 > 0. Then C is a bounded self-adjoint operator on L 2 (M ). For each s ∈ R, the Sobolev space H s (M ) is a real Hilbert space, defined as completion of C ∞ c (M ) in the norm (2.1) f 2 s = f, C -s f . The inclusion H s ֒→ H s+k for k > 0 is Hilbert-Schmidt. Define S := s<0 H s (M ) and S ′ := s>0 H s (M ). Then S ⊂ H -1 (M ) ⊂ S ′ form a Gelfand triple, and S is a nuclear space. Recall that S ′ has a natural σ-algebra of measurable sets (see for instance [7, 8, 17] ). There is a unique Gaussian probability measure µ with mean zero and covariance C defined on the cylinder sets in S ′ (see [7])."
},
{
"section_type": "BACKGROUND",
"section_title": "QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 3",
"text": "More generally, one may consider a non-Gaussian, countably-additive measure µ on S ′ and the space E := L 2 (S ′ , µ).\n\nWe are interested in the case that the monomials of the form A(Φ) = Φ(f 1 ) . . . Φ(f n ) for f i ∈ S are all elements of E , and for which their span is dense in E . This is of course true if µ is the Gaussian measure with covariance C.\n\nFor an open set Ω ⊂ M , let E Ω denote the closure in E of the set of monomials A(Φ) = i Φ(f i ) where supp(f i ) ⊂ Ω for all i. Of particular importance for Euclidean quantum field theory is the positive-time subspace E + := E Ω + .\n\n2.2. The Operator Induced by an Isometry. Isometries of the underlying space-time manifold act on a Hilbert space of classical fields arising in the study of a classical field theory. For f ∈ C ∞ (M ) and ψ : M → M an isometry, define f ψ ≡ (ψ -1 ) * f = f • ψ -1 .\n\nSince det(dψ) = 1, the operation f → f ψ extends to a bounded operator on H ±1 (M ) or on L 2 (M ). A treatment of isometries for static space-times appears in [11] . Definition 2.2. Let ψ be an isometry, and A(Φ) = Φ(f 1 ) . . . Φ(f n ) ∈ E a monomial. Define the induced operator (2.2) Γ(ψ)A ≡ Φ(f 1 ψ ) . . . Φ(f n ψ ) , and extend Γ(ψ) by linearity to the domain of polynomials in the fields, which is dense in E .\n\n3. Osterwalder-Schrader Quantization 3.1. Quantization of Vectors (The Hilbert Space H of Quantum Theory). In this section we define the quantization map E + → H , where H is the Hilbert space of quantum theory. The existence of the quantization map relies on a condition known as Osterwalder-Schrader (or reflection) positivity. A probability measure µ on S ′ is said to be reflection positive if (3.1) Γ(θ)F F dµ ≥ 0 for all F in the positive-time subspace E + ⊂ E . Let Θ = Γ(θ) be the reflection on E induced by θ. Define the sesquilinear form (A, B) on E + × E + as (A, B) = ΘA, B E , so (3.1) states that (F, F ) ≥ 0. Assumption 1 (O-S Positivity). Any measure dµ that we consider is reflection positive with respect to the time-reflection Θ.\n\n4 ARTHUR JAFFE AND GORDON RITTER Definition 3.1 (OS-Quantization). Given a reflection-positive measure dµ, the Hilbert space H of quantum theory is the completion of E + /N with respect to the inner product given by the sesquilinear form (A, B). Denote the quantization map Π for vectors E + → H by Π(A) = Â, and write (3.2) Â, B H = (A, B) = ΘA, B E for A, B ∈ E + .\n\n3.2. Quantization of Operators. The basic quantization theorem gives a sufficient condition to map a (possibly unbounded) linear operator T on E to its quantization, a linear operator T on H . Consider a densely-defined operator T on E , the unitary time-reflection Θ, and the adjoint T + = ΘT * Θ. A preliminary version of the following was also given in [10] .\n\nDefinition 3.2 (Quantization Condition I). The operator T satisfies QC-I if: i. The operator T has a domain D(T ) dense in E . ii. There is a subdomain\n\nD 0 ⊂ E + ∩ D(T ) ∩ D(T + ), for which D 0 ⊂ H is dense.\n\niii. The transformations T and T + both map D 0 into E + .\n\nTheorem 3.3 (Quantization I). If T satisfies QC-I, then i. The operators T ↾D 0 and T + ↾D 0 have quantizations T and T + with domain D0 . ii. The operators T * = T ↾ D0 * and T + agree on D0 .\n\niii. The operator T ↾D 0 has a closure, namely T * * .\n\nProof. We wish to define the quantization T with the putative domain D0 by (3.3) T  = T A .\n\nFor any vector A ∈ D 0 and for any B ∈ (D 0 ∩ N ), it is the case that  = A + B. The transformation T is defined by (3.3) iff T A = T (A + B) = T A + T B. Hence one needs to verify that T : D 0 ∩ N → N , which we now do.\n\nThe assumption D 0 ⊂ D(T + ), along with the fact that Θ is unitary, ensures that ΘD 0 ⊂ D(T * ). Therefore for any\n\nF ∈ D 0 , (3.4) ΘF, T B E = T * ΘF, B E = Θ (ΘT * ΘF ) , B E = ΘT + F, B E = T + F , B H .\n\nIn the last step we use the fact assumed in part (iii) of QC-I that T + : D 0 → E + , yielding the inner product of two vectors in H . We infer from the Schwarz inequality in H that\n\n| ΘF, T B E | ≤ T + F H B H = 0 . As ΘF, T B E = F , T B H , this means that T B ⊥ D0 .\n\nAs D0 is dense in H by QC-I.ii, we infer T B = 0. In other words, T B ∈ N as required to define T ."
},
{
"section_type": "BACKGROUND",
"section_title": "QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 5",
"text": "In order show that D0 ⊂ D( T * ), perform a similar calculation to (3.4) with arbitrary A ∈ D 0 replacing B, namely (3.5)\n\nF , T  H = ΘF, T A E = Θ (ΘT * ΘF ) , A E = ΘT + F , A E = T + F ,  H .\n\nThe right side is continuous in  ∈ H , and therefore F ∈ D(T * ). Furthermore T * F = T + F . This identity shows that if F ∈ N , then T + F = 0. Hence T + ↾D 0 has a quantization T + , and we may write (3.5) as (3.6) T * F = T + F , for all F ∈ D 0 .\n\nIn particular T * is densely defined so T has a closure. This completes the proof. Definition 3.4 (Quantization Condition II). The operator T satisfies QC-II if i. Both the operator T and its adjoint T * have dense domains D(T ), D(T * ) ⊂ E . ii. There is a domain D 0 ⊂ E + in the common domain of T , T + , T + T , and T T + . iii. Each operator T , T + , T + T , and T T + maps D 0 into E + .\n\nTheorem 3.5 (Quantization II). If T satisfies QC-II, then i. The operators T ↾D 0 and T + ↾D 0 have quantizations T and T + with domain D0 . ii. If A, B ∈ D 0 , one has B, T  H = T + B,  H ."
},
{
"section_type": "OTHER",
"section_title": "Remarks.",
"text": "i. In Theorem 3.5 we drop the assumption that the domain D0 is dense, obtaining quantizations T and T + whose domains are not necessarily dense. In order to compensate for this, we assume more properties concerning the domain and the range of T + on E . ii. As D0 need not be dense in H , the adjoint of T need not be defined.\n\nNevertheless, one calls the operator T symmetric in case one has\n\n(3.7) B, T Â H = T B, Â H , for all A, B ∈ D 0 .\n\niii. If Ŝ ⊃ T is a densely-defined extension of T , then Ŝ * = T + on the domain D0 .\n\nProof. We define the quantization T with the putative domain D0 . As in the proof of Theorem 3.3, this quantization T is well-defined iff it is the case that T :\n\nD 0 ∩ N → N . For any F ∈ D 0 ∩ N , by definition F H = 0. Also T F, T F H = (T F, T F ) = ΘT F, T F E = F, T * ΘT F E ,\n\nwhere one uses the fact that\n\nD 0 ⊂ D(T + T ). Thus T F, T F H = ΘF, T + T F E = F, T + T F H ."
},
{
"section_type": "OTHER",
"section_title": "ARTHUR JAFFE AND GORDON RITTER",
"text": "Here we use the fact that T + T maps D 0 to E + . Thus one can use the Schwarz inequality on H to obtain\n\nT F, T F H ≤ F H T + T F H = 0 . Hence T : D 0 ∩ N → N , and T has a quantization T with domain D0 .\n\nIn order verify that T + ↾D 0 has a quantization, one needs to show that T + : D 0 ∩ N ⊂ N . Repeat the argument above with T + replacing T . The assumption T T + :\n\nD 0 → E + yields for F ∈ D 0 ∩ N , T + F, T + F H = T * ΘF, T + F E = ΘF, T T + F E = F , T T + F H .\n\nUse the Schwarz inequality in H to obtain the desired result that\n\nT + F, T + F H ≤ F H T T + F H = 0 . Hence T + has a quantization T + with domain D0 , and for B ∈ D 0 one has T + B = T + B. In order to establish (ii), assume that A, B ∈ D 0 . Then B, T Â H = ΘB, T A E = Θ (ΘT * ΘB) , A E = ΘT + B, A E = T + B, Â H = T + B, Â H . (3.8)\n\nThis completes the proof."
},
{
"section_type": "OTHER",
"section_title": "Structure and Representation of the Lie Algebra of Killing Fields",
"text": "For the remainder of this paper we assume the following, which is clearly true in the Gaussian case as the Laplacian commutes with the isometry group G. (A further explanation was given in [11] .) Assumption 2. The isometry groups G that we consider leave the measure dµ invariant, in the sense that Γ, defined above, is a unitary representation of G on E ."
},
{
"section_type": "OTHER",
"section_title": "4.1. The Representation of g on E .",
"text": "The Representation of g on E . Lemma 4.1. Let G i be an analytic group with Lie algebra g i (i = 1, 2), and let λ : g 1 → g 2 be a homomorphism. There cannot exist more than one analytic homomorphism π : G 1 → G 2 for which dπ = λ. If G 1 is simply connected then there is always one such π.\n\nLet D = d/dt denote the canonical unit vector field on R. Let G be a real Lie group with algebra g, and let X ∈ g. The map tD → tX(t ∈ R) is a homomorphism of Lie(R) → g, so by the Lemma there is a unique analytic homomorphism\n\nξ X : R → G such that dξ X (D) = X. Conversely, if η is an analytic homomorphism of R → G, and if we let X = dη(D), it is obvious that η = ξ X . Thus X → ξ X is a bijection of g onto the set of analytic homomorphisms R → G.\n\nThe exponential map is defined by 1) . For complex Lie groups, the same argument applies, replacing R with C throughout. Since g is connected, so is exp(g). Hence exp(g) ⊆ G 0 , where G 0 denotes the connected component of the identity in G. It need not be the case for a general Lie group that exp(g) = G 0 , but for a large class of examples (the so-called exponential groups) this does hold. For any Lie group, exp(g) contains an open neighborhood of the identity, so the subgroup generated by exp(g) always coincides with G 0 .\n\nQUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 7 exp(X) := ξ X (\n\nWe will apply the above results with G = Iso(M ), the isometry group of M , and g = Lie(G) the algebra of global Killing fields. Thus we have a bijective correspondence between Killing fields and 1-parameter groups of isometries. This correspondence has a geometric realization: the 1-parameter group of isometries\n\nφ s = ξ X (s) = exp(sX)\n\ncorresponding to X ∈ g is the flow generated by X.\n\nConsider the two different 1-parameter groups of unitary operators: (1) the unitary group φ * s on L 2 (M ), and (2) the unitary group Γ(φ s ) on E .\n\nStone's theorem applies to both of these unitary groups to yield denselydefined self-adjoint operators on the respective Hilbert spaces.\n\nIn the first case, the relevant self-adjoint operator is simply an extension of -iX, viewed as a differential operator on C ∞ c (M ). This is because for f ∈ C ∞ c (M ) and p ∈ M , we have:\n\nX p f = (L X f )(p) = d ds f (φ s (p))| s=0 .\n\nThus -iX is a densely-defined symmetric operator on L 2 (M ), and Stone's theorem implies that -iX has self-adjoint extensions. In the second case, the unitary group Γ(φ s ) on E also has a self-adjoint generator Γ(X), which can be calculated explicitly. By definition,\n\ne -isΓ(X) n i=1 Φ(f i ) = n i=1 Φ(f i • φ -s ).\n\nNow replace s → -s and calculate d/ds| s=0 applied to both sides of the last equation to see that\n\nΓ(X) n i=1 Φ(f i ) = n j=1 Φ(f 1 ) . . . Φ(-iXf j )Φ(f j+1 ) . . . Φ(f n ) .\n\nOne may check that Γ is a Lie algebra representation of g,\n\ni.e. Γ([X, Y ]) = [Γ(X), Γ(Y )].\n\n4.2. The Cartan Decomposition of g. For each ξ ∈ g, there exists some dense domain in E on which Γ(ξ) is self-adjoint, as discussed previously.\n\n8 ARTHUR JAFFE AND GORDON RITTER However, the quantizations Γ(ξ) acting on H may be hermitian, antihermitian, or neither depending on whether there holds a relation of the form (4.1) Γ(ξ)Θ = ±ΘΓ(ξ), with one of the two possible signs, or whether no such relation holds. Even if (4.1) holds, to complete the construction of a unitary representation one must prove that there exists a dense domain in H on which Γ(ξ) is self-adjoint or skew-adjoint. This nontrivial problem will be dealt with in a later section using Theorems 3.3 and 3.5 and the theory of symmetric local semigroups [12, 4] . Presently we determine which elements within g satisfy relations of the form (4.1).\n\nLet ϑ := θ * as an operator on C ∞ (M ), and consider a Killing field X ∈ g also as an operator on C ∞ (M ). Define T : g → g by (4.2) T (X) := ϑXϑ.\n\nFrom (4.2) it is not obvious that the range of T is contained in g. To prove this, we recall some geometric constructions.\n\nLet M, N be manifolds, let ψ : M → N be a diffeomorphism, and X ∈ Vect(M ). Then\n\n(4.3) ψ -1 * Xψ * = X(• • ψ) • ψ -1 .\n\ndefines an operator on C ∞ (N ). One may check that this operator is a derivation, thus (4.3) defines a vector field on N . The vector field (4.3) is usually denoted ψ * X = dψ(X ψ -1 (p) ) and referred to as the push-forward of X.\n\nWe now wish to show that g = g + ⊕ g -, where g ± are the ±1-eigenspaces of T . This is proven by introducing an inner product (X, Y ) g on g with respect to which T is self-adjoint.\n\nTheorem 4.2. Consider g as a Hilbert space with inner product (X, Y ) g . The operator T : g → g is self-adjoint with T 2 = I; hence (4.4) g = g + ⊕ g - as an orthogonal direct sum of Hilbert spaces, where g ± are the ±1-eigenspaces of T . Further, ∂ t ∈ g -hence dim(g -) ≥ 1. Elements of g -have hermitian quantizations, while elements of g + have anti-hermitian quantizations. 2 Proof. Write (4.2) as (4.5) T (X) = θ -1 * Xθ * = θ * X .\n\nThus T is the operator of push-forward by θ. The push-forward of a Killing field by an isometry is another Killing field, hence the range of T is contained 2 It is not the case that g-consists only of ∂t. In particular, dim(g-) = 2 for M = H 2.\n\nIt can occur that dim g+ = 0.\n\nQUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 9 in g. Also, T must have a trivial kernel since T 2 = I, and this implies that T is surjective. It follows from (4.5) that T is a Hermitian operator on g. Hence T is diagonalizable and has real eigenvalues which are square roots of 1. This establishes the decomposition (4.4). That elements of g -have hermitian quantizations, while elements of g + have anti-hermitian quantizations follows from Theorem 3.3.\n\nA Cartan involution is a Lie algebra homomorphism g → g which squares to the identity. It follows from (4.2) that T is a Lie algebra homomorphism; thus, Theorem 4.2 implies that T is a Cartan involution of g. This implies that the eigenspaces (g + , g -) form a Cartan pair, meaning that\n\n(4.6) [g + , g + ] ⊂ g + , [g + , g -] ⊂ g -, and [g -, g -] ⊂ g + .\n\nClearly g + is a subalgebra while g -is not, and any subalgebra contained in g -is abelian."
},
{
"section_type": "OTHER",
"section_title": "Reflection-Invariant and Reflected Isometries",
"text": "Let G = Iso(M ) denote the isometry group of M , as above. Then G has a Z 2 subgroup containing {1, θ}. This subgroup acts on G by conjugation, which is just the action ψ → ψ θ := θψθ. Conjugation is an (inner) automorphism of the group, so\n\n(ψφ) θ = ψ θ φ θ , (ψ θ ) -1 = (ψ -1 ) θ .\n\nDefinition 5.1. We say that ψ ∈ G is reflection-invariant if\n\nψ θ = ψ,\n\nand that ψ is reflected if ψ θ = ψ -1 .\n\nLet G RI denote the subgroup of G consisting of reflection-invariant elements, and let G R denote the subset of reflected elements.\n\nNote that G RI is the stabilizer of the Z 2 action, hence a subgroup. An alternate proof of this proceeds using G RI = exp(g + ). Although G R is closed under the taking of inverses and does contain the identity, the product of two reflected isometries is no longer reflected unless they commute. Generally, the product of an element of G R with an element of G RI is neither an element of G R nor of G RI . The only isometry that is both reflection-invariant and reflected is θ itself. Thus we have:\n\nG R ∩ G RI = {1, θ} ⊂ G R ∪ G RI G.\n\nTheorem 5.2. Let G 0 denote the connected component of the identity in G. Then G 0 is generated by G R ∪ G RI . (This is a form of the Cartan decomposition for G.)\n\n10 ARTHUR JAFFE AND GORDON RITTER Proof. Since g = g + ⊕ g -as a direct sum of vector spaces (though not of Lie algebras), we have\n\nG 0 = exp(g) = exp(g + ) ∪ exp(g -) .\n\nChoose bases {ξ ±,i } i=1,...,n ± for g ± respectively. Then we have:\n\nG 0 = {exp(sξ +,i ) : 1 ≤ i ≤ n + , s ∈ R} ∪ {exp(sξ -,j ) : 1 ≤ j ≤ n -, s ∈ R} .\n\nFurthermore, exp(sξ -,i ) is reflected, while exp(sξ +,i ) is reflection-invariant, completing the proof.\n\nCorollary 5.3. The Lie algebra of the subgroup G RI is g + .\n\nTo summarize, the isometry group of a static space-time can always be generated by a collection of n (= dim g) one-parameter subgroups, each of which consists either of reflected isometries, or reflection-invariant isometries."
},
{
"section_type": "OTHER",
"section_title": "Construction of Unitary Representations",
"text": "6.1. Self-adjointness of Semigroups. In this section, we recall several known results on self-adjointness of semigroups. Roughly speaking, these results imply that if a one-parameter family S α of unbounded symmetric operators satisfies a semigroup condition of the form S α S β = S α+β , then under suitable conditions one may conclude essential self-adjointness.\n\nA theorem of this type appeared in a 1970 paper of Nussbaum [14], who assumed that the semigroup operators have a common dense domain. The result was rediscovered independently by Fröhlich, who applied it to quantum field theory in several important papers [5, 3] . For our intended application to quantum field theory, it turns out to be very convenient to drop the assumption that ∃ a such that the S α all have a common dense domain for |α| < a, in favor of the weaker assumption that α>0 D(S α ) is dense.\n\nA generalization of Nussbaum's theorem which allows the domains of the semigroup operators to vary with the parameter, and which only requires the union of the domains to be dense, was later formulated and two independent proofs were given: one by Fröhlich [4], and another by Klein and Landau [12] . The latter also used this theorem in their construction of representations of the Euclidean group and the corresponding analytic continuation to the Lorentz group [13] .\n\nIn order to keep the present article self-contained, we first define symmetric local semigroups and then recall the refined self-adjointness theorem of Fröhlich, and Klein and Landau. Definition 6.1. Let H be a Hilbert space, let T > 0 and for each α ∈ [0, T ], let S α be a symmetric linear operator on the domain\n\nD α ⊂ H , such that: (i) D α ⊃ D β if α ≤ β and D := 0<α≤T D α is dense in H , (ii) α → S α is weakly continuous, (iii) S 0 = I, S β (D α ) ⊂ D α-β for 0 ≤ β ≤ α ≤ T , and\n\nQUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 11\n\n(iv) S α S β = S α+β on D α+β for α, β, α + β ∈ [0, T ]. In this situation, we say that (S α , D α , T ) is a symmetric local semigroup.\n\nIt is important that D α is not required to be dense in H for each α; the only density requirement is (i).\n\nTheorem 6.2 ([12, 4]). For each symmetric local semigroup (S α , D α , T ), there exists a unique self-adjoint operator A such that 3\n\nD α ⊂ D(e -αA ) and S α = e -αA | Dα for all α ∈ [0, T ].\n\nAlso, A ≥ -c if and only if S α f ≤ e cα f for all f ∈ D α and 0 < α < T .\n\n6.2. Reflection-Invariant Isometries. Lemma 6.3. Let ψ be a reflection-invariant isometry and assume ∃ p ∈ Ω + such that ψ(p) ∈ Ω + . Then ψ preserves the positive-time subspace, i.e. ψ(Ω + ) ⊆ Ω + .\n\nProof. We first prove that ψ(Σ) ⊆ Σ. Suppose not; then ∃ p ∈ Σ with ψ(p) ∈ Σ. Assume ψ(p) ∈ Ω + (without loss of generality: we could repeat the same argument with ψ(p) ∈ Ω -). Then Ω + contains (θψθ)(p) = θψ(p) ∈ Ω -, a contradiction since Ω -∩Ω + = ∅. We used the fact that θ| Σ = id so θ(p) = p. Hence ψ restricts to an isometry of Σ. It follows that the restriction of ψ to M ′ = M \\ Σ is also an isometry. However, M ′ = Ω -⊔ Ω + , where ⊔ denotes the disjoint union. Therefore ψ(Ω + ) is wholly contained in either Ω + or Ω -, since ψ is a homeomorphism and so ψ(Ω + ) is connected. The possibility that ψ(Ω + ) ⊆ Ω -is ruled out by our assumption, so ψ(Ω + ) ⊆ Ω + . Lemma 6.3 has the immediate consequence that if ξ ∈ g + then the oneparameter group associated to ξ is positive-time-invariant. This result plays a key role in the proof of Theorem 6.4."
},
{
"section_type": "OTHER",
"section_title": "Construction of Unitary Representations.",
"text": "The rest of this section is devoted to proving that the theory of symmetric local semigroups can be applied to the quantized operators on H corresponding to each of a set of 1-parameter subgroups of G = Iso(M ). The proof relies upon Lemma 6.3, and Theorems 3.3, 3.5 and 6.2. Theorem 6.4. Let (M, g ab ) be a quantizable static space-time. Let ξ be a Killing field which lies in g + or g -, with associated one-parameter group of isometries {φ α } α∈R . Then there exists a densely-defined self-adjoint operator A ξ on H such that\n\nΓ(φ α ) = e -αA ξ , if ξ ∈ g - e iαA ξ if ξ ∈ g + .\n\n3 The authors of [4, 12] also showed that\n\nb D := [ 0<α≤S h [ 0<β<α S β (Dα) i , where 0 < S ≤ T,\n\nis a core for A, i.e. (A, b D) is essentially self-adjoint.\n\n12 ARTHUR JAFFE AND GORDON RITTER\n\nProof. First suppose that ξ ∈ g -, which implies that the isometries φ α are reflected, and so Γ(φ α ) + = Γ(φ α ). Define Ω ξ,α := φ -1 α (Ω + ). For all α in some neighborhood of zero, Ω ξ,α is a nonempty open subset of Ω + , and moreover, as α → 0 + , Ω ξ,α increases to fill Ω + with Ω ξ,0 = Ω + . These statements follow immediately from the fact that, for each p ∈ Ω + , φ α (p) is continuous with respect to α, and φ 0 is the identity map. Since φ α (Ω ξ,α ) ⊆ Ω + , we infer that Γ(φ α )E Ω ξ,α ⊆ E + . By Theorem 3.5, Γ(φ α ) has a quantization which is a symmetric operator on the domain\n\nD ξ,α := Π(E Ω ξ,α ).\n\nNote that D ξ,α is not necessarily dense in H . 4 We now show that Theorem 6.2 can be applied.\n\nFix some positive constant a with Ω ξ,a nonempty. Note that 0<α≤a\n\nΩ ξ,α = Ω + ⇒ 0<α≤a E Ω ξ,α = E + . It follows that D ξ := 0<α≤a D ξ,α\n\nis dense in H . This establishes condition (i) of Definition 6.1, and the other conditions are routine verifications. Theorem 6.2 implies existence of a densely-defined self-adjoint operator A ξ on H , such that Γ(φ α ) = exp(-αA ξ ) for all α ∈ [0, a] .\n\nThis proves the theorem in case ξ ∈ g -. Now suppose that ξ ∈ g + , implying that the isometries φ α are reflectioninvariant, and Γ(φ α\n\n) + = Γ(φ α ) -1 = Γ(φ -α ) on E . Lemma 6.3 implies that Γ(φ α )E + ⊆ E + . By Theorem 3.3, Γ(φ α ) has a quantization Γ(φ α ) which is defined and satisfies Γ(φ α ) * = Γ(φ α ) -1 on the domain Π(E + ), which is dense in H by definition.\n\nIn this case we do not need Theorem 6.2; for each α, Γ(φ α ) extends by continuity to a oneparameter unitary group defined on all of H (not only for a dense subspace). By Stone's theorem, Γ(φ α ) = exp(iαA ξ ) for A ξ self-adjoint and for all α ∈ R. The proof is complete.\n\n4 Density of D ξ,α would be implied by a Reeh-Schlieder theorem, which we do not prove except in the free case. Theorem 6.2 removes the need for a Reeh-Schlieder theorem in this argument.\n\nQUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 13"
},
{
"section_type": "OTHER",
"section_title": "Analytic Continuation",
"text": "Each Riemannian static space-time (M, g ab ) has a Lorentzian continuation M lor , which we construct as follows. In adapted coordinates, the metric g ab on M takes the form (7.1)\n\nds 2 = F (x)dt 2 + G µν (x)dx µ dx ν .\n\nThe analytic continuation t → -it of (7.1) is standard and gives a metric of Lorentz signature, ds 2 lor = -F dt 2 + G dx 2 , by which we define the Lorentzian space-time M lor . Einstein's equation Ric g = k g is preserved by the analytic continuation, but we do not use this fact anywhere in the present paper. Let {ξ\n\n(±) i : 1 ≤ i ≤ n ± } be bases of g ± , respectively. Let A (±) i = A ξ (±) i be the densely-defined self-adjoint operators on H , constructed by Theorem 6.4. Let (7.2) U (±) i (α) = exp(iαA (±) i ) , for 1 ≤ i ≤ n ± be the associated one-parameter unitary groups on H .\n\nWe claim that the group generated by the n = n + + n -one-parameter unitary groups (7.2) is isomorphic to the identity component of\n\nG lor := Iso(M lor ),\n\nthe group of Lorentzian isometries. Since locally, the group structure is determined by its Lie algebra, it suffices to check that the generators satisfy the defining relations of g lor := Lie(G lor ).\n\nSince quantization of operators preserves multiplication, we have\n\n(7.3) X, Y, Z ∈ g, [X, Y ] = Z ⇒ [ Γ(X), Γ(Y )] = Γ(Z).\n\nIn what follows, we will use the notation\n\ng ± for { Γ(X) : X ∈ g ± }.\n\nQuantization converts the elements of g -from skew operators into Hermitian operators; i.e. elements of g -are Hermitian on H and hence, elements of i g -are skew-symmetric on H . Thus g + ⊕ i g -is a Lie algebra represented by skew-symmetric operators on H .\n\nTheorem 7.1. We have an isomorphism of Lie algebras:\n\n(7.4) g lor ∼ = g + ⊕ i g -.\n\nProof. Let M C be the manifold obtained by allowing the t coordinate to take values in C. Define ψ :\n\nM C → M C by t → -it. Then g lor is generated by {ξ (+) i } 1≤i≤n + ∪ {η j } 1≤j≤n -,\n\nwhere η j := iψ * ξ (-) j .\n\nIt is possible to define a set of real structure constants f ijk such that (7.5) [ξ\n\n(-) i , ξ (-) j ] = n + k=1 f ijk ξ (+) k .\n\n14 ARTHUR JAFFE AND GORDON RITTER Applying ψ * to both sides of (7.5), the commutation relations of g lor are seen to be\n\n(7.6) [η i , η j ] = -f ijk ξ\n\n(+) k , together with the same relations for g + as before. Now (7.3) implies that (7.6) are the precisely the commutation relations of g + ⊕ i g -, completing the proof of (7.4).\n\nCorollary 7.2. Let (M, g ab ) be a quantizable static space-time. The unitary groups (7.2) determine a unitary representation of G 0 lor on H .\n\n7.1. Conclusions. We have obtained the following conclusions. There is a unitary representation of the group G 0 lor on the physical Hilbert space H of quantum field theory on the static space-time M . This representation maps the time-translation subgroup into the unitary group exp(itH), where the energy H ≥ 0 is a positive, densely-defined self-adjoint operator corresponding to the Hamiltonian of the theory. The Hilbert space H contains a ground state Ψ 0 = 1 which is such that HΨ 0 = 0 and Ψ 0 is invariant under the action of all spacetime symmetries. We obtain these results via analytic continuation from the Euclidean path integral, under mild assumptions on the measure which should include all physically interesting examples. This is done without introducing the field operators; nonetheless, Theorems 3.3 and 3.5 do suffice to construct field operators. In the special case M = R d with G = SO(4), we obtain a unitary representation of the proper orthochronous Lorentz group, G 0 lor = L ↑ + = SO 0 (3, 1)."
},
{
"section_type": "OTHER",
"section_title": "Hyperbolic Space and Anti-de Sitter Space",
"text": "Consider Euclidean quantum field theory on M = H d . The metric is\n\nds 2 = r -2 d i=1 dx 2 i ,\n\nwhere we define r = x d for convenience. The Laplacian is\n\n(8.1) ∆ H d = (2 -d)r ∂ ∂r + r 2 ∆ R d .\n\nThe d -1 coordinate vector fields {∂/∂x i : i = d} are all static Killing fields, and any one of the coordinates x i (i = d) is a satisfactory representation of time in this space-time. It is convenient to define t = x 1 as before, and to identify t with time. The time-zero slice is M 0 = H d-1 . From d, 1) and the orientation-preserving isometry group is SO + (d, 1).\n\nH d = {v ∈ R d,1 | v, v = -1, v 0 > 0} it follows that Isom(H d ) = O + (\n\nQUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 15\n\nFigure 1. Flow lines of the Killing field ζ = (t 2r 2\n\n)∂ t + 2tr ∂ r on H d .\n\nFor constant curvature spaces, one may solve Killing's equation L K g = 0 explicitly. Let us illustrate the solutions and their quantizations for d = 2. The three Killing fields\n\n(8.2) ξ = ∂ t , η = t∂ t + r∂ r , ζ = (t 2 -r 2 )∂ t + 2tr ∂ r\n\nare a convenient basis for g. Any d-dimensional manifold satisfies dim g ≤ d(d + 1)/2, manifolds saturating the bound are said to be maximally symmetric, and H d is maximally symmetric. Now, ∂ t f (-t) = -f ′ (-t) so ∂ t Θ = -Θ∂ t , hence ∂ t ∈ g -. Similar calculations show [Θ, η] = 0 and Θζ = -ζΘ. Thus η spans g + , while ∂ t , ζ span g -. The commutation relations 5 for g are:\n\n[η, ζ] = ζ, [η, ∂ t ] = -∂ t , [ζ, ∂ t ] = -2η.\n\nThese calculations verify that (g + , g -) forms a Cartan pair, as defined in (4.6) .\n\nThe flows associated to (8.2) are easily visualized: ξ is a right-translation, and η flow-lines are radially outward from the Euclidean origin. The flows of ζ are Euclidean circles, indicated by the darker lines in Figure 1 . Hence the flows of η are defined on all of E + , while the flows of ζ are analogous to space-time rotations in R 2 , and hence, must be defined on a wedge of the form W α = {(t, r) : t, r > 0, tan -1 (r/t) < α}.\n\nThe simple geometric idea of Section 6.2 is nicely confirmed in this case: the flows of η (the generator of g + ) preserve the t = 0 plane, and are separately isometries of Ω + and Ω -. Corollary 7.2 implies that the procedure outlined above defines a unitary representation of the identity component of Iso(AdS 2 ) on the physical Hilbert space H for quantum field theory on this background, including theories with interactions that preserve the symmetry. Since Iso(AdS d+1 ) = 5 Note that quite generally [g-, g-] ⊆ g+ so it's automatic that [ζ, ∂t] is proportional to η.\n\n16 ARTHUR JAFFE AND GORDON RITTER SO(d, 2), we have a unitary representation of SO 0 (1, 2). The latter is a noncompact, semisimple real Lie group, and thus it has no finite-dimensional unitary representations, but a host of interesting infinite-dimensional ones."
},
{
"section_type": "OTHER",
"section_title": "Appendix A. Euclidean Reeh-Schlieder Theorem",
"text": "We prove the Euclidean Reeh-Schlieder property for free theories on curved backgrounds. It is reasonable to expect this property to extend to interacting theories on curved backgrounds, but it would have to be established for each such model since it depends explicitly on the two-point function.\n\nThe Reeh-Schlieder theorem guarantees the existence of a dense quantization domain based on any open subset of Ω + . For this reason, one could use the Reeh-Schlieder (RS) theorem with Nussbaum's theorem [14] to construct a second proof of Theorem 6.4 under the additional assumption that M is real-analytic.\n\nFortunately, our proof of Theorem 6.4 is completely independent of the Reeh-Schlieder property. This has two advantages: we do not have to assume M is a real-analytic manifold and, more importantly, our proof of Theorem 6.4 generalizes immediately and transparently to interacting theories as long as the Hilbert space H is not modified by the interaction.\n\nWe state and prove this using the one-particle space; however, the result clearly extends to the quantum-field Hilbert space.\n\nTheorem A.1. Let M be a quantizable static space-time endowed with a real-analytic structure, and assume that g ab is real-analytic. Let O ⊂ Ω + and\n\nD = C ∞ (O) ⊂ L 2 (Ω + ). Then D ⊥ = {0}. Proof. Let f ∈ L 2 (Ω + ) with f ⊥ D. For x ∈ Ω + , define η(x) := f , δx H = Θf, Cδ x L 2 .\n\nReal-analyticity of η(x) follows from the real-analyticity of (the integral kernel of) C, which in turn follows from the elliptic regularity theorem in the real-analytic category (see for instance [1, Sec. II.1.3]). Now by assumption, for any g ∈ C ∞ c (O), we have 0 = ĝ, f H = Θf, Cg L 2 (M ) .\n\nLet g → δ x for x ∈ O. Then 0 = Θf, Cδ x L 2 ≡ η(x). Since η| O = 0, by real-analyticity we infer the vanishing of η on Ω + , completing the proof.\n\nAcknowledgements. We are grateful to Hanno Gottschalk and Alexander Strohmaier for helpful discussions, and G.R. is grateful to the Universität Bonn for their hospitality during February 2007.\n\nReferences [1] Lipman Bers, Fritz John, and Martin Schechter. Partial differential equations. American Mathematical Society, Providence, R.I., 1979. Lectures in Applied Mathematics 3.\n\nQUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES 17 [2] N. D. Birrell and P. C. W. Davies. Quantum fields in curved space, volume 7 of Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1982. [3] W. Driessler and J. Fröhlich. The reconstruction of local observable algebras from the Euclidean Green's functions of relativistic quantum field theory. Ann. Inst. H. Poincaré Sect. A (N.S.), 27(3):221-236, 1977. [4] J. Fröhlich. Unbounded, symmetric semigroups on a separable Hilbert space are essentially selfadjoint. Adv. in Appl. Math., 1(3):237-256, 1980. [5] Jürg Fröhlich. The pure phases, the irreducible quantum fields, and dynamical symmetry breaking in Symanzik-Nelson positive quantum field theories. Ann. Physics, 97(1):1-54, 1976. [6] Stephen A. Fulling. Aspects of quantum field theory in curved spacetime, volume 17 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1989. [7] I. M. Gel ′ fand and N. Ya. Vilenkin. Generalized functions. Vol. 4. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977]. Applications of harmonic analysis, Translated from the Russian by Amiel Feinstein. [8] James Glimm and Arthur Jaffe. Quantum physics. Springer-Verlag, New York, second edition, 1987. A functional integral point of view. [9] Arthur Jaffe. Constructive quantum field theory. In Mathematical physics 2000, pages 111-127. Imp. Coll. Press, London, 2000. [10] Arthur Jaffe. Introduction to Quantum Field Theory. 2005. Lecture notes from Harvard Physics 289r, available in part online at http://www.arthurjaffe.com/Assets/pdf/IntroQFT.pdf .\n\n[11] Arthur Jaffe and Gordon Ritter. Quantum field theory on curved backgrounds. i. the euclidean functional integral. Comm. Math. Phys., 270(2):545-572, 2007. [12] Abel Klein and Lawrence J. Landau. Construction of a unique selfadjoint generator for a symmetric local semigroup. J. Funct. Anal., 44(2):121-137, 1981. [13] Abel Klein and Lawrence J. Landau. From the Euclidean group to the Poincaré group via Osterwalder-Schrader positivity. Comm. Math. Phys., 87(4):469-484, 1983. [14] A. E. Nussbaum. Spectral representation of certain one-parametric families of symmetric operators in Hilbert space. Trans. Amer. Math. Soc., 152:419-429, 1970. [15] Konrad Osterwalder and Robert Schrader. Axioms for Euclidean Green's functions. Comm. Math. Phys., 31:83-112, 1973. [16] Konrad Osterwalder and Robert Schrader. Axioms for Euclidean Green's functions. II. Comm. Math. Phys., 42:281-305, 1975. With an appendix by Stephen Summers.\n\n18 ARTHUR JAFFE AND GORDON RITTER [17] Barry Simon. The P (φ) 2 Euclidean (quantum) field theory. Princeton University Press, Princeton, N.J., 1974. Princeton Series in Physics. [18] Robert M. Wald. Quantum field theory in curved space-time. In Gravitation et quantifications (Les Houches, 1992), pages 63-167. North-Holland, Amsterdam, 1995."
}
] | ||
arxiv:0704.0053 | 0704.0053 | 1 | 39613cb738fb14c9fcd290d9a7ddb7c314a984abeab7f7e78d81855d1e36fb82 | A Global Approach to the Theory of Special Finsler Manifolds | The aim of the present paper is to provide a global presentation of the
theory of special Finsler manifolds. We introduce and investigate globally (or
intrinsically, free from local coordinates) many of the most important and most
commonly used special Finsler manifolds: locally Minkowskian, Berwald,
Landesberg, general Landesberg, $P$-reducible, $C$-reducible,
semi-$C$-reducible, quasi-$C$-reducible, $P^{*}$-Finsler, $C^{h}$-recurrent,
$C^{v}$-recurrent, $C^{0}$-recurrent, $S^{v}$-recurrent, $S^{v}$-recurrent of
the second order, $C_{2}$-like, $S_{3}$-like, $S_{4}$-like, $P_{2}$-like,
$R_{3}$-like, $P$-symmetric, $h$-isotropic, of scalar curvature, of constant
curvature, of $p$-scalar curvature, of $s$-$ps$-curvature. The global
definitions of these special Finsler manifolds are introduced. Various
relationships between the different types of the considered special Finsler
manifolds are found. Many local results, known in the literature, are proved
globally and several new results are obtained. As a by-product, interesting
identities and properties concerning the torsion tensor fields and the
curvature tensor fields are deduced. Although our investigation is entirely
global, we provide; for comparison reasons, an appendix presenting a local
counterpart of our global approach and the local definitions of the special
Finsler spaces considered. | [
"Nabil L. Youssef",
"S. H. Abed and A. Soleiman"
] | [
"math.DG",
"gr-qc"
] | math.DG | [] | 2007-03-31 | 2026-02-26 | The aim of the present paper is to provide a global presentation of the theory of special Finsler manifolds. We introduce and investigate globally (or intrinsically, free from local coordinates) many of the most important and most commonly used special Finsler manifolds : locally Minkowskian, Berwald, Landesberg, general Landesberg, P -reducible, C-reducible, semi-C-reducible, quasi-C-reducible, P * -Finsler, C h -recurrent, C v -recurrent, C 0 -recurrent, S v -recurrent, S v -recurrent of the second order, C 2 -like, S 3 -like, S 4 -like, P 2 -like, R 3 -like, P -symmetric, h-isotropic, of scalar curvature, of constant curvature, of p-scalar curvature, of s-ps-curvature. The global definitions of these special Finsler manifolds are introduced. Various relationships between the different types of the considered special Finsler manifolds are found. Many local results, known in the literature, are proved globally and several new results are obtained. As a by-product, interesting identities and properties concerning the torsion tensor fields and the curvature tensor fields are deduced. Although our investigation is entirely global, we provide; for comparison reasons, an appendix presenting a local counterpart of our global approach and the local definitions of the special Finsler spaces considered. 1
In Finsler geometry all geometric objects depend not only on positional coordinates, as in Riemannian geometry, but also on directional arguments. In Riemannian geometry there is a canonical linear connection on the manifold M, while in Finsler geometry there is a corresponding canonical linear connection, due to E. Cartan, which is not a connection on M but is a connection on π -1 (T M), the pullback of the tangent bundle T M by π : T M -→ M (the pullback approach). Moreover, in Riemannian geometry there is one curvature tensor and one torsion tensor associated with a given linear connection on the manifold M, whereas in Finsler geometry there are three curvature tensors and five torsion tensors associated with a given linear connection on π -1 (T M).
Most of the special spaces in Finsler geometry are derived from the fact that the π-tensor fields (torsions and curvatures) associated with the Cartan connection satisfy special forms. Consequently, special spaces of Finsler geometry are more numerous than those of Riemannian geometry. Special Finsler spaces are investigated locally (using local coordinates) by many authors: M. Matsumoto [16] , [18] , [15] , [14] and others [6] , [19] , [8] , [7] . On the other hand, the global (or intrinsic, free from local coordinates) investigation of such spaces is very rare in the literature. Some considerable contributions in this direction are due to A. Tamim [24] , [25] .
In the present paper, we provide a global presentation of the theory of special Finsler manifolds. We introduce and investigate globally many of the most important and most commonly used special Finsler manifolds : locally Minkowskian, Berwald, Landesberg, general Landesberg, P -reducible, C-reducible, semi-C-reducible, quasi-C-reducible, P * -Finsler, C h -recurrent, C v -recurrent, C 0 -recurrent, S v -recurrent, S vrecurrent of the second order, C 2 -like, S 3 -like, S 4 -like, P 2 -like, R 3 -like, P -symmetric, h-isotropic, of scalar curvature, of constant curvature, of p-scalar curvature, of s-pscurvature.
The paper consists of two parts, preceded by a preliminary section ( §1), which provides a brief account of the basic concepts of the pullback approach to Finsler geometry necessary to this work. For more detail, the reader is referred to [1] , [3] , [5] and [24] .
In the first part ( §2), we introduce the global definitions of the aforementioned special Finsler manifolds in such a way that, when localized, they yield the usual local definitions current in the literature (see the appendix). The definitions are arranged according to the type of the defining property of the special Finsler manifold concerned.
In the second part ( §3), various relationships between the different types of the considered special Finsler manifolds are found. Many local results, known in the literature, are proved globally and several new results are obtained. As a by-product of some of the obtained results, interesting identities and properties concerning the torsion tensor fields and the curvature tensor fields are deduced, which in turn play a key role in obtaining other results.
Among the obtained results are: a characterization of Riemannian manifolds, a characterization of S v -recurrent manifolds, a characterization of P -symmetric manifolds, a characterization of Berwald manifolds (in certain cases), the equivalence of Landsberg and general Landsberg manifolds under certain conditions, a classifica-tion of h-isotropic C h -recurrent manifolds and a presentation of different conditions under which an R 3 -like Finsler manifold becomes a Finsler manifold of s-ps curvature. The above results are just a non-exhaustive sample of the global results obtained in this paper.
It should finally be noted that some important results of [8] , [9] , [11] , [13] , [19] , [20] ,...,etc. (obtained in local coordinates) are immediately derived from the obtained global results (when localized).
Although our investigation is entirely global, we conclude the paper with an appendix presenting a local counterpart of our global approach and the local definitions of the special Finsler spaces considered. This is done to facilitate comparison and to make the paper more self-contained.
In this section, we give a brief account of the basic concepts of the pullback formalism of Finsler geometry necessary for this work. For more details refer to [1] , [3] , [5] and [24] . We make the general assumption that all geometric objects we consider are of class C ∞ . The following notations will be used throughout this paper: M: a real differentiable manifold of finite dimension n and of class C ∞ , F(M): the R-algebra of differentiable functions on M, X(M): the F(M)-module of vector fields on M, π M : T M -→ M: the tangent bundle of M, π : T M -→ M: the subbundle of nonzero vectors tangent to M, V (T M): the vertical subbundle of the bundle T T M, P : π -1 (T M) -→ T M : the pullback of the tangent bundle T M by π, P * : π -1 (T * M) -→ T M : the pullback of the cotangent bundle T * M by π, X(π(M)): the F(T M)-module of differentiable sections of π -1 (T M).
Elements of X(π(M)) will be called π-vector fields and will be denoted by barred letters X. Tensor fields on π -1 (T M) will be called π-tensor fields. The fundamental π-vector field is the π-vector field η defined by η(u) = (u, u) for all u ∈ T M. The lift to π -1 (T M) of a vector field X on M is the π-vector field X defined by
X(u) = (u, X(π(u))). The lift to π -1 (T M) of a 1-form ω on M is the π-form ω defined by ω(u) = (u, ω(π(u))).
The tangent bundle T (T M) is related to the pullback bundle π -1 (T M) by the short exact sequence 0
-→ π -1 (T M) γ -→ T (T M) ρ -→ π -1 (T M) -→ 0,
where the bundle morphisms ρ and γ are defined respectively by ρ = (π T M , dπ) and γ(u, v) = j u (v), where j u is the natural isomorphism j u :
T π M (v) M -→ T u (T π M (v) M).
Let ∇ be a linear connection (or simply a connection) in the pullback bundle π -1 (T M). We associate to ∇ the map
K : T T M -→ π -1 (T M) : X -→ ∇ X η, called the connection (or the deflection) map of ∇. A tangent vector X ∈ T u (T M) is said to be horizontal if K(X) = 0 . The vector space H u (T M) = {X ∈ T u (T M) : K(X) = 0} of the horizontal vectors at u ∈ T M is called the horizontal space to M at u . The connection ∇ is said to be regular if T u (T M) = V u (T M) ⊕ H u (T M) ∀u ∈ T M.
If M is endowed with a regular connection, then the vector bundle maps
γ : π -1 (T M) -→ V (T M), ρ| H(T M ) : H(T M) -→ π -1 (T M), K| V (T M ) : V (T M) -→ π -1 (T M) are vector bundle isomorphisms. Let us denote β = (ρ| H(T M ) ) -1 , then ρoβ = id π -1 (T M ) , βoρ = id H(T M ) on H(TM) 0 on V(TM) (1.1)
For a regular connection ∇ we define two covariant derivatives 1 ∇ and
∇ as follows: For every vector (1)π-form A, we have
( 1 ∇ A)(øX, øY ) := (∇ βøX A)(øY ) , ( 2 ∇ A)(øX, øY ) := (∇ γøX A)(øY ).
The classical torsion tensor T of the connection ∇ is defined by
T(X, Y ) = ∇ X ρY -∇ Y ρX -ρ[X, Y ] ∀ X, Y ∈ X(T M).
The horizontal ((h)h-) and mixed ((h)hv-) torsion tensors, denoted respectively by Q and T , are defined by
Q(X, Y ) = T(βXβY ), T (X, Y ) = T(γX, βY ) ∀ X, Y ∈ X(π(M)).
The classical curvature tensor K of the connection ∇ is defined by
K(X, Y )ρZ = -∇ X ∇ Y ρZ + ∇ Y ∇ X ρZ + ∇ [X,Y ] ρZ ∀ X, Y, Z ∈ X(T M).
The horizontal (h-), mixed (hv-) and vertical (v-) curvature tensors, denoted respectively by R, P and S, are defined by
R(X, Y )øZ = K(βXβY )øZ, P (X, Y )øZ = K(βX, γY )øZ, S(X, Y )øZ = K(γX, γY )øZ.
We also have the (v)h-, (v)hv-and (v)v-torsion tensors, denoted respectively by R, P and S, defined by
R(X, Y ) = R(X, Y )øη, P (X, Y ) = P (X, Y )øη, S(X, Y ) = S(X, Y )øη.
Theorem 1.1. [25] Let (M, L) be a Finsler manifold. There exists a unique regular connection
∇ in π -1 (T M) such that (a) ∇ is metric : ∇g = 0, (b) The horizontal torsion of ∇ vanishes : Q = 0, (c) The mixed torsion T of ∇ satisfies g(T (X, Y ), Z) = g(T (X, Z), Y ).
Such a connection is called the Cartan connection associated to the Finsler manifold (M, L).
One can show that the torsion T of the Cartan connection has the property that T (X, η) = 0 for all X ∈ X(π(M)) and associated to T we have: Definition 1.2. [25] Let ∇ be the Cartan connection associated to (M, L). The torsion tensor field T of the connection ∇ induces a π-tensor field of type (0, 3), called the Cartan tensor and denoted again T , defined by :
T (X, Y , Z) = g(T (X, Y ), Z), for all X, Y , Z ∈ X(T M).
It also induces a π-form C, called the contracted torsion, defined by : C(X) := T r{Y -→ T (X, Y )}, for all X ∈ X(T M). Definition 1.3. [25] With respect to the Cartan connection ∇ associated to (M, L), we have -The horizontal and vertical Ricci tensors Ric h and Ric v are defined respectively by:
Ric h (X, Y ) := T r{Z -→ R(X, Z)Y }, for all X, Y ∈ X(T M), Ric v (X, Y ) := T r{Z -→ S(X, Z)Y }, for all X, Y ∈ X(T M).
-The horizontal and vertical Ricci maps Ric h 0 and Ric v 0 are defined respectively by:
g(Ric h 0 (X), Y ) := Ric h (X, Y ), for all X, Y ∈ X(T M), g(Ric v 0 (X), Y ) := Ric v (X, Y ), for all X, Y ∈ X(T M).
-The horizontal and vertical scalar curvatures Sc h , Sc v are defined respectively by:
Sc h := Tr(Ric h 0 ), Sc v := Tr(Ric v 0 )
, where R and S are respectively the horizontal and vertical curvature tensors of ∇.
Proposition 1.4. [12] Let (M, L) be a Finsler manifold. The vector field G determined by i G Ω = -dE is a spray, called the canonical spray associated to the energy E, where E := 1 2 L 2 and Ω := dd J E. One can show, in this case, that G = βoη, and G is thus horizontal with respect to the Cartan connection ∇.
In this section, we introduce the global definitions of the most important and commonly used special Finsler spaces in such a way that, when localized, they yield the usual local definitions existing in the literature (see the Appendix). Here we simply set the definitions, postponing investigation of the mutual relationships between these special Finsler spaces to the next section. The definitions are arranged according to the type of defining property of the special Finsler space concerned.
Throughout the paper, g, g, ∇ and D denote respectively the Finsler metric in π -1 (T M), the induced metric in π -1 (T * M), the Cartan connection and the Berwald connection associated to a given Finsler manifold (M, L). Also, T denotes the torsion tensor of the Cartan connection (or the Cartan tensor) and R, P and S denote respectively the horizontal curvature, the mixed curvature and the vertical curvature of the Cartan connection. (a) Riemannian if the metric tensor g(x, y) is independent of y or, equivalently, if
T (X, Y ) = 0, for all X, Y ∈ X(π(M)).
(b) locally Minkowskian if the metric tensor g(x, y) is independent of x or, equivalently, if ∇ βX T = 0 and R = 0.
Definition 2.2. A Finsler manifold (M, L) is said to be :
(a) Berwald [24] if the torsion tensor T is horizontally parallel. That is,
∇ βX T = 0.
(b) C h -recurrent if the torsion tensor T satisfies the condition
∇ βX T = λ o (X) T,
where λ o is a π-form of order one.
(c) P * -Finsler manifold if the π-tensor field ∇ βη T is expressed in the form
∇ βη T = λ(x, y) T, where λ(x, y) = b g(∇ βη C,C) C 2 = g(∇ βη øC,øC) C 2
and C 2 := g(C, C) = C(C) = 0; C being the π-vector field defined by g(C, X) = C(X). Definition 2.3. A Finsler manifold (M, L) is said to be:
(a) C v -recurrent if the torsion tensor T satisfies the condition (∇ γX T )(Y , Z) = λ o (X)T (Y , Z).
(b) C 0 -recurrent if the torsion tensor T satisfies the condition
(D γX T )(Y , Z) = λ o (X)T (Y , Z).
A Finsler manifold (M, L) is said to be :
(a) semi-C-reducible if dimM ≥ 3 and the Cartan tensor T has the form
T (X, Y , Z) = µ n + 1 { (X, Y )C(Z) + (Y , Z)C(X) + (Z, X)C(Y )}+ + τ C 2 C(X)C(Y )C(Z),
where µ and τ are scalar functions satisfying µ + τ = 1, = g -ℓ ⊗ ℓ and ℓ(X) := L -1 g(X, η).
(b) C-reducible if dimM ≥ 3 and the Cartan tensor T has the form
T (X, Y , Z) = 1 n + 1 { (X, Y )C(Z) + (Y , Z)C(X) + (Z, X)C(Y )}.
(c) C 2 -like if dimM ≥ 2 and the Cartan tensor T has the form
T (X, Y , Z) = 1 C 2 C(X)C(Y )C(Z).
Definition 2.5. A Finsler manifold (M, L), where dimM ≥ 3, is said to be quasi-Creducible if the Cartan tensor T is written as :
T (X, Y , Z) = A(X, Y )C(Z) + A(Y , Z)C(X) + A(Z, X)C(Y ),
where A is a symmetric indicatory (2) π-form (A(X, η) = 0 for all X).
Definition 2.6. [25] A Finsler manifold (M, L) is said to be :
(a) S 3 -like if dim(M) ≥ 4 and the vertical curvature tensor S(X, Y , Z, W ) := g(S(X, Y )Z, W ) has the form :
S(X, Y , Z, W ) = Sc v (n -1)(n -2) { (X, Z) (Y , W ) -(X, W ) (Y , Z)}.
(b) S 4 -like if dim(M) ≥ 5 and the vertical curvature tensor S(X, Y , Z, W ) has the form :
S(X, Y , Z, W ) = (X, Z)F(Y , W ) -(Y , Z)F(X, W )+ + (Y , W )F(X, Z) -(X, W )F(Y , Z), (2.1)
where
F is the (2)π-form defined by F = 1 n -3 {Ric v - Sc v 2(n -2)
}.
Definition 2.7. A Finsler manifold (M, L) is said to be :
(a) S v -recurrent if the v-curvature tensor S satisfies the condition
(∇ γX S)(Y , Z, W ) = λ(X)S(Y , Z)W ,
where λ is a π-form of order one.
(b) S v -recurrent of the second order if the v-curvature tensor S satisfies the condition
( 2 ∇ 2 ∇ S)(øY, øX, Z, W , U) = Θ(X, Y )S(Z, W )U,
where Θ is a π-form of order two.
A Finsler manifold (M, L) is said to be :
(a) a Landsberg manifold if
P (X, Y ) = P (X, Y )η = 0 ∀ X, Y ∈ X(π(M)), or equivalently ∇ βη T = 0. (b) a general Landsberg manifold if T r{Y -→ P (X, Y )} = 0 ∀ X, ∈ X(π(M)), or equivalently ∇ βη C = 0.
Definition 2.9. A Finsler manifold (M, L) is said to be P -symmetric if the mixed curvature tensor P satisfies
P (X, Y )Z = P (Y , X)Z, ∀ øX, øY, øZ ∈ X(π(M)).
Definition 2.10. A Finsler manifold (M, L), where dimM ≥ 3, is said to be P 2 -like if the mixed curvature tensor P has the form :
P (X, Y , Z, øW ) = α(Z)T (X, Y , øW ) -α(W ) T (X, øY, Z),
where α is a (1) π-form (positively homogeneous of degree 0).
A Finsler manifold (M, L), where dimM ≥ 3, is said to be P -reducible if the π-tensor field P (X, Y , Z) := g(P (X, Y )η, Z) can be expressed in the form :
P (X, Y , Z) = δ(X) (Y , Z) + δ(Y ) (Z, X) + δ(Z) (X, Y ),
where δ is a (1) π-form satisfying δ(øη) = 0.
Definition 2.12.
[2] A Finsler manifold (M, L), where dimM ≥ 3, is said to be h-isotropic if there exists a scalar k o such that the horizontal curvature tensor R has the form
R(X, Y )Z = k o {g(Y , Z)X -g(X, Z)Y }. Definition 2.13. [2] A Finsler manifold (M, L)
, where dimM ≥ 3, is said to be :
(a) of scalar curvature if there exists a scalar function k :
T M -→ R such that the horizontal curvature tensor R(X, Y , Z, W ) := g(R(X, Y )Z, W ) satisfies the relation R(η, X, η, Y ) = kL 2 (X, Y ). (b) of constant curvature if the function k in (a) is constant. Definition 2.14. A Finsler manifold (M, L) is said to be R 3 -like if dimM ≥ 4 and the horizontal curvature tensor R(X, Y , Z, W ) is expressed in the form R(X, Y , Z, W ) =g(X, Z)F (Y , W ) -g(Y , Z)F (X, W )+ + g(Y , W )F (X, Z) -g(X, W )F (Y , Z), (2.2)
where
F is the (2)π-form defined by F = 1 n-2 {Ric h -Sc h g 2(n-1) }.
This section is devoted to global investigation of some mutual relationships between the special Finsler spaces introduced in the preceding section. Some consequences are also drawn from these relationships.
We start with some immediate consequences from the definitions: (a) A Locally Minkowskian manifold is a Berwald manifold.
(b) A Berwald manifold is a Landsberg manifold. (c) A Landsberg manifold is a general Landsberg manifold. (d) A Berwald manifold is C h -recurrent (resp. P * -Finsler). (e) A P * -manifold is a Landsberg manifold. (f) A C-reducible (resp. C 2 -like) manifold is semi-C-reducible. (g) A semi-C-reducible manifold is quasi-C-reducible. (h) A Finsler manifold of constant curvature is of scalar curvature.
The following two lemmas are useful for subsequent use. where ℓ is the π-form given by ℓ(X) = L -1 g(X, η), then we have:
(a) (øX, øY ) = g(φ(øX), øY ), (b) φ(øη) = 0, (c) φ o φ = φ, (d) T r(φ) = n -1, (e) ∇ βøX φ = 0, (f) ∇ βøX = 0.
As we have seen, a Landsberg manifold is general Landsberg. The converse is not true. Nevertheless, we have
Proposition 3.3. A C-reducible general Landsberg manifold (M, L) is a Landsberg manifold.
Proof. Since (M, L) is a C-reducible manifold, then, by Definition 2.4, Lemma 3.2, the symmetry of and the non-degeneracy of g, we get
T (øX, øY ) = 1 n + 1 { (øX, øY )øC + C(øX)φ(øY ) + C(øY )φ(øX)},
where øC is the π-vector field defined by g(øC, øX) := C(øX). Taking the h-covariant derivative ∇ βøZ of both sides of the above equation, we obtain
(∇ βøZ T )(øX, øY ) = 1 n + 1 {(∇ βøZ )(øX, øY )øC + (øX, øY )∇ βøZ øC + C(øX)(∇ βøZ φ)(øY ) + +(∇ βøZ C)(øX)φ(øY ) + C(øY )(∇ βøZ φ)(øX) + (∇ βøZ C)(øY )φ(øX)},
from which, by setting øZ = øη and taking into account the fact that ∇ βøZ = 0 and that ∇ βøZ φ = 0 ( Lemma 3.2), we get Combining the above two Propositions, we obtain the more powerful result :
(∇ βøη T )(øX, øY ) = 1 n + 1 { (øX, øY )∇ βøη øC+(∇ βøη C)(øX)φ(øY )+(∇ βøη C)(øY )φ(øX)}.
Proposition 3.5. A C-reducible general Landsberg manifold (M, L) is a Berwald manifold.
Summing up, we get: Theorem 3.6. Let (M, L) be a C-reducible Finsler manifold. The following assertion are equivalent :
(a) (M, L) is a Berwald manifold. (b) (M, L) is a Landsberg manifold. (c) (M, L) is a general Landsberg manifold.
We retrieve here a result of Matsumuoto [15] , namely Corollary 3.7. If the h-curvature tensor R and hv-curvature tensor P of a Creducible manifold vanish, then the manifold is Locally Minkowskian. Remark 3.8. [15] It may be conjectured that a Finsler manifold will be Minkowskian if the h-curvature tensor R and hv-curvature tensor P vanish. As above seen the conjecture is verified already under somewhat strong condition " C-reducibility". Theorem 3.9. Let (M, L) be a Finsler manifold. Then we have : Applying the h-covariant derivative ∇ βøW on both sides of the above equation, taking into account the fact that (∇ βøW T )(øX, øY, øZ) = g((∇ βøW T )(øX, øY ), øZ) and that ∇ βøW = 0, we obtain
g((∇ βøW T )(øX, øY ), øZ) = 1 n + 1 S øX,øY,øZ { (øX, øY )(∇ βøW C)(øZ)}.
From which, by setting øW = øη and noting that P (øX, øY )øη = (∇ βøη T )(øX, øY ), the result follows.
(b) Since (M, L) is a P -reducible manifold, then by Definition 2.11, taking into account the fact that g is nondegenerate, we obtain
P (øX, øY )øη = δ(øX)φ(øY ) + δ(øY )φ(øX) + ø (øX, øY ) øζ, (3.2)
where øζ is the π-vector field defined by g(øζ, øX) := δ(øX).
Since δ(øη) = 0, then T r{øY -→ δ(øY )φ(øX) + (øX, øY ) øζ} = 2δ(øX). Taking the trace of both sides of (3.2), using the fact that P (øX, øY )øη = (∇ βøη T )(øX, øY ) (Lemma 3.1) and that T r{øY -→ (∇ βøη T )(øX, øY )} = (∇ βøη C)(øX), we get
δ(øX) = 1 n + 1 (∇ βøη C)(øX). (3.3)
Now, from Equations (3.2) and (3.3), we have
g(P (øX, øY )øη, øZ) = 1 n + 1 S øX,øY,øZ { (øX, øY )(∇ βøη C)(øZ)}. (3.4)
According to the given assumption that the manifold is general Landsberg, then ∇ βøη C = 0. Therefore, from (3.4), we get P (øX, øY )øη = 0 and hence the manifold is Landsberg.
Proposition 3.10.
(a) A C h -recurrent manifold is a P * -Finsler manifold.
(b) A general Landsberg P * -Finsler manifold is a Landsberg manifold.
Proof. The proof is straightforward and we omit it.
(∇ βøZ T )(øX, øY ) = -∇ βøZ C(øC) C 4 C(øX)C(øY )øC + 1 C(øC) (∇ βøZ C)(øX)C(øY )øC + + 1 C(øC) (∇ βøZ C)(øY )C(øX)øC + 1 C(øC) C(øX)C(øY )∇ βøZ øC.
In view of this relation, ∇ βøZ T = 0 if, and only if, ∇ βøZ C = 0. Hence the result.
Corollary 3.12. A C 2 -like general Landsberg manifold is a Landsberg manifold.
In view of the above Theorems, we have:
Corollary 3.13. The two notions of being Landsberg and general Landsberg coincide in the case of C-reducibility, P -reducibility, C 2 -likeness or P * -Finsler.
As we know, a C-reducible Landsberg manifold is a Berwald manifold (Proposition 3.4 ). Moreover, A C 2 -like Finsler manifold is a Berwald manifold if, and only if, the π-tensor field C is horizontally parallel (Proposition 3.11). We shall try to generalize these results to the case of semi-C-reduciblity. Taking the h-covariant derivative of both sides, noting that ∇ βøX = 0, we get
(∇ βøW T )(øX, øY, øZ) = 1 n + 1 S øX,øY,øZ { (øX, øY ){µ(∇ βøW C)(øZ) + (∇ βøW µ)C(øZ)}} + + τ C 2 S øX,øY,øZ {(∇ βøW C)(øX)C(øY )C(øZ)} - -{ ∇ βøW µ C 2 + τ ∇ βøW C(øC) C 4 }C(øX)C(øY )C(øZ).
Now, if the characteristic scalar µ and the π-tensor field C are horizontally parallel, then ∇ βøW T = 0 and (M, L) is a Berwald manifold.
Conversely, if (M, L) is a Berwald manifold, then ∇ βøX T = 0 and hence ∇ βøX C = 0, ∇ βøX øC = 0. These, together with the above equation, give
∇ βøW µ{ 1 n + 1 S øX,øY,øZ { (øX, øY )C(øZ)} - 1 C 2 C(øX)C(øY )C(øZ)} = 0, which implies immediately that ∇ βøW µ = 0.
The following lemmas are useful for subsequent use Lemma 3.15. For all X, Y ∈ X(π(M)), we have :
(a) [γX, γY ] = γ(∇ γX Y -∇ γY X) (b) [γX, βY ] = -γ(P (Y , X)η + ∇ βY X) + β(∇ γX Y -T (X, Y )) (c) [βX, βY ] = γ(R(X, Y )η) + β(∇ βX Y -∇ βY X)
Lemma 3.16. For all øX, øY, øZ, øW ∈ X(π(M)) and W ∈ X(T M), we have : (a) g((∇ W T )(øX, øY ), øZ) = g((∇ W T )(øX, øZ), øY ), (b) g(S(øX, øY )øZ, øW ) = -g(S(øX, øY )øW, øZ).
From the definition of the covariant derivative, we get g((
∇ W T )(øX, øY ), øZ) = g(∇ W T (øX, øY ), øZ) -g(T (∇ W øX, øY ), øZ)- -g(T (øX, ∇ W øY ), øZ). (3.5) Now, we have g(∇ W T (øX, øY ), øZ) = W • g(T (øX, øY ), øZ) -g(T (øX, øY ), ∇ W øZ) = W • g(T (øX, øY ), øZ) -g(T (øX, ∇ W øZ), øY ), Similarly, g(T (øX, ∇ W øY ), øZ) = W • g(T (øX, øZ), øY ) -g(∇ W T (øX, øZ), øY ).
Substituting these two equations into (3.5), noting the property that g(T (∇ W øX, øY ), øZ) = g(T (∇ W øX, øZ), øY ) (cf. §1), the result follows.
(b) follows directly from the general formula (which can be easily proved) g(K(X, Y )øZ, øW ) + g(K(X, Y )øW, øZ) = 0 by setting X = γøX and Y = γøY , where K is the classical curvature tensor of the Cartan connection as a linear connection in the pull-back bundle (cf. §1). Proposition 3.17. Let (M, L) be a C h -recurrent Finsler manifold (∇ βøX T = λ 0 (øX)T ). Then, we have: ) P (øX, øY ).
Proof.
(a) The hv-curvature tensor P can be written in the form [25] :
P (øX, øY, øZ, øW ) = g((∇ βøZ T )(øX, øY ), øW ) -g((∇ βøW T )(øX, øY ), øZ)+ +g(T (øX, øZ), P (øW, øY )) -g(T (øX, øW ), P (øZ, øY )).
Then, by using P (øX, øY ) = (∇ βøη T )(øX, øY ) (Lemma 3. Then, (b) follows from the above two equations. Theorem 3.18. Assume that (M, L) is C h -recurrent. Then, the v-curvature tensor S is recurrent with respect to the h-covariant differentiation : ∇ βøX S = θ(øX)S, where θ is a π-form of order one.
Proof. One can easily show that : For all X, Y, Z ∈ X(T M), From which, since g(T (øX, øY ), øZ) = g(T (øX, øZ), øY ), we have g(S(øX, øY )øZ, øW ) = g((∇ γøY T )(øX, øZ), øW ) -g((∇ γøX T )(øY, øZ), øW )+ +g(T (øX, øW ), T (øY, øZ)) -g(T (øY, øW ), T (øX, øZ)).
S X,Y,Z {K(X, Y )ρZ + ∇ X T(Y, Z) + T(X, [Y, Z])} = 0.
Similarly, g(S(øX, øY )øW, øZ) = g((∇ γøY T )(øX, øW ), øZ) -g((∇ γøX T )(øY, øW ), øZ)+ +g(T (øX, øZ), T (øY, øW )) -g(T (øY, øZ), T (øX, øW )).
The above two equations, together with Lemma 3.16, yield g((∇ γøX T )(øY, øZ), øW ) = g((∇ γøY T )(øX, øZ), øW ). Now, using the given assumption that the manifold is C h -recurrent, Equation (3.8) implies that (∇ βøX S)(øY, øZ, øV, øW ) = ∇ βøX S(øY, øZ, øV, øW ) --S(∇ βøX øY, øZ, øV, øW ) -S(øY, ∇ βøX øZ, øV, øW ) --S(øY, øZ, ∇ βøX øV, øW ) -S(øY, øZ, øV, ∇ βøX øW ). = +∇ βøX g(T (øY, øW ), T (øZ, øV )) -∇ βøX g(T (øZ, øW ), T (øY, øV )) --g(T (∇ βøX øY, øW ), T (øZ, øV )) + g(T (øZ, øW ), T (∇ βøX øY, øV )) --g(T (øY, øW ), T (∇ βøX øZ, øV )) + g(T (∇ βøX øZ, øW ), T (øY, øV )) --g(T (øY, øW ), T (øZ, ∇ βøX øV )) + g(T (øZ, øW ), T (øY, ∇ βøX øV )) --g(T (øY, ∇ βøX øW ), T (øZ, øV )) + g(T (øZ, ∇ βøX øW ), T (øY, øV )). = g((∇ βøX T )(øY, øW ), T (øZ, øV )) + g(T (øY, øW ), (∇ βøX T )(øZ, øV )) --g((∇ βøX T )(øZ, øW ), T (øY, øV )) -g(T (øZ, øW ), (∇ βøX T )(øY, øV )). = 2λ o (øX)S(øY, øZ, øV, øW ) =: θ(øX)S(øY, øZ, øV, øW ).
Hence, the result follows.
Ric v (øX, øY ) = (3 -n) (n + 1) 2 C(øX)C(øY ) - (n -1) (n + 1) 2 C 2 (øX, øY ).
(c) the vertical scalar curvature Sc v has the form
Sc v = (2 -n) (n + 1) C 2 .
Theorem 3.22. A Finsler manifold (M, L) is P -Symmetric if, and only if, the v-curvature tensor S satisfies the equation ∇ βøη S = 0.
Proof. One can show that: For all X, Y, Z ∈ X(T M), Setting øZ = øη and using the fact that T (øX, øη) = 0 and that K o γ = id X(π(M )) , the result follows.
S X,Y,Z {∇ Z K(X, Y ) -K(X, Y )∇ Z -K([X, Y ], Z)} = 0. ( 3
(b) Follows from (a) together with the relation T (øX, øη) = 0.
(c) Setting X = γøX, Y = γøY and Z = γøZ in (3.9) and using Lemma 3.15, we get S øX,øY,øZ (∇ γøX S)(øY, øZ, øW ) = 0. Again, setting øW = øη in the above equation and using the fact that S(øX, øY )øη = 0 and that K o γ = id X(π(M )) , the result follows. Proof. Taking the v-covariant derivative of both sides of the relation in Corollary 3.19(b) and, then, using the assumption that ∇ γX T = λ 0 (X)T , we get (∇ γøX S)(øY, øZ, øV, øW ) = 2λ o (øX)S(øY, øZ, øV, øW ) =: ψ(øX)S(øY, øZ, øV, øW ), which shows that S is v-recurrent. Now, setting øV = øη in the last equation, using the properties of S and noting that K o γ = id X(π(M )) , we conclude that S = 0.
The following result gives a characterization of Riemannian manifolds in terms of C v -recurrence and C 0 -recurrence.
Theorem 3.28. (a) A C v -recurrent Finsler manifold is Riemannian, (b) A C 0 -recurrent Finsler manifold is Riemannian. Proof. (a) Since (M, L) is C v -recurrent, then (∇ γX T )(Y , Z) = λ o (X)T (Y , Z),
from which, by setting øX = øη and noting that ∇ γøη T = -T , we get
T (Y , Z) = -λ o (η)T (Y , Z). (3.12)
But since (∇ γøX T )(øY, øZ) = (∇ γøY T )(øX, øZ) (Corollary 3.19), then λ o (øX)T (øY, øZ) = λ o (øY )T (øX, øZ). Hence,
λ o (η)T (Y , Z) = 0. (3.13)
Then, the result follows from (3.12) and (3.13).
(b) can be proved similarly.
Theorem 3.29. For a Finsler manifold (M, L), the following assertions are equivalent :
(a) (M, L) is S v -recurrent.
(b) The v-curvature tensor S vanishes identically.
(c) (M, L) is S v -recurrent
2 ∇ 2 ∇ S)(øY, øX, øZ, øV, øW ) = ∇ γøY (∇ γøX S)(øZ, øV, øW ) -(∇ γ∇ γøY øX S)(øZ, øV, øW )- -(∇ γøX S)(∇ γøY øZ, øV, øW ) -(∇ γøX S)(øZ, ∇ γøY øV, øW )- -(∇ γøX S)(øZ, øV, ∇ γøY øW ).
By substituting øZ = øη = øW in the above equation and using Lemma 3.25 and the fact that S(øX, øY )øη = 0, we get S(øX, øY )øZ = -S(øZ, øY )øX and S(øX, øY )øZ = -S(øX, øZ)øY.
From this, together with the identity S øX,øY,øZ S(øX, øY )øZ = 0, the v-curvature tensor S vanishes identically.
In view of the above theorem we have :
Corollary 3.30.
(a) An S v -recurrent (resp. a second order S v -recurrent) manifold (M, L) is S 3 -like, provided that dim M ≥ 4.
(b) An S v -recurrent (resp. a second order S v -recurrent) manifold (M, L) is S 4 -like, provided that dim M ≥ 5.
Theorem 3.31. If (M, L) is a P 2 -like Finsler manifold, then the v-curvature tensor S vanishes or the hv-curvature tensor P vanishes. In the later case, the h-covariant derivative of S vanishes.
Proof. As (M, L) is P 2 -like, then P (X, Y , η, øW ) = α(η)T (X, Y , øW ) =: α o T (X, Y , øW ) and hence P (øX, øY ) = α o T (X, Y ). (3.14)
Now, setting øW = øη into (3.10), we get (∇ γøY P )(øZ, øX) -(∇ γøX P )(øZ, øY ) -P (øZ, øX)øY + P (øZ, øY )øX--P (T (øX, øZ), øY ) + P (T (øY, øZ), øX) = 0.
Hence, g((∇ γøY P )(øZ, øX), øW ) -g((∇ γøX P )(øZ, øY ), øW ) -P (øZ, øX, øY, øW )+ +P (øZ, øY, øX, øW ) -g( P (T (øX, øZ), øY ), øW ) + g( P (T (øY, øZ), øX), øW ) = 0.
From which, together with (3. It is to be observed that the left-hand side of the above equation is symmetric in the arguments øZ and øW while the right-hand side is skew-symmetric in the same arguments. Hence we have
α o S(øX, øY, øW, øZ) = 0, (3.15)
ε(øY )T (øX, øZ, øW ) -ε(øX)T (øY, øZ, øW ) = 0, (3.16) where ε is the π-form defined by ε(øY ) := (∇ γøY α)(øη). Now, If ε = 0, it follows from (3.16) that there exists a scalar function Υ such that T (øX, øY, øZ) = Υ ε(øX)ε(øY )ε(øZ). Consequently, T (øX, øY ) = Υ ε(øX)ε(øY )øε, where g(øε, øX) := ε(øX). From which S(øX, øY, øZ, øW ) = g(T (øX, øW ), T (øY, øZ)) -g(T (øY, øW ), T (øX, øZ)) = Υ ε(øX)ε(øY )ε(øZ)ε(øW )g(øε, øε) -Υ ε(øX)ε(øY )ε(øZ)ε(øW )g(øε, øε) = 0.
On the other hand, if the v-curvature tensor S = 0, then it follows from (3.15) that ε = 0 and α(øη) = 0. Hence, α = 0 and the hv-curvature tensor P vanishes. In this case, it follows from the identity (3.10) that ∇ βøX S = 0. Proposition 3.32. A P 2 -like Finsler manifold (M, L) is a P * -Finsler manifold.
Proof. As (M, L) is P 2 -like, then from (3.14), we have P (X, Y ) = α o T (X, Y ). Using Lemma 3.1, we get (∇ βøη T )(øX, øY ) = α 0 T (øX, øY ), from which, by taking the trace,
∇ βøη C = α 0 T , where α 0 = b g(∇ βη C,C) C 2
. Hence the result.
The next definition will be useful in the sequel. Deicke theorem [4] can be formulated globally as follows:
Lemma 3.36. Let (M, L) be a Finsler manifold. The following assertions are equivalent: Now, we focus our attention to the interesting case (c) of the above theorem. In this case, the h-curvature tensor R = 0 and hence the (v)h-torsion tensor R = 0. Therefore, the equation (deduced from (3.9)) (∇ γøX R)(øY, øZ, øW ) + (∇ βøY P )(øZ, øX, øW ) -(∇ βøZ P )(øY, øX, øW )--P (øZ, P (øY, øX)øη)øW + R(T (øX, øY ), øZ)øW -S(R(øY, øZ)øη, øX)øW + +P (øY, P (øZ, øX)øη)øW -R(T (øX, øZ), øY )øW = 0. reduces to (∇ βøY P )(øZ, øX, øW ) -(∇ βøZ P )(øY, øX, øW )--P (øZ, P (øY, øX))øW + P (øY, P (øZ, øX))øW = 0.
(a) (M, L) is Riemannian, (
Setting øW = øη, we get (∇ βøY P )(øZ, øX) -(∇ βøZ P )(øY, øX) -P (øZ, P (øY, øX)) + P (øY, P (øZ, øX)) = 0.
(3.22) Since (M, L) is C h -recurrent, then, by Proposition 3.17, the (v)hv-torsion tensor P satisfies the relations (∇ βøZ P )(øX, øY ) = (K o λ o (øZ) + ∇ βøZ K o )T (øX, øY ) and P (øX, øY ) = λ o (øη)T (øX, øY ) = K o T (øX, øY ). From these, together with (3.22), we get On the other hand, if K o = 0, then the v-curvature tensor S vanishes from (3.23) . Next, it is seen from (3.24) that, if V(øY ) := K o λ o (øY ) + ∇ βøY K o = 0, then there exists a scalar function Υ = T (øX,øZ,øW )T (øX,øY,øZ)T (øY,øZ,øW ) (T (øX,øY,øW )) 2 (V(øZ)) 3 such that T (øX, øY, øW ) = Υ V(øX)V(øY )V(øW ).
(K o λ o (øY ) + ∇ βøY K o )T (øZ, øX) -(K o λ o (øZ) + ∇ βøZ K o )T (øX, øY )- -K 2 o T (
Summing up, we have Theorem 3.38. Let (M, L) be a Finsler manifold of dimensions n ≥ 3.
If (M, L) is h-isotropic and C h -recurrent, then (
P • ω)(øX, øY ) = φ(ω(φ(øX), φ(øY ))) = φ{ω(øX -L -1 ℓ(øX)øη, øY -L -1 ℓ(øY )øη)} = φ{ω(øX, øY ) -L -1 ℓ(øY )ω(øX, øη)- -L -1 ℓ(øX)ω(øη, øY ) + L -2 ℓ(øX)ℓ(øY )ω(øη, øη)} = ω(øX, øY ) -L -2 g(ω(øX, øY ), øη)øη -φ{L -1 ℓ(øY )ω(øX, øη)+ +L -1 ℓ(øX)ω(øη, øY ) -L -2 ℓ(øX)ℓ(øY )ω(øη, øη)} (3.
25) Now, if ω(øX, øη) = 0 = ω(øη, øX) and g(ω(øX, øY ), øη) = 0, then (3.25) implies that (P • ω)(øX, øY ) = ω(øX, øY ) and hence ω is indicatory.
On the other hand, if ω is indicatory, then ω(øX, øY ) = φ(ω(φ(øX), φ(øY ))). From which, setting øX = øη (resp. øY = øη) and taking into account the fact that φ(øη) = 0 (Lemma 3.2), we get ω(øη, øY ) = 0 (resp. ω(øX, øη) = 0). From this, together with (P•ω)(øX, øY ) = ω(øX, øY ), Equation (3.25) implies that L -2 g(ω(øX, øY ), øη)øη = 0. Consequently, g(ω(øX, øY ), øη) = 0. (b) The proof is similar to that of (a) and we omit it. From which, using the fact that g(F o (øX), øY ) = F (øX, øY ) and that the Finsler metric g is non-degenerate, the result follows. The tensor field Ψ in the above theorem being of the same form as the Weyl conformal tensor in Riemannian geometry, we draw the following Theorem 3.56. An R 3 -like Riemannian manifold is conformally flat. Remark 3.57. It should be noted that some important results of [8] , [9] , [11] , [13] , [19] , [20] ,...,etc. (obtained in local coordinates) are retrieved from the above mentioned global results (when localized).
F : F (X, Y ) := 1 n-2 {Ric h (X, Y ) -Sc h g(X,Y ) 2(n-1) }, F o : g(F o (øX), øY ) := F (øX, øY ),
• Second order S v -recurrent manifold [20] , [11] : S hijk | m | n = Θ mn S hijk , where Θ ij (x, y) is a covariant tensor field.
• Landsberg manifold [7] : P h kji y k = 0 ⇐⇒ ( ∂i Γ h jk )y k = 0 ⇐⇒ C h ij|k y k = 0.
• General Landsberg manifold [10] : P r ijr y i = 0 ⇐⇒ C j|o = 0.
• P -symmetric manifold [19] : P hijk = P hikj .
• P 2 -like manifold (dim M ≥ 3) [14] :
P hijk = α h C ijk -α i C
hjk , where α k (x, y) is a covariant vector field.
• P -reducible manifold (dim M ≥ 3) [19] : P ijk = 1 n+1 ( ij P k + jk P i + ki P j ), where P ijk = g hi P h jk .
• h-isotropic manifold (dim M ≥ 3) [13] : R hijk = k o {g hj g ik -g hk g ij }, for some scalar k o , where R hijk = g il R l
hjk .
• Manifold of scalar curvature [21] : R ijkl y i y k = kL 2 jl , for some function k : T M -→ R .
• Manifold of constant curvature [21] : the function k in the above definition is constant.
• Manifold of perpendicular scalar (or of p-scalar ) curvature [8] , [9] :
P • R hijk := l h m i n j r k R lmnr = R o { ik hj -ij hk }
, where R o is a function called a perpendicular scalar curvature.
• Manifold of s-ps curvature [8] , [9] : (M, L) is both of scalar curvature and of p-scalar curvature.
• R 3 -like manifold (dim M ≥ 4) [8] : R hijk = g hj F ik -g hk F ij + g ik F hj -g ij F hk , where
F ij := 1 n-2 {R ij -1 2 r g ij }; R ij := R h ijh , r := 1 n-1 R i i . | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The aim of the present paper is to provide a global presentation of the theory of special Finsler manifolds. We introduce and investigate globally (or intrinsically, free from local coordinates) many of the most important and most commonly used special Finsler manifolds : locally Minkowskian, Berwald, Landesberg, general Landesberg, P -reducible, C-reducible, semi-C-reducible, quasi-C-reducible, P * -Finsler, C h -recurrent, C v -recurrent, C 0 -recurrent, S v -recurrent, S v -recurrent of the second order, C 2 -like, S 3 -like, S 4 -like, P 2 -like, R 3 -like, P -symmetric, h-isotropic, of scalar curvature, of constant curvature, of p-scalar curvature, of s-ps-curvature. The global definitions of these special Finsler manifolds are introduced. Various relationships between the different types of the considered special Finsler manifolds are found. Many local results, known in the literature, are proved globally and several new results are obtained. As a by-product, interesting identities and properties concerning the torsion tensor fields and the curvature tensor fields are deduced. Although our investigation is entirely global, we provide; for comparison reasons, an appendix presenting a local counterpart of our global approach and the local definitions of the special Finsler spaces considered. 1"
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "In Finsler geometry all geometric objects depend not only on positional coordinates, as in Riemannian geometry, but also on directional arguments. In Riemannian geometry there is a canonical linear connection on the manifold M, while in Finsler geometry there is a corresponding canonical linear connection, due to E. Cartan, which is not a connection on M but is a connection on π -1 (T M), the pullback of the tangent bundle T M by π : T M -→ M (the pullback approach). Moreover, in Riemannian geometry there is one curvature tensor and one torsion tensor associated with a given linear connection on the manifold M, whereas in Finsler geometry there are three curvature tensors and five torsion tensors associated with a given linear connection on π -1 (T M).\n\nMost of the special spaces in Finsler geometry are derived from the fact that the π-tensor fields (torsions and curvatures) associated with the Cartan connection satisfy special forms. Consequently, special spaces of Finsler geometry are more numerous than those of Riemannian geometry. Special Finsler spaces are investigated locally (using local coordinates) by many authors: M. Matsumoto [16] , [18] , [15] , [14] and others [6] , [19] , [8] , [7] . On the other hand, the global (or intrinsic, free from local coordinates) investigation of such spaces is very rare in the literature. Some considerable contributions in this direction are due to A. Tamim [24] , [25] .\n\nIn the present paper, we provide a global presentation of the theory of special Finsler manifolds. We introduce and investigate globally many of the most important and most commonly used special Finsler manifolds : locally Minkowskian, Berwald, Landesberg, general Landesberg, P -reducible, C-reducible, semi-C-reducible, quasi-C-reducible, P * -Finsler, C h -recurrent, C v -recurrent, C 0 -recurrent, S v -recurrent, S vrecurrent of the second order, C 2 -like, S 3 -like, S 4 -like, P 2 -like, R 3 -like, P -symmetric, h-isotropic, of scalar curvature, of constant curvature, of p-scalar curvature, of s-pscurvature.\n\nThe paper consists of two parts, preceded by a preliminary section ( §1), which provides a brief account of the basic concepts of the pullback approach to Finsler geometry necessary to this work. For more detail, the reader is referred to [1] , [3] , [5] and [24] .\n\nIn the first part ( §2), we introduce the global definitions of the aforementioned special Finsler manifolds in such a way that, when localized, they yield the usual local definitions current in the literature (see the appendix). The definitions are arranged according to the type of the defining property of the special Finsler manifold concerned.\n\nIn the second part ( §3), various relationships between the different types of the considered special Finsler manifolds are found. Many local results, known in the literature, are proved globally and several new results are obtained. As a by-product of some of the obtained results, interesting identities and properties concerning the torsion tensor fields and the curvature tensor fields are deduced, which in turn play a key role in obtaining other results.\n\nAmong the obtained results are: a characterization of Riemannian manifolds, a characterization of S v -recurrent manifolds, a characterization of P -symmetric manifolds, a characterization of Berwald manifolds (in certain cases), the equivalence of Landsberg and general Landsberg manifolds under certain conditions, a classifica-tion of h-isotropic C h -recurrent manifolds and a presentation of different conditions under which an R 3 -like Finsler manifold becomes a Finsler manifold of s-ps curvature. The above results are just a non-exhaustive sample of the global results obtained in this paper.\n\nIt should finally be noted that some important results of [8] , [9] , [11] , [13] , [19] , [20] ,...,etc. (obtained in local coordinates) are immediately derived from the obtained global results (when localized).\n\nAlthough our investigation is entirely global, we conclude the paper with an appendix presenting a local counterpart of our global approach and the local definitions of the special Finsler spaces considered. This is done to facilitate comparison and to make the paper more self-contained."
},
{
"section_type": "OTHER",
"section_title": "Notation and Preliminaries",
"text": "In this section, we give a brief account of the basic concepts of the pullback formalism of Finsler geometry necessary for this work. For more details refer to [1] , [3] , [5] and [24] . We make the general assumption that all geometric objects we consider are of class C ∞ . The following notations will be used throughout this paper: M: a real differentiable manifold of finite dimension n and of class C ∞ , F(M): the R-algebra of differentiable functions on M, X(M): the F(M)-module of vector fields on M, π M : T M -→ M: the tangent bundle of M, π : T M -→ M: the subbundle of nonzero vectors tangent to M, V (T M): the vertical subbundle of the bundle T T M, P : π -1 (T M) -→ T M : the pullback of the tangent bundle T M by π, P * : π -1 (T * M) -→ T M : the pullback of the cotangent bundle T * M by π, X(π(M)): the F(T M)-module of differentiable sections of π -1 (T M).\n\nElements of X(π(M)) will be called π-vector fields and will be denoted by barred letters X. Tensor fields on π -1 (T M) will be called π-tensor fields. The fundamental π-vector field is the π-vector field η defined by η(u) = (u, u) for all u ∈ T M. The lift to π -1 (T M) of a vector field X on M is the π-vector field X defined by\n\nX(u) = (u, X(π(u))). The lift to π -1 (T M) of a 1-form ω on M is the π-form ω defined by ω(u) = (u, ω(π(u))).\n\nThe tangent bundle T (T M) is related to the pullback bundle π -1 (T M) by the short exact sequence 0\n\n-→ π -1 (T M) γ -→ T (T M) ρ -→ π -1 (T M) -→ 0,\n\nwhere the bundle morphisms ρ and γ are defined respectively by ρ = (π T M , dπ) and γ(u, v) = j u (v), where j u is the natural isomorphism j u :\n\nT π M (v) M -→ T u (T π M (v) M).\n\nLet ∇ be a linear connection (or simply a connection) in the pullback bundle π -1 (T M). We associate to ∇ the map\n\nK : T T M -→ π -1 (T M) : X -→ ∇ X η, called the connection (or the deflection) map of ∇. A tangent vector X ∈ T u (T M) is said to be horizontal if K(X) = 0 . The vector space H u (T M) = {X ∈ T u (T M) : K(X) = 0} of the horizontal vectors at u ∈ T M is called the horizontal space to M at u . The connection ∇ is said to be regular if T u (T M) = V u (T M) ⊕ H u (T M) ∀u ∈ T M.\n\nIf M is endowed with a regular connection, then the vector bundle maps\n\nγ : π -1 (T M) -→ V (T M), ρ| H(T M ) : H(T M) -→ π -1 (T M), K| V (T M ) : V (T M) -→ π -1 (T M) are vector bundle isomorphisms. Let us denote β = (ρ| H(T M ) ) -1 , then ρoβ = id π -1 (T M ) , βoρ = id H(T M ) on H(TM) 0 on V(TM) (1.1)\n\nFor a regular connection ∇ we define two covariant derivatives 1 ∇ and\n\n∇ as follows: For every vector (1)π-form A, we have\n\n( 1 ∇ A)(øX, øY ) := (∇ βøX A)(øY ) , ( 2 ∇ A)(øX, øY ) := (∇ γøX A)(øY ).\n\nThe classical torsion tensor T of the connection ∇ is defined by\n\nT(X, Y ) = ∇ X ρY -∇ Y ρX -ρ[X, Y ] ∀ X, Y ∈ X(T M).\n\nThe horizontal ((h)h-) and mixed ((h)hv-) torsion tensors, denoted respectively by Q and T , are defined by\n\nQ(X, Y ) = T(βXβY ), T (X, Y ) = T(γX, βY ) ∀ X, Y ∈ X(π(M)).\n\nThe classical curvature tensor K of the connection ∇ is defined by\n\nK(X, Y )ρZ = -∇ X ∇ Y ρZ + ∇ Y ∇ X ρZ + ∇ [X,Y ] ρZ ∀ X, Y, Z ∈ X(T M).\n\nThe horizontal (h-), mixed (hv-) and vertical (v-) curvature tensors, denoted respectively by R, P and S, are defined by\n\nR(X, Y )øZ = K(βXβY )øZ, P (X, Y )øZ = K(βX, γY )øZ, S(X, Y )øZ = K(γX, γY )øZ.\n\nWe also have the (v)h-, (v)hv-and (v)v-torsion tensors, denoted respectively by R, P and S, defined by\n\nR(X, Y ) = R(X, Y )øη, P (X, Y ) = P (X, Y )øη, S(X, Y ) = S(X, Y )øη.\n\nTheorem 1.1. [25] Let (M, L) be a Finsler manifold. There exists a unique regular connection\n\n∇ in π -1 (T M) such that (a) ∇ is metric : ∇g = 0, (b) The horizontal torsion of ∇ vanishes : Q = 0, (c) The mixed torsion T of ∇ satisfies g(T (X, Y ), Z) = g(T (X, Z), Y ).\n\nSuch a connection is called the Cartan connection associated to the Finsler manifold (M, L).\n\nOne can show that the torsion T of the Cartan connection has the property that T (X, η) = 0 for all X ∈ X(π(M)) and associated to T we have: Definition 1.2. [25] Let ∇ be the Cartan connection associated to (M, L). The torsion tensor field T of the connection ∇ induces a π-tensor field of type (0, 3), called the Cartan tensor and denoted again T , defined by :\n\nT (X, Y , Z) = g(T (X, Y ), Z), for all X, Y , Z ∈ X(T M).\n\nIt also induces a π-form C, called the contracted torsion, defined by : C(X) := T r{Y -→ T (X, Y )}, for all X ∈ X(T M). Definition 1.3. [25] With respect to the Cartan connection ∇ associated to (M, L), we have -The horizontal and vertical Ricci tensors Ric h and Ric v are defined respectively by:\n\nRic h (X, Y ) := T r{Z -→ R(X, Z)Y }, for all X, Y ∈ X(T M), Ric v (X, Y ) := T r{Z -→ S(X, Z)Y }, for all X, Y ∈ X(T M).\n\n-The horizontal and vertical Ricci maps Ric h 0 and Ric v 0 are defined respectively by:\n\ng(Ric h 0 (X), Y ) := Ric h (X, Y ), for all X, Y ∈ X(T M), g(Ric v 0 (X), Y ) := Ric v (X, Y ), for all X, Y ∈ X(T M).\n\n-The horizontal and vertical scalar curvatures Sc h , Sc v are defined respectively by:\n\nSc h := Tr(Ric h 0 ), Sc v := Tr(Ric v 0 )\n\n, where R and S are respectively the horizontal and vertical curvature tensors of ∇.\n\nProposition 1.4. [12] Let (M, L) be a Finsler manifold. The vector field G determined by i G Ω = -dE is a spray, called the canonical spray associated to the energy E, where E := 1 2 L 2 and Ω := dd J E. One can show, in this case, that G = βoη, and G is thus horizontal with respect to the Cartan connection ∇."
},
{
"section_type": "OTHER",
"section_title": "Special Finsler spaces",
"text": "In this section, we introduce the global definitions of the most important and commonly used special Finsler spaces in such a way that, when localized, they yield the usual local definitions existing in the literature (see the Appendix). Here we simply set the definitions, postponing investigation of the mutual relationships between these special Finsler spaces to the next section. The definitions are arranged according to the type of defining property of the special Finsler space concerned.\n\nThroughout the paper, g, g, ∇ and D denote respectively the Finsler metric in π -1 (T M), the induced metric in π -1 (T * M), the Cartan connection and the Berwald connection associated to a given Finsler manifold (M, L). Also, T denotes the torsion tensor of the Cartan connection (or the Cartan tensor) and R, P and S denote respectively the horizontal curvature, the mixed curvature and the vertical curvature of the Cartan connection. (a) Riemannian if the metric tensor g(x, y) is independent of y or, equivalently, if\n\nT (X, Y ) = 0, for all X, Y ∈ X(π(M)).\n\n(b) locally Minkowskian if the metric tensor g(x, y) is independent of x or, equivalently, if ∇ βX T = 0 and R = 0.\n\nDefinition 2.2. A Finsler manifold (M, L) is said to be :\n\n(a) Berwald [24] if the torsion tensor T is horizontally parallel. That is,\n\n∇ βX T = 0.\n\n(b) C h -recurrent if the torsion tensor T satisfies the condition\n\n∇ βX T = λ o (X) T,\n\nwhere λ o is a π-form of order one.\n\n(c) P * -Finsler manifold if the π-tensor field ∇ βη T is expressed in the form\n\n∇ βη T = λ(x, y) T, where λ(x, y) = b g(∇ βη C,C) C 2 = g(∇ βη øC,øC) C 2\n\nand C 2 := g(C, C) = C(C) = 0; C being the π-vector field defined by g(C, X) = C(X). Definition 2.3. A Finsler manifold (M, L) is said to be:\n\n(a) C v -recurrent if the torsion tensor T satisfies the condition (∇ γX T )(Y , Z) = λ o (X)T (Y , Z).\n\n(b) C 0 -recurrent if the torsion tensor T satisfies the condition\n\n(D γX T )(Y , Z) = λ o (X)T (Y , Z)."
},
{
"section_type": "OTHER",
"section_title": "Definition 2.4. [25]",
"text": "A Finsler manifold (M, L) is said to be :\n\n(a) semi-C-reducible if dimM ≥ 3 and the Cartan tensor T has the form\n\nT (X, Y , Z) = µ n + 1 { (X, Y )C(Z) + (Y , Z)C(X) + (Z, X)C(Y )}+ + τ C 2 C(X)C(Y )C(Z),\n\nwhere µ and τ are scalar functions satisfying µ + τ = 1, = g -ℓ ⊗ ℓ and ℓ(X) := L -1 g(X, η).\n\n(b) C-reducible if dimM ≥ 3 and the Cartan tensor T has the form\n\nT (X, Y , Z) = 1 n + 1 { (X, Y )C(Z) + (Y , Z)C(X) + (Z, X)C(Y )}.\n\n(c) C 2 -like if dimM ≥ 2 and the Cartan tensor T has the form\n\nT (X, Y , Z) = 1 C 2 C(X)C(Y )C(Z).\n\nDefinition 2.5. A Finsler manifold (M, L), where dimM ≥ 3, is said to be quasi-Creducible if the Cartan tensor T is written as :\n\nT (X, Y , Z) = A(X, Y )C(Z) + A(Y , Z)C(X) + A(Z, X)C(Y ),\n\nwhere A is a symmetric indicatory (2) π-form (A(X, η) = 0 for all X).\n\nDefinition 2.6. [25] A Finsler manifold (M, L) is said to be :\n\n(a) S 3 -like if dim(M) ≥ 4 and the vertical curvature tensor S(X, Y , Z, W ) := g(S(X, Y )Z, W ) has the form :\n\nS(X, Y , Z, W ) = Sc v (n -1)(n -2) { (X, Z) (Y , W ) -(X, W ) (Y , Z)}.\n\n(b) S 4 -like if dim(M) ≥ 5 and the vertical curvature tensor S(X, Y , Z, W ) has the form :\n\nS(X, Y , Z, W ) = (X, Z)F(Y , W ) -(Y , Z)F(X, W )+ + (Y , W )F(X, Z) -(X, W )F(Y , Z), (2.1)\n\nwhere\n\nF is the (2)π-form defined by F = 1 n -3 {Ric v - Sc v 2(n -2)\n\n}.\n\nDefinition 2.7. A Finsler manifold (M, L) is said to be :\n\n(a) S v -recurrent if the v-curvature tensor S satisfies the condition\n\n(∇ γX S)(Y , Z, W ) = λ(X)S(Y , Z)W ,\n\nwhere λ is a π-form of order one.\n\n(b) S v -recurrent of the second order if the v-curvature tensor S satisfies the condition\n\n( 2 ∇ 2 ∇ S)(øY, øX, Z, W , U) = Θ(X, Y )S(Z, W )U,\n\nwhere Θ is a π-form of order two."
},
{
"section_type": "OTHER",
"section_title": "Definition 2.8. [24]",
"text": "A Finsler manifold (M, L) is said to be :\n\n(a) a Landsberg manifold if\n\nP (X, Y ) = P (X, Y )η = 0 ∀ X, Y ∈ X(π(M)), or equivalently ∇ βη T = 0. (b) a general Landsberg manifold if T r{Y -→ P (X, Y )} = 0 ∀ X, ∈ X(π(M)), or equivalently ∇ βη C = 0.\n\nDefinition 2.9. A Finsler manifold (M, L) is said to be P -symmetric if the mixed curvature tensor P satisfies\n\nP (X, Y )Z = P (Y , X)Z, ∀ øX, øY, øZ ∈ X(π(M)).\n\nDefinition 2.10. A Finsler manifold (M, L), where dimM ≥ 3, is said to be P 2 -like if the mixed curvature tensor P has the form :\n\nP (X, Y , Z, øW ) = α(Z)T (X, Y , øW ) -α(W ) T (X, øY, Z),\n\nwhere α is a (1) π-form (positively homogeneous of degree 0)."
},
{
"section_type": "OTHER",
"section_title": "Definition 2.11. [25]",
"text": "A Finsler manifold (M, L), where dimM ≥ 3, is said to be P -reducible if the π-tensor field P (X, Y , Z) := g(P (X, Y )η, Z) can be expressed in the form :\n\nP (X, Y , Z) = δ(X) (Y , Z) + δ(Y ) (Z, X) + δ(Z) (X, Y ),\n\nwhere δ is a (1) π-form satisfying δ(øη) = 0.\n\nDefinition 2.12.\n\n[2] A Finsler manifold (M, L), where dimM ≥ 3, is said to be h-isotropic if there exists a scalar k o such that the horizontal curvature tensor R has the form\n\nR(X, Y )Z = k o {g(Y , Z)X -g(X, Z)Y }. Definition 2.13. [2] A Finsler manifold (M, L)\n\n, where dimM ≥ 3, is said to be :\n\n(a) of scalar curvature if there exists a scalar function k :\n\nT M -→ R such that the horizontal curvature tensor R(X, Y , Z, W ) := g(R(X, Y )Z, W ) satisfies the relation R(η, X, η, Y ) = kL 2 (X, Y ). (b) of constant curvature if the function k in (a) is constant. Definition 2.14. A Finsler manifold (M, L) is said to be R 3 -like if dimM ≥ 4 and the horizontal curvature tensor R(X, Y , Z, W ) is expressed in the form R(X, Y , Z, W ) =g(X, Z)F (Y , W ) -g(Y , Z)F (X, W )+ + g(Y , W )F (X, Z) -g(X, W )F (Y , Z), (2.2)\n\nwhere\n\nF is the (2)π-form defined by F = 1 n-2 {Ric h -Sc h g 2(n-1) }."
},
{
"section_type": "OTHER",
"section_title": "Relationships between different types of special Finsler spaces",
"text": "This section is devoted to global investigation of some mutual relationships between the special Finsler spaces introduced in the preceding section. Some consequences are also drawn from these relationships.\n\nWe start with some immediate consequences from the definitions: (a) A Locally Minkowskian manifold is a Berwald manifold.\n\n(b) A Berwald manifold is a Landsberg manifold. (c) A Landsberg manifold is a general Landsberg manifold. (d) A Berwald manifold is C h -recurrent (resp. P * -Finsler). (e) A P * -manifold is a Landsberg manifold. (f) A C-reducible (resp. C 2 -like) manifold is semi-C-reducible. (g) A semi-C-reducible manifold is quasi-C-reducible. (h) A Finsler manifold of constant curvature is of scalar curvature.\n\nThe following two lemmas are useful for subsequent use. where ℓ is the π-form given by ℓ(X) = L -1 g(X, η), then we have:\n\n(a) (øX, øY ) = g(φ(øX), øY ), (b) φ(øη) = 0, (c) φ o φ = φ, (d) T r(φ) = n -1, (e) ∇ βøX φ = 0, (f) ∇ βøX = 0.\n\nAs we have seen, a Landsberg manifold is general Landsberg. The converse is not true. Nevertheless, we have\n\nProposition 3.3. A C-reducible general Landsberg manifold (M, L) is a Landsberg manifold.\n\nProof. Since (M, L) is a C-reducible manifold, then, by Definition 2.4, Lemma 3.2, the symmetry of and the non-degeneracy of g, we get\n\nT (øX, øY ) = 1 n + 1 { (øX, øY )øC + C(øX)φ(øY ) + C(øY )φ(øX)},\n\nwhere øC is the π-vector field defined by g(øC, øX) := C(øX). Taking the h-covariant derivative ∇ βøZ of both sides of the above equation, we obtain\n\n(∇ βøZ T )(øX, øY ) = 1 n + 1 {(∇ βøZ )(øX, øY )øC + (øX, øY )∇ βøZ øC + C(øX)(∇ βøZ φ)(øY ) + +(∇ βøZ C)(øX)φ(øY ) + C(øY )(∇ βøZ φ)(øX) + (∇ βøZ C)(øY )φ(øX)},\n\nfrom which, by setting øZ = øη and taking into account the fact that ∇ βøZ = 0 and that ∇ βøZ φ = 0 ( Lemma 3.2), we get Combining the above two Propositions, we obtain the more powerful result :\n\n(∇ βøη T )(øX, øY ) = 1 n + 1 { (øX, øY )∇ βøη øC+(∇ βøη C)(øX)φ(øY )+(∇ βøη C)(øY )φ(øX)}.\n\nProposition 3.5. A C-reducible general Landsberg manifold (M, L) is a Berwald manifold.\n\nSumming up, we get: Theorem 3.6. Let (M, L) be a C-reducible Finsler manifold. The following assertion are equivalent :\n\n(a) (M, L) is a Berwald manifold. (b) (M, L) is a Landsberg manifold. (c) (M, L) is a general Landsberg manifold.\n\nWe retrieve here a result of Matsumuoto [15] , namely Corollary 3.7. If the h-curvature tensor R and hv-curvature tensor P of a Creducible manifold vanish, then the manifold is Locally Minkowskian. Remark 3.8. [15] It may be conjectured that a Finsler manifold will be Minkowskian if the h-curvature tensor R and hv-curvature tensor P vanish. As above seen the conjecture is verified already under somewhat strong condition \" C-reducibility\". Theorem 3.9. Let (M, L) be a Finsler manifold. Then we have : Applying the h-covariant derivative ∇ βøW on both sides of the above equation, taking into account the fact that (∇ βøW T )(øX, øY, øZ) = g((∇ βøW T )(øX, øY ), øZ) and that ∇ βøW = 0, we obtain\n\ng((∇ βøW T )(øX, øY ), øZ) = 1 n + 1 S øX,øY,øZ { (øX, øY )(∇ βøW C)(øZ)}.\n\nFrom which, by setting øW = øη and noting that P (øX, øY )øη = (∇ βøη T )(øX, øY ), the result follows.\n\n(b) Since (M, L) is a P -reducible manifold, then by Definition 2.11, taking into account the fact that g is nondegenerate, we obtain\n\nP (øX, øY )øη = δ(øX)φ(øY ) + δ(øY )φ(øX) + ø (øX, øY ) øζ, (3.2)\n\nwhere øζ is the π-vector field defined by g(øζ, øX) := δ(øX).\n\nSince δ(øη) = 0, then T r{øY -→ δ(øY )φ(øX) + (øX, øY ) øζ} = 2δ(øX). Taking the trace of both sides of (3.2), using the fact that P (øX, øY )øη = (∇ βøη T )(øX, øY ) (Lemma 3.1) and that T r{øY -→ (∇ βøη T )(øX, øY )} = (∇ βøη C)(øX), we get\n\nδ(øX) = 1 n + 1 (∇ βøη C)(øX). (3.3)\n\nNow, from Equations (3.2) and (3.3), we have\n\ng(P (øX, øY )øη, øZ) = 1 n + 1 S øX,øY,øZ { (øX, øY )(∇ βøη C)(øZ)}. (3.4)\n\nAccording to the given assumption that the manifold is general Landsberg, then ∇ βøη C = 0. Therefore, from (3.4), we get P (øX, øY )øη = 0 and hence the manifold is Landsberg.\n\nProposition 3.10.\n\n(a) A C h -recurrent manifold is a P * -Finsler manifold.\n\n(b) A general Landsberg P * -Finsler manifold is a Landsberg manifold.\n\nProof. The proof is straightforward and we omit it.\n\n(∇ βøZ T )(øX, øY ) = -∇ βøZ C(øC) C 4 C(øX)C(øY )øC + 1 C(øC) (∇ βøZ C)(øX)C(øY )øC + + 1 C(øC) (∇ βøZ C)(øY )C(øX)øC + 1 C(øC) C(øX)C(øY )∇ βøZ øC.\n\nIn view of this relation, ∇ βøZ T = 0 if, and only if, ∇ βøZ C = 0. Hence the result.\n\nCorollary 3.12. A C 2 -like general Landsberg manifold is a Landsberg manifold.\n\nIn view of the above Theorems, we have:\n\nCorollary 3.13. The two notions of being Landsberg and general Landsberg coincide in the case of C-reducibility, P -reducibility, C 2 -likeness or P * -Finsler.\n\nAs we know, a C-reducible Landsberg manifold is a Berwald manifold (Proposition 3.4 ). Moreover, A C 2 -like Finsler manifold is a Berwald manifold if, and only if, the π-tensor field C is horizontally parallel (Proposition 3.11). We shall try to generalize these results to the case of semi-C-reduciblity. Taking the h-covariant derivative of both sides, noting that ∇ βøX = 0, we get\n\n(∇ βøW T )(øX, øY, øZ) = 1 n + 1 S øX,øY,øZ { (øX, øY ){µ(∇ βøW C)(øZ) + (∇ βøW µ)C(øZ)}} + + τ C 2 S øX,øY,øZ {(∇ βøW C)(øX)C(øY )C(øZ)} - -{ ∇ βøW µ C 2 + τ ∇ βøW C(øC) C 4 }C(øX)C(øY )C(øZ).\n\nNow, if the characteristic scalar µ and the π-tensor field C are horizontally parallel, then ∇ βøW T = 0 and (M, L) is a Berwald manifold.\n\nConversely, if (M, L) is a Berwald manifold, then ∇ βøX T = 0 and hence ∇ βøX C = 0, ∇ βøX øC = 0. These, together with the above equation, give\n\n∇ βøW µ{ 1 n + 1 S øX,øY,øZ { (øX, øY )C(øZ)} - 1 C 2 C(øX)C(øY )C(øZ)} = 0, which implies immediately that ∇ βøW µ = 0.\n\nThe following lemmas are useful for subsequent use Lemma 3.15. For all X, Y ∈ X(π(M)), we have :\n\n(a) [γX, γY ] = γ(∇ γX Y -∇ γY X) (b) [γX, βY ] = -γ(P (Y , X)η + ∇ βY X) + β(∇ γX Y -T (X, Y )) (c) [βX, βY ] = γ(R(X, Y )η) + β(∇ βX Y -∇ βY X)\n\nLemma 3.16. For all øX, øY, øZ, øW ∈ X(π(M)) and W ∈ X(T M), we have : (a) g((∇ W T )(øX, øY ), øZ) = g((∇ W T )(øX, øZ), øY ), (b) g(S(øX, øY )øZ, øW ) = -g(S(øX, øY )øW, øZ)."
},
{
"section_type": "OTHER",
"section_title": "Proof. (a)",
"text": "From the definition of the covariant derivative, we get g((\n\n∇ W T )(øX, øY ), øZ) = g(∇ W T (øX, øY ), øZ) -g(T (∇ W øX, øY ), øZ)- -g(T (øX, ∇ W øY ), øZ). (3.5) Now, we have g(∇ W T (øX, øY ), øZ) = W • g(T (øX, øY ), øZ) -g(T (øX, øY ), ∇ W øZ) = W • g(T (øX, øY ), øZ) -g(T (øX, ∇ W øZ), øY ), Similarly, g(T (øX, ∇ W øY ), øZ) = W • g(T (øX, øZ), øY ) -g(∇ W T (øX, øZ), øY ).\n\nSubstituting these two equations into (3.5), noting the property that g(T (∇ W øX, øY ), øZ) = g(T (∇ W øX, øZ), øY ) (cf. §1), the result follows.\n\n(b) follows directly from the general formula (which can be easily proved) g(K(X, Y )øZ, øW ) + g(K(X, Y )øW, øZ) = 0 by setting X = γøX and Y = γøY , where K is the classical curvature tensor of the Cartan connection as a linear connection in the pull-back bundle (cf. §1). Proposition 3.17. Let (M, L) be a C h -recurrent Finsler manifold (∇ βøX T = λ 0 (øX)T ). Then, we have: ) P (øX, øY ).\n\nProof.\n\n(a) The hv-curvature tensor P can be written in the form [25] :\n\nP (øX, øY, øZ, øW ) = g((∇ βøZ T )(øX, øY ), øW ) -g((∇ βøW T )(øX, øY ), øZ)+ +g(T (øX, øZ), P (øW, øY )) -g(T (øX, øW ), P (øZ, øY )).\n\nThen, by using P (øX, øY ) = (∇ βøη T )(øX, øY ) (Lemma 3. Then, (b) follows from the above two equations. Theorem 3.18. Assume that (M, L) is C h -recurrent. Then, the v-curvature tensor S is recurrent with respect to the h-covariant differentiation : ∇ βøX S = θ(øX)S, where θ is a π-form of order one.\n\nProof. One can easily show that : For all X, Y, Z ∈ X(T M), From which, since g(T (øX, øY ), øZ) = g(T (øX, øZ), øY ), we have g(S(øX, øY )øZ, øW ) = g((∇ γøY T )(øX, øZ), øW ) -g((∇ γøX T )(øY, øZ), øW )+ +g(T (øX, øW ), T (øY, øZ)) -g(T (øY, øW ), T (øX, øZ)).\n\nS X,Y,Z {K(X, Y )ρZ + ∇ X T(Y, Z) + T(X, [Y, Z])} = 0.\n\nSimilarly, g(S(øX, øY )øW, øZ) = g((∇ γøY T )(øX, øW ), øZ) -g((∇ γøX T )(øY, øW ), øZ)+ +g(T (øX, øZ), T (øY, øW )) -g(T (øY, øZ), T (øX, øW )).\n\nThe above two equations, together with Lemma 3.16, yield g((∇ γøX T )(øY, øZ), øW ) = g((∇ γøY T )(øX, øZ), øW ). Now, using the given assumption that the manifold is C h -recurrent, Equation (3.8) implies that (∇ βøX S)(øY, øZ, øV, øW ) = ∇ βøX S(øY, øZ, øV, øW ) --S(∇ βøX øY, øZ, øV, øW ) -S(øY, ∇ βøX øZ, øV, øW ) --S(øY, øZ, ∇ βøX øV, øW ) -S(øY, øZ, øV, ∇ βøX øW ). = +∇ βøX g(T (øY, øW ), T (øZ, øV )) -∇ βøX g(T (øZ, øW ), T (øY, øV )) --g(T (∇ βøX øY, øW ), T (øZ, øV )) + g(T (øZ, øW ), T (∇ βøX øY, øV )) --g(T (øY, øW ), T (∇ βøX øZ, øV )) + g(T (∇ βøX øZ, øW ), T (øY, øV )) --g(T (øY, øW ), T (øZ, ∇ βøX øV )) + g(T (øZ, øW ), T (øY, ∇ βøX øV )) --g(T (øY, ∇ βøX øW ), T (øZ, øV )) + g(T (øZ, ∇ βøX øW ), T (øY, øV )). = g((∇ βøX T )(øY, øW ), T (øZ, øV )) + g(T (øY, øW ), (∇ βøX T )(øZ, øV )) --g((∇ βøX T )(øZ, øW ), T (øY, øV )) -g(T (øZ, øW ), (∇ βøX T )(øY, øV )). = 2λ o (øX)S(øY, øZ, øV, øW ) =: θ(øX)S(øY, øZ, øV, øW ).\n\nHence, the result follows.\n\nRic v (øX, øY ) = (3 -n) (n + 1) 2 C(øX)C(øY ) - (n -1) (n + 1) 2 C 2 (øX, øY ).\n\n(c) the vertical scalar curvature Sc v has the form\n\nSc v = (2 -n) (n + 1) C 2 .\n\nTheorem 3.22. A Finsler manifold (M, L) is P -Symmetric if, and only if, the v-curvature tensor S satisfies the equation ∇ βøη S = 0.\n\nProof. One can show that: For all X, Y, Z ∈ X(T M), Setting øZ = øη and using the fact that T (øX, øη) = 0 and that K o γ = id X(π(M )) , the result follows.\n\nS X,Y,Z {∇ Z K(X, Y ) -K(X, Y )∇ Z -K([X, Y ], Z)} = 0. ( 3\n\n(b) Follows from (a) together with the relation T (øX, øη) = 0.\n\n(c) Setting X = γøX, Y = γøY and Z = γøZ in (3.9) and using Lemma 3.15, we get S øX,øY,øZ (∇ γøX S)(øY, øZ, øW ) = 0. Again, setting øW = øη in the above equation and using the fact that S(øX, øY )øη = 0 and that K o γ = id X(π(M )) , the result follows. Proof. Taking the v-covariant derivative of both sides of the relation in Corollary 3.19(b) and, then, using the assumption that ∇ γX T = λ 0 (X)T , we get (∇ γøX S)(øY, øZ, øV, øW ) = 2λ o (øX)S(øY, øZ, øV, øW ) =: ψ(øX)S(øY, øZ, øV, øW ), which shows that S is v-recurrent. Now, setting øV = øη in the last equation, using the properties of S and noting that K o γ = id X(π(M )) , we conclude that S = 0.\n\nThe following result gives a characterization of Riemannian manifolds in terms of C v -recurrence and C 0 -recurrence.\n\nTheorem 3.28. (a) A C v -recurrent Finsler manifold is Riemannian, (b) A C 0 -recurrent Finsler manifold is Riemannian. Proof. (a) Since (M, L) is C v -recurrent, then (∇ γX T )(Y , Z) = λ o (X)T (Y , Z),\n\nfrom which, by setting øX = øη and noting that ∇ γøη T = -T , we get\n\nT (Y , Z) = -λ o (η)T (Y , Z). (3.12)\n\nBut since (∇ γøX T )(øY, øZ) = (∇ γøY T )(øX, øZ) (Corollary 3.19), then λ o (øX)T (øY, øZ) = λ o (øY )T (øX, øZ). Hence,\n\nλ o (η)T (Y , Z) = 0. (3.13)\n\nThen, the result follows from (3.12) and (3.13).\n\n(b) can be proved similarly.\n\nTheorem 3.29. For a Finsler manifold (M, L), the following assertions are equivalent :\n\n(a) (M, L) is S v -recurrent.\n\n(b) The v-curvature tensor S vanishes identically.\n\n(c) (M, L) is S v -recurrent\n\n2 ∇ 2 ∇ S)(øY, øX, øZ, øV, øW ) = ∇ γøY (∇ γøX S)(øZ, øV, øW ) -(∇ γ∇ γøY øX S)(øZ, øV, øW )- -(∇ γøX S)(∇ γøY øZ, øV, øW ) -(∇ γøX S)(øZ, ∇ γøY øV, øW )- -(∇ γøX S)(øZ, øV, ∇ γøY øW ).\n\nBy substituting øZ = øη = øW in the above equation and using Lemma 3.25 and the fact that S(øX, øY )øη = 0, we get S(øX, øY )øZ = -S(øZ, øY )øX and S(øX, øY )øZ = -S(øX, øZ)øY.\n\nFrom this, together with the identity S øX,øY,øZ S(øX, øY )øZ = 0, the v-curvature tensor S vanishes identically.\n\nIn view of the above theorem we have :\n\nCorollary 3.30.\n\n(a) An S v -recurrent (resp. a second order S v -recurrent) manifold (M, L) is S 3 -like, provided that dim M ≥ 4.\n\n(b) An S v -recurrent (resp. a second order S v -recurrent) manifold (M, L) is S 4 -like, provided that dim M ≥ 5.\n\nTheorem 3.31. If (M, L) is a P 2 -like Finsler manifold, then the v-curvature tensor S vanishes or the hv-curvature tensor P vanishes. In the later case, the h-covariant derivative of S vanishes.\n\nProof. As (M, L) is P 2 -like, then P (X, Y , η, øW ) = α(η)T (X, Y , øW ) =: α o T (X, Y , øW ) and hence P (øX, øY ) = α o T (X, Y ). (3.14)\n\nNow, setting øW = øη into (3.10), we get (∇ γøY P )(øZ, øX) -(∇ γøX P )(øZ, øY ) -P (øZ, øX)øY + P (øZ, øY )øX--P (T (øX, øZ), øY ) + P (T (øY, øZ), øX) = 0.\n\nHence, g((∇ γøY P )(øZ, øX), øW ) -g((∇ γøX P )(øZ, øY ), øW ) -P (øZ, øX, øY, øW )+ +P (øZ, øY, øX, øW ) -g( P (T (øX, øZ), øY ), øW ) + g( P (T (øY, øZ), øX), øW ) = 0.\n\nFrom which, together with (3. It is to be observed that the left-hand side of the above equation is symmetric in the arguments øZ and øW while the right-hand side is skew-symmetric in the same arguments. Hence we have\n\nα o S(øX, øY, øW, øZ) = 0, (3.15)\n\nε(øY )T (øX, øZ, øW ) -ε(øX)T (øY, øZ, øW ) = 0, (3.16) where ε is the π-form defined by ε(øY ) := (∇ γøY α)(øη). Now, If ε = 0, it follows from (3.16) that there exists a scalar function Υ such that T (øX, øY, øZ) = Υ ε(øX)ε(øY )ε(øZ). Consequently, T (øX, øY ) = Υ ε(øX)ε(øY )øε, where g(øε, øX) := ε(øX). From which S(øX, øY, øZ, øW ) = g(T (øX, øW ), T (øY, øZ)) -g(T (øY, øW ), T (øX, øZ)) = Υ ε(øX)ε(øY )ε(øZ)ε(øW )g(øε, øε) -Υ ε(øX)ε(øY )ε(øZ)ε(øW )g(øε, øε) = 0.\n\nOn the other hand, if the v-curvature tensor S = 0, then it follows from (3.15) that ε = 0 and α(øη) = 0. Hence, α = 0 and the hv-curvature tensor P vanishes. In this case, it follows from the identity (3.10) that ∇ βøX S = 0. Proposition 3.32. A P 2 -like Finsler manifold (M, L) is a P * -Finsler manifold.\n\nProof. As (M, L) is P 2 -like, then from (3.14), we have P (X, Y ) = α o T (X, Y ). Using Lemma 3.1, we get (∇ βøη T )(øX, øY ) = α 0 T (øX, øY ), from which, by taking the trace,\n\n∇ βøη C = α 0 T , where α 0 = b g(∇ βη C,C) C 2\n\n. Hence the result.\n\nThe next definition will be useful in the sequel. Deicke theorem [4] can be formulated globally as follows:\n\nLemma 3.36. Let (M, L) be a Finsler manifold. The following assertions are equivalent: Now, we focus our attention to the interesting case (c) of the above theorem. In this case, the h-curvature tensor R = 0 and hence the (v)h-torsion tensor R = 0. Therefore, the equation (deduced from (3.9)) (∇ γøX R)(øY, øZ, øW ) + (∇ βøY P )(øZ, øX, øW ) -(∇ βøZ P )(øY, øX, øW )--P (øZ, P (øY, øX)øη)øW + R(T (øX, øY ), øZ)øW -S(R(øY, øZ)øη, øX)øW + +P (øY, P (øZ, øX)øη)øW -R(T (øX, øZ), øY )øW = 0. reduces to (∇ βøY P )(øZ, øX, øW ) -(∇ βøZ P )(øY, øX, øW )--P (øZ, P (øY, øX))øW + P (øY, P (øZ, øX))øW = 0.\n\n(a) (M, L) is Riemannian, (\n\nSetting øW = øη, we get (∇ βøY P )(øZ, øX) -(∇ βøZ P )(øY, øX) -P (øZ, P (øY, øX)) + P (øY, P (øZ, øX)) = 0.\n\n(3.22) Since (M, L) is C h -recurrent, then, by Proposition 3.17, the (v)hv-torsion tensor P satisfies the relations (∇ βøZ P )(øX, øY ) = (K o λ o (øZ) + ∇ βøZ K o )T (øX, øY ) and P (øX, øY ) = λ o (øη)T (øX, øY ) = K o T (øX, øY ). From these, together with (3.22), we get On the other hand, if K o = 0, then the v-curvature tensor S vanishes from (3.23) . Next, it is seen from (3.24) that, if V(øY ) := K o λ o (øY ) + ∇ βøY K o = 0, then there exists a scalar function Υ = T (øX,øZ,øW )T (øX,øY,øZ)T (øY,øZ,øW ) (T (øX,øY,øW )) 2 (V(øZ)) 3 such that T (øX, øY, øW ) = Υ V(øX)V(øY )V(øW ).\n\n(K o λ o (øY ) + ∇ βøY K o )T (øZ, øX) -(K o λ o (øZ) + ∇ βøZ K o )T (øX, øY )- -K 2 o T (\n\nSumming up, we have Theorem 3.38. Let (M, L) be a Finsler manifold of dimensions n ≥ 3.\n\nIf (M, L) is h-isotropic and C h -recurrent, then (\n\nP • ω)(øX, øY ) = φ(ω(φ(øX), φ(øY ))) = φ{ω(øX -L -1 ℓ(øX)øη, øY -L -1 ℓ(øY )øη)} = φ{ω(øX, øY ) -L -1 ℓ(øY )ω(øX, øη)- -L -1 ℓ(øX)ω(øη, øY ) + L -2 ℓ(øX)ℓ(øY )ω(øη, øη)} = ω(øX, øY ) -L -2 g(ω(øX, øY ), øη)øη -φ{L -1 ℓ(øY )ω(øX, øη)+ +L -1 ℓ(øX)ω(øη, øY ) -L -2 ℓ(øX)ℓ(øY )ω(øη, øη)} (3.\n\n25) Now, if ω(øX, øη) = 0 = ω(øη, øX) and g(ω(øX, øY ), øη) = 0, then (3.25) implies that (P • ω)(øX, øY ) = ω(øX, øY ) and hence ω is indicatory.\n\nOn the other hand, if ω is indicatory, then ω(øX, øY ) = φ(ω(φ(øX), φ(øY ))). From which, setting øX = øη (resp. øY = øη) and taking into account the fact that φ(øη) = 0 (Lemma 3.2), we get ω(øη, øY ) = 0 (resp. ω(øX, øη) = 0). From this, together with (P•ω)(øX, øY ) = ω(øX, øY ), Equation (3.25) implies that L -2 g(ω(øX, øY ), øη)øη = 0. Consequently, g(ω(øX, øY ), øη) = 0. (b) The proof is similar to that of (a) and we omit it. From which, using the fact that g(F o (øX), øY ) = F (øX, øY ) and that the Finsler metric g is non-degenerate, the result follows. The tensor field Ψ in the above theorem being of the same form as the Weyl conformal tensor in Riemannian geometry, we draw the following Theorem 3.56. An R 3 -like Riemannian manifold is conformally flat. Remark 3.57. It should be noted that some important results of [8] , [9] , [11] , [13] , [19] , [20] ,...,etc. (obtained in local coordinates) are retrieved from the above mentioned global results (when localized).\n\nF : F (X, Y ) := 1 n-2 {Ric h (X, Y ) -Sc h g(X,Y ) 2(n-1) }, F o : g(F o (øX), øY ) := F (øX, øY ),\n\n• Second order S v -recurrent manifold [20] , [11] : S hijk | m | n = Θ mn S hijk , where Θ ij (x, y) is a covariant tensor field.\n\n• Landsberg manifold [7] : P h kji y k = 0 ⇐⇒ ( ∂i Γ h jk )y k = 0 ⇐⇒ C h ij|k y k = 0.\n\n• General Landsberg manifold [10] : P r ijr y i = 0 ⇐⇒ C j|o = 0.\n\n• P -symmetric manifold [19] : P hijk = P hikj .\n\n• P 2 -like manifold (dim M ≥ 3) [14] :\n\nP hijk = α h C ijk -α i C\n\nhjk , where α k (x, y) is a covariant vector field.\n\n• P -reducible manifold (dim M ≥ 3) [19] : P ijk = 1 n+1 ( ij P k + jk P i + ki P j ), where P ijk = g hi P h jk .\n\n• h-isotropic manifold (dim M ≥ 3) [13] : R hijk = k o {g hj g ik -g hk g ij }, for some scalar k o , where R hijk = g il R l\n\nhjk .\n\n• Manifold of scalar curvature [21] : R ijkl y i y k = kL 2 jl , for some function k : T M -→ R .\n\n• Manifold of constant curvature [21] : the function k in the above definition is constant.\n\n• Manifold of perpendicular scalar (or of p-scalar ) curvature [8] , [9] :\n\nP • R hijk := l h m i n j r k R lmnr = R o { ik hj -ij hk }\n\n, where R o is a function called a perpendicular scalar curvature.\n\n• Manifold of s-ps curvature [8] , [9] : (M, L) is both of scalar curvature and of p-scalar curvature.\n\n• R 3 -like manifold (dim M ≥ 4) [8] : R hijk = g hj F ik -g hk F ij + g ik F hj -g ij F hk , where\n\nF ij := 1 n-2 {R ij -1 2 r g ij }; R ij := R h ijh , r := 1 n-1 R i i ."
}
] | ||
arxiv:0704.0064 | 0704.0064 | 1 | 10.1142/S0217751X08041591 | 8f2258c6a1705a57c7308510858d8a12ea1d4ff794e5ecc2bdc0558696d75389 | Nilpotent symmetry invariance in the superfield formulation: the
(non-)Abelian 1-form gauge theories | We capture the off-shell as well as the on-shell nilpotent
Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry invariance of the
Lagrangian densities of the four (3 + 1)-dimensional (4D) (non-)Abelian 1-form
gauge theories within the framework of the superfield formalism. In particular,
we provide the geometrical interpretations for (i) the above nilpotent symmetry
invariance, and (ii) the above Lagrangian densities, in the language of the
specific quantities defined in the domain of the above superfield formalism.
Some of the subtle points, connected with the 4D (non-)Abelian 1-form gauge
theories, are clarified within the framework of the above superfield formalism
where the 4D ordinary gauge theories are considered on the (4, 2)-dimensional
supermanifold parametrized by the four spacetime coordinates x^\mu (with \mu =
0, 1, 2, 3) and a pair of Grassmannian variables \theta and \bar\theta. One of
the key results of our present investigation is a great deal of simplification
in the geometrical understanding of the nilpotent (anti-)BRST symmetry
invariance. | [
"R. P. Malik (Bhu)"
] | [
"hep-th"
] | hep-th | [] | 2007-04-01 | 2026-02-26 | The geometrical superfield approach [1] [2] [3] [4] [5] [6] [7] [8] to Becchi-Rouet-Stora-Tyutin (BRST) formalism is one of the most attractive and intuitive approaches which enables us to gain some physical insights into the beautiful (but abstract mathematical) structures that are associated with the nilpotent (anti-)BRST symmetry transformations and their corresponding generators. The latter quantities play a very decisive role in (i) the covariant canonical quantization of the gauge theories, (ii) the proof of the unitarity of the "quantum" gauge theories at any arbitrary order of perturbative computations for a given physical process (that is allowed by the theory), (iii) the definition of the physical states of the "quantum" gauge theories in the quantum Hilbert space, and (iv) the cohomological description of the physical states of the quantum Hilbert space w.r.t. the conserved and nilpotent BRST charge.
To be specific, in the superfield formulation [1-8] of the 4D 1-form gauge theories, one defines the super curvature 2-form F (2) = d Ã(1) + i Ã(1) ∧ Ã(1) in terms of the super exterior derivative d = dx µ ∂ µ + dθ∂ θ + d θ∂θ (with d2 = 0) and the super 1-form connection Ã(1) on a (4, 2)-dimensional supermanifold parametrized by the usual spacetime variables x µ (with µ = 0, 1, 2, 3) and a pair of anticommuting (i.e. θ 2 = θ2 = 0, θ θ + θθ = 0) Grassmannian variables θ and θ. The above super 2-form is subsequently equated, due to the so-called horizontality condition [1-8], to the ordinary curvature 2-form F (2) = dA (1) + iA (1) ∧ A (1) defined on the ordinary 4D flat Minkowski spacetime manifold in terms of the ordinary exterior derivative d = dx µ ∂ µ (with d 2 = 0) and the 1-form connection A (1)
= dx µ A µ .
The above super exterior derivative d and super 1-form connection Ã(1) are the generalization of the 4D ordinary exterior derivative d and 1-form connection A (1) to the (4, 2)-dimensional supermanifold because d → d, Ã(1) → A (1) in the limit (θ, θ) → 0.
The above horizontality condition (HC) has been referred to as the soul-flatness condition in [9] which amounts to setting equal to zero all the Grassmannian components of the (anti)symmetric second-rank super tensor that constitutes the super curvature 2-form F (2) on the (4, 2) -dimensional supermanifold. The key consequences, that emerge from the HC, are (i) the derivation of the nilpotent (anti-)BRST symmetry transformations for the gauge and (anti-)ghost fields of a given 4D 1-form gauge theory, (ii) the geometrical interpretation of the (anti-)BRST symmetry transformations for the 4D local fields as the translation of the corresponding superfields along the Grassmannian directions of the supermanifold, (iii) the geometrical interpretation of the nilpotency property as a pair of successive translations of the superfield along a particular Grassmannian direction of the supermanifold, and (iv) the geometrical interpretation of the anticommutativity property of the (anti-)BRST symmetry transformations for a 4D local field as the sum of (a) the translation of the corresponding superfield first along the θ-direction followed by the translation along the θ-direction, and (b) the translation of the same superfield first along the θ-direction followed by the translation along the θ-direction.
It will be noted that the above HC (i.e. F (2) = F (2) ) is valid for the non-Abelian (i.e. 2 A (1)(n) ∧ A (1)(n) = 0) 1-form gauge theory as well as the Abelian (i.e. A (1) ∧ A (1) = 0) 1-form gauge theory. As expected, for both types of theories, the HC leads to the derivation of the nilpotent (anti-)BRST symmetry transformations for the gauge and (anti-)ghost fields of the respective theories. We lay emphasis on the fact that the HC does not shed any light on the derivation of the nilpotent (anti-)BRST symmetry transformations associated with the matter fields of the interacting 4D (non-)Abelian 1-form gauge theories.
In a recent set of papers [10] [11] [12] [13] [14] [15] [16] [17] , the above HC condition has been generalized, in a consistent manner, so as to compute the nilpotent (anti-)BRST symmetry transformations associated with the matter fields of a given 4D interacting 1-form gauge theory (along with the well-known nilpotent transformations for the gauge and (anti-)ghost fields) without spoiling the cute geometrical interpretations of the (anti-)BRST symmetry transformations (and their corresponding generators) that emerge from the HC alone. The latter approach has been christened as the augmented superfield approach to BRST formalism where the restrictions imposed on the (4, 2)-dimensional superfields are (i) the HC plus the invariance of the (super) matter Noether conserved currents [10-14], (ii) the HC plus the equality of any (super) conserved quantities [15], (iii) the HC plus a restriction that owes its origin to the gauge invariance and the (super) covariant derivatives on the matter (super)fields [16, 17] , and (iv) an alternative to the HC where the gauge invariance and the property of a pair of (super) covariant derivatives on the (super) matter fields (and their intimate connection with the (super) curvatures) play a crucial role [18] [19] [20] .
In all the above approaches [1-20], however, the invariance of the Lagrangian densities of the 4D (non-)Abelian 1-form gauge theories, under the nilpotent (anti-)BRST symmetry transformations, has not yet been discussed at all. Some attempts in this direction have been made in our earlier works where the specific topological features [21, 22] of the 2D free (non-)Abelian 1-form gauge theories have been captured in the superfield formulation [23] [24] [25] . In particular, the invariance of the Lagrangian density under the nilpotent and anticommuting (anti-)BRST and (anti-)co-BRST symmetry transformations has been expressed in terms of the superfields and the Grassmannian derivatives on them. These are, however, a bit more involved in nature because of the existence of a new set of nilpotent (anti-)co-BRST symmetries in the theory. The geometrical interpretations for the Lagrangian densities and the symmetric energy-momentum tensor (for the above topological theory) have also been provided within the framework of the superfield formulation.
The purpose of our present paper is to capture the (anti-)BRST symmetry invariance of the Lagrangian density of the 4D (non-)Abelian 1-form gauge theories within the framework of the superfield approach to BRST formalism and to demonstrate that the above symmetry invariance could be understood in a very simple manner in terms of the translational generators along the Grassmannian directions of the (4, 2)-dimensional supermanifold on which the above 4D ordinary gauge theories are considered. In addition, the reason behind the existence (or non-existence) of any specific nilpotent symmetry transformation could also be explained within the framework of the above superfield approach. We demonstrate 3 the uniqueness of the existence of the nilpotent (anti-)BRST symmetry transformations for the Lagrangian density of a U(1) Abelian 1-form gauge theory. We go a step further and show the existence of the nilpotent BRST symmetry transformations for the specific Lagrangian densities (cf. (4.1) and (4.4) below) of the 4D non-Abelian 1-form gauge theory and clarify the non-existence of the anti-BRST symmetry transformations for these specific Lagrangian densities within the framework of the superfield formulation (cf. section 5 below). Finally, we provide the geometrical basis for the existence of the off-shell nilpotent and anticommuting (anti-)BRST symmetry transformations (and their corresponding generators) for the specifically defined Lagrangian densities (cf. (4.7) and/or (4.8) below) of the 4D non-Abelian 1-form gauge theory in the Feynman gauge.
The motivating factors that have propelled us to pursue our present investigation are as follows. First and foremost, to the best of our knowledge, the property of the symmetry invariance of a given Lagrangian density has not yet been captured in the language of the superfield approach to BRST formalism. Second, the above (anti-)BRST invariance of the theory has never been shown, in as simplified fashion, as we demonstrate in our present endeavour. The geometrical interpretations for (i) the existence of the above nilpotent (anti-)BRST symmetry invariance, and (ii) the on-shell conditions of the on-shell nilpotent (anti-)BRST symmetries, turn out to be quite transparent in our present work. Third, we establish the uniqueness of the existence of the (anti-)BRST symmetry invariance in their various forms. The non-existence of the specific symmetry transformation is also explained within the framework of the superfield approach to BRST formalism. Finally, our present investigation is the first modest step in the direction to gain some insights into the existence of the nilpotent symmetry transformations and their invariance for the higher form (e.g. 2-form, 3-form, etc.) gauge theories within the framework of the superfield formulation.
The contents of our present paper are organized as follows. In section 2, we recapitulate some of the key points connected with the nilpotent (anti-)BRST symmetry transformations for the free 4D Abelian 1-form gauge theory (having no interaction with matter fields) in the Lagrangian formulation. The above symmetry transformations as well as the symmetry invariance of the Lagrangian densities are captured in the geometrical superfield approach to BRST formalism in section 3 where the HC on the gauge superfield plays a crucial role. Section 4 deals with the bare essentials of the nilpotent (anti-)BRST symmetry transformations for the 4D non-Abelian 1-form gauge theory in the Lagrangian formulation. The subject matter of section 5 concerns itself with the superfield formulation of the symmetry invariance of the appropriate Lagrangian densities of the above 4D non-Abelian 1-form gauge theory. Finally, in section 6, we summarize our key results, make some concluding remarks and point out a few future directions for further investigations.
Let us begin with the following (anti-)BRST invariant Lagrangian density of the 4D Abelian 4 1-form gauge theory * in the Feynman gauge [26, 27, 9]
L (a) B = - 1 4 F µν F µν + B (∂ µ A µ ) + 1 2 B 2 -i ∂ µ C ∂ µ C, (2.1) where F µν = ∂ µ A ν -∂ ν A µ is the antisymmetric (F µν = -F νµ
) curvature tensor that constitutes the Abelian 2-form F (2)
= dA (1) ≡ 1 2! (dx µ ∧ dx ν )F µν , B
is the Nakanishi-Lautrup auxiliary multiplier field and ( C)C are the anticommuting (i.e. C 2 = C2 = 0, C C+ CC = 0) (anti-)ghost fields of the theory. The above Lagrangian density respects the off-shell nilpotent (s 2 (a)b = 0) (anti-)BRST symmetry transformations s (a)b (with s b s ab + s ab s b = 0) †
s b A µ = ∂ µ C, s b C = 0, s b C = iB, s b B = 0, s b F µν = 0, s ab A µ = ∂ µ C, s ab C = 0,
s ab C = -iB, s ab B = 0, s ab F µν = 0. (2.2) It is clear that, under the nilpotent (anti-)BRST symmetry transformations s (a)b , the curvature tensor F µν is found to be invariant. In other words, the 2-form F (2) , owing its origin to the cohomological operator d = dx µ ∂ µ , is an (anti-)BRST invariant object for the Abelian U(1) 1-form gauge theory and is, therefore, a physically meaningful (i.e. gaugeinvariant) quantity. These observations will play an important role in our discussion on the horizontality condition that would be exploited in the context of our superfield approach to (anti-)BRST invariance of the Lagrangian densities in sections 3 and 5 (see below). A noteworthy point, at this stage, is the observation that the gauge-fixing and Faddeev-Popov ghost terms can be written, modulo a total derivative, in the following fashion
s b -i C {(∂ µ A µ ) + 1 2 B}], s ab +i C {(∂ µ A µ ) + 1 2 B} , s b s ab i 2 A µ A µ + 1 2 C C . (2.
3) The above equation establishes, in a very simple manner, the (anti-)BRST invariance of the 4D Lagrangian density (2.1). The simplicity ensues due to (i) the nilpotency s 2 (a)b = 0 of the (anti-)BRST symmetry transformations, (ii) the anticommutativity property (i.e. and (iii) the invariance of the F µν term under s (a)b .
s b s ab + s ab s b = 0) of s (a)b ,
As a side remark, it is interesting to note that the following on-shell (i.e. ✷C = ✷ C = 0) nilpotent (s 2 (a)b = 0) (anti-)BRST symmetry transformations (with sb sab + sab sb = 0)
sb A µ = ∂ µ C, sb C = 0, sb C = -i(∂ µ A µ ), sb F µν = 0, sab A µ = ∂ µ C, sab C = 0, sab C = +i(∂ µ A µ ), sab F µν = 0, (2.4) 5
are the symmetry transformations for the following Lagrangian density L (a) b = -
1 4 F µν F µν - 1 2 (∂ µ A µ ) 2 -i ∂ µ C ∂ µ C. (2.5)
The above transformations (2.4) and the Lagrangian density (2.5) have been derived from (2.2) and (2.1) by the substitution B = -(∂ µ A µ ). An interesting point, connected with the on-shell nilpotent symmetry transformations, is to express the analogue of (2.3) as ‡
sb + i 2 C (∂ µ A µ ) + i A µ ∂ µ C], sab - i 2 C (∂ µ A µ ) -i A µ ∂ µ C , sb sab i 2 A µ A µ + 1 2 C C . (2.6)
It should be noted that, in the above precise computation, one has to take into account the on-shell (✷C = ✷ C = 0) conditions so that, for all practical purposes s(a)b (∂ µ A µ ) = 0. The above nilpotent (anti-)BRST symmetry transformations (i.e. s r , sr with r = b, ab) are connected with the conserved and nilpotent generators (i.e. Q r , Qr with r = b, ab). This statement can be succinctly expressed, in the mathematical form, as
s r Ω = -i [ Ω, Q r ] (±) , sr Ω = -i [ Ω, Qr ] (±) , r = b, ab, (2.7)
where the subscripts (with the signatures (±)) on the square bracket stand for the bracket to be an (anti)commutator, for the generic fields Ω = A µ , C, C, B and Ω = A µ , C, C (of the Lagrangian densities (2.1) and (2.5)), being (fermionic)bosonic in nature. The above charges Q r , Qr are found to be anticommuting (i.e. Q b Q ab +Q ab Q b = 0, Qb Qab + Qab Qb = 0) and off-shell as well as on-shell nilpotent (Q 2 (a)b = 0, Q2 (a)b = 0) in nature, respectively.
In this section, we exploit the geometrical superfield approach to BRST formalism, endowed with the theoretical arsenal of the horizontality condition, to express the (anti-)BRST symmetry transformations and the Lagrangian densities (cf. (2.1) and (2.5)) in terms of the superfields defined on the (4, 2)-dimensional supermanifold. The latter is parametrized by the spacetime coordinates x µ (with µ = 0, 1, 2, 3) and a pair of Grassmannian variables θ and θ. As a consequence, the generalization of the 4D ordinary exterior derivative d = dx µ ∂ µ and the 1-form connection A (1) = dx µ A µ (x) on the (4, 2)-dimensional supermanifold, are
d → d = dx µ ∂ µ + dθ ∂ θ + d θ ∂θ, d2 = 0, A (1) → Ã(1) = dx µ B µ (x, θ, θ) + dθ F (x, θ, θ) + d θ F (x, θ, θ), (3.1)
where the mapping from the 4D local fields to the superfields are: ) . The super-expansion of the superfields, in terms ‡ We lay emphasis on the fact that (2.6) cannot be derived directly from (2.3) by the simple substitution B = -(∂ µ A µ ). One has to be judicious to deduce the precise expression for (2.6). The logical reasons behind the derivation of (2.6) are encoded in the superfield formulation (cf. (3.9) below).
A µ (x) → B µ (x, θ, θ), C(x) → F (x, θ, θ) and C(x) → F (x, θ, θ
of the basic fields as well as the secondary fields, are (see, e.g., [4-7, 10-12]):
B µ (x, θ, θ) = A µ (x) + θ Rµ (x) + θ R µ (x) + i θ θ S µ (x), F (x, θ, θ) = C(x) + i θ B1 (x) + i θ B 1 (x) + i θ θ s(x), F (x, θ, θ) = C(x) + i θ B2 (x) + i θ B 2 (x) + i θ θ s(x).
(3.2) It can be readily seen that, in the limiting case of (θ, θ) → 0, we get back our 4D basic fields (A µ , C, C). Furthermore, on the r.h.s. of the above super expansion, the bosonic (i.e. A µ , S µ , B 1 , B1 , B 2 , B2 ) and the fermionic (R µ , Rµ , C, C, s, s) fields do match. At this juncture, we have to recall our observations after equation (2.2). The nilpotent (anti-)BRST symmetry transformations basically owe their origin to the cohomological operator d. This is capitalized in the horizontality condition where we impose the restriction d Ã(1) = dA (1) on the super 1-form connection Ã(1) that contains the superfields defined on the (4, 2)-dimensional supermanifold. The latter condition yields the following relationships (see, e.g., for details, in our earlier works [21-25]):
B 1 = B2 = s = s = 0, B1 + B 2 = 0, (3.3)
where we are free to choose the secondary fields (B 2 , B1 ) (i.e. B 2 = B ⇒ B1 = -B) in terms of the Nakanishi-Lautrup auxiliary field B of the BRST invariant Lagrangian density (2.1). The other relations, that emerge from the above HC (i.e. d Ã(1) = dA (1) ), are
R µ = ∂ µ C, Rµ = ∂ µ C, S µ = ∂ µ B, ∂ µ B ν -∂ ν B µ = ∂ µ A ν -∂ ν A µ .
(3.4) At this stage, the super-curvature tensor Fµν =
∂ µ B ν -∂ ν B µ is not equal to the ordinary curvature tensor F µν = ∂ µ A ν -∂ ν A µ
as the former contains Grassmannian dependent terms. The substitution of the above values (cf. (3.3),(3.4)) of the secondary fields, in terms of the basic and auxiliary fields of the Lagrangian density (2.1), leads to
B (h) µ (x, θ, θ) = A µ + θ ∂ µ C + θ ∂ µ C + i θ θ ∂ µ B, F (h) (x, θ, θ) = C -i θ B, F (h) (x, θ, θ) = C + i θ B, (3.5)
where the superscript (h) has been used to denote that the above expansions have been obtained after the application of the HC. It can be seen that, due to (3.5), we get
∂ µ B (h) ν -∂ ν B (h) µ = ∂ µ A ν -∂ ν A µ , (3.6)
where there is no Grassmannian θ and θ dependence on the l.h.s. In the language of the geometry on the (4, 2)-dimensional supermanifold, the expansions (3.5) imply that the (anti-)BRST symmetry transformations s (a)b (and their corresponding generators Q (a)b ) for the 4D local fields (cf. (2.7)) are connected with the translational generators (∂/∂θ, ∂/∂ θ) because the translation of the corresponding (4, 2)-dimensional superfields, along the Grassmannian directions of the supermanifold, produces it. Thus, the Grassmannian independence of the super curvature tensor F (h)
µν = ∂ µ B (h) ν -∂ ν B (h)
µ implies that the 4D curvature tensor F µν is an (anti-)BRST (i.e. gauge) invariant physical quantity.
In terms of the superfields, equations (2.3) can be expressed as
Lim θ→0 ∂ ∂ θ -i F (h) { (∂ µ B (h) µ + 1 2 B) } , Limθ →0 ∂ ∂θ + iF (h) { (∂ µ B (h) µ + 1 2 B) } , ∂ ∂ θ ∂ ∂θ i 2 B µ(h) B (h) µ + 1 2 F (h) F (h) . (3.7)
These equations are unique because there is no other way to express the above equations in terms of the derivatives w.r.t. Grassmannian variables θ and θ. Thus, besides (2.3), there is no other possibility to express the gauge-fixing and the Faddeev-Popov ghost terms in the language of the off-shell nilpotent (anti-)BRST symmetry transformations (2.2). The superfield approach to BRST formulation, therefore, establishes the uniqueness of (2.3).
To express (2.6) in terms of the superfields, one has to substitute B = -(∂ µ A µ ) in (3.5). Thus, the expansion (3.5), in terms of the transformations (2.4), becomes §
B (h) µ(o) (x, θ, θ) = A µ + θ ∂ µ C + θ ∂ µ C -i θ θ ∂ µ (∂ ρ A ρ ), ≡ A µ + θ (s ab A µ ) + θ (s b A µ ) + θ θ(s b sab A µ ), F (h) (o) (x, θ, θ) = C + i θ (∂ µ A µ ) ≡ C + θ (s ab C), F (h) (o) (x, θ, θ) = C -i θ (∂ µ A µ ) ≡ C + θ (s b C). (3.8)
We note that (3.5) and (3.8) are the super expansions (after the application of the HC) which lead to the derivation of the off-shell nilpotent (anti-)BRST symmetry transformations s (a)b as well as the on-shell nilpotent (anti-)BRST symmetry transformations s(a)b , respectively, for the basic fields A µ , C and C of the theory. The gauge-fixing and Faddeev-Popov ghost terms of the Lagrangian density (2.5) can also be expressed in terms of the superfields (3.8). In other words, (vis-à-vis (3.7)), we have the following equations that are the analogue of (2.6), namely;
Lim θ→0 ∂ ∂ θ + i 2 F (h) (o) (∂ µ A µ ) + i B (h) µ(o) ∂ µ F (h) (o) ) , Limθ →0 ∂ ∂θ - i 2 F (h) (o) (∂ µ A µ ) -i B (h) µ(o) ∂ µ F (h) (o) ) , ∂ ∂ θ ∂ ∂θ i 2 B µ(h) (o) B (h) µ(o) + 1 2 F (h) (o) F (h) (o) . (3.9)
We know that, for all practical computational purposes, it is essential to take into account s(a)b (∂ µ A µ ) = 0 because of the on-shell conditions ✷C = ✷ C = 0. The logical reason behind such a restriction (i.e. s(a)b (∂ µ A µ ) = 0) in (2.6) is encoded in the superfield approach to BRST formalism as can be seen from a close look at (3.9). The Lagrangian density (2.1) can be expressed, in terms of the (4, 2)-dimensional superfields, in the following distinct and different forms
L(1) B = - 1 4 F (h) µν F µν(h) + Lim θ→0 ∂ ∂ θ -i F (h) (∂ µ B (h) µ + 1 2 B) , (3.10) §
The on-shell nilpotent (anti-)BRST symmetry transformations s(a)b can also be obtained by invoking the (anti-)chiral superfields on the appropriately chosen supermanifolds (see, e.g. [23] for details).
L(2) B = - 1 4 F (h) µν F µν(h) + Limθ →0 ∂ ∂θ +i F (h) (∂ µ B (h) µ + 1 2 B) , (3.11) L(3) B = - 1 4 F (h) µν F µν(h) + ∂ ∂ θ ∂ ∂θ + i 2 B µ(h) B (h) µ + 1 2 F (h) F (h) . (3.12)
It would be noted that the kinetic energy term -(1/4) F (h) µν F µν(h) is independent of the variables θ and θ because F (h) µν = F µν . In exactly similar fashion, the Lagrangian density of (2.5) can be expressed, with the help of the super expansion (3.8), as
L(1) b = - 1 4 F (h) µν(o) F µν(h) (o) + Lim θ→0 ∂ ∂ θ + i 2 F (h) (o) (∂ µ A µ ) + i B (h) µ(o) ∂ µ F (h) (o) ) , (3.13) L(2) b = - 1 4 F (h) µν(o) F µν(h) (o) + Limθ →0 ∂ ∂θ - i 2 F (h) (o) (∂ µ A µ ) -i B (h) µ(0) ∂ µ F (h) (o) ) , (3.14) L(3) b = - 1 4 F (h) µν(o) F µν(h) (o) + ∂ ∂ θ ∂ ∂θ + i 2 B µ(h) (o) B (h) µ(o) + 1 2 F (h) (o) F (h) (o) . (3.15)
The form of the Lagrangian densities (e.g. from (3.10) to (3.15)) simplify the proof for the (anti-)BRST invariance of the Lagrangian densities in (2.1) and (2.5).
In the above forms (e.g. from (3.10) to (3.12)) of the Lagrangian density, the BRST invariance s b L B = 0 and the anti-BRST invariance s ab L B = 0 become very transparent and simple because the following equalities and mappings exist, namely;
s b L (a) B = 0 ⇒ Lim θ→0 ∂ ∂ θ L(1) B = 0, s b ⇔ Lim θ→0 ∂ ∂ θ , s 2 b = 0 ⇔ ∂ ∂ θ 2 = 0, (3.16) s ab L (a) B = 0 ⇒ Limθ →0 ∂ ∂θ L(2) B = 0, s ab ⇔ Limθ →0 ∂ ∂θ , s 2 ab = 0 ⇔ ∂ ∂θ 2 = 0. (3.17)
Similarly, the most beautiful relation (3.12), leads to the (anti-)BRST invariance together.
Here one has to use the anticommutativity property s b s ab + s ab s b = 0 in the language of the translational generators (i.e. (∂/∂ θ), (∂/∂θ)) along the Grassmannian directions of the supermanifold, for its proof. This statement can be mathematically expressed as
s (a)b L (a) B = 0 ⇒ ∂ ∂θ ∂ ∂ θ L(3) B = 0, s b s ab + s ab s b = 0 ⇔ ∂ ∂θ ∂ ∂ θ + ∂ ∂ θ ∂ ∂θ = 0. (3.18)
In exactly similar fashion, the on-shell nilpotent (anti-)BRST symmetry invariance (i.e. s(a)b L (a) b = 0) of the Lagrangian density (2.5) can also be captured in the language of the superfields if we exploit the expressions (3.13) to (3.15) for the Lagrangian density. In the latter case, the on-shell nilpotent (anti-)BRST invariance turns out to be like (3.16), (3.17) and (3.18) with the replacements: s (a)b → s(a)b , L 1, 2, 3) b . Mathematically, the (anti-)BRST invariance of the Lagrangian density (2.1) is captured in the equations (3.16) to (3.18) . In the language of geometry on the (4, 2)-dimensional supermanifold, the (anti-)BRST invariance corresponds to the Grassmannian independence of the supersymmetric versions of the Lagrangian density (2.1). In other words, the translation of the super Lagrangian densities (i.e. (3.10) to (3.12)), along the (θ) θ directions of 9 the supermanifold, is zero. This observation captures the (anti-)BRST invariance of (2.1).
(a) B → L (a) b , L(1,2, 3
) B → L(
We begin with the following BRST-invariant Lagrangian density, in the Feynman gauge, for the four (3 + 1)-dimensional non-Abelian 1-form gauge theory ¶ (see, e.g. [26, 27, 9]
) L (n) B = - 1 4 F µν • F µν + B • (∂ µ A µ ) + 1 2 B • B -i∂ µ C • D µ C, (4.1)
where the curvature tensor (F µν ) is defined through the 2-form F (2) (1)(n) . Here the non-Abelian 1-form gauge connection is A (1)
(n) = dA (1)(n) +iA (1)(n) ∧ A
(n) = dx µ (A µ • T
) and the exterior derivative is d = dx µ ∂ µ . The Nakanishi-Lautrup auxiliary field B = B • T is required for the linearization of the gauge-fixing term and the (anti-)ghost fields ( C)C are essential for the proof of the unitarity in the theory. The latter fields are fermionic (i.e. (C a ) 2 = 0, ( Ca ) 2 = 0, C a C b + C b C a = 0, C a Cb + Cb C a = 0, etc.) in nature.
The above Lagrangian density respects the following off-shell nilpotent ((s
(n) b ) 2 = 0) BRST symmetry transformations s (n) b , namely; s (n) b A µ = D µ C, s (n) b C = - i 2 (C × C), s (n) b C = iB, s (n) b B = 0, s (n) b F µν = i(F µν × C). (4.2)
It will be noted that (i) the curvature tensor F µν • T transforms here under the BRST symmetry transformation. However, it can be checked explicitly that the kinetic energy term -(1/4)F µν • F µν remains invariant under the BRST symmetry transformations, (ii) the nilpotent anti-BRST symmetry transformations corresponding to the above BRST symmetry transformations (4.2) cannot be defined for the Lagrangian density (4.1), and (iii) the on-shell nilpotent version of the above BRST symmetry transformations is also possible if we substitute, in the above symmetry transformations,
B = -(∂ µ A µ ). The ensuing on-shell (i.e. ∂ µ D µ C = 0) nilpotent BRST symmetry transformations s(n) b are s(n) b A µ = D µ C, s(n) b C = - i 2 (C × C), s(n) b C = -i(∂ µ A µ ), s(n) b F µν = i(F µν × C). (4.3)
The above on-shell nilpotent transformations leave the following Lagrangian density
L (n) b = - 1 4 F µν • F µν - 1 2 (∂ µ A µ ) • (∂ ρ A ρ ) -i∂ µ C • D µ C, (4.4)
¶ For the non-Abelian 1-form gauge theory, the notations used in the Lie algebraic space are:
A • B = A a B a , (A × B) a = f abc A b B c , D µ C a = ∂ µ C a + if abc A b µ C c ≡ ∂ µ C a + i(A µ × C) a , F µν = ∂ µ A ν -∂ ν A µ + iA µ × A ν , A µ = A µ • T, [T a , T b ] = f abc T c
where the Latin indices a, b, c = 1, 2, 3....N are in the SU (N ) Lie algebraic space. The structure constant f abc be chosen to be totally antisymmetric for any arbitrary semi simple Lie algebra that includes SU (N ), too (see, e.g., [27]).
10 quasi-invariant because it transforms to a total derivative.
The gauge-fixing and Faddeev-Popov ghost terms of the Lagrangian densities (4.1) and (4.4) can be written, modulo a total derivative, as a BRST-exact quantity in terms of the off-shell and on-shell nilpotent BRST symmetry transformations (4.2) and (4.3). This statement can be mathematically expressed as follows
s (n) b -i C • {(∂ µ A µ ) + 1 2 B} = B • (∂ µ A µ ) + 1 2 B • B -i ∂ µ C • D µ C, (4.5) s(n) b + i 2 C • (∂ µ A µ ) + i A µ • ∂ µ C = - 1 2 (∂ µ A µ ) • (∂ ρ A ρ ) -i ∂ µ C • D µ C. (4.6)
It will be noted that one has to take into account s(n) b (∂ µ A µ ) = ∂ µ D µ C = 0 in the above proof of the exactness of the expression in (4.6) .
The Lagrangian densities that respect the off-shell nilpotent (i.e. (s (n) (a)b ) 2 = 0) and anticommuting (s
(n) b s (n) ab + s (n) ab s (n) b = 0) (anti-)BRST symmetry transformations are L (1)(n) b = - 1 4 F µν • F µν + B • (∂ µ A µ ) + 1 2 (B • B + B • B) -i∂ µ C • D µ C, (4.7) L (2)(n) b = - 1 4 F µν • F µν -B • (∂ µ A µ ) + 1 2 (B • B + B • B) -iD µ C • ∂ µ C. (4.8)
Here auxiliary fields B and B satisfy the Curci-Ferrari condition B + B = -(C × C) [28, 29] . It is also evident, from this relation, that
B•(∂ µ A µ )-i∂ µ C •D µ C = -B•(∂ µ A µ )-iD µ C •∂ µ C.
Furthermore, it should be re-emphasized that the Lagrangian densities (4.1) and (4.4) do not respect the anti-BRST symmetry transformations of any kind. The BRST and anti-BRST symmetry transformations, for the above Lagrangian densities, are
s (n) b A µ = D µ C, s (n) b C = - i 2 (C × C), s (n) b C = iB, s (n) b B = 0, s (n) b F µν = i(F µν × C), s (n) b B = i( B × C), (4.9) s (n) ab A µ = D µ C, s (n) ab C = - i 2 ( C × C), s (n) ab C = i B, s (n) ab B = 0, s (n) ab F µν = i(F µν × C), s (n) ab B = i(B × C). (4.10)
The above off-shell nilpotent (anti-)BRST symmetry transformations leave the Lagrangian densities (4.7) as well as (4.8) quasi-invariant as they transform to some total derivatives. The gauge-fixing and Faddeev-Popov ghost terms of the Lagrangian densities (4.7) and (4.8) can be written, in a symmetrical fashion with respect to s (n) b and s
(n) ab , as s (n) b s (n) ab i 2 A µ • A µ + C • C = B • (∂ µ A µ ) + 1 2 (B • B + B • B) -i∂ µ C • D µ C, ≡ -B • (∂ µ A µ ) + 1 2 (B • B + B • B) -iD µ C • ∂ µ C. (4.11)
This demonstrates the key fact that the above gauge-fixing and Faddeev-Popov ghost terms are (anti-)BRST invariant together because of the nilpotency and anticommutativity of the 11 (anti-)BRST symmetry transformations s (n) (a)b that are present in the theory.
5 (Anti-)BRST invariance in non-Abelian theory: superfield approach
To capture (i) the off-shell as well as the on-shell nilpotent (anti-)BRST symmetry transformations, and (ii) the invariance of the Lagrangian densities, in the language of the superfield approach to BRST formalism, we have to consider the 4D 1-form non-Abelian gauge theory on a (4, 2)-dimensional supermanifold. As a consequence, we have the following mappings:
d → d = dx µ ∂ µ + dθ ∂ θ + d θ ∂θ, d2 = 0, A (1)(n) → Ã(1)(n) = dx µ (B µ • T )(x, θ, θ) + dθ( F • T )(x, θ, θ) + d θ(F • T )(x, θ, θ), (5.1)
where the (4, 2)-dimensional superfields (B µ •T, F •T, F •T ) are the generalizations of the 4D basic local fields (A µ • T, C • T, C • T ) of the Lagrangian density (4.1), (4.7) and (4.8). These superfields can be expanded along the Grassmannian directions of the supermanifold, in terms of the basic 4D fields, auxiliary fields and secondary fields as [4,16,19] .2) To determine the exact expressions for the secondary fields, in terms of the basic and auxiliary fields of the theory, we have to exploit the HC. The horizontality condition, for the non-Abelian gauge theory is the requirement of the equality of the Maurer-Cartan equation on the (super) manifolds. In other words, the covariant reduction of the super 2-form curvature F (2)(n) to the ordinary 2-form curvature (i.e. d Ã(1)(n) + i Ã(1)(n) ∧ Ã(1)(n) = dA (1)(n) + iA (1)(n) ∧A (1)(n) ) leads to the determination of the secondary fields in terms of the basic and auxiliary fields of the theory. The ensuing expansions, in terms of the basic and auxiliary fields, lead to (i) the derivation of the (anti-)BRST symmetry transformations for the basic fields of the theory, and (ii) the geometrical interpretations of the nilpotent (anti-)BRST symmetry transformations (and their corresponding nilpotent generators) for the basic fields of the theory as the translations of the corresponding superfields along the Grassmannian directions of the (4, 2)-dimensional supermanifold (see, e.g., [16, 19] ).
(B µ • T )(x, θ, θ) = (A µ • T )(x) + θ ( Rµ • T )(x) + θ (R µ • T )(x) + i θ θ (S µ • T )(x), (F • T )(x, θ, θ) = (C • T )(x) + i θ ( B1 • T )(x) + i θ (B 1 • T )(x) + i θ θ (s • T )(x), ( F • T )(x, θ, θ) = ( C • T )(x) + i θ ( B2 • T )(x) + i θ (B 2 • T )(x) + i θ θ (s • T )(x). (5
With the identifications B 2 = B and B1 = B, the following relationships emerge after the application of the horizontality condition (see, e.g., [16]):
R µ = D µ C, Rµ = D µ C, B + B = -(C × C), s = i( B × C), S µ = D µ B + D µ C × C ≡ -D µ B -D µ C × C, s = -i(B × C), B 1 = - 1 2 (C × C), B2 = - 1 2 ( C × C). (5.3)
In the rest of our present text, we shall be using the short-hand notations for all the fields e.g.:
A µ • T = A µ , C • T = C, B • T = B, etc.
, for the sake of brevity.
The substitution of the above expressions, which are obtained after the application of the horizontality condition, leads to the following expansions
B (h) µ (x, θ, θ) = A µ + θ D µ C + θ D µ C + i θ θ (D µ B + D µ C × C), F (h) (x, θ, θ) = C + i θ B - i 2 θ (C × C) -θ θ ( B × C), F (h) (x, θ, θ) = C - i 2 θ ( C × C) + i θ B + θ θ (B × C). (5.4)
The above expansions (see, e.g., our earlier works [16, 19] ) can be expressed in terms of the off-shell nilpotent (anti-)BRST symmetry transformations (4.9) and (4.10).
With the above expansion at our disposal, the gauge-fixing and Faddeev-Popov terms of the Lagrangian density (4.1) can be written, modulo a total ordinary derivative, as
Lim θ→0 ∂ ∂ θ -i F (h) • ∂ µ B (h) µ - i 2 F (h) • B = B • (∂ µ A µ ) + 1 2 B • B -i ∂ µ C • D µ C. (5.5)
Furthermore, it can be seen that, due to the validity and consequences of the horizontality condition, the super curvature tensor Fµν has the following form [16,4]
F (h) µν = F µν + iθ(F µν × C) + i θ(F µν × C) -θ θ (F µν × B + F µν × C × C). (5.6)
It is clear from the above relationship that the kinetic energy term of the present 4D non-Abelian 1-form gauge theory remains invariant, namely;
- 1 4 F (h) µν • F µν(h) = - 1 4 F µν • F µν . (5.7)
The Grassmannian independence of the l.h.s. of (5.7) has deep meaning as far as physics is concerned. It implies immediately that the kinetic energy term of the non-Abelian gauge theory is an (anti-)BRST (i.e. gauge) invariant physical quantity. At this juncture, it is worthwhile to point out that one can also capture the equation (4.6) in the superfield approach to BRST formalism where the on-shell nilpotent version of the BRST symmetry transformations (i.e. s(n) b ) plays an important role. For this purpose, we have to express the superfield expansion (5.4) for the on-shell nilpotent BRST symmetry transformation where one has to exploit the replacement B = -(∂ µ A µ ). With this substitution, the equation (5.4) for the superfield expansion becomes
B (h) µ(o) (x, θ, θ) = A µ + θ D µ C + θ D µ C + i θ θ [-D µ (∂ ρ A ρ ) + D µ C × C], F (h) (o) (x, θ, θ) = C + i θ B - i 2 θ (C × C) -θ θ ( B × C), F (h) (o) (x, θ, θ) = C - i 2 θ ( C × C) -i θ (∂ µ A µ ) -θ θ [(∂ µ A µ ) × C)]. (5.8)
Now, the equation (4.6) can be expressed in terms of the above superfields, as:
Lim θ→0 ∂ ∂ θ i 2 F (h) (o) • (∂ µ A µ ) + i B (h) µ(o) • ∂ µ F (h) (o) = - 1 2 (∂ µ A µ ) • (∂ ρ A ρ ) -i ∂ µ C • D µ C. (5.9) 13
Furthermore, it will be noted that the analogue of (5.6), for the on-shell nilpotent BRST symmetry transformation (i.e. F (h) µν(o) ), can be obtained by the replacement B = -(∂ µ A µ ). Once again, the equality (5.7) would remain intact even if we take into account the on-shell nilpotent BRST symmetry transformations. Thus, we note that the kinetic energy term (i.e.
(-(1/4)F µν • F µν = -(1/4) F µν(h) (o) • F (h) µν(o)
) of the non-Abelian gauge theory remains independent of the Grassmannian variables θ and θ after the application of the HC. This statement is true for the off-shell as well as the on-shell nilpotent (anti-)BRST symmetry transformations. Physically, it implies that the kinetic energy term for the gauge field of the non-Abelian theory is an (anti-)BRST (i.e. gauge) invariant quantity.
The above key observation helps in expressing the Lagrangian density (4.1) and (4.4) in terms of the superfields (obtained after the application of HC), as
L(n) B = - 1 4 F (h) µν • F µν(h) + Lim θ→0 ∂ ∂ θ -i F (h) • ∂ µ B (h) µ - i 2 F (h) • B , L(n) b = - 1 4 F (h) µν(o) • F µν(h) (o) + Lim θ→0 ∂ ∂ θ i 2 F (h) (o) • (∂ µ A µ ) + i B (h) µ(o) • ∂ µ F (h) (o) . (5.10)
This result, in turn, simplifies the BRST invariance of the above Lagrangian density (4.1) and (4.4) (describing the 4D 1-form non-Abelian gauge theory) as follows
Lim θ→0 ∂ ∂ θ L(n) B = 0 ⇒ s (n) b L (n) B = 0, Lim θ→0 ∂ ∂ θ L(n) b = 0 ⇒ s(n) b L (n) b = 0. (5.11)
This is a great simplification because the total super Lagrangian densities (5.10) remain independent of the Grassmannian variable θ. This key result is encoded in the mapping (s
(n) b , s(n) b ) ⇔ Lim θ→0 (∂/∂ θ) and the nilpotency (s (n) b ) 2 = 0, (s (n) b ) 2 = 0, (∂/∂ θ) 2 = 0.
It can be readily checked that the analogues of (5.5) and (5.9) cannot be expressed as the derivative w.r.t. the Grassmannian variable θ. To check this, one has to exploit the super expansions (5.4) and (5.8) obtained after the application of the HC (in the context of the derivation of the off-shell as well as the on-shell nilpotent BRST symmetry transformations s (n) b and s(n) b ). It can be clearly seen that the operation of the derivative w.r.t. the Grassmannian variable θ, on any combination of the superfields from the expansions (5.4) and (5.8), does not lead to the derivation of the r.h.s. of (5.5) and (5.9). In the language of the superfield approach to BRST formalism, this is the reason behind the non-existence of the anti-BRST symmetry transformations for the Lagrangian densities (4.1) and (4.4).
The form of the gauge-fixing and Faddeev-Popov terms (4.11), expressed in terms of the BRST and anti-BRST symmetry transformations together, can be represented in the language of the superfields (obtained after the application of HC), as
∂ ∂ θ ∂ ∂θ i 2 B (h) µ • B µ(h) + F (h) • F (h) = B • (∂ µ A µ ) + 1 2 (B • B + B • B) -i∂ µ C • D µ C.
(5.12) As a consequence of the above expression, the Lagrangian densities (4.7) (as well as (4.8)) can be presented, in terms of the superfields, as
L(1,2)(n) b = - 1 4 F µν(h) • F (h) µν + ∂ ∂ θ ∂ ∂θ i 2 B (h) µ • B µ(h) + F (h) • F (h) . (5.13) 14
The BRST and anti-BRST invariance of the above super Lagrangian density (and that of the ordinary 4D Lagrangian densities (4.7) and (4.8)) is encoded in the following simple equations that are expressed in terms of the translational generators along the Grassmannian directions of the (4, 2)-dimensional supermanifold, namely;
Lim θ→0 ∂ ∂ θ L(1,2)(n) b = 0 ⇒ s (n) b L (1)(n) b = 0, Limθ →0 ∂ ∂θ L(1,2)(n) b = 0 ⇒ s (n) ab L ( 2
)(n) b = 0. (5.14) This is a tremendous simplification of the (anti-)BRST invariance of the Lagrangian densities (4.7) and (4.8) in the language of the superfield approach to BRST formalism. In other words, if one is able to show the Grassmannian independence of the super Lagrangian densities of the theory, the (anti-)BRST invariance of the 4D theory follows automatically.
In the language of the geometry on the supermanifold, the (anti-)BRST invariance of a 4D Lagrangian density is equivalent to the statement that the translation of the super version of the above Lagrangian density, along the Grassmannian directions of the (4, 2)dimensional supermanifold, is zero. Thus, the super Lagrangian density of an (anti-)BRST invariant 4D theory is a Lorentz scalar, constructed with the help of (4, 2)-dimensional superfields (obtained after the application of HC), such that, when the partial derivatives w.r.t. the Grassmannian variables (θ and θ) operate on it, the result is zero.
The nilpotency and anticommutativity properties (that are associated with the conserved (anti-)BRST charges and (anti-)BRST symmetry transformations) are found to be captured very naturally (cf. (3.16)-(3.18)) when we consider the superfield formulation of the (anti-)BRST invariance of the Lagrangian density of a given 1-form gauge theory. We mention, in passing, that one could also derive the analogue of the equations (3.16), (3.17) and (3.18) for the 4D non-Abelian 1-form gauge theory in a straightforward manner.
In our present investigation, we have concentrated mainly on the (anti-)BRST invariance of the Lagrangian densities of the free 4D (non-)Abelian 1-form gauge theories (having no interaction with matter fields) within the framework of the superfield approach to BRST formalism. We have been able to provide the geometrical basis for the existence of the (anti-)BRST invariance in the above 4D theories. To be more specific, we have been able to show that the Grassmannian independence of the (4, 2)-dimensional super Lagrangian density, expressed in terms of the appropriate superfields, is a clear-cut proof that there is an (anti-)BRST invariance (cf. (3.16), (3.17), (3.18), (5.11), (5.14 )) in the 4D theory.
If the super Lagrangian density could be expressed as a sum of (i) a Grassmannian independent term, and (ii) a derivative w.r.t. the Grassmannian variable, then, the corresponding 4D Lagrangian density will automatically respect BRST and/or anti-BRST invariance. In the latter piece of the above super Lagrangian density, the derivative could be either w.r.t. θ or w.r.t. θ or w.r.t. both of them put together. More specifically, (i) if the derivative is w.r.t. θ, the nilpotent symmetry would correspond to the BRST, 15
(ii) if the derivative is w.r.t. θ, the nilpotent symmetry would be that of the anti-BRST type, and (iii) if both the derivatives are present together, both the nilpotent (anti-)BRST symmetries would be present together (and they would turn out to be anticommuting).
For the 4D (non-)Abelian 1-form gauge theories, that are considered on the (4, 2)dimensional supermanifold, it is the HC on the 1-form super connection Ã(1) that plays a very important role in the derivation of the (anti-)BRST symmetry transformations. The cohomological origin for the above HC lies in the (super) exterior derivatives ( d)d. This point has been made quite clear in our discussions after the off-shell as well as the on-shell nilpotent (anti-)BRST symmetry transformations (2.2), (2.4), (4.2), (4.3), (4.9) and (4.10).
In fact, it is the full kinetic energy term of the above theories (owing its origin to the cohomological operator d = dx µ ∂ µ ) that remains invariant under the above on-shell as well the off-shell nilpotent (anti-)BRST symmetry transformations.
The HC produces specifically the nilpotent (anti-)BRST symmetry transformations for the gauge and (anti-)ghost fields because of the fact that the super 1-form connection Ã(1) / Ã(1)(n) (cf. (3.1) and (5.1)) is constructed with a super vector multiplet (B µ , F , F) which is the generalization of the gauge field A µ and the (anti-)ghost fields ( C)C (of the ordinary 4D (non-)Abelian 1-form gauge theories) to the (4, 2)-dimensional supermanifold.
As a consequence, only the nilpotent and anticommuting (anti-)BRST symmetry transformations for the 4D local fields A µ , C and C are obtained when the full potential of the HC is exploited within the framework of the above superfield formulation.
It is worthwhile to point out that geometrically the super Lagrangian densities, expressed in terms of the (4, 2)-dimensional superfields, are equivalent to the sum of the kinetic energy term and the translations of some composite superfields (obtained after the application of the HC) along the Grassmannian directions (i.e. θ and/or θ) of the (4, 2)dimensional supermanifold. This observation is distinctly different from our earlier works on the superfield approach to 2D (non-)Abelian 1-form gauge theories [24, 25, 23] which are found to correspond to the topological field theories. In fact, for the latter theories, the total super Lagrangian density turns out to be a total derivative w.r.t. the Grassmannian variables (θ and/or θ). That is to say, even the kinetic energy term of the latter theories, is able to be expressed as the total derivative w.r.t. the variables θ and/or θ.
In our present endeavour, within the framework of the superfield approach to BRST formalism, we have been able to provide (i) the logical reason behind the non-existence of the anti-BRST symmetry transformations for the Lagrangian densities (4.1) and (4.4) for the 4D non-Abelian 1-form gauge theory, (ii) the explicit explanation for the uniqueness of the equations (2.3) and (2.6) for the 4D Abelian 1-form gauge theory, (iii) the convincing proof for the on-shell nilpotent (anti-)BRST invariance of the gauge-fixing term (i.e. s(a)b (∂ µ A µ ) = 0, s(n) (a)b (∂ µ A µ ) = 0) for the (non-)Abelian 1-form gauge theories, and (iv) the compelling arguments for the non-existence of the exact analogue(s) of (2.3) and (2.6) for the non-Abelian 1-form gauge theory. To the best of our knowledge, the logical explanations for the above subtle points (connected with the 1-form gauge theories) are completely 16 new. Thus, the results of our present work are simple, beautiful and original.
It is worthwhile to mention that our superfield construction and its ensuing geometrical interpretations are not specific to the Feynman gauge (which has been taken into account in our present endeavor). To corroborate this assertion, we take the simple case of the 4D Abelian 1-form gauge theory and write the Lagrangian density (2.1) in the arbitrary gauge
L (a,ξ) B = - 1 4 F µν F µν + B (∂ µ A µ ) + ξ 2 B 2 -i ∂ µ C ∂ µ C, (6.1)
where ξ is the gauge parameter. It is elementary to check that, in the limit ξ → 1, we get back our Lagrangian density (2.1) for the Abelian theory in the Feynman gauge. The analogue of the equation (2.3) (for the gauge-fixing and Faddeev-Popov ghost terms in the case of the arbitrary gauge) can be expressed as
s b -i C {(∂ µ A µ ) + ξ 2 B}], s ab +i C {(∂ µ A µ ) + ξ 2 B} , s b s ab i 2 A µ A µ + ξ 2 C C . (6.2)
The above expression can be easily generalized to the analogues of the equations (3.10)-(3.12) in terms of the superfields by taking the help of (3.8). Thus, the geometrical interpretations remain intact even in the case of the arbitrary gauge. In a similar fashion, for the 4D non-Abelian 1-form gauge theory, the equations (4.5), (4.6) and (4.11) can be generalized to the case of arbitrary gauge and, subsequently, can be expressed in terms of superfields as the analogues of (5.5), (5.9) and (5.12). Finally, we can obtain the analogues of (5. 7) , ( 5.10) and (5.13) which will lead to the derivation of the analogues of (5.11) and (5.14). Thus, we note that geometrical interpretations, in the arbitrary gauge, remain the same for the 4D (non-)Abelian 1-form gauge theory within the framework of our superfield approach to BRST formalism.
Our present work can be generalized to the case of the interacting 4D (non-)Abelian 1-form gauge theories where there exists an explicit coupling between the gauge field and the matter fields. In fact, our earlier works [14-18] might turn out to be quite handy in attempting the above problems. It seems to us that it is the combination of the HC and the restrictions, owing their origin to the (super) covariant derivative on the matter (super) fields and their intimate connection with the (super) curvatures, that would play a decisive role in proving the existence of the (anti-)BRST invariance for the above gauge theories.
It is gratifying to state that we have accomplished the above goals in our very recent endeavours [30] [31] [32] . In fact, we have been able to provide the geometrical basis for the existence of the (anti-)BRST invariance, in the context of the interacting (non-)Abelian 1-form gauge theories with Dirac as well as complex scalar fields, within the framework of the augmented superfield approach to BRST formalism. As it turns out, here too, the super Lagrangian density is found to be independent of the Grassmannian variables.
In our earlier works [33] [34] [35] , we have been able to show the existence of the nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations for the 4D free Abelian 2-form 17 gauge theory. We have also established the quasi-topological nature of it in [35] . In a recent work [36], the nilpotent (anti-)BRST symmetry transformations have been captured in the framework of the superfield formulation. It would be a very nice endeavour to study the (anti-)BRST and (anti-)co-BRST invariance of the 4D Abelian 2-form gauge theory within the framework of superfield formulation. At present, this issue and connected problems in the context of the 4D free Abelian 2-form gauge theory are under intensive investigation and our results would be reported in our forthcoming future publications [37].
Acknowledgement: Financial support from the Department of Science and Technology (DST), Government of India, under the SERC project sanction grant No: -SR/S2/HEP-23/2006, is gratefully acknowledged. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The geometrical superfield approach [1] [2] [3] [4] [5] [6] [7] [8] to Becchi-Rouet-Stora-Tyutin (BRST) formalism is one of the most attractive and intuitive approaches which enables us to gain some physical insights into the beautiful (but abstract mathematical) structures that are associated with the nilpotent (anti-)BRST symmetry transformations and their corresponding generators. The latter quantities play a very decisive role in (i) the covariant canonical quantization of the gauge theories, (ii) the proof of the unitarity of the \"quantum\" gauge theories at any arbitrary order of perturbative computations for a given physical process (that is allowed by the theory), (iii) the definition of the physical states of the \"quantum\" gauge theories in the quantum Hilbert space, and (iv) the cohomological description of the physical states of the quantum Hilbert space w.r.t. the conserved and nilpotent BRST charge.\n\nTo be specific, in the superfield formulation [1-8] of the 4D 1-form gauge theories, one defines the super curvature 2-form F (2) = d Ã(1) + i Ã(1) ∧ Ã(1) in terms of the super exterior derivative d = dx µ ∂ µ + dθ∂ θ + d θ∂θ (with d2 = 0) and the super 1-form connection Ã(1) on a (4, 2)-dimensional supermanifold parametrized by the usual spacetime variables x µ (with µ = 0, 1, 2, 3) and a pair of anticommuting (i.e. θ 2 = θ2 = 0, θ θ + θθ = 0) Grassmannian variables θ and θ. The above super 2-form is subsequently equated, due to the so-called horizontality condition [1-8], to the ordinary curvature 2-form F (2) = dA (1) + iA (1) ∧ A (1) defined on the ordinary 4D flat Minkowski spacetime manifold in terms of the ordinary exterior derivative d = dx µ ∂ µ (with d 2 = 0) and the 1-form connection A (1)\n\n= dx µ A µ .\n\nThe above super exterior derivative d and super 1-form connection Ã(1) are the generalization of the 4D ordinary exterior derivative d and 1-form connection A (1) to the (4, 2)-dimensional supermanifold because d → d, Ã(1) → A (1) in the limit (θ, θ) → 0.\n\nThe above horizontality condition (HC) has been referred to as the soul-flatness condition in [9] which amounts to setting equal to zero all the Grassmannian components of the (anti)symmetric second-rank super tensor that constitutes the super curvature 2-form F (2) on the (4, 2) -dimensional supermanifold. The key consequences, that emerge from the HC, are (i) the derivation of the nilpotent (anti-)BRST symmetry transformations for the gauge and (anti-)ghost fields of a given 4D 1-form gauge theory, (ii) the geometrical interpretation of the (anti-)BRST symmetry transformations for the 4D local fields as the translation of the corresponding superfields along the Grassmannian directions of the supermanifold, (iii) the geometrical interpretation of the nilpotency property as a pair of successive translations of the superfield along a particular Grassmannian direction of the supermanifold, and (iv) the geometrical interpretation of the anticommutativity property of the (anti-)BRST symmetry transformations for a 4D local field as the sum of (a) the translation of the corresponding superfield first along the θ-direction followed by the translation along the θ-direction, and (b) the translation of the same superfield first along the θ-direction followed by the translation along the θ-direction.\n\nIt will be noted that the above HC (i.e. F (2) = F (2) ) is valid for the non-Abelian (i.e. 2 A (1)(n) ∧ A (1)(n) = 0) 1-form gauge theory as well as the Abelian (i.e. A (1) ∧ A (1) = 0) 1-form gauge theory. As expected, for both types of theories, the HC leads to the derivation of the nilpotent (anti-)BRST symmetry transformations for the gauge and (anti-)ghost fields of the respective theories. We lay emphasis on the fact that the HC does not shed any light on the derivation of the nilpotent (anti-)BRST symmetry transformations associated with the matter fields of the interacting 4D (non-)Abelian 1-form gauge theories.\n\nIn a recent set of papers [10] [11] [12] [13] [14] [15] [16] [17] , the above HC condition has been generalized, in a consistent manner, so as to compute the nilpotent (anti-)BRST symmetry transformations associated with the matter fields of a given 4D interacting 1-form gauge theory (along with the well-known nilpotent transformations for the gauge and (anti-)ghost fields) without spoiling the cute geometrical interpretations of the (anti-)BRST symmetry transformations (and their corresponding generators) that emerge from the HC alone. The latter approach has been christened as the augmented superfield approach to BRST formalism where the restrictions imposed on the (4, 2)-dimensional superfields are (i) the HC plus the invariance of the (super) matter Noether conserved currents [10-14], (ii) the HC plus the equality of any (super) conserved quantities [15], (iii) the HC plus a restriction that owes its origin to the gauge invariance and the (super) covariant derivatives on the matter (super)fields [16, 17] , and (iv) an alternative to the HC where the gauge invariance and the property of a pair of (super) covariant derivatives on the (super) matter fields (and their intimate connection with the (super) curvatures) play a crucial role [18] [19] [20] .\n\nIn all the above approaches [1-20], however, the invariance of the Lagrangian densities of the 4D (non-)Abelian 1-form gauge theories, under the nilpotent (anti-)BRST symmetry transformations, has not yet been discussed at all. Some attempts in this direction have been made in our earlier works where the specific topological features [21, 22] of the 2D free (non-)Abelian 1-form gauge theories have been captured in the superfield formulation [23] [24] [25] . In particular, the invariance of the Lagrangian density under the nilpotent and anticommuting (anti-)BRST and (anti-)co-BRST symmetry transformations has been expressed in terms of the superfields and the Grassmannian derivatives on them. These are, however, a bit more involved in nature because of the existence of a new set of nilpotent (anti-)co-BRST symmetries in the theory. The geometrical interpretations for the Lagrangian densities and the symmetric energy-momentum tensor (for the above topological theory) have also been provided within the framework of the superfield formulation.\n\nThe purpose of our present paper is to capture the (anti-)BRST symmetry invariance of the Lagrangian density of the 4D (non-)Abelian 1-form gauge theories within the framework of the superfield approach to BRST formalism and to demonstrate that the above symmetry invariance could be understood in a very simple manner in terms of the translational generators along the Grassmannian directions of the (4, 2)-dimensional supermanifold on which the above 4D ordinary gauge theories are considered. In addition, the reason behind the existence (or non-existence) of any specific nilpotent symmetry transformation could also be explained within the framework of the above superfield approach. We demonstrate 3 the uniqueness of the existence of the nilpotent (anti-)BRST symmetry transformations for the Lagrangian density of a U(1) Abelian 1-form gauge theory. We go a step further and show the existence of the nilpotent BRST symmetry transformations for the specific Lagrangian densities (cf. (4.1) and (4.4) below) of the 4D non-Abelian 1-form gauge theory and clarify the non-existence of the anti-BRST symmetry transformations for these specific Lagrangian densities within the framework of the superfield formulation (cf. section 5 below). Finally, we provide the geometrical basis for the existence of the off-shell nilpotent and anticommuting (anti-)BRST symmetry transformations (and their corresponding generators) for the specifically defined Lagrangian densities (cf. (4.7) and/or (4.8) below) of the 4D non-Abelian 1-form gauge theory in the Feynman gauge.\n\nThe motivating factors that have propelled us to pursue our present investigation are as follows. First and foremost, to the best of our knowledge, the property of the symmetry invariance of a given Lagrangian density has not yet been captured in the language of the superfield approach to BRST formalism. Second, the above (anti-)BRST invariance of the theory has never been shown, in as simplified fashion, as we demonstrate in our present endeavour. The geometrical interpretations for (i) the existence of the above nilpotent (anti-)BRST symmetry invariance, and (ii) the on-shell conditions of the on-shell nilpotent (anti-)BRST symmetries, turn out to be quite transparent in our present work. Third, we establish the uniqueness of the existence of the (anti-)BRST symmetry invariance in their various forms. The non-existence of the specific symmetry transformation is also explained within the framework of the superfield approach to BRST formalism. Finally, our present investigation is the first modest step in the direction to gain some insights into the existence of the nilpotent symmetry transformations and their invariance for the higher form (e.g. 2-form, 3-form, etc.) gauge theories within the framework of the superfield formulation.\n\nThe contents of our present paper are organized as follows. In section 2, we recapitulate some of the key points connected with the nilpotent (anti-)BRST symmetry transformations for the free 4D Abelian 1-form gauge theory (having no interaction with matter fields) in the Lagrangian formulation. The above symmetry transformations as well as the symmetry invariance of the Lagrangian densities are captured in the geometrical superfield approach to BRST formalism in section 3 where the HC on the gauge superfield plays a crucial role. Section 4 deals with the bare essentials of the nilpotent (anti-)BRST symmetry transformations for the 4D non-Abelian 1-form gauge theory in the Lagrangian formulation. The subject matter of section 5 concerns itself with the superfield formulation of the symmetry invariance of the appropriate Lagrangian densities of the above 4D non-Abelian 1-form gauge theory. Finally, in section 6, we summarize our key results, make some concluding remarks and point out a few future directions for further investigations."
},
{
"section_type": "OTHER",
"section_title": "(Anti-)BRST symmetries in Abelian theory: Lagrangian formulation",
"text": "Let us begin with the following (anti-)BRST invariant Lagrangian density of the 4D Abelian 4 1-form gauge theory * in the Feynman gauge [26, 27, 9]\n\nL (a) B = - 1 4 F µν F µν + B (∂ µ A µ ) + 1 2 B 2 -i ∂ µ C ∂ µ C, (2.1) where F µν = ∂ µ A ν -∂ ν A µ is the antisymmetric (F µν = -F νµ\n\n) curvature tensor that constitutes the Abelian 2-form F (2)\n\n= dA (1) ≡ 1 2! (dx µ ∧ dx ν )F µν , B\n\nis the Nakanishi-Lautrup auxiliary multiplier field and ( C)C are the anticommuting (i.e. C 2 = C2 = 0, C C+ CC = 0) (anti-)ghost fields of the theory. The above Lagrangian density respects the off-shell nilpotent (s 2 (a)b = 0) (anti-)BRST symmetry transformations s (a)b (with s b s ab + s ab s b = 0) †\n\ns b A µ = ∂ µ C, s b C = 0, s b C = iB, s b B = 0, s b F µν = 0, s ab A µ = ∂ µ C, s ab C = 0,\n\ns ab C = -iB, s ab B = 0, s ab F µν = 0. (2.2) It is clear that, under the nilpotent (anti-)BRST symmetry transformations s (a)b , the curvature tensor F µν is found to be invariant. In other words, the 2-form F (2) , owing its origin to the cohomological operator d = dx µ ∂ µ , is an (anti-)BRST invariant object for the Abelian U(1) 1-form gauge theory and is, therefore, a physically meaningful (i.e. gaugeinvariant) quantity. These observations will play an important role in our discussion on the horizontality condition that would be exploited in the context of our superfield approach to (anti-)BRST invariance of the Lagrangian densities in sections 3 and 5 (see below). A noteworthy point, at this stage, is the observation that the gauge-fixing and Faddeev-Popov ghost terms can be written, modulo a total derivative, in the following fashion\n\ns b -i C {(∂ µ A µ ) + 1 2 B}], s ab +i C {(∂ µ A µ ) + 1 2 B} , s b s ab i 2 A µ A µ + 1 2 C C . (2.\n\n3) The above equation establishes, in a very simple manner, the (anti-)BRST invariance of the 4D Lagrangian density (2.1). The simplicity ensues due to (i) the nilpotency s 2 (a)b = 0 of the (anti-)BRST symmetry transformations, (ii) the anticommutativity property (i.e. and (iii) the invariance of the F µν term under s (a)b .\n\ns b s ab + s ab s b = 0) of s (a)b ,\n\nAs a side remark, it is interesting to note that the following on-shell (i.e. ✷C = ✷ C = 0) nilpotent (s 2 (a)b = 0) (anti-)BRST symmetry transformations (with sb sab + sab sb = 0)\n\nsb A µ = ∂ µ C, sb C = 0, sb C = -i(∂ µ A µ ), sb F µν = 0, sab A µ = ∂ µ C, sab C = 0, sab C = +i(∂ µ A µ ), sab F µν = 0, (2.4) 5\n\nare the symmetry transformations for the following Lagrangian density L (a) b = -\n\n1 4 F µν F µν - 1 2 (∂ µ A µ ) 2 -i ∂ µ C ∂ µ C. (2.5)\n\nThe above transformations (2.4) and the Lagrangian density (2.5) have been derived from (2.2) and (2.1) by the substitution B = -(∂ µ A µ ). An interesting point, connected with the on-shell nilpotent symmetry transformations, is to express the analogue of (2.3) as ‡\n\nsb + i 2 C (∂ µ A µ ) + i A µ ∂ µ C], sab - i 2 C (∂ µ A µ ) -i A µ ∂ µ C , sb sab i 2 A µ A µ + 1 2 C C . (2.6)\n\nIt should be noted that, in the above precise computation, one has to take into account the on-shell (✷C = ✷ C = 0) conditions so that, for all practical purposes s(a)b (∂ µ A µ ) = 0. The above nilpotent (anti-)BRST symmetry transformations (i.e. s r , sr with r = b, ab) are connected with the conserved and nilpotent generators (i.e. Q r , Qr with r = b, ab). This statement can be succinctly expressed, in the mathematical form, as\n\ns r Ω = -i [ Ω, Q r ] (±) , sr Ω = -i [ Ω, Qr ] (±) , r = b, ab, (2.7)\n\nwhere the subscripts (with the signatures (±)) on the square bracket stand for the bracket to be an (anti)commutator, for the generic fields Ω = A µ , C, C, B and Ω = A µ , C, C (of the Lagrangian densities (2.1) and (2.5)), being (fermionic)bosonic in nature. The above charges Q r , Qr are found to be anticommuting (i.e. Q b Q ab +Q ab Q b = 0, Qb Qab + Qab Qb = 0) and off-shell as well as on-shell nilpotent (Q 2 (a)b = 0, Q2 (a)b = 0) in nature, respectively."
},
{
"section_type": "OTHER",
"section_title": "(Anti-)BRST invariance in Abelian theory: superfield formalism",
"text": "In this section, we exploit the geometrical superfield approach to BRST formalism, endowed with the theoretical arsenal of the horizontality condition, to express the (anti-)BRST symmetry transformations and the Lagrangian densities (cf. (2.1) and (2.5)) in terms of the superfields defined on the (4, 2)-dimensional supermanifold. The latter is parametrized by the spacetime coordinates x µ (with µ = 0, 1, 2, 3) and a pair of Grassmannian variables θ and θ. As a consequence, the generalization of the 4D ordinary exterior derivative d = dx µ ∂ µ and the 1-form connection A (1) = dx µ A µ (x) on the (4, 2)-dimensional supermanifold, are\n\nd → d = dx µ ∂ µ + dθ ∂ θ + d θ ∂θ, d2 = 0, A (1) → Ã(1) = dx µ B µ (x, θ, θ) + dθ F (x, θ, θ) + d θ F (x, θ, θ), (3.1)\n\nwhere the mapping from the 4D local fields to the superfields are: ) . The super-expansion of the superfields, in terms ‡ We lay emphasis on the fact that (2.6) cannot be derived directly from (2.3) by the simple substitution B = -(∂ µ A µ ). One has to be judicious to deduce the precise expression for (2.6). The logical reasons behind the derivation of (2.6) are encoded in the superfield formulation (cf. (3.9) below).\n\nA µ (x) → B µ (x, θ, θ), C(x) → F (x, θ, θ) and C(x) → F (x, θ, θ\n\nof the basic fields as well as the secondary fields, are (see, e.g., [4-7, 10-12]):\n\nB µ (x, θ, θ) = A µ (x) + θ Rµ (x) + θ R µ (x) + i θ θ S µ (x), F (x, θ, θ) = C(x) + i θ B1 (x) + i θ B 1 (x) + i θ θ s(x), F (x, θ, θ) = C(x) + i θ B2 (x) + i θ B 2 (x) + i θ θ s(x).\n\n(3.2) It can be readily seen that, in the limiting case of (θ, θ) → 0, we get back our 4D basic fields (A µ , C, C). Furthermore, on the r.h.s. of the above super expansion, the bosonic (i.e. A µ , S µ , B 1 , B1 , B 2 , B2 ) and the fermionic (R µ , Rµ , C, C, s, s) fields do match. At this juncture, we have to recall our observations after equation (2.2). The nilpotent (anti-)BRST symmetry transformations basically owe their origin to the cohomological operator d. This is capitalized in the horizontality condition where we impose the restriction d Ã(1) = dA (1) on the super 1-form connection Ã(1) that contains the superfields defined on the (4, 2)-dimensional supermanifold. The latter condition yields the following relationships (see, e.g., for details, in our earlier works [21-25]):\n\nB 1 = B2 = s = s = 0, B1 + B 2 = 0, (3.3)\n\nwhere we are free to choose the secondary fields (B 2 , B1 ) (i.e. B 2 = B ⇒ B1 = -B) in terms of the Nakanishi-Lautrup auxiliary field B of the BRST invariant Lagrangian density (2.1). The other relations, that emerge from the above HC (i.e. d Ã(1) = dA (1) ), are\n\nR µ = ∂ µ C, Rµ = ∂ µ C, S µ = ∂ µ B, ∂ µ B ν -∂ ν B µ = ∂ µ A ν -∂ ν A µ .\n\n(3.4) At this stage, the super-curvature tensor Fµν =\n\n∂ µ B ν -∂ ν B µ is not equal to the ordinary curvature tensor F µν = ∂ µ A ν -∂ ν A µ\n\nas the former contains Grassmannian dependent terms. The substitution of the above values (cf. (3.3),(3.4)) of the secondary fields, in terms of the basic and auxiliary fields of the Lagrangian density (2.1), leads to\n\nB (h) µ (x, θ, θ) = A µ + θ ∂ µ C + θ ∂ µ C + i θ θ ∂ µ B, F (h) (x, θ, θ) = C -i θ B, F (h) (x, θ, θ) = C + i θ B, (3.5)\n\nwhere the superscript (h) has been used to denote that the above expansions have been obtained after the application of the HC. It can be seen that, due to (3.5), we get\n\n∂ µ B (h) ν -∂ ν B (h) µ = ∂ µ A ν -∂ ν A µ , (3.6)\n\nwhere there is no Grassmannian θ and θ dependence on the l.h.s. In the language of the geometry on the (4, 2)-dimensional supermanifold, the expansions (3.5) imply that the (anti-)BRST symmetry transformations s (a)b (and their corresponding generators Q (a)b ) for the 4D local fields (cf. (2.7)) are connected with the translational generators (∂/∂θ, ∂/∂ θ) because the translation of the corresponding (4, 2)-dimensional superfields, along the Grassmannian directions of the supermanifold, produces it. Thus, the Grassmannian independence of the super curvature tensor F (h)\n\nµν = ∂ µ B (h) ν -∂ ν B (h)\n\nµ implies that the 4D curvature tensor F µν is an (anti-)BRST (i.e. gauge) invariant physical quantity.\n\nIn terms of the superfields, equations (2.3) can be expressed as\n\nLim θ→0 ∂ ∂ θ -i F (h) { (∂ µ B (h) µ + 1 2 B) } , Limθ →0 ∂ ∂θ + iF (h) { (∂ µ B (h) µ + 1 2 B) } , ∂ ∂ θ ∂ ∂θ i 2 B µ(h) B (h) µ + 1 2 F (h) F (h) . (3.7)\n\nThese equations are unique because there is no other way to express the above equations in terms of the derivatives w.r.t. Grassmannian variables θ and θ. Thus, besides (2.3), there is no other possibility to express the gauge-fixing and the Faddeev-Popov ghost terms in the language of the off-shell nilpotent (anti-)BRST symmetry transformations (2.2). The superfield approach to BRST formulation, therefore, establishes the uniqueness of (2.3).\n\nTo express (2.6) in terms of the superfields, one has to substitute B = -(∂ µ A µ ) in (3.5). Thus, the expansion (3.5), in terms of the transformations (2.4), becomes §\n\nB (h) µ(o) (x, θ, θ) = A µ + θ ∂ µ C + θ ∂ µ C -i θ θ ∂ µ (∂ ρ A ρ ), ≡ A µ + θ (s ab A µ ) + θ (s b A µ ) + θ θ(s b sab A µ ), F (h) (o) (x, θ, θ) = C + i θ (∂ µ A µ ) ≡ C + θ (s ab C), F (h) (o) (x, θ, θ) = C -i θ (∂ µ A µ ) ≡ C + θ (s b C). (3.8)\n\nWe note that (3.5) and (3.8) are the super expansions (after the application of the HC) which lead to the derivation of the off-shell nilpotent (anti-)BRST symmetry transformations s (a)b as well as the on-shell nilpotent (anti-)BRST symmetry transformations s(a)b , respectively, for the basic fields A µ , C and C of the theory. The gauge-fixing and Faddeev-Popov ghost terms of the Lagrangian density (2.5) can also be expressed in terms of the superfields (3.8). In other words, (vis-à-vis (3.7)), we have the following equations that are the analogue of (2.6), namely;\n\nLim θ→0 ∂ ∂ θ + i 2 F (h) (o) (∂ µ A µ ) + i B (h) µ(o) ∂ µ F (h) (o) ) , Limθ →0 ∂ ∂θ - i 2 F (h) (o) (∂ µ A µ ) -i B (h) µ(o) ∂ µ F (h) (o) ) , ∂ ∂ θ ∂ ∂θ i 2 B µ(h) (o) B (h) µ(o) + 1 2 F (h) (o) F (h) (o) . (3.9)\n\nWe know that, for all practical computational purposes, it is essential to take into account s(a)b (∂ µ A µ ) = 0 because of the on-shell conditions ✷C = ✷ C = 0. The logical reason behind such a restriction (i.e. s(a)b (∂ µ A µ ) = 0) in (2.6) is encoded in the superfield approach to BRST formalism as can be seen from a close look at (3.9). The Lagrangian density (2.1) can be expressed, in terms of the (4, 2)-dimensional superfields, in the following distinct and different forms\n\nL(1) B = - 1 4 F (h) µν F µν(h) + Lim θ→0 ∂ ∂ θ -i F (h) (∂ µ B (h) µ + 1 2 B) , (3.10) §\n\nThe on-shell nilpotent (anti-)BRST symmetry transformations s(a)b can also be obtained by invoking the (anti-)chiral superfields on the appropriately chosen supermanifolds (see, e.g. [23] for details).\n\nL(2) B = - 1 4 F (h) µν F µν(h) + Limθ →0 ∂ ∂θ +i F (h) (∂ µ B (h) µ + 1 2 B) , (3.11) L(3) B = - 1 4 F (h) µν F µν(h) + ∂ ∂ θ ∂ ∂θ + i 2 B µ(h) B (h) µ + 1 2 F (h) F (h) . (3.12)\n\nIt would be noted that the kinetic energy term -(1/4) F (h) µν F µν(h) is independent of the variables θ and θ because F (h) µν = F µν . In exactly similar fashion, the Lagrangian density of (2.5) can be expressed, with the help of the super expansion (3.8), as\n\nL(1) b = - 1 4 F (h) µν(o) F µν(h) (o) + Lim θ→0 ∂ ∂ θ + i 2 F (h) (o) (∂ µ A µ ) + i B (h) µ(o) ∂ µ F (h) (o) ) , (3.13) L(2) b = - 1 4 F (h) µν(o) F µν(h) (o) + Limθ →0 ∂ ∂θ - i 2 F (h) (o) (∂ µ A µ ) -i B (h) µ(0) ∂ µ F (h) (o) ) , (3.14) L(3) b = - 1 4 F (h) µν(o) F µν(h) (o) + ∂ ∂ θ ∂ ∂θ + i 2 B µ(h) (o) B (h) µ(o) + 1 2 F (h) (o) F (h) (o) . (3.15)\n\nThe form of the Lagrangian densities (e.g. from (3.10) to (3.15)) simplify the proof for the (anti-)BRST invariance of the Lagrangian densities in (2.1) and (2.5).\n\nIn the above forms (e.g. from (3.10) to (3.12)) of the Lagrangian density, the BRST invariance s b L B = 0 and the anti-BRST invariance s ab L B = 0 become very transparent and simple because the following equalities and mappings exist, namely;\n\ns b L (a) B = 0 ⇒ Lim θ→0 ∂ ∂ θ L(1) B = 0, s b ⇔ Lim θ→0 ∂ ∂ θ , s 2 b = 0 ⇔ ∂ ∂ θ 2 = 0, (3.16) s ab L (a) B = 0 ⇒ Limθ →0 ∂ ∂θ L(2) B = 0, s ab ⇔ Limθ →0 ∂ ∂θ , s 2 ab = 0 ⇔ ∂ ∂θ 2 = 0. (3.17)\n\nSimilarly, the most beautiful relation (3.12), leads to the (anti-)BRST invariance together.\n\nHere one has to use the anticommutativity property s b s ab + s ab s b = 0 in the language of the translational generators (i.e. (∂/∂ θ), (∂/∂θ)) along the Grassmannian directions of the supermanifold, for its proof. This statement can be mathematically expressed as\n\ns (a)b L (a) B = 0 ⇒ ∂ ∂θ ∂ ∂ θ L(3) B = 0, s b s ab + s ab s b = 0 ⇔ ∂ ∂θ ∂ ∂ θ + ∂ ∂ θ ∂ ∂θ = 0. (3.18)\n\nIn exactly similar fashion, the on-shell nilpotent (anti-)BRST symmetry invariance (i.e. s(a)b L (a) b = 0) of the Lagrangian density (2.5) can also be captured in the language of the superfields if we exploit the expressions (3.13) to (3.15) for the Lagrangian density. In the latter case, the on-shell nilpotent (anti-)BRST invariance turns out to be like (3.16), (3.17) and (3.18) with the replacements: s (a)b → s(a)b , L 1, 2, 3) b . Mathematically, the (anti-)BRST invariance of the Lagrangian density (2.1) is captured in the equations (3.16) to (3.18) . In the language of geometry on the (4, 2)-dimensional supermanifold, the (anti-)BRST invariance corresponds to the Grassmannian independence of the supersymmetric versions of the Lagrangian density (2.1). In other words, the translation of the super Lagrangian densities (i.e. (3.10) to (3.12)), along the (θ) θ directions of 9 the supermanifold, is zero. This observation captures the (anti-)BRST invariance of (2.1).\n\n(a) B → L (a) b , L(1,2, 3\n\n) B → L("
},
{
"section_type": "METHOD",
"section_title": "(Anti-)BRST symmetries in non-Abelian theory: Lagrangian approach",
"text": "We begin with the following BRST-invariant Lagrangian density, in the Feynman gauge, for the four (3 + 1)-dimensional non-Abelian 1-form gauge theory ¶ (see, e.g. [26, 27, 9]\n\n) L (n) B = - 1 4 F µν • F µν + B • (∂ µ A µ ) + 1 2 B • B -i∂ µ C • D µ C, (4.1)\n\nwhere the curvature tensor (F µν ) is defined through the 2-form F (2) (1)(n) . Here the non-Abelian 1-form gauge connection is A (1)\n\n(n) = dA (1)(n) +iA (1)(n) ∧ A\n\n(n) = dx µ (A µ • T\n\n) and the exterior derivative is d = dx µ ∂ µ . The Nakanishi-Lautrup auxiliary field B = B • T is required for the linearization of the gauge-fixing term and the (anti-)ghost fields ( C)C are essential for the proof of the unitarity in the theory. The latter fields are fermionic (i.e. (C a ) 2 = 0, ( Ca ) 2 = 0, C a C b + C b C a = 0, C a Cb + Cb C a = 0, etc.) in nature.\n\nThe above Lagrangian density respects the following off-shell nilpotent ((s\n\n(n) b ) 2 = 0) BRST symmetry transformations s (n) b , namely; s (n) b A µ = D µ C, s (n) b C = - i 2 (C × C), s (n) b C = iB, s (n) b B = 0, s (n) b F µν = i(F µν × C). (4.2)\n\nIt will be noted that (i) the curvature tensor F µν • T transforms here under the BRST symmetry transformation. However, it can be checked explicitly that the kinetic energy term -(1/4)F µν • F µν remains invariant under the BRST symmetry transformations, (ii) the nilpotent anti-BRST symmetry transformations corresponding to the above BRST symmetry transformations (4.2) cannot be defined for the Lagrangian density (4.1), and (iii) the on-shell nilpotent version of the above BRST symmetry transformations is also possible if we substitute, in the above symmetry transformations,\n\nB = -(∂ µ A µ ). The ensuing on-shell (i.e. ∂ µ D µ C = 0) nilpotent BRST symmetry transformations s(n) b are s(n) b A µ = D µ C, s(n) b C = - i 2 (C × C), s(n) b C = -i(∂ µ A µ ), s(n) b F µν = i(F µν × C). (4.3)\n\nThe above on-shell nilpotent transformations leave the following Lagrangian density\n\nL (n) b = - 1 4 F µν • F µν - 1 2 (∂ µ A µ ) • (∂ ρ A ρ ) -i∂ µ C • D µ C, (4.4)\n\n¶ For the non-Abelian 1-form gauge theory, the notations used in the Lie algebraic space are:\n\nA • B = A a B a , (A × B) a = f abc A b B c , D µ C a = ∂ µ C a + if abc A b µ C c ≡ ∂ µ C a + i(A µ × C) a , F µν = ∂ µ A ν -∂ ν A µ + iA µ × A ν , A µ = A µ • T, [T a , T b ] = f abc T c\n\nwhere the Latin indices a, b, c = 1, 2, 3....N are in the SU (N ) Lie algebraic space. The structure constant f abc be chosen to be totally antisymmetric for any arbitrary semi simple Lie algebra that includes SU (N ), too (see, e.g., [27]).\n\n10 quasi-invariant because it transforms to a total derivative.\n\nThe gauge-fixing and Faddeev-Popov ghost terms of the Lagrangian densities (4.1) and (4.4) can be written, modulo a total derivative, as a BRST-exact quantity in terms of the off-shell and on-shell nilpotent BRST symmetry transformations (4.2) and (4.3). This statement can be mathematically expressed as follows\n\ns (n) b -i C • {(∂ µ A µ ) + 1 2 B} = B • (∂ µ A µ ) + 1 2 B • B -i ∂ µ C • D µ C, (4.5) s(n) b + i 2 C • (∂ µ A µ ) + i A µ • ∂ µ C = - 1 2 (∂ µ A µ ) • (∂ ρ A ρ ) -i ∂ µ C • D µ C. (4.6)\n\nIt will be noted that one has to take into account s(n) b (∂ µ A µ ) = ∂ µ D µ C = 0 in the above proof of the exactness of the expression in (4.6) .\n\nThe Lagrangian densities that respect the off-shell nilpotent (i.e. (s (n) (a)b ) 2 = 0) and anticommuting (s\n\n(n) b s (n) ab + s (n) ab s (n) b = 0) (anti-)BRST symmetry transformations are L (1)(n) b = - 1 4 F µν • F µν + B • (∂ µ A µ ) + 1 2 (B • B + B • B) -i∂ µ C • D µ C, (4.7) L (2)(n) b = - 1 4 F µν • F µν -B • (∂ µ A µ ) + 1 2 (B • B + B • B) -iD µ C • ∂ µ C. (4.8)\n\nHere auxiliary fields B and B satisfy the Curci-Ferrari condition B + B = -(C × C) [28, 29] . It is also evident, from this relation, that\n\nB•(∂ µ A µ )-i∂ µ C •D µ C = -B•(∂ µ A µ )-iD µ C •∂ µ C.\n\nFurthermore, it should be re-emphasized that the Lagrangian densities (4.1) and (4.4) do not respect the anti-BRST symmetry transformations of any kind. The BRST and anti-BRST symmetry transformations, for the above Lagrangian densities, are\n\ns (n) b A µ = D µ C, s (n) b C = - i 2 (C × C), s (n) b C = iB, s (n) b B = 0, s (n) b F µν = i(F µν × C), s (n) b B = i( B × C), (4.9) s (n) ab A µ = D µ C, s (n) ab C = - i 2 ( C × C), s (n) ab C = i B, s (n) ab B = 0, s (n) ab F µν = i(F µν × C), s (n) ab B = i(B × C). (4.10)\n\nThe above off-shell nilpotent (anti-)BRST symmetry transformations leave the Lagrangian densities (4.7) as well as (4.8) quasi-invariant as they transform to some total derivatives. The gauge-fixing and Faddeev-Popov ghost terms of the Lagrangian densities (4.7) and (4.8) can be written, in a symmetrical fashion with respect to s (n) b and s\n\n(n) ab , as s (n) b s (n) ab i 2 A µ • A µ + C • C = B • (∂ µ A µ ) + 1 2 (B • B + B • B) -i∂ µ C • D µ C, ≡ -B • (∂ µ A µ ) + 1 2 (B • B + B • B) -iD µ C • ∂ µ C. (4.11)\n\nThis demonstrates the key fact that the above gauge-fixing and Faddeev-Popov ghost terms are (anti-)BRST invariant together because of the nilpotency and anticommutativity of the 11 (anti-)BRST symmetry transformations s (n) (a)b that are present in the theory.\n\n5 (Anti-)BRST invariance in non-Abelian theory: superfield approach\n\nTo capture (i) the off-shell as well as the on-shell nilpotent (anti-)BRST symmetry transformations, and (ii) the invariance of the Lagrangian densities, in the language of the superfield approach to BRST formalism, we have to consider the 4D 1-form non-Abelian gauge theory on a (4, 2)-dimensional supermanifold. As a consequence, we have the following mappings:\n\nd → d = dx µ ∂ µ + dθ ∂ θ + d θ ∂θ, d2 = 0, A (1)(n) → Ã(1)(n) = dx µ (B µ • T )(x, θ, θ) + dθ( F • T )(x, θ, θ) + d θ(F • T )(x, θ, θ), (5.1)\n\nwhere the (4, 2)-dimensional superfields (B µ •T, F •T, F •T ) are the generalizations of the 4D basic local fields (A µ • T, C • T, C • T ) of the Lagrangian density (4.1), (4.7) and (4.8). These superfields can be expanded along the Grassmannian directions of the supermanifold, in terms of the basic 4D fields, auxiliary fields and secondary fields as [4,16,19] .2) To determine the exact expressions for the secondary fields, in terms of the basic and auxiliary fields of the theory, we have to exploit the HC. The horizontality condition, for the non-Abelian gauge theory is the requirement of the equality of the Maurer-Cartan equation on the (super) manifolds. In other words, the covariant reduction of the super 2-form curvature F (2)(n) to the ordinary 2-form curvature (i.e. d Ã(1)(n) + i Ã(1)(n) ∧ Ã(1)(n) = dA (1)(n) + iA (1)(n) ∧A (1)(n) ) leads to the determination of the secondary fields in terms of the basic and auxiliary fields of the theory. The ensuing expansions, in terms of the basic and auxiliary fields, lead to (i) the derivation of the (anti-)BRST symmetry transformations for the basic fields of the theory, and (ii) the geometrical interpretations of the nilpotent (anti-)BRST symmetry transformations (and their corresponding nilpotent generators) for the basic fields of the theory as the translations of the corresponding superfields along the Grassmannian directions of the (4, 2)-dimensional supermanifold (see, e.g., [16, 19] ).\n\n(B µ • T )(x, θ, θ) = (A µ • T )(x) + θ ( Rµ • T )(x) + θ (R µ • T )(x) + i θ θ (S µ • T )(x), (F • T )(x, θ, θ) = (C • T )(x) + i θ ( B1 • T )(x) + i θ (B 1 • T )(x) + i θ θ (s • T )(x), ( F • T )(x, θ, θ) = ( C • T )(x) + i θ ( B2 • T )(x) + i θ (B 2 • T )(x) + i θ θ (s • T )(x). (5\n\nWith the identifications B 2 = B and B1 = B, the following relationships emerge after the application of the horizontality condition (see, e.g., [16]):\n\nR µ = D µ C, Rµ = D µ C, B + B = -(C × C), s = i( B × C), S µ = D µ B + D µ C × C ≡ -D µ B -D µ C × C, s = -i(B × C), B 1 = - 1 2 (C × C), B2 = - 1 2 ( C × C). (5.3)\n\nIn the rest of our present text, we shall be using the short-hand notations for all the fields e.g.:\n\nA µ • T = A µ , C • T = C, B • T = B, etc.\n\n, for the sake of brevity.\n\nThe substitution of the above expressions, which are obtained after the application of the horizontality condition, leads to the following expansions\n\nB (h) µ (x, θ, θ) = A µ + θ D µ C + θ D µ C + i θ θ (D µ B + D µ C × C), F (h) (x, θ, θ) = C + i θ B - i 2 θ (C × C) -θ θ ( B × C), F (h) (x, θ, θ) = C - i 2 θ ( C × C) + i θ B + θ θ (B × C). (5.4)\n\nThe above expansions (see, e.g., our earlier works [16, 19] ) can be expressed in terms of the off-shell nilpotent (anti-)BRST symmetry transformations (4.9) and (4.10).\n\nWith the above expansion at our disposal, the gauge-fixing and Faddeev-Popov terms of the Lagrangian density (4.1) can be written, modulo a total ordinary derivative, as\n\nLim θ→0 ∂ ∂ θ -i F (h) • ∂ µ B (h) µ - i 2 F (h) • B = B • (∂ µ A µ ) + 1 2 B • B -i ∂ µ C • D µ C. (5.5)\n\nFurthermore, it can be seen that, due to the validity and consequences of the horizontality condition, the super curvature tensor Fµν has the following form [16,4]\n\nF (h) µν = F µν + iθ(F µν × C) + i θ(F µν × C) -θ θ (F µν × B + F µν × C × C). (5.6)\n\nIt is clear from the above relationship that the kinetic energy term of the present 4D non-Abelian 1-form gauge theory remains invariant, namely;\n\n- 1 4 F (h) µν • F µν(h) = - 1 4 F µν • F µν . (5.7)\n\nThe Grassmannian independence of the l.h.s. of (5.7) has deep meaning as far as physics is concerned. It implies immediately that the kinetic energy term of the non-Abelian gauge theory is an (anti-)BRST (i.e. gauge) invariant physical quantity. At this juncture, it is worthwhile to point out that one can also capture the equation (4.6) in the superfield approach to BRST formalism where the on-shell nilpotent version of the BRST symmetry transformations (i.e. s(n) b ) plays an important role. For this purpose, we have to express the superfield expansion (5.4) for the on-shell nilpotent BRST symmetry transformation where one has to exploit the replacement B = -(∂ µ A µ ). With this substitution, the equation (5.4) for the superfield expansion becomes\n\nB (h) µ(o) (x, θ, θ) = A µ + θ D µ C + θ D µ C + i θ θ [-D µ (∂ ρ A ρ ) + D µ C × C], F (h) (o) (x, θ, θ) = C + i θ B - i 2 θ (C × C) -θ θ ( B × C), F (h) (o) (x, θ, θ) = C - i 2 θ ( C × C) -i θ (∂ µ A µ ) -θ θ [(∂ µ A µ ) × C)]. (5.8)\n\nNow, the equation (4.6) can be expressed in terms of the above superfields, as:\n\nLim θ→0 ∂ ∂ θ i 2 F (h) (o) • (∂ µ A µ ) + i B (h) µ(o) • ∂ µ F (h) (o) = - 1 2 (∂ µ A µ ) • (∂ ρ A ρ ) -i ∂ µ C • D µ C. (5.9) 13\n\nFurthermore, it will be noted that the analogue of (5.6), for the on-shell nilpotent BRST symmetry transformation (i.e. F (h) µν(o) ), can be obtained by the replacement B = -(∂ µ A µ ). Once again, the equality (5.7) would remain intact even if we take into account the on-shell nilpotent BRST symmetry transformations. Thus, we note that the kinetic energy term (i.e.\n\n(-(1/4)F µν • F µν = -(1/4) F µν(h) (o) • F (h) µν(o)\n\n) of the non-Abelian gauge theory remains independent of the Grassmannian variables θ and θ after the application of the HC. This statement is true for the off-shell as well as the on-shell nilpotent (anti-)BRST symmetry transformations. Physically, it implies that the kinetic energy term for the gauge field of the non-Abelian theory is an (anti-)BRST (i.e. gauge) invariant quantity.\n\nThe above key observation helps in expressing the Lagrangian density (4.1) and (4.4) in terms of the superfields (obtained after the application of HC), as\n\nL(n) B = - 1 4 F (h) µν • F µν(h) + Lim θ→0 ∂ ∂ θ -i F (h) • ∂ µ B (h) µ - i 2 F (h) • B , L(n) b = - 1 4 F (h) µν(o) • F µν(h) (o) + Lim θ→0 ∂ ∂ θ i 2 F (h) (o) • (∂ µ A µ ) + i B (h) µ(o) • ∂ µ F (h) (o) . (5.10)\n\nThis result, in turn, simplifies the BRST invariance of the above Lagrangian density (4.1) and (4.4) (describing the 4D 1-form non-Abelian gauge theory) as follows\n\nLim θ→0 ∂ ∂ θ L(n) B = 0 ⇒ s (n) b L (n) B = 0, Lim θ→0 ∂ ∂ θ L(n) b = 0 ⇒ s(n) b L (n) b = 0. (5.11)\n\nThis is a great simplification because the total super Lagrangian densities (5.10) remain independent of the Grassmannian variable θ. This key result is encoded in the mapping (s\n\n(n) b , s(n) b ) ⇔ Lim θ→0 (∂/∂ θ) and the nilpotency (s (n) b ) 2 = 0, (s (n) b ) 2 = 0, (∂/∂ θ) 2 = 0.\n\nIt can be readily checked that the analogues of (5.5) and (5.9) cannot be expressed as the derivative w.r.t. the Grassmannian variable θ. To check this, one has to exploit the super expansions (5.4) and (5.8) obtained after the application of the HC (in the context of the derivation of the off-shell as well as the on-shell nilpotent BRST symmetry transformations s (n) b and s(n) b ). It can be clearly seen that the operation of the derivative w.r.t. the Grassmannian variable θ, on any combination of the superfields from the expansions (5.4) and (5.8), does not lead to the derivation of the r.h.s. of (5.5) and (5.9). In the language of the superfield approach to BRST formalism, this is the reason behind the non-existence of the anti-BRST symmetry transformations for the Lagrangian densities (4.1) and (4.4).\n\nThe form of the gauge-fixing and Faddeev-Popov terms (4.11), expressed in terms of the BRST and anti-BRST symmetry transformations together, can be represented in the language of the superfields (obtained after the application of HC), as\n\n∂ ∂ θ ∂ ∂θ i 2 B (h) µ • B µ(h) + F (h) • F (h) = B • (∂ µ A µ ) + 1 2 (B • B + B • B) -i∂ µ C • D µ C.\n\n(5.12) As a consequence of the above expression, the Lagrangian densities (4.7) (as well as (4.8)) can be presented, in terms of the superfields, as\n\nL(1,2)(n) b = - 1 4 F µν(h) • F (h) µν + ∂ ∂ θ ∂ ∂θ i 2 B (h) µ • B µ(h) + F (h) • F (h) . (5.13) 14\n\nThe BRST and anti-BRST invariance of the above super Lagrangian density (and that of the ordinary 4D Lagrangian densities (4.7) and (4.8)) is encoded in the following simple equations that are expressed in terms of the translational generators along the Grassmannian directions of the (4, 2)-dimensional supermanifold, namely;\n\nLim θ→0 ∂ ∂ θ L(1,2)(n) b = 0 ⇒ s (n) b L (1)(n) b = 0, Limθ →0 ∂ ∂θ L(1,2)(n) b = 0 ⇒ s (n) ab L ( 2\n\n)(n) b = 0. (5.14) This is a tremendous simplification of the (anti-)BRST invariance of the Lagrangian densities (4.7) and (4.8) in the language of the superfield approach to BRST formalism. In other words, if one is able to show the Grassmannian independence of the super Lagrangian densities of the theory, the (anti-)BRST invariance of the 4D theory follows automatically.\n\nIn the language of the geometry on the supermanifold, the (anti-)BRST invariance of a 4D Lagrangian density is equivalent to the statement that the translation of the super version of the above Lagrangian density, along the Grassmannian directions of the (4, 2)dimensional supermanifold, is zero. Thus, the super Lagrangian density of an (anti-)BRST invariant 4D theory is a Lorentz scalar, constructed with the help of (4, 2)-dimensional superfields (obtained after the application of HC), such that, when the partial derivatives w.r.t. the Grassmannian variables (θ and θ) operate on it, the result is zero.\n\nThe nilpotency and anticommutativity properties (that are associated with the conserved (anti-)BRST charges and (anti-)BRST symmetry transformations) are found to be captured very naturally (cf. (3.16)-(3.18)) when we consider the superfield formulation of the (anti-)BRST invariance of the Lagrangian density of a given 1-form gauge theory. We mention, in passing, that one could also derive the analogue of the equations (3.16), (3.17) and (3.18) for the 4D non-Abelian 1-form gauge theory in a straightforward manner."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusions",
"text": "In our present investigation, we have concentrated mainly on the (anti-)BRST invariance of the Lagrangian densities of the free 4D (non-)Abelian 1-form gauge theories (having no interaction with matter fields) within the framework of the superfield approach to BRST formalism. We have been able to provide the geometrical basis for the existence of the (anti-)BRST invariance in the above 4D theories. To be more specific, we have been able to show that the Grassmannian independence of the (4, 2)-dimensional super Lagrangian density, expressed in terms of the appropriate superfields, is a clear-cut proof that there is an (anti-)BRST invariance (cf. (3.16), (3.17), (3.18), (5.11), (5.14 )) in the 4D theory.\n\nIf the super Lagrangian density could be expressed as a sum of (i) a Grassmannian independent term, and (ii) a derivative w.r.t. the Grassmannian variable, then, the corresponding 4D Lagrangian density will automatically respect BRST and/or anti-BRST invariance. In the latter piece of the above super Lagrangian density, the derivative could be either w.r.t. θ or w.r.t. θ or w.r.t. both of them put together. More specifically, (i) if the derivative is w.r.t. θ, the nilpotent symmetry would correspond to the BRST, 15\n\n(ii) if the derivative is w.r.t. θ, the nilpotent symmetry would be that of the anti-BRST type, and (iii) if both the derivatives are present together, both the nilpotent (anti-)BRST symmetries would be present together (and they would turn out to be anticommuting).\n\nFor the 4D (non-)Abelian 1-form gauge theories, that are considered on the (4, 2)dimensional supermanifold, it is the HC on the 1-form super connection Ã(1) that plays a very important role in the derivation of the (anti-)BRST symmetry transformations. The cohomological origin for the above HC lies in the (super) exterior derivatives ( d)d. This point has been made quite clear in our discussions after the off-shell as well as the on-shell nilpotent (anti-)BRST symmetry transformations (2.2), (2.4), (4.2), (4.3), (4.9) and (4.10).\n\nIn fact, it is the full kinetic energy term of the above theories (owing its origin to the cohomological operator d = dx µ ∂ µ ) that remains invariant under the above on-shell as well the off-shell nilpotent (anti-)BRST symmetry transformations.\n\nThe HC produces specifically the nilpotent (anti-)BRST symmetry transformations for the gauge and (anti-)ghost fields because of the fact that the super 1-form connection Ã(1) / Ã(1)(n) (cf. (3.1) and (5.1)) is constructed with a super vector multiplet (B µ , F , F) which is the generalization of the gauge field A µ and the (anti-)ghost fields ( C)C (of the ordinary 4D (non-)Abelian 1-form gauge theories) to the (4, 2)-dimensional supermanifold.\n\nAs a consequence, only the nilpotent and anticommuting (anti-)BRST symmetry transformations for the 4D local fields A µ , C and C are obtained when the full potential of the HC is exploited within the framework of the above superfield formulation.\n\nIt is worthwhile to point out that geometrically the super Lagrangian densities, expressed in terms of the (4, 2)-dimensional superfields, are equivalent to the sum of the kinetic energy term and the translations of some composite superfields (obtained after the application of the HC) along the Grassmannian directions (i.e. θ and/or θ) of the (4, 2)dimensional supermanifold. This observation is distinctly different from our earlier works on the superfield approach to 2D (non-)Abelian 1-form gauge theories [24, 25, 23] which are found to correspond to the topological field theories. In fact, for the latter theories, the total super Lagrangian density turns out to be a total derivative w.r.t. the Grassmannian variables (θ and/or θ). That is to say, even the kinetic energy term of the latter theories, is able to be expressed as the total derivative w.r.t. the variables θ and/or θ.\n\nIn our present endeavour, within the framework of the superfield approach to BRST formalism, we have been able to provide (i) the logical reason behind the non-existence of the anti-BRST symmetry transformations for the Lagrangian densities (4.1) and (4.4) for the 4D non-Abelian 1-form gauge theory, (ii) the explicit explanation for the uniqueness of the equations (2.3) and (2.6) for the 4D Abelian 1-form gauge theory, (iii) the convincing proof for the on-shell nilpotent (anti-)BRST invariance of the gauge-fixing term (i.e. s(a)b (∂ µ A µ ) = 0, s(n) (a)b (∂ µ A µ ) = 0) for the (non-)Abelian 1-form gauge theories, and (iv) the compelling arguments for the non-existence of the exact analogue(s) of (2.3) and (2.6) for the non-Abelian 1-form gauge theory. To the best of our knowledge, the logical explanations for the above subtle points (connected with the 1-form gauge theories) are completely 16 new. Thus, the results of our present work are simple, beautiful and original.\n\nIt is worthwhile to mention that our superfield construction and its ensuing geometrical interpretations are not specific to the Feynman gauge (which has been taken into account in our present endeavor). To corroborate this assertion, we take the simple case of the 4D Abelian 1-form gauge theory and write the Lagrangian density (2.1) in the arbitrary gauge\n\nL (a,ξ) B = - 1 4 F µν F µν + B (∂ µ A µ ) + ξ 2 B 2 -i ∂ µ C ∂ µ C, (6.1)\n\nwhere ξ is the gauge parameter. It is elementary to check that, in the limit ξ → 1, we get back our Lagrangian density (2.1) for the Abelian theory in the Feynman gauge. The analogue of the equation (2.3) (for the gauge-fixing and Faddeev-Popov ghost terms in the case of the arbitrary gauge) can be expressed as\n\ns b -i C {(∂ µ A µ ) + ξ 2 B}], s ab +i C {(∂ µ A µ ) + ξ 2 B} , s b s ab i 2 A µ A µ + ξ 2 C C . (6.2)\n\nThe above expression can be easily generalized to the analogues of the equations (3.10)-(3.12) in terms of the superfields by taking the help of (3.8). Thus, the geometrical interpretations remain intact even in the case of the arbitrary gauge. In a similar fashion, for the 4D non-Abelian 1-form gauge theory, the equations (4.5), (4.6) and (4.11) can be generalized to the case of arbitrary gauge and, subsequently, can be expressed in terms of superfields as the analogues of (5.5), (5.9) and (5.12). Finally, we can obtain the analogues of (5. 7) , ( 5.10) and (5.13) which will lead to the derivation of the analogues of (5.11) and (5.14). Thus, we note that geometrical interpretations, in the arbitrary gauge, remain the same for the 4D (non-)Abelian 1-form gauge theory within the framework of our superfield approach to BRST formalism.\n\nOur present work can be generalized to the case of the interacting 4D (non-)Abelian 1-form gauge theories where there exists an explicit coupling between the gauge field and the matter fields. In fact, our earlier works [14-18] might turn out to be quite handy in attempting the above problems. It seems to us that it is the combination of the HC and the restrictions, owing their origin to the (super) covariant derivative on the matter (super) fields and their intimate connection with the (super) curvatures, that would play a decisive role in proving the existence of the (anti-)BRST invariance for the above gauge theories.\n\nIt is gratifying to state that we have accomplished the above goals in our very recent endeavours [30] [31] [32] . In fact, we have been able to provide the geometrical basis for the existence of the (anti-)BRST invariance, in the context of the interacting (non-)Abelian 1-form gauge theories with Dirac as well as complex scalar fields, within the framework of the augmented superfield approach to BRST formalism. As it turns out, here too, the super Lagrangian density is found to be independent of the Grassmannian variables.\n\nIn our earlier works [33] [34] [35] , we have been able to show the existence of the nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations for the 4D free Abelian 2-form 17 gauge theory. We have also established the quasi-topological nature of it in [35] . In a recent work [36], the nilpotent (anti-)BRST symmetry transformations have been captured in the framework of the superfield formulation. It would be a very nice endeavour to study the (anti-)BRST and (anti-)co-BRST invariance of the 4D Abelian 2-form gauge theory within the framework of superfield formulation. At present, this issue and connected problems in the context of the 4D free Abelian 2-form gauge theory are under intensive investigation and our results would be reported in our forthcoming future publications [37].\n\nAcknowledgement: Financial support from the Department of Science and Technology (DST), Government of India, under the SERC project sanction grant No: -SR/S2/HEP-23/2006, is gratefully acknowledged."
}
] | |
arxiv:0704.0066 | 0704.0066 | 1 | 5a791c216f06fd6b10882db3e3fbfbdb15a2951484e49be57da703a2d3733cbb | Lagrangian quantum field theory in momentum picture. IV. Commutation
relations for free fields | Possible (algebraic) commutation relations in the Lagrangian quantum theory
of free (scalar, spinor and vector) fields are considered from mathematical
view-point. As sources of these relations are employed the Heisenberg
equations/relations for the dynamical variables and a specific condition for
uniqueness of the operators of the dynamical variables (with respect to some
class of Lagrangians). The paracommutation relations or some their
generalizations are pointed as the most general ones that entail the validity
of all Heisenberg equations. The simultaneous fulfillment of the Heisenberg
equations and the uniqueness requirement turn to be impossible. This problem is
solved via a redefinition of the dynamical variables, similar to the normal
ordering procedure and containing it as a special case. That implies
corresponding changes in the admissible commutation relations. The introduction
of the concept of the vacuum makes narrow the class of the possible commutation
relations; in particular, the mentioned redefinition of the dynamical variables
is reduced to normal ordering. As a last restriction on that class is imposed
the requirement for existing of an effective procedure for calculating vacuum
mean values. The standard bilinear commutation relations are pointed as the
only known ones that satisfy all of the mentioned conditions and do not
contradict to the existing data. | [
"Bozhidar Z. Iliev (Institute for Nuclear Research and Nuclear Energy",
"Bulgarian Academy of Sciences",
"Sofia",
"Bulgaria)"
] | [
"hep-th"
] | hep-th | [] | 2007-04-01 | 2026-02-26 | Possible (algebraic) commutation relations in the Lagrangian quantum theory of free (scalar, spinor and vector) fields are considered from mathematical view-point. As sources of these relations are employed the Heisenberg equations/relations for the dynamical variables and a specific condition for uniqueness of the operators of the dynamical variables (with respect to some class of Lagrangians). The paracommutation relations or some their generalizations are pointed as the most general ones that entail the validity of all Heisenberg equations. The simultaneous fulfillment of the Heisenberg equations and the uniqueness requirement turn to be impossible. This problem is solved via a redefinition of the dynamical variables, similar to the normal ordering procedure and containing it as a special case. That implies corresponding changes in the admissible commutation relations. The introduction of the concept of the vacuum makes narrow the class of the possible commutation relations; in particular, the mentioned redefinition of the dynamical variables is reduced to normal ordering. As a last restriction on that class is imposed the requirement for existing of an effective procedure for calculating vacuum mean values. The standard bilinear commutation relations are pointed as the only known ones that satisfy all of the mentioned conditions and do not contradict to the existing data.
The main subject of this paper is an analysis of possible (algebraic) commutation relations in the Lagrangian quantum theory foot_0 of free fields. These relations are considered only from mathematical view-point and physical consequence of them, like the statistics of many-particle systems, are not investigated.
The canonical quantization method finds its origin in the classical Hamiltonian mechanics [9, 10] and naturally leads to the canonical (anti)commutation relations [3, 11, 12] . These relations can be obtained from different assumptions (see, e.g., [1, [13] [14] [15] ) and are one of the basic corner stones of the present-day quantum field theory.
Theoretically there are possible also non-canonical commutation relations. The best known example of them being the so-called paracommutation relations [16] [17] [18] . But, however, it seems no one of the presently known particles/fields obeys them.
In the present work is shown how different classes of commutation relations, understood in a broad sense as algebraic connections between creation and/or annihilation operators, arise from the Lagrangian formalism, when applied to three types of Lagrangians describing free scalar, spinor and vector fields. Their origin is twofold. One one hand, a requirement for uniqueness of the dynamical variables (that can be calculated from Lagrangians leading to identical Euler-Lagrange equation) entails a number of specific commutation relations. On another hand, any one of the so-called Heisenberg relations/equations [3, 11] , implies corresponding commutation relations; for example, the paracommutation relations arise from the Heisenberg equations regarding the momentum operator, when 'charge symmetric' Lagrangian is employed. 2 The combination of the both methods leads to strong, generally incompatible, restrictions on the admissible types of commutation relations.
The introduction of the concept of vacuum, combined with the mentioned uniqueness of the operators of the dynamical variables, changes the situation and requires a redefinition of these operators in a way similar to the one known as the normal ordering [1, 3, 11, 12] , which is its special case. Some natural assumptions reduce the former to the letter one; in particular, in that way are excluded the paracommutation relations. However, this does not reduce the possible commutation relations to the canonical ones. Further, the requirement to be available an effective procedure for calculating vacuum mean (expectation) values, to which reduce all predictable results in the theory, puts new restriction, whose only realistic solution at the time being seems to be the standard canonical (anti)commutation relations.
The layout of the work is as follows.
Sect. 2 gives an idea of the momentum picture of motion and discusses the relations between the creation and annihilation operators in it and in Heisenberg picture. In Sect. 3 are reviewed some basic results from [13] [14] [15] , part of which can be found also in papers like [1, 3, 11, 12] . In particular, the explicit expression of the dynamical variables via the creation and annihilation operators are presented (without assuming some commutation relations or normal ordering) and it is pointed to the existence of a family of such variables for a given system of Euler-Lagrange equations for free fields. The last fact is analyzed in Sect. 4 , where a number of its consequences, having a sense of commutation relations, are drawn. The Heisenberg relations and the commutation relations between the dynamical variables are reviewed and analyzed in Sect. 5. It is pointed that the letter should be consequences from the former ones. Arguments are presented that the Heisenberg equation concerning the angular momentum operator should be split into two independent ones, representing its 'orbital' and 'spin' parts, respectively.
Sect. 6 contains a method for assigning commutation relations to the Heisenberg equations. It is shown that the Heisenberg equation involving the 'orbital' part of the angular momentum gives rise to a differential, not algebraic, commutation relation and the one concerning the 'spin' part of the angular momentum implies a complicated integro-differential connections between the creation and annihilation operators. Special attention is paid to the paracommutation relations, whose particular kind are the ordinary ones, which ensure the validity of the Heisenberg equations concerning the momentum operator. Partially is analyzed the problem for compatibility of the different types of commutation relations derived. It is proved that some generalization of the paracommutation relations ensures the fulfillment of all of the Heisenberg relations.
Sect. 7 is devoted to consequences from the commutation relations derived in Sect. 6 under the conditions for uniqueness of the dynamical variables presented in Sect. 4. Generally, these requirements are incompatible with the commutation relations. To overcome the problem, it is proposed a redefinition of the dynamical variables via a method similar to (and generalizing) the normal ordering. This, of course, entails changes in the commutation relations, the new versions of which happen to be compatible with the uniqueness conditions and ensure the validity of the Heisenberg relations.
The concept of the vacuum is introduced in Sect. 8. It reduces (practically) the redefinition of the operators of the dynamical variables to the one obtained via the normal ordering procedure in the ordinary quantum field theory, but, without additional suppositions, does not reduce the commutation relations to the standard bilinear ones. As a last step in specifying the commutation relations as much as possible, we introduce the requirement the theory to supply an effective way for calculating vacuum mean values of (anti-normally ordered) products of creation and annihilation operators to which are reduced all predictable results, in particular the mean values of the dynamical variables. The standard bilinear commutation relation seems to be the only ones know at present that survive that last condition, however their uniqueness in this respect is not investigated.
Sect. 9 deals with the same problems as described above but for systems containing at least two different quantum fields. The main obstacle is the establishment of commutation relations between creation/annihilation operators concerning different fields. Argument is presented that they should contain commutators or anticommutators of these operators. The major of corresponding commutation relations are explicitly written and the results obtained turn to be similar to the ones just described, only in 'multifield' version.
Section 10 closes the paper by summarizing its main results.
The books [1] [2] [3] will be used as standard reference works on quantum field theory. Of course, this is more or less a random selection between the great number of (text)books and papers on the theme to which the reader is referred for more details or other points of view. For this end, e.g., [4, 12, 19] or the literature cited in [1-4, 12, 19 ] may be helpful.
Throughout this paper denotes the Planck's constant (divided by 2π), c is the velocity of light in vacuum, and i stands for the imaginary unit. The superscripts † and ⊤ mean respectively Hermitian conjugation and transposition (of operators or matrices), the superscript * denotes complex conjugation, and the symbol • denotes compositions of mappings/operators. By δ f g , or δ g f or δ f g (:= 1 for f = g, := 0 for f = g) is denoted the Kronecker δ-symbol, depending on arguments f and g, and δ n (y), y ∈ R n , stands for the n-dimensional Dirac δ-function; δ(y) := δ 1 (y) for y ∈ R.
The Minkowski spacetime is denoted by M . The Greek indices run from 0 to dim M -1 = 3. All Greek indices will be raised and lowered by means of the standard 4-dimensional Lorentz metric tensor η µν and its inverse η µν with signature (+ ---). The Latin indices a, b, . . . run from 1 to dim M -1 = 3 and, usually, label the spacial components of some object. The Einstein's summation convention over indices repeated on different levels is assumed over the whole range of their values.
At last, we ought to give an explanation why this work appears under the general title "Lagrangian quantum field theory in momentum picture" when in it all considerations are done, in fact, in Heisenberg picture with possible, but not necessary, usage of the creation and annihilation operators in momentum picture. First of all, we essentially employ the obtained in [13] [14] [15] expressions for the dynamical variables in momentum picture for three types of Lagrangians. The corresponding operators in Heisenberg picture, which in fact is used in this paper, can be obtained via a direct calculation, as it is partially done in, e.g., [1] for one of the mentioned types of Lagrangians. The important point here is that in Heisenberg picture it suffice to be used only the standard Lagrangian formalism, while in momentum picture one has to suppose the commutativity between the components of the momentum operator and the validity of the Heisenberg relations for it (see below equations (2.6 ) and (2.7)). Since for the analysis of the commutation relations we intend to do the fulfillment of these relations is not necessary (they are subsidiary restrictions on the Lagrangian formalism), the Heisenberg picture of motion is the natural one that has to be used. For this reason, the expression for the dynamical variables obtained in [13] [14] [15] will be used simply as their Heisenberg counterparts, but expressed via the creation and annihilation operators in momentum picture. The only real advantage one gets in this way is the more natural structure of the orbital angular momentum operator. As the commutation relations considered below are algebraic ones, it is inessential in what picture of motion they are written or investigated.
Since the momentum picture of motion will be used only partially in this work, below is presented only its definition and the connection between the creation/annihilation operators in it and in Heisenberg picture. Details concerning the momentum picture can be found in [20, 21] and in the corresponding sections devoted to it in [13] [14] [15] .
Let us consider a system of quantum fields, represented in Heisenberg picture of motion by field operators φi (x) : F → F, i = 1, . . . , n ∈ N, acting on the system's Hilbert space F of states and depending on a point x in Minkowski spacetime M . Here and henceforth, all quantities in Heisenberg picture will be marked by a tilde (wave) "˜" over their kernel symbols. Let Pµ denotes the system's (canonical) momentum vectorial operator, defined via the energy-momentum tensorial operator T µν of the system, viz.
Pµ := 1 c
x 0 =const T0µ (x) d 3 x.
(2.1)
Since this operator is Hermitian, P † µ = Pµ , the operator
U (x, x 0 ) = exp 1 i µ (x µ -x µ 0 ) Pµ , (2.2)
where x 0 ∈ M is arbitrarily fixed and x ∈ M , foot_2 is unitary, i.e. U † (x 0 , x) := ( U (x, x 0 )) † = U -1 (x, x 0 ) := ( U (x, x 0 )) -1 and, via the formulae X → X (x) = U (x, x 0 )( X ) (2.3) Ã(x) → A(x) = U (x, x 0 ) • ( Ã(x)) • U -1 (x, x 0 ), (2.4) realizes the transition to the momentum picture. Here X is a state vector in system's Hilbert space of states F and Ã(x) : F → F is (observable or not) operator-valued function of x ∈ M which, in particular, can be polynomial or convergent power series in the field operators φi (x); respectively X (x) and A(x) are the corresponding quantities in momentum picture.
In particular, the field operators transform as φi (x)
→ ϕ i (x) = U (x, x 0 ) • φi (x) • U -1 (x, x 0 ). (2.5)
Notice, in (2.2) the multiplier (x µx µ 0 ) is regarded as a real parameter (in which Pµ is linear). Generally, X (x) and A(x) depend also on the point x 0 and, to be quite correct, one should write X (x, x 0 ) and A(x, x 0 ) for X (x) and A(x), respectively. However, in the most situations in the present work, this dependence is not essential or, in fact, is not presented at all. For that reason, we shall not indicate it explicitly.
The momentum picture is most suitable in quantum field theories in which the components Pµ of the momentum operator commute between themselves and satisfy the Heisenberg relations/equations with the field operators, i.e. when Pµ and φi (x) satisfy the relations:
[ Pµ , Pν ] = 0 (2.6) [ φi (x), Pµ ] = i ∂ µ φi (x).
(2.7)
Here [A, B] ± := A • B ± B • A,
• being the composition of mappings sign, is the commutator/anticommutator of operators (or matrices) A and B. However, the fulfillment of the relations (2.6) and (2.7) will not be supposed in this paper until Sect. 6 (see also Sect. 5) . Let a ± s (k) and a † ± s (k) be the creation/annihilation operators of some free particular field (see Sect. 3 below for a detailed explanation of the notation). We have the connections
ã± s (k) = e ± 1 i x µ kµ U -1 (x, x 0 ) • a ± s (k) • U (x, x 0 ) ã † ± s (k) = e ± 1 i x µ kµ U -1 (x, x 0 ) • a † ± s (k) • U (x, x 0 ) k 0 = m 2 c 2 + k 2 (2.8) whose explicit form is ã± s (k) = e ± 1 i x µ 0 kµ a ± s (k) ã † ± s (k) = e ± 1 i x µ 0 kµ a † ± s (k) k 0 = m 2 c 2 + k 2 .
(2.9) Further it will be assumed ã± s (k) and ã † ± s (k) to be defined in Heisenberg picture, independently of a ± s (k) and a † ± s (k), by means of the standard Lagrangian formalism. What concerns the operators a ± s (k) and a † ± s (k), we shall regard them as defined via (2.9); this makes them independent from the momentum picture of motion. The fact that the so-defined operators a ± s (k) and a † ± s (k) coincide with the creation/annihilation operators in momentum picture (under the conditions (2.6) and (2.7)) will be inessential in the almost whole text.
In [13] [14] [15] we have investigated the Lagrangian quantum field theory of respectively scalar, spin 1 2 and vector free fields. The main Lagrangians from which it was derived are respectively (see loc. cit. or, e.g. [1, 3, 11, 12] ):
L′ sc = L′ sc ( φ, φ † ) = - 1 1 + τ ( φ) m 2 c 4 φ(x) • φ † (x) + 1 1 + τ ( φ) c 2 2 (∂ µ φ(x)) • (∂ µ φ † (x)) (3.1a) L′ sp = L′ sp ( ψ, ψ) = - 1 2 i c{ ψ⊤ (x)C -1 γ µ • (∂ µ ψ(x)) -(∂ µ ψ⊤ (x))C -1 γ µ • ψ(x)} + mc 2 ψ⊤ (x)C -1 • ψ(x) (3.1b) L′ v = L′ v ( Ũ , Ũ † ) = m 2 c 4 1 + τ ( Ũ ) Ũ † µ • Ũ µ + c 2 2 1 + τ ( Ũ ) -(∂ µ Ũ † ν ) • (∂ µ Ũ ν ) + (∂ µ Ũ µ † ) • (∂ ν Ũ ν ) (3.1c)
Here it is used the following notation: φ(x) is a scalar field, a tilde (wave) over a symbol means that it is in Heisenberg picture, the dagger † denotes Hermitian conjugation, ψ := ( ψ0 , ψ1 , ψ2 , ψ3 ) is a 4-spinor field, ψ := C ψ⊤ := C( ψ † γ 0 ) is its charge conjugate with γ µ being the Dirac gamma matrices and the matrix C satisfies the equations C -1 γ µ C = -γ µ and C ⊤ = -C, U µ is a vector field, m is the field's mass (parameter) and the function
τ (A) := 1 for A † = A (Hermitian operator) 0 for A † = A (non-Hermitian operator) , (3.2)
with A : F → F being an operator on the systems Hilbert space F of states, takes care of is the field charged (non-Hermitian) or neutral (Hermitian, uncharged). Since a spinor field is a charged one, we have τ ( ψ) = 0; sometimes below the number 0 = τ ( ψ) will be written explicitly for unification of the notation. We have explored also the consequences from the 'charge conjugate' Lagrangians
L′′ sc = L′′ sc ( φ, φ † ) := L′ sc ( φ † , φ) (3.3a) L′′ sp = L′′ sp ( ψ, ψ) := L′ sp ( ψ, ψ) (3.3b) L′′ v = L′′ v ( Ũ , Ũ † ) := L′ v ( Ũ † , Ũ ), (3.3c)
as well as from the 'charge symmetric' Lagrangians
L′′′ sc = L′′′ sc ( φ, φ † ) := 1 2 L′ sc + L′′ sc = 1 2 L′ sc ( φ, φ † ) + L′ sc ( φ † , φ) (3.4a) L′′′ sp = L′′′ sp ( ψ, ψ) := 1 2 L′ sp + L′′ sp = 1 2 L′ sp ( ψ, ψ) + L′ sp ( ψ, ψ) (3.4b) L′′′ v = L′′′ v ( Ũ , Ũ † ) := 1 2 L′ v + L′′ v = 1 2 L′ v ( Ũ , Ũ † ) + L′ v ( Ũ † , Ũ ) . (3.4c)
It is essential to be noted, for a massless, m = 0, vector field to the Lagrangian formalism are added as subsidiary conditions the Lorenz conditions
∂ µ Ũ µ = 0 ∂ µ Ũ † µ = 0 (3.5)
on the solutions of the corresponding Euler-Lagrange equations. Besides, if the opposite is not stated explicitly, no other restrictions, like the (anti)commutation relations, are supposed to be imposed on the above Lagrangians. And a technical remark, for convenience, the fields φ, ψ and Ũ and their charge conjugate φ † , ψ and Ũ † , respectively, are considered as independent field variables.
Let L′ denotes any one of the Lagrangians (3.1) and L′′ (resp. L′′′ ) the corresponding to it Lagrangian given via (3.3) (resp. (3.4) ). Physically the difference between L′ and L′′ is that the particles for L′ are antiparticles for L′′ and vice versa. Both of the Lagrangians L′ and L′′ are not charge symmetric, i.e. the arising from them theories are not invariant under the change particle↔antiparticle (or, in mathematical terms, under some of the changes φ ↔ φ † , ψ ↔ ψ, Ũ ↔ Ũ † ) unless some additional hypotheses are made. Contrary to this, the Lagrangian L′′′ is charge symmetric and, consequently, the formalism on its base is invariant under the change particle↔antiparticle. 4 The Euler-Lagrange equations for the Lagrangians L′ , L′′ and L′′′ happen to coincide [13] [14] [15] :
5 ∂ L′ ∂χ - ∂ ∂x µ ∂ L′ ∂(∂ µ χ) ≡ ∂ L′′ ∂χ - ∂ ∂x µ ∂ L′′ ∂(∂ µ χ) ≡ ∂ L′′′ ∂χ - ∂ ∂x µ ∂ L′′′ ∂(∂ µ χ) = 0, (3.6)
where χ = φ, φ † , ψ, ψ, Ũ , Ũ † for respectively scalar, spinor and vector field. Since the creation and annihilation operators are defined only on the base of Euler-Lagrange equations [1, 3, [11] [12] [13] [14] [15] , we can assert that these operators are identical for the Lagrangians L′ , L′′ and L′′′ . We shall denote these operators by a ± s (k) and a † ± s (k) with the convention that a + s (k) (resp. a † + s (k)) creates a particle (resp. antiparticle) with 4-momentum ( m 2 c 2 + k 2 , k), polarization s (see below) and charge (-q) (resp. (+q)) 6 and a † - s (k) (resp. a - s (k)) annihilates/destroys such a particle (resp. antiparticle). Here and henceforth k ∈ R 3 is interpreted as (anti)particle's 3-momentum and the values of the polarization index s depend on the field considered: s = 1 for a scalar field, s = 1 or s = 1, 2 for respectively massless (m = 0) or massive (m = 0) spinor field, and s = 1, 2, 3 for a vector field. 7 Since massless vector field's modes with s = 3 may enter only in the spin and orbital angular momenta operators [15] , we, for convenience, shall assume that the polarization indices s, t, . . . take the values from 1 to 2j + 1δ 0m (1δ 0j ), where j = 0, 1 2 , 1 is the spin for scalar, spinor and vector field, respectively, and δ 0m := 1 for m = 0 and δ 0m := 0 for m = 0; 8 if the value s = 3 is important when j = 1 and m = 0, it will be commented/considered separately. Of course, the creation and annihilation operators are different for different fields; one should write, e.g., j a ± s (k) for a ± s (k), but we shall not use such a complicated notation and will assume the dependence on j to be an implicit one. 4 Besides, under the same assumptions, the Lagrangian L′′′ does not admit quantization via anticommutators (commutators) for integer (half-integer) spin field, while L′ and L′′ do not make difference between integer and half-integer spin fields. 5 Rigorously speaking, the Euler-Lagrange equations for the Lagrangian (3.4b) are identities like 0 = 0see [22] . However, bellow we shall handle this exceptional case as pointed in [14] . 6 For a neutral field, we put q = 0.
7 For convenience, in [14] , we have set s = 0 if m = 0 and s = 1, 2 if m = 0 for a spinor field. For a massless vector field, one may set s = 1, 2, thus eliminating the 'unphysical' value s = 3 for m = 0 -see [1, 11, 15] . In [13] , for a scalar field, the notation ϕ ± 0 (k) and ϕ † ± 0 (k) is used for a ± 1 (k) and a † ± 1 (k), respectively. 8 In this way the case (j, s, m) = (1, 3, 0) is excluded from further considerations; if (j, m) = (1, 0) and q = 0, the case considered further in this work corresponds to an electromagnetic field in Coulomb gauge, as the modes with s = 3 are excluded [15] . However, if the case (j, s, m) = (1, 3, 0) is important for some reasons, the reader can easily obtain the corresponding results by applying the ones from [15] .
The following settings will be frequently used throughout this chapter:
j := 0 for scalar field 1 2
for spinor field 1 for vector field τ := 1 for q = 0 (neutral (Hermitian) field) 0 for q = 0 (charged (non-Hermitian) field)
ε := (-1) 2j = +1 for integer j (bose fields) -1 for half-integer j (fermi fields)
(3.7) [A, B] ± := [A, B] ±1 := A • B ± B • A, (3.8)
where A and B are operators on the system's Hilbert space F of states.
The dynamical variables corresponding to L′ , L′′ and L′′′ are, however, completely different, unless some additional conditions are imposed on the Lagrangian formalism [13] [14] [15] . In particular, the momentum operators Pω µ , charge operators Qω , spin operators Sω µν and orbital operators Lω µν , where ω = ′, ′′, ′′′, for these Lagrangians are [13] [14] [15] :
P′ µ = 1 1 + τ 2j+1-δ 0m (1-δ 0j ) s=1 d 3 kk µ | k 0 = √ m 2 c 2 +k 2 {a † + s (k) • a - s (k) + εa † - s (k) • a + s (k)} (3.9a) P′′ µ = 1 1 + τ 2j+1-δ 0m (1-δ 0j ) s=1 d 3 kk µ | k 0 = √ m 2 c 2 +k 2 {a + s (k) • a † - s (k) + εa - s (k) • a † + s (k)} (3.9b) P′′′ µ = 1 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 kk µ | k 0 = √ m 2 c 2 +k 2 {[a † + s (k), a - s (k)] ε + [a + s (k), a † - s (k)] ε } (3.9c) Q′ = +q 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k{a † + s (k) • a - s (k) -εa † - s (k) • a + s (k)} (3.10a) Q′′ = -q 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k{a + s (k) • a † - s (k) -εa - s (k) • a † + s (k)} (3.10b) Q′′′ = 1 2 q 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k{[a † + s (k), a - s (k)] ε -[a + s (k), a † - s (k)] ε } (3.10c) S′ µν = (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k σ ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) + σ ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) (3.11a) S′′ µν = ε (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k σ ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + σ ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) (3.11b) S′′′ µν = (-1) j-1/2 j 2(1 + τ ) 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k σ ss ′ ,- µν (k)[a † + s (k), a - s ′ (k)] ε + σ ss ′ ,+ µν (k)[a † - s (k), a + s ′ (k)] ε (3.11c) L′ µν =x 0 µ P′ ν -x 0 ν P′ µ + (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k l ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) + l ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) + i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) -εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a + s (k) k 0 = √ m 2 c 2 +k 2 (3.12a) L′′ µν =x 0 µ P′′ ν -x 0 ν P′′ µ + ε (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k l ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + l ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) + i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † - s (k) -εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † + s (k) k 0 = √ m 2 c 2 +k 2 (3.12b) L′′′ µν =x 0 µ P′′′ ν -x 0 ν P′′′ µ + (-1) j-1/2 j 2(1 + τ ) 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k l ss ′ ,- µν (k)[a † + s (k), a - s ′ (k)] ε + l ss ′ ,+ µν (k)[a † - s (k), a + s ′ (k)] ε + i 4(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) -εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † + s (k) + a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † - s (k) -εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a + s (k) k 0 = √ m 2 c 2 +k 2 .
(3.12c)
Here we have used the following notation: (-1) n+1/2 := (-1) n i for all n ∈ N and i := + √ -1,
A(k) ← ---- - → k µ ∂ ∂k ν • B(k) := -k µ ∂A(k) ∂k ν • B(k) + A(k) • k µ ∂B(k) ∂k ν = k µ A(k) ← -- -→ ∂ ∂k ν • B(k) (3.13)
for operators A(k) and B(k) having C 1 dependence on k, 9 and σ ss ′ ,± µν (k) and l ss ′ ,± µν (k) are 9 More generally, if ω : {F → F} → {F → F} is a mapping on the operator space over the system's Hilbert space, we put A
← - -→ ω • B := -ω(A) • B + A • ω(B)
for any A, B : F → F. Usually [2, 12] , this notation is used for ω = ∂µ. some functions of k such that 10
σ ss ′ ,± µν (k) = -σ ss ′ ,± νµ (k) l ss ′ ,± µν (k) = -l ss ′ ,± νµ (k) σ ss ′ ,± µν (k) = l ss ′ ,± νµ (k) = 0 for j = 0 (scalar field) σ ss ′ ,- µν (k) = -σ ss ′ ,+ µν (k) =: σ ss ′ µν (k) = -σ s ′ s µν (k) = -σ ss ′ νµ (k) for j = 1 (vector field) l ss ′ ,- µν (k) = -l ss ′ ,+ µν (k) =: l ss ′ µν (k) = -l s ′ s µν (k) = -l ss ′ νµ (k)
for j = 1 (vector field).
(3.14)
A technical remark must be make at this point. The equations (3.9)-(3.12) were derived in [13] [14] [15] under some additional conditions, represented by equations (2.6) and (2.7), which are considered bellow in Sect. 5 and ensure the effectiveness of the momentum picture of motion [21] used in [13] [14] [15] . However, as it is partially proved, e.g., in [1] , when the quantities (3.9)-(3.12) are expressed via the Heisenberg creation and annihilation operators (see (2.9)), they remain valid, up to a phase factor, and without making the mentioned assumptions, i.e. these assumptions are needless when one works entirely in Heisenberg picture. For this reason, we shall consider (3.9)-(3.12) as pure consequence of the Lagrangian formalism.
We should emphasize, in (3.11 ) and (3.12) with Sω µν and Lω µν , ω = ′, ′′, ′′′, are denoted the spin and orbital, respectively, operators for Lω , which are the spacetime-independent parts of the spin and orbital, respectively, angular momentum operators [14, 23] ; if the last operators are denoted by Sω µν and Lω µν , the total angular momentum operator of a system with Lagrangian Lω is [23]
Mω µν = Lω µν + Sω µν = Lω µν + Sω µν , ω = ′, ′′, ′′′ (3.15)
and Sω µν = Sω µν (and hence Lω µν = Lω µν ) iff Sω µν is a conserved operator or, equivalently, iff the system's canonical energy-momentum tensor is symmetric. 11 Going ahead (see Sect. 6), we would like to note that the expressions (3.9c) and, consequently, the Lagrangian L′′′ are the base from which the paracommutation relations were first derived [16] .
And a last remark. Above we have expressed the dynamical variables in Heisenberg picture via the creation and annihilation operators in momentum picture. If one works entirely in Heisenberg picture, the operators (2.9), representing the creation and annihilation operators in Heisenberg picture, should be used. Besides, by virtue of the equations
(a ± s (k)) † = a † ∓ s (k) (a † ± s (k)) † = a ∓ s (k) (3.16) ã± s (k) † = ã † ∓ s (k) ã † ± s (k) † = ã∓ s (k), (3.17)
some of the relations concerning a † ± s (k), e.g. the Euler-Lagrange and Heisenberg equations, are consequences of the similar ones regarding a ± s (k). In view of (2.9), we shall consider (3.9)-(3.12) as obtained form the corresponding expressions in Heisenberg picture by making the replacements ã± s (k) → a ± s (k) and ã † ± s (k) → a † ± s (k). So, (3.9)-(3.12) will have, up to a phase factor, a sense of dynamical variables in Heisenberg picture expressed via the creation/annihilation operators in momentum picture. 10 For the explicit form of these functions, see [13] [14] [15] ; see also equation (6.57) below. 11 In [14, 23] the spin and orbital operators are labeled with an additional left superscript •, which, for brevity, is omitted in the present work as in it only these operators, not Sω µν and Lω µν , will be considered. Notice, the operators Sω µν and Lω µν are, generally, time-dependent while the orbital and spin ones are conserved, as a result of which the total angular momentum is a conserved operator too [14, 23] .
Let D = P µ , Q, S µν , L µν denotes some dynamical variable, viz. the momentum, charge, spin, or orbital operator, of a system with Lagrangian L. Since the Euler-Lagrange equations for the Lagrangians L ′ , L ′′ and L ′′′ coincide (see (3.6 )), we can assert that any field satisfying these equations admits at least three classes of conserved operators, viz. D ′ , D ′′ and D ′′′ = 1 2 D ′ + D ′′ . Moreover, it can be proved that the Euler-Lagrange equations for the Lagrangian
L α,β := α L ′ + β L ′′ α + β = 0 (4.1)
do not depend on α, β ∈ C and coincide with (3.6) . Therefore there exists a two parameter family of conserved dynamical variables for these equations given via
D α,β := α D ′ + β D ′′ α + β = 0. ( 4
.2) Evidently L ′′′ = L 1 2 , 1 2 and D ′′′ = D 1 2 , 1 2 . Since the Euler-Lagrange equations (3.6) are linear and homogeneous (in the cases considered), we can, without a lost of generality, restrict the parameters α, β ∈ C to such that α + β = 1, (4.3) which can be achieved by an appropriate renormalization (by a factor (α+β) -1/2 ) of the field operators. Thus any field satisfying the Euler-Lagrange equations (3.6) admits the family D α,β , α + β = 1, of conserved operators. Obviously, this conclusion is valid if in (4.1) we replace the particular Lagrangians L ′ and L ′′ (see (3.1) and (3.3)) with any two Lagrangians (of one and the same field variables) which lead to identical Euler-Lagrange equations. However, the essential point in our case is that L ′ and L ′′ do not differ only by a full divergence, as a result of which the operators D α,β are different for different pairs (α, β), α + β = 1. foot_3
Since one expects a physical system to possess uniquely defined dynamical characteristics, e.g. energy and total angular momentum, and the Euler-Lagrange equations are considered (in the framework of Lagrangian formalism) as the ones governing the spacetime evolution of the system considered, the problem arises when the dynamical operators D α,β , α+β = 1, are independent of the particular choice of α and β, i.e. of the initial Lagrangian one starts off. Simple calculation show that the operators (4.2), under the condition (4.3), are independent of the particular values of the parameters α and β if and only if
D ′ = D ′′ . (4.4)
Some consequences of the condition(s) (4.4) will be considered below, as well as possible ways for satisfying these restrictions on the Lagrangian formalism. Combining (3.9)-(3.12) with (4.4), for respectively D = P µ , Q, S µν , L µν , we see that a free scalar, spinor or vector field has a uniquely defined dynamical variables if and only if the following equations are fulfilled:
2j+1-δ 0m (1-δ 0j ) s=1 d 3 k k µ k 0 = √ m 2 c 2 +k 2 a † + s (k) • a - s (k) -εa - s (k) • a † + s (k) -a + s (k) • a † - s (k) + εa † - s (k) • a + s (k) = 0 (4.5) q × 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) • a - s (k) -εa - s (k) • a † + s (k) + a + s (k) • a † - s (k) -εa † - s (k) • a + s (k) = 0 (4.6) 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k σ ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) -εσ ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) -εσ ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + σ ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) = 0 (4.7) 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k l ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) -εl ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) -εl ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + l ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) + 1 2 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a - s (k)+εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a † + s (k) -a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a † - s (k)-εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a + s (k) k 0 = √ m 2 c 2 +k 2 = 0. (4.8)
In (4.6) is retained the constant factor q as in the neutral case it is equal to zero and, consequently, the equation (4.6) reduces to identity.
Since the Euler-Lagrange equations do not impose some restrictions on the creation and annihilation operators, the equations (4.5)-(4.8) can be regarded as subsidiary conditions on the Lagrangian formalism and can serve as equations for (partial) determination of the creation and annihilation operators. The system of integral equations (4.5)-(4.8) is quite complicated and we are not going to investigate it in the general case. Below we shall restrict ourselves to analysis of only those solutions of (4.5)-(4.8), if any, for which the integrands in (4.5)-(4.8) vanish. This means that we shall replace the system of integral equations (4.5)-(4.8) with respect to creation and annihilation operators with the following system of algebraic equations (do not sum over s and s ′ in (4.12) and (4.13)!):
a † + s (k) • a - s (k) -εa - s (k) • a † + s (k) -a + s (k) • a † - s (k) + εa † - s (k) • a + s (k) = 0 (4.9) a † + s (k) • a - s (k) -εa - s (k) • a † + s (k) + a + s (k) • a † - s (k) -εa † - s (k) • a + s (k) = 0 if q = 0 (4.10) a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) + εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † + s (k) -a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a † - s (k)-εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a + s (k) k 0 = √ m 2 c 2 +k 2 = 0 (4.11) s,s ′ σ ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) -εσ ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) -εσ ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + σ ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) = 0 (4.12) s,s ′ l ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) -εl ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) -εl ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + l ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) = 0 (4.
Here: s = 1, . . . , 2j + 1δ 0m (1δ 0j ) in (4.9)-(4.11) and s, s ′ = 1, . . . , 2j + 1δ 0m (1δ 1j ) in (4.12) and (4.13). (Notice, by virtue of (3.14), the equations (4.12) and (4.13) are identically valid for j = 0, i.e. for scalar fields.) Since all polarization indices enter in (4.5) and (4.6) on equal footing, we do not sum over s in (4.9)- (4.11) . But in (4.12) and (4.13) we have retain the summation sign as the modes with definite polarization cannot be singled out in the general case. One may obtain weaker versions of (4.9)-(4.13) by summing in them over the polarization indices, but we shall not consider these conditions below regardless of the fact that they also ensure uniqueness of the dynamical variables. At first, consider the equations (4.9)- (4.11) . Since for a neutral field, q = 0, we have a † ± s (k) = a ± s (k), which physically means coincidence of field's particles and antiparticles, the equations (4.9)-(4.11) hold identically in this case.
Let consider now the case q = 0, i.e. the investigated field to be charged one. Using the standard notation (cf. (3.8))
[A, B] η := A • B + ηB • A, (4.14)
for operators A and B and η ∈ C, we rewrite (4.9) and (4.10) as
[a † + s (k), a - s (k)] -ε -[a + s (k), a † - s (k)] -ε = 0 (4.9 ′ ) [a † + s (k), a - s (k)] -ε + [a + s (k), a † - s (k)] -ε = 0 if q = 0, (4.10 ′ ) which are equivalent to [a † ± s (k), a ∓ s (k)] -ε = 0 if q = 0. ( 4
.15) Differentiating (4.15) and inserting the result into (4.11), one can verify that (4.11) is tantamount to
a † + s (k), k µ ∂ ∂k ν -k ν ∂ ∂k µ • a - s (k) -ε -a + s (k), k µ ∂ ∂k ν -k ν ∂ ∂k µ • a † - s (k) -ε k 0 = √ m 2 c 2 +k 2 = 0 if q = 0, (4.16)
Consider now (4.12) and (4.13) . By means of the shorthand (4.14), they read
s,s ′ σ ss ′ ,- µν (k)[a † + s (k), a - s ′ (k)] -ε + σ ss ′ ,+ µν (k)[a † - s (k), a + s ′ (k)] -ε = 0 (4.17) s,s ′ l ss ′ ,- µν (k)[a † + s (k), a - s ′ (k)] -ε + l ss ′ ,+ µν (k)[a † - s (k), a + s ′ (k)] -ε = 0. (4.18)
For a scalar field, j = 0, these conditions hold identically, due to (3.14) . But for j = 0 they impose new restrictions on the formalism. In particular, for vector fields, j = 1 and ε = +1 they are satisfied iff (see (3.14) )
[a † + s (k), a - s ′ (k)] -ε -[a † - s (k), a + s ′ (k)] -ε -[a † + s ′ (k), a - s (k)] -ε + [a † - s ′ (k), a + s (k)] -ε = 0. (4.19)
One can satisfy (4.17) and (4.18) if the following generalization of (4.15) holds
[a † ± s (k), a ∓ s ′ (k)] -ε = 0. ( 4.20)
For spin j = 1 2 (and hence ε = -1 -see (3.7)), the conditions (4.12) and (4.13) cannot be simplified much, but, if one requires the vanishment of the operator coefficients after σ ss ′ ,± µν (k) and l ss ′ ,± µν (k), one gets
a † ± s (k) • a ∓ s ′ (k) = 0 j = 1 2 ε = -1. (4.21)
Excluding some special cases, e.g. neutral scalar field (q = 0 and j = 0), the equations (4.15) and (4.21) are unacceptable from many viewpoints. The main of them is that they are incompatible with the ordinary (anti)commutation relations (see, e.g., e.g. [1, 11, 12, 18] or Sect. 6, in particular, equations (6.13) bellow); for example, (4.21) means that the acts of creation and annihilation of (anti)particles with identical characteristics should be mutually independent, which contradicts to the existing theory and experimental data. Now we shall try another way for achieving uniqueness of the dynamical variables for free fields. Since in (4.9)-(4.13) naturally appear (anti)commutators between creation and annihilation operators and these (anti)commutators vanish under the standard normal ordering [1, 11, 12, 18] , one may suppose that the normally ordered expressions of the dynamical variables may coincide. Let us analyze this method.
Recall [1, 3, 11, 12] , the normal ordering operator N (for free field theory) is a linear operator on the operator space of the system considered such that to a product (composition)
c 1 • • • • • c n of n ∈ N creation
and/or annihilation operators c 1 , . . . c n it assigns the operator (-1) f c α 1 • • • • c αn . Here (α 1 , . . . , α n ) is a permutation of (1, . . . , n), all creation operators stand to the left of all annihilation ones, the relative order between the creation/annihilation operators is preserved, and f is equal to the number of transpositions among the fermion operators (j = 1 2 ) needed to be achieved the just-described order ("normal order") of the operators 13 In particular this means that
c 1 • • • • • c n in c α 1 • • • • c αn .
N a + s (k) • a † - t (p) = a + s (k) • a † - t (p) N a † + s (k) • a - t (p) = a † + s (k) • a - t (p) N a - s (k) • a † + t (p) = εa † + t (p) • a - s (k) N a † - s (k) • a + t (p) = εa + t (p) • a † - s (k) (4.22)
and, consequently, we have
N [a † ± s (k), a ∓ t (p)] -ε = 0 N [a ± s (k), a † ∓ t (p)] -ε = 0, (4.23)
due to ε := (-1) 2j = ±1 (see (3.7) ). (In fact, below only the equalities (4.22) and (4.23), not the general definition of a normal product, will be applied.) Applying the normal ordering operator to (4.9 ′ ), (4.10 ′ ), (4.17) and (4.18), we, in view of (4.23), get the identity 0 = 0, which means that the conditions (4.9), (4.10), (4.12) and (4.13) are identically satisfied after normal ordering. This is confirmed by the application of N to (3.9) and (3.10), which results respectively in (see (4.22) )
N ( P′ µ ) = N ( P′′ µ ) = 1 1 + τ 2j+1-δ 0m (1-δ 0j ) s=1 d 3 kk µ | k 0 = √ m 2 c 2 +k 2 {a † + s (k) • a - s (k) + a + s (k) • a † - s (k)} (4.24)
13 We have slightly modified the definition given in [1, 3, 11, 12] because no (anti)commutation relations are presented in our exposition till the moment. In this paper we do not concern the problem for elimination of the 'unphysical' operators a ± 3 (k) and a † ± 3 (k) from the spin and orbital momentum operators when j = 1; for details, see [15] , where it is proved that, for an electromagnetic field, j = 1 and q = 0, one way to achieve this is by adding to the number f above the number of transpositions between a ± s (k), s = 1, 2, and a ± 3 (k) needed for getting normal order.
N ( Q′ ) = N ( Q′′ ) = 1 1 + τ 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k{a † + s (k) • a - s (k) -a + s (k) • a † - s (k)}.
N a + s (k) ← -- -→ ω µν • a † - s (k) = a + s (k) ← -- -→ ω µν • a † - s (k) N a † + s (k) ← -- -→ ω µν • a - s (k) = a † + s (k) ← -- -→ ω µν • a - s (k) N a - s (k) ← -- -→ ω µν • a † + s (k) = -εa † + s (k) ← -- -→ ω µν • a - s (k) N a † - s (k) ← -- -→ ω µν • a + s (k) = -εa + s (k) ← -- -→ ω µν • a † - s (k). (4.27)
As a consequence of these equalities, the action of N on the l.h.s. of (4.11) vanishes. Combining this result with the mentioned fact that the normal ordering converts (4.12) and (4.13) into identities, we see that the normal ordering procedure ensures also uniqueness of the spin and orbital operators if we redefine them respectively as:
Sµν := N ( S′ µν ) := N ( S′′ µν ) = (-1) j-1/2 j 1 + τ × 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k σ ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) + εσ ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) (4.28) Lµν := N ( L′ µν ) := N ( L′′ µν ) = x 0 µ Pν -x 0 ν Pµ + (-1) j-1/2 j 1 + τ × 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k l ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) + εl ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) + a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † - s (k) k 0 = √ m 2 c 2 +k 2 , (4.29)
where (3.14) was applied.
The conserved operators, like momentum and charge operators, are often identified with the generators of the corresponding transformations under which the action operator is invariant [1, 3, 11, 12] . This leads to a number of commutation relations between the components of these operators and between them and the field operators. The relations of the letter set are known/referred as the Heisenberg relations or equations. Both kinds of commutation relations are from pure geometric origin and, consequently, are completely external to the Lagrangian formalism; one of the reasons being that the mentioned identification is, in general, unacceptable and may be carried out only on some subset of the system's Hilbert space of states [23, 24] . Therefore their validity in a pure Lagrangian theory is questionable and should be verified [11] . However, the considered relations are weaker conditions than the identification of the corresponding operators and there are strong evidences that these relations should be valid in a realistic quantum field theory [1, 11] ; e.g., the commutativity between the momentum and charge operators (see below (5.18 )) expresses the experimental fact that the 4-momentum and charge of any system are simultaneously measurable quantities.
It is known [1, 11] , in a pure Lagrangian approach, the field equations, which are usually identified with the Euler-Lagrange, 14 are the only restrictions on the field operators. Besides, these equations do not determine uniquely the field operators and the letter can be expressed through the creation and annihilation operators. Since the last operators are left completely arbitrary by a pure Lagrangian formalism, one is free to impose on them any system of compatible restrictions. The best known examples of this kind are the famous canonical (anti)commutation relations and their generalization, the so-called paracommutation relations [16, 18] . In general, the problem for compatibility of such subsidiary to the Lagrangian formalism system of restrictions with, for instance, the Heisenberg relations is open and requires particular investigation [11] . For example, even the canonical (anti)commutation relations for electromagnetic field in Coulomb gauge are incompatible with the Heisenberg equation involving the (total) angular momentum operator unless the gauge symmetry of this field is taken into account [11, § 84] . However, the (para)commutation relations are, by construction, compatible with the Heisenberg relations regarding momentum operator (see [16] or below Subsect. 6.1). The ordinary approach is to be imposed a system of equations on the creation and annihilation operators and, then, to be checked its compatibility with, e.g., the Heisenberg relations. In the next sections we shall investigate the opposite situation: assuming the validity of (some of) the Heisenberg equations, the possible restrictions on the creation and annihilation operators will be explored. For this purpose, below we briefly review the Heisenberg relations and other ones related to them.
Consider a system of quantum fields φi (x), i = 1, . . . , N ∈ N, where φi (x) denote the components of all fields (and their Hermitian conjugates), and Pµ , Q and Mµν be its momentum, charge and (total) angular momentum operators, respectively. The Heisenberg relations/equations for these operators are [1, 3, 11, 12] [ φi (
x), Pµ ] = i ∂ φi (x) ∂x µ (5.1) [ φi (x), Q] = e( φi )q φi (x) (5.2) [ φi (x), Mµν ] = i {x µ ∂ ν φi (x) -x ν ∂ µ φi (x)} + i i ′ I j i ′ µν φi ′ (x). (5.3)
Here:
q = const is the fields' charge, e( φi ) = 0 if φ † i = φi , e( φi ) = ±1 if φ † i = φi with e( φi ) + e( φ † i ) = 0,
I i ′ iµν = -I i ′ iνµ characterize the transformation properties of the field operators under 4-rotations. (If ε( φi ) = 0, it is a convention whether to put ε( φi ) = +1 or ε( φi ) = -1 for a fixed i.)
We would like to make some comments on (5.3). Since its r.h.s. is a sum of two operators, the first (second) characterizing the pure orbital (spin) angular momentum properties of the system considered, the idea arises to split (5.3) into two independent equations, one involving the orbital angular momentum operator and another concerning the spin angular momentum operator. This is supported by the observation that, it seems, no process is known for transforming orbital angular momentum into spin one and v.v. (without destroying the system). So one may suppose the existence of operators Mor µν and Msp µν such that
[ φi (x), Mor µν ] = i {x µ ∂ ν φi (x) -x ν ∂ µ φi (x)} (5.4) [ φi (x), Msp µν ] = i i ′ I i ′ iµν φi ′ (x) (5.5) Mµν = Mor µν + Msp µν .
(5.6) However, as particular calculations demonstrate [5, 14, 15] , neither the spin (resp. orbital) nor the spin (resp. orbital) angular momentum operator is a suitable candidate for Msp µν (resp. Mor µν ). If we assume the validity of (5.1), then equations (5.4) and (5.5) can be satisfied if we choose
Mor µν (x) = Lext µν := x µ Pν -x ν Pµ (5.7) Msp µν (x) = M(0) µν (x) := Mµν -Lext µν = Sµν + Lµν -{x µ Pν -x ν Pµ } (5.8)
with Mµν satisfying (5.3). These operators are not conserved ones. Such a representation is in agreement with the equations (3.12), according to which the operator (5.7) enters additively in the expressions for the orbital operator. 15 The physical sense of the operator (5.7) is that it represents the orbital angular momentum of the system due to its movement as a whole. Respectively, the operator (5.8) describes the system's angular momentum as a result of its internal movement and/or structure. Since the spin (orbital) angular momentum is associated with the structure (movement) of a system, in the operator (5.8) are mixed the spin and orbital angular momenta. These quantities can be separated completely via the following representations of the operators M or µν and M sp µν in momentum picture (when (5.1) holds)
M or µν = x µ P ν -x µ P µ + L int µν (5.9) M sp µν = M µν -(x µ P ν -x µ P µ ) -L int µν , (5.10)
where L int µν describes the 'internal' orbital angular momentum of the system considered and depends on the Lagrangian we have started off. Generally said, L int µν is the part of the orbital angular momentum operator containing derivatives of the creation and annihilation operators. In particular, for the Lagrangians L ′ , L ′′ and L ′′′ (see Sect. 3), the explicit forms of the operators (5.9) and (5.10) respectively are:
M ′ or µν =x µ P ′ ν -x ν P ′ µ + i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) -εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a + s (k) k 0 = √ m 2 c 2 +k 2
(5.11a)
M ′′ or µν =x µ P ′′ ν -x ν P ′′ µ + i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † - s (k) -εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † + s (k) k 0 = √ m 2 c 2 +k 2
(5.11b) 15 This is evident in the momentum picture of motion, in which xµ stands for x0 µ in (3.12) -see [13] [14] [15] .
M ′′′ or µν =x µ P ′′′ ν -x ν P ′′′ µ + i 4(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) -εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † + s (k) + a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † - s (k) -εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a + s (k) k 0 = √ m 2 c 2 +k 2 .
(5.11c)
M ′ sp µν = (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k))a † + s (k) • a - s ′ (k) + (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))a † - s (k) • a + s ′ (k) (5.12a) M ′′ sp µν = ε (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))a + s ′ (k) • a † - s (k) + (σ ss ′ ,- µν (k) + σ ss ′ ,- µν (k))a - s ′ (k) • a † + s (k) (5.12b) M ′′′ sp µν = (-1) j-1/2 j 2(1 + τ ) 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k))[a † + s (k), a - s ′ (k)] ε + (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))[a † - s (k), a + s ′ (k)] ε . (5.12c)
Obviously (see Sect. 2), the equations (5.12) have the same form in Heisenberg picture in terms of the operators (2.9) (only tildes over M and a must be added), but the equations (5.11) change substantially due to the existence of derivatives of the creation and annihilation operators in them [13] [14] [15] :
M′ or µν = i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k ã † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã- s (k) -εã † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã+ s (k) k 0 = √ m 2 c 2 +k 2 (5.13a) M′′ or µν = i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k ã+ s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã † - s (k) -εã - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã † + s (k) k 0 = √ m 2 c 2 +k 2 (5.13b) M′′′ or µν = i 4(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k ã † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã- s (k) -εã - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã † + s (k) + ã+ s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã † - s (k) -εã † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã+ s (k) k 0 = √ m 2 c 2 +k 2 .
(5.13c) From (5.13) and (5.12) is clear that the operators Mor µν and Msp µν so defined are conserved (contrary to (5.7) and (5.8)) and do not depend on the validity of the Heisenberg relations (5.1) (contrary to expressions (5.11) in momentum picture).
The problem for whether the operators (5.12) and (5.13) satisfy the equations (5.4) and (5.5), respectively, will be considered in Sect. 6.
There is an essential difference between (5.4) and (5.5): the equation (5.5) depends on the particular properties of the operators φi (x) under 4-rotations via the coefficients I i ′ iµν (see (5.25) below), while (5.4) does not depend on them. This is explicitly reflected in (5.11) and (5.12): the former set of equations is valid independently of the geometrical nature of the fields considered, while the latter one depends on it via the 'spin' ('polarization') functions σ ss ′ ,± µν (k) and l ss ′ ,± µν (k). Similar remark concerns (5.3), on one hand, and (5.1) and (5.2), on another hand: the particular form of (5.3) essentially depends on the geometric properties of φi (x) under 4-rotations, the other equations being independent of them.
It should also be noted, the relation (5.3) does not hold for a canonically quantized electromagnetic field in Coulomb gauge unless some additional terms it its r.h.s., reflecting the gauge symmetry of the field, are taken into account [11, § 84] .
As it was said above, the relations (5.1)-(5.3) are from pure geometrical origin. However, the last discussion, concerning (5.4)-(5.8), reveals that the terms in braces in (5.3) should be connected with the momentum operator in the (pure) Lagrangian approach. More precisely, on the background of equations (3.11a)-(3.12c), the Heisenberg relation (5.3) should be replaced with
[ φi (x), Mµν ] = x µ [ φi (x), Pν ] -x ν [ φi (x), Pµ ] + i j I i ′ iµν φi ′ (x), (5.14)
which is equivalent to (5.3) if (5.1) is true. An advantage of the last equation is that it is valid in any picture of motion (in the same form) while (5.3) holds only in Heisenberg picture. 16 Obviously, (5.14) is equivalent to (5.5) with Msp µν defined by (5.8). The other kind of geometric relations mentioned at the beginning of this section are connected with the basic relations defining the Lie algebra of the Poincaré group [7, pp. 143-147] , [8, sect. 7.1] . They require the fulfillment of the following equations between the components Pµ of the momentum and Mµν of the angular momentum operators [3, 5, 7, 8] :
[ Pµ , Pν ] = 0 (5.15) [ Mµν , Pλ ] = -i (η λµ Pν -η λν Pµ ). (5.16) [ Mκλ , Mµν ] = -i η κµ Mλν -η λµ Mκν -η κν Mλµ + η λν Mκµ .
(5.17)
We would like to pay attention to the minus sign in the multiplier (-i ) in (5.16 ) and (5.17) with respect to the above references, where i stands instead of -i in these equations. When (a representation of) the Lie algebra of the Poincaré group is considered, this difference in the sign is insignificant as it can be absorbed into the definition of Mµν . However, the change of the sign of the angular momentum operator, Mµν → -Mµν , will result in the change i → -i in the r.h.s. of (5.3) . This means that equations (5.15), (5.16 ) and (5.3), when considered together, require a suitable choice of the signs of the multiplier i in their right hand sides as these signs change simultaneously when Mµν is replaced with -Mµν . Since equations (5.3), (5.16 ) and (5.17) hold, when Mµν is defined according to the Noether's theorem and the ordinary (anti)commutation relations are valid [13] [14] [15] , we accept these equations in the way they are written above.
To the relations (5.15)-(5.17) should be added the equations [3, p. 78 ]
[ Q, Pµ ] = 0 (5.18) [ Q, Mµν ] = 0, (5.19)
which complete the algebra of observables and express, respectively, the translational and rotational invariance of the charge operator Q; physically they mean that the charge and momentum or the charge and angular momentum are simultaneously measurable quantities.
Since the spin properties of a system are generally independent of its charge or momentum, one may also expect the validity of the relations foot_6
[ Sµν , Pµ ] = 0 (5.20) [ Sµν , Q] = 0. (5.21)
But, as the spin describes, in a sense, some of the rotational properties of the system, equality like [ Sµν , Lκλ ] = 0 is not likely to hold. Indeed, the considerations in [13] [14] [15] reveal that (5.20 (ii) Equation (5.2) implies (5.18), (5.19), (5.21), and (5.23). (iii) Equation (5.3) implies (5.16), (5.17), and (5.19). Besides, (5.3) may, possibly, entail equations like (5.17) with S or L for M , with an exception of Mµν in the l.h.s., i.e.
[ Sκλ , Mµν ] = -i η κµ Sλν -η λµ Sκν -η κν Sλµ + η λν Sκµ [ Lκλ , Mµν ] = -i η κµ Lλν -η λµ Lκν -η κν Lλµ + η λν Lκµ (5.24)
The validity of assertions (i)-(iii) above for free scalar, spinor and vector fields, when respectively φi (x) → φ(x), φ † (x)
I i ′ iµν → I µν = 0 e( φ) = -e( φ † ) = +1 (5.25a) φi (x) → ψ(x), ψ(x) I i ′ iµν → I ψµν = I ψµν = - i 2 σ µν e( ψ) = -e( ψ) = +1 (5.25b) φi (x) → Ũµ (x), Ũ † µ (x) I i ′ iµν → I σ ρµν = I † σ ρµν = δ σ µ η νρ -δ σ ν η µρ e( Ũµ ) = -e( Ũ † µ ) = +1, (5.25c)
where σ µν := i 2 [γ µ , γ ν ] with γ µ being the Dirac γ-matrices [1, 25] , is proved in [13] [14] [15] , respectively. Besides, in loc. cit. is proved that equations (5.24) hold for scalar and vector fields, but not for a spinor field. 18 Thus, we see that the Heisenberg relations (5.1)-( 5 .3) are stronger than the commutation relations (5.15)-(5.23), when imposed on the Lagrangian formalism as subsidiary restrictions.
In a broad sense, by a commutation relation we shall understand any algebraic relation between the creation and annihilation operators imposed as subsidiary restriction on the Lagrangian formalism. In a narrow sense, the commutation relations are the equations (6.13), with ε = -1, written below and satisfied by the bose creation and annihilation operators. As anticommutation relations are known the equations (6.13), with ε = +1, written below and satisfied by the fermi creation and annihilation operators. The last two types of relations are often referred as the bilinear commutation relations [18] . Theoretically are possible also trilinear commutation relations, an example being the paracommutation relations [16, 18] represented below by equations (6.18) (or (6.20)).
Generally said, the commutation relations should be postulated. Alternatively, they could be derived from (equivalent to them) different assumptions added to the Lagrangian formalism. The purpose of this section is to be explored possible classes of commutation relations, which follow from some natural restrictions on the Lagrangian formalism that are consequences from the considerations in the previous sections. Special attention will be paid on some consequences of the charge symmetric Lagrangians as the free fields possess such a symmetry [1, 3, 11, 12] .
As pointed in Sect 3, the Euler-Lagrange equations for the Lagrangians L′ , L′′ and L′′′ coincide and, in quantum field theory, the role of these equations is to be singled out the independent degrees of freedom of the fields in the form of creation and annihilation operators a ± s (k) and a † ± s (k) (which are identical for L′ , L′′ and L′′′ ). Further specialization of these operators is provided by the commutation relations (in broad sense) which play a role of field equations in this situation (with respect to the mentioned operators).
Before proceeding on, we would like to simplify our notation. As a spin variable, s say, is always coupled with a 3-momentum one, k say, we shall use the letters l, m and n to denote pairs like l = (s, k), m = (t, p) and n = (r, q). Equipped with this convention, we shall write, e.g., a ± l for a ± s (k) and a † ± l for a † ± s (k). We set δ lm := δ st δ 3 (kp) and a summation sign like l should be understood as s d 3 k, where the range of the polarization variable s will be clear from the context (see, e.g., (3.9)-(3.12)).
First of all, let us examine the consequences of the Heisenberg relation (5.1) involving the momentum operator. Since in terms of creation and annihilation operators it reads [1, [13] [14] [15] ]
[a ± s (k), P µ ] = ∓k µ a ± s (k) [a † ± s (k), P µ ] = ∓k µ a † ± s (k) k 0 = m 2 c 2 + k 2 , (6.1)
the field equations in terms of creation and annihilation operators for the Lagrangians (3.1), (3.3) and (3.4) respectively are (see [13] [14] [15] or (6.1) and (3.9)):
2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a ± s (k), a † + t (q) • a - t (q) + εa † - t (q) • a + t (q) - ± (1 + τ )a ± s (k)δ st δ 3 (k -q) d 3 q = 0 (6.2a) 2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a † ± s (k), a † + t (q) • a - t (q) + εa † - t (q) • a + t (q) - ± (1 + τ )a † ± s (k)δ st δ 3 (k -q) d 3 q = 0 (6.2b) 2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a ± s (k), a + t (q) • a † - t (q) + εa - t (q) • a † + t (q) - ± (1 + τ )a ± s (k)δ st δ 3 (k -q) d 3 q = 0 (6.3a) 2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a † ± s (k), a + t (q) • a † - t (q) + εa - t (q) • a † + t (q) - ± (1 + τ )a † ± s (k)δ st δ 3 (k -q) d 3 q = 0 (6.3b) 2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a ± s (k), [a † + t (q), a - t (q)] ε + [a + t (q), a † - t (q)] ε - ± (1 + τ )a ± s (k)δ st δ 3 (k -q) d 3 q = 0 (6.4a) 2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a † ± s (k), [a † + t (q), a - t (q)] ε + [a + t (q), a † - t (q)] ε - ± (1 + τ )a † ± s (k)δ st δ 3 (k -q) d 3 q = 0, (6.4b)
where j and ε are given via (3.7), the generalized commutation function [•, •] ε is defined by (4.14), and the polarization indices take the values
s, t = 1, . . . , 2j + 1 -δ 0m (1 -δ 0j ) =
1 for j = 0 or for j = 1 2 and m = 0 1, 2 for j = 1 2 and m = 0 or for j = 1 and m = 0 1, 2, 3 for j = 1 and m = 0 . (6.5) The "b" versions of the equations (6.2)-(6.4) are consequences of the "a" versions and the equalities
(a ± l ) † = a † ∓ l (a † ± l ) = a ∓ l (6.6) [A, B] η † = η[A † , B † ] η for [A, B] η = η[B, A] η η = ±1. (6.7)
Applying (6.2)-(6.4) and the identity
[A, B • C] = [A, B] η • C -ηB • [A, C] η for η = ±1 (6.8)
for the choice η = -1, one can prove by a direct calculation that
[ Pµ , Pν ] = 0 [ Q, Pµ ] = 0 [ Sµν , Pλ ] = 0 [ Lµν , Pλ ] = -i {η λµ Pν -η λν Pµ } [ Mµν , Pλ ] = -i {η λµ Pν -η λν Pµ }, (6.9)
where the operators Pµ , Q, Sµν , Lµν , and Mµν denote the momentum, charge, spin, orbital and total angular momentum operators, respectively, of the system considered and are calculated from one and the same initial Lagrangian. This result confirms the supposition, made in Sect. 5, that the assertion (i) before (5.24) holds for the fields investigated here.
Below we shall study only those solutions of (6.2)-(6.4) for which the integrands in them vanish, i.e. we shall replace the systems of integral equations (6.2)-(6.4) with the following systems of algebraic equations (see the above convention on the indices l and m and do not sum over indices repeated on one and the same level):
a ± l , a † + m • a - m + εa † - m • a + m -± (1 + τ )δ lm a ± l = 0 (6.10a) a † ± l , a † + m • a - m + εa † - m • a + m -± (1 + τ )δ lm a † ± l = 0 (6.10b) a ± l , a + m • a † - m + εa - m • a † + m -± (1 + τ )δ lm a ± l = 0 (6.11a) a † ± l , a + m • a † - m + εa - m • a † + m -± (1 + τ )δ lm a † ± l = 0 (6.11b) a ± l , [a † + m , a - m ] ε + [a + m , a † - m ] ε -± 2(1 + τ )δ lm a ± l = 0 (6.12a) a † ± l , [a † + m , a - m ] ε + [a + m , a † - m ] ε -± 2(1 + τ )δ lm a † ± l = 0. (6.12b)
It seems, these are the most general and sensible trilinear commutation relations one may impose on the creation and annihilation operators.
First of all, we should mentioned that the standard bilinear commutation relations, viz.
[1, 3, 11-15] [a ± l , a ± m ] -ε = 0 [a † ± l , a † ± m ] -ε = 0 [a ∓ l , a ± m ] -ε = (±1) 2j+1 τ δ lm id F [a † ∓ l , a † ± m ] -ε = (±1) 2j+1 τ δ lm id F [a ± l , a † ± m ] -ε = 0 [a † ± l , a ± m ] -ε = 0 [a ∓ l , a † ± m ] -ε = (±1) 2j+1 δ lm id F [a † ∓ l , a ± m ] -ε = (±1) 2j+1 δ lm id F , (6.13)
provide a solution of any one of the equations (6.10)-(6.12) in a sense that, due to (3.7) and (6.8), with η = -ε any set of operators satisfying (6.13) converts (6.10)-(6.12) into identities.
Besides, this conclusion remains valid also if the normal ordering is taken into account, i.e. if, in this particular case, the changes a
† - m • a + m → εa + m • a † - m and a - m • a † + m → εa † + m • a - m
are made in (6.10)-(6.12). Now we shall demonstrate how the trilinear relations (6.12) lead to the paracommutation relations. Equations (6.12) can be 'split' into different kinds of trilinear commutation relations into infinitely many ways. For example, the system of equations
a ± l , [a + m , a † - m ] ε -± (1 + τ )δ lm a ± l = 0 (6.14a) a ± l , [a † + m , a - m ] ε -± (1 + τ )δ lm a ± l = 0 (6.14b) a † ± l , [a + m , a † - m ] ε -± (1 + τ )δ lm a † ± l = 0 (6.14c) a † ± l , [a † + m , a - m ] ε -± (1 + τ )δ lm a † ± l = 0 (6.14d)
provides an evident solution of (6.12). However, it is a simple algebra to be seen that these relations are incompatible with the standard (anti)commutation relations (6.13) and, in this sense, are not suitable as subsidiary restrictions on the Lagrangian formalism. For our purpose, the equations
a + l , [a + m , a † - m ] ε -+ 2δ lm a + l = 0 (6.15a) a + l , [a † + m , a - m ] ε -+ 2τ δ lm a + l = 0 (6.15b) a - l , [a + m , a † - m ] ε --2τ δ lm a - l = 0 (6.15c) a - l , [a † + m , a - m ] ε --2δ lm a - l = 0 (6.15d)
and their Hermitian conjugate provide a solution of (6.12), which is compatible with (6.13), i.e. if (6.13) hold, the equations (6.15) are converted into identities. The idea of the paraquantization is in the following generalization of (6.15)
a + l , [a + m , a † - n ] ε -+ 2δ ln a + m = 0 (6.16a) a + l , [a † + m , a - n ] ε -+ 2τ δ ln a + m = 0 (6.16b) a - l , [a + m , a † - n ] ε --2τ δ lm a - n = 0 (6.16c) a - l , [a † + m , a - n ] ε --2δ lm a - n = 0 (6.16d)
which reduces to (6.15) for n = m and is a generalization of (6.13) in a sense that any set of operators satisfying (6.13) converts (6.16) into identities, the opposite being generally not valid. 19 Suppose that the field considered consists of a single sort of particles, e.g. electrons or photons, created by b † l := a † l and annihilated by b l := a † - l . Then the equation Hermitian conjugated to (6.15a) reads
[b l , [b † m , b m ] ε ] = 2δ lm b m . (6.17)
This is the main relation from which the paper [16] starts. The basic paracommutation relations are [16] [17] [18] 26] :
[b l , [b † m , b n ] ε ] = 2δ lm b n (6.18a) [b l , [b m , b n ] ε ] = 0. (6.18b)
The first of them is a generalization (stronger version) of (6.17) by replacing the second index m with an arbitrary one, say n, and the second one is added (by "hands") in the theory as an additional assumption. Obviously, (6.18) are a solution of (6.15) and therefore of (6.12) in the considered case of a field consisting of only one sort of particles. The equations (6.15) contain also the relativistic version of the paracommutation relations, when the existence of antiparticles must be respected [18, sec. 18.1] . Indeed, noticing that the field's particles (resp. antiparticles) are created by b † l := a + l (resp. c † l := a † + l ) and annihilated by b l := a † - l (resp. c l := a - l ), from (6.15) and the Hermitian conjugate to them equations, we get
[b l , [b † m , b m ] ε ] = 2δ lm b m [c l , [c † m , c m ] ε ] = 2δ lm c m (6.19a) [b † l , [c † m , c m ] ε ] = -2τ δ lm b † m [c † l , [b † m , b m ] ε ] = -2τ δ lm c † m . (6.19b)
Generalizing these equations in a way similar to the transition from (6.17) to (6.18), we obtain the relativistic paracommutation relations as (cf. (6.16))
[b l , [b † m , b n ] ε ] = 2δ lm b n [b l , [b m , b n ] ε ] = 0 (6.20a) [c l , [c † m , c n ] ε ] = 2δ lm c n [c l , [c m , c n ] ε ] = 0 (6.20b) [b † l , [c † m , c n ] ε ] = -2τ δ ln b † m [c † l , [b † m , b n ] ε ] = -2τ δ ln c † m . (6.20c)
The equations (6.20a) (resp. (6.20b)) represent the paracommutation relations for the field's particles (resp. antiparticles) as independent objects, while (6.20c) describe a pure relativistic effect of some "interaction" (or its absents) between field's particles and antiparticles and fixes the paracommutation relations involving the b l 's and c l 's, as pointed in [18, p. 207] (where b l is denoted by a l and c l by b l ). The relations (6.17) and (6.20) for ε = +1 (resp. ε = -1) are referred as the parabose (resp. parafermi ) commutation relations [18] . This terminology is a natural one also with respect to the commutation relations (6.16), which will be referred as the paracommutation relations too.
As first noted in [16] , the equations (6.13) provide a solution of (6.20) (or (6.18) in the nonrelativistic case) but the latter equations admit also an infinite number of other solutions. Besides, by taking Hermitian conjugations of (some of) the equations (6.18) or (6.20) and applying generalized Jacobi identities, like
α[[A, B] ξ , C] η + ξη[[A, C] -α/ξ , B] -α/η -α 2 [[B, C] ξη/α , A] 1/α = 0 αξη = 0 β[A, [B, C] α , ] -βγ + γ[B, [C, A] β , ] -γα + α[C, [A, B] γ , ] -αβ = 0 α, β, γ = ±1 [[A, B] η , C] -+ [[B, C] η , A] -+ [[C, A] η , B] -= 0 η = ±1 [[A, B] ξ , [C, D] η ] -= [[A, B] ξ , C] -, D] η + η[[A, B] ξ , D] -, C] 1/η η = 0, (6.21)
one can obtain a number of other (para)commutation relations for which the reader is referred to [16, 18, 26] .
Of course, the paracommutation relations (6.16), in particular (6.18) and (6.20) as their stronger versions, do not give the general solution of the trilinear relations (6.12). For instance, one may replace (6.12) with the equations
a + l , [a † + m , a - n ] ε + [a + m , a † - n ] ε -+ 2(1 + τ )δ ln a + m = 0 (6.22a) a - l , [a † + m , a - n ] ε + [a + m , a † - n ] ε --2(1 + τ )δ lm a - n = 0. (6.22b)
and their Hermitian conjugate, which in terms of the operators b l and c l introduced above read
[b l , [b † m , b n ] ε + [c † m , c m ] ε ] = 2(1 + τ )δ lm b n (6.23a) [c l , [b † m , b n ] ε + [c † m , c m ] ε ] = 2(1 + τ )δ lm c n , (6.23b)
and supplement these relations with equations like (6.18b). Obviously, equations (6.16) convert (6.22) into identities and, consequently, the (standard) paracommutation relations (6.20) provide a solution of (6.23). On the base of (6.23) or other similar equations that can be obtained by generalizing the ones in (6.10)-(6.12), further research on particular classes of trilinear commutation relations can be done, but, however, this is not a subject of the present work.
Let us now pay attention to the fact that equations (6.10), (6.11) and (6.12) are generally different (regardless of existence of some connections between their solutions). The cause for this being that the momentum operators for the Lagrangians L ′ , L ′′ and L ′′′ are generally different unless some additional restrictions are added to the Lagrangian formalism (see Sect. 4). A necessary and sufficient condition for (6.10)-(6.12) to be identical is
[a ± l , [a † + m , a - m ] -ε -[a + m , a † - m ] -ε ] = 0, (6.24)
which certainly is valid if the condition (4.9 ′ ), viz.
[
a † + m , a - m ] -ε -[a + m , a † - m ] -ε = 0, ( 6.25)
ensuring the uniqueness of the momentum operator are, holds. If one adopts the standard bilinear commutation relations (6.13), then (6.25), and hence (6.24), is identically valid, but in the framework of, e.g., the paracommutation relations (6.16) (or (6.20) in other form) the equations (6.25) should be postulated to ensure uniqueness of the momentum operator and therefore of the field equations. On the base of (6.10) or (6.11) one may invent other types of commutation relations, which will not be investigated in this paper because we shall be interested mainly in the case when (6.10), (6.11) and (6.12) are identical (see (6.24) ) or, more generally, when the dynamical variables are unique in the sense pointed in Sect. 4.
The consequences of the Heisenberg relations (5.2), involving the charge operator for a charged field, q = 0 (and hence τ = 0 -see (3.7)), will be examined in this subsection. In terms of creation and annihilation operators it is equivalent to [1, [13] [14] [15] ]
[a ± s (k), Q] = qa ± s (k) [a † ± s (k), Q] = -qa † ± s (k), (6.26)
the values of the polarization indices being specified by (6.5). Substituting here (3.10), we see that, for a charged field, the field equations for the Lagrangians L ′ , L ′′ and L ′′′ (see Sect. 3) respectively are:
2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a ± s (k), a † + t (p) • a - t (p) -εa † - t (p) • a + t (p)] -a ± s (k)δ st δ 3 (k -p)} = 0 (6.27a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a † ± s (k), a † + t (p) • a - t (p) -εa † - t (p) • a + t (p)] + a † ± s (k)δ st δ 3 (k -p)} = 0 (6.27b) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a ± s (k), a + t (p) • a † - t (p) -εa - t (p) • a † + t (p)] + a ± s (k)δ st δ 3 (k -p)} = 0 (6.28a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a † ± s (k), a + t (p) • a † - t (p) -εa - t (p) • a † + t (p)] -a † ± s (k)δ st δ 3 (k -p)} = 0 (6.28b) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a ± s (k), [a † + t (p), a - t (p)] ε -[a + t (p), a † - t (p) ε ] -2a ± s (k)δ st δ 3 (k -p)} = 0 (6.29a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a † ± s (k), [a † + t (p), a - t (p)] ε -[a + t (p), a † - t (p) ε ] + 2a † ± s (k)δ st δ 3 (k -p)} = 0. (6.29b)
Using (6.27)-(6.29) and (6.8), with η = ε = -1, or simply (6.26), one can easily verify the validity of the equations
[ Pµ , Q] = 0 [ Lµν , Q] = 0 [ Sµν , Q] = 0 [ Mµν , Q] = 0, (6.30)
where the operators Pµ , Q, Sµν , Lµν and Mµν are calculated from one and the same initial Lagrangian according to (3.9)-(3.12). This result confirms the validity of assertion (ii) before (5.24) for the fields considered. Following the above considerations, concerning the momentum operator, we shall now replace the systems of integral equations (6.27)-(6.29) with respectively the following stronger systems of algebraic equations (by equating to zero the integrands in (6.27)-(6.29)):
a ± l , a † + m • a - m -εa † - m • a + m --δ lm a ± l = 0 (6.31a) a † ± l , a † + m • a - m -εa † - m • a + m -+ δ lm a † ± l = 0 (6.31b) a ± l , a + m • a † - m -εa - m • a † + m -+ δ lm a ± l = 0 (6.32a) a † ± l , a + m • a † - m -εa - m • a † + m --δ lm a † ± l = 0 (6.32b) a ± l , [a † + m , a - m ] ε -[a + m , a † - m ] ε --2δ lm a ± l = 0 (6.33a) a † ± l , [a † + m , a - m ] ε -[a + m , a † - m ] ε -+ 2δ lm a † ± l = 0. (6.33b)
These trilinear commutation relations are similar to (6.10)-(6.12) and, consequently, can be treated in analogous way. By invoking (6.8), it is a simple algebra to be proved that the standard bilinear commutation relations (6.13) convert (6.31)-(6.33) into identities. Thus (6.13) are stronger version of (6.31)-(6.33) and, in this sense, any type of commutation relations, which provide a solution of (6.31)-(6.33) and is compatible with (6.13), is a suitable candidate for generalizing (6.13). To illustrate that idea, we shall proceed with (6.33) in a way similar to the 'derivation' of the paracommutation relations from (6.12).
Obviously, the equations (cf. (6.14) with τ = 0, as now q = 0)
[a ± l , [a + m , a † - m ] ε ] + δ lm a ± m = 0 (6.34a) [a ± l , [a † + m , a - m ] ε ] -δ lm a ± m = 0 (6.34b)
and their Hermitian conjugate provide a solution of (6.33), but, as a direct calculations shows, they do not agree with the standard (anti)commutation relations (6.13). A solution of (6.33) compatible with (6.13) is given by the equations (6.15), with τ = 0 as the field considered is charged one -see (3.7). Therefore equations (6.16), with τ = 0, also provide a compatible with (6.13) solution of (6.33), from where immediately follows that the paracommutation relations (6.20), with τ = 0, convert (6.33) into identities. To conclude, we can say that the paracommutation relations (6.20), in particular their special case (6.13), ensure the simultaneous validity of the Heisenberg relations (5.1) and (5.2) for free scalar, spinor and vector fields.
Similarly to (6.22), one may generalize (6.33) to
a + l , [a † + m , a - n ] ε -[a + m , a † - n ] ε --2δ ln a + m = 0 (6.35a) a - l , [a † + m , a - n ] ε -[a + m , a † - n ] ε --2δ lm a - n = 0. (6.35b)
which equations agree with (6.13), (6.15), (6.16) and (6.20), but generally do not agree with (6.22), with τ = 0, unless the equations (6.16), with τ = 0, hold. More generally, we can assert that (6.33) and (6.12), with τ = 0, hold simultaneously if and only if (6.15), with τ = 0, is fulfilled. From here, again, it follows that the paracommutation relations ensure the simultaneous validity of (5.1) and (5.2).
Let us say now some words on the uniqueness problem for the Heisenberg equations involving the charge operator. The systems of equations (6.31)-(6.33) are identical iff
a ± l , [a † + m , a - m ] -ε + [a + m , a † - m ] -ε -= 0, (6.36) which, in particular, is satisfied if the condition [a † + m , a - m ] -ε + [a + m , a † - m ] -ε = 0, (6.37)
ensuring the uniqueness of the charge operator (see (4.10 ′ )), is valid. Evidently, equations (6.36) and (6.24) are compatible iff
a + l , [a † ± m , a ∓ m ] -ε -= 0 a - l , [a † ± m , a ∓ m ] -ε -= 0 (6.38)
which is a weaker form of (4.15) ensuring simultaneous uniqueness of the momentum and charge operator.
It is now turn to be investigated the restrictions on the creation and annihilation operators that follow from the Heisenberg relations (5.3) concerning the angular momentum operator. They can be obtained by inserting the equations (3.11) and (3.12) into (5.3). As pointed in Sect. 5, the resulting equalities, however, depend not only on the particular Lagrangian employed, but also on the geometric nature of the field considered; the last dependence being explicitly given via (5.25) and the polarization functions σ ss ′ m± µν (k) and l ss ′ m± µν (k) (see also (3.14) ).
Consider the terms containing derivatives in (5.3),
Lor µν := i x µ ∂ ∂x ν -x ν ∂ ∂x µ φi (x). (6.39)
If φi (k) denotes the Fourier image of φi (x), i.e.
φi (x) = Λ d 4 ke -1 i k µ xµ φi (k), (6.40)
with Λ being a normalization constant, then the Fourier image of (6.39) is
Lor µν = i k µ ∂ ∂k ν -k ν ∂ ∂k µ φi (k). ( 6
Comparing this expression with equations (3.12), we see that the terms containing derivatives in (3.12) should be responsible for the term (6.39) in (5.3). 20 For this reason, we shall suppose that the momentum operator Mµν admits a representation Mµν = Mor µν + Msp µν (6.42) such that the operators Mor µν and Msp µν satisfy the relations (5.4) and (5.5), respectively. Thus we shall replace (5.3) with the stronger system of equations (5.4)-(5.5). Besides, we shall admit that the explicit form of the operators Mor µν and Msp µν are given via (5.13) and (5.12) for the fields investigated in the present work.
Let us consider at first the 'orbital' Heisenberg relations (5.4), which is independent of the particular geometrical nature of the fields studied. Substituting (5.13) and (6.40) into (5.4), using that φi (±k), with k 2 = m 2 c 2 , is a linear combination of ã± s (k) with classical, not operator-valued, functions of k as coefficients [1, [13] [14] [15] and introducing for brevity the operator
ω µν (k) := k µ ∂ ∂k ν -k ν ∂ ∂k µ , (6.43)
we arrive to the following integro-differential systems of equations:
2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã ± s (k), ã † + t (p) • ã- t (q) -εã † - t (p) • ã+ t (q)] ) q=p p 0 = √ m 2 c 2 +p 2 = 2(1 + τ )ω µν (k)(ã ± s (k)) (6.44a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã † ± s (k), ã † + t (p) • ã- t (q) -εã † - t (p) • ã+ t (q)] ) q=p p 0 = √ m 2 c 2 +p 2 = 2(1 + τ )ω µν (k)(ã † ± s (k)) (6.44b) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã ± s (k), ã+ t (p) • ã † - t (q) -εã - t (p) • ã † + t (q)] ) q=p p 0 = √ m 2 c 2 +p 2 = 2(1 + τ )ω µν (k)(ã ± s (k)) (6.45a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã † ± s (k), ã+ t (p) • ã † - t (q) -εã - t (p) • ã † + t (q)] ) q=p p 0 = √ m 2 c 2 +p 2 = 2(1 + τ )ω µν (k)(ã † ± s (k)) (6.45b) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã ± s (k), [ã † + t (p), ã- t (q)] ε + [ã + t (p), ã † - t (q)] ε ] ) q=p p 0 = √ m 2 c 2 +p 2 = 4(1 + τ )ω µν (k)(ã ± s (k)) (6.46a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã † ± s (k), [ã † + t (p), ã- t (q)] ε + [ã + t (p), ã † - t (q)] ε ] ) q=p p 0 = √ m 2 c 2 +p 2 = 4(1 + τ )ω µν (k)(ã † ± s (k)), (6.46b)
where k 0 = m 2 c 2 + k 2 is set after the differentiations are performed (see (6.43)). Following the procedure of the previous considerations, we replace the integro-differential equations (6.44)-(6.46) with the following differential ones:
(-ω • µν (m) + ω • µν (n))([ã ± l , ã † + m • ã- n -εã † - m • ã+ n ] ) n=m = 2(1 + τ )δ lm ω • µν (l)(ã ± l ) (6.47a) (-ω • µν (m) + ω • µν (n))([ã † ± l , ã † + m • ã- n -εã † - m • ã+ n ] ) n=m = 2(1 + τ )δ lm ω • µν (l)(ã † ± l ) (6.47b) (-ω • µν (m) + ω • µν (n))([ã ± l , ã+ m • ã † - n -εã - m • ã † + n ] ) n=m = 2(1 + τ )δ lm ω • µν (l)(ã ± l ) (6.48a) (-ω • µν (m) + ω • µν (n))([ã † ± l , ã+ m • ã † - n -εã - m • ã † + n ] ) n=m = 2(1 + τ )δ lm ω • µν (l)(ã † ± l ) (6.48b) (-ω • µν (m) + ω • µν (n))([ã ± l , [ã † + m , ã- n ] ε + [ã + m , ã † - n ] ε ] ) n=m = 4(1 + τ )δ lm ω • µν (l)(ã ± l ) (6.49a) (-ω • µν (m) + ω • µν (n))([ã † ± l , [ã † + m , ã- n ] ε + [ã + m , ã † - n ] ε ] ) n=m = 4(1 + τ )δ lm ω • µν (l)(ã † ± l ), ( 6
.49b) where we have set (cf. (6.43))
ω • µν (l) := ω µν (k) = k µ ∂ ∂k ν -k ν ∂ ∂k µ if l = (s, k) (6.50)
and k 0 = m 2 c 2 + k 2 is set after the differentiations are performed.
Remark. Instead of (6.47)-(6.49) one can write similar equations in which the operator -ω • µν (m) or +ω • µν (n) is deleted and the factor + 1 2 or -1 2 , respectively, is added on their right hand sides. These manipulations correspond to an integration by parts of some of the terms in (6.44)-(6.46).
The main difference of the obtained trilinear relations with respect to the previous ones considered above is that they are partial differential equations of first order.
The relations (6.49) agree with the equations (6.16) in a sense that if (6.16) hold, then (6.49) become identically valid. Indeed, since
(-ω • µν (m) + ω • µν (n))(ã ± m δ ln ) n=m = -2δ lm ω • µν (m)(ã ± m ) (-ω • µν (m) + ω • µν (n))(ã ± n δ lm ) n=m = +2δ lm ω • µν (m)(ã ± m ), (6.51)
due to (6.50), (6.43) and the equality dδ(x) dx f (x) = -δ(x) df (x) dx for a C 1 function f , the application of the operator (-ω • µν (m) + ω • µν (n)) to (6.16) and subsequent setting n = m entails (6.49). In particular, this means that the paracommutation relations (6.20) and, moreover, the standard (anti)commutation relations (6.13) convert (6.49) into identities. Therefore the 'orbital' Heisenberg relations (5.4) hold for scalar, spinor and vector fields satisfying the bilinear or para commutation relations.
It should be noted, the paracommutation relations are not the only trilinear commutation relations that are solutions of (6.49). As an example, we shall present the trilinear relations
a + l , [a + m , a † - n ] ε -= a + l , [a † + m , a - n ] ε -= -(1 + τ )δ ln a + m (6.52a) a - l , [a + m , a † - n ] ε -= a - l , [a † + m , a - n ] ε -= +(1 + τ )δ lm a + n , (6.52b)
which reduce to (6.14) for n = m, do not agree with (6.13), but convert (6.49) into identities (see (6.51)). Other example is provided by the equations (6.22), which are compatible with the paracommutation relations and, as a result of (6.51), convert (6.49) into identities. Prima facie one may suppose that any solution of (6.12) provides a solution of (6.49), but this is not the general case. A counterexample is provided by the commutation relations
a ± l , [a † + m , a - n ] ε + [a + m , a † - n ] ε -± 2(1 + τ )δ ln a ± m = 0, (6.53)
which reduce to (6.12) for n = m, satisfy (6.49) with ã+ l for ã± l , and do not satisfy (6.49) with ãl for ã± l (see (6.51) and cf. (6.22)). From (5.13) follows that the operator Mor µν is independent of the Lagrangian L ′ , L ′′ or L ′′′ one starts off if and only if (see (4.11))
(-ω • µν (m) + ω • µν (n)) [ã † + m , ã- n ] -ε -[ã + m , ã † - n ] -ε n=m = 0. (6.54)
This condition ensures the coincidence of the systems of equations (6.47), (6.48) and (6.49) too. However, the following necessary and sufficient condition for the coincidence of these systems is expressed by the weaker equations
(-ω • µν (m) + ω • µν (n)) ã± l , [ã † + m , ã- n ] -ε -[ã + m , ã † - n ] -ε -n=m = 0. (6.55)
It is now turn to be considered the 'spin' Heisenberg relations (5.5).
Recall, the field operators ϕ i for the fields considered here admit a representation [13] [14] [15]
ϕ i = Λ t d 3 p v t,+ i (p)a + t (p) + v t,- i (p)a - t (p) , (6.56)
where Λ is a normalization constant and v t,± i (p) are classical, not operator-valued, complex or real functions which are linearly independent. The particular definition of v t,± i (p) depends on the geometrical nature of ϕ i and can be found in [13] [14] [15] (see also [1] ), where the reader can find also a number of relations satisfied by v t,± i (p). Here we shall mention only that v t,± i (p) = 1 for a scalar field and v t,+ i (p) = v t,- i (p) =: v t i (p) = (v t i (p)) * for a vector field. The explicit form of the polarization functions σ ss ′ ,± µν (k) and l ss ′ ,± µν (k) (see Sect. 3, in particular (3.14)) through v t,± i (k) are [13] [14] [15] :
σ ss ′ ,± µν (k) = (-1) j j + δ j0 i,i ′ (v s,± i (k)) * I i i ′ µν v t,± i ′ (k) l ss ′ ,± µν (k) = (-1) j 2j + δ j0 i (v s,± i (k)) * ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ v t,± i (k), (6.57)
with an exception that σ ss ′ ,± 0a (k) = σ ss ′ ,± a0 (k) = 0, a = 1, 2, 3, for a spinor field, j = 1 2 , [14] . Evidently, the equations (3.14) follow from the mentioned facts (see also (5.25) ).
Substituting (6.56) and (5.12) into (5.5), we obtain the following systems of integral equations (corresponding respectively to the Lagrangians L ′ , L ′′ and L ′′′ ):
(-1) j+1 j 1 + τ s,s ′ ,t d 3 k d 3 pv t,± i (p) (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k))[a ± t (p), a † + s (k) • a - s ′ (k)] + (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))[a ± t (p), a † - s (k) • a + s ′ (k)] = i ′ t d 3 pI i ′ iµν v t,± i ′ (p)a ± t (p) (6.58) ε (-1) j+1 j 1 + τ s,s ′ ,t d 3 k d 3 pv t,± i (p) (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))[a ± t (p), a + s ′ (k) • a † - s (k)] + (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k))[a ± t (p), a - s ′ (k) • a † + s (k)] = i ′ t d 3 pI i ′ iµν v t,± i ′ (p)a ± t (p) (6.59) (-1) j+1 j 2(1 + τ ) s,s ′ ,t d 3 k d 3 pv t,± i (p) (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k)) a ± t (p), [a † + s (k), a - s ′ (k)] ε - + (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k)) a ± t (p), [a † - s (k), a + s ′ (k)] ε -= i ′ t d 3 pI i ′ iµν v t,± i ′ (p)a ± t (p). (6.60)
For the difference of all previously considered systems of integral equations, like (6.2)-(6.4), (6.27)-(6.29) and (6.44)-(6.46), the systems (6.58)-(6.60) cannot be replaced by ones consisting of algebraic (or differential) equations. The cause for this state of affairs is that in (6.58)-(6.60) enter polarization modes with arbitrary s and s ′ and, generally, one cannot 'diagonalize' the integrand(s) with respect to s and s ′ ; moreover, for a vector field, the modes with s = s ′ are not presented at all (see (3.14) ). That is why no commutation relations can be extracted from (6.58)-(6.60) unless further assumptions are made. Without going into details, below we shall sketch the proof of the assertion that the commutation relations (6.16) convert (6.60) into identities for massive spinor and vector fields. 21 In particular, this entails that the paracommutation and the bilinear commutation relations provide solutions of (6.60).
Let (6.16) holds. Combining it with (6.60), we see that the latter splits into the equations 21 The equations (6.58)-(6.60) are identities for scalar fields as for them Iµν = 0 and v t,± i (k) = 1, which reflects the absents of spin for these fields.
(-1) j j 1 + τ s,t d 3 pv t,+ i (p) τ (σ st,- µν (p) + l st,- µν (p)) + ε(σ ts,+ µν (p) + l ts,+ µν (p)) a + s (p), = i ′ I i ′ iµν s d 3 pv s,+ i ′ (p)a + s (p) (6.61a) (-1) j+1 j 1 + τ s,t d 3 pv t,- i (p) (σ ts,- µν (p) + l ts,- µν (p)) + ετ (σ st,+ µν (p) + l st,+ µν (p)) a - s (p), = i ′ I i ′ iµν s d 3 pv s,- i ′ (p)a - s (p). (6.61b)
Inserting here (6.57), we see that one needs the explicit definition of v s,± i (k) and formulae for sums like ρ ii ′ (k) := s v s,± i (k)(v s,± i ′ (k)) * , which are specific for any particular field and can be found in [13] [14] [15] . In this way, applying (5.25), (3.7) and the mentioned results from [13] [14] [15] , one can check the validity of (6.61) for massive fields in a way similar to the proof of (5.3) in [13] [14] [15] for scalar, spinor and vector fields, respectively.
We shall end the present subsection with the remark that the equations (4.17) and (4.18), which together with (4.15) ensure the uniqueness of the spin and orbital operators, are sufficient conditions for the coincidence of the equations (6.58), (6.59) and (6.60).
To begin with, let us summarize the major conclusions from Sect. 6. Each of the Heisenberg equations (5.1)-(5.3), the equations (5.3) being split into (5.4) and (5.5), induces in a natural way some relations that the creation and annihilation operators should satisfy. These relations can be chosen as algebraic trilinear ones in a case of (5.1) and (5.2) (see (6.10)-(6.12) and (6.31)-(6.33), respectively). But for (5.4) and (5.5) they need not to be algebraic and are differential ones in the case of (5.4) (see (6.47)-(6.49)) and integral equations in the case of (5.5) (see (6.58)-(6.60)). It was pointed that the cited relations depend on the initial Lagrangian from which the theory is derived, unless some explicitly written conditions hold (see (6.24), (6.37) and (6.55)); in particular, these conditions are true if the equations (4.9)-(4.13), ensuring the uniqueness of the corresponding dynamical operators, are valid. Since the 'charge symmetric' Lagrangians (3.4) seem to be the ones that best describe free fields, the arising from them (commutation) relations (6.12), (6.33), (6.49) and (6.60) were studied in more details. It was proved that the trilinear commutation relations (6.16) convert them into identities, as a result of which the same property possess the paracommutation relations (6.20) and, in particular, the bilinear commutation relations (6.13). Examples of trilinear commutation relations, which are neither ordinary nor para ones, were presented; some of them, like (6.14), (6.34) and (6.52), do not agree with (6.13) and other ones, like (6.16), (6.22) and (6.35), generalize (6.20) and hence are compatible with (6.13). At last, it was demonstrated that the commutators between the dynamical variables (see (5.15)-(5.23)) are uniquely defined if a Heisenberg relation for one of the operators entering in it is postulated.
The chief aim of the present section is to be explored the problem whether all of the reasonable conditions, mentioned in the previous sections and that can be imposed on the creation and annihilation operators, can hold or not hold simultaneously. This problem is suggested by the strong evidences that the relations (5.1)-(5.3) and (5.15)-(5.23), with a possible exception of (5.3) (more precisely, of (5.5)) in the massless case, should be valid in a realistic quantum field theory [1, 3, 7, 8, 11, 12] . Besides, to the arguments in loc. cit., we shall add the requirement for uniqueness of the dynamical variables (see Sect. 4).
As it was shown in Sect. 6, the relations (5.1), (5.2), (5.4) and (5.5) are compatible if one starts from a charge symmetric Lagrangian (see (3.4 )), which best describes a free field theory; in particular, the commutation relations (6.16) (and hence (6.20) and (6.13)) ensure their simultaneous validity. 22 For that reason, we shall investigate below only commutation relations for which (5.1), (5.2), (5.4) and (5.5) hold. It will be assumed that they should be such that the equations (6.10)-(6.12), (6.31)-(6.33), (6.47)-(6.49) and (6.58)-(6.60), respectively, hold.
Consider now the problem for the uniqueness of the dynamical variables and its consistency with the commutation relations just mentioned for a charged field. It will be assumed that this uniqueness is ensured via the equations (4.9)-(4.11).
The equation (4.15), viz.
[
a † ± m , a ∓ m ] -ε = 0, ( 7.1)
is a necessary and sufficient conditions for the uniqueness of the momentum and charge operators (see Sect. 4 and the notation introduced at the beginning of Sect. 6). Before commenting on this relation, we would like to derive some consequences of it. Applying consequently (6.8) for η = -ε, (7.1) and the identity
[A, B • C] + = [A, B] η • C -ηB • [A, C] -η η = ±1 (7.2)
for η = +ε, -ε, we, in view of (7.1), obtain
[a + m , [a + m , a † - m ] ε ] = [a † - m , [a + m , a + m ] -ε ] + = (1 -ε)[a † - m , a + m ] ε • a + m [a - m , [a † + m , a - m ] ε ] = ε[a † + m , [a - m , a - m ] -ε ] + = ε(1 -ε)[a † + m , a - m ] ε • a - m . (7.3)
Forming the sum and difference of (6.12a), for τ = 0, and (6.33a), we see that the system of equations they form is equivalent to
[a + l , [a † + m , a - m ] ε ] = 0 [a - l , [a + m , a † - m ] ε ] = 0 (7.4a) [a + l , [a + m , a † - m ] ε ] + 2δ lm a + l = 0 [a - l , [a † + m , a - m ] ε ] -2δ lm a - l = 0. (7.4b)
Combining (7.4b), for l = m, with (7.3), we get
(1 -ε)[a † - m , a + m ] ε • a + m + 2a + m = 0 ε(1 -ε)[a † + m , a - m ] ε • a - m -2a - m = 0. (7.5)
Obviously, these equations reduce to a ± m = 0 (7.6)
for bose fields as for them ε = +1 (see (3.7)). Since the operators (7.6) describe a completely unobservable field, or, more precisely, an absence of a field at all, the obtained result means that the theory considered cannot describe any really existing physical field with spin j = 0, 1. Such a conclusion should be regarded as a contradiction in the theory. For fermi fields, j = 1 2 and ε = -1, the equations (7.5) have solutions different from (7.6) iff a ± m are degenerate operators, i.e. with no inverse ones, in which case (7.4a) is a consequence of (7.5) and (7.1) (see (6.8 ) and ( 7 .3) too).
The source of the above contradiction is in the equation (7.1), which does not agree with the bilinear commutation relations (6.13) and contradicts to the existing correlation between creation and annihilation of particles with identical characteristics (m = (t, p) in our case) as (7.1) can be interpreted physically as mutual independence of the acts of creation and annihilation of such particles [1, § 10.1] .
At this point, there are two ways for 'repairing' of the theory. On one hand, one can forget about the uniqueness of the dynamical variables (in a sense of Sect. 4), after which the formalism can be developed by choosing, e.g., the charge symmetric Lagrangians (3.4) and following the usual Lagrangian formalism; in fact, this is the way the parafield theory is build [16, 18] . On another hand, one may try to change something at the ground of the theory in such a way that the uniqueness of the dynamical variables to be ensured automatically. We shall follow the second method. As a guiding idea, we shall have in mind that the bilinear commutation relations (6.13) and the related to them normal ordering procedure provide a base for the present-day quantum field theory, which describes sufficiently well the discovered elementary particles/fields. On this background, an extensive exploration of commutation relations which are incompatible with (6.13) is justified only if there appear some evidences for fields/particles that can be described via them. In that connection it should be recalled [17, 18] , it seems that all known particles/fields are described via (6.13) and no one of them is a para particle/field.
Using the notation introduced at the beginning of Sect. 4, we shall look for a linear mapping (operator) E on the operator space over the system's Hilbert space F of states such that
E( D ′ ) = E( D ′′ ). (7.7)
As it was shown in Sect. 4, an example of an operator E is provided by the normal ordering operator N . Therefore an operator satisfying (7.7) always exists. To any such operator E there corresponds a set of dynamical variables defined via
D = E( D ′ ). (7.8)
Let us examine the properties of the mapping E that it should possess due to the requirement (7.7) .
First of all, as the operators of the dynamical variables should be Hermitian, we shall require
E( B) † = E( B † ) (7.9)
for any operator B, which entails
D † = D, (7.10)
due to (3.9)-(3.12) and (7.8). As in Sect. 4, we shall replace the so-arising integral equations with corresponding algebraic ones. Thus the equations (4.5)-(4.20) remain valid if the operator E is applied to their left hand sides.
Consider the general case of a charged field, q = 0. So, the analogue of (4.15) reads .11) which equation ensures the uniqueness of the momentum and charge operators. Respectively, the condition (4.11) transforms into
E [a † ± m , a ∓ m ] -ε = 0, ( 7
(-ω • µν (m) + ω • µν (n)) E([a † + m , a - n ] -ε ) -E([a + m , a † - n ] -ε ) n=m = 0, (7.12)
which, by means of (7.11) can be rewritten as (cf. (4.16))
ω • µν (n) E([a † + m , a - n ] -ε ) -E([a + m , a † - n ] -ε ) n=m = 0. (7.13)
At the end, equations (4.17) and (4.18) now should be written as
s,s ′ σ ss ′ ,- µν (k) E [a † + s (k), a - s ′ (k)] -ε + σ ss ′ ,+ µν (k) E [a † - s (k), a + s ′ (k)] -ε = 0 (7.14) s,s ′ l ss ′ ,- µν (k) E [a † + s (k), a - s ′ (k)] -ε + l ss ′ ,+ µν (k) E [a † - s (k), a + s ′ (k)] -ε = 0. (7.15)
These equations can be satisfied if we generalize (7.11) to (cf. (4.20) )
E [a † ± s (k), a ∓ s ′ (k)] -ε = 0 (7.16)
for any s and s ′ . At last, the following stronger version of (7.16 )
E [a † ± m , a ∓ n ] -ε = 0, (7.17)
for any m = (t, p) and n = (r, q), ensures the validity of (7.14) and (7.15) and thus of the uniqueness of all dynamical variables. It is time now to call attention to the possible commutation relations. The replacement
D ′ , D ′′ , D ′′′ → D := E( D ′ ) = E( D ′′ ) = E( D ′′′
) results in corresponding changes in the whole of the material of Sect. 6. In particular, the systems of commutation relations (6.10)-(6.12), (6.31)-(6.33), (6.47)-( 6 .49) and (6.58)-( 6 .60) should be replaced respectively with:
23 a ± l , E(a † + m • a - m ) + ε E(a † - m • a + m ) -± (1 + τ )δ lm a ± l = 0 (7.18) a ± l , E(a † + m • a - m ) -ε E(a † - m • a + m ) --δ lm a ± l = 0 (7.19) (-ω • µν (m) + ω • µν (n))([ã ± l , E(ã † + m • ã- n ) -ε E(ã † - m • ã+ n )] ) n=m = 2(1 + τ )δ lm ω • µν (l)(ã ± l ) (7.20) (-1) j+1 j 1 + τ s,s ′ ,t d 3 k d 3 pv t,± i (p) (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k))[a ± t (p), E(a † + s (k) • a - s ′ (k))] + (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))[a ± t (p), E(a † - s (k) • a + s ′ (k))] = i ′ t d 3 pI i ′ iµν v t,± i ′ (p)a ± t (p). (7.21)
Due to the uniqueness conditions (7.11)-( 7 .14), one can rewrite the terms E(a † ± m • a ∓ m ) in (7.18)-(7.21) in a number of equivalent ways; e.g. (see (7.11 ))
E(a † ± m • a ∓ m ) = ε E(a ∓ m • a † ± m ) = 1 2 E([a † ± m , a ∓ m ] ε ). (7.22)
Consider the general case of a charged field, q = 0 (and hence τ = 0). The system of equations (7.18)- (7.19) is then equivalent to
a ± l , E(a † ± m • a ∓ m ) -= 0 (7.23a) a + l , E(a † - m • a + m ) -+ εδ lm a + l = 0 (7.23b) a - l , E(a † + m • a - m ) --δ lm a - l = 0. (7.23c)
These (commutation) relations ensure the simultaneous fulfillment of the Heisenberg relations (5.1) and (5.2) involving the momentum and charge operators, respectively. To ensure also the validity of (7.20), with τ = 0, and, consequently, of (5.4), we generalize (7.23) to
a ± l , E(a † ± m • a ∓ n ) -= 0 (7.24a) a + l , E(a † - m • a + n ) -+ εδ lm a + n = 0 (7.24b) a - l , E(a † + m • a - n ) --δ lm a - n = 0, (7.24c)
for any l = (s, k), m = (t, p) and n = (t, q) (see also (6.51) ). In the way pointed in Sect. 6, one can verify that (7.24) for any l = (s, k), m = (t, p) and n = (r, p) entails (7.21) and hence (5.5) . At last, to ensure the validity of all of the mentioned conditions and a suitable transition to a case of Hermitian field, for which q = 0 and τ = 1 (see (3.7)), we generalize (7.24) to
a + l , E(a † + m • a - n ) -+ τ δ ln a + m = 0 (7.25a) a - l , E(a † - m • a + n ) --ετ δ ln a - m = 0 (7.25b) a + l , E(a † - m • a + n ) -+ εδ lm a + n = 0, (7.25c) a - l , E(a † + m • a - n ) --δ lm a - n = 0 (7.25d)
where l, m and n are arbitrary. As a result of (7.17), which we assume to hold, and τ a † ± l = τ a ± l (see (3.7)), the equations (7.25a) and (7.25c) (resp. (7.25b ) and (7.25d)) become identical when τ = 1 (and hence a † ± l = a ± l ); for τ = 0 the system (7.25) reduces to (7.24) . Recalling that ε = (-1) 2j (see (3.7)), we can rewrite (7.25) in a more compact form as
a ± l , E(a † ± m • a ∓ n ) -+ (±1) 2j+1 τ δ ln a ± m = 0 (7.26a) a ± l , E(a † ∓ m • a ± n ) --(∓1) 2j+1 τ δ lm a ± n = 0. (7.26b)
Since the last equation is equivalent to (see (7.17) ) and use that ε = (-1) 2j )
a ± l , E(a ± m • a † ∓ n ) -+ (±1) 2j+1 δ ln a ± m = 0, (7.26b ′ )
it is evident that the equations (7.26a) and (7.26b) coincide for a neutral field.
Let us draw the main moral from the above considerations: the equations (7.17) are sufficient conditions for the uniqueness of the dynamical variables, while (7.26) are such conditions for the validity of the Heisenberg relations (5.1)-(5.5), in which the dynamical variables are redefined according to (7.8) . So, any set of operators a ± l and E, which are simultaneous solutions of (7.17) and (7.26) , ensure uniqueness of the dynamical variables and at the same time the validity of the Heisenberg relations.
Consider the uniqueness problem for the solutions of the system of equations consisting of (7.17)and (7.26) . Writing (7.17) as
E(a † ± m • a ∓ n ) = ε E(a ∓ n • a † ± m ) = 1 2 E([a † ± m , a ∓ n ] ε ), (7.27)
which reduces to (7.22) for n = m, and using ε = (-1) 2j (see (3.7)), one can verify that (7.26) is equivalent to
a + l , E([a + m , a † - n ] ε ) -+ 2δ ln a + m = 0 (7.28a) a + l , E([a † + m , a - n ] ε ) -+ 2τ δ ln a + m = 0 (7.28b) a - l , E([a + m , a † - n ] ε ) --2τ δ lm a - n = 0 (7.28c) a - l , E([a † + m , a - n ] ε ) --2δ lm a - n = 0. (7.28d)
The similarity between this system of equations and (6.16) is more than evident: (7.28) can be obtained from (6.16) by replacing
[•, •] ε with E([•, •] ε ).
As it was said earlier, the bilinear commutation relations (6.13) and the identification of E with the normal ordering operator N , E = N , (7.29) convert (7.27)-(7.28) into identities; by invoking (6.8), for η = -ε, the reader can check this via a direct calculation (see also (4.23) ). However, this is not the only possible solution of (7.27)-(7.28). For example, if, in the particular case, one defines an 'anti-normal' ordering operator A as a linear mapping such that
A(a + m • a † - n ) := εa † - n • a + m A(a † + m • a - n ) := εa - n • a † + m A(a - m • a † + n ) := a - m • a † + n A(a † - m • a + n ) := a † - m • a + n , (7.30)
then the bilinear commutation relations (6.13) and the setting E = A provide a solution of (7.27)-(7.28); to prove this, apply (6.8) for η = -ε. Evidently, a linear combination of N and A, together with (6.13), also provides a solution of (7.27)-(7.28). 24 Other solution of the same system of equations is given by E = id and operators a ± l satisfying (6.16), in particular the paracommutation relations (6.20), and
a † ± m • a ,∓ n = εa ,∓ n • a † ± m .
The problem for the general solution of (7.27)-(7.28) with respect to E and a ± l is open at present. Let us introduce the particle and antiparticle number operators respectively by (see (7.27 ), (7.9) and (3.16))
N l := 1 2 E [a + l , a † - l ] = E(a + l • a † - l ) = ( N l ) † =: N † l † N l := 1 2 E [a † + l , a - l ] = E(a † + l • a - l ) = ( † N l ) † =: † N l † . (7.31)
As a result of the commutation relations (7.28), with n = m, they satisfy the equations 25
[ N l , a + m ] -= δ lm a + l (7.32a) [ † N l , a + m ] -= τ δ lm a + l (7.32b) [ N l , a † + m ] -= τ δ lm a † + l (7.32c) [ † N l , a † + m ] -= δ lm a † + l . (7.32d)
Combining (3.9)-(3.12) and (5.11)-(5.13) with (7.8), (7.27) and ( 7 .31), we get the following expressions for the operators of the (redefined) dynamical variables:
Pµ = 1 1 + τ l k µ | k 0 = √ m 2 c 2 +k 2 ( N l + † N l ) l = (s, k) (7.33) Q = q l (-N l + † N l ) (7.34) Sµν = (-1) j-1/2 j 1 + τ m,n {εσ mn,+ µν N nm + σ mn,- µν † N mn )} m=(s,k) n=(s ′ ,k) (7.35) Lµν = x 0 µ Pν -x 0 ν Pµ + (-1) j-1/2 j 1 + τ m,n {εl mn,+ µν N nm + l mn,- µν † N mn )} m=(s,k) n=(s ′ ,k) + i 2(1 + τ ) l -ω • µν (l) + ω • µν (m) ( N l + † N l ) m=l=(s,k) (7.36) Msp µν = (-1) j-1/2 j 1 + τ m,n {ε(σ mn,+ µν + l mn,+ µν ) N nm + (σ mn,- µν + l mn,- µν ) † N mn )} m=(s,k) n=(s ′ ,k) (7.37) Mor µν = i 2(1 + τ ) l -ω • µν (l) + ω • µν (m) ( N l + † N l ) m=l=(s,k) .
(7.38) 24 If we admit a ± l to satisfy the 'anomalous" bilinear commutation relations (8.27) (see below), i.e. (6.13) with ε for -ε and (±1) 2j for (±1) 2j+1 , then E = N , A also provides a solution of (7.27)-(7.28). However, as it was demonstrated in [13] [14] [15] , the anomalous commutation relations are rejected if one works with the charge symmetric Lagrangians (3.4). 25 The equations (7.32a) and (7.32b) correspond to (7.28a) and (7.28b), respectively, and (7.32c) and (7.32d) correspond to the Hermitian conjugate to (7.28c) and (7.28d), respectively.
Here ω • µν (l) is defined via (6.50), we have set σ mn,± µν := σ ss ′ ,± µν (k) l mn,± µν := l ss ′ ,± µν (k) for m = (s, k) and n = (s ′ , k), (7.39) and (see (7.27) )
N lm := 1 2 E [a + l , a † - m ] = E(a + l • a † - m ) = ( N ml ) † =: N † ml † N lm := 1 2 E [a † + l , a - m ] = E(a † + l • a - m ) = ( † N ml ) † =: † N ml † (7.40)
are respectively the particle and antiparticle transition operators (cf. [26, sec. 1] in a case of parafields). Obviously, we have
N l = N ll † N l = † N ll . (7.41)
The choice (7.29), evidently, reduces (7.33)-( 7 .36) to (4.24), (4.25), (4.28) and (4.29), respectively.
In terms of the operators (7.38), the commutation relations (7.28) can equivalently be rewritten as (see also (7.9 ))
[ N lm , a + n ] -= δ mn a + l (7.42a) [ † N lm , a + n ] -= τ δ mn a + l (7.42b) [ N lm , a † + n ] -= τ δ mn a † + l (7.42c) [ † N lm , a † + n ] -= δ mn a † + l . (7.42d)
If m = l, these relations reduce to (7.32), due to (7.39). We shall end this section with the remark that the conditions for the uniqueness of the dynamical variables and the validity of the Heisenberg relations are quite general and are not enough for fixing some commutation relations regardless of a number of additional assumptions made to reduce these conditions to the system of equations (7.27)-(7.28).
Until now we have looked on the commutation relations only from pure mathematical viewpoint. In this way, making a number of assumptions, we arrived to the system (7.27)-(7.28) of commutation relations. Further specialization of this system is, however, almost impossible without making contact with physics. For the purpose, we have to recall [1, 3, 11, 12] that the physically measurable quantities are the mean (expectation) values of the dynamical variables (in some state) and the transition amplitudes between different states. To make some conclusions from these basic assumption of the quantum theory, we must rigorously said how the states are described as vectors in system's Hilbert space F of states, on which all operators considered act.
For the purpose, we shall need the notion of the vacuum or, more precisely, the assumption of the existence of unique vacuum state (vector) (known also as the no-particle condition). Before defining rigorously this state, which will be denoted by X 0 , we shall heuristically analyze the properties it should possess.
First of all, the vacuum state vector X 0 should represent a state of the field without any particles. From here two conclusions may be drawn: (i) as a field is thought as a collection of particles and a 'missing' particle should have vanishing dynamical variables, those of the vacuum should vanish too (or, more generally, to be finite constants, which can be set equal to zero by rescaling some theory's parameters) and (ii) since the operators a - l and a † - l are interpreted as ones that annihilate a particle characterize by l = (s, k) and charge -q or +q, respectively, and one cannot destroy an 'absent' particle, these operators should transform the vacuum into the zero vector, which may be interpreted as a complete absents of the field. Thus, we can expect that
D( X 0 ) = 0 (8.1a) a - l ( X 0 ) = 0 a † - l ( X 0 ) = 0. (8.1b)
Further, as the operators a + l and a † + l are interpreted as ones creating a particle characterize by l = (s, k) and charge -q or +q, respectively, state vectors like a + l ( X 0 ) and a † + l ( X 0 ) should correspond to 1-particle states. Of course, a necessary condition for this is
X 0 = 0, (8.2)
due to which the vacuum can be normalize to unit,
X 0 | X 0 = 1, (8.3)
where
•|• : F × F → C is the Hermitian scalar (inner) product of F. More generally, if M(a + l 1 , a † + l 2 , .
. .) is a monomial only in i ∈ N creation operators, the vector
ψ l 1 l 2 ... := M(a + l 1 , a † + l 2 , . . .)( X 0 ) (8.4)
may be expected to describe an i-particle state (with i 1 particles and i 2 antiparticles, i 1 +i 2 = i, where i 1 and i 2 are the number of operators a + l and a † + l , respectively, in M(a + l 1 , a † + l 2 , . . .)). Moreover, as a free field is intuitively thought as a collection of particles and antiparticles, it is natural to suppose that the vectors (8.4) form a basis in the Hilbert space F. But the validity of this assumption depends on the accepted commutation relations; for its proof, when the paracommutation relations are adopted, see the proof of [18, p. 26, theorem I-1] .
Accepting the last assumption and recalling that the transition amplitude between two states is represented via the scalar product of the corresponding to them state vectors, it is clear that for the calculation of such an amplitude is needed an effective procedure for calculation of scalar products of the form
ψ l 1 l 2 ... |ϕ m 1 m 2 ... := X 0 |( M(a + l 1 , a † + l 2 , . . .)) † • M ′ (a + m 1 , a † + m 2 , . . .) X 0 , (8.5)
with M and M ′ being monomials only in the creation operators. Similarly, for computation of the mean value of some dynamical operator D in a certain state, one should be equipped with a method for calculation of scalar products like
ψ l 1 l 2 ... | Dϕ m 1 m 2 ... := X 0 |( M(a + l 1 , a † + l 2 , . . .)) † • D • M ′ (a + m 1 , a † + m 2 , . . .) X 0 . (8.6)
Supposing, for the moment, the vacuum to be defined via (8.1), let us analyze (8.1)-(8.6). Besides, the validity of (7.27)-(7.28) will be assumed.
From the expressions (7.8) and (3.9)-(3.12) for the dynamical variables, it is clear that the condition (8.1a) can be satisfied if
E(a † ± m • a ∓ n )( X 0 ) = 0, (8.7)
which, in view of (7.27), is equivalent to any one of the equations
E(a ± m • a † ∓ n )( X 0 ) = 0 (8.8a) E([a ± m , a † ∓ n ] ε )( X 0 ) = 0. (8.8b)
Equation (8.7) is quite natural as it expresses the vanishment of all modes of the vacuum corresponding to different polarizations, 4-momentum and charge. It will be accepted hereafter. By means of (8.8) and the commutation relations (7.28) in the form (7.42), in particular (7.32), one can explicitly calculate the action of any one of the operators (7.33)-(7.38) on the vectors (8.4) : for the purpose one should simply to commute the operators N lm (or N l = N ll ) with the creation operators in (8.4) according to (7.42) (resp. (7.32 )) until they act on the vacuum and, hence, giving zero, as a result of (8.8) and (7.42) (resp. (7.32)). In particular, we have the equations
(k 0 = m 2 c 2 + k 2 ): Pµ a + l ( X 0 ) = k µ a + l ( X 0 ) Pµ a † + l ( X 0 ) = k µ a † + l ( X 0 ) l = (s, k) (8.9) Q a + l ( X 0 ) = -qa + l ( X 0 ) Q a † + l ( X 0 ) = +qa † + l ( X 0 ) (8.10) Sµν a + l l=(s,k) ( X 0 ) = (-1) j-1/2 j 1 + τ t {εσ lm,+ µν + τ σ ml,- µν } m=(t,k) a + m | m=(t,k) ( X 0 ) Sµν a † + l l=(s,k) ( X 0 ) = (-1) j-1/2 j 1 + τ t {ετ σ lm,+ µν + σ ml,- µν } m=(t,k) a † + m | m=(t,k) ( X 0 ) (8.11) Lµν a + l l=(s,k) ( X 0 ) = (x 0 µ k ν -x 0 ν k µ )(a + l )( X 0 ) -i ω • µν (l)(a + l ) ( X 0 ) + (-1) j-1/2 j 1 + τ t {εl lm,+ µν + τ l ml,- µν } m=(t,k) a + m | m=(t,k) ( X 0 ) Lµν a † + l l=(s,k) ( X 0 ) = (x 0 µ k ν -x 0 ν k µ )(a † + l )( X 0 ) -i ω • µν (l)(a † + l ) ( X 0 ) + (-1) j-1/2 j 1 + τ t {ετ l lm,+ µν + l ml,- µν } m=(t,k) a † + m | m=(t,k) ( X 0 ) (8.12)
Msp µν a + l l=(s,k) ( X 0 ) = (-1) j-1/2 j 1 + τ t {ε(σ lm,+ µν + l lm,+ µν ) + τ (σ ml,- µν + l ml,- µν )} m=(t,k) a + m | m=(t,k) ( X 0 ) Msp µν a † + l l=(s,k) ( X 0 ) = (-1) j-1/2 j 1 + τ t {ετ (σ lm,+ µν + l lm,+ µν ) + (σ ml,- µν + l ml,- µν )} m=(t,k) a † + m | m=(t,k) ( X 0 ) (8.13) Mor µν ã+ l ( X 0 ) = -i ω • µν (l)(ã + l ) ( X 0 ) Mor µν ã † + l ( X 0 ) = -i ω • µν (l)(ã † + l ) ( X 0
). (8.14) These equations and similar, but more complicated, ones with an arbitrary monomial in the creation operators for a + l or a † + l are the base for the particle interpretation of the quantum theory of free fields. For instance, in view of (8.9) and (8.10), the state vectors a + l ( X 0 ) and a † + l ( X 0 ) are interpreted as ones representing particles with 4-momentum ( m 2 c 2 + k 2 , k) and charges -q and +q, respectively; similar multiparticle interpretation can be given to the general vectors (8.4) too.
The equations (8.9)-(8.12) completely agree with similar ones obtained in [13] [14] [15] on the base of the bilinear commutation relations (6.13) .
By means of (8.7), the expression (8.6) can be represented as a linear combination of terms like (8.5) . Indeed, as D is a linear combinations of terms like E(a † ± m • a ∓ n ), by means of the relations (7.28) we can commute each of these terms with the creation (resp. annihilation) operators in the monomial
M ′ (a + m 1 , a † + m 2 , . . .) (resp. ( M(a + l 1 , a † + l 2 , . . .)) † = M ′′ (a † - l 1 , a - l 2 , . . .)
) and thus moving them to the right (resp. left) until they act on the vacuum X 0 , giving the zero vector -see (8.7) . In this way the matrix elements of the dynamical variables, in particular their mean values, can be expressed as linear combinations of scalar products of the form (8.5) . Therefore the supposition (8.7) reduces the computation of mean values of dynamical variables to the one of the vacuum mean value of a product (composition) of creation and annihilation operators in which the former operators stand to the right of the latter ones. (Such a product of creation and annihilation operators can be called their 'antinormal' product; cf. the properties (7.30) of the antinormal ordering operator A.)
The calculation of such mean values, like (8.5) for states ψ, ϕ = X 0 , however, cannot be done (on the base of (7.27)-(7.28), (8.7) and (8.1a)) unless additional assumption are made. For the purpose one needs some kind of commutation relations by means of which the creation (resp. annihilation) operators on the r.h.s. of (8.5) to be moved to the left (resp. right) until they act on the left (resp. right) vacuum vector X 0 ; as a result of this operation, the expressions between the two vacuum vectors in (8.5) should transform into a linear combination of constant terms and such with no contribution in (8.5) . (Examples of the last type of terms are E(a † ± m • a ∓ ) and normally ordered products of creation and annihilation operators.) An alternative procedure may consists in defining axiomatically the values of all or some of the mean values (8.5) or, more stronger, the explicit action of all or some of the operators, entering in the r.h.s. of (8.5), on the vacuum. 26 It is clear, both proposed schemes should be consistent with the relations (7.27)-(7.28), (8.1b) and (8.7)- (8.8) .
Let us summarize the problem before us: the operator E in (7.27)-( 7 .28) has to be fixed and a method for computation of scalar products like (8.5) should be given provided the vacuum vector X 0 satisfies (8.1b), (8.2), (8.3) and (8.7). Two possible ways for exploration of this problem were indicated above.
Consider the operator E. Supposing E(a † ± m • a ∓ n ) to be a function only of a † ± m and a ∓ n , we, in view of (8.1b), can write
E(a † ± m • a ∓ n ) = f ± (a † ± m • a ∓ n ) • b with b = a - n (upper sign) or b = a † -
m (lower sign) and some functions f ± . Applying (7.27), we obtain (do not sum over l)
E(a † + m • a - l ) = f + (a † + m , a - l ) • a - l E(a + m • a † - l ) = f -(a + m , a † - l ) • a † - l E(a - l • a † + m ) = εf + (a † + m , a - l ) • a - l E(a † - l • a + m ) = εf -(a + m , a † - l ) • a † - l .
Since E is a linear operator, the expression E(a † ± m •a ∓ n ) turns to be a linear and homogeneous function of a † ± m and a ∓ n , which immediately implies f ± (A, B) = λ ± A for operators A and B and some constants λ ± ∈ C. For future convenience, we assume λ ± = 1, which can be achieved via a suitable renormalization of the creation and annihilation operators. 27 Thus, the last equations reduce to
E(a † + m • a - l ) = a † + m • a - l E(a + m • a † - l ) = a + m • a † - l (8.15a) E(a - l • a † + m ) = εa † + m • a - l E(a † - l • a + m ) = εa + m • a † - l . (8.15b)
Evidently, these equations convert (7.27), (8.7) and (8.8) into identities. Comparing (8.15) and (4.22), we see that the identification
E = N (8.16)
of the operator E with the normal ordering operator N is quite natural. However, for our purposes, this identification is not necessary as only the equations (8.15) , not the general definition of N , will be employed.
As a result of (8.15), the commutation relations (7.28) now read:
[a + l , a + m • a † - n ] + δ ln a + m = 0 (8.17a) [a + l , a † + m • a - n ] + τ δ ln a + m = 0 (8.17b) [a - l , a + m • a † - n ] -τ δ lm a - n = 0 (8.17c) [a - l , a † + m • a - n ] -δ lm a - n = 0. (8.17d)
(In a sense, these relations are 'one half' of the (para)commutation relations (6.16): the latter are a sum of the former and the ones obtained from (8.17) via the changes
a + m • a † - n → εa † - n • a + m and a † + m • a - n → εa - n • a † + m ;
the last relations correspond to (7.28) with E = A, A being the antinormal ordering operator -see (7.30) . Said differently, up to the replacement a ± i → √ 2a ± l for all l, the relations (8.17) are identical with (6.16) for ε = 0; as noted in [26, the remarks following theorem 2 in sec. 1], this is a quite exceptional case from the view-point of parastatistics theory.) By means of (6.8) for η = -ε, one can verify that equations (8.17) agree with the bilinear commutation relations (6.13), i.e. (6.13) convert (8.17) into identities.
The equations (8.15) imply the following explicit forms of the number operators (7.31) and the transition operators (7.40):
N l = a + l • a † - l † N l = a † + l • a - l (8.18)
N lm = a + l • a † - m † N lm = a † + l • a - m . (8.19)
As a result of them, the equations (7.33)-(7.36) are simply a different form of writing of (
Let us return to the problem of calculation of vacuum mean values of antinormal ordered products like (8.5) . In view of (8.1b) and (8.3), the simplest of them are
X 0 |λ id F ( X 0 ) = λ X 0 | M ± ( X 0 ) = 0 (8.20)
where λ ∈ C and M + (resp. M -) is any monomial of degree not less than 1 only in the creation (resp. annihilation) operators; e.g.
M ± = a ± l , a † ± l , a ± l 1 •a ± l 2 , a ± l 1 •a † ± l 2 .
These equations, with λ = 1, are another form of what is called the stability of the vacuum: if X i denotes an i-particle state, i ∈ N ∪ {0}, then, by virtue of (8.20) and the particle interpretation of (8.4), we have
X i | X 0 = δ i0 , (8.21)
i.e. the only non-forbidden transition into (from) the vacuum is from (into) the vacuum. More generally, if X i ′ ,0 and X 0,j ′′ denote respectively i ′ -particle and j ′′ -antiparticle states, with X 0,0 := X 0 , then
X i ′ ,0 | X 0,j ′′ = δ i ′ 0 δ 0j ′′ , (8.22)
i.e. transitions between two states consisting entirely of particles and antiparticles, respectively, are forbidden unless both states coincide with the vacuum. Since we are dealing with free fields, one can expect that the amplitude of a transitions from an (i ′ -particle + j ′ -antiparticle) state X i ′ ,j ′ into an (i ′′ -particle + j ′′ -antiparticle) state X i ′′ ,j ′′ is
X i ′ ,j ′ | X i ′′ ,j ′′ = δ i ′ i ′′ δ j ′ j ′′ , (8.23)
but, however, the proof of this hypothesis requires new assumptions (vide infra).
Let us try to employ (8.17) for calculation of expressions like (8.5) . Acting with (8.17) and their Hermitian conjugate on the vacuum, in view of (8.1b), we get
a + m • (-a † - n • a + l + δ ln id F )( X 0 ) = 0 a † + n • (a - m • a † + l -δ lm id F )( X 0 ) = 0 a † + m • (-a - n • a + l + τ δ ln id F )( X 0 ) = 0 a + n • (a † - m • a † + l -τ δ lm id F )( X 0 ) = 0. (8.24)
These equalities, as well as (8.17), cannot help directly to compute vacuum mean values of antinormally ordered products of creation and annihilation operators. But the equations (8.24) suggest the restrictions 28
a † - l • a + m ( X 0 ) = δ lm X 0 a - l • a † + m ( X 0 ) = δ lm X 0 a - l • a + m ( X 0 ) = τ δ lm X 0 a † - l • a † + m ( X 0 ) = τ δ lm X 0 (8.25)
to be added to the definition of the vacuum. These conditions convert (8.24) into identities and, in this sense agree with (8.17) and, consequently, with the bilinear commutation relations (6.13) . Recall [16, 18] , the relations (8.25) are similar to ones accepted in the parafield theory and coincide with that for parastatistics of order p = 1; however, here we do not suppose the validity of the paracommutation relations (6.20) (or (6.16)). Equipped with (8.25) , one is able to calculate the r.h.s. of (8.5) for any monomial M (resp. M ′ ) and monomials M ′ (resp. M) of degree 1, deg M ′ = 1 (resp. deg M = 1). 29 Indeed, (8.25), (8.1b ) and (8.3) entail:
X 0 |a † - l • a + m ( X 0 ) = X 0 |a - l • a † + m ( X 0 ) = δ lm X 0 |a - l • a + m ( X 0 ) = X 0 |a † - l • a † + m ( X 0 ) = τ δ lm X 0 |( M(a + l 1 , a † + l 2 , • • • )) † • a + m ( X 0 ) = X 0 |( M(a + l 1 , a † + l 2 , • • • )) † • a † + m ( X 0 ) = 0 deg M ≥ 2 X 0 |a - l • M(a + m 1 , a † + m 2 , • • • )( X 0 ) = X 0 |a † - l • M(a + m 1 , a † + m 2 , • • • )( X 0 ) = 0 deg M ≥ 2. (8.26)
Hereof the equation (8.23) for i ′ + j ′ = 1 (resp. i ′′ + j ′′ = 1) and arbitrary i ′′ and j ′′ (resp. i ′ and j ′ ) follows. However, it is not difficult to be realized, the calculation of (8.5) in cases more general than (8.20) and (8.26) is not possible on the base of the assumptions made until now. 30 At this point, one is free so set in an arbitrary way the r.h.s. of (8.5) in the mentioned general case or to add to (8.17) (and, possibly, (8.25)) other (commutation) relations by means of which the r.h.s. of (8.5) to be calculated explicitly; other approaches, e.g. some mixture of the just pointed ones, for finding the explicit form of (8.5) are evidently also possible. Since expressions like (8.5) are directly connected with observable experimental results, the only criterion for solving the problem for calculating the r.h.s. of (8.5) in the general case can be the agreement with the existing experimental data. As it is known [1, 3, 11, 12] , at present (almost?) all of them are satisfactory described within the framework of the bilinear commutation relations (6.13) . This means that, from physical point of view, the theory should be considered as realistic one if the r.h.s. of (8.5) is the same as if (6.13) are valid or is reducible to it for some particular realization of an accepted method of calculation, e.g. if one accepts some commutation relations, like the paracommutation ones, which are a generalization of (6.13) and reduce to them as a special case (see, e.g., (6.20) ). It should be noted, the conditions (8.1b)-(8.3) and (8.25) are enough for calculating (8.5) if (6.16), or its versions (6.17) or (6.20) , are accepted (cf. [16] ). The causes for that difference are replacements like [a + m , a † - n ] → 2a + m •a † - n , when one passes from (6.16) to (8.17); the existence of terms like a † - n • a + m a + l in (6.16) are responsible for the possibility to calculate (8.5).
28 Since the operators a ± l and a † ± l are, generally, degenerate (with no inverse ones), we cannot say that (8.24) implies (8.25) .
29 For deg M ′ = 0 (resp. deg M ′ = 0) -see (8.20) . 30 It should be noted, the conditions (8.1b)-(8.3) and (8.25) are enough for calculating (8.5) if the relations (6.16) , or their version (6.20) , are accepted (cf. [16] ). The cause for that difference is in replacements like [a + m , a †- n ] → 2a + m • a † - n , when one passes from (6.16) to (8.17); the existence of terms like a † - n • a + m • a + l in (6. 16 ) is responsible for the possibility to calculate (8.5), in case (6.16) hold.
If evidences appear for events for which (8.5) takes other values, one should look, e.g., for other commutation relations leading to desired mean values. As an example of the last type can be pointed the following anomalous bilinear commutation relations (cf. (6.13))
[a ± l , a ± m ] ε = 0 [a † ± l , a † ± m ] ε = 0 [a ∓ l , a ± m ] ε = (±1) 2j τ δ lm id F [a † ∓ l , a † ± m ] ε = (±1) 2j τ δ lm id F [a ± l , a † ± m ] ε = 0 [a † ± l , a ± m ] ε = 0 [a ∓ l , a † ± m ] ε = (±1) 2j δ lm id F [a † ∓ l , a ± m ] ε = (±1) 2j δ lm id F , (8.27)
which should be imposed after expressions like E(a † ± m • a ∓ n ) are explicitly calculated. These relations convert (8.17) and (8.25) into identities and by their means the r.h.s. of (8.5) can be calculated explicitly, but, as it is well known [1, 3, 11, 12, 27] they lead to deep contradictions in the theory, due to which should be rejected. 31 At present, it seems, the bilinear commutation relations (6.13) are the only known commutation relations which satisfy all of the mentioned conditions and simultaneously provide an evident procedure for effective calculation of all expressions of the form (8.5). (Besides, for them and for the paracommutation relations the vectors (8.4) form a base, the Fock base, for the system's Hilbert space of states [18] .) In this connection, we want to mention that the paracommutation relations (6.16) (or their conventional version (6.20)), if imposed as additional restrictions to the theory together with (8.17), reduce in this particular case to (6.13) as the conditions (8.25) show that we are dealing with a parafield of order p = 1, i.e. with an ordinary field [17, 18] . 32 Ending this section, let us return to the definition of the vacuum X 0 . It, generally, depends on the adopted commutation relations. For instance, in a case of the bilinear commutation relations (6.13) it consists of the equations (8.1a)-(8.3), while in a case of the paracommutation relations (6.16) (or other ones generalizing (6.13)) it includes (8.1a)-(8.3) and (8.25).
Until now we have considered commutation relations for a single free field, which can be scalar, or spinor or vector one. The present section is devoted to similar treatment of a system consisting of several, not less than two, different free fields. In our context, the fields may differ by their masses and/or charges and/or spins; e.g., the system may consist of charged scalar field, neutral scalar field, massless spinor field, massive spinor field and massless neural vector field. It is a priori evident, the commutation relations regarding only one field of the system should be as discussed in the previous sections. The problem is to be derived/postulated commutation relations concerning different fields. It will be shown, the developed Lagrangian formalism provides a natural base for such an investigation and makes superfluous some of the assumptions made, for example, in [17, p. B 1159, left column] or in [18, sec. 12.1] , where systems of different parafields are explored.
To begin with, let us introduce suitable notation. With the indices α, β, γ = 1, 2, . . . , N will be distinguished the different fields of the system, with N ∈ N, N ≥ 2, being their number, and the corresponding to them quantities. Let q α and j α be respectively the charge and spin of the α-th field. Similarly to (3.7), we define
j α := 0 for scalar α-th field 1 2
for spinor α-th field 1 for vector α-th field τ α := 1 for q α = 0 (neutral (Hermitian) field) 0 for q α = 0 (charged (non-Hermitian) field)
ε α := (-1) 2j α = +1 for integer j α (bose fields) -1 for half-integer j α (fermi fields) .
(9.1) Suppose L α is the Lagrangian of the α-field. For definiteness, we assume L α for all α to be given by one and the same set of equations, viz. (3.1), or (3.3) or (3.4) . To save some space, below the case (3.4), corresponding to charge symmetric Lagrangians, will be considered in more details; the reader can explore other cases as exercises.
Since the Lagrangian of our system of free fields is
L := α L α , (9.2)
the dynamical variables are
D = α D α (9.3)
and the corresponding system of Euler-Lagrange equations consists of the independent equations for each of the fields of the system (see (3.6) with L α for L). This allows an introduction of independent creation and annihilation operators for each field. The ones for the α-th field will be denoted by a ± α,s α (k) and a † ± α,s α (k); notice, the values of the polarization variables generally depend on the field considered and, therefore, they also are labeled with index α for the α-th field. For brevity, we shall use the collective indices l α , m α and n α , with l α := (α, s α , k) etc., in terms of which the last operators are a ± l α and a † ± l α , respectively. The particular expressions for the dynamical operators D α are given via (3.9)-(3.12) in which the following changes should be made:
τ → τ α j → j α ε → ε α s → s α s ′ → s ′ α σ ss ′ ,± µν (k) → σ s α s ′ α ,± µν (k) l ss ′ ,± µν (k) → l s α s ′ α ,± µν (k). (9.4)
The content of sections 4 and 5 remains valid mutatis mutandis, viz. provided the just pointed changes (9.4) are made and the (integral) dynamical variables are understood in conformity with (9.3).
In sections 6-8, however, substantial changes occur; for instance, when one passes from (6.12) or (6.15) to (6.16) . We shall consider them briefly in a case when one starts from the charge symmetric Lagrangians (3.4) . The basic relations (6.12), which arise from the Heisenberg relation (5.1) concerning the momentum operator, now read (here and below, do not sum over α, and/or β and/or γ if the opposite is not indicated explicitly!)
a ± l α , [a † + m β , a - m β ] ε β + [a + m β , a † - m β ] ε β -± (1 + τ )δ l α m β a ± l α = 0 (9.5a) a † ± l α , [a † + m β , a - m β ] ε β + [a + m β , a † - m β ] ε β -± (1 + τ )δ l α m β a † ± l α = 0. (9.5b)
It is trivial to be seen, the following generalizations of respectively (6.14) and (6.15)
a ± l α , [a + m β , a † - m β ] ε β -± (1 + τ β )δ l α m β a ± l α = 0 (9.6a) a ± l α , [a † + m β , a - m β ] ε β -± (1 + τ β )δ l α m β a ± l α = 0 (9.6b) a † ± l α , [a + m β , a † - m β ] ε β -± (1 + τ β )δ l α m β a † ± l α = 0 (9.6c) a † ± l α , [a † + m β , a - m β ] ε β -± (1 + τ β )δ l α m β a † ± l α = 0 (9.6d) a + l α , [a + m β , a † - m β ] ε β -+ 2δ l α m β a + l α = 0 (9.7a) a + l α , [a † + m β , a - m β ] ε β -+ 2τ β δ l α m β a + l α = 0 (9.7b) a - l α , [a + m β , a † - m β ] ε β --2τ β δ l α m β a - l α = 0 (9.7c) a - l α , [a † + m β , a - m β ] ε β --2δ l α m β a - l α = 0 (9.7d)
provide a solution of (9.5) in a sense that they convert it into identity. As it was said in Sect. 6, the equations (9.6) (resp. (9.7)) for a single field, i.e. for β = α, agree (resp. disagree) with the bilinear commutation relations (6.13).
The only problem arises when one tries to generalize, e.g., the relations (9.7) in a way similar to the transition from (6.15) to (6.16 ). Its essence is in the generalization of expressions like [a † ± m β , a ∓ m β ] ε β and τ β δ l α m β a ± l α . When passing from (6.15) to (6.16), the indices l and m are changed so that the obtained equations to be consistent with (6.13); of course, the numbers ε and τ are preserved because this change does not concern the field regarded. But the situation with (9.7) is different in two directions:
(i) If we change the pair (m
β , m β ) in [a † ± m β , a ∓ m β ] ε β with (m β , n γ ),
then with what the number ε β should be replace? With ε β , or ε γ or with something else? Similarly, if the mentioned changed is performed, with what the multiplier τ β in τ β δ l α m β a ± l α should be replaced? The problem is that the numbers ε β and τ β are related to terms like a † ± m β • a ∓ m β and a ± m β • a † ∓ m β , in the momentum operator, as a whole and we cannot say whether the index β in ε β and τ β originates from the first of second index m β in these expressions.
(ii) When writing (m β , n γ ) for (m β , m β ) (see (i) above), then shall we replace δ l α m β a ± l α with δ l α m β a ± n γ , or δ l α n γ a ± m β , or δ m β n γ a ± l α ? For a single field, γ = β = α, this problem is solved by requiring an agreement of the resulting generalization (of (6.16) in the particular case) with the bilinear commutation relations (6.13). So, how shall (6.13) be generalized for several, not less than two, different fields? Obviously, here we meet an obstacle similar to the one described in (i) above, with the only change that -ε β should stand for ε β .
Let b l α and c l α denote some creation or annihilation operator of the α-field. Consider the problem for generalizing the (anti)commutator [b l α , c l α ] ±ε α . This means that we are looking for a replacement
[b l α , c l α ] ±ε α → f ± (b l α , c m β ; α, β), (9.8)
where the functions f ± are such that
f ± (b l α , c m β ; α, β) β=α = [b l α , c l α ] ±ε α . (9.9)
Unfortunately, the condition (9.9) is the only restriction on f ± that the theory of free fields can provide. Thus the functions f ± , subjected to equation (9.9), become new free parameters of the quantum theory of different free fields and it is a matter of convention how to choose/fix them.
It is generally accepted [18, appendix F], the functions f ± to have forms 'maximum' similar to the (anti)commutators they generalize. More precisely, the functions
f ± (b l α , c m β ; α, β) = [b l α , c m β ] ±ε αβ (9.10)
where ε αβ ∈ C are such that
ε αα = ε α , (9.11)
are usually considered as the only candidates for f ± . Notice, in (9.10), ε αβ are functions in α and β, not in l α and/or m β . Besides, if we assume ε αβ to be function only in ε α and ε β , then the general form of ε αβ is
ε αβ = u αβ ε α + (1 -u αβ )ε β + v αβ (1 -ε α ε β ) u αβ , v αβ ∈ C, (9.12)
due to (9.1) and (9.11) . (In view of (6.13), the value ε αβ = +1 (resp. ε αβ = -1) corresponds to quantization via commutators (resp. anticommutators) of the corresponding fields.) Call attention now on the numbers τ α which originate and are associated with each term [b l α , c m α ] ±ε α . With every change (9.8) one can associate a replacement
τ α → g(b l α , c m β ; α, β), (9.13)
where the function g is such that
g(b l α , c m β ; α, β) β=α = τ α . (9.14)
Of course, the last condition does not define g uniquely and, consequently, the function g, satisfying (9.14), enters in the theory as a new free parameter. Suppose, as a working hypothesis similar to (9.10)-(9.11), that g is of the form
g(b l α , c m β ; α, β) = τ αβ , (9.15)
where τ αβ are complex numbers that may depend only on α and β and are such that
τ αα = τ α . (9.16)
Besides, if we suppose τ αβ to be functions only in τ α and τ β , then
τ αβ = x αβ τ α + y αβ τ β + (1 -x αβ -y αβ )τ α τ β x αβ , y αβ ∈ C, (9.17)
as a result of (9.1) and (9.16).
Let us summarize the above discussion. If we suppose a preservation of the algebraic structure of the bilinear commutation relations (6.13) for a system of different free fields, then the replacements
[b l α , c l α ] ±ε α → [b l α , c m β ] ±ε αβ ε αα = ε α (9.18a) τ α → τ αα τ αα = τ α (9.18b)
should be made; accordingly, the relations (6.13) transform into:
[a ± l α , a ± m β ] -ε αβ = 0 [a † ± l α , a † ± m β ] -ε αβ = 0 [a ∓ l α , a ± m β ] -ε αβ = τ αβ δ l α m β id F × 1 -ε αβ [a † ∓ l α , a † ± m β ] -ε αβ = τ αβ δ l α m β id F × 1 -ε αβ [a ± l α , a † ± m β ] -ε αβ = 0 [a † ± l α , a ± m β ] -ε αβ = 0 [a ∓ l α , a † ± m β ] -ε αβ = δ l α m β id F × 1 -ε αβ [a † ∓ l α , a ± m β ] -ε αβ = δ l α m β id F × 1 -ε αβ , (9.19)
where 1 (resp. -ε αβ ) in 1 -ε αβ corresponds to the choice of the upper (resp. lower) signs. If we suppose additionally ε αβ (resp. τ αβ ) to be a function only in ε α and ε β (resp. in τ α and τ β ), then these numbers are defined up to two sets of complex parameters:
ε αβ = u αβ ε α + (1 -u αβ )ε β + v αβ (1 -ε α ε β ) u αβ , v αβ ∈ C (9.20a) τ αβ = x αβ τ α + y αβ τ β + (1 -x αβ -y αβ )τ α τ β x αβ , y αβ ∈ C. (9.20b)
A reasonable further specialization of ε αβ and τ αβ may be the assumption their ranges to coincide with those of ε α and τ α , respectively. As a result of (9.1), this supposition is equivalent to
v αβ = -u αβ , -u αβ + 1, u αβ -1, u αβ u αβ ∈ C (9.21a) (x αβ , y αβ ) = (0, 0), (0, 1), (1, 0), (1, 1). (9.21b)
Other admissible restriction on (9.20) may be the requirement ε αβ and τ αβ to be symmetric, viz.
ε αβ (ε α , ε β ) = ε βα (ε α , ε β ) = ε αβ (ε β , ε α ) (9.22a) τ αβ (τ α , τ β ) = τ βα (τ α , τ β ) = τ αβ (τ β , τ α ), (9.22b)
which means that the α-th and β-th fields are treated on equal footing and there is no a priori way to number some of them as the 'first' or 'second' one. 33 In view of (9.20), the conditions (9.22) are equivalent to
u αβ = 1 2 v αβ ∈ C (9.23a)
y αβ = x αβ . (9.23b)
If both of the restrictions (9.21) and (9.23) are imposed on (9.20), then the arbitrariness of the parameters in (9.20) is reduced to:
(u αβ , u αβ ) = 1 2 , - 1 2 , 1 2 , 1 2 (9.24a)
(x αβ , y αβ ) = (0, 0), (1, 1) (9.24b) and, for any fixed pair (α, β), we are left with the following candidates for respectively ε αβ and τ αβ :
ε αβ + := 1 2 (+1 + ε α + ε β -ε α ε β ) (9.25a) ε αβ -:= 1 2 (-1 + ε α + ε β + ε α ε β ) (9.25b) τ αβ 0 := τ α + τ β (9.25c) τ αβ 1 := τ α + τ β -τ α τ β . (9.25d)
When free fields are considered, as in our case, no further arguments from mathematical or physical nature can help for choosing a particular combination (ε αβ , τ αβ ) from the four possible ones according to (9.25) for a fixed pair (α, β). To end the above considerations of ε αβ and τ αβ , we have to say that the choice
(ε αβ , τ αβ ) = (ε αβ + , τ αβ 0 ) = 1 2 (+1 + ε α + ε β -ε α ε β ), τ α + τ β (9.26)
33 However, nothing can prevent us to make other choices, compatible with (9.18), in the theory of free fields; for instance, one may set
ε αβ = ε α ε β ε βα and τ αβ = 1 2 (τ α + τ β )τ βα .
is known as the normal case [18, appendix F] ; in it the relative behavior of bose (resp. fermi) fields is as in the case of a single field, i.e. they are quantized via commutators (resp. anticommutators) as (ε αβ , τ αβ ) = (+1, 0) (resp. (ε αβ , τ αβ ) = (-1, 0)), and the one of bose and fermi field is as in the case of a single fermi field, viz. the quantization is via commutators as (ε αβ , τ αβ ) = (+1, 0). All combinations between ε αβ ± and τ αβ 0,1 different from (9.26) are referred as anomalous cases. Above we supposed the pair (α, β) to be fixed. If α and β are arbitrary, the only essential change this implies is in (9.25) , where the choice of the subscripts +, -, 0 and 1 may depend on α and β. In this general situation, the normal case is defined as the one when (9.26) holds for all α and β. All other combinations are referred as anomalous cases; such are, for instance, the ones when some fermi and bose operators satisfy anticommutation relations, e.g. (9.19) with ε αβ = -1 for ε α + ε β = 0, or some fermi fields are subjected to commutation relations, like (9.19) with ε αβ = +1 for ε α = ε β = -1. For some details on this topic, see, for instance, [18, appendix F], [7, chapter 20] and [27, sect 4-4] . Fields/operators for which ε αβ = +1 (resp. ε αβ = -1), with β = α, are referred as relative parabose (resp. parafermi ) in the parafield theory [17, 18] . One can transfer this terminology in the general case and call the fields/operators for which ε αβ = +1 (resp. ε αβ = -1), with β = α, relative bose (resp. fermi ) fields/operators.
Further the relations (9.19) will be referred as the multifield bilinear commutation relations and it will be assumed that they represent the generalization of the bilinear commutation relations (6.13) when we are dealing with several, not less than two, different quantum fields. The particular values of ε αβ and τ αβ in them are insignificant in the following; if one likes, one can fix them as in the normal case (9.26) . Moreover, even the definition (9.19) of τ αβ is completely inessential at all, as τ αβ always appears in combinations like τ αβ δ l α m β (see (9.19) or similar relations, like (9.27), below), which are non-vanishing if β = α, but then τ αα = τ α ; so one can freely write τ α for τ αβ in all such cases.
Equipped with (9.19) and (9.18), we can generalize (9.7) in different ways. For example, the straightforward generalization of (6.16) is:
a + l α , [a + m β , a † - n γ ] ε βγ -+ 2δ l α n γ a + m β = 0 (9.27a) a + l α , [a † + m β , a - n γ ] ε βγ -+ 2τ αγ δ l α n γ a + m β = 0 (9.27b) a - l α , [a + m β , a † - n γ ] ε βγ --2τ αβ δ l α m β a - n γ = 0 (9.27c) a - l α , [a † + m β , a - n γ ] ε βγ --2δ l α m β a - n γ = 0. ( 9
.27d) However, generally, the relations (9.19) do not convert (9.27) into identities. The reason is that an equality/identity like (cf. (6.8)) [
b l α , c m β • d n γ ] = [b l α , c m β ] -ε αβ • d n γ + λ αβγ c m β • [b l α , d n γ ] -ε αγ , (9.28)
where b l α , c m β and d n γ are some creation/annihilation operators and λ αβγ ∈ C, can be valid only for
λ αβγ = ε αβ ε αγ = 1/ε αβ (ε αβ = 0), ( 9
.29) which, in particular, is fulfilled if γ = β and ε αβ = ±1. So, the agreement between (9.19) and (9.27) depends on the concrete choice of the numbers ε αβ . There exist cases when even the normal case (9.26) cannot ensure (9.19) to convert (9.27) into identities; e.g. when the α-th field and β-th fields are fermion ones and the γ-th field is a boson one. Moreover, it can be proved that (9.19) and (9.27) are compatible in the general case if unacceptable equalities like a ± l • a ± m = 0 hold. One may call (9.27) the multifield paracommutation relations as from them a corresponding generalization of (6.18) and/or (6.20) can be derived. For completeness, we shall record the multifield version of (6.20):
[b l α , [b † m β , b n γ ] ε βγ ] = 2δ l α m β b n γ [b l α , [b m β , b n γ ] ε βγ ] = 0 (9.30a) [c l α , [c † m β , c n γ ] ε βγ ] = 2δ l α m β c n γ [c l α , [c m β , c n γ ] ε βγ ] = 0 (9.30b) [b † l α , [c † m β , c n γ ] ε βγ ] = -2τ αγ δ l α n γ b † m β [c † l α , [b † m β , b n γ ] ε βγ ] = -2τ αγ δ l α n γ c † m β . (9.30c)
For details regarding these multifield paracommutation relations, the reader is referred to [17, 18] , where the case τ α = τ β = τ αβ = 0 is considered. We leave to the reader as exercise to write down the multifield versions of the commutation relations (6.22) or (6.23), which provide examples of generalizations of (9.7) and hence of (9.19) and (9.27).
In a case of several, not less than two, different fields, the basic trilinear commutation relations (6.33), which ensure the validity of the Heisenberg relation (5.2) concerning the charge operator, read:
a ± l α , [a † + m β , a - m β ] ε β -[a + m β , a † - m β ] ε β --2δ l α m β a ± l α = 0 (9.31a) a † ± l α , [a † + m β , a - m β ] ε β -[a + m β , a † - m β ] ε β -+ 2δ l α m β a † ± l α = 0. (9.31b)
Of course, these relations hold only for those fields which have non-vanishing charges, i.e. in (9.31) is supposed (see (9.1))
τ α = 0 τ β = 0 ( ⇐⇒ q α q β = 0). (9.32)
The problem for generalizing (9.31) for these fields is similar to the one for (9.7) in the case of non-vanishing charges, τ β = 0. Without repeating the discussion of Subsect. 9.1, we shall adopt the rule (9.18) for generalizing (anti)commutation relations between creation/annihilation operators of a single field. By its means one can obtain different generalizations of (9.31). For instance, the commutation relations.
a + l α , [a † + m β , a - n γ ] ε βγ -[a + m β , a † - n γ ] ε βγ --2δ l α n γ a + m β = 0 (9.33a) a - l α , [a † + m β , a - n γ ] ε βγ -[a + m β , a † - n γ ] ε βγ --2δ l α m β a - n γ = 0 (9.33b)
and their Hermitian conjugate contain (9.31) and (6.35) as evident special cases and agree with (9.19) if γ = β and ε αβ ε βγ = +1. Besides, the multifield paracommutation relations (9.27) for charged fields, τ α = τ β = τ γ = 0, convert (9.33) into identities and, in this sense, (9.33) agree with (contain as special case) (9.27) for charged fields. As an example of commutation relations that do not agree with (9.27) for charged fields and, consequently, with (9.33), we shall point the following ones:
a ± l α , [a + m β , a † - n γ ] ε βγ -+ δ l α n γ a ± m β = 0 (9.34a) a ± l α , [a † + m β , a - n γ ] ε βγ ---δ l α n γ a ± m β = 0, (9.34b)
which are a multifield generalization of (6.34).
The consideration of commutation relations originating from the 'orbital' Heisenberg equation (5.4) is analogous to the one of the same relations regarding the charge operator. The multifield version of (6.49) is:
(-ω • µν (m β ) + ω • µν (n γ ))([ã ± l α , [ã † + m β , ã- n γ ] ε βγ + [ã + m β , ã † - n γ ] ε βγ ] ) n γ =m β = 4(1 + τ αβ )δ l α m β ω • µν (l α )(ã ± l α ) (9.35a) (-ω • µν (m β ) + ω • µν (n γ ))([ã † ± l α , [ã † + m β , ã- n γ ] ε βγ + [ã + m β , ã † - n γ ] ε βγ ] ) n γ =m β = 4(1 + τ αβ )δ l α m β ω • µν (l α )(ã † ± l α ) (9.35b) where ω • µν (l α ) := ω µν (k) = k µ ∂ ∂k ν -k ν ∂ ∂k µ if l α = (α, s α , k). (9.36)
Applying (6.51), with m β for m and n γ for n, one can check that the multifield paracommutation relations (9.27) convert (9.35) into identities and hence provide a solution of (9.35) and ensure the validity of (5.4), when system of different free fields is considered. An example of a solution of (9.35) which does not agree with (9.27) is provided by the following multifield generalization of (6.52):
a + l α , [a + m β , a † - n γ ] ε βγ -= a + l α , [a † + m β , a - n γ ] ε βγ -= -(1 + τ αγ )δ l α n γ a + m β (9.37a) a - l α , [a + m β , a † - n γ ] ε βγ -= a - l α , [a † + m β , a - n γ ] ε βγ -= +(1 + τ αβ )δ l α m β a + n γ , (9.37b)
which provides a solution of (9.5). Notice, the evident multifield version of (6.53) agrees with (9.5), but disagrees with (9.35) when the lower signs are used. At last, the multifield exploration of the 'spin' Heisenberg relations (5.5) is a mutatis mutandis (see (9.35 )) version of the corresponding considerations in the second part of Subsect. 6.3. The main result here is that the multifield bilinear commutation relations (9.19), as well as their para counterparts (9.27), ensure the validity of (5.5).
The aim of this subsection is to be discussed/proved the commutation relations (5.15)-(5.24) for a system of at least two different quantum fields from the view-point of the commutation relations considered in subsections 9.1 and 9.2.
To begin with, we rewrite the Heisenberg relations (5.1), (5.2) and (5.4) in terms of creation and annihilation operators for a multifield system [1, 11] : where l α = (α, s α , k), ω • (l α ) is defined by (9.36) and k 0 = m 2 c 2 + k 2 is set in (9.38) and (9.40) (after the differentiations are performed in the last case). The corresponding version of (5.5) is more complicated and depends on the particular field considered (do not sum over s α !):
f s α [a ± α,s α (k), M sp µν ] = i g α t α
± σ s α t α ,+ µν (k)a + α,t α (k) + ± σ s α t α ,- µν (k)a - α,t α (k)
f s α [a † ± α,s α (k), M sp µν ] = i h α t α ± σ s α t α ,- µν (k)a † + α,t α (k) + ± σ s α t α ,+ µν (k)a † - α,t α (k) , (9.41)
where f s α = -1, 0, +1 (depending on the particular field), g α := -h α := 1 j α +δ j α 0 (-1) j α +1 and ± σ s α t α ,+ µν (k) and ± σ s α t α ,- µν (k) are some functions which strongly depend on the particular field considered, with ± σ s α t α ,± µν (k) being related to the spin (polarization) functions σ s α t α ,± µν (k) (see (3.14) and (3.11) ). 34 As a result of (5.6), (9.40) and (9.41), one can easily write the Heisenberg relations (5.3) in a form similar to (9.38)-(9.41).
l α ← -- -→ ω • µν (l α ) • c ∓ m β , P µ ] = ±2(k µ η νλ -k ν η µλ )b ± l α • c ∓ m β , (9.46b)
where b ± l α , c ± l α = a ± l α , a † ± l α and ← ---→ ω • µν (l α ) is defined via (9.36) and (3.13), the verification of (9.45) reduces to almost trivial algebraic calculations. Further, we assert that any system of commutation relations considered in Subsect. 9.1 entails (9.45): as these relations always imply (9.5) (or similar multifield versions of (6.10) and (6.11) in the case of the Lagrangians (3.1) or (3.3), respectively) and, on its turn, (9.5) implies (5.1), the required result follows from the last assertion and the remark that (5.1) and (9.38) are equivalent. As an additional verification of the validity of (9.45), the reader can prove them by invoking the identity (6.8) and any system of commutation relations mentioned in Subsect. 9.1, in particular (9.19) and (9.27) .
The commutation relations concerning the charge operator read:
[ P µ , Q] = 0 [ Q, Q] = 0 [ L µν , Q] = [ S µν , Q] = 0 [ M or µν , Q] = [ M sp µν , Q] = [ M µν , Q] = 0.
(9.47)
These equations are trivial corollaries from (3.9)-(3.12) and (5.11)-(5.13) and the observation that (9.39) implies
[a † ± l α • a ∓ m β , Q] = [a ± l α • a † ∓ m β , Q] = 0 if q α = q β , (9.48)
due to (6.8) for η = -1. Since any one of the systems of commutation relations mentioned in Subsect. 9.2 entails (9.31) (or systems of similar multifield versions of (6.31) and (6.32), if the Lagrangians (3.1) or (3.3) are employed), which is equivalent to (9.39), the equations (9.47) hold if some of these systems is valid. Alternatively, one can prove via a direct calculation that the commutation relations arising from the charge operator entail the validity of (9.47);
where v s α ,± i (k) are linearly independent functions normalize via the condition
X i `vs α ,± i (k) ´ * v t α ,± i (k) = δ s α t α f s α , (9.43)
with f s α = 1 for j α = 0, 1 2 and f s α = 0, -1 for (j α , s α ) = (1, 3) or (j α , s α ) = (1, 1), (1, 2), respectively, then
+ σ s α t α ,± µν (k) := 1 gα X i,i ′ `vs α ,+ i (k) ´ * I i ′ iµν v t α ,± i ′ (k) -σ s α t α ,± µν (k) := 1 gα X i,i ′ `vs α ,- i (k) ´ * I i ′ iµν v t α ,± i ′ (k), (9.44)
with I i ′ iµν given via (5.25) . Besides, σ s α t α ,± µν (k) = ± σ s α t α ,± µν (k) with an exception that σ s α t α ,± µν (k) = 0 for j α = 1 2 and (µ, ν) = (a, 0), (0, a) with a = 1, 2, 3.
for the purpose the identity (6.8) and the explicit expressions for the dynamical variables via the creation and annihilation operators should be applied. At last, consider the commutation relations involving the different angular momentum operators: with b l α , c l α = a + l α , a - l α , a † + l α , a † - l α , and similar, but more complicated, ones involving the other angular momentum operators. It, generally, depends on the particular field considered and will be omitted.
As it was said in Subsect. 6.3, the Heisenberg relations concerning the angular momentum operator(s) do not give rise to some (algebraic) commutation relations for the creation and annihilation operators. For this reason, the only problem is which of the commutation relations discussed in subsections 9.1 and 9.2 imply the validity of the equations (9.49) (or part of them). The general answer of this problem is not known but, however, a direct calculation by means of (9.7), if it holds, and (6.8) shows the validity of (9.49). Since (9.19) and (9.27) imply (9.7), this means that the multifield bilinear and para commutation relations are sufficient for the fulfillment of (9.49).
To conclude, let us draw the major moral of the above material: the multifield bilinear commutation relations (9.19) and the multifield paracommutation relations (9.27) ensure the validity of all 'standard' commutation relations (9.45), (9.47) and (9.49) between the operators of the dynamical variables characterizing free scalar, spinor and vector fields.
As it was said at the end of the introduction to this section, the replacements (9.4) ensure the validity of the material of Sect. 4 in the multifield case. Correspondingly, the considerations in Sect. 7 remain valid in this case provided the changes The multifield version of (7.27)-(7.28) is:
l → l α m → m β n → n γ τ δ lm → τ αβ δ l α m β = τ α δ l α m β [b m , b m ] ε → [b m β , b m β ] ε β [b m , b n ] ε → [b m β , b n γ ] ε βγ ,
E(a † ± m β • a ∓ n γ ) = ε βγ E(a ∓ n γ • a † ± m β ) = 1 2 E([a † ± m β ,
a ∓ n γ ] ε βγ ) (9.52) a + l α , E([a + m β , a † - n γ ] ε βγ ) -+ 2δ l α n γ a + m β = 0 (9.53a) a + l α , E([a † + m β , a - n γ ] ε βγ ) -+ 2τ αγ δ l α n γ a + m β = 0 (9.53b) a - l α , E([a + m β , a † - n γ ] ε βγ ) --2τ αβ δ l α m β a - n γ = 0 (9.53c) a - l α , E([a † + m β , a - n γ ] ε βγ ) --2δ l α m β a - n γ = 0 (9.53d) γ =β. (9.53e) As one can expect, the relations (9.53a)-(9.53d) can be obtained from the multifield paracommutation relations (9.27) via the replacement [•, •] ε → E([•, •] ε βγ ). It should be paid special attention on the equation (9.53e). It is due to the fact that in the expressions for the dynamical variables do not enter 'cross-field-products', like a † + l α •a - m β for β = α, and it corresponds to the condition (ii) in [17, p. B 1159]. The equality (9.53e) is quite important as it selects only that part of the ' E-transformed' multifield paracommutation relations (9.27) which is compatible with the bilinear commutation relations (9.19) (see (9.28) and (9.29)). Besides, (9.53e) makes (9.53a)-(9.53d) independent of the particular definition of ε αβ (see (9.11)). The equations (9.52) are the only restrictions on the operator E; examples of this operator are provided by the normal (resp. antinormal) ordering operator N (resp. A), which has the properties (cf. (4.22) (resp. (7.30))
N a + m β • a † - n γ := a + m β • a † - n γ N a † + m β • a - n γ := a † + m β • a - n γ N a - m β • a † + n γ := ε βγ a † + n γ • a - m β N a † - m β • a + n γ := ε βγ a + n γ • a † - m β (9.54) A a + m β • a † - n γ := ε βγ a † - n γ • a + m β A a † + m β • a - n γ := ε βγ a - n γ • a † + m β A a - m β • a † + n γ := a - m β • a † + n γ A a † - m β • a + n γ := a † - m β • a + n γ . (9.55)
The material of Sect. 8 has also a multifield variant that can be obtained via the replacements (9.51) and (9.4) . Here is a brief summary of the main results found in that way.
The operator E should possess the properties (9.54) and, in this sense, can be identified with the normal ordering operator, E = N . (9.56)
As a result of this fact and ε ββ = ε β (see (9.11) ), the commutation relations (9.53) take the final form:
a + l α , a + m β • a † - n β -+ δ l α n β a + m β = 0 (9.57a) a + l α , a † + m β • a - n β -+ τ αβ δ l α n β a + m β = 0 (9.57b) a - l α , a + m β • a † - n β --τ αβ δ l α m β a - n β = 0 (9.57c) a - l α , a † + m β • a - n β --δ l α m β a - n β = 0 (9.57d)
which is the multifield version of (8.17) and corresponds, up to the replacement a ± l α → √ 2a ± l α , to (9.27) with ε βγ = 0.
The vacuum state vector X 0 is supposed to be uniquely defined by the following equations (cf. (8.1b)-(8.3)): a - l α X 0 = 0 a † - l α X 0 = 0 (9.58a) X 0 = 0 (9.58b) X 0 | X 0 = 1 (9.58c)
a † - l α • a + m β ( X 0 ) = δ l α m β X 0 a - l α • a † + m β ( X 0 ) = δ l α m β X 0 a - l α • a + m β ( X 0 ) = τ αβ δ l α m β X 0 a - l α • a + m β ( X 0 ) = τ αβ δ l α m β X 0 .
(9.58d)
Further constraints on the possible commutation relations follow from the definition/introduction of the concept of the vacuum (vacuum state vector). They practically reduce the redefined dynamical variables to the ones obtained via normal ordering procedure, which results in the explicit form (8.17) of the admissible commutation relations. In a sense, they happen to be 'one half' of the paracommutation ones. As a last argument in the way for finding the 'unique true' commutation relations, we require the existence of procedure for calculation of vacuum mean values of anti-normally ordered products of creation and annihilation operators, to which the mean values of the dynamical variables and the transition amplitudes between different states are reduced. We have pointed that the standard bilinear commutation relations are, at present, the only known ones that satisfy all of the conditions imposed and do not contradict to the existing experimental data.
The consideration of a system of at least two different quantum free fields meets a new problem: the general relations between creation/annihilation operators belonging to different fields turn to be undefined. The cause for this is that the commutation relations for any fixed field are well defined only on the corresponding to it Hilbert subspace of the system's Hilbert space of states and their extension on the whole space, as well as the inclusion in them of creation/annihilation operators of other fields, is a matter of convention (when free fields are concerned); formally this is reflected in the structure of the dynamical variables which are sums of those of the individual fields included in the system under consideration. We have, however, presented argument by means of which the a priori existing arbitrariness in the commutation relations involving different field operators can be reduced to the 'standard' one: these relations should contain either commutators or anticommutators of the creation/annihilation operators belonging to different fields. A free field theory cannot make difference between these two possibilities. Accepting these possibilities, the admissible commutation relations (9.57) for system of several different fields are considered. They turn to be corresponding multifield versions of the ones regarding a single field. Similarly to the single field case, the standard multifield bilinear commutation relations seem to be the only known ones that satisfy all of the imposed restrictions and are in agreement with the existing data. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Possible (algebraic) commutation relations in the Lagrangian quantum theory of free (scalar, spinor and vector) fields are considered from mathematical view-point. As sources of these relations are employed the Heisenberg equations/relations for the dynamical variables and a specific condition for uniqueness of the operators of the dynamical variables (with respect to some class of Lagrangians). The paracommutation relations or some their generalizations are pointed as the most general ones that entail the validity of all Heisenberg equations. The simultaneous fulfillment of the Heisenberg equations and the uniqueness requirement turn to be impossible. This problem is solved via a redefinition of the dynamical variables, similar to the normal ordering procedure and containing it as a special case. That implies corresponding changes in the admissible commutation relations. The introduction of the concept of the vacuum makes narrow the class of the possible commutation relations; in particular, the mentioned redefinition of the dynamical variables is reduced to normal ordering. As a last restriction on that class is imposed the requirement for existing of an effective procedure for calculating vacuum mean values. The standard bilinear commutation relations are pointed as the only known ones that satisfy all of the mentioned conditions and do not contradict to the existing data."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The main subject of this paper is an analysis of possible (algebraic) commutation relations in the Lagrangian quantum theory foot_0 of free fields. These relations are considered only from mathematical view-point and physical consequence of them, like the statistics of many-particle systems, are not investigated.\n\nThe canonical quantization method finds its origin in the classical Hamiltonian mechanics [9, 10] and naturally leads to the canonical (anti)commutation relations [3, 11, 12] . These relations can be obtained from different assumptions (see, e.g., [1, [13] [14] [15] ) and are one of the basic corner stones of the present-day quantum field theory.\n\nTheoretically there are possible also non-canonical commutation relations. The best known example of them being the so-called paracommutation relations [16] [17] [18] . But, however, it seems no one of the presently known particles/fields obeys them.\n\nIn the present work is shown how different classes of commutation relations, understood in a broad sense as algebraic connections between creation and/or annihilation operators, arise from the Lagrangian formalism, when applied to three types of Lagrangians describing free scalar, spinor and vector fields. Their origin is twofold. One one hand, a requirement for uniqueness of the dynamical variables (that can be calculated from Lagrangians leading to identical Euler-Lagrange equation) entails a number of specific commutation relations. On another hand, any one of the so-called Heisenberg relations/equations [3, 11] , implies corresponding commutation relations; for example, the paracommutation relations arise from the Heisenberg equations regarding the momentum operator, when 'charge symmetric' Lagrangian is employed. 2 The combination of the both methods leads to strong, generally incompatible, restrictions on the admissible types of commutation relations.\n\nThe introduction of the concept of vacuum, combined with the mentioned uniqueness of the operators of the dynamical variables, changes the situation and requires a redefinition of these operators in a way similar to the one known as the normal ordering [1, 3, 11, 12] , which is its special case. Some natural assumptions reduce the former to the letter one; in particular, in that way are excluded the paracommutation relations. However, this does not reduce the possible commutation relations to the canonical ones. Further, the requirement to be available an effective procedure for calculating vacuum mean (expectation) values, to which reduce all predictable results in the theory, puts new restriction, whose only realistic solution at the time being seems to be the standard canonical (anti)commutation relations.\n\nThe layout of the work is as follows.\n\nSect. 2 gives an idea of the momentum picture of motion and discusses the relations between the creation and annihilation operators in it and in Heisenberg picture. In Sect. 3 are reviewed some basic results from [13] [14] [15] , part of which can be found also in papers like [1, 3, 11, 12] . In particular, the explicit expression of the dynamical variables via the creation and annihilation operators are presented (without assuming some commutation relations or normal ordering) and it is pointed to the existence of a family of such variables for a given system of Euler-Lagrange equations for free fields. The last fact is analyzed in Sect. 4 , where a number of its consequences, having a sense of commutation relations, are drawn. The Heisenberg relations and the commutation relations between the dynamical variables are reviewed and analyzed in Sect. 5. It is pointed that the letter should be consequences from the former ones. Arguments are presented that the Heisenberg equation concerning the angular momentum operator should be split into two independent ones, representing its 'orbital' and 'spin' parts, respectively.\n\nSect. 6 contains a method for assigning commutation relations to the Heisenberg equations. It is shown that the Heisenberg equation involving the 'orbital' part of the angular momentum gives rise to a differential, not algebraic, commutation relation and the one concerning the 'spin' part of the angular momentum implies a complicated integro-differential connections between the creation and annihilation operators. Special attention is paid to the paracommutation relations, whose particular kind are the ordinary ones, which ensure the validity of the Heisenberg equations concerning the momentum operator. Partially is analyzed the problem for compatibility of the different types of commutation relations derived. It is proved that some generalization of the paracommutation relations ensures the fulfillment of all of the Heisenberg relations.\n\nSect. 7 is devoted to consequences from the commutation relations derived in Sect. 6 under the conditions for uniqueness of the dynamical variables presented in Sect. 4. Generally, these requirements are incompatible with the commutation relations. To overcome the problem, it is proposed a redefinition of the dynamical variables via a method similar to (and generalizing) the normal ordering. This, of course, entails changes in the commutation relations, the new versions of which happen to be compatible with the uniqueness conditions and ensure the validity of the Heisenberg relations.\n\nThe concept of the vacuum is introduced in Sect. 8. It reduces (practically) the redefinition of the operators of the dynamical variables to the one obtained via the normal ordering procedure in the ordinary quantum field theory, but, without additional suppositions, does not reduce the commutation relations to the standard bilinear ones. As a last step in specifying the commutation relations as much as possible, we introduce the requirement the theory to supply an effective way for calculating vacuum mean values of (anti-normally ordered) products of creation and annihilation operators to which are reduced all predictable results, in particular the mean values of the dynamical variables. The standard bilinear commutation relation seems to be the only ones know at present that survive that last condition, however their uniqueness in this respect is not investigated.\n\nSect. 9 deals with the same problems as described above but for systems containing at least two different quantum fields. The main obstacle is the establishment of commutation relations between creation/annihilation operators concerning different fields. Argument is presented that they should contain commutators or anticommutators of these operators. The major of corresponding commutation relations are explicitly written and the results obtained turn to be similar to the ones just described, only in 'multifield' version.\n\nSection 10 closes the paper by summarizing its main results.\n\nThe books [1] [2] [3] will be used as standard reference works on quantum field theory. Of course, this is more or less a random selection between the great number of (text)books and papers on the theme to which the reader is referred for more details or other points of view. For this end, e.g., [4, 12, 19] or the literature cited in [1-4, 12, 19 ] may be helpful.\n\nThroughout this paper denotes the Planck's constant (divided by 2π), c is the velocity of light in vacuum, and i stands for the imaginary unit. The superscripts † and ⊤ mean respectively Hermitian conjugation and transposition (of operators or matrices), the superscript * denotes complex conjugation, and the symbol • denotes compositions of mappings/operators. By δ f g , or δ g f or δ f g (:= 1 for f = g, := 0 for f = g) is denoted the Kronecker δ-symbol, depending on arguments f and g, and δ n (y), y ∈ R n , stands for the n-dimensional Dirac δ-function; δ(y) := δ 1 (y) for y ∈ R.\n\nThe Minkowski spacetime is denoted by M . The Greek indices run from 0 to dim M -1 = 3. All Greek indices will be raised and lowered by means of the standard 4-dimensional Lorentz metric tensor η µν and its inverse η µν with signature (+ ---). The Latin indices a, b, . . . run from 1 to dim M -1 = 3 and, usually, label the spacial components of some object. The Einstein's summation convention over indices repeated on different levels is assumed over the whole range of their values.\n\nAt last, we ought to give an explanation why this work appears under the general title \"Lagrangian quantum field theory in momentum picture\" when in it all considerations are done, in fact, in Heisenberg picture with possible, but not necessary, usage of the creation and annihilation operators in momentum picture. First of all, we essentially employ the obtained in [13] [14] [15] expressions for the dynamical variables in momentum picture for three types of Lagrangians. The corresponding operators in Heisenberg picture, which in fact is used in this paper, can be obtained via a direct calculation, as it is partially done in, e.g., [1] for one of the mentioned types of Lagrangians. The important point here is that in Heisenberg picture it suffice to be used only the standard Lagrangian formalism, while in momentum picture one has to suppose the commutativity between the components of the momentum operator and the validity of the Heisenberg relations for it (see below equations (2.6 ) and (2.7)). Since for the analysis of the commutation relations we intend to do the fulfillment of these relations is not necessary (they are subsidiary restrictions on the Lagrangian formalism), the Heisenberg picture of motion is the natural one that has to be used. For this reason, the expression for the dynamical variables obtained in [13] [14] [15] will be used simply as their Heisenberg counterparts, but expressed via the creation and annihilation operators in momentum picture. The only real advantage one gets in this way is the more natural structure of the orbital angular momentum operator. As the commutation relations considered below are algebraic ones, it is inessential in what picture of motion they are written or investigated."
},
{
"section_type": "OTHER",
"section_title": "The momentum picture",
"text": "Since the momentum picture of motion will be used only partially in this work, below is presented only its definition and the connection between the creation/annihilation operators in it and in Heisenberg picture. Details concerning the momentum picture can be found in [20, 21] and in the corresponding sections devoted to it in [13] [14] [15] .\n\nLet us consider a system of quantum fields, represented in Heisenberg picture of motion by field operators φi (x) : F → F, i = 1, . . . , n ∈ N, acting on the system's Hilbert space F of states and depending on a point x in Minkowski spacetime M . Here and henceforth, all quantities in Heisenberg picture will be marked by a tilde (wave) \"˜\" over their kernel symbols. Let Pµ denotes the system's (canonical) momentum vectorial operator, defined via the energy-momentum tensorial operator T µν of the system, viz.\n\nPµ := 1 c\n\nx 0 =const T0µ (x) d 3 x.\n\n(2.1)\n\nSince this operator is Hermitian, P † µ = Pµ , the operator\n\nU (x, x 0 ) = exp 1 i µ (x µ -x µ 0 ) Pµ , (2.2)\n\nwhere x 0 ∈ M is arbitrarily fixed and x ∈ M , foot_2 is unitary, i.e. U † (x 0 , x) := ( U (x, x 0 )) † = U -1 (x, x 0 ) := ( U (x, x 0 )) -1 and, via the formulae X → X (x) = U (x, x 0 )( X ) (2.3) Ã(x) → A(x) = U (x, x 0 ) • ( Ã(x)) • U -1 (x, x 0 ), (2.4) realizes the transition to the momentum picture. Here X is a state vector in system's Hilbert space of states F and Ã(x) : F → F is (observable or not) operator-valued function of x ∈ M which, in particular, can be polynomial or convergent power series in the field operators φi (x); respectively X (x) and A(x) are the corresponding quantities in momentum picture.\n\nIn particular, the field operators transform as φi (x)\n\n→ ϕ i (x) = U (x, x 0 ) • φi (x) • U -1 (x, x 0 ). (2.5)\n\nNotice, in (2.2) the multiplier (x µx µ 0 ) is regarded as a real parameter (in which Pµ is linear). Generally, X (x) and A(x) depend also on the point x 0 and, to be quite correct, one should write X (x, x 0 ) and A(x, x 0 ) for X (x) and A(x), respectively. However, in the most situations in the present work, this dependence is not essential or, in fact, is not presented at all. For that reason, we shall not indicate it explicitly.\n\nThe momentum picture is most suitable in quantum field theories in which the components Pµ of the momentum operator commute between themselves and satisfy the Heisenberg relations/equations with the field operators, i.e. when Pµ and φi (x) satisfy the relations:\n\n[ Pµ , Pν ] = 0 (2.6) [ φi (x), Pµ ] = i ∂ µ φi (x).\n\n(2.7)\n\nHere [A, B] ± := A • B ± B • A,\n\n• being the composition of mappings sign, is the commutator/anticommutator of operators (or matrices) A and B. However, the fulfillment of the relations (2.6) and (2.7) will not be supposed in this paper until Sect. 6 (see also Sect. 5) . Let a ± s (k) and a † ± s (k) be the creation/annihilation operators of some free particular field (see Sect. 3 below for a detailed explanation of the notation). We have the connections\n\nã± s (k) = e ± 1 i x µ kµ U -1 (x, x 0 ) • a ± s (k) • U (x, x 0 ) ã † ± s (k) = e ± 1 i x µ kµ U -1 (x, x 0 ) • a † ± s (k) • U (x, x 0 ) k 0 = m 2 c 2 + k 2 (2.8) whose explicit form is ã± s (k) = e ± 1 i x µ 0 kµ a ± s (k) ã † ± s (k) = e ± 1 i x µ 0 kµ a † ± s (k) k 0 = m 2 c 2 + k 2 .\n\n(2.9) Further it will be assumed ã± s (k) and ã † ± s (k) to be defined in Heisenberg picture, independently of a ± s (k) and a † ± s (k), by means of the standard Lagrangian formalism. What concerns the operators a ± s (k) and a † ± s (k), we shall regard them as defined via (2.9); this makes them independent from the momentum picture of motion. The fact that the so-defined operators a ± s (k) and a † ± s (k) coincide with the creation/annihilation operators in momentum picture (under the conditions (2.6) and (2.7)) will be inessential in the almost whole text."
},
{
"section_type": "OTHER",
"section_title": "Lagrangians, Euler-Lagrange equations and dynamical variables",
"text": "In [13] [14] [15] we have investigated the Lagrangian quantum field theory of respectively scalar, spin 1 2 and vector free fields. The main Lagrangians from which it was derived are respectively (see loc. cit. or, e.g. [1, 3, 11, 12] ):\n\nL′ sc = L′ sc ( φ, φ † ) = - 1 1 + τ ( φ) m 2 c 4 φ(x) • φ † (x) + 1 1 + τ ( φ) c 2 2 (∂ µ φ(x)) • (∂ µ φ † (x)) (3.1a) L′ sp = L′ sp ( ψ, ψ) = - 1 2 i c{ ψ⊤ (x)C -1 γ µ • (∂ µ ψ(x)) -(∂ µ ψ⊤ (x))C -1 γ µ • ψ(x)} + mc 2 ψ⊤ (x)C -1 • ψ(x) (3.1b) L′ v = L′ v ( Ũ , Ũ † ) = m 2 c 4 1 + τ ( Ũ ) Ũ † µ • Ũ µ + c 2 2 1 + τ ( Ũ ) -(∂ µ Ũ † ν ) • (∂ µ Ũ ν ) + (∂ µ Ũ µ † ) • (∂ ν Ũ ν ) (3.1c)\n\nHere it is used the following notation: φ(x) is a scalar field, a tilde (wave) over a symbol means that it is in Heisenberg picture, the dagger † denotes Hermitian conjugation, ψ := ( ψ0 , ψ1 , ψ2 , ψ3 ) is a 4-spinor field, ψ := C ψ⊤ := C( ψ † γ 0 ) is its charge conjugate with γ µ being the Dirac gamma matrices and the matrix C satisfies the equations C -1 γ µ C = -γ µ and C ⊤ = -C, U µ is a vector field, m is the field's mass (parameter) and the function\n\nτ (A) := 1 for A † = A (Hermitian operator) 0 for A † = A (non-Hermitian operator) , (3.2)\n\nwith A : F → F being an operator on the systems Hilbert space F of states, takes care of is the field charged (non-Hermitian) or neutral (Hermitian, uncharged). Since a spinor field is a charged one, we have τ ( ψ) = 0; sometimes below the number 0 = τ ( ψ) will be written explicitly for unification of the notation. We have explored also the consequences from the 'charge conjugate' Lagrangians\n\nL′′ sc = L′′ sc ( φ, φ † ) := L′ sc ( φ † , φ) (3.3a) L′′ sp = L′′ sp ( ψ, ψ) := L′ sp ( ψ, ψ) (3.3b) L′′ v = L′′ v ( Ũ , Ũ † ) := L′ v ( Ũ † , Ũ ), (3.3c)\n\nas well as from the 'charge symmetric' Lagrangians\n\nL′′′ sc = L′′′ sc ( φ, φ † ) := 1 2 L′ sc + L′′ sc = 1 2 L′ sc ( φ, φ † ) + L′ sc ( φ † , φ) (3.4a) L′′′ sp = L′′′ sp ( ψ, ψ) := 1 2 L′ sp + L′′ sp = 1 2 L′ sp ( ψ, ψ) + L′ sp ( ψ, ψ) (3.4b) L′′′ v = L′′′ v ( Ũ , Ũ † ) := 1 2 L′ v + L′′ v = 1 2 L′ v ( Ũ , Ũ † ) + L′ v ( Ũ † , Ũ ) . (3.4c)\n\nIt is essential to be noted, for a massless, m = 0, vector field to the Lagrangian formalism are added as subsidiary conditions the Lorenz conditions\n\n∂ µ Ũ µ = 0 ∂ µ Ũ † µ = 0 (3.5)\n\non the solutions of the corresponding Euler-Lagrange equations. Besides, if the opposite is not stated explicitly, no other restrictions, like the (anti)commutation relations, are supposed to be imposed on the above Lagrangians. And a technical remark, for convenience, the fields φ, ψ and Ũ and their charge conjugate φ † , ψ and Ũ † , respectively, are considered as independent field variables.\n\nLet L′ denotes any one of the Lagrangians (3.1) and L′′ (resp. L′′′ ) the corresponding to it Lagrangian given via (3.3) (resp. (3.4) ). Physically the difference between L′ and L′′ is that the particles for L′ are antiparticles for L′′ and vice versa. Both of the Lagrangians L′ and L′′ are not charge symmetric, i.e. the arising from them theories are not invariant under the change particle↔antiparticle (or, in mathematical terms, under some of the changes φ ↔ φ † , ψ ↔ ψ, Ũ ↔ Ũ † ) unless some additional hypotheses are made. Contrary to this, the Lagrangian L′′′ is charge symmetric and, consequently, the formalism on its base is invariant under the change particle↔antiparticle. 4 The Euler-Lagrange equations for the Lagrangians L′ , L′′ and L′′′ happen to coincide [13] [14] [15] :\n\n5 ∂ L′ ∂χ - ∂ ∂x µ ∂ L′ ∂(∂ µ χ) ≡ ∂ L′′ ∂χ - ∂ ∂x µ ∂ L′′ ∂(∂ µ χ) ≡ ∂ L′′′ ∂χ - ∂ ∂x µ ∂ L′′′ ∂(∂ µ χ) = 0, (3.6)\n\nwhere χ = φ, φ † , ψ, ψ, Ũ , Ũ † for respectively scalar, spinor and vector field. Since the creation and annihilation operators are defined only on the base of Euler-Lagrange equations [1, 3, [11] [12] [13] [14] [15] , we can assert that these operators are identical for the Lagrangians L′ , L′′ and L′′′ . We shall denote these operators by a ± s (k) and a † ± s (k) with the convention that a + s (k) (resp. a † + s (k)) creates a particle (resp. antiparticle) with 4-momentum ( m 2 c 2 + k 2 , k), polarization s (see below) and charge (-q) (resp. (+q)) 6 and a † - s (k) (resp. a - s (k)) annihilates/destroys such a particle (resp. antiparticle). Here and henceforth k ∈ R 3 is interpreted as (anti)particle's 3-momentum and the values of the polarization index s depend on the field considered: s = 1 for a scalar field, s = 1 or s = 1, 2 for respectively massless (m = 0) or massive (m = 0) spinor field, and s = 1, 2, 3 for a vector field. 7 Since massless vector field's modes with s = 3 may enter only in the spin and orbital angular momenta operators [15] , we, for convenience, shall assume that the polarization indices s, t, . . . take the values from 1 to 2j + 1δ 0m (1δ 0j ), where j = 0, 1 2 , 1 is the spin for scalar, spinor and vector field, respectively, and δ 0m := 1 for m = 0 and δ 0m := 0 for m = 0; 8 if the value s = 3 is important when j = 1 and m = 0, it will be commented/considered separately. Of course, the creation and annihilation operators are different for different fields; one should write, e.g., j a ± s (k) for a ± s (k), but we shall not use such a complicated notation and will assume the dependence on j to be an implicit one. 4 Besides, under the same assumptions, the Lagrangian L′′′ does not admit quantization via anticommutators (commutators) for integer (half-integer) spin field, while L′ and L′′ do not make difference between integer and half-integer spin fields. 5 Rigorously speaking, the Euler-Lagrange equations for the Lagrangian (3.4b) are identities like 0 = 0see [22] . However, bellow we shall handle this exceptional case as pointed in [14] . 6 For a neutral field, we put q = 0.\n\n7 For convenience, in [14] , we have set s = 0 if m = 0 and s = 1, 2 if m = 0 for a spinor field. For a massless vector field, one may set s = 1, 2, thus eliminating the 'unphysical' value s = 3 for m = 0 -see [1, 11, 15] . In [13] , for a scalar field, the notation ϕ ± 0 (k) and ϕ † ± 0 (k) is used for a ± 1 (k) and a † ± 1 (k), respectively. 8 In this way the case (j, s, m) = (1, 3, 0) is excluded from further considerations; if (j, m) = (1, 0) and q = 0, the case considered further in this work corresponds to an electromagnetic field in Coulomb gauge, as the modes with s = 3 are excluded [15] . However, if the case (j, s, m) = (1, 3, 0) is important for some reasons, the reader can easily obtain the corresponding results by applying the ones from [15] .\n\nThe following settings will be frequently used throughout this chapter:\n\nj := 0 for scalar field 1 2\n\nfor spinor field 1 for vector field τ := 1 for q = 0 (neutral (Hermitian) field) 0 for q = 0 (charged (non-Hermitian) field)\n\nε := (-1) 2j = +1 for integer j (bose fields) -1 for half-integer j (fermi fields)\n\n(3.7) [A, B] ± := [A, B] ±1 := A • B ± B • A, (3.8)\n\nwhere A and B are operators on the system's Hilbert space F of states.\n\nThe dynamical variables corresponding to L′ , L′′ and L′′′ are, however, completely different, unless some additional conditions are imposed on the Lagrangian formalism [13] [14] [15] . In particular, the momentum operators Pω µ , charge operators Qω , spin operators Sω µν and orbital operators Lω µν , where ω = ′, ′′, ′′′, for these Lagrangians are [13] [14] [15] :\n\nP′ µ = 1 1 + τ 2j+1-δ 0m (1-δ 0j ) s=1 d 3 kk µ | k 0 = √ m 2 c 2 +k 2 {a † + s (k) • a - s (k) + εa † - s (k) • a + s (k)} (3.9a) P′′ µ = 1 1 + τ 2j+1-δ 0m (1-δ 0j ) s=1 d 3 kk µ | k 0 = √ m 2 c 2 +k 2 {a + s (k) • a † - s (k) + εa - s (k) • a † + s (k)} (3.9b) P′′′ µ = 1 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 kk µ | k 0 = √ m 2 c 2 +k 2 {[a † + s (k), a - s (k)] ε + [a + s (k), a † - s (k)] ε } (3.9c) Q′ = +q 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k{a † + s (k) • a - s (k) -εa † - s (k) • a + s (k)} (3.10a) Q′′ = -q 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k{a + s (k) • a † - s (k) -εa - s (k) • a † + s (k)} (3.10b) Q′′′ = 1 2 q 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k{[a † + s (k), a - s (k)] ε -[a + s (k), a † - s (k)] ε } (3.10c) S′ µν = (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k σ ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) + σ ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) (3.11a) S′′ µν = ε (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k σ ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + σ ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) (3.11b) S′′′ µν = (-1) j-1/2 j 2(1 + τ ) 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k σ ss ′ ,- µν (k)[a † + s (k), a - s ′ (k)] ε + σ ss ′ ,+ µν (k)[a † - s (k), a + s ′ (k)] ε (3.11c) L′ µν =x 0 µ P′ ν -x 0 ν P′ µ + (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k l ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) + l ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) + i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) -εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a + s (k) k 0 = √ m 2 c 2 +k 2 (3.12a) L′′ µν =x 0 µ P′′ ν -x 0 ν P′′ µ + ε (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k l ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + l ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) + i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † - s (k) -εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † + s (k) k 0 = √ m 2 c 2 +k 2 (3.12b) L′′′ µν =x 0 µ P′′′ ν -x 0 ν P′′′ µ + (-1) j-1/2 j 2(1 + τ ) 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k l ss ′ ,- µν (k)[a † + s (k), a - s ′ (k)] ε + l ss ′ ,+ µν (k)[a † - s (k), a + s ′ (k)] ε + i 4(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) -εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † + s (k) + a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † - s (k) -εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a + s (k) k 0 = √ m 2 c 2 +k 2 .\n\n(3.12c)\n\nHere we have used the following notation: (-1) n+1/2 := (-1) n i for all n ∈ N and i := + √ -1,\n\nA(k) ← ---- - → k µ ∂ ∂k ν • B(k) := -k µ ∂A(k) ∂k ν • B(k) + A(k) • k µ ∂B(k) ∂k ν = k µ A(k) ← -- -→ ∂ ∂k ν • B(k) (3.13)\n\nfor operators A(k) and B(k) having C 1 dependence on k, 9 and σ ss ′ ,± µν (k) and l ss ′ ,± µν (k) are 9 More generally, if ω : {F → F} → {F → F} is a mapping on the operator space over the system's Hilbert space, we put A\n\n← - -→ ω • B := -ω(A) • B + A • ω(B)\n\nfor any A, B : F → F. Usually [2, 12] , this notation is used for ω = ∂µ. some functions of k such that 10\n\nσ ss ′ ,± µν (k) = -σ ss ′ ,± νµ (k) l ss ′ ,± µν (k) = -l ss ′ ,± νµ (k) σ ss ′ ,± µν (k) = l ss ′ ,± νµ (k) = 0 for j = 0 (scalar field) σ ss ′ ,- µν (k) = -σ ss ′ ,+ µν (k) =: σ ss ′ µν (k) = -σ s ′ s µν (k) = -σ ss ′ νµ (k) for j = 1 (vector field) l ss ′ ,- µν (k) = -l ss ′ ,+ µν (k) =: l ss ′ µν (k) = -l s ′ s µν (k) = -l ss ′ νµ (k)\n\nfor j = 1 (vector field).\n\n(3.14)\n\nA technical remark must be make at this point. The equations (3.9)-(3.12) were derived in [13] [14] [15] under some additional conditions, represented by equations (2.6) and (2.7), which are considered bellow in Sect. 5 and ensure the effectiveness of the momentum picture of motion [21] used in [13] [14] [15] . However, as it is partially proved, e.g., in [1] , when the quantities (3.9)-(3.12) are expressed via the Heisenberg creation and annihilation operators (see (2.9)), they remain valid, up to a phase factor, and without making the mentioned assumptions, i.e. these assumptions are needless when one works entirely in Heisenberg picture. For this reason, we shall consider (3.9)-(3.12) as pure consequence of the Lagrangian formalism.\n\nWe should emphasize, in (3.11 ) and (3.12) with Sω µν and Lω µν , ω = ′, ′′, ′′′, are denoted the spin and orbital, respectively, operators for Lω , which are the spacetime-independent parts of the spin and orbital, respectively, angular momentum operators [14, 23] ; if the last operators are denoted by Sω µν and Lω µν , the total angular momentum operator of a system with Lagrangian Lω is [23]\n\nMω µν = Lω µν + Sω µν = Lω µν + Sω µν , ω = ′, ′′, ′′′ (3.15)\n\nand Sω µν = Sω µν (and hence Lω µν = Lω µν ) iff Sω µν is a conserved operator or, equivalently, iff the system's canonical energy-momentum tensor is symmetric. 11 Going ahead (see Sect. 6), we would like to note that the expressions (3.9c) and, consequently, the Lagrangian L′′′ are the base from which the paracommutation relations were first derived [16] .\n\nAnd a last remark. Above we have expressed the dynamical variables in Heisenberg picture via the creation and annihilation operators in momentum picture. If one works entirely in Heisenberg picture, the operators (2.9), representing the creation and annihilation operators in Heisenberg picture, should be used. Besides, by virtue of the equations\n\n(a ± s (k)) † = a † ∓ s (k) (a † ± s (k)) † = a ∓ s (k) (3.16) ã± s (k) † = ã † ∓ s (k) ã † ± s (k) † = ã∓ s (k), (3.17)\n\nsome of the relations concerning a † ± s (k), e.g. the Euler-Lagrange and Heisenberg equations, are consequences of the similar ones regarding a ± s (k). In view of (2.9), we shall consider (3.9)-(3.12) as obtained form the corresponding expressions in Heisenberg picture by making the replacements ã± s (k) → a ± s (k) and ã † ± s (k) → a † ± s (k). So, (3.9)-(3.12) will have, up to a phase factor, a sense of dynamical variables in Heisenberg picture expressed via the creation/annihilation operators in momentum picture. 10 For the explicit form of these functions, see [13] [14] [15] ; see also equation (6.57) below. 11 In [14, 23] the spin and orbital operators are labeled with an additional left superscript •, which, for brevity, is omitted in the present work as in it only these operators, not Sω µν and Lω µν , will be considered. Notice, the operators Sω µν and Lω µν are, generally, time-dependent while the orbital and spin ones are conserved, as a result of which the total angular momentum is a conserved operator too [14, 23] ."
},
{
"section_type": "OTHER",
"section_title": "On the uniqueness of the dynamical variables",
"text": "Let D = P µ , Q, S µν , L µν denotes some dynamical variable, viz. the momentum, charge, spin, or orbital operator, of a system with Lagrangian L. Since the Euler-Lagrange equations for the Lagrangians L ′ , L ′′ and L ′′′ coincide (see (3.6 )), we can assert that any field satisfying these equations admits at least three classes of conserved operators, viz. D ′ , D ′′ and D ′′′ = 1 2 D ′ + D ′′ . Moreover, it can be proved that the Euler-Lagrange equations for the Lagrangian\n\nL α,β := α L ′ + β L ′′ α + β = 0 (4.1)\n\ndo not depend on α, β ∈ C and coincide with (3.6) . Therefore there exists a two parameter family of conserved dynamical variables for these equations given via\n\nD α,β := α D ′ + β D ′′ α + β = 0. ( 4\n\n.2) Evidently L ′′′ = L 1 2 , 1 2 and D ′′′ = D 1 2 , 1 2 . Since the Euler-Lagrange equations (3.6) are linear and homogeneous (in the cases considered), we can, without a lost of generality, restrict the parameters α, β ∈ C to such that α + β = 1, (4.3) which can be achieved by an appropriate renormalization (by a factor (α+β) -1/2 ) of the field operators. Thus any field satisfying the Euler-Lagrange equations (3.6) admits the family D α,β , α + β = 1, of conserved operators. Obviously, this conclusion is valid if in (4.1) we replace the particular Lagrangians L ′ and L ′′ (see (3.1) and (3.3)) with any two Lagrangians (of one and the same field variables) which lead to identical Euler-Lagrange equations. However, the essential point in our case is that L ′ and L ′′ do not differ only by a full divergence, as a result of which the operators D α,β are different for different pairs (α, β), α + β = 1. foot_3\n\nSince one expects a physical system to possess uniquely defined dynamical characteristics, e.g. energy and total angular momentum, and the Euler-Lagrange equations are considered (in the framework of Lagrangian formalism) as the ones governing the spacetime evolution of the system considered, the problem arises when the dynamical operators D α,β , α+β = 1, are independent of the particular choice of α and β, i.e. of the initial Lagrangian one starts off. Simple calculation show that the operators (4.2), under the condition (4.3), are independent of the particular values of the parameters α and β if and only if\n\nD ′ = D ′′ . (4.4)\n\nSome consequences of the condition(s) (4.4) will be considered below, as well as possible ways for satisfying these restrictions on the Lagrangian formalism. Combining (3.9)-(3.12) with (4.4), for respectively D = P µ , Q, S µν , L µν , we see that a free scalar, spinor or vector field has a uniquely defined dynamical variables if and only if the following equations are fulfilled:\n\n2j+1-δ 0m (1-δ 0j ) s=1 d 3 k k µ k 0 = √ m 2 c 2 +k 2 a † + s (k) • a - s (k) -εa - s (k) • a † + s (k) -a + s (k) • a † - s (k) + εa † - s (k) • a + s (k) = 0 (4.5) q × 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) • a - s (k) -εa - s (k) • a † + s (k) + a + s (k) • a † - s (k) -εa † - s (k) • a + s (k) = 0 (4.6) 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k σ ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) -εσ ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) -εσ ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + σ ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) = 0 (4.7) 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k l ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) -εl ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) -εl ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + l ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) + 1 2 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a - s (k)+εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a † + s (k) -a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a † - s (k)-εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a + s (k) k 0 = √ m 2 c 2 +k 2 = 0. (4.8)\n\nIn (4.6) is retained the constant factor q as in the neutral case it is equal to zero and, consequently, the equation (4.6) reduces to identity.\n\nSince the Euler-Lagrange equations do not impose some restrictions on the creation and annihilation operators, the equations (4.5)-(4.8) can be regarded as subsidiary conditions on the Lagrangian formalism and can serve as equations for (partial) determination of the creation and annihilation operators. The system of integral equations (4.5)-(4.8) is quite complicated and we are not going to investigate it in the general case. Below we shall restrict ourselves to analysis of only those solutions of (4.5)-(4.8), if any, for which the integrands in (4.5)-(4.8) vanish. This means that we shall replace the system of integral equations (4.5)-(4.8) with respect to creation and annihilation operators with the following system of algebraic equations (do not sum over s and s ′ in (4.12) and (4.13)!):\n\na † + s (k) • a - s (k) -εa - s (k) • a † + s (k) -a + s (k) • a † - s (k) + εa † - s (k) • a + s (k) = 0 (4.9) a † + s (k) • a - s (k) -εa - s (k) • a † + s (k) + a + s (k) • a † - s (k) -εa † - s (k) • a + s (k) = 0 if q = 0 (4.10) a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) + εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † + s (k) -a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a † - s (k)-εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ •a + s (k) k 0 = √ m 2 c 2 +k 2 = 0 (4.11) s,s ′ σ ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) -εσ ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) -εσ ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + σ ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) = 0 (4.12) s,s ′ l ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) -εl ss ′ ,- µν (k)a - s ′ (k) • a † + s (k) -εl ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + l ss ′ ,+ µν (k)a † - s (k) • a + s ′ (k) = 0 (4.\n\nHere: s = 1, . . . , 2j + 1δ 0m (1δ 0j ) in (4.9)-(4.11) and s, s ′ = 1, . . . , 2j + 1δ 0m (1δ 1j ) in (4.12) and (4.13). (Notice, by virtue of (3.14), the equations (4.12) and (4.13) are identically valid for j = 0, i.e. for scalar fields.) Since all polarization indices enter in (4.5) and (4.6) on equal footing, we do not sum over s in (4.9)- (4.11) . But in (4.12) and (4.13) we have retain the summation sign as the modes with definite polarization cannot be singled out in the general case. One may obtain weaker versions of (4.9)-(4.13) by summing in them over the polarization indices, but we shall not consider these conditions below regardless of the fact that they also ensure uniqueness of the dynamical variables. At first, consider the equations (4.9)- (4.11) . Since for a neutral field, q = 0, we have a † ± s (k) = a ± s (k), which physically means coincidence of field's particles and antiparticles, the equations (4.9)-(4.11) hold identically in this case.\n\nLet consider now the case q = 0, i.e. the investigated field to be charged one. Using the standard notation (cf. (3.8))\n\n[A, B] η := A • B + ηB • A, (4.14)\n\nfor operators A and B and η ∈ C, we rewrite (4.9) and (4.10) as\n\n[a † + s (k), a - s (k)] -ε -[a + s (k), a † - s (k)] -ε = 0 (4.9 ′ ) [a † + s (k), a - s (k)] -ε + [a + s (k), a † - s (k)] -ε = 0 if q = 0, (4.10 ′ ) which are equivalent to [a † ± s (k), a ∓ s (k)] -ε = 0 if q = 0. ( 4\n\n.15) Differentiating (4.15) and inserting the result into (4.11), one can verify that (4.11) is tantamount to\n\na † + s (k), k µ ∂ ∂k ν -k ν ∂ ∂k µ • a - s (k) -ε -a + s (k), k µ ∂ ∂k ν -k ν ∂ ∂k µ • a † - s (k) -ε k 0 = √ m 2 c 2 +k 2 = 0 if q = 0, (4.16)\n\nConsider now (4.12) and (4.13) . By means of the shorthand (4.14), they read\n\ns,s ′ σ ss ′ ,- µν (k)[a † + s (k), a - s ′ (k)] -ε + σ ss ′ ,+ µν (k)[a † - s (k), a + s ′ (k)] -ε = 0 (4.17) s,s ′ l ss ′ ,- µν (k)[a † + s (k), a - s ′ (k)] -ε + l ss ′ ,+ µν (k)[a † - s (k), a + s ′ (k)] -ε = 0. (4.18)\n\nFor a scalar field, j = 0, these conditions hold identically, due to (3.14) . But for j = 0 they impose new restrictions on the formalism. In particular, for vector fields, j = 1 and ε = +1 they are satisfied iff (see (3.14) )\n\n[a † + s (k), a - s ′ (k)] -ε -[a † - s (k), a + s ′ (k)] -ε -[a † + s ′ (k), a - s (k)] -ε + [a † - s ′ (k), a + s (k)] -ε = 0. (4.19)\n\nOne can satisfy (4.17) and (4.18) if the following generalization of (4.15) holds\n\n[a † ± s (k), a ∓ s ′ (k)] -ε = 0. ( 4.20)\n\nFor spin j = 1 2 (and hence ε = -1 -see (3.7)), the conditions (4.12) and (4.13) cannot be simplified much, but, if one requires the vanishment of the operator coefficients after σ ss ′ ,± µν (k) and l ss ′ ,± µν (k), one gets\n\na † ± s (k) • a ∓ s ′ (k) = 0 j = 1 2 ε = -1. (4.21)\n\nExcluding some special cases, e.g. neutral scalar field (q = 0 and j = 0), the equations (4.15) and (4.21) are unacceptable from many viewpoints. The main of them is that they are incompatible with the ordinary (anti)commutation relations (see, e.g., e.g. [1, 11, 12, 18] or Sect. 6, in particular, equations (6.13) bellow); for example, (4.21) means that the acts of creation and annihilation of (anti)particles with identical characteristics should be mutually independent, which contradicts to the existing theory and experimental data. Now we shall try another way for achieving uniqueness of the dynamical variables for free fields. Since in (4.9)-(4.13) naturally appear (anti)commutators between creation and annihilation operators and these (anti)commutators vanish under the standard normal ordering [1, 11, 12, 18] , one may suppose that the normally ordered expressions of the dynamical variables may coincide. Let us analyze this method.\n\nRecall [1, 3, 11, 12] , the normal ordering operator N (for free field theory) is a linear operator on the operator space of the system considered such that to a product (composition)\n\nc 1 • • • • • c n of n ∈ N creation\n\nand/or annihilation operators c 1 , . . . c n it assigns the operator (-1) f c α 1 • • • • c αn . Here (α 1 , . . . , α n ) is a permutation of (1, . . . , n), all creation operators stand to the left of all annihilation ones, the relative order between the creation/annihilation operators is preserved, and f is equal to the number of transpositions among the fermion operators (j = 1 2 ) needed to be achieved the just-described order (\"normal order\") of the operators 13 In particular this means that\n\nc 1 • • • • • c n in c α 1 • • • • c αn .\n\nN a + s (k) • a † - t (p) = a + s (k) • a † - t (p) N a † + s (k) • a - t (p) = a † + s (k) • a - t (p) N a - s (k) • a † + t (p) = εa † + t (p) • a - s (k) N a † - s (k) • a + t (p) = εa + t (p) • a † - s (k) (4.22)\n\nand, consequently, we have\n\nN [a † ± s (k), a ∓ t (p)] -ε = 0 N [a ± s (k), a † ∓ t (p)] -ε = 0, (4.23)\n\ndue to ε := (-1) 2j = ±1 (see (3.7) ). (In fact, below only the equalities (4.22) and (4.23), not the general definition of a normal product, will be applied.) Applying the normal ordering operator to (4.9 ′ ), (4.10 ′ ), (4.17) and (4.18), we, in view of (4.23), get the identity 0 = 0, which means that the conditions (4.9), (4.10), (4.12) and (4.13) are identically satisfied after normal ordering. This is confirmed by the application of N to (3.9) and (3.10), which results respectively in (see (4.22) )\n\nN ( P′ µ ) = N ( P′′ µ ) = 1 1 + τ 2j+1-δ 0m (1-δ 0j ) s=1 d 3 kk µ | k 0 = √ m 2 c 2 +k 2 {a † + s (k) • a - s (k) + a + s (k) • a † - s (k)} (4.24)\n\n13 We have slightly modified the definition given in [1, 3, 11, 12] because no (anti)commutation relations are presented in our exposition till the moment. In this paper we do not concern the problem for elimination of the 'unphysical' operators a ± 3 (k) and a † ± 3 (k) from the spin and orbital momentum operators when j = 1; for details, see [15] , where it is proved that, for an electromagnetic field, j = 1 and q = 0, one way to achieve this is by adding to the number f above the number of transpositions between a ± s (k), s = 1, 2, and a ± 3 (k) needed for getting normal order.\n\nN ( Q′ ) = N ( Q′′ ) = 1 1 + τ 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k{a † + s (k) • a - s (k) -a + s (k) • a † - s (k)}.\n\nN a + s (k) ← -- -→ ω µν • a † - s (k) = a + s (k) ← -- -→ ω µν • a † - s (k) N a † + s (k) ← -- -→ ω µν • a - s (k) = a † + s (k) ← -- -→ ω µν • a - s (k) N a - s (k) ← -- -→ ω µν • a † + s (k) = -εa † + s (k) ← -- -→ ω µν • a - s (k) N a † - s (k) ← -- -→ ω µν • a + s (k) = -εa + s (k) ← -- -→ ω µν • a † - s (k). (4.27)\n\nAs a consequence of these equalities, the action of N on the l.h.s. of (4.11) vanishes. Combining this result with the mentioned fact that the normal ordering converts (4.12) and (4.13) into identities, we see that the normal ordering procedure ensures also uniqueness of the spin and orbital operators if we redefine them respectively as:\n\nSµν := N ( S′ µν ) := N ( S′′ µν ) = (-1) j-1/2 j 1 + τ × 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k σ ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) + εσ ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) (4.28) Lµν := N ( L′ µν ) := N ( L′′ µν ) = x 0 µ Pν -x 0 ν Pµ + (-1) j-1/2 j 1 + τ × 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k l ss ′ ,- µν (k)a † + s (k) • a - s ′ (k) + εl ss ′ ,+ µν (k)a + s ′ (k) • a † - s (k) + i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) + a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † - s (k) k 0 = √ m 2 c 2 +k 2 , (4.29)\n\nwhere (3.14) was applied."
},
{
"section_type": "OTHER",
"section_title": "Heisenberg relations",
"text": "The conserved operators, like momentum and charge operators, are often identified with the generators of the corresponding transformations under which the action operator is invariant [1, 3, 11, 12] . This leads to a number of commutation relations between the components of these operators and between them and the field operators. The relations of the letter set are known/referred as the Heisenberg relations or equations. Both kinds of commutation relations are from pure geometric origin and, consequently, are completely external to the Lagrangian formalism; one of the reasons being that the mentioned identification is, in general, unacceptable and may be carried out only on some subset of the system's Hilbert space of states [23, 24] . Therefore their validity in a pure Lagrangian theory is questionable and should be verified [11] . However, the considered relations are weaker conditions than the identification of the corresponding operators and there are strong evidences that these relations should be valid in a realistic quantum field theory [1, 11] ; e.g., the commutativity between the momentum and charge operators (see below (5.18 )) expresses the experimental fact that the 4-momentum and charge of any system are simultaneously measurable quantities.\n\nIt is known [1, 11] , in a pure Lagrangian approach, the field equations, which are usually identified with the Euler-Lagrange, 14 are the only restrictions on the field operators. Besides, these equations do not determine uniquely the field operators and the letter can be expressed through the creation and annihilation operators. Since the last operators are left completely arbitrary by a pure Lagrangian formalism, one is free to impose on them any system of compatible restrictions. The best known examples of this kind are the famous canonical (anti)commutation relations and their generalization, the so-called paracommutation relations [16, 18] . In general, the problem for compatibility of such subsidiary to the Lagrangian formalism system of restrictions with, for instance, the Heisenberg relations is open and requires particular investigation [11] . For example, even the canonical (anti)commutation relations for electromagnetic field in Coulomb gauge are incompatible with the Heisenberg equation involving the (total) angular momentum operator unless the gauge symmetry of this field is taken into account [11, § 84] . However, the (para)commutation relations are, by construction, compatible with the Heisenberg relations regarding momentum operator (see [16] or below Subsect. 6.1). The ordinary approach is to be imposed a system of equations on the creation and annihilation operators and, then, to be checked its compatibility with, e.g., the Heisenberg relations. In the next sections we shall investigate the opposite situation: assuming the validity of (some of) the Heisenberg equations, the possible restrictions on the creation and annihilation operators will be explored. For this purpose, below we briefly review the Heisenberg relations and other ones related to them.\n\nConsider a system of quantum fields φi (x), i = 1, . . . , N ∈ N, where φi (x) denote the components of all fields (and their Hermitian conjugates), and Pµ , Q and Mµν be its momentum, charge and (total) angular momentum operators, respectively. The Heisenberg relations/equations for these operators are [1, 3, 11, 12] [ φi (\n\nx), Pµ ] = i ∂ φi (x) ∂x µ (5.1) [ φi (x), Q] = e( φi )q φi (x) (5.2) [ φi (x), Mµν ] = i {x µ ∂ ν φi (x) -x ν ∂ µ φi (x)} + i i ′ I j i ′ µν φi ′ (x). (5.3)\n\nHere:\n\nq = const is the fields' charge, e( φi ) = 0 if φ † i = φi , e( φi ) = ±1 if φ † i = φi with e( φi ) + e( φ † i ) = 0,"
},
{
"section_type": "OTHER",
"section_title": "and the constants",
"text": "I i ′ iµν = -I i ′ iνµ characterize the transformation properties of the field operators under 4-rotations. (If ε( φi ) = 0, it is a convention whether to put ε( φi ) = +1 or ε( φi ) = -1 for a fixed i.)\n\nWe would like to make some comments on (5.3). Since its r.h.s. is a sum of two operators, the first (second) characterizing the pure orbital (spin) angular momentum properties of the system considered, the idea arises to split (5.3) into two independent equations, one involving the orbital angular momentum operator and another concerning the spin angular momentum operator. This is supported by the observation that, it seems, no process is known for transforming orbital angular momentum into spin one and v.v. (without destroying the system). So one may suppose the existence of operators Mor µν and Msp µν such that\n\n[ φi (x), Mor µν ] = i {x µ ∂ ν φi (x) -x ν ∂ µ φi (x)} (5.4) [ φi (x), Msp µν ] = i i ′ I i ′ iµν φi ′ (x) (5.5) Mµν = Mor µν + Msp µν .\n\n(5.6) However, as particular calculations demonstrate [5, 14, 15] , neither the spin (resp. orbital) nor the spin (resp. orbital) angular momentum operator is a suitable candidate for Msp µν (resp. Mor µν ). If we assume the validity of (5.1), then equations (5.4) and (5.5) can be satisfied if we choose\n\nMor µν (x) = Lext µν := x µ Pν -x ν Pµ (5.7) Msp µν (x) = M(0) µν (x) := Mµν -Lext µν = Sµν + Lµν -{x µ Pν -x ν Pµ } (5.8)\n\nwith Mµν satisfying (5.3). These operators are not conserved ones. Such a representation is in agreement with the equations (3.12), according to which the operator (5.7) enters additively in the expressions for the orbital operator. 15 The physical sense of the operator (5.7) is that it represents the orbital angular momentum of the system due to its movement as a whole. Respectively, the operator (5.8) describes the system's angular momentum as a result of its internal movement and/or structure. Since the spin (orbital) angular momentum is associated with the structure (movement) of a system, in the operator (5.8) are mixed the spin and orbital angular momenta. These quantities can be separated completely via the following representations of the operators M or µν and M sp µν in momentum picture (when (5.1) holds)\n\nM or µν = x µ P ν -x µ P µ + L int µν (5.9) M sp µν = M µν -(x µ P ν -x µ P µ ) -L int µν , (5.10)\n\nwhere L int µν describes the 'internal' orbital angular momentum of the system considered and depends on the Lagrangian we have started off. Generally said, L int µν is the part of the orbital angular momentum operator containing derivatives of the creation and annihilation operators. In particular, for the Lagrangians L ′ , L ′′ and L ′′′ (see Sect. 3), the explicit forms of the operators (5.9) and (5.10) respectively are:\n\nM ′ or µν =x µ P ′ ν -x ν P ′ µ + i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) -εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a + s (k) k 0 = √ m 2 c 2 +k 2\n\n(5.11a)\n\nM ′′ or µν =x µ P ′′ ν -x ν P ′′ µ + i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † - s (k) -εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † + s (k) k 0 = √ m 2 c 2 +k 2\n\n(5.11b) 15 This is evident in the momentum picture of motion, in which xµ stands for x0 µ in (3.12) -see [13] [14] [15] .\n\nM ′′′ or µν =x µ P ′′′ ν -x ν P ′′′ µ + i 4(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k a † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a - s (k) -εa - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † + s (k) + a + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a † - s (k) -εa † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • a + s (k) k 0 = √ m 2 c 2 +k 2 .\n\n(5.11c)\n\nM ′ sp µν = (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k))a † + s (k) • a - s ′ (k) + (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))a † - s (k) • a + s ′ (k) (5.12a) M ′′ sp µν = ε (-1) j-1/2 j 1 + τ 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))a + s ′ (k) • a † - s (k) + (σ ss ′ ,- µν (k) + σ ss ′ ,- µν (k))a - s ′ (k) • a † + s (k) (5.12b) M ′′′ sp µν = (-1) j-1/2 j 2(1 + τ ) 2j+1-δ 0m (1-δ 1j ) s,s ′ =1 d 3 k (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k))[a † + s (k), a - s ′ (k)] ε + (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))[a † - s (k), a + s ′ (k)] ε . (5.12c)\n\nObviously (see Sect. 2), the equations (5.12) have the same form in Heisenberg picture in terms of the operators (2.9) (only tildes over M and a must be added), but the equations (5.11) change substantially due to the existence of derivatives of the creation and annihilation operators in them [13] [14] [15] :\n\nM′ or µν = i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k ã † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã- s (k) -εã † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã+ s (k) k 0 = √ m 2 c 2 +k 2 (5.13a) M′′ or µν = i 2(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k ã+ s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã † - s (k) -εã - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã † + s (k) k 0 = √ m 2 c 2 +k 2 (5.13b) M′′′ or µν = i 4(1 + τ ) 2j+1-δ 0m (1-δ 0j ) s=1 d 3 k ã † + s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã- s (k) -εã - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã † + s (k) + ã+ s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã † - s (k) -εã † - s (k) ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ • ã+ s (k) k 0 = √ m 2 c 2 +k 2 .\n\n(5.13c) From (5.13) and (5.12) is clear that the operators Mor µν and Msp µν so defined are conserved (contrary to (5.7) and (5.8)) and do not depend on the validity of the Heisenberg relations (5.1) (contrary to expressions (5.11) in momentum picture).\n\nThe problem for whether the operators (5.12) and (5.13) satisfy the equations (5.4) and (5.5), respectively, will be considered in Sect. 6.\n\nThere is an essential difference between (5.4) and (5.5): the equation (5.5) depends on the particular properties of the operators φi (x) under 4-rotations via the coefficients I i ′ iµν (see (5.25) below), while (5.4) does not depend on them. This is explicitly reflected in (5.11) and (5.12): the former set of equations is valid independently of the geometrical nature of the fields considered, while the latter one depends on it via the 'spin' ('polarization') functions σ ss ′ ,± µν (k) and l ss ′ ,± µν (k). Similar remark concerns (5.3), on one hand, and (5.1) and (5.2), on another hand: the particular form of (5.3) essentially depends on the geometric properties of φi (x) under 4-rotations, the other equations being independent of them.\n\nIt should also be noted, the relation (5.3) does not hold for a canonically quantized electromagnetic field in Coulomb gauge unless some additional terms it its r.h.s., reflecting the gauge symmetry of the field, are taken into account [11, § 84] .\n\nAs it was said above, the relations (5.1)-(5.3) are from pure geometrical origin. However, the last discussion, concerning (5.4)-(5.8), reveals that the terms in braces in (5.3) should be connected with the momentum operator in the (pure) Lagrangian approach. More precisely, on the background of equations (3.11a)-(3.12c), the Heisenberg relation (5.3) should be replaced with\n\n[ φi (x), Mµν ] = x µ [ φi (x), Pν ] -x ν [ φi (x), Pµ ] + i j I i ′ iµν φi ′ (x), (5.14)\n\nwhich is equivalent to (5.3) if (5.1) is true. An advantage of the last equation is that it is valid in any picture of motion (in the same form) while (5.3) holds only in Heisenberg picture. 16 Obviously, (5.14) is equivalent to (5.5) with Msp µν defined by (5.8). The other kind of geometric relations mentioned at the beginning of this section are connected with the basic relations defining the Lie algebra of the Poincaré group [7, pp. 143-147] , [8, sect. 7.1] . They require the fulfillment of the following equations between the components Pµ of the momentum and Mµν of the angular momentum operators [3, 5, 7, 8] :\n\n[ Pµ , Pν ] = 0 (5.15) [ Mµν , Pλ ] = -i (η λµ Pν -η λν Pµ ). (5.16) [ Mκλ , Mµν ] = -i η κµ Mλν -η λµ Mκν -η κν Mλµ + η λν Mκµ .\n\n(5.17)\n\nWe would like to pay attention to the minus sign in the multiplier (-i ) in (5.16 ) and (5.17) with respect to the above references, where i stands instead of -i in these equations. When (a representation of) the Lie algebra of the Poincaré group is considered, this difference in the sign is insignificant as it can be absorbed into the definition of Mµν . However, the change of the sign of the angular momentum operator, Mµν → -Mµν , will result in the change i → -i in the r.h.s. of (5.3) . This means that equations (5.15), (5.16 ) and (5.3), when considered together, require a suitable choice of the signs of the multiplier i in their right hand sides as these signs change simultaneously when Mµν is replaced with -Mµν . Since equations (5.3), (5.16 ) and (5.17) hold, when Mµν is defined according to the Noether's theorem and the ordinary (anti)commutation relations are valid [13] [14] [15] , we accept these equations in the way they are written above.\n\nTo the relations (5.15)-(5.17) should be added the equations [3, p. 78 ]\n\n[ Q, Pµ ] = 0 (5.18) [ Q, Mµν ] = 0, (5.19)\n\nwhich complete the algebra of observables and express, respectively, the translational and rotational invariance of the charge operator Q; physically they mean that the charge and momentum or the charge and angular momentum are simultaneously measurable quantities.\n\nSince the spin properties of a system are generally independent of its charge or momentum, one may also expect the validity of the relations foot_6\n\n[ Sµν , Pµ ] = 0 (5.20) [ Sµν , Q] = 0. (5.21)\n\nBut, as the spin describes, in a sense, some of the rotational properties of the system, equality like [ Sµν , Lκλ ] = 0 is not likely to hold. Indeed, the considerations in [13] [14] [15] reveal that (5.20 (ii) Equation (5.2) implies (5.18), (5.19), (5.21), and (5.23). (iii) Equation (5.3) implies (5.16), (5.17), and (5.19). Besides, (5.3) may, possibly, entail equations like (5.17) with S or L for M , with an exception of Mµν in the l.h.s., i.e.\n\n[ Sκλ , Mµν ] = -i η κµ Sλν -η λµ Sκν -η κν Sλµ + η λν Sκµ [ Lκλ , Mµν ] = -i η κµ Lλν -η λµ Lκν -η κν Lλµ + η λν Lκµ (5.24)\n\nThe validity of assertions (i)-(iii) above for free scalar, spinor and vector fields, when respectively φi (x) → φ(x), φ † (x)\n\nI i ′ iµν → I µν = 0 e( φ) = -e( φ † ) = +1 (5.25a) φi (x) → ψ(x), ψ(x) I i ′ iµν → I ψµν = I ψµν = - i 2 σ µν e( ψ) = -e( ψ) = +1 (5.25b) φi (x) → Ũµ (x), Ũ † µ (x) I i ′ iµν → I σ ρµν = I † σ ρµν = δ σ µ η νρ -δ σ ν η µρ e( Ũµ ) = -e( Ũ † µ ) = +1, (5.25c)\n\nwhere σ µν := i 2 [γ µ , γ ν ] with γ µ being the Dirac γ-matrices [1, 25] , is proved in [13] [14] [15] , respectively. Besides, in loc. cit. is proved that equations (5.24) hold for scalar and vector fields, but not for a spinor field. 18 Thus, we see that the Heisenberg relations (5.1)-( 5 .3) are stronger than the commutation relations (5.15)-(5.23), when imposed on the Lagrangian formalism as subsidiary restrictions."
},
{
"section_type": "OTHER",
"section_title": "Types of possible commutation relations",
"text": "In a broad sense, by a commutation relation we shall understand any algebraic relation between the creation and annihilation operators imposed as subsidiary restriction on the Lagrangian formalism. In a narrow sense, the commutation relations are the equations (6.13), with ε = -1, written below and satisfied by the bose creation and annihilation operators. As anticommutation relations are known the equations (6.13), with ε = +1, written below and satisfied by the fermi creation and annihilation operators. The last two types of relations are often referred as the bilinear commutation relations [18] . Theoretically are possible also trilinear commutation relations, an example being the paracommutation relations [16, 18] represented below by equations (6.18) (or (6.20)).\n\nGenerally said, the commutation relations should be postulated. Alternatively, they could be derived from (equivalent to them) different assumptions added to the Lagrangian formalism. The purpose of this section is to be explored possible classes of commutation relations, which follow from some natural restrictions on the Lagrangian formalism that are consequences from the considerations in the previous sections. Special attention will be paid on some consequences of the charge symmetric Lagrangians as the free fields possess such a symmetry [1, 3, 11, 12] .\n\nAs pointed in Sect 3, the Euler-Lagrange equations for the Lagrangians L′ , L′′ and L′′′ coincide and, in quantum field theory, the role of these equations is to be singled out the independent degrees of freedom of the fields in the form of creation and annihilation operators a ± s (k) and a † ± s (k) (which are identical for L′ , L′′ and L′′′ ). Further specialization of these operators is provided by the commutation relations (in broad sense) which play a role of field equations in this situation (with respect to the mentioned operators).\n\nBefore proceeding on, we would like to simplify our notation. As a spin variable, s say, is always coupled with a 3-momentum one, k say, we shall use the letters l, m and n to denote pairs like l = (s, k), m = (t, p) and n = (r, q). Equipped with this convention, we shall write, e.g., a ± l for a ± s (k) and a † ± l for a † ± s (k). We set δ lm := δ st δ 3 (kp) and a summation sign like l should be understood as s d 3 k, where the range of the polarization variable s will be clear from the context (see, e.g., (3.9)-(3.12))."
},
{
"section_type": "BACKGROUND",
"section_title": "Restrictions related to the momentum operator",
"text": "First of all, let us examine the consequences of the Heisenberg relation (5.1) involving the momentum operator. Since in terms of creation and annihilation operators it reads [1, [13] [14] [15] ]\n\n[a ± s (k), P µ ] = ∓k µ a ± s (k) [a † ± s (k), P µ ] = ∓k µ a † ± s (k) k 0 = m 2 c 2 + k 2 , (6.1)\n\nthe field equations in terms of creation and annihilation operators for the Lagrangians (3.1), (3.3) and (3.4) respectively are (see [13] [14] [15] or (6.1) and (3.9)):\n\n2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a ± s (k), a † + t (q) • a - t (q) + εa † - t (q) • a + t (q) - ± (1 + τ )a ± s (k)δ st δ 3 (k -q) d 3 q = 0 (6.2a) 2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a † ± s (k), a † + t (q) • a - t (q) + εa † - t (q) • a + t (q) - ± (1 + τ )a † ± s (k)δ st δ 3 (k -q) d 3 q = 0 (6.2b) 2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a ± s (k), a + t (q) • a † - t (q) + εa - t (q) • a † + t (q) - ± (1 + τ )a ± s (k)δ st δ 3 (k -q) d 3 q = 0 (6.3a) 2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a † ± s (k), a + t (q) • a † - t (q) + εa - t (q) • a † + t (q) - ± (1 + τ )a † ± s (k)δ st δ 3 (k -q) d 3 q = 0 (6.3b) 2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a ± s (k), [a † + t (q), a - t (q)] ε + [a + t (q), a † - t (q)] ε - ± (1 + τ )a ± s (k)δ st δ 3 (k -q) d 3 q = 0 (6.4a) 2j+1-δ 0m (1-δ 0j ) t=1 q µ q 0 = √ m 2 c 2 +q 2 a † ± s (k), [a † + t (q), a - t (q)] ε + [a + t (q), a † - t (q)] ε - ± (1 + τ )a † ± s (k)δ st δ 3 (k -q) d 3 q = 0, (6.4b)\n\nwhere j and ε are given via (3.7), the generalized commutation function [•, •] ε is defined by (4.14), and the polarization indices take the values\n\ns, t = 1, . . . , 2j + 1 -δ 0m (1 -δ 0j ) = \n\n1 for j = 0 or for j = 1 2 and m = 0 1, 2 for j = 1 2 and m = 0 or for j = 1 and m = 0 1, 2, 3 for j = 1 and m = 0 . (6.5) The \"b\" versions of the equations (6.2)-(6.4) are consequences of the \"a\" versions and the equalities\n\n(a ± l ) † = a † ∓ l (a † ± l ) = a ∓ l (6.6) [A, B] η † = η[A † , B † ] η for [A, B] η = η[B, A] η η = ±1. (6.7)\n\nApplying (6.2)-(6.4) and the identity\n\n[A, B • C] = [A, B] η • C -ηB • [A, C] η for η = ±1 (6.8)\n\nfor the choice η = -1, one can prove by a direct calculation that\n\n[ Pµ , Pν ] = 0 [ Q, Pµ ] = 0 [ Sµν , Pλ ] = 0 [ Lµν , Pλ ] = -i {η λµ Pν -η λν Pµ } [ Mµν , Pλ ] = -i {η λµ Pν -η λν Pµ }, (6.9)\n\nwhere the operators Pµ , Q, Sµν , Lµν , and Mµν denote the momentum, charge, spin, orbital and total angular momentum operators, respectively, of the system considered and are calculated from one and the same initial Lagrangian. This result confirms the supposition, made in Sect. 5, that the assertion (i) before (5.24) holds for the fields investigated here.\n\nBelow we shall study only those solutions of (6.2)-(6.4) for which the integrands in them vanish, i.e. we shall replace the systems of integral equations (6.2)-(6.4) with the following systems of algebraic equations (see the above convention on the indices l and m and do not sum over indices repeated on one and the same level):\n\na ± l , a † + m • a - m + εa † - m • a + m -± (1 + τ )δ lm a ± l = 0 (6.10a) a † ± l , a † + m • a - m + εa † - m • a + m -± (1 + τ )δ lm a † ± l = 0 (6.10b) a ± l , a + m • a † - m + εa - m • a † + m -± (1 + τ )δ lm a ± l = 0 (6.11a) a † ± l , a + m • a † - m + εa - m • a † + m -± (1 + τ )δ lm a † ± l = 0 (6.11b) a ± l , [a † + m , a - m ] ε + [a + m , a † - m ] ε -± 2(1 + τ )δ lm a ± l = 0 (6.12a) a † ± l , [a † + m , a - m ] ε + [a + m , a † - m ] ε -± 2(1 + τ )δ lm a † ± l = 0. (6.12b)\n\nIt seems, these are the most general and sensible trilinear commutation relations one may impose on the creation and annihilation operators.\n\nFirst of all, we should mentioned that the standard bilinear commutation relations, viz.\n\n[1, 3, 11-15] [a ± l , a ± m ] -ε = 0 [a † ± l , a † ± m ] -ε = 0 [a ∓ l , a ± m ] -ε = (±1) 2j+1 τ δ lm id F [a † ∓ l , a † ± m ] -ε = (±1) 2j+1 τ δ lm id F [a ± l , a † ± m ] -ε = 0 [a † ± l , a ± m ] -ε = 0 [a ∓ l , a † ± m ] -ε = (±1) 2j+1 δ lm id F [a † ∓ l , a ± m ] -ε = (±1) 2j+1 δ lm id F , (6.13)\n\nprovide a solution of any one of the equations (6.10)-(6.12) in a sense that, due to (3.7) and (6.8), with η = -ε any set of operators satisfying (6.13) converts (6.10)-(6.12) into identities.\n\nBesides, this conclusion remains valid also if the normal ordering is taken into account, i.e. if, in this particular case, the changes a\n\n† - m • a + m → εa + m • a † - m and a - m • a † + m → εa † + m • a - m\n\nare made in (6.10)-(6.12). Now we shall demonstrate how the trilinear relations (6.12) lead to the paracommutation relations. Equations (6.12) can be 'split' into different kinds of trilinear commutation relations into infinitely many ways. For example, the system of equations\n\na ± l , [a + m , a † - m ] ε -± (1 + τ )δ lm a ± l = 0 (6.14a) a ± l , [a † + m , a - m ] ε -± (1 + τ )δ lm a ± l = 0 (6.14b) a † ± l , [a + m , a † - m ] ε -± (1 + τ )δ lm a † ± l = 0 (6.14c) a † ± l , [a † + m , a - m ] ε -± (1 + τ )δ lm a † ± l = 0 (6.14d)\n\nprovides an evident solution of (6.12). However, it is a simple algebra to be seen that these relations are incompatible with the standard (anti)commutation relations (6.13) and, in this sense, are not suitable as subsidiary restrictions on the Lagrangian formalism. For our purpose, the equations\n\na + l , [a + m , a † - m ] ε -+ 2δ lm a + l = 0 (6.15a) a + l , [a † + m , a - m ] ε -+ 2τ δ lm a + l = 0 (6.15b) a - l , [a + m , a † - m ] ε --2τ δ lm a - l = 0 (6.15c) a - l , [a † + m , a - m ] ε --2δ lm a - l = 0 (6.15d)\n\nand their Hermitian conjugate provide a solution of (6.12), which is compatible with (6.13), i.e. if (6.13) hold, the equations (6.15) are converted into identities. The idea of the paraquantization is in the following generalization of (6.15)\n\na + l , [a + m , a † - n ] ε -+ 2δ ln a + m = 0 (6.16a) a + l , [a † + m , a - n ] ε -+ 2τ δ ln a + m = 0 (6.16b) a - l , [a + m , a † - n ] ε --2τ δ lm a - n = 0 (6.16c) a - l , [a † + m , a - n ] ε --2δ lm a - n = 0 (6.16d)\n\nwhich reduces to (6.15) for n = m and is a generalization of (6.13) in a sense that any set of operators satisfying (6.13) converts (6.16) into identities, the opposite being generally not valid. 19 Suppose that the field considered consists of a single sort of particles, e.g. electrons or photons, created by b † l := a † l and annihilated by b l := a † - l . Then the equation Hermitian conjugated to (6.15a) reads\n\n[b l , [b † m , b m ] ε ] = 2δ lm b m . (6.17)\n\nThis is the main relation from which the paper [16] starts. The basic paracommutation relations are [16] [17] [18] 26] :\n\n[b l , [b † m , b n ] ε ] = 2δ lm b n (6.18a) [b l , [b m , b n ] ε ] = 0. (6.18b)\n\nThe first of them is a generalization (stronger version) of (6.17) by replacing the second index m with an arbitrary one, say n, and the second one is added (by \"hands\") in the theory as an additional assumption. Obviously, (6.18) are a solution of (6.15) and therefore of (6.12) in the considered case of a field consisting of only one sort of particles. The equations (6.15) contain also the relativistic version of the paracommutation relations, when the existence of antiparticles must be respected [18, sec. 18.1] . Indeed, noticing that the field's particles (resp. antiparticles) are created by b † l := a + l (resp. c † l := a † + l ) and annihilated by b l := a † - l (resp. c l := a - l ), from (6.15) and the Hermitian conjugate to them equations, we get\n\n[b l , [b † m , b m ] ε ] = 2δ lm b m [c l , [c † m , c m ] ε ] = 2δ lm c m (6.19a) [b † l , [c † m , c m ] ε ] = -2τ δ lm b † m [c † l , [b † m , b m ] ε ] = -2τ δ lm c † m . (6.19b)\n\nGeneralizing these equations in a way similar to the transition from (6.17) to (6.18), we obtain the relativistic paracommutation relations as (cf. (6.16))\n\n[b l , [b † m , b n ] ε ] = 2δ lm b n [b l , [b m , b n ] ε ] = 0 (6.20a) [c l , [c † m , c n ] ε ] = 2δ lm c n [c l , [c m , c n ] ε ] = 0 (6.20b) [b † l , [c † m , c n ] ε ] = -2τ δ ln b † m [c † l , [b † m , b n ] ε ] = -2τ δ ln c † m . (6.20c)\n\nThe equations (6.20a) (resp. (6.20b)) represent the paracommutation relations for the field's particles (resp. antiparticles) as independent objects, while (6.20c) describe a pure relativistic effect of some \"interaction\" (or its absents) between field's particles and antiparticles and fixes the paracommutation relations involving the b l 's and c l 's, as pointed in [18, p. 207] (where b l is denoted by a l and c l by b l ). The relations (6.17) and (6.20) for ε = +1 (resp. ε = -1) are referred as the parabose (resp. parafermi ) commutation relations [18] . This terminology is a natural one also with respect to the commutation relations (6.16), which will be referred as the paracommutation relations too.\n\nAs first noted in [16] , the equations (6.13) provide a solution of (6.20) (or (6.18) in the nonrelativistic case) but the latter equations admit also an infinite number of other solutions. Besides, by taking Hermitian conjugations of (some of) the equations (6.18) or (6.20) and applying generalized Jacobi identities, like\n\nα[[A, B] ξ , C] η + ξη[[A, C] -α/ξ , B] -α/η -α 2 [[B, C] ξη/α , A] 1/α = 0 αξη = 0 β[A, [B, C] α , ] -βγ + γ[B, [C, A] β , ] -γα + α[C, [A, B] γ , ] -αβ = 0 α, β, γ = ±1 [[A, B] η , C] -+ [[B, C] η , A] -+ [[C, A] η , B] -= 0 η = ±1 [[A, B] ξ , [C, D] η ] -= [[A, B] ξ , C] -, D] η + η[[A, B] ξ , D] -, C] 1/η η = 0, (6.21)\n\none can obtain a number of other (para)commutation relations for which the reader is referred to [16, 18, 26] .\n\nOf course, the paracommutation relations (6.16), in particular (6.18) and (6.20) as their stronger versions, do not give the general solution of the trilinear relations (6.12). For instance, one may replace (6.12) with the equations\n\na + l , [a † + m , a - n ] ε + [a + m , a † - n ] ε -+ 2(1 + τ )δ ln a + m = 0 (6.22a) a - l , [a † + m , a - n ] ε + [a + m , a † - n ] ε --2(1 + τ )δ lm a - n = 0. (6.22b)\n\nand their Hermitian conjugate, which in terms of the operators b l and c l introduced above read\n\n[b l , [b † m , b n ] ε + [c † m , c m ] ε ] = 2(1 + τ )δ lm b n (6.23a) [c l , [b † m , b n ] ε + [c † m , c m ] ε ] = 2(1 + τ )δ lm c n , (6.23b)\n\nand supplement these relations with equations like (6.18b). Obviously, equations (6.16) convert (6.22) into identities and, consequently, the (standard) paracommutation relations (6.20) provide a solution of (6.23). On the base of (6.23) or other similar equations that can be obtained by generalizing the ones in (6.10)-(6.12), further research on particular classes of trilinear commutation relations can be done, but, however, this is not a subject of the present work.\n\nLet us now pay attention to the fact that equations (6.10), (6.11) and (6.12) are generally different (regardless of existence of some connections between their solutions). The cause for this being that the momentum operators for the Lagrangians L ′ , L ′′ and L ′′′ are generally different unless some additional restrictions are added to the Lagrangian formalism (see Sect. 4). A necessary and sufficient condition for (6.10)-(6.12) to be identical is\n\n[a ± l , [a † + m , a - m ] -ε -[a + m , a † - m ] -ε ] = 0, (6.24)\n\nwhich certainly is valid if the condition (4.9 ′ ), viz.\n\n[\n\na † + m , a - m ] -ε -[a + m , a † - m ] -ε = 0, ( 6.25)\n\nensuring the uniqueness of the momentum operator are, holds. If one adopts the standard bilinear commutation relations (6.13), then (6.25), and hence (6.24), is identically valid, but in the framework of, e.g., the paracommutation relations (6.16) (or (6.20) in other form) the equations (6.25) should be postulated to ensure uniqueness of the momentum operator and therefore of the field equations. On the base of (6.10) or (6.11) one may invent other types of commutation relations, which will not be investigated in this paper because we shall be interested mainly in the case when (6.10), (6.11) and (6.12) are identical (see (6.24) ) or, more generally, when the dynamical variables are unique in the sense pointed in Sect. 4."
},
{
"section_type": "BACKGROUND",
"section_title": "Restrictions related to the charge operator",
"text": "The consequences of the Heisenberg relations (5.2), involving the charge operator for a charged field, q = 0 (and hence τ = 0 -see (3.7)), will be examined in this subsection. In terms of creation and annihilation operators it is equivalent to [1, [13] [14] [15] ]\n\n[a ± s (k), Q] = qa ± s (k) [a † ± s (k), Q] = -qa † ± s (k), (6.26)\n\nthe values of the polarization indices being specified by (6.5). Substituting here (3.10), we see that, for a charged field, the field equations for the Lagrangians L ′ , L ′′ and L ′′′ (see Sect. 3) respectively are:\n\n2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a ± s (k), a † + t (p) • a - t (p) -εa † - t (p) • a + t (p)] -a ± s (k)δ st δ 3 (k -p)} = 0 (6.27a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a † ± s (k), a † + t (p) • a - t (p) -εa † - t (p) • a + t (p)] + a † ± s (k)δ st δ 3 (k -p)} = 0 (6.27b) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a ± s (k), a + t (p) • a † - t (p) -εa - t (p) • a † + t (p)] + a ± s (k)δ st δ 3 (k -p)} = 0 (6.28a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a † ± s (k), a + t (p) • a † - t (p) -εa - t (p) • a † + t (p)] -a † ± s (k)δ st δ 3 (k -p)} = 0 (6.28b) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a ± s (k), [a † + t (p), a - t (p)] ε -[a + t (p), a † - t (p) ε ] -2a ± s (k)δ st δ 3 (k -p)} = 0 (6.29a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p{[a † ± s (k), [a † + t (p), a - t (p)] ε -[a + t (p), a † - t (p) ε ] + 2a † ± s (k)δ st δ 3 (k -p)} = 0. (6.29b)\n\nUsing (6.27)-(6.29) and (6.8), with η = ε = -1, or simply (6.26), one can easily verify the validity of the equations\n\n[ Pµ , Q] = 0 [ Lµν , Q] = 0 [ Sµν , Q] = 0 [ Mµν , Q] = 0, (6.30)\n\nwhere the operators Pµ , Q, Sµν , Lµν and Mµν are calculated from one and the same initial Lagrangian according to (3.9)-(3.12). This result confirms the validity of assertion (ii) before (5.24) for the fields considered. Following the above considerations, concerning the momentum operator, we shall now replace the systems of integral equations (6.27)-(6.29) with respectively the following stronger systems of algebraic equations (by equating to zero the integrands in (6.27)-(6.29)):\n\na ± l , a † + m • a - m -εa † - m • a + m --δ lm a ± l = 0 (6.31a) a † ± l , a † + m • a - m -εa † - m • a + m -+ δ lm a † ± l = 0 (6.31b) a ± l , a + m • a † - m -εa - m • a † + m -+ δ lm a ± l = 0 (6.32a) a † ± l , a + m • a † - m -εa - m • a † + m --δ lm a † ± l = 0 (6.32b) a ± l , [a † + m , a - m ] ε -[a + m , a † - m ] ε --2δ lm a ± l = 0 (6.33a) a † ± l , [a † + m , a - m ] ε -[a + m , a † - m ] ε -+ 2δ lm a † ± l = 0. (6.33b)\n\nThese trilinear commutation relations are similar to (6.10)-(6.12) and, consequently, can be treated in analogous way. By invoking (6.8), it is a simple algebra to be proved that the standard bilinear commutation relations (6.13) convert (6.31)-(6.33) into identities. Thus (6.13) are stronger version of (6.31)-(6.33) and, in this sense, any type of commutation relations, which provide a solution of (6.31)-(6.33) and is compatible with (6.13), is a suitable candidate for generalizing (6.13). To illustrate that idea, we shall proceed with (6.33) in a way similar to the 'derivation' of the paracommutation relations from (6.12).\n\nObviously, the equations (cf. (6.14) with τ = 0, as now q = 0)\n\n[a ± l , [a + m , a † - m ] ε ] + δ lm a ± m = 0 (6.34a) [a ± l , [a † + m , a - m ] ε ] -δ lm a ± m = 0 (6.34b)\n\nand their Hermitian conjugate provide a solution of (6.33), but, as a direct calculations shows, they do not agree with the standard (anti)commutation relations (6.13). A solution of (6.33) compatible with (6.13) is given by the equations (6.15), with τ = 0 as the field considered is charged one -see (3.7). Therefore equations (6.16), with τ = 0, also provide a compatible with (6.13) solution of (6.33), from where immediately follows that the paracommutation relations (6.20), with τ = 0, convert (6.33) into identities. To conclude, we can say that the paracommutation relations (6.20), in particular their special case (6.13), ensure the simultaneous validity of the Heisenberg relations (5.1) and (5.2) for free scalar, spinor and vector fields.\n\nSimilarly to (6.22), one may generalize (6.33) to\n\na + l , [a † + m , a - n ] ε -[a + m , a † - n ] ε --2δ ln a + m = 0 (6.35a) a - l , [a † + m , a - n ] ε -[a + m , a † - n ] ε --2δ lm a - n = 0. (6.35b)\n\nwhich equations agree with (6.13), (6.15), (6.16) and (6.20), but generally do not agree with (6.22), with τ = 0, unless the equations (6.16), with τ = 0, hold. More generally, we can assert that (6.33) and (6.12), with τ = 0, hold simultaneously if and only if (6.15), with τ = 0, is fulfilled. From here, again, it follows that the paracommutation relations ensure the simultaneous validity of (5.1) and (5.2).\n\nLet us say now some words on the uniqueness problem for the Heisenberg equations involving the charge operator. The systems of equations (6.31)-(6.33) are identical iff\n\na ± l , [a † + m , a - m ] -ε + [a + m , a † - m ] -ε -= 0, (6.36) which, in particular, is satisfied if the condition [a † + m , a - m ] -ε + [a + m , a † - m ] -ε = 0, (6.37)\n\nensuring the uniqueness of the charge operator (see (4.10 ′ )), is valid. Evidently, equations (6.36) and (6.24) are compatible iff\n\na + l , [a † ± m , a ∓ m ] -ε -= 0 a - l , [a † ± m , a ∓ m ] -ε -= 0 (6.38)\n\nwhich is a weaker form of (4.15) ensuring simultaneous uniqueness of the momentum and charge operator."
},
{
"section_type": "BACKGROUND",
"section_title": "Restrictions related to the angular momentum operator(s)",
"text": "It is now turn to be investigated the restrictions on the creation and annihilation operators that follow from the Heisenberg relations (5.3) concerning the angular momentum operator. They can be obtained by inserting the equations (3.11) and (3.12) into (5.3). As pointed in Sect. 5, the resulting equalities, however, depend not only on the particular Lagrangian employed, but also on the geometric nature of the field considered; the last dependence being explicitly given via (5.25) and the polarization functions σ ss ′ m± µν (k) and l ss ′ m± µν (k) (see also (3.14) ).\n\nConsider the terms containing derivatives in (5.3),\n\nLor µν := i x µ ∂ ∂x ν -x ν ∂ ∂x µ φi (x). (6.39)\n\nIf φi (k) denotes the Fourier image of φi (x), i.e.\n\nφi (x) = Λ d 4 ke -1 i k µ xµ φi (k), (6.40)\n\nwith Λ being a normalization constant, then the Fourier image of (6.39) is\n\nLor µν = i k µ ∂ ∂k ν -k ν ∂ ∂k µ φi (k). ( 6\n\nComparing this expression with equations (3.12), we see that the terms containing derivatives in (3.12) should be responsible for the term (6.39) in (5.3). 20 For this reason, we shall suppose that the momentum operator Mµν admits a representation Mµν = Mor µν + Msp µν (6.42) such that the operators Mor µν and Msp µν satisfy the relations (5.4) and (5.5), respectively. Thus we shall replace (5.3) with the stronger system of equations (5.4)-(5.5). Besides, we shall admit that the explicit form of the operators Mor µν and Msp µν are given via (5.13) and (5.12) for the fields investigated in the present work.\n\nLet us consider at first the 'orbital' Heisenberg relations (5.4), which is independent of the particular geometrical nature of the fields studied. Substituting (5.13) and (6.40) into (5.4), using that φi (±k), with k 2 = m 2 c 2 , is a linear combination of ã± s (k) with classical, not operator-valued, functions of k as coefficients [1, [13] [14] [15] and introducing for brevity the operator\n\nω µν (k) := k µ ∂ ∂k ν -k ν ∂ ∂k µ , (6.43)\n\nwe arrive to the following integro-differential systems of equations:\n\n2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã ± s (k), ã † + t (p) • ã- t (q) -εã † - t (p) • ã+ t (q)] ) q=p p 0 = √ m 2 c 2 +p 2 = 2(1 + τ )ω µν (k)(ã ± s (k)) (6.44a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã † ± s (k), ã † + t (p) • ã- t (q) -εã † - t (p) • ã+ t (q)] ) q=p p 0 = √ m 2 c 2 +p 2 = 2(1 + τ )ω µν (k)(ã † ± s (k)) (6.44b) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã ± s (k), ã+ t (p) • ã † - t (q) -εã - t (p) • ã † + t (q)] ) q=p p 0 = √ m 2 c 2 +p 2 = 2(1 + τ )ω µν (k)(ã ± s (k)) (6.45a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã † ± s (k), ã+ t (p) • ã † - t (q) -εã - t (p) • ã † + t (q)] ) q=p p 0 = √ m 2 c 2 +p 2 = 2(1 + τ )ω µν (k)(ã † ± s (k)) (6.45b) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã ± s (k), [ã † + t (p), ã- t (q)] ε + [ã + t (p), ã † - t (q)] ε ] ) q=p p 0 = √ m 2 c 2 +p 2 = 4(1 + τ )ω µν (k)(ã ± s (k)) (6.46a) 2j+1-δ 0m (1-δ 0j ) t=1 d 3 p (-ω µν (p) + ω µν (q))([ã † ± s (k), [ã † + t (p), ã- t (q)] ε + [ã + t (p), ã † - t (q)] ε ] ) q=p p 0 = √ m 2 c 2 +p 2 = 4(1 + τ )ω µν (k)(ã † ± s (k)), (6.46b)\n\nwhere k 0 = m 2 c 2 + k 2 is set after the differentiations are performed (see (6.43)). Following the procedure of the previous considerations, we replace the integro-differential equations (6.44)-(6.46) with the following differential ones:\n\n(-ω • µν (m) + ω • µν (n))([ã ± l , ã † + m • ã- n -εã † - m • ã+ n ] ) n=m = 2(1 + τ )δ lm ω • µν (l)(ã ± l ) (6.47a) (-ω • µν (m) + ω • µν (n))([ã † ± l , ã † + m • ã- n -εã † - m • ã+ n ] ) n=m = 2(1 + τ )δ lm ω • µν (l)(ã † ± l ) (6.47b) (-ω • µν (m) + ω • µν (n))([ã ± l , ã+ m • ã † - n -εã - m • ã † + n ] ) n=m = 2(1 + τ )δ lm ω • µν (l)(ã ± l ) (6.48a) (-ω • µν (m) + ω • µν (n))([ã † ± l , ã+ m • ã † - n -εã - m • ã † + n ] ) n=m = 2(1 + τ )δ lm ω • µν (l)(ã † ± l ) (6.48b) (-ω • µν (m) + ω • µν (n))([ã ± l , [ã † + m , ã- n ] ε + [ã + m , ã † - n ] ε ] ) n=m = 4(1 + τ )δ lm ω • µν (l)(ã ± l ) (6.49a) (-ω • µν (m) + ω • µν (n))([ã † ± l , [ã † + m , ã- n ] ε + [ã + m , ã † - n ] ε ] ) n=m = 4(1 + τ )δ lm ω • µν (l)(ã † ± l ), ( 6\n\n.49b) where we have set (cf. (6.43))\n\nω • µν (l) := ω µν (k) = k µ ∂ ∂k ν -k ν ∂ ∂k µ if l = (s, k) (6.50)\n\nand k 0 = m 2 c 2 + k 2 is set after the differentiations are performed.\n\nRemark. Instead of (6.47)-(6.49) one can write similar equations in which the operator -ω • µν (m) or +ω • µν (n) is deleted and the factor + 1 2 or -1 2 , respectively, is added on their right hand sides. These manipulations correspond to an integration by parts of some of the terms in (6.44)-(6.46).\n\nThe main difference of the obtained trilinear relations with respect to the previous ones considered above is that they are partial differential equations of first order.\n\nThe relations (6.49) agree with the equations (6.16) in a sense that if (6.16) hold, then (6.49) become identically valid. Indeed, since\n\n(-ω • µν (m) + ω • µν (n))(ã ± m δ ln ) n=m = -2δ lm ω • µν (m)(ã ± m ) (-ω • µν (m) + ω • µν (n))(ã ± n δ lm ) n=m = +2δ lm ω • µν (m)(ã ± m ), (6.51)\n\ndue to (6.50), (6.43) and the equality dδ(x) dx f (x) = -δ(x) df (x) dx for a C 1 function f , the application of the operator (-ω • µν (m) + ω • µν (n)) to (6.16) and subsequent setting n = m entails (6.49). In particular, this means that the paracommutation relations (6.20) and, moreover, the standard (anti)commutation relations (6.13) convert (6.49) into identities. Therefore the 'orbital' Heisenberg relations (5.4) hold for scalar, spinor and vector fields satisfying the bilinear or para commutation relations.\n\nIt should be noted, the paracommutation relations are not the only trilinear commutation relations that are solutions of (6.49). As an example, we shall present the trilinear relations\n\na + l , [a + m , a † - n ] ε -= a + l , [a † + m , a - n ] ε -= -(1 + τ )δ ln a + m (6.52a) a - l , [a + m , a † - n ] ε -= a - l , [a † + m , a - n ] ε -= +(1 + τ )δ lm a + n , (6.52b)\n\nwhich reduce to (6.14) for n = m, do not agree with (6.13), but convert (6.49) into identities (see (6.51)). Other example is provided by the equations (6.22), which are compatible with the paracommutation relations and, as a result of (6.51), convert (6.49) into identities. Prima facie one may suppose that any solution of (6.12) provides a solution of (6.49), but this is not the general case. A counterexample is provided by the commutation relations\n\na ± l , [a † + m , a - n ] ε + [a + m , a † - n ] ε -± 2(1 + τ )δ ln a ± m = 0, (6.53)\n\nwhich reduce to (6.12) for n = m, satisfy (6.49) with ã+ l for ã± l , and do not satisfy (6.49) with ãl for ã± l (see (6.51) and cf. (6.22)). From (5.13) follows that the operator Mor µν is independent of the Lagrangian L ′ , L ′′ or L ′′′ one starts off if and only if (see (4.11))\n\n(-ω • µν (m) + ω • µν (n)) [ã † + m , ã- n ] -ε -[ã + m , ã † - n ] -ε n=m = 0. (6.54)\n\nThis condition ensures the coincidence of the systems of equations (6.47), (6.48) and (6.49) too. However, the following necessary and sufficient condition for the coincidence of these systems is expressed by the weaker equations\n\n(-ω • µν (m) + ω • µν (n)) ã± l , [ã † + m , ã- n ] -ε -[ã + m , ã † - n ] -ε -n=m = 0. (6.55)\n\nIt is now turn to be considered the 'spin' Heisenberg relations (5.5).\n\nRecall, the field operators ϕ i for the fields considered here admit a representation [13] [14] [15]\n\nϕ i = Λ t d 3 p v t,+ i (p)a + t (p) + v t,- i (p)a - t (p) , (6.56)\n\nwhere Λ is a normalization constant and v t,± i (p) are classical, not operator-valued, complex or real functions which are linearly independent. The particular definition of v t,± i (p) depends on the geometrical nature of ϕ i and can be found in [13] [14] [15] (see also [1] ), where the reader can find also a number of relations satisfied by v t,± i (p). Here we shall mention only that v t,± i (p) = 1 for a scalar field and v t,+ i (p) = v t,- i (p) =: v t i (p) = (v t i (p)) * for a vector field. The explicit form of the polarization functions σ ss ′ ,± µν (k) and l ss ′ ,± µν (k) (see Sect. 3, in particular (3.14)) through v t,± i (k) are [13] [14] [15] :\n\nσ ss ′ ,± µν (k) = (-1) j j + δ j0 i,i ′ (v s,± i (k)) * I i i ′ µν v t,± i ′ (k) l ss ′ ,± µν (k) = (-1) j 2j + δ j0 i (v s,± i (k)) * ← ---- -→ k µ ∂ ∂k ν - ← ---- -→ k ν ∂ ∂k µ v t,± i (k), (6.57)\n\nwith an exception that σ ss ′ ,± 0a (k) = σ ss ′ ,± a0 (k) = 0, a = 1, 2, 3, for a spinor field, j = 1 2 , [14] . Evidently, the equations (3.14) follow from the mentioned facts (see also (5.25) ).\n\nSubstituting (6.56) and (5.12) into (5.5), we obtain the following systems of integral equations (corresponding respectively to the Lagrangians L ′ , L ′′ and L ′′′ ):\n\n(-1) j+1 j 1 + τ s,s ′ ,t d 3 k d 3 pv t,± i (p) (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k))[a ± t (p), a † + s (k) • a - s ′ (k)] + (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))[a ± t (p), a † - s (k) • a + s ′ (k)] = i ′ t d 3 pI i ′ iµν v t,± i ′ (p)a ± t (p) (6.58) ε (-1) j+1 j 1 + τ s,s ′ ,t d 3 k d 3 pv t,± i (p) (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))[a ± t (p), a + s ′ (k) • a † - s (k)] + (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k))[a ± t (p), a - s ′ (k) • a † + s (k)] = i ′ t d 3 pI i ′ iµν v t,± i ′ (p)a ± t (p) (6.59) (-1) j+1 j 2(1 + τ ) s,s ′ ,t d 3 k d 3 pv t,± i (p) (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k)) a ± t (p), [a † + s (k), a - s ′ (k)] ε - + (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k)) a ± t (p), [a † - s (k), a + s ′ (k)] ε -= i ′ t d 3 pI i ′ iµν v t,± i ′ (p)a ± t (p). (6.60)\n\nFor the difference of all previously considered systems of integral equations, like (6.2)-(6.4), (6.27)-(6.29) and (6.44)-(6.46), the systems (6.58)-(6.60) cannot be replaced by ones consisting of algebraic (or differential) equations. The cause for this state of affairs is that in (6.58)-(6.60) enter polarization modes with arbitrary s and s ′ and, generally, one cannot 'diagonalize' the integrand(s) with respect to s and s ′ ; moreover, for a vector field, the modes with s = s ′ are not presented at all (see (3.14) ). That is why no commutation relations can be extracted from (6.58)-(6.60) unless further assumptions are made. Without going into details, below we shall sketch the proof of the assertion that the commutation relations (6.16) convert (6.60) into identities for massive spinor and vector fields. 21 In particular, this entails that the paracommutation and the bilinear commutation relations provide solutions of (6.60).\n\nLet (6.16) holds. Combining it with (6.60), we see that the latter splits into the equations 21 The equations (6.58)-(6.60) are identities for scalar fields as for them Iµν = 0 and v t,± i (k) = 1, which reflects the absents of spin for these fields.\n\n(-1) j j 1 + τ s,t d 3 pv t,+ i (p) τ (σ st,- µν (p) + l st,- µν (p)) + ε(σ ts,+ µν (p) + l ts,+ µν (p)) a + s (p), = i ′ I i ′ iµν s d 3 pv s,+ i ′ (p)a + s (p) (6.61a) (-1) j+1 j 1 + τ s,t d 3 pv t,- i (p) (σ ts,- µν (p) + l ts,- µν (p)) + ετ (σ st,+ µν (p) + l st,+ µν (p)) a - s (p), = i ′ I i ′ iµν s d 3 pv s,- i ′ (p)a - s (p). (6.61b)\n\nInserting here (6.57), we see that one needs the explicit definition of v s,± i (k) and formulae for sums like ρ ii ′ (k) := s v s,± i (k)(v s,± i ′ (k)) * , which are specific for any particular field and can be found in [13] [14] [15] . In this way, applying (5.25), (3.7) and the mentioned results from [13] [14] [15] , one can check the validity of (6.61) for massive fields in a way similar to the proof of (5.3) in [13] [14] [15] for scalar, spinor and vector fields, respectively.\n\nWe shall end the present subsection with the remark that the equations (4.17) and (4.18), which together with (4.15) ensure the uniqueness of the spin and orbital operators, are sufficient conditions for the coincidence of the equations (6.58), (6.59) and (6.60)."
},
{
"section_type": "OTHER",
"section_title": "Inferences",
"text": "To begin with, let us summarize the major conclusions from Sect. 6. Each of the Heisenberg equations (5.1)-(5.3), the equations (5.3) being split into (5.4) and (5.5), induces in a natural way some relations that the creation and annihilation operators should satisfy. These relations can be chosen as algebraic trilinear ones in a case of (5.1) and (5.2) (see (6.10)-(6.12) and (6.31)-(6.33), respectively). But for (5.4) and (5.5) they need not to be algebraic and are differential ones in the case of (5.4) (see (6.47)-(6.49)) and integral equations in the case of (5.5) (see (6.58)-(6.60)). It was pointed that the cited relations depend on the initial Lagrangian from which the theory is derived, unless some explicitly written conditions hold (see (6.24), (6.37) and (6.55)); in particular, these conditions are true if the equations (4.9)-(4.13), ensuring the uniqueness of the corresponding dynamical operators, are valid. Since the 'charge symmetric' Lagrangians (3.4) seem to be the ones that best describe free fields, the arising from them (commutation) relations (6.12), (6.33), (6.49) and (6.60) were studied in more details. It was proved that the trilinear commutation relations (6.16) convert them into identities, as a result of which the same property possess the paracommutation relations (6.20) and, in particular, the bilinear commutation relations (6.13). Examples of trilinear commutation relations, which are neither ordinary nor para ones, were presented; some of them, like (6.14), (6.34) and (6.52), do not agree with (6.13) and other ones, like (6.16), (6.22) and (6.35), generalize (6.20) and hence are compatible with (6.13). At last, it was demonstrated that the commutators between the dynamical variables (see (5.15)-(5.23)) are uniquely defined if a Heisenberg relation for one of the operators entering in it is postulated.\n\nThe chief aim of the present section is to be explored the problem whether all of the reasonable conditions, mentioned in the previous sections and that can be imposed on the creation and annihilation operators, can hold or not hold simultaneously. This problem is suggested by the strong evidences that the relations (5.1)-(5.3) and (5.15)-(5.23), with a possible exception of (5.3) (more precisely, of (5.5)) in the massless case, should be valid in a realistic quantum field theory [1, 3, 7, 8, 11, 12] . Besides, to the arguments in loc. cit., we shall add the requirement for uniqueness of the dynamical variables (see Sect. 4).\n\nAs it was shown in Sect. 6, the relations (5.1), (5.2), (5.4) and (5.5) are compatible if one starts from a charge symmetric Lagrangian (see (3.4 )), which best describes a free field theory; in particular, the commutation relations (6.16) (and hence (6.20) and (6.13)) ensure their simultaneous validity. 22 For that reason, we shall investigate below only commutation relations for which (5.1), (5.2), (5.4) and (5.5) hold. It will be assumed that they should be such that the equations (6.10)-(6.12), (6.31)-(6.33), (6.47)-(6.49) and (6.58)-(6.60), respectively, hold.\n\nConsider now the problem for the uniqueness of the dynamical variables and its consistency with the commutation relations just mentioned for a charged field. It will be assumed that this uniqueness is ensured via the equations (4.9)-(4.11).\n\nThe equation (4.15), viz.\n\n[\n\na † ± m , a ∓ m ] -ε = 0, ( 7.1)\n\nis a necessary and sufficient conditions for the uniqueness of the momentum and charge operators (see Sect. 4 and the notation introduced at the beginning of Sect. 6). Before commenting on this relation, we would like to derive some consequences of it. Applying consequently (6.8) for η = -ε, (7.1) and the identity\n\n[A, B • C] + = [A, B] η • C -ηB • [A, C] -η η = ±1 (7.2)\n\nfor η = +ε, -ε, we, in view of (7.1), obtain\n\n[a + m , [a + m , a † - m ] ε ] = [a † - m , [a + m , a + m ] -ε ] + = (1 -ε)[a † - m , a + m ] ε • a + m [a - m , [a † + m , a - m ] ε ] = ε[a † + m , [a - m , a - m ] -ε ] + = ε(1 -ε)[a † + m , a - m ] ε • a - m . (7.3)\n\nForming the sum and difference of (6.12a), for τ = 0, and (6.33a), we see that the system of equations they form is equivalent to\n\n[a + l , [a † + m , a - m ] ε ] = 0 [a - l , [a + m , a † - m ] ε ] = 0 (7.4a) [a + l , [a + m , a † - m ] ε ] + 2δ lm a + l = 0 [a - l , [a † + m , a - m ] ε ] -2δ lm a - l = 0. (7.4b)\n\nCombining (7.4b), for l = m, with (7.3), we get\n\n(1 -ε)[a † - m , a + m ] ε • a + m + 2a + m = 0 ε(1 -ε)[a † + m , a - m ] ε • a - m -2a - m = 0. (7.5)\n\nObviously, these equations reduce to a ± m = 0 (7.6)\n\nfor bose fields as for them ε = +1 (see (3.7)). Since the operators (7.6) describe a completely unobservable field, or, more precisely, an absence of a field at all, the obtained result means that the theory considered cannot describe any really existing physical field with spin j = 0, 1. Such a conclusion should be regarded as a contradiction in the theory. For fermi fields, j = 1 2 and ε = -1, the equations (7.5) have solutions different from (7.6) iff a ± m are degenerate operators, i.e. with no inverse ones, in which case (7.4a) is a consequence of (7.5) and (7.1) (see (6.8 ) and ( 7 .3) too).\n\nThe source of the above contradiction is in the equation (7.1), which does not agree with the bilinear commutation relations (6.13) and contradicts to the existing correlation between creation and annihilation of particles with identical characteristics (m = (t, p) in our case) as (7.1) can be interpreted physically as mutual independence of the acts of creation and annihilation of such particles [1, § 10.1] .\n\nAt this point, there are two ways for 'repairing' of the theory. On one hand, one can forget about the uniqueness of the dynamical variables (in a sense of Sect. 4), after which the formalism can be developed by choosing, e.g., the charge symmetric Lagrangians (3.4) and following the usual Lagrangian formalism; in fact, this is the way the parafield theory is build [16, 18] . On another hand, one may try to change something at the ground of the theory in such a way that the uniqueness of the dynamical variables to be ensured automatically. We shall follow the second method. As a guiding idea, we shall have in mind that the bilinear commutation relations (6.13) and the related to them normal ordering procedure provide a base for the present-day quantum field theory, which describes sufficiently well the discovered elementary particles/fields. On this background, an extensive exploration of commutation relations which are incompatible with (6.13) is justified only if there appear some evidences for fields/particles that can be described via them. In that connection it should be recalled [17, 18] , it seems that all known particles/fields are described via (6.13) and no one of them is a para particle/field.\n\nUsing the notation introduced at the beginning of Sect. 4, we shall look for a linear mapping (operator) E on the operator space over the system's Hilbert space F of states such that\n\nE( D ′ ) = E( D ′′ ). (7.7)\n\nAs it was shown in Sect. 4, an example of an operator E is provided by the normal ordering operator N . Therefore an operator satisfying (7.7) always exists. To any such operator E there corresponds a set of dynamical variables defined via\n\nD = E( D ′ ). (7.8)\n\nLet us examine the properties of the mapping E that it should possess due to the requirement (7.7) .\n\nFirst of all, as the operators of the dynamical variables should be Hermitian, we shall require\n\nE( B) † = E( B † ) (7.9)\n\nfor any operator B, which entails\n\nD † = D, (7.10)\n\ndue to (3.9)-(3.12) and (7.8). As in Sect. 4, we shall replace the so-arising integral equations with corresponding algebraic ones. Thus the equations (4.5)-(4.20) remain valid if the operator E is applied to their left hand sides.\n\nConsider the general case of a charged field, q = 0. So, the analogue of (4.15) reads .11) which equation ensures the uniqueness of the momentum and charge operators. Respectively, the condition (4.11) transforms into\n\nE [a † ± m , a ∓ m ] -ε = 0, ( 7\n\n(-ω • µν (m) + ω • µν (n)) E([a † + m , a - n ] -ε ) -E([a + m , a † - n ] -ε ) n=m = 0, (7.12)\n\nwhich, by means of (7.11) can be rewritten as (cf. (4.16))\n\nω • µν (n) E([a † + m , a - n ] -ε ) -E([a + m , a † - n ] -ε ) n=m = 0. (7.13)\n\nAt the end, equations (4.17) and (4.18) now should be written as\n\ns,s ′ σ ss ′ ,- µν (k) E [a † + s (k), a - s ′ (k)] -ε + σ ss ′ ,+ µν (k) E [a † - s (k), a + s ′ (k)] -ε = 0 (7.14) s,s ′ l ss ′ ,- µν (k) E [a † + s (k), a - s ′ (k)] -ε + l ss ′ ,+ µν (k) E [a † - s (k), a + s ′ (k)] -ε = 0. (7.15)\n\nThese equations can be satisfied if we generalize (7.11) to (cf. (4.20) )\n\nE [a † ± s (k), a ∓ s ′ (k)] -ε = 0 (7.16)\n\nfor any s and s ′ . At last, the following stronger version of (7.16 )\n\nE [a † ± m , a ∓ n ] -ε = 0, (7.17)\n\nfor any m = (t, p) and n = (r, q), ensures the validity of (7.14) and (7.15) and thus of the uniqueness of all dynamical variables. It is time now to call attention to the possible commutation relations. The replacement\n\nD ′ , D ′′ , D ′′′ → D := E( D ′ ) = E( D ′′ ) = E( D ′′′\n\n) results in corresponding changes in the whole of the material of Sect. 6. In particular, the systems of commutation relations (6.10)-(6.12), (6.31)-(6.33), (6.47)-( 6 .49) and (6.58)-( 6 .60) should be replaced respectively with:\n\n23 a ± l , E(a † + m • a - m ) + ε E(a † - m • a + m ) -± (1 + τ )δ lm a ± l = 0 (7.18) a ± l , E(a † + m • a - m ) -ε E(a † - m • a + m ) --δ lm a ± l = 0 (7.19) (-ω • µν (m) + ω • µν (n))([ã ± l , E(ã † + m • ã- n ) -ε E(ã † - m • ã+ n )] ) n=m = 2(1 + τ )δ lm ω • µν (l)(ã ± l ) (7.20) (-1) j+1 j 1 + τ s,s ′ ,t d 3 k d 3 pv t,± i (p) (σ ss ′ ,- µν (k) + l ss ′ ,- µν (k))[a ± t (p), E(a † + s (k) • a - s ′ (k))] + (σ ss ′ ,+ µν (k) + l ss ′ ,+ µν (k))[a ± t (p), E(a † - s (k) • a + s ′ (k))] = i ′ t d 3 pI i ′ iµν v t,± i ′ (p)a ± t (p). (7.21)\n\nDue to the uniqueness conditions (7.11)-( 7 .14), one can rewrite the terms E(a † ± m • a ∓ m ) in (7.18)-(7.21) in a number of equivalent ways; e.g. (see (7.11 ))\n\nE(a † ± m • a ∓ m ) = ε E(a ∓ m • a † ± m ) = 1 2 E([a † ± m , a ∓ m ] ε ). (7.22)\n\nConsider the general case of a charged field, q = 0 (and hence τ = 0). The system of equations (7.18)- (7.19) is then equivalent to\n\na ± l , E(a † ± m • a ∓ m ) -= 0 (7.23a) a + l , E(a † - m • a + m ) -+ εδ lm a + l = 0 (7.23b) a - l , E(a † + m • a - m ) --δ lm a - l = 0. (7.23c)\n\nThese (commutation) relations ensure the simultaneous fulfillment of the Heisenberg relations (5.1) and (5.2) involving the momentum and charge operators, respectively. To ensure also the validity of (7.20), with τ = 0, and, consequently, of (5.4), we generalize (7.23) to\n\na ± l , E(a † ± m • a ∓ n ) -= 0 (7.24a) a + l , E(a † - m • a + n ) -+ εδ lm a + n = 0 (7.24b) a - l , E(a † + m • a - n ) --δ lm a - n = 0, (7.24c)\n\nfor any l = (s, k), m = (t, p) and n = (t, q) (see also (6.51) ). In the way pointed in Sect. 6, one can verify that (7.24) for any l = (s, k), m = (t, p) and n = (r, p) entails (7.21) and hence (5.5) . At last, to ensure the validity of all of the mentioned conditions and a suitable transition to a case of Hermitian field, for which q = 0 and τ = 1 (see (3.7)), we generalize (7.24) to\n\na + l , E(a † + m • a - n ) -+ τ δ ln a + m = 0 (7.25a) a - l , E(a † - m • a + n ) --ετ δ ln a - m = 0 (7.25b) a + l , E(a † - m • a + n ) -+ εδ lm a + n = 0, (7.25c) a - l , E(a † + m • a - n ) --δ lm a - n = 0 (7.25d)\n\nwhere l, m and n are arbitrary. As a result of (7.17), which we assume to hold, and τ a † ± l = τ a ± l (see (3.7)), the equations (7.25a) and (7.25c) (resp. (7.25b ) and (7.25d)) become identical when τ = 1 (and hence a † ± l = a ± l ); for τ = 0 the system (7.25) reduces to (7.24) . Recalling that ε = (-1) 2j (see (3.7)), we can rewrite (7.25) in a more compact form as\n\na ± l , E(a † ± m • a ∓ n ) -+ (±1) 2j+1 τ δ ln a ± m = 0 (7.26a) a ± l , E(a † ∓ m • a ± n ) --(∓1) 2j+1 τ δ lm a ± n = 0. (7.26b)\n\nSince the last equation is equivalent to (see (7.17) ) and use that ε = (-1) 2j )\n\na ± l , E(a ± m • a † ∓ n ) -+ (±1) 2j+1 δ ln a ± m = 0, (7.26b ′ )\n\nit is evident that the equations (7.26a) and (7.26b) coincide for a neutral field.\n\nLet us draw the main moral from the above considerations: the equations (7.17) are sufficient conditions for the uniqueness of the dynamical variables, while (7.26) are such conditions for the validity of the Heisenberg relations (5.1)-(5.5), in which the dynamical variables are redefined according to (7.8) . So, any set of operators a ± l and E, which are simultaneous solutions of (7.17) and (7.26) , ensure uniqueness of the dynamical variables and at the same time the validity of the Heisenberg relations.\n\nConsider the uniqueness problem for the solutions of the system of equations consisting of (7.17)and (7.26) . Writing (7.17) as\n\nE(a † ± m • a ∓ n ) = ε E(a ∓ n • a † ± m ) = 1 2 E([a † ± m , a ∓ n ] ε ), (7.27)\n\nwhich reduces to (7.22) for n = m, and using ε = (-1) 2j (see (3.7)), one can verify that (7.26) is equivalent to\n\na + l , E([a + m , a † - n ] ε ) -+ 2δ ln a + m = 0 (7.28a) a + l , E([a † + m , a - n ] ε ) -+ 2τ δ ln a + m = 0 (7.28b) a - l , E([a + m , a † - n ] ε ) --2τ δ lm a - n = 0 (7.28c) a - l , E([a † + m , a - n ] ε ) --2δ lm a - n = 0. (7.28d)\n\nThe similarity between this system of equations and (6.16) is more than evident: (7.28) can be obtained from (6.16) by replacing\n\n[•, •] ε with E([•, •] ε ).\n\nAs it was said earlier, the bilinear commutation relations (6.13) and the identification of E with the normal ordering operator N , E = N , (7.29) convert (7.27)-(7.28) into identities; by invoking (6.8), for η = -ε, the reader can check this via a direct calculation (see also (4.23) ). However, this is not the only possible solution of (7.27)-(7.28). For example, if, in the particular case, one defines an 'anti-normal' ordering operator A as a linear mapping such that\n\nA(a + m • a † - n ) := εa † - n • a + m A(a † + m • a - n ) := εa - n • a † + m A(a - m • a † + n ) := a - m • a † + n A(a † - m • a + n ) := a † - m • a + n , (7.30)\n\nthen the bilinear commutation relations (6.13) and the setting E = A provide a solution of (7.27)-(7.28); to prove this, apply (6.8) for η = -ε. Evidently, a linear combination of N and A, together with (6.13), also provides a solution of (7.27)-(7.28). 24 Other solution of the same system of equations is given by E = id and operators a ± l satisfying (6.16), in particular the paracommutation relations (6.20), and\n\na † ± m • a ,∓ n = εa ,∓ n • a † ± m .\n\nThe problem for the general solution of (7.27)-(7.28) with respect to E and a ± l is open at present. Let us introduce the particle and antiparticle number operators respectively by (see (7.27 ), (7.9) and (3.16))\n\nN l := 1 2 E [a + l , a † - l ] = E(a + l • a † - l ) = ( N l ) † =: N † l † N l := 1 2 E [a † + l , a - l ] = E(a † + l • a - l ) = ( † N l ) † =: † N l † . (7.31)\n\nAs a result of the commutation relations (7.28), with n = m, they satisfy the equations 25\n\n[ N l , a + m ] -= δ lm a + l (7.32a) [ † N l , a + m ] -= τ δ lm a + l (7.32b) [ N l , a † + m ] -= τ δ lm a † + l (7.32c) [ † N l , a † + m ] -= δ lm a † + l . (7.32d)\n\nCombining (3.9)-(3.12) and (5.11)-(5.13) with (7.8), (7.27) and ( 7 .31), we get the following expressions for the operators of the (redefined) dynamical variables:\n\nPµ = 1 1 + τ l k µ | k 0 = √ m 2 c 2 +k 2 ( N l + † N l ) l = (s, k) (7.33) Q = q l (-N l + † N l ) (7.34) Sµν = (-1) j-1/2 j 1 + τ m,n {εσ mn,+ µν N nm + σ mn,- µν † N mn )} m=(s,k) n=(s ′ ,k) (7.35) Lµν = x 0 µ Pν -x 0 ν Pµ + (-1) j-1/2 j 1 + τ m,n {εl mn,+ µν N nm + l mn,- µν † N mn )} m=(s,k) n=(s ′ ,k) + i 2(1 + τ ) l -ω • µν (l) + ω • µν (m) ( N l + † N l ) m=l=(s,k) (7.36) Msp µν = (-1) j-1/2 j 1 + τ m,n {ε(σ mn,+ µν + l mn,+ µν ) N nm + (σ mn,- µν + l mn,- µν ) † N mn )} m=(s,k) n=(s ′ ,k) (7.37) Mor µν = i 2(1 + τ ) l -ω • µν (l) + ω • µν (m) ( N l + † N l ) m=l=(s,k) .\n\n(7.38) 24 If we admit a ± l to satisfy the 'anomalous\" bilinear commutation relations (8.27) (see below), i.e. (6.13) with ε for -ε and (±1) 2j for (±1) 2j+1 , then E = N , A also provides a solution of (7.27)-(7.28). However, as it was demonstrated in [13] [14] [15] , the anomalous commutation relations are rejected if one works with the charge symmetric Lagrangians (3.4). 25 The equations (7.32a) and (7.32b) correspond to (7.28a) and (7.28b), respectively, and (7.32c) and (7.32d) correspond to the Hermitian conjugate to (7.28c) and (7.28d), respectively.\n\nHere ω • µν (l) is defined via (6.50), we have set σ mn,± µν := σ ss ′ ,± µν (k) l mn,± µν := l ss ′ ,± µν (k) for m = (s, k) and n = (s ′ , k), (7.39) and (see (7.27) )\n\nN lm := 1 2 E [a + l , a † - m ] = E(a + l • a † - m ) = ( N ml ) † =: N † ml † N lm := 1 2 E [a † + l , a - m ] = E(a † + l • a - m ) = ( † N ml ) † =: † N ml † (7.40)\n\nare respectively the particle and antiparticle transition operators (cf. [26, sec. 1] in a case of parafields). Obviously, we have\n\nN l = N ll † N l = † N ll . (7.41)\n\nThe choice (7.29), evidently, reduces (7.33)-( 7 .36) to (4.24), (4.25), (4.28) and (4.29), respectively.\n\nIn terms of the operators (7.38), the commutation relations (7.28) can equivalently be rewritten as (see also (7.9 ))\n\n[ N lm , a + n ] -= δ mn a + l (7.42a) [ † N lm , a + n ] -= τ δ mn a + l (7.42b) [ N lm , a † + n ] -= τ δ mn a † + l (7.42c) [ † N lm , a † + n ] -= δ mn a † + l . (7.42d)\n\nIf m = l, these relations reduce to (7.32), due to (7.39). We shall end this section with the remark that the conditions for the uniqueness of the dynamical variables and the validity of the Heisenberg relations are quite general and are not enough for fixing some commutation relations regardless of a number of additional assumptions made to reduce these conditions to the system of equations (7.27)-(7.28)."
},
{
"section_type": "OTHER",
"section_title": "State vectors, vacuum and mean values",
"text": "Until now we have looked on the commutation relations only from pure mathematical viewpoint. In this way, making a number of assumptions, we arrived to the system (7.27)-(7.28) of commutation relations. Further specialization of this system is, however, almost impossible without making contact with physics. For the purpose, we have to recall [1, 3, 11, 12] that the physically measurable quantities are the mean (expectation) values of the dynamical variables (in some state) and the transition amplitudes between different states. To make some conclusions from these basic assumption of the quantum theory, we must rigorously said how the states are described as vectors in system's Hilbert space F of states, on which all operators considered act.\n\nFor the purpose, we shall need the notion of the vacuum or, more precisely, the assumption of the existence of unique vacuum state (vector) (known also as the no-particle condition). Before defining rigorously this state, which will be denoted by X 0 , we shall heuristically analyze the properties it should possess.\n\nFirst of all, the vacuum state vector X 0 should represent a state of the field without any particles. From here two conclusions may be drawn: (i) as a field is thought as a collection of particles and a 'missing' particle should have vanishing dynamical variables, those of the vacuum should vanish too (or, more generally, to be finite constants, which can be set equal to zero by rescaling some theory's parameters) and (ii) since the operators a - l and a † - l are interpreted as ones that annihilate a particle characterize by l = (s, k) and charge -q or +q, respectively, and one cannot destroy an 'absent' particle, these operators should transform the vacuum into the zero vector, which may be interpreted as a complete absents of the field. Thus, we can expect that\n\nD( X 0 ) = 0 (8.1a) a - l ( X 0 ) = 0 a † - l ( X 0 ) = 0. (8.1b)\n\nFurther, as the operators a + l and a † + l are interpreted as ones creating a particle characterize by l = (s, k) and charge -q or +q, respectively, state vectors like a + l ( X 0 ) and a † + l ( X 0 ) should correspond to 1-particle states. Of course, a necessary condition for this is\n\nX 0 = 0, (8.2)\n\ndue to which the vacuum can be normalize to unit,\n\nX 0 | X 0 = 1, (8.3)\n\nwhere\n\n•|• : F × F → C is the Hermitian scalar (inner) product of F. More generally, if M(a + l 1 , a † + l 2 , .\n\n. .) is a monomial only in i ∈ N creation operators, the vector\n\nψ l 1 l 2 ... := M(a + l 1 , a † + l 2 , . . .)( X 0 ) (8.4)\n\nmay be expected to describe an i-particle state (with i 1 particles and i 2 antiparticles, i 1 +i 2 = i, where i 1 and i 2 are the number of operators a + l and a † + l , respectively, in M(a + l 1 , a † + l 2 , . . .)). Moreover, as a free field is intuitively thought as a collection of particles and antiparticles, it is natural to suppose that the vectors (8.4) form a basis in the Hilbert space F. But the validity of this assumption depends on the accepted commutation relations; for its proof, when the paracommutation relations are adopted, see the proof of [18, p. 26, theorem I-1] .\n\nAccepting the last assumption and recalling that the transition amplitude between two states is represented via the scalar product of the corresponding to them state vectors, it is clear that for the calculation of such an amplitude is needed an effective procedure for calculation of scalar products of the form\n\nψ l 1 l 2 ... |ϕ m 1 m 2 ... := X 0 |( M(a + l 1 , a † + l 2 , . . .)) † • M ′ (a + m 1 , a † + m 2 , . . .) X 0 , (8.5)\n\nwith M and M ′ being monomials only in the creation operators. Similarly, for computation of the mean value of some dynamical operator D in a certain state, one should be equipped with a method for calculation of scalar products like\n\nψ l 1 l 2 ... | Dϕ m 1 m 2 ... := X 0 |( M(a + l 1 , a † + l 2 , . . .)) † • D • M ′ (a + m 1 , a † + m 2 , . . .) X 0 . (8.6)\n\nSupposing, for the moment, the vacuum to be defined via (8.1), let us analyze (8.1)-(8.6). Besides, the validity of (7.27)-(7.28) will be assumed.\n\nFrom the expressions (7.8) and (3.9)-(3.12) for the dynamical variables, it is clear that the condition (8.1a) can be satisfied if\n\nE(a † ± m • a ∓ n )( X 0 ) = 0, (8.7)\n\nwhich, in view of (7.27), is equivalent to any one of the equations\n\nE(a ± m • a † ∓ n )( X 0 ) = 0 (8.8a) E([a ± m , a † ∓ n ] ε )( X 0 ) = 0. (8.8b)\n\nEquation (8.7) is quite natural as it expresses the vanishment of all modes of the vacuum corresponding to different polarizations, 4-momentum and charge. It will be accepted hereafter. By means of (8.8) and the commutation relations (7.28) in the form (7.42), in particular (7.32), one can explicitly calculate the action of any one of the operators (7.33)-(7.38) on the vectors (8.4) : for the purpose one should simply to commute the operators N lm (or N l = N ll ) with the creation operators in (8.4) according to (7.42) (resp. (7.32 )) until they act on the vacuum and, hence, giving zero, as a result of (8.8) and (7.42) (resp. (7.32)). In particular, we have the equations\n\n(k 0 = m 2 c 2 + k 2 ): Pµ a + l ( X 0 ) = k µ a + l ( X 0 ) Pµ a † + l ( X 0 ) = k µ a † + l ( X 0 ) l = (s, k) (8.9) Q a + l ( X 0 ) = -qa + l ( X 0 ) Q a † + l ( X 0 ) = +qa † + l ( X 0 ) (8.10) Sµν a + l l=(s,k) ( X 0 ) = (-1) j-1/2 j 1 + τ t {εσ lm,+ µν + τ σ ml,- µν } m=(t,k) a + m | m=(t,k) ( X 0 ) Sµν a † + l l=(s,k) ( X 0 ) = (-1) j-1/2 j 1 + τ t {ετ σ lm,+ µν + σ ml,- µν } m=(t,k) a † + m | m=(t,k) ( X 0 ) (8.11) Lµν a + l l=(s,k) ( X 0 ) = (x 0 µ k ν -x 0 ν k µ )(a + l )( X 0 ) -i ω • µν (l)(a + l ) ( X 0 ) + (-1) j-1/2 j 1 + τ t {εl lm,+ µν + τ l ml,- µν } m=(t,k) a + m | m=(t,k) ( X 0 ) Lµν a † + l l=(s,k) ( X 0 ) = (x 0 µ k ν -x 0 ν k µ )(a † + l )( X 0 ) -i ω • µν (l)(a † + l ) ( X 0 ) + (-1) j-1/2 j 1 + τ t {ετ l lm,+ µν + l ml,- µν } m=(t,k) a † + m | m=(t,k) ( X 0 ) (8.12)\n\nMsp µν a + l l=(s,k) ( X 0 ) = (-1) j-1/2 j 1 + τ t {ε(σ lm,+ µν + l lm,+ µν ) + τ (σ ml,- µν + l ml,- µν )} m=(t,k) a + m | m=(t,k) ( X 0 ) Msp µν a † + l l=(s,k) ( X 0 ) = (-1) j-1/2 j 1 + τ t {ετ (σ lm,+ µν + l lm,+ µν ) + (σ ml,- µν + l ml,- µν )} m=(t,k) a † + m | m=(t,k) ( X 0 ) (8.13) Mor µν ã+ l ( X 0 ) = -i ω • µν (l)(ã + l ) ( X 0 ) Mor µν ã † + l ( X 0 ) = -i ω • µν (l)(ã † + l ) ( X 0\n\n). (8.14) These equations and similar, but more complicated, ones with an arbitrary monomial in the creation operators for a + l or a † + l are the base for the particle interpretation of the quantum theory of free fields. For instance, in view of (8.9) and (8.10), the state vectors a + l ( X 0 ) and a † + l ( X 0 ) are interpreted as ones representing particles with 4-momentum ( m 2 c 2 + k 2 , k) and charges -q and +q, respectively; similar multiparticle interpretation can be given to the general vectors (8.4) too.\n\nThe equations (8.9)-(8.12) completely agree with similar ones obtained in [13] [14] [15] on the base of the bilinear commutation relations (6.13) .\n\nBy means of (8.7), the expression (8.6) can be represented as a linear combination of terms like (8.5) . Indeed, as D is a linear combinations of terms like E(a † ± m • a ∓ n ), by means of the relations (7.28) we can commute each of these terms with the creation (resp. annihilation) operators in the monomial\n\nM ′ (a + m 1 , a † + m 2 , . . .) (resp. ( M(a + l 1 , a † + l 2 , . . .)) † = M ′′ (a † - l 1 , a - l 2 , . . .)\n\n) and thus moving them to the right (resp. left) until they act on the vacuum X 0 , giving the zero vector -see (8.7) . In this way the matrix elements of the dynamical variables, in particular their mean values, can be expressed as linear combinations of scalar products of the form (8.5) . Therefore the supposition (8.7) reduces the computation of mean values of dynamical variables to the one of the vacuum mean value of a product (composition) of creation and annihilation operators in which the former operators stand to the right of the latter ones. (Such a product of creation and annihilation operators can be called their 'antinormal' product; cf. the properties (7.30) of the antinormal ordering operator A.)\n\nThe calculation of such mean values, like (8.5) for states ψ, ϕ = X 0 , however, cannot be done (on the base of (7.27)-(7.28), (8.7) and (8.1a)) unless additional assumption are made. For the purpose one needs some kind of commutation relations by means of which the creation (resp. annihilation) operators on the r.h.s. of (8.5) to be moved to the left (resp. right) until they act on the left (resp. right) vacuum vector X 0 ; as a result of this operation, the expressions between the two vacuum vectors in (8.5) should transform into a linear combination of constant terms and such with no contribution in (8.5) . (Examples of the last type of terms are E(a † ± m • a ∓ ) and normally ordered products of creation and annihilation operators.) An alternative procedure may consists in defining axiomatically the values of all or some of the mean values (8.5) or, more stronger, the explicit action of all or some of the operators, entering in the r.h.s. of (8.5), on the vacuum. 26 It is clear, both proposed schemes should be consistent with the relations (7.27)-(7.28), (8.1b) and (8.7)- (8.8) .\n\nLet us summarize the problem before us: the operator E in (7.27)-( 7 .28) has to be fixed and a method for computation of scalar products like (8.5) should be given provided the vacuum vector X 0 satisfies (8.1b), (8.2), (8.3) and (8.7). Two possible ways for exploration of this problem were indicated above.\n\nConsider the operator E. Supposing E(a † ± m • a ∓ n ) to be a function only of a † ± m and a ∓ n , we, in view of (8.1b), can write\n\nE(a † ± m • a ∓ n ) = f ± (a † ± m • a ∓ n ) • b with b = a - n (upper sign) or b = a † -\n\nm (lower sign) and some functions f ± . Applying (7.27), we obtain (do not sum over l)\n\nE(a † + m • a - l ) = f + (a † + m , a - l ) • a - l E(a + m • a † - l ) = f -(a + m , a † - l ) • a † - l E(a - l • a † + m ) = εf + (a † + m , a - l ) • a - l E(a † - l • a + m ) = εf -(a + m , a † - l ) • a † - l .\n\nSince E is a linear operator, the expression E(a † ± m •a ∓ n ) turns to be a linear and homogeneous function of a † ± m and a ∓ n , which immediately implies f ± (A, B) = λ ± A for operators A and B and some constants λ ± ∈ C. For future convenience, we assume λ ± = 1, which can be achieved via a suitable renormalization of the creation and annihilation operators. 27 Thus, the last equations reduce to\n\nE(a † + m • a - l ) = a † + m • a - l E(a + m • a † - l ) = a + m • a † - l (8.15a) E(a - l • a † + m ) = εa † + m • a - l E(a † - l • a + m ) = εa + m • a † - l . (8.15b)\n\nEvidently, these equations convert (7.27), (8.7) and (8.8) into identities. Comparing (8.15) and (4.22), we see that the identification\n\nE = N (8.16)\n\nof the operator E with the normal ordering operator N is quite natural. However, for our purposes, this identification is not necessary as only the equations (8.15) , not the general definition of N , will be employed.\n\nAs a result of (8.15), the commutation relations (7.28) now read:\n\n[a + l , a + m • a † - n ] + δ ln a + m = 0 (8.17a) [a + l , a † + m • a - n ] + τ δ ln a + m = 0 (8.17b) [a - l , a + m • a † - n ] -τ δ lm a - n = 0 (8.17c) [a - l , a † + m • a - n ] -δ lm a - n = 0. (8.17d)\n\n(In a sense, these relations are 'one half' of the (para)commutation relations (6.16): the latter are a sum of the former and the ones obtained from (8.17) via the changes\n\na + m • a † - n → εa † - n • a + m and a † + m • a - n → εa - n • a † + m ;\n\nthe last relations correspond to (7.28) with E = A, A being the antinormal ordering operator -see (7.30) . Said differently, up to the replacement a ± i → √ 2a ± l for all l, the relations (8.17) are identical with (6.16) for ε = 0; as noted in [26, the remarks following theorem 2 in sec. 1], this is a quite exceptional case from the view-point of parastatistics theory.) By means of (6.8) for η = -ε, one can verify that equations (8.17) agree with the bilinear commutation relations (6.13), i.e. (6.13) convert (8.17) into identities.\n\nThe equations (8.15) imply the following explicit forms of the number operators (7.31) and the transition operators (7.40):\n\nN l = a + l • a † - l † N l = a † + l • a - l (8.18)\n\nN lm = a + l • a † - m † N lm = a † + l • a - m . (8.19)\n\nAs a result of them, the equations (7.33)-(7.36) are simply a different form of writing of (\n\nLet us return to the problem of calculation of vacuum mean values of antinormal ordered products like (8.5) . In view of (8.1b) and (8.3), the simplest of them are\n\nX 0 |λ id F ( X 0 ) = λ X 0 | M ± ( X 0 ) = 0 (8.20)\n\nwhere λ ∈ C and M + (resp. M -) is any monomial of degree not less than 1 only in the creation (resp. annihilation) operators; e.g.\n\nM ± = a ± l , a † ± l , a ± l 1 •a ± l 2 , a ± l 1 •a † ± l 2 .\n\nThese equations, with λ = 1, are another form of what is called the stability of the vacuum: if X i denotes an i-particle state, i ∈ N ∪ {0}, then, by virtue of (8.20) and the particle interpretation of (8.4), we have\n\nX i | X 0 = δ i0 , (8.21)\n\ni.e. the only non-forbidden transition into (from) the vacuum is from (into) the vacuum. More generally, if X i ′ ,0 and X 0,j ′′ denote respectively i ′ -particle and j ′′ -antiparticle states, with X 0,0 := X 0 , then\n\nX i ′ ,0 | X 0,j ′′ = δ i ′ 0 δ 0j ′′ , (8.22)\n\ni.e. transitions between two states consisting entirely of particles and antiparticles, respectively, are forbidden unless both states coincide with the vacuum. Since we are dealing with free fields, one can expect that the amplitude of a transitions from an (i ′ -particle + j ′ -antiparticle) state X i ′ ,j ′ into an (i ′′ -particle + j ′′ -antiparticle) state X i ′′ ,j ′′ is\n\nX i ′ ,j ′ | X i ′′ ,j ′′ = δ i ′ i ′′ δ j ′ j ′′ , (8.23)\n\nbut, however, the proof of this hypothesis requires new assumptions (vide infra).\n\nLet us try to employ (8.17) for calculation of expressions like (8.5) . Acting with (8.17) and their Hermitian conjugate on the vacuum, in view of (8.1b), we get\n\na + m • (-a † - n • a + l + δ ln id F )( X 0 ) = 0 a † + n • (a - m • a † + l -δ lm id F )( X 0 ) = 0 a † + m • (-a - n • a + l + τ δ ln id F )( X 0 ) = 0 a + n • (a † - m • a † + l -τ δ lm id F )( X 0 ) = 0. (8.24)\n\nThese equalities, as well as (8.17), cannot help directly to compute vacuum mean values of antinormally ordered products of creation and annihilation operators. But the equations (8.24) suggest the restrictions 28\n\na † - l • a + m ( X 0 ) = δ lm X 0 a - l • a † + m ( X 0 ) = δ lm X 0 a - l • a + m ( X 0 ) = τ δ lm X 0 a † - l • a † + m ( X 0 ) = τ δ lm X 0 (8.25)\n\nto be added to the definition of the vacuum. These conditions convert (8.24) into identities and, in this sense agree with (8.17) and, consequently, with the bilinear commutation relations (6.13) . Recall [16, 18] , the relations (8.25) are similar to ones accepted in the parafield theory and coincide with that for parastatistics of order p = 1; however, here we do not suppose the validity of the paracommutation relations (6.20) (or (6.16)). Equipped with (8.25) , one is able to calculate the r.h.s. of (8.5) for any monomial M (resp. M ′ ) and monomials M ′ (resp. M) of degree 1, deg M ′ = 1 (resp. deg M = 1). 29 Indeed, (8.25), (8.1b ) and (8.3) entail:\n\nX 0 |a † - l • a + m ( X 0 ) = X 0 |a - l • a † + m ( X 0 ) = δ lm X 0 |a - l • a + m ( X 0 ) = X 0 |a † - l • a † + m ( X 0 ) = τ δ lm X 0 |( M(a + l 1 , a † + l 2 , • • • )) † • a + m ( X 0 ) = X 0 |( M(a + l 1 , a † + l 2 , • • • )) † • a † + m ( X 0 ) = 0 deg M ≥ 2 X 0 |a - l • M(a + m 1 , a † + m 2 , • • • )( X 0 ) = X 0 |a † - l • M(a + m 1 , a † + m 2 , • • • )( X 0 ) = 0 deg M ≥ 2. (8.26)\n\nHereof the equation (8.23) for i ′ + j ′ = 1 (resp. i ′′ + j ′′ = 1) and arbitrary i ′′ and j ′′ (resp. i ′ and j ′ ) follows. However, it is not difficult to be realized, the calculation of (8.5) in cases more general than (8.20) and (8.26) is not possible on the base of the assumptions made until now. 30 At this point, one is free so set in an arbitrary way the r.h.s. of (8.5) in the mentioned general case or to add to (8.17) (and, possibly, (8.25)) other (commutation) relations by means of which the r.h.s. of (8.5) to be calculated explicitly; other approaches, e.g. some mixture of the just pointed ones, for finding the explicit form of (8.5) are evidently also possible. Since expressions like (8.5) are directly connected with observable experimental results, the only criterion for solving the problem for calculating the r.h.s. of (8.5) in the general case can be the agreement with the existing experimental data. As it is known [1, 3, 11, 12] , at present (almost?) all of them are satisfactory described within the framework of the bilinear commutation relations (6.13) . This means that, from physical point of view, the theory should be considered as realistic one if the r.h.s. of (8.5) is the same as if (6.13) are valid or is reducible to it for some particular realization of an accepted method of calculation, e.g. if one accepts some commutation relations, like the paracommutation ones, which are a generalization of (6.13) and reduce to them as a special case (see, e.g., (6.20) ). It should be noted, the conditions (8.1b)-(8.3) and (8.25) are enough for calculating (8.5) if (6.16), or its versions (6.17) or (6.20) , are accepted (cf. [16] ). The causes for that difference are replacements like [a + m , a † - n ] → 2a + m •a † - n , when one passes from (6.16) to (8.17); the existence of terms like a † - n • a + m a + l in (6.16) are responsible for the possibility to calculate (8.5).\n\n28 Since the operators a ± l and a † ± l are, generally, degenerate (with no inverse ones), we cannot say that (8.24) implies (8.25) .\n\n29 For deg M ′ = 0 (resp. deg M ′ = 0) -see (8.20) . 30 It should be noted, the conditions (8.1b)-(8.3) and (8.25) are enough for calculating (8.5) if the relations (6.16) , or their version (6.20) , are accepted (cf. [16] ). The cause for that difference is in replacements like [a + m , a †- n ] → 2a + m • a † - n , when one passes from (6.16) to (8.17); the existence of terms like a † - n • a + m • a + l in (6. 16 ) is responsible for the possibility to calculate (8.5), in case (6.16) hold.\n\nIf evidences appear for events for which (8.5) takes other values, one should look, e.g., for other commutation relations leading to desired mean values. As an example of the last type can be pointed the following anomalous bilinear commutation relations (cf. (6.13))\n\n[a ± l , a ± m ] ε = 0 [a † ± l , a † ± m ] ε = 0 [a ∓ l , a ± m ] ε = (±1) 2j τ δ lm id F [a † ∓ l , a † ± m ] ε = (±1) 2j τ δ lm id F [a ± l , a † ± m ] ε = 0 [a † ± l , a ± m ] ε = 0 [a ∓ l , a † ± m ] ε = (±1) 2j δ lm id F [a † ∓ l , a ± m ] ε = (±1) 2j δ lm id F , (8.27)\n\nwhich should be imposed after expressions like E(a † ± m • a ∓ n ) are explicitly calculated. These relations convert (8.17) and (8.25) into identities and by their means the r.h.s. of (8.5) can be calculated explicitly, but, as it is well known [1, 3, 11, 12, 27] they lead to deep contradictions in the theory, due to which should be rejected. 31 At present, it seems, the bilinear commutation relations (6.13) are the only known commutation relations which satisfy all of the mentioned conditions and simultaneously provide an evident procedure for effective calculation of all expressions of the form (8.5). (Besides, for them and for the paracommutation relations the vectors (8.4) form a base, the Fock base, for the system's Hilbert space of states [18] .) In this connection, we want to mention that the paracommutation relations (6.16) (or their conventional version (6.20)), if imposed as additional restrictions to the theory together with (8.17), reduce in this particular case to (6.13) as the conditions (8.25) show that we are dealing with a parafield of order p = 1, i.e. with an ordinary field [17, 18] . 32 Ending this section, let us return to the definition of the vacuum X 0 . It, generally, depends on the adopted commutation relations. For instance, in a case of the bilinear commutation relations (6.13) it consists of the equations (8.1a)-(8.3), while in a case of the paracommutation relations (6.16) (or other ones generalizing (6.13)) it includes (8.1a)-(8.3) and (8.25)."
},
{
"section_type": "OTHER",
"section_title": "Commutation relations for several coexisting different free fields",
"text": "Until now we have considered commutation relations for a single free field, which can be scalar, or spinor or vector one. The present section is devoted to similar treatment of a system consisting of several, not less than two, different free fields. In our context, the fields may differ by their masses and/or charges and/or spins; e.g., the system may consist of charged scalar field, neutral scalar field, massless spinor field, massive spinor field and massless neural vector field. It is a priori evident, the commutation relations regarding only one field of the system should be as discussed in the previous sections. The problem is to be derived/postulated commutation relations concerning different fields. It will be shown, the developed Lagrangian formalism provides a natural base for such an investigation and makes superfluous some of the assumptions made, for example, in [17, p. B 1159, left column] or in [18, sec. 12.1] , where systems of different parafields are explored.\n\nTo begin with, let us introduce suitable notation. With the indices α, β, γ = 1, 2, . . . , N will be distinguished the different fields of the system, with N ∈ N, N ≥ 2, being their number, and the corresponding to them quantities. Let q α and j α be respectively the charge and spin of the α-th field. Similarly to (3.7), we define\n\nj α := 0 for scalar α-th field 1 2\n\nfor spinor α-th field 1 for vector α-th field τ α := 1 for q α = 0 (neutral (Hermitian) field) 0 for q α = 0 (charged (non-Hermitian) field)\n\nε α := (-1) 2j α = +1 for integer j α (bose fields) -1 for half-integer j α (fermi fields) .\n\n(9.1) Suppose L α is the Lagrangian of the α-field. For definiteness, we assume L α for all α to be given by one and the same set of equations, viz. (3.1), or (3.3) or (3.4) . To save some space, below the case (3.4), corresponding to charge symmetric Lagrangians, will be considered in more details; the reader can explore other cases as exercises.\n\nSince the Lagrangian of our system of free fields is\n\nL := α L α , (9.2)\n\nthe dynamical variables are\n\nD = α D α (9.3)\n\nand the corresponding system of Euler-Lagrange equations consists of the independent equations for each of the fields of the system (see (3.6) with L α for L). This allows an introduction of independent creation and annihilation operators for each field. The ones for the α-th field will be denoted by a ± α,s α (k) and a † ± α,s α (k); notice, the values of the polarization variables generally depend on the field considered and, therefore, they also are labeled with index α for the α-th field. For brevity, we shall use the collective indices l α , m α and n α , with l α := (α, s α , k) etc., in terms of which the last operators are a ± l α and a † ± l α , respectively. The particular expressions for the dynamical operators D α are given via (3.9)-(3.12) in which the following changes should be made:\n\nτ → τ α j → j α ε → ε α s → s α s ′ → s ′ α σ ss ′ ,± µν (k) → σ s α s ′ α ,± µν (k) l ss ′ ,± µν (k) → l s α s ′ α ,± µν (k). (9.4)\n\nThe content of sections 4 and 5 remains valid mutatis mutandis, viz. provided the just pointed changes (9.4) are made and the (integral) dynamical variables are understood in conformity with (9.3)."
},
{
"section_type": "OTHER",
"section_title": "Commutation relations connected with the momentum operator. Problems and their possible solutions",
"text": "In sections 6-8, however, substantial changes occur; for instance, when one passes from (6.12) or (6.15) to (6.16) . We shall consider them briefly in a case when one starts from the charge symmetric Lagrangians (3.4) . The basic relations (6.12), which arise from the Heisenberg relation (5.1) concerning the momentum operator, now read (here and below, do not sum over α, and/or β and/or γ if the opposite is not indicated explicitly!)\n\na ± l α , [a † + m β , a - m β ] ε β + [a + m β , a † - m β ] ε β -± (1 + τ )δ l α m β a ± l α = 0 (9.5a) a † ± l α , [a † + m β , a - m β ] ε β + [a + m β , a † - m β ] ε β -± (1 + τ )δ l α m β a † ± l α = 0. (9.5b)\n\nIt is trivial to be seen, the following generalizations of respectively (6.14) and (6.15)\n\na ± l α , [a + m β , a † - m β ] ε β -± (1 + τ β )δ l α m β a ± l α = 0 (9.6a) a ± l α , [a † + m β , a - m β ] ε β -± (1 + τ β )δ l α m β a ± l α = 0 (9.6b) a † ± l α , [a + m β , a † - m β ] ε β -± (1 + τ β )δ l α m β a † ± l α = 0 (9.6c) a † ± l α , [a † + m β , a - m β ] ε β -± (1 + τ β )δ l α m β a † ± l α = 0 (9.6d) a + l α , [a + m β , a † - m β ] ε β -+ 2δ l α m β a + l α = 0 (9.7a) a + l α , [a † + m β , a - m β ] ε β -+ 2τ β δ l α m β a + l α = 0 (9.7b) a - l α , [a + m β , a † - m β ] ε β --2τ β δ l α m β a - l α = 0 (9.7c) a - l α , [a † + m β , a - m β ] ε β --2δ l α m β a - l α = 0 (9.7d)\n\nprovide a solution of (9.5) in a sense that they convert it into identity. As it was said in Sect. 6, the equations (9.6) (resp. (9.7)) for a single field, i.e. for β = α, agree (resp. disagree) with the bilinear commutation relations (6.13).\n\nThe only problem arises when one tries to generalize, e.g., the relations (9.7) in a way similar to the transition from (6.15) to (6.16 ). Its essence is in the generalization of expressions like [a † ± m β , a ∓ m β ] ε β and τ β δ l α m β a ± l α . When passing from (6.15) to (6.16), the indices l and m are changed so that the obtained equations to be consistent with (6.13); of course, the numbers ε and τ are preserved because this change does not concern the field regarded. But the situation with (9.7) is different in two directions:\n\n(i) If we change the pair (m\n\nβ , m β ) in [a † ± m β , a ∓ m β ] ε β with (m β , n γ ),\n\nthen with what the number ε β should be replace? With ε β , or ε γ or with something else? Similarly, if the mentioned changed is performed, with what the multiplier τ β in τ β δ l α m β a ± l α should be replaced? The problem is that the numbers ε β and τ β are related to terms like a † ± m β • a ∓ m β and a ± m β • a † ∓ m β , in the momentum operator, as a whole and we cannot say whether the index β in ε β and τ β originates from the first of second index m β in these expressions.\n\n(ii) When writing (m β , n γ ) for (m β , m β ) (see (i) above), then shall we replace δ l α m β a ± l α with δ l α m β a ± n γ , or δ l α n γ a ± m β , or δ m β n γ a ± l α ? For a single field, γ = β = α, this problem is solved by requiring an agreement of the resulting generalization (of (6.16) in the particular case) with the bilinear commutation relations (6.13). So, how shall (6.13) be generalized for several, not less than two, different fields? Obviously, here we meet an obstacle similar to the one described in (i) above, with the only change that -ε β should stand for ε β .\n\nLet b l α and c l α denote some creation or annihilation operator of the α-field. Consider the problem for generalizing the (anti)commutator [b l α , c l α ] ±ε α . This means that we are looking for a replacement\n\n[b l α , c l α ] ±ε α → f ± (b l α , c m β ; α, β), (9.8)\n\nwhere the functions f ± are such that\n\nf ± (b l α , c m β ; α, β) β=α = [b l α , c l α ] ±ε α . (9.9)\n\nUnfortunately, the condition (9.9) is the only restriction on f ± that the theory of free fields can provide. Thus the functions f ± , subjected to equation (9.9), become new free parameters of the quantum theory of different free fields and it is a matter of convention how to choose/fix them.\n\nIt is generally accepted [18, appendix F], the functions f ± to have forms 'maximum' similar to the (anti)commutators they generalize. More precisely, the functions\n\nf ± (b l α , c m β ; α, β) = [b l α , c m β ] ±ε αβ (9.10)\n\nwhere ε αβ ∈ C are such that\n\nε αα = ε α , (9.11)\n\nare usually considered as the only candidates for f ± . Notice, in (9.10), ε αβ are functions in α and β, not in l α and/or m β . Besides, if we assume ε αβ to be function only in ε α and ε β , then the general form of ε αβ is\n\nε αβ = u αβ ε α + (1 -u αβ )ε β + v αβ (1 -ε α ε β ) u αβ , v αβ ∈ C, (9.12)\n\ndue to (9.1) and (9.11) . (In view of (6.13), the value ε αβ = +1 (resp. ε αβ = -1) corresponds to quantization via commutators (resp. anticommutators) of the corresponding fields.) Call attention now on the numbers τ α which originate and are associated with each term [b l α , c m α ] ±ε α . With every change (9.8) one can associate a replacement\n\nτ α → g(b l α , c m β ; α, β), (9.13)\n\nwhere the function g is such that\n\ng(b l α , c m β ; α, β) β=α = τ α . (9.14)\n\nOf course, the last condition does not define g uniquely and, consequently, the function g, satisfying (9.14), enters in the theory as a new free parameter. Suppose, as a working hypothesis similar to (9.10)-(9.11), that g is of the form\n\ng(b l α , c m β ; α, β) = τ αβ , (9.15)\n\nwhere τ αβ are complex numbers that may depend only on α and β and are such that\n\nτ αα = τ α . (9.16)\n\nBesides, if we suppose τ αβ to be functions only in τ α and τ β , then\n\nτ αβ = x αβ τ α + y αβ τ β + (1 -x αβ -y αβ )τ α τ β x αβ , y αβ ∈ C, (9.17)\n\nas a result of (9.1) and (9.16).\n\nLet us summarize the above discussion. If we suppose a preservation of the algebraic structure of the bilinear commutation relations (6.13) for a system of different free fields, then the replacements\n\n[b l α , c l α ] ±ε α → [b l α , c m β ] ±ε αβ ε αα = ε α (9.18a) τ α → τ αα τ αα = τ α (9.18b)\n\nshould be made; accordingly, the relations (6.13) transform into:\n\n[a ± l α , a ± m β ] -ε αβ = 0 [a † ± l α , a † ± m β ] -ε αβ = 0 [a ∓ l α , a ± m β ] -ε αβ = τ αβ δ l α m β id F × 1 -ε αβ [a † ∓ l α , a † ± m β ] -ε αβ = τ αβ δ l α m β id F × 1 -ε αβ [a ± l α , a † ± m β ] -ε αβ = 0 [a † ± l α , a ± m β ] -ε αβ = 0 [a ∓ l α , a † ± m β ] -ε αβ = δ l α m β id F × 1 -ε αβ [a † ∓ l α , a ± m β ] -ε αβ = δ l α m β id F × 1 -ε αβ , (9.19)\n\nwhere 1 (resp. -ε αβ ) in 1 -ε αβ corresponds to the choice of the upper (resp. lower) signs. If we suppose additionally ε αβ (resp. τ αβ ) to be a function only in ε α and ε β (resp. in τ α and τ β ), then these numbers are defined up to two sets of complex parameters:\n\nε αβ = u αβ ε α + (1 -u αβ )ε β + v αβ (1 -ε α ε β ) u αβ , v αβ ∈ C (9.20a) τ αβ = x αβ τ α + y αβ τ β + (1 -x αβ -y αβ )τ α τ β x αβ , y αβ ∈ C. (9.20b)\n\nA reasonable further specialization of ε αβ and τ αβ may be the assumption their ranges to coincide with those of ε α and τ α , respectively. As a result of (9.1), this supposition is equivalent to\n\nv αβ = -u αβ , -u αβ + 1, u αβ -1, u αβ u αβ ∈ C (9.21a) (x αβ , y αβ ) = (0, 0), (0, 1), (1, 0), (1, 1). (9.21b)\n\nOther admissible restriction on (9.20) may be the requirement ε αβ and τ αβ to be symmetric, viz.\n\nε αβ (ε α , ε β ) = ε βα (ε α , ε β ) = ε αβ (ε β , ε α ) (9.22a) τ αβ (τ α , τ β ) = τ βα (τ α , τ β ) = τ αβ (τ β , τ α ), (9.22b)\n\nwhich means that the α-th and β-th fields are treated on equal footing and there is no a priori way to number some of them as the 'first' or 'second' one. 33 In view of (9.20), the conditions (9.22) are equivalent to\n\nu αβ = 1 2 v αβ ∈ C (9.23a)\n\ny αβ = x αβ . (9.23b)\n\nIf both of the restrictions (9.21) and (9.23) are imposed on (9.20), then the arbitrariness of the parameters in (9.20) is reduced to:\n\n(u αβ , u αβ ) = 1 2 , - 1 2 , 1 2 , 1 2 (9.24a)\n\n(x αβ , y αβ ) = (0, 0), (1, 1) (9.24b) and, for any fixed pair (α, β), we are left with the following candidates for respectively ε αβ and τ αβ :\n\nε αβ + := 1 2 (+1 + ε α + ε β -ε α ε β ) (9.25a) ε αβ -:= 1 2 (-1 + ε α + ε β + ε α ε β ) (9.25b) τ αβ 0 := τ α + τ β (9.25c) τ αβ 1 := τ α + τ β -τ α τ β . (9.25d)\n\nWhen free fields are considered, as in our case, no further arguments from mathematical or physical nature can help for choosing a particular combination (ε αβ , τ αβ ) from the four possible ones according to (9.25) for a fixed pair (α, β). To end the above considerations of ε αβ and τ αβ , we have to say that the choice\n\n(ε αβ , τ αβ ) = (ε αβ + , τ αβ 0 ) = 1 2 (+1 + ε α + ε β -ε α ε β ), τ α + τ β (9.26)\n\n33 However, nothing can prevent us to make other choices, compatible with (9.18), in the theory of free fields; for instance, one may set\n\nε αβ = ε α ε β ε βα and τ αβ = 1 2 (τ α + τ β )τ βα .\n\nis known as the normal case [18, appendix F] ; in it the relative behavior of bose (resp. fermi) fields is as in the case of a single field, i.e. they are quantized via commutators (resp. anticommutators) as (ε αβ , τ αβ ) = (+1, 0) (resp. (ε αβ , τ αβ ) = (-1, 0)), and the one of bose and fermi field is as in the case of a single fermi field, viz. the quantization is via commutators as (ε αβ , τ αβ ) = (+1, 0). All combinations between ε αβ ± and τ αβ 0,1 different from (9.26) are referred as anomalous cases. Above we supposed the pair (α, β) to be fixed. If α and β are arbitrary, the only essential change this implies is in (9.25) , where the choice of the subscripts +, -, 0 and 1 may depend on α and β. In this general situation, the normal case is defined as the one when (9.26) holds for all α and β. All other combinations are referred as anomalous cases; such are, for instance, the ones when some fermi and bose operators satisfy anticommutation relations, e.g. (9.19) with ε αβ = -1 for ε α + ε β = 0, or some fermi fields are subjected to commutation relations, like (9.19) with ε αβ = +1 for ε α = ε β = -1. For some details on this topic, see, for instance, [18, appendix F], [7, chapter 20] and [27, sect 4-4] . Fields/operators for which ε αβ = +1 (resp. ε αβ = -1), with β = α, are referred as relative parabose (resp. parafermi ) in the parafield theory [17, 18] . One can transfer this terminology in the general case and call the fields/operators for which ε αβ = +1 (resp. ε αβ = -1), with β = α, relative bose (resp. fermi ) fields/operators.\n\nFurther the relations (9.19) will be referred as the multifield bilinear commutation relations and it will be assumed that they represent the generalization of the bilinear commutation relations (6.13) when we are dealing with several, not less than two, different quantum fields. The particular values of ε αβ and τ αβ in them are insignificant in the following; if one likes, one can fix them as in the normal case (9.26) . Moreover, even the definition (9.19) of τ αβ is completely inessential at all, as τ αβ always appears in combinations like τ αβ δ l α m β (see (9.19) or similar relations, like (9.27), below), which are non-vanishing if β = α, but then τ αα = τ α ; so one can freely write τ α for τ αβ in all such cases.\n\nEquipped with (9.19) and (9.18), we can generalize (9.7) in different ways. For example, the straightforward generalization of (6.16) is:\n\na + l α , [a + m β , a † - n γ ] ε βγ -+ 2δ l α n γ a + m β = 0 (9.27a) a + l α , [a † + m β , a - n γ ] ε βγ -+ 2τ αγ δ l α n γ a + m β = 0 (9.27b) a - l α , [a + m β , a † - n γ ] ε βγ --2τ αβ δ l α m β a - n γ = 0 (9.27c) a - l α , [a † + m β , a - n γ ] ε βγ --2δ l α m β a - n γ = 0. ( 9\n\n.27d) However, generally, the relations (9.19) do not convert (9.27) into identities. The reason is that an equality/identity like (cf. (6.8)) [\n\nb l α , c m β • d n γ ] = [b l α , c m β ] -ε αβ • d n γ + λ αβγ c m β • [b l α , d n γ ] -ε αγ , (9.28)\n\nwhere b l α , c m β and d n γ are some creation/annihilation operators and λ αβγ ∈ C, can be valid only for\n\nλ αβγ = ε αβ ε αγ = 1/ε αβ (ε αβ = 0), ( 9\n\n.29) which, in particular, is fulfilled if γ = β and ε αβ = ±1. So, the agreement between (9.19) and (9.27) depends on the concrete choice of the numbers ε αβ . There exist cases when even the normal case (9.26) cannot ensure (9.19) to convert (9.27) into identities; e.g. when the α-th field and β-th fields are fermion ones and the γ-th field is a boson one. Moreover, it can be proved that (9.19) and (9.27) are compatible in the general case if unacceptable equalities like a ± l • a ± m = 0 hold. One may call (9.27) the multifield paracommutation relations as from them a corresponding generalization of (6.18) and/or (6.20) can be derived. For completeness, we shall record the multifield version of (6.20):\n\n[b l α , [b † m β , b n γ ] ε βγ ] = 2δ l α m β b n γ [b l α , [b m β , b n γ ] ε βγ ] = 0 (9.30a) [c l α , [c † m β , c n γ ] ε βγ ] = 2δ l α m β c n γ [c l α , [c m β , c n γ ] ε βγ ] = 0 (9.30b) [b † l α , [c † m β , c n γ ] ε βγ ] = -2τ αγ δ l α n γ b † m β [c † l α , [b † m β , b n γ ] ε βγ ] = -2τ αγ δ l α n γ c † m β . (9.30c)\n\nFor details regarding these multifield paracommutation relations, the reader is referred to [17, 18] , where the case τ α = τ β = τ αβ = 0 is considered. We leave to the reader as exercise to write down the multifield versions of the commutation relations (6.22) or (6.23), which provide examples of generalizations of (9.7) and hence of (9.19) and (9.27)."
},
{
"section_type": "OTHER",
"section_title": "Commutation relations connected with the charge and angular momentum operators",
"text": "In a case of several, not less than two, different fields, the basic trilinear commutation relations (6.33), which ensure the validity of the Heisenberg relation (5.2) concerning the charge operator, read:\n\na ± l α , [a † + m β , a - m β ] ε β -[a + m β , a † - m β ] ε β --2δ l α m β a ± l α = 0 (9.31a) a † ± l α , [a † + m β , a - m β ] ε β -[a + m β , a † - m β ] ε β -+ 2δ l α m β a † ± l α = 0. (9.31b)\n\nOf course, these relations hold only for those fields which have non-vanishing charges, i.e. in (9.31) is supposed (see (9.1))\n\nτ α = 0 τ β = 0 ( ⇐⇒ q α q β = 0). (9.32)\n\nThe problem for generalizing (9.31) for these fields is similar to the one for (9.7) in the case of non-vanishing charges, τ β = 0. Without repeating the discussion of Subsect. 9.1, we shall adopt the rule (9.18) for generalizing (anti)commutation relations between creation/annihilation operators of a single field. By its means one can obtain different generalizations of (9.31). For instance, the commutation relations.\n\na + l α , [a † + m β , a - n γ ] ε βγ -[a + m β , a † - n γ ] ε βγ --2δ l α n γ a + m β = 0 (9.33a) a - l α , [a † + m β , a - n γ ] ε βγ -[a + m β , a † - n γ ] ε βγ --2δ l α m β a - n γ = 0 (9.33b)\n\nand their Hermitian conjugate contain (9.31) and (6.35) as evident special cases and agree with (9.19) if γ = β and ε αβ ε βγ = +1. Besides, the multifield paracommutation relations (9.27) for charged fields, τ α = τ β = τ γ = 0, convert (9.33) into identities and, in this sense, (9.33) agree with (contain as special case) (9.27) for charged fields. As an example of commutation relations that do not agree with (9.27) for charged fields and, consequently, with (9.33), we shall point the following ones:\n\na ± l α , [a + m β , a † - n γ ] ε βγ -+ δ l α n γ a ± m β = 0 (9.34a) a ± l α , [a † + m β , a - n γ ] ε βγ ---δ l α n γ a ± m β = 0, (9.34b)\n\nwhich are a multifield generalization of (6.34).\n\nThe consideration of commutation relations originating from the 'orbital' Heisenberg equation (5.4) is analogous to the one of the same relations regarding the charge operator. The multifield version of (6.49) is:\n\n(-ω • µν (m β ) + ω • µν (n γ ))([ã ± l α , [ã † + m β , ã- n γ ] ε βγ + [ã + m β , ã † - n γ ] ε βγ ] ) n γ =m β = 4(1 + τ αβ )δ l α m β ω • µν (l α )(ã ± l α ) (9.35a) (-ω • µν (m β ) + ω • µν (n γ ))([ã † ± l α , [ã † + m β , ã- n γ ] ε βγ + [ã + m β , ã † - n γ ] ε βγ ] ) n γ =m β = 4(1 + τ αβ )δ l α m β ω • µν (l α )(ã † ± l α ) (9.35b) where ω • µν (l α ) := ω µν (k) = k µ ∂ ∂k ν -k ν ∂ ∂k µ if l α = (α, s α , k). (9.36)\n\nApplying (6.51), with m β for m and n γ for n, one can check that the multifield paracommutation relations (9.27) convert (9.35) into identities and hence provide a solution of (9.35) and ensure the validity of (5.4), when system of different free fields is considered. An example of a solution of (9.35) which does not agree with (9.27) is provided by the following multifield generalization of (6.52):\n\na + l α , [a + m β , a † - n γ ] ε βγ -= a + l α , [a † + m β , a - n γ ] ε βγ -= -(1 + τ αγ )δ l α n γ a + m β (9.37a) a - l α , [a + m β , a † - n γ ] ε βγ -= a - l α , [a † + m β , a - n γ ] ε βγ -= +(1 + τ αβ )δ l α m β a + n γ , (9.37b)\n\nwhich provides a solution of (9.5). Notice, the evident multifield version of (6.53) agrees with (9.5), but disagrees with (9.35) when the lower signs are used. At last, the multifield exploration of the 'spin' Heisenberg relations (5.5) is a mutatis mutandis (see (9.35 )) version of the corresponding considerations in the second part of Subsect. 6.3. The main result here is that the multifield bilinear commutation relations (9.19), as well as their para counterparts (9.27), ensure the validity of (5.5)."
},
{
"section_type": "OTHER",
"section_title": "Commutation relations between the dynamical variables",
"text": "The aim of this subsection is to be discussed/proved the commutation relations (5.15)-(5.24) for a system of at least two different quantum fields from the view-point of the commutation relations considered in subsections 9.1 and 9.2.\n\nTo begin with, we rewrite the Heisenberg relations (5.1), (5.2) and (5.4) in terms of creation and annihilation operators for a multifield system [1, 11] : where l α = (α, s α , k), ω • (l α ) is defined by (9.36) and k 0 = m 2 c 2 + k 2 is set in (9.38) and (9.40) (after the differentiations are performed in the last case). The corresponding version of (5.5) is more complicated and depends on the particular field considered (do not sum over s α !):\n\nf s α [a ± α,s α (k), M sp µν ] = i g α t α\n\n± σ s α t α ,+ µν (k)a + α,t α (k) + ± σ s α t α ,- µν (k)a - α,t α (k)\n\nf s α [a † ± α,s α (k), M sp µν ] = i h α t α ± σ s α t α ,- µν (k)a † + α,t α (k) + ± σ s α t α ,+ µν (k)a † - α,t α (k) , (9.41)\n\nwhere f s α = -1, 0, +1 (depending on the particular field), g α := -h α := 1 j α +δ j α 0 (-1) j α +1 and ± σ s α t α ,+ µν (k) and ± σ s α t α ,- µν (k) are some functions which strongly depend on the particular field considered, with ± σ s α t α ,± µν (k) being related to the spin (polarization) functions σ s α t α ,± µν (k) (see (3.14) and (3.11) ). 34 As a result of (5.6), (9.40) and (9.41), one can easily write the Heisenberg relations (5.3) in a form similar to (9.38)-(9.41).\n\nl α ← -- -→ ω • µν (l α ) • c ∓ m β , P µ ] = ±2(k µ η νλ -k ν η µλ )b ± l α • c ∓ m β , (9.46b)\n\nwhere b ± l α , c ± l α = a ± l α , a † ± l α and ← ---→ ω • µν (l α ) is defined via (9.36) and (3.13), the verification of (9.45) reduces to almost trivial algebraic calculations. Further, we assert that any system of commutation relations considered in Subsect. 9.1 entails (9.45): as these relations always imply (9.5) (or similar multifield versions of (6.10) and (6.11) in the case of the Lagrangians (3.1) or (3.3), respectively) and, on its turn, (9.5) implies (5.1), the required result follows from the last assertion and the remark that (5.1) and (9.38) are equivalent. As an additional verification of the validity of (9.45), the reader can prove them by invoking the identity (6.8) and any system of commutation relations mentioned in Subsect. 9.1, in particular (9.19) and (9.27) .\n\nThe commutation relations concerning the charge operator read:\n\n[ P µ , Q] = 0 [ Q, Q] = 0 [ L µν , Q] = [ S µν , Q] = 0 [ M or µν , Q] = [ M sp µν , Q] = [ M µν , Q] = 0.\n\n(9.47)\n\nThese equations are trivial corollaries from (3.9)-(3.12) and (5.11)-(5.13) and the observation that (9.39) implies\n\n[a † ± l α • a ∓ m β , Q] = [a ± l α • a † ∓ m β , Q] = 0 if q α = q β , (9.48)\n\ndue to (6.8) for η = -1. Since any one of the systems of commutation relations mentioned in Subsect. 9.2 entails (9.31) (or systems of similar multifield versions of (6.31) and (6.32), if the Lagrangians (3.1) or (3.3) are employed), which is equivalent to (9.39), the equations (9.47) hold if some of these systems is valid. Alternatively, one can prove via a direct calculation that the commutation relations arising from the charge operator entail the validity of (9.47);\n\nwhere v s α ,± i (k) are linearly independent functions normalize via the condition\n\nX i `vs α ,± i (k) ´ * v t α ,± i (k) = δ s α t α f s α , (9.43)\n\nwith f s α = 1 for j α = 0, 1 2 and f s α = 0, -1 for (j α , s α ) = (1, 3) or (j α , s α ) = (1, 1), (1, 2), respectively, then\n\n+ σ s α t α ,± µν (k) := 1 gα X i,i ′ `vs α ,+ i (k) ´ * I i ′ iµν v t α ,± i ′ (k) -σ s α t α ,± µν (k) := 1 gα X i,i ′ `vs α ,- i (k) ´ * I i ′ iµν v t α ,± i ′ (k), (9.44)\n\nwith I i ′ iµν given via (5.25) . Besides, σ s α t α ,± µν (k) = ± σ s α t α ,± µν (k) with an exception that σ s α t α ,± µν (k) = 0 for j α = 1 2 and (µ, ν) = (a, 0), (0, a) with a = 1, 2, 3.\n\nfor the purpose the identity (6.8) and the explicit expressions for the dynamical variables via the creation and annihilation operators should be applied. At last, consider the commutation relations involving the different angular momentum operators: with b l α , c l α = a + l α , a - l α , a † + l α , a † - l α , and similar, but more complicated, ones involving the other angular momentum operators. It, generally, depends on the particular field considered and will be omitted.\n\nAs it was said in Subsect. 6.3, the Heisenberg relations concerning the angular momentum operator(s) do not give rise to some (algebraic) commutation relations for the creation and annihilation operators. For this reason, the only problem is which of the commutation relations discussed in subsections 9.1 and 9.2 imply the validity of the equations (9.49) (or part of them). The general answer of this problem is not known but, however, a direct calculation by means of (9.7), if it holds, and (6.8) shows the validity of (9.49). Since (9.19) and (9.27) imply (9.7), this means that the multifield bilinear and para commutation relations are sufficient for the fulfillment of (9.49).\n\nTo conclude, let us draw the major moral of the above material: the multifield bilinear commutation relations (9.19) and the multifield paracommutation relations (9.27) ensure the validity of all 'standard' commutation relations (9.45), (9.47) and (9.49) between the operators of the dynamical variables characterizing free scalar, spinor and vector fields."
},
{
"section_type": "OTHER",
"section_title": "Commutation relations under the uniqueness conditions",
"text": "As it was said at the end of the introduction to this section, the replacements (9.4) ensure the validity of the material of Sect. 4 in the multifield case. Correspondingly, the considerations in Sect. 7 remain valid in this case provided the changes The multifield version of (7.27)-(7.28) is:\n\nl → l α m → m β n → n γ τ δ lm → τ αβ δ l α m β = τ α δ l α m β [b m , b m ] ε → [b m β , b m β ] ε β [b m , b n ] ε → [b m β , b n γ ] ε βγ ,\n\nE(a † ± m β • a ∓ n γ ) = ε βγ E(a ∓ n γ • a † ± m β ) = 1 2 E([a † ± m β ,\n\na ∓ n γ ] ε βγ ) (9.52) a + l α , E([a + m β , a † - n γ ] ε βγ ) -+ 2δ l α n γ a + m β = 0 (9.53a) a + l α , E([a † + m β , a - n γ ] ε βγ ) -+ 2τ αγ δ l α n γ a + m β = 0 (9.53b) a - l α , E([a + m β , a † - n γ ] ε βγ ) --2τ αβ δ l α m β a - n γ = 0 (9.53c) a - l α , E([a † + m β , a - n γ ] ε βγ ) --2δ l α m β a - n γ = 0 (9.53d) γ =β. (9.53e) As one can expect, the relations (9.53a)-(9.53d) can be obtained from the multifield paracommutation relations (9.27) via the replacement [•, •] ε → E([•, •] ε βγ ). It should be paid special attention on the equation (9.53e). It is due to the fact that in the expressions for the dynamical variables do not enter 'cross-field-products', like a † + l α •a - m β for β = α, and it corresponds to the condition (ii) in [17, p. B 1159]. The equality (9.53e) is quite important as it selects only that part of the ' E-transformed' multifield paracommutation relations (9.27) which is compatible with the bilinear commutation relations (9.19) (see (9.28) and (9.29)). Besides, (9.53e) makes (9.53a)-(9.53d) independent of the particular definition of ε αβ (see (9.11)). The equations (9.52) are the only restrictions on the operator E; examples of this operator are provided by the normal (resp. antinormal) ordering operator N (resp. A), which has the properties (cf. (4.22) (resp. (7.30))\n\nN a + m β • a † - n γ := a + m β • a † - n γ N a † + m β • a - n γ := a † + m β • a - n γ N a - m β • a † + n γ := ε βγ a † + n γ • a - m β N a † - m β • a + n γ := ε βγ a + n γ • a † - m β (9.54) A a + m β • a † - n γ := ε βγ a † - n γ • a + m β A a † + m β • a - n γ := ε βγ a - n γ • a † + m β A a - m β • a † + n γ := a - m β • a † + n γ A a † - m β • a + n γ := a † - m β • a + n γ . (9.55)\n\nThe material of Sect. 8 has also a multifield variant that can be obtained via the replacements (9.51) and (9.4) . Here is a brief summary of the main results found in that way.\n\nThe operator E should possess the properties (9.54) and, in this sense, can be identified with the normal ordering operator, E = N . (9.56)\n\nAs a result of this fact and ε ββ = ε β (see (9.11) ), the commutation relations (9.53) take the final form:\n\na + l α , a + m β • a † - n β -+ δ l α n β a + m β = 0 (9.57a) a + l α , a † + m β • a - n β -+ τ αβ δ l α n β a + m β = 0 (9.57b) a - l α , a + m β • a † - n β --τ αβ δ l α m β a - n β = 0 (9.57c) a - l α , a † + m β • a - n β --δ l α m β a - n β = 0 (9.57d)\n\nwhich is the multifield version of (8.17) and corresponds, up to the replacement a ± l α → √ 2a ± l α , to (9.27) with ε βγ = 0.\n\nThe vacuum state vector X 0 is supposed to be uniquely defined by the following equations (cf. (8.1b)-(8.3)): a - l α X 0 = 0 a † - l α X 0 = 0 (9.58a) X 0 = 0 (9.58b) X 0 | X 0 = 1 (9.58c)\n\na † - l α • a + m β ( X 0 ) = δ l α m β X 0 a - l α • a † + m β ( X 0 ) = δ l α m β X 0 a - l α • a + m β ( X 0 ) = τ αβ δ l α m β X 0 a - l α • a + m β ( X 0 ) = τ αβ δ l α m β X 0 .\n\n(9.58d)\n\nFurther constraints on the possible commutation relations follow from the definition/introduction of the concept of the vacuum (vacuum state vector). They practically reduce the redefined dynamical variables to the ones obtained via normal ordering procedure, which results in the explicit form (8.17) of the admissible commutation relations. In a sense, they happen to be 'one half' of the paracommutation ones. As a last argument in the way for finding the 'unique true' commutation relations, we require the existence of procedure for calculation of vacuum mean values of anti-normally ordered products of creation and annihilation operators, to which the mean values of the dynamical variables and the transition amplitudes between different states are reduced. We have pointed that the standard bilinear commutation relations are, at present, the only known ones that satisfy all of the conditions imposed and do not contradict to the existing experimental data.\n\nThe consideration of a system of at least two different quantum free fields meets a new problem: the general relations between creation/annihilation operators belonging to different fields turn to be undefined. The cause for this is that the commutation relations for any fixed field are well defined only on the corresponding to it Hilbert subspace of the system's Hilbert space of states and their extension on the whole space, as well as the inclusion in them of creation/annihilation operators of other fields, is a matter of convention (when free fields are concerned); formally this is reflected in the structure of the dynamical variables which are sums of those of the individual fields included in the system under consideration. We have, however, presented argument by means of which the a priori existing arbitrariness in the commutation relations involving different field operators can be reduced to the 'standard' one: these relations should contain either commutators or anticommutators of the creation/annihilation operators belonging to different fields. A free field theory cannot make difference between these two possibilities. Accepting these possibilities, the admissible commutation relations (9.57) for system of several different fields are considered. They turn to be corresponding multifield versions of the ones regarding a single field. Similarly to the single field case, the standard multifield bilinear commutation relations seem to be the only known ones that satisfy all of the imposed restrictions and are in agreement with the existing data."
}
] | ||
arxiv:0704.0078 | 0704.0078 | 1 | 10.1088/0264-9381/24/14/008 | e7905f135b445b554e49d9dccd83f4cd5695b8ade30e0b980cc4cc285a3a05ec | Linear perturbations of matched spacetimes: the gauge problem and
background symmetries | We present a critical review about the study of linear perturbations of
matched spacetimes including gauge problems. We analyse the freedom introduced
in the perturbed matching by the presence of background symmetries and revisit
the particular case of spherically symmetry in n-dimensions. This analysis
includes settings with boundary layers such as brane world models and shell
cosmologies. | [
"Marc Mars",
"Filipe C. Mena",
"Raul Vera"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-01 | 2026-02-26 | We present a critical review about the study of linear perturbations of matched spacetimes including gauge problems. We analyse the freedom introduced in the perturbed matching by the presence of background symmetries and revisit the particular case of spherically symmetry in n-dimensions. This analysis includes settings with boundary layers such as brane world models and shell cosmologies.
An important aspect in any geometric gravitational theory is the analysis of how to match two spacetimes. This is true in particular for General Relativity and its perturbation theory. Despite the relevance and maturity of the matching theory one often finds papers where the matching conditions are not properly used. Most of the difficulties arise from the fact that the matching conditions are imposed in specific coordinate systems in a manner which is not completely coordinate independent. More specifically, matching two spacetimes requires identifying the boundaries pointwise, and sometimes this identification is done implicitly by fixing spacetime coordinates, without paying enough attention to the fact that solving the matching involves finding an identification of the boundary and that this should not be fixed a priori.
In perturbation theory this problem also arises, and it gets complicated by the fact that the fields to be matched (as the perturbed metric) are gauge dependent. So, in addition to a priori choices of identifications of the boundary, there is also the problem that particular gauges are often used. It may be argued that the matching theory must be gauge independent and therefore it can be performed in any gauge. This is true, but only when due care is taken to ensure that the choice of gauge does not restrict, a priori, the perturbed identification of the boundaries.
A complete description of the linearized matching conditions has been achieved only recently by Carter and Battye [5] and independently by Mukohyama [6] . To second order, the matching conditions have been recently found in [7] . Despite these papers, we believe that some confusion still lingers in the field, in particular with respect to the existing gauge invariant formulations. The aim of this paper is to try to clarify these issues. In order to do that, we will critically discuss some of the approaches proposed in the literature trying to make clear which are the implicit assumptions made and to what extent are they justified.
The first papers discussing the perturbed matching theory are, as far as we know, the classic papers by Gerlach and Sengupta [2, 3] . However, as explained below, their description of the perturbed matching theory contains imprecisions, and we will therefore start discussing their approach pointing out the difficulties they encounter. A first attempt to justify the claims in [2, 3] is due to Martín-García and Gundlach [4] , who propose a different but nevertheless closely related set of linearized matching conditions. Pointing out the implicit assumption made by these authors will also help us to try to explain the subtleties inherent to the perturbed matching theory.
In [6] the linearized matching conditions are described for arbitrary backgrounds, perturbations and matching hypersurfaces, and then applied to the case of two background spacetimes with a high degree of symmetry, namely those which admit a maximal group of isometries acting on codimension two spacelike submanifolds (e.g. spherically symmetric spacetimes). In order to simplify the matching conditions, Mukohyama derives a set of matching conditions for so-called doubly gauge invariants. However, a gap arises in his final conclusions as the presented set of conditions for doubly gauge invariant quantities for the linearized matching of spacetimes are only shown to be necessary conditions. Analysing sufficiency touches directly on the issue we are trying to emphasize in this paper, so we devote one section to clarify this point, where we show how these conditions are, stricktly speaking, not sufficient. Since the matching conditions in terms of doubly gauge invariants are widely used in the literature, we consider important to close this gap. Moreover, the constructions of gauge invariant quantities using spherical harmonic decompositons leaves out the l = 0 and l = 1 sectors. We will discuss this issue and its consequences.
The paper is organized as follows. We start by summarising the perturbed matching conditions in Section 2, where we also describe the gauge freedom involved. Then, the procedures used in the classic papers [2, 3] together with the justifications and further developments in [4] are reviewed in Section 3. Section 4 focuses on the consequences of the existence of symmetries in the background configuration, which will have relevance in our final discussion. Section 5 has three subsections. The first one is devoted to present briefly the procedure and results discussed in [6] particularised to the case of spherically symmetric backgrounds. In the second subsection we analyse the sufficiency of the doubly gauge invariant matching conditions in [6] . The last subsection is devoted to the study of the freedom left in the perturbation of the matching hypersurface once the metric perturbations have been fixed at both sides. We finish with an appendix where explicit expressions for the discontinuities of the perturbed second fundamental forms in the spherical case are given. Some of these expressions are used in the main text.
In this section we describe the gauge freedom involved in the linearised spacetime matching and summarise the perturbed matching conditions.
The purpose of the matching theory is to construct a new spacetime out of two spacetimes M ± with boundary by finding a suitable diffeomorphism between the boundaries which allows for their pointwise identification. In particular, the matched spacetime cannot be thought to exist beforehand. Another aspect to bear in mind is that the matching conditions involve exclusively tensors on the identified boundary Σ and hence any coordinate system in M ± is equally valid. This is well-known but it is still source of confusion sometimes.
In perturbed matching theory, not only the metrics are perturbed but also the matching hypersurfaces may be deformed. Furthermore, as for the metric, the "deviation" of the matching hypersurface is also a gauge dependent quantity. This can be best understood by viewing perturbations as ε-derivatives (at ε = 0) of a one-parameter family of spacetimes (M + ε , g + ε ) with boundary Σ + ε . It is convenient to embed M + ε within a larger manifold (without boundary) V + ε to clarify the discussion. A priori, the manifolds (M + ε , g ε ) are completely distinct so it makes no direct sense to talk about ε-derivatives. It is necessary to identify first the different manifolds so that a single point p refers to one point on each of the manifolds. Obviously, there are infinite ways to identify the manifolds, all of them equally valid a priori. This freedom leads to the gauge dependence of the perturbed metric (and of any other geometrically defined tensor). The identification above may, or may not, map the boundaries Σ + ε among themselves. A priori, a point in Σ + 0 may be mapped, for ε = 0, to a point on Σ + ε , to a point interior to M + ε or to a point exterior to M + ε (within the extension V + ε ) which is not part of the manifold. How can we then take derivatives with respect to ε at those later points? Since only derivatives at ε = 0 are needed, restricting to infinitesimal values of ε entails no loss of generality. Then, if for some small ε, a point q ∈ Σ + 0 is mapped to the exterior of M + ε , it follows from differentiability with respect to ε that q is mapped, for the reverse value -ε, to a point interior to M + ε . Thus, perturbations can be defined at the boundary by taking one sided derivatives, i.e. to take limits ε → 0, with a sign restriction on ε (c.f. [7] for an alternative discussion).
However, an important issue remains: How do we describe the deformation of the boundary Σ + 0 ? As a set of points each boundary Σ + ε maps, with the above identification, into a hypersurface of the background spacetime, which we call Σ+ ε . In general, this hypersurface will not coincide with Σ + 0 and may well touch it or cross it. This gives us an idea of how the boundary is deformed, but only as a subset, not pointwise. In order to know how the boundary actually moves within the background, we need to prescribe a priori a pointwise identification of Σ + 0 with Σ + ε . This identification is completely different and independent from the one described above involving spacetime points, and involves only the points on the boundaries. As before, there are infinitely many ways to identify the boundaries, and this defines a second gauge freedom, which involves objects intrinsically defined on the boundary. This gauge freedom will be referred as hypersurface gauge, as opposed to the usual spacetime gauge described above.
With both identifications chosen, the deformation of the boundary within the background can already be described: Fix a point q on the background boundary Σ + 0 . The identification of the boundaries defines a point q ε on Σ + ε , for each ε. The spacetime identification takes this point q ε and maps it into a point qε of the background M + 0 (perhaps after a sign restriction on ε). Obviously qε belongs to the perturbed hypersurface Σ+ ε . We have therefore not only a deformation of the background hypersurface as a set of points, but also pointwise information. It only remains to take the tangent vector of the curve qε at ε = 0, i.e. Z + = dqε dε | ε=0 which encodes completely the deformation of the matching hypersurface as seen from the background spacetime. Two final remarks are in order: (i) Z + is defined exclusively on Σ + 0 , no extension thereof is defined or required and (ii) Z + depends on both the spacetime and hypersurface gauges, since its defining curve is constructed using both identifications. However, decomposing Z + = Q + n 0 + + T + , where n 0 + is the unit normal of Σ + 0 (assumed non-null anywhere) and T + is tangent to it, it turns out that Q + depends on the spacetime gauge but not on the hypersurface gauge. This is because changing the hypersurface gauge reorganizes the points within each Σ+ ε , but cannot modify any of them as a set of points.
Tensors defined intrinsically on the boundaries Σ + ε are completely unaffected by the spacetime identification, and are therefore invariant under spacetime gauge transformations. Recall that the matching conditions involve only objects intrinsic to the matching hypersurfaces. Since the perturbed matching conditions are, formally, just their εderivatives, it follows by construction that the perturbed matching conditions must be gauge invariant under spacetime gauge transformations. This may seem surprising at first sight since the matching conditions must involve the perturbed metric, which is obviously gauge dependent. However, the conditions turn out to be gauge independent because they also involve the deformation vector Z + , which is spacetime gauge dependent. This vector is therefore of fundamental importance and must be taken into account in any sensible approach to the problem, as we shall see next.
Let (M ± 0 , g ± 0 ) be n-dimensional spacetimes with non-null boundaries Σ ± 0 . Matching them requires an identification of the boundaries, i.e. a pair of embeddings Φ
± : Σ 0 -→ M ± 0 with Φ ± (Σ 0 ) = Σ ± 0
, where Σ 0 is an abstract copy of any of the boundaries. Let ξ i (i, j, . . . = 1, . . . , n -1) be a coordinate system on Σ 0 . Tangent vectors to Σ ± 0 are obtained by e ±α i = ∂Φ α ± ∂ξ i (α, β, . . . = 0, . . . , n -1). There are also unique (up to orientation) unit normal vectors n (0) ± α to the boundaries. We choose them so that if n (0) + α points towards M + then n (0) -α points outside of M -or viceversa. The first and second fundamental are simply q (0)
ij ± ≡ e ±α i e ±β j g (0) αβ | Σ ± , K (0) ij ± = -n (0) ±α e ±β i ∇ ± β e ±α j .
The matching conditions (in the absence of shells) require the equality of the first and second fundamental forms on Σ ± 0 , i.e.
q (0) ij + = q (0) ij -, K (0) ij + = K (0) ij -. (1)
Under a perturbation of the background metric g ± pert = g (0)± + g (1)± and of the matching hypersurfaces via Z ± = Q ± n (0) ± + T ± , the matching conditions will be perturbatively satisfied if and only if [6] q
(1)+ ij = q (1)- ij , K (1)+ ij = K (1)- ij , (2)
with
q (1) ij ± = L T ± q (0) ij ± + 2Q ± K (0) ij ± + e ±α i e ±β j g (1) αβ ± , (3)
K (1) ij ± = L T ± K (0) ij ± -ǫD i D j Q ± + Q ± (-n (0) ± µ n (0) ± ν R (0)± αµβν e ±α i e ±β j + K (0) il ± K (0) l j ± ) + ǫ 2 g (1) αβ ± n (0) ± α n (0) ± β K (0) ij ± -n (0) ± µ S (1)±µ αβ e ±α i e ±β j , (4)
where ǫ = n (0)
α n (0)α , D is the covariant derivative of (Σ, q (0)± ) and S (1)
±α βγ ≡ 1 2 (∇ ± β g (1)±α γ + ∇ ± γ g (1)±α β -∇ ± α g (1)± βγ ). 1
In these equations, Q ± and T ± are a priori unknown quantities and fulfilling the matching conditions requires showing that two vectors Z ± exist such that (2) are satisfied. The spacetime gauge freedom can be exploited to fix either or both vectors Z ± a priori, but this should be avoided (or at least carefully analysed) if additional spacetime gauge choices are made, in order not to restrict a priori the possible matchings. Regarding the hypersurface gauge, this can be used to fix one of the vectors T + or T -, but not both.
As already stressed the linearized matching conditions are by construction spacetime gauge invariant (in fact each of the tensors q 2 ) are hypersurface gauge invariant, provided the background is properly matched, since [6] under such a gauge transformation given by the vector ζ on Σ 0 , q ( foot_0 ) ij transforms as q (1) ij + L ζ q (0) ij , and similarly for K (1) ij .
(1) ij ± , K (1) ij ± is). Moreover, the set of conditions (
The first attempt to derive a general formalism for the matching conditions in linearized gravity is, to our knowledge, due to Gerlach and Sengupta [2] . Their approach is based on the description of the matching hypersurface Σ as a level set of a function f defined on the spacetime. Assuming the level sets {f = const} to be timelike, a field of spacelike unit normals is defined as n µ = (g αβ f ,α f ,β ) -1/2 f ,µ . The unperturbed matching conditions correspond to the continuity everywhere (in particular across Σ) of the tensors
q αβ ≡ g αβ -n α n β , K αβ ≡ q α µ q β ν ∇ µ n ν , (5)
which are the spacetime versions of the first and second fundamental forms introduced above. Being f defined everywhere, it makes sense to perturb it in order to describe the variation of the matching hypersurface. Obviously, by perturbing f one also perturbs n µ . The perturbed matching conditions proposed in [2] read
q µ α q ν β △(q αβ ) + = q µ α q ν β △(q αβ ) -, q µ α q ν β △(K αβ ) + = q µ α q ν β △(K αβ ) -, (6)
where q β α is the projector onto Σ, △ stands for perturbation and + and -denote the quantities as computed from either side of the matching hypersurface Σ. These expressions involve the projections of the perturbations of q αβ and K αβ onto Σ. The need of considering only the projected components is justified in [2] since the matching conditions need to be intrinsic to the matching hypersurfaces. However, Gerlach and Sengupta themselves note that conditions (6) are not gauge foot_1 invariant.
Since the main interest in [2, 3] refers to spherically symmetric backgrounds, this "ambiguity" is fixed in that case by finding suitable gauge invariant combinations of the linearized matching conditions, which turn out to give a correct set of necessary perturbed matching conditions in spherical symmetry. It should be stressed however, that the authors consider these gauge invariant subset to be sufficient also, with no further justification.
We know from the discussion in Sect. 2.1 above that ( 6 ) cannot be correct as it leads to a set of gauge dependent conditions. Since, on the other hand the proposal (6) may look plausible, it is of interest to point out where, and in which sense, it fails to be correct.
The first source of problems comes from assuming that the matched spacetime is given beforehand. Indeed, q αβ and K αβ are spacetime tensors and they can only exist (and be continuous) once the matched spacetime is constructed. But this is precisely the purpose of the matching conditions, so the conditions become circular. Another aspect of the same problem is that one can only talk about continuity once the pointwise identification of the boundaries is chosen. But a level set of a function defines only a set of points and not the way those points must be identified. A third instance of the same issue is that tensor components must be expressed in some basis, e.g. a common coordinate system covering both sides of Σ. But again this cannot be assumed a priori. It needs to be constructed.
Let us however mention that once the pointwise identification of the boundaries is chosen, the use of spacetime tensors is allowed provided they are finally projected onto the hypersurface. In that sense, and when properly used, using spacetime indices may simplify some calculations notably (see Carter and Battye, [5] where this notation is used to derive the perturbed matching conditions).
Besides this aspect (which already affects the background matching) the perturbed equations ( 6 ) suffer from one extra problem. The perturbations △(q αβ )(p) and △(K αβ )(p) at a point p in the background can be defined by taking ε-derivatives at fixed p and ε = 0 of the corresponding tensors (defined by g αβ (ε) and f ε ). For each value of ε, the matching conditions impose the continuity of q αβ (ε) and K αβ (ε) everywhere (with the caveat already mentioned regarding the identification of the boundaries). However, continuity of △(q αβ ) and △(K αβ ) at p would only follow if derivatives of continuous functions with respect to an external parameter were necessarily continuous (in our case, the derivative with respect to ε), which is not true in general. A trivial example is given by the function u(ε, x) = |x + ε|, whith x ∈ R. For each ε this function is continuous. However, the derivative with respect to ε does not even exist at x = 0, ε = 0. This reflects the fact that subtracting continuous tensors at a fixed spacetime point p leads to objects that need not be continuous. This is in fact the main problem of (6) as linearized matching conditions.
An immediate question arises: Why is the gauge invariant subset of matching conditions found in [2, 3] for spherically symmetric backgrounds correct? In order to understand this, let us rewrite (6) using the formalism of section 2.2. First of all, since △(n α n β ) will contain, at least, one free n (0) α , we have
q µ α q ν β △(q αβ ) ± = q µ α q ν β g (1) ± αβ . (7)
Moreover, a simple calculation gives
△(∇ α n β ) = ∇ α (△n β ) -S (1)µ αβ n (0) µ and △(q α β ) = -g (1) αµ n (0) µ n (0) β + g (0)αµ △(n µ )n (0)
β + n (0)α △(n β ). These, together with standard properties of the projector, lead to
q µ α q ν β △(K αβ ) ± = a (0) ν q µ α △(n α ) + q µ α q ν β ∇ α (△n β ) -q µ α q ν β S (1)ρ αβ n (0) ρ ± , (8)
where a
(0) ν ≡ n (0)α ∇ α n (0)
ν . In general, these expressions do not agree with (3) and ( 4 ). However, when the gauges are chosen so that Z ± = 0, then △f ≡ 0 on Σ because the matching hypersurface is unperturbed as seen from the background. Consequently
∂ α (△f ) ∝ n (0) α on Σ, which implies △(n α ) Σ = hn (0)
α for some function h. Imposing n(ε) to be unit for all ε fixes h = ǫ 2 g (1) αβ n
(0) α n (0)
β . Inserting into (8) the matching conditions (6) become
q µ α q ν β g (1)
αβ
+ = q µ α q ν β g (1) αβ -, (9)
ǫ 2 g (1) αβ n (0) α n (0) β K µν -q µ α q ν β S (1)ρ αβ n (0) ρ + = ǫ 2 g (1) αβ n (0) α n (0) β K µν -q µ α q ν β S (1)ρ αβ n (0) ρ -,
which agree with (2) (with the exception that (9) refers to spacetime tensors and ( 2 ) are defined on Σ). Since Gerlach and Sengupta derive a subset of gauge invariant matching conditions out of (6) in the spherically symmetric case and their conditions are correct in one gauge, it follows that the invariant subset is correct in any gauge. This is the reason why the results in [2, 3] involving spherically symmetric backgrounds turn out to be fine. Substantial progress in the linearized matching problem was made by Martín-García and Gundlach [4] . These authors pointed out the lack of justification in [2, 3] for the choice of (6) as matching conditions. It was also argued that for spacetimes with boundary it only makes sense to define perturbations by using gauges where the perturbed matching hypersurface is mapped onto the background matching hypersurface. Perturbations in this gauge, called "surface gauge" (not to be confused with hypersurface gauge) are denoted by △, and its defining property is △f = 0. The idea was to write down the matching conditions in this gauge and then transform into any other gauge if necessary. As noticed by the authors, the surface gauge is not unique since there are still three degrees of freedom left, which correspond to the three directions tangent to Σ.
A relevant observation made in [4] was that the continuity of tensorial perturbations may depend on the index position in the tensors. The authors argue that the tensors truly intrinsic to the hypersurfaces are q αβ , K αβ (with indices upstairs) and propose the following perturbed matching conditions
△(q αβ ) + = △(q αβ ) -, △(K αβ ) + = △(K αβ ) -, (10)
which are demonstrated to become exactly (9) . This shows the equivalence of both proposals in the surface gauge, as explicitly stated in [4] . This justifies partially the validity of both approaches in the surface gauge. However, the justification is not complete because of the issue we discuss next. Indeed, conditions (10) still carry one implicit assumption that needs to be clarified. As already stressed the perturbed matching conditions have two inherent and independent degrees of gauge freedom. The approach by Martín-García and Gundlach involves only spacetime objects, and therefore can only notice the spacetime gauge freedom. This leads to an incorrect statement in [4] , as it is not true that the linearized matching conditions read (10) in any surface gauge. Conditions (10) will only be valid when the spacetime gauge maps pairs of background points (identified, via the background matching) to pairs of points on the perturbed boundaries Σ ± ε which are also identified through the matching. Notice that not all surface gauges have this property. In explicit terms, this means that the vectors Z ± must (i) only have tangential components (so that we are in surface gauge) and (ii) have the same components when written in terms of an intrinsic basis of Σ 0 . In less precise, but more intuitive terms, condition (ii) states that Z + and Z -are the same vector, i.e. that the gauges in both regions are chosen such that the displacement of one fixed point of the background hypersurface is identical in both regions (the displaced point, of course, stays on the unperturbed hypersurface, due to the choice of surface gauge). Observe finally that if Q ± = 0 and T + = T -, then the linearized matching conditions (2) truly reduce to conditions (9) , once the latter are projected on Σ. This shows the correctness of the approaches by Gerlach and Sengupta and Martín-García and Gundlach in special gauges.
We devote this section to the study of the consequences of the existence of background symmetries on perturbed spacetime matchings.
The existence of symmetries in the background configuration introduces two issues which are important to take into consideration: the first corresponds to the freedom introduced by the matching procedure, when preserving the symmetries, at the background level [9] , c.f. [10] for an application. The second issue corresponds to the consequences that the symmetries in the background configuration may have on the perturbation of the matching.
It must be stressed here that the arbitrariness introduced by the presence of symmetries in the background configuration is completely independent from both the hypersurface and spacetime gauge freedoms. However, that arbitrariness is gauge dependent and therefore a gauge choice can be made to remove it. As we will show, an isometry in the background implies that there is a direction along which the difference [ T ] ≡ T + -T -cannot be determined by the perturbed matching equations. But, as we have discussed at the end of section 2, one could eventually choose part of the spacetime gauges (if there is any freedom left) to fix [ T ]. Note, finally, that a change of hypersurface gauge leaves [ T ] invariant.
We shall now consider the presence of isometries in the background configuration. So, let us assume that one of the sides, say (M + 0 , g (0)+ ), admits an isometry generated by the Killing vector field ξ + tangent to the boundary Σ + 0 . The commutation of the Lie derivative and the pull-back implies [9]
L ξ + q (0) ij + = e +α i e +β j L ξ + g (0)
αβ
+ | Σ 0 = 0,
which means that ξ + is a Killing vector of (Σ 0 , q (0) ij + ). This implies from expression (3) that q (1) ij + is invariant under the transformation
T + → T + + ξ + | Σ 0 .
As for K (1) ij + , from its expression (4), it is again clear that the previous transformations of T + will leave K (1) ij
+ invariant provided L ξ + K (0) ij + = 0. But this is precisely the case since ξ + is a Killing vector orthogonal to n (0) + , which implies L ξ + n (0) + β | Σ + 0 = 0, and hence L ξ + K (0) ij + = e +α i e +β j L ξ + (∇n (0) + ) αβ | Σ 0 = e +α i e +β j ∇ α L ξ + n (0) + β | Σ 0 = 0.
Of course, all this discussion also applies to the (-) side.
The combination of the invariance of q (1) ij ± and K (1) ij ± leads to the fact that the first order perturbed matching conditions are invariant under a change of the vectors T ± along the direction of any isometry of the background configuration (preserved by the matching). Then, as expected, when symmetries are present the linearized matching conditions cannot determine the difference [ T ] completely: they leave undetermined the relative (between the two sides) deformation of the hypersurface along the direction of the symmetry. Note that, still, the shape of the perturbed hypersurface is completely determined, since that is driven by
Q ± .
The overall picture is as follows: at the background level we have the arbitrariness of the identification of Σ + 0 with Σ - 0 [9] , which can be seen as a "sliding" between Σ + 0 and Σ - 0 . The perturbation adds to this an arbitrary shift of the deformation of the matching hypersurface at each side along the orbits of the isometry group. As an example, in the description of stationary and axisymmetric compact bodies discussed in [10, 9] , the background sliding corresponds to an arbitrary constant rotation of the interior with respect to the exterior. Note that, in that case, this rotation is only relevant because the exterior is taken to be asymptotically flat. As a result, two identical interiors can, in principle, give rise to two exteriors that differ by a constant rate rotation [10] . The shift of the surface deformation would, in principle, lead to an arbitrary constant rotation along the axial coordinate of the surface deformation of the body. Likewise, two identical perturbations in the interior of the body may produce two different perturbations in the exterior, which may differ by a relative constant rate rotation. A choice of spacetime gauge could be used to relate the deformations inside and outside. However, this may interfere with other gauge fixings that may have been made.
In this section we shall revisit Mukohyama's theory for linearized matching in the special case of spherical symmetry. Similar results [6] hold for backgrounds admitting isometry groups of dimension (n-1)(n-2)/2 acting on non-null codimension-two orbits of arbitrary topology (strictly speaking the orbits need to be compact).
Concentrating on one of the two spacetimes to be matched, either + or -, we consider a spherically symmetric background metric of the form
g (0) αβ dx α dx β = γ ab dx a dx b + r 2 Ω AB dθ A dθ B , (11)
where γ ab (a, b, .. = 0, 1) is a Lorentzian two-dimensional metric (depending only on {x a }), r > 0 is a function of {x a }, and Ω AB dθ A dθ B is the n -2 dimensional unit sphere metric with coordinates {θ A } (A, B, . . . = 2, 3, . . . , n -1).
A general spherically symmetric background hypersurface can be given in parametric form as Σ 0 := {x 0 = Z (0)0 (λ),
x 1 = Z (0)1 (λ), θ A = ϑ A }, (12)
where {ξ i } = {λ, ϑ A } is a coordinate system in Σ 0 adapted to the spherical symmetry.
The tangent vectors to Σ 0 read
e λ = Ż (0) 0 ∂ x 0 + Ż (0) 1 ∂ x 1 Σ 0 , e ϑ A = ∂ θ A | Σ 0 , (13)
where dot is derivative w.r.t. λ. With N 2 ≡ -ǫe λ a e λ b γ ab | Σ 0 , so that ǫ = 1 (ǫ = -1) corresponds to a timelike (spacelike) hypersurface, the unit normal to Σ 0 reads
n (0) = √ -det γ N -Ż (0) 1 dx 0 + Ż (0) 0 dx 1 Σ 0 , (14)
where the sign choice of N corresponds to the choice of orientation of the normal. The background induced metric and second fundamental form on Σ 0 read
q (0) ij dξ i dξ j = -ǫN 2 dλ 2 + r 2 | Σ 0 Ω AB | Σ 0 dϑ A dϑ B , (15)
K (0) ij dξ i dξ j = N 2 Kdλ 2 + r 2 K| Σ 0 Ω AB | Σ 0 dϑ A dϑ B , (16)
where
K ≡ N -2 e λ a e λ b ∇ a n (0)
b , K = n (0)a ∂ x a ln r. It follows that the background matching conditions (1) are
N 2 + = N 2 -, r 2 + | Σ 0 = r 2 -| Σ 0 , K + = K -, K+ = K-. (17)
Using ( 3 ) and ( 4 ) we could now compute the first order perturbations q (1) ij and K (1) ij in terms of the above quantities and Z (or equivalently Q and T ), c.f. Eqs. ( 45 ) and (46) in [6] . Let us recall (see subsection 2.2) that while the individual tensors q (1) ij and K (1) ij are not hypersurface gauge invariant, their respective differences from the + and -sides (i.e. the linearized matching conditions) are. Those tensors depend of the hypersurface gauge through the tangent vectors T + and T -, which under a gauge change transform simply by adding the gauge vector. It follows that only their difference [ T ] can appear in the linearized matching conditions. Consequently there are three degrees of freedom that cannot be fixed by the equations, but can be fixed by choosing the hypersurface gauge, for instance to set T + . Thus, the linearized matching conditions can be looked at as equations for the difference [ T ] as well as for Q + and Q -, i.e. for five objects. If these equations admit solutions, then the linearized matching is possible and it is impossible otherwise.
Mukohyama emphasizes the convenience to look for doubly gauge invariant quantities to write down the linearized matching conditions, however the matching conditions are already gauge invariant (both for the spacetime and hypersurfaces gauges). Looking for gauge invariant combinations on each side amounts to writing equations where the difference vector [ T ] simply drops. Indeed, in many cases, knowing the value of such vector in a specific matching is not interesting. In that sense, using doubly gauge invariant quantities is useful as it lowers the number of equations to analyse. However, we want to stress that this is not related to obtaining gauge invariant linearized matching equations.
It is just related to not solving for superfluous information. In fact, a set of equations where also Q + and Q -have disappeared would be even more convenient from this point of view, provided one is not interested in knowing how the hypersurfaces are deformed in the specific spacetime gauge being used.
Since the use of doubly gauge invariant matching conditions is used extensively, let us recall its main ingredients in order to discuss if they really are equivalent to the full set of linearized matching equations and in which sense.
To that aim Mukohyama [6] , decomposes the perturbation tensors q (1) ij and K (1) ij in terms of scalar Y , vector V A and tensor harmonics T AB on the sphere, as foot_2
q (1) ij dξ i dξ j = ∞ l=0 (σ 00 Y dλ 2 + σ (Y ) T (Y )AB dϑ A dϑ B ) + ∞ l=1 2(σ (T )0 V (T )A + σ (L)0 V (L)A )dλdϑ A + ∞ l=2 (σ (T ) T (T )AB + σ (LT ) T (LT )AB + σ (LL) T (LL)AB )dϑ A dϑ B , (18)
K (1) ij dξ i dξ j = ∞ l=0 (κ 00 Y dλ 2 + κ (Y ) T (Y )AB dϑ A dϑ B ) + ∞ l=1 2(κ (T )0 V (T )A + κ (L)0 V (L)A )dλdϑ A + ∞ l=2 (κ (T ) T (T )AB + κ (LT ) T (LT )AB + κ (LL) T (LL)AB )dϑ A dϑ B , (19)
where all the scalar coefficients depend only on λ. Each coefficient in the decomposition has indices l and m which have been dropped for notational simplicity. Notice that each coefficient σ and κ is defined in the range of l's appearing in the corresponding summatory. By construction, each of the σ and κ are spacetime-gauge invariant (but not hypersurface-gauge invariant). For l ≥ 2 they can even be written down [6] explicity in terms of spacetime-gauge invariant quantities. In a similar way, the doubly gaugeinvariant quantities presented in [6] , are only defined for l ≥ 2 (except k (T )0 , which is also defined for l = 1), and read l ≥ 2 :
f 00 ≡ σ 00 -2N∂ λ N -1 χ , l ≥ 2 : f ≡ σ (Y ) + ǫN -2 χ∂ λ r 2 | Σ 0 + 2 n -2 k 2 l σ (LL) , l ≥ 2 : f 0 ≡ σ (T )0 -r 2 | Σ 0 ∂ λ r -2 | Σ 0 σ (LT ) , l ≥ 2 : f (T ) ≡ σ (T ) , l ≥ 2 : k 00 ≡ κ 00 + ǫKσ 00 + ǫχ∂ λ K, l ≥ 1 : k (T )0 ≡ κ (T )0 -Kσ (T )0 , (20)
l ≥ 2 : k (L)0 ≡ κ (L)0 + 1 2 (ǫK -K)σ (L)0 + 1 2 (ǫK + K) χ -r 2 | Σ 0 ∂ λ (r -2 | Σ 0 σ (LL) ) , l ≥ 2 : k (LT ) ≡ κ (LT ) -Kσ (LT ) , l ≥ 2 : k (LL) ≡ κ (LL) -Kσ (LL) , l ≥ 2 : k (Y ) ≡ κ (Y ) -Kσ (Y ) + ǫN -2 r 2 | Σ 0 χ∂ λ K, l ≥ 2 : k (T ) ≡ κ (T ) -Kσ (T ) ,
where k 2 l = l(l + n -3) and
l ≥ 2 : χ ≡ σ (L)0 -r 2 | Σ 0 ∂ λ (r -2 | Σ 0 σ (LL) ).
The orthogonality properties of the scalar, vector and tensor harmonics imply that the equalities of the coefficients σ and κ for each l and m is equivalent to the equality of the perturbation tensors (18) and ( 19 ) at both sides of Σ 0 . Thus, recalling the notation
[f ] ≡ f + | Σ 0 -f -| Σ 0 , the equations l ≥ 0 : [σ 00 ] = [σ (Y ) ] = 0 l ≥ 1 : [σ (L)0 ] = [σ (T )0 ] = 0 l ≥ 2 : [σ (T ) ] = [σ (LT ) ] = [σ (LL) ] = 0 (21) l ≥ 0 : [κ 00 ] = [κ (Y ) ] = 0 l ≥ 1 : [κ (L)0 ] = [κ (T )0 ] = 0 l ≥ 2 : [κ (T ) ] = [κ (LT ) ] = [κ (LL) ] = 0 (22)
are equivalent to (2) and therefore correspond exactly to the linearized matching conditions in this setting. Notice that each of the equalities in ( 21 ) and ( 22 ) is in fact one equation for each l and m in the appropriate range. We will however refer to them simply as equations.
The full linearized matching conditions obviously imply the following equalities in terms of the doubly-gauge invariant quantities (20),
l ≥ 2 : [f 00 ] = [f ] = [f 0 ] = [f (T ) ] = 0 (23) l ≥ 1 : [k (T )0 ] = 0 l ≥ 2 : [k 00 ] = [k (Y ) ] = [k (L)0 ] = [k (LL) ] = [k (LT ) ] = [k (T ) ] = 0. ( 24
)
Whether these equations can be regarded as the full set of linearized matching conditions or not requires studying their sufficiency, i.e. whether they imply (21)-( 22 ) or not. This point was not mentioned in [6] and in fact the answer turns out to be negative, although in a mild way, as we discuss in the next subsection.
Let us recall that fulfilling the matching conditions requires finding two Z ± such that (21)-( 22 ) are satisfied. The key issue for the matching is therefore to show existence of deformation vectors Z ± so that all the equations hold. A plausibility argument in favour of the sufficiency of (23)-(24) comes from simple equation counting. Indeed, as already discussed, the linearized matching conditions are spacetime and hypersurface gauge invariant and therefore can only involve the difference vector [ T ], i.e. three quantities. Since constructing double gauge invariant quantities on each side eliminates this vector, the number of equations should be reduced exactly by three if they are to remain equivalent to the original set. This is precisely what happens as we go from the original forteen equations in (21)-( 22 ) down to eleven equations in (23)-( 24 ). This argument however is not conclusive, both because it is not rigorous and because each equation in those expressions is, in fact, a set of equations depending on l and m, and the range of l's changes with the equations. Let us therefore analyse this issue and [z (T ) ], [z (L) ] for l ≥ 1. Now, the explicit expressions (27), ( 29 ), (28) show that (30) determine uniquely [z λ ], [z (T ) ] and [z (L) ] for l ≥ 2. So, restricted to the sector l ≥ 2 Mukoyama's doubly gauge invariant matching conditions can be regarded as equivalent to the full set of matching conditions. Taking all l's into account, however, the equations turn out not to be sufficient. To show this, it is enough to display one equation involving the discontinuity of the background metric perturbations and [Q] (but not [ T ]) which holds as a consequence of the full set of matching conditions (21)-( 22 ) but not as a consequence of ( 23 )-(24). Using the fact that each l = 1 expression refers to n -1 objects (one for each m), the number of equations in (31)-( 32 ) is 7n -3, while the number of unkowns in [ T ] not yet determined by (30), i.e. [z λ ] for l = 0, 1 and [z (T ) ], [z (L) ] for l = 1 is 3n -2, which is smaller. It is to be expected, therefore, that (31), (32) imply conditions where these variables do not appear. This can be made explicit, for instance, by combining [σ 00 ] l=0 = 0 with [σ (Y ) ] l=0 = 0 which yields
l = 0 : [h λλ ] + 2[Q]N 2 K + 2N∂ λ ǫN ∂ λ (r 2 | Σ 0 ) [h (Y ) ] + 2[Q]r 2 | Σ 0 K = 0, whenever ∂ λ (r 2 | Σ 0 ) = 0. (If ∂ λ (r 2 | Σ 0 ) = 0 it is enough to consider [σ (Y ) ] l=0 = 0.
) This relation is enough to show that the continuity of the doubly-gauge invariant variables of Mukohyama is not sufficient to ensure the existence of the perturbed matching. Of course, this does not invalidate Mukohyama's approach in any way, which remains interesting and useful. It only means that, when using this approach to solve linearized matchings, one still needs to look more carefully into the l = 0 and l = 1 sector to make sure that the remaining equations ( 31 ) and (32) hold. On the other hand, equations (31), (32) do not completely determine [ T ]. The variable [z (T ) ] l=1 only appears in [σ (T )0 ] l=1 = 0, in the term ∂ λ (r -2 | Σ 0 [z (T ) ]). As a result, the matching conditions do not fix [z (T ) ] l=1 completely, but up to a constant factor times r 2 | Σ 0 (for each m). Recalling that V (T )A dϑ A for l = 1 correspond to the three Killing vectors on the sphere, this arbitrary constant (for each m) accounts for the addition to [ T ] of an arbitrary Killing vector of the sphere. This is in accordance with the discussion in Section 4. We devote the following subsection to complete the study of the freedom left in the matching.
As already emphasized, solving the linearized matching amounts to finding perturbation vectors Z + and Z -. Assume now that a linearized matching between two given backgrounds and perturbations has been done. It is natural to ask what is the most general matching between those two spaces, i.e. what is the most general solution for Z + and Z - of the matching conditions. Geometrically, this means finding all the possible deformations of the matching hypersurface Σ 0 which allow the two spaces to be matched.
Since this problem is of interest not only when the full matching conditions are imposed but also in situations where layers of matter are present (e.g. in brane-world or shell cosmologies) so that jumps in the second fundamental forms are allowed, we will analyse this issue in two steps. First, we will study the equations involving the perturbed first fundamental forms and will determine the freedom they admit. On a second step we will write down the extra conditions coming from the equality of the second fundamental forms.
to adding Killing vectors on the sphere, something already discussed in Section 4. The rest of terms involve combinations (with functions) of the conformal Killing vectors on the sphere and tangential vectors along λ. Notice that the coefficients of the conformal Killing (i.e. < z (L) > l=1,m ) determine all the rest of the l = 1 coefficients. In particular when < z (L) > l=1,m vanishes, then all the l = 1 terms vanish and the freedom becomes radially symmetric.
We now add to the analysis the difference of the equations in (22). Due to the fact that all coefficients in < Z > vanish for l ≥ 2 we only need to consider the equations for l = 0, 1, i.e. (32). We refer the reader to Appendix A for the explicit expressions of (32) in terms of the metric perturbations and Z. For the sake of completeness we also include all the explicit expressions of (22) in Appendix A. The difference of equations (32), see (44)-(46), whenever < g (1) >= 0 read l = 0, 1 :
< κ 00 >= 0 ⇔ (41)
-< QR (γ) dbac > n (0)d n (0)a e λ b e λ c -ǫ∂ 2 λ < Q > + ǫ 2N 2 ∂ λ N 2 ∂ λ < Q > -ǫ < Q > K 2 N 2 -ǫKN 2 ∂ λ (N -2 < z λ >) -ǫ∂ λ (K < z λ >) = 0, l = 1 : < κ (L)0 >= 0 ⇔ (42) -ǫ∂ λ < Q > +ǫK < z λ > +ǫ < Q > ∂ λ ln(r| Σ 0 ) + r 2 | Σ 0 K∂ λ (r -2 | Σ 0 < z (L) >) = 0 l = 0, 1 : < κ (Y ) >= 0 ⇔ (43) + 1 2 N -2 ∂ λ (r 2 | Σ 0 ) (∂ λ < Q > +K < z λ >) + 1 2 < Qn (0)a n (0)b ∇ a ∇ b r 2 > - ǫ 2 N -2 e λ a < z λ n (0)b ∇ b ∇ a r 2 > + l(l + n -3) n -2 ǫ < Q > -2 K < z (L) > = 0.
It can be checked that in general these equations overdetermine the previous equations, i.e. (39) and (40), although there may be particular cases for which they are compatible. Therefore, generically, they will imply that < z (L) > l=1,m = 0 and < z λ > l=0 = 0, and thence all the rest of the variables vanish, < z λ > l=1,m =< Q > l=1,m = 0, < z λ > l=0 =< Q > l=0 = 0, so that the only freedom left is given by
[ Z] ′ -[ Z] = a m V m (T ) .
Finding in which particular cases equations (39)-(43) are compatible is straightforward but tedious and will not be carried out explicitly here. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We present a critical review about the study of linear perturbations of matched spacetimes including gauge problems. We analyse the freedom introduced in the perturbed matching by the presence of background symmetries and revisit the particular case of spherically symmetry in n-dimensions. This analysis includes settings with boundary layers such as brane world models and shell cosmologies."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "An important aspect in any geometric gravitational theory is the analysis of how to match two spacetimes. This is true in particular for General Relativity and its perturbation theory. Despite the relevance and maturity of the matching theory one often finds papers where the matching conditions are not properly used. Most of the difficulties arise from the fact that the matching conditions are imposed in specific coordinate systems in a manner which is not completely coordinate independent. More specifically, matching two spacetimes requires identifying the boundaries pointwise, and sometimes this identification is done implicitly by fixing spacetime coordinates, without paying enough attention to the fact that solving the matching involves finding an identification of the boundary and that this should not be fixed a priori.\n\nIn perturbation theory this problem also arises, and it gets complicated by the fact that the fields to be matched (as the perturbed metric) are gauge dependent. So, in addition to a priori choices of identifications of the boundary, there is also the problem that particular gauges are often used. It may be argued that the matching theory must be gauge independent and therefore it can be performed in any gauge. This is true, but only when due care is taken to ensure that the choice of gauge does not restrict, a priori, the perturbed identification of the boundaries.\n\nA complete description of the linearized matching conditions has been achieved only recently by Carter and Battye [5] and independently by Mukohyama [6] . To second order, the matching conditions have been recently found in [7] . Despite these papers, we believe that some confusion still lingers in the field, in particular with respect to the existing gauge invariant formulations. The aim of this paper is to try to clarify these issues. In order to do that, we will critically discuss some of the approaches proposed in the literature trying to make clear which are the implicit assumptions made and to what extent are they justified.\n\nThe first papers discussing the perturbed matching theory are, as far as we know, the classic papers by Gerlach and Sengupta [2, 3] . However, as explained below, their description of the perturbed matching theory contains imprecisions, and we will therefore start discussing their approach pointing out the difficulties they encounter. A first attempt to justify the claims in [2, 3] is due to Martín-García and Gundlach [4] , who propose a different but nevertheless closely related set of linearized matching conditions. Pointing out the implicit assumption made by these authors will also help us to try to explain the subtleties inherent to the perturbed matching theory.\n\nIn [6] the linearized matching conditions are described for arbitrary backgrounds, perturbations and matching hypersurfaces, and then applied to the case of two background spacetimes with a high degree of symmetry, namely those which admit a maximal group of isometries acting on codimension two spacelike submanifolds (e.g. spherically symmetric spacetimes). In order to simplify the matching conditions, Mukohyama derives a set of matching conditions for so-called doubly gauge invariants. However, a gap arises in his final conclusions as the presented set of conditions for doubly gauge invariant quantities for the linearized matching of spacetimes are only shown to be necessary conditions. Analysing sufficiency touches directly on the issue we are trying to emphasize in this paper, so we devote one section to clarify this point, where we show how these conditions are, stricktly speaking, not sufficient. Since the matching conditions in terms of doubly gauge invariants are widely used in the literature, we consider important to close this gap. Moreover, the constructions of gauge invariant quantities using spherical harmonic decompositons leaves out the l = 0 and l = 1 sectors. We will discuss this issue and its consequences.\n\nThe paper is organized as follows. We start by summarising the perturbed matching conditions in Section 2, where we also describe the gauge freedom involved. Then, the procedures used in the classic papers [2, 3] together with the justifications and further developments in [4] are reviewed in Section 3. Section 4 focuses on the consequences of the existence of symmetries in the background configuration, which will have relevance in our final discussion. Section 5 has three subsections. The first one is devoted to present briefly the procedure and results discussed in [6] particularised to the case of spherically symmetric backgrounds. In the second subsection we analyse the sufficiency of the doubly gauge invariant matching conditions in [6] . The last subsection is devoted to the study of the freedom left in the perturbation of the matching hypersurface once the metric perturbations have been fixed at both sides. We finish with an appendix where explicit expressions for the discontinuities of the perturbed second fundamental forms in the spherical case are given. Some of these expressions are used in the main text."
},
{
"section_type": "OTHER",
"section_title": "Linearized matching",
"text": "In this section we describe the gauge freedom involved in the linearised spacetime matching and summarise the perturbed matching conditions."
},
{
"section_type": "OTHER",
"section_title": "Gauge freedom",
"text": "The purpose of the matching theory is to construct a new spacetime out of two spacetimes M ± with boundary by finding a suitable diffeomorphism between the boundaries which allows for their pointwise identification. In particular, the matched spacetime cannot be thought to exist beforehand. Another aspect to bear in mind is that the matching conditions involve exclusively tensors on the identified boundary Σ and hence any coordinate system in M ± is equally valid. This is well-known but it is still source of confusion sometimes.\n\nIn perturbed matching theory, not only the metrics are perturbed but also the matching hypersurfaces may be deformed. Furthermore, as for the metric, the \"deviation\" of the matching hypersurface is also a gauge dependent quantity. This can be best understood by viewing perturbations as ε-derivatives (at ε = 0) of a one-parameter family of spacetimes (M + ε , g + ε ) with boundary Σ + ε . It is convenient to embed M + ε within a larger manifold (without boundary) V + ε to clarify the discussion. A priori, the manifolds (M + ε , g ε ) are completely distinct so it makes no direct sense to talk about ε-derivatives. It is necessary to identify first the different manifolds so that a single point p refers to one point on each of the manifolds. Obviously, there are infinite ways to identify the manifolds, all of them equally valid a priori. This freedom leads to the gauge dependence of the perturbed metric (and of any other geometrically defined tensor). The identification above may, or may not, map the boundaries Σ + ε among themselves. A priori, a point in Σ + 0 may be mapped, for ε = 0, to a point on Σ + ε , to a point interior to M + ε or to a point exterior to M + ε (within the extension V + ε ) which is not part of the manifold. How can we then take derivatives with respect to ε at those later points? Since only derivatives at ε = 0 are needed, restricting to infinitesimal values of ε entails no loss of generality. Then, if for some small ε, a point q ∈ Σ + 0 is mapped to the exterior of M + ε , it follows from differentiability with respect to ε that q is mapped, for the reverse value -ε, to a point interior to M + ε . Thus, perturbations can be defined at the boundary by taking one sided derivatives, i.e. to take limits ε → 0, with a sign restriction on ε (c.f. [7] for an alternative discussion).\n\nHowever, an important issue remains: How do we describe the deformation of the boundary Σ + 0 ? As a set of points each boundary Σ + ε maps, with the above identification, into a hypersurface of the background spacetime, which we call Σ+ ε . In general, this hypersurface will not coincide with Σ + 0 and may well touch it or cross it. This gives us an idea of how the boundary is deformed, but only as a subset, not pointwise. In order to know how the boundary actually moves within the background, we need to prescribe a priori a pointwise identification of Σ + 0 with Σ + ε . This identification is completely different and independent from the one described above involving spacetime points, and involves only the points on the boundaries. As before, there are infinitely many ways to identify the boundaries, and this defines a second gauge freedom, which involves objects intrinsically defined on the boundary. This gauge freedom will be referred as hypersurface gauge, as opposed to the usual spacetime gauge described above.\n\nWith both identifications chosen, the deformation of the boundary within the background can already be described: Fix a point q on the background boundary Σ + 0 . The identification of the boundaries defines a point q ε on Σ + ε , for each ε. The spacetime identification takes this point q ε and maps it into a point qε of the background M + 0 (perhaps after a sign restriction on ε). Obviously qε belongs to the perturbed hypersurface Σ+ ε . We have therefore not only a deformation of the background hypersurface as a set of points, but also pointwise information. It only remains to take the tangent vector of the curve qε at ε = 0, i.e. Z + = dqε dε | ε=0 which encodes completely the deformation of the matching hypersurface as seen from the background spacetime. Two final remarks are in order: (i) Z + is defined exclusively on Σ + 0 , no extension thereof is defined or required and (ii) Z + depends on both the spacetime and hypersurface gauges, since its defining curve is constructed using both identifications. However, decomposing Z + = Q + n 0 + + T + , where n 0 + is the unit normal of Σ + 0 (assumed non-null anywhere) and T + is tangent to it, it turns out that Q + depends on the spacetime gauge but not on the hypersurface gauge. This is because changing the hypersurface gauge reorganizes the points within each Σ+ ε , but cannot modify any of them as a set of points.\n\nTensors defined intrinsically on the boundaries Σ + ε are completely unaffected by the spacetime identification, and are therefore invariant under spacetime gauge transformations. Recall that the matching conditions involve only objects intrinsic to the matching hypersurfaces. Since the perturbed matching conditions are, formally, just their εderivatives, it follows by construction that the perturbed matching conditions must be gauge invariant under spacetime gauge transformations. This may seem surprising at first sight since the matching conditions must involve the perturbed metric, which is obviously gauge dependent. However, the conditions turn out to be gauge independent because they also involve the deformation vector Z + , which is spacetime gauge dependent. This vector is therefore of fundamental importance and must be taken into account in any sensible approach to the problem, as we shall see next."
},
{
"section_type": "OTHER",
"section_title": "Matching conditions",
"text": "Let (M ± 0 , g ± 0 ) be n-dimensional spacetimes with non-null boundaries Σ ± 0 . Matching them requires an identification of the boundaries, i.e. a pair of embeddings Φ\n\n± : Σ 0 -→ M ± 0 with Φ ± (Σ 0 ) = Σ ± 0\n\n, where Σ 0 is an abstract copy of any of the boundaries. Let ξ i (i, j, . . . = 1, . . . , n -1) be a coordinate system on Σ 0 . Tangent vectors to Σ ± 0 are obtained by e ±α i = ∂Φ α ± ∂ξ i (α, β, . . . = 0, . . . , n -1). There are also unique (up to orientation) unit normal vectors n (0) ± α to the boundaries. We choose them so that if n (0) + α points towards M + then n (0) -α points outside of M -or viceversa. The first and second fundamental are simply q (0)\n\nij ± ≡ e ±α i e ±β j g (0) αβ | Σ ± , K (0) ij ± = -n (0) ±α e ±β i ∇ ± β e ±α j .\n\nThe matching conditions (in the absence of shells) require the equality of the first and second fundamental forms on Σ ± 0 , i.e.\n\nq (0) ij + = q (0) ij -, K (0) ij + = K (0) ij -. (1)\n\nUnder a perturbation of the background metric g ± pert = g (0)± + g (1)± and of the matching hypersurfaces via Z ± = Q ± n (0) ± + T ± , the matching conditions will be perturbatively satisfied if and only if [6] q\n\n(1)+ ij = q (1)- ij , K (1)+ ij = K (1)- ij , (2)\n\nwith\n\nq (1) ij ± = L T ± q (0) ij ± + 2Q ± K (0) ij ± + e ±α i e ±β j g (1) αβ ± , (3)\n\nK (1) ij ± = L T ± K (0) ij ± -ǫD i D j Q ± + Q ± (-n (0) ± µ n (0) ± ν R (0)± αµβν e ±α i e ±β j + K (0) il ± K (0) l j ± ) + ǫ 2 g (1) αβ ± n (0) ± α n (0) ± β K (0) ij ± -n (0) ± µ S (1)±µ αβ e ±α i e ±β j , (4)\n\nwhere ǫ = n (0)\n\nα n (0)α , D is the covariant derivative of (Σ, q (0)± ) and S (1)\n\n±α βγ ≡ 1 2 (∇ ± β g (1)±α γ + ∇ ± γ g (1)±α β -∇ ± α g (1)± βγ ). 1\n\nIn these equations, Q ± and T ± are a priori unknown quantities and fulfilling the matching conditions requires showing that two vectors Z ± exist such that (2) are satisfied. The spacetime gauge freedom can be exploited to fix either or both vectors Z ± a priori, but this should be avoided (or at least carefully analysed) if additional spacetime gauge choices are made, in order not to restrict a priori the possible matchings. Regarding the hypersurface gauge, this can be used to fix one of the vectors T + or T -, but not both.\n\nAs already stressed the linearized matching conditions are by construction spacetime gauge invariant (in fact each of the tensors q 2 ) are hypersurface gauge invariant, provided the background is properly matched, since [6] under such a gauge transformation given by the vector ζ on Σ 0 , q ( foot_0 ) ij transforms as q (1) ij + L ζ q (0) ij , and similarly for K (1) ij .\n\n(1) ij ± , K (1) ij ± is). Moreover, the set of conditions ("
},
{
"section_type": "OTHER",
"section_title": "On previous spacetime gauge invariant formalisms",
"text": "The first attempt to derive a general formalism for the matching conditions in linearized gravity is, to our knowledge, due to Gerlach and Sengupta [2] . Their approach is based on the description of the matching hypersurface Σ as a level set of a function f defined on the spacetime. Assuming the level sets {f = const} to be timelike, a field of spacelike unit normals is defined as n µ = (g αβ f ,α f ,β ) -1/2 f ,µ . The unperturbed matching conditions correspond to the continuity everywhere (in particular across Σ) of the tensors\n\nq αβ ≡ g αβ -n α n β , K αβ ≡ q α µ q β ν ∇ µ n ν , (5)\n\nwhich are the spacetime versions of the first and second fundamental forms introduced above. Being f defined everywhere, it makes sense to perturb it in order to describe the variation of the matching hypersurface. Obviously, by perturbing f one also perturbs n µ . The perturbed matching conditions proposed in [2] read\n\nq µ α q ν β △(q αβ ) + = q µ α q ν β △(q αβ ) -, q µ α q ν β △(K αβ ) + = q µ α q ν β △(K αβ ) -, (6)\n\nwhere q β α is the projector onto Σ, △ stands for perturbation and + and -denote the quantities as computed from either side of the matching hypersurface Σ. These expressions involve the projections of the perturbations of q αβ and K αβ onto Σ. The need of considering only the projected components is justified in [2] since the matching conditions need to be intrinsic to the matching hypersurfaces. However, Gerlach and Sengupta themselves note that conditions (6) are not gauge foot_1 invariant.\n\nSince the main interest in [2, 3] refers to spherically symmetric backgrounds, this \"ambiguity\" is fixed in that case by finding suitable gauge invariant combinations of the linearized matching conditions, which turn out to give a correct set of necessary perturbed matching conditions in spherical symmetry. It should be stressed however, that the authors consider these gauge invariant subset to be sufficient also, with no further justification.\n\nWe know from the discussion in Sect. 2.1 above that ( 6 ) cannot be correct as it leads to a set of gauge dependent conditions. Since, on the other hand the proposal (6) may look plausible, it is of interest to point out where, and in which sense, it fails to be correct.\n\nThe first source of problems comes from assuming that the matched spacetime is given beforehand. Indeed, q αβ and K αβ are spacetime tensors and they can only exist (and be continuous) once the matched spacetime is constructed. But this is precisely the purpose of the matching conditions, so the conditions become circular. Another aspect of the same problem is that one can only talk about continuity once the pointwise identification of the boundaries is chosen. But a level set of a function defines only a set of points and not the way those points must be identified. A third instance of the same issue is that tensor components must be expressed in some basis, e.g. a common coordinate system covering both sides of Σ. But again this cannot be assumed a priori. It needs to be constructed.\n\nLet us however mention that once the pointwise identification of the boundaries is chosen, the use of spacetime tensors is allowed provided they are finally projected onto the hypersurface. In that sense, and when properly used, using spacetime indices may simplify some calculations notably (see Carter and Battye, [5] where this notation is used to derive the perturbed matching conditions).\n\nBesides this aspect (which already affects the background matching) the perturbed equations ( 6 ) suffer from one extra problem. The perturbations △(q αβ )(p) and △(K αβ )(p) at a point p in the background can be defined by taking ε-derivatives at fixed p and ε = 0 of the corresponding tensors (defined by g αβ (ε) and f ε ). For each value of ε, the matching conditions impose the continuity of q αβ (ε) and K αβ (ε) everywhere (with the caveat already mentioned regarding the identification of the boundaries). However, continuity of △(q αβ ) and △(K αβ ) at p would only follow if derivatives of continuous functions with respect to an external parameter were necessarily continuous (in our case, the derivative with respect to ε), which is not true in general. A trivial example is given by the function u(ε, x) = |x + ε|, whith x ∈ R. For each ε this function is continuous. However, the derivative with respect to ε does not even exist at x = 0, ε = 0. This reflects the fact that subtracting continuous tensors at a fixed spacetime point p leads to objects that need not be continuous. This is in fact the main problem of (6) as linearized matching conditions.\n\nAn immediate question arises: Why is the gauge invariant subset of matching conditions found in [2, 3] for spherically symmetric backgrounds correct? In order to understand this, let us rewrite (6) using the formalism of section 2.2. First of all, since △(n α n β ) will contain, at least, one free n (0) α , we have\n\nq µ α q ν β △(q αβ ) ± = q µ α q ν β g (1) ± αβ . (7)\n\nMoreover, a simple calculation gives\n\n△(∇ α n β ) = ∇ α (△n β ) -S (1)µ αβ n (0) µ and △(q α β ) = -g (1) αµ n (0) µ n (0) β + g (0)αµ △(n µ )n (0)\n\nβ + n (0)α △(n β ). These, together with standard properties of the projector, lead to\n\nq µ α q ν β △(K αβ ) ± = a (0) ν q µ α △(n α ) + q µ α q ν β ∇ α (△n β ) -q µ α q ν β S (1)ρ αβ n (0) ρ ± , (8)\n\nwhere a\n\n(0) ν ≡ n (0)α ∇ α n (0)\n\nν . In general, these expressions do not agree with (3) and ( 4 ). However, when the gauges are chosen so that Z ± = 0, then △f ≡ 0 on Σ because the matching hypersurface is unperturbed as seen from the background. Consequently\n\n∂ α (△f ) ∝ n (0) α on Σ, which implies △(n α ) Σ = hn (0)\n\nα for some function h. Imposing n(ε) to be unit for all ε fixes h = ǫ 2 g (1) αβ n\n\n(0) α n (0)\n\nβ . Inserting into (8) the matching conditions (6) become\n\nq µ α q ν β g (1)\n\nαβ\n\n+ = q µ α q ν β g (1) αβ -, (9)\n\nǫ 2 g (1) αβ n (0) α n (0) β K µν -q µ α q ν β S (1)ρ αβ n (0) ρ + = ǫ 2 g (1) αβ n (0) α n (0) β K µν -q µ α q ν β S (1)ρ αβ n (0) ρ -,\n\nwhich agree with (2) (with the exception that (9) refers to spacetime tensors and ( 2 ) are defined on Σ). Since Gerlach and Sengupta derive a subset of gauge invariant matching conditions out of (6) in the spherically symmetric case and their conditions are correct in one gauge, it follows that the invariant subset is correct in any gauge. This is the reason why the results in [2, 3] involving spherically symmetric backgrounds turn out to be fine. Substantial progress in the linearized matching problem was made by Martín-García and Gundlach [4] . These authors pointed out the lack of justification in [2, 3] for the choice of (6) as matching conditions. It was also argued that for spacetimes with boundary it only makes sense to define perturbations by using gauges where the perturbed matching hypersurface is mapped onto the background matching hypersurface. Perturbations in this gauge, called \"surface gauge\" (not to be confused with hypersurface gauge) are denoted by △, and its defining property is △f = 0. The idea was to write down the matching conditions in this gauge and then transform into any other gauge if necessary. As noticed by the authors, the surface gauge is not unique since there are still three degrees of freedom left, which correspond to the three directions tangent to Σ.\n\nA relevant observation made in [4] was that the continuity of tensorial perturbations may depend on the index position in the tensors. The authors argue that the tensors truly intrinsic to the hypersurfaces are q αβ , K αβ (with indices upstairs) and propose the following perturbed matching conditions\n\n△(q αβ ) + = △(q αβ ) -, △(K αβ ) + = △(K αβ ) -, (10)\n\nwhich are demonstrated to become exactly (9) . This shows the equivalence of both proposals in the surface gauge, as explicitly stated in [4] . This justifies partially the validity of both approaches in the surface gauge. However, the justification is not complete because of the issue we discuss next. Indeed, conditions (10) still carry one implicit assumption that needs to be clarified. As already stressed the perturbed matching conditions have two inherent and independent degrees of gauge freedom. The approach by Martín-García and Gundlach involves only spacetime objects, and therefore can only notice the spacetime gauge freedom. This leads to an incorrect statement in [4] , as it is not true that the linearized matching conditions read (10) in any surface gauge. Conditions (10) will only be valid when the spacetime gauge maps pairs of background points (identified, via the background matching) to pairs of points on the perturbed boundaries Σ ± ε which are also identified through the matching. Notice that not all surface gauges have this property. In explicit terms, this means that the vectors Z ± must (i) only have tangential components (so that we are in surface gauge) and (ii) have the same components when written in terms of an intrinsic basis of Σ 0 . In less precise, but more intuitive terms, condition (ii) states that Z + and Z -are the same vector, i.e. that the gauges in both regions are chosen such that the displacement of one fixed point of the background hypersurface is identical in both regions (the displaced point, of course, stays on the unperturbed hypersurface, due to the choice of surface gauge). Observe finally that if Q ± = 0 and T + = T -, then the linearized matching conditions (2) truly reduce to conditions (9) , once the latter are projected on Σ. This shows the correctness of the approaches by Gerlach and Sengupta and Martín-García and Gundlach in special gauges."
},
{
"section_type": "OTHER",
"section_title": "Freedom in matching due to symmetries",
"text": "We devote this section to the study of the consequences of the existence of background symmetries on perturbed spacetime matchings.\n\nThe existence of symmetries in the background configuration introduces two issues which are important to take into consideration: the first corresponds to the freedom introduced by the matching procedure, when preserving the symmetries, at the background level [9] , c.f. [10] for an application. The second issue corresponds to the consequences that the symmetries in the background configuration may have on the perturbation of the matching.\n\nIt must be stressed here that the arbitrariness introduced by the presence of symmetries in the background configuration is completely independent from both the hypersurface and spacetime gauge freedoms. However, that arbitrariness is gauge dependent and therefore a gauge choice can be made to remove it. As we will show, an isometry in the background implies that there is a direction along which the difference [ T ] ≡ T + -T -cannot be determined by the perturbed matching equations. But, as we have discussed at the end of section 2, one could eventually choose part of the spacetime gauges (if there is any freedom left) to fix [ T ]. Note, finally, that a change of hypersurface gauge leaves [ T ] invariant."
},
{
"section_type": "OTHER",
"section_title": "Isometries",
"text": "We shall now consider the presence of isometries in the background configuration. So, let us assume that one of the sides, say (M + 0 , g (0)+ ), admits an isometry generated by the Killing vector field ξ + tangent to the boundary Σ + 0 . The commutation of the Lie derivative and the pull-back implies [9]\n\nL ξ + q (0) ij + = e +α i e +β j L ξ + g (0)\n\nαβ\n\n+ | Σ 0 = 0,\n\nwhich means that ξ + is a Killing vector of (Σ 0 , q (0) ij + ). This implies from expression (3) that q (1) ij + is invariant under the transformation\n\nT + → T + + ξ + | Σ 0 .\n\nAs for K (1) ij + , from its expression (4), it is again clear that the previous transformations of T + will leave K (1) ij\n\n+ invariant provided L ξ + K (0) ij + = 0. But this is precisely the case since ξ + is a Killing vector orthogonal to n (0) + , which implies L ξ + n (0) + β | Σ + 0 = 0, and hence L ξ + K (0) ij + = e +α i e +β j L ξ + (∇n (0) + ) αβ | Σ 0 = e +α i e +β j ∇ α L ξ + n (0) + β | Σ 0 = 0.\n\nOf course, all this discussion also applies to the (-) side.\n\nThe combination of the invariance of q (1) ij ± and K (1) ij ± leads to the fact that the first order perturbed matching conditions are invariant under a change of the vectors T ± along the direction of any isometry of the background configuration (preserved by the matching). Then, as expected, when symmetries are present the linearized matching conditions cannot determine the difference [ T ] completely: they leave undetermined the relative (between the two sides) deformation of the hypersurface along the direction of the symmetry. Note that, still, the shape of the perturbed hypersurface is completely determined, since that is driven by\n\nQ ± .\n\nThe overall picture is as follows: at the background level we have the arbitrariness of the identification of Σ + 0 with Σ - 0 [9] , which can be seen as a \"sliding\" between Σ + 0 and Σ - 0 . The perturbation adds to this an arbitrary shift of the deformation of the matching hypersurface at each side along the orbits of the isometry group. As an example, in the description of stationary and axisymmetric compact bodies discussed in [10, 9] , the background sliding corresponds to an arbitrary constant rotation of the interior with respect to the exterior. Note that, in that case, this rotation is only relevant because the exterior is taken to be asymptotically flat. As a result, two identical interiors can, in principle, give rise to two exteriors that differ by a constant rate rotation [10] . The shift of the surface deformation would, in principle, lead to an arbitrary constant rotation along the axial coordinate of the surface deformation of the body. Likewise, two identical perturbations in the interior of the body may produce two different perturbations in the exterior, which may differ by a relative constant rate rotation. A choice of spacetime gauge could be used to relate the deformations inside and outside. However, this may interfere with other gauge fixings that may have been made."
},
{
"section_type": "BACKGROUND",
"section_title": "n-dimensional spherically symmetric backgrounds",
"text": "In this section we shall revisit Mukohyama's theory for linearized matching in the special case of spherical symmetry. Similar results [6] hold for backgrounds admitting isometry groups of dimension (n-1)(n-2)/2 acting on non-null codimension-two orbits of arbitrary topology (strictly speaking the orbits need to be compact)."
},
{
"section_type": "METHOD",
"section_title": "The approach of Mukohyama",
"text": "Concentrating on one of the two spacetimes to be matched, either + or -, we consider a spherically symmetric background metric of the form\n\ng (0) αβ dx α dx β = γ ab dx a dx b + r 2 Ω AB dθ A dθ B , (11)\n\nwhere γ ab (a, b, .. = 0, 1) is a Lorentzian two-dimensional metric (depending only on {x a }), r > 0 is a function of {x a }, and Ω AB dθ A dθ B is the n -2 dimensional unit sphere metric with coordinates {θ A } (A, B, . . . = 2, 3, . . . , n -1).\n\nA general spherically symmetric background hypersurface can be given in parametric form as Σ 0 := {x 0 = Z (0)0 (λ),\n\nx 1 = Z (0)1 (λ), θ A = ϑ A }, (12)\n\nwhere {ξ i } = {λ, ϑ A } is a coordinate system in Σ 0 adapted to the spherical symmetry.\n\nThe tangent vectors to Σ 0 read\n\ne λ = Ż (0) 0 ∂ x 0 + Ż (0) 1 ∂ x 1 Σ 0 , e ϑ A = ∂ θ A | Σ 0 , (13)\n\nwhere dot is derivative w.r.t. λ. With N 2 ≡ -ǫe λ a e λ b γ ab | Σ 0 , so that ǫ = 1 (ǫ = -1) corresponds to a timelike (spacelike) hypersurface, the unit normal to Σ 0 reads\n\nn (0) = √ -det γ N -Ż (0) 1 dx 0 + Ż (0) 0 dx 1 Σ 0 , (14)\n\nwhere the sign choice of N corresponds to the choice of orientation of the normal. The background induced metric and second fundamental form on Σ 0 read\n\nq (0) ij dξ i dξ j = -ǫN 2 dλ 2 + r 2 | Σ 0 Ω AB | Σ 0 dϑ A dϑ B , (15)\n\nK (0) ij dξ i dξ j = N 2 Kdλ 2 + r 2 K| Σ 0 Ω AB | Σ 0 dϑ A dϑ B , (16)\n\nwhere\n\nK ≡ N -2 e λ a e λ b ∇ a n (0)\n\nb , K = n (0)a ∂ x a ln r. It follows that the background matching conditions (1) are\n\nN 2 + = N 2 -, r 2 + | Σ 0 = r 2 -| Σ 0 , K + = K -, K+ = K-. (17)\n\nUsing ( 3 ) and ( 4 ) we could now compute the first order perturbations q (1) ij and K (1) ij in terms of the above quantities and Z (or equivalently Q and T ), c.f. Eqs. ( 45 ) and (46) in [6] . Let us recall (see subsection 2.2) that while the individual tensors q (1) ij and K (1) ij are not hypersurface gauge invariant, their respective differences from the + and -sides (i.e. the linearized matching conditions) are. Those tensors depend of the hypersurface gauge through the tangent vectors T + and T -, which under a gauge change transform simply by adding the gauge vector. It follows that only their difference [ T ] can appear in the linearized matching conditions. Consequently there are three degrees of freedom that cannot be fixed by the equations, but can be fixed by choosing the hypersurface gauge, for instance to set T + . Thus, the linearized matching conditions can be looked at as equations for the difference [ T ] as well as for Q + and Q -, i.e. for five objects. If these equations admit solutions, then the linearized matching is possible and it is impossible otherwise.\n\nMukohyama emphasizes the convenience to look for doubly gauge invariant quantities to write down the linearized matching conditions, however the matching conditions are already gauge invariant (both for the spacetime and hypersurfaces gauges). Looking for gauge invariant combinations on each side amounts to writing equations where the difference vector [ T ] simply drops. Indeed, in many cases, knowing the value of such vector in a specific matching is not interesting. In that sense, using doubly gauge invariant quantities is useful as it lowers the number of equations to analyse. However, we want to stress that this is not related to obtaining gauge invariant linearized matching equations.\n\nIt is just related to not solving for superfluous information. In fact, a set of equations where also Q + and Q -have disappeared would be even more convenient from this point of view, provided one is not interested in knowing how the hypersurfaces are deformed in the specific spacetime gauge being used.\n\nSince the use of doubly gauge invariant matching conditions is used extensively, let us recall its main ingredients in order to discuss if they really are equivalent to the full set of linearized matching equations and in which sense.\n\nTo that aim Mukohyama [6] , decomposes the perturbation tensors q (1) ij and K (1) ij in terms of scalar Y , vector V A and tensor harmonics T AB on the sphere, as foot_2\n\nq (1) ij dξ i dξ j = ∞ l=0 (σ 00 Y dλ 2 + σ (Y ) T (Y )AB dϑ A dϑ B ) + ∞ l=1 2(σ (T )0 V (T )A + σ (L)0 V (L)A )dλdϑ A + ∞ l=2 (σ (T ) T (T )AB + σ (LT ) T (LT )AB + σ (LL) T (LL)AB )dϑ A dϑ B , (18)\n\nK (1) ij dξ i dξ j = ∞ l=0 (κ 00 Y dλ 2 + κ (Y ) T (Y )AB dϑ A dϑ B ) + ∞ l=1 2(κ (T )0 V (T )A + κ (L)0 V (L)A )dλdϑ A + ∞ l=2 (κ (T ) T (T )AB + κ (LT ) T (LT )AB + κ (LL) T (LL)AB )dϑ A dϑ B , (19)\n\nwhere all the scalar coefficients depend only on λ. Each coefficient in the decomposition has indices l and m which have been dropped for notational simplicity. Notice that each coefficient σ and κ is defined in the range of l's appearing in the corresponding summatory. By construction, each of the σ and κ are spacetime-gauge invariant (but not hypersurface-gauge invariant). For l ≥ 2 they can even be written down [6] explicity in terms of spacetime-gauge invariant quantities. In a similar way, the doubly gaugeinvariant quantities presented in [6] , are only defined for l ≥ 2 (except k (T )0 , which is also defined for l = 1), and read l ≥ 2 :\n\nf 00 ≡ σ 00 -2N∂ λ N -1 χ , l ≥ 2 : f ≡ σ (Y ) + ǫN -2 χ∂ λ r 2 | Σ 0 + 2 n -2 k 2 l σ (LL) , l ≥ 2 : f 0 ≡ σ (T )0 -r 2 | Σ 0 ∂ λ r -2 | Σ 0 σ (LT ) , l ≥ 2 : f (T ) ≡ σ (T ) , l ≥ 2 : k 00 ≡ κ 00 + ǫKσ 00 + ǫχ∂ λ K, l ≥ 1 : k (T )0 ≡ κ (T )0 -Kσ (T )0 , (20)\n\nl ≥ 2 : k (L)0 ≡ κ (L)0 + 1 2 (ǫK -K)σ (L)0 + 1 2 (ǫK + K) χ -r 2 | Σ 0 ∂ λ (r -2 | Σ 0 σ (LL) ) , l ≥ 2 : k (LT ) ≡ κ (LT ) -Kσ (LT ) , l ≥ 2 : k (LL) ≡ κ (LL) -Kσ (LL) , l ≥ 2 : k (Y ) ≡ κ (Y ) -Kσ (Y ) + ǫN -2 r 2 | Σ 0 χ∂ λ K, l ≥ 2 : k (T ) ≡ κ (T ) -Kσ (T ) ,\n\nwhere k 2 l = l(l + n -3) and\n\nl ≥ 2 : χ ≡ σ (L)0 -r 2 | Σ 0 ∂ λ (r -2 | Σ 0 σ (LL) ).\n\nThe orthogonality properties of the scalar, vector and tensor harmonics imply that the equalities of the coefficients σ and κ for each l and m is equivalent to the equality of the perturbation tensors (18) and ( 19 ) at both sides of Σ 0 . Thus, recalling the notation\n\n[f ] ≡ f + | Σ 0 -f -| Σ 0 , the equations l ≥ 0 : [σ 00 ] = [σ (Y ) ] = 0 l ≥ 1 : [σ (L)0 ] = [σ (T )0 ] = 0 l ≥ 2 : [σ (T ) ] = [σ (LT ) ] = [σ (LL) ] = 0 (21) l ≥ 0 : [κ 00 ] = [κ (Y ) ] = 0 l ≥ 1 : [κ (L)0 ] = [κ (T )0 ] = 0 l ≥ 2 : [κ (T ) ] = [κ (LT ) ] = [κ (LL) ] = 0 (22)\n\nare equivalent to (2) and therefore correspond exactly to the linearized matching conditions in this setting. Notice that each of the equalities in ( 21 ) and ( 22 ) is in fact one equation for each l and m in the appropriate range. We will however refer to them simply as equations.\n\nThe full linearized matching conditions obviously imply the following equalities in terms of the doubly-gauge invariant quantities (20),\n\nl ≥ 2 : [f 00 ] = [f ] = [f 0 ] = [f (T ) ] = 0 (23) l ≥ 1 : [k (T )0 ] = 0 l ≥ 2 : [k 00 ] = [k (Y ) ] = [k (L)0 ] = [k (LL) ] = [k (LT ) ] = [k (T ) ] = 0. ( 24\n\n)\n\nWhether these equations can be regarded as the full set of linearized matching conditions or not requires studying their sufficiency, i.e. whether they imply (21)-( 22 ) or not. This point was not mentioned in [6] and in fact the answer turns out to be negative, although in a mild way, as we discuss in the next subsection."
},
{
"section_type": "OTHER",
"section_title": "On the sufficiency of the continuity of the doubly-gauge invariants",
"text": "Let us recall that fulfilling the matching conditions requires finding two Z ± such that (21)-( 22 ) are satisfied. The key issue for the matching is therefore to show existence of deformation vectors Z ± so that all the equations hold. A plausibility argument in favour of the sufficiency of (23)-(24) comes from simple equation counting. Indeed, as already discussed, the linearized matching conditions are spacetime and hypersurface gauge invariant and therefore can only involve the difference vector [ T ], i.e. three quantities. Since constructing double gauge invariant quantities on each side eliminates this vector, the number of equations should be reduced exactly by three if they are to remain equivalent to the original set. This is precisely what happens as we go from the original forteen equations in (21)-( 22 ) down to eleven equations in (23)-( 24 ). This argument however is not conclusive, both because it is not rigorous and because each equation in those expressions is, in fact, a set of equations depending on l and m, and the range of l's changes with the equations. Let us therefore analyse this issue and [z (T ) ], [z (L) ] for l ≥ 1. Now, the explicit expressions (27), ( 29 ), (28) show that (30) determine uniquely [z λ ], [z (T ) ] and [z (L) ] for l ≥ 2. So, restricted to the sector l ≥ 2 Mukoyama's doubly gauge invariant matching conditions can be regarded as equivalent to the full set of matching conditions. Taking all l's into account, however, the equations turn out not to be sufficient. To show this, it is enough to display one equation involving the discontinuity of the background metric perturbations and [Q] (but not [ T ]) which holds as a consequence of the full set of matching conditions (21)-( 22 ) but not as a consequence of ( 23 )-(24). Using the fact that each l = 1 expression refers to n -1 objects (one for each m), the number of equations in (31)-( 32 ) is 7n -3, while the number of unkowns in [ T ] not yet determined by (30), i.e. [z λ ] for l = 0, 1 and [z (T ) ], [z (L) ] for l = 1 is 3n -2, which is smaller. It is to be expected, therefore, that (31), (32) imply conditions where these variables do not appear. This can be made explicit, for instance, by combining [σ 00 ] l=0 = 0 with [σ (Y ) ] l=0 = 0 which yields\n\nl = 0 : [h λλ ] + 2[Q]N 2 K + 2N∂ λ ǫN ∂ λ (r 2 | Σ 0 ) [h (Y ) ] + 2[Q]r 2 | Σ 0 K = 0, whenever ∂ λ (r 2 | Σ 0 ) = 0. (If ∂ λ (r 2 | Σ 0 ) = 0 it is enough to consider [σ (Y ) ] l=0 = 0.\n\n) This relation is enough to show that the continuity of the doubly-gauge invariant variables of Mukohyama is not sufficient to ensure the existence of the perturbed matching. Of course, this does not invalidate Mukohyama's approach in any way, which remains interesting and useful. It only means that, when using this approach to solve linearized matchings, one still needs to look more carefully into the l = 0 and l = 1 sector to make sure that the remaining equations ( 31 ) and (32) hold. On the other hand, equations (31), (32) do not completely determine [ T ]. The variable [z (T ) ] l=1 only appears in [σ (T )0 ] l=1 = 0, in the term ∂ λ (r -2 | Σ 0 [z (T ) ]). As a result, the matching conditions do not fix [z (T ) ] l=1 completely, but up to a constant factor times r 2 | Σ 0 (for each m). Recalling that V (T )A dϑ A for l = 1 correspond to the three Killing vectors on the sphere, this arbitrary constant (for each m) accounts for the addition to [ T ] of an arbitrary Killing vector of the sphere. This is in accordance with the discussion in Section 4. We devote the following subsection to complete the study of the freedom left in the matching."
},
{
"section_type": "OTHER",
"section_title": "Freedom in the matching",
"text": "As already emphasized, solving the linearized matching amounts to finding perturbation vectors Z + and Z -. Assume now that a linearized matching between two given backgrounds and perturbations has been done. It is natural to ask what is the most general matching between those two spaces, i.e. what is the most general solution for Z + and Z - of the matching conditions. Geometrically, this means finding all the possible deformations of the matching hypersurface Σ 0 which allow the two spaces to be matched.\n\nSince this problem is of interest not only when the full matching conditions are imposed but also in situations where layers of matter are present (e.g. in brane-world or shell cosmologies) so that jumps in the second fundamental forms are allowed, we will analyse this issue in two steps. First, we will study the equations involving the perturbed first fundamental forms and will determine the freedom they admit. On a second step we will write down the extra conditions coming from the equality of the second fundamental forms.\n\nto adding Killing vectors on the sphere, something already discussed in Section 4. The rest of terms involve combinations (with functions) of the conformal Killing vectors on the sphere and tangential vectors along λ. Notice that the coefficients of the conformal Killing (i.e. < z (L) > l=1,m ) determine all the rest of the l = 1 coefficients. In particular when < z (L) > l=1,m vanishes, then all the l = 1 terms vanish and the freedom becomes radially symmetric.\n\nWe now add to the analysis the difference of the equations in (22). Due to the fact that all coefficients in < Z > vanish for l ≥ 2 we only need to consider the equations for l = 0, 1, i.e. (32). We refer the reader to Appendix A for the explicit expressions of (32) in terms of the metric perturbations and Z. For the sake of completeness we also include all the explicit expressions of (22) in Appendix A. The difference of equations (32), see (44)-(46), whenever < g (1) >= 0 read l = 0, 1 :\n\n< κ 00 >= 0 ⇔ (41)\n\n-< QR (γ) dbac > n (0)d n (0)a e λ b e λ c -ǫ∂ 2 λ < Q > + ǫ 2N 2 ∂ λ N 2 ∂ λ < Q > -ǫ < Q > K 2 N 2 -ǫKN 2 ∂ λ (N -2 < z λ >) -ǫ∂ λ (K < z λ >) = 0, l = 1 : < κ (L)0 >= 0 ⇔ (42) -ǫ∂ λ < Q > +ǫK < z λ > +ǫ < Q > ∂ λ ln(r| Σ 0 ) + r 2 | Σ 0 K∂ λ (r -2 | Σ 0 < z (L) >) = 0 l = 0, 1 : < κ (Y ) >= 0 ⇔ (43) + 1 2 N -2 ∂ λ (r 2 | Σ 0 ) (∂ λ < Q > +K < z λ >) + 1 2 < Qn (0)a n (0)b ∇ a ∇ b r 2 > - ǫ 2 N -2 e λ a < z λ n (0)b ∇ b ∇ a r 2 > + l(l + n -3) n -2 ǫ < Q > -2 K < z (L) > = 0.\n\nIt can be checked that in general these equations overdetermine the previous equations, i.e. (39) and (40), although there may be particular cases for which they are compatible. Therefore, generically, they will imply that < z (L) > l=1,m = 0 and < z λ > l=0 = 0, and thence all the rest of the variables vanish, < z λ > l=1,m =< Q > l=1,m = 0, < z λ > l=0 =< Q > l=0 = 0, so that the only freedom left is given by\n\n[ Z] ′ -[ Z] = a m V m (T ) .\n\nFinding in which particular cases equations (39)-(43) are compatible is straightforward but tedious and will not be carried out explicitly here."
}
] | |
arxiv:0704.0081 | 0704.0081 | 1 | 778900414d1a1f08fb8e88b60222a7cb368a7f0c24b131c30158994a60c01f60 | Quantum Deformations of Relativistic Symmetries | We discussed quantum deformations of D=4 Lorentz and Poincare algebras. In
the case of Poincare algebra it is shown that almost all classical r-matrices
of S. Zakrzewski classification correspond to twisted deformations of Abelian
and Jordanian types. A part of twists corresponding to the r-matrices of
Zakrzewski classification are given in explicit form. | [
"V.N. Tolstoy (INP",
"Moscow State University)"
] | [
"math.QA",
"hep-th",
"math-ph",
"math.MP",
"math.RT"
] | math.QA | [] | 2007-04-02 | 2026-02-26 | We discussed quantum deformations of D = 4 Lorentz and Poincaré algebras. In the case of Poincaré algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski classification are given in explicit form.
The quantum deformations of relativistic symmetries are described by Hopf-algebraic deformations of Lorentz and Poincaré algebras. Such quantum deformations are classified by Lorentz and Poincaré Poisson structures. These Poisson structures given by classical r-matrices were classified already some time ago by S. Zakrzewski in [1] for the Lorentz algebra and in [2] for the Poincaré algebra. In the case of the Lorentz algebra a complete list of classical r-matrices involves the four independent formulas and the corresponding quantum deformations in different forms were already discussed in literature (see [3, 4, 5, 6, 7] ). In the case of Poincaré algebra the total list of the classical r-matrices, which satisfy the homogeneous classical Yang-Baxter equation, consists of 20 cases which have various numbers of free parameters. Analysis of these twenty solutions shows that each of them can be presented as a sum of subordinated r-matrices which almost all are of Abelian and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski classification are given in explicit form.
Let r be a classical r-matrix of a Lie algebra g, i.e. r ∈ where Ω is g-invariant element, Ω ∈ ( 3 ∧ g) g . We consider two types of the classical r-matrices and corresponding twists.
Let the classical r-matrix r = r A has the form
r A = n i=1 x i ∧ y i , (2.2)
where all elements x i , y i (i = 1, . . . , n) commute among themselves. Such an r-matrix is called of Abelian type. The corresponding twist is given as follows
F r A = exp r A 2 = exp 1 2 n i=1 x i ∧ y i . (2.3)
This twisting two-tensor F := F r A satisfies the cocycle equation
F 12 (∆ ⊗ id)(F ) = F 23 (id ⊗ ∆)(F ) , (2.4)
and the "unital" normalization condition
(ǫ ⊗ id)(F ) = (id ⊗ ǫ)(F ) = 1 . (2.5)
The twisting element F defines a deformation of the universal enveloping algebra U(g) considered as a Hopf algebra. The new deformed coproduct and antipode are given as follows
∆ (F ) (a) = F ∆(a)F -1 , S (F ) (a) = uS(a)u -1 (2.6)
for any a ∈ U(g), where ∆(a) is a co-product before twisting, and
u = i f (1)
i S(f
(2) i ) if F = i f (1) i ⊗ f (2) i .
Let the classical r-matrix r = r J (ξ) has the form foot_0
r J (ξ) = ξ n ν=0 x ν ∧ y ν , (2.7)
where the elements x ν , y ν (ν = 0, 1, . . . , n) satisfy the relations foot_1
[x 0 , y 0 ] = y 0 , [x 0 , x i ] = (1 -t i )x i , [x 0 , y i ] = t i y i , [x i , y j ] = δ ij y 0 , [x i , x j ] = [y i , y j ] = 0 , [y 0 , x j ] = [y 0 , y j ] = 0 , (2.8)
(i, j = 1, . . . , n), (t i ∈ C). Such an r-matrix is called of Jordanian type. The corresponding twist is given as follows [8, 9 ]
F r J = exp ξ n i=1 x i ⊗ y i e -2t i σ exp(2x 0 ⊗ σ) , (2.9)
where σ := 1 2 ln(1 + ξy 0 ). foot_2 Let r be an arbitrary r-matrix of g. We denote a support of r by Sup(r) 4 . The following definition is useful. Definition 2.1 Let r 1 and r 2 be two arbitrary classical r-matrices. We say that r 2 is subordinated to
r 1 , r 1 ≻ r 2 , if δ r 1 (Sup(r 2 )) = 0, i.e. δ r 1 (x) := [x ⊗ 1 + 1 ⊗ x, r 1 ] = 0 , ∀x ∈ Sup(r 2 ) .
(2.10)
If r 1 ≻ r 2 then r = r 1 + r 2 is also a classical r-matrix (see [15] ). The subordination enables us to construct a correct sequence of quantizations. For instance, if the r-matrix of Jordanian type (2.7) is subordinated to the r-matrix of Abelian type (2.2), r A ≻ r J , then the total twist corresponding to the resulting r-matrix r = r A + r J is given as follows
F r = F r J F r A . (2.11)
The further definition is also useful.
Definition 2.2 A twisting two-tensor F r (ξ) of a Hopf algebra, satisfying the conditions (2.4) and (2.5), is called locally r-symmetric if the expansion of F r (ξ) in powers of the parameter deformation ξ has the form
F r (ξ) = 1 + c r + O(ξ 2 ) . . . (2.12)
where r is a classical r-matrix, and c is a numerical coefficient, c = 0.
It is evident that the Abelian twist (2.3) is globally r-symmetric and the twist of Jordanian type (2.9) does not satisfy the relation (2.12), i.e. it is not locally r-symmetric.
The results of this section in different forms were already discussed in literature (see [3, 4, 5, 6, 7] ).
The classical canonical basis of the D = 4 Lorentz algebra, o(3, 1), can be described by anti-Hermitian six generators (h, e ± , h ′ , e ′ ± ) satisfying the following non-vanishing commutation relations foot_4 :
[h, e ± ] = ±e ± , [e + , e -] = 2h , (3.1) [h, e ′ ± ] = ±e ′ ± , [h ′ , e ± ] = ±e ′ ± , [e ± , e ′ ∓ ] = ±2h ′ , (3.2) [h ′ , e ′ ± ] = ∓e ± , [e ′ + , e ′ -] = -2h , (3.3)
and moreover
x * = -x (∀ x ∈ o(3, 1)) . (3.4)
A complete list of classical r-matrices which describe all Poisson structures and generate quantum deformations for o(3, 1) involve the four independent formulas [1] :
r 1 = α e + ∧ h , (3.5
)
r 2 = α (e + ∧ h -e ′ + ∧ h ′ ) + 2β e ′ + ∧ e + , (3.6
)
r 3 = α (e ′ + ∧ e -+ e + ∧ e ′ -) + β (e + ∧ e --e ′ + ∧ e ′ -) -2γ h ∧ h ′ , (3.7
) The first two r-matrices (3.5) and (3.6) satisfy the homogeneous CYBE and they are of Jordanian type. If we assume (3.10), the corresponding quantum deformations were described detailed in the paper [6] and they are entire defined by the twist of Jordanian type:
r 4 = α e ′ + ∧ e -+ e + ∧ e ′ --2h ∧ h ′ ± e + ∧ e ′ + . (3
F r 1 = exp (h ⊗ σ) , σ = 1 2 ln(1 + αe + ) (3.12)
for the r-matrix (3.5), and
F r 2 = exp ıβ α 2 σ ∧ ϕ exp (h ⊗ σ -h ′ ⊗ ϕ) , (3.13)
σ = 1 2 ln (1 + αe + ) 2 + (αe ′ + ) 2 , ϕ = arctan αe ′ + 1 + αe + (3.14)
for the r-matrix (3.6). It should be recalled that the twists (3.12) and (3.13) are not locally r-symmetric. A locally r-symmetric twist for the r-matrix (3.5) was obtained in [14] and it has the following complicated formula:
F ′ r 1 = exp 1 2 ∆(h) - 1 2 h sinh αe + αe + ⊗ e -αe + + e αe + ⊗ h sinh αe + αe + α∆(e + ) sinh α∆(e + ) , (3.15)
where ∆ is a primitive coproduct. The last two r-matrices (3.7) and (3.8) satisfy the non-homogeneous (modified) CYBE and they can be easily obtained from the solutions of the complex algebra o(4, C) ≃ sl(2, C) ⊕ sl(2, C) which describes the complexification of o(3, 1). Indeed, let us introduce the complex basis of Lorentz algebra (o(3, 1) ≃ sl(2; C) ⊕ sl(2, C)) described by two commuting sets of complex generators:
H 1 = 1 2 (h + ıh ′ ) , E 1± = 1 2 (e ± + ıe ′ ± ) , (3.16
)
H 2 = 1 2 (h -ıh ′ ) , E 2± = 1 2 (e ± -ıe ′ ± ) , (3.17)
which satisfy the relations (compare with (3.1))
[H k , E k± ] = ±E k± , [E k+ , E k-] = 2H k (k = 1, 2) . (3.18)
The * -operation describing the real structure acts on the generators H k , and
E k± (k = 1, 2)
as follows
H * 1 = -H 2 , E * 1± = -E 2± , H * 2 = -H 1 , E * 2± = -E 1± . (3.19)
The classical r-matrix r 3 , (3.7), and r 4 , (3.8), in terms of the complex basis (3.16), (3.17) take the form
r 3 = r ′ 1 + r ′′ 1 , r ′ 3 := 2(β + ıα)E 1+ ∧ E 1-+ 2(β -ıα)E 2+ ∧ E 2-, r ′′ 3 := 4ıγ H 2 ∧ H 1 , (3.20)
and
r 4 = r ′ 4 + r ′′ 4 , r ′ 4 := 2ıα(E 1+ ∧ E 1--E 2+ ∧ E 2--2H 1 ∧ H 2 ) , r ′′ 4 := 4ıλ E 1+ ∧ E 2+ (3.21)
For the sake of convenience we introduce parameter foot_5 λ in r ′′ 4 . It should be noted that r ′ 3 , r ′′ 3 and r ′ 4 , r ′′ 4 are themselves classical r-matrices. We see that the r-matrix r ′ 3 is simply a sum of two standard r-matrices of sl(2; C), satisfying the anti-Hermitian condition r * = -r. Analogously, it is not hard to see that the r-matrix r 4 corresponds to a Belavin-Drinfeld triple [15] for the Lie algebra sl(2; C) ⊕ sl(2, C)). Indeed, applying the Cartan automorphism E 2± → E 2∓ , H 2 → -H 2 we see that this is really correct (see also [16] ).
We firstly describe quantum deformation corresponding to the classical r-matrix r 3 (3.20). Since the r-matrix r ′′ 3 is Abelian and it is subordinated to r ′ 3 therefore the algebra o(3, 1) is firstly quantized in the direction r ′ 3 and then an Abelian twist corresponding to the r-matrix r ′′ 3 is applied. We introduce the complex notations z ± := β ± ıα. It should be noted that z -= z * + if the parameters α and β are real, and z -= -z * + if the parameters α and β are pure imaginary. From structure of the classical r-matrix r ′ 3 it follows that a quantum deformation U r ′ 1 (o(3, 1)) is a combination of two q-analogs of U(sl(2; C)) with the parameter q z + and q z -, where q z ± := exp z ± . Thus U r ′ 3 (o(3, 1)) ∼ = U q z + (sl(2; C))⊗U q z -(sl(2; C)) and the standard generators q ±H 1 z + , E 1± and q ±H 2 z -, E 2± satisfy
The physical generators of the Lorentz algebra, M i , N i (i = 1, 2, 3), are related with the canonical basis h, h ′ , e ± , e ′ ± as follows
h = ıN 3 , e ± = ı(N 1 ± M 2 ), (4.4
)
h ′ = -ıM 3 , e ′ ± = ı(±N 2 -M 1 ). (4.5)
The subalgebra generated by the four-momenta P j , P 0 (j = 1, 2, 3) will be denoted by P and we also set P ± := P 0 ± P 3 . S. Zakrzewski has shown in [2] that each classical r-matrix, r ∈ P(3, Here [[•, •]] means the Schouten bracket. Moreover a total list of the classical r-matrices for the case c = 0 and also for the case c = 0, t = 0 was found. 7 It was shown that there are fifteen solutions for the case c = 0, t = 0, and six solutions for the case c = 0 where there is only one solution for t = 0. Thus Zakrzewski found twenty r-matrices which satisfy the homogeneous classical Yang-Baxter equation (t = 0 in (4.9)). Analysis of these twenty solutions shows that each of them can be presented as a sum of subordinated rmatrices which almost all are of Abelian and Jordanian types. Therefore these r-matrices correspond to twisted deformations of the Poincaré algebra P(3, 1). We present here r-matrices only for the case c = 0, t = 0:
r 1 = γh ′ ∧
h + α(P + ∧ P --P 1 ∧ P 2 ) , (4.11) r 2 = γe ′ + ∧ e + + β 1 (e + ∧ P 1e ′ + ∧ P 2 + h ∧ P + ) + β 2 h ′ ∧ P + , (4.12) r 3 = γe ′ + ∧ e + + β(e + ∧ P 1e ′ + ∧ P 2 + h ∧ P + ) + αP 1 ∧ P + , (4.13) r 4 = γ(e ′ + ∧ e + + e + ∧ P 1 + e ′ + ∧ P 2 -P 1 ∧ P 2 ) + P + ∧ (α 1 P 1 + α 2 P 2 ) , (4.14) r 5 = γ 1 (h ∧ e +h ′ ∧ e ′ + ) + γ 2 e + ∧ e ′ + . (4.15)
The first r-matrix r 1 is a sum of two subordinated Abelian r-matrices r 1 := r ′ 1 + r ′′ 1 , r ′ 1 ≻ r ′′ 1 , r ′ 1 = α(P + ∧ P --P 1 ∧ P 2 ) , r ′′ 1 := γh ′ ∧ h .
(4.16)
Therefore the total twist defining quantization in the direction to this r-matrix is the ordered product of two the Abelian twits F r 1 = F r ′′ 1 F r ′ 1 = exp γh ′ ∧ h exp α(P + ∧ P --P 1 ∧ P 2 ) . (4.17)
7 Classification of the r-matrices for the case c = 0, t = 0 is an open problem up to now. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We discussed quantum deformations of D = 4 Lorentz and Poincaré algebras. In the case of Poincaré algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski classification are given in explicit form."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The quantum deformations of relativistic symmetries are described by Hopf-algebraic deformations of Lorentz and Poincaré algebras. Such quantum deformations are classified by Lorentz and Poincaré Poisson structures. These Poisson structures given by classical r-matrices were classified already some time ago by S. Zakrzewski in [1] for the Lorentz algebra and in [2] for the Poincaré algebra. In the case of the Lorentz algebra a complete list of classical r-matrices involves the four independent formulas and the corresponding quantum deformations in different forms were already discussed in literature (see [3, 4, 5, 6, 7] ). In the case of Poincaré algebra the total list of the classical r-matrices, which satisfy the homogeneous classical Yang-Baxter equation, consists of 20 cases which have various numbers of free parameters. Analysis of these twenty solutions shows that each of them can be presented as a sum of subordinated r-matrices which almost all are of Abelian and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski classification are given in explicit form."
},
{
"section_type": "OTHER",
"section_title": "Preliminaries",
"text": "Let r be a classical r-matrix of a Lie algebra g, i.e. r ∈ where Ω is g-invariant element, Ω ∈ ( 3 ∧ g) g . We consider two types of the classical r-matrices and corresponding twists.\n\nLet the classical r-matrix r = r A has the form\n\nr A = n i=1 x i ∧ y i , (2.2)\n\nwhere all elements x i , y i (i = 1, . . . , n) commute among themselves. Such an r-matrix is called of Abelian type. The corresponding twist is given as follows\n\nF r A = exp r A 2 = exp 1 2 n i=1 x i ∧ y i . (2.3)\n\nThis twisting two-tensor F := F r A satisfies the cocycle equation\n\nF 12 (∆ ⊗ id)(F ) = F 23 (id ⊗ ∆)(F ) , (2.4)\n\nand the \"unital\" normalization condition\n\n(ǫ ⊗ id)(F ) = (id ⊗ ǫ)(F ) = 1 . (2.5)\n\nThe twisting element F defines a deformation of the universal enveloping algebra U(g) considered as a Hopf algebra. The new deformed coproduct and antipode are given as follows\n\n∆ (F ) (a) = F ∆(a)F -1 , S (F ) (a) = uS(a)u -1 (2.6)\n\nfor any a ∈ U(g), where ∆(a) is a co-product before twisting, and\n\nu = i f (1)\n\ni S(f\n\n(2) i ) if F = i f (1) i ⊗ f (2) i .\n\nLet the classical r-matrix r = r J (ξ) has the form foot_0\n\nr J (ξ) = ξ n ν=0 x ν ∧ y ν , (2.7)\n\nwhere the elements x ν , y ν (ν = 0, 1, . . . , n) satisfy the relations foot_1\n\n[x 0 , y 0 ] = y 0 , [x 0 , x i ] = (1 -t i )x i , [x 0 , y i ] = t i y i , [x i , y j ] = δ ij y 0 , [x i , x j ] = [y i , y j ] = 0 , [y 0 , x j ] = [y 0 , y j ] = 0 , (2.8)\n\n(i, j = 1, . . . , n), (t i ∈ C). Such an r-matrix is called of Jordanian type. The corresponding twist is given as follows [8, 9 ]\n\nF r J = exp ξ n i=1 x i ⊗ y i e -2t i σ exp(2x 0 ⊗ σ) , (2.9)\n\nwhere σ := 1 2 ln(1 + ξy 0 ). foot_2 Let r be an arbitrary r-matrix of g. We denote a support of r by Sup(r) 4 . The following definition is useful. Definition 2.1 Let r 1 and r 2 be two arbitrary classical r-matrices. We say that r 2 is subordinated to\n\nr 1 , r 1 ≻ r 2 , if δ r 1 (Sup(r 2 )) = 0, i.e. δ r 1 (x) := [x ⊗ 1 + 1 ⊗ x, r 1 ] = 0 , ∀x ∈ Sup(r 2 ) .\n\n(2.10)\n\nIf r 1 ≻ r 2 then r = r 1 + r 2 is also a classical r-matrix (see [15] ). The subordination enables us to construct a correct sequence of quantizations. For instance, if the r-matrix of Jordanian type (2.7) is subordinated to the r-matrix of Abelian type (2.2), r A ≻ r J , then the total twist corresponding to the resulting r-matrix r = r A + r J is given as follows\n\nF r = F r J F r A . (2.11)\n\nThe further definition is also useful.\n\nDefinition 2.2 A twisting two-tensor F r (ξ) of a Hopf algebra, satisfying the conditions (2.4) and (2.5), is called locally r-symmetric if the expansion of F r (ξ) in powers of the parameter deformation ξ has the form\n\nF r (ξ) = 1 + c r + O(ξ 2 ) . . . (2.12)\n\nwhere r is a classical r-matrix, and c is a numerical coefficient, c = 0.\n\nIt is evident that the Abelian twist (2.3) is globally r-symmetric and the twist of Jordanian type (2.9) does not satisfy the relation (2.12), i.e. it is not locally r-symmetric."
},
{
"section_type": "OTHER",
"section_title": "Quantum deformations of Lorentz algebra",
"text": "The results of this section in different forms were already discussed in literature (see [3, 4, 5, 6, 7] ).\n\nThe classical canonical basis of the D = 4 Lorentz algebra, o(3, 1), can be described by anti-Hermitian six generators (h, e ± , h ′ , e ′ ± ) satisfying the following non-vanishing commutation relations foot_4 :\n\n[h, e ± ] = ±e ± , [e + , e -] = 2h , (3.1) [h, e ′ ± ] = ±e ′ ± , [h ′ , e ± ] = ±e ′ ± , [e ± , e ′ ∓ ] = ±2h ′ , (3.2) [h ′ , e ′ ± ] = ∓e ± , [e ′ + , e ′ -] = -2h , (3.3)\n\nand moreover\n\nx * = -x (∀ x ∈ o(3, 1)) . (3.4)\n\nA complete list of classical r-matrices which describe all Poisson structures and generate quantum deformations for o(3, 1) involve the four independent formulas [1] :\n\nr 1 = α e + ∧ h , (3.5\n\n)\n\nr 2 = α (e + ∧ h -e ′ + ∧ h ′ ) + 2β e ′ + ∧ e + , (3.6\n\n)\n\nr 3 = α (e ′ + ∧ e -+ e + ∧ e ′ -) + β (e + ∧ e --e ′ + ∧ e ′ -) -2γ h ∧ h ′ , (3.7\n\n) The first two r-matrices (3.5) and (3.6) satisfy the homogeneous CYBE and they are of Jordanian type. If we assume (3.10), the corresponding quantum deformations were described detailed in the paper [6] and they are entire defined by the twist of Jordanian type:\n\nr 4 = α e ′ + ∧ e -+ e + ∧ e ′ --2h ∧ h ′ ± e + ∧ e ′ + . (3\n\nF r 1 = exp (h ⊗ σ) , σ = 1 2 ln(1 + αe + ) (3.12)\n\nfor the r-matrix (3.5), and\n\nF r 2 = exp ıβ α 2 σ ∧ ϕ exp (h ⊗ σ -h ′ ⊗ ϕ) , (3.13)\n\nσ = 1 2 ln (1 + αe + ) 2 + (αe ′ + ) 2 , ϕ = arctan αe ′ + 1 + αe + (3.14)\n\nfor the r-matrix (3.6). It should be recalled that the twists (3.12) and (3.13) are not locally r-symmetric. A locally r-symmetric twist for the r-matrix (3.5) was obtained in [14] and it has the following complicated formula:\n\nF ′ r 1 = exp 1 2 ∆(h) - 1 2 h sinh αe + αe + ⊗ e -αe + + e αe + ⊗ h sinh αe + αe + α∆(e + ) sinh α∆(e + ) , (3.15)\n\nwhere ∆ is a primitive coproduct. The last two r-matrices (3.7) and (3.8) satisfy the non-homogeneous (modified) CYBE and they can be easily obtained from the solutions of the complex algebra o(4, C) ≃ sl(2, C) ⊕ sl(2, C) which describes the complexification of o(3, 1). Indeed, let us introduce the complex basis of Lorentz algebra (o(3, 1) ≃ sl(2; C) ⊕ sl(2, C)) described by two commuting sets of complex generators:\n\nH 1 = 1 2 (h + ıh ′ ) , E 1± = 1 2 (e ± + ıe ′ ± ) , (3.16\n\n)\n\nH 2 = 1 2 (h -ıh ′ ) , E 2± = 1 2 (e ± -ıe ′ ± ) , (3.17)\n\nwhich satisfy the relations (compare with (3.1))\n\n[H k , E k± ] = ±E k± , [E k+ , E k-] = 2H k (k = 1, 2) . (3.18)\n\nThe * -operation describing the real structure acts on the generators H k , and\n\nE k± (k = 1, 2)\n\nas follows\n\nH * 1 = -H 2 , E * 1± = -E 2± , H * 2 = -H 1 , E * 2± = -E 1± . (3.19)\n\nThe classical r-matrix r 3 , (3.7), and r 4 , (3.8), in terms of the complex basis (3.16), (3.17) take the form\n\nr 3 = r ′ 1 + r ′′ 1 , r ′ 3 := 2(β + ıα)E 1+ ∧ E 1-+ 2(β -ıα)E 2+ ∧ E 2-, r ′′ 3 := 4ıγ H 2 ∧ H 1 , (3.20)\n\nand\n\nr 4 = r ′ 4 + r ′′ 4 , r ′ 4 := 2ıα(E 1+ ∧ E 1--E 2+ ∧ E 2--2H 1 ∧ H 2 ) , r ′′ 4 := 4ıλ E 1+ ∧ E 2+ (3.21)\n\nFor the sake of convenience we introduce parameter foot_5 λ in r ′′ 4 . It should be noted that r ′ 3 , r ′′ 3 and r ′ 4 , r ′′ 4 are themselves classical r-matrices. We see that the r-matrix r ′ 3 is simply a sum of two standard r-matrices of sl(2; C), satisfying the anti-Hermitian condition r * = -r. Analogously, it is not hard to see that the r-matrix r 4 corresponds to a Belavin-Drinfeld triple [15] for the Lie algebra sl(2; C) ⊕ sl(2, C)). Indeed, applying the Cartan automorphism E 2± → E 2∓ , H 2 → -H 2 we see that this is really correct (see also [16] ).\n\nWe firstly describe quantum deformation corresponding to the classical r-matrix r 3 (3.20). Since the r-matrix r ′′ 3 is Abelian and it is subordinated to r ′ 3 therefore the algebra o(3, 1) is firstly quantized in the direction r ′ 3 and then an Abelian twist corresponding to the r-matrix r ′′ 3 is applied. We introduce the complex notations z ± := β ± ıα. It should be noted that z -= z * + if the parameters α and β are real, and z -= -z * + if the parameters α and β are pure imaginary. From structure of the classical r-matrix r ′ 3 it follows that a quantum deformation U r ′ 1 (o(3, 1)) is a combination of two q-analogs of U(sl(2; C)) with the parameter q z + and q z -, where q z ± := exp z ± . Thus U r ′ 3 (o(3, 1)) ∼ = U q z + (sl(2; C))⊗U q z -(sl(2; C)) and the standard generators q ±H 1 z + , E 1± and q ±H 2 z -, E 2± satisfy\n\nThe physical generators of the Lorentz algebra, M i , N i (i = 1, 2, 3), are related with the canonical basis h, h ′ , e ± , e ′ ± as follows\n\nh = ıN 3 , e ± = ı(N 1 ± M 2 ), (4.4\n\n)\n\nh ′ = -ıM 3 , e ′ ± = ı(±N 2 -M 1 ). (4.5)\n\nThe subalgebra generated by the four-momenta P j , P 0 (j = 1, 2, 3) will be denoted by P and we also set P ± := P 0 ± P 3 . S. Zakrzewski has shown in [2] that each classical r-matrix, r ∈ P(3, Here [[•, •]] means the Schouten bracket. Moreover a total list of the classical r-matrices for the case c = 0 and also for the case c = 0, t = 0 was found. 7 It was shown that there are fifteen solutions for the case c = 0, t = 0, and six solutions for the case c = 0 where there is only one solution for t = 0. Thus Zakrzewski found twenty r-matrices which satisfy the homogeneous classical Yang-Baxter equation (t = 0 in (4.9)). Analysis of these twenty solutions shows that each of them can be presented as a sum of subordinated rmatrices which almost all are of Abelian and Jordanian types. Therefore these r-matrices correspond to twisted deformations of the Poincaré algebra P(3, 1). We present here r-matrices only for the case c = 0, t = 0:\n\nr 1 = γh ′ ∧\n\nh + α(P + ∧ P --P 1 ∧ P 2 ) , (4.11) r 2 = γe ′ + ∧ e + + β 1 (e + ∧ P 1e ′ + ∧ P 2 + h ∧ P + ) + β 2 h ′ ∧ P + , (4.12) r 3 = γe ′ + ∧ e + + β(e + ∧ P 1e ′ + ∧ P 2 + h ∧ P + ) + αP 1 ∧ P + , (4.13) r 4 = γ(e ′ + ∧ e + + e + ∧ P 1 + e ′ + ∧ P 2 -P 1 ∧ P 2 ) + P + ∧ (α 1 P 1 + α 2 P 2 ) , (4.14) r 5 = γ 1 (h ∧ e +h ′ ∧ e ′ + ) + γ 2 e + ∧ e ′ + . (4.15)\n\nThe first r-matrix r 1 is a sum of two subordinated Abelian r-matrices r 1 := r ′ 1 + r ′′ 1 , r ′ 1 ≻ r ′′ 1 , r ′ 1 = α(P + ∧ P --P 1 ∧ P 2 ) , r ′′ 1 := γh ′ ∧ h .\n\n(4.16)\n\nTherefore the total twist defining quantization in the direction to this r-matrix is the ordered product of two the Abelian twits F r 1 = F r ′′ 1 F r ′ 1 = exp γh ′ ∧ h exp α(P + ∧ P --P 1 ∧ P 2 ) . (4.17)\n\n7 Classification of the r-matrices for the case c = 0, t = 0 is an open problem up to now."
}
] | ||
arxiv:0704.0083 | 0704.0083 | 1 | 10.1103/PhysRevLett.99.071301 | 3f56a48bd29e65a018261e0299f2f784310472a5b7f577b43b2f2cc6a41bf6d7 | Why there is something rather than nothing (out of everything)? | The path integral over Euclidean geometries for the recently suggested
density matrix of the Universe is shown to describe a microcanonical ensemble
in quantum cosmology. This ensemble corresponds to a uniform (weight one)
distribution in phase space of true physical variables, but in terms of the
observable spacetime geometry it is peaked about complex saddle-points of the
{\em Lorentzian} path integral. They are represented by the recently obtained
cosmological instantons limited to a bounded range of the cosmological
constant. Inflationary cosmologies generated by these instantons at late stages
of expansion undergo acceleration whose low-energy scale can be attained within
the concept of dynamically evolving extra dimensions. Thus, together with the
bounded range of the early cosmological constant, this cosmological ensemble
suggests the mechanism of constraining the landscape of string vacua and,
simultaneously, a possible solution to the dark energy problem in the form of
the quasi-equilibrium decay of the microcanonical state of the Universe. | [
"A.O.Barvinsky"
] | [
"hep-th"
] | hep-th | [] | 2007-04-01 | 2026-02-26 | The path integral over Euclidean geometries for the recently suggested density matrix of the Universe is shown to describe a microcanonical ensemble in quantum cosmology. This ensemble corresponds to a uniform (weight one) distribution in phase space of true physical variables, but in terms of the observable spacetime geometry it is peaked about complex saddle-points of the Lorentzian path integral. They are represented by the recently obtained cosmological instantons limited to a bounded range of the cosmological constant. Inflationary cosmologies generated by these instantons at late stages of expansion undergo acceleration whose low-energy scale can be attained within the concept of dynamically evolving extra dimensions. Thus, together with the bounded range of the early cosmological constant, this cosmological ensemble suggests the mechanism of constraining the landscape of string vacua and, simultaneously, a possible solution to the dark energy problem in the form of the quasi-equilibrium decay of the microcanonical state of the Universe.
Euclidean quantum gravity (EQG) is a lame duck in modern particle physics and cosmology. After its summit in early and late eighties (in the form of the cosmological wavefunction proposals [1, 2] and baby universes boom [3] ) the interest in this theory gradually declined, especially, in cosmological context, where the problem of quantum initial conditions was superseded by the concept of stochastic inflation [4] . EQG could not stand the burden of indefiniteness of the Euclidean gravitational action [5] and the cosmology debate of the tunneling vs no-boundary proposals [6] .
Thus, a recently suggested EQG density matrix of the Universe [7] is hardly believed to be a viable candidate for the initial state of the Universe, even though it avoids the infrared catastrophe of small cosmological constant Λ, generates an ensemble of universes in the limited range of Λ, and suggests a strong selection mechanism for the landscape of string vacua [7, 8] . Here we want to justify this result by deriving it from first principles of Lorentzian quantum gravity applied to a microcanonical ensemble of closed cosmological models.
Thermal properties in quantum cosmology [9] are incorporated by a mixed physical state, which is dynamically more preferable than a pure state of the Hartle-Hawking type. This follows from the path integral for the EQG statistical sum [7, 8] . It can be cast into the form of the integral over a minisuperspace of the lapse function N (τ ) and scale factor a(τ ) of spatially closed FRW metric
ds 2 = N 2 (τ ) dτ 2 + a 2 (τ ) d 2 Ω (3) , e -Γ = periodic D[ a, N ] e -ΓE [ a, N ] , (1)
e -ΓE [ a, N ] = periodic Dφ(x) e -SE [ a, N ; φ(x) ] . (2)
Here Γ E [ a, N ] is the Euclidean effective action of all inhomogeneous "matter" fields which include also metric perturbations on minisuperspace background Φ(x) = (φ(x), ψ(x), A µ (x), h µν (x), ...). S E [a, N ; φ(x)] is the clas-sical Eucidean action, and the integration runs over periodic fields on the Euclidean spacetime with a compactified time τ (of S 1 × S 3 topology).
For free matter fields φ(x) conformally coupled to gravity (which are assumed to be dominating in the system) the effective action has the form [7]
Γ E [ a, N ] = dτ N L(a, a ′ ) + F (η), a ′ ≡ da/N dτ .
Here N L(a, a ′ ) is the effective Lagrangian of its local part including the classical Einstein term (with the cosmological constant Λ = 3H 2 ) and the contribution of the conformal anomaly of quantum fields and their vacuum (Casimir) energy,
L(a, a ′ ) = -aa ′2 -a + H 2 a 3 + B a ′2 a - a ′4 6a + 1 2a . (3)
F (η) is the free energy of their quasi-equilibrium excitations with the temperature given by the inverse of the conformal time η = dτ N/a. This is a typical boson or fermion sum F (η) = ± ω ln 1 ∓ e -ωη over field oscillators with energies ω on a unit 3-sphere. We work in units of m P = (3π/4G) 1/2 , and B is the constant determined by the coefficient of the Gauss-Bonnet term in the overall conformal anomaly of all fields φ(x).
Semiclassically the integral (1) is dominated by the saddle points -solutions of the Friedmann equation
a ′2 a 2 + B 1 2 a ′4 a 4 - a ′2 a 4 = 1 a 2 -H 2 - C a 4 , (4)
modified by the quantum B-term and the radiation term C/a 4 with the constant C satisfying the bootstrap equation C = B/2 + dF (η)/dη. Such solutions represent garland-type instantons which exist only in the limited range 0 < Λ min < Λ < 3m 2 P /B [7, 8] and eliminate the infrared catastrophe of Λ = 0. The period of these quasithermal instantons is not a freely specifiable parameter, but as a function of Λ follows from this bootstrap. Therefore this is not a canonical ensemble.
Contrary to the construction above, the density matrix that we advocate here is given by the canonical path integral of Lorentzian quantum gravity. Its kernel in the representation of 3-metrics and matter fields denoted below as q reads ρ(q + , q -) = e Γ q(t±)= q± D[ q, p, N ] e i R t + tdt (p q-N µ Hµ) , (5) where the integration runs over histories of phase-space variables (q(t), p(t)) interpolating between q ± at t ± and the Lagrange multipliers of the gravitational constraints H µ = H µ (q, p) -lapse and shift functions N (t) = N µ (t). The measure D[ q, p, N ] includes the gauge-fixing factor containing the delta function δ(χ) = µ δ(χ µ ) of gauge conditions χ µ and the ghost factor [10, 11] (condensed index µ includes also continuous spatial labels). It is important that the integration range of
N µ -∞ < N < +∞, (6)
is such that it generates in the integrand the deltafunctions of these constraints δ(H) = µ δ(H µ ). As a consequence the kernel (5) satisfies the set of quantum Dirac constraints -Wheeler-DeWitt equations Ĥµ q, ∂/i∂q ρ( q, q -) = 0, (7) and the density matrix ( 5 ) can be regarded as an operator delta-function of these constraints
ρ ∼ " µ δ( Ĥµ ) ". (8)
This notation should not be understood literally because this multiple delta-function is not well defined, for the operators Ĥµ do not commute and form a quasi-algebra with nonvanishing structure functions. Moreover, exact operator realization Ĥµ is not known except the first two orders of a semiclassical -expansion [12] . Fortunately, we do not need a precise form of these constraints, because we will proceed with their path-integral solutions well adjusted to the semiclassical perturbation theory. This strategy does not extend beyond typical field-theoretic considerations when the path integral is regarded more fundamental than the Schrodinger equation marred with the problems of divergent equal-time commutators, operator ordering, etc. The very essence of our proposal is the interpretation of ( 5 ) and ( 8 ) as the density matrix of a microcanonical ensemble in spatially closed quantum cosmology. A simplest analogy is the density matrix of an unconstrained system having a conserved Hamiltonian Ĥ in the microcanonical state with a fixed energy E, ρ ∼ δ( Ĥ -E). Major distinction of (8) from this case is that spatially closed cosmology does not have freely specifiable constants of motion like the energy or other global charges.
Rather it has as constants of motion the Hamiltonian and momentum constraints H µ , all having a particular value -zero. Therefore, the expression (8) can be considered as a most general and natural candidate for the quantum state of the closed Universe. Below we confirm this fact by showing that in the physical sector the corresponding statistical sum is just a uniformly distributed (with a unit weight) integral over entire phase space of true physical degrees of freedom. Thus, this is a sum over Everything. However, in terms of the observable quantities, like spacetime geometry, this distribution turns out to be nontrivially peaked around a particular set of universes. Semiclassically this distribution is given by the EQG density matrix and the saddle-point instantons of the above type [7] .
From the normalization of the density matrix in the physical Hilbert space the statistical sum follows as the path integral 1 = Tr phys ρ = dq µ q, ∂/i∂q ρ(q, q ′ )
q ′ =q = e Γ periodic D[ q, p, N ] e i R dt(p q-N µ Hµ) , (9)
where the integration runs over periodic in time histories of q = q(t). Here µ q, ∂/i∂q = μ is the measure which distinguishes the physical inner product in the space of solutions of the Wheeler-DeWitt equations (ψ 1 |ψ 2 ) = ψ 1 |μ|ψ 2 from that of the space of squareintegrable functions, ψ 1 |ψ 2 = dq ψ * 1 ψ 2 . This measure includes the delta-function of unitary gauge conditions and an operator factor built with the aid of the relevant ghost determinant [12] .
On the other hand, in terms of the physical phase space variables the Faddeev-Popov path integral equals [10, 11]
D[ q, p, N ] e i R dt (p q-N µ Hµ) = Dq phys Dp phys e i R dt (p phys qphys -H phys (t)) = Tr phys T e -i R dt Ĥphys (t) , (10)
where T denotes the chronological ordering. Here the physical Hamiltonian H phys (t) and its operator realization Ĥphys (t) are nonvanishing only in unitary gauges explicitly depending on time [12] , ∂ t χ µ (q, p, t) = 0. In static gauges, ∂ t χ µ = 0, they identically vanish, because in spatially closed cosmology the full Hamiltonian reduces to the combination of constraints.
The path integral ( 10 ) is gauge-independent on-shell [10, 11] and coincides with that in the static gauge. Therefore, from Eqs.( 9 )- (10) with Ĥphys = 0, the statistical sum of our microcanonical ensemble equals e -Γ = Tr phys I phys = dq phys dp phys = sum over Everything. (11) This ultimate equipartition, not modulated by any nontrivial density of states, is a result of general covariance and closed nature of the Universe lacking any freely specifiable constants of motion. The volume integral of entire physical phase space, whose structure and topology is not known, is very nontrivial. However, below we show that semiclassically it is determined by EQG methods and supported by instantons of [7] spanning a bounded range of the cosmological constant.
Integration over momenta in (9) yields a Lagrangian path integral with a relevant measure and action
e -Γ = D[ q, N ] e iSL[ q, N ] . (12)
Integration runs over periodic fields (not indicated explicitly but assumed everywhere below) even despite the Lorentzian signature of the underlying spacetime. Similarly to the procedure of [7, 8] leading to (1)-( 2 ) in the Euclidean case, we decompose [ q, N ] into a minisuperspace [ a L (t), N L (t) ] and the "matter" φ L (x) variables, the subscript L indicating their Lorentzian nature. With a relevant decomposition of the measure D[ q, N ] = D[ a L , N L ]×Dφ L (x), the microcanonical sum takes the form
e -Γ = D[ a L , N L ] e iΓL[ aL, NL ] , (13)
e iΓL[ aL, NL ] = Dφ L (x) e iSL[ aL, NL; φL(x)] , (
where Γ L [ a L , N L ] is a Lorentzian effective action. The stationary point of ( 13 ) is a solution of the effective equation δΓ L /δN L (t) = 0. In the gauge N L = 1 it reads as a Lorentzian version of the Euclidean equation ( 4 ) and the bootstrap equation for the radiation constant C with the Wick rotated τ = it, a(τ ) = a L (t), η = i dt/a L (t). However, with these identifications C turns out to be purely imaginary (in view of the complex nature of the free energy F (i dt/a L )). Therefore, no periodic solutions exist in spacetime with a real Lorentzian metric. On the contrary, such solutions exist in the Euclidean spacetime. Alternatively, the latter can be obtained with the time variable unchanged t = τ , a L (t) = a(τ ), but with the Wick rotated lapse function (15) In the gauge N = 1 (N L = -i) these solutions exactly coincide with the instantons of [7] . The corresponding saddle points of ( 13 ) can be attained by deforming the integration contour (6) of N L into the complex plane to pass through the point N L = -i and relabeling the real Lorentzian t with the Euclidean τ . In terms of the Euclidean N (τ ), a(τ ) and φ(x) the integrals ( 13 ) and ( 14 ) take the form of the path integrals (1) and ( 2
N L = -iN, iS L [ a L , N L ; φ L ] = -S E [ a, N ; φ ].
) in EQG, iΓ L [ a L , N L ] = -Γ E [ a, N ]. ( 16
)
However, the integration contour for the Euclidean N (τ ) runs from -i∞ to +i∞ through the saddle point N = 1. This is the source of the conformal rotation in Euclidean quantum gravity, which is called to resolve the problem of unboundedness of the gravitational action and effectively renders the instantons a thermal nature, even though they originate from the microcanonical ensemble. This mechanism implements the justification of EQG from canonical quantization of gravity [14] (see also [15] in black hole context).
To show this we calculate (1) in the one-loop approximation with the measure inherited from the canonical path integral (5)
D[ a, N ] = Da DN µ[ a, N ] δ[ χ ] Det Q.
Here µ[ a, N ] is a local measure determined by the Lagrangian N L(a, a ′ ), (3), in the local part of Γ E [ a, N ],
µ 1-loop = τ ∂ 2 (N L) ∂ ȧ ∂ ȧ 1/2 = τ D N a 2 a ′2 1/2 , D = a a ′2 (a 2 -B + B a ′2 ). ( 17
)
The Faddeev-Popov factor δ
[ χ ] Det Q contains a gauge condition χ = χ(a, N ) fixing the one-dimensional dif- feomorphism, τ → τ = τ -f /N , which for infinitesi- mal f = f (τ ) has the form ∆ f N ≡ N (τ ) -N (τ ) = ḟ , ∆ f a ≡ ā(τ ) -a(τ ) = ȧ f /N , and Q = Q(d/dτ ) is a ghost operator determined by the gauge transformation of χ(a, N ), ∆ f χ = Q(d/dτ ) f (τ ).
The conformal mode σ of the perturbations δa = σa 0 and δN = σN 0 on the saddle-point background (labeled below by zero, a = a 0 + δa, N = N 0 + δN ) originates from imposing the background gauge χ(a, N ) = δN -(N 0 /a 0 ) δa. In this gauge Q = a(d/dτ )a -1 , and the quadratic part of Γ E takes the form [13]
δ 2 σ Γ E = - 3πm 2 P 2 dτ N D σ a ′ ′ 2 , ( 18
)
where D is given by (17) . As is known from [7] for the background instantons a 2 0 (τ ) ≥ a 2 -> B (a -is the turning point with the smallest value of a 0 (τ )), so that D > 0 everywhere except the turning points where D degenerates to zero. Therefore δ 2 σ Γ E < 0 for real σ, but the Gaussian integration runs along the imaginary axes and yields the functional determinant of the positive operator -the kernel of the quadratic form (18)
e -Γ 1-loop = e -Γ0 Det Q 0 Dσ τ D/a ′2 1/2 e -1 2 δ 2 σ ΓE = e -Γ0 × Det d dτ Det - 1 √ D d dτ D d dτ 1 √ D -1/2
.
In view of periodic boundary conditions for both operators their determinants cancel each other (their zero modes to be eliminated because they correspond to the conformal Killing symmetry of FRW instantons) [13] .
Therefore, the contribution of the conformal mode reduces to the selection of instantons with a fixed time period, effectively endowing them with a thermal nature.
As suggested in [7, 8, 16] these instantons serve as initial conditions for inflationary universes which evolve according to the Lorentzian version of Eq.( 4 ) and, at late stages, have two branches of a cosmological acceleration with Hubble scales
H 2 ± = (m 2 P /B)(1 ± (1 -2BH 2 ) 1/2
). If the initial Λ = 3H 2 is a composite inflaton field decaying at the end of inflation, then one of the branches undergoes acceleration with H 2 + = 2m 2 P /B. This is determined by the new quantum gravity scale suggested in [8] -the upper bound of the instanton Λ-range, Λ max = 3m 2 P /B. To match the model with inflation and the dark energy phenomenon, one needs B of the order of the inflation scale in the very early Universe and B ∼ 10 120 now, so that this parameter should effectively be a growing function of time.
This picture seems to fit into string theory at its lowenergy field-theoretic level. Then, with a bounded range of Λ, it might constrain the landscape of string vacua [7, 8] . Moreover, string theory has a qualitative mechanism to promote the constant B to the level of a moduli variable indefinitely growing with the evolving size R(t) of extra dimensions. Indeed B as a coefficient in the overall conformal anomaly of 4-dimensional quantum fields basically counts their number N , B ∼ N . In the Kaluza-Klein (KK) theory and string theory the effective 4-dimensional fields arise as KK and winding modes with the masses [17]
m 2 n,w = n 2 R 2 + w 2 α ′2 R 2 (19)
(enumerated by the KK and winding numbers), which break their conformal symmetry. These modes remain approximately conformally invariant as long as their masses are much smaller than the spacetime curvature, m 2 n,w ≪ H 2 + ∼ m 2 P /N . Therefore the number of conformally invariant modes changes with R. Simple estimates show that for pure KK modes (w = 0, n ≤ N ) their number grows with R as N ∼ (m P R) 2/3 , whereas for pure winding modes (n = 0, w ≤ N ) their number grows with the decreasing R as N ∼ (m P α ′ /R) 2/3 . Thus, the effect of indefinitely growing B is possible for both scenarios with expanding or contracting extra dimensions. In the first case this is the growing tower of superhorizon KK modes which make the horizon scale H + = m P 2/B ∼ m P /(m P R) 1/3 indefinitely decreasing with R → ∞. In the second case this is the tower of superhorizon winding modes which make this acceleration scale decrease with the decreasing R as H + ∼ m P (R/m P α ′ ) 1/3 . This effect is flexible enough to accommodate the present day acceleration scale H 0 ∼ 10 -60 m P (though, by the price of fine-tuning an enormous coefficient of expansion or contraction of R). This gives a new dark energy mechanism driven by the conformal anomaly and transcending the inflationary and matter-domination stages starting with the state of the microcanonical distribution.
To summarize, within a minimum set of assumptions (the equipartition in the physical phase space ( 11 )), we seem to have the mechanism of constraining the landscape of string vacua and get the full evolution of the Universe as a quasi-equilibrium decay of its initial microcanonical state. Thus, contrary to anticipations of Sidney Coleman, "there is Nothing rather than Something" [3] , one can say that Something (rather than Nothing) comes from Everything.
The author thanks O.Andreev, C.Deffayet, A.Kamenshchik, J.Khoury, H.Tye, A.Tseytlin, I.Tyutin and B.Voronov for thought provoking discussions and especially Andrei Linde, this work having appeared as an unintended response to his discontent with EQG initial conditions. The work was supported by the RFBR grant 05-02-17661, the grant LSS 4401.2006.2 and SFB 375 grant at the Ludwig-Maximilians University in Munich. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The path integral over Euclidean geometries for the recently suggested density matrix of the Universe is shown to describe a microcanonical ensemble in quantum cosmology. This ensemble corresponds to a uniform (weight one) distribution in phase space of true physical variables, but in terms of the observable spacetime geometry it is peaked about complex saddle-points of the Lorentzian path integral. They are represented by the recently obtained cosmological instantons limited to a bounded range of the cosmological constant. Inflationary cosmologies generated by these instantons at late stages of expansion undergo acceleration whose low-energy scale can be attained within the concept of dynamically evolving extra dimensions. Thus, together with the bounded range of the early cosmological constant, this cosmological ensemble suggests the mechanism of constraining the landscape of string vacua and, simultaneously, a possible solution to the dark energy problem in the form of the quasi-equilibrium decay of the microcanonical state of the Universe."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Euclidean quantum gravity (EQG) is a lame duck in modern particle physics and cosmology. After its summit in early and late eighties (in the form of the cosmological wavefunction proposals [1, 2] and baby universes boom [3] ) the interest in this theory gradually declined, especially, in cosmological context, where the problem of quantum initial conditions was superseded by the concept of stochastic inflation [4] . EQG could not stand the burden of indefiniteness of the Euclidean gravitational action [5] and the cosmology debate of the tunneling vs no-boundary proposals [6] .\n\nThus, a recently suggested EQG density matrix of the Universe [7] is hardly believed to be a viable candidate for the initial state of the Universe, even though it avoids the infrared catastrophe of small cosmological constant Λ, generates an ensemble of universes in the limited range of Λ, and suggests a strong selection mechanism for the landscape of string vacua [7, 8] . Here we want to justify this result by deriving it from first principles of Lorentzian quantum gravity applied to a microcanonical ensemble of closed cosmological models.\n\nThermal properties in quantum cosmology [9] are incorporated by a mixed physical state, which is dynamically more preferable than a pure state of the Hartle-Hawking type. This follows from the path integral for the EQG statistical sum [7, 8] . It can be cast into the form of the integral over a minisuperspace of the lapse function N (τ ) and scale factor a(τ ) of spatially closed FRW metric\n\nds 2 = N 2 (τ ) dτ 2 + a 2 (τ ) d 2 Ω (3) , e -Γ = periodic D[ a, N ] e -ΓE [ a, N ] , (1)\n\ne -ΓE [ a, N ] = periodic Dφ(x) e -SE [ a, N ; φ(x) ] . (2)\n\nHere Γ E [ a, N ] is the Euclidean effective action of all inhomogeneous \"matter\" fields which include also metric perturbations on minisuperspace background Φ(x) = (φ(x), ψ(x), A µ (x), h µν (x), ...). S E [a, N ; φ(x)] is the clas-sical Eucidean action, and the integration runs over periodic fields on the Euclidean spacetime with a compactified time τ (of S 1 × S 3 topology).\n\nFor free matter fields φ(x) conformally coupled to gravity (which are assumed to be dominating in the system) the effective action has the form [7]\n\nΓ E [ a, N ] = dτ N L(a, a ′ ) + F (η), a ′ ≡ da/N dτ .\n\nHere N L(a, a ′ ) is the effective Lagrangian of its local part including the classical Einstein term (with the cosmological constant Λ = 3H 2 ) and the contribution of the conformal anomaly of quantum fields and their vacuum (Casimir) energy,\n\nL(a, a ′ ) = -aa ′2 -a + H 2 a 3 + B a ′2 a - a ′4 6a + 1 2a . (3)\n\nF (η) is the free energy of their quasi-equilibrium excitations with the temperature given by the inverse of the conformal time η = dτ N/a. This is a typical boson or fermion sum F (η) = ± ω ln 1 ∓ e -ωη over field oscillators with energies ω on a unit 3-sphere. We work in units of m P = (3π/4G) 1/2 , and B is the constant determined by the coefficient of the Gauss-Bonnet term in the overall conformal anomaly of all fields φ(x).\n\nSemiclassically the integral (1) is dominated by the saddle points -solutions of the Friedmann equation\n\na ′2 a 2 + B 1 2 a ′4 a 4 - a ′2 a 4 = 1 a 2 -H 2 - C a 4 , (4)\n\nmodified by the quantum B-term and the radiation term C/a 4 with the constant C satisfying the bootstrap equation C = B/2 + dF (η)/dη. Such solutions represent garland-type instantons which exist only in the limited range 0 < Λ min < Λ < 3m 2 P /B [7, 8] and eliminate the infrared catastrophe of Λ = 0. The period of these quasithermal instantons is not a freely specifiable parameter, but as a function of Λ follows from this bootstrap. Therefore this is not a canonical ensemble.\n\nContrary to the construction above, the density matrix that we advocate here is given by the canonical path integral of Lorentzian quantum gravity. Its kernel in the representation of 3-metrics and matter fields denoted below as q reads ρ(q + , q -) = e Γ q(t±)= q± D[ q, p, N ] e i R t + tdt (p q-N µ Hµ) , (5) where the integration runs over histories of phase-space variables (q(t), p(t)) interpolating between q ± at t ± and the Lagrange multipliers of the gravitational constraints H µ = H µ (q, p) -lapse and shift functions N (t) = N µ (t). The measure D[ q, p, N ] includes the gauge-fixing factor containing the delta function δ(χ) = µ δ(χ µ ) of gauge conditions χ µ and the ghost factor [10, 11] (condensed index µ includes also continuous spatial labels). It is important that the integration range of\n\nN µ -∞ < N < +∞, (6)\n\nis such that it generates in the integrand the deltafunctions of these constraints δ(H) = µ δ(H µ ). As a consequence the kernel (5) satisfies the set of quantum Dirac constraints -Wheeler-DeWitt equations Ĥµ q, ∂/i∂q ρ( q, q -) = 0, (7) and the density matrix ( 5 ) can be regarded as an operator delta-function of these constraints\n\nρ ∼ \" µ δ( Ĥµ ) \". (8)\n\nThis notation should not be understood literally because this multiple delta-function is not well defined, for the operators Ĥµ do not commute and form a quasi-algebra with nonvanishing structure functions. Moreover, exact operator realization Ĥµ is not known except the first two orders of a semiclassical -expansion [12] . Fortunately, we do not need a precise form of these constraints, because we will proceed with their path-integral solutions well adjusted to the semiclassical perturbation theory. This strategy does not extend beyond typical field-theoretic considerations when the path integral is regarded more fundamental than the Schrodinger equation marred with the problems of divergent equal-time commutators, operator ordering, etc. The very essence of our proposal is the interpretation of ( 5 ) and ( 8 ) as the density matrix of a microcanonical ensemble in spatially closed quantum cosmology. A simplest analogy is the density matrix of an unconstrained system having a conserved Hamiltonian Ĥ in the microcanonical state with a fixed energy E, ρ ∼ δ( Ĥ -E). Major distinction of (8) from this case is that spatially closed cosmology does not have freely specifiable constants of motion like the energy or other global charges.\n\nRather it has as constants of motion the Hamiltonian and momentum constraints H µ , all having a particular value -zero. Therefore, the expression (8) can be considered as a most general and natural candidate for the quantum state of the closed Universe. Below we confirm this fact by showing that in the physical sector the corresponding statistical sum is just a uniformly distributed (with a unit weight) integral over entire phase space of true physical degrees of freedom. Thus, this is a sum over Everything. However, in terms of the observable quantities, like spacetime geometry, this distribution turns out to be nontrivially peaked around a particular set of universes. Semiclassically this distribution is given by the EQG density matrix and the saddle-point instantons of the above type [7] .\n\nFrom the normalization of the density matrix in the physical Hilbert space the statistical sum follows as the path integral 1 = Tr phys ρ = dq µ q, ∂/i∂q ρ(q, q ′ )\n\nq ′ =q = e Γ periodic D[ q, p, N ] e i R dt(p q-N µ Hµ) , (9)\n\nwhere the integration runs over periodic in time histories of q = q(t). Here µ q, ∂/i∂q = μ is the measure which distinguishes the physical inner product in the space of solutions of the Wheeler-DeWitt equations (ψ 1 |ψ 2 ) = ψ 1 |μ|ψ 2 from that of the space of squareintegrable functions, ψ 1 |ψ 2 = dq ψ * 1 ψ 2 . This measure includes the delta-function of unitary gauge conditions and an operator factor built with the aid of the relevant ghost determinant [12] .\n\nOn the other hand, in terms of the physical phase space variables the Faddeev-Popov path integral equals [10, 11]\n\nD[ q, p, N ] e i R dt (p q-N µ Hµ) = Dq phys Dp phys e i R dt (p phys qphys -H phys (t)) = Tr phys T e -i R dt Ĥphys (t) , (10)\n\nwhere T denotes the chronological ordering. Here the physical Hamiltonian H phys (t) and its operator realization Ĥphys (t) are nonvanishing only in unitary gauges explicitly depending on time [12] , ∂ t χ µ (q, p, t) = 0. In static gauges, ∂ t χ µ = 0, they identically vanish, because in spatially closed cosmology the full Hamiltonian reduces to the combination of constraints.\n\nThe path integral ( 10 ) is gauge-independent on-shell [10, 11] and coincides with that in the static gauge. Therefore, from Eqs.( 9 )- (10) with Ĥphys = 0, the statistical sum of our microcanonical ensemble equals e -Γ = Tr phys I phys = dq phys dp phys = sum over Everything. (11) This ultimate equipartition, not modulated by any nontrivial density of states, is a result of general covariance and closed nature of the Universe lacking any freely specifiable constants of motion. The volume integral of entire physical phase space, whose structure and topology is not known, is very nontrivial. However, below we show that semiclassically it is determined by EQG methods and supported by instantons of [7] spanning a bounded range of the cosmological constant.\n\nIntegration over momenta in (9) yields a Lagrangian path integral with a relevant measure and action\n\ne -Γ = D[ q, N ] e iSL[ q, N ] . (12)\n\nIntegration runs over periodic fields (not indicated explicitly but assumed everywhere below) even despite the Lorentzian signature of the underlying spacetime. Similarly to the procedure of [7, 8] leading to (1)-( 2 ) in the Euclidean case, we decompose [ q, N ] into a minisuperspace [ a L (t), N L (t) ] and the \"matter\" φ L (x) variables, the subscript L indicating their Lorentzian nature. With a relevant decomposition of the measure D[ q, N ] = D[ a L , N L ]×Dφ L (x), the microcanonical sum takes the form\n\ne -Γ = D[ a L , N L ] e iΓL[ aL, NL ] , (13)\n\ne iΓL[ aL, NL ] = Dφ L (x) e iSL[ aL, NL; φL(x)] , (\n\nwhere Γ L [ a L , N L ] is a Lorentzian effective action. The stationary point of ( 13 ) is a solution of the effective equation δΓ L /δN L (t) = 0. In the gauge N L = 1 it reads as a Lorentzian version of the Euclidean equation ( 4 ) and the bootstrap equation for the radiation constant C with the Wick rotated τ = it, a(τ ) = a L (t), η = i dt/a L (t). However, with these identifications C turns out to be purely imaginary (in view of the complex nature of the free energy F (i dt/a L )). Therefore, no periodic solutions exist in spacetime with a real Lorentzian metric. On the contrary, such solutions exist in the Euclidean spacetime. Alternatively, the latter can be obtained with the time variable unchanged t = τ , a L (t) = a(τ ), but with the Wick rotated lapse function (15) In the gauge N = 1 (N L = -i) these solutions exactly coincide with the instantons of [7] . The corresponding saddle points of ( 13 ) can be attained by deforming the integration contour (6) of N L into the complex plane to pass through the point N L = -i and relabeling the real Lorentzian t with the Euclidean τ . In terms of the Euclidean N (τ ), a(τ ) and φ(x) the integrals ( 13 ) and ( 14 ) take the form of the path integrals (1) and ( 2\n\nN L = -iN, iS L [ a L , N L ; φ L ] = -S E [ a, N ; φ ].\n\n) in EQG, iΓ L [ a L , N L ] = -Γ E [ a, N ]. ( 16\n\n)\n\nHowever, the integration contour for the Euclidean N (τ ) runs from -i∞ to +i∞ through the saddle point N = 1. This is the source of the conformal rotation in Euclidean quantum gravity, which is called to resolve the problem of unboundedness of the gravitational action and effectively renders the instantons a thermal nature, even though they originate from the microcanonical ensemble. This mechanism implements the justification of EQG from canonical quantization of gravity [14] (see also [15] in black hole context).\n\nTo show this we calculate (1) in the one-loop approximation with the measure inherited from the canonical path integral (5)\n\nD[ a, N ] = Da DN µ[ a, N ] δ[ χ ] Det Q.\n\nHere µ[ a, N ] is a local measure determined by the Lagrangian N L(a, a ′ ), (3), in the local part of Γ E [ a, N ],\n\nµ 1-loop = τ ∂ 2 (N L) ∂ ȧ ∂ ȧ 1/2 = τ D N a 2 a ′2 1/2 , D = a a ′2 (a 2 -B + B a ′2 ). ( 17\n\n)\n\nThe Faddeev-Popov factor δ\n\n[ χ ] Det Q contains a gauge condition χ = χ(a, N ) fixing the one-dimensional dif- feomorphism, τ → τ = τ -f /N , which for infinitesi- mal f = f (τ ) has the form ∆ f N ≡ N (τ ) -N (τ ) = ḟ , ∆ f a ≡ ā(τ ) -a(τ ) = ȧ f /N , and Q = Q(d/dτ ) is a ghost operator determined by the gauge transformation of χ(a, N ), ∆ f χ = Q(d/dτ ) f (τ ).\n\nThe conformal mode σ of the perturbations δa = σa 0 and δN = σN 0 on the saddle-point background (labeled below by zero, a = a 0 + δa, N = N 0 + δN ) originates from imposing the background gauge χ(a, N ) = δN -(N 0 /a 0 ) δa. In this gauge Q = a(d/dτ )a -1 , and the quadratic part of Γ E takes the form [13]\n\nδ 2 σ Γ E = - 3πm 2 P 2 dτ N D σ a ′ ′ 2 , ( 18\n\n)\n\nwhere D is given by (17) . As is known from [7] for the background instantons a 2 0 (τ ) ≥ a 2 -> B (a -is the turning point with the smallest value of a 0 (τ )), so that D > 0 everywhere except the turning points where D degenerates to zero. Therefore δ 2 σ Γ E < 0 for real σ, but the Gaussian integration runs along the imaginary axes and yields the functional determinant of the positive operator -the kernel of the quadratic form (18)\n\ne -Γ 1-loop = e -Γ0 Det Q 0 Dσ τ D/a ′2 1/2 e -1 2 δ 2 σ ΓE = e -Γ0 × Det d dτ Det - 1 √ D d dτ D d dτ 1 √ D -1/2\n\n.\n\nIn view of periodic boundary conditions for both operators their determinants cancel each other (their zero modes to be eliminated because they correspond to the conformal Killing symmetry of FRW instantons) [13] .\n\nTherefore, the contribution of the conformal mode reduces to the selection of instantons with a fixed time period, effectively endowing them with a thermal nature.\n\nAs suggested in [7, 8, 16] these instantons serve as initial conditions for inflationary universes which evolve according to the Lorentzian version of Eq.( 4 ) and, at late stages, have two branches of a cosmological acceleration with Hubble scales\n\nH 2 ± = (m 2 P /B)(1 ± (1 -2BH 2 ) 1/2\n\n). If the initial Λ = 3H 2 is a composite inflaton field decaying at the end of inflation, then one of the branches undergoes acceleration with H 2 + = 2m 2 P /B. This is determined by the new quantum gravity scale suggested in [8] -the upper bound of the instanton Λ-range, Λ max = 3m 2 P /B. To match the model with inflation and the dark energy phenomenon, one needs B of the order of the inflation scale in the very early Universe and B ∼ 10 120 now, so that this parameter should effectively be a growing function of time.\n\nThis picture seems to fit into string theory at its lowenergy field-theoretic level. Then, with a bounded range of Λ, it might constrain the landscape of string vacua [7, 8] . Moreover, string theory has a qualitative mechanism to promote the constant B to the level of a moduli variable indefinitely growing with the evolving size R(t) of extra dimensions. Indeed B as a coefficient in the overall conformal anomaly of 4-dimensional quantum fields basically counts their number N , B ∼ N . In the Kaluza-Klein (KK) theory and string theory the effective 4-dimensional fields arise as KK and winding modes with the masses [17]\n\nm 2 n,w = n 2 R 2 + w 2 α ′2 R 2 (19)\n\n(enumerated by the KK and winding numbers), which break their conformal symmetry. These modes remain approximately conformally invariant as long as their masses are much smaller than the spacetime curvature, m 2 n,w ≪ H 2 + ∼ m 2 P /N . Therefore the number of conformally invariant modes changes with R. Simple estimates show that for pure KK modes (w = 0, n ≤ N ) their number grows with R as N ∼ (m P R) 2/3 , whereas for pure winding modes (n = 0, w ≤ N ) their number grows with the decreasing R as N ∼ (m P α ′ /R) 2/3 . Thus, the effect of indefinitely growing B is possible for both scenarios with expanding or contracting extra dimensions. In the first case this is the growing tower of superhorizon KK modes which make the horizon scale H + = m P 2/B ∼ m P /(m P R) 1/3 indefinitely decreasing with R → ∞. In the second case this is the tower of superhorizon winding modes which make this acceleration scale decrease with the decreasing R as H + ∼ m P (R/m P α ′ ) 1/3 . This effect is flexible enough to accommodate the present day acceleration scale H 0 ∼ 10 -60 m P (though, by the price of fine-tuning an enormous coefficient of expansion or contraction of R). This gives a new dark energy mechanism driven by the conformal anomaly and transcending the inflationary and matter-domination stages starting with the state of the microcanonical distribution.\n\nTo summarize, within a minimum set of assumptions (the equipartition in the physical phase space ( 11 )), we seem to have the mechanism of constraining the landscape of string vacua and get the full evolution of the Universe as a quasi-equilibrium decay of its initial microcanonical state. Thus, contrary to anticipations of Sidney Coleman, \"there is Nothing rather than Something\" [3] , one can say that Something (rather than Nothing) comes from Everything.\n\nThe author thanks O.Andreev, C.Deffayet, A.Kamenshchik, J.Khoury, H.Tye, A.Tseytlin, I.Tyutin and B.Voronov for thought provoking discussions and especially Andrei Linde, this work having appeared as an unintended response to his discontent with EQG initial conditions. The work was supported by the RFBR grant 05-02-17661, the grant LSS 4401.2006.2 and SFB 375 grant at the Ludwig-Maximilians University in Munich."
}
] | |
arxiv:0704.0100 | 0704.0100 | 1 | 10.1143/PTP.118.715 | 99f97029a358be0732fb87926175d30da9524c34fa9f6637aa15c218dc49eb70 | Topology Change of Black Holes | The topological structure of the event horizon has been investigated in terms
of the Morse theory. The elementary process of topological evolution can be
understood as a handle attachment. It has been found that there are certain
constraints on the nature of black hole topological evolution: (i) There are n
kinds of handle attachments in (n+1)-dimensional black hole space-times. (ii)
Handles are further classified as either of black or white type, and only black
handles appear in real black hole space-times. (iii) The spatial section of an
exterior of the black hole region is always connected. As a corollary, it is
shown that the formation of a black hole with an S**(n-2) x S**1 horizon from
that with an S**(n-1) horizon must be non-axisymmetric in asymptotically flat
space-times. | [
"Daisuke Ida and Masaru Siino"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | Black holes in space-times of greater than or equal to five dimensions have rich topological structure. According to the well-known results of Hawking concerning the topology of black holes in four-dimensional space-time, the apparent horizon or the spatial section of the stationary event horizon is necessarily diffeomorphic to a 2-sphere. [1, 2] This follows from the fact that the total curvature, which is the integral of the intrinsic scalar curvature over the horizon, is positive under the dominant energy condition and from the Gauss-Bonnet theorem. Alternative and improved proofs of Hawking's theorem have been given by several authors. [3, 4, 5, 6] However in higher dimensional space-times, an apparent horizon or the spatial section of the stationary event horizon may not be a topological sphere, [7, 8, 9, 10] because the Gauss-Bonnet theorem does not hold in such cases. Nevertheless, the positivity of the total curvature of the horizon still holds. This puts certain topological restrictions on the black hole topology, though they are rather weak. For example, the apparent horizon in five-dimensional space-time can consist of finitely many connected sums of copies of S 3 /Γ and copies of S 2 × S 1 . In fact, exact solutions representing a black hole space-time possessing a horizon of nonspherical topology have recently been found in five-dimensional general relativity. When such black holes with nontrivial topologies are regarded as being formed in the course of gravitational collapse, questions regarding the evolution of the topology of black holes naturally arise. Our purpose here is to understand the time evolution of the topology of event horizons in a general setting. The relation between the crease set, where the event horizon is nondifferentiable, and the topology of the event horizon is studied in Refs. [11, 12, 13] for four-dimensional space-times. In the present work, we carry out a systematic investigation and find useful rules to determine admissible processes of topological evolution for time slicing of a black hole.
Our approach is to utilize the Morse theory [14, 15] in differential topology. The Morse theory is useful for the purpose of understanding the topology of smooth 1 2 manifolds. The basic tool used in this approach is a smooth function on a differentiable manifold. The event horizon, however, is not a differentiable manifold but has a wedge-like structure at the past endpoints of the null geodesic generators of the horizon. For this reason, we first smooth the wedge. Then, the smooth time function which is assumed to exist plays the role of the Morse function on the smoothed event horizon. According to the Morse theory, the topological evolution of the event horizon can then be decomposed into elementary processes called "handle attachments." In such a process, starting with a spherical horizon, one adds several handles, each characterized by the index of the critical points of the Morse function, which is an integer ranging from 0 to n (the dimension of the smoothed horizon as a differentiable manifold).
The purpose of the present article is to show that there are several constraints on the handle attachments for real black hole space-times.
Let M be an (n + 1)-dimensional asymptotically flat space-time. We require the existence of a global time function t : M → R that is smooth and has an everywhere time-like and future-pointing gradient. The event horizon H is defined as the boundary of the causal past of the future null infinity H = ∂J -(I + ). [2] We treat the event horizon defined with respect to a single asymptotic end, unless otherwise stated. In other words, the future null infinity, I + , is assumed to be connected. The black hole region B is defined as the interior region of H, specifically, as B = M \ J -(I + ), and the exterior region E of the black hole region is its complement, E = int(J -(I + )). We refer to the intersection of the black hole region and the time slice Σ(t 0 ) = {t = t 0 } as the black hole B(t 0 ) = B ∩ Σ(t 0 ) at time t = t 0 . The exterior region at time t = t 0 is, accordingly, written E (t 0 ) = E ∩ Σ(t 0 ). One of most basic properties of the event horizon is that it is generated by null geodesics without future endpoints. In general, the event horizon is not smoothly imbedded into the space-time manifold M , but it has a wedge-like structure at the past endpoints of the null geodesic generators, where distinct null geodesic generators intersect. We call the set of past endpoints of null geodesic generators of H, from which two or more null geodesic generators emanate, the crease set S. [11, 12] When no crease set S exists between the time slices t = t 1 and t = t 2 , the null geodesic generators of H naturally define a diffeomorphism ∂B(t 1 ) ≈ ∂B(t 2 ). Hence, the topological evolution of a black hole can take place only when the time slice intersects the crease set S. Of course, the event horizon itself is a gaugeindependent object. Nevertheless, we often understand the dynamics of space-time by scanning it along time slices. Thus, the topological evolution of a black hole depends on the time function.
It is expected that Morse theory [14] provides useful techniques to analyze such a process of topological evolution. Because the Morse theory is concerned with functions on smooth manifolds, we first regularize H around the crease set S. The event horizon is not necessarily smooth, even on H \ S, in the case that the future null infinity I + has a pathological structure. [16] Here it is assumed that H is smooth on H \ S. Then, small deformations of H near the crease set S will make H a smooth hypersurface H in M , while B(t 0 ) remains deformed in such a manner that ∂ B(t 0 ) = H ∪ Σ(t 0 ) holds and B(t 0 ) remains homeomorphic to the original black hole for all t 0 ∈ R. This deformation is assumed to be such that the time 3 Figure 1. An example in which no smoothing procedure makes t| e H a Morse function on H. Here, the intersection of the crease set S of the event horizon and t = t 0 hypersurface has an accumulation point. function t| e H , which is the restriction of t on H, gives a Morse function on H that has only nondegenerate critical points, where the gradient of t| e H defined on H becomes zero and where also the Hessian matrix (∂ i ∂ j t| e H ) of t| e H is nondegenerate. Though this assumption should hold for a wide class of systems, it does not always hold. Figure 1 gives an example for which no smoothing procedure makes the induced time function t| e H a Morse function on H, because the intersection of the crease set S of the event horizon and the t = t 0 hypersurface has an accumulation point. It is highly nontrivial to determine whether such a smoothing procedure is generically possible. It is, however, not easy nor the primary purpose of this article to assertain the realm of validity of the assumption, and therefore we make this assumption without inquiring into its validity.
According to the Morse Lemma, there is a local coordinate system {x 1 , • • • , x n } on H in the neighborhood of the critical point p ∈ H such that the restriction t| e H of the time function t on H takes the form
t| e H (x 1 , • • • , x n ) = t(p) -(x 1 ) 2 -• • • -(x λ ) 2 + (x λ+1 ) 2 + • • • + (x n ) 2 .
The integer λ, ranging from 0 to n, is called the index of the critical point p. The topology of the black hole B(t) changes when Σ(t) pass through critical points, or equivalently, when the time function t takes critical values. This implies that critical points appears only near the crease set S.
The gradient-like vector field for t| e H is defined to be the tangent vector field X on H such that Xt| e H > 0 holds on H, except for critical points, and has the form
X = -2x 1 ∂ ∂x 1 -• • • -2x λ ∂ ∂x λ + 2x λ+1 ∂ ∂x λ+1 + • • • + 2x n ∂ ∂x n near
the critical point of index λ, in terms of the standard coordinate system appearing in the Morse Lemma. We choose a gradient-like vector field X such that it coincides with the future-directed tangent vector field of null geodesic generators of H, except in a small neighborhood of the crease set S (Fig. 2 ). The effect of a critical point p of index λ is equivalent to the attachment of a λ-handle. [14, 15] The handlebody is just a topological n-disk 0, 1] ), but it is regarded as the product space D n ≈ D λ × D n-λ (Fig. 3 ). The λ-handle attachment to an ndimensional manifold N with a boundary consists of the set
D n ≈ I n (I = [
h λ = (D λ × D n-λ , f ),
where the attaching map f induces the imbedding of 4 ). The new manifold obtained through the λ-handle attachment to N is 4 Figure 2. The smoothing procedure of the event horizon H. The gradient-like vector field on H can be constructed through a slight deformation of the null geodesic generators of H. Here, the effect of the crease set S has been replaced by that of the critical points p 1 , p 2 and p 3 .
∂D λ × D n-λ ⊂ ∂D n into ∂N (Fig.
Figure 3. The local structure around the critical point p of index λ. It can be seen that H t(p)+ǫ is homeomorphic to H t(p)-ǫ with a λ-handle attached. given by
N ∪ h λ = N ∪ (D λ × D n-λ )/(x ∼ f (x)), (x ∈ ∂D λ × D n-λ ).
Let us denote by H t0 the t ≤ t 0 part of H. Then, H t(p)+ǫ (ǫ > 0) just above the critical point p of index λ is homeomorphic (in fact diffeomorphic, taking account of the smoothing procedure) to that just below p, H t(p)-ǫ attached with a λ-handle,
H t(p)+ǫ ≈ H t(p)-ǫ ∪ h λ ,
if there are no other critical points satisfying t(p)-ǫ ≤ t ≤ t(p)+ǫ. The handlebody itself is denoted by h λ as well.
Let us consider several examples. The 0-handle attachment does not need an attaching map f . It simply corresponds to the emergence of the (n -1)-sphere S n-1 ≈ ∂D n as a black hole horizon ∂B(t). A typical example is the creation of a black hole (Fig. 5 ): A black hole always emerges as 0-handle attachment. The other 5 Figure 4. The attachment of a 1-handle and a 2-handle to a 3-manifold N creates a new 3-manifold N ∪ h 1 ∪ h 2 .
E B Figure 5. The emergence of a black hole through a 0-handle attachment.
E B Figure 6. The emergence of a bubble in the black hole region by 0-handle attachment, which does not occur in the real black hole space-times.
E B Figure 7. The collision of a pair of black holes, creating a single black hole, is realized through 1-handle attachment.
possiblity is the creation of a bubble that is subset of J -(I + ) in a black hole region (Fig. 6 ). One might think that this corresponds to wormhole creation between the internal and external regions of the event horizon. Although in the framework of the standard Morse theory on H, these two examples are indistinguishable, we below see that the latter process is in fact impossible.
Next, we consider 1-handle attachment. A typical example is the collision of two black holes. A 1-handle serves as a bridge connecting black holes, or it corresponds to taking the connected sum of each component of multiple black holes (Fig. 7 ).
Figure 8. The bifurcation of one black hole into two is represeted by an (n -1)-handle attachment. This, however, never occurs in real black hole space-times.
λ λ λ λ Figure 9. The structure of λ-handle. The core D λ × {0} corresponds to the stable submanifold with respect to the flow generated by the gradient-like vector field, and the co-core {0} × D n-λ corresponds to the unstable submanifold.
The time reversal of the collision of black holes consists of the bifurcation of one black hole into two. This would be realized through an (n -1)-handle attachment, if such a process were possible (Fig. 8 ). It is, however, well known that such a process is forbidden. [2] In general, the time reversal of the λ-handle attachment corresponds to (n -λ)-handle attachment. Before discussing general cases, let us consider the structure of a handlebody. Recall that a λ-handle consists of the product space D λ × D n-λ . The subset D λ × {0} ⊂ D λ × D n-λ is called the core of the handlebody, and {0} × D n-λ ⊂ D λ × D n-λ is called the co-core. The core and co-core intersect transversely at a point. This point can be regarded as a critical point p.
Let us refer to the subset
W s (p) of H (1) W s (p) = {q ∈ M | lim t→+∞ exp q tX = p}
which consists of points that converge to p along the flow generated by the gradientlike vector field X, as the stable manifold with respect to the critical point p. The stable manifold W s (p) is homeomorphic to R λ if the index of p is given by λ. [17] Similarly, let us refer to the subset W u (p) ⊂ H consisting of points which converge to p along the flow generated by (-X) as the unstable manifold with respect to p. For the unstable manifold, W u (p) ≈ R n-λ holds. The portions of the stable and unstable manifolds in the handlebody can be regarded as corresponding to the core and co-core, respectively.
The effect of smoothing the event horizon H to H is to deform the null vector field generating H into a gradient-like vector field X. The primary difference between the null geodesic generators and the flow generated by X is that the former does not have future endpoints, but the latter can. Thus, there are admissible and inadmissible processes for the smoothed manifold H. An admissible process is given by H, which is obtained from an in priciple realizable event horizon, while an inadmissible one is constructed from a spurious event horizon, i.e., one that consists of the null hypersurface containing null geodesic generators with a future endpoint.
The spatial topology of a black hole changes only when the time function takes a critical value. The time evolution of the black hole topology can be understood by considering its local structure around critical points. To determine whether a given topological change is admissible or inadmissible, it is not sufficient to consider only the intrinsic structure of the event horizon. Rather, it is required to take account of its imbedding structure relative to the space-time.
In a time slice, any point separate from H belongs to either of the black hole or the exterior of the black hole region. It is useful to consider the local behavior of the black hole region or the exterior region near the critical point p. Let us call the exterior E of the black hole region simply the exterior region, for brevity. The exterior region is slightly deformed by the smoothing procedure. The deformed exterior region is denoted by E , and the deformed exterior region at the time t by
E (t) = E ∩ Σ(t) = Σ(t) \ B(t). ( 2
)
The 0-handle is placed at some t ≥ t(p). Such an attachment describes the emergence of the black hole region at the critical point p and its expansion with time. The emergence of a bubble, which consists of a part of J -(I + ), in the background of the black hole region would also be described by a 0-handle attachment. This, however, never occurs, as we explain below in detail. Hence, a 0-handle attachment always describes the creation of a black hole homeomorphic to the n-disk.
An n-handle attachment corresponds to the time reversal of a 0-handle attachement. This process, however, never occurs in real black hole space-time. An nhandle is defined for t ≤ t(p), which means that it terminates at the critical point p. The crease set is isolated into critical points during the course of the smoothing procedure. The gradient-like vector field, which can be regarded as being tangent to the generator of the deformed event horizon H, may have several inward (converging) directions at the critical point due to this smoothing procedure, while the original null generator of the event horizon does not have an inward direction at the crease set. In the case of the n-handle, all the directions become inward at the critical point. This implies that the null generators of the event horizon H must have future endpoints at the critical point, which is, of course, impossible. It is thus seen that an n-handle attachment never occurs in real black hole space-times.
The remaining cases are λ-handle attachments for 1 ≤ λ ≤ n -1. In these cases, the λ-handle lies on either side of the critical point p both in the future [t > t(p)] and in the past [t < t(p)]. Then, we consider the case in which the handle exists during the sufficiently small time interval t ∈ [t(p) -δ, t(p) + δ] (δ > 0), to understand the topological change of the black hole region at the critical point p.
8 Figure 10. The neighborhood U of p is separated by h λ into the future region, U + , and the past region, U -.
First, we introduce a coordinate system {t, x i } (i = 1, • • • , n) in the neighborhood U of p, where t is a given function of time, and {x i } is the extension over U of the cannonical coordinate appearing in the Morse Lemma such that each curve (x 1 , • • • , x n ) = [const] is timelike in U . We assume that U is the solid cylinder given by t ∈ [t(p) -δ, t(p) + δ], (x i ) 2 ≤ δ. In this coordinate system, the λ-handle h λ is given by the saddle surface
t = t(p) -(x 1 ) 2 -• • • -(x λ ) 2 + (x λ+1 ) 2 + • • • + (x n ) 2
in U , which is an acausal set if the constant δ is taken sufficiently small, since h λ is tangent to the space-like hypersurface t = t(p) at p. Therefore, h λ separates U into two open subsets, the future and past regions U + and U -of U , where U + and U -are the subsets lying chronological future and past, respectively, of
h λ : U ± = I ± (h λ ) ∩ U .
Explicitly, the future and past regions U ± are the regions satisfying 10 ).
t ≷ t(p) -(x 1 ) 2 -• • • -(x λ ) 2 + (x λ+1 ) 2 + • • • + (x n ) 2 in U , respectively (Fig.
Because the λ-handle is a subset of the black hole boundary H, one of U ± is contained in the black hole region, B, and the other in the exterior region, E . However, the future region U + of U is always included in the black hole region, i.e. U + ⊂ B, and hence we have U -⊂ E , since the horizon is the boundary of the past set, J -(I + ). Therefore, the black hole region B(t(p) -ǫ) ∩ U in U at the time t = t(p) -ǫ just before the critical time is given by
(x 1 ) 2 + • • • + (x λ ) 2 > (x λ+1 ) 2 + • • • + (x n ) 2 + ǫ,
which is homotopic to the (λ -1)-sphere S λ-1 . (For λ = 1, S 0 simply consists of two points.) Similarly, B(t(p) + ǫ) ∩ U just after the critical time is given by
(x 1 ) 2 + • • • + (x λ ) 2 + ǫ > (x λ+1 ) 2 + • • • + (x n ) 2 , 9
which is homotopic to the n-disk. In this way, the black hole region restricted to the small neighborhood of the critical point p is initially homotopic to a sphere. Then, the internal region of the sphere is filled up at the critical time t = t(p) and eventually becomes homotopically trivial. The exterior region, E (t) ∩ U , in U is initially homotopic to an n-disk for t = t(p) -ǫ. Then, its (n -λ)-dimensional direction is penetrated by the black hole region at t = t(p), and thus it becomes homotopic to an (n -λ -1)-sphere S n-λ-1 for t = t(p) + ǫ. If the spurious event horizon is also taken into account, the future region U + might be a subset of E , and therefore the past region U -might be a subset of B. Then, the black hole region in the λ-handle might be homotopic to an n-disk initially and become homotopic to an (n -λ -1)-sphere finally, and vice versa for the exterior region. Let us refer to such a topological change of the black hole region B(t) ∩ U from a region homotopic to a sphere to a region homotopic to a disk as a black handle attachment, and that from a region homotopic to a disk to a region homotopic to the sphere as a white handle attachment. The above observation shows that only a black handle attachment occurs if a sufficiently small neighborhood of the critical point is considered. For example, a collision of black holes corresponds to a black 1-handle attachment, while the bifurcation of a black hole corresponds to a white (n -1)-handle attachment in the sense that the homotopy type of the exterior region E (t) ∩ U changes from that of S n-2 to that of D n . This local argument also elucidates te reason that a black hole collision is admissible while a black hole bifurcation, which is its time reversal, is inadmissible. We also note that the effect of time reversal is to convert a black λ-handle attachment into a white (n -λ)-handle attachment.
It is appropriate to refer to the 0-handle attachment corresponding to the creation of a black hole as a black 0-handle attachment. Then, the proposition above also applies to a 0-handle attachment.
There also exist processes that are unrealizable due to global conditions. Let us, for a moment, consider the event horizon in maximally extended Schwarzschild space-time. Though we are interested in the event horizon defined with respect to a specific asymptotic end, for the purpose of explanation, we examine the event horizon defined with respect to a pair of asymptotic ends in Schwarzschild spacetime (Fig. 11 ).
Let I + 1 and I + 2 be the pair of future null infinities of the maximally extended Schwarzschild space-time. The event horizon here is defined by H = ∂J -(I + 1 ∪ I + 2 ), which is nondifferentiable at the bifurcate horizon F = ∂J -(I + 1 )∩∂J -(I + 2 ). Let t be a global time function and χ be a global radial coordinate function such that each two-surface t, χ = [const] is invariant under the SO(3) isometry. These coordinates are chosen such that the bifurcation surface F is located at t = χ = 0 and the event horizon H is determined by t = |χ| around F . The smoothed event horizon H is also taken to be invariant under the SO(3) isometry. Due to the symmetry of the configuration, the time function t has critical points of degenerate type. In fact, any point on bifurcate horizon F is critical. Here, we are not interested in such a nongeneric situation. Instead, we consider a slightly different time slicing 10 determined by the new time function
t ′ = t + ǫ sin 2 ϑ 2 ,
where ǫ > 0 is a sufficiently small positive constant and ϑ, which satisfies 0 ≤ ϑ ≤ π, is the usual polar coordinate of the 2-sphere. Then, there appears only a pair of isolated critical points at the north pole (ϑ = 0) and the south pole (ϑ = π) on the bifurcate horizon F , and the time function t ′ becomes the Morse function on H. At the time t ′ = 0, the black hole appears at the north pole. This is the 0-handle attachment. The black hole formed there grows into a geometrically thick spherical shell with a hole at the south pole, which is nevertheless a topological 3-disk. At the time t ′ = ǫ, the puncture at the south pole is filled, and the black hole region becomes topologically S 2 × [0, 1]. The deformed event horizon H splits into a disjoint union of a pair of 2-spheres. This is the 2-handle attachment. This kind of 2-handle attachment occurs because the event horizon is defined with respect to the two asymptotic ends, which is in general inadmissible if the future null infinity is connected, as we assume from this point. To understand the above statement, it should be noted that there is no process through which the several connected components of the exterior region E (t) = E ∩ Σ(t) at time t merge together at a later time because such a handle attachment is not admissible. It is also seen that no connected component of E (t) disappears, because possible n-handle attachments are inadmissible. These facts imply that the number of connected components of the exterior region E (t) cannot decrease with the time function t. On the other hand, there is only one connected component of the exterior region E (t) for sufficiently large t, because of the connectedness of I + . This observation shows that the exterior region E (t) remains connected in any process.
The only possible process through which the number of connected components of the exterior region E (t) changes is an (n -1)-handle attachment, as constructed above in the Schwarzschild space-time. This is because the subset
D λ × ∂D n-λ of H 2 H 1 i + 1 i + 2 t' = t' (p) p B (t'(p)) 111 I I 1 2 + + F Figure 11.
The figure on the left is a conformal diagram of the maximally extended Schwarzschild space-time. The structure of the event horizon defined with respect to the two asymptotic ends is depicted on the right, with one dimension omitted. The shaded region represents the black hole region at the critical time t = t(p). This corresponds to the 2-handle attachment, where the exterior region is separated into a pair of connected components.
the boundary of the λ-handle
∂h λ ≈ (∂D λ × D n-λ ) ∪ (D λ × ∂D n-λ ),
namely the part of ∂h λ which is the complement of the preimage of the attaching map f :
∂h λ ⊃ ∂D λ × D n-λ → H t ,
is disconnected only when λ = n -1. In this case, the homotopy type of the exterior region E (t) changes from that of an n-disk to that of S 0 , namely two points. Note, however, that this does not imply that the exterior region E (t) is always separated into two disconnected parts through the (n -1)-handle attachment. For example, a transition from the black ring horizon ≈ S n-2 × S 1 to the spherical black hole horizon ≈ S n-1 is realized through a black (n -1)-handle attachment, which pinches the longitude {a point} × S 1 ⊂ S n-2 × S 1 into a point. The exterior region E (t) remains connected all the while. Thus, there are both admissible and inadmissible processes for (n-1)-handle attachments. An (n-1)-handle attachment is inadmissible if it separates the exterior region E (t).
The arguments given in this paper are summerized by the following rules. Assume that (i) an (n + 1)-dimensional space-time M is asymptotically flat and the future null infinity I + is connected, or the event horizon H = ∂J -(I + ) is defined with respect to a single asymptotic end, (ii) the space-time M admits a smooth global time function t, (iii) the event horizon H can be deformed so that the black hole B(t) deformed accordingly at each time t is smooth and homeomorphic to original one B(t) at each time t and the time function t becomes the Morse function on H. Then, the topological evolution of the event horizon can be regarded as a λ-handle attachment (0 ≤ λ ≤ n) subject to the following rules: (1) The n-handle attachment is inadmissible. (2) Only the black λ-handle attachment (0 ≤ λ ≤ n -1), where the black hole region in the neighborhood of the critical point varies from the region homotopic to the sphere S λ-1 (regarded as the empty set for λ = 0) to the n-disk D n , is admissible. (3) The (n -1)-handle attachment which separates the spatial section of the exterior region of the black hole is inadmissible.
The first rule simply states that no connected component of a black hole disappears. It also implies that if a bubble of the exterior region forms within the black hole region, it does not vanish.
The second rule is concerned with the imbedding structure of the event horizon relative to the space-time manifold. The neighborhood of the critical point is separated into two regions by the event horizon. One changes homotopically from a sphere to a disk and the other from a disk to a sphere. We call it a black handle attachment when the former corresponds to the black hole region and a white handle attachment otherwise. Then, the second rule states that a white handle attachment never occurs. The reverse process, in which a black hole region homotopically changes from a disk to a sphere, is ruled out. A white 0-handle attachment, which 12 Figure 12. Black ring formation from a spherical black hole must be non-axisymmetric in real black hole space-times.
describes the emergence of the exterior region, is also forbidden. This gives another reason for the well-known result that a black hole cannot bifurcate, because it corresponds to a white (n -1)-handle attachment.
The second rule applies to more general situations. For example, let us consider the topological evolution of the event horizon from S n-1 to S n-2 × S 1 in (n + 1)dimensional space-time (n ≥ 3). When it is realized with a single critical point, it corresponds to a 1-handle attachment. Here, one might expect two possibilities if the second rule is not considered. One possibility is that the 1-handle is attached in the exterior region of the black hole. This is locally equivalent to the merging of a pair of black holes, where these two black holes are connected elsewhere irrelevant. The other possibility is that it is attached from the inside such that the 1-handle pierces the black hole region. In asymptotically flat space-times, only the latter includes axisymmetric configurations such that a spherical black hole is pinched out along the symmetric axis; here the axisymmetric configuration is such that the space-time possesses the SO(n -1) isometry and the time slicing respects this symmetry. However, this latter possibility corresponds to a white 1-handle attachment, which is impossible, and only the former, which corresponds to a black 1-handle attachment, is possible. In particular, a transition from a spherical event horizon (≈ S n-1 ) to a black ring horizon (≈ S n-2 ×S 1 ) in asymptotically flat spacetimes is always non-axisymmetric in the sense that such a configuration cannot possess SO(n -1) symmetry (Fig. 12 ).
While the apparent horizon must be diffeomorphic to a two-sphere in fourdimensional space-times under the dominant energy condition, a torus event horizon may appear, even under the dominant energy condition, via a black 1-handle attachment to the spherical horizon. More generally, an event horizon with an arbitrary number of genura may be formed by several black 1-handle attachments.
The third rule is not directly determined by the local structure of the critical point. It states that the exterior region E (t) = E ∩ Σ(t) at each time is always connected under the assumption that I + is connected. Thus, the possibility that there forms a bubble of the exterior region inside the black hole horizon is ruled out. It should, however, be noted that such a process is possible if I + consists of several connected components. This may also be related to the topological censorship theorem. [19] The topological censorship theorem states that all causal curves from I -to I + are homotopic under the null energy condition. This also forbids the formation of a bubble of the exterior region inside the black hole, because otherwise there would be two nonhomotopic causal curves from I -to I + , one 13 passing inside the horizon and the other outside. Our argument, however, does not depend on energy conditions. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Black holes in space-times of greater than or equal to five dimensions have rich topological structure. According to the well-known results of Hawking concerning the topology of black holes in four-dimensional space-time, the apparent horizon or the spatial section of the stationary event horizon is necessarily diffeomorphic to a 2-sphere. [1, 2] This follows from the fact that the total curvature, which is the integral of the intrinsic scalar curvature over the horizon, is positive under the dominant energy condition and from the Gauss-Bonnet theorem. Alternative and improved proofs of Hawking's theorem have been given by several authors. [3, 4, 5, 6] However in higher dimensional space-times, an apparent horizon or the spatial section of the stationary event horizon may not be a topological sphere, [7, 8, 9, 10] because the Gauss-Bonnet theorem does not hold in such cases. Nevertheless, the positivity of the total curvature of the horizon still holds. This puts certain topological restrictions on the black hole topology, though they are rather weak. For example, the apparent horizon in five-dimensional space-time can consist of finitely many connected sums of copies of S 3 /Γ and copies of S 2 × S 1 . In fact, exact solutions representing a black hole space-time possessing a horizon of nonspherical topology have recently been found in five-dimensional general relativity. When such black holes with nontrivial topologies are regarded as being formed in the course of gravitational collapse, questions regarding the evolution of the topology of black holes naturally arise. Our purpose here is to understand the time evolution of the topology of event horizons in a general setting. The relation between the crease set, where the event horizon is nondifferentiable, and the topology of the event horizon is studied in Refs. [11, 12, 13] for four-dimensional space-times. In the present work, we carry out a systematic investigation and find useful rules to determine admissible processes of topological evolution for time slicing of a black hole.\n\nOur approach is to utilize the Morse theory [14, 15] in differential topology. The Morse theory is useful for the purpose of understanding the topology of smooth 1 2 manifolds. The basic tool used in this approach is a smooth function on a differentiable manifold. The event horizon, however, is not a differentiable manifold but has a wedge-like structure at the past endpoints of the null geodesic generators of the horizon. For this reason, we first smooth the wedge. Then, the smooth time function which is assumed to exist plays the role of the Morse function on the smoothed event horizon. According to the Morse theory, the topological evolution of the event horizon can then be decomposed into elementary processes called \"handle attachments.\" In such a process, starting with a spherical horizon, one adds several handles, each characterized by the index of the critical points of the Morse function, which is an integer ranging from 0 to n (the dimension of the smoothed horizon as a differentiable manifold).\n\nThe purpose of the present article is to show that there are several constraints on the handle attachments for real black hole space-times."
},
{
"section_type": "OTHER",
"section_title": "The Morse theory for event horizons",
"text": "Let M be an (n + 1)-dimensional asymptotically flat space-time. We require the existence of a global time function t : M → R that is smooth and has an everywhere time-like and future-pointing gradient. The event horizon H is defined as the boundary of the causal past of the future null infinity H = ∂J -(I + ). [2] We treat the event horizon defined with respect to a single asymptotic end, unless otherwise stated. In other words, the future null infinity, I + , is assumed to be connected. The black hole region B is defined as the interior region of H, specifically, as B = M \\ J -(I + ), and the exterior region E of the black hole region is its complement, E = int(J -(I + )). We refer to the intersection of the black hole region and the time slice Σ(t 0 ) = {t = t 0 } as the black hole B(t 0 ) = B ∩ Σ(t 0 ) at time t = t 0 . The exterior region at time t = t 0 is, accordingly, written E (t 0 ) = E ∩ Σ(t 0 ). One of most basic properties of the event horizon is that it is generated by null geodesics without future endpoints. In general, the event horizon is not smoothly imbedded into the space-time manifold M , but it has a wedge-like structure at the past endpoints of the null geodesic generators, where distinct null geodesic generators intersect. We call the set of past endpoints of null geodesic generators of H, from which two or more null geodesic generators emanate, the crease set S. [11, 12] When no crease set S exists between the time slices t = t 1 and t = t 2 , the null geodesic generators of H naturally define a diffeomorphism ∂B(t 1 ) ≈ ∂B(t 2 ). Hence, the topological evolution of a black hole can take place only when the time slice intersects the crease set S. Of course, the event horizon itself is a gaugeindependent object. Nevertheless, we often understand the dynamics of space-time by scanning it along time slices. Thus, the topological evolution of a black hole depends on the time function.\n\nIt is expected that Morse theory [14] provides useful techniques to analyze such a process of topological evolution. Because the Morse theory is concerned with functions on smooth manifolds, we first regularize H around the crease set S. The event horizon is not necessarily smooth, even on H \\ S, in the case that the future null infinity I + has a pathological structure. [16] Here it is assumed that H is smooth on H \\ S. Then, small deformations of H near the crease set S will make H a smooth hypersurface H in M , while B(t 0 ) remains deformed in such a manner that ∂ B(t 0 ) = H ∪ Σ(t 0 ) holds and B(t 0 ) remains homeomorphic to the original black hole for all t 0 ∈ R. This deformation is assumed to be such that the time 3 Figure 1. An example in which no smoothing procedure makes t| e H a Morse function on H. Here, the intersection of the crease set S of the event horizon and t = t 0 hypersurface has an accumulation point. function t| e H , which is the restriction of t on H, gives a Morse function on H that has only nondegenerate critical points, where the gradient of t| e H defined on H becomes zero and where also the Hessian matrix (∂ i ∂ j t| e H ) of t| e H is nondegenerate. Though this assumption should hold for a wide class of systems, it does not always hold. Figure 1 gives an example for which no smoothing procedure makes the induced time function t| e H a Morse function on H, because the intersection of the crease set S of the event horizon and the t = t 0 hypersurface has an accumulation point. It is highly nontrivial to determine whether such a smoothing procedure is generically possible. It is, however, not easy nor the primary purpose of this article to assertain the realm of validity of the assumption, and therefore we make this assumption without inquiring into its validity.\n\nAccording to the Morse Lemma, there is a local coordinate system {x 1 , • • • , x n } on H in the neighborhood of the critical point p ∈ H such that the restriction t| e H of the time function t on H takes the form\n\nt| e H (x 1 , • • • , x n ) = t(p) -(x 1 ) 2 -• • • -(x λ ) 2 + (x λ+1 ) 2 + • • • + (x n ) 2 .\n\nThe integer λ, ranging from 0 to n, is called the index of the critical point p. The topology of the black hole B(t) changes when Σ(t) pass through critical points, or equivalently, when the time function t takes critical values. This implies that critical points appears only near the crease set S.\n\nThe gradient-like vector field for t| e H is defined to be the tangent vector field X on H such that Xt| e H > 0 holds on H, except for critical points, and has the form\n\nX = -2x 1 ∂ ∂x 1 -• • • -2x λ ∂ ∂x λ + 2x λ+1 ∂ ∂x λ+1 + • • • + 2x n ∂ ∂x n near\n\nthe critical point of index λ, in terms of the standard coordinate system appearing in the Morse Lemma. We choose a gradient-like vector field X such that it coincides with the future-directed tangent vector field of null geodesic generators of H, except in a small neighborhood of the crease set S (Fig. 2 ). The effect of a critical point p of index λ is equivalent to the attachment of a λ-handle. [14, 15] The handlebody is just a topological n-disk 0, 1] ), but it is regarded as the product space D n ≈ D λ × D n-λ (Fig. 3 ). The λ-handle attachment to an ndimensional manifold N with a boundary consists of the set\n\nD n ≈ I n (I = [\n\nh λ = (D λ × D n-λ , f ),\n\nwhere the attaching map f induces the imbedding of 4 ). The new manifold obtained through the λ-handle attachment to N is 4 Figure 2. The smoothing procedure of the event horizon H. The gradient-like vector field on H can be constructed through a slight deformation of the null geodesic generators of H. Here, the effect of the crease set S has been replaced by that of the critical points p 1 , p 2 and p 3 .\n\n∂D λ × D n-λ ⊂ ∂D n into ∂N (Fig.\n\nFigure 3. The local structure around the critical point p of index λ. It can be seen that H t(p)+ǫ is homeomorphic to H t(p)-ǫ with a λ-handle attached. given by\n\nN ∪ h λ = N ∪ (D λ × D n-λ )/(x ∼ f (x)), (x ∈ ∂D λ × D n-λ ).\n\nLet us denote by H t0 the t ≤ t 0 part of H. Then, H t(p)+ǫ (ǫ > 0) just above the critical point p of index λ is homeomorphic (in fact diffeomorphic, taking account of the smoothing procedure) to that just below p, H t(p)-ǫ attached with a λ-handle,\n\nH t(p)+ǫ ≈ H t(p)-ǫ ∪ h λ ,\n\nif there are no other critical points satisfying t(p)-ǫ ≤ t ≤ t(p)+ǫ. The handlebody itself is denoted by h λ as well.\n\nLet us consider several examples. The 0-handle attachment does not need an attaching map f . It simply corresponds to the emergence of the (n -1)-sphere S n-1 ≈ ∂D n as a black hole horizon ∂B(t). A typical example is the creation of a black hole (Fig. 5 ): A black hole always emerges as 0-handle attachment. The other 5 Figure 4. The attachment of a 1-handle and a 2-handle to a 3-manifold N creates a new 3-manifold N ∪ h 1 ∪ h 2 .\n\nE B Figure 5. The emergence of a black hole through a 0-handle attachment.\n\nE B Figure 6. The emergence of a bubble in the black hole region by 0-handle attachment, which does not occur in the real black hole space-times.\n\nE B Figure 7. The collision of a pair of black holes, creating a single black hole, is realized through 1-handle attachment.\n\npossiblity is the creation of a bubble that is subset of J -(I + ) in a black hole region (Fig. 6 ). One might think that this corresponds to wormhole creation between the internal and external regions of the event horizon. Although in the framework of the standard Morse theory on H, these two examples are indistinguishable, we below see that the latter process is in fact impossible.\n\nNext, we consider 1-handle attachment. A typical example is the collision of two black holes. A 1-handle serves as a bridge connecting black holes, or it corresponds to taking the connected sum of each component of multiple black holes (Fig. 7 ).\n\nFigure 8. The bifurcation of one black hole into two is represeted by an (n -1)-handle attachment. This, however, never occurs in real black hole space-times.\n\nλ λ λ λ Figure 9. The structure of λ-handle. The core D λ × {0} corresponds to the stable submanifold with respect to the flow generated by the gradient-like vector field, and the co-core {0} × D n-λ corresponds to the unstable submanifold.\n\nThe time reversal of the collision of black holes consists of the bifurcation of one black hole into two. This would be realized through an (n -1)-handle attachment, if such a process were possible (Fig. 8 ). It is, however, well known that such a process is forbidden. [2] In general, the time reversal of the λ-handle attachment corresponds to (n -λ)-handle attachment. Before discussing general cases, let us consider the structure of a handlebody. Recall that a λ-handle consists of the product space D λ × D n-λ . The subset D λ × {0} ⊂ D λ × D n-λ is called the core of the handlebody, and {0} × D n-λ ⊂ D λ × D n-λ is called the co-core. The core and co-core intersect transversely at a point. This point can be regarded as a critical point p.\n\nLet us refer to the subset\n\nW s (p) of H (1) W s (p) = {q ∈ M | lim t→+∞ exp q tX = p}\n\nwhich consists of points that converge to p along the flow generated by the gradientlike vector field X, as the stable manifold with respect to the critical point p. The stable manifold W s (p) is homeomorphic to R λ if the index of p is given by λ. [17] Similarly, let us refer to the subset W u (p) ⊂ H consisting of points which converge to p along the flow generated by (-X) as the unstable manifold with respect to p. For the unstable manifold, W u (p) ≈ R n-λ holds. The portions of the stable and unstable manifolds in the handlebody can be regarded as corresponding to the core and co-core, respectively.\n\nThe effect of smoothing the event horizon H to H is to deform the null vector field generating H into a gradient-like vector field X. The primary difference between the null geodesic generators and the flow generated by X is that the former does not have future endpoints, but the latter can. Thus, there are admissible and inadmissible processes for the smoothed manifold H. An admissible process is given by H, which is obtained from an in priciple realizable event horizon, while an inadmissible one is constructed from a spurious event horizon, i.e., one that consists of the null hypersurface containing null geodesic generators with a future endpoint."
},
{
"section_type": "OTHER",
"section_title": "The structure of the critical points",
"text": "The spatial topology of a black hole changes only when the time function takes a critical value. The time evolution of the black hole topology can be understood by considering its local structure around critical points. To determine whether a given topological change is admissible or inadmissible, it is not sufficient to consider only the intrinsic structure of the event horizon. Rather, it is required to take account of its imbedding structure relative to the space-time.\n\nIn a time slice, any point separate from H belongs to either of the black hole or the exterior of the black hole region. It is useful to consider the local behavior of the black hole region or the exterior region near the critical point p. Let us call the exterior E of the black hole region simply the exterior region, for brevity. The exterior region is slightly deformed by the smoothing procedure. The deformed exterior region is denoted by E , and the deformed exterior region at the time t by\n\nE (t) = E ∩ Σ(t) = Σ(t) \\ B(t). ( 2\n\n)\n\nThe 0-handle is placed at some t ≥ t(p). Such an attachment describes the emergence of the black hole region at the critical point p and its expansion with time. The emergence of a bubble, which consists of a part of J -(I + ), in the background of the black hole region would also be described by a 0-handle attachment. This, however, never occurs, as we explain below in detail. Hence, a 0-handle attachment always describes the creation of a black hole homeomorphic to the n-disk.\n\nAn n-handle attachment corresponds to the time reversal of a 0-handle attachement. This process, however, never occurs in real black hole space-time. An nhandle is defined for t ≤ t(p), which means that it terminates at the critical point p. The crease set is isolated into critical points during the course of the smoothing procedure. The gradient-like vector field, which can be regarded as being tangent to the generator of the deformed event horizon H, may have several inward (converging) directions at the critical point due to this smoothing procedure, while the original null generator of the event horizon does not have an inward direction at the crease set. In the case of the n-handle, all the directions become inward at the critical point. This implies that the null generators of the event horizon H must have future endpoints at the critical point, which is, of course, impossible. It is thus seen that an n-handle attachment never occurs in real black hole space-times.\n\nThe remaining cases are λ-handle attachments for 1 ≤ λ ≤ n -1. In these cases, the λ-handle lies on either side of the critical point p both in the future [t > t(p)] and in the past [t < t(p)]. Then, we consider the case in which the handle exists during the sufficiently small time interval t ∈ [t(p) -δ, t(p) + δ] (δ > 0), to understand the topological change of the black hole region at the critical point p.\n\n8 Figure 10. The neighborhood U of p is separated by h λ into the future region, U + , and the past region, U -.\n\nFirst, we introduce a coordinate system {t, x i } (i = 1, • • • , n) in the neighborhood U of p, where t is a given function of time, and {x i } is the extension over U of the cannonical coordinate appearing in the Morse Lemma such that each curve (x 1 , • • • , x n ) = [const] is timelike in U . We assume that U is the solid cylinder given by t ∈ [t(p) -δ, t(p) + δ], (x i ) 2 ≤ δ. In this coordinate system, the λ-handle h λ is given by the saddle surface\n\nt = t(p) -(x 1 ) 2 -• • • -(x λ ) 2 + (x λ+1 ) 2 + • • • + (x n ) 2\n\nin U , which is an acausal set if the constant δ is taken sufficiently small, since h λ is tangent to the space-like hypersurface t = t(p) at p. Therefore, h λ separates U into two open subsets, the future and past regions U + and U -of U , where U + and U -are the subsets lying chronological future and past, respectively, of\n\nh λ : U ± = I ± (h λ ) ∩ U .\n\nExplicitly, the future and past regions U ± are the regions satisfying 10 ).\n\nt ≷ t(p) -(x 1 ) 2 -• • • -(x λ ) 2 + (x λ+1 ) 2 + • • • + (x n ) 2 in U , respectively (Fig.\n\nBecause the λ-handle is a subset of the black hole boundary H, one of U ± is contained in the black hole region, B, and the other in the exterior region, E . However, the future region U + of U is always included in the black hole region, i.e. U + ⊂ B, and hence we have U -⊂ E , since the horizon is the boundary of the past set, J -(I + ). Therefore, the black hole region B(t(p) -ǫ) ∩ U in U at the time t = t(p) -ǫ just before the critical time is given by\n\n(x 1 ) 2 + • • • + (x λ ) 2 > (x λ+1 ) 2 + • • • + (x n ) 2 + ǫ,\n\nwhich is homotopic to the (λ -1)-sphere S λ-1 . (For λ = 1, S 0 simply consists of two points.) Similarly, B(t(p) + ǫ) ∩ U just after the critical time is given by\n\n(x 1 ) 2 + • • • + (x λ ) 2 + ǫ > (x λ+1 ) 2 + • • • + (x n ) 2 , 9\n\nwhich is homotopic to the n-disk. In this way, the black hole region restricted to the small neighborhood of the critical point p is initially homotopic to a sphere. Then, the internal region of the sphere is filled up at the critical time t = t(p) and eventually becomes homotopically trivial. The exterior region, E (t) ∩ U , in U is initially homotopic to an n-disk for t = t(p) -ǫ. Then, its (n -λ)-dimensional direction is penetrated by the black hole region at t = t(p), and thus it becomes homotopic to an (n -λ -1)-sphere S n-λ-1 for t = t(p) + ǫ. If the spurious event horizon is also taken into account, the future region U + might be a subset of E , and therefore the past region U -might be a subset of B. Then, the black hole region in the λ-handle might be homotopic to an n-disk initially and become homotopic to an (n -λ -1)-sphere finally, and vice versa for the exterior region. Let us refer to such a topological change of the black hole region B(t) ∩ U from a region homotopic to a sphere to a region homotopic to a disk as a black handle attachment, and that from a region homotopic to a disk to a region homotopic to the sphere as a white handle attachment. The above observation shows that only a black handle attachment occurs if a sufficiently small neighborhood of the critical point is considered. For example, a collision of black holes corresponds to a black 1-handle attachment, while the bifurcation of a black hole corresponds to a white (n -1)-handle attachment in the sense that the homotopy type of the exterior region E (t) ∩ U changes from that of S n-2 to that of D n . This local argument also elucidates te reason that a black hole collision is admissible while a black hole bifurcation, which is its time reversal, is inadmissible. We also note that the effect of time reversal is to convert a black λ-handle attachment into a white (n -λ)-handle attachment.\n\nIt is appropriate to refer to the 0-handle attachment corresponding to the creation of a black hole as a black 0-handle attachment. Then, the proposition above also applies to a 0-handle attachment."
},
{
"section_type": "OTHER",
"section_title": "Connectedness of the exterior region",
"text": "There also exist processes that are unrealizable due to global conditions. Let us, for a moment, consider the event horizon in maximally extended Schwarzschild space-time. Though we are interested in the event horizon defined with respect to a specific asymptotic end, for the purpose of explanation, we examine the event horizon defined with respect to a pair of asymptotic ends in Schwarzschild spacetime (Fig. 11 ).\n\nLet I + 1 and I + 2 be the pair of future null infinities of the maximally extended Schwarzschild space-time. The event horizon here is defined by H = ∂J -(I + 1 ∪ I + 2 ), which is nondifferentiable at the bifurcate horizon F = ∂J -(I + 1 )∩∂J -(I + 2 ). Let t be a global time function and χ be a global radial coordinate function such that each two-surface t, χ = [const] is invariant under the SO(3) isometry. These coordinates are chosen such that the bifurcation surface F is located at t = χ = 0 and the event horizon H is determined by t = |χ| around F . The smoothed event horizon H is also taken to be invariant under the SO(3) isometry. Due to the symmetry of the configuration, the time function t has critical points of degenerate type. In fact, any point on bifurcate horizon F is critical. Here, we are not interested in such a nongeneric situation. Instead, we consider a slightly different time slicing 10 determined by the new time function\n\nt ′ = t + ǫ sin 2 ϑ 2 ,\n\nwhere ǫ > 0 is a sufficiently small positive constant and ϑ, which satisfies 0 ≤ ϑ ≤ π, is the usual polar coordinate of the 2-sphere. Then, there appears only a pair of isolated critical points at the north pole (ϑ = 0) and the south pole (ϑ = π) on the bifurcate horizon F , and the time function t ′ becomes the Morse function on H. At the time t ′ = 0, the black hole appears at the north pole. This is the 0-handle attachment. The black hole formed there grows into a geometrically thick spherical shell with a hole at the south pole, which is nevertheless a topological 3-disk. At the time t ′ = ǫ, the puncture at the south pole is filled, and the black hole region becomes topologically S 2 × [0, 1]. The deformed event horizon H splits into a disjoint union of a pair of 2-spheres. This is the 2-handle attachment. This kind of 2-handle attachment occurs because the event horizon is defined with respect to the two asymptotic ends, which is in general inadmissible if the future null infinity is connected, as we assume from this point. To understand the above statement, it should be noted that there is no process through which the several connected components of the exterior region E (t) = E ∩ Σ(t) at time t merge together at a later time because such a handle attachment is not admissible. It is also seen that no connected component of E (t) disappears, because possible n-handle attachments are inadmissible. These facts imply that the number of connected components of the exterior region E (t) cannot decrease with the time function t. On the other hand, there is only one connected component of the exterior region E (t) for sufficiently large t, because of the connectedness of I + . This observation shows that the exterior region E (t) remains connected in any process.\n\nThe only possible process through which the number of connected components of the exterior region E (t) changes is an (n -1)-handle attachment, as constructed above in the Schwarzschild space-time. This is because the subset\n\nD λ × ∂D n-λ of H 2 H 1 i + 1 i + 2 t' = t' (p) p B (t'(p)) 111 I I 1 2 + + F Figure 11.\n\nThe figure on the left is a conformal diagram of the maximally extended Schwarzschild space-time. The structure of the event horizon defined with respect to the two asymptotic ends is depicted on the right, with one dimension omitted. The shaded region represents the black hole region at the critical time t = t(p). This corresponds to the 2-handle attachment, where the exterior region is separated into a pair of connected components.\n\nthe boundary of the λ-handle\n\n∂h λ ≈ (∂D λ × D n-λ ) ∪ (D λ × ∂D n-λ ),\n\nnamely the part of ∂h λ which is the complement of the preimage of the attaching map f :\n\n∂h λ ⊃ ∂D λ × D n-λ → H t ,\n\nis disconnected only when λ = n -1. In this case, the homotopy type of the exterior region E (t) changes from that of an n-disk to that of S 0 , namely two points. Note, however, that this does not imply that the exterior region E (t) is always separated into two disconnected parts through the (n -1)-handle attachment. For example, a transition from the black ring horizon ≈ S n-2 × S 1 to the spherical black hole horizon ≈ S n-1 is realized through a black (n -1)-handle attachment, which pinches the longitude {a point} × S 1 ⊂ S n-2 × S 1 into a point. The exterior region E (t) remains connected all the while. Thus, there are both admissible and inadmissible processes for (n-1)-handle attachments. An (n-1)-handle attachment is inadmissible if it separates the exterior region E (t)."
},
{
"section_type": "OTHER",
"section_title": "Concluding remarks",
"text": "The arguments given in this paper are summerized by the following rules. Assume that (i) an (n + 1)-dimensional space-time M is asymptotically flat and the future null infinity I + is connected, or the event horizon H = ∂J -(I + ) is defined with respect to a single asymptotic end, (ii) the space-time M admits a smooth global time function t, (iii) the event horizon H can be deformed so that the black hole B(t) deformed accordingly at each time t is smooth and homeomorphic to original one B(t) at each time t and the time function t becomes the Morse function on H. Then, the topological evolution of the event horizon can be regarded as a λ-handle attachment (0 ≤ λ ≤ n) subject to the following rules: (1) The n-handle attachment is inadmissible. (2) Only the black λ-handle attachment (0 ≤ λ ≤ n -1), where the black hole region in the neighborhood of the critical point varies from the region homotopic to the sphere S λ-1 (regarded as the empty set for λ = 0) to the n-disk D n , is admissible. (3) The (n -1)-handle attachment which separates the spatial section of the exterior region of the black hole is inadmissible.\n\nThe first rule simply states that no connected component of a black hole disappears. It also implies that if a bubble of the exterior region forms within the black hole region, it does not vanish.\n\nThe second rule is concerned with the imbedding structure of the event horizon relative to the space-time manifold. The neighborhood of the critical point is separated into two regions by the event horizon. One changes homotopically from a sphere to a disk and the other from a disk to a sphere. We call it a black handle attachment when the former corresponds to the black hole region and a white handle attachment otherwise. Then, the second rule states that a white handle attachment never occurs. The reverse process, in which a black hole region homotopically changes from a disk to a sphere, is ruled out. A white 0-handle attachment, which 12 Figure 12. Black ring formation from a spherical black hole must be non-axisymmetric in real black hole space-times.\n\ndescribes the emergence of the exterior region, is also forbidden. This gives another reason for the well-known result that a black hole cannot bifurcate, because it corresponds to a white (n -1)-handle attachment.\n\nThe second rule applies to more general situations. For example, let us consider the topological evolution of the event horizon from S n-1 to S n-2 × S 1 in (n + 1)dimensional space-time (n ≥ 3). When it is realized with a single critical point, it corresponds to a 1-handle attachment. Here, one might expect two possibilities if the second rule is not considered. One possibility is that the 1-handle is attached in the exterior region of the black hole. This is locally equivalent to the merging of a pair of black holes, where these two black holes are connected elsewhere irrelevant. The other possibility is that it is attached from the inside such that the 1-handle pierces the black hole region. In asymptotically flat space-times, only the latter includes axisymmetric configurations such that a spherical black hole is pinched out along the symmetric axis; here the axisymmetric configuration is such that the space-time possesses the SO(n -1) isometry and the time slicing respects this symmetry. However, this latter possibility corresponds to a white 1-handle attachment, which is impossible, and only the former, which corresponds to a black 1-handle attachment, is possible. In particular, a transition from a spherical event horizon (≈ S n-1 ) to a black ring horizon (≈ S n-2 ×S 1 ) in asymptotically flat spacetimes is always non-axisymmetric in the sense that such a configuration cannot possess SO(n -1) symmetry (Fig. 12 ).\n\nWhile the apparent horizon must be diffeomorphic to a two-sphere in fourdimensional space-times under the dominant energy condition, a torus event horizon may appear, even under the dominant energy condition, via a black 1-handle attachment to the spherical horizon. More generally, an event horizon with an arbitrary number of genura may be formed by several black 1-handle attachments.\n\nThe third rule is not directly determined by the local structure of the critical point. It states that the exterior region E (t) = E ∩ Σ(t) at each time is always connected under the assumption that I + is connected. Thus, the possibility that there forms a bubble of the exterior region inside the black hole horizon is ruled out. It should, however, be noted that such a process is possible if I + consists of several connected components. This may also be related to the topological censorship theorem. [19] The topological censorship theorem states that all causal curves from I -to I + are homotopic under the null energy condition. This also forbids the formation of a bubble of the exterior region inside the black hole, because otherwise there would be two nonhomotopic causal curves from I -to I + , one 13 passing inside the horizon and the other outside. Our argument, however, does not depend on energy conditions."
}
] | |
arxiv:0704.0101 | 0704.0101 | 1 | fe528113aa79866c45eb45e2b6d5f7f386f0359c1df414aa2148395b56badb6c | The birth of string theory | In this contribution we go through the developments that in the years 1968 to
1974 led from the Veneziano model to the bosonic string. | [
"Paolo Di Vecchia"
] | [
"hep-th"
] | hep-th | [] | 2007-04-01 | 2026-02-26 | The sixties was a period in which strong interacting processes were studied in detail using the newly constructed accelerators at Cern and other places. Many new hadronic states were found that appeared as resonant peaks in various cross sections and hadronic cross sections were measured with increasing accuracy. In general, the experimental data for strongly interacting processes were rather well understood in terms of resonance exchanges in the direct channel at low energy and by the exchange of Regge poles in the transverse channel at higher energy. Field theory that had been very successful in describing QED seemed useless for strong interactions given the big number of hadrons to accomodate in a Lagrangian and the strength of the pion-nucleon coupling constant that did not allow perturbative calculations. The only domain in which field theoretical techniques were successfully used was current algebra. Here, assuming that strong interactions were described by an almost chiral invariant Lagrangian, that chiral symmetry was spontaneously broken and that the pion was the corresponding Goldstone boson, field theoretical methods gave rather good predictions for scattering amplitudes involving pions at very low energy. Going to higher energy was, however, not possible with these methods.
Because of this, many people started to think that field theory was useless to describe strong interactions and tried to describe strong interacting 2 Fig. 1. Duality diagram for the scattering of four mesons
Starting from these ideas Veneziano [4] was able to construct an S matrix for the scattering of four mesons that, at the same time, had an infinite number of zero width resonances lying on linearly rising Regge trajectories and Regge behaviour at high energy. Veneziano originally constructed the model for the 1 For a discussion of S matrix theory see Ref.s [1] The birth of string theory 3 process ππ → πω, but it was immediately extended to the scattering of four scalar particles.
In the case of four identical scalar particles, the crossing symmetric scattering amplitude found by Veneziano consists of a sum of three terms:
A(s, t, u) = A(s, t) + A(s, u) + A(t, u) ( 1
)
where A(s, t) = Γ (-α(s))Γ (-α(t)) Γ (-α(s) -α(t)) = 1 0 dxx -α(s)-1 (1 -x) -α(t)-1 (2) with linearly rising Regge trajectories
α(s) = α 0 + α ′ s ( 3
)
This was a very important property to implement in a model because it was in agreement with the experimental data in a wide range of energies. s, t and u are the Mandelstam variables:
s = -(p 1 + p 2 ) 2 , t = -(p 3 + p 2 ) 2 , u = -(p 1 + p 3 ) 2 ( 4
)
The three terms in Eq. ( 1 ) correspond to the three orderings of the four particles that are not related by a cyclic or anticyclic 2 permutation of the external legs. They correspond, respectively, to the three permutations: (1234), (1243) and (1324) of the four external legs. They have only simple pole singularities.
The first one has only poles in the s and t channels, the second only in the s and u channels and the third only in the t and u channels. This property follows directly from the duality diagram that is associated to each inequivalent permutation of the external legs. In fact, at that time one used to associate to each of the three inequivalent permutations a duality diagram where each particle was drawn as consisting of two lines that rappresented the quark and antiquark making up a meson. Furthermore, the diagram was supposed to have only poles singularities in the planar channels which are those involving adjacent external lines. This means that, for instance, the duality diagram corresponding to the permutation (1234) has only poles in the s and t channels as one can see by deforming the diagram in the plane in the two possible ways shown in figure (2) . This was a very important property of the duality diagram that makes it qualitatively different from a Feynman diagram in field theory where each diagram has only a pole in one of the three s, t and u channels and not simultaneously in two of them. If we accept the idea that each term of the sum in Eq. (1) is described by a duality diagram, then it is clear that we 2 An anticyclic permutation corresponding, for instance, to the ordering (1234) is obtained by taking the reverse of the original ordering (4321) and then performing a cyclic permutation.
Fig. 2. The duality diagram contains both s and t channel poles do not need to add terms corresponding to equivalent diagrams because the corresponding duality diagram is the same and has the same singularities. It is now clear that it was in some way implicit in this picture the fact that the Veneziano model corresponds to the scattering of relativistic strings. But at that time the connection was not obvious at all. The only S matrix property that the Veneziano model failed to satisfy was the unitarity of the S matrix. because it contained only zero width resonances and did not have the various cuts required by unitarity. We will see how this property will be implemented. Immediately after the formulation of the Veneziano model, Virasoro [5] proposed another crossing symmetric four-point amplitude for scalar particles that consisted of a unique piece given by:
A(s, t, u) ∼ Γ (-α(u) 2 )Γ (-α(s) 2 )Γ (-α(t) 2 ) Γ (1 + α(u) 2 )Γ (1 + α(s) 2 )Γ (1 + α(t) 2 ) ( 5
)
where
α(s) = α 0 + α ′ s ( 6
)
The model had poles in all three s, t and u channels and could not be written as sum of three terms having poles only in planar diagrams. In conclusion, the Veneziano model satisfies the principle of planar duality being a crossing symmetric combination of three contributions each having poles only in the planar channels. On the other hand, the Virasoro model consists of a unique crossing symmetric term having poles in both planar and non-planar channels. The attempts to construct consistent models that were in good agreement with the strong interaction phenomenology of the sixties boosted enormously the activity in this research field. The generalization of the Veneziano model to the scattering of N scalar particles was built, an operator formalism consisting of an infinite number of harmonic oscillators was constructed and the complete spectrum of mesons was determined. It turned out that the degeneracy of states grew up exponentially with the mass. It was also found that the N point amplitude had states with negative norm (ghosts) unless the intercept of the Regge trajectory was α 0 = 1 [6] . In this case it turned out that the model was free of ghosts but the lowest state was a tachyon. The model was called in the literature the "dual resonance model".
The birth of string theory 5 The model was not unitary because all the states were zero width resonances and the various cuts required by unitarity were absent. The unitarity was implemented in a perturbative way by adding loop diagrams obtained by sewing some of the external legs together after the insertion of a propagator. The multiloop amplitudes showed a structure of Riemann surfaces. This became obvious only later when the dual resonance model was recognized to correspond to scattering of strings.
But the main problem was that the model had a tachyon if α 0 = 1 or had ghosts for other values of α 0 and was not in agreement with the experimental data: α 0 was not equal to about 1 2 as required by experiments for the ρ Regge trajectory and the external scalar particles did not behave as pions satisfying the current algebra requirements. Many attempts were made to construct more realistic dual resonance models, but the main result of these attempts was the construction of the Neveu-Schwarz [7] and the Ramond [8] models, respectively, for mesons and fermions. They were constructed as two independent models and only later were recognized to be two sectors of the same model. The Neveu-Schwarz model still contained a tachyon that only in 1976 through the GSO projection was eliminated from the physical spectrum. Furthermore, it was not properly describing the properties of the physical pions.
Actually a model describing ππ scattering in a rather satisfactory way was proposed by Lovelace and Shapiro [9] 3 . According to this model the three isospin amplitudes for pion-pion scattering are given by:
A 0 = 3 2 [A(s, t) + A(s, u)] - 1 2 A(t, u) A 1 = A(s, t) -A(s, u) A 2 = A(t, u) ( 7
)
where
A(s, t) = β Γ (1 -α(s))Γ (1 -α(t)) Γ (1 -α(t) -α(s)) ; α(s) = α 0 + α ′ s ( 8
)
The amplitudes in eq.( 7 ) provide a model for ππ scattering with linearly rising Regge trajectories containing three parameters: the intercept of the ρ Regge trajectory α 0 , the Regge slope α ′ and β. The first two can be determined by imposing the Adler's self-consistency condition, that requires the vanishing of the amplitude when s = t = u = m 2 π and one of the pions is massless, and the fact that the Regge trajectory must give the spin of the ρ meson that is equal to 1 when √ s is equal to the mass of the ρ meson m ρ . These two conditions determine the Regge trajectory to be:
α(s) = 1 2 1 + s -m 2 π m 2 ρ -m π 2 = 0.48 + 0.885s ( 9
)
3 See also Ref. [10] .
Having fixed the parameters of the Regge trajectory the model predicts the masses and the couplings of the resonances that decay in ππ in terms of a unique parameter β. The values obtained are in reasonable agreement with the experiments. Moreover, one can compute the ππ scattering lenghts:
a 0 = 0.395β a 2 = -0.103β ( 10
)
and one finds that their ratio is within 10% of the current algebra ratio given by a 0 /a 2 = -7/2. The amplitude in eq.(8) has exactly the same form as that for four tachyons of the Neveu-Schwarz model with the only apparently minor difference that α 0 = 1/2 (for m π = 0) instead of 1 as in the Neveu-Schwarz model. This difference, however, implies that the critical space-time dimension of this model is d = 4 4 and not d = 10 as in the Neveu-Schwarz model. In conclusion this model seems to be a perfectly reasonable model for describing low-energy ππ scattering. The problem is, however, that nobody has been able to generalize it to the multipion scattering and therefore to get the complete meson spectrum.
As we have seen the S matrix of the dual resonance model was constructed using ideas and tools of hadron phenomenology of the end of the sixties. Although it did not seem possible to write a realistic dual resonance model describing the pions , it was nevertheless such a source of fascination for those who actively worked in this field at that time for its beautiful internal structure and consistency that a lot of energy was used to investigate its properties and for understanding its basic structure. It turned out with great surprise that the underlying structure was that of a quantum relativistic string.
The aim of this contribution is to explain the logic of the work that was done in the years from 1968 to 1974 5 in order to uncover the deep properties of this model that appeared from the beginning to be so beautiful and consistent to deserve an intensive study.
This seems to me a very good way of celebrating the 65th anniversary of Gabriele who is the person who started and also contributed to develop the whole thing with his deep physical intuition.
We have seen that the construction of the four-point amplitude is not sufficient to get information on the full hadronic spectrum because it contains only those hadrons that couple to two ground state mesons and does not see those intermediate states which only couple to three or to an higher number of ground state mesons [12] . Therefore, it was very important to construct the N -point amplitude involving identical scalar particles. The construction of 4 This can be checked by computing the coupling of the spinless particle at the level α(s) = 2 and seeing that it vanishes for d = 4. 5 Reviews from this period can be found in Ref. [11] The birth of string theory 7 the N -point amplitude was done in Ref. [13] (extending the work of Ref. [14] ) by requiring the same principles that have led to the construction of the Veneziano model, namely the fact that the axioms of S-matrix theory be satisfied by an infinite number of zero width resonances lying on linearly rising Regge trajectories and planar duality.
The fully crossing symmetric scattering amplitude of N identical scalar particles is given by a sum of terms corresponding to the inequivalent permutations of the external legs:
A = Np n=1 A n ( 11
)
Also in this case two permutations of the external legs are inequivalent if they are not related by a cyclic or anticyclic permutation. N p is the number of inequivalent permutations of the external legs and is equal to N p = (N -1)! 2 and each term has only simple pole singularities in the planar channels. Each planar channel is described by two indices (i, j), to mean that it includes the legs i, i + 1, i + 2 . . . j -1, j, by the Mandelstam variable
s ij = -(p i + p i+1 + . . . + p j ) 2 ( 12
)
and by an additional variable u ij whose role will become clear soon. It is clear that the channels (ij) and (j + 1, i -1) 6 are identical and they should be counted only once. In the case of N identical scalar particles the number of planar channels is equal to N (N -3)
. This can be obtained as follows. The independent planar diagrams involving the particle 1 are of the type (1, i) where i = 2 . . . N -2. Their number is N -3. This is also the number of planar diagrams involving the particle 2 and not the 1. The number of planar diagrams involving the particle 3 and not the particles 1 and 2 is equal to N -4. In general the number of planar diagrams involving the particle i and not the previous ones from 1 to i-1 is equal to N -1 -i. This means that the total number of planar diagram is equal to: 2(N -3) + N -2 i=3 (N -1 -i) = 2(N -3) + N -4 i=1 i = = 2(N -3) + (N -4)(N -3) 2 = N (N -3) 2 (13) If one writes down the duality diagram corresponding to a certain planar ordering of the external particles, it is easy to see that the diagram can have simultaneous pole singularities only in N -3 channels. The channels that allow simultaneous pole singularities are called compatible channels, the other 6 This channel includes the particles (j + 1, . . . , N, 1, . . . i -1).
8 Paolo Di Vecchia are called incompatible. Two channels (i,j) and (h,k) are incompatible if the following inequalities are satisfied:
i ≤ h ≤ j ; j + 1 ≤ k ≤ i -1 ( 14
)
The aim is to construct the scattering amplitude for each inequivalent permutation of the external legs that has only pole singularities in the N (N -3) 2 planar channels. We have also to impose that the amplitude has simultaneous poles only in N -3 compatible channels. In order to gain intuition on how to proceed we rewrite the four-point amplitude in Eq. (2) as follows:
A(s, t) = 1 0 du 12 1 0 du 23 u
-α(s12)-1 12 u -α(s23)-1 23 δ(u 12 + u 23 -1) (15)
where u 12 and u 23 are the variables corresponding to the two planar channels (12) and (23) and the cancellation of simultaneous poles in incompatible channels is provided by the δ-function which forbids u 12 and u 23 to vanish simultaneously. We will now extend this procedure to the N -point amplitude. But for the sake of clarity let us start with the case of N = 5 [14] . In this case we have 5 planar channels described by u 12 , u 13 , u 23 , u 24 and u 34 . Since we have only two compatible channels only two of the previous five variables are independent. We can choose them to be u 12 and u 13 . In order to determine the dependence of the other three variables on the two independent ones, we exclude simultaneous poles in incompatible channels. This can be done by imposing relations that prevent variables corresponding to incompatible channels to vanish simultaneously. A sufficient condition for excluding simultaneous poles in incompatible channels is to impose the conditions:
u P = 1 - P u P ( 16
)
where the product is over the variables P corresponding to channels that are incompatible with P . In the case of the five-point amplitude we get the following relations: u 23 = 1 -u 34 u 12 ; u 24 = 1 -u 13 u 12 u 13 = 1 -u 34 u 24 ; u 34 = 1 -u 23 u 13 ; u 12 = 1 -u 24 u 23 (17) Solving them in terms of the two independent ones we get:
u 23 = 1 -u 12 1 -u 12 u 13 ; u 34 = 1 -u 13 1 -u 12 u 13 ; u 24 = 1 -u 12 u 13 ( 18
)
In analogy with what we have done for the four-point amplitude in Eq. ( 15 ) we write the five-point amplitude as follows:
The birth of string theory 9 1 0 du 12 1 0 du 13 1 0 du 23 1 0 du 24 1 0 du 34 u
-α(s12)-1 12 u -α(s13)-1 13 × ×u -α(s24)-1 24 u -α(s23)-1 23 u -α(s34)-1 34 × δ(u 23 + u 12 u 34 -1)δ(u 24 + u 12 u 13 -1)δ(u 34 + u 13 u 23 -1) (19) Performing the integral over the variables u 23 , u 24 and u 34 we get:
1 0 du 12 1 0 du 13 u
-α(s12)-1 12 u -α(s13)-1 13 × × (1 -u 12 ) -α(s23)-1 (1 -u 13 ) -α(s13)-1 (1 -u 12 u 13 ) -α(s24)+α(s23)+α(s34) (20)
We have implicitly assumed that the Regge trajectory is the same in all channels and that the external scalar particles have the same common mass m and are the lowest lying states on the Regge trajectory. This means that their mass is given by:
α 0 -α ′ p 2 i = 0 ; p 2 i ≡ -m 2 ( 21
)
Using then the relation:
α(s 23 ) + α(s 34 ) -α(s 24 ) = 2α ′ p 2 • p 4 ( 22
)
we can rewrite Eq. (20) as follows:
B 5 = 1 0 du 2 1 0 du 3 u
-α(s2)-1 2 u -α(s3)-1 3 (1 -u 2 ) -α(s23)-1 × × (1 -u 3 ) -α(s34)-1
2 i=2 4 j=4 (1 -x ij ) 2α ′ pi•pj ( 23
)
where
s i ≡ s 1i , u i ≡ u 1i ; i = 2, 3 ; x ij = u i u i+1 . . . u j-1 . ( 24
)
We are now ready to construct the N -point function [13] . In analogy with what has been done for the four and five-point amplitudes we can write the N -point amplitude as follows:
B N = 1 0 . . . 1 0 P [u -α(sP )-1 P ] Q δ(u Q -1 + Q u Q) ( 25
)
10 Paolo Di Vecchia
where the first product is over the N (N -3) 2 variables corresponding to all planar channels, while the second one is over the (N -3)(N -2) 2 independent δ-functions. The product in the δ-function is defined in Eq. (16) .
The solution of all the non-independent linear relations imposed by the δ-functions is given by
u ij = (1 -x ij )(1 -x i-1,j+1 ) (1 -x i-1,j )(1 -x i,j+1 ) ( 26
)
where the variables x ij are given in Eq. ( 24 ). Eliminating the δ-function from Eq. (25) one gets:
B N = N -2 i=2 1 0 du i u
-α(si)-1 i (1 -u i ) -α(si,i+1)-1 N -3 i=2 N -1 j=i+2 (1 -x ij ) -γij (27) where γ ij = α(s ij ) + α(s i+1;j-1 ) -α(s i;j-1 ) -α(s i+1;j ) ; j ≥ i + 2 (28)
It is easy to see that
α(s i,i+1 ) = -α 0 -2α ′ p i • p i+1 ; γ ij = -2α ′ p i • p j ; j ≥ i + 2 ( 29
)
Inserting them in Eq. (27) we get:
B N = N -2 i=2 1 0 du i u
-α(si)-1 i (1 -u i ) α0-1 N -2 i=2 N -1
j=i+1 (1 -x ij ) 2α ′ pi•pj ( 30
)
This is the form of the N -point amplitude that was originally constructed. Then Koba and Nielsen [15] put it in the form that is more known nowadays. They constructed it using the following rules. They associated a real variable z i to each leg i. Then they associated to each channel (i, j) an anharmonic ratio constructed from the variables z i , z i-1 , z j , z j+1 in the following way
(z i , z i+1 , z j , z j+1 ) -α(sij )-1 = (z i -z j )(z i-1 -z j+1 ) (z i-1 -z j )(z i -z j+1 )
-α(sij )-1 31) and finally they gave the following expression for the N -point amplitude:
(
B N = ∞ -∞ dV (z)
(i,j)
(z i , z i+1 , z j , z j+1 ) -α(sij )-1 ( 32
)
where
dV (z) = N 1 [θ(z i -z i+1 )dz i ] N i=1 (z i -z i+2 )dV abc ; dV abc = dz a dz b dz c (z b -z a )(z c -z b )(z a -z c ) ( 33
)
The birth of string theory 11 and the variables z i are integrated along the real axis in a cyclically ordered way: z 1 ≥ z 2 . . . ≥ z N with a, b, c arbitrarily chosen. The integrand of the N -point amplitude is invariant under projective transformations acting on the leg variables z i :
z i → αz i + β γz i + δ ; i = 1 . . . N ; αδ -βγ = 1 ( 34
)
This is because both the anharmonic ratio in Eq. ( 31 ) and the measure dV abc are invariant under a projective transformation. Since a projective transformation depends on three real parameters, then the integrand of the N -point amplitude depends only on N -3 variables z i . In order to avoid infinities, one has then to divide the integration volume with the factor dV abc that is also invariant under the projective transformations. The fact that the integrand depends only on N -3 variables is in agreement with the fact that N -3 is also the maximal number of simultaneous poles allowed in the amplitude. It is convenient to write the N -point amplitude in a form that involves the scalar product of the external momenta rather than the Regge trajectories. We distinguish three kinds of channels. The first one is when the particles i and j of the channel (i, j) are separated by at least two particles. In this case the channels that contribute to the exponent of the factor (z i -z j ) are the channels (i, j) with exponent equal to -α(s ij ) -1, (i + 1, j -1) with exponent -α(s i+1,j-1 ) -1, (i + 1, j) with exponent α(s i+1,j ) + 1 and (i, j -1) with exponent α(s i,j-1 ) + 1. Adding these four contributions one gets for the channels where i and j are separated by at least two particles
-α(s ij ) -α(s i+1,j-1 ) + α(s i+1,j ) + α(s i,j-1 ) = 2α ′ p i • p j ( 35
)
The second one comes from the channels that are separated by only one particle. In this case only three of the previous four channels contribute. For instance if j = i + 2 the channel (i + 1, j -1) consists of only one particle and therefore should not be included. This means that we would get:
-α(s i;i+2 ) -1 + α(s 1+1;i+2 ) + 1 + α(s i;i+1 + 1) = 1 + 2α ′ p i • p i+2 ( 36
)
Finally the third one that comes from the channels whose particles are adjacent, gets only contribution from:
-α(s i;i+1 ) -1 = α 0 -1 + 2α ′ p i • p i+1 ( 37
)
Putting all these three terms together in Eq. ( 32 ) and remembering the factor in the denominator in the first equation of (33) we get:
B N = ∞ -∞ N 1 dz i θ(z i -z i+1 ) dV abc N i=1 (z i -z i+1 ) α0-1 j>i (z i -z j ) 2α ′ pi•pj ( 38
)
A convenient choice for the three variables to keep fixed is:
12 Paolo Di Vecchia
z a = z 1 = ∞ ; z b = z 2 = 1 ; z c = z N = 0 ( 39
)
With this choice the previous equation becomes:
B N = N -1 i=3 1 0 dz i θ(z i -z i+1 ) N -1 i=2 (z i -z i+1 ) α0-1 × × N -1 i=2 N j=i+1 (z i -z j ) 2α ′ pi•pj ( 40
)
We now want to show that this amplitude is identical to the one given in Eq. ( 30 ). This can be done by performing the following change of variables:
u i = z i+1 z i ; i = 2, 3 . . . N -2 ( 41
) that implies z i = u 2 u 3 . . . u i-1 ; i = 3, 4 . . . N -1 ( 42
)
Taking into account that the Jacobian is equal to:
det ∂z ∂u = N -2 i=3 z i = N -3 i=2 u N -2-i i ( 43
)
using the following two relations:
det ∂z ∂u N -1 i=2 (z i -z i+1 ) α0-1 = N -2 i=2 u (N -1-i)α0-1 i N -2 i=2 (1 -u i ) α0-1 ( 44
) and N -1 i=2 N j=i+1 (z j -z i ) 2α ′ pi•pj = = N -2 i=2 N -1 j=i+1 (1 -x ij ) 2α ′ pi•pj N -2 i=2 u -α(si)-(N -i-1)α0 i ( 45
)
and the conservation of momentum
N i=1 p i = 0 ( 46
)
together with Eq. ( 21 ), one can easily see that Eq.s (30) and (40) are equal.
The birth of string theory 13 The N -point amplitude that we have constructed in this section corresponds to the scattering of N spinless particles with no internal degrees of freedom. On the other hand it was known that the mesons were classified according to multiplets of an SU (3) flavour symmetry. This was implemented by Chan and Paton [16] by multiplying the N -point amplitude with a factor, called Chan-Paton factor, given by T r(λ a1 λ a2 . . . λ aN ) (
) 47
where the λ's are matrices of a unitary group in the fundamental representation. Including the Chan-Paton factors the total scattering amplitude is given by: P T r(λ a1 λ a2 . . . λ aN
)B N (p 1 , p 2 , . . . p N ) ( 48
)
where the sum is extended to the (N -1)! permutations of the external legs, that are not related by a cyclic permutations. Originally when the dual resonance model was supposed to describe strongly interacting mesons, this factor was introduced to represent their flavour degrees of freedom. Nowadays the interpretation is different and the Chan-Paton factor represents the colour degrees of freedom of the gauge bosons and the other massive particles of the spectrum. The N -point amplitude B N that we have constructed in this section contains only simple pole singularities in all possible planar channels. They correspond to zero width resonances located at non-negative integer values n of the Regge trajectory α(M 2 ) = n. The lowest state located at α(m 2 ) = 0 corresponds to the particles on the external legs of B N . The spectrum of excited particles can be obtained by factorizing the N -point amplitude in the most general channel with any number of particles. This was done in Ref.s [17] and [18] finding a spectrum of states rising exponentially with the mass M . Being the model relativistic invariant it was found that many states obtained by factorizing the N -point amplitude were "ghosts", namely states with negative norm as one finds in QED when one quantizes the electromagnetic field in a covariant gauge. The consistency of the model requires the existence of relations satisfied by the scattering amplitudes that are similar to those obtained through gauge invariance in QED. If the model is consistent they must decouple the negative norm states leaving us with a physical spectrum of positive norm states. In order to study in a simple way these issues, we discuss in the next section the operator formalism introduced already in 1969 [19, 20, 21] .
Before concluding this section let us go back to the non-planar four-point amplitude in Eq. ( 5 ) and discuss its generalization to an N -point amplitude. Using the technique of the electrostatic analogue on the sphere instead of on the disk Shapiro [22] was able to obtain a N -point amplitude that reduces to the four-point amplitude in Eq. ( 5 ) with intercept α 0 = 2. The N -point amplitude found in Ref. [22] is:
14 Paolo Di Vecchia
N i=1 d 2 z i dV abc i<j |z i -z j | α ′ pi•pj ( 49
)
where
dV abc = d 2 z a d 2 z b d 2 z c |z a -z b | 2 |z a -z c | 2 |z b -z c | 2 ( 50
)
The integral in Eq. ( 49 ) is performed in the entire complex plane.
The factorization properties of the dual resonance model were first studied by factorizing by brute force the N-point amplitude at the various poles [17, 18] . The number of terms that factorize the residue of the pole at α(s) = n, increases rapidly with the value of n. In order to find their degeneracy it turned out to be convenient to first rewrite the N-point amplitude in an operator formalism. In this section we introduce the operator formalism and we rewrite the N -point amplitude derived in the previous section in this formalism. The key idea [19, 20, 21] is to introduce an infinite set of harmonic oscillators and a position and momentum operators 7 which satisfy the following commutation relations:
[a nµ , a † mν ] = η µν δ nm ; [q µ , pν ] = iη µν ( 51
)
where η µν is the flat Minkowski metric that we take to be η µν = (-1, 1, . . . 1). A state with momentum p is constructed in terms of a state with zero momentum as follows:
p|p ≡ pe ip•q |0 = p|p ; p |0 = 0 ( 52
) normalized as 8 p|p ′ = (2π) d δ (d) (p + p ′ ) ( 53
)
In order to avoid minus signs we use the convention that
p| = 0|e ip•q ( 54
)
A complete and orthonormal basis of vectors in the harmonic oscillator space is given by
|λ 1 , λ 2 , . . . λ i ; p = n (a † µn;n ) λn;µ n λ n,µn ! e ipq |0, 0 ( 55
)
7 Actually the position and momentum operators were introduced in Ref. [23] . 8 Although we now use an arbitrary d we want to remind you that all original calculations were done for d = 4.
The birth of string theory 15
where the first |0 corresponds to the one annihilated by all annihilation operators and the second one to the state of zero momentum:
a µn;n |0, 0 = p|0, 0 = 0 ( 56
)
Notice that Lorentz invariance forces to introduce also oscillators that create states with negative norm due to the minus sign in the flat Minkowski metric. This implies that the space spanned by the states in Eq. (55) is not positive definite. This is, however, not allowed in a quantum theory and therefore if the dual resonance model is a consistent quantum-relavistic theory we expect the presence of relations of the kind of those provided by gauge invariance in QED.
Let us introduce the Fubini-Veneziano [23] operator:
Q µ (z) = Q (+) µ (z) + Q (0) µ (z) + Q (-) µ (z) ( 57
)
where
Q (+) = i √ 2α ′ ∞ n=1 a n √ n z -n ; Q (-) = -i √ 2α ′ ∞ n=1 a † n √ n z n Q (0) = q -2iα ′ p log z ( 58
)
In terms of Q we introduce the vertex operator corresponding to the external leg with momentum p: V (z; p) =: e ip•Q(z) :≡ e ip•Q (-)
(z) e ipq e +2α ′ p•p log z e ip•Q (+) (z) ( 59
)
and compute the following vacuum expectation value:
0, 0| N i=1 V (z i , p i )|0, 0 ( 60
)
It can be easily computed using the Baker-Haussdorf relation
e A e B = e B e A e [A,B] ( 61
)
that is valid if the commutator, as in our case, [A, B] is a c-number. In our case the commutation relations to be used are:
[Q (+) (z), Q (-) (w)] = -2α ′ log 1 - w z ( 62
)
and the second one in Eq. ( 51 ). Using them one gets:
V (z; p)V (w; k) =: V (z; p)V (w; k) : (z -w) 2α ′ p•k ( 63
) and 16
Paolo Di Vecchia 0, 0| N i=1 V (z i , p i )|0, 0 = i>j (z i -z j ) 2α ′ pi•pj (2π) d δ (d) ( N i=1 p i ) ( 64
)
where the normal ordering requires that all creation operators be put on the left of the annihilation one and the momentum operator p be put on the right of the position operator q. This means that
(2π) d δ (d) ( N i=1 p i )B N = ∞ -∞ N 1 dz i θ(z i -z i+1 ) dV abc N i=1 (z i -z i+1 ) α0-1 × × 0, 0| N i=1 V (z i , p i )|0, 0 ( 65
)
By choosing the three variables z a , z b and z c as in Eq. ( 39 ) we can rewrite the previous equation as follows:
(2π) d δ (d) ( N i=1 p i )B N = 1 0 N -1 i=3 dz i N -1 i=2 θ(z i -z i+1 )× × N -1 i=2 (z i -z i+1 ) α0-1 0, p 1 | N -1 i=2 V (z i ; p i )|0, p N ( 66
)
where we have taken z 2 = 1 and we have defined (α 0 ≡ α ′ p 2 i ; i = 1 . . . N ) :
lim zN →0 V (z N ; p N )|0, 0 ≡ |0; p N ; 0; 0| lim z1→∞ z 2α0 1 V (z 1 ; p 1 ) = 0, p 1 | ( 67
)
Before proceeding to factorize the N -point amplitude let us study the properties under the projective group of the operators that we have introduced. We have already seen that the projective group leaves the integrand of the Koba-Nielsen representation of the N -point amplitude invariant. The projective group has three generators L 0 , L 1 and L -1 corresponding respectively to dilatations, inversions and translations. Assuming that the Fubini-Veneziano fields Q(z) transforms as a field with weight 0 (as a scalar) we can immediately write the commutation relations that Q(z) must satisfy. This means in fact that, under a projective transformation, Q(z) transforms as follows:
Q(z) → Q T (z) = Q αz + β γz + δ ; αδ -βγ = 1 ( 68
)
Expanding for small values of the parameters we get:
Q T (z) = Q(z) + (ǫ 1 + ǫ 2 z + ǫ 3 z 2 ) dQ(z) dz + o(ǫ 2 ) ( 69
)
The birth of string theory 17 This means that the three generators of the projective group must satisfy the following commutation relations with Q(z):
[L 0 , Q(z)] = z dQ dz ; [L -1 , Q(z)] = dQ dz ; [L 1 , Q(z)] = z 2 dQ dz ( 70
)
They are given by the following expressions in terms of the harmonic oscillators:
L 0 = α ′ p2 + ∞ n=1 na † n • a n ; L 1 = √ 2α ′ p • a 1 + ∞ n=1 n(n + 1)a n+1 • a † n ( 71
)
and
L -1 = L † 1 = √ 2α ′ p • a † 1 + ∞ n=1 n(n + 1)a † n+1 • a n ( 72
)
They annihilate the vacuum
L 0 |0, 0 = L 1 |0, 0 = L -1 |0, 0 = 0 ( 73
)
that is therefore called the projective invariant vacuum, and satisfy the algebra that is called Gliozzi algebra [24] 9 :
[L 0 , L 1 ] = -L 1 ; [L 0 , L -1 ] = L -1 ; [L 1 , L -1 ] = 2L 0 ( 74
)
The vertex operator with momentum p is a projective field with weight equal to α 0 = α ′ p 2 . It transforms in fact as follows under the projective group:
[L n , V (z, p)] = z n+1 dV (z, p) dz + α 0 (n + 1)z n V (z, p) ; n = 0, ±1 ( 75
)
or in finite form as follows:
U V (z, p)U -1 = 1 (γz + δ) 2α0 V αz + β γz + δ , p ( 76
)
where U is the generator of an arbitrary finite projective transformation. Since U leaves the vacuum invariant, by using Eq. (76) it is easy to show that:
0, 0| N i=1 V (z ′ i , p)|0, 0 = N i=1 (γz i + δ) 2α0 0, 0| N i=1 V (z i , p)|0, 0 ( 77
)
that together with the following equation:
N i=1 dz ′ i N i=1 (z ′ i -z ′ i+1 ) α0-1 = N i=1 dz i N -1 i=1 (z i -z i+1 ) α0-1 N i=1 (γz i + δ) -2α0 ( 78
)
9 See also Ref. [25] .
implies that the integrand of the N -point amplitude in Eq. (65) is invariant under projective transformations. We are now ready to factorize the N -point amplitude and find the spectrum of mesons.
From Eq.s (75) and (76) it is easy to derive the transformation of the vertex operator under a finite dilatation:
z L0 V (1, p)z -L0 = V (z, p)z α0 ( 79
)
Changing the integration variables as follows:
x i = z i+1 z i ; i = 2, 3 . . . N -2 ; det ∂z i ∂x j = z 3 z 4 . . . z N -2 ( 80
)
where the last term is the jacobian of the trasformation from z i to x i , we get from Eq.(66) the following expression:
A N ≡ 0, p 1 |V (1, p 2 )DV (1, p 3 ) . . . DV (1, p N -1 )|0, p N ( 81
)
where the propagator D is equal to:
D = 1 0 dxx L0-1-α0 (1 -x) α0-1 = Γ (L 0 -α 0 )Γ (α 0 ) Γ (L 0 ) ( 82
)
and
A N = (2π) d δ (d) N i=1 p i B N ( 83
)
The factorization properties of the amplitude can be studied by inserting in the channel (1, M ) or equivalently in the channel (M + 1, N ) described by the Mandelstam variable
s = -(p 1 + p 2 + . . . p M ) 2 = -(p M+1 + p M+2 . . . + p N ) 2 ≡ -P 2 ( 84
)
the complete set of states given in Eq. ( 55 ):
A N = λ,µ p (1,M) |λ, P λ, P |D|µ, P µ, P |p (M+1,N ) ( 85
)
where
p (1,M) | = 0, p 1 |V (1, p 2 )DV (1, p 3 ) . . . V (1, p M ) ( 86
) and |p (M+1,N ) = V (1, p M+1 )D . . . V (1, p N -1 )|p N , 0 ( 87
)
Introducing the quantity:
The birth of string theory 19 R =
∞ n=1 na † n • a n ( 88
)
it is possible to rewrite
λ, P |D|µ, P = ∞ m=0 λ, P | (-1) m α 0 -1 m R + m -α(s) |µ, P ( 89
)
where s is the variable defined in Eq. ( 84 ). Using this equation we can rewrite Eq. (85) as follows
A N = λ,µ p (1,M) |λ, P ∞ m=0 λ, P | (-1) m α 0 -1 m R + m -α(s) |µ, P µ, P |p (M+1,N ) ( 90
)
This expression shows that amplitude A N has a pole in the channel (1, M ) when α(s) is equal to an integer n ≥ 0 and the states |λ that contribute to its residue are those satisfying the relation:
R|λ = (n -m)|λ ; m = 0, 1 . . . n ( 91
)
The number of independent states |λ contributing to the residue gives the degeneracy of states for each level n. Because of manifest relativistic invariance the space spanned by the complete system of states in Eq. (55) contains states with negative norm corresponding to those states having an odd number of oscillators with timelike directions (see Eq. (51)). This is not consistent in a quantum theory where the states of a system must span a positive definite Hilbert space. This means that there must exist a number of relations satisfied by the external states that decouple a number of states leaving with a positive definite Hilbert space. In order to find these relations we rewrite the state in Eq. (87) going back to the Koba-Nielsen variables:
|p (1,M) = M-1 i=2 [ dz i θ(z i -z i+1 )] M-1 i=1 (z i -z i+1 ) α0-1 × × V (1, p 1 )V (z 2 , p 2 ) . . . V (z M-1 , p M-1 )|0, p M ( 92
)
Let us consider the operator U (α) that generate the projective transformation that leaves the points z = 0, 1 invariant:
z ′ = z 1 -α(z -1) = z + α(z 2 -z) + o(α 2 ) ( 93
)
From the transformation properties of the vertex operators in Eq. (76) it is easy to see that the previous transformation leaves the state in Eq. (92) invariant:
20 Paolo Di Vecchia
U (α)|p (1,M) = |p (1,M) ( 94
)
This means that the generator of the previous transformation annihilates the state in Eq. (92):
W 1 |p (1,M) = 0 ; W 1 = L 1 -L 0 ( 95
)
The explicit form of W 1 follows from the infinitesimal form of the transformation in Eq. ( 93 ). This condition that is of the same kind of the relations that on shell amplitudes with the emission of photons satisfy as a consequence of gauge invariance, implies that the residue at the pole in Eq. ( 90 ) can be factorized with a smaller number of states. It turns out, however, that a detailed analysis of the spectrum shows that negative norm states are still present. This can be qualitatively understood as follows. Due to the Lorentz metric we have a negative norm component for each oscillator. In order to be able to decouple all negative norm states we need to have a gauge condition of the type as in Eq. (95) for each oscillator. But the number of oscillators is infinite and, therefore, we need an infinite number of conditions of the type as in Eq. ( 95 ). It was found in Ref. [6] that, if we take α 0 = 1, then one can easily construct an infinite number of operators that leave the state in Eq. (92) invariant. In the next section we will concentrate on this case.
4 The case α 0 = 1
If we take α 0 = 1 many of the formulae given in the previous section simplify. The N -point amplitude in Eq. (38) becomes:
B N = ∞ -∞ N 1 dz i θ(z i -z i+1 ) dV abc j>i (z i -z j ) 2α ′ pi•pj ( 96
)
that can be rewritten in the operator formalism as follows:
(2π) 4 δ( N i=1 p i )B N = ∞ -∞ N 1 dz i θ(z i -z i+1 ) dV abc 0, 0| N i=1 V (z i , p i )|0, 0 ( 97
) By choosing z 1 = ∞, z 2 = 1 and z N = 0 it becomes (2π) 4 δ( N i=1 p i )B N = = 1 0 N -1 i=3 dz i N -1 i=2 θ(z i -z i+1 ) 0, p 1 | N -1 i=2 V (z i ; p i )|0, p N ( 98
)
The birth of string theory 21 where lim
zN →0 V (z N ; p N )|0, 0 ≡ |0; p N ; 0; 0| lim z1→∞ z 2 1 V (z 1 ; p 1 ) = 0, p 1 | ( 99
)
Eq. (81) is as before, but now the propagator becomes:
D = dxx L0-2 = 1 L 0 -1 ( 100
)
This means that Eq. (89) becomes:
λ, P |D|µ, P = λ, P | 1 L 0 -1 |µ, P ( 101
)
and Eq. ( 90 ) has the simpler form:
B N = λ p (1,M) |λ, P λ, P | 1 R -α(s) |λ, P λ, P |p (M+1,N ) ( 102
)
B N has a pole in the channel (1, M ) when α(s) is equal to an integer n ≥ 0 and the states |λ that contribute to its residue are those satisfying the relation:
R|λ = n|λ ( 103
)
Their number gives the degeneracy of the states contributing to the pole at α(s) = n. The N -point amplitude can be written as:
B N = p (1,M) |D|p (M+1,N ) ( 104
) where |p (1,M) = M-1 i=2 [dz i θ(z i -z i+1 )] × × V (1, p 1 )V (z 2 , p 2 ) . . . V (z M-1 , p M-1 |0, p M ( 105
)
Using Eq. ( 79 ) and changing variables from
z i , i = 2 . . . M -1 to x i = zi+1 zi , i = 1 . . . M -2 with z 1 = 1
we can rewrite the previous equation as follows:
|p (1,M) = V (1, p 1 )DV (1, p 2 ) . . . DV (1, p M-1 )|0, p M ( 106
)
where the propagator D is defined in Eq. ( 100 ). We want now to show that the state in Eq.s (105) and (106) is not only annihilated by the operator in Eq. ( 95 ), but, if α 0 = 1 [6], by an infinite set of operators whose lowest one is the one in Eq. (95) . We will derive this by using the formalism developed in Ref. [26] and we will follow closely their derivation.
Starting from Eq.s (70) Fubini and Veneziano realized that the generators of the projective group acting on a function of z are given by:
22 Paolo Di Vecchia
L 0 = -z d dz ; L -1 = - d dz ; L 1 = -z 2 d dz ( 107
)
They generalized the previous generators to an arbitrary conformal transformation by introducing the following operators, called Virasoro operators:
L n = -z n+1 d dz ( 108
)
that satisfy the algebra:
[L n , L m ] = (n -m)L n+m ( 109
)
that does not contain the term with the central charge! They also showed that the Virasoro operators satisfy the following commutation relations with the vertex operator:
[L n , V (z, p)] = d dz z n+1 V (z, p) ( 110
)
More in general actually they define an operator L f corresponding to an arbitrary function f (ξ) and L f = L n if we choose f (ξ) = ξ n . In this case the commutation relation in Eq. (110) becomes:
[L f , V (z, p)] = d dz (zf (z)V (z, p)) ( 111
)
By introducing the variable:
y = z A dξ ξf (ξ) ( 112
)
where A is an arbitrary constant, one can rewrite Eq. (111) in the following form:
[L f , zf (z)V (z, p)] = d dy (zf (z)V (z, p)) ( 113
)
This implies that, under an arbitrary conformal transformation z → f (z), generated by U = e αL f , the vertex operator transforms as:
e αL f V (z, p) zf (z) e -αL f = V (z ′ , p)z ′ f (z ′ ) ( 114
)
where the parameter α is given by:
α = z ′ z dξ ξf (ξ) ( 115
)
On the other hand, this equation implies:
dz zf (z) = dz ′ z ′ f (z ′ ) ( 116
)
The birth of string theory 23 that, inserted in Eq. ( 114 ), implies that the quantity V (z, p) dz is left invariant by the transformation z → f (z):
e αL f V (z, p)dze -αL f = V (z ′ , p)dz ′ ( 117
)
Let us now act with the previous conformal transformation on the state in Eq. ( 105 ). We get:
e αL f |p (1,M) = 1 0 M-1 i=2 [dz i θ(z i -z i+1 )] e αL f V (1, p 1 )e -αL f × ×e αL f V (z 2 , p 2 )e -αL f . . . . . . e αL f V (z M-1 , p M-1 )e -αL f e αL f |0, p M = = 1 0 M-1 i=2 θ(z i -z i+1 ) × e αL f V (1, p 1 )e -αL f × × V (z ′ 2 , p 2 )dz ′ 2 . . . V (z ′ M-1 , p M-1 )dz ′ M-1 e αL f |0, p M ( 118
)
where we have used Eq. ( 117 ). The previous transformation leaves the state invariant if both z = 0 and z = 1 are fixed points of the conformal transformation. This happens if the denominator in Eq. (115) vanishes when ξ = 0, 1. This requires the following conditions:
f (1) = 0 ; lim ξ→0 ξf (ξ) = 0 ( 119
)
Expanding ξ near the poinr ξ = 1 we can determine the relation between z and z ′ near z = z ′ = 1. We get:
z ′ = ze -αf ′ (1) 1 -z + ze -αf ′ (1) ( 120
)
and from it we can determine the conformal factor:
dz ′ dz = e -αf ′ (1) (1 -z + ze -αf ′ (1) ) 2 → e αf ′ (1) ( 121
)
in the limit z → 1. Proceeding in the same near the point z = z ′ = 0 we get:
z ′ = zf (0)e αf (0) f (0) + zf ′ (0)(1 -e αf (0) → ze αf (0) ( 122
)
in the limit z → 0. This means that Eq. (118) becomes
e α(L f -f ′ (1)-f (0)) |p (1,M) = |p (1,M) ( 123
)
A choice of f that satisfies Eq.s (119) is the following:
24 Paolo Di Vecchia
f (ξ) = ξ n -1 ( 124
)
that gives the following gauge operator:
W n = L n -L 0 -(n -1) ( 125
)
that annihilates the state in Eq. (105):
W n |p 1...M = 0 ; n = 1 . . . ∞ ( 126
)
These are the Virasoro conditions found in Ref. [6] . There is one condition for each negative norm oscillator and, therefore, in this case there is the possibility that the physical subspace is positive definite. An alternative more direct derivation of Eq. ( 126 ) can be obtained by acting with W n on the state in Eq. ( 106 ) and using the following identities:
W n V (1, p) = V (1, p)(W n + n) ; (W n + n)D = [L 0 + n -1] -1 W n ( 127
)
The second equation is a consequence of the following equation:
L n x L0 = x L0+n L n ( 128
)
Eq.s (127) imply
W n V (1, p)D = V (1, p)[L 0 + n -1] -1 W n ( 129
)
This shows that the operator W n goes unchanged through all the product of terms V D until it arrives in front of the term V (1, p M-1 )|0, p M . Going through the vertex operator it becomes L n -L 0 + 1 that then annihilate the state
(L n -L 0 + 1)|p M , 0 = 0 ( 130
)
This proves Eq. ( 126 ). Using the representation of the Virasoro operators given in Eq. (108) Fubini and Veneziano showed that they satisfy the algebra given in eq. (109) without the central charge. The presence of the central charge was recognized by Joe Weis 10 in 1970 and never published. Unlike Fubini and Veneziano [26] he used the expression of the L n operators in terms of the harmonic oscillators:
L n = √ 2α ′ np • a n + ∞ m=1 m(n + m)a n+m • a m + + 1 2 n-1 m=1 m(n -m)a m-n • a m ; n ≥ 0 L n = L † n ( 131
)
10 See noted added in proof in Ref. [26] .
The birth of string theory 25 He got the following algebra:
[L n , L m ] = (n -m)L n+m + d 24 n(n 2 -1)δ n+m;0 ( 132
)
where d is the dimension of the Minkowski space-time. We write here d for the dimension of the Minkowski space, but we want to remind you that almost everybody working in a model for mesons at that time took for granted that the dimension of the space-time was d = 4. As far as I remember the first paper where a dimension d = 4 was introduced was Ref. [27] where it was shown that the unitarity violating cuts in the non-planar loop become poles that were consistent with unitarity if d = 26. In the last part of this section we will generalize the factorization procedure to the Shapiro-Virasoro model whose N -point amplitude is given in Eq. (49) .
In this case we must introduce two sets of harmonic oscillators commuting with each other and only one set of zero modes satisfying the algebra [28] :
[a nµ , a † mν ] = [ã nµ , ã † mν ] = η µν δ nm ; [q µ , pν ] = iη µν ( 133
)
In terms of them we can introduce the Fubini-Veneziano operator
Q(z, z) = q -2α ′ p log(z z) + i √ 2α ′ 2 ∞ n=1 1 √ n a n z -n -a † n z n + + i √ 2α ′ 2 ∞ n=1 1 √ n ãn z-n -ã † n zn ( 134
)
We can then introduce the vertex operator: V (z, z; p) =: e ip•Q(z,z) : (135) and write the N -point amplitude in Eq. (95) in the following factorized form:
N i=1 d 2 z i dV abc 0|R N i=1 V (z i , zi , p i )) |0 = = (2π) 4 δ (4) ( N i=1 p i ) N i=1 d 2 z i dV abc i<j |z i -z j | α ′ pi•pj ( 136
)
where the radial ordered product is given by
R N i=1 V (z i , zi , p i )) = N i=1 V (z i , zi , p i )) N -1 i=1 θ(|z i | -|z i+1 |) + . . . ( 137
)
26 Paolo Di Vecchia and the dots indicate a sum over all permutations of the vertex operators. By fixing z 1 = ∞, z 2 = 1, z N = 0 we can rewrite the previous expression as follows:
N -1
i=3 d 2 z i 0, p 1 |R N -1 i=2 V (z i , zi , p i )) |0, p N ( 138
)
For the sake of simplicity let us consider the term corresponding to the permutation 1, 2, . . . N . In this case the Koba-Nielsen variables are ordered in such a way
that |z i | ≥ |z i+1 | for i = 1, . . . N -1.
We can then use the formula:
V (z i , zi , p i )) = z L0-1 i z L0-1 i V (1, 1, p i )z -L0 i z-L0 i ( 139
)
and change variables:
w i = z i+1 z i ; |w i | ≤ 1 ( 140
)
to rewrite Eq. (138) as follows:
0, p 1 |V (1, 1, p i 1)DV (1, 1, p 2 )D . . . V (1, 1, p N -1 )|0, p N ( 141
) where D = d 2 w |w| 2 w L0-1 w L0-1 = 2 L 0 + L0 -2 • sin π(L 0 -L0 ) L 0 -L0 ( 142
)
We can now follow the same procedure for all permutations arriving at the following expression:
0, p 1 |P [V (1, 1, p 2 )DV (1, 1, p 3 )D . . . V (1, 1, p N -1 )]|0, p N ( 143
)
where P means a sum of all permutations of the particles. If we want to consider the factorization of the amplitude on the pole at s = -(p 1 + . . . p M ) 2 we get only the following contribution:
p (1...M) |D|p (M+1...N ) ( 144
) where |p (M+1...N ) = P [V (1, 1, p M+1 )D . . . V (1, 1, p N -1 ]|0, p N ( 145
) and p (1...M) | = 0, p 1 |P [V (1, 1, p 2 )D . . . V (1, 1, p M )] ( 146
)
The amplitude is factorized by introducing a complete set of states and rewriting Eq. (141) as follows:
The birth of string theory 27 λ, λ p 1...M |λ, λ 2π λ, λ|δ L0, L0 |λ, λ L 0 + L0 -2 λ, λ|p (M+1,...N ) (147) By writing
L 0 = α ′ 4 p2 + R ; L0 = α ′ 4 p2 + R ( 148
) with R = ∞ n=1 na † n • a n ; R = ∞ n=1 nã † n • ãn ( 149
)
we can rewrite Eq. (147) as follows
λ, λ p 1...M |λ, λ 2π λ, λ|δ R, R|λ, λ R + R -α(s) λ, λ|p (M+1,...N ) ( 150
)
We see that the amplitude for the Shapiro-Virasoro model has simple poles only for even integer values of α SV (s) = 2 + α ′ 2 s = 2n ≥ 0 and the residue at the poles factorizes in a sum with a finite number of terms. Notice that the Regge trajectory of the Shapiro-Virasoro model has double intercept and half slope of that of the generalized Veneziano model.
In the previous section, we have seen that the residue at the poles of the Npoint amplitudes factorizes in a sum of a finite number of terms. We have also seen that some of these terms, due to the Lorentz metric, correspond to states with negative norm. We have also derived a number of "Ward identities" given in Eq. (126) that imply that some of the terms of the residue decouple. The question to be answered now is: Is the space spanned by the physical states a positive norm Hilbert space? In order to answer this question we need first to find the conditions that characterize the on shell physical states |λ, P and then to determine which are the states that contribute to the residue of the pole at α(s = -P 2 ) = n. In other words, we have to find a way of characterizing the physical states and of eliminating the spurious states that decouple in Eq. (102) as a consequence of Eq.s (126) . A state |λ.P contributes at the residue of the pole in Eq.(102) for α(s = -P 2 ) = n if it is on shell, namely if it satisfies the following equations:
R|λ, P = n|λ, P ; α(-P 2 ) = 1 -α ′ P 2 = n ( 151
)
that can be written in a unique equation:
28 Paolo Di Vecchia
(L 0 -1)|λ, P = 0 ( 152
)
Because of Eq. ( 126 ) we also know that a state of the type:
|s, P = W † m |µ, P ( 153
)
is not going to contribute to the residue of the pole. We call it a spurious or unphysical state. We start constructing the subspace of spurious states that are on shell at the level n. Let us consider the set of orthogonal states |µ, P such that
R|µ, P = n µ |µ, P ; L 0 |µ, P = (1 -m)|µ, P ; 1 -α ′ P 2 = n ( 154
)
where
m = n + n µ ( 155
)
In terms of these states we can construct the most general spurious state that is on shell at the level n. It is given by
|s, P = W † m |µ, P ; (L 0 -1)|s, P = 0 ( 156
)
per any positive integer m. Using Eq. ( 154 ), eq. (156) becomes:
|s, P = L † m |µ, P ( 157
)
where |µ, P is an arbitrary state satisfying Eq.s (154). A physical state |λ, P is defined as the one that is orthogonal to all spurious states appearing at a certain level n. This means that it must satisfy the following equation:
λ.P |L † ℓ |µ, P = 0 ( 158
)
for any state |µ, P satisfying Eq.s (154). In conclusion, the on shell physical states at the level n are characterized by the fact that they satisfy the following conditions:
L m |λ, P = (L 0 -1)|λ, P = 0 ; 1 -α ′ P 2 = n ( 159
)
These conditions characterizing the physical subspace were first found by Del Giudice and Di Vecchia [28] where the analysis described here was done. In order to find the physical subspace one starts writing the most general on shell state contributing to the residue of the pole at level n in Eq. ( 154 ). Then one imposes Eq.s (159) and determines the states that span the physical subspace. Actually, among these states one finds also a set of zero norm states that are physical and spurious at the same time. Those states are of the form given in Eq. ( 157 ), but also satisfy Eq.s (159). It is easy to see that they are not really physical because they are not contributing to the residue of the pole
The birth of string theory 29 at the level n. This follows from the form of the unit operator given in the space of the physical states by:
1 = norm =0 |λ, P λ, P | + zero [|λ 0 , P µ 0 , P | + |µ 0 , P λ 0 , P |] ( 160
)
where |λ 0 , P is a zero norm physical and spurious state and |µ 0 , P its conjugate state. A conjugate state of a zero norm state is obtained by changing the sign of the oscillators with timelike direction. Since |λ 0 , P is a spurious state when we insert the unit operator, given in Eq. ( 160 ), in Eq. ( 102 ) we see that the zero norm states never contribute to the residue because their contribution is annihilated either from the state p (1,M) | or from the state |p (M+1,N ) . In conclusion, the physical subspace contains only the states in the first term in the r.h.s. of Eq. ( 160 ).
Let us analyze the first two excited levels. The first excited level corresponds to a massless gauge field. It is spanned by the states ǫ µ a † 1µ |0, P . In this case the only condition that we must impose is:
L 1 ǫ µ a † 1µ |0, P = 0 =⇒ P • ǫ = 0 ( 161
)
Choosing a frame of reference where the momentum of the photon is given by P µ ≡ (P, 0....0, P ) , Eq. ( 161 ) implies that the only physical states are:
ǫ i a + † 1i |0, P + ǫ(a † 1;0 -a † 1;d-1 )|0, P ; i = 1 . . . d -2 ( 162
)
where ǫ i and ǫ are arbitrary parameters. The state in Eq. ( 162 ) is the most general state of the level N = 1 satisfying the conditions in Eq. ( 159 ). The first state in eq. (162) has positive norm, while the second one has zero norm that is orthogonal to all other physical states since it can be written as follows:
(a † 1;0 -a † 1;D-1 )|0, P = L † 1 |0, P ( 163
)
in the frame of reference where P µ ≡ (P, ...0, P ). Because of the previous property it is decoupled from the physical states together with its conjugate:
(a † 1,0 + a † 1,d-1 )|0, P ( 164
)
In conclusion, we are left only with the transverse d -2 states corresponding to the physical degrees of freedom of a massless spin 1 state. At the next level n = 2 the most general state is given by:
[α µν a † 1,µ a † 1,ν + β µ a † 2,µ ]|0, P ( 165
)
If we work in the center of mass frame where P µ = (M, 0) we get the following most general physical state:
|P hys >= α ij [a † 1,i a † 1,j - 1 (d -1) δ ij d-1 k=1 a † 1,k a † 1,k ]|0, P + 30 Paolo Di Vecchia +β i [a † 2,i + a † 1,0 a † 1,i ]|0, P > + + d-1 i=1 α ii d-1 i=1 a † 1,i a † 1,i + d -1 5 (a †2 1,0 -2a † 2,0 ) |0, P ( 166
)
where the indices i, j run over the d -1 space components. The first term in (166) corresponds to a spin 2 in (d -1) dimensional space and has a positive norm being made with space indices. The second term has zero norm and is orthogonal to the other physical states since it can be written as L + 1 a + 1,i |0, P . Therefore it must be eliminated from the physical spectrum together with its conjugate, as explained above. Finally, the last state in (166) is spinless and has a norm given by: 2(d -1)(26
-d) ( 167
)
If d < 26 it corresponds to a physical spin zero particle with positive norm. If d > 26 it is a ghost. Finally, if d = 26 it has a zero norm and is also orthogonal to the other physical states since it can be written in the form:
(2L † 2 + 3L †2 1 )|0 > ( 168
)
It does not belong, therefore, to the physical spectrum. The analysis of this level was done in Ref. [29] with d = 4. This did not allow the authors of Ref. [29] to see that there was a critical dimension. The analysis of the physical states can be easily extended [28] to the Shapiro-Virasoro model. In this case the physical conditions given in Eq. (159) for the open string, become [28]:
L m |λ, λ = Lm |λ, λ = (L 0 -1)|λ, λ = ( L0 -1)|λ, λ = 0 ( 169
)
for any positive integer m. It can be easily seen from the previous equations that the lowest state of the Shapiro-Virasoro model is the vacuum |0 a , 0 ã, p corresponding to a tachyon with mass α ′ p 2 = 4, while the next level described by the state a † 1µ ã † 1ν |0 a , 0 ã, p contains massless states corresponding to the graviton, a dilaton and a two-index antisymmetric tensor B µν .
Having characterized the physical subspace one can go on and construct a N -point scattering amplitude involving arbitrary physical states. This was done by Campagna, Fubini, Napolitano and Sciuto [30] where the vertex operator for an arbitrary physical state was constructed in analogy with what has been done for the ground tachyonic state. They associated to each physical state |α, P a vertex operator V α (z, P ) that is a conformal field with conformal dimension equal to 1:
[L n , V α (z, p)] = d dz z n+1 V α (z, p) ( 170
)
and reproduces the corresponding state acting on the vacuum as follows:
lim z→0 V α (z; p)|0, 0 ≡ |α; p ; 0; 0| lim z→∞ z 2 V α (z; p) = α, p| ( 171
)
The birth of string theory 31 It satisfies, in addition, the hermiticity relation:
V † α (z, P ) = V α ( 1 z , -P )(-1) α(-P 2 ) ( 172
)
An excited vertex that will play an important role in the next section is the one associated to the massless gauge field. It is given by:
V ǫ (z, k) ≡ ǫ • dQ(z) dz e ik•Q(z) ; k • ǫ = k 2 = 0 ( 173
)
Because of the last two conditions in Eq. ( 173 ) the normal order is not necessary. It is convenient to give the expression of dQ(z) dz in terms of the harmonic oscillators:
P (z) ≡ dQ(z) dz = -i √ 2α ′ ∞ n=-∞ α n z -n-1 ( 174
)
It is a conformal field with conformal dimension equal to 1. The rescaled oscillators α n are given by:
α n = √ na n ; α -n = √ na † n ; n > 0 ; α 0 = √ 2α ′ p ( 175
)
In terms of the vertex operators previously introduced the most general amplitude involving arbitrary physical states is given by [30]:
(2π) 4 δ( N i=1 p i )B ex N = ∞ -∞ N 1 dz i θ(z i -z i+1 ) dV abc 0, 0| N i=1 V αi (z i , p i )|0, 0 ( 176
)
In the case of the Shapiro-Virasoro model the tachyon vertex operator is given in Eq. (135) . By rewriting Eq. (134) as follows:
Q(z, z) = Q(z) + Q(z) ( 177
)
where
Q(z) = 1 2 q -2α ′ p log(z) + i √ 2α ′ ∞ n=1 1 √ n a n z -n -a † n z n ( 178
)
and
Q(z) = 1 2 q -2α ′ p log(z) + i √ 2α ′ ∞ n=1 1 √ n ãn z-n -ã † n zn ( 179
)
we can write the tachyon vertex operator in the following way:
V (z, z, p) =: e ip•Q(z) e ip• Q(z) : ( 180
)
32 Paolo Di Vecchia
This shows that the vertex operator corresponding to the tachyon of the Shapiro-Virasoro model can be written as the product of two vertex operators corresponding each to the tachyon of the generalized Veneziano model. Analogously the vertex operator corresponding to an arbitrary physical state of the Shapiro-Virasoro model can always be written as a product of two vertex operators of the generalized Veneziano model:
V α,β (z, z, p) = V α (z, p 2 )V β (z, p 2 ) ( 181
)
The first one contains only the oscillators α n , while the second one only the oscillators αn . They both contain only half of the total momentum p and the same zero modes p and q. The two vertex operators of the generalized Veneziano model are both conformal fields with conformal dimension equal to 1. If they correspond to physical states at the level 2n, they satisfy the following relation (n = ñ):
α ′ p 2 4 + n = 1 ( 182
)
They lie on the following Regge trajectory:
2 - α ′ 2 p 2 ≡ α SV (-p 2 ) = 2n ( 183
)
as we have already seen by factorizing the amplitude in Eq. ( 150 ).
In the previous section we have derived the equations that characterize the physical states and their corresponding vertex operators. In this section we will explicitly construct an infinite number of orthonormal physical states with positive norm. The starting point is the DDF operator introduced by Del Giudice, Di Vecchia and Fubini [31] and defined in terms of the vertex operator corresponding to the massless gauge field introduced in eq. (173):
A i,n = i √ 2α ′ 0 dzǫ µ i P µ (z)e ik•Q(z) ( 184
)
where the index i runs over the d-2 transverse directions, that are orthogonal to the momentum k. We have also taken 0 dz z = 1. Because of the log z term appearing in the zero mode part of the exponential, the integral in Eq. ( 184 ), that is performed around the origin z = 0, is well defined only if we constrain the momentum of the state, on which A i,n acts, to satisfy the relation:
2α ′ p • k = n ( 185
)
The birth of string theory 33 where n is a non-vanishing integer. The operator in Eq. (184) will generate physical states because it commutes with the gauge operators L m :
[L m , A n;i ] = 0 ( 186
)
since the vertex operator transforms as a primary field with conformal dimension equal to 1 as it follows from Eq. (170). On the other hand it also satisfies the algebra of the harmonic oscillator as we are now going to show. From Eq. (184) we get:
[A n,i , A m,j ] = - 1 2α ′ 0 dζ ζ dzǫ i • P (z)e ik•Q(ζ) ǫ j • P (ζ)e ik ′ •Q(ζ) ( 187
) where 2α ′ p • k = n ; 2α ′ p • k ′ = m ( 188
)
and k and k ′ are supposed to be in the same direction, namely
k µ = n kµ ; k ′ µ = m kµ ( 189
) with 2α ′ p • k = 1 ( 190
)
Finally the polarizations are normalized as:
ǫ i • ǫ j = δ ij ( 191
)
Since k • ǫ i = k • ǫ j = k2 = 0 a singularity for z = ζ can appear only from the contraction of the two terms P (ζ) and P ((z) that is given by:
0, 0|ǫ i • P (z)ǫ j • P (ζ)|0, 0 = - 2α ′ δ ij (z -ζ) 2 ( 192
)
Inserting it in Eq. (187) we get:
[A n,i , A m,j ] = δ ij in 0 dζ k • P (ζ)e -i(n+m)) k•Q(ζ) = = inδ ij δ n+m;0 0 dζ k • P (ζ) ( 193
)
where we have used the fact that the integrand is a total derivative and therefore one gets a vanishing contribution unless n + m = 0. If n + m = 0 from Eq.s (174) and (190) we get:
[A n,i , A m,j ] = nδ ij δ n+m;0 ; i, j = 1 . . . d -2 ( 194
)
34 Paolo Di Vecchia
Eq. ( 194 ) shows that the DDF operators satisfy the harmonic oscillator algebra. In terms of this infinite set of transverse oscillators we can construct an orthonormal set of states:
|i 1 , N 1 ; i 2 , N 2 ; . . . i m , N m = h 1 √ λ h ! m k=1 A i k ,-N k √ N k |0, p ( 195
)
where λ h is the multiplicity of the operator A i h ,-N h in the product in Eq. ( 195 ) and the momentum of the state in Eq. ( 195 ) is given by
P = p + m i=1 kN i ( 196
)
They were constructed in four dimensions where they were not a complete system of states 11 and it took some time to realize that in fact they were a complete system of states if d = 26 [32, 33] 12 . Brower [32] and Goddard and Thorn [33] showed also that the dual resonance model was ghost free for any dimension d ≤ 26. In d = 26 this follows from the fact that the DDF operators obviously span a positive definite Hilbert space (See Eq. ( 194 )). For d < 26 there are extra states called Brower states [32] . The first of these states is the last state in Eq. (166) that becomes a zero norm state for d = 26. But also for d < 26 there is no negative norm state among the physical states. The proof of the no-ghost theorem in the case α 0 = 1 is a very important step because it shows that the dual resonance model constructed generalizing the four-point Veneziano formula, is a fully consistent quantum-relativistic theory! This is not quite true because, when the intercept α 0 = 1, the lowest state of the spectrum corresponding to the pole in the N -point amplitude for α(s) = 0, is a tachyon with mass m 2 = -1 α ′ . A lot of effort was then made to construct a model without tachyon and with a meson spectrum consistent with the experimental data. The only reasonably consistent models that came out from these attempts, were the Neveu-Schwarz [7] for mesons and the Ramond model [8] for fermions that only later were recognized to be part of a unique model that nowadays is called the Neveu-Schwarz-Ramond model. But this model was not really more consistent than the original dual resonance 11 Because of this Fubini did not want to publish our result, but then he went to a meeting in Israel in spring 1971 giving a talk on our work where he found that the audience was very interested in our result and when he came back to MIT we decided to publish our result. 12 I still remember Charles Thorn coming into my office at Cern and telling me:
Paolo, do you know that your DDF states are complete if d = 26? I quickly redid the analysis done in Ref. [29] with an arbitrary value of the space-time dimension obtaining Eq.s (166) and (167) that show that the spinless state at the level α(s) = 2 is decoupled if d = 26. I strongly regretted not to have used an arbitrary space-time dimension d in the analysis of Ref. [29] .
The birth of string theory 35 model because it still had a tachyon with mass m 2 = -1 2α ′ . The tachyon was eliminated from the spectrum only in 1976 through the GSO projection proposed by Gliozzi, Scherk and Olive [34] .
Having realized that, at least for the critical value of the space-time dimension d = 26, the physical states are described by the DDF states having only d -2 = 24 independent components, open the way to Brink and Nielsen [35] to compute the value α 0 = 1 of the Regge trajectory with a very physical argument. They related the intercept of the Regge trajectory to the zero point energy of a system with an infinite number of oscillators having only d -2 independent components:
α 0 = - d -2 2 ∞ n=1 n ( 197
)
This quantity is obviously infinite and, in order to make sense of it, they introduced a cutoff on the frequencies of the harmonic oscillators obtaining an infinite term that they eliminated by renormalizing the speed of light and a finite universal constant term that gave the intercept of the Regge trajectory. Instead of following their original approach we discuss here an alternative approach due to Gliozzi [36] that uses the ζ-function regularization. He rewrites Eq. (197) as follows:
α 0 = - d -2 2 ∞ n=1 n = - d -2 2 lim s→-1 ∞ n=1 n -s = - d -2 2 ζ R (-1) = 1 ( 198
)
where in the last equation we have used the identity ζ R (-1) = -1 12 and we have put d = 26. Since the Shapiro-Virasoro model has two sets of transverse harmonic oscillators it is obvious that its intercept is twice that of the generalized Veneziano model.
Using the rules discussed in the previous section we can construct the vertex operator corresponding to the state in Eq. (195) . It is given by:
V (i;Ni) (z, P ) = m i=1 z dz i ǫ i • P (z i )e iNi k•Q(zi) : e ip•Q(z) : ( 199
)
where the integral on the variable z i is evaluated along a curve of the complex plane z i containing the point z. The singularity of the integrand for z i = z is a pole provided that the following condition is satisfied.
2α ′ p • k = 1 ( 200
)
The last vertex in Eq. (199) is the vertex operator corresponding to the ground tachyonic state given in Eq. ( 59 ) with α ′ p 2 = 1. Using the general form of the vertex one can compute the three-point amplitude involving three arbitrary DDF vertex operators. This calculation 36 Paolo Di Vecchia has been performed in Ref. [37] and since the vertex operators are conformal fields with dimension equal to 1 one gets:
0, 0|V (i (1) k 1 ;N (1) k 1 ) (z 1 , P 1 )V (i (2) k 2 ;N ( 2
) k (2) ) (z 2 , P 2 )V (i (3) k 3 ;N ( 3
) k (3) ) (z 3 , P 3 )|0, 0 = = C 123 (z 1 -z 2 )(z 1 -z 3 )(z 2 -z 3 ) ( 201
)
where the explicit form of the coefficient C 123 is given by:
C 123 = 1 0, 0| 2 0, 0| 3 0, 0|e 1 2 3 r.s=1 ∞ n,m=1 A (r) -n;i N rs nm A (s) -m;i + 3 i=1 Pi• ∞ n=1 A (r) -n;i × × e τ0 3 r=1 (α ′ Π 2 r -1) |N ( 1
) k1 , i ( 1
) k1 1 |N ( 2
) k2 , i ( 2
) k2 2 |N ( 3
) k3 , i (3) k3 3 ( 202
)
where
N rs nm = -N r n N s m nmα 1 α 2 α 3 nα s + mα r ; N r n = Γ (-n αr+1 αr ) α r n!Γ (1 -n αr+1 αr -n) ( 203
) with Π = P r+1 α r -P r α r+1 ; r = 1, 2, 3 ( 204
)
Π is independent on the value of r chosen as a consequence of the equations:
3 r=1 α r = 3 r=1 P r = 0 ( 205
)
7 The zero slope limit
In the introduction we have seen that the dual resonance model has been constructed using rules that are different from those used in field theory. For instance, we have seen that planar duality implies that the amplitude corresponding to a certain duality diagram, contains poles in both s and t channels, while the amplitude corresponding to a Feynman diagram in field theory contains only a pole in one of the two channels. Furthermore, the scattering amplitude in the dual resonance model contains an infinite number of resonant states that, at high energy, average out to give Regge behaviour. Also this property is not observed in field theory. The question that was natural to ask, was then: is there any relation between the dual resonance model and field theory? It turned out, to the surprise of many, that the dual resonance model was not in contradiction with field theory, but was instead an extension of a certain number of field theories. We will see that the limit in
The birth of string theory 37 which a field theory is obtained from the dual resonance model corresponds to taking the slope of the Regge trajectory α ′ to zero. Let us consider the scattering amplitude of four ground state particles in Eq. (1) that we rewrite here with the correct normalization factor:
A(s, t, u) = C 0 N 4 0 (A(s, t) + A(s, u) + A(t, u)) ( 206
)
where
N 0 = √ 2g(2α ′ ) d-2 4 ( 207
)
is the correct normalization factor for each external leg, g is the dimensionless open string coupling constant that we have constantly ignored in the previous sections and C 0 is determined by the following relation:
C 0 N 2 0 α ′ = 1 ( 208
)
that is obtained by requiring the factorization of the amplitude at the pole corresponding to the ground state particle whose mass is given in Eq. (21). Using Eq. (21) in order to rewrite the intercept of the Regge trajectory in terms of the mass of the ground state particle m 2 and the following relation satisfied by the Γ -function:
Γ (1 + z) = zΓ (z) ( 209
)
we can easily perform the limit for α ′ → 0 of A(s, t) obtaining:
lim α ′ →0 A(s, t) = 1 α ′ 1 m 2 -s + 1 m 2 -s ( 210
)
Performing the same limit on the other two planar amplitudes we get the following expression for the total amplitude in Eq. ( 206 ):
lim α ′ →0 A(s, t, u) = √ 2g(2α ′ ) d-2 4 2 2 (α ′ ) 2 1 m 2 -s + 1 m 2 -s + 1 m 2 -u ( 211
)
By introducing the coupling constant:
g 3 = 4g(2α ′ ) d-6 4 ( 212
) Eq. (211) becomes lim α ′ →0 A(s, t, u) = g 2 3 1 m 2 -s + 1 m 2 -s + 1 m 2 -u ( 213
)
that is equal to the sum of the tree diagrams for the scattering of four particles with mass m of Φ 3 theory with coupling constant equal to g 3 . We have shown that, by keeping g 3 fixed in the limit α ′ → 0, the scattering amplitude of four 38 Paolo Di Vecchia ground state particles of the dual resonance model is equal to the tree diagrams of Φ 3 theory. This proof can be extended to the scattering of N ground state particles recovering also in this case the tree diagrams of Φ 3 theory. It is also valid for loop diagrams that we will discuss in the next section. In conclusion, the dual resonance model reduces in the zero slope limit to Φ 3 theory. The proof that we have presented here is due to J. Scherk [38] 13 A more interesting case to study is the one with intercept α 0 = 1. We will see that, in this case, one will obtain the tree diagrams of Yang-Mills theory, as shown by Neveu and Scherk [40] 14 .
Let us consider the three-point amplitude involving three massless gauge particles described by the vertex operator in Eq. ( 173 ). It is given by the sum of two planar diagrams. The first one corresponding to the ordering (123) is given by:
C 0 N 3 0 i 3 T r (λ a1 λ a2 λ a3 ) 0, 0|V ǫ1 (z 1 , p 1 )V ǫ2 (z 2 , p 2 )V ǫ3 (z 3 , p 3 )|0, 0 [(z 1 -z 2 )(z 2 -z 3 )(z 1 -z 3 )] -1 ( 214
)
Using momentum conservation p 1 + p 2 + p 3 = 0 and the mass shell conditions p 2 i = p i • ǫ i = 0 one can rewrite the previous equation as follows:
C 0 N 3 0 T r(λ a1 λ a2 λ a3 ) √ 2α ′ × × [(ǫ 1 • ǫ 2 )(p 1 • ǫ 3 ) + (ǫ 1 • ǫ 3 )(p 3 • ǫ 2 ) + (ǫ 2 • ǫ 3 )(p 2 • ǫ 1 )] ( 215
)
The second contribution comes from the ordering 132 that can be obtained from the previous one by the substitution T r(λ a1 λ a2 λ a3 ) → -T r(λ a1 λ a3 λ a2 ) (216)
Summing the two contributions one gets
C 0 N 3 o T r(λ a1 [λ a2 , λ a3 ]) √ 2α ′ × × [(ǫ 1 • ǫ 2 )(p 1 • ǫ 3 ) + (ǫ 1 • ǫ 3 )(p 3 • ǫ 2 ) + (ǫ 2 • ǫ 3 )(p 2 • ǫ 1 )] ( 217
)
The factor
N 0 = 2g(2α ′ ) (d-2)/4 ( 218
)
is the correct normalization factor for each vertex operator if we normalize the generators of the Chan-Paton group as follows:
T r λ i λ j = 1 2 δ ij ( 219
)
13 See also Ref. [39] . 14 See also Ref. [41] .
The birth of string theory 39 It is related to C 0 through the relation 15 :
C 0 N 2 o α ′ = 2 ( 220
)
g is the dimensionless open string coupling constant. Notice that Eq.s (218) and (220) differ from Eq.s (207) and (208) because of the presence of the Chan-Paton factors that we did not include in the case of Φ 3 theory. By using the commutation relations:
[λ a , λ b ] = if abc λ c ( 221
)
and the previous normalization factors we get for the three-gluon amplitude:
ig Y M f a1a2a3 [(ǫ 1 • ǫ 2 )((p 1 -p 2 ) • ǫ 3 + +(ǫ 1 • ǫ 3 )((p 3 -p 1 ) • ǫ 2 ) + (ǫ 2 • ǫ 3 )((p 2 -p 3 ) • ǫ 1 )] ( 222
)
that is equal to the 3-gluon vertex that one obtains from the Yang-Mills action
L Y M = - 1 4 F a αβ F αβ a , F a αβ = ∂ α A a β -∂ β A a α + g Y M f abc A b α A c β ( 223
)
where
g Y M = 2g(2α ′ ) d-4 4 ( 224
)
The previous procedure can be extended to the scattering of N gluons finding the same result that one gets from the tree diagrams of Yang-Mills theory.
In the next section, we will discuss the loop diagrams. Also, in this case one finds that the h-loop diagrams involving N external gluons reproduces in the zero slope limit the sum of the h-loop diagrams with N external gluons of Yang-Mills theory. We conclude this section mentioning that one can also take the zero slope limit of a scattering amplitude involving three and four gravitons obtaining agreement with what one gets from the Einstein Lagrangian of general relativity. This has been shown by Yoneya [43] .
The N -point amplitude previously constructed satisfies all the axioms of Smatrix theory except unitarity because its only singularities are simple poles corresponding to zero width resonances lying on the real axis of the Mandelstam variables and does not contain the various cuts required by unitarity [1]. 15 The determination of the previous normalization factors can be found in the Appendix of Ref. [42] .
In order to eliminate this problem it was proposed already in the early days of dual theories to assume, in analogy with what happens for instance in perturbative field theory, that the N -point amplitude was only the lowest order (the tree diagram) of a perturbative expansion and, in order to implement unitarity, it was necessary to include loop diagrams. Then, the one-loop diagrams were constructed from the propagator and vertices that we have introduced in the previous sections [44] . The planar one-loop amplitude with M external particles was computed by starting from a (M + 2)-point tree amplitude and then by sewing two external legs together after the insertion of a propagator D given in Eq. (100) . In this way one gets:
d d P (2α ′ ) d/2 (2π) d λ P, λ|V (1, p 1 )DV (1, p 2 ) . . . V (1, p N )D|P, λ ( 225
)
where the sum over λ corresponds to the trace in the space of the harmonic oscillators and the integral in d d P corresponds to integrate over the momentum circulating in the loop. The previous expression for the one-loop amplitude cannot be quite correct because all states of the space generated by the oscillators in Eq. ( 51 ) are circulating in the loop, while we know that we should include only the physical ones. This was achieved first by cancelling by hand the time and one of the space components of the harmonic oscillators reducing the degrees of freedom of each oscillator from d to d -2 as suggested by the DDF operators at least for d = 26. This procedure was then shown to be correct by Brink and Olive [45] . They constructed the operator that projects over the physical states and, by inserting it in the loop, showed that the reduction of the degrees of freedom of the oscillators from d to d -2 was indeed correct. This was, at that time, the only procedure available to let only the physical states circulate in the loop because the BRST procedure was discovered a bit later also in the framework of the gauge field theories! To be more explicit let us compute the trace in Eq. ( 225 ) adding also the Chan-Paton factor. We get:
(2π) d δ (d) M i=1 p i N T r(λ a1 . . . λ aM ) (8π 2 α ′ ) d/2 N M 0 ∞ 0 dτ τ d/2+1 [f 1 (k)] 2-d k d-26 12 (2π) M × × 1 0 dν M νM 0 dν M-1 . . . ν3 0 dν 2 τ M i<j e G(νji) 2α ′ pi•pj ; k ≡ e -πτ ( 226
) where ν ji ≡ ν j -ν i , G(ν) = log ie -πν 2 τ Θ 1 (iντ |iτ ) f 3 1 (k) ; f 1 (k) = k 1/12 ∞ n=1 (1 -k 2n ) ( 227
)
The birth of string theory 41 Θ 1 (ν|iτ ) = -2k 1/4 sin πν
∞ n=1 1 -e 2iπν k 2n 1 -e -2iπν k 2n (1 -k 2n )( 228
)
Finally the normalization factor N 0 is given in Eq. ( 218 ). We have performed the calculation for an arbitrary value of the space-time dimension d. However, in this way one gets also the extra factor of k d-26 12
appearing in the first line of Eq. (226) that implies that our calculation is actually only consistent if d = 26. In fact, the presence of this factor does not allow one to rewrite the amplitude, originally obtained in the Reggeon sector, in the Pomeron sector as explained below. In the following we neglect this extra factor, implicitly assuming that d = 26, but, on the other hand, still keeping an arbitrary d.
Using the relations:
f 1 (k) = √ tf 1 (q) ; Θ 1 (iντ |iτ ) = iΘ 1 (ν|it)t 1/2 e πν 2 /t ( 229
)
where t = 1 τ and q ≡ e -πt , we can rewrite the one-loop planar diagram in the Pomeron channel. We get:
(2π) d δ (d) M i=1 p i N T r(λ a1 . . . λ aM ) (8π 2 α ′ ) d/2 N M 0 ∞ 0 dt[f 1 (q)] 2-d (2π) M × × 1 0 dν M νM 0 dν M-1 . . . ν3 0 dν 2 i<j - Θ 1 (ν ji |it) f 3 1 (q) 2α ′ pi•pj ( 230
)
Notice that, by factorizing the planar loop in the Pomeron channel, one constructed for the first time what we now call the boundary state [46] 16 . This can be easily seen in the way that we are now going to describe. First of all, notice that the last quantity in Eq. ( 230 ) can be written as follows:
i<j - Θ 1 (ν ji |it) f 3 1 (q) 2α ′ pi•pj = = i<j -2 sin(πν ji ) ∞ n=1 1 -q 2n e 2πiνji 1 -q 2n e -2πiνji (1 -q 2n ) 2 2α ′ pi•pj ( 231
)
This equation can be rewritten as follows:
T r p = 0|q 2R M i=1 :
e ipi•Q(e 2iπν i ) : |p = 0 i M T r ( p = 0|q 2N |p = 0 ) ; R = ∞ n=1 na † n • a n ( 232
) 16
See also the first paper in Ref. [47] .
where the trace is taken only over the non-zero modes and momentum conservation has been used. It must also be stressed that the normal ordering of the vertex operators in the previous equation is such that the zero modes are taken to be both in the same exponential instead of being ordered as in Eq. ( 59 ). By bringing all annihilation operators on the left of the creation ones, from the expression in Eq. (232) one gets (z i ≡ e 2πiνi ):
(2π) d δ (d) ∞ i=1 p i i<j (-2 sin πν ji ) 2α ′ pi•pj × × i.j ∞ n=1 T r q 2na † n •an e √ 2α ′ pj • a † n √ n z n j e - √ 2α ′ pi• an √ n z -n i T r ( p = 0|q 2N |p = 0 ) ( 233
)
The trace can be computed by using the completeness relation involving coherent states |f = e f a † |0 :
d 2 f π e -|f | 2 |f f | = 1 ( 234
)
Inserting the previous identity operator in Eq. ( 233 ) one gets after some calculation:
(2π) d δ (d) ∞ i=1 p i i<j (-2 sin πν ji ) 2α ′ pi•pj × × M i.j=1 ∞ n=1 e -2α ′ pi•pje 2πinν ji q 2n n(1-q 2n ) ( 235
)
Expanding the denominator in the last exponent and performing the sum over n one gets:
(2π) d δ (d) ∞ i=1 p i i<j (-2 sin πν ji ) 2α ′ pi•pj × × i.j e 2α ′ pi•pj ∞ m=0 log(1-e 2πiν ji q 2(m+1) ) ( 236
)
that is equal to the last line of Eq. (231) apart from the δ-function for momentum conservation. In conclusion, we have shown that Eq.s (231) and (232) are equal. Using Eq. (231) we can rewrite Eq. (230) as follows:
N N M 0 T r(λ a1 . . . λ aM ) (8π 2 α ′ ) d/2 ∞ 0 dt[f 1 (q)] 2-d (2πi) M 1 0 dν M νM 0 dν M-1 . . .
The birth of string theory 43 . . .
ν3 0 dν 2 λ p = 0, λ|q 2R M i=1 : e ipi•Q(e 2iπν i ) : |p = 0, λ λ p = 0, λ|q 2N |p = 0, λ ( 237
)
where the sum over any state |λ corresponds to taking the trace over the non-zero modes. If d = 26 we can rewrite Eq. (237) in a simpler form:
N N M 0 T r(λ a1 . . . λ aM ) (8π 2 α ′ ) d/2 ∞ 0 dt (2πi) M 1 0 dν M νM 0 dν M-1 . . . ν3 0 dν 2 × × λ p = 0, λ|q 2R-2 M i=1 : e ipi•Q(e 2iπν i ) : |p = 0, λ ( 238
)
The previous equation contains the factor dtq 2R-2 that is like the propagator of the Shapiro-Virasoro model, but with only one set of oscillators as in the generalized Veneziano model. In the following we will rewrite it completely with the formalism of the Shapiro-Virasoro model. This can be done by introducing the Pomeron propagator:
∞ 0 dt q 2N -2 = 2 πα ′ D ; D ≡ α ′ 4π d 2 z |z| 2 z L0-1 z L0 -1 ; |z| ≡ q = e -πt ( 239
)
and rewriting the planar loop in the following compact form:
B 0 | D|B M ; |B 0 ≡ T d-1 2 N ∞ n=1 e a † n •ã † n |p = 0, 0 a , 0 ã ( 240
)
where |B 0 is the boundary state without any Reggeon on it,
T d-1 = √ π 2 (d-10)/4 (2π √ α ′ ) -d/2-1 ( 241
)
and |B M is instead the one with M Reggeons given by:
|B M = N M 0 T r(λ a1 . . . λ aM )(2πi) M 1 0 dν M νM 0 dν M-1 . . . ν3 0 dν 2 × × M i=1 : e ipi•Q(e 2iπν i ) : |B 0 ( 242
)
We want to stress once more that the normal ordering in the previous equation is defined by taking the zero modes in the same exponential. Both the boundary states and the propagator are now states of the Shapiro-Virasoro model. This means that we have rewritten the one-loop planar diagram, where the states of the generalized Veneziano model circulate in the loop, as a tree
44 Paolo Di Vecchia
diagram of the Shapiro-Virasoro model involving two boundary states and a propagator. This is what nowadays is called open/closed string duality. Besides the one-loop planar diagram in Eq. ( 225 ), that is nowadays called the annulus diagram, also the non-planar and the non-orientable diagrams were constructed and studied. In particular the non-planar one, that is obtained as the planar one in Eq. (225) but with two propagators multiplied with the twist operator
Ω = e L-1 (-1) R , ( 243
)
had unitarity violating cuts that disappeared [27] if the dimension of the space-time d = 26, leaving behind additional pole singularities. The explicit form of the non-planar loop can be obtained following the same steps done for the planar loop. One gets for the non-planar loop the following amplitude:
B R | D|B M ( 244
)
where now both boundary states contain, respectively, R and M Reggeon states. The additional poles found in the non-planar loop were called Pomerons because they occur in the Pomeron sector, that today is called the closed string channel, to distinguish them from the Reggeons that instead occur in the Reggeon sector, that today is called the open string sector of the planar and non-planar loop diagrams. At that time in fact, the states of the generalized Veneziano models were called Reggeons, while the additional ones appearing in the non-planar loop were called Pomerons. The Reggeons correspond nowadays to open string states, while the Pomerons to closed string states. These things are obvious now, but at that time it took a while to show that the additional states appearing in the Pomeron sector have to be identified with those of the Shapiro-Virasoro model. The proof that the spectrum was the same came rather early. This was obtained by factorizing the non-planar diagram in the Pomeron channel [46] as we have done in Eq. ( 244 ). It was found that the states of the Pomeron channel lie on a linear Regge trajectory that has double intercept and half slope of the one of the Reggeons. This follows immediately from the propagator D in Eq. (239) that has poles for values of the momentum of the Pomeron exchanged given by:
2 - α ′ 2 p 2 = 2n ( 245
)
that are exactly the values of the masses of the states of the Shapiro-Virasoro model [48] , while the Reggeon propagator in Eq. (100) has poles for values of momentum equal to:
1 -α ′ p 2 = n ( 246
)
However, it was still not clear that the Pomeron states interact among themselves as the states of the Shapiro-Virasoro model. To show this it was first
The birth of string theory 45 necessary to construct tree amplitudes containing both states of the generalized Veneziano model and of the Shapiro-Virasoro model [49] . They reduced to the amplitudes of the generalized Veneziano (Shapiro-Virasoro) model if we have only external states of the generalized Veneziano (Shapiro-Virasoro) model. Those amplitudes are called today disk amplitudes containing both open and closed string states. They were constructed [49] by using for the Reggeon states the vertex operators that we have discussed in Sect. (5) involving one set of harmonic oscillators and for the Pomeron states the vertex operators given in Eq. (181) that we rewrite here:
V α,β (z, z, p) = V α (z, p 2 )V β (z, p 2 ) ( 247
)
because now both component vertices contain the same set of harmonic oscillators as in the generalized Veneziano model. Furthermore, each of the two vertices is separately normal ordered, but their product is nor normal ordered. The amplitude involving both kinds of states is then constructed by taking the product of all vertices between the projective invariant vacuum and integrating the Reggeons on the real axis in an ordered way and the Pomerons in the upper half plane, as one does for a disk amplitude. We have mentioned above that the two vertices are separately normal ordered, but their product is not normal ordered. When we normal order them we get, for instance for the tachyon of the Pomeron sector, a factor (z -z) α ′ p 2 /2 that describes the Reggeon-Pomeron transition. This implies a direct coupling [51] between the U (1) part of gauge field and the two-index antisymmetric field B µν , called Kalb-Ramond field [50], of the Pomeron sector, that makes the gauge field massive [51] .
It was then shown that, by factorizing the non-planal loop in the Pomeron channel, one reproduced the scattering amplitude containing one state of the Shapiro-Virasoro and a number of states of the generalized Veneziano model [52] . If we have also external states belonging to the generalized Shapiro-Virasoro model, then by factorizing the non-planar one loop amplitude in the pure Pomeron channel, one would obtain the tree amplitudes of the Shapiro-Virasoro model [52] .
All this implies that the generalized Veneziano model and the Shapiro-Virasoro model are not two independent models, but they are part of the same and unique model. In fact, if one started with the generalized Veneziano model and added loop diagrams to implement unitarity, one found the appearence in the non-planar loop of additional states that had the same mass and interaction of those of the Shapiro-Virasoro model. The planar diagram, written in Eq. (230) in the closed string channel, is divergent for large values of t. This divergence was recognized to be due to exchange, in the Pomeron channel, of the tachyon of the Shapiro-Virasoro model and of the dilaton [47] . They correspond, respectively, to the first two terms of the expansion:
[f 1 (q)] -24 = e 2πt + 24 + O e -2πt ( 248
)
46 Paolo Di Vecchia
The first one could be cancelled by an analytic continuation, while the second one could be eliminated through a renormalization of the slope of the Regge trajectory α ′ [47] . We conclude the discussion of the one-loop diagrams by mentioning that the one-loop diagram for the Shapiro-Virasoro model was computed by Shapiro [53] who also found that the integrand was modular invariant.
The computation of multiloop diagrams requires a more advanced technology that was also developed in the early days of the dual resonance model few years before the discovery of its connection to string theory. In order to compute multiloop diagrams one needs first to construct an object that was called the N -Reggeon vertex and that has the properties of containing N sets of harmonic oscillators, one for each external leg, and is such that, when we saturate it with N physical states, we get the corresponding N -point amplitude. In the following we will discuss how to determine the N -Reggeon vertex.
The first step toward the N -Reggeon vertex is the Sciuto-Della Selva-Saito [54] vertex that includes two sets of harmonic oscillators that we denote with the indices 1 and 2. It is equal to:
V SDS = 2 x = 0, 0| : exp - 1 2α ′ 0 dzX ′ 2 (z) • X 1 (1 -z) : ( 249
)
where X is the quantity that we have called Q in Eq. ( 57 ) and the prime denotes a derivative with respect to z. It satisfies the important property of giving the vertex operator V α (z = 1) of an arbitrary state |α when we saturate it with the corresponding state:
V SDS |α 2 = V α (z = 1) ( 250
)
A shortcoming of this vertex is that it is not invariant under a cyclic permutation of the three legs. A cyclic symmetric vertex has been constructed by Caneschi, Schwimmer and Veneziano [55] by inserting the twist operator in Eq. ( 243 ). But the 3-Reggeon vertex is not enough if we want to compute an arbitrary multiloop amplitude. We must generalize it to an arbitrary number of external legs. Such a vertex, that can be obtained from the one in Eq. (249) with a very direct procedure, or that can also be obtained by sewing together three-Reggeon vertices, has been written in its final form by Lovelace [56] 17 .
Here we do not derive it, but we give directly its expression written in Ref. [56] :
V N,0 = N i=1 dz i dV abc N i=1 [V ′ i (0)] N i=1 [ i < x = 0, O a |] δ( N i=1 p i ) N i,j=1 i =j exp - 1 2 ∞ n,m=0 a (i) n D nm (Γ V -1 i V j ) a (j) m ( 251
)
17 See also Ref. [57] . Earlier papers on the N -Reggeon can be found in Ref.s [58] .
The birth of string theory 47 where a (i) 0 ≡ α i 0 = √ 2α ′ pi is the momentum of particle i and the infinite matrix:
D nm (γ) = 1 m! m n ∂ m z [γ(z)] n | z=0 ; n, m = 1.. : D 00 (γ) = -log | D √ AD -BC | D n0 = 1 √ n ( B D ) n ; D 0n = 1 √ n (- C D ) n ; γ(z) = Az + B Cz + D ( 252
)
is a "representation" of the projective group corresponding to the conformal weight ∆ = 0, that satisfies the eqs.:
D nm (γ 1 γ 2 ) = ∞ l=1 D nl (γ 1 )D lm (γ 2 ) + D n0 (γ 1 )δ 0m + D 0m (γ 2 )δ n0 ( 253
)
and
D nm (γ) = D mn (Γ γ -1 Γ ) Γ (z) = 1 z ( 254
)
Finally V i is a projective transformation that maps 0, 1 and ∞ into z i-1 , z i and z i+1 . The previous vertex can be written in a more elegant form as follows:
V N,0 = N i=1 dz i dV abc N i=1 [V ′ i (0)] N i=1 [ i < x = 0, O a |] δ( N i=1 p i ) exp i 4α ′ dz∂X (i) (z)p i log V ′ i (z) exp - 1 2 N i,j=1 i =j dz dy∂X (i) (z) log[V i (z) -V j (y)]∂X (j) (y) ( 255
)
where the quantities X (i) are what we called Q, namely the Fubini-Veneziano field, in the previous sections. The N -Reggeon vertex that satisfies the important property of giving the scattering amplitude of N physical particle when we saturate it with their corresponding states, is the fundamental object for computing the multiloop amplitudes. In fact, if we want to compute a M -loop amplitude with N external states, we need to start from the (N +2M )-Reggeon vertex and then we have to sew the M pairs together after having inserted a propagator D. In this way we obtain an amplitude that is not only integrated over the punctures z i (i = 1 . . . N ) of the N external states, but also over the additional 3h -3 moduli corresponding to the punctures variables of the 48 Paolo Di Vecchia states that we sew together and the integration variable of the M propagators. h is the number of loops. The multiloop amplitudes have been obtained in this way already in 1970 [59, 60, 61] and, through the sewing procedure, one obtained functions, as the period matrix, the abelian differentials, the prime form, etc., that are well defined on Riemann surface! The only thing that was missing, was the correct measure of integrations over the 3h -3 variables because it was technically not possible to let only the physical states to circulate in the loops. This problem was solved only much later [62, 63] when a BRST invariant formulation of string theory and the light-cone functional integral could be used for computing multiloops. They are two very different approaches that, however, gave the same result. For the sake of completeness we write here the planar h-loop amplitude involving M tachyons:
A (h) M (p 1 , . . . , p M ) = N h Tr(λ a1 • • • λ aM ) C h 2g s (2α ′ ) (d-2)/4 M × [dm] M h i<j exp G (h) (z i , z j ) V ′ i (0) V ′ j (0) 2α ′ pi•pj , ( 256
)
where N h Tr(λ a1 • • • λ aM ) is the appropriate U (N ) Chan-Paton factor, g is the dimensionless open string coupling constant, C h is a normalization factor given by
C h = 1 (2π) dh g 2h-2 s 1 (2α ′ ) d/2 , ( 257
)
and G (h) is the h-loop bosonic Green function
G (h) (z i , z j ) = log E (h) (z i , z j ) - 1 2 zj zi ω µ (2πImτ µν ) -1 zj zi ω ν , ( 258
)
with E (h) (z i , z j ) being the prime form, ω µ (µ = 1, . . . , h) the abelian differentials and τ µν the period matrix. All these objects, as well as the measure on moduli space [dm] M h , can be explicitly written in the Schottky parametrization of the Riemann surface, and their expressions for arbitrary h can be found for example in Ref. [64] . It is given by
[dm] M h = 1 dV abc M i=1 dz i V ′ i (0) h µ=1 dk µ dξ µ dη µ k 2 µ (ξ µ -η µ ) 2 (1 -k µ ) 2 ( 259
) × [det (-iτ µν )] -d/2 α ′ ∞ n=1 (1 -k n α ) -d ∞ n=2 (1 -k n α ) 2 .
where k µ are the multipliers, ξ µ and η µ are the fixed points of the generators of the Schottky group,
The birth of string theory 49
The approach presented in the previous sections is a real bottom-up approach. The experimental data were the driving force in the construction of the Veneziano model and of its generalization to N external legs. The rest of the work that we have described above consisted in deriving its properties. The result is, except for a tachyon, a fully consistent quantum-relativistic model that was a source of fascination for those who worked in the field. Although the model grew out of S-matrix theory where the scattering amplitude is the only observable object, while the action or the Lagrangian have not a central role, some people nevertheless started to investigate what was the underlying microscopic structure that gave rise to such a consistent and beautiful model. It turned out, as we know today, that this underlying structure is that of a quantum-relativistic string. However, the process of connecting the dual resonance model (actually two of them the generalized Veneziano and the Shapiro-Virasoro model) to string theory took several years from the original idea to a complete and convincing proof of the conjecture. The original conjecture was independently formulated by Nambu [20, 65] , Nielsen [66] and Susskind [21] 18 . If we look at it in retrospective, it was at that time a fantastic idea that shows the enormous physical intuition of those who formulated it. On the other hand, it took several years to digest it before one was able to derive from it all the deep features of the dual resonance model. Because of this, the idea that the underlying structure was that of a relativistic string, did not really influence most of the research in the field up to 1973. Let me try to explain why. A common feature of the work of Ref.s [20, 66, 21] is the suggestion that the infinite number of oscillators, that one got through the factorization of the dual resonance model, naturally comes out from a two-dimensional free Lagrangian for the coordinate X µ (τ, σ) of a one-dimensional string, that is an obvious generalization of the Lagrangian that one writes for the coordinate X µ (τ ) of a pointlike object in the proper-time gauge:
L ∼ 1 2 dX dτ • dX dτ =⇒ L ∼ 1 2 dX dτ • dX dτ - dX dσ • dX dσ ( 260
)
Being this theory conformal invariant the Virasoro operators were also constructed together with their algebra. In this very first formulation, however, the Virasoro generators L n were just the generators associated to the conformal symmetry of the string world-sheet Lagrangian given in Eq. (260) as in any conformal field theory. It was not clear at all why they should imply the gauge conditions found by Virasoro or, in modern terms, why they should be zero classically. The basic ingredient to solve this problem was provided by Nambu [65] and Goto [68] who wrote the non-linear Lagrangian proportional 18 See also Ref. [67] .
to the area spanned by the string in the external target space. They proceeded in analogy with the point particle and wrote the following action:
S ∼ -dσ µν dσ µν ( 261
) where dσ µν = ∂X µ ∂ζ α ∂X ν ∂ζ β dζ α ∧ dζ β = ∂X µ ∂ζ α ∂X ν ∂ζ β ǫ αβ dσdτ ( 262
)
X µ (σ, τ ) is the string coordinate and ζ 0 = τ and ζ 1 = σ are the coordinates of the string worldsheet. ǫ αβ is an antisymmetric tensor with ǫ 01 = 1. Inserting eq. (262) in (261) and fixing the proportionality constant one gets the Nambu-Goto action [65, 68]:
S = -cT τ f τi dτ π 0 dσ ( Ẋ • X ′ ) 2 -Ẋ2 X ′ 2 ( 263
) where Ẋµ ≡ ∂X µ ∂τ X ′ µ ≡ ∂X µ ∂σ ( 264
)
and T ≡ 1 2πα ′ is the string tension, that replaces the mass appearing in the case of a point particle. In this formulation, the string Lagrangian is invariant under any reparametrization of the world-sheet coordinates σ and τ and not only under the conformal transformations. This, in fact, implies that the twodimensional world-sheet energy-momentum tensor of the string is actually zero as we will show later on. But it took still a few years to connect the Nambu-Goto action to the properties of the dual resonance model. In the meantime an analogue model was formulated [69] that reproduced the tree and loop amplitudes of the generalized Veneziano model. This approach anticipated by several years the path integral derivation of dual amplitudes. It was very closely related to the functional integral formulation of Ref.s [70] . However, one needed to wait until 1973 with the paper of Goddard, Goldstone, Rebbi and Thorn [71] , where the Nambu-Goto action was correctly treated, all its consequences were derived and it became completely clear that the structure underlying the dual resonance model was that of a quantum-relativistic string. The equation of motion for the string were derived from the action in Eq. (263) by imposing δS = 0 for variations such that δX µ (τ i ) = δX µ (τ f ) = 0. One gets:
δS = τ f τi π 0 dσ - ∂ ∂τ ∂L ∂ Ẋµ - ∂ ∂σ ∂L ∂X ′ µ δX µ + ∂L ∂X ′ µ δX µ | σ=π σ=0 = 0 (265)
where L is the Lagrangian in Eq. (263) . Since δX µ is arbitrary, from eq. (265) one gets the Euler-Lagrange equation of motion
The birth of string theory 51
∂ ∂τ ∂L ∂ Ẋµ + ∂ ∂σ ∂L ∂X ′ µ ≡ ∂ ∂ζ α ∂L ∂( ∂X µ ∂ζ α ) = 0 ( 266
)
and the boundary conditions
∂L ∂X ′ µ = 0 or δX µ = 0 at σ = 0, π ( 267
)
for an open string and
X µ (τ, 0) = X µ (τ, π) ( 268
)
for a closed string. In the case of an open string, the first kind of boundary condition in Eq.(267) corresponds to Neumann boundary conditions, while the second one to Dirichlet boundary conditions. Only the Neumann boundary conditions preserve the translation invariance of the theory and, therefore, they were mostly used in the early days of string theory. It must be stressed, however, that Dirichlet boundary conditions were already discussed and used in the early days of string theory for constructing models with offshell states [72]. From Eq. (263) one can compute the momentum density along the string:
∂L ∂ Ẋµ ≡ P µ = cT Ẋµ X ′ 2 -X ′ µ ( Ẋ • X ′ ) ( Ẋ • X ′ ) 2 -Ẋ2 X ′ 2 ( 269
)
and obtain the following constraints between the dynamical variables X µ and P µ :
c 2 T 2 x ′ 2 + P 2 = x ′ • P = 0 ( 270
)
They are a consequence of the reparametrization invariance of the string Lagrangian. Because of this one can choose the orthonormal gauge specified by the conditions:
Ẋ2 + X ′ 2 = Ẋ • X ′ = 0 ( 271
)
that nowadays is called conformal gauge. In this gauge eq. (269) becomes:
P µ = cT Ẋµ ∂L ∂X ′ µ = -cT X ′ µ ( 272
)
and therefore the eq. of motion in eq.(266) becomes:
Ẍµ -X ′′ µ = 0 ( 273
)
while the boundary condition in eq.(267) becomes:
X ′ µ (σ = 0, π) = 0 ( 274
)
52 Paolo Di Vecchia
The most general solution of the eq. of motion and of the boundary conditions can be written as follows:
X µ (τ, σ) = q µ + 2α ′ p µ τ + i √ 2α ′ ∞ n=1 [a µ n e -inτ -a +µ n e inτ ] cosnσ √ n ( 275
)
for an open string and
X µ (τ, σ) = q µ + 2α ′ p µ τ + i 2 √ 2α ′ ∞ n=1 [ã µ n e -2in(τ +σ) -ã+µ n e 2in(τ +σ) ] 1 √ n + + i 2 √ 2α ′ ∞ n=1 [a µ n e -2in(τ -σ) -a +µ n e 2in(τ -σ) ] 1 √ n ( 276
)
for a closed string. This procedure really shows that, starting from the Nambu-Goto action, one can choose a gauge (the orthonormal or conformal gauge) where the equation of motion of the string becomes the twodimensional D'Alembert equation in Eq. ( 273 ). Furthermore, the invariance under reparametrization of the Nambu-Goto action implies that the twodimensional energy-momentum tensor is identically zero at the classical level (See Eq. (271)).
As the Lorentz gauge in QED the orthonormal gauge does not fix completely the gauge. We can still perform reparametrizations that leave in the conformal gauge: they are conformal transformatiuons. Introducing the variable z = e iτ the generators of the conformal transformations for the open string can be written as follows:
L n = 1 2πi dzz n+1 - 1 4α ′ ∂X µ ∂z 2 = 1 2 ∞ m=-∞ α n-m • α m = 0 ( 277
)
where
α µ n = √ na µ n if n > 0 √ 2α ′ p µ if n = 0 √ na †µ n if n < 0 ( 278
)
They are zero as a consequence of Eq.s (270) that in the conformal gauge become Eq.s (271). In the case of a closed string we get instead:
Ln = 1 2πi dzz n+1 - 1 α ′ ∂X µ ∂z 2 = 0 ( 279
) L n = 1 2πi dz zn+1 - 1 α ′ ∂X µ ∂ z 2 = 0 ( 280
)
The birth of string theory 53 In terms of the harmonic oscillators introduced in eq. (276) we get
L n = 1 2 ∞ m=-∞ α m • α n-m = 0 ; Ln = 1 2 ∞ m=-∞ αm • αn-m = 0 ( 281
)
where for the non-zero modes we have used the convention in (278), while the zero mode is given by:
α µ 0 = αµ 0 = √ 2α ′ p µ 2 ( 282
)
In conclusion, the fact that we have reparametrization invariance implies that the Virasoro generators are classically identically zero. When we quantize the theory one cannot and also does not need to impose that they are vanishing at the operator level. They are imposed as conditions characterizing the physical states.
P hys ′ |L n |P hys = P hys ′ |(L 0 -1)|P hys = 0 ; n = 0 ( 283
)
These equations are satisfied if we require:
L n |P hys >= (L 0 -1)|P hys >= 0 ( 284
)
The extra factor -1 in the previous equations comes from the normal ordering as explained in Eq. ( 198 ). The authors of Ref. [71] further specified the gauge by fixing it completely. They introduced the light-cone gauge specified by imposing the condition:
X + = 2α ′ p + τ ( 285
)
where
X ± = X 0 ± X d-1 √ 2 X ± = X 0 ± X d-1 √ 2 ( 286
)
In this gauge the only physical degrees of freedom are the transverse ones.
In fact the components along the directions 0 and d -1 can be expressed in terms of the transverse ones by inserting Eq. ( 285 ) in the constraints in Eq. ( 271 ) and getting:
Ẋ-= 1 4α ′ p + ( Ẋ2 i + X ′ 2 i ) X ′ -= 1 2α ′ p + Ẋi • X ′ i ( 287
)
that up to a constant of integration determine completely X -as a function of X i . In terms of oscillators we get
α + n = 0 ; √ 2α ′ α - n = 1 2p + ∞ m=-∞ α i n-m α i m n = 0 ( 288
)
54 Paolo Di Vecchia
for an open string and
α + n = α+ n = 0 n = 0 ( 289
) together with √ 2α ′ α - n = 1 2p + ∞ m=-∞ α i n-m α i m √ 2α ′ α- n = 1 2p + ∞ m=-∞ αi n-m αi m ( 290
)
in the case of a closed string.
This shows that the physical states are described only by the transverse oscillators having only d -2 components. Those transverse oscillators correspond to the transverse DDF operators that we have discussed in Section 6. The authors of Ref. [71] also constructed the Lorentz generators only in terms of the transverse oscillators and they showed that they satisfy the correct Lorentz algebra only if the space-time dimension is d = 26. In this way the spectrum of the dual resonance model was completely reproduced starting from the Nambu-Goto action if d = 26! On the other hand, the choice of d = 26 is a necessity if we want to keep Lorentz invariance! Immediately after this, the interaction was also included either by adding a term describing the interaction of the string with an external gauge field [73] or by using a functional formalism [74, 75] .
In the following we will give some detail only of the first approach for the case of an open string. A way to describe the string interaction is by adding to the free string action an additional term that describes the interaction of the string with an external field.
S IN T = d D yΦ L (y)J L (y) ( 291
)
where Φ L (y) is the external field and J L is the current generated by the string. The index L stands for possible Lorentz indices that are saturated in order to have a Lorentz invariant action.
In the case of a point particle, such an interaction term will not give any information on the self-interaction of a particle.
In the case of a string, instead, we will see that S IN T will describe the interaction among strings because the external fields that can consistently interact with a string are only those that correspond to the various states of the string, as it will become clear in the discussion below.
This is a consequence of the fact that, for the sake of consistency, we must put the following restrictions on S IN T :
• It must be a well defined operator in the space spanned by the string oscillators.
The birth of string theory 55 • It must preserve the invariances of the free string theory. In particular, in the "conformal gauge" it must be conformal invariant. • In the case of an open string, the interaction occurs at the end point of a string (say at σ = 0). This follows from the fact that two open strings interact attaching to each other at the end points.
The simplest scalar current generated by the motion of a string can be written as follows
J(y) = dτ dσδ(σ)δ (d) [y µ -x µ (τ, σ)] ( 292
)
where δ(σ) has been introduced because the interaction occurs at the end of the string. For the sake of simplicity we omit to write a coupling constant g in (292). Inserting (292) in (291) and using for the scalar external field Φ(y) = e ik•y a plane wave, we get the following interaction:
S IN T = dτ : e ik•X(τ,0) : ( 293
)
where the normal ordering has been introduced in order to have a well defined operator. The invariance of (293) under a conformal transformation τ → w(τ ) requires the following identity:
S IN T = dτ : e ik•X(τ,0) : = dw : e ik•X(w,0) : ( 294
)
or, in other words, that : e ik•X(τ,0) :=⇒ w ′ (τ ) : e ik•X(w,0) : (
) 295
This means that the integrand in Eq. ( 294 ) must be a conformal field with conformal dimension equal to one and this happens only if α ′ k 2 = 1. The external field corresponds then to the tachyonic lowest state of the open string. Another simple current generated by the string is given by:
J µ (y) = dτ dσδ(σ) Ẋµ (τ, σ)δ (d) (y -X(τ, σ)) ( 296
)
Inserting (296) in (291) we get
S IN T = dτ Ẋµ (τ, 0)ǫ µ e ik•X(τ,0) ( 297
)
if we use a plane wave for Φ µ (y) = ǫ µ e ik•y . The vertex operator in eq. ( 297 ) is conformal invariant only if
k 2 = ǫ • k = 0 ( 298
)
56 Paolo Di Vecchia and, therefore, the external vector must be the massless photon state of the string. We can generalize this procedure to an arbitrary external field and the result is that we can only use external fields that correspond to on shell physical states of the string. This procedure has been extended in Ref. [73] to the case of external gravitons by introducing in the Nambu-Goto action a target space metric and obtaining the vertex operator for the graviton that is a massless state in the closed string theory. Remember that, at that time, this could have been done only with the Nambu-Goto action because the σ-model action was introduced only in 1976 first for the point particle [76] and then for the string [77]. As in the case of the photon it turned out that the external field corresponding to the graviton was required to be on shell. This condition is the precursor of the equations of motion that one obtains from the σ-model action requiring the vanishing of the β-function [78] .
One can then compute the probability amplitude for the emission of a number of string states corresponding to the various external fields, from an initial string state to a final one. This amplitude gives precisely the N -point amplitude that we discussed in the previous sections [73] . In particular, one learns that, in the case of the open string, the Fubini-Veneziano field is just the string coordinate computed at σ = 0:
Q µ (z) ≡ X µ (z, σ = 0) ; z = e iτ ( 299
)
In the case of a closed string we get instead: z, z) ; z = e 2i(τ -σ) , z = e 2i(τ +σ) (300) Finally, let me mention that with the functional approach Mandelstam [74] and Cremmer and Gervais [79] computed the interaction between three arbitrary physical string states and reproduced in this way the coupling of three DDF states given in Eq. ( 202 ) and obtained in Ref. [37] by using the operator formalism. At this point it was completely clear that the structure underlying the generalized Veneziano model was that of an open relativistic string, while that underlying the Shapiro-Virasoro model was that of a closed relativistic string. Furthermore, these two theories are not independent because, if one starts from an open string theory, one gets automatically closed strings by loop corrections.
Q µ (z, z) ≡ X µ (
In this contribution, we have gone through the developments that led from the construction of the dual resonance model to the bosonic string theory trying as much as possible to include all the necessary technical details. This is because we believe that they are not only important from an historical point of view, but are also still part of the formalism that one uses today in many
The birth of string theory 57 string calculations. We have tried to be as complete and objective as possible, but it could very well be that some of those who participated in the research of these years, will not agree with some or even many of the statements we made. We apologize to those we have forgotten to mention or we have not mentioned as they would have liked.
Finally, after having gone through the developments of these years, my thoughts go to Sergio Fubini who shared with me and Gabriele many of the ideas described here and who is deeply missed, and to my friends from Florence, Naples and Turin for a pleasant collaboration in many papers discussed here.
I thank R. Marotta and I. Pesando for a critical reading of the manuscript. References 1. G.F. Chew, The analytic S matrix, W.A.Benjamin, Inc. (1966). R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S matrix, Cambridge University Press (1966). 2. R. Dolen, D. Horn and C. Schmid, Phys. Rev. 166, 1768 (1968). C. Schmid, Phys. Rev. Letters 20, 689 (1968). 3. H. Harari, Phys. Rev. Letters 22, 562 (1969). J.L. Rosner, Phys. Rev. Letters 22, 689 (1969). 4. G. Veneziano, Nuovo Cimento A 57, 190 (1968). 5. M. A. Virasoro, Phys. Rev. 177, 2309 (1969). 6. M.A. Virasoro, Phys. Rev. D 1, 2933 (1970). 7. A. Neveu and J.H. Schwarz, Nucl. Phys. B 31, 86 (1971) and Phys. Rev. D 4, 1109 (1971). 8. P. Ramond, Phys. Rev. D 3, 2415 (1971). 9. C. Lovelace, Phys. Lett. B 28, 265 (1968). J. Shapiro, Phys. Rev. 179, 1345 (1969). 10. P.H. Frampton, Phys. Lett. B 41, 364 (1972). 11. V. Alessandrini, D. Amati, M. Le Bellac and D. Olive, Phys. Rep. C 1, 269 (1971). G. Veneziano, Phys. Rep. C 9, 199 (1974). S. Mandelstam, Phys. Rep. C 13, 259 (1974). C. Rebbi, Phys. Rep. C 12, 1 (1974). J. Scherk, Rev. Mod. Phys. 47, 123 (1975). 12. F. Gliozzi, Lett. Nuovo Cimento 2, 1160 (1970). 13. K. Bardakçi and H. Ruegg, Phys. Rev. 181, 1884 (1969). C.G. Goebel and B. Sakita, Phys. Rev. Letters 22, 257 (1969). Chan Hong-Mo and T.S. Tsun, Phys. Lett. B 28, 485 (1969). Z. Koba and H.B.Nielsen, Nucl. Phys. B 10, 633 (1969). 14. K. Bardakçi and H. Ruegg, Phys. Lett.B 28, 671 (1969). M.A. Virasoro, Phys. Rev. Lett. 22, 37 (1969). 58 Paolo Di Vecchia 15. Z. Koba and H.B.Nielsen, Nucl. Phys. B 12, 517 (1969). 16. H. M. Chan and J.E. Paton, Nucl. Phys. B 10, 516 (1969). 17. S. Fubini and G. Veneziano, Nuovo Cimento A 64, 811 (1969). 18. Bardakçi and S. Mandelstam, Phys. Rev. 184, 1640 (1969). 19. S. Fubini, D. Gordon and G. Veneziano, Phys. Lett. B 29, 679 (1969) 20. Y. Nambu, Proc. Int. Conf. on Symmetries and Quark Models, Wayne State University 1969 (Gordon and Breach, 1970) p. 269. 21. L. Susskind, Nuovo Cimento A 69, 457 (1970) and Phys. Rev. Letter 23, 545 (1969). 22. J. Shapiro, Phys. Lett. B 33, 361 (1970). 23. S. Fubini and G. Veneziano, Nuovo Cimento A 67, 29 (1970). 24. F. Gliozzi, Lettere al Nuovo Cimento 2, 846 (1969). 25. C.B. Chiu, S. Matsuda and C. Rebbi, Phys. Rev. Lett. 23, 1526 (1969). C.B. Thorn, Phys. Rev. D 1, 1963 (1970). 26. S. Fubini and G. Veneziano, Annals of Physics 63, 12 (1971). 27. C. Lovelace, Phys. Lett. B 34, 500 (1971). 28. E. Del Giudice and P. Di Vecchia, Nuovo Cimento A 5, 90 (1971). M. Yoshimura, Phys. Lett. B 34, 79 (1971). 29. E. Del Giudice and P. Di Vecchia, Nuovo Cimento A 70, 579 (1970). 30. P. Campagna, S. Fubini, E Napolitano and S. Sciuto, Nuovo Cimento A 2, 911 (1971). 31. E. Del Giudice, P. Di Vecchia and S. Fubini, Annals of Physics, 70, 378 (1972). 32. R.C. Brower, Phys. Rev. D 6, 1655 (1972). 33. P. Goddard and C.B. Thorn, Phys. Lett. B 40, 235 (1972). 34. F. Gliozzi, J. Scherk and D. Olive, Phys. Lett. B 65, 282 (1976) ; Nucl. Phys. B 122, 253 (1977). 35. L. Brink and H.B. Nielsen, Phys. Lett. B 45, 332 (1973). 36. F. Gliozzi, unpublished.
See also P. Di Vecchia in Many Degrees of Freedom in Particle Physics, Edited by H. Satz, Plenum Publishing Corporation, 1978, p. 493. 37. M. Ademollo, E. Del Giudice, P. Di Vecchia and S. Fubini, Nuovo Cimento A 19, 181 (1974). 38. J. Scherk, Nucl. Phys. B 31, 222 (1971). 39. N. Nakanishi, Prog. Theor. Phys. 48, 355 (1972) . P.H. Frampton and K.C. Wali, Phys. Rev. D 8, 1879 (1973). 40. A. Neveu and J. Scherk, Nucl. Phys.B 36, 155 (1973). 41. A. Neveu and J.L. Gervais, Nucl. Phys. B 46, 381 (1972). 42. P. Di Vecchia, A. Lerda, L. Magnea, R. Marotta and R. Russo, Nucl. Phys. B 469, 235 (1996) 43. T. Yoneya, Prog. of Theor. Phys. 51, 1907 (1974). 44. K. Kikkawa, B. Sakita and M. Virasoro, Phys. Rev. 184, 1701 (1969). K. Bardakçi, M.B. Halpern and J. Shapiro, Phys. Rev. 185, 1910 (1969). D. Amati, C. Bouchiat and J.L. Gervais, Lett. al Nuovo Cimento 2, 399 (1969). A. Neveu and J. Scherk, Phys. Rev. D 1, 2355 (1970). G. Frye and L. Susskind, Phys. Lett. B 31, 537 (1970). D.J. Gross, A. Neveu, J. Scherk and J.H. Schwarz, Phys. Rev. D 2, 697 (1970). 45. L. Brink and D. Olive, Nucl. Phys. B 56, 253 (1973) and Nucl. Phys. B 58, 237 (1973).
The birth of string theory 59 46. E. Cremmer and J. Scherk, Nucl. Phys. B 50, 222 (1972). L. Clavelli and J. Shapiro, Nucl. Phys. B 57, 490 (1973). L. Brink, D.I. Olive and J. Scherk, Nucl. Phys. B 61, 173 (1973). 47. M. Ademollo, A. D'Adda, R. D'Auria, F. Gliozzi, E. Napolitano, S. Sciuto and P. Di Vecchia, Nucl. Phys. B 94, 221 (1975). J. Shapiro, Phys. Rev. D 11, 2937 (1975). 48. D.I.Olive and J. Scherk, Phys. Lett. B 44, 296 (1973). 49. M. Ademollo, A. D'Adda, R. D'Auria, E. Napolitano, P. Di Vecchia, F. Gliozzi and S. Sciuto, Nucl. Phys. B 77, 189 (1974). 50. M. Kalb and P. Ramond, Phys. Rev. D 9, 2273 (1974). 51. E, Cremmer and J. Scherk, Nucl. Phys. B 72, 117 (1974). 52. A. D'Adda, R. D'Auria, E. Napolitano, P. Di Vecchia, F. Gliozzi and S. Sciuto, Phys. Lett. B 68, 81 (1977). 53. J. Shapiro, Phys. Rev. D 5, 1945 (1975). 54. S. Sciuto, Lettere al Nuovo Cimento 2, 411 (1969).
A. Della Selva and S. Saito, Lett. al Nuovo Cimento 4, 689 (1970). 55. L. Caneschi, A. Schwimmer and G. Veneziano, Phys. Lett.B 30, 356 (1969). L. Caneschi and A. Schwimmer, Lettere al Nuovo Cimento 3, 213 (1970). 56. C. Lovelace, Phys. Lett. B 32, 490 (1970). 57. D.I. Olive, Nuovo Cimento A 3, 399 (1971). 58. I. Drummond, Nuovo Cimento A 67, 71 (1970). G. Carbone and S. Sciuto, Lett. Nuovo Cimento 3, 246 (1970). L. Kosterlitz and D. Wray, Lett. al Nuovo Cimento 3, 491 (1970). D. Collop, Nuovo Cimento A 1, 217 (1971). L.P. Yu, Phys. Rev. D 2, 1010 (!970); Phys. Rev. D 2, 2256 (!970). E. Corrigan and C. Montonen, Nucl. Phys. B 36, 58 (1972). J.L. Gervais and B. Sakita, Phys. Rev. D 4, 2291 (1971). 59. C. Lovelace, Phys. Lett. B 32, 703 (1970). 60. V. Alessandrini, Nuovo Cimento A 2, 321 (1971). 61. D. Amati and V. Alessandrini, Nuovo Cimento A 4, 793 (1971). 62. P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, Phys. Lett. B 199, 49 (1987). J.L. Petersen and J. Sidenius, Nucl. Phys. B 301, 247 (1988). 63. S. Mandelstam, In "Unified String Theories", edited by M. Green and D. Gross, World Scientific, p. 46. 64. P. Di Vecchia, F. Pezzella, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, Nucl.
Phys. B 322, 317 (1989). 65. Y. Nambu, Lectures at the Copenhagen Symposium, 1970, unpublished. 66. H. B. Nielsen, Paper submitted to the 15th Int. Conf. on High Energy Physics, Kiev, 1970 and Nordita preprint (1969). 67. T. Takabayasi, Progr. Theor. Phys. 44 (1970) 1117. O. Hara, Progr. Theor. Phys. 46, 1549 (1971). L.N. Chang and J. Mansouri, Phys. Rev. D 5, 2535 (1972). J. Mansouri and Y. Nambu, Phys. Lett. B 39, 357 (1972). M. Minami, Prog. Theor. Phys. 48, 1308 (1972). 68. T. Goto, Progr. Theor. Phys. 46 (1971) 1560. 69. D. Fairlie and H.B. Nielsen, Nucl. Phys. B 20, 637 (1970) and 22, 525 (1970). 70. C.S. Hsue, B. Sakita and M.A. Virasoro, Phys. Rev. 2, 2857 (1970). J.L. Gervais and B. Sakita, Phys. Rev. D 4, 2291 (1971). 71. P. Goddard, J. Goldstone, C. Rebbi and C. Thorn, Nucl. Phys. B 56, 109 (1973). 60 Paolo Di Vecchia 72. E.F. Corrigan and D.B. Fairlie, Nucl. Phys. B 91, 527 (1975). 73. M. Ademollo, A. D'Adda, R. D'Auria, P. Di Vecchia, F. Gliozzi, R. Musto, E. Napolitano, F. Nicodemi and S. Sciuto, Nuovo Cimento A 21, 77 (1974). 74. S. Mandelstam, Nucl. Phys. B 64, 205 (1973). 75. J.L. Gervais and B. Sakita, Phys. Rev. Lett. 30, 716 (1973). 76. L. Brink, P. Di Vecchia, P. Howe, S. Deser and B. Zumino, Phys. Lett. B 64, 435 (1976). 77. L. Brink, P. Di Vecchia and P. Howe, Phys. Lett. B 65, 471 (1976). S. Deser and B. Zumino, Phys. Lett. B 65, 369 (1976). 78. C. Lovelace, Phys. Lett. B 136, 75 (1984). C.G. Callan, E.J.Martinec, M.J. Perry and D. Friedan, Nucl. Phys. B 262, 593 (1985). 79. E. Cremmer and J.L. Gervais, Nucl. Phys. 76, 209 (1974). | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The sixties was a period in which strong interacting processes were studied in detail using the newly constructed accelerators at Cern and other places. Many new hadronic states were found that appeared as resonant peaks in various cross sections and hadronic cross sections were measured with increasing accuracy. In general, the experimental data for strongly interacting processes were rather well understood in terms of resonance exchanges in the direct channel at low energy and by the exchange of Regge poles in the transverse channel at higher energy. Field theory that had been very successful in describing QED seemed useless for strong interactions given the big number of hadrons to accomodate in a Lagrangian and the strength of the pion-nucleon coupling constant that did not allow perturbative calculations. The only domain in which field theoretical techniques were successfully used was current algebra. Here, assuming that strong interactions were described by an almost chiral invariant Lagrangian, that chiral symmetry was spontaneously broken and that the pion was the corresponding Goldstone boson, field theoretical methods gave rather good predictions for scattering amplitudes involving pions at very low energy. Going to higher energy was, however, not possible with these methods.\n\nBecause of this, many people started to think that field theory was useless to describe strong interactions and tried to describe strong interacting 2 Fig. 1. Duality diagram for the scattering of four mesons\n\nStarting from these ideas Veneziano [4] was able to construct an S matrix for the scattering of four mesons that, at the same time, had an infinite number of zero width resonances lying on linearly rising Regge trajectories and Regge behaviour at high energy. Veneziano originally constructed the model for the 1 For a discussion of S matrix theory see Ref.s [1] The birth of string theory 3 process ππ → πω, but it was immediately extended to the scattering of four scalar particles.\n\nIn the case of four identical scalar particles, the crossing symmetric scattering amplitude found by Veneziano consists of a sum of three terms:\n\nA(s, t, u) = A(s, t) + A(s, u) + A(t, u) ( 1\n\n)\n\nwhere A(s, t) = Γ (-α(s))Γ (-α(t)) Γ (-α(s) -α(t)) = 1 0 dxx -α(s)-1 (1 -x) -α(t)-1 (2) with linearly rising Regge trajectories\n\nα(s) = α 0 + α ′ s ( 3\n\n)\n\nThis was a very important property to implement in a model because it was in agreement with the experimental data in a wide range of energies. s, t and u are the Mandelstam variables:\n\ns = -(p 1 + p 2 ) 2 , t = -(p 3 + p 2 ) 2 , u = -(p 1 + p 3 ) 2 ( 4\n\n)\n\nThe three terms in Eq. ( 1 ) correspond to the three orderings of the four particles that are not related by a cyclic or anticyclic 2 permutation of the external legs. They correspond, respectively, to the three permutations: (1234), (1243) and (1324) of the four external legs. They have only simple pole singularities.\n\nThe first one has only poles in the s and t channels, the second only in the s and u channels and the third only in the t and u channels. This property follows directly from the duality diagram that is associated to each inequivalent permutation of the external legs. In fact, at that time one used to associate to each of the three inequivalent permutations a duality diagram where each particle was drawn as consisting of two lines that rappresented the quark and antiquark making up a meson. Furthermore, the diagram was supposed to have only poles singularities in the planar channels which are those involving adjacent external lines. This means that, for instance, the duality diagram corresponding to the permutation (1234) has only poles in the s and t channels as one can see by deforming the diagram in the plane in the two possible ways shown in figure (2) . This was a very important property of the duality diagram that makes it qualitatively different from a Feynman diagram in field theory where each diagram has only a pole in one of the three s, t and u channels and not simultaneously in two of them. If we accept the idea that each term of the sum in Eq. (1) is described by a duality diagram, then it is clear that we 2 An anticyclic permutation corresponding, for instance, to the ordering (1234) is obtained by taking the reverse of the original ordering (4321) and then performing a cyclic permutation."
},
{
"section_type": "OTHER",
"section_title": "Paolo Di Vecchia",
"text": "Fig. 2. The duality diagram contains both s and t channel poles do not need to add terms corresponding to equivalent diagrams because the corresponding duality diagram is the same and has the same singularities. It is now clear that it was in some way implicit in this picture the fact that the Veneziano model corresponds to the scattering of relativistic strings. But at that time the connection was not obvious at all. The only S matrix property that the Veneziano model failed to satisfy was the unitarity of the S matrix. because it contained only zero width resonances and did not have the various cuts required by unitarity. We will see how this property will be implemented. Immediately after the formulation of the Veneziano model, Virasoro [5] proposed another crossing symmetric four-point amplitude for scalar particles that consisted of a unique piece given by:\n\nA(s, t, u) ∼ Γ (-α(u) 2 )Γ (-α(s) 2 )Γ (-α(t) 2 ) Γ (1 + α(u) 2 )Γ (1 + α(s) 2 )Γ (1 + α(t) 2 ) ( 5\n\n)\n\nwhere\n\nα(s) = α 0 + α ′ s ( 6\n\n)\n\nThe model had poles in all three s, t and u channels and could not be written as sum of three terms having poles only in planar diagrams. In conclusion, the Veneziano model satisfies the principle of planar duality being a crossing symmetric combination of three contributions each having poles only in the planar channels. On the other hand, the Virasoro model consists of a unique crossing symmetric term having poles in both planar and non-planar channels. The attempts to construct consistent models that were in good agreement with the strong interaction phenomenology of the sixties boosted enormously the activity in this research field. The generalization of the Veneziano model to the scattering of N scalar particles was built, an operator formalism consisting of an infinite number of harmonic oscillators was constructed and the complete spectrum of mesons was determined. It turned out that the degeneracy of states grew up exponentially with the mass. It was also found that the N point amplitude had states with negative norm (ghosts) unless the intercept of the Regge trajectory was α 0 = 1 [6] . In this case it turned out that the model was free of ghosts but the lowest state was a tachyon. The model was called in the literature the \"dual resonance model\".\n\nThe birth of string theory 5 The model was not unitary because all the states were zero width resonances and the various cuts required by unitarity were absent. The unitarity was implemented in a perturbative way by adding loop diagrams obtained by sewing some of the external legs together after the insertion of a propagator. The multiloop amplitudes showed a structure of Riemann surfaces. This became obvious only later when the dual resonance model was recognized to correspond to scattering of strings.\n\nBut the main problem was that the model had a tachyon if α 0 = 1 or had ghosts for other values of α 0 and was not in agreement with the experimental data: α 0 was not equal to about 1 2 as required by experiments for the ρ Regge trajectory and the external scalar particles did not behave as pions satisfying the current algebra requirements. Many attempts were made to construct more realistic dual resonance models, but the main result of these attempts was the construction of the Neveu-Schwarz [7] and the Ramond [8] models, respectively, for mesons and fermions. They were constructed as two independent models and only later were recognized to be two sectors of the same model. The Neveu-Schwarz model still contained a tachyon that only in 1976 through the GSO projection was eliminated from the physical spectrum. Furthermore, it was not properly describing the properties of the physical pions.\n\nActually a model describing ππ scattering in a rather satisfactory way was proposed by Lovelace and Shapiro [9] 3 . According to this model the three isospin amplitudes for pion-pion scattering are given by:\n\nA 0 = 3 2 [A(s, t) + A(s, u)] - 1 2 A(t, u) A 1 = A(s, t) -A(s, u) A 2 = A(t, u) ( 7\n\n)\n\nwhere\n\nA(s, t) = β Γ (1 -α(s))Γ (1 -α(t)) Γ (1 -α(t) -α(s)) ; α(s) = α 0 + α ′ s ( 8\n\n)\n\nThe amplitudes in eq.( 7 ) provide a model for ππ scattering with linearly rising Regge trajectories containing three parameters: the intercept of the ρ Regge trajectory α 0 , the Regge slope α ′ and β. The first two can be determined by imposing the Adler's self-consistency condition, that requires the vanishing of the amplitude when s = t = u = m 2 π and one of the pions is massless, and the fact that the Regge trajectory must give the spin of the ρ meson that is equal to 1 when √ s is equal to the mass of the ρ meson m ρ . These two conditions determine the Regge trajectory to be:\n\nα(s) = 1 2 1 + s -m 2 π m 2 ρ -m π 2 = 0.48 + 0.885s ( 9\n\n)\n\n3 See also Ref. [10] ."
},
{
"section_type": "OTHER",
"section_title": "Paolo Di Vecchia",
"text": "Having fixed the parameters of the Regge trajectory the model predicts the masses and the couplings of the resonances that decay in ππ in terms of a unique parameter β. The values obtained are in reasonable agreement with the experiments. Moreover, one can compute the ππ scattering lenghts:\n\na 0 = 0.395β a 2 = -0.103β ( 10\n\n)\n\nand one finds that their ratio is within 10% of the current algebra ratio given by a 0 /a 2 = -7/2. The amplitude in eq.(8) has exactly the same form as that for four tachyons of the Neveu-Schwarz model with the only apparently minor difference that α 0 = 1/2 (for m π = 0) instead of 1 as in the Neveu-Schwarz model. This difference, however, implies that the critical space-time dimension of this model is d = 4 4 and not d = 10 as in the Neveu-Schwarz model. In conclusion this model seems to be a perfectly reasonable model for describing low-energy ππ scattering. The problem is, however, that nobody has been able to generalize it to the multipion scattering and therefore to get the complete meson spectrum.\n\nAs we have seen the S matrix of the dual resonance model was constructed using ideas and tools of hadron phenomenology of the end of the sixties. Although it did not seem possible to write a realistic dual resonance model describing the pions , it was nevertheless such a source of fascination for those who actively worked in this field at that time for its beautiful internal structure and consistency that a lot of energy was used to investigate its properties and for understanding its basic structure. It turned out with great surprise that the underlying structure was that of a quantum relativistic string.\n\nThe aim of this contribution is to explain the logic of the work that was done in the years from 1968 to 1974 5 in order to uncover the deep properties of this model that appeared from the beginning to be so beautiful and consistent to deserve an intensive study.\n\nThis seems to me a very good way of celebrating the 65th anniversary of Gabriele who is the person who started and also contributed to develop the whole thing with his deep physical intuition."
},
{
"section_type": "OTHER",
"section_title": "Construction of the N -point amplitude",
"text": "We have seen that the construction of the four-point amplitude is not sufficient to get information on the full hadronic spectrum because it contains only those hadrons that couple to two ground state mesons and does not see those intermediate states which only couple to three or to an higher number of ground state mesons [12] . Therefore, it was very important to construct the N -point amplitude involving identical scalar particles. The construction of 4 This can be checked by computing the coupling of the spinless particle at the level α(s) = 2 and seeing that it vanishes for d = 4. 5 Reviews from this period can be found in Ref. [11] The birth of string theory 7 the N -point amplitude was done in Ref. [13] (extending the work of Ref. [14] ) by requiring the same principles that have led to the construction of the Veneziano model, namely the fact that the axioms of S-matrix theory be satisfied by an infinite number of zero width resonances lying on linearly rising Regge trajectories and planar duality.\n\nThe fully crossing symmetric scattering amplitude of N identical scalar particles is given by a sum of terms corresponding to the inequivalent permutations of the external legs:\n\nA = Np n=1 A n ( 11\n\n)\n\nAlso in this case two permutations of the external legs are inequivalent if they are not related by a cyclic or anticyclic permutation. N p is the number of inequivalent permutations of the external legs and is equal to N p = (N -1)! 2 and each term has only simple pole singularities in the planar channels. Each planar channel is described by two indices (i, j), to mean that it includes the legs i, i + 1, i + 2 . . . j -1, j, by the Mandelstam variable\n\ns ij = -(p i + p i+1 + . . . + p j ) 2 ( 12\n\n)\n\nand by an additional variable u ij whose role will become clear soon. It is clear that the channels (ij) and (j + 1, i -1) 6 are identical and they should be counted only once. In the case of N identical scalar particles the number of planar channels is equal to N (N -3)\n\n. This can be obtained as follows. The independent planar diagrams involving the particle 1 are of the type (1, i) where i = 2 . . . N -2. Their number is N -3. This is also the number of planar diagrams involving the particle 2 and not the 1. The number of planar diagrams involving the particle 3 and not the particles 1 and 2 is equal to N -4. In general the number of planar diagrams involving the particle i and not the previous ones from 1 to i-1 is equal to N -1 -i. This means that the total number of planar diagram is equal to: 2(N -3) + N -2 i=3 (N -1 -i) = 2(N -3) + N -4 i=1 i = = 2(N -3) + (N -4)(N -3) 2 = N (N -3) 2 (13) If one writes down the duality diagram corresponding to a certain planar ordering of the external particles, it is easy to see that the diagram can have simultaneous pole singularities only in N -3 channels. The channels that allow simultaneous pole singularities are called compatible channels, the other 6 This channel includes the particles (j + 1, . . . , N, 1, . . . i -1).\n\n8 Paolo Di Vecchia are called incompatible. Two channels (i,j) and (h,k) are incompatible if the following inequalities are satisfied:\n\ni ≤ h ≤ j ; j + 1 ≤ k ≤ i -1 ( 14\n\n)\n\nThe aim is to construct the scattering amplitude for each inequivalent permutation of the external legs that has only pole singularities in the N (N -3) 2 planar channels. We have also to impose that the amplitude has simultaneous poles only in N -3 compatible channels. In order to gain intuition on how to proceed we rewrite the four-point amplitude in Eq. (2) as follows:\n\nA(s, t) = 1 0 du 12 1 0 du 23 u\n\n-α(s12)-1 12 u -α(s23)-1 23 δ(u 12 + u 23 -1) (15)\n\nwhere u 12 and u 23 are the variables corresponding to the two planar channels (12) and (23) and the cancellation of simultaneous poles in incompatible channels is provided by the δ-function which forbids u 12 and u 23 to vanish simultaneously. We will now extend this procedure to the N -point amplitude. But for the sake of clarity let us start with the case of N = 5 [14] . In this case we have 5 planar channels described by u 12 , u 13 , u 23 , u 24 and u 34 . Since we have only two compatible channels only two of the previous five variables are independent. We can choose them to be u 12 and u 13 . In order to determine the dependence of the other three variables on the two independent ones, we exclude simultaneous poles in incompatible channels. This can be done by imposing relations that prevent variables corresponding to incompatible channels to vanish simultaneously. A sufficient condition for excluding simultaneous poles in incompatible channels is to impose the conditions:\n\nu P = 1 - P u P ( 16\n\n)\n\nwhere the product is over the variables P corresponding to channels that are incompatible with P . In the case of the five-point amplitude we get the following relations: u 23 = 1 -u 34 u 12 ; u 24 = 1 -u 13 u 12 u 13 = 1 -u 34 u 24 ; u 34 = 1 -u 23 u 13 ; u 12 = 1 -u 24 u 23 (17) Solving them in terms of the two independent ones we get:\n\nu 23 = 1 -u 12 1 -u 12 u 13 ; u 34 = 1 -u 13 1 -u 12 u 13 ; u 24 = 1 -u 12 u 13 ( 18\n\n)\n\nIn analogy with what we have done for the four-point amplitude in Eq. ( 15 ) we write the five-point amplitude as follows:\n\nThe birth of string theory 9 1 0 du 12 1 0 du 13 1 0 du 23 1 0 du 24 1 0 du 34 u\n\n-α(s12)-1 12 u -α(s13)-1 13 × ×u -α(s24)-1 24 u -α(s23)-1 23 u -α(s34)-1 34 × δ(u 23 + u 12 u 34 -1)δ(u 24 + u 12 u 13 -1)δ(u 34 + u 13 u 23 -1) (19) Performing the integral over the variables u 23 , u 24 and u 34 we get:\n\n1 0 du 12 1 0 du 13 u\n\n-α(s12)-1 12 u -α(s13)-1 13 × × (1 -u 12 ) -α(s23)-1 (1 -u 13 ) -α(s13)-1 (1 -u 12 u 13 ) -α(s24)+α(s23)+α(s34) (20)\n\nWe have implicitly assumed that the Regge trajectory is the same in all channels and that the external scalar particles have the same common mass m and are the lowest lying states on the Regge trajectory. This means that their mass is given by:\n\nα 0 -α ′ p 2 i = 0 ; p 2 i ≡ -m 2 ( 21\n\n)\n\nUsing then the relation:\n\nα(s 23 ) + α(s 34 ) -α(s 24 ) = 2α ′ p 2 • p 4 ( 22\n\n)\n\nwe can rewrite Eq. (20) as follows:\n\nB 5 = 1 0 du 2 1 0 du 3 u\n\n-α(s2)-1 2 u -α(s3)-1 3 (1 -u 2 ) -α(s23)-1 × × (1 -u 3 ) -α(s34)-1\n\n2 i=2 4 j=4 (1 -x ij ) 2α ′ pi•pj ( 23\n\n)\n\nwhere\n\ns i ≡ s 1i , u i ≡ u 1i ; i = 2, 3 ; x ij = u i u i+1 . . . u j-1 . ( 24\n\n)\n\nWe are now ready to construct the N -point function [13] . In analogy with what has been done for the four and five-point amplitudes we can write the N -point amplitude as follows:\n\nB N = 1 0 . . . 1 0 P [u -α(sP )-1 P ] Q δ(u Q -1 + Q u Q) ( 25\n\n)\n\n10 Paolo Di Vecchia\n\nwhere the first product is over the N (N -3) 2 variables corresponding to all planar channels, while the second one is over the (N -3)(N -2) 2 independent δ-functions. The product in the δ-function is defined in Eq. (16) .\n\nThe solution of all the non-independent linear relations imposed by the δ-functions is given by\n\nu ij = (1 -x ij )(1 -x i-1,j+1 ) (1 -x i-1,j )(1 -x i,j+1 ) ( 26\n\n)\n\nwhere the variables x ij are given in Eq. ( 24 ). Eliminating the δ-function from Eq. (25) one gets:\n\nB N = N -2 i=2 1 0 du i u\n\n-α(si)-1 i (1 -u i ) -α(si,i+1)-1 N -3 i=2 N -1 j=i+2 (1 -x ij ) -γij (27) where γ ij = α(s ij ) + α(s i+1;j-1 ) -α(s i;j-1 ) -α(s i+1;j ) ; j ≥ i + 2 (28)\n\nIt is easy to see that\n\nα(s i,i+1 ) = -α 0 -2α ′ p i • p i+1 ; γ ij = -2α ′ p i • p j ; j ≥ i + 2 ( 29\n\n)\n\nInserting them in Eq. (27) we get:\n\nB N = N -2 i=2 1 0 du i u\n\n-α(si)-1 i (1 -u i ) α0-1 N -2 i=2 N -1\n\nj=i+1 (1 -x ij ) 2α ′ pi•pj ( 30\n\n)\n\nThis is the form of the N -point amplitude that was originally constructed. Then Koba and Nielsen [15] put it in the form that is more known nowadays. They constructed it using the following rules. They associated a real variable z i to each leg i. Then they associated to each channel (i, j) an anharmonic ratio constructed from the variables z i , z i-1 , z j , z j+1 in the following way\n\n(z i , z i+1 , z j , z j+1 ) -α(sij )-1 = (z i -z j )(z i-1 -z j+1 ) (z i-1 -z j )(z i -z j+1 )\n\n-α(sij )-1 31) and finally they gave the following expression for the N -point amplitude:\n\n(\n\nB N = ∞ -∞ dV (z)\n\n(i,j)\n\n(z i , z i+1 , z j , z j+1 ) -α(sij )-1 ( 32\n\n)\n\nwhere\n\ndV (z) = N 1 [θ(z i -z i+1 )dz i ] N i=1 (z i -z i+2 )dV abc ; dV abc = dz a dz b dz c (z b -z a )(z c -z b )(z a -z c ) ( 33\n\n)\n\nThe birth of string theory 11 and the variables z i are integrated along the real axis in a cyclically ordered way: z 1 ≥ z 2 . . . ≥ z N with a, b, c arbitrarily chosen. The integrand of the N -point amplitude is invariant under projective transformations acting on the leg variables z i :\n\nz i → αz i + β γz i + δ ; i = 1 . . . N ; αδ -βγ = 1 ( 34\n\n)\n\nThis is because both the anharmonic ratio in Eq. ( 31 ) and the measure dV abc are invariant under a projective transformation. Since a projective transformation depends on three real parameters, then the integrand of the N -point amplitude depends only on N -3 variables z i . In order to avoid infinities, one has then to divide the integration volume with the factor dV abc that is also invariant under the projective transformations. The fact that the integrand depends only on N -3 variables is in agreement with the fact that N -3 is also the maximal number of simultaneous poles allowed in the amplitude. It is convenient to write the N -point amplitude in a form that involves the scalar product of the external momenta rather than the Regge trajectories. We distinguish three kinds of channels. The first one is when the particles i and j of the channel (i, j) are separated by at least two particles. In this case the channels that contribute to the exponent of the factor (z i -z j ) are the channels (i, j) with exponent equal to -α(s ij ) -1, (i + 1, j -1) with exponent -α(s i+1,j-1 ) -1, (i + 1, j) with exponent α(s i+1,j ) + 1 and (i, j -1) with exponent α(s i,j-1 ) + 1. Adding these four contributions one gets for the channels where i and j are separated by at least two particles\n\n-α(s ij ) -α(s i+1,j-1 ) + α(s i+1,j ) + α(s i,j-1 ) = 2α ′ p i • p j ( 35\n\n)\n\nThe second one comes from the channels that are separated by only one particle. In this case only three of the previous four channels contribute. For instance if j = i + 2 the channel (i + 1, j -1) consists of only one particle and therefore should not be included. This means that we would get:\n\n-α(s i;i+2 ) -1 + α(s 1+1;i+2 ) + 1 + α(s i;i+1 + 1) = 1 + 2α ′ p i • p i+2 ( 36\n\n)\n\nFinally the third one that comes from the channels whose particles are adjacent, gets only contribution from:\n\n-α(s i;i+1 ) -1 = α 0 -1 + 2α ′ p i • p i+1 ( 37\n\n)\n\nPutting all these three terms together in Eq. ( 32 ) and remembering the factor in the denominator in the first equation of (33) we get:\n\nB N = ∞ -∞ N 1 dz i θ(z i -z i+1 ) dV abc N i=1 (z i -z i+1 ) α0-1 j>i (z i -z j ) 2α ′ pi•pj ( 38\n\n)\n\nA convenient choice for the three variables to keep fixed is:\n\n12 Paolo Di Vecchia\n\nz a = z 1 = ∞ ; z b = z 2 = 1 ; z c = z N = 0 ( 39\n\n)\n\nWith this choice the previous equation becomes:\n\nB N = N -1 i=3 1 0 dz i θ(z i -z i+1 ) N -1 i=2 (z i -z i+1 ) α0-1 × × N -1 i=2 N j=i+1 (z i -z j ) 2α ′ pi•pj ( 40\n\n)\n\nWe now want to show that this amplitude is identical to the one given in Eq. ( 30 ). This can be done by performing the following change of variables:\n\nu i = z i+1 z i ; i = 2, 3 . . . N -2 ( 41\n\n) that implies z i = u 2 u 3 . . . u i-1 ; i = 3, 4 . . . N -1 ( 42\n\n)\n\nTaking into account that the Jacobian is equal to:\n\ndet ∂z ∂u = N -2 i=3 z i = N -3 i=2 u N -2-i i ( 43\n\n)\n\nusing the following two relations:\n\ndet ∂z ∂u N -1 i=2 (z i -z i+1 ) α0-1 = N -2 i=2 u (N -1-i)α0-1 i N -2 i=2 (1 -u i ) α0-1 ( 44\n\n) and N -1 i=2 N j=i+1 (z j -z i ) 2α ′ pi•pj = = N -2 i=2 N -1 j=i+1 (1 -x ij ) 2α ′ pi•pj N -2 i=2 u -α(si)-(N -i-1)α0 i ( 45\n\n)\n\nand the conservation of momentum\n\nN i=1 p i = 0 ( 46\n\n)\n\ntogether with Eq. ( 21 ), one can easily see that Eq.s (30) and (40) are equal.\n\nThe birth of string theory 13 The N -point amplitude that we have constructed in this section corresponds to the scattering of N spinless particles with no internal degrees of freedom. On the other hand it was known that the mesons were classified according to multiplets of an SU (3) flavour symmetry. This was implemented by Chan and Paton [16] by multiplying the N -point amplitude with a factor, called Chan-Paton factor, given by T r(λ a1 λ a2 . . . λ aN ) (\n\n) 47\n\nwhere the λ's are matrices of a unitary group in the fundamental representation. Including the Chan-Paton factors the total scattering amplitude is given by: P T r(λ a1 λ a2 . . . λ aN\n\n)B N (p 1 , p 2 , . . . p N ) ( 48\n\n)\n\nwhere the sum is extended to the (N -1)! permutations of the external legs, that are not related by a cyclic permutations. Originally when the dual resonance model was supposed to describe strongly interacting mesons, this factor was introduced to represent their flavour degrees of freedom. Nowadays the interpretation is different and the Chan-Paton factor represents the colour degrees of freedom of the gauge bosons and the other massive particles of the spectrum. The N -point amplitude B N that we have constructed in this section contains only simple pole singularities in all possible planar channels. They correspond to zero width resonances located at non-negative integer values n of the Regge trajectory α(M 2 ) = n. The lowest state located at α(m 2 ) = 0 corresponds to the particles on the external legs of B N . The spectrum of excited particles can be obtained by factorizing the N -point amplitude in the most general channel with any number of particles. This was done in Ref.s [17] and [18] finding a spectrum of states rising exponentially with the mass M . Being the model relativistic invariant it was found that many states obtained by factorizing the N -point amplitude were \"ghosts\", namely states with negative norm as one finds in QED when one quantizes the electromagnetic field in a covariant gauge. The consistency of the model requires the existence of relations satisfied by the scattering amplitudes that are similar to those obtained through gauge invariance in QED. If the model is consistent they must decouple the negative norm states leaving us with a physical spectrum of positive norm states. In order to study in a simple way these issues, we discuss in the next section the operator formalism introduced already in 1969 [19, 20, 21] .\n\nBefore concluding this section let us go back to the non-planar four-point amplitude in Eq. ( 5 ) and discuss its generalization to an N -point amplitude. Using the technique of the electrostatic analogue on the sphere instead of on the disk Shapiro [22] was able to obtain a N -point amplitude that reduces to the four-point amplitude in Eq. ( 5 ) with intercept α 0 = 2. The N -point amplitude found in Ref. [22] is:\n\n14 Paolo Di Vecchia\n\nN i=1 d 2 z i dV abc i<j |z i -z j | α ′ pi•pj ( 49\n\n)\n\nwhere\n\ndV abc = d 2 z a d 2 z b d 2 z c |z a -z b | 2 |z a -z c | 2 |z b -z c | 2 ( 50\n\n)\n\nThe integral in Eq. ( 49 ) is performed in the entire complex plane."
},
{
"section_type": "OTHER",
"section_title": "Operator formalism and factorization",
"text": "The factorization properties of the dual resonance model were first studied by factorizing by brute force the N-point amplitude at the various poles [17, 18] . The number of terms that factorize the residue of the pole at α(s) = n, increases rapidly with the value of n. In order to find their degeneracy it turned out to be convenient to first rewrite the N-point amplitude in an operator formalism. In this section we introduce the operator formalism and we rewrite the N -point amplitude derived in the previous section in this formalism. The key idea [19, 20, 21] is to introduce an infinite set of harmonic oscillators and a position and momentum operators 7 which satisfy the following commutation relations:\n\n[a nµ , a † mν ] = η µν δ nm ; [q µ , pν ] = iη µν ( 51\n\n)\n\nwhere η µν is the flat Minkowski metric that we take to be η µν = (-1, 1, . . . 1). A state with momentum p is constructed in terms of a state with zero momentum as follows:\n\np|p ≡ pe ip•q |0 = p|p ; p |0 = 0 ( 52\n\n) normalized as 8 p|p ′ = (2π) d δ (d) (p + p ′ ) ( 53\n\n)\n\nIn order to avoid minus signs we use the convention that\n\np| = 0|e ip•q ( 54\n\n)\n\nA complete and orthonormal basis of vectors in the harmonic oscillator space is given by\n\n|λ 1 , λ 2 , . . . λ i ; p = n (a † µn;n ) λn;µ n λ n,µn ! e ipq |0, 0 ( 55\n\n)\n\n7 Actually the position and momentum operators were introduced in Ref. [23] . 8 Although we now use an arbitrary d we want to remind you that all original calculations were done for d = 4.\n\nThe birth of string theory 15\n\nwhere the first |0 corresponds to the one annihilated by all annihilation operators and the second one to the state of zero momentum:\n\na µn;n |0, 0 = p|0, 0 = 0 ( 56\n\n)\n\nNotice that Lorentz invariance forces to introduce also oscillators that create states with negative norm due to the minus sign in the flat Minkowski metric. This implies that the space spanned by the states in Eq. (55) is not positive definite. This is, however, not allowed in a quantum theory and therefore if the dual resonance model is a consistent quantum-relavistic theory we expect the presence of relations of the kind of those provided by gauge invariance in QED.\n\nLet us introduce the Fubini-Veneziano [23] operator:\n\nQ µ (z) = Q (+) µ (z) + Q (0) µ (z) + Q (-) µ (z) ( 57\n\n)\n\nwhere\n\nQ (+) = i √ 2α ′ ∞ n=1 a n √ n z -n ; Q (-) = -i √ 2α ′ ∞ n=1 a † n √ n z n Q (0) = q -2iα ′ p log z ( 58\n\n)\n\nIn terms of Q we introduce the vertex operator corresponding to the external leg with momentum p: V (z; p) =: e ip•Q(z) :≡ e ip•Q (-)\n\n(z) e ipq e +2α ′ p•p log z e ip•Q (+) (z) ( 59\n\n)\n\nand compute the following vacuum expectation value:\n\n0, 0| N i=1 V (z i , p i )|0, 0 ( 60\n\n)\n\nIt can be easily computed using the Baker-Haussdorf relation\n\ne A e B = e B e A e [A,B] ( 61\n\n)\n\nthat is valid if the commutator, as in our case, [A, B] is a c-number. In our case the commutation relations to be used are:\n\n[Q (+) (z), Q (-) (w)] = -2α ′ log 1 - w z ( 62\n\n)\n\nand the second one in Eq. ( 51 ). Using them one gets:\n\nV (z; p)V (w; k) =: V (z; p)V (w; k) : (z -w) 2α ′ p•k ( 63\n\n) and 16\n\nPaolo Di Vecchia 0, 0| N i=1 V (z i , p i )|0, 0 = i>j (z i -z j ) 2α ′ pi•pj (2π) d δ (d) ( N i=1 p i ) ( 64\n\n)\n\nwhere the normal ordering requires that all creation operators be put on the left of the annihilation one and the momentum operator p be put on the right of the position operator q. This means that\n\n(2π) d δ (d) ( N i=1 p i )B N = ∞ -∞ N 1 dz i θ(z i -z i+1 ) dV abc N i=1 (z i -z i+1 ) α0-1 × × 0, 0| N i=1 V (z i , p i )|0, 0 ( 65\n\n)\n\nBy choosing the three variables z a , z b and z c as in Eq. ( 39 ) we can rewrite the previous equation as follows:\n\n(2π) d δ (d) ( N i=1 p i )B N = 1 0 N -1 i=3 dz i N -1 i=2 θ(z i -z i+1 )× × N -1 i=2 (z i -z i+1 ) α0-1 0, p 1 | N -1 i=2 V (z i ; p i )|0, p N ( 66\n\n)\n\nwhere we have taken z 2 = 1 and we have defined (α 0 ≡ α ′ p 2 i ; i = 1 . . . N ) :\n\nlim zN →0 V (z N ; p N )|0, 0 ≡ |0; p N ; 0; 0| lim z1→∞ z 2α0 1 V (z 1 ; p 1 ) = 0, p 1 | ( 67\n\n)\n\nBefore proceeding to factorize the N -point amplitude let us study the properties under the projective group of the operators that we have introduced. We have already seen that the projective group leaves the integrand of the Koba-Nielsen representation of the N -point amplitude invariant. The projective group has three generators L 0 , L 1 and L -1 corresponding respectively to dilatations, inversions and translations. Assuming that the Fubini-Veneziano fields Q(z) transforms as a field with weight 0 (as a scalar) we can immediately write the commutation relations that Q(z) must satisfy. This means in fact that, under a projective transformation, Q(z) transforms as follows:\n\nQ(z) → Q T (z) = Q αz + β γz + δ ; αδ -βγ = 1 ( 68\n\n)\n\nExpanding for small values of the parameters we get:\n\nQ T (z) = Q(z) + (ǫ 1 + ǫ 2 z + ǫ 3 z 2 ) dQ(z) dz + o(ǫ 2 ) ( 69\n\n)\n\nThe birth of string theory 17 This means that the three generators of the projective group must satisfy the following commutation relations with Q(z):\n\n[L 0 , Q(z)] = z dQ dz ; [L -1 , Q(z)] = dQ dz ; [L 1 , Q(z)] = z 2 dQ dz ( 70\n\n)\n\nThey are given by the following expressions in terms of the harmonic oscillators:\n\nL 0 = α ′ p2 + ∞ n=1 na † n • a n ; L 1 = √ 2α ′ p • a 1 + ∞ n=1 n(n + 1)a n+1 • a † n ( 71\n\n)\n\nand\n\nL -1 = L † 1 = √ 2α ′ p • a † 1 + ∞ n=1 n(n + 1)a † n+1 • a n ( 72\n\n)\n\nThey annihilate the vacuum\n\nL 0 |0, 0 = L 1 |0, 0 = L -1 |0, 0 = 0 ( 73\n\n)\n\nthat is therefore called the projective invariant vacuum, and satisfy the algebra that is called Gliozzi algebra [24] 9 :\n\n[L 0 , L 1 ] = -L 1 ; [L 0 , L -1 ] = L -1 ; [L 1 , L -1 ] = 2L 0 ( 74\n\n)\n\nThe vertex operator with momentum p is a projective field with weight equal to α 0 = α ′ p 2 . It transforms in fact as follows under the projective group:\n\n[L n , V (z, p)] = z n+1 dV (z, p) dz + α 0 (n + 1)z n V (z, p) ; n = 0, ±1 ( 75\n\n)\n\nor in finite form as follows:\n\nU V (z, p)U -1 = 1 (γz + δ) 2α0 V αz + β γz + δ , p ( 76\n\n)\n\nwhere U is the generator of an arbitrary finite projective transformation. Since U leaves the vacuum invariant, by using Eq. (76) it is easy to show that:\n\n0, 0| N i=1 V (z ′ i , p)|0, 0 = N i=1 (γz i + δ) 2α0 0, 0| N i=1 V (z i , p)|0, 0 ( 77\n\n)\n\nthat together with the following equation:\n\nN i=1 dz ′ i N i=1 (z ′ i -z ′ i+1 ) α0-1 = N i=1 dz i N -1 i=1 (z i -z i+1 ) α0-1 N i=1 (γz i + δ) -2α0 ( 78\n\n)\n\n9 See also Ref. [25] ."
},
{
"section_type": "OTHER",
"section_title": "Paolo Di Vecchia",
"text": "implies that the integrand of the N -point amplitude in Eq. (65) is invariant under projective transformations. We are now ready to factorize the N -point amplitude and find the spectrum of mesons.\n\nFrom Eq.s (75) and (76) it is easy to derive the transformation of the vertex operator under a finite dilatation:\n\nz L0 V (1, p)z -L0 = V (z, p)z α0 ( 79\n\n)\n\nChanging the integration variables as follows:\n\nx i = z i+1 z i ; i = 2, 3 . . . N -2 ; det ∂z i ∂x j = z 3 z 4 . . . z N -2 ( 80\n\n)\n\nwhere the last term is the jacobian of the trasformation from z i to x i , we get from Eq.(66) the following expression:\n\nA N ≡ 0, p 1 |V (1, p 2 )DV (1, p 3 ) . . . DV (1, p N -1 )|0, p N ( 81\n\n)\n\nwhere the propagator D is equal to:\n\nD = 1 0 dxx L0-1-α0 (1 -x) α0-1 = Γ (L 0 -α 0 )Γ (α 0 ) Γ (L 0 ) ( 82\n\n)\n\nand\n\nA N = (2π) d δ (d) N i=1 p i B N ( 83\n\n)\n\nThe factorization properties of the amplitude can be studied by inserting in the channel (1, M ) or equivalently in the channel (M + 1, N ) described by the Mandelstam variable\n\ns = -(p 1 + p 2 + . . . p M ) 2 = -(p M+1 + p M+2 . . . + p N ) 2 ≡ -P 2 ( 84\n\n)\n\nthe complete set of states given in Eq. ( 55 ):\n\nA N = λ,µ p (1,M) |λ, P λ, P |D|µ, P µ, P |p (M+1,N ) ( 85\n\n)\n\nwhere\n\np (1,M) | = 0, p 1 |V (1, p 2 )DV (1, p 3 ) . . . V (1, p M ) ( 86\n\n) and |p (M+1,N ) = V (1, p M+1 )D . . . V (1, p N -1 )|p N , 0 ( 87\n\n)\n\nIntroducing the quantity:\n\nThe birth of string theory 19 R =\n\n∞ n=1 na † n • a n ( 88\n\n)\n\nit is possible to rewrite\n\nλ, P |D|µ, P = ∞ m=0 λ, P | (-1) m α 0 -1 m R + m -α(s) |µ, P ( 89\n\n)\n\nwhere s is the variable defined in Eq. ( 84 ). Using this equation we can rewrite Eq. (85) as follows\n\nA N = λ,µ p (1,M) |λ, P ∞ m=0 λ, P | (-1) m α 0 -1 m R + m -α(s) |µ, P µ, P |p (M+1,N ) ( 90\n\n)\n\nThis expression shows that amplitude A N has a pole in the channel (1, M ) when α(s) is equal to an integer n ≥ 0 and the states |λ that contribute to its residue are those satisfying the relation:\n\nR|λ = (n -m)|λ ; m = 0, 1 . . . n ( 91\n\n)\n\nThe number of independent states |λ contributing to the residue gives the degeneracy of states for each level n. Because of manifest relativistic invariance the space spanned by the complete system of states in Eq. (55) contains states with negative norm corresponding to those states having an odd number of oscillators with timelike directions (see Eq. (51)). This is not consistent in a quantum theory where the states of a system must span a positive definite Hilbert space. This means that there must exist a number of relations satisfied by the external states that decouple a number of states leaving with a positive definite Hilbert space. In order to find these relations we rewrite the state in Eq. (87) going back to the Koba-Nielsen variables:\n\n|p (1,M) = M-1 i=2 [ dz i θ(z i -z i+1 )] M-1 i=1 (z i -z i+1 ) α0-1 × × V (1, p 1 )V (z 2 , p 2 ) . . . V (z M-1 , p M-1 )|0, p M ( 92\n\n)\n\nLet us consider the operator U (α) that generate the projective transformation that leaves the points z = 0, 1 invariant:\n\nz ′ = z 1 -α(z -1) = z + α(z 2 -z) + o(α 2 ) ( 93\n\n)\n\nFrom the transformation properties of the vertex operators in Eq. (76) it is easy to see that the previous transformation leaves the state in Eq. (92) invariant:\n\n20 Paolo Di Vecchia\n\nU (α)|p (1,M) = |p (1,M) ( 94\n\n)\n\nThis means that the generator of the previous transformation annihilates the state in Eq. (92):\n\nW 1 |p (1,M) = 0 ; W 1 = L 1 -L 0 ( 95\n\n)\n\nThe explicit form of W 1 follows from the infinitesimal form of the transformation in Eq. ( 93 ). This condition that is of the same kind of the relations that on shell amplitudes with the emission of photons satisfy as a consequence of gauge invariance, implies that the residue at the pole in Eq. ( 90 ) can be factorized with a smaller number of states. It turns out, however, that a detailed analysis of the spectrum shows that negative norm states are still present. This can be qualitatively understood as follows. Due to the Lorentz metric we have a negative norm component for each oscillator. In order to be able to decouple all negative norm states we need to have a gauge condition of the type as in Eq. (95) for each oscillator. But the number of oscillators is infinite and, therefore, we need an infinite number of conditions of the type as in Eq. ( 95 ). It was found in Ref. [6] that, if we take α 0 = 1, then one can easily construct an infinite number of operators that leave the state in Eq. (92) invariant. In the next section we will concentrate on this case.\n\n4 The case α 0 = 1\n\nIf we take α 0 = 1 many of the formulae given in the previous section simplify. The N -point amplitude in Eq. (38) becomes:\n\nB N = ∞ -∞ N 1 dz i θ(z i -z i+1 ) dV abc j>i (z i -z j ) 2α ′ pi•pj ( 96\n\n)\n\nthat can be rewritten in the operator formalism as follows:\n\n(2π) 4 δ( N i=1 p i )B N = ∞ -∞ N 1 dz i θ(z i -z i+1 ) dV abc 0, 0| N i=1 V (z i , p i )|0, 0 ( 97\n\n) By choosing z 1 = ∞, z 2 = 1 and z N = 0 it becomes (2π) 4 δ( N i=1 p i )B N = = 1 0 N -1 i=3 dz i N -1 i=2 θ(z i -z i+1 ) 0, p 1 | N -1 i=2 V (z i ; p i )|0, p N ( 98\n\n)\n\nThe birth of string theory 21 where lim\n\nzN →0 V (z N ; p N )|0, 0 ≡ |0; p N ; 0; 0| lim z1→∞ z 2 1 V (z 1 ; p 1 ) = 0, p 1 | ( 99\n\n)\n\nEq. (81) is as before, but now the propagator becomes:\n\nD = dxx L0-2 = 1 L 0 -1 ( 100\n\n)\n\nThis means that Eq. (89) becomes:\n\nλ, P |D|µ, P = λ, P | 1 L 0 -1 |µ, P ( 101\n\n)\n\nand Eq. ( 90 ) has the simpler form:\n\nB N = λ p (1,M) |λ, P λ, P | 1 R -α(s) |λ, P λ, P |p (M+1,N ) ( 102\n\n)\n\nB N has a pole in the channel (1, M ) when α(s) is equal to an integer n ≥ 0 and the states |λ that contribute to its residue are those satisfying the relation:\n\nR|λ = n|λ ( 103\n\n)\n\nTheir number gives the degeneracy of the states contributing to the pole at α(s) = n. The N -point amplitude can be written as:\n\nB N = p (1,M) |D|p (M+1,N ) ( 104\n\n) where |p (1,M) = M-1 i=2 [dz i θ(z i -z i+1 )] × × V (1, p 1 )V (z 2 , p 2 ) . . . V (z M-1 , p M-1 |0, p M ( 105\n\n)\n\nUsing Eq. ( 79 ) and changing variables from\n\nz i , i = 2 . . . M -1 to x i = zi+1 zi , i = 1 . . . M -2 with z 1 = 1\n\nwe can rewrite the previous equation as follows:\n\n|p (1,M) = V (1, p 1 )DV (1, p 2 ) . . . DV (1, p M-1 )|0, p M ( 106\n\n)\n\nwhere the propagator D is defined in Eq. ( 100 ). We want now to show that the state in Eq.s (105) and (106) is not only annihilated by the operator in Eq. ( 95 ), but, if α 0 = 1 [6], by an infinite set of operators whose lowest one is the one in Eq. (95) . We will derive this by using the formalism developed in Ref. [26] and we will follow closely their derivation.\n\nStarting from Eq.s (70) Fubini and Veneziano realized that the generators of the projective group acting on a function of z are given by:\n\n22 Paolo Di Vecchia\n\nL 0 = -z d dz ; L -1 = - d dz ; L 1 = -z 2 d dz ( 107\n\n)\n\nThey generalized the previous generators to an arbitrary conformal transformation by introducing the following operators, called Virasoro operators:\n\nL n = -z n+1 d dz ( 108\n\n)\n\nthat satisfy the algebra:\n\n[L n , L m ] = (n -m)L n+m ( 109\n\n)\n\nthat does not contain the term with the central charge! They also showed that the Virasoro operators satisfy the following commutation relations with the vertex operator:\n\n[L n , V (z, p)] = d dz z n+1 V (z, p) ( 110\n\n)\n\nMore in general actually they define an operator L f corresponding to an arbitrary function f (ξ) and L f = L n if we choose f (ξ) = ξ n . In this case the commutation relation in Eq. (110) becomes:\n\n[L f , V (z, p)] = d dz (zf (z)V (z, p)) ( 111\n\n)\n\nBy introducing the variable:\n\ny = z A dξ ξf (ξ) ( 112\n\n)\n\nwhere A is an arbitrary constant, one can rewrite Eq. (111) in the following form:\n\n[L f , zf (z)V (z, p)] = d dy (zf (z)V (z, p)) ( 113\n\n)\n\nThis implies that, under an arbitrary conformal transformation z → f (z), generated by U = e αL f , the vertex operator transforms as:\n\ne αL f V (z, p) zf (z) e -αL f = V (z ′ , p)z ′ f (z ′ ) ( 114\n\n)\n\nwhere the parameter α is given by:\n\nα = z ′ z dξ ξf (ξ) ( 115\n\n)\n\nOn the other hand, this equation implies:\n\ndz zf (z) = dz ′ z ′ f (z ′ ) ( 116\n\n)\n\nThe birth of string theory 23 that, inserted in Eq. ( 114 ), implies that the quantity V (z, p) dz is left invariant by the transformation z → f (z):\n\ne αL f V (z, p)dze -αL f = V (z ′ , p)dz ′ ( 117\n\n)\n\nLet us now act with the previous conformal transformation on the state in Eq. ( 105 ). We get:\n\ne αL f |p (1,M) = 1 0 M-1 i=2 [dz i θ(z i -z i+1 )] e αL f V (1, p 1 )e -αL f × ×e αL f V (z 2 , p 2 )e -αL f . . . . . . e αL f V (z M-1 , p M-1 )e -αL f e αL f |0, p M = = 1 0 M-1 i=2 θ(z i -z i+1 ) × e αL f V (1, p 1 )e -αL f × × V (z ′ 2 , p 2 )dz ′ 2 . . . V (z ′ M-1 , p M-1 )dz ′ M-1 e αL f |0, p M ( 118\n\n)\n\nwhere we have used Eq. ( 117 ). The previous transformation leaves the state invariant if both z = 0 and z = 1 are fixed points of the conformal transformation. This happens if the denominator in Eq. (115) vanishes when ξ = 0, 1. This requires the following conditions:\n\nf (1) = 0 ; lim ξ→0 ξf (ξ) = 0 ( 119\n\n)\n\nExpanding ξ near the poinr ξ = 1 we can determine the relation between z and z ′ near z = z ′ = 1. We get:\n\nz ′ = ze -αf ′ (1) 1 -z + ze -αf ′ (1) ( 120\n\n)\n\nand from it we can determine the conformal factor:\n\ndz ′ dz = e -αf ′ (1) (1 -z + ze -αf ′ (1) ) 2 → e αf ′ (1) ( 121\n\n)\n\nin the limit z → 1. Proceeding in the same near the point z = z ′ = 0 we get:\n\nz ′ = zf (0)e αf (0) f (0) + zf ′ (0)(1 -e αf (0) → ze αf (0) ( 122\n\n)\n\nin the limit z → 0. This means that Eq. (118) becomes\n\ne α(L f -f ′ (1)-f (0)) |p (1,M) = |p (1,M) ( 123\n\n)\n\nA choice of f that satisfies Eq.s (119) is the following:\n\n24 Paolo Di Vecchia\n\nf (ξ) = ξ n -1 ( 124\n\n)\n\nthat gives the following gauge operator:\n\nW n = L n -L 0 -(n -1) ( 125\n\n)\n\nthat annihilates the state in Eq. (105):\n\nW n |p 1...M = 0 ; n = 1 . . . ∞ ( 126\n\n)\n\nThese are the Virasoro conditions found in Ref. [6] . There is one condition for each negative norm oscillator and, therefore, in this case there is the possibility that the physical subspace is positive definite. An alternative more direct derivation of Eq. ( 126 ) can be obtained by acting with W n on the state in Eq. ( 106 ) and using the following identities:\n\nW n V (1, p) = V (1, p)(W n + n) ; (W n + n)D = [L 0 + n -1] -1 W n ( 127\n\n)\n\nThe second equation is a consequence of the following equation:\n\nL n x L0 = x L0+n L n ( 128\n\n)\n\nEq.s (127) imply\n\nW n V (1, p)D = V (1, p)[L 0 + n -1] -1 W n ( 129\n\n)\n\nThis shows that the operator W n goes unchanged through all the product of terms V D until it arrives in front of the term V (1, p M-1 )|0, p M . Going through the vertex operator it becomes L n -L 0 + 1 that then annihilate the state\n\n(L n -L 0 + 1)|p M , 0 = 0 ( 130\n\n)\n\nThis proves Eq. ( 126 ). Using the representation of the Virasoro operators given in Eq. (108) Fubini and Veneziano showed that they satisfy the algebra given in eq. (109) without the central charge. The presence of the central charge was recognized by Joe Weis 10 in 1970 and never published. Unlike Fubini and Veneziano [26] he used the expression of the L n operators in terms of the harmonic oscillators:\n\nL n = √ 2α ′ np • a n + ∞ m=1 m(n + m)a n+m • a m + + 1 2 n-1 m=1 m(n -m)a m-n • a m ; n ≥ 0 L n = L † n ( 131\n\n)\n\n10 See noted added in proof in Ref. [26] .\n\nThe birth of string theory 25 He got the following algebra:\n\n[L n , L m ] = (n -m)L n+m + d 24 n(n 2 -1)δ n+m;0 ( 132\n\n)\n\nwhere d is the dimension of the Minkowski space-time. We write here d for the dimension of the Minkowski space, but we want to remind you that almost everybody working in a model for mesons at that time took for granted that the dimension of the space-time was d = 4. As far as I remember the first paper where a dimension d = 4 was introduced was Ref. [27] where it was shown that the unitarity violating cuts in the non-planar loop become poles that were consistent with unitarity if d = 26. In the last part of this section we will generalize the factorization procedure to the Shapiro-Virasoro model whose N -point amplitude is given in Eq. (49) .\n\nIn this case we must introduce two sets of harmonic oscillators commuting with each other and only one set of zero modes satisfying the algebra [28] :\n\n[a nµ , a † mν ] = [ã nµ , ã † mν ] = η µν δ nm ; [q µ , pν ] = iη µν ( 133\n\n)\n\nIn terms of them we can introduce the Fubini-Veneziano operator\n\nQ(z, z) = q -2α ′ p log(z z) + i √ 2α ′ 2 ∞ n=1 1 √ n a n z -n -a † n z n + + i √ 2α ′ 2 ∞ n=1 1 √ n ãn z-n -ã † n zn ( 134\n\n)\n\nWe can then introduce the vertex operator: V (z, z; p) =: e ip•Q(z,z) : (135) and write the N -point amplitude in Eq. (95) in the following factorized form:\n\nN i=1 d 2 z i dV abc 0|R N i=1 V (z i , zi , p i )) |0 = = (2π) 4 δ (4) ( N i=1 p i ) N i=1 d 2 z i dV abc i<j |z i -z j | α ′ pi•pj ( 136\n\n)\n\nwhere the radial ordered product is given by\n\nR N i=1 V (z i , zi , p i )) = N i=1 V (z i , zi , p i )) N -1 i=1 θ(|z i | -|z i+1 |) + . . . ( 137\n\n)\n\n26 Paolo Di Vecchia and the dots indicate a sum over all permutations of the vertex operators. By fixing z 1 = ∞, z 2 = 1, z N = 0 we can rewrite the previous expression as follows:\n\nN -1\n\ni=3 d 2 z i 0, p 1 |R N -1 i=2 V (z i , zi , p i )) |0, p N ( 138\n\n)\n\nFor the sake of simplicity let us consider the term corresponding to the permutation 1, 2, . . . N . In this case the Koba-Nielsen variables are ordered in such a way\n\nthat |z i | ≥ |z i+1 | for i = 1, . . . N -1.\n\nWe can then use the formula:\n\nV (z i , zi , p i )) = z L0-1 i z L0-1 i V (1, 1, p i )z -L0 i z-L0 i ( 139\n\n)\n\nand change variables:\n\nw i = z i+1 z i ; |w i | ≤ 1 ( 140\n\n)\n\nto rewrite Eq. (138) as follows:\n\n0, p 1 |V (1, 1, p i 1)DV (1, 1, p 2 )D . . . V (1, 1, p N -1 )|0, p N ( 141\n\n) where D = d 2 w |w| 2 w L0-1 w L0-1 = 2 L 0 + L0 -2 • sin π(L 0 -L0 ) L 0 -L0 ( 142\n\n)\n\nWe can now follow the same procedure for all permutations arriving at the following expression:\n\n0, p 1 |P [V (1, 1, p 2 )DV (1, 1, p 3 )D . . . V (1, 1, p N -1 )]|0, p N ( 143\n\n)\n\nwhere P means a sum of all permutations of the particles. If we want to consider the factorization of the amplitude on the pole at s = -(p 1 + . . . p M ) 2 we get only the following contribution:\n\np (1...M) |D|p (M+1...N ) ( 144\n\n) where |p (M+1...N ) = P [V (1, 1, p M+1 )D . . . V (1, 1, p N -1 ]|0, p N ( 145\n\n) and p (1...M) | = 0, p 1 |P [V (1, 1, p 2 )D . . . V (1, 1, p M )] ( 146\n\n)\n\nThe amplitude is factorized by introducing a complete set of states and rewriting Eq. (141) as follows:\n\nThe birth of string theory 27 λ, λ p 1...M |λ, λ 2π λ, λ|δ L0, L0 |λ, λ L 0 + L0 -2 λ, λ|p (M+1,...N ) (147) By writing\n\nL 0 = α ′ 4 p2 + R ; L0 = α ′ 4 p2 + R ( 148\n\n) with R = ∞ n=1 na † n • a n ; R = ∞ n=1 nã † n • ãn ( 149\n\n)\n\nwe can rewrite Eq. (147) as follows\n\nλ, λ p 1...M |λ, λ 2π λ, λ|δ R, R|λ, λ R + R -α(s) λ, λ|p (M+1,...N ) ( 150\n\n)\n\nWe see that the amplitude for the Shapiro-Virasoro model has simple poles only for even integer values of α SV (s) = 2 + α ′ 2 s = 2n ≥ 0 and the residue at the poles factorizes in a sum with a finite number of terms. Notice that the Regge trajectory of the Shapiro-Virasoro model has double intercept and half slope of that of the generalized Veneziano model."
},
{
"section_type": "OTHER",
"section_title": "Physical states and their vertex operators",
"text": "In the previous section, we have seen that the residue at the poles of the Npoint amplitudes factorizes in a sum of a finite number of terms. We have also seen that some of these terms, due to the Lorentz metric, correspond to states with negative norm. We have also derived a number of \"Ward identities\" given in Eq. (126) that imply that some of the terms of the residue decouple. The question to be answered now is: Is the space spanned by the physical states a positive norm Hilbert space? In order to answer this question we need first to find the conditions that characterize the on shell physical states |λ, P and then to determine which are the states that contribute to the residue of the pole at α(s = -P 2 ) = n. In other words, we have to find a way of characterizing the physical states and of eliminating the spurious states that decouple in Eq. (102) as a consequence of Eq.s (126) . A state |λ.P contributes at the residue of the pole in Eq.(102) for α(s = -P 2 ) = n if it is on shell, namely if it satisfies the following equations:\n\nR|λ, P = n|λ, P ; α(-P 2 ) = 1 -α ′ P 2 = n ( 151\n\n)\n\nthat can be written in a unique equation:\n\n28 Paolo Di Vecchia\n\n(L 0 -1)|λ, P = 0 ( 152\n\n)\n\nBecause of Eq. ( 126 ) we also know that a state of the type:\n\n|s, P = W † m |µ, P ( 153\n\n)\n\nis not going to contribute to the residue of the pole. We call it a spurious or unphysical state. We start constructing the subspace of spurious states that are on shell at the level n. Let us consider the set of orthogonal states |µ, P such that\n\nR|µ, P = n µ |µ, P ; L 0 |µ, P = (1 -m)|µ, P ; 1 -α ′ P 2 = n ( 154\n\n)\n\nwhere\n\nm = n + n µ ( 155\n\n)\n\nIn terms of these states we can construct the most general spurious state that is on shell at the level n. It is given by\n\n|s, P = W † m |µ, P ; (L 0 -1)|s, P = 0 ( 156\n\n)\n\nper any positive integer m. Using Eq. ( 154 ), eq. (156) becomes:\n\n|s, P = L † m |µ, P ( 157\n\n)\n\nwhere |µ, P is an arbitrary state satisfying Eq.s (154). A physical state |λ, P is defined as the one that is orthogonal to all spurious states appearing at a certain level n. This means that it must satisfy the following equation:\n\nλ.P |L † ℓ |µ, P = 0 ( 158\n\n)\n\nfor any state |µ, P satisfying Eq.s (154). In conclusion, the on shell physical states at the level n are characterized by the fact that they satisfy the following conditions:\n\nL m |λ, P = (L 0 -1)|λ, P = 0 ; 1 -α ′ P 2 = n ( 159\n\n)\n\nThese conditions characterizing the physical subspace were first found by Del Giudice and Di Vecchia [28] where the analysis described here was done. In order to find the physical subspace one starts writing the most general on shell state contributing to the residue of the pole at level n in Eq. ( 154 ). Then one imposes Eq.s (159) and determines the states that span the physical subspace. Actually, among these states one finds also a set of zero norm states that are physical and spurious at the same time. Those states are of the form given in Eq. ( 157 ), but also satisfy Eq.s (159). It is easy to see that they are not really physical because they are not contributing to the residue of the pole\n\nThe birth of string theory 29 at the level n. This follows from the form of the unit operator given in the space of the physical states by:\n\n1 = norm =0 |λ, P λ, P | + zero [|λ 0 , P µ 0 , P | + |µ 0 , P λ 0 , P |] ( 160\n\n)\n\nwhere |λ 0 , P is a zero norm physical and spurious state and |µ 0 , P its conjugate state. A conjugate state of a zero norm state is obtained by changing the sign of the oscillators with timelike direction. Since |λ 0 , P is a spurious state when we insert the unit operator, given in Eq. ( 160 ), in Eq. ( 102 ) we see that the zero norm states never contribute to the residue because their contribution is annihilated either from the state p (1,M) | or from the state |p (M+1,N ) . In conclusion, the physical subspace contains only the states in the first term in the r.h.s. of Eq. ( 160 ).\n\nLet us analyze the first two excited levels. The first excited level corresponds to a massless gauge field. It is spanned by the states ǫ µ a † 1µ |0, P . In this case the only condition that we must impose is:\n\nL 1 ǫ µ a † 1µ |0, P = 0 =⇒ P • ǫ = 0 ( 161\n\n)\n\nChoosing a frame of reference where the momentum of the photon is given by P µ ≡ (P, 0....0, P ) , Eq. ( 161 ) implies that the only physical states are:\n\nǫ i a + † 1i |0, P + ǫ(a † 1;0 -a † 1;d-1 )|0, P ; i = 1 . . . d -2 ( 162\n\n)\n\nwhere ǫ i and ǫ are arbitrary parameters. The state in Eq. ( 162 ) is the most general state of the level N = 1 satisfying the conditions in Eq. ( 159 ). The first state in eq. (162) has positive norm, while the second one has zero norm that is orthogonal to all other physical states since it can be written as follows:\n\n(a † 1;0 -a † 1;D-1 )|0, P = L † 1 |0, P ( 163\n\n)\n\nin the frame of reference where P µ ≡ (P, ...0, P ). Because of the previous property it is decoupled from the physical states together with its conjugate:\n\n(a † 1,0 + a † 1,d-1 )|0, P ( 164\n\n)\n\nIn conclusion, we are left only with the transverse d -2 states corresponding to the physical degrees of freedom of a massless spin 1 state. At the next level n = 2 the most general state is given by:\n\n[α µν a † 1,µ a † 1,ν + β µ a † 2,µ ]|0, P ( 165\n\n)\n\nIf we work in the center of mass frame where P µ = (M, 0) we get the following most general physical state:\n\n|P hys >= α ij [a † 1,i a † 1,j - 1 (d -1) δ ij d-1 k=1 a † 1,k a † 1,k ]|0, P + 30 Paolo Di Vecchia +β i [a † 2,i + a † 1,0 a † 1,i ]|0, P > + + d-1 i=1 α ii d-1 i=1 a † 1,i a † 1,i + d -1 5 (a †2 1,0 -2a † 2,0 ) |0, P ( 166\n\n)\n\nwhere the indices i, j run over the d -1 space components. The first term in (166) corresponds to a spin 2 in (d -1) dimensional space and has a positive norm being made with space indices. The second term has zero norm and is orthogonal to the other physical states since it can be written as L + 1 a + 1,i |0, P . Therefore it must be eliminated from the physical spectrum together with its conjugate, as explained above. Finally, the last state in (166) is spinless and has a norm given by: 2(d -1)(26\n\n-d) ( 167\n\n)\n\nIf d < 26 it corresponds to a physical spin zero particle with positive norm. If d > 26 it is a ghost. Finally, if d = 26 it has a zero norm and is also orthogonal to the other physical states since it can be written in the form:\n\n(2L † 2 + 3L †2 1 )|0 > ( 168\n\n)\n\nIt does not belong, therefore, to the physical spectrum. The analysis of this level was done in Ref. [29] with d = 4. This did not allow the authors of Ref. [29] to see that there was a critical dimension. The analysis of the physical states can be easily extended [28] to the Shapiro-Virasoro model. In this case the physical conditions given in Eq. (159) for the open string, become [28]:\n\nL m |λ, λ = Lm |λ, λ = (L 0 -1)|λ, λ = ( L0 -1)|λ, λ = 0 ( 169\n\n)\n\nfor any positive integer m. It can be easily seen from the previous equations that the lowest state of the Shapiro-Virasoro model is the vacuum |0 a , 0 ã, p corresponding to a tachyon with mass α ′ p 2 = 4, while the next level described by the state a † 1µ ã † 1ν |0 a , 0 ã, p contains massless states corresponding to the graviton, a dilaton and a two-index antisymmetric tensor B µν .\n\nHaving characterized the physical subspace one can go on and construct a N -point scattering amplitude involving arbitrary physical states. This was done by Campagna, Fubini, Napolitano and Sciuto [30] where the vertex operator for an arbitrary physical state was constructed in analogy with what has been done for the ground tachyonic state. They associated to each physical state |α, P a vertex operator V α (z, P ) that is a conformal field with conformal dimension equal to 1:\n\n[L n , V α (z, p)] = d dz z n+1 V α (z, p) ( 170\n\n)\n\nand reproduces the corresponding state acting on the vacuum as follows:\n\nlim z→0 V α (z; p)|0, 0 ≡ |α; p ; 0; 0| lim z→∞ z 2 V α (z; p) = α, p| ( 171\n\n)\n\nThe birth of string theory 31 It satisfies, in addition, the hermiticity relation:\n\nV † α (z, P ) = V α ( 1 z , -P )(-1) α(-P 2 ) ( 172\n\n)\n\nAn excited vertex that will play an important role in the next section is the one associated to the massless gauge field. It is given by:\n\nV ǫ (z, k) ≡ ǫ • dQ(z) dz e ik•Q(z) ; k • ǫ = k 2 = 0 ( 173\n\n)\n\nBecause of the last two conditions in Eq. ( 173 ) the normal order is not necessary. It is convenient to give the expression of dQ(z) dz in terms of the harmonic oscillators:\n\nP (z) ≡ dQ(z) dz = -i √ 2α ′ ∞ n=-∞ α n z -n-1 ( 174\n\n)\n\nIt is a conformal field with conformal dimension equal to 1. The rescaled oscillators α n are given by:\n\nα n = √ na n ; α -n = √ na † n ; n > 0 ; α 0 = √ 2α ′ p ( 175\n\n)\n\nIn terms of the vertex operators previously introduced the most general amplitude involving arbitrary physical states is given by [30]:\n\n(2π) 4 δ( N i=1 p i )B ex N = ∞ -∞ N 1 dz i θ(z i -z i+1 ) dV abc 0, 0| N i=1 V αi (z i , p i )|0, 0 ( 176\n\n)\n\nIn the case of the Shapiro-Virasoro model the tachyon vertex operator is given in Eq. (135) . By rewriting Eq. (134) as follows:\n\nQ(z, z) = Q(z) + Q(z) ( 177\n\n)\n\nwhere\n\nQ(z) = 1 2 q -2α ′ p log(z) + i √ 2α ′ ∞ n=1 1 √ n a n z -n -a † n z n ( 178\n\n)\n\nand\n\nQ(z) = 1 2 q -2α ′ p log(z) + i √ 2α ′ ∞ n=1 1 √ n ãn z-n -ã † n zn ( 179\n\n)\n\nwe can write the tachyon vertex operator in the following way:\n\nV (z, z, p) =: e ip•Q(z) e ip• Q(z) : ( 180\n\n)\n\n32 Paolo Di Vecchia\n\nThis shows that the vertex operator corresponding to the tachyon of the Shapiro-Virasoro model can be written as the product of two vertex operators corresponding each to the tachyon of the generalized Veneziano model. Analogously the vertex operator corresponding to an arbitrary physical state of the Shapiro-Virasoro model can always be written as a product of two vertex operators of the generalized Veneziano model:\n\nV α,β (z, z, p) = V α (z, p 2 )V β (z, p 2 ) ( 181\n\n)\n\nThe first one contains only the oscillators α n , while the second one only the oscillators αn . They both contain only half of the total momentum p and the same zero modes p and q. The two vertex operators of the generalized Veneziano model are both conformal fields with conformal dimension equal to 1. If they correspond to physical states at the level 2n, they satisfy the following relation (n = ñ):\n\nα ′ p 2 4 + n = 1 ( 182\n\n)\n\nThey lie on the following Regge trajectory:\n\n2 - α ′ 2 p 2 ≡ α SV (-p 2 ) = 2n ( 183\n\n)\n\nas we have already seen by factorizing the amplitude in Eq. ( 150 )."
},
{
"section_type": "OTHER",
"section_title": "The DDF states and absence of ghosts",
"text": "In the previous section we have derived the equations that characterize the physical states and their corresponding vertex operators. In this section we will explicitly construct an infinite number of orthonormal physical states with positive norm. The starting point is the DDF operator introduced by Del Giudice, Di Vecchia and Fubini [31] and defined in terms of the vertex operator corresponding to the massless gauge field introduced in eq. (173):\n\nA i,n = i √ 2α ′ 0 dzǫ µ i P µ (z)e ik•Q(z) ( 184\n\n)\n\nwhere the index i runs over the d-2 transverse directions, that are orthogonal to the momentum k. We have also taken 0 dz z = 1. Because of the log z term appearing in the zero mode part of the exponential, the integral in Eq. ( 184 ), that is performed around the origin z = 0, is well defined only if we constrain the momentum of the state, on which A i,n acts, to satisfy the relation:\n\n2α ′ p • k = n ( 185\n\n)\n\nThe birth of string theory 33 where n is a non-vanishing integer. The operator in Eq. (184) will generate physical states because it commutes with the gauge operators L m :\n\n[L m , A n;i ] = 0 ( 186\n\n)\n\nsince the vertex operator transforms as a primary field with conformal dimension equal to 1 as it follows from Eq. (170). On the other hand it also satisfies the algebra of the harmonic oscillator as we are now going to show. From Eq. (184) we get:\n\n[A n,i , A m,j ] = - 1 2α ′ 0 dζ ζ dzǫ i • P (z)e ik•Q(ζ) ǫ j • P (ζ)e ik ′ •Q(ζ) ( 187\n\n) where 2α ′ p • k = n ; 2α ′ p • k ′ = m ( 188\n\n)\n\nand k and k ′ are supposed to be in the same direction, namely\n\nk µ = n kµ ; k ′ µ = m kµ ( 189\n\n) with 2α ′ p • k = 1 ( 190\n\n)\n\nFinally the polarizations are normalized as:\n\nǫ i • ǫ j = δ ij ( 191\n\n)\n\nSince k • ǫ i = k • ǫ j = k2 = 0 a singularity for z = ζ can appear only from the contraction of the two terms P (ζ) and P ((z) that is given by:\n\n0, 0|ǫ i • P (z)ǫ j • P (ζ)|0, 0 = - 2α ′ δ ij (z -ζ) 2 ( 192\n\n)\n\nInserting it in Eq. (187) we get:\n\n[A n,i , A m,j ] = δ ij in 0 dζ k • P (ζ)e -i(n+m)) k•Q(ζ) = = inδ ij δ n+m;0 0 dζ k • P (ζ) ( 193\n\n)\n\nwhere we have used the fact that the integrand is a total derivative and therefore one gets a vanishing contribution unless n + m = 0. If n + m = 0 from Eq.s (174) and (190) we get:\n\n[A n,i , A m,j ] = nδ ij δ n+m;0 ; i, j = 1 . . . d -2 ( 194\n\n)\n\n34 Paolo Di Vecchia\n\nEq. ( 194 ) shows that the DDF operators satisfy the harmonic oscillator algebra. In terms of this infinite set of transverse oscillators we can construct an orthonormal set of states:\n\n|i 1 , N 1 ; i 2 , N 2 ; . . . i m , N m = h 1 √ λ h ! m k=1 A i k ,-N k √ N k |0, p ( 195\n\n)\n\nwhere λ h is the multiplicity of the operator A i h ,-N h in the product in Eq. ( 195 ) and the momentum of the state in Eq. ( 195 ) is given by\n\nP = p + m i=1 kN i ( 196\n\n)\n\nThey were constructed in four dimensions where they were not a complete system of states 11 and it took some time to realize that in fact they were a complete system of states if d = 26 [32, 33] 12 . Brower [32] and Goddard and Thorn [33] showed also that the dual resonance model was ghost free for any dimension d ≤ 26. In d = 26 this follows from the fact that the DDF operators obviously span a positive definite Hilbert space (See Eq. ( 194 )). For d < 26 there are extra states called Brower states [32] . The first of these states is the last state in Eq. (166) that becomes a zero norm state for d = 26. But also for d < 26 there is no negative norm state among the physical states. The proof of the no-ghost theorem in the case α 0 = 1 is a very important step because it shows that the dual resonance model constructed generalizing the four-point Veneziano formula, is a fully consistent quantum-relativistic theory! This is not quite true because, when the intercept α 0 = 1, the lowest state of the spectrum corresponding to the pole in the N -point amplitude for α(s) = 0, is a tachyon with mass m 2 = -1 α ′ . A lot of effort was then made to construct a model without tachyon and with a meson spectrum consistent with the experimental data. The only reasonably consistent models that came out from these attempts, were the Neveu-Schwarz [7] for mesons and the Ramond model [8] for fermions that only later were recognized to be part of a unique model that nowadays is called the Neveu-Schwarz-Ramond model. But this model was not really more consistent than the original dual resonance 11 Because of this Fubini did not want to publish our result, but then he went to a meeting in Israel in spring 1971 giving a talk on our work where he found that the audience was very interested in our result and when he came back to MIT we decided to publish our result. 12 I still remember Charles Thorn coming into my office at Cern and telling me:\n\nPaolo, do you know that your DDF states are complete if d = 26? I quickly redid the analysis done in Ref. [29] with an arbitrary value of the space-time dimension obtaining Eq.s (166) and (167) that show that the spinless state at the level α(s) = 2 is decoupled if d = 26. I strongly regretted not to have used an arbitrary space-time dimension d in the analysis of Ref. [29] .\n\nThe birth of string theory 35 model because it still had a tachyon with mass m 2 = -1 2α ′ . The tachyon was eliminated from the spectrum only in 1976 through the GSO projection proposed by Gliozzi, Scherk and Olive [34] .\n\nHaving realized that, at least for the critical value of the space-time dimension d = 26, the physical states are described by the DDF states having only d -2 = 24 independent components, open the way to Brink and Nielsen [35] to compute the value α 0 = 1 of the Regge trajectory with a very physical argument. They related the intercept of the Regge trajectory to the zero point energy of a system with an infinite number of oscillators having only d -2 independent components:\n\nα 0 = - d -2 2 ∞ n=1 n ( 197\n\n)\n\nThis quantity is obviously infinite and, in order to make sense of it, they introduced a cutoff on the frequencies of the harmonic oscillators obtaining an infinite term that they eliminated by renormalizing the speed of light and a finite universal constant term that gave the intercept of the Regge trajectory. Instead of following their original approach we discuss here an alternative approach due to Gliozzi [36] that uses the ζ-function regularization. He rewrites Eq. (197) as follows:\n\nα 0 = - d -2 2 ∞ n=1 n = - d -2 2 lim s→-1 ∞ n=1 n -s = - d -2 2 ζ R (-1) = 1 ( 198\n\n)\n\nwhere in the last equation we have used the identity ζ R (-1) = -1 12 and we have put d = 26. Since the Shapiro-Virasoro model has two sets of transverse harmonic oscillators it is obvious that its intercept is twice that of the generalized Veneziano model.\n\nUsing the rules discussed in the previous section we can construct the vertex operator corresponding to the state in Eq. (195) . It is given by:\n\nV (i;Ni) (z, P ) = m i=1 z dz i ǫ i • P (z i )e iNi k•Q(zi) : e ip•Q(z) : ( 199\n\n)\n\nwhere the integral on the variable z i is evaluated along a curve of the complex plane z i containing the point z. The singularity of the integrand for z i = z is a pole provided that the following condition is satisfied.\n\n2α ′ p • k = 1 ( 200\n\n)\n\nThe last vertex in Eq. (199) is the vertex operator corresponding to the ground tachyonic state given in Eq. ( 59 ) with α ′ p 2 = 1. Using the general form of the vertex one can compute the three-point amplitude involving three arbitrary DDF vertex operators. This calculation 36 Paolo Di Vecchia has been performed in Ref. [37] and since the vertex operators are conformal fields with dimension equal to 1 one gets:\n\n0, 0|V (i (1) k 1 ;N (1) k 1 ) (z 1 , P 1 )V (i (2) k 2 ;N ( 2\n\n) k (2) ) (z 2 , P 2 )V (i (3) k 3 ;N ( 3\n\n) k (3) ) (z 3 , P 3 )|0, 0 = = C 123 (z 1 -z 2 )(z 1 -z 3 )(z 2 -z 3 ) ( 201\n\n)\n\nwhere the explicit form of the coefficient C 123 is given by:\n\nC 123 = 1 0, 0| 2 0, 0| 3 0, 0|e 1 2 3 r.s=1 ∞ n,m=1 A (r) -n;i N rs nm A (s) -m;i + 3 i=1 Pi• ∞ n=1 A (r) -n;i × × e τ0 3 r=1 (α ′ Π 2 r -1) |N ( 1\n\n) k1 , i ( 1\n\n) k1 1 |N ( 2\n\n) k2 , i ( 2\n\n) k2 2 |N ( 3\n\n) k3 , i (3) k3 3 ( 202\n\n)\n\nwhere\n\nN rs nm = -N r n N s m nmα 1 α 2 α 3 nα s + mα r ; N r n = Γ (-n αr+1 αr ) α r n!Γ (1 -n αr+1 αr -n) ( 203\n\n) with Π = P r+1 α r -P r α r+1 ; r = 1, 2, 3 ( 204\n\n)\n\nΠ is independent on the value of r chosen as a consequence of the equations:\n\n3 r=1 α r = 3 r=1 P r = 0 ( 205\n\n)\n\n7 The zero slope limit\n\nIn the introduction we have seen that the dual resonance model has been constructed using rules that are different from those used in field theory. For instance, we have seen that planar duality implies that the amplitude corresponding to a certain duality diagram, contains poles in both s and t channels, while the amplitude corresponding to a Feynman diagram in field theory contains only a pole in one of the two channels. Furthermore, the scattering amplitude in the dual resonance model contains an infinite number of resonant states that, at high energy, average out to give Regge behaviour. Also this property is not observed in field theory. The question that was natural to ask, was then: is there any relation between the dual resonance model and field theory? It turned out, to the surprise of many, that the dual resonance model was not in contradiction with field theory, but was instead an extension of a certain number of field theories. We will see that the limit in\n\nThe birth of string theory 37 which a field theory is obtained from the dual resonance model corresponds to taking the slope of the Regge trajectory α ′ to zero. Let us consider the scattering amplitude of four ground state particles in Eq. (1) that we rewrite here with the correct normalization factor:\n\nA(s, t, u) = C 0 N 4 0 (A(s, t) + A(s, u) + A(t, u)) ( 206\n\n)\n\nwhere\n\nN 0 = √ 2g(2α ′ ) d-2 4 ( 207\n\n)\n\nis the correct normalization factor for each external leg, g is the dimensionless open string coupling constant that we have constantly ignored in the previous sections and C 0 is determined by the following relation:\n\nC 0 N 2 0 α ′ = 1 ( 208\n\n)\n\nthat is obtained by requiring the factorization of the amplitude at the pole corresponding to the ground state particle whose mass is given in Eq. (21). Using Eq. (21) in order to rewrite the intercept of the Regge trajectory in terms of the mass of the ground state particle m 2 and the following relation satisfied by the Γ -function:\n\nΓ (1 + z) = zΓ (z) ( 209\n\n)\n\nwe can easily perform the limit for α ′ → 0 of A(s, t) obtaining:\n\nlim α ′ →0 A(s, t) = 1 α ′ 1 m 2 -s + 1 m 2 -s ( 210\n\n)\n\nPerforming the same limit on the other two planar amplitudes we get the following expression for the total amplitude in Eq. ( 206 ):\n\nlim α ′ →0 A(s, t, u) = √ 2g(2α ′ ) d-2 4 2 2 (α ′ ) 2 1 m 2 -s + 1 m 2 -s + 1 m 2 -u ( 211\n\n)\n\nBy introducing the coupling constant:\n\ng 3 = 4g(2α ′ ) d-6 4 ( 212\n\n) Eq. (211) becomes lim α ′ →0 A(s, t, u) = g 2 3 1 m 2 -s + 1 m 2 -s + 1 m 2 -u ( 213\n\n)\n\nthat is equal to the sum of the tree diagrams for the scattering of four particles with mass m of Φ 3 theory with coupling constant equal to g 3 . We have shown that, by keeping g 3 fixed in the limit α ′ → 0, the scattering amplitude of four 38 Paolo Di Vecchia ground state particles of the dual resonance model is equal to the tree diagrams of Φ 3 theory. This proof can be extended to the scattering of N ground state particles recovering also in this case the tree diagrams of Φ 3 theory. It is also valid for loop diagrams that we will discuss in the next section. In conclusion, the dual resonance model reduces in the zero slope limit to Φ 3 theory. The proof that we have presented here is due to J. Scherk [38] 13 A more interesting case to study is the one with intercept α 0 = 1. We will see that, in this case, one will obtain the tree diagrams of Yang-Mills theory, as shown by Neveu and Scherk [40] 14 .\n\nLet us consider the three-point amplitude involving three massless gauge particles described by the vertex operator in Eq. ( 173 ). It is given by the sum of two planar diagrams. The first one corresponding to the ordering (123) is given by:\n\nC 0 N 3 0 i 3 T r (λ a1 λ a2 λ a3 ) 0, 0|V ǫ1 (z 1 , p 1 )V ǫ2 (z 2 , p 2 )V ǫ3 (z 3 , p 3 )|0, 0 [(z 1 -z 2 )(z 2 -z 3 )(z 1 -z 3 )] -1 ( 214\n\n)\n\nUsing momentum conservation p 1 + p 2 + p 3 = 0 and the mass shell conditions p 2 i = p i • ǫ i = 0 one can rewrite the previous equation as follows:\n\nC 0 N 3 0 T r(λ a1 λ a2 λ a3 ) √ 2α ′ × × [(ǫ 1 • ǫ 2 )(p 1 • ǫ 3 ) + (ǫ 1 • ǫ 3 )(p 3 • ǫ 2 ) + (ǫ 2 • ǫ 3 )(p 2 • ǫ 1 )] ( 215\n\n)\n\nThe second contribution comes from the ordering 132 that can be obtained from the previous one by the substitution T r(λ a1 λ a2 λ a3 ) → -T r(λ a1 λ a3 λ a2 ) (216)\n\nSumming the two contributions one gets\n\nC 0 N 3 o T r(λ a1 [λ a2 , λ a3 ]) √ 2α ′ × × [(ǫ 1 • ǫ 2 )(p 1 • ǫ 3 ) + (ǫ 1 • ǫ 3 )(p 3 • ǫ 2 ) + (ǫ 2 • ǫ 3 )(p 2 • ǫ 1 )] ( 217\n\n)\n\nThe factor\n\nN 0 = 2g(2α ′ ) (d-2)/4 ( 218\n\n)\n\nis the correct normalization factor for each vertex operator if we normalize the generators of the Chan-Paton group as follows:\n\nT r λ i λ j = 1 2 δ ij ( 219\n\n)\n\n13 See also Ref. [39] . 14 See also Ref. [41] .\n\nThe birth of string theory 39 It is related to C 0 through the relation 15 :\n\nC 0 N 2 o α ′ = 2 ( 220\n\n)\n\ng is the dimensionless open string coupling constant. Notice that Eq.s (218) and (220) differ from Eq.s (207) and (208) because of the presence of the Chan-Paton factors that we did not include in the case of Φ 3 theory. By using the commutation relations:\n\n[λ a , λ b ] = if abc λ c ( 221\n\n)\n\nand the previous normalization factors we get for the three-gluon amplitude:\n\nig Y M f a1a2a3 [(ǫ 1 • ǫ 2 )((p 1 -p 2 ) • ǫ 3 + +(ǫ 1 • ǫ 3 )((p 3 -p 1 ) • ǫ 2 ) + (ǫ 2 • ǫ 3 )((p 2 -p 3 ) • ǫ 1 )] ( 222\n\n)\n\nthat is equal to the 3-gluon vertex that one obtains from the Yang-Mills action\n\nL Y M = - 1 4 F a αβ F αβ a , F a αβ = ∂ α A a β -∂ β A a α + g Y M f abc A b α A c β ( 223\n\n)\n\nwhere\n\ng Y M = 2g(2α ′ ) d-4 4 ( 224\n\n)\n\nThe previous procedure can be extended to the scattering of N gluons finding the same result that one gets from the tree diagrams of Yang-Mills theory.\n\nIn the next section, we will discuss the loop diagrams. Also, in this case one finds that the h-loop diagrams involving N external gluons reproduces in the zero slope limit the sum of the h-loop diagrams with N external gluons of Yang-Mills theory. We conclude this section mentioning that one can also take the zero slope limit of a scattering amplitude involving three and four gravitons obtaining agreement with what one gets from the Einstein Lagrangian of general relativity. This has been shown by Yoneya [43] ."
},
{
"section_type": "OTHER",
"section_title": "Loop diagrams",
"text": "The N -point amplitude previously constructed satisfies all the axioms of Smatrix theory except unitarity because its only singularities are simple poles corresponding to zero width resonances lying on the real axis of the Mandelstam variables and does not contain the various cuts required by unitarity [1]. 15 The determination of the previous normalization factors can be found in the Appendix of Ref. [42] ."
},
{
"section_type": "OTHER",
"section_title": "Paolo Di Vecchia",
"text": "In order to eliminate this problem it was proposed already in the early days of dual theories to assume, in analogy with what happens for instance in perturbative field theory, that the N -point amplitude was only the lowest order (the tree diagram) of a perturbative expansion and, in order to implement unitarity, it was necessary to include loop diagrams. Then, the one-loop diagrams were constructed from the propagator and vertices that we have introduced in the previous sections [44] . The planar one-loop amplitude with M external particles was computed by starting from a (M + 2)-point tree amplitude and then by sewing two external legs together after the insertion of a propagator D given in Eq. (100) . In this way one gets:\n\nd d P (2α ′ ) d/2 (2π) d λ P, λ|V (1, p 1 )DV (1, p 2 ) . . . V (1, p N )D|P, λ ( 225\n\n)\n\nwhere the sum over λ corresponds to the trace in the space of the harmonic oscillators and the integral in d d P corresponds to integrate over the momentum circulating in the loop. The previous expression for the one-loop amplitude cannot be quite correct because all states of the space generated by the oscillators in Eq. ( 51 ) are circulating in the loop, while we know that we should include only the physical ones. This was achieved first by cancelling by hand the time and one of the space components of the harmonic oscillators reducing the degrees of freedom of each oscillator from d to d -2 as suggested by the DDF operators at least for d = 26. This procedure was then shown to be correct by Brink and Olive [45] . They constructed the operator that projects over the physical states and, by inserting it in the loop, showed that the reduction of the degrees of freedom of the oscillators from d to d -2 was indeed correct. This was, at that time, the only procedure available to let only the physical states circulate in the loop because the BRST procedure was discovered a bit later also in the framework of the gauge field theories! To be more explicit let us compute the trace in Eq. ( 225 ) adding also the Chan-Paton factor. We get:\n\n(2π) d δ (d) M i=1 p i N T r(λ a1 . . . λ aM ) (8π 2 α ′ ) d/2 N M 0 ∞ 0 dτ τ d/2+1 [f 1 (k)] 2-d k d-26 12 (2π) M × × 1 0 dν M νM 0 dν M-1 . . . ν3 0 dν 2 τ M i<j e G(νji) 2α ′ pi•pj ; k ≡ e -πτ ( 226\n\n) where ν ji ≡ ν j -ν i , G(ν) = log ie -πν 2 τ Θ 1 (iντ |iτ ) f 3 1 (k) ; f 1 (k) = k 1/12 ∞ n=1 (1 -k 2n ) ( 227\n\n)\n\nThe birth of string theory 41 Θ 1 (ν|iτ ) = -2k 1/4 sin πν\n\n∞ n=1 1 -e 2iπν k 2n 1 -e -2iπν k 2n (1 -k 2n )( 228\n\n)\n\nFinally the normalization factor N 0 is given in Eq. ( 218 ). We have performed the calculation for an arbitrary value of the space-time dimension d. However, in this way one gets also the extra factor of k d-26 12\n\nappearing in the first line of Eq. (226) that implies that our calculation is actually only consistent if d = 26. In fact, the presence of this factor does not allow one to rewrite the amplitude, originally obtained in the Reggeon sector, in the Pomeron sector as explained below. In the following we neglect this extra factor, implicitly assuming that d = 26, but, on the other hand, still keeping an arbitrary d.\n\nUsing the relations:\n\nf 1 (k) = √ tf 1 (q) ; Θ 1 (iντ |iτ ) = iΘ 1 (ν|it)t 1/2 e πν 2 /t ( 229\n\n)\n\nwhere t = 1 τ and q ≡ e -πt , we can rewrite the one-loop planar diagram in the Pomeron channel. We get:\n\n(2π) d δ (d) M i=1 p i N T r(λ a1 . . . λ aM ) (8π 2 α ′ ) d/2 N M 0 ∞ 0 dt[f 1 (q)] 2-d (2π) M × × 1 0 dν M νM 0 dν M-1 . . . ν3 0 dν 2 i<j - Θ 1 (ν ji |it) f 3 1 (q) 2α ′ pi•pj ( 230\n\n)\n\nNotice that, by factorizing the planar loop in the Pomeron channel, one constructed for the first time what we now call the boundary state [46] 16 . This can be easily seen in the way that we are now going to describe. First of all, notice that the last quantity in Eq. ( 230 ) can be written as follows:\n\ni<j - Θ 1 (ν ji |it) f 3 1 (q) 2α ′ pi•pj = = i<j -2 sin(πν ji ) ∞ n=1 1 -q 2n e 2πiνji 1 -q 2n e -2πiνji (1 -q 2n ) 2 2α ′ pi•pj ( 231\n\n)\n\nThis equation can be rewritten as follows:\n\nT r p = 0|q 2R M i=1 :\n\ne ipi•Q(e 2iπν i ) : |p = 0 i M T r ( p = 0|q 2N |p = 0 ) ; R = ∞ n=1 na † n • a n ( 232\n\n) 16\n\nSee also the first paper in Ref. [47] ."
},
{
"section_type": "OTHER",
"section_title": "Paolo Di Vecchia",
"text": "where the trace is taken only over the non-zero modes and momentum conservation has been used. It must also be stressed that the normal ordering of the vertex operators in the previous equation is such that the zero modes are taken to be both in the same exponential instead of being ordered as in Eq. ( 59 ). By bringing all annihilation operators on the left of the creation ones, from the expression in Eq. (232) one gets (z i ≡ e 2πiνi ):\n\n(2π) d δ (d) ∞ i=1 p i i<j (-2 sin πν ji ) 2α ′ pi•pj × × i.j ∞ n=1 T r q 2na † n •an e √ 2α ′ pj • a † n √ n z n j e - √ 2α ′ pi• an √ n z -n i T r ( p = 0|q 2N |p = 0 ) ( 233\n\n)\n\nThe trace can be computed by using the completeness relation involving coherent states |f = e f a † |0 :\n\nd 2 f π e -|f | 2 |f f | = 1 ( 234\n\n)\n\nInserting the previous identity operator in Eq. ( 233 ) one gets after some calculation:\n\n(2π) d δ (d) ∞ i=1 p i i<j (-2 sin πν ji ) 2α ′ pi•pj × × M i.j=1 ∞ n=1 e -2α ′ pi•pje 2πinν ji q 2n n(1-q 2n ) ( 235\n\n)\n\nExpanding the denominator in the last exponent and performing the sum over n one gets:\n\n(2π) d δ (d) ∞ i=1 p i i<j (-2 sin πν ji ) 2α ′ pi•pj × × i.j e 2α ′ pi•pj ∞ m=0 log(1-e 2πiν ji q 2(m+1) ) ( 236\n\n)\n\nthat is equal to the last line of Eq. (231) apart from the δ-function for momentum conservation. In conclusion, we have shown that Eq.s (231) and (232) are equal. Using Eq. (231) we can rewrite Eq. (230) as follows:\n\nN N M 0 T r(λ a1 . . . λ aM ) (8π 2 α ′ ) d/2 ∞ 0 dt[f 1 (q)] 2-d (2πi) M 1 0 dν M νM 0 dν M-1 . . .\n\nThe birth of string theory 43 . . .\n\nν3 0 dν 2 λ p = 0, λ|q 2R M i=1 : e ipi•Q(e 2iπν i ) : |p = 0, λ λ p = 0, λ|q 2N |p = 0, λ ( 237\n\n)\n\nwhere the sum over any state |λ corresponds to taking the trace over the non-zero modes. If d = 26 we can rewrite Eq. (237) in a simpler form:\n\nN N M 0 T r(λ a1 . . . λ aM ) (8π 2 α ′ ) d/2 ∞ 0 dt (2πi) M 1 0 dν M νM 0 dν M-1 . . . ν3 0 dν 2 × × λ p = 0, λ|q 2R-2 M i=1 : e ipi•Q(e 2iπν i ) : |p = 0, λ ( 238\n\n)\n\nThe previous equation contains the factor dtq 2R-2 that is like the propagator of the Shapiro-Virasoro model, but with only one set of oscillators as in the generalized Veneziano model. In the following we will rewrite it completely with the formalism of the Shapiro-Virasoro model. This can be done by introducing the Pomeron propagator:\n\n∞ 0 dt q 2N -2 = 2 πα ′ D ; D ≡ α ′ 4π d 2 z |z| 2 z L0-1 z L0 -1 ; |z| ≡ q = e -πt ( 239\n\n)\n\nand rewriting the planar loop in the following compact form:\n\nB 0 | D|B M ; |B 0 ≡ T d-1 2 N ∞ n=1 e a † n •ã † n |p = 0, 0 a , 0 ã ( 240\n\n)\n\nwhere |B 0 is the boundary state without any Reggeon on it,\n\nT d-1 = √ π 2 (d-10)/4 (2π √ α ′ ) -d/2-1 ( 241\n\n)\n\nand |B M is instead the one with M Reggeons given by:\n\n|B M = N M 0 T r(λ a1 . . . λ aM )(2πi) M 1 0 dν M νM 0 dν M-1 . . . ν3 0 dν 2 × × M i=1 : e ipi•Q(e 2iπν i ) : |B 0 ( 242\n\n)\n\nWe want to stress once more that the normal ordering in the previous equation is defined by taking the zero modes in the same exponential. Both the boundary states and the propagator are now states of the Shapiro-Virasoro model. This means that we have rewritten the one-loop planar diagram, where the states of the generalized Veneziano model circulate in the loop, as a tree\n\n44 Paolo Di Vecchia\n\ndiagram of the Shapiro-Virasoro model involving two boundary states and a propagator. This is what nowadays is called open/closed string duality. Besides the one-loop planar diagram in Eq. ( 225 ), that is nowadays called the annulus diagram, also the non-planar and the non-orientable diagrams were constructed and studied. In particular the non-planar one, that is obtained as the planar one in Eq. (225) but with two propagators multiplied with the twist operator\n\nΩ = e L-1 (-1) R , ( 243\n\n)\n\nhad unitarity violating cuts that disappeared [27] if the dimension of the space-time d = 26, leaving behind additional pole singularities. The explicit form of the non-planar loop can be obtained following the same steps done for the planar loop. One gets for the non-planar loop the following amplitude:\n\nB R | D|B M ( 244\n\n)\n\nwhere now both boundary states contain, respectively, R and M Reggeon states. The additional poles found in the non-planar loop were called Pomerons because they occur in the Pomeron sector, that today is called the closed string channel, to distinguish them from the Reggeons that instead occur in the Reggeon sector, that today is called the open string sector of the planar and non-planar loop diagrams. At that time in fact, the states of the generalized Veneziano models were called Reggeons, while the additional ones appearing in the non-planar loop were called Pomerons. The Reggeons correspond nowadays to open string states, while the Pomerons to closed string states. These things are obvious now, but at that time it took a while to show that the additional states appearing in the Pomeron sector have to be identified with those of the Shapiro-Virasoro model. The proof that the spectrum was the same came rather early. This was obtained by factorizing the non-planar diagram in the Pomeron channel [46] as we have done in Eq. ( 244 ). It was found that the states of the Pomeron channel lie on a linear Regge trajectory that has double intercept and half slope of the one of the Reggeons. This follows immediately from the propagator D in Eq. (239) that has poles for values of the momentum of the Pomeron exchanged given by:\n\n2 - α ′ 2 p 2 = 2n ( 245\n\n)\n\nthat are exactly the values of the masses of the states of the Shapiro-Virasoro model [48] , while the Reggeon propagator in Eq. (100) has poles for values of momentum equal to:\n\n1 -α ′ p 2 = n ( 246\n\n)\n\nHowever, it was still not clear that the Pomeron states interact among themselves as the states of the Shapiro-Virasoro model. To show this it was first\n\nThe birth of string theory 45 necessary to construct tree amplitudes containing both states of the generalized Veneziano model and of the Shapiro-Virasoro model [49] . They reduced to the amplitudes of the generalized Veneziano (Shapiro-Virasoro) model if we have only external states of the generalized Veneziano (Shapiro-Virasoro) model. Those amplitudes are called today disk amplitudes containing both open and closed string states. They were constructed [49] by using for the Reggeon states the vertex operators that we have discussed in Sect. (5) involving one set of harmonic oscillators and for the Pomeron states the vertex operators given in Eq. (181) that we rewrite here:\n\nV α,β (z, z, p) = V α (z, p 2 )V β (z, p 2 ) ( 247\n\n)\n\nbecause now both component vertices contain the same set of harmonic oscillators as in the generalized Veneziano model. Furthermore, each of the two vertices is separately normal ordered, but their product is nor normal ordered. The amplitude involving both kinds of states is then constructed by taking the product of all vertices between the projective invariant vacuum and integrating the Reggeons on the real axis in an ordered way and the Pomerons in the upper half plane, as one does for a disk amplitude. We have mentioned above that the two vertices are separately normal ordered, but their product is not normal ordered. When we normal order them we get, for instance for the tachyon of the Pomeron sector, a factor (z -z) α ′ p 2 /2 that describes the Reggeon-Pomeron transition. This implies a direct coupling [51] between the U (1) part of gauge field and the two-index antisymmetric field B µν , called Kalb-Ramond field [50], of the Pomeron sector, that makes the gauge field massive [51] .\n\nIt was then shown that, by factorizing the non-planal loop in the Pomeron channel, one reproduced the scattering amplitude containing one state of the Shapiro-Virasoro and a number of states of the generalized Veneziano model [52] . If we have also external states belonging to the generalized Shapiro-Virasoro model, then by factorizing the non-planar one loop amplitude in the pure Pomeron channel, one would obtain the tree amplitudes of the Shapiro-Virasoro model [52] .\n\nAll this implies that the generalized Veneziano model and the Shapiro-Virasoro model are not two independent models, but they are part of the same and unique model. In fact, if one started with the generalized Veneziano model and added loop diagrams to implement unitarity, one found the appearence in the non-planar loop of additional states that had the same mass and interaction of those of the Shapiro-Virasoro model. The planar diagram, written in Eq. (230) in the closed string channel, is divergent for large values of t. This divergence was recognized to be due to exchange, in the Pomeron channel, of the tachyon of the Shapiro-Virasoro model and of the dilaton [47] . They correspond, respectively, to the first two terms of the expansion:\n\n[f 1 (q)] -24 = e 2πt + 24 + O e -2πt ( 248\n\n)\n\n46 Paolo Di Vecchia\n\nThe first one could be cancelled by an analytic continuation, while the second one could be eliminated through a renormalization of the slope of the Regge trajectory α ′ [47] . We conclude the discussion of the one-loop diagrams by mentioning that the one-loop diagram for the Shapiro-Virasoro model was computed by Shapiro [53] who also found that the integrand was modular invariant.\n\nThe computation of multiloop diagrams requires a more advanced technology that was also developed in the early days of the dual resonance model few years before the discovery of its connection to string theory. In order to compute multiloop diagrams one needs first to construct an object that was called the N -Reggeon vertex and that has the properties of containing N sets of harmonic oscillators, one for each external leg, and is such that, when we saturate it with N physical states, we get the corresponding N -point amplitude. In the following we will discuss how to determine the N -Reggeon vertex.\n\nThe first step toward the N -Reggeon vertex is the Sciuto-Della Selva-Saito [54] vertex that includes two sets of harmonic oscillators that we denote with the indices 1 and 2. It is equal to:\n\nV SDS = 2 x = 0, 0| : exp - 1 2α ′ 0 dzX ′ 2 (z) • X 1 (1 -z) : ( 249\n\n)\n\nwhere X is the quantity that we have called Q in Eq. ( 57 ) and the prime denotes a derivative with respect to z. It satisfies the important property of giving the vertex operator V α (z = 1) of an arbitrary state |α when we saturate it with the corresponding state:\n\nV SDS |α 2 = V α (z = 1) ( 250\n\n)\n\nA shortcoming of this vertex is that it is not invariant under a cyclic permutation of the three legs. A cyclic symmetric vertex has been constructed by Caneschi, Schwimmer and Veneziano [55] by inserting the twist operator in Eq. ( 243 ). But the 3-Reggeon vertex is not enough if we want to compute an arbitrary multiloop amplitude. We must generalize it to an arbitrary number of external legs. Such a vertex, that can be obtained from the one in Eq. (249) with a very direct procedure, or that can also be obtained by sewing together three-Reggeon vertices, has been written in its final form by Lovelace [56] 17 .\n\nHere we do not derive it, but we give directly its expression written in Ref. [56] :\n\nV N,0 = N i=1 dz i dV abc N i=1 [V ′ i (0)] N i=1 [ i < x = 0, O a |] δ( N i=1 p i ) N i,j=1 i =j exp - 1 2 ∞ n,m=0 a (i) n D nm (Γ V -1 i V j ) a (j) m ( 251\n\n)\n\n17 See also Ref. [57] . Earlier papers on the N -Reggeon can be found in Ref.s [58] .\n\nThe birth of string theory 47 where a (i) 0 ≡ α i 0 = √ 2α ′ pi is the momentum of particle i and the infinite matrix:\n\nD nm (γ) = 1 m! m n ∂ m z [γ(z)] n | z=0 ; n, m = 1.. : D 00 (γ) = -log | D √ AD -BC | D n0 = 1 √ n ( B D ) n ; D 0n = 1 √ n (- C D ) n ; γ(z) = Az + B Cz + D ( 252\n\n)\n\nis a \"representation\" of the projective group corresponding to the conformal weight ∆ = 0, that satisfies the eqs.:\n\nD nm (γ 1 γ 2 ) = ∞ l=1 D nl (γ 1 )D lm (γ 2 ) + D n0 (γ 1 )δ 0m + D 0m (γ 2 )δ n0 ( 253\n\n)\n\nand\n\nD nm (γ) = D mn (Γ γ -1 Γ ) Γ (z) = 1 z ( 254\n\n)\n\nFinally V i is a projective transformation that maps 0, 1 and ∞ into z i-1 , z i and z i+1 . The previous vertex can be written in a more elegant form as follows:\n\nV N,0 = N i=1 dz i dV abc N i=1 [V ′ i (0)] N i=1 [ i < x = 0, O a |] δ( N i=1 p i ) exp i 4α ′ dz∂X (i) (z)p i log V ′ i (z) exp - 1 2 N i,j=1 i =j dz dy∂X (i) (z) log[V i (z) -V j (y)]∂X (j) (y) ( 255\n\n)\n\nwhere the quantities X (i) are what we called Q, namely the Fubini-Veneziano field, in the previous sections. The N -Reggeon vertex that satisfies the important property of giving the scattering amplitude of N physical particle when we saturate it with their corresponding states, is the fundamental object for computing the multiloop amplitudes. In fact, if we want to compute a M -loop amplitude with N external states, we need to start from the (N +2M )-Reggeon vertex and then we have to sew the M pairs together after having inserted a propagator D. In this way we obtain an amplitude that is not only integrated over the punctures z i (i = 1 . . . N ) of the N external states, but also over the additional 3h -3 moduli corresponding to the punctures variables of the 48 Paolo Di Vecchia states that we sew together and the integration variable of the M propagators. h is the number of loops. The multiloop amplitudes have been obtained in this way already in 1970 [59, 60, 61] and, through the sewing procedure, one obtained functions, as the period matrix, the abelian differentials, the prime form, etc., that are well defined on Riemann surface! The only thing that was missing, was the correct measure of integrations over the 3h -3 variables because it was technically not possible to let only the physical states to circulate in the loops. This problem was solved only much later [62, 63] when a BRST invariant formulation of string theory and the light-cone functional integral could be used for computing multiloops. They are two very different approaches that, however, gave the same result. For the sake of completeness we write here the planar h-loop amplitude involving M tachyons:\n\nA (h) M (p 1 , . . . , p M ) = N h Tr(λ a1 • • • λ aM ) C h 2g s (2α ′ ) (d-2)/4 M × [dm] M h i<j exp G (h) (z i , z j ) V ′ i (0) V ′ j (0) 2α ′ pi•pj , ( 256\n\n)\n\nwhere N h Tr(λ a1 • • • λ aM ) is the appropriate U (N ) Chan-Paton factor, g is the dimensionless open string coupling constant, C h is a normalization factor given by\n\nC h = 1 (2π) dh g 2h-2 s 1 (2α ′ ) d/2 , ( 257\n\n)\n\nand G (h) is the h-loop bosonic Green function\n\nG (h) (z i , z j ) = log E (h) (z i , z j ) - 1 2 zj zi ω µ (2πImτ µν ) -1 zj zi ω ν , ( 258\n\n)\n\nwith E (h) (z i , z j ) being the prime form, ω µ (µ = 1, . . . , h) the abelian differentials and τ µν the period matrix. All these objects, as well as the measure on moduli space [dm] M h , can be explicitly written in the Schottky parametrization of the Riemann surface, and their expressions for arbitrary h can be found for example in Ref. [64] . It is given by\n\n[dm] M h = 1 dV abc M i=1 dz i V ′ i (0) h µ=1 dk µ dξ µ dη µ k 2 µ (ξ µ -η µ ) 2 (1 -k µ ) 2 ( 259\n\n) × [det (-iτ µν )] -d/2 α ′ ∞ n=1 (1 -k n α ) -d ∞ n=2 (1 -k n α ) 2 .\n\nwhere k µ are the multipliers, ξ µ and η µ are the fixed points of the generators of the Schottky group,\n\nThe birth of string theory 49"
},
{
"section_type": "OTHER",
"section_title": "From dual models to string theory",
"text": "The approach presented in the previous sections is a real bottom-up approach. The experimental data were the driving force in the construction of the Veneziano model and of its generalization to N external legs. The rest of the work that we have described above consisted in deriving its properties. The result is, except for a tachyon, a fully consistent quantum-relativistic model that was a source of fascination for those who worked in the field. Although the model grew out of S-matrix theory where the scattering amplitude is the only observable object, while the action or the Lagrangian have not a central role, some people nevertheless started to investigate what was the underlying microscopic structure that gave rise to such a consistent and beautiful model. It turned out, as we know today, that this underlying structure is that of a quantum-relativistic string. However, the process of connecting the dual resonance model (actually two of them the generalized Veneziano and the Shapiro-Virasoro model) to string theory took several years from the original idea to a complete and convincing proof of the conjecture. The original conjecture was independently formulated by Nambu [20, 65] , Nielsen [66] and Susskind [21] 18 . If we look at it in retrospective, it was at that time a fantastic idea that shows the enormous physical intuition of those who formulated it. On the other hand, it took several years to digest it before one was able to derive from it all the deep features of the dual resonance model. Because of this, the idea that the underlying structure was that of a relativistic string, did not really influence most of the research in the field up to 1973. Let me try to explain why. A common feature of the work of Ref.s [20, 66, 21] is the suggestion that the infinite number of oscillators, that one got through the factorization of the dual resonance model, naturally comes out from a two-dimensional free Lagrangian for the coordinate X µ (τ, σ) of a one-dimensional string, that is an obvious generalization of the Lagrangian that one writes for the coordinate X µ (τ ) of a pointlike object in the proper-time gauge:\n\nL ∼ 1 2 dX dτ • dX dτ =⇒ L ∼ 1 2 dX dτ • dX dτ - dX dσ • dX dσ ( 260\n\n)\n\nBeing this theory conformal invariant the Virasoro operators were also constructed together with their algebra. In this very first formulation, however, the Virasoro generators L n were just the generators associated to the conformal symmetry of the string world-sheet Lagrangian given in Eq. (260) as in any conformal field theory. It was not clear at all why they should imply the gauge conditions found by Virasoro or, in modern terms, why they should be zero classically. The basic ingredient to solve this problem was provided by Nambu [65] and Goto [68] who wrote the non-linear Lagrangian proportional 18 See also Ref. [67] ."
},
{
"section_type": "OTHER",
"section_title": "Paolo Di Vecchia",
"text": "to the area spanned by the string in the external target space. They proceeded in analogy with the point particle and wrote the following action:\n\nS ∼ -dσ µν dσ µν ( 261\n\n) where dσ µν = ∂X µ ∂ζ α ∂X ν ∂ζ β dζ α ∧ dζ β = ∂X µ ∂ζ α ∂X ν ∂ζ β ǫ αβ dσdτ ( 262\n\n)\n\nX µ (σ, τ ) is the string coordinate and ζ 0 = τ and ζ 1 = σ are the coordinates of the string worldsheet. ǫ αβ is an antisymmetric tensor with ǫ 01 = 1. Inserting eq. (262) in (261) and fixing the proportionality constant one gets the Nambu-Goto action [65, 68]:\n\nS = -cT τ f τi dτ π 0 dσ ( Ẋ • X ′ ) 2 -Ẋ2 X ′ 2 ( 263\n\n) where Ẋµ ≡ ∂X µ ∂τ X ′ µ ≡ ∂X µ ∂σ ( 264\n\n)\n\nand T ≡ 1 2πα ′ is the string tension, that replaces the mass appearing in the case of a point particle. In this formulation, the string Lagrangian is invariant under any reparametrization of the world-sheet coordinates σ and τ and not only under the conformal transformations. This, in fact, implies that the twodimensional world-sheet energy-momentum tensor of the string is actually zero as we will show later on. But it took still a few years to connect the Nambu-Goto action to the properties of the dual resonance model. In the meantime an analogue model was formulated [69] that reproduced the tree and loop amplitudes of the generalized Veneziano model. This approach anticipated by several years the path integral derivation of dual amplitudes. It was very closely related to the functional integral formulation of Ref.s [70] . However, one needed to wait until 1973 with the paper of Goddard, Goldstone, Rebbi and Thorn [71] , where the Nambu-Goto action was correctly treated, all its consequences were derived and it became completely clear that the structure underlying the dual resonance model was that of a quantum-relativistic string. The equation of motion for the string were derived from the action in Eq. (263) by imposing δS = 0 for variations such that δX µ (τ i ) = δX µ (τ f ) = 0. One gets:\n\nδS = τ f τi π 0 dσ - ∂ ∂τ ∂L ∂ Ẋµ - ∂ ∂σ ∂L ∂X ′ µ δX µ + ∂L ∂X ′ µ δX µ | σ=π σ=0 = 0 (265)\n\nwhere L is the Lagrangian in Eq. (263) . Since δX µ is arbitrary, from eq. (265) one gets the Euler-Lagrange equation of motion\n\nThe birth of string theory 51\n\n∂ ∂τ ∂L ∂ Ẋµ + ∂ ∂σ ∂L ∂X ′ µ ≡ ∂ ∂ζ α ∂L ∂( ∂X µ ∂ζ α ) = 0 ( 266\n\n)\n\nand the boundary conditions\n\n∂L ∂X ′ µ = 0 or δX µ = 0 at σ = 0, π ( 267\n\n)\n\nfor an open string and\n\nX µ (τ, 0) = X µ (τ, π) ( 268\n\n)\n\nfor a closed string. In the case of an open string, the first kind of boundary condition in Eq.(267) corresponds to Neumann boundary conditions, while the second one to Dirichlet boundary conditions. Only the Neumann boundary conditions preserve the translation invariance of the theory and, therefore, they were mostly used in the early days of string theory. It must be stressed, however, that Dirichlet boundary conditions were already discussed and used in the early days of string theory for constructing models with offshell states [72]. From Eq. (263) one can compute the momentum density along the string:\n\n∂L ∂ Ẋµ ≡ P µ = cT Ẋµ X ′ 2 -X ′ µ ( Ẋ • X ′ ) ( Ẋ • X ′ ) 2 -Ẋ2 X ′ 2 ( 269\n\n)\n\nand obtain the following constraints between the dynamical variables X µ and P µ :\n\nc 2 T 2 x ′ 2 + P 2 = x ′ • P = 0 ( 270\n\n)\n\nThey are a consequence of the reparametrization invariance of the string Lagrangian. Because of this one can choose the orthonormal gauge specified by the conditions:\n\nẊ2 + X ′ 2 = Ẋ • X ′ = 0 ( 271\n\n)\n\nthat nowadays is called conformal gauge. In this gauge eq. (269) becomes:\n\nP µ = cT Ẋµ ∂L ∂X ′ µ = -cT X ′ µ ( 272\n\n)\n\nand therefore the eq. of motion in eq.(266) becomes:\n\nẌµ -X ′′ µ = 0 ( 273\n\n)\n\nwhile the boundary condition in eq.(267) becomes:\n\nX ′ µ (σ = 0, π) = 0 ( 274\n\n)\n\n52 Paolo Di Vecchia\n\nThe most general solution of the eq. of motion and of the boundary conditions can be written as follows:\n\nX µ (τ, σ) = q µ + 2α ′ p µ τ + i √ 2α ′ ∞ n=1 [a µ n e -inτ -a +µ n e inτ ] cosnσ √ n ( 275\n\n)\n\nfor an open string and\n\nX µ (τ, σ) = q µ + 2α ′ p µ τ + i 2 √ 2α ′ ∞ n=1 [ã µ n e -2in(τ +σ) -ã+µ n e 2in(τ +σ) ] 1 √ n + + i 2 √ 2α ′ ∞ n=1 [a µ n e -2in(τ -σ) -a +µ n e 2in(τ -σ) ] 1 √ n ( 276\n\n)\n\nfor a closed string. This procedure really shows that, starting from the Nambu-Goto action, one can choose a gauge (the orthonormal or conformal gauge) where the equation of motion of the string becomes the twodimensional D'Alembert equation in Eq. ( 273 ). Furthermore, the invariance under reparametrization of the Nambu-Goto action implies that the twodimensional energy-momentum tensor is identically zero at the classical level (See Eq. (271)).\n\nAs the Lorentz gauge in QED the orthonormal gauge does not fix completely the gauge. We can still perform reparametrizations that leave in the conformal gauge: they are conformal transformatiuons. Introducing the variable z = e iτ the generators of the conformal transformations for the open string can be written as follows:\n\nL n = 1 2πi dzz n+1 - 1 4α ′ ∂X µ ∂z 2 = 1 2 ∞ m=-∞ α n-m • α m = 0 ( 277\n\n)\n\nwhere\n\nα µ n = √ na µ n if n > 0 √ 2α ′ p µ if n = 0 √ na †µ n if n < 0 ( 278\n\n)\n\nThey are zero as a consequence of Eq.s (270) that in the conformal gauge become Eq.s (271). In the case of a closed string we get instead:\n\nLn = 1 2πi dzz n+1 - 1 α ′ ∂X µ ∂z 2 = 0 ( 279\n\n) L n = 1 2πi dz zn+1 - 1 α ′ ∂X µ ∂ z 2 = 0 ( 280\n\n)\n\nThe birth of string theory 53 In terms of the harmonic oscillators introduced in eq. (276) we get\n\nL n = 1 2 ∞ m=-∞ α m • α n-m = 0 ; Ln = 1 2 ∞ m=-∞ αm • αn-m = 0 ( 281\n\n)\n\nwhere for the non-zero modes we have used the convention in (278), while the zero mode is given by:\n\nα µ 0 = αµ 0 = √ 2α ′ p µ 2 ( 282\n\n)\n\nIn conclusion, the fact that we have reparametrization invariance implies that the Virasoro generators are classically identically zero. When we quantize the theory one cannot and also does not need to impose that they are vanishing at the operator level. They are imposed as conditions characterizing the physical states.\n\nP hys ′ |L n |P hys = P hys ′ |(L 0 -1)|P hys = 0 ; n = 0 ( 283\n\n)\n\nThese equations are satisfied if we require:\n\nL n |P hys >= (L 0 -1)|P hys >= 0 ( 284\n\n)\n\nThe extra factor -1 in the previous equations comes from the normal ordering as explained in Eq. ( 198 ). The authors of Ref. [71] further specified the gauge by fixing it completely. They introduced the light-cone gauge specified by imposing the condition:\n\nX + = 2α ′ p + τ ( 285\n\n)\n\nwhere\n\nX ± = X 0 ± X d-1 √ 2 X ± = X 0 ± X d-1 √ 2 ( 286\n\n)\n\nIn this gauge the only physical degrees of freedom are the transverse ones.\n\nIn fact the components along the directions 0 and d -1 can be expressed in terms of the transverse ones by inserting Eq. ( 285 ) in the constraints in Eq. ( 271 ) and getting:\n\nẊ-= 1 4α ′ p + ( Ẋ2 i + X ′ 2 i ) X ′ -= 1 2α ′ p + Ẋi • X ′ i ( 287\n\n)\n\nthat up to a constant of integration determine completely X -as a function of X i . In terms of oscillators we get\n\nα + n = 0 ; √ 2α ′ α - n = 1 2p + ∞ m=-∞ α i n-m α i m n = 0 ( 288\n\n)\n\n54 Paolo Di Vecchia\n\nfor an open string and\n\nα + n = α+ n = 0 n = 0 ( 289\n\n) together with √ 2α ′ α - n = 1 2p + ∞ m=-∞ α i n-m α i m √ 2α ′ α- n = 1 2p + ∞ m=-∞ αi n-m αi m ( 290\n\n)\n\nin the case of a closed string.\n\nThis shows that the physical states are described only by the transverse oscillators having only d -2 components. Those transverse oscillators correspond to the transverse DDF operators that we have discussed in Section 6. The authors of Ref. [71] also constructed the Lorentz generators only in terms of the transverse oscillators and they showed that they satisfy the correct Lorentz algebra only if the space-time dimension is d = 26. In this way the spectrum of the dual resonance model was completely reproduced starting from the Nambu-Goto action if d = 26! On the other hand, the choice of d = 26 is a necessity if we want to keep Lorentz invariance! Immediately after this, the interaction was also included either by adding a term describing the interaction of the string with an external gauge field [73] or by using a functional formalism [74, 75] .\n\nIn the following we will give some detail only of the first approach for the case of an open string. A way to describe the string interaction is by adding to the free string action an additional term that describes the interaction of the string with an external field.\n\nS IN T = d D yΦ L (y)J L (y) ( 291\n\n)\n\nwhere Φ L (y) is the external field and J L is the current generated by the string. The index L stands for possible Lorentz indices that are saturated in order to have a Lorentz invariant action.\n\nIn the case of a point particle, such an interaction term will not give any information on the self-interaction of a particle.\n\nIn the case of a string, instead, we will see that S IN T will describe the interaction among strings because the external fields that can consistently interact with a string are only those that correspond to the various states of the string, as it will become clear in the discussion below.\n\nThis is a consequence of the fact that, for the sake of consistency, we must put the following restrictions on S IN T :\n\n• It must be a well defined operator in the space spanned by the string oscillators.\n\nThe birth of string theory 55 • It must preserve the invariances of the free string theory. In particular, in the \"conformal gauge\" it must be conformal invariant. • In the case of an open string, the interaction occurs at the end point of a string (say at σ = 0). This follows from the fact that two open strings interact attaching to each other at the end points.\n\nThe simplest scalar current generated by the motion of a string can be written as follows\n\nJ(y) = dτ dσδ(σ)δ (d) [y µ -x µ (τ, σ)] ( 292\n\n)\n\nwhere δ(σ) has been introduced because the interaction occurs at the end of the string. For the sake of simplicity we omit to write a coupling constant g in (292). Inserting (292) in (291) and using for the scalar external field Φ(y) = e ik•y a plane wave, we get the following interaction:\n\nS IN T = dτ : e ik•X(τ,0) : ( 293\n\n)\n\nwhere the normal ordering has been introduced in order to have a well defined operator. The invariance of (293) under a conformal transformation τ → w(τ ) requires the following identity:\n\nS IN T = dτ : e ik•X(τ,0) : = dw : e ik•X(w,0) : ( 294\n\n)\n\nor, in other words, that : e ik•X(τ,0) :=⇒ w ′ (τ ) : e ik•X(w,0) : (\n\n) 295\n\nThis means that the integrand in Eq. ( 294 ) must be a conformal field with conformal dimension equal to one and this happens only if α ′ k 2 = 1. The external field corresponds then to the tachyonic lowest state of the open string. Another simple current generated by the string is given by:\n\nJ µ (y) = dτ dσδ(σ) Ẋµ (τ, σ)δ (d) (y -X(τ, σ)) ( 296\n\n)\n\nInserting (296) in (291) we get\n\nS IN T = dτ Ẋµ (τ, 0)ǫ µ e ik•X(τ,0) ( 297\n\n)\n\nif we use a plane wave for Φ µ (y) = ǫ µ e ik•y . The vertex operator in eq. ( 297 ) is conformal invariant only if\n\nk 2 = ǫ • k = 0 ( 298\n\n)\n\n56 Paolo Di Vecchia and, therefore, the external vector must be the massless photon state of the string. We can generalize this procedure to an arbitrary external field and the result is that we can only use external fields that correspond to on shell physical states of the string. This procedure has been extended in Ref. [73] to the case of external gravitons by introducing in the Nambu-Goto action a target space metric and obtaining the vertex operator for the graviton that is a massless state in the closed string theory. Remember that, at that time, this could have been done only with the Nambu-Goto action because the σ-model action was introduced only in 1976 first for the point particle [76] and then for the string [77]. As in the case of the photon it turned out that the external field corresponding to the graviton was required to be on shell. This condition is the precursor of the equations of motion that one obtains from the σ-model action requiring the vanishing of the β-function [78] .\n\nOne can then compute the probability amplitude for the emission of a number of string states corresponding to the various external fields, from an initial string state to a final one. This amplitude gives precisely the N -point amplitude that we discussed in the previous sections [73] . In particular, one learns that, in the case of the open string, the Fubini-Veneziano field is just the string coordinate computed at σ = 0:\n\nQ µ (z) ≡ X µ (z, σ = 0) ; z = e iτ ( 299\n\n)\n\nIn the case of a closed string we get instead: z, z) ; z = e 2i(τ -σ) , z = e 2i(τ +σ) (300) Finally, let me mention that with the functional approach Mandelstam [74] and Cremmer and Gervais [79] computed the interaction between three arbitrary physical string states and reproduced in this way the coupling of three DDF states given in Eq. ( 202 ) and obtained in Ref. [37] by using the operator formalism. At this point it was completely clear that the structure underlying the generalized Veneziano model was that of an open relativistic string, while that underlying the Shapiro-Virasoro model was that of a closed relativistic string. Furthermore, these two theories are not independent because, if one starts from an open string theory, one gets automatically closed strings by loop corrections.\n\nQ µ (z, z) ≡ X µ ("
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusions",
"text": "In this contribution, we have gone through the developments that led from the construction of the dual resonance model to the bosonic string theory trying as much as possible to include all the necessary technical details. This is because we believe that they are not only important from an historical point of view, but are also still part of the formalism that one uses today in many\n\nThe birth of string theory 57 string calculations. We have tried to be as complete and objective as possible, but it could very well be that some of those who participated in the research of these years, will not agree with some or even many of the statements we made. We apologize to those we have forgotten to mention or we have not mentioned as they would have liked.\n\nFinally, after having gone through the developments of these years, my thoughts go to Sergio Fubini who shared with me and Gabriele many of the ideas described here and who is deeply missed, and to my friends from Florence, Naples and Turin for a pleasant collaboration in many papers discussed here."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
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D 9, 2273 (1974). 51. E, Cremmer and J. Scherk, Nucl. Phys. B 72, 117 (1974). 52. A. D'Adda, R. D'Auria, E. Napolitano, P. Di Vecchia, F. Gliozzi and S. Sciuto, Phys. Lett. B 68, 81 (1977). 53. J. Shapiro, Phys. Rev. D 5, 1945 (1975). 54. S. Sciuto, Lettere al Nuovo Cimento 2, 411 (1969).\n\nA. Della Selva and S. Saito, Lett. al Nuovo Cimento 4, 689 (1970). 55. L. Caneschi, A. Schwimmer and G. Veneziano, Phys. Lett.B 30, 356 (1969). L. Caneschi and A. Schwimmer, Lettere al Nuovo Cimento 3, 213 (1970). 56. C. Lovelace, Phys. Lett. B 32, 490 (1970). 57. D.I. Olive, Nuovo Cimento A 3, 399 (1971). 58. I. Drummond, Nuovo Cimento A 67, 71 (1970). G. Carbone and S. Sciuto, Lett. Nuovo Cimento 3, 246 (1970). L. Kosterlitz and D. Wray, Lett. al Nuovo Cimento 3, 491 (1970). D. Collop, Nuovo Cimento A 1, 217 (1971). L.P. Yu, Phys. Rev. D 2, 1010 (!970); Phys. Rev. D 2, 2256 (!970). E. Corrigan and C. Montonen, Nucl. Phys. B 36, 58 (1972). J.L. Gervais and B. Sakita, Phys. Rev. D 4, 2291 (1971). 59. C. Lovelace, Phys. Lett. B 32, 703 (1970). 60. V. Alessandrini, Nuovo Cimento A 2, 321 (1971). 61. D. Amati and V. Alessandrini, Nuovo Cimento A 4, 793 (1971). 62. P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, Phys. Lett. B 199, 49 (1987). J.L. Petersen and J. Sidenius, Nucl. Phys. B 301, 247 (1988). 63. S. Mandelstam, In \"Unified String Theories\", edited by M. Green and D. Gross, World Scientific, p. 46. 64. P. Di Vecchia, F. Pezzella, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, Nucl.\n\nPhys. B 322, 317 (1989). 65. Y. Nambu, Lectures at the Copenhagen Symposium, 1970, unpublished. 66. H. B. Nielsen, Paper submitted to the 15th Int. Conf. on High Energy Physics, Kiev, 1970 and Nordita preprint (1969). 67. T. Takabayasi, Progr. Theor. Phys. 44 (1970) 1117. O. Hara, Progr. Theor. Phys. 46, 1549 (1971). L.N. Chang and J. Mansouri, Phys. Rev. D 5, 2535 (1972). J. Mansouri and Y. Nambu, Phys. Lett. B 39, 357 (1972). M. Minami, Prog. Theor. 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}
] | ||
arxiv:0704.0103 | 0704.0103 | 1 | 10.1007/s10773-007-9450-y | b5569420bcf9c3028ac0cd9a97fe5e5d178d687547076d0da25a18a46e491bbf | Generalized regularly discontinuous solutions of the Einstein equations | The physical consistency of the match of piecewise-$C^0$ metrics is
discussed. The mathematical theory of gravitational discontinuity hypersurfaces
is generalized to cover the match of regularly discontinuous metrics. The
mean-value differential geometry framework on a hypersurface is introduced, and
corresponding compatibility conditions are deduced. Examples of generalized
boundary layers, gravitational shock waves and thin shells are studied. | [
"Gianluca Gemelli"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-01 | 2026-02-26 | The physical consistency of the match of piecewise-C 0 metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value differential geometry framework on a hypersurface is introduced, and corresponding compatibility conditions are deduced. Examples of generalized boundary layers, gravitational shock waves and thin shells are studied.
Is it possible to define weak solutions of the Einstein equations of class piecewise-C 0 , i.e. to generalize the compatibility conditions which replace the field equations on a singular hypersurface to the case when the metric is regularly discontinuous?
To reach this goal would probably mean to define the most general class of regularly discontinuous weak solutions of the Einstein equations. It seems that this problem was never studied before in the literature. But, before we proceed, we need to discuss whether we are talking of something mathematically and physically consistent or not.
A fundamental concept of Riemannian geometry is that at any point of a submanifold there are coordinate choices for which the metric reduces to the Minkowski flat metric. Clearly, if this choice is made on both sides of the discontinuity surface, any "jump" in the metric disappears. Thus, the metric discontinuity appears as a coordinate dependent concept, which is neither geometrically (nor physically) acceptable in the context of General Relativity.
But we also have to consider that regularity of the global coordinates plays an important role in our approach, which is that of [1] and of the literature cited therein. In particular, since here the spacetime is only C 0 , we are led to considering (C 0 , piecewise C 1 ) coordinate transformations. If the metric is discontinuous in some globally C 0 chart, it is in general impossible to obtain the vanishing of the metric jump on both sides of a hypersurface with a C 0 coordinate transformation (see section 2). Moeover in the following we are led in a natural way to considering C 1 coordinate transformations; the metric discontinuity is a tensor with respect to such coordinate changes!
In other words the jump of the metric has a precise mathematical meaning, if we consider it in connection with global regular coordinates.
In a well consolidated procedure, the assumption of continuity for the metric across a gravitational interface is usually taken for granted; however it follows from the limiting process of the thin sandwich modelization, in consequence of the hypothesis that the external derivatives of the metric are bounded [2] . Yet in this paper we are going to see that, even removing the assumption of continuity, it is still possible to define a generalized inner geometry of the discontinuity hypersurface; one thus can consistently find a corresponding generalized set of compatibility conditions, which obviously reduces to the usual ones when the continuity hypothesis is restored.
Yet, which are the physical motivations to move to such generalization? Actually gravitational shock waves and thin shells are usually defined by the presence of singular curvature with a "delta" component concentrated on a hypersurface, situation which is well cast within the classic C 0 piecewise-C 1 match of metrics [1, 3] .
We were originally led to consider solutions of class piecewise-C 0 , as possible generalizations of shock waves and thin shells, by the sake of mathematical completeness, with the idea that phisical interpretation would follow. We actually found a reacher framework than the usual one, with some interesting new features (and even some rather undesiderable ones), which we display in this paper.
There are two main theories in the literature for solutions of class C 0 piecewise-C 1 , i.e. that in terms of the second fundamental form (heuristic theory, see e.g. [4, 5] ) and that in terms of the curvature tensor-distribution (axiomatic theory, see e.g. [6, 1] ); such are equivalent through appropiate extensions (for a self-contained overview see e.g. [1] ).
The axiomatic theory appears to be inappropriate to the study of generalized solutions, since the theory of distributions is basically linear. Even if we could in principle replace the discontinuous metric with its associated distribution g D , then it would be impossible to define, for example, replacements for the Christoffel symbols, since this would involve product of distributions, which, as it is generally believed, is impossible to define. In fact it was proved by Schwarz [7] that, under reasonable hypothesis, there can be no definition of commutative and associative operation on distributions which reduce to ordinary multiplication on integrable distributions (say on regular functions); thus in a word it is impossible to define product of distributions.
Or is it? Colombeau [8, 9, 10] developed a theory which apparently contradicts Schwarz's result. He introduced a very broad space of generalized functions, which extends the usual space of distributions, a subspace of which corresponds, in a certain sense (the correspondance is not 1 to 1), to usual distributions. Colombeau's formalism permits multiplication of generalized functions; but the contradiction with the impossibility theorem is only apparent, in fact Schwarz's hypothesis are violated, since the operation does not coincide with ordinary multiplication on regular functions nor with multiplication of a regular function times a distribution (although it does at least for C ∞ functions).
Such theory, however, does not fit in a natural way in general relativity, since it is impossible to define covariantly invariant geometrical objects; in fact Colombeau's space is not invariant for smooth coordinate transformations, unless they are linear. Such difficulty, however, seems to have been overcome in subsequent adjustments of the theory, with the introduction of a richer mathematical framework [11, 12] , so that the generalized functions current apparatus can be used in general relativity, and indeed it has been applied at least to the calculation of singular curvatures of the spacetimes of Kerr [13] , Reissner-Nordstrom [14] , and so-called cosmic-string spacetime [15] . In such literature Colombeau's theory is adapted to the handling of curvature when the metric has a singularity in the sense of functions, i.e. the ordinary curvature would explode, at a singular event-point or at a singular worldline. There seems to be no particular reason to forbid Colombeau's method also for defining the match of piecewise-C 0 regularly discontinuous metrics at a singular hypersurface; however, as far as the author is aware, no attempt has been made yet to use it in this framework.
The direct method we will introduce in the following sections, however, is so conceptually simple that we prefer not to experiment with Colombeau's generalized functions, which would instead mean introducing a far more complicated and unfamiliar mathematical apparatus.
In this paper in fact we propose a new, generalized theory for regularly discontinuous solutions, covering also the match of piecewise-C 0 metrics. Our theory is heuristic, as it is constructed in a way similar to the heuristic theory of C 0 , piecewise-C 1 solutions originated from the studies of Israel, but we completely avoid the traditional or projectional Gauss-Codazzi framework (which either does not include the lightlike case [4, 5] , or needs a special adaptation for it [16, 1] ) and introduce what we called "mean-value differential geometry" framework, instead (see section 3). This is conceptually very simple, and permits to construct in a natural way a generalized theory, where the main role (which used to be that of the jump of the secund fundamental form) is here played by the jump of the Christoffel symbols. The theory is an extension of the theory of gravitational discontinuity hypersurfaces we have studied in [1] , to which it reduces when the metric is C 0 . Even if we should restrict to C 0 solution, by adding the traditional assumption of continuity for the metric, our theory would undoubtedly have at least the good qualities of not needing the timelike and the lightlike case to be distinguished (different from usual heuristic theory), and of just requiring C 0 continuity for the coordinates (different from the axiomatic theory). Moreover, it is completely cast in the framework of general coordinates of the ambient (glued) spacetime, with no use of parametric equations of the hypersurface, nor of inner coordinates and holonomic 3-basis, which could be considered a good quality in some applications as well.
Piecewise-C 0 weak solutions of the Einstein equations, as far as the author is aware, have never been considered previously in the literature. They generalize the corresponding C 0 solutions, as examples in this paper show; however there is more. Apparently in fact the theory allows the propagation of free gravitational discontinuity at lower speed than the speed of light (section 8); or rather, we still have no general proof that the absence of stress energy concentrated on Σ should, in the timelike case, necessarily imply the degeneracy of a generalized solution to a boundary layer, although it does at least for a wide class of spherical matchs (see section 6). Moreover, nonsimmetric stress-energy is allowed on the hypersurface (section 9), like e.g. in Einstein-Cartan dynamics. This possible link to classical unification theories is surprising, since in our framework we have nothing similar to Einstein-Cartan torsion. We therefore see a lot of space for future investigation.
Let us suppose V 4 an oriented differentiable manifold of dimension 4, of class (C 0 , piecewise C 2 ), provided with a strictly hyperbolic metric of signature -+++ and class piecewise-C 0 . Let Ω ⊂ V 4 be an open connected subset with compact closure. Let units be chosen in order to have the speed of light in empty space c ≡ 1. Greek indices run from 0 to 3.
Let Σ ⊂ Ω be a regular hypersuperface of equation f (x) = 0; let Ω + and Ω -denote the subdomains distinguished by the sign of f . We suppose the metric and its first and second partial derivatives to be regularly discontinuous on Σ in all charts of class C 0 (Ω). Let f ∈ C 0 (Ω) ∩ C 2 (Ω\Σ), and let second and third derivatives of f be regularly discontinuous on Σ. Finally, let ℓ α ≡ ∂ α f denote the gradient of f .
Let the metric be a solution of the ordinary Einstein equations on each of the two domains Ω + and Ω -. In this situation Σ is the interface between two general relativistic spacetimes and it is called a (generalized) gravitational discontinuity hypersurface.
In the following we will develope a theory to justify the introduction of suitable generalized compatibility conditions to replace the Einstein equations on Σ (section 5); if such conditions are satisfied the match across the generalized gravitational hypersurface Σ will be called a generalized regularly discontinuous solution of the Einstein equations. Now let us briefly recall some basics notions on regularly discontinuous fields, which we will use as tools. In any case, for notation and terminology we refer to [1] .
A field ϕ is said to be regularly discontinuous on Σ if its restrictions to the two subdomains Ω + and Ω -both have a finite limit for f -→ 0; we denote such limits by ϕ + and ϕ -, respectively.
In this case the jump [ϕ] across Σ and its arithmetic mean value ϕ are well defined on the hypersurface:
[ϕ] = ϕ + -ϕ - ϕ = (1/2)(ϕ + + ϕ -) (1)
If ϕ is continuous across Σ, we obviously have: [ϕ] = 0, ϕ = ϕ. We also have the converse formulae:
ϕ + = ϕ + (1/2)[ϕ] ϕ -= ϕ -(1/2)[ϕ]. (2)
As for the product of two functions ϕ and ψ, we have:
[ϕψ] = [ϕ]ψ + ϕ[ψ] ϕψ = ϕψ + (1/4)[ϕ][ψ] (3)
If a field ϕ is regularly discontinuous on Σ, its jump [ϕ] is sometimes called its discontinuity of order 0. The jump of a regularly discontinuous function has support on Σ, but in general, the partial derivative of the jump is well defined as the jump of the derivative of the function (see [17, 18] ). In particular, the derivative of the jump of a continuous field is not null, unless the field is also C 1 .
Similarly, we define the partial derivative of the mean value as the mean value of the partial derivative. We can also use regular prolongations to extend, in a sense, the definition of ϕ and [ϕ] to the whole domain Ω. Thus they can be regarded as regular and derivable fields in Ω, but their values (and those of their derivatives) are well defined only on Σ, while in Ω\Σ they depend on the choice of the prolongation. For details on the method of regular prolongations see e.g. [17, 18] .
We moreover define the covariant derivative of a field with support on Σ by means of the mean value Γ βρ σ of the Christoffel symbols. For the jump of a regularly discontinuous vector, for example, with this definition one has that the jump of the covariant derivative is different than the covariant derivative of the jump. Thus, by definition, we have:
∇ α [V β ] = ∂ α [V β ] + Γ ασ β [V σ ] (4)
and in consequence of (3):
∇ α [V β ] = [∇ α V β ] -[Γ ασ β ]V σ , (5)
and similarly for the jump of any regularly discontinuous tensor. Since the spacetime is only C 0 , we are led to considering (C 0 , piecewise C 1 ) coordinate transformations, with regularly discontinuous first derivatives; the metric discontinuity [g αβ ] is not a tensor with respect to such changes of coordinates. In fact we have:
[g αβ ] = [g α ′ β ′ ] dx α ′ dx α • dx β ′ dx β + q αβ ′ dx β ′ dx β + q α ′ β dx α ′ dx α (6)
where:
q α ′ β = 1 8 [g α ′ β ′ ] dx β ′ dx β + ḡα ′ β ′ dx β ′ dx β (7)
We therefore can simulate all (C 0 , piecewise C 1 ) coordinate changes by combining C 1 changes with metric gauge changes:
[g αβ ] ←→ [g αβ ] + q αβ ′ dx β ′ dx β + q α ′ β dx α ′ dx α (8)
which generalize usual gravitational gauge changes of the theory of (C 0 , piecewise C 1 ) solutions [1] .
Is it always possible to make [g αβ ] vanish with an appropriate C 0 transformation? Clearly the answer is negative. In fact it suffices to consider the case when [g αβ ] and ḡαβ are both definite positive in a given chart to see that the equation obtained from ( 6 ) by replacing the first hand side with 0 has no solution for [∂x α ′ /∂x α ] and ∂x α ′ /∂x α . Thus the set of effective generalized gravitational discontinuity hypersurfaces is non empty.
Moreover it will occur in many applications to have ℓ α ∈ C 0 . Therefore it will be often desiderable to work in the framework of (C 1 , piecewise C 2 ) coordinate transformations, which preserve such condition. The metric discontinuity is a tensor with respect to such changes of coordinates, but the jump of the Christoffel symbols, which appear to play a main role in the following, is not; we have in fact:
[Γ αβ σ ] = [Γ α ′ β ′ σ ′ ] ∂x α ′ ∂x α ∂x β ′ ∂x β ∂x σ ∂x σ ′ + ∂ 2 x σ ′ ∂x α ∂x β ∂x σ ∂x σ ′ (9)
If the coordinates are C 0 and so is the form ℓ α we can write:
∂ 2 x σ ′ ∂x α ∂x β = ℓ α ℓ β ∂ 2 x σ ′ ( 10
)
where ∂ 2 denotes the weak discontinuity of order 2 (see e.g. [17, 18] ). Thus on Σ we can generate all (C 1 , piecewise C 2 ) transformations for [Γ] combining C 2 transformations (with respect to which Γ is a tensor) and Christoffel gauges transformations, i.e. of the kind:
[Γ αβ σ ] ↔ [Γ αβ σ ] + ℓ α ℓ β Q σ (11)
with some analogy with the case of C 0 metrics (where the main role is played by the first order metric discontinuity ∂g, see [1] section 3).
In any case neither the mean value of the metric g or its jump [g] now have null covariant derivatives. Consider in fact the identity ∇ α g βρ = 0 in the domain Ω + ; from the limit f -→ 0 + , on Σ we have:
∂ α g + βρ -(Γ αβ ν ) + g + νρ -(Γ αρ ν ) + g + βν = 0 (12)
Here, with obvious meaning of the symbols, we denote: g + βρ = (g βρ ) + , g βρ = g βρ , etc. Consequently on Σ, from (2) 1 we have:
∂ α g βρ + (1/2)∂ α [g βρ ] -Γ αβ ν g νρ -Γ αρ ν g νβ + -(1/2)([Γ αβ ν ]g νρ + Γ αβ ν [g ρν ] + [Γ αρ ν ]g νβ + Γ αρ ν [g βν ])+ -(1/4)([Γ αβ ν ][g νρ ] + [Γ αρ ν ][g βν ]) = 0 (13)
Similarly, from the limit f -→ 0 -and from (2) 2 we also have on Σ:
∂ α g βρ -(1/2)∂ α [g βρ ] -Γ αβ ν g νρ -Γ αρ ν g νβ + +(1/2)([Γ αβ ν ]g νρ + Γ αβ ν [g ρν ] + [Γ αρ ν ]g νβ + Γ αρ ν [g βν ])+ -(1/4)([Γ αβ ν ][g νρ ] + [Γ αρ ν ][g βν ]) = 0 (14)
From the sum of expressions ( 13 ) and ( 14 ) we thus have:
∇ α g βρ = (1/4)([Γ αβ ν ][g ν ρ ] + [Γ αρ ν ][g βν ]) (15)
and, from difference:
∂ α [g βρ ] = [Γ αβρ ] + [Γ αρβ ] (16)
From ( 16 ), (3), and from the definition of covariant derivative over Σ, we then have:
∇ α [g βρ ] = [Γ αβ ν ]g νρ + [Γ αρ ν ]g βν (17)
As for the jump and the mean value of the Christoffel symbols we have, from
:
Γ αβ ν = (1/2){g νσ (∂ α g βσ + ∂ β g σα -∂ σ g αβ )+ +(1/4)[g νσ ](∂ α [g βσ ] + ∂ β [g σα ] -∂ σ [g αβ ])} (18)
and
[Γ αβ ν ] = (1/2){g νσ (∂ α [g βσ ] + ∂ β [g σα ] -∂ σ [g αβ ])+ +[g νσ ](∂ α g βσ + ∂ β g σα -∂ σ g αβ ) (19)
or, from ( 15 ) and ( 17 ):
[Γ αβ ν ]g νρ = (1/2)(∇ α [g βρ ] + ∇ β [g ρα ] -∇ ρ [g αβ ]) [Γ αβ ν ][g νρ ] = 2(∇ α g βρ + ∇ β g ρα -∇ ρ g αβ ) ( 20
)
3 Mean-value geometry on a hypersurface
Let us consider a 4-vector V α , regularly discontinuous on Σ, the jump and the mean value of which will work as a prototype of vectors with Σ as support.
We have, by definition:
[∇ β ∇ α V σ ] = ∇ β [∇ α V σ ] -[Γ βα ν ]∇ ν V σ + [∇ β ν σ ]∇ α V ν (21)
where
[∇ α V σ ] = ∇ α [V σ ] + [Γ αν σ ]V
ν and where, again by definition, we have:
∇ ν V σ = 1 2 {∂ ν (V + ) σ + (Γ + ) νλ σ (V + ) λ + ∂ ν (V -) σ + (Γ -) νλ σ (V -) λ } (22)
Thus, from (2) we have:
∇ ν V σ = ∇ ν V σ + (1/4)[Γ νλ σ ][V λ ], (23)
which, incidentally, is the same result we could get from the formal application of (3), wich actually can be applied to covariant derivatives, provided one interpretes ∇ = ∇. We therefore have:
[∇ α ∇ β V σ ] = ∇ α ∇ β [V σ ] + ∇ β [Γ αν σ ]V ν + [Γ αν σ ]∇ β V ν + -[Γ βα ν ]∇ ν V σ -(1/4)[Γ βα ν][Γ νλ σ ][V λ ]+ +[Γ βν σ ]∇ α V ν + (1/4)[Γ βν σ ][Γ αλ ν ][V λ ] (24)
and thus, by antisymmetrization:
[∇ [β ∇ α] V σ ] = ∇ [β ∇ α] [V σ ] + ∇ [β [Γ α]ν σ ]V ν + 1 4 [Γ ν[β σ ][Γ α]λ ν ][V λ ] (25)
Now, from the Ricci identity we have:
[∇ [β ∇ α] V σ ] = [R αβρ σ V ρ ] and then, by (3)
: [∇ [β ∇ α] V σ ] = [R αβρ σ ]V ρ + R αβρ σ [V ρ ], (26)
and thus from a well known identity which follows from (3) as a consequence our definition (5) for the covariant derivative on Σ, i.e. (see [1] ):
[R αβρ σ ] = ∇ β [Γ αρ σ ] -∇ α [Γ βρ σ ] ( 27
)
we have that the commutator of the covariant derivatives of the jump of a generic regularly discontinuous vector obeys the following Ricci-like formula:
∇ [βα] [V σ ] = (1/2)R αβρ σ -(1/4)[Γ ν[β σ ][Γ α]ρ ν ] [V ρ ]. ( 28
)
Not surprisingly, working in a similar way starting from ∇ β ∇ α V σ and antisymmetrizing, we find again:
∇ [βα] V σ = (1/2)R αβρ σ -(1/4)[Γ ν[β σ ][Γ α]ρ ν ] V ρ ; (29)
in fact any given field with support on Σ can be considered, by the help of suitable prolongations, as the jump (or as the mean value of) some regularly discontinuous field. Thus, for any vector V with support on Σ, we can introduce the following mean-value geometry Ricci-like formula on Σ:
(∇ [β ∇ α] )V σ = (R Σ ) αβρ σ V ρ ; ( 30
)
where we have introduced the mean-value geometry curvature (R Σ ), defined by the following mean-value geometry first Gauss-Codazzi identity:
(R Σ ) αβρ σ = R αβρ σ -(1/4)([Γ βν σ ][Γ αρ ν ] -[Γ αν σ ][Γ βρ ν ]) (31)
Notice that, for the sake of simplicity, we have introduced a slight abuse of notation, since in [1] and [16] the same symbol R Σ instead denotes the inner curvature defined with the help of projections. Actually anything goes like in [1] section 4 with the Gauss-Codazzi framework, with the difference that here we don't have to make projections, which would involve product times a discontinuous tangent metric. Moreover here we don't even have to distinguish between the cases of Σ timelike or lightlike. In other words our mean-value differential geometry on a hypersurface is a very simple, in conceptual terms, analogue of the Gauss-Codazzi apparatus. Thus, with the heuristic theory of [1] section 6 (see also [4] for the timelike case) in mind as a prototype, we expect the jump of the Christoffel symbols to play the main role, in place of the secund fundamental form, in the definition of compatibility conditions for very weak solutions of the Einstein equations. Indeed, this happens, as it will be shown in the following.
The metric being dicontinuous on Σ, we are missing the fundamental tool to rise and lower indices, and to construct curvature in the traditional way. This is the reason why sometimes one is tempted to introduce some hybrid metric object on Σ to replace the metric, even in the (C 0 , piecewise C 1 ) case (see e.g. [5] ). It is reassuring to find out that the framework of the preceeding section can be confirmed by such a kind of approach.
It would be desiderable to simply replace g with g on Σ, but it is easy to check that g has not the necessary algebraic requisites; in particular we have g αβ g αρ = δ β ρ . Consider instead:
gαβ = g αβ + i(1/2)[g αβ ], gαβ = g αβ -i(1/2)[g αβ ] ( 32
)
where i is the imaginary unit (i.e. we have i 2 = -1). It is easy to check, with the help of (3), that we have:
gαβ gαρ = δ α ρ + i[g αβ ]g αρ (33)
i.e., in particular: ℜ(g αβ gαρ ) = δ β ρ . For the sake of brevity in the following we will denote A ≈ B the relation ℜ(A) = ℜ(B). Thus the pair gαβ and gαβ is a good candidate replacement for the metric on Σ, for the purposes of rising and lowering indices. Now, similar to (32) let us introduce:
Γαβν = Γ αβν + i(1/2)[Γ αβν ], Γαβ σ = Γ αβ σ -i(1/2)[Γ αβ σ ] (34)
so that we have: Γαβ σ ≈ Γαβν gσν and conversely: Γαβν ≈ Γαβ σ gνσ . Let us now introduce the differential operator ∇ on Σ, which makes use of Γ in place of Γ. As we could expect we have:
∇ρ gαβ ≈ 0, ∇ρ gαβ ≈ 0 (35)
which is the replacement on Σ for the covariant conservation of the metric tensor.
Now let us construct on Σ the complex curvature tensor R in the familiar way, but with Γ in place of the ordinary Christoffel symbols (which are undefined on Σ):
Rαβρ σ = ∂ β Γαρ σ -∂ α Γβρ σ + Γβµ σ Γαρ µ -Γαµ σ Γβρ µ (36)
We rather unespectedly find out that
Rαβρ σ = (R Σ ) αβρ σ + i(1/2)[R αβρ σ ] (37)
i.e. in particular we have:
Rαβρ σ ≈ (R Σ ) αβρ σ
, where R Σ is given by (31). This is just another reason for identifying R Σ as the replacement for the curvature tensor of Σ, which is the first step of our path to the generalized compatibility conditions.
Let us now consider limit f → 0 + of the curvature tensor of the subdomain Ω + ; by (2) we have:
(R αβρ σ ) + = R αβρ σ + (1/2)[R αβρ σ ] ( 38
)
and, by ( 27 ):
(R αβρ σ ) + = R αβρ σ + ∇ [β [Γ α]ρ σ ] ( 39
)
We also have, by (31):
(R αβρ σ ) + = (R Σ ) αβρ σ + ∇ [β [Γ α]ρ σ ] + [Γ ν[β σ ][Γ α]ρ ν ] (40)
We see that R and R Σ only differ by terms proportional to [Γ], and not involving derivatives of it. Thus, in view of neglecting these tems, in the following we will consider R instead of R Σ ; this simply avoids the introduction of the symbol " ∼ = ", with the meaning of equality but for terms not involving derivatives of [Γ] (which here replaces the second fundamental form K) as in [1] section 6. Then for the Ricci tensor R βρ = R αβρ α we have:
(R βρ ) + = R βρ + (1/2)∇ µ δ β µ [Γ ν ρ ν ] -[Γ βρ µ ] ( 41
)
and for the curvature scalar R = R α α :
R + = R + (1/2)∇ µ [Γ ν µν ] -[Γ ν νµ ] (42)
Now, to construct the Einstein tensor G + we have to remember that, since the metric is also discontinuous:
(g αβ ) + = g αβ + (1/2)[g αβ ] ( 43
)
so that we have:
(G βρ ) + = G βρ + (1/2)∇ µ H βρ µ -(1/8)[g βρ ] [Γ ν µν ] -[Γ ν ν µ ] ( 44
)
where we have denoted, for the sake of brevity:
H βρ µ = δ β µ [Γ νρ ν ] -[Γ βρ µ ] -(1/2)g βρ [Γ ν µν ] -[Γ ν ν µ ] ( 45
)
Let us fix a coordinate chart and consider a generic (for the moment) regular prolongation for G, so that its mean value is defined in the whole Ω. Now consider the Riemann 4-volume integral of G + over the domain Ω + ; from the Green theorem we obtain (for the general definition of integral on a hypersurface see [6] p. 6):
Ω + G βρ = Ω\Σ G βρ + (1/2) Σ ℓ + µ H βρ µ -(1/8) Σ ℓ + µ [g βρ ] [Γ ν µν ] -[Γ ν ν µ ]
(46) The analogous formula for Ω -involves -ℓ -as the outgoing normal vector and the metric g - αβ = g αβ -(1/2)[g αβ ], so we have:
Ω - G βρ = Ω\Σ G βρ + (1/2) Σ ℓ - µ H βρ µ + (1/8) Σ ℓ - µ [g βρ ] [Γ ν µν ] -[Γ ν ν µ ]
(47) and consequently we have:
Ω G βρ = Ω\Σ G βρ + Σ ℓ µ H βρ µ (48)
Thus reasons similar to those of the heuristic theory (see [4] and [1] section 6) lead to the reasonable hypothesis that G remain bounded in the neighbourhood of Σ, for any admissible prolongation, so that from the volume with dΩ 2 = dθ 2 + sin 2 θdϕ 2 , across a spherical admissible gravitational discontinuity hypersurface Σ of equation r = ρ(t), with ρ(t) ∈ C 1 . Therefore the form ℓ α = δ α r -ρδ α t is continuous (while ℓ β = g βα ℓ α in general is not). We suppose globally C 0 coordinates, the same form of the metric in both domains Ω + and Ω -, and the identification t + = t -, r + = r -, θ + = θ -, ϕ + = ϕ -on Σ. Leta, b > 0 and let a, b ∈ piecewise-C 0 be regularly discontinuous on Σ and with regularly discontinuous first derivatives. Let us denote by a dot the partial derivative with respect to t, and by a prime that with respect to r. Let moreover condition a -b ρ > 0, i.e. (ℓ • ℓ) > 0, hold on both sides on Σ.
We have:
[g αβ ] = -[a]δ α t δ β t + [b]δ α r δ β r (60)
Now let us define the match as a generalized regularly discontinuous solution by (51), with T = 0, i.e. in the absence of stress-energy concentrated on Σ.
In this case our compatibility conditions reduce to:
ℓ β [Γ µρ µ ] -ℓ µ [Γ βρ µ ] = 0 (61)
which, for a match of metrics of the kind (59), are equivalent to the following system:
ρ[ ḃb -1 ] + [a ′ b -1 ] = 0 ρ[ ḃa -1 ] + [a ′ a -1 ] = 0 ρ[a ′ a -1 ] + [ ȧa -1 ] = 0 ρ[b ′ b -1 ] + [ ḃb -1 ] = 0 [b -1 ] = 0 (62)
i.e. we have [b] = 0 and consequently:
ρ[ ḃ] + [a ′ ] = 0 ρ[ ḃa -1 ] + [a ′ a -1 ] = 0 ρ[a ′ a -1 ] + [ ȧa -1 ] = 0 ρ[b ′ ] + [ ḃ] = 0 (63)
and from (3):
ρ[ ḃ] + [a ′ ] = 0 ( ρḃ + a ′ )[a -1 ] = 0 ( ρ a ′ + ȧ)[a -1 ] + ( ρ[a ′ ] + [ ȧ])a -1 = 0 ρ[b ′ ] + [ ḃ] = 0 (64)
Now if we had both ρ[ ḃ] + [a ′ ] = 0 and ρḃ + a ′ = 0, by (2) we would have ρḃ + a ′ = 0 on both sides of the hypersurface. We discard for the moment this singular situation, and from (64) 2 we conclude that [a -1 ] = 0. Thus in this case our generalized compatibility conditions imply [a] = [b] = 0, i.e. they force the match to be C 0 , piecewise-C 1 .
In [1] we have already studied some examples of C 0 , piecewise-C 1 matchs of metrics of the kind (59) at a hypersurface of constant radius r = r b , with ℓ α = δ α r . Namely, we have considered: external Schwarzschild -internal Schwarzschild; external Schwarzschild -Tolman VI; external Schwarzschild -Tolman V. Such matchs obviously have ℓ α ∈ C 0 ; moreover condition ρ ḃ+a ′ = 0 reduce in this case to a ′ = 0, which is obviously satisfied. In each case we have verified that condition [a] = [b] = 0 imply ∂a = 0 (where ∂ denotes first order discontinuity), which then define the match as a boundary layer [1] (it actually also imply ∂b = 0, as one can verify). Such is a general result, since for a metric of the kind (59) the completely temporal and radial components of the Einstein tensor are independent from the second derivatives of the metric:
G tt = -a(b ′ r + b 2 -b)/r 2 b 2 G rr = -(a ′ r -ab + a)/ar 2 (65)
so that the corresponding vacuum Einstein equations reduce to:
b ′ r + b 2 -b = 0, a ′ r -ab + a = 0. (66)
Now, since in the match of (59) vacuum solutions equations (66) are satisfied on each side of the interface Σ, their jump is in particular null, and from (3) we have:
r b [b ′ ] + (2b -1)[b] = 0 r b [a ′ ] -a[b] -(b + 1)[a] = 0 (67) from which it clearly follows that conditions [a] = [b] = 0 imply [a ′ ] = [b ′ ] = 0, i.e. ∂a = ∂b = 0.
Summarizing, for the match of two piecewise-C 0 regularly discontinuous spherical solutions, in the above hypothesis, generalized compatibility conditions (51) imply [a] = [b] = 0 i.e. they force the match to be C 0 . On the other hand conditions [a] = [b] = 0 imply that Σ is a boundary layer.
Therefore for such spherical matchs generalized compatibility conditions (51) are necessary and sufficient for the match to be a boundary layer.
Let us consider the match of two plane wave metrics of the form
ds 2 = -2dξdη + F (ξ) 2 dx 2 + G(ξ) 2 dy 2 (68)
across a hypersurface Σ of equation ξ = 0. Here ξ and η are the two null coordinates. We suppose continuously matching coordinates and F, G regularly discontinuous, together with their first and second derivatives. The gradient vector of Σ is the continuous characteristic (on each side of Σ) vector ℓ α = δ α ξ . Generalized compatibility conditions (51) in the case T = 0 (i.e. no stressenergy concentrated on the hypersurface) reduce to the following single scalar equation:
[F -1 F ′ + G -1 G ′ ] = 0 (69)
which characterize the generalized gravitational shock wave. Let us now study compatibility of (69) with the Einstein Equations. Einstein vacuum equations also reduce to a single scalar equation:
F -1 F ′′ + G -1 G ′′ = 0 (70)
which we suppose to hold on each side of the hypersurface Σ; thus replacing F + and G + by their expressions in terms of F , [F ], G and [G] according to (2) gives rise to the following couple scalar conditions:
(2F ′′ + [F ′′ ])(2G + [G]) + (2G ′′ + [G ′′ ])(2F + [F ]) = 0 (71) (2F ′′ -[F ′′ ])(2G -[G]) + (2G ′′ -[G ′′ ])(2F -[F ]) = 0 (72)
Equations ( 71 )-(72) are compatible with (69), i.e. the three equations set can be solved algebraically with respect to F , [G] and to any member of the pair (F , G), and the solution is not necessarily trivial. Finally let us notice that, if the additional condition [F ] = [G] = 0 holds, i.e. if the solution is C 0 , condition (69) reduces to
F -1 [F ′ ] + G -1 [G ′ ] = 0 i.e.: F -1 ∂F + G -1 ∂G = 0 (73)
which is the analogous condition for the ordinary shock wave, according to [1] section 10.5.
Let us start trying to match two vacuum solutions of the kind (68) across the timelike (on both sides) hypersurface Σ of equation ξ = ζ. Again we suppose continuously matching coordinates, F, G regularly discontinuous together with their first and second derivatives, and T = 0. This times generalized compatibility conditions include (69) and the following two additional scalar conditions:
[F F ′ ] = 0, [GG ′ ] = 0 (74)
i.e., in terms of F , [F ], G and [G], according to (3):
F [F ′ ] + [F ]F ′ = 0 (75) G[G ′ ] + [G]G ′ = 0 (76)
It is easy to check that the system (75)-( 76 ) is not compatible with (71)-(72), in the sense that the whole system does not admit non-trivial solutions for F , [F ], G and [G].
On the other hand we have proved in section 6 that a wide class of generalized spherical matchs at a hypersurface of constant radius necessarily degenerate to a C 0 match.
Other examples of degeneracy have not been included in the paper for the sake of brevity, but at least it seems to be a hard task to construct a nontrivial generalized match across a timelike (on each side) hypersurface, with no stress-energy content. Such difficulty is certainly not a proof that this is an impossible task, but it makes us wonder whether such a solution should necessarily degenerate to a boundary layer, just like it happens for ordinary C 0 solutions (see e.g. [1] ). This would forbid the existence of generalized solutions which propagate at a speed slower than light. Such would be a desiderable prohibition under certain respect, since one could expect that gravitational interactions in vacuo must necessarily propagate at the speed of light also in a generalized theory.
In general terms, since for generalized solutions the metric is discontinuous, a hypersurface can in principle have different signatures on the different sides; for this reason we cannot simply distinguish between the timelike and the lightlike case, as for usual C 0 solutions. We should rather distinguish between three cases: timelike-timelike, timelike-lightlike (or conversely) and lightlike-lightlike.
In any case it is legitimate to expect that, at least in the timelike-timelike case, similar to the timelike case of (C 0 , piecewise C 1 ) solutions, absence of stress-energy concentrated on Σ should imply the solution to degenerate to a boundary layer [1] .
Unfortunately for generalized solutions we still have no proof that absence of stress energy concentrated on Σ does necessarily imply the degeneracy of the solution to a boundary layer.
Therefore, although the examples considered in this paper seem to suggest that such property could hold true also in the generalized case, for the moment such result is still a conjecture; we thus have to admit that the theory in principle allows propagation of generalized gravitational shock waves at lower speed than the speed of light. We would call such waves "slow generalized gravitational shock waves". It would be reasonable to forbid this situation as unphysical, but for now this can only be done ad hoc, by means of a corresponding additional hypothesis.
Notice that ℓ µ H βρ µ is not necessarily symmetric; from identity:
Γ να ν = (1/2)g -1 ∂ α g (77)
where g denotes the determinant of the contravariant metric, we have:
ℓ µ H [βρ] µ = (1/4)(ℓ β [g -1 ∂ ρ g] -ℓ ρ [g -1 ∂ β g]) (78)
Thus the generalized scheme allows in principle the presence of non symmetric stress-energy on the discontinuity hypersurface. We will display nontrivial examples of non-symmetry in the following section. Notice that the right hand side of (78) is identically null in case g ∈ C 0 and ℓ α ∈ C 0 , since in this case we have [g -1 ∂ β g] = ℓ β g -1 ∂g.
A non-symmetric Einstein tensor is a feature of Einstein-Cartan theory of gravitation (see [19] , see also [3] section 7.2), where it is due to the presence of torsion in the non-symmetric connection used to construct generalized curvature. Thus the generalized theory can be interpreted, at least to some extent, as introducing a torsion equivalent tool on the shell only, even if there are no geometrical objects in our theory which can be directly interpretated as torsion. However, Einstein-Cartan theory also has a spin -angular momentum field equation in addition to the Einstein equations, which here is missing.
In the literature, compatibility conditions for C 0 solutions of boundary layers [20] , and recently of shock waves and thin shells [21] , have been studied also in the framework of Einstein-Cartan theory; actually this can lead to non-symmetric stress-energy on the shell. But in that theory this feature is inherited from the ambient spacetime, which is not here: non-symmetric stress-energy arises on the shell only, in consequence of the theory. This interesting feauture probably is worth investigating.
Now let us consider a more general form of the spherical metric: ds 2 = -a(r, t)dt 2 + b(r, t)dr 2 + c(r, t)dΩ 2 (79)
Let us consider a match of two spherical solutions of the Einstein equations across a timelike (on each side) hypersurface of equation r = ρ(t). Again we suppose ρ(t) ∈ C 1 and therefore ℓ α = δ α r -ρδ α t ∈ C 0 . Let the coordinates be C 0 globally, and let the metric have the same form (79) in both domains Ω + and Ω -, with the identification t + = t -, r + = r -, θ + = θ -, ϕ + = ϕ -on Σ.
Let moreover a, b, c > 0 and let a, b, c ∈ piecewise-C 0 be regularly discontinuous on Σ and with regularly discontinuous first derivatives. Again we denote by a dot the partial derivative with respect to t, and by a prime that with respect to r.
In this case for the left hand side of the generalized compatibility conditions ℓ µ H βρ µ we obtain: The slightly more general case of ȧ = ḃ = ċ = 0, but ρ = 0, displays non-symmetric terms in (80); however it is not difficult to see that the perfect magnetofluid interpretation still holds, provided such additional nonsymmetric terms are interpreted, or neglected. In fact in this case we have:
ρ 0 = χ -1 a -1 [a ′ b -1 ]/2 + χ -1 c -1 (b b -1 /2 -aa -1 + 1)[c ′ b -1 ]/2+ -χ -1 [b -1 ](a ′ a -1 /2 + c ′ c -1 ) -(3/2)χ -1 b -1 [a ′ a -1 /2 + c ′ c -1 ] p = χ -1 c -1 (bb -1 /2 -1)[c ′ b -1 /2] + (1/2)χ -1 b -1 [a ′ a -1 /2 + c ′ c -1 ]+ +χ -1 [b -1 ](a ′ a -1 /2 + c ′ c -1 ) h α = ± b -1 χ -1 ([a ′ a -1 /2 + c ′ c -1 ] -bc -1 [c ′ b -1 /2])δ α
ℓ µ H βρ µ = (-[a ′ b -1 /2] + a c -1 [c ′ b -1 /2])δ β t δ ρ t + +([a ′ a -1 /2 + c ′ c -1 ] -bc -1 [c ′ b -1 /2])δ β r δ ρ r + +(1/2) ρ[a ′ a -1 /2 -b ′ b -1 /2 -c ′ c -1 ](δ β r δ ρ t + δ β t δ ρ r )+ +(1/2) ρ[a ′ a -1 /2 + b ′ b -1 /2 + c ′ c -1 ](δ β r δ ρ t -δ β t δ ρ r )+ + c -1 [c ′ b -1 /2] -[b -1 (a ′ a -1 /2 + c ′ c -1 )] g βρ (86)
Now let us consider, for the sake of brevity, the following quantities:
α = 1 4 ρ2 [ a ′ 2a -b ′ 2b -c ′ c ] 2 b -1 -( a c [ c ′ 2b ] -[ a ′ 2b ]) 2 a -1 a c [ c ′ 2b ] -[ a ′ 2b ] (87)
β = a ′ 2a + c ′ c - b c c ′ 2b + 1 4 ρ2 [ a ′ 2a -b ′ 2b -c ′ c ] 2 a c [ c ′ 2b ] -[ a ′ 2b ] (88)
and let us suppose that inequality α < 0 holds, which is necessary for the physical interpretation. In fact in this case the following vector:
U α = ( a c [ c ′ 2b ] -[ a ′ 2b ])δ α t + 1 2 ρ[ a ′ 2a -b ′ 2b -c ′ c ]δ α r -α( a c [ c ′ 2b ] -[ a ′ 2b ]) (89)
is a unit timelike vector on Σ, in the sense that U α U β g αβ = -1. Rearranging terms, (86) now reads: | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The physical consistency of the match of piecewise-C 0 metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value differential geometry framework on a hypersurface is introduced, and corresponding compatibility conditions are deduced. Examples of generalized boundary layers, gravitational shock waves and thin shells are studied."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Is it possible to define weak solutions of the Einstein equations of class piecewise-C 0 , i.e. to generalize the compatibility conditions which replace the field equations on a singular hypersurface to the case when the metric is regularly discontinuous?\n\nTo reach this goal would probably mean to define the most general class of regularly discontinuous weak solutions of the Einstein equations. It seems that this problem was never studied before in the literature. But, before we proceed, we need to discuss whether we are talking of something mathematically and physically consistent or not.\n\nA fundamental concept of Riemannian geometry is that at any point of a submanifold there are coordinate choices for which the metric reduces to the Minkowski flat metric. Clearly, if this choice is made on both sides of the discontinuity surface, any \"jump\" in the metric disappears. Thus, the metric discontinuity appears as a coordinate dependent concept, which is neither geometrically (nor physically) acceptable in the context of General Relativity.\n\nBut we also have to consider that regularity of the global coordinates plays an important role in our approach, which is that of [1] and of the literature cited therein. In particular, since here the spacetime is only C 0 , we are led to considering (C 0 , piecewise C 1 ) coordinate transformations. If the metric is discontinuous in some globally C 0 chart, it is in general impossible to obtain the vanishing of the metric jump on both sides of a hypersurface with a C 0 coordinate transformation (see section 2). Moeover in the following we are led in a natural way to considering C 1 coordinate transformations; the metric discontinuity is a tensor with respect to such coordinate changes!\n\nIn other words the jump of the metric has a precise mathematical meaning, if we consider it in connection with global regular coordinates.\n\nIn a well consolidated procedure, the assumption of continuity for the metric across a gravitational interface is usually taken for granted; however it follows from the limiting process of the thin sandwich modelization, in consequence of the hypothesis that the external derivatives of the metric are bounded [2] . Yet in this paper we are going to see that, even removing the assumption of continuity, it is still possible to define a generalized inner geometry of the discontinuity hypersurface; one thus can consistently find a corresponding generalized set of compatibility conditions, which obviously reduces to the usual ones when the continuity hypothesis is restored.\n\nYet, which are the physical motivations to move to such generalization? Actually gravitational shock waves and thin shells are usually defined by the presence of singular curvature with a \"delta\" component concentrated on a hypersurface, situation which is well cast within the classic C 0 piecewise-C 1 match of metrics [1, 3] .\n\nWe were originally led to consider solutions of class piecewise-C 0 , as possible generalizations of shock waves and thin shells, by the sake of mathematical completeness, with the idea that phisical interpretation would follow. We actually found a reacher framework than the usual one, with some interesting new features (and even some rather undesiderable ones), which we display in this paper.\n\nThere are two main theories in the literature for solutions of class C 0 piecewise-C 1 , i.e. that in terms of the second fundamental form (heuristic theory, see e.g. [4, 5] ) and that in terms of the curvature tensor-distribution (axiomatic theory, see e.g. [6, 1] ); such are equivalent through appropiate extensions (for a self-contained overview see e.g. [1] ).\n\nThe axiomatic theory appears to be inappropriate to the study of generalized solutions, since the theory of distributions is basically linear. Even if we could in principle replace the discontinuous metric with its associated distribution g D , then it would be impossible to define, for example, replacements for the Christoffel symbols, since this would involve product of distributions, which, as it is generally believed, is impossible to define. In fact it was proved by Schwarz [7] that, under reasonable hypothesis, there can be no definition of commutative and associative operation on distributions which reduce to ordinary multiplication on integrable distributions (say on regular functions); thus in a word it is impossible to define product of distributions.\n\nOr is it? Colombeau [8, 9, 10] developed a theory which apparently contradicts Schwarz's result. He introduced a very broad space of generalized functions, which extends the usual space of distributions, a subspace of which corresponds, in a certain sense (the correspondance is not 1 to 1), to usual distributions. Colombeau's formalism permits multiplication of generalized functions; but the contradiction with the impossibility theorem is only apparent, in fact Schwarz's hypothesis are violated, since the operation does not coincide with ordinary multiplication on regular functions nor with multiplication of a regular function times a distribution (although it does at least for C ∞ functions).\n\nSuch theory, however, does not fit in a natural way in general relativity, since it is impossible to define covariantly invariant geometrical objects; in fact Colombeau's space is not invariant for smooth coordinate transformations, unless they are linear. Such difficulty, however, seems to have been overcome in subsequent adjustments of the theory, with the introduction of a richer mathematical framework [11, 12] , so that the generalized functions current apparatus can be used in general relativity, and indeed it has been applied at least to the calculation of singular curvatures of the spacetimes of Kerr [13] , Reissner-Nordstrom [14] , and so-called cosmic-string spacetime [15] . In such literature Colombeau's theory is adapted to the handling of curvature when the metric has a singularity in the sense of functions, i.e. the ordinary curvature would explode, at a singular event-point or at a singular worldline. There seems to be no particular reason to forbid Colombeau's method also for defining the match of piecewise-C 0 regularly discontinuous metrics at a singular hypersurface; however, as far as the author is aware, no attempt has been made yet to use it in this framework.\n\nThe direct method we will introduce in the following sections, however, is so conceptually simple that we prefer not to experiment with Colombeau's generalized functions, which would instead mean introducing a far more complicated and unfamiliar mathematical apparatus.\n\nIn this paper in fact we propose a new, generalized theory for regularly discontinuous solutions, covering also the match of piecewise-C 0 metrics. Our theory is heuristic, as it is constructed in a way similar to the heuristic theory of C 0 , piecewise-C 1 solutions originated from the studies of Israel, but we completely avoid the traditional or projectional Gauss-Codazzi framework (which either does not include the lightlike case [4, 5] , or needs a special adaptation for it [16, 1] ) and introduce what we called \"mean-value differential geometry\" framework, instead (see section 3). This is conceptually very simple, and permits to construct in a natural way a generalized theory, where the main role (which used to be that of the jump of the secund fundamental form) is here played by the jump of the Christoffel symbols. The theory is an extension of the theory of gravitational discontinuity hypersurfaces we have studied in [1] , to which it reduces when the metric is C 0 . Even if we should restrict to C 0 solution, by adding the traditional assumption of continuity for the metric, our theory would undoubtedly have at least the good qualities of not needing the timelike and the lightlike case to be distinguished (different from usual heuristic theory), and of just requiring C 0 continuity for the coordinates (different from the axiomatic theory). Moreover, it is completely cast in the framework of general coordinates of the ambient (glued) spacetime, with no use of parametric equations of the hypersurface, nor of inner coordinates and holonomic 3-basis, which could be considered a good quality in some applications as well.\n\nPiecewise-C 0 weak solutions of the Einstein equations, as far as the author is aware, have never been considered previously in the literature. They generalize the corresponding C 0 solutions, as examples in this paper show; however there is more. Apparently in fact the theory allows the propagation of free gravitational discontinuity at lower speed than the speed of light (section 8); or rather, we still have no general proof that the absence of stress energy concentrated on Σ should, in the timelike case, necessarily imply the degeneracy of a generalized solution to a boundary layer, although it does at least for a wide class of spherical matchs (see section 6). Moreover, nonsimmetric stress-energy is allowed on the hypersurface (section 9), like e.g. in Einstein-Cartan dynamics. This possible link to classical unification theories is surprising, since in our framework we have nothing similar to Einstein-Cartan torsion. We therefore see a lot of space for future investigation."
},
{
"section_type": "OTHER",
"section_title": "Discontinuous metrics",
"text": "Let us suppose V 4 an oriented differentiable manifold of dimension 4, of class (C 0 , piecewise C 2 ), provided with a strictly hyperbolic metric of signature -+++ and class piecewise-C 0 . Let Ω ⊂ V 4 be an open connected subset with compact closure. Let units be chosen in order to have the speed of light in empty space c ≡ 1. Greek indices run from 0 to 3.\n\nLet Σ ⊂ Ω be a regular hypersuperface of equation f (x) = 0; let Ω + and Ω -denote the subdomains distinguished by the sign of f . We suppose the metric and its first and second partial derivatives to be regularly discontinuous on Σ in all charts of class C 0 (Ω). Let f ∈ C 0 (Ω) ∩ C 2 (Ω\\Σ), and let second and third derivatives of f be regularly discontinuous on Σ. Finally, let ℓ α ≡ ∂ α f denote the gradient of f .\n\nLet the metric be a solution of the ordinary Einstein equations on each of the two domains Ω + and Ω -. In this situation Σ is the interface between two general relativistic spacetimes and it is called a (generalized) gravitational discontinuity hypersurface.\n\nIn the following we will develope a theory to justify the introduction of suitable generalized compatibility conditions to replace the Einstein equations on Σ (section 5); if such conditions are satisfied the match across the generalized gravitational hypersurface Σ will be called a generalized regularly discontinuous solution of the Einstein equations. Now let us briefly recall some basics notions on regularly discontinuous fields, which we will use as tools. In any case, for notation and terminology we refer to [1] .\n\nA field ϕ is said to be regularly discontinuous on Σ if its restrictions to the two subdomains Ω + and Ω -both have a finite limit for f -→ 0; we denote such limits by ϕ + and ϕ -, respectively.\n\nIn this case the jump [ϕ] across Σ and its arithmetic mean value ϕ are well defined on the hypersurface:\n\n[ϕ] = ϕ + -ϕ - ϕ = (1/2)(ϕ + + ϕ -) (1)\n\nIf ϕ is continuous across Σ, we obviously have: [ϕ] = 0, ϕ = ϕ. We also have the converse formulae:\n\nϕ + = ϕ + (1/2)[ϕ] ϕ -= ϕ -(1/2)[ϕ]. (2)\n\nAs for the product of two functions ϕ and ψ, we have:\n\n[ϕψ] = [ϕ]ψ + ϕ[ψ] ϕψ = ϕψ + (1/4)[ϕ][ψ] (3)\n\nIf a field ϕ is regularly discontinuous on Σ, its jump [ϕ] is sometimes called its discontinuity of order 0. The jump of a regularly discontinuous function has support on Σ, but in general, the partial derivative of the jump is well defined as the jump of the derivative of the function (see [17, 18] ). In particular, the derivative of the jump of a continuous field is not null, unless the field is also C 1 .\n\nSimilarly, we define the partial derivative of the mean value as the mean value of the partial derivative. We can also use regular prolongations to extend, in a sense, the definition of ϕ and [ϕ] to the whole domain Ω. Thus they can be regarded as regular and derivable fields in Ω, but their values (and those of their derivatives) are well defined only on Σ, while in Ω\\Σ they depend on the choice of the prolongation. For details on the method of regular prolongations see e.g. [17, 18] .\n\nWe moreover define the covariant derivative of a field with support on Σ by means of the mean value Γ βρ σ of the Christoffel symbols. For the jump of a regularly discontinuous vector, for example, with this definition one has that the jump of the covariant derivative is different than the covariant derivative of the jump. Thus, by definition, we have:\n\n∇ α [V β ] = ∂ α [V β ] + Γ ασ β [V σ ] (4)\n\nand in consequence of (3):\n\n∇ α [V β ] = [∇ α V β ] -[Γ ασ β ]V σ , (5)\n\nand similarly for the jump of any regularly discontinuous tensor. Since the spacetime is only C 0 , we are led to considering (C 0 , piecewise C 1 ) coordinate transformations, with regularly discontinuous first derivatives; the metric discontinuity [g αβ ] is not a tensor with respect to such changes of coordinates. In fact we have:\n\n[g αβ ] = [g α ′ β ′ ] dx α ′ dx α • dx β ′ dx β + q αβ ′ dx β ′ dx β + q α ′ β dx α ′ dx α (6)\n\nwhere:\n\nq α ′ β = 1 8 [g α ′ β ′ ] dx β ′ dx β + ḡα ′ β ′ dx β ′ dx β (7)\n\nWe therefore can simulate all (C 0 , piecewise C 1 ) coordinate changes by combining C 1 changes with metric gauge changes:\n\n[g αβ ] ←→ [g αβ ] + q αβ ′ dx β ′ dx β + q α ′ β dx α ′ dx α (8)\n\nwhich generalize usual gravitational gauge changes of the theory of (C 0 , piecewise C 1 ) solutions [1] .\n\nIs it always possible to make [g αβ ] vanish with an appropriate C 0 transformation? Clearly the answer is negative. In fact it suffices to consider the case when [g αβ ] and ḡαβ are both definite positive in a given chart to see that the equation obtained from ( 6 ) by replacing the first hand side with 0 has no solution for [∂x α ′ /∂x α ] and ∂x α ′ /∂x α . Thus the set of effective generalized gravitational discontinuity hypersurfaces is non empty.\n\nMoreover it will occur in many applications to have ℓ α ∈ C 0 . Therefore it will be often desiderable to work in the framework of (C 1 , piecewise C 2 ) coordinate transformations, which preserve such condition. The metric discontinuity is a tensor with respect to such changes of coordinates, but the jump of the Christoffel symbols, which appear to play a main role in the following, is not; we have in fact:\n\n[Γ αβ σ ] = [Γ α ′ β ′ σ ′ ] ∂x α ′ ∂x α ∂x β ′ ∂x β ∂x σ ∂x σ ′ + ∂ 2 x σ ′ ∂x α ∂x β ∂x σ ∂x σ ′ (9)\n\nIf the coordinates are C 0 and so is the form ℓ α we can write:\n\n∂ 2 x σ ′ ∂x α ∂x β = ℓ α ℓ β ∂ 2 x σ ′ ( 10\n\n)\n\nwhere ∂ 2 denotes the weak discontinuity of order 2 (see e.g. [17, 18] ). Thus on Σ we can generate all (C 1 , piecewise C 2 ) transformations for [Γ] combining C 2 transformations (with respect to which Γ is a tensor) and Christoffel gauges transformations, i.e. of the kind:\n\n[Γ αβ σ ] ↔ [Γ αβ σ ] + ℓ α ℓ β Q σ (11)\n\nwith some analogy with the case of C 0 metrics (where the main role is played by the first order metric discontinuity ∂g, see [1] section 3).\n\nIn any case neither the mean value of the metric g or its jump [g] now have null covariant derivatives. Consider in fact the identity ∇ α g βρ = 0 in the domain Ω + ; from the limit f -→ 0 + , on Σ we have:\n\n∂ α g + βρ -(Γ αβ ν ) + g + νρ -(Γ αρ ν ) + g + βν = 0 (12)\n\nHere, with obvious meaning of the symbols, we denote: g + βρ = (g βρ ) + , g βρ = g βρ , etc. Consequently on Σ, from (2) 1 we have:\n\n∂ α g βρ + (1/2)∂ α [g βρ ] -Γ αβ ν g νρ -Γ αρ ν g νβ + -(1/2)([Γ αβ ν ]g νρ + Γ αβ ν [g ρν ] + [Γ αρ ν ]g νβ + Γ αρ ν [g βν ])+ -(1/4)([Γ αβ ν ][g νρ ] + [Γ αρ ν ][g βν ]) = 0 (13)\n\nSimilarly, from the limit f -→ 0 -and from (2) 2 we also have on Σ:\n\n∂ α g βρ -(1/2)∂ α [g βρ ] -Γ αβ ν g νρ -Γ αρ ν g νβ + +(1/2)([Γ αβ ν ]g νρ + Γ αβ ν [g ρν ] + [Γ αρ ν ]g νβ + Γ αρ ν [g βν ])+ -(1/4)([Γ αβ ν ][g νρ ] + [Γ αρ ν ][g βν ]) = 0 (14)\n\nFrom the sum of expressions ( 13 ) and ( 14 ) we thus have:\n\n∇ α g βρ = (1/4)([Γ αβ ν ][g ν ρ ] + [Γ αρ ν ][g βν ]) (15)\n\nand, from difference:\n\n∂ α [g βρ ] = [Γ αβρ ] + [Γ αρβ ] (16)\n\nFrom ( 16 ), (3), and from the definition of covariant derivative over Σ, we then have:\n\n∇ α [g βρ ] = [Γ αβ ν ]g νρ + [Γ αρ ν ]g βν (17)\n\nAs for the jump and the mean value of the Christoffel symbols we have, from\n\n:\n\nΓ αβ ν = (1/2){g νσ (∂ α g βσ + ∂ β g σα -∂ σ g αβ )+ +(1/4)[g νσ ](∂ α [g βσ ] + ∂ β [g σα ] -∂ σ [g αβ ])} (18)\n\nand\n\n[Γ αβ ν ] = (1/2){g νσ (∂ α [g βσ ] + ∂ β [g σα ] -∂ σ [g αβ ])+ +[g νσ ](∂ α g βσ + ∂ β g σα -∂ σ g αβ ) (19)\n\nor, from ( 15 ) and ( 17 ):\n\n[Γ αβ ν ]g νρ = (1/2)(∇ α [g βρ ] + ∇ β [g ρα ] -∇ ρ [g αβ ]) [Γ αβ ν ][g νρ ] = 2(∇ α g βρ + ∇ β g ρα -∇ ρ g αβ ) ( 20\n\n)\n\n3 Mean-value geometry on a hypersurface\n\nLet us consider a 4-vector V α , regularly discontinuous on Σ, the jump and the mean value of which will work as a prototype of vectors with Σ as support.\n\nWe have, by definition:\n\n[∇ β ∇ α V σ ] = ∇ β [∇ α V σ ] -[Γ βα ν ]∇ ν V σ + [∇ β ν σ ]∇ α V ν (21)\n\nwhere\n\n[∇ α V σ ] = ∇ α [V σ ] + [Γ αν σ ]V\n\nν and where, again by definition, we have:\n\n∇ ν V σ = 1 2 {∂ ν (V + ) σ + (Γ + ) νλ σ (V + ) λ + ∂ ν (V -) σ + (Γ -) νλ σ (V -) λ } (22)\n\nThus, from (2) we have:\n\n∇ ν V σ = ∇ ν V σ + (1/4)[Γ νλ σ ][V λ ], (23)\n\nwhich, incidentally, is the same result we could get from the formal application of (3), wich actually can be applied to covariant derivatives, provided one interpretes ∇ = ∇. We therefore have:\n\n[∇ α ∇ β V σ ] = ∇ α ∇ β [V σ ] + ∇ β [Γ αν σ ]V ν + [Γ αν σ ]∇ β V ν + -[Γ βα ν ]∇ ν V σ -(1/4)[Γ βα ν][Γ νλ σ ][V λ ]+ +[Γ βν σ ]∇ α V ν + (1/4)[Γ βν σ ][Γ αλ ν ][V λ ] (24)\n\nand thus, by antisymmetrization:\n\n[∇ [β ∇ α] V σ ] = ∇ [β ∇ α] [V σ ] + ∇ [β [Γ α]ν σ ]V ν + 1 4 [Γ ν[β σ ][Γ α]λ ν ][V λ ] (25)\n\nNow, from the Ricci identity we have:\n\n[∇ [β ∇ α] V σ ] = [R αβρ σ V ρ ] and then, by (3)\n\n: [∇ [β ∇ α] V σ ] = [R αβρ σ ]V ρ + R αβρ σ [V ρ ], (26)\n\nand thus from a well known identity which follows from (3) as a consequence our definition (5) for the covariant derivative on Σ, i.e. (see [1] ):\n\n[R αβρ σ ] = ∇ β [Γ αρ σ ] -∇ α [Γ βρ σ ] ( 27\n\n)\n\nwe have that the commutator of the covariant derivatives of the jump of a generic regularly discontinuous vector obeys the following Ricci-like formula:\n\n∇ [βα] [V σ ] = (1/2)R αβρ σ -(1/4)[Γ ν[β σ ][Γ α]ρ ν ] [V ρ ]. ( 28\n\n)\n\nNot surprisingly, working in a similar way starting from ∇ β ∇ α V σ and antisymmetrizing, we find again:\n\n∇ [βα] V σ = (1/2)R αβρ σ -(1/4)[Γ ν[β σ ][Γ α]ρ ν ] V ρ ; (29)\n\nin fact any given field with support on Σ can be considered, by the help of suitable prolongations, as the jump (or as the mean value of) some regularly discontinuous field. Thus, for any vector V with support on Σ, we can introduce the following mean-value geometry Ricci-like formula on Σ:\n\n(∇ [β ∇ α] )V σ = (R Σ ) αβρ σ V ρ ; ( 30\n\n)\n\nwhere we have introduced the mean-value geometry curvature (R Σ ), defined by the following mean-value geometry first Gauss-Codazzi identity:\n\n(R Σ ) αβρ σ = R αβρ σ -(1/4)([Γ βν σ ][Γ αρ ν ] -[Γ αν σ ][Γ βρ ν ]) (31)\n\nNotice that, for the sake of simplicity, we have introduced a slight abuse of notation, since in [1] and [16] the same symbol R Σ instead denotes the inner curvature defined with the help of projections. Actually anything goes like in [1] section 4 with the Gauss-Codazzi framework, with the difference that here we don't have to make projections, which would involve product times a discontinuous tangent metric. Moreover here we don't even have to distinguish between the cases of Σ timelike or lightlike. In other words our mean-value differential geometry on a hypersurface is a very simple, in conceptual terms, analogue of the Gauss-Codazzi apparatus. Thus, with the heuristic theory of [1] section 6 (see also [4] for the timelike case) in mind as a prototype, we expect the jump of the Christoffel symbols to play the main role, in place of the secund fundamental form, in the definition of compatibility conditions for very weak solutions of the Einstein equations. Indeed, this happens, as it will be shown in the following."
},
{
"section_type": "OTHER",
"section_title": "Complex mean-value formalism",
"text": "The metric being dicontinuous on Σ, we are missing the fundamental tool to rise and lower indices, and to construct curvature in the traditional way. This is the reason why sometimes one is tempted to introduce some hybrid metric object on Σ to replace the metric, even in the (C 0 , piecewise C 1 ) case (see e.g. [5] ). It is reassuring to find out that the framework of the preceeding section can be confirmed by such a kind of approach.\n\nIt would be desiderable to simply replace g with g on Σ, but it is easy to check that g has not the necessary algebraic requisites; in particular we have g αβ g αρ = δ β ρ . Consider instead:\n\ngαβ = g αβ + i(1/2)[g αβ ], gαβ = g αβ -i(1/2)[g αβ ] ( 32\n\n)\n\nwhere i is the imaginary unit (i.e. we have i 2 = -1). It is easy to check, with the help of (3), that we have:\n\ngαβ gαρ = δ α ρ + i[g αβ ]g αρ (33)\n\ni.e., in particular: ℜ(g αβ gαρ ) = δ β ρ . For the sake of brevity in the following we will denote A ≈ B the relation ℜ(A) = ℜ(B). Thus the pair gαβ and gαβ is a good candidate replacement for the metric on Σ, for the purposes of rising and lowering indices. Now, similar to (32) let us introduce:\n\nΓαβν = Γ αβν + i(1/2)[Γ αβν ], Γαβ σ = Γ αβ σ -i(1/2)[Γ αβ σ ] (34)\n\nso that we have: Γαβ σ ≈ Γαβν gσν and conversely: Γαβν ≈ Γαβ σ gνσ . Let us now introduce the differential operator ∇ on Σ, which makes use of Γ in place of Γ. As we could expect we have:\n\n∇ρ gαβ ≈ 0, ∇ρ gαβ ≈ 0 (35)\n\nwhich is the replacement on Σ for the covariant conservation of the metric tensor.\n\nNow let us construct on Σ the complex curvature tensor R in the familiar way, but with Γ in place of the ordinary Christoffel symbols (which are undefined on Σ):\n\nRαβρ σ = ∂ β Γαρ σ -∂ α Γβρ σ + Γβµ σ Γαρ µ -Γαµ σ Γβρ µ (36)\n\nWe rather unespectedly find out that\n\nRαβρ σ = (R Σ ) αβρ σ + i(1/2)[R αβρ σ ] (37)\n\ni.e. in particular we have:\n\nRαβρ σ ≈ (R Σ ) αβρ σ\n\n, where R Σ is given by (31). This is just another reason for identifying R Σ as the replacement for the curvature tensor of Σ, which is the first step of our path to the generalized compatibility conditions."
},
{
"section_type": "OTHER",
"section_title": "Generalized compatibility conditions",
"text": "Let us now consider limit f → 0 + of the curvature tensor of the subdomain Ω + ; by (2) we have:\n\n(R αβρ σ ) + = R αβρ σ + (1/2)[R αβρ σ ] ( 38\n\n)\n\nand, by ( 27 ):\n\n(R αβρ σ ) + = R αβρ σ + ∇ [β [Γ α]ρ σ ] ( 39\n\n)\n\nWe also have, by (31):\n\n(R αβρ σ ) + = (R Σ ) αβρ σ + ∇ [β [Γ α]ρ σ ] + [Γ ν[β σ ][Γ α]ρ ν ] (40)\n\nWe see that R and R Σ only differ by terms proportional to [Γ], and not involving derivatives of it. Thus, in view of neglecting these tems, in the following we will consider R instead of R Σ ; this simply avoids the introduction of the symbol \" ∼ = \", with the meaning of equality but for terms not involving derivatives of [Γ] (which here replaces the second fundamental form K) as in [1] section 6. Then for the Ricci tensor R βρ = R αβρ α we have:\n\n(R βρ ) + = R βρ + (1/2)∇ µ δ β µ [Γ ν ρ ν ] -[Γ βρ µ ] ( 41\n\n)\n\nand for the curvature scalar R = R α α :\n\nR + = R + (1/2)∇ µ [Γ ν µν ] -[Γ ν νµ ] (42)\n\nNow, to construct the Einstein tensor G + we have to remember that, since the metric is also discontinuous:\n\n(g αβ ) + = g αβ + (1/2)[g αβ ] ( 43\n\n)\n\nso that we have:\n\n(G βρ ) + = G βρ + (1/2)∇ µ H βρ µ -(1/8)[g βρ ] [Γ ν µν ] -[Γ ν ν µ ] ( 44\n\n)\n\nwhere we have denoted, for the sake of brevity:\n\nH βρ µ = δ β µ [Γ νρ ν ] -[Γ βρ µ ] -(1/2)g βρ [Γ ν µν ] -[Γ ν ν µ ] ( 45\n\n)\n\nLet us fix a coordinate chart and consider a generic (for the moment) regular prolongation for G, so that its mean value is defined in the whole Ω. Now consider the Riemann 4-volume integral of G + over the domain Ω + ; from the Green theorem we obtain (for the general definition of integral on a hypersurface see [6] p. 6):\n\nΩ + G βρ = Ω\\Σ G βρ + (1/2) Σ ℓ + µ H βρ µ -(1/8) Σ ℓ + µ [g βρ ] [Γ ν µν ] -[Γ ν ν µ ]\n\n(46) The analogous formula for Ω -involves -ℓ -as the outgoing normal vector and the metric g - αβ = g αβ -(1/2)[g αβ ], so we have:\n\nΩ - G βρ = Ω\\Σ G βρ + (1/2) Σ ℓ - µ H βρ µ + (1/8) Σ ℓ - µ [g βρ ] [Γ ν µν ] -[Γ ν ν µ ]\n\n(47) and consequently we have:\n\nΩ G βρ = Ω\\Σ G βρ + Σ ℓ µ H βρ µ (48)\n\nThus reasons similar to those of the heuristic theory (see [4] and [1] section 6) lead to the reasonable hypothesis that G remain bounded in the neighbourhood of Σ, for any admissible prolongation, so that from the volume with dΩ 2 = dθ 2 + sin 2 θdϕ 2 , across a spherical admissible gravitational discontinuity hypersurface Σ of equation r = ρ(t), with ρ(t) ∈ C 1 . Therefore the form ℓ α = δ α r -ρδ α t is continuous (while ℓ β = g βα ℓ α in general is not). We suppose globally C 0 coordinates, the same form of the metric in both domains Ω + and Ω -, and the identification t + = t -, r + = r -, θ + = θ -, ϕ + = ϕ -on Σ. Leta, b > 0 and let a, b ∈ piecewise-C 0 be regularly discontinuous on Σ and with regularly discontinuous first derivatives. Let us denote by a dot the partial derivative with respect to t, and by a prime that with respect to r. Let moreover condition a -b ρ > 0, i.e. (ℓ • ℓ) > 0, hold on both sides on Σ.\n\nWe have:\n\n[g αβ ] = -[a]δ α t δ β t + [b]δ α r δ β r (60)\n\nNow let us define the match as a generalized regularly discontinuous solution by (51), with T = 0, i.e. in the absence of stress-energy concentrated on Σ.\n\nIn this case our compatibility conditions reduce to:\n\nℓ β [Γ µρ µ ] -ℓ µ [Γ βρ µ ] = 0 (61)\n\nwhich, for a match of metrics of the kind (59), are equivalent to the following system:\n\nρ[ ḃb -1 ] + [a ′ b -1 ] = 0 ρ[ ḃa -1 ] + [a ′ a -1 ] = 0 ρ[a ′ a -1 ] + [ ȧa -1 ] = 0 ρ[b ′ b -1 ] + [ ḃb -1 ] = 0 [b -1 ] = 0 (62)\n\ni.e. we have [b] = 0 and consequently:\n\nρ[ ḃ] + [a ′ ] = 0 ρ[ ḃa -1 ] + [a ′ a -1 ] = 0 ρ[a ′ a -1 ] + [ ȧa -1 ] = 0 ρ[b ′ ] + [ ḃ] = 0 (63)\n\nand from (3):\n\nρ[ ḃ] + [a ′ ] = 0 ( ρḃ + a ′ )[a -1 ] = 0 ( ρ a ′ + ȧ)[a -1 ] + ( ρ[a ′ ] + [ ȧ])a -1 = 0 ρ[b ′ ] + [ ḃ] = 0 (64)\n\nNow if we had both ρ[ ḃ] + [a ′ ] = 0 and ρḃ + a ′ = 0, by (2) we would have ρḃ + a ′ = 0 on both sides of the hypersurface. We discard for the moment this singular situation, and from (64) 2 we conclude that [a -1 ] = 0. Thus in this case our generalized compatibility conditions imply [a] = [b] = 0, i.e. they force the match to be C 0 , piecewise-C 1 .\n\nIn [1] we have already studied some examples of C 0 , piecewise-C 1 matchs of metrics of the kind (59) at a hypersurface of constant radius r = r b , with ℓ α = δ α r . Namely, we have considered: external Schwarzschild -internal Schwarzschild; external Schwarzschild -Tolman VI; external Schwarzschild -Tolman V. Such matchs obviously have ℓ α ∈ C 0 ; moreover condition ρ ḃ+a ′ = 0 reduce in this case to a ′ = 0, which is obviously satisfied. In each case we have verified that condition [a] = [b] = 0 imply ∂a = 0 (where ∂ denotes first order discontinuity), which then define the match as a boundary layer [1] (it actually also imply ∂b = 0, as one can verify). Such is a general result, since for a metric of the kind (59) the completely temporal and radial components of the Einstein tensor are independent from the second derivatives of the metric:\n\nG tt = -a(b ′ r + b 2 -b)/r 2 b 2 G rr = -(a ′ r -ab + a)/ar 2 (65)\n\nso that the corresponding vacuum Einstein equations reduce to:\n\nb ′ r + b 2 -b = 0, a ′ r -ab + a = 0. (66)\n\nNow, since in the match of (59) vacuum solutions equations (66) are satisfied on each side of the interface Σ, their jump is in particular null, and from (3) we have:\n\nr b [b ′ ] + (2b -1)[b] = 0 r b [a ′ ] -a[b] -(b + 1)[a] = 0 (67) from which it clearly follows that conditions [a] = [b] = 0 imply [a ′ ] = [b ′ ] = 0, i.e. ∂a = ∂b = 0.\n\nSummarizing, for the match of two piecewise-C 0 regularly discontinuous spherical solutions, in the above hypothesis, generalized compatibility conditions (51) imply [a] = [b] = 0 i.e. they force the match to be C 0 . On the other hand conditions [a] = [b] = 0 imply that Σ is a boundary layer.\n\nTherefore for such spherical matchs generalized compatibility conditions (51) are necessary and sufficient for the match to be a boundary layer."
},
{
"section_type": "OTHER",
"section_title": "Generalized gravitational shock waves",
"text": "Let us consider the match of two plane wave metrics of the form\n\nds 2 = -2dξdη + F (ξ) 2 dx 2 + G(ξ) 2 dy 2 (68)\n\nacross a hypersurface Σ of equation ξ = 0. Here ξ and η are the two null coordinates. We suppose continuously matching coordinates and F, G regularly discontinuous, together with their first and second derivatives. The gradient vector of Σ is the continuous characteristic (on each side of Σ) vector ℓ α = δ α ξ . Generalized compatibility conditions (51) in the case T = 0 (i.e. no stressenergy concentrated on the hypersurface) reduce to the following single scalar equation:\n\n[F -1 F ′ + G -1 G ′ ] = 0 (69)\n\nwhich characterize the generalized gravitational shock wave. Let us now study compatibility of (69) with the Einstein Equations. Einstein vacuum equations also reduce to a single scalar equation:\n\nF -1 F ′′ + G -1 G ′′ = 0 (70)\n\nwhich we suppose to hold on each side of the hypersurface Σ; thus replacing F + and G + by their expressions in terms of F , [F ], G and [G] according to (2) gives rise to the following couple scalar conditions:\n\n(2F ′′ + [F ′′ ])(2G + [G]) + (2G ′′ + [G ′′ ])(2F + [F ]) = 0 (71) (2F ′′ -[F ′′ ])(2G -[G]) + (2G ′′ -[G ′′ ])(2F -[F ]) = 0 (72)\n\nEquations ( 71 )-(72) are compatible with (69), i.e. the three equations set can be solved algebraically with respect to F , [G] and to any member of the pair (F , G), and the solution is not necessarily trivial. Finally let us notice that, if the additional condition [F ] = [G] = 0 holds, i.e. if the solution is C 0 , condition (69) reduces to\n\nF -1 [F ′ ] + G -1 [G ′ ] = 0 i.e.: F -1 ∂F + G -1 ∂G = 0 (73)\n\nwhich is the analogous condition for the ordinary shock wave, according to [1] section 10.5."
},
{
"section_type": "OTHER",
"section_title": "Slow generalized gravitational waves",
"text": "Let us start trying to match two vacuum solutions of the kind (68) across the timelike (on both sides) hypersurface Σ of equation ξ = ζ. Again we suppose continuously matching coordinates, F, G regularly discontinuous together with their first and second derivatives, and T = 0. This times generalized compatibility conditions include (69) and the following two additional scalar conditions:\n\n[F F ′ ] = 0, [GG ′ ] = 0 (74)\n\ni.e., in terms of F , [F ], G and [G], according to (3):\n\nF [F ′ ] + [F ]F ′ = 0 (75) G[G ′ ] + [G]G ′ = 0 (76)\n\nIt is easy to check that the system (75)-( 76 ) is not compatible with (71)-(72), in the sense that the whole system does not admit non-trivial solutions for F , [F ], G and [G].\n\nOn the other hand we have proved in section 6 that a wide class of generalized spherical matchs at a hypersurface of constant radius necessarily degenerate to a C 0 match.\n\nOther examples of degeneracy have not been included in the paper for the sake of brevity, but at least it seems to be a hard task to construct a nontrivial generalized match across a timelike (on each side) hypersurface, with no stress-energy content. Such difficulty is certainly not a proof that this is an impossible task, but it makes us wonder whether such a solution should necessarily degenerate to a boundary layer, just like it happens for ordinary C 0 solutions (see e.g. [1] ). This would forbid the existence of generalized solutions which propagate at a speed slower than light. Such would be a desiderable prohibition under certain respect, since one could expect that gravitational interactions in vacuo must necessarily propagate at the speed of light also in a generalized theory.\n\nIn general terms, since for generalized solutions the metric is discontinuous, a hypersurface can in principle have different signatures on the different sides; for this reason we cannot simply distinguish between the timelike and the lightlike case, as for usual C 0 solutions. We should rather distinguish between three cases: timelike-timelike, timelike-lightlike (or conversely) and lightlike-lightlike.\n\nIn any case it is legitimate to expect that, at least in the timelike-timelike case, similar to the timelike case of (C 0 , piecewise C 1 ) solutions, absence of stress-energy concentrated on Σ should imply the solution to degenerate to a boundary layer [1] .\n\nUnfortunately for generalized solutions we still have no proof that absence of stress energy concentrated on Σ does necessarily imply the degeneracy of the solution to a boundary layer.\n\nTherefore, although the examples considered in this paper seem to suggest that such property could hold true also in the generalized case, for the moment such result is still a conjecture; we thus have to admit that the theory in principle allows propagation of generalized gravitational shock waves at lower speed than the speed of light. We would call such waves \"slow generalized gravitational shock waves\". It would be reasonable to forbid this situation as unphysical, but for now this can only be done ad hoc, by means of a corresponding additional hypothesis."
},
{
"section_type": "OTHER",
"section_title": "Non-symmetric stress-energy",
"text": "Notice that ℓ µ H βρ µ is not necessarily symmetric; from identity:\n\nΓ να ν = (1/2)g -1 ∂ α g (77)\n\nwhere g denotes the determinant of the contravariant metric, we have:\n\nℓ µ H [βρ] µ = (1/4)(ℓ β [g -1 ∂ ρ g] -ℓ ρ [g -1 ∂ β g]) (78)\n\nThus the generalized scheme allows in principle the presence of non symmetric stress-energy on the discontinuity hypersurface. We will display nontrivial examples of non-symmetry in the following section. Notice that the right hand side of (78) is identically null in case g ∈ C 0 and ℓ α ∈ C 0 , since in this case we have [g -1 ∂ β g] = ℓ β g -1 ∂g.\n\nA non-symmetric Einstein tensor is a feature of Einstein-Cartan theory of gravitation (see [19] , see also [3] section 7.2), where it is due to the presence of torsion in the non-symmetric connection used to construct generalized curvature. Thus the generalized theory can be interpreted, at least to some extent, as introducing a torsion equivalent tool on the shell only, even if there are no geometrical objects in our theory which can be directly interpretated as torsion. However, Einstein-Cartan theory also has a spin -angular momentum field equation in addition to the Einstein equations, which here is missing.\n\nIn the literature, compatibility conditions for C 0 solutions of boundary layers [20] , and recently of shock waves and thin shells [21] , have been studied also in the framework of Einstein-Cartan theory; actually this can lead to non-symmetric stress-energy on the shell. But in that theory this feature is inherited from the ambient spacetime, which is not here: non-symmetric stress-energy arises on the shell only, in consequence of the theory. This interesting feauture probably is worth investigating."
},
{
"section_type": "OTHER",
"section_title": "Generalized thin shells",
"text": "Now let us consider a more general form of the spherical metric: ds 2 = -a(r, t)dt 2 + b(r, t)dr 2 + c(r, t)dΩ 2 (79)\n\nLet us consider a match of two spherical solutions of the Einstein equations across a timelike (on each side) hypersurface of equation r = ρ(t). Again we suppose ρ(t) ∈ C 1 and therefore ℓ α = δ α r -ρδ α t ∈ C 0 . Let the coordinates be C 0 globally, and let the metric have the same form (79) in both domains Ω + and Ω -, with the identification t + = t -, r + = r -, θ + = θ -, ϕ + = ϕ -on Σ.\n\nLet moreover a, b, c > 0 and let a, b, c ∈ piecewise-C 0 be regularly discontinuous on Σ and with regularly discontinuous first derivatives. Again we denote by a dot the partial derivative with respect to t, and by a prime that with respect to r.\n\nIn this case for the left hand side of the generalized compatibility conditions ℓ µ H βρ µ we obtain: The slightly more general case of ȧ = ḃ = ċ = 0, but ρ = 0, displays non-symmetric terms in (80); however it is not difficult to see that the perfect magnetofluid interpretation still holds, provided such additional nonsymmetric terms are interpreted, or neglected. In fact in this case we have:\n\nρ 0 = χ -1 a -1 [a ′ b -1 ]/2 + χ -1 c -1 (b b -1 /2 -aa -1 + 1)[c ′ b -1 ]/2+ -χ -1 [b -1 ](a ′ a -1 /2 + c ′ c -1 ) -(3/2)χ -1 b -1 [a ′ a -1 /2 + c ′ c -1 ] p = χ -1 c -1 (bb -1 /2 -1)[c ′ b -1 /2] + (1/2)χ -1 b -1 [a ′ a -1 /2 + c ′ c -1 ]+ +χ -1 [b -1 ](a ′ a -1 /2 + c ′ c -1 ) h α = ± b -1 χ -1 ([a ′ a -1 /2 + c ′ c -1 ] -bc -1 [c ′ b -1 /2])δ α\n\nℓ µ H βρ µ = (-[a ′ b -1 /2] + a c -1 [c ′ b -1 /2])δ β t δ ρ t + +([a ′ a -1 /2 + c ′ c -1 ] -bc -1 [c ′ b -1 /2])δ β r δ ρ r + +(1/2) ρ[a ′ a -1 /2 -b ′ b -1 /2 -c ′ c -1 ](δ β r δ ρ t + δ β t δ ρ r )+ +(1/2) ρ[a ′ a -1 /2 + b ′ b -1 /2 + c ′ c -1 ](δ β r δ ρ t -δ β t δ ρ r )+ + c -1 [c ′ b -1 /2] -[b -1 (a ′ a -1 /2 + c ′ c -1 )] g βρ (86)\n\nNow let us consider, for the sake of brevity, the following quantities:\n\nα = 1 4 ρ2 [ a ′ 2a -b ′ 2b -c ′ c ] 2 b -1 -( a c [ c ′ 2b ] -[ a ′ 2b ]) 2 a -1 a c [ c ′ 2b ] -[ a ′ 2b ] (87)\n\nβ = a ′ 2a + c ′ c - b c c ′ 2b + 1 4 ρ2 [ a ′ 2a -b ′ 2b -c ′ c ] 2 a c [ c ′ 2b ] -[ a ′ 2b ] (88)\n\nand let us suppose that inequality α < 0 holds, which is necessary for the physical interpretation. In fact in this case the following vector:\n\nU α = ( a c [ c ′ 2b ] -[ a ′ 2b ])δ α t + 1 2 ρ[ a ′ 2a -b ′ 2b -c ′ c ]δ α r -α( a c [ c ′ 2b ] -[ a ′ 2b ]) (89)\n\nis a unit timelike vector on Σ, in the sense that U α U β g αβ = -1. Rearranging terms, (86) now reads:"
}
] | |
arxiv:0704.0116 | 0704.0116 | 1 | 10.1103/PhysRevD.75.127902 | 9f85542fe308a852df10252f186431c073464294ec78899d772556fdbc3036a9 | Stringy Jacobi fields in Morse theory | We consider the variation of the surface spanned by closed strings in a
spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic
surface equation, the geodesic surface deviation equation which yields a Jacobi
field, and we define the index form of a geodesic surface as in the case of
point particles to discuss conjugate strings on the geodesic surface. | [
"Yong Seung Cho",
"Soon-Tae Hong"
] | [
"math-ph",
"hep-th",
"math.MP"
] | math-ph | [] | 2007-04-02 | 2026-02-26 | We consider the variation of the surface spanned by closed strings in a spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic surface equation, the geodesic surface deviation equation which yields a Jacobi field, and we define the index form of a geodesic surface as in the case of point particles to discuss conjugate strings on the geodesic surface.
It is well known that string theory [1, 2] is one of the best candidates for a consistent quantum theory of gravity to yield a unification theory of all the four basic forces in nature. In D-brane models [2] , closed strings represent gravitons propagating on a curved manifold, while open strings describe gauge bosons such as photons, or matter attached on the D-branes. Moreover, because the two ends of an open string can always meet and connect, forming a closed string, there are no string theories without closed strings.
On the other hand, the supersymmetric quantum mechanics has been exploited by Witten [3] to discuss the Morse inequalities [4, 5, 6] . The Morse indices for pair of critical points of the symplectic action function have been also investigated based on the spectral flow of the Hessian of the symplectic function [7] , and on the Hilbert spaces the Morse homology [8] has been considered to discuss the critical points associated with the Morse index [9] . The string topology was initiated in the seminal work of Chas and Sullivan [10] . Using the Morse theoretic techniques, Cohen in Ref. [11] constructs string topology operations on the loop space of a manifold and relates the string topology operations to the counting of pseudoholomorphic curves in the cotangent bundle. He also speculates the relation between the Gromov-Witten invariant [12] of the cotangent bundle and the string topology of the underlying manifold. Recently, the Jacobi fields and their eigenvalues of the Sturm-Liouville operator associated with the particle geodesics on a curved manifold have been investigated [13] , to relate the phase factor of the partition function to the eta invariant of Atiyah [14, 15] .
In this paper, we will exploit the Nambu-Goto string action to investigate the geodesic surface equation and the geodesic surface deviation equation associated with a Jacobi field. The index form of a geodesic surface will be also discussed for the closed strings on the curved manifold.
In Section II, the string action will be introduced to investigate the geodesic surface equation in terms of the world sheet currents associated with τ and σ world sheet coordinate directions. By taking the second variation of the surface spanned by closed strings, the geodesic surface deviation equation will be discussed for the closed strings on the curved manifold. In Section III, exploiting the orthonormal gauge, the index form of a geodesic surface will be also investigated together with breaks on the string tubes. The geodesic surface deviation equation in the orthonormal gauge will be exploited to discuss the Jacobi field on the geodesic surface.
In analogy of the relativistic action of a point particle, the action for a string is proportional to the area of the surface spanned in spacetime manifold M by the evolution of the string. In order to define the action on the curved manifold, let (M, g ab ) be a n-dimensional manifold associated with the metric g ab . Given g ab , we can have a unique covariant derivative ∇ a satisfying [6]
∇ a g bc = 0, ∇ a ω b = ∂ a ω b + Γ b ac ω c and (∇ a ∇ b -∇ b ∇ a )ω c = R d abc ω d . (2.1)
We parameterize the closed string by two world sheet coordinates τ and σ, and then we have the corresponding vector fields ξ a = (∂/∂τ ) a and ζ a = (∂/∂σ) a . The Nambu-Goto string action is then given by [1, 2, 16 ]
S = - dτ dσf (τ, σ) (2.2)
where the coordinates τ and σ have ranges 0 ≤ τ ≤ T and 0 ≤ σ ≤ 2π respectively and
f (τ, σ) = [(ξ • ζ) 2 -(ξ • ξ)(ζ • ζ)] 1/2 . (2.3)
We now perform an infinitesimal variation of the tubes γ α (τ, σ) traced by the closed string during its evolution in order to find the geodesic surface equation from the least action principle. Here we impose the restriction that the length of the string circumference is τ independent. Let the vector field η a = (∂/∂α) a be the deviation vector which represents the displacement to an infinitesimally nearby tube, and let Σ denote the three-dimensional submanifold spanned by the tubes γ α (τ, σ). We then may choose τ , σ and α as coordinates of Σ to yield the commutator relations,
£ ξ η a = ξ b ∇ b η a -η b ∇ b ξ a = 0, £ ζ η a = ζ b ∇ b η a -η b ∇ b ζ a = 0, £ ξ ζ a = ξ b ∇ b ζ a -ζ b ∇ b ξ a = 0. (2.4)
Now we find the first variation as follows [17]
dS dα = dτ dσ η b (ξ a ∇ a P b τ + ζ a ∇ a P b σ ) -dσ P b τ η b | τ =T τ =0 -dτ P b σ η b | σ=2π σ=0 , (2.5)
where world sheet currents associated with τ and σ directions are respectively given by [17]
P a τ = 1 f [(ξ • ζ)ζ a -(ζ • ζ)ξ a ], P a σ = 1 f [(ξ • ζ)ξ a -(ξ • ξ)ζ a ]. (2.6)
Using the endpoint conditions η a (0) = η a (T ) = 0 and periodic condition η a (σ +2π) = η a (σ), we have the geodesic surface equation [17]
ξ a ∇ a P b τ + ζ a ∇ a P b σ = 0, (2.7)
and the constraint identities [17]
P τ • ζ = 0, P τ • P τ + ζ • ζ = 0, P σ • ξ = 0, P σ • P σ + ξ • ξ = 0. (2.8)
Let γ α (τ, σ) denote a smooth one-parameter family of geodesic surfaces: for each α ∈ R, the tube γ α is a geodesic surface parameterized by affine parameters τ and σ. For an infinitesimally nearby geodesic surface in the family, we then have the following geodesic surface deviation equation
ξ b ∇ b (η c ∇ c P a τ ) + ζ b ∇ b (η c ∇ c P a σ ) +R a bcd (ξ b P d τ + ζ b P d σ )η c ≡ (Λη) a = 0. (2.9)
For a small variation η a , our goal is to compare S(α) with S(0) of the string. The second variation d 2 S/dα 2 (0) is then needed only when dS/dα(0) = 0. Explicitly, the second variation is given by
d 2 S dα 2 | α=0 = - dτ dσ (η c ∇ c P b τ )(ξ a ∇ a η b ) +(η c ∇ c P b σ )(ζ a ∇ a η b ) -R d acb (ξ a P b τ + ζ a P b σ )η c η d -dσ P b τ η a ∇ a η b | τ =T τ =0 -dτ P b σ η a ∇ a η b | σ=2π σ=0 .
(2.10)
Here the boundary terms vanish for the fixed endpoint and the periodic conditions, even though on the geodesic surface we have breaks which we will explain later. After some algebra using the geodesic surface deviation equation, we have
d 2 S dα 2 | α=0 = dτ dσ η a (Λη) a .
(2.11)
The string action and the corresponding equations of motion are invariant under reparameterization σ = σ(τ, σ) and τ = τ (τ, σ). We have then gauge degrees of freedom so that we can choose the orthonormal gauge as follows [17] ξ
• ζ = 0, ξ • ξ + ζ • ζ = 0, (3.1)
where the plus sign in the second equation is due to the fact that ξ•ξ is timelike and ζ•ζ is spacelike. Note that the gauge fixing (3.1) for the world sheet coordinates means that the tangent vectors are orthonormal everywhere up to a local scale factor [17] . In this parameterization the world sheet currents (2.6) satisfying the constraints (2.8) are of the form
P a τ = -ξ a , P a σ = ζ a . (3.2)
The geodesic surface equation and the geodesic surface deviation equation read
-ξ a ∇ a ξ b + ζ a ∇ a ζ b = 0, (3.3)
and
-ξ b ∇ b (ξ c ∇ c η a ) + ζ b ∇ b (ζ c ∇ c η a ) -R a bcd (ξ b ξ d -ζ b ζ d )η c = (Λη) a = 0. (3.4)
We now restrict ourselves to strings on constant scalar curvature manifold such as S n . We take an ansatz that on this manifold the string shape on the geodesic surface γ 0 is the same as that on a nearby geodesic surface γ α at a given time τ . We can thus construct the variation vectors η a (τ ) as vectors associated with the centers of the string of the two nearby geodesic surfaces at the given time τ . We then introduce an orthonormal basis of spatial vectors e a i (i = 1, 2, ..., n-2) orthogonal to ξ a and ζ a and parallelly propagated along the geodesic surface. The geodesic surface deviation equation (3.4) then yields for i, j = 1, 2, ..., n -2
d 2 η i dτ 2 + (R i τ jτ -R i σjσ )η j = 0. (3.5)
The value of η i at time τ must depend linearly on the initial data η i (0) and dη i dτ (0) at τ = 0. Since by construction η i (0) = 0 for the family of geodesic surfaces, we must have
η i (τ ) = A i j (τ ) dη j dτ (0). (3.6)
Inserting (3.6) into (3.5) we have the differential equation for A i j (τ )
d 2 A i j dτ 2 + (R i τ kτ -R i σkσ )A k j = 0, (3.7)
with the initial conditions
A i j (0) = 0, dA i j dτ (0) = δ i j . (3.8)
Note that in (3.7) we have the last term originated from the contribution of string property.
Next we consider the second variation equation (2.10) under the above restrictions
d 2 S dα 2 | α=0 = dτ dσ dη i dτ η i dτ -(R i τ jτ -R i σjσ )η j η i .
(3.9) We define the index form I γ of a geodesic surface γ as the unique symmetric bilinear form
I γ : T γ × T γ → R such that I γ (V, V ) = d 2 S dα 2 | α=0 (3.10)
for V ∈ T γ . From (3.9) we can easily find
I γ (V, W ) = dτ dσ dW m dτ dV m dτ -(R m τ jτ -R m σjσ )W j V m . (3.11) If we have breaks 0 = τ 0 < • • • < τ k+1 = T ,
and the restriction of γ to each set [τ i-1 , τ i ] is smooth, then the tube γ is piecewise smooth. The variation vector field V of γ is always piecewise smooth. However dV /dτ will generally have a discontinuity at each break τ i (1 ≤ i ≤ k). This discontinuity is measured by
∆ dV dτ (τ i ) = dV dτ (τ + i ) - dV dτ (τ - i ), (3.12)
where the first term derives from the restrictions γ|[τ i , τ i+1 ] and the second from
γ|[τ i-1 , τ i ]. If γ and V ∈ T γ have the breaks τ 1 < • • • < τ k , we have k i=0 τi+1 τi d dτ V m dW m dτ dτ = - k i=0 V m ∆ dW m dτ (τ i ) (3.13)
to yield
I γ (V, W ) = - dτ dσ V m d 2 W m dτ 2 (3.14) +(R m τ jτ -R m σjσ )W j - k i=0 dσ V m ∆ dW m dτ (τ i ). (3.15)
Here note that if we do not have the breaks, (3.9) yields
d 2 S dα 2 | α=0 = - dτ dσ η i d 2 η i dτ 2 + (R i τ jτ -R i σjσ )η j .
(3.16) A solution η a of the geodesic surface deviation equation (3.5) is called a Jacobi field on the geodesic surface γ. A pair of strings p, q ⊂ γ defined by the centers of the closed strings on the geodesic surface is then conjugate if there exists a Jacobi field η a which is not identically zero but vanishes at both strings p and q. Roughly speaking, p and q are conjugate if an infinitesimally nearby geodesic surface intersects γ at both p and q. From (3.6), q will be conjugate to p if and only if there exists nontrivial initial data: dη i /dτ (0) = 0, for which η i = 0 at q. This occurs if and only if det A i j = 0 at q, and thus det A i j = 0 is the necessary and sufficient condition for a conjugate string to p. Note that between conjugate strings, we have det A i j = 0 and thus the inverse of A i j exists. Using (3.7) we can easily see that
d dτ dA ij dτ A i k -A ij dA i k dτ = 0. (3.17)
In addition, the quantity in parenthesis of (3.17) vanishes at p, since A i j (0) = 0. Along a geodesic surface γ, we thus find
dA ij dτ A i k -A ij dA i k dτ = 0. (3.18)
If γ is a geodesic surface with no string conjugate to p between p and q, then A i j defined above will be nonsingular between p and q. We can then define Y i = (A -1 ) i j η j or η i = A i j Y j . From (3.16) and (3.18), we can easily verify
d 2 S dα 2 | α=0 = dτ dσ A ij dY j dτ 2 ≥ 0. (3.19)
Locally γ minimizes the Nambu-Goto string action, if γ is a geodesic surface with no string conjugate to p between p and q.
On the other hand, if γ is a geodesic surface but has a conjugate string r between strings p and q, then we have a non-zero Jacobi field J i along γ which vanishes at p and r. Extend J i to q by putting it zero in [r, q]. Then dJ i /dτ (r -) = 0, since J i is nonzero. But dJ i /dτ (r + ) = 0 to yield
∆ dJ i dτ (r) = - dJ i dτ (r -) = 0. (3.20)
We choose any k i ∈ T γ such that
k i ∆ dJ i dτ (r) = c, (3.21)
with a positive constant c. Let η i be η i = ǫk i + ǫ -1 J i where ǫ is some constant, then we have
I γ (η, η) = ǫ 2 I γ (k, k) + 2I γ (k, J) + ǫ -2 I γ (J, J). ( 3
d 2 S dα 2 | α=0 = -4πc, (3.23)
which is negative definite. From the above arguments, we conclude that given a smooth timelike tube γ connecting two strings p, q ⊂ M , the necessary and sufficient condition that γ locally minimizes the surface of the closed string tube between p and q over smooth one parameter variations is that γ is a geodesic surface with no string conjugate to p between p and q. It is also interesting to see that on S n , the first non-minimal geodesic surface has n -1 conjugate strings as in the case of point particle. Moreover, on the Riemannian manifold with the constant sectional curvature K, the geodesic surfaces have no conjugate strings for K < 0 or K = 0, while conjugate strings occur for K > 0 [18] .
The Nambu-Goto string action has been introduced to study the geodesic surface equation in terms of the world sheet currents associated with τ and σ directions. By constructing the second variation of the surface spanned by closed strings, the geodesic surface deviation equation has been discussed for the closed strings on the curved manifold.
Exploiting the orthonormal gauge, the index form of a geodesic surface has been defined together with breaks on the string tubes. The geodesic surface deviation equation in this orthonormal gauge has been derived to find the Jacobi field on the geodesic surface. Given a smooth timelike tube connecting two strings on the manifold, the condition that the tube locally minimizes the surface of the closed string tube between the two strings over smooth one parameter variations has been also discussed in terms of the conjugate strings on the geodesic surface.
In the Morse theoretic approach to the string theory, one could consider the physical implications associated with geodesic surface congruences and their expansion, shear and twist. It would be also desirable if the string topology and the Gromov-Witten invariant can be investigated by exploiting the Morse theoretic techniques. These works are in progress and will be reported elsewhere. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We consider the variation of the surface spanned by closed strings in a spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic surface equation, the geodesic surface deviation equation which yields a Jacobi field, and we define the index form of a geodesic surface as in the case of point particles to discuss conjugate strings on the geodesic surface."
},
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "It is well known that string theory [1, 2] is one of the best candidates for a consistent quantum theory of gravity to yield a unification theory of all the four basic forces in nature. In D-brane models [2] , closed strings represent gravitons propagating on a curved manifold, while open strings describe gauge bosons such as photons, or matter attached on the D-branes. Moreover, because the two ends of an open string can always meet and connect, forming a closed string, there are no string theories without closed strings.\n\nOn the other hand, the supersymmetric quantum mechanics has been exploited by Witten [3] to discuss the Morse inequalities [4, 5, 6] . The Morse indices for pair of critical points of the symplectic action function have been also investigated based on the spectral flow of the Hessian of the symplectic function [7] , and on the Hilbert spaces the Morse homology [8] has been considered to discuss the critical points associated with the Morse index [9] . The string topology was initiated in the seminal work of Chas and Sullivan [10] . Using the Morse theoretic techniques, Cohen in Ref. [11] constructs string topology operations on the loop space of a manifold and relates the string topology operations to the counting of pseudoholomorphic curves in the cotangent bundle. He also speculates the relation between the Gromov-Witten invariant [12] of the cotangent bundle and the string topology of the underlying manifold. Recently, the Jacobi fields and their eigenvalues of the Sturm-Liouville operator associated with the particle geodesics on a curved manifold have been investigated [13] , to relate the phase factor of the partition function to the eta invariant of Atiyah [14, 15] .\n\nIn this paper, we will exploit the Nambu-Goto string action to investigate the geodesic surface equation and the geodesic surface deviation equation associated with a Jacobi field. The index form of a geodesic surface will be also discussed for the closed strings on the curved manifold.\n\nIn Section II, the string action will be introduced to investigate the geodesic surface equation in terms of the world sheet currents associated with τ and σ world sheet coordinate directions. By taking the second variation of the surface spanned by closed strings, the geodesic surface deviation equation will be discussed for the closed strings on the curved manifold. In Section III, exploiting the orthonormal gauge, the index form of a geodesic surface will be also investigated together with breaks on the string tubes. The geodesic surface deviation equation in the orthonormal gauge will be exploited to discuss the Jacobi field on the geodesic surface.\n\nIn analogy of the relativistic action of a point particle, the action for a string is proportional to the area of the surface spanned in spacetime manifold M by the evolution of the string. In order to define the action on the curved manifold, let (M, g ab ) be a n-dimensional manifold associated with the metric g ab . Given g ab , we can have a unique covariant derivative ∇ a satisfying [6]\n\n∇ a g bc = 0, ∇ a ω b = ∂ a ω b + Γ b ac ω c and (∇ a ∇ b -∇ b ∇ a )ω c = R d abc ω d . (2.1)\n\nWe parameterize the closed string by two world sheet coordinates τ and σ, and then we have the corresponding vector fields ξ a = (∂/∂τ ) a and ζ a = (∂/∂σ) a . The Nambu-Goto string action is then given by [1, 2, 16 ]\n\nS = - dτ dσf (τ, σ) (2.2)\n\nwhere the coordinates τ and σ have ranges 0 ≤ τ ≤ T and 0 ≤ σ ≤ 2π respectively and\n\nf (τ, σ) = [(ξ • ζ) 2 -(ξ • ξ)(ζ • ζ)] 1/2 . (2.3)\n\nWe now perform an infinitesimal variation of the tubes γ α (τ, σ) traced by the closed string during its evolution in order to find the geodesic surface equation from the least action principle. Here we impose the restriction that the length of the string circumference is τ independent. Let the vector field η a = (∂/∂α) a be the deviation vector which represents the displacement to an infinitesimally nearby tube, and let Σ denote the three-dimensional submanifold spanned by the tubes γ α (τ, σ). We then may choose τ , σ and α as coordinates of Σ to yield the commutator relations,\n\n£ ξ η a = ξ b ∇ b η a -η b ∇ b ξ a = 0, £ ζ η a = ζ b ∇ b η a -η b ∇ b ζ a = 0, £ ξ ζ a = ξ b ∇ b ζ a -ζ b ∇ b ξ a = 0. (2.4)\n\nNow we find the first variation as follows [17]\n\ndS dα = dτ dσ η b (ξ a ∇ a P b τ + ζ a ∇ a P b σ ) -dσ P b τ η b | τ =T τ =0 -dτ P b σ η b | σ=2π σ=0 , (2.5)\n\nwhere world sheet currents associated with τ and σ directions are respectively given by [17]\n\nP a τ = 1 f [(ξ • ζ)ζ a -(ζ • ζ)ξ a ], P a σ = 1 f [(ξ • ζ)ξ a -(ξ • ξ)ζ a ]. (2.6)\n\nUsing the endpoint conditions η a (0) = η a (T ) = 0 and periodic condition η a (σ +2π) = η a (σ), we have the geodesic surface equation [17]\n\nξ a ∇ a P b τ + ζ a ∇ a P b σ = 0, (2.7)\n\nand the constraint identities [17]\n\nP τ • ζ = 0, P τ • P τ + ζ • ζ = 0, P σ • ξ = 0, P σ • P σ + ξ • ξ = 0. (2.8)\n\nLet γ α (τ, σ) denote a smooth one-parameter family of geodesic surfaces: for each α ∈ R, the tube γ α is a geodesic surface parameterized by affine parameters τ and σ. For an infinitesimally nearby geodesic surface in the family, we then have the following geodesic surface deviation equation\n\nξ b ∇ b (η c ∇ c P a τ ) + ζ b ∇ b (η c ∇ c P a σ ) +R a bcd (ξ b P d τ + ζ b P d σ )η c ≡ (Λη) a = 0. (2.9)\n\nFor a small variation η a , our goal is to compare S(α) with S(0) of the string. The second variation d 2 S/dα 2 (0) is then needed only when dS/dα(0) = 0. Explicitly, the second variation is given by\n\nd 2 S dα 2 | α=0 = - dτ dσ (η c ∇ c P b τ )(ξ a ∇ a η b ) +(η c ∇ c P b σ )(ζ a ∇ a η b ) -R d acb (ξ a P b τ + ζ a P b σ )η c η d -dσ P b τ η a ∇ a η b | τ =T τ =0 -dτ P b σ η a ∇ a η b | σ=2π σ=0 .\n\n(2.10)\n\nHere the boundary terms vanish for the fixed endpoint and the periodic conditions, even though on the geodesic surface we have breaks which we will explain later. After some algebra using the geodesic surface deviation equation, we have\n\nd 2 S dα 2 | α=0 = dτ dσ η a (Λη) a .\n\n(2.11)"
},
{
"section_type": "OTHER",
"section_title": "III. JACOBI FIELDS IN ORTHONORMAL GAUGE",
"text": "The string action and the corresponding equations of motion are invariant under reparameterization σ = σ(τ, σ) and τ = τ (τ, σ). We have then gauge degrees of freedom so that we can choose the orthonormal gauge as follows [17] ξ\n\n• ζ = 0, ξ • ξ + ζ • ζ = 0, (3.1)\n\nwhere the plus sign in the second equation is due to the fact that ξ•ξ is timelike and ζ•ζ is spacelike. Note that the gauge fixing (3.1) for the world sheet coordinates means that the tangent vectors are orthonormal everywhere up to a local scale factor [17] . In this parameterization the world sheet currents (2.6) satisfying the constraints (2.8) are of the form\n\nP a τ = -ξ a , P a σ = ζ a . (3.2)\n\nThe geodesic surface equation and the geodesic surface deviation equation read\n\n-ξ a ∇ a ξ b + ζ a ∇ a ζ b = 0, (3.3)\n\nand\n\n-ξ b ∇ b (ξ c ∇ c η a ) + ζ b ∇ b (ζ c ∇ c η a ) -R a bcd (ξ b ξ d -ζ b ζ d )η c = (Λη) a = 0. (3.4)\n\nWe now restrict ourselves to strings on constant scalar curvature manifold such as S n . We take an ansatz that on this manifold the string shape on the geodesic surface γ 0 is the same as that on a nearby geodesic surface γ α at a given time τ . We can thus construct the variation vectors η a (τ ) as vectors associated with the centers of the string of the two nearby geodesic surfaces at the given time τ . We then introduce an orthonormal basis of spatial vectors e a i (i = 1, 2, ..., n-2) orthogonal to ξ a and ζ a and parallelly propagated along the geodesic surface. The geodesic surface deviation equation (3.4) then yields for i, j = 1, 2, ..., n -2\n\nd 2 η i dτ 2 + (R i τ jτ -R i σjσ )η j = 0. (3.5)\n\nThe value of η i at time τ must depend linearly on the initial data η i (0) and dη i dτ (0) at τ = 0. Since by construction η i (0) = 0 for the family of geodesic surfaces, we must have\n\nη i (τ ) = A i j (τ ) dη j dτ (0). (3.6)\n\nInserting (3.6) into (3.5) we have the differential equation for A i j (τ )\n\nd 2 A i j dτ 2 + (R i τ kτ -R i σkσ )A k j = 0, (3.7)\n\nwith the initial conditions\n\nA i j (0) = 0, dA i j dτ (0) = δ i j . (3.8)\n\nNote that in (3.7) we have the last term originated from the contribution of string property.\n\nNext we consider the second variation equation (2.10) under the above restrictions\n\nd 2 S dα 2 | α=0 = dτ dσ dη i dτ η i dτ -(R i τ jτ -R i σjσ )η j η i .\n\n(3.9) We define the index form I γ of a geodesic surface γ as the unique symmetric bilinear form\n\nI γ : T γ × T γ → R such that I γ (V, V ) = d 2 S dα 2 | α=0 (3.10)\n\nfor V ∈ T γ . From (3.9) we can easily find\n\nI γ (V, W ) = dτ dσ dW m dτ dV m dτ -(R m τ jτ -R m σjσ )W j V m . (3.11) If we have breaks 0 = τ 0 < • • • < τ k+1 = T ,\n\nand the restriction of γ to each set [τ i-1 , τ i ] is smooth, then the tube γ is piecewise smooth. The variation vector field V of γ is always piecewise smooth. However dV /dτ will generally have a discontinuity at each break τ i (1 ≤ i ≤ k). This discontinuity is measured by\n\n∆ dV dτ (τ i ) = dV dτ (τ + i ) - dV dτ (τ - i ), (3.12)\n\nwhere the first term derives from the restrictions γ|[τ i , τ i+1 ] and the second from\n\nγ|[τ i-1 , τ i ]. If γ and V ∈ T γ have the breaks τ 1 < • • • < τ k , we have k i=0 τi+1 τi d dτ V m dW m dτ dτ = - k i=0 V m ∆ dW m dτ (τ i ) (3.13)\n\nto yield\n\nI γ (V, W ) = - dτ dσ V m d 2 W m dτ 2 (3.14) +(R m τ jτ -R m σjσ )W j - k i=0 dσ V m ∆ dW m dτ (τ i ). (3.15)\n\nHere note that if we do not have the breaks, (3.9) yields\n\nd 2 S dα 2 | α=0 = - dτ dσ η i d 2 η i dτ 2 + (R i τ jτ -R i σjσ )η j .\n\n(3.16) A solution η a of the geodesic surface deviation equation (3.5) is called a Jacobi field on the geodesic surface γ. A pair of strings p, q ⊂ γ defined by the centers of the closed strings on the geodesic surface is then conjugate if there exists a Jacobi field η a which is not identically zero but vanishes at both strings p and q. Roughly speaking, p and q are conjugate if an infinitesimally nearby geodesic surface intersects γ at both p and q. From (3.6), q will be conjugate to p if and only if there exists nontrivial initial data: dη i /dτ (0) = 0, for which η i = 0 at q. This occurs if and only if det A i j = 0 at q, and thus det A i j = 0 is the necessary and sufficient condition for a conjugate string to p. Note that between conjugate strings, we have det A i j = 0 and thus the inverse of A i j exists. Using (3.7) we can easily see that\n\nd dτ dA ij dτ A i k -A ij dA i k dτ = 0. (3.17)\n\nIn addition, the quantity in parenthesis of (3.17) vanishes at p, since A i j (0) = 0. Along a geodesic surface γ, we thus find\n\ndA ij dτ A i k -A ij dA i k dτ = 0. (3.18)\n\nIf γ is a geodesic surface with no string conjugate to p between p and q, then A i j defined above will be nonsingular between p and q. We can then define Y i = (A -1 ) i j η j or η i = A i j Y j . From (3.16) and (3.18), we can easily verify\n\nd 2 S dα 2 | α=0 = dτ dσ A ij dY j dτ 2 ≥ 0. (3.19)\n\nLocally γ minimizes the Nambu-Goto string action, if γ is a geodesic surface with no string conjugate to p between p and q.\n\nOn the other hand, if γ is a geodesic surface but has a conjugate string r between strings p and q, then we have a non-zero Jacobi field J i along γ which vanishes at p and r. Extend J i to q by putting it zero in [r, q]. Then dJ i /dτ (r -) = 0, since J i is nonzero. But dJ i /dτ (r + ) = 0 to yield\n\n∆ dJ i dτ (r) = - dJ i dτ (r -) = 0. (3.20)\n\nWe choose any k i ∈ T γ such that\n\nk i ∆ dJ i dτ (r) = c, (3.21)\n\nwith a positive constant c. Let η i be η i = ǫk i + ǫ -1 J i where ǫ is some constant, then we have\n\nI γ (η, η) = ǫ 2 I γ (k, k) + 2I γ (k, J) + ǫ -2 I γ (J, J). ( 3\n\nd 2 S dα 2 | α=0 = -4πc, (3.23)\n\nwhich is negative definite. From the above arguments, we conclude that given a smooth timelike tube γ connecting two strings p, q ⊂ M , the necessary and sufficient condition that γ locally minimizes the surface of the closed string tube between p and q over smooth one parameter variations is that γ is a geodesic surface with no string conjugate to p between p and q. It is also interesting to see that on S n , the first non-minimal geodesic surface has n -1 conjugate strings as in the case of point particle. Moreover, on the Riemannian manifold with the constant sectional curvature K, the geodesic surfaces have no conjugate strings for K < 0 or K = 0, while conjugate strings occur for K > 0 [18] ."
},
{
"section_type": "CONCLUSION",
"section_title": "IV. CONCLUSIONS",
"text": "The Nambu-Goto string action has been introduced to study the geodesic surface equation in terms of the world sheet currents associated with τ and σ directions. By constructing the second variation of the surface spanned by closed strings, the geodesic surface deviation equation has been discussed for the closed strings on the curved manifold.\n\nExploiting the orthonormal gauge, the index form of a geodesic surface has been defined together with breaks on the string tubes. The geodesic surface deviation equation in this orthonormal gauge has been derived to find the Jacobi field on the geodesic surface. Given a smooth timelike tube connecting two strings on the manifold, the condition that the tube locally minimizes the surface of the closed string tube between the two strings over smooth one parameter variations has been also discussed in terms of the conjugate strings on the geodesic surface.\n\nIn the Morse theoretic approach to the string theory, one could consider the physical implications associated with geodesic surface congruences and their expansion, shear and twist. It would be also desirable if the string topology and the Gromov-Witten invariant can be investigated by exploiting the Morse theoretic techniques. These works are in progress and will be reported elsewhere."
}
] | |
arxiv:0704.0117 | 0704.0117 | 1 | 10.1088/0953-4075/40/11/002 | ab49ae90fd17c08af50d23a7e91bdb7703f6901f0b714d66f75e6faf5bbb44ee | Lower ground state due to counter-rotating wave interaction in trapped
ion system | We consider a single ion confined in a trap under radiation of two traveling
waves of lasers. In the strong-excitation regime and without the restriction of
Lamb-Dicke limit, the Hamiltonian of the system is similar to a driving
Jaynes-Cummings model without rotating wave approximation (RWA). The approach
we developed enables us to present a complete eigensolutions, which makes it
available to compare with the solutions under the RWA. We find that, the ground
state in our non-RWA solution is energically lower than the counterpart under
the RWA. If we have the ion in the ground state, it is equivalent to a spin
dependent force on the trapped ion. Discussion is made for the difference
between the solutions with and without the RWA, and for the relevant
experimental test, as well as for the possible application in quantum
information processing. | [
"T. Liu",
"K.L. Wang",
"M. Feng"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | Ultracold ions trapped as a line are considered as a promising system for quantum information processing [1] . Since the first quantum gate performed in the ion trap [2] , there have been a series of experiments with trapped ions to achieve nonclassical states [3], simple quantum algorithm [4] , and quantum communication [5] .
There have been also a number of proposals to employ trapped ions for quantum computing, most of which work only in the weak excitation regime (WER), i.e., the Rabi frequency smaller than the trap frequency. While as bigger Rabi frequency would lead to faster quantum gating, some proposals [6, 7, 8] have aimed to achieve operations in the case of the Rabi frequency larger than the trap frequency, i.e., the so called strong excitation regime (SER). The difference of the WER from the SER is mathematically reflected in the employment of the rotating wave approximation (RWA), which averages out the fast oscillating terms in the interaction Hamiltonian. As the RWA is less valid with the larger Rabi frequency, the treatment for the SER was complicated, imcomplete [9] , and sometimes resorted to numerics [10] .
In addition, the Lamb-Dicke limit strongly restricts the application of the trapped ions due to technical challenge and the slow quantum gating. We have noticed some ideas [11, 12] to remove the Lamb-Dicke limit in designing quantum gates, which are achieved by using some complicated laser pulse sequences.
In the present work, we investigate, from another research angle, the system mentioned above in SER and in the absence of the Lamb-Dicke limit. The main idea, based on an analytical approach we have developed, is to check the eigenvectors and the eigenenergies of such a system, with which we hope to obtain new insight into the system for more application. The main result in our work is a newly found ground state, energically lower than the ground state calculated by standard Jaynes-Cummings model. We will also present the analytical forms of the eigenvectors and the variance of the eigenenergies with respect to the parameters of the system, which might be used in understanding the time evolution of the system.
The paper is organized as follows. In Section II we will solve the system in the absence of the RWA. Then some numerical results will be presented in comparison with the RWA solutions in Section III. We will discuss about the new results for their possible application. More extensive discussion and the conclusion are made in Section IV. Some analytical deduction details could be found in Appendix.
As shown in Fig. 1 , we consider a Raman Λ-type configuration, which corresponds to the actual process in NIST experiments. Like in [13] , we will employ some unitary transformations to get rid of the assumption of Lamb-Dicke limit and the WER. So our solution is more general than most of the previous work [14] . For a single trapped 2 ion experiencing two off-resonant counter-propagating traveling wave lasers with frequencies ω 1 and ω 2 , respectively, and in the case of a large detuning δ, we have an effective two-level system with the lasers driving the electric-dipole forbidden transition |g ↔ |e by the effective laser frequency ω L = ω 1 -ω 2 . So we have the dimensionless Hamiltonian
H = ∆ 2 σ z + a † a + Ω 2 (σ + e iη x + σ -e -iη x), ( 1
)
in the frame rotating with ω L , where ∆ = (ω 0 -ω L )/ν, ω 0 and ν are the resonant frequency of the two levels of the ion and the trap frequency, respectively. Ω is the dimensionless Rabi frequency in units of ν and η the Lamb-Dicke parameter. σ ±,z are usual Pauli operators, and we have x = a † + a for the dimensionless position operator of the ion with a † and a being operators of creation and annihilation of the phonon field, respectively. We suppose that both Ω and ν are much larger than the atomic decay rate and the phonon dissipative rate so that no dissipation is considered below. Like in [13] , we first carry out some unitary transformations on Eq. ( 1 ) to avoid the expansion of the exponentials. So we have
H I = U HU † = Ω 2 σ z + a † a + g(a † + a)σ x + ǫσ x + g 2 , ( 2
)
where
U = 1 √ 2 e iπa † a/2 F † (η) F (η) -F † (η) F (η) ,
with F (η) = exp [iη(a † + a)/2], g = η/2, and ǫ = -∆/2. Eq. ( 2 ) is a typical driving Jaynes-Cummings model including the counter-rotating wave terms. In contrast to the usual treatments to consider the Lamb-Dicke limit by using the RWA in a frame rotation, we remain the counter-rotating wave interaction in the third term of the right-hand side of Eq. ( 2 ) in our case. To go on our treatment, we make a further rotation with V = exp (iπσ y /4), yielding
H ′ = V H I V † = - Ω 2 σ x + a † a + g(a † + a)σ z + ǫσ z + g 2 , ( 3
)
where we have used exp (iθσ y )σ x exp (-iθσ y ) = cos(2θ)σ x + sin(2θ)σ z , and exp (iθσ y )σ z exp (-iθσ y ) = cos(2θ)σ zsin(2θ)σ x . For convenience of our following treatment, we rewrite Eq. ( 3 ) to be
H ′ = ǫ(|e e| -|g g|) - Ω 2 (|e g| + |g e|) + a † a + g(a † + a)(|e e| -|g g|) + g 2 . ( 4
)
Using Schrödinger equation, and the orthogonality between |e and |g , we suppose
| = |ϕ 1 |e + |ϕ 2 |g , ( 5
) which yields ǫ|ϕ 1 + a † a|ϕ 1 + g(a † + a)|ϕ 1 - Ω 2 |ϕ 2 + g 2 |ϕ 1 = E|ϕ 1 , ( 6
) -ǫ|ϕ 2 + a † a|ϕ 2 -g(a † + a)|ϕ 2 - Ω 2 |ϕ 1 + g 2 |ϕ 2 = E|ϕ 2 . ( 7
)
To make the above equations concise, we apply the displacement operator D(g
) = exp [g(a † -a)] on a † and a, which gives A = D(g) † a D(g) = a+g, A † = D(g) † a † D(g) = a † +g, B = D(-g) † a D(-g) = a-g, and B † = D(-g) † a † D(-g) = a † -g. So we have (A † A + ǫ)|ϕ 1 - Ω 2 |ϕ 2 = E|ϕ 1 , ( 8
) (B † B -ǫ)|ϕ 2 - Ω 2 |ϕ 1 = E|ϕ 2 . ( 9
)
3 Obvious, the new operators work in different subspaces, which leads to different evolutions regarding different internal levels |g and |e . We will later refer to this feature to be relevant to spin-dependent force. The solution of the two equations above can be simply set as
|ϕ 1 = N n=0 c n |n A , ( 10
) |ϕ 2 = N n=0 d n |n B , ( 11
)
with N a large integer to be determined later
, |n A = 1 √ n! (a † + g) n |0 A = 1 √ n! (a † + g) n D(g) † |0 = 1 √ n! (a † + g) n exp{-ga † -g 2 /2}|0 , and |n B = 1 √ n! (a † -g) n |0 B = 1 √ n! (a † -g) n D(-g) † |0 = 1 √ n! (a † -g) n exp{ga † -g 2 /2}|0 .
Taking Eqs. (10) and (11) into Eqs. (8) and (9), respectively, and multiplying by A m| and B m|, respectively, we have,
(m + ǫ)c m - Ω 2 N n=0 (-1) n D mn d n = Ec m , ( 12
) (m -ǫ)d m - Ω 2 N n=0 (-1) m D mn c n = Ed m , ( 13
)
where we have set (-1
) n D mn = A m|n B and (-1) m D mn = B m|n A ,
whose deduction can be found in Appendix. Diagonizing the relevant determinants, we may have the eigenenergies E i and the eigenvectors regarding c i n and d i n
(n = 0, • • • , N, i = 0, • • • , N
). Therefore, as long as we could find a closed subspace with c i N +1 and d i N +1 approaching zero for a certain big integer N, we may have a complete eigensolution of the system.
Before doing numerics, we first consider a treatment by involving the RWA. As the RWA solution could present complete eigenenergy spectra, it is interesting to make a comparison between the RWA solution and our non-RWA one. We consider a rotation in Eq. ( 2 ) with respect to exp{-i[(Ω/2)σ z + a † a]t}, which results in
H A = Ω 2 σ z + a † a + g(aσ + + a † σ -) + g 2 , ( 14
)
where the RWA has been made by setting Ω = 1, and we have corresponding eigenenergies
E ± n = (n + g 2 + 1/2) ± g √ n + 1. ( 15
)
So the system is degenerate in the case of η = 0 and there are two eigenenergy spectra corresponding to E ± n as long as η = 0. Figs. 2(a) and 2(b) demonstrate two spectra, respectively, and in each figure we compare the differences between the RWA and non-RWA solutions [15] . In contrast to the two spectra in the RWA solution, the non-RWA solution includes only one spectrum. Comparing the two eigensolutions, we find that the even-number and odd-number excited levels in the non-RWA case correspond to E + n and E - n of the RWA case, respectively, and the difference becomes bigger and bigger with the increase of η. It is physically understandable for these differences because the RWA solution, valid only for small η, does not work beyond the Lamb-Dicke regime. Above comparison also demonstrates the change of the ion trap system from an integrable case (i.e., with RWA validity) to the non-integrable case (i.e., without RWA validity). But besides these differences, we find an unusual result in this comparison, i.e., a new level without the counterpart in RWA solution appearing in our solution, which is lower than the ground state in RWA solution by ν + xη with x a η-dependent coefficient. In the viewpoint of physics, due to additional counter-rotating wave interaction involved, it is reasonable to have something more in our solution than the RWA case, although this does not surely lead to a new level lower than the previous ground state. Anyway, this is a good news for quantum information processing with trapped ions. As the situation in SER and beyond the Lamb-Dicke limit involves more instability, a stable 4 confinement of the ion requires a stronger trapping condition. In this sense, our solution, with the possibility to have the ion stay in an energically lower state, gives a hope in this respect. We will come to this point again later. Since no report of the new ground state had been found either theoretically or experimentally in previous publications, we suggest to check it experimentally by resonant absorption spectrum. As shown above, in the case of non-zero Lamb-Dicke parameter, the degeneracy of the neighboring level spacing is released, and the bigger the η, the larger the spacing difference between the neighboring levels. Therefore, an experimental test of the newly found ground state should be available by resonant transition between the ground and the first excited states in Fig. 2 , once the SER is reached. We have noticed that the SER could be achieved by first cooling the ions within the Lamb-Dicke limit and under the WER, and then by decreasing the trap frequency by opening the trap adiabatically [6] .
Since it is lower in energy than the previously recognized ground states, the new ground state we found is more stable, and thereby more suitable to store quantum information. Once the trapped ion is cooled down to the ground state in the SER, it is, as shown in Eq. ( 5 ) with n = 0, actually equivalent to the effect of a spin-dependent force on the trapped ion [16] . If we make Hadamard gate on the ion by |g → (|g + |e )/ √ 2 and |e → (|g -|e )/ √ 2, we reach a Schrödinger cat state, i.e., (1/2){[
D † (g)|0 + D † (-g)|0 ]|g -[D † (g)|0 -D † (-g)|0 ]|e }.
Two ions confined in a trap in above situation will yield two-qubit gates without really exciting the vibrational mode [11] . It is also the way with this spin-dependent force towards scalable quantum information processing [12] . As in SER, we may have larger Rabi frequency than in WER, the quantum gate could be in principle carried out faster in the SER.
In addition, as it is convergent throughout the parameter subspace, our complete eigensolution enables us to accurately write down the state of the system at an arbitrary evolution time, provided that we have known the initial state. This would be useful for future experiments in preparing non-classical states and in designing any desired quantum gates with trapped ions in the SER and beyond the Lamb-Dicke limit. Moreover, as shown in Figs 3(a), 3(b) and 3(c), our present solution is helpful for us to understand the particular solutions in previous publication [13] . The comparison in the figures shows that the results in [13] are actually mixtures of different eigensolutions. For example, the lowest level in Fig. 2 in [ 13] , corresponding to Ω = 2 and η = 0.2, is actually constituted at least by the third, the fourth, and the fifth excited states of the eigensolution.
The observation of the counter-rotating effects is an interesting topic discussed previously. In [17], a standard method is used to study the observable effects regarding the rotating and the counter-rotating terms in the Jaynes-Cummings model, including to observe Bloch-Siegert shift [18] and quantum chaos in a cavity QED by using differently polarized lights. A recent work [19] for a two-photon Jaynes-Cummings model has also investigated the observability of the counter-rotating terms. By using perturbation theory, the authors claimed that the counter-rotating effects, although very small, can be in principle observed by measuring the energy of the atom going through the cavity. Actually, for the cavity QED system without any external source involved, it is generally thought that the counter-rotating terms only make contribution in some virtual fluctuations of the energy in the weak coupling regime. While the interference between the rotating and counter-rotating contributions could result in some phase dependent effects [20] . Anyway, if there is an external source, for example, the laser radiating a trapped ultracold ion, the counter-rotating terms will show their effects, e.g., related to heating in the case of WER [21] . In this sense, our result is somewhat amazing because the counter-rotating interaction in the SER, making entanglement between internal and vibrational states of the trapped ion, plays positive role in the ion trapping.
We argue that our approach is applicable to different physical processes involving counter-rotating interaction. Since the counter-rotating terms result in energy nonconservation in single quanta processes, usual techniques cannot solve the Hamiltonian with eigenstates spanning in an open form. In this case, path-integral approach [22] and perturbation approach [20] , assisted by numerical techniques were employed in the weak coupling regime of the Jaynes-Cummings model. In contrast, our method, based on the diagonalization of the coherent-state subspace, could in principle study the Jaynes-Cummings model without the RWA in any cases. We have also noticed a recent publication [23] to treat a strongly coupled two-level system to a quntum oscillator under an adiabatic approximation, in which something is similar to our work in the solution of the Hamiltonian in the absence of the RWA. But due to the different features in their system from our atomic case, the two-level splitting term, much smaller compared to other terms, can be taken as a perturbation. So the advantage of that treatment is the possibility to analytically obtain good approximate solutions. In contrast, not any approximation is used in our solution, which should be more efficient to do the relevant job.
In summary, we have investigated the eigensolution of the system with a single trapped ion, experiencing two traveling waves of lasers, in the SER and in the absence of the Lamb-Dicke limit. We have found the ground state in the non-RWA case to be energically lower than the counterpart of the solution with RWA, which would be useful for quantum information storage and for quantum computing. The analytical forms of the eigenfunction and the 5 complete set of the eigensolutions would be helpful for us to understand a trapped ion in the SER and with a large Lamb-Dicke parameter. We argue that our work would be applied to different systems in dealing with strong coupling problems.
This work is supported in part by NNSFC No. 10474118, by Hubei Provincial Funding for Distinguished Young Scholars, and by Sichuan Provincial Funding.
We give the deduction of A m|n B and B m|n A below,
A m|n B = 1 √ m!n! 0|e -ga-g 2 /2 (a + g) m (a † -g) n e ga † -g 2 /2 |0 = 1 √ m!n! e -2g 2 0|(a + g) m e ga † e -ga (a † -g) n |0 = 1 √ m!n! e -2g 2 0|(a + 2g) m (a † -2g) n |0 = (-1) n D mn , with D mn = e -2g 2 min[m,n] i=0 (-1) -i √ m!n!(2g) m+n-2i (m -i)!(n -i)!i! .
It is easily proven following a similar step to above that
B m|n A = 1 √ m!n! 0|e ga-g 2 /2 (a -g) m (a † + g) n e -ga † -g 2 /2 |0 ,
would finally get to (-1) m D mn .
[1] Cirac J I, Zoller P 1995 Phys. Rev. Lett. 74 4091 [2] Monroe C, Meekhof D M, King B E, Itano W M, Wineland D J 1995 Phys. Rev. Lett. 75 4714 [3] Turchette Q A, Wood C S, King B E, Myatt C J, Leibfried D, Itano W M, Monroe C, Wineland D J 1998 Phys. Rev. Lett. 81 3631; Sackett C A, Kielpinski D, King B E, Langer C, Meyer V, Myatt C J, Rowe M, Turchette Q A, Itano W M, Wineland D J, Monroe C 2000 Nature 404 256 [4] Gulde S, Riebe M, Lancaster G P T, Becher C, Eschner J, Haeffner H, Schmidt-Kaler F, Chuang I L, Blatt R 2003 Nature 421 48 [5] Riebe M, Haeffner H, Roos C F, Haensel W, Benhelm J, Lancaster G P T, Koerber T W, Becher C, Schmidt-Kaler F, James D F V, Blatt R 2004 Nature 429 734; Barrett M D, Chiaverini J, Schaetz T, Britton J, Itano W M, Jost J D, Knill E, Langer C, Leibfried D, Ozeri R, Wineland D J 2004 Nature 429 737 [6] Poyators J F, Cirac J I, Blatt R, Zoller P 1996 Phys. Rev. A 54 1532; Poyatos J F, Cirac J I, Zoller P 1998 Phys. Rev.
Lett. 81 1322 [7] Zheng S, Zhu X W, Feng M 2000 Phys. Rev. A 62 033807 [8] Feng M 2004 Eur. Phys. J. D 29 189 [9] Feng M, Zhu X, Fang X, Yan M, Shi L 1999 J. Phys. B 32 701; Feng M 2002 Eur. Phys. J. D 18 371 [10] Zeng H, Lin F, Wang Y, Segawa Y 1999 Phys. Rev. A 59 4589 [11] Garcia-Ripoll J J, Zoller P and Cirac J I 2003 Phys. Rev. Lett. 91 157901; [12] Duan L -M 2004 Phys. Rev. Lett. 93 100502 [13] Feng M 2001 J. Phys. B 34 451 6 [14] Most of the previous work in this respect were carried out by cuting off the expansion of the exponentials regarding the quantized phonon operators, which is only reasonable in the WER and within the Lamb-Dicke limit. In contrast, our treatment can be used in both the SER and the WER cases. [15] We take throughout this paper N = 40 in which the coefficients c i 41 and d i 41 with i = 0, 1, ..40 are negligible in the case of Ω = 1 and 2. Although with the increase of values of Ω the diagonalization space has to be enlarged, our analytical method generally works well in a wide range of parameters. [16] Haljan P C, Brickman K -A, Deslauriers L, Lee P J and Monroe C 2005 Phys. Rev. Lett. 94 153602 [17] Crisp M D 1991 Phys. Rev. A 43 2430 [18] Bloch F and Siegert A 1940 Phys. Rev. 57 522 [19] Janowicz M and Orlowski A 2004 Rep.Math. Phys. 54 71 [20] Phoenix S J D 1989 J. Mod. Optics 3 127 [21] Leibfrid D, Blatt R, Monroe C, and Wineland D J 2003 Rev. Mod. Phys. 75 281 [22] Zaheer K and Zubairy M S 1998 Phys. Rev. A 37 1628 [23] Irish E K, Gea-Banacloche J, Martin I, and Schwab K C 2005 Phys. Rev. B 72 195410
The captions of the figures
Fig. 1 Schematic of a single trapped ion under radiation of two traveling wave lasers, where ω 1 and ω 2 are frequencies regarding the two lasers, respectively, ω 0 is the resonant frequency between |g and |e , and δ and ∆ are relevant detunings. This is a typical Raman process employed in NIST experiments, with for example Be + , for quantum computing.
Fig. 2 The eigenenergy spectra with Ω = 1, where (a) and (b) correspond to two different sets of eigenenergies with respect to Lamb-Dicke parameter. In (a) the comparison is made between E + n in the RWA case (dashed-dotted curves) and E n with n = even numbers in the non-RWA case (star curves for n = 0 and solid curves for others); In (b) the comparison is for E - n in the RWA case (dashed-dotted curves) to E n with n = odd numbers in the non-RWA case (solid curves).
Fig. 3 The eigenenergy with respect to the detuning ∆, where for convenience of comparison we have used the same parameter numbers as in [13] . For clarity, we plot the different levels with different lines. The parameter numbers are Ω = 2, and (a) η = 0.2; (b) η = 0.4; (c) η = 0.6. | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Ultracold ions trapped as a line are considered as a promising system for quantum information processing [1] . Since the first quantum gate performed in the ion trap [2] , there have been a series of experiments with trapped ions to achieve nonclassical states [3], simple quantum algorithm [4] , and quantum communication [5] .\n\nThere have been also a number of proposals to employ trapped ions for quantum computing, most of which work only in the weak excitation regime (WER), i.e., the Rabi frequency smaller than the trap frequency. While as bigger Rabi frequency would lead to faster quantum gating, some proposals [6, 7, 8] have aimed to achieve operations in the case of the Rabi frequency larger than the trap frequency, i.e., the so called strong excitation regime (SER). The difference of the WER from the SER is mathematically reflected in the employment of the rotating wave approximation (RWA), which averages out the fast oscillating terms in the interaction Hamiltonian. As the RWA is less valid with the larger Rabi frequency, the treatment for the SER was complicated, imcomplete [9] , and sometimes resorted to numerics [10] .\n\nIn addition, the Lamb-Dicke limit strongly restricts the application of the trapped ions due to technical challenge and the slow quantum gating. We have noticed some ideas [11, 12] to remove the Lamb-Dicke limit in designing quantum gates, which are achieved by using some complicated laser pulse sequences.\n\nIn the present work, we investigate, from another research angle, the system mentioned above in SER and in the absence of the Lamb-Dicke limit. The main idea, based on an analytical approach we have developed, is to check the eigenvectors and the eigenenergies of such a system, with which we hope to obtain new insight into the system for more application. The main result in our work is a newly found ground state, energically lower than the ground state calculated by standard Jaynes-Cummings model. We will also present the analytical forms of the eigenvectors and the variance of the eigenenergies with respect to the parameters of the system, which might be used in understanding the time evolution of the system.\n\nThe paper is organized as follows. In Section II we will solve the system in the absence of the RWA. Then some numerical results will be presented in comparison with the RWA solutions in Section III. We will discuss about the new results for their possible application. More extensive discussion and the conclusion are made in Section IV. Some analytical deduction details could be found in Appendix."
},
{
"section_type": "OTHER",
"section_title": "II. THE ANALYTICAL SOLUTION OF THE SYSTEM",
"text": "As shown in Fig. 1 , we consider a Raman Λ-type configuration, which corresponds to the actual process in NIST experiments. Like in [13] , we will employ some unitary transformations to get rid of the assumption of Lamb-Dicke limit and the WER. So our solution is more general than most of the previous work [14] . For a single trapped 2 ion experiencing two off-resonant counter-propagating traveling wave lasers with frequencies ω 1 and ω 2 , respectively, and in the case of a large detuning δ, we have an effective two-level system with the lasers driving the electric-dipole forbidden transition |g ↔ |e by the effective laser frequency ω L = ω 1 -ω 2 . So we have the dimensionless Hamiltonian\n\nH = ∆ 2 σ z + a † a + Ω 2 (σ + e iη x + σ -e -iη x), ( 1\n\n)\n\nin the frame rotating with ω L , where ∆ = (ω 0 -ω L )/ν, ω 0 and ν are the resonant frequency of the two levels of the ion and the trap frequency, respectively. Ω is the dimensionless Rabi frequency in units of ν and η the Lamb-Dicke parameter. σ ±,z are usual Pauli operators, and we have x = a † + a for the dimensionless position operator of the ion with a † and a being operators of creation and annihilation of the phonon field, respectively. We suppose that both Ω and ν are much larger than the atomic decay rate and the phonon dissipative rate so that no dissipation is considered below. Like in [13] , we first carry out some unitary transformations on Eq. ( 1 ) to avoid the expansion of the exponentials. So we have\n\nH I = U HU † = Ω 2 σ z + a † a + g(a † + a)σ x + ǫσ x + g 2 , ( 2\n\n)\n\nwhere\n\nU = 1 √ 2 e iπa † a/2 F † (η) F (η) -F † (η) F (η) ,\n\nwith F (η) = exp [iη(a † + a)/2], g = η/2, and ǫ = -∆/2. Eq. ( 2 ) is a typical driving Jaynes-Cummings model including the counter-rotating wave terms. In contrast to the usual treatments to consider the Lamb-Dicke limit by using the RWA in a frame rotation, we remain the counter-rotating wave interaction in the third term of the right-hand side of Eq. ( 2 ) in our case. To go on our treatment, we make a further rotation with V = exp (iπσ y /4), yielding\n\nH ′ = V H I V † = - Ω 2 σ x + a † a + g(a † + a)σ z + ǫσ z + g 2 , ( 3\n\n)\n\nwhere we have used exp (iθσ y )σ x exp (-iθσ y ) = cos(2θ)σ x + sin(2θ)σ z , and exp (iθσ y )σ z exp (-iθσ y ) = cos(2θ)σ zsin(2θ)σ x . For convenience of our following treatment, we rewrite Eq. ( 3 ) to be\n\nH ′ = ǫ(|e e| -|g g|) - Ω 2 (|e g| + |g e|) + a † a + g(a † + a)(|e e| -|g g|) + g 2 . ( 4\n\n)\n\nUsing Schrödinger equation, and the orthogonality between |e and |g , we suppose\n\n| = |ϕ 1 |e + |ϕ 2 |g , ( 5\n\n) which yields ǫ|ϕ 1 + a † a|ϕ 1 + g(a † + a)|ϕ 1 - Ω 2 |ϕ 2 + g 2 |ϕ 1 = E|ϕ 1 , ( 6\n\n) -ǫ|ϕ 2 + a † a|ϕ 2 -g(a † + a)|ϕ 2 - Ω 2 |ϕ 1 + g 2 |ϕ 2 = E|ϕ 2 . ( 7\n\n)\n\nTo make the above equations concise, we apply the displacement operator D(g\n\n) = exp [g(a † -a)] on a † and a, which gives A = D(g) † a D(g) = a+g, A † = D(g) † a † D(g) = a † +g, B = D(-g) † a D(-g) = a-g, and B † = D(-g) † a † D(-g) = a † -g. So we have (A † A + ǫ)|ϕ 1 - Ω 2 |ϕ 2 = E|ϕ 1 , ( 8\n\n) (B † B -ǫ)|ϕ 2 - Ω 2 |ϕ 1 = E|ϕ 2 . ( 9\n\n)\n\n3 Obvious, the new operators work in different subspaces, which leads to different evolutions regarding different internal levels |g and |e . We will later refer to this feature to be relevant to spin-dependent force. The solution of the two equations above can be simply set as\n\n|ϕ 1 = N n=0 c n |n A , ( 10\n\n) |ϕ 2 = N n=0 d n |n B , ( 11\n\n)\n\nwith N a large integer to be determined later\n\n, |n A = 1 √ n! (a † + g) n |0 A = 1 √ n! (a † + g) n D(g) † |0 = 1 √ n! (a † + g) n exp{-ga † -g 2 /2}|0 , and |n B = 1 √ n! (a † -g) n |0 B = 1 √ n! (a † -g) n D(-g) † |0 = 1 √ n! (a † -g) n exp{ga † -g 2 /2}|0 .\n\nTaking Eqs. (10) and (11) into Eqs. (8) and (9), respectively, and multiplying by A m| and B m|, respectively, we have,\n\n(m + ǫ)c m - Ω 2 N n=0 (-1) n D mn d n = Ec m , ( 12\n\n) (m -ǫ)d m - Ω 2 N n=0 (-1) m D mn c n = Ed m , ( 13\n\n)\n\nwhere we have set (-1\n\n) n D mn = A m|n B and (-1) m D mn = B m|n A ,\n\nwhose deduction can be found in Appendix. Diagonizing the relevant determinants, we may have the eigenenergies E i and the eigenvectors regarding c i n and d i n\n\n(n = 0, • • • , N, i = 0, • • • , N\n\n). Therefore, as long as we could find a closed subspace with c i N +1 and d i N +1 approaching zero for a certain big integer N, we may have a complete eigensolution of the system."
},
{
"section_type": "DISCUSSION",
"section_title": "III. DISCUSSION BASED ON NUMERICS",
"text": "Before doing numerics, we first consider a treatment by involving the RWA. As the RWA solution could present complete eigenenergy spectra, it is interesting to make a comparison between the RWA solution and our non-RWA one. We consider a rotation in Eq. ( 2 ) with respect to exp{-i[(Ω/2)σ z + a † a]t}, which results in\n\nH A = Ω 2 σ z + a † a + g(aσ + + a † σ -) + g 2 , ( 14\n\n)\n\nwhere the RWA has been made by setting Ω = 1, and we have corresponding eigenenergies\n\nE ± n = (n + g 2 + 1/2) ± g √ n + 1. ( 15\n\n)\n\nSo the system is degenerate in the case of η = 0 and there are two eigenenergy spectra corresponding to E ± n as long as η = 0. Figs. 2(a) and 2(b) demonstrate two spectra, respectively, and in each figure we compare the differences between the RWA and non-RWA solutions [15] . In contrast to the two spectra in the RWA solution, the non-RWA solution includes only one spectrum. Comparing the two eigensolutions, we find that the even-number and odd-number excited levels in the non-RWA case correspond to E + n and E - n of the RWA case, respectively, and the difference becomes bigger and bigger with the increase of η. It is physically understandable for these differences because the RWA solution, valid only for small η, does not work beyond the Lamb-Dicke regime. Above comparison also demonstrates the change of the ion trap system from an integrable case (i.e., with RWA validity) to the non-integrable case (i.e., without RWA validity). But besides these differences, we find an unusual result in this comparison, i.e., a new level without the counterpart in RWA solution appearing in our solution, which is lower than the ground state in RWA solution by ν + xη with x a η-dependent coefficient. In the viewpoint of physics, due to additional counter-rotating wave interaction involved, it is reasonable to have something more in our solution than the RWA case, although this does not surely lead to a new level lower than the previous ground state. Anyway, this is a good news for quantum information processing with trapped ions. As the situation in SER and beyond the Lamb-Dicke limit involves more instability, a stable 4 confinement of the ion requires a stronger trapping condition. In this sense, our solution, with the possibility to have the ion stay in an energically lower state, gives a hope in this respect. We will come to this point again later. Since no report of the new ground state had been found either theoretically or experimentally in previous publications, we suggest to check it experimentally by resonant absorption spectrum. As shown above, in the case of non-zero Lamb-Dicke parameter, the degeneracy of the neighboring level spacing is released, and the bigger the η, the larger the spacing difference between the neighboring levels. Therefore, an experimental test of the newly found ground state should be available by resonant transition between the ground and the first excited states in Fig. 2 , once the SER is reached. We have noticed that the SER could be achieved by first cooling the ions within the Lamb-Dicke limit and under the WER, and then by decreasing the trap frequency by opening the trap adiabatically [6] .\n\nSince it is lower in energy than the previously recognized ground states, the new ground state we found is more stable, and thereby more suitable to store quantum information. Once the trapped ion is cooled down to the ground state in the SER, it is, as shown in Eq. ( 5 ) with n = 0, actually equivalent to the effect of a spin-dependent force on the trapped ion [16] . If we make Hadamard gate on the ion by |g → (|g + |e )/ √ 2 and |e → (|g -|e )/ √ 2, we reach a Schrödinger cat state, i.e., (1/2){[\n\nD † (g)|0 + D † (-g)|0 ]|g -[D † (g)|0 -D † (-g)|0 ]|e }.\n\nTwo ions confined in a trap in above situation will yield two-qubit gates without really exciting the vibrational mode [11] . It is also the way with this spin-dependent force towards scalable quantum information processing [12] . As in SER, we may have larger Rabi frequency than in WER, the quantum gate could be in principle carried out faster in the SER.\n\nIn addition, as it is convergent throughout the parameter subspace, our complete eigensolution enables us to accurately write down the state of the system at an arbitrary evolution time, provided that we have known the initial state. This would be useful for future experiments in preparing non-classical states and in designing any desired quantum gates with trapped ions in the SER and beyond the Lamb-Dicke limit. Moreover, as shown in Figs 3(a), 3(b) and 3(c), our present solution is helpful for us to understand the particular solutions in previous publication [13] . The comparison in the figures shows that the results in [13] are actually mixtures of different eigensolutions. For example, the lowest level in Fig. 2 in [ 13] , corresponding to Ω = 2 and η = 0.2, is actually constituted at least by the third, the fourth, and the fifth excited states of the eigensolution."
},
{
"section_type": "DISCUSSION",
"section_title": "IV. FURTHER DISCUSSION AND CONCLUSION",
"text": "The observation of the counter-rotating effects is an interesting topic discussed previously. In [17], a standard method is used to study the observable effects regarding the rotating and the counter-rotating terms in the Jaynes-Cummings model, including to observe Bloch-Siegert shift [18] and quantum chaos in a cavity QED by using differently polarized lights. A recent work [19] for a two-photon Jaynes-Cummings model has also investigated the observability of the counter-rotating terms. By using perturbation theory, the authors claimed that the counter-rotating effects, although very small, can be in principle observed by measuring the energy of the atom going through the cavity. Actually, for the cavity QED system without any external source involved, it is generally thought that the counter-rotating terms only make contribution in some virtual fluctuations of the energy in the weak coupling regime. While the interference between the rotating and counter-rotating contributions could result in some phase dependent effects [20] . Anyway, if there is an external source, for example, the laser radiating a trapped ultracold ion, the counter-rotating terms will show their effects, e.g., related to heating in the case of WER [21] . In this sense, our result is somewhat amazing because the counter-rotating interaction in the SER, making entanglement between internal and vibrational states of the trapped ion, plays positive role in the ion trapping.\n\nWe argue that our approach is applicable to different physical processes involving counter-rotating interaction. Since the counter-rotating terms result in energy nonconservation in single quanta processes, usual techniques cannot solve the Hamiltonian with eigenstates spanning in an open form. In this case, path-integral approach [22] and perturbation approach [20] , assisted by numerical techniques were employed in the weak coupling regime of the Jaynes-Cummings model. In contrast, our method, based on the diagonalization of the coherent-state subspace, could in principle study the Jaynes-Cummings model without the RWA in any cases. We have also noticed a recent publication [23] to treat a strongly coupled two-level system to a quntum oscillator under an adiabatic approximation, in which something is similar to our work in the solution of the Hamiltonian in the absence of the RWA. But due to the different features in their system from our atomic case, the two-level splitting term, much smaller compared to other terms, can be taken as a perturbation. So the advantage of that treatment is the possibility to analytically obtain good approximate solutions. In contrast, not any approximation is used in our solution, which should be more efficient to do the relevant job.\n\nIn summary, we have investigated the eigensolution of the system with a single trapped ion, experiencing two traveling waves of lasers, in the SER and in the absence of the Lamb-Dicke limit. We have found the ground state in the non-RWA case to be energically lower than the counterpart of the solution with RWA, which would be useful for quantum information storage and for quantum computing. The analytical forms of the eigenfunction and the 5 complete set of the eigensolutions would be helpful for us to understand a trapped ion in the SER and with a large Lamb-Dicke parameter. We argue that our work would be applied to different systems in dealing with strong coupling problems."
},
{
"section_type": "OTHER",
"section_title": "V. ACKNOWLEDGMENTS",
"text": "This work is supported in part by NNSFC No. 10474118, by Hubei Provincial Funding for Distinguished Young Scholars, and by Sichuan Provincial Funding."
},
{
"section_type": "OTHER",
"section_title": "VI. APPENDIX",
"text": "We give the deduction of A m|n B and B m|n A below,\n\nA m|n B = 1 √ m!n! 0|e -ga-g 2 /2 (a + g) m (a † -g) n e ga † -g 2 /2 |0 = 1 √ m!n! e -2g 2 0|(a + g) m e ga † e -ga (a † -g) n |0 = 1 √ m!n! e -2g 2 0|(a + 2g) m (a † -2g) n |0 = (-1) n D mn , with D mn = e -2g 2 min[m,n] i=0 (-1) -i √ m!n!(2g) m+n-2i (m -i)!(n -i)!i! .\n\nIt is easily proven following a similar step to above that\n\nB m|n A = 1 √ m!n! 0|e ga-g 2 /2 (a -g) m (a † + g) n e -ga † -g 2 /2 |0 ,\n\nwould finally get to (-1) m D mn .\n\n[1] Cirac J I, Zoller P 1995 Phys. Rev. Lett. 74 4091 [2] Monroe C, Meekhof D M, King B E, Itano W M, Wineland D J 1995 Phys. Rev. Lett. 75 4714 [3] Turchette Q A, Wood C S, King B E, Myatt C J, Leibfried D, Itano W M, Monroe C, Wineland D J 1998 Phys. Rev. Lett. 81 3631; Sackett C A, Kielpinski D, King B E, Langer C, Meyer V, Myatt C J, Rowe M, Turchette Q A, Itano W M, Wineland D J, Monroe C 2000 Nature 404 256 [4] Gulde S, Riebe M, Lancaster G P T, Becher C, Eschner J, Haeffner H, Schmidt-Kaler F, Chuang I L, Blatt R 2003 Nature 421 48 [5] Riebe M, Haeffner H, Roos C F, Haensel W, Benhelm J, Lancaster G P T, Koerber T W, Becher C, Schmidt-Kaler F, James D F V, Blatt R 2004 Nature 429 734; Barrett M D, Chiaverini J, Schaetz T, Britton J, Itano W M, Jost J D, Knill E, Langer C, Leibfried D, Ozeri R, Wineland D J 2004 Nature 429 737 [6] Poyators J F, Cirac J I, Blatt R, Zoller P 1996 Phys. Rev. A 54 1532; Poyatos J F, Cirac J I, Zoller P 1998 Phys. Rev.\n\nLett. 81 1322 [7] Zheng S, Zhu X W, Feng M 2000 Phys. Rev. A 62 033807 [8] Feng M 2004 Eur. Phys. J. D 29 189 [9] Feng M, Zhu X, Fang X, Yan M, Shi L 1999 J. Phys. B 32 701; Feng M 2002 Eur. Phys. J. D 18 371 [10] Zeng H, Lin F, Wang Y, Segawa Y 1999 Phys. Rev. A 59 4589 [11] Garcia-Ripoll J J, Zoller P and Cirac J I 2003 Phys. Rev. Lett. 91 157901; [12] Duan L -M 2004 Phys. Rev. Lett. 93 100502 [13] Feng M 2001 J. Phys. B 34 451 6 [14] Most of the previous work in this respect were carried out by cuting off the expansion of the exponentials regarding the quantized phonon operators, which is only reasonable in the WER and within the Lamb-Dicke limit. In contrast, our treatment can be used in both the SER and the WER cases. [15] We take throughout this paper N = 40 in which the coefficients c i 41 and d i 41 with i = 0, 1, ..40 are negligible in the case of Ω = 1 and 2. Although with the increase of values of Ω the diagonalization space has to be enlarged, our analytical method generally works well in a wide range of parameters. [16] Haljan P C, Brickman K -A, Deslauriers L, Lee P J and Monroe C 2005 Phys. Rev. Lett. 94 153602 [17] Crisp M D 1991 Phys. Rev. A 43 2430 [18] Bloch F and Siegert A 1940 Phys. Rev. 57 522 [19] Janowicz M and Orlowski A 2004 Rep.Math. Phys. 54 71 [20] Phoenix S J D 1989 J. Mod. Optics 3 127 [21] Leibfrid D, Blatt R, Monroe C, and Wineland D J 2003 Rev. Mod. Phys. 75 281 [22] Zaheer K and Zubairy M S 1998 Phys. Rev. A 37 1628 [23] Irish E K, Gea-Banacloche J, Martin I, and Schwab K C 2005 Phys. Rev. B 72 195410\n\nThe captions of the figures\n\nFig. 1 Schematic of a single trapped ion under radiation of two traveling wave lasers, where ω 1 and ω 2 are frequencies regarding the two lasers, respectively, ω 0 is the resonant frequency between |g and |e , and δ and ∆ are relevant detunings. This is a typical Raman process employed in NIST experiments, with for example Be + , for quantum computing.\n\nFig. 2 The eigenenergy spectra with Ω = 1, where (a) and (b) correspond to two different sets of eigenenergies with respect to Lamb-Dicke parameter. In (a) the comparison is made between E + n in the RWA case (dashed-dotted curves) and E n with n = even numbers in the non-RWA case (star curves for n = 0 and solid curves for others); In (b) the comparison is for E - n in the RWA case (dashed-dotted curves) to E n with n = odd numbers in the non-RWA case (solid curves).\n\nFig. 3 The eigenenergy with respect to the detuning ∆, where for convenience of comparison we have used the same parameter numbers as in [13] . For clarity, we plot the different levels with different lines. The parameter numbers are Ω = 2, and (a) η = 0.2; (b) η = 0.4; (c) η = 0.6."
}
] | |
arxiv:0704.0121 | 0704.0121 | 1 | 10.1088/0264-9381/25/7/075001 | d667e8a785fe3fccabc03f16e745f748f169faf7f75bfc37ac9c853839a39ce5 | Meta-Stable Brane Configuration of Product Gauge Groups | Starting from the N=1 SU(N_c) x SU(N_c') gauge theory with fundamental and
bifundamental flavors, we apply the Seiberg dual to the first gauge group and
obtain the N=1 dual gauge theory with dual matters including the gauge
singlets. By analyzing the F-term equations of the superpotential, we describe
the intersecting type IIA brane configuration for the meta-stable
nonsupersymmetric vacua of this gauge theory. By introducing an orientifold
6-plane, we generalize to the case for N=1 SU(N_c) x SO(N_c') gauge theory with
fundamental and bifundamental flavors. Finally, the N=1 SU(N_c) x Sp(N_c')
gauge theory with matters is also described very briefly. | [
"Changhyun Ahn"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | Starting from the N = 1 SU(N c ) × SU(N ′ c ) gauge theory with fundamental and bifundamental flavors, we apply the Seiberg dual to the first gauge group and obtain the N = 1 dual gauge theory with dual matters including the gauge singlets. By analyzing the F-term equations of the superpotential, we describe the intersecting type IIA brane configuration for the meta-stable nonsupersymmetric vacua of this gauge theory. By introducing an orientifold 6-plane, we generalize to the case for N = 1 SU(N c )×SO(N ′ c ) gauge theory with fundamental and bifundamental flavors. Finally, the N = 1 SU(N c ) × Sp(N ′ c ) gauge theory with matters is also described very briefly.
It is well-known that the N = 1 SU(N c ) QCD with fundamental flavors has a vanishing superpotential before we deform this theory by mass term for quarks. The vanishing superpotential in the electric theory makes it easier to analyze its nonvanishing dual magnetic superpotential. Sometimes by tuning the various rotation angles between NS5-branes and D6-branes appropriately, even if the electric theory has nonvanishing superpotential, one can make the nonzero superpotential to vanish in the electric theory. Two procedures, deforming the electric gauge theory by adding the mass for the quarks and taking the Seiberg dual magnetic theory from the electric theory, are crucial to find out meta-stable supersymmetry breaking vacua in the context of dynamical supersymmetry breaking [1, 2] . Some models of dynamical supersymmetry breaking can be studied by gauging the subgroup of the flavor symmetry group by either field theory analysis or using the brane configuration 1 .
In this paper, starting from the known N = 1 supersymmetric electric gauge theories, we construct the N = 1 supersymmetric magnetic gauge theories by brane motion and linking number counting. The dual gauge group appears only on the first gauge group. Based on their particular limits of corresponding magnetic brane configurations in the sense that their electric theories do not have any superpotentials except the mass deformations for the quarks, we describe the intersecting brane configurations of type IIA string theory for the meta-stable nonsupersymmetric vacua of these gauge theories.
We focus on the cases where the whole gauge group is given by a product of two gauge groups. One example can be realized by three NS5-branes with D4-and D6-branes, and the other by four NS5-branes with D4-and D6-branes. For the latter, the appropriate orientifold 6-plane should be located at the center of this brane configuration in order to have two gauge groups. Of course, it is also possible, without changing the number of gauge groups, to have the brane configuration consisting of five NS5-branes and orientifold 6-plane, at which the extra NS5-brane is located, with D4-and D6-branes, but we'll not do this particular case in this paper.
In section 2, we review the type IIA brane configuration that contains three NS5-branes, corresponding to the electric theory based on the N = 1 SU(N c ) × SU(N ′ c ) gauge theory [4, 5, 6] with matter contents and deform this theory by adding the mass term for the quarks. Then we construct the Seiberg dual magnetic theory which is N = 1 SU( N c ) × SU(N ′ c ) gauge theory with corresponding dual matters as well as various gauge singlets, by brane motion and linking number counting. We do not touch the part of second gauge group SU(N ′ c ) in this dual process.
In section 3, we consider the nonsupersymmetric meta-stable minimum by looking at the magnetic brane configuration we obtained in section 2 and present the corresponding intersecting brane configuration of type IIA string theory, along the line of [7, 8, 9, 10, 11] (see also [12, 13, 14] ) and we describe M-theory lift of this supersymmetry breaking type IIA brane configuration.
In section 4, we describe the type IIA brane configuration that contains four NS5-branes, corresponding to the electric theory based on the N = 1 SU(N c ) × SO(N ′ c ) gauge theory [15] with matter contents and deform this theory by adding the mass term for the quarks.
Then we take the Seiberg dual magnetic theory which is N = 1 SU( N c ) × SO(N ′ c ) gauge theory with corresponding dual matters as well as various gauge singlets, by brane motion and linking number counting. The part of second gauge group SO(N ′ c ) in this dual process is not changed under this process.
In section 5, the nonsupersymmetric meta-stable minimum by looking at the magnetic brane configuration we obtained in section 4 is constructed and we present the corresponding intersecting brane configuration of type IIA string theory and describe M-theory lift of this supersymmetry breaking type IIA brane configuration, as we did in section 3.
In section 6, we describe the similar application to the N = 1 SU(N c ) × Sp(N ′ c ) gauge theory [15] briefly and make some comments for the future directions. 2 The N = 1 supersymmetric brane configuration of SU (N c ) × SU (N ′ c ) gauge theory
After reviewing the type IIA brane configuration corresponding to the electric theory based on the N = 1 SU(N c )×SU(N ′ c ) gauge theory [4, 5, 6] , we construct the Seiberg dual magnetic theory which is N = 1 SU( N c ) × SU(N ′ c ) gauge theory.
(N c ) × SU (N ′ c ) gauge group
The gauge group is given by SU(N c ) × SU(N ′ c ) and the matter contents [4, 5, 6] • The flavor singlet field X is in the bifundamental representation (N c , N ′ c ) under the gauge group and its complex conjugate field X is in the bifundamental representation (N c , N ′ c ) under the gauge group
In the electric theory, since there exist N f quarks Q, N f quarks Q, one bifundamental field X which will give rise to the contribution of N ′ c and its complex conjugate X which will give rise to the contribution of N ′ c , the coefficient of the beta function of the first gauge group factor is
b SU (Nc) = 3N c -N f -N ′ c
and similarly since there exist
N ′ f quarks Q ′ , N ′ f quarks Q ′ ,
one bifundamental field X which will give rise to the contribution of N c and its complex conjugate X which will give rise to the contribution of N c , the coefficient of the beta function of the second gauge group factor is
b SU (N ′ c ) = 3N ′ c -N ′ f -N c .
The anomaly free global symmetry is given by L -brane) is located at the left hand side of a middle NS5-brane along the x 6 direction and there exist N ′ c color D4-branes suspended between them, with N ′ f D6-branes which have zero (45) directions. These are shown in Figure 1 explicitly. See also [3] for the brane configuration.
[SU(N f ) × SU(N ′ f )] 2 × U(1) 3 × U(1) R [4,
By realizing that the two outer NS5 ′ L,R -branes are perpendicular to a middle NS5-brane and the fact that N f D6-branes are parallel to NS5 ′ R -brane and N ′ f D6-branes are parallel to NS5 ′ L -brane, the classical superpotential vanishes. However, one can deform this theory. Then the classical superpotential by deforming this theory by adding the mass term for the quarks Q and Q, along the lines of [1, 11, 10, 9, 8, 7] , is given by
W = mQ Q (2.1)
and this type IIA brane configuration can be summarized as follows foot_1 :
• One middle NS5-brane with worldvolume (012345).
• Two NS5'-branes with worldvolume (012389).
• N f D6-branes with worldvolume (0123789) located at the positive region in v direction.
• N c color D4-branes with worldvolume (01236). These are suspended between a middle NS5-brane and NS5 ′ R -brane. • N ′ c color D4-branes with worldvolume (01236). These are suspended between NS5 ′ Lbrane and a middle NS5-brane. Now we draw this electric brane configuration in Figure 1 and we put the coincident N f D6-branes in the nonzero v direction. If we ignore the left NS5 ′ L -brane, N ′ c D4-branes and N ′ f D6-branes, then this brane configuration corresponds to the standard N = 1 SQCD with the gauge group SU(N c ) with N f massive flavors. The electric quarks Q and Q correspond to strings stretching between the N c color D4-branes with N f D6-branes, the electric quarks Q ′ and Q ′ correspond to strings between the N ′ c color D4-branes with N ′ f D6-branes and the bifundamentals X and X correspond to strings stretching between the N c color D4-branes with N ′ c color D4-branes.
Figure 1 :
The N = 1 supersymmetric electric brane configuration of SU(N c ) × SU(N ′ c ) with N f chiral multiplets Q, N f chiral multiplets Q, N ′ f chiral multiplets Q ′ , N ′ f chiral multiplets Q ′ ,
the flavor singlet bifundamental field X and its complex conjugate bifundamental field X. The N f D6-branes have nonzero v coordinates where v = m for equal massive case of quarks Q, Q while Q ′ and Q ′ are massless.
( N c ) × SU (N ′ c ) gauge group
One can consider dualizing one of the gauge groups regarding as the other gauge group as a spectator. One takes the Seiberg dual for the first gauge group factor SU(N c ) while remaining the second gauge group factor SU(N ′ c ) unchanged. Also we consider the case where Λ 1 >> Λ 2 , in other words, the dualized group's dynamical scale is far above that of the other spectator group.
Let us move a middle NS5-brane to the right all the way past the right NS5 ′ R -brane. For example, see [12, 13, 14, 11, 10, 9, 8, 7] . After this brane motion, one arrives at the Figure 2 . Note that there exists a creation of N f D4-branes connecting N f D6-branes and NS5 ′ R -brane. Recall that the N f D6-branes are perpendicular to a middle NS5-brane in Figure 1 . The linking number [16] of NS5-brane from Figure 2
is L 5 = N f 2 -N c .
On the other hand, the linking number of NS5-brane from Figure 1 is
L 5 = - N f 2 + N c -N ′ c .
Due to the connection of N ′ c D4-branes with NS5 ′ R -brane, the presence of N ′ c in the linking number arises. From these two relations, one obtains the number of colors of dual magnetic theory
N c = N f + N ′ c -N c . (2.2)
The linking number counting looks similar to the one in [7] where there exists a contribution from O4-plane. Let us draw this magnetic brane configuration in Figure 2 and recall that we put the coincident N f D6-branes in the nonzero v directions in the electric theory, along the lines of [12, 13, 14, 11, 10, 9, 8, 7] . The N f created D4-branes connecting between D6-branes and NS5 ′ R -brane can move freely in the w direction. Moreover since N ′ c D4-branes are suspending between two equal NS5 ′ L,R -branes located at different x 6 coordinate, these D4-branes can slide along the w direction also. If we ignore the left NS5 ′ L -brane, N ′ c D4-branes and N ′ f D6-branes(detaching these from Figure 2 ), then this brane configuration corresponds to the standard N = 1 SQCD with the magnetic gauge group SU( N c = N f -N c ) with N f massive flavors [12, 13, 14] . The dual magnetic gauge group is given by SU( N c ) × SU(N ′ c ) and the matter contents are given by • N f chiral multiplets q are in the fundamental representation under the SU( N c ), N f chiral multiplets q are in the antifundamental representation under the SU( N c ) and then q are in the representation ( N c , 1) while q are in the representation ( N c , 1) under the gauge group
• N ′ f chiral multiplets Q ′ are
c = N f + N ′ c - N c ) × SU(N ′ c ) with N f chiral multiplets q, N f chiral multiplets q, N ′ f chiral multiplets Q ′ , N ′
f chiral multiplets Q ′ , the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Y as well as N f fields F ′ , its complex conjugate N f fields F ′ , N 2 f fields M and the gauge singlet Φ. There exist N f flavor D4-branes connecting D6-branes and • The N ′2 c -fields Φ is in the representation (1, N ′2 c -1) ⊕ (1, 1) under the gauge group This corresponds to the SU(N c ) chiral meson X X and note that X has a representation
NS5 ′ R -brane.
N ′ c of SU(N ′ c ) while X has a representation N ′ c of SU(N ′ c
). The fluctuations of the singlet Φ correspond to the motion of N ′ c D4-branes suspended two NS5 ′ L,R -branes along the (789) directions in Figure 2 .
In the dual theory, since there exist N f quarks q, N f quarks q, one bifundamental field Y which will give rise to the contribution of N ′ c and its complex conjugate Y which will give rise to the contribution of N ′ c , the coefficient of the beta function for the first gauge group factor [6] is
b mag SU ( e Nc) = 3 N c -N f -N ′ c = 2N f + 2N ′ c -3N c
where we inserted the number of colors given in (2.2) in the second equality and since there exist
N ′ f quarks Q ′ , N ′ f quarks Q ′ ,
one bifundamental field Y which will give rise to the contribution of N c , its complex conjugate Y which will give rise to the contribution of N c , N f fields F ′ , its complex conjugate N f fields F ′ and the singlet Φ which will give rise to N ′ c , the coefficient of the beta function of second gauge group factor [6] is
b mag SU (N ′ c ) = 3N ′ c -N ′ f -N c -N f -N ′ c = N ′ c + N c -2N f -N ′ f .
Therefore
SU (N ′ c ) -b mag SU (N ′ c ) > 0, SU(N ′ c ) is more asymptotically free than SU(N ′ c ) mag [6]. Neglecting the SU(N ′ c ) dynamics, the magnetic SU( N c ) is IR free when N f + N ′ c < 3 2 N c [6]
. The dual magnetic superpotential, by adding the mass term (2.1) for Q and Q in the electric theory which is equal to put a linear term in M in the dual magnetic theory, is given by
W dual = Mq q + Y F ′ q + Y q F ′ + ΦY Y + mM (2.3)
where the mesons in terms of the fields defined in the electric theory are
M ≡ Q Q, Φ ≡ X X, F ′ ≡ XQ, F ′ ≡ X Q.
By orientifolding procedure(O4-plane) into this brane configuration, the q(Q) and q( Q) are equivalent to each other, the Y (X) and Y ( X) become identical and F ′ and F ′ become the same. Then the reduced superpotential is identical with the one in [7] . Here q and q are fundamental and antifundamental for the gauge group index respectively and antifundamentals for the flavor index. Then, q q has rank N c while m has a rank N f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M, cannot be satisfied if the rank N f exceeds N c . This is so-called rank condition and the supersymmetry is broken.
Other F-term equations are satisfied by taking the vacuum expectation values of Y, Y , F ′ and F ′ to vanish.
The classical moduli space of vacua can be obtained from F-term equations
q q + m = 0, qM + F ′ Y = 0, Mq + Y F ′ = 0, F ′ q + Y Φ = 0, qY = 0, q F ′ + ΦY = 0, Y q = 0, Y Y = 0.
Then, it is easy to see that there exist three reduced equations qM = 0 = Mq, q q + m = 0 and other F-term equations are satisfied if one takes the zero vacuum expectation values for the fields Y, Y , F ′ and F ′ . Then the solutions can be written as follows:
< q > = √ me φ 1 e Nc 0 , < q >= √ me -φ 1 e Nc 0 , < M >= 0 0 0 Φ 0 1 N f -e Nc < Y > = < Y >=< F ′ >=< F ′ >= 0.
(2.4)
Let us expand around a point on (2.4), as done in [1] . Then the remaining relevant terms of superpotential are given by W rel dual = Φ 0 (δϕ δ ϕ + m) + δZ δϕ q 0 + δ Z q 0 δ ϕ by following the same fluctuations for the various fields as in [9] :
q = q 0 1 e Nc + 1 √ 2 (δχ + + δχ -)1 e Nc δϕ , q = q 0 1 e Nc + 1 √ 2 (δχ + -δχ -)1 e Nc δ ϕ , M = δY δZ δ Z Φ 0 1 N f -e
as well as the fluctuations of Y, Y , F ′ and F ′ . Note that there exist also three kinds of terms, the vacuum < q > multiplied by δ Y δ F ′ , the vacuum < q > multiplied by δF ′ δY , and the vacuum < Φ > multiplied by δY δ Y . However, by redefining these, they do not enter the contributions for the one loop result, up to quadratic order. As done in [17] , one gets that m 2 Φ 0 will contain (log 4 -1) > 0 implying that these are stable. Then the minimal energy supersymmetry breaking brane configuration is shown in Figure 3 , along the lines of [12, 13, 14, 11, 10, 9, 8, 7] . If we ignore the left NS5 ′ L -brane, N ′ c D4branes and N ′ f D6-branes(detaching these from Figure 3 ), as observed already, then this brane configuration corresponds to the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group SU( N c = N f -N c ) with N f massive flavors [12, 13, 14] .
c = N f + N ′ c -N c ) × SU(N ′ c ) with N f chiral multiplets q, N f chiral multiplets q, N ′ f chiral multiplets Q ′ , N ′ f chiral multiplets Q ′ ,
the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Y and various gauge singlets.
The type IIA/M-theory brane construction for the N = 2 gauge theory was described by [18] and after lifting the type IIA description to M-theory, the corresponding magnetic M5-brane configuration foot_2 with equal mass for the quarks where the gauge group is given by
SU( N c )×SU(N ′ c ), in a background space of xt = v N ′ f N f k=1 (v-e k )
where this four dimensional space replaces (45610) directions, is described by
t 3 + (v e Nc + • • • )t 2 + (v N ′ c + • • • )t + v N ′ f N f k=1 (v -e k ) = 0 (3.1)
where e k is the position of the D6-branes in the v direction(for equal massive case, we can write e k = m) and we have ignored the lower power terms in v in t 2 and t denoted by • • • and the scales for the gauge groups in front of the first term and the last term, for simplicity. For fixed x, the coordinate t corresponds to y.
From this curve (3.1) of cubic equation for t above, the asymptotic regions for three NS5branes can be classified by looking at the first two terms providing NS5-brane asymptotic region, next two terms providing NS5 ′ R -brane asymptotic region and the final two terms giving NS5 ′ L -brane asymptotic region as follows 1. v → ∞ limit implies
w → 0, y ∼ v e Nc + • • • NS asymptotic region. 2. w → ∞ limit implies v → m, y ∼ w N f +N ′ f -N ′ c + • • • NS ′ L asymptotic region, v → m, y ∼ w N ′ c -e Nc + • • • NS ′ R asymptotic region.
Here the two NS5 ′ L,R -branes are moving in the +v direction holding everything else fixed instead of moving D6-branes in the +v direction, in the spirit of [14] . The harmonic function sourced by the D6-branes can be written explicitly by summing over two contributions from the N f and N ′ f D6-branes and similar analysis to both solve the differential equation and find out the nonholomorphic curve can be done [14, 10, 9, 8, 7] . An instability from a new M5-brane mode arises. 4 The N = 1 supersymmetric brane configuration of
SU (N c ) × SO(N ′ c ) gauge theory
After reviewing the type IIA brane configuration corresponding to the electric theory based on the N = 1 SU(N c ) × SO(N ′ c ) gauge theory [15] , we describe the Seiberg dual magnetic theory which is N = 1 SU( N c ) × SO(N ′ c ) gauge theory.
(N c ) × SO(N ′ c ) gauge group
The gauge group is given by SU(N c ) × SO(N ′ c ) and the matter contents [15] (similar matter contents are found in [4] ) are given by In the electric theory, since there exist N f quarks Q, N f quarks Q, one bifundamental field X which will give rise to the contribution of N ′ c and its complex conjugate X which will give rise to the contribution of N ′ c , the coefficient of the beta function of the first gauge group factor is
• N f chiral multiplets Q are in
b SU (Nc) = 3N c -N f -N ′ c
and similarly, since there exist 2N ′ f quarks Q ′ , one bifundamental field X which will give rise to the contribution of N c and its complex conjugate X which will give rise to the contribution of N c , the coefficient of the beta function of the second gauge group factor is
b SO(N ′ c ) = 3(N ′ c -2) -2N ′ f -2N c .
The anomaly free global symmetry is given by SU 0123789 ) and an orientifold 6 plane (0123789) of positive Ramond charge foot_3 . According to Z 2 symmetry of orientifold 6-plane(O6-plane) sitting at v = 0 and x 6 = 0, the coordinates (x 4 , x 5 , x 6 ) transform as -(x 4 , x 5 , x 6 ), as usual. See also [3] for the discussion of O6-plane.
(N f ) 2 × SU(2N ′ f ) × U(1) 2 × U(1)
By rotating the third and fourth NS5-branes which are located at the right hand side of O6-plane, from v direction toward -w and +w directions respectively, one obtains N = 1 theory. Their mirrors, the first and second NS5-branes which are located at the left hand side of O6-plane, can be rotated in a Z 2 symmetric manner due to the presence of O6-plane simultaneously. That is, if the first NS5-brane rotates by an angle -ω in (v, w) plane, denoted by NS5 -ω -brane [3] , then the mirror image of this NS5-brane, the fourth NS5-brane, is rotated by an angle ω in the same plane, denoted by NS5 ω -brane. If the second NS5-brane rotates by an angle θ in (v, w) plane, denoted by NS5 θ -brane [3] , then the mirror image of this NS5-brane, the third NS5-brane, is rotated by an angle -θ in the same plane, denoted by NS5 -θ -brane. For more details, see the Figure 4 5 .
We also rotate the N ′ f D6-branes which are located between the second NS5-brane and an O6-plane and make them be parallel to NS5 θ -brane and denote them as D6 θ -brane with zero v coordinate(the angle between the unrotated D6-branes and D6 θ -branes is equal to π 2θ) and its mirrors N ′ f D6-branes appear as D6 -θ -branes between the O6-plane and third NS5-brane. There is no coupling between the adjoint field and the quarks since the rotated D6 θ -branes are parallel to the rotated NS5 θ -brane [5, 3] . Similarly, the N f D6-branes which are located between the third NS5-brane and the fourth NS5-brane can be rotated and we can make them be parallel to NS5 ω -brane and denote them as D6 ω -branes with nonzero v coordinate(the angle between the unrotated D6-branes and D6 ω -branes is equal to π 2ω) and its mirrors N f D6-branes appear as D6 -ω -branes between the first NS5-brane and the second NS5-brane.
Moreover the N c D4-branes are suspended between the first NS5-brane and the second NS5-brane(and its mirrors) and the N ′ c D4-branes are suspended between the second NS5brane and the third NS5-brane.
For this brane setup 6 , the classical superpotential is given by [15]
W = - 1 4 1 4 tan(ω -θ) + 1 tan 2θ tr(X X) 2 + tr X X XX 4 sin 2θ + (tr X X) 2 4N c tan(ω -θ) . (4.1)
It is easy to see that when θ approaches 0 and ω approaches π 2 , then this superpotential vanishes. 5 The angles of θ 1 and θ 2 in [15] are related to the angles θ and ω as follows: θ = θ 1 and ω = θ 2 . 6 For arbitrary angles θ and ω, the superpotential for the SU (N c ) sector is given by W = Xφ X + tan(ωθ) tr φ 2 where φ ia an adjoint field for SU (N c ). There is no coupling between φ and N f quarks because D6 ±ω -branes are parallel to N S5 ±ω -branes. The superpotential for the SO(N ′ c ) sector is given by W = Xφ A X + Xφ S X + tan θ tr φ 2 A -1 tan θ tr φ 2 S where φ A and φ S are an adjoint field and a symmetric tensor for SO(N ′ c ) [25] . After integrating out φ, φ A and φ S , the whole superpotential can be written as in (4.1).
Now one summarizes the supersymmetric electric brane configuration with their worldvolumes in type IIA string theory as follows.
• NS5 -ω -brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with negative x 6 .
• NS5 θ -brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with negative x 6 .
• NS5 -θ -brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with positive x 6 .
• NS5 ω -brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with positive x 6 .
• N ′ f D6 θ -branes with worldvolume by both (01237) and two spatial dimensions in (v, w) plane and with negative x 6 and v = 0.
• N ′ f D6 -θ -branes with worldvolume by both (01237) and two space dimensions in (v, w) plane and with positive x 6 and v = 0.
• N f D6 ω -branes with worldvolume by both (01237) and two spatial dimensions in (v, w) plane and with positive x 6 . Before the rotation, the distance from N c color D4-branes in the +v direction is nonzero.
• N f D6 -ω -branes with worldvolume by both (01237) and two space dimensions in (v, w) plane and with negative x 6 . Before the rotation, the distance from N c color D4-branes in the -v direction is nonzero.
• O6-plane with worldvolume (0123789) with v = 0 = x 6 .
• N c D4-branes connecting NS5 -ω -brane and NS5 θ -brane, with worldvolume (01236) with v = 0 = w(and its mirrors).
• N ′ c D4-branes connecting NS5 θ -brane and NS5 -θ -brane, with worldvolume (01236) with v = 0 = w.
We draw the type IIA electric brane configuration in Figure 4 which was basically given in [15] already but the only difference is to put N f D6-branes in the nonzero v direction in order to obtain nonzero masses for the quarks which are necessary to obtain the meta-stable vacua.
( N c ) × SO(N ′ c ) gauge group
One takes the Seiberg dual for the first gauge group factor SU(N c ) while remaining the second gauge group factor SO(N ′ c ), as in previous case. Also we consider the case where Λ 1 >> Λ 2 , in other words, the dualized group's dynamical scale is far above that of the other spectator group.
N f chiral multiplets Q, N f chiral multiplets Q, 2N ′
f chiral multiplets Q ′ , the flavor singlet bifundamental field X and its complex conjugate bifundamental field X. The N f D6 ω -branes have nonzero v coordinates where v = m(and its mirrors) for equal massive case of quarks Q, Q while Q ′ is massless.
Let us move the NS5 -θ -brane to the right all the way past the right NS5 ω -brane(and its mirrors to the left). After this brane motion, one arrives at the Figure 5 . Note that there exists a creation of N f D4-branes connecting N f D6 ω -branes and NS5 ω -brane(and its mirrors). Recall that the N f D6 ω -branes are not parallel to the NS5 -θ -brane in Figure 4 (and its mirrors). The linking number of NS5 -θ -brane from Figure 5 is L 5 = N f 2 -N c . On the other hand, the linking number of NS5 -θ -brane from Figure 4 is
L 5 = - N f 2 + N c -N ′ c
. From these, one gets the number of colors in dual magnetic theory
N c = N f + N ′ c -N c . (4.2)
Let us draw this magnetic brane configuration in Figure 5 and remember that we put the coincident N f D6 ω -branes in the nonzero v directions(and its mirrors). The N f created D4branes connecting between D6 ω -branes and NS5 ω -brane can move freely in the w direction, as in previous case. Moreover, since N ′ c D4-branes are suspending between two unequal NS5 ±ω -branes located at different x 6 coordinate, these D4-branes cannot slide along the w direction, for arbitrary rotation angles. If we are detaching all the branes except NS5 ω -brane, NS5 -θ -brane, D6 ω -branes, N f D4-branes and N c D4-branes from Figure 5 , then this brane configuration corresponds to N = 1 SQCD with the magnetic gauge group SU( N c = N f -N c ) with N f massive flavors with tilted NS5-branes.
The dual magnetic gauge group is given by SU( N c ) × SO(N ′ c ) and the matter contents are given by
c = N f + N ′ c - N c ) × SO(N ′ c
) with N f chiral multiplets q, N f chiral multiplets q, 2N ′ f chiral multiplets Q ′ , the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Y as well as N f fields F ′ , its complex conjugate N f fields F ′ , N 2 f fields M and the gauge singlet Φ. There exist N f flavor D4-branes connecting D6 ω -branes and NS5 ω -brane(and its mirrors).
• N f chiral multiplets q are in the fundamental representation under the SU( N c ), N f chiral multiplets q are in the antifundamental representation under the SU( N c ) and then q are in the representation ( N c , 1) while q are in the representation ( N c , 1) under the gauge group These additional 2N f SO(N ′ c ) vectors are originating from the SU(N c ) chiral mesons XQ and X Q respectively. It is easy to see that from the Figure 5 , since the D6 -ω -branes are parallel to the NS5 -ω -brane, the newly created N f D4-branes can slide along the plane consisting of D6 -ω -branes and NS5 -ω -brane arbitrarily(and its mirrors). Then strings connecting the N f D6 -ω -branes and N ′ c D4-branes will give rise to these additional 2N f SO(N ′ c ) vectors.
• 2N ′ f chiral multiplets Q ′ are
• N 2 f -fields M are in the representation (1, 1) under the gauge group This corresponds to the SU(N c ) chiral meson Q Q and the fluctuations of the singlet M correspond to the motion of N f flavor D4-branes along (789) directions in Figure 5 .
• The N ′ 2 c singlet Φ is in the representation (1, adj) ⊕ (1, symm) under the gauge group This corresponds to the SU(N c ) chiral meson X X and note that both X and X have representation N ′ c of SO(N ′ c ). In general, the fluctuations of the singlet Φ correspond to the motion of N ′ c D4-branes suspended two NS5 ±ω -branes along the (789) directions in Figure 5 . In the dual theory, since there exist N f quarks q, N f quarks q, one bifundamental field Y which will give rise to the contribution of N ′ c and its complex conjugate Y which will give rise to the contribution of N ′ c , the coefficient of the beta function of the first gauge group factor with (4.2) is
b mag SU ( e Nc) = 3 N c -N f -N ′ c = 2N f + 2N ′ c -3N c
and since there exist 2N ′ f quarks Q ′ , one bifundamental field Y which will give rise to the contribution of N c , its complex conjugate Y which will give rise to the contribution of N c , N f fields F ′ , its complex conjugate N f fields F ′ and the singlet Φ which will give rise to N ′ c , the coefficient of the beta function is
b mag SO(N ′ c ) = 3(N ′ c -2) -2N ′ f -2 N c -2N f -2N ′ c = -N ′ c + 2N c -4N f -2N ′ f -6.
SO(N ′ c ) -b mag SO(N ′ c ) > 0, SO(N ′ c ) is more asymptotically free than SO(N ′ c ) mag . Neglecting the SO(N ′ c ) dynamics, the magnetic SU( N c ) is IR free when N f + N ′ c < 3 2 N c
, as in previous case. The dual magnetic superpotential, by adding the mass term for Q and Q in the electric theory which is equal to put a linear term in M in the dual magnetic theory, is given by 7
W dual = (Φ 2 + • • • ) + Q ′ ΦQ ′ + Mq q + Y F ′ q + Y qF ′ + ΦY Y + mM (4.3)
where the mesons in terms of the fields defined in the electric theory are
M ≡ Q Q, Φ ≡ X X, F ′ ≡ XQ, F ′ ≡ X Q.
7 There appears a mismatch between the number of colors from field theory analysis and those from brane motion when we take the full dual process on the two gauge group factors simultaneously [15] . By adding 4N ′ f D4-branes to the dual brane configuration without affecting the linking number counting, this mismatch can be removed. Similar phenomena occurred in [5, 26] . Then this turned out that there exists a deformation ∆W generated by the meson Q ′ X XQ ′ . This is exactly the second term, Q ′ ΦQ ′ , in (4.3). In previous example, there is no such deformation term in (2.3) .
We abbreviated all the relevant terms and coefficients appearing in the quartic superpotential for the bifundamentals in electric theory (4.1) and denote them here by Φ 2 + • • • . Here q and q are fundamental and antifundamental for the gauge group index respectively and antifundamentals for the flavor index. Then, q q has rank N c and m has a rank N f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M, cannot be satisfied if the rank N f exceeds N c and the supersymmetry is broken. Other F-term equations are satisfied by taking the vacuum expectation values of Y, Y , F ′ , F ′ and Q ′ to vanish.
The classical moduli space of vacua can be obtained from F-term equations and one gets
q q + m = 0, qM + F ′ Y = 0, Mq + Y F ′ = 0, F ′ q + Y Φ = 0, qY = 0, qF ′ + ΦY = 0, Y q = 0, Q ′ Q ′ + Y Y = 0, ΦQ ′ = 0.
Then, it is easy to see that there exists a solution qM = 0 = Mq, q q + m = 0.
Other F-term equations are satisfied if one takes the zero vacuum expectation values for the fields Y, Y , F ′ , Q ′ and F ′ . Then the solutions can be written as
< q > = √ me φ 1 e Nc 0 , < q >= √ me -φ 1 e Nc 0 , < M >= 0 0 0 Φ 0 1 N f -e Nc < Y > = < Y >=< F ′ >=< F ′ >=< Q ′ >= 0. ( 4.4)
Let us expand around a point on (4.4), as done in [1] . Then the remaining relevant terms of superpotential are given by W rel dual = Φ 0 (δϕ δ ϕ + m) + δZ δϕ q 0 + δ Z q 0 δ ϕ by following the similar fluctuations for the various fields as in [9] . Note that there exist also four kinds of terms, the vacuum < q > multiplied by δ Y δF ′ , the vacuum < q > multiplied by δ F ′ δY , the vacuum < Φ > multiplied by δY δ Y , and the vacuum < Φ > multiplied by δQ ′ δQ ′ . However, by redefining these, they do not enter the contributions for the one loop result, up to quadratic order. As done in [17] , one gets that m 2 Φ 0 will contain (log 4 -1) > 0 implying that these are stable.
5 Nonsupersymmetric meta-stable brane configuration of SU (N c ) × SO(N ′ c ) gauge theory
Since the electric superpotential (4.1) vanishes for θ = 0 and ω = π 2 , the corresponding magnetic superpotential in (4.3) does not contain the terms Φ 2 + • • • and it becomes
W dual = Q ′ ΦQ ′ + Mq q + Y F ′ q + Y qF ′ + ΦY Y + mM.
Now we recombine N c D4-branes among N f flavor D4-branes connecting between D6 ω= π 2 = D6-branes and NS5 ω= π 2 = NS5 ′ R -brane with those connecting between NS5 ′ R -brane and NS5 -θ=0 = NS5 R -brane(and its mirrors) and push them in +v direction from Figure 5 . Of course their mirrors will move to -v direction in a Z 2 symmetric manner due to the O6 + -plane. After this procedure, there are no color D4-branes between NS5 ′ R -brane and NS5 R -brane. For the flavor D4-branes, we are left with only (N f -N c ) D4-branes(and its mirrors).
Then the minimal energy supersymmetry breaking brane configuration is shown in Figure 6 . If we ignore all the branes except NS5 ′ R -brane, NS5 R -brane, D6-branes, (N f -N c ) D4branes and N c D4-branes, as observed already, then this brane configuration corresponds to the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group SU( N c ) with N f massive flavors [12, 13, 14] . Note that N ′ c D4-branes can slide w direction for this brane configuration.
The type IIA/M-theory brane construction for the N = 2 gauge theory was described by [19] and after lifting the type IIA description we explained so far to M-theory, the corresponding magnetic M5-brane configuration with equal mass for the quarks where the gauge group is given by SU( N c ) × SO(N ′ c ), in a background space of xt = (-1)
N f +N ′ f v 2N ′ f +4 N f k=1 (v 2 -e 2 k
) where this four dimensional space replaces (45610) directions, is characterized by
t 4 + (v e Nc + • • • )t 3 + (v N ′ c + • • • )t 2 + (v e Nc + • • • )t + v 2N ′ f +4 N f k=1 (v 2 -e 2 k ) = 0.
From this curve of quartic equation for t above, the asymptotic regions can be classified by looking at the first two terms providing NS5 R -brane asymptotic region, next two terms providing NS5 ′ R -brane asymptotic region, next two terms providing NS5 ′ L -brane asymptotic region, and the final two terms giving NS5 L -brane asymptotic region as follows:
1. v → ∞ limit implies
w → 0, y ∼ v e Nc + • • • NS5 R asymptotic region, w → 0, y ∼ v 2N f +2N ′ f -e Nc+4 + • • • NS5 L asymptotic region.
c = N f + N ′ c -N c ) × SO(N ′ c
) with N f chiral multiplets q, N f chiral multiplets q, 2N ′ f chiral multiplets Q ′ , the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Y and gauge singlets. The N ′ c D4-branes and 2(N f -N c ) D4-branes can slide w direction freely in a Z 2 symmetric way.
2. w → ∞ limit implies v → -m, y ∼ w e Nc-N ′ c + • • • NS5 ′ L asymptotic region, v → +m, y ∼ w N ′ c -e Nc + • • • NS5 ′ R asymptotic region.
Now the two NS5 ′ L,R -branes are moving in the ±v direction holding everything else fixed instead of moving D6-branes in the ±v direction. Then the mirrors of D4-branes are moved appropriately. The harmonic function sourced by the D6-branes can be written explicitly by summing of three contributions from the N f and N ′ f D6-branes(and its mirrors) plus an O6plane, and similar analysis to solve the differential equation and find out the nonholomorphic curve can be done [14, 10, 9, 8, 7] . In this case also, we expect an instability from a new M5-brane mode.
So far, we have dualized only the first gauge group factor in the gauge group SU(N c )×SO(N ′ c ). What happens if we dualize the second gauge group factor SO(N ′ c )?(For the case SU(N c ) × SU(N ′ c ), the behavior of dual for the second gauge group will be the same as when we take the dual for the first gauge group factor.) This can be done by moving the NS5 θ -brane and N ′ f D6 θ -branes that can be located at the nonzero v coordinate for massive quarks Q ′ , to the right passing through O6-plane(and their mirrors to the left). According to the linking number counting, one obtains the dual gauge group SU(N c ) × SO( N ′ c = 2N c + 2N ′ f -N ′ c + 4). One can easily see that there is a creation of N ′ f D4-branes connecting NS5 θ -brane and D6 θ -branes(and its mirrors). Then from the brane configuration, there exist the additional 2N ′ f SU(N c ) quarks originating from the SO(N ′ c ) chiral mesons Q ′ X ≡ F ′ and Q ′ X ≡ F ′ . The deformed superpotential ∆W = Q ′ X XQ ′ can be interpreted as the mass term of F ′ F ′ . Then one can write dual magnetic superpotential in this case. However, it is not clear how the recombination of color and flavor D4-branes and splitting procedure between them in the construction of meta-stable vacua arises since there is no extra NS5-brane between two NS5 ±θ -branes. If there exists an extra NS5-brane at the origin of our brane configuration(then the gauge group and matter contents will change), it would be possible to construct the corresponding meta-stable brane configuration. It would be interesting to study these more in the future.
As already mentioned in [8] and section 4, the matter contents in [4] are different from the ones in section 4 with the same gauge group. In other words, the theory of SU(N c ) × SO(N ′ c ) with X, which transform as fundamental in SU(N c ) and vector in SO(N ′ c ), a antisymmetric tensor A in SU(N c ), as well as fundamentals for SU(N c ) and vectors for SO(N ′ c ) can confine either SU(N c ) factor or SO(N ′ c ) factor. This theory can be described by the web of branes in the presence of O4 --plane and orbifold fixed points. With two NS5-branes and O4 --plane, by modding out Z 3 symmetry acting on (v, w) as (v, w) → (v exp( 2πi 3 ), w exp( 2πi 3 )), the resulting gauge group will be SU(N c )×SO(N c +4) with above matter contents [27] . Similar analysis for SU(N c )×Sp( Nc 2 -2) gauge group with opposite O4 + -plane can be done. Then in this case, the matter in SU(N c ) will be a symmetric tensor S and other matter contents are present also. It would be interesting to see whether this gauge theory and corresponding brane configuration will provide a meta-stable vacuum. The type IIA brane configuration of an electric theory is exactly the same as the Figure 4 except the RR charge O6-plane with negative sign. The classical superpotential 8 is given by [15]
W = - 1 4 1 4 tan(ω -θ) + 1 tan 2θ tr(X X) 2 - tr X X XX 4 sin 2θ + (tr X X) 2 4N c tan(ω -θ) . (6.1)
In this case, when θ approaches π 2 and ω approaches 0, then this superpotential vanishes. The dual magnetic gauge group is given by SU
( N c = N f + 2N ′ c -N c ) × Sp(N ′ c )
with the same number of colors of dual theory as those in previous cases and the matter contents are given by
• N f chiral multiplets q are in the fundamental representation under the SU( N c ), N f chiral multiplets q are in the antifundamental representation under the SU( N c ) and then q are in the representation ( N c , 1) while q are in the representation ( N c , 1) under the gauge group The dual magnetic superpotential for arbitrary angles is given by (4.3) with appropriate Sp(N ′ c ) invariant metric J. The stability analysis can be done similarly. 8 The superpotential for the Sp(N ′ c ) sector is given by W = Xφ A X + Xφ S X + tan θ tr φ 2 S -1 tan θ tr φ 2 A where φ S and φ A are an adjoint field(symmetric tensor) and an antisymmetric tensor for Sp(N ′ c ) [25] . Note that there is a sign change in the second trace term of the superpotential in (6.1), compared to (4.1).
• 2N ′ f chiral multiplets Q ′ are
After following the procedure from Figure 4 to Figure 5 with opposite RR charge for O6plane and by taking the limit where θ → π 2 and ω → 0, the minimal energy supersymmetry breaking brane configuration is shown in Figure 7 .
c = N f + 2N ′ c -N c ) × Sp(N ′ c ) with N f chiral multiplets q, N f chiral multiplets q, 2N ′ f chiral multiplets Q ′ ,
the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Y and gauge singlets. Note the RR charge of O6-plane is negative and its charge is equivalent to -4 D6-branes. The 2N ′ c D4-branes and 2(N f -N c ) D4-branes can slide w direction freely in a Z 2 symmetric way.
Compared to the previous nonsupersymmetric brane configuration in Figure 6 , the role of NS5-brane and NS5'-brane is interchanged to each other: undoing the Seiberg dual in the context of [13] . This kind of feature of recombination and splitting between color D4-branes and flavor D4-branes occurs in [8] . At the electric brane configuration, N f D6-branes are perpendicular to NS5-brane and this leads to the coupling between the quarks and adjoint in the superpotential. However, the overall coefficient function including this extra terms vanishes and eventually the whole electric superpotential will vanish according to the above limit we take.
From the quartic equation with the presence of opposite RR charge for O6-plane, in a background space of xt = (-1)
N f +N ′ f v 2N ′ f -4 N f k=1 (v 2 -e 2 k ), t 4 + (v e Nc + • • • )t 3 + (v N ′ c + • • • )t 2 + (v e Nc + • • • )t + v 2N ′ f -4 N f k=1 (v 2 -e 2 k ) = 0,
the asymptotic regions can be classified as follows: On the other hand, the models SU(2N c + 1) × SU(2) have its brane box model description in [29] where the above examples correspond to N c = 3 and N c = 4 respectively. In particular, the case where N c = 1(the gauge group is SU(3) × SU(2), i.e., (3, 2) model [30] )
was described by brane box model with superpotential or without superpotential. Then it would be interesting to obtain the Seiberg dual for these models using brane box model and look for the possibility of having meta-stable vacua for these models. Moreover, this gauge theory was generalized to SU(2N c + 1) × Sp(N ′ c ) model with a bifundamental and 2N ′ c antifundamentals for SU(2N c + 1) and a fundamental for Sp(N ′ c ) and its dual description SU(2N c + 1) × Sp( N ′ c = N c -N ′ c -1) with a bifundamental and 2N ′ c antifundamentals for SU(2N c + 1) and a fundamental for Sp(N ′ c ) as well as two gauge singlets [28] . For the particular range of N c , the dual theory is IR free, not asymptotically free.
According to [31] , SU(2N c ) with antisymmetric tensor and antifundamentals can be described by two gauge groups Sp(2N c -4)×SU(2N c ) with bifundamental and antifundamentals for SU(2N c ). Some of the brane realization with zero superpotential was given in the brane box model in [29] . Similarly from the result of [32] by following the method of [31] , the dual description for SU(2N c + 1) with antisymmetric tensor and fundamentals can be represented by two gauge group factors. This dual theory breaks the supersymmetry at the tree level. Similar discussions are present in [33] . Then it would be interesting to construct the corresponding Seigerg dual and see how the electric theory and its magnetic theory can be mapped into each other in the brane box model.
Ther are also different directions concerning on the meta-stable vacua in different contexts and some of the relevant works are present in [34] - [43] where some of them use anti D-branes and some of them describe the type IIB theory and it would be interesting to find out how similarities if any appear and what are the differences in what sense between the present work and those works. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Starting from the N = 1 SU(N c ) × SU(N ′ c ) gauge theory with fundamental and bifundamental flavors, we apply the Seiberg dual to the first gauge group and obtain the N = 1 dual gauge theory with dual matters including the gauge singlets. By analyzing the F-term equations of the superpotential, we describe the intersecting type IIA brane configuration for the meta-stable nonsupersymmetric vacua of this gauge theory. By introducing an orientifold 6-plane, we generalize to the case for N = 1 SU(N c )×SO(N ′ c ) gauge theory with fundamental and bifundamental flavors. Finally, the N = 1 SU(N c ) × Sp(N ′ c ) gauge theory with matters is also described very briefly."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "It is well-known that the N = 1 SU(N c ) QCD with fundamental flavors has a vanishing superpotential before we deform this theory by mass term for quarks. The vanishing superpotential in the electric theory makes it easier to analyze its nonvanishing dual magnetic superpotential. Sometimes by tuning the various rotation angles between NS5-branes and D6-branes appropriately, even if the electric theory has nonvanishing superpotential, one can make the nonzero superpotential to vanish in the electric theory. Two procedures, deforming the electric gauge theory by adding the mass for the quarks and taking the Seiberg dual magnetic theory from the electric theory, are crucial to find out meta-stable supersymmetry breaking vacua in the context of dynamical supersymmetry breaking [1, 2] . Some models of dynamical supersymmetry breaking can be studied by gauging the subgroup of the flavor symmetry group by either field theory analysis or using the brane configuration 1 .\n\nIn this paper, starting from the known N = 1 supersymmetric electric gauge theories, we construct the N = 1 supersymmetric magnetic gauge theories by brane motion and linking number counting. The dual gauge group appears only on the first gauge group. Based on their particular limits of corresponding magnetic brane configurations in the sense that their electric theories do not have any superpotentials except the mass deformations for the quarks, we describe the intersecting brane configurations of type IIA string theory for the meta-stable nonsupersymmetric vacua of these gauge theories.\n\nWe focus on the cases where the whole gauge group is given by a product of two gauge groups. One example can be realized by three NS5-branes with D4-and D6-branes, and the other by four NS5-branes with D4-and D6-branes. For the latter, the appropriate orientifold 6-plane should be located at the center of this brane configuration in order to have two gauge groups. Of course, it is also possible, without changing the number of gauge groups, to have the brane configuration consisting of five NS5-branes and orientifold 6-plane, at which the extra NS5-brane is located, with D4-and D6-branes, but we'll not do this particular case in this paper.\n\nIn section 2, we review the type IIA brane configuration that contains three NS5-branes, corresponding to the electric theory based on the N = 1 SU(N c ) × SU(N ′ c ) gauge theory [4, 5, 6] with matter contents and deform this theory by adding the mass term for the quarks. Then we construct the Seiberg dual magnetic theory which is N = 1 SU( N c ) × SU(N ′ c ) gauge theory with corresponding dual matters as well as various gauge singlets, by brane motion and linking number counting. We do not touch the part of second gauge group SU(N ′ c ) in this dual process.\n\nIn section 3, we consider the nonsupersymmetric meta-stable minimum by looking at the magnetic brane configuration we obtained in section 2 and present the corresponding intersecting brane configuration of type IIA string theory, along the line of [7, 8, 9, 10, 11] (see also [12, 13, 14] ) and we describe M-theory lift of this supersymmetry breaking type IIA brane configuration.\n\nIn section 4, we describe the type IIA brane configuration that contains four NS5-branes, corresponding to the electric theory based on the N = 1 SU(N c ) × SO(N ′ c ) gauge theory [15] with matter contents and deform this theory by adding the mass term for the quarks.\n\nThen we take the Seiberg dual magnetic theory which is N = 1 SU( N c ) × SO(N ′ c ) gauge theory with corresponding dual matters as well as various gauge singlets, by brane motion and linking number counting. The part of second gauge group SO(N ′ c ) in this dual process is not changed under this process.\n\nIn section 5, the nonsupersymmetric meta-stable minimum by looking at the magnetic brane configuration we obtained in section 4 is constructed and we present the corresponding intersecting brane configuration of type IIA string theory and describe M-theory lift of this supersymmetry breaking type IIA brane configuration, as we did in section 3.\n\nIn section 6, we describe the similar application to the N = 1 SU(N c ) × Sp(N ′ c ) gauge theory [15] briefly and make some comments for the future directions. 2 The N = 1 supersymmetric brane configuration of SU (N c ) × SU (N ′ c ) gauge theory\n\nAfter reviewing the type IIA brane configuration corresponding to the electric theory based on the N = 1 SU(N c )×SU(N ′ c ) gauge theory [4, 5, 6] , we construct the Seiberg dual magnetic theory which is N = 1 SU( N c ) × SU(N ′ c ) gauge theory."
},
{
"section_type": "OTHER",
"section_title": "Electric theory with SU",
"text": "(N c ) × SU (N ′ c ) gauge group\n\nThe gauge group is given by SU(N c ) × SU(N ′ c ) and the matter contents [4, 5, 6] • The flavor singlet field X is in the bifundamental representation (N c , N ′ c ) under the gauge group and its complex conjugate field X is in the bifundamental representation (N c , N ′ c ) under the gauge group\n\nIn the electric theory, since there exist N f quarks Q, N f quarks Q, one bifundamental field X which will give rise to the contribution of N ′ c and its complex conjugate X which will give rise to the contribution of N ′ c , the coefficient of the beta function of the first gauge group factor is\n\nb SU (Nc) = 3N c -N f -N ′ c\n\nand similarly since there exist\n\nN ′ f quarks Q ′ , N ′ f quarks Q ′ ,\n\none bifundamental field X which will give rise to the contribution of N c and its complex conjugate X which will give rise to the contribution of N c , the coefficient of the beta function of the second gauge group factor is\n\nb SU (N ′ c ) = 3N ′ c -N ′ f -N c .\n\nThe anomaly free global symmetry is given by L -brane) is located at the left hand side of a middle NS5-brane along the x 6 direction and there exist N ′ c color D4-branes suspended between them, with N ′ f D6-branes which have zero (45) directions. These are shown in Figure 1 explicitly. See also [3] for the brane configuration.\n\n[SU(N f ) × SU(N ′ f )] 2 × U(1) 3 × U(1) R [4,\n\nBy realizing that the two outer NS5 ′ L,R -branes are perpendicular to a middle NS5-brane and the fact that N f D6-branes are parallel to NS5 ′ R -brane and N ′ f D6-branes are parallel to NS5 ′ L -brane, the classical superpotential vanishes. However, one can deform this theory. Then the classical superpotential by deforming this theory by adding the mass term for the quarks Q and Q, along the lines of [1, 11, 10, 9, 8, 7] , is given by\n\nW = mQ Q (2.1)\n\nand this type IIA brane configuration can be summarized as follows foot_1 :\n\n• One middle NS5-brane with worldvolume (012345).\n\n• Two NS5'-branes with worldvolume (012389).\n\n• N f D6-branes with worldvolume (0123789) located at the positive region in v direction.\n\n• N c color D4-branes with worldvolume (01236). These are suspended between a middle NS5-brane and NS5 ′ R -brane. • N ′ c color D4-branes with worldvolume (01236). These are suspended between NS5 ′ Lbrane and a middle NS5-brane. Now we draw this electric brane configuration in Figure 1 and we put the coincident N f D6-branes in the nonzero v direction. If we ignore the left NS5 ′ L -brane, N ′ c D4-branes and N ′ f D6-branes, then this brane configuration corresponds to the standard N = 1 SQCD with the gauge group SU(N c ) with N f massive flavors. The electric quarks Q and Q correspond to strings stretching between the N c color D4-branes with N f D6-branes, the electric quarks Q ′ and Q ′ correspond to strings between the N ′ c color D4-branes with N ′ f D6-branes and the bifundamentals X and X correspond to strings stretching between the N c color D4-branes with N ′ c color D4-branes.\n\nFigure 1 :\n\nThe N = 1 supersymmetric electric brane configuration of SU(N c ) × SU(N ′ c ) with N f chiral multiplets Q, N f chiral multiplets Q, N ′ f chiral multiplets Q ′ , N ′ f chiral multiplets Q ′ ,\n\nthe flavor singlet bifundamental field X and its complex conjugate bifundamental field X. The N f D6-branes have nonzero v coordinates where v = m for equal massive case of quarks Q, Q while Q ′ and Q ′ are massless."
},
{
"section_type": "OTHER",
"section_title": "Magnetic theory with SU",
"text": "( N c ) × SU (N ′ c ) gauge group\n\nOne can consider dualizing one of the gauge groups regarding as the other gauge group as a spectator. One takes the Seiberg dual for the first gauge group factor SU(N c ) while remaining the second gauge group factor SU(N ′ c ) unchanged. Also we consider the case where Λ 1 >> Λ 2 , in other words, the dualized group's dynamical scale is far above that of the other spectator group.\n\nLet us move a middle NS5-brane to the right all the way past the right NS5 ′ R -brane. For example, see [12, 13, 14, 11, 10, 9, 8, 7] . After this brane motion, one arrives at the Figure 2 . Note that there exists a creation of N f D4-branes connecting N f D6-branes and NS5 ′ R -brane. Recall that the N f D6-branes are perpendicular to a middle NS5-brane in Figure 1 . The linking number [16] of NS5-brane from Figure 2\n\nis L 5 = N f 2 -N c .\n\nOn the other hand, the linking number of NS5-brane from Figure 1 is\n\nL 5 = - N f 2 + N c -N ′ c .\n\nDue to the connection of N ′ c D4-branes with NS5 ′ R -brane, the presence of N ′ c in the linking number arises. From these two relations, one obtains the number of colors of dual magnetic theory\n\nN c = N f + N ′ c -N c . (2.2)\n\nThe linking number counting looks similar to the one in [7] where there exists a contribution from O4-plane. Let us draw this magnetic brane configuration in Figure 2 and recall that we put the coincident N f D6-branes in the nonzero v directions in the electric theory, along the lines of [12, 13, 14, 11, 10, 9, 8, 7] . The N f created D4-branes connecting between D6-branes and NS5 ′ R -brane can move freely in the w direction. Moreover since N ′ c D4-branes are suspending between two equal NS5 ′ L,R -branes located at different x 6 coordinate, these D4-branes can slide along the w direction also. If we ignore the left NS5 ′ L -brane, N ′ c D4-branes and N ′ f D6-branes(detaching these from Figure 2 ), then this brane configuration corresponds to the standard N = 1 SQCD with the magnetic gauge group SU( N c = N f -N c ) with N f massive flavors [12, 13, 14] . The dual magnetic gauge group is given by SU( N c ) × SU(N ′ c ) and the matter contents are given by • N f chiral multiplets q are in the fundamental representation under the SU( N c ), N f chiral multiplets q are in the antifundamental representation under the SU( N c ) and then q are in the representation ( N c , 1) while q are in the representation ( N c , 1) under the gauge group\n\n• N ′ f chiral multiplets Q ′ are\n\nc = N f + N ′ c - N c ) × SU(N ′ c ) with N f chiral multiplets q, N f chiral multiplets q, N ′ f chiral multiplets Q ′ , N ′\n\nf chiral multiplets Q ′ , the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Y as well as N f fields F ′ , its complex conjugate N f fields F ′ , N 2 f fields M and the gauge singlet Φ. There exist N f flavor D4-branes connecting D6-branes and • The N ′2 c -fields Φ is in the representation (1, N ′2 c -1) ⊕ (1, 1) under the gauge group This corresponds to the SU(N c ) chiral meson X X and note that X has a representation\n\nNS5 ′ R -brane.\n\nN ′ c of SU(N ′ c ) while X has a representation N ′ c of SU(N ′ c\n\n). The fluctuations of the singlet Φ correspond to the motion of N ′ c D4-branes suspended two NS5 ′ L,R -branes along the (789) directions in Figure 2 .\n\nIn the dual theory, since there exist N f quarks q, N f quarks q, one bifundamental field Y which will give rise to the contribution of N ′ c and its complex conjugate Y which will give rise to the contribution of N ′ c , the coefficient of the beta function for the first gauge group factor [6] is\n\nb mag SU ( e Nc) = 3 N c -N f -N ′ c = 2N f + 2N ′ c -3N c\n\nwhere we inserted the number of colors given in (2.2) in the second equality and since there exist\n\nN ′ f quarks Q ′ , N ′ f quarks Q ′ ,\n\none bifundamental field Y which will give rise to the contribution of N c , its complex conjugate Y which will give rise to the contribution of N c , N f fields F ′ , its complex conjugate N f fields F ′ and the singlet Φ which will give rise to N ′ c , the coefficient of the beta function of second gauge group factor [6] is\n\nb mag SU (N ′ c ) = 3N ′ c -N ′ f -N c -N f -N ′ c = N ′ c + N c -2N f -N ′ f .\n\nTherefore\n\nSU (N ′ c ) -b mag SU (N ′ c ) > 0, SU(N ′ c ) is more asymptotically free than SU(N ′ c ) mag [6]. Neglecting the SU(N ′ c ) dynamics, the magnetic SU( N c ) is IR free when N f + N ′ c < 3 2 N c [6]\n\n. The dual magnetic superpotential, by adding the mass term (2.1) for Q and Q in the electric theory which is equal to put a linear term in M in the dual magnetic theory, is given by\n\nW dual = Mq q + Y F ′ q + Y q F ′ + ΦY Y + mM (2.3)\n\nwhere the mesons in terms of the fields defined in the electric theory are\n\nM ≡ Q Q, Φ ≡ X X, F ′ ≡ XQ, F ′ ≡ X Q.\n\nBy orientifolding procedure(O4-plane) into this brane configuration, the q(Q) and q( Q) are equivalent to each other, the Y (X) and Y ( X) become identical and F ′ and F ′ become the same. Then the reduced superpotential is identical with the one in [7] . Here q and q are fundamental and antifundamental for the gauge group index respectively and antifundamentals for the flavor index. Then, q q has rank N c while m has a rank N f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M, cannot be satisfied if the rank N f exceeds N c . This is so-called rank condition and the supersymmetry is broken.\n\nOther F-term equations are satisfied by taking the vacuum expectation values of Y, Y , F ′ and F ′ to vanish.\n\nThe classical moduli space of vacua can be obtained from F-term equations\n\nq q + m = 0, qM + F ′ Y = 0, Mq + Y F ′ = 0, F ′ q + Y Φ = 0, qY = 0, q F ′ + ΦY = 0, Y q = 0, Y Y = 0.\n\nThen, it is easy to see that there exist three reduced equations qM = 0 = Mq, q q + m = 0 and other F-term equations are satisfied if one takes the zero vacuum expectation values for the fields Y, Y , F ′ and F ′ . Then the solutions can be written as follows:\n\n< q > = √ me φ 1 e Nc 0 , < q >= √ me -φ 1 e Nc 0 , < M >= 0 0 0 Φ 0 1 N f -e Nc < Y > = < Y >=< F ′ >=< F ′ >= 0.\n\n(2.4)\n\nLet us expand around a point on (2.4), as done in [1] . Then the remaining relevant terms of superpotential are given by W rel dual = Φ 0 (δϕ δ ϕ + m) + δZ δϕ q 0 + δ Z q 0 δ ϕ by following the same fluctuations for the various fields as in [9] :\n\nq = q 0 1 e Nc + 1 √ 2 (δχ + + δχ -)1 e Nc δϕ , q = q 0 1 e Nc + 1 √ 2 (δχ + -δχ -)1 e Nc δ ϕ , M = δY δZ δ Z Φ 0 1 N f -e\n\nas well as the fluctuations of Y, Y , F ′ and F ′ . Note that there exist also three kinds of terms, the vacuum < q > multiplied by δ Y δ F ′ , the vacuum < q > multiplied by δF ′ δY , and the vacuum < Φ > multiplied by δY δ Y . However, by redefining these, they do not enter the contributions for the one loop result, up to quadratic order. As done in [17] , one gets that m 2 Φ 0 will contain (log 4 -1) > 0 implying that these are stable. Then the minimal energy supersymmetry breaking brane configuration is shown in Figure 3 , along the lines of [12, 13, 14, 11, 10, 9, 8, 7] . If we ignore the left NS5 ′ L -brane, N ′ c D4branes and N ′ f D6-branes(detaching these from Figure 3 ), as observed already, then this brane configuration corresponds to the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group SU( N c = N f -N c ) with N f massive flavors [12, 13, 14] .\n\nc = N f + N ′ c -N c ) × SU(N ′ c ) with N f chiral multiplets q, N f chiral multiplets q, N ′ f chiral multiplets Q ′ , N ′ f chiral multiplets Q ′ ,\n\nthe flavor singlet bifundamental field Y and its complex conjugate bifundamental field Y and various gauge singlets.\n\nThe type IIA/M-theory brane construction for the N = 2 gauge theory was described by [18] and after lifting the type IIA description to M-theory, the corresponding magnetic M5-brane configuration foot_2 with equal mass for the quarks where the gauge group is given by\n\nSU( N c )×SU(N ′ c ), in a background space of xt = v N ′ f N f k=1 (v-e k )\n\nwhere this four dimensional space replaces (45610) directions, is described by\n\nt 3 + (v e Nc + • • • )t 2 + (v N ′ c + • • • )t + v N ′ f N f k=1 (v -e k ) = 0 (3.1)\n\nwhere e k is the position of the D6-branes in the v direction(for equal massive case, we can write e k = m) and we have ignored the lower power terms in v in t 2 and t denoted by • • • and the scales for the gauge groups in front of the first term and the last term, for simplicity. For fixed x, the coordinate t corresponds to y.\n\nFrom this curve (3.1) of cubic equation for t above, the asymptotic regions for three NS5branes can be classified by looking at the first two terms providing NS5-brane asymptotic region, next two terms providing NS5 ′ R -brane asymptotic region and the final two terms giving NS5 ′ L -brane asymptotic region as follows 1. v → ∞ limit implies\n\nw → 0, y ∼ v e Nc + • • • NS asymptotic region. 2. w → ∞ limit implies v → m, y ∼ w N f +N ′ f -N ′ c + • • • NS ′ L asymptotic region, v → m, y ∼ w N ′ c -e Nc + • • • NS ′ R asymptotic region.\n\nHere the two NS5 ′ L,R -branes are moving in the +v direction holding everything else fixed instead of moving D6-branes in the +v direction, in the spirit of [14] . The harmonic function sourced by the D6-branes can be written explicitly by summing over two contributions from the N f and N ′ f D6-branes and similar analysis to both solve the differential equation and find out the nonholomorphic curve can be done [14, 10, 9, 8, 7] . An instability from a new M5-brane mode arises. 4 The N = 1 supersymmetric brane configuration of\n\nSU (N c ) × SO(N ′ c ) gauge theory\n\nAfter reviewing the type IIA brane configuration corresponding to the electric theory based on the N = 1 SU(N c ) × SO(N ′ c ) gauge theory [15] , we describe the Seiberg dual magnetic theory which is N = 1 SU( N c ) × SO(N ′ c ) gauge theory."
},
{
"section_type": "OTHER",
"section_title": "Electric theory with SU",
"text": "(N c ) × SO(N ′ c ) gauge group\n\nThe gauge group is given by SU(N c ) × SO(N ′ c ) and the matter contents [15] (similar matter contents are found in [4] ) are given by In the electric theory, since there exist N f quarks Q, N f quarks Q, one bifundamental field X which will give rise to the contribution of N ′ c and its complex conjugate X which will give rise to the contribution of N ′ c , the coefficient of the beta function of the first gauge group factor is\n\n• N f chiral multiplets Q are in\n\nb SU (Nc) = 3N c -N f -N ′ c\n\nand similarly, since there exist 2N ′ f quarks Q ′ , one bifundamental field X which will give rise to the contribution of N c and its complex conjugate X which will give rise to the contribution of N c , the coefficient of the beta function of the second gauge group factor is\n\nb SO(N ′ c ) = 3(N ′ c -2) -2N ′ f -2N c .\n\nThe anomaly free global symmetry is given by SU 0123789 ) and an orientifold 6 plane (0123789) of positive Ramond charge foot_3 . According to Z 2 symmetry of orientifold 6-plane(O6-plane) sitting at v = 0 and x 6 = 0, the coordinates (x 4 , x 5 , x 6 ) transform as -(x 4 , x 5 , x 6 ), as usual. See also [3] for the discussion of O6-plane.\n\n(N f ) 2 × SU(2N ′ f ) × U(1) 2 × U(1)\n\nBy rotating the third and fourth NS5-branes which are located at the right hand side of O6-plane, from v direction toward -w and +w directions respectively, one obtains N = 1 theory. Their mirrors, the first and second NS5-branes which are located at the left hand side of O6-plane, can be rotated in a Z 2 symmetric manner due to the presence of O6-plane simultaneously. That is, if the first NS5-brane rotates by an angle -ω in (v, w) plane, denoted by NS5 -ω -brane [3] , then the mirror image of this NS5-brane, the fourth NS5-brane, is rotated by an angle ω in the same plane, denoted by NS5 ω -brane. If the second NS5-brane rotates by an angle θ in (v, w) plane, denoted by NS5 θ -brane [3] , then the mirror image of this NS5-brane, the third NS5-brane, is rotated by an angle -θ in the same plane, denoted by NS5 -θ -brane. For more details, see the Figure 4 5 .\n\nWe also rotate the N ′ f D6-branes which are located between the second NS5-brane and an O6-plane and make them be parallel to NS5 θ -brane and denote them as D6 θ -brane with zero v coordinate(the angle between the unrotated D6-branes and D6 θ -branes is equal to π 2θ) and its mirrors N ′ f D6-branes appear as D6 -θ -branes between the O6-plane and third NS5-brane. There is no coupling between the adjoint field and the quarks since the rotated D6 θ -branes are parallel to the rotated NS5 θ -brane [5, 3] . Similarly, the N f D6-branes which are located between the third NS5-brane and the fourth NS5-brane can be rotated and we can make them be parallel to NS5 ω -brane and denote them as D6 ω -branes with nonzero v coordinate(the angle between the unrotated D6-branes and D6 ω -branes is equal to π 2ω) and its mirrors N f D6-branes appear as D6 -ω -branes between the first NS5-brane and the second NS5-brane.\n\nMoreover the N c D4-branes are suspended between the first NS5-brane and the second NS5-brane(and its mirrors) and the N ′ c D4-branes are suspended between the second NS5brane and the third NS5-brane.\n\nFor this brane setup 6 , the classical superpotential is given by [15]\n\nW = - 1 4 1 4 tan(ω -θ) + 1 tan 2θ tr(X X) 2 + tr X X XX 4 sin 2θ + (tr X X) 2 4N c tan(ω -θ) . (4.1)\n\nIt is easy to see that when θ approaches 0 and ω approaches π 2 , then this superpotential vanishes. 5 The angles of θ 1 and θ 2 in [15] are related to the angles θ and ω as follows: θ = θ 1 and ω = θ 2 . 6 For arbitrary angles θ and ω, the superpotential for the SU (N c ) sector is given by W = Xφ X + tan(ωθ) tr φ 2 where φ ia an adjoint field for SU (N c ). There is no coupling between φ and N f quarks because D6 ±ω -branes are parallel to N S5 ±ω -branes. The superpotential for the SO(N ′ c ) sector is given by W = Xφ A X + Xφ S X + tan θ tr φ 2 A -1 tan θ tr φ 2 S where φ A and φ S are an adjoint field and a symmetric tensor for SO(N ′ c ) [25] . After integrating out φ, φ A and φ S , the whole superpotential can be written as in (4.1).\n\nNow one summarizes the supersymmetric electric brane configuration with their worldvolumes in type IIA string theory as follows.\n\n• NS5 -ω -brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with negative x 6 .\n\n• NS5 θ -brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with negative x 6 .\n\n• NS5 -θ -brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with positive x 6 .\n\n• NS5 ω -brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with positive x 6 .\n\n• N ′ f D6 θ -branes with worldvolume by both (01237) and two spatial dimensions in (v, w) plane and with negative x 6 and v = 0.\n\n• N ′ f D6 -θ -branes with worldvolume by both (01237) and two space dimensions in (v, w) plane and with positive x 6 and v = 0.\n\n• N f D6 ω -branes with worldvolume by both (01237) and two spatial dimensions in (v, w) plane and with positive x 6 . Before the rotation, the distance from N c color D4-branes in the +v direction is nonzero.\n\n• N f D6 -ω -branes with worldvolume by both (01237) and two space dimensions in (v, w) plane and with negative x 6 . Before the rotation, the distance from N c color D4-branes in the -v direction is nonzero.\n\n• O6-plane with worldvolume (0123789) with v = 0 = x 6 .\n\n• N c D4-branes connecting NS5 -ω -brane and NS5 θ -brane, with worldvolume (01236) with v = 0 = w(and its mirrors).\n\n• N ′ c D4-branes connecting NS5 θ -brane and NS5 -θ -brane, with worldvolume (01236) with v = 0 = w.\n\nWe draw the type IIA electric brane configuration in Figure 4 which was basically given in [15] already but the only difference is to put N f D6-branes in the nonzero v direction in order to obtain nonzero masses for the quarks which are necessary to obtain the meta-stable vacua."
},
{
"section_type": "OTHER",
"section_title": "Magnetic theory with SU",
"text": "( N c ) × SO(N ′ c ) gauge group\n\nOne takes the Seiberg dual for the first gauge group factor SU(N c ) while remaining the second gauge group factor SO(N ′ c ), as in previous case. Also we consider the case where Λ 1 >> Λ 2 , in other words, the dualized group's dynamical scale is far above that of the other spectator group.\n\nN f chiral multiplets Q, N f chiral multiplets Q, 2N ′\n\nf chiral multiplets Q ′ , the flavor singlet bifundamental field X and its complex conjugate bifundamental field X. The N f D6 ω -branes have nonzero v coordinates where v = m(and its mirrors) for equal massive case of quarks Q, Q while Q ′ is massless.\n\nLet us move the NS5 -θ -brane to the right all the way past the right NS5 ω -brane(and its mirrors to the left). After this brane motion, one arrives at the Figure 5 . Note that there exists a creation of N f D4-branes connecting N f D6 ω -branes and NS5 ω -brane(and its mirrors). Recall that the N f D6 ω -branes are not parallel to the NS5 -θ -brane in Figure 4 (and its mirrors). The linking number of NS5 -θ -brane from Figure 5 is L 5 = N f 2 -N c . On the other hand, the linking number of NS5 -θ -brane from Figure 4 is\n\nL 5 = - N f 2 + N c -N ′ c\n\n. From these, one gets the number of colors in dual magnetic theory\n\nN c = N f + N ′ c -N c . (4.2)\n\nLet us draw this magnetic brane configuration in Figure 5 and remember that we put the coincident N f D6 ω -branes in the nonzero v directions(and its mirrors). The N f created D4branes connecting between D6 ω -branes and NS5 ω -brane can move freely in the w direction, as in previous case. Moreover, since N ′ c D4-branes are suspending between two unequal NS5 ±ω -branes located at different x 6 coordinate, these D4-branes cannot slide along the w direction, for arbitrary rotation angles. If we are detaching all the branes except NS5 ω -brane, NS5 -θ -brane, D6 ω -branes, N f D4-branes and N c D4-branes from Figure 5 , then this brane configuration corresponds to N = 1 SQCD with the magnetic gauge group SU( N c = N f -N c ) with N f massive flavors with tilted NS5-branes.\n\nThe dual magnetic gauge group is given by SU( N c ) × SO(N ′ c ) and the matter contents are given by\n\nc = N f + N ′ c - N c ) × SO(N ′ c\n\n) with N f chiral multiplets q, N f chiral multiplets q, 2N ′ f chiral multiplets Q ′ , the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Y as well as N f fields F ′ , its complex conjugate N f fields F ′ , N 2 f fields M and the gauge singlet Φ. There exist N f flavor D4-branes connecting D6 ω -branes and NS5 ω -brane(and its mirrors).\n\n• N f chiral multiplets q are in the fundamental representation under the SU( N c ), N f chiral multiplets q are in the antifundamental representation under the SU( N c ) and then q are in the representation ( N c , 1) while q are in the representation ( N c , 1) under the gauge group These additional 2N f SO(N ′ c ) vectors are originating from the SU(N c ) chiral mesons XQ and X Q respectively. It is easy to see that from the Figure 5 , since the D6 -ω -branes are parallel to the NS5 -ω -brane, the newly created N f D4-branes can slide along the plane consisting of D6 -ω -branes and NS5 -ω -brane arbitrarily(and its mirrors). Then strings connecting the N f D6 -ω -branes and N ′ c D4-branes will give rise to these additional 2N f SO(N ′ c ) vectors.\n\n• 2N ′ f chiral multiplets Q ′ are\n\n• N 2 f -fields M are in the representation (1, 1) under the gauge group This corresponds to the SU(N c ) chiral meson Q Q and the fluctuations of the singlet M correspond to the motion of N f flavor D4-branes along (789) directions in Figure 5 .\n\n• The N ′ 2 c singlet Φ is in the representation (1, adj) ⊕ (1, symm) under the gauge group This corresponds to the SU(N c ) chiral meson X X and note that both X and X have representation N ′ c of SO(N ′ c ). In general, the fluctuations of the singlet Φ correspond to the motion of N ′ c D4-branes suspended two NS5 ±ω -branes along the (789) directions in Figure 5 . In the dual theory, since there exist N f quarks q, N f quarks q, one bifundamental field Y which will give rise to the contribution of N ′ c and its complex conjugate Y which will give rise to the contribution of N ′ c , the coefficient of the beta function of the first gauge group factor with (4.2) is\n\nb mag SU ( e Nc) = 3 N c -N f -N ′ c = 2N f + 2N ′ c -3N c\n\nand since there exist 2N ′ f quarks Q ′ , one bifundamental field Y which will give rise to the contribution of N c , its complex conjugate Y which will give rise to the contribution of N c , N f fields F ′ , its complex conjugate N f fields F ′ and the singlet Φ which will give rise to N ′ c , the coefficient of the beta function is\n\nb mag SO(N ′ c ) = 3(N ′ c -2) -2N ′ f -2 N c -2N f -2N ′ c = -N ′ c + 2N c -4N f -2N ′ f -6.\n\nSO(N ′ c ) -b mag SO(N ′ c ) > 0, SO(N ′ c ) is more asymptotically free than SO(N ′ c ) mag . Neglecting the SO(N ′ c ) dynamics, the magnetic SU( N c ) is IR free when N f + N ′ c < 3 2 N c\n\n, as in previous case. The dual magnetic superpotential, by adding the mass term for Q and Q in the electric theory which is equal to put a linear term in M in the dual magnetic theory, is given by 7\n\nW dual = (Φ 2 + • • • ) + Q ′ ΦQ ′ + Mq q + Y F ′ q + Y qF ′ + ΦY Y + mM (4.3)\n\nwhere the mesons in terms of the fields defined in the electric theory are\n\nM ≡ Q Q, Φ ≡ X X, F ′ ≡ XQ, F ′ ≡ X Q.\n\n7 There appears a mismatch between the number of colors from field theory analysis and those from brane motion when we take the full dual process on the two gauge group factors simultaneously [15] . By adding 4N ′ f D4-branes to the dual brane configuration without affecting the linking number counting, this mismatch can be removed. Similar phenomena occurred in [5, 26] . Then this turned out that there exists a deformation ∆W generated by the meson Q ′ X XQ ′ . This is exactly the second term, Q ′ ΦQ ′ , in (4.3). In previous example, there is no such deformation term in (2.3) .\n\nWe abbreviated all the relevant terms and coefficients appearing in the quartic superpotential for the bifundamentals in electric theory (4.1) and denote them here by Φ 2 + • • • . Here q and q are fundamental and antifundamental for the gauge group index respectively and antifundamentals for the flavor index. Then, q q has rank N c and m has a rank N f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M, cannot be satisfied if the rank N f exceeds N c and the supersymmetry is broken. Other F-term equations are satisfied by taking the vacuum expectation values of Y, Y , F ′ , F ′ and Q ′ to vanish.\n\nThe classical moduli space of vacua can be obtained from F-term equations and one gets\n\nq q + m = 0, qM + F ′ Y = 0, Mq + Y F ′ = 0, F ′ q + Y Φ = 0, qY = 0, qF ′ + ΦY = 0, Y q = 0, Q ′ Q ′ + Y Y = 0, ΦQ ′ = 0.\n\nThen, it is easy to see that there exists a solution qM = 0 = Mq, q q + m = 0.\n\nOther F-term equations are satisfied if one takes the zero vacuum expectation values for the fields Y, Y , F ′ , Q ′ and F ′ . Then the solutions can be written as\n\n< q > = √ me φ 1 e Nc 0 , < q >= √ me -φ 1 e Nc 0 , < M >= 0 0 0 Φ 0 1 N f -e Nc < Y > = < Y >=< F ′ >=< F ′ >=< Q ′ >= 0. ( 4.4)\n\nLet us expand around a point on (4.4), as done in [1] . Then the remaining relevant terms of superpotential are given by W rel dual = Φ 0 (δϕ δ ϕ + m) + δZ δϕ q 0 + δ Z q 0 δ ϕ by following the similar fluctuations for the various fields as in [9] . Note that there exist also four kinds of terms, the vacuum < q > multiplied by δ Y δF ′ , the vacuum < q > multiplied by δ F ′ δY , the vacuum < Φ > multiplied by δY δ Y , and the vacuum < Φ > multiplied by δQ ′ δQ ′ . However, by redefining these, they do not enter the contributions for the one loop result, up to quadratic order. As done in [17] , one gets that m 2 Φ 0 will contain (log 4 -1) > 0 implying that these are stable.\n\n5 Nonsupersymmetric meta-stable brane configuration of SU (N c ) × SO(N ′ c ) gauge theory\n\nSince the electric superpotential (4.1) vanishes for θ = 0 and ω = π 2 , the corresponding magnetic superpotential in (4.3) does not contain the terms Φ 2 + • • • and it becomes\n\nW dual = Q ′ ΦQ ′ + Mq q + Y F ′ q + Y qF ′ + ΦY Y + mM.\n\nNow we recombine N c D4-branes among N f flavor D4-branes connecting between D6 ω= π 2 = D6-branes and NS5 ω= π 2 = NS5 ′ R -brane with those connecting between NS5 ′ R -brane and NS5 -θ=0 = NS5 R -brane(and its mirrors) and push them in +v direction from Figure 5 . Of course their mirrors will move to -v direction in a Z 2 symmetric manner due to the O6 + -plane. After this procedure, there are no color D4-branes between NS5 ′ R -brane and NS5 R -brane. For the flavor D4-branes, we are left with only (N f -N c ) D4-branes(and its mirrors).\n\nThen the minimal energy supersymmetry breaking brane configuration is shown in Figure 6 . If we ignore all the branes except NS5 ′ R -brane, NS5 R -brane, D6-branes, (N f -N c ) D4branes and N c D4-branes, as observed already, then this brane configuration corresponds to the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group SU( N c ) with N f massive flavors [12, 13, 14] . Note that N ′ c D4-branes can slide w direction for this brane configuration.\n\nThe type IIA/M-theory brane construction for the N = 2 gauge theory was described by [19] and after lifting the type IIA description we explained so far to M-theory, the corresponding magnetic M5-brane configuration with equal mass for the quarks where the gauge group is given by SU( N c ) × SO(N ′ c ), in a background space of xt = (-1)\n\nN f +N ′ f v 2N ′ f +4 N f k=1 (v 2 -e 2 k\n\n) where this four dimensional space replaces (45610) directions, is characterized by\n\nt 4 + (v e Nc + • • • )t 3 + (v N ′ c + • • • )t 2 + (v e Nc + • • • )t + v 2N ′ f +4 N f k=1 (v 2 -e 2 k ) = 0.\n\nFrom this curve of quartic equation for t above, the asymptotic regions can be classified by looking at the first two terms providing NS5 R -brane asymptotic region, next two terms providing NS5 ′ R -brane asymptotic region, next two terms providing NS5 ′ L -brane asymptotic region, and the final two terms giving NS5 L -brane asymptotic region as follows:\n\n1. v → ∞ limit implies\n\nw → 0, y ∼ v e Nc + • • • NS5 R asymptotic region, w → 0, y ∼ v 2N f +2N ′ f -e Nc+4 + • • • NS5 L asymptotic region.\n\nc = N f + N ′ c -N c ) × SO(N ′ c\n\n) with N f chiral multiplets q, N f chiral multiplets q, 2N ′ f chiral multiplets Q ′ , the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Y and gauge singlets. The N ′ c D4-branes and 2(N f -N c ) D4-branes can slide w direction freely in a Z 2 symmetric way.\n\n2. w → ∞ limit implies v → -m, y ∼ w e Nc-N ′ c + • • • NS5 ′ L asymptotic region, v → +m, y ∼ w N ′ c -e Nc + • • • NS5 ′ R asymptotic region.\n\nNow the two NS5 ′ L,R -branes are moving in the ±v direction holding everything else fixed instead of moving D6-branes in the ±v direction. Then the mirrors of D4-branes are moved appropriately. The harmonic function sourced by the D6-branes can be written explicitly by summing of three contributions from the N f and N ′ f D6-branes(and its mirrors) plus an O6plane, and similar analysis to solve the differential equation and find out the nonholomorphic curve can be done [14, 10, 9, 8, 7] . In this case also, we expect an instability from a new M5-brane mode."
},
{
"section_type": "DISCUSSION",
"section_title": "Discussions",
"text": "So far, we have dualized only the first gauge group factor in the gauge group SU(N c )×SO(N ′ c ). What happens if we dualize the second gauge group factor SO(N ′ c )?(For the case SU(N c ) × SU(N ′ c ), the behavior of dual for the second gauge group will be the same as when we take the dual for the first gauge group factor.) This can be done by moving the NS5 θ -brane and N ′ f D6 θ -branes that can be located at the nonzero v coordinate for massive quarks Q ′ , to the right passing through O6-plane(and their mirrors to the left). According to the linking number counting, one obtains the dual gauge group SU(N c ) × SO( N ′ c = 2N c + 2N ′ f -N ′ c + 4). One can easily see that there is a creation of N ′ f D4-branes connecting NS5 θ -brane and D6 θ -branes(and its mirrors). Then from the brane configuration, there exist the additional 2N ′ f SU(N c ) quarks originating from the SO(N ′ c ) chiral mesons Q ′ X ≡ F ′ and Q ′ X ≡ F ′ . The deformed superpotential ∆W = Q ′ X XQ ′ can be interpreted as the mass term of F ′ F ′ . Then one can write dual magnetic superpotential in this case. However, it is not clear how the recombination of color and flavor D4-branes and splitting procedure between them in the construction of meta-stable vacua arises since there is no extra NS5-brane between two NS5 ±θ -branes. If there exists an extra NS5-brane at the origin of our brane configuration(then the gauge group and matter contents will change), it would be possible to construct the corresponding meta-stable brane configuration. It would be interesting to study these more in the future.\n\nAs already mentioned in [8] and section 4, the matter contents in [4] are different from the ones in section 4 with the same gauge group. In other words, the theory of SU(N c ) × SO(N ′ c ) with X, which transform as fundamental in SU(N c ) and vector in SO(N ′ c ), a antisymmetric tensor A in SU(N c ), as well as fundamentals for SU(N c ) and vectors for SO(N ′ c ) can confine either SU(N c ) factor or SO(N ′ c ) factor. This theory can be described by the web of branes in the presence of O4 --plane and orbifold fixed points. With two NS5-branes and O4 --plane, by modding out Z 3 symmetry acting on (v, w) as (v, w) → (v exp( 2πi 3 ), w exp( 2πi 3 )), the resulting gauge group will be SU(N c )×SO(N c +4) with above matter contents [27] . Similar analysis for SU(N c )×Sp( Nc 2 -2) gauge group with opposite O4 + -plane can be done. Then in this case, the matter in SU(N c ) will be a symmetric tensor S and other matter contents are present also. It would be interesting to see whether this gauge theory and corresponding brane configuration will provide a meta-stable vacuum. The type IIA brane configuration of an electric theory is exactly the same as the Figure 4 except the RR charge O6-plane with negative sign. The classical superpotential 8 is given by [15]\n\nW = - 1 4 1 4 tan(ω -θ) + 1 tan 2θ tr(X X) 2 - tr X X XX 4 sin 2θ + (tr X X) 2 4N c tan(ω -θ) . (6.1)\n\nIn this case, when θ approaches π 2 and ω approaches 0, then this superpotential vanishes. The dual magnetic gauge group is given by SU\n\n( N c = N f + 2N ′ c -N c ) × Sp(N ′ c )\n\nwith the same number of colors of dual theory as those in previous cases and the matter contents are given by\n\n• N f chiral multiplets q are in the fundamental representation under the SU( N c ), N f chiral multiplets q are in the antifundamental representation under the SU( N c ) and then q are in the representation ( N c , 1) while q are in the representation ( N c , 1) under the gauge group The dual magnetic superpotential for arbitrary angles is given by (4.3) with appropriate Sp(N ′ c ) invariant metric J. The stability analysis can be done similarly. 8 The superpotential for the Sp(N ′ c ) sector is given by W = Xφ A X + Xφ S X + tan θ tr φ 2 S -1 tan θ tr φ 2 A where φ S and φ A are an adjoint field(symmetric tensor) and an antisymmetric tensor for Sp(N ′ c ) [25] . Note that there is a sign change in the second trace term of the superpotential in (6.1), compared to (4.1).\n\n• 2N ′ f chiral multiplets Q ′ are\n\nAfter following the procedure from Figure 4 to Figure 5 with opposite RR charge for O6plane and by taking the limit where θ → π 2 and ω → 0, the minimal energy supersymmetry breaking brane configuration is shown in Figure 7 .\n\nc = N f + 2N ′ c -N c ) × Sp(N ′ c ) with N f chiral multiplets q, N f chiral multiplets q, 2N ′ f chiral multiplets Q ′ ,\n\nthe flavor singlet bifundamental field Y and its complex conjugate bifundamental field Y and gauge singlets. Note the RR charge of O6-plane is negative and its charge is equivalent to -4 D6-branes. The 2N ′ c D4-branes and 2(N f -N c ) D4-branes can slide w direction freely in a Z 2 symmetric way.\n\nCompared to the previous nonsupersymmetric brane configuration in Figure 6 , the role of NS5-brane and NS5'-brane is interchanged to each other: undoing the Seiberg dual in the context of [13] . This kind of feature of recombination and splitting between color D4-branes and flavor D4-branes occurs in [8] . At the electric brane configuration, N f D6-branes are perpendicular to NS5-brane and this leads to the coupling between the quarks and adjoint in the superpotential. However, the overall coefficient function including this extra terms vanishes and eventually the whole electric superpotential will vanish according to the above limit we take.\n\nFrom the quartic equation with the presence of opposite RR charge for O6-plane, in a background space of xt = (-1)\n\nN f +N ′ f v 2N ′ f -4 N f k=1 (v 2 -e 2 k ), t 4 + (v e Nc + • • • )t 3 + (v N ′ c + • • • )t 2 + (v e Nc + • • • )t + v 2N ′ f -4 N f k=1 (v 2 -e 2 k ) = 0,\n\nthe asymptotic regions can be classified as follows: On the other hand, the models SU(2N c + 1) × SU(2) have its brane box model description in [29] where the above examples correspond to N c = 3 and N c = 4 respectively. In particular, the case where N c = 1(the gauge group is SU(3) × SU(2), i.e., (3, 2) model [30] )\n\nwas described by brane box model with superpotential or without superpotential. Then it would be interesting to obtain the Seiberg dual for these models using brane box model and look for the possibility of having meta-stable vacua for these models. Moreover, this gauge theory was generalized to SU(2N c + 1) × Sp(N ′ c ) model with a bifundamental and 2N ′ c antifundamentals for SU(2N c + 1) and a fundamental for Sp(N ′ c ) and its dual description SU(2N c + 1) × Sp( N ′ c = N c -N ′ c -1) with a bifundamental and 2N ′ c antifundamentals for SU(2N c + 1) and a fundamental for Sp(N ′ c ) as well as two gauge singlets [28] . For the particular range of N c , the dual theory is IR free, not asymptotically free.\n\nAccording to [31] , SU(2N c ) with antisymmetric tensor and antifundamentals can be described by two gauge groups Sp(2N c -4)×SU(2N c ) with bifundamental and antifundamentals for SU(2N c ). Some of the brane realization with zero superpotential was given in the brane box model in [29] . Similarly from the result of [32] by following the method of [31] , the dual description for SU(2N c + 1) with antisymmetric tensor and fundamentals can be represented by two gauge group factors. This dual theory breaks the supersymmetry at the tree level. Similar discussions are present in [33] . Then it would be interesting to construct the corresponding Seigerg dual and see how the electric theory and its magnetic theory can be mapped into each other in the brane box model.\n\nTher are also different directions concerning on the meta-stable vacua in different contexts and some of the relevant works are present in [34] - [43] where some of them use anti D-branes and some of them describe the type IIB theory and it would be interesting to find out how similarities if any appear and what are the differences in what sense between the present work and those works."
}
] | |
arxiv:0704.0135 | 0704.0135 | 1 | 10.1103/PhysRevA.76.052105 | 17b1b58b19c72b2e4c86cbc9c53a65e08eed391855e30408198cb0ce7a198ac5 | A Single Trapped Ion as a Time-Dependent Harmonic Oscillator | We show how a single trapped ion may be used to test a variety of important
physical models realized as time-dependent harmonic oscillators. The ion itself
functions as its own motional detector through laser-induced electronic
transitions. Alsing et al. [Phys. Rev. Lett. 94, 220401 (2005)] proposed that
an exponentially decaying trap frequency could be used to simulate (thermal)
Gibbons-Hawking radiation in an expanding universe, but the Hamiltonian used
was incorrect. We apply our general solution to this experimental proposal,
correcting the result for a single ion and showing that while the actual
spectrum is different from the Gibbons-Hawking case, it nevertheless shares an
important experimental signature with this result. | [
"Nicolas C. Menicucci and G. J. Milburn"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | The time-dependent quantum harmonic oscillator has long served as a paradigm for nonadiabatic timedependent Hamiltonian systems and has been applied to a wide range of physical problems by choosing the mass, the frequency, or both, to be time-dependent. The earliest application is to squeezed state generation in quantum optics [1, 2, 3] , in which the effect of a second-order optical nonlinearity on a single-mode field can be modeled by a harmonic oscillator with a frequency that is harmonically modulated at twice the bare oscillator frequency. It was subsequently shown that any modulation of the frequency could produce squeezing [4] , and thus the same model could be used to approximately describe the generation of photons in a cavity with a time-dependent boundary [5, 6] .
The model has been used in a number of quantum cosmological models. In Ref. [7] , a time-dependent frequency has been used to explain entropy production in a quantum mini-superspace model. The model, with both mass and frequency time-dependent, has been particularly important in developing an understanding of how quantum fluctuations in a scalar field can drive classical metric fluctuation during inflation [8, 9] . In a cosmological setting the time-dependence is not harmonic and is usually exponential. In all physical applications, of course, the model is only an approximation to the true physics, and its validity can be tested only with considerable difficulty, especially in the cosmological setting. Here we propose a realistic experimental context in which the time-dependent quantum harmonic oscillator can be studied directly.
Many decades of effort to refine spectroscopic measurements for time standards now enable a single ion to be confined in three dimensions, its vibrational motion restricted effectively to one dimension, and the ion cooled * Electronic address: nmen@princeton.edu
to the vibrational ground state with a probability greater than 99% [10] . Laser cooling is based on the ability to couple an internal electronic transition to the vibrational motion of the ion [11] . These methods can easily be extended to more than one ion and their collective normal modes of vibration [12] . Indeed so carefully can the coupling between the electronic and vibrational states be engineered that is is possible to realise simple quantum information processing tasks [13, 14] . We use the control of trapping potential afforded by ion traps, together with the ability to reach quantum limited motion, to propose a simple experimental test of quantum harmonic oscillators with time-dependent frequencies. We also make use of the ability to make highly efficient quantum measurements, based on fluorescent shelving [10] , to propose a practical means to test our predictions.
In this paper, we calculate the excitation probability of a trapped ion in a general time-dependent potential. When beginning in the vibrational ground state of the unchirped trap and starting the chirping process adiabatically, the excitation probability is simply related to the Fourier transform of the solution of the Heisenberg equations of motion (which is also the same as the trajectory of the equivalent classical oscillator). We compare our result with that of Ref. [15] for the case of a single ion undergoing an exponential frequency chirp. The cited work attempts to use this experimental setup to model a massless scalar field during an inflating (i.e., de Sitter) universe, which would give a thermal excitation spectrum as a function of the detector response frequency [16] . The analysis is incorrect, however, because the wrong Hamiltonian was used. Nevertheless, the corrected calculation presented here also gives an excitation spectrum with a thermal signature, although the particular functional form is different.
The quantum Hamiltonian for a single ion in a timedependent harmonic trap can be well-approximated in 2 one dimension by
H = p 2 2M + M 2 ν(t) 2 q 2 , ( 1
)
where ν(t) is time-dependent but always assumed to be much slower than the timescale of the micromotion [10] . For emphasis, we have indicated the explicit time-dependence of the frequency ν; we will often omit this from now on. Working in the Heisenberg picture, we get the following equations of motion for q and p:
q = p M , ( 2
) ṗ = -M ν 2 q . ( 3
)
Dots indicate total derivatives with respect to time. Differentiating again and plugging in these results gives
0 = q + ν 2 q , ( 4
) 0 = p -2 ν ν ṗ + ν 2 p . ( 5
)
As we shall see, only Eq. ( 4 ) is necessary for calculating excitation probabilities, so we will focus only on it. These equations are operator equations, but they are identical to the classical equations of motion for the analogous classical system. Interpreting them as such, we will label the two linearly independent c-number solutions as h(t) and g(t), where the following initial conditions are satisfied: h(0) = ġ(0) = 1 and ḣ(0) = g(0) = 0 , (6) Writing q(0) = q 0 and p(0) = p 0 , the unique solution for q to the initial value problem above is
q(t) = q 0 h(t) + p 0 M g(t) . ( 7
)
By differentiating and using the relations above, we know also that
p(t) = M q 0 ḣ(t) + p 0 ġ(t) . ( 8
)
To check our math, we can verify that [q(t), p(t)] = i , which is fulfilled if and only if the Wronskian W (h, g) of the two solutions is one for all times-specifically,
h ġ -ḣg = 1 , ( 9
)
where we have assumed that [q 0 , p 0 ] = i . Moreover, if the initial state at t = 0 is symmetric with respect to phase-space rotations, then we have additional rotational freedom in choosing the initial quadratures. (This would be the case, for instance, if we start in the instantaneous ground state.) Notice that Eq. ( 7 ) can be written as the inner product of two vectors:
q(t) = q 0 , p 0 M ν 0 • h(t), ν 0 g(t) ( 10
)
(and similarly for Eq. ( 8 )), where we have normalized the quadrature operators to have the same units. As an inner product, this expression is invariant under simultaneous rotations of both vectors. Thus, if the initial state possesses rotational symmetry in the phase plane, then the rotated quadratures are equally as valid as the original ones for representing the initial state, which means that an arbitrary rotation can be applied to the second vector above without changing any measurable property of the system. This freedom can be used, for instance, to define new functions h ′ (t) and g ′ (t) that are more convenient for calculations, where the linear transformation between them and the original ones (with prefactors as in Eq. ( 10 )) is a rotation. We will use this freedom in the next section. One reason why ion traps have become a leading implementation for quantum information processing is the ability to efficiently read out the internal electronic state using a fluorescence shelving scheme [10] . As the internal state can become correlated with the vibrational motion of the ion, this scheme can be configured as a way to measure the vibrational state directly [17] . To correlate the internal electronic state with the motion of the ion, an external laser can be used to drive an electronic transition between two levels |g and |e , separated in energy by ω A . The interaction between an external classical laser field and the ion is described, in the dipole and rotating-wave approximation, by the interaction-picture Hamiltonian [10]
H L = -i Ω 0 σ + (t)e ik cos θq(t) -σ -(t)e -ik cos θq(t) , ( 11
)
where Ω 0 is the Rabi frequency for the laser-atom interaction, ω L is the laser frequency, k is the magnitude of the wave vector k, which makes an angle θ with the trap axis, q(t) is given in Eq. ( 7 ), and
σ ± (t) = e ±i∆t σ ± . ( 12
)
The electronic-state raising and lowering operators are defined as σ + = |e g| and σ -= |g e|, respectively, and
∆ = ω A -ω L ( 13
)
is the detuning of the laser below the atomic transition. We can construct a meaningful quantity that characterizes the "size" of q(t) based on the width of the ground-state wave packet for an oscillator with frequency ν(t), namely /2M ν(t). As long as this quantity is much smaller than k cos θ throughout the chirping process, then we can expand the exponentials in Eq. (11) to first order and define the interaction Hamiltonian H I between the electronic states and vibrational motion (still in the interaction picture) by
H I = Ω 0 k cos θq(t) e -i∆t σ -+ e +i∆t σ + . ( 14
)
where we have assumed that ω L is far off-resonance, and thus ∆ ≃ 0. 3 Using first-order time-dependent perturbation theory, the probability to find the ion in the excited state is
P (1) = 1 2 T 0 dt 1 T 0 dt 2 H I (t 1 )P e H I (t 2 ) = Ω 2 0 k 2 cos 2 θ T 0 dt 1 T 0 dt 2 e -i∆(t1-t2) q(t 1 )q(t 2 ) , ( 15
)
where P e = 1 vib ⊗ |e e| is the projector onto the excited electronic state (and the identity on the vibrational subspace). We always assume that the ion begins in the electronic ground state. If the ion also starts out in the instantaneous vibrational ground state for a static trap of frequency ν 0 = ν(0) at t = 0 (which is most useful when the chirping begins in the adiabatic regime), then we can evaluate the two-time correlation function as
q(t 1 )q(t 2 ) ground = q 2 0 h(t 1 )h(t 2 ) + p 2 0 M 2 g(t 1 )g(t 2 ) + q 0 p 0 M h(t 1 )g(t 2 ) -h(t 2 )g(t 1 ) = 2M ν 0 h(t 1 ) -iν 0 g(t 1 ) h(t 2 ) + iν 0 g(t 2 ) = 2M ν 0 f (t 1 )f * (t 2 ) , ( 16
)
where we have used the facts that for the vibrational ground state, q 2 0
= (p 0 /M ν 0 ) 2 = /2M ν 0 and q 0 p 0 = 1 2 {q 0 , p 0 } + [q 0 , p 0 ] = i /2,
and we have defined the complex function
f (t) = h(t) -iν 0 g(t) , ( 17
)
which is the solution to Eq. ( 4 ) with initial the conditions, f (0) = 1 and ḟ (0) = -iν 0 . Plugging this into Eq. ( 15 ) gives, quite simply,
P (1) → (Ω 0 η 0 ) 2 |F ¸|2 , ( 18
) where F ¸= T 0 dt e -i∆t f (t) , ( 19
)
and we have defined the unitless, time-dependent Lamb-Dicke parameter [10] as 0) . Recalling that f (t) can be considered a complex c-number solution to the equations of motion for the equivalent classical Hamiltonian, Eq. (18) shows that the excitation probability is simply related to the Fourier transform of the classical trajectories when beginning in the vibrational ground state.
η(t) = k 2 cos 2 θ 2M ν(t) , ( 20
)
and η 0 = η(
III. EXPONENTIAL CHIRPING Recent work [15] has suggested that an exponentially decaying trap frequency has the same effect on the phonon modes of a string of ions as an expanding (i.e., de Sitter) spacetime does on a one-dimensional scalar field [18] . An inertial detector that responds to such an expanding scalar field would register a thermal bath of particles, called Gibbons-Hawking radiation [16]. Ref. [15] suggests that the acoustic analog [19] of this radiation could be seen in an ion trap, causing each ion to be excited with a thermal spectrum with temperature κ/2πk B , as a function of the detuning ∆, where κ is the trap-frequency decay rate. The analysis used an incorrect Hamiltonian that neglected squeezing and source terms that have no analog in the expanding scalar field model but which are present when considering trapped ions in this way, and the results are incorrect. In this section, we revisit this problem and calculate the excitation probability for a single ion in an exponentially decaying harmonic potential, as a function of the detuning ∆.
We write the time-dependent frequency as [20] ν(t) = ν 0 e -κt . (21) This results in q + ν 2 0 e -2κt q = 0 . (22) Solutions with initial conditions (6) are
h(t) = πν 0 2κ J 1 ν 0 κ Y 0 ν κ -Y 1 ν 0 κ J 0 ν κ , ( 23
) g(t) = π 2κ -J 0 ν 0 κ Y 0 ν κ + Y 0 ν 0 κ J 0 ν κ , ( 24
)
where the time dependence is carried in ν = ν(t) from Eq. (21), and J n and Y n are Bessel functions. We could plug these directly into the formulas from the last section, but we will simplify the calculations by considering the limits of slow and long-time frequency decay, represented by
ν 0 ≫ κ and ν 0 e -κT ≪ κ , ( 25
)
respectively. This allows us to do several things. First, it allows us to use the usual ground state of the unchirped trap at frequency ν 0 as a good approximation to the ground state of the expanding trap at t = 0, since at that time the system is being chirped adiabatically. This is important because it allows the experiment to begin with a static potential, which is useful for cooling. Second, it allows us to simplify h(t) and g(t) using the phase-space rotation freedom discussed above. Using asymptotic approximations for the Bessel functions in the coefficients,
J 0 ν 0 κ ≃ -Y 1 ν 0 κ ≃ 2κ πν 0 cos ν 0 κ - π 4 , ( 26
) J 1 ν 0 κ ≃ Y 0 ν 0 κ ≃ 2κ πν 0 sin ν 0 κ - π 4 , ( 27
) 4 we get h(t) ≃ πν 0 2κ sin ϕ Y 0 ν κ + cos ϕ J 0 ν κ , ( 28
) ν 0 g(t) ≃ πν 0 2κ -cos ϕ Y 0 ν κ + sin ϕ J 0 ν κ . ( 29
)
where ϕ = ν 0 /κ -π/4. Since we are taking the initial state to be the ground state, which is symmetric with respect to phase-space rotations, we can use the freedom discussed in the previous section to undo the rotation represented by Eqs. (28) and (29) and define the simpler functions
h(t) → h ′ (t) = πν 0 2κ Y 0 ν κ , ( 30
) g(t) → g ′ (t) = π 2κν 0 J 0 ν κ . ( 31
)
The primes are unnecessary due to the symmetry of the initial state, so we drop them from now on and plug directly into Eq. (17):
f (t) = πν 0 2κ Y 0 ν κ -iJ 0 ν κ = -i πν 0 2κ H (1) 0 ν κ , ( 32
)
where H (1) n is a Hankel function of the first kind. The integral in Eq. ( 19 ) can be evaluated in the limits (25) using techniques similar to those used in Ref. [15] . First, define e α = ν κ , τ = α -κt , u = e τ , and x = ∆/κ .
The integral in question then becomes (neglecting the prefactor)
T 0 dt e -i∆t H (1) 0 ν κ = T 0 dt e -i∆t H ( 1
) 0 (e α-κt ) = 1 κ α α-κT dτ e -ix(α-τ ) H (1) 0 (e τ ) → e -ixα κ ∞ -∞ dτ e ixτ H (1) 0 (e τ ) = e -ixα κ ∞ 0 du u ix-1 H (1) 0 (u) . ( 34
)
Inserting a convergence factor with x → x -iǫ, and then taking the limit ǫ → 0 + , we can use the formula
∞ 0 du u ix-1 H (1) 0 (u) = -2 ix Γ(ix/2) (e πx -1)Γ(1 -ix/2) ( 35
) to evaluate |F ¸|2 = πν 0 2κ 1 κ 2 Γ(ix/2) Γ(1 -ix/2) 2 1 (e πx -1) 2 = 2πν 0 κ 3 x 2 1 (e πx -1) 2 . ( 36
)
When plugging in for the dummy variables (33), this gives
P (1) = (Ω 0 η 0 ) 2 2πν 0 κ∆ 2 1 (e π∆/κ -1) 2 . ( 37
)
The calculated result from Ref. [15] for a single ion is
P ( 1
) GH = (Ω 0 η 0 ) 2 2π κ∆ 1 e 2π∆/κ -1 , ( 38
)
which contains a Planck factor with Gibbons-Hawking [16] temperature T = κ/2πk B but is different from the actual result for a single ion, given by Eq. (37) .
Several things should be noted about these functions. First, they both break down as ∆ → 0 because of the approximation made in obtaining Eq. ( 14 ). They also fail if the time-dependent Lamb-Dicke parameter (20) ever becomes too large throughout the chirping process. Furthermore, most cases of interest will be ∆ ≃ ν 0 (the first red sideband) and near ∆ ≃ -ν 0 (the first blue sideband), which means that |∆| ≫ κ, since ν 0 ≫ κ. The first red sideband represents a detector that requires the absorption of one phonon (plus one laser photon) in order to excite the atom-the usual thing we mean by "particle detector" when the particles are phonons. The first blue sideband, on the other hand, represents a detector that emits a phonon in order to excite the atom (along with absorbing one laser photon).
There are a couple of ways to compare these functions. First, we can take the ratio of the two for both the redand blue-sideband cases. In both cases, we obtain
P (1) P (1) GH ≃ ν 0 |∆| (1 + 2e -π|∆|/κ ) ( 39
)
plus terms of order O(e -2π|∆|/κ ). Since |∆| ≃ ν 0 , the prefactor is close to one, and the second term is very small (since ν 0 ≫ κ). Furthermore, it is cumbersome to directly compare the measured probability to the full function (with all the prefactors). It is often easier instead to make measurements on both the first red sideband and the first blue sideband and then take the ratio of the two. The constant prefactors disappear in this calculation, and both functions then have the same experimental signature:
P (1) (∆) P (1) (-∆) = P ( 1
) GH (∆) P (1) GH (-∆) = e -2π∆/κ , ( 40
)
which is that of a thermal distribution with temperature T = κ/2πk B , which is of the Gibbons-Hawking form [16] with the expansion rate given by κ. Therefore, although the Hamiltonian used in the calculations in Ref. [15] was missing terms, the intuition (at least for a single ion) was correct in that the actual experimental signature in this case matches that of an ion undergoing 5 thermal motion in a static trap, where the temperature is proportional to κ. To see whether this experiment is feasible, we must examine the validity of our approximations. For a typical trap, we expect that ν 0 ≃ 1 MHz, and thus if we take κ ≃ 1 kHZ, we easily satisfy the first of conditions (25), namely ν 0 ≫ κ. The second of these conditions gives a constraint on the modulation time T . For these parameters we expect that T ≃ a few msec. This is compatible with typical cooling and readout time scales and is less than those for heating due to fluctuating patch potentials [10] . Thus, this is a realizable experiment with current technology.
We have shown that a single trapped ion in a modulated trapping potential can serve as an experimentally accessible implementation of a quantum harmonic oscillator with time-dependent frequency, including robust control over state preparation, manipulation, and measure-ment. The ion itself serves both as the oscillating particle and as the local detector of vibrational motion via coupling to internal electronic states by an external laser. For the case of a general time-dependent trap frequency, we calculated the first-order excitation probability for the ion in terms of the solution to the classical equations of motion for the equivalent classical oscillator. We applied this general result to the case of exponential chirping and corrected the calculation in Ref. [15] for a single ion. We found that while the results from the two calculations differ, the experimental signature in both cases is the same and equivalent to that of a thermal ion in a static trap.
We thank Dave Kielpinski for invaluable help with the experimental details. We also thank Paul Alsing, Bill Unruh, John Preskill, Jeff Kimble, Greg Ver Steeg, and Michael Nielsen for useful discussions and suggestions. NCM extends much appreciation to the faculty and staff of the Caltech Institute for Quantum Information for their hospitality during his visit, which helped bring this work to fruition. NCM was supported by the United States Department of Defense, and GJM acknowledges support from the Australian Research Council.
[1] D. Stoler, Phys. Rev. D 1, 3217 (1970). [2] H. P. Yuen, Phys. Rev. A 13, 2226 (1976). [3] J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). [4] X. Ma and W. Rhodes, Phys. Rev. A 39, 1941 (1989). [5] V. V. Dodonov and A. B. Klimov, Phys. Rev. A 53, 2664 (1996). [6] G. T. Moore, J. Math. Phys. 11, 2679 (1970). [7] S. P. Kim and S.-W. Kim, Phys. Rev. D 51, 4254 (1995). [8] D. Polarski and A. A. Starobinsky, Classical and Quantum Gravity 13, 377 (1996). [9] C. Kiefer, J. Lesgourgues, D. Polarski, and A. A. Starobinsky, Classical and Quantum Gravity 15, L67 (1998). [10] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev.
Mod. Phys. 75, 281 (2003). [11] C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. L. Gould, Phys. Rev. Lett. 75, 4011 (1995). [12] D. F. V. James, Applied Physics B: Lasers and Optics 66, 181 (1998). [13] D. Leibfried, B. De Marco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenkovic, C. Langer, T. Rosenband, et al., Nature 422, 412 (2003). [14] F. Schmidt-Kaler, H. Häffner, M. Riebe, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003). [15] P. M. Alsing, J. P. Dowling, and G. J. Milburn, Phys.
Rev. Lett. 94, 220401 (2005). [16] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977). [17] S. Wallentowitz and W. Vogel, Phys. Rev. A 54, 3322 (1996). [18] A. M. de M. Carvalho, C. Furtado, and I. A. Pedrosa, Phys. Rev. D 70, 123523 (pages 6) (2004). [19] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). [20] The authors of Ref. [15] consider both signs in the exponential, but we will restrict ourselves to the case that allows us to begin chirping in the adiabatic limit. | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The time-dependent quantum harmonic oscillator has long served as a paradigm for nonadiabatic timedependent Hamiltonian systems and has been applied to a wide range of physical problems by choosing the mass, the frequency, or both, to be time-dependent. The earliest application is to squeezed state generation in quantum optics [1, 2, 3] , in which the effect of a second-order optical nonlinearity on a single-mode field can be modeled by a harmonic oscillator with a frequency that is harmonically modulated at twice the bare oscillator frequency. It was subsequently shown that any modulation of the frequency could produce squeezing [4] , and thus the same model could be used to approximately describe the generation of photons in a cavity with a time-dependent boundary [5, 6] .\n\nThe model has been used in a number of quantum cosmological models. In Ref. [7] , a time-dependent frequency has been used to explain entropy production in a quantum mini-superspace model. The model, with both mass and frequency time-dependent, has been particularly important in developing an understanding of how quantum fluctuations in a scalar field can drive classical metric fluctuation during inflation [8, 9] . In a cosmological setting the time-dependence is not harmonic and is usually exponential. In all physical applications, of course, the model is only an approximation to the true physics, and its validity can be tested only with considerable difficulty, especially in the cosmological setting. Here we propose a realistic experimental context in which the time-dependent quantum harmonic oscillator can be studied directly.\n\nMany decades of effort to refine spectroscopic measurements for time standards now enable a single ion to be confined in three dimensions, its vibrational motion restricted effectively to one dimension, and the ion cooled * Electronic address: nmen@princeton.edu\n\nto the vibrational ground state with a probability greater than 99% [10] . Laser cooling is based on the ability to couple an internal electronic transition to the vibrational motion of the ion [11] . These methods can easily be extended to more than one ion and their collective normal modes of vibration [12] . Indeed so carefully can the coupling between the electronic and vibrational states be engineered that is is possible to realise simple quantum information processing tasks [13, 14] . We use the control of trapping potential afforded by ion traps, together with the ability to reach quantum limited motion, to propose a simple experimental test of quantum harmonic oscillators with time-dependent frequencies. We also make use of the ability to make highly efficient quantum measurements, based on fluorescent shelving [10] , to propose a practical means to test our predictions.\n\nIn this paper, we calculate the excitation probability of a trapped ion in a general time-dependent potential. When beginning in the vibrational ground state of the unchirped trap and starting the chirping process adiabatically, the excitation probability is simply related to the Fourier transform of the solution of the Heisenberg equations of motion (which is also the same as the trajectory of the equivalent classical oscillator). We compare our result with that of Ref. [15] for the case of a single ion undergoing an exponential frequency chirp. The cited work attempts to use this experimental setup to model a massless scalar field during an inflating (i.e., de Sitter) universe, which would give a thermal excitation spectrum as a function of the detector response frequency [16] . The analysis is incorrect, however, because the wrong Hamiltonian was used. Nevertheless, the corrected calculation presented here also gives an excitation spectrum with a thermal signature, although the particular functional form is different."
},
{
"section_type": "OTHER",
"section_title": "II. GENERAL SOLUTION",
"text": "The quantum Hamiltonian for a single ion in a timedependent harmonic trap can be well-approximated in 2 one dimension by\n\nH = p 2 2M + M 2 ν(t) 2 q 2 , ( 1\n\n)\n\nwhere ν(t) is time-dependent but always assumed to be much slower than the timescale of the micromotion [10] . For emphasis, we have indicated the explicit time-dependence of the frequency ν; we will often omit this from now on. Working in the Heisenberg picture, we get the following equations of motion for q and p:\n\nq = p M , ( 2\n\n) ṗ = -M ν 2 q . ( 3\n\n)\n\nDots indicate total derivatives with respect to time. Differentiating again and plugging in these results gives\n\n0 = q + ν 2 q , ( 4\n\n) 0 = p -2 ν ν ṗ + ν 2 p . ( 5\n\n)\n\nAs we shall see, only Eq. ( 4 ) is necessary for calculating excitation probabilities, so we will focus only on it. These equations are operator equations, but they are identical to the classical equations of motion for the analogous classical system. Interpreting them as such, we will label the two linearly independent c-number solutions as h(t) and g(t), where the following initial conditions are satisfied: h(0) = ġ(0) = 1 and ḣ(0) = g(0) = 0 , (6) Writing q(0) = q 0 and p(0) = p 0 , the unique solution for q to the initial value problem above is\n\nq(t) = q 0 h(t) + p 0 M g(t) . ( 7\n\n)\n\nBy differentiating and using the relations above, we know also that\n\np(t) = M q 0 ḣ(t) + p 0 ġ(t) . ( 8\n\n)\n\nTo check our math, we can verify that [q(t), p(t)] = i , which is fulfilled if and only if the Wronskian W (h, g) of the two solutions is one for all times-specifically,\n\nh ġ -ḣg = 1 , ( 9\n\n)\n\nwhere we have assumed that [q 0 , p 0 ] = i . Moreover, if the initial state at t = 0 is symmetric with respect to phase-space rotations, then we have additional rotational freedom in choosing the initial quadratures. (This would be the case, for instance, if we start in the instantaneous ground state.) Notice that Eq. ( 7 ) can be written as the inner product of two vectors:\n\nq(t) = q 0 , p 0 M ν 0 • h(t), ν 0 g(t) ( 10\n\n)\n\n(and similarly for Eq. ( 8 )), where we have normalized the quadrature operators to have the same units. As an inner product, this expression is invariant under simultaneous rotations of both vectors. Thus, if the initial state possesses rotational symmetry in the phase plane, then the rotated quadratures are equally as valid as the original ones for representing the initial state, which means that an arbitrary rotation can be applied to the second vector above without changing any measurable property of the system. This freedom can be used, for instance, to define new functions h ′ (t) and g ′ (t) that are more convenient for calculations, where the linear transformation between them and the original ones (with prefactors as in Eq. ( 10 )) is a rotation. We will use this freedom in the next section. One reason why ion traps have become a leading implementation for quantum information processing is the ability to efficiently read out the internal electronic state using a fluorescence shelving scheme [10] . As the internal state can become correlated with the vibrational motion of the ion, this scheme can be configured as a way to measure the vibrational state directly [17] . To correlate the internal electronic state with the motion of the ion, an external laser can be used to drive an electronic transition between two levels |g and |e , separated in energy by ω A . The interaction between an external classical laser field and the ion is described, in the dipole and rotating-wave approximation, by the interaction-picture Hamiltonian [10]\n\nH L = -i Ω 0 σ + (t)e ik cos θq(t) -σ -(t)e -ik cos θq(t) , ( 11\n\n)\n\nwhere Ω 0 is the Rabi frequency for the laser-atom interaction, ω L is the laser frequency, k is the magnitude of the wave vector k, which makes an angle θ with the trap axis, q(t) is given in Eq. ( 7 ), and\n\nσ ± (t) = e ±i∆t σ ± . ( 12\n\n)\n\nThe electronic-state raising and lowering operators are defined as σ + = |e g| and σ -= |g e|, respectively, and\n\n∆ = ω A -ω L ( 13\n\n)\n\nis the detuning of the laser below the atomic transition. We can construct a meaningful quantity that characterizes the \"size\" of q(t) based on the width of the ground-state wave packet for an oscillator with frequency ν(t), namely /2M ν(t). As long as this quantity is much smaller than k cos θ throughout the chirping process, then we can expand the exponentials in Eq. (11) to first order and define the interaction Hamiltonian H I between the electronic states and vibrational motion (still in the interaction picture) by\n\nH I = Ω 0 k cos θq(t) e -i∆t σ -+ e +i∆t σ + . ( 14\n\n)\n\nwhere we have assumed that ω L is far off-resonance, and thus ∆ ≃ 0. 3 Using first-order time-dependent perturbation theory, the probability to find the ion in the excited state is\n\nP (1) = 1 2 T 0 dt 1 T 0 dt 2 H I (t 1 )P e H I (t 2 ) = Ω 2 0 k 2 cos 2 θ T 0 dt 1 T 0 dt 2 e -i∆(t1-t2) q(t 1 )q(t 2 ) , ( 15\n\n)\n\nwhere P e = 1 vib ⊗ |e e| is the projector onto the excited electronic state (and the identity on the vibrational subspace). We always assume that the ion begins in the electronic ground state. If the ion also starts out in the instantaneous vibrational ground state for a static trap of frequency ν 0 = ν(0) at t = 0 (which is most useful when the chirping begins in the adiabatic regime), then we can evaluate the two-time correlation function as\n\nq(t 1 )q(t 2 ) ground = q 2 0 h(t 1 )h(t 2 ) + p 2 0 M 2 g(t 1 )g(t 2 ) + q 0 p 0 M h(t 1 )g(t 2 ) -h(t 2 )g(t 1 ) = 2M ν 0 h(t 1 ) -iν 0 g(t 1 ) h(t 2 ) + iν 0 g(t 2 ) = 2M ν 0 f (t 1 )f * (t 2 ) , ( 16\n\n)\n\nwhere we have used the facts that for the vibrational ground state, q 2 0\n\n= (p 0 /M ν 0 ) 2 = /2M ν 0 and q 0 p 0 = 1 2 {q 0 , p 0 } + [q 0 , p 0 ] = i /2,\n\nand we have defined the complex function\n\nf (t) = h(t) -iν 0 g(t) , ( 17\n\n)\n\nwhich is the solution to Eq. ( 4 ) with initial the conditions, f (0) = 1 and ḟ (0) = -iν 0 . Plugging this into Eq. ( 15 ) gives, quite simply,\n\nP (1) → (Ω 0 η 0 ) 2 |F ¸|2 , ( 18\n\n) where F ¸= T 0 dt e -i∆t f (t) , ( 19\n\n)\n\nand we have defined the unitless, time-dependent Lamb-Dicke parameter [10] as 0) . Recalling that f (t) can be considered a complex c-number solution to the equations of motion for the equivalent classical Hamiltonian, Eq. (18) shows that the excitation probability is simply related to the Fourier transform of the classical trajectories when beginning in the vibrational ground state.\n\nη(t) = k 2 cos 2 θ 2M ν(t) , ( 20\n\n)\n\nand η 0 = η(\n\nIII. EXPONENTIAL CHIRPING Recent work [15] has suggested that an exponentially decaying trap frequency has the same effect on the phonon modes of a string of ions as an expanding (i.e., de Sitter) spacetime does on a one-dimensional scalar field [18] . An inertial detector that responds to such an expanding scalar field would register a thermal bath of particles, called Gibbons-Hawking radiation [16]. Ref. [15] suggests that the acoustic analog [19] of this radiation could be seen in an ion trap, causing each ion to be excited with a thermal spectrum with temperature κ/2πk B , as a function of the detuning ∆, where κ is the trap-frequency decay rate. The analysis used an incorrect Hamiltonian that neglected squeezing and source terms that have no analog in the expanding scalar field model but which are present when considering trapped ions in this way, and the results are incorrect. In this section, we revisit this problem and calculate the excitation probability for a single ion in an exponentially decaying harmonic potential, as a function of the detuning ∆.\n\nWe write the time-dependent frequency as [20] ν(t) = ν 0 e -κt . (21) This results in q + ν 2 0 e -2κt q = 0 . (22) Solutions with initial conditions (6) are\n\nh(t) = πν 0 2κ J 1 ν 0 κ Y 0 ν κ -Y 1 ν 0 κ J 0 ν κ , ( 23\n\n) g(t) = π 2κ -J 0 ν 0 κ Y 0 ν κ + Y 0 ν 0 κ J 0 ν κ , ( 24\n\n)\n\nwhere the time dependence is carried in ν = ν(t) from Eq. (21), and J n and Y n are Bessel functions. We could plug these directly into the formulas from the last section, but we will simplify the calculations by considering the limits of slow and long-time frequency decay, represented by\n\nν 0 ≫ κ and ν 0 e -κT ≪ κ , ( 25\n\n)\n\nrespectively. This allows us to do several things. First, it allows us to use the usual ground state of the unchirped trap at frequency ν 0 as a good approximation to the ground state of the expanding trap at t = 0, since at that time the system is being chirped adiabatically. This is important because it allows the experiment to begin with a static potential, which is useful for cooling. Second, it allows us to simplify h(t) and g(t) using the phase-space rotation freedom discussed above. Using asymptotic approximations for the Bessel functions in the coefficients,\n\nJ 0 ν 0 κ ≃ -Y 1 ν 0 κ ≃ 2κ πν 0 cos ν 0 κ - π 4 , ( 26\n\n) J 1 ν 0 κ ≃ Y 0 ν 0 κ ≃ 2κ πν 0 sin ν 0 κ - π 4 , ( 27\n\n) 4 we get h(t) ≃ πν 0 2κ sin ϕ Y 0 ν κ + cos ϕ J 0 ν κ , ( 28\n\n) ν 0 g(t) ≃ πν 0 2κ -cos ϕ Y 0 ν κ + sin ϕ J 0 ν κ . ( 29\n\n)\n\nwhere ϕ = ν 0 /κ -π/4. Since we are taking the initial state to be the ground state, which is symmetric with respect to phase-space rotations, we can use the freedom discussed in the previous section to undo the rotation represented by Eqs. (28) and (29) and define the simpler functions\n\nh(t) → h ′ (t) = πν 0 2κ Y 0 ν κ , ( 30\n\n) g(t) → g ′ (t) = π 2κν 0 J 0 ν κ . ( 31\n\n)\n\nThe primes are unnecessary due to the symmetry of the initial state, so we drop them from now on and plug directly into Eq. (17):\n\nf (t) = πν 0 2κ Y 0 ν κ -iJ 0 ν κ = -i πν 0 2κ H (1) 0 ν κ , ( 32\n\n)\n\nwhere H (1) n is a Hankel function of the first kind. The integral in Eq. ( 19 ) can be evaluated in the limits (25) using techniques similar to those used in Ref. [15] . First, define e α = ν κ , τ = α -κt , u = e τ , and x = ∆/κ .\n\nThe integral in question then becomes (neglecting the prefactor)\n\nT 0 dt e -i∆t H (1) 0 ν κ = T 0 dt e -i∆t H ( 1\n\n) 0 (e α-κt ) = 1 κ α α-κT dτ e -ix(α-τ ) H (1) 0 (e τ ) → e -ixα κ ∞ -∞ dτ e ixτ H (1) 0 (e τ ) = e -ixα κ ∞ 0 du u ix-1 H (1) 0 (u) . ( 34\n\n)\n\nInserting a convergence factor with x → x -iǫ, and then taking the limit ǫ → 0 + , we can use the formula\n\n∞ 0 du u ix-1 H (1) 0 (u) = -2 ix Γ(ix/2) (e πx -1)Γ(1 -ix/2) ( 35\n\n) to evaluate |F ¸|2 = πν 0 2κ 1 κ 2 Γ(ix/2) Γ(1 -ix/2) 2 1 (e πx -1) 2 = 2πν 0 κ 3 x 2 1 (e πx -1) 2 . ( 36\n\n)\n\nWhen plugging in for the dummy variables (33), this gives\n\nP (1) = (Ω 0 η 0 ) 2 2πν 0 κ∆ 2 1 (e π∆/κ -1) 2 . ( 37\n\n)\n\nThe calculated result from Ref. [15] for a single ion is\n\nP ( 1\n\n) GH = (Ω 0 η 0 ) 2 2π κ∆ 1 e 2π∆/κ -1 , ( 38\n\n)\n\nwhich contains a Planck factor with Gibbons-Hawking [16] temperature T = κ/2πk B but is different from the actual result for a single ion, given by Eq. (37) .\n\nSeveral things should be noted about these functions. First, they both break down as ∆ → 0 because of the approximation made in obtaining Eq. ( 14 ). They also fail if the time-dependent Lamb-Dicke parameter (20) ever becomes too large throughout the chirping process. Furthermore, most cases of interest will be ∆ ≃ ν 0 (the first red sideband) and near ∆ ≃ -ν 0 (the first blue sideband), which means that |∆| ≫ κ, since ν 0 ≫ κ. The first red sideband represents a detector that requires the absorption of one phonon (plus one laser photon) in order to excite the atom-the usual thing we mean by \"particle detector\" when the particles are phonons. The first blue sideband, on the other hand, represents a detector that emits a phonon in order to excite the atom (along with absorbing one laser photon).\n\nThere are a couple of ways to compare these functions. First, we can take the ratio of the two for both the redand blue-sideband cases. In both cases, we obtain\n\nP (1) P (1) GH ≃ ν 0 |∆| (1 + 2e -π|∆|/κ ) ( 39\n\n)\n\nplus terms of order O(e -2π|∆|/κ ). Since |∆| ≃ ν 0 , the prefactor is close to one, and the second term is very small (since ν 0 ≫ κ). Furthermore, it is cumbersome to directly compare the measured probability to the full function (with all the prefactors). It is often easier instead to make measurements on both the first red sideband and the first blue sideband and then take the ratio of the two. The constant prefactors disappear in this calculation, and both functions then have the same experimental signature:\n\nP (1) (∆) P (1) (-∆) = P ( 1\n\n) GH (∆) P (1) GH (-∆) = e -2π∆/κ , ( 40\n\n)\n\nwhich is that of a thermal distribution with temperature T = κ/2πk B , which is of the Gibbons-Hawking form [16] with the expansion rate given by κ. Therefore, although the Hamiltonian used in the calculations in Ref. [15] was missing terms, the intuition (at least for a single ion) was correct in that the actual experimental signature in this case matches that of an ion undergoing 5 thermal motion in a static trap, where the temperature is proportional to κ. To see whether this experiment is feasible, we must examine the validity of our approximations. For a typical trap, we expect that ν 0 ≃ 1 MHz, and thus if we take κ ≃ 1 kHZ, we easily satisfy the first of conditions (25), namely ν 0 ≫ κ. The second of these conditions gives a constraint on the modulation time T . For these parameters we expect that T ≃ a few msec. This is compatible with typical cooling and readout time scales and is less than those for heating due to fluctuating patch potentials [10] . Thus, this is a realizable experiment with current technology."
},
{
"section_type": "CONCLUSION",
"section_title": "IV. CONCLUSION",
"text": "We have shown that a single trapped ion in a modulated trapping potential can serve as an experimentally accessible implementation of a quantum harmonic oscillator with time-dependent frequency, including robust control over state preparation, manipulation, and measure-ment. The ion itself serves both as the oscillating particle and as the local detector of vibrational motion via coupling to internal electronic states by an external laser. For the case of a general time-dependent trap frequency, we calculated the first-order excitation probability for the ion in terms of the solution to the classical equations of motion for the equivalent classical oscillator. We applied this general result to the case of exponential chirping and corrected the calculation in Ref. [15] for a single ion. We found that while the results from the two calculations differ, the experimental signature in both cases is the same and equivalent to that of a thermal ion in a static trap.\n\nWe thank Dave Kielpinski for invaluable help with the experimental details. We also thank Paul Alsing, Bill Unruh, John Preskill, Jeff Kimble, Greg Ver Steeg, and Michael Nielsen for useful discussions and suggestions. NCM extends much appreciation to the faculty and staff of the Caltech Institute for Quantum Information for their hospitality during his visit, which helped bring this work to fruition. NCM was supported by the United States Department of Defense, and GJM acknowledges support from the Australian Research Council.\n\n[1] D. Stoler, Phys. Rev. D 1, 3217 (1970). [2] H. P. Yuen, Phys. Rev. A 13, 2226 (1976). [3] J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). [4] X. Ma and W. Rhodes, Phys. Rev. A 39, 1941 (1989). [5] V. V. Dodonov and A. B. Klimov, Phys. Rev. A 53, 2664 (1996). [6] G. T. Moore, J. Math. Phys. 11, 2679 (1970). [7] S. P. Kim and S.-W. Kim, Phys. Rev. D 51, 4254 (1995). [8] D. Polarski and A. A. Starobinsky, Classical and Quantum Gravity 13, 377 (1996). [9] C. Kiefer, J. Lesgourgues, D. Polarski, and A. A. Starobinsky, Classical and Quantum Gravity 15, L67 (1998). [10] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev.\n\nMod. Phys. 75, 281 (2003). [11] C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. L. Gould, Phys. Rev. Lett. 75, 4011 (1995). [12] D. F. V. James, Applied Physics B: Lasers and Optics 66, 181 (1998). [13] D. Leibfried, B. De Marco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenkovic, C. Langer, T. Rosenband, et al., Nature 422, 412 (2003). [14] F. Schmidt-Kaler, H. Häffner, M. Riebe, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003). [15] P. M. Alsing, J. P. Dowling, and G. J. Milburn, Phys.\n\nRev. Lett. 94, 220401 (2005). [16] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977). [17] S. Wallentowitz and W. Vogel, Phys. Rev. A 54, 3322 (1996). [18] A. M. de M. Carvalho, C. Furtado, and I. A. Pedrosa, Phys. Rev. D 70, 123523 (pages 6) (2004). [19] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). [20] The authors of Ref. [15] consider both signs in the exponential, but we will restrict ourselves to the case that allows us to begin chirping in the adiabatic limit."
}
] | |
arxiv:0704.0137 | 0704.0137 | 1 | f75d6ea4258f3061a36baa4625616d5304a5b4f7168f59319f746f367c8447aa | Topological defects, geometric phases, and the angular momentum of light | Recent reports on the intriguing features of vector vortex bearing beams are
analyzed using geometric phases in optics. It is argued that the spin
redirection phase induced circular birefringence is the origin of topological
phase singularities arising in the inhomogeneous polarization patterns. A
unified picture of recent results is presented based on this proposition.
Angular momentum shift within the light beam (OAM) has exact equivalence with
the angular momentum holonomy associated with the geometric phase consistent
with our conjecture. | [
"S C Tiwari"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | Recent reports on the intriguing features of vector vortex bearing beams are analyzed using geometric phases in optics. It is argued that the spin redirection phase induced circular birefringence is the origin of topological phase singularities arising in the inhomogeneous polarization patterns. A unified picture of recent results is presented based on this proposition. Angular momentum shift within the light beam (OAM) has exact equivalence with the angular momentum holonomy associated with the geometric phase consistent with our conjecture.
Topological defects in continuous field theoretic framework are usually associated with field singularities, however in analogy with crystal defects wavefront dislocations for scalar waves [1] and disclinations for vector waves [2] have been discussed in the literature. An important advancement was the realization that topological charge was related with the orbital angular momentum (OAM) of finite sized (transverse) light beams: typically for the Laguerre-Gaussian (LG) beams helicoidal spatial structure of the wavefront with azimuthal phase exp(ilφ) gives rise to OAM per photon of lh where l is the topological charge, see review [3] . Adopting the fluid dynamical paradigm topological defects in optics are termed vortices; singularities in the polarization patterns are called vector vortices [4] .
The aim of this Letter is to present a unified picture of the underlying physics of intriguing aspects of recent reports [5, 6, 7] in terms of the transformation of topological charges due to spin redirection phase (SRP) such that OAM is exchanged within the beams [8] . The role of Pancharatnam phase (PP) invoked in [4, 6, 7] is also critically examined.
For the sake of clarity we briefly review the essentials of geometric phases (GP) in optics which are primarily of two types, see [8, 9] for details and original references. Rytov-Vladimirskii phase rediscovered by Chiao and Wu in 1986 (inspired by the Berry phase) arises in the wave vector or momentum space of light. The unit wave vector κ and polarization vector ǫ(κ) describe the intrinsic properties of the light wave. A plane wave propagating along a slowly varying path in the real space can be mapped on to the surface of a unit sphere in the wave vector space, and under parallel transport along a curve in this space preserving the spin helicity, ǫ.κ, the polarization vector is found to be rotated after the completion of a closed cycle on the sphere. The magnitude of the rotation is given by the solid angle enclosed by the cycle, and the sign is determined by the initial polarization state. Since left circular |L > and right circular |R > polarization states acquire equal but opposite geometric phases, Berry terms this effect as geometric circular birefringence [10] .
A polarized light wave propagating in a fixed direction passed through optical elements traversing a polariza-tion cycle on the Poincare sphere acquires Pancharatnam phase equal to half of the solid angle of the cycle. Berry points out that [11] Pancharatnam actually made two important contributions. One,a notion of Pancharatnam connection was introduced for the phase difference between two arbitrary nonorthogonal polarizations which can be written as arg(E 1 * .E 2 ) for complex electric field vectors. Secondly this connection is nontransitive resulting into the Pancharatnam phase for a geodesic triangle on the sphere. Note that a parallel transport on the Poincare sphere is made with fixed direction of propagation for the occurrence of PP.
In the case of space varying polarization patterns, defining a direction of propagation is not easy, however Nye [12] has used Pancharatnam connection to define propagation vector k δ as a gradient of phase difference between fields at spatial locations r 1 and r 2 given by
dδ = Im(E * .dE)/|E| 2 (1)
In [4] authors correctly use Pancharatnam connection to obtain the phase difference of light at two locations in the space varying polarization plane, however the GP involved is not Pancharatnam phase as one cannot complete polarization cycle without changing the wave vector direction. As discussed above we have to construct appropriate wave vector space for the case of vector vortices. For an initial beam propagating along z-axis, at each point (r, φ) on the inhomogeneous polarization plane there will correspond a k-space, and spin helicity preserving parallel transport will give SRP for closed cycles. It is known that for a linearly polarized plane wave represented by
|P >= e -iα |R > +e iα |L > (2)
the SRP corresponding to a cycle with solid angle Ω results into [10]
|P t >= e -i(α+Ω) |R > +e i(α+Ω) |L > (3)
For the optical vortices this equation has to be generalized: we suggest a spatially evolving GP embodied in the solid angle as a function of (r, φ). This is one of the main contributions of this Letter leading to Eq.( 4 ) below. For the special case in which only the azimuthal angle dependence matters we obtain Ω in the following way. In the transverse plane consider a point A on a circle specified by φ, then the area enclosed by the great circles in k-space corresponding to this point and the reference point O specified by φ = 0 would subtend a solid angle which varies linearly with φ; the solid angle will vary from 0 to 4π as φ varies from 0 to 2π. Thus we obtain the generalization of Eq.( 3 ) to
|P v >= e -i(α+2φ) |R > +e i(α+2φ |L > (4)
Spatially evolving SRP [13] is crucial to understand interesting features observerd in vector vortices; we state our second principal contribution in the form of a proposition.
Proposition: Geometric phase induced circular birefringence is the origin of topological charge transformation in vector vortex carrying beams, and angular momentum holonomy is manifested as OAM.
We demonstrate in the following that this proposition offers transparent physical mechanism to explain the recent reports on inhomogeneous polarization patterns.
Backscattered polarization [5] : Theory and experiments on the backscattered light for linearly polarized light from random media have been of current interest. Interesting features for the backscattering geometry have been observed. Authors of [5] give insightful treatment of the observations invoking GP in wave vector space, and this is in agreement with our proposition. Note that backscattered light wave vector could be treated similar to the discrete transformation for reflection from a mirror, see discussion in [9] one can envisage an adiabatic path in a modified k-space. It may be noted that essentially spatially evolving SRP is used in [5] . It seems the term 'geometrical phase vortex' introduced by them for scalar vortices appearing in space varying polarization pattern is quite revealing.
The q-plate experiment [6] : An inhomogeneous anisotropic optical element called q-plate has novel addition to HWP : inhomogeneity is introduced orienting the fast (or slow) optical axis making an angle of α with the x-axis in the xy-plane for a planar slab given by α(r, φ) = qφ+ α 0 . Jones calculus applied at each point of the q-plate shows that the output beam for an incident |L > state is not only converted to |R > state but it also acquires an azimuthal phase factor of exp(2iqφ). Similar to the LG beams this phase is interpreted as OAM of 2qh per photon in the output wave. Experiment is carried out using nematic liquid crystal planar cell for q = 1 and the measurements on the interference pattern formed by the superposition of the output beam with the reference beam display the wave front singularities and helical modes in the output beam.
We argue that in the light of our proposition SRP induced circular birefringence is the origin of topological phase singularity and OAM in q-plates. We picture helicity preserving transformation in the wave vector space defined by k δ . Since the polarization variation is confined in the transverse plane for the q-plates the constraint of the spin helicity preserving process in the q-plate with the azimuthal dependence of α would lead to a spiral path for the wave vector. In q=1 plate the circular plus linear propagation along z-axis will result into a helical path and the width of the plate ensures that the input and output ends of the helix are parallel. On the unit sphere in the wave vector space this will correspond to a great circle, and the solid angle would be 2π. Since the SRP equals the solid angle for the evolving paths, our Eq.( 4 ) above is in agreement with Eq.( 3 ) of [6] . The important observation emphasized by the authors that the incident polarization controls the sign of the orbital helicity or topological charge is easily explained in view of the property of the geometric birefingence in which handedness decides the sign of the phase. Thus both magnitude and sign of the azimuthal phase have been explained in accordance with our proposition.
Tightly focused beams [7] : Analysis of the light field at the focal plane of a high numerical aperture lens for the incoming circularly polarized plane wave shows the existence of inhomogeneous polarization pattern. Pancharatnam connection at two different points on the circle around the focus shows φ-dependence of the phase difference. The field can be decomposed into |L > and |R > states, Eq.( 7 ) in [7] , and it is found that the component with the spin same as that in the object plane does not change phase while the one with opposite spin acquires an azimuthal phase of 2φ, i. e. topological charge 2 and OAM of 2h per photon. Application of Eq. ( 4 ) immediately leads to this result in conformity with our proposition. We may remark that the construction used by the authors to derive PP, namely the geodesic triangle on the Poincare sphere formed by the pole, E(r, 0), and E(r, φ) cannot be completed with a fixed direction of propagation for space varying polarization pattern, and therefore it is SRP not PP that arises.
Having established first part of the proposition, we discuss the role of angular momentum (AM) holonomy conjecture [8, 9] . Transfer of spin AM of light to matter was measured long ago by Beth [14] , and there are many reports of OAM transfer to particles in recent years [3] . Since polarization cycle for PP requires spin exchange with optical elements, it is natural to envisage a role of AM in GP; however it would be trivial. In the AM holonomy conjecture, we argued AM level shifts within the light beams as physical mechanism for GP. This implies exact equivalence between AM shift and GP. Indirectly the backscattered light experiments and their interpretation in terms of SRP supports our conjecture. In the context of the AM conservation [5] the redistribution of total AM within the beam is also indicative of AM level shifts. The q=1 plate is a special optical element in which no transfer of AM to the crystal takes place, and total AM is conserved within the light beam. We argue that spin is intrinsic, and the OAM is a manifestation of the GP with exact equivalence between them in this case. The counter-intuitive interpretation in terms of spin to OAM conversion claimed in [6] is clearly ruled out. In fact, Marrucci et al experiment offers first direct evidence in support of our conjecture. It is remarkable that the light field structure calculated for tightly focused beams shows strong resemblence with the action of q-plates on light wave, and offers another setting to test our proposition.
To conclude the Letter we make few observations. First let us note that even without the existence of phase singularities it should be possible to exchange AM within the light beam accompanied with GP: as argued earlier transverse shifts in the beam would account for the change in OAM [9] . Secondly the interplay of evolving GP in space and time domains could be of interest. A simple rotating q-plate experiment is suggested: polarized light beam after traversing the q-plate is made to pass through a rotating HWP. Another variant with nonintegral q for this arrangement, i.e. q-plate plus rotating HWP, is also suggested. Analysis of the emerging beams may delineate the role of SRP and PP as well as provide further test to AM holonomy conjecture.
The physical mechanism proposed here for space varying polarization pattern of light could find important application in 'all optical information processing'. The angular momentum holonomy associated with GP, and the strong evidence of its proof discussed here will have significance in the context of the controversy surrounding 'the hidden momentum' and Aharanov-Bohm effect. We believe present ideas also hold promise to address some fundamental questions in physics. An important recent example is that of birefringence of the vacuum in quantum electrodynamics in strong external magnetic field. Though this has been known since long, last year PVLAS experiment reported polarization rotation [15] apparently very much in excess than the expected one. This has led to a controversy on the interpretation of QED birefringence in external rotating magnetic field, see [16] for a short review. As remarked by Adler essentially it involves light wave propagation in a nontrivial refracive media, and he finds that to first order there should be no rotation of the polarization of light. Could there be a role of GP in this case? It would be interesting to use Pancharatnam connection to calculate the phase of propagating light, and see if evolving GP in time domain will arise due to rotating magnetic field. It is interesting to note that the magnetic field direction rotates in the plane transverse to the direction of the propagation of the light. Obviously it would give additional polarization rotation. This problem is being investigated, and will be reported elewhere.
The Library facility at Banaras Hindu University is acknowledged. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Recent reports on the intriguing features of vector vortex bearing beams are analyzed using geometric phases in optics. It is argued that the spin redirection phase induced circular birefringence is the origin of topological phase singularities arising in the inhomogeneous polarization patterns. A unified picture of recent results is presented based on this proposition. Angular momentum shift within the light beam (OAM) has exact equivalence with the angular momentum holonomy associated with the geometric phase consistent with our conjecture."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Topological defects in continuous field theoretic framework are usually associated with field singularities, however in analogy with crystal defects wavefront dislocations for scalar waves [1] and disclinations for vector waves [2] have been discussed in the literature. An important advancement was the realization that topological charge was related with the orbital angular momentum (OAM) of finite sized (transverse) light beams: typically for the Laguerre-Gaussian (LG) beams helicoidal spatial structure of the wavefront with azimuthal phase exp(ilφ) gives rise to OAM per photon of lh where l is the topological charge, see review [3] . Adopting the fluid dynamical paradigm topological defects in optics are termed vortices; singularities in the polarization patterns are called vector vortices [4] .\n\nThe aim of this Letter is to present a unified picture of the underlying physics of intriguing aspects of recent reports [5, 6, 7] in terms of the transformation of topological charges due to spin redirection phase (SRP) such that OAM is exchanged within the beams [8] . The role of Pancharatnam phase (PP) invoked in [4, 6, 7] is also critically examined.\n\nFor the sake of clarity we briefly review the essentials of geometric phases (GP) in optics which are primarily of two types, see [8, 9] for details and original references. Rytov-Vladimirskii phase rediscovered by Chiao and Wu in 1986 (inspired by the Berry phase) arises in the wave vector or momentum space of light. The unit wave vector κ and polarization vector ǫ(κ) describe the intrinsic properties of the light wave. A plane wave propagating along a slowly varying path in the real space can be mapped on to the surface of a unit sphere in the wave vector space, and under parallel transport along a curve in this space preserving the spin helicity, ǫ.κ, the polarization vector is found to be rotated after the completion of a closed cycle on the sphere. The magnitude of the rotation is given by the solid angle enclosed by the cycle, and the sign is determined by the initial polarization state. Since left circular |L > and right circular |R > polarization states acquire equal but opposite geometric phases, Berry terms this effect as geometric circular birefringence [10] .\n\nA polarized light wave propagating in a fixed direction passed through optical elements traversing a polariza-tion cycle on the Poincare sphere acquires Pancharatnam phase equal to half of the solid angle of the cycle. Berry points out that [11] Pancharatnam actually made two important contributions. One,a notion of Pancharatnam connection was introduced for the phase difference between two arbitrary nonorthogonal polarizations which can be written as arg(E 1 * .E 2 ) for complex electric field vectors. Secondly this connection is nontransitive resulting into the Pancharatnam phase for a geodesic triangle on the sphere. Note that a parallel transport on the Poincare sphere is made with fixed direction of propagation for the occurrence of PP.\n\nIn the case of space varying polarization patterns, defining a direction of propagation is not easy, however Nye [12] has used Pancharatnam connection to define propagation vector k δ as a gradient of phase difference between fields at spatial locations r 1 and r 2 given by\n\ndδ = Im(E * .dE)/|E| 2 (1)\n\nIn [4] authors correctly use Pancharatnam connection to obtain the phase difference of light at two locations in the space varying polarization plane, however the GP involved is not Pancharatnam phase as one cannot complete polarization cycle without changing the wave vector direction. As discussed above we have to construct appropriate wave vector space for the case of vector vortices. For an initial beam propagating along z-axis, at each point (r, φ) on the inhomogeneous polarization plane there will correspond a k-space, and spin helicity preserving parallel transport will give SRP for closed cycles. It is known that for a linearly polarized plane wave represented by\n\n|P >= e -iα |R > +e iα |L > (2)\n\nthe SRP corresponding to a cycle with solid angle Ω results into [10]\n\n|P t >= e -i(α+Ω) |R > +e i(α+Ω) |L > (3)\n\nFor the optical vortices this equation has to be generalized: we suggest a spatially evolving GP embodied in the solid angle as a function of (r, φ). This is one of the main contributions of this Letter leading to Eq.( 4 ) below. For the special case in which only the azimuthal angle dependence matters we obtain Ω in the following way. In the transverse plane consider a point A on a circle specified by φ, then the area enclosed by the great circles in k-space corresponding to this point and the reference point O specified by φ = 0 would subtend a solid angle which varies linearly with φ; the solid angle will vary from 0 to 4π as φ varies from 0 to 2π. Thus we obtain the generalization of Eq.( 3 ) to\n\n|P v >= e -i(α+2φ) |R > +e i(α+2φ |L > (4)\n\nSpatially evolving SRP [13] is crucial to understand interesting features observerd in vector vortices; we state our second principal contribution in the form of a proposition.\n\nProposition: Geometric phase induced circular birefringence is the origin of topological charge transformation in vector vortex carrying beams, and angular momentum holonomy is manifested as OAM.\n\nWe demonstrate in the following that this proposition offers transparent physical mechanism to explain the recent reports on inhomogeneous polarization patterns.\n\nBackscattered polarization [5] : Theory and experiments on the backscattered light for linearly polarized light from random media have been of current interest. Interesting features for the backscattering geometry have been observed. Authors of [5] give insightful treatment of the observations invoking GP in wave vector space, and this is in agreement with our proposition. Note that backscattered light wave vector could be treated similar to the discrete transformation for reflection from a mirror, see discussion in [9] one can envisage an adiabatic path in a modified k-space. It may be noted that essentially spatially evolving SRP is used in [5] . It seems the term 'geometrical phase vortex' introduced by them for scalar vortices appearing in space varying polarization pattern is quite revealing.\n\nThe q-plate experiment [6] : An inhomogeneous anisotropic optical element called q-plate has novel addition to HWP : inhomogeneity is introduced orienting the fast (or slow) optical axis making an angle of α with the x-axis in the xy-plane for a planar slab given by α(r, φ) = qφ+ α 0 . Jones calculus applied at each point of the q-plate shows that the output beam for an incident |L > state is not only converted to |R > state but it also acquires an azimuthal phase factor of exp(2iqφ). Similar to the LG beams this phase is interpreted as OAM of 2qh per photon in the output wave. Experiment is carried out using nematic liquid crystal planar cell for q = 1 and the measurements on the interference pattern formed by the superposition of the output beam with the reference beam display the wave front singularities and helical modes in the output beam.\n\nWe argue that in the light of our proposition SRP induced circular birefringence is the origin of topological phase singularity and OAM in q-plates. We picture helicity preserving transformation in the wave vector space defined by k δ . Since the polarization variation is confined in the transverse plane for the q-plates the constraint of the spin helicity preserving process in the q-plate with the azimuthal dependence of α would lead to a spiral path for the wave vector. In q=1 plate the circular plus linear propagation along z-axis will result into a helical path and the width of the plate ensures that the input and output ends of the helix are parallel. On the unit sphere in the wave vector space this will correspond to a great circle, and the solid angle would be 2π. Since the SRP equals the solid angle for the evolving paths, our Eq.( 4 ) above is in agreement with Eq.( 3 ) of [6] . The important observation emphasized by the authors that the incident polarization controls the sign of the orbital helicity or topological charge is easily explained in view of the property of the geometric birefingence in which handedness decides the sign of the phase. Thus both magnitude and sign of the azimuthal phase have been explained in accordance with our proposition.\n\nTightly focused beams [7] : Analysis of the light field at the focal plane of a high numerical aperture lens for the incoming circularly polarized plane wave shows the existence of inhomogeneous polarization pattern. Pancharatnam connection at two different points on the circle around the focus shows φ-dependence of the phase difference. The field can be decomposed into |L > and |R > states, Eq.( 7 ) in [7] , and it is found that the component with the spin same as that in the object plane does not change phase while the one with opposite spin acquires an azimuthal phase of 2φ, i. e. topological charge 2 and OAM of 2h per photon. Application of Eq. ( 4 ) immediately leads to this result in conformity with our proposition. We may remark that the construction used by the authors to derive PP, namely the geodesic triangle on the Poincare sphere formed by the pole, E(r, 0), and E(r, φ) cannot be completed with a fixed direction of propagation for space varying polarization pattern, and therefore it is SRP not PP that arises.\n\nHaving established first part of the proposition, we discuss the role of angular momentum (AM) holonomy conjecture [8, 9] . Transfer of spin AM of light to matter was measured long ago by Beth [14] , and there are many reports of OAM transfer to particles in recent years [3] . Since polarization cycle for PP requires spin exchange with optical elements, it is natural to envisage a role of AM in GP; however it would be trivial. In the AM holonomy conjecture, we argued AM level shifts within the light beams as physical mechanism for GP. This implies exact equivalence between AM shift and GP. Indirectly the backscattered light experiments and their interpretation in terms of SRP supports our conjecture. In the context of the AM conservation [5] the redistribution of total AM within the beam is also indicative of AM level shifts. The q=1 plate is a special optical element in which no transfer of AM to the crystal takes place, and total AM is conserved within the light beam. We argue that spin is intrinsic, and the OAM is a manifestation of the GP with exact equivalence between them in this case. The counter-intuitive interpretation in terms of spin to OAM conversion claimed in [6] is clearly ruled out. In fact, Marrucci et al experiment offers first direct evidence in support of our conjecture. It is remarkable that the light field structure calculated for tightly focused beams shows strong resemblence with the action of q-plates on light wave, and offers another setting to test our proposition.\n\nTo conclude the Letter we make few observations. First let us note that even without the existence of phase singularities it should be possible to exchange AM within the light beam accompanied with GP: as argued earlier transverse shifts in the beam would account for the change in OAM [9] . Secondly the interplay of evolving GP in space and time domains could be of interest. A simple rotating q-plate experiment is suggested: polarized light beam after traversing the q-plate is made to pass through a rotating HWP. Another variant with nonintegral q for this arrangement, i.e. q-plate plus rotating HWP, is also suggested. Analysis of the emerging beams may delineate the role of SRP and PP as well as provide further test to AM holonomy conjecture.\n\nThe physical mechanism proposed here for space varying polarization pattern of light could find important application in 'all optical information processing'. The angular momentum holonomy associated with GP, and the strong evidence of its proof discussed here will have significance in the context of the controversy surrounding 'the hidden momentum' and Aharanov-Bohm effect. We believe present ideas also hold promise to address some fundamental questions in physics. An important recent example is that of birefringence of the vacuum in quantum electrodynamics in strong external magnetic field. Though this has been known since long, last year PVLAS experiment reported polarization rotation [15] apparently very much in excess than the expected one. This has led to a controversy on the interpretation of QED birefringence in external rotating magnetic field, see [16] for a short review. As remarked by Adler essentially it involves light wave propagation in a nontrivial refracive media, and he finds that to first order there should be no rotation of the polarization of light. Could there be a role of GP in this case? It would be interesting to use Pancharatnam connection to calculate the phase of propagating light, and see if evolving GP in time domain will arise due to rotating magnetic field. It is interesting to note that the magnetic field direction rotates in the plane transverse to the direction of the propagation of the light. Obviously it would give additional polarization rotation. This problem is being investigated, and will be reported elewhere.\n\nThe Library facility at Banaras Hindu University is acknowledged."
}
] | ||
arxiv:0704.0138 | 0704.0138 | 1 | 10.1103/PhysRevD.76.044007 | 3ba301ce5e70e09a4143cd5f8753dd327f2ff619fe397cd40de8fcf206841f95 | Circular and non-circular nearly horizon-skimming orbits in Kerr
spacetimes | We have performed a detailed analysis of orbital motion in the vicinity of a
nearly extremal Kerr black hole. For very rapidly rotating black holes (spin
a=J/M>0.9524M) we have found a class of very strong field eccentric orbits
whose angular momentum L_z increases with the orbit's inclination with respect
to the equatorial plane, while keeping latus rectum and eccentricity fixed.
This behavior is in contrast with Newtonian intuition, and is in fact opposite
to the "normal" behavior of black hole orbits. Such behavior was noted
previously for circular orbits; since it only applies to orbits very close to
the black hole, they were named "nearly horizon-skimming orbits". Our analysis
generalizes this result, mapping out the full generic (inclined and eccentric)
family of nearly horizon-skimming orbits. The earlier work on circular orbits
reported that, under gravitational radiation emission, nearly horizon-skimming
orbits tend to evolve to smaller orbit inclination, toward prograde equatorial
configuration. Normal orbits, by contrast, always demonstrate slowly growing
orbit inclination (orbits evolve toward the retrograde equatorial
configuration). Using up-to-date Teukolsky-fluxes, we have concluded that the
earlier result was incorrect: all circular orbits, including nearly
horizon-skimming ones, exhibit growing orbit inclination. Using kludge fluxes
based on a Post-Newtonian expansion corrected with fits to circular and to
equatorial Teukolsky-fluxes, we argue that the inclination grows also for
eccentric nearly horizon-skimming orbits. We also find that the inclination
change is, in any case, very small. As such, we conclude that these orbits are
not likely to have a clear and peculiar imprint on the gravitational waveforms
expected to be measured by the space-based detector LISA. | [
"Enrico Barausse",
"Scott A. Hughes",
"Luciano Rezzolla"
] | [
"gr-qc",
"astro-ph"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | The space-based gravitational-wave detector LISA [1] will be a unique tool to probe the nature of supermassive black holes (SMBHs), making it possible to map in detail their spacetimes. This goal is expected to be achieved by observing gravitational waves emitted by compact stars or black holes with masses µ ≈ 1 -100 M ⊙ spiraling into the SMBHs which reside in the center of galaxies [2] (particularly the low end of the galactic center black hole mass function, M ≈ 10 5 -10 7 M ⊙ ). Such events are known as extreme mass ratio inspirals (EMRIs) . Current wisdom suggests that several tens to perhaps of order a thousand such events could be measured per year by LISA [3] .
Though the distribution of spins for observed astrophysical black holes is not very well understood at present, very rapid spin is certainly plausible, as accretion tends to spinup SMBHs [4] . Most models for quasi-periodic oscillations (QPOs) suggest this is indeed the case in all low-mass x-ray binaries for which data is available [5] . On the other hand, continuum spectral fitting of some high-mass x-ray binaries indicates that modest spins (spin parameter a/M ≡ J/M 2 ∼ 0.6 -0.8) are likewise plausible [6] . The continuum-fit technique does find an extremely high spin of a/M 0.98 for the galactic "microquasar" GRS1915+105 [7] . This argues for a wide variety of possible spins, depending on the detailed birth and growth history of a given black hole.
In the mass range corresponding to black holes in galactic centers, measurements of the broad iron Kα emission line in active galactic nuclei suggest that SMBHs can be very rapidly rotating (see Ref. [8] for a recent review). For instance, in the case of MCG-6-30-15, for which highly accurate observations are available, a has been found to be larger than 0.987M at 90% confidence [9] . Because gravitational waves from EM-RIs are expected to yield a very precise determination of the spins of SMBHs [10] , it is interesting to investigate whether EMRIs around very rapidly rotating black holes may possess peculiar features which would be observable by LISA. Should such features exist, they would provide unambiguous information on the spin of SMBHs and thus on the mechanisms leading to their formation [11] .
For extremal Kerr black holes (a = M ), the existence of a special class of "circular" orbits was pointed out long ago by Wilkins [12] , who named them "horizon-skimming" orbits. ("Circular" here means that the orbits are of constant Boyer-Lindquist coordinate radius r.) These orbits have varying inclination angle with respect to the equatorial plane and have the same coordinate radius as the horizon, r = M . De-2 spite this seemingly hazardous location, it can be shown that all these r = M orbits have finite separation from one another and from the event horizon [13] . Their somewhat pathological description is due to a singularity in the Boyer-Lindquist coordinates, which collapses a finite span of the spacetime into r = M . Besides being circular and "horizon-skimming", these orbits also show peculiar behavior in their relation of angular momentum to inclination. In Newtonian gravity, a generic orbit has L z = |L| cos ι, where ι is the inclination angle relative to the equatorial plane (going from ι = 0 for equatorial prograde orbits to ι = π for equatorial retrograde orbits, passing through ι = π/2 for polar orbits), and L is the orbital angular momentum vector. As a result, ∂L z (r, ι)/∂ι < 0, and the angular momentum in the z-direction always decreases with increasing inclination if the orbit's radius is kept constant. This intuitively reasonable decrease of L z with ι is seen for almost all black hole orbits as well. Horizon-skimming orbits, by contrast, exhibit exactly the opposite behavior: L z increases with inclination angle.
Reference [14] asked whether the behavior ∂L z /∂ι > 0 could be extended to a broader class of circular orbits than just those at the radius r = M for the spin value a = M . It was found that this condition is indeed more general, and extended over a range of radius from the "innermost stable circular orbit" to r ≃ 1.8M for black holes with a > 0.9524M . Orbits that show this property have been named "nearly horizonskimming". The Newtonian behavior ∂L z (r, ι)/∂ι < 0 is recovered for all orbits at r 1.8M [14] .
A qualitative understanding of this behavior comes from recalling that very close to the black hole all physical processes become "locked" to the hole's event horizon [15] , with the orbital motion of point particles coupling to the horizon's spin. This locking dominates the "Keplerian" tendency of an orbit to move more quickly at smaller radii, forcing an orbiting particle to slow down in the innermost orbits. Locking is particularly strong for the most-bound (equatorial) orbits; the least-bound orbits (which have the largest inclination) do not strongly lock to the black hole's spin until they have very nearly reached the innermost orbit [14] . The property ∂L z (r, ι)/∂ι > 0 reflects the different efficiency of nearly horizon-skimming orbits to lock with the horizon.
Reference [14] argued that this behavior could have observational consequences. It is well-known that the inclination angle of an inspiraling body generally increases due to gravitational-wave emission [16, 17] . Since dL z /dt < 0 because of the positive angular momentum carried away by the gravitational waves, and since "normal" orbits have ∂L z /∂ι < 0, one would expect dι/dt > 0. However, if during an evolution ∂L z /∂ι switches sign, then dι/dt might switch sign as well: An inspiraling body could evolve towards an equatorial orbit, signalling the presence of an "almostextremal" Kerr black hole.
It should be emphasized that this argument is not rigorous. In particular, one needs to consider the joint evolution of orbital radius and inclination angle; and, one must include the dependence of these two quantities on orbital energy as well as angular momentum 1 . As such, dι/dt depends not only on dL z /dt and ∂L z /∂ι, but also on dE/dt, ∂E/∂ι, ∂E/∂r and ∂L z /∂r.
In this sense, the argument made in Ref. [14] should be seen as claiming that the contribution coming from dL z /dt and ∂L z /∂ι are simply the dominant ones. Using the numerical code described in [17] to compute the fluxes dL z /dt and dE/dt, it was then found that a test-particle on a circular orbit passing through the nearly horizon-skimming region of a Kerr black hole with a = 0.998M (the value at which a hole's spin tends to be buffered due to photon capture from thin disk accretion [19]) had its inclination angle decreased by δι ≈ 1 • -2 • [14] in the adiabatic approximation [20] . It should be noted at this point that the rate of change of inclination angle, dι/dt, appears as the difference of two relatively small and expensive to compute rates of change [cf. Eq. (3.8) of Ref. [17] ]. As such, small relative errors in those rates of change can lead to large relative errors in dι/dt. Finally, in Ref. [14] it was speculated that the decrease could be even larger for eccentric orbits satisfying the condition ∂L z /∂ι > 0, possibly leading to an observable imprint on EMRI gravitational waveforms.
The main purpose of this paper is to extend Ref. [14] 's analysis of nearly horizon-skimming orbits to include the effect of orbital eccentricity, and to thereby test the speculation that there may be an observable imprint on EMRI waveforms of nearly horizon-skimming behavior. In doing so, we have revisited all the calculations of Ref. [14] using a more accurate Teukolsky solver which serves as the engine for the analysis presented in Ref. [21] .
We have found that the critical spin value for circular nearly horizon-skimming orbits, a > 0.9524M , also delineates a family of eccentric orbits for which the condition ∂L z (p, e, ι)/∂ι > 0 holds. (More precisely, we consider variation with respect to an angle θ inc that is easier to work with in the extreme strong field, but that is easily related to ι.) The parameters p and e are the orbit's latus rectum and eccentricity, defined precisely in Sec. II. These generic nearly horizonskimming orbits all have p 2M , deep in the black hole's extreme strong field.
We next study the evolution of these orbits under gravitational-wave emission in the adiabatic approximation. We first revisited the evolution of circular, nearly horizonskimming orbits using the improved Teukolsky solver which was used for the analysis of Ref. [21] . The results of this analysis were somewhat surprising: Just as for "normal" orbits, we found that orbital inclination always increases during inspiral, even in the nearly horizon-skimming regime. This is in stark contrast to the claims of Ref. [14] . As noted above, the inclination's rate of change depends on the difference of two expensive and difficult to compute numbers, and thus can be strongly impacted by small relative errors in those numbers. 1 In the general case, one must also include the dependence on "Carter's constant" Q [18], the third integral of black hole orbits (described more carefully in Sec. II). For circular orbits, Q = Q(E, Lz): knowledge of E and Lz completely determines Q.
3 A primary result of this paper is thus to retract the claim of Ref. [14] that an important dynamic signature of the nearly horizon-skimming region is a reversal in the sign of inclination angle evolution: The inclination always grows under gravitational radiation emission. We next extended this analysis to study the evolution of generic nearly horizon-skimming orbits. The Teukolsky code to which we have direct access can, at this point, only compute the radiated fluxes of energy E and angular momentum L z ; results for the evolution of the Carter constant Q are just now beginning to be understood [22] , and have not yet been incorporated into this code. We instead use "kludge" expressions for dE/dt, dL z /dt, and dQ/dt which were inspired by Refs. [23, 24] . These expression are based on post-Newtonian flux formulas, modified in such a way that they fit strong-field radiation reaction results obtained from a Teukolsky integrator; see Ref. [24] for further discussion. Our analysis indicates that, just as in the circular limit, the result dι/dt > 0 holds for generic nearly horizon-skimming orbits. Furthermore, and contrary to the speculation of Ref. [14] , we do not find a large amplification of dι/dt as orbits are made more eccentric.
Our conclusion is that the nearly horizon-skimming regime, though an interesting curiosity of strong-field orbits of nearly extremal black holes, will not imprint any peculiar observational signature on EMRI waveforms.
The remainder of this paper is organized as follows. In Sec. II, we review the properties of bound stable orbits in Kerr, providing expressions for the constants of motion which we will use in Sec. III to generalize nearly horizon-skimming orbits to the non-circular case. In Sec. IV, we study the evolution of the inclination angle for circular nearly horizon-skimming orbits using Teukolsky-based fluxes; in Sec. V we do the same for non-circular orbits and using kludge fluxes. We present and discuss our detailed conclusions in Sec. VI. The fits and post-Newtonian fluxes used for the kludge fluxes are presented in the Appendix. Throughout the paper we have used units in which G = c = 1.
The line element of a Kerr spacetime, written in Boyer-Lindquist coordinates reads [25]
ds 2 = -1 - 2M r Σ dt 2 + Σ ∆ dr 2 + Σ dθ 2 + r 2 + a 2 + 2M a 2 r Σ sin 2 θ sin 2 θ dφ 2 - 4M ar Σ sin 2 θ dt dφ, ( 1
)
where
Σ ≡ r 2 + a 2 cos 2 θ, ∆ ≡ r 2 -2M r + a 2 . ( 2
)
Up to initial conditions, geodesics can then be labelled by four constants of motion: the mass µ of the test particle, its energy E and angular momentum L z as measured by an observer at infinity and the Carter constant Q [18] . The presence of these four conserved quantities makes the geodesic equations separable in Boyer-Lindquist coordinates. Introducing the Carter time λ, defined by
dτ dλ ≡ Σ , ( 3
)
the geodesic equations become
µ dr dλ 2 = V r (r), µ dt dλ = V t (r, θ), µ dθ dλ 2 = V θ (θ), µ dφ dλ = V φ (r, θ) , ( 4
) with V t (r, θ) ≡ E ̟ 4 ∆ -a 2 sin 2 θ + aL z 1 - ̟ 2 ∆ , ( 5a
) V r (r) ≡ E̟ 2 -aL z 2 -∆ µ 2 r 2 + (L z -aE) 2 + Q , ( 5b
) V θ (θ) ≡ Q -L 2 z cot 2 θ -a 2 (µ 2 -E 2 ) cos 2 θ, ( 5c
) V φ (r, θ) ≡ L z csc 2 θ + aE ̟ 2 ∆ -1 - a 2 L z ∆ , ( 5d
)
where we have defined
̟ 2 ≡ r 2 + a 2 . ( 6
)
The conserved parameters E, L z , and Q can be remapped to other parameters that describe the geometry of the orbit. We have found it useful to describe the orbit in terms of an angle θ min -the minimum polar angle reached by the orbit -as well as the latus rectum p and the eccentricity e. In the weakfield limit, p and e correspond exactly to the latus rectum and eccentricity used to describe orbits in Newtonian gravity; in the strong field, they are essentially just a convenient remapping of the orbit's apoastron and periastron:
r ap ≡ p 1 -e , r peri ≡ p 1 + e . ( 7
)
Finally, in much of our analysis, it is useful to refer to 4 ) to impose dr/dλ = 0 at r = r ap and r = r peri , and to impose dθ/dλ = 0 at θ = θ min . (Note that for a circular orbit, r ap = r peri = r 0 . In this case, one must apply the rules dr/dλ = 0 and d 2 r/dλ 2 = 0 at r = r 0 .) Following this approach, Schmidt [26] was able to derive explicit expressions for E, L z and Q in terms of p, e and z -. We now briefly review Schmidt's results.
z -≡ cos 2 θ min , ( 8
) rather than to θ min directly. To map (E, L z , Q) to (p, e, z -), use Eq. (
Let us first introduce the dimensionless quantities
Ẽ ≡ E/µ , Lz ≡ L z /(µM ) , Q ≡ Q/(µM ) 2 , ( 9
) ã ≡ a/M , r ≡ r/M , ∆ ≡ ∆/M 2 , ( 10
)
4 Figure 1: Left panel: Inclination angles θinc for which bound stable orbits exist for a black hole with spin a = 0.998 M . The allowed range for θinc goes from θinc = 0 to the curve corresponding to the eccentricity under consideration, θinc = θ max inc . Right panel: Same as left but for an extremal black hole, a = M . Note that in this case θ max inc never reaches zero.
and the functions
f (r) ≡ r4 + ã2 r(r + 2) + z -∆ , ( 11
) g(r) ≡ 2 ã r , ( 12
) h(r) ≡ r(r -2) + z - 1 -z - ∆ , ( 13
) d(r) ≡ (r 2 + ã2 z -) ∆ . ( 14
)
Let us further define the set of functions
(f 1 , g 1 , h 1 , d 1 ) ≡ (f (r p ), g(r p ), h(r p ), d(r p )) if e > 0 , (f (r 0 ), g(r 0 ), h(r 0 ), d(r 0 )) if e = 0 , ( 15
) (f 2 , g 2 , h 2 , d 2 ) ≡ (f (r a ), g(r a ), h(r a ), d(r a )) if e > 0 , (f ′ (r 0 ), g ′ (r 0 ), h ′ (r 0 ), d ′ (r 0 )) if e = 0 , ( 16
)
and the determinants
κ ≡ d 1 h 2 -d 2 h 1 , (17) ε ≡ d 1 g 2 -d 2 g 1 , (18) ρ ≡ f 1 h 2 -f 2 h 1 , (19) η ≡ f 1 g 2 -f 2 g 1 , (20) σ ≡ g 1 h 2 -g 2 h 1 . ( 21
)
The energy of the particle can then be written
Ẽ = κρ + 2ǫσ -2D σ(σǫ 2 + ρǫκ -ηκ 2 ) ρ 2 + 4ησ . ( 22
)
The parameter D takes the values ±1. The angular momentum is a solution of the system
f 1 Ẽ2 -2g 1 Ẽ Lz -h 1 L2 z -d 1 = 0 , (23) f 2 Ẽ2 -2g 2 Ẽ Lz -h 2 L2 z -d 2 = 0 . ( 24
)
By eliminating the L2 z terms in these equations, one finds the solution
Lz = ρ Ẽ2 -κ 2 Ẽσ ( 25
)
for the angular momentum. Using dθ/dλ = 0 at θ = θ min , the Carter constant can be written
Q = z - ã2 (1 -Ẽ2 ) + L2 z 1 -z - . ( 26
)
Additional constraints on p, e, z -are needed for the orbits to be stable. Inspection of Eq. ( 4 ) shows that an eccentric orbit is stable only if
∂V r ∂r (r peri ) > 0 . ( 27
)
It is marginally stable if ∂V r /∂r = 0 at r = r peri . Similarly, the stability condition for circular orbits is
∂ 2 V r ∂r 2 (r 0 ) < 0 ; ( 28
)
marginally stable orbits are set by
∂ 2 V r /∂r 2 = 0 at r = r 0 .
5 Finally, we note that one can massage the above solutions for the conserved orbital quantities of bound stable orbits to rewrite the solution for Lz as
Lz = - g 1 Ẽ h 1 + D h 1 g 2 1 Ẽ2 + (f 1 Ẽ2 -d 1 )h 1 . ( 29
)
From this solution, we see that it is quite natural to refer to orbits with D = 1 as prograde and to orbits with D = -1 as retrograde. Note also that Eq. (29) is a more useful form than the corresponding expression, Eq. ( A4 ), of Ref. [21] . In that expression, the factor 1/h 1 has been squared and moved inside the square root. This obscures the fact that h 1 changes sign for very strong field orbits. Differences between Eq. ( 29 ) and Eq. (A4) of [21] are apparent for a 0.835, although only for orbits close to the separatrix (i.e., the surface in the space of parameters (p, e, ι) where marginally stable bound orbits lie).
With explicit expressions for E, L z and Q as functions of p, e and z -, we now examine how to generalize the condition ∂L z (r, ι)/∂ι > 0, which defined circular nearly horizonskimming orbits in Ref. [14] , to encompass the non-circular case. We recall that the inclination angle ι is defined as [14]
cos ι = L z Q + L 2 z . ( 30
)
Such a definition is not always easy to handle in the case of eccentric orbits. In addition, ι does not have an obvious physical interpretation (even in the circular limit), but rather was introduced essentially to generalize (at least formally) the definition of inclination for Schwarzschild black hole orbits. In that case, one has
Q = L 2 x + L 2 y and therefore L z = |L| cos ι.
A more useful definition for the inclination angle in Kerr was introduced in Ref. [21] :
θ inc = π 2 -D θ min , ( 31
)
where θ min is the minimum reached by θ during the orbital motion. This angle is trivially related to z -(z -= sin 2 θ inc ) and ranges from 0 to π/2 for prograde orbits and from π/2 to π for retrograde orbits. It is a simple numerical calculation to convert between ι and θ inc ; doing so shows that the differences between ι and θ inc are very small, with the two coinciding for a = 0, and with a difference that is less than 2.6 • for a = M and circular orbits with r = M . Bearing all this in mind, the condition we have adopted to generalize nearly horizon-skimming orbits is
∂L z (p, e, θ inc ) ∂θ inc > 0 . ( 32
)
We have found that certain parts of this calculation, particular the analysis of strong-field geodesic orbits, are best done using the angle θ inc ; other parts are more simply done using the angle ι, particularly the "kludge" computation of fluxes described in Sec. V. (This is because the kludge fluxes are based on an extension of post-Newtonian formulas to the strongfield regime, and these formulas use ι for inclination angle.) Accordingly, we often switch back and forth between these two notions of inclination, and in fact present our final results for inclination evolution using both dι/dt and dθ inc /dt. Before mapping out the region corresponding to nearly horizon-skimming orbits, it is useful to examine stable orbits more generally in the strong field of rapidly rotating black holes. We first fix a value for a, and then discretize the space of parameters (p, e, θ inc ). We next identify the points in this space corresponding to bound stable geodesic orbits. Sufficiently close to the horizon, the bound stable orbits with specified values of p and e have an inclination angle θ inc ranging from 0 (equatorial orbit) to a maximum value θ max inc . For given p and e, θ max inc defines the separatrix between stable and unstable orbits.
Example separatrices are shown in Fig. 1 for a = 0.998M and a = M . This figure shows the behavior of θ max inc as a function of the latus rectum for the different values of the eccentricity indicated by the labels. Note that for a = 0.998M the angle θ max inc eventually goes to zero. This is the general behavior for a < M . On the other hand, for an extremal black hole, a = M , θ max inc never goes to zero. The orbits which reside at r = M (the circular limit) are the "horizon-skimming orbits" identified by Wilkins [12] ; the a = M separatrix has a similar shape even for eccentric orbits. As expected, we find that for given latus rectum and eccentricity the orbit with θ inc = 0 is the one with the lowest energy E (and hence is the most-bound orbit), whereas the orbit with θ inc = θ max inc has the highest E (and is least bound).
Having mapped out stable orbits in (p, e, θ inc ) space, we then computed the partial derivative ∂L z (p, e, θ inc )/∂θ inc and identified the following three overlapping regions:
• Region A: The portion of the (p, e) plane for which ∂L z (p, e, θ inc )/∂θ inc > 0 for 0 ≤ θ inc ≤ θ max inc . This region is illustrated in Fig. 2 as the area under the heavy solid line and to the left of the dot-dashed line (green in the color version).
• Region B: The portion of the (p, e) plane for which (L z ) most bound (p, e) is smaller than (L z ) least bound (p, e). In other words,
L z (p, e, 0) < L z (p, e, θ max inc ) ( 33
)
in Region B. Note that Region B contains Region A. It is illustrated in Fig. 2 as the area under the heavy solid line and to the left of the dotted line (red in the color version).
• Region C: The portion of the (p, e) plane for which ∂L z (p, e, θ inc )/∂θ inc > 0 for at least one angle θ inc between 0 and θ max inc . Region C contains Region B, and is illustrated in Fig. 2 as the area under the heavy solid line and to the left of the dashed line (blue in the color version). 6 Figure 2: Left panel: Non-circular nearly horizon-skimming orbits for a = 0.998M . The heavy solid line indicates the separatrix between stable and unstable orbits for equatorial orbits (ι = θinc = 0). All orbits above and to the left of this line are unstable. The dot-dashed line (green in the color version) bounds the region of the (p, e)-plane where ∂Lz/∂θinc > 0 for all allowed inclination angles ("Region A"). All orbits between this line and the separatrix belong to Region A. The dotted line (red in the color version) bounds the region (Lz) most bound < (Lz) least bound ("Region B"). Note that B includes A. The dashed line (blue in the color version) bounds the region where ∂Lz/∂θinc > 0 for at least one inclination angle ("Region C"); note that C includes B. All three of these regions are candidate generalizations of the notion of nearly horizon-skimming orbits. Right panel: Same as the left panel, but for the extreme spin case, a = M . In this case the separatrix between stable and unstable equatorial orbits is given by the line p/M = 1 + e.
Orbits in any of these three regions give possible generalizations of the nearly horizon-skimming circular orbits presented in Ref. [14] . Notice, as illustrated in Fig. 2 , that the size of these regions depends rather strongly on the spin of the black hole. All three regions disappear altogether for a < 0.9524M (in agreement with [14] ); their sizes grow with a, reaching maximal extent for a = M . These regions never extend beyond p ≃ 2M .
As we shall see, the difference between these three regions is not terribly important for assessing whether there is a strong signature of the nearly horizon-skimming regime on the inspiral dynamics. As such, it is perhaps most useful to use Region C as our definition, since it is the most inclusive.
To ascertain whether nearly horizon-skimming orbits can affect an EMRI in such a way as to leave a clear imprint in the gravitational-wave signal, we have studied the time evolution of the inclination angle θ inc . To this purpose we have used the so-called adiabatic approximation [20] , in which the infalling body moves along a geodesic with slowly changing parameters. The evolution of the orbital parameters is computed using the time-averaged fluxes dE/dt, dL z /dt and dQ/dt due to gravitational-wave emission ("radiation reaction"). As discussed in Sec. II, E, L z and Q can be expressed in terms of p, e, and θ inc . Given rates of change of E, L z and Q, it is then straightforward [23] to calculate dp/dt, de/dt, and dθ inc /dt (or dι/dt).
We should note that although perfectly well-behaved for all bound stable geodesics, the adiabatic approximation breaks down in a small region of the orbital parameters space very close to the separatrix, where the transition from an inspiral to a plunging orbit takes place [27] . However, since this region is expected to be very small 2 and its impact on LISA waveforms rather hard to detect [27] , we expect our results to be at least qualitatively correct also in this region of the space of parameters.
Accurate calculation of dE/dt and dL z /dt in the adiabatic approximation involves solving the Teukolsky and Sasaki-Nakamura equations [28, 29] . For generic orbits this has been done for the first time in Ref. [21] . The calculation of dQ/dt for generic orbits is more involved. A formula for dQ/dt has been recently derived [22] , but has not yet been implemented (at least in a code to which we have access).
On the other hand, it is well-known that a circular orbit will remain circular under radiation reaction [30, 31, 32] . This constraint means that Teukolsky-based fluxes for E and L z 2 Its width in p/M is expected to be of the order of ∆p/M ∼ (µ/M ) 2/5 , where µ is the mass of the infalling body [27].
7 are sufficient to compute dQ/dt. Considering this limit, the rate of change dQ/dt can be expressed in terms of dE/dt and dL/dt as
dQ dt circ = - N 1 (p, ι) N 5 (p, ι) dE dt circ - N 4 (p, ι) N 5 (p, ι) dL z dt circ (34) where N 1 (p, ι) ≡ E(p, ι) p 4 + a 2 E(p, ι) p 2 -2 a M (L z (p, ι) -a E(p, ι)) p , ( 35
) N 4 (p, ι) ≡ (2 M p -p 2 ) L z (p, ι) -2 M a E(p, ι) p , ( 36
) N 5 (p, ι) ≡ (2 M p -p 2 -a 2 )/2 . ( 37
)
(These quantities are for a circular orbit of radius p.) Using this, it is simple to compute dθ inc /dt (or dι/dt). This procedure was followed in Ref. [14] , using the code presented in Ref. [17] , to determine the evolution of ι; this analysis indicated that dι/dt < 0 for circular nearly horizonskimming orbits. As a first step to our more general analysis, we have repeated this calculation but using the improved Sasaki-Nakamura-Teukolsky code presented in Ref. [21] ; we focused on the case a = 0.998M .
Rather to our surprise, we discovered that the fluxes dE/dt and dL z /dt computed with this more accurate code indicate that dι/dt > 0 (and dθ inc /dt > 0) for all circular nearly horizon-skimming orbits -in stark contrast with what was found in Ref. [14] . As mentioned in the introduction, the rate of change of inclination angle appears as the difference of two quantities. These quantities nearly cancel (and indeed cancel exactly in the limit a = 0); as such, small relative errors in their values can lead to large relative error in the inferred inclination evolution. Values for dE/dt, dL z /dt, dι/dt, and dθ inc /dt computed using the present code are shown in Table I in the columns with the header "Teukolsky".
The corrected behavior of circular nearly horizonskimming orbits has naturally led us to investigate the evolution of non-circular nearly horizon-skimming orbits. Since our code cannot be used to compute dQ/dt, we have resorted to a "kludge" approach, based on those described in Refs. [23, 24] . In particular, we mostly follow the procedure developed by Gair & Glampedakis [24] , though (as described below) importantly modified.
The basic idea of the "kludge" procedure is to use the functional form of 2PN fluxes E, L z and Q, but to correct the circular part of these fluxes using fits to circular Teukolsky data. As developed in Ref. [24] , the fluxes are written
dE dt GG = (1 -e 2 ) 3/2 (1 -e 2 ) -3/2 dE dt 2PN (p, e, ι) - dE dt 2PN (p, 0, ι) + dE dt fit circ (p, ι) , ( 38
)
dL z dt GG = (1 -e 2 ) 3/2 (1 -e 2 ) -3/2 dL z dt 2PN (p, e, ι) - dL z dt 2PN (p, 0, ι) + dL z dt fit circ (p, ι) , ( 39
)
dQ dt GG = (1 -e 2 ) 3/2 Q(p, e, ι) × (1 -e 2 ) -3/2 dQ/dt √ Q 2PN (p, e, ι) - dQ/dt √ Q 2PN (p, 0, ι) + dQ/dt √ Q fit circ (p, ι) . ( 40
)
The post-Newtonian fluxes (dE/dt) 2PN , (dL z /dt) 2PN and (dQ/dt) 2PN are given in the Appendix [particularly Eqs. (A.1), (A.2), and (A.3)]. Since for circular orbits the fluxes dE/dt, dL z /dt and dQ/dt are related through Eq. (34) , only two fits to circular Teukolsky data are needed. One possible choice is to fit dL z /dt and dι/dt, and then use the circularity constraint to obtain 3 [24]
dQ/dt √ Q fit circ (p, ι) = 2 tan ι dL z dt fit circ + Q(p, 0, ι) sin 2 ι dι dt fit circ , ( 41
) dE dt fit circ (p, ι) = - N 4 (p, ι) N 1 (p, ι) dL z dt fit circ (p, ι) - N 5 (p, ι) N 1 (p, ι) Q(p, 0, ι) dQ/dt √ Q fit circ (p, ι) . ( 42
)
As stressed in Ref. [24] , one does not expect these fluxes to work well in the strong field, both because the post-Newtonian approximation breaks down close to the black hole, and because the circular Teukolsky data used for the fits in Ref. [24] was computed for 3M ≤ p ≤ 30M . As a first attempt to improve their behavior in the nearly horizon-skimming region, we have made fits using circular Teukolsky data for orbits with M < p ≤ 2M . In particular, for a black hole with a = 0.998M , we computed the circular Teukolsky-based fluxes dL z /dt and dι/dt listed in Table I (columns 8 and 10). These results were fit (with error 0.2%); see Eqs. (A.4) and (A.6) in the Appendix. 3 This choice might seem more involved than fitting directly dLz/dt and dQ/dt, but, as noted by Gair & Glampedakis, ensures more sensible results for the evolution of the inclination angle. This generates more physically realistic inspirals [24].
8 Despite using strong-field Teukolsky fluxes for our fit, we found fairly poor behavior of these rates of change, particularly as a function of eccentricity. To compensate for this, we introduced a kludge-type fit to correct the equatorial part of the flux, in addition to the circular part. We fit, as a function of p and e, Teukolsky-based fluxes for dE/dt and dL z /dt for orbits in the equatorial plane, and then introduce the following kludge fluxes for E and L z :
dE dt (p, e, ι) = dE dt GG (p, e, ι) - dE dt GG (p, e, 0) + dE dt fit eq (p, e) ( 43
)
dL z dt (p, e, ι) = dL z dt GG (p, e, ι) - dL z dt GG (p, e, 0) + dL z dt fit eq (p, e) . ( 44
)
[Note that Eq. (40) for dQ/dt is not modified by this procedure since dQ/dt = 0 for equatorial orbits.] Using equatorial non-circular Teukolsky data provided by Drasco [21, 33] for a = 0.998 and M < p ≤ 2M (the ι = 0 "Teukolsky" data in Tables II, III and IV), we found fits (with error 1.5%); see Eqs. (A.9) and (A.10). Note that the fits for equatorial fluxes are significantly less accurate than the fits for circular fluxes. This appears to be due to the fact that, close to the black hole, many harmonics are needed in order for the Teukolsky-based fluxes to converge, especially for eccentric orbits (cf. Figs. 2 and 3 of Ref. [21] , noting the number of radial harmonics that have significant contribution to the flux). Truncation of these sums is likely a source of some error in the fluxes themselves, making it difficult to make a fit of as high quality as we could in the circular case. These fits were then finally used in Eqs. (43) and (44) to calculate the kludge fluxes dE/dt and dL z /dt for generic orbits. This kludge reproduces to high accuracy our fits to the Teukolsky-based fluxes for circular orbits (e = 0) or equatorial orbits (ι = 0). Some residual error remains because the ι = 0 limit of the circular fits do not precisely equal the e = 0 limit of the equatorial fits.
Table I compares our kludge to Teukolsky-based fluxes for circular orbits; the two methods agree to several digits. Tables II, III and IV compare our kludge to the generic Teukolskybased fluxes for dE/dt and dL z /dt provided by Drasco [21, 33] . In all cases, the kludge fluxes dE/dt and dL z /dt have the correct qualitative behavior, being negative for all the orbital parameters under consideration (a = 0.998M ,
1 < p/M ≤ 2, 0 ≤ e ≤ 0.5 and 0 • ≤ ι ≤ 41 • ).
The relative difference between the kludge and Teukolsky fluxes is always less than 25% for e = 0 and e = 0.1 (even for orbits very close to separatrix). The accuracy remains good at larger eccentricity, though it degrades somewhat as orbits come close to the separatrix.
Tables I, II, III and IV also present the kludge values of the fluxes dι/dt and dθ inc /dt as computed using Eqs. (43) and (44) for dE/dt and dL z /dt, plus Eq. (40) for dQ/dt. Though certainly not the last word on inclination evolution (pending rigorous computation of dQ/dt), these rates of change probably represent a better approximation than results published to date in the literature. (Indeed, prior work has often used the crude approximation dι/dt = 0 [21] to estimate dQ/dt given dE/dt and dL z /dt.) Most significantly, we find that (dι/dt) kludge > 0 and (dθ inc /dt) kludge > 0 for all of the orbital parameters we consider. In other words, we find that dι/dt and dθ inc /dt do not ever change sign.
Finally, in Table V we compute the changes in θ inc and ι for the inspiral with mass ratio µ/M = 10 -6 . In all cases, we start at p/M = 1.9. The small body then inspirals through the nearly horizon-skimming region until it reaches the separatrix; at this point, the small body will fall into the large black hole on a dynamical timescale ∼ M , so we terminate the calculation. The evolution of circular orbits is computed using our fits to the circular-Teukolsky fluxes of E and L z ; for eccentric orbits we use the kludge fluxes (40), (43) and (44). As this exercise demonstrates, the change in inclination during inspiral is never larger than a few degrees. Not only is there no unique sign change in the nearly horizon-skimming region, but the magnitude of the inclination change remains puny. This leaves little room for the possibility that this class of orbits may have a clear observational imprint on the EMRIwaveforms to be detected by LISA.
We have performed a detailed analysis of the orbital motion near the horizon of near-extremal Kerr black holes. We have demonstrated the existence of a class of orbits, which we have named "non-circular nearly horizon-skimming orbits", for which the angular momentum L z increases with the orbit's inclination, while keeping latus rectum and eccentricity fixed. This behavior, in stark contrast to that of Newtonian orbits, generalizes earlier results for circular orbits [14] .
Furthermore, to assess whether this class of orbits can produce a unique imprint on EMRI waveforms (an important source for future LISA observations), we have studied, in the adiabatic approximation, the radiative evolution of inclination angle for a small body orbiting in the nearly horizonskimming region. For circular orbits, we have re-examined the analysis of Ref. [14] using an improved code for computing Teukolsky-based fluxes of the energy and angular momentum. Significantly correcting Ref. [14] 's results, we found no decrease in the orbit's inclination angle. Inclination always increases during inspiral.
We next carried out such an analysis for eccentric nearly horizon-skimming orbits. In this case, we used "kludge" fluxes to evolve the constants of motion E, L z and Q [24] . We find that these fluxes are fairly accurate when compared with the available Teukolsky-based fluxes, indicating that they should provide at least qualitatively correct information regarding inclination evolution. As for circular orbits, we find that the orbit's inclination never decreases. For both circular and non-circular configurations, we find that the magnitude of the inclination change is quite paltry -only a few degrees at 9 most. Quite generically, therefore, we found that the inclination angle of both circular and eccentric nearly horizon-skimming orbits never decreases during the inspiral. Revising the results obtained in Ref. [14] , we thus conclude that such orbits are not likely to yield a peculiar, unique imprint on the EMRIwaveforms detectable by LISA.
It is a pleasure to thank Kostas Glampedakis for enlightening comments and advice, and Steve Drasco for useful discussions and for also providing the non-circular Teukolsky data that we used in this paper. The supercomputers used in this investigation were provided by funding from the JPL Office of the Chief Information Officer. This work was supported in part by the DFG grant SFB TR/7, by NASA Grant NNG05G105G, and by NSF Grant PHY-0449884. SAH gratefully acknowledges support from the MIT Class of 1956 Career Development Fund.
In this Appendix we report the expressions for the post-Newtonian fluxes and the fits to the Teukolsky data necessary to compute the kludge fluxes introduced in Sec. V. In particular the 2PN fluxes are given by [24]
dE dt 2PN = - 32 5 µ 2 M 2 M p 5 (1 -e 2 ) 3/2 g 1 (e) -ã M p 3/2 g 2 (e) cos ι - M p g 3 (e) + π M p 3/2 g 4 (e) - M p 2 g 5 (e) + ã2 M p 2 g 6 (e) - 527 96 ã2 M p 2 sin 2 ι , (A.1) dL z dt 2PN = - 32 5 µ 2 M M p 7/2 (1 -e 2 ) 3/2 g 9 (e) cos ι + ã M p 3/2 (g a 10 (e) -cos 2 ιg b 10 (e)) - M p g 11 (e) cos ι +π M p 3/2 g 12 (e) cos ι - M p 2 g 13 (e) cos ι + ã2 M p 2 cos ι g 14 (e) - 45 8 sin 2 ι , (A.2) dQ dt 2PN = - 64 5 µ 2 M M p 7/2 Q sin ι (1 -e 2 ) 3/2 g 9 (e) -ã M p 3/2 cos ιg b 10 (e) - M p g 11 (e) +π M p 3/2 g 12 (e) - M p 2 g 13 (e) + ã2 M p 2 × g 14 (e) - 45 8 sin 2 ι , (A.3)
where µ is the mass of the infalling body and where the various e-dependent coefficients are
g 1 (e) ≡ 1 + 73 24 e 2 + 37 96 e 4 , g 2 (e) ≡ 73 12 + 823 24 e 2 + 949 32 e 4 + 491 192 e 6 , g 3 (e) ≡ 1247 336 + 9181 672 e 2 , g 4 (e) ≡ 4 + 1375 48 e 2 , g 5 (e) ≡ 44711 9072 + 172157 2592 e 2 , g 6 (e) ≡ 33 16 + 359 32 e 2 , g 9 (e) ≡ 1 + 7 8 e 2 , g a 10 (e) ≡ 61 24 + 63 8 e 2 + 95 64 e 4 , g b 10 (e) ≡ 61 8 + 91 4 e 2 + 461 64 e 4 , g 11 (e) ≡ 1247 336 + 425 336 e 2 , g 12 (e) ≡ 4 + 97 8 e 2 , g 13 (e) ≡ 44711 9072 + 302893 6048 e 2 , g 14 (e) ≡ 33 16 + 95 16 e 2 ,
The fits to the circular-Teukolsky data of Table I are instead given by
dL z dt fit circ (p, ι ) = - 32 5 µ 2 M M p 7/2 cos ι + M p 3/2 61 24 - 61 8 cos 2 ι + 4π cos ι - 1247 336 M p cos ι + M p 2 cos ι - 1625 567 - 45 8 sin 2 ι + M p 5 2 d 1 (p/M ) + d 2 (p/M ) cos ι + d 3 (p/M ) cos 2 ι + d 4 (p/M ) cos 3 ι + d 5 (p/M ) cos 4 ι + d 6 (p/M ) cos 5 ι + cos ι M p 3/2 A + B cos 2 ι , (A.4) (A.5)
10 dι dt fit circ (p, ι ) = 32 5/2 h 1 (p/M ) + cos 2 ι h 2 (p/M ) , (A.6) where
d i (x) ≡ a i d + b i d x -1/2 + c i d x -1 , i = 1, . . . , 8, h i (x) ≡ a i h + b i h x -1/2 , i = 1, 2 (A.7)
and the numerical coefficients are given by a 1 h = -278.9387 , b 1 h = 84.1414 , a 2 h = 8.6679 , b 2 h = -9.2401 , A = -18.3362 , B = 24.9034 , (A.8) and by the following table i 1 2 3 4 5 6 7 8 a i d 15.8363 445.4418 -2027.7797 3089.1709 -2045.2248 498.6411 -8.7220 50.8345 b i d -55.6777 -1333.2461 5940.4831 -9103.4472 6113.1165 -1515.8506 -50.8950 -131.6422 c i d 38.6405 1049.5637 -4513.0879 6926.3191 -4714.9633 1183.5875 251.4025 83.0834 Note that the functional form of these fits was obtained from Eqs. (57) and (58) of Ref. [24] by setting ã (i.e., q in their notation) to 1. Finally, we give expressions for the fits to the equatorial Teukolsky data of tables II, III and IV (data with ι = 0, columns with header "Teukolsky"):
dE dt fit eq (p, e) = dE dt 2P N (p, e, 0) - 32 5 µ M 2 M p 5 (1 -e 2 ) 3/2 g 1 (e) + g 2 (e) M p 1/2 + g 3 (e) M p + g 4 (e) M p 3/2 + g 5 (e) M p 2 , (A.9) L z dt fit eq (p, e) = L z dt 2P N (p, e, 0) - 32 5 µ 2 M M p 7/2 (1 -e 2 ) 3/2 f 1 (e) + f 2 (e) M p 1/2 + f 3 (e) M p + f 4 (e) M p 3/2 + f 5 (e) M p 2 , (A.10) g i (e) ≡ a i g + b i g e 2 + c i g e 4 + d i g e 6 , f i (e) ≡ a i f + b i f e 2 + c i f e 4 + d i f e 6 , i = 1, . . . , 5 (A.11)
where the numerical coefficients are given by the following table
i a i g b i g c i g d i g a i f b i f c i f d i f
1 6.4590 -2038.7301 6639.9843 227709.2187 5.4577 -3116.4034 4711.7065 214332.2907 2 -31.2215 10390.6778 -27505.7295 -1224376.5294 -26.6519 15958.6191 -16390.4868 -1147201.4687 3 57.1208 -19800.4891 39527.8397 2463977.3622 50.4374 -30579.3129 15749.9411 2296989.5466 4 -49.7051 16684.4629 -21714.7941 -2199231.9494 -46.7816 25968.8743 656.3460 -2038650.9838 5 16.4697 -5234.2077 2936.2391 734454.5696 15.6660 -8226.3892 -4903.9260 676553.2755 11 p M e θinc ι
dE dt × M 2 µ 2 dE dt × M 2 µ 2 dLz dt × M µ 2 dLz dt × M µ 2 dι dt × M µ 2 dι dt × M µ 2 dθ inc dt × M µ 2 dθ inc dt × M µ 2
(deg.) (deg.) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (Teukolsky) 1.3 0 0 0 -9.108×10 -2 -9.109×10 -2 -2.258×10 -1 -2.259×10 -1 0 0 0 0 1.3 0 10.4870 11.6773 -9.328×10 -2 -9.332×10 -2 -2.304×10 -1 -2.306×10 -1 1.837×10 -2 1.839×10 -2 6.462×10 -3 6.475×10 -3 1.3 0 14.6406 16.1303 -9.588×10 -2 -9.588×10 -2 -2.359×10 -1 -2.360×10 -1 2.397×10 -2 2.400×10 -2 8.645×10 -3 8.667×10 -3 1.3 0 17.7000 19.3172 -9.875×10 -2 -9.876×10 -2 -2.420×10 -1 -2.421×10 -1 2.728×10 -2 2.731×10 -2 1.007×10 -2 1.010×10 -2 1.3 0 20.1636 21.8210 -1.019×10 -1 -1.019×10 -1 -2.486×10 -1 -2.488×10 -1 2.943×10 -2 2.950×10 -2 1.111×10 -2 1.117×10 -2 1.4 0 0 0 -8.700×10 -2 -8.709×10 -2 -2.311×10 -1 -2.312×10 -1 0 0 0 0 1.4 0 14.5992 16.0005 -9.062×10 -2 -9.070×10 -2 -2.386×10 -1 -2.386×10 -1 2.316×10 -2 2.319×10 -2 8.823×10 -3 8.848×10 -3 1.4 0 20.1756 21.7815 -9.520×10 -2 -9.526×10 -2 -2.482×10 -1 -2.482×10 -1 2.875×10 -2 2.877×10 -2 1.141×10 -2 1.143×10 -2 1.4 0 24.1503 25.7517 -1.006×10 -1 -1.007×10 -1 -2.595×10 -1 -2.596×10 -1 3.140×10 -2 3.141×10 -2 1.289×10 -2 1.288×10 -2 1.4 0 27.2489 28.7604 -1.067×10 -1 -1.068×10 -1 -2.725×10 -1 -2.725×10 -1 3.274×10 -2 3.275×10 -2 1.378×10 -2 1.377×10 -2 1.5 0 0 0 -8.009×10 -2 -7.989×10 -2 -2.270×10 -1 -2.265×10 -1 0 0 0 0 1.5 0 16.7836 18.1857 -8.401×10 -2 -8.383×10 -2 -2.348×10 -1 -2.343×10 -1 2.360×10 -2 2.351×10 -2 9.602×10 -3 9.545×10 -3 1.5 0 23.0755 24.6167 -8.917×10 -2 -8.897×10 -2 -2.
454×10 -1 -2.449×10 -1 2.872×10 -2 2.863×10 -2 1.228×10 -2 1.222×10 -2 1.5 0 27.4892 28.9670 -9.537×10 -2 -9.516×10 -2 -2.583×10 -1 -2.579×10 -1 3.091×10 -2 3.082×10 -2 1.372×10 -2 1.367×10 -2 1.5 0 30.8795 32.2231 -1.025×10 -1 -1.023×10 -1 -2.733×10 -1 -2.728×10 -1 3.184×10 -2 3.173×10 -2 1.452×10 -2 1.443×10 -2 1.6 0 0 0 -7.181×10 -2 -7.156×10 -2 -2.168×10 -1 -2.162×10 -1 0 0 0 0 1.6 0 18.3669 19.7220 -7.568×10 -2 -7.545×10 -2 -2.242×10 -1 -2.237×10 -1 2.240×10 -2 2.229×10 -2 9.600×10 -3 9.515×10 -3 1.6 0 25.1720 26.6245 -8.084×10 -2 -8.062×10 -2 -2.346×10 -1 -2.341×10 -1 2.701×10 -2 2.685×10 -2 1.223×10 -2 1.210×10 -2 1.6 0 29.9014 31.2625 -8.708×10 -2 -8.687×10 -2 -2.474×10 -1 -2.470×10 -1 2.889×10 -2 2.872×10 -2 1.363×10 -2 1.349×10 -2 1.6 0 33.5053 34.7164 -9.425×10 -2 -9.399×10 -2 -2.622×10 -1 -2.616×10 -1 2.964×10 -2 2.951×10 -2 1.441×10 -2 1.432×10 -2 1.7 0 0 0 -6.332×10 -2 -6.317×10 -2 -2.034×10 -1 -2.031×10 -1 0 0 0 0 1.7 0 19.6910 20.9859 -6.702×10 -2 -6.687×10 -2 -2.101×10 -1 -2.098×10 -1 2.057×10 -2 2.052×10 -2 9.202×10 -3 9.171×10 -3 1.7 0 26.9252 28.2884 -7.197×10 -2 -7.184×10 -2 -2.199×10 -1 -2.196×10 -1 2.467×10 -2 2.456×10 -2 1.170×10 -2 1.162×10 -2 1.7 0 31.9218 33.1786 -7.794×10 -2 -7.782×10 -2 -2.319×10 -1 -2.316×10 -1 2.632×10 -2 2.620×10 -2 1.306×10 -2 1.296×10 -2 1.7 0 35.7100 36.8118 -8.475×10 -2 -8.465×10 -2 -2.457×10 -1 -2.455×10 -1 2.698×10 -2 2.686×10 -2 1.384×10 -2 1.373×10 -2 1.8 0 0 0 -5.531×10 -2 -5.528×10 -2 -1.888×10 -1 -1.887×10 -1 0 0 0 0 1.8 0 20.8804 22.1128 -5.879×10 -2 -5.874×10 -2 -1.948×10 -1 -1.946×10 -1 1.858×10 -2 1.858×10 -2 8.635×10 -3 8.639×10 -3 1.8 0 28.5007 29.7791 -6.343×10 -2 -6.336×10 -2 -2.036×10 -1 -2.035×10 -1 2.221×10 -2 2.223×10 -2 1.098×10 -2 1.101×10 -2 1.8 0 33.7400 34.9034 -6.901×10 -2 -6.894×10 -2 -2.146×10 -1 -2.144×10 -1 2.368×10 -2 2.371×10 -2 1.228×10 -2 1.232×10 -2 1.8 0 37.6985 38.7065 -7.533×10 -2 -7.533×10 -2 -2.271×10 -1 -2.271×10 -1 2.429×10 -2 2.427×10 -2 1.306×10 -2 1.303×10 -2 1.9 0 0 0 -4.809×10 -2 -4.811×10 -2 -1.740×10 -1 -1.740×10 -1 0 0 0 0 1.9 0 21.9900 23.1615 -5.132×10 -2 -5.134×10 -2 -1.792×10 -1 -1.793×10 -1 1.666×10 -2 1.664×10 -2 8.022×10 -3 8.007×10 -3 1.9 0 29.9708 31.1702 -5.562×10 -2 -5.564×10 -2 -1.872×10 -1 -1.872×10 -1 1.986×10 -2 1.987×10 -2 1.019×10 -2 1.020×10 -2 1.9 0 35.4385 36.5176 -6.078×10 -2 -6.077×10 -2 -1.971×10 -1 -1.970×10 -1 2.118×10 -2 2.122×10 -2 1.143×10 -2 1.148×10 -2 1.9 0 39.5592 40.4847 -6.659×10 -2 -6.658×10 -2 -2.082×10 -1 -2.082×10 -1 2.177×10 -2 2.182×10 -2 1.222×10 -2 1.228×10 -2 2.0 0 0 0 -4.174×10 -2 -4.175×10 -2 -1.598×10 -1 -1.598×10 -1 0 0 0 0 2.0 0 23.0471 24.1605 -4.471×10 -2 -4.472×10 -2 -1.643×10 -1 -1.643×10 -1 1.489×10 -2 1.489×10 -2 7.425×10 -3 7.424×10 -3 2.0 0 31.3715 32.4978 -4.867×10 -2 -4.871×10 -2 -1.713×10 -1 -1.714×10 -1 1.773×10 -2 1.770×10 -2 9.436×10 -3 9.411×10 -3 2.0 0 37.0583 38.0608 -5.341×10 -2 -5.345×10 -2 -1.801×10 -1 -1.801×10 -1 1.893×10 -2 1.889×10 -2 1.062×10 -2 1.057×10 -2 2.0 0 41.3358 42.1876 -5.873×10 -2 -5.875×10 -2 -1.900×10 -1 -1.900×10 -1 1.950×10 -2 1.948×10 -2 1.141×10 -2 1.138×10 -2
Table I: Teukolsky-based fluxes and kludge fluxes [computed using Eqs. (40), (43) and (44)] for circular orbits about a hole with a = 0.998M ; µ represents the mass of the infalling body. The Teukolsky-based fluxes have an accuracy of 10 -6 . [1] http://lisa.nasa.gov/ ; http://sci.esa.int/home/lisa/ [2] J. Kormendy and D. Richstone, Ann. Rev. Astron. Astrophys.
33, 581 (1995). [3] J. R. Gair, L. Barack, T. Creighton, C. Cutler, S. L. Larson, E. S. Phinney, and M. Vallisneri, Class. Quantum Grav. 21, S1595 (2004). [4] S. L. Shapiro, Astrophys. J. 620, 59 (2005). [5] L. Rezzolla, T. W. Maccarone, S. Yoshida, and O. Zanotti, Mon.
Not. Roy. Astron. Soc 344, L37 (2003). [6] R. Shafee, J. E. McClintock, R. Narayan, S. W. Davis, L.-X. Li, and R. A. Remilland, Astrophys. J. 636, L113 (2006).
[7] J. E. McClintock, R. Shafee, R. Narayan, R. A. Remilland, S. W. Davis, and L.-X. Li, Astrophys. J. 652, 518 (2006). [8] A. C. Fabian and G. Miniutti, G. 2005, to appear in Kerr Spacetime: Rotating Black Holes in General Relativity, edited by D. L. Wiltshire, M. Visser, and S. M. Scott; astro-ph/0507409. [9] L. W. Brenneman and C. S. Reynolds, Astrophys. J. 652, 1028 (2006). [10] L. Barack and C. Cutler, Phys. Rev. D 69, 082005 (2004). [11] M. Volonteri, P. Madau, E. Quataert, and M. J. Rees, Astrophys. J. 620, 69 (2005). [12] D. C. Wilkins, Phys. Rev. D 5, 814 (1972).
12 p M e θinc ι dE dt × M 2 µ 2 dE dt × M 2 µ 2 dLz dt × M µ 2 dLz dt × M µ 2 dι dt × M µ 2 dθ inc dt × M µ 2
(deg.) (deg.) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (kludge) 1.3 0.1 0 0 -8.804×10 -2 -8.804×10 -2 -2.098×10 -1 -2.098×10 -1 0 0 1.4 0.1 0 0 -8.728×10 -2 -8.719×10 -2 -2.274×10 -1 -2.275×10 -1 0 0 1.4 0.1 8 8.8664 -9.110×10 -2 -8.736×10 -2 -2.355×10 -1 -2.273×10 -1 4.066×10 -2 2.938×10 -2 1.4 0.1 16 17.4519 -1.030×10 -1 -8.958×10 -2 -2.602×10 -1 -2.309×10 -1 7.428×10 -2 5.475×10 -2 1.4 0.1 24 25.5784 -1.243×10 -1 -9.771×10 -2 -3.037×10 -1 -2.415×10 -1 9.663×10 -2 7.316×10 -2 1.5 0.1 0 0 -8.069×10 -2 -8.095×10 -2 -2.255×10 -1 -2.260×10 -1 0 0 1.5 0.1 8 8.7910 -8.323×10 -2 -8.133×10 -2 -2.310×10 -1 -2.264×10 -1 2.996×10 -2 2.070×10 -2 1.5 0.1 16 17.3490 -9.121×10 -2 -8.395×10 -2 -2.483×10 -1 -2.314×10 -1 5.512×10 -2 3.888×10 -2 1.5 0.1 24 25.5197 -1.059×10 -1 -8.980×10 -2 -2.792×10 -1 -2.423×10 -1 7.255×10 -2 5.264×10 -2 1.6 0.1 0 0 -7.255×10 -2 -7.281×10 -2 -2.161×10 -1 -2.168×10 -1 0 0 1.6 0.1 8 8.7195 -7.430×10 -2 -7.321×10 -2 -2.201×10 -1 -2.173×10 -1 2.258×10 -2 1.502×10 -2 1.6 0.1 16 17.2437 -7.986×10 -2 -7.533×10 -2 -2.323×10 -1 -2.212×10 -1 4.179×10 -2 2.839×10 -2 1.6 0.1 24 25.4388 -9.025×10 -2 -8.040×10 -2 -2.547×10 -1 -2.309×10 -1 5.554×10 -2 3.886×10 -2 1.6 0.1 32 33.2683 -1.082×10 -1 -9.435×10 -2 -2.920×10 -1 -2.551×10 -1 6.316×10 -2 4.559×10 -2 1.7 0.1 0 0 -6.427×10 -2 -6.440×10 -2 -2.036×10 -1 -2.040×10 -1 0 0 1.7 0.1 8 8.6555 -6.552×10 -2 -6.478×10 -2 -2.065×10 -1 -2.045×10 -1 1.742×10 -2 1.124×10 -2 1.7 0.1 16 17.1454 -6.953×10 -2 -6.651×10 -2 -2.154×10 -1 -2.075×10 -1 3.240×10 -2 2.134×10 -2 1.7 0.1 24 25.3531 -7.707×10 -2 -7.052×10 -2 -2.317×10 -1 -2.150×10 -1 4.342×10 -2 2.948×10 -2 1.7 0.1 32 33.2416 -9.009×10 -2 -7.959×10 -2 -2.590×10 -1 -2.324×10 -1 4.998×10 -2 3.512×10 -2 1.8 0.1 0 0 -5.640×10 -2 -5.640×10 -2 -1.897×10 -1 -1.897×10 -1 0 0 1.8 0.1 8 8.5991 -5.732×10 -2 -5.676×10 -2 -1.918×10 -1 -1.902×10 -1 1.371×10 -2 8.640×10 -3 1.8 0.1 16 17.0562 -6.028×10 -2 -5.817×10 -2 -1.984×10 -1 -1.925×10 -1 2.562×10 -2 1.647×10 -2 1.8 0.1 24 25.2693 -6.588×10 -2 -6.139×10 -2 -2.105×10 -1 -1.983×10 -1 3.456×10 -2 2.291×10 -2 1.8 0.1 32 33.2018 -7.555×10 -2 -6.849×10 -2 -2.307×10 -1 -2.120×10 -1 4.020×10 -2 2.765×10 -2 1.9 0.1 0 0 -4.915×10 -2 -4.911×10 -2 -1.753×10 -1 -1.751×10 -1 0 0 1.9 0.1 8 8.5494 -4.985×10 -2 -4.945×10 -2 -1.768×10 -1 -1.755×10 -1 1.097×10 -2 6.791×10 -3 1.9 0.1 16 16.9760 -5.208×10 -2 -5.064×10 -2 -1.817×10 -1 -1.774×10 -1 2.055×10 -2 1.298×10 -2 1.9 0.1 24 25.1898 -5.633×10 -2 -5.328×10 -2 -1.908×10 -1 -1.819×10 -1 2.788×10 -2 1.816×10 -2 1.9 0.1 32 33.1555 -6.364×10 -2 -5.870×10 -2 -2.059×10 -1 -1.920×10 -1 3.272×10 -2 2.214×10 -2 2.0 0.1 0 0 -4.263×10 -2 -4.264×10 -2 -1.607×10 -1 -1.608×10 -1 0 0 2.0 0.1 8 8.5057 -4.316×10 -2 -4.292×10 -2 -1.619×10 -1 -1.611×10 -1 8.862×10 -3 5.424×10 -3 2.0 0.1 16 16.9042 -4.488×10 -2 -4.390×10 -2 -1.656×10 -1 -1.625×10 -1 1.666×10 -2 1.039×10 -2 2.0 0.1 24 25.1156 -4.815×10 -2 -4.604×10 -2 -1.724×10 -1 -1.660×10 -1 2.271×10 -2 1.459×10 -2 2.0 0.1 32 33.1064 -5.376×10 -2 -5.031×10 -2 -1.838×10 -1 -1.736×10 -1 2.684×10 -2 1.793×10 -2 2.0 0.1 40 40.8954 -6.339×10 -2 -6.236×10 -2 -2.027×10 -1 -1.967×10 -1 2.917×10 -2 2.036×10 -2
Table II: As in Table I but for non-circular orbits; the Teukolsky-based fluxes for E and Lz have an accuracy of 10 -3 . Note that our code, as all the Teukolsky-based code that we are aware of, presently does not have the capability to compute inclination angle evolution for generic orbits. [13] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J. 178, 347 (1972). [14] S. A. Hughes, Phys. Rev. D 63, 064016 (2001). [15] K. S. Thorne, R. H. Price, and D. A. MacDonald, Black Holes: The Membrane Paradigm (Yale University Press, New Haven, CT, 1986). [16] F. D. Ryan, Phys. Rev. D 52, R3159 (1995). [17] S. A. Hughes, Phys. Rev. D 61, 084004 (2000). [18] B. Carter, Phys. Rev. 174, 1559 (1968). [19] K. S. Thorne, Astrophys. J. 191, 507 (1974). [20] Y. Mino, Phys. Rev. D 67, 084027 (2003) [21] S. Drasco and S. A. Hughes, Phys. Rev. D 73, 024027 (2006). [22] N. Sago, T. Tanaka, W. Hikida, and H. Nakano, Prog. Theor. Phys. 114, 509 (2005); N. Sago, T. Tanaka, W. Hikida, K. Ganz, and H. Nakano, Prog. Theor. Phys. 115, 873 (2006). [23] K. Glampedakis, S. A. Hughes, and D. Kennefick, Phys. Rev.
D 66, 064005 (2002). [24] J. R. Gair and K. Glampedakis, Phys. Rev. D 73, 064037 (2006). [25] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973). [26] W. Schmidt, Class. Quantum Grav. 19, 2743 (2002). [27] A. Ori and K. S. Thorne, Phys. Rev. D 62, 124022 (2000) [28] S. A. Teukolsky, Astrophys. J. 185, 635 (1973). [29] M. Sasaki and T. Nakamura, Prog. Theor. Phys. 67, 1788 (1982). [30] F. D. Ryan, Phys. Ref. D 53, 3064 (1996) . [31] D. Kennefick and A. Ori, Phys. Rev. D 53, 4319 (1996). [32] Y. Mino, unpublished Ph. D. thesis, Kyoto University, 1996. [33] Data available at http://gmunu.mit.edu/sdrasco/snapshots/ 13 p M e θinc ι dE dt × M 2 µ 2 dE dt × M 2 µ 2 dLz dt × M µ 2 dLz dt × M µ 2 dι dt × M µ 2 dθ inc dt × M µ 2 (deg.) (deg.) (kludge) (Teukolsky ) (kludge) (Teukolsky) (kludge) (kludge) 1.4 0.2 0 0 -8.636×10 -2 -8.642×10 -2 -2.119×10 -1 -2.121×10 -1 0 0 1.4 0.2 8 8.8215 -9.853×10 -2 -8.240×10 -2 -2.374×10 -1 -2.015×10 -1 1.148×10 -1 9.714×10 -2 1.5 0.2 0 0 -8.362×10 -2 -8.349×10 -2 -2.236×10 -1 -2.230×10 -1 0 0 1.5 0.2 8 8.7595 -9.141×10 -2 -8.276×10 -2 -2.410×10 -1 -2.206×10 -1 7.893×10 -2 6.549×10 -2 1.5 0.2 16 17.2957 -1.145×10 -1 -8.394×10 -2 -2.915×10 -1 -2.215×10 -1 1.466×10 -1 1.230×10 -1 1.5 0.2 24 25.4608 -1.524×10 -1 -9.230×10 -2 -3.712×10 -1 -2.357×10 -1 1.952×10 -1 1.661×10 -1 1.6 0.2 0 0 -7.596×10 -2 -7.616×10 -2 -2.171×10 -1 -2.176×10 -1 0 0 1.6 0.2 8 8.6935 -8.111×10 -2 -7.641×10 -2 -2.292×10 -1 -2.177×10 -1 5.520×10 -2 4.502×10 -2 1.6 0.2 16 17.1994 -9.649×10 -2 -7.798×10 -2 -2.647×10 -1 -2.198×10 -1 1.032×10 -1 8.500×10 -2 1.6 0.2 24 25.3891 -1.221×10 -1 -8.314×10 -2 -3.212×10 -1 -2.288×10 -1 1.388×10 -1 1.160×10 -1 1.7 0.2 0 0 -6.765×10 -2 -6.799×10 -2 -2.057×10 -1 -2.068×10 -1 0 0 1.7 0.2 8 8.6329 -7.116×10 -2 -6.813×10 -2 -2.144×10 -1 -2.066×10 -1 3.963×10 -2 3.176×10 -2 1.7 0.2 16 17.1064 -8.171×10 -2 -6.995×10 -2 -2.398×10 -1 -2.096×10 -1 7.441×10 -2 6.024×10 -2 1.7 0.2 24 25.3085 -9.948×10 -2 -7.443×10 -2 -2.806×10 -1 -2.178×10 -1 1.009×10 -1 8.290×10 -2 1.7 0.2 32 33.2037 -1.257×10 -1 -8.558×10 -2 -3.371×10 -1 -2.366×10 -1 1.175×10 -1 9.806×10 -2 1.8 0.2 0 0 -5.965×10 -2 -5.962×10 -2 -1.927×10 -1 -1.926×10 -1 0 0 1.8 0.2 8 8.5789 -6.211×10 -2 -5.997×10 -2 -1.990×10 -1 -1.930×10 -1 2.919×10 -2 2.300×10 -2 1.8 0.2 16 17.0211 -6.953×10 -2 -6.147×10 -2 -2.175×10 -1 -1.954×10 -1 5.504×10 -2 4.380×10 -2 1.8 0.2 24 25.2283 -8.216×10 -2 -6.502×10 -2 -2.474×10 -1 -2.016×10 -1 7.515×10 -2 6.068×10 -2 1.8 0.2 32 33.1656 -1.009×10 -1 -7.410×10 -2 -2.890×10 -1 -2.190×10 -1 8.839×10 -2 7.258×10 -2 1.9 0.2 0 0 -5.218×10 -2 -5.210×10 -2 -1.786×10 -1 -1.783×10 -1 0 0 1.9 0.2 8 8.5312 -5.394×10 -2 -5.244×10 -2 -1.833×10 -1 -1.787×10 -1 2.197×10 -2 1.704×10 -2 1.9 0.2 16 16.9441 -5.928×10 -2 -5.373×10 -2 -1.970×10 -1 -1.807×10 -1 4.156×10 -2 3.254×10 -2 1.9 0.2 24 25.1518 -6.843×10 -2 -5.669×10 -2 -2.192×10 -1 -1.858×10 -1 5.706×10 -2 4.535×10 -2 1.9 0.2 32 33.1207 -8.213×10 -2 -6.277×10 -2 -2.502×10 -1 -1.966×10 -1 6.767×10 -2 5.475×10 -2 2.0 0.2 0 0 -4.528×10 -2 -4.530×10 -2 -1.637×10 -1 -1.638×10 -1 0 0 2.0 0.2 8 8.4891 -4.657×10 -2 -4.557×10 -2 -1.671×10 -1 -1.641×10 -1 1.679×10 -2 1.283×10 -2 2.0 0.2 16 16.8749 -5.049×10 -2 -4.664×10 -2 -1.774×10 -1 -1.657×10 -1 3.184×10 -2 2.457×10 -2 2.0 0.2 24 25.0802 -5.725×10 -2 -4.904×10 -2 -1.941×10 -1 -1.696×10 -1 4.391×10 -2 3.440×10 -2 2.0 0.2 32 33.0730 -6.743×10 -2 -5.427×10 -2 -2.175×10 -1 -1.793×10 -1 5.243×10 -2 4.184×10 -2 1.5 0.3 0 0 -8.481×10 -2 -8.478×10 -2 -2.094×10 -1 -2.094×10 -1 0 0 1.5 0.3 8 8.7037 -1.006×10 -1 -7.824×10 -2 -2.442×10 -1 -1.934×10 -1 1.484×10 -1 1.301×10 -1 1.5 0.3 16 17.2003 -1.469×10 -1 -7.811×10 -2 -3.435×10 -1 -1.864×10 -1 2.766×10 -1 2.440×10 -1 1.6 0.3 0 0 -8.144×10 -2 -8.123×10 -2 -2.183×10 -1 -2.178×10 -1 0 0 1.6 0.3 8 8.6498 -9.182×10 -2 -7.807×10 -2 -2.426×10 -1 -2.095×10 -1 1.028×10 -1 8.918×10 -2 1.6 0.3 16 17.1246 -1.
223×10 -1 -8.089×10 -2 -3.122×10 -1 -2.144×10 -1 1.928×10 -1 1.683×10 -1 1.6 0.3 24 25.3046 -1.716×10 -1 -8.666×10 -2 -4.197×10 -1 -2.229×10 -1 2.607×10 -1 2.295×10 -1 1.7 0.3 0 0 -7.362×10 -2 -7.314×10 -2 -2.104×10 -1 -2.095×10 -1 0 0 1.7 0.3 8 8.5953 -8.060×10 -2 -7.224×10 -2 -2.277×10 -1 -2.065×10 -1 7.240×10 -2 6.224×10 -2 1.7 0.3 16 17.0415 -1.013×10 -1 -7.369×10 -2 -2.774×10 -1 -2.084×10 -1 1.365×10 -1 1.180×10 -1 1.7 0.3 24 25.2339 -1.349×10 -1 -7.800×10 -2 -3.547×10 -1 -2.153×10 -1 1.861×10 -1 1.622×10 -1 1.8 0.3 0 0 -6.488×10 -2 -6.484×10 -2 -1.973×10 -1 -1.972×10 -1 0 0 1.8 0.3 8 8.5454 -6.970×10 -2 -6.480×10 -2 -2.099×10 -1 -1.966×10 -1 5.206×10 -2 4.436×10 -2 1.8 0.3 16 16.9628 -8.402×10 -2 -6.671×10 -2 -2.461×10 -1 -1.998×10 -1 9.857×10 -2 8.445×10 -2 1.8 0.3 24 25.1601 -1.075×10 -1 -7.030×10 -2 -3.026×10 -1 -2.056×10 -1 1.353×10 -1 1.169×10 -1 1.8 0.3 32 33.1047 -1.404×10 -1 -8.153×10 -2 -3.762×10 -1 -2.255×10 -1 1.600×10 -1 1.394×10 -1 1.9 0.3 0 0 -5.669×10 -2 -5.690×10 -2 -1.829×10 -1 -1.832×10 -1 0 0 1.9 0.3 8 8.5010 -6.010×10 -2 -5.683×10 -2 -1.922×10 -1 -1.824×10 -1 3.823×10 -2 3.229×10 -2 1.9 0.3 16 16.8911 -7.025×10 -2 -5.818×10 -2 -2.189×10 -1 -1.844×10 -1 7.263×10 -2 6.165×10 -2 1.9 0.3 24 25.0887 -8.701×10 -2 -6.054×10 -2 -2.609×10 -1 -1.874×10 -1 1.003×10 -1 8.579×10 -2 1.9 0.3 32 33.0624 -1.106×10 -1 -6.912×10 -2 -3.157×10 -1 -2.034×10 -1 1.195×10 -1 1.032×10 -1 2.0 0.3 0 0 -4.953×10 -2 -4.946×10 -2 -1.683×10 -1 -1.683×10 -1 0 0 2.0 0.3 8 8.4616 -5.199×10 -2 -4.970×10 -2 -1.753×10 -1 -1.685×10 -1 2.862×10 -2 2.395×10 -2 2.0 0.3 16 16.8262 -5.932×10 -2 -5.079×10 -2 -1.954×10 -1 -1.699×10 -1 5.452×10 -2 4.585×10 -2 2.0 0.3 24 25.0215 -7.150×10 -2 -5.328×10 -2 -2.269×10 -1 -1.737×10 -1 7.564×10 -2 6.411×10 -2 2.0 0.3 32 33.0172 -8.878×10 -2 -6.003×10 -2 -2.682×10 -1 -1.864×10 -1 9.077×10 -2 7.771×10 -2
Table III: As in Table II , but for additional values of eccentricity e; the Teukolsky-based fluxes for E and Lz have an accuracy of 10 -3 . 14 p M e θinc ι dE dt × M 2 µ 2 dE dt × M 2 µ 2 dLz dt × M µ 2 dLz dt × M µ 2 dι dt × M µ 2 dθ inc dt × M µ 2 (deg.) (deg.) (kludge) (Teukolsky ) (kludge) (Teukolsky) (kludge) (kludge) 1.6 0.4 0 0 -7.766×10 -2 -7.772×10 -2 -1.918×10 -1 -1.919×10 -1 0 0 1.6 0.4 8 8.5863 -9.433×10 -2 -7.645×10 -2 -2.297×10 -1 -1.881×10 -1 1.528×10 -1 1.370×10 -1 1.6 0.4 16 17.0151 -1.432×10 -1 -7.651×10 -2 -3.382×10 -1 -1.837×10 -1 2.873×10 -1 2.584×10 -1 1.7 0.4 0 0 -7.882×10 -2 -7.953×10 -2 -2.097×10 -1 -2.115×10 -1 0 0 1.7 0.4 8 8.5426 -9.002×10 -2 -7.408×10 -2 -2.367×10 -1 -1.978×10 -1 1.087×10 -1 9.656×10 -2 1.7 0.4 16 16.9502 -1.229×10 -1 -7.682×10 -2 -3.143×10 -1 -2.025×10 -1 2.054×10 -1 1.830×10 -1 1.7 0.4 24 25.1282 -1.760×10 -1 -8.090×10 -2 -4.336×10 -1 -2.075×10 -1 2.809×10 -1 2.514×10 -1 1.8 0.4 0 0 -7.107×10 -2 -7.007×10 -2 -2.013×10 -1 -1.988×10 -1 0 0 1.8 0.4 8 8.4989 -7.877×10 -2 -7.001×10 -2 -2.209×10 -1 -1.981×10 -1 7.788×10 -2 6.879×10 -2 1.8 0.4 16 16.8817 -1.015×10 -1 -7.009×10 -2 -2.774×10 -1 -1.965×10 -1 1.478×10 -1 1.309×10 -1 1.8 0.4 24 25.0646 -1.383×10 -1 -7.314×10 -2 -3.646×10 -1 -2.003×10 -1 2.036×10 -1 1.810×10 -1 1.8 0.4 32 33.0184 -1.887×10 -1 -9.193×10 -2 -4.755×10 -1 -2.319×10 -1 2.414×10 -1 2.156×10 -1 1.9 0.4 0 0 -6.187×10 -2 -6.267×10 -2 -1.861×10 -1 -1.881×10 -1 0 0 1.9 0.4 8 8.4591 -6.728×10 -2 -6.216×10 -2 -2.006×10 -1 -1.861×10 -1 5.666×10 -2 4.980×10 -2 1.9 0.4 16 16.8173 -8.328×10 -2 -6.222×10 -2 -2.424×10 -1 -1.844×10 -1 1.079×10 -1 9.506×10 -2 1.9 0.4 24 25.0006 -1.094×10 -1 -6.486×10 -2 -3.071×10 -1 -1.878×10 -1 1.495×10 -1 1.322×10 -1 1.9 0.4 32 32.9804 -1.452×10 -1 -7.884×10 -2 -3.896×10 -1 -2.158×10 -1 1.787×10 -1 1.588×10 -1 2.0 0.4 0 0 -5.483×10 -2 -5.457×10 -2 -1.735×10 -1 -1.729×10 -1 0 0 2.0 0.4 8 8.4235 -5.871×10 -2 -5.445×10 -2 -1.844×10 -1 -1.720×10 -1 4.222×10 -2 3.686×10 -2 2.0 0.4 16 16.7586 -7.020×10 -2 -5.555×10 -2 -2.158×10 -1 -1.733×10 -1 8.064×10 -2 7.057×10 -2 2.0 0.4 24 24.9396 -8.902×10 -2 -5.844×10 -2 -2.645×10 -1 -1.778×10 -1 1.122×10 -1 9.860×10 -2 2.0 0.4 32 32.9389 -1.150×10 -1 -6.536×10 -2 -3.267×10 -1 -1.896×10 -1 1.351×10 -1 1.193×10 -1 1.7 0.5 0 0 -7.421×10 -2 -7.401×10 -2 -1.815×10 -1 -1.810×10 -1 0 0 1.7 0.5 8 8.4736 -8.957×10 -2 -7.168×10 -2 -2.173×10 -1 -1.750×10 -1 1.379×10 -1 1.256×10 -1 1.7 0.5 16 16.8300 -1.347×10 -1 -6.999×10 -2 -3.201×10 -1 -1.676×10 -1 2.611×10 -1 2.378×10 -1 1.8 0.5 0 0 -7.589×10 -2 -7.620×10 -2 -1.993×10 -1 -2.000×10 -1 0 0 1.8 0.5 8 8.4395 -8.644×10 -2 -6.929×10 -2 -2.254×10 -1 -1.829×10 -1 1.005×10 -1 9.076×10 -2 1.8 0.5 16 16.7776 -1.175×10 -1 -7.210×10 -2 -3.004×10 -1 -1.880×10 -1 1.911×10 -1 1.726×10 -1 1.8 0.5 24 24.9413 -1.678×10 -1 -7.395×10 -2 -4.158×10 -1 -1.881×10 -1 2.638×10 -1 2.385×10 -1 1.9 0.5 0 0 -6.646×10 -2 -6.620×10 -2 -1.855×10 -1 -1.849×10 -1 0 0 1.9 0.5 8 8.4059 -7.386×10 -2 -6.320×10 -2 -2.048×10 -1 -1.768×10 -1 7.312×10 -2 6.579×10 -2 1.9 0.5 16 16.7233 -9.572×10 -2 -6.551×10 -2 -2.603×10 -1 -1.809×10 -1 1.395×10 -1 1.255×10 -1 1.9 0.5 24 24.8877 -1.312×10 -1 -7.087×10 -2 -3.461×10 -1 -1.909×10 -1 1.937×10 -1 1.744×10 -1 1.9 0.5 32 32.8741 -1.795×10 -1 -8.247×10 -2 -4.544×10 -1 -2.091×10 -1 2.320×10 -1 2.092×10 -1 2.0 0.5 0 0 -5.987×10 -2 -5.995×10 -2 -1.761×10 -1 -1.763×10 -1 0 0 2.0 0.5 8 8.3750 -6.516×10 -2 -5.918×10 -2 -1.906×10 -1 -1.738×10 -1 5.456×10 -2 4.882×10 -2 2.0 0.5 16 16.6725 -8.081×10 -2 -5.817×10 -2 -2.324×10 -1 -1.694×10 -1 1.044×10 -1 9.343×10 -2 2.0 0.5 24 24.8347 -1.063×10 -1 -6.254×10 -2 -2.970×10 -1 -1.776×10 -1 1.456×10 -1 1.304×10 -1 2.0 0.5 32 32.8378 -1.412×10 -1 -6.993×10 -2 -3.787×10 -1 -1.893×10 -1 1.756×10 -1 1.576×10 -1
Table IV: As in Tables II and III, but for different values of eccentricity e; the Teukolsky-based fluxes for E and Lz have an accuracy of 10 -3 .
e θinc ι ∆t/M ∆θinc ∆ι (deg.) (deg.) (deg.) (deg.) 0 0 0 1.250×10 6 0 0 0 5 5.355510 1.217×10 6 1.949×10 -1 4.954×10 -1 0 10 10.679331 1.118×10 6 3.468×10 -1 8.631×10 -1 0 15 15.943192 9.574×10 5 4.236×10 -1 1.019 0 20 21.125167 7.446×10 5 4.109×10 -1 9.440×10 -1 0 25 26.211779 4.981×10 5 3.158×10 -1 6.860×10 -1 0 30 31.199048 2.528×10 5 1.732×10 -1 3.527×10 -1 0 35 36.092514 6.584×10 4 4.636×10 -2 8.806×10 -2 0.1 0 0 1.228×10 6 0 0 0.1 5 5.351602 1.198×10 6 4.517×10 -1 7.766×10 -1 0.1 10 10.671900 1.103×10 6 6.900×10 -1 1.236 0.1 15 15.932962 9.426×10 5 7.283×10 -1 1.344 0.1 20 21.113129 7.315×10 5 6.433×10 -1 1.187 0.1 25 26.199088 4.900×10 5 4.780×10 -1 8.547×10 -1 0.1 30 31.186915 2.513×10 5 2.730×10 -1 4.585×10 -1 0.1 35 36.082095 6.589×10 4 8.385×10 -2 1.279×10 -1 0.2 0 0 1.173×10 6 0 0 0.2 5 5.339916 1.150×10 6 1.204 1.598 0.2 10 10.649670 1.064×10 6 1.698 2.331 0.2 15 15.902348 9.043×10 5 1.618 2.293 0.2 20 21.077081 6.980×10 5 1.324 1.900 0.2 25 26.161046 4.693×10 5 9.545×10 -1 1.351 0.2 30 31.150481 2.486×10 5 5.674×10 -1 7.711×10 -1 0.2 35 36.050712 7.562×10 4 2.070×10 -1 2.648×10 -1 0.3 0 0 1.087×10 6 0 0 0.3 5 5.320559 1.069×10 6 2.307 2.788 0.3 10 10.612831 1.001×10 6 3.256 4.007 0.3 15 15.851572 8.454×10 5 2.984 3.741 0.3 20 21.017212 6.483×10 5 2.375 2.998 0.3 25 26.097732 4.408×10 5 1.700 2.129 0.3 30 31.089639 2.493×10 5 1.040 1.276 0.3 35 35.997987 1.108×10 5 4.626×10 -1 5.569×10 -1
Table V: Variation in the inclination angles ι and θinc as well as time needed to reach the separatrix for several inspirals through the nearly horizon-skimming regime. In all of these cases, the binary's mass ratio was fixed to µ/M = 10 -6 , the large black hole's spin was fixed to a = 0.998M , and the orbits were begun at p = 1.9M . The time interval ∆t is the total accumulated time it takes for the inspiralling body to reach the separatrix (at which time it rapidly plunges into the black hole). The angles ∆θinc and ∆ι are the total integrated change in these inclination angles that we compute. For the e = 0 cases, inspirals are computed using fits to the circular-Teukolsky fluxes of E and Lz; for eccentric orbits we use the kludge fluxes (40), (43) and (44) . Notice that ∆θinc and ∆ι are always positive -the inclination angle always increases during the inspiral through the nearly horizon-skimming region. The magnitude of this increase never exceeds a few degrees. | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The space-based gravitational-wave detector LISA [1] will be a unique tool to probe the nature of supermassive black holes (SMBHs), making it possible to map in detail their spacetimes. This goal is expected to be achieved by observing gravitational waves emitted by compact stars or black holes with masses µ ≈ 1 -100 M ⊙ spiraling into the SMBHs which reside in the center of galaxies [2] (particularly the low end of the galactic center black hole mass function, M ≈ 10 5 -10 7 M ⊙ ). Such events are known as extreme mass ratio inspirals (EMRIs) . Current wisdom suggests that several tens to perhaps of order a thousand such events could be measured per year by LISA [3] .\n\nThough the distribution of spins for observed astrophysical black holes is not very well understood at present, very rapid spin is certainly plausible, as accretion tends to spinup SMBHs [4] . Most models for quasi-periodic oscillations (QPOs) suggest this is indeed the case in all low-mass x-ray binaries for which data is available [5] . On the other hand, continuum spectral fitting of some high-mass x-ray binaries indicates that modest spins (spin parameter a/M ≡ J/M 2 ∼ 0.6 -0.8) are likewise plausible [6] . The continuum-fit technique does find an extremely high spin of a/M 0.98 for the galactic \"microquasar\" GRS1915+105 [7] . This argues for a wide variety of possible spins, depending on the detailed birth and growth history of a given black hole.\n\nIn the mass range corresponding to black holes in galactic centers, measurements of the broad iron Kα emission line in active galactic nuclei suggest that SMBHs can be very rapidly rotating (see Ref. [8] for a recent review). For instance, in the case of MCG-6-30-15, for which highly accurate observations are available, a has been found to be larger than 0.987M at 90% confidence [9] . Because gravitational waves from EM-RIs are expected to yield a very precise determination of the spins of SMBHs [10] , it is interesting to investigate whether EMRIs around very rapidly rotating black holes may possess peculiar features which would be observable by LISA. Should such features exist, they would provide unambiguous information on the spin of SMBHs and thus on the mechanisms leading to their formation [11] .\n\nFor extremal Kerr black holes (a = M ), the existence of a special class of \"circular\" orbits was pointed out long ago by Wilkins [12] , who named them \"horizon-skimming\" orbits. (\"Circular\" here means that the orbits are of constant Boyer-Lindquist coordinate radius r.) These orbits have varying inclination angle with respect to the equatorial plane and have the same coordinate radius as the horizon, r = M . De-2 spite this seemingly hazardous location, it can be shown that all these r = M orbits have finite separation from one another and from the event horizon [13] . Their somewhat pathological description is due to a singularity in the Boyer-Lindquist coordinates, which collapses a finite span of the spacetime into r = M . Besides being circular and \"horizon-skimming\", these orbits also show peculiar behavior in their relation of angular momentum to inclination. In Newtonian gravity, a generic orbit has L z = |L| cos ι, where ι is the inclination angle relative to the equatorial plane (going from ι = 0 for equatorial prograde orbits to ι = π for equatorial retrograde orbits, passing through ι = π/2 for polar orbits), and L is the orbital angular momentum vector. As a result, ∂L z (r, ι)/∂ι < 0, and the angular momentum in the z-direction always decreases with increasing inclination if the orbit's radius is kept constant. This intuitively reasonable decrease of L z with ι is seen for almost all black hole orbits as well. Horizon-skimming orbits, by contrast, exhibit exactly the opposite behavior: L z increases with inclination angle.\n\nReference [14] asked whether the behavior ∂L z /∂ι > 0 could be extended to a broader class of circular orbits than just those at the radius r = M for the spin value a = M . It was found that this condition is indeed more general, and extended over a range of radius from the \"innermost stable circular orbit\" to r ≃ 1.8M for black holes with a > 0.9524M . Orbits that show this property have been named \"nearly horizonskimming\". The Newtonian behavior ∂L z (r, ι)/∂ι < 0 is recovered for all orbits at r 1.8M [14] .\n\nA qualitative understanding of this behavior comes from recalling that very close to the black hole all physical processes become \"locked\" to the hole's event horizon [15] , with the orbital motion of point particles coupling to the horizon's spin. This locking dominates the \"Keplerian\" tendency of an orbit to move more quickly at smaller radii, forcing an orbiting particle to slow down in the innermost orbits. Locking is particularly strong for the most-bound (equatorial) orbits; the least-bound orbits (which have the largest inclination) do not strongly lock to the black hole's spin until they have very nearly reached the innermost orbit [14] . The property ∂L z (r, ι)/∂ι > 0 reflects the different efficiency of nearly horizon-skimming orbits to lock with the horizon.\n\nReference [14] argued that this behavior could have observational consequences. It is well-known that the inclination angle of an inspiraling body generally increases due to gravitational-wave emission [16, 17] . Since dL z /dt < 0 because of the positive angular momentum carried away by the gravitational waves, and since \"normal\" orbits have ∂L z /∂ι < 0, one would expect dι/dt > 0. However, if during an evolution ∂L z /∂ι switches sign, then dι/dt might switch sign as well: An inspiraling body could evolve towards an equatorial orbit, signalling the presence of an \"almostextremal\" Kerr black hole.\n\nIt should be emphasized that this argument is not rigorous. In particular, one needs to consider the joint evolution of orbital radius and inclination angle; and, one must include the dependence of these two quantities on orbital energy as well as angular momentum 1 . As such, dι/dt depends not only on dL z /dt and ∂L z /∂ι, but also on dE/dt, ∂E/∂ι, ∂E/∂r and ∂L z /∂r.\n\nIn this sense, the argument made in Ref. [14] should be seen as claiming that the contribution coming from dL z /dt and ∂L z /∂ι are simply the dominant ones. Using the numerical code described in [17] to compute the fluxes dL z /dt and dE/dt, it was then found that a test-particle on a circular orbit passing through the nearly horizon-skimming region of a Kerr black hole with a = 0.998M (the value at which a hole's spin tends to be buffered due to photon capture from thin disk accretion [19]) had its inclination angle decreased by δι ≈ 1 • -2 • [14] in the adiabatic approximation [20] . It should be noted at this point that the rate of change of inclination angle, dι/dt, appears as the difference of two relatively small and expensive to compute rates of change [cf. Eq. (3.8) of Ref. [17] ]. As such, small relative errors in those rates of change can lead to large relative errors in dι/dt. Finally, in Ref. [14] it was speculated that the decrease could be even larger for eccentric orbits satisfying the condition ∂L z /∂ι > 0, possibly leading to an observable imprint on EMRI gravitational waveforms.\n\nThe main purpose of this paper is to extend Ref. [14] 's analysis of nearly horizon-skimming orbits to include the effect of orbital eccentricity, and to thereby test the speculation that there may be an observable imprint on EMRI waveforms of nearly horizon-skimming behavior. In doing so, we have revisited all the calculations of Ref. [14] using a more accurate Teukolsky solver which serves as the engine for the analysis presented in Ref. [21] .\n\nWe have found that the critical spin value for circular nearly horizon-skimming orbits, a > 0.9524M , also delineates a family of eccentric orbits for which the condition ∂L z (p, e, ι)/∂ι > 0 holds. (More precisely, we consider variation with respect to an angle θ inc that is easier to work with in the extreme strong field, but that is easily related to ι.) The parameters p and e are the orbit's latus rectum and eccentricity, defined precisely in Sec. II. These generic nearly horizonskimming orbits all have p 2M , deep in the black hole's extreme strong field.\n\nWe next study the evolution of these orbits under gravitational-wave emission in the adiabatic approximation. We first revisited the evolution of circular, nearly horizonskimming orbits using the improved Teukolsky solver which was used for the analysis of Ref. [21] . The results of this analysis were somewhat surprising: Just as for \"normal\" orbits, we found that orbital inclination always increases during inspiral, even in the nearly horizon-skimming regime. This is in stark contrast to the claims of Ref. [14] . As noted above, the inclination's rate of change depends on the difference of two expensive and difficult to compute numbers, and thus can be strongly impacted by small relative errors in those numbers. 1 In the general case, one must also include the dependence on \"Carter's constant\" Q [18], the third integral of black hole orbits (described more carefully in Sec. II). For circular orbits, Q = Q(E, Lz): knowledge of E and Lz completely determines Q.\n\n3 A primary result of this paper is thus to retract the claim of Ref. [14] that an important dynamic signature of the nearly horizon-skimming region is a reversal in the sign of inclination angle evolution: The inclination always grows under gravitational radiation emission. We next extended this analysis to study the evolution of generic nearly horizon-skimming orbits. The Teukolsky code to which we have direct access can, at this point, only compute the radiated fluxes of energy E and angular momentum L z ; results for the evolution of the Carter constant Q are just now beginning to be understood [22] , and have not yet been incorporated into this code. We instead use \"kludge\" expressions for dE/dt, dL z /dt, and dQ/dt which were inspired by Refs. [23, 24] . These expression are based on post-Newtonian flux formulas, modified in such a way that they fit strong-field radiation reaction results obtained from a Teukolsky integrator; see Ref. [24] for further discussion. Our analysis indicates that, just as in the circular limit, the result dι/dt > 0 holds for generic nearly horizon-skimming orbits. Furthermore, and contrary to the speculation of Ref. [14] , we do not find a large amplification of dι/dt as orbits are made more eccentric.\n\nOur conclusion is that the nearly horizon-skimming regime, though an interesting curiosity of strong-field orbits of nearly extremal black holes, will not imprint any peculiar observational signature on EMRI waveforms.\n\nThe remainder of this paper is organized as follows. In Sec. II, we review the properties of bound stable orbits in Kerr, providing expressions for the constants of motion which we will use in Sec. III to generalize nearly horizon-skimming orbits to the non-circular case. In Sec. IV, we study the evolution of the inclination angle for circular nearly horizon-skimming orbits using Teukolsky-based fluxes; in Sec. V we do the same for non-circular orbits and using kludge fluxes. We present and discuss our detailed conclusions in Sec. VI. The fits and post-Newtonian fluxes used for the kludge fluxes are presented in the Appendix. Throughout the paper we have used units in which G = c = 1."
},
{
"section_type": "OTHER",
"section_title": "II. BOUND STABLE ORBITS IN KERR SPACETIMES",
"text": "The line element of a Kerr spacetime, written in Boyer-Lindquist coordinates reads [25]\n\nds 2 = -1 - 2M r Σ dt 2 + Σ ∆ dr 2 + Σ dθ 2 + r 2 + a 2 + 2M a 2 r Σ sin 2 θ sin 2 θ dφ 2 - 4M ar Σ sin 2 θ dt dφ, ( 1\n\n)\n\nwhere\n\nΣ ≡ r 2 + a 2 cos 2 θ, ∆ ≡ r 2 -2M r + a 2 . ( 2\n\n)\n\nUp to initial conditions, geodesics can then be labelled by four constants of motion: the mass µ of the test particle, its energy E and angular momentum L z as measured by an observer at infinity and the Carter constant Q [18] . The presence of these four conserved quantities makes the geodesic equations separable in Boyer-Lindquist coordinates. Introducing the Carter time λ, defined by\n\ndτ dλ ≡ Σ , ( 3\n\n)\n\nthe geodesic equations become\n\nµ dr dλ 2 = V r (r), µ dt dλ = V t (r, θ), µ dθ dλ 2 = V θ (θ), µ dφ dλ = V φ (r, θ) , ( 4\n\n) with V t (r, θ) ≡ E ̟ 4 ∆ -a 2 sin 2 θ + aL z 1 - ̟ 2 ∆ , ( 5a\n\n) V r (r) ≡ E̟ 2 -aL z 2 -∆ µ 2 r 2 + (L z -aE) 2 + Q , ( 5b\n\n) V θ (θ) ≡ Q -L 2 z cot 2 θ -a 2 (µ 2 -E 2 ) cos 2 θ, ( 5c\n\n) V φ (r, θ) ≡ L z csc 2 θ + aE ̟ 2 ∆ -1 - a 2 L z ∆ , ( 5d\n\n)\n\nwhere we have defined\n\n̟ 2 ≡ r 2 + a 2 . ( 6\n\n)\n\nThe conserved parameters E, L z , and Q can be remapped to other parameters that describe the geometry of the orbit. We have found it useful to describe the orbit in terms of an angle θ min -the minimum polar angle reached by the orbit -as well as the latus rectum p and the eccentricity e. In the weakfield limit, p and e correspond exactly to the latus rectum and eccentricity used to describe orbits in Newtonian gravity; in the strong field, they are essentially just a convenient remapping of the orbit's apoastron and periastron:\n\nr ap ≡ p 1 -e , r peri ≡ p 1 + e . ( 7\n\n)\n\nFinally, in much of our analysis, it is useful to refer to 4 ) to impose dr/dλ = 0 at r = r ap and r = r peri , and to impose dθ/dλ = 0 at θ = θ min . (Note that for a circular orbit, r ap = r peri = r 0 . In this case, one must apply the rules dr/dλ = 0 and d 2 r/dλ 2 = 0 at r = r 0 .) Following this approach, Schmidt [26] was able to derive explicit expressions for E, L z and Q in terms of p, e and z -. We now briefly review Schmidt's results.\n\nz -≡ cos 2 θ min , ( 8\n\n) rather than to θ min directly. To map (E, L z , Q) to (p, e, z -), use Eq. (\n\nLet us first introduce the dimensionless quantities\n\nẼ ≡ E/µ , Lz ≡ L z /(µM ) , Q ≡ Q/(µM ) 2 , ( 9\n\n) ã ≡ a/M , r ≡ r/M , ∆ ≡ ∆/M 2 , ( 10\n\n)\n\n4 Figure 1: Left panel: Inclination angles θinc for which bound stable orbits exist for a black hole with spin a = 0.998 M . The allowed range for θinc goes from θinc = 0 to the curve corresponding to the eccentricity under consideration, θinc = θ max inc . Right panel: Same as left but for an extremal black hole, a = M . Note that in this case θ max inc never reaches zero.\n\nand the functions\n\nf (r) ≡ r4 + ã2 r(r + 2) + z -∆ , ( 11\n\n) g(r) ≡ 2 ã r , ( 12\n\n) h(r) ≡ r(r -2) + z - 1 -z - ∆ , ( 13\n\n) d(r) ≡ (r 2 + ã2 z -) ∆ . ( 14\n\n)\n\nLet us further define the set of functions\n\n(f 1 , g 1 , h 1 , d 1 ) ≡ (f (r p ), g(r p ), h(r p ), d(r p )) if e > 0 , (f (r 0 ), g(r 0 ), h(r 0 ), d(r 0 )) if e = 0 , ( 15\n\n) (f 2 , g 2 , h 2 , d 2 ) ≡ (f (r a ), g(r a ), h(r a ), d(r a )) if e > 0 , (f ′ (r 0 ), g ′ (r 0 ), h ′ (r 0 ), d ′ (r 0 )) if e = 0 , ( 16\n\n)\n\nand the determinants\n\nκ ≡ d 1 h 2 -d 2 h 1 , (17) ε ≡ d 1 g 2 -d 2 g 1 , (18) ρ ≡ f 1 h 2 -f 2 h 1 , (19) η ≡ f 1 g 2 -f 2 g 1 , (20) σ ≡ g 1 h 2 -g 2 h 1 . ( 21\n\n)\n\nThe energy of the particle can then be written\n\nẼ = κρ + 2ǫσ -2D σ(σǫ 2 + ρǫκ -ηκ 2 ) ρ 2 + 4ησ . ( 22\n\n)\n\nThe parameter D takes the values ±1. The angular momentum is a solution of the system\n\nf 1 Ẽ2 -2g 1 Ẽ Lz -h 1 L2 z -d 1 = 0 , (23) f 2 Ẽ2 -2g 2 Ẽ Lz -h 2 L2 z -d 2 = 0 . ( 24\n\n)\n\nBy eliminating the L2 z terms in these equations, one finds the solution\n\nLz = ρ Ẽ2 -κ 2 Ẽσ ( 25\n\n)\n\nfor the angular momentum. Using dθ/dλ = 0 at θ = θ min , the Carter constant can be written\n\nQ = z - ã2 (1 -Ẽ2 ) + L2 z 1 -z - . ( 26\n\n)\n\nAdditional constraints on p, e, z -are needed for the orbits to be stable. Inspection of Eq. ( 4 ) shows that an eccentric orbit is stable only if\n\n∂V r ∂r (r peri ) > 0 . ( 27\n\n)\n\nIt is marginally stable if ∂V r /∂r = 0 at r = r peri . Similarly, the stability condition for circular orbits is\n\n∂ 2 V r ∂r 2 (r 0 ) < 0 ; ( 28\n\n)\n\nmarginally stable orbits are set by\n\n∂ 2 V r /∂r 2 = 0 at r = r 0 .\n\n5 Finally, we note that one can massage the above solutions for the conserved orbital quantities of bound stable orbits to rewrite the solution for Lz as\n\nLz = - g 1 Ẽ h 1 + D h 1 g 2 1 Ẽ2 + (f 1 Ẽ2 -d 1 )h 1 . ( 29\n\n)\n\nFrom this solution, we see that it is quite natural to refer to orbits with D = 1 as prograde and to orbits with D = -1 as retrograde. Note also that Eq. (29) is a more useful form than the corresponding expression, Eq. ( A4 ), of Ref. [21] . In that expression, the factor 1/h 1 has been squared and moved inside the square root. This obscures the fact that h 1 changes sign for very strong field orbits. Differences between Eq. ( 29 ) and Eq. (A4) of [21] are apparent for a 0.835, although only for orbits close to the separatrix (i.e., the surface in the space of parameters (p, e, ι) where marginally stable bound orbits lie)."
},
{
"section_type": "OTHER",
"section_title": "III. NON-CIRCULAR NEARLY HORIZON-SKIMMING ORBITS",
"text": "With explicit expressions for E, L z and Q as functions of p, e and z -, we now examine how to generalize the condition ∂L z (r, ι)/∂ι > 0, which defined circular nearly horizonskimming orbits in Ref. [14] , to encompass the non-circular case. We recall that the inclination angle ι is defined as [14]\n\ncos ι = L z Q + L 2 z . ( 30\n\n)\n\nSuch a definition is not always easy to handle in the case of eccentric orbits. In addition, ι does not have an obvious physical interpretation (even in the circular limit), but rather was introduced essentially to generalize (at least formally) the definition of inclination for Schwarzschild black hole orbits. In that case, one has\n\nQ = L 2 x + L 2 y and therefore L z = |L| cos ι.\n\nA more useful definition for the inclination angle in Kerr was introduced in Ref. [21] :\n\nθ inc = π 2 -D θ min , ( 31\n\n)\n\nwhere θ min is the minimum reached by θ during the orbital motion. This angle is trivially related to z -(z -= sin 2 θ inc ) and ranges from 0 to π/2 for prograde orbits and from π/2 to π for retrograde orbits. It is a simple numerical calculation to convert between ι and θ inc ; doing so shows that the differences between ι and θ inc are very small, with the two coinciding for a = 0, and with a difference that is less than 2.6 • for a = M and circular orbits with r = M . Bearing all this in mind, the condition we have adopted to generalize nearly horizon-skimming orbits is\n\n∂L z (p, e, θ inc ) ∂θ inc > 0 . ( 32\n\n)\n\nWe have found that certain parts of this calculation, particular the analysis of strong-field geodesic orbits, are best done using the angle θ inc ; other parts are more simply done using the angle ι, particularly the \"kludge\" computation of fluxes described in Sec. V. (This is because the kludge fluxes are based on an extension of post-Newtonian formulas to the strongfield regime, and these formulas use ι for inclination angle.) Accordingly, we often switch back and forth between these two notions of inclination, and in fact present our final results for inclination evolution using both dι/dt and dθ inc /dt. Before mapping out the region corresponding to nearly horizon-skimming orbits, it is useful to examine stable orbits more generally in the strong field of rapidly rotating black holes. We first fix a value for a, and then discretize the space of parameters (p, e, θ inc ). We next identify the points in this space corresponding to bound stable geodesic orbits. Sufficiently close to the horizon, the bound stable orbits with specified values of p and e have an inclination angle θ inc ranging from 0 (equatorial orbit) to a maximum value θ max inc . For given p and e, θ max inc defines the separatrix between stable and unstable orbits.\n\nExample separatrices are shown in Fig. 1 for a = 0.998M and a = M . This figure shows the behavior of θ max inc as a function of the latus rectum for the different values of the eccentricity indicated by the labels. Note that for a = 0.998M the angle θ max inc eventually goes to zero. This is the general behavior for a < M . On the other hand, for an extremal black hole, a = M , θ max inc never goes to zero. The orbits which reside at r = M (the circular limit) are the \"horizon-skimming orbits\" identified by Wilkins [12] ; the a = M separatrix has a similar shape even for eccentric orbits. As expected, we find that for given latus rectum and eccentricity the orbit with θ inc = 0 is the one with the lowest energy E (and hence is the most-bound orbit), whereas the orbit with θ inc = θ max inc has the highest E (and is least bound).\n\nHaving mapped out stable orbits in (p, e, θ inc ) space, we then computed the partial derivative ∂L z (p, e, θ inc )/∂θ inc and identified the following three overlapping regions:\n\n• Region A: The portion of the (p, e) plane for which ∂L z (p, e, θ inc )/∂θ inc > 0 for 0 ≤ θ inc ≤ θ max inc . This region is illustrated in Fig. 2 as the area under the heavy solid line and to the left of the dot-dashed line (green in the color version).\n\n• Region B: The portion of the (p, e) plane for which (L z ) most bound (p, e) is smaller than (L z ) least bound (p, e). In other words,\n\nL z (p, e, 0) < L z (p, e, θ max inc ) ( 33\n\n)\n\nin Region B. Note that Region B contains Region A. It is illustrated in Fig. 2 as the area under the heavy solid line and to the left of the dotted line (red in the color version).\n\n• Region C: The portion of the (p, e) plane for which ∂L z (p, e, θ inc )/∂θ inc > 0 for at least one angle θ inc between 0 and θ max inc . Region C contains Region B, and is illustrated in Fig. 2 as the area under the heavy solid line and to the left of the dashed line (blue in the color version). 6 Figure 2: Left panel: Non-circular nearly horizon-skimming orbits for a = 0.998M . The heavy solid line indicates the separatrix between stable and unstable orbits for equatorial orbits (ι = θinc = 0). All orbits above and to the left of this line are unstable. The dot-dashed line (green in the color version) bounds the region of the (p, e)-plane where ∂Lz/∂θinc > 0 for all allowed inclination angles (\"Region A\"). All orbits between this line and the separatrix belong to Region A. The dotted line (red in the color version) bounds the region (Lz) most bound < (Lz) least bound (\"Region B\"). Note that B includes A. The dashed line (blue in the color version) bounds the region where ∂Lz/∂θinc > 0 for at least one inclination angle (\"Region C\"); note that C includes B. All three of these regions are candidate generalizations of the notion of nearly horizon-skimming orbits. Right panel: Same as the left panel, but for the extreme spin case, a = M . In this case the separatrix between stable and unstable equatorial orbits is given by the line p/M = 1 + e.\n\nOrbits in any of these three regions give possible generalizations of the nearly horizon-skimming circular orbits presented in Ref. [14] . Notice, as illustrated in Fig. 2 , that the size of these regions depends rather strongly on the spin of the black hole. All three regions disappear altogether for a < 0.9524M (in agreement with [14] ); their sizes grow with a, reaching maximal extent for a = M . These regions never extend beyond p ≃ 2M .\n\nAs we shall see, the difference between these three regions is not terribly important for assessing whether there is a strong signature of the nearly horizon-skimming regime on the inspiral dynamics. As such, it is perhaps most useful to use Region C as our definition, since it is the most inclusive."
},
{
"section_type": "OTHER",
"section_title": "IV. EVOLUTION OF θinc: CIRCULAR ORBITS",
"text": "To ascertain whether nearly horizon-skimming orbits can affect an EMRI in such a way as to leave a clear imprint in the gravitational-wave signal, we have studied the time evolution of the inclination angle θ inc . To this purpose we have used the so-called adiabatic approximation [20] , in which the infalling body moves along a geodesic with slowly changing parameters. The evolution of the orbital parameters is computed using the time-averaged fluxes dE/dt, dL z /dt and dQ/dt due to gravitational-wave emission (\"radiation reaction\"). As discussed in Sec. II, E, L z and Q can be expressed in terms of p, e, and θ inc . Given rates of change of E, L z and Q, it is then straightforward [23] to calculate dp/dt, de/dt, and dθ inc /dt (or dι/dt).\n\nWe should note that although perfectly well-behaved for all bound stable geodesics, the adiabatic approximation breaks down in a small region of the orbital parameters space very close to the separatrix, where the transition from an inspiral to a plunging orbit takes place [27] . However, since this region is expected to be very small 2 and its impact on LISA waveforms rather hard to detect [27] , we expect our results to be at least qualitatively correct also in this region of the space of parameters.\n\nAccurate calculation of dE/dt and dL z /dt in the adiabatic approximation involves solving the Teukolsky and Sasaki-Nakamura equations [28, 29] . For generic orbits this has been done for the first time in Ref. [21] . The calculation of dQ/dt for generic orbits is more involved. A formula for dQ/dt has been recently derived [22] , but has not yet been implemented (at least in a code to which we have access).\n\nOn the other hand, it is well-known that a circular orbit will remain circular under radiation reaction [30, 31, 32] . This constraint means that Teukolsky-based fluxes for E and L z 2 Its width in p/M is expected to be of the order of ∆p/M ∼ (µ/M ) 2/5 , where µ is the mass of the infalling body [27].\n\n7 are sufficient to compute dQ/dt. Considering this limit, the rate of change dQ/dt can be expressed in terms of dE/dt and dL/dt as\n\ndQ dt circ = - N 1 (p, ι) N 5 (p, ι) dE dt circ - N 4 (p, ι) N 5 (p, ι) dL z dt circ (34) where N 1 (p, ι) ≡ E(p, ι) p 4 + a 2 E(p, ι) p 2 -2 a M (L z (p, ι) -a E(p, ι)) p , ( 35\n\n) N 4 (p, ι) ≡ (2 M p -p 2 ) L z (p, ι) -2 M a E(p, ι) p , ( 36\n\n) N 5 (p, ι) ≡ (2 M p -p 2 -a 2 )/2 . ( 37\n\n)\n\n(These quantities are for a circular orbit of radius p.) Using this, it is simple to compute dθ inc /dt (or dι/dt). This procedure was followed in Ref. [14] , using the code presented in Ref. [17] , to determine the evolution of ι; this analysis indicated that dι/dt < 0 for circular nearly horizonskimming orbits. As a first step to our more general analysis, we have repeated this calculation but using the improved Sasaki-Nakamura-Teukolsky code presented in Ref. [21] ; we focused on the case a = 0.998M .\n\nRather to our surprise, we discovered that the fluxes dE/dt and dL z /dt computed with this more accurate code indicate that dι/dt > 0 (and dθ inc /dt > 0) for all circular nearly horizon-skimming orbits -in stark contrast with what was found in Ref. [14] . As mentioned in the introduction, the rate of change of inclination angle appears as the difference of two quantities. These quantities nearly cancel (and indeed cancel exactly in the limit a = 0); as such, small relative errors in their values can lead to large relative error in the inferred inclination evolution. Values for dE/dt, dL z /dt, dι/dt, and dθ inc /dt computed using the present code are shown in Table I in the columns with the header \"Teukolsky\"."
},
{
"section_type": "OTHER",
"section_title": "V. EVOLUTION OF θinc: NON-CIRCULAR ORBITS",
"text": "The corrected behavior of circular nearly horizonskimming orbits has naturally led us to investigate the evolution of non-circular nearly horizon-skimming orbits. Since our code cannot be used to compute dQ/dt, we have resorted to a \"kludge\" approach, based on those described in Refs. [23, 24] . In particular, we mostly follow the procedure developed by Gair & Glampedakis [24] , though (as described below) importantly modified.\n\nThe basic idea of the \"kludge\" procedure is to use the functional form of 2PN fluxes E, L z and Q, but to correct the circular part of these fluxes using fits to circular Teukolsky data. As developed in Ref. [24] , the fluxes are written\n\ndE dt GG = (1 -e 2 ) 3/2 (1 -e 2 ) -3/2 dE dt 2PN (p, e, ι) - dE dt 2PN (p, 0, ι) + dE dt fit circ (p, ι) , ( 38\n\n)\n\ndL z dt GG = (1 -e 2 ) 3/2 (1 -e 2 ) -3/2 dL z dt 2PN (p, e, ι) - dL z dt 2PN (p, 0, ι) + dL z dt fit circ (p, ι) , ( 39\n\n)\n\ndQ dt GG = (1 -e 2 ) 3/2 Q(p, e, ι) × (1 -e 2 ) -3/2 dQ/dt √ Q 2PN (p, e, ι) - dQ/dt √ Q 2PN (p, 0, ι) + dQ/dt √ Q fit circ (p, ι) . ( 40\n\n)\n\nThe post-Newtonian fluxes (dE/dt) 2PN , (dL z /dt) 2PN and (dQ/dt) 2PN are given in the Appendix [particularly Eqs. (A.1), (A.2), and (A.3)]. Since for circular orbits the fluxes dE/dt, dL z /dt and dQ/dt are related through Eq. (34) , only two fits to circular Teukolsky data are needed. One possible choice is to fit dL z /dt and dι/dt, and then use the circularity constraint to obtain 3 [24]\n\ndQ/dt √ Q fit circ (p, ι) = 2 tan ι dL z dt fit circ + Q(p, 0, ι) sin 2 ι dι dt fit circ , ( 41\n\n) dE dt fit circ (p, ι) = - N 4 (p, ι) N 1 (p, ι) dL z dt fit circ (p, ι) - N 5 (p, ι) N 1 (p, ι) Q(p, 0, ι) dQ/dt √ Q fit circ (p, ι) . ( 42\n\n)\n\nAs stressed in Ref. [24] , one does not expect these fluxes to work well in the strong field, both because the post-Newtonian approximation breaks down close to the black hole, and because the circular Teukolsky data used for the fits in Ref. [24] was computed for 3M ≤ p ≤ 30M . As a first attempt to improve their behavior in the nearly horizon-skimming region, we have made fits using circular Teukolsky data for orbits with M < p ≤ 2M . In particular, for a black hole with a = 0.998M , we computed the circular Teukolsky-based fluxes dL z /dt and dι/dt listed in Table I (columns 8 and 10). These results were fit (with error 0.2%); see Eqs. (A.4) and (A.6) in the Appendix. 3 This choice might seem more involved than fitting directly dLz/dt and dQ/dt, but, as noted by Gair & Glampedakis, ensures more sensible results for the evolution of the inclination angle. This generates more physically realistic inspirals [24].\n\n8 Despite using strong-field Teukolsky fluxes for our fit, we found fairly poor behavior of these rates of change, particularly as a function of eccentricity. To compensate for this, we introduced a kludge-type fit to correct the equatorial part of the flux, in addition to the circular part. We fit, as a function of p and e, Teukolsky-based fluxes for dE/dt and dL z /dt for orbits in the equatorial plane, and then introduce the following kludge fluxes for E and L z :\n\ndE dt (p, e, ι) = dE dt GG (p, e, ι) - dE dt GG (p, e, 0) + dE dt fit eq (p, e) ( 43\n\n)\n\ndL z dt (p, e, ι) = dL z dt GG (p, e, ι) - dL z dt GG (p, e, 0) + dL z dt fit eq (p, e) . ( 44\n\n)\n\n[Note that Eq. (40) for dQ/dt is not modified by this procedure since dQ/dt = 0 for equatorial orbits.] Using equatorial non-circular Teukolsky data provided by Drasco [21, 33] for a = 0.998 and M < p ≤ 2M (the ι = 0 \"Teukolsky\" data in Tables II, III and IV), we found fits (with error 1.5%); see Eqs. (A.9) and (A.10). Note that the fits for equatorial fluxes are significantly less accurate than the fits for circular fluxes. This appears to be due to the fact that, close to the black hole, many harmonics are needed in order for the Teukolsky-based fluxes to converge, especially for eccentric orbits (cf. Figs. 2 and 3 of Ref. [21] , noting the number of radial harmonics that have significant contribution to the flux). Truncation of these sums is likely a source of some error in the fluxes themselves, making it difficult to make a fit of as high quality as we could in the circular case. These fits were then finally used in Eqs. (43) and (44) to calculate the kludge fluxes dE/dt and dL z /dt for generic orbits. This kludge reproduces to high accuracy our fits to the Teukolsky-based fluxes for circular orbits (e = 0) or equatorial orbits (ι = 0). Some residual error remains because the ι = 0 limit of the circular fits do not precisely equal the e = 0 limit of the equatorial fits.\n\nTable I compares our kludge to Teukolsky-based fluxes for circular orbits; the two methods agree to several digits. Tables II, III and IV compare our kludge to the generic Teukolskybased fluxes for dE/dt and dL z /dt provided by Drasco [21, 33] . In all cases, the kludge fluxes dE/dt and dL z /dt have the correct qualitative behavior, being negative for all the orbital parameters under consideration (a = 0.998M ,\n\n1 < p/M ≤ 2, 0 ≤ e ≤ 0.5 and 0 • ≤ ι ≤ 41 • ).\n\nThe relative difference between the kludge and Teukolsky fluxes is always less than 25% for e = 0 and e = 0.1 (even for orbits very close to separatrix). The accuracy remains good at larger eccentricity, though it degrades somewhat as orbits come close to the separatrix.\n\nTables I, II, III and IV also present the kludge values of the fluxes dι/dt and dθ inc /dt as computed using Eqs. (43) and (44) for dE/dt and dL z /dt, plus Eq. (40) for dQ/dt. Though certainly not the last word on inclination evolution (pending rigorous computation of dQ/dt), these rates of change probably represent a better approximation than results published to date in the literature. (Indeed, prior work has often used the crude approximation dι/dt = 0 [21] to estimate dQ/dt given dE/dt and dL z /dt.) Most significantly, we find that (dι/dt) kludge > 0 and (dθ inc /dt) kludge > 0 for all of the orbital parameters we consider. In other words, we find that dι/dt and dθ inc /dt do not ever change sign.\n\nFinally, in Table V we compute the changes in θ inc and ι for the inspiral with mass ratio µ/M = 10 -6 . In all cases, we start at p/M = 1.9. The small body then inspirals through the nearly horizon-skimming region until it reaches the separatrix; at this point, the small body will fall into the large black hole on a dynamical timescale ∼ M , so we terminate the calculation. The evolution of circular orbits is computed using our fits to the circular-Teukolsky fluxes of E and L z ; for eccentric orbits we use the kludge fluxes (40), (43) and (44). As this exercise demonstrates, the change in inclination during inspiral is never larger than a few degrees. Not only is there no unique sign change in the nearly horizon-skimming region, but the magnitude of the inclination change remains puny. This leaves little room for the possibility that this class of orbits may have a clear observational imprint on the EMRIwaveforms to be detected by LISA."
},
{
"section_type": "CONCLUSION",
"section_title": "VI. CONCLUSIONS",
"text": "We have performed a detailed analysis of the orbital motion near the horizon of near-extremal Kerr black holes. We have demonstrated the existence of a class of orbits, which we have named \"non-circular nearly horizon-skimming orbits\", for which the angular momentum L z increases with the orbit's inclination, while keeping latus rectum and eccentricity fixed. This behavior, in stark contrast to that of Newtonian orbits, generalizes earlier results for circular orbits [14] .\n\nFurthermore, to assess whether this class of orbits can produce a unique imprint on EMRI waveforms (an important source for future LISA observations), we have studied, in the adiabatic approximation, the radiative evolution of inclination angle for a small body orbiting in the nearly horizonskimming region. For circular orbits, we have re-examined the analysis of Ref. [14] using an improved code for computing Teukolsky-based fluxes of the energy and angular momentum. Significantly correcting Ref. [14] 's results, we found no decrease in the orbit's inclination angle. Inclination always increases during inspiral.\n\nWe next carried out such an analysis for eccentric nearly horizon-skimming orbits. In this case, we used \"kludge\" fluxes to evolve the constants of motion E, L z and Q [24] . We find that these fluxes are fairly accurate when compared with the available Teukolsky-based fluxes, indicating that they should provide at least qualitatively correct information regarding inclination evolution. As for circular orbits, we find that the orbit's inclination never decreases. For both circular and non-circular configurations, we find that the magnitude of the inclination change is quite paltry -only a few degrees at 9 most. Quite generically, therefore, we found that the inclination angle of both circular and eccentric nearly horizon-skimming orbits never decreases during the inspiral. Revising the results obtained in Ref. [14] , we thus conclude that such orbits are not likely to yield a peculiar, unique imprint on the EMRIwaveforms detectable by LISA."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "It is a pleasure to thank Kostas Glampedakis for enlightening comments and advice, and Steve Drasco for useful discussions and for also providing the non-circular Teukolsky data that we used in this paper. The supercomputers used in this investigation were provided by funding from the JPL Office of the Chief Information Officer. This work was supported in part by the DFG grant SFB TR/7, by NASA Grant NNG05G105G, and by NSF Grant PHY-0449884. SAH gratefully acknowledges support from the MIT Class of 1956 Career Development Fund."
},
{
"section_type": "OTHER",
"section_title": "Appendix",
"text": "In this Appendix we report the expressions for the post-Newtonian fluxes and the fits to the Teukolsky data necessary to compute the kludge fluxes introduced in Sec. V. In particular the 2PN fluxes are given by [24]\n\ndE dt 2PN = - 32 5 µ 2 M 2 M p 5 (1 -e 2 ) 3/2 g 1 (e) -ã M p 3/2 g 2 (e) cos ι - M p g 3 (e) + π M p 3/2 g 4 (e) - M p 2 g 5 (e) + ã2 M p 2 g 6 (e) - 527 96 ã2 M p 2 sin 2 ι , (A.1) dL z dt 2PN = - 32 5 µ 2 M M p 7/2 (1 -e 2 ) 3/2 g 9 (e) cos ι + ã M p 3/2 (g a 10 (e) -cos 2 ιg b 10 (e)) - M p g 11 (e) cos ι +π M p 3/2 g 12 (e) cos ι - M p 2 g 13 (e) cos ι + ã2 M p 2 cos ι g 14 (e) - 45 8 sin 2 ι , (A.2) dQ dt 2PN = - 64 5 µ 2 M M p 7/2 Q sin ι (1 -e 2 ) 3/2 g 9 (e) -ã M p 3/2 cos ιg b 10 (e) - M p g 11 (e) +π M p 3/2 g 12 (e) - M p 2 g 13 (e) + ã2 M p 2 × g 14 (e) - 45 8 sin 2 ι , (A.3)\n\nwhere µ is the mass of the infalling body and where the various e-dependent coefficients are\n\ng 1 (e) ≡ 1 + 73 24 e 2 + 37 96 e 4 , g 2 (e) ≡ 73 12 + 823 24 e 2 + 949 32 e 4 + 491 192 e 6 , g 3 (e) ≡ 1247 336 + 9181 672 e 2 , g 4 (e) ≡ 4 + 1375 48 e 2 , g 5 (e) ≡ 44711 9072 + 172157 2592 e 2 , g 6 (e) ≡ 33 16 + 359 32 e 2 , g 9 (e) ≡ 1 + 7 8 e 2 , g a 10 (e) ≡ 61 24 + 63 8 e 2 + 95 64 e 4 , g b 10 (e) ≡ 61 8 + 91 4 e 2 + 461 64 e 4 , g 11 (e) ≡ 1247 336 + 425 336 e 2 , g 12 (e) ≡ 4 + 97 8 e 2 , g 13 (e) ≡ 44711 9072 + 302893 6048 e 2 , g 14 (e) ≡ 33 16 + 95 16 e 2 ,\n\nThe fits to the circular-Teukolsky data of Table I are instead given by\n\ndL z dt fit circ (p, ι ) = - 32 5 µ 2 M M p 7/2 cos ι + M p 3/2 61 24 - 61 8 cos 2 ι + 4π cos ι - 1247 336 M p cos ι + M p 2 cos ι - 1625 567 - 45 8 sin 2 ι + M p 5 2 d 1 (p/M ) + d 2 (p/M ) cos ι + d 3 (p/M ) cos 2 ι + d 4 (p/M ) cos 3 ι + d 5 (p/M ) cos 4 ι + d 6 (p/M ) cos 5 ι + cos ι M p 3/2 A + B cos 2 ι , (A.4) (A.5)\n\n10 dι dt fit circ (p, ι ) = 32 5/2 h 1 (p/M ) + cos 2 ι h 2 (p/M ) , (A.6) where\n\nd i (x) ≡ a i d + b i d x -1/2 + c i d x -1 , i = 1, . . . , 8, h i (x) ≡ a i h + b i h x -1/2 , i = 1, 2 (A.7)\n\nand the numerical coefficients are given by a 1 h = -278.9387 , b 1 h = 84.1414 , a 2 h = 8.6679 , b 2 h = -9.2401 , A = -18.3362 , B = 24.9034 , (A.8) and by the following table i 1 2 3 4 5 6 7 8 a i d 15.8363 445.4418 -2027.7797 3089.1709 -2045.2248 498.6411 -8.7220 50.8345 b i d -55.6777 -1333.2461 5940.4831 -9103.4472 6113.1165 -1515.8506 -50.8950 -131.6422 c i d 38.6405 1049.5637 -4513.0879 6926.3191 -4714.9633 1183.5875 251.4025 83.0834 Note that the functional form of these fits was obtained from Eqs. (57) and (58) of Ref. [24] by setting ã (i.e., q in their notation) to 1. Finally, we give expressions for the fits to the equatorial Teukolsky data of tables II, III and IV (data with ι = 0, columns with header \"Teukolsky\"):\n\ndE dt fit eq (p, e) = dE dt 2P N (p, e, 0) - 32 5 µ M 2 M p 5 (1 -e 2 ) 3/2 g 1 (e) + g 2 (e) M p 1/2 + g 3 (e) M p + g 4 (e) M p 3/2 + g 5 (e) M p 2 , (A.9) L z dt fit eq (p, e) = L z dt 2P N (p, e, 0) - 32 5 µ 2 M M p 7/2 (1 -e 2 ) 3/2 f 1 (e) + f 2 (e) M p 1/2 + f 3 (e) M p + f 4 (e) M p 3/2 + f 5 (e) M p 2 , (A.10) g i (e) ≡ a i g + b i g e 2 + c i g e 4 + d i g e 6 , f i (e) ≡ a i f + b i f e 2 + c i f e 4 + d i f e 6 , i = 1, . . . , 5 (A.11)\n\nwhere the numerical coefficients are given by the following table\n\ni a i g b i g c i g d i g a i f b i f c i f d i f\n\n1 6.4590 -2038.7301 6639.9843 227709.2187 5.4577 -3116.4034 4711.7065 214332.2907 2 -31.2215 10390.6778 -27505.7295 -1224376.5294 -26.6519 15958.6191 -16390.4868 -1147201.4687 3 57.1208 -19800.4891 39527.8397 2463977.3622 50.4374 -30579.3129 15749.9411 2296989.5466 4 -49.7051 16684.4629 -21714.7941 -2199231.9494 -46.7816 25968.8743 656.3460 -2038650.9838 5 16.4697 -5234.2077 2936.2391 734454.5696 15.6660 -8226.3892 -4903.9260 676553.2755 11 p M e θinc ι\n\ndE dt × M 2 µ 2 dE dt × M 2 µ 2 dLz dt × M µ 2 dLz dt × M µ 2 dι dt × M µ 2 dι dt × M µ 2 dθ inc dt × M µ 2 dθ inc dt × M µ 2\n\n(deg.) (deg.) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (Teukolsky) 1.3 0 0 0 -9.108×10 -2 -9.109×10 -2 -2.258×10 -1 -2.259×10 -1 0 0 0 0 1.3 0 10.4870 11.6773 -9.328×10 -2 -9.332×10 -2 -2.304×10 -1 -2.306×10 -1 1.837×10 -2 1.839×10 -2 6.462×10 -3 6.475×10 -3 1.3 0 14.6406 16.1303 -9.588×10 -2 -9.588×10 -2 -2.359×10 -1 -2.360×10 -1 2.397×10 -2 2.400×10 -2 8.645×10 -3 8.667×10 -3 1.3 0 17.7000 19.3172 -9.875×10 -2 -9.876×10 -2 -2.420×10 -1 -2.421×10 -1 2.728×10 -2 2.731×10 -2 1.007×10 -2 1.010×10 -2 1.3 0 20.1636 21.8210 -1.019×10 -1 -1.019×10 -1 -2.486×10 -1 -2.488×10 -1 2.943×10 -2 2.950×10 -2 1.111×10 -2 1.117×10 -2 1.4 0 0 0 -8.700×10 -2 -8.709×10 -2 -2.311×10 -1 -2.312×10 -1 0 0 0 0 1.4 0 14.5992 16.0005 -9.062×10 -2 -9.070×10 -2 -2.386×10 -1 -2.386×10 -1 2.316×10 -2 2.319×10 -2 8.823×10 -3 8.848×10 -3 1.4 0 20.1756 21.7815 -9.520×10 -2 -9.526×10 -2 -2.482×10 -1 -2.482×10 -1 2.875×10 -2 2.877×10 -2 1.141×10 -2 1.143×10 -2 1.4 0 24.1503 25.7517 -1.006×10 -1 -1.007×10 -1 -2.595×10 -1 -2.596×10 -1 3.140×10 -2 3.141×10 -2 1.289×10 -2 1.288×10 -2 1.4 0 27.2489 28.7604 -1.067×10 -1 -1.068×10 -1 -2.725×10 -1 -2.725×10 -1 3.274×10 -2 3.275×10 -2 1.378×10 -2 1.377×10 -2 1.5 0 0 0 -8.009×10 -2 -7.989×10 -2 -2.270×10 -1 -2.265×10 -1 0 0 0 0 1.5 0 16.7836 18.1857 -8.401×10 -2 -8.383×10 -2 -2.348×10 -1 -2.343×10 -1 2.360×10 -2 2.351×10 -2 9.602×10 -3 9.545×10 -3 1.5 0 23.0755 24.6167 -8.917×10 -2 -8.897×10 -2 -2.\n\n454×10 -1 -2.449×10 -1 2.872×10 -2 2.863×10 -2 1.228×10 -2 1.222×10 -2 1.5 0 27.4892 28.9670 -9.537×10 -2 -9.516×10 -2 -2.583×10 -1 -2.579×10 -1 3.091×10 -2 3.082×10 -2 1.372×10 -2 1.367×10 -2 1.5 0 30.8795 32.2231 -1.025×10 -1 -1.023×10 -1 -2.733×10 -1 -2.728×10 -1 3.184×10 -2 3.173×10 -2 1.452×10 -2 1.443×10 -2 1.6 0 0 0 -7.181×10 -2 -7.156×10 -2 -2.168×10 -1 -2.162×10 -1 0 0 0 0 1.6 0 18.3669 19.7220 -7.568×10 -2 -7.545×10 -2 -2.242×10 -1 -2.237×10 -1 2.240×10 -2 2.229×10 -2 9.600×10 -3 9.515×10 -3 1.6 0 25.1720 26.6245 -8.084×10 -2 -8.062×10 -2 -2.346×10 -1 -2.341×10 -1 2.701×10 -2 2.685×10 -2 1.223×10 -2 1.210×10 -2 1.6 0 29.9014 31.2625 -8.708×10 -2 -8.687×10 -2 -2.474×10 -1 -2.470×10 -1 2.889×10 -2 2.872×10 -2 1.363×10 -2 1.349×10 -2 1.6 0 33.5053 34.7164 -9.425×10 -2 -9.399×10 -2 -2.622×10 -1 -2.616×10 -1 2.964×10 -2 2.951×10 -2 1.441×10 -2 1.432×10 -2 1.7 0 0 0 -6.332×10 -2 -6.317×10 -2 -2.034×10 -1 -2.031×10 -1 0 0 0 0 1.7 0 19.6910 20.9859 -6.702×10 -2 -6.687×10 -2 -2.101×10 -1 -2.098×10 -1 2.057×10 -2 2.052×10 -2 9.202×10 -3 9.171×10 -3 1.7 0 26.9252 28.2884 -7.197×10 -2 -7.184×10 -2 -2.199×10 -1 -2.196×10 -1 2.467×10 -2 2.456×10 -2 1.170×10 -2 1.162×10 -2 1.7 0 31.9218 33.1786 -7.794×10 -2 -7.782×10 -2 -2.319×10 -1 -2.316×10 -1 2.632×10 -2 2.620×10 -2 1.306×10 -2 1.296×10 -2 1.7 0 35.7100 36.8118 -8.475×10 -2 -8.465×10 -2 -2.457×10 -1 -2.455×10 -1 2.698×10 -2 2.686×10 -2 1.384×10 -2 1.373×10 -2 1.8 0 0 0 -5.531×10 -2 -5.528×10 -2 -1.888×10 -1 -1.887×10 -1 0 0 0 0 1.8 0 20.8804 22.1128 -5.879×10 -2 -5.874×10 -2 -1.948×10 -1 -1.946×10 -1 1.858×10 -2 1.858×10 -2 8.635×10 -3 8.639×10 -3 1.8 0 28.5007 29.7791 -6.343×10 -2 -6.336×10 -2 -2.036×10 -1 -2.035×10 -1 2.221×10 -2 2.223×10 -2 1.098×10 -2 1.101×10 -2 1.8 0 33.7400 34.9034 -6.901×10 -2 -6.894×10 -2 -2.146×10 -1 -2.144×10 -1 2.368×10 -2 2.371×10 -2 1.228×10 -2 1.232×10 -2 1.8 0 37.6985 38.7065 -7.533×10 -2 -7.533×10 -2 -2.271×10 -1 -2.271×10 -1 2.429×10 -2 2.427×10 -2 1.306×10 -2 1.303×10 -2 1.9 0 0 0 -4.809×10 -2 -4.811×10 -2 -1.740×10 -1 -1.740×10 -1 0 0 0 0 1.9 0 21.9900 23.1615 -5.132×10 -2 -5.134×10 -2 -1.792×10 -1 -1.793×10 -1 1.666×10 -2 1.664×10 -2 8.022×10 -3 8.007×10 -3 1.9 0 29.9708 31.1702 -5.562×10 -2 -5.564×10 -2 -1.872×10 -1 -1.872×10 -1 1.986×10 -2 1.987×10 -2 1.019×10 -2 1.020×10 -2 1.9 0 35.4385 36.5176 -6.078×10 -2 -6.077×10 -2 -1.971×10 -1 -1.970×10 -1 2.118×10 -2 2.122×10 -2 1.143×10 -2 1.148×10 -2 1.9 0 39.5592 40.4847 -6.659×10 -2 -6.658×10 -2 -2.082×10 -1 -2.082×10 -1 2.177×10 -2 2.182×10 -2 1.222×10 -2 1.228×10 -2 2.0 0 0 0 -4.174×10 -2 -4.175×10 -2 -1.598×10 -1 -1.598×10 -1 0 0 0 0 2.0 0 23.0471 24.1605 -4.471×10 -2 -4.472×10 -2 -1.643×10 -1 -1.643×10 -1 1.489×10 -2 1.489×10 -2 7.425×10 -3 7.424×10 -3 2.0 0 31.3715 32.4978 -4.867×10 -2 -4.871×10 -2 -1.713×10 -1 -1.714×10 -1 1.773×10 -2 1.770×10 -2 9.436×10 -3 9.411×10 -3 2.0 0 37.0583 38.0608 -5.341×10 -2 -5.345×10 -2 -1.801×10 -1 -1.801×10 -1 1.893×10 -2 1.889×10 -2 1.062×10 -2 1.057×10 -2 2.0 0 41.3358 42.1876 -5.873×10 -2 -5.875×10 -2 -1.900×10 -1 -1.900×10 -1 1.950×10 -2 1.948×10 -2 1.141×10 -2 1.138×10 -2\n\nTable I: Teukolsky-based fluxes and kludge fluxes [computed using Eqs. (40), (43) and (44)] for circular orbits about a hole with a = 0.998M ; µ represents the mass of the infalling body. The Teukolsky-based fluxes have an accuracy of 10 -6 . [1] http://lisa.nasa.gov/ ; http://sci.esa.int/home/lisa/ [2] J. Kormendy and D. Richstone, Ann. Rev. Astron. Astrophys.\n\n33, 581 (1995). [3] J. R. Gair, L. Barack, T. Creighton, C. Cutler, S. L. Larson, E. S. Phinney, and M. Vallisneri, Class. Quantum Grav. 21, S1595 (2004). [4] S. L. Shapiro, Astrophys. J. 620, 59 (2005). [5] L. Rezzolla, T. W. Maccarone, S. Yoshida, and O. Zanotti, Mon.\n\nNot. Roy. Astron. Soc 344, L37 (2003). [6] R. Shafee, J. E. McClintock, R. Narayan, S. W. Davis, L.-X. Li, and R. A. Remilland, Astrophys. J. 636, L113 (2006).\n\n[7] J. E. McClintock, R. Shafee, R. Narayan, R. A. Remilland, S. W. Davis, and L.-X. Li, Astrophys. J. 652, 518 (2006). [8] A. C. Fabian and G. Miniutti, G. 2005, to appear in Kerr Spacetime: Rotating Black Holes in General Relativity, edited by D. L. Wiltshire, M. Visser, and S. M. Scott; astro-ph/0507409. [9] L. W. Brenneman and C. S. Reynolds, Astrophys. J. 652, 1028 (2006). [10] L. Barack and C. Cutler, Phys. Rev. D 69, 082005 (2004). [11] M. Volonteri, P. Madau, E. Quataert, and M. J. Rees, Astrophys. J. 620, 69 (2005). [12] D. C. Wilkins, Phys. Rev. D 5, 814 (1972).\n\n12 p M e θinc ι dE dt × M 2 µ 2 dE dt × M 2 µ 2 dLz dt × M µ 2 dLz dt × M µ 2 dι dt × M µ 2 dθ inc dt × M µ 2\n\n(deg.) (deg.) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (kludge) 1.3 0.1 0 0 -8.804×10 -2 -8.804×10 -2 -2.098×10 -1 -2.098×10 -1 0 0 1.4 0.1 0 0 -8.728×10 -2 -8.719×10 -2 -2.274×10 -1 -2.275×10 -1 0 0 1.4 0.1 8 8.8664 -9.110×10 -2 -8.736×10 -2 -2.355×10 -1 -2.273×10 -1 4.066×10 -2 2.938×10 -2 1.4 0.1 16 17.4519 -1.030×10 -1 -8.958×10 -2 -2.602×10 -1 -2.309×10 -1 7.428×10 -2 5.475×10 -2 1.4 0.1 24 25.5784 -1.243×10 -1 -9.771×10 -2 -3.037×10 -1 -2.415×10 -1 9.663×10 -2 7.316×10 -2 1.5 0.1 0 0 -8.069×10 -2 -8.095×10 -2 -2.255×10 -1 -2.260×10 -1 0 0 1.5 0.1 8 8.7910 -8.323×10 -2 -8.133×10 -2 -2.310×10 -1 -2.264×10 -1 2.996×10 -2 2.070×10 -2 1.5 0.1 16 17.3490 -9.121×10 -2 -8.395×10 -2 -2.483×10 -1 -2.314×10 -1 5.512×10 -2 3.888×10 -2 1.5 0.1 24 25.5197 -1.059×10 -1 -8.980×10 -2 -2.792×10 -1 -2.423×10 -1 7.255×10 -2 5.264×10 -2 1.6 0.1 0 0 -7.255×10 -2 -7.281×10 -2 -2.161×10 -1 -2.168×10 -1 0 0 1.6 0.1 8 8.7195 -7.430×10 -2 -7.321×10 -2 -2.201×10 -1 -2.173×10 -1 2.258×10 -2 1.502×10 -2 1.6 0.1 16 17.2437 -7.986×10 -2 -7.533×10 -2 -2.323×10 -1 -2.212×10 -1 4.179×10 -2 2.839×10 -2 1.6 0.1 24 25.4388 -9.025×10 -2 -8.040×10 -2 -2.547×10 -1 -2.309×10 -1 5.554×10 -2 3.886×10 -2 1.6 0.1 32 33.2683 -1.082×10 -1 -9.435×10 -2 -2.920×10 -1 -2.551×10 -1 6.316×10 -2 4.559×10 -2 1.7 0.1 0 0 -6.427×10 -2 -6.440×10 -2 -2.036×10 -1 -2.040×10 -1 0 0 1.7 0.1 8 8.6555 -6.552×10 -2 -6.478×10 -2 -2.065×10 -1 -2.045×10 -1 1.742×10 -2 1.124×10 -2 1.7 0.1 16 17.1454 -6.953×10 -2 -6.651×10 -2 -2.154×10 -1 -2.075×10 -1 3.240×10 -2 2.134×10 -2 1.7 0.1 24 25.3531 -7.707×10 -2 -7.052×10 -2 -2.317×10 -1 -2.150×10 -1 4.342×10 -2 2.948×10 -2 1.7 0.1 32 33.2416 -9.009×10 -2 -7.959×10 -2 -2.590×10 -1 -2.324×10 -1 4.998×10 -2 3.512×10 -2 1.8 0.1 0 0 -5.640×10 -2 -5.640×10 -2 -1.897×10 -1 -1.897×10 -1 0 0 1.8 0.1 8 8.5991 -5.732×10 -2 -5.676×10 -2 -1.918×10 -1 -1.902×10 -1 1.371×10 -2 8.640×10 -3 1.8 0.1 16 17.0562 -6.028×10 -2 -5.817×10 -2 -1.984×10 -1 -1.925×10 -1 2.562×10 -2 1.647×10 -2 1.8 0.1 24 25.2693 -6.588×10 -2 -6.139×10 -2 -2.105×10 -1 -1.983×10 -1 3.456×10 -2 2.291×10 -2 1.8 0.1 32 33.2018 -7.555×10 -2 -6.849×10 -2 -2.307×10 -1 -2.120×10 -1 4.020×10 -2 2.765×10 -2 1.9 0.1 0 0 -4.915×10 -2 -4.911×10 -2 -1.753×10 -1 -1.751×10 -1 0 0 1.9 0.1 8 8.5494 -4.985×10 -2 -4.945×10 -2 -1.768×10 -1 -1.755×10 -1 1.097×10 -2 6.791×10 -3 1.9 0.1 16 16.9760 -5.208×10 -2 -5.064×10 -2 -1.817×10 -1 -1.774×10 -1 2.055×10 -2 1.298×10 -2 1.9 0.1 24 25.1898 -5.633×10 -2 -5.328×10 -2 -1.908×10 -1 -1.819×10 -1 2.788×10 -2 1.816×10 -2 1.9 0.1 32 33.1555 -6.364×10 -2 -5.870×10 -2 -2.059×10 -1 -1.920×10 -1 3.272×10 -2 2.214×10 -2 2.0 0.1 0 0 -4.263×10 -2 -4.264×10 -2 -1.607×10 -1 -1.608×10 -1 0 0 2.0 0.1 8 8.5057 -4.316×10 -2 -4.292×10 -2 -1.619×10 -1 -1.611×10 -1 8.862×10 -3 5.424×10 -3 2.0 0.1 16 16.9042 -4.488×10 -2 -4.390×10 -2 -1.656×10 -1 -1.625×10 -1 1.666×10 -2 1.039×10 -2 2.0 0.1 24 25.1156 -4.815×10 -2 -4.604×10 -2 -1.724×10 -1 -1.660×10 -1 2.271×10 -2 1.459×10 -2 2.0 0.1 32 33.1064 -5.376×10 -2 -5.031×10 -2 -1.838×10 -1 -1.736×10 -1 2.684×10 -2 1.793×10 -2 2.0 0.1 40 40.8954 -6.339×10 -2 -6.236×10 -2 -2.027×10 -1 -1.967×10 -1 2.917×10 -2 2.036×10 -2\n\nTable II: As in Table I but for non-circular orbits; the Teukolsky-based fluxes for E and Lz have an accuracy of 10 -3 . Note that our code, as all the Teukolsky-based code that we are aware of, presently does not have the capability to compute inclination angle evolution for generic orbits. [13] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J. 178, 347 (1972). [14] S. A. Hughes, Phys. Rev. D 63, 064016 (2001). [15] K. S. Thorne, R. H. Price, and D. A. MacDonald, Black Holes: The Membrane Paradigm (Yale University Press, New Haven, CT, 1986). [16] F. D. Ryan, Phys. Rev. D 52, R3159 (1995). [17] S. A. Hughes, Phys. Rev. D 61, 084004 (2000). [18] B. Carter, Phys. Rev. 174, 1559 (1968). [19] K. S. Thorne, Astrophys. J. 191, 507 (1974). [20] Y. Mino, Phys. Rev. D 67, 084027 (2003) [21] S. Drasco and S. A. Hughes, Phys. Rev. D 73, 024027 (2006). [22] N. Sago, T. Tanaka, W. Hikida, and H. Nakano, Prog. Theor. Phys. 114, 509 (2005); N. Sago, T. Tanaka, W. Hikida, K. Ganz, and H. Nakano, Prog. Theor. Phys. 115, 873 (2006). [23] K. Glampedakis, S. A. Hughes, and D. Kennefick, Phys. Rev.\n\nD 66, 064005 (2002). [24] J. R. Gair and K. Glampedakis, Phys. Rev. D 73, 064037 (2006). [25] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973). [26] W. Schmidt, Class. Quantum Grav. 19, 2743 (2002). [27] A. Ori and K. S. Thorne, Phys. Rev. D 62, 124022 (2000) [28] S. A. Teukolsky, Astrophys. J. 185, 635 (1973). [29] M. Sasaki and T. Nakamura, Prog. Theor. Phys. 67, 1788 (1982). [30] F. D. Ryan, Phys. Ref. D 53, 3064 (1996) . [31] D. Kennefick and A. Ori, Phys. Rev. D 53, 4319 (1996). [32] Y. Mino, unpublished Ph. D. thesis, Kyoto University, 1996. [33] Data available at http://gmunu.mit.edu/sdrasco/snapshots/ 13 p M e θinc ι dE dt × M 2 µ 2 dE dt × M 2 µ 2 dLz dt × M µ 2 dLz dt × M µ 2 dι dt × M µ 2 dθ inc dt × M µ 2 (deg.) (deg.) (kludge) (Teukolsky ) (kludge) (Teukolsky) (kludge) (kludge) 1.4 0.2 0 0 -8.636×10 -2 -8.642×10 -2 -2.119×10 -1 -2.121×10 -1 0 0 1.4 0.2 8 8.8215 -9.853×10 -2 -8.240×10 -2 -2.374×10 -1 -2.015×10 -1 1.148×10 -1 9.714×10 -2 1.5 0.2 0 0 -8.362×10 -2 -8.349×10 -2 -2.236×10 -1 -2.230×10 -1 0 0 1.5 0.2 8 8.7595 -9.141×10 -2 -8.276×10 -2 -2.410×10 -1 -2.206×10 -1 7.893×10 -2 6.549×10 -2 1.5 0.2 16 17.2957 -1.145×10 -1 -8.394×10 -2 -2.915×10 -1 -2.215×10 -1 1.466×10 -1 1.230×10 -1 1.5 0.2 24 25.4608 -1.524×10 -1 -9.230×10 -2 -3.712×10 -1 -2.357×10 -1 1.952×10 -1 1.661×10 -1 1.6 0.2 0 0 -7.596×10 -2 -7.616×10 -2 -2.171×10 -1 -2.176×10 -1 0 0 1.6 0.2 8 8.6935 -8.111×10 -2 -7.641×10 -2 -2.292×10 -1 -2.177×10 -1 5.520×10 -2 4.502×10 -2 1.6 0.2 16 17.1994 -9.649×10 -2 -7.798×10 -2 -2.647×10 -1 -2.198×10 -1 1.032×10 -1 8.500×10 -2 1.6 0.2 24 25.3891 -1.221×10 -1 -8.314×10 -2 -3.212×10 -1 -2.288×10 -1 1.388×10 -1 1.160×10 -1 1.7 0.2 0 0 -6.765×10 -2 -6.799×10 -2 -2.057×10 -1 -2.068×10 -1 0 0 1.7 0.2 8 8.6329 -7.116×10 -2 -6.813×10 -2 -2.144×10 -1 -2.066×10 -1 3.963×10 -2 3.176×10 -2 1.7 0.2 16 17.1064 -8.171×10 -2 -6.995×10 -2 -2.398×10 -1 -2.096×10 -1 7.441×10 -2 6.024×10 -2 1.7 0.2 24 25.3085 -9.948×10 -2 -7.443×10 -2 -2.806×10 -1 -2.178×10 -1 1.009×10 -1 8.290×10 -2 1.7 0.2 32 33.2037 -1.257×10 -1 -8.558×10 -2 -3.371×10 -1 -2.366×10 -1 1.175×10 -1 9.806×10 -2 1.8 0.2 0 0 -5.965×10 -2 -5.962×10 -2 -1.927×10 -1 -1.926×10 -1 0 0 1.8 0.2 8 8.5789 -6.211×10 -2 -5.997×10 -2 -1.990×10 -1 -1.930×10 -1 2.919×10 -2 2.300×10 -2 1.8 0.2 16 17.0211 -6.953×10 -2 -6.147×10 -2 -2.175×10 -1 -1.954×10 -1 5.504×10 -2 4.380×10 -2 1.8 0.2 24 25.2283 -8.216×10 -2 -6.502×10 -2 -2.474×10 -1 -2.016×10 -1 7.515×10 -2 6.068×10 -2 1.8 0.2 32 33.1656 -1.009×10 -1 -7.410×10 -2 -2.890×10 -1 -2.190×10 -1 8.839×10 -2 7.258×10 -2 1.9 0.2 0 0 -5.218×10 -2 -5.210×10 -2 -1.786×10 -1 -1.783×10 -1 0 0 1.9 0.2 8 8.5312 -5.394×10 -2 -5.244×10 -2 -1.833×10 -1 -1.787×10 -1 2.197×10 -2 1.704×10 -2 1.9 0.2 16 16.9441 -5.928×10 -2 -5.373×10 -2 -1.970×10 -1 -1.807×10 -1 4.156×10 -2 3.254×10 -2 1.9 0.2 24 25.1518 -6.843×10 -2 -5.669×10 -2 -2.192×10 -1 -1.858×10 -1 5.706×10 -2 4.535×10 -2 1.9 0.2 32 33.1207 -8.213×10 -2 -6.277×10 -2 -2.502×10 -1 -1.966×10 -1 6.767×10 -2 5.475×10 -2 2.0 0.2 0 0 -4.528×10 -2 -4.530×10 -2 -1.637×10 -1 -1.638×10 -1 0 0 2.0 0.2 8 8.4891 -4.657×10 -2 -4.557×10 -2 -1.671×10 -1 -1.641×10 -1 1.679×10 -2 1.283×10 -2 2.0 0.2 16 16.8749 -5.049×10 -2 -4.664×10 -2 -1.774×10 -1 -1.657×10 -1 3.184×10 -2 2.457×10 -2 2.0 0.2 24 25.0802 -5.725×10 -2 -4.904×10 -2 -1.941×10 -1 -1.696×10 -1 4.391×10 -2 3.440×10 -2 2.0 0.2 32 33.0730 -6.743×10 -2 -5.427×10 -2 -2.175×10 -1 -1.793×10 -1 5.243×10 -2 4.184×10 -2 1.5 0.3 0 0 -8.481×10 -2 -8.478×10 -2 -2.094×10 -1 -2.094×10 -1 0 0 1.5 0.3 8 8.7037 -1.006×10 -1 -7.824×10 -2 -2.442×10 -1 -1.934×10 -1 1.484×10 -1 1.301×10 -1 1.5 0.3 16 17.2003 -1.469×10 -1 -7.811×10 -2 -3.435×10 -1 -1.864×10 -1 2.766×10 -1 2.440×10 -1 1.6 0.3 0 0 -8.144×10 -2 -8.123×10 -2 -2.183×10 -1 -2.178×10 -1 0 0 1.6 0.3 8 8.6498 -9.182×10 -2 -7.807×10 -2 -2.426×10 -1 -2.095×10 -1 1.028×10 -1 8.918×10 -2 1.6 0.3 16 17.1246 -1.\n\n223×10 -1 -8.089×10 -2 -3.122×10 -1 -2.144×10 -1 1.928×10 -1 1.683×10 -1 1.6 0.3 24 25.3046 -1.716×10 -1 -8.666×10 -2 -4.197×10 -1 -2.229×10 -1 2.607×10 -1 2.295×10 -1 1.7 0.3 0 0 -7.362×10 -2 -7.314×10 -2 -2.104×10 -1 -2.095×10 -1 0 0 1.7 0.3 8 8.5953 -8.060×10 -2 -7.224×10 -2 -2.277×10 -1 -2.065×10 -1 7.240×10 -2 6.224×10 -2 1.7 0.3 16 17.0415 -1.013×10 -1 -7.369×10 -2 -2.774×10 -1 -2.084×10 -1 1.365×10 -1 1.180×10 -1 1.7 0.3 24 25.2339 -1.349×10 -1 -7.800×10 -2 -3.547×10 -1 -2.153×10 -1 1.861×10 -1 1.622×10 -1 1.8 0.3 0 0 -6.488×10 -2 -6.484×10 -2 -1.973×10 -1 -1.972×10 -1 0 0 1.8 0.3 8 8.5454 -6.970×10 -2 -6.480×10 -2 -2.099×10 -1 -1.966×10 -1 5.206×10 -2 4.436×10 -2 1.8 0.3 16 16.9628 -8.402×10 -2 -6.671×10 -2 -2.461×10 -1 -1.998×10 -1 9.857×10 -2 8.445×10 -2 1.8 0.3 24 25.1601 -1.075×10 -1 -7.030×10 -2 -3.026×10 -1 -2.056×10 -1 1.353×10 -1 1.169×10 -1 1.8 0.3 32 33.1047 -1.404×10 -1 -8.153×10 -2 -3.762×10 -1 -2.255×10 -1 1.600×10 -1 1.394×10 -1 1.9 0.3 0 0 -5.669×10 -2 -5.690×10 -2 -1.829×10 -1 -1.832×10 -1 0 0 1.9 0.3 8 8.5010 -6.010×10 -2 -5.683×10 -2 -1.922×10 -1 -1.824×10 -1 3.823×10 -2 3.229×10 -2 1.9 0.3 16 16.8911 -7.025×10 -2 -5.818×10 -2 -2.189×10 -1 -1.844×10 -1 7.263×10 -2 6.165×10 -2 1.9 0.3 24 25.0887 -8.701×10 -2 -6.054×10 -2 -2.609×10 -1 -1.874×10 -1 1.003×10 -1 8.579×10 -2 1.9 0.3 32 33.0624 -1.106×10 -1 -6.912×10 -2 -3.157×10 -1 -2.034×10 -1 1.195×10 -1 1.032×10 -1 2.0 0.3 0 0 -4.953×10 -2 -4.946×10 -2 -1.683×10 -1 -1.683×10 -1 0 0 2.0 0.3 8 8.4616 -5.199×10 -2 -4.970×10 -2 -1.753×10 -1 -1.685×10 -1 2.862×10 -2 2.395×10 -2 2.0 0.3 16 16.8262 -5.932×10 -2 -5.079×10 -2 -1.954×10 -1 -1.699×10 -1 5.452×10 -2 4.585×10 -2 2.0 0.3 24 25.0215 -7.150×10 -2 -5.328×10 -2 -2.269×10 -1 -1.737×10 -1 7.564×10 -2 6.411×10 -2 2.0 0.3 32 33.0172 -8.878×10 -2 -6.003×10 -2 -2.682×10 -1 -1.864×10 -1 9.077×10 -2 7.771×10 -2\n\nTable III: As in Table II , but for additional values of eccentricity e; the Teukolsky-based fluxes for E and Lz have an accuracy of 10 -3 . 14 p M e θinc ι dE dt × M 2 µ 2 dE dt × M 2 µ 2 dLz dt × M µ 2 dLz dt × M µ 2 dι dt × M µ 2 dθ inc dt × M µ 2 (deg.) (deg.) (kludge) (Teukolsky ) (kludge) (Teukolsky) (kludge) (kludge) 1.6 0.4 0 0 -7.766×10 -2 -7.772×10 -2 -1.918×10 -1 -1.919×10 -1 0 0 1.6 0.4 8 8.5863 -9.433×10 -2 -7.645×10 -2 -2.297×10 -1 -1.881×10 -1 1.528×10 -1 1.370×10 -1 1.6 0.4 16 17.0151 -1.432×10 -1 -7.651×10 -2 -3.382×10 -1 -1.837×10 -1 2.873×10 -1 2.584×10 -1 1.7 0.4 0 0 -7.882×10 -2 -7.953×10 -2 -2.097×10 -1 -2.115×10 -1 0 0 1.7 0.4 8 8.5426 -9.002×10 -2 -7.408×10 -2 -2.367×10 -1 -1.978×10 -1 1.087×10 -1 9.656×10 -2 1.7 0.4 16 16.9502 -1.229×10 -1 -7.682×10 -2 -3.143×10 -1 -2.025×10 -1 2.054×10 -1 1.830×10 -1 1.7 0.4 24 25.1282 -1.760×10 -1 -8.090×10 -2 -4.336×10 -1 -2.075×10 -1 2.809×10 -1 2.514×10 -1 1.8 0.4 0 0 -7.107×10 -2 -7.007×10 -2 -2.013×10 -1 -1.988×10 -1 0 0 1.8 0.4 8 8.4989 -7.877×10 -2 -7.001×10 -2 -2.209×10 -1 -1.981×10 -1 7.788×10 -2 6.879×10 -2 1.8 0.4 16 16.8817 -1.015×10 -1 -7.009×10 -2 -2.774×10 -1 -1.965×10 -1 1.478×10 -1 1.309×10 -1 1.8 0.4 24 25.0646 -1.383×10 -1 -7.314×10 -2 -3.646×10 -1 -2.003×10 -1 2.036×10 -1 1.810×10 -1 1.8 0.4 32 33.0184 -1.887×10 -1 -9.193×10 -2 -4.755×10 -1 -2.319×10 -1 2.414×10 -1 2.156×10 -1 1.9 0.4 0 0 -6.187×10 -2 -6.267×10 -2 -1.861×10 -1 -1.881×10 -1 0 0 1.9 0.4 8 8.4591 -6.728×10 -2 -6.216×10 -2 -2.006×10 -1 -1.861×10 -1 5.666×10 -2 4.980×10 -2 1.9 0.4 16 16.8173 -8.328×10 -2 -6.222×10 -2 -2.424×10 -1 -1.844×10 -1 1.079×10 -1 9.506×10 -2 1.9 0.4 24 25.0006 -1.094×10 -1 -6.486×10 -2 -3.071×10 -1 -1.878×10 -1 1.495×10 -1 1.322×10 -1 1.9 0.4 32 32.9804 -1.452×10 -1 -7.884×10 -2 -3.896×10 -1 -2.158×10 -1 1.787×10 -1 1.588×10 -1 2.0 0.4 0 0 -5.483×10 -2 -5.457×10 -2 -1.735×10 -1 -1.729×10 -1 0 0 2.0 0.4 8 8.4235 -5.871×10 -2 -5.445×10 -2 -1.844×10 -1 -1.720×10 -1 4.222×10 -2 3.686×10 -2 2.0 0.4 16 16.7586 -7.020×10 -2 -5.555×10 -2 -2.158×10 -1 -1.733×10 -1 8.064×10 -2 7.057×10 -2 2.0 0.4 24 24.9396 -8.902×10 -2 -5.844×10 -2 -2.645×10 -1 -1.778×10 -1 1.122×10 -1 9.860×10 -2 2.0 0.4 32 32.9389 -1.150×10 -1 -6.536×10 -2 -3.267×10 -1 -1.896×10 -1 1.351×10 -1 1.193×10 -1 1.7 0.5 0 0 -7.421×10 -2 -7.401×10 -2 -1.815×10 -1 -1.810×10 -1 0 0 1.7 0.5 8 8.4736 -8.957×10 -2 -7.168×10 -2 -2.173×10 -1 -1.750×10 -1 1.379×10 -1 1.256×10 -1 1.7 0.5 16 16.8300 -1.347×10 -1 -6.999×10 -2 -3.201×10 -1 -1.676×10 -1 2.611×10 -1 2.378×10 -1 1.8 0.5 0 0 -7.589×10 -2 -7.620×10 -2 -1.993×10 -1 -2.000×10 -1 0 0 1.8 0.5 8 8.4395 -8.644×10 -2 -6.929×10 -2 -2.254×10 -1 -1.829×10 -1 1.005×10 -1 9.076×10 -2 1.8 0.5 16 16.7776 -1.175×10 -1 -7.210×10 -2 -3.004×10 -1 -1.880×10 -1 1.911×10 -1 1.726×10 -1 1.8 0.5 24 24.9413 -1.678×10 -1 -7.395×10 -2 -4.158×10 -1 -1.881×10 -1 2.638×10 -1 2.385×10 -1 1.9 0.5 0 0 -6.646×10 -2 -6.620×10 -2 -1.855×10 -1 -1.849×10 -1 0 0 1.9 0.5 8 8.4059 -7.386×10 -2 -6.320×10 -2 -2.048×10 -1 -1.768×10 -1 7.312×10 -2 6.579×10 -2 1.9 0.5 16 16.7233 -9.572×10 -2 -6.551×10 -2 -2.603×10 -1 -1.809×10 -1 1.395×10 -1 1.255×10 -1 1.9 0.5 24 24.8877 -1.312×10 -1 -7.087×10 -2 -3.461×10 -1 -1.909×10 -1 1.937×10 -1 1.744×10 -1 1.9 0.5 32 32.8741 -1.795×10 -1 -8.247×10 -2 -4.544×10 -1 -2.091×10 -1 2.320×10 -1 2.092×10 -1 2.0 0.5 0 0 -5.987×10 -2 -5.995×10 -2 -1.761×10 -1 -1.763×10 -1 0 0 2.0 0.5 8 8.3750 -6.516×10 -2 -5.918×10 -2 -1.906×10 -1 -1.738×10 -1 5.456×10 -2 4.882×10 -2 2.0 0.5 16 16.6725 -8.081×10 -2 -5.817×10 -2 -2.324×10 -1 -1.694×10 -1 1.044×10 -1 9.343×10 -2 2.0 0.5 24 24.8347 -1.063×10 -1 -6.254×10 -2 -2.970×10 -1 -1.776×10 -1 1.456×10 -1 1.304×10 -1 2.0 0.5 32 32.8378 -1.412×10 -1 -6.993×10 -2 -3.787×10 -1 -1.893×10 -1 1.756×10 -1 1.576×10 -1\n\nTable IV: As in Tables II and III, but for different values of eccentricity e; the Teukolsky-based fluxes for E and Lz have an accuracy of 10 -3 .\n\ne θinc ι ∆t/M ∆θinc ∆ι (deg.) (deg.) (deg.) (deg.) 0 0 0 1.250×10 6 0 0 0 5 5.355510 1.217×10 6 1.949×10 -1 4.954×10 -1 0 10 10.679331 1.118×10 6 3.468×10 -1 8.631×10 -1 0 15 15.943192 9.574×10 5 4.236×10 -1 1.019 0 20 21.125167 7.446×10 5 4.109×10 -1 9.440×10 -1 0 25 26.211779 4.981×10 5 3.158×10 -1 6.860×10 -1 0 30 31.199048 2.528×10 5 1.732×10 -1 3.527×10 -1 0 35 36.092514 6.584×10 4 4.636×10 -2 8.806×10 -2 0.1 0 0 1.228×10 6 0 0 0.1 5 5.351602 1.198×10 6 4.517×10 -1 7.766×10 -1 0.1 10 10.671900 1.103×10 6 6.900×10 -1 1.236 0.1 15 15.932962 9.426×10 5 7.283×10 -1 1.344 0.1 20 21.113129 7.315×10 5 6.433×10 -1 1.187 0.1 25 26.199088 4.900×10 5 4.780×10 -1 8.547×10 -1 0.1 30 31.186915 2.513×10 5 2.730×10 -1 4.585×10 -1 0.1 35 36.082095 6.589×10 4 8.385×10 -2 1.279×10 -1 0.2 0 0 1.173×10 6 0 0 0.2 5 5.339916 1.150×10 6 1.204 1.598 0.2 10 10.649670 1.064×10 6 1.698 2.331 0.2 15 15.902348 9.043×10 5 1.618 2.293 0.2 20 21.077081 6.980×10 5 1.324 1.900 0.2 25 26.161046 4.693×10 5 9.545×10 -1 1.351 0.2 30 31.150481 2.486×10 5 5.674×10 -1 7.711×10 -1 0.2 35 36.050712 7.562×10 4 2.070×10 -1 2.648×10 -1 0.3 0 0 1.087×10 6 0 0 0.3 5 5.320559 1.069×10 6 2.307 2.788 0.3 10 10.612831 1.001×10 6 3.256 4.007 0.3 15 15.851572 8.454×10 5 2.984 3.741 0.3 20 21.017212 6.483×10 5 2.375 2.998 0.3 25 26.097732 4.408×10 5 1.700 2.129 0.3 30 31.089639 2.493×10 5 1.040 1.276 0.3 35 35.997987 1.108×10 5 4.626×10 -1 5.569×10 -1\n\nTable V: Variation in the inclination angles ι and θinc as well as time needed to reach the separatrix for several inspirals through the nearly horizon-skimming regime. In all of these cases, the binary's mass ratio was fixed to µ/M = 10 -6 , the large black hole's spin was fixed to a = 0.998M , and the orbits were begun at p = 1.9M . The time interval ∆t is the total accumulated time it takes for the inspiralling body to reach the separatrix (at which time it rapidly plunges into the black hole). The angles ∆θinc and ∆ι are the total integrated change in these inclination angles that we compute. For the e = 0 cases, inspirals are computed using fits to the circular-Teukolsky fluxes of E and Lz; for eccentric orbits we use the kludge fluxes (40), (43) and (44) . Notice that ∆θinc and ∆ι are always positive -the inclination angle always increases during the inspiral through the nearly horizon-skimming region. The magnitude of this increase never exceeds a few degrees."
}
] | |
arxiv:0704.0140 | 0704.0140 | 1 | 10.1016/j.physletb.2007.08.026 | 666aaac79baf722a17a45b8f72128f29e25f1b9f014cf65105319dc3ee7806f0 | Entanglement entropy of two-dimensional Anti-de Sitter black holes | Using the AdS/CFT correspondence we derive a formula for the entanglement
entropy of the anti-de Sitter black hole in two spacetime dimensions. The
leading term in the large black hole mass expansion of our formula reproduces
exactly the Bekenstein-Hawking entropy S_{BH}, whereas the subleading term
behaves as ln S_{BH}. This subleading term has the universal form typical for
the entanglement entropy of physical systems described by effective conformal
fields theories (e.g. one-dimensional statistical models at the critical
point). The well-known form of the entanglement entropy for a two-dimensional
conformal field theory is obtained as analytic continuation of our result and
is related with the entanglement entropy of a black hole with negative mass. | [
"Mariano Cadoni"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | Using the AdS/CFT correspondence we derive a formula for the entanglement entropy of the anti-de Sitter black hole in two spacetime dimensions. The leading term in the large black hole mass expansion of our formula reproduces exactly the Bekenstein-Hawking entropy SBH , whereas the subleading term behaves as ln SBH . This subleading term has the universal form typical for the entanglement entropy of physical systems described by effective conformal fields theories (e.g. one-dimensional statistical models at the critical point). The well-known form of the entanglement entropy for a two-dimensional conformal field theory is obtained as analytic continuation of our result and is related with the entanglement entropy of a black hole with negative mass.
Quantum entanglement is a fundamental feature of quantum systems. It is related to the existence of correlations between parts of the system. The degree of entanglement of a quantum system is measured by the entanglement entropy S ent . In quantum field theory (QFT), or more in general in many body systems, we can localize observable and unobservable degrees of freedom in spatially separated regions Q and R. S ent is then defined as the von Neumann entropy of the system when the degrees of freedom in the region R are traced over, S ent = -T r Q ρQ ln ρQ , where the trace is taken over states in the observable region Q and the reduced density matrix ρQ = T r R ρ is obtained by tracing the density matrix ρ over states in the region R.
Investigation of the entanglement entropy (EE) has become relevant in many research areas. Apart from quantum information theory, the field that gave birth to the notion of entanglement entropy, it plays a crucial role in condensed matter systems, where it helps to understand quantum phases of matter (e.g spin chains and quantum liquids) [1, 2, 3, 4, 5] . Entanglement (geometric) entropy is also an useful concept for investigating general features of QFT, in particular two-dimensional conformal field theory (CFT) and the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence [6, 7, 8, 9, 10, 11, 12] . Last but not least entanglement may held the key for unraveling the mystery of black hole entropy [13, 14, 15, 16, 17, 18, 19, 20, 21, 22] .
We will be mainly concerned with the entanglement entropy of two-dimensional (2D) CFT and its relationship with the entropy of 2D black holes. It is an old idea that black hole entropy may be explained in terms of the EE of the quantum state of matter fields in the black hole geometry [13] . The main support to this conjecture comes from the fact that both the EE of matter fields and the Bekenstein-Hawking (BH) entropy depend on the area of the boundary region. On the other hand any attempt to explain the BH entropy as originating from quantum entanglement has to solve conceptual and technical difficulties.
The usual statistical paradigm explains the BH entropy in terms of a microstate gas. This is conceptually different from the EE that measures the observer's lack of information about the quantum state of the system in a inaccessible region of spacetime. Moreover, the EE depends both on the number of species n s of the matter fields, whose entanglement should reproduce the BH entropy, and on the value of the UV cutoff δ arising owing to the presence of a sharp boundary between the accessible and inaccessible regions of the spacetime. Conversely, the BH entropy is meant to be universal, hence independent from n s and δ. Some conceptual difficulties can be solved using Sakharov's induced gravity approach [23, 24, 25] , but the problem of the dependence on n s and δ still remains unsolved.
In this letter we will show that in the case of two-dimensional AdS black hole these difficulties can be completely solved. We will derive an expression for the black hole EE that in the large black hole mass limit reproduces exactly the BH entropy. Moreover, we will show that the subleading term has the universal behavior typical for CFTs and in particular for critical phenomena. The reason of this success is related to the peculiarities of 2D AdS gravity, namely the existence of an AdS/CFT correspondence and the fact that 2D Newton constant can be considered as wholly induced by quantum fluctuations of the dual CFT.
Most of the progress in understanding the EE in QFT has been achieved in the case of 2D CFT. Conformal invariance in two space-time dimension is a powerful tool that allows us to compute the EE in closed form. The entanglement entropy for the ground state of a 2D CFT originated from tracing over correlations between spacelike separated points has been calculated by Holzhey, Larsen and Wilckzek [6] . Introducing an infrared cutoff Λ the spacelike coordinate of our 2D universe will belong to C = [0, Λ[. The subsystem where measurements are performed is Q = [0, Σ[, whereas the outside region where the degrees of freedom are traced over is R = [Σ, Λ[. Because of the contribution of localized excitations arbitrarily near to the boundary the entanglement entropy diverges. Introducing an ultraviolet cutoff δ, the regularized entanglement entropy turns out to be [6]
S ent = c + c 6 ln Λ δπ sin πΣ Λ , (1)
where c and c are the central charges of the 2D CFT. The expression (1) emphasizes the characterizing features of the entanglement entropy, namely subadditivity and invariance under the transformation which exchanges the inside and outside regions
Σ → Λ -Σ. (2)
Moreover, S ent is not a monotonic function of Σ, but increases and reaches its maximum for Σ = Λ/2 and then decreases as Σ increases further. This behavior has an obvious explanation. When the subsystem begins to fill most of the universe there is lesser information to be lost and the entanglement entropy decreases.
Let us now consider 2D AdS black holes. As classical solutions of a 2D gravity theory they are endowed with a non-constant scalar field, the dilaton Φ. In the Schwarzschild gauge the 2D AdS black hole solutions are [26] ,
ds 2 = - r 2 L 2 -a 2 dt 2 + r 2 L 2 -a 2 -1 dr 2 , Φ = Φ 0 r L , (3)
where the length L is related to cosmological constant of the AdS spacetime (λ = 1/L 2 ), Φ 0 is the dimensionless 2D inverse Newton constant and a is an integration constants related to the black hole mass M and horizon radius r h by
a = r h L = 2M L Φ 0 . (4)
The thermodynamical, Bekenstein-Hawking, entropy of the black hole is [26]
S BH = 2πΦ 0 a = 2π 2Φ 0 M L, (5)
whereas the black hole temperature is T = a/2πL. Setting a = 0 in Eq. ( 3 ) we have the AdS black hole ground state ( in the following called AdS 0 ) with zero mass, temperature and entropy. The AdS black hole (3) can be considered as the thermalization of the AdS 0 solution at temperature a/2πL [26] .
It has been shown that the 2D black hole has a dual description in terms of a CFT with central charge [27, 28, 29, 30 ]
c = 12Φ 0 . (6)
The dual CFT can have both the form of a 2D [29, 30] or a 1D [27, 28] conformal field theory. This AdS 2 /CFT 2 ( or AdS 2 /CFT 1 ) correspondence has been used to give a microscopical meaning to the thermodynamical entropy of 2D AdS black holes. Eq. ( 5 ) has been reproduced by counting states in the dual CFT. In Ref. [16] (see also Refs. [24, 25, 31] ) it was observed that in two dimensions black hole entropy can be ascribed to quantum entanglement if 2D Newton constant is wholly induced by quantum fluctuations of matter fields. On the other hand the AdS 2 /CFT 2 correspondence, and in particular Eq. ( 6 ), tells us that the 2D Newton constant is induced by quantum fluctuations of the dual CFT. It follows that the black hole entropy (5) should be explained as the entanglement of the vacuum of the 2D CFT of central charge given by Eq. ( 6 ) in the gravitational black hole background (3) .
At first sight one is tempted to use Eq. ( 1 ) to calculate the entanglement entropy of the vacuum of the dual CFT. The exterior region of the 2D black hole can be easily identified with the region Q, whereas the black hole interior has to be identified with the R region where the degrees of freedom are traced over. There are two obstacles that prevents direct application of Eq. (1). First, Eq. ( 1 ) holds for a 2D flat spacetime, whereas we are dealing with a curved 2D background. Second, the calculations leading to Eq. ( 1 ) are performed for spacelike slice Q, whereas in our case the coordinate singularities at r = r h (the horizon) and r = ∞ (the timelike asymptotic boundary of the AdS spacetime) do not allow for a global notion of spacelike coordinate (a coordinate system covering the whole black hole spacetime in which the metric is non-singular and static). Owing to these geometrical features, in the black hole case we cannot give a direct meaning to both the measures Σ and (Λ -Σ) of the subsystems Q, R. As a consequence invariance under the transformation (2) is meaningless in the black hole case.
The second difficulty can be circumvented using appropriate coordinate system and regularization procedure, the first using instead of Eq. ( 1 ) the formula derived by Fiola et al. [16] , which gives the EE of the vacuum of matter fields in the case of a curved gravitational background. In the coordinate system used to define the vacuum of scalar fields in AdS 2 , the 2D black hole metric (3) is [26]
ds 2 = a 2 sinh 2 ( aσ L ) -dt 2 + dσ 2 . (7)
The coordinate system (t, σ) covers only the black hole exterior. The black hole horizon corresponds to σ = ∞ where the conformal factor of the metric vanishes. The asymptotic r = ∞ timelike conformal boundary of the AdS 2 spacetime is located at σ = 0, where the conformal factor diverges. The entanglement entropy of the CFT vacuum in the curved background (7) can be calculated, using the formula of Ref. [16] as the half line entanglement entropy seen by an observer in the 0 < σ < ∞ region. From the CFT point of view the AdS black hole has to be considered as the AdS 0 vacuum seen by the observer using the black hole coordinates (7) [26] . Moreover, this observer sees the the AdS 0 vacuum as filled with thermal radiation with negative flux [26] . It follows that the black hole entanglement entropy is given by the formula of Ref. [16] with reversed sign,
S (bh) ent = - c 6 ρ(σ = 0) -ln δ Λ , (8)
where ρ defines the conformal factor of the metric in the conformal gauge (ds 2 = exp(2ρ)(-dt 2 + dσ 2 )), c is the central charge given by Eq. ( 6 ) and δ, Λ are respectively UV and IR cutoffs. Notice that in Eq. ( 8 ) we have only contributions from only one sector (e.g. right movers) of the CFT. In Ref. [29, 30] it has been shown that the 2D AdS black hole is dual to an open string with appropriate boundary conditions. These boundary conditions are such that only one sector of the CFT 2 is present. The same is obviously true for the AdS 2 /CFT 1 realization of the correspondence [27, 28] . The conformal factor of the metric (7) , hence the entanglement entropy (8) blows up on the σ = 0 boundary of the AdS spacetime. The simplest regularization procedure that solves this problem is to consider a regularized boundary at σ = ǫ. Notice that ǫ plays the role of a UV cutoff for the coordinate σ, which is the natural spacelike coordinate of the dual CFT. ǫ is an IR cutoff for the coordinate r, which is the natural spacelike coordinate for the AdS 2 black hole. The regularized euclidean instanton corresponding to the black hole (7) is shown in figure (1) . The regularizing parameter ǫ can be set equal to the UV cutoff, δ = ǫ. Moreover, the regularized boundary is at finite proper distance from the horizon so that ǫ acts also as IR regulator, making the presence of the IR cutoff Λ in Eq. ( 8 ) redundant. It follows that the regularized EE is given by S (bh) ent = -c 6 ρ(ǫ) -ln ǫ L , which using equations ( 7 ) and ( 4 ) becomes
S (bh) ent = c 6 ln L 2 r h ǫ sinh ǫr h L 2 . ( 9
)
As a check of the validity of our formula we note that in the case of AdS 0 (r h = 0) the entanglement entropy vanishes. The AdS/CFT correspondence enable us to identify the cutoff ǫ as the UV cutoff of the CFT : ǫ ∝ L. The proportionality factor can be determined by requiring that the analytical continuation of Eq. ( 9 ) is invariant under the transformation (2) (see later). This requirement fixes ǫ = πL. With this position we get
S (bh) ent = c 6 ln L πr h sinh πr h L . ( 10
)
This formula is our main result, it gives the entanglement entropy of the 2D AdS black hole. This entanglement entropy has the expected behavior as a function of the horizon radius r h or, equivalently, of the black hole mass M . S
ent becomes zero in the AdS 0 ground state, r h = 0 (M = 0), whereas it grows monotonically for r h > 0 (M > 0). In order to compare the black hole EE (10) with the BH entropy (5) let us consider the limit of macroscopic black holes, that is the limit a → ∞ or equivalently r h >> L or also M >> 1/L. Expanding Eq. ( 10 ) and using Eqs. ( 4 ) and ( 6 ) we get
S (bh) ent = 2π 2Φ 0 M L -Φ 0 ln LM + O(1) = S BH -2Φ 0 ln S BH + O(1). (11)
We have obtained the remarkable result that the leading term in the large mass expansion of the black hole entanglement entropy reproduces exactly the Bekenstein-Hawking entropy. Moreover, the subleading term behaves as the logarithm of the BH entropy and describes quantum corrections to S BH . It is an universally accepted result that the quantum corrections to the BH entropy behave as ln S BH [32, 33,
. However, there is no general consensus about the value of the prefactor of this term. For the microcanonical ensemble this term has to be negative, whereas there are positive contributions coming from thermal fluctuation. Equation ( 11 ) fixes the prefactor of ln S BH in terms of the 2D Newton constant. This result contradicts some previous results supporting a Φ 0 -independent value of the prefactor. Our result is consistent with the approach followed in this paper, which considers 2D gravity as induced from the quantum fluctuations of a CFT with central charge 12Φ 0 . The first (Bekenstein-Hawking) term in Eq. ( 11 ) is the induced entanglement entropy, whereas the second term, -(c/6) ln(r h /L), is determined by the conformal symmetry. It gives the entanglement entropy (1) of a CFT in 2D flat spacetime with central charge 12Φ 0 and Σ = r h in the limit Σ << Λ [6] . The subleading term in Eq. ( 11 ) represents therefore an universal behavior shared with other systems described by 2D QFTs, such as one-dimensional statistical models near to the critical point (with the black hole radius r h corresponding to the correlation length) or free scalars fields [7, 9] . Eq. (10) shows a close resemblance with the CFT entanglement entropy (1). Eqs. ( 10 ) and (1) differs in two main points: the absence in the black hole case of something corresponding to the measure of the whole space (the parameter Λ in Eq. ( 1 )) and the appearance of hyperbolic instead of trigonometric functions. These are expected features for the entanglement entropy of a black hole. They solve the problems concerning the application of formula (1) to the black hole case. For a black hole one cannot define a measure of the whole space analogue to Λ. For static solutions the coordinate system covers only the black hole exterior. The appearance of hyperbolic instead of trigonometric functions allows for monotonic increasing of S (bh) ent (r h ), eliminating the unphysical decreasing behavior of S ent (Σ) in the region Σ > Λ/2.
It is interesting to see how Eq. ( 1 ) can be obtained as the analytic continuation r h → ir h of our formula (10), i.e by considering an AdS black hole with negative mass. The analytically continued black hole solution is given by Eq. (3) with a 2 < 0. In the conformal gauge the solution reads now ds 2 = [a 2 / sin 2 (aσ/L)](-dt 2 + dσ 2 ). The range of the spacelike coordinate, corresponding to 0 < r < ∞, is now 0 < σ < πL/2a. Regularizing the solution at σ = 0 by introducing the cutoff ǫ we get the euclidean instanton shown in Fig. (2) . In terms of the 2D CFT we have to trace over the degrees of freedom outside the spacelike slice ǫ < σ < πL/2a. The related entanglement entropy can be calculated using the formula of Ref. [16] in the case of a spacelike slice with two boundary points: S ent = -c/6[ρ(ǫ) + ρ(πL/2a) -ln(δ/Λ)]. Applying this formula to the case of the black hole solution of negative mass, identifying ǫ in terms of the IR cutoff Λ, ǫ = πL 2 /Λ, and redefining appropriately the UV cutoff δ, we get
S ent = c 6 ln Λ πδ sin πr h Λ . (12)
σ σ= πL/2a σ=ε t σ=0 FIG. 2 : Regularized euclidean instanton corresponding to the 2D AdS black hole with negative mass. The euclidean time is periodic. The point σ = πL/2a corresponds to the black hole singularity at r = 0. σ = 0 corresponds to the asymptotic timelike boundary of AdS2.
Thus, the entanglement entropy of the 2D CFT in the curved background given by the AdS black hole of negative mass has exactly the form given by Eq. ( 1 ) with the horizon radius r h playing the role of Σ. Notice that the presence of the factor π in the argument of the sin-function is necessary if one wants invariance under the transformation (2) . The requirement that equation (12) is the analytic continuation of Eq. ( 10 ) fixes, as previously anticipated, the proportionality factor between ǫ and L in the calculations leading to Eq. (10) .
In this letter we have derived a formula for the entanglement entropy of 2D AdS black holes that has nice striking features. The leading term in the large black hole mass expansion reproduces exactly the BH entropy. The subleading term has the right ln S BH , behavior of the quantum corrections to the BH formula and represents an universal term typical of CFTs. Analytic continuation to negative black hole masses give exactly the entanglement entropy of 2D CFT with the black hole radius playing the role of the measure of the observable spacelike slice in the CFT. Our results rely heavily on peculiarities of 2D AdS gravity, namely the existence of an AdS/CFT correspondence and on the fact that 2D Newton constant arises from quantum fluctuation of the dual CFT. The generalization of our approach to higher dimensional gravity theories is therefore far from being trivial. A related problem is the form of the coefficient of the ln S BH term. In the 2D context our result, stating that this coefficient is given in terms of the 2D Newton constant (or equivalently the central charge of the dual CFT) is rather natural. For higher dimensional gravity theories this is again a rather subtle point. I thank G. D'Appollonio for discussions and valuable comments. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Using the AdS/CFT correspondence we derive a formula for the entanglement entropy of the anti-de Sitter black hole in two spacetime dimensions. The leading term in the large black hole mass expansion of our formula reproduces exactly the Bekenstein-Hawking entropy SBH , whereas the subleading term behaves as ln SBH . This subleading term has the universal form typical for the entanglement entropy of physical systems described by effective conformal fields theories (e.g. one-dimensional statistical models at the critical point). The well-known form of the entanglement entropy for a two-dimensional conformal field theory is obtained as analytic continuation of our result and is related with the entanglement entropy of a black hole with negative mass."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Quantum entanglement is a fundamental feature of quantum systems. It is related to the existence of correlations between parts of the system. The degree of entanglement of a quantum system is measured by the entanglement entropy S ent . In quantum field theory (QFT), or more in general in many body systems, we can localize observable and unobservable degrees of freedom in spatially separated regions Q and R. S ent is then defined as the von Neumann entropy of the system when the degrees of freedom in the region R are traced over, S ent = -T r Q ρQ ln ρQ , where the trace is taken over states in the observable region Q and the reduced density matrix ρQ = T r R ρ is obtained by tracing the density matrix ρ over states in the region R.\n\nInvestigation of the entanglement entropy (EE) has become relevant in many research areas. Apart from quantum information theory, the field that gave birth to the notion of entanglement entropy, it plays a crucial role in condensed matter systems, where it helps to understand quantum phases of matter (e.g spin chains and quantum liquids) [1, 2, 3, 4, 5] . Entanglement (geometric) entropy is also an useful concept for investigating general features of QFT, in particular two-dimensional conformal field theory (CFT) and the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence [6, 7, 8, 9, 10, 11, 12] . Last but not least entanglement may held the key for unraveling the mystery of black hole entropy [13, 14, 15, 16, 17, 18, 19, 20, 21, 22] .\n\nWe will be mainly concerned with the entanglement entropy of two-dimensional (2D) CFT and its relationship with the entropy of 2D black holes. It is an old idea that black hole entropy may be explained in terms of the EE of the quantum state of matter fields in the black hole geometry [13] . The main support to this conjecture comes from the fact that both the EE of matter fields and the Bekenstein-Hawking (BH) entropy depend on the area of the boundary region. On the other hand any attempt to explain the BH entropy as originating from quantum entanglement has to solve conceptual and technical difficulties.\n\nThe usual statistical paradigm explains the BH entropy in terms of a microstate gas. This is conceptually different from the EE that measures the observer's lack of information about the quantum state of the system in a inaccessible region of spacetime. Moreover, the EE depends both on the number of species n s of the matter fields, whose entanglement should reproduce the BH entropy, and on the value of the UV cutoff δ arising owing to the presence of a sharp boundary between the accessible and inaccessible regions of the spacetime. Conversely, the BH entropy is meant to be universal, hence independent from n s and δ. Some conceptual difficulties can be solved using Sakharov's induced gravity approach [23, 24, 25] , but the problem of the dependence on n s and δ still remains unsolved.\n\nIn this letter we will show that in the case of two-dimensional AdS black hole these difficulties can be completely solved. We will derive an expression for the black hole EE that in the large black hole mass limit reproduces exactly the BH entropy. Moreover, we will show that the subleading term has the universal behavior typical for CFTs and in particular for critical phenomena. The reason of this success is related to the peculiarities of 2D AdS gravity, namely the existence of an AdS/CFT correspondence and the fact that 2D Newton constant can be considered as wholly induced by quantum fluctuations of the dual CFT.\n\nMost of the progress in understanding the EE in QFT has been achieved in the case of 2D CFT. Conformal invariance in two space-time dimension is a powerful tool that allows us to compute the EE in closed form. The entanglement entropy for the ground state of a 2D CFT originated from tracing over correlations between spacelike separated points has been calculated by Holzhey, Larsen and Wilckzek [6] . Introducing an infrared cutoff Λ the spacelike coordinate of our 2D universe will belong to C = [0, Λ[. The subsystem where measurements are performed is Q = [0, Σ[, whereas the outside region where the degrees of freedom are traced over is R = [Σ, Λ[. Because of the contribution of localized excitations arbitrarily near to the boundary the entanglement entropy diverges. Introducing an ultraviolet cutoff δ, the regularized entanglement entropy turns out to be [6]\n\nS ent = c + c 6 ln Λ δπ sin πΣ Λ , (1)\n\nwhere c and c are the central charges of the 2D CFT. The expression (1) emphasizes the characterizing features of the entanglement entropy, namely subadditivity and invariance under the transformation which exchanges the inside and outside regions\n\nΣ → Λ -Σ. (2)\n\nMoreover, S ent is not a monotonic function of Σ, but increases and reaches its maximum for Σ = Λ/2 and then decreases as Σ increases further. This behavior has an obvious explanation. When the subsystem begins to fill most of the universe there is lesser information to be lost and the entanglement entropy decreases.\n\nLet us now consider 2D AdS black holes. As classical solutions of a 2D gravity theory they are endowed with a non-constant scalar field, the dilaton Φ. In the Schwarzschild gauge the 2D AdS black hole solutions are [26] ,\n\nds 2 = - r 2 L 2 -a 2 dt 2 + r 2 L 2 -a 2 -1 dr 2 , Φ = Φ 0 r L , (3)\n\nwhere the length L is related to cosmological constant of the AdS spacetime (λ = 1/L 2 ), Φ 0 is the dimensionless 2D inverse Newton constant and a is an integration constants related to the black hole mass M and horizon radius r h by\n\na = r h L = 2M L Φ 0 . (4)\n\nThe thermodynamical, Bekenstein-Hawking, entropy of the black hole is [26]\n\nS BH = 2πΦ 0 a = 2π 2Φ 0 M L, (5)\n\nwhereas the black hole temperature is T = a/2πL. Setting a = 0 in Eq. ( 3 ) we have the AdS black hole ground state ( in the following called AdS 0 ) with zero mass, temperature and entropy. The AdS black hole (3) can be considered as the thermalization of the AdS 0 solution at temperature a/2πL [26] .\n\nIt has been shown that the 2D black hole has a dual description in terms of a CFT with central charge [27, 28, 29, 30 ]\n\nc = 12Φ 0 . (6)\n\nThe dual CFT can have both the form of a 2D [29, 30] or a 1D [27, 28] conformal field theory. This AdS 2 /CFT 2 ( or AdS 2 /CFT 1 ) correspondence has been used to give a microscopical meaning to the thermodynamical entropy of 2D AdS black holes. Eq. ( 5 ) has been reproduced by counting states in the dual CFT. In Ref. [16] (see also Refs. [24, 25, 31] ) it was observed that in two dimensions black hole entropy can be ascribed to quantum entanglement if 2D Newton constant is wholly induced by quantum fluctuations of matter fields. On the other hand the AdS 2 /CFT 2 correspondence, and in particular Eq. ( 6 ), tells us that the 2D Newton constant is induced by quantum fluctuations of the dual CFT. It follows that the black hole entropy (5) should be explained as the entanglement of the vacuum of the 2D CFT of central charge given by Eq. ( 6 ) in the gravitational black hole background (3) .\n\nAt first sight one is tempted to use Eq. ( 1 ) to calculate the entanglement entropy of the vacuum of the dual CFT. The exterior region of the 2D black hole can be easily identified with the region Q, whereas the black hole interior has to be identified with the R region where the degrees of freedom are traced over. There are two obstacles that prevents direct application of Eq. (1). First, Eq. ( 1 ) holds for a 2D flat spacetime, whereas we are dealing with a curved 2D background. Second, the calculations leading to Eq. ( 1 ) are performed for spacelike slice Q, whereas in our case the coordinate singularities at r = r h (the horizon) and r = ∞ (the timelike asymptotic boundary of the AdS spacetime) do not allow for a global notion of spacelike coordinate (a coordinate system covering the whole black hole spacetime in which the metric is non-singular and static). Owing to these geometrical features, in the black hole case we cannot give a direct meaning to both the measures Σ and (Λ -Σ) of the subsystems Q, R. As a consequence invariance under the transformation (2) is meaningless in the black hole case.\n\nThe second difficulty can be circumvented using appropriate coordinate system and regularization procedure, the first using instead of Eq. ( 1 ) the formula derived by Fiola et al. [16] , which gives the EE of the vacuum of matter fields in the case of a curved gravitational background. In the coordinate system used to define the vacuum of scalar fields in AdS 2 , the 2D black hole metric (3) is [26]\n\nds 2 = a 2 sinh 2 ( aσ L ) -dt 2 + dσ 2 . (7)\n\nThe coordinate system (t, σ) covers only the black hole exterior. The black hole horizon corresponds to σ = ∞ where the conformal factor of the metric vanishes. The asymptotic r = ∞ timelike conformal boundary of the AdS 2 spacetime is located at σ = 0, where the conformal factor diverges. The entanglement entropy of the CFT vacuum in the curved background (7) can be calculated, using the formula of Ref. [16] as the half line entanglement entropy seen by an observer in the 0 < σ < ∞ region. From the CFT point of view the AdS black hole has to be considered as the AdS 0 vacuum seen by the observer using the black hole coordinates (7) [26] . Moreover, this observer sees the the AdS 0 vacuum as filled with thermal radiation with negative flux [26] . It follows that the black hole entanglement entropy is given by the formula of Ref. [16] with reversed sign,\n\nS (bh) ent = - c 6 ρ(σ = 0) -ln δ Λ , (8)\n\nwhere ρ defines the conformal factor of the metric in the conformal gauge (ds 2 = exp(2ρ)(-dt 2 + dσ 2 )), c is the central charge given by Eq. ( 6 ) and δ, Λ are respectively UV and IR cutoffs. Notice that in Eq. ( 8 ) we have only contributions from only one sector (e.g. right movers) of the CFT. In Ref. [29, 30] it has been shown that the 2D AdS black hole is dual to an open string with appropriate boundary conditions. These boundary conditions are such that only one sector of the CFT 2 is present. The same is obviously true for the AdS 2 /CFT 1 realization of the correspondence [27, 28] . The conformal factor of the metric (7) , hence the entanglement entropy (8) blows up on the σ = 0 boundary of the AdS spacetime. The simplest regularization procedure that solves this problem is to consider a regularized boundary at σ = ǫ. Notice that ǫ plays the role of a UV cutoff for the coordinate σ, which is the natural spacelike coordinate of the dual CFT. ǫ is an IR cutoff for the coordinate r, which is the natural spacelike coordinate for the AdS 2 black hole. The regularized euclidean instanton corresponding to the black hole (7) is shown in figure (1) . The regularizing parameter ǫ can be set equal to the UV cutoff, δ = ǫ. Moreover, the regularized boundary is at finite proper distance from the horizon so that ǫ acts also as IR regulator, making the presence of the IR cutoff Λ in Eq. ( 8 ) redundant. It follows that the regularized EE is given by S (bh) ent = -c 6 ρ(ǫ) -ln ǫ L , which using equations ( 7 ) and ( 4 ) becomes\n\nS (bh) ent = c 6 ln L 2 r h ǫ sinh ǫr h L 2 . ( 9\n\n)\n\nAs a check of the validity of our formula we note that in the case of AdS 0 (r h = 0) the entanglement entropy vanishes. The AdS/CFT correspondence enable us to identify the cutoff ǫ as the UV cutoff of the CFT : ǫ ∝ L. The proportionality factor can be determined by requiring that the analytical continuation of Eq. ( 9 ) is invariant under the transformation (2) (see later). This requirement fixes ǫ = πL. With this position we get\n\nS (bh) ent = c 6 ln L πr h sinh πr h L . ( 10\n\n)\n\nThis formula is our main result, it gives the entanglement entropy of the 2D AdS black hole. This entanglement entropy has the expected behavior as a function of the horizon radius r h or, equivalently, of the black hole mass M . S\n\nent becomes zero in the AdS 0 ground state, r h = 0 (M = 0), whereas it grows monotonically for r h > 0 (M > 0). In order to compare the black hole EE (10) with the BH entropy (5) let us consider the limit of macroscopic black holes, that is the limit a → ∞ or equivalently r h >> L or also M >> 1/L. Expanding Eq. ( 10 ) and using Eqs. ( 4 ) and ( 6 ) we get\n\nS (bh) ent = 2π 2Φ 0 M L -Φ 0 ln LM + O(1) = S BH -2Φ 0 ln S BH + O(1). (11)\n\nWe have obtained the remarkable result that the leading term in the large mass expansion of the black hole entanglement entropy reproduces exactly the Bekenstein-Hawking entropy. Moreover, the subleading term behaves as the logarithm of the BH entropy and describes quantum corrections to S BH . It is an universally accepted result that the quantum corrections to the BH entropy behave as ln S BH [32, 33,\n\n. However, there is no general consensus about the value of the prefactor of this term. For the microcanonical ensemble this term has to be negative, whereas there are positive contributions coming from thermal fluctuation. Equation ( 11 ) fixes the prefactor of ln S BH in terms of the 2D Newton constant. This result contradicts some previous results supporting a Φ 0 -independent value of the prefactor. Our result is consistent with the approach followed in this paper, which considers 2D gravity as induced from the quantum fluctuations of a CFT with central charge 12Φ 0 . The first (Bekenstein-Hawking) term in Eq. ( 11 ) is the induced entanglement entropy, whereas the second term, -(c/6) ln(r h /L), is determined by the conformal symmetry. It gives the entanglement entropy (1) of a CFT in 2D flat spacetime with central charge 12Φ 0 and Σ = r h in the limit Σ << Λ [6] . The subleading term in Eq. ( 11 ) represents therefore an universal behavior shared with other systems described by 2D QFTs, such as one-dimensional statistical models near to the critical point (with the black hole radius r h corresponding to the correlation length) or free scalars fields [7, 9] . Eq. (10) shows a close resemblance with the CFT entanglement entropy (1). Eqs. ( 10 ) and (1) differs in two main points: the absence in the black hole case of something corresponding to the measure of the whole space (the parameter Λ in Eq. ( 1 )) and the appearance of hyperbolic instead of trigonometric functions. These are expected features for the entanglement entropy of a black hole. They solve the problems concerning the application of formula (1) to the black hole case. For a black hole one cannot define a measure of the whole space analogue to Λ. For static solutions the coordinate system covers only the black hole exterior. The appearance of hyperbolic instead of trigonometric functions allows for monotonic increasing of S (bh) ent (r h ), eliminating the unphysical decreasing behavior of S ent (Σ) in the region Σ > Λ/2.\n\nIt is interesting to see how Eq. ( 1 ) can be obtained as the analytic continuation r h → ir h of our formula (10), i.e by considering an AdS black hole with negative mass. The analytically continued black hole solution is given by Eq. (3) with a 2 < 0. In the conformal gauge the solution reads now ds 2 = [a 2 / sin 2 (aσ/L)](-dt 2 + dσ 2 ). The range of the spacelike coordinate, corresponding to 0 < r < ∞, is now 0 < σ < πL/2a. Regularizing the solution at σ = 0 by introducing the cutoff ǫ we get the euclidean instanton shown in Fig. (2) . In terms of the 2D CFT we have to trace over the degrees of freedom outside the spacelike slice ǫ < σ < πL/2a. The related entanglement entropy can be calculated using the formula of Ref. [16] in the case of a spacelike slice with two boundary points: S ent = -c/6[ρ(ǫ) + ρ(πL/2a) -ln(δ/Λ)]. Applying this formula to the case of the black hole solution of negative mass, identifying ǫ in terms of the IR cutoff Λ, ǫ = πL 2 /Λ, and redefining appropriately the UV cutoff δ, we get\n\nS ent = c 6 ln Λ πδ sin πr h Λ . (12)\n\nσ σ= πL/2a σ=ε t σ=0 FIG. 2 : Regularized euclidean instanton corresponding to the 2D AdS black hole with negative mass. The euclidean time is periodic. The point σ = πL/2a corresponds to the black hole singularity at r = 0. σ = 0 corresponds to the asymptotic timelike boundary of AdS2.\n\nThus, the entanglement entropy of the 2D CFT in the curved background given by the AdS black hole of negative mass has exactly the form given by Eq. ( 1 ) with the horizon radius r h playing the role of Σ. Notice that the presence of the factor π in the argument of the sin-function is necessary if one wants invariance under the transformation (2) . The requirement that equation (12) is the analytic continuation of Eq. ( 10 ) fixes, as previously anticipated, the proportionality factor between ǫ and L in the calculations leading to Eq. (10) .\n\nIn this letter we have derived a formula for the entanglement entropy of 2D AdS black holes that has nice striking features. The leading term in the large black hole mass expansion reproduces exactly the BH entropy. The subleading term has the right ln S BH , behavior of the quantum corrections to the BH formula and represents an universal term typical of CFTs. Analytic continuation to negative black hole masses give exactly the entanglement entropy of 2D CFT with the black hole radius playing the role of the measure of the observable spacelike slice in the CFT. Our results rely heavily on peculiarities of 2D AdS gravity, namely the existence of an AdS/CFT correspondence and on the fact that 2D Newton constant arises from quantum fluctuation of the dual CFT. The generalization of our approach to higher dimensional gravity theories is therefore far from being trivial. A related problem is the form of the coefficient of the ln S BH term. In the 2D context our result, stating that this coefficient is given in terms of the 2D Newton constant (or equivalently the central charge of the dual CFT) is rather natural. For higher dimensional gravity theories this is again a rather subtle point. I thank G. D'Appollonio for discussions and valuable comments."
}
] | |
arxiv:0704.0144 | 0704.0144 | 1 | 10.1088/1475-7516/2009/02/007 | 502f365bbcbc68e856ad4bde1d13bfe2bd6767db20ac4b4d59c7d8c9f51fa3d9 | Eternal inflation and localization on the landscape | We model the essential features of eternal inflation on the landscape of a
dense discretuum of vacua by the potential $V(\phi)=V_{0}+\delta V(\phi)$,
where $|\delta V(\phi)|\ll V_{0}$ is random. We find that the diffusion of the
distribution function $\rho(\phi,t)$ of the inflaton expectation value in
different Hubble patches may be suppressed due to the effect analogous to the
Anderson localization in disordered quantum systems. At $t \to \infty$ only the
localized part of the distribution function $\rho (\phi, t)$ survives which
leads to dynamical selection principle on the landscape. The probability to
measure any but a small value of the cosmological constant in a given Hubble
patch on the landscape is exponentially suppressed at $t\to \infty$. | [
"D. Podolsky",
"K. Enqvist"
] | [
"hep-th",
"astro-ph",
"gr-qc"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | We model the essential features of eternal inflation on the landscape of a dense discretuum of vacua by the potential V (φ) = V0 + δV (φ), where |δV (φ)| ≪ V0 is random. We find that the diffusion of the distribution function ρ(φ, t) of the inflaton expectation value in different Hubble patches may be suppressed due to the effect analogous to the Anderson localization in disordered quantum systems. At t → ∞ only the localized part of the distribution function ρ(φ, t) survives which leads to dynamical selection principle on the landscape. The probability to measure any but a small value of the cosmological constant in a given Hubble patch on the landscape is exponentially suppressed at t → ∞.
String theory is believed to imply a wide landscape [1] of both metastable vacua with a positive cosmological constant and true vacua with a vanishing or a negative cosmological constant; the latter are called anti-de Sitter or AdS vacua, where space-time collapses into a singularity. In regions with positive cosmological constant, or in de Sitter (dS) vacua, the universe inflates, and because of the possibility of tunneling between different de Sitter vacua inflation is eternal.
The problem of calculating statistical distributions of the landscape vacua is very complicated [2] and is even considered to be NP-hard [3] (the total number of vacua on the landscape is estimated to be of order 10 100 ÷ 10 1000 ). Our aim is to consider how eternal inflation proceeds on the landscape by using the mere fact that the number of vacua within the landscape is extremely large, so that their distribution can have significant disorder. The dynamics of eternal inflation is then described by the Fokker-Planck equations in the disordered effective potential. 1 In that case, the landscape dynamics may have some interesting parallels in solid state physics, as we will discuss in the present paper.
Eternal inflation on the landscape can be modeled as follows [5, 6] . Let us numerate vacua on the landscape by the discrete index i and define P i (t) as the probability to measure a given (positive) value of the cosmological constant Λ i in a given Hubble patch. If the rates of tunneling between the metastable minima i and j on the landscape are given by the time independent matrix Γ ij , then the probabilities P i satisfy the system of "vacuum dynamics" equations [7] Ṗi =
j =i (Γ ji P j -Γ ij P i ) -Γ is P i . (1)
The last term in this equation corresponds to tunneling * On leave from Landau Institute for Theoretical Physics, 119940, Moscow, Russia. 1 An approach somewhat similar to ours was also presented in [4] .
between the metastable de Sitter vacuum i and a true vacuum with a negative cosmological constant (an AdS vacuum), i.e. tunneling into a collapsing AdS space-time [8] . The collapse time
t col ∼ M P /V 1/2
AdS is much shorter than the characteristic time t rec ∼ exp M 4 P /V dS for tunneling back into a de Sitter metastable vacuum, so that the AdS true vacua effectively play the role of sinks for the probability current (1) describing eternal inflation on the landscape [5] .
In what follows we will assume that the effect of the AdS sinks is relatively small; otherwise the landscape will be divided into almost unconnected "islands" of vacua [6] , preventing the population of the whole landscape by eternal inflation.
In the limit of weak tunneling only the vacua closest to each other are important. It is convenient to classify parts (islands) of the landscape according to the typical number of adjacent vacua within each part. Technically, the landscape of vacua of the string theory can be represented as a graph with 10 100 ÷10 1000 nodes and a number of connections between them of the same order. By an island on the landscape, we mean a subgraph relatively weakly connected to the major "tree". The dimensionality of the island can then be defined as the Hausdorff dimension N H of the corresponding subgraph [17] . For example, if there are only two adjacent vacua for any vacuum in a given island, then N H = 1 for this island and we denote it as quasi-one-dimensional; a domain of vacua with N H = 2 is quasi-two-dimensional, and so on.
In the quasi-one-dimensional case (neglecting the AdS sinks) the system (1) reduces to Ṗi = -Γ i,i+1
P i + Γ i+1,i P i+1 -Γ i,i-1 P i + Γ i-1,i P i-1 . (2)
While in general Γ ij = Γ ji , we will take Γ ij = Γ ji on the average. 2 Furthermore, suppose that the initial condition for Eq. ( 2 ) is P i (0) = 1, P j =i (0) = 0.
(3) so that the initial state is well localized. Naively, one may expect that the distribution function P i (t) would start to spread out according to the usual diffusion law and the system of vacua would exponentially quickly reach a "thermal" equilibrium distribution of probabilities for a given Hubble patch to be in a given dS vacuum. However, there exists a well known theorem [10] from the theory of diffusion on random lattices stating that the distribution function P i remains localized near the initial distribution peak for a very long time, with its characteristic width behaving as
i 2 (t) ∼ log 4 t . (4)
This is a surprising result when applied to eternal inflation where the general lore (see for example [11] ) is that the initial conditions for eternal inflation will be forgotten almost immediately after its beginning. Instead, in what follows we will argue that the memory about the initial conditions may survive during a very long time on the quasi-one-dimensional islands of the landscape.
We will model the landscape by a continuous inflaton potential
(φ) = V 0 + δV (φ), (5)
where V 0 is constant, and δV (φ) is a random contribution such that |δV (φ)| ≪ V 0 , and φ is the inflaton or the order parameter describing the transitions. As in stochastic inflation [16] , in different causally connected regions fluctuations have a randomly distributed amplitude and observers living in different Hubble patches see different expectation values of the inflaton. When stochastic fluctuations of the inflaton are large enough, the expectation value of the inflaton in a given Hubble patch is determined by the Langevin equation [16] φ = -
1 3H 0 ∂δV ∂φ + f (t), (6)
where the stochastic force f (φ, t) is Gaussian with correlation properties
f (t)f (t ′ ) = H 3 0 4π 2 δ(t -t ′ ). (7)
From ( 6 ) one can derive the Fokker-Planck equation, which controls the evolution of the probability distribution ρ(φ, t) describing how the values of φ are distributed of the effective cosmological constant. However, the spectrum of states on Bousso-Polchinski landscape is not disordered, so that the analysis based on averaging over disorder is not applicable. Disorder appears in more realistic multithroat models of the string theory landscape.
among different Hubble patches in the multiverse. One finds [16] ∂ρ(φ, t)
∂t = H 3 0 8π 2 ∂ 2 ρ ∂φ 2 + 1 3H 0 ∂ ∂φ ∂V ∂φ ρ . (8)
The general solution to Eq. ( 8 ) is given by
ρ = e -4π 2 δV (φ) 3H 4 0 n c n ψ n (φ)e -EnH 3 0 (t-t 0 ) 4π 2 , ( 9
)
where ψ n and E n are respectively the eigenfunctions and the eigenvalues of the effective Hamiltonian Ĥ = -1 2
∂ 2 ∂φ 2 + W (φ). ( 10
)
Here
W (φ) = 8π 4 9H 8 0 ∂δV ∂φ 2 - 2π 2 3H 4 0 ∂ 2 δV ∂φ 2 (11)
is a functional of the scalar field potential V (φ). It is often denoted as the superpotential due to its "supersymmetric" form: the Hamiltonian ( 10 ) can be rewrit-
ten as Ĥ = Q † Q, where Q = -∂/∂φ + v ′ (φ) with v(φ) = 4π 2 δV (φ)/(3H 4 0
). The eigenfunctions of the Hamiltonian (10) satisfy the Schrödinger equation 1 2
∂ 2 ψ n ∂φ 2 + (E n -W (φ))ψ n = 0, ( 12
)
and its solutions have the following well known features [16] :
1. The eigenvalues of the Hamiltonian ( 10 ) are all positive definite.
2. The contributions from eigenfunctions of excited states ψ n>0 (φ) to the solution Eq. ( 9 ) become exponentially quickly damped with time. However, if one is interested in what happens at time scales ∆t 1/E n , the first n eigenfunctions should be taken into account. In particular, if the spectrum of the Hamiltonian ( 10 ) is very dense, as in the case of the string theory landscape, knowing the ground state alone is not enough for complete understanding dynamics of eternal inflation.
We now recall that the potential V (φ) is a random function of the inflaton field and has extremely large number of minima. This allows us to draw several conclusions about the form of the eigenfunctions ψ n (φ) using the formal analogy between Eq. ( 12 ) and the time-independent Schrödinger equation describing the motion of carriers in disordered quantum systems such as semiconductors with impurities. The physical quantities in disordered systems can be calculated by averaging over the random potential of the impurities. 3 A famous consequence of the random potential generated by impurities in crystalline materials is the strong suppression of the conductivity, known as Anderson localization [12, 13] . This effect is essential in dimensions lower than 3 and completely defines the kinetics of carriers in one-dimensional systems. There, impurities create a random potential for Bloch waves with the correlation properties
u(r)u(r ′ ) = 1 ντ δ(r -r ′ ), u(r) = 0, ( 13
)
where τ is the mean free path for electrons and ν is the density of states per one spin degree of freedom of the electron gas at the Fermi surface. As a consequence, in the one-dimensional case all eigenstates of the electron hamiltonian become localized with
ψ n (r) ∼ exp - |r -r n | L ( 14
)
at t → ∞, where r n are the positions of localization centers, and the localization length L is of the order of the mean free path l τ = v τ . As a result, the probability density ρ(R, t) to find electron at the point R at time t asymptotically approaches the limit ρ(R)
∼ exp(-R/L) for R ≫ L, or ρ(R) ∼ Const for R ≪ L at t → ∞.
The one-dimensional Anderson localization takes place for an arbitrarily weak disorder and arbitrary correlation properties of the random potential u(r) [13] . Also, in a two-dimensional case all the electron eigenstates in a random potential remain localized. However, the localization length grows exponentially with energy, the rate of growth being related to the strength of the disorder. In three-dimensional case, the localization properties of eigenstates are defined by the Ioffe-Regel-Mott criterion: if the corresponding eigenvalue of the Hamiltonian of electrons E n satisfies the condition E n < E g where E g is so called mobility edge, then the eigenstate is localized. The mobility edge E g is a function of the strength of the disorder. In higher dimensional cases the situation is unknown.
Let us now return to the discussion of eternal inflation described by the Fokker-Planck equation (8) . Since the localization is the property of the eigenfunctions of the time-independent hamiltonian (10) , it is also a natural consequence of the effective randomness of the potential of the string theory landscape. 4 The diffusion of the probability distribution ( 4 ) is suppressed due to the localization of the eigenfunctions ψ n (φ) contributing to the overall solution (9) . This counteracts the general wisdom that eternal inflation rapidly washes out any information of the initial conditions. Indeed,in the quasione-dimensional case all the wave functions ψ n (φ) are localized, i.e., for a particular realization of disorder they behave as
ψ n (φ) ∼ exp - |φ -φ n | L . ( 15
)
where φ n define the "localization centers" as in the Eq. ( 14 ), and L is the localization length which is of the same order of magnitude as the "mean free path" related to the strength of the disorder in the superpotential W (φ).
Let us now discuss how eternal inflation proceeds on islands where the typical number of adjacent vacua is larger than two. In the quasi-two-dimensional case the network of vacua within a given island is described by a composite index i = (i, j). The distribution function ρ for finding a given value of the cosmological constant in a given Hubble patch is a two-dimensional matrix. Again, all the eigenstates of the corresponding tunneling hamiltonian Ĥ are localized. However, since the localization length grows exponentially with energy, the distribution function effectively spreads out almost linearly with
i 2 (t) ∼ t 1 + c 1 1 log α t + • • • , (16)
where α > 0 are constants depending on the correlation properties of the disorder on the landscape [18] . The low energy eigenstates (namely, the states with E < E g where E g is the mobility edge) are localized with a relatively small localization length.
In the quasi-higher-dimensional cases the distribution function spreads out according to the linear diffusion law at intermediate times. Again, there exists a mobility edge E g such that the eigenstates of the tunneling Hamiltonian with energies E < E g are localized. These low energy eigenstates define the asymptotics of the distribution function ρ at
t ≫ E -1 g . ( 17
)
The value of the mobility edge E g strongly depends on the dimensionality of the island and the strength of the disorder, and the higher is the dimensionality, the lower is the mobility edge [17] . Localization of the low energy eigenstates in two-and higher-dimensional cases introduces an effective dynamical selection principle for different vacua on the landscape (5): in the asymptotic future, not all of them will be populated, but only those near the localization centers φ n , and the probability to populate other minima will be suppressed exponentially according to the Eq. (15) .
It is interesting to note that in condensed matter systems the localization centers are typically located near the points where the effective potential has its deepest minima [13] . In the case of eternal inflation, it means that the probability to measure any but very low value of the cosmological constant in a given Hubble patch will be exponentially suppressed in the asymptotic future [17] .
Finally, we discuss the effect of sinks on the dynamics of tunneling between the vacua. On the string theory landscape, dS metastable vacua are typically realized by uplifting stable AdS vacua (as, for example, in the well known KKLT model [19] ). The probability to tunnel from the uplifted dS state i back into the AdS vacuum is related to the value of gravitino mass m 3/2 in the dS state [8] and given by
t AdS ∼ Γ -1 is ∼ exp Const.M 2 P m 2 3/2,i . (18)
The gravitino mass after uplifting [20] has the order of magnitude m 3/2,i ∼ |V AdS,i | 1/2 /M P . Since at long time scales V AdS,i can also be regarded as a random quantity, our analysis of the general solution of "vacuum dynamics" equations (1) does not have to be modified in any essential way [17] .
In addition to AdS sinks, Hubble patches where eternal inflation has ended (stochastic fluctuations of the inflaton expectation value became smaller than the effect of classical force) also effectively play a role of sinks for the probability current described by the Eq. ( 8 ). In particular, the Hubble patch we live in is one of such sinks. Related to the effect of sinks, there exists a time scale t end for eternal inflation on the landscape (5) such that the unitarity of the evolution of the probability distribution ρ breaks down at t ≫ t end [17] . Our discussion remains valid if t ≪ t end . It is unclear whether the probability distribution ρ has achieved the late time asymptotics in the corner of the landscape we live in.
In summary, we have argued that eternal inflation on the landscape may lead to a strong localization of the inflaton distribution function among different Hubble patches. This is a consequence of the high density of the vacua, which effectively implies a random potential for the order parameter responsible for inflation. We found that the inflaton motion is analogous to the motion of carriers in disordered quantum systems, and there exists an analogue of the Anderson localization for eternal inflation on the landscape. Physically, this means that not all the vacua on the landscape are populated by eternal inflation in the asymptotic future, but only those near the localization centers of the inflaton effective potential. They are located near the deepest minima of the potential, which implies that the probability to measure any but very low value of the cosmological constant in a given Hubble patch is exponentially suppressed at late times. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We model the essential features of eternal inflation on the landscape of a dense discretuum of vacua by the potential V (φ) = V0 + δV (φ), where |δV (φ)| ≪ V0 is random. We find that the diffusion of the distribution function ρ(φ, t) of the inflaton expectation value in different Hubble patches may be suppressed due to the effect analogous to the Anderson localization in disordered quantum systems. At t → ∞ only the localized part of the distribution function ρ(φ, t) survives which leads to dynamical selection principle on the landscape. The probability to measure any but a small value of the cosmological constant in a given Hubble patch on the landscape is exponentially suppressed at t → ∞."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "String theory is believed to imply a wide landscape [1] of both metastable vacua with a positive cosmological constant and true vacua with a vanishing or a negative cosmological constant; the latter are called anti-de Sitter or AdS vacua, where space-time collapses into a singularity. In regions with positive cosmological constant, or in de Sitter (dS) vacua, the universe inflates, and because of the possibility of tunneling between different de Sitter vacua inflation is eternal.\n\nThe problem of calculating statistical distributions of the landscape vacua is very complicated [2] and is even considered to be NP-hard [3] (the total number of vacua on the landscape is estimated to be of order 10 100 ÷ 10 1000 ). Our aim is to consider how eternal inflation proceeds on the landscape by using the mere fact that the number of vacua within the landscape is extremely large, so that their distribution can have significant disorder. The dynamics of eternal inflation is then described by the Fokker-Planck equations in the disordered effective potential. 1 In that case, the landscape dynamics may have some interesting parallels in solid state physics, as we will discuss in the present paper.\n\nEternal inflation on the landscape can be modeled as follows [5, 6] . Let us numerate vacua on the landscape by the discrete index i and define P i (t) as the probability to measure a given (positive) value of the cosmological constant Λ i in a given Hubble patch. If the rates of tunneling between the metastable minima i and j on the landscape are given by the time independent matrix Γ ij , then the probabilities P i satisfy the system of \"vacuum dynamics\" equations [7] Ṗi =\n\nj =i (Γ ji P j -Γ ij P i ) -Γ is P i . (1)\n\nThe last term in this equation corresponds to tunneling * On leave from Landau Institute for Theoretical Physics, 119940, Moscow, Russia. 1 An approach somewhat similar to ours was also presented in [4] .\n\nbetween the metastable de Sitter vacuum i and a true vacuum with a negative cosmological constant (an AdS vacuum), i.e. tunneling into a collapsing AdS space-time [8] . The collapse time\n\nt col ∼ M P /V 1/2\n\nAdS is much shorter than the characteristic time t rec ∼ exp M 4 P /V dS for tunneling back into a de Sitter metastable vacuum, so that the AdS true vacua effectively play the role of sinks for the probability current (1) describing eternal inflation on the landscape [5] .\n\nIn what follows we will assume that the effect of the AdS sinks is relatively small; otherwise the landscape will be divided into almost unconnected \"islands\" of vacua [6] , preventing the population of the whole landscape by eternal inflation.\n\nIn the limit of weak tunneling only the vacua closest to each other are important. It is convenient to classify parts (islands) of the landscape according to the typical number of adjacent vacua within each part. Technically, the landscape of vacua of the string theory can be represented as a graph with 10 100 ÷10 1000 nodes and a number of connections between them of the same order. By an island on the landscape, we mean a subgraph relatively weakly connected to the major \"tree\". The dimensionality of the island can then be defined as the Hausdorff dimension N H of the corresponding subgraph [17] . For example, if there are only two adjacent vacua for any vacuum in a given island, then N H = 1 for this island and we denote it as quasi-one-dimensional; a domain of vacua with N H = 2 is quasi-two-dimensional, and so on.\n\nIn the quasi-one-dimensional case (neglecting the AdS sinks) the system (1) reduces to Ṗi = -Γ i,i+1\n\nP i + Γ i+1,i P i+1 -Γ i,i-1 P i + Γ i-1,i P i-1 . (2)\n\nWhile in general Γ ij = Γ ji , we will take Γ ij = Γ ji on the average. 2 Furthermore, suppose that the initial condition for Eq. ( 2 ) is P i (0) = 1, P j =i (0) = 0.\n\n(3) so that the initial state is well localized. Naively, one may expect that the distribution function P i (t) would start to spread out according to the usual diffusion law and the system of vacua would exponentially quickly reach a \"thermal\" equilibrium distribution of probabilities for a given Hubble patch to be in a given dS vacuum. However, there exists a well known theorem [10] from the theory of diffusion on random lattices stating that the distribution function P i remains localized near the initial distribution peak for a very long time, with its characteristic width behaving as\n\ni 2 (t) ∼ log 4 t . (4)\n\nThis is a surprising result when applied to eternal inflation where the general lore (see for example [11] ) is that the initial conditions for eternal inflation will be forgotten almost immediately after its beginning. Instead, in what follows we will argue that the memory about the initial conditions may survive during a very long time on the quasi-one-dimensional islands of the landscape.\n\nWe will model the landscape by a continuous inflaton potential\n\n(φ) = V 0 + δV (φ), (5)\n\nwhere V 0 is constant, and δV (φ) is a random contribution such that |δV (φ)| ≪ V 0 , and φ is the inflaton or the order parameter describing the transitions. As in stochastic inflation [16] , in different causally connected regions fluctuations have a randomly distributed amplitude and observers living in different Hubble patches see different expectation values of the inflaton. When stochastic fluctuations of the inflaton are large enough, the expectation value of the inflaton in a given Hubble patch is determined by the Langevin equation [16] φ = -\n\n1 3H 0 ∂δV ∂φ + f (t), (6)\n\nwhere the stochastic force f (φ, t) is Gaussian with correlation properties\n\nf (t)f (t ′ ) = H 3 0 4π 2 δ(t -t ′ ). (7)\n\nFrom ( 6 ) one can derive the Fokker-Planck equation, which controls the evolution of the probability distribution ρ(φ, t) describing how the values of φ are distributed of the effective cosmological constant. However, the spectrum of states on Bousso-Polchinski landscape is not disordered, so that the analysis based on averaging over disorder is not applicable. Disorder appears in more realistic multithroat models of the string theory landscape.\n\namong different Hubble patches in the multiverse. One finds [16] ∂ρ(φ, t)\n\n∂t = H 3 0 8π 2 ∂ 2 ρ ∂φ 2 + 1 3H 0 ∂ ∂φ ∂V ∂φ ρ . (8)\n\nThe general solution to Eq. ( 8 ) is given by\n\nρ = e -4π 2 δV (φ) 3H 4 0 n c n ψ n (φ)e -EnH 3 0 (t-t 0 ) 4π 2 , ( 9\n\n)\n\nwhere ψ n and E n are respectively the eigenfunctions and the eigenvalues of the effective Hamiltonian Ĥ = -1 2\n\n∂ 2 ∂φ 2 + W (φ). ( 10\n\n)\n\nHere\n\nW (φ) = 8π 4 9H 8 0 ∂δV ∂φ 2 - 2π 2 3H 4 0 ∂ 2 δV ∂φ 2 (11)\n\nis a functional of the scalar field potential V (φ). It is often denoted as the superpotential due to its \"supersymmetric\" form: the Hamiltonian ( 10 ) can be rewrit-\n\nten as Ĥ = Q † Q, where Q = -∂/∂φ + v ′ (φ) with v(φ) = 4π 2 δV (φ)/(3H 4 0\n\n). The eigenfunctions of the Hamiltonian (10) satisfy the Schrödinger equation 1 2\n\n∂ 2 ψ n ∂φ 2 + (E n -W (φ))ψ n = 0, ( 12\n\n)\n\nand its solutions have the following well known features [16] :\n\n1. The eigenvalues of the Hamiltonian ( 10 ) are all positive definite.\n\n2. The contributions from eigenfunctions of excited states ψ n>0 (φ) to the solution Eq. ( 9 ) become exponentially quickly damped with time. However, if one is interested in what happens at time scales ∆t 1/E n , the first n eigenfunctions should be taken into account. In particular, if the spectrum of the Hamiltonian ( 10 ) is very dense, as in the case of the string theory landscape, knowing the ground state alone is not enough for complete understanding dynamics of eternal inflation.\n\nWe now recall that the potential V (φ) is a random function of the inflaton field and has extremely large number of minima. This allows us to draw several conclusions about the form of the eigenfunctions ψ n (φ) using the formal analogy between Eq. ( 12 ) and the time-independent Schrödinger equation describing the motion of carriers in disordered quantum systems such as semiconductors with impurities. The physical quantities in disordered systems can be calculated by averaging over the random potential of the impurities. 3 A famous consequence of the random potential generated by impurities in crystalline materials is the strong suppression of the conductivity, known as Anderson localization [12, 13] . This effect is essential in dimensions lower than 3 and completely defines the kinetics of carriers in one-dimensional systems. There, impurities create a random potential for Bloch waves with the correlation properties\n\nu(r)u(r ′ ) = 1 ντ δ(r -r ′ ), u(r) = 0, ( 13\n\n)\n\nwhere τ is the mean free path for electrons and ν is the density of states per one spin degree of freedom of the electron gas at the Fermi surface. As a consequence, in the one-dimensional case all eigenstates of the electron hamiltonian become localized with\n\nψ n (r) ∼ exp - |r -r n | L ( 14\n\n)\n\nat t → ∞, where r n are the positions of localization centers, and the localization length L is of the order of the mean free path l τ = v τ . As a result, the probability density ρ(R, t) to find electron at the point R at time t asymptotically approaches the limit ρ(R)\n\n∼ exp(-R/L) for R ≫ L, or ρ(R) ∼ Const for R ≪ L at t → ∞.\n\nThe one-dimensional Anderson localization takes place for an arbitrarily weak disorder and arbitrary correlation properties of the random potential u(r) [13] . Also, in a two-dimensional case all the electron eigenstates in a random potential remain localized. However, the localization length grows exponentially with energy, the rate of growth being related to the strength of the disorder. In three-dimensional case, the localization properties of eigenstates are defined by the Ioffe-Regel-Mott criterion: if the corresponding eigenvalue of the Hamiltonian of electrons E n satisfies the condition E n < E g where E g is so called mobility edge, then the eigenstate is localized. The mobility edge E g is a function of the strength of the disorder. In higher dimensional cases the situation is unknown.\n\nLet us now return to the discussion of eternal inflation described by the Fokker-Planck equation (8) . Since the localization is the property of the eigenfunctions of the time-independent hamiltonian (10) , it is also a natural consequence of the effective randomness of the potential of the string theory landscape. 4 The diffusion of the probability distribution ( 4 ) is suppressed due to the localization of the eigenfunctions ψ n (φ) contributing to the overall solution (9) . This counteracts the general wisdom that eternal inflation rapidly washes out any information of the initial conditions. Indeed,in the quasione-dimensional case all the wave functions ψ n (φ) are localized, i.e., for a particular realization of disorder they behave as\n\nψ n (φ) ∼ exp - |φ -φ n | L . ( 15\n\n)\n\nwhere φ n define the \"localization centers\" as in the Eq. ( 14 ), and L is the localization length which is of the same order of magnitude as the \"mean free path\" related to the strength of the disorder in the superpotential W (φ).\n\nLet us now discuss how eternal inflation proceeds on islands where the typical number of adjacent vacua is larger than two. In the quasi-two-dimensional case the network of vacua within a given island is described by a composite index i = (i, j). The distribution function ρ for finding a given value of the cosmological constant in a given Hubble patch is a two-dimensional matrix. Again, all the eigenstates of the corresponding tunneling hamiltonian Ĥ are localized. However, since the localization length grows exponentially with energy, the distribution function effectively spreads out almost linearly with\n\ni 2 (t) ∼ t 1 + c 1 1 log α t + • • • , (16)\n\nwhere α > 0 are constants depending on the correlation properties of the disorder on the landscape [18] . The low energy eigenstates (namely, the states with E < E g where E g is the mobility edge) are localized with a relatively small localization length.\n\nIn the quasi-higher-dimensional cases the distribution function spreads out according to the linear diffusion law at intermediate times. Again, there exists a mobility edge E g such that the eigenstates of the tunneling Hamiltonian with energies E < E g are localized. These low energy eigenstates define the asymptotics of the distribution function ρ at\n\nt ≫ E -1 g . ( 17\n\n)\n\nThe value of the mobility edge E g strongly depends on the dimensionality of the island and the strength of the disorder, and the higher is the dimensionality, the lower is the mobility edge [17] . Localization of the low energy eigenstates in two-and higher-dimensional cases introduces an effective dynamical selection principle for different vacua on the landscape (5): in the asymptotic future, not all of them will be populated, but only those near the localization centers φ n , and the probability to populate other minima will be suppressed exponentially according to the Eq. (15) .\n\nIt is interesting to note that in condensed matter systems the localization centers are typically located near the points where the effective potential has its deepest minima [13] . In the case of eternal inflation, it means that the probability to measure any but very low value of the cosmological constant in a given Hubble patch will be exponentially suppressed in the asymptotic future [17] .\n\nFinally, we discuss the effect of sinks on the dynamics of tunneling between the vacua. On the string theory landscape, dS metastable vacua are typically realized by uplifting stable AdS vacua (as, for example, in the well known KKLT model [19] ). The probability to tunnel from the uplifted dS state i back into the AdS vacuum is related to the value of gravitino mass m 3/2 in the dS state [8] and given by\n\nt AdS ∼ Γ -1 is ∼ exp Const.M 2 P m 2 3/2,i . (18)\n\nThe gravitino mass after uplifting [20] has the order of magnitude m 3/2,i ∼ |V AdS,i | 1/2 /M P . Since at long time scales V AdS,i can also be regarded as a random quantity, our analysis of the general solution of \"vacuum dynamics\" equations (1) does not have to be modified in any essential way [17] .\n\nIn addition to AdS sinks, Hubble patches where eternal inflation has ended (stochastic fluctuations of the inflaton expectation value became smaller than the effect of classical force) also effectively play a role of sinks for the probability current described by the Eq. ( 8 ). In particular, the Hubble patch we live in is one of such sinks. Related to the effect of sinks, there exists a time scale t end for eternal inflation on the landscape (5) such that the unitarity of the evolution of the probability distribution ρ breaks down at t ≫ t end [17] . Our discussion remains valid if t ≪ t end . It is unclear whether the probability distribution ρ has achieved the late time asymptotics in the corner of the landscape we live in.\n\nIn summary, we have argued that eternal inflation on the landscape may lead to a strong localization of the inflaton distribution function among different Hubble patches. This is a consequence of the high density of the vacua, which effectively implies a random potential for the order parameter responsible for inflation. We found that the inflaton motion is analogous to the motion of carriers in disordered quantum systems, and there exists an analogue of the Anderson localization for eternal inflation on the landscape. Physically, this means that not all the vacua on the landscape are populated by eternal inflation in the asymptotic future, but only those near the localization centers of the inflaton effective potential. They are located near the deepest minima of the potential, which implies that the probability to measure any but very low value of the cosmological constant in a given Hubble patch is exponentially suppressed at late times."
}
] | |
arxiv:0704.0145 | 0704.0145 | 1 | bb697fd8f846fc1c08d49ce99684bc4e78b602b4ac1a88ae22206ad50b4209aa | Singularity Resolution in Isotropic Loop Quantum Cosmology: Recent
Developments | Since the past Iagrg meeting in December 2004, new developments in loop
quantum cosmology have taken place, especially with regards to the resolution
of the Big Bang singularity in the isotropic models. The singularity resolution
issue has been discussed in terms of physical quantities (expectation values of
Dirac observables) and there is also an ``improved'' quantization of the
Hamiltonian constraint. These developments are briefly discussed.
This is an expanded version of the review talk given at the
24$^{\mathrm{th}}$ IAGRG meeting in February 2007. | [
"Ghanashyam Date"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | Since the past Iagrg meeting in December 2004, new developments in loop quantum cosmology have taken place, especially with regards to the resolution of the Big Bang singularity in the isotropic models. The singularity resolution issue has been discussed in terms of physical quantities (expectation values of Dirac observables) and there is also an "improved" quantization of the Hamiltonian constraint. These developments are briefly discussed. This is an expanded version of the review talk given at the 24 th IAGRG meeting in February 2007.
Our current understanding of large scale properties of the universe is summarised by the so called Λ-CDM Big Bang model -homogeneous and isotropic, spatially flat spacetime geometry with a positive cosmological constant and cold dark matter. Impressive as it is, the model is based on an Einsteinian description of space-time geometry which has the Big Bang singularity. The existence of cosmological singularities is in fact much more general. There are homogeneous but anisotropic solution space-times which are singular and even in the inhomogeneous context there is a general solution which is singular [1] .
The singularity theorems give a very general argument for the existence of initial singularity for an everywhere expanding universe with normal matter content, the singularity being characterized as incompleteness of causal geodesic in the past. Secondly, in conjunction with an inflationary scenario, one imagines the origin of the smaller scale structure to be attributed to quantum mechanical fluctuations of matter and geometry. On account of both the features, a role for the quantum nature of matter and geometry is indicated.
Quantum mechanical models for cosmological context were in fact constructed, albeit formally. For the homogeneous and isotropic sector, the geometry is described by just the scale factor and the extrinsic curvature of the homogeneous spatial slices. In the gravitational sector, a quantum mechanical wave function is a function of the scale factor i.e. a function on the (mini-) superspace of gravity. The scale factor being positive, the minisuperspace has a boundary and wave functions need to satisfy a suitable boundary condition. Furthermore, the singularity was not resolved in that the Wheeler-De Witt equation (or the Hamiltonian constraint) which is a differential equation with respect to the scale factor, had singular coefficient due to the matter density diverging near the boundary. Thus quantization per se does not necessarily give a satisfactory replacement of the Big Bang singularity.
Meanwhile, over the past 20 years, a background independent, non-perturbative quantum theory of gravity is being constructed starting from a (gauge-) connection formulation of classical general relativity [2] . The background independence provided strong constraints on the construction of the quantum theory already at the kinematical level (i.e. before imposition of the constraints) and in particular revealed a discrete and non-commutative nature of quantum (three dimensional Riemannian) geometry. The full theory is still quite unwieldy.
Martin Bojowald took to step of restricting to homogeneous geometries and quantizing such models in a loopy way. Being inherited from the connection formulation, the geometry is described in terms of densitized triad which for the homogeneous and isotropic context is described by p ∼ sgn(p)a 2 which can also take negative values (encoding the orientation of the triad). This means that the classical singularity (at p = 0) now lies in the interior of the superspace. Classically, the singularity prevents any relation between the two regions of positive and negative values of p. Quantum mechanically however, the wave functions in these two regions, could be related. One question that becomes relevant in a quantum theory is that if a wave function, specified for one orientation and satisfying the Hamiltonian constraint, can be unambiguously extended to the other orientation while continuing to satisfy the Hamiltonian constraint. Second main implication of loop quantization is the necessity of using holonomies -exponentials of connection variable c -as well defined operators. This makes the Hamiltonian constraint a difference equation on the one hand and also requires an indirect definition for inverse triad (and inverse volume) operators entering in the definition of the matter Hamiltonian or densities and pressures. Quite interestingly, the Hamiltonian constraint equation turns out to be non-singular (i.e. deterministic) and in the effective classical approximation suggests interesting phenomenological implication quite naturally. These two features in fact made LQC an attractive field.
We will briefly summarise the results prior to 2005 and then turn to more recent developments. An extensive review of LQC is available in [3] . For simplicity and definiteness, we will focus on the spatially flat isotropic models.
Classical model: Using coordinates adapted to the spatially homogeneous slicing of the space-time, the metric and the extrinsic curvature are given by,
ds 2 := -dt 2 + a 2 (t) (dr 2 + r 2 dΩ 2 .
(1)
Starting from the usual Einstein-Hilbert action and scalar matter for definiteness, one can get to the Hamiltonian as,
S := dt cell dx 3 |detg µν | R(g) 16πG + 1 2 φ2 -V (φ) = V 0 dt 3 8πG (-a ȧ2 ) + 1 2 a 3 φ2 -V (φ)a 3 (2)
p a = - 3V 0 4πG a ȧ , p φ = V 0 a 3 φ , V 0 := cell d 3 x ; H(a, p a , φ, p φ ) = H grav + H matter = - 2πG 3 p 2 a V 0 a + 1 2 p 2 φ a 3 V 0 + a 3 V 0 V (φ) (3) = 3V 0 a 3 8πG - ȧ2 a 2 + 8πG 3
H matter V 0 a 3 (4)
Thus, H = 0 ↔ Friedmann Equation. For the spatially flat model, one has to choose a fiducial cell whose fiducial volume is denoted by V 0 .
In the connection formulation, instead of the metric one uses the densitized triad i.e.
instead of the scale factor a one has p, |p| := a 2 /4 while the connection variable is related to the extrinsic curvature as: c := γ ȧ/2 (the spin connections is absent for the flat model).
Their Poisson bracket is given by {c, p} = (8πGγ)/(3V 0 ). The arbitrary fiducial volume can be absorbed away by defining
c := V 1/3 0 c, p := V 2/3 0 p.
Here, γ is the Barbero-Immirzi parameter which is dimensionless and is determined from the Black hole entropy computations to be approximately 0.23 [4] . From now on we put 8πG := κ. The classical Hamiltonian is then given by,
H = - 3 κ γ -2 c 2 |p| + 1 2 |p| -3/2 p 2 φ + |p| 3/2 V (φ) . (5)
For future comparison, we now take the potential for the scalar field, V (φ) to be zero as well.
One can obtain the Hamilton's equations of motion and solve them easily. On the constrained surface (H = 0), eliminating c in favour of p and p φ , one has,
c = ± γ κ 6 |p φ | |p| , ṗ = ± κ 6 |p φ ||p| -1/2 . φ = p φ |p| -3/2 , ṗφ = 0 , (6)
dp dφ = ± 2κ 3 |p| ⇒ p(φ) = p * e ± √ 2κ 3 (φ-φ * ) (7)
Since φ is a monotonic function of the synchronous time t, it can be taken as a new "time"
variable. The solution is determined by p(φ) which is (i) independent of the constant p φ and (ii) passes through p = 0 as φ → ±∞ (expanding/contracting solutions). It is immediate that, along these curves, p(φ), the energy density and the extrinsic curvature diverge as p → 0. Furthermore, the divergence of the density implies that φ(t) is incomplete i.e. t ranges over a semi-infinite interval as φ ranges over the full real line foot_0 . Thus a singularity is signalled by a solution p(φ) passing through p = 0 in finite synchronous time (or equivalently by the density diverging somewhere along the solution). A natural way to ensure that all solutions are non-singular is to ensure that either of the two terms in the Hamiltonian constraint are bounded. Question is: does a quantum theory replace the Big Bang singularity by something non-singular?.
There are at least two ways to explore this question. One can imagine computing corrections to the Hamiltonian constraint such that individual terms in the effective constraint are bounded. Alternatively and more satisfactorily, one should be able to define suitable observables whose expectation values will generate the analogue of p(φ) curves along which physical quantities such as energy density, remain bounded. The former method was used pre-2005 because it could be used for more general models (non-zero potential, anisotropy etc). The latter has been elaborated in 2006, for the special case of vanishing potential.
Both methods imply that classical singularity is resolved in LQC but not in Wheeler-De Witt quantum cosmology. We will first discuss the issue in terms of effective Hamiltonian because it is easier and then discuss it in terms of the expectation values.
In the standard Schrodinger quantization, one can introduce wave functions of p, φ and quantize the Hamiltonian operator by c → i κγ/3∂ p , p φ → -i ∂ φ , in equation (5) . With a choice of operator ordering, ĤΨ(p, φ) = 0 leads to the Wheeler-De Witt partial differential equation which has singular coefficients.
The background independent quantization of Loop Quantum Gravity however suggest a different quantization of the isotropic model. One should look for a Hilbert space on which only exponentials of c (holonomies of the connection) are well defined operators and not ĉ.
Such a Hilbert space consists of almost periodic functions of c which implies that the triad operator has every real number as a proper eigenvalue: p|µ := 1 6 γℓ 2 P µ|µ , ∀µ ∈ R , ℓ 2 P := κ . This has a major implication: inverses of positive powers of triad operators do not exist [5] .
These have to be defined by using alternative classical expressions and promoting them to quantum operators. This can be done with at least one parameter worth of freedom, eg.
|p| -1 = 3 8πGγl {c, |p| l } 1/(1-l)
, l ∈ (0, 1) .
Only positive powers of |p| appear now. However, this still cannot be used for quantization since there is no ĉ operator. One must use holonomies: h j (c) := e µ 0 cΛ i τ i , where τ i are anti-hermitian generators of SU(2) in the j th representation satisfying Tr j (τ i τ j ) = -1 3 j(j + 1)(2j + 1)δ ij , Λ i is a unit vector specifying a direction in the Lie algebra of SU(2) and µ 0 is the coordinate length of the loop used in defining the holonomy. Using the holonomies,
|p| -1 = (8πGµ 0 γl) 1 l-1 3 j(j + 1)(2j + 1) Tr j Λ • τ h j {h -1 j , |p| l } 1 1-l , (9)
which can be promoted to an operator. Two parameters, µ 0 ∈ R and j ∈ N/2, have crept in and we have a three parameter family of inverse triad operators. The definitions are:
|p| -1 (jl) |µ = 2jµ 0 6 γℓ 2 P -1 (F l (q)) 1 1-l |µ , q := µ 2µ 0 j := p 2jp 0 , F l (q) := 3 2l 1 l + 2 (q + 1) l+2 -|q -1| l+2 - 1 l + 1 q (q + 1) l+1 -sgn(q -1)|q -1| l+1 (10)
F l (q ≫ 1) ≈ q -1 1-l , F l (q ≈ 0) ≈ 3q l + 1 . (11)
All these operators obviously commute with p and their eigenvalues are bounded above. This implies that the matter densities (and also intrinsic curvatures for more general homogeneous models), remain bounded at the classically singular region. Most of the phenomenological novelties are consequences of this particular feature predominantly anchored in the matter sector. We have thus two scales: p 0 := 1 6 µ 0 ℓ 2 P and 2jp 0 := 1 6 µ 0 (2j)ℓ 2 P . The regime |p| ≪ p 0 is termed the deep quantum regime, p ≫ 2jp 0 is termed the classical regime and p 0 |p| 2jp 0 is termed the semiclassical regime. The modifications due to the inverse triad defined above are strong in the semiclassical and the deep quantum regimes. For j = 1/2 the semiclassical regime is absent. Note that such scales are not available for the Schrodinger quantization.
The necessity of using holonomies also imparts a non-trivial structure for the gravitational Hamiltonian. The expression obtained is:
H grav = - 4 8πGγ 3 µ 3 0 ijk ǫ ijk Tr h i h j h -1 i h -1 j h k {h -1 k , V } (12)
In the above, we have used j = 1/2 representation for the holonomies and V denotes the volume function. In the limit µ 0 → 0 one gets back the classical expression.
While promoting this expression to operators, there is a choice of factor ordering involved and many are possible. We will present two choices of ordering: the non-symmetric one which keeps the holonomies on the left as used in the existing choice for the full theory, and the particular symmetric one used in [6] .
Ĥnon-sym grav = 24i γ 3 µ 3 0 ℓ 2 P sin 2 µ 0 c sin µ 0 c 2 V cos µ 0 c 2 -cos µ 0 c 2 V sin µ 0 c 2 (13) Ĥsym grav = 24i(sgn(p)) γ 3 µ 3 0 ℓ 2 P sinµ 0 c sin µ 0 c 2 V cos µ 0 c 2 -cos µ 0 c 2 V sin µ 0 c 2 sinµ 0 c (14)
At the quantum level, µ 0 cannot be taken to zero since ĉ operator does not exist. The action of the Hamiltonian operators on |µ is obtained as, Ĥnon-sym
grav |µ = 3 µ 3 0 γ 3 ℓ 2 P (V µ+µ 0 -V µ-µ 0 ) (|µ + 4µ 0 -2|µ + |µ -4µ 0 ) (15)
Ĥsym
grav |µ = 3 µ 3 0 γ 3 ℓ 2 P [|V µ+3µ 0 -V µ+µ 0 | |µ + 4µ 0 + |V µ-µ 0 -V µ-3µ 0 | |µ -4µ 0 -{|V µ+3µ 0 -V µ+µ 0 | + |V µ-µ 0 -V µ-3µ 0 |} |µ ] (16)
where V µ := ( 1 6 γℓ 2 P |µ|) 3/2 denotes the eigenvalue of V . Denoting quantum wave function by Ψ(µ, φ) the Wheeler-De Witt equation now becomes a difference equation. For the nonsymmetric one we get,
A(µ + 4µ 0 )ψ(µ + 4µ 0 , φ) -2A(µ)ψ(µ, φ) + A(µ -4µ 0 )ψ(µ -4µ 0 , φ) = - 2κ 3 µ 3 0 γ 3 ℓ 2 P H matter (µ)ψ(µ, φ) (17)
where, A(µ) := V µ+µ 0 -V µ-µ 0 and vanishes for µ = 0.
For the symmetric operator one gets, For the non-symmetric case, the highest (lowest) A coefficients vanish for their argument equal to zero thus leaving the corresponding ψ component undetermined. However, this undetermined component is decoupled from the others. Thus apart from admitting the trivial solution ψ(µ, φ) := Φ(φ)δ µ,0 , ∀µ, all other non-trivial solutions are completely determined by giving two consecutive components: ψ(μ, φ), ψ(μ + 4µ 0 , φ).
f + (µ)ψ(µ + 4µ 0 , φ) + f 0 (µ)ψ(µ, φ) + f -(µ)ψ(µ -4µ 0 , φ) = - 2κ 3 µ 3 0 γ 3 ℓ 2 P H matter (µ)ψ(µ, φ) where, (18)
f + (µ) := |V µ+3µ 0 -V µ+µ 0 | , f -(µ) := f + (µ -4µ 0 ) , f 0 := -f + (µ) -f -(µ) . Notice that f + (-2µ 0 ) = 0 = f -(2µ 0 ), but f 0 (µ)
For the symmetric case, due to these properties of the f ±,0 (µ), it looks as if the difference
equation is non-deterministic if µ = 2µ 0 + 4µ 0 n, n ∈ Z. This is because for µ = -2µ 0 ,
ψ(2µ 0 , φ) is undetermined by the lower order ψ's and this coefficient enters in the determination of ψ(2µ 0 , φ). However, the symmetric operator also commutes with the parity operator: (Πψ)(µ, φ) := ψ(-µ, φ). Consequently, ψ(2µ 0 , φ) is determined by ψ(-2µ 0 , φ).
Thus, we can restrict to µ = 2µ 0 + 4kµ 0 , k ≥ 0 where the equation is deterministic.
In both cases then, the space of solutions of the constraint equation, is completely determined by giving appropriate data for large |µ| i.e. in the classical regime. Such a deterministic nature of the constraint equation has been taken as a necessary condition for non-singularity at the quantum level foot_1 . As such this could be viewed as a criterion to limit the choice of factor ordering.
By introducing an interpolating, slowly varying smooth function, Ψ(p(µ) := 1 6 γℓ 2 P ), and keeping only the first non-vanishing terms, one deduces the Wheeler-De Witt differential equation (with a modified matter Hamiltonian) from the above difference equation. Making a WKB approximation, one infers an effective Hamiltonian which matches with the classical Hamiltonian for large volume (µ ≫ µ 0 ) and small extrinsic curvature (derivative of the WKB phase is small). There are terms of o( 0 ) which contain arbitrary powers of the first derivative of the phase which can all be summed up. The resulting effective Hamiltonian now contains modifications of the classical gravitational Hamiltonian, apart from the modifications in the matter Hamiltonian due to the inverse powers of the triad. The largest possible domain of validity of effective Hamiltonian so deduced must have |p| p 0 [7, 8] .
An effective Hamiltonian can alternatively obtained by computing expectation values of the Hamiltonian operator in semiclassical states peaked in classical regimes [9] . The leading order effective Hamiltonian that one obtains is (spatially flat case):
H non-sym eff = - 1 16πG 6 µ 3 0 γ 3 ℓ 2 P B + (p)sin 2 (µ 0 c) + A(p) - 1 2 B + (p) + H matter ; B + (p) := A(p + 4p 0 ) + A(p -4p 0 ) , A(p) := (|p + p 0 | 3/2 -|p -p 0 | 3/2 ) , (19)
p := 1 6 γℓ 2 P µ , p 0 := 1 6 γℓ 2 P µ 0 .
For the symmetric operator, the effective Hamiltonian is the same as above except that
B + (p) → f + (p) + f -(p) and 2A(p) → f + (p) + f -(p).
The second bracket in the square bracket, is the quantum geometry potential which is negative and higher order in ℓ P but is important in the small volume regime and plays a role in the genericness of bounce deduced from the effective Hamiltonian [10] . This term is absent in effective Hamiltonian deduced from the symmetric constraint. The matter Hamiltonian will typically have the eigenvalues of powers of inverse triad operator which depend on the ambiguity parameters j, l.
We already see that the quantum modifications are such that both the matter and the gravitational parts in the effective Hamiltonian, are rendered bounded and effective dynamics must be non-singular.
For large values of the triad,
p ≫ p 0 , B + (p) ∼ 6p 0 √ p -o(p -3/2 ) while A(p) ∼ 3p 0 √ p - o(p -3/2
). In this regime, the effective Hamiltonians deduced from both symmetric and nonsymmetric ordering are the same. The classical Hamiltonian is obtained for µ 0 → 0. From this, one can obtain the equations of motion and by computing the left hand side of the Friedmann equation, infer the effective energy density. For p ≫ p 0 one obtains foot_2 ,
3 8πG ȧ2 a 2 := ρ eff = H matter p 3/2 1 - 8πGµ 2 0 γ 2 3 p H matter p 3/2 , p := a 2 /4 . ( 20
)
The effective density is quadratic in the classical density, ρ cl := H matter p -3/2 . This modification is due to the quantum correction in the gravitational Hamiltonian (due to the sin 2 feature). This is over and above the corrections hidden in the matter Hamiltonian (due to the "inverse volume" modifications). As noted before, we have two scales: p 0 controlled by µ 0 in the gravitational part and 2p 0 j in the matter part. For large j it is possible that we can have p 0 ≪ p ≪ 2p 0 j in which case the above expressions will hold with j dependent corrections in the matter Hamiltonian. In this semiclassical regime, the corrections from sin 2 term are smaller in comparison to those from inverse volume. If p ≫ 2p 0 j then the matter Hamiltonian is also the classical expression. For j = 1/2, there is only the p ≫ p 0 regime and ρ cl is genuinely the classical density.
Let us quickly note a comparison of the two quantizations as reflected in the corresponding effective Hamiltonians, particularly with regards to the extrema of p(t). For this, we will assume same ambiguity parameters (j, l) in the matter Hamiltonian, (1/2)p 2 φ F it is necessary that the quantum geometry potential is present. Thus, for the symmetric ordering, case (A) cannot be realised -it will imply p φ = 0.
An extremum determined by case (A): It is a bounce if p * is in in the semiclassical regime;
p * varies inversely with p φ while the corresponding density varies directly. p * being limited to the semiclassical regime implies that p φ is also bounded both above and below, for such an extremum to occur. It turns out that p * can be in the classical regime, provided p φ ∼ ℓ 2 P . Thus, the non-symmetric constraint, at the effective level, can accommodate a bounce only in the semiclassical regime and with large densities.
An extremum determined by case (B): It is bounce if p * is in the classical regime; p * varies directly with p φ and the corresponding density varies inversely. p * being limited to the classical regime implies that p φ must be bounded below but can be arbitrarily large and thus the density can be arbitrarily small. This is quite unreasonable and has been sited as one of the reasons for considering the "improved" quantization (more on this later). If p * is in the semiclassical regime, it has to be a re-collapse with p φ ∼ ℓ 2 P .
In the early works, one worked with the non-symmetric constraint operator and the sin 2 corrections were not incorporated (i.e. µ 0 c ≪ π/2 was assumed) and the phenomenological implications were entirely due to the modified matter Hamiltonian. These already implied genericness of inflation and genericness of bounce. These results were discussed at the previous IAGRG meeting in Jaipur. implications: its discrete nature of quantum geometry leads to bounded energy densities and bounded extrinsic and intrinsic curvatures (for the anisotropic models). These two features are construed as "resolving the classical singularity". Quite un-expectedly, the effective dynamics incorporating quantum corrections is also singularity-free (via a bounce), accommodates an inflationary phase rather naturally and is well behaved with regards to perturbations. Although there are many ambiguity parameters, these results are robust with respect to their values.
Despite many attractive features of LQC, many points need to be addressed further:
• LQC being a constrained theory, it would be more appropriate if singularity resolution is formulated and demonstrated in terms of physical expectation values of physical (Dirac) operators i.e. in terms of "gauge invariant quantities". This can be done at present with self-adjoint constraint i.e. a symmetric ordering and for free, massless scalar matter.
• There are at least three distinct ambiguity parameters: µ 0 related to the fiducial length of the loop used in writing the holonomies; j entering in the choice of SU (2) representation which is chosen to be 1/2 in the gravitational sector and some large value in the matter sector; l entering in writing the inverse powers in terms of Poisson brackets. The first one was thought to be determined by the area gap from the full theory. The j = 1/2 in the gravitational Hamiltonian seems needed to avoid high order difference equation and larger j values are hinted to be problematic in the study of a three dimensional model [11] . Given this, the choice of a high value of j in the matter Hamiltonian seems unnatural foot_3 . For phenomenology however the higher values allowing for a larger semiclassical regime are preferred. The l does not play as significant a role.
• The bounce scale and density at the bounce, implied by the effective Hamiltonian (from symmetric ordering), is dependent on the parameters of the matter Hamiltonian and can be arranged such that the bounce density is arbitrarily small. This is a highly undesirable feature. Furthermore, the largest possible domain of validity of WKB approximation is given by the turning points (eg the bounce scale). However, the approximation could break down even before reaching the turning point. An independent check on the domain of validity of effective Hamiltonian is thus desirable.
• A systematic derivation of LQC from LQG is expected to tighten the ambiguity parameters. However, such a derivation is not yet available.
When the Hamiltonian is a constraint, at the classical level itself, the notion of dynamics in terms of the 'time translations' generated by the Hamiltonian is devoid of any physical meaning. Furthermore, at the quantum level when one attempts to impose the constraint as Ĥ|Ψ = 0, typically one finds that there are no solutions in the Hilbert space on which Ĥ is defined -the solutions are generically distributional. One then has to consider the space of all distributional solutions, define a new physical inner product to turn it into a Hilbert space (the physical Hilbert space), define operators on the space of solutions (which must thus act invariantly) which are self-adjoint (physical operators) and compute expectation values, uncertainties etc of these operators to make physical predictions. Clearly, the space of solutions depends on the quantization of the constraint and there is an arbitrariness in the choice of physical inner product. This is usually chosen so that a complete set of Dirac observables (as deduced from the classical theory) are self-adjoint. This is greatly simplified if the constraint has a separable form with respect to some degree of freedom foot_4 . For LQC (and also for the Wheeler-De Witt quantum cosmology), such a simplification is available for a free, massless scalar matter: H matter (φ, p φ ) := 1 2 p 2 φ |p| -3/2 . Let us sketch the steps schematically, focusing on the spatially flat model for simplicity [6, 13] .
The classical constraint equations is:
- 6 γ 2 c 2 |p| + 8πG p 2 φ |p| -3/2 = 0 = C grav + C matter ; (21)
The corresponding quantum equation for the wave function, Ψ(p, φ) is:
8πGp 2 φ Ψ(p, φ) = [ B(p)] -1 Ĉgrav Ψ(p, φ) , [ B(p)] is eigenvalue of |p| -3/2 ; ( 22
)
Putting pφ = -i ∂ φ , p := ) -3/2 B(µ), the equation can be written in a separated form as,
∂ 2 Ψ(µ, φ) ∂φ 2 = [B(µ)] -1 8πG γ 6 3/2 ℓ -1 P Ĉgrav Ψ(µ, φ) := -Θ(µ)Ψ(µ, φ). (23)
The Θ operator for different quantizations is different. For Schrodinger quantization (Wheeler-De Witt), with a particular factor ordering suggested by the continuum limit of the difference equation, the operator Θ(µ) is given by,
ΘSch (µ)Ψ(µ, φ) = - 16πG 3 |µ| 3/2 ∂ µ √ µ ∂ µ Ψ(µ, φ) (24)
while for LQC, with symmetric ordering, it is given by,
ΘLQC (µ)Ψ(µ, φ) = -[B(µ)] -1 C + (µ)Ψ(µ + 4µ 0 , φ) + C 0 (µ)Ψ(µ, φ)+ C -(µ)Ψ(µ -4µ 0 , φ) , C + (µ) := πG 9µ 3 0 |µ + 3µ 0 | 3/2 -|µ + µ 0 | 3/2 , (25)
C -(µ) := C + (µ -4µ 0 ) , C 0 (µ) := -C + (µ) -C -(µ) .
Note that in the Schrodinger quantization, the B Sch (µ) = |µ| -3/2 diverges at µ = 0 while in LQC, B LQC (µ) vanishes for all allowed choices of ambiguity parameters. In both cases, B(µ) ∼ |µ| -3/2 as |µ| → ∞.
The operator Θ turns out to be a self-adjoint, positive definite operator on the space of functions Ψ(µ, φ) for each fixed φ with an inner product scaled by B(µ). That is, for the Schrodinger quantization, it is an operator on L 2 (R, B Sch (µ)dµ) while for LQC it is an operator on L 2 (R Bohr , B Bohr (µ)dµ Bohr ). Because of this, the operator has a complete set of eigenvectors: Θe
k (µ) = ω 2 (k)e k (µ), k ∈ R, e k |e k ′ = δ(k, k ′ ), and
the general solution of the fundamental constraint equation can be expressed as
Ψ(µ, φ) = dk Ψ+ (k)e k (µ)e iωφ + Ψ-(k)ē k (µ)e -iωφ . ( 26
)
The orthonormality relations among the e k (µ) are in the corresponding Hilbert spaces.
Different quantizations differ in the form of the eigenfunctions, possibly the spectrum itself and of course ω(k). In general, these solutions are not normalizable in L 2 (R Bohr × R, dµ Bohr × dµ), i.e. these are distributional.
Since the classical kinematical phase space is 4 dimensional and we have a single first class constraint, the phase space of physical states (reduced phase space) is two dimensional and we need two functions to coordinatize this space. We should thus look for two (classical) Dirac observables: functions on the kinematical phase space whose
Poisson bracket with the Hamiltonian constraint vanishes on the constraint surface.
It is easy to see that p φ is a Dirac observable. For the second one, we choose a one parameter family of functions µ(φ) satisfying {µ(φ), C(µ, c, φ, p φ )} ≈ 0. The corresponding quantum definitions, with the operators acting on the solutions, are:
pφ Ψ(µ, φ) := -i ∂ φ Ψ(µ, φ) , ( 27
)
|µ| φ 0 Ψ(µ, φ) := e i √ Θ(φ-φ 0 ) |µ|Ψ + (µ, φ 0 ) + e -i √ Θ(φ-φ 0 ) |µ|Ψ -(µ, φ 0 ) (28)
On an initial datum, Ψ(µ, φ 0 ), these operators act as,
|µ| φ 0 Ψ(µ, φ 0 ) = |µ|Ψ(µ, φ 0 ) , pφ Ψ(µ, φ 0 ) = ΘΨ(µ, φ 0 ) . ( 29
)
It follows that the Dirac operators defined on the space of solutions are self-adjoint if we define a physical inner product on the space of solutions as:
Ψ|Ψ ′ phys := " φ=φ 0 dµB(µ)" Ψ(µ, φ)Ψ ′ (µ, φ) . ( 30
)
Thus the eigenvalues of the inverse volume operator crucially enter the definition of the physical inner product. For Schrodinger quantization, the integral is really an integral while for LQC it is actually a sum over µ taking values in a lattice. The inner product is independent of the choice of φ 0 .
A complete set of physical operators and physical inner product has now been specified
and physical questions can be phrased in terms of (physical) expectation values of functions of these operators.
To discuss semiclassical regime, typically one defines semiclassical states: physical states such that a chosen set of self-adjoint operators have specified expectation values with uncertainties bounded by specified tolerances. A natural choice of operators for us are the two Dirac operators defined above. It is easy to construct semiclassical states with respect to these operators. For example, a state peaked around, p φ = p * φ and |µ| φ 0 = µ * is given by (in Schrodinger quantization for instance),
Ψ semi (µ, φ 0 ) := dke -(k-k * ) 2 2σ 2
e k (µ)e iω(φ 0 -φ * ) (31)
k * = -3/2κ -1 p * φ , φ * = φ 0 + -3/2κℓn|µ * | . ( 32
)
For LQC, the e k (µ) functions are different and the physical expectation values are to be evaluated using the physical inner product defined in the LQC context.
Since one knows the general solution of the constraint equation, Ψ(µ, φ), given Ψ(µ, φ 0 ), one can compute the physical expectation values in the semiclassical solution, Ψ semi (µ, φ) and track the position of the peak as a function of φ as well as the uncertainties as a function of φ.
A classical solution is obtained as a curve in (µ, φ) plane, different curves being labelled by the points (µ * , φ * ) in the plane. The curves are independent of the constant value of p * φ These curves are already given in (7) .
Quantum mechanically, we first select a semiclassical solution, Ψ semi (p * φ , µ * : φ) in which the expectation values of the Dirac operators, at φ = φ 0 , are p * φ and µ * respectively. These values serve as labels for the semiclassical solution. The former one continues to be p * φ for all φ whereas |µ| φ 0 (φ) =: |µ| p * φ ,µ * (φ), determines a curve in the (µ, φ) plane. In general one expects this curve to be different from the classical curve in the region of small µ (small volume).
The result of the computations is that Schrodinger quantization, the curve |µ| p * φ ,µ * (φ), does approach the µ = 0 axis asymptotically. However for LQC, the curve bounces away from the µ = 0 axis. In this sense -and now inferred in terms of physical quantities -the Big Bang singularity is resolved in LQC. It also turns out that for large enough values of p * φ , the quantum trajectories constructed by the above procedure are well approximated by the trajectories by the effective Hamiltonian. All these statements are for semiclassical solutions which are peaked at large µ * at late times.
Two further features are noteworthy as they corroborate the suggestions from the effective Hamiltonian analysis.
First one is revealed by computing expectation value of the matter density operator,
ρ matter := 1 2 (p * φ ) 2 |p| -3
, at the bounce value of |p|. It turns out that this value is sensitive to the value of p * φ and can be made arbitrarily small by choosing p * φ to be large. Physically this is unsatisfactory as quantum effects are not expected to be significant for matter density very small compared to the Planck density. This is traced to the quantization of the gravitational Hamiltonian, in particular to the step which introduces the ambiguity parameter µ 0 . A novel solution proposed in the "improved quantization", removes this undesirable feature.
The second one refers to the role of quantum modifications in the gravitational Hamiltonian compared to those in the matter Hamiltonian (the inverse volume modification or B(µ)). The former is much more significant than the latter. So much so, that even if one uses the B(µ) from the Schrodinger quantization (i.e. switch-off the inverse volume modifications), one still obtains the bounce. So bounce is seen as the consequence of Θ being different and as far as qualitative singularity resolution is concerned, the inverse volume modifications are un-important. As the effective picture (for symmetric constraint) showed, the bounce occurs in the classical region (for j = 1/2) where the inverse volume corrections can be neglected. For an exact model which seeks to understand as to why the bounces are seen, please see [14] .
The undesirable features of the bounce coming from the classical region, can be seen readily using the effective Hamiltonian, as remarked earlier. To see the effects of modifications from the gravitational Hamiltonian, choose j = 1/2 and consider the Friedmann equation derived from the effective Hamiltonian (20) , with matter Hamiltonian given by
H matter = 1 2 p 2 φ |p| -3/2
. The positivity of the effective density implies that p ≥ p * with p * determined by vanishing of the effective energy density:
ρ * := ρ cl (p * ) = ( 8πGµ 2 0 γ 2 3 p * ) -1 . This leads to |p * | = 4πGµ 2 0 γ 2 3 |p φ | and ρ * = √ 2( 8πGµ 2 0 γ 2 3
) -3/2 |p φ | -1 . One sees that for large |p φ |, the bounce scale |p * | can be large and the maximum density -density at bounce -could be small. Thus, within the model, there exist a possibility of seeing quantum effects (bounce) even when neither the energy density nor the bounce scale are comparable to the corresponding Planck quantities and this is an undesirable feature of the model. This feature is independent of factor ordering as long as the bounce occurs in the classical regime.
One may notice that if we replace µ 0 → μ(p) := ∆/|p| where ∆ is a constant, then the effective density vanishes when ρ cl equals the critical value ρ crit := ( 8πG∆γ 2ρ crit ) 1/3 . Now although the bounce scale can again be large depending upon p φ , the density at bounce is always the universal value determined by ∆. This is a rather nice feature in that quantum geometry effects are revealed when matter density (which couples to gravity) reaches a universal, critical value regardless of the dynamical variables describing matter. For a suitable choice of ∆ one can ensure that a bounce always happen when the energy density becomes comparable to the Planck density. In this manner, one can retain the good feature (bounce) even for j = 1/2 thus "effectively fixing" an ambiguity parameter and also trade another ambiguity parameter µ 0 for ∆. This is precisely what is achieved by the "improved quantization" of the gravitational Hamiltonian [15] .
The place where the quantization procedure is modified is when one expresses the cur-vature in terms of the holonomies along a loop around a "plaquette". One shrinks the plaquette in the limiting procedure. One now makes an important departure: the plaquette should be shrunk only till the physical area (as distinct from a fiducial one) reaches its minimum possible value which is given by the area gap in the known spectrum of area operator in quantum geometry: ∆ = 2 √ 3πγG . Since the plaquette is a square of fiducial length µ 0 , its physical area is µ 2 0 |p| and this should set be to ∆. Since |p| is a dynamical variable, µ 0 cannot be a constant and is to be thought of a function on the phase space, μ(p) := ∆/|p|. It turns out that even with such a change which makes the curvature to be a function of both connection and triad, the form of both the gravitational constraint and inverse volume operator appearing in the matter Hamiltonian, remains the same with just doing the replacement, µ 0 → μ defined above, in the holonomies. The expressions simplify by using eigenfunctions of the volume operator V := | p| 3/2 , instead of those of the triad.
The relevant expressions are: the the parity symmetry again saves the day); the densities continue to be bounded aboveand now with a bound independent of matter parameters; the effective picture continues to be singularity free and with undesirable features removed and the classical Big Bang being replaced by a quantum bounce is established in terms of physical quantities.
v := Ksgn(µ)|µ| 3/2 , K := 2 √ 2 3 3 √ 3 ; (33)
V |v = γ 6 3/2 ℓ 3 P K |v||v , (34)
e ik μ 2 c Ψ(v) := Ψ(v + k) , (35)
|p| -1/2 j=1/2,l=3/4 Ψ(v) = 3 2
γℓ 2 P 6 -1/2 K 1/3 |v| 1/3 |v + 1| 1/3 -|v -1| 1/3 Ψ(v) (36) B(v) = 3 2 3/2 K|v| |v + 1| 1/3 -|v -1| 1/3 3 (37) ΘImproved Ψ(v, φ) = -[B(v)] -1 C + (v)Ψ(v + 4, φ) + C 0 (v)Ψ(v, φ)+ C -(v)Ψ(v -4, φ) , (38)
C + (v) := 3πKG 8 |v + 2| | |v + 1| -|v + 3|| , (39)
C -(v) := C + (v -4) , C 0 (v) := -C + (v) -C -(v) . (40
While close model seems phenomenologically disfavoured, it provides further testing ground for quantization of the Hamiltonian constraint. Because of the intrinsic (spatial) curvature, the plaquettes used in expressing the F ij in terms of holonomies, are not bounded by just four edges -a fifth one is necessary. This was attempted and was found to lead to an "unstable" quantization. This difficulty was bypassed by using the holonomies of the extrinsic curvature instead of the gauge connection which is permissible in the homogeneous context. The corresponding, non-symmetric constraint and its difference equation was analysed for the massless scalar matter. Green and Unruh, found that solutions of the difference equation was always diverging (at least for one orientation) for large volumes. Further, the divergence seemed to set in just where one expected a re-collapse from the classical theory.
In the absence of physical inner product and physical interpretation of the solutions, it was concluded that this version of LQC for close model is unlikely to accommodate classical re-collapse even though it avoided the Big Bang/Big Crunch singularities.
Recently, this model has been revisited [16] . One went back to using the gauge connection and the fifth edge difficulty was circumvented by using both the left-invariant and the right-invariant vector fields to define the plaquette. In addition, the symmetric ordering was chosen and finally the µ 0 → μ improvement was also incorporated. Without the improvement, there were still the problems of getting bounce for low energy density and also not getting a reasonable re-collapse (either re-collapse is absent or the scale is marginally larger than the bounce scale). With the improvement, the bounces and re-collapses are neatly accommodated and one gets a cyclic evolution. In this case also, the scalar field serves as a good clock variable as it continues to be monotonic with the synchronous time.
I have focussed on the singularity resolution issue in this talk. Other developments have also taken place in the past couple of years. I will just list these giving references.
1. Effective models and their properties: The effective picture was shown to be non-singular and since this is based on the usual framework of GR, it follows that energy conditions must be violated (and indeed they are thanks to the inverse volume modifications). This raised questions regarding stability of matter and causal propagation of perturbations. Golam Hossain showed that despite the energy conditions violations, neither of the above pathologies result [17] .
Minimally coupled scalar has been used in elaborating inflationary scenarios. However non-minimally coupled scalars are also conceivable models. The singularity resolution and inflationary scenarios continue to hold also in this case. Furthermore sufficient e-foldings are also admissible [18] .
In the improved quantization, one sets the ambiguity parameter j = 1/2 and shifts the dominant effects to the the gravitational Hamiltonian. All the previous phenomenological implications however were driven by the inverse volume modifications in the matter sector. Consequently, it is necessary to check if and how the phenomenology works with the improved quantization. This has been explored in [19] .
Using the effective dynamics for the homogeneous mode, density perturbations were explored and power spectra were computed with the required small amplitude [20, 21] .
As many of the phenomenology oriented questions have been explored using effective
Hamiltonian which incorporate quantum corrections from various sources (gravity, matter etc). This motivates a some what systematic approach to constructing effective approximations. This has been initiated in [22] .
2. Anisotropic models: The anisotropic models provide further testing grounds for loop quantization. At the difference equation level, the non-singularity has been checked also for these models in the non-symmetric scheme. For the vacuum Bianchi I model, there is no place for the inverse volume type corrections to appear at an effective Hamiltonian level and the effective dynamics would continue to be singular. However, once the gravitational corrections (sin 2 ) are incorporated, the effective dynamics again is non-singular and one can obtain the non-singular version of the (singular) Kasner solution [23] . More recently, the Bianchi I model with a free, massless scalar is also analysed in the improved quantization [24] . A perturbative treatment of anisotropies has been explored in [25] .
3. Inhomogeneities: Inhomogeneities are a fact of nature although these are small in the early universe. This suggests a perturbative approach to incorporate inhomogeneities.
On the one hand one can study their evolution in the homogeneous, isotropic background (cosmological perturbation theory). One can also begin with a (simplified) inhomogeneous model and try to see how a homogeneous approximation can become viable. The work on the former has already begun. For the latter part, Bojowald has discussed a simplified lattice model to draw some lessons for the homogeneous models.
In particular he has given an alternative argument for the µ 0 → μ modification which does not appeal to the area operator [12] .
In summary, over the past two years, we have seen how to phrase and understand the fate of Big Bang singularity in a quantum framework.
Firstly, with the help of a minimally coupled, free, massless scalar which serves as a good clock variable in the isotropic context, one can define physical inner product, a complete set of Dirac observables and their physical matrix elements. At present this can be done only for self-adjoint Hamiltonian constraint. Using these, one can construct trajectories in the (p, p φ ) plane which are followed by the peak of a semiclassical state as well as the uncertainties in the Dirac observables. It so happens that these trajectories do not pass through the zero volume -Big Bang is replaced by a Bounce. For close isotropic model, the Big Crunch is also replaced by a bounce while retaining classically understood re-collapse. In conjunction with the μ improvement, the gravitational Hamiltonian can be given the the main role in generating the bounce. A corresponding treatment in Schrodinger quantization (Wheeler-De Witt theory), does not generate a bounce nor does it render the density, curvatures bounded.
Thus, quantum representation plays a significant role in the singularity resolution.
Secondly, the improved quantization motivated by the regulation of the F ij invoking the area operator from the full theory (or by the argument from the inhomogeneous lattice model), also leads the bounce to be "triggered" when the energy density reaches a critical value (∼ 0.82ρ Planck ) which is independent of the values of the dynamical variables. Close model also gives the same critical value.
While the improvement is demonstrated to be viable in the isotropic context, the proce-dure differs from that followed in the full theory. One may either view this as something special to the mini-superspace model(s) or view it as providing hints for newer approaches in the full theory.
A general criteria for "non-singularity" is not in sight yet and so also a systematic derivation of the mini-superspace model(s) from a larger, full theory. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Since the past Iagrg meeting in December 2004, new developments in loop quantum cosmology have taken place, especially with regards to the resolution of the Big Bang singularity in the isotropic models. The singularity resolution issue has been discussed in terms of physical quantities (expectation values of Dirac observables) and there is also an \"improved\" quantization of the Hamiltonian constraint. These developments are briefly discussed. This is an expanded version of the review talk given at the 24 th IAGRG meeting in February 2007."
},
{
"section_type": "OTHER",
"section_title": "I. COSMOLOGY, QUANTUM COSMOLOGY, LOOP QUANTUM COSMOLOGY",
"text": "Our current understanding of large scale properties of the universe is summarised by the so called Λ-CDM Big Bang model -homogeneous and isotropic, spatially flat spacetime geometry with a positive cosmological constant and cold dark matter. Impressive as it is, the model is based on an Einsteinian description of space-time geometry which has the Big Bang singularity. The existence of cosmological singularities is in fact much more general. There are homogeneous but anisotropic solution space-times which are singular and even in the inhomogeneous context there is a general solution which is singular [1] .\n\nThe singularity theorems give a very general argument for the existence of initial singularity for an everywhere expanding universe with normal matter content, the singularity being characterized as incompleteness of causal geodesic in the past. Secondly, in conjunction with an inflationary scenario, one imagines the origin of the smaller scale structure to be attributed to quantum mechanical fluctuations of matter and geometry. On account of both the features, a role for the quantum nature of matter and geometry is indicated.\n\nQuantum mechanical models for cosmological context were in fact constructed, albeit formally. For the homogeneous and isotropic sector, the geometry is described by just the scale factor and the extrinsic curvature of the homogeneous spatial slices. In the gravitational sector, a quantum mechanical wave function is a function of the scale factor i.e. a function on the (mini-) superspace of gravity. The scale factor being positive, the minisuperspace has a boundary and wave functions need to satisfy a suitable boundary condition. Furthermore, the singularity was not resolved in that the Wheeler-De Witt equation (or the Hamiltonian constraint) which is a differential equation with respect to the scale factor, had singular coefficient due to the matter density diverging near the boundary. Thus quantization per se does not necessarily give a satisfactory replacement of the Big Bang singularity.\n\nMeanwhile, over the past 20 years, a background independent, non-perturbative quantum theory of gravity is being constructed starting from a (gauge-) connection formulation of classical general relativity [2] . The background independence provided strong constraints on the construction of the quantum theory already at the kinematical level (i.e. before imposition of the constraints) and in particular revealed a discrete and non-commutative nature of quantum (three dimensional Riemannian) geometry. The full theory is still quite unwieldy.\n\nMartin Bojowald took to step of restricting to homogeneous geometries and quantizing such models in a loopy way. Being inherited from the connection formulation, the geometry is described in terms of densitized triad which for the homogeneous and isotropic context is described by p ∼ sgn(p)a 2 which can also take negative values (encoding the orientation of the triad). This means that the classical singularity (at p = 0) now lies in the interior of the superspace. Classically, the singularity prevents any relation between the two regions of positive and negative values of p. Quantum mechanically however, the wave functions in these two regions, could be related. One question that becomes relevant in a quantum theory is that if a wave function, specified for one orientation and satisfying the Hamiltonian constraint, can be unambiguously extended to the other orientation while continuing to satisfy the Hamiltonian constraint. Second main implication of loop quantization is the necessity of using holonomies -exponentials of connection variable c -as well defined operators. This makes the Hamiltonian constraint a difference equation on the one hand and also requires an indirect definition for inverse triad (and inverse volume) operators entering in the definition of the matter Hamiltonian or densities and pressures. Quite interestingly, the Hamiltonian constraint equation turns out to be non-singular (i.e. deterministic) and in the effective classical approximation suggests interesting phenomenological implication quite naturally. These two features in fact made LQC an attractive field.\n\nWe will briefly summarise the results prior to 2005 and then turn to more recent developments. An extensive review of LQC is available in [3] . For simplicity and definiteness, we will focus on the spatially flat isotropic models."
},
{
"section_type": "OTHER",
"section_title": "II. SUMMARY OF PRE 2005 LQC",
"text": "Classical model: Using coordinates adapted to the spatially homogeneous slicing of the space-time, the metric and the extrinsic curvature are given by,\n\nds 2 := -dt 2 + a 2 (t) (dr 2 + r 2 dΩ 2 .\n\n(1)\n\nStarting from the usual Einstein-Hilbert action and scalar matter for definiteness, one can get to the Hamiltonian as,\n\nS := dt cell dx 3 |detg µν | R(g) 16πG + 1 2 φ2 -V (φ) = V 0 dt 3 8πG (-a ȧ2 ) + 1 2 a 3 φ2 -V (φ)a 3 (2)\n\np a = - 3V 0 4πG a ȧ , p φ = V 0 a 3 φ , V 0 := cell d 3 x ; H(a, p a , φ, p φ ) = H grav + H matter = - 2πG 3 p 2 a V 0 a + 1 2 p 2 φ a 3 V 0 + a 3 V 0 V (φ) (3) = 3V 0 a 3 8πG - ȧ2 a 2 + 8πG 3\n\nH matter V 0 a 3 (4)\n\nThus, H = 0 ↔ Friedmann Equation. For the spatially flat model, one has to choose a fiducial cell whose fiducial volume is denoted by V 0 .\n\nIn the connection formulation, instead of the metric one uses the densitized triad i.e.\n\ninstead of the scale factor a one has p, |p| := a 2 /4 while the connection variable is related to the extrinsic curvature as: c := γ ȧ/2 (the spin connections is absent for the flat model).\n\nTheir Poisson bracket is given by {c, p} = (8πGγ)/(3V 0 ). The arbitrary fiducial volume can be absorbed away by defining\n\nc := V 1/3 0 c, p := V 2/3 0 p.\n\nHere, γ is the Barbero-Immirzi parameter which is dimensionless and is determined from the Black hole entropy computations to be approximately 0.23 [4] . From now on we put 8πG := κ. The classical Hamiltonian is then given by,\n\nH = - 3 κ γ -2 c 2 |p| + 1 2 |p| -3/2 p 2 φ + |p| 3/2 V (φ) . (5)\n\nFor future comparison, we now take the potential for the scalar field, V (φ) to be zero as well.\n\nOne can obtain the Hamilton's equations of motion and solve them easily. On the constrained surface (H = 0), eliminating c in favour of p and p φ , one has,\n\nc = ± γ κ 6 |p φ | |p| , ṗ = ± κ 6 |p φ ||p| -1/2 . φ = p φ |p| -3/2 , ṗφ = 0 , (6)\n\ndp dφ = ± 2κ 3 |p| ⇒ p(φ) = p * e ± √ 2κ 3 (φ-φ * ) (7)\n\nSince φ is a monotonic function of the synchronous time t, it can be taken as a new \"time\"\n\nvariable. The solution is determined by p(φ) which is (i) independent of the constant p φ and (ii) passes through p = 0 as φ → ±∞ (expanding/contracting solutions). It is immediate that, along these curves, p(φ), the energy density and the extrinsic curvature diverge as p → 0. Furthermore, the divergence of the density implies that φ(t) is incomplete i.e. t ranges over a semi-infinite interval as φ ranges over the full real line foot_0 . Thus a singularity is signalled by a solution p(φ) passing through p = 0 in finite synchronous time (or equivalently by the density diverging somewhere along the solution). A natural way to ensure that all solutions are non-singular is to ensure that either of the two terms in the Hamiltonian constraint are bounded. Question is: does a quantum theory replace the Big Bang singularity by something non-singular?.\n\nThere are at least two ways to explore this question. One can imagine computing corrections to the Hamiltonian constraint such that individual terms in the effective constraint are bounded. Alternatively and more satisfactorily, one should be able to define suitable observables whose expectation values will generate the analogue of p(φ) curves along which physical quantities such as energy density, remain bounded. The former method was used pre-2005 because it could be used for more general models (non-zero potential, anisotropy etc). The latter has been elaborated in 2006, for the special case of vanishing potential.\n\nBoth methods imply that classical singularity is resolved in LQC but not in Wheeler-De Witt quantum cosmology. We will first discuss the issue in terms of effective Hamiltonian because it is easier and then discuss it in terms of the expectation values.\n\nIn the standard Schrodinger quantization, one can introduce wave functions of p, φ and quantize the Hamiltonian operator by c → i κγ/3∂ p , p φ → -i ∂ φ , in equation (5) . With a choice of operator ordering, ĤΨ(p, φ) = 0 leads to the Wheeler-De Witt partial differential equation which has singular coefficients.\n\nThe background independent quantization of Loop Quantum Gravity however suggest a different quantization of the isotropic model. One should look for a Hilbert space on which only exponentials of c (holonomies of the connection) are well defined operators and not ĉ.\n\nSuch a Hilbert space consists of almost periodic functions of c which implies that the triad operator has every real number as a proper eigenvalue: p|µ := 1 6 γℓ 2 P µ|µ , ∀µ ∈ R , ℓ 2 P := κ . This has a major implication: inverses of positive powers of triad operators do not exist [5] .\n\nThese have to be defined by using alternative classical expressions and promoting them to quantum operators. This can be done with at least one parameter worth of freedom, eg.\n\n|p| -1 = 3 8πGγl {c, |p| l } 1/(1-l)\n\n, l ∈ (0, 1) .\n\nOnly positive powers of |p| appear now. However, this still cannot be used for quantization since there is no ĉ operator. One must use holonomies: h j (c) := e µ 0 cΛ i τ i , where τ i are anti-hermitian generators of SU(2) in the j th representation satisfying Tr j (τ i τ j ) = -1 3 j(j + 1)(2j + 1)δ ij , Λ i is a unit vector specifying a direction in the Lie algebra of SU(2) and µ 0 is the coordinate length of the loop used in defining the holonomy. Using the holonomies,\n\n|p| -1 = (8πGµ 0 γl) 1 l-1 3 j(j + 1)(2j + 1) Tr j Λ • τ h j {h -1 j , |p| l } 1 1-l , (9)\n\nwhich can be promoted to an operator. Two parameters, µ 0 ∈ R and j ∈ N/2, have crept in and we have a three parameter family of inverse triad operators. The definitions are:\n\n|p| -1 (jl) |µ = 2jµ 0 6 γℓ 2 P -1 (F l (q)) 1 1-l |µ , q := µ 2µ 0 j := p 2jp 0 , F l (q) := 3 2l 1 l + 2 (q + 1) l+2 -|q -1| l+2 - 1 l + 1 q (q + 1) l+1 -sgn(q -1)|q -1| l+1 (10)\n\nF l (q ≫ 1) ≈ q -1 1-l , F l (q ≈ 0) ≈ 3q l + 1 . (11)\n\nAll these operators obviously commute with p and their eigenvalues are bounded above. This implies that the matter densities (and also intrinsic curvatures for more general homogeneous models), remain bounded at the classically singular region. Most of the phenomenological novelties are consequences of this particular feature predominantly anchored in the matter sector. We have thus two scales: p 0 := 1 6 µ 0 ℓ 2 P and 2jp 0 := 1 6 µ 0 (2j)ℓ 2 P . The regime |p| ≪ p 0 is termed the deep quantum regime, p ≫ 2jp 0 is termed the classical regime and p 0 |p| 2jp 0 is termed the semiclassical regime. The modifications due to the inverse triad defined above are strong in the semiclassical and the deep quantum regimes. For j = 1/2 the semiclassical regime is absent. Note that such scales are not available for the Schrodinger quantization.\n\nThe necessity of using holonomies also imparts a non-trivial structure for the gravitational Hamiltonian. The expression obtained is:\n\nH grav = - 4 8πGγ 3 µ 3 0 ijk ǫ ijk Tr h i h j h -1 i h -1 j h k {h -1 k , V } (12)\n\nIn the above, we have used j = 1/2 representation for the holonomies and V denotes the volume function. In the limit µ 0 → 0 one gets back the classical expression.\n\nWhile promoting this expression to operators, there is a choice of factor ordering involved and many are possible. We will present two choices of ordering: the non-symmetric one which keeps the holonomies on the left as used in the existing choice for the full theory, and the particular symmetric one used in [6] .\n\nĤnon-sym grav = 24i γ 3 µ 3 0 ℓ 2 P sin 2 µ 0 c sin µ 0 c 2 V cos µ 0 c 2 -cos µ 0 c 2 V sin µ 0 c 2 (13) Ĥsym grav = 24i(sgn(p)) γ 3 µ 3 0 ℓ 2 P sinµ 0 c sin µ 0 c 2 V cos µ 0 c 2 -cos µ 0 c 2 V sin µ 0 c 2 sinµ 0 c (14)\n\nAt the quantum level, µ 0 cannot be taken to zero since ĉ operator does not exist. The action of the Hamiltonian operators on |µ is obtained as, Ĥnon-sym\n\ngrav |µ = 3 µ 3 0 γ 3 ℓ 2 P (V µ+µ 0 -V µ-µ 0 ) (|µ + 4µ 0 -2|µ + |µ -4µ 0 ) (15)\n\nĤsym\n\ngrav |µ = 3 µ 3 0 γ 3 ℓ 2 P [|V µ+3µ 0 -V µ+µ 0 | |µ + 4µ 0 + |V µ-µ 0 -V µ-3µ 0 | |µ -4µ 0 -{|V µ+3µ 0 -V µ+µ 0 | + |V µ-µ 0 -V µ-3µ 0 |} |µ ] (16)\n\nwhere V µ := ( 1 6 γℓ 2 P |µ|) 3/2 denotes the eigenvalue of V . Denoting quantum wave function by Ψ(µ, φ) the Wheeler-De Witt equation now becomes a difference equation. For the nonsymmetric one we get,\n\nA(µ + 4µ 0 )ψ(µ + 4µ 0 , φ) -2A(µ)ψ(µ, φ) + A(µ -4µ 0 )ψ(µ -4µ 0 , φ) = - 2κ 3 µ 3 0 γ 3 ℓ 2 P H matter (µ)ψ(µ, φ) (17)\n\nwhere, A(µ) := V µ+µ 0 -V µ-µ 0 and vanishes for µ = 0.\n\nFor the symmetric operator one gets, For the non-symmetric case, the highest (lowest) A coefficients vanish for their argument equal to zero thus leaving the corresponding ψ component undetermined. However, this undetermined component is decoupled from the others. Thus apart from admitting the trivial solution ψ(µ, φ) := Φ(φ)δ µ,0 , ∀µ, all other non-trivial solutions are completely determined by giving two consecutive components: ψ(μ, φ), ψ(μ + 4µ 0 , φ).\n\nf + (µ)ψ(µ + 4µ 0 , φ) + f 0 (µ)ψ(µ, φ) + f -(µ)ψ(µ -4µ 0 , φ) = - 2κ 3 µ 3 0 γ 3 ℓ 2 P H matter (µ)ψ(µ, φ) where, (18)\n\nf + (µ) := |V µ+3µ 0 -V µ+µ 0 | , f -(µ) := f + (µ -4µ 0 ) , f 0 := -f + (µ) -f -(µ) . Notice that f + (-2µ 0 ) = 0 = f -(2µ 0 ), but f 0 (µ)\n\nFor the symmetric case, due to these properties of the f ±,0 (µ), it looks as if the difference\n\nequation is non-deterministic if µ = 2µ 0 + 4µ 0 n, n ∈ Z. This is because for µ = -2µ 0 ,\n\nψ(2µ 0 , φ) is undetermined by the lower order ψ's and this coefficient enters in the determination of ψ(2µ 0 , φ). However, the symmetric operator also commutes with the parity operator: (Πψ)(µ, φ) := ψ(-µ, φ). Consequently, ψ(2µ 0 , φ) is determined by ψ(-2µ 0 , φ).\n\nThus, we can restrict to µ = 2µ 0 + 4kµ 0 , k ≥ 0 where the equation is deterministic.\n\nIn both cases then, the space of solutions of the constraint equation, is completely determined by giving appropriate data for large |µ| i.e. in the classical regime. Such a deterministic nature of the constraint equation has been taken as a necessary condition for non-singularity at the quantum level foot_1 . As such this could be viewed as a criterion to limit the choice of factor ordering.\n\nBy introducing an interpolating, slowly varying smooth function, Ψ(p(µ) := 1 6 γℓ 2 P ), and keeping only the first non-vanishing terms, one deduces the Wheeler-De Witt differential equation (with a modified matter Hamiltonian) from the above difference equation. Making a WKB approximation, one infers an effective Hamiltonian which matches with the classical Hamiltonian for large volume (µ ≫ µ 0 ) and small extrinsic curvature (derivative of the WKB phase is small). There are terms of o( 0 ) which contain arbitrary powers of the first derivative of the phase which can all be summed up. The resulting effective Hamiltonian now contains modifications of the classical gravitational Hamiltonian, apart from the modifications in the matter Hamiltonian due to the inverse powers of the triad. The largest possible domain of validity of effective Hamiltonian so deduced must have |p| p 0 [7, 8] .\n\nAn effective Hamiltonian can alternatively obtained by computing expectation values of the Hamiltonian operator in semiclassical states peaked in classical regimes [9] . The leading order effective Hamiltonian that one obtains is (spatially flat case):\n\nH non-sym eff = - 1 16πG 6 µ 3 0 γ 3 ℓ 2 P B + (p)sin 2 (µ 0 c) + A(p) - 1 2 B + (p) + H matter ; B + (p) := A(p + 4p 0 ) + A(p -4p 0 ) , A(p) := (|p + p 0 | 3/2 -|p -p 0 | 3/2 ) , (19)\n\np := 1 6 γℓ 2 P µ , p 0 := 1 6 γℓ 2 P µ 0 .\n\nFor the symmetric operator, the effective Hamiltonian is the same as above except that\n\nB + (p) → f + (p) + f -(p) and 2A(p) → f + (p) + f -(p).\n\nThe second bracket in the square bracket, is the quantum geometry potential which is negative and higher order in ℓ P but is important in the small volume regime and plays a role in the genericness of bounce deduced from the effective Hamiltonian [10] . This term is absent in effective Hamiltonian deduced from the symmetric constraint. The matter Hamiltonian will typically have the eigenvalues of powers of inverse triad operator which depend on the ambiguity parameters j, l.\n\nWe already see that the quantum modifications are such that both the matter and the gravitational parts in the effective Hamiltonian, are rendered bounded and effective dynamics must be non-singular.\n\nFor large values of the triad,\n\np ≫ p 0 , B + (p) ∼ 6p 0 √ p -o(p -3/2 ) while A(p) ∼ 3p 0 √ p - o(p -3/2\n\n). In this regime, the effective Hamiltonians deduced from both symmetric and nonsymmetric ordering are the same. The classical Hamiltonian is obtained for µ 0 → 0. From this, one can obtain the equations of motion and by computing the left hand side of the Friedmann equation, infer the effective energy density. For p ≫ p 0 one obtains foot_2 ,\n\n3 8πG ȧ2 a 2 := ρ eff = H matter p 3/2 1 - 8πGµ 2 0 γ 2 3 p H matter p 3/2 , p := a 2 /4 . ( 20\n\n)\n\nThe effective density is quadratic in the classical density, ρ cl := H matter p -3/2 . This modification is due to the quantum correction in the gravitational Hamiltonian (due to the sin 2 feature). This is over and above the corrections hidden in the matter Hamiltonian (due to the \"inverse volume\" modifications). As noted before, we have two scales: p 0 controlled by µ 0 in the gravitational part and 2p 0 j in the matter part. For large j it is possible that we can have p 0 ≪ p ≪ 2p 0 j in which case the above expressions will hold with j dependent corrections in the matter Hamiltonian. In this semiclassical regime, the corrections from sin 2 term are smaller in comparison to those from inverse volume. If p ≫ 2p 0 j then the matter Hamiltonian is also the classical expression. For j = 1/2, there is only the p ≫ p 0 regime and ρ cl is genuinely the classical density.\n\nLet us quickly note a comparison of the two quantizations as reflected in the corresponding effective Hamiltonians, particularly with regards to the extrema of p(t). For this, we will assume same ambiguity parameters (j, l) in the matter Hamiltonian, (1/2)p 2 φ F it is necessary that the quantum geometry potential is present. Thus, for the symmetric ordering, case (A) cannot be realised -it will imply p φ = 0.\n\nAn extremum determined by case (A): It is a bounce if p * is in in the semiclassical regime;\n\np * varies inversely with p φ while the corresponding density varies directly. p * being limited to the semiclassical regime implies that p φ is also bounded both above and below, for such an extremum to occur. It turns out that p * can be in the classical regime, provided p φ ∼ ℓ 2 P . Thus, the non-symmetric constraint, at the effective level, can accommodate a bounce only in the semiclassical regime and with large densities.\n\nAn extremum determined by case (B): It is bounce if p * is in the classical regime; p * varies directly with p φ and the corresponding density varies inversely. p * being limited to the classical regime implies that p φ must be bounded below but can be arbitrarily large and thus the density can be arbitrarily small. This is quite unreasonable and has been sited as one of the reasons for considering the \"improved\" quantization (more on this later). If p * is in the semiclassical regime, it has to be a re-collapse with p φ ∼ ℓ 2 P .\n\nIn the early works, one worked with the non-symmetric constraint operator and the sin 2 corrections were not incorporated (i.e. µ 0 c ≪ π/2 was assumed) and the phenomenological implications were entirely due to the modified matter Hamiltonian. These already implied genericness of inflation and genericness of bounce. These results were discussed at the previous IAGRG meeting in Jaipur. implications: its discrete nature of quantum geometry leads to bounded energy densities and bounded extrinsic and intrinsic curvatures (for the anisotropic models). These two features are construed as \"resolving the classical singularity\". Quite un-expectedly, the effective dynamics incorporating quantum corrections is also singularity-free (via a bounce), accommodates an inflationary phase rather naturally and is well behaved with regards to perturbations. Although there are many ambiguity parameters, these results are robust with respect to their values."
},
{
"section_type": "OTHER",
"section_title": "III. POST 2004 ISOTROPIC LQC",
"text": "Despite many attractive features of LQC, many points need to be addressed further:\n\n• LQC being a constrained theory, it would be more appropriate if singularity resolution is formulated and demonstrated in terms of physical expectation values of physical (Dirac) operators i.e. in terms of \"gauge invariant quantities\". This can be done at present with self-adjoint constraint i.e. a symmetric ordering and for free, massless scalar matter.\n\n• There are at least three distinct ambiguity parameters: µ 0 related to the fiducial length of the loop used in writing the holonomies; j entering in the choice of SU (2) representation which is chosen to be 1/2 in the gravitational sector and some large value in the matter sector; l entering in writing the inverse powers in terms of Poisson brackets. The first one was thought to be determined by the area gap from the full theory. The j = 1/2 in the gravitational Hamiltonian seems needed to avoid high order difference equation and larger j values are hinted to be problematic in the study of a three dimensional model [11] . Given this, the choice of a high value of j in the matter Hamiltonian seems unnatural foot_3 . For phenomenology however the higher values allowing for a larger semiclassical regime are preferred. The l does not play as significant a role.\n\n• The bounce scale and density at the bounce, implied by the effective Hamiltonian (from symmetric ordering), is dependent on the parameters of the matter Hamiltonian and can be arranged such that the bounce density is arbitrarily small. This is a highly undesirable feature. Furthermore, the largest possible domain of validity of WKB approximation is given by the turning points (eg the bounce scale). However, the approximation could break down even before reaching the turning point. An independent check on the domain of validity of effective Hamiltonian is thus desirable.\n\n• A systematic derivation of LQC from LQG is expected to tighten the ambiguity parameters. However, such a derivation is not yet available."
},
{
"section_type": "OTHER",
"section_title": "A. Physical quantities and Singularity Resolution",
"text": "When the Hamiltonian is a constraint, at the classical level itself, the notion of dynamics in terms of the 'time translations' generated by the Hamiltonian is devoid of any physical meaning. Furthermore, at the quantum level when one attempts to impose the constraint as Ĥ|Ψ = 0, typically one finds that there are no solutions in the Hilbert space on which Ĥ is defined -the solutions are generically distributional. One then has to consider the space of all distributional solutions, define a new physical inner product to turn it into a Hilbert space (the physical Hilbert space), define operators on the space of solutions (which must thus act invariantly) which are self-adjoint (physical operators) and compute expectation values, uncertainties etc of these operators to make physical predictions. Clearly, the space of solutions depends on the quantization of the constraint and there is an arbitrariness in the choice of physical inner product. This is usually chosen so that a complete set of Dirac observables (as deduced from the classical theory) are self-adjoint. This is greatly simplified if the constraint has a separable form with respect to some degree of freedom foot_4 . For LQC (and also for the Wheeler-De Witt quantum cosmology), such a simplification is available for a free, massless scalar matter: H matter (φ, p φ ) := 1 2 p 2 φ |p| -3/2 . Let us sketch the steps schematically, focusing on the spatially flat model for simplicity [6, 13] ."
},
{
"section_type": "OTHER",
"section_title": "Fundamental constraint equation:",
"text": "The classical constraint equations is:\n\n- 6 γ 2 c 2 |p| + 8πG p 2 φ |p| -3/2 = 0 = C grav + C matter ; (21)\n\nThe corresponding quantum equation for the wave function, Ψ(p, φ) is:\n\n8πGp 2 φ Ψ(p, φ) = [ B(p)] -1 Ĉgrav Ψ(p, φ) , [ B(p)] is eigenvalue of |p| -3/2 ; ( 22\n\n)\n\nPutting pφ = -i ∂ φ , p := ) -3/2 B(µ), the equation can be written in a separated form as,\n\n∂ 2 Ψ(µ, φ) ∂φ 2 = [B(µ)] -1 8πG γ 6 3/2 ℓ -1 P Ĉgrav Ψ(µ, φ) := -Θ(µ)Ψ(µ, φ). (23)\n\nThe Θ operator for different quantizations is different. For Schrodinger quantization (Wheeler-De Witt), with a particular factor ordering suggested by the continuum limit of the difference equation, the operator Θ(µ) is given by,\n\nΘSch (µ)Ψ(µ, φ) = - 16πG 3 |µ| 3/2 ∂ µ √ µ ∂ µ Ψ(µ, φ) (24)\n\nwhile for LQC, with symmetric ordering, it is given by,\n\nΘLQC (µ)Ψ(µ, φ) = -[B(µ)] -1 C + (µ)Ψ(µ + 4µ 0 , φ) + C 0 (µ)Ψ(µ, φ)+ C -(µ)Ψ(µ -4µ 0 , φ) , C + (µ) := πG 9µ 3 0 |µ + 3µ 0 | 3/2 -|µ + µ 0 | 3/2 , (25)\n\nC -(µ) := C + (µ -4µ 0 ) , C 0 (µ) := -C + (µ) -C -(µ) .\n\nNote that in the Schrodinger quantization, the B Sch (µ) = |µ| -3/2 diverges at µ = 0 while in LQC, B LQC (µ) vanishes for all allowed choices of ambiguity parameters. In both cases, B(µ) ∼ |µ| -3/2 as |µ| → ∞."
},
{
"section_type": "OTHER",
"section_title": "Inner product and General solution:",
"text": "The operator Θ turns out to be a self-adjoint, positive definite operator on the space of functions Ψ(µ, φ) for each fixed φ with an inner product scaled by B(µ). That is, for the Schrodinger quantization, it is an operator on L 2 (R, B Sch (µ)dµ) while for LQC it is an operator on L 2 (R Bohr , B Bohr (µ)dµ Bohr ). Because of this, the operator has a complete set of eigenvectors: Θe\n\nk (µ) = ω 2 (k)e k (µ), k ∈ R, e k |e k ′ = δ(k, k ′ ), and\n\nthe general solution of the fundamental constraint equation can be expressed as\n\nΨ(µ, φ) = dk Ψ+ (k)e k (µ)e iωφ + Ψ-(k)ē k (µ)e -iωφ . ( 26\n\n)\n\nThe orthonormality relations among the e k (µ) are in the corresponding Hilbert spaces.\n\nDifferent quantizations differ in the form of the eigenfunctions, possibly the spectrum itself and of course ω(k). In general, these solutions are not normalizable in L 2 (R Bohr × R, dµ Bohr × dµ), i.e. these are distributional."
},
{
"section_type": "OTHER",
"section_title": "Choice of Dirac observables:",
"text": "Since the classical kinematical phase space is 4 dimensional and we have a single first class constraint, the phase space of physical states (reduced phase space) is two dimensional and we need two functions to coordinatize this space. We should thus look for two (classical) Dirac observables: functions on the kinematical phase space whose\n\nPoisson bracket with the Hamiltonian constraint vanishes on the constraint surface.\n\nIt is easy to see that p φ is a Dirac observable. For the second one, we choose a one parameter family of functions µ(φ) satisfying {µ(φ), C(µ, c, φ, p φ )} ≈ 0. The corresponding quantum definitions, with the operators acting on the solutions, are:\n\npφ Ψ(µ, φ) := -i ∂ φ Ψ(µ, φ) , ( 27\n\n)\n\n|µ| φ 0 Ψ(µ, φ) := e i √ Θ(φ-φ 0 ) |µ|Ψ + (µ, φ 0 ) + e -i √ Θ(φ-φ 0 ) |µ|Ψ -(µ, φ 0 ) (28)\n\nOn an initial datum, Ψ(µ, φ 0 ), these operators act as,\n\n|µ| φ 0 Ψ(µ, φ 0 ) = |µ|Ψ(µ, φ 0 ) , pφ Ψ(µ, φ 0 ) = ΘΨ(µ, φ 0 ) . ( 29\n\n)"
},
{
"section_type": "OTHER",
"section_title": "Physical inner product:",
"text": "It follows that the Dirac operators defined on the space of solutions are self-adjoint if we define a physical inner product on the space of solutions as:\n\nΨ|Ψ ′ phys := \" φ=φ 0 dµB(µ)\" Ψ(µ, φ)Ψ ′ (µ, φ) . ( 30\n\n)\n\nThus the eigenvalues of the inverse volume operator crucially enter the definition of the physical inner product. For Schrodinger quantization, the integral is really an integral while for LQC it is actually a sum over µ taking values in a lattice. The inner product is independent of the choice of φ 0 .\n\nA complete set of physical operators and physical inner product has now been specified\n\nand physical questions can be phrased in terms of (physical) expectation values of functions of these operators."
},
{
"section_type": "OTHER",
"section_title": "Semiclassical states:",
"text": "To discuss semiclassical regime, typically one defines semiclassical states: physical states such that a chosen set of self-adjoint operators have specified expectation values with uncertainties bounded by specified tolerances. A natural choice of operators for us are the two Dirac operators defined above. It is easy to construct semiclassical states with respect to these operators. For example, a state peaked around, p φ = p * φ and |µ| φ 0 = µ * is given by (in Schrodinger quantization for instance),\n\nΨ semi (µ, φ 0 ) := dke -(k-k * ) 2 2σ 2\n\ne k (µ)e iω(φ 0 -φ * ) (31)\n\nk * = -3/2κ -1 p * φ , φ * = φ 0 + -3/2κℓn|µ * | . ( 32\n\n)\n\nFor LQC, the e k (µ) functions are different and the physical expectation values are to be evaluated using the physical inner product defined in the LQC context."
},
{
"section_type": "OTHER",
"section_title": "Evolution of physical quantities:",
"text": "Since one knows the general solution of the constraint equation, Ψ(µ, φ), given Ψ(µ, φ 0 ), one can compute the physical expectation values in the semiclassical solution, Ψ semi (µ, φ) and track the position of the peak as a function of φ as well as the uncertainties as a function of φ."
},
{
"section_type": "OTHER",
"section_title": "Resolution of Big Bang Singularity:",
"text": "A classical solution is obtained as a curve in (µ, φ) plane, different curves being labelled by the points (µ * , φ * ) in the plane. The curves are independent of the constant value of p * φ These curves are already given in (7) .\n\nQuantum mechanically, we first select a semiclassical solution, Ψ semi (p * φ , µ * : φ) in which the expectation values of the Dirac operators, at φ = φ 0 , are p * φ and µ * respectively. These values serve as labels for the semiclassical solution. The former one continues to be p * φ for all φ whereas |µ| φ 0 (φ) =: |µ| p * φ ,µ * (φ), determines a curve in the (µ, φ) plane. In general one expects this curve to be different from the classical curve in the region of small µ (small volume).\n\nThe result of the computations is that Schrodinger quantization, the curve |µ| p * φ ,µ * (φ), does approach the µ = 0 axis asymptotically. However for LQC, the curve bounces away from the µ = 0 axis. In this sense -and now inferred in terms of physical quantities -the Big Bang singularity is resolved in LQC. It also turns out that for large enough values of p * φ , the quantum trajectories constructed by the above procedure are well approximated by the trajectories by the effective Hamiltonian. All these statements are for semiclassical solutions which are peaked at large µ * at late times.\n\nTwo further features are noteworthy as they corroborate the suggestions from the effective Hamiltonian analysis.\n\nFirst one is revealed by computing expectation value of the matter density operator,\n\nρ matter := 1 2 (p * φ ) 2 |p| -3\n\n, at the bounce value of |p|. It turns out that this value is sensitive to the value of p * φ and can be made arbitrarily small by choosing p * φ to be large. Physically this is unsatisfactory as quantum effects are not expected to be significant for matter density very small compared to the Planck density. This is traced to the quantization of the gravitational Hamiltonian, in particular to the step which introduces the ambiguity parameter µ 0 . A novel solution proposed in the \"improved quantization\", removes this undesirable feature.\n\nThe second one refers to the role of quantum modifications in the gravitational Hamiltonian compared to those in the matter Hamiltonian (the inverse volume modification or B(µ)). The former is much more significant than the latter. So much so, that even if one uses the B(µ) from the Schrodinger quantization (i.e. switch-off the inverse volume modifications), one still obtains the bounce. So bounce is seen as the consequence of Θ being different and as far as qualitative singularity resolution is concerned, the inverse volume modifications are un-important. As the effective picture (for symmetric constraint) showed, the bounce occurs in the classical region (for j = 1/2) where the inverse volume corrections can be neglected. For an exact model which seeks to understand as to why the bounces are seen, please see [14] ."
},
{
"section_type": "OTHER",
"section_title": "B. Improved Quantization",
"text": "The undesirable features of the bounce coming from the classical region, can be seen readily using the effective Hamiltonian, as remarked earlier. To see the effects of modifications from the gravitational Hamiltonian, choose j = 1/2 and consider the Friedmann equation derived from the effective Hamiltonian (20) , with matter Hamiltonian given by\n\nH matter = 1 2 p 2 φ |p| -3/2\n\n. The positivity of the effective density implies that p ≥ p * with p * determined by vanishing of the effective energy density:\n\nρ * := ρ cl (p * ) = ( 8πGµ 2 0 γ 2 3 p * ) -1 . This leads to |p * | = 4πGµ 2 0 γ 2 3 |p φ | and ρ * = √ 2( 8πGµ 2 0 γ 2 3\n\n) -3/2 |p φ | -1 . One sees that for large |p φ |, the bounce scale |p * | can be large and the maximum density -density at bounce -could be small. Thus, within the model, there exist a possibility of seeing quantum effects (bounce) even when neither the energy density nor the bounce scale are comparable to the corresponding Planck quantities and this is an undesirable feature of the model. This feature is independent of factor ordering as long as the bounce occurs in the classical regime.\n\nOne may notice that if we replace µ 0 → μ(p) := ∆/|p| where ∆ is a constant, then the effective density vanishes when ρ cl equals the critical value ρ crit := ( 8πG∆γ 2ρ crit ) 1/3 . Now although the bounce scale can again be large depending upon p φ , the density at bounce is always the universal value determined by ∆. This is a rather nice feature in that quantum geometry effects are revealed when matter density (which couples to gravity) reaches a universal, critical value regardless of the dynamical variables describing matter. For a suitable choice of ∆ one can ensure that a bounce always happen when the energy density becomes comparable to the Planck density. In this manner, one can retain the good feature (bounce) even for j = 1/2 thus \"effectively fixing\" an ambiguity parameter and also trade another ambiguity parameter µ 0 for ∆. This is precisely what is achieved by the \"improved quantization\" of the gravitational Hamiltonian [15] .\n\nThe place where the quantization procedure is modified is when one expresses the cur-vature in terms of the holonomies along a loop around a \"plaquette\". One shrinks the plaquette in the limiting procedure. One now makes an important departure: the plaquette should be shrunk only till the physical area (as distinct from a fiducial one) reaches its minimum possible value which is given by the area gap in the known spectrum of area operator in quantum geometry: ∆ = 2 √ 3πγG . Since the plaquette is a square of fiducial length µ 0 , its physical area is µ 2 0 |p| and this should set be to ∆. Since |p| is a dynamical variable, µ 0 cannot be a constant and is to be thought of a function on the phase space, μ(p) := ∆/|p|. It turns out that even with such a change which makes the curvature to be a function of both connection and triad, the form of both the gravitational constraint and inverse volume operator appearing in the matter Hamiltonian, remains the same with just doing the replacement, µ 0 → μ defined above, in the holonomies. The expressions simplify by using eigenfunctions of the volume operator V := | p| 3/2 , instead of those of the triad.\n\nThe relevant expressions are: the the parity symmetry again saves the day); the densities continue to be bounded aboveand now with a bound independent of matter parameters; the effective picture continues to be singularity free and with undesirable features removed and the classical Big Bang being replaced by a quantum bounce is established in terms of physical quantities.\n\nv := Ksgn(µ)|µ| 3/2 , K := 2 √ 2 3 3 √ 3 ; (33)\n\nV |v = γ 6 3/2 ℓ 3 P K |v||v , (34)\n\ne ik μ 2 c Ψ(v) := Ψ(v + k) , (35)\n\n|p| -1/2 j=1/2,l=3/4 Ψ(v) = 3 2\n\nγℓ 2 P 6 -1/2 K 1/3 |v| 1/3 |v + 1| 1/3 -|v -1| 1/3 Ψ(v) (36) B(v) = 3 2 3/2 K|v| |v + 1| 1/3 -|v -1| 1/3 3 (37) ΘImproved Ψ(v, φ) = -[B(v)] -1 C + (v)Ψ(v + 4, φ) + C 0 (v)Ψ(v, φ)+ C -(v)Ψ(v -4, φ) , (38)\n\nC + (v) := 3πKG 8 |v + 2| | |v + 1| -|v + 3|| , (39)\n\nC -(v) := C + (v -4) , C 0 (v) := -C + (v) -C -(v) . (40"
},
{
"section_type": "OTHER",
"section_title": "C. Close Isotropic Model",
"text": "While close model seems phenomenologically disfavoured, it provides further testing ground for quantization of the Hamiltonian constraint. Because of the intrinsic (spatial) curvature, the plaquettes used in expressing the F ij in terms of holonomies, are not bounded by just four edges -a fifth one is necessary. This was attempted and was found to lead to an \"unstable\" quantization. This difficulty was bypassed by using the holonomies of the extrinsic curvature instead of the gauge connection which is permissible in the homogeneous context. The corresponding, non-symmetric constraint and its difference equation was analysed for the massless scalar matter. Green and Unruh, found that solutions of the difference equation was always diverging (at least for one orientation) for large volumes. Further, the divergence seemed to set in just where one expected a re-collapse from the classical theory.\n\nIn the absence of physical inner product and physical interpretation of the solutions, it was concluded that this version of LQC for close model is unlikely to accommodate classical re-collapse even though it avoided the Big Bang/Big Crunch singularities.\n\nRecently, this model has been revisited [16] . One went back to using the gauge connection and the fifth edge difficulty was circumvented by using both the left-invariant and the right-invariant vector fields to define the plaquette. In addition, the symmetric ordering was chosen and finally the µ 0 → μ improvement was also incorporated. Without the improvement, there were still the problems of getting bounce for low energy density and also not getting a reasonable re-collapse (either re-collapse is absent or the scale is marginally larger than the bounce scale). With the improvement, the bounces and re-collapses are neatly accommodated and one gets a cyclic evolution. In this case also, the scalar field serves as a good clock variable as it continues to be monotonic with the synchronous time.\n\nI have focussed on the singularity resolution issue in this talk. Other developments have also taken place in the past couple of years. I will just list these giving references.\n\n1. Effective models and their properties: The effective picture was shown to be non-singular and since this is based on the usual framework of GR, it follows that energy conditions must be violated (and indeed they are thanks to the inverse volume modifications). This raised questions regarding stability of matter and causal propagation of perturbations. Golam Hossain showed that despite the energy conditions violations, neither of the above pathologies result [17] .\n\nMinimally coupled scalar has been used in elaborating inflationary scenarios. However non-minimally coupled scalars are also conceivable models. The singularity resolution and inflationary scenarios continue to hold also in this case. Furthermore sufficient e-foldings are also admissible [18] .\n\nIn the improved quantization, one sets the ambiguity parameter j = 1/2 and shifts the dominant effects to the the gravitational Hamiltonian. All the previous phenomenological implications however were driven by the inverse volume modifications in the matter sector. Consequently, it is necessary to check if and how the phenomenology works with the improved quantization. This has been explored in [19] .\n\nUsing the effective dynamics for the homogeneous mode, density perturbations were explored and power spectra were computed with the required small amplitude [20, 21] .\n\nAs many of the phenomenology oriented questions have been explored using effective\n\nHamiltonian which incorporate quantum corrections from various sources (gravity, matter etc). This motivates a some what systematic approach to constructing effective approximations. This has been initiated in [22] .\n\n2. Anisotropic models: The anisotropic models provide further testing grounds for loop quantization. At the difference equation level, the non-singularity has been checked also for these models in the non-symmetric scheme. For the vacuum Bianchi I model, there is no place for the inverse volume type corrections to appear at an effective Hamiltonian level and the effective dynamics would continue to be singular. However, once the gravitational corrections (sin 2 ) are incorporated, the effective dynamics again is non-singular and one can obtain the non-singular version of the (singular) Kasner solution [23] . More recently, the Bianchi I model with a free, massless scalar is also analysed in the improved quantization [24] . A perturbative treatment of anisotropies has been explored in [25] .\n\n3. Inhomogeneities: Inhomogeneities are a fact of nature although these are small in the early universe. This suggests a perturbative approach to incorporate inhomogeneities.\n\nOn the one hand one can study their evolution in the homogeneous, isotropic background (cosmological perturbation theory). One can also begin with a (simplified) inhomogeneous model and try to see how a homogeneous approximation can become viable. The work on the former has already begun. For the latter part, Bojowald has discussed a simplified lattice model to draw some lessons for the homogeneous models.\n\nIn particular he has given an alternative argument for the µ 0 → μ modification which does not appeal to the area operator [12] ."
},
{
"section_type": "OTHER",
"section_title": "IV. OPEN ISSUES AND OUT LOOK",
"text": "In summary, over the past two years, we have seen how to phrase and understand the fate of Big Bang singularity in a quantum framework.\n\nFirstly, with the help of a minimally coupled, free, massless scalar which serves as a good clock variable in the isotropic context, one can define physical inner product, a complete set of Dirac observables and their physical matrix elements. At present this can be done only for self-adjoint Hamiltonian constraint. Using these, one can construct trajectories in the (p, p φ ) plane which are followed by the peak of a semiclassical state as well as the uncertainties in the Dirac observables. It so happens that these trajectories do not pass through the zero volume -Big Bang is replaced by a Bounce. For close isotropic model, the Big Crunch is also replaced by a bounce while retaining classically understood re-collapse. In conjunction with the μ improvement, the gravitational Hamiltonian can be given the the main role in generating the bounce. A corresponding treatment in Schrodinger quantization (Wheeler-De Witt theory), does not generate a bounce nor does it render the density, curvatures bounded.\n\nThus, quantum representation plays a significant role in the singularity resolution.\n\nSecondly, the improved quantization motivated by the regulation of the F ij invoking the area operator from the full theory (or by the argument from the inhomogeneous lattice model), also leads the bounce to be \"triggered\" when the energy density reaches a critical value (∼ 0.82ρ Planck ) which is independent of the values of the dynamical variables. Close model also gives the same critical value.\n\nWhile the improvement is demonstrated to be viable in the isotropic context, the proce-dure differs from that followed in the full theory. One may either view this as something special to the mini-superspace model(s) or view it as providing hints for newer approaches in the full theory.\n\nA general criteria for \"non-singularity\" is not in sight yet and so also a systematic derivation of the mini-superspace model(s) from a larger, full theory."
}
] | ||
arxiv:0704.0147 | 0704.0147 | 1 | f4e3ecd27fdf2db85fc052451f1350a72df99a3fcaae6faa64830b02aeba16fe | A POVM view of the ensemble approach to polarization optics | Statistical ensemble formalism of Kim, Mandel and Wolf (J. Opt. Soc. Am. A 4,
433 (1987)) offers a realistic model for characterizing the effect of
stochastic non-image forming optical media on the state of polarization of
transmittedlight. With suitable choice of the Jones ensemble, various Mueller
transformations - some of which have been unknown so far - are deduced. It is
observed that the ensemble approach is formally identical to the positive
operator valued measures (POVM) on the quantum density matrix. This
observation, in combination with the recent suggestion by Ahnert and Payne
(Phys. Rev. A 71, 012330, (2005)) - in the context of generalized quantum
measurement on single photon polarization states - that linear optics elements
can be employed in setting up all possible POVMs, enables us to propose a way
of realizing different types of Mueller devices. | [
"Sudha",
"A. V. Gopala Rao",
"A. R. Usha Devi and A. K. Rajagopal"
] | [
"physics.optics",
"physics.class-ph",
"quant-ph"
] | physics.optics | [] | 2007-04-02 | 2026-02-26 | The intensity and polarization of a beam of light passing through an isolated optical device undergoes a linear transformation. But this is an ideal situation because, in general, the optical system is embedded in some media such as atmosphere or other ambient material, which further modifies the polarization properties of the light beam passing through it. A statistical ensemble model describing random linear optical media was formulated two decades ago by Kim, Mandel and Wolf [1] , but is not examined in any detail in the literature, to the best of our knowledge. The purpose of the present paper is to pursue this avenue in a new way arising from the realization of a relationship, presented here, with the positive operator valued measures (POVM) of quantum measurement theory. This is because the transformation of the polarization states of a light beam propagating through an ensemble of deterministic optical devices exhibits a structural similarity with the POVM transformation of quantum density matrices. This connection motivates, in view of the recent interest in the implementations of POVMs on single photon density matrix employing linear optics elements [2] , identification of experimental schemes to realize various kinds of Muller transformations. The properties of the transformation of the polarization states of light form a much studied topic in literature [3 -17] . Thus the power of the ensemble approach becomes evident in elucidating the known optical devices as well as some hitherto unknown types [17] , which had remained only a mathematical possibility.
The contents of this paper are organized as follows. In Sec. 2, a concise formulation of the Jones and Mueller matrix theory, along with a summary of main results of Gopala Rao et al. [17] is given. Based on the approach of Kim, Mandel and Wolf [1] suitable Jones ensembles, corresponding to various types of Mueller transformations are identified in Sec. 3. In Sec. 4, a structural equivalence between Jones ensemble and POVMs of quantum measurement theory is established. Following the linear optics scheme of Ahnert and Payne [2] for the implementation of POVMs on single photon density matrix, experimental setup for realizing Mueller matrices of types I and II are suggested in Sec. 5. The final section has some concluding remarks. 2 2. Brief summary of known results on the Jones and the Mueller formalism.
Following the standard procedure, let E 1 and E 2 , defined here as a column matrix E = E 1 E 2 , denote two components of the transverse electric field vector associated with a light beam. The coherency matrix (or the polarization matrix) of the light beam is a positive semidefinite 2x2 hermitian matrix defined by, defined by Eq. (3) is the well known Stokes vector, which represents the state of polarization of the light beam. Because C is hermitian, the Stokes vector is real. The positive semidefiniteness of C implies that the Stokes vector must satisfy the properties
s 0 > 0, s 2 0 -| s| 2 ≥ 0 ( 4
)
A 2x2 complex matrix J, called the Jones matrix, represents the so-called deterministic optical device [18] or medium. When a light beam represented by E passes through such a medium, the transformed light beam is given by E ′ = JE. Correspondingly, the coherency matrix C transforms as
C ′ = JCJ † ( 5
)
(Here J † is the hermitian conjugate of J.) 3 Alternatively, instead of the 2 × 2 matrix transformation of the coherency matrix, as given by Eq. ( 5 ), a transformation
S ′ = MS ( 6
)
of the four componental Stokes column S through a real 4x4 matrix M, called the Mueller matrix, is found be more useful [18]. Using Eq. ( 3 ) and Eq. (5 we have, s ′ i = Tr(C ′ σ i ) = Tr(JCJ † σ i ) = 1 2 3 j=0 Tr(J † σ i Jσ j )s j which leads to the well-known relationship [1]
M ij = 1 2 Tr(J † σ i Jσ j )
between the elements of a Jones matrix and that of corresponding Mueller matrix.
But in the case where medium cannot be represented by a Jones matrix, it is not possible to characterize the change in the state of polarization of the light beam through Eq. ( 5 ).
In such a situation, Mueller formalism provides a general approach for the polarization transformation of the light beam. The Mueller matrix M is said to be non-deterministic when it has no corresponding Jones characterization.
Mathematically, a Mueller device can be represented by any 4 × 4 matrix such that the Stokes parameters of the outgoing light beam satisfy the physical constraint Eq. ( 4 ). In other words, a Mueller matrix is any 4 ×4 real matrix that transforms a Stokes vector into another Stokes vector. There are many aspects of the relationships between these two formulations of the polarization optics and a complete characterization of Mueller matrices has been the subject matter of Ref. [1, [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] . It was Gopala Rao et al. [17] who presented a complete set of necessary and sufficient conditions for any 4x4 real matrix to be a Mueller matrix. In so doing, they found that there are two algebraic types of Mueller matrices called type I and type II; and it has been shown [17] that only a subset of the type-I Mueller matrices -called deterministic or pure Mueller matrices -have corresponding Jones characterization. All the known polarizing optical devices such as retarders, polarizers, analyzers, optical rotators are pure Mueller type and are well understood. Mueller matrices of the Type II variety are yet to be physically realized and have remained as mere mathematical possibility. For the sake of completeness, we present here the characterization as well as categorization of these two 4 types of Mueller matrices as is given in Ref. [17] . This will enable us to show that both Type I and II Mueller devices are realizable in an unified manner in terms of the proposed ensemble approach [1].
I. A 4 × 4 real matrix M is called a type-I Mueller matrix iff
(i) M 00 ≥ 0
(ii) The G-eigenvalues ρ 0 , ρ 1 , ρ 2 , ρ 3 of the matrix N= MGM are all real. (Here, M stands for the transpose of M; G-eigenvalues are the eigenvalues of the matrix GN,
with G = diag(1, -1, -1, -1)).
iii) The largest G-eigenvalue ρ 0 possesses a time-like G-eigenvector and the G-eigenspace of N contains one time-like and three space-like G-eigenvectors.
II. A 4 × 4 real matrix M is called a type-II Mueller matrix iff
(i) M 00 > 0.
(ii) The G-eigenvalues ρ 0 , ρ 1 , ρ 2 , ρ 3 of N= MGM are all real.
(iii) The largest G-eigenvalue ρ 0 possesses a null G-eigenvector and the G-eigenspace of N contains one null and two space-like G-eigenvectors.
(iv) If X 0 = e 0 + e 1 is the null G-eigenvector of N such that e 0 is a time-like vector with positive zeroth component, e 1 is a space-like vector G-orthogonal to e 0 then
e 0 Ne 0 > 0.
Despite the knowledge of these new category of Mueller matrices [15, 17], not much attention is paid for realizing the corresponding devices. An experimental arrangement involving a parallel combination of deterministic (pure Mueller) optical devices is proposed in Ref. [17] for realizing type-II Mueller devices. The physical situations, where the beam of light is subjected to the influence of a medium such as atmosphere was addressed in Ref. [1] .
In the next section, we discuss this ensemble approach for random optical media, proposed by Kim, Mandel and Wolf [1] . 5 3. Mueller matrices as ensemble of Jones devices Kim et. al. [1] associate a set of probabilities {p e , p e = 1} to describe the stochastic medium. Then a Jones device J e associated with each element e of the ensemble gives a corresponding coherency matrix C ′ e = J e CJ † e . The ensemble averaged coherency matrix
C av = e p e C ′ e = e p e (J e CJ † e ) ( 7
)
then describes the effects of the medium on the beam of light. In a similar fashion, the corresponding ensemble of Mueller matrices {M e } associated with the ensemble of Jones matrices {J e } is constructed and its ensemble averaged Mueller matrix is similarly formed as M av = e p e M e . Since a linear combination of Mueller matrices with non-negative coefficients is also a Mueller matrix, the ensemble averaged Mueller matrix M av is a Mueller matrix 1 .
We now turn to the question of constructing an appropriate ensemble designed to describe a given physical situation. The simplest example of an ensemble is one where the elements are chosen entirely randomly, i.e., the system is described by a chaotic ensemble where the probabilities are all equal, p e = 1 n , where n denotes the number of elements in the ensemble. The coherency matrix C av of the light beam passing through such a chaotic assembly is just an arithmetic average of the coherency matrices C ′ e = J e CJ † e and hence
C av = 1 n n e=1 J e CJ † e ( 8
)
More general models can be constructed depending on the medium for the propagation of the beam of light. For example, one may employ various types of filters or solid state systems through which the light passes; the assignment of the Jones matrices and the corresponding probabilities will then differ depending on the weights placed on these elements.
Restricting ourselves to an ensemble consisting of only two Jones devices which occur with equal probability p 1 = 1/2, p 2 = 1/2, we have found out that the resultant Mueller matrices can either be deterministic or non-deterministic. We give in the foregoing (see Table I ) some examples of Mueller matrices corresponding to different choices of Jones matrices in an ensemble J e , e = 1, 2, for some representative cases. This will also serve to show the 1 This is because, each Mueller matrix M e transforms an initial Stokes vector into a final Stokes vector and a linear combination of Stokes vectors with non-negative coefficients p e is again a Stokes vector.
6 generality of the ensemble procedure in capturing the physical realizations for the Mueller devices discussed in Ref. [17] .
Table 1. Mueller matrices resulting from 2-element Jones ensemble.
J 1 J 2 M = p 1 M 1 + p 2 M 2 , Type of M p 1 = p 2 = 1 2 . 1. 1 √ 6 1 1 -i 1 + i -1 , 1 √ 6 1 1 -i 1 + i -1 1 3 3 0 0 0 0 -1 2 2 0 2 -1 2 0 2 2 -1 Pure Mueller 2. 1 0 0 0 0 1 0 0 1 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
Type-I 3.
1 √ 2 0 1 1 0 1 √ 2 1 0 0 -1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 Type-I 4. 1 √ 3 0 1 1 0 1 √ 3 1 -i i -1 1 6 3 0 0 0 0 -1 0 2 0 0 -1 0 0 2 0 -1 Type-I 5. 1 √ 5 1 1 -i 1 + i -1 1 √ 5 1 -i i -1 1 10 5 0 0 0 0 -1 2 4 0 2 -3 2 0 4 2 -1 Type-I 6. 0 1 0 0 0 0 0 1 1 2 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Type-II 7.
1 2 1 1 1 -1 1 2 1 -i i -1 1 4 2 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 Type-II
In Table I , the Jones matrices chosen are so as to give pure Mueller (deterministic) and non-deterministic type-I, type-II matrices respectively. We observe that an assembly of Jones matrices can result in a pure Mueller matrix if and only if all elements of the assembly correspond to the same optical device. This is because, with all J e 's are same, a transformation of the form C av = e p e (J e CJ † e ) is equivalent to a transformation of the Stokes vector S through a Mueller matrix M av = p e M e = M pure . When the medium is represented by a pure Mueller matrix, the outgoing light beam will have the same degree of 7 polarization as the incoming light beam. In fact, pure Mueller matrix is the simplest among type-I Mueller matrices. Not all type-I Mueller matrices preserve the degree of polarization of the incident light beam. To see this, note that the type-I matrix of example 2 (see Table I ) converts any incident light beam into a linearly polarized light beam; the other three type-I matrices (examples 3 to 5) transform completely polarized light beams into partially polarized light beams. Similarly, type-II Mueller matrices do not, in general, preserve the degree of polarization of the incident light beam. It may be seen that the type-II Mueller matrix of example 7 is a depolarizer matrix, since it converts any incident light beam into an unpolarized light beam.
Though one cannot a priori state which choices of Jones matrices result in type-I or type-II, it is interesting to observe that all types of Mueller matrices result -even in 2-element ensembles. It is not difficult to conclude that an ensemble, with more Jones devices and with different weight factors, can give rise to a variety of Mueller matrices of all possible algebraic types. It would certainly be interesting to physically realize such systems.
In the following section, a connection between the ensemble approach for optical devices and the POVMs of quantum measurement theory is established.
We will now show that the phenomenology of the ensemble construction of Kim, Mandel and Wolf [1] described above has a fundamental theoretical underpinning, if we make a formal identification of the coherency matrix with the density matrix description of the subsystem of a composite quantum system. The coherency matrix defined by Eqs. (1) and (2) resembles a quantum density matrix in that both describe a physical system by a hermitian, trace-class, and positive semi-definite matrix. While the quantum density matrix has unit trace, the coherency matrix has intensity of the beam as the value of the trace. The Jones matrix transformation is a general transformation of the coherency matrix, which preserves its hermiticity and positive semi-definiteness -but changes the values of the elements of the coherency matrix. The most general transformation of the density matrix ρ, which preserves its hermiticity, positive semi-definiteness and also the unit trace is the positive operator valued measures (POVM) [19]:
ρ ′ = n i=1 V i ρV † i ; n i=1 V † i V i = I ( 9
)
8 where V i 's are general matrices and I is the unit element in the Hilbert space. More generally, one could relax the condition of preservation of the unit trace of the density matrix by examining the possibility of a contracting transformation, where the unit matrix condition on the POVM operators is replaced by an inequality.
This mathematical theorem has a physical basis in the Kraus operator formalism [19] when we consider the Hamiltonian description of a composite interacting system A, B described by a density matrix ρ(A, B) and deduce the subsystem density matrix of A given by, ρ(A) = Tr B ρ(A, B). In this case, the Kraus operators are the explicit expressions of the POVM operators and contain the effects of interaction between the systems A and B in the description of the subsystem A. It is thus clear that the phenomenology of Ref. [1] has a correspondence with the Kraus formulation and the POVM theory. In order to make this association complete, we compare Eq. ( 9 ) with the expression given by Eq. ( 7 ). Apart from a phase factor, the Kraus operators {V i }, associated with POVMs, may be related to the Jones assembly {J i }, chosen in the form
V i = √ p i J i , n i=1 V † i V i = n i=1 p i J † i J i ( 10
)
In the construction of the Table I presented earlier, a simple model was proposed where all probabilities were chosen to be equal and the condition on the sum over the Jones matrix combinations was set equal to unit matrix. In such cases, the intensity of the beam gets reduced by 1/n and the polarization properties of the beam gets changed as was described earlier. With this identification, we have provided here an important interpretation and meaning to the phenomenology of the ensemble approach of Kim et al.[1].
Recently Ahnert and Payne [2] proposed an experimental scheme to implement all possible POVMs on single photon polarization states using linear optical elements. In view of the connection between the ensemble formalism for Jones and Mueller matrices with the POVMs, a possible experimental realization of the two types of Mueller matrices is suggested in the next section.
We first observe that the density matrix of a single photon polarization state,
ρ = ρ HH |H H| + ρ HV |H V | + ρ * HV |V H| + ρ V V |V V | ( 11
) 9
is nothing but the coherency matrix of the photon [20]
C = â † H âH â † H âV â † V âH â † V âV , ( 12
)
where âH and âV are the creation operators of the polarization states of the single photon;
{|H , |V } denote the transverse orthogonal polarization states of photon. This is seen explicitly by noting that the average values of the Stokes operators are obtained as,
s 0 = Ŝ0 = (â † H âH + â † V âV ) = ρ HH + ρ V V = Tr(ρ), s 1 = Ŝ1 = (â † H âV + â † V âH ) = ρ HV + ρ * HV = Tr(ρ σ 1 ), s 2 = Ŝ2 = i (â † V âH -â † H âV ) = i (ρ HV -ρ * HV ) = Tr(ρ σ 2 ), s 3 = Ŝ3 = (â † H âH -â † V âV ) = ρ HH -ρ V V = Tr(ρ σ 3 ). ( 13
)
Hence the proposed setup [2], involving only linear optics elements such as polarizing beam splitters, rotators and phase shifters, that promises to implement all possible POVMs on a single photon polarization state leads to all possible ensemble realizations for the Mueller matrices. More specifically, this provides a general experimental scheme to realize varieties of Mueller matrices -including the hitherto unreported type-II Mueller matrices. We briefly describe the scheme proposed in Ref. [2] and illustrate, by way of examples, how it leads to both type-I and type-II Mueller matrices.
In Ref. [2] , a module corresponds to an arrangement having polarization beam splitters, polarization rotators, phase shifters and unitary operators. For an n element POVM, a setup involving n -1 modules are needed. That means, a single module is enough for a 2 element POVM; a setup involving two modules is required for a 3 element POVM and so on.
We describe two, three element POVMs by specifying the optical elements in the respective modules and by specifying the corresponding Kraus operators in terms of these elements.
For any two operator POVM, the Kraus operators
V 1 , V 2 are given by V 1 = U ′ D 1 U and V 2 = U ′′ D 2 U. Here U, U ′ , U ′′
are the three unitary operators in a single module.
Denoting θ, φ as the angles of rotation of the two variable polarization rotators and γ, ξ, the angles of the two variable phase shifters in the module, the diagonal matrices D 1 , D 2 are given by,
D 1 = e iγ cos θ 0 0 cos φ , D 2 = e iξ sin θ 0 0 sin φ ( 14
)
10 The POVM elements
F 1 = V † 1 V 1 = U † D † 1 D 1 U, F 2 = V † 2 V 2 = U † D † 2 D 2 U ( 15
) satisfy the condition i=1,2 F i = F 1 + F 2 = I.
For any three operator POVM, the Kraus operators are given by
V 1 = U ′ I D I U I , V 2 = U ′ II D II U II U ′′ I D ′ I U I , V 3 = U ′′ II D ′ II U II U ′′ I D ′ I U I , ( 16
)
Here, the diagonal D matrices are
D I = e iγ I cos θ I 0 0 cos φ I , D ′ I = e iξ I sin θ I 0 0 sin φ I ( 17
)
and
D II = e iγ II cos θ II 0 0 cos φ II , D ′ II = e iξ II sin θ II 0 0 sin φ II ( 18
) (θ I , φ I ), (γ I , ξ I
) are respectively the pair of angles corresponding to variable polarization rotators and variable phase shifters in the first module. Similarly, (θ II , φ II ), (γ II , ξ II ) are the pairs of angles corresponding to variable polarization rotators and variable phase shifters respectively in the second module. U I , U ′ I , U ′′ I are the unitary operators used in the first module and U II , U ′ II , U ′′ II are the unitary operators used in the second module. (Notice that all the unitary operators in the above schemes are arbitrary and a particular choice of the associated unitary operators gives rise to a different experimental arrangement). The extension of this scheme to n operator POVM involving n-1 modules is quite similar and is given in [2] .
We had identified, in Sec. 3, that an ensemble average of Jones devices will lead to all possible types of Mueller matrices, some examples of which are given in Table 1 . We now show that the experimental set up proposed in Ref. [2] can also be used to realize varieties of Mueller devices. To substantiate our claim, we identify here the linear optical elements needed in the single module set up of Ahnert and Payne [2] , which lead to the physical realization of two typical Mueller matrices given in Table 1 .
To obtain the type-I Mueller matrix M= 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 of example 3 (see Table I ), we
use U = I, U ′ =
These two examples illustrate that the experimental set up given in Ref. [2] may be utilized to realize the required non-determinisitc Mueller devices. In fact Mueller matrices corresponding to an ensemble with more than two Jones devices may also be realized by employing larger number of modules as given in the experimental scheme proposed by Ref. [2].
We have established here a connection between the phenomenological ensemble approach [1] for the coherency matrix and the POVM transformation of quantum density matrix. This opens up a fresh avenue to physically realize types I and II of the Mueller matrix classification of Ref. [17] . We have also given experimental setup to implement Mueller transformations corresponding to ensemble average of Jones devices by employing the POVM scheme on the single photon density matrix suggested in Ref. [2] , in the context of quantum measurement theory. It is gratifying to note that two decades after the introduction of the ensemble approach, which had remained obscure and only received passing reference in textbooks such as [20] , its value is revealed in this paper through its connection with the new developments 12 in quantum measurement theory. We plan on exploring further the POVM transformation in the description of quantum polarization optics. References 1. K. Kim, L. Mandel, and E. Wolf, "Relationship between Jones and Mueller matrices for random media", J. Opt. Soc. Am. A 4, 433-437 (1987).
2. S. E. Ahnert and M. C. Payne, "General implementation of all possible positiveoperator-value measurements of single photon polarization states", Phys. Rev. A 71, 012330-33, (2005). 3. R. Barakat, "Bilinear constraints between elements of the 4 × 4 Mueller-Jones transfer matrix of polarization theory", Opt. Commun. 38,159-161 (1981). 4. R. Simon, "The connection between Mueller and Jones matrices of Polarization Optics", Opt. Commun. 42, 293-297 (1982). 5. A. B. Kostinski, B. James, and W. M. Boerner, "Optimal reception of partially polarized waves" J. Opt. Soc. Am. A 5, 58-64 (1988). 6. A. B. Kostinski, Depolarization criterion for incoherent scattering" Appl. Optics 31, -3508 (1992) . 7. J. J. Gil, and E. Bernabeau, "A depolarization criterion in Mueller matrices" Optica Acta, 32, 259-261 (1985). 8. R. Simon," Mueller matrices and depolarization criteria" J. Mod. Optics 34, 569-575 (1987). 9. R. Simon, "Non-depolarizing systems and degree of polarization" Opt. Commun. 77, -354 (1990) 10. M. Sanjay Kumar, and R. Simon, "Characterization of Mueller matrices in Polarizatio Optics", Optics Commun. 88, 464-470 (1992). 11. R. Sridhar and R. Simon, "Normal form for Mueller matrices in Polarization Optics" J. Mod. Optics 41, 1903-1915 (1994). 12. D. G. M. Anderson, and R. Barakat,"Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix" J. Opt. Soc. Am. A 11, 2305-2319 (1994). 13. C. V. M. van der Mee, and J. W. Hovenier, "Structure of matrices transforming Stokes parameters", J. Math. Phys. 33, 3574-3584 (1992).
14. C. R. Givens, and A. B. Kostinski, "A simple necessary and sufficient criterion on 13 physically realizable Mueller matrices", J. Mod. Opt. 40, 471-481 (1993). 15. C. V. M. van der Mee, "An eigenvalue criterion for matrices transforming Stokes parameters", J. Math. Phys. 34, 5072-5088 (1993). 16. S. R. Cloude, "Group Theory and Polarization algebra", Optik 75, 26-36 (1986). 17. A. V. Gopala Rao, K. S. Mallesh, and Sudha, "On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix" J. Mod. Optics, 45, 955-987 (1998). 18. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized light, (North Holland Publishing Co., Amsterdam, 1977) 19. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2002). 20. L. Mandel and E. Wolf, Quantum Coherence and quantum optics, (Cambridge University Press, Cambridge, 1995).
14 | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The intensity and polarization of a beam of light passing through an isolated optical device undergoes a linear transformation. But this is an ideal situation because, in general, the optical system is embedded in some media such as atmosphere or other ambient material, which further modifies the polarization properties of the light beam passing through it. A statistical ensemble model describing random linear optical media was formulated two decades ago by Kim, Mandel and Wolf [1] , but is not examined in any detail in the literature, to the best of our knowledge. The purpose of the present paper is to pursue this avenue in a new way arising from the realization of a relationship, presented here, with the positive operator valued measures (POVM) of quantum measurement theory. This is because the transformation of the polarization states of a light beam propagating through an ensemble of deterministic optical devices exhibits a structural similarity with the POVM transformation of quantum density matrices. This connection motivates, in view of the recent interest in the implementations of POVMs on single photon density matrix employing linear optics elements [2] , identification of experimental schemes to realize various kinds of Muller transformations. The properties of the transformation of the polarization states of light form a much studied topic in literature [3 -17] . Thus the power of the ensemble approach becomes evident in elucidating the known optical devices as well as some hitherto unknown types [17] , which had remained only a mathematical possibility.\n\nThe contents of this paper are organized as follows. In Sec. 2, a concise formulation of the Jones and Mueller matrix theory, along with a summary of main results of Gopala Rao et al. [17] is given. Based on the approach of Kim, Mandel and Wolf [1] suitable Jones ensembles, corresponding to various types of Mueller transformations are identified in Sec. 3. In Sec. 4, a structural equivalence between Jones ensemble and POVMs of quantum measurement theory is established. Following the linear optics scheme of Ahnert and Payne [2] for the implementation of POVMs on single photon density matrix, experimental setup for realizing Mueller matrices of types I and II are suggested in Sec. 5. The final section has some concluding remarks. 2 2. Brief summary of known results on the Jones and the Mueller formalism.\n\nFollowing the standard procedure, let E 1 and E 2 , defined here as a column matrix E = E 1 E 2 , denote two components of the transverse electric field vector associated with a light beam. The coherency matrix (or the polarization matrix) of the light beam is a positive semidefinite 2x2 hermitian matrix defined by, defined by Eq. (3) is the well known Stokes vector, which represents the state of polarization of the light beam. Because C is hermitian, the Stokes vector is real. The positive semidefiniteness of C implies that the Stokes vector must satisfy the properties\n\ns 0 > 0, s 2 0 -| s| 2 ≥ 0 ( 4\n\n)\n\nA 2x2 complex matrix J, called the Jones matrix, represents the so-called deterministic optical device [18] or medium. When a light beam represented by E passes through such a medium, the transformed light beam is given by E ′ = JE. Correspondingly, the coherency matrix C transforms as\n\nC ′ = JCJ † ( 5\n\n)\n\n(Here J † is the hermitian conjugate of J.) 3 Alternatively, instead of the 2 × 2 matrix transformation of the coherency matrix, as given by Eq. ( 5 ), a transformation\n\nS ′ = MS ( 6\n\n)\n\nof the four componental Stokes column S through a real 4x4 matrix M, called the Mueller matrix, is found be more useful [18]. Using Eq. ( 3 ) and Eq. (5 we have, s ′ i = Tr(C ′ σ i ) = Tr(JCJ † σ i ) = 1 2 3 j=0 Tr(J † σ i Jσ j )s j which leads to the well-known relationship [1]\n\nM ij = 1 2 Tr(J † σ i Jσ j )\n\nbetween the elements of a Jones matrix and that of corresponding Mueller matrix.\n\nBut in the case where medium cannot be represented by a Jones matrix, it is not possible to characterize the change in the state of polarization of the light beam through Eq. ( 5 ).\n\nIn such a situation, Mueller formalism provides a general approach for the polarization transformation of the light beam. The Mueller matrix M is said to be non-deterministic when it has no corresponding Jones characterization.\n\nMathematically, a Mueller device can be represented by any 4 × 4 matrix such that the Stokes parameters of the outgoing light beam satisfy the physical constraint Eq. ( 4 ). In other words, a Mueller matrix is any 4 ×4 real matrix that transforms a Stokes vector into another Stokes vector. There are many aspects of the relationships between these two formulations of the polarization optics and a complete characterization of Mueller matrices has been the subject matter of Ref. [1, [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] . It was Gopala Rao et al. [17] who presented a complete set of necessary and sufficient conditions for any 4x4 real matrix to be a Mueller matrix. In so doing, they found that there are two algebraic types of Mueller matrices called type I and type II; and it has been shown [17] that only a subset of the type-I Mueller matrices -called deterministic or pure Mueller matrices -have corresponding Jones characterization. All the known polarizing optical devices such as retarders, polarizers, analyzers, optical rotators are pure Mueller type and are well understood. Mueller matrices of the Type II variety are yet to be physically realized and have remained as mere mathematical possibility. For the sake of completeness, we present here the characterization as well as categorization of these two 4 types of Mueller matrices as is given in Ref. [17] . This will enable us to show that both Type I and II Mueller devices are realizable in an unified manner in terms of the proposed ensemble approach [1].\n\nI. A 4 × 4 real matrix M is called a type-I Mueller matrix iff\n\n(i) M 00 ≥ 0\n\n(ii) The G-eigenvalues ρ 0 , ρ 1 , ρ 2 , ρ 3 of the matrix N= MGM are all real. (Here, M stands for the transpose of M; G-eigenvalues are the eigenvalues of the matrix GN,\n\nwith G = diag(1, -1, -1, -1)).\n\niii) The largest G-eigenvalue ρ 0 possesses a time-like G-eigenvector and the G-eigenspace of N contains one time-like and three space-like G-eigenvectors.\n\nII. A 4 × 4 real matrix M is called a type-II Mueller matrix iff\n\n(i) M 00 > 0.\n\n(ii) The G-eigenvalues ρ 0 , ρ 1 , ρ 2 , ρ 3 of N= MGM are all real.\n\n(iii) The largest G-eigenvalue ρ 0 possesses a null G-eigenvector and the G-eigenspace of N contains one null and two space-like G-eigenvectors.\n\n(iv) If X 0 = e 0 + e 1 is the null G-eigenvector of N such that e 0 is a time-like vector with positive zeroth component, e 1 is a space-like vector G-orthogonal to e 0 then\n\ne 0 Ne 0 > 0.\n\nDespite the knowledge of these new category of Mueller matrices [15, 17], not much attention is paid for realizing the corresponding devices. An experimental arrangement involving a parallel combination of deterministic (pure Mueller) optical devices is proposed in Ref. [17] for realizing type-II Mueller devices. The physical situations, where the beam of light is subjected to the influence of a medium such as atmosphere was addressed in Ref. [1] .\n\nIn the next section, we discuss this ensemble approach for random optical media, proposed by Kim, Mandel and Wolf [1] . 5 3. Mueller matrices as ensemble of Jones devices Kim et. al. [1] associate a set of probabilities {p e , p e = 1} to describe the stochastic medium. Then a Jones device J e associated with each element e of the ensemble gives a corresponding coherency matrix C ′ e = J e CJ † e . The ensemble averaged coherency matrix\n\nC av = e p e C ′ e = e p e (J e CJ † e ) ( 7\n\n)\n\nthen describes the effects of the medium on the beam of light. In a similar fashion, the corresponding ensemble of Mueller matrices {M e } associated with the ensemble of Jones matrices {J e } is constructed and its ensemble averaged Mueller matrix is similarly formed as M av = e p e M e . Since a linear combination of Mueller matrices with non-negative coefficients is also a Mueller matrix, the ensemble averaged Mueller matrix M av is a Mueller matrix 1 .\n\nWe now turn to the question of constructing an appropriate ensemble designed to describe a given physical situation. The simplest example of an ensemble is one where the elements are chosen entirely randomly, i.e., the system is described by a chaotic ensemble where the probabilities are all equal, p e = 1 n , where n denotes the number of elements in the ensemble. The coherency matrix C av of the light beam passing through such a chaotic assembly is just an arithmetic average of the coherency matrices C ′ e = J e CJ † e and hence\n\nC av = 1 n n e=1 J e CJ † e ( 8\n\n)\n\nMore general models can be constructed depending on the medium for the propagation of the beam of light. For example, one may employ various types of filters or solid state systems through which the light passes; the assignment of the Jones matrices and the corresponding probabilities will then differ depending on the weights placed on these elements.\n\nRestricting ourselves to an ensemble consisting of only two Jones devices which occur with equal probability p 1 = 1/2, p 2 = 1/2, we have found out that the resultant Mueller matrices can either be deterministic or non-deterministic. We give in the foregoing (see Table I ) some examples of Mueller matrices corresponding to different choices of Jones matrices in an ensemble J e , e = 1, 2, for some representative cases. This will also serve to show the 1 This is because, each Mueller matrix M e transforms an initial Stokes vector into a final Stokes vector and a linear combination of Stokes vectors with non-negative coefficients p e is again a Stokes vector.\n\n6 generality of the ensemble procedure in capturing the physical realizations for the Mueller devices discussed in Ref. [17] .\n\nTable 1. Mueller matrices resulting from 2-element Jones ensemble.\n\nJ 1 J 2 M = p 1 M 1 + p 2 M 2 , Type of M p 1 = p 2 = 1 2 . 1. 1 √ 6 1 1 -i 1 + i -1 , 1 √ 6 1 1 -i 1 + i -1 1 3 3 0 0 0 0 -1 2 2 0 2 -1 2 0 2 2 -1 Pure Mueller 2. 1 0 0 0 0 1 0 0 1 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 \n\nType-I 3.\n\n1 √ 2 0 1 1 0 1 √ 2 1 0 0 -1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 Type-I 4. 1 √ 3 0 1 1 0 1 √ 3 1 -i i -1 1 6 3 0 0 0 0 -1 0 2 0 0 -1 0 0 2 0 -1 Type-I 5. 1 √ 5 1 1 -i 1 + i -1 1 √ 5 1 -i i -1 1 10 5 0 0 0 0 -1 2 4 0 2 -3 2 0 4 2 -1 Type-I 6. 0 1 0 0 0 0 0 1 1 2 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \n\nType-II 7.\n\n1 2 1 1 1 -1 1 2 1 -i i -1 1 4 2 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 Type-II\n\nIn Table I , the Jones matrices chosen are so as to give pure Mueller (deterministic) and non-deterministic type-I, type-II matrices respectively. We observe that an assembly of Jones matrices can result in a pure Mueller matrix if and only if all elements of the assembly correspond to the same optical device. This is because, with all J e 's are same, a transformation of the form C av = e p e (J e CJ † e ) is equivalent to a transformation of the Stokes vector S through a Mueller matrix M av = p e M e = M pure . When the medium is represented by a pure Mueller matrix, the outgoing light beam will have the same degree of 7 polarization as the incoming light beam. In fact, pure Mueller matrix is the simplest among type-I Mueller matrices. Not all type-I Mueller matrices preserve the degree of polarization of the incident light beam. To see this, note that the type-I matrix of example 2 (see Table I ) converts any incident light beam into a linearly polarized light beam; the other three type-I matrices (examples 3 to 5) transform completely polarized light beams into partially polarized light beams. Similarly, type-II Mueller matrices do not, in general, preserve the degree of polarization of the incident light beam. It may be seen that the type-II Mueller matrix of example 7 is a depolarizer matrix, since it converts any incident light beam into an unpolarized light beam.\n\nThough one cannot a priori state which choices of Jones matrices result in type-I or type-II, it is interesting to observe that all types of Mueller matrices result -even in 2-element ensembles. It is not difficult to conclude that an ensemble, with more Jones devices and with different weight factors, can give rise to a variety of Mueller matrices of all possible algebraic types. It would certainly be interesting to physically realize such systems.\n\nIn the following section, a connection between the ensemble approach for optical devices and the POVMs of quantum measurement theory is established."
},
{
"section_type": "OTHER",
"section_title": "A connection to Positive Operator Valued Measures",
"text": "We will now show that the phenomenology of the ensemble construction of Kim, Mandel and Wolf [1] described above has a fundamental theoretical underpinning, if we make a formal identification of the coherency matrix with the density matrix description of the subsystem of a composite quantum system. The coherency matrix defined by Eqs. (1) and (2) resembles a quantum density matrix in that both describe a physical system by a hermitian, trace-class, and positive semi-definite matrix. While the quantum density matrix has unit trace, the coherency matrix has intensity of the beam as the value of the trace. The Jones matrix transformation is a general transformation of the coherency matrix, which preserves its hermiticity and positive semi-definiteness -but changes the values of the elements of the coherency matrix. The most general transformation of the density matrix ρ, which preserves its hermiticity, positive semi-definiteness and also the unit trace is the positive operator valued measures (POVM) [19]:\n\nρ ′ = n i=1 V i ρV † i ; n i=1 V † i V i = I ( 9\n\n)\n\n8 where V i 's are general matrices and I is the unit element in the Hilbert space. More generally, one could relax the condition of preservation of the unit trace of the density matrix by examining the possibility of a contracting transformation, where the unit matrix condition on the POVM operators is replaced by an inequality.\n\nThis mathematical theorem has a physical basis in the Kraus operator formalism [19] when we consider the Hamiltonian description of a composite interacting system A, B described by a density matrix ρ(A, B) and deduce the subsystem density matrix of A given by, ρ(A) = Tr B ρ(A, B). In this case, the Kraus operators are the explicit expressions of the POVM operators and contain the effects of interaction between the systems A and B in the description of the subsystem A. It is thus clear that the phenomenology of Ref. [1] has a correspondence with the Kraus formulation and the POVM theory. In order to make this association complete, we compare Eq. ( 9 ) with the expression given by Eq. ( 7 ). Apart from a phase factor, the Kraus operators {V i }, associated with POVMs, may be related to the Jones assembly {J i }, chosen in the form\n\nV i = √ p i J i , n i=1 V † i V i = n i=1 p i J † i J i ( 10\n\n)\n\nIn the construction of the Table I presented earlier, a simple model was proposed where all probabilities were chosen to be equal and the condition on the sum over the Jones matrix combinations was set equal to unit matrix. In such cases, the intensity of the beam gets reduced by 1/n and the polarization properties of the beam gets changed as was described earlier. With this identification, we have provided here an important interpretation and meaning to the phenomenology of the ensemble approach of Kim et al.[1].\n\nRecently Ahnert and Payne [2] proposed an experimental scheme to implement all possible POVMs on single photon polarization states using linear optical elements. In view of the connection between the ensemble formalism for Jones and Mueller matrices with the POVMs, a possible experimental realization of the two types of Mueller matrices is suggested in the next section."
},
{
"section_type": "METHOD",
"section_title": "Possible experimental realization of types I and II Mueller matrices.",
"text": "We first observe that the density matrix of a single photon polarization state,\n\nρ = ρ HH |H H| + ρ HV |H V | + ρ * HV |V H| + ρ V V |V V | ( 11\n\n) 9\n\nis nothing but the coherency matrix of the photon [20]\n\nC = â † H âH â † H âV â † V âH â † V âV , ( 12\n\n)\n\nwhere âH and âV are the creation operators of the polarization states of the single photon;\n\n{|H , |V } denote the transverse orthogonal polarization states of photon. This is seen explicitly by noting that the average values of the Stokes operators are obtained as,\n\ns 0 = Ŝ0 = (â † H âH + â † V âV ) = ρ HH + ρ V V = Tr(ρ), s 1 = Ŝ1 = (â † H âV + â † V âH ) = ρ HV + ρ * HV = Tr(ρ σ 1 ), s 2 = Ŝ2 = i (â † V âH -â † H âV ) = i (ρ HV -ρ * HV ) = Tr(ρ σ 2 ), s 3 = Ŝ3 = (â † H âH -â † V âV ) = ρ HH -ρ V V = Tr(ρ σ 3 ). ( 13\n\n)\n\nHence the proposed setup [2], involving only linear optics elements such as polarizing beam splitters, rotators and phase shifters, that promises to implement all possible POVMs on a single photon polarization state leads to all possible ensemble realizations for the Mueller matrices. More specifically, this provides a general experimental scheme to realize varieties of Mueller matrices -including the hitherto unreported type-II Mueller matrices. We briefly describe the scheme proposed in Ref. [2] and illustrate, by way of examples, how it leads to both type-I and type-II Mueller matrices.\n\nIn Ref. [2] , a module corresponds to an arrangement having polarization beam splitters, polarization rotators, phase shifters and unitary operators. For an n element POVM, a setup involving n -1 modules are needed. That means, a single module is enough for a 2 element POVM; a setup involving two modules is required for a 3 element POVM and so on.\n\nWe describe two, three element POVMs by specifying the optical elements in the respective modules and by specifying the corresponding Kraus operators in terms of these elements.\n\nFor any two operator POVM, the Kraus operators\n\nV 1 , V 2 are given by V 1 = U ′ D 1 U and V 2 = U ′′ D 2 U. Here U, U ′ , U ′′\n\nare the three unitary operators in a single module.\n\nDenoting θ, φ as the angles of rotation of the two variable polarization rotators and γ, ξ, the angles of the two variable phase shifters in the module, the diagonal matrices D 1 , D 2 are given by,\n\nD 1 = e iγ cos θ 0 0 cos φ , D 2 = e iξ sin θ 0 0 sin φ ( 14\n\n)\n\n10 The POVM elements\n\nF 1 = V † 1 V 1 = U † D † 1 D 1 U, F 2 = V † 2 V 2 = U † D † 2 D 2 U ( 15\n\n) satisfy the condition i=1,2 F i = F 1 + F 2 = I.\n\nFor any three operator POVM, the Kraus operators are given by\n\nV 1 = U ′ I D I U I , V 2 = U ′ II D II U II U ′′ I D ′ I U I , V 3 = U ′′ II D ′ II U II U ′′ I D ′ I U I , ( 16\n\n)\n\nHere, the diagonal D matrices are\n\nD I = e iγ I cos θ I 0 0 cos φ I , D ′ I = e iξ I sin θ I 0 0 sin φ I ( 17\n\n)\n\nand\n\nD II = e iγ II cos θ II 0 0 cos φ II , D ′ II = e iξ II sin θ II 0 0 sin φ II ( 18\n\n) (θ I , φ I ), (γ I , ξ I\n\n) are respectively the pair of angles corresponding to variable polarization rotators and variable phase shifters in the first module. Similarly, (θ II , φ II ), (γ II , ξ II ) are the pairs of angles corresponding to variable polarization rotators and variable phase shifters respectively in the second module. U I , U ′ I , U ′′ I are the unitary operators used in the first module and U II , U ′ II , U ′′ II are the unitary operators used in the second module. (Notice that all the unitary operators in the above schemes are arbitrary and a particular choice of the associated unitary operators gives rise to a different experimental arrangement). The extension of this scheme to n operator POVM involving n-1 modules is quite similar and is given in [2] .\n\nWe had identified, in Sec. 3, that an ensemble average of Jones devices will lead to all possible types of Mueller matrices, some examples of which are given in Table 1 . We now show that the experimental set up proposed in Ref. [2] can also be used to realize varieties of Mueller devices. To substantiate our claim, we identify here the linear optical elements needed in the single module set up of Ahnert and Payne [2] , which lead to the physical realization of two typical Mueller matrices given in Table 1 .\n\nTo obtain the type-I Mueller matrix M= 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 of example 3 (see Table I ), we\n\nuse U = I, U ′ = \n\nThese two examples illustrate that the experimental set up given in Ref. [2] may be utilized to realize the required non-determinisitc Mueller devices. In fact Mueller matrices corresponding to an ensemble with more than two Jones devices may also be realized by employing larger number of modules as given in the experimental scheme proposed by Ref. [2]."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusion",
"text": "We have established here a connection between the phenomenological ensemble approach [1] for the coherency matrix and the POVM transformation of quantum density matrix. This opens up a fresh avenue to physically realize types I and II of the Mueller matrix classification of Ref. [17] . We have also given experimental setup to implement Mueller transformations corresponding to ensemble average of Jones devices by employing the POVM scheme on the single photon density matrix suggested in Ref. [2] , in the context of quantum measurement theory. It is gratifying to note that two decades after the introduction of the ensemble approach, which had remained obscure and only received passing reference in textbooks such as [20] , its value is revealed in this paper through its connection with the new developments 12 in quantum measurement theory. We plan on exploring further the POVM transformation in the description of quantum polarization optics. References 1. K. Kim, L. Mandel, and E. Wolf, \"Relationship between Jones and Mueller matrices for random media\", J. Opt. Soc. Am. A 4, 433-437 (1987).\n\n2. S. E. Ahnert and M. C. Payne, \"General implementation of all possible positiveoperator-value measurements of single photon polarization states\", Phys. Rev. A 71, 012330-33, (2005). 3. R. Barakat, \"Bilinear constraints between elements of the 4 × 4 Mueller-Jones transfer matrix of polarization theory\", Opt. Commun. 38,159-161 (1981). 4. R. Simon, \"The connection between Mueller and Jones matrices of Polarization Optics\", Opt. Commun. 42, 293-297 (1982). 5. A. B. Kostinski, B. James, and W. M. Boerner, \"Optimal reception of partially polarized waves\" J. Opt. Soc. Am. A 5, 58-64 (1988). 6. A. B. Kostinski, Depolarization criterion for incoherent scattering\" Appl. Optics 31, -3508 (1992) . 7. J. J. Gil, and E. Bernabeau, \"A depolarization criterion in Mueller matrices\" Optica Acta, 32, 259-261 (1985). 8. R. Simon,\" Mueller matrices and depolarization criteria\" J. Mod. Optics 34, 569-575 (1987). 9. R. Simon, \"Non-depolarizing systems and degree of polarization\" Opt. Commun. 77, -354 (1990) 10. M. Sanjay Kumar, and R. Simon, \"Characterization of Mueller matrices in Polarizatio Optics\", Optics Commun. 88, 464-470 (1992). 11. R. Sridhar and R. Simon, \"Normal form for Mueller matrices in Polarization Optics\" J. Mod. Optics 41, 1903-1915 (1994). 12. D. G. M. Anderson, and R. Barakat,\"Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix\" J. Opt. Soc. Am. A 11, 2305-2319 (1994). 13. C. V. M. van der Mee, and J. W. Hovenier, \"Structure of matrices transforming Stokes parameters\", J. Math. Phys. 33, 3574-3584 (1992).\n\n14. C. R. Givens, and A. B. Kostinski, \"A simple necessary and sufficient criterion on 13 physically realizable Mueller matrices\", J. Mod. Opt. 40, 471-481 (1993). 15. C. V. M. van der Mee, \"An eigenvalue criterion for matrices transforming Stokes parameters\", J. Math. Phys. 34, 5072-5088 (1993). 16. S. R. Cloude, \"Group Theory and Polarization algebra\", Optik 75, 26-36 (1986). 17. A. V. Gopala Rao, K. S. Mallesh, and Sudha, \"On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix\" J. Mod. Optics, 45, 955-987 (1998). 18. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized light, (North Holland Publishing Co., Amsterdam, 1977) 19. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2002). 20. L. Mandel and E. Wolf, Quantum Coherence and quantum optics, (Cambridge University Press, Cambridge, 1995).\n\n14"
}
] | ||
arxiv:0704.0149 | 0704.0149 | 1 | 10.1088/1742-6596/91/1/012001 | babd34aae0c4bd479db9f42c5c013c84b85c0036741df9993f8ea22f00ffc55a | Construction of initial data for 3+1 numerical relativity | This lecture is devoted to the problem of computing initial data for the
Cauchy problem of 3+1 general relativity. The main task is to solve the
constraint equations. The conformal technique, introduced by Lichnerowicz and
enhanced by York, is presented. Two standard methods, the conformal
transverse-traceless one and the conformal thin sandwich, are discussed and
illustrated by some simple examples. Finally a short review regarding initial
data for binary systems (black holes and neutron stars) is given. | [
"Eric Gourgoulhon (LUTH",
"CNRS / Observatoire de Paris / Univ. Paris 7)"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | The 3+1 formalism is the basis of most modern numerical relativity and has lead, along with alternative approaches [82] , to the recent successes in the binary black hole merger problem [6, 7, 99, 25, 26, 27, 28] (see [24, 69, 86] for a review). Thanks to the 3+1 formalism, the resolution of Einstein equation amounts to solving a Cauchy problem, namely to evolve "forward in time" some initial data. However this is a Cauchy problem with constraints. This makes the set up of initial data a non trivial task, because these data must fulfill the constraints. In this lecture, we present the most wide spread methods to deal with this problem. Notice that we do not discuss the numerical techniques employed to solve the constraints (see e.g. Choptuik's lecture for finite differences [32] and Grandclément and Novak's review for spectral methods [58] ).
Standard reviews about the initial data problem are the articles by York [106] and Choquet-Bruhat and York [36] . Recent reviews are the articles by Cook [37] , Pfeiffer [79] and Bartnik and Isenberg [10] .
In this lecture, we consider a spacetime (M, g), where M is a four-dimensional smooth manifold and g a Lorentzian metric on M. We assume that (M, g) is globally Construction of initial data for 3+1 numerical relativity 2 hyperbolic, i.e. that M can be foliated by a family (Σ t ) t∈R of spacelike hypersurfaces. We denote by γ the (Riemannian) metric induced by g on each hypersurface Σ t and K the extrinsic curvature of Σ t , with the same sign convention as that used in the numerical relativity community, i.e. for any pair of vector fields (u, v) tangent to Σ t , g(u, ∇ v n) = -K(u, v), where n is the future directed unit normal to Σ t and ∇ is the Levi-Civita connection associated with g.
The 3+1 decomposition of Einstein equation with respect to the foliation (Σ t ) t∈R leads to three sets of equations: (i) the evolution equations of the Cauchy problem (full projection of Einstein equation onto Σ t ), (ii) the Hamiltonian constraint (full projection of Einstein equation along the normal n), (iii) the momentum constraint (mixed projection: once onto Σ t , once along n). The latter two sets of equations do not contain any second derivative of the metric with respect to t. They are written ‡
R + K 2 -K ij K ij = 16πE (Hamiltonian constraint), ( 1
) D j K ij -D i K = 8πp i (momentum constraint), ( 2
)
where R is the Ricci scalar (also called scalar curvature) associated with the 3-metric γ, K is the trace of K with respect to γ: K = γ ij K ij , D stands for the Levi-Civita connection associated with the 3-metric γ, and E and p i are respectively the energy density and linear momentum of matter, both measured by the observer of 4-velocity n (Eulerian observer). In terms of the matter energy-momentum tensor T they are expressed as
E = T µν n µ n ν and p i = -T µν n µ γ ν i . ( 3
) Notice that Eqs. (1)-(2) involve a single hypersurface Σ 0 , not a foliation (Σ t ) t∈R . In particular, neither the lapse function nor the shift vector appear in these equations.
In order to get valid initial data for the Cauchy problem, one must find solutions to the constraints (1) and (2). Actually one may distinguish two problems:
• The mathematical problem: given some hypersurface Σ 0 , find a Riemannian metric γ, a symmetric bilinear form K and some matter distribution (E, p) on Σ 0 such that the Hamiltonian constraint (1) and the momentum constraint (2) are satisfied. In addition, the matter distribution (E, p) may have some constraints from its own. We shall not discuss them here.
• The astrophysical problem: make sure that the solution to the constraint equations has something to do with the physical system that one wish to study.
Facing the constraint equations (1) and (2), a naive way to proceed would be to choose freely the metric γ, thereby fixing the connection D and the scalar curvature R, and to solve Eqs. (1)-(2) for K. Indeed, for fixed γ, E, and p, Eqs. (1)-(2) form a quasi-linear system of first order for the components K ij . However, as discussed by Choquet-Bruhat [45] , this approach is not satisfactory because we have only four equations for six unknowns K ij and there is no natural prescription for choosing arbitrarily two among the six components K ij . In 1944, Lichnerowicz [70] has shown that a much more satisfactory split of the initial data (γ, K) between freely choosable parts and parts obtained by solving Construction of initial data for 3+1 numerical relativity 3 Eqs. (1)-(2) is provided by a conformal decomposition of the metric γ. Lichnerowicz method has been extended by Choquet-Bruhat (1956 , 1971) [45, 33] , by York and Ó Murchadha (1972, 1974, 1979) [103, 104, 76, 106] and more recently by York and Pfeiffer (1999, 2003) [107, 80] . Actually, conformal decompositions are by far the most widely spread techniques to get initial data for the 3+1 Cauchy problem. Alternative methods exist, such as the quasi-spherical ansatz introduced by Bartnik in 1993 [8] or a procedure developed by Corvino (2000) [39] and by Isenberg, Mazzeo and Pollack (2002) [63] for gluing together known solutions of the constraints, thereby producing new ones. Here we shall limit ourselves to the conformal methods.
In the conformal approach initiated by Lichnerowicz [70] , one introduces a conformal metric γ and a conformal factor Ψ such that the (physical) metric γ induced by the spacetime metric on the hypersurface Σ t is
γ ij = Ψ 4 γij . ( 4
)
We could fix some degree of freedom by demanding that det γij = 1. This would imply Ψ = (det γ ij ) 1/12 . However, in this case γ and Ψ would be tensor densities. Moreover the condition det γij = 1 has a meaning only for Cartesian-like coordinates. In order to deal with tensor fields and to allow for any type of coordinates, we proceed differently and introduce a background Riemannian metric f on Σ t . If the topology of Σ t allows it, we shall demand that f is flat. Then we replace the condition det γij = 1 by det γij = det f ij . This fixes Ψ = det γ ij det f ij 1/12 . (5) Ψ is then a genuine scalar field on Σ t (as a quotient of two determinants). Consequently γ is a tensor field and not a tensor density. Associated with the above conformal transformation, there are two decompositions of the traceless part A ij of the extrinsic curvature, the latter being defined by
K ij =: A ij + 1 3 Kγ ij . ( 6
)
These two decompositions are A ij =: Ψ -10 Âij , (
) 7
A ij =: Ψ -4 Ãij . ( 8
)
The choice -10 for the exponent of Ψ in Eq. ( 7 ) is motivated by the following identity, valid for any symmetric and traceless tensor field,
D j A ij = Ψ -10 Dj Ψ 10 A ij , ( 9
)
where Dj denotes the covariant derivative associated with the conformal metric γ. This choice is well adapted to the momentum constraint, because the latter involves the divergence of K. The alternative choice, i.e. Eq. ( 8 ), is motivated by time evolution considerations, as we shall discuss below. For the time being, we limit ourselves to the decomposition ( 7 ), having in mind to simplify the writing of the momentum constraint.
Construction of initial data for 3+1 numerical relativity 4 By means of the decompositions (4), (6) and (7), the Hamiltonian constraint (1) and the momentum constraint (2) are rewritten as (see Ref. [51] for details) Di Di Ψ -1 8 RΨ + 1 8 Âij Âij Ψ -7 + 2π ẼΨ -3 -1 12 K 2 Ψ 5 = 0, (10) Dj Âij -2 3 Ψ 6 Di K = 8π pi , (
) 11
where R is the Ricci scalar associated with the conformal metric γ and we have introduced the rescaled matter quantities Ẽ := Ψ 8 E and pi := Ψ 10 p i . (12) Equation ( 10) is known as Lichnerowicz equation, or sometimes Lichnerowicz-York equation. The definition of pi is such that there is no Ψ factor in the right-hand side of Eq. ( 11 ). On the contrary the power 8 in the definition of Ẽ is not the only possible choice. As we shall see in § 3.4, it is chosen (i) to guarantee a negative power of Ψ in the Ẽ term in Eq. ( 10 ), resulting in some uniqueness property of the solution and (ii) to allow for an easy implementation of the dominant energy condition.
In order to solve the system (10)-(11), York (1973,1979) [104, 105, 106] has decomposed Âij into a longitudinal part and a transverse one, setting Âij = ( LX) ij + Âij TT , (
) 13
where Âij TT is both traceless and transverse (i.e. divergence-free) with respect to the metric γ:
γij Âij TT = 0 and Dj Âij TT = 0, (14) and ( LX) ij is the conformal Killing operator associated with the metric γ and acting on the vector field X:
( LX) ij := Di X j + Dj X i - 2 3 Dk X k γij . ( 15
)
( LX) ij is by construction traceless: γij ( LX) ij = 0 (16) (it must be so because in Eq. (13) both Âij and Âij TT are traceless). The kernel of L is made of the conformal Killing vectors of the metric γ, i.e. the generators of the conformal isometries (see e.g. Ref. [51] for more details). The symmetric tensor ( LX) ij is called the longitudinal part of Âij , whereas Âij TT is called the transverse part. Given Âij , the vector X is determined by taking the divergence of Eq. (13): taking into account property (14), we get Dj ( LX) ij = Dj Âij . (
) 17
The second order operator Dj ( LX) ij acting on the vector X is the conformal vector Laplacian ∆L :
∆L X i := Dj ( LX) ij = Dj Dj X i + 1 3 Di Dj X j + Ri j X j , ( 18
)
Construction of initial data for 3+1 numerical relativity 5 where the second equality follows from the Ricci identity applied to the connection D, Rij being the associated Ricci tensor. The operator ∆L is elliptic and its kernel is, in practice, reduced to the conformal Killing vectors of γ, if any. We rewrite Eq. (17) as ∆L X i = Dj Âij . (19) The existence and uniqueness of the longitudinal/transverse decomposition (13) depend on the existence and uniqueness of solutions X to Eq. (19). We shall consider two cases:
• Σ 0 is a closed manifold, i.e. is compact without boundary; • (Σ 0 , γ) is an asymptotically flat manifold, i.e. is such that the background metric f is flat (except possibly on a compact sub-domain B of Σ t ) and there exists a coordinate system (x i ) = (x, y, z) on Σ t such that outside B, the components of f are f ij = diag(1, 1, 1) ("Cartesian-type coordinates") and the variable r := x 2 + y 2 + z 2 can take arbitrarily large values on Σ t ; then when r → +∞, the components of γ and K with respect to the coordinates (x i ) satisfy γ ij = f ij + O(r -1 ) and ∂γ ij ∂x k = O(r -2 ), (20) K ij = O(r -2 ) and ∂K ij ∂x k = O(r -3 ). (
) 21
In the case of a closed manifold, one can show (see Appendix B of Ref. [51] for details) that solutions to Eq. (19) exist provided that the source Dj Âij is orthogonal to all conformal Killing vectors of γ, in the sense that ∀C ∈ ker L, Σ γij C i Dk Âjk γ d 3 x = 0. (22) But the above property is easy to verify: using the fact that the source is a pure divergence and that Σ 0 is closed, we may integrate the left-hand side by parts and get, for any vector field C, Σ0 γij C i Dk Âjk γ d 3 x = -1 2 Σ0 γij γkl ( LC) ik Âjl γ d 3 x. (23)
Then, obviously, when C is a conformal Killing vector, the right-hand side of the above equation vanishes. So there exists a solution to Eq. ( 19 ) and this solution is unique up to the addition of a conformal Killing vector. However, given a solution X, for any conformal Killing vector C, the solution X + C yields to the same value of LX, since C is by definition in the kernel of L. Therefore we conclude that the decomposition (13) of Âij is unique, although the vector X may not be if (Σ 0 , γ) admits some conformal isometries. In the case of an asymptotically flat manifold, the existence and uniqueness is guaranteed by a theorem proved by Cantor in 1979 [30] (see also Appendix B of Ref. [87] as well as Refs. [35, 51] ). This theorem requires the decay condition ∂ 2 γij ∂x k ∂x l = O(r -3 ) (24) in addition to the asymptotic flatness conditions (20) . This guarantees that Rij = O(r -3 ). (25) Then all conditions are fulfilled to conclude that Eq. (19) admits a unique solution X which vanishes at infinity. To summarize, for all considered cases (asymptotic flatness and closed manifold), any symmetric and traceless tensor Âij (decaying as O(r -2 ) in the asymptotically flat case) admits a unique longitudinal/transverse decomposition of the form (13).
Construction of initial data for 3+1 numerical relativity 6 3.2. Conformal transverse-traceless form of the constraints Inserting the longitudinal/transverse decomposition (13) into the constraint equations (10) and (11) and making use of Eq. (19) yields to the system Di Di Ψ -1 8 RΨ + 1 8 ( LX) ij + ÂTT ij
( LX) ij + Âij TT Ψ -7 + 2π ẼΨ -3 - 1 12 K 2 Ψ 5 = 0, ( 26
) ∆L X i - 2 3 Ψ 6 Di K = 8π pi , ( 27
)
where ( LX) ij := γik γjl ( LX) kl and ÂTT ij := γik γjl Âkl TT . (
) 28
With the constraint equations written as (26) and (27), we see clearly which part of the initial data on Σ 0 can be freely chosen and which part is "constrained":
• free data:
conformal metric γ; -symmetric traceless and transverse tensor Âij TT (traceless and transverse are meant with respect to γ: γij Âij TT = 0 and Dj Âij TT = 0); -scalar field K; -conformal matter variables: ( Ẽ, pi ); • constrained data (or "determined data"):
conformal factor Ψ, obeying the non-linear elliptic equation (26) (Lichnerowicz equation) -vector X, obeying the linear elliptic equation (27) .
Accordingly the general strategy to get valid initial data for the Cauchy problem is to choose (γ ij , Âij TT , K, Ẽ, pi ) on Σ 0 and solve the system (26)-(27) to get Ψ and X i . Then one constructs
γ ij = Ψ 4 γij ( 29
) K ij = Ψ -10 ( LX) ij + Âij TT + 1 3 Ψ -4 K γij ( 30
) E = Ψ -8 Ẽ (31) p i = Ψ -10 pi ( 32
)
and obtains a set (γ, K, E, p) which satisfies the constraint equations (1)-(2). This method has been proposed by York (1979) [106] and is naturally called the conformal transverse traceless (CTT ) method. Equations (26) and (27) are coupled, but we notice that if, among the free data, we choose K to be a constant field on Σ 0 ,
K = const, ( 33
)
then they decouple partially : condition (33) implies Di K = 0, so that the momentum constraint (27) becomes independent of Ψ:
∆L X i = 8π pi (K = const). ( 34
)
Construction of initial data for 3+1 numerical relativity 7 The condition (33) on the extrinsic curvature of Σ 0 defines what is called a constant mean curvature (CMC ) hypersurface. Indeed let us recall that K is nothing but (minus three times) the mean curvature of (Σ 0 , γ) embedded in (M, g). A maximal hypersurface, having K = 0, is of course a special case of a CMC hypersurface. On a CMC hypersurface, the task of obtaining initial data is greatly simplified: one has first to solve the linear elliptic equation (34) to get X and plug the solution into Eq. (26) to form an equation for Ψ. Equation (34) is the conformal vector Poisson equation discussed above (Eq. (19), with Dj Âij replaced by 8π pi ). We know then that it is solvable for the two cases of interest mentioned in Sec. 3.1: closed or asymptotically flat manifold. Moreover, the solutions X are such that the value of LX is unique.
Taking into account the CMC decoupling, the difficult problem is to solve Eq. (26) for Ψ. This equation is elliptic and highly non-linear §. It has been first studied by Lichnerowicz [70, 71] in the case K = 0 (Σ 0 maximal) and Ẽ = 0 (vacuum). Lichnerowicz has shown that given the value of Ψ at the boundary of a bounded domain of Σ 0 (Dirichlet problem), there exists at most one solution to Eq. (26) . Besides, he showed the existence of a solution provided that Âij Âij is not too large. These early results have been much improved since then. In particular Cantor [29] has shown that in the asymptotically flat case, still with K = 0 and Ẽ = 0, Eq. (26) is solvable if and only if the metric γ is conformal to a metric with vanishing scalar curvature (one says then that γ belongs to the positive Yamabe class) (see also Ref. [74] ). In the case of closed manifolds, the complete analysis of the CMC case has been achieved by Isenberg (1995) [62] .
For more details and further references, we recommend the review articles by Choquet-Bruhat and York [36] and Bartnik and Isenberg [10] . Here we shall simply repeat the argument of York [107] to justify the rescaling (12) of E. This rescaling is indeed related to the uniqueness of solutions to the Lichnerowicz equation. Consider a solution Ψ 0 to Eq. (26) in the case K = 0, to which we restrict ourselves. Another solution close to
Ψ 0 can be written Ψ = Ψ 0 + ǫ, with |ǫ| ≪ Ψ 0 : Di Di (Ψ 0 + ǫ) - R 8 (Ψ 0 + ǫ) + 1 8 Âij Âij (Ψ 0 + ǫ) -7 + 2π Ẽ(Ψ 0 + ǫ) -3 = 0. ( 35
)
Expanding to the first order in ǫ/Ψ 0 leads to the following linear equation for ǫ:
Di Di ǫ -αǫ = 0, ( 36
) with α := 1 8 R + 7 8 Âij Âij Ψ -8 0 + 6π ẼΨ -4 0 . ( 37
)
Now, if α ≥ 0, one can show, by means of the maximum principle, that the solution of (36) which vanishes at spatial infinity is necessarily ǫ = 0 (see Ref. [34] or § B.1 of Ref. [35] ). We therefore conclude that the solution Ψ 0 to Eq. (26) is unique (at least locally) in this case. On the contrary, if α < 0, non trivial oscillatory solutions of Eq. (36) exist, making the solution Ψ 0 not unique. The key point is that the scaling (12) of E yields the term +6π ẼΨ -4 0 in Eq. ( 37 ), which contributes to make α positive. If we had not rescaled E, i.e. had considered the original Hamiltonian constraint, the § although it is quasi-linear in the technical sense, i.e. linear with respect to the highest-order derivatives Construction of initial data for 3+1 numerical relativity 8 contribution to α would have been instead -10πEΨ 4 0 , i.e. would have been negative. Actually, any rescaling Ẽ = Ψ s E with s > 5 would have work to make α positive. The choice s = 8 in Eq. (12) is motivated by the fact that if the conformal data ( Ẽ, pi ) obey the "conformal" dominant energy condition Ẽ ≥ γij pi pj , (38) then, via the scaling (12) of p i , the reconstructed physical data (E, p i ) will automatically obey the dominant energy condition
E ≥ γ ij p i p j . ( 39
)
4. Conformally flat initial data by the CTT method
In this section we search for asymptotically flat initial data (Σ 0 , γ, K) by the CTT method exposed above. As a purpose of illustration, we shall start by the simplest case one may think of, namely choose the freely specifiable data (γ ij , Âij TT , K, Ẽ, pi ) to be a flat metric:
γij = f ij , ( 40
)
a vanishing transverse-traceless part of the extrinsic curvature: Âij TT = 0, (41) a vanishing mean curvature (maximal hypersurface)
K = 0, ( 42
)
and a vacuum spacetime: Ẽ = 0, pi = 0. (43)
Then Di = D i , where D denotes the Levi-Civita connection associated with f , R = 0 (f is flat) and the constraint equations (26)-(27) reduce to
∆Ψ + 1 8 (LX) ij (LX) ij Ψ -7 = 0 (44) ∆ L X i = 0, ( 45
)
where ∆ and ∆ L are respectively the scalar Laplacian and the conformal vector Laplacian associated with the flat metric f :
∆ := D i D i and ∆ L X i := D j D j X i + 1 3 D i D j X j . ( 46
)
Equations (44)-(45) must be solved with the boundary conditions
Ψ = 1 when r → ∞ ( 47
) X = 0 when r → ∞, ( 48
)
which follow from the asymptotic flatness requirement. The solution depends on the topology of Σ 0 , since the latter may introduce some inner boundary conditions in addition to (47)-(48).
Let us start with the simplest case: Σ 0 = R 3 . Then the unique solution of Eq. (45) subject to the boundary condition (48) is
X = 0. ( 49
)
Construction of initial data for 3+1 numerical relativity 9 Figure 1. Hypersurface Σ 0 as R 3 minus a ball, displayed via an embedding diagram based on the metric γ, which coincides with the Euclidean metric on R 3 . Hence Σ 0 appears to be flat. The unit normal of the inner boundary S with respect to the metric γ is s. Notice that D • s > 0.
Consequently (LX) ij = 0, so that Eq. ( 44 ) reduces to Laplace equation for Ψ:
∆Ψ = 0. ( 50
)
With the boundary condition (47), there is a unique regular solution on R 3 :
Ψ = 1. ( 51
)
The initial data reconstructed from Eqs. (29)-(30) is then
γ = f ( 52
) K = 0. ( 53
)
These data correspond to a spacelike hyperplane of Minkowski spacetime. Geometrically the condition K = 0 is that of a totally geodesic hypersurface [i.e. all the geodesics of (Σ t , γ) are geodesics of (M, g)]. Physically data with K = 0 are said to be momentarily static or time symmetric. Indeed, if we consider a foliation with unit lapse around Σ 0 (geodesic slicing), the following relation holds: L n g = -2K, where L n denotes the Lie derivative along the unit normal n. So if K = 0, L n g = 0. This means that, locally (i.e. on Σ 0 ), n is a spacetime Killing vector. This vector being timelike, the configuration is then stationary. Moreover, the Killing vector n being orthogonal to some hypersurface (i.e. Σ 0 ), the stationary configuration is called static. Of course, this staticity properties holds a priori only on Σ 0 since there is no guarantee that the time development of Cauchy data with K = 0 at t = 0 maintains K = 0 at t > 0. Hence the qualifier 'momentarily' in the expression 'momentarily static' for data with K = 0.
To get something less trivial than a slice of Minkowski spacetime, let us consider a slightly more complicated topology for Σ 0 , namely R 3 minus a ball (cf. Fig. 1 ). The sphere S delimiting the ball is then the inner boundary of Σ 0 and we must provide boundary conditions for Ψ and X on S to solve Eqs. (44)-(45). For simplicity, let us choose
X| S = 0. ( 54
)
Altogether with the outer boundary condition (48) , this leads to X being identically zero as the unique solution of Eq. ( 45 ). So, again, the Hamiltonian constraint reduces to Laplace equation
∆Ψ = 0. ( 55
)
Construction of initial data for 3+1 numerical relativity 10 Figure 2. Same hypersurface Σ 0 as in Fig. 1 but displayed via an embedding diagram based on the metric γ instead of γ. The unit normal of the inner boundary S with respect to that metric is s. Notice that D • s = 0, which means that S is a minimal surface of (Σ 0 , γ).
If we choose the boundary condition Ψ| S = 1, then the unique solution is Ψ = 1 and we are back to the previous example (slice of Minkowski spacetime). In order to have something non trivial, i.e. to ensure that the metric γ will not be flat, let us demand that γ admits a closed minimal surface, that we will choose to be S. This will necessarily translate as a boundary condition for Ψ since all the information on the metric is encoded in Ψ (let us recall that from the choice (40), γ = Ψ 4 f ). S is a minimal surface of (Σ 0 , γ) iff its mean curvature vanishes, or equivalently if its unit normal s is divergence-free (cf. Fig. 2 ):
D i s i S = 0. ( 56
)
This is the analog of ∇ • n = 0 for maximal hypersurfaces, the change from minimal to maximal being due to the change of metric signature, from the Riemannian to the Lorentzian one. Expressed in term of the connection D = D (recall that in the present case γ = f ), condition (56) is equivalent to
D i (Ψ 6 s i ) S = 0. ( 57
)
Let us rewrite this expression in terms of the unit vector s normal to S with respect to the metric γ (cf. Fig. 1 ); we have
s = Ψ -2 s, ( 58
) since γ(s, s) = Ψ -4 γ(s, s) = γ(s, s) = 1. Thus Eq. (57) becomes D i (Ψ 4 si ) S = 1 √ f ∂ ∂x i f Ψ 4 si S = 0. ( 59
)
Let us introduce on Σ 0 a coordinate system of spherical type, (x i ) = (r, θ, ϕ), such that (i) f ij = diag(1, r 2 , r 2 sin 2 θ) and (ii) S is the sphere r = a, where a is some positive constant. Since in these coordinates √ f = r 2 sin θ and si = (1, 0, 0), the minimal surface condition (59) is written as
1 r 2 ∂ ∂r Ψ 4 r 2 r=a = 0, ( 60
)
Construction of initial data for 3+1 numerical relativity 11 Figure 3. Extended hypersurface Σ ′ 0 obtained by gluing a copy of Σ 0 at the minimal surface S; it defines an Einstein-Rosen bridge between two asymptotically flat regions.
∂Ψ ∂r + Ψ 2r r=a = 0 (61) This is a boundary condition of mixed Newmann/Dirichlet type for Ψ. The unique solution of the Laplace equation (55) which satisfies boundary conditions (47) and (61) is
Ψ = 1 + a r . ( 62
)
The parameter a is then easily related to the ADM mass m of the hypersurface Σ 0 . Indeed for a conformally flat 3-metric (and more generally in the quasi-isotropic gauge, cf. Chap. 7 of Ref. [51] ), the ADM mass m is given by the flux of the gradient of the conformal factor at spatial infinity:
m = - 1 2π lim r→∞ r=const ∂Ψ ∂r r 2 sin θ dθ dϕ = - 1 2π lim r→∞ 4πr 2 ∂ ∂r 1 + a r = 2a. ( 63
)
Hence a = m/2 and we may write
Ψ = 1 + m 2r . ( 64
)
Therefore, in terms of the coordinates (r, θ, ϕ), the obtained initial data (γ, K) are
γ ij = 1 + m 2r 4 diag(1, r 2 , r 2 sin θ) ( 65
) K ij = 0. ( 66
)
So, as above, the initial data are momentarily static. Actually, we recognize on (65)-(66) a slice t = const of Schwarzschild spacetime in isotropic coordinates. The isotropic coordinates (r, θ, ϕ) covering the manifold Σ 0 are such that the range of r is [m/2, +∞). But thanks to the minimal character of the inner boundary S, we can extend (Σ
0 , γ) to a larger Riemannian manifold (Σ ′ 0 , γ ′ ) with γ ′ | Σ0 = γ
and γ ′ smooth at S. This is made possible by gluing a copy of Σ 0 at S (cf. Fig. 3 ).
Construction of initial data for 3+1 numerical relativity 12 Figure 4.
Extended hypersurface Σ ′ 0 depicted in the Kruskal-Szekeres representation of Schwarzschild spacetime. R stands for Schwarzschild radial coordinate and r for the isotropic radial coordinate. R = 0 is the singularity and R = 2m the event horizon. Σ ′ 0 is nothing but a hypersurface t = const, where t is the Schwarzschild time coordinate. In this diagram, these hypersurfaces are straight lines and the Einstein-Rosen bridge S is reduced to a point.
The topology of Σ ′ 0 is S 2 × R and the range of r in Σ ′ 0 is (0, +∞). The extended metric γ ′ keeps exactly the same form as (65):
γ ′ ij dx i dx j = 1 + m 2r 4 dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 . ( 67
)
By the change of variable
r → r ′ = m 2 4r ( 68
)
it is easily shown that the region r → 0 does not correspond to some "center" but is actually a second asymptotically flat region (the lower one in Fig. 3 ). Moreover the transformation (68), with θ and ϕ kept fixed, is an isometry of γ ′ . It maps a point p of Σ 0 to the point located at the vertical of p in Fig. 3 . The minimal sphere S is invariant under this isometry. The region around S is called an Einstein-Rosen bridge. (Σ ′ 0 , γ ′ ) is still a slice of Schwarzschild spacetime. It connects two asymptotically flat regions without entering below the event horizon, as shown in the Kruskal-Szekeres diagram of Fig. 4.
Let us select the same simple free data as above, namely γij = f ij , Âij TT = 0, K = 0, Ẽ = 0 and pi = 0. (69) For the hypersurface Σ 0 , instead of R 3 minus a ball, we choose R 3 minus a point:
Σ 0 = R 3 \{O}. ( 70
)
The removed point O is called a puncture [21] . The topology of Σ 0 is S 2 × R; it differs from the topology considered in Sec. 4.1 (R 3 minus a ball); actually it is the same topology as that of the extended manifold Σ ′ 0 (cf. Fig. 3 ).
Construction of initial data for 3+1 numerical relativity 13 Thanks to the choice (69), the system to be solved is still (44)-(45). If we choose the trivial solution X = 0 for Eq. ( 45 ), we are back to the slice of Schwarzschild spacetime considered in Sec. 4.1, except that now Σ 0 is the extended manifold previously denoted Σ ′ 0 . Bowen and York [20] have obtained a simple non-trivial solution to the momentum constraint (45) (see also Ref. [15] ). Given a Cartesian coordinate system (x i ) = (x, y, z) on Σ 0 (i.e. a coordinate system such that f ij = diag (1, 1, 1) ) with respect to which the coordinates of the puncture O are (0, 0, 0), this solution writes
X i = - 1 4r 7f ij P j + P j x j x i r 2 - 1 r 3 ǫ ij k S j x k , ( 71
)
where r := x 2 + y 2 + z 2 , ǫ ij k is the Levi-Civita alternating tensor associated with the flat metric f and
(P i , S j ) = (P 1 , P 2 , P 3 , S 1 , S 2 , S 3
) are six real numbers, which constitute the six parameters of the Bowen-York solution. Notice that since r = 0 on Σ 0 , the Bowen-York solution is a regular and smooth solution on the entire Σ 0 .
The conformal traceless extrinsic curvature corresponding to the solution (71) is deduced from formula (13), which in the present case reduces to Âij = (LX) ij ; one gets
Âij = 3 2r 3 x i P j + x j P i -f ij - x i x j r 2 P k x k + 3 r 5 ǫ ik l S k x l x j + ǫ jk l S k x l x i , ( 72
)
where P i := f ij P j . The tensor Âij given by Eq. (72) is called the Bowen-York extrinsic curvature. Notice that the P i part of Âij decays asymptotically as O(r -2 ), whereas the S i part decays as O(r -3 ).
Remark : Actually the expression of Âij given in the original Bowen-York article [20] contains an additional term with respect to Eq. ( 72 ), but the role of this extra term is only to ensure that the solution is isometric through an inversion across some sphere. We are not interested by such a property here, so we have dropped this term. Therefore, strictly speaking, we should name expression (72) the simplified Bowen-York extrinsic curvature.
The Bowen-York extrinsic curvature provides an analytical solution of the momentum constraint (45) but there remains to solve the Hamiltonian constraint (44) for Ψ, with the asymptotic flatness boundary condition Ψ = 1 when r → ∞. Since X = 0, Eq. (44) is no longer a simple Laplace equation, as in Sec. 4.1, but a non-linear elliptic equation. There is no hope to get any analytical solution and one must solve Eq. (44) numerically to get Ψ and reconstruct the full initial data (γ, K) via Eqs. (29)- (30) .
The parameters P i of the Bowen-York solution are nothing but the three components of the ADM linear momentum of the hypersurface Σ 0 Similarly, the parameters S i of the Bowen-York solution are nothing but the three components of the angular momentum of the hypersurface Σ 0 , the latter being defined relatively to the quasi-isotropic gauge, in the absence of any axial symmetry (see e.g. [51]).
Remark : The Bowen-York solution with P i = 0 and S i = 0 reduces to the momentarily static solution found in Sec. 4.1, i.e. is a slice t = const of the Schwarzschild spacetime (t being the Schwarzschild time coordinate). However Bowen-York initial data with P i = 0 and S i = 0 do not constitute a slice of Kerr spacetime. Indeed, it has been shown [47] that there does not exist any foliation of Kerr spacetime by hypersurfaces which (i) are axisymmetric, (ii) smoothly Construction of initial data for 3+1 numerical relativity 14 reduce in the non-rotating limit to the hypersurfaces of constant Schwarzschild time and (iii) are conformally flat, i.e. have induced metric γ = f , as the Bowen-York hypersurfaces have. This means that a Bowen-York solution with S i = 0 does represent initial data for a rotating black hole, but this black hole is not stationary: it is "surrounded" by gravitational radiation, as demonstrated by the time development of these initial data [22, 49] .
An alternative to the conformal transverse-traceless method for computing initial data has been introduced by York in 1999 [107] . The starting point is the identity
K = - 1 2N L N n γ = - 1 2N ∂ ∂t -L β γ, ( 73
)
where N is the lapse function and β is the shift vector associated with some 3+1 coordinates (t, x i ). The traceless part of Eq. (73) leads to
Ãij = 1 2N ∂ ∂t -L β γij - 2 3 Dk β k γij , ( 74
)
where Ãij is defined by Eq. ( 8 ). Noticing that
-L β γij = ( Lβ) ij + 2 3 Dk β k , ( 75
)
and introducing the short-hand notation γij := ∂ ∂t γij , (76) we can rewrite Eq. (74) as Ãij = 1 2N γij + ( Lβ
) ij . ( 77
)
The relation between Ãij and Âij is [cf. Eqs. ( 7
)-(8)] Âij = Ψ 6 Ãij . ( 78
) Accordingly, Eq. (77) yields Âij = 1 2 Ñ γij + ( Lβ) ij , ( 79
)
where we have introduced the conformal lapse Ñ := Ψ -6 N. (80) Equation (79) constitutes a decomposition of Âij alternative to the longitudinal/transverse decomposition (13). Instead of expressing Âij in terms of a vector X and a TT tensor Âij TT , it expresses it in terms of the shift vector β, the time derivative of the conformal metric, γij , and the conformal lapse Ñ . The Hamiltonian constraint, written as the Lichnerowicz equation (10), takes the same form as before:
Di Di Ψ - R 8 Ψ + 1 8 Âij Âij Ψ -7 + 2π ẼΨ -3 - K 2 12 Ψ 5 = 0, ( 81
)
Construction of initial data for 3+1 numerical relativity 15 except that now Âij is to be understood as the combination (79) of β i , γij and Ñ . On the other side, the momentum constraint (11) becomes, once expression (79) is substituted for Âij , Dj 1 Ñ ( Lβ) ij + Dj
1 Ñ γij - 4 3 Ψ 6 Di K = 16π pi . ( 82
)
In view of the system (81)-(82), the method to compute initial data consists in choosing freely γij , γij , K, Ñ , Ẽ and pi on Σ 0 and solving (81)-(82) to get Ψ and β i . This method is called conformal thin sandwich (CTS ), because one input is the time derivative γij , which can be obtained from the value of the conformal metric on two neighbouring hypersurfaces Σ t and Σ t+δt ("thin sandwich" view point).
Remark : The term "thin sandwich" originates from a previous method devised in the early sixties by Wheeler and his collaborators [4, 101] . Contrary to the methods exposed here, the thin sandwich method was not based on a conformal decomposition: it considered the constraint equations (1)-(2) as a system to be solved for the lapse N and the shift vector β, given the metric γ and its time derivative. The extrinsic curvature which appears in (1)-(2) was then considered as the function of γ, ∂γ/∂t, N and β given by Eq. (73). However, this method does not work in general [9] . On the contrary the conformal thin sandwich method introduced by York [107] and exposed above was shown to work [35] .
As for the conformal transverse-traceless method treated in Sec. 3, on CMC hypersurfaces, Eq. (82) decouples from Eq. ( 81 ) and becomes an elliptic linear equation for β.
An input of the above method is the conformal lapse Ñ . Considering the astrophysical problem stated in Sec. 2.2, it is not clear how to pick a relevant value for Ñ . Instead of choosing an arbitrary value, Pfeiffer and York [80] have suggested to compute Ñ from the Einstein equation giving the time derivative of the trace K of the extrinsic curvature, i.e.
∂ ∂t -L β K = -Ψ -4 Di Di N + 2 Di ln Ψ Di N + N 4π(E + S) + Ãij Ãij + K 2 3 , ( 83
)
where S is the trace of the matter stress tensor as measured by the Eulerian observer: S = γ µν T µν . This amounts to add this equation to the initial data system. More precisely, Pfeiffer and York [80] suggested to combine Eq. ( 83 ) with the Hamiltonian constraint to get an equation involving the quantity N Ψ = Ñ Ψ 7 and containing no scalar products of gradients as the Di ln Ψ Di N term in Eq. ( 83 ), thanks to the identity
Di Di N + 2 Di ln Ψ Di N = Ψ -1 Di Di (N Ψ) + N Di Di Ψ . ( 84
)
Expressing the left-hand side of the above equation in terms of Eq. ( 83 ) and substituting Di Di Ψ in the right-hand side by its expression deduced from Eq. ( 81 ), Construction of initial data for 3+1 numerical relativity 16 we get
Di Di ( Ñ Ψ 7 ) -( Ñ Ψ 7 ) 1 8 R + 5 12 K 2 Ψ 4 + 7 8 Âij Âij Ψ -8 + 2π( Ẽ + 2 S)Ψ -4 + K -β i Di K Ψ 5 = 0, ( 85
)
where we have used the short-hand notation K := ∂K ∂t (86) and have set S := Ψ 8 S. (87) Adding Eq. (85) to Eqs. (81) and (82), the initial data system becomes
Di Di Ψ - R 8 Ψ + 1 8 Âij Âij Ψ -7 + 2π ẼΨ -3 - K 2 12 Ψ 5 = 0 ( 88
) Dj 1 Ñ ( Lβ) ij + Dj 1 Ñ γij - 4 3 Ψ 6 Di K = 16π pi ( 89
) Di Di ( Ñ Ψ 7 ) -( Ñ Ψ 7 ) R 8 + 5 12 K 2 Ψ 4 + 7 8 Âij Âij Ψ -8 + 2π( Ẽ + 2 S)Ψ -4 + K -β i Di K Ψ 5 = 0, ( 90
)
where Âij is the function of Ñ , β i , γij and γij defined by Eq. (79). Equations (88)-(90) constitute the extended conformal thin sandwich (XCTS ) system for the initial data problem. The free data are the conformal metric γ, its coordinate time derivative γ, the extrinsic curvature trace K, its coordinate time derivative K, and the rescaled matter variables Ẽ, S and pi . The constrained data are the conformal factor Ψ, the conformal lapse Ñ and the shift vector β.
Remark : The XCTS system (88)-(90) is a coupled system. Contrary to the CTT system (26)-(27), the assumption of constant mean curvature, and in particular of maximal slicing, does not allow to decouple it.
Let us illustrate the extended conformal thin sandwich method on a simple example. Take for the hypersurface Σ 0 the punctured manifold considered in Sec. 4.3, namely
Σ 0 = R 3 \{O}. ( 91
)
For the free data, let us perform the simplest choice:
γij = f ij , γij = 0, K = 0, K = 0, Ẽ = 0, S = 0, and pi = 0, ( 92
)
i.e. we are searching for vacuum initial data on a maximal and conformally flat hypersurface with all the freely specifiable time derivatives set to zero. Thanks to (92), the XCTS system (88)-(90) reduces to
∆Ψ + 1 8 Âij Âij Ψ -7 = 0 ( 93
) D j 1 Ñ (Lβ) ij = 0 ( 94
) ∆( Ñ Ψ 7 ) - 7 8 Âij Âij Ψ -1 Ñ = 0. ( 95
)
Construction of initial data for 3+1 numerical relativity 17 Aiming at finding the simplest solution, we notice that β = 0 (96) is a solution of Eq. ( 94 ). Together with γij = 0, it leads to [cf. Eq. (79)] Âij = 0. (97)
The system (93)-(95) reduces then further:
∆Ψ = 0 ( 98
) ∆( Ñ Ψ 7 ) = 0. ( 99
)
Hence we have only two Laplace equations to solve. Moreover Eq. (98) decouples from Eq. ( 99 ). For simplicity, let us assume spherical symmetry around the puncture O. We introduce an adapted spherical coordinate system (x i ) = (r, θ, ϕ) on Σ 0 . The puncture O is then at r = 0. The simplest non-trivial solution of (98) which obeys the asymptotic flatness condition Ψ → 1 as r → +∞ is
Ψ = 1 + m 2r , ( 100
)
where as in Sec. 4.1, the constant m is the ADM mass of Σ 0 [cf. Eq. (63)]. Notice that since r = 0 is excluded from Σ 0 , Ψ is a perfectly regular solution on the entire manifold Σ 0 . Let us recall that the Riemannian manifold (Σ 0 , γ) corresponding to this value of Ψ via γ = Ψ 4 f is the Riemannian manifold denoted (Σ ′ 0 , γ) in Sec. 4.1 and depicted in Fig. 3 . In particular it has two asymptotically flat ends: r → +∞ and r → 0 (the puncture).
As for Eq. ( 98 ), the simplest solution of Eq. (99) obeying the asymptotic flatness requirement Ñ Ψ 7 → 1 as r → +∞ is
Ñ Ψ 7 = 1 + a r , ( 101
)
where a is some constant. Let us determine a from the value of the lapse function at the second asymptotically flat end r → 0. The lapse being related to Ñ via Eq. (80), Eq. ( 101 ) is equivalent to
N = 1 + a r Ψ -1 = 1 + a r 1 + m 2r -1 = r + a r + m/2 . ( 102
) Hence lim r→0 N = 2a m . ( 103
)
There are two natural choices for lim r→0 N . The first one is lim r→0
N = 1, ( 104
)
yielding a = m/2. Then, from Eq. (102) N = 1 everywhere on Σ 0 . This value of N corresponds to a geodesic slicing. The second choice is
lim r→0 N = -1. ( 105
)
This choice is compatible with asymptotic flatness: it simply means that the coordinate time t is running "backward" near the asymptotic flat end r → 0. This contradicts the assumption N > 0 in the standard definition of the lapse function. However, we shall generalize here the definition of the lapse to allow for negative values: whereas the unit vector n is always future-oriented, the scalar field t is allowed to decrease towards the future. Such a situation has already been encountered for the Construction of initial data for 3+1 numerical relativity 18 part of the slices t = const located on the left side of Fig. 4. Once reported into Eq. ( 103 ), the choice (105) yields a = -m/2, so that
N = 1 - m 2r 1 + m 2r -1 . ( 106
)
Gathering relations (96), (100) and (106), we arrive at the following expression of the spacetime metric components:
g µν dx µ dx ν = - 1 -m 2r 1 + m 2r 2 dt 2 + 1 + m 2r 4 dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ) . ( 107
)
We recognize the line element of Schwarzschild spacetime in isotropic coordinates. Hence we recover the same initial data as in Sec. 4.1 and depicted in Figs. 3 and 4. The bonus is that we have the complete expression of the metric g on Σ 0 , and not only the induced metric γ.
Remark : The choices (104) and (105) for the asymptotic value of the lapse both lead to a momentarily static initial slice in Schwarzschild spacetime. The difference is that the time development corresponding to choice (104) (geodesic slicing) will depend on t, whereas the time development corresponding to choice (105) will not, since in the latter case t coincides with the standard Schwarzschild time coordinate, which makes ∂ t a Killing vector.
Recently, Pfeiffer and York [81] have exhibited a choice of vacuum free data (γ ij , γij , K, K) for which the solution (Ψ, Ñ , β i ) to the XCTS system (88)-(90) is not unique (actually two solutions are found). The conformal metric γ is the flat metric plus a linearized quadrupolar gravitational wave, as obtained by Teukolsky [92], with a tunable amplitude.
γij corresponds to the time derivative of this wave, and both K and K are chosen to zero. On the contrary, for the same free data, with K = 0 substituted by Ñ = 1, Pfeiffer and York have shown that the original conformal thin sandwich method as described in Sec. 5.1 leads to a unique solution (or no solution at all if the amplitude of the wave is two large).
Baumgarte, Ó Murchadha and Pfeiffer [14] have argued that the lack of uniqueness for the XCTS system may be due to the term
-( Ñ Ψ 7 ) 7 8 Âij Âij Ψ -8 = - 7 32 Ψ 6 γik γjl γij + ( Lβ) ij γkl + ( Lβ) kl ( Ñ Ψ 7 ) -1 ( 108
)
in Eq. ( 90 ). Indeed, if we proceed as for the analysis of Lichnerowicz equation in Sec. 3.4, we notice that this term, with the minus sign and the negative power of ( Ñ Ψ 7 ) -1 , makes the linearization of Eq. (90) of the type Di Di ǫ + αǫ = σ, with α > 0. This "wrong" sign of α prevents the application of the maximum principle to guarantee the uniqueness of the solution.
The non-uniqueness of solution of the XCTS system for certain choice of free data has been confirmed by Walsh [100] by means of bifurcation theory.
The conformal transverse traceless (CTT) method exposed in Sec. 3 and the (extended) conformal thin sandwich (XCTS) method considered here differ by the choice of free data: whereas both methods use the conformal metric γ and the trace Construction of initial data for 3+1 numerical relativity 19 of the extrinsic curvature K as free data, CTT employs in addition Âij TT , whereas for CTS (resp. XCTS) the additional free data is γij , as well as Ñ (resp. K). Since Âij TT is directly related to the extrinsic curvature and the latter is linked to the canonical momentum of the gravitational field in the Hamiltonian formulation of general relativity, the CTT method can be considered as the approach to the initial data problem in the Hamiltonian representation. On the other side, γij being the "velocity" of γij , the (X)CTS method constitutes the approach in the Lagrangian representation [108].
Remark : The (X)CTS method assumes that the conformal metric is unimodular:
det(γ ij ) = f (since Eq. (79) follows from this assumption), whereas the CTT method can be applied with any conformal metric.
The advantage of CTT is that its mathematical theory is well developed, yielding existence and uniqueness theorems, at least for constant mean curvature (CMC) slices. The mathematical theory of CTS is very close to CTT. In particular, the momentum constraint decouples from the Hamiltonian constraint on CMC slices. On the contrary, XCTS has a much more involved mathematical structure. In particular the CMC condition does not yield to any decoupling. The advantage of XCTS is then to be better suited to the description of quasi-stationary spacetimes, since γij = 0 and K = 0 are necessary conditions for ∂ t to be a Killing vector. This makes XCTS the method to be used in order to prepare initial data in quasi-equilibrium. For instance, it has been shown [57, 43] that XCTS yields orbiting binary black hole configurations in much better agreement with post-Newtonian computations than the CTT treatment based on a superposition of two Bowen-York solutions. Indeed, except when they are very close and about to merge, the orbits of binary black holes evolve very slowly, so that it is a very good approximation to consider that the system is in quasi-equilibrium. XCTS takes this fully into account, while CTT relies on a technical simplification (Bowen-York analytical solution of the momentum constraint), with no direct relation to the quasi-equilibrium state.
A detailed comparison of CTT and XCTS for a single spinning or boosted black hole has been performed by Laguna [68].
A major topic of contemporary numerical relativity is the computation of the merger of a binary system of black holes [24] or neutron stars [84] , for such systems are among the most promising sources of gravitational radiation for the interferometric detectors either groundbased (LIGO, VIRGO, GEO600, TAMA) or in space (LISA). The problem of preparing initial data for these systems has therefore received a lot of attention in the past decade.
Due to the gravitational-radiation reaction, a relativistic binary system has an inspiral motion, leading to the merger of the two components. However, when the two bodies are sufficiently far apart, one may approximate the spiraling orbits by closed ones. Moreover, it is well known that gravitational radiation circularizes the orbits very efficiently, at least for comparable mass systems [18] . We may then consider that the motion is described by a sequence of closed circular orbits.
Construction of initial data for 3+1 numerical relativity 20 Figure 5. Action of the helical symmetry group, with Killing vector ℓ. χτ (P ) is the displacement of the point P by the member of the symmetry group of parameter τ . N and β are respectively the lapse function and the shift vector associated with coordinates adapted to the symmetry, i.e. coordinates (t, x i ) such that ∂t = ℓ.
The geometrical translation of this physical assumption is that the spacetime (M, g) is endowed with some symmetry, called helical symmetry. Indeed exactly circular orbits imply the existence of a one-parameter symmetry group such that the associated Killing vector ℓ obeys the following properties [46]: (i) ℓ is timelike near the system, (ii) far from it, ℓ is spacelike but there exists a smaller number T > 0 such that the separation between any point P and its image χ T (P ) under the symmetry group is timelike (cf. Fig. 5 ). ℓ is called a helical Killing vector, its field lines in a spacetime diagram being helices (cf. Fig. 5 ).
Helical symmetry is exact in theories of gravity where gravitational radiation does not exist, namely:
• in Newtonian gravity,
• in post-Newtonian gravity, up to the second order, • in the Isenberg-Wilson-Mathews (IWM) approximation to general relativity, based on the assumptions γ = f and K = 0 [61, 102].
Moreover helical symmetry can be exact in full general relativity for a nonaxisymmetric system (such as a binary) with standing gravitational waves [44] . But notice that a spacetime with helical symmetry and standing gravitational waves cannot be asymptotically flat [48] .
To treat helically symmetric spacetimes, it is natural to choose coordinates (t, x i ) that are adapted to the symmetry, i.e. such that
∂ t = ℓ. ( 109
)
Then all the fields are independent of the coordinate t. In particular, γij = 0 and K = 0. (
) 110
Construction of initial data for 3+1 numerical relativity 21 If we employ the XCTS formalism to compute initial data, we therefore get some definite prescription for the free data γij and K. On the contrary, the requirements (110) do not have any immediate translation in the CTT formalism.
Remark : Helical symmetry can also be useful to treat binary black holes outside the scope of the 3+1 formalism, as shown by Klein [67], who developed a quotient space formalism to reduce the problem to a three dimensional SL(2, R)/SO(1, 1) sigma model.
Taking into account (110) and choosing maximal slicing (K = 0), the XCTS system (88)-(90) becomes
Di Di Ψ - R 8 Ψ + 1 8 Âij Âij Ψ -7 + 2π ẼΨ -3 = 0 ( 111
) Dj 1 Ñ ( Lβ) ij -16π pi = 0 ( 112
) Di Di ( Ñ Ψ 7 ) -( Ñ Ψ 7 ) R 8 + 7 8 Âij Âij Ψ -8 + 2π( Ẽ + 2 S)Ψ -4 = 0, ( 113
)
where [cf. Eq. (79)] Âij = 1 2 Ñ ( Lβ) ij . (114)
If we choose, as part of the free data, the conformal metric to be flat,
γij = f ij , ( 115
)
then the helically symmetric XCTS system (111)-(113) reduces to
∆Ψ + 1 8 Âij Âij Ψ -7 + 2π ẼΨ -3 = 0 ( 116
) ∆β i + 1 3 D i D j β j -(Lβ) ij D j ln Ñ = 16π Ñ pi ( 117
) ∆( Ñ Ψ 7 ) -( Ñ Ψ 7 ) 7 8 Âij Âij Ψ -8 + 2π( Ẽ + 2 S)Ψ -4 = 0, ( 118
)
where Âij = 1 2 Ñ (Lβ) ij (119) and D is the connection associated with the flat metric f , ∆ := D i D i is the flat Laplacian [Eq. (46)], and (Lβ 15 ) with Di = D i ]. We remark that the system (116)-(118) is identical to the system defining the Isenberg-Wilson-Mathews approximation to general relativity [61, 102] (see e.g. Sec. 6.6 of Ref. [51] ). This means that, within helical symmetry, the XCTS system with the choice K = 0 and γ = f is equivalent to the IWM system.
) ij := D i β j + D j β i -2 3 D k β k f ij [Eq. (
Remark : Contrary to IWM, XCTS is not some approximation to general relativity: it provides exact initial data. The only thing that may be questioned is the astrophysical relevance of the XCTS data with γ = f .
Construction of initial data for 3+1 numerical relativity 22 6.3. Initial data for orbiting binary black holes
The concept of helical symmetry for generating orbiting binary black hole initial data has been introduced in 2002 by Gourgoulhon, Grandclément and Bonazzola [52, 57]. The system of equations that these authors have derived is equivalent to the XCTS system with γ = f , their work being previous to the formulation of the XCTS method by Pfeiffer and York (2003) [80]. Since then other groups have combined XCTS with helical symmetry to compute binary black hole initial data [38, 1, 2, 31] . Since all these studies are using a flat conformal metric [choice (115)], the PDE system to be solved is (116)-(118), with the additional simplification Ẽ = 0 and pi = 0 (vacuum). The initial data manifold Σ 0 is chosen to be R 3 minus two balls:
Σ 0 = R 3 \(B 1 ∪ B 2 ). ( 120
)
In addition to the asymptotic flatness conditions, some boundary conditions must be provided on the surfaces S 1 and S 2 of B 1 and B 2 . One choose boundary conditions corresponding to a non-expanding horizon, since this concept characterizes black holes in equilibrium. We shall not detail these boundary conditions here; they can be found in Refs. [38, 40, 41, 54, 65] . The condition of non-expanding horizon provides 3 among the 5 required boundary conditions [for the 5 components (Ψ, Ñ , β i )]. The two remaining boundary conditions are given by (i) the choice of the foliation (choice of the value of N at S 1 and S 2 ) and (ii) the choice of the rotation state of each black hole ("individual spin"), as explained in Ref. [31] . Numerical codes for solving the above system have been constructed by
• Grandclément, Gourgoulhon and Bonazzola (2002) [57] for corotating binary black holes; Cook, Pfeiffer, Caudill and Grigsby (2004, 2006) [38, 31] for corotating and irrotational binary black holes; Ansorg (2005 Ansorg ( , 2007) [1, 2] ) [1, 2] for corotating binary black holes.
•
•
Detailed comparisons with post-Newtonian initial data (either from the standard post-Newtonian formalism [17] or from the Effective One-Body approach [23, 42] ) have revealed a very good agreement, as shown in Refs. [43, 31] . An alternative to (120) for the initial data manifold would be to consider the twice-punctured R 3 :
Σ 0 = R 3 \{O 1 , O 2 }, ( 121
)
where O 1 and O 2 are two points of R 3 . This would constitute some extension to the two bodies case of the punctured initial data discussed in Sec. 5.3. However, as shown by Hannam, Evans, Cook and Baumgarte in 2003 [60] , it is not possible to find a solution of the helically symmetric XCTS system with a regular lapse in this case . For this reason, initial data based on the puncture manifold (121) are computed within the CTT framework discussed in Sec. 3. As already mentioned, there is no natural way to implement helical symmetry in this framework. One instead selects the free data Âij TT to vanish identically, as in the single black hole case treated in Secs. 4.1 and 4.3. Then Âij = ( LX) ij . (
) 122
see however Ref. [59] for some attempt to circumvent this
Construction of initial data for 3+1 numerical relativity 23 The vector X must obey Eq. ( 45 ), which arises from the momentum constraint. Since this equation is linear, one may choose for X a linear superposition of two Bowen-York solutions (Sec. 4.3):
X = X (P (1) ,S (1) ) + X (P (2) ,S (2) ) , ( 123
)
where X (P (a) ,S (a) ) (a = 1, 2) is the Bowen-York solution (71) centered on O a . This method has been first implemented by Baumgarte in 2000 [11] . It has been since then used by Baker, Campanelli, Lousto and Takashi (2002) [5] and Ansorg, Brügmann and Tichy (2004) [3]. The initial data hence obtained are closed from helically symmetric XCTS initial data at large separation but deviate significantly from them, as well as from post-Newtonian initial data, when the two black holes are very close. This means that the Bowen-York extrinsic curvature is bad for close binary systems in quasi-equilibrium (see discussion in Ref. [43] ).
Remark : Despite of this, CTT Bowen-York configurations have been used as initial data for the recent binary black hole inspiral and merger computations by Baker et al. [6, 7, 99] and Campanelli et al. [25, 26, 27, 28] . Fortunately, these initial data had a relative large separation, so that they differed only slightly from the helically symmetric XCTS ones.
Instead of choosing somewhat arbitrarily the free data of the CTT and XCTS methods, notably setting γ = f , one may deduce them from post-Newtonian results. This has been done for the binary black hole problem by Tichy, Brügmann, Campanelli and Diener (2003) [94], who have used the CTT method with the free data (γ ij , Âij TT ) given by the second order post-Newtonian (2PN) metric. This work has been improved recently by Kelly, Tichy, Campanelli and Whiting (2007) [66] . In the same spirit, Nissanke (2006) [75] has provided 2PN free data for both the CTT and XCTS methods.
For computing initial data corresponding to orbiting binary neutron stars, one must solve equations for the fluid motion in addition to the Einstein constraints. Basically this amounts to solving ∇ ν T µν = 0 in the context of helical symmetry. One can then show that a first integral of motion exists in two cases: (i) the stars are corotating, i.e. the fluid 4-velocity is colinear to the helical Killing vector (rigid motion), (ii) the stars are irrotational, i.e. the fluid vorticity vanishes. The most straightforward way to get the first integral of motion is by means of the Carter-Lichnerowicz formulation of relativistic hydrodynamics, as shown in Sec. 7 of Ref. [50] . Other derivations have been obtained in 1998 by Teukolsky [93] and Shibata [83] .
From the astrophysical point of view, the irrotational motion is much more interesting than the corotating one, because the viscosity of neutron star matter is far too low to ensure the synchronization of the stellar spins with the orbital motion. On the other side, the irrotational state is a very good approximation for neutron stars that are not millisecond rotators. Indeed, for these stars the spin frequency is much lower than the orbital frequency at the late stages of the inspiral and thus can be neglected.
The first initial data for binary neutron stars on circular orbits have been computed by Baumgarte, Cook, Scheel, Shapiro and Teukolsky in 1997 [12, 13] in the corotating case, and by Bonazzola, Gourgoulhon and Marck in 1999 [19] in the irrotational case. These results were based on a polytropic equation of state. Since then configurations in the irrotational regime have been obtained Construction of initial data for 3+1 numerical relativity 24 • for a polytropic equation of state [73, 96, 97, 53, 90, 91] (the configurations obtained in Ref. [91] have been used as initial data by Shibata [84] to compute the merger of binary neutron stars); • for nuclear matter equations of state issued from recent nuclear physics computations [16, 77]; • for strange quark matter [78, 72] .
All these computation are based on a flat conformal metric [choice (115)], by solving the helically symmetric XCTS system (116)-(118), supplemented by an elliptic equation for the velocity potential. Only very recently, configurations based on a non flat conformal metric have been obtained by Uryu, Limousin, Friedman, Gourgoulhon and Shibata [98] . The conformal metric is then deduced from a waveless approximation developed by Shibata, Uryu and Friedman [85] and which goes beyond the IWM approximation.
Let us mention briefly that initial data for a mixed binary system, i.e. a system composed of a black hole and a neutron star, have been obtained very recently by Grandclément [55] and Taniguchi, Baumgarte, Faber and Shapiro [88, 89] . Codes aiming at computing such systems have also been presented by Ansorg [2] and Tsokaros and Uryu [95] .
I warmly thank the organizers of the VII Mexican school, namely Miguel Alcubierre, Hugo Garcia-Compean and Luis Urena, for their support and the success of the school. I also express my gratitude to Marcelo Salgado for his help and many discussions and to Nicolas Vasset for the careful reading of the manuscript.
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{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The 3+1 formalism is the basis of most modern numerical relativity and has lead, along with alternative approaches [82] , to the recent successes in the binary black hole merger problem [6, 7, 99, 25, 26, 27, 28] (see [24, 69, 86] for a review). Thanks to the 3+1 formalism, the resolution of Einstein equation amounts to solving a Cauchy problem, namely to evolve \"forward in time\" some initial data. However this is a Cauchy problem with constraints. This makes the set up of initial data a non trivial task, because these data must fulfill the constraints. In this lecture, we present the most wide spread methods to deal with this problem. Notice that we do not discuss the numerical techniques employed to solve the constraints (see e.g. Choptuik's lecture for finite differences [32] and Grandclément and Novak's review for spectral methods [58] ).\n\nStandard reviews about the initial data problem are the articles by York [106] and Choquet-Bruhat and York [36] . Recent reviews are the articles by Cook [37] , Pfeiffer [79] and Bartnik and Isenberg [10] ."
},
{
"section_type": "OTHER",
"section_title": "3+1 decomposition of Einstein equation",
"text": "In this lecture, we consider a spacetime (M, g), where M is a four-dimensional smooth manifold and g a Lorentzian metric on M. We assume that (M, g) is globally Construction of initial data for 3+1 numerical relativity 2 hyperbolic, i.e. that M can be foliated by a family (Σ t ) t∈R of spacelike hypersurfaces. We denote by γ the (Riemannian) metric induced by g on each hypersurface Σ t and K the extrinsic curvature of Σ t , with the same sign convention as that used in the numerical relativity community, i.e. for any pair of vector fields (u, v) tangent to Σ t , g(u, ∇ v n) = -K(u, v), where n is the future directed unit normal to Σ t and ∇ is the Levi-Civita connection associated with g.\n\nThe 3+1 decomposition of Einstein equation with respect to the foliation (Σ t ) t∈R leads to three sets of equations: (i) the evolution equations of the Cauchy problem (full projection of Einstein equation onto Σ t ), (ii) the Hamiltonian constraint (full projection of Einstein equation along the normal n), (iii) the momentum constraint (mixed projection: once onto Σ t , once along n). The latter two sets of equations do not contain any second derivative of the metric with respect to t. They are written ‡\n\nR + K 2 -K ij K ij = 16πE (Hamiltonian constraint), ( 1\n\n) D j K ij -D i K = 8πp i (momentum constraint), ( 2\n\n)\n\nwhere R is the Ricci scalar (also called scalar curvature) associated with the 3-metric γ, K is the trace of K with respect to γ: K = γ ij K ij , D stands for the Levi-Civita connection associated with the 3-metric γ, and E and p i are respectively the energy density and linear momentum of matter, both measured by the observer of 4-velocity n (Eulerian observer). In terms of the matter energy-momentum tensor T they are expressed as\n\nE = T µν n µ n ν and p i = -T µν n µ γ ν i . ( 3\n\n) Notice that Eqs. (1)-(2) involve a single hypersurface Σ 0 , not a foliation (Σ t ) t∈R . In particular, neither the lapse function nor the shift vector appear in these equations."
},
{
"section_type": "OTHER",
"section_title": "Constructing initial data",
"text": "In order to get valid initial data for the Cauchy problem, one must find solutions to the constraints (1) and (2). Actually one may distinguish two problems:\n\n• The mathematical problem: given some hypersurface Σ 0 , find a Riemannian metric γ, a symmetric bilinear form K and some matter distribution (E, p) on Σ 0 such that the Hamiltonian constraint (1) and the momentum constraint (2) are satisfied. In addition, the matter distribution (E, p) may have some constraints from its own. We shall not discuss them here.\n\n• The astrophysical problem: make sure that the solution to the constraint equations has something to do with the physical system that one wish to study.\n\nFacing the constraint equations (1) and (2), a naive way to proceed would be to choose freely the metric γ, thereby fixing the connection D and the scalar curvature R, and to solve Eqs. (1)-(2) for K. Indeed, for fixed γ, E, and p, Eqs. (1)-(2) form a quasi-linear system of first order for the components K ij . However, as discussed by Choquet-Bruhat [45] , this approach is not satisfactory because we have only four equations for six unknowns K ij and there is no natural prescription for choosing arbitrarily two among the six components K ij . In 1944, Lichnerowicz [70] has shown that a much more satisfactory split of the initial data (γ, K) between freely choosable parts and parts obtained by solving Construction of initial data for 3+1 numerical relativity 3 Eqs. (1)-(2) is provided by a conformal decomposition of the metric γ. Lichnerowicz method has been extended by Choquet-Bruhat (1956 , 1971) [45, 33] , by York and Ó Murchadha (1972, 1974, 1979) [103, 104, 76, 106] and more recently by York and Pfeiffer (1999, 2003) [107, 80] . Actually, conformal decompositions are by far the most widely spread techniques to get initial data for the 3+1 Cauchy problem. Alternative methods exist, such as the quasi-spherical ansatz introduced by Bartnik in 1993 [8] or a procedure developed by Corvino (2000) [39] and by Isenberg, Mazzeo and Pollack (2002) [63] for gluing together known solutions of the constraints, thereby producing new ones. Here we shall limit ourselves to the conformal methods."
},
{
"section_type": "OTHER",
"section_title": "Conformal decomposition of the constraints",
"text": "In the conformal approach initiated by Lichnerowicz [70] , one introduces a conformal metric γ and a conformal factor Ψ such that the (physical) metric γ induced by the spacetime metric on the hypersurface Σ t is\n\nγ ij = Ψ 4 γij . ( 4\n\n)\n\nWe could fix some degree of freedom by demanding that det γij = 1. This would imply Ψ = (det γ ij ) 1/12 . However, in this case γ and Ψ would be tensor densities. Moreover the condition det γij = 1 has a meaning only for Cartesian-like coordinates. In order to deal with tensor fields and to allow for any type of coordinates, we proceed differently and introduce a background Riemannian metric f on Σ t . If the topology of Σ t allows it, we shall demand that f is flat. Then we replace the condition det γij = 1 by det γij = det f ij . This fixes Ψ = det γ ij det f ij 1/12 . (5) Ψ is then a genuine scalar field on Σ t (as a quotient of two determinants). Consequently γ is a tensor field and not a tensor density. Associated with the above conformal transformation, there are two decompositions of the traceless part A ij of the extrinsic curvature, the latter being defined by\n\nK ij =: A ij + 1 3 Kγ ij . ( 6\n\n)\n\nThese two decompositions are A ij =: Ψ -10 Âij , (\n\n) 7\n\nA ij =: Ψ -4 Ãij . ( 8\n\n)\n\nThe choice -10 for the exponent of Ψ in Eq. ( 7 ) is motivated by the following identity, valid for any symmetric and traceless tensor field,\n\nD j A ij = Ψ -10 Dj Ψ 10 A ij , ( 9\n\n)\n\nwhere Dj denotes the covariant derivative associated with the conformal metric γ. This choice is well adapted to the momentum constraint, because the latter involves the divergence of K. The alternative choice, i.e. Eq. ( 8 ), is motivated by time evolution considerations, as we shall discuss below. For the time being, we limit ourselves to the decomposition ( 7 ), having in mind to simplify the writing of the momentum constraint.\n\nConstruction of initial data for 3+1 numerical relativity 4 By means of the decompositions (4), (6) and (7), the Hamiltonian constraint (1) and the momentum constraint (2) are rewritten as (see Ref. [51] for details) Di Di Ψ -1 8 RΨ + 1 8 Âij Âij Ψ -7 + 2π ẼΨ -3 -1 12 K 2 Ψ 5 = 0, (10) Dj Âij -2 3 Ψ 6 Di K = 8π pi , (\n\n) 11\n\nwhere R is the Ricci scalar associated with the conformal metric γ and we have introduced the rescaled matter quantities Ẽ := Ψ 8 E and pi := Ψ 10 p i . (12) Equation ( 10) is known as Lichnerowicz equation, or sometimes Lichnerowicz-York equation. The definition of pi is such that there is no Ψ factor in the right-hand side of Eq. ( 11 ). On the contrary the power 8 in the definition of Ẽ is not the only possible choice. As we shall see in § 3.4, it is chosen (i) to guarantee a negative power of Ψ in the Ẽ term in Eq. ( 10 ), resulting in some uniqueness property of the solution and (ii) to allow for an easy implementation of the dominant energy condition."
},
{
"section_type": "OTHER",
"section_title": "Longitudinal/transverse decomposition of Âij",
"text": "In order to solve the system (10)-(11), York (1973,1979) [104, 105, 106] has decomposed Âij into a longitudinal part and a transverse one, setting Âij = ( LX) ij + Âij TT , (\n\n) 13\n\nwhere Âij TT is both traceless and transverse (i.e. divergence-free) with respect to the metric γ:\n\nγij Âij TT = 0 and Dj Âij TT = 0, (14) and ( LX) ij is the conformal Killing operator associated with the metric γ and acting on the vector field X:\n\n( LX) ij := Di X j + Dj X i - 2 3 Dk X k γij . ( 15\n\n)\n\n( LX) ij is by construction traceless: γij ( LX) ij = 0 (16) (it must be so because in Eq. (13) both Âij and Âij TT are traceless). The kernel of L is made of the conformal Killing vectors of the metric γ, i.e. the generators of the conformal isometries (see e.g. Ref. [51] for more details). The symmetric tensor ( LX) ij is called the longitudinal part of Âij , whereas Âij TT is called the transverse part. Given Âij , the vector X is determined by taking the divergence of Eq. (13): taking into account property (14), we get Dj ( LX) ij = Dj Âij . (\n\n) 17\n\nThe second order operator Dj ( LX) ij acting on the vector X is the conformal vector Laplacian ∆L :\n\n∆L X i := Dj ( LX) ij = Dj Dj X i + 1 3 Di Dj X j + Ri j X j , ( 18\n\n)\n\nConstruction of initial data for 3+1 numerical relativity 5 where the second equality follows from the Ricci identity applied to the connection D, Rij being the associated Ricci tensor. The operator ∆L is elliptic and its kernel is, in practice, reduced to the conformal Killing vectors of γ, if any. We rewrite Eq. (17) as ∆L X i = Dj Âij . (19) The existence and uniqueness of the longitudinal/transverse decomposition (13) depend on the existence and uniqueness of solutions X to Eq. (19). We shall consider two cases:\n\n• Σ 0 is a closed manifold, i.e. is compact without boundary; • (Σ 0 , γ) is an asymptotically flat manifold, i.e. is such that the background metric f is flat (except possibly on a compact sub-domain B of Σ t ) and there exists a coordinate system (x i ) = (x, y, z) on Σ t such that outside B, the components of f are f ij = diag(1, 1, 1) (\"Cartesian-type coordinates\") and the variable r := x 2 + y 2 + z 2 can take arbitrarily large values on Σ t ; then when r → +∞, the components of γ and K with respect to the coordinates (x i ) satisfy γ ij = f ij + O(r -1 ) and ∂γ ij ∂x k = O(r -2 ), (20) K ij = O(r -2 ) and ∂K ij ∂x k = O(r -3 ). (\n\n) 21\n\nIn the case of a closed manifold, one can show (see Appendix B of Ref. [51] for details) that solutions to Eq. (19) exist provided that the source Dj Âij is orthogonal to all conformal Killing vectors of γ, in the sense that ∀C ∈ ker L, Σ γij C i Dk Âjk γ d 3 x = 0. (22) But the above property is easy to verify: using the fact that the source is a pure divergence and that Σ 0 is closed, we may integrate the left-hand side by parts and get, for any vector field C, Σ0 γij C i Dk Âjk γ d 3 x = -1 2 Σ0 γij γkl ( LC) ik Âjl γ d 3 x. (23)\n\nThen, obviously, when C is a conformal Killing vector, the right-hand side of the above equation vanishes. So there exists a solution to Eq. ( 19 ) and this solution is unique up to the addition of a conformal Killing vector. However, given a solution X, for any conformal Killing vector C, the solution X + C yields to the same value of LX, since C is by definition in the kernel of L. Therefore we conclude that the decomposition (13) of Âij is unique, although the vector X may not be if (Σ 0 , γ) admits some conformal isometries. In the case of an asymptotically flat manifold, the existence and uniqueness is guaranteed by a theorem proved by Cantor in 1979 [30] (see also Appendix B of Ref. [87] as well as Refs. [35, 51] ). This theorem requires the decay condition ∂ 2 γij ∂x k ∂x l = O(r -3 ) (24) in addition to the asymptotic flatness conditions (20) . This guarantees that Rij = O(r -3 ). (25) Then all conditions are fulfilled to conclude that Eq. (19) admits a unique solution X which vanishes at infinity. To summarize, for all considered cases (asymptotic flatness and closed manifold), any symmetric and traceless tensor Âij (decaying as O(r -2 ) in the asymptotically flat case) admits a unique longitudinal/transverse decomposition of the form (13).\n\nConstruction of initial data for 3+1 numerical relativity 6 3.2. Conformal transverse-traceless form of the constraints Inserting the longitudinal/transverse decomposition (13) into the constraint equations (10) and (11) and making use of Eq. (19) yields to the system Di Di Ψ -1 8 RΨ + 1 8 ( LX) ij + ÂTT ij\n\n( LX) ij + Âij TT Ψ -7 + 2π ẼΨ -3 - 1 12 K 2 Ψ 5 = 0, ( 26\n\n) ∆L X i - 2 3 Ψ 6 Di K = 8π pi , ( 27\n\n)\n\nwhere ( LX) ij := γik γjl ( LX) kl and ÂTT ij := γik γjl Âkl TT . (\n\n) 28\n\nWith the constraint equations written as (26) and (27), we see clearly which part of the initial data on Σ 0 can be freely chosen and which part is \"constrained\":\n\n• free data:\n\nconformal metric γ; -symmetric traceless and transverse tensor Âij TT (traceless and transverse are meant with respect to γ: γij Âij TT = 0 and Dj Âij TT = 0); -scalar field K; -conformal matter variables: ( Ẽ, pi ); • constrained data (or \"determined data\"):\n\nconformal factor Ψ, obeying the non-linear elliptic equation (26) (Lichnerowicz equation) -vector X, obeying the linear elliptic equation (27) .\n\nAccordingly the general strategy to get valid initial data for the Cauchy problem is to choose (γ ij , Âij TT , K, Ẽ, pi ) on Σ 0 and solve the system (26)-(27) to get Ψ and X i . Then one constructs\n\nγ ij = Ψ 4 γij ( 29\n\n) K ij = Ψ -10 ( LX) ij + Âij TT + 1 3 Ψ -4 K γij ( 30\n\n) E = Ψ -8 Ẽ (31) p i = Ψ -10 pi ( 32\n\n)\n\nand obtains a set (γ, K, E, p) which satisfies the constraint equations (1)-(2). This method has been proposed by York (1979) [106] and is naturally called the conformal transverse traceless (CTT ) method. Equations (26) and (27) are coupled, but we notice that if, among the free data, we choose K to be a constant field on Σ 0 ,"
},
{
"section_type": "OTHER",
"section_title": "Decoupling on hypersurfaces of constant mean curvature",
"text": "K = const, ( 33\n\n)\n\nthen they decouple partially : condition (33) implies Di K = 0, so that the momentum constraint (27) becomes independent of Ψ:\n\n∆L X i = 8π pi (K = const). ( 34\n\n)\n\nConstruction of initial data for 3+1 numerical relativity 7 The condition (33) on the extrinsic curvature of Σ 0 defines what is called a constant mean curvature (CMC ) hypersurface. Indeed let us recall that K is nothing but (minus three times) the mean curvature of (Σ 0 , γ) embedded in (M, g). A maximal hypersurface, having K = 0, is of course a special case of a CMC hypersurface. On a CMC hypersurface, the task of obtaining initial data is greatly simplified: one has first to solve the linear elliptic equation (34) to get X and plug the solution into Eq. (26) to form an equation for Ψ. Equation (34) is the conformal vector Poisson equation discussed above (Eq. (19), with Dj Âij replaced by 8π pi ). We know then that it is solvable for the two cases of interest mentioned in Sec. 3.1: closed or asymptotically flat manifold. Moreover, the solutions X are such that the value of LX is unique."
},
{
"section_type": "OTHER",
"section_title": "Lichnerowicz equation",
"text": "Taking into account the CMC decoupling, the difficult problem is to solve Eq. (26) for Ψ. This equation is elliptic and highly non-linear §. It has been first studied by Lichnerowicz [70, 71] in the case K = 0 (Σ 0 maximal) and Ẽ = 0 (vacuum). Lichnerowicz has shown that given the value of Ψ at the boundary of a bounded domain of Σ 0 (Dirichlet problem), there exists at most one solution to Eq. (26) . Besides, he showed the existence of a solution provided that Âij Âij is not too large. These early results have been much improved since then. In particular Cantor [29] has shown that in the asymptotically flat case, still with K = 0 and Ẽ = 0, Eq. (26) is solvable if and only if the metric γ is conformal to a metric with vanishing scalar curvature (one says then that γ belongs to the positive Yamabe class) (see also Ref. [74] ). In the case of closed manifolds, the complete analysis of the CMC case has been achieved by Isenberg (1995) [62] .\n\nFor more details and further references, we recommend the review articles by Choquet-Bruhat and York [36] and Bartnik and Isenberg [10] . Here we shall simply repeat the argument of York [107] to justify the rescaling (12) of E. This rescaling is indeed related to the uniqueness of solutions to the Lichnerowicz equation. Consider a solution Ψ 0 to Eq. (26) in the case K = 0, to which we restrict ourselves. Another solution close to\n\nΨ 0 can be written Ψ = Ψ 0 + ǫ, with |ǫ| ≪ Ψ 0 : Di Di (Ψ 0 + ǫ) - R 8 (Ψ 0 + ǫ) + 1 8 Âij Âij (Ψ 0 + ǫ) -7 + 2π Ẽ(Ψ 0 + ǫ) -3 = 0. ( 35\n\n)\n\nExpanding to the first order in ǫ/Ψ 0 leads to the following linear equation for ǫ:\n\nDi Di ǫ -αǫ = 0, ( 36\n\n) with α := 1 8 R + 7 8 Âij Âij Ψ -8 0 + 6π ẼΨ -4 0 . ( 37\n\n)\n\nNow, if α ≥ 0, one can show, by means of the maximum principle, that the solution of (36) which vanishes at spatial infinity is necessarily ǫ = 0 (see Ref. [34] or § B.1 of Ref. [35] ). We therefore conclude that the solution Ψ 0 to Eq. (26) is unique (at least locally) in this case. On the contrary, if α < 0, non trivial oscillatory solutions of Eq. (36) exist, making the solution Ψ 0 not unique. The key point is that the scaling (12) of E yields the term +6π ẼΨ -4 0 in Eq. ( 37 ), which contributes to make α positive. If we had not rescaled E, i.e. had considered the original Hamiltonian constraint, the § although it is quasi-linear in the technical sense, i.e. linear with respect to the highest-order derivatives Construction of initial data for 3+1 numerical relativity 8 contribution to α would have been instead -10πEΨ 4 0 , i.e. would have been negative. Actually, any rescaling Ẽ = Ψ s E with s > 5 would have work to make α positive. The choice s = 8 in Eq. (12) is motivated by the fact that if the conformal data ( Ẽ, pi ) obey the \"conformal\" dominant energy condition Ẽ ≥ γij pi pj , (38) then, via the scaling (12) of p i , the reconstructed physical data (E, p i ) will automatically obey the dominant energy condition\n\nE ≥ γ ij p i p j . ( 39\n\n)\n\n4. Conformally flat initial data by the CTT method"
},
{
"section_type": "OTHER",
"section_title": "Momentarily static initial data",
"text": "In this section we search for asymptotically flat initial data (Σ 0 , γ, K) by the CTT method exposed above. As a purpose of illustration, we shall start by the simplest case one may think of, namely choose the freely specifiable data (γ ij , Âij TT , K, Ẽ, pi ) to be a flat metric:\n\nγij = f ij , ( 40\n\n)\n\na vanishing transverse-traceless part of the extrinsic curvature: Âij TT = 0, (41) a vanishing mean curvature (maximal hypersurface)\n\nK = 0, ( 42\n\n)\n\nand a vacuum spacetime: Ẽ = 0, pi = 0. (43)\n\nThen Di = D i , where D denotes the Levi-Civita connection associated with f , R = 0 (f is flat) and the constraint equations (26)-(27) reduce to\n\n∆Ψ + 1 8 (LX) ij (LX) ij Ψ -7 = 0 (44) ∆ L X i = 0, ( 45\n\n)\n\nwhere ∆ and ∆ L are respectively the scalar Laplacian and the conformal vector Laplacian associated with the flat metric f :\n\n∆ := D i D i and ∆ L X i := D j D j X i + 1 3 D i D j X j . ( 46\n\n)\n\nEquations (44)-(45) must be solved with the boundary conditions\n\nΨ = 1 when r → ∞ ( 47\n\n) X = 0 when r → ∞, ( 48\n\n)\n\nwhich follow from the asymptotic flatness requirement. The solution depends on the topology of Σ 0 , since the latter may introduce some inner boundary conditions in addition to (47)-(48).\n\nLet us start with the simplest case: Σ 0 = R 3 . Then the unique solution of Eq. (45) subject to the boundary condition (48) is\n\nX = 0. ( 49\n\n)\n\nConstruction of initial data for 3+1 numerical relativity 9 Figure 1. Hypersurface Σ 0 as R 3 minus a ball, displayed via an embedding diagram based on the metric γ, which coincides with the Euclidean metric on R 3 . Hence Σ 0 appears to be flat. The unit normal of the inner boundary S with respect to the metric γ is s. Notice that D • s > 0.\n\nConsequently (LX) ij = 0, so that Eq. ( 44 ) reduces to Laplace equation for Ψ:\n\n∆Ψ = 0. ( 50\n\n)\n\nWith the boundary condition (47), there is a unique regular solution on R 3 :\n\nΨ = 1. ( 51\n\n)\n\nThe initial data reconstructed from Eqs. (29)-(30) is then\n\nγ = f ( 52\n\n) K = 0. ( 53\n\n)\n\nThese data correspond to a spacelike hyperplane of Minkowski spacetime. Geometrically the condition K = 0 is that of a totally geodesic hypersurface [i.e. all the geodesics of (Σ t , γ) are geodesics of (M, g)]. Physically data with K = 0 are said to be momentarily static or time symmetric. Indeed, if we consider a foliation with unit lapse around Σ 0 (geodesic slicing), the following relation holds: L n g = -2K, where L n denotes the Lie derivative along the unit normal n. So if K = 0, L n g = 0. This means that, locally (i.e. on Σ 0 ), n is a spacetime Killing vector. This vector being timelike, the configuration is then stationary. Moreover, the Killing vector n being orthogonal to some hypersurface (i.e. Σ 0 ), the stationary configuration is called static. Of course, this staticity properties holds a priori only on Σ 0 since there is no guarantee that the time development of Cauchy data with K = 0 at t = 0 maintains K = 0 at t > 0. Hence the qualifier 'momentarily' in the expression 'momentarily static' for data with K = 0."
},
{
"section_type": "OTHER",
"section_title": "Slice of Schwarzschild spacetime",
"text": "To get something less trivial than a slice of Minkowski spacetime, let us consider a slightly more complicated topology for Σ 0 , namely R 3 minus a ball (cf. Fig. 1 ). The sphere S delimiting the ball is then the inner boundary of Σ 0 and we must provide boundary conditions for Ψ and X on S to solve Eqs. (44)-(45). For simplicity, let us choose\n\nX| S = 0. ( 54\n\n)\n\nAltogether with the outer boundary condition (48) , this leads to X being identically zero as the unique solution of Eq. ( 45 ). So, again, the Hamiltonian constraint reduces to Laplace equation\n\n∆Ψ = 0. ( 55\n\n)\n\nConstruction of initial data for 3+1 numerical relativity 10 Figure 2. Same hypersurface Σ 0 as in Fig. 1 but displayed via an embedding diagram based on the metric γ instead of γ. The unit normal of the inner boundary S with respect to that metric is s. Notice that D • s = 0, which means that S is a minimal surface of (Σ 0 , γ).\n\nIf we choose the boundary condition Ψ| S = 1, then the unique solution is Ψ = 1 and we are back to the previous example (slice of Minkowski spacetime). In order to have something non trivial, i.e. to ensure that the metric γ will not be flat, let us demand that γ admits a closed minimal surface, that we will choose to be S. This will necessarily translate as a boundary condition for Ψ since all the information on the metric is encoded in Ψ (let us recall that from the choice (40), γ = Ψ 4 f ). S is a minimal surface of (Σ 0 , γ) iff its mean curvature vanishes, or equivalently if its unit normal s is divergence-free (cf. Fig. 2 ):\n\nD i s i S = 0. ( 56\n\n)\n\nThis is the analog of ∇ • n = 0 for maximal hypersurfaces, the change from minimal to maximal being due to the change of metric signature, from the Riemannian to the Lorentzian one. Expressed in term of the connection D = D (recall that in the present case γ = f ), condition (56) is equivalent to\n\nD i (Ψ 6 s i ) S = 0. ( 57\n\n)\n\nLet us rewrite this expression in terms of the unit vector s normal to S with respect to the metric γ (cf. Fig. 1 ); we have\n\ns = Ψ -2 s, ( 58\n\n) since γ(s, s) = Ψ -4 γ(s, s) = γ(s, s) = 1. Thus Eq. (57) becomes D i (Ψ 4 si ) S = 1 √ f ∂ ∂x i f Ψ 4 si S = 0. ( 59\n\n)\n\nLet us introduce on Σ 0 a coordinate system of spherical type, (x i ) = (r, θ, ϕ), such that (i) f ij = diag(1, r 2 , r 2 sin 2 θ) and (ii) S is the sphere r = a, where a is some positive constant. Since in these coordinates √ f = r 2 sin θ and si = (1, 0, 0), the minimal surface condition (59) is written as\n\n1 r 2 ∂ ∂r Ψ 4 r 2 r=a = 0, ( 60\n\n)\n\nConstruction of initial data for 3+1 numerical relativity 11 Figure 3. Extended hypersurface Σ ′ 0 obtained by gluing a copy of Σ 0 at the minimal surface S; it defines an Einstein-Rosen bridge between two asymptotically flat regions.\n\n∂Ψ ∂r + Ψ 2r r=a = 0 (61) This is a boundary condition of mixed Newmann/Dirichlet type for Ψ. The unique solution of the Laplace equation (55) which satisfies boundary conditions (47) and (61) is\n\nΨ = 1 + a r . ( 62\n\n)\n\nThe parameter a is then easily related to the ADM mass m of the hypersurface Σ 0 . Indeed for a conformally flat 3-metric (and more generally in the quasi-isotropic gauge, cf. Chap. 7 of Ref. [51] ), the ADM mass m is given by the flux of the gradient of the conformal factor at spatial infinity:\n\nm = - 1 2π lim r→∞ r=const ∂Ψ ∂r r 2 sin θ dθ dϕ = - 1 2π lim r→∞ 4πr 2 ∂ ∂r 1 + a r = 2a. ( 63\n\n)\n\nHence a = m/2 and we may write\n\nΨ = 1 + m 2r . ( 64\n\n)\n\nTherefore, in terms of the coordinates (r, θ, ϕ), the obtained initial data (γ, K) are\n\nγ ij = 1 + m 2r 4 diag(1, r 2 , r 2 sin θ) ( 65\n\n) K ij = 0. ( 66\n\n)\n\nSo, as above, the initial data are momentarily static. Actually, we recognize on (65)-(66) a slice t = const of Schwarzschild spacetime in isotropic coordinates. The isotropic coordinates (r, θ, ϕ) covering the manifold Σ 0 are such that the range of r is [m/2, +∞). But thanks to the minimal character of the inner boundary S, we can extend (Σ\n\n0 , γ) to a larger Riemannian manifold (Σ ′ 0 , γ ′ ) with γ ′ | Σ0 = γ\n\nand γ ′ smooth at S. This is made possible by gluing a copy of Σ 0 at S (cf. Fig. 3 ).\n\nConstruction of initial data for 3+1 numerical relativity 12 Figure 4.\n\nExtended hypersurface Σ ′ 0 depicted in the Kruskal-Szekeres representation of Schwarzschild spacetime. R stands for Schwarzschild radial coordinate and r for the isotropic radial coordinate. R = 0 is the singularity and R = 2m the event horizon. Σ ′ 0 is nothing but a hypersurface t = const, where t is the Schwarzschild time coordinate. In this diagram, these hypersurfaces are straight lines and the Einstein-Rosen bridge S is reduced to a point.\n\nThe topology of Σ ′ 0 is S 2 × R and the range of r in Σ ′ 0 is (0, +∞). The extended metric γ ′ keeps exactly the same form as (65):\n\nγ ′ ij dx i dx j = 1 + m 2r 4 dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 . ( 67\n\n)\n\nBy the change of variable\n\nr → r ′ = m 2 4r ( 68\n\n)\n\nit is easily shown that the region r → 0 does not correspond to some \"center\" but is actually a second asymptotically flat region (the lower one in Fig. 3 ). Moreover the transformation (68), with θ and ϕ kept fixed, is an isometry of γ ′ . It maps a point p of Σ 0 to the point located at the vertical of p in Fig. 3 . The minimal sphere S is invariant under this isometry. The region around S is called an Einstein-Rosen bridge. (Σ ′ 0 , γ ′ ) is still a slice of Schwarzschild spacetime. It connects two asymptotically flat regions without entering below the event horizon, as shown in the Kruskal-Szekeres diagram of Fig. 4."
},
{
"section_type": "OTHER",
"section_title": "Bowen-York initial data",
"text": "Let us select the same simple free data as above, namely γij = f ij , Âij TT = 0, K = 0, Ẽ = 0 and pi = 0. (69) For the hypersurface Σ 0 , instead of R 3 minus a ball, we choose R 3 minus a point:\n\nΣ 0 = R 3 \\{O}. ( 70\n\n)\n\nThe removed point O is called a puncture [21] . The topology of Σ 0 is S 2 × R; it differs from the topology considered in Sec. 4.1 (R 3 minus a ball); actually it is the same topology as that of the extended manifold Σ ′ 0 (cf. Fig. 3 ).\n\nConstruction of initial data for 3+1 numerical relativity 13 Thanks to the choice (69), the system to be solved is still (44)-(45). If we choose the trivial solution X = 0 for Eq. ( 45 ), we are back to the slice of Schwarzschild spacetime considered in Sec. 4.1, except that now Σ 0 is the extended manifold previously denoted Σ ′ 0 . Bowen and York [20] have obtained a simple non-trivial solution to the momentum constraint (45) (see also Ref. [15] ). Given a Cartesian coordinate system (x i ) = (x, y, z) on Σ 0 (i.e. a coordinate system such that f ij = diag (1, 1, 1) ) with respect to which the coordinates of the puncture O are (0, 0, 0), this solution writes\n\nX i = - 1 4r 7f ij P j + P j x j x i r 2 - 1 r 3 ǫ ij k S j x k , ( 71\n\n)\n\nwhere r := x 2 + y 2 + z 2 , ǫ ij k is the Levi-Civita alternating tensor associated with the flat metric f and\n\n(P i , S j ) = (P 1 , P 2 , P 3 , S 1 , S 2 , S 3\n\n) are six real numbers, which constitute the six parameters of the Bowen-York solution. Notice that since r = 0 on Σ 0 , the Bowen-York solution is a regular and smooth solution on the entire Σ 0 .\n\nThe conformal traceless extrinsic curvature corresponding to the solution (71) is deduced from formula (13), which in the present case reduces to Âij = (LX) ij ; one gets\n\nÂij = 3 2r 3 x i P j + x j P i -f ij - x i x j r 2 P k x k + 3 r 5 ǫ ik l S k x l x j + ǫ jk l S k x l x i , ( 72\n\n)\n\nwhere P i := f ij P j . The tensor Âij given by Eq. (72) is called the Bowen-York extrinsic curvature. Notice that the P i part of Âij decays asymptotically as O(r -2 ), whereas the S i part decays as O(r -3 ).\n\nRemark : Actually the expression of Âij given in the original Bowen-York article [20] contains an additional term with respect to Eq. ( 72 ), but the role of this extra term is only to ensure that the solution is isometric through an inversion across some sphere. We are not interested by such a property here, so we have dropped this term. Therefore, strictly speaking, we should name expression (72) the simplified Bowen-York extrinsic curvature.\n\nThe Bowen-York extrinsic curvature provides an analytical solution of the momentum constraint (45) but there remains to solve the Hamiltonian constraint (44) for Ψ, with the asymptotic flatness boundary condition Ψ = 1 when r → ∞. Since X = 0, Eq. (44) is no longer a simple Laplace equation, as in Sec. 4.1, but a non-linear elliptic equation. There is no hope to get any analytical solution and one must solve Eq. (44) numerically to get Ψ and reconstruct the full initial data (γ, K) via Eqs. (29)- (30) .\n\nThe parameters P i of the Bowen-York solution are nothing but the three components of the ADM linear momentum of the hypersurface Σ 0 Similarly, the parameters S i of the Bowen-York solution are nothing but the three components of the angular momentum of the hypersurface Σ 0 , the latter being defined relatively to the quasi-isotropic gauge, in the absence of any axial symmetry (see e.g. [51]).\n\nRemark : The Bowen-York solution with P i = 0 and S i = 0 reduces to the momentarily static solution found in Sec. 4.1, i.e. is a slice t = const of the Schwarzschild spacetime (t being the Schwarzschild time coordinate). However Bowen-York initial data with P i = 0 and S i = 0 do not constitute a slice of Kerr spacetime. Indeed, it has been shown [47] that there does not exist any foliation of Kerr spacetime by hypersurfaces which (i) are axisymmetric, (ii) smoothly Construction of initial data for 3+1 numerical relativity 14 reduce in the non-rotating limit to the hypersurfaces of constant Schwarzschild time and (iii) are conformally flat, i.e. have induced metric γ = f , as the Bowen-York hypersurfaces have. This means that a Bowen-York solution with S i = 0 does represent initial data for a rotating black hole, but this black hole is not stationary: it is \"surrounded\" by gravitational radiation, as demonstrated by the time development of these initial data [22, 49] ."
},
{
"section_type": "METHOD",
"section_title": "The original conformal thin sandwich method",
"text": "An alternative to the conformal transverse-traceless method for computing initial data has been introduced by York in 1999 [107] . The starting point is the identity\n\nK = - 1 2N L N n γ = - 1 2N ∂ ∂t -L β γ, ( 73\n\n)\n\nwhere N is the lapse function and β is the shift vector associated with some 3+1 coordinates (t, x i ). The traceless part of Eq. (73) leads to\n\nÃij = 1 2N ∂ ∂t -L β γij - 2 3 Dk β k γij , ( 74\n\n)\n\nwhere Ãij is defined by Eq. ( 8 ). Noticing that\n\n-L β γij = ( Lβ) ij + 2 3 Dk β k , ( 75\n\n)\n\nand introducing the short-hand notation γij := ∂ ∂t γij , (76) we can rewrite Eq. (74) as Ãij = 1 2N γij + ( Lβ\n\n) ij . ( 77\n\n)\n\nThe relation between Ãij and Âij is [cf. Eqs. ( 7\n\n)-(8)] Âij = Ψ 6 Ãij . ( 78\n\n) Accordingly, Eq. (77) yields Âij = 1 2 Ñ γij + ( Lβ) ij , ( 79\n\n)\n\nwhere we have introduced the conformal lapse Ñ := Ψ -6 N. (80) Equation (79) constitutes a decomposition of Âij alternative to the longitudinal/transverse decomposition (13). Instead of expressing Âij in terms of a vector X and a TT tensor Âij TT , it expresses it in terms of the shift vector β, the time derivative of the conformal metric, γij , and the conformal lapse Ñ . The Hamiltonian constraint, written as the Lichnerowicz equation (10), takes the same form as before:\n\nDi Di Ψ - R 8 Ψ + 1 8 Âij Âij Ψ -7 + 2π ẼΨ -3 - K 2 12 Ψ 5 = 0, ( 81\n\n)\n\nConstruction of initial data for 3+1 numerical relativity 15 except that now Âij is to be understood as the combination (79) of β i , γij and Ñ . On the other side, the momentum constraint (11) becomes, once expression (79) is substituted for Âij , Dj 1 Ñ ( Lβ) ij + Dj\n\n1 Ñ γij - 4 3 Ψ 6 Di K = 16π pi . ( 82\n\n)\n\nIn view of the system (81)-(82), the method to compute initial data consists in choosing freely γij , γij , K, Ñ , Ẽ and pi on Σ 0 and solving (81)-(82) to get Ψ and β i . This method is called conformal thin sandwich (CTS ), because one input is the time derivative γij , which can be obtained from the value of the conformal metric on two neighbouring hypersurfaces Σ t and Σ t+δt (\"thin sandwich\" view point).\n\nRemark : The term \"thin sandwich\" originates from a previous method devised in the early sixties by Wheeler and his collaborators [4, 101] . Contrary to the methods exposed here, the thin sandwich method was not based on a conformal decomposition: it considered the constraint equations (1)-(2) as a system to be solved for the lapse N and the shift vector β, given the metric γ and its time derivative. The extrinsic curvature which appears in (1)-(2) was then considered as the function of γ, ∂γ/∂t, N and β given by Eq. (73). However, this method does not work in general [9] . On the contrary the conformal thin sandwich method introduced by York [107] and exposed above was shown to work [35] .\n\nAs for the conformal transverse-traceless method treated in Sec. 3, on CMC hypersurfaces, Eq. (82) decouples from Eq. ( 81 ) and becomes an elliptic linear equation for β."
},
{
"section_type": "METHOD",
"section_title": "Extended conformal thin sandwich method",
"text": "An input of the above method is the conformal lapse Ñ . Considering the astrophysical problem stated in Sec. 2.2, it is not clear how to pick a relevant value for Ñ . Instead of choosing an arbitrary value, Pfeiffer and York [80] have suggested to compute Ñ from the Einstein equation giving the time derivative of the trace K of the extrinsic curvature, i.e.\n\n∂ ∂t -L β K = -Ψ -4 Di Di N + 2 Di ln Ψ Di N + N 4π(E + S) + Ãij Ãij + K 2 3 , ( 83\n\n)\n\nwhere S is the trace of the matter stress tensor as measured by the Eulerian observer: S = γ µν T µν . This amounts to add this equation to the initial data system. More precisely, Pfeiffer and York [80] suggested to combine Eq. ( 83 ) with the Hamiltonian constraint to get an equation involving the quantity N Ψ = Ñ Ψ 7 and containing no scalar products of gradients as the Di ln Ψ Di N term in Eq. ( 83 ), thanks to the identity\n\nDi Di N + 2 Di ln Ψ Di N = Ψ -1 Di Di (N Ψ) + N Di Di Ψ . ( 84\n\n)\n\nExpressing the left-hand side of the above equation in terms of Eq. ( 83 ) and substituting Di Di Ψ in the right-hand side by its expression deduced from Eq. ( 81 ), Construction of initial data for 3+1 numerical relativity 16 we get\n\nDi Di ( Ñ Ψ 7 ) -( Ñ Ψ 7 ) 1 8 R + 5 12 K 2 Ψ 4 + 7 8 Âij Âij Ψ -8 + 2π( Ẽ + 2 S)Ψ -4 + K -β i Di K Ψ 5 = 0, ( 85\n\n)\n\nwhere we have used the short-hand notation K := ∂K ∂t (86) and have set S := Ψ 8 S. (87) Adding Eq. (85) to Eqs. (81) and (82), the initial data system becomes\n\nDi Di Ψ - R 8 Ψ + 1 8 Âij Âij Ψ -7 + 2π ẼΨ -3 - K 2 12 Ψ 5 = 0 ( 88\n\n) Dj 1 Ñ ( Lβ) ij + Dj 1 Ñ γij - 4 3 Ψ 6 Di K = 16π pi ( 89\n\n) Di Di ( Ñ Ψ 7 ) -( Ñ Ψ 7 ) R 8 + 5 12 K 2 Ψ 4 + 7 8 Âij Âij Ψ -8 + 2π( Ẽ + 2 S)Ψ -4 + K -β i Di K Ψ 5 = 0, ( 90\n\n)\n\nwhere Âij is the function of Ñ , β i , γij and γij defined by Eq. (79). Equations (88)-(90) constitute the extended conformal thin sandwich (XCTS ) system for the initial data problem. The free data are the conformal metric γ, its coordinate time derivative γ, the extrinsic curvature trace K, its coordinate time derivative K, and the rescaled matter variables Ẽ, S and pi . The constrained data are the conformal factor Ψ, the conformal lapse Ñ and the shift vector β.\n\nRemark : The XCTS system (88)-(90) is a coupled system. Contrary to the CTT system (26)-(27), the assumption of constant mean curvature, and in particular of maximal slicing, does not allow to decouple it."
},
{
"section_type": "OTHER",
"section_title": "XCTS at work: static black hole example",
"text": "Let us illustrate the extended conformal thin sandwich method on a simple example. Take for the hypersurface Σ 0 the punctured manifold considered in Sec. 4.3, namely\n\nΣ 0 = R 3 \\{O}. ( 91\n\n)\n\nFor the free data, let us perform the simplest choice:\n\nγij = f ij , γij = 0, K = 0, K = 0, Ẽ = 0, S = 0, and pi = 0, ( 92\n\n)\n\ni.e. we are searching for vacuum initial data on a maximal and conformally flat hypersurface with all the freely specifiable time derivatives set to zero. Thanks to (92), the XCTS system (88)-(90) reduces to\n\n∆Ψ + 1 8 Âij Âij Ψ -7 = 0 ( 93\n\n) D j 1 Ñ (Lβ) ij = 0 ( 94\n\n) ∆( Ñ Ψ 7 ) - 7 8 Âij Âij Ψ -1 Ñ = 0. ( 95\n\n)\n\nConstruction of initial data for 3+1 numerical relativity 17 Aiming at finding the simplest solution, we notice that β = 0 (96) is a solution of Eq. ( 94 ). Together with γij = 0, it leads to [cf. Eq. (79)] Âij = 0. (97)\n\nThe system (93)-(95) reduces then further:\n\n∆Ψ = 0 ( 98\n\n) ∆( Ñ Ψ 7 ) = 0. ( 99\n\n)\n\nHence we have only two Laplace equations to solve. Moreover Eq. (98) decouples from Eq. ( 99 ). For simplicity, let us assume spherical symmetry around the puncture O. We introduce an adapted spherical coordinate system (x i ) = (r, θ, ϕ) on Σ 0 . The puncture O is then at r = 0. The simplest non-trivial solution of (98) which obeys the asymptotic flatness condition Ψ → 1 as r → +∞ is\n\nΨ = 1 + m 2r , ( 100\n\n)\n\nwhere as in Sec. 4.1, the constant m is the ADM mass of Σ 0 [cf. Eq. (63)]. Notice that since r = 0 is excluded from Σ 0 , Ψ is a perfectly regular solution on the entire manifold Σ 0 . Let us recall that the Riemannian manifold (Σ 0 , γ) corresponding to this value of Ψ via γ = Ψ 4 f is the Riemannian manifold denoted (Σ ′ 0 , γ) in Sec. 4.1 and depicted in Fig. 3 . In particular it has two asymptotically flat ends: r → +∞ and r → 0 (the puncture).\n\nAs for Eq. ( 98 ), the simplest solution of Eq. (99) obeying the asymptotic flatness requirement Ñ Ψ 7 → 1 as r → +∞ is\n\nÑ Ψ 7 = 1 + a r , ( 101\n\n)\n\nwhere a is some constant. Let us determine a from the value of the lapse function at the second asymptotically flat end r → 0. The lapse being related to Ñ via Eq. (80), Eq. ( 101 ) is equivalent to\n\nN = 1 + a r Ψ -1 = 1 + a r 1 + m 2r -1 = r + a r + m/2 . ( 102\n\n) Hence lim r→0 N = 2a m . ( 103\n\n)\n\nThere are two natural choices for lim r→0 N . The first one is lim r→0\n\nN = 1, ( 104\n\n)\n\nyielding a = m/2. Then, from Eq. (102) N = 1 everywhere on Σ 0 . This value of N corresponds to a geodesic slicing. The second choice is\n\nlim r→0 N = -1. ( 105\n\n)\n\nThis choice is compatible with asymptotic flatness: it simply means that the coordinate time t is running \"backward\" near the asymptotic flat end r → 0. This contradicts the assumption N > 0 in the standard definition of the lapse function. However, we shall generalize here the definition of the lapse to allow for negative values: whereas the unit vector n is always future-oriented, the scalar field t is allowed to decrease towards the future. Such a situation has already been encountered for the Construction of initial data for 3+1 numerical relativity 18 part of the slices t = const located on the left side of Fig. 4. Once reported into Eq. ( 103 ), the choice (105) yields a = -m/2, so that\n\nN = 1 - m 2r 1 + m 2r -1 . ( 106\n\n)\n\nGathering relations (96), (100) and (106), we arrive at the following expression of the spacetime metric components:\n\ng µν dx µ dx ν = - 1 -m 2r 1 + m 2r 2 dt 2 + 1 + m 2r 4 dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ) . ( 107\n\n)\n\nWe recognize the line element of Schwarzschild spacetime in isotropic coordinates. Hence we recover the same initial data as in Sec. 4.1 and depicted in Figs. 3 and 4. The bonus is that we have the complete expression of the metric g on Σ 0 , and not only the induced metric γ.\n\nRemark : The choices (104) and (105) for the asymptotic value of the lapse both lead to a momentarily static initial slice in Schwarzschild spacetime. The difference is that the time development corresponding to choice (104) (geodesic slicing) will depend on t, whereas the time development corresponding to choice (105) will not, since in the latter case t coincides with the standard Schwarzschild time coordinate, which makes ∂ t a Killing vector."
},
{
"section_type": "OTHER",
"section_title": "Uniqueness of solutions",
"text": "Recently, Pfeiffer and York [81] have exhibited a choice of vacuum free data (γ ij , γij , K, K) for which the solution (Ψ, Ñ , β i ) to the XCTS system (88)-(90) is not unique (actually two solutions are found). The conformal metric γ is the flat metric plus a linearized quadrupolar gravitational wave, as obtained by Teukolsky [92], with a tunable amplitude.\n\nγij corresponds to the time derivative of this wave, and both K and K are chosen to zero. On the contrary, for the same free data, with K = 0 substituted by Ñ = 1, Pfeiffer and York have shown that the original conformal thin sandwich method as described in Sec. 5.1 leads to a unique solution (or no solution at all if the amplitude of the wave is two large).\n\nBaumgarte, Ó Murchadha and Pfeiffer [14] have argued that the lack of uniqueness for the XCTS system may be due to the term\n\n-( Ñ Ψ 7 ) 7 8 Âij Âij Ψ -8 = - 7 32 Ψ 6 γik γjl γij + ( Lβ) ij γkl + ( Lβ) kl ( Ñ Ψ 7 ) -1 ( 108\n\n)\n\nin Eq. ( 90 ). Indeed, if we proceed as for the analysis of Lichnerowicz equation in Sec. 3.4, we notice that this term, with the minus sign and the negative power of ( Ñ Ψ 7 ) -1 , makes the linearization of Eq. (90) of the type Di Di ǫ + αǫ = σ, with α > 0. This \"wrong\" sign of α prevents the application of the maximum principle to guarantee the uniqueness of the solution.\n\nThe non-uniqueness of solution of the XCTS system for certain choice of free data has been confirmed by Walsh [100] by means of bifurcation theory."
},
{
"section_type": "OTHER",
"section_title": "Comparing CTT, CTS and XCTS",
"text": "The conformal transverse traceless (CTT) method exposed in Sec. 3 and the (extended) conformal thin sandwich (XCTS) method considered here differ by the choice of free data: whereas both methods use the conformal metric γ and the trace Construction of initial data for 3+1 numerical relativity 19 of the extrinsic curvature K as free data, CTT employs in addition Âij TT , whereas for CTS (resp. XCTS) the additional free data is γij , as well as Ñ (resp. K). Since Âij TT is directly related to the extrinsic curvature and the latter is linked to the canonical momentum of the gravitational field in the Hamiltonian formulation of general relativity, the CTT method can be considered as the approach to the initial data problem in the Hamiltonian representation. On the other side, γij being the \"velocity\" of γij , the (X)CTS method constitutes the approach in the Lagrangian representation [108].\n\nRemark : The (X)CTS method assumes that the conformal metric is unimodular:\n\ndet(γ ij ) = f (since Eq. (79) follows from this assumption), whereas the CTT method can be applied with any conformal metric.\n\nThe advantage of CTT is that its mathematical theory is well developed, yielding existence and uniqueness theorems, at least for constant mean curvature (CMC) slices. The mathematical theory of CTS is very close to CTT. In particular, the momentum constraint decouples from the Hamiltonian constraint on CMC slices. On the contrary, XCTS has a much more involved mathematical structure. In particular the CMC condition does not yield to any decoupling. The advantage of XCTS is then to be better suited to the description of quasi-stationary spacetimes, since γij = 0 and K = 0 are necessary conditions for ∂ t to be a Killing vector. This makes XCTS the method to be used in order to prepare initial data in quasi-equilibrium. For instance, it has been shown [57, 43] that XCTS yields orbiting binary black hole configurations in much better agreement with post-Newtonian computations than the CTT treatment based on a superposition of two Bowen-York solutions. Indeed, except when they are very close and about to merge, the orbits of binary black holes evolve very slowly, so that it is a very good approximation to consider that the system is in quasi-equilibrium. XCTS takes this fully into account, while CTT relies on a technical simplification (Bowen-York analytical solution of the momentum constraint), with no direct relation to the quasi-equilibrium state.\n\nA detailed comparison of CTT and XCTS for a single spinning or boosted black hole has been performed by Laguna [68]."
},
{
"section_type": "OTHER",
"section_title": "Initial data for binary systems",
"text": "A major topic of contemporary numerical relativity is the computation of the merger of a binary system of black holes [24] or neutron stars [84] , for such systems are among the most promising sources of gravitational radiation for the interferometric detectors either groundbased (LIGO, VIRGO, GEO600, TAMA) or in space (LISA). The problem of preparing initial data for these systems has therefore received a lot of attention in the past decade."
},
{
"section_type": "OTHER",
"section_title": "Helical symmetry",
"text": "Due to the gravitational-radiation reaction, a relativistic binary system has an inspiral motion, leading to the merger of the two components. However, when the two bodies are sufficiently far apart, one may approximate the spiraling orbits by closed ones. Moreover, it is well known that gravitational radiation circularizes the orbits very efficiently, at least for comparable mass systems [18] . We may then consider that the motion is described by a sequence of closed circular orbits.\n\nConstruction of initial data for 3+1 numerical relativity 20 Figure 5. Action of the helical symmetry group, with Killing vector ℓ. χτ (P ) is the displacement of the point P by the member of the symmetry group of parameter τ . N and β are respectively the lapse function and the shift vector associated with coordinates adapted to the symmetry, i.e. coordinates (t, x i ) such that ∂t = ℓ.\n\nThe geometrical translation of this physical assumption is that the spacetime (M, g) is endowed with some symmetry, called helical symmetry. Indeed exactly circular orbits imply the existence of a one-parameter symmetry group such that the associated Killing vector ℓ obeys the following properties [46]: (i) ℓ is timelike near the system, (ii) far from it, ℓ is spacelike but there exists a smaller number T > 0 such that the separation between any point P and its image χ T (P ) under the symmetry group is timelike (cf. Fig. 5 ). ℓ is called a helical Killing vector, its field lines in a spacetime diagram being helices (cf. Fig. 5 ).\n\nHelical symmetry is exact in theories of gravity where gravitational radiation does not exist, namely:\n\n• in Newtonian gravity,\n\n• in post-Newtonian gravity, up to the second order, • in the Isenberg-Wilson-Mathews (IWM) approximation to general relativity, based on the assumptions γ = f and K = 0 [61, 102].\n\nMoreover helical symmetry can be exact in full general relativity for a nonaxisymmetric system (such as a binary) with standing gravitational waves [44] . But notice that a spacetime with helical symmetry and standing gravitational waves cannot be asymptotically flat [48] .\n\nTo treat helically symmetric spacetimes, it is natural to choose coordinates (t, x i ) that are adapted to the symmetry, i.e. such that\n\n∂ t = ℓ. ( 109\n\n)\n\nThen all the fields are independent of the coordinate t. In particular, γij = 0 and K = 0. (\n\n) 110\n\nConstruction of initial data for 3+1 numerical relativity 21 If we employ the XCTS formalism to compute initial data, we therefore get some definite prescription for the free data γij and K. On the contrary, the requirements (110) do not have any immediate translation in the CTT formalism.\n\nRemark : Helical symmetry can also be useful to treat binary black holes outside the scope of the 3+1 formalism, as shown by Klein [67], who developed a quotient space formalism to reduce the problem to a three dimensional SL(2, R)/SO(1, 1) sigma model.\n\nTaking into account (110) and choosing maximal slicing (K = 0), the XCTS system (88)-(90) becomes\n\nDi Di Ψ - R 8 Ψ + 1 8 Âij Âij Ψ -7 + 2π ẼΨ -3 = 0 ( 111\n\n) Dj 1 Ñ ( Lβ) ij -16π pi = 0 ( 112\n\n) Di Di ( Ñ Ψ 7 ) -( Ñ Ψ 7 ) R 8 + 7 8 Âij Âij Ψ -8 + 2π( Ẽ + 2 S)Ψ -4 = 0, ( 113\n\n)\n\nwhere [cf. Eq. (79)] Âij = 1 2 Ñ ( Lβ) ij . (114)"
},
{
"section_type": "OTHER",
"section_title": "Helical symmetry and IWM approximation",
"text": "If we choose, as part of the free data, the conformal metric to be flat,\n\nγij = f ij , ( 115\n\n)\n\nthen the helically symmetric XCTS system (111)-(113) reduces to\n\n∆Ψ + 1 8 Âij Âij Ψ -7 + 2π ẼΨ -3 = 0 ( 116\n\n) ∆β i + 1 3 D i D j β j -(Lβ) ij D j ln Ñ = 16π Ñ pi ( 117\n\n) ∆( Ñ Ψ 7 ) -( Ñ Ψ 7 ) 7 8 Âij Âij Ψ -8 + 2π( Ẽ + 2 S)Ψ -4 = 0, ( 118\n\n)\n\nwhere Âij = 1 2 Ñ (Lβ) ij (119) and D is the connection associated with the flat metric f , ∆ := D i D i is the flat Laplacian [Eq. (46)], and (Lβ 15 ) with Di = D i ]. We remark that the system (116)-(118) is identical to the system defining the Isenberg-Wilson-Mathews approximation to general relativity [61, 102] (see e.g. Sec. 6.6 of Ref. [51] ). This means that, within helical symmetry, the XCTS system with the choice K = 0 and γ = f is equivalent to the IWM system.\n\n) ij := D i β j + D j β i -2 3 D k β k f ij [Eq. (\n\nRemark : Contrary to IWM, XCTS is not some approximation to general relativity: it provides exact initial data. The only thing that may be questioned is the astrophysical relevance of the XCTS data with γ = f .\n\nConstruction of initial data for 3+1 numerical relativity 22 6.3. Initial data for orbiting binary black holes\n\nThe concept of helical symmetry for generating orbiting binary black hole initial data has been introduced in 2002 by Gourgoulhon, Grandclément and Bonazzola [52, 57]. The system of equations that these authors have derived is equivalent to the XCTS system with γ = f , their work being previous to the formulation of the XCTS method by Pfeiffer and York (2003) [80]. Since then other groups have combined XCTS with helical symmetry to compute binary black hole initial data [38, 1, 2, 31] . Since all these studies are using a flat conformal metric [choice (115)], the PDE system to be solved is (116)-(118), with the additional simplification Ẽ = 0 and pi = 0 (vacuum). The initial data manifold Σ 0 is chosen to be R 3 minus two balls:\n\nΣ 0 = R 3 \\(B 1 ∪ B 2 ). ( 120\n\n)\n\nIn addition to the asymptotic flatness conditions, some boundary conditions must be provided on the surfaces S 1 and S 2 of B 1 and B 2 . One choose boundary conditions corresponding to a non-expanding horizon, since this concept characterizes black holes in equilibrium. We shall not detail these boundary conditions here; they can be found in Refs. [38, 40, 41, 54, 65] . The condition of non-expanding horizon provides 3 among the 5 required boundary conditions [for the 5 components (Ψ, Ñ , β i )]. The two remaining boundary conditions are given by (i) the choice of the foliation (choice of the value of N at S 1 and S 2 ) and (ii) the choice of the rotation state of each black hole (\"individual spin\"), as explained in Ref. [31] . Numerical codes for solving the above system have been constructed by\n\n• Grandclément, Gourgoulhon and Bonazzola (2002) [57] for corotating binary black holes; Cook, Pfeiffer, Caudill and Grigsby (2004, 2006) [38, 31] for corotating and irrotational binary black holes; Ansorg (2005 Ansorg ( , 2007) [1, 2] ) [1, 2] for corotating binary black holes.\n\n•\n\n•\n\nDetailed comparisons with post-Newtonian initial data (either from the standard post-Newtonian formalism [17] or from the Effective One-Body approach [23, 42] ) have revealed a very good agreement, as shown in Refs. [43, 31] . An alternative to (120) for the initial data manifold would be to consider the twice-punctured R 3 :\n\nΣ 0 = R 3 \\{O 1 , O 2 }, ( 121\n\n)\n\nwhere O 1 and O 2 are two points of R 3 . This would constitute some extension to the two bodies case of the punctured initial data discussed in Sec. 5.3. However, as shown by Hannam, Evans, Cook and Baumgarte in 2003 [60] , it is not possible to find a solution of the helically symmetric XCTS system with a regular lapse in this case . For this reason, initial data based on the puncture manifold (121) are computed within the CTT framework discussed in Sec. 3. As already mentioned, there is no natural way to implement helical symmetry in this framework. One instead selects the free data Âij TT to vanish identically, as in the single black hole case treated in Secs. 4.1 and 4.3. Then Âij = ( LX) ij . (\n\n) 122\n\nsee however Ref. [59] for some attempt to circumvent this\n\nConstruction of initial data for 3+1 numerical relativity 23 The vector X must obey Eq. ( 45 ), which arises from the momentum constraint. Since this equation is linear, one may choose for X a linear superposition of two Bowen-York solutions (Sec. 4.3):\n\nX = X (P (1) ,S (1) ) + X (P (2) ,S (2) ) , ( 123\n\n)\n\nwhere X (P (a) ,S (a) ) (a = 1, 2) is the Bowen-York solution (71) centered on O a . This method has been first implemented by Baumgarte in 2000 [11] . It has been since then used by Baker, Campanelli, Lousto and Takashi (2002) [5] and Ansorg, Brügmann and Tichy (2004) [3]. The initial data hence obtained are closed from helically symmetric XCTS initial data at large separation but deviate significantly from them, as well as from post-Newtonian initial data, when the two black holes are very close. This means that the Bowen-York extrinsic curvature is bad for close binary systems in quasi-equilibrium (see discussion in Ref. [43] ).\n\nRemark : Despite of this, CTT Bowen-York configurations have been used as initial data for the recent binary black hole inspiral and merger computations by Baker et al. [6, 7, 99] and Campanelli et al. [25, 26, 27, 28] . Fortunately, these initial data had a relative large separation, so that they differed only slightly from the helically symmetric XCTS ones.\n\nInstead of choosing somewhat arbitrarily the free data of the CTT and XCTS methods, notably setting γ = f , one may deduce them from post-Newtonian results. This has been done for the binary black hole problem by Tichy, Brügmann, Campanelli and Diener (2003) [94], who have used the CTT method with the free data (γ ij , Âij TT ) given by the second order post-Newtonian (2PN) metric. This work has been improved recently by Kelly, Tichy, Campanelli and Whiting (2007) [66] . In the same spirit, Nissanke (2006) [75] has provided 2PN free data for both the CTT and XCTS methods."
},
{
"section_type": "OTHER",
"section_title": "Initial data for orbiting binary neutron stars",
"text": "For computing initial data corresponding to orbiting binary neutron stars, one must solve equations for the fluid motion in addition to the Einstein constraints. Basically this amounts to solving ∇ ν T µν = 0 in the context of helical symmetry. One can then show that a first integral of motion exists in two cases: (i) the stars are corotating, i.e. the fluid 4-velocity is colinear to the helical Killing vector (rigid motion), (ii) the stars are irrotational, i.e. the fluid vorticity vanishes. The most straightforward way to get the first integral of motion is by means of the Carter-Lichnerowicz formulation of relativistic hydrodynamics, as shown in Sec. 7 of Ref. [50] . Other derivations have been obtained in 1998 by Teukolsky [93] and Shibata [83] .\n\nFrom the astrophysical point of view, the irrotational motion is much more interesting than the corotating one, because the viscosity of neutron star matter is far too low to ensure the synchronization of the stellar spins with the orbital motion. On the other side, the irrotational state is a very good approximation for neutron stars that are not millisecond rotators. Indeed, for these stars the spin frequency is much lower than the orbital frequency at the late stages of the inspiral and thus can be neglected.\n\nThe first initial data for binary neutron stars on circular orbits have been computed by Baumgarte, Cook, Scheel, Shapiro and Teukolsky in 1997 [12, 13] in the corotating case, and by Bonazzola, Gourgoulhon and Marck in 1999 [19] in the irrotational case. These results were based on a polytropic equation of state. Since then configurations in the irrotational regime have been obtained Construction of initial data for 3+1 numerical relativity 24 • for a polytropic equation of state [73, 96, 97, 53, 90, 91] (the configurations obtained in Ref. [91] have been used as initial data by Shibata [84] to compute the merger of binary neutron stars); • for nuclear matter equations of state issued from recent nuclear physics computations [16, 77]; • for strange quark matter [78, 72] .\n\nAll these computation are based on a flat conformal metric [choice (115)], by solving the helically symmetric XCTS system (116)-(118), supplemented by an elliptic equation for the velocity potential. Only very recently, configurations based on a non flat conformal metric have been obtained by Uryu, Limousin, Friedman, Gourgoulhon and Shibata [98] . The conformal metric is then deduced from a waveless approximation developed by Shibata, Uryu and Friedman [85] and which goes beyond the IWM approximation."
},
{
"section_type": "OTHER",
"section_title": "Initial data for black hole -neutron star binaries",
"text": "Let us mention briefly that initial data for a mixed binary system, i.e. a system composed of a black hole and a neutron star, have been obtained very recently by Grandclément [55] and Taniguchi, Baumgarte, Faber and Shapiro [88, 89] . Codes aiming at computing such systems have also been presented by Ansorg [2] and Tsokaros and Uryu [95] ."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "I warmly thank the organizers of the VII Mexican school, namely Miguel Alcubierre, Hugo Garcia-Compean and Luis Urena, for their support and the success of the school. I also express my gratitude to Marcelo Salgado for his help and many discussions and to Nicolas Vasset for the careful reading of the manuscript."
},
{
"section_type": "OTHER",
"section_title": "Compositio",
"text": "Rev. D 70, 104016 (2004). [39] J. Corvino : Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Commun. Math. Phys. 214, 137 (2000). [40] S. Dain : Trapped surfaces as boundaries for the constraint equations, Class. Quantum Grav.\n\n21, 555 (2004); errata in Class. Quantum Grav. 22, 769 (2005). [41] S. Dain, J.L. Jaramillo, and B. Krishnan : On the existence of initial data containing isolated\n\nConstruction of initial data for 3+1 numerical relativity 26 black holes, Phys.Rev. D 71, 064003 (2005). [42] T. Damour : Coalescence of two spinning black holes: An effective one-body approach, Phys.\n\nRev. D 64, 124013 (2001). [43] T. Damour, E. Gourgoulhon, and P. Grandclément : Circular orbits of corotating binary black holes: comparison between analytical and numerical results, Phys. Rev. D 66, 024007 (2002). [44] S. Detweiler : Periodic solutions of the Einstein equations for binary systems, Phys. Rev. D 50, 4929 (1994). [45] Y. Fourès-Bruhat (Y. Choquet-Bruhat) : Sur l'Intégration des Équations de la Relativité Générale, J. Rational Mech. Anal. 5, 951 (1956). [46] J.L. Friedman, K. Uryu and M. Shibata : Thermodynamics of binary black holes and neutron stars, Phys. Rev. D 65, 064035 (2002); erratum in Phys. Rev. D 70, 129904(E) (2004). [47] A. Garat and R.H. Price : Nonexistence of conformally flat slices of the Kerr spacetime, Phys.\n\nRev. D 61, 124011 (2000). [48] G.W. Gibbons and J.M. Stewart : Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity, in Classical General Relativity, Eds. W.B. Bonnor, J.N. Islam and M.A.H. MacCallum Cambridge University Press, Cambridge (1983), p. 77. [49] R.J. Gleiser, C.O. Nicasio, R.H. Price, and J. Pullin : Evolving the Bowen-York initial data for spinning black holes, Phys. Rev. D 57, 3401 (1998). [50] E. 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}
] | |
arxiv:0704.0166 | 0704.0166 | 1 | 10.1088/1126-6708/2007/05/047 | 969f036fa8b412543185b0b02e81a1874555556c0056a68af5f5029d7028cad8 | Supersymmetry breaking metastable vacua in runaway quiver gauge theories | In this paper we consider quiver gauge theories with fractional branes whose
infrared dynamics removes the classical supersymmetric vacua (DSB branes). We
show that addition of flavors to these theories (via additional non-compact
branes) leads to local meta-stable supersymmetry breaking minima, closely
related to those of SQCD with massive flavors. We simplify the study of the
one-loop lifting of the accidental classical flat directions by direct
computation of the pseudomoduli masses via Feynman diagrams. This new approach
allows to obtain analytic results for all these theories. This work extends the
results for the $dP_1$ theory in hep-th/0607218. The new approach allows to
generalize the computation to general examples of DSB branes, and for arbitrary
values of the superpotential couplings. | [
"Inaki Garcia-Etxebarria",
"Fouad Saad",
"Angel M. Uranga"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | Systems of D-branes at singularities provide a very interesting setup to realize and study diverse non-perturbative gauge dynamics phenomena in string theory. In the context of N = 1 supersymmetric gauge field theories, systems of D3-branes at Calabi-Yau singularities lead to interesting families of tractable 4d strongly coupled conformal field theories, which extend the AdS/CFT correspondence [1, 2, 3] to theories with reduced (super)symmetry [4, 5, 6] and enable non-trivial precision tests of the correspondence (see for instance [7, 8] ). Addition of fractional branes leads to families of non-conformal gauge theories, with intricate RG flows involving cascades of Seiberg dualities [9, 10, 11, 12, 13] , and strong dynamics effects in the infrared.
For instance, fractional branes associated to complex deformations of the singular geometry (denoted deformation fractional branes in [12] ), correspond to supersymmetric confinement of one or several gauge factors in the gauge theory [9, 12] . The generic case of fractional branes associated to obstructed complex deformations (denoted DSB branes in [12] ), corresponds to gauge theories developing a non-perturbative Affleck-Dine-Seiberg superpotential, which removes the classical supersymmetric vacua [14, 15, 16] . As shown in [15] (see also [17, 18] ), assuming canonical Kahler potential leads to a runaway potential for the theory, along a baryonic direction. A natural suggestion to stop this runaway has been proposed for the particular example of the dP 1 theory (the theory on fractional branes at the complex cone over dP 1 ) in [19] . It was shown that, upon the addition of D7-branes to the configuration (which introduce massive flavors), the theory develops a meta-stable minimum (closely related to the Intriligator-Seiberg-Shih (ISS) model [20] ), parametrically long-lived against decay to the runaway regime (see [21] for an alternative suggestion to stop the runaway, in compact models).
In this paper we show that the appearance of meta-stable minima in gauge theories on DSB fractional branes, in the presence of additional massless flavors, is much more general (and possibly valid in full generality). We use the tools of [15] to introduce D7-branes on general toric singularities, and give masses to the corresponding flavors.
Since quiver gauge theories are rather involved, we develop new techniques to efficiently analyze the one-loop stability of the meta-stable minima, via the direct computation of Feynman diagrams. These tools can be used to argue that the results plausibly hold for general systems of DSB fractional branes at toric singularities. It is very satisfactory to verify the correspondence between the existence of meta-stable vacua and the geometric property of having obstructed complex deformations.
The present work thus enlarges the class of string models realizing dynamical supersymmetry breaking in meta-stable vacua (see [22, 23, 24, 25, 26] for other proposed realizations, and [27, 28, 29] for models of dynamical supersymmetry breaking in orientifold theories). Although we will not discuss it in the present paper, these results can be applied to the construction of models of gauge mediation in string theory as in [30] (based on the additional tools in [31] ), in analogy with [32] . This is another motivation for the present work.
The paper is organized as follows. In Section 2 we review the ISS model, evaluating one-loop pseudomoduli masses directly in terms of Feynman diagrams. In Section 3 we study the theory of DSB branes at the dP 1 and dP 2 singularities upon the addition of flavors, and we find that metastable vacua exist for these theories. In Section 4 we extend this analysis to the general case of DSB branes at toric singularities with massive flavors, and we illustrate the results by showing the existence of metastable vacua for DSB branes at some well known families of toric singularities. Finally, the Appendix provides some technical details that we have omitted from the main text in order to improve the legibility.
In this Section we review the ISS meta-stable minima in SQCD, and propose that the analysis of the relevant piece of the one-loop potential (the quadratic terms around the maximal symmetry point) is most simply carried out by direct evaluation of Feynman diagrams. This new tool will be most useful in the study of the more involved examples of quiver gauge theories.
The ISS model [20] (see also [33] for a review of these and other models) is given by N = 1 SU(N c ) theory with N f flavors, with small masses W electric = mTr φ φ, (2.1)
where φ and φ are the quarks of the theory. The number of colors and flavors are chosen so as to be in the free magnetic phase:
N c + 1 ≤ N f < 3 2 N c . (2.2) 2
This condition guarantees that the Seiberg dual is infrared free. This Seiberg dual is the SU(N) theory (with N = N f -N c ) with N f flavors of dual quarks q and q and the meson M. The dual superpotential is given by rewriting (2.1) in terms of the mesons and adding the usual coupling between the meson and the dual quarks:
W magnetic = h (Tr qMqµ 2 Tr M), (2.3) where h and µ can be expressed in terms of the parameters m and Λ, and some (unknown) information about the dual Kähler metric 1 . It was also argued in [20] that it is possible to study the supersymmetry breaking minimum in the origin of (dual) field space without taking into account the gauge dynamics (their main effect in this discussion consists of restoring supersymmetry dynamically far in field space). In the following we will assume that this is always the case, and we will forget completely about the gauge dynamics of the dual.
Once we forget about gauge dynamics, studying the vacua of the dual theory becomes a matter of solving the F-term equations coming from the superpotential (2.3).
The mesonic F-term equation reads:
-F M ij = hq i • q jhµ 2 δ ij = 0, (2.4)
where i and j are flavor indices and the dot denotes color contraction. This has no solution, since the identity matrix δ ij has rank N f while qi • q j has rank N = N f -N c . Thus this theory breaks supersymmetry spontaneously at tree level. This mechanism for F-term supersymmetry breaking is called the rank condition.
The classical scalar potential has a continuous set of minima, but the one-loop potential lifts all of the non-Goldstone directions, which are usually called pseudomoduli. The usual approach to study the one-loop stabilization is the computation of the complete one-loop effective potential over all pseudomoduli space via the Coleman-Weinberg formula [34]: V = 1 64π 2 Tr M 4 B log M 2 B Λ 2 -M 4 F log M 2 F Λ 2 . (2.5) This approach has the advantage that it allows the determination of the one-loop minimum, without a priori information about its location, and moreover it provides the full potential around it, including higher terms. However, it has the disadvantage 1 The exact expressions can be found in (5.7) in [20] , but we will not need them for our analysis.
We just take all masses in the electric description to be small enough for the analysis of the metastable vacuum to be reliable.
of requiring the diagonalization of the mass matrix, which very often does not admit a closed expression, e.g. for the theories we are interested in.
In fact, we would like to point out that to determine the existence of a meta-stable minimum there exists a computationally much simpler approach. In our situation, we have a good ansatz for the location of the one-loop minimum, and are interested just in the one-loop pseudomoduli masses around such point. This information can be directly obtained by computing the one-loop masses via the relevant Feynman diagrams. This technique is extremely economical, and provides results in closed form in full generality, e.g. for general values of the couplings, etc. The correctness of the original ansatz for the vacuum can eventually be confirmed by the results of the computation (namely positive one-loop squared masses, and negligible tadpoles for the classically massive fields 2 ).
Hence, our strategy to study the one-loop stabilization in this paper is as follows:
• First we choose an ansatz for the classical minimum to become the one-loop vacuum. It is natural to propose a point of maximal enhanced symmetry (in particular, close to the origin in the space of vevs for M there exist and Rsymmetry, whose breaking by gauge interactions (via anomalies) is negligible in that region). Hence the natural candidate for the one-loop minimum is q = qT = µ 0 , (2.6) with the rest of the fields set to 0. This initial ansatz for the one-loop minimum is eventually confirmed by the positive square masses at one-loop resulting from the computations described below. In our more general discussion of meta-stable minima in runaway quiver gauge theories, our ansatz for the one-loop minimum is a direct generalization of the above (and is similarly eventually confirmed by the one-loop mass computation).
• Then we expand the field linearly around this vacuum, and identify the set of classically massless fields. We refer to these as pseudomoduli (with some abuse of language, since there could be massless fields which are not classically flat directions due to higher potential terms) 2 Since supersymmetry is spontaneously broken the effective potential will get renormalized by quantum effects, and thus classically massive fields might shift slightly. This appears as a one loop tadpole which can be encoded as a small shift of µ. This will enter in the two loop computation of the pseudomoduli masses, which are beyond the scope of the present paper.
• As a final step we compute one-loop masses for these pseudomoduli by evaluating their two-point functions via conventional Feynman diagrams, as explained in more detail in appendix A.1 and illustrated below in several examples.
The ISS model is a simple example where this technique can be illustrated. Considering the above ansatz for the vacuum, we expand the fields around this point as:
q = µ + 1 √ 2 (ξ + + ξ -) 1 √ 2 (ρ + + ρ -) , qT = µ + 1 √ 2 (ξ + -ξ -) 1 √ 2 (ρ + -ρ -) , M = Y Z ZT Φ , (2.7)
where we have taken linear combinations of the fields in such a way that the bosonic mass matrix is diagonal. This will also be convenient in section 2.2, where we discuss the Goldstone bosons in greater detail.
We now expand the superpotential (2.3) to get
W = √ 2µξ + Y + 1 √ 2 µZρ + + 1 √ 2 µZρ -+ 1 √ 2 µρ + Z - 1 √ 2 µρ - Z + 1 2 ρ 2 + Φ - 1 2 ρ 2 -Φ -µ 2 Φ + . . . , (2.8)
where we have not displayed terms of order three or higher in the fluctuations, unless they contain Φ, since they are irrelevant for the one loop computation we will perform.
Note also that we have set h = 1 and we have removed the trace (the matricial structure is easy to restore later on, here we just set N f = 2 for simplicity). The massless bosonic fluctuations are given by Re ρ + , Im ρ -, Φ and ξ -. The first two together with Im ξ -are Goldstone bosons, as explained in section 2.2. Thus the pseudomoduli we are interested in are given by Φ and Re ξ -. Let us focus on Φ (the case of Re ξ -admits a similar discussion). In this case the relevant terms in the superpotential simplify further, and just the following superpotential contributes:
W = µZ 1 √ 2 (ρ + + ρ -) + µ Z 1 √ 2 (ρ + -ρ -) + 1 2 ρ 2 + Φ - 1 2 ρ 2 -Φ -µ 2 Φ + . . . ,
which we recognize, up to a field redefinition, as the symmetric model of appendix A.2.
We can thus directly read the result
δm 2 Φ = |h| 4 µ 2 8π 2 (log 4 -1). (2.9)
This matches the value given in [20] , which was found using the Coleman-Weinberg potential. 5
One aspect of our technique that merits some additional explanation concerns the Goldstone bosons. The one-loop computation of the masses for the fluctuations associated to the symmetries broken by the vacuum, using just the interactions described in appendix A.1, leads to a non-vanishing result. This puzzle is however easily solved by realizing that certain (classically massive) fields have a one-loop tadpole. This leads to a new contribution to the one-loop Goldstone two-point amplitude, given by the diagram in Figure 1 . Adding this contribution the total one-loop mass for the Goldstone bosons is indeed vanishing, as expected. This tadpole does not affect the computation of the one-loop pseudomoduli masses (except for Re ξ + , but its mass remains positive) as it is straightforward to check.
Im ξ - Im ξ - Re ξ + Figure 1: Schematic tadpole contribution to the Im ξ -two point function. Both bosons and fermions run in the loop.
The structure of this cancellation can be understood by using the derivation of the Goldstone theorem for the 1PI effective potential, as we now discuss. The proof can be found in slightly more detail, together with other proofs, in [35] . Let us denote by V the 1PI effective potential. Invariance of the action under a given symmetry implies
that δV δφ i ∆φ i = 0, (2.10)
where we denote by ∆φ i the variation of the field φ i under the symmetry, which will in general be a function of all the fields in the theory. Taking the derivative of this equation with respect to some other field φ k
δ 2 V δφ i δφ k ∆φ i + δV δφ i • δ∆φ i δφ k = 0. (2.11)
Let us consider how this applies to our case. At tree level, there is no tadpole and the above equation (truncated at tree level) states that for each symmetry generator broken by the vacuum, the value of ∆φ i gives a nonvanishing eigenvector of the mass matrix with zero eigenvalue. This is the classical version of the Goldstone theorem, which allows the identification of the Goldstone bosons of the theory.
For instance, in the ISS model in the previous section (for N f = 2), there are three global symmetry generators broken at the minimum described around (2.6). The 6 SU(2) × U(1) symmetry of the potential gets broken down to a U(1) ′ , which can be understood as a combination of the original U(1) and the t z generator of SU(2). The Goldstone bosons can be taken to be the ones associated to the three generators of SU(2), and correspond (for µ real) to Im ξ -, Im ρ -and Re ρ + , in the parametrization of the fields given by equation (2.7).
Even in the absence of tree-level tadpoles, there could still be a one-loop tadpole.
When this happens, there should also be a non-trivial contribution to the mass term for the Goldstone bosons in the one-loop 1PI potential, related to the tadpole by the one-loop version of (2.11). This relation guarantees that the mass term in the physical (i.e. Wilsonian) effective potential, which includes the 1PI contribution, plus those of the diagram in Figure 1 , vanishes, as we described above.
In fact, in the ISS example, there is a non-vanishing one-loop tadpole for the real part of ξ + (and no tadpole for other fields). The calculation of the tadpole at one loop is straightforward, and we will only present here the result iM = -i|h| 4 µ 3 (4π) 2 (2 log 2). (2.12)
The 1PI one-loop contribution to the Goldstone boson mass is also simple to calculate, giving the result iM = -i|h| 4 µ 2 (4π) 2 (log 2). (2.13) Using the variations of the relevant fields under the symmetry generator, e.g. for t z ,
∆Re ξ + = -Im ξ - (2.14) ∆Im ξ -= Re ξ + + 2µ. (2.15)
we find that the (2.11) is satisfied at one-loop.
δ 2 V δφ i δφ k ∆φ i + δV δφ i • δ∆φ i δφ k = m 2 Im ξ -• 2µ + (Re ξ + tadpole) • (-1) = 0. (2.16)
A very similar discussion applies to t x and t y .
The above discussion of Goldstone bosons can be similarly carried out in all examples of this paper. Hence, it will be enough to carry out the computation of the 1PI diagrams discussed in appendix A.1, and verify that they lead to positive squared masses for all classically massless fields (with Goldstone bosons rendered massless by the additional diagrams involving the tadpole). 7 3 Meta-stable vacua in quiver gauge theories with DSB branes
In this section we show the existence of a meta-stable vacuum in a few examples of gauge theories on DSB branes, upon the addition of massive flavors. As already discussed in [19] , the choice of fractional branes of DSB kind is crucial in the result.
The reason is that in order to have the ISS structure, and in particular supersymmetry breaking by the rank condition, one needs a node such that its Seiberg dual satisfies
N f > N, with N = N f -N c with N c
, N f the number of colors, flavors of that gauge factor. Denoting N f,0 , N f,1 the number of massless and massive flavors (namely flavors arising from bi-fundamentals of the original D3-brane quiver, or introduced by the D7branes), the condition is equivalent to N f,0 < N c . This is precisely the condition that an ADS superpotential is generated, and is the prototypical behavior of DSB branes [14, 15, 16, 18] .
Another important general comment, also discussed in [19] , is that theories on DSB branes generically contain one or more chiral multiplets which do not appear in the superpotential. Being decoupled, such fields remain as accidental flat directions at one-loop, so that the one-loop minimum is not isolated. The proper treatment of these flat directions is beyond the reach of present tools, so they remain an open question.
However, it is plausible that they do not induce a runaway behavior to infinity, since they parametrize a direction orthogonal to the fields parametrizing the runaway of DSB fractional branes.
In this section we describe the most familiar example of quiver gauge theory with DSB fractional branes, the dP 1 theory. In this theory, a non-perturbative superpotential removes the classical supersymmetric vacua [14, 15, 16] . Assuming canonical Kähler potential the theory has a runaway behavior [15, 17] . In this section, we revisit with our techniques the result in [19] that the addition of massive flavors can induce the appearance of meta-stable supersymmetry breaking minima, long-lived against tunneling to the runaway regime. As we show in coming sections, this behavior is prototypical and extends to many other theories with DSB fractional branes. The example is also representative of the computations for a general quiver coming from a brane at a toric singularity, and illustrates the usefulness of the direct Feynman diagram evaluation of one-loop masses.
Consider the dP 1 theory, realized on a set of M fractional D3-branes at the complex cone over dP 1 . In order to introduce additional flavors, we introduce sets of N f,1 D7-branes wrapping non-compact 4-cycles on the geometry and passing through the singular point. We refer the reader to [19] , and also to later sections, for more details on the construction of the theory, and in particular on the introduction of the D7-branes.
Its quiver is shown in Figure 2 , and its superpotential is
W = λ(X 23 X 31 Y 12 -X 23 Y 31 X 12 ) + λ ′ (Q 3i Qi2 X 23 + Q 2j Qj1 X 12 + Q 1k Qk3 X 31 ) + m 3 Q 3i Qk3 δ ik + m 2 Q 2j Qi2 δ ji + m 1 Q 1k Qj1 δ kj , (3.1)
where the subindices denote the groups under which the field is charged. The first line is the superpotential of the theory of fractional brane, the second line describes 77-73-37 couplings between the flavor branes and the fractional brane, and the last line gives the flavor masses. Note that there is a massless field, denoted Z 12 in [19] , that does not appear in the superpotential. This is one of the decoupled fields mentioned above, and we leave its treatment as an open question. 1 3 2 i j k SU(3M) SU(2M) SU(M) PSfrag replacements
Q 3i Qi2 Q 2j Qj1 Q 1k
Qk3 Figure 2: Extended quiver diagram for a dP 1 theory with flavors, from [19].
We are interested in gauge factors in the free magnetic phase. This is the case for the SU(3M) gauge factor in the regime
M + 1 ≤ N f,1 < 5 2 M. (3.2)
To apply Seiberg duality on node 3, we introduce the dual mesons:
M 21 = 1 Λ X 23 X 31 ; N k1 = 1 Λ Qk3 X 31 M ′ 21 = 1 Λ X 23 Y 31 ; N ′ k1 = 1 Λ Qk3 Y 31 N 2i = 1 Λ X 23 Q 3i ; Φ ki = 1 Λ Qk3 Q 3i (3.3) 9
and we also replace the electric quarks
Q 3i , Qk3 , X 23 , X 31 , Y 31 by their magnetic duals Qi3 , Q 3k , X 32 , X 13 , Y 13 .
The magnetic superpotential is given by rewriting the confined fields in terms of the mesons and adding the coupling between the mesons and the dual quarks,
W = h ( M 21 X 13 X 32 + M ′ 21 Y 13 X 32 + N 2i Qi3 X 32 + N k1 X 13 Q 3k + N ′ k1 Y 13 Q 3k + Φ ki Qi3 Q 3k ) + hµ 0 ( M 21 Y 12 -M ′ 21 X 12 ) + µ ′ Q 1k N k1 + µ ′ N 2i Qi2 -hµ 2 Tr Φ + λ ′ Q 2j Qj1 X 12 + m 2 Q 2i Qi2 + m 1 Q 1i Qi1 . (3.4)
This is the theory we want to study. In order to simplify the treatment of this example we will disregard any subleading terms in m i /µ ′ , and effectively integrate out N k1 and N 2i by substituting them by 0. This is not necessary, and indeed the computations in the next sections are exact. We do it here in order to compare results with [19].
As in the ISS model, this theory breaks supersymmetry via the rank condition. The fields Qi3 , Q 3k and Φ ki are the analogs of q, q and M in the ISS case discussed above.
This motivates a vacuum ansatz analogous to (2.6) and the following linear expansion:
Φ = φ 00 φ 01 φ 10 φ 11 ; Qi3 = µe θ + Q 3,1 Q3,2 ; Q T 3i = µe -θ + Q 3,1 Q 3,2 Qk1 = Q1,1 y ; Q 2j = Q 2,11 x Q 2,21 x ′ ; M 21 = M 21,1 M 21,2 Y 13 = (Y 13 ) ; X T 12 = X 12,1 X 12,2 ; X T 32 = X 32,1 X 32,2 Y T 12 = Y 12,1 Y 12,2 ; N ′ k1 = N ′ k1,1 z ; M ′ 21 = λ ′ hµ 0 M ′ 21,1 M ′ 21,2 X 13 = (X 13 ) .
(3.5) Note that we have chosen to introduce the nonlinear expansion in θ in order to reproduce the results found in the literature in their exact form 3 . Note also that for the sake of clarity we have not been explicit about the ranks of the different matrices.
They can be easily worked out (or for this case, looked up in [19] ), and we will restrict ourselves to the 2 flavor case where the matrix structure is trivial. As a last remark, we are not being explicit either about the definitions of the different couplings in terms of the electric theory. This can be done easily (and as in the ISS case they involve 3 A linear expansion would lead to identical conclusions concerning the existence of the meta-stable
vacua, but to one-loop masses not directly amenable to comparison with results in the literature.
10 an unknown coefficient in the Kähler potential), but in any event, the existence of the meta-stable vacua can be established for general values of the coefficients in the superpotential. Hence we skip this more detailed but not very relevant discussion.
The next step consists in expanding the superpotential and identifying the massless fields. We get the following quadratic contributions to the superpotential:
W mass = 2hµφ 00 Q3,1 + hµφ 01 Q3,2 + hµφ 10 Q 3,2 + hµ 0 M 21,1 Y 12,1 + hµ 0 M 21,2 Y 12,2 -λ ′ M ′ 21,1 X 12,1 -λ ′ M ′ 21,2 X 12,2 + hµN ′ k1,1 Y 13 -h 1 µ Q1,1 X 13 -h 2 µQ 2,11 X 32,1 -h 2 µQ 2,21 X 32,2 . (3.6)
The fields massless at tree level are x, x ′ , y, z, φ 11 , θ, Q 3,2 and Q3,2 . Three of these are Goldstone bosons as described in the previous section. For real µ they are Im θ, Re ( Q3,2 + Q 3,2 ) and Im ( Q3,2 -Q 3,2 ). We now show that all other classically massless fields get masses at one loop (with positive squared masses).
As a first step towards finding the one-loop correction, notice that the supersymmetry breaking mechanism is extremely similar to the one in the ISS model before, in particular it comes only from the following couplings in the superpotential:
W rank = hQ 3,2 Q3,2 φ 11 -hµ 2 φ 11 + . . . (3.7)
This breaks the spectrum degeneracy in the multiplets Q 3,2 and Q3,2 at tree level, so we refer to them as the fields with broken supersymmetry.
Let us compute now the correction for the mass of x, for example. For the one-loop computation we just need the cubic terms involving one pseudomodulus and at least one of the broken supersymmetry fields, and any quadratic term involving fields present in the previous set of couplings. From the complete expansion one finds the following supersymmetry breaking sector:
W symm. = hφ 11 Q 3,2 Q3,2 + hµφ 01 Q3,2 + hµφ 10 Q 3,2 -hµ 2 φ 11 . (3.8)
The only cubic term involving the pseudomodulus x and the broken supersymmetry fields is
W cubic = -h 2 x Q3,2 X 32,1 , (3.9)
and there is a quadratic term involving the field X 32,1
W mass coupling = -h 2 µQ 2,11 X 32,1 . (3.10)
Assembling the three previous equations, the resulting superpotential corresponds to the asymmetric model in appendix A.2, so we can directly obtain the one-loop mass 11 for x:
δm 2 x = 1 16π 2 |h| 4 µ 2 C |h 2 | 2 |h| 2 . (3.11)
Proceeding in a similar way, the one-loop masses for φ 11 , x ′ , y and z are:
δm 2 φ 11 = 1 8π 2 |h| 4 µ 2 (log 4 -1) δm 2 x ′ = 1 16π 2 |h| 4 µ 2 C |h 2 | 2 |h| 2 , δm 2 y = 1 16π 2 |h| 4 µ 2 C |h 1 | 2 |h| 2 δm 2 z = 1 16π 2 |h| 4 µ 2 (log 4 -1).
(3.12) There is just one pseudomodulus left, Re θ, which is qualitatively different to the others. With similar reasoning, one concludes that it is necessary to study a superpotential of the form
W = h(Xφ 1 φ 2 + µe θ φ 1 φ 3 + µe -θ φ 2 φ 4 -µ 2 X). (3.13)
Due to the non-linear parametrization, the expansion in θ shows that there is a term quadratic in θ which contributes to the one-loop mass via a vertex with two bosons and two fermions, the relevant diagram is shown in Figure 16d . The result is a vanishing mass for Im θ, as expected for a Goldstone boson (the one-loop tadpole vanishes in this case), and a non-vanishing mass for Re θ
δm 2 Re θ = 1 4π 2 |h| 4 µ 4 (log 4 -1). (3.14)
We conclude by mentioning that all squared masses are positive, thus confirming that the proposed point in field space is the one-loop minimum. As shown in [19] , this minimum is parametrically long-lived against tunneling to the runaway regime.
Let us apply these techniques to consider new examples. In this section we consider a DSB fractional brane in the complex cone over dP 2 , which provides another quiver theory with runaway behavior [15] . The quiver diagram for dP 2 is given in Figure 3 ,
with superpotential W = X 34 X 45 X 53 -X 53 Y 31 X 15 -X 34 X 42 Y 23 + Y 23 X 31 X 15 X 52 + X 42 X 23 Y 31 X 14 -X 23 X 31 X 14 X 45 X 52 ( 3
.15) 12 1 2 3 5 4 Figure 3: Quiver diagram for the dP 2 theory.
We consider a set of M DSB fractional branes, corresponding to choosing ranks (M, 0, M, 0, 2M) for the corresponding gauge factors. The resulting quiver is shown in Figure 4 , with superpotential
W = -λX 53 Y 31 X 15 (3.16) U(2M) U(M) U(M) 3 1 5
Figure 4: Quiver diagram for the dP 2 theory with M DSB fractional branes.
Following [19] and appendix B, one can introduce D7-branes leading to D3-D7 open strings providing (possibly massive) flavors for all gauge factors, and having cubic couplings with diverse D3-D3 bifundamental chiral multiplets. We obtain the quiver in Figure 5 . Adding the cubic 33-37-73 coupling superpotential, and the flavor masses, the complete superpotential reads
W total = -λX 53 Y 31 X 15 -λ ′ (Q 1i Qi3 Y 31 + Q 3j Qj5 X 53 + Q 5k Qk1 X 15 ) + m 1 Q 1i Qk1 + m 2 Q 3j Qi3 + m 5 Q 5k Qj5 (3.17)
where 1, 2, 3 are the gauge group indices and i, j, k are the flavor indices.
We consider the U(2M) node in the free magnetic phase, namely
M + 1 ≤ N f,1 < 2M (3.18) 13 U(M) U(M) U( 2M
)
PSfrag replacements
Q 1i Q i3 Q 3j Q j5 Q 5k Q k1
Figure 5: Quiver for the dP 2 theory with M fractional branes and flavors.
After Seiberg Duality the dual gauge factor is SU(N) with N = N f,1 -M and dynamical scale Λ. To get the matter content in the dual, we replace the microscopic flavors
Q 5k , Qj5 , X 53 , X 15 by the dual flavors Qk5 , Q 5j , X 35 , X 51 respectively.
We also have the mesons related to the fields in the electric theory by
M 1k = 1 Λ X 15 Q 5K ; Ñj3 = 1 Λ Qj5 X 53 M 13 = 1 Λ X 15 X 53 ; Φjk = 1 Λ Qj5 Q 5k (3.19)
There is a cubic superpotential coupling the mesons and the dual flavors
W mes. = h ( M 1k Qk5 X 51 + M 13 X 35 X 51 + Ñj3 X 35 Q 5j + Φjk Qk5 Q 5j ) (3.20)
where h = Λ/ Λ with Λ given by Λ 3Nc-N f elect Λ 3(N f -Nc)-N f = ΛN f , where Λ elect is the dynamical scale of the electric theory. Writing the classical superpotential terms of the new fields gives
W clas. = -h µ 0 M 13 Y 31 + λ ′ Q 1i Qi3 Y 31 + µ ′ Ñj3 Q 3j + µ ′ M 1k Qk1 + m 1 Q 1i Qk1 + m 3 Q 3j Qi3 -hµ 2 Tr Φ (3.21) where µ 0 = λΛ, µ ′ = λ ′ Λ, and µ 2 = -m 5 Λ.
So the complete superpotential in the Seiberg dual is
W dual = -h µ 0 M 13 Y 31 + λ ′ Q 1i Qi3 Y 31 + µ ′ Ñj3 Q 3j + µ ′ M 1k Qk1 + m 1 Q 1i Qk1 + m 3 Q 3j Qi3 -hµ 2 Tr Φ + h ( M 1k Qk5 X 51 + M 13 X 35 X 51 + Ñj3 X 35 Q 5j + Φjk Qk5 Q 5j ) (3.22)
This superpotential has a sector completely analogous to the ISS model, triggering supersymmetry breaking by the rank condition. This suggests the following ansatz for 14 the point to become the one-loop vacuum
Q 5k = Q T 5k = µ 0 , (3.23)
with all other vevs set to zero. Following our technique as explained above, we expand fields at linear order around this point. Focusing on N f,1 = 2 and N c = 1 for simplicity (the general case can be easily recovered), we have
Qk5 = µ + δ Q5,1 δ Q5,2 ; Q 5k = (µ + δQ 5,1 ; δQ 5,2 ) ; Φ = δΦ 0,0 δΦ 0,1 δΦ 1,0 δΦ 1,1 Qk1 = δ Q1,1 δ Q1,2 ; Q 1i = (δQ 1,1 ; δQ 1,2 ) ; Qi3 = δ Q3,1 δ Q3,2 ; Q 3j = (δQ 3,1 ; δQ 3,2 ) Ñj3 = δ Ñ3,1 δ Ñ3,2 ; M 1k = (δM 1,1 ; δM 1,2 ) ; M 13 = δM 13 ; Y 31 = δY 31 ; X 51 = δX 51 X 35 = δX 35 (3.24)
Inserting this into equation (3.22) gives
W dual = -h µ 0 δM 13 δY 31 + λ ′ δQ 1,1 δ Q3,1 δY 31 + λ ′ δQ 1,2 δ Q3,2 δY 31 + µ ′ δ Ñ3,1 δQ 3,1 + µ ′ δ Ñ3,2 δQ 3,2 + µ ′ δM 1,1 δ Q1,1 + µ ′ δM 1,2 δ Q1,2 + m 1 δQ 1,1 δ Q1,1 + m 1 δQ 1,2 δ Q1,2 + m 3 δQ 3,1 δ Q3,1 + m 3 δQ 3,2 δ Q3,2 -hµ 2 δΦ 11 + h ( µδM 1,1 δX 51 + δM 1,1 δ Q5,1 δX 51 + δM 1,2 δ Q5,2 δX 51 + δM 13 δX 35 δX 51 + µδX 35 δ Ñ3,1 + δX 35 δ Ñ3,1 δQ 5,1 + δX 35 δ Ñ3,2 δQ 5,2 + µδ Q5,1 δΦ 00 + µδQ 5,1 δΦ 00 + δQ 5,1 δ Q5,1 δΦ 00 + µδΦ 01 δ Q5,2 + δQ 5,1 δΦ 01 δ Q5,2 + µδΦ 10 δQ 5,2 + δ Q5,1 δΦ 10 δQ 5,2 + δ Q5,2 δΦ 11 δQ 5,2 ).
We now need to identify the pseudomoduli, in other words the massless fluctuations at tree level. We focus then just on the quadratic terms in the superpotential
W mass = -h µ 0 δM 13 δY 31 + µ ′ δ Ñ3,1 δQ 3,1 + m 3 δQ 3,1 δ Q3,1 + hµδX 35 δ Ñ3,1 + µ ′ δ Ñ3,2 δQ 3,2 + m 3 δQ 3,2 δ Q3,2 + µ ′ δM 1,1 δ Q1,1 + m 1 δQ 1,1 δ Q1,1 + hµδM 1,1 δX 51 + µ ′ δM 1,2 δ Q1,2 + m 1 δQ 1,2 δ Q1,2 + hµδ Q5,1 δΦ 00 + hµδQ 5,1 δΦ 00 + hµδΦ 01 δ Q5,2 + µδΦ 10 δQ 5,2 . (3.25) 15
We have displayed the superpotential so that fields mixing at the quadratic level appear in the same line. In order to identify the pseudomoduli we have to diagonalize 4 these fields. Note that the structure of the mass terms corresponds to the one in appendix C, in particular around equation (C.9). From the analysis performed there we know that upon diagonalization, fields mixing in groups of four (i.e., three mixing terms in the superpotential, for example the δM 1,
1 , δ Q1,1 , δQ 1,1 , δX 51 mixing) get nonzero masses,
while fields mixing in groups of three (two mixing terms in the superpotential, for example δM 1,2 , δ Q1,2 and δQ 1,2 ) give rise to two massive perturbations and a massless one, a pseudomodulus. We then just need to study the fate of the pseudomoduli. From the analysis in appendix C, the pseudomoduli coming from the mixing terms are
Y 1 = m 3 δ Ñ3,2 -µ ′ δ Q3,2 , Y 2 = m 1 δM 1,2 -µ ′ δQ 1,2 , Y 3 = hµ(δQ 5,1 -δ Q5,1 ) . (3.26)
In order to continue the analysis, one just needs to change basis to the diagonal fields and notice that the one loop contributions to the pseudomoduli are described again by the asymmetric model of appendix A.2, so they receive positive definite contributions.
The exact analytic expressions can be easily found with the help of some computer algebra program, but we omit them here since they are quite unwieldy.
In the previous section we showed that several examples of quiver gauge theories on DSB fractional branes have metastable vacua once additional flavors are included.
In this section we generalize the arguments for general DSB branes. We will show how to add D7-branes in a specific manner so as to generate the appropriate cubic flavor couplings and mass terms. Once this is achieved, we describe the structure of the Seiberg dual theory. The results of our analysis show that, with the specified configuration of D7-branes, the determination of metastability is greatly simplified and only involves looking at the original superpotential. Thus, although we do not prove that DSB branes on arbitrary singularities generate metastable vacua, we show how one can determine the existence of metastability in a very simple and systematic 4 As a technical remark, let us note that it is possible to set all the mass terms to be real by an appropriate redefinition of the fields, so we are diagonalizing a real symmetric matrix.
manner. Using this analysis we show further examples of metastable vacua on systems of DSB branes.
Consider a general quiver gauge theory arising from branes at singularities. As we have argued previously, we focus on DSB branes, so that there is a gauge factor satisfying N f,0 < N c , which can lead to supersymmetry breaking by the rank condition in its Seiberg dual. To make the general analysis more concrete, let us consider a quiver like that in Figure 6 , which is characteristic enough, and let us assume that the gauge factor to be dualized corresponds to node 2. In what follows we analyze the structure of the fields and couplings in the Seiberg dual, and reduce the problem of studying the meta-stability of the theory with flavors to analyzing the structure of the theory in the absence of flavors.
2 1 3 5 4 PSfrag replacements X 21 Y 21 X 32 Y 32 Z 32 X 14 X 43 Y 43
Figure 6: Quiver diagram used to illustrate general results. It does not correspond to any geometry in particular.
The first step is the introduction of flavors in the theory. As discussed in [19] , for any bi-fundamental X ab of the D3-brane quiver gauge theory there exist a supersymmetric D7-brane leading to flavors Q bi , Qia in the fundamental (antifundamental) of the b th (a th ) gauge factor. There is also a cubic coupling X ab Q bi Qia . Let us now specify a concrete set of D7-branes to introduce flavors in our quiver gauge theory. Consider a superpotential coupling of the D3-brane quiver gauge theory, involving fields charged under the node to be dualized. This corresponds to a loop in the quiver, involving node 2, for instance X 32 X 21 X 14 Y 43 in Figure 6 . For any bi-fundamental chiral multiplet in 17 this coupling, we introduce a set of N f,1 of the corresponding D7-brane. This leads to a set of flavors for the different gauge factors, in a way consistent with anomaly cancellation, such as that shown in Figure 7 . The description of this system of D7branes in terms of dimer diagrams is carried out in Appendix B. The cubic couplings described above lead to the superpotential terms 5
W f lavor = λ ′ ( X 32 Q 2b Q b3 + X 21 Q 1a Q a2 + X 14 Q 4d Q d1 + Y 43 Q 3c Q c4 ) (4.1)
Finally, we introduce mass terms for all flavors of all involved gauge factors:
W mass = m 2 Q a2 Q 2b + m 3 Q b3 Q 3c + m 4 Q c4 Q 4d + m 1 Q d1 Q 1a (4.2)
These mass terms break the flavor group into a diagonal subgroup. 2 1 3 5 4 a b c d PSfrag replacements
X 21 Y 21 X 32 Y 32 Z 32 X 14 X 43 Y 43 Q 1a Q a2 Q 2b Q b3 Q 3c Q c4 Q 4d Q d1
Figure 7: Quiver diagram with flavors. White nodes denote flavor groups.
We consider introducing a number of massive flavors such that node 2 is in the free magnetic phase, and consider its Seiberg dual. The only relevant fields in this case are those charged under gauge factor 2, as shown if Figure 8 . The Seiberg dual gives us Figure 9 where the M's are mesons with indices in the gauge groups, R's and S's are 5 Here we assume the same coupling, but the conclusions hold for arbitrary non-zero couplings.
18 2 a b 1 3 PSfrag replacements X 21 Y 21 X 32 Y 32 Z 32 Q a2 Q 2b
Figure 8: Relevant part of quiver before Seiberg duality. a b 1 3 2 PSfrag replacements X12 Ỹ12 X23 Ỹ23 Z23 Qb2 Q2a
X ab R 1 R 2 S 1 S 2 S 3 M 1 , .
mesons with only one index in the flavor group, and X ab is a meson with both indices in the flavor groups. The original cubic superpotential and flavor mass superpotentials become
W f lavor dual = λ ′ ( S 1 3b Q b3 + R 1 a1 Q 1a + X 14 Q 4d Q d1 + Y 43 Q 3c Q c4 ) W mass dual = m 2 X ab + m 3 Q b3 Q 3c + m 4 Q c4 Q 4d + m 1 Q d1 Q 1a (4.3)
In addition we have the extra meson superpotential
W mesons = h ( X ab Qb2 Q2a + R 1 a1 X12 Q2a + R 2 a1 Ỹ12 Q2a + S 1 3b Qb2 X23 + S 2 3b Qb2 Ỹ23 + S 3 3b Qb2 Z23 + M 1 31 X12 X23 + M 2 31 X12 Ỹ23 + M 3 31 X12 Z23 + M 4 31 Ỹ12 X23 + M 5 31 Ỹ12 Ỹ23 + M 6 31 Ỹ12 Z23 ). (4.4)
The crucial point is that we always obtain terms of the kind underlined above, namely a piece of the superpotential reading m 2 X ab + hX ab Qb2 Q2a . This leads to tree level supersymmetry breaking by the rank condition, as announced. Moreover the superpotential fits in the structure of the generalized asymmetric O'Raifeartaigh model studied in appendix A.2, with X ab , Qb2 , Q2a corresponding to X, φ 1 , φ 2 respectively. The multiplets Qb2 and Q2a are split at tree level, and X ab is massive at 1-loop. From our study of the generalized asymmetric case, any field which has a cubic coupling to the supersymmetry breaking fields Qb2 or Q2a is one-loop massive as well. Using the general structure of W mesons , a little thought shows that all dual quarks with no flavor index (e.g. X, Ỹ ) and all mesons with one flavor index (e.g. R or S) couple to the supersymmetry breaking fields.
Thus they all get one-loop masses (with positive squared mass). Finally, the flavors of other gauge factors (e.g. Q b3 ) are massive at tree level from W mass .
The bottom line is that the only fields which do not get mass from these interactions are the mesons with no flavor index, and the bi-fundamentals which do not get dualized (uncharged under node 2). All these fields are related to the theory in the absence of extra flavors, so they can be already stabilized at tree-level from the original superpotential. So, the criteria for a metastable vacua is that the original theory, in the absence of flavors leads, after dualization of the node with N f < N c , to masses for all these fields (or more mildly that they correspond to directions stabilized by mass terms, or perhaps higher order superpotential terms).
For example, if we apply this criteria to the dP 2 case studied previously, the original superpotential for the fractional DSB brane is
W = -λX 53 Y 31 X 15 (4.5)
20 so after dualization we get W = -λM 13 Y 31 (4.6) which makes these fields massive. Hence this fractional brane, after adding the D7branes in the appropriate configuration, will generate a metastable vacua will all moduli stabilized.
The argument is completely general, and leads to an enormous simplification in the study of the theories. In the next section we describe several examples. A more rigorous and elaborate proof is provided in the appendix where we take into account the matricial structure, and show that all fields, except for Goldstone bosons, get positive squared masses at tree-level or at one-loop.
Figure 9: Relevant part of the quiver after Seiberg duality on node 2.
4.2.1 The dP 3 case Let us consider the complex cone over dP 3 , and introduce fractional DSB branes of the kind considered in [15] . The quiver is shown in Figure 10 and the superpotential is
W = X 13 X 35 X 51 (4.7)
Node 1 has N f < N c so upon addition of massive flavors and dualization will lead to supersymmetry breaking by the rank condition. Following the procedure of the previous section, we add N f,1 flavors coupling to the bi-fundamentals X 13 , X 35 and X 51 . Node 1 is in the free magnetic phase for P + 1 ≤ N f,1 < 3 2 P + 1 2 . Dualizing node 1, the above superpotential becomes
W = X 35 M 53 (4.8)
where M 53 is the meson X 51 X 13 . So, following the results of the previous section, we can conclude that this DSB fractional brane generates a metastable vacua with all pseudomoduli lifted.
Let us consider the P dP 4 theory, and introduce the DSB fractional brane of the kind considered in [15] . The quiver is shown in Figure 11 . The superpotential is
W = -X 25 X 51 X 12 (4.9) 21 U(P) U( 1
) 5 3 U(1) 1 4 U(P+1)
Figure 10: Quiver diagram for the dP 3 theory with a DSB fractional brane.
U(P) 5 1 4 U(M) 2 U(M) U(M+P)
Figure 11: Quiver diagram for the dP 4 theory with a DSB fractional branes.
22 Node 1 has N f < N c and will lead to supersymmetry breaking by the rank condition in the dual. Following the procedure of the previous section, we add N f,1 flavors coupling to the bi-fundamentals X 12 , X 25 and X 51 . Node 1 is in the free magnetic phase for P + 2 ≤ M + N f,1 < 3 2 (M + P ). Dualizing node 1, the above superpotential becomes
W = X 25 M 52 , where M 53 is the meson X 51 X 12 .
Again we conclude that this DSB fractional brane generates a metastable vacua with all pseudomoduli lifted.
The Y p,q family Consider D3-branes at the real cones over the Y p,q Sasaki-Einstein manifolds [36, 37, 38, 39] , whose field theory were determined in [8] . The theory admits a fractional brane [13] of DSB kind, which namely breaks supersymmetry and lead to runaway behavior [15, 18] . The analysis of metastability upon addition of massive flavors for arbitrary Y p,q 's is much more involved than previous examples. Already the description of the field theory on the fractional brane is complicated. Even for the simpler cases of Y p,q and Y p,p-1 the superpotential contains many terms. In this section we do not provide a general proof of metastability, but rather consider the more modest aim of showing that all directions related to the runaway behavior in the absence of flavors are stabilized by the addition of flavors. We expect that this will guarantee full metastability, since the fields not involved in our analysis parametrize directions orthogonal to the runaway at infinity. The dimer for Y p,q is shown in Figure 12 and consists of a column of n hexagons and 2 m quadrilaterals which are just halved hexagons [18] . The labels (n, m) are related
to (p, q) by n = 2q ; m = p -q (4.10)
• The Y p,1 case
The dimer for the theory on the DSB fractional brane in the Y p,1 case is shown in Figure 13 , a periodic array of a column of two full hexagons, followed by p -1 cut hexagons (the shaded quadrilateral has N c = 0). As shown in [18] , the top quadrilateral which has N f < N c , and induces the ADS superpotential triggering the runaway. The relevant part of the dimer is shown in Figure 14 , where V 1 and V 2 are the fields that run to infinity [18] . This node will lead to supersymmetry breaking by the rank condition in the dual. It is in the free magnetic phase for M + 1 ≤ N f,1 < pM + M 2 . The piece 23 n n n n+1 n+2 n+m 1 n+1 n+2 n+m 3 n 2 2 1 3 1 2 3 n+1 n+2 n+m 1 1 2 3 n+1 n+2 n+m z w Figure 12: The generic dimer for Y p,q , from [18].
of the superpotential involving the V 1 and V 2 terms is
W = Y U 2 V 2 -Y U 1 V 1 . (4.11)
In the dual theory, the dual superpotential makes the fields massive. Hence, the theory has a metastable vacua where the runaway fields are stabilized.
00 00 11 11 00 00 11 11 00 00 11 11
Figure 13: The dimer for Y p,1 . 24 1 2 2 1 pM (p-1)M (p-2)M Z Y (2p-1)M (p-1)M pM (p+1)M V U V U Figure 14: Top part of the dimer for Y p,1 . The hexagons are labeled by the ranks of the respective gauge groups
• The Y p,p-1 case The analysis for Y p,p-1 is similar but in this case it is the bottom quadrilateral which has the highest rank and thus gives the ADS superpotential [18] . The relevant part of the dimer is shown in Figure 15 , and the runaway direction is described by the fields V 1 and V 2 . Upon addition of N f,1 flavors, the relevant node in the in the free magnetic phase for M + 1 ≤ N f,1 < pM + M 2 Considering the superpotential, it is straightforward to show that the runaway fields become massive. Complementing this with our analysis in previous section, we conclude that the theory has a metastable vacua where the runaway fields are stabilized.
We have thus shown that we can obtain metastable vacua for fractional branes at cones over the Y p,1 and Y p,p-1 geometries. Although there is no obvious generalization for arbitrary Y p,q 's, our results strongly suggest that the existence of metastable vacua extends to the complete family.
The present work introduces techniques and computations which suggest that the existence of metastable supersymmetry breaking vacua is a general property of quiver gauge theories on DSB fractional branes, namely fractional branes associated to obstructed complex deformations. It is very satisfactory to verify the correlation between a non-trivial dynamical property in gauge theories and a geometric property in their 25
(p-1)M V U2 U2 U1 V1 (p-1)M Y Y (2p-1)M (p-2)M (2p-2)M (2p-2)M 2 Figure 15: Bottom part of the dimer for Y p,p-1 . The hexagons are labeled by the ranks of the respective gauge groups string theory realization. The existence of such correlation fits nicely with the remarkable properties of gauge theories on D-branes at singularities, and the gauge/gravity correspondence for fractional branes.
Beyond the fact that our arguments do not constitute a general proof, our analysis has left a number of interesting open questions. In fact, as we have mentioned, all theories on DSB fractional branes contain one or several fields which do not appear in the superpotential. We expect the presence of these fields to have a direct physical interpretation, which has not been uncovered hitherto. It would be interesting to find a natural explanation for them.
Finally, a possible extension of our results concerns D-branes at orientifold singularities, which can lead to supersymmetry breaking and runaway as in [27] . Interestingly, in this case the field theory analysis is more challenging, since they would require Seiberg dualities of gauge factors with matter in two-index tensors. It is very possible that the string theory realization, and the geometry of the singularity provide a much more powerful tool to study the system.
Overall, we expect other surprises and interesting relations to come up from further study of D-branes at singularities.
We thank S. Franco for useful discussions. A.U. thanks M. González for encouragement and support. This work has been supported by the European Commission under RTN European Programs MRTN-CT-2004-503369, MRTN-CT-2004-005105, by the CICYT (Spain), and by the Comunidad de Madrid under project HEPHACOS P-ESP-00346.
The research by I.G.-E. is supported by the Gobierno Vasco PhD fellowship program.
The research of F.S is supported by the Ministerio de Educación y Ciencia through an FPU grant. I.G.-E. and F.S. thank the CERN Theory Division for hospitality during the completion of this work.
A Technical details about the calculation via Feynman diagrams A.1 The basic amplitudes
In the main text we are interested in computing two point functions for the pseudomoduli at one loop, and in section 2.2 also tadpole diagrams. There are just a few kinds of diagrams entering in the calculation, which we will present now for the two-point function, see Figure 16 . The (real) bosonic fields are denoted by φ i and the (Weyl) fermions by ψ i . The pseudomodulus we are interested in is denoted by ϕ. c) d) a) b)
ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ φ 2 φ 1 φ ψ 2 ψ ψ 1
Figure 16: Feynman diagrams contributing to the one-loop two point function. The dashed line denotes bosons and the solid one fermions.
These come from two terms in the Lagrangian. First there is a diagram coming from terms of the form (Figure 16b ):
L = . . . + λϕ 2 φ 2 - 1 2 m 2 φ 2 , (A.1)
giving an amplitude (we will be using dimensional regularization)
iM = -2iλ (4π) 2 m 2 1 ǫ -γ + 1 + log 4π -log m 2 . (A.2)
The other contribution comes from the diagram in Figure 16a :
L = . . . + λϕφ 1 φ 2 - 1 2 m 2 1 φ 2 1 - 1 2 m 2 2 φ 2 2 , (A.
3) which contributes to the two point function with an amplitude:
iM = iλ 2 (4π) 2 1 ǫ -γ + log 4π - 1 0 dx log ∆ , (A.4)
where here and in the following we denote ∆ ≡ xm 2 1 + (1x)m 2 2 .
The relevant vertices here are again of two possible kinds, one of which is nonrenormalizable. The cubic interaction comes from terms in the Lagrangian given by the diagram in Figure 16c :
L = . . . + ϕ(aψ 1 ψ 2 + a * ψ1 ψ2 ) + 1 2 m 1 (ψ 2 1 + ψ2 1 ) + 1 2 m 2 (ψ 2 2 + ψ2 2 ). (A.5)
We are assuming real masses for the fermions here, in the configurations we study this can always be achieved by an appropriate field redefinition. The contribution from such vertices is given by:
iM = 1 0 dx -2im 1 m 2 (4π) 2 (a 2 + (a 2 ) * ) 1 ǫ -γ + log 4π -log ∆ - 8i|a| 2 (4π) 2 ∆ 1 ǫ -γ + log 4π + 1 2 -log ∆ . (A.6)
The other fermionic contribution, which one does not need as long as one is dealing with renormalizable interactions only (but we will need in the main text when analyzing the pseudomodulus θ), is given by terms in the Lagrangian of the form (Figure 16d ):
L = . . . + λϕ 2 (ψ 2 + ψ2 ) + 1 2 m(ψ 2 + ψ2 ), (A.7)
which contributes to the total amplitude with:
iM = 8λmi (4π) 2 m 2 1 ǫ -γ + 1 + log 4π -log m 2 . (A.8)
The previous amplitudes are the basic ingredients entering the computation, but in general the number of diagrams contributing to the two point amplitudes is quite big, so calculating all the contributions by hand can get quite involved in particular examples 6 . Happily, one finds that complicated models (such as dP 1 or dP 2 , studied in the main text) reduce to performing the analysis for only two different superpotentials, which we analyze in this section.
We want to study in this section a superpotential of the form:
W = h(Xφ 1 φ 2 + µφ 1 φ 3 + µφ 2 φ 4 -µ 2 X). (A.9)
6 The authors wrote the computer program in http://cern.ch/inaki/pm.tar.gz which helped greatly in the process of computing the given amplitudes for the relevant models.
This model is a close cousin of the basic O'Raifeartaigh model. We are interested in the one loop contribution to the two point function of X, which is massless at tree level.
From the (F-term) bosonic potential one obtains the following terms entering the one loop computation:
V = |hXφ 2 | 2 + |h| 2 µ(Xφ 2 φ * 3 + X * φ * 2 φ 3 ) + |h| 2 µ(Xφ 1 φ * 4 + X * φ * 1 φ 4 ) + |h| 2 µ 2 (φ 1 φ 2 + φ * 1 φ * 2 ) + 4 i=1 |h| 2 µ 2 |φ i | 2 (A.10)
In order to do the computation it is useful to diagonalize the mass matrix by introducing φ + and φ -such that:
φ 1 = 1 √ 2 (φ + + iφ -) φ 2 = 1 √ 2 (φ + -iφ -) (A.11)
and φ a , φ b such that:
φ * 3 = 1 √ 2 (φ a + iφ b ) φ * 4 = 1 √ 2 (φ a -iφ b ). (A.12)
With these redefinitions the bosonic scalar potential decouples into identical φ + and φ -sectors, giving two decoupled copies of:
V = |h| 2 |X| 2 |φ + | 2 + |h| 2 µ 2 (|φ + | 2 + |φ a | 2 ) +|h| 2 µ(Xφ + φ a + X * φ * + φ * a ) - |h| 2 µ 2 2 φ 2 + + (φ 2 + ) * . (A.13)
Calculating the amplitude consists simply of constructing the (very few) two point diagrams from the potential above and plugging the formulas above for each diagram (the fermionic part is even simpler in this case). The final answer is that in this model the one loop correction to the mass squared of X is given by:
δm 2 X = |h 4 |µ 2 8π 2 (log 4 -1). (A.14)
The generalized asymmetric case
The next case is slightly more complicated, but will suffice to analyze completely all the models we encounter. We will be interested in the one loop contribution to the mass of the pseudomoduli Y in a theory with superpotential:
W = h(Xφ 1 φ 2 + µφ 1 φ 3 + µφ 2 φ 4 -µ 2 X) + k(rY φ 1 φ 5 + µφ 5 φ 7 ), (A.15) 30
with k and r arbitrary complex numbers. The procedure is straightforward as above, so we will just quote the result. We obtain an amplitude given by:
iM = -i (4π) 2 |h 2 rµ| 2 C |k| 2 |h| 2 , (A.16)
where we have defined C(t) as: C(t) = t 2t log 4 -t t -1 log t . (A.17) Note that this is a positive definite function, meaning that the one loop correction to the mass is always positive, and the pseudomoduli get stabilized for any (nonzero) value of the parameters. Also note that the limit of vanishing t with |r| 2 t fixed (i.e., vanishing masses for φ 5 and φ 7 , but nonvanishing coupling of Y to the supersymmetry breaking sector) gives a nonvanishing contribution to the mass of Y .
The gauge theory of D3-branes at toric singularities can be encoded in a dimer diagram [40, 41, 42, 43, 44] . This corresponds to a bi-partite tiling of T 2 , where faces correspond to gauge groups, edges correspond to bi-fundamentals, and nodes correspond to superpotential terms. As an example, the dimer diagram of D3-branes on the cone over dP 2 is shown in Figure 17 . As shown in [43] , D3-branes on a toric singularity are mirror to D6-branes on intersecting 3-cycles in a geometry given by a fibration of a Riemann surface Σ with punctures. This Riemann surface is just a thickening of the web diagram of the toric singularity [45, 46, 47] , with punctures associated to external legs of the web diagram. The mirror D6-branes wrap non-trivial 1-cycles on this Riemann surface, with their intersections giving rise to bi-fundamental chiral multiplets, and superpotential terms arising from closed discs bounded by the D6-branes. In [19], it was shown that D7-branes passing through the singular point can be described in the mirror Riemann surface Σ by non-compact 1-cycles which come from infinity at one puncture and go to infinity at another. Figure 18 shows the 1-cycles corresponding to some D3-and D7-branes in the Riemann surface in the geometry mirror to the complex cone over dP 2 . A D7-brane leads to flavors for the two D3-brane gauge factors whose 1-cycles are intersected by the D7-brane 1-cycle, and there is a cubic coupling among the three fields (related to the disk bounded by the three 1-cycles in the Riemann surface).
31 Figure 17: Dimer diagram for D3-branes at a dP 2 singularity. 1 00 00 00 11 11 11 00 00 00 00 11 11 11 11 000 000 000 000 111 111 111 111 E C F E B A E C F D7 5 5 3 C 3 1 D7 D7 Figure 18: Riemann surface in the geometry mirror to the complex cone over dP 2 , shown as a tiling of a T 2 with punctures (denoted by capital letters). The figure shows the noncompact 1-cycles extending between punctures, corresponding to D7-branes, and a piece of the 1-cycles that correspond to the mirror of the D3-branes.
U(M) U(M) U(2M)
PSfrag replacements
Q 1i Q i3 Q 3j Q j5 Q 5k Q k1
Figure 19: Quiver for the dP 2 theory with M fractional branes and flavors.
As stated in Section 4, given a gauge theory of D3-branes at a toric singularity, we introduce flavors for some of the gauge factors in a specific way. We pick a term in the superpotential, and we introduce flavors for all the involved gauge factors, and coupling to all the involved bifundamental multiplets. For example, the quiver with flavors for the dP 2 theory is shown in Figure 19 .
On the Riemann surface, this procedure amounts to picking a node and introducing D7-branes crossing all the edges ending on the node, see Figure 18 . In this example we obtain the superpotential terms
W f lavor = λ ′ (Q 1i Qi3 Y 31 + Q 3j Qj5 X 53 + Q 5k Qk1 X 15 ) (B.1)
In addition we introduce mass terms
W mass = m 1 Q 1i Qk1 + m 2 Q 3j Qi3 + m 5 Q 5k Qj5 (B.2)
This procedure is completely general and applies to all gauge theories for branes at toric singularities 7 .
Recall that in Section 4 we considered the illustrative example of the gauge theory given by the quiver in Figure 20 . Since node 2 is the one we wish to dualize, the only relevant part of the diagram is shown in Figure 21 . We show the Seiberg dual in Figure 22 . The above choice of D7-branes, which we showed in appendix B can be applied to arbitrary toric singularities, gives us the superpotential terms
W f lavor = λ ′ ( X 32 Q 2b Q b3 + X 21 Q 1a Q a2 + X 14 Q 4d Q d1 + Y 43 Q 3c Q c4 ) W mass = m 2 Q a2 Q 2b + m 3 Q b3 Q 3c + m 4 Q c4 Q 4d + m 1 Q d1 Q 1a (C.1)
Taking the Seiberg dual of node 2 gives
W f lavor dual = λ ′ ( S 1 3b Q b3 + R 1 a1 Q 1a + X 14 Q 4d Q d1 + Y 43 Q 3c Q c4 ) W mass dual = m 2 X ab + m 3 Q b3 Q 3c + m 4 Q c4 Q 4d + m 1 Q d1 Q 1a W mesons = h ( X ab Qb2 Q2a + R 1 a1 X12 Q2a + R 2 a1 Ỹ12 Q2a
7 This procedure does not apply if the superpotential (regarded as a loop in the quiver) passes twice through the node which is eventually dualized in the derivation of the metastable vacua. However we have found no example of this for any DSB fractional branes.
33 2 1 3 5 4 a b c d X 21 Y 21 X 32 Y 32 Z 32 X 14 X 43 Y 43 Q 1a Q a2 Q 2b Q b3 Q 3c Q c4 Q 4d Q d1
Figure 20: Quiver diagram with flavors. White nodes denote flavor groups 2 a b 1 3 PSfrag replacements
X 21 Y 21 X 32 Y 32 Z 32 Q a2 Q 2b
Figure 21: Relevant part of quiver before Seiberg duality. a b 1 3 2 PSfrag replacements X12 Ỹ12 X23 Ỹ23 Z23 Qb2 Q2a
X ab R 1 R 2 S 1 S 2 S 3 M 1 , . . . , M 6
Figure 22: Relevant part of the quiver after Seiberg duality on node 2.
34 + S 1 3b Qb2 X23 + S 2 3b Qb2 Ỹ23 + S 3 3b Qb2 Z23 + M 1 31 X12 X23 + M 2 31 X12 Ỹ23 + M 3 31 X12 Z23 + M 4 31 Ỹ12 X23 + M 5 31 Ỹ12 Ỹ23 + M 6 31 Ỹ12 Z23 ) (C.2)
where we have not included the original superpotential. The crucial point is that the underlined terms appear for any quiver gauge theory with flavors introduced as described in appendix B. As described in the main text, supersymmetry is broken by the rank condition due to the F-term of the dual meson associated to the massive flavors. Our vacuum ansatz is (we take N f = 2 and N c = 1 for simplicity; this does not affect our conclusions) Qb2 = µ1 Nc 0 ; Q2a = (µ1 Nc ; 0) (C.3) with all other vevs set to zero. We parametrize the perturbations around this minimum as
Qb2 = µ + φ 1 φ 2 ; Q2a = (µ + φ 3 ; φ 4 ) ; X ab = X 00 X 01 X 10 X 11 (C.4)
and the underlined terms give
hX ab Qb2 Q2a -hµ 2 X ab = hX 11 φ 2 φ 4 -hµ 2 X 11 + hµ φ 2 X 01 + hµ φ 4 X 10 + hµ φ 1 X 00 + hµ φ 3 X 00 + h φ 1 φ 3 X 00 + h φ 2 φ 3 X 01 + h φ 1 φ 4 X 10 (C.5)
It is important to note that all the fields in (C.4) will have quadratic couplings only in the underlined term (C.5). Thus, one can safely study this term, and the conclusions are independent of the other terms in the superpotential. Diagonalizing (C.5) gives
hX ab Qb2 Q2a -hµ 2 X ab = hX 11 φ 2 φ 4 -hµ 2 X 11 + hµ φ 2 X 01 + hµ φ 4 X 10 + √ 2hµ φ + X 00 + h 2 φ 2 + X 00 - h 2 φ 2 -X 00 + h √ 2 (ξ + -ξ -) φ 2 X 01 + h √ 2 (ξ + + ξ -) φ 4 X 10 (C.6) where ξ + = 1 √ 2 (φ 1 + φ 3 ) ; ξ -= 1 √ 2 (φ 1 -φ 3 ) (C.7)
This term is similar to the generalized asymmetric case studied in appendix A.2 with
X 11 → X ; φ 4 → φ 1 ; φ 2 → φ 2 ; X 10 → φ 3 ; X 01 → φ 4 (C.8) 35
So here X 11 is the linear term that breaks supersymmetry, and φ 2 , φ 4 are the broken supersymmetry fields. In (C.6), the only massless fields at tree-level are X 11 and ξ -. Comparing to the ISS case in Section 2.1 shows that Im ξ -is a Goldstone boson and X 11 , Re ξ -get mass at tree-level. As for φ 2 and φ 4 , setting ρ + = 1 √ 2 (φ 2 + φ 4 ) and ρ -= 1 √ 2 (φ 2 -φ 4 ) gives us Re(ρ + ) and Im (ρ -) massless and the rest massive. Following the discussion in Section 2.1, Re(ρ + ) and Im (ρ -) are just the Goldstone bosons of the broken SU(N f ) symmetry 8 . We have thus shown that the dualized flavors (e.g. Qb2 , Q2a ) and the meson with two flavor indices (e.g. X ab ) get mass at tree-level or at 1-loop unless they are Goldstone bosons. Now, we need to verify that this is the case for the remaining fields.
a b 1 2 3 5 4 d c PSfrag replacements X 14 X 43 Y 43 Q 1a Q b3 Q 3c Q c4 Q 4d Q d1 X12 Ỹ12 X23 Ỹ23 Z23 Qb2 Q2a X ab R 1 R 2 S 1 S 2 S 3 M 1 ..M 6
Figure 23: Quiver after Seiberg duality on node 2.
The Seiberg dual of the original quiver diagram is shown in Figure 23 . The dualized bi-fundamentals come in two classes. The first are the ones that initially (before dualizing) had cubic flavor couplings, there will always be only two of those (e.g. X12 , X23 ). The second are those that did not initially have cubic couplings to flavors, there is an arbitrary number of those (e.g. Ỹ12 , Ỹ23 , Z23 ). Figure 24 shows the relevant part of the quiver for the first class. Recalling the superpotential terms (C.2), there are several possible sources of tree-level masses. For instance, these can arise in W f lavor dual and W mass dual . Also, remembering our assignation of vevs in (C.3), tree-level masses can also arise in W mesons from cubic couplings involving the broken supersymmetry fields (e.g. Qb2 , Q2a ). The first class of bi-fundamentals (e.g. X12 , X23 ) only appear in W mesons coupled to their respective mesons (e.g. R 1 , S 1 ). In turn these mesons will ap-8 In the case where the flavor group is SU (2), these Goldstone bosons are associated to the generators t x and t y .
36 a 1 2 3 d c b X12 X23 Qb2 Q2a
X ab R 1 R 2 S 1 S 2 S 3 M 1 , . . . , M 6 Q 1a Q b3 Q 3c Q d1
Figure 24: Relevant part of dual quiver for first class of bi-fundamentals.
pear in quadratic terms in W f lavor dual coupled to flavors (e.g. S 1 3b Q b3 and R 1 a1 Q 1a ), and these flavors each appear in one term in W mass . Thus there are two sets of three terms which are coupled at tree-level and which always couple in the same way. Consider for instance the term
λ ′ S 1 3b Q b3 + m 3 Q b3 Q 3c + h S 1 3b Qb2 X23 = λ ′ (S 1 S 2 ) B 1 B 2 + m 1 (C 1 C 2 ) B 1 B 2 + h (S 1 S 2 ) µ + φ 1 φ 2 X23 = λ ′ (S 1 B 1 + S 2 B 2 ) + m 1 (B 1 C 1 + B 2 C 2 ) + hµ S 1 X23 + h S 1 φ 1 X23 + h S 2 φ 2 X23 (C.9)
where S i , B i , C i and X23 are the perturbations around the minimum. Diagonalizing (which can be done analytically for any values of the couplings), we get that all terms except one get tree-level masses, the massless field being:
Y = m 1 S 2 -λ ′ C 2 (C.10)
This massless field has a cubic coupling to φ 2 X23 and gets mass at 1-loop since φ 2 is a broken supersymmetry field, as described in appendix A.2.
Figure 25 shows the relevant part of the quiver for the second class of bi-fundamentals (i.e. those that are dualized but do not have cubic flavor couplings).
These fields and their mesons only appear in one term, so will always couple in the same way. Taking as an example h R 2 a1 Ỹ12 Q2a = R 1
R 2 Ỹ12 (µ + φ 3 ; φ 4 ) = µR 1 Ỹ12 + R 1 φ 3 Ỹ12 + R 2 φ 4 Ỹ12 (C.11) 37 a b 1 2 3 d c 4 Ỹ12 Ỹ23 Z23 Qb2 Q2a X ab R 1 R 2 S 1 S 2 S 3 M 1 , . . . , M 6 Q c4 Q 4d
Figure 25: Relevant part of dual quiver for second class of bi-fundamentals.
This shows that R 1 and Ỹ12 get tree-level masses and R 2 gets a mass at 1-loop since it couples to the broken supersymmetry field φ 4 . The only remaining fields are flavors like Q c4 , Q 4d , which do not transform in a gauge group adjacent to the dualized node (i.e. not adjacent in the quiver loop corresponding to the superpotential term used to introduce flavors). These are directly massive from the tree-level W mass term. So, as stated, all fields except those that appear in the original superpotential (i.e.
mesons with gauge indices and bi-fundamentals which are not dualized) get masses either at tree-level or at one-loop. So we only need to check the dualized original superpotential to see if we have a metastable vacua. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Systems of D-branes at singularities provide a very interesting setup to realize and study diverse non-perturbative gauge dynamics phenomena in string theory. In the context of N = 1 supersymmetric gauge field theories, systems of D3-branes at Calabi-Yau singularities lead to interesting families of tractable 4d strongly coupled conformal field theories, which extend the AdS/CFT correspondence [1, 2, 3] to theories with reduced (super)symmetry [4, 5, 6] and enable non-trivial precision tests of the correspondence (see for instance [7, 8] ). Addition of fractional branes leads to families of non-conformal gauge theories, with intricate RG flows involving cascades of Seiberg dualities [9, 10, 11, 12, 13] , and strong dynamics effects in the infrared.\n\nFor instance, fractional branes associated to complex deformations of the singular geometry (denoted deformation fractional branes in [12] ), correspond to supersymmetric confinement of one or several gauge factors in the gauge theory [9, 12] . The generic case of fractional branes associated to obstructed complex deformations (denoted DSB branes in [12] ), corresponds to gauge theories developing a non-perturbative Affleck-Dine-Seiberg superpotential, which removes the classical supersymmetric vacua [14, 15, 16] . As shown in [15] (see also [17, 18] ), assuming canonical Kahler potential leads to a runaway potential for the theory, along a baryonic direction. A natural suggestion to stop this runaway has been proposed for the particular example of the dP 1 theory (the theory on fractional branes at the complex cone over dP 1 ) in [19] . It was shown that, upon the addition of D7-branes to the configuration (which introduce massive flavors), the theory develops a meta-stable minimum (closely related to the Intriligator-Seiberg-Shih (ISS) model [20] ), parametrically long-lived against decay to the runaway regime (see [21] for an alternative suggestion to stop the runaway, in compact models).\n\nIn this paper we show that the appearance of meta-stable minima in gauge theories on DSB fractional branes, in the presence of additional massless flavors, is much more general (and possibly valid in full generality). We use the tools of [15] to introduce D7-branes on general toric singularities, and give masses to the corresponding flavors.\n\nSince quiver gauge theories are rather involved, we develop new techniques to efficiently analyze the one-loop stability of the meta-stable minima, via the direct computation of Feynman diagrams. These tools can be used to argue that the results plausibly hold for general systems of DSB fractional branes at toric singularities. It is very satisfactory to verify the correspondence between the existence of meta-stable vacua and the geometric property of having obstructed complex deformations.\n\nThe present work thus enlarges the class of string models realizing dynamical supersymmetry breaking in meta-stable vacua (see [22, 23, 24, 25, 26] for other proposed realizations, and [27, 28, 29] for models of dynamical supersymmetry breaking in orientifold theories). Although we will not discuss it in the present paper, these results can be applied to the construction of models of gauge mediation in string theory as in [30] (based on the additional tools in [31] ), in analogy with [32] . This is another motivation for the present work.\n\nThe paper is organized as follows. In Section 2 we review the ISS model, evaluating one-loop pseudomoduli masses directly in terms of Feynman diagrams. In Section 3 we study the theory of DSB branes at the dP 1 and dP 2 singularities upon the addition of flavors, and we find that metastable vacua exist for these theories. In Section 4 we extend this analysis to the general case of DSB branes at toric singularities with massive flavors, and we illustrate the results by showing the existence of metastable vacua for DSB branes at some well known families of toric singularities. Finally, the Appendix provides some technical details that we have omitted from the main text in order to improve the legibility."
},
{
"section_type": "OTHER",
"section_title": "The ISS model revisited",
"text": "In this Section we review the ISS meta-stable minima in SQCD, and propose that the analysis of the relevant piece of the one-loop potential (the quadratic terms around the maximal symmetry point) is most simply carried out by direct evaluation of Feynman diagrams. This new tool will be most useful in the study of the more involved examples of quiver gauge theories."
},
{
"section_type": "OTHER",
"section_title": "The ISS metastable minimum",
"text": "The ISS model [20] (see also [33] for a review of these and other models) is given by N = 1 SU(N c ) theory with N f flavors, with small masses W electric = mTr φ φ, (2.1)\n\nwhere φ and φ are the quarks of the theory. The number of colors and flavors are chosen so as to be in the free magnetic phase:\n\nN c + 1 ≤ N f < 3 2 N c . (2.2) 2\n\nThis condition guarantees that the Seiberg dual is infrared free. This Seiberg dual is the SU(N) theory (with N = N f -N c ) with N f flavors of dual quarks q and q and the meson M. The dual superpotential is given by rewriting (2.1) in terms of the mesons and adding the usual coupling between the meson and the dual quarks:\n\nW magnetic = h (Tr qMqµ 2 Tr M), (2.3) where h and µ can be expressed in terms of the parameters m and Λ, and some (unknown) information about the dual Kähler metric 1 . It was also argued in [20] that it is possible to study the supersymmetry breaking minimum in the origin of (dual) field space without taking into account the gauge dynamics (their main effect in this discussion consists of restoring supersymmetry dynamically far in field space). In the following we will assume that this is always the case, and we will forget completely about the gauge dynamics of the dual.\n\nOnce we forget about gauge dynamics, studying the vacua of the dual theory becomes a matter of solving the F-term equations coming from the superpotential (2.3).\n\nThe mesonic F-term equation reads:\n\n-F M ij = hq i • q jhµ 2 δ ij = 0, (2.4)\n\nwhere i and j are flavor indices and the dot denotes color contraction. This has no solution, since the identity matrix δ ij has rank N f while qi • q j has rank N = N f -N c . Thus this theory breaks supersymmetry spontaneously at tree level. This mechanism for F-term supersymmetry breaking is called the rank condition.\n\nThe classical scalar potential has a continuous set of minima, but the one-loop potential lifts all of the non-Goldstone directions, which are usually called pseudomoduli. The usual approach to study the one-loop stabilization is the computation of the complete one-loop effective potential over all pseudomoduli space via the Coleman-Weinberg formula [34]: V = 1 64π 2 Tr M 4 B log M 2 B Λ 2 -M 4 F log M 2 F Λ 2 . (2.5) This approach has the advantage that it allows the determination of the one-loop minimum, without a priori information about its location, and moreover it provides the full potential around it, including higher terms. However, it has the disadvantage 1 The exact expressions can be found in (5.7) in [20] , but we will not need them for our analysis.\n\nWe just take all masses in the electric description to be small enough for the analysis of the metastable vacuum to be reliable.\n\nof requiring the diagonalization of the mass matrix, which very often does not admit a closed expression, e.g. for the theories we are interested in.\n\nIn fact, we would like to point out that to determine the existence of a meta-stable minimum there exists a computationally much simpler approach. In our situation, we have a good ansatz for the location of the one-loop minimum, and are interested just in the one-loop pseudomoduli masses around such point. This information can be directly obtained by computing the one-loop masses via the relevant Feynman diagrams. This technique is extremely economical, and provides results in closed form in full generality, e.g. for general values of the couplings, etc. The correctness of the original ansatz for the vacuum can eventually be confirmed by the results of the computation (namely positive one-loop squared masses, and negligible tadpoles for the classically massive fields 2 ).\n\nHence, our strategy to study the one-loop stabilization in this paper is as follows:\n\n• First we choose an ansatz for the classical minimum to become the one-loop vacuum. It is natural to propose a point of maximal enhanced symmetry (in particular, close to the origin in the space of vevs for M there exist and Rsymmetry, whose breaking by gauge interactions (via anomalies) is negligible in that region). Hence the natural candidate for the one-loop minimum is q = qT = µ 0 , (2.6) with the rest of the fields set to 0. This initial ansatz for the one-loop minimum is eventually confirmed by the positive square masses at one-loop resulting from the computations described below. In our more general discussion of meta-stable minima in runaway quiver gauge theories, our ansatz for the one-loop minimum is a direct generalization of the above (and is similarly eventually confirmed by the one-loop mass computation).\n\n• Then we expand the field linearly around this vacuum, and identify the set of classically massless fields. We refer to these as pseudomoduli (with some abuse of language, since there could be massless fields which are not classically flat directions due to higher potential terms) 2 Since supersymmetry is spontaneously broken the effective potential will get renormalized by quantum effects, and thus classically massive fields might shift slightly. This appears as a one loop tadpole which can be encoded as a small shift of µ. This will enter in the two loop computation of the pseudomoduli masses, which are beyond the scope of the present paper.\n\n• As a final step we compute one-loop masses for these pseudomoduli by evaluating their two-point functions via conventional Feynman diagrams, as explained in more detail in appendix A.1 and illustrated below in several examples.\n\nThe ISS model is a simple example where this technique can be illustrated. Considering the above ansatz for the vacuum, we expand the fields around this point as:\n\nq = µ + 1 √ 2 (ξ + + ξ -) 1 √ 2 (ρ + + ρ -) , qT = µ + 1 √ 2 (ξ + -ξ -) 1 √ 2 (ρ + -ρ -) , M = Y Z ZT Φ , (2.7)\n\nwhere we have taken linear combinations of the fields in such a way that the bosonic mass matrix is diagonal. This will also be convenient in section 2.2, where we discuss the Goldstone bosons in greater detail.\n\nWe now expand the superpotential (2.3) to get\n\nW = √ 2µξ + Y + 1 √ 2 µZρ + + 1 √ 2 µZρ -+ 1 √ 2 µρ + Z - 1 √ 2 µρ - Z + 1 2 ρ 2 + Φ - 1 2 ρ 2 -Φ -µ 2 Φ + . . . , (2.8)\n\nwhere we have not displayed terms of order three or higher in the fluctuations, unless they contain Φ, since they are irrelevant for the one loop computation we will perform.\n\nNote also that we have set h = 1 and we have removed the trace (the matricial structure is easy to restore later on, here we just set N f = 2 for simplicity). The massless bosonic fluctuations are given by Re ρ + , Im ρ -, Φ and ξ -. The first two together with Im ξ -are Goldstone bosons, as explained in section 2.2. Thus the pseudomoduli we are interested in are given by Φ and Re ξ -. Let us focus on Φ (the case of Re ξ -admits a similar discussion). In this case the relevant terms in the superpotential simplify further, and just the following superpotential contributes:\n\nW = µZ 1 √ 2 (ρ + + ρ -) + µ Z 1 √ 2 (ρ + -ρ -) + 1 2 ρ 2 + Φ - 1 2 ρ 2 -Φ -µ 2 Φ + . . . ,\n\nwhich we recognize, up to a field redefinition, as the symmetric model of appendix A.2.\n\nWe can thus directly read the result\n\nδm 2 Φ = |h| 4 µ 2 8π 2 (log 4 -1). (2.9)\n\nThis matches the value given in [20] , which was found using the Coleman-Weinberg potential. 5"
},
{
"section_type": "OTHER",
"section_title": "The Goldstone bosons",
"text": "One aspect of our technique that merits some additional explanation concerns the Goldstone bosons. The one-loop computation of the masses for the fluctuations associated to the symmetries broken by the vacuum, using just the interactions described in appendix A.1, leads to a non-vanishing result. This puzzle is however easily solved by realizing that certain (classically massive) fields have a one-loop tadpole. This leads to a new contribution to the one-loop Goldstone two-point amplitude, given by the diagram in Figure 1 . Adding this contribution the total one-loop mass for the Goldstone bosons is indeed vanishing, as expected. This tadpole does not affect the computation of the one-loop pseudomoduli masses (except for Re ξ + , but its mass remains positive) as it is straightforward to check.\n\nIm ξ - Im ξ - Re ξ + Figure 1: Schematic tadpole contribution to the Im ξ -two point function. Both bosons and fermions run in the loop.\n\nThe structure of this cancellation can be understood by using the derivation of the Goldstone theorem for the 1PI effective potential, as we now discuss. The proof can be found in slightly more detail, together with other proofs, in [35] . Let us denote by V the 1PI effective potential. Invariance of the action under a given symmetry implies\n\nthat δV δφ i ∆φ i = 0, (2.10)\n\nwhere we denote by ∆φ i the variation of the field φ i under the symmetry, which will in general be a function of all the fields in the theory. Taking the derivative of this equation with respect to some other field φ k\n\nδ 2 V δφ i δφ k ∆φ i + δV δφ i • δ∆φ i δφ k = 0. (2.11)\n\nLet us consider how this applies to our case. At tree level, there is no tadpole and the above equation (truncated at tree level) states that for each symmetry generator broken by the vacuum, the value of ∆φ i gives a nonvanishing eigenvector of the mass matrix with zero eigenvalue. This is the classical version of the Goldstone theorem, which allows the identification of the Goldstone bosons of the theory.\n\nFor instance, in the ISS model in the previous section (for N f = 2), there are three global symmetry generators broken at the minimum described around (2.6). The 6 SU(2) × U(1) symmetry of the potential gets broken down to a U(1) ′ , which can be understood as a combination of the original U(1) and the t z generator of SU(2). The Goldstone bosons can be taken to be the ones associated to the three generators of SU(2), and correspond (for µ real) to Im ξ -, Im ρ -and Re ρ + , in the parametrization of the fields given by equation (2.7).\n\nEven in the absence of tree-level tadpoles, there could still be a one-loop tadpole.\n\nWhen this happens, there should also be a non-trivial contribution to the mass term for the Goldstone bosons in the one-loop 1PI potential, related to the tadpole by the one-loop version of (2.11). This relation guarantees that the mass term in the physical (i.e. Wilsonian) effective potential, which includes the 1PI contribution, plus those of the diagram in Figure 1 , vanishes, as we described above.\n\nIn fact, in the ISS example, there is a non-vanishing one-loop tadpole for the real part of ξ + (and no tadpole for other fields). The calculation of the tadpole at one loop is straightforward, and we will only present here the result iM = -i|h| 4 µ 3 (4π) 2 (2 log 2). (2.12)\n\nThe 1PI one-loop contribution to the Goldstone boson mass is also simple to calculate, giving the result iM = -i|h| 4 µ 2 (4π) 2 (log 2). (2.13) Using the variations of the relevant fields under the symmetry generator, e.g. for t z ,\n\n∆Re ξ + = -Im ξ - (2.14) ∆Im ξ -= Re ξ + + 2µ. (2.15)\n\nwe find that the (2.11) is satisfied at one-loop.\n\nδ 2 V δφ i δφ k ∆φ i + δV δφ i • δ∆φ i δφ k = m 2 Im ξ -• 2µ + (Re ξ + tadpole) • (-1) = 0. (2.16)\n\nA very similar discussion applies to t x and t y .\n\nThe above discussion of Goldstone bosons can be similarly carried out in all examples of this paper. Hence, it will be enough to carry out the computation of the 1PI diagrams discussed in appendix A.1, and verify that they lead to positive squared masses for all classically massless fields (with Goldstone bosons rendered massless by the additional diagrams involving the tadpole). 7 3 Meta-stable vacua in quiver gauge theories with DSB branes\n\nIn this section we show the existence of a meta-stable vacuum in a few examples of gauge theories on DSB branes, upon the addition of massive flavors. As already discussed in [19] , the choice of fractional branes of DSB kind is crucial in the result.\n\nThe reason is that in order to have the ISS structure, and in particular supersymmetry breaking by the rank condition, one needs a node such that its Seiberg dual satisfies\n\nN f > N, with N = N f -N c with N c\n\n, N f the number of colors, flavors of that gauge factor. Denoting N f,0 , N f,1 the number of massless and massive flavors (namely flavors arising from bi-fundamentals of the original D3-brane quiver, or introduced by the D7branes), the condition is equivalent to N f,0 < N c . This is precisely the condition that an ADS superpotential is generated, and is the prototypical behavior of DSB branes [14, 15, 16, 18] .\n\nAnother important general comment, also discussed in [19] , is that theories on DSB branes generically contain one or more chiral multiplets which do not appear in the superpotential. Being decoupled, such fields remain as accidental flat directions at one-loop, so that the one-loop minimum is not isolated. The proper treatment of these flat directions is beyond the reach of present tools, so they remain an open question.\n\nHowever, it is plausible that they do not induce a runaway behavior to infinity, since they parametrize a direction orthogonal to the fields parametrizing the runaway of DSB fractional branes."
},
{
"section_type": "OTHER",
"section_title": "The complex cone over dP 1",
"text": "In this section we describe the most familiar example of quiver gauge theory with DSB fractional branes, the dP 1 theory. In this theory, a non-perturbative superpotential removes the classical supersymmetric vacua [14, 15, 16] . Assuming canonical Kähler potential the theory has a runaway behavior [15, 17] . In this section, we revisit with our techniques the result in [19] that the addition of massive flavors can induce the appearance of meta-stable supersymmetry breaking minima, long-lived against tunneling to the runaway regime. As we show in coming sections, this behavior is prototypical and extends to many other theories with DSB fractional branes. The example is also representative of the computations for a general quiver coming from a brane at a toric singularity, and illustrates the usefulness of the direct Feynman diagram evaluation of one-loop masses.\n\nConsider the dP 1 theory, realized on a set of M fractional D3-branes at the complex cone over dP 1 . In order to introduce additional flavors, we introduce sets of N f,1 D7-branes wrapping non-compact 4-cycles on the geometry and passing through the singular point. We refer the reader to [19] , and also to later sections, for more details on the construction of the theory, and in particular on the introduction of the D7-branes.\n\nIts quiver is shown in Figure 2 , and its superpotential is\n\nW = λ(X 23 X 31 Y 12 -X 23 Y 31 X 12 ) + λ ′ (Q 3i Qi2 X 23 + Q 2j Qj1 X 12 + Q 1k Qk3 X 31 ) + m 3 Q 3i Qk3 δ ik + m 2 Q 2j Qi2 δ ji + m 1 Q 1k Qj1 δ kj , (3.1)\n\nwhere the subindices denote the groups under which the field is charged. The first line is the superpotential of the theory of fractional brane, the second line describes 77-73-37 couplings between the flavor branes and the fractional brane, and the last line gives the flavor masses. Note that there is a massless field, denoted Z 12 in [19] , that does not appear in the superpotential. This is one of the decoupled fields mentioned above, and we leave its treatment as an open question. 1 3 2 i j k SU(3M) SU(2M) SU(M) PSfrag replacements\n\nQ 3i Qi2 Q 2j Qj1 Q 1k\n\nQk3 Figure 2: Extended quiver diagram for a dP 1 theory with flavors, from [19].\n\nWe are interested in gauge factors in the free magnetic phase. This is the case for the SU(3M) gauge factor in the regime\n\nM + 1 ≤ N f,1 < 5 2 M. (3.2)\n\nTo apply Seiberg duality on node 3, we introduce the dual mesons:\n\nM 21 = 1 Λ X 23 X 31 ; N k1 = 1 Λ Qk3 X 31 M ′ 21 = 1 Λ X 23 Y 31 ; N ′ k1 = 1 Λ Qk3 Y 31 N 2i = 1 Λ X 23 Q 3i ; Φ ki = 1 Λ Qk3 Q 3i (3.3) 9\n\nand we also replace the electric quarks\n\nQ 3i , Qk3 , X 23 , X 31 , Y 31 by their magnetic duals Qi3 , Q 3k , X 32 , X 13 , Y 13 .\n\nThe magnetic superpotential is given by rewriting the confined fields in terms of the mesons and adding the coupling between the mesons and the dual quarks,\n\nW = h ( M 21 X 13 X 32 + M ′ 21 Y 13 X 32 + N 2i Qi3 X 32 + N k1 X 13 Q 3k + N ′ k1 Y 13 Q 3k + Φ ki Qi3 Q 3k ) + hµ 0 ( M 21 Y 12 -M ′ 21 X 12 ) + µ ′ Q 1k N k1 + µ ′ N 2i Qi2 -hµ 2 Tr Φ + λ ′ Q 2j Qj1 X 12 + m 2 Q 2i Qi2 + m 1 Q 1i Qi1 . (3.4)\n\nThis is the theory we want to study. In order to simplify the treatment of this example we will disregard any subleading terms in m i /µ ′ , and effectively integrate out N k1 and N 2i by substituting them by 0. This is not necessary, and indeed the computations in the next sections are exact. We do it here in order to compare results with [19].\n\nAs in the ISS model, this theory breaks supersymmetry via the rank condition. The fields Qi3 , Q 3k and Φ ki are the analogs of q, q and M in the ISS case discussed above.\n\nThis motivates a vacuum ansatz analogous to (2.6) and the following linear expansion:\n\nΦ = φ 00 φ 01 φ 10 φ 11 ; Qi3 = µe θ + Q 3,1 Q3,2 ; Q T 3i = µe -θ + Q 3,1 Q 3,2 Qk1 = Q1,1 y ; Q 2j = Q 2,11 x Q 2,21 x ′ ; M 21 = M 21,1 M 21,2 Y 13 = (Y 13 ) ; X T 12 = X 12,1 X 12,2 ; X T 32 = X 32,1 X 32,2 Y T 12 = Y 12,1 Y 12,2 ; N ′ k1 = N ′ k1,1 z ; M ′ 21 = λ ′ hµ 0 M ′ 21,1 M ′ 21,2 X 13 = (X 13 ) .\n\n(3.5) Note that we have chosen to introduce the nonlinear expansion in θ in order to reproduce the results found in the literature in their exact form 3 . Note also that for the sake of clarity we have not been explicit about the ranks of the different matrices.\n\nThey can be easily worked out (or for this case, looked up in [19] ), and we will restrict ourselves to the 2 flavor case where the matrix structure is trivial. As a last remark, we are not being explicit either about the definitions of the different couplings in terms of the electric theory. This can be done easily (and as in the ISS case they involve 3 A linear expansion would lead to identical conclusions concerning the existence of the meta-stable\n\nvacua, but to one-loop masses not directly amenable to comparison with results in the literature.\n\n10 an unknown coefficient in the Kähler potential), but in any event, the existence of the meta-stable vacua can be established for general values of the coefficients in the superpotential. Hence we skip this more detailed but not very relevant discussion.\n\nThe next step consists in expanding the superpotential and identifying the massless fields. We get the following quadratic contributions to the superpotential:\n\nW mass = 2hµφ 00 Q3,1 + hµφ 01 Q3,2 + hµφ 10 Q 3,2 + hµ 0 M 21,1 Y 12,1 + hµ 0 M 21,2 Y 12,2 -λ ′ M ′ 21,1 X 12,1 -λ ′ M ′ 21,2 X 12,2 + hµN ′ k1,1 Y 13 -h 1 µ Q1,1 X 13 -h 2 µQ 2,11 X 32,1 -h 2 µQ 2,21 X 32,2 . (3.6)\n\nThe fields massless at tree level are x, x ′ , y, z, φ 11 , θ, Q 3,2 and Q3,2 . Three of these are Goldstone bosons as described in the previous section. For real µ they are Im θ, Re ( Q3,2 + Q 3,2 ) and Im ( Q3,2 -Q 3,2 ). We now show that all other classically massless fields get masses at one loop (with positive squared masses).\n\nAs a first step towards finding the one-loop correction, notice that the supersymmetry breaking mechanism is extremely similar to the one in the ISS model before, in particular it comes only from the following couplings in the superpotential:\n\nW rank = hQ 3,2 Q3,2 φ 11 -hµ 2 φ 11 + . . . (3.7)\n\nThis breaks the spectrum degeneracy in the multiplets Q 3,2 and Q3,2 at tree level, so we refer to them as the fields with broken supersymmetry.\n\nLet us compute now the correction for the mass of x, for example. For the one-loop computation we just need the cubic terms involving one pseudomodulus and at least one of the broken supersymmetry fields, and any quadratic term involving fields present in the previous set of couplings. From the complete expansion one finds the following supersymmetry breaking sector:\n\nW symm. = hφ 11 Q 3,2 Q3,2 + hµφ 01 Q3,2 + hµφ 10 Q 3,2 -hµ 2 φ 11 . (3.8)\n\nThe only cubic term involving the pseudomodulus x and the broken supersymmetry fields is\n\nW cubic = -h 2 x Q3,2 X 32,1 , (3.9)\n\nand there is a quadratic term involving the field X 32,1\n\nW mass coupling = -h 2 µQ 2,11 X 32,1 . (3.10)\n\nAssembling the three previous equations, the resulting superpotential corresponds to the asymmetric model in appendix A.2, so we can directly obtain the one-loop mass 11 for x:\n\nδm 2 x = 1 16π 2 |h| 4 µ 2 C |h 2 | 2 |h| 2 . (3.11)\n\nProceeding in a similar way, the one-loop masses for φ 11 , x ′ , y and z are:\n\nδm 2 φ 11 = 1 8π 2 |h| 4 µ 2 (log 4 -1) δm 2 x ′ = 1 16π 2 |h| 4 µ 2 C |h 2 | 2 |h| 2 , δm 2 y = 1 16π 2 |h| 4 µ 2 C |h 1 | 2 |h| 2 δm 2 z = 1 16π 2 |h| 4 µ 2 (log 4 -1).\n\n(3.12) There is just one pseudomodulus left, Re θ, which is qualitatively different to the others. With similar reasoning, one concludes that it is necessary to study a superpotential of the form\n\nW = h(Xφ 1 φ 2 + µe θ φ 1 φ 3 + µe -θ φ 2 φ 4 -µ 2 X). (3.13)\n\nDue to the non-linear parametrization, the expansion in θ shows that there is a term quadratic in θ which contributes to the one-loop mass via a vertex with two bosons and two fermions, the relevant diagram is shown in Figure 16d . The result is a vanishing mass for Im θ, as expected for a Goldstone boson (the one-loop tadpole vanishes in this case), and a non-vanishing mass for Re θ\n\nδm 2 Re θ = 1 4π 2 |h| 4 µ 4 (log 4 -1). (3.14)\n\nWe conclude by mentioning that all squared masses are positive, thus confirming that the proposed point in field space is the one-loop minimum. As shown in [19] , this minimum is parametrically long-lived against tunneling to the runaway regime."
},
{
"section_type": "OTHER",
"section_title": "Additional examples: The dP 2 case",
"text": "Let us apply these techniques to consider new examples. In this section we consider a DSB fractional brane in the complex cone over dP 2 , which provides another quiver theory with runaway behavior [15] . The quiver diagram for dP 2 is given in Figure 3 ,\n\nwith superpotential W = X 34 X 45 X 53 -X 53 Y 31 X 15 -X 34 X 42 Y 23 + Y 23 X 31 X 15 X 52 + X 42 X 23 Y 31 X 14 -X 23 X 31 X 14 X 45 X 52 ( 3\n\n.15) 12 1 2 3 5 4 Figure 3: Quiver diagram for the dP 2 theory.\n\nWe consider a set of M DSB fractional branes, corresponding to choosing ranks (M, 0, M, 0, 2M) for the corresponding gauge factors. The resulting quiver is shown in Figure 4 , with superpotential\n\nW = -λX 53 Y 31 X 15 (3.16) U(2M) U(M) U(M) 3 1 5\n\nFigure 4: Quiver diagram for the dP 2 theory with M DSB fractional branes.\n\nFollowing [19] and appendix B, one can introduce D7-branes leading to D3-D7 open strings providing (possibly massive) flavors for all gauge factors, and having cubic couplings with diverse D3-D3 bifundamental chiral multiplets. We obtain the quiver in Figure 5 . Adding the cubic 33-37-73 coupling superpotential, and the flavor masses, the complete superpotential reads\n\nW total = -λX 53 Y 31 X 15 -λ ′ (Q 1i Qi3 Y 31 + Q 3j Qj5 X 53 + Q 5k Qk1 X 15 ) + m 1 Q 1i Qk1 + m 2 Q 3j Qi3 + m 5 Q 5k Qj5 (3.17)\n\nwhere 1, 2, 3 are the gauge group indices and i, j, k are the flavor indices.\n\nWe consider the U(2M) node in the free magnetic phase, namely\n\nM + 1 ≤ N f,1 < 2M (3.18) 13 U(M) U(M) U( 2M\n\n)\n\nPSfrag replacements\n\nQ 1i Q i3 Q 3j Q j5 Q 5k Q k1\n\nFigure 5: Quiver for the dP 2 theory with M fractional branes and flavors.\n\nAfter Seiberg Duality the dual gauge factor is SU(N) with N = N f,1 -M and dynamical scale Λ. To get the matter content in the dual, we replace the microscopic flavors\n\nQ 5k , Qj5 , X 53 , X 15 by the dual flavors Qk5 , Q 5j , X 35 , X 51 respectively.\n\nWe also have the mesons related to the fields in the electric theory by\n\nM 1k = 1 Λ X 15 Q 5K ; Ñj3 = 1 Λ Qj5 X 53 M 13 = 1 Λ X 15 X 53 ; Φjk = 1 Λ Qj5 Q 5k (3.19)\n\nThere is a cubic superpotential coupling the mesons and the dual flavors\n\nW mes. = h ( M 1k Qk5 X 51 + M 13 X 35 X 51 + Ñj3 X 35 Q 5j + Φjk Qk5 Q 5j ) (3.20)\n\nwhere h = Λ/ Λ with Λ given by Λ 3Nc-N f elect Λ 3(N f -Nc)-N f = ΛN f , where Λ elect is the dynamical scale of the electric theory. Writing the classical superpotential terms of the new fields gives\n\nW clas. = -h µ 0 M 13 Y 31 + λ ′ Q 1i Qi3 Y 31 + µ ′ Ñj3 Q 3j + µ ′ M 1k Qk1 + m 1 Q 1i Qk1 + m 3 Q 3j Qi3 -hµ 2 Tr Φ (3.21) where µ 0 = λΛ, µ ′ = λ ′ Λ, and µ 2 = -m 5 Λ.\n\nSo the complete superpotential in the Seiberg dual is\n\nW dual = -h µ 0 M 13 Y 31 + λ ′ Q 1i Qi3 Y 31 + µ ′ Ñj3 Q 3j + µ ′ M 1k Qk1 + m 1 Q 1i Qk1 + m 3 Q 3j Qi3 -hµ 2 Tr Φ + h ( M 1k Qk5 X 51 + M 13 X 35 X 51 + Ñj3 X 35 Q 5j + Φjk Qk5 Q 5j ) (3.22)\n\nThis superpotential has a sector completely analogous to the ISS model, triggering supersymmetry breaking by the rank condition. This suggests the following ansatz for 14 the point to become the one-loop vacuum\n\nQ 5k = Q T 5k = µ 0 , (3.23)\n\nwith all other vevs set to zero. Following our technique as explained above, we expand fields at linear order around this point. Focusing on N f,1 = 2 and N c = 1 for simplicity (the general case can be easily recovered), we have\n\nQk5 = µ + δ Q5,1 δ Q5,2 ; Q 5k = (µ + δQ 5,1 ; δQ 5,2 ) ; Φ = δΦ 0,0 δΦ 0,1 δΦ 1,0 δΦ 1,1 Qk1 = δ Q1,1 δ Q1,2 ; Q 1i = (δQ 1,1 ; δQ 1,2 ) ; Qi3 = δ Q3,1 δ Q3,2 ; Q 3j = (δQ 3,1 ; δQ 3,2 ) Ñj3 = δ Ñ3,1 δ Ñ3,2 ; M 1k = (δM 1,1 ; δM 1,2 ) ; M 13 = δM 13 ; Y 31 = δY 31 ; X 51 = δX 51 X 35 = δX 35 (3.24)\n\nInserting this into equation (3.22) gives\n\nW dual = -h µ 0 δM 13 δY 31 + λ ′ δQ 1,1 δ Q3,1 δY 31 + λ ′ δQ 1,2 δ Q3,2 δY 31 + µ ′ δ Ñ3,1 δQ 3,1 + µ ′ δ Ñ3,2 δQ 3,2 + µ ′ δM 1,1 δ Q1,1 + µ ′ δM 1,2 δ Q1,2 + m 1 δQ 1,1 δ Q1,1 + m 1 δQ 1,2 δ Q1,2 + m 3 δQ 3,1 δ Q3,1 + m 3 δQ 3,2 δ Q3,2 -hµ 2 δΦ 11 + h ( µδM 1,1 δX 51 + δM 1,1 δ Q5,1 δX 51 + δM 1,2 δ Q5,2 δX 51 + δM 13 δX 35 δX 51 + µδX 35 δ Ñ3,1 + δX 35 δ Ñ3,1 δQ 5,1 + δX 35 δ Ñ3,2 δQ 5,2 + µδ Q5,1 δΦ 00 + µδQ 5,1 δΦ 00 + δQ 5,1 δ Q5,1 δΦ 00 + µδΦ 01 δ Q5,2 + δQ 5,1 δΦ 01 δ Q5,2 + µδΦ 10 δQ 5,2 + δ Q5,1 δΦ 10 δQ 5,2 + δ Q5,2 δΦ 11 δQ 5,2 ).\n\nWe now need to identify the pseudomoduli, in other words the massless fluctuations at tree level. We focus then just on the quadratic terms in the superpotential\n\nW mass = -h µ 0 δM 13 δY 31 + µ ′ δ Ñ3,1 δQ 3,1 + m 3 δQ 3,1 δ Q3,1 + hµδX 35 δ Ñ3,1 + µ ′ δ Ñ3,2 δQ 3,2 + m 3 δQ 3,2 δ Q3,2 + µ ′ δM 1,1 δ Q1,1 + m 1 δQ 1,1 δ Q1,1 + hµδM 1,1 δX 51 + µ ′ δM 1,2 δ Q1,2 + m 1 δQ 1,2 δ Q1,2 + hµδ Q5,1 δΦ 00 + hµδQ 5,1 δΦ 00 + hµδΦ 01 δ Q5,2 + µδΦ 10 δQ 5,2 . (3.25) 15\n\nWe have displayed the superpotential so that fields mixing at the quadratic level appear in the same line. In order to identify the pseudomoduli we have to diagonalize 4 these fields. Note that the structure of the mass terms corresponds to the one in appendix C, in particular around equation (C.9). From the analysis performed there we know that upon diagonalization, fields mixing in groups of four (i.e., three mixing terms in the superpotential, for example the δM 1,\n\n1 , δ Q1,1 , δQ 1,1 , δX 51 mixing) get nonzero masses,\n\nwhile fields mixing in groups of three (two mixing terms in the superpotential, for example δM 1,2 , δ Q1,2 and δQ 1,2 ) give rise to two massive perturbations and a massless one, a pseudomodulus. We then just need to study the fate of the pseudomoduli. From the analysis in appendix C, the pseudomoduli coming from the mixing terms are\n\nY 1 = m 3 δ Ñ3,2 -µ ′ δ Q3,2 , Y 2 = m 1 δM 1,2 -µ ′ δQ 1,2 , Y 3 = hµ(δQ 5,1 -δ Q5,1 ) . (3.26)\n\nIn order to continue the analysis, one just needs to change basis to the diagonal fields and notice that the one loop contributions to the pseudomoduli are described again by the asymmetric model of appendix A.2, so they receive positive definite contributions.\n\nThe exact analytic expressions can be easily found with the help of some computer algebra program, but we omit them here since they are quite unwieldy."
},
{
"section_type": "OTHER",
"section_title": "The general case",
"text": "In the previous section we showed that several examples of quiver gauge theories on DSB fractional branes have metastable vacua once additional flavors are included.\n\nIn this section we generalize the arguments for general DSB branes. We will show how to add D7-branes in a specific manner so as to generate the appropriate cubic flavor couplings and mass terms. Once this is achieved, we describe the structure of the Seiberg dual theory. The results of our analysis show that, with the specified configuration of D7-branes, the determination of metastability is greatly simplified and only involves looking at the original superpotential. Thus, although we do not prove that DSB branes on arbitrary singularities generate metastable vacua, we show how one can determine the existence of metastability in a very simple and systematic 4 As a technical remark, let us note that it is possible to set all the mass terms to be real by an appropriate redefinition of the fields, so we are diagonalizing a real symmetric matrix.\n\nmanner. Using this analysis we show further examples of metastable vacua on systems of DSB branes."
},
{
"section_type": "OTHER",
"section_title": "Construction of the flavored theories",
"text": "Consider a general quiver gauge theory arising from branes at singularities. As we have argued previously, we focus on DSB branes, so that there is a gauge factor satisfying N f,0 < N c , which can lead to supersymmetry breaking by the rank condition in its Seiberg dual. To make the general analysis more concrete, let us consider a quiver like that in Figure 6 , which is characteristic enough, and let us assume that the gauge factor to be dualized corresponds to node 2. In what follows we analyze the structure of the fields and couplings in the Seiberg dual, and reduce the problem of studying the meta-stability of the theory with flavors to analyzing the structure of the theory in the absence of flavors.\n\n2 1 3 5 4 PSfrag replacements X 21 Y 21 X 32 Y 32 Z 32 X 14 X 43 Y 43\n\nFigure 6: Quiver diagram used to illustrate general results. It does not correspond to any geometry in particular.\n\nThe first step is the introduction of flavors in the theory. As discussed in [19] , for any bi-fundamental X ab of the D3-brane quiver gauge theory there exist a supersymmetric D7-brane leading to flavors Q bi , Qia in the fundamental (antifundamental) of the b th (a th ) gauge factor. There is also a cubic coupling X ab Q bi Qia . Let us now specify a concrete set of D7-branes to introduce flavors in our quiver gauge theory. Consider a superpotential coupling of the D3-brane quiver gauge theory, involving fields charged under the node to be dualized. This corresponds to a loop in the quiver, involving node 2, for instance X 32 X 21 X 14 Y 43 in Figure 6 . For any bi-fundamental chiral multiplet in 17 this coupling, we introduce a set of N f,1 of the corresponding D7-brane. This leads to a set of flavors for the different gauge factors, in a way consistent with anomaly cancellation, such as that shown in Figure 7 . The description of this system of D7branes in terms of dimer diagrams is carried out in Appendix B. The cubic couplings described above lead to the superpotential terms 5\n\nW f lavor = λ ′ ( X 32 Q 2b Q b3 + X 21 Q 1a Q a2 + X 14 Q 4d Q d1 + Y 43 Q 3c Q c4 ) (4.1)\n\nFinally, we introduce mass terms for all flavors of all involved gauge factors:\n\nW mass = m 2 Q a2 Q 2b + m 3 Q b3 Q 3c + m 4 Q c4 Q 4d + m 1 Q d1 Q 1a (4.2)\n\nThese mass terms break the flavor group into a diagonal subgroup. 2 1 3 5 4 a b c d PSfrag replacements\n\nX 21 Y 21 X 32 Y 32 Z 32 X 14 X 43 Y 43 Q 1a Q a2 Q 2b Q b3 Q 3c Q c4 Q 4d Q d1\n\nFigure 7: Quiver diagram with flavors. White nodes denote flavor groups."
},
{
"section_type": "OTHER",
"section_title": "Seiberg duality and one-loop masses - . . , M 6",
"text": "We consider introducing a number of massive flavors such that node 2 is in the free magnetic phase, and consider its Seiberg dual. The only relevant fields in this case are those charged under gauge factor 2, as shown if Figure 8 . The Seiberg dual gives us Figure 9 where the M's are mesons with indices in the gauge groups, R's and S's are 5 Here we assume the same coupling, but the conclusions hold for arbitrary non-zero couplings.\n\n18 2 a b 1 3 PSfrag replacements X 21 Y 21 X 32 Y 32 Z 32 Q a2 Q 2b\n\nFigure 8: Relevant part of quiver before Seiberg duality. a b 1 3 2 PSfrag replacements X12 Ỹ12 X23 Ỹ23 Z23 Qb2 Q2a\n\nX ab R 1 R 2 S 1 S 2 S 3 M 1 , .\n\nmesons with only one index in the flavor group, and X ab is a meson with both indices in the flavor groups. The original cubic superpotential and flavor mass superpotentials become\n\nW f lavor dual = λ ′ ( S 1 3b Q b3 + R 1 a1 Q 1a + X 14 Q 4d Q d1 + Y 43 Q 3c Q c4 ) W mass dual = m 2 X ab + m 3 Q b3 Q 3c + m 4 Q c4 Q 4d + m 1 Q d1 Q 1a (4.3)\n\nIn addition we have the extra meson superpotential\n\nW mesons = h ( X ab Qb2 Q2a + R 1 a1 X12 Q2a + R 2 a1 Ỹ12 Q2a + S 1 3b Qb2 X23 + S 2 3b Qb2 Ỹ23 + S 3 3b Qb2 Z23 + M 1 31 X12 X23 + M 2 31 X12 Ỹ23 + M 3 31 X12 Z23 + M 4 31 Ỹ12 X23 + M 5 31 Ỹ12 Ỹ23 + M 6 31 Ỹ12 Z23 ). (4.4)\n\nThe crucial point is that we always obtain terms of the kind underlined above, namely a piece of the superpotential reading m 2 X ab + hX ab Qb2 Q2a . This leads to tree level supersymmetry breaking by the rank condition, as announced. Moreover the superpotential fits in the structure of the generalized asymmetric O'Raifeartaigh model studied in appendix A.2, with X ab , Qb2 , Q2a corresponding to X, φ 1 , φ 2 respectively. The multiplets Qb2 and Q2a are split at tree level, and X ab is massive at 1-loop. From our study of the generalized asymmetric case, any field which has a cubic coupling to the supersymmetry breaking fields Qb2 or Q2a is one-loop massive as well. Using the general structure of W mesons , a little thought shows that all dual quarks with no flavor index (e.g. X, Ỹ ) and all mesons with one flavor index (e.g. R or S) couple to the supersymmetry breaking fields.\n\nThus they all get one-loop masses (with positive squared mass). Finally, the flavors of other gauge factors (e.g. Q b3 ) are massive at tree level from W mass .\n\nThe bottom line is that the only fields which do not get mass from these interactions are the mesons with no flavor index, and the bi-fundamentals which do not get dualized (uncharged under node 2). All these fields are related to the theory in the absence of extra flavors, so they can be already stabilized at tree-level from the original superpotential. So, the criteria for a metastable vacua is that the original theory, in the absence of flavors leads, after dualization of the node with N f < N c , to masses for all these fields (or more mildly that they correspond to directions stabilized by mass terms, or perhaps higher order superpotential terms).\n\nFor example, if we apply this criteria to the dP 2 case studied previously, the original superpotential for the fractional DSB brane is\n\nW = -λX 53 Y 31 X 15 (4.5)\n\n20 so after dualization we get W = -λM 13 Y 31 (4.6) which makes these fields massive. Hence this fractional brane, after adding the D7branes in the appropriate configuration, will generate a metastable vacua will all moduli stabilized.\n\nThe argument is completely general, and leads to an enormous simplification in the study of the theories. In the next section we describe several examples. A more rigorous and elaborate proof is provided in the appendix where we take into account the matricial structure, and show that all fields, except for Goldstone bosons, get positive squared masses at tree-level or at one-loop.\n\nFigure 9: Relevant part of the quiver after Seiberg duality on node 2."
},
{
"section_type": "OTHER",
"section_title": "Additional examples",
"text": "4.2.1 The dP 3 case Let us consider the complex cone over dP 3 , and introduce fractional DSB branes of the kind considered in [15] . The quiver is shown in Figure 10 and the superpotential is\n\nW = X 13 X 35 X 51 (4.7)\n\nNode 1 has N f < N c so upon addition of massive flavors and dualization will lead to supersymmetry breaking by the rank condition. Following the procedure of the previous section, we add N f,1 flavors coupling to the bi-fundamentals X 13 , X 35 and X 51 . Node 1 is in the free magnetic phase for P + 1 ≤ N f,1 < 3 2 P + 1 2 . Dualizing node 1, the above superpotential becomes\n\nW = X 35 M 53 (4.8)\n\nwhere M 53 is the meson X 51 X 13 . So, following the results of the previous section, we can conclude that this DSB fractional brane generates a metastable vacua with all pseudomoduli lifted."
},
{
"section_type": "OTHER",
"section_title": "Phase 1 of P dP 4",
"text": "Let us consider the P dP 4 theory, and introduce the DSB fractional brane of the kind considered in [15] . The quiver is shown in Figure 11 . The superpotential is\n\nW = -X 25 X 51 X 12 (4.9) 21 U(P) U( 1\n\n) 5 3 U(1) 1 4 U(P+1)\n\nFigure 10: Quiver diagram for the dP 3 theory with a DSB fractional brane.\n\nU(P) 5 1 4 U(M) 2 U(M) U(M+P)\n\nFigure 11: Quiver diagram for the dP 4 theory with a DSB fractional branes.\n\n22 Node 1 has N f < N c and will lead to supersymmetry breaking by the rank condition in the dual. Following the procedure of the previous section, we add N f,1 flavors coupling to the bi-fundamentals X 12 , X 25 and X 51 . Node 1 is in the free magnetic phase for P + 2 ≤ M + N f,1 < 3 2 (M + P ). Dualizing node 1, the above superpotential becomes\n\nW = X 25 M 52 , where M 53 is the meson X 51 X 12 .\n\nAgain we conclude that this DSB fractional brane generates a metastable vacua with all pseudomoduli lifted."
},
{
"section_type": "OTHER",
"section_title": "4.2.3",
"text": "The Y p,q family Consider D3-branes at the real cones over the Y p,q Sasaki-Einstein manifolds [36, 37, 38, 39] , whose field theory were determined in [8] . The theory admits a fractional brane [13] of DSB kind, which namely breaks supersymmetry and lead to runaway behavior [15, 18] . The analysis of metastability upon addition of massive flavors for arbitrary Y p,q 's is much more involved than previous examples. Already the description of the field theory on the fractional brane is complicated. Even for the simpler cases of Y p,q and Y p,p-1 the superpotential contains many terms. In this section we do not provide a general proof of metastability, but rather consider the more modest aim of showing that all directions related to the runaway behavior in the absence of flavors are stabilized by the addition of flavors. We expect that this will guarantee full metastability, since the fields not involved in our analysis parametrize directions orthogonal to the runaway at infinity. The dimer for Y p,q is shown in Figure 12 and consists of a column of n hexagons and 2 m quadrilaterals which are just halved hexagons [18] . The labels (n, m) are related\n\nto (p, q) by n = 2q ; m = p -q (4.10)\n\n• The Y p,1 case\n\nThe dimer for the theory on the DSB fractional brane in the Y p,1 case is shown in Figure 13 , a periodic array of a column of two full hexagons, followed by p -1 cut hexagons (the shaded quadrilateral has N c = 0). As shown in [18] , the top quadrilateral which has N f < N c , and induces the ADS superpotential triggering the runaway. The relevant part of the dimer is shown in Figure 14 , where V 1 and V 2 are the fields that run to infinity [18] . This node will lead to supersymmetry breaking by the rank condition in the dual. It is in the free magnetic phase for M + 1 ≤ N f,1 < pM + M 2 . The piece 23 n n n n+1 n+2 n+m 1 n+1 n+2 n+m 3 n 2 2 1 3 1 2 3 n+1 n+2 n+m 1 1 2 3 n+1 n+2 n+m z w Figure 12: The generic dimer for Y p,q , from [18].\n\nof the superpotential involving the V 1 and V 2 terms is\n\nW = Y U 2 V 2 -Y U 1 V 1 . (4.11)\n\nIn the dual theory, the dual superpotential makes the fields massive. Hence, the theory has a metastable vacua where the runaway fields are stabilized.\n\n00 00 11 11 00 00 11 11 00 00 11 11\n\nFigure 13: The dimer for Y p,1 . 24 1 2 2 1 pM (p-1)M (p-2)M Z Y (2p-1)M (p-1)M pM (p+1)M V U V U Figure 14: Top part of the dimer for Y p,1 . The hexagons are labeled by the ranks of the respective gauge groups\n\n• The Y p,p-1 case The analysis for Y p,p-1 is similar but in this case it is the bottom quadrilateral which has the highest rank and thus gives the ADS superpotential [18] . The relevant part of the dimer is shown in Figure 15 , and the runaway direction is described by the fields V 1 and V 2 . Upon addition of N f,1 flavors, the relevant node in the in the free magnetic phase for M + 1 ≤ N f,1 < pM + M 2 Considering the superpotential, it is straightforward to show that the runaway fields become massive. Complementing this with our analysis in previous section, we conclude that the theory has a metastable vacua where the runaway fields are stabilized.\n\nWe have thus shown that we can obtain metastable vacua for fractional branes at cones over the Y p,1 and Y p,p-1 geometries. Although there is no obvious generalization for arbitrary Y p,q 's, our results strongly suggest that the existence of metastable vacua extends to the complete family."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusions and outlook",
"text": "The present work introduces techniques and computations which suggest that the existence of metastable supersymmetry breaking vacua is a general property of quiver gauge theories on DSB fractional branes, namely fractional branes associated to obstructed complex deformations. It is very satisfactory to verify the correlation between a non-trivial dynamical property in gauge theories and a geometric property in their 25\n\n(p-1)M V U2 U2 U1 V1 (p-1)M Y Y (2p-1)M (p-2)M (2p-2)M (2p-2)M 2 Figure 15: Bottom part of the dimer for Y p,p-1 . The hexagons are labeled by the ranks of the respective gauge groups string theory realization. The existence of such correlation fits nicely with the remarkable properties of gauge theories on D-branes at singularities, and the gauge/gravity correspondence for fractional branes.\n\nBeyond the fact that our arguments do not constitute a general proof, our analysis has left a number of interesting open questions. In fact, as we have mentioned, all theories on DSB fractional branes contain one or several fields which do not appear in the superpotential. We expect the presence of these fields to have a direct physical interpretation, which has not been uncovered hitherto. It would be interesting to find a natural explanation for them.\n\nFinally, a possible extension of our results concerns D-branes at orientifold singularities, which can lead to supersymmetry breaking and runaway as in [27] . Interestingly, in this case the field theory analysis is more challenging, since they would require Seiberg dualities of gauge factors with matter in two-index tensors. It is very possible that the string theory realization, and the geometry of the singularity provide a much more powerful tool to study the system.\n\nOverall, we expect other surprises and interesting relations to come up from further study of D-branes at singularities."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "We thank S. Franco for useful discussions. A.U. thanks M. González for encouragement and support. This work has been supported by the European Commission under RTN European Programs MRTN-CT-2004-503369, MRTN-CT-2004-005105, by the CICYT (Spain), and by the Comunidad de Madrid under project HEPHACOS P-ESP-00346.\n\nThe research by I.G.-E. is supported by the Gobierno Vasco PhD fellowship program.\n\nThe research of F.S is supported by the Ministerio de Educación y Ciencia through an FPU grant. I.G.-E. and F.S. thank the CERN Theory Division for hospitality during the completion of this work.\n\nA Technical details about the calculation via Feynman diagrams A.1 The basic amplitudes\n\nIn the main text we are interested in computing two point functions for the pseudomoduli at one loop, and in section 2.2 also tadpole diagrams. There are just a few kinds of diagrams entering in the calculation, which we will present now for the two-point function, see Figure 16 . The (real) bosonic fields are denoted by φ i and the (Weyl) fermions by ψ i . The pseudomodulus we are interested in is denoted by ϕ. c) d) a) b)\n\nϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ φ 2 φ 1 φ ψ 2 ψ ψ 1\n\nFigure 16: Feynman diagrams contributing to the one-loop two point function. The dashed line denotes bosons and the solid one fermions."
},
{
"section_type": "OTHER",
"section_title": "Bosonic contributions",
"text": "These come from two terms in the Lagrangian. First there is a diagram coming from terms of the form (Figure 16b ):\n\nL = . . . + λϕ 2 φ 2 - 1 2 m 2 φ 2 , (A.1)\n\ngiving an amplitude (we will be using dimensional regularization)\n\niM = -2iλ (4π) 2 m 2 1 ǫ -γ + 1 + log 4π -log m 2 . (A.2)\n\nThe other contribution comes from the diagram in Figure 16a :\n\nL = . . . + λϕφ 1 φ 2 - 1 2 m 2 1 φ 2 1 - 1 2 m 2 2 φ 2 2 , (A.\n\n3) which contributes to the two point function with an amplitude:\n\niM = iλ 2 (4π) 2 1 ǫ -γ + log 4π - 1 0 dx log ∆ , (A.4)\n\nwhere here and in the following we denote ∆ ≡ xm 2 1 + (1x)m 2 2 ."
},
{
"section_type": "OTHER",
"section_title": "Fermionic contributions",
"text": "The relevant vertices here are again of two possible kinds, one of which is nonrenormalizable. The cubic interaction comes from terms in the Lagrangian given by the diagram in Figure 16c :\n\nL = . . . + ϕ(aψ 1 ψ 2 + a * ψ1 ψ2 ) + 1 2 m 1 (ψ 2 1 + ψ2 1 ) + 1 2 m 2 (ψ 2 2 + ψ2 2 ). (A.5)\n\nWe are assuming real masses for the fermions here, in the configurations we study this can always be achieved by an appropriate field redefinition. The contribution from such vertices is given by:\n\niM = 1 0 dx -2im 1 m 2 (4π) 2 (a 2 + (a 2 ) * ) 1 ǫ -γ + log 4π -log ∆ - 8i|a| 2 (4π) 2 ∆ 1 ǫ -γ + log 4π + 1 2 -log ∆ . (A.6)\n\nThe other fermionic contribution, which one does not need as long as one is dealing with renormalizable interactions only (but we will need in the main text when analyzing the pseudomodulus θ), is given by terms in the Lagrangian of the form (Figure 16d ):\n\nL = . . . + λϕ 2 (ψ 2 + ψ2 ) + 1 2 m(ψ 2 + ψ2 ), (A.7)\n\nwhich contributes to the total amplitude with:\n\niM = 8λmi (4π) 2 m 2 1 ǫ -γ + 1 + log 4π -log m 2 . (A.8)"
},
{
"section_type": "OTHER",
"section_title": "A.2 The basic superpotentials",
"text": "The previous amplitudes are the basic ingredients entering the computation, but in general the number of diagrams contributing to the two point amplitudes is quite big, so calculating all the contributions by hand can get quite involved in particular examples 6 . Happily, one finds that complicated models (such as dP 1 or dP 2 , studied in the main text) reduce to performing the analysis for only two different superpotentials, which we analyze in this section."
},
{
"section_type": "OTHER",
"section_title": "The symmetric case",
"text": "We want to study in this section a superpotential of the form:\n\nW = h(Xφ 1 φ 2 + µφ 1 φ 3 + µφ 2 φ 4 -µ 2 X). (A.9)\n\n6 The authors wrote the computer program in http://cern.ch/inaki/pm.tar.gz which helped greatly in the process of computing the given amplitudes for the relevant models.\n\nThis model is a close cousin of the basic O'Raifeartaigh model. We are interested in the one loop contribution to the two point function of X, which is massless at tree level.\n\nFrom the (F-term) bosonic potential one obtains the following terms entering the one loop computation:\n\nV = |hXφ 2 | 2 + |h| 2 µ(Xφ 2 φ * 3 + X * φ * 2 φ 3 ) + |h| 2 µ(Xφ 1 φ * 4 + X * φ * 1 φ 4 ) + |h| 2 µ 2 (φ 1 φ 2 + φ * 1 φ * 2 ) + 4 i=1 |h| 2 µ 2 |φ i | 2 (A.10)\n\nIn order to do the computation it is useful to diagonalize the mass matrix by introducing φ + and φ -such that:\n\nφ 1 = 1 √ 2 (φ + + iφ -) φ 2 = 1 √ 2 (φ + -iφ -) (A.11)\n\nand φ a , φ b such that:\n\nφ * 3 = 1 √ 2 (φ a + iφ b ) φ * 4 = 1 √ 2 (φ a -iφ b ). (A.12)\n\nWith these redefinitions the bosonic scalar potential decouples into identical φ + and φ -sectors, giving two decoupled copies of:\n\nV = |h| 2 |X| 2 |φ + | 2 + |h| 2 µ 2 (|φ + | 2 + |φ a | 2 ) +|h| 2 µ(Xφ + φ a + X * φ * + φ * a ) - |h| 2 µ 2 2 φ 2 + + (φ 2 + ) * . (A.13)\n\nCalculating the amplitude consists simply of constructing the (very few) two point diagrams from the potential above and plugging the formulas above for each diagram (the fermionic part is even simpler in this case). The final answer is that in this model the one loop correction to the mass squared of X is given by:\n\nδm 2 X = |h 4 |µ 2 8π 2 (log 4 -1). (A.14)\n\nThe generalized asymmetric case\n\nThe next case is slightly more complicated, but will suffice to analyze completely all the models we encounter. We will be interested in the one loop contribution to the mass of the pseudomoduli Y in a theory with superpotential:\n\nW = h(Xφ 1 φ 2 + µφ 1 φ 3 + µφ 2 φ 4 -µ 2 X) + k(rY φ 1 φ 5 + µφ 5 φ 7 ), (A.15) 30\n\nwith k and r arbitrary complex numbers. The procedure is straightforward as above, so we will just quote the result. We obtain an amplitude given by:\n\niM = -i (4π) 2 |h 2 rµ| 2 C |k| 2 |h| 2 , (A.16)\n\nwhere we have defined C(t) as: C(t) = t 2t log 4 -t t -1 log t . (A.17) Note that this is a positive definite function, meaning that the one loop correction to the mass is always positive, and the pseudomoduli get stabilized for any (nonzero) value of the parameters. Also note that the limit of vanishing t with |r| 2 t fixed (i.e., vanishing masses for φ 5 and φ 7 , but nonvanishing coupling of Y to the supersymmetry breaking sector) gives a nonvanishing contribution to the mass of Y ."
},
{
"section_type": "OTHER",
"section_title": "B D7-branes in the Riemann surface",
"text": "The gauge theory of D3-branes at toric singularities can be encoded in a dimer diagram [40, 41, 42, 43, 44] . This corresponds to a bi-partite tiling of T 2 , where faces correspond to gauge groups, edges correspond to bi-fundamentals, and nodes correspond to superpotential terms. As an example, the dimer diagram of D3-branes on the cone over dP 2 is shown in Figure 17 . As shown in [43] , D3-branes on a toric singularity are mirror to D6-branes on intersecting 3-cycles in a geometry given by a fibration of a Riemann surface Σ with punctures. This Riemann surface is just a thickening of the web diagram of the toric singularity [45, 46, 47] , with punctures associated to external legs of the web diagram. The mirror D6-branes wrap non-trivial 1-cycles on this Riemann surface, with their intersections giving rise to bi-fundamental chiral multiplets, and superpotential terms arising from closed discs bounded by the D6-branes. In [19], it was shown that D7-branes passing through the singular point can be described in the mirror Riemann surface Σ by non-compact 1-cycles which come from infinity at one puncture and go to infinity at another. Figure 18 shows the 1-cycles corresponding to some D3-and D7-branes in the Riemann surface in the geometry mirror to the complex cone over dP 2 . A D7-brane leads to flavors for the two D3-brane gauge factors whose 1-cycles are intersected by the D7-brane 1-cycle, and there is a cubic coupling among the three fields (related to the disk bounded by the three 1-cycles in the Riemann surface).\n\n31 Figure 17: Dimer diagram for D3-branes at a dP 2 singularity. 1 00 00 00 11 11 11 00 00 00 00 11 11 11 11 000 000 000 000 111 111 111 111 E C F E B A E C F D7 5 5 3 C 3 1 D7 D7 Figure 18: Riemann surface in the geometry mirror to the complex cone over dP 2 , shown as a tiling of a T 2 with punctures (denoted by capital letters). The figure shows the noncompact 1-cycles extending between punctures, corresponding to D7-branes, and a piece of the 1-cycles that correspond to the mirror of the D3-branes.\n\nU(M) U(M) U(2M)\n\nPSfrag replacements\n\nQ 1i Q i3 Q 3j Q j5 Q 5k Q k1\n\nFigure 19: Quiver for the dP 2 theory with M fractional branes and flavors.\n\nAs stated in Section 4, given a gauge theory of D3-branes at a toric singularity, we introduce flavors for some of the gauge factors in a specific way. We pick a term in the superpotential, and we introduce flavors for all the involved gauge factors, and coupling to all the involved bifundamental multiplets. For example, the quiver with flavors for the dP 2 theory is shown in Figure 19 .\n\nOn the Riemann surface, this procedure amounts to picking a node and introducing D7-branes crossing all the edges ending on the node, see Figure 18 . In this example we obtain the superpotential terms\n\nW f lavor = λ ′ (Q 1i Qi3 Y 31 + Q 3j Qj5 X 53 + Q 5k Qk1 X 15 ) (B.1)\n\nIn addition we introduce mass terms\n\nW mass = m 1 Q 1i Qk1 + m 2 Q 3j Qi3 + m 5 Q 5k Qj5 (B.2)\n\nThis procedure is completely general and applies to all gauge theories for branes at toric singularities 7 ."
},
{
"section_type": "OTHER",
"section_title": "C Detailed proof of Section 4",
"text": "Recall that in Section 4 we considered the illustrative example of the gauge theory given by the quiver in Figure 20 . Since node 2 is the one we wish to dualize, the only relevant part of the diagram is shown in Figure 21 . We show the Seiberg dual in Figure 22 . The above choice of D7-branes, which we showed in appendix B can be applied to arbitrary toric singularities, gives us the superpotential terms\n\nW f lavor = λ ′ ( X 32 Q 2b Q b3 + X 21 Q 1a Q a2 + X 14 Q 4d Q d1 + Y 43 Q 3c Q c4 ) W mass = m 2 Q a2 Q 2b + m 3 Q b3 Q 3c + m 4 Q c4 Q 4d + m 1 Q d1 Q 1a (C.1)\n\nTaking the Seiberg dual of node 2 gives\n\nW f lavor dual = λ ′ ( S 1 3b Q b3 + R 1 a1 Q 1a + X 14 Q 4d Q d1 + Y 43 Q 3c Q c4 ) W mass dual = m 2 X ab + m 3 Q b3 Q 3c + m 4 Q c4 Q 4d + m 1 Q d1 Q 1a W mesons = h ( X ab Qb2 Q2a + R 1 a1 X12 Q2a + R 2 a1 Ỹ12 Q2a\n\n7 This procedure does not apply if the superpotential (regarded as a loop in the quiver) passes twice through the node which is eventually dualized in the derivation of the metastable vacua. However we have found no example of this for any DSB fractional branes.\n\n33 2 1 3 5 4 a b c d X 21 Y 21 X 32 Y 32 Z 32 X 14 X 43 Y 43 Q 1a Q a2 Q 2b Q b3 Q 3c Q c4 Q 4d Q d1\n\nFigure 20: Quiver diagram with flavors. White nodes denote flavor groups 2 a b 1 3 PSfrag replacements\n\nX 21 Y 21 X 32 Y 32 Z 32 Q a2 Q 2b\n\nFigure 21: Relevant part of quiver before Seiberg duality. a b 1 3 2 PSfrag replacements X12 Ỹ12 X23 Ỹ23 Z23 Qb2 Q2a\n\nX ab R 1 R 2 S 1 S 2 S 3 M 1 , . . . , M 6\n\nFigure 22: Relevant part of the quiver after Seiberg duality on node 2.\n\n34 + S 1 3b Qb2 X23 + S 2 3b Qb2 Ỹ23 + S 3 3b Qb2 Z23 + M 1 31 X12 X23 + M 2 31 X12 Ỹ23 + M 3 31 X12 Z23 + M 4 31 Ỹ12 X23 + M 5 31 Ỹ12 Ỹ23 + M 6 31 Ỹ12 Z23 ) (C.2)\n\nwhere we have not included the original superpotential. The crucial point is that the underlined terms appear for any quiver gauge theory with flavors introduced as described in appendix B. As described in the main text, supersymmetry is broken by the rank condition due to the F-term of the dual meson associated to the massive flavors. Our vacuum ansatz is (we take N f = 2 and N c = 1 for simplicity; this does not affect our conclusions) Qb2 = µ1 Nc 0 ; Q2a = (µ1 Nc ; 0) (C.3) with all other vevs set to zero. We parametrize the perturbations around this minimum as\n\nQb2 = µ + φ 1 φ 2 ; Q2a = (µ + φ 3 ; φ 4 ) ; X ab = X 00 X 01 X 10 X 11 (C.4)\n\nand the underlined terms give\n\nhX ab Qb2 Q2a -hµ 2 X ab = hX 11 φ 2 φ 4 -hµ 2 X 11 + hµ φ 2 X 01 + hµ φ 4 X 10 + hµ φ 1 X 00 + hµ φ 3 X 00 + h φ 1 φ 3 X 00 + h φ 2 φ 3 X 01 + h φ 1 φ 4 X 10 (C.5)\n\nIt is important to note that all the fields in (C.4) will have quadratic couplings only in the underlined term (C.5). Thus, one can safely study this term, and the conclusions are independent of the other terms in the superpotential. Diagonalizing (C.5) gives\n\nhX ab Qb2 Q2a -hµ 2 X ab = hX 11 φ 2 φ 4 -hµ 2 X 11 + hµ φ 2 X 01 + hµ φ 4 X 10 + √ 2hµ φ + X 00 + h 2 φ 2 + X 00 - h 2 φ 2 -X 00 + h √ 2 (ξ + -ξ -) φ 2 X 01 + h √ 2 (ξ + + ξ -) φ 4 X 10 (C.6) where ξ + = 1 √ 2 (φ 1 + φ 3 ) ; ξ -= 1 √ 2 (φ 1 -φ 3 ) (C.7)\n\nThis term is similar to the generalized asymmetric case studied in appendix A.2 with\n\nX 11 → X ; φ 4 → φ 1 ; φ 2 → φ 2 ; X 10 → φ 3 ; X 01 → φ 4 (C.8) 35\n\nSo here X 11 is the linear term that breaks supersymmetry, and φ 2 , φ 4 are the broken supersymmetry fields. In (C.6), the only massless fields at tree-level are X 11 and ξ -. Comparing to the ISS case in Section 2.1 shows that Im ξ -is a Goldstone boson and X 11 , Re ξ -get mass at tree-level. As for φ 2 and φ 4 , setting ρ + = 1 √ 2 (φ 2 + φ 4 ) and ρ -= 1 √ 2 (φ 2 -φ 4 ) gives us Re(ρ + ) and Im (ρ -) massless and the rest massive. Following the discussion in Section 2.1, Re(ρ + ) and Im (ρ -) are just the Goldstone bosons of the broken SU(N f ) symmetry 8 . We have thus shown that the dualized flavors (e.g. Qb2 , Q2a ) and the meson with two flavor indices (e.g. X ab ) get mass at tree-level or at 1-loop unless they are Goldstone bosons. Now, we need to verify that this is the case for the remaining fields.\n\na b 1 2 3 5 4 d c PSfrag replacements X 14 X 43 Y 43 Q 1a Q b3 Q 3c Q c4 Q 4d Q d1 X12 Ỹ12 X23 Ỹ23 Z23 Qb2 Q2a X ab R 1 R 2 S 1 S 2 S 3 M 1 ..M 6\n\nFigure 23: Quiver after Seiberg duality on node 2.\n\nThe Seiberg dual of the original quiver diagram is shown in Figure 23 . The dualized bi-fundamentals come in two classes. The first are the ones that initially (before dualizing) had cubic flavor couplings, there will always be only two of those (e.g. X12 , X23 ). The second are those that did not initially have cubic couplings to flavors, there is an arbitrary number of those (e.g. Ỹ12 , Ỹ23 , Z23 ). Figure 24 shows the relevant part of the quiver for the first class. Recalling the superpotential terms (C.2), there are several possible sources of tree-level masses. For instance, these can arise in W f lavor dual and W mass dual . Also, remembering our assignation of vevs in (C.3), tree-level masses can also arise in W mesons from cubic couplings involving the broken supersymmetry fields (e.g. Qb2 , Q2a ). The first class of bi-fundamentals (e.g. X12 , X23 ) only appear in W mesons coupled to their respective mesons (e.g. R 1 , S 1 ). In turn these mesons will ap-8 In the case where the flavor group is SU (2), these Goldstone bosons are associated to the generators t x and t y .\n\n36 a 1 2 3 d c b X12 X23 Qb2 Q2a\n\nX ab R 1 R 2 S 1 S 2 S 3 M 1 , . . . , M 6 Q 1a Q b3 Q 3c Q d1\n\nFigure 24: Relevant part of dual quiver for first class of bi-fundamentals.\n\npear in quadratic terms in W f lavor dual coupled to flavors (e.g. S 1 3b Q b3 and R 1 a1 Q 1a ), and these flavors each appear in one term in W mass . Thus there are two sets of three terms which are coupled at tree-level and which always couple in the same way. Consider for instance the term\n\nλ ′ S 1 3b Q b3 + m 3 Q b3 Q 3c + h S 1 3b Qb2 X23 = λ ′ (S 1 S 2 ) B 1 B 2 + m 1 (C 1 C 2 ) B 1 B 2 + h (S 1 S 2 ) µ + φ 1 φ 2 X23 = λ ′ (S 1 B 1 + S 2 B 2 ) + m 1 (B 1 C 1 + B 2 C 2 ) + hµ S 1 X23 + h S 1 φ 1 X23 + h S 2 φ 2 X23 (C.9)\n\nwhere S i , B i , C i and X23 are the perturbations around the minimum. Diagonalizing (which can be done analytically for any values of the couplings), we get that all terms except one get tree-level masses, the massless field being:\n\nY = m 1 S 2 -λ ′ C 2 (C.10)\n\nThis massless field has a cubic coupling to φ 2 X23 and gets mass at 1-loop since φ 2 is a broken supersymmetry field, as described in appendix A.2.\n\nFigure 25 shows the relevant part of the quiver for the second class of bi-fundamentals (i.e. those that are dualized but do not have cubic flavor couplings).\n\nThese fields and their mesons only appear in one term, so will always couple in the same way. Taking as an example h R 2 a1 Ỹ12 Q2a = R 1\n\nR 2 Ỹ12 (µ + φ 3 ; φ 4 ) = µR 1 Ỹ12 + R 1 φ 3 Ỹ12 + R 2 φ 4 Ỹ12 (C.11) 37 a b 1 2 3 d c 4 Ỹ12 Ỹ23 Z23 Qb2 Q2a X ab R 1 R 2 S 1 S 2 S 3 M 1 , . . . , M 6 Q c4 Q 4d\n\nFigure 25: Relevant part of dual quiver for second class of bi-fundamentals.\n\nThis shows that R 1 and Ỹ12 get tree-level masses and R 2 gets a mass at 1-loop since it couples to the broken supersymmetry field φ 4 . The only remaining fields are flavors like Q c4 , Q 4d , which do not transform in a gauge group adjacent to the dualized node (i.e. not adjacent in the quiver loop corresponding to the superpotential term used to introduce flavors). These are directly massive from the tree-level W mass term. So, as stated, all fields except those that appear in the original superpotential (i.e.\n\nmesons with gauge indices and bi-fundamentals which are not dualized) get masses either at tree-level or at one-loop. So we only need to check the dualized original superpotential to see if we have a metastable vacua."
}
] | |
arxiv:0704.0175 | 0704.0175 | 1 | 10.1088/1475-7516/2007/10/004 | ddc73a689f5b1abc0a2024a40159971b1b2edd52d945335f605bb7ffa6400bf3 | Solar System Constraints on Gauss-Bonnet Mediated Dark Energy | Although the Gauss-Bonnet term is a topological invariant for general
relativity, it couples naturally to a quintessence scalar field, modifying
gravity at solar system scales. We determine the solar system constraints due
to this term by evaluating the post-Newtonian metric for a distributional
source. We find a mass dependent, 1/r^7 correction to the Newtonian potential,
and also deviations from the Einstein gravity prediction for light-bending. We
constrain the parameters of the theory using planetary orbits, the Cassini
spacecraft data, and a laboratory test of Newton's law, always finding
extremely tight bounds on the energy associated to the Gauss-Bonnet term. We
discuss the relevance of these constraints to late-time cosmological
acceleration. | [
"Luca Amendola",
"Christos Charmousis and Stephen C. Davis"
] | [
"astro-ph",
"gr-qc",
"hep-th"
] | astro-ph | [] | 2007-04-02 | 2026-02-26 | Although the Gauss-Bonnet term is a topological invariant for general relativity, it couples naturally to a quintessence scalar field, modifying gravity at solar system scales. We determine the solar system constraints due to this term by evaluating the post-Newtonian metric for a distributional source. We find a mass dependent, 1/r 7 correction to the Newtonian potential, and also deviations from the Einstein gravity prediction for light-bending. We constrain the parameters of the theory using planetary orbits, the Cassini spacecraft data, and a laboratory test of Newton's law, always finding extremely tight bounds on the energy associated to the Gauss-Bonnet term. We discuss the relevance of these constraints to late-time cosmological acceleration.
Supernovae measurements [1] indicate that our universe has entered a phase of latetime acceleration. One can question the magnitude of the acceleration and its equation of state, although given the concordance of different cosmological data, acceleration seems a robust observation (although see [2] for criticisms). Commonly, in order to explain this phenomenon one postulates the existence of a minute cosmological constant Λ ∼ 10 -12 eV 4 . This fits the data well and is the most economic explanation in terms of parameter(s). However such a tiny value is extremely unnatural from a particle physics point of view [3] . Given the theoretical problems of a cosmological constant, one hopes that the intriguing phenomenon of acceleration is a window to new observable physics. This could be in the matter sector, in the form of dark energy [4, 5] , or in the gravity sector, in the form of a large distance modification of Einstein gravity [6, 7, 8, 9] . Scalar field driven dark energy, or quintessence [4] is one of the most popular of the former possibilities. However these models have important drawbacks, such as the fine tuning of the mass of the quintessence field (which has to be smaller than the actual Hubble parameter, H 0 ∼ 10 -33 eV), and stability of radiative corrections from the matter sector [10] (see however [11] ). Modified gravity models have the potential to avoid these problems, and can give a more profound explanation of the acceleration. However, these are far more difficult to obtain since Einstein's theory is experimentally well established [12] , and the required modifications happen at very low (classical) energy scales which are (supposed to be) theoretically well understood. Furthermore, many apparently successful modified gravity models suffer from instabilities or are incompatible with gravity experiments. For example the self-accelerating solutions of DGP [8] suffer from perturbative ghosts [13] , and f (R) gravity theories [9] can conflict with solar system measurements and present instabilities [14] .
In this paper we will consider observational constraints on a class of gravity theories which feature both dark energy and modified gravity. Specifically, we will examine solar system and laboratory constraints resulting from the response of gravity to a quintessence-like scalar field, which couples to quadratic order curvature terms such as the Gauss-Bonnet term. Such couplings arise naturally [15] , and modify gravity at local and cosmological scales [15, 16] . Although the Gauss-Bonnet invariant shares many of the properties of the Einstein-Hilbert term, the resulting theory can have substantially different features, see for example [17] . It is a promising candidate for a consistent explanation of cosmological acceleration, but as we will show, can also produce undesirable effects at solar system scales.
In particular, we will determine constraints from deviations in planetary orbits around the sun, the frequency shift of signals from the Cassini probe, and tabletop experiments. In contrast to some previous efforts in the field [18] , we will not suppose a priori the order of the Gauss-Bonnet correction or the scalar field potential. Instead we will calculate leading-order gravity corrections for each of them, and obtain constraints on the relevant coupling constants (checking they fall within the validity of our perturbative expansion). Hence our analysis will apply for large couplings, which as we will see, are in accord with Gauss-Bonnet driven effective dark energy models. In this way we will show such models generally produce significant deviations from general relativity at local scales. We also include higher-order scalar field kinetic terms, although for the solutions we consider they turn out to be subdominant.
In the next section we will present the theory in question and calculate the corrections to a post-Newtonian metric for a distributional point mass source. In section 3, we derive constraints from planetary motion, the Cassini probe, and a tabletop experiment. For the Cassini constraint, we have to explicitly derive the predicted frequency shift for our theory, as it does not fall within the usual Parametrised Post-Newtonian (PPN) analysis. We discuss the implications of our results in section 4.
We will consider a theory with the second-order gravitational Lagrangian
L = √ -g R -(∇φ) 2 -2V (φ) + α ξ 1 (φ)L GB + ξ 2 (φ)G µν ∇ µ φ∇ ν φ + ξ 3 (φ)(∇φ) 2 ∇ 2 φ + ξ 4 (φ)(∇φ) 4 , (1)
which includes the Gauss-Bonnet term L GB = R 2 -4R µν R µν + R µνρσ R µνρσ . Note for example that such a Lagrangian with given ξ's arises naturally from higher dimensional compactification of a pure gravitational theory [15] . On its own, in four dimensions, the Gauss-Bonnet term does not contribute to the gravitational field equations. However we emphasise that when coupled to a scalar field (as above), it has a non-trivial effect. Throughout this paper we take the dimensionless couplings ξ i and their derivatives to be O (1) . There is then only one scale for the higher curvature part of the action, given by the parameter α, with dimensions of length squared. Similarly we assume that all derivatives of the potential V are of O(V ), which in our conventions has dimensions of inverse length squared. These two simplifying assumptions will hold for a wide range of theories, including those in which ξ i and V arise from a toroidal compactification of a higher dimensional space [15] . On the other hand it is perfectly conceivable that they do not apply for our universe, in which case the corresponding gravity theories will not be covered by the analysis in this article.
Using the post-Newtonian limit, the metric for the solar system can be written [12]
ds 2 = -(1 -h 00 )(c dt) 2 + (δ ij + h ij )dx i dx j + O(ǫ 3/2 ) . (2)
with h 00 , h ij = O(ǫ). The dimensionless parameter ǫ is the typical gravitational strength, given by ǫ = Gm/(rc 2 ) where m is the typical mass scale and r the typical length scale (see below). For the solar system ǫ is at most 10 -5 , while for cosmology, or close to the event horizon of a black hole, it is of order unity. The scale of planetary velocities v, is of order ǫ 1/2 , and so the h 0i components of the metric are O(ǫ 3/2 ), as are ∂ t h 00 and
∂ t h ij .
In what follows, we will take φ = φ 0 + O(ǫ). For the linearised approximation we are using, we can adopt a post-Newtonian gauge in which the off-diagonal components of h ij are zero. We can then write
h ij = -2Ψδ ij , h 00 = -2Φ , (3)
and so c 2 Φ is the Newtonian potential.
In this paper we will consider the leading-order corrections in ǫ without assumptions on the magnitude of V and α. To leading order in ǫ, the Einstein equations take the nice compact form,
∆Φ = 4πG 0 c 2 ρ m -V -2αξ ′ 1 D(Φ + Ψ, φ) + O(ǫ 2 , αǫ 3 /r 2 , V ǫr 2 ) (4) ∆Ψ = 4πG 0 c 2 ρ m + V 2 -α 2ξ ′ 1 D(Ψ, φ) + ξ 2 4 D(φ, φ) + O(ǫ 2 , αǫ 3 /r 2 , V ǫr 2 ) ( 5
)
where primes denote ∂/∂φ, and V , ξ ′ 1 , etc. are evaluated at φ = φ 0 . The matter energy density in the solar system is ρ m , and G 0 is its bare coupling strength (without quadratic gravity corrections). Other components of the energy-momentum tensor are higher order in ǫ. The scalar field equation is
∆φ = V ′ -α [4ξ ′ 1 D(Φ, Ψ) + ξ 2 D(Φ -Ψ, φ) + ξ 3 D(φ, φ)] + O(ǫ 2 , αǫ 3 /r 2 , V ǫr 2 ) . ( 6
)
We have defined the operators
∆X = i X ,ii , D(X, Y ) = i,j X ,ij Y ,ij -∆X∆Y . (7)
with i, j = 1, 2, 3 where to leading order, the Gauss-Bonnet term is then L GB = 8D(Φ, Ψ). For standard Einstein gravity (V = α = 0), the solution of the above equations is
Φ = Ψ = -U m , φ = φ 0 , (8)
where
U m = 4πG 0 c 2 d 3 x ′ ρ m ( x ′ , t) | x -x ′ | . ( 9
)
We will now study solutions which are close to the post-Newtonian limit of general relativity, and take
Φ = -U m + δΦ , Ψ = -U m + δΨ , φ = φ 0 + δφ , (10)
where δφ, etc. are the leading-order α-and V -dependent corrections.
Note that the Laplacian carries a distribution and therefore we have to be careful with the implementation of the D operator. We see that δφ is O(V, αǫ 2 ), and so, to leading order, we have
∆ δφ = V ′ -4αξ ′ 1 D(U m , U m ) . (11)
Having calculated δφ, we obtain
∆ δΦ = -V + 4αξ ′ 1 D(U m , δφ) (12)
∆ δΨ = V 2 + 2αξ ′ 1 D(U m , δφ) . (13)
In the case of a spherical distributional source ρ m = mδ (3) (x),
U m = G 0 m c 2 r . (14)
In accordance to our estimations for ǫ the solar system Newtonian potentials are U m 10 -5 , and the velocities satisfy v 2 U m . For planets we have U m 10 -7 (with the maximum attained by Mercury).
With the aid of the relation
D(r -n , r -m ) = 2nm n + m + 2 ∆r -(n+m+2) (15)
the above expressions evaluate, at leading order, to
φ = φ 0 + r 2 V ′ 6 -2ξ ′ 1 α(G 0 m) 2 c 4 r 4 (16) Φ = - G 0 m c 2 r 1 + 8ξ ′ 1 3 αV ′ - r 2 V 6 - 64(ξ ′ 1 ) 2 7 α 2 (G 0 m) 3 c 6 r 7 (17) Ψ = - G 0 m c 2 r 1 + 4ξ ′ 1 3 αV ′ + r 2 V 12 - 32(ξ ′ 1 ) 2 7 α 2 (G 0 m) 3 c 6 r 7 . ( 18
)
We find that there are now non-standard corrections to the Newtonian potential which do not follow the usual parametrised expansion, in agreement with [19] , but not [18] (which uses different assumptions on the form of the theory). First of all note that the Gauss-Bonnet coupling α couples to the running of the dark energy potential V ′ , giving a 1/r contribution to the modified Newtonian potential (17) . We absorb this into the gravitational coupling,
G = G 0 1 + 8ξ ′ 1 3 αV ′ . ( 19
)
The corresponding term in (18) gives a constant contribution to the effective γ PPN parameter. The r 2 V terms in ( 17 ), ( 18 ) are typical of a theory with a cosmological constant, whereas the final, 1/r 7 terms are the leading pure Gauss-Bonnet correction, which is enhanced at small distances. If we take the usual expression for the PPN parameter γ = Ψ/Φ, we see that it is r dependent. In using the Cassini constraint on γ we must be careful to calculate the frequency shift from scratch.
For the above derivation we have assumed δφ ≪ U m , which implies V ≪ U m /r 2 and α ≪ r 2 /U m . This will hold in the solar system if
V ≪ 10 -36 m -2
and α ≪ 10 23 m 2 (everywhere) 10 29 m 2 (planets only) (20) in geometrised units. Note that strictly speaking there is also a lower bound on our coupling constants, if the above analysis is to be valid. Indeed, if we were to find corrections of order ǫ 2 ∼ 10 -14 , then it would imply that higher-order corrections from general relativity were just as important as the ones appearing in (17) , (18) .
Deviations from the usual Newtonian potential will affect planetary motions, which provides a way of bounding them. This idea has been used to bound dark matter in the solar system [20] , and also the value of the cosmological constant [21] . We will apply the same arguments to our theory. From the above gravitational potential (17) , we obtain the Newtonian acceleration
g acc (r) = -c 2 dΦ dr = - Gm r 2 1 - V r 3 3r g - 64(αξ ′ 1 ) 2 r 2 g r 6 ≡ - Gm eff r 2 (21)
where r g ≡ Gm/c 2 is gravitational radius of the mass m. The above expression gives the effective mass m eff felt by a body at distance r. If the test body is a planet with semi-major axis a, we can use this formula at r ≈ a. Its mean motion n ≡ Gm/a 3 will then be changed by δn = (n/2)(δm eff /m). By evaluating the statistical errors of the mean motions of the planets, δn = -(3n/2)δa/a, we can derive a bound on δm eff and hence our deviations from general relativity 1 3
δm eff m = - V a 3 9r g - 64(αξ ′ 1 ) 2 r 2 g 3a 6 < δa a . ( 22
)
The values of a for the planets are determined using Kepler's third law, with a constant sun's mass m ⊙ . Constraints on δΦ then follow from the errors δa, in the measure of a. These can be found in [22] , and are also listed in the appendix for convenience. Given their different r-dependence, the two corrections to δm eff are unlikely to cancel. We will therefore bound them separately, giving constraints on α and V .
The strongest bound on the combination ξ ′ 1 α comes from Mercury, with δa a 1.8 × 10 -12 .
Neglecting the cosmological constant term, and using a ≈ 5.8 × 10 7 km and r g ≈ 1.5 km, we find
|ξ ′ 1 α| (3a 5 δa) 1/2 8r g ≈ 3.8 × 10 22 m 2 . ( 24
)
We see that this is within range of validity (20) for our perturbative treatment of gravity.
In cosmology, the density fraction corresponding to the Gauss-Bonnet term is [15] Ω
GB = 4ξ ′ 1 αH dφ dt . ( 25
)
If this is to play the role of dark energy in our universe, it needs to take, along with the contribution of the potential, a value around 0.7 at cosmological length scales (and for redshift z ∼ 1).
If we wish to accurately apply the bound on α (24) to cosmological scales, details of the dynamical evolution of φ will be required. These will depend on the form of V and the ξ i , and are expected to involve complex numerical analysis, all of which is beyond the scope of this work. Here we will instead assume that the cosmological value of φ is also φ 0 , which, while crude, will allow us to estimate the significance of the above result. Given the hierarchy between cosmological and solar system scales it is natural to question this assumption but we will make it here, and discuss it in more detail in the concluding section.
Making the further, and less controversial, assumption that dφ/dt ≈ H, we obtain a very stringent constraint on Ω GB :
|Ω GB | ≈ 4|ξ ′ 1 α|H 2 0 8.8 × 10 -30 . ( 26
)
Hence we see that solar system constraints on Gauss-Bonnet fraction of the dark energy are potentially very significant, despite the fact that the Gauss-bonnet term is quadratic in curvature.
Since we are assuming that all the ξ i are of the same order, the above bound also applies to the dark energy fractions arising from the final three terms in (1) . Clearly there are effective dark energy models for which the analysis leading to the above bound (26) does not apply. However any successful model will require a huge variation of ξ 1 between local and cosmological scales, or a very substantial violation of one of our other assumptions.
For comparison, we apply similar arguments to obtain a constraint on the potential. The strongest bound comes from the motion of Mars [21] , and is
|V | 9r g δa a 4 ♂ ≈ 1.2 × 10 -40 m -2 . ( 27
)
This suggests Ω V = V /(3H 2 0 ) 7.3×10 11 , which is vastly weaker than the corresponding cosmological constraint (Ω V 1). Hence planetary orbits tell us little of significance about dark energy arising from a potential, in sharp contrast to the situation for Gauss-Bonnet dark energy.
The most stringent constraint on the PPN parameter γ was obtained from the Cassini spacecraft in 2002 while on its way to Saturn. The signals between the spacecraft and the earth pass close to the sun, whose gravitational field produces a time delay. The smallest value of r on the light ray's path defines the impact parameter b. A small impact parameter maximises the light delay. During that year's superior solar conjunction the spacecraft was r e = 8.43 AU = 1.26 × 10 12 m away from the sun, and the impact parameter dropped as low as b min = 1.6R ⊙ . A PPN analysis of the system produced the strong constraint
δγ ≡ γ -1 = (2.1 ± 2.3) × 10 -5 . ( 28
)
Given that our theory is not PPN we have to undertake the calculation from scratch.
The above constraint comes from considering a round trip, in which the light travels from earth, grazes the sun's 'surface', reaches the spacecraft, and then returns by the same route. We take the path of the photon to be the straight line between the earth and the spacecraft, x = (x, b, 0) with x varying from -x e to x ⊕ . For a round trip (there and back), the additional time delay for a light ray due to the gravitational field of the sun is then
c∆t = 2 x ⊕ -xe h 00 (r) + h xx (r) 2 dx = -2 x ⊕ -xe (Φ + Ψ)| r= √ x 2 +b 2 dx . ( 29
)
For the solution ( 17 ) and ( 18 ), this evaluates to
c∆t = 4r g 1 - 2αξ ′ 1 V ′ 3 ln a ⊕ r e 4b 2 + a 3 ⊕ + r 3 e 3 + b 2 (a ⊕ + r e ) V 6 + 1024(αξ ′ 1 ) 2 r 3 g b 6 , ( 30
)
where we have assumed x ⊕ ≈ a ⊕ ≫ b, and similarly for the spacecraft. Rather than directly measure ∆t, the Cassini experiment actually found the frequency shift in the signal [23]
y gr = d∆t dt ≈ d∆t db db dt . ( 31
)
The results obtained were
y gr = - 10 -5 s b db dt (2 + δγ) . ( 32
)
If gravitation were to be described by the standard PPN formalism, then δγ would be the possible deviation of the PPN parameter γ from the general relativity value of 1.
From ( 30 ) we obtain
y gr = -2 - b 2 V (a ⊕ + r e ) 12r g + 1536(αξ ′ 1 ) 2 r 2 g b 6 - 4αξ ′ 1 V ′ 3 4r g cb db dt . (33)
Requiring that the corrections are within the errors (28) of ( 32 ), implies
|ξ ′ 1 α| √ 6 δγ 96 b 3 r g 1.6 × 10 20 m 2 . ( 34
)
This suggests the dark energy bound
|Ω GB | 3.6 × 10 -32 , (35)
although obtaining this bound from solar system data requires major assumptions about the cosmological behaviour of φ, as we will point out in section 4.
The data obtained by the spacecraft were actually for a range of impact parameters b, but we have just used the most conservative value b = b min = 1.6R ⊙ . The above constraint is even stronger than (24) , which was obtained for planetary motion. This is because the experiment involved smaller r, and so the possible Gauss-Bonnet effects were larger.
Taking the above expression for y gr (33) at face value, we can also constrain the potential to be |V | 10 -22 m 2 and the cross-term |αξ ′ 1 V ′ | 10 -5 . However these are of little interest as they are much weaker than the planetary motion constraints ( 24 ), (27) , and also the former is far outside the range of validity (20) of our analysis.
Laboratory experiments can also be used to obtain bounds on deviations from Newton's law. For illustration we will consider the table-top experiment described in [24] . It consists of a 60 cm copper bar, suspended at its midpoint by a tungsten wire. Two 7.3 kg masses are placed on carts far (105 cm) from the bar, and another mass of m ≈ 43 g is placed near (5 cm) to the side of bar. Moving the masses to the opposite sides of the bar changes in the torque felt by it. The experiment measures the torques N 105 and -N 5 produced respectively by the far and near masses. The masses and distances are chosen so that the two torques roughly cancel. The ratio R = N 105 /N 5 is then determined, and compared with the theoretical value. The deviation from the Newtonian result is
δ R = R expt R Newton -1 = (1.2 ± 7) × 10 -4 . (36)
In fact, to help reduce errors, additional measurements were taken. To account for the gravitational field of the carts that the far masses sit on, the experiment was repeated with only the carts and a m ′ ≈ 3 g near mass. The measured torque was then subtracted from the result for the loaded carts.
The Gauss-Bonnet corrections to the Newton potential (17) will alter the torques produced by all four masses, as well as the carts. Furthermore, since δΦ is non-linear in mass, there will be further corrections coming from cross terms. The expressions derived in section 2 are just for the gravitational field of a single mass, and so will not fully describe the above table-top experiment. However, we find that the contribution from the mass m will dominate the other corrections, and so we can get a good estimate of the Gauss-Bonnet contribution to the ratio R by just considering m.
The torque experienced by the copper bar, due to a point mass at X = (X, Y, Z) is
N = bar d 3 x ( x ∧ F ) z = ρ Cu bar d 3 x yX -xY r c 2 dΦ dr r=| X-x| , (37)
where ρ Cu is the bar's density. A full list of parameters for the experiment is given in table I of [24] . The bar's dimensions are 60 cm × 1.5 cm × 0.65 cm. Working in coordinates with the origin at the centre of the bar, the mass m = 43.58 g is at X = (24.42, -4.77, -0.03) cm. Treating m as a point mass, Newtonian gravity implies a torque of N 5 ≈ (8.2 cm 2 ) Gmρ Cu is produced. The Gauss-Bonnet correction is
δN 5 = ρ Cu 64G 3 m 3 (αξ ′ 1 ) 2 c 4 bar d 3 x yX -xY | X -x| 9 ≈ -(0.025 cm -4 ) (Gm) 3 (αξ ′ 1 ) 2 ρ Cu c 4 . ( 38
)
To be consistent with the bound (36), we require δN 5 /N 5 to be within the range of δ R . This implies
|αξ ′ 1 | (18 cm 3 ) c 2 δ 1/2 R Gm 1.3 × 10 22 m 2 , (39)
which is comparable to the planetary constraint (24) . Extrapolating it to cosmological scales gives
|Ω GB | 3.1 × 10 -30 . ( 40
)
There are of course many more recent laboratory tests of gravity, and we expect that stronger constraints can be obtained from them. Table-top experiments frequently involve multiple gravitational sources, or gravitational fields which cannot reasonably be treated as point masses. A more detailed calculation than the one presented in section 2 will then be required. For example, the gravitational field inside a sphere or cylinder will not receive corrections of the form (17) , and so any experiment involving a test mass moving in such a field requires a different analysis.
We have shown that significant constraints on Gauss-Bonnet gravity can be derived from both solar system measurements and table-top laboratory experiments (note that further constraints arise when imposing theoretical constraints like absence of superluminal or ghost modes, see [25] ). The fact that the corrections to Einstein gravity are second order in curvature suggests they will automatically be small. However this does not take into account the fact that the dimensionfull coupling of the Gauss-Bonnet term must be large if it is to have any hope of producing effective dark energy. Additional constraints will come from the perihelion precession of Mercury, although the linearised analysis we have used is inadequate to determine this, and higher-order (in ǫ) effects will need to be calculated.
Performing an extrapolation of our results to cosmological scales suggests that the density fraction Ω GB will be far too small to explain the accelerated expansion of our universe. This agrees with the conclusions of [19] . Hence if Gauss-Bonnet gravity is to be a viable dark energy candidate, one needs to find a loophole in the above arguments. This is not too difficult, and we will now turn to this question.
In particular, we have assumed no spatial or temporal evolution of the field φ between cosmological and solar system scales, even though the supernova measurements correspond to a higher redshift and a far different typical distance scale. A varying φ would of course imply that different values of ξ i , and their derivatives, would be perceived by supernovas and the planets. It is interesting to note that the size of the bound we have found ( 26 ) is of order the square of the ratio of the solar system and the cosmological horizon scales, s = (1 AU H 0 ) 2 ∼ 10 -30 . Therefore one could reasonably argue that the small number appearing in (26) could in fact be due to the hierarchy scale, s, rather than a very stringent constraint on Ω GB . This could perhaps be concretely realised with something similar to the chameleon effect [26] giving some constraint on the running of the quintessence theory. One other possibility is that the baryons (which make up the solar system) and dark matter (which is dominant at cosmological scales) have different couplings to φ [27] . Again, this would alter the relation between local and cosmological constraints.
Alternatively, it may be that our assumptions on the form of the theory should be changed. The scalar field could be coupled directly to the Einstein-Hilbert term, as in Brans-Dicke gravity. Additionally, the couplings ξ i and their derivatives could be of different orders. The same could be true of the potential. In particular, if φ were to have a significant mass, this would suppress the quadratic curvature effects, as they operate via the scalar field. This would be similar to the situation in scalar-tensor gravity with a potential, where the strong constraints on the theory can be avoided by giving the scalar a large mass (which, however, would inhibit acceleration).
Finally, the behaviour of the scalar field could be radically different. We took it to be O(ǫ), like the metric perturbations. However since our constraints are on the metric, and not φ, this need not be true. Furthermore, since the theory is quadratic, there may well be alternative solutions of the field equations, and not just the one we studied.
Hence to obtain a viable Gauss-Bonnet dark energy model, which is compatible with solar system constraints, at least one of the above assumptions must be broken. For many of the above ideas the higher-order scalar kinetic terms will play a significant role. This then opens up the possibility that the higher-gravity corrections will cancel each other, further weakening the constraints. We hope to address some of these issues in the near future. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Although the Gauss-Bonnet term is a topological invariant for general relativity, it couples naturally to a quintessence scalar field, modifying gravity at solar system scales. We determine the solar system constraints due to this term by evaluating the post-Newtonian metric for a distributional source. We find a mass dependent, 1/r 7 correction to the Newtonian potential, and also deviations from the Einstein gravity prediction for light-bending. We constrain the parameters of the theory using planetary orbits, the Cassini spacecraft data, and a laboratory test of Newton's law, always finding extremely tight bounds on the energy associated to the Gauss-Bonnet term. We discuss the relevance of these constraints to late-time cosmological acceleration."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Supernovae measurements [1] indicate that our universe has entered a phase of latetime acceleration. One can question the magnitude of the acceleration and its equation of state, although given the concordance of different cosmological data, acceleration seems a robust observation (although see [2] for criticisms). Commonly, in order to explain this phenomenon one postulates the existence of a minute cosmological constant Λ ∼ 10 -12 eV 4 . This fits the data well and is the most economic explanation in terms of parameter(s). However such a tiny value is extremely unnatural from a particle physics point of view [3] . Given the theoretical problems of a cosmological constant, one hopes that the intriguing phenomenon of acceleration is a window to new observable physics. This could be in the matter sector, in the form of dark energy [4, 5] , or in the gravity sector, in the form of a large distance modification of Einstein gravity [6, 7, 8, 9] . Scalar field driven dark energy, or quintessence [4] is one of the most popular of the former possibilities. However these models have important drawbacks, such as the fine tuning of the mass of the quintessence field (which has to be smaller than the actual Hubble parameter, H 0 ∼ 10 -33 eV), and stability of radiative corrections from the matter sector [10] (see however [11] ). Modified gravity models have the potential to avoid these problems, and can give a more profound explanation of the acceleration. However, these are far more difficult to obtain since Einstein's theory is experimentally well established [12] , and the required modifications happen at very low (classical) energy scales which are (supposed to be) theoretically well understood. Furthermore, many apparently successful modified gravity models suffer from instabilities or are incompatible with gravity experiments. For example the self-accelerating solutions of DGP [8] suffer from perturbative ghosts [13] , and f (R) gravity theories [9] can conflict with solar system measurements and present instabilities [14] .\n\nIn this paper we will consider observational constraints on a class of gravity theories which feature both dark energy and modified gravity. Specifically, we will examine solar system and laboratory constraints resulting from the response of gravity to a quintessence-like scalar field, which couples to quadratic order curvature terms such as the Gauss-Bonnet term. Such couplings arise naturally [15] , and modify gravity at local and cosmological scales [15, 16] . Although the Gauss-Bonnet invariant shares many of the properties of the Einstein-Hilbert term, the resulting theory can have substantially different features, see for example [17] . It is a promising candidate for a consistent explanation of cosmological acceleration, but as we will show, can also produce undesirable effects at solar system scales.\n\nIn particular, we will determine constraints from deviations in planetary orbits around the sun, the frequency shift of signals from the Cassini probe, and tabletop experiments. In contrast to some previous efforts in the field [18] , we will not suppose a priori the order of the Gauss-Bonnet correction or the scalar field potential. Instead we will calculate leading-order gravity corrections for each of them, and obtain constraints on the relevant coupling constants (checking they fall within the validity of our perturbative expansion). Hence our analysis will apply for large couplings, which as we will see, are in accord with Gauss-Bonnet driven effective dark energy models. In this way we will show such models generally produce significant deviations from general relativity at local scales. We also include higher-order scalar field kinetic terms, although for the solutions we consider they turn out to be subdominant.\n\nIn the next section we will present the theory in question and calculate the corrections to a post-Newtonian metric for a distributional point mass source. In section 3, we derive constraints from planetary motion, the Cassini probe, and a tabletop experiment. For the Cassini constraint, we have to explicitly derive the predicted frequency shift for our theory, as it does not fall within the usual Parametrised Post-Newtonian (PPN) analysis. We discuss the implications of our results in section 4."
},
{
"section_type": "OTHER",
"section_title": "Quadratic Curvature Gravity",
"text": "We will consider a theory with the second-order gravitational Lagrangian\n\nL = √ -g R -(∇φ) 2 -2V (φ) + α ξ 1 (φ)L GB + ξ 2 (φ)G µν ∇ µ φ∇ ν φ + ξ 3 (φ)(∇φ) 2 ∇ 2 φ + ξ 4 (φ)(∇φ) 4 , (1)\n\nwhich includes the Gauss-Bonnet term L GB = R 2 -4R µν R µν + R µνρσ R µνρσ . Note for example that such a Lagrangian with given ξ's arises naturally from higher dimensional compactification of a pure gravitational theory [15] . On its own, in four dimensions, the Gauss-Bonnet term does not contribute to the gravitational field equations. However we emphasise that when coupled to a scalar field (as above), it has a non-trivial effect. Throughout this paper we take the dimensionless couplings ξ i and their derivatives to be O (1) . There is then only one scale for the higher curvature part of the action, given by the parameter α, with dimensions of length squared. Similarly we assume that all derivatives of the potential V are of O(V ), which in our conventions has dimensions of inverse length squared. These two simplifying assumptions will hold for a wide range of theories, including those in which ξ i and V arise from a toroidal compactification of a higher dimensional space [15] . On the other hand it is perfectly conceivable that they do not apply for our universe, in which case the corresponding gravity theories will not be covered by the analysis in this article.\n\nUsing the post-Newtonian limit, the metric for the solar system can be written [12]\n\nds 2 = -(1 -h 00 )(c dt) 2 + (δ ij + h ij )dx i dx j + O(ǫ 3/2 ) . (2)\n\nwith h 00 , h ij = O(ǫ). The dimensionless parameter ǫ is the typical gravitational strength, given by ǫ = Gm/(rc 2 ) where m is the typical mass scale and r the typical length scale (see below). For the solar system ǫ is at most 10 -5 , while for cosmology, or close to the event horizon of a black hole, it is of order unity. The scale of planetary velocities v, is of order ǫ 1/2 , and so the h 0i components of the metric are O(ǫ 3/2 ), as are ∂ t h 00 and\n\n∂ t h ij .\n\nIn what follows, we will take φ = φ 0 + O(ǫ). For the linearised approximation we are using, we can adopt a post-Newtonian gauge in which the off-diagonal components of h ij are zero. We can then write\n\nh ij = -2Ψδ ij , h 00 = -2Φ , (3)\n\nand so c 2 Φ is the Newtonian potential.\n\nIn this paper we will consider the leading-order corrections in ǫ without assumptions on the magnitude of V and α. To leading order in ǫ, the Einstein equations take the nice compact form,\n\n∆Φ = 4πG 0 c 2 ρ m -V -2αξ ′ 1 D(Φ + Ψ, φ) + O(ǫ 2 , αǫ 3 /r 2 , V ǫr 2 ) (4) ∆Ψ = 4πG 0 c 2 ρ m + V 2 -α 2ξ ′ 1 D(Ψ, φ) + ξ 2 4 D(φ, φ) + O(ǫ 2 , αǫ 3 /r 2 , V ǫr 2 ) ( 5\n\n)\n\nwhere primes denote ∂/∂φ, and V , ξ ′ 1 , etc. are evaluated at φ = φ 0 . The matter energy density in the solar system is ρ m , and G 0 is its bare coupling strength (without quadratic gravity corrections). Other components of the energy-momentum tensor are higher order in ǫ. The scalar field equation is\n\n∆φ = V ′ -α [4ξ ′ 1 D(Φ, Ψ) + ξ 2 D(Φ -Ψ, φ) + ξ 3 D(φ, φ)] + O(ǫ 2 , αǫ 3 /r 2 , V ǫr 2 ) . ( 6\n\n)\n\nWe have defined the operators\n\n∆X = i X ,ii , D(X, Y ) = i,j X ,ij Y ,ij -∆X∆Y . (7)\n\nwith i, j = 1, 2, 3 where to leading order, the Gauss-Bonnet term is then L GB = 8D(Φ, Ψ). For standard Einstein gravity (V = α = 0), the solution of the above equations is\n\nΦ = Ψ = -U m , φ = φ 0 , (8)\n\nwhere\n\nU m = 4πG 0 c 2 d 3 x ′ ρ m ( x ′ , t) | x -x ′ | . ( 9\n\n)\n\nWe will now study solutions which are close to the post-Newtonian limit of general relativity, and take\n\nΦ = -U m + δΦ , Ψ = -U m + δΨ , φ = φ 0 + δφ , (10)\n\nwhere δφ, etc. are the leading-order α-and V -dependent corrections.\n\nNote that the Laplacian carries a distribution and therefore we have to be careful with the implementation of the D operator. We see that δφ is O(V, αǫ 2 ), and so, to leading order, we have\n\n∆ δφ = V ′ -4αξ ′ 1 D(U m , U m ) . (11)\n\nHaving calculated δφ, we obtain\n\n∆ δΦ = -V + 4αξ ′ 1 D(U m , δφ) (12)\n\n∆ δΨ = V 2 + 2αξ ′ 1 D(U m , δφ) . (13)\n\nIn the case of a spherical distributional source ρ m = mδ (3) (x),\n\nU m = G 0 m c 2 r . (14)\n\nIn accordance to our estimations for ǫ the solar system Newtonian potentials are U m 10 -5 , and the velocities satisfy v 2 U m . For planets we have U m 10 -7 (with the maximum attained by Mercury).\n\nWith the aid of the relation\n\nD(r -n , r -m ) = 2nm n + m + 2 ∆r -(n+m+2) (15)\n\nthe above expressions evaluate, at leading order, to\n\nφ = φ 0 + r 2 V ′ 6 -2ξ ′ 1 α(G 0 m) 2 c 4 r 4 (16) Φ = - G 0 m c 2 r 1 + 8ξ ′ 1 3 αV ′ - r 2 V 6 - 64(ξ ′ 1 ) 2 7 α 2 (G 0 m) 3 c 6 r 7 (17) Ψ = - G 0 m c 2 r 1 + 4ξ ′ 1 3 αV ′ + r 2 V 12 - 32(ξ ′ 1 ) 2 7 α 2 (G 0 m) 3 c 6 r 7 . ( 18\n\n)\n\nWe find that there are now non-standard corrections to the Newtonian potential which do not follow the usual parametrised expansion, in agreement with [19] , but not [18] (which uses different assumptions on the form of the theory). First of all note that the Gauss-Bonnet coupling α couples to the running of the dark energy potential V ′ , giving a 1/r contribution to the modified Newtonian potential (17) . We absorb this into the gravitational coupling,\n\nG = G 0 1 + 8ξ ′ 1 3 αV ′ . ( 19\n\n)\n\nThe corresponding term in (18) gives a constant contribution to the effective γ PPN parameter. The r 2 V terms in ( 17 ), ( 18 ) are typical of a theory with a cosmological constant, whereas the final, 1/r 7 terms are the leading pure Gauss-Bonnet correction, which is enhanced at small distances. If we take the usual expression for the PPN parameter γ = Ψ/Φ, we see that it is r dependent. In using the Cassini constraint on γ we must be careful to calculate the frequency shift from scratch.\n\nFor the above derivation we have assumed δφ ≪ U m , which implies V ≪ U m /r 2 and α ≪ r 2 /U m . This will hold in the solar system if\n\nV ≪ 10 -36 m -2\n\nand α ≪ 10 23 m 2 (everywhere) 10 29 m 2 (planets only) (20) in geometrised units. Note that strictly speaking there is also a lower bound on our coupling constants, if the above analysis is to be valid. Indeed, if we were to find corrections of order ǫ 2 ∼ 10 -14 , then it would imply that higher-order corrections from general relativity were just as important as the ones appearing in (17) , (18) ."
},
{
"section_type": "OTHER",
"section_title": "Planetary motion",
"text": "Deviations from the usual Newtonian potential will affect planetary motions, which provides a way of bounding them. This idea has been used to bound dark matter in the solar system [20] , and also the value of the cosmological constant [21] . We will apply the same arguments to our theory. From the above gravitational potential (17) , we obtain the Newtonian acceleration\n\ng acc (r) = -c 2 dΦ dr = - Gm r 2 1 - V r 3 3r g - 64(αξ ′ 1 ) 2 r 2 g r 6 ≡ - Gm eff r 2 (21)\n\nwhere r g ≡ Gm/c 2 is gravitational radius of the mass m. The above expression gives the effective mass m eff felt by a body at distance r. If the test body is a planet with semi-major axis a, we can use this formula at r ≈ a. Its mean motion n ≡ Gm/a 3 will then be changed by δn = (n/2)(δm eff /m). By evaluating the statistical errors of the mean motions of the planets, δn = -(3n/2)δa/a, we can derive a bound on δm eff and hence our deviations from general relativity 1 3\n\nδm eff m = - V a 3 9r g - 64(αξ ′ 1 ) 2 r 2 g 3a 6 < δa a . ( 22\n\n)\n\nThe values of a for the planets are determined using Kepler's third law, with a constant sun's mass m ⊙ . Constraints on δΦ then follow from the errors δa, in the measure of a. These can be found in [22] , and are also listed in the appendix for convenience. Given their different r-dependence, the two corrections to δm eff are unlikely to cancel. We will therefore bound them separately, giving constraints on α and V .\n\nThe strongest bound on the combination ξ ′ 1 α comes from Mercury, with δa a 1.8 × 10 -12 .\n\nNeglecting the cosmological constant term, and using a ≈ 5.8 × 10 7 km and r g ≈ 1.5 km, we find\n\n|ξ ′ 1 α| (3a 5 δa) 1/2 8r g ≈ 3.8 × 10 22 m 2 . ( 24\n\n)\n\nWe see that this is within range of validity (20) for our perturbative treatment of gravity.\n\nIn cosmology, the density fraction corresponding to the Gauss-Bonnet term is [15] Ω\n\nGB = 4ξ ′ 1 αH dφ dt . ( 25\n\n)\n\nIf this is to play the role of dark energy in our universe, it needs to take, along with the contribution of the potential, a value around 0.7 at cosmological length scales (and for redshift z ∼ 1).\n\nIf we wish to accurately apply the bound on α (24) to cosmological scales, details of the dynamical evolution of φ will be required. These will depend on the form of V and the ξ i , and are expected to involve complex numerical analysis, all of which is beyond the scope of this work. Here we will instead assume that the cosmological value of φ is also φ 0 , which, while crude, will allow us to estimate the significance of the above result. Given the hierarchy between cosmological and solar system scales it is natural to question this assumption but we will make it here, and discuss it in more detail in the concluding section.\n\nMaking the further, and less controversial, assumption that dφ/dt ≈ H, we obtain a very stringent constraint on Ω GB :\n\n|Ω GB | ≈ 4|ξ ′ 1 α|H 2 0 8.8 × 10 -30 . ( 26\n\n)\n\nHence we see that solar system constraints on Gauss-Bonnet fraction of the dark energy are potentially very significant, despite the fact that the Gauss-bonnet term is quadratic in curvature.\n\nSince we are assuming that all the ξ i are of the same order, the above bound also applies to the dark energy fractions arising from the final three terms in (1) . Clearly there are effective dark energy models for which the analysis leading to the above bound (26) does not apply. However any successful model will require a huge variation of ξ 1 between local and cosmological scales, or a very substantial violation of one of our other assumptions.\n\nFor comparison, we apply similar arguments to obtain a constraint on the potential. The strongest bound comes from the motion of Mars [21] , and is\n\n|V | 9r g δa a 4 ♂ ≈ 1.2 × 10 -40 m -2 . ( 27\n\n)\n\nThis suggests Ω V = V /(3H 2 0 ) 7.3×10 11 , which is vastly weaker than the corresponding cosmological constraint (Ω V 1). Hence planetary orbits tell us little of significance about dark energy arising from a potential, in sharp contrast to the situation for Gauss-Bonnet dark energy."
},
{
"section_type": "OTHER",
"section_title": "Cassini spacecraft",
"text": "The most stringent constraint on the PPN parameter γ was obtained from the Cassini spacecraft in 2002 while on its way to Saturn. The signals between the spacecraft and the earth pass close to the sun, whose gravitational field produces a time delay. The smallest value of r on the light ray's path defines the impact parameter b. A small impact parameter maximises the light delay. During that year's superior solar conjunction the spacecraft was r e = 8.43 AU = 1.26 × 10 12 m away from the sun, and the impact parameter dropped as low as b min = 1.6R ⊙ . A PPN analysis of the system produced the strong constraint\n\nδγ ≡ γ -1 = (2.1 ± 2.3) × 10 -5 . ( 28\n\n)\n\nGiven that our theory is not PPN we have to undertake the calculation from scratch.\n\nThe above constraint comes from considering a round trip, in which the light travels from earth, grazes the sun's 'surface', reaches the spacecraft, and then returns by the same route. We take the path of the photon to be the straight line between the earth and the spacecraft, x = (x, b, 0) with x varying from -x e to x ⊕ . For a round trip (there and back), the additional time delay for a light ray due to the gravitational field of the sun is then\n\nc∆t = 2 x ⊕ -xe h 00 (r) + h xx (r) 2 dx = -2 x ⊕ -xe (Φ + Ψ)| r= √ x 2 +b 2 dx . ( 29\n\n)\n\nFor the solution ( 17 ) and ( 18 ), this evaluates to\n\nc∆t = 4r g 1 - 2αξ ′ 1 V ′ 3 ln a ⊕ r e 4b 2 + a 3 ⊕ + r 3 e 3 + b 2 (a ⊕ + r e ) V 6 + 1024(αξ ′ 1 ) 2 r 3 g b 6 , ( 30\n\n)\n\nwhere we have assumed x ⊕ ≈ a ⊕ ≫ b, and similarly for the spacecraft. Rather than directly measure ∆t, the Cassini experiment actually found the frequency shift in the signal [23]\n\ny gr = d∆t dt ≈ d∆t db db dt . ( 31\n\n)\n\nThe results obtained were\n\ny gr = - 10 -5 s b db dt (2 + δγ) . ( 32\n\n)\n\nIf gravitation were to be described by the standard PPN formalism, then δγ would be the possible deviation of the PPN parameter γ from the general relativity value of 1.\n\nFrom ( 30 ) we obtain\n\ny gr = -2 - b 2 V (a ⊕ + r e ) 12r g + 1536(αξ ′ 1 ) 2 r 2 g b 6 - 4αξ ′ 1 V ′ 3 4r g cb db dt . (33)\n\nRequiring that the corrections are within the errors (28) of ( 32 ), implies\n\n|ξ ′ 1 α| √ 6 δγ 96 b 3 r g 1.6 × 10 20 m 2 . ( 34\n\n)\n\nThis suggests the dark energy bound\n\n|Ω GB | 3.6 × 10 -32 , (35)\n\nalthough obtaining this bound from solar system data requires major assumptions about the cosmological behaviour of φ, as we will point out in section 4.\n\nThe data obtained by the spacecraft were actually for a range of impact parameters b, but we have just used the most conservative value b = b min = 1.6R ⊙ . The above constraint is even stronger than (24) , which was obtained for planetary motion. This is because the experiment involved smaller r, and so the possible Gauss-Bonnet effects were larger.\n\nTaking the above expression for y gr (33) at face value, we can also constrain the potential to be |V | 10 -22 m 2 and the cross-term |αξ ′ 1 V ′ | 10 -5 . However these are of little interest as they are much weaker than the planetary motion constraints ( 24 ), (27) , and also the former is far outside the range of validity (20) of our analysis."
},
{
"section_type": "METHOD",
"section_title": "A table-top experiment",
"text": "Laboratory experiments can also be used to obtain bounds on deviations from Newton's law. For illustration we will consider the table-top experiment described in [24] . It consists of a 60 cm copper bar, suspended at its midpoint by a tungsten wire. Two 7.3 kg masses are placed on carts far (105 cm) from the bar, and another mass of m ≈ 43 g is placed near (5 cm) to the side of bar. Moving the masses to the opposite sides of the bar changes in the torque felt by it. The experiment measures the torques N 105 and -N 5 produced respectively by the far and near masses. The masses and distances are chosen so that the two torques roughly cancel. The ratio R = N 105 /N 5 is then determined, and compared with the theoretical value. The deviation from the Newtonian result is\n\nδ R = R expt R Newton -1 = (1.2 ± 7) × 10 -4 . (36)\n\nIn fact, to help reduce errors, additional measurements were taken. To account for the gravitational field of the carts that the far masses sit on, the experiment was repeated with only the carts and a m ′ ≈ 3 g near mass. The measured torque was then subtracted from the result for the loaded carts.\n\nThe Gauss-Bonnet corrections to the Newton potential (17) will alter the torques produced by all four masses, as well as the carts. Furthermore, since δΦ is non-linear in mass, there will be further corrections coming from cross terms. The expressions derived in section 2 are just for the gravitational field of a single mass, and so will not fully describe the above table-top experiment. However, we find that the contribution from the mass m will dominate the other corrections, and so we can get a good estimate of the Gauss-Bonnet contribution to the ratio R by just considering m.\n\nThe torque experienced by the copper bar, due to a point mass at X = (X, Y, Z) is\n\nN = bar d 3 x ( x ∧ F ) z = ρ Cu bar d 3 x yX -xY r c 2 dΦ dr r=| X-x| , (37)\n\nwhere ρ Cu is the bar's density. A full list of parameters for the experiment is given in table I of [24] . The bar's dimensions are 60 cm × 1.5 cm × 0.65 cm. Working in coordinates with the origin at the centre of the bar, the mass m = 43.58 g is at X = (24.42, -4.77, -0.03) cm. Treating m as a point mass, Newtonian gravity implies a torque of N 5 ≈ (8.2 cm 2 ) Gmρ Cu is produced. The Gauss-Bonnet correction is\n\nδN 5 = ρ Cu 64G 3 m 3 (αξ ′ 1 ) 2 c 4 bar d 3 x yX -xY | X -x| 9 ≈ -(0.025 cm -4 ) (Gm) 3 (αξ ′ 1 ) 2 ρ Cu c 4 . ( 38\n\n)\n\nTo be consistent with the bound (36), we require δN 5 /N 5 to be within the range of δ R . This implies\n\n|αξ ′ 1 | (18 cm 3 ) c 2 δ 1/2 R Gm 1.3 × 10 22 m 2 , (39)\n\nwhich is comparable to the planetary constraint (24) . Extrapolating it to cosmological scales gives\n\n|Ω GB | 3.1 × 10 -30 . ( 40\n\n)\n\nThere are of course many more recent laboratory tests of gravity, and we expect that stronger constraints can be obtained from them. Table-top experiments frequently involve multiple gravitational sources, or gravitational fields which cannot reasonably be treated as point masses. A more detailed calculation than the one presented in section 2 will then be required. For example, the gravitational field inside a sphere or cylinder will not receive corrections of the form (17) , and so any experiment involving a test mass moving in such a field requires a different analysis."
},
{
"section_type": "DISCUSSION",
"section_title": "Discussion",
"text": "We have shown that significant constraints on Gauss-Bonnet gravity can be derived from both solar system measurements and table-top laboratory experiments (note that further constraints arise when imposing theoretical constraints like absence of superluminal or ghost modes, see [25] ). The fact that the corrections to Einstein gravity are second order in curvature suggests they will automatically be small. However this does not take into account the fact that the dimensionfull coupling of the Gauss-Bonnet term must be large if it is to have any hope of producing effective dark energy. Additional constraints will come from the perihelion precession of Mercury, although the linearised analysis we have used is inadequate to determine this, and higher-order (in ǫ) effects will need to be calculated.\n\nPerforming an extrapolation of our results to cosmological scales suggests that the density fraction Ω GB will be far too small to explain the accelerated expansion of our universe. This agrees with the conclusions of [19] . Hence if Gauss-Bonnet gravity is to be a viable dark energy candidate, one needs to find a loophole in the above arguments. This is not too difficult, and we will now turn to this question.\n\nIn particular, we have assumed no spatial or temporal evolution of the field φ between cosmological and solar system scales, even though the supernova measurements correspond to a higher redshift and a far different typical distance scale. A varying φ would of course imply that different values of ξ i , and their derivatives, would be perceived by supernovas and the planets. It is interesting to note that the size of the bound we have found ( 26 ) is of order the square of the ratio of the solar system and the cosmological horizon scales, s = (1 AU H 0 ) 2 ∼ 10 -30 . Therefore one could reasonably argue that the small number appearing in (26) could in fact be due to the hierarchy scale, s, rather than a very stringent constraint on Ω GB . This could perhaps be concretely realised with something similar to the chameleon effect [26] giving some constraint on the running of the quintessence theory. One other possibility is that the baryons (which make up the solar system) and dark matter (which is dominant at cosmological scales) have different couplings to φ [27] . Again, this would alter the relation between local and cosmological constraints.\n\nAlternatively, it may be that our assumptions on the form of the theory should be changed. The scalar field could be coupled directly to the Einstein-Hilbert term, as in Brans-Dicke gravity. Additionally, the couplings ξ i and their derivatives could be of different orders. The same could be true of the potential. In particular, if φ were to have a significant mass, this would suppress the quadratic curvature effects, as they operate via the scalar field. This would be similar to the situation in scalar-tensor gravity with a potential, where the strong constraints on the theory can be avoided by giving the scalar a large mass (which, however, would inhibit acceleration).\n\nFinally, the behaviour of the scalar field could be radically different. We took it to be O(ǫ), like the metric perturbations. However since our constraints are on the metric, and not φ, this need not be true. Furthermore, since the theory is quadratic, there may well be alternative solutions of the field equations, and not just the one we studied.\n\nHence to obtain a viable Gauss-Bonnet dark energy model, which is compatible with solar system constraints, at least one of the above assumptions must be broken. For many of the above ideas the higher-order scalar kinetic terms will play a significant role. This then opens up the possibility that the higher-gravity corrections will cancel each other, further weakening the constraints. We hope to address some of these issues in the near future."
}
] | |
arxiv:0704.0179 | 0704.0179 | 1 | 10.1103/PhysRevA.75.052106 | d39eb69b08b240bbbd706fc5d1cd420d2aa1e7fda46b24b7ff1f6682613fd534 | Experimental nonclassicality of single-photon-added thermal light states | We report the experimental realization and tomographic analysis of novel
quantum light states obtained by exciting a classical thermal field by a single
photon. Such states, although completely incoherent, possess a tunable degree
of quantumness which is here exploited to put to a stringent experimental test
some of the criteria proposed for the proof and the measurement of state
non-classicality. The quantum character of the states is also given in quantum
information terms by evaluating the amount of entanglement that they can
produce. | [
"Alessandro Zavatta",
"Valentina Parigi",
"Marco Bellini"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | The definition and the measurement of the nonclassicality of a quantum light state is a hot and widely discussed topic in the physics community; nonclassical light is the starting point for generating even more nonclassical states [1, 2] or producing the entanglement which is essential to implement quantum information protocols with continuous variables [3, 4] . A quantum state is said to be nonclassical when it cannot be written as a mixture of coherent states. In terms of the Glauber-Sudarshan P representation [5, 6] , the P function of a nonclassical state is highly singular or not positive, i.e. it cannot be interpreted as a classical probability distribution. In general however, since the P function can be badly behaved, it cannot be connected to any observable quantity. In recent years, a nonclassicality criterion based on the measurable quadrature distributions obtained from homodyne detection has been proposed by Richter and Vogel [7] . Moreover, a variety of nonclassical states has recently been characterized by means of the negativeness of their Wigner function [8, 9, 10, 11] , this however being just a sufficient and not necessary condition for nonclassicality [12] . It is still an open question which is the universal way to experimentally characterize the nonclassicality of a quantum state.
A conceptually simple way to generate a quantum light state with a varying degree of nonclassicality consists in adding a single photon to any completely classical one. This is quite different from photon subtraction which, on the other hand, produces a nonclassical state only when starting from an already nonclassical one [13, 14] .
In this Letter we report the generation and the analysis of single-photon-added thermal states (SPATSs), i.e., completely classical states excited by a single photon, first described by Agarwal and Tara in 1992 [15] . We use the techniques of conditioned parametric amplification recently demonstrated by our group [10, 11] to generate such states, and we employ ultrafast pulsed homodyne detection and quantum tomography to investigate their character. The peculiar nonclassical behavior of SPATSs has recently triggered an interesting debate [7, 16] and has been described in several theoretical papers [14, 15, 16, 17, 18] ; their experimental generation has already been proposed, although with more complex schemes [14, 18, 19] , but never realized. Thanks to their adjustable degree of quantumness, these states are an ideal benchmark to test the different experimental criteria of nonclassicality recently proposed, and to investigate the possibility of multi-photon entanglement generation. The nonclassicality of SPATSs is here analyzed by reconstructing their negative-valued Wigner functions, by using the quadrature-based Richter-Vogel (RV) criterion, and finally comparing these with two other methods based on quantum tomography. In particular, we show that the so-called entanglement potential [20] is a sensitive measurement of nonclassicality, and that it provides quantitative data about the possible use of the states for quantum information applications in terms of the entanglement that they would generate once sent to a 50-50 beam-splitter.
The main source of our apparatus is a mode-locked Ti:Sa laser which emits 1.5 ps pulses with a repetition rate of 82 MHz. The pulse train is frequency-doubled to 393 nm by second harmonic generation in a LBO crystal. The spatially-cleaned UV beam then serves as a pump for a type-I BBO crystal which generates spontaneous parametric down-conversion (SPDC) at the same wavelength of the laser source. Pairs of SPDC photons are emitted in two distinct spatial channels called signal and idler. Along the idler channel the photons are strongly filtered 2 in the spectral and spatial domain by means of etalon cavities and by a single-mode fiber which is directly connected to a single-photon-counting module (further details are given in [9, 11] ). The signal field is mixed with a strong local oscillator (LO, an attenuated portion of the main laser source) by means of a 50% beam-splitter (BS). The BS outputs are detected by two photodiodes connected to a wide-bandwidth amplifier which provides the difference (homodyne) signal between the two photocurrents on a pulse-to-pulse basis [21] . Whenever a single photon is detected in the idler channel, an homodyne measurement is performed on the correlated spatiotemporal mode of the signal channel by storing the corresponding electrical signal (proportional to the quadrature operator value) on a digital scope.
FIG. 1: (color online) Experimental setup. HR (HT) is a high reflectivity (transmittivity) beam splitter; SPCM is a singlephoton-counting module; all other symbols are defined in the text. The mode-cleaning fiber used to inject the thermal state coming from the rotating ground glass disk (RD) into the parametric crystal is not shown here for clarity.
When no field is injected in the SPDC crystal, conditioned single-photon Fock states are generated from spontaneous emission in the signal channel [8, 9] . We have recently shown that, if the SPDC crystal is injected with a coherent state, stimulated emission comes into play and single-photon excitation of such a pure state is obtained [10, 11] . However, a coherent state is still at the border between the quantum and the classical regimes; it is therefore extremely interesting to use a truly classical state, like the thermal one, as the input, and to observe its degaussification [13] . In order to avoid the technical problems connected to the handling of a true high-temperature thermal source, we use pseudo-thermal one, obtained by inserting a rotating ground glass disk (RD) in a portion of the laser beam (see Fig. 1 ). By coupling a fraction (much smaller than the typical speckle size) of the randomly scattered light into a single-mode fiber, at the output we obtain a clean spatial mode with random amplitude and phase yielding the photon distribution typical of a thermal source [22] which is then used to inject the parametric amplifier.
In order to describe the state generated in our experiment, we give a general treatment of photon addition based on conditioned parametric amplification. By firstorder perturbation theory, the output of the parametric amplifier when a pure state |ϕ m is injected along the signal channel is given by
|ψ m = [1 + (gâ † s â † i -g * âs âi )] |ϕ m s |0 i , ( 1
)
where g accounts for the coupling and the amplitude of the pump and â, â † are the usual noncommuting annihilation and creation operators. For a generic signal input, the output state of the parametric amplifier can be written as ρout =
m P m |ψ m ψ m | ( 2
)
where the input mixed state is ρs = P m |ϕ m ϕ m | and P m is the probability for the state |ϕ m . If we condition the preparation of the signal state to single-photon detection on the idler channel, we obtain the prepared state
ρ = Tr i (ρ out |1 i 1| i ) = |g| 2 â † s ρs âs . ( 3
)
When the input state ρs is a thermal state with mean photon number n, we obtain that the single-photonadded thermal state is described by the following density operator expressed in the Fock base:
ρ = 1 n(n + 1) ∞ n=0 n 1 + n n n |n n|. ( 4
)
The lack of the vacuum term and the rescaling of higher excited terms is evident in this expression. The P phasespace representation can be easily calculated and is given by (see also [15] )
P (α) = 1 πn 3 [(1 + n)|α| 2 -n]e -|α| 2 /n , ( 5
)
while the corresponding Wigner function reads as
W (α) = 2 π |2α| 2 (1 + n) -(1 + 2n) (1 + 2n) 3 e -2|α| 2 /(1+2n) ( 6
)
where α = x + iy. SPATSs have a well-behaved P function which is always negative around α = 0; this feature is also present in the Wigner function and assures their nonclassicality, however both P (0) and W (0) tend to zero in the limit of n → ∞.
After the acquisition of about 10 5 quadrature values with random phases, we have performed the reconstruction of the diagonal density matrix elements using the 3 maximum likelihood estimation [23] . This method gives the density matrix that most likely represents the measured homodyne data. Firstly, we build the likelihood function contracted for a density matrix truncated to 25 diagonal elements (with the constraints of Hermiticity, positivity and normalization), then the function is maximized by an iterative procedure [24, 25] and the errors on the reconstructed density matrix elements are evaluated using the Fisher information [25] . The results are shown in Fig. 2 , together with the corresponding reconstructed [11] Wigner functions for two different temper-FIG. 2: (color online) Experimentally reconstructed diagonal density matrix elements (reconstruction errors of statistical origin are of the order of 1%) and Wigner functions for thermal states (left) and SPATSs (right): a) n = 0.08; b) n = 1.15. Filled circles indicate the density matrix elements calculated for thermal states and SPATSs with the expected efficiencies.
atures of the injected thermal state. Since in the lowgain regime the count rate in the idler channel is given by n = Tr(ρ out â † i âi ) = |g| 2 (1 + n), the mean photon number values n reported in Fig. 2 and in the following are obtained from the ratio between the trigger count rates when the thermal injection is present and when it is blocked (see Ref. [11] and references therein).
The finite experimental efficiency in the preparation and homodyne detection of SPATSs is fully accounted for by a loss mechanism which can be modeled by the transmission of the ideal state ρ of Eq.(4) through a beam splitter of trasmittivity η coupling vacuum into the detection mode, such that the detected state ρη is finally found as:
ρη = Tr R {U η (ρ |0 0|)U † η } ( 7
)
where U η is the beam splitter operator acting on two input modes containing the state ρ and the vacuum, and the states of the reflected mode (indicated by R) are traced out. In the case of finite efficiency the expression for the Wigner function thus results:
W η (α) = 2 π 1 + 2η[n + 2(1 + n)|α| 2 -2nη -1] (1 + 2nη) 3 e -2|α| 2 1+2nη .
(8) It should be noted that the value of experimental efficiency which best fits the data is the same (η = 0.62) as that obtained for single-photon Fock states (i.e., without injection), and implies that only a portion of vacuum due to losses enters the mode during the generation of SPATS. Thanks to a very low rate of dark counts in the trigger detector, the portion of the injected thermal state which survives the conditional preparation procedure and contributes to degradation of the SPATSs is in fact completely negligible. However, since the nonclassical features of the state get weaker for large n, a limited efficiency (η < 1) has the effect of progressively hiding them among unwanted vacuum components.
Indeed, the measured negativity of the Wigner function at the origin (see Fig. 3a and b ) rapidly gets smaller as the mean photon number of the input thermal state is increased. With the current level of efficiency and reconstruction accuracy we are able to prove the nonclassicality of all the generated states (up to n = 1.15), but one may expect to experimentally detect negativity above the reconstruction noise, and thus prove state nonclassicality, up to about n ≈ 1.5 (also see Fig. 6a ). It should be noted that, even for a single-photon Fock state, the Wigner function loses its negativity for efficiencies lower than 50%, so that surpassing this experimental threshold is an essential requisite in order to use this nonclassicality criterion.
After having experimentally proved the nonclassicality of the states for all the investigated values of n, it is interesting to verify the nonclassical character of the measured SPATSs also using different criteria.
The first one has been recently proposed by Richter and Vogel [7] and is based on the characteristic function G(k, θ) = e ikx(θ) of the quadratures (i.e., the Fourier transform of the quadrature distribution), where x(θ) = (âe -iθ + â † e iθ )/2 is the phase-dependent quadrature operator. At the first-order, the criterion defines a phase-independent state as nonclassical if there is a value of k) , where 4 -2 -1 0 1 2 -0.1 0.0 0.1 0.2 0.0 0.4 0.8 1.2 -0.16 -0.12 -0.08 -0.04 0.00 x W(x) 0.08 0.34 0.70 1.15 a) W(0) n Classical limit b) FIG. 3: (color online) a) Sections of the experimentally reconstructed Wigner functions for SPATSs with different n; b) Experimental values for the minimum of the Wigner function W (0) as a function of n for SPATSs (solid squares) and for single-photon Fock states (empty circles) obtained by blocking the injection; the values calculated from Eq.( 8 ) for η = 0.62 (solid curves) are in very good agreement with experimental data and clearly show the appropriateness of the model. Negativity of the Wigner function is a sufficient condition for affirming the nonclassical character of the state.
k such that |G(k, θ)| ≡ |G(k)| > G gr (
G gr (k) is the characteristic function for the vacuum measured when the signal beam is blocked before homodyne detection. In other words, the evidence of structures narrower than those associated to vacuum in the quadrature distribution is a sufficient condition to define a nonclassical state [12] . However, it has been shown that nonclassical states exist (as pointed out by Diósi [16] for a vacuum-lacking thermal state [17], which is very similar to SPATSs) which fail to fulfil such inequality; when this happens, the first-order Richter-Vogel (RV) criterion has to be extended to higher orders: the second-order RV inequality reads as
2G 2 (k/2)G gr (k/ √ 2) -G(k) > G gr (k). ( 9
)
It is evident that, as higher orders are investigated, the increasing sensitivity to experimental and statistical noise may soon become unmanageable.
The measured |G(k)| and left hand side of Eq. (9) are plotted in Fig. 4a ) and b), together with the G gr (k) characteristic function, also obtained from the experimental quadrature distribution of vacuum. While the detected 0 2 4 6 8 0.0 0.5 1.0 0 2 4 6 8 0.0 0.5 1.0 a) k G gr (k) 0.53 0.70 0.90 1.15 b) 4 6 8 0.00 0.05 0.10 G gr (k) 0.08 0.34 FIG. 4: (color online) Experimental characteristic functions involved in the RV nonclassicality criterion for the detected SPATSs: a) first order; b) second order (the inset shows a magnified view of the region where the state with n = 0.53 is just slightly fulfilling the criterion).
SPATSs satisfy the nonclassical first-order RV criterion only for the two lowest values of n, it is necessary to extend the criterion to the second order to just barely show nonclassicality at large values of k for n = 0.53 (see the inset of Fig. 4b , where the shaded region indicates the error area of the experimental G gr (k)). At higher temperatures, no sign of nonclassical behavior is experimentally evident with this approach, although the Wigner function of the corresponding states still clearly exhibits a measurable negativity (see Fig. 3 ). It should be noted that the second-order RV criterion for the ideal state of Eq. ( 4 ) is expected to prove the nonclassicality of SPATSs up to n ≈ 0.6 [7]; however, when the limited experimental efficiency and the statistical noise is taken into account, it will start to fail even earlier.
The tomographic reconstruction of the state that was earlier used for the nonclassicality test based on the negativity of the Wigner function, can also be exploited to 5 test alternative criteria: for example by reconstructing the photon-number distribution ρ n = n| ρmeas |n and then looking for strong modulations in neighboring photon probabilities by the following relationship [26, 27]
B(n) ≡ (n + 2)ρ n ρ n+2 -(n + 1)ρ 2 n+1 < 0, ( 10
)
introduced by Klyshko in 1996, which is known to hold for nonclassical states. In the ideal situation of unit efficiency SPATSs should always give B(0) < 0 due to the absence of the vacuum term ρ 0 , in agreement with Ref. [17] . The experimental results obtained for B(0) by using the reconstructed density matrix ρmeas are presented in Fig. 5a ) together with those calculated for the state described by ρη (see Eq.( 7 )) with η = 0.62. The agreement between the experimental data and the expected ones is again very satisfactory, showing that our model state ρη well represents the experimental one. Our current efficiency should in principle allow us to find negative values of B(0) even for much larger values of n; however, if one takes the current reconstruction errors due to statistical noise into account, the maximum n for which the corresponding SPATS can be safely declared nonclassical is of the order of 2. It should be noted that, differently from the Wigner function approach, here the nonclassicality can be proved even for experimental efficiencies much lower than 50%, as far as the mean photon number of the thermal state is not too high (see Fig. 6b ).
Finally, it is particularly interesting to measure the entanglement potential (EP) of our states as recently proposed by Asboth et al. [20] . This measurement is based on the fact that, when a nonclassical state is mixed with vacuum on a 50-50 beam splitter, some amount of entanglement (depending on the nonclassicality of the input state) appears between the BS outputs. No entanglement can be produced by a classical initial state. For a given single-mode density operator ρ, one calculates the entanglement of the bipartite state at the BS outputs ρ′ = U BS (ρ|0 0|)U † BS by means of the logarithmic negativity E N (ρ ′ ) based on the Peres separability criterion and defined in [28] , where U BS is the 50-50 BS transformation. The computed entanglement potentials for the reconstructed SPATS density matrices ρmeas are shown in Fig. 5b ) together with those expected at the experimentally-evaluated efficiency (i.e., obtained from ρη with η = 0.62). The EP is definitely greater than zero (by more than 13σ) for all the detected states, thus confirming that they are indeed nonclassical, in agreement with the findings obtained by the measurement of B(0) and W (0). As a comparison, the EP would be equal to unity for a pure single-photon Fock state, while it would reduce to 0.43 for a single-photon state mixed with vac-
uum ρ = (1 -η) |0 0| + η |1 1| with η = 0.62.
To summarize, the three tomographic approaches to test nonclassicality have all been able to experimentally prove it for all the generated states (i.e., SPATSs with 0.0 0.4 0.8 1.2 -0.3 -0.2 -0.1 0.0 0.0 0.4 0.8 1.2 0.0 0.1 0.2 0.3 0.4 Classical limit B(0) a) n Classical limit EP b)
FIG. 5: (color online) a) Experimental data (squares) and calculated values (solid curve) of B(0) as a function of n; negative values indicate nonclassicality of the state. b) The same as above for the entanglement potential (EP) of the SPATSs; here nonclassicality is demonstrated by EP values greater than zero.
an average number of photons in the seed thermal state up to n = 1.15) for a global experimental efficiency of η = 0.62. In order to gain a better view of the range of values for n and for the global experimental efficiency η which allow to prove the nonclassical character of singlephoton-added thermal states under realistic experimental conditions, we have calculated the indicators W (0), B(0), and EP from the model state described by ρη . The results are shown in Fig. 6 : the contour plots define the regions of parameters where the detected state is classical (white areas), where it would result nonclassical if the reconstruction errors coming from statistical noise could be neglected (grey areas) and, finally, where it is definitely nonclassical even with the current level of noise (black areas). From such plots it is evident that, as already noted, the Wigner function negativity only works for sufficiently high efficiencies, while both B(0) and EP are able to detect nonclassical behavior even for η < 50%.
In particular, the entanglement potential is clearly seen to be the most powerful criterion, at least for these particular states, and to allow for an experimental proof of 6 h n n a) b) n c) W(0) EP B(0) FIG. 6: Calculated regions of nonclassical behavior of SPATSs as a function of n and η according to: a) the negativity of the Wigner function at the origin W (0); b) the Klyshko criterion B(0); c) the entanglement potential EP. White areas indicate classical behavior; grey areas indicate where a potentially nonclassical character is not measurable due to experimental reconstruction noise (estimated as the average error on the experimentally reconstructed parameters); black areas indicate regions where the nonclassical character is measurable given the current statistical uncertainties.
nonclassicality for all combinations of n and η, as long as reconstruction errors can be neglected. Also considering the current experimental parameters, EP should show the quantum character of SPATSs even for n > 3, thus demonstrating its higher immunity to noise.
Although at a different degree, all three indicators are however very sensitive to the presence of reconstruction noise of statistical origin which may completely mask the nonclassical character of the states, even for relatively low values of n or for low efficiencies. In order to unambiguously prove the quantum character of higher-temperature SPATSs in these circumstances the only possibility is to reduce the "grey zone" by significantly increasing the number of quadrature measurements.
In conclusion, we have generated a completely incoherent light state possessing an adjustable degree of quantumness which has been used to experimentally test and compare different criteria of nonclassicality. Although the direct analysis of quadrature distributions, done following the criterion proposed by Richter and Vogel, has been able to show the nonclassical character of some of the states with lower mean photon numbers, quantum tomography, with the reconstruction of the density matrix and the Wigner function from the homodyne data, has allowed us to unambiguously show the nonclassical character of all the generated states: three different criteria, the negativity of the Wigner function, the Klyshko criterion and the entanglement potential, have been used with varying degree of effectiveness in revealing nonclassicality. Besides being a useful tool for the measurement of nonclassicality through the definition of the entanglement potential, the combination of nonclassical field states -such as those generated here -with a beam-splitter, can be viewed as a simple entangling device generating multiphoton states with varying degree of purity and entanglement and allowing the future investigation of continuousvariable mixed entangled states [29] .
The authors gratefully acknowledge Koji Usami for giving the initial stimulus for this work and Milena D'Angelo and Girish Agarwal for useful discussions and comments. This work was partially supported by Ente Cassa di Risparmio di Firenze and MIUR, under the PRIN initiative and FIRB contract RBNE01KZ94.
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{
"section_type": "BACKGROUND",
"section_title": "INTRODUCTION",
"text": "The definition and the measurement of the nonclassicality of a quantum light state is a hot and widely discussed topic in the physics community; nonclassical light is the starting point for generating even more nonclassical states [1, 2] or producing the entanglement which is essential to implement quantum information protocols with continuous variables [3, 4] . A quantum state is said to be nonclassical when it cannot be written as a mixture of coherent states. In terms of the Glauber-Sudarshan P representation [5, 6] , the P function of a nonclassical state is highly singular or not positive, i.e. it cannot be interpreted as a classical probability distribution. In general however, since the P function can be badly behaved, it cannot be connected to any observable quantity. In recent years, a nonclassicality criterion based on the measurable quadrature distributions obtained from homodyne detection has been proposed by Richter and Vogel [7] . Moreover, a variety of nonclassical states has recently been characterized by means of the negativeness of their Wigner function [8, 9, 10, 11] , this however being just a sufficient and not necessary condition for nonclassicality [12] . It is still an open question which is the universal way to experimentally characterize the nonclassicality of a quantum state.\n\nA conceptually simple way to generate a quantum light state with a varying degree of nonclassicality consists in adding a single photon to any completely classical one. This is quite different from photon subtraction which, on the other hand, produces a nonclassical state only when starting from an already nonclassical one [13, 14] .\n\nIn this Letter we report the generation and the analysis of single-photon-added thermal states (SPATSs), i.e., completely classical states excited by a single photon, first described by Agarwal and Tara in 1992 [15] . We use the techniques of conditioned parametric amplification recently demonstrated by our group [10, 11] to generate such states, and we employ ultrafast pulsed homodyne detection and quantum tomography to investigate their character. The peculiar nonclassical behavior of SPATSs has recently triggered an interesting debate [7, 16] and has been described in several theoretical papers [14, 15, 16, 17, 18] ; their experimental generation has already been proposed, although with more complex schemes [14, 18, 19] , but never realized. Thanks to their adjustable degree of quantumness, these states are an ideal benchmark to test the different experimental criteria of nonclassicality recently proposed, and to investigate the possibility of multi-photon entanglement generation. The nonclassicality of SPATSs is here analyzed by reconstructing their negative-valued Wigner functions, by using the quadrature-based Richter-Vogel (RV) criterion, and finally comparing these with two other methods based on quantum tomography. In particular, we show that the so-called entanglement potential [20] is a sensitive measurement of nonclassicality, and that it provides quantitative data about the possible use of the states for quantum information applications in terms of the entanglement that they would generate once sent to a 50-50 beam-splitter."
},
{
"section_type": "METHOD",
"section_title": "EXPERIMENTAL",
"text": "The main source of our apparatus is a mode-locked Ti:Sa laser which emits 1.5 ps pulses with a repetition rate of 82 MHz. The pulse train is frequency-doubled to 393 nm by second harmonic generation in a LBO crystal. The spatially-cleaned UV beam then serves as a pump for a type-I BBO crystal which generates spontaneous parametric down-conversion (SPDC) at the same wavelength of the laser source. Pairs of SPDC photons are emitted in two distinct spatial channels called signal and idler. Along the idler channel the photons are strongly filtered 2 in the spectral and spatial domain by means of etalon cavities and by a single-mode fiber which is directly connected to a single-photon-counting module (further details are given in [9, 11] ). The signal field is mixed with a strong local oscillator (LO, an attenuated portion of the main laser source) by means of a 50% beam-splitter (BS). The BS outputs are detected by two photodiodes connected to a wide-bandwidth amplifier which provides the difference (homodyne) signal between the two photocurrents on a pulse-to-pulse basis [21] . Whenever a single photon is detected in the idler channel, an homodyne measurement is performed on the correlated spatiotemporal mode of the signal channel by storing the corresponding electrical signal (proportional to the quadrature operator value) on a digital scope.\n\nFIG. 1: (color online) Experimental setup. HR (HT) is a high reflectivity (transmittivity) beam splitter; SPCM is a singlephoton-counting module; all other symbols are defined in the text. The mode-cleaning fiber used to inject the thermal state coming from the rotating ground glass disk (RD) into the parametric crystal is not shown here for clarity.\n\nWhen no field is injected in the SPDC crystal, conditioned single-photon Fock states are generated from spontaneous emission in the signal channel [8, 9] . We have recently shown that, if the SPDC crystal is injected with a coherent state, stimulated emission comes into play and single-photon excitation of such a pure state is obtained [10, 11] . However, a coherent state is still at the border between the quantum and the classical regimes; it is therefore extremely interesting to use a truly classical state, like the thermal one, as the input, and to observe its degaussification [13] . In order to avoid the technical problems connected to the handling of a true high-temperature thermal source, we use pseudo-thermal one, obtained by inserting a rotating ground glass disk (RD) in a portion of the laser beam (see Fig. 1 ). By coupling a fraction (much smaller than the typical speckle size) of the randomly scattered light into a single-mode fiber, at the output we obtain a clean spatial mode with random amplitude and phase yielding the photon distribution typical of a thermal source [22] which is then used to inject the parametric amplifier."
},
{
"section_type": "OTHER",
"section_title": "PROPERTIES OF SPATSS",
"text": "In order to describe the state generated in our experiment, we give a general treatment of photon addition based on conditioned parametric amplification. By firstorder perturbation theory, the output of the parametric amplifier when a pure state |ϕ m is injected along the signal channel is given by\n\n|ψ m = [1 + (gâ † s â † i -g * âs âi )] |ϕ m s |0 i , ( 1\n\n)\n\nwhere g accounts for the coupling and the amplitude of the pump and â, â † are the usual noncommuting annihilation and creation operators. For a generic signal input, the output state of the parametric amplifier can be written as ρout =\n\nm P m |ψ m ψ m | ( 2\n\n)\n\nwhere the input mixed state is ρs = P m |ϕ m ϕ m | and P m is the probability for the state |ϕ m . If we condition the preparation of the signal state to single-photon detection on the idler channel, we obtain the prepared state\n\nρ = Tr i (ρ out |1 i 1| i ) = |g| 2 â † s ρs âs . ( 3\n\n)\n\nWhen the input state ρs is a thermal state with mean photon number n, we obtain that the single-photonadded thermal state is described by the following density operator expressed in the Fock base:\n\nρ = 1 n(n + 1) ∞ n=0 n 1 + n n n |n n|. ( 4\n\n)\n\nThe lack of the vacuum term and the rescaling of higher excited terms is evident in this expression. The P phasespace representation can be easily calculated and is given by (see also [15] )\n\nP (α) = 1 πn 3 [(1 + n)|α| 2 -n]e -|α| 2 /n , ( 5\n\n)\n\nwhile the corresponding Wigner function reads as\n\nW (α) = 2 π |2α| 2 (1 + n) -(1 + 2n) (1 + 2n) 3 e -2|α| 2 /(1+2n) ( 6\n\n)\n\nwhere α = x + iy. SPATSs have a well-behaved P function which is always negative around α = 0; this feature is also present in the Wigner function and assures their nonclassicality, however both P (0) and W (0) tend to zero in the limit of n → ∞."
},
{
"section_type": "DISCUSSION",
"section_title": "DATA ANALYSIS AND DISCUSSION",
"text": "After the acquisition of about 10 5 quadrature values with random phases, we have performed the reconstruction of the diagonal density matrix elements using the 3 maximum likelihood estimation [23] . This method gives the density matrix that most likely represents the measured homodyne data. Firstly, we build the likelihood function contracted for a density matrix truncated to 25 diagonal elements (with the constraints of Hermiticity, positivity and normalization), then the function is maximized by an iterative procedure [24, 25] and the errors on the reconstructed density matrix elements are evaluated using the Fisher information [25] . The results are shown in Fig. 2 , together with the corresponding reconstructed [11] Wigner functions for two different temper-FIG. 2: (color online) Experimentally reconstructed diagonal density matrix elements (reconstruction errors of statistical origin are of the order of 1%) and Wigner functions for thermal states (left) and SPATSs (right): a) n = 0.08; b) n = 1.15. Filled circles indicate the density matrix elements calculated for thermal states and SPATSs with the expected efficiencies.\n\natures of the injected thermal state. Since in the lowgain regime the count rate in the idler channel is given by n = Tr(ρ out â † i âi ) = |g| 2 (1 + n), the mean photon number values n reported in Fig. 2 and in the following are obtained from the ratio between the trigger count rates when the thermal injection is present and when it is blocked (see Ref. [11] and references therein).\n\nThe finite experimental efficiency in the preparation and homodyne detection of SPATSs is fully accounted for by a loss mechanism which can be modeled by the transmission of the ideal state ρ of Eq.(4) through a beam splitter of trasmittivity η coupling vacuum into the detection mode, such that the detected state ρη is finally found as:\n\nρη = Tr R {U η (ρ |0 0|)U † η } ( 7\n\n)\n\nwhere U η is the beam splitter operator acting on two input modes containing the state ρ and the vacuum, and the states of the reflected mode (indicated by R) are traced out. In the case of finite efficiency the expression for the Wigner function thus results:\n\nW η (α) = 2 π 1 + 2η[n + 2(1 + n)|α| 2 -2nη -1] (1 + 2nη) 3 e -2|α| 2 1+2nη .\n\n(8) It should be noted that the value of experimental efficiency which best fits the data is the same (η = 0.62) as that obtained for single-photon Fock states (i.e., without injection), and implies that only a portion of vacuum due to losses enters the mode during the generation of SPATS. Thanks to a very low rate of dark counts in the trigger detector, the portion of the injected thermal state which survives the conditional preparation procedure and contributes to degradation of the SPATSs is in fact completely negligible. However, since the nonclassical features of the state get weaker for large n, a limited efficiency (η < 1) has the effect of progressively hiding them among unwanted vacuum components.\n\nIndeed, the measured negativity of the Wigner function at the origin (see Fig. 3a and b ) rapidly gets smaller as the mean photon number of the input thermal state is increased. With the current level of efficiency and reconstruction accuracy we are able to prove the nonclassicality of all the generated states (up to n = 1.15), but one may expect to experimentally detect negativity above the reconstruction noise, and thus prove state nonclassicality, up to about n ≈ 1.5 (also see Fig. 6a ). It should be noted that, even for a single-photon Fock state, the Wigner function loses its negativity for efficiencies lower than 50%, so that surpassing this experimental threshold is an essential requisite in order to use this nonclassicality criterion.\n\nAfter having experimentally proved the nonclassicality of the states for all the investigated values of n, it is interesting to verify the nonclassical character of the measured SPATSs also using different criteria.\n\nThe first one has been recently proposed by Richter and Vogel [7] and is based on the characteristic function G(k, θ) = e ikx(θ) of the quadratures (i.e., the Fourier transform of the quadrature distribution), where x(θ) = (âe -iθ + â † e iθ )/2 is the phase-dependent quadrature operator. At the first-order, the criterion defines a phase-independent state as nonclassical if there is a value of k) , where 4 -2 -1 0 1 2 -0.1 0.0 0.1 0.2 0.0 0.4 0.8 1.2 -0.16 -0.12 -0.08 -0.04 0.00 x W(x) 0.08 0.34 0.70 1.15 a) W(0) n Classical limit b) FIG. 3: (color online) a) Sections of the experimentally reconstructed Wigner functions for SPATSs with different n; b) Experimental values for the minimum of the Wigner function W (0) as a function of n for SPATSs (solid squares) and for single-photon Fock states (empty circles) obtained by blocking the injection; the values calculated from Eq.( 8 ) for η = 0.62 (solid curves) are in very good agreement with experimental data and clearly show the appropriateness of the model. Negativity of the Wigner function is a sufficient condition for affirming the nonclassical character of the state.\n\nk such that |G(k, θ)| ≡ |G(k)| > G gr (\n\nG gr (k) is the characteristic function for the vacuum measured when the signal beam is blocked before homodyne detection. In other words, the evidence of structures narrower than those associated to vacuum in the quadrature distribution is a sufficient condition to define a nonclassical state [12] . However, it has been shown that nonclassical states exist (as pointed out by Diósi [16] for a vacuum-lacking thermal state [17], which is very similar to SPATSs) which fail to fulfil such inequality; when this happens, the first-order Richter-Vogel (RV) criterion has to be extended to higher orders: the second-order RV inequality reads as\n\n2G 2 (k/2)G gr (k/ √ 2) -G(k) > G gr (k). ( 9\n\n)\n\nIt is evident that, as higher orders are investigated, the increasing sensitivity to experimental and statistical noise may soon become unmanageable.\n\nThe measured |G(k)| and left hand side of Eq. (9) are plotted in Fig. 4a ) and b), together with the G gr (k) characteristic function, also obtained from the experimental quadrature distribution of vacuum. While the detected 0 2 4 6 8 0.0 0.5 1.0 0 2 4 6 8 0.0 0.5 1.0 a) k G gr (k) 0.53 0.70 0.90 1.15 b) 4 6 8 0.00 0.05 0.10 G gr (k) 0.08 0.34 FIG. 4: (color online) Experimental characteristic functions involved in the RV nonclassicality criterion for the detected SPATSs: a) first order; b) second order (the inset shows a magnified view of the region where the state with n = 0.53 is just slightly fulfilling the criterion).\n\nSPATSs satisfy the nonclassical first-order RV criterion only for the two lowest values of n, it is necessary to extend the criterion to the second order to just barely show nonclassicality at large values of k for n = 0.53 (see the inset of Fig. 4b , where the shaded region indicates the error area of the experimental G gr (k)). At higher temperatures, no sign of nonclassical behavior is experimentally evident with this approach, although the Wigner function of the corresponding states still clearly exhibits a measurable negativity (see Fig. 3 ). It should be noted that the second-order RV criterion for the ideal state of Eq. ( 4 ) is expected to prove the nonclassicality of SPATSs up to n ≈ 0.6 [7]; however, when the limited experimental efficiency and the statistical noise is taken into account, it will start to fail even earlier.\n\nThe tomographic reconstruction of the state that was earlier used for the nonclassicality test based on the negativity of the Wigner function, can also be exploited to 5 test alternative criteria: for example by reconstructing the photon-number distribution ρ n = n| ρmeas |n and then looking for strong modulations in neighboring photon probabilities by the following relationship [26, 27]\n\nB(n) ≡ (n + 2)ρ n ρ n+2 -(n + 1)ρ 2 n+1 < 0, ( 10\n\n)\n\nintroduced by Klyshko in 1996, which is known to hold for nonclassical states. In the ideal situation of unit efficiency SPATSs should always give B(0) < 0 due to the absence of the vacuum term ρ 0 , in agreement with Ref. [17] . The experimental results obtained for B(0) by using the reconstructed density matrix ρmeas are presented in Fig. 5a ) together with those calculated for the state described by ρη (see Eq.( 7 )) with η = 0.62. The agreement between the experimental data and the expected ones is again very satisfactory, showing that our model state ρη well represents the experimental one. Our current efficiency should in principle allow us to find negative values of B(0) even for much larger values of n; however, if one takes the current reconstruction errors due to statistical noise into account, the maximum n for which the corresponding SPATS can be safely declared nonclassical is of the order of 2. It should be noted that, differently from the Wigner function approach, here the nonclassicality can be proved even for experimental efficiencies much lower than 50%, as far as the mean photon number of the thermal state is not too high (see Fig. 6b ).\n\nFinally, it is particularly interesting to measure the entanglement potential (EP) of our states as recently proposed by Asboth et al. [20] . This measurement is based on the fact that, when a nonclassical state is mixed with vacuum on a 50-50 beam splitter, some amount of entanglement (depending on the nonclassicality of the input state) appears between the BS outputs. No entanglement can be produced by a classical initial state. For a given single-mode density operator ρ, one calculates the entanglement of the bipartite state at the BS outputs ρ′ = U BS (ρ|0 0|)U † BS by means of the logarithmic negativity E N (ρ ′ ) based on the Peres separability criterion and defined in [28] , where U BS is the 50-50 BS transformation. The computed entanglement potentials for the reconstructed SPATS density matrices ρmeas are shown in Fig. 5b ) together with those expected at the experimentally-evaluated efficiency (i.e., obtained from ρη with η = 0.62). The EP is definitely greater than zero (by more than 13σ) for all the detected states, thus confirming that they are indeed nonclassical, in agreement with the findings obtained by the measurement of B(0) and W (0). As a comparison, the EP would be equal to unity for a pure single-photon Fock state, while it would reduce to 0.43 for a single-photon state mixed with vac-\n\nuum ρ = (1 -η) |0 0| + η |1 1| with η = 0.62.\n\nTo summarize, the three tomographic approaches to test nonclassicality have all been able to experimentally prove it for all the generated states (i.e., SPATSs with 0.0 0.4 0.8 1.2 -0.3 -0.2 -0.1 0.0 0.0 0.4 0.8 1.2 0.0 0.1 0.2 0.3 0.4 Classical limit B(0) a) n Classical limit EP b)\n\nFIG. 5: (color online) a) Experimental data (squares) and calculated values (solid curve) of B(0) as a function of n; negative values indicate nonclassicality of the state. b) The same as above for the entanglement potential (EP) of the SPATSs; here nonclassicality is demonstrated by EP values greater than zero.\n\nan average number of photons in the seed thermal state up to n = 1.15) for a global experimental efficiency of η = 0.62. In order to gain a better view of the range of values for n and for the global experimental efficiency η which allow to prove the nonclassical character of singlephoton-added thermal states under realistic experimental conditions, we have calculated the indicators W (0), B(0), and EP from the model state described by ρη . The results are shown in Fig. 6 : the contour plots define the regions of parameters where the detected state is classical (white areas), where it would result nonclassical if the reconstruction errors coming from statistical noise could be neglected (grey areas) and, finally, where it is definitely nonclassical even with the current level of noise (black areas). From such plots it is evident that, as already noted, the Wigner function negativity only works for sufficiently high efficiencies, while both B(0) and EP are able to detect nonclassical behavior even for η < 50%.\n\nIn particular, the entanglement potential is clearly seen to be the most powerful criterion, at least for these particular states, and to allow for an experimental proof of 6 h n n a) b) n c) W(0) EP B(0) FIG. 6: Calculated regions of nonclassical behavior of SPATSs as a function of n and η according to: a) the negativity of the Wigner function at the origin W (0); b) the Klyshko criterion B(0); c) the entanglement potential EP. White areas indicate classical behavior; grey areas indicate where a potentially nonclassical character is not measurable due to experimental reconstruction noise (estimated as the average error on the experimentally reconstructed parameters); black areas indicate regions where the nonclassical character is measurable given the current statistical uncertainties.\n\nnonclassicality for all combinations of n and η, as long as reconstruction errors can be neglected. Also considering the current experimental parameters, EP should show the quantum character of SPATSs even for n > 3, thus demonstrating its higher immunity to noise.\n\nAlthough at a different degree, all three indicators are however very sensitive to the presence of reconstruction noise of statistical origin which may completely mask the nonclassical character of the states, even for relatively low values of n or for low efficiencies. In order to unambiguously prove the quantum character of higher-temperature SPATSs in these circumstances the only possibility is to reduce the \"grey zone\" by significantly increasing the number of quadrature measurements."
},
{
"section_type": "CONCLUSION",
"section_title": "CONCLUSIONS",
"text": "In conclusion, we have generated a completely incoherent light state possessing an adjustable degree of quantumness which has been used to experimentally test and compare different criteria of nonclassicality. Although the direct analysis of quadrature distributions, done following the criterion proposed by Richter and Vogel, has been able to show the nonclassical character of some of the states with lower mean photon numbers, quantum tomography, with the reconstruction of the density matrix and the Wigner function from the homodyne data, has allowed us to unambiguously show the nonclassical character of all the generated states: three different criteria, the negativity of the Wigner function, the Klyshko criterion and the entanglement potential, have been used with varying degree of effectiveness in revealing nonclassicality. Besides being a useful tool for the measurement of nonclassicality through the definition of the entanglement potential, the combination of nonclassical field states -such as those generated here -with a beam-splitter, can be viewed as a simple entangling device generating multiphoton states with varying degree of purity and entanglement and allowing the future investigation of continuousvariable mixed entangled states [29] ."
},
{
"section_type": "OTHER",
"section_title": "ACKNOWLEDGMENTS",
"text": "The authors gratefully acknowledge Koji Usami for giving the initial stimulus for this work and Milena D'Angelo and Girish Agarwal for useful discussions and comments. This work was partially supported by Ente Cassa di Risparmio di Firenze and MIUR, under the PRIN initiative and FIRB contract RBNE01KZ94.\n\n[1] A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim, Phys.\n\nRev. A 70, 020101(R) (2004). [2] H. Jeong, A. P. Lund, and T. C. Ralph, Phys. Rev. A 72, 013801 (2005). [3] M. S. Kim, W. Son, V. Bužek, and P. L. Knight, Phys.\n\nRev. A 65, 032323 (2002). [4] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005). [5] R. J. Glauber, Phys. Rev. 131, 2766 (1963). [6] E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963). [7] W. Vogel, Phys. Rev. Lett. 84, 1849 (2000). [8] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, Phys. Rev. Lett. 87, 050402 (2001). [9] A. Zavatta, S. Viciani, and M. Bellini, Phys. Rev. A 70, 053821 (2004). [10] A. Zavatta, S. Viciani, and M. Bellini, Science 306, 660 (2004). [11] A. Zavatta, S. Viciani, and M. Bellini, Phys. Rev. A 72, 023820 (2005). 7 [12] A. I. Lvovsky and J. H. Shapiro, Phys. Rev. A 65, 033830 (2002). [13] J. Wenger, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 92, 153601 (2004). [14] M. S. Kim, E. Park, P. L. Knight, and H. Jeong, Phys.\n\nRev. A 71, 043805 (2005). [15] G. S. Agarwal and K. Tara, Phys. Rev. A 46, 485 (1992). [16] L. Diósi, Phys. Rev. Lett. 85, 2841 (2000). [17] C. T. Lee, Phys. Rev. A 52, 3374 (1995). [18] G. N. Jones, J. Haight, and C. T. Lee, Quantum Semiclass. Opt. 9, 411 (1997). [19] M. Dakna, L. Knöll, and D.-G. Welsch, Eur. Phys. J. D 3, 295 (1998). [20] J. K. Asboth, J. Calsamiglia, and H. Ritsch, Phys. Rev.\n\nLett. 94, 173602 (2005). [21] A. Zavatta, M. Bellini, P. L. Ramazza, F. Marin, and F. T. Arecchi, J. Opt. Soc. Am. B 19, 1189 (2002). [22] F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965). [23] K. Banaszek, G. M. D'Ariano, M. G. A. Paris, and M. F. Sacchi, Phys. Rev. A 61, 010304 (1999). [24] A. I. Lvovsky, J. Opt. B: Quantum Semiclass. Opt. 6, 556 (2004). [25] Z. Hradil, D. Mogilevtsev, and J. Rehacek, Phys. Rev. Lett. 96, 230401 (2006). [26] D. N. Klyshko, Phys. Lett. A 231, 7 (1996). [27] G. M. D'Ariano, M. F. Sacchi, and P. Kumar, Phys. Rev. A 59, 826 (1999). [28] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002). [29] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.\n\nRev. Lett. 80, 5239 (1998)."
}
] | |
arxiv:0704.0194 | 0704.0194 | 1 | 10.1103/PhysRevB.77.075325 | 02598565965bfa7af91a44c7c2b2e5ee6dda1cbe143ab6d535bfb77e73fe21eb | Quantum mechanical approach to decoherence and relaxation generated by
fluctuating environment | We consider an electrostatic qubit, interacting with a fluctuating charge of
single electron transistor (SET) in the framework of exactly solvable model.
The SET plays a role of the fluctuating environment affecting the qubit's
parameters in a controllable way. We derive the rate equations describing
dynamics of the entire system for both weak and strong qubit-SET coupling.
Solving these equation we obtain decoherence and relaxation rates of the qubit,
as well as the spectral density of the fluctuating qubit's parameters. We found
that in the weak coupling regime the decoherence and relaxation rates are
directly related to the spectral density taken at Rabi or at zero frequency,
depending on what a particular qubit's parameters is fluctuating. This relation
holds also in the presence of weak back-action of the qubit on the fluctuating
environment. In the case of strong back-action, such simple relationship no
longer holds, even if the qubit-SET coupling is small. It does not hold either
in the strong-coupling regime, even in the absence of the back-action. In
addition, we found that our model predicts localization of the qubit in the
strong-coupling regime, resembling that of the spin-boson model. | [
"S.A. Gurvitz and D. Mozyrsky"
] | [
"cond-mat.mes-hall",
"quant-ph"
] | cond-mat.mes-hall | [] | 2007-04-02 | 2026-02-26 | The influence of environment on a single quantum system is the issue of crucial importance in quantum information science. It is mainly associated with decoherence, or dephasing, which transforms any pure state of a quantum system into a statistical mixture. Despite a large body of theoretical work devoted to decoherence, its mechanism has not been clarified enough. For instance, how decoherence is related to environmental noise, in particular in the presence of back-action of the system on the environment (quantum measurements). Moreover, decoherence is often intermixed with relaxation. Although each of them represents an irreversible process, decoherence and relaxation affect quantum systems in quite different ways.
In order to establish a relation between the fluctuation spectrum and decoherence and relaxation rates one needs a model that describes the effects of decoherence and relaxation in a consistent quantum mechanical way. An obvious candidate is the spin-boson model 1,2 which represents the environment as a bath of harmonic oscillators at equilibrium, where the fluctuations obey Gaussian statistics 3 . Despite its apparent simplicity, the spinboson model cannot be solved exactly 2 . Also, it is hard to manipulate the fluctuation spectrum in the framework of this model. In addition, mesoscopic structures may couple only to a few isolated fluctuators, like spins, local currents, background charge fluctuations, etc. This would require models of the environment, different from Electrodes (a) (b) E 1 E 2 0 Ω FIG. 1: Electrostatic qubit, realized by an electron trapped in a coupled-dot system (a), and its schematic representation by a double-well (b). Ω0 denotes the coupling between the two dots.
the spin-boson model (see for instace 4, 5, 6, 7, 8, 9, 10, 11, 12 ). In general, the environment can be out of equilibrium, like a steady-state fluctuating current, interacting with the qubit 13, 14, 15, 16 . This for instance, takes place in the continuous measurement (monitoring) of quantum systems 17 and in the "control dephasing" experiments 18, 19, 20 . All these types on non-Gaussian and non-equilibrium environments attracted recently a great deal of attention 21 .
In this paper we consider an electrostatic qubit, which can be viewed as a generic example of two-state systems. It is realized by an electron trapped in coupled quantum dots 22, 23, 24 , Fig. 1. Here E 1 and E 2 denote energies of the electron states in each of the dots and Ω 0 is a coupling between these states. It is reasonable to assume that the decoherence of a qubit is associated with fluctu-
2 E 1 E 2 E r R µ L µ 0 E E l
Single Electron Transistor (a) (b) I I FIG. 2: Qubit near Single Electron Transistor. Here E l,r and E0 denote the energy levels in the left (right) reservoirs and in the quantum dot, respectively, and µL,R are the corresponding chemical potentials. The electric current I generates fluctuations of the electrostatic opening between two dots (a), or it fluctuates the energy level of the nearest dot (b).
ations of the qubit parameters, E 1,2 and Ω 0 , generated by the environment. Indeed, a stochastic averaging of the Schrödinger equation over these fluctuations parameters results in the qubit's decoherence, which transfers any qubit state into a statistical mixture 25, 26 . In general, one can expect that the fluctuating environment should result in the qubit's relaxation, as well, as for instance in the phenomenological Redfield's description of relaxation in the magnetic resonance 27 .
As a quantum mechanical model of the environment we consider a Single Electron Transistor (SET) capacitively coupled to the qubit, e.g., Fig. 2. Such setup has been contemplated in numerous solid state quantum computing architectures where SET plays role of a readout device 16, 17, 28, 29 and contains most of the generic features of a fluctuating non-equilibrium environment. The discreteness of the electron charge creates fluctuations in the electrostatic field near the SET. If the electrostatic qubit is placed near the SET, this fluctuating field should affect the qubit behavior as shown in Fig. 2 . It can produce fluctuations of the tunneling coupling between the dots (off-diagonal coupling) by narrowing the electrostatic opening connecting these dots, as in Fig. 2a , or make the energy levels of the dots fluctuate, as shown schematically in Fig. 2b . Note that while in some regimes the SET operates as a measuring device 16, 17 , in other regimes it corresponds purely to a source of noise. Indeed, if the energy level E 0 , Fig. 2 , is deeply inside the voltage bias -the case we consider in the beginning, the SET current is not modulated by the qubit electron. In this case the SET represents only the fluctuating environment affecting the qubit behavior ("pure environment" 30 ).
A similar model of the fluctuating environment has been studied mostly for small bias (linear response) or for the environment in an equilibrium. Here, however, we consider strongly non-equilibrium case where the bias voltage applied on the SET (V = µ Lµ R ) is much larger than the levels widths and the coupling between the SET and the qubit. In this limit our model can be solved exactly for both weak and strong coupling (but is still smaller than the bias voltage). This constitutes an essential advantage with regard to perturbative treatments of similar models. For instance, the results of our model can be compared in different regimes with phenomenological descriptions used in the literature. Such a comparison would allow us to determine the regions where these phenomenological models are valid.
Since our model is very simple in treatment, the decoherence and relaxation rates can be extracted from the exact solution analytically, as well as the time-correlator of the electric charge inside the SET. This would make it possible to establish a relation between the frequencydependent fluctuation spectrum of the environment and the decoherence and the relaxation rates of the qubit, and to determine how far this relation can be extended. We expect that such a relation should not depend on a source of fluctuations. This point can be verified by a comparison with a similar results obtained for equilibrium environment in the framework of the spin-boson model 1, 2 .
It is also important to understand how the decoherence and relaxation rates depend on the frequency of the environmental fluctuations. This problem has been investigated in many phenomenological approaches for "classical" environments at equilibrium. Yet, there still exists an ambiguity in the literature related to this point for non-equilibrium environment. For instance, it was found by Levinson that the decoherence rate, generated by fluctuations of the energy level in a single quantum dot is proportional to the spectral density of fluctuations at zero frequency 31 . The same result, but for a double-dot system has been obtained by Rabenstein et al. 32 . On the other hand, it follows from the Redfied's approach that the corresponding decoherence rate is proportional to the spectral density at the frequency of the qubit's oscillations (the Rabi frequency) 27 . Since our model is the exactly solvable one, we can resolve this ambiguity and establish the appropriate physical conditions that can result in different relations of decoherence rate to the environmental fluctuations.
The most important results of our study are related to the situation when back-action of the qubit on the environment takes place. This problem did not receive such a considerable amount of attention in the literature as, for example, the case of "inert" environment. This is in spite of a fact that the back-action always takes place in the presence of measurement. There are many questions related to the effects of a back-action. For instance, what would be a relation between decoherence (relaxation) of the qubit and the noise spectrum of the environment? Or, how decoherence is affected by a strong response of environment? We believe that our model appears to be 3 more suitable for studying these and other problems related to the back-action than most of the other existing approaches. The plan of this paper is as follows: Sect. II presents a phenomenological description of decoherence and relaxation in the framework of Bloch equations, applied to the electrostatic qubit. Sect. III contains description of the model and the quantum rate-equation formalism, used for its solution. Detailed quantum-mechanical derivation of these equations for a specific example is presented in Appendix A. Sect. IV deals with a configuration where the SET can generate only decoherence of the qubit. We consider separately the situations when SET produces fluctuations of the tunneling coupling (Rabi frequency) or of the energy levels. The results are compared with the SET fluctuation spectrum, evaluated in Appendix B. Sect. V deals with a configuration where the SET generates both decoherence and relaxation of the qubit. Sect. VI is summary.
In this section we describe in a general phenomenological framework the effect of decoherence and relaxation on the qubit behavior. Although the results are known, there still exists some confusion in the literature in this issue. We therefore need to define precisely these quantities and demonstrate how the corresponding decoherenece and relaxation rates can be extracted from the qubit density matrix.
Let us consider an electrostatic qubit, realized by an electron trapped in coupled quantum dots, Fig. 1. This system is described by the following tunneling Hamiltonian
H qb = E 1 a † 1 a 1 + E 2 a † 2 a 2 -Ω 0 (a † 2 a 1 + a † 1 a 2 ) ( 1
)
where a † 1,2 , a 1,2 are the creation and annihilation operators of the electron in the first or in the second dot. For simplicity we consider electrons as spinless fermions. In addition, we assume that a † 1 a 1 + a † 2 a 2 = 1, so that only one electron is present in the double-dot. The electron wave function can be written as |Ψ(t) = b (1) (t)a † 1 + b (2) (t)a † 2 |0 ¯ (2) where b (1, 2) (t) are the probability amplitudes for finding the electron in the first or second well, obtained from the Schrödinger equation i∂ t |Ψ(t) = H qb |Ψ(t) (we adopt the units where = 1 and the electron charge e = 1). The corresponding density matrix, σ jj ′ (t) = b (j) (t)b (j ′ ) * (t), with j, j ′ = {1, 2}, is obtained from the equation i∂ t σ = [H, σ]. This can be written explicitly as
σ11 = iΩ 0 (σ 21 -σ 12 ) (3a) σ12 = -iǫσ 12 + iΩ 0 (1 -2σ 11 ) , ( 3b
)
where σ 22 (t) = 1σ 11 (t), σ 21 (t) = σ * 12 (t) and ǫ = E 1 -E 2 . Solving these equations one easily finds that the electron oscillates between the two dots (Rabi oscillations) with frequency ω R = 4Ω 2 0 + ǫ 2 . For instance, for the initial conditions σ 11 (0) = 1 and σ 12 (0) = 1, the probability of finding the electron in the second dot is σ 22 (t) = 2(Ω 0 /ω R ) 2 (1cos ω R t). This result shows that for ǫ ≫ Ω 0 the amplitude of the Rabi oscillations is small, so the electron remains localized in its initial state.
The situation is different when the qubit interacts with the environment. In this case the (reduced) density matrix of the qubit σ(t) is obtained by tracing out the environment variables from the total density matrix. The question is how to modify Eqs. (3), written for an isolated qubit, in order to obtain the reduced density matrix of the qubit, σ(t). In general one expects that the environment could affect the qubit in two different ways. First, it can destroy the off-diagonal elements of the qubit density matrix. This process is usually referred to as decoherence (or dephasing). It can be accounted for phenomenologically by introducing an additional (damping) term in Eq. ( 3b ),
σ12 = -iǫσ 12 + iΩ 0 (1 -2σ 11 ) - Γ d 2 σ 12 ( 4
)
where Γ d is the decoherence rate. As a result the qubit density-matrix σ(t) becomes a statistical mixture in the stationary limit,
σ(t) t→∞ -→ 1/2 0 0 1/2 . ( 5
)
This happens for any initial conditions and even for large level displacement, ǫ ≫ Ω 0 , Γ d (provided that Ω 0 = 0). Note that the statistical mixture (5) is proportional to the unity matrix and therefore it remains the same in any basis. Secondly, the environment can put the qubit in its ground state, for instance via photon or phonon emission. This process is usually referred to as relaxation.
For a symmetric qubit we would have σ(t) t→∞ -→ 1/2 1/2 1/2 1/2 . (6) In contrast with decoherence, Eq. ( 5 ), the relaxation process puts the qubit into a pure state. That implies that the corresponding density matrix can be always written as δ 1i δ 1j in a certain basis (the basis of the qubit eigenstates). This is in fact the essential difference between decoherence and relaxation. With respect to elimination of the off-diagonal density matrix elements, note that relaxation would eliminate these terms only in the qubit's eigenstates basis. In contrast, decoherence eliminates the off-diagonal density matrix element in any basis (Eq. (5)). In fact, if the environment has some energy, it can put the qubit into an exited state. However, if the qubit is finally in a pure state, such excitation process generated by the environment affects the qubit in the same way as 4 relaxation: it eliminates the off-diagonal density matrix elements only in a certain qubit's basis. Therefore excitation of the qubit can be described phenomenologically on the same footing as relaxation.
It is often claimed that decoherence is associated with an absence of energy transfer between the system and the environment, in contrast with relaxation (excitation). This distinction is not generally valid. For instance, if the initial qubit state corresponds to the electron in the state |E 2 , Fig. 1 , the final state after decoherence corresponds to an equal distribution between the two dots, E = (E 1 + E 2 )/2. In the case of E 1 ≫ E 2 , this process would require a large energy transfer between the qubit and the environment. Therefore decoherence can be consistently defines as a process leading to a statistical mixture, where all states of the system have equal probabilities (as in Eq. (5)).
The relaxation (excitation) process can be described most simply by diagonalizing the qubit Hamiltonian, Eqs. ( 1
), to obtain H qb = E + a † + a + + E -a † -a -, where
the operators a ± are obtained by the corresponding rotation of the operators a 1,2 30 . Here E + and E -are the ground (symmetric) and excited (antisymmetric) state energies. Then the relaxation process can be described phenomenologically in the new qubit basis
|± = a † ± |0 ¯ as σ--(t) = -Γ r σ --(t) ( 7a
) σ+-(t) = i(E --E + )σ +-(t) - Γ r 2 σ +-(t) , ( 7b
)
where σ ++ (t) = 1σ --(t), σ -+ (t) = σ * +-(t) and Γ r is the relaxation rate.
In order to add decoherence, we return to the original qubit basis |1, 2 = a † 1,2 |0 ¯ and add the damping term to the equation for the off-diagonal matrix elements, Eq. ( 4 ). We arrive at the quantum rate equation describing the qubit's behavior in the presence of both decoherence and relaxation 30, 33 ,
σ11 = iΩ 0 (σ 21 -σ 12 ) -Γ r κǫ 2ǫ (σ 12 + σ 21 ) - Γ r 4 1 + ǫ ǫ 2 (2σ 11 -1) + Γ r ǫ 2ǫ ( 8a
) σ12 = -iǫσ 12 + iΩ 0 + Γ r κǫ 2ǫ (1 -2σ 11 ) + Γ r κ - 1 2 σ 12 -κ 2 (σ 12 + σ 21 ) - Γ d 2 σ 12 , ( 8b
)
where ǫ = (ǫ 2 + 4Ω 2 0 ) 1/2 and κ = Ω 0 /ǫ. In fact, these equations can be derived in the framework of a particular model, representing an electrostatic qubit interacting with the point-contact detector and the environment, described by the Lee model Hamiltonian 33 .
Equations (8) can be rewritten in a simpler form by mapping the qubit density matrix σ = {σ 11 , σ 12 , σ 21 } to a "polarization" vector S(t) via σ(t) = [1 + τ • S(t)]/2, where τ x,y,z are the Pauli matrices. For instance, one obtains for the symmetric case, ǫ = 0,
Ṡz = - Γ r 2 S z -2Ω 0 S y ( 9a
) Ṡy = 2Ω 0 S z - Γ d + Γ r 2 S y ( 9b
) Ṡx = - Γ d + 2Γ r 2 (S x -Sx ) ( 9c
)
where Sx = S x (t → ∞) = 2Γ r /(Γ d +2Γ r ). One finds that Eqs. (9) have a form of the Bloch equations for spinprecession in the magnetic field 27 , where the effect of environment is accounted for by two relaxation times for the different spin components: the longitudinal T 1 and the transverse T 2 , related to Γ d and 2Γ r as
T -1 1 = Γ d + 2Γ r 2 , and T -1 2 = Γ d + Γ r 2 , ( 10
)
The corresponding damping rates, the so-called "depolarization" (Γ 1 = 1/T 1 ) and the "dephasing" (Γ 2 = 1/T 2 ) are used for phenomenological description of two-level systems 34 . However, neither Γ 1 nor Γ 2 taken alone would drive the qubit density matrix into a statistical mixture Eq. ( 5 ) or into a pure state Eq. ( 6 ). In contrast, our definition of decoherence and relaxation (excitation) is associated with two opposite effects of the environment on the qubit: the first drives it into a statistical mixture, whereas the second drives it into a pure state. We expect therefore that such a natural distinction between decoherence and relaxation would be more useful for finding a relation between these quantities and the environmental behavior than other alternative definitions of these quantities existing in the literature.
In general, the two rates, Γ d,r , introduced in phenomenological equations (8), (9), are consistent with our definitions of decoherence and relaxation. The only exception is the case of Γ r = 0 and Ω 0 = 0, where are no transitions between the qubit's states even in the presence of the environment ("static" qubit). One easily finds from Eqs. (3a), (4) that σ 12 (t) → 0 for t → ∞, whereas the diagonal density-matrix elements of the qubit remain unchanged (so-called "pure dephasing" 5,34 ):
σ(t) t→∞ -→ σ 11 (0) 0 0 σ 22 (0) . ( 11
)
5 Thus, if the initial probabilities of finding the qubit in each of its states are not equal, σ 11 (0) = σ 22 (0), then the final qubit state is neither a mixture nor a pure state, but a combination of the both. It implies that Γ d in Eqs. (8) would also generate relaxation (excitation) of the qubit. Note that in this case the off-diagonal density-matrix elements, absent in Eq.( 11 ), would reappear in a different basis. This implies that the "pure dephasing" 5,34 occurs only in a particular basis.
Let us evaluate the probability of finding the electron in the first dot, σ 11 (t). Solving Eqs. (9) for the initial conditions σ 11 (0) = 1, σ 12 (0) = 0, we find 33 :
σ 11 (t) = 1 2 + e -Γrt/2 4 C 1 e -e-t + C 2 e -e+t ( 12
)
where e ± = 1 4 (Γ d ± Ω), Ω = Γ 2 d -64Ω 2 0 and C 1,2 = 1±(Γ d / Ω). Solving the same equations in the limit of t → ∞, we find that the steady-state qubit density matrix is
σ(t) t→∞ -→ 1/2 Γ r /(Γ d + 2Γ r ) Γ r /(Γ d + 2Γ r ) 1/2 . ( 13
)
Thus the off-diagonal elements of the density matrix can provide us with a ratio of relaxation to decoherence rates 33 .
Consider the setup shown in Fig. 2 . The entire system can be described by the following tunneling Hamiltonian, represented by a sum of the qubit and SET Hamiltonians and the interaction term, 1 ) and describes the qubit. The second term, H SET , describes the single-electron transistor. It can be written as
H = H qb + H SET + H int . Here H qb is given by Eq. (
H SET = l E l c † l c l + r E r c † r c r + E 0 c † 0 c 0 + l,r (Ω l c † l c 0 + Ω r c † r c 0 + H.c.) , ( 14
)
where c † l,r and c l,r are the creation and annihilation electron operators in the state E l,r of the right or left reservoir; c † 0 and c 0 are those for the level E 0 inside the quantum dot; and Ω l,r are the couplings between the level E 0 and the level E l,r in the left (right) reservoir. In order to avoid too lengthy formulaes, our summation indices l, r indicate simultaneously the left and the right leads of the SET, where the corresponding summation is carried out. As follows from the Hamiltonian (14), the quantum dot of the SET contains only one level (E 0 ). This assumption has been implied only for the sake of simplicity for our presentation, although our approach is well suited for a case of n levels inside the SET,
E 0 c † 0 c 0 → n E n c † n c n ,
and even when the interaction between these levels is included (providing that the latter is much less or much larger than the bias V ) 35, 36 . We also assumed a weak energy dependence of the couplings Ω l,r ≃ Ω L,R .
The interaction between the qubit and the SET, H int , depends on a position of the SET with respect to the qubit. If the SET is placed near the middle of the qubit, Fig. 2a , then the tunneling coupling between two dots of the qubit in Eq. ( 1 ) decreases, Ω 0 → Ω 0 -δΩ 0 , whenever the quantum dot of the SET is occupied by an electron. This is due to the electron's repulsive field. In this case the interaction term can be written as
H int = δΩ c † 0 c 0 (a † 1 a 2 + a † 2 a 1 ) . ( 15
)
On the other hand, in the configuration shown in Fig. 2b where the SET is placed near one of the dots of the qubit, the electron repulsive field displaces the qubit energy levels by ∆E = U . The interaction terms in this case can be written as
H int = U a † 1 a 1 c † 0 c 0 . ( 16
)
Consider the initial state where all the levels in the left and the right reservoirs are filled with electrons up to the Fermi levels µ L,R respectively. This state will be called the "vacuum" state |0 ¯ . The wave function for the entire system can be written as
|Ψ(t) = b (1) (t)a † 1 + l b ( 1
) 0l (t)a † 1 c † 0 c l + l,r b ( 1
) rl (t)a † 1 c † r c l + l<l ′ ,r b ( 1
) 0rll ′ (t)a † 1 c † 0 c † r c l c l ′ + • • • +b (2) (t)a † 2 + l b ( 2
) 0l (t)a † 2 c † 0 c l + l,r b ( 2
) rl (t)a † 2 c † r c l + l<l ′ ,r b ( 2
) 0rll ′ (t)a † 2 c † 0 c † r c l c l ′ + . . . |0 ¯ , ( 17
)
where b (j) (t), b (j) α (t) are the probability amplitudes to find the entire system in the state described by the cor-responding creation and annihilation operators. These amplitudes are obtained from the Schrödinger equation 6 i| Ψ(t) = H|Ψ(t) , supplemented with the initial condition b (1) (0) = p 1 , b (2) (0) = p 2 , and b (j) α (0) = 0, where p 1,2 are the amplitudes of the initial qubit state.
Note that Eq. (17) implies a fixed electron number (N ) in the reservoirs. At the first sight it would lead to depletion of the left reservoir of electrons over the time. Yet in the limit of N → ∞ (infinite reservoirs) the dynamics of an entire system reaches its steady state before such a depletion takes place 37, 38 .
The behavior of the qubit and the SET is given by the reduced density matrix, σ ss ′ (t). It is obtained from the entire system's density matrix |Ψ(t) Ψ(t)| by tracing out the (continuum) reservoir states. The space of such a reduced density matrix consists of four discrete states s, s ′ = a, b, c, d, shown schematically in Fig. 3 for the setup of Fig. 2a . The corresponding density-matrix elements are directly related to the amplitudes b(t), for instance,
σ aa (t) = |b (1) (t)| 2 + l,r |b (1) lr (t)| 2 + l<l ′ ,r<r ′ |b ( 1
) rr ′ ll ′ (t)| 2 + • • • ( 18a
)
σ dd (t) = l |b ( 2
) 0l (t)| 2 + l<l ′ ,r |b ( 2
) 0rll ′ (t)| 2 + l<l ′ <l ′′ ,r<r ′ |b ( 2
) 0rr ′ ll ′ (t)| 2 + • • • ( 18b
)
σ bd (t) = l b ( 1
) 0l (t)b (2) * 0l (t) + l<l ′ ,r b ( 1
)
0rll ′ (t)b (2) * 0rll ′ (t) +
l<l ′ <l ′′ ,r<r ′ b ( 1
) 0rr ′ ll ′ (t)b (2) * 0rr ′ ll ′ (t) + • • • . ( 18c
)
In was shown in 37, 38 that the trace over the reservoir states in the system's density matrix can be performed in the large bias limit (strong non-equilibrium limit)
V = µ L -µ R ≫ Γ, Ω 0 , U ( 19
)
where the level (levels) of the SET carrying the current are far away from the chemical potentials, and Γ is the width of the level E 0 . In this derivation we assumed only weak energy dependence of the transition amplitudes Ω l,r ≡ Ω L,R and the density of the reservoir states, ρ(E l,r ) = ρ L,R . As a result we arrive at Bloch-type rate equations for the reduced density matrix without any additional assumptions. The general form of these equations is 36,38
σjj ′ = i(E j ′ -E j )σ jj ′ + i k σ jk Ωk→j ′ -Ωj→k σ kj ′ - k,k ′ P 2 πρ(σ jk Ω k→k ′ Ω k ′ →j ′ + σ kj ′ Ω k→k ′ Ω k ′ →j ) + k,k ′ P 2 πρ (Ω k→j Ω k ′ →j ′ + Ω k→j ′ Ω k ′ →j )σ kk ′ ( 20
)
Here Ω k→k ′ denotes the single-electron hopping amplitude that generates the k → k ′ transition. We distinguish between the amplitudes Ω describing single-electron hopping between isolated states and Ω describing transitions between isolated and continuum states. The latter can generate transitions between the isolated states of the system, but only indirectly, via two consecutive jumps of an electron, into and out of the continuum reservoir states (with the density of states ρ). These transitions are represented by the third and the fourth terms of Eq. ( 20 ). The third term describes the transitions (k
→ k ′ → j) or (k → k ′ → j ′ )
, which cannot change the number of electrons in the collector. The fourth term describes the transitions (k → j and k ′ → j ′ ) or (k → j ′ and k ′ → j) which increase the number of electrons in the collector by one. These two terms of Eq. (20) are analogues of the "loss" (negative) and the "gain" (positive) terms in the classical rate equations, respectively. The factor P 2 = ±1 in front of these terms is due anti-commutation of the fermions, so that P 2 = -1 whenever the loss or the gain terms in Eq. (20) proceed through a two-fermion state of the dot. Otherwise P 2 = 1. Note that the reduction of the time-dependent Schrödinger equation, i| Ψ(t) = H|Ψ(t) , to Eqs. (20) is performed in the limit of large bias without explicit use of any Markov-type or weak coupling approximations. The accuracy of these equations is respectively max
(Γ, Ω 0 , U, T )/|µ L,R -E j |.
A detailed example of this derivation is presented in Appendix A for the case of resonant tunneling through a single level. The derivation there and in Refs. 37, 38 were performed by assuming zero temperature in the leads, T = 0. Yet, this assumption is not important in the case of large bias, providing the levels carrying the current are far away from the Fermi
energies, |µ L,R -E j | ≫ T .
A. Fluctuation of the tunneling coupling Now we apply Eqs. (20) to investigate the qubit's behavior in the configurations shown in Fig. 2 . First we consider the SET placed near the middle of the qubit, Figs. 2a,3. In this case the electron current through the SET will influence the coupling between two dots of the 7 (d) (b) (a) (c)
R µ L µ R E 0 Γ L Γ E E 1 2 Ω 0 Ω 0 Ω 0 Ω 0 ' ' FIG. 3:
The available discrete states of the entire system corresponding to the setup of Fig. 2a . ΓL,R denote the tunneling rates to the corresponding reservoirs and Ω ′ 0 = Ω0 -δΩ.
qubit, making it fluctuate between the values Ω 0 and Ω ′ 0 = Ω 0 -δΩ. The corresponding rate equations can be written straightforwardly from Eqs. (20). One finds,
σaa = -Γ L σ aa + Γ R σ bb -iΩ 0 (σ ac -σ ca ), (21a) σbb = -Γ R σ bb + Γ L σ aa -iΩ ′ 0 (σ bd -σ db ), (21b) σcc = -Γ L σ cc + Γ R σ dd -iΩ 0 (σ ca -σ ac ), (21c) σdd = -Γ R σ dd + Γ L σ cc -iΩ ′ 0 (σ db -σ bd ), (21d) σac = -iǫ 0 σ ac -iΩ 0 (σ aa -σ cc ) -Γ L σ ac + Γ R σ bd , (21e) σbd = -iǫ 0 σ bd -iΩ ′ 0 (σ bb -σ dd ) -Γ R σ bd + Γ L σ ac , ( 21f
) where Γ L,R = 2π|Ω L,R | 2 ρ L,R
are the tunneling rates from the reservoirs and ǫ 0 = E 1 -E 2 . These equations display explicitly the time evolution of the SET and the qubit. The evolution of the former is driven by the first two terms in Eqs. (21a)-(21d). They generate charge-fluctuations inside the quantum dot of the SET (the transitions a←→b and c←→d), described by the "classical" Boltzmann-type dynamics. The qubit's evolution is described by the Bloch-type terms (c.f. Eqs. (3)), generating the qubit transitions (a←→c and b←→d). Thus Eqs. (21) are quite general, since they described fluctuations of the tunneling coupling driven by the Boltzmann-type dynamics.
The resulting time evolution of the qubit is given by the qubit (reduced) density matrix:
σ 11 (t) = σ aa (t) + σ bb (t) , (22a) σ 12 (t) = σ ac (t) + σ bd (t) , ( 22b
) and σ 22 (t) = 1 -σ 11 (t).
Similarly, the charge fluctuations of SET are determined by the probability of finding the SET occupied,
P 1 (t) = σ bb (t) + σ dd (t) . ( 23
)
It is given by the equation
Ṗ1 (t) = Γ L -ΓP 1 (t) , ( 24
)
obtained straightforwardly from Eqs. (21). Here Γ = Γ L + Γ R is the total width. The same equation for P 1 (t) can be obtained if the qubit is decoupled from the SET (δΩ = 0). Thus there is no back-action of the qubit on the charge fluctuations inside the SET in the limit of large bias voltage. Consider first the stationary limit, t → ∞, where Ṗ1 (t) → 0 and σ(t) → 0. It follows from Eq. (24) that the probability of finding the SET occupied in this limit is P1 = Γ L /Γ. This implies that the fluctuations of the coupling Ω 0 , induced by the SET, would take place around the average value Ω = Ω 0 -P1 δΩ.
With respect to the qubit in the stationary limit, one easily obtains from Eqs. (21) that the qubit density matrix always becomes the statistical mixture (5), when t → ∞. This takes place for any initial conditions and any values of the qubit and the SET parameters. Therefore the effect of the fluctuating charge inside the SET does not lead to relaxation of the qubit, but rather to its decoherence.
It is important to note, however, that for the aligned qubit, ǫ = 0, the decoherence due to fluctuations of the tunneling coupling Ω 0 is not complete. Indeed, it follows from Eqs. (21) that d/dt[Re σ 12 (t)] = 0. The reason is that the corresponding operator, a † 14) and (15) 0).
1 a 2 + a † 2 a 1 commutes with the total Hamiltonian H = H qb + H SET + H int , Eqs. (1), (
, for E 1 = E 2 . As a result, Re σ 12 (t) = Re σ 12 (
In order to determine the decoherence rate analytically, we perform a Laplace transform on the density matrix, σ(E) = ∞ 0 σ(t) exp(-iEt)dE. Then solving Eq. (21) we can determine the decoherence rate from the locations of the poles of σ(E) in the complex E-plane. Consider for instance the case of ǫ 0 = 0 and the symmetric SET, Γ L = Γ R = Γ/2. One finds from Eqs. (21) and (22a) that
σ11 (E) = i 2E + i(E -2Ω + iΓ) 4(E -2Ω + iΓ/2) 2 + Γ 2 -(2 δΩ) 2 + i(E + 2Ω + iΓ) 4(E + 2Ω + iΓ/2) 2 + Γ 2 -(2 δΩ) 2 . ( 25
)
Upon performing the inverse Laplace transform,
σ 11 (t) = ∞+i0 -∞+i0 σ11 (E) e -iEt dE 2πi , ( 26
)
and closing the integration contour around the poles of the integrand, we obtain for Γ > 2δΩ and t ≫ 1/Γ
σ 11 (t) -(1/2) ∝ e -(Γ- √ Γ 2 -4δΩ 2 )t/2 sin(2Ω t). ( 27
)
Comparing this result with Eq. ( 12 ) we find that the decoherence rate is
Γ d = 2 Γ -Γ 2 -4δΩ 2 Γ≫δΩ -→ (2δΩ) 2 /Γ . ( 28
) For ǫ 0 = 0 and ǫ 0 , Γ ≪ Ω the decoherece rate Γ d is multiplied by an additional factor [1 -(ǫ 0 /2Ω) 2 ].
8 In a general case, Γ L = Γ R , we obtain in the same limit (Γ L,R ≫ δΩ) for the decoherence rate:
Γ d = (4 δΩ) 2 1 + ǫ 0 2Ω 2 Γ L Γ R (Γ L + Γ R ) 3 ( 29
)
It is interesting to compare this result with the fluctuation spectrum of the charge inside the SET, Eq. ( B8 ), Appendix B. We find
Γ d = 2 (δω R ) 2 S Q (0) , ( 30
)
where ω R = 4Ω 2 + ǫ 2 0 is the Rabi frequency. The latter represents the energy splitting in the diagonalized qubit Hamiltonian. Thus δω R corresponds to the amplitude of energy level fluctuations in a single dot. Although Eq. (30) has been obtained for small fluctuations δω R , it might be approximately correct even if δω R is of the order of Γ. It is demonstrated in Fig. 4 , where we compare σ 11 (t) and σ 12 (t), obtained from Eqs. (21) and (22) (solid line) with those from Eqs. (3a) and (4) (dashed line) for the decoherence rate Γ d given by Eq. (30). The initial conditions correspond to
σ 11 (0) = 1 and σ 12 (0) = 0 (respectively, σ aa (0) = Γ R /Γ and σ bb (0) = Γ R /Γ).
In the case of aligned qubit, however, Re σ 12 (t) = Re σ 12 (0), as was explained above. On the other hand, one always obtains from (3a) and (4) that Re [σ 12 (t → ∞)] = 0. Therefore the phenomenological Bloch equations are not applicable for evaluation of Re [σ 12 (t)], even in the weak coupling limit (besides the case of Re [σ 12 (t = 0)] = 0).
In the large coupling regime (δΩ ≫ Γ) the phenomenological Bloch equations, Eqs. (3a) and (4), cannot be used, as well. Consider for simplicity the case of ǫ = 0 and Γ L,R = Γ/2. Then one finds from Eq. (27) that the damping oscillations between the two dots take place at two different frequencies, 2Ω ± (δΩ) 2 -(Γ/2) 2 , instead of the one frequency, ω R = 2Ω, given the Bloch equations. Moreover, Eq. (30) does not reproduce the decoherence (damping) rate in this limit. Indeed, one obtains from Eq. (28) that the decoherence rate Γ d = 2Γ for δΩ > Γ/2, so Γ d does not depend on the coupling (δΩ) at all.
Consider the SET placed near one of the qubit dots, as shown in Fig. 2b . In this case the qubit-SET interaction term is given by Eq. ( 16 ). As a result the energy level E 1 will fluctuate under the influence of the fluctuations of the electron charge inside the SET. The available discrete states of the entire system are shown in Fig. 5 . Using Eqs. (20) we can write the rate equations, similar 10 20 30 40 50 t 0.4 0.2 0 0.2 0.4 (a) (b) 10 20 30 40 50 t 0.4 0.2 0 0.2 0.4 0.6 0.8 1 t t t Re 12 Im 12 11 σ ( ) σ ( ) σ ( ) FIG. 4: The occupation probability of the first dot of the qubit for ǫ = 2Ω, ΓL = Ω, ΓR = 2Ω and δΩ = 0.5Ω. The solid line is the exact result, whereas the dashed line is obtained from the Bloch-type rate equations with the decoherence rate given by Eq. (30).
(d) (b) (a) (c) ' ' ' '
Γ L Γ R Γ L Γ R E 0 E 0 R Γ L Γ L Γ R Γ 0 Ω 0 Ω 0 µ L µ R E 1 E 2 E 2 E 1 E 2 E 1 E 2 E +U 1 Ω 0 Ω FIG. 5:
The available discrete states of the entire system for the configuration shown in Fig. 2b . Here U is the repulsion energy between the electrons.
to Eqs. (21),
σaa = -Γ ′ L σ aa + Γ ′ R σ bb -iΩ 0 (σ ac -σ ca ), (31a) σbb = -Γ ′ R σ bb + Γ ′ L σ aa -iΩ 0 (σ bd -σ db ), (31b) σcc = -Γ L σ cc + Γ R σ dd -iΩ 0 (σ ca -σ ac ), (31c) σdd = -Γ R σ dd + Γ L σ cc -iΩ 0 (σ db -σ bd ), ( 31d
) σac = -iǫ 0 σ ac -iΩ 0 (σ aa -σ cc ) - Γ L + Γ ′ L 2 σ ac + Γ R Γ ′ R σ bd , ( 31e
) σbd = -i(ǫ 0 + U )σ bd -iΩ 0 (σ bb -σ dd ) - Γ R + Γ ′ R 2 σ bd + Γ L Γ ′ L σ ac , ( 31f
)
where Γ ′ L,R are the tunneling rate at the energy E 0 +U 39 . Let us assume that Γ ′ L,R = Γ L,R . Then it follows from Eqs. (31) that the behavior of the charge inside the SET is not affected by the qubit, the same as in the previous case of the Rabi frequency fluctuations. Also the qubit density matrix becomes the mixture (5) in the stationary state for any values of the qubit and the SET parameters. Hence, there is no qubit relaxation in this case either (except for the static qubit, Ω 0 = 0, and
σ 11 (0) = σ 22 (0), Eq. (11)).
Since according to Eq. ( 24 ), the probability of finding an electron inside the SET in the stationary state is P1 = Γ L /Γ, the energy level E 1 of the qubit is shifted by P1 U . Therefore it is useful to define the "renormalized" level displacement, ǫ = ǫ 0 + P1 U . 9 As in the previous case we use the Laplace transform, σ(t) → σ(E), in order to determine the decoherence rate analytically. In the case of Γ L = Γ R = Γ/2 and ǫ = 0 we obtain from Eqs. (31) σ11 (E) = i 2E + i 2E + 32(E + iΓ)Ω 2 0
U 2 -4E(E + iΓ) . ( 32
)
The position of the pole in the second term of this expression determines the decoherence rate. In contrast with Eq. (25), however, the exact analytical expression for the decoherence rate (Γ d ) is complicated, since it is given by a cubic equation. We therefore evaluate Γ d in a different way, by substituting E = ±2Ω 0iγ in the second term of Eq. ( 32 ) and then expanding the latter in powers of γ by keeping only the first two terms of this expansion. The decoherence rate Γ d is related to γ by Γ d = 4γ, as follows from Eq. ( 12 ). Then we obtain:
Γ d = U 2 Γ 2(Γ 2 + 4Ω 2 0 ) for U ≪ (Ω 2 0 + ΓΩ 0 ) 1/2 64ΓΩ 2 0 U 2 + 16Ω 2 0 for U ≫ (Ω 2 0 + ΓΩ 0 ) 1/2 ( 33
)
In general, if Γ L = Γ R , one finds from Eqs. (31) that
Γ d = 2U 2 Γ L Γ R /[Γ(Γ 2 + 4Ω 2 0 )] for U ≪ (Ω 2 0 + ΓΩ 0 ) 1/2 .
The same as in the previous case, Eq. ( 30 ), the decoherence rate in a weak coupling limit is related to the fluctuation spectrum of the SET, S Q (ω), Eq. (B8), but now taken at a different frequency, ω = 2Ω 0 . The latter corresponds to the level splitting of the diagonalized qubit's Hamiltonian, ω R . Thus,
Γ d = U 2 S Q (ω R ) , ( 34
)
which can be applied also for ǫ = 0. This is illustrated by Fig. 6 which shows σ 11 (t) obtained from Eqs. (31) and (22) (solid line) with Eqs. (3a) and (4) (dashed line) for the decoherence rate Γ d given by Eq. ( 34 ). As in the previous case, shown in Fig. 4 , the initial conditions correspond to σ 11 (0) = 1 and σ 12 (0) = 0 (respectively, σ aa (0) = Γ R /Γ and σ bb (0) = Γ R /Γ). One finds from Fig. 6 that Eq. (34) can be used for an estimation of Γ d even for U ∼ Γ, Ω 0 . In contrast with the tunneling-coupling fluctuations, Eq. (30), where the decoherence rate is given by S Q (0), the fluctuations of the qubit's energy level generate the decoherence rate, determined by the fluctuation spectrum at Rabi frequency, S Q (ω R ), Eq. (34). A similar distinction between the decoherence rates generated by different components of the fluctuating field, exists in a phenomenological description of magnetic resonance 27 . One can understand this distinction by diagonalizing the qubit's Hamiltonian. In this case the Rabi frequency, ω R , becomes the level splitting of the qubit's states |± = (|1 ± |2 )/ √ 2 (for ǫ = 0). So in this basis, the tunneling-coupling fluctuations correspond to simultaneous fluctuations of the energy levels in the both dots.
10 20 30 40 50 t 0 0.4 0.2 0 0.2 0.4 (a) (b) 10 20 30 40 50 t 0 0.4 0.2 0 0.2 0.4 0.6 0.8 1 t t t Re 12 Im 12 11 σ ( ) σ ( ) σ ( ) FIG. 6: The probability of finding the electron in the first dot of the qubit for ǫ = 2Ω0, ΓL = Ω0, ΓR = 2Ω0 and U = 0.5Ω0. The solid line is the exact result, whereas the dashed line is obtained from the Bloch-type rate equations with the decoherence rate given by Eq. (34).
Since these fluctuations are "in phase", we could expect that the corresponding dephasing rate is determined by spectral density at zero frequency. In fact, it looks like as fluctuations of a single dot state, considered by Levinson in a weak coupling limit 31 . On the other hand by fluctuating the energy level in one of the dots only, one can anticipate that the corresponding dephasing rate is determined by the fluctuation spectrum at the Rabi frequency, ω R , Eq. ( 34 ), which is a frequency of the inter-dot transitions. Since ω R can be controlled by the qubit's levels displacement, ǫ, the relation (34) can be implied by using qubit for a measurement of the shot-noise spectrum of the environment 18, 19, 40 . For instance, it can be done by attaching a qubit to reservoirs at different chemical potentials. The corresponding resonant current which would flow through the qubit in this case, can be evaluated via a simple analytical expression 13 that includes explicitly the decoherence rate, Eq. (34). Thus by measuring this current for different level displacement of the qubit (ǫ 0 ), one can extract the spectral density of the fluctuating environment acting on the qubit 18 . Although Eq. ( 34 ) for the decoherence rate has been obtained by using a particular mechanism for fluctuations of the qubit's energy levels, we suggest that this mechanism is quite general. Indeed, the rate equations (31) can describe any fluctuating media near a qubit, driven by the Boltzmann type of equations. Therefore it is rather natural to assume that Eq. (34) would be valid for any type of such (classical) environment in weak coupling limit. This implies that the decoherence rate is always determined via the spectral density of a fluctuating qubit's level, whereas the nature of a particular medium inducing these fluctuations would be irrelevant. In order to substantiate this point it is important to compare Eq. ( 34 ) with the corresponding decoherence rate induced by the thermal environment in the framework of the spin-boson model. In a weak damping limit this model predicts 1,2 T -1 1 = T -1 2 = (q 2 0 /2)S(ω R ) , where q 0 is a coupling of the medium with the qubit levels (q 0 corresponds to U in our case) and S(ω) is a spectral density. Using Eq. (10) one finds that this result coincides with Eq. (34).
10 0 5 10 15 20 t 0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 t 0 0.2 0.4 0.6 0.8 1.0 t t 11 11 0 0 U/ =100 Ω U/ =10 σ ( ) Ω σ ( ) (a) (b)
FIG. 7: The probability of finding the electron in the first dot of the qubit for ǫ = 0, ΓL = ΓR = Ω0 and U , as given by Eqs. (31) (solid line) and from the Bloch-type equations (dashed line) with the decoherence rate given by Eq. (33).
Let us consider the limit of U ≫ (Ω 2 0 + ΓΩ 0 ) 1/2 . Our rate equation (31) are perfectly valid in this region, providing only that E 0 + U is deeply inside of the potential bias, Eq. (19). We find from Eq. (33) that the decoherence rate is not directly related to the spectrum of fluctuations in strong coupling limit. In addition, the effective frequency of the qubit's Rabi oscillations (ω ef f R ) decreases in this limit. Indeed, by using Eqs. (32), (26), one finds that the main contribution to σ 11 (t), is coming from a pole of σ11 (E), which lies on the imaginary axis. This implies that the effective frequency of Rabi oscillations strongly decreases when U ≫ (Ω 2 0 + ΓΩ 0 ) 1/2 . In addition, the decoherence rate Γ d → 0 in the same limit, Eq. (33). As a result, the electron would localize in the initial qubit state, Fig. 7 .
The results displayed in this figure show that the solution of the Bloch-type rate equations, with the decoherence rate given by Eq. (33), represents damped oscillations (dashed line). It is very far from the exact result (solid line), obtained from Eqs. (31) and corresponding to the electron localization in the first dot. The latter is a result of an effective decrease of the Rabi frequency for large U that slows down electron transitions between the dots. Thus such an environment-induced localization is different from the Zeno-type effect (unlike an assumption of Ref. 12 ). Indeed, the Zeno effect takes place whenever the decoherence rate is much larger then the coupling between the qubit's states 13, 33 . However, the decoherence rate in the strong coupling limit is much smaller then the coupling Ω 0 . In fact, the localization shown in Fig. 7 is rather similar to that in the spin-boson model 1, 2 . It shows that in spite of their defferences, both models trace the same physics of the back-action of the environment (SET) on the qubit.
V. BACK-ACTION OF THE QUBIT ON THE ENVIRONMENT A. Weak back-action effect
Now we investigate a weak dependence of the width's Γ L,R on the energy U , Fig. 5 . We keep only the linear term, Γ ′ L,R = Γ L,R +α L,R U , by assuming that U is small. (A similar model has been considered in 28, 41 ). In contrast with the previous examples, where the widths have not been dependent on the energy, the qubit's oscillation would affect the SET current and its charge correlator. A more interesting case corresponds to α L = α R . Let us take for simplicity α L = 0 and α R = α = 0. Similarly to the previous case we introduce the "renormalized" level displacement, ǫ = ǫ 0 -(Γ L /Γ)U , where ǫ = 0 corresponds to the aligned qubit. Solving Eqs. (31) in the steady-state limit, σ = σ(t → ∞), and keeping only the first term in expansion in powers of U , we find for the reduced density matrix of the qubit, Eqs. (22):
σ = 1 2 - α ǫ 4Γ R αΩ 0 (1 + c α U ) 2Γ R αΩ 0 (1 + c α U ) 2Γ R 1 2 + α ǫ 4Γ R , ( 35
)
where c = (αǫ -2Γ)/(4Γ R Γ). It follows from Eqs. (35) that the qubit's density matrix in the steady-state is no longer a mixture, Eq. ( 5 ) . Indeed, the probability to occupy the lowest level is always larger than 1/2 and σ12 = 0. This implies that relaxation takes place together with decoherence. The ratio of the relaxation and decoherence rates is given by the off-diagonal terms of the reduced density matrix of the qubit. For ǫ = 0 one finds from Eq. (13) that Γ d /Γ r = σ-1 12 -2. In order to find a relation between the decoherence and relaxation rates, Γ d,r , and the fluctuation spectrum of the qubit energy level, S Q (ω), we first evaluate the damping rate of the qubit's oscillations (γ). Using Eq. ( 12 ) we find that this quantity is related to the decoherence and relaxation rates by γ = (Γ d + 2Γ r )/4. The same as in the previous case the rate γ determined by poles of Laplace transformed density matrix σ(t) → σ(E) in the complex E-plane. Consider for simplicity the case of ǫ = 0 and Γ L = Γ R = Γ/2. Performing the Laplace transform of Eqs. (31) we look for the poles of σ 11 (E) at E = ±2Ω 0iγ by assuming that γ is small. We obtain
Γ d + 2Γ r = U 2 2(Γ 2 + 4Ω 2 0 ) Γ -α U Γ 2 -4Ω 2 0 2 (Γ 2 + 4Ω 2 0 ) ( 36
)
for U ≪ Ω 0 . Now we evaluate the correlator of the charge inside the SET, S Q (ω) which induces the energy-level fluctuations of the qubit. Using Eqs. (31) and (B6) we find,
S Q (ω) = Γ 2 (Γ 2 + ω 2 ) -α U Γ 2 -ω 2 4 (Γ 2 + ω 2 ) 2 ( 37
)
11 for αU ≪ Γ. Therefore in the limit of U ≪ Ω 0 and α U ≪ Γ the total damping rate of the qubit's oscillations is directly related to the spectral density of the fluctuations spectrum taken at the Rabi frequency,
Γ d + 2Γ r = U 2 S Q (2Ω 0 ). ( 38
)
This represents a generalization of Eq. (34) for the case of a weak back-action of qubit oscillations on the spectral density of the environment. As a result, the qubit displays relaxation together with decoherence. It is remarkable that the total qubit's damping rate is still given by the fluctuation spectrum of the SET (environment) modulated by the qubit. Note that Eq. ( 38 ) can be applied only if the modulation of the tunneling rate through the SET (tunneling current) is small α U ≪ Γ, in addition to a weak distortion of the qubit (U ≪ Ω 0 ). In the case of strong back-action of the qubit on the environment the decorerence and relaxation rates of the qubit are not directly related to the fluctuation spectrum of the environment, even if the distortion of the qubit is small. This point is illustrated by the following example.
Until now we considered the case where E 0 + U ≪ µ L , so that the interacting electron of the SET remains deeply inside the voltage bias. If however, the interaction U between the qubit and the SET is such that E 0 + U ≫ µ L , the qubit's oscillation would strongly affect the fluctuation of charge inside the SET. Indeed, the current through the SET is blocked whenever the level E 1 of the qubit is occupied, Fig. 8 . In fact, this case can be treated with small modification of the rate equations (31)
, if only µ L -E 0 ≫ Γ and E 0 + U -µ L ≫ Γ, where E 0 is a level
of the SET carrying the current.
The corresponding quantum rate equations describing the system are obtained directly from Eqs. (20). Assuming that the widths Γ are energy independent we find 16
σaa = (Γ L + Γ R )σ bb -iΩ 0 (σ ac -σ ca ), (39a) σbb = -(Γ R + Γ L )σ bb -iΩ 0 (σ bd -σ db ), (39b) σcc = -Γ L σ cc + Γ R σ dd -iΩ 0 (σ ca -σ ac ), (39c) σdd = -Γ R σ dd + Γ L σ cc -iΩ 0 (σ db -σ bd ), ( 39d
) σac = -iǫ 0 σ ac -iΩ 0 (σ aa -σ cc ) - Γ L 2 σ ac + Γ R σ bd , (39e) σbd = -i(ǫ 0 + U )σ bd -iΩ 0 (σ bb -σ dd ) -Γ R + Γ L 2 σ bd . ( 39f
)
Solving Eqs. (39) in the stationary limit, σ = σ(t → ∞) and introducing the "renormalized" level displacement, ǫ = ǫ 0 -U Γ L /(2Γ), we obtain for the qubit's den-
(d) (b) (a) (c) 0 E 0 E 2 E R µ L µ R Γ L 1 Ω 0 E 2 E +U 1 Ω 0 E 1 Ω 0 E 2 E 1 Ω 0 E 2 E Γ FIG. 8:
The available discrete states of the entire system when the electron-electron repulsive interaction U breaks off the current through the SET.
sity matrix, Eqs. (22) in the steady state:
σ11 = 1 2 - 8ǫU 16ǫ 2 + 8U ǫ + 48Ω 2 0 + 9(U 2 + Γ 2 ) ,( 40a
) σ12 = 12U Ω 0 16ǫ 2 + 8U ǫ + 48Ω 2 0 + 9 (U 2 + Γ 2 ) , ( 40b
)
where for simplicity we considered the symmetric case, Γ L = Γ R = Γ/2. It follows from Eqs. (40) that similarly to the previous example, the qubit's density matrix is no longer a mixture (5). The relaxation takes place together with decoherence in this case too. Let us consider weak distortion of the qubit by the SET, U < Ω 0 . Although the values of U are restricted from below (U ≫ Γ+µ L -E 0 ), this limit can be achieved if the level E 0 is close to the Fermi energy, providing only that µ L -E 0 ≫ Γ, and Γ ≪ U . Now we evaluate σ 11 (t) with the rate equations (39) and then compare it with the same quantity obtained from the Bloch equations, Eq. ( 12 ), where Γ d,r are given by Eqs. (34)and (13). The corresponding charge-correlator, S Q (ω R ), is evaluated by Eqs. (B6) and (39). As an example, we take symmetric qubit with aligned levels, ǫ = 0, Γ L = Γ R = 0.05Ω 0 and U = 0.5Ω 0 . The decoherence and relaxation rates, corresponding to these parameters are respectively:
Γ d /Ω 0 = 0.0038 and Γ r /Ω 0 = 0.00059.
The results are presented in Fig. 9a . The solid line shows σ 11 (t), obtained from the rate equations (39), where the dashed line is the same quantity obtained from Eq. ( 12 ). We find that Eq. (34) (or (38)) underestimates the actual damping rate of σ 11 (t) by an order of magnitude). This lies in a sharp contrast with the previous case, where the energy level of the SET is not distorted by the qubit, Γ ′ L,R = Γ L,R , Fig. 5 . Indeed, in this case σ 11 (t) obtained Eq. (12) with Γ d given by Eq. ( 34 ) and Γ r = 0, agrees very well with that obtained from the rate equations (31), as shown in Fig. 9b . Such an example clearly illustrates that the decoherence is not related to the fluctuation spectrum of the environment, whenever the environment is strongly affected by the qubit, even if the coupling with a qubit is small. This is a typical case of measurement, corresponding to a noticeable response of the environment to the qubit's state (a "signal"). 12 (a) 20 40 60 80 100 t 0 0.2 0.4 0.6 0.8 1 20 40 60 80 100 t 0 0.2 0.4 0.6 0.8 1 (b) t t 11 11 σ ( ) σ ( ) FIG. 9: (a) The probability of finding the electron in the first dot of the qubit for ǫ = 0, ΓL = ΓR = 0.05Ω0 and U = 0.5Ω0. The solid line is obtained from Eqs. (39), whereas the dashed line corresponds to the Eq. ( 12 ) with Γ d given by Eq. (34); (b) the same for the case, shown in Fig. 5 , where the solid line corresponds to Eqs. (31).
In this paper we propose a simple model describing a qubit interacting with fluctuating environment. The latter is represented by a single electron transistor (SET) in close proximity of the qubit. Then the fluctuations of the charge inside the SET generate fluctuating field acting on the qubit. In the limit of large bias voltage, the Schrödinger equation for the entire system is reduced to the Bloch-type rate equations. The resulting equations are very simple, so that one can easily analyze the limits of weak and strong coupling of the qubit with the SET.
We considered separately two different cases: (a) there is no back-action of the qubit on the SET behavior, so that the latter represents a "pure environment"; and (b) the SET behavior depends on the qubit's state. In the latter case the SET can "measure" the qubit. The setup corresponding to the "pure environment" is realized when the energy level of the SET carrying the current lies deeply inside the potential bias. The second (measurement) regime of the SET is realized when the tunneling widths of the SET are energy dependent, or when the energy level of the SET carrying the current is close enough to the Fermi level of the corresponding reservoir. Then the electron-electron interaction between the qubit and the SET modulates the electron current through the SET.
In the case of the "pure environment" ("nomeasurement" regime) we investigate separately two different configurations of the qubit with respect to the SET. In the first one the SET produces fluctuations of the off-diagonal coupling (Rabi frequency) between two qubit's states. In the second configuration the SET produces fluctuations of the qubit's energy levels. In the both cases we find no relaxation of the qubit, despite the energy transfer between the qubit and the SET can take place. As a result the qubit always turns asymptotically to the statistical mixture. We also found that in both cases the decoherence rate of the qubit in the weak coupling limit is given by the spectral density of the cor-responding fluctuating parameter. The difference is that in the case of the off-diagonal coupling fluctuations the spectral density is taken at zero frequency, whereas in the case of the energy level fluctuations it is taken at the Rabi-frequency.
In the case of the strong coupling limit, however, the decoherence rate is not related to the fluctuation spectrum. Moreover we found that the electron in the qubit is localized in this limit due to an effective decrease of the off-diagonal coupling. This phenomenon may resemble the localization in the spin-boson model in the strong coupling limit.
If the charge correlator and the total SET current are affected by the qubit (back-action effect), we found that the off-diagonal density-matrix elements of the qubit survive in the steady-state limit and therefore the relaxation rate is not zero. We concentrated on the case of weak coupling, when the Coulomb repulsion between the qubit and the SET is smaller then the Rabi frequency. The back-action of the qubit on the SET, however, can be weak or strong. In the first case we found that the total damping rate of the qubit due to decoherence and relaxation is again given by the spectral density of the SET charge fluctuations, modulated by the qubit. This relation, however, is not working if the back-action is strong. Indeed, we found that the damping rate of the qubit in this case is larger by an order of magnitude than that given by the spectral density of the corresponding fluctuating parameter.
This looks like that in the strong back-action of the qubit on the SET the major component of decoherence is not coming from the fluctuation spectrum of the qubit's parameters only, but also from the measurement "signal" of the SET. On the first sight it could agree with an analysis of Ref. 30 , suggesting that the decoherence rate contains two components, generated by a measurement and by a "pure environment" (environmental fluctuations). The latter therefore represents an unavoidable decoherence, generated by any environment. Yet, in a weak coupling regime such a separation seems not working. In this case the damping (decoherence) rate is totally determined by the environment fluctuations, even so modulated by the qubit.
Although our model deals with a particular setup, it bears the main physics of a fluctuating environment, acting on a qubit. Indeed, the Bloch-type rate equations, which we used in our analysis have a pronounced physical meaning: they relate the variation of qubit parameters with a nearby fluctuating field described by rate equations. A particular mechanism, generated this field should not be relevant for an evaluations of the decoherence and relaxation rates, but only its fluctuation spectrum. Indeed, in the weak coupling limit our result for the decorence rate coincides with that obtained in a framework of the spin-boson model. Thus our model can be considered as a generic one. Its main advantage is that it can be easily extended to multiple coupled qubits. Such an analysis would allow to determine how decoher-13 ence scales with number of qubits 42 , which is extremely important for a realization of quantum computations.
In addition, our model can be extended to a more complicated fluctuating environments, such as containing characteristic frequencies in its spectrum. It would formally correspond to a replacement of the SET in Fig. 2 by a double-dot (DD) coupled to the reservoirs 43 . All these situations, however, must be a subject of a separate investigation.
One of us (S.G.) thanks T. Brandes and C. Emary for helpful discussions and important suggestions. S.G is also grateful to the Max Planck Institute for the Physics of Complex Systems, Dresden, Germany, and to NTT Basic Research Laboratories, Atsugi-shi, Kanagawa, Japan, for kind hospitality.
APPENDIX A: QUANTUM-MECHANICAL DERIVATION OF RATE EQUATIONS FOR QUANTUM TRANSPORT
Consider the resonant tunneling through the SET, shown schematically in Fig. 10 . The entire system is described by the Hamiltonian H SET , given by Eq. ( 14 ). The wave function can be written in the same way as Eq. ( 17 ), where the variables related to the qubit are omitted,
|Ψ(t) = b(t) + l b 0l (t)c † 0 c l + l,r b rl (t)c † r c l + l<l ′ ,r b 0rll ′ (t)c † 0 c † r c l c l ′ + • • • |0 ¯ . (A1)
Substituting |Ψ(t) into the time-dependent Schrödinger equation, i∂ t |Ψ(t) = H SET |Ψ(t) , and performing the Laplace transform, b(E) = ∞ 0 exp(iEt) b(t)dt, we obtain the following infinite set of algebraic equations for the L µ E l E r 0 E Ω Ω µ R l r FIG. 10: Resonant tunneling through a single dot. µL,R are the Fermi energies in the collector and emitter, respectively.
amplitudes b(E):
E b(E) - l Ω l b0l (E) = i ( A2a
) (E + E l -E 0 ) b0l (E) -Ω l b(E) - r Ω r blr (E) = 0 (A2b) (E + E l -E r ) blr (E) -Ω r b0l (E) - l ′ Ω l ′ b0ll ′ r (E) = 0 (A2c) (E + E l + E l ′ -E 0 -E r ) b0ll ′ r (E) -Ω l ′ blr (E) + Ω l bl ′ r (E) - r ′ Ω r ′ bll ′ rr ′ (E) = 0 (A2d) • • • • • • • • •
(The r.h.s of Eq. (A2a) reflects the initial condition.) Let us replace the amplitude b in the term Ω b of each of the equations (A2) by its expression obtained from the subsequent equation. For example, substituting b0l (E) from Eq. (A2b) into Eq. (A2a) we obtain
E - l Ω 2 l E + E l -E 0 b(E) - l,r Ω l Ω r E + E l -E 0 blr (E) = i. ( A3
)
Since the states in the reservoirs are very dense (continuum), one can replace the sums over l and r by integrals, for instance l → ρ L (E l ) dE l , where ρ L (E l ) is the density of states in the emitter, and Ω l,r → Ω L,R (E l,r ). Consider the first term
S 1 = µL -Λ Ω 2 L (E l ) E + E l -E 0 ρ L (E l )dE l ( A4
)
where Λ is the cut-off parameter. Assuming weak energy dependence of the couplings Ω L,R and the density of states ρ L,R , we find in the limit of high bias,
µ L = Λ → ∞ S 1 = -iπΩ 2 L (E 0 -E)ρ L (E 0 -E) = -i Γ L 2 . ( A5
)
14 Consider now the second sum in Eq. (A3).
S 2 = Λ -Λ ρ R (E r )dE r × Λ -Λ Ω L (E l )Ω R (E r ) blr (E, E l , E r ) E + E l -E 0 ρ L (E l )dE l , ( A6
)
where we replaced blr (E) by b(E, E l , E r ) and took µ L = Λ, µ R = -Λ. In contrast with the first term of Eq. ( A3 ), the amplitude b is not factorized out the integral (A6). We refer to this type of terms as "cross-terms". Fortunately, all "cross-terms" vanish in the limit of large bias, Λ → ∞. This greatly simplifies the problem and is very crucial for a transformation of the Schrödinger to the rate equations. The reason is that the poles of the integrand in the E l (E r )-variable in the "cross-terms" are on the same side of the integration contour. One can find it by using a perturbation series the amplitudes b in powers of Ω. For instance, from iterations of Eqs. (A2) one finds
b(E, E l , E r ) = iΩ L Ω R E(E + E l -E r )(E + E l -E 0 ) + • • • ( A7
)
The higher order powers of Ω have the same structure. Since E → E + iǫ in the Laplace transform, all poles of the amplitude b(E, E l , E r ) in the E l -variable are below the real axis. In this case, substituting Eq. (A7) into Eq. (A6) we find lim Λ→∞ Λ -Λ
Ω L Ω R (E + iǫ)(E + E 0 -E 1 + iǫ) 2 (E + E 0 -E r + iǫ) + • • • dE l = 0 , ( A8
)
Thus, S 2 → 0 in the limit of µ L → ∞, µ R → -∞.
Applying analogous considerations to the other equations of the system (A2), we finally arrive at the following set of equations:
(E + iΓ L /2) b(E) = i ( A9a
) (E + E l -E 0 + iΓ R /2) b0l (E) -Ω l b(E) = 0 ( A9b
) (E + E l -E r + iΓ L /2) blr (E) -Ω r b0l (E) = 0 ( A9c
) (E + E l + E l ′ -E 0 -E r + iΓ R /2) b0ll ′ r (E) -Ω l ′ blr (E) + Ω l bl ′ r (E) = 0 (A9d) • • • • • • • • •
Eqs. (A9) can be transformed directly to the reduced density matrix σ (n,n ′ ) jj ′ (t), where j = 0, 1 denote the state of the SET with an unoccupied or occupied dot and n denotes the number of electrons which have arrived at the collector by time t. In fact, as follows from our derivation, the diagonal density-matrix elements, j = j ′ and n = n ′ , form a closed system in the case of resonant tunneling through one level, Fig. 10 . The off-diagonal elements, j = j ′ , appear in the equation of motion whenever more than one discrete level of the system carry the transport (see Eq. ( 20 ). Therefore we concentrate below on the diagonal density-matrix elements only, σ (n) 00 (t) ≡ σ (n,n) 00 (t) and σ (n) 11 (t) ≡ σ (n,n) 11 (t). Applying the inverse Laplace transform on finds σ (n) 00 (t) = l...,r...
′ 4π 2 bl • • • n r • • • n (E) b * l • • • n r • • • n (E ′ )e i(E ′ -E)t ( A10a
) σ (n) 11 (t) =
l...,r...
dEdE ′ 4π 2 b0l • • • n+1 r • • • n (E) b * 0 l • • • n+1 r • • • n (E ′ )e i(E ′ -E)t ( A10b
)
Consider, for instance, the term σ A9b ) by b * 0l (E ′ ) and then subtracting the complex conjugated equation with the interchange
(0) 11 (t) = l |b 0l (t)| 2 . Multiplying Eq. (
E ↔ E ′ we obtain dEdE ′ 4π 2 l (E ′ -E -iΓ R ) b0l (E) b * 0l (E ′ ) -2Im l Ω l b0l (E) b * (E ′ ) e i(E ′ -E)t = 0 (A11)
15 Using Eq. (A10b) one easily finds that the first integral in Eq. (A11) equals to -i[ σ(0) 11 (t) + Γ R σ (0) 11 (t)]. Next, substituting b0l (E) = Ω l b(E) E + E l -E 0 + iΓ R /2 (A12) from Eq. ( A9b ) into the second term of Eq. (A11), and replacing a sum by an integral, one can perform the E lintegration in the large bias limit, µ L → ∞, µ R → -∞.
Then using again Eq. (A10b) one reduces the second term of Eq. (A11) to iΓ L σ (0) 00 (t). Finally, Eq. (A11) reads σ(0)
11 (t) = Γ L σ (0) 00 (t) -Γ R σ (0) 11 (t).
The same algebra can be applied for all other amplitudes bα (t). For instance, by using Eq. ( A10a ) one easily finds that Eq. (A9c) is converted to the following rate equation σ(1) 00 (t) = -Γ L σ (1) 00 (t) + Γ R σ (0) 11 (t). With respect to the states involving more than one electron (hole) in the reservoirs (the amplitudes like b0ll ′ r (E) and so on), the corresponding equations contain the Pauli exchange terms. By converting these equations into those for the density matrix using our procedure, one finds the "cross terms", like Ω l bl ′ r (E)Ω l ′ b * lr (E ′ ), generated by Eq. (A9d). Yet, these terms vanish after an integration over E l(r) in the large bias limit, as the second term in Eq. ( A3 ). The rest of the algebra remains the same, as described above. Finally we arrive at the following infinite system of the chain equations for the diagonal elements, σ (n) 00 and σ (n) 11 , of the density matrix,
σ(0) 00 (t) = -Γ L σ ( 0
) 00 (t) , ( A13a
) σ(0) 11 (t) = Γ L σ ( 0
) 00 (t) -Γ R σ ( 0
) 11 (t) , ( A13b
) σ(1) 00 (t) = -Γ L σ ( 1
) 00 (t) + Γ R σ ( 0
) 11 (t) , (A13c) σ(1) 11 (t) = Γ L σ ( 1
) 00 (t) -Γ R σ ( 1
) 11 (t) , (A13d) • • • • • • • • •
Summing over n in Eqs. (A13) we find for the reduced density matrix of the SET, σ(t) = n σ (n) (t), the following "classical" rate equations,
σ00 (t) = -Γ L σ 00 (t) + Γ R σ 11 (t) (A14a) σ11 (t) = Γ L σ 00 (t) -Γ R σ 11 (t) ( A14b
)
These equations represent a particular case of our general quantum rate equations (20), which are derived using the above described technique 37,38 .
APPENDIX B: CORRELATOR OF ELECTRIC CHARGE INSIDE THE SET.
The charge correlator inside the SET is given by S Q (ω) = SQ (ω) + SQ (-ω), where
SQ (ω) = ∞ 0 δ Q(0)δ Q(t) e iωt dt . ( B1
)
Here δ Q(t) = c † 0 (t)c 0 (t)q and q = P1 = P 1 (t → ∞) is the average charge inside the dot. Since the initial state, t = 0 in Eq. (B1) corresponds to the steady state, one can represent the time-correlator as δ Q(0)δ Q(t) = q=0,1
P q (0)(q -q)( Q q (t) -q) , (B2)
where P q (0) is the probability of finding the charge q = 0, 1 inside the quantum dot in the steady state, such that P 1 (0) = q and P 0 (0) = 1q, and Q q (t) = P (q) 1 (t) is the average charge in the dot at time t, starting with the initial condition P (q) 1 (0) = q. Substituting Eq. (B2) into Eq. (B1) we finally obtain
SQ (ω) = q(1 -q)[ P (1) 1 (ω) - P (0) 1 (ω)] , ( B3
)
where P (q) 1 (ω) is a Laplace transform of P (q) 1 (t). These quantities are obtained directly from the rate equations, such that q = σbb + σdd and P (q) 1 (ω) = σ(q) bb (ω) + σ(q) dd (ω), where σ = σ(t → ∞) and σ(q) (ω) is the Laplace transform σ (q) (t) with the initial conditions corresponding to the occupied (q = 1) or unoccupied (q = 0) SET. In order to find these quantities it is useful to rewrite the rate equations in the matrix form, σ(t) = M σ(t), representing σ(t) as the eight-vector, σ = {σ aa , σ bb , σ cc , σ dd , σ ac , σ ca , σ bd , σ db } and M as the corresponding 8 × 8-matrix. Applying the Laplace transform we find the following matrix equation, (i ω I + M )σ (q) (ω) = -σ (q)
(0) , ( B4
)
where I is the unit matrix and σ (q) (0) is the initial condition for the density-matrix obtained by projecting the total wave function (17) on occupied (q = 1) and unoccupied (q = 0) states of the SET in the limit of t → ∞, σ (1) (0) = N 1 {0, σbb , 0, σdd , 0, 0, σbd , σdb } , (B5a)
σ (0) (0) = N 0 {σ aa , 0, σcc , 0, σac , σca , 0, 0} , ( B5b
)
and N 1 = 1/q and N 0 = 1/(1q) are the corresponding normalization factors. Finally one obtains:
S Q (ω) = 2q(1 -q)Re [σ ( 1
) bb (ω) + σ(1) dd (ω)
σ(0) bb (ω)σ(0) dd (ω)]. (B6) In the case shown in Fig. 2 one finds from Eqs. (21) or Eqs. (31) for Γ ′ L,R = Γ L,R that σac = σ bd = 0, q = Γ L /Γ and σ(q) bb (ω) + σ(q) dd (ω) = P (q) 1 (ω). The latter equation is given by (iω -Γ) P (q) 1 (ω) = -q + iΓ L ω . (B7) Substituting Eq. (B7) into Eq. (B3) one obtains:
S Q (ω) = 2Γ L Γ R Γ(ω 2 + Γ 2 ) . ( B8
)
16 Obviously, for a more general case when Γ ′ L,R = Γ L,R , or when the electron-electron interaction excites the electron inside the SET above the Fermi level, Fig. 8 , the ex-pressions for S Q (ω), obtained from Eq. (B6) have a more complicated than Eq. (B8). * Electronic address: shmuel.gurvitz@weizmann.ac.il 1 A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987). 2 U. Weiss, Quantum Dissipative Systems (World Scientific, Singapure, 2000). 3 A. Shnirman, Y. Makhlin, and G. Schoön, Phys. Scr. T102, 147 (2002). 4 H. Gassmann, F. Marquardt, and C. Bruder, Phys. Rev. E66, 041111 (2002). 5 E. Paladino, L. Faoro, G. Falci, and R. Fazio, Phys. Rev.
Lett. 88, 228304 (2002). 6 T. Itakura and Y. Tokura, Phys. Rev. B67, 195320 (2003). 7 J.Q. You, X. Hu, and F. Nori, Phys. Rev. B 72, 144529 (2005). 8 A. Grishin, I.V. Yurkevich and I.V. Lerner, Phys. Rev. B 72, 060509(R) (2005). 9 J. Schriefl1, Y. Makhlin, A. Shnirman, and Gerd Schön, New J. Phys. 8, 1 (2006). 10 Y.M. Galperin, B.L. Altshuler, J. Bergli, and D.V. Shantsev, Phys. Rev. Lett., 96, 097009 (2006). 11 S. Ashhab, J.R. Johansson, and F. Nori, Phys. Rev. A74, 052330 (2006); ibid, Physica C 444, 45 (2006); ibid, New J. Phys. 8, 103 (2006). 12 U. Hartmann and F.K. Wilhelm, Phys. Rev. B 75, 165308 (2007). 13 S.A. Gurvitz, Phys. Rev. B56, 15215 (1997). 14 S. Pilgram and M. Büttiker, Phys. Rev. Lett., 89, 200401 (2002). 15 A.A. Clerk, S.M. Girvin, and A.D. Stone, Phys. Rev. B67, 165324 (2003). 16 S.A. Gurvitz and G.P. Berman, Phys. Rev. B72, 073303 (2005). 17 A. Käck, G. Wendin, and G. Johansson, Phys. Rev. B 67, 035301 (2003). 18 R. Aguado and L. P. Kouwenhoven, Phys. Rev. Lett., 84, 1986 (2000). 19 E. Onac, F. Balestro, L.H. Willems van Beveren, U. Hartmann, Y.V. Nazarov, and L.P. Kouwenhoven, Phys. Rev. Lett. 96, 176601 (2006). 20 I. Neder, M. Heiblum, D. Mahalu, and V. Umansky, Phys.
Rev. Lett. 98, 036803 (2007). 21 I. Neder and F. Marquardt, New J. Phys. 9, 112 (2007), and references therein. 22 W.G. van der Wiel, T. Fujisawa, S. Tarucha, L.P. Kouwenhoven, Japanese Jour. Appl. Phys. 40, 2100 (2001). 23 J. M. Elzerman, R. Hanson, J. S. Greidanus, L. H. W.
van Beveren, S. De Franceschi, L. M. K. Vandersypen, S.
Tarucha, L. P. Kouwenhoven, Physica E 25, 135 (2004). 24 T. Hayashi, T. Fujisawa, H.D. Cheong, Y.H. Jeong, Y.
Hirayama, Phys. Rev. Lett. 91, 226804 (2003). 25 J. Shao, C. Zerbe, and P. Hanggi, Chem. Phys. 235, 81 (1998). 26 X.R. Wang, Y.S. Zheng, and S. Yin, Phys. Rev. B72, 121303(R) (2005). 27 C.P. Slichter, Principles of Magnetic Resonance, (Springer-Verlag, 1980). 28 Y. Makhlin, G. Schoön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). 29 H.S. Goan, Quantum Information and Computation, 2, 121 (2003); ibid, Phys. Rev. B 70, 075305 (2004). 30 A.N. Korotkov, Phys. Rev. B63, 085312 (2001); ibid, Phys.
Rev. B63, 115403 (2001). 31 Y. Levinson, Phys. Rev. B61, 4748 (2000). 32 K. Rabenstein, V.A. Sverdlov, and D.V. Averin, JETP Lett. 79, 646 (2004). 33 S.A. Gurvitz, L. Fedichkin, D. Mozyrsky and G.P. Berman, Phys. Rev. Lett., 91, 066801 (2003). 34 G. Ithier, E. Collin, P. Joyez, P.J. Meeson, D. Vion, D.
Esteve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl, and G. Schön, Phys. Rev. B72, 134519 (2005). 35 S.A. Gurvitz, IEEE Transactions on Nanotechnology 4, 45 (2005). 36 S.A. Gurvitz, D. Mozyrsky, and G.P. Berman, Phys.Rev. B72, 205341 (2005). 37 S.A. Gurvitz and Ya.S. Prager, Phys. Rev. B53, 15932 (1996). 38 S.A. Gurvitz, Phys. Rev. B57, 6602, (1998). 39 In a strict sense the quantum rate equations (20) were derived by assuming constant widths Γ. Yet these equation are also valid when the widths are weakly energy dependent, as follows from their derivations (see 37, 38 and Appendix A). 40 R.J. Schoelkopf, A.A. Clerk, S.M. Girvin, K.W. Lehnert and M.H. Devoret, Quantum Noise in Mesoscopic Physics, edited by Yu.V. Nazarov, (Springer, 2003). 41 Y. Makhlin, G. Schoön, and A. Shnirman, in Exploring the Quantum-Classical Frontier, edited by J.R. Friedman and S. Han (Nova Science, Commack, New York, 2002). 42 A.M. Zagoskin, S. Ashhab, J.R. Johansson, and F. Nori, Phys. Rev. Lett. 97, 077001 (2006). 43 T. Gilad and S.A. Gurvitz, Phys. Rev. Lett. 97, 116806 (2006); H.J. Jiao, X.Q. Li, and J.Y. Luo, Phys. Rev. B75, 155333 (2007). | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The influence of environment on a single quantum system is the issue of crucial importance in quantum information science. It is mainly associated with decoherence, or dephasing, which transforms any pure state of a quantum system into a statistical mixture. Despite a large body of theoretical work devoted to decoherence, its mechanism has not been clarified enough. For instance, how decoherence is related to environmental noise, in particular in the presence of back-action of the system on the environment (quantum measurements). Moreover, decoherence is often intermixed with relaxation. Although each of them represents an irreversible process, decoherence and relaxation affect quantum systems in quite different ways.\n\nIn order to establish a relation between the fluctuation spectrum and decoherence and relaxation rates one needs a model that describes the effects of decoherence and relaxation in a consistent quantum mechanical way. An obvious candidate is the spin-boson model 1,2 which represents the environment as a bath of harmonic oscillators at equilibrium, where the fluctuations obey Gaussian statistics 3 . Despite its apparent simplicity, the spinboson model cannot be solved exactly 2 . Also, it is hard to manipulate the fluctuation spectrum in the framework of this model. In addition, mesoscopic structures may couple only to a few isolated fluctuators, like spins, local currents, background charge fluctuations, etc. This would require models of the environment, different from Electrodes (a) (b) E 1 E 2 0 Ω FIG. 1: Electrostatic qubit, realized by an electron trapped in a coupled-dot system (a), and its schematic representation by a double-well (b). Ω0 denotes the coupling between the two dots.\n\nthe spin-boson model (see for instace 4, 5, 6, 7, 8, 9, 10, 11, 12 ). In general, the environment can be out of equilibrium, like a steady-state fluctuating current, interacting with the qubit 13, 14, 15, 16 . This for instance, takes place in the continuous measurement (monitoring) of quantum systems 17 and in the \"control dephasing\" experiments 18, 19, 20 . All these types on non-Gaussian and non-equilibrium environments attracted recently a great deal of attention 21 .\n\nIn this paper we consider an electrostatic qubit, which can be viewed as a generic example of two-state systems. It is realized by an electron trapped in coupled quantum dots 22, 23, 24 , Fig. 1. Here E 1 and E 2 denote energies of the electron states in each of the dots and Ω 0 is a coupling between these states. It is reasonable to assume that the decoherence of a qubit is associated with fluctu-\n\n2 E 1 E 2 E r R µ L µ 0 E E l"
},
{
"section_type": "OTHER",
"section_title": "Electrostatic Qubit",
"text": "Single Electron Transistor (a) (b) I I FIG. 2: Qubit near Single Electron Transistor. Here E l,r and E0 denote the energy levels in the left (right) reservoirs and in the quantum dot, respectively, and µL,R are the corresponding chemical potentials. The electric current I generates fluctuations of the electrostatic opening between two dots (a), or it fluctuates the energy level of the nearest dot (b).\n\nations of the qubit parameters, E 1,2 and Ω 0 , generated by the environment. Indeed, a stochastic averaging of the Schrödinger equation over these fluctuations parameters results in the qubit's decoherence, which transfers any qubit state into a statistical mixture 25, 26 . In general, one can expect that the fluctuating environment should result in the qubit's relaxation, as well, as for instance in the phenomenological Redfield's description of relaxation in the magnetic resonance 27 .\n\nAs a quantum mechanical model of the environment we consider a Single Electron Transistor (SET) capacitively coupled to the qubit, e.g., Fig. 2. Such setup has been contemplated in numerous solid state quantum computing architectures where SET plays role of a readout device 16, 17, 28, 29 and contains most of the generic features of a fluctuating non-equilibrium environment. The discreteness of the electron charge creates fluctuations in the electrostatic field near the SET. If the electrostatic qubit is placed near the SET, this fluctuating field should affect the qubit behavior as shown in Fig. 2 . It can produce fluctuations of the tunneling coupling between the dots (off-diagonal coupling) by narrowing the electrostatic opening connecting these dots, as in Fig. 2a , or make the energy levels of the dots fluctuate, as shown schematically in Fig. 2b . Note that while in some regimes the SET operates as a measuring device 16, 17 , in other regimes it corresponds purely to a source of noise. Indeed, if the energy level E 0 , Fig. 2 , is deeply inside the voltage bias -the case we consider in the beginning, the SET current is not modulated by the qubit electron. In this case the SET represents only the fluctuating environment affecting the qubit behavior (\"pure environment\" 30 ).\n\nA similar model of the fluctuating environment has been studied mostly for small bias (linear response) or for the environment in an equilibrium. Here, however, we consider strongly non-equilibrium case where the bias voltage applied on the SET (V = µ Lµ R ) is much larger than the levels widths and the coupling between the SET and the qubit. In this limit our model can be solved exactly for both weak and strong coupling (but is still smaller than the bias voltage). This constitutes an essential advantage with regard to perturbative treatments of similar models. For instance, the results of our model can be compared in different regimes with phenomenological descriptions used in the literature. Such a comparison would allow us to determine the regions where these phenomenological models are valid.\n\nSince our model is very simple in treatment, the decoherence and relaxation rates can be extracted from the exact solution analytically, as well as the time-correlator of the electric charge inside the SET. This would make it possible to establish a relation between the frequencydependent fluctuation spectrum of the environment and the decoherence and the relaxation rates of the qubit, and to determine how far this relation can be extended. We expect that such a relation should not depend on a source of fluctuations. This point can be verified by a comparison with a similar results obtained for equilibrium environment in the framework of the spin-boson model 1, 2 .\n\nIt is also important to understand how the decoherence and relaxation rates depend on the frequency of the environmental fluctuations. This problem has been investigated in many phenomenological approaches for \"classical\" environments at equilibrium. Yet, there still exists an ambiguity in the literature related to this point for non-equilibrium environment. For instance, it was found by Levinson that the decoherence rate, generated by fluctuations of the energy level in a single quantum dot is proportional to the spectral density of fluctuations at zero frequency 31 . The same result, but for a double-dot system has been obtained by Rabenstein et al. 32 . On the other hand, it follows from the Redfied's approach that the corresponding decoherence rate is proportional to the spectral density at the frequency of the qubit's oscillations (the Rabi frequency) 27 . Since our model is the exactly solvable one, we can resolve this ambiguity and establish the appropriate physical conditions that can result in different relations of decoherence rate to the environmental fluctuations.\n\nThe most important results of our study are related to the situation when back-action of the qubit on the environment takes place. This problem did not receive such a considerable amount of attention in the literature as, for example, the case of \"inert\" environment. This is in spite of a fact that the back-action always takes place in the presence of measurement. There are many questions related to the effects of a back-action. For instance, what would be a relation between decoherence (relaxation) of the qubit and the noise spectrum of the environment? Or, how decoherence is affected by a strong response of environment? We believe that our model appears to be 3 more suitable for studying these and other problems related to the back-action than most of the other existing approaches. The plan of this paper is as follows: Sect. II presents a phenomenological description of decoherence and relaxation in the framework of Bloch equations, applied to the electrostatic qubit. Sect. III contains description of the model and the quantum rate-equation formalism, used for its solution. Detailed quantum-mechanical derivation of these equations for a specific example is presented in Appendix A. Sect. IV deals with a configuration where the SET can generate only decoherence of the qubit. We consider separately the situations when SET produces fluctuations of the tunneling coupling (Rabi frequency) or of the energy levels. The results are compared with the SET fluctuation spectrum, evaluated in Appendix B. Sect. V deals with a configuration where the SET generates both decoherence and relaxation of the qubit. Sect. VI is summary."
},
{
"section_type": "OTHER",
"section_title": "II. DECOHERENCE AND RELAXATION OF A QUBIT",
"text": "In this section we describe in a general phenomenological framework the effect of decoherence and relaxation on the qubit behavior. Although the results are known, there still exists some confusion in the literature in this issue. We therefore need to define precisely these quantities and demonstrate how the corresponding decoherenece and relaxation rates can be extracted from the qubit density matrix.\n\nLet us consider an electrostatic qubit, realized by an electron trapped in coupled quantum dots, Fig. 1. This system is described by the following tunneling Hamiltonian\n\nH qb = E 1 a † 1 a 1 + E 2 a † 2 a 2 -Ω 0 (a † 2 a 1 + a † 1 a 2 ) ( 1\n\n)\n\nwhere a † 1,2 , a 1,2 are the creation and annihilation operators of the electron in the first or in the second dot. For simplicity we consider electrons as spinless fermions. In addition, we assume that a † 1 a 1 + a † 2 a 2 = 1, so that only one electron is present in the double-dot. The electron wave function can be written as |Ψ(t) = b (1) (t)a † 1 + b (2) (t)a † 2 |0 ¯ (2) where b (1, 2) (t) are the probability amplitudes for finding the electron in the first or second well, obtained from the Schrödinger equation i∂ t |Ψ(t) = H qb |Ψ(t) (we adopt the units where = 1 and the electron charge e = 1). The corresponding density matrix, σ jj ′ (t) = b (j) (t)b (j ′ ) * (t), with j, j ′ = {1, 2}, is obtained from the equation i∂ t σ = [H, σ]. This can be written explicitly as\n\nσ11 = iΩ 0 (σ 21 -σ 12 ) (3a) σ12 = -iǫσ 12 + iΩ 0 (1 -2σ 11 ) , ( 3b\n\n)\n\nwhere σ 22 (t) = 1σ 11 (t), σ 21 (t) = σ * 12 (t) and ǫ = E 1 -E 2 . Solving these equations one easily finds that the electron oscillates between the two dots (Rabi oscillations) with frequency ω R = 4Ω 2 0 + ǫ 2 . For instance, for the initial conditions σ 11 (0) = 1 and σ 12 (0) = 1, the probability of finding the electron in the second dot is σ 22 (t) = 2(Ω 0 /ω R ) 2 (1cos ω R t). This result shows that for ǫ ≫ Ω 0 the amplitude of the Rabi oscillations is small, so the electron remains localized in its initial state.\n\nThe situation is different when the qubit interacts with the environment. In this case the (reduced) density matrix of the qubit σ(t) is obtained by tracing out the environment variables from the total density matrix. The question is how to modify Eqs. (3), written for an isolated qubit, in order to obtain the reduced density matrix of the qubit, σ(t). In general one expects that the environment could affect the qubit in two different ways. First, it can destroy the off-diagonal elements of the qubit density matrix. This process is usually referred to as decoherence (or dephasing). It can be accounted for phenomenologically by introducing an additional (damping) term in Eq. ( 3b ),\n\nσ12 = -iǫσ 12 + iΩ 0 (1 -2σ 11 ) - Γ d 2 σ 12 ( 4\n\n)\n\nwhere Γ d is the decoherence rate. As a result the qubit density-matrix σ(t) becomes a statistical mixture in the stationary limit,\n\nσ(t) t→∞ -→ 1/2 0 0 1/2 . ( 5\n\n)\n\nThis happens for any initial conditions and even for large level displacement, ǫ ≫ Ω 0 , Γ d (provided that Ω 0 = 0). Note that the statistical mixture (5) is proportional to the unity matrix and therefore it remains the same in any basis. Secondly, the environment can put the qubit in its ground state, for instance via photon or phonon emission. This process is usually referred to as relaxation.\n\nFor a symmetric qubit we would have σ(t) t→∞ -→ 1/2 1/2 1/2 1/2 . (6) In contrast with decoherence, Eq. ( 5 ), the relaxation process puts the qubit into a pure state. That implies that the corresponding density matrix can be always written as δ 1i δ 1j in a certain basis (the basis of the qubit eigenstates). This is in fact the essential difference between decoherence and relaxation. With respect to elimination of the off-diagonal density matrix elements, note that relaxation would eliminate these terms only in the qubit's eigenstates basis. In contrast, decoherence eliminates the off-diagonal density matrix element in any basis (Eq. (5)). In fact, if the environment has some energy, it can put the qubit into an exited state. However, if the qubit is finally in a pure state, such excitation process generated by the environment affects the qubit in the same way as 4 relaxation: it eliminates the off-diagonal density matrix elements only in a certain qubit's basis. Therefore excitation of the qubit can be described phenomenologically on the same footing as relaxation.\n\nIt is often claimed that decoherence is associated with an absence of energy transfer between the system and the environment, in contrast with relaxation (excitation). This distinction is not generally valid. For instance, if the initial qubit state corresponds to the electron in the state |E 2 , Fig. 1 , the final state after decoherence corresponds to an equal distribution between the two dots, E = (E 1 + E 2 )/2. In the case of E 1 ≫ E 2 , this process would require a large energy transfer between the qubit and the environment. Therefore decoherence can be consistently defines as a process leading to a statistical mixture, where all states of the system have equal probabilities (as in Eq. (5)).\n\nThe relaxation (excitation) process can be described most simply by diagonalizing the qubit Hamiltonian, Eqs. ( 1\n\n), to obtain H qb = E + a † + a + + E -a † -a -, where\n\nthe operators a ± are obtained by the corresponding rotation of the operators a 1,2 30 . Here E + and E -are the ground (symmetric) and excited (antisymmetric) state energies. Then the relaxation process can be described phenomenologically in the new qubit basis\n\n|± = a † ± |0 ¯ as σ--(t) = -Γ r σ --(t) ( 7a\n\n) σ+-(t) = i(E --E + )σ +-(t) - Γ r 2 σ +-(t) , ( 7b\n\n)\n\nwhere σ ++ (t) = 1σ --(t), σ -+ (t) = σ * +-(t) and Γ r is the relaxation rate.\n\nIn order to add decoherence, we return to the original qubit basis |1, 2 = a † 1,2 |0 ¯ and add the damping term to the equation for the off-diagonal matrix elements, Eq. ( 4 ). We arrive at the quantum rate equation describing the qubit's behavior in the presence of both decoherence and relaxation 30, 33 ,\n\nσ11 = iΩ 0 (σ 21 -σ 12 ) -Γ r κǫ 2ǫ (σ 12 + σ 21 ) - Γ r 4 1 + ǫ ǫ 2 (2σ 11 -1) + Γ r ǫ 2ǫ ( 8a\n\n) σ12 = -iǫσ 12 + iΩ 0 + Γ r κǫ 2ǫ (1 -2σ 11 ) + Γ r κ - 1 2 σ 12 -κ 2 (σ 12 + σ 21 ) - Γ d 2 σ 12 , ( 8b\n\n)\n\nwhere ǫ = (ǫ 2 + 4Ω 2 0 ) 1/2 and κ = Ω 0 /ǫ. In fact, these equations can be derived in the framework of a particular model, representing an electrostatic qubit interacting with the point-contact detector and the environment, described by the Lee model Hamiltonian 33 .\n\nEquations (8) can be rewritten in a simpler form by mapping the qubit density matrix σ = {σ 11 , σ 12 , σ 21 } to a \"polarization\" vector S(t) via σ(t) = [1 + τ • S(t)]/2, where τ x,y,z are the Pauli matrices. For instance, one obtains for the symmetric case, ǫ = 0,\n\nṠz = - Γ r 2 S z -2Ω 0 S y ( 9a\n\n) Ṡy = 2Ω 0 S z - Γ d + Γ r 2 S y ( 9b\n\n) Ṡx = - Γ d + 2Γ r 2 (S x -Sx ) ( 9c\n\n)\n\nwhere Sx = S x (t → ∞) = 2Γ r /(Γ d +2Γ r ). One finds that Eqs. (9) have a form of the Bloch equations for spinprecession in the magnetic field 27 , where the effect of environment is accounted for by two relaxation times for the different spin components: the longitudinal T 1 and the transverse T 2 , related to Γ d and 2Γ r as\n\nT -1 1 = Γ d + 2Γ r 2 , and T -1 2 = Γ d + Γ r 2 , ( 10\n\n)\n\nThe corresponding damping rates, the so-called \"depolarization\" (Γ 1 = 1/T 1 ) and the \"dephasing\" (Γ 2 = 1/T 2 ) are used for phenomenological description of two-level systems 34 . However, neither Γ 1 nor Γ 2 taken alone would drive the qubit density matrix into a statistical mixture Eq. ( 5 ) or into a pure state Eq. ( 6 ). In contrast, our definition of decoherence and relaxation (excitation) is associated with two opposite effects of the environment on the qubit: the first drives it into a statistical mixture, whereas the second drives it into a pure state. We expect therefore that such a natural distinction between decoherence and relaxation would be more useful for finding a relation between these quantities and the environmental behavior than other alternative definitions of these quantities existing in the literature.\n\nIn general, the two rates, Γ d,r , introduced in phenomenological equations (8), (9), are consistent with our definitions of decoherence and relaxation. The only exception is the case of Γ r = 0 and Ω 0 = 0, where are no transitions between the qubit's states even in the presence of the environment (\"static\" qubit). One easily finds from Eqs. (3a), (4) that σ 12 (t) → 0 for t → ∞, whereas the diagonal density-matrix elements of the qubit remain unchanged (so-called \"pure dephasing\" 5,34 ):\n\nσ(t) t→∞ -→ σ 11 (0) 0 0 σ 22 (0) . ( 11\n\n)\n\n5 Thus, if the initial probabilities of finding the qubit in each of its states are not equal, σ 11 (0) = σ 22 (0), then the final qubit state is neither a mixture nor a pure state, but a combination of the both. It implies that Γ d in Eqs. (8) would also generate relaxation (excitation) of the qubit. Note that in this case the off-diagonal density-matrix elements, absent in Eq.( 11 ), would reappear in a different basis. This implies that the \"pure dephasing\" 5,34 occurs only in a particular basis.\n\nLet us evaluate the probability of finding the electron in the first dot, σ 11 (t). Solving Eqs. (9) for the initial conditions σ 11 (0) = 1, σ 12 (0) = 0, we find 33 :\n\nσ 11 (t) = 1 2 + e -Γrt/2 4 C 1 e -e-t + C 2 e -e+t ( 12\n\n)\n\nwhere e ± = 1 4 (Γ d ± Ω), Ω = Γ 2 d -64Ω 2 0 and C 1,2 = 1±(Γ d / Ω). Solving the same equations in the limit of t → ∞, we find that the steady-state qubit density matrix is\n\nσ(t) t→∞ -→ 1/2 Γ r /(Γ d + 2Γ r ) Γ r /(Γ d + 2Γ r ) 1/2 . ( 13\n\n)\n\nThus the off-diagonal elements of the density matrix can provide us with a ratio of relaxation to decoherence rates 33 ."
},
{
"section_type": "OTHER",
"section_title": "III. DESCRIPTION OF THE MODEL",
"text": "Consider the setup shown in Fig. 2 . The entire system can be described by the following tunneling Hamiltonian, represented by a sum of the qubit and SET Hamiltonians and the interaction term, 1 ) and describes the qubit. The second term, H SET , describes the single-electron transistor. It can be written as\n\nH = H qb + H SET + H int . Here H qb is given by Eq. (\n\nH SET = l E l c † l c l + r E r c † r c r + E 0 c † 0 c 0 + l,r (Ω l c † l c 0 + Ω r c † r c 0 + H.c.) , ( 14\n\n)\n\nwhere c † l,r and c l,r are the creation and annihilation electron operators in the state E l,r of the right or left reservoir; c † 0 and c 0 are those for the level E 0 inside the quantum dot; and Ω l,r are the couplings between the level E 0 and the level E l,r in the left (right) reservoir. In order to avoid too lengthy formulaes, our summation indices l, r indicate simultaneously the left and the right leads of the SET, where the corresponding summation is carried out. As follows from the Hamiltonian (14), the quantum dot of the SET contains only one level (E 0 ). This assumption has been implied only for the sake of simplicity for our presentation, although our approach is well suited for a case of n levels inside the SET,\n\nE 0 c † 0 c 0 → n E n c † n c n ,\n\nand even when the interaction between these levels is included (providing that the latter is much less or much larger than the bias V ) 35, 36 . We also assumed a weak energy dependence of the couplings Ω l,r ≃ Ω L,R .\n\nThe interaction between the qubit and the SET, H int , depends on a position of the SET with respect to the qubit. If the SET is placed near the middle of the qubit, Fig. 2a , then the tunneling coupling between two dots of the qubit in Eq. ( 1 ) decreases, Ω 0 → Ω 0 -δΩ 0 , whenever the quantum dot of the SET is occupied by an electron. This is due to the electron's repulsive field. In this case the interaction term can be written as\n\nH int = δΩ c † 0 c 0 (a † 1 a 2 + a † 2 a 1 ) . ( 15\n\n)\n\nOn the other hand, in the configuration shown in Fig. 2b where the SET is placed near one of the dots of the qubit, the electron repulsive field displaces the qubit energy levels by ∆E = U . The interaction terms in this case can be written as\n\nH int = U a † 1 a 1 c † 0 c 0 . ( 16\n\n)\n\nConsider the initial state where all the levels in the left and the right reservoirs are filled with electrons up to the Fermi levels µ L,R respectively. This state will be called the \"vacuum\" state |0 ¯ . The wave function for the entire system can be written as\n\n|Ψ(t) = b (1) (t)a † 1 + l b ( 1\n\n) 0l (t)a † 1 c † 0 c l + l,r b ( 1\n\n) rl (t)a † 1 c † r c l + l<l ′ ,r b ( 1\n\n) 0rll ′ (t)a † 1 c † 0 c † r c l c l ′ + • • • +b (2) (t)a † 2 + l b ( 2\n\n) 0l (t)a † 2 c † 0 c l + l,r b ( 2\n\n) rl (t)a † 2 c † r c l + l<l ′ ,r b ( 2\n\n) 0rll ′ (t)a † 2 c † 0 c † r c l c l ′ + . . . |0 ¯ , ( 17\n\n)\n\nwhere b (j) (t), b (j) α (t) are the probability amplitudes to find the entire system in the state described by the cor-responding creation and annihilation operators. These amplitudes are obtained from the Schrödinger equation 6 i| Ψ(t) = H|Ψ(t) , supplemented with the initial condition b (1) (0) = p 1 , b (2) (0) = p 2 , and b (j) α (0) = 0, where p 1,2 are the amplitudes of the initial qubit state.\n\nNote that Eq. (17) implies a fixed electron number (N ) in the reservoirs. At the first sight it would lead to depletion of the left reservoir of electrons over the time. Yet in the limit of N → ∞ (infinite reservoirs) the dynamics of an entire system reaches its steady state before such a depletion takes place 37, 38 .\n\nThe behavior of the qubit and the SET is given by the reduced density matrix, σ ss ′ (t). It is obtained from the entire system's density matrix |Ψ(t) Ψ(t)| by tracing out the (continuum) reservoir states. The space of such a reduced density matrix consists of four discrete states s, s ′ = a, b, c, d, shown schematically in Fig. 3 for the setup of Fig. 2a . The corresponding density-matrix elements are directly related to the amplitudes b(t), for instance,\n\nσ aa (t) = |b (1) (t)| 2 + l,r |b (1) lr (t)| 2 + l<l ′ ,r<r ′ |b ( 1\n\n) rr ′ ll ′ (t)| 2 + • • • ( 18a\n\n)\n\nσ dd (t) = l |b ( 2\n\n) 0l (t)| 2 + l<l ′ ,r |b ( 2\n\n) 0rll ′ (t)| 2 + l<l ′ <l ′′ ,r<r ′ |b ( 2\n\n) 0rr ′ ll ′ (t)| 2 + • • • ( 18b\n\n)\n\nσ bd (t) = l b ( 1\n\n) 0l (t)b (2) * 0l (t) + l<l ′ ,r b ( 1\n\n)\n\n0rll ′ (t)b (2) * 0rll ′ (t) +\n\nl<l ′ <l ′′ ,r<r ′ b ( 1\n\n) 0rr ′ ll ′ (t)b (2) * 0rr ′ ll ′ (t) + • • • . ( 18c\n\n)\n\nIn was shown in 37, 38 that the trace over the reservoir states in the system's density matrix can be performed in the large bias limit (strong non-equilibrium limit)\n\nV = µ L -µ R ≫ Γ, Ω 0 , U ( 19\n\n)\n\nwhere the level (levels) of the SET carrying the current are far away from the chemical potentials, and Γ is the width of the level E 0 . In this derivation we assumed only weak energy dependence of the transition amplitudes Ω l,r ≡ Ω L,R and the density of the reservoir states, ρ(E l,r ) = ρ L,R . As a result we arrive at Bloch-type rate equations for the reduced density matrix without any additional assumptions. The general form of these equations is 36,38\n\nσjj ′ = i(E j ′ -E j )σ jj ′ + i k σ jk Ωk→j ′ -Ωj→k σ kj ′ - k,k ′ P 2 πρ(σ jk Ω k→k ′ Ω k ′ →j ′ + σ kj ′ Ω k→k ′ Ω k ′ →j ) + k,k ′ P 2 πρ (Ω k→j Ω k ′ →j ′ + Ω k→j ′ Ω k ′ →j )σ kk ′ ( 20\n\n)\n\nHere Ω k→k ′ denotes the single-electron hopping amplitude that generates the k → k ′ transition. We distinguish between the amplitudes Ω describing single-electron hopping between isolated states and Ω describing transitions between isolated and continuum states. The latter can generate transitions between the isolated states of the system, but only indirectly, via two consecutive jumps of an electron, into and out of the continuum reservoir states (with the density of states ρ). These transitions are represented by the third and the fourth terms of Eq. ( 20 ). The third term describes the transitions (k\n\n→ k ′ → j) or (k → k ′ → j ′ )\n\n, which cannot change the number of electrons in the collector. The fourth term describes the transitions (k → j and k ′ → j ′ ) or (k → j ′ and k ′ → j) which increase the number of electrons in the collector by one. These two terms of Eq. (20) are analogues of the \"loss\" (negative) and the \"gain\" (positive) terms in the classical rate equations, respectively. The factor P 2 = ±1 in front of these terms is due anti-commutation of the fermions, so that P 2 = -1 whenever the loss or the gain terms in Eq. (20) proceed through a two-fermion state of the dot. Otherwise P 2 = 1. Note that the reduction of the time-dependent Schrödinger equation, i| Ψ(t) = H|Ψ(t) , to Eqs. (20) is performed in the limit of large bias without explicit use of any Markov-type or weak coupling approximations. The accuracy of these equations is respectively max\n\n(Γ, Ω 0 , U, T )/|µ L,R -E j |.\n\nA detailed example of this derivation is presented in Appendix A for the case of resonant tunneling through a single level. The derivation there and in Refs. 37, 38 were performed by assuming zero temperature in the leads, T = 0. Yet, this assumption is not important in the case of large bias, providing the levels carrying the current are far away from the Fermi\n\nenergies, |µ L,R -E j | ≫ T ."
},
{
"section_type": "OTHER",
"section_title": "IV. NO BACK-ACTION ON THE ENVIRONMENT",
"text": "A. Fluctuation of the tunneling coupling Now we apply Eqs. (20) to investigate the qubit's behavior in the configurations shown in Fig. 2 . First we consider the SET placed near the middle of the qubit, Figs. 2a,3. In this case the electron current through the SET will influence the coupling between two dots of the 7 (d) (b) (a) (c)\n\nR µ L µ R E 0 Γ L Γ E E 1 2 Ω 0 Ω 0 Ω 0 Ω 0 ' ' FIG. 3:\n\nThe available discrete states of the entire system corresponding to the setup of Fig. 2a . ΓL,R denote the tunneling rates to the corresponding reservoirs and Ω ′ 0 = Ω0 -δΩ.\n\nqubit, making it fluctuate between the values Ω 0 and Ω ′ 0 = Ω 0 -δΩ. The corresponding rate equations can be written straightforwardly from Eqs. (20). One finds,\n\nσaa = -Γ L σ aa + Γ R σ bb -iΩ 0 (σ ac -σ ca ), (21a) σbb = -Γ R σ bb + Γ L σ aa -iΩ ′ 0 (σ bd -σ db ), (21b) σcc = -Γ L σ cc + Γ R σ dd -iΩ 0 (σ ca -σ ac ), (21c) σdd = -Γ R σ dd + Γ L σ cc -iΩ ′ 0 (σ db -σ bd ), (21d) σac = -iǫ 0 σ ac -iΩ 0 (σ aa -σ cc ) -Γ L σ ac + Γ R σ bd , (21e) σbd = -iǫ 0 σ bd -iΩ ′ 0 (σ bb -σ dd ) -Γ R σ bd + Γ L σ ac , ( 21f\n\n) where Γ L,R = 2π|Ω L,R | 2 ρ L,R\n\nare the tunneling rates from the reservoirs and ǫ 0 = E 1 -E 2 . These equations display explicitly the time evolution of the SET and the qubit. The evolution of the former is driven by the first two terms in Eqs. (21a)-(21d). They generate charge-fluctuations inside the quantum dot of the SET (the transitions a←→b and c←→d), described by the \"classical\" Boltzmann-type dynamics. The qubit's evolution is described by the Bloch-type terms (c.f. Eqs. (3)), generating the qubit transitions (a←→c and b←→d). Thus Eqs. (21) are quite general, since they described fluctuations of the tunneling coupling driven by the Boltzmann-type dynamics.\n\nThe resulting time evolution of the qubit is given by the qubit (reduced) density matrix:\n\nσ 11 (t) = σ aa (t) + σ bb (t) , (22a) σ 12 (t) = σ ac (t) + σ bd (t) , ( 22b\n\n) and σ 22 (t) = 1 -σ 11 (t).\n\nSimilarly, the charge fluctuations of SET are determined by the probability of finding the SET occupied,\n\nP 1 (t) = σ bb (t) + σ dd (t) . ( 23\n\n)\n\nIt is given by the equation\n\nṖ1 (t) = Γ L -ΓP 1 (t) , ( 24\n\n)\n\nobtained straightforwardly from Eqs. (21). Here Γ = Γ L + Γ R is the total width. The same equation for P 1 (t) can be obtained if the qubit is decoupled from the SET (δΩ = 0). Thus there is no back-action of the qubit on the charge fluctuations inside the SET in the limit of large bias voltage. Consider first the stationary limit, t → ∞, where Ṗ1 (t) → 0 and σ(t) → 0. It follows from Eq. (24) that the probability of finding the SET occupied in this limit is P1 = Γ L /Γ. This implies that the fluctuations of the coupling Ω 0 , induced by the SET, would take place around the average value Ω = Ω 0 -P1 δΩ.\n\nWith respect to the qubit in the stationary limit, one easily obtains from Eqs. (21) that the qubit density matrix always becomes the statistical mixture (5), when t → ∞. This takes place for any initial conditions and any values of the qubit and the SET parameters. Therefore the effect of the fluctuating charge inside the SET does not lead to relaxation of the qubit, but rather to its decoherence.\n\nIt is important to note, however, that for the aligned qubit, ǫ = 0, the decoherence due to fluctuations of the tunneling coupling Ω 0 is not complete. Indeed, it follows from Eqs. (21) that d/dt[Re σ 12 (t)] = 0. The reason is that the corresponding operator, a † 14) and (15) 0).\n\n1 a 2 + a † 2 a 1 commutes with the total Hamiltonian H = H qb + H SET + H int , Eqs. (1), (\n\n, for E 1 = E 2 . As a result, Re σ 12 (t) = Re σ 12 (\n\nIn order to determine the decoherence rate analytically, we perform a Laplace transform on the density matrix, σ(E) = ∞ 0 σ(t) exp(-iEt)dE. Then solving Eq. (21) we can determine the decoherence rate from the locations of the poles of σ(E) in the complex E-plane. Consider for instance the case of ǫ 0 = 0 and the symmetric SET, Γ L = Γ R = Γ/2. One finds from Eqs. (21) and (22a) that\n\nσ11 (E) = i 2E + i(E -2Ω + iΓ) 4(E -2Ω + iΓ/2) 2 + Γ 2 -(2 δΩ) 2 + i(E + 2Ω + iΓ) 4(E + 2Ω + iΓ/2) 2 + Γ 2 -(2 δΩ) 2 . ( 25\n\n)\n\nUpon performing the inverse Laplace transform,\n\nσ 11 (t) = ∞+i0 -∞+i0 σ11 (E) e -iEt dE 2πi , ( 26\n\n)\n\nand closing the integration contour around the poles of the integrand, we obtain for Γ > 2δΩ and t ≫ 1/Γ\n\nσ 11 (t) -(1/2) ∝ e -(Γ- √ Γ 2 -4δΩ 2 )t/2 sin(2Ω t). ( 27\n\n)\n\nComparing this result with Eq. ( 12 ) we find that the decoherence rate is\n\nΓ d = 2 Γ -Γ 2 -4δΩ 2 Γ≫δΩ -→ (2δΩ) 2 /Γ . ( 28\n\n) For ǫ 0 = 0 and ǫ 0 , Γ ≪ Ω the decoherece rate Γ d is multiplied by an additional factor [1 -(ǫ 0 /2Ω) 2 ].\n\n8 In a general case, Γ L = Γ R , we obtain in the same limit (Γ L,R ≫ δΩ) for the decoherence rate:\n\nΓ d = (4 δΩ) 2 1 + ǫ 0 2Ω 2 Γ L Γ R (Γ L + Γ R ) 3 ( 29\n\n)\n\nIt is interesting to compare this result with the fluctuation spectrum of the charge inside the SET, Eq. ( B8 ), Appendix B. We find\n\nΓ d = 2 (δω R ) 2 S Q (0) , ( 30\n\n)\n\nwhere ω R = 4Ω 2 + ǫ 2 0 is the Rabi frequency. The latter represents the energy splitting in the diagonalized qubit Hamiltonian. Thus δω R corresponds to the amplitude of energy level fluctuations in a single dot. Although Eq. (30) has been obtained for small fluctuations δω R , it might be approximately correct even if δω R is of the order of Γ. It is demonstrated in Fig. 4 , where we compare σ 11 (t) and σ 12 (t), obtained from Eqs. (21) and (22) (solid line) with those from Eqs. (3a) and (4) (dashed line) for the decoherence rate Γ d given by Eq. (30). The initial conditions correspond to\n\nσ 11 (0) = 1 and σ 12 (0) = 0 (respectively, σ aa (0) = Γ R /Γ and σ bb (0) = Γ R /Γ).\n\nIn the case of aligned qubit, however, Re σ 12 (t) = Re σ 12 (0), as was explained above. On the other hand, one always obtains from (3a) and (4) that Re [σ 12 (t → ∞)] = 0. Therefore the phenomenological Bloch equations are not applicable for evaluation of Re [σ 12 (t)], even in the weak coupling limit (besides the case of Re [σ 12 (t = 0)] = 0).\n\nIn the large coupling regime (δΩ ≫ Γ) the phenomenological Bloch equations, Eqs. (3a) and (4), cannot be used, as well. Consider for simplicity the case of ǫ = 0 and Γ L,R = Γ/2. Then one finds from Eq. (27) that the damping oscillations between the two dots take place at two different frequencies, 2Ω ± (δΩ) 2 -(Γ/2) 2 , instead of the one frequency, ω R = 2Ω, given the Bloch equations. Moreover, Eq. (30) does not reproduce the decoherence (damping) rate in this limit. Indeed, one obtains from Eq. (28) that the decoherence rate Γ d = 2Γ for δΩ > Γ/2, so Γ d does not depend on the coupling (δΩ) at all."
},
{
"section_type": "OTHER",
"section_title": "B. Fluctuation of the energy level",
"text": "Consider the SET placed near one of the qubit dots, as shown in Fig. 2b . In this case the qubit-SET interaction term is given by Eq. ( 16 ). As a result the energy level E 1 will fluctuate under the influence of the fluctuations of the electron charge inside the SET. The available discrete states of the entire system are shown in Fig. 5 . Using Eqs. (20) we can write the rate equations, similar 10 20 30 40 50 t 0.4 0.2 0 0.2 0.4 (a) (b) 10 20 30 40 50 t 0.4 0.2 0 0.2 0.4 0.6 0.8 1 t t t Re 12 Im 12 11 σ ( ) σ ( ) σ ( ) FIG. 4: The occupation probability of the first dot of the qubit for ǫ = 2Ω, ΓL = Ω, ΓR = 2Ω and δΩ = 0.5Ω. The solid line is the exact result, whereas the dashed line is obtained from the Bloch-type rate equations with the decoherence rate given by Eq. (30).\n\n(d) (b) (a) (c) ' ' ' '\n\nΓ L Γ R Γ L Γ R E 0 E 0 R Γ L Γ L Γ R Γ 0 Ω 0 Ω 0 µ L µ R E 1 E 2 E 2 E 1 E 2 E 1 E 2 E +U 1 Ω 0 Ω FIG. 5:\n\nThe available discrete states of the entire system for the configuration shown in Fig. 2b . Here U is the repulsion energy between the electrons.\n\nto Eqs. (21),\n\nσaa = -Γ ′ L σ aa + Γ ′ R σ bb -iΩ 0 (σ ac -σ ca ), (31a) σbb = -Γ ′ R σ bb + Γ ′ L σ aa -iΩ 0 (σ bd -σ db ), (31b) σcc = -Γ L σ cc + Γ R σ dd -iΩ 0 (σ ca -σ ac ), (31c) σdd = -Γ R σ dd + Γ L σ cc -iΩ 0 (σ db -σ bd ), ( 31d\n\n) σac = -iǫ 0 σ ac -iΩ 0 (σ aa -σ cc ) - Γ L + Γ ′ L 2 σ ac + Γ R Γ ′ R σ bd , ( 31e\n\n) σbd = -i(ǫ 0 + U )σ bd -iΩ 0 (σ bb -σ dd ) - Γ R + Γ ′ R 2 σ bd + Γ L Γ ′ L σ ac , ( 31f\n\n)\n\nwhere Γ ′ L,R are the tunneling rate at the energy E 0 +U 39 . Let us assume that Γ ′ L,R = Γ L,R . Then it follows from Eqs. (31) that the behavior of the charge inside the SET is not affected by the qubit, the same as in the previous case of the Rabi frequency fluctuations. Also the qubit density matrix becomes the mixture (5) in the stationary state for any values of the qubit and the SET parameters. Hence, there is no qubit relaxation in this case either (except for the static qubit, Ω 0 = 0, and\n\nσ 11 (0) = σ 22 (0), Eq. (11)).\n\nSince according to Eq. ( 24 ), the probability of finding an electron inside the SET in the stationary state is P1 = Γ L /Γ, the energy level E 1 of the qubit is shifted by P1 U . Therefore it is useful to define the \"renormalized\" level displacement, ǫ = ǫ 0 + P1 U . 9 As in the previous case we use the Laplace transform, σ(t) → σ(E), in order to determine the decoherence rate analytically. In the case of Γ L = Γ R = Γ/2 and ǫ = 0 we obtain from Eqs. (31) σ11 (E) = i 2E + i 2E + 32(E + iΓ)Ω 2 0\n\nU 2 -4E(E + iΓ) . ( 32\n\n)\n\nThe position of the pole in the second term of this expression determines the decoherence rate. In contrast with Eq. (25), however, the exact analytical expression for the decoherence rate (Γ d ) is complicated, since it is given by a cubic equation. We therefore evaluate Γ d in a different way, by substituting E = ±2Ω 0iγ in the second term of Eq. ( 32 ) and then expanding the latter in powers of γ by keeping only the first two terms of this expansion. The decoherence rate Γ d is related to γ by Γ d = 4γ, as follows from Eq. ( 12 ). Then we obtain:\n\nΓ d = U 2 Γ 2(Γ 2 + 4Ω 2 0 ) for U ≪ (Ω 2 0 + ΓΩ 0 ) 1/2 64ΓΩ 2 0 U 2 + 16Ω 2 0 for U ≫ (Ω 2 0 + ΓΩ 0 ) 1/2 ( 33\n\n)\n\nIn general, if Γ L = Γ R , one finds from Eqs. (31) that\n\nΓ d = 2U 2 Γ L Γ R /[Γ(Γ 2 + 4Ω 2 0 )] for U ≪ (Ω 2 0 + ΓΩ 0 ) 1/2 .\n\nThe same as in the previous case, Eq. ( 30 ), the decoherence rate in a weak coupling limit is related to the fluctuation spectrum of the SET, S Q (ω), Eq. (B8), but now taken at a different frequency, ω = 2Ω 0 . The latter corresponds to the level splitting of the diagonalized qubit's Hamiltonian, ω R . Thus,\n\nΓ d = U 2 S Q (ω R ) , ( 34\n\n)\n\nwhich can be applied also for ǫ = 0. This is illustrated by Fig. 6 which shows σ 11 (t) obtained from Eqs. (31) and (22) (solid line) with Eqs. (3a) and (4) (dashed line) for the decoherence rate Γ d given by Eq. ( 34 ). As in the previous case, shown in Fig. 4 , the initial conditions correspond to σ 11 (0) = 1 and σ 12 (0) = 0 (respectively, σ aa (0) = Γ R /Γ and σ bb (0) = Γ R /Γ). One finds from Fig. 6 that Eq. (34) can be used for an estimation of Γ d even for U ∼ Γ, Ω 0 . In contrast with the tunneling-coupling fluctuations, Eq. (30), where the decoherence rate is given by S Q (0), the fluctuations of the qubit's energy level generate the decoherence rate, determined by the fluctuation spectrum at Rabi frequency, S Q (ω R ), Eq. (34). A similar distinction between the decoherence rates generated by different components of the fluctuating field, exists in a phenomenological description of magnetic resonance 27 . One can understand this distinction by diagonalizing the qubit's Hamiltonian. In this case the Rabi frequency, ω R , becomes the level splitting of the qubit's states |± = (|1 ± |2 )/ √ 2 (for ǫ = 0). So in this basis, the tunneling-coupling fluctuations correspond to simultaneous fluctuations of the energy levels in the both dots.\n\n10 20 30 40 50 t 0 0.4 0.2 0 0.2 0.4 (a) (b) 10 20 30 40 50 t 0 0.4 0.2 0 0.2 0.4 0.6 0.8 1 t t t Re 12 Im 12 11 σ ( ) σ ( ) σ ( ) FIG. 6: The probability of finding the electron in the first dot of the qubit for ǫ = 2Ω0, ΓL = Ω0, ΓR = 2Ω0 and U = 0.5Ω0. The solid line is the exact result, whereas the dashed line is obtained from the Bloch-type rate equations with the decoherence rate given by Eq. (34).\n\nSince these fluctuations are \"in phase\", we could expect that the corresponding dephasing rate is determined by spectral density at zero frequency. In fact, it looks like as fluctuations of a single dot state, considered by Levinson in a weak coupling limit 31 . On the other hand by fluctuating the energy level in one of the dots only, one can anticipate that the corresponding dephasing rate is determined by the fluctuation spectrum at the Rabi frequency, ω R , Eq. ( 34 ), which is a frequency of the inter-dot transitions. Since ω R can be controlled by the qubit's levels displacement, ǫ, the relation (34) can be implied by using qubit for a measurement of the shot-noise spectrum of the environment 18, 19, 40 . For instance, it can be done by attaching a qubit to reservoirs at different chemical potentials. The corresponding resonant current which would flow through the qubit in this case, can be evaluated via a simple analytical expression 13 that includes explicitly the decoherence rate, Eq. (34). Thus by measuring this current for different level displacement of the qubit (ǫ 0 ), one can extract the spectral density of the fluctuating environment acting on the qubit 18 . Although Eq. ( 34 ) for the decoherence rate has been obtained by using a particular mechanism for fluctuations of the qubit's energy levels, we suggest that this mechanism is quite general. Indeed, the rate equations (31) can describe any fluctuating media near a qubit, driven by the Boltzmann type of equations. Therefore it is rather natural to assume that Eq. (34) would be valid for any type of such (classical) environment in weak coupling limit. This implies that the decoherence rate is always determined via the spectral density of a fluctuating qubit's level, whereas the nature of a particular medium inducing these fluctuations would be irrelevant. In order to substantiate this point it is important to compare Eq. ( 34 ) with the corresponding decoherence rate induced by the thermal environment in the framework of the spin-boson model. In a weak damping limit this model predicts 1,2 T -1 1 = T -1 2 = (q 2 0 /2)S(ω R ) , where q 0 is a coupling of the medium with the qubit levels (q 0 corresponds to U in our case) and S(ω) is a spectral density. Using Eq. (10) one finds that this result coincides with Eq. (34).\n\n10 0 5 10 15 20 t 0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 t 0 0.2 0.4 0.6 0.8 1.0 t t 11 11 0 0 U/ =100 Ω U/ =10 σ ( ) Ω σ ( ) (a) (b)\n\nFIG. 7: The probability of finding the electron in the first dot of the qubit for ǫ = 0, ΓL = ΓR = Ω0 and U , as given by Eqs. (31) (solid line) and from the Bloch-type equations (dashed line) with the decoherence rate given by Eq. (33)."
},
{
"section_type": "OTHER",
"section_title": "C. Strong-coupling limit and localization",
"text": "Let us consider the limit of U ≫ (Ω 2 0 + ΓΩ 0 ) 1/2 . Our rate equation (31) are perfectly valid in this region, providing only that E 0 + U is deeply inside of the potential bias, Eq. (19). We find from Eq. (33) that the decoherence rate is not directly related to the spectrum of fluctuations in strong coupling limit. In addition, the effective frequency of the qubit's Rabi oscillations (ω ef f R ) decreases in this limit. Indeed, by using Eqs. (32), (26), one finds that the main contribution to σ 11 (t), is coming from a pole of σ11 (E), which lies on the imaginary axis. This implies that the effective frequency of Rabi oscillations strongly decreases when U ≫ (Ω 2 0 + ΓΩ 0 ) 1/2 . In addition, the decoherence rate Γ d → 0 in the same limit, Eq. (33). As a result, the electron would localize in the initial qubit state, Fig. 7 .\n\nThe results displayed in this figure show that the solution of the Bloch-type rate equations, with the decoherence rate given by Eq. (33), represents damped oscillations (dashed line). It is very far from the exact result (solid line), obtained from Eqs. (31) and corresponding to the electron localization in the first dot. The latter is a result of an effective decrease of the Rabi frequency for large U that slows down electron transitions between the dots. Thus such an environment-induced localization is different from the Zeno-type effect (unlike an assumption of Ref. 12 ). Indeed, the Zeno effect takes place whenever the decoherence rate is much larger then the coupling between the qubit's states 13, 33 . However, the decoherence rate in the strong coupling limit is much smaller then the coupling Ω 0 . In fact, the localization shown in Fig. 7 is rather similar to that in the spin-boson model 1, 2 . It shows that in spite of their defferences, both models trace the same physics of the back-action of the environment (SET) on the qubit.\n\nV. BACK-ACTION OF THE QUBIT ON THE ENVIRONMENT A. Weak back-action effect\n\nNow we investigate a weak dependence of the width's Γ L,R on the energy U , Fig. 5 . We keep only the linear term, Γ ′ L,R = Γ L,R +α L,R U , by assuming that U is small. (A similar model has been considered in 28, 41 ). In contrast with the previous examples, where the widths have not been dependent on the energy, the qubit's oscillation would affect the SET current and its charge correlator. A more interesting case corresponds to α L = α R . Let us take for simplicity α L = 0 and α R = α = 0. Similarly to the previous case we introduce the \"renormalized\" level displacement, ǫ = ǫ 0 -(Γ L /Γ)U , where ǫ = 0 corresponds to the aligned qubit. Solving Eqs. (31) in the steady-state limit, σ = σ(t → ∞), and keeping only the first term in expansion in powers of U , we find for the reduced density matrix of the qubit, Eqs. (22):\n\nσ = 1 2 - α ǫ 4Γ R αΩ 0 (1 + c α U ) 2Γ R αΩ 0 (1 + c α U ) 2Γ R 1 2 + α ǫ 4Γ R , ( 35\n\n)\n\nwhere c = (αǫ -2Γ)/(4Γ R Γ). It follows from Eqs. (35) that the qubit's density matrix in the steady-state is no longer a mixture, Eq. ( 5 ) . Indeed, the probability to occupy the lowest level is always larger than 1/2 and σ12 = 0. This implies that relaxation takes place together with decoherence. The ratio of the relaxation and decoherence rates is given by the off-diagonal terms of the reduced density matrix of the qubit. For ǫ = 0 one finds from Eq. (13) that Γ d /Γ r = σ-1 12 -2. In order to find a relation between the decoherence and relaxation rates, Γ d,r , and the fluctuation spectrum of the qubit energy level, S Q (ω), we first evaluate the damping rate of the qubit's oscillations (γ). Using Eq. ( 12 ) we find that this quantity is related to the decoherence and relaxation rates by γ = (Γ d + 2Γ r )/4. The same as in the previous case the rate γ determined by poles of Laplace transformed density matrix σ(t) → σ(E) in the complex E-plane. Consider for simplicity the case of ǫ = 0 and Γ L = Γ R = Γ/2. Performing the Laplace transform of Eqs. (31) we look for the poles of σ 11 (E) at E = ±2Ω 0iγ by assuming that γ is small. We obtain\n\nΓ d + 2Γ r = U 2 2(Γ 2 + 4Ω 2 0 ) Γ -α U Γ 2 -4Ω 2 0 2 (Γ 2 + 4Ω 2 0 ) ( 36\n\n)\n\nfor U ≪ Ω 0 . Now we evaluate the correlator of the charge inside the SET, S Q (ω) which induces the energy-level fluctuations of the qubit. Using Eqs. (31) and (B6) we find,\n\nS Q (ω) = Γ 2 (Γ 2 + ω 2 ) -α U Γ 2 -ω 2 4 (Γ 2 + ω 2 ) 2 ( 37\n\n)\n\n11 for αU ≪ Γ. Therefore in the limit of U ≪ Ω 0 and α U ≪ Γ the total damping rate of the qubit's oscillations is directly related to the spectral density of the fluctuations spectrum taken at the Rabi frequency,\n\nΓ d + 2Γ r = U 2 S Q (2Ω 0 ). ( 38\n\n)\n\nThis represents a generalization of Eq. (34) for the case of a weak back-action of qubit oscillations on the spectral density of the environment. As a result, the qubit displays relaxation together with decoherence. It is remarkable that the total qubit's damping rate is still given by the fluctuation spectrum of the SET (environment) modulated by the qubit. Note that Eq. ( 38 ) can be applied only if the modulation of the tunneling rate through the SET (tunneling current) is small α U ≪ Γ, in addition to a weak distortion of the qubit (U ≪ Ω 0 ). In the case of strong back-action of the qubit on the environment the decorerence and relaxation rates of the qubit are not directly related to the fluctuation spectrum of the environment, even if the distortion of the qubit is small. This point is illustrated by the following example."
},
{
"section_type": "OTHER",
"section_title": "B. Strong back-action",
"text": "Until now we considered the case where E 0 + U ≪ µ L , so that the interacting electron of the SET remains deeply inside the voltage bias. If however, the interaction U between the qubit and the SET is such that E 0 + U ≫ µ L , the qubit's oscillation would strongly affect the fluctuation of charge inside the SET. Indeed, the current through the SET is blocked whenever the level E 1 of the qubit is occupied, Fig. 8 . In fact, this case can be treated with small modification of the rate equations (31)\n\n, if only µ L -E 0 ≫ Γ and E 0 + U -µ L ≫ Γ, where E 0 is a level\n\nof the SET carrying the current.\n\nThe corresponding quantum rate equations describing the system are obtained directly from Eqs. (20). Assuming that the widths Γ are energy independent we find 16\n\nσaa = (Γ L + Γ R )σ bb -iΩ 0 (σ ac -σ ca ), (39a) σbb = -(Γ R + Γ L )σ bb -iΩ 0 (σ bd -σ db ), (39b) σcc = -Γ L σ cc + Γ R σ dd -iΩ 0 (σ ca -σ ac ), (39c) σdd = -Γ R σ dd + Γ L σ cc -iΩ 0 (σ db -σ bd ), ( 39d\n\n) σac = -iǫ 0 σ ac -iΩ 0 (σ aa -σ cc ) - Γ L 2 σ ac + Γ R σ bd , (39e) σbd = -i(ǫ 0 + U )σ bd -iΩ 0 (σ bb -σ dd ) -Γ R + Γ L 2 σ bd . ( 39f\n\n)\n\nSolving Eqs. (39) in the stationary limit, σ = σ(t → ∞) and introducing the \"renormalized\" level displacement, ǫ = ǫ 0 -U Γ L /(2Γ), we obtain for the qubit's den-\n\n(d) (b) (a) (c) 0 E 0 E 2 E R µ L µ R Γ L 1 Ω 0 E 2 E +U 1 Ω 0 E 1 Ω 0 E 2 E 1 Ω 0 E 2 E Γ FIG. 8:\n\nThe available discrete states of the entire system when the electron-electron repulsive interaction U breaks off the current through the SET.\n\nsity matrix, Eqs. (22) in the steady state:\n\nσ11 = 1 2 - 8ǫU 16ǫ 2 + 8U ǫ + 48Ω 2 0 + 9(U 2 + Γ 2 ) ,( 40a\n\n) σ12 = 12U Ω 0 16ǫ 2 + 8U ǫ + 48Ω 2 0 + 9 (U 2 + Γ 2 ) , ( 40b\n\n)\n\nwhere for simplicity we considered the symmetric case, Γ L = Γ R = Γ/2. It follows from Eqs. (40) that similarly to the previous example, the qubit's density matrix is no longer a mixture (5). The relaxation takes place together with decoherence in this case too. Let us consider weak distortion of the qubit by the SET, U < Ω 0 . Although the values of U are restricted from below (U ≫ Γ+µ L -E 0 ), this limit can be achieved if the level E 0 is close to the Fermi energy, providing only that µ L -E 0 ≫ Γ, and Γ ≪ U . Now we evaluate σ 11 (t) with the rate equations (39) and then compare it with the same quantity obtained from the Bloch equations, Eq. ( 12 ), where Γ d,r are given by Eqs. (34)and (13). The corresponding charge-correlator, S Q (ω R ), is evaluated by Eqs. (B6) and (39). As an example, we take symmetric qubit with aligned levels, ǫ = 0, Γ L = Γ R = 0.05Ω 0 and U = 0.5Ω 0 . The decoherence and relaxation rates, corresponding to these parameters are respectively:\n\nΓ d /Ω 0 = 0.0038 and Γ r /Ω 0 = 0.00059.\n\nThe results are presented in Fig. 9a . The solid line shows σ 11 (t), obtained from the rate equations (39), where the dashed line is the same quantity obtained from Eq. ( 12 ). We find that Eq. (34) (or (38)) underestimates the actual damping rate of σ 11 (t) by an order of magnitude). This lies in a sharp contrast with the previous case, where the energy level of the SET is not distorted by the qubit, Γ ′ L,R = Γ L,R , Fig. 5 . Indeed, in this case σ 11 (t) obtained Eq. (12) with Γ d given by Eq. ( 34 ) and Γ r = 0, agrees very well with that obtained from the rate equations (31), as shown in Fig. 9b . Such an example clearly illustrates that the decoherence is not related to the fluctuation spectrum of the environment, whenever the environment is strongly affected by the qubit, even if the coupling with a qubit is small. This is a typical case of measurement, corresponding to a noticeable response of the environment to the qubit's state (a \"signal\"). 12 (a) 20 40 60 80 100 t 0 0.2 0.4 0.6 0.8 1 20 40 60 80 100 t 0 0.2 0.4 0.6 0.8 1 (b) t t 11 11 σ ( ) σ ( ) FIG. 9: (a) The probability of finding the electron in the first dot of the qubit for ǫ = 0, ΓL = ΓR = 0.05Ω0 and U = 0.5Ω0. The solid line is obtained from Eqs. (39), whereas the dashed line corresponds to the Eq. ( 12 ) with Γ d given by Eq. (34); (b) the same for the case, shown in Fig. 5 , where the solid line corresponds to Eqs. (31)."
},
{
"section_type": "OTHER",
"section_title": "VI. SUMMARY",
"text": "In this paper we propose a simple model describing a qubit interacting with fluctuating environment. The latter is represented by a single electron transistor (SET) in close proximity of the qubit. Then the fluctuations of the charge inside the SET generate fluctuating field acting on the qubit. In the limit of large bias voltage, the Schrödinger equation for the entire system is reduced to the Bloch-type rate equations. The resulting equations are very simple, so that one can easily analyze the limits of weak and strong coupling of the qubit with the SET.\n\nWe considered separately two different cases: (a) there is no back-action of the qubit on the SET behavior, so that the latter represents a \"pure environment\"; and (b) the SET behavior depends on the qubit's state. In the latter case the SET can \"measure\" the qubit. The setup corresponding to the \"pure environment\" is realized when the energy level of the SET carrying the current lies deeply inside the potential bias. The second (measurement) regime of the SET is realized when the tunneling widths of the SET are energy dependent, or when the energy level of the SET carrying the current is close enough to the Fermi level of the corresponding reservoir. Then the electron-electron interaction between the qubit and the SET modulates the electron current through the SET.\n\nIn the case of the \"pure environment\" (\"nomeasurement\" regime) we investigate separately two different configurations of the qubit with respect to the SET. In the first one the SET produces fluctuations of the off-diagonal coupling (Rabi frequency) between two qubit's states. In the second configuration the SET produces fluctuations of the qubit's energy levels. In the both cases we find no relaxation of the qubit, despite the energy transfer between the qubit and the SET can take place. As a result the qubit always turns asymptotically to the statistical mixture. We also found that in both cases the decoherence rate of the qubit in the weak coupling limit is given by the spectral density of the cor-responding fluctuating parameter. The difference is that in the case of the off-diagonal coupling fluctuations the spectral density is taken at zero frequency, whereas in the case of the energy level fluctuations it is taken at the Rabi-frequency.\n\nIn the case of the strong coupling limit, however, the decoherence rate is not related to the fluctuation spectrum. Moreover we found that the electron in the qubit is localized in this limit due to an effective decrease of the off-diagonal coupling. This phenomenon may resemble the localization in the spin-boson model in the strong coupling limit.\n\nIf the charge correlator and the total SET current are affected by the qubit (back-action effect), we found that the off-diagonal density-matrix elements of the qubit survive in the steady-state limit and therefore the relaxation rate is not zero. We concentrated on the case of weak coupling, when the Coulomb repulsion between the qubit and the SET is smaller then the Rabi frequency. The back-action of the qubit on the SET, however, can be weak or strong. In the first case we found that the total damping rate of the qubit due to decoherence and relaxation is again given by the spectral density of the SET charge fluctuations, modulated by the qubit. This relation, however, is not working if the back-action is strong. Indeed, we found that the damping rate of the qubit in this case is larger by an order of magnitude than that given by the spectral density of the corresponding fluctuating parameter.\n\nThis looks like that in the strong back-action of the qubit on the SET the major component of decoherence is not coming from the fluctuation spectrum of the qubit's parameters only, but also from the measurement \"signal\" of the SET. On the first sight it could agree with an analysis of Ref. 30 , suggesting that the decoherence rate contains two components, generated by a measurement and by a \"pure environment\" (environmental fluctuations). The latter therefore represents an unavoidable decoherence, generated by any environment. Yet, in a weak coupling regime such a separation seems not working. In this case the damping (decoherence) rate is totally determined by the environment fluctuations, even so modulated by the qubit.\n\nAlthough our model deals with a particular setup, it bears the main physics of a fluctuating environment, acting on a qubit. Indeed, the Bloch-type rate equations, which we used in our analysis have a pronounced physical meaning: they relate the variation of qubit parameters with a nearby fluctuating field described by rate equations. A particular mechanism, generated this field should not be relevant for an evaluations of the decoherence and relaxation rates, but only its fluctuation spectrum. Indeed, in the weak coupling limit our result for the decorence rate coincides with that obtained in a framework of the spin-boson model. Thus our model can be considered as a generic one. Its main advantage is that it can be easily extended to multiple coupled qubits. Such an analysis would allow to determine how decoher-13 ence scales with number of qubits 42 , which is extremely important for a realization of quantum computations.\n\nIn addition, our model can be extended to a more complicated fluctuating environments, such as containing characteristic frequencies in its spectrum. It would formally correspond to a replacement of the SET in Fig. 2 by a double-dot (DD) coupled to the reservoirs 43 . All these situations, however, must be a subject of a separate investigation."
},
{
"section_type": "OTHER",
"section_title": "VII. ACKNOWLEDGEMENT",
"text": "One of us (S.G.) thanks T. Brandes and C. Emary for helpful discussions and important suggestions. S.G is also grateful to the Max Planck Institute for the Physics of Complex Systems, Dresden, Germany, and to NTT Basic Research Laboratories, Atsugi-shi, Kanagawa, Japan, for kind hospitality.\n\nAPPENDIX A: QUANTUM-MECHANICAL DERIVATION OF RATE EQUATIONS FOR QUANTUM TRANSPORT\n\nConsider the resonant tunneling through the SET, shown schematically in Fig. 10 . The entire system is described by the Hamiltonian H SET , given by Eq. ( 14 ). The wave function can be written in the same way as Eq. ( 17 ), where the variables related to the qubit are omitted,\n\n|Ψ(t) = b(t) + l b 0l (t)c † 0 c l + l,r b rl (t)c † r c l + l<l ′ ,r b 0rll ′ (t)c † 0 c † r c l c l ′ + • • • |0 ¯ . (A1)\n\nSubstituting |Ψ(t) into the time-dependent Schrödinger equation, i∂ t |Ψ(t) = H SET |Ψ(t) , and performing the Laplace transform, b(E) = ∞ 0 exp(iEt) b(t)dt, we obtain the following infinite set of algebraic equations for the L µ E l E r 0 E Ω Ω µ R l r FIG. 10: Resonant tunneling through a single dot. µL,R are the Fermi energies in the collector and emitter, respectively.\n\namplitudes b(E):\n\nE b(E) - l Ω l b0l (E) = i ( A2a\n\n) (E + E l -E 0 ) b0l (E) -Ω l b(E) - r Ω r blr (E) = 0 (A2b) (E + E l -E r ) blr (E) -Ω r b0l (E) - l ′ Ω l ′ b0ll ′ r (E) = 0 (A2c) (E + E l + E l ′ -E 0 -E r ) b0ll ′ r (E) -Ω l ′ blr (E) + Ω l bl ′ r (E) - r ′ Ω r ′ bll ′ rr ′ (E) = 0 (A2d) • • • • • • • • •\n\n(The r.h.s of Eq. (A2a) reflects the initial condition.) Let us replace the amplitude b in the term Ω b of each of the equations (A2) by its expression obtained from the subsequent equation. For example, substituting b0l (E) from Eq. (A2b) into Eq. (A2a) we obtain\n\nE - l Ω 2 l E + E l -E 0 b(E) - l,r Ω l Ω r E + E l -E 0 blr (E) = i. ( A3\n\n)\n\nSince the states in the reservoirs are very dense (continuum), one can replace the sums over l and r by integrals, for instance l → ρ L (E l ) dE l , where ρ L (E l ) is the density of states in the emitter, and Ω l,r → Ω L,R (E l,r ). Consider the first term\n\nS 1 = µL -Λ Ω 2 L (E l ) E + E l -E 0 ρ L (E l )dE l ( A4\n\n)\n\nwhere Λ is the cut-off parameter. Assuming weak energy dependence of the couplings Ω L,R and the density of states ρ L,R , we find in the limit of high bias,\n\nµ L = Λ → ∞ S 1 = -iπΩ 2 L (E 0 -E)ρ L (E 0 -E) = -i Γ L 2 . ( A5\n\n)\n\n14 Consider now the second sum in Eq. (A3).\n\nS 2 = Λ -Λ ρ R (E r )dE r × Λ -Λ Ω L (E l )Ω R (E r ) blr (E, E l , E r ) E + E l -E 0 ρ L (E l )dE l , ( A6\n\n)\n\nwhere we replaced blr (E) by b(E, E l , E r ) and took µ L = Λ, µ R = -Λ. In contrast with the first term of Eq. ( A3 ), the amplitude b is not factorized out the integral (A6). We refer to this type of terms as \"cross-terms\". Fortunately, all \"cross-terms\" vanish in the limit of large bias, Λ → ∞. This greatly simplifies the problem and is very crucial for a transformation of the Schrödinger to the rate equations. The reason is that the poles of the integrand in the E l (E r )-variable in the \"cross-terms\" are on the same side of the integration contour. One can find it by using a perturbation series the amplitudes b in powers of Ω. For instance, from iterations of Eqs. (A2) one finds\n\nb(E, E l , E r ) = iΩ L Ω R E(E + E l -E r )(E + E l -E 0 ) + • • • ( A7\n\n)\n\nThe higher order powers of Ω have the same structure. Since E → E + iǫ in the Laplace transform, all poles of the amplitude b(E, E l , E r ) in the E l -variable are below the real axis. In this case, substituting Eq. (A7) into Eq. (A6) we find lim Λ→∞ Λ -Λ\n\nΩ L Ω R (E + iǫ)(E + E 0 -E 1 + iǫ) 2 (E + E 0 -E r + iǫ) + • • • dE l = 0 , ( A8\n\n)\n\nThus, S 2 → 0 in the limit of µ L → ∞, µ R → -∞.\n\nApplying analogous considerations to the other equations of the system (A2), we finally arrive at the following set of equations:\n\n(E + iΓ L /2) b(E) = i ( A9a\n\n) (E + E l -E 0 + iΓ R /2) b0l (E) -Ω l b(E) = 0 ( A9b\n\n) (E + E l -E r + iΓ L /2) blr (E) -Ω r b0l (E) = 0 ( A9c\n\n) (E + E l + E l ′ -E 0 -E r + iΓ R /2) b0ll ′ r (E) -Ω l ′ blr (E) + Ω l bl ′ r (E) = 0 (A9d) • • • • • • • • •\n\nEqs. (A9) can be transformed directly to the reduced density matrix σ (n,n ′ ) jj ′ (t), where j = 0, 1 denote the state of the SET with an unoccupied or occupied dot and n denotes the number of electrons which have arrived at the collector by time t. In fact, as follows from our derivation, the diagonal density-matrix elements, j = j ′ and n = n ′ , form a closed system in the case of resonant tunneling through one level, Fig. 10 . The off-diagonal elements, j = j ′ , appear in the equation of motion whenever more than one discrete level of the system carry the transport (see Eq. ( 20 ). Therefore we concentrate below on the diagonal density-matrix elements only, σ (n) 00 (t) ≡ σ (n,n) 00 (t) and σ (n) 11 (t) ≡ σ (n,n) 11 (t). Applying the inverse Laplace transform on finds σ (n) 00 (t) = l...,r...\n\n′ 4π 2 bl • • • n r • • • n (E) b * l • • • n r • • • n (E ′ )e i(E ′ -E)t ( A10a\n\n) σ (n) 11 (t) =\n\nl...,r...\n\ndEdE ′ 4π 2 b0l • • • n+1 r • • • n (E) b * 0 l • • • n+1 r • • • n (E ′ )e i(E ′ -E)t ( A10b\n\n)\n\nConsider, for instance, the term σ A9b ) by b * 0l (E ′ ) and then subtracting the complex conjugated equation with the interchange\n\n(0) 11 (t) = l |b 0l (t)| 2 . Multiplying Eq. (\n\nE ↔ E ′ we obtain dEdE ′ 4π 2 l (E ′ -E -iΓ R ) b0l (E) b * 0l (E ′ ) -2Im l Ω l b0l (E) b * (E ′ ) e i(E ′ -E)t = 0 (A11)\n\n15 Using Eq. (A10b) one easily finds that the first integral in Eq. (A11) equals to -i[ σ(0) 11 (t) + Γ R σ (0) 11 (t)]. Next, substituting b0l (E) = Ω l b(E) E + E l -E 0 + iΓ R /2 (A12) from Eq. ( A9b ) into the second term of Eq. (A11), and replacing a sum by an integral, one can perform the E lintegration in the large bias limit, µ L → ∞, µ R → -∞.\n\nThen using again Eq. (A10b) one reduces the second term of Eq. (A11) to iΓ L σ (0) 00 (t). Finally, Eq. (A11) reads σ(0)\n\n11 (t) = Γ L σ (0) 00 (t) -Γ R σ (0) 11 (t).\n\nThe same algebra can be applied for all other amplitudes bα (t). For instance, by using Eq. ( A10a ) one easily finds that Eq. (A9c) is converted to the following rate equation σ(1) 00 (t) = -Γ L σ (1) 00 (t) + Γ R σ (0) 11 (t). With respect to the states involving more than one electron (hole) in the reservoirs (the amplitudes like b0ll ′ r (E) and so on), the corresponding equations contain the Pauli exchange terms. By converting these equations into those for the density matrix using our procedure, one finds the \"cross terms\", like Ω l bl ′ r (E)Ω l ′ b * lr (E ′ ), generated by Eq. (A9d). Yet, these terms vanish after an integration over E l(r) in the large bias limit, as the second term in Eq. ( A3 ). The rest of the algebra remains the same, as described above. Finally we arrive at the following infinite system of the chain equations for the diagonal elements, σ (n) 00 and σ (n) 11 , of the density matrix,\n\nσ(0) 00 (t) = -Γ L σ ( 0\n\n) 00 (t) , ( A13a\n\n) σ(0) 11 (t) = Γ L σ ( 0\n\n) 00 (t) -Γ R σ ( 0\n\n) 11 (t) , ( A13b\n\n) σ(1) 00 (t) = -Γ L σ ( 1\n\n) 00 (t) + Γ R σ ( 0\n\n) 11 (t) , (A13c) σ(1) 11 (t) = Γ L σ ( 1\n\n) 00 (t) -Γ R σ ( 1\n\n) 11 (t) , (A13d) • • • • • • • • •\n\nSumming over n in Eqs. (A13) we find for the reduced density matrix of the SET, σ(t) = n σ (n) (t), the following \"classical\" rate equations,\n\nσ00 (t) = -Γ L σ 00 (t) + Γ R σ 11 (t) (A14a) σ11 (t) = Γ L σ 00 (t) -Γ R σ 11 (t) ( A14b\n\n)\n\nThese equations represent a particular case of our general quantum rate equations (20), which are derived using the above described technique 37,38 .\n\nAPPENDIX B: CORRELATOR OF ELECTRIC CHARGE INSIDE THE SET.\n\nThe charge correlator inside the SET is given by S Q (ω) = SQ (ω) + SQ (-ω), where\n\nSQ (ω) = ∞ 0 δ Q(0)δ Q(t) e iωt dt . ( B1\n\n)\n\nHere δ Q(t) = c † 0 (t)c 0 (t)q and q = P1 = P 1 (t → ∞) is the average charge inside the dot. Since the initial state, t = 0 in Eq. (B1) corresponds to the steady state, one can represent the time-correlator as δ Q(0)δ Q(t) = q=0,1\n\nP q (0)(q -q)( Q q (t) -q) , (B2)\n\nwhere P q (0) is the probability of finding the charge q = 0, 1 inside the quantum dot in the steady state, such that P 1 (0) = q and P 0 (0) = 1q, and Q q (t) = P (q) 1 (t) is the average charge in the dot at time t, starting with the initial condition P (q) 1 (0) = q. Substituting Eq. (B2) into Eq. (B1) we finally obtain\n\nSQ (ω) = q(1 -q)[ P (1) 1 (ω) - P (0) 1 (ω)] , ( B3\n\n)\n\nwhere P (q) 1 (ω) is a Laplace transform of P (q) 1 (t). These quantities are obtained directly from the rate equations, such that q = σbb + σdd and P (q) 1 (ω) = σ(q) bb (ω) + σ(q) dd (ω), where σ = σ(t → ∞) and σ(q) (ω) is the Laplace transform σ (q) (t) with the initial conditions corresponding to the occupied (q = 1) or unoccupied (q = 0) SET. In order to find these quantities it is useful to rewrite the rate equations in the matrix form, σ(t) = M σ(t), representing σ(t) as the eight-vector, σ = {σ aa , σ bb , σ cc , σ dd , σ ac , σ ca , σ bd , σ db } and M as the corresponding 8 × 8-matrix. Applying the Laplace transform we find the following matrix equation, (i ω I + M )σ (q) (ω) = -σ (q)\n\n(0) , ( B4\n\n)\n\nwhere I is the unit matrix and σ (q) (0) is the initial condition for the density-matrix obtained by projecting the total wave function (17) on occupied (q = 1) and unoccupied (q = 0) states of the SET in the limit of t → ∞, σ (1) (0) = N 1 {0, σbb , 0, σdd , 0, 0, σbd , σdb } , (B5a)\n\nσ (0) (0) = N 0 {σ aa , 0, σcc , 0, σac , σca , 0, 0} , ( B5b\n\n)\n\nand N 1 = 1/q and N 0 = 1/(1q) are the corresponding normalization factors. Finally one obtains:\n\nS Q (ω) = 2q(1 -q)Re [σ ( 1\n\n) bb (ω) + σ(1) dd (ω)\n\nσ(0) bb (ω)σ(0) dd (ω)]. (B6) In the case shown in Fig. 2 one finds from Eqs. (21) or Eqs. (31) for Γ ′ L,R = Γ L,R that σac = σ bd = 0, q = Γ L /Γ and σ(q) bb (ω) + σ(q) dd (ω) = P (q) 1 (ω). The latter equation is given by (iω -Γ) P (q) 1 (ω) = -q + iΓ L ω . (B7) Substituting Eq. (B7) into Eq. (B3) one obtains:\n\nS Q (ω) = 2Γ L Γ R Γ(ω 2 + Γ 2 ) . ( B8\n\n)\n\n16 Obviously, for a more general case when Γ ′ L,R = Γ L,R , or when the electron-electron interaction excites the electron inside the SET above the Fermi level, Fig. 8 , the ex-pressions for S Q (ω), obtained from Eq. (B6) have a more complicated than Eq. (B8). * Electronic address: shmuel.gurvitz@weizmann.ac.il 1 A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987). 2 U. Weiss, Quantum Dissipative Systems (World Scientific, Singapure, 2000). 3 A. Shnirman, Y. Makhlin, and G. Schoön, Phys. Scr. T102, 147 (2002). 4 H. Gassmann, F. Marquardt, and C. Bruder, Phys. Rev. E66, 041111 (2002). 5 E. Paladino, L. Faoro, G. Falci, and R. Fazio, Phys. 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Lett., 84, 1986 (2000). 19 E. Onac, F. Balestro, L.H. Willems van Beveren, U. Hartmann, Y.V. Nazarov, and L.P. Kouwenhoven, Phys. Rev. Lett. 96, 176601 (2006). 20 I. Neder, M. Heiblum, D. Mahalu, and V. Umansky, Phys.\n\nRev. Lett. 98, 036803 (2007). 21 I. Neder and F. Marquardt, New J. Phys. 9, 112 (2007), and references therein. 22 W.G. van der Wiel, T. Fujisawa, S. Tarucha, L.P. Kouwenhoven, Japanese Jour. Appl. Phys. 40, 2100 (2001). 23 J. M. Elzerman, R. Hanson, J. S. Greidanus, L. H. W.\n\nvan Beveren, S. De Franceschi, L. M. K. Vandersypen, S.\n\nTarucha, L. P. Kouwenhoven, Physica E 25, 135 (2004). 24 T. Hayashi, T. Fujisawa, H.D. Cheong, Y.H. Jeong, Y.\n\nHirayama, Phys. Rev. Lett. 91, 226804 (2003). 25 J. Shao, C. Zerbe, and P. Hanggi, Chem. Phys. 235, 81 (1998). 26 X.R. Wang, Y.S. Zheng, and S. Yin, Phys. Rev. B72, 121303(R) (2005). 27 C.P. Slichter, Principles of Magnetic Resonance, (Springer-Verlag, 1980). 28 Y. Makhlin, G. Schoön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). 29 H.S. Goan, Quantum Information and Computation, 2, 121 (2003); ibid, Phys. Rev. B 70, 075305 (2004). 30 A.N. Korotkov, Phys. Rev. B63, 085312 (2001); ibid, Phys.\n\nRev. B63, 115403 (2001). 31 Y. Levinson, Phys. Rev. B61, 4748 (2000). 32 K. Rabenstein, V.A. Sverdlov, and D.V. Averin, JETP Lett. 79, 646 (2004). 33 S.A. Gurvitz, L. Fedichkin, D. Mozyrsky and G.P. Berman, Phys. Rev. Lett., 91, 066801 (2003). 34 G. Ithier, E. Collin, P. Joyez, P.J. Meeson, D. Vion, D.\n\nEsteve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl, and G. Schön, Phys. Rev. B72, 134519 (2005). 35 S.A. Gurvitz, IEEE Transactions on Nanotechnology 4, 45 (2005). 36 S.A. Gurvitz, D. Mozyrsky, and G.P. Berman, Phys.Rev. B72, 205341 (2005). 37 S.A. Gurvitz and Ya.S. Prager, Phys. Rev. B53, 15932 (1996). 38 S.A. Gurvitz, Phys. Rev. B57, 6602, (1998). 39 In a strict sense the quantum rate equations (20) were derived by assuming constant widths Γ. Yet these equation are also valid when the widths are weakly energy dependent, as follows from their derivations (see 37, 38 and Appendix A). 40 R.J. Schoelkopf, A.A. Clerk, S.M. Girvin, K.W. Lehnert and M.H. Devoret, Quantum Noise in Mesoscopic Physics, edited by Yu.V. Nazarov, (Springer, 2003). 41 Y. Makhlin, G. Schoön, and A. Shnirman, in Exploring the Quantum-Classical Frontier, edited by J.R. Friedman and S. Han (Nova Science, Commack, New York, 2002). 42 A.M. Zagoskin, S. Ashhab, J.R. Johansson, and F. Nori, Phys. Rev. Lett. 97, 077001 (2006). 43 T. Gilad and S.A. Gurvitz, Phys. Rev. Lett. 97, 116806 (2006); H.J. Jiao, X.Q. Li, and J.Y. Luo, Phys. Rev. B75, 155333 (2007)."
}
] | |
arxiv:0704.0202 | 0704.0202 | 1 | 10.1088/1367-2630/9/6/206 | aec1b49ed6698e9bc7096ed01dbf802374bfe48e10ea829f1e872a3c24923292 | Towards Minimal Resources of Measurement-based Quantum Computation | We improve the upper bound on the minimal resources required for
measurement-based quantum computation. Minimizing the resources required for
this model is a key issue for experimental realization of a quantum computer
based on projective measurements. This new upper bound allows also to reply in
the negative to the open question about the existence of a trade-off between
observable and ancillary qubits in measurement-based quantum computation. | [
"Simon Perdrix"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | The discovery of new models of quantum computation (QC), such as the one-way quantum computer [7] and the projective measurement-based model [4] , have opened up new experimental avenues toward the realisation of a quantum computer in laboratories. At the same time they have challenged the traditional view about the nature of quantum computation.
Since the introduction of the quantum Turing machine by Deutsch [1], unitary transformations plays a central rôle in QC. However, it is known that the action of unitary gates can be simulated using the process of quantum teleportation based on projective measurements-only [4] . Characterizing the minimal resources that are sufficient for this model to be universal, is a key issue.
Resources of measurement-based quantum computations are composed of two ingredients: (i) a universal family of observables, which describes the measurements that can be performed (ii) the number of ancillary qubits used to simulate any unitary transformation. Successive improvements of the upper bounds on these minimal resources have been made by Leung and others [2, 3] . These bounds have been significantly reduced when the state transfer (which is a restricted form of teleportation) has been introduced: one two-qubit observable (Z ⊗ X) and three one-qubit observables (X, Z and (X + Y )/ √ 2), associated with only one ancillary qubit, are sufficient for simulating any unitary-based QC [6] . Are these resources minimal ? In [5], a sub-family of observables (Z ⊗ X, Z, and (X -Y )/ √ 2) is proved to be universal, however two ancillary qubits are used to make this sub-family universal.
These two results lead to an open question : is there a trade-off between observables and ancillary qubits in measurement-based QC ? In this paper, we reply in the negative to this
Towards Minimal Resources of Measurement-based Quantum Computation 2 open question by proving that the sub-family {Z ⊗ X, Z, (X -Y )/ √ 2} is universal using only one ancillary qubit, improving the upper bound on the minimal resources required for measurement-based QC.
The simulation of a given unitary transformation U by means of projective measurements can be decomposed into:
• (Step of simulation) First, U is probabilistically simulated up to a Pauli operator, leading to σU, where σ is either identity or a Pauli operator σ x , σ y , or σ z .
• (Correction) Then, a corrective strategy consisting in combinig conditionally steps of simulation is used to obtain a non-probabilistic simulation of U.
Teleportation can be realized by two successive Bell measurements (figure 1 ), where a Bell measurement is a projective measurement in the basis of the Bell states {|B ij } i,j∈{0,1} , where |B ij = 1 √ 2 (σ i z ⊗ σ j x )(|00 + |11 ). A step of simulation of U is obtained by replacing the second measurement by a measurement in the basis {(U † ⊗ Id)|B ij } i,j∈{0,1} (figure 2 ). a Bell Bell Φ Φ σ b c Figure 1. Bell measurement-based teleportation c Bell Bell Φ Φ U σ Φ U σ Φ b a c B Bell U U b a Figure 2. Simulation of U up to a Pauli operator If a step of simulation is represented as a probabilistic black box (figure 3 , left) , there exists a corrective strategy (figure 3 , right) which leads to a full simulation of U. This strategy consists in conditionally composing steps of simulation of U, but also of each Pauli operator. A similar step of simulation and strategy are given for the two-qubit unitary transformation ΛX (Controlled-X) in [4] . Notice that this simulation uses four ancillary qubits.
As a consequence, since any unitary transformation can be decomposed into ΛX and one-qubit unitary transformations, any unitary transformation can be simulated by means of projective measurements only. More precisely, for any circuit C of size n -with basis ΛX and all one-qubit unitary transformations -and for any ǫ > 0, O(n log(n/ǫ)) projective measurements are enough to simulate C with probability greater than 1 -ǫ.
Towards Minimal Resources of Measurement-based Quantum Computation
3 U Φ U Φ U Φ U Φ σ x σ x Φ Φ σ y σ z σ z σ y Φ U U U
Figure 3. Left: step of simulation abstracted into a probabilistic black box representation -Rigth: conditional composition of steps of simulation.
Approximative universality, based on a finite family of projective measurements, can also be considered. Leung [3] has shown that a family composed of five observables
F 0 = {Z, X ⊗ X, Z ⊗ Z, X ⊗ Z, 1 √ 2 (X -Y )
⊗ X} is approximatively universal, using four ancillary qubits. It means that for any unitary transformation U, any ǫ > 0 and any δ > 0, there exists a conditional composition of projective measurements from F 0 leading to the simulation of a unitary transformation Ũ with probability greater than 1 -ǫ and such that ||U -Ũ|| < δ.
σ Z Φ Φ b a X X Z Figure 4. State transfer XU Z Φ a VZV U Z Φ U σ V b a X X Φ U ZU Φ U σ V VXV b U V Figure 5.
Step of simulation based on state transfer
In order to decrease the number of two-qubit measurements -four in F 0 -and the number of ancillary, an new scheme called state transfer has been introduced [6] . The state transfer (figure 4 ) replaces the teleportation scheme for realizing a step of simulation. Composed of one two-qubit measurements, two one-qubit measurements, and using only one ancillary qubit, the state transfer can be used to simulate any one-qubit unitary transformation up to a Pauli operator (figure 5 ). For instance, simulations of H and HT -see section 3 for definitions of H and T -are represented in figure 6 . Moreover a scheme composed of two two-qubit measurements, two one-qubit measurements, and using only one ancillary qubit can be used to simulated ΛX up to a Pauli operator (figure 7 ). Since {H, T, ΛX} is a universal family of unitary transformations, the family
F 1 = {Z ⊗ X, X, Z, 1 √ 2 (X -Y )} of observables is
approximatively universal, using one ancillary qubit [6] . This result improves the result by Leung, since only one two-qubit measurement and one ancillary qubit are used, instead of four two-qubit measurements and four ancillary qubits. Moreover, one can prove that at least one Towards Minimal Resources of Measurement-based Quantum Computation 4 two-qubit measurement and one ancillary qubit are required for approximative universality. Thus, it turns out that the upper bound on the minimal resources for measurement-based QC differs form the lower bound, on the number of one-qubit measurements only.
X Φ Φ σH a b Z X Z X-Y/ 2 Φ Φ HT σ a b Z X Z
Figure 6. Simulation of H and HT up to a Pauli operator.
X Z X Z Φ Φ c Z X b a σ Λ X
Figure 7. Simulation of ΛX up to a Pauli operator.
In [5], it has been shown that the number of these one-qubit measurements can be decreased, since the family
F 2 = {Z ⊗ X, Z, 1 √ 2 (X -Y )
}, composed of one two-qubit only two one-qubit measurements, is also approximatively universal, using two ancillary qubit. The proof is based on the simulation of X-measurements by means of Z and Z ⊗ X measurements (figure 8 ). This result leads to a possible trade-off between the number of one-qubit measurements and the number of ancillary qubits required for approximative universality. X X Z Z Figure 8. X-measurement simulation
In this paper, we meanly prove that the family F 2 is approximatively universal, using only one ancillary qubit. Thus, the upper bound on the minimal resources required for approximative universality is improved, and moreover we answer the open question of the trade-off between observables and ancillary qubits. Notice that we prove that the trade-off conjectured in [5] does not exist, but another trade-off between observables and ancillary qubits may exist since the bounds on the minimal resources for measurement-based quantum computation are not tight.
There exist several universal families of unitary transformations, for instance {H, T, ΛX} is one of them:
Towards Minimal Resources of Measurement-based Quantum Computation 5
H = 1 √ 2 1 1 1 -1 , T = 1 0 0 e iπ 4 , ΛX = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ΛZ = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1
We prove that the family {HT, σ y , ΛZ} is also approximatively universal:
Theorem 1 U = {HT, σ y , ΛZ} is approximatively universal.
The proof is based on the following properties. Let R n (α) be the rotation of the Bloch sphere about the axis n through an angle α. Proposition 1 If n = (a, b, c) is a real unit vector, then for any α, R n (α) = cos(α/2)Ii sin(α/2)(aσ x + bσ y + cσ z ). Proposition 2 For a given vector n of the Bloch sphere, if θ is an irrational multiple of π, then for any α and any ǫ > 0, there exists k such that
||R n (α) -R n (θ) k )|| < ǫ/3
Proposition 3 If n and m are non parallel vectors of the Bloch sphere, then for any onequbit unitary transformation U, there exists α, β, γ, δ such that:
U = e iα R n (β)R m (γ)R n (δ)
Proposition 4 (Włodarski [8]) If α is not an integer multiple of π/4 and cos β = cos 2 α, then either α or β is an irrational multiple of π.
First we prove that any 1-qubit unitary transformation can be approximated by HT and σ y HT . Consider the unitary transformations U 1 = T , U 2 = HT H, U 3 = σ y HT Hσ y . Notice that T is, up to an unimportant global phase, a rotation by π/4 radians around z axis on the Block sphere:
U 1 = T = e -iπ/8 (cos(π/8)I -i sin(π/8)σ z ) U 2 = HT H = e -iπ/8 (cos(π/8)I -i sin(π/8)σ x ) U 3 = σ y HT Hσ y = e -iπ/8 (cos(π/8)I + i sin(π/8)σ x )
Composing U 1 and U 2 gives, up to a global phase:
U 2 U 1 = (cos(π/8)I -i sin(π/8)σ x )(cos(π/8)I -i sin(π/8)σ z ) = cos 2 (π/8)I -i[cos(π/8)(σ x + σ z ) -sin(π/8)σ y ] sin(π/8)
Towards Minimal Resources of Measurement-based Quantum Computation 6 According to proposition 1, U 2 U 1 is a rotation of the Bloch sphere about an axis along n = (cos(π/8), -sin(π/8), cos(π/8)) and through an angle θ defined as a solution of cos(θ/2) = cos 2 (π/8). Since π/8 is not an integer multiple of π/4 but a rational multiple of π, according to proposition 4, a such θ is an irrational multiple of π. This irrationality implies that for any angle α, the rotation around n about angle α can be approximated to arbitrary accuracy by repeating rotations around n about angle θ (see proposition 3). For any α and any ǫ > 0, there exists k such that
||R n (α) -R n (θ) k )|| < ǫ/3
Moreover, composing U 1 and U 3 gives, up to a global phase:
U 3 U 1 = (cos(π/8)I + i sin(π/8)σ x )(cos(π/8)I -i sin(π/8)σ z ) = cos 2 (π/8)I -i[cos(π/8)(-σ x + σ z ) + sin(π/8)σ y ] sin(π/8) U 3 U 1
is a rotation of the Bloch sphere about an axis along m = (-cos(π/8), sin(π/8), cos(π/8)) and through the angle θ. Thus, for any α and any ǫ > 0, there exists k such that
||R m (α) -R m (θ) k )|| < ǫ/3
Since n and m are non-parallel, any one-qubit unitary transformation U, according to proposition 2, can be decomposed into rotations around n and m : There exist α, β, γ, δ such that U = e iα R n (β)R m (γ)R n (δ)
Finally, for any U and ǫ > 0, there exist k 1 , k 2 , k 3 such that
||U -R n (θ) k 1 R m (θ) k 2 R n (θ) k 3 || < ǫ
Thus, any one-qubit unitary transformation can be approximated by means of
U 2 U 1 , and U 3 U 1 . Since U 2 U 1 = (HT )(HT ) and U 3 U 1 = σ y HT Hσ y T = -(σ y HT )(σ y HT ), the family {HT, σ y } approximates any one-qubit unitary transformation.
With the additional ΛZ gate, the family U is approximatively universal.
In [5], the family of observables F 2 = {Z ⊗ X, Z, X-Y √ 2 } is proved to be approximatively universal using two ancillary qubits. We prove that this family requires only one ancillary qubit to be universal:
Theorem 2 F 2 = {Z ⊗ X, Z, X-Y √
2 } is approximatively universal, using one ancillary qubit only.
The proof consists in simulating the unitary transformations of the universal family U. First, one can notice that HT can be simulated up to a Pauli operator, using measurements of F 2 , as it is depicted in figure 6 . So, the universality is reduced to the ability to simulate ΛZ and the Pauli operators -Pauli operators are needed to simulated σ y ∈ F , but also to perform the corrections required by the corrective strategy (figure 3 ).
7 Lemma 5 For a given 2-qubit register a, b and one ancillary qubit c, the sequence of measurements according to Z c , Z a ⊗X c , Z c ⊗X b , and Z b (see figure 9 ) simulates ΛZ(Id⊗H) on qubits a, b, up to a Pauli operator. The resulting state is located on qubits a and c.
(Id H) Z X Z Φ Φ c Z X b a σ Z Λ
Figure 9. Simulation of ΛZ(Id ⊗ H) Proof: One can show that, if the state of the register a, b is |Φ before the sequence of measurements, the state of the register a, b after the measurements is σΛZ(Id ⊗H)|Φ , where
σ = σ s 1 z ⊗ σ s 3 x σ s 2 +s 4 z
and s i 's are the classical outcomes of the sequence of measurements. In order to simulate Pauli operators, a new scheme, different from the state transfer, is introduced.
Lemma 6 For a given qubit b and one ancillary qubit a, the sequence of measurements Z a , X a ⊗ Z b , and Z a (figure 10 ) simulates, on qubit b, the application of σ z with probability 1/2 and Id with probability 1/2. k Φ Φ b a X Z σ k Z Z z Figure 10. Simulation of σ z Proof: Let |Φ = α|0 + β|1 be the state of qubit b. After the first measurement, the state of the register a,
b is |ψ 1 = (σ s 1 x ⊗ Id)|0 ⊗ |Φ where s 1 ∈ {0, 1}
is the classical outcome of the measurement. Since ψ 1 |X ⊗ Z|ψ 1 = 0, the state of the register a, b is:
|ψ 2 = √ 2 2 (σ s 1 x ⊗ Id)(Id + (-1) s 2 X ⊗ Z)|0 ⊗ |Φ = √ 2 2 (σ s 1 x σ s 2 z ⊗ Id)(|0 ⊗ |Φ + |1 ⊗ (σ z |Φ ) Let s 3 ∈ {0, 1}
be the outcome of the last measurement, on qubit a. If s 1 = s 3 then state of the qubit b is |Φ , and σ z |Φ otherwise. One can prove that these two possibilities occur with equal probabilities.
Lemma 7 For a given qubit b and one ancillary qubit a, the sequence of measurements 11 ) simulates, on qubit b, the application of σ x with probability 1/2 and Id with probability 1/2.
X-Y √ 2 a , Z a ⊗ X b , and X-Y √ 2 a (figure
Towards Minimal Resources of Measurement-based Quantum Computation 8 k Φ Φ (X-Y)/ 2 (X-Y)/ 2 b a Z X σ x Figure 11. Simulation of σ x
The proof of lemma 7 is similar to the proof of lemma 6. Proof of theorem 2: First notice that the family of unitary transformations U ′ = {HT, σ y , ΛZ(I ⊗ H)} is approximatively universal since U = {HT, σ y , ΛZ} is universal.
HT and ΛZ(I ⊗H) can be simulated up to a Pauli operator (lemmas 5). The universality of the family of observables F 2 = {Z ⊗ X, Z, X-Y √ 2 } is reduced to the ability to simulate any Pauli operators. Lemma 7 (resp. lemma 6), shows that σ x (σ z ) can be simulated with probability 1/2, moreover if the simulation fails, the resulting state is same as the original one. Thus, this simulation can be repeated until a full simulation of σ x (σ z ). Finally, σ y = iσ z σ x can be simulated, up to a global phase, by combining simulations of σ x and σ z . Thus,
F 2 = {Z ⊗ X, Z, X-Y √
2 } is approximatively universal using only one ancillary qubit.
We have proved a new upper bound on the minimal resources required for measurementbased QC: one two-qubit, and two one-qubit observables are universal, using one ancillary qubit only. This new upper bound has experimental applications, but allows also to prove that the trade-off between observables and ancillary qubits, conjectured in [5] , does not exist. This new upper bound is not tight since the lower bound on the minimal resources for this model is one two-qubit observable and one ancillary qubit. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The discovery of new models of quantum computation (QC), such as the one-way quantum computer [7] and the projective measurement-based model [4] , have opened up new experimental avenues toward the realisation of a quantum computer in laboratories. At the same time they have challenged the traditional view about the nature of quantum computation.\n\nSince the introduction of the quantum Turing machine by Deutsch [1], unitary transformations plays a central rôle in QC. However, it is known that the action of unitary gates can be simulated using the process of quantum teleportation based on projective measurements-only [4] . Characterizing the minimal resources that are sufficient for this model to be universal, is a key issue.\n\nResources of measurement-based quantum computations are composed of two ingredients: (i) a universal family of observables, which describes the measurements that can be performed (ii) the number of ancillary qubits used to simulate any unitary transformation. Successive improvements of the upper bounds on these minimal resources have been made by Leung and others [2, 3] . These bounds have been significantly reduced when the state transfer (which is a restricted form of teleportation) has been introduced: one two-qubit observable (Z ⊗ X) and three one-qubit observables (X, Z and (X + Y )/ √ 2), associated with only one ancillary qubit, are sufficient for simulating any unitary-based QC [6] . Are these resources minimal ? In [5], a sub-family of observables (Z ⊗ X, Z, and (X -Y )/ √ 2) is proved to be universal, however two ancillary qubits are used to make this sub-family universal.\n\nThese two results lead to an open question : is there a trade-off between observables and ancillary qubits in measurement-based QC ? In this paper, we reply in the negative to this\n\nTowards Minimal Resources of Measurement-based Quantum Computation 2 open question by proving that the sub-family {Z ⊗ X, Z, (X -Y )/ √ 2} is universal using only one ancillary qubit, improving the upper bound on the minimal resources required for measurement-based QC."
},
{
"section_type": "OTHER",
"section_title": "Measurement-based QC",
"text": "The simulation of a given unitary transformation U by means of projective measurements can be decomposed into:\n\n• (Step of simulation) First, U is probabilistically simulated up to a Pauli operator, leading to σU, where σ is either identity or a Pauli operator σ x , σ y , or σ z .\n\n• (Correction) Then, a corrective strategy consisting in combinig conditionally steps of simulation is used to obtain a non-probabilistic simulation of U.\n\nTeleportation can be realized by two successive Bell measurements (figure 1 ), where a Bell measurement is a projective measurement in the basis of the Bell states {|B ij } i,j∈{0,1} , where |B ij = 1 √ 2 (σ i z ⊗ σ j x )(|00 + |11 ). A step of simulation of U is obtained by replacing the second measurement by a measurement in the basis {(U † ⊗ Id)|B ij } i,j∈{0,1} (figure 2 ). a Bell Bell Φ Φ σ b c Figure 1. Bell measurement-based teleportation c Bell Bell Φ Φ U σ Φ U σ Φ b a c B Bell U U b a Figure 2. Simulation of U up to a Pauli operator If a step of simulation is represented as a probabilistic black box (figure 3 , left) , there exists a corrective strategy (figure 3 , right) which leads to a full simulation of U. This strategy consists in conditionally composing steps of simulation of U, but also of each Pauli operator. A similar step of simulation and strategy are given for the two-qubit unitary transformation ΛX (Controlled-X) in [4] . Notice that this simulation uses four ancillary qubits.\n\nAs a consequence, since any unitary transformation can be decomposed into ΛX and one-qubit unitary transformations, any unitary transformation can be simulated by means of projective measurements only. More precisely, for any circuit C of size n -with basis ΛX and all one-qubit unitary transformations -and for any ǫ > 0, O(n log(n/ǫ)) projective measurements are enough to simulate C with probability greater than 1 -ǫ.\n\nTowards Minimal Resources of Measurement-based Quantum Computation\n\n3 U Φ U Φ U Φ U Φ σ x σ x Φ Φ σ y σ z σ z σ y Φ U U U\n\nFigure 3. Left: step of simulation abstracted into a probabilistic black box representation -Rigth: conditional composition of steps of simulation.\n\nApproximative universality, based on a finite family of projective measurements, can also be considered. Leung [3] has shown that a family composed of five observables\n\nF 0 = {Z, X ⊗ X, Z ⊗ Z, X ⊗ Z, 1 √ 2 (X -Y )\n\n⊗ X} is approximatively universal, using four ancillary qubits. It means that for any unitary transformation U, any ǫ > 0 and any δ > 0, there exists a conditional composition of projective measurements from F 0 leading to the simulation of a unitary transformation Ũ with probability greater than 1 -ǫ and such that ||U -Ũ|| < δ.\n\nσ Z Φ Φ b a X X Z Figure 4. State transfer XU Z Φ a VZV U Z Φ U σ V b a X X Φ U ZU Φ U σ V VXV b U V Figure 5.\n\nStep of simulation based on state transfer\n\nIn order to decrease the number of two-qubit measurements -four in F 0 -and the number of ancillary, an new scheme called state transfer has been introduced [6] . The state transfer (figure 4 ) replaces the teleportation scheme for realizing a step of simulation. Composed of one two-qubit measurements, two one-qubit measurements, and using only one ancillary qubit, the state transfer can be used to simulate any one-qubit unitary transformation up to a Pauli operator (figure 5 ). For instance, simulations of H and HT -see section 3 for definitions of H and T -are represented in figure 6 . Moreover a scheme composed of two two-qubit measurements, two one-qubit measurements, and using only one ancillary qubit can be used to simulated ΛX up to a Pauli operator (figure 7 ). Since {H, T, ΛX} is a universal family of unitary transformations, the family\n\nF 1 = {Z ⊗ X, X, Z, 1 √ 2 (X -Y )} of observables is\n\napproximatively universal, using one ancillary qubit [6] . This result improves the result by Leung, since only one two-qubit measurement and one ancillary qubit are used, instead of four two-qubit measurements and four ancillary qubits. Moreover, one can prove that at least one Towards Minimal Resources of Measurement-based Quantum Computation 4 two-qubit measurement and one ancillary qubit are required for approximative universality. Thus, it turns out that the upper bound on the minimal resources for measurement-based QC differs form the lower bound, on the number of one-qubit measurements only.\n\nX Φ Φ σH a b Z X Z X-Y/ 2 Φ Φ HT σ a b Z X Z\n\nFigure 6. Simulation of H and HT up to a Pauli operator.\n\nX Z X Z Φ Φ c Z X b a σ Λ X\n\nFigure 7. Simulation of ΛX up to a Pauli operator.\n\nIn [5], it has been shown that the number of these one-qubit measurements can be decreased, since the family\n\nF 2 = {Z ⊗ X, Z, 1 √ 2 (X -Y )\n\n}, composed of one two-qubit only two one-qubit measurements, is also approximatively universal, using two ancillary qubit. The proof is based on the simulation of X-measurements by means of Z and Z ⊗ X measurements (figure 8 ). This result leads to a possible trade-off between the number of one-qubit measurements and the number of ancillary qubits required for approximative universality. X X Z Z Figure 8. X-measurement simulation\n\nIn this paper, we meanly prove that the family F 2 is approximatively universal, using only one ancillary qubit. Thus, the upper bound on the minimal resources required for approximative universality is improved, and moreover we answer the open question of the trade-off between observables and ancillary qubits. Notice that we prove that the trade-off conjectured in [5] does not exist, but another trade-off between observables and ancillary qubits may exist since the bounds on the minimal resources for measurement-based quantum computation are not tight."
},
{
"section_type": "OTHER",
"section_title": "Universal family of unitary transformations",
"text": "There exist several universal families of unitary transformations, for instance {H, T, ΛX} is one of them:\n\nTowards Minimal Resources of Measurement-based Quantum Computation 5\n\nH = 1 √ 2 1 1 1 -1 , T = 1 0 0 e iπ 4 , ΛX = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ΛZ = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1 \n\nWe prove that the family {HT, σ y , ΛZ} is also approximatively universal:\n\nTheorem 1 U = {HT, σ y , ΛZ} is approximatively universal.\n\nThe proof is based on the following properties. Let R n (α) be the rotation of the Bloch sphere about the axis n through an angle α. Proposition 1 If n = (a, b, c) is a real unit vector, then for any α, R n (α) = cos(α/2)Ii sin(α/2)(aσ x + bσ y + cσ z ). Proposition 2 For a given vector n of the Bloch sphere, if θ is an irrational multiple of π, then for any α and any ǫ > 0, there exists k such that\n\n||R n (α) -R n (θ) k )|| < ǫ/3\n\nProposition 3 If n and m are non parallel vectors of the Bloch sphere, then for any onequbit unitary transformation U, there exists α, β, γ, δ such that:\n\nU = e iα R n (β)R m (γ)R n (δ)\n\nProposition 4 (Włodarski [8]) If α is not an integer multiple of π/4 and cos β = cos 2 α, then either α or β is an irrational multiple of π."
},
{
"section_type": "OTHER",
"section_title": "Proof of theorem 1:",
"text": "First we prove that any 1-qubit unitary transformation can be approximated by HT and σ y HT . Consider the unitary transformations U 1 = T , U 2 = HT H, U 3 = σ y HT Hσ y . Notice that T is, up to an unimportant global phase, a rotation by π/4 radians around z axis on the Block sphere:\n\nU 1 = T = e -iπ/8 (cos(π/8)I -i sin(π/8)σ z ) U 2 = HT H = e -iπ/8 (cos(π/8)I -i sin(π/8)σ x ) U 3 = σ y HT Hσ y = e -iπ/8 (cos(π/8)I + i sin(π/8)σ x )\n\nComposing U 1 and U 2 gives, up to a global phase:\n\nU 2 U 1 = (cos(π/8)I -i sin(π/8)σ x )(cos(π/8)I -i sin(π/8)σ z ) = cos 2 (π/8)I -i[cos(π/8)(σ x + σ z ) -sin(π/8)σ y ] sin(π/8)\n\nTowards Minimal Resources of Measurement-based Quantum Computation 6 According to proposition 1, U 2 U 1 is a rotation of the Bloch sphere about an axis along n = (cos(π/8), -sin(π/8), cos(π/8)) and through an angle θ defined as a solution of cos(θ/2) = cos 2 (π/8). Since π/8 is not an integer multiple of π/4 but a rational multiple of π, according to proposition 4, a such θ is an irrational multiple of π. This irrationality implies that for any angle α, the rotation around n about angle α can be approximated to arbitrary accuracy by repeating rotations around n about angle θ (see proposition 3). For any α and any ǫ > 0, there exists k such that\n\n||R n (α) -R n (θ) k )|| < ǫ/3\n\nMoreover, composing U 1 and U 3 gives, up to a global phase:\n\nU 3 U 1 = (cos(π/8)I + i sin(π/8)σ x )(cos(π/8)I -i sin(π/8)σ z ) = cos 2 (π/8)I -i[cos(π/8)(-σ x + σ z ) + sin(π/8)σ y ] sin(π/8) U 3 U 1\n\nis a rotation of the Bloch sphere about an axis along m = (-cos(π/8), sin(π/8), cos(π/8)) and through the angle θ. Thus, for any α and any ǫ > 0, there exists k such that\n\n||R m (α) -R m (θ) k )|| < ǫ/3\n\nSince n and m are non-parallel, any one-qubit unitary transformation U, according to proposition 2, can be decomposed into rotations around n and m : There exist α, β, γ, δ such that U = e iα R n (β)R m (γ)R n (δ)\n\nFinally, for any U and ǫ > 0, there exist k 1 , k 2 , k 3 such that\n\n||U -R n (θ) k 1 R m (θ) k 2 R n (θ) k 3 || < ǫ\n\nThus, any one-qubit unitary transformation can be approximated by means of\n\nU 2 U 1 , and U 3 U 1 . Since U 2 U 1 = (HT )(HT ) and U 3 U 1 = σ y HT Hσ y T = -(σ y HT )(σ y HT ), the family {HT, σ y } approximates any one-qubit unitary transformation.\n\nWith the additional ΛZ gate, the family U is approximatively universal."
},
{
"section_type": "OTHER",
"section_title": "Universal family of projective measurements",
"text": "In [5], the family of observables F 2 = {Z ⊗ X, Z, X-Y √ 2 } is proved to be approximatively universal using two ancillary qubits. We prove that this family requires only one ancillary qubit to be universal:\n\nTheorem 2 F 2 = {Z ⊗ X, Z, X-Y √\n\n2 } is approximatively universal, using one ancillary qubit only.\n\nThe proof consists in simulating the unitary transformations of the universal family U. First, one can notice that HT can be simulated up to a Pauli operator, using measurements of F 2 , as it is depicted in figure 6 . So, the universality is reduced to the ability to simulate ΛZ and the Pauli operators -Pauli operators are needed to simulated σ y ∈ F , but also to perform the corrections required by the corrective strategy (figure 3 )."
},
{
"section_type": "OTHER",
"section_title": "Towards Minimal Resources of Measurement-based Quantum Computation",
"text": "7 Lemma 5 For a given 2-qubit register a, b and one ancillary qubit c, the sequence of measurements according to Z c , Z a ⊗X c , Z c ⊗X b , and Z b (see figure 9 ) simulates ΛZ(Id⊗H) on qubits a, b, up to a Pauli operator. The resulting state is located on qubits a and c.\n\n(Id H) Z X Z Φ Φ c Z X b a σ Z Λ\n\nFigure 9. Simulation of ΛZ(Id ⊗ H) Proof: One can show that, if the state of the register a, b is |Φ before the sequence of measurements, the state of the register a, b after the measurements is σΛZ(Id ⊗H)|Φ , where\n\nσ = σ s 1 z ⊗ σ s 3 x σ s 2 +s 4 z\n\nand s i 's are the classical outcomes of the sequence of measurements. In order to simulate Pauli operators, a new scheme, different from the state transfer, is introduced.\n\nLemma 6 For a given qubit b and one ancillary qubit a, the sequence of measurements Z a , X a ⊗ Z b , and Z a (figure 10 ) simulates, on qubit b, the application of σ z with probability 1/2 and Id with probability 1/2. k Φ Φ b a X Z σ k Z Z z Figure 10. Simulation of σ z Proof: Let |Φ = α|0 + β|1 be the state of qubit b. After the first measurement, the state of the register a,\n\nb is |ψ 1 = (σ s 1 x ⊗ Id)|0 ⊗ |Φ where s 1 ∈ {0, 1}\n\nis the classical outcome of the measurement. Since ψ 1 |X ⊗ Z|ψ 1 = 0, the state of the register a, b is:\n\n|ψ 2 = √ 2 2 (σ s 1 x ⊗ Id)(Id + (-1) s 2 X ⊗ Z)|0 ⊗ |Φ = √ 2 2 (σ s 1 x σ s 2 z ⊗ Id)(|0 ⊗ |Φ + |1 ⊗ (σ z |Φ ) Let s 3 ∈ {0, 1}\n\nbe the outcome of the last measurement, on qubit a. If s 1 = s 3 then state of the qubit b is |Φ , and σ z |Φ otherwise. One can prove that these two possibilities occur with equal probabilities.\n\nLemma 7 For a given qubit b and one ancillary qubit a, the sequence of measurements 11 ) simulates, on qubit b, the application of σ x with probability 1/2 and Id with probability 1/2.\n\nX-Y √ 2 a , Z a ⊗ X b , and X-Y √ 2 a (figure\n\nTowards Minimal Resources of Measurement-based Quantum Computation 8 k Φ Φ (X-Y)/ 2 (X-Y)/ 2 b a Z X σ x Figure 11. Simulation of σ x\n\nThe proof of lemma 7 is similar to the proof of lemma 6. Proof of theorem 2: First notice that the family of unitary transformations U ′ = {HT, σ y , ΛZ(I ⊗ H)} is approximatively universal since U = {HT, σ y , ΛZ} is universal.\n\nHT and ΛZ(I ⊗H) can be simulated up to a Pauli operator (lemmas 5). The universality of the family of observables F 2 = {Z ⊗ X, Z, X-Y √ 2 } is reduced to the ability to simulate any Pauli operators. Lemma 7 (resp. lemma 6), shows that σ x (σ z ) can be simulated with probability 1/2, moreover if the simulation fails, the resulting state is same as the original one. Thus, this simulation can be repeated until a full simulation of σ x (σ z ). Finally, σ y = iσ z σ x can be simulated, up to a global phase, by combining simulations of σ x and σ z . Thus,\n\nF 2 = {Z ⊗ X, Z, X-Y √\n\n2 } is approximatively universal using only one ancillary qubit."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusion",
"text": "We have proved a new upper bound on the minimal resources required for measurementbased QC: one two-qubit, and two one-qubit observables are universal, using one ancillary qubit only. This new upper bound has experimental applications, but allows also to prove that the trade-off between observables and ancillary qubits, conjectured in [5] , does not exist. This new upper bound is not tight since the lower bound on the minimal resources for this model is one two-qubit observable and one ancillary qubit."
}
] | |
arxiv:0704.0207 | 0704.0207 | 1 | 10.1088/0954-3899/34/8/S10 | e619f6f2484f53a0c62c1275bf154e2d99f7cc13d034107653ef12838f5e51c6 | Quark matter and the astrophysics of neutron stars | Some of the means through which the possible presence of nearly deconfined
quarks in neutron stars can be detected by astrophysical observations of
neutron stars from their birth to old age are highlighted. | [
"M Prakash"
] | [
"astro-ph",
"gr-qc",
"nucl-th"
] | astro-ph | [] | 2007-04-02 | 2026-02-26 | Utilizing the asymptotic freedom of QCD, Collins and Perry [1] first noted that the dense cores of neutron stars may consist of deconfined quarks instead of hadrons. The crucial question is whether observations of neutron stars from their birth to death through neutrino, photon and gravity-wave emissions can unequivocably reveal the presence of nearly deconfined quarks instead of other possibilities such as only nucleons or other exotica such as strangeness-bearing hyperons or Bose (pion and kaon) condensates.
The birth of a neutron star is heralded by the arrival of neutrinos on earth as confirmed by IMB and Kamiokande neutrino detectors in the case of supernova SN 1987A. Nearly all of the gravitational binding energy (of order 300 bethes, where 1 bethe ≡ 10 51 erg) released in the progenitor star's white dwarf-like core is carried off by neutrinos and antineutrinos of all flavors in roughly equal proportions. The remarkable fact that the weakly interacting neutrinos are trapped in matter prior to their release as a burst is due to their short mean free paths in matter, λ ≈ (σn) -1 ≈ 10 cm, (here σ ≈ 10 -40 cm 2 is the neutrino-matter cross section and n ≈ 2 to 3 n s , where n s ≃ 0.16 fm -3 is the reference nuclear equilibrium density), which is much less than the proto-neutron star radius, which exceeds 20 km. Should a core-collapse supernova occur in their lifetimes, current neutrino detectors, such as SK, SNO, LVD's, AMANDA, etc., offer a great opportunity for understanding a proto-neutron star's birth and propagation of neutrinos in dense matter insofar as they can detect tens of thousands of neutrinos in contrast to the tens of neutrinos detected by IMB and Kamiokande.
The appearence of quarks inside a neutron star leads to a decrease in the maximum mass that matter can support, implying metastability of the star. This would occur if the proto-neutron star's mass, which must be less than the maximum mass of the
Quark matter and the astrophysics of neutron stars 2 Figure 1. Evolutions of the central baryon density n B , ν concentration Y ν , quark volume fraction χ and temperature T for different baryon masses M B . Solid lines show stable stars whereas dashed lines showing stars with larger masses are metastable. Diamonds indicate when quarks appear at the star's center, and asterisks denote when metastable stars become gravitationally unstable. Figure after Ref. [2] hot, lepton-rich matter is greater than the maximum mass of hot, lepton-poor matter. For matter with nucleons only, such a metastability is denied (see, e.g., [3] ). Figure 1 shows the evolution of some thermodynamic quantities at the center of stars of various fixed baryonic masses. With the equation of state used (see [2] for details), stars with M B ∼ < 1.1 M ⊙ do not contain quarks and those with M B ∼ 1.7 M ⊙ are metastable. The subsequent collapse to a black hole could be observed as a cessation in the neutrino signals well above the sensitivity limits of the current detectors (Figure 2 ).
Multiwavelength photon observations of neutron stars, the bread and butter affair of astronomy, has yielded estimates of the surface tempeartures and ages of several neutron stars (Fig. 3 ). As neutron stars cool principally through neutrino emission from their cores, the possibility exists that the interior composition can be determined. The star continuosly emits photons, dominantly in x-rays, with an effective temperature T ef f that tracks the interior temperature but that is smaller by a factor ∼ 100. The dominant neutrino cooling reactions are of a general type, known as Urca processes [4], in which thermally excited particles undergo beta and inverse-beta decays. Each
Quark matter and the astrophysics of neutron stars 3 Figure 2. The evolution of the total neutrino luminosity for stars of indicated baryon masses. Shaded bands illustrate the limiting luninosities corresponding to a count rate of 0.2 Hz in all detectors assuming 50 kpc for IMB and and Kamioka, 8.5 kpc for SNO, SuperK, and UNO. Shaded regions represent uncertanities in the average neutrino energy from the use of a diffusion scheme for neutrino transport in matter.
Figure after Ref. [2].
reaction produces a neutrino or anti-neutrino, and thermal energy is thus continuously lost. Depending upon the proton-fraction of matter, which in turn depends on the nature of strong interactions at high density, direct Urca processes involving nucleons, hyperons or quarks lead to enhanced cooling compared to modified Urca processes in which an additional particle is required to conserve momentum. However, effects of superfluidity abates cooling as sufficient thermal energy is required to break paired fermions. In addition, the poorly known envelope composition also plays a role in the inferred surface temperature (Fig. 3 ). The multitude of high density phases, cooling mechanisms, effects of superfluidity, and unknown envelope composition have thus far prevented definitive conclusions to be drawn (see, e.g., [5]).
Several recent observations of neutron stars have direct bearing on the determination of the maximum mass. The most accurately measured masses are from timing observations of the radio binary pulsars. As shown in Fig. 4 , which is compilation of the measured neutron star masses as of November 2006, observations include pulsars orbiting another Quark matter and the astrophysics of neutron stars 4 Figure 3. Observational estimates of neutron star temperatures and ages together with theoretical cooling simulations for M = 1.4 M ⊙ . Models and data are described in [6] . Orange error boxes (see online) indicate sources from which both X-ray and optical emissions have been observed. Simulations are for models with Fe or H envelopes, with and without the effects of superfluidity, and allowing or forbidding direct Urca processes. Models forbidding direct Urca processes are relatively independent of M and superfluid properties. Trajectories for models with enhanced cooling (direct Urca processes) and superfluidity lie within the yellow region, the exact location depending upon M as well as superfluid and Urca properties. Figure adapted from Ref. [7].
neutron star, a white dwarf or a main-sequence star.
One significant development concerns mass determinations in binaries with white dwarf companions, which show a broader range of neutron star masses than binary neutron star pulsars. Perhaps a rather narrow set of evolutionary circumstances conspire to form double neutron star binaries, leading to a restricted range of neutron star masses [9]. This restriction is likely relaxed for other neutron star binaries. A few of the white dwarf binaries may contain neutron stars larger than the canonical 1.4 M ⊙ value, including the intriguing case [10] of PSR J0751+1807 in which the estimated mass with 1σ error bars is 2.1 ± 0.2 M ⊙ . In addition, to 95% confidence, one of the two pulsars Ter 5 I and J has a reported mass larger than 1.68 M ⊙ [11] .
Whereas the observed simple mean mass of neutron stars with white dwarf companions exceeds those with neutron star companions by 0.25 M ⊙ , the weighted means of the two groups are virtually the same. The 2.1 M ⊙ neutron star, PSR J0751+1807, is about 4σ from the canonical value of 1.4 M ⊙ . It is furthermore the case that the 2σ errors of all but two systems extend into the range below 1.45 M ⊙ , so caution should be exercised before concluding that firm evidence of large neutron star masses exists. Continued observations, which will reduce the observational errors, are necessary to clarify this situation.
Masses can also be estimated for another handful of binaries which contain an
Quark matter and the astrophysics of neutron stars 5 Figure 4. Measured and estimated masses of neutron stars in binarie pulsars (gold, silver and blue regions online) and in x-ray accreting binaries (green). For each region, simple averages are shown as dotted lines; error weighted averages are shown as dashed lines. For labels and other details, consult Ref. [8].
accreting neutron star emitting x-rays. Some of these systems are characterized by relatively large masses, but the estimated errors are also large. The system of Vela X-1 is noteworthy because its lower mass limit (1.6 to 1.7M ⊙ ) is at least mildly constrained by geometry [12]. Raising the limit for the neutron star maximum mass could eliminate entire families of EOS's, especially those in which substantial softening begins around 2 to 3n s . This could be extremely significant, since exotica (hyperons, Bose condensates, or quarks) generally reduce the maximum mass appreciably.
Measurements of neutron star masses can set an upper limit to the maximum possible energy density in any compact object. It has been found [13] that no causal EOS has a central density, for a given mass, greater than that for the Tolman VII [14] analytic solution. This solution corresponds to a quadratic mass-energy density ρ dependence Quark matter and the astrophysics of neutron stars 6 Figure 5. Model predictions are compared with results from the Tolman IV and VII analytic solutions of general relativistic stucture equations. NR refers to nonrelativistic potential models, R are field-theoretical models, and Exotica refers to NR or R models in which strong softening occurs, due to hyperons, a Bose condensate, or quark matter as well as self-bound strange quark matter. Constraints from a possible redshift measurement of z = 0.35 is also shown. The dashed lines for 1.44 and 2.2 M ⊙ serve to guide the eye. Figure taken from Ref. [13] .
on r, ρ = ρ c [1 -(r/R) 2 ], where the central density is ρ c . For this solution,
ρ c,T V II = 2.5ρ c,Inc ≃ 1.5 × 10 16 M ⊙ M 2 g cm -3 . ( 1
)
A measured mass of 2.2 M ⊙ would imply ρ max < 3.1 × 10 15 g cm -3 , or about 8n s .
Figure 5 displays maximum masses and accompanying central densities for a wide wariety of neutron star EOS's, including models containing significant softening due to "exotica", such as strange quark matter. The upper limit to the density could be lowered if the causal constraint is not approached in practice. For example, at high densities in which quark asymptotic freedom is realized, the sound speed is limited to c/ √ 3. Using this as a strict limit at all densities, the Rhoades & Ruffini [15] mass limit is reduced by approximately 1/ √ 3 and the compactness limit GM/Rc 2 = 1/2.94 is reduced by a factor 3 -1/4 to 1/3.8 [16] . In this extreme case, the maximum density would be reduced by a factor of 3 -1/4 from that of Eq. ( 1 ). A 2.2 M ⊙ measured mass would imply a maximum density of about 4.2n s .
Mergers of compact objects in binary systems, such as a pair of neutron stars (NS-NS), a neutron star and a black hole (NS-BH), or two black holes (BH-BH), are expected to be prominent sources of gravitational radiation [17] . The gravitationalwave signature of such systems is primarily determined by the chirp mass M chirp =
Quark matter and the astrophysics of neutron stars 7 Figure 6. Physical and observational variables in mergers between low-mass black holes and neutron stars or self-bound quark stars. The total system mass is 6 M ⊙ and the initial mass ratio is q = 1/3 in both cases. The initial radii of the neutron star and quark star were assumed to be equal. The time scales have arbitrary zero points. Upper panel displays semi-major axis a (thick lines) and component mass M N S , M QS (thin lines) evolution. Lower panel displays orbital frequency ν (thick lines) and strain amplitude |h + r| evolution. Solid curves refer to the neutron star simulation and dashed curves to the quark star simulations. Figure taken from Ref. [8]. 1/5 , where M 1 and M 2 are the masses of the coalescing objects. The radiation of gravitational waves removes energy which causes the mutual orbits to decay. For example, the binary pulsar PSR B1913+16 has a merger timescale of about 250 million years, and the pulsar binary PSR J0737-3039 has a merger timescale of about 85 million years [18] , so there is ample reason to expect that many such decaying compact binaries exist in the Galaxy. Besides emitting copious amounts of gravitational radiation, binary mergers have been proposed as a source of the r-process elements [19] and the origin of the shorter-duration gamma ray bursters [20] .
(M 1 M 2 ) 3/5 (M 1 + M 2 ) -
Observations of gravity waves from merger events can simultaneously measure masses and radii of neutron stars, and could set firm limits on the neutron star maximum mass [21, 22] . Binary mergers for the two cases of a black hole and a normal neutron star and a black hole and a self-bound strange quark matter star (Fig. 6 ) illustrate the unique opportunity afforded by gravitational wave detectors due to begin operation over the next decade, including LIGO, VIRGO, GEO600, and TAMA.
A careful analysis of the gravitational waveform during inspiral yields values for not only the chirp mass M chirp , but for also the reduced mass M BH M N S /M, so that both M BH and M N S can be found [23] . The onset of mass transfer can be determined by the peak in ω, and the value of ω there gives a. A general relativistic analysis of mass transfer conditions then allows the determination of the star's radius [22] . Thus a point on the mass-radius diagram can be estimated [24] . The combination h + ω -1/3 depends only on a function of q, so the ratio of that combination and knowledge of q i
Quark matter and the astrophysics of neutron stars 8 should allow determination of q f . From the Roche condition and knowledge of a f from ω f , another mass-radius combination can be found.
The sharp contrast between the evolutions during stable mass transfer of a normal neutron star and a strange quark star should make these cases distinguishable. For strange quark matter stars, the differences in the height of the frequency peak and the plateau in the frequency values at later times are related to the differences in radii of the stars at these two epochs. It could be an indirect indicator of the maximum mass of the star: the closer is the star's mass before mass transfer to the maximum mass, the greater is the difference between these frequency values, because the radius change will be larger. Together with radius information, the value of the maximum mass remains the most important unknown that could reveal the true equation of state at high densities.
This work was supported in part by the U.S. Department of Energy under the grant DOE/DE-FG02-93ER40756. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Utilizing the asymptotic freedom of QCD, Collins and Perry [1] first noted that the dense cores of neutron stars may consist of deconfined quarks instead of hadrons. The crucial question is whether observations of neutron stars from their birth to death through neutrino, photon and gravity-wave emissions can unequivocably reveal the presence of nearly deconfined quarks instead of other possibilities such as only nucleons or other exotica such as strangeness-bearing hyperons or Bose (pion and kaon) condensates."
},
{
"section_type": "OTHER",
"section_title": "Neutrino signals during the birth of a neutron star",
"text": "The birth of a neutron star is heralded by the arrival of neutrinos on earth as confirmed by IMB and Kamiokande neutrino detectors in the case of supernova SN 1987A. Nearly all of the gravitational binding energy (of order 300 bethes, where 1 bethe ≡ 10 51 erg) released in the progenitor star's white dwarf-like core is carried off by neutrinos and antineutrinos of all flavors in roughly equal proportions. The remarkable fact that the weakly interacting neutrinos are trapped in matter prior to their release as a burst is due to their short mean free paths in matter, λ ≈ (σn) -1 ≈ 10 cm, (here σ ≈ 10 -40 cm 2 is the neutrino-matter cross section and n ≈ 2 to 3 n s , where n s ≃ 0.16 fm -3 is the reference nuclear equilibrium density), which is much less than the proto-neutron star radius, which exceeds 20 km. Should a core-collapse supernova occur in their lifetimes, current neutrino detectors, such as SK, SNO, LVD's, AMANDA, etc., offer a great opportunity for understanding a proto-neutron star's birth and propagation of neutrinos in dense matter insofar as they can detect tens of thousands of neutrinos in contrast to the tens of neutrinos detected by IMB and Kamiokande.\n\nThe appearence of quarks inside a neutron star leads to a decrease in the maximum mass that matter can support, implying metastability of the star. This would occur if the proto-neutron star's mass, which must be less than the maximum mass of the\n\nQuark matter and the astrophysics of neutron stars 2 Figure 1. Evolutions of the central baryon density n B , ν concentration Y ν , quark volume fraction χ and temperature T for different baryon masses M B . Solid lines show stable stars whereas dashed lines showing stars with larger masses are metastable. Diamonds indicate when quarks appear at the star's center, and asterisks denote when metastable stars become gravitationally unstable. Figure after Ref. [2] hot, lepton-rich matter is greater than the maximum mass of hot, lepton-poor matter. For matter with nucleons only, such a metastability is denied (see, e.g., [3] ). Figure 1 shows the evolution of some thermodynamic quantities at the center of stars of various fixed baryonic masses. With the equation of state used (see [2] for details), stars with M B ∼ < 1.1 M ⊙ do not contain quarks and those with M B ∼ 1.7 M ⊙ are metastable. The subsequent collapse to a black hole could be observed as a cessation in the neutrino signals well above the sensitivity limits of the current detectors (Figure 2 )."
},
{
"section_type": "OTHER",
"section_title": "Photon signals during the thermal evolution of a neutron star",
"text": "Multiwavelength photon observations of neutron stars, the bread and butter affair of astronomy, has yielded estimates of the surface tempeartures and ages of several neutron stars (Fig. 3 ). As neutron stars cool principally through neutrino emission from their cores, the possibility exists that the interior composition can be determined. The star continuosly emits photons, dominantly in x-rays, with an effective temperature T ef f that tracks the interior temperature but that is smaller by a factor ∼ 100. The dominant neutrino cooling reactions are of a general type, known as Urca processes [4], in which thermally excited particles undergo beta and inverse-beta decays. Each\n\nQuark matter and the astrophysics of neutron stars 3 Figure 2. The evolution of the total neutrino luminosity for stars of indicated baryon masses. Shaded bands illustrate the limiting luninosities corresponding to a count rate of 0.2 Hz in all detectors assuming 50 kpc for IMB and and Kamioka, 8.5 kpc for SNO, SuperK, and UNO. Shaded regions represent uncertanities in the average neutrino energy from the use of a diffusion scheme for neutrino transport in matter.\n\nFigure after Ref. [2].\n\nreaction produces a neutrino or anti-neutrino, and thermal energy is thus continuously lost. Depending upon the proton-fraction of matter, which in turn depends on the nature of strong interactions at high density, direct Urca processes involving nucleons, hyperons or quarks lead to enhanced cooling compared to modified Urca processes in which an additional particle is required to conserve momentum. However, effects of superfluidity abates cooling as sufficient thermal energy is required to break paired fermions. In addition, the poorly known envelope composition also plays a role in the inferred surface temperature (Fig. 3 ). The multitude of high density phases, cooling mechanisms, effects of superfluidity, and unknown envelope composition have thus far prevented definitive conclusions to be drawn (see, e.g., [5])."
},
{
"section_type": "OTHER",
"section_title": "Mesured masses and their implications",
"text": "Several recent observations of neutron stars have direct bearing on the determination of the maximum mass. The most accurately measured masses are from timing observations of the radio binary pulsars. As shown in Fig. 4 , which is compilation of the measured neutron star masses as of November 2006, observations include pulsars orbiting another Quark matter and the astrophysics of neutron stars 4 Figure 3. Observational estimates of neutron star temperatures and ages together with theoretical cooling simulations for M = 1.4 M ⊙ . Models and data are described in [6] . Orange error boxes (see online) indicate sources from which both X-ray and optical emissions have been observed. Simulations are for models with Fe or H envelopes, with and without the effects of superfluidity, and allowing or forbidding direct Urca processes. Models forbidding direct Urca processes are relatively independent of M and superfluid properties. Trajectories for models with enhanced cooling (direct Urca processes) and superfluidity lie within the yellow region, the exact location depending upon M as well as superfluid and Urca properties. Figure adapted from Ref. [7].\n\nneutron star, a white dwarf or a main-sequence star.\n\nOne significant development concerns mass determinations in binaries with white dwarf companions, which show a broader range of neutron star masses than binary neutron star pulsars. Perhaps a rather narrow set of evolutionary circumstances conspire to form double neutron star binaries, leading to a restricted range of neutron star masses [9]. This restriction is likely relaxed for other neutron star binaries. A few of the white dwarf binaries may contain neutron stars larger than the canonical 1.4 M ⊙ value, including the intriguing case [10] of PSR J0751+1807 in which the estimated mass with 1σ error bars is 2.1 ± 0.2 M ⊙ . In addition, to 95% confidence, one of the two pulsars Ter 5 I and J has a reported mass larger than 1.68 M ⊙ [11] .\n\nWhereas the observed simple mean mass of neutron stars with white dwarf companions exceeds those with neutron star companions by 0.25 M ⊙ , the weighted means of the two groups are virtually the same. The 2.1 M ⊙ neutron star, PSR J0751+1807, is about 4σ from the canonical value of 1.4 M ⊙ . It is furthermore the case that the 2σ errors of all but two systems extend into the range below 1.45 M ⊙ , so caution should be exercised before concluding that firm evidence of large neutron star masses exists. Continued observations, which will reduce the observational errors, are necessary to clarify this situation.\n\nMasses can also be estimated for another handful of binaries which contain an\n\nQuark matter and the astrophysics of neutron stars 5 Figure 4. Measured and estimated masses of neutron stars in binarie pulsars (gold, silver and blue regions online) and in x-ray accreting binaries (green). For each region, simple averages are shown as dotted lines; error weighted averages are shown as dashed lines. For labels and other details, consult Ref. [8].\n\naccreting neutron star emitting x-rays. Some of these systems are characterized by relatively large masses, but the estimated errors are also large. The system of Vela X-1 is noteworthy because its lower mass limit (1.6 to 1.7M ⊙ ) is at least mildly constrained by geometry [12]. Raising the limit for the neutron star maximum mass could eliminate entire families of EOS's, especially those in which substantial softening begins around 2 to 3n s . This could be extremely significant, since exotica (hyperons, Bose condensates, or quarks) generally reduce the maximum mass appreciably."
},
{
"section_type": "OTHER",
"section_title": "Ultimate energy density of observable cold baryonic matter",
"text": "Measurements of neutron star masses can set an upper limit to the maximum possible energy density in any compact object. It has been found [13] that no causal EOS has a central density, for a given mass, greater than that for the Tolman VII [14] analytic solution. This solution corresponds to a quadratic mass-energy density ρ dependence Quark matter and the astrophysics of neutron stars 6 Figure 5. Model predictions are compared with results from the Tolman IV and VII analytic solutions of general relativistic stucture equations. NR refers to nonrelativistic potential models, R are field-theoretical models, and Exotica refers to NR or R models in which strong softening occurs, due to hyperons, a Bose condensate, or quark matter as well as self-bound strange quark matter. Constraints from a possible redshift measurement of z = 0.35 is also shown. The dashed lines for 1.44 and 2.2 M ⊙ serve to guide the eye. Figure taken from Ref. [13] .\n\non r, ρ = ρ c [1 -(r/R) 2 ], where the central density is ρ c . For this solution,\n\nρ c,T V II = 2.5ρ c,Inc ≃ 1.5 × 10 16 M ⊙ M 2 g cm -3 . ( 1\n\n)\n\nA measured mass of 2.2 M ⊙ would imply ρ max < 3.1 × 10 15 g cm -3 , or about 8n s .\n\nFigure 5 displays maximum masses and accompanying central densities for a wide wariety of neutron star EOS's, including models containing significant softening due to \"exotica\", such as strange quark matter. The upper limit to the density could be lowered if the causal constraint is not approached in practice. For example, at high densities in which quark asymptotic freedom is realized, the sound speed is limited to c/ √ 3. Using this as a strict limit at all densities, the Rhoades & Ruffini [15] mass limit is reduced by approximately 1/ √ 3 and the compactness limit GM/Rc 2 = 1/2.94 is reduced by a factor 3 -1/4 to 1/3.8 [16] . In this extreme case, the maximum density would be reduced by a factor of 3 -1/4 from that of Eq. ( 1 ). A 2.2 M ⊙ measured mass would imply a maximum density of about 4.2n s ."
},
{
"section_type": "OTHER",
"section_title": "Gravitational wave signals during mergers of binary stars",
"text": "Mergers of compact objects in binary systems, such as a pair of neutron stars (NS-NS), a neutron star and a black hole (NS-BH), or two black holes (BH-BH), are expected to be prominent sources of gravitational radiation [17] . The gravitationalwave signature of such systems is primarily determined by the chirp mass M chirp =\n\nQuark matter and the astrophysics of neutron stars 7 Figure 6. Physical and observational variables in mergers between low-mass black holes and neutron stars or self-bound quark stars. The total system mass is 6 M ⊙ and the initial mass ratio is q = 1/3 in both cases. The initial radii of the neutron star and quark star were assumed to be equal. The time scales have arbitrary zero points. Upper panel displays semi-major axis a (thick lines) and component mass M N S , M QS (thin lines) evolution. Lower panel displays orbital frequency ν (thick lines) and strain amplitude |h + r| evolution. Solid curves refer to the neutron star simulation and dashed curves to the quark star simulations. Figure taken from Ref. [8]. 1/5 , where M 1 and M 2 are the masses of the coalescing objects. The radiation of gravitational waves removes energy which causes the mutual orbits to decay. For example, the binary pulsar PSR B1913+16 has a merger timescale of about 250 million years, and the pulsar binary PSR J0737-3039 has a merger timescale of about 85 million years [18] , so there is ample reason to expect that many such decaying compact binaries exist in the Galaxy. Besides emitting copious amounts of gravitational radiation, binary mergers have been proposed as a source of the r-process elements [19] and the origin of the shorter-duration gamma ray bursters [20] .\n\n(M 1 M 2 ) 3/5 (M 1 + M 2 ) -\n\nObservations of gravity waves from merger events can simultaneously measure masses and radii of neutron stars, and could set firm limits on the neutron star maximum mass [21, 22] . Binary mergers for the two cases of a black hole and a normal neutron star and a black hole and a self-bound strange quark matter star (Fig. 6 ) illustrate the unique opportunity afforded by gravitational wave detectors due to begin operation over the next decade, including LIGO, VIRGO, GEO600, and TAMA.\n\nA careful analysis of the gravitational waveform during inspiral yields values for not only the chirp mass M chirp , but for also the reduced mass M BH M N S /M, so that both M BH and M N S can be found [23] . The onset of mass transfer can be determined by the peak in ω, and the value of ω there gives a. A general relativistic analysis of mass transfer conditions then allows the determination of the star's radius [22] . Thus a point on the mass-radius diagram can be estimated [24] . The combination h + ω -1/3 depends only on a function of q, so the ratio of that combination and knowledge of q i\n\nQuark matter and the astrophysics of neutron stars 8 should allow determination of q f . From the Roche condition and knowledge of a f from ω f , another mass-radius combination can be found.\n\nThe sharp contrast between the evolutions during stable mass transfer of a normal neutron star and a strange quark star should make these cases distinguishable. For strange quark matter stars, the differences in the height of the frequency peak and the plateau in the frequency values at later times are related to the differences in radii of the stars at these two epochs. It could be an indirect indicator of the maximum mass of the star: the closer is the star's mass before mass transfer to the maximum mass, the greater is the difference between these frequency values, because the radius change will be larger. Together with radius information, the value of the maximum mass remains the most important unknown that could reveal the true equation of state at high densities."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "This work was supported in part by the U.S. Department of Energy under the grant DOE/DE-FG02-93ER40756."
}
] | |
arxiv:0704.0214 | 0704.0214 | 1 | 060470b4dbeb72f041d1d9f0eb467fa26eb61cf66c63cb9dae16a7a3a8c0e5bc | A schematic model of scattering in PT-symmetric Quantum Mechanics | One-dimensional scattering problem admitting a complex, PT-symmetric
short-range potential V(x) is considered. Using a Runge-Kutta-discretized
version of Schroedinger equation we derive the formulae for the reflection and
transmission coefficients and emphasize that the only innovation emerges in
fact via a complexification of one of the potential-characterizing parameters. | [
"Miloslav Znojil"
] | [
"quant-ph",
"hep-th"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | One-dimensional scattering problem admitting a complex, PT -symmetric shortrange potential V (x) is considered. Using a Runge-Kutta-discretized version of Schrödinger equation we derive the formulae for the reflection and transmission coefficients and emphasize that the only innovation emerges in fact via a complexification of one of the potential-characterizing parameters.
Standard textbooks describe the stationary one-dimensional motion of a quantum particle in a real potential well V (x) by the ordinary differential Schrödinger equation
- d 2 dx 2 + V (x) ψ(x) = E ψ(x) , x ∈ (-∞, ∞) (1)
which may be considered and solved in the bound-state regime at E < V (∞) ≤ +∞ or in the scattering regime with, say, E = κ 2 > V (∞) = 0. In this way one either employs the boundary conditions ψ(±∞) = 0 and determines the spectrum of bound states or, alternatively, switches to the different boundary conditions, say,
ψ(x) = A e iκx + B e -iκx , x ≪ -1 , C e iκx , x ≫ 1 .
(
) 2
Under the conventional choice of A = 1 the latter problem specifies the reflection and transmission coefficients B and C, respectively [1] .
The conventional approach to the quantum bound state problem has recently been, fairly unexpectedly, generalized to many unconventional and manifestly non-Hermitian Hamiltonians H = H † which are merely quasi-Hermitian, i.e., which are Hermitian only in the sense of an identity H † = Θ H Θ -1 which contains a nontrivial "metric" operator Θ = Θ † > 0 as introduced, e.g., in ref. [2] . The key ideas and sources of the latter new development in Quantum Mechanics incorporate the so called PT -symmetry of the Hamiltonians and have been summarized in the very fresh review by Carl Bender [3] . This text may be complemented by a sample [4] of the dedicated conference proceedings.
In this context we intend to pay attention to a very simple PT -symmetric scattering model where
V (x) = Z(x) + i Y (x) , Z(-x) = Z(x) = real , Y (-x) = -Y (x) = real
and where the ordinary differential equation ( 1 ) is replaced by its Runge-Kuttadiscretized, difference-equation representation
- ψ(x k-1 ) -2 ψ(x k ) + ψ(x k+1 ) h 2 + V (x k ) ψ(x k ) = E ψ(x k ) (3)
with
x k = k h , h > 0 , k = 0, ±1, . . .
as employed, in the context of the bound-state problem, in refs. [5] .
Once we assume, for the sake of simplicity, that the potential in eq. ( 3 ) vanishes beyond certain distance from the origin,
V (x ±j ) = 0 j = M, M + 1, . . . ,
we may abbreviate 3 ),
ψ k = ψ(x k ), V k = h 2 V (x k ) and 2 cos ϕ = 2 -h 2 E in eq. (
-ψ k-1 + (2 cos ϕ + V k ) ψ k -ψ k+1 = 0 . (4)
In the region of |k| ≥ M with vanishing potential V k = 0 the two independent solutions of our difference Schrödinger eq. ( 4 ) are easily found, via a suitable ansatz, as elementary functions of the new "energy" variable ϕ,
ψ k = const • ̺ k =⇒ ̺ = ̺ ± = exp(±i ϕ) .
This enables us to replace the standard boundary conditions ( 2 ) by their discrete scattering version
ψ(x m ) = A e i m ϕ + B e -i m ϕ , m ≤ -(M -1) , C e i m ϕ , m ≥ M -1 (5)
with a conventional choice of A = 1.
Two comments may be added here. Firstly, one notices that the condition of the reality of the new energy variable ϕ imposes the constraint upon the original energy itself, -2 ≤ 2 -h 2 E ≤ 2, i.e., E ∈ (0, 4/h 2 ). At any finite choice of the lattice step h > 0 this inequality is intuitively reminiscent of the spectra in relativistic quantum systems. Via an explicit display of the higher O(h 4 ) corrections in eq. ( 3 ), this connection has been given a more quantitative interpretation in ref. [6] .
The second eligible way of dealing with the uncertainty represented by the O(h 4 ) discrepancy between the difference-and differential-operator representation of the Schrödinger's kinetic energy is more standard and lies in its disappearance in the limit h → 0. This is a purely numerical recipe known as the Runge-Kutta method [7] . In the present context of scattering one has to keep in mind that the two "small" parameters h and 1/M may and, in order to achieve the quickest convergence, should be chosen and varied independently.
3 The matching method of solution 3.1 The simplest model of the scattering with M = 1
Once we are given the boundary conditions ( 5 ) the process of the construction of the solutions is straightforward. Let us first illustrate its key technical ingredients on the model with the first nontrivial choice of the cutoff M = 1. In this case our difference Schrödinger eq. ( 4 ) degenerates to the mere three nontrivial relations,
-ψ -2 + 2 cos ϕ ψ -1 -ψ (-) 0 = 0 -ψ -1 + (2 cos ϕ + Z 0 ) ψ 0 -ψ 1 = 0 -ψ (+) 0 + 2 cos ϕ ψ 1 -ψ 2 = 0 (6)
where we may insert, from eq. ( 5 ),
ψ -1 = e -i ϕ + B e i ϕ , ψ (-) 0 = 1 + B , ψ (+) 0 = C , ψ 1 = C e i ϕ (7)
and where we have to demand, subsequently,
ψ (-) 0 = 1 + B = ψ (+) 0 = C = ψ 0 , -e -i ϕ -B e i ϕ + (2 cos ϕ + Z 0 ) C -C e i ϕ = 0 . (8)
Thus, at an arbitrary "energy" ϕ one identifies B = C -1 and gets the solution
C = 2i sin ϕ 2i sin ϕ -Z 0 , B = Z 0 2i sin ϕ -Z 0 .
Of course, as long as we deal just with the real "interaction term" Z 0 , our M = 1 toy problem remains Hermitian since no PT -symmetry has entered the scene yet.
3.2 PT -symmetry and the scattering at M = 2
In the next, M = 2 version of our model we have to insert the four known quantities
ψ -2 = e -2 i ϕ + B e 2 i ϕ , ψ -1 = e -i ϕ + B e i ϕ , ψ 1 = C e i ϕ , ψ 2 = C e 2 i ϕ
in the triplet of relations
-ψ -2 + (2 cos ϕ + Z -1 -i Y -1 ) ψ -1 -ψ (-) 0 = 0 -ψ -1 + (2 cos ϕ + Z 0 ) ψ 0 -ψ 1 = 0 -ψ (+) 0 + (2 cos ϕ + Z -1 + i Y -1 ) ψ 1 -ψ 2 = 0 (9)
where the three symbols ψ 0 , ψ (-) 0 and ψ (+) 0
defined by these respective equations should represent the same quantity and must be equal to each other, therefore.
Having this in mind we introduce ξ
(-) 0 = 1 + B and ξ (+) 0 = C and decompose ψ (-) 0 = ξ (-) 0 + χ (-) 0 , ψ (+) 0 = ξ (+) 0 + χ (+) 0 .
This enables us eliminate
χ (-) 0 = V -1 ψ -1 , χ (+) 0 = V 1 ψ 1
and eq. ( 9 ) becomes reduced to the pair of conditions,
1 + B + V -1 ψ -1 = C + V 1 ψ 1 = ψ 0 , -ψ -1 + (2 cos ϕ + Z 0 ) ψ 0 -ψ 1 = 0 (10)
They lead to the two-dimensional linear algebraic problem which defines the reflection and transmission coefficients B and C at any input energy ϕ. The same conclusion applies to all the models with the larger M.
Let us now re-write our difference Schrödinger eq. ( 4 ) as a doubly infinite system of linear algebraic equations
. . . . . . . . . . . . . . . S -1 -1 0 . . . . . . -1 S 0 -1 . . . . . . 0 -1 S 1 . . . . . . . . . . . . . . . . . . ψ -1 ψ 0 ψ 1 . . . = = 0 , (11)
where
S k ( ≡ S * -k ) = 2 cos ϕ + Z k + i Y k sign k , |k| < M , 2 cos ϕ , |k| ≥ M (12)
and where the majority of the elements of the "eigenvector" are prescribed, in advance, by the boundary conditions (5) . Once we denote all of them by a different symbol,
ψ(x m ) = A e i m ϕ + B e -i m ϕ ≡ ξ (-) m , m ≤ -(M -1) , C e i m ϕ ≡ ξ (+) m , m ≥ M -1 , (13)
we may reduce eq. ( 11 ) to a finite-dimensional and tridiagonal non-square-matrix problem
-1 S * (M -1) -1 . . . . . . . . . -1 S * 1 -1 -1 S 0 -1 -1 S 1 -1 . . . . . . . . . . . . . . . -1 S (M -1) -1 ξ (-) -M ξ (-) -(M -1)
ψ -(M -2) . . .
ψ M -2 ξ (+) M -1 ξ (+) M =
= 0 (14) or, better, to a non-homogeneous system of 2M -1 equations
T • ξ (-) -(M -1) ψ -(M -2) . . . ψ M -2 ξ (+) M -1 = ξ (-) -M 0 . . . 0 ξ (+) M (15)
where the (2M -1)-dimensional square-matrix of the system can be partitioned as follows,
T = S * (M -1) -1 -1 S * (M -2) -1 -1 . . . . . . . . . S 0 . . . . . . . . . -1 -1 S (M -2) -1 -1 S (M -1) . ( 16
)
Whenever this matrix proves non-singular, it may assigned the inverse matrix R=T -1 , the knowledge of which enables us to re-write eq. ( 15 ), in the same partitioning, as follows,
ξ (-) -(M -1) Ψ ξ (+) M -1 = R • ξ (-) -M 0 ξ (+) M , Ψ = ψ -(M -2) . . . ψ M -2 . ( 17
)
In the next step we deduce that the matrix R has the following partitioned form
R = α * t T β u Q v β w T α
.
We may summarize that in the light of the overall partitioned structure of eq. ( 17 ), the knowledge of the (2M -3)-dimensional submatrix Q as well as of the two (2M -3)-dimensional row vectors t T and w T (where T denotes transposition) is entirely redundant. Moreover, the knowledge of the other two column vectors u and v only helps us to eliminate the "wavefunction" components ψ
-(M -2) , ψ -(M -3) , . . . , ψ M -3 , ψ M -2 .
In this sense, equation ( 15 ) degenerates to the mere two scalar relations ξ (-)
-(M -1) -α * ξ (-) -(M ) -β ξ (+) M = 0 , ξ (+) M -1 -β ξ (-) -M -α ξ (+) M = 0 . ( 18
)
Once we insert the explicit definitions from eq. ( 13 ) we get the final pair of linear equations
e -i (M -1) ϕ + B e i (M -1) ϕ -α * e -i M ϕ + B e i M ϕ -C β e i M ϕ = 0 , C e i (M -1) ϕ -β e -i M ϕ + B e i M ϕ -C α e i M ϕ = 0 (19)
which are solved by the elimination of
B = -e -2iM ϕ + C β e -iϕ -α (20)
and, subsequently, of
C = 2iβe -2iM ϕ sin ϕ β 2 -(e -iϕ -α * ) (e -iϕ -α) . (21)
This is our present main result.
Our final scattering-determining formulae ( 20 ) and ( 21 ) indicate that the complex coefficient α and the real coefficient β carry all the "dynamical input" information.
At any given energy parameter ϕ these matrix elements are, by construction, rational functions of our 2M -1 real coupling constants Z 0 , Z 1 , . . . , Z M -1 and Y 1 , . . . , Y M -1 .
In particular, β is equal to 1/ det T and α has the same denominator of course. An explicit algebraic determination of the determinant det T and of the numerator (say, γ) of α is less easy. Let us illustrate this assertion on a few examples.
5.1 M = 2 once more
det T = Z 0 Z 1 2 -2 Z 1 + Y 1 2 Z 0 Re γ = Z 0 Z 1 -1 Im γ = -Z 0 Y 1 5.2 M = 3 det T = Z 0 Z 1 2 Z 2 2 -2 Z 0 Z 1 Z 2 -2 Z 1 Z 2 2 + +Y 1 2 Z 0 Z 2 2 + 2 Z 2 + Y 2 2 Z 0 Z 1 2 -2 Y 2 2 Z 1 + Y 2 2 Y 1 2 Z 0 + 2 Z 0 Y 1 Y 2 + Z 0 Re γ = Z 0 Z 1 2 Z 2 -2 Z 1 Z 2 + Y 1 2 Z 0 Z 2 -Z 1 Z 0 + 1 Im γ = -Z 0 Z 1 2 Y 2 + 2 Z 1 Y 2 -Y 1 2 Z 0 Y 2 -Y 1 Z 0 5.3 M = 4
The growth of complexity of the formulae occurs already at the dimension as low as M = 4. The determinant det T and the real and imaginary parts of γ are them represented by the sums of 15 and 14 and 32 products of couplings, respectively.
A simplification is only encountered in the weak coupling regime where one finds just two terms in the determinant which are linear in the couplings, det T = -2 Z 3 -2 Z 1 + . . . being followed by the 10 triple-product terms,
. . . + Y 1 2 Z 0 + 4 Z 1 Z 2 Z 3 -4 Z 1 Y 2 Y 3 + Z 1 2 Z 0 + 2 Y 3 2 Z 2 + 2 Y 1 Z 0 Y 3 + +2 Z 1 Z 0 Z 3 + Z 3 2 Z 0 + Y 3 2 Z 0 + 2 Z 3 2 Z 2 + . . .
etc. Similarly, we may decompose, in the even-number products,
Re γ = -1 + 2 Z 2 Z 3 + Z 0 Z 1 + Z 0 Z 3 + 2 Z 2 Z 1 + . . . and continue . . . + -2 Z 0 Z 1 Z 2 Z 3 + 2 Z 0 Y 1 Y 2 Z 3 - -Z 2 Y 1 2 Z 0 -2 Z 2 2 Z 1 Z 3 -Z 2 Z 1 2 Z 0 -2 Y 2 2 Z 1 Z 3 + . . . etc, plus Im γ = -2 Z 2 Y 3 + 2 Y 2 Z 1 -Z 0 Y 1 -Z 0 Y 3 -. . . with a continuation . . . -Y 2 Y 1 2 Z 0 -Y 2 Z 1 2 Z 0 + +2 Y 2 2 Z 1 Y 3 + 2 Z 0 Z 1 Z 2 Y 3 + 2 Z 2 2 Z 1 Y 3 -2 Z 0 Y 1 Y 2 Y 3 + . . .
etc. Symbolic manipulations on a computer should be employed at all the higher dimensions M ≥ 4 in general.
The main inspiration of the activities and attention paid to the PT -symmetry originates from the pioneering 1998 letter by Bender and Boettcher [8] where the the definition of the transmission and reflection coefficients has the same form [cf., e.g., eq. ( 21 )], with all the differences represented by the differences in the form of the "dynamical input" information. It has been shown to be encoded, in both the Hermitian and non-Hermitian cases, in the two functions α and β of the lattice potentials, with the vanishing or non-vanishing coefficients Y k , respectively (cf. a few samples of the concrete form of α and β in section 5).
On the level of physics we would like to emphasize that one of the main distinguishing features of the scattering problem in PT -symmetric quantum mechanics lies in the manifest asymmetry between the "in" and "out" states [22] . In its present solvable exemplification we showed that such an asymmetry is merely formal and that the problem remains tractable by the standard, non-matching and nonrecurrent techniques of linear algebra. A key to the success proved to lie in the partitioning of the Schrödinger equation which enabled us to separate its essential and inessential components and to reduce the construction of the amplitudes to the mere two-dimensional matrix inversion [cf. eq. ( 18 )] where all the dynamical input is represented by the four corners of the inverse matrix R = T -1 [cf. the definition (16) ].
We believe that the merits of the present discrete model were not exhausted by its present short analysis and that its further study might throw new light, e.g., on the non-Hermitian versions of the inverse problem of scattering. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "One-dimensional scattering problem admitting a complex, PT -symmetric shortrange potential V (x) is considered. Using a Runge-Kutta-discretized version of Schrödinger equation we derive the formulae for the reflection and transmission coefficients and emphasize that the only innovation emerges in fact via a complexification of one of the potential-characterizing parameters."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Standard textbooks describe the stationary one-dimensional motion of a quantum particle in a real potential well V (x) by the ordinary differential Schrödinger equation\n\n- d 2 dx 2 + V (x) ψ(x) = E ψ(x) , x ∈ (-∞, ∞) (1)\n\nwhich may be considered and solved in the bound-state regime at E < V (∞) ≤ +∞ or in the scattering regime with, say, E = κ 2 > V (∞) = 0. In this way one either employs the boundary conditions ψ(±∞) = 0 and determines the spectrum of bound states or, alternatively, switches to the different boundary conditions, say,\n\nψ(x) = A e iκx + B e -iκx , x ≪ -1 , C e iκx , x ≫ 1 .\n\n(\n\n) 2\n\nUnder the conventional choice of A = 1 the latter problem specifies the reflection and transmission coefficients B and C, respectively [1] .\n\nThe conventional approach to the quantum bound state problem has recently been, fairly unexpectedly, generalized to many unconventional and manifestly non-Hermitian Hamiltonians H = H † which are merely quasi-Hermitian, i.e., which are Hermitian only in the sense of an identity H † = Θ H Θ -1 which contains a nontrivial \"metric\" operator Θ = Θ † > 0 as introduced, e.g., in ref. [2] . The key ideas and sources of the latter new development in Quantum Mechanics incorporate the so called PT -symmetry of the Hamiltonians and have been summarized in the very fresh review by Carl Bender [3] . This text may be complemented by a sample [4] of the dedicated conference proceedings.\n\nIn this context we intend to pay attention to a very simple PT -symmetric scattering model where\n\nV (x) = Z(x) + i Y (x) , Z(-x) = Z(x) = real , Y (-x) = -Y (x) = real\n\nand where the ordinary differential equation ( 1 ) is replaced by its Runge-Kuttadiscretized, difference-equation representation\n\n- ψ(x k-1 ) -2 ψ(x k ) + ψ(x k+1 ) h 2 + V (x k ) ψ(x k ) = E ψ(x k ) (3)\n\nwith\n\nx k = k h , h > 0 , k = 0, ±1, . . .\n\nas employed, in the context of the bound-state problem, in refs. [5] ."
},
{
"section_type": "OTHER",
"section_title": "Runge-Kutta scattering",
"text": "Once we assume, for the sake of simplicity, that the potential in eq. ( 3 ) vanishes beyond certain distance from the origin,\n\nV (x ±j ) = 0 j = M, M + 1, . . . ,\n\nwe may abbreviate 3 ),\n\nψ k = ψ(x k ), V k = h 2 V (x k ) and 2 cos ϕ = 2 -h 2 E in eq. (\n\n-ψ k-1 + (2 cos ϕ + V k ) ψ k -ψ k+1 = 0 . (4)\n\nIn the region of |k| ≥ M with vanishing potential V k = 0 the two independent solutions of our difference Schrödinger eq. ( 4 ) are easily found, via a suitable ansatz, as elementary functions of the new \"energy\" variable ϕ,\n\nψ k = const • ̺ k =⇒ ̺ = ̺ ± = exp(±i ϕ) .\n\nThis enables us to replace the standard boundary conditions ( 2 ) by their discrete scattering version\n\nψ(x m ) = A e i m ϕ + B e -i m ϕ , m ≤ -(M -1) , C e i m ϕ , m ≥ M -1 (5)\n\nwith a conventional choice of A = 1.\n\nTwo comments may be added here. Firstly, one notices that the condition of the reality of the new energy variable ϕ imposes the constraint upon the original energy itself, -2 ≤ 2 -h 2 E ≤ 2, i.e., E ∈ (0, 4/h 2 ). At any finite choice of the lattice step h > 0 this inequality is intuitively reminiscent of the spectra in relativistic quantum systems. Via an explicit display of the higher O(h 4 ) corrections in eq. ( 3 ), this connection has been given a more quantitative interpretation in ref. [6] .\n\nThe second eligible way of dealing with the uncertainty represented by the O(h 4 ) discrepancy between the difference-and differential-operator representation of the Schrödinger's kinetic energy is more standard and lies in its disappearance in the limit h → 0. This is a purely numerical recipe known as the Runge-Kutta method [7] . In the present context of scattering one has to keep in mind that the two \"small\" parameters h and 1/M may and, in order to achieve the quickest convergence, should be chosen and varied independently.\n\n3 The matching method of solution 3.1 The simplest model of the scattering with M = 1\n\nOnce we are given the boundary conditions ( 5 ) the process of the construction of the solutions is straightforward. Let us first illustrate its key technical ingredients on the model with the first nontrivial choice of the cutoff M = 1. In this case our difference Schrödinger eq. ( 4 ) degenerates to the mere three nontrivial relations,\n\n-ψ -2 + 2 cos ϕ ψ -1 -ψ (-) 0 = 0 -ψ -1 + (2 cos ϕ + Z 0 ) ψ 0 -ψ 1 = 0 -ψ (+) 0 + 2 cos ϕ ψ 1 -ψ 2 = 0 (6)\n\nwhere we may insert, from eq. ( 5 ),\n\nψ -1 = e -i ϕ + B e i ϕ , ψ (-) 0 = 1 + B , ψ (+) 0 = C , ψ 1 = C e i ϕ (7)\n\nand where we have to demand, subsequently,\n\nψ (-) 0 = 1 + B = ψ (+) 0 = C = ψ 0 , -e -i ϕ -B e i ϕ + (2 cos ϕ + Z 0 ) C -C e i ϕ = 0 . (8)\n\nThus, at an arbitrary \"energy\" ϕ one identifies B = C -1 and gets the solution\n\nC = 2i sin ϕ 2i sin ϕ -Z 0 , B = Z 0 2i sin ϕ -Z 0 .\n\nOf course, as long as we deal just with the real \"interaction term\" Z 0 , our M = 1 toy problem remains Hermitian since no PT -symmetry has entered the scene yet.\n\n3.2 PT -symmetry and the scattering at M = 2\n\nIn the next, M = 2 version of our model we have to insert the four known quantities\n\nψ -2 = e -2 i ϕ + B e 2 i ϕ , ψ -1 = e -i ϕ + B e i ϕ , ψ 1 = C e i ϕ , ψ 2 = C e 2 i ϕ\n\nin the triplet of relations\n\n-ψ -2 + (2 cos ϕ + Z -1 -i Y -1 ) ψ -1 -ψ (-) 0 = 0 -ψ -1 + (2 cos ϕ + Z 0 ) ψ 0 -ψ 1 = 0 -ψ (+) 0 + (2 cos ϕ + Z -1 + i Y -1 ) ψ 1 -ψ 2 = 0 (9)\n\nwhere the three symbols ψ 0 , ψ (-) 0 and ψ (+) 0\n\ndefined by these respective equations should represent the same quantity and must be equal to each other, therefore.\n\nHaving this in mind we introduce ξ\n\n(-) 0 = 1 + B and ξ (+) 0 = C and decompose ψ (-) 0 = ξ (-) 0 + χ (-) 0 , ψ (+) 0 = ξ (+) 0 + χ (+) 0 .\n\nThis enables us eliminate\n\nχ (-) 0 = V -1 ψ -1 , χ (+) 0 = V 1 ψ 1\n\nand eq. ( 9 ) becomes reduced to the pair of conditions,\n\n1 + B + V -1 ψ -1 = C + V 1 ψ 1 = ψ 0 , -ψ -1 + (2 cos ϕ + Z 0 ) ψ 0 -ψ 1 = 0 (10)\n\nThey lead to the two-dimensional linear algebraic problem which defines the reflection and transmission coefficients B and C at any input energy ϕ. The same conclusion applies to all the models with the larger M."
},
{
"section_type": "METHOD",
"section_title": "The matrix-inversion method of solution",
"text": "Let us now re-write our difference Schrödinger eq. ( 4 ) as a doubly infinite system of linear algebraic equations\n\n . . . . . . . . . . . . . . . S -1 -1 0 . . . . . . -1 S 0 -1 . . . . . . 0 -1 S 1 . . . . . . . . . . . . . . . . . . ψ -1 ψ 0 ψ 1 . . . = = 0 , (11)\n\nwhere\n\nS k ( ≡ S * -k ) = 2 cos ϕ + Z k + i Y k sign k , |k| < M , 2 cos ϕ , |k| ≥ M (12)\n\nand where the majority of the elements of the \"eigenvector\" are prescribed, in advance, by the boundary conditions (5) . Once we denote all of them by a different symbol,\n\nψ(x m ) = A e i m ϕ + B e -i m ϕ ≡ ξ (-) m , m ≤ -(M -1) , C e i m ϕ ≡ ξ (+) m , m ≥ M -1 , (13)\n\nwe may reduce eq. ( 11 ) to a finite-dimensional and tridiagonal non-square-matrix problem\n\n -1 S * (M -1) -1 . . . . . . . . . -1 S * 1 -1 -1 S 0 -1 -1 S 1 -1 . . . . . . . . . . . . . . . -1 S (M -1) -1 ξ (-) -M ξ (-) -(M -1)\n\nψ -(M -2) . . .\n\nψ M -2 ξ (+) M -1 ξ (+) M = \n\n= 0 (14) or, better, to a non-homogeneous system of 2M -1 equations\n\nT • ξ (-) -(M -1) ψ -(M -2) . . . ψ M -2 ξ (+) M -1 = ξ (-) -M 0 . . . 0 ξ (+) M (15)\n\nwhere the (2M -1)-dimensional square-matrix of the system can be partitioned as follows,\n\nT = S * (M -1) -1 -1 S * (M -2) -1 -1 . . . . . . . . . S 0 . . . . . . . . . -1 -1 S (M -2) -1 -1 S (M -1) . ( 16\n\n)\n\nWhenever this matrix proves non-singular, it may assigned the inverse matrix R=T -1 , the knowledge of which enables us to re-write eq. ( 15 ), in the same partitioning, as follows,\n\n ξ (-) -(M -1) Ψ ξ (+) M -1 = R • ξ (-) -M 0 ξ (+) M , Ψ = ψ -(M -2) . . . ψ M -2 . ( 17\n\n)\n\nIn the next step we deduce that the matrix R has the following partitioned form\n\nR = α * t T β u Q v β w T α \n\n.\n\nWe may summarize that in the light of the overall partitioned structure of eq. ( 17 ), the knowledge of the (2M -3)-dimensional submatrix Q as well as of the two (2M -3)-dimensional row vectors t T and w T (where T denotes transposition) is entirely redundant. Moreover, the knowledge of the other two column vectors u and v only helps us to eliminate the \"wavefunction\" components ψ\n\n-(M -2) , ψ -(M -3) , . . . , ψ M -3 , ψ M -2 .\n\nIn this sense, equation ( 15 ) degenerates to the mere two scalar relations ξ (-)\n\n-(M -1) -α * ξ (-) -(M ) -β ξ (+) M = 0 , ξ (+) M -1 -β ξ (-) -M -α ξ (+) M = 0 . ( 18\n\n)\n\nOnce we insert the explicit definitions from eq. ( 13 ) we get the final pair of linear equations\n\ne -i (M -1) ϕ + B e i (M -1) ϕ -α * e -i M ϕ + B e i M ϕ -C β e i M ϕ = 0 , C e i (M -1) ϕ -β e -i M ϕ + B e i M ϕ -C α e i M ϕ = 0 (19)\n\nwhich are solved by the elimination of\n\nB = -e -2iM ϕ + C β e -iϕ -α (20)\n\nand, subsequently, of\n\nC = 2iβe -2iM ϕ sin ϕ β 2 -(e -iϕ -α * ) (e -iϕ -α) . (21)\n\nThis is our present main result."
},
{
"section_type": "OTHER",
"section_title": "Coefficients α and β",
"text": "Our final scattering-determining formulae ( 20 ) and ( 21 ) indicate that the complex coefficient α and the real coefficient β carry all the \"dynamical input\" information.\n\nAt any given energy parameter ϕ these matrix elements are, by construction, rational functions of our 2M -1 real coupling constants Z 0 , Z 1 , . . . , Z M -1 and Y 1 , . . . , Y M -1 .\n\nIn particular, β is equal to 1/ det T and α has the same denominator of course. An explicit algebraic determination of the determinant det T and of the numerator (say, γ) of α is less easy. Let us illustrate this assertion on a few examples.\n\n5.1 M = 2 once more\n\ndet T = Z 0 Z 1 2 -2 Z 1 + Y 1 2 Z 0 Re γ = Z 0 Z 1 -1 Im γ = -Z 0 Y 1 5.2 M = 3 det T = Z 0 Z 1 2 Z 2 2 -2 Z 0 Z 1 Z 2 -2 Z 1 Z 2 2 + +Y 1 2 Z 0 Z 2 2 + 2 Z 2 + Y 2 2 Z 0 Z 1 2 -2 Y 2 2 Z 1 + Y 2 2 Y 1 2 Z 0 + 2 Z 0 Y 1 Y 2 + Z 0 Re γ = Z 0 Z 1 2 Z 2 -2 Z 1 Z 2 + Y 1 2 Z 0 Z 2 -Z 1 Z 0 + 1 Im γ = -Z 0 Z 1 2 Y 2 + 2 Z 1 Y 2 -Y 1 2 Z 0 Y 2 -Y 1 Z 0 5.3 M = 4\n\nThe growth of complexity of the formulae occurs already at the dimension as low as M = 4. The determinant det T and the real and imaginary parts of γ are them represented by the sums of 15 and 14 and 32 products of couplings, respectively.\n\nA simplification is only encountered in the weak coupling regime where one finds just two terms in the determinant which are linear in the couplings, det T = -2 Z 3 -2 Z 1 + . . . being followed by the 10 triple-product terms,\n\n. . . + Y 1 2 Z 0 + 4 Z 1 Z 2 Z 3 -4 Z 1 Y 2 Y 3 + Z 1 2 Z 0 + 2 Y 3 2 Z 2 + 2 Y 1 Z 0 Y 3 + +2 Z 1 Z 0 Z 3 + Z 3 2 Z 0 + Y 3 2 Z 0 + 2 Z 3 2 Z 2 + . . .\n\netc. Similarly, we may decompose, in the even-number products,\n\nRe γ = -1 + 2 Z 2 Z 3 + Z 0 Z 1 + Z 0 Z 3 + 2 Z 2 Z 1 + . . . and continue . . . + -2 Z 0 Z 1 Z 2 Z 3 + 2 Z 0 Y 1 Y 2 Z 3 - -Z 2 Y 1 2 Z 0 -2 Z 2 2 Z 1 Z 3 -Z 2 Z 1 2 Z 0 -2 Y 2 2 Z 1 Z 3 + . . . etc, plus Im γ = -2 Z 2 Y 3 + 2 Y 2 Z 1 -Z 0 Y 1 -Z 0 Y 3 -. . . with a continuation . . . -Y 2 Y 1 2 Z 0 -Y 2 Z 1 2 Z 0 + +2 Y 2 2 Z 1 Y 3 + 2 Z 0 Z 1 Z 2 Y 3 + 2 Z 2 2 Z 1 Y 3 -2 Z 0 Y 1 Y 2 Y 3 + . . .\n\netc. Symbolic manipulations on a computer should be employed at all the higher dimensions M ≥ 4 in general."
},
{
"section_type": "DISCUSSION",
"section_title": "Discussion",
"text": "The main inspiration of the activities and attention paid to the PT -symmetry originates from the pioneering 1998 letter by Bender and Boettcher [8] where the the definition of the transmission and reflection coefficients has the same form [cf., e.g., eq. ( 21 )], with all the differences represented by the differences in the form of the \"dynamical input\" information. It has been shown to be encoded, in both the Hermitian and non-Hermitian cases, in the two functions α and β of the lattice potentials, with the vanishing or non-vanishing coefficients Y k , respectively (cf. a few samples of the concrete form of α and β in section 5).\n\nOn the level of physics we would like to emphasize that one of the main distinguishing features of the scattering problem in PT -symmetric quantum mechanics lies in the manifest asymmetry between the \"in\" and \"out\" states [22] . In its present solvable exemplification we showed that such an asymmetry is merely formal and that the problem remains tractable by the standard, non-matching and nonrecurrent techniques of linear algebra. A key to the success proved to lie in the partitioning of the Schrödinger equation which enabled us to separate its essential and inessential components and to reduce the construction of the amplitudes to the mere two-dimensional matrix inversion [cf. eq. ( 18 )] where all the dynamical input is represented by the four corners of the inverse matrix R = T -1 [cf. the definition (16) ].\n\nWe believe that the merits of the present discrete model were not exhausted by its present short analysis and that its further study might throw new light, e.g., on the non-Hermitian versions of the inverse problem of scattering."
}
] | ||
arxiv:0704.0221 | 0704.0221 | 1 | 10.1007/s10714-007-0472-9 10.1142/S0218271808012449 | a130963edee5fcad5bf3d395b233f9ef589cee0efc87491104a122e6065970a1 | The Return of a Static Universe and the End of Cosmology | We demonstrate that as we extrapolate the current $\Lambda$CDM universe
forward in time, all evidence of the Hubble expansion will disappear, so that
observers in our "island universe" will be fundamentally incapable of
determining the true nature of the universe, including the existence of the
highly dominant vacuum energy, the existence of the CMB, and the primordial
origin of light elements. With these pillars of the modern Big Bang gone, this
epoch will mark the end of cosmology and the return of a static universe. In
this sense, the coordinate system appropriate for future observers will perhaps
fittingly resemble the static coordinate system in which the de Sitter universe
was first presented. | [
"Lawrence M. Krauss (1",
"2) and Robert J. Scherrer (2) ((1) Case Western\n Reserve University",
"(2) Vanderbilt University)"
] | [
"astro-ph",
"gr-qc",
"hep-ph",
"hep-th"
] | astro-ph | [] | 2007-04-02 | 2026-02-26 | We demonstrate that as we extrapolate the current ΛCDM universe forward in time, all evidence of the Hubble expansion will disappear, so that observers in our "island universe" will be fundamentally incapable of determining the true nature of the universe, including the existence of the highly dominant vacuum energy, the existence of the CMB, and the primordial origin of light elements. With these pillars of the modern Big Bang gone, this epoch will mark the end of cosmology and the return of a static universe. In this sense, the coordinate system appropriate for future observers will perhaps fittingly resemble the static coordinate system in which the de Sitter universe was first presented.
1 Shortly after Einstein's development of general relativity, the Dutch astronomer Willem de Sitter proposed a static model of the universe containing no matter, which he thought might be a reasonable approximation to our low density universe. One can define a coordinate system in which the de Sitter metric takes a static form by defining de Sitter spacetime with a cosmological constant Λ as a four dimensional hyperboloid S Λ : η AB ξ A ξ B = -R 2 , R 2 = 3Λ -1 embedded in a 5d Minkowski spacetime with ds 2 = η AB dξ A dξ B , and (η AB ) = diag(1, -1, -1, -1, -1), A, B = 0, • • • , 4. The static form of the de Sitter metric is then
ds 2 s = (1 -r 2 s /R 2 )dt 2 s - dr 2 s 1 -r 2 s /R 2 -r 2 s dΩ 2 ,
which can be obtained by setting ξ 0 = (R 2 -r 2 s ) 1/2 sinh(t s /R), ξ 1 = r s sin θ cos ϕ, ξ 2 = r s sin θ sin ϕ, ξ 3 = r s cos θ, ξ 4 = (R 2 -r 2 s ) 1/2 cosh(t s /R). In this case the metric only corresponds to the section of de Sitter space within a cosmological horizon at R = rs.
In fact de Sitter's model wasn't globally static, but eternally expanding, as can be seen by a coordinate transformation which explicitly incorporates the time dependence of the scale factor R(t) = exp(Ht). While spatially flat, it actually incorporated Einstein's cosmological term, which is of course now understood to be equivalent to a vacuum energy density, leading to a redshift proportional to distance.
The de Sitter model languished for much of the last century, once the Hubble expansion had been discovered, and the cosmological term abandoned. However, all present observational evidence is consistent with a ΛCDM flat universe consisting of roughly 30% matter (both dark matter and baryonic matter) and 70% dark energy [1, 2, 3] , with the latter having a density that appears constant with time. All cosmological models with a non-zero cosmological constant will approach a de Sitter universe in the far future, and many of the implications of this fact have been explored in the literature [4, 5, 6, 7, 8, 9, 10, 11, 12, 13 ].
Here we re-examine the practical significance of the ultimate de Sitter expansion and point out a new eschatological physical consequence: from the perspective of any observer within a bound gravitational system in the far future, the static version of de Sitter space outside of that system will eventually become the appropriate physical coordinate system. Put more succinctly, in a time comparable to the age of the longest lived stars, observers will not be able to perform any observation or experiment that infers either the existence of an expanding universe dominated by a cosmological constant, or that there was a hot Big Bang. Observers will be able to infer a finite age for their island universe, but beyond that cosmology will effectively be over. The static universe, with which cosmology at the turn of the last century began, will have returned with a vengeance.
Modern cosmology is built on integrating general relativity and three observational pillars: the observed Hubble expansion, detection of the cosmic microwave background radiation, and the determination of the abundance of elements produced in the early universe. We describe next in detail how these observables will disappear for an observer in the far future, and how this will be likely to affect the theoretical conclusions one might derive about the universe.
The most basic component of modern cosmology is the expansion of the universe, firmly established by Hubble in 1929. Currently, galaxies and galaxy clusters are gravitationally bound and have dropped out of the Hubble flow, but structures on larger length scales are observed to obey the Hubble expansion law. Now consider what happens in the far future of the universe. Both analytic [7] and numerical [10] calculations indicate that the Local Group remains gravitationally bound in the face of the accelerated Hubble expansion. All more distant structures will be driven outside of the de Sitter event horizon in a timescale on the order of 100 billion years ( [4] , see also Refs. [8, 9] ). While objects will not be observed to cross the event horizon, light from them will be exponentially redshifted, so that within a time frame comparable to the longest lived main sequence stars all objects outside of our local cluster will truly become invisible [4] .
Since the only remaining visible objects will in fact be gravitationally bound and decoupled from the underlying Hubble expansion, any local observer in the far future will see a single galaxy (the merger product of the Milky Way and Andromeda and other remnants of the Local Group) and will have no observational evidence of the Hubble expansion. Lacking such evidence, one may wonder whether such an observer will postulate the correct cosmological model. We would argue that in fact, such an observer will conclude the existence of a static "island universe," precisely the standard model of the universe c. 1900. This will be true in spite of the fact that the dominant energy in this universe will not be due to matter, but due to dark energy, with ρ M /ρ Λ ∼ 10 -12 inside the horizon volume [9] . The irony, of course, is that the denizens of this static universe will have no idea of the existence of the dark energy, much less of its magnitude, since they will have no probes of the length scales over which Λ dominates gravitational dynamics. It appears that dark energy is undetectable not only in the limit where ρ Λ ≪ ρ M , but also when ρ Λ ≫ ρ M .
Even if there were no direct evidence of the Hubble expansion, we might expect three other bits of evidence, two observational and one theoretical, to lead physicists in the future to ascertain the underlying nature of cosmology. However, we next describe how this is unlikely to be the case.
The existence of a Cosmic Microwave Background was the key observation that convinced most physicists and astronomers that there was in fact a hot big bang, which essentially implies a Hubble expansion today. But even if skeptical observers in the future were inclined to undertake a search for this afterglow of the Big Bang, they would come up emptyhanded. At t ≈ 100 Gyr, the peak wavelength of the cosmic microwave background will be redshifted to roughly λ ≈ 1 m, or a frequency of roughly 300 MHz. While a uniform radio background at this frequency would in principle be observable, the intensity of the CMB will also be redshifted by about 12 orders of magnitude. At much later times, the CMB becomes unobservable even in principle, as the peak wavelength is driven to a length larger than the horizon [4] . Well before then, however, the microwave background peak will redshift below the plasma frequency of the interstellar medium, and so will be screened from any observer within the galaxy. Recall that the plasma frequency is given by
ν p = n e e 2 πm e 1/2
,
where n e and m e are the electron number density and mass, respectively. Observations of dispersion in pulsar signals give [14] n e ≈ 0.03 cm -3 in the interstellar medium, which corresponds to a plasma frequency of ν p ≈ 1 kHz, or a wavelength of λ p ≈ 3 × 10 7 cm.
This corresponds to an expansion factor ∼ 10 8 relative to the present-day peak of the CMB.
Assuming an exponential expansion, dominated by dark energy, this expansion factor will be reached when the universe is less than 50 times its present age, well below the lifetime of the longest-lived main sequence stars.
After this time, even if future residents of our island universe set out to measure a universal radiation background, they would be unable to do so. The wealth of information about early universe cosmology that can be derived from fluctuations in the CMB would be even further out of reach.
We may assume that theoretical physicists in the future will infer that gravitation is described by general relativity, using observations of planetary dynamics, and ground-based tests of such phenomena as gravitational time dilation. Will they then not be led to a Big Bang expansion, and a beginning in a Big Bang singularity, independent of data, as Lemaitre was? Indeed, is not a static universe incompatible with general relativity?
The answer is no. The inference that the universe must be expanding or contracting is dependent upon the cosmological hypothesis that we live in an isotropic and homogeneous universe. For future observers, this will manifestly not be the case. Outside of our local cluster, the universe will appear to be empty and static. Nothing is inconsistent with the temporary existence of a non-singular isolated self-gravitating object in such a universe, governed by general relativity. Physicists will infer that this system must ultimately collapse into a future singularity, but only as we presently conclude our galaxy must ultimately coalesce into a large black hole. Outside of this region, an empty static universe can prevail.
While physicists in the island universe will therefore conclude that their island has a finite future, the question will naturally arise as to whether it had a finite beginning. As we next describe, observers will in fact be able to determine the age of their local cluster, but not the nature of the beginning.
The theory of Big Bang Nucleosynthesis reached a fully-developed state [15] only after the discovery of the CMB (despite early abortive attempts by Gamow and his collaborators [16] ). Thus, it is unlikely that the residents of the static universe would have any motivation to explore the possibility of primordial nucleosynthesis. However, even if they did, the evidence for BBN rests crucially on the fact that relic abundances of deuterium remain observable at the present day, while helium-4 has been enhanced by only a few percent since it was produced in the early universe. Extrapolating forward by 100 Gyr, we expect significantly more contamination of the helium-4 abundance, and concomitant destruction of the relic deuterium. It has been argued [17] that the ultimate extrapolation of light elemental abundances, following many generations of stellar evolution, is a mass fraction of helium given by Y = 0.6. The primordial helium mass fraction of Y = 0.25 will be a relatively small fraction of this abundance. It is unlikely that much deuterium could survive this degree of processing. Of course, the current "smoking gun" deuterium abundance is provided by Lyman-α absorption systems, back-lit by QSOs (see, e.g., Ref. [18] ). Such systems will be unavailable to our observers of the future, as both the QSOs and the Lyman-α systems will have redshifted outside of the horizon.
Astute observers will be able to determine a lower limit on the age of their system, however, using standard stellar evolution analyses of their own local stars. They will be able to examine the locus of all stars and extrapolate to the oldest such stars to estimate a lower bound on the age of the galaxy. They will be able to determine an upper limit as well, by determining how long it would take for all of the observed helium to be generated by stellar nucleosynthesis. However, without any way to detect primordial elemental abundances, such as the aforementioned possibility of measuring deuterium in distant intergalactic clouds that currently absorb radiation from distant quasars and allow a determination of the deuterium abundance in these pre-stellar systems, and with the primordial helium abundance dwarfed by that produced in stars, inferring the original BBN abundances will be difficult, and probably not well motivated.
Thus, while physicists of the future will be able to infer that their island universe has not been eternal, it is unlikely that they will be able to infer that the beginning involved a Big
Bang.
The remarkable cosmic coincidence that we happen to live at the only time in the history of the universe when the magnitude of dark energy and dark matter densities are comparable has been a source of great current speculation, leading to a resurgence of interest in possible anthropic arguments limiting the value of the vacuum energy (see, e.g., Ref. [19] ). But this coincidence endows our current epoch with another special feature, namely that we can actually infer both the existence of the cosmological expansion, and the existence of dark energy. Thus, we live in a very special time in the evolution of the universe: the time at which we can observationally verify that we live in a very special time in the evolution of the universe! Observers when the universe was an order of magnitude younger would not have been able to discern any effects of dark energy on the expansion, and observers when the universe is more than an order of magnitude older will be hard pressed to know that they live in an expanding universe at all, or that the expansion is dominated by dark energy. By the time the longest lived main sequence stars are nearing the end of their lives, for all intents and purposes, the universe will appear static, and all evidence that now forms the basis of our current understanding of cosmology will have disappeared.
Note added in proof: After this paper was submitted we learned of a prescient 1987 paper [20] , written before the discovery of dark energy and other cosmological observables that are central to our analysis, which nevertheless raised the general question of whether there would be epochs in the Universe when observational cosmology, as we now understand it, would not be possible. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We demonstrate that as we extrapolate the current ΛCDM universe forward in time, all evidence of the Hubble expansion will disappear, so that observers in our \"island universe\" will be fundamentally incapable of determining the true nature of the universe, including the existence of the highly dominant vacuum energy, the existence of the CMB, and the primordial origin of light elements. With these pillars of the modern Big Bang gone, this epoch will mark the end of cosmology and the return of a static universe. In this sense, the coordinate system appropriate for future observers will perhaps fittingly resemble the static coordinate system in which the de Sitter universe was first presented."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "1 Shortly after Einstein's development of general relativity, the Dutch astronomer Willem de Sitter proposed a static model of the universe containing no matter, which he thought might be a reasonable approximation to our low density universe. One can define a coordinate system in which the de Sitter metric takes a static form by defining de Sitter spacetime with a cosmological constant Λ as a four dimensional hyperboloid S Λ : η AB ξ A ξ B = -R 2 , R 2 = 3Λ -1 embedded in a 5d Minkowski spacetime with ds 2 = η AB dξ A dξ B , and (η AB ) = diag(1, -1, -1, -1, -1), A, B = 0, • • • , 4. The static form of the de Sitter metric is then\n\nds 2 s = (1 -r 2 s /R 2 )dt 2 s - dr 2 s 1 -r 2 s /R 2 -r 2 s dΩ 2 ,\n\nwhich can be obtained by setting ξ 0 = (R 2 -r 2 s ) 1/2 sinh(t s /R), ξ 1 = r s sin θ cos ϕ, ξ 2 = r s sin θ sin ϕ, ξ 3 = r s cos θ, ξ 4 = (R 2 -r 2 s ) 1/2 cosh(t s /R). In this case the metric only corresponds to the section of de Sitter space within a cosmological horizon at R = rs.\n\nIn fact de Sitter's model wasn't globally static, but eternally expanding, as can be seen by a coordinate transformation which explicitly incorporates the time dependence of the scale factor R(t) = exp(Ht). While spatially flat, it actually incorporated Einstein's cosmological term, which is of course now understood to be equivalent to a vacuum energy density, leading to a redshift proportional to distance.\n\nThe de Sitter model languished for much of the last century, once the Hubble expansion had been discovered, and the cosmological term abandoned. However, all present observational evidence is consistent with a ΛCDM flat universe consisting of roughly 30% matter (both dark matter and baryonic matter) and 70% dark energy [1, 2, 3] , with the latter having a density that appears constant with time. All cosmological models with a non-zero cosmological constant will approach a de Sitter universe in the far future, and many of the implications of this fact have been explored in the literature [4, 5, 6, 7, 8, 9, 10, 11, 12, 13 ].\n\nHere we re-examine the practical significance of the ultimate de Sitter expansion and point out a new eschatological physical consequence: from the perspective of any observer within a bound gravitational system in the far future, the static version of de Sitter space outside of that system will eventually become the appropriate physical coordinate system. Put more succinctly, in a time comparable to the age of the longest lived stars, observers will not be able to perform any observation or experiment that infers either the existence of an expanding universe dominated by a cosmological constant, or that there was a hot Big Bang. Observers will be able to infer a finite age for their island universe, but beyond that cosmology will effectively be over. The static universe, with which cosmology at the turn of the last century began, will have returned with a vengeance.\n\nModern cosmology is built on integrating general relativity and three observational pillars: the observed Hubble expansion, detection of the cosmic microwave background radiation, and the determination of the abundance of elements produced in the early universe. We describe next in detail how these observables will disappear for an observer in the far future, and how this will be likely to affect the theoretical conclusions one might derive about the universe."
},
{
"section_type": "OTHER",
"section_title": "A. The disappearance of the Hubble Expansion",
"text": "The most basic component of modern cosmology is the expansion of the universe, firmly established by Hubble in 1929. Currently, galaxies and galaxy clusters are gravitationally bound and have dropped out of the Hubble flow, but structures on larger length scales are observed to obey the Hubble expansion law. Now consider what happens in the far future of the universe. Both analytic [7] and numerical [10] calculations indicate that the Local Group remains gravitationally bound in the face of the accelerated Hubble expansion. All more distant structures will be driven outside of the de Sitter event horizon in a timescale on the order of 100 billion years ( [4] , see also Refs. [8, 9] ). While objects will not be observed to cross the event horizon, light from them will be exponentially redshifted, so that within a time frame comparable to the longest lived main sequence stars all objects outside of our local cluster will truly become invisible [4] .\n\nSince the only remaining visible objects will in fact be gravitationally bound and decoupled from the underlying Hubble expansion, any local observer in the far future will see a single galaxy (the merger product of the Milky Way and Andromeda and other remnants of the Local Group) and will have no observational evidence of the Hubble expansion. Lacking such evidence, one may wonder whether such an observer will postulate the correct cosmological model. We would argue that in fact, such an observer will conclude the existence of a static \"island universe,\" precisely the standard model of the universe c. 1900. This will be true in spite of the fact that the dominant energy in this universe will not be due to matter, but due to dark energy, with ρ M /ρ Λ ∼ 10 -12 inside the horizon volume [9] . The irony, of course, is that the denizens of this static universe will have no idea of the existence of the dark energy, much less of its magnitude, since they will have no probes of the length scales over which Λ dominates gravitational dynamics. It appears that dark energy is undetectable not only in the limit where ρ Λ ≪ ρ M , but also when ρ Λ ≫ ρ M .\n\nEven if there were no direct evidence of the Hubble expansion, we might expect three other bits of evidence, two observational and one theoretical, to lead physicists in the future to ascertain the underlying nature of cosmology. However, we next describe how this is unlikely to be the case."
},
{
"section_type": "OTHER",
"section_title": "B. Vanishing CMB",
"text": "The existence of a Cosmic Microwave Background was the key observation that convinced most physicists and astronomers that there was in fact a hot big bang, which essentially implies a Hubble expansion today. But even if skeptical observers in the future were inclined to undertake a search for this afterglow of the Big Bang, they would come up emptyhanded. At t ≈ 100 Gyr, the peak wavelength of the cosmic microwave background will be redshifted to roughly λ ≈ 1 m, or a frequency of roughly 300 MHz. While a uniform radio background at this frequency would in principle be observable, the intensity of the CMB will also be redshifted by about 12 orders of magnitude. At much later times, the CMB becomes unobservable even in principle, as the peak wavelength is driven to a length larger than the horizon [4] . Well before then, however, the microwave background peak will redshift below the plasma frequency of the interstellar medium, and so will be screened from any observer within the galaxy. Recall that the plasma frequency is given by\n\nν p = n e e 2 πm e 1/2\n\n,\n\nwhere n e and m e are the electron number density and mass, respectively. Observations of dispersion in pulsar signals give [14] n e ≈ 0.03 cm -3 in the interstellar medium, which corresponds to a plasma frequency of ν p ≈ 1 kHz, or a wavelength of λ p ≈ 3 × 10 7 cm.\n\nThis corresponds to an expansion factor ∼ 10 8 relative to the present-day peak of the CMB.\n\nAssuming an exponential expansion, dominated by dark energy, this expansion factor will be reached when the universe is less than 50 times its present age, well below the lifetime of the longest-lived main sequence stars.\n\nAfter this time, even if future residents of our island universe set out to measure a universal radiation background, they would be unable to do so. The wealth of information about early universe cosmology that can be derived from fluctuations in the CMB would be even further out of reach."
},
{
"section_type": "OTHER",
"section_title": "C. General Relativity Gives No Assistance",
"text": "We may assume that theoretical physicists in the future will infer that gravitation is described by general relativity, using observations of planetary dynamics, and ground-based tests of such phenomena as gravitational time dilation. Will they then not be led to a Big Bang expansion, and a beginning in a Big Bang singularity, independent of data, as Lemaitre was? Indeed, is not a static universe incompatible with general relativity?\n\nThe answer is no. The inference that the universe must be expanding or contracting is dependent upon the cosmological hypothesis that we live in an isotropic and homogeneous universe. For future observers, this will manifestly not be the case. Outside of our local cluster, the universe will appear to be empty and static. Nothing is inconsistent with the temporary existence of a non-singular isolated self-gravitating object in such a universe, governed by general relativity. Physicists will infer that this system must ultimately collapse into a future singularity, but only as we presently conclude our galaxy must ultimately coalesce into a large black hole. Outside of this region, an empty static universe can prevail.\n\nWhile physicists in the island universe will therefore conclude that their island has a finite future, the question will naturally arise as to whether it had a finite beginning. As we next describe, observers will in fact be able to determine the age of their local cluster, but not the nature of the beginning."
},
{
"section_type": "OTHER",
"section_title": "D. Polluted Elemental Abundances",
"text": "The theory of Big Bang Nucleosynthesis reached a fully-developed state [15] only after the discovery of the CMB (despite early abortive attempts by Gamow and his collaborators [16] ). Thus, it is unlikely that the residents of the static universe would have any motivation to explore the possibility of primordial nucleosynthesis. However, even if they did, the evidence for BBN rests crucially on the fact that relic abundances of deuterium remain observable at the present day, while helium-4 has been enhanced by only a few percent since it was produced in the early universe. Extrapolating forward by 100 Gyr, we expect significantly more contamination of the helium-4 abundance, and concomitant destruction of the relic deuterium. It has been argued [17] that the ultimate extrapolation of light elemental abundances, following many generations of stellar evolution, is a mass fraction of helium given by Y = 0.6. The primordial helium mass fraction of Y = 0.25 will be a relatively small fraction of this abundance. It is unlikely that much deuterium could survive this degree of processing. Of course, the current \"smoking gun\" deuterium abundance is provided by Lyman-α absorption systems, back-lit by QSOs (see, e.g., Ref. [18] ). Such systems will be unavailable to our observers of the future, as both the QSOs and the Lyman-α systems will have redshifted outside of the horizon.\n\nAstute observers will be able to determine a lower limit on the age of their system, however, using standard stellar evolution analyses of their own local stars. They will be able to examine the locus of all stars and extrapolate to the oldest such stars to estimate a lower bound on the age of the galaxy. They will be able to determine an upper limit as well, by determining how long it would take for all of the observed helium to be generated by stellar nucleosynthesis. However, without any way to detect primordial elemental abundances, such as the aforementioned possibility of measuring deuterium in distant intergalactic clouds that currently absorb radiation from distant quasars and allow a determination of the deuterium abundance in these pre-stellar systems, and with the primordial helium abundance dwarfed by that produced in stars, inferring the original BBN abundances will be difficult, and probably not well motivated.\n\nThus, while physicists of the future will be able to infer that their island universe has not been eternal, it is unlikely that they will be able to infer that the beginning involved a Big\n\nBang."
},
{
"section_type": "CONCLUSION",
"section_title": "E. Conclusion",
"text": "The remarkable cosmic coincidence that we happen to live at the only time in the history of the universe when the magnitude of dark energy and dark matter densities are comparable has been a source of great current speculation, leading to a resurgence of interest in possible anthropic arguments limiting the value of the vacuum energy (see, e.g., Ref. [19] ). But this coincidence endows our current epoch with another special feature, namely that we can actually infer both the existence of the cosmological expansion, and the existence of dark energy. Thus, we live in a very special time in the evolution of the universe: the time at which we can observationally verify that we live in a very special time in the evolution of the universe! Observers when the universe was an order of magnitude younger would not have been able to discern any effects of dark energy on the expansion, and observers when the universe is more than an order of magnitude older will be hard pressed to know that they live in an expanding universe at all, or that the expansion is dominated by dark energy. By the time the longest lived main sequence stars are nearing the end of their lives, for all intents and purposes, the universe will appear static, and all evidence that now forms the basis of our current understanding of cosmology will have disappeared.\n\nNote added in proof: After this paper was submitted we learned of a prescient 1987 paper [20] , written before the discovery of dark energy and other cosmological observables that are central to our analysis, which nevertheless raised the general question of whether there would be epochs in the Universe when observational cosmology, as we now understand it, would not be possible."
}
] | |
arxiv:0704.0228 | 0704.0228 | 1 | 10.1209/0295-5075/78/10010 | 22a164d3e9d3ebb9f8267fe76e523e592d71abee6945767ca79ae21df8db21a5 | Einstein vs Maxwell: Is gravitation a curvature of space, a field in
flat space, or both? | Starting with a field theoretic approach in Minkowski space, the
gravitational energy momentum tensor is derived from the Einstein equations in
a straightforward manner. This allows to present them as {\it acceleration
tensor} = const. $\times$ {\it total energy momentum tensor}. For flat space
cosmology the gravitational energy is negative and cancels the material energy.
In the relativistic theory of gravitation a bimetric coupling between the
Riemann and Minkowski metrics breaks general coordinate invariance. The case of
a positive cosmological constant is considered. A singularity free version of
the Schwarzschild black hole is solved analytically. In the interior the
components of the metric tensor quickly die out, but do not change sign,
leaving the role of time as usual. For cosmology the $\Lambda$CDM model is
covered, while there appears a form of inflation at early times. Here both the
total energy and the zero point energy vanish. | [
"Theo M. Nieuwenhuizen"
] | [
"gr-qc",
"astro-ph",
"quant-ph"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | Starting with a field theoretic approach in Minkowski space, the gravitational energy momentum tensor is derived from the Einstein equations in a straightforward manner. This allows to present them as acceleration tensor = const. × total energy momentum tensor. For flat space cosmology the gravitational energy is negative and cancels the material energy. In the relativistic theory of gravitation a bimetric coupling between the Riemann and Minkowski metrics breaks general coordinate invariance. The case of a positive cosmological constant is considered. A singularity free version of the Schwarzschild black hole is solved analytically. In the interior the components of the metric tensor quickly die out, but do not change sign, leaving the role of time as usual. For cosmology the ΛCDM model is covered, while there appears a form of inflation at early times. Here both the total energy and the zero point energy vanish.
It is said that in introducing the general theory of relativity (GTR), Einstein made the step that Lorentz and Poincaré had failed to make: to go from flat space to curved space. Technically, this arises from the group of general coordinate transformations [1, 2] . One fundamental difficulty is then how to deal with the physics of gravitation itself, since there is only a quasi energy-momentum tensor [3] . For gravitational wave detection, e.g., this leaves open the question as to how energy can be faithfully transferred from the wave to the detector. The proper energy momentum tensor of gravitation was derived only recently by Babak and Grishchuk [4] , who start with a field theoretic approach to gravitation, in terms of a tensor field h µν in a Minkowski background space-time. The metric of the latter, η µν = diag(1, -1, -1, -1), is denoted in arbitrary coordinates by γ µν = (γ µν ) -1 . The Riemann metric tensor g µν = (g µν ) -1 , is then defined by
g γ g µν = γ µν + h µν ≡ k µν , g γ = det(g µν ) det(γ µν ) . (1)
It is just a way to code the gravitational field, allowing to expresses distances by ds 2 = g µν dx µ dx ν . Such a nonlinear way to code distances in a flat space is not uncommon. For diffuse light transport through clouds, one may express distances in the optical thickness, the number of extinction lengths. If the cloud is not homogeneous, points at the same physical distance are described by a different optical distance and, vice versa. The Maxwell view that gravitation is a field in flat space, was actually the starting point for Einstein, and reappeared regularly. Nathan Rosen [5] , coauthor of the Einstein-Podolsky-Rosen paper that led the basis for quantum information, considers a bimetric theory, involving the Minkowski metric and the Riemann metric. Bimetrism is quite natural, with η µν entering e.g. particle physics, and g µν e.g. cosmology. Rosen considers covariant derivatives D µ of Minkowski space, with Christoffel symbols γ λ
•µν vanishing in Cartesian coordinates. When replacing in the Riemann Christoffel symbols partial derivatives by Minkowski covariant ones,
Γ λ •µν = 1 2 g λσ (∂ µ g νσ + ∂ ν g µσ -∂ σ g µν ) → G λ •µν = 1 2 g λσ (D µ g νσ + D ν g µσ -D σ g µν ), (2)
the obtained Christoffel-type symbols G λ •µν are tensors in Minkowski space. Inspired by the Landau-Lifshitz and Babak-Grishchuk results, we may define the acceleration tensor
A µν = 1 2 D α D β (k µν k αβ -k µα k νβ ), (3)
where k µν = γ µν + h µν and in which the γγ terms do not contribute. Then we can calculate the combination
τ µν = c 4 8πG γ g A µν -(R µν - 1 2 g µν R) . (4)
In doing so, we make use of Rosen's observation that R µν remain unchanged if one replaces all partial derivatives by covariant ones in Minkowski space [5] . It appears that all second order derivatives drop out from (4), leaving a bilinear form in first order covariant derivatives,
τ µν = c 4 γ 8πG g 1 2 h µν • • :λ h λρ • • :ρ - 1 2 h µλ • • :λ h νρ • • :ρ + 1 2 h µλ:ρ h ν •λ:ρ + 1 4 k µν h λρ:σ h λσ:ρ - 1 2 h λρ:µ h ν •λ:ρ - 1 2 h µλ:ρ h • • :ν λρ + 1 4 h λρ:µ h • • :ν λρ - 1 8 h •λ:µ λ h •ρ:ν ρ - 1 8 k µν h λρ:σ h λρ:σ + 1 16 k µν h •ρ:λ ρ h σ •σ:λ . ( 5
)
in which X :µ ≡ D µ X and raising (lowering) of indices of h µν • • :ρ is performed with k µν (k µν ). τ µν is a tensor in Minkowski space. For Cartesian coordinates, it coincides with the Landau-Lifshitz quasi-tensor. In general, it coincides with the Babak-Grishchuk tensor γt µν Inclusion of matter is now much easier than in [4] . Inserting the Einstein equations in the right hand side of (4), we may write the Einstein equations in the Newton shape: acceleration=mass -1 ×force,
A µν = 8πG c 4 Θ µν , Θ µν = g γ θ µν , θ µν ≡ τ µν + T µν . ( 6
)
Θ µν is the total energy momentum tensor of gravitation and matter. It is conserved, D ν Θ µν = 0, since Eq. (3) implies D ν A µν = 0, because covariant Minkowski derivatives commute.
As an application, let us consider cosmology, described by the Friedman-Lemaitre-Robertson-Walker (FLRW) metric,
ds 2 = U (t)c 2 dt 2 -V (t) dr 2 1 -kr 2 + r 2 dΩ 2 , ( 7
)
dΩ 2 = dθ 2 + sin 2 θdφ 2 .
Let us consider flat space, k = 0, and U = 1, V (t) = a 2 (t) with a the scale factor. Then ds 2 = c 2 dt 2 -a 2 (t)dr 2 is space-independent, implying that A 00 = 0, due to the shape (3). According to (6) it then follows that the total energy density is zero, because the gravitational energy density, τ 00 = -3c 4 ȧ2 /(8πGa 2 ), is negative and cancels the one of matter, T 00 = ρ, due to the Friedman equation. In other words, such a universe contains no overall energy.
So far we have discussed an alternative, field theoretic formulation of GTR. If we consider a local energy momentum density as a sine qua non property, then we are led to consider Minkowski space as a fixed "pre-space", that exist already without matter, just as a region of space ahead of the earth's orbit is right now almost empty (Minkowskian), and when the earth arrives, there will be more gravitational and matter fields, but, in our view, no change of space. Also for cosmology there is a different interpretation. In GTR coordinates are fixed to clusters of galaxies, this is called "coordinate space", but due to the increasing scale factor galaxies are said to move away from each other: physical space (i.e. Riemann space) is said to expand. Here we are led to another view: Coordinate space is physical space, so clusters of galaxies do not move away from each other in time. [6] However, the cosmic speed of light dr/dt = c/a(t), which was very large at early times, keeps on decreasing, thus causing a redshift, till a is infinite, when galaxies are invisible.
Relativistic Theory of Gravitation, RTG. Let us move on to an extension of GTR, giving up general coordinate invariance. Discarding a total derivative of the Hilbert-Einstein action, Rosen expresses the gravitational action
S R = d 3 xdt √ -g L R in terms of [5] L R = c 4 g µν 16πG (G λ •µν G σ •λσ -G λ •µσ G σ •νλ ) = c 4 γ/g 128πG (8)
× (2h µν:ρ h µν:ρ -4h µν:ρ h µρ:ν -h ν •ν:µ h •ρ:µ ρ ).
Involving only Minkowski covariant first order derivatives, it is close to general approaches in field theory. Logunov and coworkers continue on this [6] . The subgroup of gauge transformations that transform h µν but leave coordinates invariant, allows three extra terms [6] ,
L g = L R -ρ Λ + 1 2 ρ bi γ µν g µν -ρ 0 γ/g. (9)
Here ρ Λ is the familiar energy related to a cosmological constant. The ρ 0 term describes a harmless shift of the zero level of energy, δS = -d 3 xdt √ -γρ 0 . The bimetric term ρ bi couples the Minkowski and the Riemann metrics. It acts like a mass term, because it breaks general coordinate invariance, and has some analogy to a mass term in massive electrodynamics. Logunov then imposes the relation
ρ Λ = ρ bi = ρ 0 , (10)
which, in the absence of matter, keeps space flat, h µν = 0, g µν = γ µν and also L g = 0. Thus one free parameter remains. Logunov's choice ρ bi ≡ -m 2 c 4 /(16πG) < 0 leads to an inverse length m and, in quantum language, a graviton mass hm/c. The negative cosmological constant can be counteracted by an inflaton field [7] . The obtained theory has some drawbacks, such as self-repulsive properties for matter falling onto a black hole, and a minimal and a maximal size of the scale factor in cosmology [6] [7] . For a related approach to finite range gravity, based on a generalized Fierz-Pauli coupling, see [8] .
We shall focus on the opposite choice, a positive cosmological constant Λ, [9] [10]
ρ Λ ≡ Λc 4 8πG = 3c 2 8πG Ω v,0 H 2 0 = 3c 2 8πG 0.74 0.71 9.78Gyr 2 , ρ bi ≡ Λ bi c 4 8πG = ρ Λ . (11)
Now the graviton has an "imaginary mass", m = h√ -2Λ bi /c, it is a "tachyon": Gravitational waves are unstable at today's Hubble scale. But this is of no concern, since on that scale, not single gravitational waves but the whole Universe matters, being unstable (expanding) anyhow.
Though we take ρ bi = ρ Λ , Λ bi = Λ, our further notation is valid for the general case
ρ bi = ρ Λ , Λ bi = Λ.
The Einstein equations that couple the Riemann metric to matter read
R µν - 1 2 g µν R = 8πG c T µν tot , (12)
T µν tot = T µν + ρ Λ g µν + ρ bi γ ρσ (g µρ g σν - 1 2 g µν g ρσ ).
Conservation of energy momentum, T µν tot;ν = 0, imposes a constraint due to the ρ bi terms, [6]
D ν g γ g µν = 0, or D ν h µν = 0, (13)
which for Cartesian coordinates coincides with the GTR harmonic condition ∂ ν ( √ -gg µν ) = 0 [2] . Thus the theory automatically demands the harmonic constraint for g µν , or, equivalently, the Lorentz gauge for h µν , thereby severely reducing the gauge invariance of GTR.
Changes of Einstein's GTR have mostly met deep troubles with one or another established property, though not all proposals are ruled out [1, 11] . The present one is rather subtle and promising. For most applications, the Hubblesize ρ Λ = ρ bi terms in Eq. (11, 12) are too small to be relevant, so known results from general relativity can be reproduced. Indeed, viewed from a GTR standpoint, Eq. ( 13 ) is only a particular gauge, and actually often considered, while the cosmological constant only plays a role in cosmology. Logunov checked a number of effects in the solar system: deflection of light rays by the sun, the delay of a radio signal, the shift of Mercury's perihelion, the precession of a gyroscope, and the gravitational shift of spectral lines. [6] Likewise, we expect agreement for binary pulsars. [11] Differences between GTR and RTG may arise, though, for large gravitational fields, that we consider now.
Black holes. It is known that true black holes, objects that have a horizon, do not occur in the RTG with ρ Λ , ρ bi → 0. [5] But there are solutions very similar to it, that might be named "grey holes", but we just call them "black holes". The Minkowski line element in spherical coordinates is simply
γ µν dx µ dx ν = c 2 dt 2 -dr 2 -r 2 dΩ 2 .
The one of Riemann space is
ds 2 = g µν dx µ dx ν = U (r)c 2 dt 2 -V (r)dr 2 -W 2 (r)dΩ 2 . (14)
In harmonic coordinates, the Schwarzschild black hole is described by [2]
U s = 1 V s = r -r h r + r h , W s = r + r h , r h = GM c 2 . ( 15
)
The horizon radius r h equals half the Schwarzschild radius.
Let us scale r → rr h , and define
U = e u , V = e v , W = 2r h e w , (16)
so that w is small near the horizon. The dimensionless small parameter arising from
ρ bi = ρ Λ , is very small, λ ≡ r h √ 2Λ = 2.38 10 -23 M M ⊙ . μ ≡ r h 2Λ bi = λ. ( 17
)
The sum and difference of the (t, t) and (r, r) Einstein equations give
1 2 e v-2w -w ′ (u ′ -v ′ + 4w ′ ) -2w ′′ = e v ( λ2 - 1 4 μ2 r 2 e -2w ) + 8πGr 2 h c 4 e v (ρ -p), (18) w ′ (u ′ + v ′ -2w ′ ) -2w ′′ = 1 2 μ2 (e v-u -1) + 8πGr 2 h c 4 e v (ρ + p), (19)
respectively. The harmonic condition imposes
u ′ -v ′ + 4w ′ = r exp(v -2w).
In the Schwarzschild black hole of GTR, there is no matter outside the origin. We shall focus on that situation. A parametric solution of these equations then reads
r = 1 + η(e ξ + ξ + log η + r 0 ) 1 -η(e ξ + ξ + log η + r 0 ) , (20)
u = ξ + log η, v = ξ -ln η -2 log(e ξ + 1), (21)
w = ηe ξ + μ2 (ξ + log η + w 0 ).
where ξ is the running variable and η is a small scale. Corrections of next order in η can be expressed in dilogarithms, but they are not needed since μ is very small. To fix the scale η, we note that energy momentum conservation implies, as in GTR, (ρ + p)u ′ + 2p ′ = 0. In the stationary state all matter is located at the origin, which is only possible if p(r) ≡ 0, implying ρ(r)u ′ (r) = 0. This is obeyed for r = 0 since ρ = 0 there, but since ρ(0) > 0 (it is infinite), we have to demand u ′ (0) = 0. Let us define a factor α by α = μ2 /η. The above solution brings w ′ (r) = ∂ ξ w/∂ ξ r = (e ξ + α)/[2(e ξ + 1)], so in the interior w ′ = 1 2 α. Since e v ≪ 1 there, Eqs. (18,19) confirm that w ′′ = 0, and with w(1) = O(η) this solves w(r) = 1 2 α (r -1). Moreover, from the harmonic constraint (20) we have in the interior u(r) -v(r) + 4w(r) = const = 2 ln η, implying that Eq. ( 19 ) yields in the interior
u ′ (r) = {exp[2α(r -1)] -η 2 -μ2 }/(2η). From u ′ (0) = 0 we can now solve α, α = log 1 μ , η = μ2 ln 1/μ . ( 22
)
As seen in fig. 1 , our solution (16,20,-22) coincides with Schwarzschild's for ξ ≫ 1. In the regime ξ = O(1), there is a transition towards the interior ξ ≪ -1, where exponential corrections can be neglected. Both U = ηe ξ and V = e ξ /η are very small there, but, contrary to the Schwarzschild case, they remain positive: The behavior in the interior of the RTG black hole is not qualitatively different from usual, be it that the gravitational field is large.
Width of the brick wall. The transition layer ξ = O(1) acts like 't Hooft's brick wall, [12] of characteristic width ℓ ⋆ = ηr h . Comparing to the Planck length ℓ P = hG/c 3 , we get
ℓ * ℓ P = 0.977 10 -9 1 + 0.019 log(M/M ⊙ ) M 3 M 3 ⊙ . ( 23
)
If quantum physics sets in at the Planck scale, our approach makes sense only for M > 10 3 M ⊙ . Motion of test particles. For RTG with a negative cosmological constant, [6] it was claimed that an incoming spherical shell of matter is scattered off from a black hole, a counter-intuitive finding. Let us reconsider this issue. The motion of a test body occurs along a geodesic
dv µ ds + Γ µ νρ v ν v ρ = 0, v µ = dx µ ds . (24)
For spherical shells of in-falling matter one needs Γ 0 01 = U ′ /(2U ). This brings dt/ds = v 0 = 1/(C i U ), for some C i . Solving v 1 = dr/ds from g µν v µ v ν = 1, we then get dr/dt = (ds/dt)(dr/ds) = -c U (1 -C 2 i U )/V . We can now fix C i at the initial position r = r i , where the spherical shell is assumed to have a speed dr i /dt
= v i = β i c U i /V i , viz. C i = (1 -β 2 i )/U i , with |β i | ≤ 1.
The differential proper time dτ = √ U dt and length dℓ = √ V dr bring in the particle's rest frame dℓ/dτ = V /U dr/dt, yielding
dℓ dτ = -c 1 - U (r(τ )) U (r i ) (1 -β 2 i ). (25)
The extreme case is when
β i = 0 at r i = ∞, dℓ/dτ = -c √ 1 -U .
To have |dℓ/dτ | < c, it thus suffices that 0 < U ≤ 1, which is the case. Near the horizon, |dℓ/dτ | is almost equal to c and the more the shell penetrates the interior, the closer its speed gets to c. For an outside observer, the time to see it hit the center of the hole, T = dr/| ṙ| equals (r h /c)
1 0 dr exp[ 1 2 (v -u)].
It is finite and predominantly comes from the horizon, T = r h /cμ 2 = 2.74 × 10 32 M/M ⊙ yr.
The approaches [6] [7] [8] have a similar a black hole. While [8] properly has U ′ (0) = 0, in Logunov's case one has μ2 < 0, so w ′ = 1 2 α < 0 in the interior. This seems to solve the paradox of "matter reflected by the black hole": In-falling matter just enters, but the Logunov coordinate x = exp(w) -1 is non-monotonic (x ′ < 0 in the interior). However, the situation is more severe: For α < 0, the theory does not allow a solution with u ′ (0) = 0, depriving that theory of a proper black hole. This condition can neither be obeyed in GTR: If the central mass is slightly smeared, the Schwarzschild black hole cannot obey energymomentum conservation in GTR.
Cosmology. Starting from the FLRW metric, the harmonic condition brings two relations: U ∼ V 3 and k = 0: Minkowski space filled homogeneously with matter remains flat [6] . We may thus put U = a 6 (t)/a 4 * , V = a 2 (t). Going from cosmic time t to conformal time τ = a 3 a -2 * dt yields the familiar Einstein equations, extended by Λ bi terms, ȧ2
a 2 c 2 = 8πG 3c 4 ρ + Λ 3 - Λ bi 2a 2 + Λ bi a 4 * 6a 6 , (26)
ä a c 2 = - 4πG 3c 4 (ρ + 3p) + Λ 3 - Λ bi 3 a 4 * a 6 .
The first is the modified Friedman equation, the second corresponds to the first law d(ρ tot a 3 ) = -p tot da 3 provided we define ρ tot = ρ+ρ Λ +ρ 2 +ρ 6 and p tot = p-ρ Λ -1 3 ρ 2 +ρ 6 , with ρ 2 = -3ρ bi /2a 2 and ρ 6 = ρ bi a 4 * /2a 6 . Note that ρ 2 acts as a positive curvature term.
The scale factor has an absolute meaning. If we assume that a ≫ 1 and a ≫ a 2/3 * , Eq. (26) just coincides with the ΛCDM model (cosmological constant plus cold dark matter), that gives the best fit of the observations [9] [10] . The ρ 2 term allows a positive curvaturetype contribution. At large times, there is the exponential growth a(τ
) = C exp(H ∞ τ ) with H ∞ = c Λ/3. In cos- mic time this reads a(t) = a 2/3 * [3H ∞ (t 0 -t)] -1/3
, where t 0 is "the end of time", the moment where the scale factor has become infinite. The minimal scale factor is zero: in this theory a big bang can occur since ρ bi > 0. Without including an inflaton field, Eq. ( 26 ) yields an initial growth of the expansion a = (a 2 * c τ 3Λ bi /2) 1/3 . In cosmic time this reads a = a 1 exp(ct Λ bi /6), i. e., a certain inflation scenario starting at t = -∞.
Also in RTG the gravitational energy precisely compensates the other energy contributions at all times. The vacuum energy also vanishes: In empty space, the cosmological constant energy ρ Λ cancels the ρ bi terms, due to Eq. (10) . See Eq. ( 12 ) for g µν = γ µν .
In conclusion, we have first written the Einstein equation in a form that involves the gravitational energy momentum tensor. An underlying Minkowski space is needed, in which gravitation is a field. The metric tensor is a way to deal with it, but the equations for the field itself exist too, see Eq. ( 6 ). For flat cosmology it follows that the total energy vanishes.
Next we have broken general coordinate invariance by going to the bimetric theory of Logunov, called Relativistic Theory of Gravitation. We have shown that the choice of a positive bimetric constant allows to regularize the interior of the Schwarzschild black hole: time keeps its standard role and escape is, in principle, possible. While neither the Schwarzschild nor the Logunov black hole survives smearing of the central mass by a tiny pressure in the equation of state, ours does. Our modification of the Einstein equations involves the cosmological constant, so it is of Hubble size, immaterial for solar problems. In cosmology, the theory directly leads to the ΛCDM model, while it could accommodate a positive curvature-like term. At short times, there is a form of inflation. The gravitational energy exactly compensates the material energy. The zero point energy vanishes ("again"), though the cosmological constant is finite and positive: It is canceled by the bimetric terms.
Euclidean space, a special case of Riemann geometry, seems to be invoked by Nature, at least far away from bodies and in cosmology. Our approach supports the following space-time interpretation: curvature is a geometric description of the gravitational field in flat space. Clusters of galaxies do not move away from each other, but the speed of light changes with cosmic time, dr/dt = [a(t)/a * ] 2 c, while the conformal speed is dr/dτ = c/a(τ ) as usual.
An empirical way to establish the Minkowski metric is to present the Einstein equations as (c 4 /8πG)R µν -T µν + 1 2 g µν T + ρ Λ g µν = ρ bi γ µν , and to measure the left hand side, which in the geometric view is considered to consist of curved space properties alone. [6] As in the standard model of elementary particles, the separation of curved space into flat space and the gravitational field has the following implication: the quantum version of RTG -if it exists -will involve quantization of fields, but not of space.
Finally we answer the question posed in the title. The field theoretic approach to gravitation is by itself equivalent to a curved space description, so both views apply, describing the same physics from a different angle. But when the theory is extended to the relativistic theory of gravitation, the bimetrism forces to describe the Minkowski metric separately, and then we see it as most natural to view gravitation as a field in flat space, which is Maxwell's view.
Topics such as a realistic equation of state for black holes and classical tunneling of its radiation, regularization of other singularities, as well as aspects of the inflation and of inhomogeneous cosmology are under study.
Discussion with Martin Nieuwenhuizen and Armen Allahverdyan is gratefully remembered. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Starting with a field theoretic approach in Minkowski space, the gravitational energy momentum tensor is derived from the Einstein equations in a straightforward manner. This allows to present them as acceleration tensor = const. × total energy momentum tensor. For flat space cosmology the gravitational energy is negative and cancels the material energy. In the relativistic theory of gravitation a bimetric coupling between the Riemann and Minkowski metrics breaks general coordinate invariance. The case of a positive cosmological constant is considered. A singularity free version of the Schwarzschild black hole is solved analytically. In the interior the components of the metric tensor quickly die out, but do not change sign, leaving the role of time as usual. For cosmology the ΛCDM model is covered, while there appears a form of inflation at early times. Here both the total energy and the zero point energy vanish."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "It is said that in introducing the general theory of relativity (GTR), Einstein made the step that Lorentz and Poincaré had failed to make: to go from flat space to curved space. Technically, this arises from the group of general coordinate transformations [1, 2] . One fundamental difficulty is then how to deal with the physics of gravitation itself, since there is only a quasi energy-momentum tensor [3] . For gravitational wave detection, e.g., this leaves open the question as to how energy can be faithfully transferred from the wave to the detector. The proper energy momentum tensor of gravitation was derived only recently by Babak and Grishchuk [4] , who start with a field theoretic approach to gravitation, in terms of a tensor field h µν in a Minkowski background space-time. The metric of the latter, η µν = diag(1, -1, -1, -1), is denoted in arbitrary coordinates by γ µν = (γ µν ) -1 . The Riemann metric tensor g µν = (g µν ) -1 , is then defined by\n\ng γ g µν = γ µν + h µν ≡ k µν , g γ = det(g µν ) det(γ µν ) . (1)\n\nIt is just a way to code the gravitational field, allowing to expresses distances by ds 2 = g µν dx µ dx ν . Such a nonlinear way to code distances in a flat space is not uncommon. For diffuse light transport through clouds, one may express distances in the optical thickness, the number of extinction lengths. If the cloud is not homogeneous, points at the same physical distance are described by a different optical distance and, vice versa. The Maxwell view that gravitation is a field in flat space, was actually the starting point for Einstein, and reappeared regularly. Nathan Rosen [5] , coauthor of the Einstein-Podolsky-Rosen paper that led the basis for quantum information, considers a bimetric theory, involving the Minkowski metric and the Riemann metric. Bimetrism is quite natural, with η µν entering e.g. particle physics, and g µν e.g. cosmology. Rosen considers covariant derivatives D µ of Minkowski space, with Christoffel symbols γ λ\n\n•µν vanishing in Cartesian coordinates. When replacing in the Riemann Christoffel symbols partial derivatives by Minkowski covariant ones,\n\nΓ λ •µν = 1 2 g λσ (∂ µ g νσ + ∂ ν g µσ -∂ σ g µν ) → G λ •µν = 1 2 g λσ (D µ g νσ + D ν g µσ -D σ g µν ), (2)\n\nthe obtained Christoffel-type symbols G λ •µν are tensors in Minkowski space. Inspired by the Landau-Lifshitz and Babak-Grishchuk results, we may define the acceleration tensor\n\nA µν = 1 2 D α D β (k µν k αβ -k µα k νβ ), (3)\n\nwhere k µν = γ µν + h µν and in which the γγ terms do not contribute. Then we can calculate the combination\n\nτ µν = c 4 8πG γ g A µν -(R µν - 1 2 g µν R) . (4)\n\nIn doing so, we make use of Rosen's observation that R µν remain unchanged if one replaces all partial derivatives by covariant ones in Minkowski space [5] . It appears that all second order derivatives drop out from (4), leaving a bilinear form in first order covariant derivatives,\n\nτ µν = c 4 γ 8πG g 1 2 h µν • • :λ h λρ • • :ρ - 1 2 h µλ • • :λ h νρ • • :ρ + 1 2 h µλ:ρ h ν •λ:ρ + 1 4 k µν h λρ:σ h λσ:ρ - 1 2 h λρ:µ h ν •λ:ρ - 1 2 h µλ:ρ h • • :ν λρ + 1 4 h λρ:µ h • • :ν λρ - 1 8 h •λ:µ λ h •ρ:ν ρ - 1 8 k µν h λρ:σ h λρ:σ + 1 16 k µν h •ρ:λ ρ h σ •σ:λ . ( 5\n\n)\n\nin which X :µ ≡ D µ X and raising (lowering) of indices of h µν • • :ρ is performed with k µν (k µν ). τ µν is a tensor in Minkowski space. For Cartesian coordinates, it coincides with the Landau-Lifshitz quasi-tensor. In general, it coincides with the Babak-Grishchuk tensor γt µν Inclusion of matter is now much easier than in [4] . Inserting the Einstein equations in the right hand side of (4), we may write the Einstein equations in the Newton shape: acceleration=mass -1 ×force,\n\nA µν = 8πG c 4 Θ µν , Θ µν = g γ θ µν , θ µν ≡ τ µν + T µν . ( 6\n\n)\n\nΘ µν is the total energy momentum tensor of gravitation and matter. It is conserved, D ν Θ µν = 0, since Eq. (3) implies D ν A µν = 0, because covariant Minkowski derivatives commute.\n\nAs an application, let us consider cosmology, described by the Friedman-Lemaitre-Robertson-Walker (FLRW) metric,\n\nds 2 = U (t)c 2 dt 2 -V (t) dr 2 1 -kr 2 + r 2 dΩ 2 , ( 7\n\n)\n\ndΩ 2 = dθ 2 + sin 2 θdφ 2 .\n\nLet us consider flat space, k = 0, and U = 1, V (t) = a 2 (t) with a the scale factor. Then ds 2 = c 2 dt 2 -a 2 (t)dr 2 is space-independent, implying that A 00 = 0, due to the shape (3). According to (6) it then follows that the total energy density is zero, because the gravitational energy density, τ 00 = -3c 4 ȧ2 /(8πGa 2 ), is negative and cancels the one of matter, T 00 = ρ, due to the Friedman equation. In other words, such a universe contains no overall energy.\n\nSo far we have discussed an alternative, field theoretic formulation of GTR. If we consider a local energy momentum density as a sine qua non property, then we are led to consider Minkowski space as a fixed \"pre-space\", that exist already without matter, just as a region of space ahead of the earth's orbit is right now almost empty (Minkowskian), and when the earth arrives, there will be more gravitational and matter fields, but, in our view, no change of space. Also for cosmology there is a different interpretation. In GTR coordinates are fixed to clusters of galaxies, this is called \"coordinate space\", but due to the increasing scale factor galaxies are said to move away from each other: physical space (i.e. Riemann space) is said to expand. Here we are led to another view: Coordinate space is physical space, so clusters of galaxies do not move away from each other in time. [6] However, the cosmic speed of light dr/dt = c/a(t), which was very large at early times, keeps on decreasing, thus causing a redshift, till a is infinite, when galaxies are invisible.\n\nRelativistic Theory of Gravitation, RTG. Let us move on to an extension of GTR, giving up general coordinate invariance. Discarding a total derivative of the Hilbert-Einstein action, Rosen expresses the gravitational action\n\nS R = d 3 xdt √ -g L R in terms of [5] L R = c 4 g µν 16πG (G λ •µν G σ •λσ -G λ •µσ G σ •νλ ) = c 4 γ/g 128πG (8)\n\n× (2h µν:ρ h µν:ρ -4h µν:ρ h µρ:ν -h ν •ν:µ h •ρ:µ ρ ).\n\nInvolving only Minkowski covariant first order derivatives, it is close to general approaches in field theory. Logunov and coworkers continue on this [6] . The subgroup of gauge transformations that transform h µν but leave coordinates invariant, allows three extra terms [6] ,\n\nL g = L R -ρ Λ + 1 2 ρ bi γ µν g µν -ρ 0 γ/g. (9)\n\nHere ρ Λ is the familiar energy related to a cosmological constant. The ρ 0 term describes a harmless shift of the zero level of energy, δS = -d 3 xdt √ -γρ 0 . The bimetric term ρ bi couples the Minkowski and the Riemann metrics. It acts like a mass term, because it breaks general coordinate invariance, and has some analogy to a mass term in massive electrodynamics. Logunov then imposes the relation\n\nρ Λ = ρ bi = ρ 0 , (10)\n\nwhich, in the absence of matter, keeps space flat, h µν = 0, g µν = γ µν and also L g = 0. Thus one free parameter remains. Logunov's choice ρ bi ≡ -m 2 c 4 /(16πG) < 0 leads to an inverse length m and, in quantum language, a graviton mass hm/c. The negative cosmological constant can be counteracted by an inflaton field [7] . The obtained theory has some drawbacks, such as self-repulsive properties for matter falling onto a black hole, and a minimal and a maximal size of the scale factor in cosmology [6] [7] . For a related approach to finite range gravity, based on a generalized Fierz-Pauli coupling, see [8] .\n\nWe shall focus on the opposite choice, a positive cosmological constant Λ, [9] [10]\n\nρ Λ ≡ Λc 4 8πG = 3c 2 8πG Ω v,0 H 2 0 = 3c 2 8πG 0.74 0.71 9.78Gyr 2 , ρ bi ≡ Λ bi c 4 8πG = ρ Λ . (11)\n\nNow the graviton has an \"imaginary mass\", m = h√ -2Λ bi /c, it is a \"tachyon\": Gravitational waves are unstable at today's Hubble scale. But this is of no concern, since on that scale, not single gravitational waves but the whole Universe matters, being unstable (expanding) anyhow.\n\nThough we take ρ bi = ρ Λ , Λ bi = Λ, our further notation is valid for the general case\n\nρ bi = ρ Λ , Λ bi = Λ.\n\nThe Einstein equations that couple the Riemann metric to matter read\n\nR µν - 1 2 g µν R = 8πG c T µν tot , (12)\n\nT µν tot = T µν + ρ Λ g µν + ρ bi γ ρσ (g µρ g σν - 1 2 g µν g ρσ ).\n\nConservation of energy momentum, T µν tot;ν = 0, imposes a constraint due to the ρ bi terms, [6]\n\nD ν g γ g µν = 0, or D ν h µν = 0, (13)\n\nwhich for Cartesian coordinates coincides with the GTR harmonic condition ∂ ν ( √ -gg µν ) = 0 [2] . Thus the theory automatically demands the harmonic constraint for g µν , or, equivalently, the Lorentz gauge for h µν , thereby severely reducing the gauge invariance of GTR.\n\nChanges of Einstein's GTR have mostly met deep troubles with one or another established property, though not all proposals are ruled out [1, 11] . The present one is rather subtle and promising. For most applications, the Hubblesize ρ Λ = ρ bi terms in Eq. (11, 12) are too small to be relevant, so known results from general relativity can be reproduced. Indeed, viewed from a GTR standpoint, Eq. ( 13 ) is only a particular gauge, and actually often considered, while the cosmological constant only plays a role in cosmology. Logunov checked a number of effects in the solar system: deflection of light rays by the sun, the delay of a radio signal, the shift of Mercury's perihelion, the precession of a gyroscope, and the gravitational shift of spectral lines. [6] Likewise, we expect agreement for binary pulsars. [11] Differences between GTR and RTG may arise, though, for large gravitational fields, that we consider now.\n\nBlack holes. It is known that true black holes, objects that have a horizon, do not occur in the RTG with ρ Λ , ρ bi → 0. [5] But there are solutions very similar to it, that might be named \"grey holes\", but we just call them \"black holes\". The Minkowski line element in spherical coordinates is simply\n\nγ µν dx µ dx ν = c 2 dt 2 -dr 2 -r 2 dΩ 2 .\n\nThe one of Riemann space is\n\nds 2 = g µν dx µ dx ν = U (r)c 2 dt 2 -V (r)dr 2 -W 2 (r)dΩ 2 . (14)\n\nIn harmonic coordinates, the Schwarzschild black hole is described by [2]\n\nU s = 1 V s = r -r h r + r h , W s = r + r h , r h = GM c 2 . ( 15\n\n)\n\nThe horizon radius r h equals half the Schwarzschild radius.\n\nLet us scale r → rr h , and define\n\nU = e u , V = e v , W = 2r h e w , (16)\n\nso that w is small near the horizon. The dimensionless small parameter arising from\n\nρ bi = ρ Λ , is very small, λ ≡ r h √ 2Λ = 2.38 10 -23 M M ⊙ . μ ≡ r h 2Λ bi = λ. ( 17\n\n)\n\nThe sum and difference of the (t, t) and (r, r) Einstein equations give\n\n1 2 e v-2w -w ′ (u ′ -v ′ + 4w ′ ) -2w ′′ = e v ( λ2 - 1 4 μ2 r 2 e -2w ) + 8πGr 2 h c 4 e v (ρ -p), (18) w ′ (u ′ + v ′ -2w ′ ) -2w ′′ = 1 2 μ2 (e v-u -1) + 8πGr 2 h c 4 e v (ρ + p), (19)\n\nrespectively. The harmonic condition imposes\n\nu ′ -v ′ + 4w ′ = r exp(v -2w).\n\nIn the Schwarzschild black hole of GTR, there is no matter outside the origin. We shall focus on that situation. A parametric solution of these equations then reads\n\nr = 1 + η(e ξ + ξ + log η + r 0 ) 1 -η(e ξ + ξ + log η + r 0 ) , (20)\n\nu = ξ + log η, v = ξ -ln η -2 log(e ξ + 1), (21)\n\nw = ηe ξ + μ2 (ξ + log η + w 0 ).\n\nwhere ξ is the running variable and η is a small scale. Corrections of next order in η can be expressed in dilogarithms, but they are not needed since μ is very small. To fix the scale η, we note that energy momentum conservation implies, as in GTR, (ρ + p)u ′ + 2p ′ = 0. In the stationary state all matter is located at the origin, which is only possible if p(r) ≡ 0, implying ρ(r)u ′ (r) = 0. This is obeyed for r = 0 since ρ = 0 there, but since ρ(0) > 0 (it is infinite), we have to demand u ′ (0) = 0. Let us define a factor α by α = μ2 /η. The above solution brings w ′ (r) = ∂ ξ w/∂ ξ r = (e ξ + α)/[2(e ξ + 1)], so in the interior w ′ = 1 2 α. Since e v ≪ 1 there, Eqs. (18,19) confirm that w ′′ = 0, and with w(1) = O(η) this solves w(r) = 1 2 α (r -1). Moreover, from the harmonic constraint (20) we have in the interior u(r) -v(r) + 4w(r) = const = 2 ln η, implying that Eq. ( 19 ) yields in the interior\n\nu ′ (r) = {exp[2α(r -1)] -η 2 -μ2 }/(2η). From u ′ (0) = 0 we can now solve α, α = log 1 μ , η = μ2 ln 1/μ . ( 22\n\n)\n\nAs seen in fig. 1 , our solution (16,20,-22) coincides with Schwarzschild's for ξ ≫ 1. In the regime ξ = O(1), there is a transition towards the interior ξ ≪ -1, where exponential corrections can be neglected. Both U = ηe ξ and V = e ξ /η are very small there, but, contrary to the Schwarzschild case, they remain positive: The behavior in the interior of the RTG black hole is not qualitatively different from usual, be it that the gravitational field is large.\n\nWidth of the brick wall. The transition layer ξ = O(1) acts like 't Hooft's brick wall, [12] of characteristic width ℓ ⋆ = ηr h . Comparing to the Planck length ℓ P = hG/c 3 , we get\n\nℓ * ℓ P = 0.977 10 -9 1 + 0.019 log(M/M ⊙ ) M 3 M 3 ⊙ . ( 23\n\n)\n\nIf quantum physics sets in at the Planck scale, our approach makes sense only for M > 10 3 M ⊙ . Motion of test particles. For RTG with a negative cosmological constant, [6] it was claimed that an incoming spherical shell of matter is scattered off from a black hole, a counter-intuitive finding. Let us reconsider this issue. The motion of a test body occurs along a geodesic\n\ndv µ ds + Γ µ νρ v ν v ρ = 0, v µ = dx µ ds . (24)\n\nFor spherical shells of in-falling matter one needs Γ 0 01 = U ′ /(2U ). This brings dt/ds = v 0 = 1/(C i U ), for some C i . Solving v 1 = dr/ds from g µν v µ v ν = 1, we then get dr/dt = (ds/dt)(dr/ds) = -c U (1 -C 2 i U )/V . We can now fix C i at the initial position r = r i , where the spherical shell is assumed to have a speed dr i /dt\n\n= v i = β i c U i /V i , viz. C i = (1 -β 2 i )/U i , with |β i | ≤ 1.\n\nThe differential proper time dτ = √ U dt and length dℓ = √ V dr bring in the particle's rest frame dℓ/dτ = V /U dr/dt, yielding\n\ndℓ dτ = -c 1 - U (r(τ )) U (r i ) (1 -β 2 i ). (25)\n\nThe extreme case is when\n\nβ i = 0 at r i = ∞, dℓ/dτ = -c √ 1 -U .\n\nTo have |dℓ/dτ | < c, it thus suffices that 0 < U ≤ 1, which is the case. Near the horizon, |dℓ/dτ | is almost equal to c and the more the shell penetrates the interior, the closer its speed gets to c. For an outside observer, the time to see it hit the center of the hole, T = dr/| ṙ| equals (r h /c)\n\n1 0 dr exp[ 1 2 (v -u)].\n\nIt is finite and predominantly comes from the horizon, T = r h /cμ 2 = 2.74 × 10 32 M/M ⊙ yr.\n\nThe approaches [6] [7] [8] have a similar a black hole. While [8] properly has U ′ (0) = 0, in Logunov's case one has μ2 < 0, so w ′ = 1 2 α < 0 in the interior. This seems to solve the paradox of \"matter reflected by the black hole\": In-falling matter just enters, but the Logunov coordinate x = exp(w) -1 is non-monotonic (x ′ < 0 in the interior). However, the situation is more severe: For α < 0, the theory does not allow a solution with u ′ (0) = 0, depriving that theory of a proper black hole. This condition can neither be obeyed in GTR: If the central mass is slightly smeared, the Schwarzschild black hole cannot obey energymomentum conservation in GTR.\n\nCosmology. Starting from the FLRW metric, the harmonic condition brings two relations: U ∼ V 3 and k = 0: Minkowski space filled homogeneously with matter remains flat [6] . We may thus put U = a 6 (t)/a 4 * , V = a 2 (t). Going from cosmic time t to conformal time τ = a 3 a -2 * dt yields the familiar Einstein equations, extended by Λ bi terms, ȧ2\n\na 2 c 2 = 8πG 3c 4 ρ + Λ 3 - Λ bi 2a 2 + Λ bi a 4 * 6a 6 , (26)\n\nä a c 2 = - 4πG 3c 4 (ρ + 3p) + Λ 3 - Λ bi 3 a 4 * a 6 .\n\nThe first is the modified Friedman equation, the second corresponds to the first law d(ρ tot a 3 ) = -p tot da 3 provided we define ρ tot = ρ+ρ Λ +ρ 2 +ρ 6 and p tot = p-ρ Λ -1 3 ρ 2 +ρ 6 , with ρ 2 = -3ρ bi /2a 2 and ρ 6 = ρ bi a 4 * /2a 6 . Note that ρ 2 acts as a positive curvature term.\n\nThe scale factor has an absolute meaning. If we assume that a ≫ 1 and a ≫ a 2/3 * , Eq. (26) just coincides with the ΛCDM model (cosmological constant plus cold dark matter), that gives the best fit of the observations [9] [10] . The ρ 2 term allows a positive curvaturetype contribution. At large times, there is the exponential growth a(τ\n\n) = C exp(H ∞ τ ) with H ∞ = c Λ/3. In cos- mic time this reads a(t) = a 2/3 * [3H ∞ (t 0 -t)] -1/3\n\n, where t 0 is \"the end of time\", the moment where the scale factor has become infinite. The minimal scale factor is zero: in this theory a big bang can occur since ρ bi > 0. Without including an inflaton field, Eq. ( 26 ) yields an initial growth of the expansion a = (a 2 * c τ 3Λ bi /2) 1/3 . In cosmic time this reads a = a 1 exp(ct Λ bi /6), i. e., a certain inflation scenario starting at t = -∞.\n\nAlso in RTG the gravitational energy precisely compensates the other energy contributions at all times. The vacuum energy also vanishes: In empty space, the cosmological constant energy ρ Λ cancels the ρ bi terms, due to Eq. (10) . See Eq. ( 12 ) for g µν = γ µν .\n\nIn conclusion, we have first written the Einstein equation in a form that involves the gravitational energy momentum tensor. An underlying Minkowski space is needed, in which gravitation is a field. The metric tensor is a way to deal with it, but the equations for the field itself exist too, see Eq. ( 6 ). For flat cosmology it follows that the total energy vanishes.\n\nNext we have broken general coordinate invariance by going to the bimetric theory of Logunov, called Relativistic Theory of Gravitation. We have shown that the choice of a positive bimetric constant allows to regularize the interior of the Schwarzschild black hole: time keeps its standard role and escape is, in principle, possible. While neither the Schwarzschild nor the Logunov black hole survives smearing of the central mass by a tiny pressure in the equation of state, ours does. Our modification of the Einstein equations involves the cosmological constant, so it is of Hubble size, immaterial for solar problems. In cosmology, the theory directly leads to the ΛCDM model, while it could accommodate a positive curvature-like term. At short times, there is a form of inflation. The gravitational energy exactly compensates the material energy. The zero point energy vanishes (\"again\"), though the cosmological constant is finite and positive: It is canceled by the bimetric terms.\n\nEuclidean space, a special case of Riemann geometry, seems to be invoked by Nature, at least far away from bodies and in cosmology. Our approach supports the following space-time interpretation: curvature is a geometric description of the gravitational field in flat space. Clusters of galaxies do not move away from each other, but the speed of light changes with cosmic time, dr/dt = [a(t)/a * ] 2 c, while the conformal speed is dr/dτ = c/a(τ ) as usual.\n\nAn empirical way to establish the Minkowski metric is to present the Einstein equations as (c 4 /8πG)R µν -T µν + 1 2 g µν T + ρ Λ g µν = ρ bi γ µν , and to measure the left hand side, which in the geometric view is considered to consist of curved space properties alone. [6] As in the standard model of elementary particles, the separation of curved space into flat space and the gravitational field has the following implication: the quantum version of RTG -if it exists -will involve quantization of fields, but not of space.\n\nFinally we answer the question posed in the title. The field theoretic approach to gravitation is by itself equivalent to a curved space description, so both views apply, describing the same physics from a different angle. But when the theory is extended to the relativistic theory of gravitation, the bimetrism forces to describe the Minkowski metric separately, and then we see it as most natural to view gravitation as a field in flat space, which is Maxwell's view.\n\nTopics such as a realistic equation of state for black holes and classical tunneling of its radiation, regularization of other singularities, as well as aspects of the inflation and of inhomogeneous cosmology are under study.\n\nDiscussion with Martin Nieuwenhuizen and Armen Allahverdyan is gratefully remembered."
}
] | |
arxiv:0704.0232 | 0704.0232 | 1 | 4a56b2e7ec34ce8c00d29a9d0df3c84e728944549eb85e55ec38a4d0b73e8f38 | New algebraic aspects of perturbative and non-perturbative Quantum Field
Theory | In this expository article we review recent advances in our understanding of
the combinatorial and algebraic structure of perturbation theory in terms of
Feynman graphs, and Dyson-Schwinger equations. Starting from Lie and Hopf
algebras of Feynman graphs, perturbative renormalization is rephrased
algebraically. The Hochschild cohomology of these Hopf algebras leads the way
to Slavnov-Taylor identities and Dyson-Schwinger equations. We discuss recent
progress in solving simple Dyson-Schwinger equations in the high energy sector
using the algebraic machinery. Finally there is a short account on a relation
to algebraic geometry and number theory: understanding Feynman integrals as
periods of mixed (Tate) motives. | [
"Christoph Bergbauer",
"Dirk Kreimer"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | As elements of perturbative expansions of Quantum field theories, Feynman graphs have been playing and still play a key role both for our conceptual understanding and for state-of-the-art computations in particle physics. This 1 article is concerned with several aspects of Feynman graphs: First, the combinatorics of perturbative renormalization give rise to Hopf algebras of rooted trees and Feynman graphs. These Hopf algebras come with a cohomology theory and structure maps that help understand important physical notions, such as locality of counterterms, the beta function, certain symmetries, or Dyson-Schwinger equations from a unified mathematical point of view. This point of view is about self-similarity and recursion. The atomic (primitive) elements in this combinatorial approach are divergent graphs without subdivergences. They must be studied by additional means, be it analytic methods or algebraic geometry and number theory, and this is a significantly more difficult task. However, the Hopf algebra structure of graphs for renormalization is in this sense a substructure of the Hopf algebra structure underlying the relative cohomology of graph hypersurfaces needed to understand the number-theoretic properties of field theory amplitudes [6, 5] .
Given a Feynman graph Γ with several divergent subgraphs, the Bogoliubov recursion and Zimmermann's forest formula tell how Γ must be renormalized in order to obtain a finite conceptual result, using only local counterterms. This has an analytic (regularization/extension of distributions) and a combinatorial aspect. The basic combinatorial question of perturbative renormalization is to find a good model which describes disentanglement of graphs into subdivergent pieces, or dually insertion of divergent pieces one into each other, from the point of view of renormalized Feynman rules. It has been known now for several years that commutative Hopf algebras and (dual) Lie algebras provide such a framework [26, 14, 15] with many ramifications in pure mathematics. From the physical side, it is important to know that, for example, recovering aspects of gauge/BRST symmetry [39, 37, 30, 38] and the transition to nonperturbative equations of motion [12, 28, 29, 36, 3, 35, 32, 34, 4] are conveniently possible in this framework, as will be discussed in subsequent sections.
In order to introduce these Lie and Hopf algebras, let us now fix a renormalizable quantum field theory (in the sense of perturbation theory), given by a local Lagrangian. A convenient first example is massless φ 3 theory in 6 dimensions. We look at its perturbative expansion in terms of 1PI Feynman graphs. Each 1PI graph Γ comes with two integers, |Γ| = rank H 1 (Γ), its number of loops, and sdd(Γ), its superficial degree of divergence. As usual, vacuum and tadpole graphs need not be considered, and the only remaining superficial divergent graphs have exactly two or three external edges, a feature of renormalizability. Graphs without subdivergences are called primitive. Here are two examples.
Both are superficially divergent as they have three external edges. The first one has two subdivergences, the second one is primitive. Note that there are infinitely many primitive graphs with three external edges. In particular, for every n ∈ N one finds a primitive Γ such that |Γ| = n.
Let now L be the Q-vector space generated by all the superficially divergent (sdd ≥ 0) 1PI graphs of our theory, graded by the number of loops | • |. There is an operation on L given by insertion of graphs into each other: Let γ 1 , γ 2 be two generators of L. Then γ 1 ⋆ γ 2 := Γ n(γ 1 , γ 2 , Γ) where n(γ 1 , γ 2 , Γ) is the number of times that γ 1 shows up as a subgraph of Γ and Γ/γ 1 ∼ = γ 2 . Here are two examples: ⋆ = + + ⋆ = 2 This definition is extended bilinearly onto all of L. Note that ⋆ respects the grading as
|γ 1 ⋆ γ 2 | = |γ 1 | + |γ 2 |.
The operation ⋆ is not in general associative. Indeed, it is pre-Lie [14, 17]:
(γ 1 ⋆ γ 2 ) ⋆ γ 3 -γ 1 ⋆ (γ 2 ⋆ γ 3 ) = (γ 1 ⋆ γ 3 ) ⋆ γ 2 -γ 1 ⋆ (γ 3 ⋆ γ 3 ). ( 1
)
To see that (1) holds observe that on both sides nested insertions cancel. What remains are disjoint insertions of γ 2 and γ 3 into γ 1 which do obviously not depend on the order of γ 2 and γ 3 . One defines a Lie bracket on L : [γ 1 , γ 2 ] := γ 1 ⋆ γ 2 -γ 2 ⋆ γ 1 .
The Jacobi identity for [•, •] is satisfied as a consequence of the pre-Lie property (1) of ⋆. This makes L a graded Lie algebra. The bracket is defined by mutual insertions of graphs. As usual, U(L), the universal envelopping algebra of L is a cocommutative Hopf algebra. Its graded dual, in the sense of Milnor-Moore, is therefore a commutative Hopf algebra H. As an algebra, H is free commutative, generated by the vector space L and an adjoined unit I. By duality, one expects the coproduct of H to disentangle its argument into subdivergent pieces. Indeed, one finds ∆(Γ) = I ⊗ Γ + Γ ⊗ I +
γ Γ γ ⊗ Γ/γ. ( 2
)
The relation γ Γ refers to disjoint unions γ of 1PI superficially divergent subgraphs of Γ. Disjoint unions of graphs are in turn identified with their product in H. For example, ∆ = I ⊗ + ⊗ I + 2 ⊗ .
The coproduct respects the grading by the loop number, as does the product (by definition). Therefore H = ∞ n=0 H n is a graded Hopf algebra. Since H 0 ∼ = Q it is connected. The counit ǫ vanishes on the subspace ∞ n=1 H n , called augmentation ideal, and ǫ(I) = 1. As usual, if ∆(x) = I ⊗ x + x ⊗ I, the element x is called primitive. The linear subspace of primitive elements is denoted Prim H.
The interest in H and L arises from the fact that the Bogoliubov recursion is essentially solved by the antipode of H. In any connected graded bialgebra, the antipode S is given by
S(x) = -x - S(x ′ )x ′′ , x / ∈ H 0 ( 3
)
in Sweedler's notation. Let now V be a C-algebra. The space of linear maps
L Q (H, V ) is equipped with a convolution product (f, g) → f * g = m V (f ⊗ g)∆ where m V is the product in V.
Relevant examples for V are suggested by regularization schemes such as the algebra V = C[[ǫ, ǫ -1 ] of Laurent series with finite pole part for dimensional regularization (space-time dimension D = 6+2ǫ.) The (unrenormalized) Feynman rules provide then an algebra homomorphism φ : H → V mapping Feynman graphs to Feynman integrals in 6+2ǫ dimensions. On V there is a linear endomorphism R (renormalization scheme) defined, for example minimal subtraction R(ǫ n ) = 0 if n ≥ 0, R(ǫ n ) = ǫ n if n < 0. If Γ is primitive, as defined above, then φ(Γ) has only a simple pole in ǫ, hence (1 -R)φ(Γ) is a good renormalized value for Γ. If Γ does have subdivergences, the situation is more complicated. However, the map
S φ R : H → V S φ R (Γ) = -R φ(Γ) - S φ R (Γ ′ )φ(Γ ′′ )
provides the counterterm prescribed by the Bogoliubov recursion, and (S φ R * φ)(Γ) yields the renormalized value of Γ. The map S φ R is a recursive deformation of φ • S by R, compare its definition with (3). These are results obtained by one of the authors in collaboration with Connes [26, 14, 15] . For S φ R to be an algebra homomorphism again, one requires R to be a Rota-Baxter operator, studied in a more general setting by Ebrahimi-Fard, Guo and one of the authors in [20, 22, 21] . The Rota-Baxter property is at the algebraic origin of the Birkhoff decomposition introduced in [15, 16] . In the presence of mass terms, or gauge symmetries etc. in the Lagrangian, φ, S φ R and S φ R ⋆ φ may contribute to several form factors in the usual way. This can be resolved by considering a slight extension of the Hopf algebra containing projections onto single structure functions, as discussed for example in [15, 32] . For the case of gauge theories, a precise definition of the coefficients n(γ 1 , γ 2 , Γ) is given in [30] .
The Hopf algebra H arises from the simple insertion of graphs into each other in a completely canonical way. Indeed, the pre-Lie product determines the coproduct, and the coproduct determines the antipode. Like this, each quantum field theory gives rise to such a Hopf algebra H based on its 1PI graphs. It is 4 no surprise then that there is an even more universal Hopf algebra behind all of them: The Hopf algebra H rt of rooted trees [26, 14] . In order to see this, imagine a purely nested situation of subdivergences like which can be represented by the rooted tree
• • • ❆ ❆ ✁ ✁ .
To account for each single graph of this kind, the tree's vertices should actually be labeled according to which primitive graph they correspond to (plus some gluing data) which we will suppress for the sake of simplicity. The coproduct on H rt -corresponding to the one (2) of H -is
∆(τ ) = I ⊗ τ + τ ⊗ I + adm.c P c (τ ) ⊗ R c (τ )
where the sum runs over all admissible cuts of the tree τ. A cut of τ is a nonempty subset of its edges which are to be removed. A cut c(τ ) is defined to be admissible, if for each leaf l of τ at most one edge on the path from l to the root is cut. The product of subtrees which fall down when those edges are removed is denoted P c (τ ). The part which remains connected with the root is denoted R c (τ ). Here is an example:
∆ • • • • ✁ ✁ ❆ ❆ = • • • • ✁ ✁ ❆ ❆ ⊗ I + I ⊗ • • • • ✁ ✁ ❆ ❆ + 2 • ⊗ • • • + + • • ⊗ • • + • • • ❆ ❆ ✁ ✁ ⊗ •.
Compared to H rt , the advantage of H is however that overlapping divergences are resolved automatically. To achieve this in H rt requires some care [27].
There is a natural cohomology theory on H and H rt whose non-exact 1-cocycles play an important "operadic" role in the sense that they drive the recursion that define the full 1PI Green's functions in terms of primitve graphs. In order to introduce this cohomology theory, let A be any bialgebra. We view A as a bicomodule over itself with right coaction (id ⊗ ǫ)∆. Then the Hochschild cohomology of A (with respect to the coalgebra part) is defined as follows [14]: 5 Linear maps L : A → A ⊗n are considered as n-cochains. The operator b, defined as
bL := (id ⊗ L)∆ + n i=1 (-1) i ∆ i L + (-1) n+1 L ⊗ I ( 4
)
furnishes a codifferential: b 2 = 0. Here ∆ denotes the coproduct of A and ∆ i the coproduct applied to the i-th factor in A ⊗n . The map L ⊗ I is given by x → L(x) ⊗ I. Clearly this codifferential encodes only information about the coalgebra (as opposed to the algebra) part of A. The resulting cohomology is denoted HH • ǫ (A). For n = 1, the cocycle condition bL = 0 is simply
∆L = (id ⊗ L)∆ + L ⊗ I ( 5
)
for L a linear endomorphism of A. In the Hopf algebra H rt of rooted trees (where things are often simpler), a 1-cocycle is quickly found: the grafting operator B + , defined by
B + (I) = • B + (τ 1 . . . τ n ) = • ❅ ❅ ❆ ❆ ✁ ✁ τ 1 . . . τ n
for trees τ i joining all the roots of its argument to a newly created root. Clearly, B + reminds of an operad multiplication. It is easily seen that B + is not exact and therefore a generator (among others) of HH 1 ǫ (H rt ). Foissy [23] showed that L → L(I) is an onto map HH 1 ǫ (H rt ) → Prim H rt . The higher Hochschild cohomology (n ≥ 2) of H rt is known to vanish [23] . The pair (H rt , B + ) is the universal model for all Hopf algebras of Feynman graphs and their 1-cocycles [14] . Let us now turn to those 1-cocycles of H. Clearly, every primitve graph γ gives rise to a 1-cocycle B γ + defined as the operator which inserts its argument, a product of graphs, into γ in all possible ways. Here is a simple example:
B + = 1 2
See [30] for the general definition involving some combinatorics of insertion places and symmetries.
It is an important consequence of the B γ + satisfying the cocycle condition (5) that (S φ R * φ
)B + = (1 -R) B+ (S φ R * φ) ( 6
)
where B+ is the push-forward of B + along the Feynman rules φ. In other words, Bγ + is the integral operator corresponding to the skeleton graph γ. This is the combinatorial key to the proof of locality of counterterms and finiteness of renormalization [13, 28, 2, 3] . Indeed, equation (6) says that after treating all subdivergences, an overall subtraction (1-R) suffices. The only analytic ingredient is 6 Weinberg's theorem applied to the primitive graphs. In [2] it is emphasized that H is actually generated (and determined) by the action of prescribed 1-cocycles and the multiplication. A version of (6) with decorated trees is available which describes renormalization in coordinate space [2] .
The 1-cocycles B γ + give rise to a number of useful Hopf subalgebras of H. Many of them are isomorphic. They are studied in [3] on the model of decorated rooted trees, and we will come back to them in the next section. In [30] one of the authors showed that in nonabelian gauge theories, the existence of a certain Hopf subalgebra, generated by 1-cocycles, is closely related to the Slavnov-Taylor identities for the couplings to hold. In a similar spirit, van Suijlekom showed that, in QED, Ward-Takahashi identities, and in nonabelian Yang-Mills theories, the Slavnov-Taylor identities for the couplings generate Hopf ideals I of H such that the quotients H/I are defined and the Feynman rules factor through them [37, 38] . The Hopf algebra H for QED had been studied before in [11, 33, 39] .
The ultimate application of the Hochschild 1-cocycles introduced in the previous section aims at non-perturbative results. Dyson-Schwinger equations, reorganized using the correspondence Prim H → HH 1 ǫ (H), become recursive equations in H[[α]], α the coupling constant, with contributions from (degree 1) 1-cocycles. The Feynman rules connect them to the usual integral kernel representation. We remain in the massless φ 3 theory in 6 dimensions for the moment. Let Γ be the full 1PI vertex function,
Γ = I + res Γ= α |Γ| Γ Sym Γ ( 7
)
(normalized such that the tree level contribution equals 1). This is a formal power series in α with values in H. Here res Γ is the result of collapsing all internal lines of Γ. The graph res Γ is called the residue of Γ. In a renormalizable theory, res can be seen as a map from the set of generators of H to the terms in the Lagrangian. For instance, in the φ 3 theory, vertex graphs have residue , and self energy graphs have residue -. The number Sym Γ denotes the order of the group of automorphisms of Γ, defined in detail for example in [30, 38] . Similarly, the full inverse propagator Γ -is represented by
Γ -= I - res Γ=- α |Γ| Γ Sym Γ . ( 8
)
These series can be reorganized by summing only over primitive graphs, with all possible insertions into these primitive graphs. In H, the insertions are afforded 7 by the corresponding Hochschild 1-cocycles. Indeed,
Γ = I + γ∈Prim H,res γ= α |γ| B γ + (Γ Q |γ| ) Sym γ Γ -= I -
γ∈Prim H,res γ=-
α |γ| B γ + (Γ -Q |γ| ) Sym γ . ( 9
)
The universal invariant charge Q is a monomial in the Γ r and their inverses, where r are residues (terms in the Lagrangian) provided by the theory. In φ 3 theory we have Q = (Γ ) 2 (Γ -) -3 . In φ 3 theory, the universality of Q (i. e. the fact that the same Q is good for all Dyson-Schwinger equations of the theory) comes from a simple topological argument. In nonabelian gauge theories however, the universality of Q takes care that the solution of the corresponding system of coupled Dyson-Schwinger equations gives rise to a Hopf subalgebra and therefore amounts to the Slavnov-Taylor identities for the couplings [30].
The system (9) of coupled Dyson-Schwinger equations has (7,8) as its solution. Note that in the first equation of (9) an infinite number of cocycles contributes as there are infinitely many primitive vertex graphs in φ 3 6 theory -the second equation has only finitely many contributions -here one. Before we describe how to actually attempt to solve equations of this kind analytically (application of the Feynman rules φ), we discuss the combinatorial ramifications of this construction in the Hopf algebra. It makes sense to call all (systems of) recursive equations of the form
X 1 = I ± n α k 1 n B d 1 n + (M 1 n ) . . . X s = I ± n α k s n B d s n + (M s n )
combinatorial Dyson-Schwinger equations, and to study their combinatorics. Here, the B dn + are non-exact Hochschild 1-cocycles and the M n are monomials in the 3] we studied a large class of single (uncoupled) combinatorial Dyson-Schwinger equations in a decorated version of H rt as a model for vertex insertions:
X 1 . . . X s . In [
X = I + ∞ n=1 α n w n B dn + (X n+1 ) where the w n ∈ Q. For example, X = I + αB + (X 2 ) + α 2 B + (X 3 ) is in this class.
It turns out [28, 3] that the coefficients c n of X, defined by X = ∞ n=0 α n c n generate a Hopf subalgebra themselves:
∆(c n ) = n k=0 P n k ⊗ c k . 8
The P n k are homogeneous polynomials of degree n-k in the c l , l ≤ n. These polynomials have been worked out explicitly in [3] . One notices in particular that the P n k are independent of the w n and B dn + , and hence that under mild assumptions (on the algebraic independence of the c n ) the Hopf subalgebras generated this way are actually isomorphic. For example,
X = I + αB + (X 2 ) + α 2 B + (X 3 ) and X = I + αB + (X 2
) yield isomorphic Hopf subalgebras. This is an aspect of the fact that truncation of Dyson-Schwinger equations -considering only a finite instead of an infinite number of contributing cocycles -does make (at least combinatorial) sense. Indeed, the combinatorics remain invariant. Similar results hold for Dyson-Schwinger equations in the true Hopf algebra of graphs H where things are a bit more difficult though as the cocycles there involve some bookkeeping of insertion places.
The simplest nontrivial Dyson-Schwinger equation one can think of is the linear one:
X = I + αB + (X). Its solution is given by X = ∞ n=0 α n (B + ) n (I).
In this case X is grouplike and the corresponding Hopf subalgebra of c n s is cocommutative [25] . A typical and important non-linear Dyson-Schwinger equation arises from propagator insertions: X = I -αB + (1/X), for example the massless fermion propagator in Yukawa theory where only the fermion line obtains radiative corrections (other corrections are ignored). This problem has been studied and solved by Broadhurst and one of the authors in [12] and revisited recently by one of the authors and Yeats [35] . As we now turn to the analytic aspects of Dyson-Schwinger equations, we briefly sketch the general approach presented in [35] on how to successfully treat the nonlinearity of Dyson-Schwinger equations. Indeed, the linear Dyson-Schwinger equations can be solved by a simple scaling ansatz [25] . In any case, let γ be a primitive graph. The following works for amplitudes which depend on a single scale, so let us assume a massless situation with only one non-zero external momentumhow more than one external momentum (vertex insertions) are incorporated by enlarging the set of primitive elements is sketched in [32] . The grafting operator B γ + associated to γ translates to an integral operator under the (renormalized) Feynman rules
φ R (B γ + )(I)(p 2 /µ 2 ) = (I γ (k, p) -I γ (k, µ))dk
where I γ is the integral kernel corresponding to γ, the internal momenta are denoted by k, the external momentum by p, and µ is the fixed momentum at which we subtract: R(x) = x| p 2 =µ 2 .
In the following we stick to the special case discussed in [35] where only one internal edge is allowed to receive corrections. The integral kernel φ(B γ + ) defines 9 a Mellin transform
F (ρ) = I γ (k, µ)(k 2 i ) -ρ dk
where k i is the momentum of the internal edge of γ at which insertions may take place (here the fermion line). If there are several insertion sites, obvious multiple Mellin transforms become necessary. The case of two (propagator) insertion places has been studied, at the same example, in [35] .
The function F (ρ) has a simple pole in ρ at 0. We write [35] is that, even in the difficult nonlinear situation, the anomalous dimension γ 1 is implicitly defined by the residue r and Taylor coefficients f n of the Mellin transform F. On the other hand, all the γ n for n ≥ 2, are recursively defined in terms of the γ i , i < n. This last statement amounts to a renormalization group argument that is afforded in the Hopf algebra by the scattering formula of [16] . Curiously, for this argument only a linearized part of the coproduct is needed. We refer to [35] for the actual algorithm. For a linear Dyson-Schwinger equation, the situation is considerably simpler as the γ n = 0 for n ≥ 2 since X is grouplike [25].
F (ρ) = r ρ + ∞ n=0 f n ρ n We denote L = log p 2 /µ 2 . Clearly φ R (X) = 1 + n γ n L n . An important result of
Let us restate the results for the high energy sector of non-linear Dyson-Schwinger equations [12, 35] : Primitive graphs γ define Mellin transforms via their integral kernels Bγ + . The anomalous dimension γ 1 is implicitly determined order by order from the coefficients of those Mellin transforms. All non-leading log coefficients γ n are recursively determined by γ 1 , thanks to the renormalization group. This reduces, in principle, the problem to a study of all the primitive graphs and the intricacies of insertion places.
Finding useful representations of those Mellin transforms -even one-dimensional ones -of higher loop order skeleton graphs is difficult. However, the two-loop primitive vertex in massless Yukawa theory has been worked out by Bierenbaum, Weinzierl and one of the authors in [4] , a result that can be applied to other theories as well. Combined with the algebraic treatment [12, 3, 35] sketched in the previous paragraphs and new geometric insight on primitive graphs (see section 5), there is reasonable hope that actual solutions of Dyson-Schwinger equations will be more accessible in the future.
Using the Dyson-Schwinger analysis, one of the authors and Yeats [34] were able to deduce a bound for the convergence of superficially divergent amplitudes/structure functions from the (desirable) existence of a bound for the superficially convergent amplitudes. 10 5 Feynman integrals and periods of mixed (Tate) Hodge structures
A primitive graph Γ ∈ Prim H defines a real number r Γ , called the residue of Γ, which is independent of the renormalization scheme. In the case that Γ is massless and has one external momentum p, the residue r Γ is the coefficient of log p 2 /µ 2 in φ R (Γ) = (1 -R)φ(Γ). It coincides with the coefficient r of the Mellin transform introduced in the previous section. One may ask what kind of a number r is, for example if it is rational or algebraic. The origin of this question is that the irrational or transcendental numbers that show up for various Γ strongly suggest a motivic interpretation of the r Γ . Indeed, explicit calculations [9, 10, 8] display patterns of Riemann zeta and multiple zeta values that are known to be periods of mixed Tate Hodge structures -here the periods are provided by the Feynman rules which produce Γ → r Γ . By disproving a related conjecture of Kontsevich, Belkale and Brosnan [1] have shown that not all these Feynman motives must be mixed Tate, so one may expect a larger class of Feynman periods than multiple zeta values. Our detailed understanding of these phenomena is still far from complete, and only some very first steps have been made in the last few years. However, techniques developed in recent work by Bloch, Esnault and one of the authors [7] do permit reasonable insight for some special cases which we briefly sketch in the following.
Let Γ be a logarithmically divergent massless primitive graph with one external momentum p. It is convenient to work in the "Schwinger" parametric representation [24] obtained by the usual trick of replacing propagators
1 k 2 = ∞ 0 dae -ak 2 ,
and performing the loop integrations (Gaussian integrals) first which leaves us with a (divergent) integral over various Schwinger parameters a. It is a classical exercise [24, 7, 6] to show that in four dimensions, up to some powers of i and 2π,
φ(Γ) = ∞ 0 da 1 . . . da n e -QΓ(a,p 2 )/ΨΓ(a) Ψ 2 Γ (a)
where n is the number of edges of Γ. Q Γ and Ψ Γ are graph polynomials of Γ, where Ψ Γ , sometimes called Symanzik or Kirchhoff polynomial, is defined as follows: Let T (Γ) be the set of spanning trees of Γ, i. e. the set of connected simply connected subgraphs which meet all vertices of Γ. We think of the edges e of Γ as being numbered from 1 to n. Then Ψ Γ = t∈T (Γ) e ∈t a e 11 This is a homogeneous polynomial in the
a i of degree |H 1 (Γ)|. It is easily seen (scaling behaviour of Q Γ and Ψ Γ ) that r Γ = ∂φR(Γ)
∂ log p 2 /µ 2 is extracted from φ(Γ) by considering the a i as homogeneous coordinates of P n-1 (R) and evaluating at p 2 = 0 :
r Γ = σ⊂P n-1 (R) Ω Ψ 2 Γ ( 10
)
where σ = {[a 1 , . . . , a n ] : all a i can be choosen ≥ 0} and Ω is a volume form on
P n-1 . Let X Γ := {Ψ Γ = 0} ⊂ P n-1 . If |H 1 (Γ)| = 1
, the integrand in (10) has no poles. If |H 1 (Γ)| > 1, poles will show up on the union ∆ = γ Γ,H1(γ) =0 L γ of coordinate linear spaces γ = {a e = 0 for e edge of γ} -these need to be separated from the chain of integration by blowing up. The blowups being understood, the Feynman motive is, by abuse of notation,
H n-1 (P n-1 -X Γ , ∆ -∆ ∩ X Γ )
with Feynman period given by (10) . See [7, 6] for details. Some particularly accessible examples are the wheel with n spokes graphs Γ n := studied extensively in [7] . The corresponding Feynman periods (10) yield rational multiples of zeta values [9] r Γn ∈ ζ(2n -3)Q.
Due to the simple topology of the Γ n , the geometry of the pairs (X Γn , ∆ Γn ) are well understood and the corresponding motives have been worked out explicitly [7] . The methods used are however nontrivial and not immediately applicable to more general situations.
When confronted with non-primitive graphs, i. e. graphs with subdivergences, there are more than one period to consider. In the Schwinger parameter picture, subdivergences arise when poles appear along exceptional divisors as pieces of ∆ are blown up. This situation can be understood using limiting mixed Hodge structures [6] , see also [31, 36] for a toy model approach to the combinatorics involved. In [6] it is also shown how the Hopf algebra H of graphs lifts to the category of motives. For the motivic role of solutions of Dyson-Schwinger equations we refer to work in progress. Finally we mention that there is related work by Connes and Marcolli [18, 19] who attack the problem via Riemann-Hilbert correspondences and motivic Galois theory.
Acknowledgements. We thank Spencer Bloch and Karen Yeats for discussion on the subject of this review. The first named author (C. B.) thanks the organizers of the ICMP 2006 and the IHES for general support. His research is 12 supported by the Deutsche Forschungsgemeinschaft. The IHES, Boston University and the Erwin-Schrödinger-Institute are gratefully acknowledged for their kind hospitality. At the time of writing this article, C. B. is visiting the ESI as a Junior Research Fellow. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "As elements of perturbative expansions of Quantum field theories, Feynman graphs have been playing and still play a key role both for our conceptual understanding and for state-of-the-art computations in particle physics. This 1 article is concerned with several aspects of Feynman graphs: First, the combinatorics of perturbative renormalization give rise to Hopf algebras of rooted trees and Feynman graphs. These Hopf algebras come with a cohomology theory and structure maps that help understand important physical notions, such as locality of counterterms, the beta function, certain symmetries, or Dyson-Schwinger equations from a unified mathematical point of view. This point of view is about self-similarity and recursion. The atomic (primitive) elements in this combinatorial approach are divergent graphs without subdivergences. They must be studied by additional means, be it analytic methods or algebraic geometry and number theory, and this is a significantly more difficult task. However, the Hopf algebra structure of graphs for renormalization is in this sense a substructure of the Hopf algebra structure underlying the relative cohomology of graph hypersurfaces needed to understand the number-theoretic properties of field theory amplitudes [6, 5] ."
},
{
"section_type": "OTHER",
"section_title": "Lie and Hopf algebras of Feynman graphs",
"text": "Given a Feynman graph Γ with several divergent subgraphs, the Bogoliubov recursion and Zimmermann's forest formula tell how Γ must be renormalized in order to obtain a finite conceptual result, using only local counterterms. This has an analytic (regularization/extension of distributions) and a combinatorial aspect. The basic combinatorial question of perturbative renormalization is to find a good model which describes disentanglement of graphs into subdivergent pieces, or dually insertion of divergent pieces one into each other, from the point of view of renormalized Feynman rules. It has been known now for several years that commutative Hopf algebras and (dual) Lie algebras provide such a framework [26, 14, 15] with many ramifications in pure mathematics. From the physical side, it is important to know that, for example, recovering aspects of gauge/BRST symmetry [39, 37, 30, 38] and the transition to nonperturbative equations of motion [12, 28, 29, 36, 3, 35, 32, 34, 4] are conveniently possible in this framework, as will be discussed in subsequent sections.\n\nIn order to introduce these Lie and Hopf algebras, let us now fix a renormalizable quantum field theory (in the sense of perturbation theory), given by a local Lagrangian. A convenient first example is massless φ 3 theory in 6 dimensions. We look at its perturbative expansion in terms of 1PI Feynman graphs. Each 1PI graph Γ comes with two integers, |Γ| = rank H 1 (Γ), its number of loops, and sdd(Γ), its superficial degree of divergence. As usual, vacuum and tadpole graphs need not be considered, and the only remaining superficial divergent graphs have exactly two or three external edges, a feature of renormalizability. Graphs without subdivergences are called primitive. Here are two examples.\n\nBoth are superficially divergent as they have three external edges. The first one has two subdivergences, the second one is primitive. Note that there are infinitely many primitive graphs with three external edges. In particular, for every n ∈ N one finds a primitive Γ such that |Γ| = n.\n\nLet now L be the Q-vector space generated by all the superficially divergent (sdd ≥ 0) 1PI graphs of our theory, graded by the number of loops | • |. There is an operation on L given by insertion of graphs into each other: Let γ 1 , γ 2 be two generators of L. Then γ 1 ⋆ γ 2 := Γ n(γ 1 , γ 2 , Γ) where n(γ 1 , γ 2 , Γ) is the number of times that γ 1 shows up as a subgraph of Γ and Γ/γ 1 ∼ = γ 2 . Here are two examples: ⋆ = + + ⋆ = 2 This definition is extended bilinearly onto all of L. Note that ⋆ respects the grading as\n\n|γ 1 ⋆ γ 2 | = |γ 1 | + |γ 2 |.\n\nThe operation ⋆ is not in general associative. Indeed, it is pre-Lie [14, 17]:\n\n(γ 1 ⋆ γ 2 ) ⋆ γ 3 -γ 1 ⋆ (γ 2 ⋆ γ 3 ) = (γ 1 ⋆ γ 3 ) ⋆ γ 2 -γ 1 ⋆ (γ 3 ⋆ γ 3 ). ( 1\n\n)\n\nTo see that (1) holds observe that on both sides nested insertions cancel. What remains are disjoint insertions of γ 2 and γ 3 into γ 1 which do obviously not depend on the order of γ 2 and γ 3 . One defines a Lie bracket on L : [γ 1 , γ 2 ] := γ 1 ⋆ γ 2 -γ 2 ⋆ γ 1 .\n\nThe Jacobi identity for [•, •] is satisfied as a consequence of the pre-Lie property (1) of ⋆. This makes L a graded Lie algebra. The bracket is defined by mutual insertions of graphs. As usual, U(L), the universal envelopping algebra of L is a cocommutative Hopf algebra. Its graded dual, in the sense of Milnor-Moore, is therefore a commutative Hopf algebra H. As an algebra, H is free commutative, generated by the vector space L and an adjoined unit I. By duality, one expects the coproduct of H to disentangle its argument into subdivergent pieces. Indeed, one finds ∆(Γ) = I ⊗ Γ + Γ ⊗ I +\n\nγ Γ γ ⊗ Γ/γ. ( 2\n\n)\n\nThe relation γ Γ refers to disjoint unions γ of 1PI superficially divergent subgraphs of Γ. Disjoint unions of graphs are in turn identified with their product in H. For example, ∆ = I ⊗ + ⊗ I + 2 ⊗ .\n\nThe coproduct respects the grading by the loop number, as does the product (by definition). Therefore H = ∞ n=0 H n is a graded Hopf algebra. Since H 0 ∼ = Q it is connected. The counit ǫ vanishes on the subspace ∞ n=1 H n , called augmentation ideal, and ǫ(I) = 1. As usual, if ∆(x) = I ⊗ x + x ⊗ I, the element x is called primitive. The linear subspace of primitive elements is denoted Prim H.\n\nThe interest in H and L arises from the fact that the Bogoliubov recursion is essentially solved by the antipode of H. In any connected graded bialgebra, the antipode S is given by\n\nS(x) = -x - S(x ′ )x ′′ , x / ∈ H 0 ( 3\n\n)\n\nin Sweedler's notation. Let now V be a C-algebra. The space of linear maps\n\nL Q (H, V ) is equipped with a convolution product (f, g) → f * g = m V (f ⊗ g)∆ where m V is the product in V.\n\nRelevant examples for V are suggested by regularization schemes such as the algebra V = C[[ǫ, ǫ -1 ] of Laurent series with finite pole part for dimensional regularization (space-time dimension D = 6+2ǫ.) The (unrenormalized) Feynman rules provide then an algebra homomorphism φ : H → V mapping Feynman graphs to Feynman integrals in 6+2ǫ dimensions. On V there is a linear endomorphism R (renormalization scheme) defined, for example minimal subtraction R(ǫ n ) = 0 if n ≥ 0, R(ǫ n ) = ǫ n if n < 0. If Γ is primitive, as defined above, then φ(Γ) has only a simple pole in ǫ, hence (1 -R)φ(Γ) is a good renormalized value for Γ. If Γ does have subdivergences, the situation is more complicated. However, the map\n\nS φ R : H → V S φ R (Γ) = -R φ(Γ) - S φ R (Γ ′ )φ(Γ ′′ )\n\nprovides the counterterm prescribed by the Bogoliubov recursion, and (S φ R * φ)(Γ) yields the renormalized value of Γ. The map S φ R is a recursive deformation of φ • S by R, compare its definition with (3). These are results obtained by one of the authors in collaboration with Connes [26, 14, 15] . For S φ R to be an algebra homomorphism again, one requires R to be a Rota-Baxter operator, studied in a more general setting by Ebrahimi-Fard, Guo and one of the authors in [20, 22, 21] . The Rota-Baxter property is at the algebraic origin of the Birkhoff decomposition introduced in [15, 16] . In the presence of mass terms, or gauge symmetries etc. in the Lagrangian, φ, S φ R and S φ R ⋆ φ may contribute to several form factors in the usual way. This can be resolved by considering a slight extension of the Hopf algebra containing projections onto single structure functions, as discussed for example in [15, 32] . For the case of gauge theories, a precise definition of the coefficients n(γ 1 , γ 2 , Γ) is given in [30] .\n\nThe Hopf algebra H arises from the simple insertion of graphs into each other in a completely canonical way. Indeed, the pre-Lie product determines the coproduct, and the coproduct determines the antipode. Like this, each quantum field theory gives rise to such a Hopf algebra H based on its 1PI graphs. It is 4 no surprise then that there is an even more universal Hopf algebra behind all of them: The Hopf algebra H rt of rooted trees [26, 14] . In order to see this, imagine a purely nested situation of subdivergences like which can be represented by the rooted tree\n\n• • • ❆ ❆ ✁ ✁ .\n\nTo account for each single graph of this kind, the tree's vertices should actually be labeled according to which primitive graph they correspond to (plus some gluing data) which we will suppress for the sake of simplicity. The coproduct on H rt -corresponding to the one (2) of H -is\n\n∆(τ ) = I ⊗ τ + τ ⊗ I + adm.c P c (τ ) ⊗ R c (τ )\n\nwhere the sum runs over all admissible cuts of the tree τ. A cut of τ is a nonempty subset of its edges which are to be removed. A cut c(τ ) is defined to be admissible, if for each leaf l of τ at most one edge on the path from l to the root is cut. The product of subtrees which fall down when those edges are removed is denoted P c (τ ). The part which remains connected with the root is denoted R c (τ ). Here is an example:\n\n∆ • • • • ✁ ✁ ❆ ❆ = • • • • ✁ ✁ ❆ ❆ ⊗ I + I ⊗ • • • • ✁ ✁ ❆ ❆ + 2 • ⊗ • • • + + • • ⊗ • • + • • • ❆ ❆ ✁ ✁ ⊗ •.\n\nCompared to H rt , the advantage of H is however that overlapping divergences are resolved automatically. To achieve this in H rt requires some care [27]."
},
{
"section_type": "OTHER",
"section_title": "From Hochschild cohomology to physics",
"text": "There is a natural cohomology theory on H and H rt whose non-exact 1-cocycles play an important \"operadic\" role in the sense that they drive the recursion that define the full 1PI Green's functions in terms of primitve graphs. In order to introduce this cohomology theory, let A be any bialgebra. We view A as a bicomodule over itself with right coaction (id ⊗ ǫ)∆. Then the Hochschild cohomology of A (with respect to the coalgebra part) is defined as follows [14]: 5 Linear maps L : A → A ⊗n are considered as n-cochains. The operator b, defined as\n\nbL := (id ⊗ L)∆ + n i=1 (-1) i ∆ i L + (-1) n+1 L ⊗ I ( 4\n\n)\n\nfurnishes a codifferential: b 2 = 0. Here ∆ denotes the coproduct of A and ∆ i the coproduct applied to the i-th factor in A ⊗n . The map L ⊗ I is given by x → L(x) ⊗ I. Clearly this codifferential encodes only information about the coalgebra (as opposed to the algebra) part of A. The resulting cohomology is denoted HH • ǫ (A). For n = 1, the cocycle condition bL = 0 is simply\n\n∆L = (id ⊗ L)∆ + L ⊗ I ( 5\n\n)\n\nfor L a linear endomorphism of A. In the Hopf algebra H rt of rooted trees (where things are often simpler), a 1-cocycle is quickly found: the grafting operator B + , defined by\n\nB + (I) = • B + (τ 1 . . . τ n ) = • ❅ ❅ ❆ ❆ ✁ ✁ τ 1 . . . τ n\n\nfor trees τ i joining all the roots of its argument to a newly created root. Clearly, B + reminds of an operad multiplication. It is easily seen that B + is not exact and therefore a generator (among others) of HH 1 ǫ (H rt ). Foissy [23] showed that L → L(I) is an onto map HH 1 ǫ (H rt ) → Prim H rt . The higher Hochschild cohomology (n ≥ 2) of H rt is known to vanish [23] . The pair (H rt , B + ) is the universal model for all Hopf algebras of Feynman graphs and their 1-cocycles [14] . Let us now turn to those 1-cocycles of H. Clearly, every primitve graph γ gives rise to a 1-cocycle B γ + defined as the operator which inserts its argument, a product of graphs, into γ in all possible ways. Here is a simple example:\n\nB + = 1 2\n\nSee [30] for the general definition involving some combinatorics of insertion places and symmetries.\n\nIt is an important consequence of the B γ + satisfying the cocycle condition (5) that (S φ R * φ\n\n)B + = (1 -R) B+ (S φ R * φ) ( 6\n\n)\n\nwhere B+ is the push-forward of B + along the Feynman rules φ. In other words, Bγ + is the integral operator corresponding to the skeleton graph γ. This is the combinatorial key to the proof of locality of counterterms and finiteness of renormalization [13, 28, 2, 3] . Indeed, equation (6) says that after treating all subdivergences, an overall subtraction (1-R) suffices. The only analytic ingredient is 6 Weinberg's theorem applied to the primitive graphs. In [2] it is emphasized that H is actually generated (and determined) by the action of prescribed 1-cocycles and the multiplication. A version of (6) with decorated trees is available which describes renormalization in coordinate space [2] .\n\nThe 1-cocycles B γ + give rise to a number of useful Hopf subalgebras of H. Many of them are isomorphic. They are studied in [3] on the model of decorated rooted trees, and we will come back to them in the next section. In [30] one of the authors showed that in nonabelian gauge theories, the existence of a certain Hopf subalgebra, generated by 1-cocycles, is closely related to the Slavnov-Taylor identities for the couplings to hold. In a similar spirit, van Suijlekom showed that, in QED, Ward-Takahashi identities, and in nonabelian Yang-Mills theories, the Slavnov-Taylor identities for the couplings generate Hopf ideals I of H such that the quotients H/I are defined and the Feynman rules factor through them [37, 38] . The Hopf algebra H for QED had been studied before in [11, 33, 39] ."
},
{
"section_type": "OTHER",
"section_title": "Dyson-Schwinger equations",
"text": "The ultimate application of the Hochschild 1-cocycles introduced in the previous section aims at non-perturbative results. Dyson-Schwinger equations, reorganized using the correspondence Prim H → HH 1 ǫ (H), become recursive equations in H[[α]], α the coupling constant, with contributions from (degree 1) 1-cocycles. The Feynman rules connect them to the usual integral kernel representation. We remain in the massless φ 3 theory in 6 dimensions for the moment. Let Γ be the full 1PI vertex function,\n\nΓ = I + res Γ= α |Γ| Γ Sym Γ ( 7\n\n)\n\n(normalized such that the tree level contribution equals 1). This is a formal power series in α with values in H. Here res Γ is the result of collapsing all internal lines of Γ. The graph res Γ is called the residue of Γ. In a renormalizable theory, res can be seen as a map from the set of generators of H to the terms in the Lagrangian. For instance, in the φ 3 theory, vertex graphs have residue , and self energy graphs have residue -. The number Sym Γ denotes the order of the group of automorphisms of Γ, defined in detail for example in [30, 38] . Similarly, the full inverse propagator Γ -is represented by\n\nΓ -= I - res Γ=- α |Γ| Γ Sym Γ . ( 8\n\n)\n\nThese series can be reorganized by summing only over primitive graphs, with all possible insertions into these primitive graphs. In H, the insertions are afforded 7 by the corresponding Hochschild 1-cocycles. Indeed,\n\nΓ = I + γ∈Prim H,res γ= α |γ| B γ + (Γ Q |γ| ) Sym γ Γ -= I -\n\nγ∈Prim H,res γ=-\n\nα |γ| B γ + (Γ -Q |γ| ) Sym γ . ( 9\n\n)\n\nThe universal invariant charge Q is a monomial in the Γ r and their inverses, where r are residues (terms in the Lagrangian) provided by the theory. In φ 3 theory we have Q = (Γ ) 2 (Γ -) -3 . In φ 3 theory, the universality of Q (i. e. the fact that the same Q is good for all Dyson-Schwinger equations of the theory) comes from a simple topological argument. In nonabelian gauge theories however, the universality of Q takes care that the solution of the corresponding system of coupled Dyson-Schwinger equations gives rise to a Hopf subalgebra and therefore amounts to the Slavnov-Taylor identities for the couplings [30].\n\nThe system (9) of coupled Dyson-Schwinger equations has (7,8) as its solution. Note that in the first equation of (9) an infinite number of cocycles contributes as there are infinitely many primitive vertex graphs in φ 3 6 theory -the second equation has only finitely many contributions -here one. Before we describe how to actually attempt to solve equations of this kind analytically (application of the Feynman rules φ), we discuss the combinatorial ramifications of this construction in the Hopf algebra. It makes sense to call all (systems of) recursive equations of the form\n\nX 1 = I ± n α k 1 n B d 1 n + (M 1 n ) . . . X s = I ± n α k s n B d s n + (M s n )\n\ncombinatorial Dyson-Schwinger equations, and to study their combinatorics. Here, the B dn + are non-exact Hochschild 1-cocycles and the M n are monomials in the 3] we studied a large class of single (uncoupled) combinatorial Dyson-Schwinger equations in a decorated version of H rt as a model for vertex insertions:\n\nX 1 . . . X s . In [\n\nX = I + ∞ n=1 α n w n B dn + (X n+1 ) where the w n ∈ Q. For example, X = I + αB + (X 2 ) + α 2 B + (X 3 ) is in this class.\n\nIt turns out [28, 3] that the coefficients c n of X, defined by X = ∞ n=0 α n c n generate a Hopf subalgebra themselves:\n\n∆(c n ) = n k=0 P n k ⊗ c k . 8\n\nThe P n k are homogeneous polynomials of degree n-k in the c l , l ≤ n. These polynomials have been worked out explicitly in [3] . One notices in particular that the P n k are independent of the w n and B dn + , and hence that under mild assumptions (on the algebraic independence of the c n ) the Hopf subalgebras generated this way are actually isomorphic. For example,\n\nX = I + αB + (X 2 ) + α 2 B + (X 3 ) and X = I + αB + (X 2\n\n) yield isomorphic Hopf subalgebras. This is an aspect of the fact that truncation of Dyson-Schwinger equations -considering only a finite instead of an infinite number of contributing cocycles -does make (at least combinatorial) sense. Indeed, the combinatorics remain invariant. Similar results hold for Dyson-Schwinger equations in the true Hopf algebra of graphs H where things are a bit more difficult though as the cocycles there involve some bookkeeping of insertion places.\n\nThe simplest nontrivial Dyson-Schwinger equation one can think of is the linear one:\n\nX = I + αB + (X). Its solution is given by X = ∞ n=0 α n (B + ) n (I).\n\nIn this case X is grouplike and the corresponding Hopf subalgebra of c n s is cocommutative [25] . A typical and important non-linear Dyson-Schwinger equation arises from propagator insertions: X = I -αB + (1/X), for example the massless fermion propagator in Yukawa theory where only the fermion line obtains radiative corrections (other corrections are ignored). This problem has been studied and solved by Broadhurst and one of the authors in [12] and revisited recently by one of the authors and Yeats [35] . As we now turn to the analytic aspects of Dyson-Schwinger equations, we briefly sketch the general approach presented in [35] on how to successfully treat the nonlinearity of Dyson-Schwinger equations. Indeed, the linear Dyson-Schwinger equations can be solved by a simple scaling ansatz [25] . In any case, let γ be a primitive graph. The following works for amplitudes which depend on a single scale, so let us assume a massless situation with only one non-zero external momentumhow more than one external momentum (vertex insertions) are incorporated by enlarging the set of primitive elements is sketched in [32] . The grafting operator B γ + associated to γ translates to an integral operator under the (renormalized) Feynman rules\n\nφ R (B γ + )(I)(p 2 /µ 2 ) = (I γ (k, p) -I γ (k, µ))dk\n\nwhere I γ is the integral kernel corresponding to γ, the internal momenta are denoted by k, the external momentum by p, and µ is the fixed momentum at which we subtract: R(x) = x| p 2 =µ 2 .\n\nIn the following we stick to the special case discussed in [35] where only one internal edge is allowed to receive corrections. The integral kernel φ(B γ + ) defines 9 a Mellin transform\n\nF (ρ) = I γ (k, µ)(k 2 i ) -ρ dk\n\nwhere k i is the momentum of the internal edge of γ at which insertions may take place (here the fermion line). If there are several insertion sites, obvious multiple Mellin transforms become necessary. The case of two (propagator) insertion places has been studied, at the same example, in [35] .\n\nThe function F (ρ) has a simple pole in ρ at 0. We write [35] is that, even in the difficult nonlinear situation, the anomalous dimension γ 1 is implicitly defined by the residue r and Taylor coefficients f n of the Mellin transform F. On the other hand, all the γ n for n ≥ 2, are recursively defined in terms of the γ i , i < n. This last statement amounts to a renormalization group argument that is afforded in the Hopf algebra by the scattering formula of [16] . Curiously, for this argument only a linearized part of the coproduct is needed. We refer to [35] for the actual algorithm. For a linear Dyson-Schwinger equation, the situation is considerably simpler as the γ n = 0 for n ≥ 2 since X is grouplike [25].\n\nF (ρ) = r ρ + ∞ n=0 f n ρ n We denote L = log p 2 /µ 2 . Clearly φ R (X) = 1 + n γ n L n . An important result of\n\nLet us restate the results for the high energy sector of non-linear Dyson-Schwinger equations [12, 35] : Primitive graphs γ define Mellin transforms via their integral kernels Bγ + . The anomalous dimension γ 1 is implicitly determined order by order from the coefficients of those Mellin transforms. All non-leading log coefficients γ n are recursively determined by γ 1 , thanks to the renormalization group. This reduces, in principle, the problem to a study of all the primitive graphs and the intricacies of insertion places.\n\nFinding useful representations of those Mellin transforms -even one-dimensional ones -of higher loop order skeleton graphs is difficult. However, the two-loop primitive vertex in massless Yukawa theory has been worked out by Bierenbaum, Weinzierl and one of the authors in [4] , a result that can be applied to other theories as well. Combined with the algebraic treatment [12, 3, 35] sketched in the previous paragraphs and new geometric insight on primitive graphs (see section 5), there is reasonable hope that actual solutions of Dyson-Schwinger equations will be more accessible in the future.\n\nUsing the Dyson-Schwinger analysis, one of the authors and Yeats [34] were able to deduce a bound for the convergence of superficially divergent amplitudes/structure functions from the (desirable) existence of a bound for the superficially convergent amplitudes. 10 5 Feynman integrals and periods of mixed (Tate) Hodge structures\n\nA primitive graph Γ ∈ Prim H defines a real number r Γ , called the residue of Γ, which is independent of the renormalization scheme. In the case that Γ is massless and has one external momentum p, the residue r Γ is the coefficient of log p 2 /µ 2 in φ R (Γ) = (1 -R)φ(Γ). It coincides with the coefficient r of the Mellin transform introduced in the previous section. One may ask what kind of a number r is, for example if it is rational or algebraic. The origin of this question is that the irrational or transcendental numbers that show up for various Γ strongly suggest a motivic interpretation of the r Γ . Indeed, explicit calculations [9, 10, 8] display patterns of Riemann zeta and multiple zeta values that are known to be periods of mixed Tate Hodge structures -here the periods are provided by the Feynman rules which produce Γ → r Γ . By disproving a related conjecture of Kontsevich, Belkale and Brosnan [1] have shown that not all these Feynman motives must be mixed Tate, so one may expect a larger class of Feynman periods than multiple zeta values. Our detailed understanding of these phenomena is still far from complete, and only some very first steps have been made in the last few years. However, techniques developed in recent work by Bloch, Esnault and one of the authors [7] do permit reasonable insight for some special cases which we briefly sketch in the following.\n\nLet Γ be a logarithmically divergent massless primitive graph with one external momentum p. It is convenient to work in the \"Schwinger\" parametric representation [24] obtained by the usual trick of replacing propagators\n\n1 k 2 = ∞ 0 dae -ak 2 ,\n\nand performing the loop integrations (Gaussian integrals) first which leaves us with a (divergent) integral over various Schwinger parameters a. It is a classical exercise [24, 7, 6] to show that in four dimensions, up to some powers of i and 2π,\n\nφ(Γ) = ∞ 0 da 1 . . . da n e -QΓ(a,p 2 )/ΨΓ(a) Ψ 2 Γ (a)\n\nwhere n is the number of edges of Γ. Q Γ and Ψ Γ are graph polynomials of Γ, where Ψ Γ , sometimes called Symanzik or Kirchhoff polynomial, is defined as follows: Let T (Γ) be the set of spanning trees of Γ, i. e. the set of connected simply connected subgraphs which meet all vertices of Γ. We think of the edges e of Γ as being numbered from 1 to n. Then Ψ Γ = t∈T (Γ) e ∈t a e 11 This is a homogeneous polynomial in the\n\na i of degree |H 1 (Γ)|. It is easily seen (scaling behaviour of Q Γ and Ψ Γ ) that r Γ = ∂φR(Γ)\n\n∂ log p 2 /µ 2 is extracted from φ(Γ) by considering the a i as homogeneous coordinates of P n-1 (R) and evaluating at p 2 = 0 :\n\nr Γ = σ⊂P n-1 (R) Ω Ψ 2 Γ ( 10\n\n)\n\nwhere σ = {[a 1 , . . . , a n ] : all a i can be choosen ≥ 0} and Ω is a volume form on\n\nP n-1 . Let X Γ := {Ψ Γ = 0} ⊂ P n-1 . If |H 1 (Γ)| = 1\n\n, the integrand in (10) has no poles. If |H 1 (Γ)| > 1, poles will show up on the union ∆ = γ Γ,H1(γ) =0 L γ of coordinate linear spaces γ = {a e = 0 for e edge of γ} -these need to be separated from the chain of integration by blowing up. The blowups being understood, the Feynman motive is, by abuse of notation,\n\nH n-1 (P n-1 -X Γ , ∆ -∆ ∩ X Γ )\n\nwith Feynman period given by (10) . See [7, 6] for details. Some particularly accessible examples are the wheel with n spokes graphs Γ n := studied extensively in [7] . The corresponding Feynman periods (10) yield rational multiples of zeta values [9] r Γn ∈ ζ(2n -3)Q.\n\nDue to the simple topology of the Γ n , the geometry of the pairs (X Γn , ∆ Γn ) are well understood and the corresponding motives have been worked out explicitly [7] . The methods used are however nontrivial and not immediately applicable to more general situations.\n\nWhen confronted with non-primitive graphs, i. e. graphs with subdivergences, there are more than one period to consider. In the Schwinger parameter picture, subdivergences arise when poles appear along exceptional divisors as pieces of ∆ are blown up. This situation can be understood using limiting mixed Hodge structures [6] , see also [31, 36] for a toy model approach to the combinatorics involved. In [6] it is also shown how the Hopf algebra H of graphs lifts to the category of motives. For the motivic role of solutions of Dyson-Schwinger equations we refer to work in progress. Finally we mention that there is related work by Connes and Marcolli [18, 19] who attack the problem via Riemann-Hilbert correspondences and motivic Galois theory.\n\nAcknowledgements. We thank Spencer Bloch and Karen Yeats for discussion on the subject of this review. The first named author (C. B.) thanks the organizers of the ICMP 2006 and the IHES for general support. His research is 12 supported by the Deutsche Forschungsgemeinschaft. The IHES, Boston University and the Erwin-Schrödinger-Institute are gratefully acknowledged for their kind hospitality. At the time of writing this article, C. B. is visiting the ESI as a Junior Research Fellow."
}
] | ||
arxiv:0704.0240 | 0704.0240 | 1 | 10.1146/annurev.nucl.57.090506.123120 | 8f74bfa02fac7122abfb05d8312aa2a50217ef778782bdd953a27059735c323b | Viscosity, Black Holes, and Quantum Field Theory | We review recent progress in applying the AdS/CFT correspondence to
finite-temperature field theory. In particular, we show how the hydrodynamic
behavior of field theory is reflected in the low-momentum limit of correlation
functions computed through a real-time AdS/CFT prescription, which we
formulate. We also show how the hydrodynamic modes in field theory correspond
to the low-lying quasinormal modes of the AdS black p-brane metric. We provide
a proof of the universality of the viscosity/entropy ratio within a class of
theories with gravity duals and formulate a viscosity bound conjecture.
Possible implications for real systems are mentioned. | [
"D. T. Son",
"A. O. Starinets"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | 1 2 Son, Starinets
This review is about the recently emerging connection, through the gauge/gravity correspondence, between hydrodynamics and black hole physics. The study of quantum field theory at high temperature has a long history. It was first motivated by the Big Bang cosmology when it was hoped that early phase transitions might leave some imprints on the Universe [1] . One of those phase transitions is the QCD phase transitions (which could actually be a crossover) which happened at a temperature around T c ∼ 200 MeV, when matter turned from a gas of quarks and gluons (the quark-gluon plasma, or QGP) into a gas of hadrons.
An experimental program was designed to create and study the QGP by colliding two heavy atomic nuclei. Most recent experiments are conducted at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory. Although significant circumstantial evidence for the QGP was accumulated [2] , a theoretical interpretation of most of the experimental data proved difficult, because the QGP created at RHIC is far from being a weakly coupled gas of quarks and gluons. Indeed, the temperature of the plasma, as inferred from the spectrum of final particles, is only approximately 170 MeV, near the confinement scale of QCD. This is deep in the nonperturbative regime of QCD, where reliable theoretical tools are lacking. Most notably, the kinetic coefficients of the QGP, which enter the hydrodynamic equations (reviewed in Sec. 2), are not theoretically computable at these temperatures.
The paucity of information about the kinetic coefficients of the QGP in particular and of strongly coupled thermal quantum field theories in general is one of the main reasons for our interest in their computation in a class of strongly coupled field theories, even though this class does not include QCD. The necessary technological tool is the anti-de Sitter-conformal field theory (AdS/CFT) correspondence [3, 4, 5] , discovered in the investigation of D-branes in string theory. This correspondence allows one to describe the thermal plasmas in these theories in terms of black holes in AdS space. The AdS/CFT correspondence is reviewed in Sec. 3. The first calculation of this type, that of the shear viscosity in N = 4 supersymmetric Yang-Mills (SYM) theory [6] , is followed by the theoretical work to establish the rules of real-time finite-temperature AdS/CFT correspondence [7, 8] . Applications of these rules to various special cases [9, 10, 11, 12] clearly show that even very exotic field theories, when heated up to finite temperature, behave hydrodynamically at large distances and time scales (provided that the number of space-time dimensions is 2+1 or higher). This development is reviewed in Sec. 4. Moreover, the way AdS/CFT works reveals deep connections to properties of black holes in classical gravity. For example, the hydrodynamic modes of a thermal medium are mapped, through the correspondence, to the low-lying quasi-normal modes of a black-brane metric. It seems that our understanding of the connection between hydrodynamics and black hole physics is still incomplete; we may understand more about gravity by studying thermal field theories. One idea along this direction is reviewed in Sec. 5.
From the point of view of heavy-ion (QGP) physics, a particularly interesting finding has been the formulation of a conjecture on the lowest possible value of the ratio of viscosity and volume density of entropy. This conjecture was motivated by the universality of this ratio in theories with gravity duals. This is reviewed in Sec. 6.
Viscosity, Black Holes, and QFT 3 2 HYDRODYNAMICS
From the modern perspective, hydrodynamics [13] is best thought of as an effective theory, describing the dynamics at large distances and time-scales. Unlike the familiar effective field theories (for example, the chiral perturbation theory), it is normally formulated in the language of equations of motion instead of an action principle. The reason for this is the presence of dissipation in thermal media.
In the simplest case, the hydrodynamic equations are just the laws of conservation of energy and momentum,
∂ µ T µν = 0 . ( 1
)
To close the system of equations, we must reduce the number of independent elements of T µν . This is done through the assumption of local thermal equilibrium: If perturbations have long wavelengths, the state of the system, at a given time, is determined by the temperature as a function of coordinates T (x) and the local fluid velocity u µ , which is also a function of coordinates u µ (x). Because u µ u µ = -1, only three components of u µ are independent. The number of hydrodynamic variables is four, equal to the number of equations.
In hydrodynamics we express T µν through T (x) and u µ (x) through the so-called constitutive equations. Following the standard procedure of effective field theories, we expand in powers of spatial derivatives. To zeroth order, T µν is given by the familiar formula for ideal fluids,
T µν = (ǫ + P )u µ u ν + P g µν , ( 2
)
where ǫ is the energy density, and P is the pressure. Normally one would stop at this leading order, but qualitatively new effects necessitate going to the next order. Indeed, from Eq. 2 and the thermodynamic relations dǫ = T dS, dP = sdT , and ǫ + P = T s (s is the entropy per unit volume), one finds that entropy is conserved [14] ∂ µ (su µ ) = 0 . (3) Thus, to have entropy production, one needs to go to the next order in the derivative expansion. At the next order, we write
T µν = (ǫ + P )u µ u ν + P g µν -σ µν , ( 4
)
where σ µν is proportional to derivatives of T (x) and u µ (x) and is termed the dissipative part of T µν . To write these terms, let us first fix a point x and go to the local rest frame where u i (x) = 0. 4 Son, Starinets In this frame, in principle one can have dissipative corrections to the energy-momentum density T 0µ . However, one recalls that the choice of T and u µ is arbitrary, and thus one can always redefine them so that these corrections vanish, σ 00 = σ 0i = 0, and so at a point x,
T 00 = ǫ, T 0i = 0 . ( 5
)
The only nonzero elements of the dissipative energy-momentum tensor are σ ij . To the next-toleading order there are extra contributions whose forms are dictated by rotational symmetry:
σ ij = η ∂ i u j + ∂ j u i - 2 3 δ ij ∂ k u k + ζδ ij ∂ k u k . ( 6
)
Going back to the general frame, we can now write the dissipative part of the energy-momentum tensor as
σ µν = P µα P νβ η ∂ α u β + ∂ β u α - 2 3 g αβ ∂ λ u λ + ζg αβ ∂ λ u λ , ( 7
)
where P µν = g µν + u µ u ν is the projection operator onto the directions perpendicular to u µ . If the system contains a conserved current, there is an additional hydrodynamic equation related to the current conservation,
∂ µ j µ = 0 . ( 8
)
The constitutive equation contains two terms:
j µ = ρu µ -DP µν ∂ ν α , ( 9
)
where ρ is the charge density in the fluid rest frame and D is some constant. The first term corresponds to convection, the second one to diffusion. In the fluid rest frame, j = -D∇ρ, which is Fick's law of diffusion, with D being the diffusion constant.
As mentioned above, the hydrodynamic equations can be thought of as an effective theory describing the dynamics of the system at large lengths and time scales. Therefore one should be able to use these equations to extract information about the low-momentum behavior of Green's functions in the original theory. For example, let us recall how the two-point correlation functions can be extracted. If we couple sources J a (x) to a set of (bosonic) operators O a (x), so that the new action is
S = S 0 + x J a (x)O a (x) , ( 10
)
then the source will introduce a perturbation of the system. In particular, the average values of O a will differ from the equilibrium values, which we assume to be zero. If J a are small, the perturbations are given by the linear response theory as
O a (x) = - y G R ab (x -y)J b (y) , ( 11
)
where G R ab is the retarded Green's function
iG R ab (x -y) = θ(x 0 -y 0 ) [O a (x), O b (y)] . ( 12
)
Viscosity, Black Holes, and QFT 5 The fact that the linear response is determined by the retarded (and not by any other) Green's function is obvious from causality: The source can influence the system only after it has been turned on. Thus, to determine the correlation functions of T µν , we need to couple a weak source to T µν and determine the average value of T µν after this source is turned on. To find these correlators at low momenta, we can use the hydrodynamic theory. So far in our treatment of hydrodynamics we have included no source coupled to T µν . This deficiency can be easily corrected, as the source of the energy-momentum tensor is the metric g µν . One must generalize the hydrodynamic equations to curved space-time and from it determine the response of the thermal medium to a weak perturbation of the metric. This procedure is rather straightforward and in the interest of space is left as an exercise to the reader.
Here we concentrate on a particular case when the metric perturbation is homogeneous in space but time dependent:
g ij (t, x) = δ ij + h ij (t), h ij ≪ 1 ( 13
)
g 00 (t, x) = -1, g 0i (t, x) = 0 . (14)
Moreover, we assume the perturbation to be traceless, h ii = 0. Because the perturbation is spatially homogeneous, if the fluid moves, it can only move uniformly: u i = u i (t). However, this possibility can be ruled out by parity, so the fluid must remain at rest all the time: u µ = (1, 0, 0, 0). We now compute the dissipative part of the stress-energy tensor. The generalization of Eq. 7 to curved space-time is
σ µν = P µα P νβ η(∇ α u β + ∇ β u α ) + ζ - 2 3 η g αβ ∇ • u . ( 15
)
Substituting u µ = (1, 0, 0, 0) and g µν from Eq. 13, we find only contributions to the traceless spatial components, and these contributions come entirely from the Christoffel symbols in the covariant derivatives. For example,
σ xy = 2ηΓ 0 xy = η∂ 0 h xy . ( 16
)
By comparison with the expectation from the linear response theory, this equation means that we have found the zero spatial momentum, low-frequency limit of the retarded Green's function of
T xy : G R xy,xy (ω, 0) = dt dx e iωt θ(t) [T xy (t, x), T xy (0, 0)] = -iηω + O(ω 2 ) ( 17
)
(modulo contact terms). We have, in essence, derived the Kubo's formula relating the shear viscosity and a Green's function:
η = -lim ω→0 1 ω Im G R xy,xy (ω, 0) . ( 18
)
There is a similar Kubo's relation for the charge diffusion constant D.
If one is interested only in the locations of the poles of the correlators, one can simply look for the normal modes of the linearized hydrodynamic equations, that is, solutions that behave as e -iωt+ik•x . Owing to dissipation, the frequency ω(k) is complex. For example, the equation of charge diffusion,
∂ t ρ -D∇ 2 ρ = 0, ( 19
)
6 Son, Starinets corresponds to a pole in the current-current correlator at ω = -iDk 2 .
To find the poles in the correlators between elements of the stress-energy tensor one can, without loss of generality, choose the coordinate system so that k is aligned along the x 3 -axis: k = (0, 0, k).
Then one can distinguish two types of normal modes: 1. Shear modes correspond to the fluctuations of pairs of components T 0a and T 3a , where a = 1, 2. The constitutive equation is
T 3a = -η∂ 3 u a = - η ǫ + P ∂ 3 T 0a , ( 20
)
and the equation for
T 0a is ∂ t T 0a - η ǫ + P ∂ 2 3 T 0a = 0 . ( 21
)
That is, it has the form of a diffusion equation for T 0a . Substituting e -iωt+ikx 3 into the equation, one finds the dispersion law
ω = -i η ǫ + P k 2 . ( 22
)
2. Sound modes are fluctuations of T 00 , T 03 , and T 33 . There are now two conservation equations, and by diagonalizing them one finds the dispersion law
ω = c s k - i 2 4 3 η + ζ k 2 ǫ + P , ( 23
)
where c s = dP/dǫ. This is simply the sound wave, which involves the fluctuation of the energy density. It propagates with velocity c s , and its damping is related to a linear combination of shear and bulk viscosities.
In CFTs it is possible to use conformal Ward identities to show that the bulk viscosity vanishes: ζ = 0. Hence, we shall concentrate our attention on the shear viscosity η.
We now briefly consider the behavior of the shear viscosity in weakly coupled field theories, with the λφ 4 theory as a concrete example. At weak coupling, there is a separation between two length scales: The mean free path of particles is much larger than the distance scales over which scatterings occur. Each scattering event takes a time of order T -1 (which can be thought of as the time required for final particles to become on-shell). The mean free path ℓ mfp can be estimated from the formula
ℓ mfp ∼ 1 nσv , ( 24
)
where n is the density of particles, σ is the typical scattering cross section, and v is the typical particle velocity. Inserting the values for thermal λφ 4 theory, n ∼ T 3 , σ ∼ λ 2 T -2 , and v ∼ 1, one finds
ℓ mfp ∼ 1 λ 2 T ≫ 1 T . ( 25
)
The viscosity can be estimated from kinetic theory to be
η ∼ ǫℓ mfp , ( 26
)
Viscosity, Black Holes, and QFT 7 where ǫ is the energy density. From ǫ ∼ T 4 and the estimate of ℓ mft , one finds
η ∼ T 3 λ 2 . ( 27
)
In particular, the weaker the coupling λ, the larger the viscosity η. This behavior is explained by the fact that the viscosity measures the rate of momentum diffusion. The smaller λ is, the longer a particle travels before colliding with another one, and the easier the momentum transfer. It may appear counterintuitive that viscosity tends to infinity in the limit of zero coupling λ → 0: At zero coupling there is no dissipation, so should the viscosity be zero? The confusion arises owing to the fact that the hydrodynamic theory, and hence the notion of viscosity, makes sense only on distances much larger than the mean free path of particles. If one takes λ → 0, then to measure the viscosity one has to do the experiment at larger and larger length scales. If one fixes the size of the experiment and takes λ → 0, dissipation disappears, but it does not tell us anything about the viscosity.
As will become apparent below, a particularly interesting ratio to consider is the ratio of shear viscosity and entropy density s. The latter is proportional to T 3 ; thus η s ∼ 1 λ 2 . (28) One has η/s ≫ 1 for λ ≪ 1. This is a common feature of weakly coupled field theories. Extrapolating to λ ∼ 1, one finds η/s ∼ 1. We shall see that theories with gravity duals are strongly coupled, and η/s is of order one. More surprisingly, this ratio is the same for all theories with gravity duals.
To compute rather than estimate the viscosity, one can use Kubo's formula. It turns out that one has to sum an infinite number of Feynman graphs to even find the viscosity to leading order. Another way that leads to the same result is to first formulate a kinetic Boltzmann equation for the quasi-particles as an intermediate effective description, and then derive hydrodynamics by taking the limit of very long lengths and time scales in the kinetic equation. Interested readers should consult Refs. [15, 16] for more details.
3 AdS/CFT CORRESPONDENCE
This section briefly reviews the AdS/CFT correspondence at zero temperature. It contains only the minimal amount of materials required to understand the rest of the review. Further information can be found in existing reviews and lecture notes [17, 18] .
The original example of AdS/CFT correspondence is between N = 4 supersymmetric Yang-Mills (SYM) theory and type IIB string theory on AdS 5 ×S 5 space. Let us describe the two sides of the correspondence in some more detail.
The N = 4 SYM theory is a gauge theory with a gauge field, four Weyl fermions, and six real scalars, all in the adjoint representation of the color group. Its Lagrangian can be written down explicitly, but is not very important for our purposes. It has a vanishing beta function and is a conformal field theory (CFT) (thus the CFT in AdS/CFT). In our further discussion, we frequently use the generic terms "field theory" or CFT for the N = 4 SYM theory.
8 Son, Starinets On the string theory side, we have type IIB string theory, which contains a finite number of massless fields, including the graviton, the dilaton Φ, some other fields (forms) and their fermionic superpartners, and an infinite number of massive string excitations. It has two parameters: the string length l s (related to the slope parameter α ′ by α ′ = l 2 s ) and the string coupling g s . In the long-wavelength limit, when all fields vary over length scales much larger than l s , the massive modes decouple and one is left with type IIB supergravity in 10 dimensions, which can be described by an action [19]
S SUGRA = 1 2κ 2 10 d 10 x √ -g e -2Φ (R + 4 ∂ µ Φ∂ µ Φ + • • •) , ( 29
)
where κ 10 is the 10-dimensional gravitational constant,
κ 10 = √ 8πG = 8π 7/2 g s l 4 s , ( 30
)
and • • • stay for the contributions from fields other than the metric and the dilaton. One of these fields is the five-form F 5 , which is constrained to be self-dual. The type IIB string theory lives is a 10-dimensional space-time with the following metric:
ds 2 = r 2 R 2 (-dt 2 + dx 2 ) + R 2 r 2 dr 2 + R 2 dΩ 2 5 . ( 31
)
The metric is a direct product of a five-dimensional sphere (dΩ 2 5 ) and another five-dimensional space-time spanned by t, x, and r. An alternative form of the metric is obtained from Eq. (31) by a change of variable z = R 2 /r,
ds 2 = R 2 z 2 (-dt 2 + dx 2 + dz 2 ) + R 2 dΩ 2 5 . ( 32
)
Both coordinates r and z are known as the radial coordinate. The limiting value r = ∞ (or z = 0) is the boundary of the AdS space. It is a simple exercise to check that the (t, x, r) part of the metric is a space with constant negative curvature, or an anti de-Sitter (AdS) space. To support the metric (31) (i.e., to satisfy the Einstein equation) there must be some background matter field that gives a stress-energy tensor in the form of a negative cosmological constant in AdS 5 and a positive one in S 5 . Such a field is the self-dual five-form field F 5 mentioned above.
Field theory has two parameters: the number of colors N and the gauge coupling g. When the number of colors is large, it is the 't Hooft coupling λ = g 2 N that controls the perturbation theory. On the string theory side, the parameters are g s , l s , and radius R of the AdS space. String theory and field theory each have two dimensionless parameters which map to each other through the following relations:
g 2 = 4πg s , ( 33
)
g 2 N c = R 4 l 4 s . ( 34
)
Equation ( 33) tells us that, if one wants to keep string theory weakly interacting, then the gauge coupling in field theory must be small. Equation (34) is particularly interesting. It says that the large 't Hooft coupling limit in field theory corresponds to the limit when the curvature radius of Viscosity, Black Holes, and QFT 9 space-time is much larger than the string length l s . In this limit, one can reliably decouple the massive string modes and reduce string theory to supergravity. In the limit g s ≪ 1 and R ≫ l s , one has classical supergravity instead of string theory. The practical utility of the AdS/CFT correspondence comes, in large part, from its ability to deal with the strong coupling limit in gauge theory. One can perform a Kaluza-Klein reduction [20] by expanding all fields in S 5 harmonics. Keeping only the lowest harmonics, one finds a five-dimensional theory with the massless dilaton, SO(6) gauge bosons, and gravitons [21]:
S 5D = N 2 8π 2 R 3 d 5 x R 5D -2Λ - 1 2 ∂ µ Φ∂ µ Φ - R 2 8 F a µν F aµν + • • • . ( 35
)
In AdS/CFT, an operator O of field theory is put in a correspondence with a field φ ("bulk" field) in supergravity. We elaborate on this correspondence below; here we keep the operator and the field unspecified. In the supergravity approximation, the mathematical statement of the correspondence is
Z 4D [J] = e iS[φ cl ] . ( 36
)
On the left is the partition function of a field theory, where the source J coupled to the operator O is included:
Z 4D [J] = Dφ exp iS + i d 4 x JO . ( 37
)
On the right, S[φ cl ] is the classical action of the classical solution φ cl to the field equation with the boundary condition:
lim z→0 φ cl (z, x) z ∆ = J(x) . ( 38
)
Here ∆ is a constant that depends on the nature of the operator O (namely, on its spin and dimension). In the simplest case, ∆ = 0, and the boundary condition becomes φ cl (z=0) = J. Differentiating Eq. (36) with respect to J, one can find the correlation functions of O. For example, the two-point Green's function of O is obtained by differentiating S cl [φ] twice with respect to the boundary value of φ,
G(x -y) = -i T O(x)O(y) = - δ 2 S[φ cl ] δJ(x)δJ(y) φ(z=0)=J . ( 39
)
The AdS/CFT correspondence thus maps the problem of finding quantum correlation functions in field theory to a classical problem in gravity. Moreover, to find two-point correlation functions in field theory, one can be limited to the quadratic part of the classical action on the gravity side. The complete operator to field mapping can be found in Refs. [5, 17] . For our purpose, the following is sufficient:
• The dilaton Φ corresponds to O = -L = 1 4 F 2 µν + • • •,
where L is the Lagrangian density. • The gauge field A a µ corresponds to the conserved R-charge current J aµ of field theory. • The metric tensor corresponds to the stress-energy tensor T µν . More precisely, the partition function of the four-dimensional field theory in an external metric g 0 µν is equal to
Z 4D [g 0 µν ] = exp(iS cl [g µν ]) , ( 40
)
10 Son, Starinets where the five-dimensional metric g µν satisfies the Einstein's equations and has the following asymptotics at z = 0:
ds 2 = g µν dx µ dx ν = R 2 z 2 (dz 2 + g 0 µν dx µ dx ν ) . ( 41
)
From the point of view of hydrodynamics, the operator 1 4 F 2 is not very interesting because its correlator does not have a hydrodynamic pole. In contrast, we find the correlators of the R-charge current and the stress-energy tensor to contain hydrodynamic information.
We simplify the graviton part of the action further. Our two-point functions are functions of the momentum p = (ω, k). We can choose spatial coordinates so that k points along the x 3 -axis. This corresponds to perturbations that propagate along the x 3 direction: h µν = h µν (t, r, x 3 ). These perturbations can be classified according to the representations of the O(2) symmetry of the (x 1 , x 2 ) plane. Owing to that symmetry, only certain components can mix; for example, h 12 does not mix with any other components, whereas components h 01 and h 31 mix only with each other. We assume that only these three metric components are nonzero and introduce shorthand notations
φ = h 1 2 , a 0 = h 1 0 , a 3 = h 1 3 . ( 42
)
The quadratic part of the graviton action acquires a very simple form in terms of these fields:
S quad = N 2 8π 2 R 3 d 4 x dr √ -g - 1 2 g µν ∂ µ φ∂ ν φ - 1 4g 2 eff g µα g νβ f µν f αβ , ( 43
) where f µν = ∂ µ a ν -∂ ν a µ , and g 2 eff = g xx .
In deriving Eq. ( 43 ), our only assumption about the metric is that it has a diagonal form,
ds 2 = g tt dt 2 + g rr dr 2 + g xx dx 2 , ( 44
)
so it can also be used below for the finite-temperature metric. As a simple example, let us compute the two-point correlation function of T xy , which corresponds to φ in gravity. The field equation for φ is
∂ µ ( √ -g g µν ∂ ν φ) = 0 . ( 45
)
The solution to this equation, with the boundary condition φ(p, z = 0) = φ 0 (p), can be written as
φ(p, z) = f p (z)φ 0 (p) , ( 46
)
where the mode function f p (z) satisfies the equation
f ′ p z 3 ′ - p 2 z 3 f p = 0 ( 47
)
with the boundary condition f p (0) = 1. The mode equation (47) can be solved exactly. Assuming p is spacelike, p 2 > 0, the exact solution and its expansion around z = 0 is
f p (z) = 1 2 (pz) 2 K 2 (pz) = 1 - 1 4 (pz) 2 - 1 16 (pz) 4 ln(pz) + O((pz) 4 ) . ( 48
)
The second solution to Eq. ( 47 ), (pz) 2 I 2 (pz), is ruled out because it blows up at z → ∞.
Viscosity, Black Holes, and QFT 11 We now substitute the solution into the quadratic action. Using the field equation, one can perform integration by parts and write the action as a boundary integral at z = 0. One finds
S = N 2 16π 2 d 4 x 1 z 3 φ(x, z)φ ′ (x, z)| z→0 = d 4 p (2π) 4 φ 0 (-p)F(p, z)φ 0 (p)| z→0 , ( 49
)
where
F(p, z) = N 2 16π 2 1 z 3 f -p (z)∂ z f p (z) . ( 50
)
Differentiating the action twice with respect to the boundary value φ 0 one finds
T xy T xy p = -2 lim z→0 F(p, z) = N 2 64π 2 p 4 ln(p 2 ) . ( 51
)
Note that we have dropped the term ∼ p 4 ln z, which, although singular in the limit z → 0, is a contact term [i.e., a term proportional to a derivative of δ(x) after Fourier transform]. Removing such terms by adding local counter terms to the supergravity action is known as the holographic renormalization [22] . It is, in a sense, a holographic counterpart to the standard renormalization procedure in quantum field theory, here applied to composite operators. For time-like p, p 2 < 0, there are two solutions to Eq. (47) which involve Hankel functions H (1) (z) and H (2) (z) instead of K 2 (z). Neither function blows up at z → ∞, and it is not clear which should be picked. Here we encounter, for the first time, a subtlety of Minkowski-space AdS/CFT, which is discussed in great length in subsequent sections. At zero temperature this problem can be overcome by an analytic continuation from space-like p. However, this will not work at nonzero temperatures.
At nonzero temperatures, the metric dual to N = 4 SYM theory is the black three-brane metric,
ds 2 = r 2 R 2 (-f dt 2 + dx 2 ) + R 2 r 2 f dr 2 + R 2 dΩ 2 5 , ( 52
) with f = 1 -r 4 0 /r 4 .
The event horizon is located at r = r 0 , where f = 0. In contrast to the usual Schwarzschild black hole, the horizon has three flat directions x. The metric (52) is thus called a black three-brane metric.
We frequently use an alternative radial coordinate u, defined as u = r 2 0 /r 2 . In terms of u, the boundary is at u = 0, the horizon at u = 1, and the metric is
ds 2 = (πT R) 2 u 2 (-f (u)dt 2 + dx 2 ) + R 2 4u 2 f (u) du 2 + R 2 dΩ 2 5 . ( 53
)
The Hawking temperature is determined completely by the behavior of the metric near the horizon. Let us concentrate on the (t, r) part of the metric,
ds 2 = - 4r 0 R 2 (r -r 0 )dt 2 + R 2 4r 0 (r -r 0 ) dr 2 . ( 54
)
Changing the radial variable from r to ρ,
r = r 0 + ρ 2 r 0 , ( 55
)
12 Son, Starinets and the metric components become nonsingular:
ds 2 = R 2 r 2 0 dρ 2 - 4r 2 0 R 2 ρ 2 dt 2 . ( 56
)
Note also that after a Wick rotation to Euclidean time τ , the metric has the form of the flat metric in cylindrical coordinates,
ds 2 ∼ dρ 2 + ρ 2 dϕ 2 , where ϕ = 2r 0 R -2 τ .
To avoid a conical singularity at ρ = 0, ϕ must be a periodic variable with periodicity 2π. This fact matches with the periodicity of the Euclidean time in thermal field theory τ ∼ τ + 1/T , from which one finds the Hawking temperature:
T H = r 0 πR 2 . ( 57
)
One of the first finite-temperature predictions of AdS/CFT correspondence is that of the thermodynamic potentials of the N = 4 SYM theory in the strong coupling regime. The entropy is given by the Bekenstein-Hawking formula S = A/(4G), where A is the area of the horizon of the metric (52); the result can then be converted to parameters of the gauge theory using Eqs. (30), (33), and (34) . One obtains
s = S V = π 2 2 N 2 T 3 , ( 58
)
which is 3/4 of the entropy density in N = 4 SYM theory at zero 't Hooft coupling. We now try to generalize the AdS/CFT prescription to finite temperature. In the Euclidean formulation of finite-temperature field theory, field theory lives in a space-time with the Euclidean time direction τ compactified. The metric is regular at r = r 0 : If one views the (τ, r) space as a cigar-shaped surface, then the horizon r = r 0 is the tip of the cigar. Thus, r 0 is the minimal radius where the space ends, and there is no point in space with r less than r 0 . The only boundary condition at r = r 0 is that fields are regular at the tip of the cigar, and the AdS/CFT correspondence is formulated as
Z 4D [J] = Z 5D [φ]| φ(z=0)→J . ( 59
)
4 REAL-TIME AdS/CFT
In many cases we must find real-time correlation functions not given directly by the Euclidean pathintegral formulation of thermal field theory. One example is the set of kinetic coefficients expressed, through Kubo's formulas, via a certain limit of real-time thermal Green's functions. Another related example appears if we want to directly find the position of the poles in the correlation functions that would correspond to the hydrodynamic modes. In principle, some real-time Green's functions can be obtained by analytic continuation of the Euclidean ones. For example, an analytic continuation of a two-point Euclidean propagator gives a retarded or advanced Green's function, depending on the way one performs the continuation. However, it is often very difficult to directly compute a quantity of interest in that way. In particular, it is very difficult to get the information about the hydrodynamic (small ω, small k) limit of real-time correlators from Euclidean propagators. The problem here is that we need to perform an analytic continuation from a discrete set of points in Euclidean frequencies (the Matsubara frequencies) ω = 2πin, where n is an integer, to the real values of ω. In the hydrodynamic limit, we are interested in real and small ω, whereas the smallest Matsubara frequency is already 2πT .
Viscosity, Black Holes, and QFT 13 Therefore, we need a real-time AdS/CFT prescription that would allow us to directly compute the real-time correlators. However, if one tries to naively generalize the AdS/CFT prescription, one immediately faces a problem. Namely, now r = r 0 is not the end of space but just the location of the horizon. Without specifying a boundary condition at r = r 0 , there is an ambiguity in defining the solution to the field equations, even as the boundary condition at r = ∞ is set.
As an example, let us consider the equation of motion of a scalar field in the black hole background, ∂ µ (g µν ∂ ν φ) = 0. The solution to this equation with the boundary condition φ = φ 0 at u = 0 is φ(p, u) = f p φ 0 (p), where f p (u) satisfies the following equation in the metric (53):
f ′′ p - 1 + u 2 uf f ′ p + w 2 uf 2 f p - q 2 uf f p = 0 . ( 60
)
Here the prime denotes differentiation with respect to u, and we have defined the dimensionless frequency and momentum:
w = ω 2πT , q = k 2πT . ( 61
)
Near u = 0 the equation has two solutions, f 1 ∼ 1 and f 2 ∼ u 2 . In the Euclidean version of thermal AdS/CFT, there is only one regular solution at the horizon u = 1, which corresponds to a particular linear combination of f 1 and f 2 . However, in Minkowski space there are two solutions, and both are finite near the horizon. One solution termed f p behaves as (1u) -iw/2 , and the other is its complex conjugate f * p ∼ (1u) iw/2 . These two solutions oscillate rapidly as u → 1, but the amplitude of the oscillations is constant. Thus, the requirement of finiteness of f p allows for any linear combination of f 1 and f 2 near the boundary, which means that there is no unique solution to Eq. (60).
Physically, the two solutions f p and f * p have very different behavior. Restoring the e -iωt phase in the wave function, one can write
e -iωt f p ∼ e -iω(t+r * ) , ( 62
)
e -iωt f * p ∼ e iω(t-r * ) , ( 63
)
where the coordinate r * = ln(1u) 4πT (64) was introduced so that Eqs. (62) and (63) looked like plane waves. In fact, Eq. (62) corresponds to a wave that moves toward the horizon (incoming wave) and Eq. ( 63 ) to a wave that moves away from the horizon (outgoing wave). The simplest idea, which is motivated by the fact that nothing should come out of a horizon, is to impose the incoming-wave boundary condition at r = r 0 and then proceed as instructed by the AdS/CFT correspondence. However, now we encounter another problem. If we write down the classical action for the bulk field, after integrating by parts we get contributions from both the boundary and the horizon:
S = d 4 p (2π) 4 φ 0 (-p)F(p, z)φ 0 (p) z=z H z=0 . ( 65
)
14 Son, Starinets If one tried to differentiate the action with respect to the boundary value φ 0 , one would find
G(p) = F(p, z)| z H 0 + F(-p, z)| z H 0 . ( 66
)
From the equation satisfied by f p and from f * p = f -p , it is easy to show that the imaginary part of F(p, z) does not depend on z; hence the quantity G(p) in Eq. (66) is real. This is clearly not what we want, as the retarded Green's functions are, in general, complex. Simply throwing away the contribution from the horizon does not help because F(-p, z) = F * (p, z) owing to the reality of the equation satisfied by f p .
A partial solution to this problem was suggested in Ref. [7] . It was postulated that the retarded Green's function is related to the function F by the same formula that was found at zero temperature: G R (p) = -2 lim z→0 F(p, z) . (67)
In particular, we throw away all contributions from the horizon. This prescription was established more rigorously in Ref. [8] (following an earlier suggestion in Ref. [23] ) as a particular case of a general real-time AdS/CFT formulation, which establishes the connection between the close-timepath formulation of real-time quantum field theory with the dynamics of fields in the whole Penrose diagram of the AdS black brane. Here we accept Eq. (67) as a postulate and proceed to extract physical results from it. It is also easy to generalize this prescription to the case when we have more than one field. In that case, the quantity F becomes a matrix F ab , whose elements are proportional to the retarded Green's function G ab .
As an illustration of the real-time AdS/CFT correspondence, we compute the correlator of T xy . First we write down the equation of motion for φ = h x y :
φ ′′ p - 1 + u 2 uf φ ′ p + w 2 -q 2 f uf 2 φ p = 0 . ( 68
)
In contrast to the zero-temperature equation, now ω and k enter the equation separately rather than through the combination ω 2k 2 . Thus the Green's function will have no Lorentz invariance. The equation cannot be solved exactly for all ω and k. However, when ω and k are both much smaller than T , or w, q ≪ 1, one can develop series expansion in powers of w and q. There are two solutions that are complex conjugates of each other. The solution that is an incoming wave at u = 1 and normalized to 1 at u = 0 is
f p (z) = (1 -u 2 ) -iw/2 + O(w 2 , q 2 ) . ( 69
)
The kinetic term in the action for φ is
S = - π 2 N 2 T 4 8 du f u φ ′2 . ( 70
)
Applying the general formula (67), one finds the retarded Green's function of T xy ,
G R xy,xy (ω, k) = - π 2 N 2 T 4 4 iw , ( 71
)
Viscosity, Black Holes, and QFT 15 and, using Kubo's formula for η, the viscosity,
η = π 8 N 2 T 3 . ( 72
)
It is instructive to compute other correlators that have poles corresponding to hydrodynamic modes. As a warm-up, let us compute the two-point correlators of the R-charge currents, which should have a pole at ω = -iDk 2 , where D is the diffusion constant. We first write down Maxwell's equations for the bulk gauge field. Let the spatial momentum be aligned along the x 3 -axis: p = (ω, 0, 0, k). Then the equations for A 0 and A 3 are coupled:
wA ′ 0 + qf A ′ 3 = 0 , ( 73
) A ′′ 0 - 1 uf (q 2 A 0 + wqA 3 ) = 0 , ( 74
) A ′′ 3 + f ′ f A ′ 3 + 1 uf 2 (w 2 A 3 + wqA 0 ) = 0 . ( 75
)
One can eliminate A 3 and write down a third-order equation for A 0 ,
A ′′′ 0 + (uf ) ′ uf A ′′ 0 + w 2 -q 2 f uf 2 A ′ 0 = 0 . ( 76
)
Near u=1 we find two independent solutions, A ′ 0 ∼ (1u) ±iw/2 , and the incoming-wave boundary condition singles out (1u) -iw/2 . One can substitute A ′ 0 = (1u) -iw/2 F (u) into Eq. (76). The resulting equation can be solved perturbatively in w and q 2 . We find
A ′ 0 = C(1 -u) -iw/2 1 + iw 2 ln 2u 2 1 + u + q 2 ln 1 + u 2u . ( 77
)
Using Eq. (74) one can express C through the boundary values of A 0 and A 3 at u = 0:
C = q 2 A 0 + wqA 3 iw -q 2 u=0 . ( 78
)
Differentiating the action with respect to the boundary values, we find, in particular,
J 0 J 0 p = N 2 T 16π k 2 iω -Dk 2 , ( 79
) where D = 1 2πT . ( 80
)
The correlator given by Eq. (79) has the expected hydrodynamic diffusive pole, and D is the R-charge diffusion constant. Similarly, one can observe the appearance of the shear mode in the correlators of the metric tensor. We note that the shear flow along the x 1 direction with velocity gradient along the x 3 direction involves T 01 and T 31 , hence the interesting metric components are a 0 = h 1 0 and a 3 = h 1 3 . Two of the field equations are
a ′ 0 - qf w a ′ 3 = 0 , ( 81
)
a ′′ 3 - 1 + u 2 uf a ′ 3 + 1 uf 2 (w 2 a 3 + wqa 0 ) = 0 . ( 82
)
16 Son, Starinets They can be combined into a single equation:
a ′′′ 0 - 2u f a ′′ 0 + 2uf -q 2 f + w 2 uf 2 a ′ 0 = 0 . ( 83
)
Again, the solution can be found perturbatively in w and q:
a ′ 0 = C(1 -u) -iw/2 u -iw 1 -u - u 2 ln 1 + u 2 + q 2 2 (1 -u) . ( 84
)
Applying the prescription, one finds the retarded Green's functions. For example,
G tx,tx (ω, k) = ξk 2 iω -Dk 2 , ( 85
) where ξ = π 8 N 2 T 3 , D = 1 4πT . ( 86
)
Thus, we found that the correlator contains a diffusive pole ω = -iDk 2 , just as anticipated from hydrodynamics. Furthermore, the magnitude of the momentum diffusion constant D also matched our expectation. Indeed, if one recalls the value of η from Eq. ( 72 ) and the entropy density from Eq. ( 58 ), one can check that D = η ǫ + P . (87)
Let us now look at the problem from a different perspective. The existence of hydrodynamic modes in thermal field theory is reflected by the existence of the poles of the retarded correlators computed from gravity. Are there direct gravity counterparts of the hydrodynamic normal modes? If the answer to this question is yes, then there must exist linear gravitational perturbations of the metric that have the dispersion relation identical to that of the shear hydrodynamic mode, ω ∼ -iq 2 , and of the sound mode, ω = c s qiγq 2 . It turns out that one can explicitly construct the gravitational counterpart of the shear mode. (It should be possible to find a similar construction for the sound mode, but it has not been done in the literature; for a recent work on the subject, see [24] .) Our discussion is physical but somewhat sketchy; for more details see Ref. [25] .
First, let us construct a gravity perturbation that corresponds to a diffusion of a conserved charge (e.g., the R-charge in N = 4 SYM theory). To keep the discussion general, we use the form of the metric (44), with the metric components unspecified. Our only assumptions are that the metric is diagonal and has a horizon at r = r 0 , near which
g 00 = -γ 0 (r -r 0 ), g rr = γ r r -r 0 . ( 88
)
The Hawking temperature can be computed by the method used to arrive at Eq. ( 57 ), and one finds T = (4π
) -1 (γ 0 /γ r ) 1/2 .
We also assume that the action of the gauge field dual to the conserved current is
S gauge = dx √ -g - 1 4g 2 eff F µν F µν , ( 89
)
Viscosity, Black Holes, and QFT 17 where g eff is an effective gauge coupling that can be a function of the radial coordinate r. For simplicity we set g eff to a constant in our derivation of the formula for D; it can be restored by replacing √ -g → √ -g/g 2 eff in the final answer. The field equations are
∂ µ 1 g 2 eff √ -g F µν = 0 . ( 90
)
We search for a solution to this equation that vanishes at the boundary and satisfies the incomingwave boundary condition at the horizon.
The first indication that one can have a hydrodynamic behavior on the gravity side is that Eq. (90) implies a conservation law on a four-dimensional surface. We define the stretched horizon as a surface with constant r just outside the horizon,
r = r h = r 0 + ε, ε ≪ r 0 , ( 91
)
and the normal vector n µ directed along the r direction (i.e., perpendicularly to the stretched horizon). Then with any solution to Eq. (90), one can associate a current on the stretched horizon:
j µ = n ν F µν r h . ( 92
)
The antisymmetry of F µν implies that j µ has no radial component, j r = 0. The field equation (90) and the constancy of n ν on the stretched horizon imply that this current is conserved: ∂ µ j µ = 0. To establish the diffusive nature of the solution, we must show the validity of the constitutive equation j i = -D∂ i j 0 . Such constitutive equation breaks time reversal and obviously must come from the absorptive boundary condition on the horizon. The situation is analogous to the propagation of plane waves to a non-reflecting surface in classical electrodynamics. In this case, we have the relation B = -n × E between electric and magnetic fields. In our case, the corresponding relation is
F ir = - γ r γ 0 F 0i r -r 0 , ( 93
)
valid when r is close to r 0 . This relates j i ∼ F ir to the parallel to the horizon component of the electric field F 0i , which is one of the main points of the "membrane paradigm" approach to black hole physics [26, 27] . We have yet to relate j i to j 0 ∼ F 0r , which is the component of the electric field normal to the horizon. To make the connection to F 0r , we use the radial gauge A r = 0, in which
F 0i ≈ -∂ i A 0 . ( 94
)
Moreover, when k is small the fields change very slowly along the horizon. Therefore, at each point on the horizon the radial dependence of the scalar potential A 0 is determined by the Poisson equation,
∂ r ( √ -g g rr g 00 ∂ r A 0 ) = 0 , ( 95
) whose solution, which satisfies A 0 (r = ∞) = 0, is A 0 (r) = C 0 ∞ r dr ′ g 00 (r ′ )g rr (r ′ ) -g(r ′ ) . ( 96
)
18 Son, Starinets This means that the ratio of the scalar potential A 0 and electric field F 0r approaches a constant near the horizon:
A 0 F 0r r=r 0 = √ -g g 00 g rr (r 0 ) ∞ r 0 dr g 00 g rr √ -g (r) . ( 97
) Combining the formulas j i ∼ F 0i ∼ ∂ i A 0 , and A 0 ∼ F 0r ∼ j 0 , we find Fick's law j i = -D∂ i j 0 , with the diffusion constant D = √ -g g xx g 2 eff √ -g 00 g rr (r 0 ) ∞ r 0 dr -g 00 g rr g 2 eff √ -g (r) . ( 98
)
Thus, we found that for a slowly varying solution to Maxwell's equations, the corresponding charge on the stretched horizon evolves according to the diffusion equation. Therefore, the gravity solution must be an overdamped one, with ω = -iDk 2 . This is an example of a quasi-normal mode. We also found the diffusion constant D directly in terms of the metric and the gauge coupling g eff .
The reader may notice that our quasinormal modes satisfy a vanishing Dirichlet condition at the boundary r=∞. This is different from the boundary condition one uses to find the retarded propagators in AdS/CFT, so the relation of the quasinormal modes to AdS/CFT correspondence may be not clear. It can be shown, however, that the quasi-normal frequencies coincide with the poles of the retarded correlators [28, 29] .
We can now apply our general formulas to the case of N = 4 SYM theory. The metric components are given by Eq. ( 52 ). For the R-charge current g eff = const, Eq. (98) gives D = 1/(2πT ), in agreement with our AdS/CFT computation. For the shear mode of the stress-energy tensor we have effectively g 2 eff = g xx , so D = 1/(4πT ), which also coincides with our previous result. In both cases, the computation is much simpler than the AdS/CFT calculation.
In all thermal field theories in the regime described by gravity duals the ratio of shear viscosity η to (volume) density of entropy s is a universal constant equal to 1/(4π) [h/(4πk B ), if one restores h, c and the Boltzmann constant k B ].
One proof of the universality is based on the relationship between graviton's absorption cross section and the imaginary part of the retarded Green's function for T xy [31] . Another way to prove the universality [32] is via the direct AdS/CFT calculation of the correlation function in Kubo's formula (18) .
We, however, follow a different method. It is based on the formula for the viscosity derived from the membrane paradigm. A similar proof was given by Buchel & Liu [30] .
The observation is that the shear gravitational perturbation with k = 0 can be found exactly by performing a Lorentz boost of the black-brane metric (52) . Consider the coordinate transformations r, t,
x i → r ′ , t ′ , x ′ i of the form r = r ′ , t = t ′ + vy ′ √ 1 -v 2 ≈ t ′ + vy ′ ,
Viscosity, Black Holes, and QFT
19 y = y ′ + vt ′ √ 1 -v 2 ≈ y ′ + vt ′ , x i = x ′ i , ( 99
)
where v < 1 is a constant parameter and the expansion on the right corresponds to v ≪ 1. In the new coordinates, the metric becomes
ds 2 = g 00 dt ′ 2 + g rr dr ′ 2 + g xx (r) p i=1 (dx ′ i ) 2 + 2v(g 00 + g xx )dt ′ dy ′ . ( 100
)
This is simply a shear fluctuation at k = 0. In our language, the corresponding gauge potential is
a 0 = vg xx (g 00 + g xx ) . ( 101
)
This field satisfies the vanishing boundary condition a 0 (r = ∞) = 0 owing to the restoration of Poincaré invariance at the boundary: g 00 /g xx → -1 when r → ∞. This clearly has a much simpler form than Eq. (96) for the solution to the generic Poisson equation. The simple form of solution (101) is valid only for the specific case of the shear gravitational mode with g 2 eff = g xx . We have also implicitly used the fact that the metric satisfies the Einstein equations, with the stress-energy tensor on the right being invariant under a Lorentz boost. Equation (97) now becomes
a 0 f 0r r→r 0 = - 1 + g xx g 00 ∂ r (g xx g 00 ) r→r 0 = g xx (r 0 ) γ 0 . ( 102
)
The shear mode diffusion constant is
D = a 0 f 0r r→r 0 √ γ 0 γ r g xx (r 0 ) = γ r γ 0 = 1 4πT . ( 103
)
Because D = η/(ǫ + P ), and ǫ + P = T s in the absence of chemical potentials, we find that
η s = 1 4π . ( 104
)
In fact, the constancy of this ratio has been checked directly for theories dual to Dp-brane [25], M -brane [11] , Klebanov-Tseytlin and Maldacena-Nunez backgrounds [30], N = 2 * SYM theory [33] and others. Curiously, the viscosity to entropy ratio is also equal to 1/4π in the pre-AdS/CFT "membrane paradigm" hydrodynamics [34]: there, for a four-dimensional Schwarzschild black hole one has η m.p. = 1/16πG N , while the Bekenstein-Hawking entropy is s = 1/4G N . As remarked in Sec. 2, the ratio η/s is much larger than the one for weakly coupled theories. The fact that we found the ratio to be parametrically of order one implies that all theories with gravity duals are strongly coupled.
In N = 4 SYM theory, the ratio η/s has been computed to the next order in the inverse 't Hooft coupling expansion [35]
η s = 1 4π 1 + 135ζ(3) 8(g 2 N ) 3/2 . ( 105
)
The sign of the correction can be guessed from the fact that in the limit of zero 't Hooft coupling g 2 N → 0, the ratio diverges, η/s → ∞.
20 Son, Starinets
From our discussion above, one can argue that η s ≥ h 4π (106) in all systems that can be obtained from a sensible relativistic quantum field theory by turning on temperatures and chemical potentials. The bound, if correct, implies that a liquid with a given volume density of entropy cannot be arbitrarily close to being a perfect fluid (which has zero viscosity). As such, it implies a lower bound on the viscosity of the QGP one may be creating at RHIC. Interestingly, some model calculations suggest that the viscosity at RHIC may be not too far away from the lower bound [36, 37] .
One place where one may think that the bound should break down is superfluids. The ability of a superfluid to flow without dissipation in a channel is sometimes described as "zero viscosity". However, within the Landau's two-fluid model, any superfluid has a measurable shear viscosity (together with three bulk viscosities). For superfluid helium, the shear viscosity has been measured in a torsion-pendulum experiment by Andronikashvili [38] . If one substitutes the experimental values, the ratio η/s for helium remains larger than h/4πk B ≈ 6.08 × 10 -13 K s for all ranges of temperatures and pressures, by a factor of at least 8.8.
As discussed in Sec. 2.3, the ratio η/s is proportional to the ratio of the mean free path and the de Broglie wavelength of particles,
η s ∼ ℓ mfp λ . ( 107
)
For the quasi-particle picture to be valid, the mean free path must be much larger than the de Broglie wavelength. Therefore, if the coupling is weak and the system can be described as a collection of quasi-particles, the ratio η/s is larger than 1. We have found is that, within the N = 4 SYM theory and, more generally, theories with gravity duals, even in the limit of infinite coupling the ratio η/s cannot be made smaller than 1/(4π).
In this review, we covered only a small part of the applications of AdS/CFT correspondence to finite-temperature quantum field theory. Here we briefly mention further developments and refer the reader to the original literature for more details.
In addition to N = 4 SYM theory, there exists a large number of other theories whose hydrodynamic behavior has been studied using the AdS/CFT correspondence, including the worldvolume theories on M2-and M5-branes [11] , theories on Dp branes [25] , and little string theory [39] . In all examples the ratio η/s is equal to 1/(4π), which is not surprising because the general proofs of Sec. 6 apply in these cases.
We have concentrated on the shear hydrodynamic mode, which has a diffusive pole (ω ∼ -ik 2 ). One can also compute correlators which have a sound-wave pole from the AdS/CFT prescription [10] . One such correlator is between the energy density T 00 at two different points in space-time.
The result confirms the existence of such a pole, with both the real part and imaginary part having exactly the values predicted by hydrodynamics (recall that in conformal field theories the bulk viscosity is zero and the sound attenuation rate is determined completely by the shear viscosity).
Viscosity, Black Holes, and QFT 21 Some of the theories listed above are conformal field theories, but many are not (e.g., the Dpbrane worldvolume theories with p = 3). The fact that η/s = 1/(4π) also in those theories implies that the constancy of this ratio is not a consequence of conformal symmetry. Theories with less than maximal number of supersymmetries have been found to have the universal value of η/s, for example, the N = 2 * theory [40], theories described by Klebanov-Tseytlin, and Maldacena-Nunez backgrounds [30] . A common feature of these theories is that they all have a gravitational dual description. The bulk viscosity has been computed for some of these theories [41, 39] . Besides viscosity, one can also compute diffusion constants of conserved charges by using the AdS/CFT correspondence. Above we presented the computation of the R-charge diffusion constant in N = 4 SYM theory; for similar calculations in some other theories see Ref. [11, 25] .
Recently, the AdS/CFT correspondence was used to compute the energy loss rate of a quark in the fundamental representation moving in a finite-temperature plasma [42, 43, 44, 45] . This quantity is of importance to the phenomenon of "jet quenching" in heavy-ion collisions.
So far, the only quantity that shows a universal behavior at the quantitative level, across all theories with gravitational duals, is the ratio of the shear viscosity and entropy density. Recently, it was found that this ratio remains constant even at nonzero chemical potentials [46, 47, 48, 49, 50] .
What have we learned from the application of AdS/CFT correspondence to thermal field theory? Although, at least at this moment, we cannot use the AdS/CFT approach to study QCD directly, we have found quite interesting facts about strongly coupled field theories. We have also learned new facts about quasi-normal modes of black branes. However, we have also found a set of puzzles: Why is the ratio of the viscosity and entropy density constant in a wide class of theories? Is there a lower bound on this ratio for all quantum field theories? Can this be understood without any reference to gravity duals? With these open questions, we conclude this review.
The work of DTS is supported in part by U.S. Department of Energy under Grant No. DE-FG02-00ER41132. Research at Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MEDT. Literature Cited 1. Kirzhnits DA, Linde AD. Macroscopic consequences of the Weinberg model. Phys. Lett. B 42:471 (1972) 2. Gyulassy M, and McLerran L, New forms of QCD matter discovered at RHIC. Nucl. Phys. A 750:30 (2005) 3. Maldacena JM. The large N limit of superconformal field theory and supergravity. Adv. Theor. Math. Phys. 2:231 (1998) 4. Gubser SS, Klebanov IR, Polyakov AM. Gauge theory correlators from noncritical string theory.
Phys. Lett. B 428:105 (1998) 5. Witten E. Anti de Sitter space and holography. Adv. Theor. Math. Phys. 2:253 (1998) 6. Policastro G, Son DT, Starinets AO. Shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma. Phys. Rev. Lett. 87:081601 (2001) 22 Son, Starinets 7. Son DT, Starinets AO. Minkowski-space correlators in AdS/CFT correspondence: Recipe and applications. J. High Energy Phys. 0209:042 (2002) 8. Herzog CP, Son DT. Schwinger-Keldysh propagators from AdS/CFT correspondence. J. High Energy Phys. 0303:046 (2003) 9. Policastro G, Son DT, Starinets AO. From AdS/CFT correspondence to hydrodynamics. J. High Energy Phys. 0209:042 (2002) 10. Policastro G, Son DT, Starinets AO. From AdS/CFT correspondence to hydrodynamics. II: Sound waves. J. High Energy Phys. 0212:054 (2002) 11. Herzog CP. The hydrodynamics of M-theory. J. High Energy Phys. 0212:026 (2002) 12. Herzog CP. The sound of M-theory. Phys. Rev. D 68:024013 (2003) 13. Forster D. Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, Reading: Benjamin (1975) 14. Landau LD, Lifshitz EM. Fluid Mechanics, Oxford: Pergamon Press (1987) 15. Jeon S. Hydrodynamic transport coefficients in relativistic scalar field theory. Phys. Rev. D 52:3591 (1995) 16. Jeon S, Yaffe LG. From quantum field theory to hydrodynamics: transport coefficients and effective kinetic theory. Phys. Rev. D 53:5799 (1996) 17. Aharony O et al. Large N field theories, string theory and gravity. Phys. Rep. 323:183 (2000) 18. Klebanov IR. TASI lectures: Introduction to the AdS/CFT correspondence. arXiv:hep-th/0009139 19. Polchinski J. String theory, Cambridge: Cambridge University Press (1998). 20. Appelquist T, Chodos A, Freund PG. Modern Kaluza-Klein theories, Menlo Park: Addison-Wesley (1987) 21. Kim HJ, Romans LJ, van Nieuwenhuizen P. The mass spectrum of chiral N = 2 D = 10 supergravity on S 5 . Phys. Rev. D 32:389 (1985) 22. Bianchi M, Freedman DZ, Skenderis K. Holographic renormalization. Nucl. Phys. B 631:159 (2002) 23. Maldacena JM. Eternal black holes in anti-de-Sitter. J. High Energy Phys. 0304:021 (2003) 24. Saremi O. Shear waves, sound waves on a shimmering horizon. arXiv:hep-th/0703170 25. Kovtun P, Son DT, Starinets AO. Holography and hydrodynamics: Diffusion on stretched horizons. J. High Energy Phys. 0310:064 (2003) 26. Damour T. Black Hole Eddy Currents. Phys. Rev. D 18: 3598 (1978) 27. Thorne, KS, Price RH, Macdonald DA. Black Hole: The Membrane Paradigm, New Haven: Yale University Press (1986) 28. Kovtun PK, Starinets AO. Quasinormal modes and holography. Phys. Rev. D 72:086009 (2005) 29. Starinets AO. Quasinormal spectrum and black hole membrane paradigm. Unpublished 30. Buchel A, Liu JT. Universality of the shear viscosity in supergravity. Phys. Rev. Lett., 93:090602 (2004) 31. Kovtun P, Son DT, Starinets AO. Viscosity in strongly interacting quantum field theories from black hole physics. Phys. Rev. Lett. 94:111601 (2004) 32. Buchel A. On universality of stress-energy tensor correlation functions in supergravity, Phys. Lett. B, 609:392 (2005). 33. Buchel A. N = 2* hydrodynamics. Nucl. Phys. B 708:451 (2005)
Viscosity, Black Holes, and QFT 23 34. Damour T. Surface effects in black hole physics. Proceedings of the Second Marcel Grossmann Meeting on General Relativity. Ed. R. Ruffini, Amsterdam: North Holland (1982) 35. Buchel A, Liu JT, Starinets AO. Coupling constant dependence of the shear viscosity in N = 4 supersymmetric Yang-Mills theory. Nucl. Phys. B 707:56 (2005) 36. Teaney D. Effect of shear viscosity on spectra, elliptic flow, and Hanbury Brown-Twiss radii.
Phys. Rev. C 68:034913 (2003) 37. Shuryak E. Why does the quark gluon plasma at RHIC behave as a nearly ideal fluid? Prog. Part. Nucl. Phys. 53:273 (2004) 38. Andronikashvili E. Zh. Eksp. Teor. Fiz. 18:429 (1948) 39. Parnachev A, Starinets A. The silence of the little strings. J. High Energy Phys. 0510:027 (2005) 40. Buchel A, Liu JT, Thermodynamics of the N = 2 * flow. J. High Energy Phys. 0311:031 (2003) 41. Benincasa P, Buchel A, Starinets AO. Sound waves in strongly coupled non-conformal gauge theory plasma. Nucl. Phys. B 733:160 (2006) 42. Herzog CP et al. Energy loss of a heavy quark moving through N = 4 supersymmetric Yang-Mills plasma. J. High Energy Phys. 0607:013 (2006) 43. Liu H, Rajagopal K, Wiedemann UA. Calculating the jet quenching parameter from AdS/CFT, Phys. Rev. Lett. 97:182301 (2006) 44. Casalderrey-Solana J, Teaney D. Heavy quark diffusion in strongly coupled N = 4 Yang Mills.
Phys. Rev. D 74:085012 (2006) 45. Gubser, SS. Drag force in AdS/CFT. Phys. Rev. D 74:126005 (2006) 46. Son DT, Starinets AO. Hydrodynamics of R-charged black holes. J. High Energy Phys. 0603:052 (2006) 47. Mas J, Shear viscosity from R-charged AdS black holes. J. High Energy Phys. 0603:016 (2006) 48. Maeda K, Natsuume M, Okamura T. Viscosity of gauge theory plasma with a chemical potential from AdS/CFT. Phys. Rev. D 73:066013 (2006) 49. Saremi O. The viscosity bound conjecture and hydrodynamics of M2-brane theory at finite chemical potential. J. High Energy Phys. 0610:083 (2006) 50. Benincasa P, Buchel A, Naryshkin R. The shear viscosity of gauge theory plasma with chemical potentials. Phys. Lett. B 645:309 (2007) | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "1 2 Son, Starinets"
},
{
"section_type": "BACKGROUND",
"section_title": "INTRODUCTION",
"text": "This review is about the recently emerging connection, through the gauge/gravity correspondence, between hydrodynamics and black hole physics. The study of quantum field theory at high temperature has a long history. It was first motivated by the Big Bang cosmology when it was hoped that early phase transitions might leave some imprints on the Universe [1] . One of those phase transitions is the QCD phase transitions (which could actually be a crossover) which happened at a temperature around T c ∼ 200 MeV, when matter turned from a gas of quarks and gluons (the quark-gluon plasma, or QGP) into a gas of hadrons.\n\nAn experimental program was designed to create and study the QGP by colliding two heavy atomic nuclei. Most recent experiments are conducted at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory. Although significant circumstantial evidence for the QGP was accumulated [2] , a theoretical interpretation of most of the experimental data proved difficult, because the QGP created at RHIC is far from being a weakly coupled gas of quarks and gluons. Indeed, the temperature of the plasma, as inferred from the spectrum of final particles, is only approximately 170 MeV, near the confinement scale of QCD. This is deep in the nonperturbative regime of QCD, where reliable theoretical tools are lacking. Most notably, the kinetic coefficients of the QGP, which enter the hydrodynamic equations (reviewed in Sec. 2), are not theoretically computable at these temperatures.\n\nThe paucity of information about the kinetic coefficients of the QGP in particular and of strongly coupled thermal quantum field theories in general is one of the main reasons for our interest in their computation in a class of strongly coupled field theories, even though this class does not include QCD. The necessary technological tool is the anti-de Sitter-conformal field theory (AdS/CFT) correspondence [3, 4, 5] , discovered in the investigation of D-branes in string theory. This correspondence allows one to describe the thermal plasmas in these theories in terms of black holes in AdS space. The AdS/CFT correspondence is reviewed in Sec. 3. The first calculation of this type, that of the shear viscosity in N = 4 supersymmetric Yang-Mills (SYM) theory [6] , is followed by the theoretical work to establish the rules of real-time finite-temperature AdS/CFT correspondence [7, 8] . Applications of these rules to various special cases [9, 10, 11, 12] clearly show that even very exotic field theories, when heated up to finite temperature, behave hydrodynamically at large distances and time scales (provided that the number of space-time dimensions is 2+1 or higher). This development is reviewed in Sec. 4. Moreover, the way AdS/CFT works reveals deep connections to properties of black holes in classical gravity. For example, the hydrodynamic modes of a thermal medium are mapped, through the correspondence, to the low-lying quasi-normal modes of a black-brane metric. It seems that our understanding of the connection between hydrodynamics and black hole physics is still incomplete; we may understand more about gravity by studying thermal field theories. One idea along this direction is reviewed in Sec. 5.\n\nFrom the point of view of heavy-ion (QGP) physics, a particularly interesting finding has been the formulation of a conjecture on the lowest possible value of the ratio of viscosity and volume density of entropy. This conjecture was motivated by the universality of this ratio in theories with gravity duals. This is reviewed in Sec. 6.\n\nViscosity, Black Holes, and QFT 3 2 HYDRODYNAMICS\n\nFrom the modern perspective, hydrodynamics [13] is best thought of as an effective theory, describing the dynamics at large distances and time-scales. Unlike the familiar effective field theories (for example, the chiral perturbation theory), it is normally formulated in the language of equations of motion instead of an action principle. The reason for this is the presence of dissipation in thermal media.\n\nIn the simplest case, the hydrodynamic equations are just the laws of conservation of energy and momentum,\n\n∂ µ T µν = 0 . ( 1\n\n)\n\nTo close the system of equations, we must reduce the number of independent elements of T µν . This is done through the assumption of local thermal equilibrium: If perturbations have long wavelengths, the state of the system, at a given time, is determined by the temperature as a function of coordinates T (x) and the local fluid velocity u µ , which is also a function of coordinates u µ (x). Because u µ u µ = -1, only three components of u µ are independent. The number of hydrodynamic variables is four, equal to the number of equations.\n\nIn hydrodynamics we express T µν through T (x) and u µ (x) through the so-called constitutive equations. Following the standard procedure of effective field theories, we expand in powers of spatial derivatives. To zeroth order, T µν is given by the familiar formula for ideal fluids,\n\nT µν = (ǫ + P )u µ u ν + P g µν , ( 2\n\n)\n\nwhere ǫ is the energy density, and P is the pressure. Normally one would stop at this leading order, but qualitatively new effects necessitate going to the next order. Indeed, from Eq. 2 and the thermodynamic relations dǫ = T dS, dP = sdT , and ǫ + P = T s (s is the entropy per unit volume), one finds that entropy is conserved [14] ∂ µ (su µ ) = 0 . (3) Thus, to have entropy production, one needs to go to the next order in the derivative expansion. At the next order, we write\n\nT µν = (ǫ + P )u µ u ν + P g µν -σ µν , ( 4\n\n)\n\nwhere σ µν is proportional to derivatives of T (x) and u µ (x) and is termed the dissipative part of T µν . To write these terms, let us first fix a point x and go to the local rest frame where u i (x) = 0. 4 Son, Starinets In this frame, in principle one can have dissipative corrections to the energy-momentum density T 0µ . However, one recalls that the choice of T and u µ is arbitrary, and thus one can always redefine them so that these corrections vanish, σ 00 = σ 0i = 0, and so at a point x,\n\nT 00 = ǫ, T 0i = 0 . ( 5\n\n)\n\nThe only nonzero elements of the dissipative energy-momentum tensor are σ ij . To the next-toleading order there are extra contributions whose forms are dictated by rotational symmetry:\n\nσ ij = η ∂ i u j + ∂ j u i - 2 3 δ ij ∂ k u k + ζδ ij ∂ k u k . ( 6\n\n)\n\nGoing back to the general frame, we can now write the dissipative part of the energy-momentum tensor as\n\nσ µν = P µα P νβ η ∂ α u β + ∂ β u α - 2 3 g αβ ∂ λ u λ + ζg αβ ∂ λ u λ , ( 7\n\n)\n\nwhere P µν = g µν + u µ u ν is the projection operator onto the directions perpendicular to u µ . If the system contains a conserved current, there is an additional hydrodynamic equation related to the current conservation,\n\n∂ µ j µ = 0 . ( 8\n\n)\n\nThe constitutive equation contains two terms:\n\nj µ = ρu µ -DP µν ∂ ν α , ( 9\n\n)\n\nwhere ρ is the charge density in the fluid rest frame and D is some constant. The first term corresponds to convection, the second one to diffusion. In the fluid rest frame, j = -D∇ρ, which is Fick's law of diffusion, with D being the diffusion constant."
},
{
"section_type": "OTHER",
"section_title": "Kubo's Formula For Viscosity",
"text": "As mentioned above, the hydrodynamic equations can be thought of as an effective theory describing the dynamics of the system at large lengths and time scales. Therefore one should be able to use these equations to extract information about the low-momentum behavior of Green's functions in the original theory. For example, let us recall how the two-point correlation functions can be extracted. If we couple sources J a (x) to a set of (bosonic) operators O a (x), so that the new action is\n\nS = S 0 + x J a (x)O a (x) , ( 10\n\n)\n\nthen the source will introduce a perturbation of the system. In particular, the average values of O a will differ from the equilibrium values, which we assume to be zero. If J a are small, the perturbations are given by the linear response theory as\n\nO a (x) = - y G R ab (x -y)J b (y) , ( 11\n\n)\n\nwhere G R ab is the retarded Green's function\n\niG R ab (x -y) = θ(x 0 -y 0 ) [O a (x), O b (y)] . ( 12\n\n)\n\nViscosity, Black Holes, and QFT 5 The fact that the linear response is determined by the retarded (and not by any other) Green's function is obvious from causality: The source can influence the system only after it has been turned on. Thus, to determine the correlation functions of T µν , we need to couple a weak source to T µν and determine the average value of T µν after this source is turned on. To find these correlators at low momenta, we can use the hydrodynamic theory. So far in our treatment of hydrodynamics we have included no source coupled to T µν . This deficiency can be easily corrected, as the source of the energy-momentum tensor is the metric g µν . One must generalize the hydrodynamic equations to curved space-time and from it determine the response of the thermal medium to a weak perturbation of the metric. This procedure is rather straightforward and in the interest of space is left as an exercise to the reader.\n\nHere we concentrate on a particular case when the metric perturbation is homogeneous in space but time dependent:\n\ng ij (t, x) = δ ij + h ij (t), h ij ≪ 1 ( 13\n\n)\n\ng 00 (t, x) = -1, g 0i (t, x) = 0 . (14)\n\nMoreover, we assume the perturbation to be traceless, h ii = 0. Because the perturbation is spatially homogeneous, if the fluid moves, it can only move uniformly: u i = u i (t). However, this possibility can be ruled out by parity, so the fluid must remain at rest all the time: u µ = (1, 0, 0, 0). We now compute the dissipative part of the stress-energy tensor. The generalization of Eq. 7 to curved space-time is\n\nσ µν = P µα P νβ η(∇ α u β + ∇ β u α ) + ζ - 2 3 η g αβ ∇ • u . ( 15\n\n)\n\nSubstituting u µ = (1, 0, 0, 0) and g µν from Eq. 13, we find only contributions to the traceless spatial components, and these contributions come entirely from the Christoffel symbols in the covariant derivatives. For example,\n\nσ xy = 2ηΓ 0 xy = η∂ 0 h xy . ( 16\n\n)\n\nBy comparison with the expectation from the linear response theory, this equation means that we have found the zero spatial momentum, low-frequency limit of the retarded Green's function of\n\nT xy : G R xy,xy (ω, 0) = dt dx e iωt θ(t) [T xy (t, x), T xy (0, 0)] = -iηω + O(ω 2 ) ( 17\n\n)\n\n(modulo contact terms). We have, in essence, derived the Kubo's formula relating the shear viscosity and a Green's function:\n\nη = -lim ω→0 1 ω Im G R xy,xy (ω, 0) . ( 18\n\n)\n\nThere is a similar Kubo's relation for the charge diffusion constant D."
},
{
"section_type": "OTHER",
"section_title": "Hydrodynamic Modes",
"text": "If one is interested only in the locations of the poles of the correlators, one can simply look for the normal modes of the linearized hydrodynamic equations, that is, solutions that behave as e -iωt+ik•x . Owing to dissipation, the frequency ω(k) is complex. For example, the equation of charge diffusion,\n\n∂ t ρ -D∇ 2 ρ = 0, ( 19\n\n)\n\n6 Son, Starinets corresponds to a pole in the current-current correlator at ω = -iDk 2 .\n\nTo find the poles in the correlators between elements of the stress-energy tensor one can, without loss of generality, choose the coordinate system so that k is aligned along the x 3 -axis: k = (0, 0, k).\n\nThen one can distinguish two types of normal modes: 1. Shear modes correspond to the fluctuations of pairs of components T 0a and T 3a , where a = 1, 2. The constitutive equation is\n\nT 3a = -η∂ 3 u a = - η ǫ + P ∂ 3 T 0a , ( 20\n\n)\n\nand the equation for\n\nT 0a is ∂ t T 0a - η ǫ + P ∂ 2 3 T 0a = 0 . ( 21\n\n)\n\nThat is, it has the form of a diffusion equation for T 0a . Substituting e -iωt+ikx 3 into the equation, one finds the dispersion law\n\nω = -i η ǫ + P k 2 . ( 22\n\n)\n\n2. Sound modes are fluctuations of T 00 , T 03 , and T 33 . There are now two conservation equations, and by diagonalizing them one finds the dispersion law\n\nω = c s k - i 2 4 3 η + ζ k 2 ǫ + P , ( 23\n\n)\n\nwhere c s = dP/dǫ. This is simply the sound wave, which involves the fluctuation of the energy density. It propagates with velocity c s , and its damping is related to a linear combination of shear and bulk viscosities.\n\nIn CFTs it is possible to use conformal Ward identities to show that the bulk viscosity vanishes: ζ = 0. Hence, we shall concentrate our attention on the shear viscosity η."
},
{
"section_type": "OTHER",
"section_title": "Viscosity In Weakly Coupled Field Theories",
"text": "We now briefly consider the behavior of the shear viscosity in weakly coupled field theories, with the λφ 4 theory as a concrete example. At weak coupling, there is a separation between two length scales: The mean free path of particles is much larger than the distance scales over which scatterings occur. Each scattering event takes a time of order T -1 (which can be thought of as the time required for final particles to become on-shell). The mean free path ℓ mfp can be estimated from the formula\n\nℓ mfp ∼ 1 nσv , ( 24\n\n)\n\nwhere n is the density of particles, σ is the typical scattering cross section, and v is the typical particle velocity. Inserting the values for thermal λφ 4 theory, n ∼ T 3 , σ ∼ λ 2 T -2 , and v ∼ 1, one finds\n\nℓ mfp ∼ 1 λ 2 T ≫ 1 T . ( 25\n\n)\n\nThe viscosity can be estimated from kinetic theory to be\n\nη ∼ ǫℓ mfp , ( 26\n\n)\n\nViscosity, Black Holes, and QFT 7 where ǫ is the energy density. From ǫ ∼ T 4 and the estimate of ℓ mft , one finds\n\nη ∼ T 3 λ 2 . ( 27\n\n)\n\nIn particular, the weaker the coupling λ, the larger the viscosity η. This behavior is explained by the fact that the viscosity measures the rate of momentum diffusion. The smaller λ is, the longer a particle travels before colliding with another one, and the easier the momentum transfer. It may appear counterintuitive that viscosity tends to infinity in the limit of zero coupling λ → 0: At zero coupling there is no dissipation, so should the viscosity be zero? The confusion arises owing to the fact that the hydrodynamic theory, and hence the notion of viscosity, makes sense only on distances much larger than the mean free path of particles. If one takes λ → 0, then to measure the viscosity one has to do the experiment at larger and larger length scales. If one fixes the size of the experiment and takes λ → 0, dissipation disappears, but it does not tell us anything about the viscosity.\n\nAs will become apparent below, a particularly interesting ratio to consider is the ratio of shear viscosity and entropy density s. The latter is proportional to T 3 ; thus η s ∼ 1 λ 2 . (28) One has η/s ≫ 1 for λ ≪ 1. This is a common feature of weakly coupled field theories. Extrapolating to λ ∼ 1, one finds η/s ∼ 1. We shall see that theories with gravity duals are strongly coupled, and η/s is of order one. More surprisingly, this ratio is the same for all theories with gravity duals.\n\nTo compute rather than estimate the viscosity, one can use Kubo's formula. It turns out that one has to sum an infinite number of Feynman graphs to even find the viscosity to leading order. Another way that leads to the same result is to first formulate a kinetic Boltzmann equation for the quasi-particles as an intermediate effective description, and then derive hydrodynamics by taking the limit of very long lengths and time scales in the kinetic equation. Interested readers should consult Refs. [15, 16] for more details.\n\n3 AdS/CFT CORRESPONDENCE"
},
{
"section_type": "OTHER",
"section_title": "Review Of AdS/CFT Correspondence At Zero Temperature",
"text": "This section briefly reviews the AdS/CFT correspondence at zero temperature. It contains only the minimal amount of materials required to understand the rest of the review. Further information can be found in existing reviews and lecture notes [17, 18] .\n\nThe original example of AdS/CFT correspondence is between N = 4 supersymmetric Yang-Mills (SYM) theory and type IIB string theory on AdS 5 ×S 5 space. Let us describe the two sides of the correspondence in some more detail.\n\nThe N = 4 SYM theory is a gauge theory with a gauge field, four Weyl fermions, and six real scalars, all in the adjoint representation of the color group. Its Lagrangian can be written down explicitly, but is not very important for our purposes. It has a vanishing beta function and is a conformal field theory (CFT) (thus the CFT in AdS/CFT). In our further discussion, we frequently use the generic terms \"field theory\" or CFT for the N = 4 SYM theory.\n\n8 Son, Starinets On the string theory side, we have type IIB string theory, which contains a finite number of massless fields, including the graviton, the dilaton Φ, some other fields (forms) and their fermionic superpartners, and an infinite number of massive string excitations. It has two parameters: the string length l s (related to the slope parameter α ′ by α ′ = l 2 s ) and the string coupling g s . In the long-wavelength limit, when all fields vary over length scales much larger than l s , the massive modes decouple and one is left with type IIB supergravity in 10 dimensions, which can be described by an action [19]\n\nS SUGRA = 1 2κ 2 10 d 10 x √ -g e -2Φ (R + 4 ∂ µ Φ∂ µ Φ + • • •) , ( 29\n\n)\n\nwhere κ 10 is the 10-dimensional gravitational constant,\n\nκ 10 = √ 8πG = 8π 7/2 g s l 4 s , ( 30\n\n)\n\nand • • • stay for the contributions from fields other than the metric and the dilaton. One of these fields is the five-form F 5 , which is constrained to be self-dual. The type IIB string theory lives is a 10-dimensional space-time with the following metric:\n\nds 2 = r 2 R 2 (-dt 2 + dx 2 ) + R 2 r 2 dr 2 + R 2 dΩ 2 5 . ( 31\n\n)\n\nThe metric is a direct product of a five-dimensional sphere (dΩ 2 5 ) and another five-dimensional space-time spanned by t, x, and r. An alternative form of the metric is obtained from Eq. (31) by a change of variable z = R 2 /r,\n\nds 2 = R 2 z 2 (-dt 2 + dx 2 + dz 2 ) + R 2 dΩ 2 5 . ( 32\n\n)\n\nBoth coordinates r and z are known as the radial coordinate. The limiting value r = ∞ (or z = 0) is the boundary of the AdS space. It is a simple exercise to check that the (t, x, r) part of the metric is a space with constant negative curvature, or an anti de-Sitter (AdS) space. To support the metric (31) (i.e., to satisfy the Einstein equation) there must be some background matter field that gives a stress-energy tensor in the form of a negative cosmological constant in AdS 5 and a positive one in S 5 . Such a field is the self-dual five-form field F 5 mentioned above.\n\nField theory has two parameters: the number of colors N and the gauge coupling g. When the number of colors is large, it is the 't Hooft coupling λ = g 2 N that controls the perturbation theory. On the string theory side, the parameters are g s , l s , and radius R of the AdS space. String theory and field theory each have two dimensionless parameters which map to each other through the following relations:\n\ng 2 = 4πg s , ( 33\n\n)\n\ng 2 N c = R 4 l 4 s . ( 34\n\n)\n\nEquation ( 33) tells us that, if one wants to keep string theory weakly interacting, then the gauge coupling in field theory must be small. Equation (34) is particularly interesting. It says that the large 't Hooft coupling limit in field theory corresponds to the limit when the curvature radius of Viscosity, Black Holes, and QFT 9 space-time is much larger than the string length l s . In this limit, one can reliably decouple the massive string modes and reduce string theory to supergravity. In the limit g s ≪ 1 and R ≫ l s , one has classical supergravity instead of string theory. The practical utility of the AdS/CFT correspondence comes, in large part, from its ability to deal with the strong coupling limit in gauge theory. One can perform a Kaluza-Klein reduction [20] by expanding all fields in S 5 harmonics. Keeping only the lowest harmonics, one finds a five-dimensional theory with the massless dilaton, SO(6) gauge bosons, and gravitons [21]:\n\nS 5D = N 2 8π 2 R 3 d 5 x R 5D -2Λ - 1 2 ∂ µ Φ∂ µ Φ - R 2 8 F a µν F aµν + • • • . ( 35\n\n)\n\nIn AdS/CFT, an operator O of field theory is put in a correspondence with a field φ (\"bulk\" field) in supergravity. We elaborate on this correspondence below; here we keep the operator and the field unspecified. In the supergravity approximation, the mathematical statement of the correspondence is\n\nZ 4D [J] = e iS[φ cl ] . ( 36\n\n)\n\nOn the left is the partition function of a field theory, where the source J coupled to the operator O is included:\n\nZ 4D [J] = Dφ exp iS + i d 4 x JO . ( 37\n\n)\n\nOn the right, S[φ cl ] is the classical action of the classical solution φ cl to the field equation with the boundary condition:\n\nlim z→0 φ cl (z, x) z ∆ = J(x) . ( 38\n\n)\n\nHere ∆ is a constant that depends on the nature of the operator O (namely, on its spin and dimension). In the simplest case, ∆ = 0, and the boundary condition becomes φ cl (z=0) = J. Differentiating Eq. (36) with respect to J, one can find the correlation functions of O. For example, the two-point Green's function of O is obtained by differentiating S cl [φ] twice with respect to the boundary value of φ,\n\nG(x -y) = -i T O(x)O(y) = - δ 2 S[φ cl ] δJ(x)δJ(y) φ(z=0)=J . ( 39\n\n)\n\nThe AdS/CFT correspondence thus maps the problem of finding quantum correlation functions in field theory to a classical problem in gravity. Moreover, to find two-point correlation functions in field theory, one can be limited to the quadratic part of the classical action on the gravity side. The complete operator to field mapping can be found in Refs. [5, 17] . For our purpose, the following is sufficient:\n\n• The dilaton Φ corresponds to O = -L = 1 4 F 2 µν + • • •,\n\nwhere L is the Lagrangian density. • The gauge field A a µ corresponds to the conserved R-charge current J aµ of field theory. • The metric tensor corresponds to the stress-energy tensor T µν . More precisely, the partition function of the four-dimensional field theory in an external metric g 0 µν is equal to\n\nZ 4D [g 0 µν ] = exp(iS cl [g µν ]) , ( 40\n\n)\n\n10 Son, Starinets where the five-dimensional metric g µν satisfies the Einstein's equations and has the following asymptotics at z = 0:\n\nds 2 = g µν dx µ dx ν = R 2 z 2 (dz 2 + g 0 µν dx µ dx ν ) . ( 41\n\n)\n\nFrom the point of view of hydrodynamics, the operator 1 4 F 2 is not very interesting because its correlator does not have a hydrodynamic pole. In contrast, we find the correlators of the R-charge current and the stress-energy tensor to contain hydrodynamic information.\n\nWe simplify the graviton part of the action further. Our two-point functions are functions of the momentum p = (ω, k). We can choose spatial coordinates so that k points along the x 3 -axis. This corresponds to perturbations that propagate along the x 3 direction: h µν = h µν (t, r, x 3 ). These perturbations can be classified according to the representations of the O(2) symmetry of the (x 1 , x 2 ) plane. Owing to that symmetry, only certain components can mix; for example, h 12 does not mix with any other components, whereas components h 01 and h 31 mix only with each other. We assume that only these three metric components are nonzero and introduce shorthand notations\n\nφ = h 1 2 , a 0 = h 1 0 , a 3 = h 1 3 . ( 42\n\n)\n\nThe quadratic part of the graviton action acquires a very simple form in terms of these fields:\n\nS quad = N 2 8π 2 R 3 d 4 x dr √ -g - 1 2 g µν ∂ µ φ∂ ν φ - 1 4g 2 eff g µα g νβ f µν f αβ , ( 43\n\n) where f µν = ∂ µ a ν -∂ ν a µ , and g 2 eff = g xx .\n\nIn deriving Eq. ( 43 ), our only assumption about the metric is that it has a diagonal form,\n\nds 2 = g tt dt 2 + g rr dr 2 + g xx dx 2 , ( 44\n\n)\n\nso it can also be used below for the finite-temperature metric. As a simple example, let us compute the two-point correlation function of T xy , which corresponds to φ in gravity. The field equation for φ is\n\n∂ µ ( √ -g g µν ∂ ν φ) = 0 . ( 45\n\n)\n\nThe solution to this equation, with the boundary condition φ(p, z = 0) = φ 0 (p), can be written as\n\nφ(p, z) = f p (z)φ 0 (p) , ( 46\n\n)\n\nwhere the mode function f p (z) satisfies the equation\n\nf ′ p z 3 ′ - p 2 z 3 f p = 0 ( 47\n\n)\n\nwith the boundary condition f p (0) = 1. The mode equation (47) can be solved exactly. Assuming p is spacelike, p 2 > 0, the exact solution and its expansion around z = 0 is\n\nf p (z) = 1 2 (pz) 2 K 2 (pz) = 1 - 1 4 (pz) 2 - 1 16 (pz) 4 ln(pz) + O((pz) 4 ) . ( 48\n\n)\n\nThe second solution to Eq. ( 47 ), (pz) 2 I 2 (pz), is ruled out because it blows up at z → ∞.\n\nViscosity, Black Holes, and QFT 11 We now substitute the solution into the quadratic action. Using the field equation, one can perform integration by parts and write the action as a boundary integral at z = 0. One finds\n\nS = N 2 16π 2 d 4 x 1 z 3 φ(x, z)φ ′ (x, z)| z→0 = d 4 p (2π) 4 φ 0 (-p)F(p, z)φ 0 (p)| z→0 , ( 49\n\n)\n\nwhere\n\nF(p, z) = N 2 16π 2 1 z 3 f -p (z)∂ z f p (z) . ( 50\n\n)\n\nDifferentiating the action twice with respect to the boundary value φ 0 one finds\n\nT xy T xy p = -2 lim z→0 F(p, z) = N 2 64π 2 p 4 ln(p 2 ) . ( 51\n\n)\n\nNote that we have dropped the term ∼ p 4 ln z, which, although singular in the limit z → 0, is a contact term [i.e., a term proportional to a derivative of δ(x) after Fourier transform]. Removing such terms by adding local counter terms to the supergravity action is known as the holographic renormalization [22] . It is, in a sense, a holographic counterpart to the standard renormalization procedure in quantum field theory, here applied to composite operators. For time-like p, p 2 < 0, there are two solutions to Eq. (47) which involve Hankel functions H (1) (z) and H (2) (z) instead of K 2 (z). Neither function blows up at z → ∞, and it is not clear which should be picked. Here we encounter, for the first time, a subtlety of Minkowski-space AdS/CFT, which is discussed in great length in subsequent sections. At zero temperature this problem can be overcome by an analytic continuation from space-like p. However, this will not work at nonzero temperatures."
},
{
"section_type": "OTHER",
"section_title": "Black Three-Brane Metric",
"text": "At nonzero temperatures, the metric dual to N = 4 SYM theory is the black three-brane metric,\n\nds 2 = r 2 R 2 (-f dt 2 + dx 2 ) + R 2 r 2 f dr 2 + R 2 dΩ 2 5 , ( 52\n\n) with f = 1 -r 4 0 /r 4 .\n\nThe event horizon is located at r = r 0 , where f = 0. In contrast to the usual Schwarzschild black hole, the horizon has three flat directions x. The metric (52) is thus called a black three-brane metric.\n\nWe frequently use an alternative radial coordinate u, defined as u = r 2 0 /r 2 . In terms of u, the boundary is at u = 0, the horizon at u = 1, and the metric is\n\nds 2 = (πT R) 2 u 2 (-f (u)dt 2 + dx 2 ) + R 2 4u 2 f (u) du 2 + R 2 dΩ 2 5 . ( 53\n\n)\n\nThe Hawking temperature is determined completely by the behavior of the metric near the horizon. Let us concentrate on the (t, r) part of the metric,\n\nds 2 = - 4r 0 R 2 (r -r 0 )dt 2 + R 2 4r 0 (r -r 0 ) dr 2 . ( 54\n\n)\n\nChanging the radial variable from r to ρ,\n\nr = r 0 + ρ 2 r 0 , ( 55\n\n)\n\n12 Son, Starinets and the metric components become nonsingular:\n\nds 2 = R 2 r 2 0 dρ 2 - 4r 2 0 R 2 ρ 2 dt 2 . ( 56\n\n)\n\nNote also that after a Wick rotation to Euclidean time τ , the metric has the form of the flat metric in cylindrical coordinates,\n\nds 2 ∼ dρ 2 + ρ 2 dϕ 2 , where ϕ = 2r 0 R -2 τ .\n\nTo avoid a conical singularity at ρ = 0, ϕ must be a periodic variable with periodicity 2π. This fact matches with the periodicity of the Euclidean time in thermal field theory τ ∼ τ + 1/T , from which one finds the Hawking temperature:\n\nT H = r 0 πR 2 . ( 57\n\n)\n\nOne of the first finite-temperature predictions of AdS/CFT correspondence is that of the thermodynamic potentials of the N = 4 SYM theory in the strong coupling regime. The entropy is given by the Bekenstein-Hawking formula S = A/(4G), where A is the area of the horizon of the metric (52); the result can then be converted to parameters of the gauge theory using Eqs. (30), (33), and (34) . One obtains\n\ns = S V = π 2 2 N 2 T 3 , ( 58\n\n)\n\nwhich is 3/4 of the entropy density in N = 4 SYM theory at zero 't Hooft coupling. We now try to generalize the AdS/CFT prescription to finite temperature. In the Euclidean formulation of finite-temperature field theory, field theory lives in a space-time with the Euclidean time direction τ compactified. The metric is regular at r = r 0 : If one views the (τ, r) space as a cigar-shaped surface, then the horizon r = r 0 is the tip of the cigar. Thus, r 0 is the minimal radius where the space ends, and there is no point in space with r less than r 0 . The only boundary condition at r = r 0 is that fields are regular at the tip of the cigar, and the AdS/CFT correspondence is formulated as\n\nZ 4D [J] = Z 5D [φ]| φ(z=0)→J . ( 59\n\n)\n\n4 REAL-TIME AdS/CFT\n\nIn many cases we must find real-time correlation functions not given directly by the Euclidean pathintegral formulation of thermal field theory. One example is the set of kinetic coefficients expressed, through Kubo's formulas, via a certain limit of real-time thermal Green's functions. Another related example appears if we want to directly find the position of the poles in the correlation functions that would correspond to the hydrodynamic modes. In principle, some real-time Green's functions can be obtained by analytic continuation of the Euclidean ones. For example, an analytic continuation of a two-point Euclidean propagator gives a retarded or advanced Green's function, depending on the way one performs the continuation. However, it is often very difficult to directly compute a quantity of interest in that way. In particular, it is very difficult to get the information about the hydrodynamic (small ω, small k) limit of real-time correlators from Euclidean propagators. The problem here is that we need to perform an analytic continuation from a discrete set of points in Euclidean frequencies (the Matsubara frequencies) ω = 2πin, where n is an integer, to the real values of ω. In the hydrodynamic limit, we are interested in real and small ω, whereas the smallest Matsubara frequency is already 2πT .\n\nViscosity, Black Holes, and QFT 13 Therefore, we need a real-time AdS/CFT prescription that would allow us to directly compute the real-time correlators. However, if one tries to naively generalize the AdS/CFT prescription, one immediately faces a problem. Namely, now r = r 0 is not the end of space but just the location of the horizon. Without specifying a boundary condition at r = r 0 , there is an ambiguity in defining the solution to the field equations, even as the boundary condition at r = ∞ is set.\n\nAs an example, let us consider the equation of motion of a scalar field in the black hole background, ∂ µ (g µν ∂ ν φ) = 0. The solution to this equation with the boundary condition φ = φ 0 at u = 0 is φ(p, u) = f p φ 0 (p), where f p (u) satisfies the following equation in the metric (53):\n\nf ′′ p - 1 + u 2 uf f ′ p + w 2 uf 2 f p - q 2 uf f p = 0 . ( 60\n\n)\n\nHere the prime denotes differentiation with respect to u, and we have defined the dimensionless frequency and momentum:\n\nw = ω 2πT , q = k 2πT . ( 61\n\n)\n\nNear u = 0 the equation has two solutions, f 1 ∼ 1 and f 2 ∼ u 2 . In the Euclidean version of thermal AdS/CFT, there is only one regular solution at the horizon u = 1, which corresponds to a particular linear combination of f 1 and f 2 . However, in Minkowski space there are two solutions, and both are finite near the horizon. One solution termed f p behaves as (1u) -iw/2 , and the other is its complex conjugate f * p ∼ (1u) iw/2 . These two solutions oscillate rapidly as u → 1, but the amplitude of the oscillations is constant. Thus, the requirement of finiteness of f p allows for any linear combination of f 1 and f 2 near the boundary, which means that there is no unique solution to Eq. (60)."
},
{
"section_type": "OTHER",
"section_title": "Prescription For Retarded Two-Point Functions",
"text": "Physically, the two solutions f p and f * p have very different behavior. Restoring the e -iωt phase in the wave function, one can write\n\ne -iωt f p ∼ e -iω(t+r * ) , ( 62\n\n)\n\ne -iωt f * p ∼ e iω(t-r * ) , ( 63\n\n)\n\nwhere the coordinate r * = ln(1u) 4πT (64) was introduced so that Eqs. (62) and (63) looked like plane waves. In fact, Eq. (62) corresponds to a wave that moves toward the horizon (incoming wave) and Eq. ( 63 ) to a wave that moves away from the horizon (outgoing wave). The simplest idea, which is motivated by the fact that nothing should come out of a horizon, is to impose the incoming-wave boundary condition at r = r 0 and then proceed as instructed by the AdS/CFT correspondence. However, now we encounter another problem. If we write down the classical action for the bulk field, after integrating by parts we get contributions from both the boundary and the horizon:\n\nS = d 4 p (2π) 4 φ 0 (-p)F(p, z)φ 0 (p) z=z H z=0 . ( 65\n\n)\n\n14 Son, Starinets If one tried to differentiate the action with respect to the boundary value φ 0 , one would find\n\nG(p) = F(p, z)| z H 0 + F(-p, z)| z H 0 . ( 66\n\n)\n\nFrom the equation satisfied by f p and from f * p = f -p , it is easy to show that the imaginary part of F(p, z) does not depend on z; hence the quantity G(p) in Eq. (66) is real. This is clearly not what we want, as the retarded Green's functions are, in general, complex. Simply throwing away the contribution from the horizon does not help because F(-p, z) = F * (p, z) owing to the reality of the equation satisfied by f p .\n\nA partial solution to this problem was suggested in Ref. [7] . It was postulated that the retarded Green's function is related to the function F by the same formula that was found at zero temperature: G R (p) = -2 lim z→0 F(p, z) . (67)\n\nIn particular, we throw away all contributions from the horizon. This prescription was established more rigorously in Ref. [8] (following an earlier suggestion in Ref. [23] ) as a particular case of a general real-time AdS/CFT formulation, which establishes the connection between the close-timepath formulation of real-time quantum field theory with the dynamics of fields in the whole Penrose diagram of the AdS black brane. Here we accept Eq. (67) as a postulate and proceed to extract physical results from it. It is also easy to generalize this prescription to the case when we have more than one field. In that case, the quantity F becomes a matrix F ab , whose elements are proportional to the retarded Green's function G ab ."
},
{
"section_type": "OTHER",
"section_title": "Calculating Hydrodynamic Quantities",
"text": "As an illustration of the real-time AdS/CFT correspondence, we compute the correlator of T xy . First we write down the equation of motion for φ = h x y :\n\nφ ′′ p - 1 + u 2 uf φ ′ p + w 2 -q 2 f uf 2 φ p = 0 . ( 68\n\n)\n\nIn contrast to the zero-temperature equation, now ω and k enter the equation separately rather than through the combination ω 2k 2 . Thus the Green's function will have no Lorentz invariance. The equation cannot be solved exactly for all ω and k. However, when ω and k are both much smaller than T , or w, q ≪ 1, one can develop series expansion in powers of w and q. There are two solutions that are complex conjugates of each other. The solution that is an incoming wave at u = 1 and normalized to 1 at u = 0 is\n\nf p (z) = (1 -u 2 ) -iw/2 + O(w 2 , q 2 ) . ( 69\n\n)\n\nThe kinetic term in the action for φ is\n\nS = - π 2 N 2 T 4 8 du f u φ ′2 . ( 70\n\n)\n\nApplying the general formula (67), one finds the retarded Green's function of T xy ,\n\nG R xy,xy (ω, k) = - π 2 N 2 T 4 4 iw , ( 71\n\n)\n\nViscosity, Black Holes, and QFT 15 and, using Kubo's formula for η, the viscosity,\n\nη = π 8 N 2 T 3 . ( 72\n\n)\n\nIt is instructive to compute other correlators that have poles corresponding to hydrodynamic modes. As a warm-up, let us compute the two-point correlators of the R-charge currents, which should have a pole at ω = -iDk 2 , where D is the diffusion constant. We first write down Maxwell's equations for the bulk gauge field. Let the spatial momentum be aligned along the x 3 -axis: p = (ω, 0, 0, k). Then the equations for A 0 and A 3 are coupled:\n\nwA ′ 0 + qf A ′ 3 = 0 , ( 73\n\n) A ′′ 0 - 1 uf (q 2 A 0 + wqA 3 ) = 0 , ( 74\n\n) A ′′ 3 + f ′ f A ′ 3 + 1 uf 2 (w 2 A 3 + wqA 0 ) = 0 . ( 75\n\n)\n\nOne can eliminate A 3 and write down a third-order equation for A 0 ,\n\nA ′′′ 0 + (uf ) ′ uf A ′′ 0 + w 2 -q 2 f uf 2 A ′ 0 = 0 . ( 76\n\n)\n\nNear u=1 we find two independent solutions, A ′ 0 ∼ (1u) ±iw/2 , and the incoming-wave boundary condition singles out (1u) -iw/2 . One can substitute A ′ 0 = (1u) -iw/2 F (u) into Eq. (76). The resulting equation can be solved perturbatively in w and q 2 . We find\n\nA ′ 0 = C(1 -u) -iw/2 1 + iw 2 ln 2u 2 1 + u + q 2 ln 1 + u 2u . ( 77\n\n)\n\nUsing Eq. (74) one can express C through the boundary values of A 0 and A 3 at u = 0:\n\nC = q 2 A 0 + wqA 3 iw -q 2 u=0 . ( 78\n\n)\n\nDifferentiating the action with respect to the boundary values, we find, in particular,\n\nJ 0 J 0 p = N 2 T 16π k 2 iω -Dk 2 , ( 79\n\n) where D = 1 2πT . ( 80\n\n)\n\nThe correlator given by Eq. (79) has the expected hydrodynamic diffusive pole, and D is the R-charge diffusion constant. Similarly, one can observe the appearance of the shear mode in the correlators of the metric tensor. We note that the shear flow along the x 1 direction with velocity gradient along the x 3 direction involves T 01 and T 31 , hence the interesting metric components are a 0 = h 1 0 and a 3 = h 1 3 . Two of the field equations are\n\na ′ 0 - qf w a ′ 3 = 0 , ( 81\n\n)\n\na ′′ 3 - 1 + u 2 uf a ′ 3 + 1 uf 2 (w 2 a 3 + wqa 0 ) = 0 . ( 82\n\n)\n\n16 Son, Starinets They can be combined into a single equation:\n\na ′′′ 0 - 2u f a ′′ 0 + 2uf -q 2 f + w 2 uf 2 a ′ 0 = 0 . ( 83\n\n)\n\nAgain, the solution can be found perturbatively in w and q:\n\na ′ 0 = C(1 -u) -iw/2 u -iw 1 -u - u 2 ln 1 + u 2 + q 2 2 (1 -u) . ( 84\n\n)\n\nApplying the prescription, one finds the retarded Green's functions. For example,\n\nG tx,tx (ω, k) = ξk 2 iω -Dk 2 , ( 85\n\n) where ξ = π 8 N 2 T 3 , D = 1 4πT . ( 86\n\n)\n\nThus, we found that the correlator contains a diffusive pole ω = -iDk 2 , just as anticipated from hydrodynamics. Furthermore, the magnitude of the momentum diffusion constant D also matched our expectation. Indeed, if one recalls the value of η from Eq. ( 72 ) and the entropy density from Eq. ( 58 ), one can check that D = η ǫ + P . (87)"
},
{
"section_type": "OTHER",
"section_title": "THE MEMBRANE PARADIGM",
"text": "Let us now look at the problem from a different perspective. The existence of hydrodynamic modes in thermal field theory is reflected by the existence of the poles of the retarded correlators computed from gravity. Are there direct gravity counterparts of the hydrodynamic normal modes? If the answer to this question is yes, then there must exist linear gravitational perturbations of the metric that have the dispersion relation identical to that of the shear hydrodynamic mode, ω ∼ -iq 2 , and of the sound mode, ω = c s qiγq 2 . It turns out that one can explicitly construct the gravitational counterpart of the shear mode. (It should be possible to find a similar construction for the sound mode, but it has not been done in the literature; for a recent work on the subject, see [24] .) Our discussion is physical but somewhat sketchy; for more details see Ref. [25] .\n\nFirst, let us construct a gravity perturbation that corresponds to a diffusion of a conserved charge (e.g., the R-charge in N = 4 SYM theory). To keep the discussion general, we use the form of the metric (44), with the metric components unspecified. Our only assumptions are that the metric is diagonal and has a horizon at r = r 0 , near which\n\ng 00 = -γ 0 (r -r 0 ), g rr = γ r r -r 0 . ( 88\n\n)\n\nThe Hawking temperature can be computed by the method used to arrive at Eq. ( 57 ), and one finds T = (4π\n\n) -1 (γ 0 /γ r ) 1/2 .\n\nWe also assume that the action of the gauge field dual to the conserved current is\n\nS gauge = dx √ -g - 1 4g 2 eff F µν F µν , ( 89\n\n)\n\nViscosity, Black Holes, and QFT 17 where g eff is an effective gauge coupling that can be a function of the radial coordinate r. For simplicity we set g eff to a constant in our derivation of the formula for D; it can be restored by replacing √ -g → √ -g/g 2 eff in the final answer. The field equations are\n\n∂ µ 1 g 2 eff √ -g F µν = 0 . ( 90\n\n)\n\nWe search for a solution to this equation that vanishes at the boundary and satisfies the incomingwave boundary condition at the horizon.\n\nThe first indication that one can have a hydrodynamic behavior on the gravity side is that Eq. (90) implies a conservation law on a four-dimensional surface. We define the stretched horizon as a surface with constant r just outside the horizon,\n\nr = r h = r 0 + ε, ε ≪ r 0 , ( 91\n\n)\n\nand the normal vector n µ directed along the r direction (i.e., perpendicularly to the stretched horizon). Then with any solution to Eq. (90), one can associate a current on the stretched horizon:\n\nj µ = n ν F µν r h . ( 92\n\n)\n\nThe antisymmetry of F µν implies that j µ has no radial component, j r = 0. The field equation (90) and the constancy of n ν on the stretched horizon imply that this current is conserved: ∂ µ j µ = 0. To establish the diffusive nature of the solution, we must show the validity of the constitutive equation j i = -D∂ i j 0 . Such constitutive equation breaks time reversal and obviously must come from the absorptive boundary condition on the horizon. The situation is analogous to the propagation of plane waves to a non-reflecting surface in classical electrodynamics. In this case, we have the relation B = -n × E between electric and magnetic fields. In our case, the corresponding relation is\n\nF ir = - γ r γ 0 F 0i r -r 0 , ( 93\n\n)\n\nvalid when r is close to r 0 . This relates j i ∼ F ir to the parallel to the horizon component of the electric field F 0i , which is one of the main points of the \"membrane paradigm\" approach to black hole physics [26, 27] . We have yet to relate j i to j 0 ∼ F 0r , which is the component of the electric field normal to the horizon. To make the connection to F 0r , we use the radial gauge A r = 0, in which\n\nF 0i ≈ -∂ i A 0 . ( 94\n\n)\n\nMoreover, when k is small the fields change very slowly along the horizon. Therefore, at each point on the horizon the radial dependence of the scalar potential A 0 is determined by the Poisson equation,\n\n∂ r ( √ -g g rr g 00 ∂ r A 0 ) = 0 , ( 95\n\n) whose solution, which satisfies A 0 (r = ∞) = 0, is A 0 (r) = C 0 ∞ r dr ′ g 00 (r ′ )g rr (r ′ ) -g(r ′ ) . ( 96\n\n)\n\n18 Son, Starinets This means that the ratio of the scalar potential A 0 and electric field F 0r approaches a constant near the horizon:\n\nA 0 F 0r r=r 0 = √ -g g 00 g rr (r 0 ) ∞ r 0 dr g 00 g rr √ -g (r) . ( 97\n\n) Combining the formulas j i ∼ F 0i ∼ ∂ i A 0 , and A 0 ∼ F 0r ∼ j 0 , we find Fick's law j i = -D∂ i j 0 , with the diffusion constant D = √ -g g xx g 2 eff √ -g 00 g rr (r 0 ) ∞ r 0 dr -g 00 g rr g 2 eff √ -g (r) . ( 98\n\n)\n\nThus, we found that for a slowly varying solution to Maxwell's equations, the corresponding charge on the stretched horizon evolves according to the diffusion equation. Therefore, the gravity solution must be an overdamped one, with ω = -iDk 2 . This is an example of a quasi-normal mode. We also found the diffusion constant D directly in terms of the metric and the gauge coupling g eff .\n\nThe reader may notice that our quasinormal modes satisfy a vanishing Dirichlet condition at the boundary r=∞. This is different from the boundary condition one uses to find the retarded propagators in AdS/CFT, so the relation of the quasinormal modes to AdS/CFT correspondence may be not clear. It can be shown, however, that the quasi-normal frequencies coincide with the poles of the retarded correlators [28, 29] .\n\nWe can now apply our general formulas to the case of N = 4 SYM theory. The metric components are given by Eq. ( 52 ). For the R-charge current g eff = const, Eq. (98) gives D = 1/(2πT ), in agreement with our AdS/CFT computation. For the shear mode of the stress-energy tensor we have effectively g 2 eff = g xx , so D = 1/(4πT ), which also coincides with our previous result. In both cases, the computation is much simpler than the AdS/CFT calculation."
},
{
"section_type": "OTHER",
"section_title": "Universality",
"text": "In all thermal field theories in the regime described by gravity duals the ratio of shear viscosity η to (volume) density of entropy s is a universal constant equal to 1/(4π) [h/(4πk B ), if one restores h, c and the Boltzmann constant k B ].\n\nOne proof of the universality is based on the relationship between graviton's absorption cross section and the imaginary part of the retarded Green's function for T xy [31] . Another way to prove the universality [32] is via the direct AdS/CFT calculation of the correlation function in Kubo's formula (18) .\n\nWe, however, follow a different method. It is based on the formula for the viscosity derived from the membrane paradigm. A similar proof was given by Buchel & Liu [30] .\n\nThe observation is that the shear gravitational perturbation with k = 0 can be found exactly by performing a Lorentz boost of the black-brane metric (52) . Consider the coordinate transformations r, t,\n\nx i → r ′ , t ′ , x ′ i of the form r = r ′ , t = t ′ + vy ′ √ 1 -v 2 ≈ t ′ + vy ′ ,\n\nViscosity, Black Holes, and QFT\n\n19 y = y ′ + vt ′ √ 1 -v 2 ≈ y ′ + vt ′ , x i = x ′ i , ( 99\n\n)\n\nwhere v < 1 is a constant parameter and the expansion on the right corresponds to v ≪ 1. In the new coordinates, the metric becomes\n\nds 2 = g 00 dt ′ 2 + g rr dr ′ 2 + g xx (r) p i=1 (dx ′ i ) 2 + 2v(g 00 + g xx )dt ′ dy ′ . ( 100\n\n)\n\nThis is simply a shear fluctuation at k = 0. In our language, the corresponding gauge potential is\n\na 0 = vg xx (g 00 + g xx ) . ( 101\n\n)\n\nThis field satisfies the vanishing boundary condition a 0 (r = ∞) = 0 owing to the restoration of Poincaré invariance at the boundary: g 00 /g xx → -1 when r → ∞. This clearly has a much simpler form than Eq. (96) for the solution to the generic Poisson equation. The simple form of solution (101) is valid only for the specific case of the shear gravitational mode with g 2 eff = g xx . We have also implicitly used the fact that the metric satisfies the Einstein equations, with the stress-energy tensor on the right being invariant under a Lorentz boost. Equation (97) now becomes\n\na 0 f 0r r→r 0 = - 1 + g xx g 00 ∂ r (g xx g 00 ) r→r 0 = g xx (r 0 ) γ 0 . ( 102\n\n)\n\nThe shear mode diffusion constant is\n\nD = a 0 f 0r r→r 0 √ γ 0 γ r g xx (r 0 ) = γ r γ 0 = 1 4πT . ( 103\n\n)\n\nBecause D = η/(ǫ + P ), and ǫ + P = T s in the absence of chemical potentials, we find that\n\nη s = 1 4π . ( 104\n\n)\n\nIn fact, the constancy of this ratio has been checked directly for theories dual to Dp-brane [25], M -brane [11] , Klebanov-Tseytlin and Maldacena-Nunez backgrounds [30], N = 2 * SYM theory [33] and others. Curiously, the viscosity to entropy ratio is also equal to 1/4π in the pre-AdS/CFT \"membrane paradigm\" hydrodynamics [34]: there, for a four-dimensional Schwarzschild black hole one has η m.p. = 1/16πG N , while the Bekenstein-Hawking entropy is s = 1/4G N . As remarked in Sec. 2, the ratio η/s is much larger than the one for weakly coupled theories. The fact that we found the ratio to be parametrically of order one implies that all theories with gravity duals are strongly coupled.\n\nIn N = 4 SYM theory, the ratio η/s has been computed to the next order in the inverse 't Hooft coupling expansion [35]\n\nη s = 1 4π 1 + 135ζ(3) 8(g 2 N ) 3/2 . ( 105\n\n)\n\nThe sign of the correction can be guessed from the fact that in the limit of zero 't Hooft coupling g 2 N → 0, the ratio diverges, η/s → ∞.\n\n20 Son, Starinets"
},
{
"section_type": "OTHER",
"section_title": "The Viscosity Bound Conjecture",
"text": "From our discussion above, one can argue that η s ≥ h 4π (106) in all systems that can be obtained from a sensible relativistic quantum field theory by turning on temperatures and chemical potentials. The bound, if correct, implies that a liquid with a given volume density of entropy cannot be arbitrarily close to being a perfect fluid (which has zero viscosity). As such, it implies a lower bound on the viscosity of the QGP one may be creating at RHIC. Interestingly, some model calculations suggest that the viscosity at RHIC may be not too far away from the lower bound [36, 37] .\n\nOne place where one may think that the bound should break down is superfluids. The ability of a superfluid to flow without dissipation in a channel is sometimes described as \"zero viscosity\". However, within the Landau's two-fluid model, any superfluid has a measurable shear viscosity (together with three bulk viscosities). For superfluid helium, the shear viscosity has been measured in a torsion-pendulum experiment by Andronikashvili [38] . If one substitutes the experimental values, the ratio η/s for helium remains larger than h/4πk B ≈ 6.08 × 10 -13 K s for all ranges of temperatures and pressures, by a factor of at least 8.8.\n\nAs discussed in Sec. 2.3, the ratio η/s is proportional to the ratio of the mean free path and the de Broglie wavelength of particles,\n\nη s ∼ ℓ mfp λ . ( 107\n\n)\n\nFor the quasi-particle picture to be valid, the mean free path must be much larger than the de Broglie wavelength. Therefore, if the coupling is weak and the system can be described as a collection of quasi-particles, the ratio η/s is larger than 1. We have found is that, within the N = 4 SYM theory and, more generally, theories with gravity duals, even in the limit of infinite coupling the ratio η/s cannot be made smaller than 1/(4π)."
},
{
"section_type": "CONCLUSION",
"section_title": "CONCLUSION",
"text": "In this review, we covered only a small part of the applications of AdS/CFT correspondence to finite-temperature quantum field theory. Here we briefly mention further developments and refer the reader to the original literature for more details.\n\nIn addition to N = 4 SYM theory, there exists a large number of other theories whose hydrodynamic behavior has been studied using the AdS/CFT correspondence, including the worldvolume theories on M2-and M5-branes [11] , theories on Dp branes [25] , and little string theory [39] . In all examples the ratio η/s is equal to 1/(4π), which is not surprising because the general proofs of Sec. 6 apply in these cases.\n\nWe have concentrated on the shear hydrodynamic mode, which has a diffusive pole (ω ∼ -ik 2 ). One can also compute correlators which have a sound-wave pole from the AdS/CFT prescription [10] . One such correlator is between the energy density T 00 at two different points in space-time.\n\nThe result confirms the existence of such a pole, with both the real part and imaginary part having exactly the values predicted by hydrodynamics (recall that in conformal field theories the bulk viscosity is zero and the sound attenuation rate is determined completely by the shear viscosity).\n\nViscosity, Black Holes, and QFT 21 Some of the theories listed above are conformal field theories, but many are not (e.g., the Dpbrane worldvolume theories with p = 3). The fact that η/s = 1/(4π) also in those theories implies that the constancy of this ratio is not a consequence of conformal symmetry. Theories with less than maximal number of supersymmetries have been found to have the universal value of η/s, for example, the N = 2 * theory [40], theories described by Klebanov-Tseytlin, and Maldacena-Nunez backgrounds [30] . A common feature of these theories is that they all have a gravitational dual description. The bulk viscosity has been computed for some of these theories [41, 39] . Besides viscosity, one can also compute diffusion constants of conserved charges by using the AdS/CFT correspondence. Above we presented the computation of the R-charge diffusion constant in N = 4 SYM theory; for similar calculations in some other theories see Ref. [11, 25] .\n\nRecently, the AdS/CFT correspondence was used to compute the energy loss rate of a quark in the fundamental representation moving in a finite-temperature plasma [42, 43, 44, 45] . This quantity is of importance to the phenomenon of \"jet quenching\" in heavy-ion collisions.\n\nSo far, the only quantity that shows a universal behavior at the quantitative level, across all theories with gravitational duals, is the ratio of the shear viscosity and entropy density. Recently, it was found that this ratio remains constant even at nonzero chemical potentials [46, 47, 48, 49, 50] .\n\nWhat have we learned from the application of AdS/CFT correspondence to thermal field theory? Although, at least at this moment, we cannot use the AdS/CFT approach to study QCD directly, we have found quite interesting facts about strongly coupled field theories. We have also learned new facts about quasi-normal modes of black branes. However, we have also found a set of puzzles: Why is the ratio of the viscosity and entropy density constant in a wide class of theories? Is there a lower bound on this ratio for all quantum field theories? Can this be understood without any reference to gravity duals? With these open questions, we conclude this review."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "The work of DTS is supported in part by U.S. Department of Energy under Grant No. DE-FG02-00ER41132. Research at Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MEDT. Literature Cited 1. Kirzhnits DA, Linde AD. Macroscopic consequences of the Weinberg model. Phys. Lett. B 42:471 (1972) 2. Gyulassy M, and McLerran L, New forms of QCD matter discovered at RHIC. Nucl. Phys. A 750:30 (2005) 3. Maldacena JM. The large N limit of superconformal field theory and supergravity. Adv. Theor. Math. Phys. 2:231 (1998) 4. Gubser SS, Klebanov IR, Polyakov AM. Gauge theory correlators from noncritical string theory.\n\nPhys. Lett. B 428:105 (1998) 5. Witten E. Anti de Sitter space and holography. Adv. Theor. Math. Phys. 2:253 (1998) 6. Policastro G, Son DT, Starinets AO. Shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma. Phys. Rev. Lett. 87:081601 (2001) 22 Son, Starinets 7. Son DT, Starinets AO. Minkowski-space correlators in AdS/CFT correspondence: Recipe and applications. J. High Energy Phys. 0209:042 (2002) 8. Herzog CP, Son DT. Schwinger-Keldysh propagators from AdS/CFT correspondence. J. High Energy Phys. 0303:046 (2003) 9. Policastro G, Son DT, Starinets AO. From AdS/CFT correspondence to hydrodynamics. J. High Energy Phys. 0209:042 (2002) 10. Policastro G, Son DT, Starinets AO. From AdS/CFT correspondence to hydrodynamics. II: Sound waves. J. High Energy Phys. 0212:054 (2002) 11. Herzog CP. The hydrodynamics of M-theory. J. High Energy Phys. 0212:026 (2002) 12. Herzog CP. The sound of M-theory. Phys. Rev. D 68:024013 (2003) 13. Forster D. Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, Reading: Benjamin (1975) 14. Landau LD, Lifshitz EM. Fluid Mechanics, Oxford: Pergamon Press (1987) 15. Jeon S. Hydrodynamic transport coefficients in relativistic scalar field theory. Phys. Rev. D 52:3591 (1995) 16. Jeon S, Yaffe LG. From quantum field theory to hydrodynamics: transport coefficients and effective kinetic theory. Phys. Rev. D 53:5799 (1996) 17. Aharony O et al. Large N field theories, string theory and gravity. Phys. Rep. 323:183 (2000) 18. Klebanov IR. TASI lectures: Introduction to the AdS/CFT correspondence. arXiv:hep-th/0009139 19. Polchinski J. String theory, Cambridge: Cambridge University Press (1998). 20. Appelquist T, Chodos A, Freund PG. Modern Kaluza-Klein theories, Menlo Park: Addison-Wesley (1987) 21. Kim HJ, Romans LJ, van Nieuwenhuizen P. The mass spectrum of chiral N = 2 D = 10 supergravity on S 5 . Phys. Rev. D 32:389 (1985) 22. Bianchi M, Freedman DZ, Skenderis K. Holographic renormalization. Nucl. Phys. B 631:159 (2002) 23. Maldacena JM. Eternal black holes in anti-de-Sitter. J. High Energy Phys. 0304:021 (2003) 24. Saremi O. Shear waves, sound waves on a shimmering horizon. arXiv:hep-th/0703170 25. Kovtun P, Son DT, Starinets AO. Holography and hydrodynamics: Diffusion on stretched horizons. J. High Energy Phys. 0310:064 (2003) 26. Damour T. Black Hole Eddy Currents. Phys. Rev. D 18: 3598 (1978) 27. Thorne, KS, Price RH, Macdonald DA. Black Hole: The Membrane Paradigm, New Haven: Yale University Press (1986) 28. Kovtun PK, Starinets AO. Quasinormal modes and holography. Phys. Rev. D 72:086009 (2005) 29. Starinets AO. Quasinormal spectrum and black hole membrane paradigm. Unpublished 30. Buchel A, Liu JT. Universality of the shear viscosity in supergravity. Phys. Rev. Lett., 93:090602 (2004) 31. Kovtun P, Son DT, Starinets AO. Viscosity in strongly interacting quantum field theories from black hole physics. Phys. Rev. Lett. 94:111601 (2004) 32. Buchel A. On universality of stress-energy tensor correlation functions in supergravity, Phys. Lett. B, 609:392 (2005). 33. Buchel A. N = 2* hydrodynamics. Nucl. Phys. B 708:451 (2005)\n\nViscosity, Black Holes, and QFT 23 34. Damour T. Surface effects in black hole physics. Proceedings of the Second Marcel Grossmann Meeting on General Relativity. Ed. R. Ruffini, Amsterdam: North Holland (1982) 35. Buchel A, Liu JT, Starinets AO. Coupling constant dependence of the shear viscosity in N = 4 supersymmetric Yang-Mills theory. Nucl. Phys. B 707:56 (2005) 36. Teaney D. Effect of shear viscosity on spectra, elliptic flow, and Hanbury Brown-Twiss radii.\n\nPhys. Rev. C 68:034913 (2003) 37. Shuryak E. Why does the quark gluon plasma at RHIC behave as a nearly ideal fluid? Prog. Part. Nucl. Phys. 53:273 (2004) 38. Andronikashvili E. Zh. Eksp. Teor. Fiz. 18:429 (1948) 39. Parnachev A, Starinets A. The silence of the little strings. J. High Energy Phys. 0510:027 (2005) 40. Buchel A, Liu JT, Thermodynamics of the N = 2 * flow. J. High Energy Phys. 0311:031 (2003) 41. Benincasa P, Buchel A, Starinets AO. Sound waves in strongly coupled non-conformal gauge theory plasma. Nucl. Phys. B 733:160 (2006) 42. Herzog CP et al. Energy loss of a heavy quark moving through N = 4 supersymmetric Yang-Mills plasma. J. High Energy Phys. 0607:013 (2006) 43. Liu H, Rajagopal K, Wiedemann UA. Calculating the jet quenching parameter from AdS/CFT, Phys. Rev. Lett. 97:182301 (2006) 44. Casalderrey-Solana J, Teaney D. Heavy quark diffusion in strongly coupled N = 4 Yang Mills.\n\nPhys. Rev. D 74:085012 (2006) 45. Gubser, SS. Drag force in AdS/CFT. Phys. Rev. D 74:126005 (2006) 46. Son DT, Starinets AO. Hydrodynamics of R-charged black holes. J. High Energy Phys. 0603:052 (2006) 47. Mas J, Shear viscosity from R-charged AdS black holes. J. High Energy Phys. 0603:016 (2006) 48. Maeda K, Natsuume M, Okamura T. Viscosity of gauge theory plasma with a chemical potential from AdS/CFT. Phys. Rev. D 73:066013 (2006) 49. Saremi O. The viscosity bound conjecture and hydrodynamics of M2-brane theory at finite chemical potential. J. High Energy Phys. 0610:083 (2006) 50. Benincasa P, Buchel A, Naryshkin R. The shear viscosity of gauge theory plasma with chemical potentials. Phys. Lett. B 645:309 (2007)"
}
] | |
arxiv:0704.0245 | 0704.0245 | 1 | 10.1088/1126-6708/2007/07/002 | 2a173fa7f0bb0e54db35e723c0ed83458928b85329dcea45e520d5d198a792a5 | One-loop MHV Rules and Pure Yang-Mills | It has been known for some time that the standard MHV diagram formulation of
perturbative Yang-Mills theory is incomplete, as it misses rational terms in
one-loop scattering amplitudes of pure Yang-Mills. We propose that certain
Lorentz violating counterterms, when expressed in the field variables which
give rise to standard MHV vertices, produce precisely these missing terms.
These counterterms appear when Yang-Mills is treated with a regulator,
introduced by Thorn and collaborators, which arises in worldsheet formulations
of Yang-Mills theory in the lightcone gauge. As an illustration of our
proposal, we show that a simple one-loop, two-point counterterm is the
generating function for the infinite sequence of one-loop, all-plus helicity
amplitudes in pure Yang-Mills, in complete agreement with known expressions. | [
"Andreas Brandhuber",
"Bill Spence",
"Gabriele Travaglini",
"Konstantinos\n Zoubos"
] | [
"hep-th",
"hep-ph"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | 1 {a.brandhuber, w.j.spence, g.travaglini, k.zoubos}@qmul.ac.uk Contents 1 Introduction 1 2 Background 3 2.1 The classical MHV Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 A four-dimensional regulator for lightcone Yang-Mills . . . . . . . . . . . . . 6 2.3 The one-loop (++++) amplitude . . . . . . . . . . . . . . . . . . . . . . . . 11 3 The all-plus amplitudes from a counterterm 12 3.1 Mansfield transformation of L CT . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 The four-point case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 The general all-plus amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 Discussion 27 A Notation 29 B Details on the four-point calculation 31 1 Introduction
One of the success stories arising from twistor string theory [1] (see [2] for a review) has been the development of new techniques in perturbative quantum field theory. These include recursion relations [3, 4] , generalised unitarity [5] and MHV methods (see [6] for a review). One of the key motivations of this work is to provide new approaches to study and derive phenomenologically relevant scattering amplitudes. In particular, this requires that one be able to deal with non-supersymmetric theories, and to include fermions, scalars, and particles with masses. A vital first step is to apply these new methods to pure Yang-Mills (YM) theory, and indeed, some of the first new results inspired by twistor string theory involved pure YM amplitudes at tree- [7, 8, 9, 10, 11, 12, 13, 14] and one-loop [15] level. 1 A recalcitrant issue in this work is the derivation of rational terms in quantum amplitudes. Unitarity-based techniques [16] and loop MHV methods [17] are successful in obtaining the cut-constructible parts of amplitudes; essentially this is because at some level they are dealing with four-dimensional cuts. In principle performing D-dimensional cuts generates all parts of amplitudes [18, 19, 20, 21] as long as only massless particles are involved, however these techniques still appear to be relatively cumbersome. Combinations of recursive techniques and unitarity have led to important progress recently [22, 23, 24, 25, 26, 27, 28, 29, 30, 31] , but it would be preferable to have a more powerful prescriptive formulation, particularly keeping in mind that applications to more general situations are sought.
A promising development from this point of view is the Lagrangian approach [32, 33, 34] . Here it has been argued that lightcone Yang-Mills theory, combined with a certain change of field variables, yields a classical action which comprises precisely the MHV vertices. A full Lagrangian description of MHV techniques would in principle give a prescription for applying such methods to diverse theories. The next step in developing this is to understand the quantum corrections in this Lagrangian approach. If one directly uses in a path integral the classical MHV action, containing only purely four-dimensional MHV vertices, then it is immediately clear that this cannot yield all known quantum amplitudes. For example, there is no way to construct one-loop amplitudes where the external gluons all have positive helicities, or where only one gluon has negative helicity, as all MHV vertices contain two negative helicity particles (this issue has been recently discussed in [35] ). These amplitudes are particular cases where the entire amplitude consists of rational terms. More generally, it seems clear that the vertices of the classical MHV Lagrangian will not yield the rational parts of amplitudes, but only the cut-constructible terms [15] . Important insights into this question can be obtained from the study of self-dual Yang-Mills theory, which has the same all-plus one-loop amplitude of full YM [36, 37, 38] as its sole quantum correction. 1 An example, relevant to the discussion in this paper, is given in [35] where it was shown how these amplitudes might be obtained from the Jacobian arising from a Bäcklund-type change of variables which takes the self-dual Yang-Mills theory to a free theory.
A discussion of the full Yang-Mills theory in the lightcone gauge has recently been given by Chakrabarti, Qiu and Thorn (CQT) in [39, 40, 41] . These papers employ an interesting regularisation which, importantly, does not change the dimension of spacetime. For this reason, we find it particularly suitable for setting the scene for the MHV diagram method, which is inherently four-dimensional in current approaches. The regularisation of CQT requires the introduction of certain counterterms, which prove to be rather simple in form. What we will show in this paper is that these simple counterterms provide a very compact and powerful way to represent the rational terms in gauge theory amplitudes; specifically, we will demonstrate that the single two-point counterterm contains all the n-point all-plus amplitudes. The way this happens is through the use of the new field variables of [32, 33, 34] . Other counterterms will combine with vertices from the Lagrangian and should generate the rational parts of more general amplitudes. Based on the discussion in this paper, we propose 1 In real Minkowski space, this is in fact its single non-vanishing amplitude.
that the counterterms, expressed in the field variables which give rise to standard MHV vertices, in combination with Lagrangian vertices, generate the rational terms previously missing from MHV diagram formulations.
The rest of the paper is organised as follows. After giving some background material in section 2, we explicitly derive in section 3 the four point all-plus amplitude from the twopoint counterterm of CQT. We follow this by showing that the n-point expression, obtained by writing the counterterm in new variables, has precisely the right collinear and soft limits required for it to be the correct all-plus n-point amplitude. We present our conclusions in section 4, and our notation and derivations of certain identities have been collected in two appendices.
In this section, we first review the classical field redefinition from the lightcone Yang-Mills Lagrangian to the MHV-rules Lagrangian. We then move on to motivate the fourdimensional regularisation scheme we will employ, and argue that it leads directly to the introduction of a certain Lorentz-violating counterterm in the Yang-Mills Lagrangian. We close the section with the remarkable observation that this counterterm provides a simple way to calculate the four-point all-plus one-loop amplitude using only tree-level combinatorics.
It seemed clear from the beginning that the MHV diagram approach to Yang-Mills theory must be closely related to lightcone gauge theory. This idea was substantiated by Mansfield [33] (see also [32] ). The starting point of [33] is the lightcone gauge-fixed YM Lagrangian for the fields corresponding to the two physical polarisations of the gluon. It was argued convincingly in [33] that a certain canonical change of the field variables re-expresses this lightcone Lagrangian as a theory containing the infinite series of MHV vertices. Some of the arguments in [33] were rather general; these were reviewed in [34] , where the change of variables was discussed in more detail, and in particular it was shown how the four-and five-point MHV vertices arise from the change of variables. In this paper we will mainly follow the notation of [34] .
The general structure of the lightcone YM Lagrangian, after integrating out unphysical degrees of freedom, is (see appendix A for more details)
L YM = L +-+ L ++-+ L --+ + L ++--, (2.1)
where the gauge condition is η µ A µ = 0 with the null vector η = (1/ √ 2, 0, 0, 1/ √ 2). Since this Lagrangian contains a + +vertex, it is not of MHV type. In [33] , Mansfield proposed to 3 eliminate this vertex through a suitably chosen field redefinition. Specifically, he performed a canonical change of variables from (A, Ā) to new fields (B, B), in such a way that L +-(A, Ā) + L ++-(A, Ā) = L +-(B, B) . (2.2)
The remarkable result is that upon inserting this change of variables into the remaining two vertices, the Lagrangian, written in terms of (B, B), becomes a sum of MHV vertices, L YM = L +-+ L +--+ L ++--+ L +++--+ . . . . (2.3) The crucial property of Mansfield's transformation that makes this possible is that, while both A and Ā are series expansions in the new B fields, A has no dependence on the B fields while Ā turns out to be linear in B. Thus, since the remaining vertices are quadratic in the B, the new interaction vertices have the helicity configuration of an MHV amplitude. Mansfield was also able to show that the explicit form of the vertices coincides with the CSW off-shell continuation of the Parke-Taylor formula for the MHV scattering amplitudes, as proposed by [7].
One of the main results of [34] was the derivation of an explicit, closed formula for the expansion of the original fields (A, Ā) in terms of the new fields (B, B). This was then used to verify that the new vertices are indeed precisely the MHV vertices of [7], at least up to the five-point level. We will now briefly review these results. First, recall that the positive helicity field A is a function of the positive helicity B field only. It is expanded as follows:
A( p ) = ∞ n=1 Σ n i=1 d 3 p i (2π) 3 ∆( p , p 1 , . . . p n ) Y( p ; 1 • • • n) B( p 1 )B( p 2 ) • • • B( p n ) , (2.4) where ∆( p , p 1 , . . . p n ) := (2π) 3 δ (3) ( pp 1 -• • •p n ). Note that the x -coordinate is common to all the fields, which is why we have restricted the transformation to the lightcone quantisation surface Σ.
By inserting this expansion into (2.2) and using the requirement that the transformation be canonical, Ettle and Morris succeeded in deriving a very simple expression for the coefficients Y. After translating to our conventions (see appendix A), they are given by:
Y( p ; 12 • • • n) = ( √ 2ig) n-1 p + p 1 + p n + 1 12 23 • • • n -1, n . (2.5)
The first few terms in (2.4) are then: 3) ( pp 1p 2p 3 ) p 1 + p 3 + 1 12 23 B( p 1 )B( p 2 )B( p 3 ) + • • • . (2.6) 4 Similarly, one can write down the expansion of the negative helicity field Ā, which, as discussed above, is linear in B, but is an infinite series in the new field B. In [34] it was shown that the coefficients in the expansion of Ā are very closely related to those for A. 2
A( p ) =B( p ) + √ 2igp + Σ d 3 p 1 d 3 p 2 (2π) 3 δ (3) ( p -p 1 -p 2 ) p 1 + p 2 + 1 12 B( p 1 )B( p 2 ) -2g 2 p + Σ d 3 p 1 d 3 p 2 d 3 p 3 (2π) 6 δ (
The expansion of B turns out to be simply Ā( p ) =-∞ n=1 n s=1 Σ n i=1 d 3 p i (2π) 3 ∆( p , p 1 , . . . , p n )
(p s + ) 2 (p + ) 2 Y( p ; 1 • • • n) B( p 1 )• • • B( p s )• • •B( p n ) = - ∞ n=1 Σ n i=1 d 3 p i (2π) 3 ∆( p , p 1 , . . . , p n ) 1 (p + ) 2 Y( p ; 1 • • • n) × n s=1 (p s + ) 2 B( p 1 ) • • • B( p s ) • • • B( p n ) . (2.7)
Thus we see that at each order in the expansion, we need to sum over all possible positions of B. Explicitly, the first few terms are: Ā( p ) = B( p ) -√ 2ig Σ d 3 p 1 d 3 p 2 (2π) 3 δ (3) ( pp 1p 2 ) 1 p + p 1 + p 2 + 1 12 × × (p 1 + ) 2 B( p 1 )B( p 2 ) + (p 2 + ) 2 B( p 1 ) B( p 2 ) + 2g 2 Σ d 3 p 1 d 3 p 2 d 3 p 3 (2π) 6 δ (3) ( pp 1p 2p 3 ) 1 p + p 1 + p 3 + 1 12 23 × × (p 1 + ) 2 B( p 1 )B( p 2 )B( p 3 )+(p 2 + ) 2 B( p 1 ) B( p 2 )B( p 3 )+(p 3 + ) 2 B( p 1 )B( p 2 ) B( p 3 ) + • • • (2.8) Using the above results, it is in principle straightforward to derive the terms that arise on inserting the Mansfield transformation into the two remaining vertices of the theory. For the simplest cases, one can see explicitly that these combine to produce MHV vertices, and some arguments were also given in [33, 34] that this must be true in general.
In supersymmetric theories, the MHV vertices are enough to reproduce complete scattering amplitudes at one loop [43] . However, as we mentioned earlier, for pure YM it is clear that the terms in the MHV Lagrangian (2.3) will not be enough to generate complete quantum amplitudes. For instance, the scattering amplitude with all gluons with positive helicity, which at one loop is finite and given by a rational term, cannot be obtained by only using MHV diagrams, for the simple reason that one cannot draw any diagram contributing to it by only resorting to MHV vertices. 3 Another amplitude which cannot be derived within conventional MHV diagrams is the amplitude with only one gluon of negative helicity. Similarly to the all-plus amplitude, this single-minus amplitude vanishes at tree level, and at one loop is given by a finite, rational function of the spinor variables.
2 This is perhaps easiest to see [42] by considering that, in the context of N = 4 SYM, A and B are part of the same lightcone superfield.
3 On the other hand, it was shown in [35] that the parity conjugate all-minus amplitude is correctly generated by using MHV diagrams.
The lesson we learn from this is that, in order to apply the MHV method to derive complete amplitudes in pure YM, one should look more closely at the change of variables in the full quantum theory. There are several possible subtleties one should pay careful attention to at the quantum level. First of all, it is possible that the canonical nature of the transformation is not preserved, leading to a non-trivial Jacobian which could provide the missing amplitudes. Another possible source of contributions could come from violations of the equivalence theorem. This theorem states that, although correlation functions of the new fields are in general different from those of the old fields, the scattering amplitudes are actually the same 4 , as long as the new fields are good interpolating fields. These issues were explored in some detail in [35] (see also [34, 42] ) where it was shown, for a different (non-canonical) field redefinition, how a careful treatment of these effects can combine to reproduce some of the amplitudes that would seem to be missing at first sight.
Another aim of [35] was to demonstrate how to reproduce one of the above-mentioned rational amplitudes, the one with all-minus helicities, in the MHV formalism. This amplitude is slightly less mysterious than the all-plus amplitude in the sense that one can write down the contributing diagrams using only MHV vertices; however a calculation without a suitable regulator in place would give a vanishing answer, despite the fact that this amplitude is finite.
In [35] , it was shown, using dimensional regularisation, that the full nonzero result arises from a slight mismatch between four-and D (= 4 -2ǫ)-dimensional momenta.
It is natural therefore to expect that dimensional regularisation will be helpful also for the problem at hand, which is to recover the rational amplitudes of pure Yang-Mills after the Mansfield transformation. Decomposing the regularised lightcone Lagrangian into a pure four-dimensional part and the remaining ǫ-dependent terms, and performing the transformation on the four-dimensional part only, will give rise to several new ǫ-dependent terms that can potentially give finite answers when forming loops.
Although this approach shows promise, it is not the one we will make use of in the following. Instead, motivated by the fact that the Mansfield transformation seems to be deeply rooted in four dimensions, we would like to look for a purely four-dimensional regularisation scheme. We now turn to a review of the particular scheme we will use.
In the above we explained why a naïve application of the Mansfield transform leads to puzzles at the quantum level, and discussed possible ways to improve the situation. The conclusion was that, since the missing amplitudes arise from subtle mismatches in regularisation, one should be careful to perform the Mansfield transform on a suitably regularised version of the lightcone Yang-Mills action. Here we will review one approach to the regularisation of lightcone Yang-Mills, which, despite several slightly unusual features, appears to be ideally 4 Modulo a trivial wave-function renormalisation.
suited for the problem at hand. The regularisation we propose to use is inspired by recent work of CQT [39, 40, 41] on Yang-Mills amplitudes in the lightcone worldsheet approach [44, 45] . This is an attempt to understand gauge-string duality which is similar in spirit to 't Hooft's original work on the planar limit of gauge theory [46] , and aims at improving on early dual model techniques [47, 48] . We recall that one of the main goals in those works is to exhibit the string worldsheet as made up of very large planar diagrams ("fishnets").
In their recent work, Thorn and collaborators make this statement more precise, using techniques that were unavailable when the original ideas were put forward. It is hoped that, by understanding how to translate a generic Yang-Mills planar diagram to a configuration of fields (with suitable boundary conditions) on the lightcone worldsheet, it will eventually become possible to perform the sum of all these diagrams. This approach to gauge-string duality is thus complementary to that using the AdS/CFT correspondence.
The field content and structure of the worldsheet theory dual to Yang-Mills theory is rather intricate (see e.g. [45] ), but for our purposes the details are not important. What is most relevant for us is that one of the principles of this approach is that all quantities on the Yang-Mills side should have a local worldsheet description. This includes the choice of regulator that needs to be introduced when calculating loop diagrams. This requirement led Thorn [49] (see also [50, 51] ) to introduce an exponential UV cutoff, which we will discuss in a short while.
Since one of the goals of this programme is to translate an arbitrary planar diagram into worldsheet form (and eventually calculate it), it is an important intermediate goal to understand how to do standard Yang-Mills perturbation theory in "worldsheet-friendly" language. In [39, 40, 41] CQT do exactly that for the simplest case, that of one-loop diagrams in Yang-Mills theory, by analysing how familiar features like renormalisation are affected by the unusual regularisation procedure and other special features of the lightcone worldsheet formalism.
To conclude this brief overview of the lightcone worldsheet formalism, the main point for our current purposes is that it provides motivation and justification for a slightly unusual regularisation of lightcone Yang-Mills, which we will now describe.
Let us momentarily focus on the self-dual part of the lightcone Yang-Mills Lagrangian:
L = L -+ + L ++-= -A z A z + 2ig[A z , ∂ + A z ](∂ + ) -1 (∂ z A z ) .
(2.9) This action provides one of the representations of self-dual Yang-Mills theory. After transforming to momentum space, we find that the only vertex in the theory is the following (suppressing the gauge index structure):
7 A 2 A 1 Ā3 = -2g p 3 + p 1 + p 2 + [p 1 + p 2 z -p 2 + p 1 z ] = - √ 2g p 3 + p 1 + p 2 + [12] . (2.10)
As for propagators, following [40], we will use the Schwinger representation:
1 p 2 = - ∞ 0 dT e +T p 2 . (2.11)
In (2.11) p 2 is understood to be the appropriate (p 2 < 0) Wick rotated version of the Minkowski space inner product. For our choice of signature, the latter is
p • q = p + q -+ p -q + -p • q = p + q -+ p -q + -(p z q z + p z q z ) , (2.12) so that p 2 = 2(p + p --p z p z ).
We will also make use of the dual or "region momentum" representation, where one assigns a momentum to each region that is bounded by a line in the planar diagram. By convention, the actual momentum of the line is given by the region momentum to its right minus that on its left, as given by the direction of momentum flow 5 . Clearly such a prescription can only be straightforwardly implemented for planar diagrams, which is the case considered in [40] . This is also sufficient for our purposes, since we are calculating the leading single-trace contribution to one-loop scattering amplitudes. Non-planar (multi-trace) contributions can be recovered from suitable permutations of the leading-trace ones (see e.g. [52] ).
To demonstrate the use of region momenta, a sample one-loop diagram is pictured in Figure 1 .
q k 1 k 2 k 3 k 4 1 2 3 4 l
Figure 1: A sample one-loop diagram indicating the labelling of region momenta. The outgoing leg momenta are
p 1 = k 1 -k 4 , p 2 = k 2 -k 1 , p 3 = k 3 -k 2 , p 4 = k 4 -k 3 , while
the loop momentum (directed as indicated) is l = qk 1 .
5 In [40] the flow of momentum is chosen to always match the flow of helicity, but we will not use this convention.
The "worldsheet-friendly" regulator that CQT employ is simply defined as follows [49]: For a general n-loop diagram, with q i being the loop region momenta, one simply inserts an exponential cutoff factor exp(-δ n i=1 q 2 i ) (2.13) in the loop integrand, where δ is positive and will be taken to zero at the end of the calculation. This clearly regulates UV divergences (from large transverse momenta), but, as we will see, has some surprising consequences since it will lead to finite values for certain Lorentzviolating processes, which therefore have to be cancelled by the introduction of appropriate counterterms.
Note that q 2 = 2q z q z has components only along the two transverse directions, hence it breaks explicitly even more Lorentz invariance than the lightcone usually does. This might seem rather unnatural from a field-theoretical point of view, however it is crucial in the lightcone worldsheet approach. Indeed, the lightcone time x -and x + (or in practice its dual momentum p + ) parametrise the worldsheet itself, and are regulated by discretisation; thus, they are necessarily treated very differently from the two transverse momenta q z , q z which appear as dynamical worldsheet scalars. Fundamentally, this is because of the need to preserve longitudinal (x + ) boost invariance (which eventually leads to conservation of discrete p + ). The fact that the regulator depends on the region momenta rather than the actual ones is a consequence of asking for it to have a local description on the worldsheet.
The main ingredient for what will follow later in this paper is the computation of the (++) one-loop gluon self-energy in the regularisation scheme discussed earlier. This is performed on page 10 of [40] , and we will briefly outline it here. This helicity-flipping gluon self-energy, which we denote by Π ++ , is the only potential self-energy contribution in self-dual Yang-Mills; in full YM we would also have Π +-and, by parity invariance, Π --.
There are two contributions to this process, corresponding to the two ways to route helicity in the loop, but they can be easily shown to be equal so we will concentrate on one of them, which is pictured in Figure 2
. A A A Ā A Ā k ′ k q p -p l p + l
Figure 2: Labelling of one of the selfenergy diagrams contributing to Π ++ .
In Figure 2, p, -p are the outgoing line momenta, l is the loop line momentum, and 9 k, k ′ , q are the region momenta, in terms of which the line momenta are given by p = k ′k, l = qk ′ . (2.14) Remembering to double the result of this diagram, we find the following expression for the self-energy:
Π ++ =8g 2 N d 4 l (2π) 4 -(p + l) + p + l + (p + l z -l + p z) × 1 l 2 (p + l) 2 × × -l + (-p + )(p + l) + ((-p + )(p z + l z ) -(p + + l + )(p z)) = g 2 N 2π 4 d 4 l 1 (p + ) 2 (p + l z -l + p z )(p + (p z + l z ) -(p + + l + )p z ) 1 l 2 (p + l) 2 . (2.15)
Although we are suppressing the colour structure, the factor of N is easy to see by thinking of the double-line representation of this diagram 6 . One of the crucial properties of (2.15) is that the factors of the loop momentum l + coming from the vertices have cancelled out, hence there are no potential subtleties in the loop integration as l + → 0. This means that, although for general loop calculations one would have to follow the DLCQ procedure and discretise l + (as is done for other processes considered in [39, 40, 41] ), this issue does not arise at all for this particular integral, and we are free to keep l + continuous.
To proceed, we convert momenta to region momenta, rewrite propagators in Schwinger representation, and regulate divergences using the regulator (2.13). Employing the unbroken shift symmetry in the + region momenta to further set k + = 0, (2.15) can be recast as:
Π ++ = g 2 N 2π 4 ∞ 0 dT 1 dT 2 d 4 q 1 (k ′ + ) 2 e T 1 (q-k) 2 +T 2 (q-k ′ ) 2 -δq 2 × × k ′ + (q z -k ′ z) -(q + -k ′ + )(k ′ z -k z) k ′ + (q z -k z ) -q + (k ′ z -k z ) .
(2.16) Since q -only appears in the exponential, the q -integration will lead to a delta function containing q + , which can be easily integrated and leads to a Gaussian-type integral for q z , q z . Performing this integral, we obtain (setting xk+(1-x)
T = T 1 + T 2 , x = T 1 /(T 1 + T 2 )) Π ++ = g 2 N 2π 2 1 0 dx ∞ 0 dT δ 2 [xk z + (1 -x)k ′ z ] 2 (T + δ) 3 e T x(1-x)p 2 -δT T +δ (
k ′ ) 2 . (2.17)
Notice that, had we not regularised using the δ regulator, we would have obtained zero at this stage. Instead, now we can see that there is a region of the T integration (where T ∼ δ) that can lead to a nonzero result. On performing the T and x integrations, and sending δ to zero at the end, we obtain the following finite answer:
Π ++ = 2 + + = g 2 N 12π 2 (k z ) 2 + (k ′ z) 2 + k z k ′ z . (2.18)
6 For simplicity, we take the gauge group to be U(N ).
10 Notice that this nonvanishing, finite result violates Lorentz invariance, since it would imply that a single gluon can flip its helicity. Also, it explicitly depends on only the z components of the region momenta. Such a term is clearly absent in the tree-level Lagrangian (unlike e.g. the Π +-contribution in full Yang-Mills theory), thus it cannot be absorbed through renormalisation -it will have to be explicitly cancelled by a counterterm. This counterterm, which will play a major rôle in the following, is defined in such a way that: + = 0 , (2.19) in other words it will cancel all insertions of Π ++ , diagram by diagram. Let us note here that, had we been doing dimensional regularisation, all bubble contributions would vanish on their own, so there would be no need to add any counterterms. So this effect is purely due to the "worldsheet-friendly" regulator (2.13).
It is also interesting to observe that in a supersymmetric theory this bubble contribution would vanish 7 so this effect is only of relevance to pure Yang-Mills theory.
Now let us look at the all-plus four-point one-loop amplitude in this theory. It is easy to see that it will receive contributions from three types of geometries: boxes, triangles and bubbles. It is a remarkable property 8 that the sum of all these geometries adds up to zero. In particular, with a suitable routing of momenta, the integrand itself is zero. Pictorially, we can state this as:
+ 4 × + 2 × + 8 × = 0 . (2.20)
The coefficients mean that we need to add that number of inequivalent orderings. So we see (and refer to [40] for the explicit calculation) that the sum of all the diagrams that one can construct from the single vertex in our theory, gives a vanishing answer. However, as discussed in the previous section, this is not everything: we need to also include the contribution of the counterterm that we are forced to add in order to preserve Lorentz invariance. Since this counterterm, by design, cancels all the bubble graph contributions, we are left with just the sum of the box and the four triangle diagrams. In pictures,
A ++++ = +4× + 2 × + 8 × + 2 × + 8 × (2.21) 7
This can in fact be derived from the results of [53] , where similar calculations were considered with fermions and scalars in the loop.
8 This observation is attributed to Zvi Bern [40].
where A ++++ is the known result [54] for the leading-trace part of the four-point all-plus amplitude:
A ++++ (A 1 A 2 A 3 A 4 ) = i g 4 N 48π 2 [12][34] 12 34 , (2.22)
and the terms in the parentheses clearly cancel among themselves. This leaves the box and triangle diagrams, which are exactly those appearing in the calculation of the parity conjugate amplitude using dimensional regularisation [35] , where the bubbles were zero to begin with.
Following [40], we make the obvious, but important for the following, observation that one can change the position of the parentheses:
A ++++ = + 4 × + 2 × + 8 × +2× +8× (2.23)
where again the terms in the parentheses are zero (by (2.20)). This demonstrates that one can compute the all-plus amplitude just from a tree-level calculation with counterterm insertions (of course, these diagrams are at the same order of the coupling constant as oneloop diagrams because of the counterterm insertion). This remarkable claim is verified in [40] , where CQT explicitly calculate the 10 counterterm diagrams and recover the correct result for the four-point amplitude (see pp. 22-23 of [40]) 9 .
This result, apart from being very appealing in that one does not have to perform any integrals (apart from the original integral that defined the counterterm) so that the calculation reduces to tree-level combinatorics, will also turn out to be a convenient starting point for performing the Mansfield transformation. Specifically, our claim is that the whole series of all-plus amplitudes will arise just from the counterterm action. In the following we will show how this works explicitly for the four-point all-plus case, and then we will argue for the n-point case that the corresponding expression derived from the counterterm has all the correct singularities (soft and collinear), giving strong evidence that the result is true in general.
Having reviewed the relevant new features that arise when doing perturbation theory with the worldsheet-motivated regulator of [49] , we now have all the necessary ingredients to perform the Mansfield change of variables on the regulated lightcone Lagrangian. In this section, we will carry out this procedure. We will first regulate lightcone self-dual Yang-Mills, which, as 9 In practice, these authors choose to insert the self-energy result (2.18) in the tree diagrams, so what they compute is minus the all-plus amplitude.
discussed, will require us to introduce an explicit counterterm in the Lagrangian. Then we will perform the Mansfield transformation on the original Lagrangian (converting it to a free theory). We will then show that, upon inserting the change of variables into the counterterm Lagrangian, we recover the all-plus amplitudes as vertices in the theory.
As we saw, the "worldsheet-friendly" regularisation requires us to add a certain counterterm to the lightcone Yang-Mills action, required in order to cancel the Lorentz-violating helicityflipping gluon selfenergy. As mentioned previously, the calculation of the all-plus amplitude can be tackled purely within the context of self-dual Yang-Mills, which we will focus on from now on. We see that, as a result of this regularisation, the complete action at the quantum level becomes:
L (r) SDYM = L +-+ L ++-+ L CT , (3.1)
where L +-+ L ++-is the classical Lagrangian for self-dual Yang-Mills introduced in (2.9). Although CQT do not write down a spacetime Lagrangian for L CT , it is easy to see that the following expression would have the right structure:
L CT = - g 2 N 12π 2 Σ d 3 k i d 3 k j A i j (k i , k j )[(k i z ) 2 + (k j z ) 2 + k i z k j z ]A j i (k j , k i ) . (3.2)
This expression depends explicitly on the dual, or region, momenta. In (3.2) we have made use of the simplest way to associate region momenta to fields, which is to assign a region momentum to each index line in double-line notation [46] , and thus a momentum k i , k j to each of the indices of the gauge field A i j (now slightly extended into a dipole, as would be natural from the worldsheet perspective, where an index is associated to each boundary). Since each line has a natural orientation, the actual momentum of each line can be taken to be the difference of the index momentum of the incoming index line and the outgoing index line. So the momentum of
A i j (k i , k j ) is taken to be p = k j -k i .
As discussed above, this assignment can only be performed consistently for planar diagrams, which is sufficient for our purposes.
Clearly, the structure of (3.2) is rather unusual. First of all, it depends only on the antiholomorphic (z) components of the region momenta, and so is clearly not (lightcone) covariant. Even more troubling is the fact that it does not depend only on differences of region momenta, but also on their sums. Since each field thus carries more information than just its momentum, L CT is a non-local object from a four-dimensional point of view (although, as shown in [40] , it can be given a perfectly local worldsheet description).
Leaving the above discussion as food for thought, we will now rewrite (3.2) in a more conventional way that is most convenient for inserting into Feynman diagrams,
L CT = - g 2 N 12π 2 Σ d 3 p d 3 p ′ δ(p + p ′ ) A i j (p ′ )((k i z) 2 + (k j z) 2 + k i z k j z)A j i (p) . (3.3) 13
In this expression, which can be thought of as the zero-mode or field theory limit of (3.2), all the region momentum dependence is confined to the polynomial factor (k i z ) 2 +(k j z) 2 +k i z k j z. This vertex, inserted into tree diagrams, would exactly reproduce the effects of the counterterm pictured in (2.19). Although (3.3) still exhibits some of the apparently undesirable features we discussed above, the calculations in [40] demonstrate that, after summing over all possible insertions of this term, the final result is covariant and correctly reproduces the all-plus amplitudes 10 . Therefore, we believe that its problematic properties are really a virtue in disguise, and (as we will see explicitly) they seem to be crucial in obtaining the full series of n-point all-plus amplitudes from the Mansfield transformation of a single term.
We are now ready to perform the Mansfield change of variables. In the spirit of the discussion earlier, we will perform the transformation on the classical part of the action only:
L +-(A, Ā) + L ++-(A, Ā) = L +-(B, B) (3.4)
Hence the classical part of the action has been converted to a free theory. Without a regulator, this would be the whole story. However we now see that, within the particular regularisation we are working with, the full Lagrangian L (r) SDYM contains one extra, one-loop piece, given by L CT in (3.3), which is quadratic in the positive helicity fields A. To complete the Mansfield transformation, we will clearly need to expand this term in the new fields B, using the Ettle-Morris coefficients (2.4).
Since L CT depends only on the holomorphic A fields, we will only need the expansion of A in terms of B given in (2.4). As a first check that L CT leads to the right kind of structure, note that since A depends only on the holomorphic B fields, all the new vertices are all-plus. Thus, the full action, when expressed in terms of the B fields, takes the schematic form:
L (r) SDYM (A, Ā) = L +-(B, B) + L ++ (B) + L +++ (B) + L ++++ (B) + • • • (3.5)
In the next section we will calculate the four-point term L ++++ and demonstrate that, when restricted on-shell, it reproduces the known form (2.22) for the all-plus amplitude.
To begin with, we focus on the derivation of the four-point all-plus vertex, whose on-shell version will give us the four-point scattering amplitude. We will thus expand the old fields A in the counterterm (3.3) (or (3.2)) up to terms containing four B-fields.
When inserting the Ettle-Morris coefficients into (3.3), one has to sum over all possible cyclic orderings with which this can be done. A complication is that now the counterterm 10 Note that similar-looking treatments using index momenta instead of line momenta for vertices, but which in the end sum up to covariant results have appeared in the context of noncommutative geometry (see e.g. [55] ). Although it is possible to write e.g. (3.2) in star-product form, at this stage it is not clear whether that is a useful reformulation.
itself depends on the ordering. In other words, we need to sum over all the ways of assigning dual momenta to the indices. Schematically, the inequivalent terms that we obtain are:
AA → ( B 1 B 2 )( B 3 B 4 ) + ( B 2 B 3 )( B 4 B 1 ) + ( B 1 B 2 B 3 )B 4 + ( B 2 B 3 B 4 )B 1 + ( B 3 B 4 B 1 )B 2 + ( B 4 B 1 B 2 )B 3 , (3.6)
where the terms on the first line arise from doing two quadratic substitutions and those on the second from doing one cubic substitution. All the other possibilities are related by cyclicity of the trace. For definiteness, let us now write down what one of these terms means explicitly: 11
( B 1 B 2 B 3 )B 4 = -2g 2 tr dpdp 4 δ(p+p 4 ) dp 1 dp 2 dp 3 δ(p-p 1 -p 2 -p 3 ) p + p 1 + p 3 + 1 12 23 × × B(p 1 )B(p 2 )B(p 3 ) (k 3 z ) 2 + (k 4 z ) 2 + k 4 z k 3 z B(p 4 ) = 2g 2 dp 1 dp 2 dp 3 dp 4 δ(p 1 + p 2 + p 3 + p 4 )× × p 4 + p 1 + p 3 + (k 3 z ) 2 + (k 4 z ) 2 + k 4 z k 3 z 12 23 tr B(p 1 )B(p 2 )B(p 3 )B(p 4 ) . (3.7)
The reason this particular combination of k z's appears here is that, given the ordering we chose, after the Mansfield transformation the counterterm ends up being on leg 4, and its line bounds the regions with momenta k 3 and k 3 . This is represented pictorially in Figure 3 .
k 1 k 2 k 3 k 4 B 1 B 2 B 4 B 3
Figure 3: One of the contributions to the four-point all-plus vertex.
Although Figure 3 might suggest that there is a propagator between the counterterm insertion and the location of the original A, which has now split into three B's, this is of course not the case since the whole expression is a vertex at the same point. We have drawn the diagram in this fashion to emphasise which leg the counterterm is located on after the transformation. On the other hand, this vertex is nonlocal (as discussed above, it was nonlocal even in the original variables, but this is now compounded by the Mansfield coefficients, which contain momenta in the denominator), so this notation serves as a useful reminder of that fact.
11 We suppress the overall factor of -g 2 N/(12π 2 ) until the end of this section. Also, the integrals are implicitly taken to be on the quantisation surface Σ.
It is interesting to note that (3.7) is essentially the same expression as the sum of the two channels with the same region momentum dependence that appear in CQT's calculation of this amplitude using tree-level diagrammatics (compare with Eq. 83 in [40] ), which we illustrate in Fig. 4 . Thus we have a picture where one post-Mansfield transform vertex (with
k 1 k 2 k 3 k 4 A 1 A 2 A 4 A 3 + k 1 k 2 k 3 k 4 A 1 A 2 A 4 A 3
Figure 4: The two diagrams with counterterm insertions on leg 4 that arise in the calculation of CQT, and, combined, add up to the contribution in Fig. 3. B's) effectively sums two tree-level pre-transformation (with A's) Feynman diagrams. This is a first indication that our calculation of the all-plus vertex can be mapped, practically one-to-one, to that of the all-plus amplitude on pp. 22-23 of [40] .
Another type of contribution to the vertex arises when we transform both of the A's in L CT . One of the two terms that we find is: (3.8) This contribution can also be mapped to one of the two terms with bubbles on internal lines in CQT.
( B 2 B 3 )( B 4 B 1 ) = -2g 2 tr dp dp ′ δ(p + p ′ ) dp 2 dp 3 δ(p -p 2 -p 3 ) p + p 2 + p 3 + 1 23 B(p 2 )B(p 3 ) × × (k 1 z ) 2 + (k 3 z ) 2 + k 1 z k 3 z dp 4 dp 1 δ(p ′ -p 4 -p 1 ) p ′ + p 4 + p 1 + 1 41 B(p 4 )B(p 1 ) = -2g 2 dp 1 • • • dp 4 δ(p 1 +p 2 +p 3 +p 4 ) (p 2 + + p 3 + )(p 1 + + p 4 + ) p 1 + p 2 + p 3 + p 4 + ((k 1 z ) 2 + (k 3 z ) 2 + k 1 z k 3 z ) 23 41 × tr B(p 1 )B(p 2 )B(p 3 )B(p 4 ) .
We can now tabulate all the terms that we obtain in this way by making the schematic form (3.6) precise. Since the delta-function and trace over B parts are the same for all these terms, in Table 1 we just list the rest of the integrand.
To obtain the final form of the vertex, we are now instructed to sum over all these contributions. Thus we can write
L ++++ (B) = 2g 2 dp 1 dp 2 dp 3 dp 4 δ(p 1 +p 2 +p 3 +p 4 ) V (4) tr[B(p 1 )B(p 2 )B(p 3 )B(p 4 )] (3.9) 16 Schematic form Pictorial form Integrand ( B 1 B 2 B 3 )B 4 p 4 + √ p 1 + p 3 + k 2 3 +k 2 4 +k 3 k 4 12 23 ( B 2 B 3 B 4 )B 1 p 1 + √ p 2 + p 4 + k 2 1 +k 2 4 +k 1 k 4 23 34 ( B 3 B 4 B 1 )B 2 p 2 + √ p 3 + p 1 + k 2 1 +k 2 2 +k 2 k 1 34 41 ( B 4 B 1 B 2 )B 3 p 3 + √ p 4 + p 2 + k 2 2 +k 2 3 +k 2 k 3 41 12 ( B 2 B 3 )( B 4 B 1 ) - (p 2 + +p 3 + )(p 1 + +p 4 + ) √ p 1 + p 2 + p 3 + p 4 + k 2 1 +k 2 3 +k 1 k 3 23 41 ( B 1 B 2 )( B 3 B 4 ) - (p 3 + +p 3 + )(p 2 + +p 1 + ) √ p 1 + p 2 + p 3 + p 4 + k 2 4 +k 2 2 +k 4 k 2 34 12
Table 1: The various contributions to the all-plus four-point vertex. Note that we use the simplifying notation k i := k i z.
where V (4) is given by the following expression: 12
V (4) = 1 p 1 + p 2 + p 3 + p 4 + 1 12 23 34 41 × × p 4 + p 2 + p 4 + (k 2 3 + k 2 4 + k 3 k 4 ) 34 41 + p 1 + p 1 + p 3 + (k 2 1 + k 2 4 + k 1 k 4 ) 12 41 + p 2 + p 2 + p 4 + (k 2 2 + k 2 1 + k 2 k 1 ) 12 23 + p 3 + p 3 + p 1 + (k 2 3 + k 2 2 + k 2 k 3 ) 23 34 -(p 2 + + p 3 + )(p 1 + + p 4 + )(k 2 1 + k 2 3 + k 1 k 3 ) 12 34 -(p 3 + + p 4 + )(p 2 + + p 1 + )(k 2 4 + k 2 2 + k 4 k 2 ) 23 41 . (3.10)
Comparing this to the expected answer (2.22), we see that the (quadratic) antiholomorphic momentum dependence should arise from the various k z factors in (3.10). In [40], CQT start from essentially the same expression and demonstrate that it gives the correct result for the all-plus amplitude. Therefore, following practically the same steps as those authors, we can easily see that we obtain the expected answer. However, since we would like to find the full vertex V, we will need to keep off-shell information, and so we will choose a slightly different route.
12 For the sake of brevity we omit a subscript z in the region momenta appearing in (3.10).
The main complication in bringing (3.10) into a manageable form is clearly the presence of the region momenta. We would like to disentangle their effects as cleanly as possible. Therefore, our derivation will proceed by the following steps: 1. First, we will show that (3.10) can be manipulated so that the quadratic dependence on region momenta drops out, leaving only terms linear in the region momenta.
2. Second, we will decompose the resulting expression into a part that depends on the region momenta and one that does not. The k-dependent part turns out to have a very simple form, and vanishes on-shell.
3. Finally, we will show that the k-independent part reduces to the known amplitude.
For the first step, we will need the following identity, which is proved in appendix B:
p 4 + p 2 + p 4 + 34 41 + p 1 + p 1 + p 3 + 12 41 + p 2 + p 2 + p 4 + 12 23 + p 3 + p 3 + p 1 + 23 34 -(p 2 + + p 3 + )(p 1 + + p 4 + ) 12 34 -(p 3 + + p 4 + )(p 2 + + p 1 + ) 23 41 = 0 (3.11)
Also, using the shorthand notation K ij := (k i z) 2 + (k j z) 2 + k i z k j z : we note the following very useful identity:
K ij = K ik + (k j z -k k z )(k i z + k j z + k k z ) = K ik + (k j z -k k z )l ijk (3.12) where 1 ≤ k ≤ n and l ijk = k i z +k j z +k k z . Noting that, for j > k, k j z -k k z = p k+1 z +p k+2 z +• • • p j z ,
we can use this to rewrite all the region momentum combinations appearing in (3.10) in the following way:
K 34 = 1 4 (K 12 + K 23 + K 34 + K 41 + (p 3 + p4 )(l 124 + l 234 ) + 2(p 2 + p3 )l 134 ) K 14 = 1 4 (K 12 + K 23 + K 34 + K 41 -(p 2 + p3 )(l 134 + l 123 ) + 2(p 3 + p4 )l 124 ) K 12 = 1 4 (K 12 + K 23 + K 34 + K 41 -(p 3 + p4 )(l 124 + l 234 ) -2(p 2 + p3 )l 123 ) K 23 = 1 4 (K 12 + K 23 + K 34 + K 41 + (p 2 + p3 )(l 134 + l 123 ) -2(p 3 + p4 )l 234 ) K 13 = 1 4 (K 12 + K 23 + K 34 + K 41 + (p 3 -p2 )l 123 + (p 1 -p4 )l 134 ) K 24 = 1 4 (K 12 + K 23 + K 34 + K 41 + (p 4 -p3 )l 234 + (p 2 -p1 )l 124 ) (3.13)
where we have introduced the notation pi = p i z . We have thus expressed all the quadratic region momentum dependence in terms of the common factor and, given (3.11) , it is clear that this contribution will vanish. 13 13 One could have chosen a different combination of the K ij 's, but we find the symmetric choice in (3.13) convenient.
K 12 + K 23 + K 34 + K 41 ,
After this step, we are left with an expression which is linear in the region momenta. We will now proceed in a similar way, and rewrite all the expressions that contain l ijk in terms of a suitably chosen common factor:
l 124 + l 234 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) - 1 2 (p 1 z + p 3 z ) l 134 + l 123 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) - 1 2 (p 2 z + p 4 z ) 2l 234 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) + 1 2 (2p 2 z + p 3 z -p 1 z ) 2l 123 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) + 1 2 (2p 1 z + p 2 z -p 4 z ) 2l 134 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) + 1 2 (2p 3 z + p 4 z -p 2 z ) 2l 124 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) + 1 2 (2p 4 z + p 1 z -p 3 z ) (3.14)
In appendix B we show that the total coefficient of the common (k
1 z + k 2 z + k 3 z + k 4 z ) factor is 3 8 p 4 + p 2 + p 4 + (+(p 3 + p4 ) + (p 2 + p3 )) 34 41 + p 1 + p 1 + p 3 + (-(p 2 + p3 ) + (p 3 + p4 )) 12 41 + p 2 + p 2 + p 4 + (-(p 3 + p4 ) -(p 2 + p3 )) 12 23 + p 3 + p 3 + p 1 + (+(p 2 + p3 ) -(p 3 + p4 )) 23 34 -(p 2 + + p 3 + )(p 1 + + p 4 + )( 1 2 (p 3 -p2 ) + 1 2 (p 1 -p4 )) 12 34 -(p 3 + + p 4 + )(p 2 + + p 1 + )( 1 2 (p 4 -p3 ) + 1 2 (p 2 -p1 )) 23 41 = = - 3 16 [(12) + (23) + (34) + (41)] 4 i=i (p i ) 2 p i + , (3.15)
where (p i ) 2 is the full covariant momentum squared, and (ij
) = p i + p j z -p j + p i z .
Thus we see that the complete dependence on the region momenta can be rewritten as follows:
V (4) k = - 3 16 (12) + (23) + (34) + (41) 12 23 34 41 4 i=1 k i z 4 i=i (p i ) 2 p i + . (3.16)
It is rather satisfying that the region momentum dependence of the vertex takes this simple form, which clearly vanishes when the external legs are on-shell, and thus will not contribute to the all-plus amplitudes.
Having completely disentangled the region momenta k z from the actual momenta p z , we will now focus on the terms containing only the latter, which were produced during the 19 decompositions in (3.14). After a few simple manipulations, they can be rewritten as 14 V (4) p = 1 8 p 4 + p 2 + p 4 + [(p 1 + p2 )(p 1 -p2 ) + (p 3 + p2 )(p 3 -p2 )] 34 41 + p 1 + p 1 + p 3 + [(p 2 + p3 )(p 2 -p3 ) + (p 4 + p3 )(p 4 -p3 )] 41 12
+ p 2 + p 2 + p 4 + [(p 3 + p4 )(p 3 -p4 ) + (p 1 + p4 )(p 1 -p4 )] 12 23 + p 3 + p 3 + p 1 + [(p 4 + p1 )(p 4 -p1 ) + (p 2 + p1 )(p 2 -p1 )] 23 34 -(p 2 + + p 3 + )(p 1 + + p 4 + )[(p 3 -p2 )(p 1 -p4 ) -(p 1 + p2 ) 2 ] 12 34 -(p 3 + + p 4 + )(p 2 + + p 1 + )[(p 4 -p3 )(p 2 -p1 ) -(p 2 + p3 ) 2 ] 23 41 .
(3.17) This expression, together with (3.16) is our proposal for the off-shell four-point all-plus vertex that should be part of the MHV-rules formalism at the quantum level. It would be very interesting to elucidate its structure and bring it into a more compact form. For the moment, however, we will be content to demonstrate that (3.17) is equal on shell to the sought-for amplitude.
To that end, we will follow a similar approach to CQT, and rewrite all the holomorphic spinor brackets in terms of the following three: 12 34 , 23 41 , 12 41 . To achieve this, we use momentum conservation and a certain cyclic identity (see appendix A) to write
p 4 + p 2 + p 4 + 34 41 = p 4 + p 4 + -p 3 + 42 -p 4 + 23 41 = -p 4 + p 3 + p 4 + 42 -(p 4 + ) 2 41 = -p 4 + p 3 + -p 1 + 12 -p 3 + 32 -(p 4 + ) 2 23 41 = p 4 + p 3 + p 1 + 12 41 -p 4 + (p 4 + + p 3 + ) 23 41 . (3.18)
In a similar way, we can show that
p 2 + p 2 + p 4 + 12 23 = p 2 + p 3 + p 1 + 12 41 -p 2 + (p 2 + + p 3 + ) 34 12 , p 3 + p 1 + p 3 + 23 34 = -p 3 + (p 3 + +p 2 + ) 12 34 -p 3 + (p 1 + +p 2 + ) 23 41 +p 3 + p 1 + p 3 + 12 14 . (3.19)
Collecting all the terms together, and manipulating the resulting expressions, it is straight- 4) .
14 We write V (4) = p 1 + p 2 + p 3 + p 4 + 12 23 34 41 V (
forward to show that (3.17) simplifies to just V (4) p = 1 4 23 41 {34}(p 1 + + p 2 + )[(p 1 -p2 ) -(p 2 + p3 )] + 12 34 {23}(p 2 + + p 3 + )[(p 1 + p2 ) + (p 1 -p4 )] + 12 41 p 3 + p 1 + (p 1 + p2 )({41} + {32})+(p 2 + p3 )({12} + {43}) , (3.20)
where we use the notation [34] ij] . Converting to the usual antiholomorphic bracket notation, we rewrite (3.20) as
{ij} = p i + p j z -p j + p i z = (1/ √ 2) p i + p j + [
V (4) p = 1 4 √ 2 23 41 p 3 + p 4 + [34](p 1 + + p 2 + )[(p 1 -p2 ) -(p 2 + p3 )] + 12 34 p 2 + p 3 + [23](p 2 + + p 3 + )[(p 1 + p2 ) + (p 1 -p4 )] + 12 41 (p 1 + p2 )(p 1 + p 3 + p 4 + [41] + p 2 + p 2 + p 1 + [32]) + (p 2 + p3 )(p 1 + p 2 + p 3 + [12] + p 3 + p 1 + p 4 + [43]) .
(3.21) Note that so far this expression is completely off shell. We will now show that on shell it reduces to the known result (2.22). In doing this we will keep track of the p 2 terms that appear when applying momentum conservation in the form
k ik [kj] = p i + p j + k (p k ) 2 p k + . (3.22)
These terms are collected in appendix B.
We start by rewriting each of the terms in the last two lines of (3.21) as follows
12 41 [41] p 1 + p 3 + p 4 + (p 1 + p2 ) = -23 41 [34] p 1 + p 3 + p 4 + (p 1 + p2 ) 12 41 [32] p 3 + p 1 + p 2 + (p 1 + p2 ) = -12 [32] 42 p 2 + p 3 + (p 1 + p2 ) -12 34 [23] p 3 + p 2 + p 3 + (p 1 + p2 ) 12 41 [12] p 1 + p 2 + p 3 + (p 2 + p3 ) = -12 34 [23] p 1 + p 2 + p 3 + (p 2 + p3 ) 12 41 [43] p 3 + p 1 + p 4 + (p 2 + p3 ) = -41 23 [34] p 3 + p 3 + p 4 + (p 2 + p3 ) -41 [43] 42 p 4 + p 3 + (p 2 + p3 ) . (3.23)
We also transform the 12 34 term using the Schouten identity and also momentum conservation,
12 34 [23] p 2 + p 3 + = 23 41 [34] p 3 + p 4 + + 14 23 [13] p 1 + p 3 + -13 42 [23] p 2 + p 3 + , (3.24) 21
and add up all contributions to the 23 41 term, which are 1 4 √ 2 23 41 [34] p 3 + p 4 + 4(p 2 + p1p 1 + p2 ) + 2(p 3 + p1p 1 + p3 ) = 1 4 √ 2 23 41 [34] p 3 + p 4 + [4{21} + 2{31}] .
(3.25) Converting to the spinor bracket, the first of these terms is
- 1 2 p 1 + p 2 + p 3 + p 4 + [12] 23 [34] 41 , (3.26)
while the remaining terms from (3.23) and (3.24) combine to give
14 23 [13] p 1 + p 3 + -13 42 [23] p 2 + p 3 + (p 2 + + p 3 + )[(p 1 + p2 ) + (p 1 -p4 )] + 12 [32] 42 p 2 + [p 2 + (p 1 + p2 ) -p 4 + (p 2 + p3 )] = -14 [13] 12 p 3 + (p 2 + + p 3 + )[(p 1 + p2 ) + (p 1 -p4 )] + 12 [32] 42 p 2 + [p 2 + (p 1 + p2 ) -p 4 + (p 2 + p3 )] = -14 [13] 12 p 3 + (2(p 2 + + p 3 + )p 1 -2p 1 + (p 2 + p3 )) = 2 14 [13] 12 p 3 + {41} (3.27)
(where we suppress an overall 1/(4 √ 2)) and we see that (3.27) cancels the second term in (3.25), thus showing that (3.26) is the complete on-shell answer. Reintroducing all the prefactors, we thus find that the amplitude is
A (4) = - g 2 N 12π 2 2g 2 p 1 + p 2 + p 3 + p 4 + 1 12 23 34 41 × - 1 2 p 1 + p 2 + p 3 + p 4 + [12] 23 [34] 41 = g 4 N 12π 2 [12][34] 12 34 .
(3.28) Now note that, as discussed in appendix A, in order to convert to the usual Yang-Mills theory normalisation we need to send g → g/ √ 2. We conclude that A (4) gives precisely the result (2.22) for the all-plus scattering amplitude.
We have just given an explicit derivation of the four point all-plus amplitude, from the two-point counterterm (3.3). We will argue in the following that this two-point counterterm contains all the all-plus amplitudes.
First, we can see immediately that the counterterm (3.3) has the right kind of structure. Consider the n-point all-plus amplitude [56] :
A (n) = 1≤i<j<k<l≤n ij [jk] kl [li] 12 • • • n1 . (3.29) 22
In terms of spinor brackets this amplitude has terms of the form 2-n [ ] 2 . A quick look at the Ettle-Morris coefficients shows that, for an n-point vertex coming from L CT , they contribute exactly 2n powers of the spinor brackets . Furthermore, there are exactly two powers of [ ] coming from the counterterm Lagrangian L CT ∼ (k 2 z )A 2 -one for each power of k. Thus the general structure of L CT is appropriate to reproduce (3.29).
Pictorially, we can represent the general n-point amplitude, arising from the counterterm in the new variables, as in Figure 5.
B i B i-1 B j+1 B j B j-1 B i+1 k i k j
Figure 5: The structure of a generic term contributing to the n-point vertex. All momenta are taken to be outgoing, and all indices are modulo n.
Thus we can write this n-point all-plus vertex as follows:
A (n) +•••+ = 1•••n δ(p + p ′ ) 1≤i<j≤n Y(p; j + 1, . . . , i) (k i z ) 2 + (k j z ) 2 + k i z k j z Y(p ′ ; i + 1, . . . , j)× × tr[B i B i+1 • • • B j B j+1 • • • B i-1 ] =( √ 2i) n-2 1•••n δ(p 1 +• • •+p n ) 1≤i<j≤n (p j+1 + + • • • + p i + ) p j+1 + p i + 1 j + 1, j + 2 • • • i -1, i × × (k i z) 2 + (k j z) 2 + k i zk j z (p i+1 + + • • • + p j + ) p i+1 + p j + 1 i + 1, i + 2 • • • j -1, j tr[B 1 • • • B n ] . (3.30)
Focusing only on the relevant part of the above expression, and ignoring all coefficients, the general structure we obtain is the following:
V (n) +•••+ = 1 12 • • • n1 × 1≤i<j≤n j, j + 1 i, i + 1 p i + p i+1 + p j + p j+1 + (k j + -k i + ) 2 ((k i z ) 2 + (k j z ) 2 + k i z k j z ) (3.31) 23
where we have extracted the denominator at the expense of introducing the two missing holomorphic factors j, j + 1 and i, i + 1 in the numerator. We also made use of the fact that
k j -k i = p i+1 + p i+2 + • • • + p j = -(p j+1 + p j+2 + • • • + p i ) , ( 3
.32) applied to the + components, to rewrite the two p + sums in the numerator in terms of the k's (this gives rise to a minus which we suppress).
It is easy to verify that, for n = 4, this sum reproduces the 6 contributions that appeared in the four-point case, and (as we explicitly showed above) combined to give the expected answer. Therefore, we would like to propose that the vertex (3.31) will reduce on-shell to an expression proportional to (3.29). We will not attempt to prove this statement here 15 , but will instead move on to study the general properties of the n-point expression (3.30).
Whilst the explicit calculation for the four point case was rather involved as we saw earlier, the study of the general properties of the n-point amplitudes proves much simpler. In particular, we will show that the collinear and soft limits of the expressions proposed for the n-point case can be very easily shown to be correct. Let us start by introducing some simplifying notation. One can write the change of variables for the A field as 34) . . . (n -1 n) (3.35) (for simplicity, we are dropping inconsequential constant factors in this discussion). This notation is similar to that of [34] . Integrations and the insertion of suitable delta functions are understood, and can be illustrated by comparing the short-hand expressions above with the full equations given earlier. It will prove convenient to define
A 1 = Y 12 B 2 + Y 123 B 2 B 3 + Y 1234 B 2 B 3 B 4 + • • • , (3.33) where Y 12 = δ 12 , Y 123 = 1 + (23) , Y 1234 = 1 + 3 + (23)(34) , (3.34) and generally Y 12...n = 1 + 3 + 4 + . . . (n -1) + (23)(
K ij = k 2 i + k 2 j + k i k j , k i := k i z . (3.36)
We will use the expression Y •12...n in the following, where the dot in the first placemark in the Y means that one substitutes in that place the negative of the sum of the other momenta. Then the result which we have proved above for the four point amplitude V 1234 can be expressed as (3.37) 15 It is perhaps interesting to remark that the proof would involve converting the double sum in (3.31) to the quadruple sum in (3.29)-a state of affairs which has appeared before in a rather different context [20].
V 1234 =K 43 Y •4 Y •123 + K 14 Y •1 Y •234 + K 21 Y •2 Y •341 + K 32 Y •3 Y •412 + K 31 Y •23 Y •41 + K 24 Y •12 Y •34 ,
or very simply
V 1234 = 1≤i<j≤4 K ij Y • j+1...i Y • i+1...j .
(3.38) It is clear that the general conjecture that all the n-point all plus amplitudes are generated from the two-point counterterm (3.3) translates into the proposal that the n-point all-plus amplitude V 12...n is given by
V 12...n = 1≤i<j≤n K ij Y • j+1...i Y • i+1...j , (3.39)
Let us now show that the expression on the right-hand side of (3.39) has precisely the same soft and collinear limits as the known amplitude on the left-hand side.
Under the collinear limit
p i → zP , p i+1 → (1 -z)P , P 2 → 0 , (3.40)
the n-point amplitude V 12...n behaves as
V 12...n → 1 z(1 -z) i + (i i + 1) V 12...i i+2...n , (3.41)
where we relabel P → p i after the limit is taken (the i + and (i i + 1) factors involve momenta rather than spinors, which is why the z-dependent factor is 1/z(1z), rather than the conventional 1/ z(1z)).
Consider the behaviour of the right-hand side of (3.39) under the limit (3.40). The first point is that if the indices i, i + 1 lie on different Y's, then there are no poles generated in this collinear limit. This is clear from the explicit expressions for the Y's in (3.35). Thus we may ignore any terms of this type. It is then immediate from the explicit forms of the Y's that (3.42) for any i = 2, . . . s -1, with s ≤ n (the first index in Y never contributes in a collinear limit, as one can see from the conjecture (3.39)). Thus we see that the Y expressions have the right sort of collinear behaviour. It is straightforward to see that the K coefficients in (3.39) also get relabelled correctly in the collinear limit; they are not explicitly involved as they refer to pairs of momenta attached to different Y fields, and as we saw, these do not contribute.
Y 12...s → 1 z(1 -z) i + (i i + 1) Y 12...i i+2...s ,
It is then immediate to see that the summation over the products of Y's in (3.39) reduces correctly in the collinear limit to the required summation over products of Y's with one fewer 25 leg in total. Hence the proposal (3.39) for the amplitude has precisely the same collinear limits as the physical amplitude.
We also find that there is a simple derivation of the soft limits of the expression in (3.39). In the soft limit p j → 0 , (3.43) the n-point amplitude V 12...n behaves as
V 12...n → S(j) V 12...j-1 j+1...n , (3.44)
where we assume cyclic ordering as usual, so that, for example, p n+1 = p 1 . The soft function S(j) is given in terms of the momentum brackets by
S(j) = j + (j -1 j + 1) (j -1 j) (j j + 1) . (3.45)
The Y functions have a simple behaviour under soft limits. One has immediately that in the soft limit p j → 0, Y 12...s → S(j) Y 12...j-1 j+1...s , (3.46) for j = 3, . . . s -1 (with s ≤ n). For the soft limits corresponding to the case missing in the above, we need the results
Y •s+1...j = Y •s+1...j-1 (j -1) + (j -1 j) , Y •j...s = Y •j+1...s (j + 1) + (j j + 1) , (3.47)
which follow from the definitions of the Y's, and
(j + 1) + (j j + 1) + (j -1) + (j -1 j) = j + (j -1 j + 1) (j -1 j) (j j + 1) = S(j) , (3.48) which follows from the cyclic identity i + (jk) + j + (ki) + k + (ij) = 0. Finally, from
relabelling the K's we have in the soft limit that K sj → K sj-1 . Then it follows that in the soft limit
K sj Y •s+1...j Y •j+1...s + K sj-1 Y •s+1...j-1 Y •j...s → S(j)K sj-1 Y •s+1...j-1 Y •j+1...s , (3.49) as required.
Again, it is then easy to see that the summation over the products of Y's in (3.39) reduces correctly in the soft limit to the required summation over products of Y's with one fewer leg in total. Hence the proposal (3.39) for the amplitude has precisely the same soft limits as the physical amplitude.
Whilst new, twistor-inspired methods for calculating amplitudes in gauge theory have led to much progress, the lack of a systematic action-based formulation which incorporates these new ideas has been an impediment to further developments. MHV diagrams have the two advantages of being closely allied to the twistor picture, as well as providing an explicit realisation of the dispersion and phase space integrals fundamental to unitaritybased methods. However, without an action formalism, standard MHV methods have so far been mainly restricted to massless theories at one-loop level, and to the cut-constructible parts of amplitudes.
The advent of a classical MHV Lagrangian for gauge theory, derived from lightcone YM theory [32, 33, 34] , provides the basis for transcending these limitations. In order for this to be realised, it is necessary to describe the quantum MHV theory. What we have done in this paper is to investigate this quantum theory. Using the regularisation methods of [39, 40, 41] , we have provided arguments that the simplest one-loop counterterm in the quantum MHV theory -a two point vertex -provides an extraordinarily concise generating function for the infinite sequence of one-loop, all-plus helicity amplitudes in YM theory. We showed this by explicit calculation for the four-point case, and then proved that the soft and collinear limits of the conjectured n-point amplitude precisely matched those of the correct answer.
We would like to emphasise that the simplicity of our approach -which reduced the calculations of the loop amplitudes we considered to tree-level algebraic manipulationsis largely due to the four-dimensional nature of the regularisation scheme we employed. By staying in four dimensions, we preserve the appealing features of the inherently fourdimensional field redefinition of [32, 33] .
Based upon this result, it is very natural to conjecture that the full quantum YM theory is correctly described by this quantum MHV Lagrangian. The correct ingredients appear to be present. For example, in the approach of [39, 40, 41] there arise one-loop counterterms with helicities (++), (+ + -), (--), (--+). We studied the (++) counterterm in this paper, arguing that when expressed in the (B, B) variables this generates the full set of allplus amplitudes. Transforming the (+ + -) counterterm to (B, B) variables will generate an infinite sequence of single-minus vertices. There will be other contributions to single-minus vertices from combinations of all-plus vertices and MHV vertices. It would be surprising if the combined contributions of these did not lead to the correct YM single-minus expressions. Certainly all of these have the correct powers of spinor brackets for this to be the case.
Transforming the (--) and (--+) counterterms to (B, B) variables will lead to new contributions to MHV vertices 16 . The MHV vertices from the classical MHV Lagrangian only generate the cut-constructible parts of YM loop amplitudes, such as the one-loop MHV 16 In the MHV case there are additional counterterms noted in [41] which may also need to be taken into account in future discussions.
amplitude. These new contributions might be expected to lead to the missing, rational parts. This would also potentially explain why in [57] the combination of all-plus vertices with MHV tree vertices did not yield the correct single-minus amplitudes -these additional MHV contributions are missing.
Further evidence for the conjecture that the quantum MHV Lagrangian is equivalent to quantum YM theory would be welcome. One could start with seeking explicit proofs of the above proposals. One can also investigate beyond massless one-loop gauge theory -an advantage of the Lagrangian approach is that the inclusion of masses, and of fermions and scalars, is in principle clear. There are other issues raised by this work. It is plausible that the potential quantum versions of the twistor space formulations of gauge theory [58, 59, 60] are most likely to be allied to the quantum theory discussed here -one simple reason for believing this is that the regularisation employed here keeps one in four dimensions. Perhaps there are simple twistor space analogues of the counterterms discussed above.
Finally, although for our purposes the lightcone worldsheet approach to perturbative gauge theory provided simply the motivation for a particular choice of regularisation scheme, we believe that it would be fruitful to further explore possible connections between that framework and the twistor string programme.
Addendum: We would like to thank Paul Mansfield and Tim Morris for having informed us that they have recently been pursuing research related to that presented in this paper. Their work, which is complementary to ours in that it employs dimensional regularisation, has now appeared in [61] .
It is a pleasure to thank Paul Heslop, Gregory Korchemsky, Paul Mansfield, Tim Morris and Adele Nasti for discussions. We would like to thank PPARC for support under the Rolling Grant PP/D507323/1 and the Special Programme Grant PP/C50426X/1. The work of GT is supported by an EPSRC Advanced Fellowship EP/C544242/1 and by an EPSRC Standard Research Grant EP/C544250/1.
Lightcone conventions Here we summarise our lightcone conventions. We start off by introducing lightcone coordinates
x ± := x 0 ± x 3 √ 2 , x z := x 1 + ix 2 √ 2 , x z := x 1 -ix 2 √ 2 . (A.1)
We also have x + = x -, x z = -x z , and so on. The scalar product between two vectors A and B is written as
A • B := A + B -+ A -B + -A z B z -A z B z . (A.2)
We choose x -as our lightcone time coordinate, therefore the lightcone gauge used in this paper is defined by A -= 0 . (A.3) This condition can be written as η • A = 0, where η is a constant null vector, chosen to have components η := (1/ √ 2, 0, 0, 1/ √ 2) (hence
η -= 1, η + = η z = η z = 0).
To any four-vector p we associate the bispinor p a ȧ defined by
p a ȧ := √ 2 p --p z -p z p + . (A.4)
We also define holomorphic and anti-holomorphic spinors as
λ a := 2 1 4 √ p + -p z p + , λȧ := 2 1 4 √ p + -p z p + , (A.5)
from which it follows that
λ a λȧ := √ 2 pzpz p + -p z -p z p + . (A.6)
This is of course consistent with the on-shell condition p -= p z p z /p + . Furthermore, comparing (A.4) and (A.6) and choosing η as specified earlier, we see that a generic off-shell vector p can be decomposed as
p = λ λ + zη , (A.7) where z = p -p + -p z p z p + η - = p 2 2(p • η) . (A.8) (A.7
) and (A.8) are the familiar decompositions of off-shell vectors in the MHV literature [62, 17, 63, 15] .
The off-shell holomorphic spinor product is defined as:
ij = √ 2 p i + p j z -p j + p i z p i + p j + , (A.9) 29
whereas for the antiholomorphic spinors we define
[ij] = √ 2 p i + p j z -p j + p i z p i + p j + . (A.10)
In these conventions, one finds
2(p i • p j ) = i j [i j] + p j + p i + (p i ) 2 + p i + p j + (p j ) 2 , (A.11)
or, in the case where p i and p j are on shell, 2(p i • p j ) = i j [i j]. In the standard QCD literature conventions it is customary to define 2(p i • p j ) = i j [j i]; this can be obtained by simply re-defining the inner product of two anti-holomorphic spinors, [i j], to be the negative of the right hand side of (A.10).
The form (A.9) is very convenient for deriving identities for ij that also involve the p + components. For instance, one has:
p i + jk + p j + ki + p k + ij = √ 2 p i + (p j + p k z -p k + p j z ) p i + p j + p k + + √ 2 p j + (p k + p i z -p i + p k z ) p i + p j + p k + + √ 2 p k + (p i + p j z -p j + p i z ) p i + p j + p k + = 0 . (A.12)
It is also easy to see how to apply momentum conservation, take say ij , and substitute p j =k =j p k (for each component). (A.13) Then we have
p j + ij = √ 2 p i + (-k =j p k z ) + ( k =j p k + )p i z p i + = - √ 2 k =j p k + p i + p k z -p k + p i z p i + p k + = k =j p k + ki .
(A.14) We have also used the momentum bracket notation from [34]
(ij) = p i + p j z -p j + p i z , {ij} = p i + p j z -p j + p i z . (A.15)
Here we give the form of the lightcone Yang-Mills action that we use in this paper. As discussed in more detail in [35] , starting from the YM Lagrangian -(1/4) trF 2 , imposing the 30 lightcone gauge (A.3), and integrating out the A + component which appears quadratically, the final lightcone theory contains only the two physical components A z and A z [64, 65, 66] , which we associate with positive and negative helicity respectively. The Lagrangian takes the simple form (2.1)
L YM = L +-+ L ++-+ L --+ + L ++--, (A.16) with L +-= -2 tr{A z (∂ + ∂ --∂ z ∂ z )A z } , L ++-= 2ig tr{[A z , ∂ + A z ](∂ + ) -1 (∂ z A z )} , L --+ = 2ig tr{[A z , ∂ + A z ](∂ + ) -1 (∂ z A z )} , L ++--= -2g 2 tr{[A z, ∂ + A z ](∂ + ) -2 [A z , ∂ + A z ]} .
(A.17) Note that, in agreement with CQT, we have used the normalisation tr{T a T b } = δ ab . In order to convert to the usual conventions for Yang-Mills theory, we therefore need to rescale g → g/ √ 2.
To compare our notation to that of [39, 40, 41] , note that we employ outgoing momenta instead of incoming, therefore the all-plus amplitudes in these works would be all-minus from our perspective, and should thus be conjugated when comparing. Also, our time evolution coordinate is taken to be x -rather that x + , which (among other changes) implies that p + of CQT becomes p + . Our metric is also taken to have opposite signature to that in CQT. Finally, CQT define momentum brackets K ∧ ij and K ∨ ij , which are just our (ij) and {ij} brackets respectively.
In this appendix we prove two results that were used in section 3 , namely equations (3.11) and (3.15) . To make the expressions more compact, instead of momentum brackets we use the following notation:
f ij = - (ij) p i + p j + = p i z p i + - p j z p j + . (B.1)
The f ij variables satisfy the simple relation:
f ij = f ik + f kj , (B.2)
while momentum conservation is applied as
p i + f ij = - p k + f kj .
(B.3) Also, to minimise clutter, in this appendix we use the notation q i := p i + .
In order to show (3.11), it is convenient to divide out by the p 1 + p 2 + p 3 + p 4 + factor (which is there anyway in (3.10)) in order to bring it to the form
q 2 4 f 34 f 41 + q 2 1 f 12 f 41 + q 2 2 f 12 f 23 + q 2 3 f 23 f 34 -(q 2 + q 3 )(q 1 + q 4 )f 12 f 34 -(q 3 + q 4 )(q 2 + q 1 )f 23 f 41 = 0 , (B.4)
Expanding out the two last terms in (B.4) as
-(q 1 q 3 + q 2 q 4 )(f 12 f 34 + f 23 f 41 ) -(q 1 q 2 + q 3 q 4 )f 12 f 34 -(q 2 q 3 + q 4 q 1 )f 23 f 41 , (B.5)
we apply momentum conservation on each of the four components of the first term of (B.5), in the following way:
-q 1 q 3 f 12 f 34 = q 1 f 12 (q 1 f 14 + q 2 f 24 ) = -q 2 1 f 12 f 41 + q 1 q 2 f 12 f 24 , -q 1 q 3 f 23 f 41 = q 3 f 23 (q 2 f 42 + q 3 f 43 ) = -q 2 3 f 23 f 34 + q 2 q 3 f 23 f 42 , -q 2 q 4 f 12 f 34 = q 4 (q 3 f 13 + q 4 f 14 )f 34 = -q 2 4 f 34 f 41 + q 3 q 4 f 13 f 34 , -q 2 q 4 f 23 f 41 = q 2 f 23 (q 2 f 21 + q 3 f 31 ) = -q 2 2 f 12 f 23 + q 2 q 3 f 31 f 23 .
(B.6) Clearly these transformations have been chosen to cancel the first four terms in (B.4). Collecting the remaining terms, we obtain
q 1 q 2 f 12 (f 24 -f 34 ) + q 2 q 3 f 23 (f 42 + f 31 -f 41 ) + q 3 q 4 f 34 (f 13 -f 12 ) -q 1 q 4 f 23 f 41 = q 1 q 2 f 12 f 23 + q 2 q 3 f 23 f 32 + q 3 q 4 f 34 f 23 + q 1 q 4 f 23 f 14 = f 23 [q 2 (q 1 f 12 + q 3 f 32 ) + q 4 (q 3 f 34 + q 1 f 14 )] = f 23 [-q 2 (q 4 f 42 ) -q 4 (q 2 f 24 )] = 0 (B.7)
thus showing (3.11).
We will now outline the proof ot the linear (in region momenta) identity (3.15). Converting it to the notation used in the appendix, and performing simple manipulations, we 32 find (suppressing the overall 3/8 factor):
X = q 2 4 ((p 3 + p4 ) + (p 2 + p3 ))f 34 f 41 + q 2 1 (-(p 2 + p3 ) + (p 3 + p4 ))f 12 f 41 + q 2 2 (-(p 3 + p4 ) -(p 2 + p3 ))f 12 f 23 + q 2 3 (+(p 2 + p3 ) -(p 3 + p4 ))f 23 f 34 - 1 2 (q 2 + q 3 )(q 1 + q 4 )[(p 3 -p2 ) + (p 1 -p4 )]f 12 f 34 - 1 2 (q 3 + q 4 )(q 1 + q 2 )[(p 4 -p3 ) + (p 2 -p1 )]f 23 f 41 = (p 3 -p1 )(q 2 4 f 34 f 41 -q 2 2 f 12 f 23 ) + (p 4 -p2 )(q 2 1 f 12 f 41 -q 2 3 f 23 f 34 ) -(q 2 + q 3 )(q 1 + q 4 )(p 3 + p1 )f 12 f 34 -(q 3 + q 4 )(q 1 + q 2 )(p 2 + p4 )f 23 f 41 = (p 3 -p1 )(q 2 4 f 34 f 41 -q 2 2 f 12 f 23 ) + (p 4 -p2 )(q 2 1 f 12 f 41 -q 2 3 f 23 f 34 ) -(p 1 + p3 )q 2 q 4 (f 12 f 34 -f 23 f 41 ) + (p 2 + p4 )q 1 q 3 (f 12 f 34 -f 23 f 41 ) -(p 1 + p3 )(q 1 q 2 + q 3 q 4 )f 12 f 34 + (p 1 + p3 )(q 2 q 3 + q 4 q 1 )f 23 f 41 .
(B.8) Similarly to the previous case, we will rewrite the second line in the final expression in such a way that we completely cancel all the terms in the first line. To do that we use
-(p 1 + p3 )q 2 q 4 (f 12 f 34 -f 23 f 41 ) =(p 3 -p1 )(q 2 2 f 12 f 23 -q 2 4 f 34 f 41 )+ + q 1 q 2 p1 f 12 f 31 -q 4 q 1 p1 f 41 f 13 + + q 3 q 4 p3 f 34 f 13 -q 2 q 3 p3 f 23 f 31 (B.9) and (p 2 + p4 )q 1 q 3 (f 12 f 34 -f 23 f 41 ) =(p 4 -p2 )(q 2 3 f 23 f 34 -q 2 1 f 12 f 41 )+ + q 2 q 3 p2 f 23 f 42 -q 1 q 2 p2 f 12 f 24 + + q 4 q 1 p4 f 41 f 24 -q 3 q 4 p4 f 34 f 42 .
(B.10) What remains after substituting these is
X = p1 q 1 f 31 (q 2 f 12 + q 4 f 41 ) + q 3 p3 f 13 (q 4 f 34 + q 2 f 23 ) + p2 q 2 f 42 (q 3 f 23 + q 1 f 12 ) + q 4 p4 f 24 (q 1 f 41 + q 3 f 34 ) -(p 1 + p3 )(q 1 q 2 + q 3 q 4 )f 12 f 34 + (p 1 + p3 )(q 2 q 3 + q 4 q 1 )f 23 f 41 = p1 q 1 q 2 f 12 f 41 + p3 q 3 q 4 f 34 f 23 + p1 q 4 q 1 f 41 f 21 + p3 q 2 q 3 f 23 f 43 + p2 q 2 f 42 (q 3 f 23 + q 1 f 12 ) + q 4 p4 f 24 (q 1 f 41 + q 3 f 34 ) -(p 1 q 3 q 4 + p3 q 1 q 2 )f 12 f 34 + (p 1 q 2 q 3 + p3 q 4 q 1 )f 23 f 41 .
(B.11) Now we collect various terms together to rewrite X as
X = p1 q 2 f 41 (q 1 f 12 + q 3 f 23 ) + p3 q 4 f 23 (q 3 f 34 + q 1 f 41 ) + p1 q 4 f 21 (q 1 f 41 + q 3 f 34 ) + p3 q 2 f 43 (q 3 f 23 + q 1 f 12 ) + p2 q 2 f 42 (q 3 f 23 + q 1 f 12 ) + p4 q 4 f 24 (q 1 f 41 + q 3 f 34 ) = p1 q 2 f 41 (2q 3 f 23 -q 4 f 42 ) + p3 q 4 f 23 (2q 1 f 41 -q 2 f 24 ) + p1 q 4 f 21 (2q 1 f 41 -q 4 f 42 ) + p3 q 2 f 43 (2q 3 f 23 -q 4 f 42 ) + p2 q 2 f 42 (2q 3 f 23 -q 4 f 42 ) + p4 q 4 f 24 (2q 1 f 41 -q 2 f 24 ) = 2[q 2 q 3 f 23 (p 1 f 41 + p3 f 43 + p2 f 42 ) + q 4 q 1 f 41 (p 3 f 23 + p1 f 21 + p4 f 24 )] + (p 1 + p2 + p3 + p4 )q 2 q 4 f 24 f 42 .
(B.12) 33 Clearly the term on the last line vanishes by momentum conservation. We now restore all labels to write the final result as
X =2 (32)[f 4 (p 1 z + p 2 z + p 3 z) -p 1 z f 1 -p 2 z f 2 -p 3 z f 3 ]+ + 2 (14)[f 2 (p 1 z + p 3 z + p 4 z ) -p 3 z f 3 -p 1 z f 1 -p 4 z f 4 ] , (B.13)
where we used that
q 2 q 3 f 23 = p 2 + p 3 + (p 2 z /q 2 + -p 3 z /p 3 + ) = p 3 + p 2 z -p 2 + p 3 z = ( 32
) (and similarly for (14)), and where f i = p i z /p i + . Using momentum conservation on both terms, we rewrite them as
X = -2[(32) + (14)] p 1 zp 1 z p 1 + + p 2 z p 2 z p 2 + + p 3 z p 3 z p 3 + + p 4 z p 4 z p 4 + . (B.14)
For each momentum we have that p 2 = 2(p + p -p z p z), therefore we can rewrite the above as
X = +[(32) + (14)] (p 1 ) 2 p 1 + + (p 2 ) 2 p 2 + + (p 3 ) 2 p 3 + + (p 4 ) 2 p 4 + + 2(p 1 -+ p 2 -+ p 3 -+ p 4 -) . (B.15)
The p -term vanishes, hence, noticing also that (32) + (14) = -1 2 ((12) + (23) + (34) + (41)), we conclude that
X = - 1 2 [(12) + (23) + (34) + (41)] 4 i=1 (p i ) 2 p i + . (B.16)
Off-shell terms in the four-point case
For completeness, we also give the form of the off-shell terms that arose in the manipulations leading to (3.26).
Using the notation
P ij = ( (p i ) 2 p i + + (p j ) 2 p j +
) they are : 3.16 ), should be added to (3.26) in order to recover a fully off-shell four-point vertex.
f (p 2 ) = 1 4 12 • • • 41 -P 13 (p 1 + p2 )(41) -P 13 (p 2 + p3 )(12) + P 24 (p 2 + p3 )( 42
) + 1 p 1 + P 12 [(p 2 + + p 3 + )(2p 1 + p2 -p3 ) -p 3 + (p 1 + p2 ) -p 1 + (p 2 + p3 )] ( 13
) + P 12 p 3 + p 1 + p 2 + [p 2 + (p 1 + p2 ) -p 4 + (p 2 + p3 )](12) -2P 13 1 p 1 + {31}(41) . (B.17) This expression, together with V (4) k in (
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{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "1 {a.brandhuber, w.j.spence, g.travaglini, k.zoubos}@qmul.ac.uk Contents 1 Introduction 1 2 Background 3 2.1 The classical MHV Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 A four-dimensional regulator for lightcone Yang-Mills . . . . . . . . . . . . . 6 2.3 The one-loop (++++) amplitude . . . . . . . . . . . . . . . . . . . . . . . . 11 3 The all-plus amplitudes from a counterterm 12 3.1 Mansfield transformation of L CT . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 The four-point case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 The general all-plus amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 Discussion 27 A Notation 29 B Details on the four-point calculation 31 1 Introduction\n\nOne of the success stories arising from twistor string theory [1] (see [2] for a review) has been the development of new techniques in perturbative quantum field theory. These include recursion relations [3, 4] , generalised unitarity [5] and MHV methods (see [6] for a review). One of the key motivations of this work is to provide new approaches to study and derive phenomenologically relevant scattering amplitudes. In particular, this requires that one be able to deal with non-supersymmetric theories, and to include fermions, scalars, and particles with masses. A vital first step is to apply these new methods to pure Yang-Mills (YM) theory, and indeed, some of the first new results inspired by twistor string theory involved pure YM amplitudes at tree- [7, 8, 9, 10, 11, 12, 13, 14] and one-loop [15] level. 1 A recalcitrant issue in this work is the derivation of rational terms in quantum amplitudes. Unitarity-based techniques [16] and loop MHV methods [17] are successful in obtaining the cut-constructible parts of amplitudes; essentially this is because at some level they are dealing with four-dimensional cuts. In principle performing D-dimensional cuts generates all parts of amplitudes [18, 19, 20, 21] as long as only massless particles are involved, however these techniques still appear to be relatively cumbersome. Combinations of recursive techniques and unitarity have led to important progress recently [22, 23, 24, 25, 26, 27, 28, 29, 30, 31] , but it would be preferable to have a more powerful prescriptive formulation, particularly keeping in mind that applications to more general situations are sought.\n\nA promising development from this point of view is the Lagrangian approach [32, 33, 34] . Here it has been argued that lightcone Yang-Mills theory, combined with a certain change of field variables, yields a classical action which comprises precisely the MHV vertices. A full Lagrangian description of MHV techniques would in principle give a prescription for applying such methods to diverse theories. The next step in developing this is to understand the quantum corrections in this Lagrangian approach. If one directly uses in a path integral the classical MHV action, containing only purely four-dimensional MHV vertices, then it is immediately clear that this cannot yield all known quantum amplitudes. For example, there is no way to construct one-loop amplitudes where the external gluons all have positive helicities, or where only one gluon has negative helicity, as all MHV vertices contain two negative helicity particles (this issue has been recently discussed in [35] ). These amplitudes are particular cases where the entire amplitude consists of rational terms. More generally, it seems clear that the vertices of the classical MHV Lagrangian will not yield the rational parts of amplitudes, but only the cut-constructible terms [15] . Important insights into this question can be obtained from the study of self-dual Yang-Mills theory, which has the same all-plus one-loop amplitude of full YM [36, 37, 38] as its sole quantum correction. 1 An example, relevant to the discussion in this paper, is given in [35] where it was shown how these amplitudes might be obtained from the Jacobian arising from a Bäcklund-type change of variables which takes the self-dual Yang-Mills theory to a free theory.\n\nA discussion of the full Yang-Mills theory in the lightcone gauge has recently been given by Chakrabarti, Qiu and Thorn (CQT) in [39, 40, 41] . These papers employ an interesting regularisation which, importantly, does not change the dimension of spacetime. For this reason, we find it particularly suitable for setting the scene for the MHV diagram method, which is inherently four-dimensional in current approaches. The regularisation of CQT requires the introduction of certain counterterms, which prove to be rather simple in form. What we will show in this paper is that these simple counterterms provide a very compact and powerful way to represent the rational terms in gauge theory amplitudes; specifically, we will demonstrate that the single two-point counterterm contains all the n-point all-plus amplitudes. The way this happens is through the use of the new field variables of [32, 33, 34] . Other counterterms will combine with vertices from the Lagrangian and should generate the rational parts of more general amplitudes. Based on the discussion in this paper, we propose 1 In real Minkowski space, this is in fact its single non-vanishing amplitude.\n\nthat the counterterms, expressed in the field variables which give rise to standard MHV vertices, in combination with Lagrangian vertices, generate the rational terms previously missing from MHV diagram formulations.\n\nThe rest of the paper is organised as follows. After giving some background material in section 2, we explicitly derive in section 3 the four point all-plus amplitude from the twopoint counterterm of CQT. We follow this by showing that the n-point expression, obtained by writing the counterterm in new variables, has precisely the right collinear and soft limits required for it to be the correct all-plus n-point amplitude. We present our conclusions in section 4, and our notation and derivations of certain identities have been collected in two appendices."
},
{
"section_type": "BACKGROUND",
"section_title": "Background",
"text": "In this section, we first review the classical field redefinition from the lightcone Yang-Mills Lagrangian to the MHV-rules Lagrangian. We then move on to motivate the fourdimensional regularisation scheme we will employ, and argue that it leads directly to the introduction of a certain Lorentz-violating counterterm in the Yang-Mills Lagrangian. We close the section with the remarkable observation that this counterterm provides a simple way to calculate the four-point all-plus one-loop amplitude using only tree-level combinatorics."
},
{
"section_type": "OTHER",
"section_title": "The classical MHV Lagrangian",
"text": "It seemed clear from the beginning that the MHV diagram approach to Yang-Mills theory must be closely related to lightcone gauge theory. This idea was substantiated by Mansfield [33] (see also [32] ). The starting point of [33] is the lightcone gauge-fixed YM Lagrangian for the fields corresponding to the two physical polarisations of the gluon. It was argued convincingly in [33] that a certain canonical change of the field variables re-expresses this lightcone Lagrangian as a theory containing the infinite series of MHV vertices. Some of the arguments in [33] were rather general; these were reviewed in [34] , where the change of variables was discussed in more detail, and in particular it was shown how the four-and five-point MHV vertices arise from the change of variables. In this paper we will mainly follow the notation of [34] .\n\nThe general structure of the lightcone YM Lagrangian, after integrating out unphysical degrees of freedom, is (see appendix A for more details)\n\nL YM = L +-+ L ++-+ L --+ + L ++--, (2.1)\n\nwhere the gauge condition is η µ A µ = 0 with the null vector η = (1/ √ 2, 0, 0, 1/ √ 2). Since this Lagrangian contains a + +vertex, it is not of MHV type. In [33] , Mansfield proposed to 3 eliminate this vertex through a suitably chosen field redefinition. Specifically, he performed a canonical change of variables from (A, Ā) to new fields (B, B), in such a way that L +-(A, Ā) + L ++-(A, Ā) = L +-(B, B) . (2.2)\n\nThe remarkable result is that upon inserting this change of variables into the remaining two vertices, the Lagrangian, written in terms of (B, B), becomes a sum of MHV vertices, L YM = L +-+ L +--+ L ++--+ L +++--+ . . . . (2.3) The crucial property of Mansfield's transformation that makes this possible is that, while both A and Ā are series expansions in the new B fields, A has no dependence on the B fields while Ā turns out to be linear in B. Thus, since the remaining vertices are quadratic in the B, the new interaction vertices have the helicity configuration of an MHV amplitude. Mansfield was also able to show that the explicit form of the vertices coincides with the CSW off-shell continuation of the Parke-Taylor formula for the MHV scattering amplitudes, as proposed by [7].\n\nOne of the main results of [34] was the derivation of an explicit, closed formula for the expansion of the original fields (A, Ā) in terms of the new fields (B, B). This was then used to verify that the new vertices are indeed precisely the MHV vertices of [7], at least up to the five-point level. We will now briefly review these results. First, recall that the positive helicity field A is a function of the positive helicity B field only. It is expanded as follows:\n\nA( p ) = ∞ n=1 Σ n i=1 d 3 p i (2π) 3 ∆( p , p 1 , . . . p n ) Y( p ; 1 • • • n) B( p 1 )B( p 2 ) • • • B( p n ) , (2.4) where ∆( p , p 1 , . . . p n ) := (2π) 3 δ (3) ( pp 1 -• • •p n ). Note that the x -coordinate is common to all the fields, which is why we have restricted the transformation to the lightcone quantisation surface Σ.\n\nBy inserting this expansion into (2.2) and using the requirement that the transformation be canonical, Ettle and Morris succeeded in deriving a very simple expression for the coefficients Y. After translating to our conventions (see appendix A), they are given by:\n\nY( p ; 12 • • • n) = ( √ 2ig) n-1 p + p 1 + p n + 1 12 23 • • • n -1, n . (2.5)\n\nThe first few terms in (2.4) are then: 3) ( pp 1p 2p 3 ) p 1 + p 3 + 1 12 23 B( p 1 )B( p 2 )B( p 3 ) + • • • . (2.6) 4 Similarly, one can write down the expansion of the negative helicity field Ā, which, as discussed above, is linear in B, but is an infinite series in the new field B. In [34] it was shown that the coefficients in the expansion of Ā are very closely related to those for A. 2\n\nA( p ) =B( p ) + √ 2igp + Σ d 3 p 1 d 3 p 2 (2π) 3 δ (3) ( p -p 1 -p 2 ) p 1 + p 2 + 1 12 B( p 1 )B( p 2 ) -2g 2 p + Σ d 3 p 1 d 3 p 2 d 3 p 3 (2π) 6 δ (\n\nThe expansion of B turns out to be simply Ā( p ) =-∞ n=1 n s=1 Σ n i=1 d 3 p i (2π) 3 ∆( p , p 1 , . . . , p n )\n\n(p s + ) 2 (p + ) 2 Y( p ; 1 • • • n) B( p 1 )• • • B( p s )• • •B( p n ) = - ∞ n=1 Σ n i=1 d 3 p i (2π) 3 ∆( p , p 1 , . . . , p n ) 1 (p + ) 2 Y( p ; 1 • • • n) × n s=1 (p s + ) 2 B( p 1 ) • • • B( p s ) • • • B( p n ) . (2.7)\n\nThus we see that at each order in the expansion, we need to sum over all possible positions of B. Explicitly, the first few terms are: Ā( p ) = B( p ) -√ 2ig Σ d 3 p 1 d 3 p 2 (2π) 3 δ (3) ( pp 1p 2 ) 1 p + p 1 + p 2 + 1 12 × × (p 1 + ) 2 B( p 1 )B( p 2 ) + (p 2 + ) 2 B( p 1 ) B( p 2 ) + 2g 2 Σ d 3 p 1 d 3 p 2 d 3 p 3 (2π) 6 δ (3) ( pp 1p 2p 3 ) 1 p + p 1 + p 3 + 1 12 23 × × (p 1 + ) 2 B( p 1 )B( p 2 )B( p 3 )+(p 2 + ) 2 B( p 1 ) B( p 2 )B( p 3 )+(p 3 + ) 2 B( p 1 )B( p 2 ) B( p 3 ) + • • • (2.8) Using the above results, it is in principle straightforward to derive the terms that arise on inserting the Mansfield transformation into the two remaining vertices of the theory. For the simplest cases, one can see explicitly that these combine to produce MHV vertices, and some arguments were also given in [33, 34] that this must be true in general.\n\nIn supersymmetric theories, the MHV vertices are enough to reproduce complete scattering amplitudes at one loop [43] . However, as we mentioned earlier, for pure YM it is clear that the terms in the MHV Lagrangian (2.3) will not be enough to generate complete quantum amplitudes. For instance, the scattering amplitude with all gluons with positive helicity, which at one loop is finite and given by a rational term, cannot be obtained by only using MHV diagrams, for the simple reason that one cannot draw any diagram contributing to it by only resorting to MHV vertices. 3 Another amplitude which cannot be derived within conventional MHV diagrams is the amplitude with only one gluon of negative helicity. Similarly to the all-plus amplitude, this single-minus amplitude vanishes at tree level, and at one loop is given by a finite, rational function of the spinor variables.\n\n2 This is perhaps easiest to see [42] by considering that, in the context of N = 4 SYM, A and B are part of the same lightcone superfield.\n\n3 On the other hand, it was shown in [35] that the parity conjugate all-minus amplitude is correctly generated by using MHV diagrams.\n\nThe lesson we learn from this is that, in order to apply the MHV method to derive complete amplitudes in pure YM, one should look more closely at the change of variables in the full quantum theory. There are several possible subtleties one should pay careful attention to at the quantum level. First of all, it is possible that the canonical nature of the transformation is not preserved, leading to a non-trivial Jacobian which could provide the missing amplitudes. Another possible source of contributions could come from violations of the equivalence theorem. This theorem states that, although correlation functions of the new fields are in general different from those of the old fields, the scattering amplitudes are actually the same 4 , as long as the new fields are good interpolating fields. These issues were explored in some detail in [35] (see also [34, 42] ) where it was shown, for a different (non-canonical) field redefinition, how a careful treatment of these effects can combine to reproduce some of the amplitudes that would seem to be missing at first sight.\n\nAnother aim of [35] was to demonstrate how to reproduce one of the above-mentioned rational amplitudes, the one with all-minus helicities, in the MHV formalism. This amplitude is slightly less mysterious than the all-plus amplitude in the sense that one can write down the contributing diagrams using only MHV vertices; however a calculation without a suitable regulator in place would give a vanishing answer, despite the fact that this amplitude is finite.\n\nIn [35] , it was shown, using dimensional regularisation, that the full nonzero result arises from a slight mismatch between four-and D (= 4 -2ǫ)-dimensional momenta.\n\nIt is natural therefore to expect that dimensional regularisation will be helpful also for the problem at hand, which is to recover the rational amplitudes of pure Yang-Mills after the Mansfield transformation. Decomposing the regularised lightcone Lagrangian into a pure four-dimensional part and the remaining ǫ-dependent terms, and performing the transformation on the four-dimensional part only, will give rise to several new ǫ-dependent terms that can potentially give finite answers when forming loops.\n\nAlthough this approach shows promise, it is not the one we will make use of in the following. Instead, motivated by the fact that the Mansfield transformation seems to be deeply rooted in four dimensions, we would like to look for a purely four-dimensional regularisation scheme. We now turn to a review of the particular scheme we will use."
},
{
"section_type": "OTHER",
"section_title": "A four-dimensional regulator for lightcone Yang-Mills",
"text": "In the above we explained why a naïve application of the Mansfield transform leads to puzzles at the quantum level, and discussed possible ways to improve the situation. The conclusion was that, since the missing amplitudes arise from subtle mismatches in regularisation, one should be careful to perform the Mansfield transform on a suitably regularised version of the lightcone Yang-Mills action. Here we will review one approach to the regularisation of lightcone Yang-Mills, which, despite several slightly unusual features, appears to be ideally 4 Modulo a trivial wave-function renormalisation.\n\nsuited for the problem at hand. The regularisation we propose to use is inspired by recent work of CQT [39, 40, 41] on Yang-Mills amplitudes in the lightcone worldsheet approach [44, 45] . This is an attempt to understand gauge-string duality which is similar in spirit to 't Hooft's original work on the planar limit of gauge theory [46] , and aims at improving on early dual model techniques [47, 48] . We recall that one of the main goals in those works is to exhibit the string worldsheet as made up of very large planar diagrams (\"fishnets\").\n\nIn their recent work, Thorn and collaborators make this statement more precise, using techniques that were unavailable when the original ideas were put forward. It is hoped that, by understanding how to translate a generic Yang-Mills planar diagram to a configuration of fields (with suitable boundary conditions) on the lightcone worldsheet, it will eventually become possible to perform the sum of all these diagrams. This approach to gauge-string duality is thus complementary to that using the AdS/CFT correspondence.\n\nThe field content and structure of the worldsheet theory dual to Yang-Mills theory is rather intricate (see e.g. [45] ), but for our purposes the details are not important. What is most relevant for us is that one of the principles of this approach is that all quantities on the Yang-Mills side should have a local worldsheet description. This includes the choice of regulator that needs to be introduced when calculating loop diagrams. This requirement led Thorn [49] (see also [50, 51] ) to introduce an exponential UV cutoff, which we will discuss in a short while.\n\nSince one of the goals of this programme is to translate an arbitrary planar diagram into worldsheet form (and eventually calculate it), it is an important intermediate goal to understand how to do standard Yang-Mills perturbation theory in \"worldsheet-friendly\" language. In [39, 40, 41] CQT do exactly that for the simplest case, that of one-loop diagrams in Yang-Mills theory, by analysing how familiar features like renormalisation are affected by the unusual regularisation procedure and other special features of the lightcone worldsheet formalism.\n\nTo conclude this brief overview of the lightcone worldsheet formalism, the main point for our current purposes is that it provides motivation and justification for a slightly unusual regularisation of lightcone Yang-Mills, which we will now describe.\n\nLet us momentarily focus on the self-dual part of the lightcone Yang-Mills Lagrangian:\n\nL = L -+ + L ++-= -A z A z + 2ig[A z , ∂ + A z ](∂ + ) -1 (∂ z A z ) .\n\n(2.9) This action provides one of the representations of self-dual Yang-Mills theory. After transforming to momentum space, we find that the only vertex in the theory is the following (suppressing the gauge index structure):\n\n7 A 2 A 1 Ā3 = -2g p 3 + p 1 + p 2 + [p 1 + p 2 z -p 2 + p 1 z ] = - √ 2g p 3 + p 1 + p 2 + [12] . (2.10)\n\nAs for propagators, following [40], we will use the Schwinger representation:\n\n1 p 2 = - ∞ 0 dT e +T p 2 . (2.11)\n\nIn (2.11) p 2 is understood to be the appropriate (p 2 < 0) Wick rotated version of the Minkowski space inner product. For our choice of signature, the latter is\n\np • q = p + q -+ p -q + -p • q = p + q -+ p -q + -(p z q z + p z q z ) , (2.12) so that p 2 = 2(p + p --p z p z ).\n\nWe will also make use of the dual or \"region momentum\" representation, where one assigns a momentum to each region that is bounded by a line in the planar diagram. By convention, the actual momentum of the line is given by the region momentum to its right minus that on its left, as given by the direction of momentum flow 5 . Clearly such a prescription can only be straightforwardly implemented for planar diagrams, which is the case considered in [40] . This is also sufficient for our purposes, since we are calculating the leading single-trace contribution to one-loop scattering amplitudes. Non-planar (multi-trace) contributions can be recovered from suitable permutations of the leading-trace ones (see e.g. [52] ).\n\nTo demonstrate the use of region momenta, a sample one-loop diagram is pictured in Figure 1 .\n\nq k 1 k 2 k 3 k 4 1 2 3 4 l\n\nFigure 1: A sample one-loop diagram indicating the labelling of region momenta. The outgoing leg momenta are\n\np 1 = k 1 -k 4 , p 2 = k 2 -k 1 , p 3 = k 3 -k 2 , p 4 = k 4 -k 3 , while\n\nthe loop momentum (directed as indicated) is l = qk 1 .\n\n5 In [40] the flow of momentum is chosen to always match the flow of helicity, but we will not use this convention.\n\nThe \"worldsheet-friendly\" regulator that CQT employ is simply defined as follows [49]: For a general n-loop diagram, with q i being the loop region momenta, one simply inserts an exponential cutoff factor exp(-δ n i=1 q 2 i ) (2.13) in the loop integrand, where δ is positive and will be taken to zero at the end of the calculation. This clearly regulates UV divergences (from large transverse momenta), but, as we will see, has some surprising consequences since it will lead to finite values for certain Lorentzviolating processes, which therefore have to be cancelled by the introduction of appropriate counterterms.\n\nNote that q 2 = 2q z q z has components only along the two transverse directions, hence it breaks explicitly even more Lorentz invariance than the lightcone usually does. This might seem rather unnatural from a field-theoretical point of view, however it is crucial in the lightcone worldsheet approach. Indeed, the lightcone time x -and x + (or in practice its dual momentum p + ) parametrise the worldsheet itself, and are regulated by discretisation; thus, they are necessarily treated very differently from the two transverse momenta q z , q z which appear as dynamical worldsheet scalars. Fundamentally, this is because of the need to preserve longitudinal (x + ) boost invariance (which eventually leads to conservation of discrete p + ). The fact that the regulator depends on the region momenta rather than the actual ones is a consequence of asking for it to have a local description on the worldsheet.\n\nThe main ingredient for what will follow later in this paper is the computation of the (++) one-loop gluon self-energy in the regularisation scheme discussed earlier. This is performed on page 10 of [40] , and we will briefly outline it here. This helicity-flipping gluon self-energy, which we denote by Π ++ , is the only potential self-energy contribution in self-dual Yang-Mills; in full YM we would also have Π +-and, by parity invariance, Π --.\n\nThere are two contributions to this process, corresponding to the two ways to route helicity in the loop, but they can be easily shown to be equal so we will concentrate on one of them, which is pictured in Figure 2\n\n. A A A Ā A Ā k ′ k q p -p l p + l\n\nFigure 2: Labelling of one of the selfenergy diagrams contributing to Π ++ .\n\nIn Figure 2, p, -p are the outgoing line momenta, l is the loop line momentum, and 9 k, k ′ , q are the region momenta, in terms of which the line momenta are given by p = k ′k, l = qk ′ . (2.14) Remembering to double the result of this diagram, we find the following expression for the self-energy:\n\nΠ ++ =8g 2 N d 4 l (2π) 4 -(p + l) + p + l + (p + l z -l + p z) × 1 l 2 (p + l) 2 × × -l + (-p + )(p + l) + ((-p + )(p z + l z ) -(p + + l + )(p z)) = g 2 N 2π 4 d 4 l 1 (p + ) 2 (p + l z -l + p z )(p + (p z + l z ) -(p + + l + )p z ) 1 l 2 (p + l) 2 . (2.15)\n\nAlthough we are suppressing the colour structure, the factor of N is easy to see by thinking of the double-line representation of this diagram 6 . One of the crucial properties of (2.15) is that the factors of the loop momentum l + coming from the vertices have cancelled out, hence there are no potential subtleties in the loop integration as l + → 0. This means that, although for general loop calculations one would have to follow the DLCQ procedure and discretise l + (as is done for other processes considered in [39, 40, 41] ), this issue does not arise at all for this particular integral, and we are free to keep l + continuous.\n\nTo proceed, we convert momenta to region momenta, rewrite propagators in Schwinger representation, and regulate divergences using the regulator (2.13). Employing the unbroken shift symmetry in the + region momenta to further set k + = 0, (2.15) can be recast as:\n\nΠ ++ = g 2 N 2π 4 ∞ 0 dT 1 dT 2 d 4 q 1 (k ′ + ) 2 e T 1 (q-k) 2 +T 2 (q-k ′ ) 2 -δq 2 × × k ′ + (q z -k ′ z) -(q + -k ′ + )(k ′ z -k z) k ′ + (q z -k z ) -q + (k ′ z -k z ) .\n\n(2.16) Since q -only appears in the exponential, the q -integration will lead to a delta function containing q + , which can be easily integrated and leads to a Gaussian-type integral for q z , q z . Performing this integral, we obtain (setting xk+(1-x)\n\nT = T 1 + T 2 , x = T 1 /(T 1 + T 2 )) Π ++ = g 2 N 2π 2 1 0 dx ∞ 0 dT δ 2 [xk z + (1 -x)k ′ z ] 2 (T + δ) 3 e T x(1-x)p 2 -δT T +δ (\n\nk ′ ) 2 . (2.17)\n\nNotice that, had we not regularised using the δ regulator, we would have obtained zero at this stage. Instead, now we can see that there is a region of the T integration (where T ∼ δ) that can lead to a nonzero result. On performing the T and x integrations, and sending δ to zero at the end, we obtain the following finite answer:\n\nΠ ++ = 2 + + = g 2 N 12π 2 (k z ) 2 + (k ′ z) 2 + k z k ′ z . (2.18)\n\n6 For simplicity, we take the gauge group to be U(N ).\n\n10 Notice that this nonvanishing, finite result violates Lorentz invariance, since it would imply that a single gluon can flip its helicity. Also, it explicitly depends on only the z components of the region momenta. Such a term is clearly absent in the tree-level Lagrangian (unlike e.g. the Π +-contribution in full Yang-Mills theory), thus it cannot be absorbed through renormalisation -it will have to be explicitly cancelled by a counterterm. This counterterm, which will play a major rôle in the following, is defined in such a way that: + = 0 , (2.19) in other words it will cancel all insertions of Π ++ , diagram by diagram. Let us note here that, had we been doing dimensional regularisation, all bubble contributions would vanish on their own, so there would be no need to add any counterterms. So this effect is purely due to the \"worldsheet-friendly\" regulator (2.13).\n\nIt is also interesting to observe that in a supersymmetric theory this bubble contribution would vanish 7 so this effect is only of relevance to pure Yang-Mills theory."
},
{
"section_type": "OTHER",
"section_title": "The one-loop (++++) amplitude",
"text": "Now let us look at the all-plus four-point one-loop amplitude in this theory. It is easy to see that it will receive contributions from three types of geometries: boxes, triangles and bubbles. It is a remarkable property 8 that the sum of all these geometries adds up to zero. In particular, with a suitable routing of momenta, the integrand itself is zero. Pictorially, we can state this as:\n\n+ 4 × + 2 × + 8 × = 0 . (2.20)\n\nThe coefficients mean that we need to add that number of inequivalent orderings. So we see (and refer to [40] for the explicit calculation) that the sum of all the diagrams that one can construct from the single vertex in our theory, gives a vanishing answer. However, as discussed in the previous section, this is not everything: we need to also include the contribution of the counterterm that we are forced to add in order to preserve Lorentz invariance. Since this counterterm, by design, cancels all the bubble graph contributions, we are left with just the sum of the box and the four triangle diagrams. In pictures,\n\nA ++++ = +4× + 2 × + 8 × + 2 × + 8 × (2.21) 7\n\nThis can in fact be derived from the results of [53] , where similar calculations were considered with fermions and scalars in the loop.\n\n8 This observation is attributed to Zvi Bern [40].\n\nwhere A ++++ is the known result [54] for the leading-trace part of the four-point all-plus amplitude:\n\nA ++++ (A 1 A 2 A 3 A 4 ) = i g 4 N 48π 2 [12][34] 12 34 , (2.22)\n\nand the terms in the parentheses clearly cancel among themselves. This leaves the box and triangle diagrams, which are exactly those appearing in the calculation of the parity conjugate amplitude using dimensional regularisation [35] , where the bubbles were zero to begin with.\n\nFollowing [40], we make the obvious, but important for the following, observation that one can change the position of the parentheses:\n\nA ++++ = + 4 × + 2 × + 8 × +2× +8× (2.23)\n\nwhere again the terms in the parentheses are zero (by (2.20)). This demonstrates that one can compute the all-plus amplitude just from a tree-level calculation with counterterm insertions (of course, these diagrams are at the same order of the coupling constant as oneloop diagrams because of the counterterm insertion). This remarkable claim is verified in [40] , where CQT explicitly calculate the 10 counterterm diagrams and recover the correct result for the four-point amplitude (see pp. 22-23 of [40]) 9 .\n\nThis result, apart from being very appealing in that one does not have to perform any integrals (apart from the original integral that defined the counterterm) so that the calculation reduces to tree-level combinatorics, will also turn out to be a convenient starting point for performing the Mansfield transformation. Specifically, our claim is that the whole series of all-plus amplitudes will arise just from the counterterm action. In the following we will show how this works explicitly for the four-point all-plus case, and then we will argue for the n-point case that the corresponding expression derived from the counterterm has all the correct singularities (soft and collinear), giving strong evidence that the result is true in general."
},
{
"section_type": "OTHER",
"section_title": "The all-plus amplitudes from a counterterm",
"text": "Having reviewed the relevant new features that arise when doing perturbation theory with the worldsheet-motivated regulator of [49] , we now have all the necessary ingredients to perform the Mansfield change of variables on the regulated lightcone Lagrangian. In this section, we will carry out this procedure. We will first regulate lightcone self-dual Yang-Mills, which, as 9 In practice, these authors choose to insert the self-energy result (2.18) in the tree diagrams, so what they compute is minus the all-plus amplitude.\n\ndiscussed, will require us to introduce an explicit counterterm in the Lagrangian. Then we will perform the Mansfield transformation on the original Lagrangian (converting it to a free theory). We will then show that, upon inserting the change of variables into the counterterm Lagrangian, we recover the all-plus amplitudes as vertices in the theory."
},
{
"section_type": "OTHER",
"section_title": "Mansfield transformation of L CT",
"text": "As we saw, the \"worldsheet-friendly\" regularisation requires us to add a certain counterterm to the lightcone Yang-Mills action, required in order to cancel the Lorentz-violating helicityflipping gluon selfenergy. As mentioned previously, the calculation of the all-plus amplitude can be tackled purely within the context of self-dual Yang-Mills, which we will focus on from now on. We see that, as a result of this regularisation, the complete action at the quantum level becomes:\n\nL (r) SDYM = L +-+ L ++-+ L CT , (3.1)\n\nwhere L +-+ L ++-is the classical Lagrangian for self-dual Yang-Mills introduced in (2.9). Although CQT do not write down a spacetime Lagrangian for L CT , it is easy to see that the following expression would have the right structure:\n\nL CT = - g 2 N 12π 2 Σ d 3 k i d 3 k j A i j (k i , k j )[(k i z ) 2 + (k j z ) 2 + k i z k j z ]A j i (k j , k i ) . (3.2)\n\nThis expression depends explicitly on the dual, or region, momenta. In (3.2) we have made use of the simplest way to associate region momenta to fields, which is to assign a region momentum to each index line in double-line notation [46] , and thus a momentum k i , k j to each of the indices of the gauge field A i j (now slightly extended into a dipole, as would be natural from the worldsheet perspective, where an index is associated to each boundary). Since each line has a natural orientation, the actual momentum of each line can be taken to be the difference of the index momentum of the incoming index line and the outgoing index line. So the momentum of\n\nA i j (k i , k j ) is taken to be p = k j -k i .\n\nAs discussed above, this assignment can only be performed consistently for planar diagrams, which is sufficient for our purposes.\n\nClearly, the structure of (3.2) is rather unusual. First of all, it depends only on the antiholomorphic (z) components of the region momenta, and so is clearly not (lightcone) covariant. Even more troubling is the fact that it does not depend only on differences of region momenta, but also on their sums. Since each field thus carries more information than just its momentum, L CT is a non-local object from a four-dimensional point of view (although, as shown in [40] , it can be given a perfectly local worldsheet description).\n\nLeaving the above discussion as food for thought, we will now rewrite (3.2) in a more conventional way that is most convenient for inserting into Feynman diagrams,\n\nL CT = - g 2 N 12π 2 Σ d 3 p d 3 p ′ δ(p + p ′ ) A i j (p ′ )((k i z) 2 + (k j z) 2 + k i z k j z)A j i (p) . (3.3) 13\n\nIn this expression, which can be thought of as the zero-mode or field theory limit of (3.2), all the region momentum dependence is confined to the polynomial factor (k i z ) 2 +(k j z) 2 +k i z k j z. This vertex, inserted into tree diagrams, would exactly reproduce the effects of the counterterm pictured in (2.19). Although (3.3) still exhibits some of the apparently undesirable features we discussed above, the calculations in [40] demonstrate that, after summing over all possible insertions of this term, the final result is covariant and correctly reproduces the all-plus amplitudes 10 . Therefore, we believe that its problematic properties are really a virtue in disguise, and (as we will see explicitly) they seem to be crucial in obtaining the full series of n-point all-plus amplitudes from the Mansfield transformation of a single term.\n\nWe are now ready to perform the Mansfield change of variables. In the spirit of the discussion earlier, we will perform the transformation on the classical part of the action only:\n\nL +-(A, Ā) + L ++-(A, Ā) = L +-(B, B) (3.4)\n\nHence the classical part of the action has been converted to a free theory. Without a regulator, this would be the whole story. However we now see that, within the particular regularisation we are working with, the full Lagrangian L (r) SDYM contains one extra, one-loop piece, given by L CT in (3.3), which is quadratic in the positive helicity fields A. To complete the Mansfield transformation, we will clearly need to expand this term in the new fields B, using the Ettle-Morris coefficients (2.4).\n\nSince L CT depends only on the holomorphic A fields, we will only need the expansion of A in terms of B given in (2.4). As a first check that L CT leads to the right kind of structure, note that since A depends only on the holomorphic B fields, all the new vertices are all-plus. Thus, the full action, when expressed in terms of the B fields, takes the schematic form:\n\nL (r) SDYM (A, Ā) = L +-(B, B) + L ++ (B) + L +++ (B) + L ++++ (B) + • • • (3.5)\n\nIn the next section we will calculate the four-point term L ++++ and demonstrate that, when restricted on-shell, it reproduces the known form (2.22) for the all-plus amplitude."
},
{
"section_type": "OTHER",
"section_title": "The four-point case",
"text": "To begin with, we focus on the derivation of the four-point all-plus vertex, whose on-shell version will give us the four-point scattering amplitude. We will thus expand the old fields A in the counterterm (3.3) (or (3.2)) up to terms containing four B-fields.\n\nWhen inserting the Ettle-Morris coefficients into (3.3), one has to sum over all possible cyclic orderings with which this can be done. A complication is that now the counterterm 10 Note that similar-looking treatments using index momenta instead of line momenta for vertices, but which in the end sum up to covariant results have appeared in the context of noncommutative geometry (see e.g. [55] ). Although it is possible to write e.g. (3.2) in star-product form, at this stage it is not clear whether that is a useful reformulation.\n\nitself depends on the ordering. In other words, we need to sum over all the ways of assigning dual momenta to the indices. Schematically, the inequivalent terms that we obtain are:\n\nAA → ( B 1 B 2 )( B 3 B 4 ) + ( B 2 B 3 )( B 4 B 1 ) + ( B 1 B 2 B 3 )B 4 + ( B 2 B 3 B 4 )B 1 + ( B 3 B 4 B 1 )B 2 + ( B 4 B 1 B 2 )B 3 , (3.6)\n\nwhere the terms on the first line arise from doing two quadratic substitutions and those on the second from doing one cubic substitution. All the other possibilities are related by cyclicity of the trace. For definiteness, let us now write down what one of these terms means explicitly: 11\n\n( B 1 B 2 B 3 )B 4 = -2g 2 tr dpdp 4 δ(p+p 4 ) dp 1 dp 2 dp 3 δ(p-p 1 -p 2 -p 3 ) p + p 1 + p 3 + 1 12 23 × × B(p 1 )B(p 2 )B(p 3 ) (k 3 z ) 2 + (k 4 z ) 2 + k 4 z k 3 z B(p 4 ) = 2g 2 dp 1 dp 2 dp 3 dp 4 δ(p 1 + p 2 + p 3 + p 4 )× × p 4 + p 1 + p 3 + (k 3 z ) 2 + (k 4 z ) 2 + k 4 z k 3 z 12 23 tr B(p 1 )B(p 2 )B(p 3 )B(p 4 ) . (3.7)\n\nThe reason this particular combination of k z's appears here is that, given the ordering we chose, after the Mansfield transformation the counterterm ends up being on leg 4, and its line bounds the regions with momenta k 3 and k 3 . This is represented pictorially in Figure 3 .\n\nk 1 k 2 k 3 k 4 B 1 B 2 B 4 B 3\n\nFigure 3: One of the contributions to the four-point all-plus vertex.\n\nAlthough Figure 3 might suggest that there is a propagator between the counterterm insertion and the location of the original A, which has now split into three B's, this is of course not the case since the whole expression is a vertex at the same point. We have drawn the diagram in this fashion to emphasise which leg the counterterm is located on after the transformation. On the other hand, this vertex is nonlocal (as discussed above, it was nonlocal even in the original variables, but this is now compounded by the Mansfield coefficients, which contain momenta in the denominator), so this notation serves as a useful reminder of that fact.\n\n11 We suppress the overall factor of -g 2 N/(12π 2 ) until the end of this section. Also, the integrals are implicitly taken to be on the quantisation surface Σ.\n\nIt is interesting to note that (3.7) is essentially the same expression as the sum of the two channels with the same region momentum dependence that appear in CQT's calculation of this amplitude using tree-level diagrammatics (compare with Eq. 83 in [40] ), which we illustrate in Fig. 4 . Thus we have a picture where one post-Mansfield transform vertex (with\n\nk 1 k 2 k 3 k 4 A 1 A 2 A 4 A 3 + k 1 k 2 k 3 k 4 A 1 A 2 A 4 A 3\n\nFigure 4: The two diagrams with counterterm insertions on leg 4 that arise in the calculation of CQT, and, combined, add up to the contribution in Fig. 3. B's) effectively sums two tree-level pre-transformation (with A's) Feynman diagrams. This is a first indication that our calculation of the all-plus vertex can be mapped, practically one-to-one, to that of the all-plus amplitude on pp. 22-23 of [40] .\n\nAnother type of contribution to the vertex arises when we transform both of the A's in L CT . One of the two terms that we find is: (3.8) This contribution can also be mapped to one of the two terms with bubbles on internal lines in CQT.\n\n( B 2 B 3 )( B 4 B 1 ) = -2g 2 tr dp dp ′ δ(p + p ′ ) dp 2 dp 3 δ(p -p 2 -p 3 ) p + p 2 + p 3 + 1 23 B(p 2 )B(p 3 ) × × (k 1 z ) 2 + (k 3 z ) 2 + k 1 z k 3 z dp 4 dp 1 δ(p ′ -p 4 -p 1 ) p ′ + p 4 + p 1 + 1 41 B(p 4 )B(p 1 ) = -2g 2 dp 1 • • • dp 4 δ(p 1 +p 2 +p 3 +p 4 ) (p 2 + + p 3 + )(p 1 + + p 4 + ) p 1 + p 2 + p 3 + p 4 + ((k 1 z ) 2 + (k 3 z ) 2 + k 1 z k 3 z ) 23 41 × tr B(p 1 )B(p 2 )B(p 3 )B(p 4 ) .\n\nWe can now tabulate all the terms that we obtain in this way by making the schematic form (3.6) precise. Since the delta-function and trace over B parts are the same for all these terms, in Table 1 we just list the rest of the integrand.\n\nTo obtain the final form of the vertex, we are now instructed to sum over all these contributions. Thus we can write\n\nL ++++ (B) = 2g 2 dp 1 dp 2 dp 3 dp 4 δ(p 1 +p 2 +p 3 +p 4 ) V (4) tr[B(p 1 )B(p 2 )B(p 3 )B(p 4 )] (3.9) 16 Schematic form Pictorial form Integrand ( B 1 B 2 B 3 )B 4 p 4 + √ p 1 + p 3 + k 2 3 +k 2 4 +k 3 k 4 12 23 ( B 2 B 3 B 4 )B 1 p 1 + √ p 2 + p 4 + k 2 1 +k 2 4 +k 1 k 4 23 34 ( B 3 B 4 B 1 )B 2 p 2 + √ p 3 + p 1 + k 2 1 +k 2 2 +k 2 k 1 34 41 ( B 4 B 1 B 2 )B 3 p 3 + √ p 4 + p 2 + k 2 2 +k 2 3 +k 2 k 3 41 12 ( B 2 B 3 )( B 4 B 1 ) - (p 2 + +p 3 + )(p 1 + +p 4 + ) √ p 1 + p 2 + p 3 + p 4 + k 2 1 +k 2 3 +k 1 k 3 23 41 ( B 1 B 2 )( B 3 B 4 ) - (p 3 + +p 3 + )(p 2 + +p 1 + ) √ p 1 + p 2 + p 3 + p 4 + k 2 4 +k 2 2 +k 4 k 2 34 12\n\nTable 1: The various contributions to the all-plus four-point vertex. Note that we use the simplifying notation k i := k i z.\n\nwhere V (4) is given by the following expression: 12\n\nV (4) = 1 p 1 + p 2 + p 3 + p 4 + 1 12 23 34 41 × × p 4 + p 2 + p 4 + (k 2 3 + k 2 4 + k 3 k 4 ) 34 41 + p 1 + p 1 + p 3 + (k 2 1 + k 2 4 + k 1 k 4 ) 12 41 + p 2 + p 2 + p 4 + (k 2 2 + k 2 1 + k 2 k 1 ) 12 23 + p 3 + p 3 + p 1 + (k 2 3 + k 2 2 + k 2 k 3 ) 23 34 -(p 2 + + p 3 + )(p 1 + + p 4 + )(k 2 1 + k 2 3 + k 1 k 3 ) 12 34 -(p 3 + + p 4 + )(p 2 + + p 1 + )(k 2 4 + k 2 2 + k 4 k 2 ) 23 41 . (3.10)\n\nComparing this to the expected answer (2.22), we see that the (quadratic) antiholomorphic momentum dependence should arise from the various k z factors in (3.10). In [40], CQT start from essentially the same expression and demonstrate that it gives the correct result for the all-plus amplitude. Therefore, following practically the same steps as those authors, we can easily see that we obtain the expected answer. However, since we would like to find the full vertex V, we will need to keep off-shell information, and so we will choose a slightly different route.\n\n12 For the sake of brevity we omit a subscript z in the region momenta appearing in (3.10).\n\nThe main complication in bringing (3.10) into a manageable form is clearly the presence of the region momenta. We would like to disentangle their effects as cleanly as possible. Therefore, our derivation will proceed by the following steps: 1. First, we will show that (3.10) can be manipulated so that the quadratic dependence on region momenta drops out, leaving only terms linear in the region momenta.\n\n2. Second, we will decompose the resulting expression into a part that depends on the region momenta and one that does not. The k-dependent part turns out to have a very simple form, and vanishes on-shell.\n\n3. Finally, we will show that the k-independent part reduces to the known amplitude.\n\nFor the first step, we will need the following identity, which is proved in appendix B:\n\np 4 + p 2 + p 4 + 34 41 + p 1 + p 1 + p 3 + 12 41 + p 2 + p 2 + p 4 + 12 23 + p 3 + p 3 + p 1 + 23 34 -(p 2 + + p 3 + )(p 1 + + p 4 + ) 12 34 -(p 3 + + p 4 + )(p 2 + + p 1 + ) 23 41 = 0 (3.11)\n\nAlso, using the shorthand notation K ij := (k i z) 2 + (k j z) 2 + k i z k j z : we note the following very useful identity:\n\nK ij = K ik + (k j z -k k z )(k i z + k j z + k k z ) = K ik + (k j z -k k z )l ijk (3.12) where 1 ≤ k ≤ n and l ijk = k i z +k j z +k k z . Noting that, for j > k, k j z -k k z = p k+1 z +p k+2 z +• • • p j z ,\n\nwe can use this to rewrite all the region momentum combinations appearing in (3.10) in the following way:\n\nK 34 = 1 4 (K 12 + K 23 + K 34 + K 41 + (p 3 + p4 )(l 124 + l 234 ) + 2(p 2 + p3 )l 134 ) K 14 = 1 4 (K 12 + K 23 + K 34 + K 41 -(p 2 + p3 )(l 134 + l 123 ) + 2(p 3 + p4 )l 124 ) K 12 = 1 4 (K 12 + K 23 + K 34 + K 41 -(p 3 + p4 )(l 124 + l 234 ) -2(p 2 + p3 )l 123 ) K 23 = 1 4 (K 12 + K 23 + K 34 + K 41 + (p 2 + p3 )(l 134 + l 123 ) -2(p 3 + p4 )l 234 ) K 13 = 1 4 (K 12 + K 23 + K 34 + K 41 + (p 3 -p2 )l 123 + (p 1 -p4 )l 134 ) K 24 = 1 4 (K 12 + K 23 + K 34 + K 41 + (p 4 -p3 )l 234 + (p 2 -p1 )l 124 ) (3.13)\n\nwhere we have introduced the notation pi = p i z . We have thus expressed all the quadratic region momentum dependence in terms of the common factor and, given (3.11) , it is clear that this contribution will vanish. 13 13 One could have chosen a different combination of the K ij 's, but we find the symmetric choice in (3.13) convenient.\n\nK 12 + K 23 + K 34 + K 41 ,\n\nAfter this step, we are left with an expression which is linear in the region momenta. We will now proceed in a similar way, and rewrite all the expressions that contain l ijk in terms of a suitably chosen common factor:\n\nl 124 + l 234 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) - 1 2 (p 1 z + p 3 z ) l 134 + l 123 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) - 1 2 (p 2 z + p 4 z ) 2l 234 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) + 1 2 (2p 2 z + p 3 z -p 1 z ) 2l 123 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) + 1 2 (2p 1 z + p 2 z -p 4 z ) 2l 134 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) + 1 2 (2p 3 z + p 4 z -p 2 z ) 2l 124 = 3 2 (k 1 z + k 2 z + k 3 z + k 4 z ) + 1 2 (2p 4 z + p 1 z -p 3 z ) (3.14)\n\nIn appendix B we show that the total coefficient of the common (k\n\n1 z + k 2 z + k 3 z + k 4 z ) factor is 3 8 p 4 + p 2 + p 4 + (+(p 3 + p4 ) + (p 2 + p3 )) 34 41 + p 1 + p 1 + p 3 + (-(p 2 + p3 ) + (p 3 + p4 )) 12 41 + p 2 + p 2 + p 4 + (-(p 3 + p4 ) -(p 2 + p3 )) 12 23 + p 3 + p 3 + p 1 + (+(p 2 + p3 ) -(p 3 + p4 )) 23 34 -(p 2 + + p 3 + )(p 1 + + p 4 + )( 1 2 (p 3 -p2 ) + 1 2 (p 1 -p4 )) 12 34 -(p 3 + + p 4 + )(p 2 + + p 1 + )( 1 2 (p 4 -p3 ) + 1 2 (p 2 -p1 )) 23 41 = = - 3 16 [(12) + (23) + (34) + (41)] 4 i=i (p i ) 2 p i + , (3.15)\n\nwhere (p i ) 2 is the full covariant momentum squared, and (ij\n\n) = p i + p j z -p j + p i z .\n\nThus we see that the complete dependence on the region momenta can be rewritten as follows:\n\nV (4) k = - 3 16 (12) + (23) + (34) + (41) 12 23 34 41 4 i=1 k i z 4 i=i (p i ) 2 p i + . (3.16)\n\nIt is rather satisfying that the region momentum dependence of the vertex takes this simple form, which clearly vanishes when the external legs are on-shell, and thus will not contribute to the all-plus amplitudes.\n\nHaving completely disentangled the region momenta k z from the actual momenta p z , we will now focus on the terms containing only the latter, which were produced during the 19 decompositions in (3.14). After a few simple manipulations, they can be rewritten as 14 V (4) p = 1 8 p 4 + p 2 + p 4 + [(p 1 + p2 )(p 1 -p2 ) + (p 3 + p2 )(p 3 -p2 )] 34 41 + p 1 + p 1 + p 3 + [(p 2 + p3 )(p 2 -p3 ) + (p 4 + p3 )(p 4 -p3 )] 41 12\n\n+ p 2 + p 2 + p 4 + [(p 3 + p4 )(p 3 -p4 ) + (p 1 + p4 )(p 1 -p4 )] 12 23 + p 3 + p 3 + p 1 + [(p 4 + p1 )(p 4 -p1 ) + (p 2 + p1 )(p 2 -p1 )] 23 34 -(p 2 + + p 3 + )(p 1 + + p 4 + )[(p 3 -p2 )(p 1 -p4 ) -(p 1 + p2 ) 2 ] 12 34 -(p 3 + + p 4 + )(p 2 + + p 1 + )[(p 4 -p3 )(p 2 -p1 ) -(p 2 + p3 ) 2 ] 23 41 .\n\n(3.17) This expression, together with (3.16) is our proposal for the off-shell four-point all-plus vertex that should be part of the MHV-rules formalism at the quantum level. It would be very interesting to elucidate its structure and bring it into a more compact form. For the moment, however, we will be content to demonstrate that (3.17) is equal on shell to the sought-for amplitude.\n\nTo that end, we will follow a similar approach to CQT, and rewrite all the holomorphic spinor brackets in terms of the following three: 12 34 , 23 41 , 12 41 . To achieve this, we use momentum conservation and a certain cyclic identity (see appendix A) to write\n\np 4 + p 2 + p 4 + 34 41 = p 4 + p 4 + -p 3 + 42 -p 4 + 23 41 = -p 4 + p 3 + p 4 + 42 -(p 4 + ) 2 41 = -p 4 + p 3 + -p 1 + 12 -p 3 + 32 -(p 4 + ) 2 23 41 = p 4 + p 3 + p 1 + 12 41 -p 4 + (p 4 + + p 3 + ) 23 41 . (3.18)\n\nIn a similar way, we can show that\n\np 2 + p 2 + p 4 + 12 23 = p 2 + p 3 + p 1 + 12 41 -p 2 + (p 2 + + p 3 + ) 34 12 , p 3 + p 1 + p 3 + 23 34 = -p 3 + (p 3 + +p 2 + ) 12 34 -p 3 + (p 1 + +p 2 + ) 23 41 +p 3 + p 1 + p 3 + 12 14 . (3.19)\n\nCollecting all the terms together, and manipulating the resulting expressions, it is straight- 4) .\n\n14 We write V (4) = p 1 + p 2 + p 3 + p 4 + 12 23 34 41 V (\n\nforward to show that (3.17) simplifies to just V (4) p = 1 4 23 41 {34}(p 1 + + p 2 + )[(p 1 -p2 ) -(p 2 + p3 )] + 12 34 {23}(p 2 + + p 3 + )[(p 1 + p2 ) + (p 1 -p4 )] + 12 41 p 3 + p 1 + (p 1 + p2 )({41} + {32})+(p 2 + p3 )({12} + {43}) , (3.20)\n\nwhere we use the notation [34] ij] . Converting to the usual antiholomorphic bracket notation, we rewrite (3.20) as\n\n{ij} = p i + p j z -p j + p i z = (1/ √ 2) p i + p j + [\n\nV (4) p = 1 4 √ 2 23 41 p 3 + p 4 + [34](p 1 + + p 2 + )[(p 1 -p2 ) -(p 2 + p3 )] + 12 34 p 2 + p 3 + [23](p 2 + + p 3 + )[(p 1 + p2 ) + (p 1 -p4 )] + 12 41 (p 1 + p2 )(p 1 + p 3 + p 4 + [41] + p 2 + p 2 + p 1 + [32]) + (p 2 + p3 )(p 1 + p 2 + p 3 + [12] + p 3 + p 1 + p 4 + [43]) .\n\n(3.21) Note that so far this expression is completely off shell. We will now show that on shell it reduces to the known result (2.22). In doing this we will keep track of the p 2 terms that appear when applying momentum conservation in the form\n\nk ik [kj] = p i + p j + k (p k ) 2 p k + . (3.22)\n\nThese terms are collected in appendix B.\n\nWe start by rewriting each of the terms in the last two lines of (3.21) as follows\n\n12 41 [41] p 1 + p 3 + p 4 + (p 1 + p2 ) = -23 41 [34] p 1 + p 3 + p 4 + (p 1 + p2 ) 12 41 [32] p 3 + p 1 + p 2 + (p 1 + p2 ) = -12 [32] 42 p 2 + p 3 + (p 1 + p2 ) -12 34 [23] p 3 + p 2 + p 3 + (p 1 + p2 ) 12 41 [12] p 1 + p 2 + p 3 + (p 2 + p3 ) = -12 34 [23] p 1 + p 2 + p 3 + (p 2 + p3 ) 12 41 [43] p 3 + p 1 + p 4 + (p 2 + p3 ) = -41 23 [34] p 3 + p 3 + p 4 + (p 2 + p3 ) -41 [43] 42 p 4 + p 3 + (p 2 + p3 ) . (3.23)\n\nWe also transform the 12 34 term using the Schouten identity and also momentum conservation,\n\n12 34 [23] p 2 + p 3 + = 23 41 [34] p 3 + p 4 + + 14 23 [13] p 1 + p 3 + -13 42 [23] p 2 + p 3 + , (3.24) 21\n\nand add up all contributions to the 23 41 term, which are 1 4 √ 2 23 41 [34] p 3 + p 4 + 4(p 2 + p1p 1 + p2 ) + 2(p 3 + p1p 1 + p3 ) = 1 4 √ 2 23 41 [34] p 3 + p 4 + [4{21} + 2{31}] .\n\n(3.25) Converting to the spinor bracket, the first of these terms is\n\n- 1 2 p 1 + p 2 + p 3 + p 4 + [12] 23 [34] 41 , (3.26)\n\nwhile the remaining terms from (3.23) and (3.24) combine to give\n\n14 23 [13] p 1 + p 3 + -13 42 [23] p 2 + p 3 + (p 2 + + p 3 + )[(p 1 + p2 ) + (p 1 -p4 )] + 12 [32] 42 p 2 + [p 2 + (p 1 + p2 ) -p 4 + (p 2 + p3 )] = -14 [13] 12 p 3 + (p 2 + + p 3 + )[(p 1 + p2 ) + (p 1 -p4 )] + 12 [32] 42 p 2 + [p 2 + (p 1 + p2 ) -p 4 + (p 2 + p3 )] = -14 [13] 12 p 3 + (2(p 2 + + p 3 + )p 1 -2p 1 + (p 2 + p3 )) = 2 14 [13] 12 p 3 + {41} (3.27)\n\n(where we suppress an overall 1/(4 √ 2)) and we see that (3.27) cancels the second term in (3.25), thus showing that (3.26) is the complete on-shell answer. Reintroducing all the prefactors, we thus find that the amplitude is\n\nA (4) = - g 2 N 12π 2 2g 2 p 1 + p 2 + p 3 + p 4 + 1 12 23 34 41 × - 1 2 p 1 + p 2 + p 3 + p 4 + [12] 23 [34] 41 = g 4 N 12π 2 [12][34] 12 34 .\n\n(3.28) Now note that, as discussed in appendix A, in order to convert to the usual Yang-Mills theory normalisation we need to send g → g/ √ 2. We conclude that A (4) gives precisely the result (2.22) for the all-plus scattering amplitude."
},
{
"section_type": "OTHER",
"section_title": "The general all-plus amplitude",
"text": "We have just given an explicit derivation of the four point all-plus amplitude, from the two-point counterterm (3.3). We will argue in the following that this two-point counterterm contains all the all-plus amplitudes.\n\nFirst, we can see immediately that the counterterm (3.3) has the right kind of structure. Consider the n-point all-plus amplitude [56] :\n\nA (n) = 1≤i<j<k<l≤n ij [jk] kl [li] 12 • • • n1 . (3.29) 22\n\nIn terms of spinor brackets this amplitude has terms of the form 2-n [ ] 2 . A quick look at the Ettle-Morris coefficients shows that, for an n-point vertex coming from L CT , they contribute exactly 2n powers of the spinor brackets . Furthermore, there are exactly two powers of [ ] coming from the counterterm Lagrangian L CT ∼ (k 2 z )A 2 -one for each power of k. Thus the general structure of L CT is appropriate to reproduce (3.29).\n\nPictorially, we can represent the general n-point amplitude, arising from the counterterm in the new variables, as in Figure 5.\n\nB i B i-1 B j+1 B j B j-1 B i+1 k i k j\n\nFigure 5: The structure of a generic term contributing to the n-point vertex. All momenta are taken to be outgoing, and all indices are modulo n.\n\nThus we can write this n-point all-plus vertex as follows:\n\nA (n) +•••+ = 1•••n δ(p + p ′ ) 1≤i<j≤n Y(p; j + 1, . . . , i) (k i z ) 2 + (k j z ) 2 + k i z k j z Y(p ′ ; i + 1, . . . , j)× × tr[B i B i+1 • • • B j B j+1 • • • B i-1 ] =( √ 2i) n-2 1•••n δ(p 1 +• • •+p n ) 1≤i<j≤n (p j+1 + + • • • + p i + ) p j+1 + p i + 1 j + 1, j + 2 • • • i -1, i × × (k i z) 2 + (k j z) 2 + k i zk j z (p i+1 + + • • • + p j + ) p i+1 + p j + 1 i + 1, i + 2 • • • j -1, j tr[B 1 • • • B n ] . (3.30)\n\nFocusing only on the relevant part of the above expression, and ignoring all coefficients, the general structure we obtain is the following:\n\nV (n) +•••+ = 1 12 • • • n1 × 1≤i<j≤n j, j + 1 i, i + 1 p i + p i+1 + p j + p j+1 + (k j + -k i + ) 2 ((k i z ) 2 + (k j z ) 2 + k i z k j z ) (3.31) 23\n\nwhere we have extracted the denominator at the expense of introducing the two missing holomorphic factors j, j + 1 and i, i + 1 in the numerator. We also made use of the fact that\n\nk j -k i = p i+1 + p i+2 + • • • + p j = -(p j+1 + p j+2 + • • • + p i ) , ( 3\n\n.32) applied to the + components, to rewrite the two p + sums in the numerator in terms of the k's (this gives rise to a minus which we suppress).\n\nIt is easy to verify that, for n = 4, this sum reproduces the 6 contributions that appeared in the four-point case, and (as we explicitly showed above) combined to give the expected answer. Therefore, we would like to propose that the vertex (3.31) will reduce on-shell to an expression proportional to (3.29). We will not attempt to prove this statement here 15 , but will instead move on to study the general properties of the n-point expression (3.30).\n\nWhilst the explicit calculation for the four point case was rather involved as we saw earlier, the study of the general properties of the n-point amplitudes proves much simpler. In particular, we will show that the collinear and soft limits of the expressions proposed for the n-point case can be very easily shown to be correct. Let us start by introducing some simplifying notation. One can write the change of variables for the A field as 34) . . . (n -1 n) (3.35) (for simplicity, we are dropping inconsequential constant factors in this discussion). This notation is similar to that of [34] . Integrations and the insertion of suitable delta functions are understood, and can be illustrated by comparing the short-hand expressions above with the full equations given earlier. It will prove convenient to define\n\nA 1 = Y 12 B 2 + Y 123 B 2 B 3 + Y 1234 B 2 B 3 B 4 + • • • , (3.33) where Y 12 = δ 12 , Y 123 = 1 + (23) , Y 1234 = 1 + 3 + (23)(34) , (3.34) and generally Y 12...n = 1 + 3 + 4 + . . . (n -1) + (23)(\n\nK ij = k 2 i + k 2 j + k i k j , k i := k i z . (3.36)\n\nWe will use the expression Y •12...n in the following, where the dot in the first placemark in the Y means that one substitutes in that place the negative of the sum of the other momenta. Then the result which we have proved above for the four point amplitude V 1234 can be expressed as (3.37) 15 It is perhaps interesting to remark that the proof would involve converting the double sum in (3.31) to the quadruple sum in (3.29)-a state of affairs which has appeared before in a rather different context [20].\n\nV 1234 =K 43 Y •4 Y •123 + K 14 Y •1 Y •234 + K 21 Y •2 Y •341 + K 32 Y •3 Y •412 + K 31 Y •23 Y •41 + K 24 Y •12 Y •34 ,\n\nor very simply\n\nV 1234 = 1≤i<j≤4 K ij Y • j+1...i Y • i+1...j .\n\n(3.38) It is clear that the general conjecture that all the n-point all plus amplitudes are generated from the two-point counterterm (3.3) translates into the proposal that the n-point all-plus amplitude V 12...n is given by\n\nV 12...n = 1≤i<j≤n K ij Y • j+1...i Y • i+1...j , (3.39)\n\nLet us now show that the expression on the right-hand side of (3.39) has precisely the same soft and collinear limits as the known amplitude on the left-hand side."
},
{
"section_type": "OTHER",
"section_title": "Collinear limits",
"text": "Under the collinear limit\n\np i → zP , p i+1 → (1 -z)P , P 2 → 0 , (3.40)\n\nthe n-point amplitude V 12...n behaves as\n\nV 12...n → 1 z(1 -z) i + (i i + 1) V 12...i i+2...n , (3.41)\n\nwhere we relabel P → p i after the limit is taken (the i + and (i i + 1) factors involve momenta rather than spinors, which is why the z-dependent factor is 1/z(1z), rather than the conventional 1/ z(1z)).\n\nConsider the behaviour of the right-hand side of (3.39) under the limit (3.40). The first point is that if the indices i, i + 1 lie on different Y's, then there are no poles generated in this collinear limit. This is clear from the explicit expressions for the Y's in (3.35). Thus we may ignore any terms of this type. It is then immediate from the explicit forms of the Y's that (3.42) for any i = 2, . . . s -1, with s ≤ n (the first index in Y never contributes in a collinear limit, as one can see from the conjecture (3.39)). Thus we see that the Y expressions have the right sort of collinear behaviour. It is straightforward to see that the K coefficients in (3.39) also get relabelled correctly in the collinear limit; they are not explicitly involved as they refer to pairs of momenta attached to different Y fields, and as we saw, these do not contribute.\n\nY 12...s → 1 z(1 -z) i + (i i + 1) Y 12...i i+2...s ,\n\nIt is then immediate to see that the summation over the products of Y's in (3.39) reduces correctly in the collinear limit to the required summation over products of Y's with one fewer 25 leg in total. Hence the proposal (3.39) for the amplitude has precisely the same collinear limits as the physical amplitude."
},
{
"section_type": "OTHER",
"section_title": "Soft limits",
"text": "We also find that there is a simple derivation of the soft limits of the expression in (3.39). In the soft limit p j → 0 , (3.43) the n-point amplitude V 12...n behaves as\n\nV 12...n → S(j) V 12...j-1 j+1...n , (3.44)\n\nwhere we assume cyclic ordering as usual, so that, for example, p n+1 = p 1 . The soft function S(j) is given in terms of the momentum brackets by\n\nS(j) = j + (j -1 j + 1) (j -1 j) (j j + 1) . (3.45)\n\nThe Y functions have a simple behaviour under soft limits. One has immediately that in the soft limit p j → 0, Y 12...s → S(j) Y 12...j-1 j+1...s , (3.46) for j = 3, . . . s -1 (with s ≤ n). For the soft limits corresponding to the case missing in the above, we need the results\n\nY •s+1...j = Y •s+1...j-1 (j -1) + (j -1 j) , Y •j...s = Y •j+1...s (j + 1) + (j j + 1) , (3.47)\n\nwhich follow from the definitions of the Y's, and\n\n(j + 1) + (j j + 1) + (j -1) + (j -1 j) = j + (j -1 j + 1) (j -1 j) (j j + 1) = S(j) , (3.48) which follows from the cyclic identity i + (jk) + j + (ki) + k + (ij) = 0. Finally, from\n\nrelabelling the K's we have in the soft limit that K sj → K sj-1 . Then it follows that in the soft limit\n\nK sj Y •s+1...j Y •j+1...s + K sj-1 Y •s+1...j-1 Y •j...s → S(j)K sj-1 Y •s+1...j-1 Y •j+1...s , (3.49) as required.\n\nAgain, it is then easy to see that the summation over the products of Y's in (3.39) reduces correctly in the soft limit to the required summation over products of Y's with one fewer leg in total. Hence the proposal (3.39) for the amplitude has precisely the same soft limits as the physical amplitude."
},
{
"section_type": "DISCUSSION",
"section_title": "4 Discussion",
"text": "Whilst new, twistor-inspired methods for calculating amplitudes in gauge theory have led to much progress, the lack of a systematic action-based formulation which incorporates these new ideas has been an impediment to further developments. MHV diagrams have the two advantages of being closely allied to the twistor picture, as well as providing an explicit realisation of the dispersion and phase space integrals fundamental to unitaritybased methods. However, without an action formalism, standard MHV methods have so far been mainly restricted to massless theories at one-loop level, and to the cut-constructible parts of amplitudes.\n\nThe advent of a classical MHV Lagrangian for gauge theory, derived from lightcone YM theory [32, 33, 34] , provides the basis for transcending these limitations. In order for this to be realised, it is necessary to describe the quantum MHV theory. What we have done in this paper is to investigate this quantum theory. Using the regularisation methods of [39, 40, 41] , we have provided arguments that the simplest one-loop counterterm in the quantum MHV theory -a two point vertex -provides an extraordinarily concise generating function for the infinite sequence of one-loop, all-plus helicity amplitudes in YM theory. We showed this by explicit calculation for the four-point case, and then proved that the soft and collinear limits of the conjectured n-point amplitude precisely matched those of the correct answer.\n\nWe would like to emphasise that the simplicity of our approach -which reduced the calculations of the loop amplitudes we considered to tree-level algebraic manipulationsis largely due to the four-dimensional nature of the regularisation scheme we employed. By staying in four dimensions, we preserve the appealing features of the inherently fourdimensional field redefinition of [32, 33] .\n\nBased upon this result, it is very natural to conjecture that the full quantum YM theory is correctly described by this quantum MHV Lagrangian. The correct ingredients appear to be present. For example, in the approach of [39, 40, 41] there arise one-loop counterterms with helicities (++), (+ + -), (--), (--+). We studied the (++) counterterm in this paper, arguing that when expressed in the (B, B) variables this generates the full set of allplus amplitudes. Transforming the (+ + -) counterterm to (B, B) variables will generate an infinite sequence of single-minus vertices. There will be other contributions to single-minus vertices from combinations of all-plus vertices and MHV vertices. It would be surprising if the combined contributions of these did not lead to the correct YM single-minus expressions. Certainly all of these have the correct powers of spinor brackets for this to be the case.\n\nTransforming the (--) and (--+) counterterms to (B, B) variables will lead to new contributions to MHV vertices 16 . The MHV vertices from the classical MHV Lagrangian only generate the cut-constructible parts of YM loop amplitudes, such as the one-loop MHV 16 In the MHV case there are additional counterterms noted in [41] which may also need to be taken into account in future discussions.\n\namplitude. These new contributions might be expected to lead to the missing, rational parts. This would also potentially explain why in [57] the combination of all-plus vertices with MHV tree vertices did not yield the correct single-minus amplitudes -these additional MHV contributions are missing.\n\nFurther evidence for the conjecture that the quantum MHV Lagrangian is equivalent to quantum YM theory would be welcome. One could start with seeking explicit proofs of the above proposals. One can also investigate beyond massless one-loop gauge theory -an advantage of the Lagrangian approach is that the inclusion of masses, and of fermions and scalars, is in principle clear. There are other issues raised by this work. It is plausible that the potential quantum versions of the twistor space formulations of gauge theory [58, 59, 60] are most likely to be allied to the quantum theory discussed here -one simple reason for believing this is that the regularisation employed here keeps one in four dimensions. Perhaps there are simple twistor space analogues of the counterterms discussed above.\n\nFinally, although for our purposes the lightcone worldsheet approach to perturbative gauge theory provided simply the motivation for a particular choice of regularisation scheme, we believe that it would be fruitful to further explore possible connections between that framework and the twistor string programme.\n\nAddendum: We would like to thank Paul Mansfield and Tim Morris for having informed us that they have recently been pursuing research related to that presented in this paper. Their work, which is complementary to ours in that it employs dimensional regularisation, has now appeared in [61] ."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgements",
"text": "It is a pleasure to thank Paul Heslop, Gregory Korchemsky, Paul Mansfield, Tim Morris and Adele Nasti for discussions. We would like to thank PPARC for support under the Rolling Grant PP/D507323/1 and the Special Programme Grant PP/C50426X/1. The work of GT is supported by an EPSRC Advanced Fellowship EP/C544242/1 and by an EPSRC Standard Research Grant EP/C544250/1."
},
{
"section_type": "OTHER",
"section_title": "A Notation",
"text": "Lightcone conventions Here we summarise our lightcone conventions. We start off by introducing lightcone coordinates\n\nx ± := x 0 ± x 3 √ 2 , x z := x 1 + ix 2 √ 2 , x z := x 1 -ix 2 √ 2 . (A.1)\n\nWe also have x + = x -, x z = -x z , and so on. The scalar product between two vectors A and B is written as\n\nA • B := A + B -+ A -B + -A z B z -A z B z . (A.2)\n\nWe choose x -as our lightcone time coordinate, therefore the lightcone gauge used in this paper is defined by A -= 0 . (A.3) This condition can be written as η • A = 0, where η is a constant null vector, chosen to have components η := (1/ √ 2, 0, 0, 1/ √ 2) (hence\n\nη -= 1, η + = η z = η z = 0).\n\nTo any four-vector p we associate the bispinor p a ȧ defined by\n\np a ȧ := √ 2 p --p z -p z p + . (A.4)\n\nWe also define holomorphic and anti-holomorphic spinors as\n\nλ a := 2 1 4 √ p + -p z p + , λȧ := 2 1 4 √ p + -p z p + , (A.5)\n\nfrom which it follows that\n\nλ a λȧ := √ 2 pzpz p + -p z -p z p + . (A.6)\n\nThis is of course consistent with the on-shell condition p -= p z p z /p + . Furthermore, comparing (A.4) and (A.6) and choosing η as specified earlier, we see that a generic off-shell vector p can be decomposed as\n\np = λ λ + zη , (A.7) where z = p -p + -p z p z p + η - = p 2 2(p • η) . (A.8) (A.7\n\n) and (A.8) are the familiar decompositions of off-shell vectors in the MHV literature [62, 17, 63, 15] .\n\nThe off-shell holomorphic spinor product is defined as:\n\nij = √ 2 p i + p j z -p j + p i z p i + p j + , (A.9) 29\n\nwhereas for the antiholomorphic spinors we define\n\n[ij] = √ 2 p i + p j z -p j + p i z p i + p j + . (A.10)\n\nIn these conventions, one finds\n\n2(p i • p j ) = i j [i j] + p j + p i + (p i ) 2 + p i + p j + (p j ) 2 , (A.11)\n\nor, in the case where p i and p j are on shell, 2(p i • p j ) = i j [i j]. In the standard QCD literature conventions it is customary to define 2(p i • p j ) = i j [j i]; this can be obtained by simply re-defining the inner product of two anti-holomorphic spinors, [i j], to be the negative of the right hand side of (A.10)."
},
{
"section_type": "OTHER",
"section_title": "Useful identities",
"text": "The form (A.9) is very convenient for deriving identities for ij that also involve the p + components. For instance, one has:\n\np i + jk + p j + ki + p k + ij = √ 2 p i + (p j + p k z -p k + p j z ) p i + p j + p k + + √ 2 p j + (p k + p i z -p i + p k z ) p i + p j + p k + + √ 2 p k + (p i + p j z -p j + p i z ) p i + p j + p k + = 0 . (A.12)\n\nIt is also easy to see how to apply momentum conservation, take say ij , and substitute p j =k =j p k (for each component). (A.13) Then we have\n\np j + ij = √ 2 p i + (-k =j p k z ) + ( k =j p k + )p i z p i + = - √ 2 k =j p k + p i + p k z -p k + p i z p i + p k + = k =j p k + ki .\n\n(A.14) We have also used the momentum bracket notation from [34]\n\n(ij) = p i + p j z -p j + p i z , {ij} = p i + p j z -p j + p i z . (A.15)"
},
{
"section_type": "OTHER",
"section_title": "Lightcone Yang-Mills action",
"text": "Here we give the form of the lightcone Yang-Mills action that we use in this paper. As discussed in more detail in [35] , starting from the YM Lagrangian -(1/4) trF 2 , imposing the 30 lightcone gauge (A.3), and integrating out the A + component which appears quadratically, the final lightcone theory contains only the two physical components A z and A z [64, 65, 66] , which we associate with positive and negative helicity respectively. The Lagrangian takes the simple form (2.1)\n\nL YM = L +-+ L ++-+ L --+ + L ++--, (A.16) with L +-= -2 tr{A z (∂ + ∂ --∂ z ∂ z )A z } , L ++-= 2ig tr{[A z , ∂ + A z ](∂ + ) -1 (∂ z A z )} , L --+ = 2ig tr{[A z , ∂ + A z ](∂ + ) -1 (∂ z A z )} , L ++--= -2g 2 tr{[A z, ∂ + A z ](∂ + ) -2 [A z , ∂ + A z ]} .\n\n(A.17) Note that, in agreement with CQT, we have used the normalisation tr{T a T b } = δ ab . In order to convert to the usual conventions for Yang-Mills theory, we therefore need to rescale g → g/ √ 2."
},
{
"section_type": "OTHER",
"section_title": "Relation to the notation of CQT",
"text": "To compare our notation to that of [39, 40, 41] , note that we employ outgoing momenta instead of incoming, therefore the all-plus amplitudes in these works would be all-minus from our perspective, and should thus be conjugated when comparing. Also, our time evolution coordinate is taken to be x -rather that x + , which (among other changes) implies that p + of CQT becomes p + . Our metric is also taken to have opposite signature to that in CQT. Finally, CQT define momentum brackets K ∧ ij and K ∨ ij , which are just our (ij) and {ij} brackets respectively."
},
{
"section_type": "OTHER",
"section_title": "B Details on the four-point calculation",
"text": "In this appendix we prove two results that were used in section 3 , namely equations (3.11) and (3.15) . To make the expressions more compact, instead of momentum brackets we use the following notation:\n\nf ij = - (ij) p i + p j + = p i z p i + - p j z p j + . (B.1)\n\nThe f ij variables satisfy the simple relation:\n\nf ij = f ik + f kj , (B.2)\n\nwhile momentum conservation is applied as\n\np i + f ij = - p k + f kj .\n\n(B.3) Also, to minimise clutter, in this appendix we use the notation q i := p i + ."
},
{
"section_type": "OTHER",
"section_title": "Proof of the quadratic identity",
"text": "In order to show (3.11), it is convenient to divide out by the p 1 + p 2 + p 3 + p 4 + factor (which is there anyway in (3.10)) in order to bring it to the form\n\nq 2 4 f 34 f 41 + q 2 1 f 12 f 41 + q 2 2 f 12 f 23 + q 2 3 f 23 f 34 -(q 2 + q 3 )(q 1 + q 4 )f 12 f 34 -(q 3 + q 4 )(q 2 + q 1 )f 23 f 41 = 0 , (B.4)\n\nExpanding out the two last terms in (B.4) as\n\n-(q 1 q 3 + q 2 q 4 )(f 12 f 34 + f 23 f 41 ) -(q 1 q 2 + q 3 q 4 )f 12 f 34 -(q 2 q 3 + q 4 q 1 )f 23 f 41 , (B.5)\n\nwe apply momentum conservation on each of the four components of the first term of (B.5), in the following way:\n\n-q 1 q 3 f 12 f 34 = q 1 f 12 (q 1 f 14 + q 2 f 24 ) = -q 2 1 f 12 f 41 + q 1 q 2 f 12 f 24 , -q 1 q 3 f 23 f 41 = q 3 f 23 (q 2 f 42 + q 3 f 43 ) = -q 2 3 f 23 f 34 + q 2 q 3 f 23 f 42 , -q 2 q 4 f 12 f 34 = q 4 (q 3 f 13 + q 4 f 14 )f 34 = -q 2 4 f 34 f 41 + q 3 q 4 f 13 f 34 , -q 2 q 4 f 23 f 41 = q 2 f 23 (q 2 f 21 + q 3 f 31 ) = -q 2 2 f 12 f 23 + q 2 q 3 f 31 f 23 .\n\n(B.6) Clearly these transformations have been chosen to cancel the first four terms in (B.4). Collecting the remaining terms, we obtain\n\nq 1 q 2 f 12 (f 24 -f 34 ) + q 2 q 3 f 23 (f 42 + f 31 -f 41 ) + q 3 q 4 f 34 (f 13 -f 12 ) -q 1 q 4 f 23 f 41 = q 1 q 2 f 12 f 23 + q 2 q 3 f 23 f 32 + q 3 q 4 f 34 f 23 + q 1 q 4 f 23 f 14 = f 23 [q 2 (q 1 f 12 + q 3 f 32 ) + q 4 (q 3 f 34 + q 1 f 14 )] = f 23 [-q 2 (q 4 f 42 ) -q 4 (q 2 f 24 )] = 0 (B.7)\n\nthus showing (3.11)."
},
{
"section_type": "OTHER",
"section_title": "Proof of the linear identity",
"text": "We will now outline the proof ot the linear (in region momenta) identity (3.15). Converting it to the notation used in the appendix, and performing simple manipulations, we 32 find (suppressing the overall 3/8 factor):\n\nX = q 2 4 ((p 3 + p4 ) + (p 2 + p3 ))f 34 f 41 + q 2 1 (-(p 2 + p3 ) + (p 3 + p4 ))f 12 f 41 + q 2 2 (-(p 3 + p4 ) -(p 2 + p3 ))f 12 f 23 + q 2 3 (+(p 2 + p3 ) -(p 3 + p4 ))f 23 f 34 - 1 2 (q 2 + q 3 )(q 1 + q 4 )[(p 3 -p2 ) + (p 1 -p4 )]f 12 f 34 - 1 2 (q 3 + q 4 )(q 1 + q 2 )[(p 4 -p3 ) + (p 2 -p1 )]f 23 f 41 = (p 3 -p1 )(q 2 4 f 34 f 41 -q 2 2 f 12 f 23 ) + (p 4 -p2 )(q 2 1 f 12 f 41 -q 2 3 f 23 f 34 ) -(q 2 + q 3 )(q 1 + q 4 )(p 3 + p1 )f 12 f 34 -(q 3 + q 4 )(q 1 + q 2 )(p 2 + p4 )f 23 f 41 = (p 3 -p1 )(q 2 4 f 34 f 41 -q 2 2 f 12 f 23 ) + (p 4 -p2 )(q 2 1 f 12 f 41 -q 2 3 f 23 f 34 ) -(p 1 + p3 )q 2 q 4 (f 12 f 34 -f 23 f 41 ) + (p 2 + p4 )q 1 q 3 (f 12 f 34 -f 23 f 41 ) -(p 1 + p3 )(q 1 q 2 + q 3 q 4 )f 12 f 34 + (p 1 + p3 )(q 2 q 3 + q 4 q 1 )f 23 f 41 .\n\n(B.8) Similarly to the previous case, we will rewrite the second line in the final expression in such a way that we completely cancel all the terms in the first line. To do that we use\n\n-(p 1 + p3 )q 2 q 4 (f 12 f 34 -f 23 f 41 ) =(p 3 -p1 )(q 2 2 f 12 f 23 -q 2 4 f 34 f 41 )+ + q 1 q 2 p1 f 12 f 31 -q 4 q 1 p1 f 41 f 13 + + q 3 q 4 p3 f 34 f 13 -q 2 q 3 p3 f 23 f 31 (B.9) and (p 2 + p4 )q 1 q 3 (f 12 f 34 -f 23 f 41 ) =(p 4 -p2 )(q 2 3 f 23 f 34 -q 2 1 f 12 f 41 )+ + q 2 q 3 p2 f 23 f 42 -q 1 q 2 p2 f 12 f 24 + + q 4 q 1 p4 f 41 f 24 -q 3 q 4 p4 f 34 f 42 .\n\n(B.10) What remains after substituting these is\n\nX = p1 q 1 f 31 (q 2 f 12 + q 4 f 41 ) + q 3 p3 f 13 (q 4 f 34 + q 2 f 23 ) + p2 q 2 f 42 (q 3 f 23 + q 1 f 12 ) + q 4 p4 f 24 (q 1 f 41 + q 3 f 34 ) -(p 1 + p3 )(q 1 q 2 + q 3 q 4 )f 12 f 34 + (p 1 + p3 )(q 2 q 3 + q 4 q 1 )f 23 f 41 = p1 q 1 q 2 f 12 f 41 + p3 q 3 q 4 f 34 f 23 + p1 q 4 q 1 f 41 f 21 + p3 q 2 q 3 f 23 f 43 + p2 q 2 f 42 (q 3 f 23 + q 1 f 12 ) + q 4 p4 f 24 (q 1 f 41 + q 3 f 34 ) -(p 1 q 3 q 4 + p3 q 1 q 2 )f 12 f 34 + (p 1 q 2 q 3 + p3 q 4 q 1 )f 23 f 41 .\n\n(B.11) Now we collect various terms together to rewrite X as\n\nX = p1 q 2 f 41 (q 1 f 12 + q 3 f 23 ) + p3 q 4 f 23 (q 3 f 34 + q 1 f 41 ) + p1 q 4 f 21 (q 1 f 41 + q 3 f 34 ) + p3 q 2 f 43 (q 3 f 23 + q 1 f 12 ) + p2 q 2 f 42 (q 3 f 23 + q 1 f 12 ) + p4 q 4 f 24 (q 1 f 41 + q 3 f 34 ) = p1 q 2 f 41 (2q 3 f 23 -q 4 f 42 ) + p3 q 4 f 23 (2q 1 f 41 -q 2 f 24 ) + p1 q 4 f 21 (2q 1 f 41 -q 4 f 42 ) + p3 q 2 f 43 (2q 3 f 23 -q 4 f 42 ) + p2 q 2 f 42 (2q 3 f 23 -q 4 f 42 ) + p4 q 4 f 24 (2q 1 f 41 -q 2 f 24 ) = 2[q 2 q 3 f 23 (p 1 f 41 + p3 f 43 + p2 f 42 ) + q 4 q 1 f 41 (p 3 f 23 + p1 f 21 + p4 f 24 )] + (p 1 + p2 + p3 + p4 )q 2 q 4 f 24 f 42 .\n\n(B.12) 33 Clearly the term on the last line vanishes by momentum conservation. We now restore all labels to write the final result as\n\nX =2 (32)[f 4 (p 1 z + p 2 z + p 3 z) -p 1 z f 1 -p 2 z f 2 -p 3 z f 3 ]+ + 2 (14)[f 2 (p 1 z + p 3 z + p 4 z ) -p 3 z f 3 -p 1 z f 1 -p 4 z f 4 ] , (B.13)\n\nwhere we used that\n\nq 2 q 3 f 23 = p 2 + p 3 + (p 2 z /q 2 + -p 3 z /p 3 + ) = p 3 + p 2 z -p 2 + p 3 z = ( 32\n\n) (and similarly for (14)), and where f i = p i z /p i + . Using momentum conservation on both terms, we rewrite them as\n\nX = -2[(32) + (14)] p 1 zp 1 z p 1 + + p 2 z p 2 z p 2 + + p 3 z p 3 z p 3 + + p 4 z p 4 z p 4 + . (B.14)\n\nFor each momentum we have that p 2 = 2(p + p -p z p z), therefore we can rewrite the above as\n\nX = +[(32) + (14)] (p 1 ) 2 p 1 + + (p 2 ) 2 p 2 + + (p 3 ) 2 p 3 + + (p 4 ) 2 p 4 + + 2(p 1 -+ p 2 -+ p 3 -+ p 4 -) . (B.15)\n\nThe p -term vanishes, hence, noticing also that (32) + (14) = -1 2 ((12) + (23) + (34) + (41)), we conclude that\n\nX = - 1 2 [(12) + (23) + (34) + (41)] 4 i=1 (p i ) 2 p i + . (B.16)\n\nOff-shell terms in the four-point case\n\nFor completeness, we also give the form of the off-shell terms that arose in the manipulations leading to (3.26).\n\nUsing the notation\n\nP ij = ( (p i ) 2 p i + + (p j ) 2 p j +\n\n) they are : 3.16 ), should be added to (3.26) in order to recover a fully off-shell four-point vertex.\n\nf (p 2 ) = 1 4 12 • • • 41 -P 13 (p 1 + p2 )(41) -P 13 (p 2 + p3 )(12) + P 24 (p 2 + p3 )( 42\n\n) + 1 p 1 + P 12 [(p 2 + + p 3 + )(2p 1 + p2 -p3 ) -p 3 + (p 1 + p2 ) -p 1 + (p 2 + p3 )] ( 13\n\n) + P 12 p 3 + p 1 + p 2 + [p 2 + (p 1 + p2 ) -p 4 + (p 2 + p3 )](12) -2P 13 1 p 1 + {31}(41) . (B.17) This expression, together with V (4) k in (\n\nReferences [1] E. Witten, Perturbative gauge theory as a string theory in twistor space, Comm. Math. Phys. 252 (2004) 189, hep-th/0312171. [2] F. Cachazo and P. Svrček, Lectures on twistor strings and perturbative Yang-Mills theory, PoS RTN2005 (2005) 004, hep-th/0504194. [3] R. Britto, F. Cachazo, and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499, hep-th/0412308. [4] R. Britto, F. Cachazo, B. Feng, and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602, hep-th/0501052. [5] R. Britto, F. Cachazo, and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275, hep-th/0412103. [6] A. Brandhuber and G. Travaglini, Quantum MHV diagrams, hep-th/0609011. [7] F. Cachazo, P. Svrček, and E. Witten, MHV vertices and tree amplitudes in gauge theory, J. High Energy Phys. 0409 (2004) 006, hep-th/0403047. [8] G. Georgiou and V. V. Khoze, Tree amplitudes in gauge theory as scalar MHV diagrams, J. High Energy Phys. 0405 (2004) 070, hep-th/0404072. [9] C.-J. Zhu, The googly amplitudes in gauge theory, J. High Energy Phys. 0404 (2004) 032, hep-th/0403115.\n\n[10] J.-B. Wu and C.-J. Zhu, MHV vertices and scattering amplitudes in gauge theory, J. High Energy Phys. 0407 (2004) 032, hep-th/0406085.\n\n[11] J.-B. Wu and C.-J. Zhu, MHV vertices and fermionic scattering amplitudes in gauge theory with quarks and gluinos, J. High Energy Phys. 0409 (2004) 063, hep-th/0406146. [12] G. Georgiou, E. W. N. Glover, and V. V. Khoze, Non-MHV tree amplitudes in gauge theory, J. High Energy Phys. 0407 (2004) 048, hep-th/0407027. [13] L. J. Dixon, E. W. N. Glover, and V. V. Khoze, MHV rules for Higgs plus multi-gluon amplitudes, J. High Energy Phys. 0412 (2004) 015, hep-th/0411092. [14] Z. Bern, D. Forde, D. A. Kosower, and P. Mastrolia, Twistor-inspired construction of electroweak vector boson currents, Phys. Rev. D 72 (2005) 025006, hep-ph/0412167. [15] J. Bedford, A. Brandhuber, B. Spence, and G. Travaglini, Non-supersymmetric loop amplitudes and MHV vertices, Nucl. Phys. B 712 (2005) 59, hep-th/0412108. [16] Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. 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}
] | |
arxiv:0704.0247 | 0704.0247 | 1 | 10.1088/1126-6708/2007/07/046 | bf8e7c8ad3d0d7319400cec91eff26db7fcb257f5a326c8e6755831de4e3278a | Geometry of four-dimensional Killing spinors | The supersymmetric solutions of N=2, D=4 minimal ungauged and gauged
supergravity are classified according to the fraction of preserved
supersymmetry using spinorial geometry techniques. Subject to a reasonable
assumption in the 1/2-supersymmetric time-like case of the gauged theory, we
derive the complete form of all supersymmetric solutions. This includes a
number of new 1/4- and 1/2-supersymmetric possibilities, like gravitational
waves on bubbles of nothing in AdS_4. | [
"S. L. Cacciatori",
"M. M. Caldarelli",
"D. Klemm",
"D. S. Mansi",
"D. Roest"
] | [
"hep-th",
"gr-qc"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | The supersymmetric solutions of N = 2, D = 4 minimal ungauged and gauged supergravity are classified according to the fraction of preserved supersymmetry using spinorial geometry techniques. Subject to a reasonable assumption in the 1/2supersymmetric time-like case of the gauged theory, we derive the complete form of all supersymmetric solutions. This includes a number of new 1/4-and 1/2-supersymmetric possibilities, like gravitational waves on bubbles of nothing in AdS 4 .
Throughout the history of string and M-theory an important part in many developments in the subject has been played by supersymmetric solutions of supergravity, i.e. by backgrounds which admit a number of Killing spinors ǫ which are parallel with respect to the supercovariant derivative foot_0 : D µ ǫ = 0. Due to their ubiquitous role it has long been realised that it would be advantageous to have classifications of all supersymmetric solutions of a given theory.
For purely gravitational backgrounds the supersymmetric possibilities follow from the Berger classification of the possible Riemannian holonomies [1] (see [2, 3] for an extension to the Lorentzian case). However, in the presence of additional force fields (carried by e. g. scalars, gauge potentials or a cosmological constant) it has proven very difficult to obtain knowledge of all supersymmetric possibilities.
The reason for the complication in the presence of additional fields lies in the holonomy of the supercurvature R µν = D [µ D ν] . For purely gravitational backgrounds the holonomy of the supercurvature is generically given by H = Spin(d -1, 1) in d dimensions, and hence coincides with the Lorentz group. In such cases the Lorentz gauge freedom allows one to choose constant Killing spinors. Another simplification is that if there is one Killing spinor with a specific stability subgroup, i.e. it is invariant under some Lorentz subgroup, all other spinors with the same stability subgroup are Killing as well.
For more general solutions including fields other than gravity, the holonomy is generically extended to a larger group H ⊃ Spin(d -1, 1). For example, in the present paper we consider gauged minimal four-dimensional N = 2 supergravity, which has H = GL(4,C) [4] . In such cases one cannot choose constant Killing spinors nor are all spinors with the same stability subgroup automatically Killing. For these reasons the classification of the backgrounds that allow for Killing spinors is more convoluted, or richer, in such cases. For a long time the only classification available was in ungauged minimal four-dimensional N = 2 supergravity [5, 6] , which has H = SL (2,H) .
A new impulse was given to the subject with the introduction of G-structures and the method of spinor bilinears to solve the Killing spinor equations [7] . In this approach, space-time forms are constructed as bilinears from a Killing spinor and one analyses the constraints that these forms imply for the background. Using this framework, a number of complete classifications [8] [9] [10] and many partial results (see e.g. [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] for an incomplete list) have been obtained. By complete we mean that the most general solutions for all possible fractions of supersymmetry have been obtained, while for partial classifications this is only available for some fractions. Note that the complete classifications mentioned above involve theories with eight supercharges and H = SL(2,H), and allow for either half-or maximally supersymmetric solutions.
An approach which exploits the linearity of the Killing spinors has been proposed [22] under the name of spinorial geometry. Its basic ingredients are an explicit oscillator basis for the spinors in terms of forms and the use of the gauge symmetry to transform them to a preferred representative of their orbit. In this way one can construct a linear system for the background fields from any (set of) Killing spinor(s) [23] . This method has proven fruitful in e.g. the challenging case of IIB supergravity [24] [25] [26] . In addition, it has been adjusted to impose 'near-maximal' supersymmetry and thus has been used to rule out certain large fractions of supersymmetry [27] [28] [29] [30] . Finally, a complete classification for type I supergravity in ten dimensions has been obtained [32] .
In the present paper we would like to address the classification of supersymmetric solutions in four-dimensional minimal N = 2 supergravity. As will also be reviewed in section 2, the ungauged case has been classified completely [5, 6] . For the gauged case, the discussion of 1/4 supersymmetry splits up in a time-like and a light-like class (depending on the causal nature of the Killing vector associated to the Killing spinor). The time-like class is completely specified by a single complex function depending on three spatial coordinates b = b(z, w, w), subject to a second-order differential equation which can not be solved in general [13] . The light-like class can be given in all generality, and in addition its restriction to 1/2-BPS solutions has been derived [16] . Furthermore, there are no backgrounds with 3/4 supersymmetry [29] and AdS 4 is the unique possibility with maximal supersymmetry. Therefore the remaining open question concerns half-supersymmetric backgrounds in the gauged theory 2 .
In the following, we will first re-analyse the 1/4-supersymmetric backgrounds using the method of spinorial geometry, and in fact find an additional possibility in the light-like case: a half supersymmetric bubble of nothing in AdS 4 and its Petrov type II generalization, a new 1/4 BPS configuration that has the interpretation of gravitational waves propagating on the bubble of nothing. This completes the analysis of the null class in all its generality. Then we will derive the constraints for halfsupersymmetric backgrounds for the timelike class. Subject to a single assumption on the time-dependence of the second Killing spinor these will be solved in general, up to a second order ordinary differential equation. The assumption will be justified by solving the full set of conditions in a number of examples which illustrate the possible spatial dependence of b. All these cases turn out to have time-dependence of the assumed form. The different examples are:
• the b = b(z) family of solutions, comprising part of the Reissner-Nordström-Taub-NUT-AdS 4 backgrounds,
• waves on the previous backgrounds with b = b(z, w),
• solutions with b imaginary and their P SL(2, R) transformed counterparts,
• solutions of the dimensionally reduced gravitational Chern-Simons model that can be embedded in the equations for a timelike Killing spinor [16] .
We determine when these backgrounds preserve 1/2 supersymmetry and provide the explicit Killing spinors. Moreover, in the subcases consisting of AdS 4 and AdS 2 ×H 2 , the action of the isometries of these backgrounds on the Killing spinors is given explicitly. The outline of this paper is as follows. In section 2, we discuss the orbits of Killing spinors and review the known classification results in the theory at hand. In section 3, we go through the complete classification of the null class. In section 4, we discuss the constraints for 1/4 and 1/2 supersymmetry in the timelike class. We derive the time-dependence of the second Killing spinor and solve the equations for the case of linear time-dependence (G 0 = 0). A number of examples of the 1/2 BPS timelike class are provided in section 5. Finally, in section 6 we present our conclusions and outlook. In appendix A we review our notation and conventions for spinors, while in appendix B the associated bilinear forms are given. Appendix C deals with the special case P ′ = 0, to be defined in section 4.4. Finally, in appendix D, we will give the details of the G 0 = 0 case.
2.1 Orbits of Dirac spinors under the gauge group In order to obtain the possible orbits of Spin (3, 1) in the space of Dirac spinors ∆ c , we first consider the most general positive chirality spinor foot_2 a1 + be 12 (a, b ∈ C) and determine its stability subgroup. This is done by solving the infinitesimal equation α cd Γ cd (a1 + be 12 ) = 0 .
(2.1)
First of all, notice that a1 + be 12 is in the same orbit as 1, which can be seen from e γΓ 13 e ψΓ 12 e δΓ 13 e hΓ 02 1 = e i(δ+γ) e h cos ψ 1 + e i(δ-γ) e h sin ψ e 12 .
This means that we can set a = 1, b = 0 in (2.1), which implies then α 02 = α 13 = 0, α 01 = -α 12 , α 03 = α 23 . The stability subgroup of 1 is thus generated by
X = Γ 01 -Γ 12 , Y = Γ 03 + Γ 23 . (2.2)
One easily verifies that X 2 = Y 2 = XY = 0, and thus exp(µX + νY ) = 1 + µX + νY , so that X, Y generate R 2 . Spinors of negative chirality are composed of odd forms, i.e. ae 1 + be 2 . One can show in a similar way that they are in the same orbit as e 1 , and the stability subgroup is again R 2 , with the above generators X, Y .
For definiteness and without loss of generality we will always assume that the first Killing spinor has a non-vanishing positive chirality component, and use (part of) the Lorentz symmetry to bring this to the form 1. Hence we can write a general spinor as 1 + ae 1 + be 2 . Now act with the stability subgroup of 1 to bring ae 1 + be 2 to a special form:
(
1 + µX + νY )(1 + ae 1 + be 2 ) = 1 + be 2 + [a + 2b(ν -iµ)]e 1 .
In the case b = 0 this spinor is invariant, so the representative is 1 + ae 1 , with isotropy group R 2 . If b = 0, one can bring the spinor to the form 1 + be 2 , with isotropy group I.
The representatives foot_3 together with the stability subgroups are summarized in table 1.
In the ungauged theory, we therefore can have the following G-invariant Killing spinors. The R 2 -invariant Killing spinors are spanned by 1 and e 1 and there can be up to four of these. The I-invariant Killing spinors are spanned by all four basis elements and there can be up to eight of these. In the first two case, the vector V a bilinear in the spinor ǫ is lightlike, whereas in the last case it is timelike, see table 1 . The existence of a globally defined Killing spinor ǫ, with isotropy group G ∈ Spin(3,1), gives rise to a G-structure. This means that we have an R 2 -structure in the null case and an identity structure in the timelike case.
In U(1) gauged supergravity, the local Spin(3,1) invariance is actually enhanced to Spin(3,1) × U(1). Thus, in order to obtain the stability subgroup, one determines the Lorentz transformations that leave a spinor invariant up to an arbitrary phase factor, which can then be gauged away using the additional U(1) symmetry. For the representative 1, one gets in this way an isotropy group generated by X, Y and Γ 13 obeying
[Γ 13 , X] = -2Y , [Γ 13 , Y ] = 2X , [X, Y ] = 0 ,
i. e. G ∼ = U(1)⋉R 2 . For ǫ = 1 + ae 1 with a = 0, the stability subgroup R 2 is not enhanced, whereas the I of the representative 1 + be 2 is promoted to U(1) generated by
Γ 13 = iΓ •• . The Lorentz transformation matrix a AB corresponding to Λ = exp(iψΓ •• ) ∈ U(1), with ΛΓ B Λ -1 = a A B Γ A , has nonvanishing components a +-= a -+ = 1 , a •• = e 2iψ , a •• = e -2iψ . (2.3)
Finally, notice that in U(1) gauged supergravity one can choose the function a in 1+ae 1 real and positive: Write a = R exp(2iδ), use
e δΓ 13 (1 + ae 1 ) = e iδ 1 + e -iδ ae 1 = e iδ (1 + Re 1 ) ,
and gauge away the phase factor exp(iδ) using the electromagnetic U (1) .
ǫ G ⊂ Spin(3,1) G ⊂ Spin(3,1) × U(1) V a = D(ǫ, Γ a ǫ) 1 R 2 U(1)⋉R 2 (1, 0, -1, 0) 1 + ae 1 R 2 R 2 (a ∈ R) (1 + |a| 2 , 0, -1 -|a| 2 , 0) 1 + be 2 I U(1) (1 + |b| 2 , 0, -1 + |b| 2 , 0)
Table 1 : The representatives ǫ of the orbits of Dirac spinors and their stability subgroups G under the gauge groups Spin(3,1) and Spin(3,1) × U(1) in the ungauged and gauged theories, respectively. The number of orbits is the same in both theories, the only difference lies in the stability subgroups and the fact that a is real in the gauged theory. In the last column we give the vectors constructed from the spinors.
In the gauged theory the classification of G-invariant spinors is therefore slightly more complicated. There can be at most two U(1)⋉R 2 -invariant Killing spinors, spanned by 1. The four R 2 -invariant spinors are spanned by 1 and e 1 . Then there are the U(1)-invariant spinors, spanned by 1 and e 2 . Finally, for generic enough Killing spinors, one does not fall in any of the above classes and the common stability subgroup is I. Note that in the gauged theory the presence of G-invariant Killing spinors will in general not lead to a G-structure on the manifold but to stronger conditions. The structure group is in fact reduced to the intersection of G with Spin(3,1), and hence is equal to the stability subgroup in the ungauged theory.
We will now consider the possible supersymmetric solutions to the equation D µ ǫ = 0 in various sectors of N = 2, D = 4 in terms of the stability subgroup G of the Killing spinors.
The supercovariant derivative of ungauged minimal N = 2 supergravity in four dimensions reads
D µ = ∂ µ + 1 4 ω ab µ Γ ab + i 4 F ab Γ ab Γ µ . (2.4)
As mentioned in the introduction, a first point to notice is that there is no complex conjugation on the Killing spinor. Therefore, the number of supersymmetries that are preserved is always even: if ǫ is Killing, then so is iǫ.
First consider purely gravitational solutions with F = 0. In this case the supercovariant connection truncates to the Levi-Civita connection and has Spin (3, 1) holonomy. This implies the following. If ǫ is Killing, then so are foot_4 Γ 3 * ǫ and Γ 012 * ǫ (where * denotes complex conjugation). Together, the operations i, Γ 3 * and Γ 012 * generate four linearly independent Killing spinors from any null spinor ǫ = 1 or ǫ = 1+ae 1 and eight from any time-like spinor ǫ = 1 + be 2 . This illustrates the general statement in the introduction: if the gauge group equals the holonomy, as in this case, then there is only one possible number of Killing spinors for every stability subgroup. Therefore there are only two classes of supersymmetric solutions, which are listed in table 2, and which consist of the gravitational wave and Minkowski space-time, respectively.
G = \ N = 4 8 R 2 √ × I × √
Table 2: Gravitational solutions with G-invariant Killing spinors in the ungauged theory.
Now let us also allow for fluxes F . The supercovariant connection no longer equals the Levi-Civita connection due to the flux term. In particular, this implies that Γ 012 * no longer commutes with D µ . However, this does still hold for the other operation: Γ 3 * ǫ is Killing provided ǫ is. The combined operations of i and Γ 3 * generate four linearly independent spinors from any null or time-like spinor. Thus the number of supersymmetries is always N = 4p, as illustrated in table 3 . Indeed the generalised holonomy of the supercovariant connection in the ungauged case is SL(2,H) [4] , consistent with the supersymmetries coming in quadruplets.
G = \ N = 4 8 R 2 √ × I √ √
Table 3: General solutions with G-invariant Killing spinors in the ungauged theory.
The half-supersymmetric solution have been classified by Tod [5] and consist of the plane wave and the Israel-Wilson-Perjes metric, respectively. The maximally supersymmetric solutions are AdS 2 × S 2 and its Penrose limits, the Hpp wave and Minkowski space-time [6] .
The supercovariant derivative of gauged minimal N = 2 supergravity in four dimensions reads
D µ = ∂ µ + 1 4 ω ab µ Γ ab -iℓ -1 A µ + 1 2 ℓ -1 Γ µ + i 4 F ab Γ ab Γ µ . (2.5)
Due to the gauging the structure of Γ-matrices is richer, but there still is no complex conjugation on the Killing spinor. Therefore, the number of supersymmetries that are preserved is always even: if ǫ is Killing, then so is iǫ. Again, we first consider the purely gravitational solutions. In this case the supercovariant derivative has SO (3, 2) holonomy. The operation Γ 012 * commutes with D µ and therefore generates additional Killing spinors. Together, the operations i and Γ 012 * generate four linearly independent Killing spinors from generic null or time-like spinors. The exception is the null spinor ǫ = 1 + e 1 , in which case ǫ and Γ 012 * are linearly dependent, and hence allows for two instead of four Killing spinors. The possibilities allowed for by this analysis of the supercovariant derivative can be found in table 4 .
However, although all these entries are allowed for by the spinor orbit structure and the crude analysis of the supercurvature above, not all of them have an actual field theoretic realisation in supergravity. In other words, there are no solutions to the Killing spinor equations for all of the above sets of Killing spinors. The lightlike cases were considered in [16] : The 1/4-BPS case is the Lobatchevski wave while imposing more supersymmetries leads to the maximally supersymmetric AdS 4 solution (with
G = \ N = 2 4 6 8 U(1) ⋉ R 2 × × × × R 2 √ • × × U(1) × • × × I × • • √
Table 4: Gravitational solutions with G-invariant Killing spinors in the gauged theory. Check marks indicate entries with actual solutions, while circles stand for allowed entries which are not realized.
). The N = 4 and G = R 2 entry is thus effectively empty. In particular, this implies that imposing a single Killing spinor 1 + ae 1 with a = 1 leads to AdS 4 . Also note that the N = 6 and G = 1 entry must be empty since any time-like spinor plus 1+e 1 leads to maximal supersymmetry, while all other Killing spinors come in groups of four. The only remaining entries are N = 4 and G = U(1) or G = I. Using the results of [13, 16] , it is straightforward to show that in these purely gravitational timelike cases the geometry is given by
ds 2 = - z 2 + n 2 ℓ 2 (dt -2n cosh θdφ) 2 + ℓ 2 dz 2 z 2 + n 2 + (z 2 + n 2 )(dθ 2 + sinh 2 θdφ 2 ) ,
where n = ±ℓ/2. But this is simply AdS 4 written as a line bundle over a threedimensional base manifold, so both N = 4 entries are empty as well. We conclude that there are no 1/2-supersymmetric gravitational solutions in the gauged theory, only the 1/4-supersymmetric Lobatchevski waves and maximally supersymmetric AdS 4 .
We now come to the general supersymmetric solutions in the gauged case. Due to the gauging and flux terms, neither Γ 012 * nor Γ 3 * commute with D µ . Therefore we have the cases as listed in table 5 . The supercovariant connection in the gauged case has generalized holonomy GL(4, C) [4] , again consistent with the supersymmetries coming in doublets.
The 1/4-BPS solutions with G = R 2 and G = U(1) were derived in [13] , and we will show there is no solution with G = U(1) ⋉ R 2 . In addition, it was shown in [16] that any additional supersymmetries in the null case are always timelike, i.e. end up in the N = 4 and G = 1 entry. Again, the N = 4 and G = R 2 entry is empty. It would be interesting to see if there is a nice explanation for this. In addition, the maximally supersymmetric case is always AdS 4 . Recently, it has been shown in [29] that the N = 6 and G = 1 entry is empty as well, because imposing three complex Killing spinors implies that the spacetime is AdS 4 and thus maximally supersymmetric.
G = \ N = 2 4 6 8 U(1) ⋉ R 2 • × × × R 2 √ • × × U(1) √ √ × × I × √ • √
Table 5: General solutions with G-invariant Killing spinors in the gauged theory. Check marks indicate entries with actual solutions, while circles stand for allowed entries which are not realized.
The most general 1/2-BPS solution in the timelike case remains an open issue and will be studied in this paper.
In minimal gauged supergravity theories with eight supercharges, the generalized holonomy group for vacua preserving N supersymmetries, where N = 0, 2, 4, 6, 8, is GL(
8-N 2 , C) ⋉ N 2 C
8-N 2 [4] . To see this, assume that there exists a Killing spinor ǫ 1 . By a local GL(4, C) transformation, ǫ 1 can be brought to the form ǫ 1 = (1, 0, 0, 0) T . This is annihilated by matrices of the form
A = 0 a T 0 A ,
that generate the affine group A(3, C) ∼ = GL(3, C) ⋉ C 3 . Now impose a second Killing spinor ǫ 2 = (ǫ 0 2 , ǫ 2 ) T . Acting with the stability subgroup of ǫ 1 yields
e A ǫ 2 = ǫ 0 2 + b T ǫ 2 e A ǫ 2
, where b T = a T A -1 (e A -1) .
We can choose A ∈ gl(3, C) such that e A ǫ 2 = (1, 0, 0) T , and b such that ǫ 0 2 + b T ǫ 2 = 0. This means that the stability subgroup of ǫ 1 can be used to bring ǫ 2 to the form ǫ 2 = (0, 1, 0, 0). The subgroup of A(3, C) that stabilizes also ǫ 2 consists of the matrices
1 0 b 2 b 3 0 1 B 12 B 13 0 0 B 22 B 23 0 0 B 32 B 33 ∈ GL(2, C) ⋉ 2C 2 .
Finally, imposing a third Killing spinor yields GL(1, C) ⋉ 3C as maximal generalized holonomy group, which is however not realized in N = 2, D = 4 minimal gauged supergravity [16, 29] . It would be interesting to better understand why such preons actually do not exist. In section 4.3, we explicitely compute the generalized holonomy group for N = 2, D = 4 minimal gauged supergravity in the case N = 2 and show that it is indeed contained in A(3, C), supporting thus the classification scheme of [4] .
In this section we will analyse the conditions coming from a single null Killing spinor. As we saw in section 2.1, there are two orbits of such spinors, one with representative ǫ = 1 and stability subgroup G = U(1)⋉R 2 and one with ǫ = 1 + ae 1 and G = R 2 .
Owing to local U(1) gauge invariance, it is always possible to choose the function a real and positive, so in the following we set a = e χ , χ ∈ R. The Killing spinor equations become
- i ℓ A + 1 2 Ω + e χ √ 2 1 ℓ + iφ E • -2iF +• E -= 0 , dχ + i ℓ A + 1 2 Ω + e -χ √ 2 1 ℓ -iφ E • + 2iF +• E -= 0 , ω -• + e χ √ 2 2iF -• E • + 1 ℓ -iφ E -= 0 , ω -• + e -χ √ 2 -2iF -• E • + 1 ℓ + iφ E -= 0 , (3.1)
where φ ≡ F +-+ F •• and Ω ≡ ω +-+ ω •• . The conditions for the special U(1)⋉R 2 -orbit with ǫ = 1 can be obtained as the singular limit χ → -∞ of the above equations. Note however that, in this limit, the second line implies the constraint ℓ -1 -iφ = 0, while the fourth line leads to ℓ -1 +iφ = 0. Clearly, for ℓ -1 = 0 this does not allow for a solution. Hence, in the gauged theory, there are no backgrounds with U(1)⋉R 2 -invariant Killing spinors.
The only null possibility is therefore given by the R 2 -invariant Killing spinor ǫ = 1 + e χ e 1 . We will now analyse the above conditions for the generic case with χ finite. In fact, we will furthermore assume it is positive. This does not constitute any loss of generality since one can flip the sign of χ by changing chirality (a spinor 1 + e χ e 1 with χ negative is gauge equivalent to a spinor e 1 + e χ1 with χ = -χ positive), and hence the resulting background will not depend on this sign.
From the last two equations one obtains the constraints
F -• = F -• = 0 , φ = - i ℓ tanh χ (3.2)
on the field strength, as well as
ω -• = ω -• = - 1 √ 2ℓ cosh χ E - (3.3)
for the spin connection. (3.2) implies F +-= 0 and F •• = -i ℓ tanh χ. The first two equations of (3.1) yield then
ω +-= 2e χ H 3 E -- 1 ℓ e 2χ cosh χ E 1 , ω •• = 2i sinh χH 1 E -+ i ℓ cosh 2χ cosh χ E 3 , A = -ℓ cosh χH 1 E --sinh χE 3 , dχ = -2 cosh χH 3 E -+ 2 ℓ sinh χE 1 , (3.4)
where
E 1 = (E • + E •)/ √ 2, iE 3 = (E • -E •)/ √
2, and we defined
F +• + F +• √ 2 = H 1 , F +• -F +• √ 2 = iH 3 .
In order to proceed, we distinguish two subcases, namely dχ = 0 and dχ = 0.
T -= dE -+ 2 ℓ E 1 ∧ E -, T + = dE + -E 1 ∧ ω +1 + E + ℓ + ω +3 ∧ E 3 , T 1 = dE 1 + E -∧ ω +1 + E + ℓ , T 3 = dE 3 + 1 ℓ E 1 ∧ E 3 -ω +3 ∧ E -. (3.5)
From T -= 0 one gets E -∧dE -= 0, so by Fröbenius' theorem there exist two functions η and u such that locally E -= ηdu .
Plugging this into T -= 0 yields
η d log η + 2 ℓ E 1 ∧ du = 0 ,
so that there exists a function ξ such that
E 1 = - ℓ 2η dη + ξdu .
The gauge field and its field strength can now be written as
A = -ℓηH 1 du , F = ℓ 2 H 1 dη ∧ du ,
and the Bianchi identity F = dA implies
dH 1 + 3 2 H 1 d log η ∧ du = 0 .
This means that H 1 η 3/2 can depend only on u,
H 1 η 3/2 = - ϕ ′ (u) ℓ ,
where the prefactor and the derivative were chosen in order to conform with the notation of [13] . Let us define a new coordinate x = -η -1/2 , so that
E 1 = ℓ x dx+ξdu, E -= x -2 du and A = -xϕ ′ (u)du . (3.6)
One can now use part of the residual gauge freedom, given by the stability subgroup R 2 of the null spinor 1 + ae 1 , in order to simplify E 1 . To this end, consider an R 2 transformation with group element
Λ = 1 + µX + νY ,
where X and Y are given in (2.2). Defining α = µ + iν, this can also be written as
Λ = 1 + αΓ +• + ᾱΓ +• . (3.7)
Given the ordering A, B = +, -, •, •, the Lorentz transformation matrix a AB corresponding to Λ ∈ R 2 ⊆ Spin(3,1) reads
a AB = 0 1 0 0 1 -4|α| 2 2 ᾱ 2α 0 -2 ᾱ 0 1 0 -2α 1 0 . ( 3.8)
The transformed vielbein
α E A = a A B E B is thus given by α E • = E • -2αE -, α E 1 = E 1 - √ 2 (α + ᾱ) E -, α E • = E • -2 ᾱE -, α E 3 = E 3 + √ 2i (α -ᾱ) E -, α E -= E -, α E + = E + + 2 ᾱE • + 2αE • -4|α| 2 E -. (3.9)
Choosing α + ᾱ = ξx foot_5 / √ 2, we can eliminate E 1 u , so one can set ξ = 0 without loss of generality. Note that this still leaves a residual gauge freedom associated to the imaginary part of α, which will be used below.
From dT 3 = 0 we get d(ω +3 /x) ∧ du = 0, and thus there exist two functions β, β such that ω +3 = -xdβ + βdu .
Plugging this into T 3 = 0 yields d(xE 3 + βdu) = 0, which is solved by
E 3 = - ℓ x dy + βdu , (3.10)
where y denotes some function that we shall use as a coordinate. Using the remaining gauge freedom (3.8) with Imα = -βx 2 /2 √ 2 allows to set also β = 0. The equation T 1 = 0 tells us that ω +1 + E + /ℓ = γdu for some function γ. Using this together with
T + = 0, one shows that d E -∧ E + = - 2 x dx ∧ E -∧ E + ,
which means that the surface described by E -and E + is integrable, so that
E + = ℓ 2 G 2 du + hdV , (3.11)
for some functions G, h, V . The metric becomes then
ds 2 = 2E -E + + E 1 2 + E 3 2 = ℓ 2 x 2 Gdu 2 + 2h ℓ 2 dudV + dx 2 + dy 2 . (3.12)
Finally, the equation T + = 0 implies
∂ x h = ∂ y h = 0 , ∂ V G = 2 ℓ 2 ∂ u h , (3.13)
γ = xℓ 2 ∂ x G , β = - xℓ 2 ∂ y G .
h can be eliminated by introducing a new coordinate v(u, V ) with
∂ V v = h/ℓ 2 and shifting G → G + 2∂ u v, which leads to ds 2 = ℓ 2 x 2 Gdu 2 + 2dudv + dx 2 + dy 2 . (3.14)
Note that, due to (3.13), G is independent of v, therefore ∂ v is a Killing vector. One easily verifies that it coincides with the Killing vector constructed from the Killing spinor as -ℓ 2
All that remains is to impose the Maxwell and Einstein equations. One finds that the former are automatically satisfied by the gauge potential (3.6) . The same holds for the Einstein equations, except for the uu-component, which gives the Siklos equation with sources
∆G - 2 x ∂ x G = - 4x 2 ℓ 2 ϕ ′ (u) 2 . (3.15)
This family of solutions enjoys a large group of diffeomorphisms which leave the solution invariant in form but change the function G. This is the Siklos-Virasoro invariance, discussed in [16, 33] . In conclusion, the geometry of solutions admitting the constant null spinor 1 + e 1 is given by the Lobachevski waves with metric (3.14) and gauge field (3.6) , where G satisfies (3.15) and ϕ(u) is arbitrary. This coincides exactly with the results of [13] , where it was shown moreover that there is a second covariantly constant spinor iff the wave profiles G and ϕ have the form
G α (x, y, u) = - x 4 ℓ 2 + 2αx 3 -α 2 ℓ 2 (x 2 + y 2 ) , ϕ(u) = u , (3.16)
up to Siklos-Virasoro transformation, with α ∈ R constant. In this case, the solution does also belong to the timelike class [13] . While the α = 0 solution only has the obvious Killing vectors ∂ v and ∂ y , the special α = 0 case is maximally symmetric with a five-dimensional isometry group.
If da and hence also dχ do not vanish, one can use the R 2 stability subgroup of the spinor 1 + e χ e 1 to eliminate the fluxes F +• and F +• . To see this, observe that under an R 2 transformation (3.8),
α F +• = F +• - 2iα ℓ tanh χ , α F •• = F •• , so by choosing α = -iℓ 2 F +• coth χ one can achieve α F +• = 0. Note that this would not be possible if χ = 0. With this gauge fixing, one has dχ = 2 ℓ sinh χ E 1 , A = -sinh χ E 3 , F = - 1 ℓ tanh χ E 1 ∧ E 3 . (3.17)
Next we impose vanishing torsion. Using (3.17 Before we come to the other torsion components, let us consider the Bianchi identity and the Maxwell equations. The gauge field strength reads
F = dχ sinh 2χ ∧ A .
Requiring it to be equal to dA implies that A/ √ tanh χ is closed, so that locally
A = tanh χdΨ . (3.19)
Note that the functions χ, u and Ψ must be independent, because otherwise E 1 , E - and E 3 would not be linearly independent. We can thus use these three functions as coordinates.
Using
* F = - 1 ℓ tanh χE -∧ E + , the Maxwell equations d * F = 0 imply d E -∧ E + + 2 dχ sinh 2χ ∧ E -∧ E + = 0 .
By Fröbenius' theorem and (3.18), E + can thus be written as
E + = K 2 du + hdV ,
where K, h and V are some functions, and we can use V as the remaining coordinate. Substituing E + into the Maxwell equations one obtains a constraint on the function h,
d h e 2χ + 1 ∧ du ∧ dV = 0 , and hence h = h 0 (u, V ) e 2χ + 1 .
In what follows, we define K = K/(e 2χ + 1) and use ω +1 = (ω
+• + ω +• )/ √ 2, ω +3 = (ω +• -ω +• )/ √ 2i.
We now come to the remaining torsion components. From T 3 = 0 and T 1 = 0 one obtains respectively
ω +3 = AE -, ω +1 = - E + ℓ cosh χ + BE -,
where A and B are some functions to be determined. Finally, T + = 0 yields
∂ V K = 2∂ u h 0 , A = - 1 2 e 4χ -1 sinh χ √ tanh χ ∂ Ψ K , B = 1 ℓ e 4χ -1 sinh χ∂ χ K .
The line element is given by
ds 2 = 2E -E + + E 1 2 + E 3 2 = coth χ Kdu 2 + 2h 0 dudV + ℓ 2 dχ 2 4 sinh 2 χ + dΨ 2 sinh χ cosh χ . (3.20)
As before, one can eliminate h 0 by introducing a new coordinate v(u, V ) with ∂ V v = h 0 and shifting K → K + 2∂ u v, whereupon the metric becomes
ds 2 = coth χ Kdu 2 + 2dudv + ℓ 2 dχ 2 4 sinh 2 χ + dΨ 2 sinh χ cosh χ . (3.21)
Notice that, owing to (3.20), K is independent of v, therefore ∂ v is a Killing vector.
It coincides with the Killing vector -√ 2D(ǫ, Γ µ ǫ) constructed from the Killing spinor. All that remains now is to impose Einstein's equations. One finds that they are all satisfied except for the uu component, which yields again a Siklos-type equation for K,
∂ 2 Ψ K + 4 tanh χ∂ 2 χ K - 2 cosh 2 χ ∂ χ K = 0 . (3.22)
In conclusion, the bosonic fields for a configuration admitting a null Killing spinor with dχ = 0 are given by (3.19) and (3.21), with K satisfying (3.22) 6 . As we will discuss in section 5.3, the K = 0 solution is of Petrov type D and represents a bubble of nothing in anti-De Sitter space-time. When K = 0, the metric becomes of Petrov type II and the Weyl scalar signalling the presence of gravitational radiation acquires a non-vanishing value. Hence the general solution represents a gravitational wave on a bubble of nothing. To our knowledge these solutions have not featured in the literature before.
In the previous subsections we have addressed the conditions for preserving one null Killing spinor of the form ǫ 1 = 1 or ǫ 1 = 1 + e χ e 1 . It is natural to enquire about the possibility of these backgrounds admitting an additional Killing spinor with the same R 2 stability subgroup, i.e. of the form ǫ 2 = c 0 1 + c 1 e 1 . Using the fact that ǫ 1 is Killing, the second Killing spinor equation D µ ǫ 2 = 0 can then be rewritten as
(c 0 -c 1 )D µ 1 + ∂ µ c 0 1 + ∂ µ c 1 e 1 = 0 , (3.23)
in the U(1)⋉R 2 case and
(c 0 -c 1 e -χ )D µ 1 + ∂ µ c 0 1 + (∂ µ c 1 -c 1 ∂ µ χ)e 1 = 0 , (3.24)
in the R 2 case. Furthermore, we can assume that (c 0c 1 ) = 0 and (c 0c 1 e -χ ) = 0 in the two cases, respectively, since otherwise the second Killing spinor would be linearly dependent on the first and there would not be any additional constraints. Hence the e 2 and e 12 components of D µ 1 have to vanish separately. In particular, this implies that ω -• = 0 (as can be seen from the third line of (3.1) in the singular limit χ → -∞). However, this is clearly incompatible with (3.3). We conclude that, in the gauged theory, there are no backgrounds with four R 2 -invariant Killing spinors. In other words, there are no half-supersymmetric backgrounds with an R 2 -structure. This is unlike the ungauged case, where the half-supersymmetric gravitational waves provide such solutions.
Therefore, the only possibility to augment the supersymmetry of the null solutions above is to add a Killing spinor which breaks the R 2 invariance, i.e. with a non-vanishing e 2 and/or e 12 component. From a linear combination of the first and second Killing spinor one can then always construct a time-like Killing spinor, and hence this brings us to the next section. For the convenience of the reader, we will already summarise how to restrict the 1/4-supersymmetric null solutions to allow for a time-like Killing spinor as well.
For the case with constant null Killing spinors, dχ = 0, the restriction was already discussed in [13] and is given in (3.16) . For the other case, with dχ = 0, it is straightforward to show that the solution (3.19), (3.21) admits a second Killing spinor iff ∂ χ G = ∂ Ψ G = 0, so that G depends only on u. By a simple diffeomorphism one can then set G = 0. The general solution to the Killing spinor equations reads in this case
ǫ = λ 1 (1 + e χ e 1 ) + λ 2 √ e 4χ -1 (e 2 + e χ e 1 ∧ e 2 ) , (3.25)
where λ 1,2 ∈ C are constants. The invariants constructed from ǫ, as defined in appendix B, are
V = √ 2 coth χ(|λ 2 | 2 dv -|λ 1 | 2 du) - 2i sinh 2χ (λ 2 λ1 -λ2 λ 1 ) dΨ , B = - √ 2(|λ 1 | 2 du + |λ 2 | 2 dv) + ℓe χ √ e 4χ -1 sinh χ ( λ1 λ 2 + λ 1 λ2 ) dχ , f = i(λ 1 λ2 -λ1 λ 2 ) tanh χ , g = ( λ1 λ 2 + λ 1 λ2 ) coth χ .
The norm of the Killing vector V is given by
V 2 = - 2 sinh 2χ ( λ1 λ 2 + λ 1 λ2 ) 2 -4|λ 1 λ 2 | 2 tanh χ .
Since χ > 0, this is negative unless λ 1 = 0 or λ 2 = 0, so indeed the solution (3.19), (3.21) with G = 0 must belong also to the timelike class. It turns out that it is identical to the bubble of nothing of section 5.3 with imaginary b and L < 0. The coordinate transformation
u = √ 2A 2 (t -Ly) - z 2 √ 2A 2 , v = - √ 2A 2 (t -Ly) - z 2 √ 2A 2 , Ψ = -2A 2 t , χ = artanh X 2 A 4 (3.26)
with A 8 = -1/4L brings the metric (3.21) (with G = 0) to (5.60), and the field strength of (3.19) to (5.61). Note that, in the new coordinates, the above invariants become V = ∂ t as a vector, and B = dz, in agreement with section 4.2.
We will now turn to the timelike case and first recover the general 1/4-BPS solutions [13] . Afterwards we will study the conditions for 1/2 supersymmetry. This will complete the classification since we already know that no 3/4-supersymmetric solutions can arise and AdS 4 is the unique maximally supersymmetric possibility.
Acting with the supercovariant derivative (2.5) on the representative 1 + be 2 yields the linear system
∂ + b + b 2 ω •• + - b 2 ω +- + - i ℓ b A + = 0 , 1 2 ω •• + + 1 2 ω +- + - i ℓ A + + b ℓ √ 2 + ib √ 2 F •• + ib √ 2 F +-= 0 , ω •- + + i √ 2b F •-= ω •+ + = 0 , (4.1)
∂ -b + b 2 ω •• -- b 2 ω +- -- i ℓ b A -+ 1 ℓ √ 2 + i √ 2 F •• - i √ 2 F +-= 0 , 1 2 ω •• -+ 1 2 ω +- -- i ℓ A -= 0 , b ω •+ -+ i √ 2 F •+ = ω •- -= 0 , (4.2)
∂ • b + b 2 ω •• • - b 2 ω +- • - i ℓ b A • -i √ 2 F •-= 0 , 1 2 ω •• • + 1 2 ω +- • - i ℓ A • -i √ 2b F •+ = 0 , ω •- • + b ℓ √ 2 - ib √ 2 F •• - ib √ 2 F +-= 0 , b ω •+ • + 1 ℓ √ 2 - i √ 2 F •• + i √ 2 F +-= 0 , (4.3)
∂ •b + b 2 ω •• • - b 2 ω +- • - i ℓ b A • = 0 , 1 2 ω •• • + 1 2 ω +- • - i ℓ A • = 0 , ω •- • = b ω •+ • = 0 . ( 4
A + = iℓ 2 ∂ + b b - ∂ + b b -ω •• + , A -= iℓ 2 ω •• -, A • = iℓ 2 (ω •• • + ω +- • ) , F +-= i √ 2 (b ω •+ • -b -1 ω •- • ) , F •+ = i b√ 2 ω +- • , F •• = i √ 2 (b ω •+ • + b -1 ω •- • ) + i ℓ , F •-= i b √ 2 ω •- + . ( 4
ω +- + = ∂ + b b + ∂ + b b , ω +- -= 0 , ω +- • = ∂ • b b , ω +• + = ω +• • = 0 , ω +• -= - ∂ •b b 2b , ω +• • = ∂ -b b + √ 2 bℓ , ω -• + = -b ∂ •b , ω -• -= ω -• • = 0 , ω -• • = ∂ + b b + b √ 2 ℓ . (4.6)
In what follows, we assume b = 0. One easily shows that b = 0 leads to ℓ -1 = 0, so this case appears only in ungauged supergravity.
In order to obtain the spacetime geometry, we consider the spinor bilinears
V µ = D(ǫ, Γ µ ǫ) , B µ = D(ǫ, Γ 5 Γ µ ǫ) ,
whose nonvanishing components are
V + = √ 2 bb , V -= - √ 2 , B + = √ 2 bb , B -= √ 2 .
As V 2 = -4 bb = -B 2 , V is timelike and B is spacelike. Using eqns. (4.1) -(4.4), it is straightforward to show that V is Killing and B is closed, i. e. ,
∂ A V B + ∂ B V A -ω C B|A V C -ω C A|B V C = 0 , ∂ A B B -∂ B B A -ω C B|A B C + ω C A|B B C = 0 .
There exists thus a function z such that B = dz locally. Let us choose coordinates (t, z, x i ) such that V = ∂ t and i = 1, 2. The metric will then be independent of t. Note also that the system (4.1) -(4.4) yields
∂ t b = √ 2 (|b| 2 ∂ --∂ + )b = 0 ,
so b is time-independent as well. In terms of the vierbein E A µ the metric is given
ds 2 = 2E + E -+ 2E • E • , (4.7)
where
E + µ = B µ + V µ 2 √ 2|b| 2 , E - µ = B µ -V µ 2 √ 2 .
From V 2 = -4|b| 2 and V = ∂ t as a vector we get V t = -4|b| 2 , so that V = -4|b| 2 (dt+σ) as a one-form, with σ t = 0. Furthermore, V • = 0 yields E • t = 0, and thus
E • = E • z dz + E • i dx i .
The component E • z can be eliminated by a diffeomorphism
x i = x i (x ′j , z) , with
E I i ∂x i ∂z = -E I z , I = •, • .
As the matrix E I i is invertible 7 , one can always solve for ∂x i /∂z. Note that the metric is invariant under
t → t + χ(x i , z) , σ → σ -dχ ,
where χ(x i , z) denotes an arbitrary function. This second gauge freedom can be used to eliminate σ z . Hence, without loss of generality , we can take σ = σ i dx i , and the metric (4.7) becomes
ds 2 = -4|b| 2 (dt + σ i dx i ) 2 + dz 2 4|b| 2 + 2E • i dx i E • j dx j . (4.8)
Next one has to impose vanishing torsion,
∂ µ E A ν -∂ ν E A µ + ω A µB E B ν -ω A νB E B µ = 0 .
One finds that some of these equations are already identically satisfied, while the remaining ones yield (using the expressions (4.6) for the spin connection) the constraints
∂ z σ i = - 1 4|b| 2 (E • i E j • -E • i E j • )∂ j ln(b/ b) , (4.9)
∂ i σ j -∂ j σ i = (E • i E • j -E • j E • i ) ∂ z ln(b/ b) + 1 bℓ - 1 bℓ , (4.10
)
ω •• t = -2|b| 2 ∂ z ln(b/ b) + 2b ℓ - 2 b ℓ , (4.11)
∂ i E • j -∂ j E • i = (E • i E • j -E • j E • i )ω •• • , (4.12)
as well as
∂ z + ω •• z + 1 2 ∂ z ln( bb) + 1 2ℓ 1 b + 1 b E • i = 0 . (4.13)
In (4.9), E i I denotes the inverse of E J j . In order to obtain the above equations, one has to make use of the inverse tetrad
E + = - 1 2 √ 2 ∂ t + √ 2|b| 2 ∂ z , E -= 1 2 √ 2|b| 2 ∂ t + √ 2 ∂ z , E • = E i • (∂ i -σ i ∂ t ) .
(4.13) can be solved to give
E • i = 1 |b| Ê• i exp -dz ω •• z - 1 2ℓ dz 1 b + 1 b , (4.14)
where Ê• i is an integration constant that depends only on the coordinates x j . At this point it is convenient to use the residual U(1) gauge freedom of a combined local Lorentz and gauge transformation to eliminate ω •• z . This is accomplished by the transformation (2.3), with
ψ = i 2 dz ω •• z .
Note that ψ is real, as it must be. Defining
Φ := - 1 2ℓ dz 1 b + 1 b , (4.15)
we have thus
E • i = 1 |b| Ê• i exp Φ . ( 4
.16) Using (4.16) in (4.12), one gets for the only remaining unknown component ω •• • of the spin connection
ω •• • = ω•• • -Êi • ∂ i |b| exp(-Φ) ,
where ω••
• denotes the spin connection following from the zweibein ÊI i . In what follows, we shall choose the conformal gauge for the two-metric
h ij = ÊIi ÊI j , i. e. , h ij = e 2ξ [(dx 1 ) 2 + (dx 2 ) 2 ] . (4.17)
with ξ depending only on the coordinates x i . Furthermore, we choose an orientation such that Ê
• i Ê• j -Ê• j Ê• i = -ie 2ξ ǫ ij ,
where ǫ 12 = 1. To be concrete, we shall take
( ÊI i ) = 1 √ 2 e ξ 1 i 1 -i .
The eqns. (4.9) and (4.10) then simplify to
∂ z σ i = - i 4|b| 2 ǫ ij ∂ j ln(b/ b) , (4.18)
∂ i σ j -∂ j σ i = - i |b| 2 e 2(Φ+ξ) ǫ ij ∂ z ln(b/ b) + 1 bℓ - 1 bℓ . (4.19)
Moreover, one has
ω •• • = -∂ • ln |b|e -Φ-ξ . (4.20)
In [13] it has been shown that in the case where the Killing vector constructed from the Killing spinor is timelike, the Einstein equations follow from the Killing spinor equations, so all that remains to do at this point is to impose the Bianchi identity and the Maxwell equations. Using the spin connection (4.6) and (4.11) in (4.5), the gauge potential and the field strength become
A = i(dt + σ)(b -b) + ℓ 2 ǫ ij ∂ j (Φ + ξ) dx i - iℓ 4 d ln(b/ b) , F = i(dt + σ) ∧ d ( b -b) + 1 4|b| 2 dz ∧ dx i ǫ ij ∂ j (b + b) + 1 2|b| 2 ∂ z (b + b) + 1 ℓ e 2(Φ+ξ) ǫ ij dx i ∧ dx j . (4.21)
The Bianchi identity F = dA yields
∆(Φ + ξ) = 2 ℓ e 2(Φ+ξ) ∂ z 1 b + ∂ z 1 b - 1 b 2 ℓ - 1 b2 ℓ + 1 bbℓ , (4.22)
with ∆ = ∂ i ∂ i denoting the flat space Laplacian in two dimensions. As for the Maxwell equations, ∂ µ ( √ -gF µν ) = 0 , the only nontrivial information comes from the t-component, which gives
4e 2(Φ+ξ) b 2 ∂ 2 z 1 b -b2 ∂ 2 z 1 b - 3b ℓ ∂ z 1 b + 3 b ℓ ∂ z 1 b + 1 bℓ 2 - 1 bℓ 2 + b 2 ∆ 1 b -b2 ∆ 1 b = 0 , (4.23)
where we used eqns. (4.18) and (4. 19 ).
Let us now show that the equations (4.22) and (4.23) are actually the same as the ones in [16] . If we set
F = - 1 ℓ b , e φ = 2e Φ+ξ , (4.24)
(4.22) yields exactly equation (2.3) of [16] . On the other hand, deriving (4.22) with respect to z and using (4.15), one obtains
∆A + e 2φ 3A∂ z A -3B∂ z B + A 3 -3AB 2 + ∂ 2 z A = 0 , (4.25)
where A and B denote the real and imaginary part of F respectively. This can be used in (4.23) to get
∆B + e 2φ ∂ 2 z B + 3B∂ z A + 3A∂ z B -B 3 + 3A 2 B = 0 ,
which, together with (4.25), yields
∆F + e 2φ F 3 + 3F ∂ z F + ∂ 2 z F = 0 , (4.26)
i. e. , equation (2.2) of [16] . For a complete identification of the present results with the ones in [16] , one also has to set σ = ω.
In conclusion, the metric of the general 1/4-supersymmetric solution is given by
ds 2 = -4|b| 2 (dt + σ) 2 + 1 4|b| 2 dz 2 + 4e
2(Φ+ξ) dw d w , (4.27) where b and φ are determined by the system (4.22), (4.23) and w = x 1 + ix 2 ≡ x + iy. The one-form σ follows then from (4.18) and (4.19), and the gauge field strength is given by (4.21). Note that (4.23) represents also the integrability condition for (4.18), (4.19). As noted in [16], this system of equations is invariant under PSL(2, R) transformations foot_8 . If we define a new coordinate z ′ through the Möbius transformation
z ′ = αz + β γz + δ , (4.28)
with α, β, γ and δ arbitrary real constants satisfying αδβγ = 1, then the functions b(z ′ , x i ) and Φ(z ′ , x i ) defined by
1 b = 1 (γz ′ -α) 2 b - 2lγ γz ′ -α , e Φ = (γz ′ -α) 2 e Φ , (4.29)
solve the system in the new coordinate system (z ′ , x i ), with the function ξ(x i ) left invariant and z seen as a function of z ′ . This symmetry allows to generate new BPS solutions from the known ones. Note however that it is only a symmetry of the equations for 1/4 supersymmetry, and if we apply it to solutions with additional Killing spinors, it will in general not preserve them, as we shall show explicitely in some examples.
We now would like to investigate the possibility of adding a second Killing spinor. Since the first Killing spinor ǫ 1 has stability subgroup 1, one cannot use Lorentz transformations to bring the second spinor to a preferred form. Therefore we use the most general form
ǫ 2 = c 0 1 + c 1 e 1 + c 2 e 2 + c 12 e 1 ∧ e 2 . (4.30)
The corresponding linear system simplifies significantly after inserting the results from ǫ 1 . These determine all the fluxes and the spin connection in terms of the functions b, ξ and their derivatives. First it is convenient to introduce the new basis
9 α = α 0 α 1 α 2 α 12 = c 0 b -1 c 2 -c 0 bc 1 c 12
, in which the Killing spinor equations for ǫ 2 read
(∂ A + M A )α = 0 , (4.31)
with the connection M A given by
M + = 0 -∂ + ln b 0 0 0 ∂ + ln b -∂ • ln b ∂ • ln b 0 0 b-b √ 2ℓ + 1 2 ∂ + ln b b - √ 2 ℓ b -∂ + ln b 0 -|b| 2 ∂ • ln b 0 b-b √ 2ℓ -1 2 ∂ + ln( bb) , M -= 0 0 |b| -2 ∂ • ln b -|b| -2 ∂ • ln b 0 ∂ -ln b -|b| -2 ∂ • ln b |b| -2 ∂ • ln b 0 ∂ • ln b b-b √ 2ℓ|b| 2 -1 2 ∂ -ln( bb) 0 0 0 - √ 2 ℓ b -∂ -ln b b-b √ 2ℓ|b| 2 + 1 2 ∂ -ln b b , M • = 0 -∂ • ln b 0 0 0 ∂ • ln( bb) 0 0 0 - √ 2 ℓ b -∂ + ln b -∂ • ln |b|e -Φ-ξ 0 0 √ 2 ℓ b + ∂ + ln b 0 -∂ • ln |b|e -Φ-ξ , M • = 0 0 -∂ -ln b ∂ -ln b + √ 2 ℓ b 0 0 ∂ -ln( bb) + √ 2 ℓb -∂ -ln( bb) - √ 2 ℓ b 0 0 ∂ • ln b b e -Φ-ξ -∂ • ln b 0 0 -∂ • ln b ∂ • ln b b e -Φ-ξ .
Let us first of all consider the simpler possibility of a second Killing spinor of the form ǫ 2 = c 0 1 + c 2 e 2 . As discussed in section 2.1, both ǫ 1 and ǫ 2 are invariant under the same U(1) symmetry, and hence this case constitutes the G = U(1) case with four supersymmetries. As can easily be seen from the above Killing spinor equations with α 1 = 0 and α 2 = α 12 = 0, this restricts the derivatives of the coefficient b to be
∂ -b = - √ 2 ℓ , ∂ + b = - √ 2b b ℓ , ∂ • b = ∂ •b = 0 . (4.32)
Hence this corresponds to ∂ z b = -1/ℓ. As will be discussed in section 5.1, this restriction uniquely leads to the half-supersymmetric anti-Nariai space-time. Hence AdS 2 ×H 2 is the only possibility for backgrounds with four U(1)-invariant Killing spinors.
In the more general case with α 2 and α 12 non-vanishing, i.e. with trivial stability subgroup, the Killing spinor equations do not so readily provide information about b and one has to resort to their integrability conditions. The first integrability conditions for the linear system (4.31) are
N µν α ≡ (∂ µ M ν -∂ ν M µ + [M µ , M ν ])α = 0 , (4.33)
where the matrices M µ = E A µ M A are given by
M t = √ 2(|b| 2 M --M + ) , M z = 1 2 √ 2|b| 2 (M + + |b| 2 M -) , M w = σ w M t + 1 √ 2|b| e Φ+ξ M • , M w = σ wM t + 1 √ 2|b| e Φ+ξ M • ,
and we introduced the complex coordinates w = x + iy, w = xiy. For halfsupersymmetric solutions, the six matrices N µν must have rank two. (As at least one Killing spinor exists, namely ǫ 1 = (1, 0, 0, 0), we already know that the N µν can have at most rank three. Rank one is not possible, because 3/4 BPS solutions cannot exist [29] . Rank zero corresponds to the maximally supersymmetric case, which implies that the spacetime geometry is AdS 4 [13] .) Let us define
ѵν ≡ SN µν T , with S = 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 , T = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1
.
The similarity transformation S corresponds to adding the first line to the second one and T adds the last column to the third one. This does not alter the rank of N µν . One finds
Ñwt = 2b∂∂ z b + 2 ℓ ∂ b -2|b| b e -Φ-ξ ∂ 2 b + 1 b ∂ b∂ b 0 +2b ∂ z b + 1 ℓ ∂ ln b 0 -2∂(Φ + ξ)∂ b 2 b∂∂ z b + 2 ℓ ∂b -2|b| b e -Φ-ξ ∂ 2 b + 1 b ∂b∂b 0 +2 b ∂ z b + 1 ℓ ∂ ln b 0 -2∂(Φ + ξ)∂b) 2|b| 3 e -Φ-ξ ∂∂ ln b 2 b∂∂ z b 0 -2|b| 3 e Φ+ξ b -2 2∂ z b + 1 ℓ ∂ z b + 1 ℓ 2 ℓ ∂b +2 b ∂ z b + 1 ℓ ∂ ln b 2|b| 3 e -Φ-ξ ∂∂ ln b 2b∂∂ z b -2 ℓ ∂ b 0 -2|b| 3 e Φ+ξ b-2 2∂ z b + 1 ℓ ∂ z b + 1 ℓ -2 ℓ ∂ b +2b ∂ z b + 1 ℓ ∂ ln b , Ñ wt = 2b ∂∂ z b -2|b|e -Φ-ξ ∂∂ ln b 0 +2b ∂ z b + 1 ℓ ∂ ln b -2 ℓ|b| e Φ+ξ 2∂ z b + 1 ℓ + 2b |b| b e Φ+ξ 2∂ z b + 1 ℓ ∂ z b + b- b ℓb 2 b ∂∂ z b -2|b|e -Φ-ξ ∂∂ ln b 0 +2 b ∂ z b + 1 ℓ ∂ ln b 2 ℓ|b| e Φ+ξ 2∂ z b + 1 ℓ + 2 b |b|b e Φ+ξ 2∂ z b + 1 ℓ ∂ z b + 1 ℓ 2|b| be -Φ-ξ ∂2 b + 1 b ∂b ∂b 2 b ∂∂ z b + 2 ℓ ∂b 0 -2 ∂(Φ + ξ) ∂b 6 ℓ ∂b +2 b ∂ z b + 1 ℓ ∂ ln b 2|b|be -Φ-ξ ∂2 b + 1 b ∂b ∂b 2b ∂∂ z b -4 ℓ ∂b 0 -2 ∂(Φ + ξ) ∂b -6 ℓ ∂b +2b ∂ z b + 1 ℓ ∂ ln b , where ∂ = ∂ w , ∂ = ∂ w.
The other four integrability conditions give no additional information, because the lines of the corresponding matrices are proportional to the lines of Ñwt and Ñ wt 10 . As the upper right 3 × 3 determinant of Ñwt must vanish, we obtain ∂b = 0 or
∂ z e -2(Φ+ξ) b∂ b ∂ e -2(Φ+ξ) b∂b -∂ z e -2(Φ+ξ) b∂b ∂ e -2(Φ+ξ) b∂ b = 0 . (4.34)
Let us assume that the expression in (4.34) does not vanish. One has then ∂b = 0 as well as ∂ b = 0 foot_11 . But then also (4.34) holds, which leads to a contradiction. Thus (4.34) must be satisfied in any case. Note that the vanishing of the first column of ѵν implies that also the first column of T -1 N µν T is zero, and thus T -1 N µν T ∈ a(3, C), hence the generalized holonomy in the case of one preserved complex supercharge is contained in the affine group A(3, C). This supports the classification scheme of [4] . Of course, depending on the particular solution, the generalized holonomy may also be a subgroup of A(3, C).
In this section we will utilize the above Killing spinor equations to derive the timedependence of the second Killing spinor. In addition, we will show that the Killing spinor equations can be completely solved when the second Killing spinor is timeindependent.
Let us first simplify the Killing spinor equations (4.31). In the following we set b = re iϕ and define ψ = Φ + ξ, ψ 1 = r 2 α 1 , ψ 2 = re -ψ α 2 , ψ 12 = re -ψ α 12 and ψ ± = ψ 2 ± ψ 12 . First of all, use the integrability conditions (4.33), that can be rewritten as ѵν T -1 α = 0. Defining P = e -2ψ b∂b, the second component for µ = w, ν = t gives
ψ 1 P ′ + ψ -∂P = 0 , (4.35)
with ′ = ∂ z . Let us assume P ′ = 0 (the case P ′ = 0 is considered in appendix C and will lead to the same conclusions). If we define g(t, z, w, w) = -ψ -/P ′ , we get
ψ -= -gP ′ , ψ 1 = g∂P .
The third component of the (w, t) integrability condition is of the form
ψ 1 f 1 + ψ 2 ∂b + ψ -f -= 0 ,
for some functions f 1 , f -that depend on z, w, w but not on t. Using the above form of ψ 1 and ψ -, this becomes
f 1 g∂P + ψ 2 ∂b -f -gP ′ = 0 . (4.36)
Now, if g = 0, the latter equation implies ψ 2 ∂b = 0, and hence (since ∂b = 0 due to P ′ = 0) ψ 2 = 0. Furthermore, ψ 1 = ψ -= 0 in this case, so there exists no other Killing spinor. Thus, g = 0 and we can write g = exp G. Dividing (4.36) by g and deriving with respect to t yields ∂ t (ψ 2 /g) = 0 and hence
ψ 2 = e G ψ 0 2 (z, w, w) .
It is then plain that ∂ t ψ i = ψ i ∂ t G, i = 1, 2, 12. The Killing spinor equations are of the form ∂ µ ψ i = M µij ψ j , for some time-independent matrices M µ . Taking the derivative of this with respect to t, one gets ∂ µ ∂ t G = 0, whence
G = G 0 t + G(z, w, w) ,
with G 0 ∈ C constant. We have thus ∂ t ψ i = G 0 ψ i and hence also ∂ t α i = G 0 α i . Furthermore, the time-dependence of α 0 can be easily deduced from the Killing spinor equations: if G 0 does not vanish it is of the same exponential form as the other components of the second Killing spinor, i.e. ∂ t α 0 = G 0 α 0 , while if G 0 vanishes there can be a linear part in t, i.e. ∂ t α 0 = c for some constant c. Hence, in terms of the basis elements, the time-dependence of the second Killing spinor takes the form foot_12
G 0 = 0 : ǫ 2 = c 0 1 + c 1 e 1 + c 2 e 2 + c 12 e 1 ∧ e 2 + ct(1 + be 2 ) , G 0 = 0 : ǫ 2 = e G 0 t (c 0 1 + c 1 e 1 + c 2 e 2 + c 12 e 1 ∧ e 2 ) , (4.37)
where c 0 , c 1 , c 2 , c 12 are time-independent functions of the spatial coordinates, and c is a constant. This was derived assuming P ′ does not vanish, but as we show in appendix C is in fact a completely general result. Hence, adding a second Killing spinor to ǫ 1 = 1 + be 2 , the Killing spinor equations imply that ǫ 2 always has the above timedependence.
Plugging this time-dependence into the subsystem of the Killing spinor equations not containing α 0 one obtains in terms of
ψ i ψ ′ 1 - G 0 4r 2 + b ′ b ψ 1 - ∂b b ψ -= 0 , (4.38)
ψ ′ 2 - G 0 4r 2 + b′ b + 1 ℓ b ψ 2 - b ′ b + 1 ℓb ψ 12 = 0 , (4.39)
ψ ′ 12 -e -2ψ ∂b b ψ 1 - G 0 4r 2 + b ′ b + b′ b + 1 ℓ b ψ 12 = 0 , (4.40)
ψ ′ 1 -- G 0 4r 2 + b′ b ψ 1 - ∂ b b ψ -= 0 , (4.41)
ψ ′ 2 + e -2ψ ∂b b ψ 1 -- G 0 4r 2 + b ′ b + b′ b + 1 ℓb ψ 2 = 0 , (4.42)
ψ ′ 12 - b′ b + 1 ℓ b ψ 2 -- G 0 4r 2 + b ′ b + 1 ℓb ψ 12 = 0 , (4.43
)
∂ψ 1 -σ w G 0 ψ 1 = 0 , (4.44) ∂ψ 2 - b ′ b + 1 ℓb ψ 1 -σ w G 0 + ∂b b + ∂ b b -2∂ψ ψ 2 = 0 , (4.45
) For G 0 = 0, these equations simplify significantly, and allow for a complete solution. As is shown in appendix D, under the additional assumption ψ -= 0, ψ 1 = 0, the metric and the field strength for half-supersymmetric solutions with G 0 = 0 are given in terms of a single real function H depending only on the combination Zww and satisfying the second order differential equation
∂ψ 12 + b′ b + 1 ℓ b ψ 1 -σ w G 0 + ∂b b + ∂ b b -2∂ψ ψ 12 = 0 , (4.46) ∂ψ 1 -σ wG 0 + ∂b b + ∂b b ψ 1 + e 2ψ b ′ b + b′ b ψ -+ 1 ℓ ψ 2 b - ψ 12 b = 0 ,(4.
2 1 + e -2H Ḧ + Ḣ2 1 - 3α 2 e 2H + 1 -α 2 = γ 2 ℓ 2 , (4.50)
where α ∈ R denotes an arbitrary constant and γ = 0, 1. The new coordinate Z is defined by Z = z for γ = 0 and Z = ℓ ln 1 + z ℓ for γ = 1. Furthermore, in the remainder of this section and in appendix D, a dot denotes a derivative with respect to Zww. Given a solution of (4.50), one defines the functions χ, ρ by where the shift vector satisfies
χ = iα √ e 2H + 1 -α 2 , 1 ℓ 2 ρ 2 = γ ℓ + Ḣ 2 -Ḣ2 χ 2 . ( 4
∂ Z σ w = 1 4 e -γZ/ℓ χ ρ 2 • , ∂σ w -∂σ w = - 1 2 e -γZ/ℓ e 2H χ ρ 2 • .
Finally, the gauge field strength is given by (4.21). Equation (4.50) is actually the Euler-Lagrange equation for the following standard action for the scalar
H S = d (Z -w -w) 1 2 M(H) Ḣ2 -V (H) , (4.53)
where
M(H) = e 2H + 1 2 (e 2H + 1 -α 2 ) 3/2 , V (H) = - γ 2 2ℓ 2 e 2H + 1 -2α 2 (e 2H + 1 -α 2 ) 1/2 . (4.54)
Thus it is possible to use the energy conservation law of that model in order to evaluate the "velocity" Ḣ in terms of H. Since dH = Ḣd (Zww) one has d dH
1 2 M(H) Ḣ2 + V (H) = 0 , (4.55)
so that there must exist a constant E such that
Ḣ = 2 M(H) [E -V (H)] = √ 2 e 2H + 1 -α 2 3/4 e 2H + 1 E + γ 2 2ℓ 2 e 2H + 1 -2α 2 √ e 2H + 1 -α 2 1/2 . ( 4
The key-point is to consider now, as a new coordinate, the function H in place of w + w13 and to write down the full solution, say metric plus gauge field, in terms of H. Using w = x + iy, the general solution is given by ds 2 = -4ρ 2 e 2γZ/ℓ dt + e -γZ/ℓ σy dy
2 + e 2H 4ρ 2 dy 2 + 1 4ρ 2 dZ 2 + e 2H dZ - dH Ḣ 2 , A = ℓ Ḣ -2iρ 2 χe γZ/ℓ dt + 1 -e 2H χ 2 dy -i ℓ 4 d log b b , (4.57)
where Ḣ is given in equation (4.56), the functions χ and ρ are defined in (4.51) and the shift vector reads σy = -i 2 e 2H χ ρ 2 .
If γ = 1, a simple example of this set of solutions can be obtained by setting α = 0, so
Ḣ = 1/ℓ , b = 1 2 1 + z ℓ . (4.58)
As will be shown in section 5.1.2, this corresponds to the maximally supersymmetric AdS 4 solution. More general γ = 1 solutions will be two-parameter deformations thereof, the parameters being α and the energy E of the associated scalar system.
Setting γ = 0 the potential V (H) vanishes and the parameter E can be fixed by a simple rescaling of the coordinates. Thus we are left with a one-parameter family of solutions. Since the metric does no more depend explicitly on Z, it is useful to replace the coordinate Z instead of x by H. Defining a new coordinate r such that r 4 ≡ 16 e 2H + 1α 2 and a new parameter Q = 4 ℓ α, the complete solution reads
ds 2 = - r 2 ℓ 2 + Q 2 r 2 dt - 2ℓ 3 Q r 4 + ℓ 2 Q 2 dy 2 + + r 2 ℓ 2 + Q 2 r 2 -1 h(r) 2 dr + 2 h(r) dx 2 + 1 4 r 4 + ℓ 2 Q 2 -16 dx 2 + dy 2 , A = - Q r dt + 2ℓ r dy -i ℓ 4 d log b b , (4.59)
where
h(r) = r 4 + ℓ 2 Q 2 r 4 + ℓ 2 Q 2 -16 . ( 4
The parameter Q can thus be interpreted as an electric charge. The Petrov type of the solution is D or simpler. If one sets Q = 4/ℓ the Petrov type is reduced to N, so that there is a gravitational wave.
In order to complete the classification of G 0 = 0 solutions, we need to study separately the cases where either ψ 1 or ψ -vanishes (it can easily be seen from (4.39) and (4.40) that there is no solution if both vanish). As one can see by looking at equations (4.38) and (4.41), the condition ψ 1 = 0 leads to b = b(z), which is studied in detail in section 5.1. The other possibility, ψ -= 0, is more involved, but as we show in appendix D it boils down to three different cases, that can be completely solved: the AdS 2 × H 2 anti-Nariai spacetime studied in section 5.1.1, the imaginary b case solved in section 5.3, and finally the half BPS solution coming from the gravitational Chern-Simons model, that we analyse in section 5.5.
We would like to remark that the assumption G 0 = 0 on the overall time-dependence of the second Killing spinor seems a reasonable choice since all known 1/2-supersymmetric solutions to be studied in the next section are contained in this class, or can be brought to this class by a general coordinate transformation. Hence we expect the G 0 = 0 class to form an important subclass of all 1/2-supersymmetric solutions.
The problem of finding all half BPS configurations in the timelike class involves the solution of the integrability conditions we obtained above. To obtain explicit examples of half BPS solutions, we shall restrict to some simple subclasses with particular b. This will determine the fraction of preserved supersymmetry for the solutions which are already known to be 1/4 supersymmetric, and will also lead to new solutions.
The timelike vector field V , constructed as a bilinear of the Killing spinor, is static if the associated one-form V = dt + σ satisfies the Fröbenius condition V ∧ dV = 0. Obviously, there can be static BPS solutions with V not being static itself, due to the choice of coordinates; we shall loosely refer the Killing spinors whose vector bilinear is static as static Killing spinors. The staticity condition, in turn, implies dσ = 0 and puts strong constraints on the function b. Indeed, equation (4.18) implies that the phase ϕ of b depends only on z. Then, (4. 19) gives the modulus r of b in terms of its phase,
r = sin ϕ(z) lϕ ′ (z) . (5.1)
As a consequence, r and therefore the complete complex function b, depend on the single variable z. The full solution is therefore determined by the single real function ϕ, which has to satisfy the equations for supersymmetry (together with the conformal factor ψ).
However, since the equations can be exactly solved for arbitrary b(z), we will stick to this more general case and eventually comment on the static subcase.
If b depends only on z, the equations of motion simplify to
Im b 2 ∂ 2 z 1 b - 3b l ∂ z 1 b + 1 bl 2 = 0 , (5.2)
e -2ξ ∆ξ = 2 l e 2Φ ∂ z 1 b + 1 b - 1 l 1 b + 1 b 2 + 3 lb b . (5.3)
Here we have used the fact that Φ, defined in (4.15), depends only on the coordinate z. In principle there is also an integration constant K(w, w) with arbitrary dependence on the transverse coordinates, but since Φ appears only in the combination Φ + ξ in all the equations, we can always absorb the (w, w) dependence into the conformal factor ξ. Now the left hand side of equation ( 5 .3) depends only on the coordinates w and w, while the right hand side depends only on z. This equation can be therefore satisfied only if both sides are equal to some constant κ. The system of equations is then ∆ξ + κ 2 e 2ξ = 0 , (5.4)
e 2Φ(z) ∂ z 1 b + 1 b - 1 l 1 b + 1 b 2 + 3 lb b = - l 4 κ . (5.5)
Note that the first one is the Liouville equation, whose solution describes the transverse two-dimensional manifold, which has therefore constant curvature κ.
Equations (5.2) and (5.5) can easily be solved [16] . Their solution is given by foot_14 b = - This solution generically belongs to the supersymmetric Reissner-Nordström-Taub-NUT-AdS 4 family of spacetimes. The values α = 0 and β 2 = 4αγ are special cases and will be treated separately in the following. Note that the coefficients α, β and γ are not three independent parameters, as they can be rescaled without changing the function b: the solutions depend only on their ratios. For example, if α = 0, one can use β/α and γ/α as independent complex parameters of the family of solutions.
αz 2 + βz + γ ℓ(2αz + β) , ( 5
The solutions with static Killing spinor form a subset of this family. For (5.6) the staticity condition (5.1) yields the condition α βᾱβ = 0. Recalling the expression for the NUT charge of these solutions,
n = i 4 β ᾱ - β α , (5.8)
this charge must vanish for non-vanishing α, as one could have guessed. On the other hand, for α = 0 the solution is anti-Nariai, as we shall see below. We conclude that the most general supersymmetric configuration with static Killing vector constructed as a Killing spinor bilinear is either of the form (5.6) -i. e. in the fourth row of table 1 of [34] -with vanishing NUT charge, or it is anti-Nariai spacetime. The supersymmetric static solutions discussed so far are generically 1/4-BPS. We want to see what further condition ensures the presence of an additional Killing spinor. Inserting the staticity ansatz b = b(z) into the integrability equations and requiring these matrices to be of rank smaller or equal to two, one finds the following condition (in particular this is obtained from the vanishing of the minor of the last row of Ñwt and the first two rows of Ñ wt )
b ′ + 1 l 2b ′ + 1 l ∂ z 1 b + 1 b - 1 l 1 b + 1 b 2 + 3 lb b = 0 , (5.9)
As an aside, note that we have only used the ansatz b = b(z) so far and not the staticity condition (5.1), i.e. the precise relation between r and ϕ. The static solutions are therefore in general still a subset of the solutions under consideration. Condition (5.9) calls for the following three different cases, corresponding to the vanishing of its three factors.
2 × H 2 space-time (α = 0)
Requiring the first factor of (5.9) to vanish leads to b = -z ℓ + ic with constant c, corresponding to α = 0 in (5.6) . We can absorb the imaginary part of c by a shift of the coordinate z and henceforth will assume c ∈ R.
In this case κ = -4 and we have a hyperbolic transverse space. As a solution of (5.4) we can take
e 2ξ = 1 2x 2 .
(5.10)
Moreover, e Φ = l|b| and σ = 0, therefore giving the metric
ds 2 = -4 z 2 ℓ 2 + c 2 dt 2 + dz 2 4 z 2 ℓ 2 + c 2 + ℓ 2 2x 2 dx 2 + dy 2 . (5.11)
This is the anti-Nariai AdS 2 × H 2 solution, with the AdS 2 factor written in Poincaré coordinates for c = 0 and in global coordinates for c = 0. The coordinate transformations between Poincaré coordinates (t P , z P ) (with c = 0) to global ones (t gl , z gl ) (with c = 0) is given by
z P = 1 2c (z gl -z 2 gl + ℓ 2 c 2 cos(4ct gl /ℓ)) , t P = - ℓ 2 z 2 gl + ℓ 2 c 2 sin(4ct gl /ℓ) z gl -z 2 gl + ℓ 2 c 2 cos(4ct gl /ℓ) .
(5.12)
The electromagnetic field strength (4.21) in this case is given by
F = - 1 ℓx 2 dx ∧ dy , (5.13)
i.e. only lives on the hyperbolic part and is independent of the coordinates of the AdS part of space-time. This solution preserves precisely 1/2 of the supersymmetries, as was already shown in [35] . To obtain the form of the Killing spinors admitted by this metric we first observe that the integrability conditions impose α 2 = α 12 = 0. Then the Killing spinor equations are easily solved, but one should treat separately the cases c = 0 and c = 0:
• If c = 0, then α 0 = λ 1 + λ 2 2t ℓ - 1 2b , α 1 = λ 2 b , (5.14)
where λ 1,2 ∈ C are integration constants. This yields the following Killing spinors, spanning a two-dimensional complex space,
ǫ = λ 1 + λ 2 2t ℓ - 1 2b 1 + b λ 1 + λ 2 2t ℓ + 1 2b e 2 .
(5.15)
Note that λ 1 = 1, λ 2 = 0 corresponds to the original Killing spinor. Also note that the constant G 0 , corresponding to the time-dependence of the second Killing spinor with λ 2 = 0, is zero. The form of the scalar invariant corresponding to the general spinor
ǫ is b = b |λ 1 | 2 + |λ 2 | 2 4t 2 ℓ 2 - 1 4b 2 + 2t ℓ λ1 λ 2 + λ 1 λ2 + 1 2 λ1 λ 2 -λ 1 λ2 . (5.16)
Here the first term is real, while the second is imaginary. Note that the latter is in fact constant. Then the Killing vector Ṽ built from ǫ will have a norm Ṽ 2 = -4| b| 2 , and will be timelike unless b vanishes. This is however not possible, because both the real and imaginary parts of b should vanish, but since λ 1,2 do not depend on the coordinates, the real part cannot vanish. Therefore, every Killing spinor of this solution belongs to the timelike class.
• If c = 0 we have
α 0 = 1 2 √ c [λ 1 -iλ 2 +(λ 1 +iλ 2 ) b |b| e -4ict/ℓ , α 1 = - i √ c |b| (λ 1 +iλ
2 )e -4ict/ℓ , (5.17)
and the most general Killing spinor is parametrized by λ1,2 ∈ C as follows
ǫ = 1 2 √ c (λ 1 -iλ 2 )(1 + be 2 ) + b 2|b| (λ 1 + iλ 2 )e -4ict/ℓ (1 + b * e 2 ) .
(5.18)
Note that the combination λ 1iλ 2 corresponds to the first Killing spinor 1 + be 2 , while the orthogonal combination λ 1 + iλ 2 gives rise to the second Killing spinor proportional to 1 + b * e 2 . Any combination with λ 2 = 0 has G 0 = -4ic/ℓ.
In this case, the real part of the invariant b is given by
Re( b) = |λ 1 | 2 2ℓc (-z + √ z 2 + ℓ 2 c 2 cos(4ct/ℓ)) + |λ 2 | 2 2ℓc (-z - √ z 2 + ℓ 2 c 2 cos(4ct/ℓ))+ + 1 2ℓc (λ 1 λ * 2 + λ 2 λ * 1 ) √ z 2 + ℓ 2 c 2 sin(4ct/ℓ)) , (5.19)
while the imaginary part is identical to that of (5.16) .
It can easily be checked that the coordinate transformation (5.12) indeed relates the complex scalar b, which is composed of spinor bilinears, in (5.16) and (5.19) to each other. Let's now check how the isometries of AdS 2 act on the Killing spinors. It is useful to do this by embedding AdS 2 with metric
ds 2 = -4 z 2 ℓ 2 + c 2 dt 2 + dz 2 4 z 2 ℓ 2 + c 2 (5.20)
into the three-dimensional flat space X a = (U, T, X) with metric
ds 2 = -dU 2 -dT 2 + dX 2 . (5.21)
Then, AdS 2 is obtained as the hyperboloid defined by
-U 2 -T 2 + X 2 = ℓ 2 4 , (5.22)
and its isometry group SO(2,1) will act as the three-dimensional Lorentz group on the embedding coordinates X a (here a is a three-dimensional Lorentz index). If c = 0, the AdS 2 metric (5.20) is in the Poincaré form, and can be seen to be the induced metric on the hyperboloid by parameterizing it with the coordinates (t, z) given by z = U + X , t = ℓT 2(U + X) .
(5.23)
Then, if one defines the 3d Lorentz vector Now, the real and imaginary part of b are independently manifestly invariant under the AdS 2 isometries, as they should be (since they transform respectively as pseudoscalar and scalar under diffeomorphism 16 ). If c = 0 we have AdS 2 in global coordinates, and the embedding is modified to
Λ a = 1 ℓ |λ 1 | 2 -|λ 2 | 2 , 1 ℓ (λ * 1 λ 2 + λ 1 λ * 2 ) , - 1 ℓ |λ 1 | 2 + |λ 2 | 2 , ( 5
U = - ℓ 2c z 2 ℓ 2 + c 2 cos 4ct ℓ , T = - ℓ 2c z 2 ℓ 2 + c 2 sin 4ct ℓ , X = z 2c .
(5.26)
The invariant (5.19) takes again the manifestly invariant form (5.25), as expected, and the isometries of AdS 2 are realized linearly on the Killing spinors through their action on Λ a . This result may be useful to study in detail quotients of AdS 2 and to see whether this operation breaks some supersymmetry. The following subcase corresponds to the vanishing of the second factor of the integrability condition (5.9). The function b is then given by b = -z 2l + ic, which can be obtained as the special case β 2 = 4αγ from (5.6) . This corresponds to AdS 4 , the only maximally supersymmetric solution of the theory. Indeed the integrability condition matrices vanish in this case.
Let's see in detail the form of the metric arising from different values of c. As in the previous case we can take the constant c to be real. If c = 0, the metric is static, σ = 0, ξ = 0 and e 2Φ = |b| 4 , and we obtain anti-de Sitter in Poincaré coordinates,
ds 2 = - z 2 ℓ 2 dt 2 -dx 2 -dy 2 + ℓ 2 z 2 dz 2 . (5.27)
On the other hand, for c = 0, the metric appears in non-static coordinates,
σ = - ℓdy 4cx 2 , e 2ξ = ℓ 2 4c 2 x 2 , e 2Φ = |b| 4 , (5.28)
which give
ds 2 = - z 2 l 2 + 4c 2 dt - ℓdy 4cx 2 2 - ℓ 2 16c 2 x 2 dx 2 + dy 2 + z 2 l 2 + 4c 2 -1 dz 2 .
(5.29)
The field strength (4.21) vanishes in this case.
We shall now obtain the form of the Killing spinors for AdS 4 , and will do this in the simpler c = 0 case. The solution of the Killing spinor equations yields
α 0 = λ 1 - t ℓ + ℓ z λ 2 + w ℓ λ 3 , α 2 = - wz 2ℓ 2 λ 2 + 1 2 1 + zt ℓ 2 λ 3 - z 2ℓ λ 4 , α 1 = 2ℓ z λ 2 , α 12 = wz 2ℓ 2 λ 2 + 1 2 1 - zt ℓ 2 λ 3 + z 2ℓ λ 4 , (5.30)
where the coefficients λ 1,...,4 span a four dimensional complex space, as expected in the case of maximal supersymmetry. In the form basis of the spinors ǫ = c 0 1 + c 1 e 1 + c 2 e 2 + c 12 e 1 ∧ e 2 , we obtain
c 0 = λ 1 - t ℓ + ℓ z λ 2 + w ℓ λ 3 , c 2 = - z 2ℓ λ 1 + z 2ℓ t ℓ - ℓ z λ 2 - z w 2ℓ 2 λ 3 , c 1 = w ℓ λ 2 - t ℓ + ℓ z λ 3 + λ 4 , c 12 = wz 2ℓ 2 λ 2 - z 2ℓ t ℓ - ℓ z λ 3 + z 2ℓ λ 4 . (5.31)
The new Killing spinors corresponding to λ 2 and λ 4 both have foot_17 G 0 = 0. To study the action of the AdS 4 isometries it is useful to embed the hyperboloid in a five-dimensional flat space (U, V, T, X, Y ) with metric
ds 2 = -dU 2 + dV 2 -dT 2 + dX 2 + dY 2 .
(5.32)
Then, AdS 4 is the hypersurface
-U 2 + V 2 -T 2 + X 2 + Y 2 = -ℓ 2 /4
and its isometries are realized as the SO(3,2) isometries of the embedding space. The relation with the Poincaré coordinates is
t ℓ = T U -V , x ℓ = X U -V , y ℓ = Y U -V , z = 2(U -V ) . (5.33)
If we define the vectors
ℓΛ a = |λ 1 | 2 -|λ 2 | 2 + |λ 3 | 2 -|λ 4 | 2 |λ 1 | 2 + |λ 2 | 2 -|λ 3 | 2 -|λ 4 | 2 λ 3 λ4 + λ3 λ 4 -λ1 λ 2 -λ 1 λ2 λ 2 λ4 + λ2 λ 4 -λ1 λ 3 -λ 1 λ3 i λ 2 λ4 -λ2 λ 4 + λ1 λ 3 -λ 1 λ3 , X a = U V T X Y , (5.34)
where the index a = 1, . . . , 5 is an SO(3,2) index raised and lowered using the metric (5.32), then
Λ a Λ a = - 1 ℓ 2 λ 3 λ4 -λ3 λ 4 + λ1 λ 2 -λ 1 λ2 2 ≥ 0 , (5.35)
and the invariant b for the Killing spinors reads
b = c * 0 c 2 + c 1 c * 12 = X a Λ a + iℓ 2 Λ a Λ a . (5.36)
The subfamily of static half BPS configurations is obtained by imposing the staticity condition ζ = 0 or equivalently vanishing NUT charge. It is parameterized by the single parameter left, δ ∈ R and the solutions are restricted to have the following charges
M = 0 , n = 0 , P = 0 , Q = - δ ℓ .
In terms of the charges, the solution is given by b
= - 1 ℓ z 2 + iℓQ 2z . (5.42)
The metric and electromagnetic field strength for this solution read
ds 2 = - Q 2 z 2 + z 2 ℓ 2 dt 2 + dz 2 Q 2 z 2 + z 2 ℓ 2 + 4ℓ 2 z 2 dwd w , (5.43)
and
F = - Q z 2 dt ∧ dz . (5.44)
This is simply the backreacted AdS 4 filled with the electric field generated by an electric charge Q placed in its center ζ = 0. The solution has a singularity there. Note that this solution was already shown to be 1/2 supersymmetric in [36] . It was also shown there that the Killing spinors are preserved if one compactifies the transverse two-dimensional plane to a two-torus. We will now discuss the Killing spinors for these metrics. The integrability conditions impose α 2 = 0 and
b ′ + 1 ℓ - b ℓ b α 3 = b ′ + 1 ℓ α 4 . (5.45)
With these constraints, the Killing spinor equations simplify, and can be solved to give
α 0 = λ 1 + 2iζ wλ 2 , α 1 = 0 , (5.46)
α 2 = z 2 + iζz + ζ 2 4 + iδ 4z 2 + ζ 2 , α 12 = α 2 - λ 2 2 4z 2 + ζ 2 , (5.47)
where λ 1,2 ∈ C parameterize the two dimensional space of Killing spinors. Then the most general Killing spinor for these metrics is
ǫ = (λ 1 + 2iζ wλ 2 ) 1 -ℓλ 2 2z + iζ 2z -iζ e 1 +b (λ 1 + 2iζ wλ 2 ) e 2 - z 2 -iζz + ζ 2 4 -iδ 4z 2 + ζ 2 λ 2 e 1 ∧ e 2 . (5.48)
are particular cases of this larger class, and are obtained for α, β and γ constant. Note that also the ∂b = 0 and ∂ b = 0 subclasses fall into this family. Let's take for definiteness α, β, γ all anti-holomorphic, then b = b(z, w). The requirement that the integrability conditions allow for an extra Killing spinor, i.e. that they are of rank ≤ 2, in this case leads to several conditions. One of these is obtained from the minor of the last three lines of Ñwt and reads
2∂ z b + 1 l ∂ z b + 1 l ∂ 2 b + 1 b ∂b∂b -2∂(Φ + ξ)∂b ∂b = 0. (5.51)
This gives three different cases to be analysed, corresponding to the vanishing of the first three factors of this equation (vanishing of the fourth factor implies b = b(z) and hence brings one back to the previous section).
The vanishing of the first factor in (5.51) implies b = -z ℓ + ic(w), where c(w) is an arbitrary holomorphic function. These are the α(w) = 0 supersymmetric Kundt solutions of Petrov type II, describing gravitational and electro-magnetic waves propagating on anti-Nariai space-time [16] .
The remaining integrability conditions however imply α 1 = α 2 = α 12 = 0, in which case there is no second Killing spinor, or ∂c = 0. Therefore there are no new half BPS solutions with non constant c. In this class c constant is the half supersymmetric anti-Nariai spacetime and the other preserve only 1/4 of the supersymmetries.
The vanishing of the second factor in (5.51) implies b = -z 2ℓ + ic(w). In this case we are considering the β 2 = 4αγ supersymmetric Kundt solutions, describing gravitational and electro-magnetic waves propagating on AdS 4 spacetime [16] .
Again the remaining integrability equations have to solutions: α 1 = α 2 = α 12 = 0 or ∂c = 0. Hence, as in the previous case, we find that there are no harmonic deformations of AdS 4 preserving half supersymmetry.
Not considering the previous two special cases, the general solution represents expanding gravitational and electro-magnetic waves propagating on a Reissner-Nordström-Taub-NUT-AdS 4 spacetime [16] . When Im(β) = 0, the solution can be put in Robinson-Trautman form and is of Petrov type II.
The vanishing of the third factor in (5.51) is given by
∂ 2 b + 1 b ∂b∂b -2∂(Φ + ξ)∂b = 0 . (5.52)
only one differential constraint which needs to be satisfied for the existence of a second Killing spinor, i. e. for the matrices of integrability conditions to have rank 2, namely
∂ 2 X -1 -2∂ξ∂X -1 = 0 .
(5.56)
The above three differential equations can be integrated to
e 2ξ = -i K( w)∂X -1 , ∂X -1 = i ℓ 2 K(w) 1 4X 4 + L , (5.57)
where K(w) is an arbitrary holomorphic function and L is a real constant. The function K(w) corresponds to the freedom to choose holomorphic coordinates on the twodimensional space, and hence it can be gauged away. A convenient gauge choice will be K(w) = iℓ. Note that, for this choice, the imaginary part of the right hand side of the last equation vanishes, and therefore that ∂ y X = 0. For L = 0, (5.57) can be integrated to give
X 3 = 3x 2ℓ , (5.58)
which is (up to a rescaling of the coordinate x) the example given above with α = 1 3 . This was already found to be 1/2 supersymmetric in [16] . Here we find that this solution is a special case of the most general possibility.
For other values of the constant L it is convenient to use X as a new coordinate instead of solving for X(x). From (4.18) and (4. 19 ) it follows that σ can be chosen to be σ = dy 4X 4 .
(5.59)
Then the metric reads
ds 2 = -4X 2 dt + dy 4X 4 2 + 1 4X 2 dz 2 + ℓ 2 dX 2 X 2 (1 + 4LX 4 ) + 1 + 4LX 4 4X 6 dy 2 .
(5.60)
Finally, from (4.21) we obtain the gauge field strength
F = 2dt ∧ dX . (5.61)
Note that the geometry (5.60) is generically of Petrov type D, and becomes of Petrov type N for L = 0. Now let us turn our attention to the form of the second Killing spinor. First of all, the integrability conditions imply that it takes the form
α T = (β 1 , β 2 , iX 3 e ξ β 2 , iX 3 e ξ β 2 ) ,
where β 1 and β 2 are arbitrary space-time dependent functions. The Killing spinor equations (4.31) yield
β 1 = λ 1 -1 2 λ 2 b -2 , β 2 = λ 2 b -2
, where λ 1 and λ 2 are integration constants. This implies that the new Killing spinor takes the form ǫ = λ 1 ǫ 1 + λ 2 ǫ 2 , where
ǫ 1 = 1 + iXe 2 , ǫ 2 = 1 2 X -2 (1 -iXe 2 ) + 1 4 X -4 + L (e 1 -iXe 1 ∧ e 2 ) .
(5.62)
Note that G 0 = 0 as well in this class.
One interesting aspect of the second Killing spinor ǫ 2 is the norm of its associated Killing vector V µ = D(ǫ 2 , Γ µ ǫ 2 ). We find V µ V µ = -4X 2 L 2 , hence the second Killing spinor is indeed null for the case L = 0, as was noticed before, while it is timelike for L = 0. In the latter case, to understand whether the solution belongs also to the null class of supersymmetric solutions, we have therefore to study the most general linear combination of the two Killing spinors. The Killing vector Ṽ constructed from
ǫ = λ 1 ǫ 1 + λ 2 ǫ 2 has norm Ṽ 2 = 1 X 2 λ1 λ 2 -λ 1 λ2 2 -4X 2 L|λ 1 | 2 + |λ 2 | 2 2 ,
which can vanish only if L ≤ 0. We have therefore three cases:
1. L > 0, pure timelike class, Petrov type D.
2. L = 0, belongs to both null and timelike classes, Petrov type N. This is the homogeneous half BPS pp-wave in AdS. (In the terminology of [16] it has a wave profile G α with α = 0).
3. L < 0, belongs to both null and timelike classes, Petrov type D.
Actually the solutions (5.60) with L > 0 can be cast into a simpler form. This is done by trading the coordinate y for a new variable ψ = Lyt. For convenience, let us also introduce the Schwarzschild coordinate r and rescale z,
r = - ℓ √ LX , ζ = 1 2 √ L z .
(5.63)
In the new coordinates, the metric and the gauge field strength read
ds 2 = - r 2 ℓ 2 + q 2 e r 2 dt 2 + dr 2 r 2 ℓ 2 + q 2 e r 2 + r 2 ℓ 2 dψ 2 + dζ 2 , F = q e r 2 dt ∧ dr , (5.64)
where we have defined q e = 2ℓ/ √ L. This is precisely the half BPS solution obtained in [36] , the massless limit of an electrically charged toroidal black hole, which forms a naked singularity. It is also interesting to note that the charge q e diverges in the L → 0 limit. This limit is naively singular in these coordinates, but it can be taken if we perform a Penrose limit [37, 38] . The existence of this limit explains why we obtained a one-parameter family of geometries (5.60) connecting the massless limit of toroidal black holes and a pp-wave. Indeed, define the new coordinates (X + , X -, R, Z) and the rescaled charge Q e by
ψ + t = 2ǫ 2 X + , ψ -t = 2X -, r = 1 ǫR , ζ = ǫZ , q e = Q e ǫ .
(5.65)
Then, the singular limit ǫ → 0 yields is a regular solution of the theory and corresponds to the half supersymmetric solution (5.60) with L = 0,
ds 2 = ℓ 2 R 2 4 dX + dX -- Q 2 e R 4 ℓ 6 dX -2 + dR 2 + dZ 2 , F = Q e ℓ 2 dX -∧ dR . (5.66)
In the procedure, we have blown up the metric in the neighborhood of a geodesic with ψ + t constant near the boundary r → ∞ of AdS. We now turn to the L < 0 case, which is both timelike and lightlike. Let us define L = -µ 2 . We can perform a coordinate transformation inspired from the previous one,
ψ = Ly -t , r = - ℓ µX , ζ = µ 2 z , (5.67)
under which the metric and the field strength become
ds 2 = r 2 ℓ 2 - q 2 e r 2 dt 2 + dr 2 r 2 ℓ 2 -q 2 e r 2 + r 2 ℓ 2 -dψ 2 + dζ 2 , F = q e r 2 dt ∧ dr , (5.68)
where we have defined q e = 2ℓ/µ. We see that this is the precisely the metric for L > 0 after the double analytic continuation t → it , ψ → iψ , q e → -iq e .
(5.69)
This solution represents therefore a bubble of nothing in AdS [39] [40] [41] [42] . Note that the metric is singular for r = √ ℓq e . One should compactify t, in such a way to eliminate the conical singularity on the (t, r) hypersurface. Then, if we compactify also ζ, this S 1 will have a minimal radius for r = √ ℓq e (the boundary of the bubble of nothing) and then grow with r. Note that for r → ∞ one locally recovers AdS spacetime, and that the L = 0 solutions can again be understood as a Penrose limit of this metric.
We can now generate new supersymmetric solutions by acting with the PSL(2, R) symmetry group (4.28)-(4.29) on the known ones. It is easy to check that the AdS 4 and AdS 2 ×H 2 solutions are invariant under this group (although it acts non trivially on the Killing spinors). Its action on the b = b(z) subfamily of the RNTN-AdS 4 solutions was studied in [16] , where it was shown that it acts non trivially on the charges, by mixing them. Here we want to apply it to the imaginary b solutions of the previous paragraph.
The new solution solution of the supersymmetry equations ( 4 .22)-(4.23) generated by the transformation (4.28)-(4.29) is b = -γ 2 Xz 2 2γ 2 ℓXz + i , e 2( Φ+ξ) = γ 4 z 4 4X 4 1 + 4LX 4 , (5.70) where, without loss of generality, we eliminated α by means of a translation of z foot_19 , and dropped the prime of the new coordinate z ′ . The shift function is then determined by solving equations (4.18) and (4.19),
σ x = 0 , σ y = 1 + 4LX 4 4γ 2 X 4 z 2 + γ 2 ℓ 2 X 2 .
(5.71)
Then, defining the new coordinates (T, σ, p, q) through
T = t 2ℓ 2 γ 2 , σ = y 2 , p = - ℓ X , q = 2ℓ 2 γ 2 z , (5.72)
the metric reads
ds 2 = - Q(q) q 2 + p 2 dT + P (p) q 2 + p 2 ℓ 2 dσ 2 + q 2 + p 2 Q(q) dq 2 + q 2 + p 2 P (p) dp 2 + 1 ℓ 4 (q 2 + p 2 )P (p) dσ 2 , (5.73)
with
Q(q) = q 4 ℓ 2 , P (p) = 1 ℓ 2 p 4 + 4Lℓ 2 , (5.74)
and the gauge field (4.21) is
F = d pq 2 ℓ(q 2 + p 2 )
∧ dT + d 4ℓLp q 2 + p 2 ∧ dσ .
(5.75)
The form of the metric suggests some connection with the Plebanski-Demianski family of solutions, and indeed these geometries are of Petrov type D for L = 0, and of Petrov type N for L = 0, but we were not able to find the precise relation. Note also that the parameter γ has been reabsorbed in the new variables, and we are left with a one-parameter (L) family of solutions.
The left hand side of the necessary condition (4.34) for the existence of a second Killing spinor reads, for this solution,
- 9iX 4 (1 + 4LX 4 ) ℓ 2 (1 + 4γ 4 ℓ 2 X 2 z 2 ) 4 γ 2 (5.
76) which clearly vanishes only for γ = 0, i.e. if the PSL(2, R) transformation is trivial. Therefore, the new solutions (5.73)-(5.75) preserves only 1/4 of the supersymmetries, and we explicitly see that the PSL(2, R) transformations can break any additional supersymmetry. Also note that if we perform the PSL(2, R) transformation adapting the original metric to a different Killing spinor, we could in principle end up with other supersymmetric solutions.
Surprisingly, we find that the L = 0 solution can be cast in the Lobatchevski wave form, even though it only has a time-like Killing spinor. This can be seen by trading the coordinates (q, p) for (x, z) defined by
x = ℓ 3 2 1 q 2 - 1 p 2 , z = ℓ 3 qp , (5.77)
in the metric (5.73) with L = 0, which becomes
ds 2 = ℓ 2 z 2 -2 dT dσ + z 2 2ℓ √ x 2 + z 2 x - √ x 2 + z 2 x + √ x 2 + z 2 dT 2 + dz 2 + dx 2 .
(5.78)
The field strength can be easily obtained from equation (5.75) but the result is not particularly enlightening and therefore we do not report it. This metric represents a 1/4 BPS Lobatchevski wave, whose Killing spinor falls in the timelike class. This does not contradict the results obtained in the null case, since the null Lobatchevski had a field strength (3.6) of the form F = φ ′ (T )dT ∧ dz, while this solution has a much more complicated gauge field. It is however interesting to note that the solutions of the null case do not exhaust all possible supersymmetric Lobatchevski waves. where we have defined k = 4αγβ 2 and ∆ = 4∂ ∂. Interestingly, as shown in [16] , this system of equations follows from the dimensionally reduced Chern-Simons action [43, 44] ,
S = d 2 x (2) g (2) Rη + η 3 , (5.81)
if we use the conformal gauge (2) g ij dx i dx j = e 2ξ (dx 2 + dy 2 ) and η is the curl of a vector potential, (2) g ǫ ij η = ∂ i A j -∂ j A i . To obtain equations (5.80) we vary the action with respect to A i and ξ. When varying the dimensionally reduced Chern-Simons action with respect to g ij there is however an additional equation to (5.80) .
Using the results of Grumiller and Kummer [48] , one obtains the most general solution to the dimensionally reduced Chern-Simons system [16]
e 2ξ = L ℓ 4 - k 2 η 2 + 1 4 η 4 , (5.82)
where L is an integration constant and dη = e 2ξ dx. Trading the coordinate x for η, we get the following configuration of the fields
ds 2 = - 4 ℓ 2 P 2 2 P ′2 2 + η 2 [dt + σ] 2 + ℓ 2 4 P ′2 2 + η 2 P 2 2 dz 2 + P 2 2 e -2ξ dη 2 + e 2ξ dy 2 , A = 2 ℓ P 2 η P ′2 2 + η 2 [dt + σ] + ℓ 4 Vdy -i ℓ 4 d log b b , (5.83)
where P 2 (z) = αz 2 + βz + γ, k is defined as above and the shift function reads As can be seen from the Poisson bracket (4.34), the only possibility to have 1/2 supersymmetry is α = 0 and hence k ≤ 0. In fact, starting from any solution with k ≤ 0, one can always obtain α = 0 by an appropriate PSL(2, R) transformation. The these equations in a number of subcases in section 5, and thereby found several new solutions, like the bubbles of nothing in AdS 4 , already obtained in the null formalism, and their PSL(2, R)-transformed configurations. Furthermore, our results showed that the generalized holonomy in the case of one preserved complex supercharge is contained in A(3, C), supporting thus the classification scheme of [4] .
σ = ℓ 2 2 αη 2 + e 2ξ P 2 dy . ( 5
In addition, the time-dependence of a second time-like Killing spinor was shown to be an overall exponential factor with coefficient G 0 in section 4.4. In the case G 0 = 0 these equations have been solved in full generality, up to a second order ordinary differential equation. We expect this class to comprise a large number of interesting 1/2-BPS solutions. Indeed, all the examples of section 5 either have vanishing G 0 or can be transformed to that case by a coordinate transformation.
There are several interesting points that remain to be understood. First of all, it would be desirable to get a deeper insight into the underlying geometric structure in the case of U(1) invariant spinors. In five dimensions, spacetime is a fibration over a four-dimensional Hyperkähler or Kähler base for ungauged and gauged supergravity respectively [8, 12] , whereas in four-dimensional ungauged supergravity one has a fibration over a three-dimensional flat space [5] . This suggests that the base for D = 4 gauged supergravity might be an odd-dimensional analogue of a Kähler manifold, i. e. , a Sasaki manifold. From the equations (4.22) and (4.23) this is not obvious. Secondly, in [16] , a surprising relationship between the equations (4.22), (4.23) governing 1/4 BPS solutions and the gravitational Chern-Simons theory [43] was found. Why such a relationship should exist is not clear at all, and deserves further investigations.
The third point concerns preons, which were conjectured in [45] to be elementary constituents of other BPS states. In type II and eleven-dimensional supergravity, it was shown that imposing 31 supersymmetries implies that the solution is locally maximally supersymmetric [27, 30, 46] . Similar results in four-and five-dimensional gauged supergravity were obtained in [28, 29] . This implies that preonic backgrounds are necessarily quotients of maximally supersymmetric solutions. While M-theory preons cannot arise by quotients [47] , it remains to be seen if 3/4 supersymmetric solutions to N = 2, D = 4 or D = 5 gauged supergravities really do not exist. The only maximally supersymmetric backgrounds in these theories are AdS 4 [13] and AdS 5 [12] respectively, so the putative preonic configurations must be quotients of AdS.
Finally, it would be interesting to apply spinorial geometry techniques to classify all supersymmetric solutions of four-dimensional N = 2 matter-coupled gauged supergravity. Work in this direction is in progress [49] . on U ⊗ C, and then extend it to ∆ c . The Spin(3,1) invariant Dirac inner product is then given by D(η, θ) = Γ 0 η, θ .
(A.3)
In many applications it is convenient to use a basis in which the gamma matrices act like creation and annihilation operators, given by
Γ + η ≡ 1 √ 2 (Γ 2 + Γ 0 ) η = √ 2 e 2 ⌋η , Γ -η ≡ 1 √ 2 (Γ 2 -Γ 0 ) η = √ 2 e 2 ∧ η , Γ • η ≡ 1 √ 2 (Γ 1 -iΓ 3 ) η = √ 2 e 1 ∧ η , Γ •η ≡ 1 √ 2 (Γ 1 + iΓ 3 ) η = √ 2 e 1 ⌋η . (A.4)
The Clifford algebra relations in this basis are {Γ A , Γ B } = 2η AB , where A, B, . . . = +, -, •, • and the nonvanishing components of the tangent space metric read η
+-= η -+ = η •• = η •• = 1.
The spinor 1 is a Clifford vacuum, Γ + 1 = Γ •1 = 0, and the representation ∆ c can be constructed by acting on 1 with the creation operators Γ + = Γ -, Γ • = Γ • , so that any spinor can be written as
η = 2 k=0 1 k! φ ā1 ...ā k Γ ā1 ...ā k 1 , ā = +, • .
The action of the Gamma matrices and the Lorentz generators Γ AB is summarized in the table 6 .
1 e 1 e 2 e 1 ∧ e 2 Γ + 0 0 √ 2 - √ 2e 1 Γ - √ 2e 2 - √ 2e 1 ∧ e 2 0 0 Γ • √ 2e 1 0 √ 2e 1 ∧ e 2 0 Γ • 0 √ 2 0 √ 2e 2 Γ +- 1 e 1 -e 2 -e 1 ∧ e 2 Γ •• 1 -e 1 e 2 -e 1 ∧ e 2 Γ +• 0 0 -2e 1 0 Γ +• 0 0 0 2 Γ -• -2e 1 ∧ e 2 0 0 0 Γ -• 0 2e 2 0 0
Table 6: The action of the Gamma matrices and the Lorentz generators Γ AB on the different basis elements.
ψ |b| c 2 c * 12 σ w -c 12 c * 0 σ w + |b| 2 (c 0 c * 1 σ w -c 1 c * 0 σ w ) + e ψ 4|b| (c 0 c * 2 + c 1 c * 12 -c 2 c * 0 -c 12 c * 1 ) dw ∧ d w . (B.6)
Given the first Killing spinor of the form ǫ 1 = 1 + be 2 and the second Killing spinor ǫ 2 = c 0 1 + c 1 e 1 + c 2 e 2 + c 12 e 1 ∧ e 2 , one can also construct mixed bilinears of the type D(ǫ 1 , Γ ••• ǫ 2 ), which verify the same differential equations as the bilinears built from the original two Killing spinors:
f = -i( bc 0 -c 2 ) , ĝ = bc 0 + c 2 , (B.7) V = 1 2b (c 2 + bc 0 ) (dt + σ) + 1 2b (c 2 -bc 0 ) dz + 1 |b| e ψ bc 1 -c 12 d w , (B.8) B = 1 2b (c 2 -bc 0 ) (dt + σ) + 1 2b (c 2 + bc 0 ) dz + 1 |b| e ψ bc 1 + c 12 d w . (B.9)
C. The case P ′ = 0
In section 4.3, we simplified the equations for the second Killing spinor under the assumption P ′ = 0, where P = e -2ψ b∂b. Here we consider the case P ′ = 0. To this end, we need the following subset of the Killing spinor equations (4.31): 3) and deriving with respect to t, one gets ∂b ∂ t ψ 1 = ∂b ∂ t ψ 1 = 0. When ∂ t ψ 1 = 0, this means that ∂b = ∂b = 0, so b = b(z), which is a case analyzed in section 5.1. If instead ∂ t ψ 1 = 0, all the ψ i are independent of t, and the Killing spinor equations reduce to the system (4.38) to (4.49) with G 0 = 0.
∂ + ψ 2 - √ 2r 2 b′ b + 1 ℓ b ψ 2 - √ 2r 2 b ′ b + 1 ℓb ψ 12 = 0 , (C.1) ∂ + ψ 12 -re -ψ ∂ • ln b ψ 1 - √ 2r 2 2 r ′ r + 1 ℓ b ψ 12 = 0 , (C.2) ∂ -ψ 2 + 1 r e -ψ ∂ • ln b ψ 1 - √ 2 2 r ′ r + 1 ℓb ψ 2 = 0 , (C.3) ∂ -ψ 12 - √ 2 b′ b + 1 ℓ b ψ 2 - √ 2 b ′ b + 1 ℓb ψ 12 = 0 , (C.4) re -ψ ∂ • 1 r 2 e 2ψ ψ 2 - √ 2 b ′ b + 1 ℓb ψ 1 = 0 , (C.5) re -ψ ∂ • 1 r 2 e 2ψ ψ 12 + √ 2 b′ b + 1 ℓ b ψ 1 = 0 . (C.6) If P ′ = 0, ( 4
In the case ∂P = 0, consider the integrability condition
ψ 1 Q ′ + ψ -∂Q = 0 , (C.8)
where Q = e -2ψ b∂ b, following from the first line of Ñwt . As long as Q ′ = 0, with the same reasoning as in section 4.3, one obtains the system (4.38) to (4.49). If Q ′ = 0, (C.8) implies ψ -= 0 or ∂Q = 0. The case ψ -= 0 was already considered above, so the only remaining case is P ′ = ∂P = Q ′ = ∂Q = 0. For P = Q = 0 we get again b = b(z), so without loss of generality we can assume P = 0 or Q = 0. Suppose that Q = 0, P = 0, so b = b(w, z). Take the logarithm of e -2ψ b∂b = P ( w), derive with respect to z, use (4.15), and apply ∂. This leads to ∂b = 0, which is a contradiction to the assumption P = 0. In the same way one shows that P = 0, Q = 0 is not possible, so that both P and Q must be nonvanishing. Now use the third row of Ñ wt , which leads to Qψ 2 = 0 and hence ψ 2 = 0. Finally, the last row of Ñ wt yields ψ -= 0, i. e. , the case already considered above. Hence, the conclusion is that in the case P ′ = 0, the second Killing spinor either has G 0 time-dependence of the form (4.37), or leads to solutions with b = b(z). The latter are treated separately in section 5.1. As can be found there, all 1/2-BPS solutions with b = b(z) also have second Killing spinors with G 0 time-dependence of the form (4.37). Hence this time-dependence is a completely general result 20 for second Killing spinors in the time-like case. We thus conclude that ψ1 ψ ′ 1 -ψ′ 1 ψ 1 = 0, and hence ψ 1 = ζ(z)e iθ 0 where θ 0 is a constant and ζ(z) is a real function. Sending ψ i → e -iθ 0 ψ i we can take ψ 1 real and non-negative without loss of generality. Let us now consider the case where both ψ 1 and ψ -are non-vanishing. This allows to introduce new coordinates Z, W, W such that where α is a real integration constant. Equations (D.9) and (D.11) are then identically satisfied. Solving (D.14) for χ and plugging into (D.13) yields finally the ordinary differential equation (4.50), which determines half-supersymmetric solutions with G 0 = 0. Putting together all our results, we obtain (4.52) for the metric. Note that in the case γ = 0 one can always set γ = 1 by rescaling the coordinates. The second Killing spinor for these backgrounds is given by α T = (α 0 , ρ -2 e -γZ/ℓ , χ + 1 2ρ e H , χ -1 2ρ e H ) , where α 0 = -2γt ℓ + α0 (Z, w, w) , and α0 is a solution of the system
∂ Z α0 = 1 ψ 1 ρ 2 ρ ρ -i φ + γ 2ℓ , ∂ α0 = - 2γ ℓ σ w + 1 ψ 1 ρ 2 - ρ ρ + i φ , (D.15) ∂ α0 = - 2γ ℓ σ w + 1 ψ 1 ρ 2 - ρ ρ + i φ + γχ e 2H ℓψ 1 ρ 2 .
It is straightforward to verify that the integrability conditions for this system are already implied by (D.9), (D.10) and (D.12). Consider now the case ψ -= 0. From the difference of equations where Y (w, w) is some real function to be determined.
We thus have to solve just for the ansatz (D.16). Equation holds. We conclude that a solution to the system (D. 19 ), (D.20) describes a 1/2-BPS configuration of the "gravitational Chern-Simons" system discussed in [16] . If C(w) = 0 then necessarily also Y = 0 so that we are left with AdS. If C(w) = 0 then we can define new variables W and W such that As what we did in the previous case, we can set C(w) = 1 using the residual gauge invariance w → W (w), ψ → ψ = ψ -1 2 ln(dW/dw) -1 2 ln(d W /d w) leaving invariant the metric e 2ψ dwd w. We can thus take W = w without loss of generality, and get where L is a real constant and k = -1 21 . We can thus use Y as a new coordinate, instead of i(ww). Call X = w + w, so that the solution reads
ds 2 = - 4 ℓ 2 z 2 1 + Y 2 dt
+ ℓ 2 2z P C (Y )dX 2 + ℓ 2 4 1 + Y 2 z 2 dz 2 + z 2 P C (Y )dX 2 + dY 2 P C (Y ) , A = 2 ℓ z Y 1 + Y 2 dt + ℓY P C (Y ) 1 + Y 2 -1 4 1 + Y 2 dX + ℓ 2 dY 1 + Y 2 . (D.32) | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The supersymmetric solutions of N = 2, D = 4 minimal ungauged and gauged supergravity are classified according to the fraction of preserved supersymmetry using spinorial geometry techniques. Subject to a reasonable assumption in the 1/2supersymmetric time-like case of the gauged theory, we derive the complete form of all supersymmetric solutions. This includes a number of new 1/4-and 1/2-supersymmetric possibilities, like gravitational waves on bubbles of nothing in AdS 4 ."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Throughout the history of string and M-theory an important part in many developments in the subject has been played by supersymmetric solutions of supergravity, i.e. by backgrounds which admit a number of Killing spinors ǫ which are parallel with respect to the supercovariant derivative foot_0 : D µ ǫ = 0. Due to their ubiquitous role it has long been realised that it would be advantageous to have classifications of all supersymmetric solutions of a given theory.\n\nFor purely gravitational backgrounds the supersymmetric possibilities follow from the Berger classification of the possible Riemannian holonomies [1] (see [2, 3] for an extension to the Lorentzian case). However, in the presence of additional force fields (carried by e. g. scalars, gauge potentials or a cosmological constant) it has proven very difficult to obtain knowledge of all supersymmetric possibilities.\n\nThe reason for the complication in the presence of additional fields lies in the holonomy of the supercurvature R µν = D [µ D ν] . For purely gravitational backgrounds the holonomy of the supercurvature is generically given by H = Spin(d -1, 1) in d dimensions, and hence coincides with the Lorentz group. In such cases the Lorentz gauge freedom allows one to choose constant Killing spinors. Another simplification is that if there is one Killing spinor with a specific stability subgroup, i.e. it is invariant under some Lorentz subgroup, all other spinors with the same stability subgroup are Killing as well.\n\nFor more general solutions including fields other than gravity, the holonomy is generically extended to a larger group H ⊃ Spin(d -1, 1). For example, in the present paper we consider gauged minimal four-dimensional N = 2 supergravity, which has H = GL(4,C) [4] . In such cases one cannot choose constant Killing spinors nor are all spinors with the same stability subgroup automatically Killing. For these reasons the classification of the backgrounds that allow for Killing spinors is more convoluted, or richer, in such cases. For a long time the only classification available was in ungauged minimal four-dimensional N = 2 supergravity [5, 6] , which has H = SL (2,H) .\n\nA new impulse was given to the subject with the introduction of G-structures and the method of spinor bilinears to solve the Killing spinor equations [7] . In this approach, space-time forms are constructed as bilinears from a Killing spinor and one analyses the constraints that these forms imply for the background. Using this framework, a number of complete classifications [8] [9] [10] and many partial results (see e.g. [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] for an incomplete list) have been obtained. By complete we mean that the most general solutions for all possible fractions of supersymmetry have been obtained, while for partial classifications this is only available for some fractions. Note that the complete classifications mentioned above involve theories with eight supercharges and H = SL(2,H), and allow for either half-or maximally supersymmetric solutions.\n\nAn approach which exploits the linearity of the Killing spinors has been proposed [22] under the name of spinorial geometry. Its basic ingredients are an explicit oscillator basis for the spinors in terms of forms and the use of the gauge symmetry to transform them to a preferred representative of their orbit. In this way one can construct a linear system for the background fields from any (set of) Killing spinor(s) [23] . This method has proven fruitful in e.g. the challenging case of IIB supergravity [24] [25] [26] . In addition, it has been adjusted to impose 'near-maximal' supersymmetry and thus has been used to rule out certain large fractions of supersymmetry [27] [28] [29] [30] . Finally, a complete classification for type I supergravity in ten dimensions has been obtained [32] .\n\nIn the present paper we would like to address the classification of supersymmetric solutions in four-dimensional minimal N = 2 supergravity. As will also be reviewed in section 2, the ungauged case has been classified completely [5, 6] . For the gauged case, the discussion of 1/4 supersymmetry splits up in a time-like and a light-like class (depending on the causal nature of the Killing vector associated to the Killing spinor). The time-like class is completely specified by a single complex function depending on three spatial coordinates b = b(z, w, w), subject to a second-order differential equation which can not be solved in general [13] . The light-like class can be given in all generality, and in addition its restriction to 1/2-BPS solutions has been derived [16] . Furthermore, there are no backgrounds with 3/4 supersymmetry [29] and AdS 4 is the unique possibility with maximal supersymmetry. Therefore the remaining open question concerns half-supersymmetric backgrounds in the gauged theory 2 .\n\nIn the following, we will first re-analyse the 1/4-supersymmetric backgrounds using the method of spinorial geometry, and in fact find an additional possibility in the light-like case: a half supersymmetric bubble of nothing in AdS 4 and its Petrov type II generalization, a new 1/4 BPS configuration that has the interpretation of gravitational waves propagating on the bubble of nothing. This completes the analysis of the null class in all its generality. Then we will derive the constraints for halfsupersymmetric backgrounds for the timelike class. Subject to a single assumption on the time-dependence of the second Killing spinor these will be solved in general, up to a second order ordinary differential equation. The assumption will be justified by solving the full set of conditions in a number of examples which illustrate the possible spatial dependence of b. All these cases turn out to have time-dependence of the assumed form. The different examples are:\n\n• the b = b(z) family of solutions, comprising part of the Reissner-Nordström-Taub-NUT-AdS 4 backgrounds,\n\n• waves on the previous backgrounds with b = b(z, w),\n\n• solutions with b imaginary and their P SL(2, R) transformed counterparts,\n\n• solutions of the dimensionally reduced gravitational Chern-Simons model that can be embedded in the equations for a timelike Killing spinor [16] .\n\nWe determine when these backgrounds preserve 1/2 supersymmetry and provide the explicit Killing spinors. Moreover, in the subcases consisting of AdS 4 and AdS 2 ×H 2 , the action of the isometries of these backgrounds on the Killing spinors is given explicitly. The outline of this paper is as follows. In section 2, we discuss the orbits of Killing spinors and review the known classification results in the theory at hand. In section 3, we go through the complete classification of the null class. In section 4, we discuss the constraints for 1/4 and 1/2 supersymmetry in the timelike class. We derive the time-dependence of the second Killing spinor and solve the equations for the case of linear time-dependence (G 0 = 0). A number of examples of the 1/2 BPS timelike class are provided in section 5. Finally, in section 6 we present our conclusions and outlook. In appendix A we review our notation and conventions for spinors, while in appendix B the associated bilinear forms are given. Appendix C deals with the special case P ′ = 0, to be defined in section 4.4. Finally, in appendix D, we will give the details of the G 0 = 0 case."
},
{
"section_type": "OTHER",
"section_title": "G-invariant Killing spinors in 4D",
"text": "2.1 Orbits of Dirac spinors under the gauge group In order to obtain the possible orbits of Spin (3, 1) in the space of Dirac spinors ∆ c , we first consider the most general positive chirality spinor foot_2 a1 + be 12 (a, b ∈ C) and determine its stability subgroup. This is done by solving the infinitesimal equation α cd Γ cd (a1 + be 12 ) = 0 .\n\n(2.1)\n\nFirst of all, notice that a1 + be 12 is in the same orbit as 1, which can be seen from e γΓ 13 e ψΓ 12 e δΓ 13 e hΓ 02 1 = e i(δ+γ) e h cos ψ 1 + e i(δ-γ) e h sin ψ e 12 .\n\nThis means that we can set a = 1, b = 0 in (2.1), which implies then α 02 = α 13 = 0, α 01 = -α 12 , α 03 = α 23 . The stability subgroup of 1 is thus generated by\n\nX = Γ 01 -Γ 12 , Y = Γ 03 + Γ 23 . (2.2)\n\nOne easily verifies that X 2 = Y 2 = XY = 0, and thus exp(µX + νY ) = 1 + µX + νY , so that X, Y generate R 2 . Spinors of negative chirality are composed of odd forms, i.e. ae 1 + be 2 . One can show in a similar way that they are in the same orbit as e 1 , and the stability subgroup is again R 2 , with the above generators X, Y .\n\nFor definiteness and without loss of generality we will always assume that the first Killing spinor has a non-vanishing positive chirality component, and use (part of) the Lorentz symmetry to bring this to the form 1. Hence we can write a general spinor as 1 + ae 1 + be 2 . Now act with the stability subgroup of 1 to bring ae 1 + be 2 to a special form:\n\n(\n\n1 + µX + νY )(1 + ae 1 + be 2 ) = 1 + be 2 + [a + 2b(ν -iµ)]e 1 .\n\nIn the case b = 0 this spinor is invariant, so the representative is 1 + ae 1 , with isotropy group R 2 . If b = 0, one can bring the spinor to the form 1 + be 2 , with isotropy group I.\n\nThe representatives foot_3 together with the stability subgroups are summarized in table 1.\n\nIn the ungauged theory, we therefore can have the following G-invariant Killing spinors. The R 2 -invariant Killing spinors are spanned by 1 and e 1 and there can be up to four of these. The I-invariant Killing spinors are spanned by all four basis elements and there can be up to eight of these. In the first two case, the vector V a bilinear in the spinor ǫ is lightlike, whereas in the last case it is timelike, see table 1 . The existence of a globally defined Killing spinor ǫ, with isotropy group G ∈ Spin(3,1), gives rise to a G-structure. This means that we have an R 2 -structure in the null case and an identity structure in the timelike case.\n\nIn U(1) gauged supergravity, the local Spin(3,1) invariance is actually enhanced to Spin(3,1) × U(1). Thus, in order to obtain the stability subgroup, one determines the Lorentz transformations that leave a spinor invariant up to an arbitrary phase factor, which can then be gauged away using the additional U(1) symmetry. For the representative 1, one gets in this way an isotropy group generated by X, Y and Γ 13 obeying\n\n[Γ 13 , X] = -2Y , [Γ 13 , Y ] = 2X , [X, Y ] = 0 ,\n\ni. e. G ∼ = U(1)⋉R 2 . For ǫ = 1 + ae 1 with a = 0, the stability subgroup R 2 is not enhanced, whereas the I of the representative 1 + be 2 is promoted to U(1) generated by\n\nΓ 13 = iΓ •• . The Lorentz transformation matrix a AB corresponding to Λ = exp(iψΓ •• ) ∈ U(1), with ΛΓ B Λ -1 = a A B Γ A , has nonvanishing components a +-= a -+ = 1 , a •• = e 2iψ , a •• = e -2iψ . (2.3)\n\nFinally, notice that in U(1) gauged supergravity one can choose the function a in 1+ae 1 real and positive: Write a = R exp(2iδ), use\n\ne δΓ 13 (1 + ae 1 ) = e iδ 1 + e -iδ ae 1 = e iδ (1 + Re 1 ) ,\n\nand gauge away the phase factor exp(iδ) using the electromagnetic U (1) .\n\nǫ G ⊂ Spin(3,1) G ⊂ Spin(3,1) × U(1) V a = D(ǫ, Γ a ǫ) 1 R 2 U(1)⋉R 2 (1, 0, -1, 0) 1 + ae 1 R 2 R 2 (a ∈ R) (1 + |a| 2 , 0, -1 -|a| 2 , 0) 1 + be 2 I U(1) (1 + |b| 2 , 0, -1 + |b| 2 , 0)\n\nTable 1 : The representatives ǫ of the orbits of Dirac spinors and their stability subgroups G under the gauge groups Spin(3,1) and Spin(3,1) × U(1) in the ungauged and gauged theories, respectively. The number of orbits is the same in both theories, the only difference lies in the stability subgroups and the fact that a is real in the gauged theory. In the last column we give the vectors constructed from the spinors.\n\nIn the gauged theory the classification of G-invariant spinors is therefore slightly more complicated. There can be at most two U(1)⋉R 2 -invariant Killing spinors, spanned by 1. The four R 2 -invariant spinors are spanned by 1 and e 1 . Then there are the U(1)-invariant spinors, spanned by 1 and e 2 . Finally, for generic enough Killing spinors, one does not fall in any of the above classes and the common stability subgroup is I. Note that in the gauged theory the presence of G-invariant Killing spinors will in general not lead to a G-structure on the manifold but to stronger conditions. The structure group is in fact reduced to the intersection of G with Spin(3,1), and hence is equal to the stability subgroup in the ungauged theory.\n\nWe will now consider the possible supersymmetric solutions to the equation D µ ǫ = 0 in various sectors of N = 2, D = 4 in terms of the stability subgroup G of the Killing spinors."
},
{
"section_type": "OTHER",
"section_title": "The ungauged theory",
"text": "The supercovariant derivative of ungauged minimal N = 2 supergravity in four dimensions reads\n\nD µ = ∂ µ + 1 4 ω ab µ Γ ab + i 4 F ab Γ ab Γ µ . (2.4)\n\nAs mentioned in the introduction, a first point to notice is that there is no complex conjugation on the Killing spinor. Therefore, the number of supersymmetries that are preserved is always even: if ǫ is Killing, then so is iǫ.\n\nFirst consider purely gravitational solutions with F = 0. In this case the supercovariant connection truncates to the Levi-Civita connection and has Spin (3, 1) holonomy. This implies the following. If ǫ is Killing, then so are foot_4 Γ 3 * ǫ and Γ 012 * ǫ (where * denotes complex conjugation). Together, the operations i, Γ 3 * and Γ 012 * generate four linearly independent Killing spinors from any null spinor ǫ = 1 or ǫ = 1+ae 1 and eight from any time-like spinor ǫ = 1 + be 2 . This illustrates the general statement in the introduction: if the gauge group equals the holonomy, as in this case, then there is only one possible number of Killing spinors for every stability subgroup. Therefore there are only two classes of supersymmetric solutions, which are listed in table 2, and which consist of the gravitational wave and Minkowski space-time, respectively.\n\nG = \\ N = 4 8 R 2 √ × I × √\n\nTable 2: Gravitational solutions with G-invariant Killing spinors in the ungauged theory.\n\nNow let us also allow for fluxes F . The supercovariant connection no longer equals the Levi-Civita connection due to the flux term. In particular, this implies that Γ 012 * no longer commutes with D µ . However, this does still hold for the other operation: Γ 3 * ǫ is Killing provided ǫ is. The combined operations of i and Γ 3 * generate four linearly independent spinors from any null or time-like spinor. Thus the number of supersymmetries is always N = 4p, as illustrated in table 3 . Indeed the generalised holonomy of the supercovariant connection in the ungauged case is SL(2,H) [4] , consistent with the supersymmetries coming in quadruplets.\n\nG = \\ N = 4 8 R 2 √ × I √ √\n\nTable 3: General solutions with G-invariant Killing spinors in the ungauged theory.\n\nThe half-supersymmetric solution have been classified by Tod [5] and consist of the plane wave and the Israel-Wilson-Perjes metric, respectively. The maximally supersymmetric solutions are AdS 2 × S 2 and its Penrose limits, the Hpp wave and Minkowski space-time [6] ."
},
{
"section_type": "OTHER",
"section_title": "The gauged theory",
"text": "The supercovariant derivative of gauged minimal N = 2 supergravity in four dimensions reads\n\nD µ = ∂ µ + 1 4 ω ab µ Γ ab -iℓ -1 A µ + 1 2 ℓ -1 Γ µ + i 4 F ab Γ ab Γ µ . (2.5)\n\nDue to the gauging the structure of Γ-matrices is richer, but there still is no complex conjugation on the Killing spinor. Therefore, the number of supersymmetries that are preserved is always even: if ǫ is Killing, then so is iǫ. Again, we first consider the purely gravitational solutions. In this case the supercovariant derivative has SO (3, 2) holonomy. The operation Γ 012 * commutes with D µ and therefore generates additional Killing spinors. Together, the operations i and Γ 012 * generate four linearly independent Killing spinors from generic null or time-like spinors. The exception is the null spinor ǫ = 1 + e 1 , in which case ǫ and Γ 012 * are linearly dependent, and hence allows for two instead of four Killing spinors. The possibilities allowed for by this analysis of the supercovariant derivative can be found in table 4 .\n\nHowever, although all these entries are allowed for by the spinor orbit structure and the crude analysis of the supercurvature above, not all of them have an actual field theoretic realisation in supergravity. In other words, there are no solutions to the Killing spinor equations for all of the above sets of Killing spinors. The lightlike cases were considered in [16] : The 1/4-BPS case is the Lobatchevski wave while imposing more supersymmetries leads to the maximally supersymmetric AdS 4 solution (with\n\nG = \\ N = 2 4 6 8 U(1) ⋉ R 2 × × × × R 2 √ • × × U(1) × • × × I × • • √\n\nTable 4: Gravitational solutions with G-invariant Killing spinors in the gauged theory. Check marks indicate entries with actual solutions, while circles stand for allowed entries which are not realized.\n\n). The N = 4 and G = R 2 entry is thus effectively empty. In particular, this implies that imposing a single Killing spinor 1 + ae 1 with a = 1 leads to AdS 4 . Also note that the N = 6 and G = 1 entry must be empty since any time-like spinor plus 1+e 1 leads to maximal supersymmetry, while all other Killing spinors come in groups of four. The only remaining entries are N = 4 and G = U(1) or G = I. Using the results of [13, 16] , it is straightforward to show that in these purely gravitational timelike cases the geometry is given by\n\nds 2 = - z 2 + n 2 ℓ 2 (dt -2n cosh θdφ) 2 + ℓ 2 dz 2 z 2 + n 2 + (z 2 + n 2 )(dθ 2 + sinh 2 θdφ 2 ) ,\n\nwhere n = ±ℓ/2. But this is simply AdS 4 written as a line bundle over a threedimensional base manifold, so both N = 4 entries are empty as well. We conclude that there are no 1/2-supersymmetric gravitational solutions in the gauged theory, only the 1/4-supersymmetric Lobatchevski waves and maximally supersymmetric AdS 4 .\n\nWe now come to the general supersymmetric solutions in the gauged case. Due to the gauging and flux terms, neither Γ 012 * nor Γ 3 * commute with D µ . Therefore we have the cases as listed in table 5 . The supercovariant connection in the gauged case has generalized holonomy GL(4, C) [4] , again consistent with the supersymmetries coming in doublets.\n\nThe 1/4-BPS solutions with G = R 2 and G = U(1) were derived in [13] , and we will show there is no solution with G = U(1) ⋉ R 2 . In addition, it was shown in [16] that any additional supersymmetries in the null case are always timelike, i.e. end up in the N = 4 and G = 1 entry. Again, the N = 4 and G = R 2 entry is empty. It would be interesting to see if there is a nice explanation for this. In addition, the maximally supersymmetric case is always AdS 4 . Recently, it has been shown in [29] that the N = 6 and G = 1 entry is empty as well, because imposing three complex Killing spinors implies that the spacetime is AdS 4 and thus maximally supersymmetric.\n\nG = \\ N = 2 4 6 8 U(1) ⋉ R 2 • × × × R 2 √ • × × U(1) √ √ × × I × √ • √\n\nTable 5: General solutions with G-invariant Killing spinors in the gauged theory. Check marks indicate entries with actual solutions, while circles stand for allowed entries which are not realized.\n\nThe most general 1/2-BPS solution in the timelike case remains an open issue and will be studied in this paper."
},
{
"section_type": "OTHER",
"section_title": "Generalized holonomy",
"text": "In minimal gauged supergravity theories with eight supercharges, the generalized holonomy group for vacua preserving N supersymmetries, where N = 0, 2, 4, 6, 8, is GL(\n\n8-N 2 , C) ⋉ N 2 C\n\n8-N 2 [4] . To see this, assume that there exists a Killing spinor ǫ 1 . By a local GL(4, C) transformation, ǫ 1 can be brought to the form ǫ 1 = (1, 0, 0, 0) T . This is annihilated by matrices of the form\n\nA = 0 a T 0 A ,\n\nthat generate the affine group A(3, C) ∼ = GL(3, C) ⋉ C 3 . Now impose a second Killing spinor ǫ 2 = (ǫ 0 2 , ǫ 2 ) T . Acting with the stability subgroup of ǫ 1 yields\n\ne A ǫ 2 = ǫ 0 2 + b T ǫ 2 e A ǫ 2\n\n, where b T = a T A -1 (e A -1) .\n\nWe can choose A ∈ gl(3, C) such that e A ǫ 2 = (1, 0, 0) T , and b such that ǫ 0 2 + b T ǫ 2 = 0. This means that the stability subgroup of ǫ 1 can be used to bring ǫ 2 to the form ǫ 2 = (0, 1, 0, 0). The subgroup of A(3, C) that stabilizes also ǫ 2 consists of the matrices\n\n 1 0 b 2 b 3 0 1 B 12 B 13 0 0 B 22 B 23 0 0 B 32 B 33 ∈ GL(2, C) ⋉ 2C 2 .\n\nFinally, imposing a third Killing spinor yields GL(1, C) ⋉ 3C as maximal generalized holonomy group, which is however not realized in N = 2, D = 4 minimal gauged supergravity [16, 29] . It would be interesting to better understand why such preons actually do not exist. In section 4.3, we explicitely compute the generalized holonomy group for N = 2, D = 4 minimal gauged supergravity in the case N = 2 and show that it is indeed contained in A(3, C), supporting thus the classification scheme of [4] ."
},
{
"section_type": "OTHER",
"section_title": "Null representative 1 + ae 1",
"text": "In this section we will analyse the conditions coming from a single null Killing spinor. As we saw in section 2.1, there are two orbits of such spinors, one with representative ǫ = 1 and stability subgroup G = U(1)⋉R 2 and one with ǫ = 1 + ae 1 and G = R 2 .\n\nOwing to local U(1) gauge invariance, it is always possible to choose the function a real and positive, so in the following we set a = e χ , χ ∈ R. The Killing spinor equations become\n\n- i ℓ A + 1 2 Ω + e χ √ 2 1 ℓ + iφ E • -2iF +• E -= 0 , dχ + i ℓ A + 1 2 Ω + e -χ √ 2 1 ℓ -iφ E • + 2iF +• E -= 0 , ω -• + e χ √ 2 2iF -• E • + 1 ℓ -iφ E -= 0 , ω -• + e -χ √ 2 -2iF -• E • + 1 ℓ + iφ E -= 0 , (3.1)\n\nwhere φ ≡ F +-+ F •• and Ω ≡ ω +-+ ω •• . The conditions for the special U(1)⋉R 2 -orbit with ǫ = 1 can be obtained as the singular limit χ → -∞ of the above equations. Note however that, in this limit, the second line implies the constraint ℓ -1 -iφ = 0, while the fourth line leads to ℓ -1 +iφ = 0. Clearly, for ℓ -1 = 0 this does not allow for a solution. Hence, in the gauged theory, there are no backgrounds with U(1)⋉R 2 -invariant Killing spinors.\n\nThe only null possibility is therefore given by the R 2 -invariant Killing spinor ǫ = 1 + e χ e 1 . We will now analyse the above conditions for the generic case with χ finite. In fact, we will furthermore assume it is positive. This does not constitute any loss of generality since one can flip the sign of χ by changing chirality (a spinor 1 + e χ e 1 with χ negative is gauge equivalent to a spinor e 1 + e χ1 with χ = -χ positive), and hence the resulting background will not depend on this sign.\n\nFrom the last two equations one obtains the constraints\n\nF -• = F -• = 0 , φ = - i ℓ tanh χ (3.2)\n\non the field strength, as well as\n\nω -• = ω -• = - 1 √ 2ℓ cosh χ E - (3.3)\n\nfor the spin connection. (3.2) implies F +-= 0 and F •• = -i ℓ tanh χ. The first two equations of (3.1) yield then\n\nω +-= 2e χ H 3 E -- 1 ℓ e 2χ cosh χ E 1 , ω •• = 2i sinh χH 1 E -+ i ℓ cosh 2χ cosh χ E 3 , A = -ℓ cosh χH 1 E --sinh χE 3 , dχ = -2 cosh χH 3 E -+ 2 ℓ sinh χE 1 , (3.4)\n\nwhere\n\nE 1 = (E • + E •)/ √ 2, iE 3 = (E • -E •)/ √\n\n2, and we defined\n\nF +• + F +• √ 2 = H 1 , F +• -F +• √ 2 = iH 3 .\n\nIn order to proceed, we distinguish two subcases, namely dχ = 0 and dχ = 0.\n\nT -= dE -+ 2 ℓ E 1 ∧ E -, T + = dE + -E 1 ∧ ω +1 + E + ℓ + ω +3 ∧ E 3 , T 1 = dE 1 + E -∧ ω +1 + E + ℓ , T 3 = dE 3 + 1 ℓ E 1 ∧ E 3 -ω +3 ∧ E -. (3.5)\n\nFrom T -= 0 one gets E -∧dE -= 0, so by Fröbenius' theorem there exist two functions η and u such that locally E -= ηdu .\n\nPlugging this into T -= 0 yields\n\nη d log η + 2 ℓ E 1 ∧ du = 0 ,\n\nso that there exists a function ξ such that\n\nE 1 = - ℓ 2η dη + ξdu .\n\nThe gauge field and its field strength can now be written as\n\nA = -ℓηH 1 du , F = ℓ 2 H 1 dη ∧ du ,\n\nand the Bianchi identity F = dA implies\n\ndH 1 + 3 2 H 1 d log η ∧ du = 0 .\n\nThis means that H 1 η 3/2 can depend only on u,\n\nH 1 η 3/2 = - ϕ ′ (u) ℓ ,\n\nwhere the prefactor and the derivative were chosen in order to conform with the notation of [13] . Let us define a new coordinate x = -η -1/2 , so that\n\nE 1 = ℓ x dx+ξdu, E -= x -2 du and A = -xϕ ′ (u)du . (3.6)\n\nOne can now use part of the residual gauge freedom, given by the stability subgroup R 2 of the null spinor 1 + ae 1 , in order to simplify E 1 . To this end, consider an R 2 transformation with group element\n\nΛ = 1 + µX + νY ,\n\nwhere X and Y are given in (2.2). Defining α = µ + iν, this can also be written as\n\nΛ = 1 + αΓ +• + ᾱΓ +• . (3.7)\n\nGiven the ordering A, B = +, -, •, •, the Lorentz transformation matrix a AB corresponding to Λ ∈ R 2 ⊆ Spin(3,1) reads\n\na AB = 0 1 0 0 1 -4|α| 2 2 ᾱ 2α 0 -2 ᾱ 0 1 0 -2α 1 0 . ( 3.8)\n\nThe transformed vielbein\n\nα E A = a A B E B is thus given by α E • = E • -2αE -, α E 1 = E 1 - √ 2 (α + ᾱ) E -, α E • = E • -2 ᾱE -, α E 3 = E 3 + √ 2i (α -ᾱ) E -, α E -= E -, α E + = E + + 2 ᾱE • + 2αE • -4|α| 2 E -. (3.9)\n\nChoosing α + ᾱ = ξx foot_5 / √ 2, we can eliminate E 1 u , so one can set ξ = 0 without loss of generality. Note that this still leaves a residual gauge freedom associated to the imaginary part of α, which will be used below.\n\nFrom dT 3 = 0 we get d(ω +3 /x) ∧ du = 0, and thus there exist two functions β, β such that ω +3 = -xdβ + βdu .\n\nPlugging this into T 3 = 0 yields d(xE 3 + βdu) = 0, which is solved by\n\nE 3 = - ℓ x dy + βdu , (3.10)\n\nwhere y denotes some function that we shall use as a coordinate. Using the remaining gauge freedom (3.8) with Imα = -βx 2 /2 √ 2 allows to set also β = 0. The equation T 1 = 0 tells us that ω +1 + E + /ℓ = γdu for some function γ. Using this together with\n\nT + = 0, one shows that d E -∧ E + = - 2 x dx ∧ E -∧ E + ,\n\nwhich means that the surface described by E -and E + is integrable, so that\n\nE + = ℓ 2 G 2 du + hdV , (3.11)\n\nfor some functions G, h, V . The metric becomes then\n\nds 2 = 2E -E + + E 1 2 + E 3 2 = ℓ 2 x 2 Gdu 2 + 2h ℓ 2 dudV + dx 2 + dy 2 . (3.12)\n\nFinally, the equation T + = 0 implies\n\n∂ x h = ∂ y h = 0 , ∂ V G = 2 ℓ 2 ∂ u h , (3.13)\n\nγ = xℓ 2 ∂ x G , β = - xℓ 2 ∂ y G .\n\nh can be eliminated by introducing a new coordinate v(u, V ) with\n\n∂ V v = h/ℓ 2 and shifting G → G + 2∂ u v, which leads to ds 2 = ℓ 2 x 2 Gdu 2 + 2dudv + dx 2 + dy 2 . (3.14)\n\nNote that, due to (3.13), G is independent of v, therefore ∂ v is a Killing vector. One easily verifies that it coincides with the Killing vector constructed from the Killing spinor as -ℓ 2\n\nAll that remains is to impose the Maxwell and Einstein equations. One finds that the former are automatically satisfied by the gauge potential (3.6) . The same holds for the Einstein equations, except for the uu-component, which gives the Siklos equation with sources\n\n∆G - 2 x ∂ x G = - 4x 2 ℓ 2 ϕ ′ (u) 2 . (3.15)\n\nThis family of solutions enjoys a large group of diffeomorphisms which leave the solution invariant in form but change the function G. This is the Siklos-Virasoro invariance, discussed in [16, 33] . In conclusion, the geometry of solutions admitting the constant null spinor 1 + e 1 is given by the Lobachevski waves with metric (3.14) and gauge field (3.6) , where G satisfies (3.15) and ϕ(u) is arbitrary. This coincides exactly with the results of [13] , where it was shown moreover that there is a second covariantly constant spinor iff the wave profiles G and ϕ have the form\n\nG α (x, y, u) = - x 4 ℓ 2 + 2αx 3 -α 2 ℓ 2 (x 2 + y 2 ) , ϕ(u) = u , (3.16)\n\nup to Siklos-Virasoro transformation, with α ∈ R constant. In this case, the solution does also belong to the timelike class [13] . While the α = 0 solution only has the obvious Killing vectors ∂ v and ∂ y , the special α = 0 case is maximally symmetric with a five-dimensional isometry group."
},
{
"section_type": "OTHER",
"section_title": "Killing spinor with da = 0",
"text": "If da and hence also dχ do not vanish, one can use the R 2 stability subgroup of the spinor 1 + e χ e 1 to eliminate the fluxes F +• and F +• . To see this, observe that under an R 2 transformation (3.8),\n\nα F +• = F +• - 2iα ℓ tanh χ , α F •• = F •• , so by choosing α = -iℓ 2 F +• coth χ one can achieve α F +• = 0. Note that this would not be possible if χ = 0. With this gauge fixing, one has dχ = 2 ℓ sinh χ E 1 , A = -sinh χ E 3 , F = - 1 ℓ tanh χ E 1 ∧ E 3 . (3.17)\n\nNext we impose vanishing torsion. Using (3.17 Before we come to the other torsion components, let us consider the Bianchi identity and the Maxwell equations. The gauge field strength reads\n\nF = dχ sinh 2χ ∧ A .\n\nRequiring it to be equal to dA implies that A/ √ tanh χ is closed, so that locally\n\nA = tanh χdΨ . (3.19)\n\nNote that the functions χ, u and Ψ must be independent, because otherwise E 1 , E - and E 3 would not be linearly independent. We can thus use these three functions as coordinates.\n\nUsing\n\n* F = - 1 ℓ tanh χE -∧ E + , the Maxwell equations d * F = 0 imply d E -∧ E + + 2 dχ sinh 2χ ∧ E -∧ E + = 0 .\n\nBy Fröbenius' theorem and (3.18), E + can thus be written as\n\nE + = K 2 du + hdV ,\n\nwhere K, h and V are some functions, and we can use V as the remaining coordinate. Substituing E + into the Maxwell equations one obtains a constraint on the function h,\n\nd h e 2χ + 1 ∧ du ∧ dV = 0 , and hence h = h 0 (u, V ) e 2χ + 1 .\n\nIn what follows, we define K = K/(e 2χ + 1) and use ω +1 = (ω\n\n+• + ω +• )/ √ 2, ω +3 = (ω +• -ω +• )/ √ 2i.\n\nWe now come to the remaining torsion components. From T 3 = 0 and T 1 = 0 one obtains respectively\n\nω +3 = AE -, ω +1 = - E + ℓ cosh χ + BE -,\n\nwhere A and B are some functions to be determined. Finally, T + = 0 yields\n\n∂ V K = 2∂ u h 0 , A = - 1 2 e 4χ -1 sinh χ √ tanh χ ∂ Ψ K , B = 1 ℓ e 4χ -1 sinh χ∂ χ K .\n\nThe line element is given by\n\nds 2 = 2E -E + + E 1 2 + E 3 2 = coth χ Kdu 2 + 2h 0 dudV + ℓ 2 dχ 2 4 sinh 2 χ + dΨ 2 sinh χ cosh χ . (3.20)\n\nAs before, one can eliminate h 0 by introducing a new coordinate v(u, V ) with ∂ V v = h 0 and shifting K → K + 2∂ u v, whereupon the metric becomes\n\nds 2 = coth χ Kdu 2 + 2dudv + ℓ 2 dχ 2 4 sinh 2 χ + dΨ 2 sinh χ cosh χ . (3.21)\n\nNotice that, owing to (3.20), K is independent of v, therefore ∂ v is a Killing vector.\n\nIt coincides with the Killing vector -√ 2D(ǫ, Γ µ ǫ) constructed from the Killing spinor. All that remains now is to impose Einstein's equations. One finds that they are all satisfied except for the uu component, which yields again a Siklos-type equation for K,\n\n∂ 2 Ψ K + 4 tanh χ∂ 2 χ K - 2 cosh 2 χ ∂ χ K = 0 . (3.22)\n\nIn conclusion, the bosonic fields for a configuration admitting a null Killing spinor with dχ = 0 are given by (3.19) and (3.21), with K satisfying (3.22) 6 . As we will discuss in section 5.3, the K = 0 solution is of Petrov type D and represents a bubble of nothing in anti-De Sitter space-time. When K = 0, the metric becomes of Petrov type II and the Weyl scalar signalling the presence of gravitational radiation acquires a non-vanishing value. Hence the general solution represents a gravitational wave on a bubble of nothing. To our knowledge these solutions have not featured in the literature before."
},
{
"section_type": "BACKGROUND",
"section_title": "Half-supersymmetric backgrounds",
"text": "In the previous subsections we have addressed the conditions for preserving one null Killing spinor of the form ǫ 1 = 1 or ǫ 1 = 1 + e χ e 1 . It is natural to enquire about the possibility of these backgrounds admitting an additional Killing spinor with the same R 2 stability subgroup, i.e. of the form ǫ 2 = c 0 1 + c 1 e 1 . Using the fact that ǫ 1 is Killing, the second Killing spinor equation D µ ǫ 2 = 0 can then be rewritten as\n\n(c 0 -c 1 )D µ 1 + ∂ µ c 0 1 + ∂ µ c 1 e 1 = 0 , (3.23)\n\nin the U(1)⋉R 2 case and\n\n(c 0 -c 1 e -χ )D µ 1 + ∂ µ c 0 1 + (∂ µ c 1 -c 1 ∂ µ χ)e 1 = 0 , (3.24)\n\nin the R 2 case. Furthermore, we can assume that (c 0c 1 ) = 0 and (c 0c 1 e -χ ) = 0 in the two cases, respectively, since otherwise the second Killing spinor would be linearly dependent on the first and there would not be any additional constraints. Hence the e 2 and e 12 components of D µ 1 have to vanish separately. In particular, this implies that ω -• = 0 (as can be seen from the third line of (3.1) in the singular limit χ → -∞). However, this is clearly incompatible with (3.3). We conclude that, in the gauged theory, there are no backgrounds with four R 2 -invariant Killing spinors. In other words, there are no half-supersymmetric backgrounds with an R 2 -structure. This is unlike the ungauged case, where the half-supersymmetric gravitational waves provide such solutions.\n\nTherefore, the only possibility to augment the supersymmetry of the null solutions above is to add a Killing spinor which breaks the R 2 invariance, i.e. with a non-vanishing e 2 and/or e 12 component. From a linear combination of the first and second Killing spinor one can then always construct a time-like Killing spinor, and hence this brings us to the next section. For the convenience of the reader, we will already summarise how to restrict the 1/4-supersymmetric null solutions to allow for a time-like Killing spinor as well.\n\nFor the case with constant null Killing spinors, dχ = 0, the restriction was already discussed in [13] and is given in (3.16) . For the other case, with dχ = 0, it is straightforward to show that the solution (3.19), (3.21) admits a second Killing spinor iff ∂ χ G = ∂ Ψ G = 0, so that G depends only on u. By a simple diffeomorphism one can then set G = 0. The general solution to the Killing spinor equations reads in this case\n\nǫ = λ 1 (1 + e χ e 1 ) + λ 2 √ e 4χ -1 (e 2 + e χ e 1 ∧ e 2 ) , (3.25)\n\nwhere λ 1,2 ∈ C are constants. The invariants constructed from ǫ, as defined in appendix B, are\n\nV = √ 2 coth χ(|λ 2 | 2 dv -|λ 1 | 2 du) - 2i sinh 2χ (λ 2 λ1 -λ2 λ 1 ) dΨ , B = - √ 2(|λ 1 | 2 du + |λ 2 | 2 dv) + ℓe χ √ e 4χ -1 sinh χ ( λ1 λ 2 + λ 1 λ2 ) dχ , f = i(λ 1 λ2 -λ1 λ 2 ) tanh χ , g = ( λ1 λ 2 + λ 1 λ2 ) coth χ .\n\nThe norm of the Killing vector V is given by\n\nV 2 = - 2 sinh 2χ ( λ1 λ 2 + λ 1 λ2 ) 2 -4|λ 1 λ 2 | 2 tanh χ .\n\nSince χ > 0, this is negative unless λ 1 = 0 or λ 2 = 0, so indeed the solution (3.19), (3.21) with G = 0 must belong also to the timelike class. It turns out that it is identical to the bubble of nothing of section 5.3 with imaginary b and L < 0. The coordinate transformation\n\nu = √ 2A 2 (t -Ly) - z 2 √ 2A 2 , v = - √ 2A 2 (t -Ly) - z 2 √ 2A 2 , Ψ = -2A 2 t , χ = artanh X 2 A 4 (3.26)\n\nwith A 8 = -1/4L brings the metric (3.21) (with G = 0) to (5.60), and the field strength of (3.19) to (5.61). Note that, in the new coordinates, the above invariants become V = ∂ t as a vector, and B = dz, in agreement with section 4.2."
},
{
"section_type": "OTHER",
"section_title": "Timelike representative 1 + be 2",
"text": "We will now turn to the timelike case and first recover the general 1/4-BPS solutions [13] . Afterwards we will study the conditions for 1/2 supersymmetry. This will complete the classification since we already know that no 3/4-supersymmetric solutions can arise and AdS 4 is the unique maximally supersymmetric possibility."
},
{
"section_type": "OTHER",
"section_title": "Conditions from the Killing spinor equations",
"text": "Acting with the supercovariant derivative (2.5) on the representative 1 + be 2 yields the linear system\n\n∂ + b + b 2 ω •• + - b 2 ω +- + - i ℓ b A + = 0 , 1 2 ω •• + + 1 2 ω +- + - i ℓ A + + b ℓ √ 2 + ib √ 2 F •• + ib √ 2 F +-= 0 , ω •- + + i √ 2b F •-= ω •+ + = 0 , (4.1)\n\n∂ -b + b 2 ω •• -- b 2 ω +- -- i ℓ b A -+ 1 ℓ √ 2 + i √ 2 F •• - i √ 2 F +-= 0 , 1 2 ω •• -+ 1 2 ω +- -- i ℓ A -= 0 , b ω •+ -+ i √ 2 F •+ = ω •- -= 0 , (4.2)\n\n∂ • b + b 2 ω •• • - b 2 ω +- • - i ℓ b A • -i √ 2 F •-= 0 , 1 2 ω •• • + 1 2 ω +- • - i ℓ A • -i √ 2b F •+ = 0 , ω •- • + b ℓ √ 2 - ib √ 2 F •• - ib √ 2 F +-= 0 , b ω •+ • + 1 ℓ √ 2 - i √ 2 F •• + i √ 2 F +-= 0 , (4.3)\n\n∂ •b + b 2 ω •• • - b 2 ω +- • - i ℓ b A • = 0 , 1 2 ω •• • + 1 2 ω +- • - i ℓ A • = 0 , ω •- • = b ω •+ • = 0 . ( 4\n\nA + = iℓ 2 ∂ + b b - ∂ + b b -ω •• + , A -= iℓ 2 ω •• -, A • = iℓ 2 (ω •• • + ω +- • ) , F +-= i √ 2 (b ω •+ • -b -1 ω •- • ) , F •+ = i b√ 2 ω +- • , F •• = i √ 2 (b ω •+ • + b -1 ω •- • ) + i ℓ , F •-= i b √ 2 ω •- + . ( 4\n\nω +- + = ∂ + b b + ∂ + b b , ω +- -= 0 , ω +- • = ∂ • b b , ω +• + = ω +• • = 0 , ω +• -= - ∂ •b b 2b , ω +• • = ∂ -b b + √ 2 bℓ , ω -• + = -b ∂ •b , ω -• -= ω -• • = 0 , ω -• • = ∂ + b b + b √ 2 ℓ . (4.6)\n\nIn what follows, we assume b = 0. One easily shows that b = 0 leads to ℓ -1 = 0, so this case appears only in ungauged supergravity."
},
{
"section_type": "OTHER",
"section_title": "Geometry of spacetime",
"text": "In order to obtain the spacetime geometry, we consider the spinor bilinears\n\nV µ = D(ǫ, Γ µ ǫ) , B µ = D(ǫ, Γ 5 Γ µ ǫ) ,\n\nwhose nonvanishing components are\n\nV + = √ 2 bb , V -= - √ 2 , B + = √ 2 bb , B -= √ 2 .\n\nAs V 2 = -4 bb = -B 2 , V is timelike and B is spacelike. Using eqns. (4.1) -(4.4), it is straightforward to show that V is Killing and B is closed, i. e. ,\n\n∂ A V B + ∂ B V A -ω C B|A V C -ω C A|B V C = 0 , ∂ A B B -∂ B B A -ω C B|A B C + ω C A|B B C = 0 .\n\nThere exists thus a function z such that B = dz locally. Let us choose coordinates (t, z, x i ) such that V = ∂ t and i = 1, 2. The metric will then be independent of t. Note also that the system (4.1) -(4.4) yields\n\n∂ t b = √ 2 (|b| 2 ∂ --∂ + )b = 0 ,\n\nso b is time-independent as well. In terms of the vierbein E A µ the metric is given\n\nds 2 = 2E + E -+ 2E • E • , (4.7)\n\nwhere\n\nE + µ = B µ + V µ 2 √ 2|b| 2 , E - µ = B µ -V µ 2 √ 2 .\n\nFrom V 2 = -4|b| 2 and V = ∂ t as a vector we get V t = -4|b| 2 , so that V = -4|b| 2 (dt+σ) as a one-form, with σ t = 0. Furthermore, V • = 0 yields E • t = 0, and thus\n\nE • = E • z dz + E • i dx i .\n\nThe component E • z can be eliminated by a diffeomorphism\n\nx i = x i (x ′j , z) , with\n\nE I i ∂x i ∂z = -E I z , I = •, • .\n\nAs the matrix E I i is invertible 7 , one can always solve for ∂x i /∂z. Note that the metric is invariant under\n\nt → t + χ(x i , z) , σ → σ -dχ ,\n\nwhere χ(x i , z) denotes an arbitrary function. This second gauge freedom can be used to eliminate σ z . Hence, without loss of generality , we can take σ = σ i dx i , and the metric (4.7) becomes\n\nds 2 = -4|b| 2 (dt + σ i dx i ) 2 + dz 2 4|b| 2 + 2E • i dx i E • j dx j . (4.8)\n\nNext one has to impose vanishing torsion,\n\n∂ µ E A ν -∂ ν E A µ + ω A µB E B ν -ω A νB E B µ = 0 .\n\nOne finds that some of these equations are already identically satisfied, while the remaining ones yield (using the expressions (4.6) for the spin connection) the constraints\n\n∂ z σ i = - 1 4|b| 2 (E • i E j • -E • i E j • )∂ j ln(b/ b) , (4.9)\n\n∂ i σ j -∂ j σ i = (E • i E • j -E • j E • i ) ∂ z ln(b/ b) + 1 bℓ - 1 bℓ , (4.10\n\n)\n\nω •• t = -2|b| 2 ∂ z ln(b/ b) + 2b ℓ - 2 b ℓ , (4.11)\n\n∂ i E • j -∂ j E • i = (E • i E • j -E • j E • i )ω •• • , (4.12)\n\nas well as\n\n∂ z + ω •• z + 1 2 ∂ z ln( bb) + 1 2ℓ 1 b + 1 b E • i = 0 . (4.13)\n\nIn (4.9), E i I denotes the inverse of E J j . In order to obtain the above equations, one has to make use of the inverse tetrad\n\nE + = - 1 2 √ 2 ∂ t + √ 2|b| 2 ∂ z , E -= 1 2 √ 2|b| 2 ∂ t + √ 2 ∂ z , E • = E i • (∂ i -σ i ∂ t ) .\n\n(4.13) can be solved to give\n\nE • i = 1 |b| Ê• i exp -dz ω •• z - 1 2ℓ dz 1 b + 1 b , (4.14)\n\nwhere Ê• i is an integration constant that depends only on the coordinates x j . At this point it is convenient to use the residual U(1) gauge freedom of a combined local Lorentz and gauge transformation to eliminate ω •• z . This is accomplished by the transformation (2.3), with\n\nψ = i 2 dz ω •• z .\n\nNote that ψ is real, as it must be. Defining\n\nΦ := - 1 2ℓ dz 1 b + 1 b , (4.15)\n\nwe have thus\n\nE • i = 1 |b| Ê• i exp Φ . ( 4\n\n.16) Using (4.16) in (4.12), one gets for the only remaining unknown component ω •• • of the spin connection\n\nω •• • = ω•• • -Êi • ∂ i |b| exp(-Φ) ,\n\nwhere ω••\n\n• denotes the spin connection following from the zweibein ÊI i . In what follows, we shall choose the conformal gauge for the two-metric\n\nh ij = ÊIi ÊI j , i. e. , h ij = e 2ξ [(dx 1 ) 2 + (dx 2 ) 2 ] . (4.17)\n\nwith ξ depending only on the coordinates x i . Furthermore, we choose an orientation such that Ê\n\n• i Ê• j -Ê• j Ê• i = -ie 2ξ ǫ ij ,\n\nwhere ǫ 12 = 1. To be concrete, we shall take\n\n( ÊI i ) = 1 √ 2 e ξ 1 i 1 -i .\n\nThe eqns. (4.9) and (4.10) then simplify to\n\n∂ z σ i = - i 4|b| 2 ǫ ij ∂ j ln(b/ b) , (4.18)\n\n∂ i σ j -∂ j σ i = - i |b| 2 e 2(Φ+ξ) ǫ ij ∂ z ln(b/ b) + 1 bℓ - 1 bℓ . (4.19)\n\nMoreover, one has\n\nω •• • = -∂ • ln |b|e -Φ-ξ . (4.20)\n\nIn [13] it has been shown that in the case where the Killing vector constructed from the Killing spinor is timelike, the Einstein equations follow from the Killing spinor equations, so all that remains to do at this point is to impose the Bianchi identity and the Maxwell equations. Using the spin connection (4.6) and (4.11) in (4.5), the gauge potential and the field strength become\n\nA = i(dt + σ)(b -b) + ℓ 2 ǫ ij ∂ j (Φ + ξ) dx i - iℓ 4 d ln(b/ b) , F = i(dt + σ) ∧ d ( b -b) + 1 4|b| 2 dz ∧ dx i ǫ ij ∂ j (b + b) + 1 2|b| 2 ∂ z (b + b) + 1 ℓ e 2(Φ+ξ) ǫ ij dx i ∧ dx j . (4.21)\n\nThe Bianchi identity F = dA yields\n\n∆(Φ + ξ) = 2 ℓ e 2(Φ+ξ) ∂ z 1 b + ∂ z 1 b - 1 b 2 ℓ - 1 b2 ℓ + 1 bbℓ , (4.22)\n\nwith ∆ = ∂ i ∂ i denoting the flat space Laplacian in two dimensions. As for the Maxwell equations, ∂ µ ( √ -gF µν ) = 0 , the only nontrivial information comes from the t-component, which gives\n\n4e 2(Φ+ξ) b 2 ∂ 2 z 1 b -b2 ∂ 2 z 1 b - 3b ℓ ∂ z 1 b + 3 b ℓ ∂ z 1 b + 1 bℓ 2 - 1 bℓ 2 + b 2 ∆ 1 b -b2 ∆ 1 b = 0 , (4.23)\n\nwhere we used eqns. (4.18) and (4. 19 ).\n\nLet us now show that the equations (4.22) and (4.23) are actually the same as the ones in [16] . If we set\n\nF = - 1 ℓ b , e φ = 2e Φ+ξ , (4.24)\n\n(4.22) yields exactly equation (2.3) of [16] . On the other hand, deriving (4.22) with respect to z and using (4.15), one obtains\n\n∆A + e 2φ 3A∂ z A -3B∂ z B + A 3 -3AB 2 + ∂ 2 z A = 0 , (4.25)\n\nwhere A and B denote the real and imaginary part of F respectively. This can be used in (4.23) to get\n\n∆B + e 2φ ∂ 2 z B + 3B∂ z A + 3A∂ z B -B 3 + 3A 2 B = 0 ,\n\nwhich, together with (4.25), yields\n\n∆F + e 2φ F 3 + 3F ∂ z F + ∂ 2 z F = 0 , (4.26)\n\ni. e. , equation (2.2) of [16] . For a complete identification of the present results with the ones in [16] , one also has to set σ = ω.\n\nIn conclusion, the metric of the general 1/4-supersymmetric solution is given by\n\nds 2 = -4|b| 2 (dt + σ) 2 + 1 4|b| 2 dz 2 + 4e\n\n2(Φ+ξ) dw d w , (4.27) where b and φ are determined by the system (4.22), (4.23) and w = x 1 + ix 2 ≡ x + iy. The one-form σ follows then from (4.18) and (4.19), and the gauge field strength is given by (4.21). Note that (4.23) represents also the integrability condition for (4.18), (4.19). As noted in [16], this system of equations is invariant under PSL(2, R) transformations foot_8 . If we define a new coordinate z ′ through the Möbius transformation\n\nz ′ = αz + β γz + δ , (4.28)\n\nwith α, β, γ and δ arbitrary real constants satisfying αδβγ = 1, then the functions b(z ′ , x i ) and Φ(z ′ , x i ) defined by\n\n1 b = 1 (γz ′ -α) 2 b - 2lγ γz ′ -α , e Φ = (γz ′ -α) 2 e Φ , (4.29)\n\nsolve the system in the new coordinate system (z ′ , x i ), with the function ξ(x i ) left invariant and z seen as a function of z ′ . This symmetry allows to generate new BPS solutions from the known ones. Note however that it is only a symmetry of the equations for 1/4 supersymmetry, and if we apply it to solutions with additional Killing spinors, it will in general not preserve them, as we shall show explicitely in some examples."
},
{
"section_type": "BACKGROUND",
"section_title": "Half-supersymmetric backgrounds",
"text": "We now would like to investigate the possibility of adding a second Killing spinor. Since the first Killing spinor ǫ 1 has stability subgroup 1, one cannot use Lorentz transformations to bring the second spinor to a preferred form. Therefore we use the most general form\n\nǫ 2 = c 0 1 + c 1 e 1 + c 2 e 2 + c 12 e 1 ∧ e 2 . (4.30)\n\nThe corresponding linear system simplifies significantly after inserting the results from ǫ 1 . These determine all the fluxes and the spin connection in terms of the functions b, ξ and their derivatives. First it is convenient to introduce the new basis\n\n9 α = α 0 α 1 α 2 α 12 = c 0 b -1 c 2 -c 0 bc 1 c 12 \n\n, in which the Killing spinor equations for ǫ 2 read\n\n(∂ A + M A )α = 0 , (4.31)\n\nwith the connection M A given by\n\nM + = 0 -∂ + ln b 0 0 0 ∂ + ln b -∂ • ln b ∂ • ln b 0 0 b-b √ 2ℓ + 1 2 ∂ + ln b b - √ 2 ℓ b -∂ + ln b 0 -|b| 2 ∂ • ln b 0 b-b √ 2ℓ -1 2 ∂ + ln( bb) , M -= 0 0 |b| -2 ∂ • ln b -|b| -2 ∂ • ln b 0 ∂ -ln b -|b| -2 ∂ • ln b |b| -2 ∂ • ln b 0 ∂ • ln b b-b √ 2ℓ|b| 2 -1 2 ∂ -ln( bb) 0 0 0 - √ 2 ℓ b -∂ -ln b b-b √ 2ℓ|b| 2 + 1 2 ∂ -ln b b , M • = 0 -∂ • ln b 0 0 0 ∂ • ln( bb) 0 0 0 - √ 2 ℓ b -∂ + ln b -∂ • ln |b|e -Φ-ξ 0 0 √ 2 ℓ b + ∂ + ln b 0 -∂ • ln |b|e -Φ-ξ , M • = 0 0 -∂ -ln b ∂ -ln b + √ 2 ℓ b 0 0 ∂ -ln( bb) + √ 2 ℓb -∂ -ln( bb) - √ 2 ℓ b 0 0 ∂ • ln b b e -Φ-ξ -∂ • ln b 0 0 -∂ • ln b ∂ • ln b b e -Φ-ξ .\n\nLet us first of all consider the simpler possibility of a second Killing spinor of the form ǫ 2 = c 0 1 + c 2 e 2 . As discussed in section 2.1, both ǫ 1 and ǫ 2 are invariant under the same U(1) symmetry, and hence this case constitutes the G = U(1) case with four supersymmetries. As can easily be seen from the above Killing spinor equations with α 1 = 0 and α 2 = α 12 = 0, this restricts the derivatives of the coefficient b to be\n\n∂ -b = - √ 2 ℓ , ∂ + b = - √ 2b b ℓ , ∂ • b = ∂ •b = 0 . (4.32)\n\nHence this corresponds to ∂ z b = -1/ℓ. As will be discussed in section 5.1, this restriction uniquely leads to the half-supersymmetric anti-Nariai space-time. Hence AdS 2 ×H 2 is the only possibility for backgrounds with four U(1)-invariant Killing spinors.\n\nIn the more general case with α 2 and α 12 non-vanishing, i.e. with trivial stability subgroup, the Killing spinor equations do not so readily provide information about b and one has to resort to their integrability conditions. The first integrability conditions for the linear system (4.31) are\n\nN µν α ≡ (∂ µ M ν -∂ ν M µ + [M µ , M ν ])α = 0 , (4.33)\n\nwhere the matrices M µ = E A µ M A are given by\n\nM t = √ 2(|b| 2 M --M + ) , M z = 1 2 √ 2|b| 2 (M + + |b| 2 M -) , M w = σ w M t + 1 √ 2|b| e Φ+ξ M • , M w = σ wM t + 1 √ 2|b| e Φ+ξ M • ,\n\nand we introduced the complex coordinates w = x + iy, w = xiy. For halfsupersymmetric solutions, the six matrices N µν must have rank two. (As at least one Killing spinor exists, namely ǫ 1 = (1, 0, 0, 0), we already know that the N µν can have at most rank three. Rank one is not possible, because 3/4 BPS solutions cannot exist [29] . Rank zero corresponds to the maximally supersymmetric case, which implies that the spacetime geometry is AdS 4 [13] .) Let us define\n\nѵν ≡ SN µν T , with S = 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 , T = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 \n\n.\n\nThe similarity transformation S corresponds to adding the first line to the second one and T adds the last column to the third one. This does not alter the rank of N µν . One finds\n\nÑwt = 2b∂∂ z b + 2 ℓ ∂ b -2|b| b e -Φ-ξ ∂ 2 b + 1 b ∂ b∂ b 0 +2b ∂ z b + 1 ℓ ∂ ln b 0 -2∂(Φ + ξ)∂ b 2 b∂∂ z b + 2 ℓ ∂b -2|b| b e -Φ-ξ ∂ 2 b + 1 b ∂b∂b 0 +2 b ∂ z b + 1 ℓ ∂ ln b 0 -2∂(Φ + ξ)∂b) 2|b| 3 e -Φ-ξ ∂∂ ln b 2 b∂∂ z b 0 -2|b| 3 e Φ+ξ b -2 2∂ z b + 1 ℓ ∂ z b + 1 ℓ 2 ℓ ∂b +2 b ∂ z b + 1 ℓ ∂ ln b 2|b| 3 e -Φ-ξ ∂∂ ln b 2b∂∂ z b -2 ℓ ∂ b 0 -2|b| 3 e Φ+ξ b-2 2∂ z b + 1 ℓ ∂ z b + 1 ℓ -2 ℓ ∂ b +2b ∂ z b + 1 ℓ ∂ ln b , Ñ wt = 2b ∂∂ z b -2|b|e -Φ-ξ ∂∂ ln b 0 +2b ∂ z b + 1 ℓ ∂ ln b -2 ℓ|b| e Φ+ξ 2∂ z b + 1 ℓ + 2b |b| b e Φ+ξ 2∂ z b + 1 ℓ ∂ z b + b- b ℓb 2 b ∂∂ z b -2|b|e -Φ-ξ ∂∂ ln b 0 +2 b ∂ z b + 1 ℓ ∂ ln b 2 ℓ|b| e Φ+ξ 2∂ z b + 1 ℓ + 2 b |b|b e Φ+ξ 2∂ z b + 1 ℓ ∂ z b + 1 ℓ 2|b| be -Φ-ξ ∂2 b + 1 b ∂b ∂b 2 b ∂∂ z b + 2 ℓ ∂b 0 -2 ∂(Φ + ξ) ∂b 6 ℓ ∂b +2 b ∂ z b + 1 ℓ ∂ ln b 2|b|be -Φ-ξ ∂2 b + 1 b ∂b ∂b 2b ∂∂ z b -4 ℓ ∂b 0 -2 ∂(Φ + ξ) ∂b -6 ℓ ∂b +2b ∂ z b + 1 ℓ ∂ ln b , where ∂ = ∂ w , ∂ = ∂ w.\n\nThe other four integrability conditions give no additional information, because the lines of the corresponding matrices are proportional to the lines of Ñwt and Ñ wt 10 . As the upper right 3 × 3 determinant of Ñwt must vanish, we obtain ∂b = 0 or\n\n∂ z e -2(Φ+ξ) b∂ b ∂ e -2(Φ+ξ) b∂b -∂ z e -2(Φ+ξ) b∂b ∂ e -2(Φ+ξ) b∂ b = 0 . (4.34)\n\nLet us assume that the expression in (4.34) does not vanish. One has then ∂b = 0 as well as ∂ b = 0 foot_11 . But then also (4.34) holds, which leads to a contradiction. Thus (4.34) must be satisfied in any case. Note that the vanishing of the first column of ѵν implies that also the first column of T -1 N µν T is zero, and thus T -1 N µν T ∈ a(3, C), hence the generalized holonomy in the case of one preserved complex supercharge is contained in the affine group A(3, C). This supports the classification scheme of [4] . Of course, depending on the particular solution, the generalized holonomy may also be a subgroup of A(3, C)."
},
{
"section_type": "OTHER",
"section_title": "Time-dependence of second Killing spinor",
"text": "In this section we will utilize the above Killing spinor equations to derive the timedependence of the second Killing spinor. In addition, we will show that the Killing spinor equations can be completely solved when the second Killing spinor is timeindependent.\n\nLet us first simplify the Killing spinor equations (4.31). In the following we set b = re iϕ and define ψ = Φ + ξ, ψ 1 = r 2 α 1 , ψ 2 = re -ψ α 2 , ψ 12 = re -ψ α 12 and ψ ± = ψ 2 ± ψ 12 . First of all, use the integrability conditions (4.33), that can be rewritten as ѵν T -1 α = 0. Defining P = e -2ψ b∂b, the second component for µ = w, ν = t gives\n\nψ 1 P ′ + ψ -∂P = 0 , (4.35)\n\nwith ′ = ∂ z . Let us assume P ′ = 0 (the case P ′ = 0 is considered in appendix C and will lead to the same conclusions). If we define g(t, z, w, w) = -ψ -/P ′ , we get\n\nψ -= -gP ′ , ψ 1 = g∂P .\n\nThe third component of the (w, t) integrability condition is of the form\n\nψ 1 f 1 + ψ 2 ∂b + ψ -f -= 0 ,\n\nfor some functions f 1 , f -that depend on z, w, w but not on t. Using the above form of ψ 1 and ψ -, this becomes\n\nf 1 g∂P + ψ 2 ∂b -f -gP ′ = 0 . (4.36)\n\nNow, if g = 0, the latter equation implies ψ 2 ∂b = 0, and hence (since ∂b = 0 due to P ′ = 0) ψ 2 = 0. Furthermore, ψ 1 = ψ -= 0 in this case, so there exists no other Killing spinor. Thus, g = 0 and we can write g = exp G. Dividing (4.36) by g and deriving with respect to t yields ∂ t (ψ 2 /g) = 0 and hence\n\nψ 2 = e G ψ 0 2 (z, w, w) .\n\nIt is then plain that ∂ t ψ i = ψ i ∂ t G, i = 1, 2, 12. The Killing spinor equations are of the form ∂ µ ψ i = M µij ψ j , for some time-independent matrices M µ . Taking the derivative of this with respect to t, one gets ∂ µ ∂ t G = 0, whence\n\nG = G 0 t + G(z, w, w) ,\n\nwith G 0 ∈ C constant. We have thus ∂ t ψ i = G 0 ψ i and hence also ∂ t α i = G 0 α i . Furthermore, the time-dependence of α 0 can be easily deduced from the Killing spinor equations: if G 0 does not vanish it is of the same exponential form as the other components of the second Killing spinor, i.e. ∂ t α 0 = G 0 α 0 , while if G 0 vanishes there can be a linear part in t, i.e. ∂ t α 0 = c for some constant c. Hence, in terms of the basis elements, the time-dependence of the second Killing spinor takes the form foot_12\n\nG 0 = 0 : ǫ 2 = c 0 1 + c 1 e 1 + c 2 e 2 + c 12 e 1 ∧ e 2 + ct(1 + be 2 ) , G 0 = 0 : ǫ 2 = e G 0 t (c 0 1 + c 1 e 1 + c 2 e 2 + c 12 e 1 ∧ e 2 ) , (4.37)\n\nwhere c 0 , c 1 , c 2 , c 12 are time-independent functions of the spatial coordinates, and c is a constant. This was derived assuming P ′ does not vanish, but as we show in appendix C is in fact a completely general result. Hence, adding a second Killing spinor to ǫ 1 = 1 + be 2 , the Killing spinor equations imply that ǫ 2 always has the above timedependence.\n\nPlugging this time-dependence into the subsystem of the Killing spinor equations not containing α 0 one obtains in terms of\n\nψ i ψ ′ 1 - G 0 4r 2 + b ′ b ψ 1 - ∂b b ψ -= 0 , (4.38)\n\nψ ′ 2 - G 0 4r 2 + b′ b + 1 ℓ b ψ 2 - b ′ b + 1 ℓb ψ 12 = 0 , (4.39)\n\nψ ′ 12 -e -2ψ ∂b b ψ 1 - G 0 4r 2 + b ′ b + b′ b + 1 ℓ b ψ 12 = 0 , (4.40)\n\nψ ′ 1 -- G 0 4r 2 + b′ b ψ 1 - ∂ b b ψ -= 0 , (4.41)\n\nψ ′ 2 + e -2ψ ∂b b ψ 1 -- G 0 4r 2 + b ′ b + b′ b + 1 ℓb ψ 2 = 0 , (4.42)\n\nψ ′ 12 - b′ b + 1 ℓ b ψ 2 -- G 0 4r 2 + b ′ b + 1 ℓb ψ 12 = 0 , (4.43\n\n)\n\n∂ψ 1 -σ w G 0 ψ 1 = 0 , (4.44) ∂ψ 2 - b ′ b + 1 ℓb ψ 1 -σ w G 0 + ∂b b + ∂ b b -2∂ψ ψ 2 = 0 , (4.45\n\n) For G 0 = 0, these equations simplify significantly, and allow for a complete solution. As is shown in appendix D, under the additional assumption ψ -= 0, ψ 1 = 0, the metric and the field strength for half-supersymmetric solutions with G 0 = 0 are given in terms of a single real function H depending only on the combination Zww and satisfying the second order differential equation\n\n∂ψ 12 + b′ b + 1 ℓ b ψ 1 -σ w G 0 + ∂b b + ∂ b b -2∂ψ ψ 12 = 0 , (4.46) ∂ψ 1 -σ wG 0 + ∂b b + ∂b b ψ 1 + e 2ψ b ′ b + b′ b ψ -+ 1 ℓ ψ 2 b - ψ 12 b = 0 ,(4.\n\n2 1 + e -2H Ḧ + Ḣ2 1 - 3α 2 e 2H + 1 -α 2 = γ 2 ℓ 2 , (4.50)\n\nwhere α ∈ R denotes an arbitrary constant and γ = 0, 1. The new coordinate Z is defined by Z = z for γ = 0 and Z = ℓ ln 1 + z ℓ for γ = 1. Furthermore, in the remainder of this section and in appendix D, a dot denotes a derivative with respect to Zww. Given a solution of (4.50), one defines the functions χ, ρ by where the shift vector satisfies\n\nχ = iα √ e 2H + 1 -α 2 , 1 ℓ 2 ρ 2 = γ ℓ + Ḣ 2 -Ḣ2 χ 2 . ( 4\n\n∂ Z σ w = 1 4 e -γZ/ℓ χ ρ 2 • , ∂σ w -∂σ w = - 1 2 e -γZ/ℓ e 2H χ ρ 2 • .\n\nFinally, the gauge field strength is given by (4.21). Equation (4.50) is actually the Euler-Lagrange equation for the following standard action for the scalar\n\nH S = d (Z -w -w) 1 2 M(H) Ḣ2 -V (H) , (4.53)\n\nwhere\n\nM(H) = e 2H + 1 2 (e 2H + 1 -α 2 ) 3/2 , V (H) = - γ 2 2ℓ 2 e 2H + 1 -2α 2 (e 2H + 1 -α 2 ) 1/2 . (4.54)\n\nThus it is possible to use the energy conservation law of that model in order to evaluate the \"velocity\" Ḣ in terms of H. Since dH = Ḣd (Zww) one has d dH\n\n1 2 M(H) Ḣ2 + V (H) = 0 , (4.55)\n\nso that there must exist a constant E such that\n\nḢ = 2 M(H) [E -V (H)] = √ 2 e 2H + 1 -α 2 3/4 e 2H + 1 E + γ 2 2ℓ 2 e 2H + 1 -2α 2 √ e 2H + 1 -α 2 1/2 . ( 4\n\nThe key-point is to consider now, as a new coordinate, the function H in place of w + w13 and to write down the full solution, say metric plus gauge field, in terms of H. Using w = x + iy, the general solution is given by ds 2 = -4ρ 2 e 2γZ/ℓ dt + e -γZ/ℓ σy dy\n\n2 + e 2H 4ρ 2 dy 2 + 1 4ρ 2 dZ 2 + e 2H dZ - dH Ḣ 2 , A = ℓ Ḣ -2iρ 2 χe γZ/ℓ dt + 1 -e 2H χ 2 dy -i ℓ 4 d log b b , (4.57)\n\nwhere Ḣ is given in equation (4.56), the functions χ and ρ are defined in (4.51) and the shift vector reads σy = -i 2 e 2H χ ρ 2 .\n\nIf γ = 1, a simple example of this set of solutions can be obtained by setting α = 0, so\n\nḢ = 1/ℓ , b = 1 2 1 + z ℓ . (4.58)\n\nAs will be shown in section 5.1.2, this corresponds to the maximally supersymmetric AdS 4 solution. More general γ = 1 solutions will be two-parameter deformations thereof, the parameters being α and the energy E of the associated scalar system.\n\nSetting γ = 0 the potential V (H) vanishes and the parameter E can be fixed by a simple rescaling of the coordinates. Thus we are left with a one-parameter family of solutions. Since the metric does no more depend explicitly on Z, it is useful to replace the coordinate Z instead of x by H. Defining a new coordinate r such that r 4 ≡ 16 e 2H + 1α 2 and a new parameter Q = 4 ℓ α, the complete solution reads\n\nds 2 = - r 2 ℓ 2 + Q 2 r 2 dt - 2ℓ 3 Q r 4 + ℓ 2 Q 2 dy 2 + + r 2 ℓ 2 + Q 2 r 2 -1 h(r) 2 dr + 2 h(r) dx 2 + 1 4 r 4 + ℓ 2 Q 2 -16 dx 2 + dy 2 , A = - Q r dt + 2ℓ r dy -i ℓ 4 d log b b , (4.59)\n\nwhere\n\nh(r) = r 4 + ℓ 2 Q 2 r 4 + ℓ 2 Q 2 -16 . ( 4\n\nThe parameter Q can thus be interpreted as an electric charge. The Petrov type of the solution is D or simpler. If one sets Q = 4/ℓ the Petrov type is reduced to N, so that there is a gravitational wave.\n\nIn order to complete the classification of G 0 = 0 solutions, we need to study separately the cases where either ψ 1 or ψ -vanishes (it can easily be seen from (4.39) and (4.40) that there is no solution if both vanish). As one can see by looking at equations (4.38) and (4.41), the condition ψ 1 = 0 leads to b = b(z), which is studied in detail in section 5.1. The other possibility, ψ -= 0, is more involved, but as we show in appendix D it boils down to three different cases, that can be completely solved: the AdS 2 × H 2 anti-Nariai spacetime studied in section 5.1.1, the imaginary b case solved in section 5.3, and finally the half BPS solution coming from the gravitational Chern-Simons model, that we analyse in section 5.5.\n\nWe would like to remark that the assumption G 0 = 0 on the overall time-dependence of the second Killing spinor seems a reasonable choice since all known 1/2-supersymmetric solutions to be studied in the next section are contained in this class, or can be brought to this class by a general coordinate transformation. Hence we expect the G 0 = 0 class to form an important subclass of all 1/2-supersymmetric solutions."
},
{
"section_type": "OTHER",
"section_title": "Timelike half-supersymmetric examples",
"text": "The problem of finding all half BPS configurations in the timelike class involves the solution of the integrability conditions we obtained above. To obtain explicit examples of half BPS solutions, we shall restrict to some simple subclasses with particular b. This will determine the fraction of preserved supersymmetry for the solutions which are already known to be 1/4 supersymmetric, and will also lead to new solutions."
},
{
"section_type": "OTHER",
"section_title": "Static Killing spinors and b = b(z)",
"text": "The timelike vector field V , constructed as a bilinear of the Killing spinor, is static if the associated one-form V = dt + σ satisfies the Fröbenius condition V ∧ dV = 0. Obviously, there can be static BPS solutions with V not being static itself, due to the choice of coordinates; we shall loosely refer the Killing spinors whose vector bilinear is static as static Killing spinors. The staticity condition, in turn, implies dσ = 0 and puts strong constraints on the function b. Indeed, equation (4.18) implies that the phase ϕ of b depends only on z. Then, (4. 19) gives the modulus r of b in terms of its phase,\n\nr = sin ϕ(z) lϕ ′ (z) . (5.1)\n\nAs a consequence, r and therefore the complete complex function b, depend on the single variable z. The full solution is therefore determined by the single real function ϕ, which has to satisfy the equations for supersymmetry (together with the conformal factor ψ).\n\nHowever, since the equations can be exactly solved for arbitrary b(z), we will stick to this more general case and eventually comment on the static subcase.\n\nIf b depends only on z, the equations of motion simplify to\n\nIm b 2 ∂ 2 z 1 b - 3b l ∂ z 1 b + 1 bl 2 = 0 , (5.2)\n\ne -2ξ ∆ξ = 2 l e 2Φ ∂ z 1 b + 1 b - 1 l 1 b + 1 b 2 + 3 lb b . (5.3)\n\nHere we have used the fact that Φ, defined in (4.15), depends only on the coordinate z. In principle there is also an integration constant K(w, w) with arbitrary dependence on the transverse coordinates, but since Φ appears only in the combination Φ + ξ in all the equations, we can always absorb the (w, w) dependence into the conformal factor ξ. Now the left hand side of equation ( 5 .3) depends only on the coordinates w and w, while the right hand side depends only on z. This equation can be therefore satisfied only if both sides are equal to some constant κ. The system of equations is then ∆ξ + κ 2 e 2ξ = 0 , (5.4)\n\ne 2Φ(z) ∂ z 1 b + 1 b - 1 l 1 b + 1 b 2 + 3 lb b = - l 4 κ . (5.5)\n\nNote that the first one is the Liouville equation, whose solution describes the transverse two-dimensional manifold, which has therefore constant curvature κ.\n\nEquations (5.2) and (5.5) can easily be solved [16] . Their solution is given by foot_14 b = - This solution generically belongs to the supersymmetric Reissner-Nordström-Taub-NUT-AdS 4 family of spacetimes. The values α = 0 and β 2 = 4αγ are special cases and will be treated separately in the following. Note that the coefficients α, β and γ are not three independent parameters, as they can be rescaled without changing the function b: the solutions depend only on their ratios. For example, if α = 0, one can use β/α and γ/α as independent complex parameters of the family of solutions.\n\nαz 2 + βz + γ ℓ(2αz + β) , ( 5\n\nThe solutions with static Killing spinor form a subset of this family. For (5.6) the staticity condition (5.1) yields the condition α βᾱβ = 0. Recalling the expression for the NUT charge of these solutions,\n\nn = i 4 β ᾱ - β α , (5.8)\n\nthis charge must vanish for non-vanishing α, as one could have guessed. On the other hand, for α = 0 the solution is anti-Nariai, as we shall see below. We conclude that the most general supersymmetric configuration with static Killing vector constructed as a Killing spinor bilinear is either of the form (5.6) -i. e. in the fourth row of table 1 of [34] -with vanishing NUT charge, or it is anti-Nariai spacetime. The supersymmetric static solutions discussed so far are generically 1/4-BPS. We want to see what further condition ensures the presence of an additional Killing spinor. Inserting the staticity ansatz b = b(z) into the integrability equations and requiring these matrices to be of rank smaller or equal to two, one finds the following condition (in particular this is obtained from the vanishing of the minor of the last row of Ñwt and the first two rows of Ñ wt )\n\nb ′ + 1 l 2b ′ + 1 l ∂ z 1 b + 1 b - 1 l 1 b + 1 b 2 + 3 lb b = 0 , (5.9)\n\nAs an aside, note that we have only used the ansatz b = b(z) so far and not the staticity condition (5.1), i.e. the precise relation between r and ϕ. The static solutions are therefore in general still a subset of the solutions under consideration. Condition (5.9) calls for the following three different cases, corresponding to the vanishing of its three factors.\n\n2 × H 2 space-time (α = 0)\n\nRequiring the first factor of (5.9) to vanish leads to b = -z ℓ + ic with constant c, corresponding to α = 0 in (5.6) . We can absorb the imaginary part of c by a shift of the coordinate z and henceforth will assume c ∈ R.\n\nIn this case κ = -4 and we have a hyperbolic transverse space. As a solution of (5.4) we can take\n\ne 2ξ = 1 2x 2 .\n\n(5.10)\n\nMoreover, e Φ = l|b| and σ = 0, therefore giving the metric\n\nds 2 = -4 z 2 ℓ 2 + c 2 dt 2 + dz 2 4 z 2 ℓ 2 + c 2 + ℓ 2 2x 2 dx 2 + dy 2 . (5.11)\n\nThis is the anti-Nariai AdS 2 × H 2 solution, with the AdS 2 factor written in Poincaré coordinates for c = 0 and in global coordinates for c = 0. The coordinate transformations between Poincaré coordinates (t P , z P ) (with c = 0) to global ones (t gl , z gl ) (with c = 0) is given by\n\nz P = 1 2c (z gl -z 2 gl + ℓ 2 c 2 cos(4ct gl /ℓ)) , t P = - ℓ 2 z 2 gl + ℓ 2 c 2 sin(4ct gl /ℓ) z gl -z 2 gl + ℓ 2 c 2 cos(4ct gl /ℓ) .\n\n(5.12)\n\nThe electromagnetic field strength (4.21) in this case is given by\n\nF = - 1 ℓx 2 dx ∧ dy , (5.13)\n\ni.e. only lives on the hyperbolic part and is independent of the coordinates of the AdS part of space-time. This solution preserves precisely 1/2 of the supersymmetries, as was already shown in [35] . To obtain the form of the Killing spinors admitted by this metric we first observe that the integrability conditions impose α 2 = α 12 = 0. Then the Killing spinor equations are easily solved, but one should treat separately the cases c = 0 and c = 0:\n\n• If c = 0, then α 0 = λ 1 + λ 2 2t ℓ - 1 2b , α 1 = λ 2 b , (5.14)\n\nwhere λ 1,2 ∈ C are integration constants. This yields the following Killing spinors, spanning a two-dimensional complex space,\n\nǫ = λ 1 + λ 2 2t ℓ - 1 2b 1 + b λ 1 + λ 2 2t ℓ + 1 2b e 2 .\n\n(5.15)\n\nNote that λ 1 = 1, λ 2 = 0 corresponds to the original Killing spinor. Also note that the constant G 0 , corresponding to the time-dependence of the second Killing spinor with λ 2 = 0, is zero. The form of the scalar invariant corresponding to the general spinor\n\nǫ is b = b |λ 1 | 2 + |λ 2 | 2 4t 2 ℓ 2 - 1 4b 2 + 2t ℓ λ1 λ 2 + λ 1 λ2 + 1 2 λ1 λ 2 -λ 1 λ2 . (5.16)\n\nHere the first term is real, while the second is imaginary. Note that the latter is in fact constant. Then the Killing vector Ṽ built from ǫ will have a norm Ṽ 2 = -4| b| 2 , and will be timelike unless b vanishes. This is however not possible, because both the real and imaginary parts of b should vanish, but since λ 1,2 do not depend on the coordinates, the real part cannot vanish. Therefore, every Killing spinor of this solution belongs to the timelike class.\n\n• If c = 0 we have\n\nα 0 = 1 2 √ c [λ 1 -iλ 2 +(λ 1 +iλ 2 ) b |b| e -4ict/ℓ , α 1 = - i √ c |b| (λ 1 +iλ\n\n2 )e -4ict/ℓ , (5.17)\n\nand the most general Killing spinor is parametrized by λ1,2 ∈ C as follows\n\nǫ = 1 2 √ c (λ 1 -iλ 2 )(1 + be 2 ) + b 2|b| (λ 1 + iλ 2 )e -4ict/ℓ (1 + b * e 2 ) .\n\n(5.18)\n\nNote that the combination λ 1iλ 2 corresponds to the first Killing spinor 1 + be 2 , while the orthogonal combination λ 1 + iλ 2 gives rise to the second Killing spinor proportional to 1 + b * e 2 . Any combination with λ 2 = 0 has G 0 = -4ic/ℓ.\n\nIn this case, the real part of the invariant b is given by\n\nRe( b) = |λ 1 | 2 2ℓc (-z + √ z 2 + ℓ 2 c 2 cos(4ct/ℓ)) + |λ 2 | 2 2ℓc (-z - √ z 2 + ℓ 2 c 2 cos(4ct/ℓ))+ + 1 2ℓc (λ 1 λ * 2 + λ 2 λ * 1 ) √ z 2 + ℓ 2 c 2 sin(4ct/ℓ)) , (5.19)\n\nwhile the imaginary part is identical to that of (5.16) .\n\nIt can easily be checked that the coordinate transformation (5.12) indeed relates the complex scalar b, which is composed of spinor bilinears, in (5.16) and (5.19) to each other. Let's now check how the isometries of AdS 2 act on the Killing spinors. It is useful to do this by embedding AdS 2 with metric\n\nds 2 = -4 z 2 ℓ 2 + c 2 dt 2 + dz 2 4 z 2 ℓ 2 + c 2 (5.20)\n\ninto the three-dimensional flat space X a = (U, T, X) with metric\n\nds 2 = -dU 2 -dT 2 + dX 2 . (5.21)\n\nThen, AdS 2 is obtained as the hyperboloid defined by\n\n-U 2 -T 2 + X 2 = ℓ 2 4 , (5.22)\n\nand its isometry group SO(2,1) will act as the three-dimensional Lorentz group on the embedding coordinates X a (here a is a three-dimensional Lorentz index). If c = 0, the AdS 2 metric (5.20) is in the Poincaré form, and can be seen to be the induced metric on the hyperboloid by parameterizing it with the coordinates (t, z) given by z = U + X , t = ℓT 2(U + X) .\n\n(5.23)\n\nThen, if one defines the 3d Lorentz vector Now, the real and imaginary part of b are independently manifestly invariant under the AdS 2 isometries, as they should be (since they transform respectively as pseudoscalar and scalar under diffeomorphism 16 ). If c = 0 we have AdS 2 in global coordinates, and the embedding is modified to\n\nΛ a = 1 ℓ |λ 1 | 2 -|λ 2 | 2 , 1 ℓ (λ * 1 λ 2 + λ 1 λ * 2 ) , - 1 ℓ |λ 1 | 2 + |λ 2 | 2 , ( 5\n\nU = - ℓ 2c z 2 ℓ 2 + c 2 cos 4ct ℓ , T = - ℓ 2c z 2 ℓ 2 + c 2 sin 4ct ℓ , X = z 2c .\n\n(5.26)\n\nThe invariant (5.19) takes again the manifestly invariant form (5.25), as expected, and the isometries of AdS 2 are realized linearly on the Killing spinors through their action on Λ a . This result may be useful to study in detail quotients of AdS 2 and to see whether this operation breaks some supersymmetry. The following subcase corresponds to the vanishing of the second factor of the integrability condition (5.9). The function b is then given by b = -z 2l + ic, which can be obtained as the special case β 2 = 4αγ from (5.6) . This corresponds to AdS 4 , the only maximally supersymmetric solution of the theory. Indeed the integrability condition matrices vanish in this case.\n\nLet's see in detail the form of the metric arising from different values of c. As in the previous case we can take the constant c to be real. If c = 0, the metric is static, σ = 0, ξ = 0 and e 2Φ = |b| 4 , and we obtain anti-de Sitter in Poincaré coordinates,\n\nds 2 = - z 2 ℓ 2 dt 2 -dx 2 -dy 2 + ℓ 2 z 2 dz 2 . (5.27)\n\nOn the other hand, for c = 0, the metric appears in non-static coordinates,\n\nσ = - ℓdy 4cx 2 , e 2ξ = ℓ 2 4c 2 x 2 , e 2Φ = |b| 4 , (5.28)\n\nwhich give\n\nds 2 = - z 2 l 2 + 4c 2 dt - ℓdy 4cx 2 2 - ℓ 2 16c 2 x 2 dx 2 + dy 2 + z 2 l 2 + 4c 2 -1 dz 2 .\n\n(5.29)\n\nThe field strength (4.21) vanishes in this case.\n\nWe shall now obtain the form of the Killing spinors for AdS 4 , and will do this in the simpler c = 0 case. The solution of the Killing spinor equations yields\n\nα 0 = λ 1 - t ℓ + ℓ z λ 2 + w ℓ λ 3 , α 2 = - wz 2ℓ 2 λ 2 + 1 2 1 + zt ℓ 2 λ 3 - z 2ℓ λ 4 , α 1 = 2ℓ z λ 2 , α 12 = wz 2ℓ 2 λ 2 + 1 2 1 - zt ℓ 2 λ 3 + z 2ℓ λ 4 , (5.30)\n\nwhere the coefficients λ 1,...,4 span a four dimensional complex space, as expected in the case of maximal supersymmetry. In the form basis of the spinors ǫ = c 0 1 + c 1 e 1 + c 2 e 2 + c 12 e 1 ∧ e 2 , we obtain\n\nc 0 = λ 1 - t ℓ + ℓ z λ 2 + w ℓ λ 3 , c 2 = - z 2ℓ λ 1 + z 2ℓ t ℓ - ℓ z λ 2 - z w 2ℓ 2 λ 3 , c 1 = w ℓ λ 2 - t ℓ + ℓ z λ 3 + λ 4 , c 12 = wz 2ℓ 2 λ 2 - z 2ℓ t ℓ - ℓ z λ 3 + z 2ℓ λ 4 . (5.31)\n\nThe new Killing spinors corresponding to λ 2 and λ 4 both have foot_17 G 0 = 0. To study the action of the AdS 4 isometries it is useful to embed the hyperboloid in a five-dimensional flat space (U, V, T, X, Y ) with metric\n\nds 2 = -dU 2 + dV 2 -dT 2 + dX 2 + dY 2 .\n\n(5.32)\n\nThen, AdS 4 is the hypersurface\n\n-U 2 + V 2 -T 2 + X 2 + Y 2 = -ℓ 2 /4\n\nand its isometries are realized as the SO(3,2) isometries of the embedding space. The relation with the Poincaré coordinates is\n\nt ℓ = T U -V , x ℓ = X U -V , y ℓ = Y U -V , z = 2(U -V ) . (5.33)\n\nIf we define the vectors\n\nℓΛ a = |λ 1 | 2 -|λ 2 | 2 + |λ 3 | 2 -|λ 4 | 2 |λ 1 | 2 + |λ 2 | 2 -|λ 3 | 2 -|λ 4 | 2 λ 3 λ4 + λ3 λ 4 -λ1 λ 2 -λ 1 λ2 λ 2 λ4 + λ2 λ 4 -λ1 λ 3 -λ 1 λ3 i λ 2 λ4 -λ2 λ 4 + λ1 λ 3 -λ 1 λ3 , X a = U V T X Y , (5.34)\n\nwhere the index a = 1, . . . , 5 is an SO(3,2) index raised and lowered using the metric (5.32), then\n\nΛ a Λ a = - 1 ℓ 2 λ 3 λ4 -λ3 λ 4 + λ1 λ 2 -λ 1 λ2 2 ≥ 0 , (5.35)\n\nand the invariant b for the Killing spinors reads\n\nb = c * 0 c 2 + c 1 c * 12 = X a Λ a + iℓ 2 Λ a Λ a . (5.36)\n\nThe subfamily of static half BPS configurations is obtained by imposing the staticity condition ζ = 0 or equivalently vanishing NUT charge. It is parameterized by the single parameter left, δ ∈ R and the solutions are restricted to have the following charges\n\nM = 0 , n = 0 , P = 0 , Q = - δ ℓ .\n\nIn terms of the charges, the solution is given by b\n\n= - 1 ℓ z 2 + iℓQ 2z . (5.42)\n\nThe metric and electromagnetic field strength for this solution read\n\nds 2 = - Q 2 z 2 + z 2 ℓ 2 dt 2 + dz 2 Q 2 z 2 + z 2 ℓ 2 + 4ℓ 2 z 2 dwd w , (5.43)\n\nand\n\nF = - Q z 2 dt ∧ dz . (5.44)\n\nThis is simply the backreacted AdS 4 filled with the electric field generated by an electric charge Q placed in its center ζ = 0. The solution has a singularity there. Note that this solution was already shown to be 1/2 supersymmetric in [36] . It was also shown there that the Killing spinors are preserved if one compactifies the transverse two-dimensional plane to a two-torus. We will now discuss the Killing spinors for these metrics. The integrability conditions impose α 2 = 0 and\n\nb ′ + 1 ℓ - b ℓ b α 3 = b ′ + 1 ℓ α 4 . (5.45)\n\nWith these constraints, the Killing spinor equations simplify, and can be solved to give\n\nα 0 = λ 1 + 2iζ wλ 2 , α 1 = 0 , (5.46)\n\nα 2 = z 2 + iζz + ζ 2 4 + iδ 4z 2 + ζ 2 , α 12 = α 2 - λ 2 2 4z 2 + ζ 2 , (5.47)\n\nwhere λ 1,2 ∈ C parameterize the two dimensional space of Killing spinors. Then the most general Killing spinor for these metrics is\n\nǫ = (λ 1 + 2iζ wλ 2 ) 1 -ℓλ 2 2z + iζ 2z -iζ e 1 +b (λ 1 + 2iζ wλ 2 ) e 2 - z 2 -iζz + ζ 2 4 -iδ 4z 2 + ζ 2 λ 2 e 1 ∧ e 2 . (5.48)\n\nare particular cases of this larger class, and are obtained for α, β and γ constant. Note that also the ∂b = 0 and ∂ b = 0 subclasses fall into this family. Let's take for definiteness α, β, γ all anti-holomorphic, then b = b(z, w). The requirement that the integrability conditions allow for an extra Killing spinor, i.e. that they are of rank ≤ 2, in this case leads to several conditions. One of these is obtained from the minor of the last three lines of Ñwt and reads\n\n2∂ z b + 1 l ∂ z b + 1 l ∂ 2 b + 1 b ∂b∂b -2∂(Φ + ξ)∂b ∂b = 0. (5.51)\n\nThis gives three different cases to be analysed, corresponding to the vanishing of the first three factors of this equation (vanishing of the fourth factor implies b = b(z) and hence brings one back to the previous section)."
},
{
"section_type": "OTHER",
"section_title": "Deformations of AdS 2 × H 2",
"text": "The vanishing of the first factor in (5.51) implies b = -z ℓ + ic(w), where c(w) is an arbitrary holomorphic function. These are the α(w) = 0 supersymmetric Kundt solutions of Petrov type II, describing gravitational and electro-magnetic waves propagating on anti-Nariai space-time [16] .\n\nThe remaining integrability conditions however imply α 1 = α 2 = α 12 = 0, in which case there is no second Killing spinor, or ∂c = 0. Therefore there are no new half BPS solutions with non constant c. In this class c constant is the half supersymmetric anti-Nariai spacetime and the other preserve only 1/4 of the supersymmetries."
},
{
"section_type": "OTHER",
"section_title": "Deformations of AdS 4",
"text": "The vanishing of the second factor in (5.51) implies b = -z 2ℓ + ic(w). In this case we are considering the β 2 = 4αγ supersymmetric Kundt solutions, describing gravitational and electro-magnetic waves propagating on AdS 4 spacetime [16] .\n\nAgain the remaining integrability equations have to solutions: α 1 = α 2 = α 12 = 0 or ∂c = 0. Hence, as in the previous case, we find that there are no harmonic deformations of AdS 4 preserving half supersymmetry."
},
{
"section_type": "OTHER",
"section_title": "Deformations of Reissner-Nordström-Taub-NUT-AdS 4",
"text": "Not considering the previous two special cases, the general solution represents expanding gravitational and electro-magnetic waves propagating on a Reissner-Nordström-Taub-NUT-AdS 4 spacetime [16] . When Im(β) = 0, the solution can be put in Robinson-Trautman form and is of Petrov type II.\n\nThe vanishing of the third factor in (5.51) is given by\n\n∂ 2 b + 1 b ∂b∂b -2∂(Φ + ξ)∂b = 0 . (5.52)\n\nonly one differential constraint which needs to be satisfied for the existence of a second Killing spinor, i. e. for the matrices of integrability conditions to have rank 2, namely\n\n∂ 2 X -1 -2∂ξ∂X -1 = 0 .\n\n(5.56)\n\nThe above three differential equations can be integrated to\n\ne 2ξ = -i K( w)∂X -1 , ∂X -1 = i ℓ 2 K(w) 1 4X 4 + L , (5.57)\n\nwhere K(w) is an arbitrary holomorphic function and L is a real constant. The function K(w) corresponds to the freedom to choose holomorphic coordinates on the twodimensional space, and hence it can be gauged away. A convenient gauge choice will be K(w) = iℓ. Note that, for this choice, the imaginary part of the right hand side of the last equation vanishes, and therefore that ∂ y X = 0. For L = 0, (5.57) can be integrated to give\n\nX 3 = 3x 2ℓ , (5.58)\n\nwhich is (up to a rescaling of the coordinate x) the example given above with α = 1 3 . This was already found to be 1/2 supersymmetric in [16] . Here we find that this solution is a special case of the most general possibility.\n\nFor other values of the constant L it is convenient to use X as a new coordinate instead of solving for X(x). From (4.18) and (4. 19 ) it follows that σ can be chosen to be σ = dy 4X 4 .\n\n(5.59)\n\nThen the metric reads\n\nds 2 = -4X 2 dt + dy 4X 4 2 + 1 4X 2 dz 2 + ℓ 2 dX 2 X 2 (1 + 4LX 4 ) + 1 + 4LX 4 4X 6 dy 2 .\n\n(5.60)\n\nFinally, from (4.21) we obtain the gauge field strength\n\nF = 2dt ∧ dX . (5.61)\n\nNote that the geometry (5.60) is generically of Petrov type D, and becomes of Petrov type N for L = 0. Now let us turn our attention to the form of the second Killing spinor. First of all, the integrability conditions imply that it takes the form\n\nα T = (β 1 , β 2 , iX 3 e ξ β 2 , iX 3 e ξ β 2 ) ,\n\nwhere β 1 and β 2 are arbitrary space-time dependent functions. The Killing spinor equations (4.31) yield\n\nβ 1 = λ 1 -1 2 λ 2 b -2 , β 2 = λ 2 b -2\n\n, where λ 1 and λ 2 are integration constants. This implies that the new Killing spinor takes the form ǫ = λ 1 ǫ 1 + λ 2 ǫ 2 , where\n\nǫ 1 = 1 + iXe 2 , ǫ 2 = 1 2 X -2 (1 -iXe 2 ) + 1 4 X -4 + L (e 1 -iXe 1 ∧ e 2 ) .\n\n(5.62)\n\nNote that G 0 = 0 as well in this class.\n\nOne interesting aspect of the second Killing spinor ǫ 2 is the norm of its associated Killing vector V µ = D(ǫ 2 , Γ µ ǫ 2 ). We find V µ V µ = -4X 2 L 2 , hence the second Killing spinor is indeed null for the case L = 0, as was noticed before, while it is timelike for L = 0. In the latter case, to understand whether the solution belongs also to the null class of supersymmetric solutions, we have therefore to study the most general linear combination of the two Killing spinors. The Killing vector Ṽ constructed from\n\nǫ = λ 1 ǫ 1 + λ 2 ǫ 2 has norm Ṽ 2 = 1 X 2 λ1 λ 2 -λ 1 λ2 2 -4X 2 L|λ 1 | 2 + |λ 2 | 2 2 ,\n\nwhich can vanish only if L ≤ 0. We have therefore three cases:\n\n1. L > 0, pure timelike class, Petrov type D.\n\n2. L = 0, belongs to both null and timelike classes, Petrov type N. This is the homogeneous half BPS pp-wave in AdS. (In the terminology of [16] it has a wave profile G α with α = 0).\n\n3. L < 0, belongs to both null and timelike classes, Petrov type D.\n\nActually the solutions (5.60) with L > 0 can be cast into a simpler form. This is done by trading the coordinate y for a new variable ψ = Lyt. For convenience, let us also introduce the Schwarzschild coordinate r and rescale z,\n\nr = - ℓ √ LX , ζ = 1 2 √ L z .\n\n(5.63)\n\nIn the new coordinates, the metric and the gauge field strength read\n\nds 2 = - r 2 ℓ 2 + q 2 e r 2 dt 2 + dr 2 r 2 ℓ 2 + q 2 e r 2 + r 2 ℓ 2 dψ 2 + dζ 2 , F = q e r 2 dt ∧ dr , (5.64)\n\nwhere we have defined q e = 2ℓ/ √ L. This is precisely the half BPS solution obtained in [36] , the massless limit of an electrically charged toroidal black hole, which forms a naked singularity. It is also interesting to note that the charge q e diverges in the L → 0 limit. This limit is naively singular in these coordinates, but it can be taken if we perform a Penrose limit [37, 38] . The existence of this limit explains why we obtained a one-parameter family of geometries (5.60) connecting the massless limit of toroidal black holes and a pp-wave. Indeed, define the new coordinates (X + , X -, R, Z) and the rescaled charge Q e by\n\nψ + t = 2ǫ 2 X + , ψ -t = 2X -, r = 1 ǫR , ζ = ǫZ , q e = Q e ǫ .\n\n(5.65)\n\nThen, the singular limit ǫ → 0 yields is a regular solution of the theory and corresponds to the half supersymmetric solution (5.60) with L = 0,\n\nds 2 = ℓ 2 R 2 4 dX + dX -- Q 2 e R 4 ℓ 6 dX -2 + dR 2 + dZ 2 , F = Q e ℓ 2 dX -∧ dR . (5.66)\n\nIn the procedure, we have blown up the metric in the neighborhood of a geodesic with ψ + t constant near the boundary r → ∞ of AdS. We now turn to the L < 0 case, which is both timelike and lightlike. Let us define L = -µ 2 . We can perform a coordinate transformation inspired from the previous one,\n\nψ = Ly -t , r = - ℓ µX , ζ = µ 2 z , (5.67)\n\nunder which the metric and the field strength become\n\nds 2 = r 2 ℓ 2 - q 2 e r 2 dt 2 + dr 2 r 2 ℓ 2 -q 2 e r 2 + r 2 ℓ 2 -dψ 2 + dζ 2 , F = q e r 2 dt ∧ dr , (5.68)\n\nwhere we have defined q e = 2ℓ/µ. We see that this is the precisely the metric for L > 0 after the double analytic continuation t → it , ψ → iψ , q e → -iq e .\n\n(5.69)\n\nThis solution represents therefore a bubble of nothing in AdS [39] [40] [41] [42] . Note that the metric is singular for r = √ ℓq e . One should compactify t, in such a way to eliminate the conical singularity on the (t, r) hypersurface. Then, if we compactify also ζ, this S 1 will have a minimal radius for r = √ ℓq e (the boundary of the bubble of nothing) and then grow with r. Note that for r → ∞ one locally recovers AdS spacetime, and that the L = 0 solutions can again be understood as a Penrose limit of this metric."
},
{
"section_type": "OTHER",
"section_title": "Action of the PSL(2, R) group on the imaginary b solutions",
"text": "We can now generate new supersymmetric solutions by acting with the PSL(2, R) symmetry group (4.28)-(4.29) on the known ones. It is easy to check that the AdS 4 and AdS 2 ×H 2 solutions are invariant under this group (although it acts non trivially on the Killing spinors). Its action on the b = b(z) subfamily of the RNTN-AdS 4 solutions was studied in [16] , where it was shown that it acts non trivially on the charges, by mixing them. Here we want to apply it to the imaginary b solutions of the previous paragraph.\n\nThe new solution solution of the supersymmetry equations ( 4 .22)-(4.23) generated by the transformation (4.28)-(4.29) is b = -γ 2 Xz 2 2γ 2 ℓXz + i , e 2( Φ+ξ) = γ 4 z 4 4X 4 1 + 4LX 4 , (5.70) where, without loss of generality, we eliminated α by means of a translation of z foot_19 , and dropped the prime of the new coordinate z ′ . The shift function is then determined by solving equations (4.18) and (4.19),\n\nσ x = 0 , σ y = 1 + 4LX 4 4γ 2 X 4 z 2 + γ 2 ℓ 2 X 2 .\n\n(5.71)\n\nThen, defining the new coordinates (T, σ, p, q) through\n\nT = t 2ℓ 2 γ 2 , σ = y 2 , p = - ℓ X , q = 2ℓ 2 γ 2 z , (5.72)\n\nthe metric reads\n\nds 2 = - Q(q) q 2 + p 2 dT + P (p) q 2 + p 2 ℓ 2 dσ 2 + q 2 + p 2 Q(q) dq 2 + q 2 + p 2 P (p) dp 2 + 1 ℓ 4 (q 2 + p 2 )P (p) dσ 2 , (5.73)\n\nwith\n\nQ(q) = q 4 ℓ 2 , P (p) = 1 ℓ 2 p 4 + 4Lℓ 2 , (5.74)\n\nand the gauge field (4.21) is\n\nF = d pq 2 ℓ(q 2 + p 2 )\n\n∧ dT + d 4ℓLp q 2 + p 2 ∧ dσ .\n\n(5.75)\n\nThe form of the metric suggests some connection with the Plebanski-Demianski family of solutions, and indeed these geometries are of Petrov type D for L = 0, and of Petrov type N for L = 0, but we were not able to find the precise relation. Note also that the parameter γ has been reabsorbed in the new variables, and we are left with a one-parameter (L) family of solutions.\n\nThe left hand side of the necessary condition (4.34) for the existence of a second Killing spinor reads, for this solution,\n\n- 9iX 4 (1 + 4LX 4 ) ℓ 2 (1 + 4γ 4 ℓ 2 X 2 z 2 ) 4 γ 2 (5.\n\n76) which clearly vanishes only for γ = 0, i.e. if the PSL(2, R) transformation is trivial. Therefore, the new solutions (5.73)-(5.75) preserves only 1/4 of the supersymmetries, and we explicitly see that the PSL(2, R) transformations can break any additional supersymmetry. Also note that if we perform the PSL(2, R) transformation adapting the original metric to a different Killing spinor, we could in principle end up with other supersymmetric solutions.\n\nSurprisingly, we find that the L = 0 solution can be cast in the Lobatchevski wave form, even though it only has a time-like Killing spinor. This can be seen by trading the coordinates (q, p) for (x, z) defined by\n\nx = ℓ 3 2 1 q 2 - 1 p 2 , z = ℓ 3 qp , (5.77)\n\nin the metric (5.73) with L = 0, which becomes\n\nds 2 = ℓ 2 z 2 -2 dT dσ + z 2 2ℓ √ x 2 + z 2 x - √ x 2 + z 2 x + √ x 2 + z 2 dT 2 + dz 2 + dx 2 .\n\n(5.78)\n\nThe field strength can be easily obtained from equation (5.75) but the result is not particularly enlightening and therefore we do not report it. This metric represents a 1/4 BPS Lobatchevski wave, whose Killing spinor falls in the timelike class. This does not contradict the results obtained in the null case, since the null Lobatchevski had a field strength (3.6) of the form F = φ ′ (T )dT ∧ dz, while this solution has a much more complicated gauge field. It is however interesting to note that the solutions of the null case do not exhaust all possible supersymmetric Lobatchevski waves. where we have defined k = 4αγβ 2 and ∆ = 4∂ ∂. Interestingly, as shown in [16] , this system of equations follows from the dimensionally reduced Chern-Simons action [43, 44] ,\n\nS = d 2 x (2) g (2) Rη + η 3 , (5.81)\n\nif we use the conformal gauge (2) g ij dx i dx j = e 2ξ (dx 2 + dy 2 ) and η is the curl of a vector potential, (2) g ǫ ij η = ∂ i A j -∂ j A i . To obtain equations (5.80) we vary the action with respect to A i and ξ. When varying the dimensionally reduced Chern-Simons action with respect to g ij there is however an additional equation to (5.80) .\n\nUsing the results of Grumiller and Kummer [48] , one obtains the most general solution to the dimensionally reduced Chern-Simons system [16]\n\ne 2ξ = L ℓ 4 - k 2 η 2 + 1 4 η 4 , (5.82)\n\nwhere L is an integration constant and dη = e 2ξ dx. Trading the coordinate x for η, we get the following configuration of the fields\n\nds 2 = - 4 ℓ 2 P 2 2 P ′2 2 + η 2 [dt + σ] 2 + ℓ 2 4 P ′2 2 + η 2 P 2 2 dz 2 + P 2 2 e -2ξ dη 2 + e 2ξ dy 2 , A = 2 ℓ P 2 η P ′2 2 + η 2 [dt + σ] + ℓ 4 Vdy -i ℓ 4 d log b b , (5.83)\n\nwhere P 2 (z) = αz 2 + βz + γ, k is defined as above and the shift function reads As can be seen from the Poisson bracket (4.34), the only possibility to have 1/2 supersymmetry is α = 0 and hence k ≤ 0. In fact, starting from any solution with k ≤ 0, one can always obtain α = 0 by an appropriate PSL(2, R) transformation. The these equations in a number of subcases in section 5, and thereby found several new solutions, like the bubbles of nothing in AdS 4 , already obtained in the null formalism, and their PSL(2, R)-transformed configurations. Furthermore, our results showed that the generalized holonomy in the case of one preserved complex supercharge is contained in A(3, C), supporting thus the classification scheme of [4] .\n\nσ = ℓ 2 2 αη 2 + e 2ξ P 2 dy . ( 5\n\nIn addition, the time-dependence of a second time-like Killing spinor was shown to be an overall exponential factor with coefficient G 0 in section 4.4. In the case G 0 = 0 these equations have been solved in full generality, up to a second order ordinary differential equation. We expect this class to comprise a large number of interesting 1/2-BPS solutions. Indeed, all the examples of section 5 either have vanishing G 0 or can be transformed to that case by a coordinate transformation.\n\nThere are several interesting points that remain to be understood. First of all, it would be desirable to get a deeper insight into the underlying geometric structure in the case of U(1) invariant spinors. In five dimensions, spacetime is a fibration over a four-dimensional Hyperkähler or Kähler base for ungauged and gauged supergravity respectively [8, 12] , whereas in four-dimensional ungauged supergravity one has a fibration over a three-dimensional flat space [5] . This suggests that the base for D = 4 gauged supergravity might be an odd-dimensional analogue of a Kähler manifold, i. e. , a Sasaki manifold. From the equations (4.22) and (4.23) this is not obvious. Secondly, in [16] , a surprising relationship between the equations (4.22), (4.23) governing 1/4 BPS solutions and the gravitational Chern-Simons theory [43] was found. Why such a relationship should exist is not clear at all, and deserves further investigations.\n\nThe third point concerns preons, which were conjectured in [45] to be elementary constituents of other BPS states. In type II and eleven-dimensional supergravity, it was shown that imposing 31 supersymmetries implies that the solution is locally maximally supersymmetric [27, 30, 46] . Similar results in four-and five-dimensional gauged supergravity were obtained in [28, 29] . This implies that preonic backgrounds are necessarily quotients of maximally supersymmetric solutions. While M-theory preons cannot arise by quotients [47] , it remains to be seen if 3/4 supersymmetric solutions to N = 2, D = 4 or D = 5 gauged supergravities really do not exist. The only maximally supersymmetric backgrounds in these theories are AdS 4 [13] and AdS 5 [12] respectively, so the putative preonic configurations must be quotients of AdS.\n\nFinally, it would be interesting to apply spinorial geometry techniques to classify all supersymmetric solutions of four-dimensional N = 2 matter-coupled gauged supergravity. Work in this direction is in progress [49] . on U ⊗ C, and then extend it to ∆ c . The Spin(3,1) invariant Dirac inner product is then given by D(η, θ) = Γ 0 η, θ .\n\n(A.3)\n\nIn many applications it is convenient to use a basis in which the gamma matrices act like creation and annihilation operators, given by\n\nΓ + η ≡ 1 √ 2 (Γ 2 + Γ 0 ) η = √ 2 e 2 ⌋η , Γ -η ≡ 1 √ 2 (Γ 2 -Γ 0 ) η = √ 2 e 2 ∧ η , Γ • η ≡ 1 √ 2 (Γ 1 -iΓ 3 ) η = √ 2 e 1 ∧ η , Γ •η ≡ 1 √ 2 (Γ 1 + iΓ 3 ) η = √ 2 e 1 ⌋η . (A.4)\n\nThe Clifford algebra relations in this basis are {Γ A , Γ B } = 2η AB , where A, B, . . . = +, -, •, • and the nonvanishing components of the tangent space metric read η\n\n+-= η -+ = η •• = η •• = 1.\n\nThe spinor 1 is a Clifford vacuum, Γ + 1 = Γ •1 = 0, and the representation ∆ c can be constructed by acting on 1 with the creation operators Γ + = Γ -, Γ • = Γ • , so that any spinor can be written as\n\nη = 2 k=0 1 k! φ ā1 ...ā k Γ ā1 ...ā k 1 , ā = +, • .\n\nThe action of the Gamma matrices and the Lorentz generators Γ AB is summarized in the table 6 .\n\n1 e 1 e 2 e 1 ∧ e 2 Γ + 0 0 √ 2 - √ 2e 1 Γ - √ 2e 2 - √ 2e 1 ∧ e 2 0 0 Γ • √ 2e 1 0 √ 2e 1 ∧ e 2 0 Γ • 0 √ 2 0 √ 2e 2 Γ +- 1 e 1 -e 2 -e 1 ∧ e 2 Γ •• 1 -e 1 e 2 -e 1 ∧ e 2 Γ +• 0 0 -2e 1 0 Γ +• 0 0 0 2 Γ -• -2e 1 ∧ e 2 0 0 0 Γ -• 0 2e 2 0 0\n\nTable 6: The action of the Gamma matrices and the Lorentz generators Γ AB on the different basis elements.\n\nψ |b| c 2 c * 12 σ w -c 12 c * 0 σ w + |b| 2 (c 0 c * 1 σ w -c 1 c * 0 σ w ) + e ψ 4|b| (c 0 c * 2 + c 1 c * 12 -c 2 c * 0 -c 12 c * 1 ) dw ∧ d w . (B.6)\n\nGiven the first Killing spinor of the form ǫ 1 = 1 + be 2 and the second Killing spinor ǫ 2 = c 0 1 + c 1 e 1 + c 2 e 2 + c 12 e 1 ∧ e 2 , one can also construct mixed bilinears of the type D(ǫ 1 , Γ ••• ǫ 2 ), which verify the same differential equations as the bilinears built from the original two Killing spinors:\n\nf = -i( bc 0 -c 2 ) , ĝ = bc 0 + c 2 , (B.7) V = 1 2b (c 2 + bc 0 ) (dt + σ) + 1 2b (c 2 -bc 0 ) dz + 1 |b| e ψ bc 1 -c 12 d w , (B.8) B = 1 2b (c 2 -bc 0 ) (dt + σ) + 1 2b (c 2 + bc 0 ) dz + 1 |b| e ψ bc 1 + c 12 d w . (B.9)\n\nC. The case P ′ = 0\n\nIn section 4.3, we simplified the equations for the second Killing spinor under the assumption P ′ = 0, where P = e -2ψ b∂b. Here we consider the case P ′ = 0. To this end, we need the following subset of the Killing spinor equations (4.31): 3) and deriving with respect to t, one gets ∂b ∂ t ψ 1 = ∂b ∂ t ψ 1 = 0. When ∂ t ψ 1 = 0, this means that ∂b = ∂b = 0, so b = b(z), which is a case analyzed in section 5.1. If instead ∂ t ψ 1 = 0, all the ψ i are independent of t, and the Killing spinor equations reduce to the system (4.38) to (4.49) with G 0 = 0.\n\n∂ + ψ 2 - √ 2r 2 b′ b + 1 ℓ b ψ 2 - √ 2r 2 b ′ b + 1 ℓb ψ 12 = 0 , (C.1) ∂ + ψ 12 -re -ψ ∂ • ln b ψ 1 - √ 2r 2 2 r ′ r + 1 ℓ b ψ 12 = 0 , (C.2) ∂ -ψ 2 + 1 r e -ψ ∂ • ln b ψ 1 - √ 2 2 r ′ r + 1 ℓb ψ 2 = 0 , (C.3) ∂ -ψ 12 - √ 2 b′ b + 1 ℓ b ψ 2 - √ 2 b ′ b + 1 ℓb ψ 12 = 0 , (C.4) re -ψ ∂ • 1 r 2 e 2ψ ψ 2 - √ 2 b ′ b + 1 ℓb ψ 1 = 0 , (C.5) re -ψ ∂ • 1 r 2 e 2ψ ψ 12 + √ 2 b′ b + 1 ℓ b ψ 1 = 0 . (C.6) If P ′ = 0, ( 4\n\nIn the case ∂P = 0, consider the integrability condition\n\nψ 1 Q ′ + ψ -∂Q = 0 , (C.8)\n\nwhere Q = e -2ψ b∂ b, following from the first line of Ñwt . As long as Q ′ = 0, with the same reasoning as in section 4.3, one obtains the system (4.38) to (4.49). If Q ′ = 0, (C.8) implies ψ -= 0 or ∂Q = 0. The case ψ -= 0 was already considered above, so the only remaining case is P ′ = ∂P = Q ′ = ∂Q = 0. For P = Q = 0 we get again b = b(z), so without loss of generality we can assume P = 0 or Q = 0. Suppose that Q = 0, P = 0, so b = b(w, z). Take the logarithm of e -2ψ b∂b = P ( w), derive with respect to z, use (4.15), and apply ∂. This leads to ∂b = 0, which is a contradiction to the assumption P = 0. In the same way one shows that P = 0, Q = 0 is not possible, so that both P and Q must be nonvanishing. Now use the third row of Ñ wt , which leads to Qψ 2 = 0 and hence ψ 2 = 0. Finally, the last row of Ñ wt yields ψ -= 0, i. e. , the case already considered above. Hence, the conclusion is that in the case P ′ = 0, the second Killing spinor either has G 0 time-dependence of the form (4.37), or leads to solutions with b = b(z). The latter are treated separately in section 5.1. As can be found there, all 1/2-BPS solutions with b = b(z) also have second Killing spinors with G 0 time-dependence of the form (4.37). Hence this time-dependence is a completely general result 20 for second Killing spinors in the time-like case. We thus conclude that ψ1 ψ ′ 1 -ψ′ 1 ψ 1 = 0, and hence ψ 1 = ζ(z)e iθ 0 where θ 0 is a constant and ζ(z) is a real function. Sending ψ i → e -iθ 0 ψ i we can take ψ 1 real and non-negative without loss of generality. Let us now consider the case where both ψ 1 and ψ -are non-vanishing. This allows to introduce new coordinates Z, W, W such that where α is a real integration constant. Equations (D.9) and (D.11) are then identically satisfied. Solving (D.14) for χ and plugging into (D.13) yields finally the ordinary differential equation (4.50), which determines half-supersymmetric solutions with G 0 = 0. Putting together all our results, we obtain (4.52) for the metric. Note that in the case γ = 0 one can always set γ = 1 by rescaling the coordinates. The second Killing spinor for these backgrounds is given by α T = (α 0 , ρ -2 e -γZ/ℓ , χ + 1 2ρ e H , χ -1 2ρ e H ) , where α 0 = -2γt ℓ + α0 (Z, w, w) , and α0 is a solution of the system\n\n∂ Z α0 = 1 ψ 1 ρ 2 ρ ρ -i φ + γ 2ℓ , ∂ α0 = - 2γ ℓ σ w + 1 ψ 1 ρ 2 - ρ ρ + i φ , (D.15) ∂ α0 = - 2γ ℓ σ w + 1 ψ 1 ρ 2 - ρ ρ + i φ + γχ e 2H ℓψ 1 ρ 2 .\n\nIt is straightforward to verify that the integrability conditions for this system are already implied by (D.9), (D.10) and (D.12). Consider now the case ψ -= 0. From the difference of equations where Y (w, w) is some real function to be determined.\n\nWe thus have to solve just for the ansatz (D.16). Equation holds. We conclude that a solution to the system (D. 19 ), (D.20) describes a 1/2-BPS configuration of the \"gravitational Chern-Simons\" system discussed in [16] . If C(w) = 0 then necessarily also Y = 0 so that we are left with AdS. If C(w) = 0 then we can define new variables W and W such that As what we did in the previous case, we can set C(w) = 1 using the residual gauge invariance w → W (w), ψ → ψ = ψ -1 2 ln(dW/dw) -1 2 ln(d W /d w) leaving invariant the metric e 2ψ dwd w. We can thus take W = w without loss of generality, and get where L is a real constant and k = -1 21 . We can thus use Y as a new coordinate, instead of i(ww). Call X = w + w, so that the solution reads\n\nds 2 = - 4 ℓ 2 z 2 1 + Y 2 dt\n\n+ ℓ 2 2z P C (Y )dX 2 + ℓ 2 4 1 + Y 2 z 2 dz 2 + z 2 P C (Y )dX 2 + dY 2 P C (Y ) , A = 2 ℓ z Y 1 + Y 2 dt + ℓY P C (Y ) 1 + Y 2 -1 4 1 + Y 2 dX + ℓ 2 dY 1 + Y 2 . (D.32)"
}
] | |
arxiv:0704.0251 | 0704.0251 | 1 | 10.1103/PhysRevA.76.042309 | 377bbca34ff04927ba11a6c996fae271f96407730fe7a625810d51b06e081bb2 | Entanglement of Subspaces and Error Correcting Codes | We introduce the notion of entanglement of subspaces as a measure that
quantify the entanglement of bipartite states in a randomly selected subspace.
We discuss its properties and in particular we show that for maximally
entangled subspaces it is additive. Furthermore, we show that maximally
entangled subspaces can play an important role in the study of quantum error
correction codes. We discuss both degenerate and non-degenerate codes and show
that the subspace spanned by the logical codewords of a non-degenerate code is
a 2k-totally (maximally) entangled subspace. As for non-degenerate codes, we
provide a mathematical definition in terms of subspaces and, as an example, we
analyze Shor's nine qubits code in terms of 22 mutually orthogonal subspaces. | [
"Gilad Gour",
"Nolan R. Wallach"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | We introduce the notion of entanglement of subspaces as a measure that quantify the entanglement of bipartite states in a randomly selected subspace. We discuss its properties and in particular we show that for maximally entangled subspaces it is additive. Furthermore, we show that maximally entangled subspaces can play an important role in the study of quantum error correction codes. We discuss both degenerate and non-degenerate codes and show that the subspace spanned by the logical codewords of a non-degenerate code is a k-totally (maximally) entangled subspace. As for non-degenerate codes, we provide a mathematical definition in terms of subspaces and, as an example, we analyze Shor's nine qubits code in terms of 22 mutually orthogonal subspaces.
Bipartite entanglement has been recognized as a crucial resource for quantum information processing tasks such as teleportation [1] and super dense coding [2] . As a result, in the last years there has been an enormous effort to understand and study the characterization, manipulation and quantification of bipartite entanglement [3] . Yet, despite a great deal of progress that was achieved, the theory on mixed bipartite entanglement is incomplete and a few central important questions such as the additivity of the entanglement of formation [4] remained open. Perhaps the richness and complexity of mixed bipartite entanglement can be found in the fact that a finite set of measures of entanglement is insufficient to completely quantify it [5] . In this paper we shed some light on mixed bipartite entanglement with the introduction of a new kind of measure of entanglement which we call entanglement of subspaces (EoS). We will see that EoS can play an important role in the study of quantum error correcting codes (QECC).
It has been shown recently [6, 7] that geometry of high-dimensional vector spaces can be counterintuitive especially when subspaces with very unique properties are more common than one intuitively expects. That is, roughly speaking, if a high dimensional subspace is selected randomly it is quite likely to have strange properties. For example, in [7] it has been demonstrated that a randomly chosen subspace of a bipartite quantum system will likely contain nothing but nearly maximally entangled states even if the dimension of the subspace is almost of the same order as the dimension of the original system. This kind of result has implications, in particular, to super-dense coding [8] and for quantum communication in general (see also [9] for other implications of randomly selected subspaces). The quantification of the entanglement of such subspaces is therefore very important and we start with its definition. Definition 1. Let H A and H B be finite dimensional Hilbert spaces and let W AB be a subspace of H A ⊗ H B . The entanglement of W AB is defined as:
E W AB ≡ min ψ AB ∈W AB E ψ AB : ψ AB = 1 , (1)
where E ψ AB is the entropy of entanglement of ψ AB . Note that if the subspace W AB contains a product state then E(W AB ) = 0. On the other hand, if, for example, W AB is orthogonal to a subspace spaned by an unextendible product basis (UPB) [11, 12] then E(W AB ) > 0.
Claim: Let d A = dim H A and d B = dim H B . If E(W AB ) > 0 then dim W AB ≤ (d A -1)(d B -1). (2)
This claim follows from [10] and also related to the fact that the number of (bipartite) states in a UPB is at least [11] . Note that for two qubits (i.e.
d A + d B -1
d A = d B = 2) E(W AB
) can be greater than zero only for one dimensional subspaces. We can use Eq. ( 1 ) to define another measure of entanglement on bipartite mixed states.
A ⊗ H B be a bipartite mixed state and let S AB ρ be the support subspace of ρ. Then, the entanglement of the support of ρ is defined as
E Support (ρ) ≡ E(S AB ρ ) .
It can be easily seen that this measure is not continuous and therefore can not be considered as a proper measure of entanglement. Nevertheless, this measure can serve as a mathematical tool to find lower bounds for other measures of entanglement that are more difficult to calculate especially in higher dimensions. For example, the entanglement of the support of ρ provides a lower bound for the entanglement of formation. It can be shown that in lower dimensions the bound is generally not tight. For example, for two qubits in a mixed state ρ, the entanglement of the support E Support (ρ) = 0 (see Eq. ( 2 )). On the other hand, in higher dimensions the bound can be very tight [6, 7] .
In this section we study some of the properties of EoS with a focus on additivity properties. The EoS provides a lower bound on the entanglement of formation and our interest in its additivity properties is due to one of the most important unresolved questions in quantum information, namely the additivity conjecture for the entanglement of formation. In particular, the additivity question of EoS is identical to the additivity conjecture of quantum channel output entropy [13] that has been shown to be equivalent to the additivity conjecture of entanglement of formation [4] . Thus, additivity properties of EoS can shed some light on this topic.
Here we consider the additivity properties of EoS. We start by showing that if U AB and V A ′ B ′ are two subspaces such that E(U AB ) > 0 and/or E(V
A ′ B ′ ) > 0 then E(U AB ⊗ V A ′ B ′ ) > 0.
Consider W = C n ⊗ C m . Let e j , j = 1, ..., n be the standard basis of C n . We will also use the notation f j for the standard basis of C m . An element of a tensor product of two vector spaces, A and B will be called a product if it is of the form a ⊗ b with a ∈ A and b ∈ B. Hence i u m ik v i is a product. Our assumption implies that it must be 0. Hence
Proposition 1. Let u 1 , ..., u d , v 1 , ..., v d ∈ W be such that if x = i b i v i is a product then x = 0. If z = i u i ⊗ v i is a product in (C n ⊗ C n ) ⊗ (C m ⊗ C m ) then z = 0. Proof. We write u i = n j=1 e j ⊗ u ij and v j = n j=1 e j ⊗ v ij . Assume that i u i ⊗v i is a product in (C n ⊗C n )⊗(C m ⊗C m ). This means that there exists z ∈ C n ⊗C n and w ∈ C m ⊗C m such that i,k,l (e k ⊗e l ) ⊗(u ik ⊗v il ) = z⊗w.
0 = i,k,m u m ik e k ⊗f m ⊗v i = i u i ⊗v i .
As was to be proved.
Note that the proposition above states that if none of the decompositions of a bipartite mixed state, ρ, contain a product state, then also none of the decompositions of ρ ⊗ σ (σ is a bipartite mixed state) contain a product state. This property is related to the additivity conjecture [4] for the entanglement of formation (and other measures) and one of the main questions that we will consider here is wether the EoS is additive. That is, does
E(U AB ⊗ V A ′ B ′ ) = E(U AB ) + E(V A ′ B ′ ) ?
Clearly, if the EoS were additive then the proposition above would have been a trivial consequence of that. However, we were not able to prove the additivity of EoS (in general) although for some special cases it has been tested numerically in [14] and no counter example has been found. The proposition below provides a lower bound.
Proposition 2. Let N = min{dim U AB , dim V A ′ B ′ }. Then E(U AB ) + E(V A ′ B ′ ) -log N ≤ E U AB ⊗ V A ′ B ′ . (3)
The equation above provides a lower bound whereas the upper bound
E U AB ⊗ V A ′ B ′ ≤ E(U AB )+E(V A ′ B ′ )
follows directly from the definition of EoS. Thus, for N = 1 the EoS is additive. Note also that even if N is small (e.g. N = 2), E(U AB ) and E(V A ′ B ′ ) can be arbitrarily large (i.e. depending on d A and d B but not on N ).
Proof. Let χ be a normalized vector in U AB ⊗ V A ′ B ′ . We can write χ in its Schmidt decomposition as follows:
χ = i √ p i u AB i ⊗ v A ′ B ′ i ,
where i p i = 1 (p i ≥ 0) and the u AB i 's (v A ′ B ′ i 's) are orthonormal. Now, from the strong subadditivity of the von-Neumann entropy we have
S(ρ A ′ ) + S(ρ B ) ≤ S(ρ AB ) + S(ρ AA ′ ) , where ρ A ≡ Tr A ′ BB ′ χ ⊗ χ * , ρ B ≡ Tr AA ′ B ′ χ ⊗ χ * , etc. Now, note that S(ρ AA ′ ) = E(χ) and S(ρ AB ) = H({p i }) ≤ log N , where H({p i }) is the Shanon entropy. Furthermore, note that ρ A ′ = i p i ω i and ρ B = i p i σ i
where
ω i ≡ Tr B ′ v A ′ B ′ i ⊗ v A ′ B ′ i * and σ i ≡ Tr A u AB i ⊗ u AB i * .
Hence, since the von-Neumann entropy is concave we have
S(ρ A ′ ) ≥ i p i S(ω i ) = i p i E(v A ′ B ′ i ) ≥ E V A ′ B ′ .
and similarly S(ρ B ) ≥ E U AB . Combining all this we get
E V A ′ B ′ + E U AB ≤ log N + E(χ) , for all χ ∈ U AB ⊗ V A ′ B ′ .
This complete the proof.
As we have seen above, if N = 1 then the EoS is clearly additive. As we will see in the next subsection, it is also additive for maximally entangled subspaces:
Definition 3. Let W be a subspace of H A ⊗ H B and let d A = dim H A and d B = dim H B . W is said to be a maximally entangled subspace in H A ⊗ H B if E(W ) = log m , (4)
where m ≡ min{d A , d B }.
The term maximally entangled subspace have been used in [6, 7] for a subspace W with E(W ) slightly less than log m. In this paper, we will call such subspaces nearly maximally entangled to distinguish from (exactly) maximally entangled subspaces as defined above.
In [15] it has been shown that the average entanglement of a pure state φ ∈ H A ⊗ H B which is chosen randomly according to the unitarily invariant measure satisfies
E(φ) ≥ log 2 d A - d A 2 ln 2d B
where without loss of generality d A ≥ d B . Later on, in [6, 7] this result has been extended to subspaces and in particular it has been shown, somewhat surprisingly, that a randomly chosen subspace of bipartite quantum system will likely be a nearly maximally entangled subspace. Thus, as nearly maximally entangled subspaces are quite common it is important to understand their structure. As a first step in this direction, in the following we study the structure of (exactly) maximally entangled subspaces.
Let φ be a state in H A ⊗H B . If e 1 , ..., e m is an orthonormal basis of H B we may write
φ = dB i=1 φ i ⊗e i . We define a d B × d B Hermitian matrix B = [ φ i |φ j ] (i.e.
B is the reduced density matrix). Let λ 1 , ..., λ dB be the set of eigenvalues of B counting multiplicity. Then the entanglement of φ is
E(φ) = - i λ i log(λ i ) .
It is easy to show that E(φ) ≤ log m and equality is attained if and only if B = 1 m P with P a projection matrix onto a d dimensional subspace of C dB . Clearly this definition of entropy is independent of the choice of basis and could also be given using an orthonormal basis of H A and analyzing the corresponding d A coefficients in H B . Under the condition of equality φ is maximally entangled, and this in particular implies that if
d A ≥ d B then φ i |φ j = 1 d B δ ij . Proposition 3. Assume that d A ≥ d B and set m = d B .
Let U AB be a maximally entangled subspace in H A ⊗H B of dimension d. If e 1 , ..., e m is an orthonormal basis of H B then there exist U 1 , ..., U m subspaces of H A such that U i |U j = 0 if i = j, dim U j = d > 0 for all j = 1, ..., m and unitary operators T i :
C d → U i i = 1, ..., m such that U AB = { i T i w⊗e i |w ∈ C d }.
Conversely, if U 1 , ..., U m are mutually orthogonal subspaces of H A such that dim U j = d > 0 for all j = 1, ..., m and we have unitary operators
T i : C d → U i i = 1, ..., m such that U AB = { i T i w⊗e i |w ∈ C d } ,
then U AB is maximally entangled.
Proof. Let ψ 1 , ..., ψ d be an orthonomal basis of U AB . Then we can write
ψ j = i ψ ij ⊗e i with ψ ij |ψ kj = 1 m δ ik . The condition on U AB is that if a ∈ C d is a unit vector then a j ψ j is maximally entangled in H A ⊗H B . This implies that d j=1 a j ψ lj d j=1 a j ψ kj = 1 m δ l,k .
Fix l = k and let p = q ≤ d be two integers. Let a = (a 1 , ..., a d ) with a j = 0 for j = p or j = q. Set a p = b, a q = c and |b| 2 + |c| 2 = 1. Then we have bψ lp + cψ lq |bψ kp + cψ kq = 0.
On the other hand we have bψ lp + cψ lq |bψ kp + cψ kq = bc ψ lp |ψ kq + cb ψ lq |ψ kp Set z = bc. We look at two cases:
first z = 1 2 (b = c = 1 √ 2 ) and second z = i √ 2 (b = 1 √ 2 , c = i √ 2
). Thus we have
ψ lp |ψ kq + ψ lq |ψ kp = 0
for the first case and
ψ lp |ψ kq -ψ lq |ψ kp = 0
for the second. Hence, ψ lp |ψ kq = ψ lq |ψ kp = 0. We set
U l = Span{ψ lp |p = 1, ..., d}. Then U l |U k = 0 if l = k.
We now consider what happens when l = k. We first note that taking a p = 1 and all other entries equal to 0 we have ψ lp |ψ lp = 1 m . Now using b, c as above for p = q we have
ψ lp |ψ lp + ψ lp |ψ lq + ψ lq |ψ lp + ψ kp |ψ kp = 2 m
and
ψ lp |ψ lp + i ψ lp |ψ lq -i ψ lq |ψ lp + ψ kp |ψ kp = 2 m .
Hence as above we find that ψ lp |ψ lq = 0 if p = q. Thus √ mψ l1 , ..., √ mψ ld is an orthonormal basis of U l . This implies that the spaces U 1 , ..., U m have the desired properties. Let u 1 , ..., u d , be the standard orthonormal basis of C d and define T i u j = √ mψ ij . With this notation in place U AB has the desired form. The converse is proved by the obvious calculation.
Corollary 4. If U AB is a maximally entangled subspace in H A ⊗H B (d A ≥ d B ), then dim U AB ≤ d A d B .
Furthermore, there always exists a maximally entangled subspace of dimension ⌊d A /d B ⌋.
A = d A ≥ dim H B = d B .
According to the first part of Proposition 4, if U AB is a maximally entangled subspace of dimension d then
d × d B ≤ d A .
On the other hand, if d ≤ ⌊d A /d B ⌋ then the second half of the statement implies that there is a maximally entangled subspace of dimension d.
In the following we find necessary and sufficient conditions for a subspace to be maximally entangled. In section III we use this to show that maximally entangled subspaces can play an important role in the study of error correcting codes. As above we consider the space
H A ⊗H B with dim H A = d A ≥ dim H B = d B
and a maximally entangled subspace U AB ⊂ H A ⊗H B . We will also consider End(H B ) to be a Hilbert space with inner product X|Y = Tr(X † Y ) for any two operators X and Y in End(H B ).
Proposition 5. Let U AB ⊂ H A ⊗H B be a subspace and
d A ≥ d B .
Then, U AB is maximally entangled if and only if the map End(H B )⊗U AB → H A ⊗H B given by X ⊗u → √ d B (I ⊗ X)u is an isometry onto its image.
Proof. Let d = dim U AB and let the notation be as in Proposition 3. Thus, if U AB is maximally entangled and if e 1 , ..., e dB an orthonormal basis of H B then an element of U AB is of the form
T (w) = dB i=1 T i (w) ⊗ e i ,
with T i a unitary operator from C d onto a subspace U i of H A and U i and U j are orthogonal for all i = j. We now calculate
(I⊗X)T (w)|(I⊗Y )T (z) = i,j T i (w)|T j (z) Xe i |Y e j .
Now, since T i (w)|T j (z) = δ ij w|z (see Proposition 3) we have:
(I ⊗ X)T (w)|(I ⊗ Y )T (z) = i,j δ ij w|z Xe i |Y e j = w|z Tr(X † Y ) .
That is, we proved that if U AB is maximally entangled then the map is an isometry. For the converse we note that we have an isometry of C d onto U AB given by
T (w) = dB i=1 T i (w) ⊗ e i .
Now, if the map defined in the proposition is an isometry then
(I ⊗ X)T (w)|(I ⊗ Y )T (z) = w|z Tr(X † Y ) . That is, i,j T i (w)|T j (z) Xe i |Y e j = w|z Tr(X † Y ) ,
for all X, Y ∈ End(H B ). Hence, we must have T i (w)|T j (z) = δ ij w|z / and from Proposition 3 the subspace U AB is maximally entangled.
We now discuss the additivity properties of maximally entangled subspaces.
Proposition 6. Let U AB ⊂ H A ⊗H B and V A ′ B ′ ⊂ H A ′ ⊗H B ′ be maximally entangled subspaces. Then, E U AB ⊗ V A ′ B ′ = E U AB + E V A ′ B ′ = log m + log m ′ , (5)
where m ≡ min{d A , d B } and m ′ ≡ min{d A ′ , d B ′ }.
Remark. From the above proposition it follows that if
d A ≥ d B and d A ′ ≥ d B ′ or d B ≥ d A and d B ′ ≥ d A ′ then U AB ⊗V A ′ B ′ is maximally entangled in (H A ⊗H A ′ )⊗(H B ⊗H B ′ ). However, if for example d A > d B and d A ′ < d B ′ then U AB ⊗V A ′ B ′ is NOT maximally entangled in (H A ⊗H A ′ )⊗(H B ⊗H B ′ ) because mm ′ < min{d A d A ′ , d B d B ′ }.
Proof. There are basically two possibilities (up to interchanging factors): the first is
d A ≥ d B and d A ′ ≥ d B ′
, and the second is
d A ≥ d B and d A ′ < d B ′ .
In the first case we have as in the statement of proposition 3 the subspaces U j and the unitaries T j :
C d → U j such that U AB = { i T i w⊗e i |w ∈ C d }.
We also have the orthonormal basis f i of H B ′ , the subspaces V j and the unitaries S j :
C d ′ → V j such that V A ′ B ′ = { i S i w ′ ⊗ f i |w ′ ∈ C d ′ }. Thus, as a subspace of (H A ⊗H A ′ )⊗(H B ⊗H B ′ ), U AB ⊗V A ′ B ′ is spanned by the elements i,j (T i w⊗S j w ′ )⊗(e i ⊗f j ).
Thus if we identify C d ⊗C d ′ with C dd ′ then the converse assertion in proposition 3 implies that U AB ⊗V A ′ B ′ is a maximally entangled space. This implies that
E U AB ⊗V A ′ B ′ = log d B + log d B ′ = log m + log m ′ .
We now consider the second case. For U AB we have exactly as above U AB = { i T i w⊗e i |w ∈ C d }. For V A ′ B ′ we denote by f j an orthonormal basis of H A ′ (not of H B ′ as above). Thus, according to proposition 3 we have
V A ′ B ′ = { i f i ⊗ S i w ′ |w ′ ∈ C d ′ }. As a subspace of (H A ⊗H A ′ )⊗(H B ⊗H B ′ ), U AB ⊗V A ′ B ′ is spanned by the elements i,j (T i w⊗f j )⊗(e i ⊗S j w ′ ).
We will assume first that d ′ ≤ d. Let w ′ 1 , ..., w ′ d ′ be an orthonormal basis of C d ′ . Thus, if φ is a state in U AB ⊗V A ′ B ′ we can write it as
φ = i,j,k (T i w k ⊗f j )⊗(e i ⊗S j w ′ k ) ,
where w k are some non-normalized vectors in C d . Furthermore,
φ|φ = d B d A ′ k u k 2 .
Hence, if φ is normalized then
k u k 2 = 1 d B d A ′ .
Now, since S j w ′ k is an orthonormal set of vectors for all j and k (see proposition 3), the entanglement of φ as an element of (H
A ⊗H A ′ )⊗(H B ⊗H B ′ ) is given by d B d A ′ S(B)
where B = [ w i |w j ] 1≤i,j≤d ′ , and if λ 1 , ..., λ d ′ are the eigenvalues of B then the von-Neumann entropy of B is
S(B) = - λ i log λ i . Now B is the most general d ′ × d ′ self
adjoint, positive semidefinate matrix with trace 1/d B d A ′ . The minimum of the entropy for such matrices is log(d a d B ) d a d B .
This proves the proposition for the case d ′ ≤ d. If d < d ′ then we can prove the proposition by using the same argument, this time with w k an orthonormal basis of C d and with B ′ = w ′ i |w ′ j 1≤i,j≤d . This completes the proof.
The above proposition also shows that the entanglement of formation is additive for bipartite states with maximally entangled support. If ρ is a mixed state in H A ⊗H B then the entanglement of formation is defined in terms of the convex roof extension:
E F (ρ) = min p i E(φ i )
where the minimum taken over all decompositions ρ = p i φ i ⊗φ * i with φ i a pure bipartite state and p i > 0 and p i = 1.
Corollary 7. Let ρ and σ be mixed states in B(H A ⊗H B ) and B(H A ′ ⊗H B ′ ), respectively. If the support subspaces S ρ and S σ are maximally entangled then
E F (ρ ⊗ σ) = E F (ρ) + E F (σ) .
The proof of this corollary follows directly from the fact that for states with maximally entangled support E F (ρ) = E(S ρ ). Note that the class of mixed states with maximally entangled support is extremely small (i.e. of measure zero). In particular, it is a much smaller class than the one found by Vidal, Dur and Cirac [16] .
We consider error correcting codes that are used to encode l qubits in n ≥ l qubits in such a way that they can correct errors on any subset of k or fewer qubits. These codes, which we call (n, l, k) error correcting codes, can be classified into two classes (for example see [17] ): degenerate and non-degenerate (or orthogonal) codes. We start with a general definition of error correcting codes that is equivalent to the definition given (for example) in [17] , but here we define the codes in terms of subspaces.
Definition 4. Let X ∈ End(⊗ k C 2 ) and 0 ≤ i 0 < i 1 < ... < i k-1 ≤ n -1. The operator X i0i1•••i k-1 on ⊗ n C 2 , that represents the errors on the k qubits i 1 , ..., i k-1 , is defined by X i0...i k-1 v = σ(X ⊗ I)σ -1 v, where (a) σ ∈ S n (acting on {0, 1, ..., n -1} by permutations) is defined such that σ(j) = i j , (b) σ can act on ⊗ n C 2 by σ(v 0 ⊗ v 1 ⊗ • • • ⊗ v n-1 ) = v σ(0) ⊗ v σ(1) ⊗ • • • ⊗ v σ(n-1) and (c) ⊗ n C 2 is viewed as (⊗ k C 2 ) ⊗ (⊗ n-k C 2 ) (
putting together the k tensor factors that correspond to the k qubits i 1 , ..., i k-1 and the rest n -k tensor factors). An (n, l, k) error correcting code is defined from its following ingredients:
I. An isometry T :
⊗ l C 2 → ⊗ n C 2 . II. Let V 0 = T (⊗ l C 2 ). There are V 1 , ..., V d mutually orthogonal subspaces of ⊗ n C 2 that are also orthogonal to V 0 .
III. For each V there is a unitary isomorphism, U j , of V j onto V 0 with U 0 = I.
IV.
X i0i1•••i k-1 V 0 ⊂ ⊕ d j=0 V j . V. Let P j be the ortogonal projection of ⊗ n C 2 onto V j then if v ∈ V 0 is a unit vector and P j (X i0i1•••i k-1 v) = 0 then U j P j (X i0i1•••i k-1 v) P j (X i0i1•••i k-1 v)
equals v up to a phase.
In the next subsection we study Shor's (9, 1, 1) error correcting code and show that it satisfies this definition. However, before that, let us introduce the notion of ktotally entangled subspaces which will play an important role in our discussion of QECC. Definition 5. Let H be the space of n qubits, ⊗ n C 2 . Corresponding to any choice of k qubits (tensor factors) we can consider
H = H A ⊗ H B with H A = ⊗ n-k C 2 and H B = ⊗ k C 2 .
For k ≤ n/2 we will say that a subspace, V , of H is k-totally entangled it is maximally entangled relative to every decomposition of H as above.
It is interesting to note that all the subspaces spanned by the logical codewords of the different non-degenerate error correcting codes given in [18, 19, 20] are 2-totally entangled subspaces. On the other hand, the subspaces spanned by the logical codewords of degenerate codes, like Shor's 9 qubits code, are in general only partially maximally entangled subspaces (i.e. maximally entangled for some choices of k qubits but not for all choices). In the following subsections we will see the reason for that.
We start with the following notations. We set u ± = 1 √ 2 (|000 ± |111 ) so that the two logical codewords in Shor's 9 qubit code are v + = u + ⊗ u + ⊗ u + and v -= u -⊗u -⊗u -. The subspace spanned by these codewords is denoted by
V 0 = Cv + ⊕ Cv -. We also denote u 0 ± = (|100 ± |011 )/ √ 2, u 1 ± = (|010 ± |101 )/ √ 2 and u 2 ± = (|001 ± |110 )/ √ 2.
Using these notations, we define 21 mutually orthogonal 2 dimensional subspaces orthogonal to V 0 :
V 1 = Cu -⊗ u + ⊗ u + ⊕ Cu + ⊗ u -⊗ u -, V 2 = Cu + ⊗ u -⊗ u + ⊕ Cu -⊗ u + ⊗ u -, V 3 = Cu + ⊗ u + ⊗ u -⊕ Cu -⊗ u -⊗ u + , V 4+i = Cu i + ⊗ u + ⊗ u + ⊕ Cu i -⊗ u -⊗ u -, for i = 0, 1, 2, V 7+i = Cu + ⊗ u i + ⊗ u + ⊕ Cu -⊗ u i -⊗ u -, for i = 0, 1, 2, V 10+i = Cu + ⊗u + ⊗u i + ⊕Cu -⊗u -⊗u i -, for i = 0, 1, 2, V 13+i = Cu i -⊗u + ⊗u + ⊕Cu i + ⊗u -⊗u -, for i = 0, 1, 2, V 16+i = Cu + ⊗u i -⊗u + ⊕Cu -⊗u i + ⊗u -, for i = 0, 1, 2, V 19+i = Cu + ⊗u + ⊗u i -⊕Cu -⊗u -⊗u i + , for i = 0, 1, 2.
If X ∈ End(C 2 ) (linear maps of C 2 to C 2 ) then we denote by X i the linear map of ⊗ 9 C 2 to itself that is the tensor product of the identity of C 2 in every tensor factor but the i-th and is X in the i-th factor thus
X 0 = X ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I, X 1 = I ⊗ X ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I etc.
Then we have (here ⌊x⌋ = max{m|m ≤ x, m ∈ Z})
X i V 0 ⊂ V 0 ⊕ V ⌊i/3⌋+1 ⊕ V i+4 ⊕ V i+13 , 0 ≤ i ≤ 8.
We choose an observable R with
R |Vi = λ i I, 0 ≤ i ≤ 21
and
R |W = µI,
where W is the orthogonal complement of ⊕ 21 i=0 V i and λ i = λ j for i = j and λ i = µ for any i. We define a unitary operator U j : V j → V 0 as follows: we denote the Pauli matrices by
A 1 = 1 0 0 -1 , A 2 = 0 1 1 0 , A 3 = 0 i -i 0 ,
and then define U 0 = I, U i = (A 1 ) 3i-1 , for i = 1, 2, 3, U i = (A 2 ) i-4 , for 4 ≤ i ≤ 12 and U i = (A 3 ) i-13 , for 13 ≤ i ≤ 21. This gives an one qubit error correcting code since if v ∈ V 0 is a state and if we have an error in the i-th position then we will have
X i v ∈ V 0 ⊕ V ⌊i/3⌋+1 ⊕ V i+4 ⊕ V i+13 .
Thus, if we measure the observable R on X i v then the measurement will yield one of λ j with j = 0, ⌊i/3⌋+1, i+4 or i + 13 and X i v will have collapsed up to a phase to U j v; hence applying U j will fix the error.
Remark. Note that the subspace V 0 is not 2-totally entangled subspace. Nevertheless, V 0 has very special properties. In particular, if we group the 9 qubits as (1, 2, 3) : (4, 5, 6) : (7, 8, 9) , then for any choice of 2 qubits that are not from the same group, the subspace V 0 is maximally entangled with respect to the decomposition between the 2 qubits and the rest 7 qubits. If the 2 qubits are chosen from the same group then the entanglement of V 0 with respect to this decomposition 1ebit. Thus, out of the 36 different decompositions, with respect to 27 of them E(V 0 ) = 2ebits and with respect to the other 9 decompositions E(V 0 ) = 1ebit.
We now consider a somewhat more intuitive class of codes known as non-degenerate codes which we also name as orthogonal codes. Definition 6. Let A 0 = I, A 1 , A 2 , A 3 the Pauli basis and define
A j0j1•••j k-1 i0i1•••i k-1 to be (A j0 ⊗ A j1 ⊗ • • • ⊗ A j k-1 ) i0•••i k-1 ,
where 0 ≤ j r ≤ 3 and 0
≤ i 0 < i 1 < ... < i k-1 ≤ n -1. Let Σ be the set of distinct operators of the form A j0j1•••j k-1 i0i1•••i k-1 .
Then an orthogonal (n, l, k) code is an (n, l, k) error correcting code such that if we label Σ as the set of d + 1 operators S 0 = I, S 1 , ..., S d then V j = S j V 0 .
Note that Σ has
d + 1 = k r=0 3 r n r
elements. Thus, a necessary condition that there exist an (n, l, k) code is the quantum Hamming bound [17] :
k r=0 3 r n r ≤ 2 n-l . Proposition 8. A 2 l dimensional subspace V of ⊗ n C 2
is the V 0 of an (n, l, k)-orthoganal error correcting code if and only if V is 2k-totally entangled.
Proof. Let V be a 2k-totally entangled subspace in H = ⊗ n C 2 , and let X : ⊗ k C 2 → ⊗ k C 2 be a linear map on k qubits. As above, for any i 0 < i 1 < ... < i k-1 (1 ≤ i l ≤ n) we denote by X i0i1...i k-1 the operation X on H, when acting on the k qubits i 0 , i 1 , ...., i k-1 (the rest of the n -k qubits are left "untouched"). Let also Z ≡ {X ∈ End ⊗ k C 2 |TrX = 0} and for any i 0 < ...
< i k-1 let U i0...i k-1 ≡ {X i0...i k-1 V |X ∈ Z}.
We define the subspace
W = V + i0<...<i k-1 U i0...i k-1 .
That is, W consists of all the possible states after an error on k or less qubits has been occurred. Now, let A 0 = I, A 1 , A 2 , A 3 be an orthonormal basis of End(C 2 ) with A i invertible (e.g. the Pauli basis of 2 × 2 matrices).
As in Definition 6, we denote by
A j0...j k-1 i0...i k-1 the operator X i0...i k-1 that corresponds to X = A j0 ⊗ • • • ⊗ A j k-1
, and the set of all such operators we denote by
Σ ≡ {A j0...j k-1 i0...i k-1 |1 ≤ i l ≤ n, 0 ≤ j l ≤ 3} .
Now, let v 1 , ..., v d be an orthonormal basis of V and define
B ≡ {Sv i S ∈ Σ, 1 ≤ i ≤ d}.
We now argue that B an orthonormal basis of W. Clearly, the vectors in B span W. It is therefore enough to show that the vectors in B are orthogonal. Let Sv i and S ′ v j be two vectors in B with S = A j0...j k-1 i0...i k-1 and
S ′ = A j ′ 0 ...j ′ k-1 i ′ 0 ...i ′ k-1
. We denote by H B the Hilbert space of the qubits i 0 , ..., i k-1 and i ′ 0 , ..., i ′ k-1 , and by H A the Hilbert space of the rest of the qubits. Note that H B consists of at most 2k qubits. Now, since V is 2k-totally entangled subspace, it is maximally entangled relative to the decomposition H = H A ⊗ H B . Thus, from Proposition 5 we clearly have
Sv i |S ′ v j = v i |v j Tr S † S′ = δ ij δ SS ′ ,
where S ≡ I A ⊗ S and S ′ ≡ I A ⊗ S′ ; that is, S and S′ are the projections of S and S ′ onto H B , respectively. Hence, B is an orthonormal basis of W.
Since B is an orthonormal basis we can construct an observable (i.e. Hermition operator) R such that for all v ∈ V R(Sv) = λ S Sv with all of the λ S distinct. We also define R to be zero on the orthogonal complement to W in H. Now, suppose that an element v has been changed by a k-qubit transformation yielding X i0...i k-1 v. We do a measurment of R and since the image is in W the outcome is λ S for some S. After the measurment, the quantum state is Sv and so we recover v by applying S -1 (actually S if we used the Pauli basis). The converse follows from the same lines in the opposite direction. This completes the proof.
Note that Corollary 4 together with the proposition above is consistent with the quantum Singleton bound [22] , n ≥ 4k + l, which also follows trivially from the quantum Hamming bound for the case of orthogonal codes that we considered in this subsection.
We introduced the notion of entanglement of subspaces as a measure that quantify the entanglement of bipartite states in a randomly selected subspace. We discussed its properties and suggested that it is additive. We were not able to prove this conjecture (which is equivalent to the additivity conjecture of the entanglement of formation) although some numerical tests [14] supports that and for maximally entangled subspaces we proved that it is additive. We then extended the definition of maximally entangled subspaces into k-totally entangled subspaces and showed that the later can play an important role in the study of quantum error correction codes.
We considered both degenerate non-degenerate codes and showed that the subspace spanned by the logical codewords of a non-degenerate code is a k-totally (maximally) entangled subspace. This observation, followed by an analysis of the degenerate Shor's nine qubits code in terms of 22 mutually orthogonal subspaces, motivated us to define a general (possible degenerate) error correcting code in terms of subspaces. We believe that further investigation in this direction would lead to a better understanding of degenerate quantum error correcting codes. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We introduce the notion of entanglement of subspaces as a measure that quantify the entanglement of bipartite states in a randomly selected subspace. We discuss its properties and in particular we show that for maximally entangled subspaces it is additive. Furthermore, we show that maximally entangled subspaces can play an important role in the study of quantum error correction codes. We discuss both degenerate and non-degenerate codes and show that the subspace spanned by the logical codewords of a non-degenerate code is a k-totally (maximally) entangled subspace. As for non-degenerate codes, we provide a mathematical definition in terms of subspaces and, as an example, we analyze Shor's nine qubits code in terms of 22 mutually orthogonal subspaces."
},
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION AND DEFINITIONS",
"text": "Bipartite entanglement has been recognized as a crucial resource for quantum information processing tasks such as teleportation [1] and super dense coding [2] . As a result, in the last years there has been an enormous effort to understand and study the characterization, manipulation and quantification of bipartite entanglement [3] . Yet, despite a great deal of progress that was achieved, the theory on mixed bipartite entanglement is incomplete and a few central important questions such as the additivity of the entanglement of formation [4] remained open. Perhaps the richness and complexity of mixed bipartite entanglement can be found in the fact that a finite set of measures of entanglement is insufficient to completely quantify it [5] . In this paper we shed some light on mixed bipartite entanglement with the introduction of a new kind of measure of entanglement which we call entanglement of subspaces (EoS). We will see that EoS can play an important role in the study of quantum error correcting codes (QECC).\n\nIt has been shown recently [6, 7] that geometry of high-dimensional vector spaces can be counterintuitive especially when subspaces with very unique properties are more common than one intuitively expects. That is, roughly speaking, if a high dimensional subspace is selected randomly it is quite likely to have strange properties. For example, in [7] it has been demonstrated that a randomly chosen subspace of a bipartite quantum system will likely contain nothing but nearly maximally entangled states even if the dimension of the subspace is almost of the same order as the dimension of the original system. This kind of result has implications, in particular, to super-dense coding [8] and for quantum communication in general (see also [9] for other implications of randomly selected subspaces). The quantification of the entanglement of such subspaces is therefore very important and we start with its definition. Definition 1. Let H A and H B be finite dimensional Hilbert spaces and let W AB be a subspace of H A ⊗ H B . The entanglement of W AB is defined as:\n\nE W AB ≡ min ψ AB ∈W AB E ψ AB : ψ AB = 1 , (1)\n\nwhere E ψ AB is the entropy of entanglement of ψ AB . Note that if the subspace W AB contains a product state then E(W AB ) = 0. On the other hand, if, for example, W AB is orthogonal to a subspace spaned by an unextendible product basis (UPB) [11, 12] then E(W AB ) > 0.\n\nClaim: Let d A = dim H A and d B = dim H B . If E(W AB ) > 0 then dim W AB ≤ (d A -1)(d B -1). (2)\n\nThis claim follows from [10] and also related to the fact that the number of (bipartite) states in a UPB is at least [11] . Note that for two qubits (i.e.\n\nd A + d B -1\n\nd A = d B = 2) E(W AB\n\n) can be greater than zero only for one dimensional subspaces. We can use Eq. ( 1 ) to define another measure of entanglement on bipartite mixed states."
},
{
"section_type": "OTHER",
"section_title": "Definition 2. Let ρ ∈ B H",
"text": "A ⊗ H B be a bipartite mixed state and let S AB ρ be the support subspace of ρ. Then, the entanglement of the support of ρ is defined as\n\nE Support (ρ) ≡ E(S AB ρ ) .\n\nIt can be easily seen that this measure is not continuous and therefore can not be considered as a proper measure of entanglement. Nevertheless, this measure can serve as a mathematical tool to find lower bounds for other measures of entanglement that are more difficult to calculate especially in higher dimensions. For example, the entanglement of the support of ρ provides a lower bound for the entanglement of formation. It can be shown that in lower dimensions the bound is generally not tight. For example, for two qubits in a mixed state ρ, the entanglement of the support E Support (ρ) = 0 (see Eq. ( 2 )). On the other hand, in higher dimensions the bound can be very tight [6, 7] ."
},
{
"section_type": "OTHER",
"section_title": "II. ENTANGLEMENT OF SUBSPACES",
"text": "In this section we study some of the properties of EoS with a focus on additivity properties. The EoS provides a lower bound on the entanglement of formation and our interest in its additivity properties is due to one of the most important unresolved questions in quantum information, namely the additivity conjecture for the entanglement of formation. In particular, the additivity question of EoS is identical to the additivity conjecture of quantum channel output entropy [13] that has been shown to be equivalent to the additivity conjecture of entanglement of formation [4] . Thus, additivity properties of EoS can shed some light on this topic."
},
{
"section_type": "OTHER",
"section_title": "A. Additivity properties of the entanglement of subspaces",
"text": "Here we consider the additivity properties of EoS. We start by showing that if U AB and V A ′ B ′ are two subspaces such that E(U AB ) > 0 and/or E(V\n\nA ′ B ′ ) > 0 then E(U AB ⊗ V A ′ B ′ ) > 0.\n\nConsider W = C n ⊗ C m . Let e j , j = 1, ..., n be the standard basis of C n . We will also use the notation f j for the standard basis of C m . An element of a tensor product of two vector spaces, A and B will be called a product if it is of the form a ⊗ b with a ∈ A and b ∈ B. Hence i u m ik v i is a product. Our assumption implies that it must be 0. Hence\n\nProposition 1. Let u 1 , ..., u d , v 1 , ..., v d ∈ W be such that if x = i b i v i is a product then x = 0. If z = i u i ⊗ v i is a product in (C n ⊗ C n ) ⊗ (C m ⊗ C m ) then z = 0. Proof. We write u i = n j=1 e j ⊗ u ij and v j = n j=1 e j ⊗ v ij . Assume that i u i ⊗v i is a product in (C n ⊗C n )⊗(C m ⊗C m ). This means that there exists z ∈ C n ⊗C n and w ∈ C m ⊗C m such that i,k,l (e k ⊗e l ) ⊗(u ik ⊗v il ) = z⊗w.\n\n0 = i,k,m u m ik e k ⊗f m ⊗v i = i u i ⊗v i .\n\nAs was to be proved.\n\nNote that the proposition above states that if none of the decompositions of a bipartite mixed state, ρ, contain a product state, then also none of the decompositions of ρ ⊗ σ (σ is a bipartite mixed state) contain a product state. This property is related to the additivity conjecture [4] for the entanglement of formation (and other measures) and one of the main questions that we will consider here is wether the EoS is additive. That is, does\n\nE(U AB ⊗ V A ′ B ′ ) = E(U AB ) + E(V A ′ B ′ ) ?\n\nClearly, if the EoS were additive then the proposition above would have been a trivial consequence of that. However, we were not able to prove the additivity of EoS (in general) although for some special cases it has been tested numerically in [14] and no counter example has been found. The proposition below provides a lower bound.\n\nProposition 2. Let N = min{dim U AB , dim V A ′ B ′ }. Then E(U AB ) + E(V A ′ B ′ ) -log N ≤ E U AB ⊗ V A ′ B ′ . (3)\n\nThe equation above provides a lower bound whereas the upper bound\n\nE U AB ⊗ V A ′ B ′ ≤ E(U AB )+E(V A ′ B ′ )\n\nfollows directly from the definition of EoS. Thus, for N = 1 the EoS is additive. Note also that even if N is small (e.g. N = 2), E(U AB ) and E(V A ′ B ′ ) can be arbitrarily large (i.e. depending on d A and d B but not on N ).\n\nProof. Let χ be a normalized vector in U AB ⊗ V A ′ B ′ . We can write χ in its Schmidt decomposition as follows:\n\nχ = i √ p i u AB i ⊗ v A ′ B ′ i ,\n\nwhere i p i = 1 (p i ≥ 0) and the u AB i 's (v A ′ B ′ i 's) are orthonormal. Now, from the strong subadditivity of the von-Neumann entropy we have\n\nS(ρ A ′ ) + S(ρ B ) ≤ S(ρ AB ) + S(ρ AA ′ ) , where ρ A ≡ Tr A ′ BB ′ χ ⊗ χ * , ρ B ≡ Tr AA ′ B ′ χ ⊗ χ * , etc. Now, note that S(ρ AA ′ ) = E(χ) and S(ρ AB ) = H({p i }) ≤ log N , where H({p i }) is the Shanon entropy. Furthermore, note that ρ A ′ = i p i ω i and ρ B = i p i σ i\n\nwhere\n\nω i ≡ Tr B ′ v A ′ B ′ i ⊗ v A ′ B ′ i * and σ i ≡ Tr A u AB i ⊗ u AB i * .\n\nHence, since the von-Neumann entropy is concave we have\n\nS(ρ A ′ ) ≥ i p i S(ω i ) = i p i E(v A ′ B ′ i ) ≥ E V A ′ B ′ .\n\nand similarly S(ρ B ) ≥ E U AB . Combining all this we get\n\nE V A ′ B ′ + E U AB ≤ log N + E(χ) , for all χ ∈ U AB ⊗ V A ′ B ′ .\n\nThis complete the proof."
},
{
"section_type": "OTHER",
"section_title": "B. Maximally entangled subspaces",
"text": "As we have seen above, if N = 1 then the EoS is clearly additive. As we will see in the next subsection, it is also additive for maximally entangled subspaces:\n\nDefinition 3. Let W be a subspace of H A ⊗ H B and let d A = dim H A and d B = dim H B . W is said to be a maximally entangled subspace in H A ⊗ H B if E(W ) = log m , (4)\n\nwhere m ≡ min{d A , d B }.\n\nThe term maximally entangled subspace have been used in [6, 7] for a subspace W with E(W ) slightly less than log m. In this paper, we will call such subspaces nearly maximally entangled to distinguish from (exactly) maximally entangled subspaces as defined above.\n\nIn [15] it has been shown that the average entanglement of a pure state φ ∈ H A ⊗ H B which is chosen randomly according to the unitarily invariant measure satisfies\n\nE(φ) ≥ log 2 d A - d A 2 ln 2d B\n\nwhere without loss of generality d A ≥ d B . Later on, in [6, 7] this result has been extended to subspaces and in particular it has been shown, somewhat surprisingly, that a randomly chosen subspace of bipartite quantum system will likely be a nearly maximally entangled subspace. Thus, as nearly maximally entangled subspaces are quite common it is important to understand their structure. As a first step in this direction, in the following we study the structure of (exactly) maximally entangled subspaces.\n\nLet φ be a state in H A ⊗H B . If e 1 , ..., e m is an orthonormal basis of H B we may write\n\nφ = dB i=1 φ i ⊗e i . We define a d B × d B Hermitian matrix B = [ φ i |φ j ] (i.e.\n\nB is the reduced density matrix). Let λ 1 , ..., λ dB be the set of eigenvalues of B counting multiplicity. Then the entanglement of φ is\n\nE(φ) = - i λ i log(λ i ) .\n\nIt is easy to show that E(φ) ≤ log m and equality is attained if and only if B = 1 m P with P a projection matrix onto a d dimensional subspace of C dB . Clearly this definition of entropy is independent of the choice of basis and could also be given using an orthonormal basis of H A and analyzing the corresponding d A coefficients in H B . Under the condition of equality φ is maximally entangled, and this in particular implies that if\n\nd A ≥ d B then φ i |φ j = 1 d B δ ij . Proposition 3. Assume that d A ≥ d B and set m = d B .\n\nLet U AB be a maximally entangled subspace in H A ⊗H B of dimension d. If e 1 , ..., e m is an orthonormal basis of H B then there exist U 1 , ..., U m subspaces of H A such that U i |U j = 0 if i = j, dim U j = d > 0 for all j = 1, ..., m and unitary operators T i :\n\nC d → U i i = 1, ..., m such that U AB = { i T i w⊗e i |w ∈ C d }.\n\nConversely, if U 1 , ..., U m are mutually orthogonal subspaces of H A such that dim U j = d > 0 for all j = 1, ..., m and we have unitary operators\n\nT i : C d → U i i = 1, ..., m such that U AB = { i T i w⊗e i |w ∈ C d } ,\n\nthen U AB is maximally entangled.\n\nProof. Let ψ 1 , ..., ψ d be an orthonomal basis of U AB . Then we can write\n\nψ j = i ψ ij ⊗e i with ψ ij |ψ kj = 1 m δ ik . The condition on U AB is that if a ∈ C d is a unit vector then a j ψ j is maximally entangled in H A ⊗H B . This implies that d j=1 a j ψ lj d j=1 a j ψ kj = 1 m δ l,k .\n\nFix l = k and let p = q ≤ d be two integers. Let a = (a 1 , ..., a d ) with a j = 0 for j = p or j = q. Set a p = b, a q = c and |b| 2 + |c| 2 = 1. Then we have bψ lp + cψ lq |bψ kp + cψ kq = 0.\n\nOn the other hand we have bψ lp + cψ lq |bψ kp + cψ kq = bc ψ lp |ψ kq + cb ψ lq |ψ kp Set z = bc. We look at two cases:\n\nfirst z = 1 2 (b = c = 1 √ 2 ) and second z = i √ 2 (b = 1 √ 2 , c = i √ 2\n\n). Thus we have\n\nψ lp |ψ kq + ψ lq |ψ kp = 0\n\nfor the first case and\n\nψ lp |ψ kq -ψ lq |ψ kp = 0\n\nfor the second. Hence, ψ lp |ψ kq = ψ lq |ψ kp = 0. We set\n\nU l = Span{ψ lp |p = 1, ..., d}. Then U l |U k = 0 if l = k.\n\nWe now consider what happens when l = k. We first note that taking a p = 1 and all other entries equal to 0 we have ψ lp |ψ lp = 1 m . Now using b, c as above for p = q we have\n\nψ lp |ψ lp + ψ lp |ψ lq + ψ lq |ψ lp + ψ kp |ψ kp = 2 m\n\nand\n\nψ lp |ψ lp + i ψ lp |ψ lq -i ψ lq |ψ lp + ψ kp |ψ kp = 2 m .\n\nHence as above we find that ψ lp |ψ lq = 0 if p = q. Thus √ mψ l1 , ..., √ mψ ld is an orthonormal basis of U l . This implies that the spaces U 1 , ..., U m have the desired properties. Let u 1 , ..., u d , be the standard orthonormal basis of C d and define T i u j = √ mψ ij . With this notation in place U AB has the desired form. The converse is proved by the obvious calculation.\n\nCorollary 4. If U AB is a maximally entangled subspace in H A ⊗H B (d A ≥ d B ), then dim U AB ≤ d A d B .\n\nFurthermore, there always exists a maximally entangled subspace of dimension ⌊d A /d B ⌋."
},
{
"section_type": "OTHER",
"section_title": "Proof. Assume that dim H",
"text": "A = d A ≥ dim H B = d B .\n\nAccording to the first part of Proposition 4, if U AB is a maximally entangled subspace of dimension d then\n\nd × d B ≤ d A .\n\nOn the other hand, if d ≤ ⌊d A /d B ⌋ then the second half of the statement implies that there is a maximally entangled subspace of dimension d.\n\nIn the following we find necessary and sufficient conditions for a subspace to be maximally entangled. In section III we use this to show that maximally entangled subspaces can play an important role in the study of error correcting codes. As above we consider the space\n\nH A ⊗H B with dim H A = d A ≥ dim H B = d B\n\nand a maximally entangled subspace U AB ⊂ H A ⊗H B . We will also consider End(H B ) to be a Hilbert space with inner product X|Y = Tr(X † Y ) for any two operators X and Y in End(H B ).\n\nProposition 5. Let U AB ⊂ H A ⊗H B be a subspace and\n\nd A ≥ d B .\n\nThen, U AB is maximally entangled if and only if the map End(H B )⊗U AB → H A ⊗H B given by X ⊗u → √ d B (I ⊗ X)u is an isometry onto its image.\n\nProof. Let d = dim U AB and let the notation be as in Proposition 3. Thus, if U AB is maximally entangled and if e 1 , ..., e dB an orthonormal basis of H B then an element of U AB is of the form\n\nT (w) = dB i=1 T i (w) ⊗ e i ,\n\nwith T i a unitary operator from C d onto a subspace U i of H A and U i and U j are orthogonal for all i = j. We now calculate\n\n(I⊗X)T (w)|(I⊗Y )T (z) = i,j T i (w)|T j (z) Xe i |Y e j .\n\nNow, since T i (w)|T j (z) = δ ij w|z (see Proposition 3) we have:\n\n(I ⊗ X)T (w)|(I ⊗ Y )T (z) = i,j δ ij w|z Xe i |Y e j = w|z Tr(X † Y ) .\n\nThat is, we proved that if U AB is maximally entangled then the map is an isometry. For the converse we note that we have an isometry of C d onto U AB given by\n\nT (w) = dB i=1 T i (w) ⊗ e i .\n\nNow, if the map defined in the proposition is an isometry then\n\n(I ⊗ X)T (w)|(I ⊗ Y )T (z) = w|z Tr(X † Y ) . That is, i,j T i (w)|T j (z) Xe i |Y e j = w|z Tr(X † Y ) ,\n\nfor all X, Y ∈ End(H B ). Hence, we must have T i (w)|T j (z) = δ ij w|z / and from Proposition 3 the subspace U AB is maximally entangled."
},
{
"section_type": "OTHER",
"section_title": "C. Additivity of maximally entangled subspaces",
"text": "We now discuss the additivity properties of maximally entangled subspaces.\n\nProposition 6. Let U AB ⊂ H A ⊗H B and V A ′ B ′ ⊂ H A ′ ⊗H B ′ be maximally entangled subspaces. Then, E U AB ⊗ V A ′ B ′ = E U AB + E V A ′ B ′ = log m + log m ′ , (5)\n\nwhere m ≡ min{d A , d B } and m ′ ≡ min{d A ′ , d B ′ }.\n\nRemark. From the above proposition it follows that if\n\nd A ≥ d B and d A ′ ≥ d B ′ or d B ≥ d A and d B ′ ≥ d A ′ then U AB ⊗V A ′ B ′ is maximally entangled in (H A ⊗H A ′ )⊗(H B ⊗H B ′ ). However, if for example d A > d B and d A ′ < d B ′ then U AB ⊗V A ′ B ′ is NOT maximally entangled in (H A ⊗H A ′ )⊗(H B ⊗H B ′ ) because mm ′ < min{d A d A ′ , d B d B ′ }.\n\nProof. There are basically two possibilities (up to interchanging factors): the first is\n\nd A ≥ d B and d A ′ ≥ d B ′\n\n, and the second is\n\nd A ≥ d B and d A ′ < d B ′ .\n\nIn the first case we have as in the statement of proposition 3 the subspaces U j and the unitaries T j :\n\nC d → U j such that U AB = { i T i w⊗e i |w ∈ C d }.\n\nWe also have the orthonormal basis f i of H B ′ , the subspaces V j and the unitaries S j :\n\nC d ′ → V j such that V A ′ B ′ = { i S i w ′ ⊗ f i |w ′ ∈ C d ′ }. Thus, as a subspace of (H A ⊗H A ′ )⊗(H B ⊗H B ′ ), U AB ⊗V A ′ B ′ is spanned by the elements i,j (T i w⊗S j w ′ )⊗(e i ⊗f j ).\n\nThus if we identify C d ⊗C d ′ with C dd ′ then the converse assertion in proposition 3 implies that U AB ⊗V A ′ B ′ is a maximally entangled space. This implies that\n\nE U AB ⊗V A ′ B ′ = log d B + log d B ′ = log m + log m ′ .\n\nWe now consider the second case. For U AB we have exactly as above U AB = { i T i w⊗e i |w ∈ C d }. For V A ′ B ′ we denote by f j an orthonormal basis of H A ′ (not of H B ′ as above). Thus, according to proposition 3 we have\n\nV A ′ B ′ = { i f i ⊗ S i w ′ |w ′ ∈ C d ′ }. As a subspace of (H A ⊗H A ′ )⊗(H B ⊗H B ′ ), U AB ⊗V A ′ B ′ is spanned by the elements i,j (T i w⊗f j )⊗(e i ⊗S j w ′ ).\n\nWe will assume first that d ′ ≤ d. Let w ′ 1 , ..., w ′ d ′ be an orthonormal basis of C d ′ . Thus, if φ is a state in U AB ⊗V A ′ B ′ we can write it as\n\nφ = i,j,k (T i w k ⊗f j )⊗(e i ⊗S j w ′ k ) ,\n\nwhere w k are some non-normalized vectors in C d . Furthermore,\n\nφ|φ = d B d A ′ k u k 2 .\n\nHence, if φ is normalized then\n\nk u k 2 = 1 d B d A ′ .\n\nNow, since S j w ′ k is an orthonormal set of vectors for all j and k (see proposition 3), the entanglement of φ as an element of (H\n\nA ⊗H A ′ )⊗(H B ⊗H B ′ ) is given by d B d A ′ S(B)\n\nwhere B = [ w i |w j ] 1≤i,j≤d ′ , and if λ 1 , ..., λ d ′ are the eigenvalues of B then the von-Neumann entropy of B is\n\nS(B) = - λ i log λ i . Now B is the most general d ′ × d ′ self\n\nadjoint, positive semidefinate matrix with trace 1/d B d A ′ . The minimum of the entropy for such matrices is log(d a d B ) d a d B .\n\nThis proves the proposition for the case d ′ ≤ d. If d < d ′ then we can prove the proposition by using the same argument, this time with w k an orthonormal basis of C d and with B ′ = w ′ i |w ′ j 1≤i,j≤d . This completes the proof.\n\nThe above proposition also shows that the entanglement of formation is additive for bipartite states with maximally entangled support. If ρ is a mixed state in H A ⊗H B then the entanglement of formation is defined in terms of the convex roof extension:\n\nE F (ρ) = min p i E(φ i )\n\nwhere the minimum taken over all decompositions ρ = p i φ i ⊗φ * i with φ i a pure bipartite state and p i > 0 and p i = 1.\n\nCorollary 7. Let ρ and σ be mixed states in B(H A ⊗H B ) and B(H A ′ ⊗H B ′ ), respectively. If the support subspaces S ρ and S σ are maximally entangled then\n\nE F (ρ ⊗ σ) = E F (ρ) + E F (σ) .\n\nThe proof of this corollary follows directly from the fact that for states with maximally entangled support E F (ρ) = E(S ρ ). Note that the class of mixed states with maximally entangled support is extremely small (i.e. of measure zero). In particular, it is a much smaller class than the one found by Vidal, Dur and Cirac [16] ."
},
{
"section_type": "OTHER",
"section_title": "A. Definitions",
"text": "We consider error correcting codes that are used to encode l qubits in n ≥ l qubits in such a way that they can correct errors on any subset of k or fewer qubits. These codes, which we call (n, l, k) error correcting codes, can be classified into two classes (for example see [17] ): degenerate and non-degenerate (or orthogonal) codes. We start with a general definition of error correcting codes that is equivalent to the definition given (for example) in [17] , but here we define the codes in terms of subspaces.\n\nDefinition 4. Let X ∈ End(⊗ k C 2 ) and 0 ≤ i 0 < i 1 < ... < i k-1 ≤ n -1. The operator X i0i1•••i k-1 on ⊗ n C 2 , that represents the errors on the k qubits i 1 , ..., i k-1 , is defined by X i0...i k-1 v = σ(X ⊗ I)σ -1 v, where (a) σ ∈ S n (acting on {0, 1, ..., n -1} by permutations) is defined such that σ(j) = i j , (b) σ can act on ⊗ n C 2 by σ(v 0 ⊗ v 1 ⊗ • • • ⊗ v n-1 ) = v σ(0) ⊗ v σ(1) ⊗ • • • ⊗ v σ(n-1) and (c) ⊗ n C 2 is viewed as (⊗ k C 2 ) ⊗ (⊗ n-k C 2 ) (\n\nputting together the k tensor factors that correspond to the k qubits i 1 , ..., i k-1 and the rest n -k tensor factors). An (n, l, k) error correcting code is defined from its following ingredients:\n\nI. An isometry T :\n\n⊗ l C 2 → ⊗ n C 2 . II. Let V 0 = T (⊗ l C 2 ). There are V 1 , ..., V d mutually orthogonal subspaces of ⊗ n C 2 that are also orthogonal to V 0 .\n\nIII. For each V there is a unitary isomorphism, U j , of V j onto V 0 with U 0 = I.\n\nIV.\n\nX i0i1•••i k-1 V 0 ⊂ ⊕ d j=0 V j . V. Let P j be the ortogonal projection of ⊗ n C 2 onto V j then if v ∈ V 0 is a unit vector and P j (X i0i1•••i k-1 v) = 0 then U j P j (X i0i1•••i k-1 v) P j (X i0i1•••i k-1 v)\n\nequals v up to a phase.\n\nIn the next subsection we study Shor's (9, 1, 1) error correcting code and show that it satisfies this definition. However, before that, let us introduce the notion of ktotally entangled subspaces which will play an important role in our discussion of QECC. Definition 5. Let H be the space of n qubits, ⊗ n C 2 . Corresponding to any choice of k qubits (tensor factors) we can consider\n\nH = H A ⊗ H B with H A = ⊗ n-k C 2 and H B = ⊗ k C 2 .\n\nFor k ≤ n/2 we will say that a subspace, V , of H is k-totally entangled it is maximally entangled relative to every decomposition of H as above.\n\nIt is interesting to note that all the subspaces spanned by the logical codewords of the different non-degenerate error correcting codes given in [18, 19, 20] are 2-totally entangled subspaces. On the other hand, the subspaces spanned by the logical codewords of degenerate codes, like Shor's 9 qubits code, are in general only partially maximally entangled subspaces (i.e. maximally entangled for some choices of k qubits but not for all choices). In the following subsections we will see the reason for that."
},
{
"section_type": "OTHER",
"section_title": "B. Analysis of Shor's 9 qubits code",
"text": "We start with the following notations. We set u ± = 1 √ 2 (|000 ± |111 ) so that the two logical codewords in Shor's 9 qubit code are v + = u + ⊗ u + ⊗ u + and v -= u -⊗u -⊗u -. The subspace spanned by these codewords is denoted by\n\nV 0 = Cv + ⊕ Cv -. We also denote u 0 ± = (|100 ± |011 )/ √ 2, u 1 ± = (|010 ± |101 )/ √ 2 and u 2 ± = (|001 ± |110 )/ √ 2.\n\nUsing these notations, we define 21 mutually orthogonal 2 dimensional subspaces orthogonal to V 0 :\n\nV 1 = Cu -⊗ u + ⊗ u + ⊕ Cu + ⊗ u -⊗ u -, V 2 = Cu + ⊗ u -⊗ u + ⊕ Cu -⊗ u + ⊗ u -, V 3 = Cu + ⊗ u + ⊗ u -⊕ Cu -⊗ u -⊗ u + , V 4+i = Cu i + ⊗ u + ⊗ u + ⊕ Cu i -⊗ u -⊗ u -, for i = 0, 1, 2, V 7+i = Cu + ⊗ u i + ⊗ u + ⊕ Cu -⊗ u i -⊗ u -, for i = 0, 1, 2, V 10+i = Cu + ⊗u + ⊗u i + ⊕Cu -⊗u -⊗u i -, for i = 0, 1, 2, V 13+i = Cu i -⊗u + ⊗u + ⊕Cu i + ⊗u -⊗u -, for i = 0, 1, 2, V 16+i = Cu + ⊗u i -⊗u + ⊕Cu -⊗u i + ⊗u -, for i = 0, 1, 2, V 19+i = Cu + ⊗u + ⊗u i -⊕Cu -⊗u -⊗u i + , for i = 0, 1, 2.\n\nIf X ∈ End(C 2 ) (linear maps of C 2 to C 2 ) then we denote by X i the linear map of ⊗ 9 C 2 to itself that is the tensor product of the identity of C 2 in every tensor factor but the i-th and is X in the i-th factor thus\n\nX 0 = X ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I, X 1 = I ⊗ X ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I etc.\n\nThen we have (here ⌊x⌋ = max{m|m ≤ x, m ∈ Z})\n\nX i V 0 ⊂ V 0 ⊕ V ⌊i/3⌋+1 ⊕ V i+4 ⊕ V i+13 , 0 ≤ i ≤ 8.\n\nWe choose an observable R with\n\nR |Vi = λ i I, 0 ≤ i ≤ 21\n\nand\n\nR |W = µI,\n\nwhere W is the orthogonal complement of ⊕ 21 i=0 V i and λ i = λ j for i = j and λ i = µ for any i. We define a unitary operator U j : V j → V 0 as follows: we denote the Pauli matrices by\n\nA 1 = 1 0 0 -1 , A 2 = 0 1 1 0 , A 3 = 0 i -i 0 ,\n\nand then define U 0 = I, U i = (A 1 ) 3i-1 , for i = 1, 2, 3, U i = (A 2 ) i-4 , for 4 ≤ i ≤ 12 and U i = (A 3 ) i-13 , for 13 ≤ i ≤ 21. This gives an one qubit error correcting code since if v ∈ V 0 is a state and if we have an error in the i-th position then we will have\n\nX i v ∈ V 0 ⊕ V ⌊i/3⌋+1 ⊕ V i+4 ⊕ V i+13 .\n\nThus, if we measure the observable R on X i v then the measurement will yield one of λ j with j = 0, ⌊i/3⌋+1, i+4 or i + 13 and X i v will have collapsed up to a phase to U j v; hence applying U j will fix the error.\n\nRemark. Note that the subspace V 0 is not 2-totally entangled subspace. Nevertheless, V 0 has very special properties. In particular, if we group the 9 qubits as (1, 2, 3) : (4, 5, 6) : (7, 8, 9) , then for any choice of 2 qubits that are not from the same group, the subspace V 0 is maximally entangled with respect to the decomposition between the 2 qubits and the rest 7 qubits. If the 2 qubits are chosen from the same group then the entanglement of V 0 with respect to this decomposition 1ebit. Thus, out of the 36 different decompositions, with respect to 27 of them E(V 0 ) = 2ebits and with respect to the other 9 decompositions E(V 0 ) = 1ebit."
},
{
"section_type": "OTHER",
"section_title": "C. Orthogonal codes",
"text": "We now consider a somewhat more intuitive class of codes known as non-degenerate codes which we also name as orthogonal codes. Definition 6. Let A 0 = I, A 1 , A 2 , A 3 the Pauli basis and define\n\nA j0j1•••j k-1 i0i1•••i k-1 to be (A j0 ⊗ A j1 ⊗ • • • ⊗ A j k-1 ) i0•••i k-1 ,\n\nwhere 0 ≤ j r ≤ 3 and 0\n\n≤ i 0 < i 1 < ... < i k-1 ≤ n -1. Let Σ be the set of distinct operators of the form A j0j1•••j k-1 i0i1•••i k-1 .\n\nThen an orthogonal (n, l, k) code is an (n, l, k) error correcting code such that if we label Σ as the set of d + 1 operators S 0 = I, S 1 , ..., S d then V j = S j V 0 .\n\nNote that Σ has\n\nd + 1 = k r=0 3 r n r\n\nelements. Thus, a necessary condition that there exist an (n, l, k) code is the quantum Hamming bound [17] :\n\nk r=0 3 r n r ≤ 2 n-l . Proposition 8. A 2 l dimensional subspace V of ⊗ n C 2\n\nis the V 0 of an (n, l, k)-orthoganal error correcting code if and only if V is 2k-totally entangled.\n\nProof. Let V be a 2k-totally entangled subspace in H = ⊗ n C 2 , and let X : ⊗ k C 2 → ⊗ k C 2 be a linear map on k qubits. As above, for any i 0 < i 1 < ... < i k-1 (1 ≤ i l ≤ n) we denote by X i0i1...i k-1 the operation X on H, when acting on the k qubits i 0 , i 1 , ...., i k-1 (the rest of the n -k qubits are left \"untouched\"). Let also Z ≡ {X ∈ End ⊗ k C 2 |TrX = 0} and for any i 0 < ...\n\n< i k-1 let U i0...i k-1 ≡ {X i0...i k-1 V |X ∈ Z}.\n\nWe define the subspace\n\nW = V + i0<...<i k-1 U i0...i k-1 .\n\nThat is, W consists of all the possible states after an error on k or less qubits has been occurred. Now, let A 0 = I, A 1 , A 2 , A 3 be an orthonormal basis of End(C 2 ) with A i invertible (e.g. the Pauli basis of 2 × 2 matrices).\n\nAs in Definition 6, we denote by\n\nA j0...j k-1 i0...i k-1 the operator X i0...i k-1 that corresponds to X = A j0 ⊗ • • • ⊗ A j k-1\n\n, and the set of all such operators we denote by\n\nΣ ≡ {A j0...j k-1 i0...i k-1 |1 ≤ i l ≤ n, 0 ≤ j l ≤ 3} .\n\nNow, let v 1 , ..., v d be an orthonormal basis of V and define\n\nB ≡ {Sv i S ∈ Σ, 1 ≤ i ≤ d}.\n\nWe now argue that B an orthonormal basis of W. Clearly, the vectors in B span W. It is therefore enough to show that the vectors in B are orthogonal. Let Sv i and S ′ v j be two vectors in B with S = A j0...j k-1 i0...i k-1 and\n\nS ′ = A j ′ 0 ...j ′ k-1 i ′ 0 ...i ′ k-1\n\n. We denote by H B the Hilbert space of the qubits i 0 , ..., i k-1 and i ′ 0 , ..., i ′ k-1 , and by H A the Hilbert space of the rest of the qubits. Note that H B consists of at most 2k qubits. Now, since V is 2k-totally entangled subspace, it is maximally entangled relative to the decomposition H = H A ⊗ H B . Thus, from Proposition 5 we clearly have\n\nSv i |S ′ v j = v i |v j Tr S † S′ = δ ij δ SS ′ ,\n\nwhere S ≡ I A ⊗ S and S ′ ≡ I A ⊗ S′ ; that is, S and S′ are the projections of S and S ′ onto H B , respectively. Hence, B is an orthonormal basis of W.\n\nSince B is an orthonormal basis we can construct an observable (i.e. Hermition operator) R such that for all v ∈ V R(Sv) = λ S Sv with all of the λ S distinct. We also define R to be zero on the orthogonal complement to W in H. Now, suppose that an element v has been changed by a k-qubit transformation yielding X i0...i k-1 v. We do a measurment of R and since the image is in W the outcome is λ S for some S. After the measurment, the quantum state is Sv and so we recover v by applying S -1 (actually S if we used the Pauli basis). The converse follows from the same lines in the opposite direction. This completes the proof.\n\nNote that Corollary 4 together with the proposition above is consistent with the quantum Singleton bound [22] , n ≥ 4k + l, which also follows trivially from the quantum Hamming bound for the case of orthogonal codes that we considered in this subsection."
},
{
"section_type": "CONCLUSION",
"section_title": "IV. SUMMERY AND CONCLUSIONS",
"text": "We introduced the notion of entanglement of subspaces as a measure that quantify the entanglement of bipartite states in a randomly selected subspace. We discussed its properties and suggested that it is additive. We were not able to prove this conjecture (which is equivalent to the additivity conjecture of the entanglement of formation) although some numerical tests [14] supports that and for maximally entangled subspaces we proved that it is additive. We then extended the definition of maximally entangled subspaces into k-totally entangled subspaces and showed that the later can play an important role in the study of quantum error correction codes.\n\nWe considered both degenerate non-degenerate codes and showed that the subspace spanned by the logical codewords of a non-degenerate code is a k-totally (maximally) entangled subspace. This observation, followed by an analysis of the degenerate Shor's nine qubits code in terms of 22 mutually orthogonal subspaces, motivated us to define a general (possible degenerate) error correcting code in terms of subspaces. We believe that further investigation in this direction would lead to a better understanding of degenerate quantum error correcting codes."
}
] | |
arxiv:0704.0258 | 0704.0258 | 1 | 10.1140/epjb/e2007-00296-x | 96b172a7d5bf4dfce0e35c8b9e85024619374d97db6d0892a0a15eee3ce76b46 | Correlation functions in the Non Perturbative Renormalization Group and
field expansion | The usual procedure of including a finite number of vertices in Non
Perturbative Renormalization Group equations in order to obtain $n$-point
correlation functions at finite momenta is analyzed. This is done by exploiting
a general method recently introduced which includes simultaneously all vertices
although approximating their momentum dependence. The study is performed using
the self-energy of the tridimensional scalar model at criticality. At least in
this example, low order truncations miss quantities as the critical exponent
$\eta$ by as much as 60%. However, if one goes to high order truncations the
procedure seems to converge rapidly. | [
"Diego Guerra",
"Ramon Mendez-Galain and Nicolas Wschebor"
] | [
"hep-th",
"cond-mat.stat-mech"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | In nearly all fields in physics, there are systems having a large number of strongly correlated constituents. These cannot be treated with usual perturbative methods. Phase transitions and critical phenomena, disordered systems, strongly correlated electrons, quantum chromodynamics at large distances, are just a few examples which demand a general and efficient method to treat non-perturbative situations. In problems as those just quoted, the calculation of correlation functions of the configuration variables is, in general, a very complicated task.
The non perturbative renormalization group (NPRG) [1, 2, 3, 4, 5] has proven to be a powerful tool to achieve this goal. It presents itself as an infinite hierarchy of flow equations relating sequentially the various n-point functions. It has been successfully applied in many different problems, either in condensed matter, particle or nuclear physics (for reviews, see e.g. [6, 7, 8] ; a pedagogical introduction can be found in [9] ). In most of these problems however, one is interested in observables dominated by long wavelength modes. In these cases, it is then possible to approximately close the infinite hierarchy of NPRG equations performing an expansion in the number of derivatives of the field. This approximation scheme is known as the derivative expansion (DE) [10] . The price to pay is that the n-point functions can be calculated only at small external momenta, i.e. smaller than the smallest mass on the problem (vanishing momenta in the case of critical phenomena).
In many other physical problems however, this is not enough: the full knowledge of the momentum dependence of correlation functions is needed in order to calculate quantities of physical interest (e.g. to get the spectrum of excitations, the shape of a Fermi surface, the scattering matrix, etc.). There have been many attempts to solve the infinite system of flow equations at finite momenta; most of them are based on various forms of an early proposal by Weinberg [11] . Although some of these attempts [12, 13, 14, 15] introduce sophisticated ansatz for the unknown correlation functions appearing in a given flow equation, most efforts simply ignore high order vertices. In all these works, only low order vertices are taken into account: usual calculations do not even include the complete flow of the 3-and 4-point functions. Moreover, it is not possible a priori to gauge the quality of such approximations schemes.
Recently, an alternative general method to get n-point functions at any finite momenta 2 within the NPRG has been proposed [16] . It has many similarities with DE. First, it is an approximation scheme that can be systematically improved. Second, the scheme yields a closed set of flow equations including simultaneously an infinite number of vertices; one thus goes far beyond schemes including a small number of vertices, as those quoted in the previous paragraph. Moreover, it has been proven [16] that in their corresponding limits, both perturvative and DE results are recovered; this remains valid at each order of the respective expansion. Finally, in the large-N limit of O(N) models, the leading-order (LO) of the approximation scheme becomes exact for all n-point functions. (The expression "leading order" means the first step in the approximation scheme; it does not refer to an expansion in a small parameter which usually does not exist in these kind of problems.).
In [17] , the method has been applied, in its leading order, to the calculation of the selfenergy of the scalar model, at criticality. That is, we have fine-tuned the bare mass of the model in order for the correlation length to be infinite, and then we have studied the full range of momenta, from the high momenta Gaussian regime to the low momenta scaling one. At this order of the approximation scheme the self-energy is expected to include all one loop contributions and to achieve DE at next-to-leading order (NLO) precision, in the corresponding limit [16] . The numerical solution found in [17] verifies these properties.
Moreover, the function has the expected physical properties in all momenta regime. First, it presents the correct scaling behavior in the infrared limit. The model reproduces critical exponent η with a level of precision comparable to the DE at NLO. Moreover, contrarily to DE, the anomalous power-law behavior can be read directly from the momentum dependence of the 2-point function. Second, it shows the expected logarithmic shape of the perturbative regime even though the coefficient in front of the logarithm, which is a 2-loops quantity, is only reproduced with an error of 8%. In order to check the quality of the solution in the intermediate momentum region, a quantity sensitive to this crossover sector has been calculated: one gets a result almost within the error bars of both Monte-Carlo and resumed 7-loops perturbative calculations. Please observe that this quantity is extremely difficult to calculate: even these sophisticated methods give an error of the order of 10%.
Another interesting similarity between DE and the method presented in [16] is that, as a price to pay in order to close the equations including an infinite number of vertices, one has to study the problem in an external constant field. Accordingly, one ends up with partial differential equations which may be difficult to solve. A useful approximation scheme, widely 3 used in DE calculations, is to perform, on top of the expansion in derivatives of the field, an extra expansion in powers of the field (see e.g. [7]), in the spirit of Weinberg proposal.
During the last 10 years, this strategy has been widely used [18, 19, 20, 21, 22, 23, 24, 25] ; in many studied situations this expansion seems to converge (generally oscillating) [19, 20, 21, 23, 24, 25] , while in many others it does not [22, 26] . In d = 2, the field expansion has been explored with no indication of convergence for critical exponents, even going to high orders [27] .
In this work we shall explore this procedure of expansion in powers of the field, in the framework of the calculation scheme presented in [16] . More precisely, we shall make a field expansion on top of the already approximated 2-point function flow equation solved in [17] .
Then we shall compare results with and without field expansion. In doing so, we have two goals. First, we shall study the apparent convergence of this procedure. This comparison is essential if one hopes to apply the scheme described in [16] to situations more complicated than that considered in [17] . For example, within DE scheme, when trying to go to higher orders or when considering more involved models, the expansion in powers of the field on top of the corresponding approximate flow equations is sometimes the only practical strategy to solve them [8, 23, 28] . The second and more important goal is the following: as we shall see in section III, truncation in powers of the field is equivalent to ignoring high order vertices in the flow equations. Thus, the comparison presented here can help to estimate the quality of the calculations made so far to get n-point functions at finite momenta neglecting high order vertices.
The article is organized as follows. In the next section we describe the basics ingredients of both the NPRG and the approximation scheme introduced in [16] . We also present the results obtained in [17] , when this scheme is used to find the 2-point function of the scalar model. In section III, we apply the expansion in the field at various orders and compare these results with those found in [17] . Finally, we present the conclusions of the study.
Let us consider a scalar field theory with the classical action
S = d d x 1 2 (∂ µ ϕ(x)) 2 + r 2 ϕ 2 (x) + u 4! ϕ 4 (x) . ( 1
)
4 Here, r and u are the microscopic mass and coupling, respectively.
The NPRG builds a family of effective actions, Γ κ [φ] (where φ(x) = ϕ(x) J is the expectation value of the field in presence of an external source J(x)), in which the magnitude of long wavelength fluctuations are controlled by an infrared regulator depending on a continuous parameter κ. One can write for Γ κ [φ] an exact flow equation [5, 14, 21, 29] :
∂ κ Γ κ [φ] = 1 2 d d q ( 2π
) d ∂ κ R κ (q 2 ) Γ (2) κ + R κ -1 q,-q , ( 2
)
where Γ (2) κ is the second functional derivative of Γ κ with respect to φ(x), and R κ denotes a family of "cut-off functions" depending on κ: R κ (q) behaves like κ 2 when q ≪ κ and it vanishes rapidly when q ≫ κ [19, 20] . The effective action Γ κ [φ] interpolates between the classical action obtained for κ = Λ (where Λ -1 is the microscopic length scale), and the full effective action obtained when κ → 0, i.e., when all fluctuations are taken into account (see e.g. [7]). By differentiating eq. (2) with respect to φ(x), and then letting the field be constant, one gets the flow equation for the n-point function Γ (n) κ in a constant background field φ. For example, for the 2-point function one gets:
∂ κ Γ (2) κ (p; φ) = d d q ( 2π
) d ∂ κ R k (q) G κ (q; φ)Γ (3) κ (p, q, -p -q; φ) ×G κ (q + p; φ)Γ (3) κ (-p, p + q, -q; φ)G κ (q; φ) - 1 2 G κ (q; φ)Γ (4) κ (p, -p, q, -q; φ)G κ (q; φ) , ( 3
)
where
G -1 κ (q; φ) ≡ Γ (2) κ (q, -q; φ) + R κ (q 2 ), ( 4
)
and we used the definition
(2π) d δ (d) i p i Γ (n) κ (p 1 , . . . , p n ; φ) = d d x 1 . . . d d x n e i P n j=1 p j x j δ n Γ κ δφ(x 1 ) . . . δφ(x n ) φ(x)≡φ . ( 5
)
The flow equation for a given n-point function involves the n+1 and n+2 point functions (see, e.g., eq. ( 3 )), so that the flow equations for all correlation functions constitute an infinite hierarchy of coupled equations.
In [16] , a general method to solve this infinite hierarchy was proposed. It exploits the smoothness of the regularized n-point functions, and the fact that the loop momentum q in 5 the right hand side of the flow equations (such as eq. (2) or eq. ( 3 )) is limited to q < ∼ κ due to the presence of ∂ κ R κ (q). The leading order of the method presented in [16] thus consists in setting
Γ (n) κ (p 1 , p 2 , ..., p n-1 + q, p n -q) ∼ Γ (n) κ (p 1 , p 2 , ..., p n-1 , p n ) ( 6
)
in the r.h.s. of the flow equations. After making this approximation, some momenta in some of the n-point functions vanish, and their expressions can then be obtained as derivatives of m-point functions (m < n) with respect to a constant background field.
Specifically, in the flow equation for the 2-point function, eq. ( 3 ), after setting q = 0 in the vertices of the r.h.s., the 3-and 4-point functions will contain one and two vanishing momenta, respectively. These can be related to the following derivatives of the 2-point function:
Γ (3) κ (p, -p, 0; φ) = ∂Γ ( 2
) κ (p, -p; φ) ∂φ , Γ (4) κ (p, -p, 0, 0; φ) = ∂ 2 Γ ( 2
) κ (p, -p; φ) ∂φ 2 . ( 7
)
One then gets a closed equation for Γ (2) κ (p; φ):
κ∂ κ Γ (2) κ (p 2 ; φ) = J ( 3
) d (p, κ; φ) ∂Γ ( 2
) κ (p, -p; φ) ∂φ 2 - 1 2 I ( 2
) d (κ; φ) ∂ 2 Γ ( 2
) κ (p, -p; φ) ∂φ 2 , ( 8
)
where
J (n) d (p; κ; φ) ≡ d d q ( 2π
) d κ∂ κ R κ (q 2 )G κ (p + q; φ)G (n-1) κ (q; φ), ( 9
)
and
I (n) d (κ; φ) ≡ d d q ( 2π
) d κ∂ κ R κ (q 2 )G n κ (q; φ). ( 10
)
In fact, in order to preserve the relation
Γ (2) κ (p = 0; φ) = ∂ 2 V κ ∂φ 2 , ( 11
) V κ (φ) = Γ κ [φ(x) ≡ φ]/Vol
being the effective potential, it is better to make the approximation (6) (followed by ( 7 )) in the flow equation for Σ κ (p; φ) defined as
Σ κ (p; φ) = Γ (2) κ (p; φ) -p 2 -Γ (2) κ (p = 0; φ). ( 12
)
The 2-point function is then obtained from Γ (2)
(p; φ) = ∂ 2 V κ (φ)/∂φ 2 + p 2 + Σ κ (p; φ), which
demands the simultaneous solution of the flow equations for V κ (φ) and Σ κ (p; φ).
As shown in [17] , even if the complete solution of these equations is a priori complicated, a simple, and still accurate, way of solving them consists in assuming in the various integrals
G -1 κ (q; φ) ≃ Z κ q 2 + ∂ 2 V κ (φ)/∂φ 2 + R κ (q 2 ), ( 13
) where Z κ ≡ Z κ (φ = 0), with Z κ (φ) ≡ 1+∂Σ κ (p; φ)/∂p 2 | p=0
. This approximation is consistent with an improved version of the Local Potential Approximation (LPA, the first order of the DE), which includes explicitly a field renormalization factor Z κ [7]. Doing so, the "p = 0" sector decouples from the p = 0 one. Here, by "p = 0" we mean the sector describing vertices and derivative of vertices at zero momenta, i.e., flow equations for V κ and Z κ . Moreover, it is useful to use the regulator [20]
R κ (q 2 ) = Z κ (κ 2 -q 2 ) Θ(κ 2 -q 2 ), ( 14
)
which allows the functions J (n) d (p; κ; φ) and I (n) d (κ; φ) to be calculated analytically. The corresponding expressions can be found in [17] . In fact, all quantities are functions of ρ ≡ φ 2 /2. The problem is then reduced to the solution of the three flow equations for V κ (ρ) and Z κ (ρ), for the p = 0 sector, and for Σ κ (p; ρ), in the p = 0 one. As only the "effective mass"
m 2 κ (ρ) ≡ ∂ 2 V κ (φ) ∂φ 2 = ∂V κ (ρ) ∂ρ + 2ρ ∂ 2 V κ (ρ) ∂ρ 2 ( 15
)
(and its derivatives with respect to ρ) enters in the p = 0 sector, it is more convenient to work with the flow equation for m 2 κ (ρ) instead of that for V κ (ρ) itself. The non-trivial fact is that by differentiating twice the flow equation for V κ (ρ) w.r.t φ, one gets a closed equation for m 2 κ (ρ). In order to make explicit the fixed point in the κ → 0 limit, it is necessary to work with dimensionless variables:
µ κ (ρ) ≡ Z -1 κ κ -2 m 2 κ (ρ) , χ κ (ρ) ≡ Z -1 κ Z κ (ρ) , ρ ≡ K -1 d Z κ κ 2-d ρ , ( 16
)
which, in the critical case, have a finite limit when κ → 0. Above, K d is a constant
conveniently taken as K -1 d ≡ d 2 d-1 π d/2 Γ(d/2) (e.g., K 3 = 1/(6π 2 )).
In the p = 0 sector, the dimensionful variable p in the self-energy flow equation makes Σ κ (p; ρ) reach a finite value when κ → 0. As discussed in [17] , the inclusion of the flow equation for the renormalization factor Z κ (ρ) is essential in order to preserve the correct scaling behavior of Γ (2) (p; ρ) in the 7 infrared limit. Doing so, in the critical case, the function Γ (2) (p; ρ)/(Z κ κ 2 ) has to reach a fixed point expression depending on ρ and p/κ, when κ, p ≪ u and ρ ∼ 1.
Putting all together, in d = 3, the three flow equations that have to be solved are
κ∂ κ µ κ (ρ) = -(2 -η κ )µ κ (ρ) + (1 + η κ )ρµ ′ κ (ρ) -1 - η κ 5 µ ′ κ (ρ) + 2ρµ ′′ κ (ρ) (1 + µ κ (ρ)) 2 - 4ρµ ′ κ (ρ) 2 (1 + µ κ (ρ)) 3 ( 17
)
and
κ∂ κ χ κ (ρ) = η κ χ κ (ρ) + (1 + η κ )ρχ ′ κ (ρ) -2ρ µ ′2 κ (ρ) (1+µκ(ρ)) 4 + 1 -ηκ 5 8ρχ ′ κ (ρ) µ ′ κ (ρ) (1+µκ(ρ)) 3 -χ ′ κ (ρ)+2ρχ ′′ κ (ρ) (1+µκ(ρ)) 2 , ( 18
) together with η κ = χ ′ κ (0) χ ′ κ (0)/5 + (1 + µ κ (0)) 2 , κ∂ κ Z κ = -η κ Z κ , ( 19
)
for the p = 0 sector, and
κ∂ κ Σ κ (p, ρ) = (1 + η κ )ρΣ ′ κ (p, ρ) + 2ρµ ′2 κ (ρ)κ 2 Zκ (1+µκ(ρ)) 2 f κ (p, ρ) -2(1-ηκ/5) (1+µκ(ρ)) 2 + 2ρfκ(p,ρ) (1+µκ(ρ)) 2 2µ ′ κ (ρ)Σ ′ κ (p, ρ) + Σ ′2 κ (p,ρ) κ 2 Zκ -(1-ηκ/5) (1+µκ(ρ)) 2 (Σ ′ κ (p, ρ) + 2ρΣ ′′ κ (p, ρ)) ( 20
)
for the p = 0 one. In these equations, the prime means ∂ ρ and we used the explicit expression for I 20 ), we introduced the dimensionless In [17] , this strategy is used to get the 2-point function of the scalar model at criticality and zero external field (i.e., Σ(p = 0, ρ = 0) = 0), in d = 3. As recalled above, the function thus obtained has the correct shape, either in the scaling, perturbative and intermediate momenta regimes.
(n) 3 = 2K 3 κ 5-2n Z 1-n κ (1-η κ /5)/(1 + µ κ (ρ)) n . In eq. (
expression f κ defined as J ( 3
) 3 (p; κ; ρ) ≡ K 3 κ -1 Z -2 κ /(1 + µ κ (ρ)) 2 × f κ (p; ρ), with p ≡ p/κ.
In this section, we shall compare the solution obtained in [17] using the procedure described above, with the solution of the same three flow equations expanded in powers of ρ and truncating up to a given order. Before doing so, let us first consider only the flow equation for the potential or, equivalently, that for the effective mass, i.e., eq. (17), with 0) . This corresponds to the pure LPA sector and it is thus independent of 8 the scheme presented in [16] . In d = 3, its expansion in powers of the field has been widely studied during the last ten years, using various regulators [22, 24] . Recently, another interesting truncation scheme has also been considered in [30] showing much better convergence properties. However, here we shall consider the simpler expansion in powers of the fields; as shall be seen bellow this is the field expansion that can be compared to usual truncation in the number of vertices. It has been shown [25] that, using the regulator we consider here (see eq. ( 14 )), this expansion seems to converge. This result follows when expanding both around finite and zero external field, although faster in the first case. In [25] the convergence in this situation has been discussed studying the critical exponent ν. In order to strengthen this conclusion, as a first step in our study, we have analyzed the effect of the expansion on the function µ κ (ρ):
Z κ ≡ 1 (η κ ≡
µ κ (ρ) = ∞ n=0 1 n! µ (n) κ ρn . ( 21
)
More precisely, we shall gauge the impact of truncating this sum on the fixed point values of the coefficients µ (n) κ , which are proportional to vertices at zero momenta and zero external field. This study is motivated by the fact that these µ (n) κ shall appear in the Σ κ (p; ρ) flow equation, eq. (20), when the later shall be expanded around ρ = 0. Results are shown in Figure 1 . The four plots present the fixed point value for the first 4 couplings, µ 3. For each coupling, we present the result which follows by solving the complete LPA equation, eq. (17), together with the result obtained with the equation expanded in powers of ρ. For example, when going only up to the first order (i.e., neglecting all µ (n) κ with n ≥ 2), the corresponding equations for µ (0) κ and µ (1) κ , are:
(n) κ=0 , n = 0, • • • ,
κ∂ κ µ (0) κ = (η κ -2)µ (0) κ - (1 -η κ /5)µ (1) κ (1 + µ ( 0
) κ ) 2 ( 22
)
and
κ∂ κ µ (1) κ = (2η κ -1)µ (1) κ + 6(1 -η κ /5)(µ ( 1
) κ ) 2 (1 + µ ( 0
) κ ) 3 . ( 23
)
which have to be solved simultaneously. (In fact, if solving just the LPA, η κ = 0; nevertheless, we have kept η κ in eqs. (22)-(23) for a later use of these equations). When going to the second order, eq. (23) acquires a new term and a new flow equation, that for µ (2) κ , appears; and so on. According to Figure 1 , an apparent convergence shows up. In all cases one observes 9 1 2 3 4 5 6 7 8 9 order -0.18 -0.15 -0.12 -0.09 µ (0) 4 5 6 7 8 9 -0.189 -0.1875 -0.186 4 5 6 7 8 9 0.246 0.2475 0.249 1 2 3 4 5 6 7 8 9 order 0.135 0.18 0.225 0.27 µ (1) 6 7 8 9 0.093 0.0945 0.096 2 3 4 5 6 7 8 9 order 0.06 0.075 0.09 µ (2) 3 4 5 6 7 8 9 order 0.0280 0.0315 0.0350 0.0385 µ (3) FIG. 1: First four dimensionless fixed point couplings at zero momenta and zero external field: results obtained by truncating the flow equation, as a function of the order; the corresponding value for the complete equation is represented by the dotted-line.
that: 1) there seems to be an oscillating convergence, 2) the value of µ (i) is found with about 1% error truncating at order i + 3.
Let us now turn to the study of the flow equation for the 2-point function coming from the scheme proposed in [16] . As the effective potential (or the effective mass), Γ (2) κ (p; ρ) can also be expanded in powers of the external field:
Γ (2) κ (p; ρ) = ∞ n=0 2 n (2n)! Γ (2n+2) κ (p, -p, 0, 0, • • • , 0; ρ)| ρ=0 ρ n , ( 24
) because Γ (m+2) κ (p, -p, 0, 0, • • • , 0; ρ) = ∂ m Γ ( 2
) κ (p; φ) ∂φ m ( 25
) 10
3,5×10 -2 3,6×10 -2 2,0×10 -11 2,5×10 -11 10 -4 10 -2 10 0 10 2 p/u 10 -18 10 -15 10 -12 10 -9 Σ(p)
FIG. 2: Comparison of the self-energy when expanding only the flow equations for the self-energy Σ κ (p; ρ) and its derivative Z κ (ρ) (strategy I): truncation is made at first (double dotted-dashed), second (dotted-dashed), third (dashed) and fourth (dotted) order; the complete solution is given by the straight line. In the figure, u = 5.9210 -4 Λ.
and we used that, at zero field, all odd vertex functions vanish. Equation (24) makes clear the point stated above: once approximation (6) is performed, truncating the expansion in powers of the external field is equivalent to neglecting high order vertices. Moreover, eqs. (24) and (25) show that the procedure proposed in [16] indeed includes all vertices, although approximately.
We have now all the ingredients to discuss the main goal of this paper: the analysis of the expansion of the three flow equations for µ κ (ρ), Z κ (ρ) and Σ κ (p; ρ), eqs. (17-20), around ρ = 0. In doing so, one can write:
Σ κ (p, ρ) = ∞ n=0 1 n! Σ (n) κ (p) ρn . ( 26
)
and
χ κ (ρ) = ∞ n=0 1 n! χ (n) κ ρn . ( 27
)
together with eq. (21). For example, when going to the first order, the six equations that 11 have to be solved are:
κ∂ κ Σ (0) κ (p) = - (1 -η κ /5)Σ ( 1
) κ (p) (1 + µ (0) κ ) 2 ( 28
)
and
κ∂ κ Σ (1) κ (p) = (1 + η κ )Σ (1) κ (p) + 2(µ ( 1
) κ ) 2 Z κ κ 2 (1 + µ ( 0
) κ ) 2 f κ (p, 0) - 2(1 -η κ /5) (1 + µ ( 0
) κ ) + 2f κ (p, 0) (1 + µ ( 0
) κ ) 2 2µ (1) κ Σ (1) κ (p) + Σ ( 1
) κ (p) 2 κ 2 Z κ + 2(1 -η κ /5)µ ( 0
) κ Σ ( 1
) κ (p) (1 + µ ( 0
) κ ) 3 , ( 29
)
which correspond to the expansion of eq. (20),
κ∂ κ χ (0) κ = η κ χ (0) κ - (1 -η κ /5)χ (1) κ (1 + µ ( 0
) κ ) 2 ( 30
)
and
κ∂ κ χ (1) κ = (1 + 2η κ )χ (1) κ - 2(µ ( 1
) κ ) 2 (1 + µ ( 0
) κ ) 4 + 10µ ( 0
) κ χ ( 1
) κ (1 -η κ /5) (1 + µ (0) κ ) 3 , ( 31
)
which correspond to the expansion of eq. (18), together with eqs. (22) and (23).
In fact, it is possible to perform two kinds of expansion. First, in order to isolate the effect of the field expansion just in the flow equations provided by the scheme presented in [16] , we shall expand only the flow equations for Σ κ (p; ρ) and its derivative at zero momenta Z κ (ρ), eqs. (20) and (18), solving exactly the differential flow equation for µ κ (ρ), eq. (17).
For example, at first order, one should solve simultaneously eqs. (28-29), (30-31), and (17). This is called "strategy I". Second, to consider all the effects, we shall make the expansion in the three flow equations. For example, at first order, one should solve simultaneously eqs. (28-29), (30-31), and (22) (23) . We call this "strategy II". Notice that, as explained in [17] , in order to get the correct scaling behavior it is mandatory to treat the equations for Z κ (ρ) and Σ κ (p; ρ) with the same approximations; it is then not possible to solve one of them completely while expanding the other one.
Figure 2 presents the self-energy one gets truncating up to fourth order, following strategy I; it is also shown the function obtained in [17] (from now on, the latter function, obtained by solving the 3 differential equations, eqs. (17), (18) and (20), shall be called the "complete solution"). Figure 3 presents the same results when following strategy II. These Figures show that, in both strategies of expansion, by truncating at first order one already gets a function with the correct shape in all momenta regimes. 12 3,5×10 -2 3,6×10 -2 1,9×10 -11 2,0×10 -11 10 -6 10 -4 10 -2 10 0 10 2 p/u 10 -18 10 -15 10 -12 10 -9 Σ(p)
FIG. 3: Comparison of the self-energy when expanding the three flow equations (strategy II): truncation is made at first (double dotted-dashed), second (dotted-dashed), third (dashed) and fourth (dotted) order; the complete solution is given by the straight line. In the figure, u = 5.9210 -4 Λ.
In order to make a quantitative evaluation of the approximate solution obtained doing the expansion, we have calculated different numbers describing the physical properties of the self-energy. First, as can be seen in both figures above, all solutions have, in the infrared (p ≪ u), the potential behavior characterizing the scaling regime: Σ(p) + p 2 ∼ p 2-η , where η is the anomalous dimension. We have checked that, at each order and in both strategies, the resulting self-energy does have scaling, and we extracted the corresponding value of η. In fact, this can be done in two different ways: either using the κ-dependence of Z κ
(η = -lim κ→0 κ∂ κ log Z κ )
or the p-dependence of Σ(p) stated above. We checked that those two values always coincide, within numerical uncertainties. Figure 4 presents the relative error for η, at each order, when compared with the value following from the complete solution. One observes: 1) in both strategies of expansion there is an apparent convergence, which is oscillatory; 2) the solution from strategy I reaches faster the correct result; 3) when following strategy I, already with a second order truncation the error is about 3% and it drops to less that 1% at the third order. Nevertheless, due to the mixed characteristic of 13 3 4 5 6 7 8 -4 -2 0 2 1 2 3 4 5 6 7 8 order -60 -50 -40 -30 -20 -10 0 10 η relative error [%]
FIG. 4: Relative error (measured in percent) for the anomalous dimension, with respect to the value coming from the complete solution, as a function of the truncation order. Full line: expanding only the flow equations for the self-energy Σ κ (p; ρ) and its derivative Z κ (ρ) (strategy I); dashed line: expanding all flow equations (strategy II).
strategy I, when using this strategy at high order numerical problems arise: indeed, this task demands the numerical evaluation of high order derivatives of µ κ (ρ), to be used in the various flow equations obtained when expanding that of Σ κ (p; ρ). If high precision in the result is required, strategy II is then numerically preferable.
It is important to observe here that the procedure which can be compared to the usual truncation including a finite number of vertices is strategy II. Moreover, the inclusion of high order vertices without performing any other approximation is difficult; for example, the complete inclusion of the 6-point vertex has never been done. Accordingly, as can be seen in fig. 4 , when including only up to the 4-point vertex, as it usually done, the error in η can be as large as 60%.
A second number to assess the quality of the approximate solution is the critical exponent ν. In order to calculate it, we extract the renormalized dimensionful mass from m 2 R = κ 2 µ κ (ρ = 0) and we relate it to the microscopic one by
m 2 R (κ = 0) ∝ (m 2 R (κ = Λ) -m 2 R,crit (κ = Λ)) 2ν , ( 32
)
where m R,crit is the critical renormalized mass. With the complete solution one gets ν = 14 3 4 5 6 7 8 -1 -0.5 0 0.5 1 2 3 4 5 6 7 8 order -10 -5 0 ν relative error [%]
FIG. 5: Relative error (measured in percent) for the critical exponent ν, with respect to the value coming from the complete solution, as a function of the truncation order. Full line: expanding only the flow equations for the self-energy Σ κ (p; ρ) and its derivative Z κ (ρ) (strategy I); dashed line: expanding all flow equations (strategy II).
0.647, to be compared with the best accepted value [31]: ν = 0.6304 ± 0.0013. Figure 5 presents the relative error of the value of ν extracted from the expansion. Once again, one observes that the convergence is much faster when following strategy I, i.e., when considering the effect of the expansion only on the self-energy equation.
The large momenta regime (p ≫ u) of the self-energy can be calculated using perturbation theory, yielding the well known logarithmic shape: Σ(p) ∼ A log(p/B), where A and B are constants. For the complete solution presented in [17] one can prove analytically that A = u 2 /9π 4 , which is only 8% away from the exact result A = u 2 /96π 2 (please observe that this coefficient is given by a 2-loop diagram for the self energy which is only approximatively included at this order). The proof of this analytical result remains valid when performing the field expansion, at any order and within both strategies. We have checked that our numerical solution always has the correct shape, with A = u 2 /9π 4 . This is due to the fact that already the first order in the expansion of Σ κ (p; ρ) around ρ = 0 contains the same 2-loop diagrams contributing to the complete solution.
In order to study the quality of the self-energy in the intermediate momenta regime, we 15 2 3 4 -1 0 1 1 2 3 4 order 0 5 10 15 20 ∆〈φ 2 〉 relative error [%]
FIG. 6: Relative error (measured in percent) for ∆ φ 2 , with respect to the value coming from the complete solution, as a function of the truncation order. Full line: expanding only the flow equations for the self-energy Σ κ (p; ρ) and its derivative Z κ (ρ) (strategy I); dashed line: expanding all flow equations (strategy II).
have calculated a quantity which is very sensitive to this cross-over region:
∆ φ 2 = d 3 p (2π) 3 1 p 2 + Σ(p) - 1 p 2 . ( 33
)
(the integrand is non zero only in the region 10 -3 < ∼ p/u < ∼ 10, see for example [15] ). This quantity received recently much attention because it has been shown [32] that for a scalar model with O(N) symmetry, in d = 3 and N = 2, it determines the shift of the critical temperature of the weakly repulsive Bose gas. It has then been widely evaluated by many methods, for different values of N, in particular, for N = 1. With the numerical solution found in [17] , one gets a number almost within the error bars of the best accepted results available in the literature, using lattice and 7 loops resumed perturbative calculations. Please observe that these errors are as large as 10%, which is an indication that this quantity is particularly difficult to calculate. In Figure 6 we plot the relative error in ∆ φ 2 , at each order of the expansion, when compared with the complete solution result found in [17] . One can appreciate that 1) for both expansion strategies there is an apparent convergence, which is also oscillatory; 2) in both strategies, already with a second order truncation the error is about 1%.
16
In this article, the inclusion of a finite number of vertices in NPRG flow equations is analyzed. An unsolved difficulty of this usual strategy (originally proposed by Weinberg) is the estimation of the error introduced at a given step. Moreover, without performing further approximations, it is very hard to reach high orders of the procedure. The study of its convergence is thus a difficult task. In the present work we analyse this problem using a different approximation scheme [16]: instead of considering a finite number of vertices, this procedure includes all of them, although approximately. Within this context, it is possible to estimate the error of the Weinberg approximation, order by order. To do so one can perform, on top of the approximation presented in [16] , the usual truncation in the number of vertices. The analysis has been done in the particular case of the 2-point function of the scalar field theory in d = 3 at criticality. It has been shown [17] that, at least in this case, the procedure proposed in [16] yields very precise results. Another interesting outcome of the present work follows from the fact that, within the approximation [16], truncation in the number of vertices is equivalent to an expansion in powers of a constant external field. The latter is usually employed in the DE context in order to deal with complicated situations. The analysis of the present paper generalizes this expansion procedure when non zero external momentum are involved.
The calculation of the 2-point function demands the study of both the p = 0 and the p = 0 sectors. While the first one is given by the well studied DE flow equations, the latter follows from the approximation scheme introduced in [16] to calculate the flow of Σ κ (p; ρ).
We used two different strategies to perform the field expansion, both of them around zero external field: either expanding only the flow equation for the self-energy (and its derivative) (strategy I), or both the effective potential and the self-energy (and its derivative) flow equations (strategy II). We have studied the convergence of various quantities measuring physical properties of the self-energy in all momenta regimes: the critical exponents η and ν of the infrared regime, the coefficient of the ultraviolet logarithm, and ∆ φ 2 which is dominated by the crossover momenta regime.
As stated in section III, the strategy that can be compared to the usual truncation which includes a finite number of vertices is strategy II. For example, including completely the 4-point vertex as it is usually done (i.e., in the language of field expansion, going only up to 17 the first order of the expansion), when describing the deep infrared regime one could make errors as big as 60% in the critical exponent η (see figure 4 ). If one wants results with less that 5% error for this quantity, the inclusion of up to 8-point vertices (i.e., going up to third order) is necessary.
However, when going to higher orders in the field expansion, the series for all considered quantities seem to converge rapidly, within both strategies. The convergence is faster when using strategy I, i.e., when making the expansion only for the approximate flow equation
resulting from the method presented in [16] . For example, using strategy I, a third order truncation introduces a relative error smaller that 1% for all studied quantities; while using strategy II, in order to reach the same error one needs 6th order for η, 4th order for ν and 2nd order for ∆ φ 2 . Nevertheless, due to numerical difficulties, if trying to go to high order expansions, it is preferable to use strategy II, i.e., expanding also the effective potential flow equation.
It is difficult to assess the generality of these results on the use of field expansion on top of the strategy proposed in [16] . Of course, there are situations where expanding in an external field is not a priori convenient. One can mention as a first example, situations where there is a physical external field (as in a broken phase or when an external source for the field is considered). A second example is two-dimensional systems where even in the DE, the field expansion does not seem to converge. Nevertheless, the short study presented in the present paper allows to consider field expansion on top of the approximation proposed in [16] as a possible strategy to deal with many involved models, as for example QCD.
[1] K. G. Wilson and J. B. Kogut, Phys. Rept. 12, 75 (1974). [2] J. Polchinski, Nucl. Phys. B 231, 269 (1984). [3] C.Wetterich, Phys. Lett., B301, 90 (1993). [4] U.Ellwanger, Z.Phys., C58, 619 (1993). [5] T.R.Morris, Int. J. Mod. Phys., A9, 2411 (1994). [6] C. Bagnuls and C. Bervillier, Phys. Rept. 348, 91 (2001). [7] J. Berges, N. Tetradis and C. Wetterich, Phys. Rept. 363 (2002) 223. [8] B. Delamotte, D. Mouhanna, M. Tissier, Phys.Rev. B69 (2004) 134413.
[9] B. Delamotte, cond-mat/0702365. [10] G.R. Golner, Phys. Rev. B33, (1986) 7863. [11] S. Weinberg, Phys. Rev. D8 (1973) 3497. [12] U. Ellwanger, M. Hirsch and A. Weber, Eur. Phys. J. C 1 (1998) 563; J. M. Pawlowski, D. F. Litim, S. Nedelko and L. von Smekal, Phys. Rev. Lett. 93 (2004) 152002; J. Kato, arXiv:hep-th/0401068; C. S. Fischer and H. Gies, JHEP 0410 (2004) 048; S. Ledowski, N. Hasselmann, P. Kopietz, Phys. Rev. A69, 061601(R) (2004); ibid, Phys. Rev. A70, 063621 (2004) J. P. Blaizot, R. Mendez Galain and N. Wschebor, Phys. Rev. E74 051116, 2006; ibid E74:051117, 2006. [13] U. Ellwanger and C. Wetterich, Nucl. Phys. B423 (1994), 137. [14] Ulrich Ellwanger, Z. Phys. C62 (1994) 503-510.
[15] J. P. Blaizot, R. Mendez Galain and N. Wschebor, Europhys. Lett., 72 (5), 705-711 (2005). [16] J. P. Blaizot, R. Mendez Galain and N. Wschebor, Phys.Lett.B632:571-578,2006. [17] J. P. Blaizot, R. Mendez Galain and N. Wschebor, Non Perturbative Renormalization Group calculation of the scalar self-energy, arXiv:hep-th/0605252, to appear in Eur. Jour. of Phys. B. [18] G. Zumbach, Nucl. Phys. B 413, 754 (1994). [19] L.Canet, B.Delamotte, D.Mouhanna and J.Vidal, Phys. Rev. D67 065004 (2003). [20] D.Litim, Phys. Lett. B486, 92 (2000); Phys. Rev. D64, 105007 (2001); Nucl. Phys. B631, 128 (2002); Int.J.Mod.Phys. A16, 2081 (2001). [21] N. Tetradis and C. Wetterich, Nucl. Phys. B 422 (1994) 541. [22] T. R. Morris, Phys. Lett. B 334 (1994) 355. [23] L. Canet, B. Delamotte, D. Mouhanna and J. Vidal, Phys. Rev. B68 (2003) 064421. [24] K. I. Aoki, K. Morikawa, W. Souma, J. I. Sumi and H. Terao, Prog. Theor. Phys. 99, 451 (1998) [25] D. F. Litim, Nucl. Phys. B 631, 128 (2002) [arXiv:hep-th/0203006]. [26] N. Tetradis and C. Wetterich, Nucl. Phys. B 383 (1992) 197.
[27] B. Delamotte, D. Mouhanna and M. Tissier, private comunication. [28] G. Tarjus, M. Tissier, Phys. Rev. Lett. 93 (2003) 267008; L. Canet, B. Delamotte, O. Deloubriere, N. Wschebor, Phys. Rev. Lett. 92 (2004) 195703; L. Canet, cond-mat/0509541. [29] T.R.Morris, Phys. Lett. B329, 241 (1994).
[30] B. Boisseau, P. Forgacs and H. Giacomini, J. Phys. A 40 (2007) F215 [arXiv:hep-th/0611306]. [31] R. Guida, J. Zinn-Justin, J. Phys. A31, 8103 (1998). [32] G. Baym, J.-P. Blaizot, M. Holzmann, F. Laloë, and D. Vautherin, Phys. Rev. Lett. 83, 1703 (1999). | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In nearly all fields in physics, there are systems having a large number of strongly correlated constituents. These cannot be treated with usual perturbative methods. Phase transitions and critical phenomena, disordered systems, strongly correlated electrons, quantum chromodynamics at large distances, are just a few examples which demand a general and efficient method to treat non-perturbative situations. In problems as those just quoted, the calculation of correlation functions of the configuration variables is, in general, a very complicated task.\n\nThe non perturbative renormalization group (NPRG) [1, 2, 3, 4, 5] has proven to be a powerful tool to achieve this goal. It presents itself as an infinite hierarchy of flow equations relating sequentially the various n-point functions. It has been successfully applied in many different problems, either in condensed matter, particle or nuclear physics (for reviews, see e.g. [6, 7, 8] ; a pedagogical introduction can be found in [9] ). In most of these problems however, one is interested in observables dominated by long wavelength modes. In these cases, it is then possible to approximately close the infinite hierarchy of NPRG equations performing an expansion in the number of derivatives of the field. This approximation scheme is known as the derivative expansion (DE) [10] . The price to pay is that the n-point functions can be calculated only at small external momenta, i.e. smaller than the smallest mass on the problem (vanishing momenta in the case of critical phenomena).\n\nIn many other physical problems however, this is not enough: the full knowledge of the momentum dependence of correlation functions is needed in order to calculate quantities of physical interest (e.g. to get the spectrum of excitations, the shape of a Fermi surface, the scattering matrix, etc.). There have been many attempts to solve the infinite system of flow equations at finite momenta; most of them are based on various forms of an early proposal by Weinberg [11] . Although some of these attempts [12, 13, 14, 15] introduce sophisticated ansatz for the unknown correlation functions appearing in a given flow equation, most efforts simply ignore high order vertices. In all these works, only low order vertices are taken into account: usual calculations do not even include the complete flow of the 3-and 4-point functions. Moreover, it is not possible a priori to gauge the quality of such approximations schemes.\n\nRecently, an alternative general method to get n-point functions at any finite momenta 2 within the NPRG has been proposed [16] . It has many similarities with DE. First, it is an approximation scheme that can be systematically improved. Second, the scheme yields a closed set of flow equations including simultaneously an infinite number of vertices; one thus goes far beyond schemes including a small number of vertices, as those quoted in the previous paragraph. Moreover, it has been proven [16] that in their corresponding limits, both perturvative and DE results are recovered; this remains valid at each order of the respective expansion. Finally, in the large-N limit of O(N) models, the leading-order (LO) of the approximation scheme becomes exact for all n-point functions. (The expression \"leading order\" means the first step in the approximation scheme; it does not refer to an expansion in a small parameter which usually does not exist in these kind of problems.).\n\nIn [17] , the method has been applied, in its leading order, to the calculation of the selfenergy of the scalar model, at criticality. That is, we have fine-tuned the bare mass of the model in order for the correlation length to be infinite, and then we have studied the full range of momenta, from the high momenta Gaussian regime to the low momenta scaling one. At this order of the approximation scheme the self-energy is expected to include all one loop contributions and to achieve DE at next-to-leading order (NLO) precision, in the corresponding limit [16] . The numerical solution found in [17] verifies these properties.\n\nMoreover, the function has the expected physical properties in all momenta regime. First, it presents the correct scaling behavior in the infrared limit. The model reproduces critical exponent η with a level of precision comparable to the DE at NLO. Moreover, contrarily to DE, the anomalous power-law behavior can be read directly from the momentum dependence of the 2-point function. Second, it shows the expected logarithmic shape of the perturbative regime even though the coefficient in front of the logarithm, which is a 2-loops quantity, is only reproduced with an error of 8%. In order to check the quality of the solution in the intermediate momentum region, a quantity sensitive to this crossover sector has been calculated: one gets a result almost within the error bars of both Monte-Carlo and resumed 7-loops perturbative calculations. Please observe that this quantity is extremely difficult to calculate: even these sophisticated methods give an error of the order of 10%.\n\nAnother interesting similarity between DE and the method presented in [16] is that, as a price to pay in order to close the equations including an infinite number of vertices, one has to study the problem in an external constant field. Accordingly, one ends up with partial differential equations which may be difficult to solve. A useful approximation scheme, widely 3 used in DE calculations, is to perform, on top of the expansion in derivatives of the field, an extra expansion in powers of the field (see e.g. [7]), in the spirit of Weinberg proposal.\n\nDuring the last 10 years, this strategy has been widely used [18, 19, 20, 21, 22, 23, 24, 25] ; in many studied situations this expansion seems to converge (generally oscillating) [19, 20, 21, 23, 24, 25] , while in many others it does not [22, 26] . In d = 2, the field expansion has been explored with no indication of convergence for critical exponents, even going to high orders [27] .\n\nIn this work we shall explore this procedure of expansion in powers of the field, in the framework of the calculation scheme presented in [16] . More precisely, we shall make a field expansion on top of the already approximated 2-point function flow equation solved in [17] .\n\nThen we shall compare results with and without field expansion. In doing so, we have two goals. First, we shall study the apparent convergence of this procedure. This comparison is essential if one hopes to apply the scheme described in [16] to situations more complicated than that considered in [17] . For example, within DE scheme, when trying to go to higher orders or when considering more involved models, the expansion in powers of the field on top of the corresponding approximate flow equations is sometimes the only practical strategy to solve them [8, 23, 28] . The second and more important goal is the following: as we shall see in section III, truncation in powers of the field is equivalent to ignoring high order vertices in the flow equations. Thus, the comparison presented here can help to estimate the quality of the calculations made so far to get n-point functions at finite momenta neglecting high order vertices.\n\nThe article is organized as follows. In the next section we describe the basics ingredients of both the NPRG and the approximation scheme introduced in [16] . We also present the results obtained in [17] , when this scheme is used to find the 2-point function of the scalar model. In section III, we apply the expansion in the field at various orders and compare these results with those found in [17] . Finally, we present the conclusions of the study."
},
{
"section_type": "OTHER",
"section_title": "II. GENERAL CONSIDERATIONS",
"text": "Let us consider a scalar field theory with the classical action\n\nS = d d x 1 2 (∂ µ ϕ(x)) 2 + r 2 ϕ 2 (x) + u 4! ϕ 4 (x) . ( 1\n\n)\n\n4 Here, r and u are the microscopic mass and coupling, respectively.\n\nThe NPRG builds a family of effective actions, Γ κ [φ] (where φ(x) = ϕ(x) J is the expectation value of the field in presence of an external source J(x)), in which the magnitude of long wavelength fluctuations are controlled by an infrared regulator depending on a continuous parameter κ. One can write for Γ κ [φ] an exact flow equation [5, 14, 21, 29] :\n\n∂ κ Γ κ [φ] = 1 2 d d q ( 2π\n\n) d ∂ κ R κ (q 2 ) Γ (2) κ + R κ -1 q,-q , ( 2\n\n)\n\nwhere Γ (2) κ is the second functional derivative of Γ κ with respect to φ(x), and R κ denotes a family of \"cut-off functions\" depending on κ: R κ (q) behaves like κ 2 when q ≪ κ and it vanishes rapidly when q ≫ κ [19, 20] . The effective action Γ κ [φ] interpolates between the classical action obtained for κ = Λ (where Λ -1 is the microscopic length scale), and the full effective action obtained when κ → 0, i.e., when all fluctuations are taken into account (see e.g. [7]). By differentiating eq. (2) with respect to φ(x), and then letting the field be constant, one gets the flow equation for the n-point function Γ (n) κ in a constant background field φ. For example, for the 2-point function one gets:\n\n∂ κ Γ (2) κ (p; φ) = d d q ( 2π\n\n) d ∂ κ R k (q) G κ (q; φ)Γ (3) κ (p, q, -p -q; φ) ×G κ (q + p; φ)Γ (3) κ (-p, p + q, -q; φ)G κ (q; φ) - 1 2 G κ (q; φ)Γ (4) κ (p, -p, q, -q; φ)G κ (q; φ) , ( 3\n\n)\n\nwhere\n\nG -1 κ (q; φ) ≡ Γ (2) κ (q, -q; φ) + R κ (q 2 ), ( 4\n\n)\n\nand we used the definition\n\n(2π) d δ (d) i p i Γ (n) κ (p 1 , . . . , p n ; φ) = d d x 1 . . . d d x n e i P n j=1 p j x j δ n Γ κ δφ(x 1 ) . . . δφ(x n ) φ(x)≡φ . ( 5\n\n)\n\nThe flow equation for a given n-point function involves the n+1 and n+2 point functions (see, e.g., eq. ( 3 )), so that the flow equations for all correlation functions constitute an infinite hierarchy of coupled equations.\n\nIn [16] , a general method to solve this infinite hierarchy was proposed. It exploits the smoothness of the regularized n-point functions, and the fact that the loop momentum q in 5 the right hand side of the flow equations (such as eq. (2) or eq. ( 3 )) is limited to q < ∼ κ due to the presence of ∂ κ R κ (q). The leading order of the method presented in [16] thus consists in setting\n\nΓ (n) κ (p 1 , p 2 , ..., p n-1 + q, p n -q) ∼ Γ (n) κ (p 1 , p 2 , ..., p n-1 , p n ) ( 6\n\n)\n\nin the r.h.s. of the flow equations. After making this approximation, some momenta in some of the n-point functions vanish, and their expressions can then be obtained as derivatives of m-point functions (m < n) with respect to a constant background field.\n\nSpecifically, in the flow equation for the 2-point function, eq. ( 3 ), after setting q = 0 in the vertices of the r.h.s., the 3-and 4-point functions will contain one and two vanishing momenta, respectively. These can be related to the following derivatives of the 2-point function:\n\nΓ (3) κ (p, -p, 0; φ) = ∂Γ ( 2\n\n) κ (p, -p; φ) ∂φ , Γ (4) κ (p, -p, 0, 0; φ) = ∂ 2 Γ ( 2\n\n) κ (p, -p; φ) ∂φ 2 . ( 7\n\n)\n\nOne then gets a closed equation for Γ (2) κ (p; φ):\n\nκ∂ κ Γ (2) κ (p 2 ; φ) = J ( 3\n\n) d (p, κ; φ) ∂Γ ( 2\n\n) κ (p, -p; φ) ∂φ 2 - 1 2 I ( 2\n\n) d (κ; φ) ∂ 2 Γ ( 2\n\n) κ (p, -p; φ) ∂φ 2 , ( 8\n\n)\n\nwhere\n\nJ (n) d (p; κ; φ) ≡ d d q ( 2π\n\n) d κ∂ κ R κ (q 2 )G κ (p + q; φ)G (n-1) κ (q; φ), ( 9\n\n)\n\nand\n\nI (n) d (κ; φ) ≡ d d q ( 2π\n\n) d κ∂ κ R κ (q 2 )G n κ (q; φ). ( 10\n\n)\n\nIn fact, in order to preserve the relation\n\nΓ (2) κ (p = 0; φ) = ∂ 2 V κ ∂φ 2 , ( 11\n\n) V κ (φ) = Γ κ [φ(x) ≡ φ]/Vol\n\nbeing the effective potential, it is better to make the approximation (6) (followed by ( 7 )) in the flow equation for Σ κ (p; φ) defined as\n\nΣ κ (p; φ) = Γ (2) κ (p; φ) -p 2 -Γ (2) κ (p = 0; φ). ( 12\n\n)\n\nThe 2-point function is then obtained from Γ (2)\n\n(p; φ) = ∂ 2 V κ (φ)/∂φ 2 + p 2 + Σ κ (p; φ), which\n\ndemands the simultaneous solution of the flow equations for V κ (φ) and Σ κ (p; φ).\n\nAs shown in [17] , even if the complete solution of these equations is a priori complicated, a simple, and still accurate, way of solving them consists in assuming in the various integrals\n\nG -1 κ (q; φ) ≃ Z κ q 2 + ∂ 2 V κ (φ)/∂φ 2 + R κ (q 2 ), ( 13\n\n) where Z κ ≡ Z κ (φ = 0), with Z κ (φ) ≡ 1+∂Σ κ (p; φ)/∂p 2 | p=0\n\n. This approximation is consistent with an improved version of the Local Potential Approximation (LPA, the first order of the DE), which includes explicitly a field renormalization factor Z κ [7]. Doing so, the \"p = 0\" sector decouples from the p = 0 one. Here, by \"p = 0\" we mean the sector describing vertices and derivative of vertices at zero momenta, i.e., flow equations for V κ and Z κ . Moreover, it is useful to use the regulator [20]\n\nR κ (q 2 ) = Z κ (κ 2 -q 2 ) Θ(κ 2 -q 2 ), ( 14\n\n)\n\nwhich allows the functions J (n) d (p; κ; φ) and I (n) d (κ; φ) to be calculated analytically. The corresponding expressions can be found in [17] . In fact, all quantities are functions of ρ ≡ φ 2 /2. The problem is then reduced to the solution of the three flow equations for V κ (ρ) and Z κ (ρ), for the p = 0 sector, and for Σ κ (p; ρ), in the p = 0 one. As only the \"effective mass\"\n\nm 2 κ (ρ) ≡ ∂ 2 V κ (φ) ∂φ 2 = ∂V κ (ρ) ∂ρ + 2ρ ∂ 2 V κ (ρ) ∂ρ 2 ( 15\n\n)\n\n(and its derivatives with respect to ρ) enters in the p = 0 sector, it is more convenient to work with the flow equation for m 2 κ (ρ) instead of that for V κ (ρ) itself. The non-trivial fact is that by differentiating twice the flow equation for V κ (ρ) w.r.t φ, one gets a closed equation for m 2 κ (ρ). In order to make explicit the fixed point in the κ → 0 limit, it is necessary to work with dimensionless variables:\n\nµ κ (ρ) ≡ Z -1 κ κ -2 m 2 κ (ρ) , χ κ (ρ) ≡ Z -1 κ Z κ (ρ) , ρ ≡ K -1 d Z κ κ 2-d ρ , ( 16\n\n)\n\nwhich, in the critical case, have a finite limit when κ → 0. Above, K d is a constant\n\nconveniently taken as K -1 d ≡ d 2 d-1 π d/2 Γ(d/2) (e.g., K 3 = 1/(6π 2 )).\n\nIn the p = 0 sector, the dimensionful variable p in the self-energy flow equation makes Σ κ (p; ρ) reach a finite value when κ → 0. As discussed in [17] , the inclusion of the flow equation for the renormalization factor Z κ (ρ) is essential in order to preserve the correct scaling behavior of Γ (2) (p; ρ) in the 7 infrared limit. Doing so, in the critical case, the function Γ (2) (p; ρ)/(Z κ κ 2 ) has to reach a fixed point expression depending on ρ and p/κ, when κ, p ≪ u and ρ ∼ 1.\n\nPutting all together, in d = 3, the three flow equations that have to be solved are\n\nκ∂ κ µ κ (ρ) = -(2 -η κ )µ κ (ρ) + (1 + η κ )ρµ ′ κ (ρ) -1 - η κ 5 µ ′ κ (ρ) + 2ρµ ′′ κ (ρ) (1 + µ κ (ρ)) 2 - 4ρµ ′ κ (ρ) 2 (1 + µ κ (ρ)) 3 ( 17\n\n)\n\nand\n\nκ∂ κ χ κ (ρ) = η κ χ κ (ρ) + (1 + η κ )ρχ ′ κ (ρ) -2ρ µ ′2 κ (ρ) (1+µκ(ρ)) 4 + 1 -ηκ 5 8ρχ ′ κ (ρ) µ ′ κ (ρ) (1+µκ(ρ)) 3 -χ ′ κ (ρ)+2ρχ ′′ κ (ρ) (1+µκ(ρ)) 2 , ( 18\n\n) together with η κ = χ ′ κ (0) χ ′ κ (0)/5 + (1 + µ κ (0)) 2 , κ∂ κ Z κ = -η κ Z κ , ( 19\n\n)\n\nfor the p = 0 sector, and\n\nκ∂ κ Σ κ (p, ρ) = (1 + η κ )ρΣ ′ κ (p, ρ) + 2ρµ ′2 κ (ρ)κ 2 Zκ (1+µκ(ρ)) 2 f κ (p, ρ) -2(1-ηκ/5) (1+µκ(ρ)) 2 + 2ρfκ(p,ρ) (1+µκ(ρ)) 2 2µ ′ κ (ρ)Σ ′ κ (p, ρ) + Σ ′2 κ (p,ρ) κ 2 Zκ -(1-ηκ/5) (1+µκ(ρ)) 2 (Σ ′ κ (p, ρ) + 2ρΣ ′′ κ (p, ρ)) ( 20\n\n)\n\nfor the p = 0 one. In these equations, the prime means ∂ ρ and we used the explicit expression for I 20 ), we introduced the dimensionless In [17] , this strategy is used to get the 2-point function of the scalar model at criticality and zero external field (i.e., Σ(p = 0, ρ = 0) = 0), in d = 3. As recalled above, the function thus obtained has the correct shape, either in the scaling, perturbative and intermediate momenta regimes.\n\n(n) 3 = 2K 3 κ 5-2n Z 1-n κ (1-η κ /5)/(1 + µ κ (ρ)) n . In eq. (\n\nexpression f κ defined as J ( 3\n\n) 3 (p; κ; ρ) ≡ K 3 κ -1 Z -2 κ /(1 + µ κ (ρ)) 2 × f κ (p; ρ), with p ≡ p/κ."
},
{
"section_type": "OTHER",
"section_title": "III. EXPANSION IN POWERS OF THE FIELD",
"text": "In this section, we shall compare the solution obtained in [17] using the procedure described above, with the solution of the same three flow equations expanded in powers of ρ and truncating up to a given order. Before doing so, let us first consider only the flow equation for the potential or, equivalently, that for the effective mass, i.e., eq. (17), with 0) . This corresponds to the pure LPA sector and it is thus independent of 8 the scheme presented in [16] . In d = 3, its expansion in powers of the field has been widely studied during the last ten years, using various regulators [22, 24] . Recently, another interesting truncation scheme has also been considered in [30] showing much better convergence properties. However, here we shall consider the simpler expansion in powers of the fields; as shall be seen bellow this is the field expansion that can be compared to usual truncation in the number of vertices. It has been shown [25] that, using the regulator we consider here (see eq. ( 14 )), this expansion seems to converge. This result follows when expanding both around finite and zero external field, although faster in the first case. In [25] the convergence in this situation has been discussed studying the critical exponent ν. In order to strengthen this conclusion, as a first step in our study, we have analyzed the effect of the expansion on the function µ κ (ρ):\n\nZ κ ≡ 1 (η κ ≡\n\nµ κ (ρ) = ∞ n=0 1 n! µ (n) κ ρn . ( 21\n\n)\n\nMore precisely, we shall gauge the impact of truncating this sum on the fixed point values of the coefficients µ (n) κ , which are proportional to vertices at zero momenta and zero external field. This study is motivated by the fact that these µ (n) κ shall appear in the Σ κ (p; ρ) flow equation, eq. (20), when the later shall be expanded around ρ = 0. Results are shown in Figure 1 . The four plots present the fixed point value for the first 4 couplings, µ 3. For each coupling, we present the result which follows by solving the complete LPA equation, eq. (17), together with the result obtained with the equation expanded in powers of ρ. For example, when going only up to the first order (i.e., neglecting all µ (n) κ with n ≥ 2), the corresponding equations for µ (0) κ and µ (1) κ , are:\n\n(n) κ=0 , n = 0, • • • ,\n\nκ∂ κ µ (0) κ = (η κ -2)µ (0) κ - (1 -η κ /5)µ (1) κ (1 + µ ( 0\n\n) κ ) 2 ( 22\n\n)\n\nand\n\nκ∂ κ µ (1) κ = (2η κ -1)µ (1) κ + 6(1 -η κ /5)(µ ( 1\n\n) κ ) 2 (1 + µ ( 0\n\n) κ ) 3 . ( 23\n\n)\n\nwhich have to be solved simultaneously. (In fact, if solving just the LPA, η κ = 0; nevertheless, we have kept η κ in eqs. (22)-(23) for a later use of these equations). When going to the second order, eq. (23) acquires a new term and a new flow equation, that for µ (2) κ , appears; and so on. According to Figure 1 , an apparent convergence shows up. In all cases one observes 9 1 2 3 4 5 6 7 8 9 order -0.18 -0.15 -0.12 -0.09 µ (0) 4 5 6 7 8 9 -0.189 -0.1875 -0.186 4 5 6 7 8 9 0.246 0.2475 0.249 1 2 3 4 5 6 7 8 9 order 0.135 0.18 0.225 0.27 µ (1) 6 7 8 9 0.093 0.0945 0.096 2 3 4 5 6 7 8 9 order 0.06 0.075 0.09 µ (2) 3 4 5 6 7 8 9 order 0.0280 0.0315 0.0350 0.0385 µ (3) FIG. 1: First four dimensionless fixed point couplings at zero momenta and zero external field: results obtained by truncating the flow equation, as a function of the order; the corresponding value for the complete equation is represented by the dotted-line.\n\nthat: 1) there seems to be an oscillating convergence, 2) the value of µ (i) is found with about 1% error truncating at order i + 3.\n\nLet us now turn to the study of the flow equation for the 2-point function coming from the scheme proposed in [16] . As the effective potential (or the effective mass), Γ (2) κ (p; ρ) can also be expanded in powers of the external field:\n\nΓ (2) κ (p; ρ) = ∞ n=0 2 n (2n)! Γ (2n+2) κ (p, -p, 0, 0, • • • , 0; ρ)| ρ=0 ρ n , ( 24\n\n) because Γ (m+2) κ (p, -p, 0, 0, • • • , 0; ρ) = ∂ m Γ ( 2\n\n) κ (p; φ) ∂φ m ( 25\n\n) 10\n\n3,5×10 -2 3,6×10 -2 2,0×10 -11 2,5×10 -11 10 -4 10 -2 10 0 10 2 p/u 10 -18 10 -15 10 -12 10 -9 Σ(p)\n\nFIG. 2: Comparison of the self-energy when expanding only the flow equations for the self-energy Σ κ (p; ρ) and its derivative Z κ (ρ) (strategy I): truncation is made at first (double dotted-dashed), second (dotted-dashed), third (dashed) and fourth (dotted) order; the complete solution is given by the straight line. In the figure, u = 5.9210 -4 Λ.\n\nand we used that, at zero field, all odd vertex functions vanish. Equation (24) makes clear the point stated above: once approximation (6) is performed, truncating the expansion in powers of the external field is equivalent to neglecting high order vertices. Moreover, eqs. (24) and (25) show that the procedure proposed in [16] indeed includes all vertices, although approximately.\n\nWe have now all the ingredients to discuss the main goal of this paper: the analysis of the expansion of the three flow equations for µ κ (ρ), Z κ (ρ) and Σ κ (p; ρ), eqs. (17-20), around ρ = 0. In doing so, one can write:\n\nΣ κ (p, ρ) = ∞ n=0 1 n! Σ (n) κ (p) ρn . ( 26\n\n)\n\nand\n\nχ κ (ρ) = ∞ n=0 1 n! χ (n) κ ρn . ( 27\n\n)\n\ntogether with eq. (21). For example, when going to the first order, the six equations that 11 have to be solved are:\n\nκ∂ κ Σ (0) κ (p) = - (1 -η κ /5)Σ ( 1\n\n) κ (p) (1 + µ (0) κ ) 2 ( 28\n\n)\n\nand\n\nκ∂ κ Σ (1) κ (p) = (1 + η κ )Σ (1) κ (p) + 2(µ ( 1\n\n) κ ) 2 Z κ κ 2 (1 + µ ( 0\n\n) κ ) 2 f κ (p, 0) - 2(1 -η κ /5) (1 + µ ( 0\n\n) κ ) + 2f κ (p, 0) (1 + µ ( 0\n\n) κ ) 2 2µ (1) κ Σ (1) κ (p) + Σ ( 1\n\n) κ (p) 2 κ 2 Z κ + 2(1 -η κ /5)µ ( 0\n\n) κ Σ ( 1\n\n) κ (p) (1 + µ ( 0\n\n) κ ) 3 , ( 29\n\n)\n\nwhich correspond to the expansion of eq. (20),\n\nκ∂ κ χ (0) κ = η κ χ (0) κ - (1 -η κ /5)χ (1) κ (1 + µ ( 0\n\n) κ ) 2 ( 30\n\n)\n\nand\n\nκ∂ κ χ (1) κ = (1 + 2η κ )χ (1) κ - 2(µ ( 1\n\n) κ ) 2 (1 + µ ( 0\n\n) κ ) 4 + 10µ ( 0\n\n) κ χ ( 1\n\n) κ (1 -η κ /5) (1 + µ (0) κ ) 3 , ( 31\n\n)\n\nwhich correspond to the expansion of eq. (18), together with eqs. (22) and (23).\n\nIn fact, it is possible to perform two kinds of expansion. First, in order to isolate the effect of the field expansion just in the flow equations provided by the scheme presented in [16] , we shall expand only the flow equations for Σ κ (p; ρ) and its derivative at zero momenta Z κ (ρ), eqs. (20) and (18), solving exactly the differential flow equation for µ κ (ρ), eq. (17).\n\nFor example, at first order, one should solve simultaneously eqs. (28-29), (30-31), and (17). This is called \"strategy I\". Second, to consider all the effects, we shall make the expansion in the three flow equations. For example, at first order, one should solve simultaneously eqs. (28-29), (30-31), and (22) (23) . We call this \"strategy II\". Notice that, as explained in [17] , in order to get the correct scaling behavior it is mandatory to treat the equations for Z κ (ρ) and Σ κ (p; ρ) with the same approximations; it is then not possible to solve one of them completely while expanding the other one.\n\nFigure 2 presents the self-energy one gets truncating up to fourth order, following strategy I; it is also shown the function obtained in [17] (from now on, the latter function, obtained by solving the 3 differential equations, eqs. (17), (18) and (20), shall be called the \"complete solution\"). Figure 3 presents the same results when following strategy II. These Figures show that, in both strategies of expansion, by truncating at first order one already gets a function with the correct shape in all momenta regimes. 12 3,5×10 -2 3,6×10 -2 1,9×10 -11 2,0×10 -11 10 -6 10 -4 10 -2 10 0 10 2 p/u 10 -18 10 -15 10 -12 10 -9 Σ(p)\n\nFIG. 3: Comparison of the self-energy when expanding the three flow equations (strategy II): truncation is made at first (double dotted-dashed), second (dotted-dashed), third (dashed) and fourth (dotted) order; the complete solution is given by the straight line. In the figure, u = 5.9210 -4 Λ.\n\nIn order to make a quantitative evaluation of the approximate solution obtained doing the expansion, we have calculated different numbers describing the physical properties of the self-energy. First, as can be seen in both figures above, all solutions have, in the infrared (p ≪ u), the potential behavior characterizing the scaling regime: Σ(p) + p 2 ∼ p 2-η , where η is the anomalous dimension. We have checked that, at each order and in both strategies, the resulting self-energy does have scaling, and we extracted the corresponding value of η. In fact, this can be done in two different ways: either using the κ-dependence of Z κ\n\n(η = -lim κ→0 κ∂ κ log Z κ )\n\nor the p-dependence of Σ(p) stated above. We checked that those two values always coincide, within numerical uncertainties. Figure 4 presents the relative error for η, at each order, when compared with the value following from the complete solution. One observes: 1) in both strategies of expansion there is an apparent convergence, which is oscillatory; 2) the solution from strategy I reaches faster the correct result; 3) when following strategy I, already with a second order truncation the error is about 3% and it drops to less that 1% at the third order. Nevertheless, due to the mixed characteristic of 13 3 4 5 6 7 8 -4 -2 0 2 1 2 3 4 5 6 7 8 order -60 -50 -40 -30 -20 -10 0 10 η relative error [%]\n\nFIG. 4: Relative error (measured in percent) for the anomalous dimension, with respect to the value coming from the complete solution, as a function of the truncation order. Full line: expanding only the flow equations for the self-energy Σ κ (p; ρ) and its derivative Z κ (ρ) (strategy I); dashed line: expanding all flow equations (strategy II).\n\nstrategy I, when using this strategy at high order numerical problems arise: indeed, this task demands the numerical evaluation of high order derivatives of µ κ (ρ), to be used in the various flow equations obtained when expanding that of Σ κ (p; ρ). If high precision in the result is required, strategy II is then numerically preferable.\n\nIt is important to observe here that the procedure which can be compared to the usual truncation including a finite number of vertices is strategy II. Moreover, the inclusion of high order vertices without performing any other approximation is difficult; for example, the complete inclusion of the 6-point vertex has never been done. Accordingly, as can be seen in fig. 4 , when including only up to the 4-point vertex, as it usually done, the error in η can be as large as 60%.\n\nA second number to assess the quality of the approximate solution is the critical exponent ν. In order to calculate it, we extract the renormalized dimensionful mass from m 2 R = κ 2 µ κ (ρ = 0) and we relate it to the microscopic one by\n\nm 2 R (κ = 0) ∝ (m 2 R (κ = Λ) -m 2 R,crit (κ = Λ)) 2ν , ( 32\n\n)\n\nwhere m R,crit is the critical renormalized mass. With the complete solution one gets ν = 14 3 4 5 6 7 8 -1 -0.5 0 0.5 1 2 3 4 5 6 7 8 order -10 -5 0 ν relative error [%]\n\nFIG. 5: Relative error (measured in percent) for the critical exponent ν, with respect to the value coming from the complete solution, as a function of the truncation order. Full line: expanding only the flow equations for the self-energy Σ κ (p; ρ) and its derivative Z κ (ρ) (strategy I); dashed line: expanding all flow equations (strategy II).\n\n0.647, to be compared with the best accepted value [31]: ν = 0.6304 ± 0.0013. Figure 5 presents the relative error of the value of ν extracted from the expansion. Once again, one observes that the convergence is much faster when following strategy I, i.e., when considering the effect of the expansion only on the self-energy equation.\n\nThe large momenta regime (p ≫ u) of the self-energy can be calculated using perturbation theory, yielding the well known logarithmic shape: Σ(p) ∼ A log(p/B), where A and B are constants. For the complete solution presented in [17] one can prove analytically that A = u 2 /9π 4 , which is only 8% away from the exact result A = u 2 /96π 2 (please observe that this coefficient is given by a 2-loop diagram for the self energy which is only approximatively included at this order). The proof of this analytical result remains valid when performing the field expansion, at any order and within both strategies. We have checked that our numerical solution always has the correct shape, with A = u 2 /9π 4 . This is due to the fact that already the first order in the expansion of Σ κ (p; ρ) around ρ = 0 contains the same 2-loop diagrams contributing to the complete solution.\n\nIn order to study the quality of the self-energy in the intermediate momenta regime, we 15 2 3 4 -1 0 1 1 2 3 4 order 0 5 10 15 20 ∆〈φ 2 〉 relative error [%]\n\nFIG. 6: Relative error (measured in percent) for ∆ φ 2 , with respect to the value coming from the complete solution, as a function of the truncation order. Full line: expanding only the flow equations for the self-energy Σ κ (p; ρ) and its derivative Z κ (ρ) (strategy I); dashed line: expanding all flow equations (strategy II).\n\nhave calculated a quantity which is very sensitive to this cross-over region:\n\n∆ φ 2 = d 3 p (2π) 3 1 p 2 + Σ(p) - 1 p 2 . ( 33\n\n)\n\n(the integrand is non zero only in the region 10 -3 < ∼ p/u < ∼ 10, see for example [15] ). This quantity received recently much attention because it has been shown [32] that for a scalar model with O(N) symmetry, in d = 3 and N = 2, it determines the shift of the critical temperature of the weakly repulsive Bose gas. It has then been widely evaluated by many methods, for different values of N, in particular, for N = 1. With the numerical solution found in [17] , one gets a number almost within the error bars of the best accepted results available in the literature, using lattice and 7 loops resumed perturbative calculations. Please observe that these errors are as large as 10%, which is an indication that this quantity is particularly difficult to calculate. In Figure 6 we plot the relative error in ∆ φ 2 , at each order of the expansion, when compared with the complete solution result found in [17] . One can appreciate that 1) for both expansion strategies there is an apparent convergence, which is also oscillatory; 2) in both strategies, already with a second order truncation the error is about 1%.\n\n16"
},
{
"section_type": "CONCLUSION",
"section_title": "IV. SUMMARY AND CONCLUSIONS",
"text": "In this article, the inclusion of a finite number of vertices in NPRG flow equations is analyzed. An unsolved difficulty of this usual strategy (originally proposed by Weinberg) is the estimation of the error introduced at a given step. Moreover, without performing further approximations, it is very hard to reach high orders of the procedure. The study of its convergence is thus a difficult task. In the present work we analyse this problem using a different approximation scheme [16]: instead of considering a finite number of vertices, this procedure includes all of them, although approximately. Within this context, it is possible to estimate the error of the Weinberg approximation, order by order. To do so one can perform, on top of the approximation presented in [16] , the usual truncation in the number of vertices. The analysis has been done in the particular case of the 2-point function of the scalar field theory in d = 3 at criticality. It has been shown [17] that, at least in this case, the procedure proposed in [16] yields very precise results. Another interesting outcome of the present work follows from the fact that, within the approximation [16], truncation in the number of vertices is equivalent to an expansion in powers of a constant external field. The latter is usually employed in the DE context in order to deal with complicated situations. The analysis of the present paper generalizes this expansion procedure when non zero external momentum are involved.\n\nThe calculation of the 2-point function demands the study of both the p = 0 and the p = 0 sectors. While the first one is given by the well studied DE flow equations, the latter follows from the approximation scheme introduced in [16] to calculate the flow of Σ κ (p; ρ).\n\nWe used two different strategies to perform the field expansion, both of them around zero external field: either expanding only the flow equation for the self-energy (and its derivative) (strategy I), or both the effective potential and the self-energy (and its derivative) flow equations (strategy II). We have studied the convergence of various quantities measuring physical properties of the self-energy in all momenta regimes: the critical exponents η and ν of the infrared regime, the coefficient of the ultraviolet logarithm, and ∆ φ 2 which is dominated by the crossover momenta regime.\n\nAs stated in section III, the strategy that can be compared to the usual truncation which includes a finite number of vertices is strategy II. For example, including completely the 4-point vertex as it is usually done (i.e., in the language of field expansion, going only up to 17 the first order of the expansion), when describing the deep infrared regime one could make errors as big as 60% in the critical exponent η (see figure 4 ). If one wants results with less that 5% error for this quantity, the inclusion of up to 8-point vertices (i.e., going up to third order) is necessary.\n\nHowever, when going to higher orders in the field expansion, the series for all considered quantities seem to converge rapidly, within both strategies. The convergence is faster when using strategy I, i.e., when making the expansion only for the approximate flow equation\n\nresulting from the method presented in [16] . For example, using strategy I, a third order truncation introduces a relative error smaller that 1% for all studied quantities; while using strategy II, in order to reach the same error one needs 6th order for η, 4th order for ν and 2nd order for ∆ φ 2 . Nevertheless, due to numerical difficulties, if trying to go to high order expansions, it is preferable to use strategy II, i.e., expanding also the effective potential flow equation.\n\nIt is difficult to assess the generality of these results on the use of field expansion on top of the strategy proposed in [16] . Of course, there are situations where expanding in an external field is not a priori convenient. One can mention as a first example, situations where there is a physical external field (as in a broken phase or when an external source for the field is considered). A second example is two-dimensional systems where even in the DE, the field expansion does not seem to converge. Nevertheless, the short study presented in the present paper allows to consider field expansion on top of the approximation proposed in [16] as a possible strategy to deal with many involved models, as for example QCD.\n\n[1] K. G. Wilson and J. B. Kogut, Phys. Rept. 12, 75 (1974). [2] J. Polchinski, Nucl. Phys. B 231, 269 (1984). [3] C.Wetterich, Phys. Lett., B301, 90 (1993). [4] U.Ellwanger, Z.Phys., C58, 619 (1993). [5] T.R.Morris, Int. J. Mod. Phys., A9, 2411 (1994). [6] C. Bagnuls and C. Bervillier, Phys. Rept. 348, 91 (2001). [7] J. Berges, N. Tetradis and C. Wetterich, Phys. Rept. 363 (2002) 223. [8] B. Delamotte, D. Mouhanna, M. Tissier, Phys.Rev. B69 (2004) 134413.\n\n[9] B. Delamotte, cond-mat/0702365. [10] G.R. Golner, Phys. Rev. B33, (1986) 7863. [11] S. Weinberg, Phys. Rev. D8 (1973) 3497. [12] U. Ellwanger, M. Hirsch and A. Weber, Eur. Phys. J. C 1 (1998) 563; J. M. Pawlowski, D. F. Litim, S. Nedelko and L. von Smekal, Phys. Rev. Lett. 93 (2004) 152002; J. Kato, arXiv:hep-th/0401068; C. S. Fischer and H. Gies, JHEP 0410 (2004) 048; S. Ledowski, N. Hasselmann, P. Kopietz, Phys. Rev. A69, 061601(R) (2004); ibid, Phys. Rev. A70, 063621 (2004) J. P. Blaizot, R. Mendez Galain and N. Wschebor, Phys. Rev. E74 051116, 2006; ibid E74:051117, 2006. [13] U. Ellwanger and C. Wetterich, Nucl. Phys. B423 (1994), 137. [14] Ulrich Ellwanger, Z. Phys. C62 (1994) 503-510.\n\n[15] J. P. Blaizot, R. Mendez Galain and N. Wschebor, Europhys. Lett., 72 (5), 705-711 (2005). [16] J. P. Blaizot, R. Mendez Galain and N. Wschebor, Phys.Lett.B632:571-578,2006. [17] J. P. Blaizot, R. Mendez Galain and N. Wschebor, Non Perturbative Renormalization Group calculation of the scalar self-energy, arXiv:hep-th/0605252, to appear in Eur. Jour. of Phys. B. [18] G. Zumbach, Nucl. Phys. B 413, 754 (1994). [19] L.Canet, B.Delamotte, D.Mouhanna and J.Vidal, Phys. Rev. D67 065004 (2003). [20] D.Litim, Phys. Lett. B486, 92 (2000); Phys. Rev. D64, 105007 (2001); Nucl. Phys. B631, 128 (2002); Int.J.Mod.Phys. A16, 2081 (2001). [21] N. Tetradis and C. Wetterich, Nucl. Phys. B 422 (1994) 541. [22] T. R. Morris, Phys. Lett. B 334 (1994) 355. [23] L. Canet, B. Delamotte, D. Mouhanna and J. Vidal, Phys. Rev. B68 (2003) 064421. [24] K. I. Aoki, K. Morikawa, W. Souma, J. I. Sumi and H. Terao, Prog. Theor. Phys. 99, 451 (1998) [25] D. F. Litim, Nucl. Phys. B 631, 128 (2002) [arXiv:hep-th/0203006]. [26] N. Tetradis and C. Wetterich, Nucl. Phys. B 383 (1992) 197.\n\n[27] B. Delamotte, D. Mouhanna and M. Tissier, private comunication. [28] G. Tarjus, M. Tissier, Phys. Rev. Lett. 93 (2003) 267008; L. Canet, B. Delamotte, O. Deloubriere, N. Wschebor, Phys. Rev. Lett. 92 (2004) 195703; L. Canet, cond-mat/0509541. [29] T.R.Morris, Phys. Lett. B329, 241 (1994).\n\n[30] B. Boisseau, P. Forgacs and H. Giacomini, J. Phys. A 40 (2007) F215 [arXiv:hep-th/0611306]. [31] R. Guida, J. Zinn-Justin, J. Phys. A31, 8103 (1998). [32] G. Baym, J.-P. Blaizot, M. Holzmann, F. Laloë, and D. Vautherin, Phys. Rev. Lett. 83, 1703 (1999)."
}
] | |
arxiv:0704.0262 | 0704.0262 | 1 | 10.1088/1126-6708/2007/06/067 | 19cd9dce875df2ee7b54bc584bd0d87527c95d622d0c83b18bce087e2c3afc07 | Stringy Instantons at Orbifold Singularities | We study the effects produced by D-brane instantons on the holomorphic
quantities of a D-brane gauge theory at an orbifold singularity. These effects
are not limited to reproducing the well known contributions of the gauge theory
instantons but also generate extra terms in the superpotential or the
prepotential. On these brane instantons there are some neutral fermionic
zero-modes in addition to the ones expected from broken supertranslations. They
are crucial in correctly reproducing effects which are dual to gauge theory
instantons, but they may make some other interesting contributions vanish. We
analyze how orientifold projections can remove these zero-modes and thus allow
for new superpotential terms. These terms contribute to the dynamics of the
effective gauge theory, for instance in the stabilization of runaway
directions. | [
"Riccardo Argurio",
"Matteo Bertolini",
"Gabriele Ferretti",
"Alberto Lerda\n and Christoffer Petersson"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | Contents 1. Introduction 1 2. Preliminaries 3 3. The N = 1 Z 2 × Z 2 orbifold 7 3.1 Instanton sector 10 3.2 Recovery of the ADS superpotential 11 3.3 Absence of exotic contributions 14 3.4 Study of the back-reaction 16
4. The N = 1 Z 2 × Z 2 orientifold 17 4.1 Instanton sector 19 5. An N = 2 example: the Z 3 orientifold 21 5.1 Instanton sector 24 6. Conclusions 25 1. Introduction
It has long been realized that instantons in string theory are often in close correspondence with instantons in gauge theories [1, 2, 3, 4, 5, 6] . Recently it was found that in some situations stringy instantons can dynamically generate some terms which from a low-energy effective point of view enter as ordinary external couplings in the superpotential of gauge theories living on space-filling branes [7, 8, 9, 10, 11, 12, 13, 14] . By instantons in string theory we generally mean instantons which are geometrically realized as Euclidean extended objects wrapped on some non-trivial cycles of the geometry. Thus, in a sense, a stringy instanton has a "life of its own", not requiring an underlying gauge theory. This opens up the possibility of having contributions originating from instantons that do not admit a standard gauge theory realization. We shall refer to these instantons as exotic.
1 There has been some debate in the recent literature about the instances where such exotic instantons can actually contribute to the gauge theory superpotential in a non-trivial manner. In this work we will contribute to such a debate by considering backgrounds where a simple CFT description is possible, such as orbifolds or orientifolds thereof.
We present various simple examples of what we believe to be a rather generic situation. Namely, the presence of extra zero-modes for these instantons, in addition to those required by the counting of broken symmetries, makes some of their contributions vanish. Such extra zero-modes should not come as a surprise, since a D-brane instanton in a CY manifold breaks a total of four out of eight supercharges, i.e. it has two extra fermionic zero-modes from the point of view of holomorphic N = 1 gauge theory quantities. We give some arguments as to why the backreaction of the space-filling branes on the geometry might not help in lifting these extra zero-modes. We further argue that only more radical changes of the background, such as the introduction of fluxes, deformations of the CY geometry or the introduction of orientifold planes, can remove these zero-modes. When this happens, exotic instantons do contribute to the gauge theory superpotential and may provide qualitative changes in the low energy effective dynamics, as for instance the stabilization of otherwise runaway directions.
We will be interested in Euclidean D-branes in type II theories. We will work with IIB fractional branes at orbifold and orientifold singularities rather than type IIA wrapped branes. The motivation for this choice of setting is two-fold. First, recent advances in the gauge/gravity correspondence require the study of exotic instantons, whose effects tend to stabilize the gauge theory rather than unstabilize it [15, 16, 9, 17] , and the gauge/gravity correspondence is more naturally defined in the context of IIB theory. Second, similar effects are used in string phenomenology to try to understand possible mechanisms for neutrino masses [7, 8, 13] . This latest activity is mainly done in the type IIA scenario, but we find it easier to address some subtle issues in the IIB orbifold case.
While working in an exact string background, our considerations will nonetheless be only local, i.e. we will not be concerned with global issues such as tadpole cancellation that arise in proper compactifications. This is perfectly acceptable in the context of the gauge/gravity correspondence where the internal manifold is non-compact but, even for string phenomenology, the results we obtain stand (locally) when properly embedded in a consistent compactification.
The paper is organized as follows: In section 2 we set up the notation and discuss some preliminary material. In section 3 we discuss our first case, namely the N = 1 Z 2 × Z 2 orbifold. After briefly recovering the usual instanton generated corrections to the superpotential we discuss the possible presence of additional exotic contributions 2 and find that they are not present because of the additional zero-modes. We conclude by giving a CFT argument on why such zero-modes are not expected to be lifted even by taking into account the backreaction of the D-branes, unless one is willing to move out the orbifold point in the CY moduli space. Sections 4 and 5 present two separate instances where exotic contributions are present after having removed the extra zero-modes by orientifolding. The first is an N = 1 orientifold, the second is an N = 2 orientifold, displaying corrections to the superpotential and the prepotential, respectively. We end with some conclusions and a discussion of further developments.
In this section we briefly review the generic setup in the well understood N = 4 situation in order to introduce the notation for the various fields and moduli and their couplings. The more interesting theories we will consider next will be suitable projections of the N = 4 theory. In fact, the exotic cases can all be reduced to orbifolds/orientifolds of this master case once the appropriate projections on the Chan-Paton factors are performed.
Since we are interested in instanton physics (for comprehensive reviews see [18] and the recent [19]) we will take the ten dimensional metric to be Euclidean. We consider a system where both D3-branes and D(-1)-branes (D-instantons) are present. To be definite, we take N D3's and k D-instantons 1 .
Quite generically we can distinguish three separate open string sectors:
• The gauge sector, made of those open strings with both ends on a D3-brane. We assume the brane world-volumes are lying along the first four coordinates x µ and are orthogonal to the last six x a . The massless fields in this sector form an N = 4 SYM multiplet [22] . We denote the bosonic components by A µ and X a . Written in N = 1 language this multiplet is formed by a gauge superfield whose field strength is denoted by W α and three chiral superfields Φ 1,2,3 . With a slight abuse of notation, the bosonic components of the chiral superfields will also be denoted by Φ, i.e. Φ 1 = X 4 + iX 5 and so on. In N = 2 language we have instead a gauge superfield A and a hypermultiplet H, all in the adjoint representation. The low energy action of these fields is a four dimensional N = 4 gauge theory. All these fields are N × N matrices for a gauge group SU(N).
1 These D3/D(-1) brane systems (and their orbifold projections) are very useful and efficient in studying instanton effects from a stringy perspective even in the presence of non-trivial closed string backgrounds, both of NS-NS type [20] and of R-R type [21] .
• The neutral sector, which comprises the zero-modes of strings with both ends on the D-instantons. It is usually referred to as the neutral sector because these modes do not transform under the gauge group. The zero-modes are easily obtained by dimensionally reducing the maximally supersymmetric gauge theory to zero dimensions. We will use an ADHM [23] inspired notation [5, 6] . We denote the bosonic fields as a µ and χ a , where the distinction between the two is made by the presence of the D3-branes. The fermionic zero-modes are denoted by M αA and λ αA , where α and α denote the (positive and negative) four dimensional chiralities and A is an SU(4) (fundamental or anti-fundamental) index denoting the chirality in the transverse six dimensions. The ten dimensional chirality of both fields is taken to be negative. In Euclidean space M and λ must be treated as independent. When needed, we will also introduce the triplet of auxiliary fields D c , directly analogous to the four dimensional D, that can be used to express the various interactions in an easier form as we will see momentarily. All these fields are k × k matrices where k is the instanton number.
• The charged sector, comprising the zero-modes of strings stretching between a D3-brane and a D-instanton. For each pair of such branes we have two conjugate sectors distinguished by the orientation of the string. In the NS sector, where the world-sheet fermions have opposite modding as the bosons, we obtain a bosonic spinor ω α in the first four directions where the GSO projection picks out the negative chirality. In the conjugate sector, we will get an independent bosonic spinor ω α of the same chirality. Similarly, in the R sector, after the GSO projection we obtain a pair of independent fermions (one for each conjugate sector) both in the fundamental of SU(4) which we denote by µ A and μA . These fields are rectangular matrices N × k and k × N.
The couplings of the fields in the gauge sector give rise to a four dimensional gauge theory. The instanton corrections to such a theory are obtained by constructing the Lagrangian describing the interaction of the gauge sector with the charged sector zeromodes while performing the integral over all zero-modes, both charged and neutral. A crucial point to notice and which will be important later is that while the neutral modes do not transform under the gauge group, their presence affects the integral because of their coupling to the charged sector.
The part of the interaction involving only the instanton moduli is well known from the ADHM construction and it is essentially the reduction of the interacting gauge Lagrangian for these modes in a specific limit where the Yukawa terms for λ and the quadratic term for D are scaled out (see [18, 6] for details). The final form of this part 4 of the interaction is: S 1 = tr -[a µ , χ a ] 2 + χ a ω αω αχ a + i 2 ( Σa ) AB μA µ B χ a -i 4 ( Σa ) AB M αA [χ a , M B α ] + i μA ω α + ω αµ A + σ µ β α[M βA , a µ ] λ α A -iD c ω α(τ c ) β αω β + iη c µν [a µ , a ν ] (2.1) where the sum over colors and instanton indices is understood. τ denotes the usual Pauli matrices, η (and η) the 't Hooft symbols and Σ (and Σ) are used to construct the six-dimensional gamma-matrices Γ a = 0 Σ a Σa 0 . (2.2) The above interactions can all be understood in terms of string diagrams on a disk with open string vertex operators inserted at the boundary in the α ′ → 0 limit. The interaction of the charged sector with the scalars of the gauge sector can be worked out in a similar way and yields
S 2 = tr ω αX a X a ω α + i 2 ( Σa ) AB μA X a µ B . (2.
3) Let us rewrite the above action in a way which will be more illuminating in the following sections. Since we will be mainly focusing on situations where we have N = 1 supersymmetry, it is useful to write explicitly all indices in SU(4) notation, and then break them into SU(3) representations. We thus write the six scalars X a as the antisymmetric representation of SU(4) as follows
X AB = -X BA ≡ ( Σa ) AB X a . (2.4)
The action S 2 then reads
S 2 = tr 1 8 ǫ ABCD ω αX AB X CD ω α + i 2 μA X AB µ B . (2.5)
Splitting now the indices A into i = 1 . . . 3 and 4, we can identify Φ † i ≡ X i4 in the 3 of SU(3) and Φ i ≡ 1 2 ǫ ijk X jk in the 3 of SU(3). Thus we can rewrite the action (2.5) as
S 2 = tr 1 2 ω α Φ i , Φ † i ω α + i 2 μi Φ † i µ 4 - i 2 μ4 Φ † i µ i - i 2 ǫ ijk μi Φ j µ k . (2.6)
In the above form, it is clear which zero-modes couple to the holomorphic superfields and which others couple to the anti-holomorphic ones. This distinction will play an important role later.
The main object of our investigation is the integral of e -S 1 -S 2 over all moduli Z = C d{a, χ, M, λ, D, ω, ω, µ, μ} e -S 1 -S 2 , (2.7)
where we have lumped all field independent normalization constants (including the instanton classical action and the appropriate powers of α ′ required by dimensional analysis) into an overall coefficient C. There are, of course, other interactions involving the fermions and the gauge bosons but, as far as the determination of the holomorphic quantities are concerned, they can be obtained from the previous ones and supersymmetry arguments. For example, a term in the superpotential is written as the integral over chiral superspace dx 4 dθ 2 of a holomorphic function of the chiral superfields, but such a function is completely specified by its value for bosonic arguments at θ = 0. Thus, if we can "factor out" a term dx 4 dθ 2 from the moduli integral (2.7), whatever is left will define the complex function to be used in the superpotential and similarly for the prepotential in the N = 2 case if we succeed in factoring out an integral over N = 2 chiral superspace dx 4 dθ 4 .
The coordinates x and θ must of course come from the (super)translations broken by the instanton and they will be associated to the center of mass motion of the D-instanton, namely, x µ = tr a µ and θ αA = tr M αA for some values of A. 2 One must pay attention however to the presence of possible additional neutral zero-modes coming either from the traceless parts of the above moduli or from the fields λ and χ. These modes must also be integrated over in (2.7) and their effects, as we shall see, can be quite dramatic. In particular, the presence of λ in some instances is crucial for the implementation of the usual ADHM fermionic constraints whereas in other circumstances it makes the whole contribution to the superpotential vanish. These extra λ zero-modes are ubiquitous in orbifold theories and generically make it difficult to obtain exotic instanton corrections for these models. As we shall see, they can however be easily projected out by an orientifold construction making the derivation of such terms possible.
In the full expression for the instanton corrections there will also be a field-independent normalization factor coming from the one-loop string diagrams and giving for instance the proper g Y M dependence in the case of the usual instanton corrections. In this paper we will only focus on the integral over the zero-modes, which gives the proper field-dependence, referring the reader to [10, 11] for a discussion of these other issues.
2 Obviously, for the case of an anti-instanton, the roles of M and λ are reversed.
3. The N = 1 Z 2 × Z 2 orbifold
In order to present a concrete example of the above discussion, let us study a simple C 3 /Z 2 × Z 2 orbifold singularity. The resulting N = 1 theory is a non-chiral four-node quiver gauge theory with matter in the bi-fundamental. Non-chirality implies that the four gauge group ranks can be chosen independently [24] . This corresponds to being able to find a basis of three independent fractional branes in the geometry (for a review on fractional branes on orbifolds see e.g. [25]).
The field content can be conveniently summarized in a quiver diagram, see Fig. 1 , which, together with the cubic superpotential
W = Φ 12 Φ 23 Φ 31 -Φ 13 Φ 32 Φ 21 + Φ 13 Φ 34 Φ 41 -Φ 14 Φ 43 Φ 31 +Φ 14 Φ 42 Φ 21 -Φ 12 Φ 24 Φ 41 + Φ 24 Φ 43 Φ 32 -Φ 23 Φ 34 Φ 42 , (3.1)
uniquely specifies the theory.
SU(N ) SU(N ) SU(N ) SU(N ) 1 2 3 4 Figure 1: Quiver diagram for the Z 2 × Z 2 orbifold theory. Round circles correspond to SU(N ℓ ) gauge factors while the lines connecting quiver nodes represent the bi-fundamental chiral superfields Φ ℓm .
A stack of N regular D3-branes amounts to having one and the same rank assignment on the quiver. The gauge group is then SU(N) 4 and the theory is an N = 1 SCFT. Fractional branes correspond instead to different (but anomlay free) rank assignments. Quite generically, fractional branes can be divided into three different classes, depending on the IR dynamics they trigger [26] . The non-chiral nature and the particularly symmetric structure of the orbifold under consideration allows one to easily construct any such instance of fractional brane class.
If we turn on a single node, we are left with a pure SU(N) SYM gauge theory, with no matter fields and no superpotential. This theory is believed to confine. The geometric dual effect is that the corresponding fractional brane leads to a geometric 7 transition where the branes disappear leaving behind a deformed geometry. Indeed, there is one such deformation in the above singularity.
Turning on two nodes leads already to more varied phenomena. There are now two bi-fundamental superfields, but still no tree level superpotential. Thus, the system is just like two coupled massless SQCD theories or, by a slightly asymmetric point of view, massless SQCD with a gauged diagonal flavor group. The low-energy behavior depends on the relative ranks of the two nodes.
If the ranks are different, the node with the highest rank is in a situation where it has less flavors than colors. Then an Affleck-Dine-Seiberg (ADS) superpotential [27, 28] should be dynamically generated, leading eventually to a runaway behavior. This set up of fractional branes is sometimes referred to as supersymmetry breaking fractional branes [29, 26, 30] .
If the ranks are the same we are in a situation similar to N f = N c SQCD for both nodes. Hence we expect to have a moduli space of SUSY vacua, which gets deformed, but not lifted, at the quantum level. This moduli space is roughly identified in the geometry with the fact that the relevant fractional branes are interpreted as D5-branes wrapped on the 2-cycle of a singularity which is locally C × (C 2 /Z 2 ). Such a fractional brane can move in the C direction. This is what is called an N = 2 fractional brane since, at least geometrically, it resembles very much the situation of fractional branes at N = 2 singularities.
In what follows we use the two-node example as a simple setting in which we can analyze the subtleties involved in the integration over the neutral modes. For the gauge theory instanton case it is known that there are extra neutral fermionic zero-modes in addition to those required to generate the superpotential. Their integration allows to recover the fermionic ADHM constraints on the moduli space of the usual field theory instantons. For such instantons, we will be able to obtain the ADS superpotential and corresponding runaway behavior in the familiar context with N c and N f fractional branes at the respective nodes, for N f = N c -1. On the other hand, we will argue that the presence of such extra zero-modes rules out the possibility of having exotic instanton effects, such as terms involving baryonic operators in the N f = N c case. It was the desire to study such possible contributions that constituted the original motivation for this investigation. We will first show that such effects are absent for this theory as it stands, and we will later discuss when and how this problem can be cured. 3 3 In a situation where the CFT description is less under control than in the setting discussed in the present paper, it has been argued in [17] that such baryonic couplings do arise in the context of fractional branes on orbifolds of the conifold, possibly at the expense of introducing O-planes. Also in a IIA set up similar to the ones of [7, 8, 10, 11, 13] it seems reasonable that one can wrap an ED2-brane along an O6-plane and produce such couplings on other intersecting D6-branes.
Our orbifold theory can be easily obtained as an orbifold projection of N = 4 SYM. The orbifolding procedure and the derivation of the superpotential (3.1) are by now standard. We briefly recall the main points in order to fix the notation and because some of the details will be useful later in describing the instantons in such a set up.
The group Z 2 × Z 2 has four elements: the identity e, the generators of the two Z 2 that we denote with g 1 and g 2 and their product, denoted by
g 3 = g 1 g 2 . If we introduce complex coordinates (z 1 , z 2 , z 3 ) ∈ C 3 z 1 = x 4 + ix 5 , z 2 = x 6 + ix 7 , z 3 = x 8 + ix 9 (3.2)
the action of the orbifold group can be defined as in Table 1 .
z 1 z 2 z 3 e z 1 z 2 z 3 g 1 z 1 -z 2 -z 3 g 2 -z 1 z 2 -z 3 g 3 -z 1 -z 2 z 3
Table 1: The action of the orbifold generators.
Let γ(g) be the regular representation of the orbifold group on the Chan-Paton factors. If the orbifold is abelian, as always in the cases we shall be interested in, we can always diagonalize all matrices γ(g). We will assume that the two generators have the following matrix representation
γ(g 1 ) = σ 3 ⊗ 1 = 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1 , γ(g 2 ) = 1 ⊗ σ 3 = 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1 (3.3)
where the 1's denote N ℓ × N ℓ unit matrices (ℓ = 1, ..., 4). Then, the orbifold projection amounts to enforcing the conditions
A µ = γ(g)A µ γ(g) -1 , Φ i = ±γ(g)Φ i γ(g) -1 (3.4)
where the sign ± must be chosen according to the action of the orbifold generators g that can be read off from Table 1 . With the choice (3.3), the vector superfields are block diagonal matrices of different size (N 1 , N 2 , N 3 , N 4 ), one for each node of the 9 quiver, while the three chiral superfields Φ i have the following form [24]
Φ 1 = 0 × 0 0 × 0 0 0 0 0 0 × 0 0 × 0 , Φ 2 = 0 0 × 0 0 0 0 × × 0 0 0 0 × 0 0 , Φ 3 = 0 0 0 × 0 0 × 0 0 × 0 0 × 0 0 0 , (3.5)
where the crosses represent the non-zero entries Φ ℓm appearing in the superpotential (3.1).
Now consider D-instantons in the above set up. Such instantons preserve half of the 4 supercharges preserved by the system of D3-branes plus orbifold. In this respect recall that the fractional branes preserve exactly the same supercharges as the regular branes. 4 Using the N = 4 construction of the previous section and the structure of the orbifold presented in eq. (3.5), we now proceed in describing the zero-modes for such instantons. The neutral sector is very similar to the gauge sector. Indeed, in the (-1) superghost picture, the vertex operators for such strings will be exactly the same, except for the e ip•X factor which is absent for the instanton. The Chan-Paton structure will also be the same, so that the same pattern of fractional D-instantons will arise as for the fractional D3-branes. In particular, the only regular D-instanton (which could be thought of as deriving from the one of N = 4 SYM) is the one with rank (instanton number) one at every node. All other situations can be thought of as fractional Dinstantons, which can be interpreted as Euclidean D1-branes wrapped on the two-cycles at the singularity, ED1 for short. Generically, we can then characterize an instanton configuration in our orbifold by (k
1 , k 2 , k 3 , k 4 ).
Following the notation introduced in section 2, the bosonic modes will comprise a 4 × 4 block diagonal matrix a µ , and six more matrix fields χ 1 , . . . χ 6 , that can be paired into three complex matrix fields χ 1 +iχ 2 , χ 3 +iχ 4 , χ 5 +iχ 6 , having the same structure as (3.5) but now where each block entry is a k ℓ × k m matrix. On the fermionic zero-modes M αA and λ αA (also matrices) the orbifold projection enforces the conditions
M αA = R(g) A B γ(g)M αB γ(g) -1 , λ αA = γ(g)λ αB γ(g) -1 R(g) B A (3.6) 4
There is another Euclidean brane which preserves two supercharges, namely the Euclidean (anti) D3-branes orthogonal to the 4 dimensions of space-time. We will be considering here only the Dinstantons, leaving the complete analysis of the other effects to future work. In this context, note that the extended brane instantons would have an infinite action (and thus a vanishing contribution) in the strict non-compact set up we are using here.
where R(g) is the orbifold action of Table 1 in the spinor representation which can be chosen as R(g 1 ) = -Γ 6789 , R(g 2 ) = -Γ 4589 . (3.7) It is easy to find an explicit representation of the Dirac matrices such that M αA and λ αA for A = 1, 2, 3 also have the structure of (3.5) while for A = 4 they are block diagonal.
Equivalently, one could write the spinor indices in the internal space in terms of the three SO(2) charges associated to the embedding SO(2) × SO(2) × SO(2) ⊂ SO(6) ≃ SU(4)
M α-++ = M α1 , M α+-+ = M α2 , M α++-= M α3 , M α---= M α4 , λ α+--= λ α1 , λ α-+-= λ α2 , λ α--+ = λ α3 , λ α+++ = λ α4 . (3.8)
The most notable difference between the neutral sector and the gauge theory on the D3-branes is that, whereas in the four-dimensional theory the U(1) gauge factors are rendered massive by a generalization of the Green-Schwarz mechanism and do not appear in the low energy action, for the instanton they are in fact present and enter crucially into the dynamics. Let us finally turn to the charged sector, describing strings going from the instantons to the D3-branes. The analysis of the spectrum and the action of the orbifold group on the Chan-Paton factors show, in particular, that the bosonic zero-modes are diagonal in the gauge factors. There are four block diagonal matrices of bosonic zeromodes ω α, ω α with entries N ℓ ×k ℓ and k ℓ ×N ℓ respectively and eight fermionic matrices µ A , μA with entries N ℓ × k m and k m × N ℓ , that again display the same structure as above -same as (3.5) for A = 1, 2, 3 and diagonal for A = 4.
The measure on the moduli space of the instantons and the ADHM constraints are simply obtained by inserting the above expressions into the moduli integral (2.7). If one chooses some of the N ℓ or k ℓ to vanish one can deduce immediately from the structure of the projection which modes will survive and which will not.
As a consistency check, one can try to reproduce the ADS correction to the superpotential [27, 28] for the theory with two nodes. Take fractional branes corresponding to a rank assignment (N c , N f , 0, 0), and consider the effect of a ED1 corresponding to instanton numbers (1, 0, 0, 0) .
The only chiral fields present are the two components of Φ 1 connecting the first 11 and second node
Φ 1 = 0 Q 0 0 Q 0 0 0 0 0 0 0 0 0 0 0 . (3.9)
Since the instanton is sitting only at one node, all off diagonal neutral modes are absent, as they connect instantons at two distinct nodes. Thus, the only massless modes present in the neutral sector are four bosons x µ , denoting the upper-left component of a µ , two fermions θ α denoting the upper-left component of M α4 and two more fermions λ α denoting the upper-left component of λ α4 . We have identified the non zero entries of a µ and M α4 with the super-coordinates x µ and θ α since they precisely correspond to the Goldstone modes of the super-translation symmetries broken by the instanton and do not appear in S 1 + S 2 (cfr. (2.1) and (2.3)). Their integration produces the integral over space-time and half of Grassmann space which precedes the superpotential term to which the instanton contributes. On the contrary, λ α appears in S 1 and when it is integrated it yields the fermionic ADHM constraint.
In the charged sector, we have bosonic zero-modes ω u α and ω αu , with u an index in the fundamental or anti-fundamental of SU(N c ). In addition, there are fermionic zero-modes µ u and μu with indices in SU(N c ), together with additional fermionic zeromodes µ ′f and μ′ f where the index f is now in the fundamental or anti-fundamental of SU(N f ). 5 Note that the µ zero-modes carry an SU(4) index 4 (being on the diagonal) while the µ ′ zero-modes carry an SU(4) index 1, since they are of the same form as Φ 1 .
All this can be conveniently summarized in a generalized quiver diagram as represented in Fig. 2 , which accounts for both the brane configuration and the instanton zero-modes.
For a single instanton, the action (2.1) greatly simplifies since many fields are vanishing as well as all commutators and one gets
S 1 = i (μ u ω u α + ω αu µ u ) λ α -iD c ω α u (τ c ) β αω u β . (3.10)
Similarly, the coupling of the charged modes to the chiral superfield can be expressed by writing eq. (2.3) as
S 2 = 1 2 ω αu Q u f Q †f v + Q †u f Qf v ω αv - i 2 μu Q †u f µ ′f + i 2 μ′ f Q †f u µ u . (3.11)
Note that it is the anti-holomorphic superfields that enter in the couplings with the fermionic zero-modes, as is clear by comparing with (2.6). The above action is exactly the same which appears in the ADHM construction as reviewed in [18] .
5 Recall that the bosonic zero-modes are diagonal in the gauge factors; therefore there are no ω f α and ω αf zero-modes.
12 c 1 E ω Q Q µ µ λ ~µ µ SU(N ) SU(N ) f _ ' _ ' ω _
Figure 2: Quiver diagram describing an ordinary instanton in a SU(N c ) × SU(N f ) theory. Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1 is wrapped on the same cycle as the color branes. All zero-modes are included except the θ's and the x µ 's, which only contribute to the measure for the integral over chiral superspace.
We are now ready to perform the integral (2.7) over all the existing zero-modes. Writing (3.12) we see that the instanton induced superpotential is
Z = dx 4 dθ 2 W ,
W = C d{λ, D, ω, ω, µ, μ} e -S 1 -S 2 . (3.13)
The integrals over D and λ enforce the bosonic and fermionic ADHM constraints, respectively. Thus
W = C d{ω, ω, µ, μ} δ(μ u ω u α + ω αu µ u ) δ(ω α u (τ c ) β αω u β ) e -S 2 . (3.14)
We essentially arrive at the point of having to evaluate an integral over a set of zeromodes which is exactly the same as the one discussed in detail in the literature, e.g. [18] . We thus quickly go to the result referring the reader to the above review for further details. First of all, it is easy to see that, due to the presence of extra µ modes in the integrand from the fermionic delta function, only when N f = N c -1 we obtain a non-vanishing result. After having integrated over the µ and µ ′ , we are left with a (constrained) gaussian integration that can be performed e.g. by going to a region 13 of the moduli space where the chiral fields are diagonal, up to a row/column of zeroes. Furthermore, the D-terms in the gauge sector constrain the quark superfields to obey QQ † = Q † Q, so that the bosonic integration brings the square of a simple determinant in the denominator. The last fermionic integration conspires to cancel the anti-holomorphic contributions and gives
W ADS = Λ 2Nc+1 det( QQ) , (3.15)
which is just the expected ADS superpotential for N f = N c -1, the only case where such non-perturbative contribution is generated by a genuine one-instanton effect and not by gaugino condensation. In (3.15) Λ is the SQCD strong coupling scale that is reconstructed by the combination of e -8π 2 /g 2 coming from the instanton action with various dimensional factors coming from the normalization of the instanton measure [18].
Until now, we have reproduced from stringy considerations the effect that is supposed to be generated also by instantons in the gauge theory. Considering a slightly different set up, we would like to study the possibility of generating other terms.
Let us consider a system with rank assignment (N c , N f , 0, 0), as before, but fractional instanton numbers (0, 0, 1, 0) . In other words, we study the effect of a single fractional instanton sitting on an unoccupied node of the gauge theory. The quiver diagram, with the relevant zero-modes structure, is given in Fig. 3 .
The neutral zero-modes of the instanton sector are the same as before. This is because the quantization of this sector does not know the whereabouts of the D3branes and thus all nodes are equivalent, in this respect. In the mixed sector, we have no bosonic zero-modes now, since the ω and ω are diagonal. Note that, although we always have four mixed (ND) boundary conditions, due to the quiver structure induced by the orbifold, here we effectively realize the same situation one has when there are eight ND directions, namely that the bosonic sector of the charged moduli is empty.
On the other hand, there are fermionic zero-modes µ u , μu , µ ′f and μ′ f , as in the previous case. Note that despite having the same name, these zero-modes correspond actually to different Chan-Paton matrix elements with respect to the previous ones, the difference being in the instanton index that is not written explicitly. In particular we can think of µ and µ ′ as carrying an SU(4) index 2 and 3 respectively.
Because of the absence of bosonic charged modes, the action (2.1) is identically 14 c E 1 λ Q Q μ µ µ µ SU(N ) f SU(N ) _ ' _ ' Figure 3: Quiver diagram describing an exotic instanton in a SU(N c ) × SU(N f ) theory. Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1 is wrapped on a different cycle with respect to both sets of quiver branes.
zero and the action (2.3) contains only the last term:
S 1 = 0 S 2 = i 2 μu Q u f µ ′f - i 2 μ′ f Qf u µ u . (3.16)
Note that in this case it is the holomorphic superfields which appear above, as is clear from (2.6) and from noticing that the diagonal fermionic zero-mode µ 4 is not present. We are thus led to consider W = C d{λ, D, µ, μ} e -S 2 . (3.17) One notices right away that the integral over the charged modes is non vanishing (only) for the case N f = N c and gives a tantalizing contribution proportional to B B, where B = det Q and B = det Q are the baryon fields of the theory. However, we must carefully analyze the integration over the remaining zero-modes of the neutral sector. Now neither D nor λ appear in the integrand. The integral over D does not raise any concern: it is, after all, an auxiliary field and its disappearance from the integrand is due to the peculiarities of the ADHM limit. Before taking this limit, D appeared quadratically in the action and could be integrated out, leaving an overall normalization constant. The integral over λ is another issue. In this case, λ is absent from the integrand even before taking the ADHM limit and its integration multiplies 15 the above result by zero, making the overall contribution of such instantons to the superpotential vanishing. Of course, the presence of such extra zero-modes should not come as a surprise since they correspond to the two extra broken supersymmetries of an instanton on a CY. Therefore we see that the neutral zero-modes contribution, in the exotic instanton case, plays a dramatic role and conspires to make everything vanishing (as opposite to the ADS case analyzed before). A natural question is to see whether these zeromodes get lifted by some effect we have not taken into account, yet. For one thing, supersymmetry arguments would make one think that taking into account the backreaction of the D3-branes might change things. However, in the following subsection we show that this seems not to be the case.
Let us stick to the case N f = N c , which is the only one where the integral (3.17) might give a non-vanishing contribution. In this case the fractional brane system is nothing but a stack of (N c ) N = 2 fractional branes. These branes couple to only one of the 3 closed string twisted sectors [24] . More specifically, they source the metric h µν , the R-R four-form potential C µνρσ and two twisted scalars b and c from the NS-NS and R-R sector respectively. This means that the disk one-point function of their vertex operators [31, 32] is non vanishing when the disk boundary is attached to such D3-branes. (Indeed in this way or, equivalently, by using the boundary-state formalism [33, 34] , one can derive the profile for these fields.) If the back-reaction of these fields on the instanton lifted the extra zero-modes λ's, this should be visible when computing the one point function of the corresponding closed string vertex operators on a disk with insertions on this boundary of the vertex operators for such moduli. To see whether such coupling is there, we first need to write down the vertex operators for the λ's in the (±1/2) superghost pictures. The vertex in the (-1/2) picture is found e.g. in [6] and reads V -1/2 λ
(z) = λ αA S α(z)S A (z)e -φ(z)/2 , (3.18)
where S α(z) and S A (z) are the spin-fields in the first four and last six directions respectively. For our argument we need to focus on the S A (z) dependence. Since the modulus that survives the orbifold projection is, with our conventions, λ α4 = λ α+++ , we write the corresponding spin-field as S +++ (z) = e iH 1 (z)/2 e iH 2 (z)/2 e iH 3 (z)/2 , (3.19) where H i (z) is the free boson used to bosonize the fermionic sector in the i-th complex direction: ψ i (z) = e iH i (z) . The vertex operator in the +1/2 picture can be obtained by 16 applying the picture-changing operator to (3.18) V 1/2 λ (z) = [Q BRST , ξV -1/2 λ (z)] . (3.20)
The crucial part in Q BRST is [31]
Q BRST = dz 2πi η e φ ψ µ ∂X µ + ψi ∂Z i + ψ i ∂ Zi + . . . (3.21)
Because of the nature of the supercurrent, we see that (3.21) flips at most one sign in (3.19), hence the product V -1/2 λ V 1/2 λ will always carry an unbalanced charge in some of the three internal SO(2) groups. On the other hand, the vertex operators for the fields sourced by the fractional D3's cannot compensate such an unbalance. Hence, their correlation function on the D-instanton with the insertion of V -1/2 λ V 1/2 λ carries a charge unbalance and therefore vanishes. Therefore, at least within the above perturbative approach, the neutral zero-modes seem not to get lifted by the back-reaction of the D3-branes.
One might consider some additional ingredients which could provide the lifting. A natural guess would be moving in the CY moduli space or adding suitable background fluxes [35, 36] . There are indeed non-vanishing background fields at the orbifold point, i.e. the b fields of the twisted sectors which the N = 2 fractional branes do not couple to. These fields, however, being not associated to geometric deformations of the internal space should be described by a CFT vertex operator uncharged under the SO(2)'s, simply because of Lorentz invariance in the internal space. Therefore, the only way to get an effective mass term for the zero-modes λ would be to move out of the orbifold point in the CY moduli space. Indeed, the other moduli of the NS-NS twisted sector, being associated to geometric blow-ups of the singularity, are charged under (some of) the internal SO(2)'s and can have a non vanishing coupling with the λ's. More generically, complicated closed string background fluxes might be suitable. This is an interesting option which however we do not pursue here, since we want to stick to situations where a CFT description is available.
A more radical thing to do is to remove the zero-modes from the very start, for instance by means of an orientifold projection [37, 38] . This is the option we are going to consider in the remainder of this work.
N = 1 Z 2 × Z 2 orientifold
In this section we supplement our orbifold background by an O3 orientifold and show that in this case exotic instanton contributions do arise and provide new terms in the 17 superpotential. We refer to e.g. [39, 40, 41] for a comprehensive discussion of N = 1 and N = 2 orientifolds.
The first ingredient we need is the action of the O3-plane on the various fields. Denote by Ω the generator of the orientifold. The action of Ω on the vertex operators for the various fields (ignoring for the time being the Chan-Paton factors) is well known. The vertex operators for the bosonic fields on the D3-brane contain, in the 0 picture, the following terms:
A µ ∼ ∂ τ x µ and Φ i ∼ ∂ σ
zi . They both change sign under Ω, the first because of the derivative ∂ τ and the second because the orientifold action for the O3-plane is always accompanied by a simultaneous reflection of all the transverse coordinates z i .
The action of the orientifold on the Chan-Paton factors is realized by means of a matrix γ(Ω) which in presence of an orbifold must satisfy the following consistency condition [39] γ(g)γ(Ω)γ(g) T = + γ(Ω) (4.1) for all orbifold generators g. This amounts to require that the orientifold projection commutes with the orbifold projection. The matrix γ(Ω) can be either symmetric or anti-symmetric. We choose to perform an anti-symmetric orientifold projection on the D3 branes and denote the corresponding matrix by γ -(Ω). This requires having an even number N ℓ of D3 branes on each node of the quiver so that we can write
γ -(Ω) = ǫ 1 0 0 0 0 ǫ 2 0 0 0 0 ǫ 3 0 0 0 0 ǫ 4 (4.2)
where the ǫ ℓ 's are N ℓ × N ℓ antisymmetric matrices obeying ǫ 2 ℓ = -1. Using (3.3) and (4.2) it is straightforward to verify that the consistency condition (4.1) is verified.
The field content of the stacks of fractional D3-branes in this orientifold model is obtained by supplementing the orbifold conditions (3.4) with the orientifold ones
A µ = -γ -(Ω)A T µ γ -(Ω) -1 , Φ l = -γ -(Ω)Φ lT γ -(Ω) -1 . (4.3) This implies that A µ = diag (A 1 µ , A 2 µ , A 3 µ , A 4 µ ) with A ℓ µ = ǫ ℓ A iT µ ǫ ℓ .
Thus, the resulting gauge theory is a USp(N
1 ) × USp(N 2 ) × USp(N 3 ) × USp(N 4 ) theory.
The chiral superfields, which after the orbifold have the structure (3.5), are such that the Φ ℓm component joining the nodes ℓ and m of the quiver, must obey the orientifold condition Φ ℓm = ǫ ℓ Φ T mℓ ǫ m . In the following, we will take N 3 = N 4 = 0 so that we are left with only two gauge groups and no tree level superpotential.
Let us now consider the instanton sector, starting by analyzing the zero-mode content in the neutral sector. There are two basic changes to the previous story. The first is that the vertex operator for a µ is now proportional to ∂ σ x µ , not to ∂ τ x µ and it remains invariant under Ω (the vertex operator for χ a still changes sign). The second is that the crucial consistency condition discussed in [38] requires that we now represent the action of Ω on the Chan-Paton factors of the neutral modes by a symmetric matrix which can be taken to be
γ + (Ω) = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , (4.4)
where the 1's are k ℓ × k ℓ unit matrices. The matrix a µ will be 4 × 4 block diagonal, e.g.
a µ = diag (a 1 µ , a 2 µ , a 3 µ , a 4 µ ), but now a ℓ µ = a ℓT µ .
The most generic situation is to have a configuration with instanton numbers (k 1 , k 2 , k 3 , k 4 ). By considering a configuration with k 3 = 1 and k 1 = k 2 = k 4 = 0, we can project out all bosonic zero-modes except for the four components a 3 µ that we denote by x µ . The scalars χ 4 . . . χ 9 are off-diagonal and we shall not consider them further.
The nice surprise comes when considering the orientifold action on the fermionic neutral zero-modes M αA and λ αA . The orbifold part of the group acts on the spinor indices as in (3.7), while the orientifold projection acts as the reflection in the transverse space, namely R(Ω) = -i Γ 456789 (4.5) Putting together the orbifold projections (3.6) with the orientifold ones
M αA = R A B (Ω)γ + (Ω)(M αB ) T γ + (Ω) -1 , λ αA = γ + (Ω)(λ αB ) T γ + (Ω) -1 R B A (Ω) (4.6)
we can find the spectrum of surviving fermionic zero-modes. Using (4.4) and (4.5), it is easy to see that (4.6) implies
M αA = (M αA ) T , λ αA = -(λ αA ) T . (4.7)
Thus, for the simple case where k 3 = 1 and k 1 = k 2 = k 4 = 0, all λ's are projected out and only two chiral M zero-modes remain: M α---, to be identified with the N = 1 chiral superspace coordinates θ α . Also the charged zero-modes are easy to discuss in this simple scenario. There are no bosonic modes since the D-instanton and the D3-branes sit at different nodes while 19 the bosonic modes are necessarily diagonal. Most of the fermionic zero-modes µ A and μA are also projected out by the orbifold condition µ A = R(g) A B γ(g)µ B γ(g) -1 , μA = R(g) A B γ(g)μ B γ(g) -1 . (4.8) Finally, the orientifold condition relates this time the fields in the conjugate sectors, allowing one to express μ as a linear combination of the µ
μA = R(Ω) A B γ + (Ω)(µ B ) T γ -(Ω) -1 . (4.9)
The only charged modes surviving these projections can be expressed, in block 4 × 4 notation, as
µ 2 = 0 0 µ 13 0 0 0 0 0 0 0 0 0 0 0 0 0 , μ2 = 0 0 0 0 0 0 0 0 μ31 0 0 0 0 0 0 0 , µ 3 = 0 0 0 0 0 0 µ 23 0 0 0 0 0 0 0 0 0 , μ3 = 0 0 0 0 0 0 0 0 0 μ32 0 0 0 0 0 0 , (4.10)
where the entries, to be thought of as column/row vectors in the fundamental/antifundamental of SU(N ℓ ) depending on their position, are such that μ31 = -µ T 13 ǫ 1 and μ32 = -µ T 23 ǫ 2 . Thus, in the case where we have fractional D3 branes (N 1 , N 2 , 0, 0) and an exotic instanton (0, 0, 1, 0) , the only surviving chiral field is Φ 12 ≡ ǫ 1 Φ T 21 ǫ 2 , the orientifold projection eliminates the offending λ's and we are left with just the neutral zero-modes x µ and θ α and the charged ones µ 13 and µ 23 . This is summarized in the generalized quiver of Fig. 4 .
In this case the instanton partition function is
Z = dx 4 dθ 2 W (4.11)
where the superpotential W is
W = C dµ e -S 1 -S 2 = C dµ 13 dµ 23 e iµ T 13 ǫ 1 Φ 12 µ 23 . (4.12)
This integral clearly vanishes unless N 1 = N 2 , in which case we have
W ∝ det(Φ 12 ) (4.13) 20 E 1 µ µ USP(N ) USP(N ) 1 2 Φ Φ 12 21 _ 32 23 13 µ 31 µ _
Figure 4: The generalized Z 2 × Z 2 orientifold quiver and the exotic instanton contribution.
We thus see that exotic instanton corrections are possible in this simple model. 6
It is interesting to note that the above correction is present in the same case (N 1 = N 2 ≡ N) where the usual ADS superpotential for USp(N) is generated [42]
W ADS = Λ 2N +3 det(Φ 12 ) (4.14)
and its presence stabilizes the runaway behavior and gives a theory with a non-trivial moduli space of supersymmetric vacua given by det(Φ 12 ) = const. Of course, the ADS superpotential for this case can also be constructed along the same lines as section 3.2, see e.g. [18] . In fact, this derivation is somewhat simpler than the one for the SU(N) gauge group since there are no ADHM constraints at all in the one instanton case. We think the above situation is not specific to the background we have been considering, but is in fact quite generic. As soon as the λ zero-modes are consistently lifted, we expect the exotic instantons to contribute new superpotential terms. As a further example, in the next section we will consider a N = 2 model, where exotic instantons will turn out to contribute to the prepotential.
Let us now consider the quiver gauge theory obtained by placing an orientifold O3-plane at a C × C 2 /Z 3 orbifold singularity. In what follows we will use N = 1 superspace notation. We first briefly repeat the steps that led to the constructions of such a quiver 6 The gauge invariant quantity above can be rewritten as the Pfaffian of a suitably defined mesonic matrix.
21 theory in the seminal paper [39] . Define ξ = e 2πi/3 and let the generator of the orbifold group act on the first two complex coordinates as g : z 1 z 2 → ξ 0 0 ξ -1 z 1 z 2 , (5.1)
while leaving the third one invariant. This preserves N = 2 SUSY. The action of the generator g on the Chan-Paton factors is given by the matrix
γ(g) = 1 0 0 0 ξ 0 0 0 ξ 2 . (5.2)
The N = 2 theory obtained this way, summarized in Fig. 5 , is a three node quiver gauge theory with gauge groups SU(N 1 ) × SU(N 2 ) × SU(N 3 ), supplemented by a cubic superpotential which is nothing but the orbifold projection of the N = 4 superpotential (its precise form is not relevant for the present purposes).
SU(N ) SU(N ) SU(N ) 1 3 2 Figure 5: The Z 3 (un-orientifolded) theory. The lines with both ends on a single node represent adjoint chiral multiplets which, together with the vector multiplets at each node constitute the N = 2 vector multiplets. Similarly, lines between nodes represent chiral multiplets which pair up into hyper-multiplets, in N = 2 language.
As for the action of Ω on the Chan-Paton factors, we choose again to perform the symplectic projection on the D3-branes. To do so, we must take N 1 to be even and N 2 = N 3 , so that we can write
γ -(Ω) = ǫ 0 0 0 0 1 0 -1 0 , (5.3) 22
where ǫ is a N 1 × N 1 antisymmetric matrix obeying ǫ 2 = -1 and the 1's denote N 2 × N 2 identity matrices. The matrices γ(g) and γ -(Ω) satisfy the usual consistency condition [38, 39] as in (4.1). The field content on the fractional D3-branes at the singularity will be given by implementing the conditions
A µ = γ(g)A µ γ(g) -1 , Φ i = ξ -i γ(g)Φ i γ(g) -1 , A µ = -γ -(Ω)A T µ γ -(Ω) -1 , Φ i = -γ -(Ω)Φ iT γ -(Ω) -1 . (5.4)
The orbifold part of these conditions forces A µ and Φ 3 to be 3 × 3 block diagonal matrices, e.g. A µ = diag (A 1 µ , A 2 µ , A 3 µ ), while the orientifold imposes that A 1 µ = ǫA 1T µ ǫ and A 2 µ = -A 3T µ . The resulting gauge theory is thus a USp(N 1 ) × SU(N 2 ) theory. It is convenient, however, to still denote A 2 µ and A 3 µ diagramatically as belonging to different nodes with the understanding that these should be identified in the above sense.
The projection on the chiral fields can be done similarly and we obtain, denoting by Φ ℓm the non-zero entries of the fields Φ 1 and Φ 2 (only one can be non-zero for each pair ℓm)
Φ 12 = -ǫΦ T 31 , Φ 13 = +ǫΦ T 21 , Φ 23 = Φ T 23 , Φ 32 = Φ T 32 . (5.5)
The field content is summarized in Table 2 .
USp(N 1 ) SU(N 2 ) Φ 12 Φ 21 Φ 13 Φ 31 Φ 23 • Φ 32 •
Table 2: Chiral fields making up the quiver gauge theory.
The theory we want to focus on in the following has rank assignment (N 1 , N 2 ) = (0, N). This yields an N = 2 SU(N) gauge theory with an hyper-multiplet in the symmetric/(conjugate)symmetric representation. We denote the N = 2 vector multiplet by A whose field content in the block 3 × 3 notation is thus
 = 0 0 0 0 A 0 0 0 -A T . (5.6) 23
In what follows we will be interested in studying corrections to the prepotential F coming from exotic instantons associated to the first node (the one that is not populated by D3-branes). Let us then analyze the structure of the stringy instanton sector of the present model, first.
The most generic situation is to have a configuration with instanton numbers (k 1 , k 2 ) (later we will be mainly concerned with a configuration with instanton numbers (1, 0)).
Let us start analyzing the zero-modes content in neutral sector. The story is pretty similar to the one discussed in the previous section. The vertex operator for a µ is proportional to ∂ σ x µ and so it remains invariant under Ω. The action on the Chan-Paton factors of these D-instantons must now be represented by a symmetric matrix which we take to be
γ + (Ω) = 1 ′ 0 0 0 0 1 0 1 0 (5.7)
where 1 ′ is a k 1 × k 1 unit matrix and the 1's are k 2 × k 2 unit matrices. Because of the different orientifold projection, the matrices of bosonic zero-modes behave slightly differently. The matrices a µ , χ 8 and χ 9 will still be 3 × 3 block diagonal, e.g. a µ = diag (a 1 µ , a 2 µ , a 3 µ ), but now a 1 µ = a 1T µ and a 2 µ = a 3T µ whereas the same relations for χ 8 and χ 9 will have an additional minus sign. The remaining fields χ 4...7 are off diagonal and we shall not consider them further since we will consider only the case of one type of instanton. By considering a configuration with k 1 = 1 and k 2 = 0, we can project out all bosonic zero-modes except for the four components a 1 µ that we denote by x µ .
Let us now consider the orientifold action on the fermionic neutral zero-modes M αA and λ αA . The orbifold part of the group acts on the internal spinor indices as a rotation
R(g) = e π 3 Γ 45 e -π 3 Γ 67 , (5.8)
while the orientifold acts through the matrix R(Ω) given in (4.5). The orbifold and orientifold projections thus require
M αA = R(g) A B γ(g)M αB γ(g) -1 , λ αA = γ(g)λ αB γ(g) -1 R(g) B A , (5.9) M αA = R(Ω) A B γ + (Ω)(M αB ) T γ + (Ω) -1 , λ αA = γ + (Ω)(λ αB ) T γ + (Ω) -1 R(Ω) B A .
Using the explicit expressions for the various matrices, we see that, for the simple case where k 1 = 1 and k 2 = 0, all λ's are projected out and only four chiral M zeromodes remain: M α---and M α++-to be identified with the N = 2 chiral superspace 24 coordinates θ 1 α and θ 2 α . Hence, also in this case the orientifold projection has cured the problem encountered in section 3 (albeit in a N = 2 context now) and we can rest assured that the integration over the charged modes will yield a contribution to the prepotential.
Let us now move to the charged zero-modes sector. Just as in the previous model, there are no bosonic modes since the D-instanton and the D3-branes sit at different nodes while the bosonic modes are necessarily diagonal. Most of the fermionic zeromodes µ A and μA are projected out by the orbifold condition which is formally the same as in (4.8), while the orientifold condition relates the fields in the conjugate sectors, giving μ as a linear combination of the µ's according to
μA = R(Ω) A B γ + (Ω)(µ B ) T γ -(Ω) -1 . (5.10)
To summarize, the only charged modes surviving the projection can be expressed, in block 3 × 3 notation as
µ 1 = 0 0 0 0 0 0 µ 0 0 , μ1 = 0 µ T 0 0 0 0 0 0 0 , µ 2 = 0 0 0 µ ′ 0 0 0 0 0 , μ2 = 0 0 -µ ′T 0 0 0 0 0 0 (5.11)
where the entries are to be thought of as column/row vectors in the fundamental/antifundamental of SU(N) depending on their position. As anticipated, the configuration we want to consider is a (0, N) fractional D3branes system together with an exotic (1, 0) instanton. The quiver structure, including the relevant moduli, is depicted in Fig. 6 . It is now easy to see that inserting the expressions (5.6) and (5.11) into Eqs. (2.1), (2.3) and (2.7) we finally obtain
Z = dx 4 dθ 4 F with F = C dµdµ ′ e iµ T Aµ ′ ∝ det A .
(5.12) It would be interesting to study the potential implications of this result in the gauge theory. There are many other simple models that could be analyzed along these lines.
In this paper we have presented some simple examples of what seem to be rather generic phenomena in the context of string instanton physics. We paid particular attention to 25 E 1 µ Φ Φ 32 23 _ µ U(N) U(N) -1,T 1 2 1 µ 2 µ _
Figure 6: The extended Z 3 orientifold theory with (0, N ) fractional D3-branes and (1, 0) instanton number. The upper node (which would represent the USp(N 1 ) gauge group and disappears when we set N 1 = 0 as in the case under consideration) is where the instanton sits. The lower nodes denote only one gauge group. The charged fermionic zero-modes follow Eq. (5.11). For simplicity we have not drawn the lines denoting the adjoint.
the study of the fermionic zero-modes and their effects on the holomorphic quantities of the theory. We have seen both examples where the instanton contributions vanish due to the presence of extra zero-modes and where they do not. In the second case, as explicitly shown in a N = 1 example, exotic instantons can have a stabilizing effect on the theory.
Although we have only considered some simple examples, we would like to stress that these results are quite generic and can be carried over to all orbifold gauge theories. A future direction would be to try to be more systematic and analyze the various possibilities encountered in more complex N = 2 and N = 1 models. In a similar spirit, one should analyze the multi-instanton contributions as well, since the total correction to the holomorphic quantities will be the sum of all such terms. The study of the zero-modes is expected to be even more relevant in this case as it will probably make many contributions vanish. With an eye to string phenomenology, one should also incorporate these models into globally consistent compactifications and study the effects of these terms there.
Lastly, it would be interesting to study the dynamical implications of some of the terms generated. We briefly touched upon this at the end of section 4 when we mentioned the stabilizing effect of the exotic instanton on the USp(N) theory. Although from the strict field theory point of view these terms are thought of as ordinary polyno-26 mial terms in the holomorphic quantities, 7 they are "special" when seen from the point of view of string theory and they might therefore induce a particular type of dynamics.
We would like to thank many people for discussions and email exchanges at various stages of this work that helped us sharpen the focus of the presentation: M. Bianchi, M. Billò, P. Di Vecchia, S. Franco, M. Frau, F. Fucito, S. Kachru, R. Marotta, L. Martucci, F. Morales, B. E. W. Nilsson, D. Persson, I. Pesando, D. Robles-Llana, R. Russo, A. Tanzini, A. Tomasiello, A. Uranga, T. Weigand and N. Wyllard. R.A., M.B. and A.L. are partially supported by the European Commission FP6 Programme MRTN-CT-2004-005104, in which R.A is associated to V.U. Brussel, M.B. to University of Padova and A.L. to University of Torino. R.A. is a Research Associate of the Fonds National de la Recherche Scientifique (Belgium). The research of R.A. is also supported by IISN -Belgium (convention 4.4505.86) and by the "Interuniversity Attraction Poles Programme -Belgian Science Policy". M.B. is also supported by Italian MIUR under contract PRIN-2005023102 and by a MIUR fellowship within the program "Rientro dei Cervelli". The research of G.F. is supported by the Swedish Research Council (Vetenskapsrådet) contracts 622-2003-1124 and 621-2002-3884. A.L. thanks the Galileo Galilei Institute for the hospitality and support during the completion of this work.
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{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Contents 1. Introduction 1 2. Preliminaries 3 3. The N = 1 Z 2 × Z 2 orbifold 7 3.1 Instanton sector 10 3.2 Recovery of the ADS superpotential 11 3.3 Absence of exotic contributions 14 3.4 Study of the back-reaction 16\n\n4. The N = 1 Z 2 × Z 2 orientifold 17 4.1 Instanton sector 19 5. An N = 2 example: the Z 3 orientifold 21 5.1 Instanton sector 24 6. Conclusions 25 1. Introduction\n\nIt has long been realized that instantons in string theory are often in close correspondence with instantons in gauge theories [1, 2, 3, 4, 5, 6] . Recently it was found that in some situations stringy instantons can dynamically generate some terms which from a low-energy effective point of view enter as ordinary external couplings in the superpotential of gauge theories living on space-filling branes [7, 8, 9, 10, 11, 12, 13, 14] . By instantons in string theory we generally mean instantons which are geometrically realized as Euclidean extended objects wrapped on some non-trivial cycles of the geometry. Thus, in a sense, a stringy instanton has a \"life of its own\", not requiring an underlying gauge theory. This opens up the possibility of having contributions originating from instantons that do not admit a standard gauge theory realization. We shall refer to these instantons as exotic.\n\n1 There has been some debate in the recent literature about the instances where such exotic instantons can actually contribute to the gauge theory superpotential in a non-trivial manner. In this work we will contribute to such a debate by considering backgrounds where a simple CFT description is possible, such as orbifolds or orientifolds thereof.\n\nWe present various simple examples of what we believe to be a rather generic situation. Namely, the presence of extra zero-modes for these instantons, in addition to those required by the counting of broken symmetries, makes some of their contributions vanish. Such extra zero-modes should not come as a surprise, since a D-brane instanton in a CY manifold breaks a total of four out of eight supercharges, i.e. it has two extra fermionic zero-modes from the point of view of holomorphic N = 1 gauge theory quantities. We give some arguments as to why the backreaction of the space-filling branes on the geometry might not help in lifting these extra zero-modes. We further argue that only more radical changes of the background, such as the introduction of fluxes, deformations of the CY geometry or the introduction of orientifold planes, can remove these zero-modes. When this happens, exotic instantons do contribute to the gauge theory superpotential and may provide qualitative changes in the low energy effective dynamics, as for instance the stabilization of otherwise runaway directions.\n\nWe will be interested in Euclidean D-branes in type II theories. We will work with IIB fractional branes at orbifold and orientifold singularities rather than type IIA wrapped branes. The motivation for this choice of setting is two-fold. First, recent advances in the gauge/gravity correspondence require the study of exotic instantons, whose effects tend to stabilize the gauge theory rather than unstabilize it [15, 16, 9, 17] , and the gauge/gravity correspondence is more naturally defined in the context of IIB theory. Second, similar effects are used in string phenomenology to try to understand possible mechanisms for neutrino masses [7, 8, 13] . This latest activity is mainly done in the type IIA scenario, but we find it easier to address some subtle issues in the IIB orbifold case.\n\nWhile working in an exact string background, our considerations will nonetheless be only local, i.e. we will not be concerned with global issues such as tadpole cancellation that arise in proper compactifications. This is perfectly acceptable in the context of the gauge/gravity correspondence where the internal manifold is non-compact but, even for string phenomenology, the results we obtain stand (locally) when properly embedded in a consistent compactification.\n\nThe paper is organized as follows: In section 2 we set up the notation and discuss some preliminary material. In section 3 we discuss our first case, namely the N = 1 Z 2 × Z 2 orbifold. After briefly recovering the usual instanton generated corrections to the superpotential we discuss the possible presence of additional exotic contributions 2 and find that they are not present because of the additional zero-modes. We conclude by giving a CFT argument on why such zero-modes are not expected to be lifted even by taking into account the backreaction of the D-branes, unless one is willing to move out the orbifold point in the CY moduli space. Sections 4 and 5 present two separate instances where exotic contributions are present after having removed the extra zero-modes by orientifolding. The first is an N = 1 orientifold, the second is an N = 2 orientifold, displaying corrections to the superpotential and the prepotential, respectively. We end with some conclusions and a discussion of further developments."
},
{
"section_type": "OTHER",
"section_title": "Preliminaries",
"text": "In this section we briefly review the generic setup in the well understood N = 4 situation in order to introduce the notation for the various fields and moduli and their couplings. The more interesting theories we will consider next will be suitable projections of the N = 4 theory. In fact, the exotic cases can all be reduced to orbifolds/orientifolds of this master case once the appropriate projections on the Chan-Paton factors are performed.\n\nSince we are interested in instanton physics (for comprehensive reviews see [18] and the recent [19]) we will take the ten dimensional metric to be Euclidean. We consider a system where both D3-branes and D(-1)-branes (D-instantons) are present. To be definite, we take N D3's and k D-instantons 1 .\n\nQuite generically we can distinguish three separate open string sectors:\n\n• The gauge sector, made of those open strings with both ends on a D3-brane. We assume the brane world-volumes are lying along the first four coordinates x µ and are orthogonal to the last six x a . The massless fields in this sector form an N = 4 SYM multiplet [22] . We denote the bosonic components by A µ and X a . Written in N = 1 language this multiplet is formed by a gauge superfield whose field strength is denoted by W α and three chiral superfields Φ 1,2,3 . With a slight abuse of notation, the bosonic components of the chiral superfields will also be denoted by Φ, i.e. Φ 1 = X 4 + iX 5 and so on. In N = 2 language we have instead a gauge superfield A and a hypermultiplet H, all in the adjoint representation. The low energy action of these fields is a four dimensional N = 4 gauge theory. All these fields are N × N matrices for a gauge group SU(N).\n\n1 These D3/D(-1) brane systems (and their orbifold projections) are very useful and efficient in studying instanton effects from a stringy perspective even in the presence of non-trivial closed string backgrounds, both of NS-NS type [20] and of R-R type [21] .\n\n• The neutral sector, which comprises the zero-modes of strings with both ends on the D-instantons. It is usually referred to as the neutral sector because these modes do not transform under the gauge group. The zero-modes are easily obtained by dimensionally reducing the maximally supersymmetric gauge theory to zero dimensions. We will use an ADHM [23] inspired notation [5, 6] . We denote the bosonic fields as a µ and χ a , where the distinction between the two is made by the presence of the D3-branes. The fermionic zero-modes are denoted by M αA and λ αA , where α and α denote the (positive and negative) four dimensional chiralities and A is an SU(4) (fundamental or anti-fundamental) index denoting the chirality in the transverse six dimensions. The ten dimensional chirality of both fields is taken to be negative. In Euclidean space M and λ must be treated as independent. When needed, we will also introduce the triplet of auxiliary fields D c , directly analogous to the four dimensional D, that can be used to express the various interactions in an easier form as we will see momentarily. All these fields are k × k matrices where k is the instanton number.\n\n• The charged sector, comprising the zero-modes of strings stretching between a D3-brane and a D-instanton. For each pair of such branes we have two conjugate sectors distinguished by the orientation of the string. In the NS sector, where the world-sheet fermions have opposite modding as the bosons, we obtain a bosonic spinor ω α in the first four directions where the GSO projection picks out the negative chirality. In the conjugate sector, we will get an independent bosonic spinor ω α of the same chirality. Similarly, in the R sector, after the GSO projection we obtain a pair of independent fermions (one for each conjugate sector) both in the fundamental of SU(4) which we denote by µ A and μA . These fields are rectangular matrices N × k and k × N.\n\nThe couplings of the fields in the gauge sector give rise to a four dimensional gauge theory. The instanton corrections to such a theory are obtained by constructing the Lagrangian describing the interaction of the gauge sector with the charged sector zeromodes while performing the integral over all zero-modes, both charged and neutral. A crucial point to notice and which will be important later is that while the neutral modes do not transform under the gauge group, their presence affects the integral because of their coupling to the charged sector.\n\nThe part of the interaction involving only the instanton moduli is well known from the ADHM construction and it is essentially the reduction of the interacting gauge Lagrangian for these modes in a specific limit where the Yukawa terms for λ and the quadratic term for D are scaled out (see [18, 6] for details). The final form of this part 4 of the interaction is: S 1 = tr -[a µ , χ a ] 2 + χ a ω αω αχ a + i 2 ( Σa ) AB μA µ B χ a -i 4 ( Σa ) AB M αA [χ a , M B α ] + i μA ω α + ω αµ A + σ µ β α[M βA , a µ ] λ α A -iD c ω α(τ c ) β αω β + iη c µν [a µ , a ν ] (2.1) where the sum over colors and instanton indices is understood. τ denotes the usual Pauli matrices, η (and η) the 't Hooft symbols and Σ (and Σ) are used to construct the six-dimensional gamma-matrices Γ a = 0 Σ a Σa 0 . (2.2) The above interactions can all be understood in terms of string diagrams on a disk with open string vertex operators inserted at the boundary in the α ′ → 0 limit. The interaction of the charged sector with the scalars of the gauge sector can be worked out in a similar way and yields\n\nS 2 = tr ω αX a X a ω α + i 2 ( Σa ) AB μA X a µ B . (2.\n\n3) Let us rewrite the above action in a way which will be more illuminating in the following sections. Since we will be mainly focusing on situations where we have N = 1 supersymmetry, it is useful to write explicitly all indices in SU(4) notation, and then break them into SU(3) representations. We thus write the six scalars X a as the antisymmetric representation of SU(4) as follows\n\nX AB = -X BA ≡ ( Σa ) AB X a . (2.4)\n\nThe action S 2 then reads\n\nS 2 = tr 1 8 ǫ ABCD ω αX AB X CD ω α + i 2 μA X AB µ B . (2.5)\n\nSplitting now the indices A into i = 1 . . . 3 and 4, we can identify Φ † i ≡ X i4 in the 3 of SU(3) and Φ i ≡ 1 2 ǫ ijk X jk in the 3 of SU(3). Thus we can rewrite the action (2.5) as\n\nS 2 = tr 1 2 ω α Φ i , Φ † i ω α + i 2 μi Φ † i µ 4 - i 2 μ4 Φ † i µ i - i 2 ǫ ijk μi Φ j µ k . (2.6)\n\nIn the above form, it is clear which zero-modes couple to the holomorphic superfields and which others couple to the anti-holomorphic ones. This distinction will play an important role later.\n\nThe main object of our investigation is the integral of e -S 1 -S 2 over all moduli Z = C d{a, χ, M, λ, D, ω, ω, µ, μ} e -S 1 -S 2 , (2.7)\n\nwhere we have lumped all field independent normalization constants (including the instanton classical action and the appropriate powers of α ′ required by dimensional analysis) into an overall coefficient C. There are, of course, other interactions involving the fermions and the gauge bosons but, as far as the determination of the holomorphic quantities are concerned, they can be obtained from the previous ones and supersymmetry arguments. For example, a term in the superpotential is written as the integral over chiral superspace dx 4 dθ 2 of a holomorphic function of the chiral superfields, but such a function is completely specified by its value for bosonic arguments at θ = 0. Thus, if we can \"factor out\" a term dx 4 dθ 2 from the moduli integral (2.7), whatever is left will define the complex function to be used in the superpotential and similarly for the prepotential in the N = 2 case if we succeed in factoring out an integral over N = 2 chiral superspace dx 4 dθ 4 .\n\nThe coordinates x and θ must of course come from the (super)translations broken by the instanton and they will be associated to the center of mass motion of the D-instanton, namely, x µ = tr a µ and θ αA = tr M αA for some values of A. 2 One must pay attention however to the presence of possible additional neutral zero-modes coming either from the traceless parts of the above moduli or from the fields λ and χ. These modes must also be integrated over in (2.7) and their effects, as we shall see, can be quite dramatic. In particular, the presence of λ in some instances is crucial for the implementation of the usual ADHM fermionic constraints whereas in other circumstances it makes the whole contribution to the superpotential vanish. These extra λ zero-modes are ubiquitous in orbifold theories and generically make it difficult to obtain exotic instanton corrections for these models. As we shall see, they can however be easily projected out by an orientifold construction making the derivation of such terms possible.\n\nIn the full expression for the instanton corrections there will also be a field-independent normalization factor coming from the one-loop string diagrams and giving for instance the proper g Y M dependence in the case of the usual instanton corrections. In this paper we will only focus on the integral over the zero-modes, which gives the proper field-dependence, referring the reader to [10, 11] for a discussion of these other issues.\n\n2 Obviously, for the case of an anti-instanton, the roles of M and λ are reversed.\n\n3. The N = 1 Z 2 × Z 2 orbifold\n\nIn order to present a concrete example of the above discussion, let us study a simple C 3 /Z 2 × Z 2 orbifold singularity. The resulting N = 1 theory is a non-chiral four-node quiver gauge theory with matter in the bi-fundamental. Non-chirality implies that the four gauge group ranks can be chosen independently [24] . This corresponds to being able to find a basis of three independent fractional branes in the geometry (for a review on fractional branes on orbifolds see e.g. [25]).\n\nThe field content can be conveniently summarized in a quiver diagram, see Fig. 1 , which, together with the cubic superpotential\n\nW = Φ 12 Φ 23 Φ 31 -Φ 13 Φ 32 Φ 21 + Φ 13 Φ 34 Φ 41 -Φ 14 Φ 43 Φ 31 +Φ 14 Φ 42 Φ 21 -Φ 12 Φ 24 Φ 41 + Φ 24 Φ 43 Φ 32 -Φ 23 Φ 34 Φ 42 , (3.1)\n\nuniquely specifies the theory.\n\nSU(N ) SU(N ) SU(N ) SU(N ) 1 2 3 4 Figure 1: Quiver diagram for the Z 2 × Z 2 orbifold theory. Round circles correspond to SU(N ℓ ) gauge factors while the lines connecting quiver nodes represent the bi-fundamental chiral superfields Φ ℓm .\n\nA stack of N regular D3-branes amounts to having one and the same rank assignment on the quiver. The gauge group is then SU(N) 4 and the theory is an N = 1 SCFT. Fractional branes correspond instead to different (but anomlay free) rank assignments. Quite generically, fractional branes can be divided into three different classes, depending on the IR dynamics they trigger [26] . The non-chiral nature and the particularly symmetric structure of the orbifold under consideration allows one to easily construct any such instance of fractional brane class.\n\nIf we turn on a single node, we are left with a pure SU(N) SYM gauge theory, with no matter fields and no superpotential. This theory is believed to confine. The geometric dual effect is that the corresponding fractional brane leads to a geometric 7 transition where the branes disappear leaving behind a deformed geometry. Indeed, there is one such deformation in the above singularity.\n\nTurning on two nodes leads already to more varied phenomena. There are now two bi-fundamental superfields, but still no tree level superpotential. Thus, the system is just like two coupled massless SQCD theories or, by a slightly asymmetric point of view, massless SQCD with a gauged diagonal flavor group. The low-energy behavior depends on the relative ranks of the two nodes.\n\nIf the ranks are different, the node with the highest rank is in a situation where it has less flavors than colors. Then an Affleck-Dine-Seiberg (ADS) superpotential [27, 28] should be dynamically generated, leading eventually to a runaway behavior. This set up of fractional branes is sometimes referred to as supersymmetry breaking fractional branes [29, 26, 30] .\n\nIf the ranks are the same we are in a situation similar to N f = N c SQCD for both nodes. Hence we expect to have a moduli space of SUSY vacua, which gets deformed, but not lifted, at the quantum level. This moduli space is roughly identified in the geometry with the fact that the relevant fractional branes are interpreted as D5-branes wrapped on the 2-cycle of a singularity which is locally C × (C 2 /Z 2 ). Such a fractional brane can move in the C direction. This is what is called an N = 2 fractional brane since, at least geometrically, it resembles very much the situation of fractional branes at N = 2 singularities.\n\nIn what follows we use the two-node example as a simple setting in which we can analyze the subtleties involved in the integration over the neutral modes. For the gauge theory instanton case it is known that there are extra neutral fermionic zero-modes in addition to those required to generate the superpotential. Their integration allows to recover the fermionic ADHM constraints on the moduli space of the usual field theory instantons. For such instantons, we will be able to obtain the ADS superpotential and corresponding runaway behavior in the familiar context with N c and N f fractional branes at the respective nodes, for N f = N c -1. On the other hand, we will argue that the presence of such extra zero-modes rules out the possibility of having exotic instanton effects, such as terms involving baryonic operators in the N f = N c case. It was the desire to study such possible contributions that constituted the original motivation for this investigation. We will first show that such effects are absent for this theory as it stands, and we will later discuss when and how this problem can be cured. 3 3 In a situation where the CFT description is less under control than in the setting discussed in the present paper, it has been argued in [17] that such baryonic couplings do arise in the context of fractional branes on orbifolds of the conifold, possibly at the expense of introducing O-planes. Also in a IIA set up similar to the ones of [7, 8, 10, 11, 13] it seems reasonable that one can wrap an ED2-brane along an O6-plane and produce such couplings on other intersecting D6-branes.\n\nOur orbifold theory can be easily obtained as an orbifold projection of N = 4 SYM. The orbifolding procedure and the derivation of the superpotential (3.1) are by now standard. We briefly recall the main points in order to fix the notation and because some of the details will be useful later in describing the instantons in such a set up.\n\nThe group Z 2 × Z 2 has four elements: the identity e, the generators of the two Z 2 that we denote with g 1 and g 2 and their product, denoted by\n\ng 3 = g 1 g 2 . If we introduce complex coordinates (z 1 , z 2 , z 3 ) ∈ C 3 z 1 = x 4 + ix 5 , z 2 = x 6 + ix 7 , z 3 = x 8 + ix 9 (3.2)\n\nthe action of the orbifold group can be defined as in Table 1 .\n\nz 1 z 2 z 3 e z 1 z 2 z 3 g 1 z 1 -z 2 -z 3 g 2 -z 1 z 2 -z 3 g 3 -z 1 -z 2 z 3\n\nTable 1: The action of the orbifold generators.\n\nLet γ(g) be the regular representation of the orbifold group on the Chan-Paton factors. If the orbifold is abelian, as always in the cases we shall be interested in, we can always diagonalize all matrices γ(g). We will assume that the two generators have the following matrix representation\n\nγ(g 1 ) = σ 3 ⊗ 1 = 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1 , γ(g 2 ) = 1 ⊗ σ 3 = 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1 (3.3)\n\nwhere the 1's denote N ℓ × N ℓ unit matrices (ℓ = 1, ..., 4). Then, the orbifold projection amounts to enforcing the conditions\n\nA µ = γ(g)A µ γ(g) -1 , Φ i = ±γ(g)Φ i γ(g) -1 (3.4)\n\nwhere the sign ± must be chosen according to the action of the orbifold generators g that can be read off from Table 1 . With the choice (3.3), the vector superfields are block diagonal matrices of different size (N 1 , N 2 , N 3 , N 4 ), one for each node of the 9 quiver, while the three chiral superfields Φ i have the following form [24]\n\nΦ 1 = 0 × 0 0 × 0 0 0 0 0 0 × 0 0 × 0 , Φ 2 = 0 0 × 0 0 0 0 × × 0 0 0 0 × 0 0 , Φ 3 = 0 0 0 × 0 0 × 0 0 × 0 0 × 0 0 0 , (3.5)\n\nwhere the crosses represent the non-zero entries Φ ℓm appearing in the superpotential (3.1)."
},
{
"section_type": "OTHER",
"section_title": "Instanton sector",
"text": "Now consider D-instantons in the above set up. Such instantons preserve half of the 4 supercharges preserved by the system of D3-branes plus orbifold. In this respect recall that the fractional branes preserve exactly the same supercharges as the regular branes. 4 Using the N = 4 construction of the previous section and the structure of the orbifold presented in eq. (3.5), we now proceed in describing the zero-modes for such instantons. The neutral sector is very similar to the gauge sector. Indeed, in the (-1) superghost picture, the vertex operators for such strings will be exactly the same, except for the e ip•X factor which is absent for the instanton. The Chan-Paton structure will also be the same, so that the same pattern of fractional D-instantons will arise as for the fractional D3-branes. In particular, the only regular D-instanton (which could be thought of as deriving from the one of N = 4 SYM) is the one with rank (instanton number) one at every node. All other situations can be thought of as fractional Dinstantons, which can be interpreted as Euclidean D1-branes wrapped on the two-cycles at the singularity, ED1 for short. Generically, we can then characterize an instanton configuration in our orbifold by (k\n\n1 , k 2 , k 3 , k 4 ).\n\nFollowing the notation introduced in section 2, the bosonic modes will comprise a 4 × 4 block diagonal matrix a µ , and six more matrix fields χ 1 , . . . χ 6 , that can be paired into three complex matrix fields χ 1 +iχ 2 , χ 3 +iχ 4 , χ 5 +iχ 6 , having the same structure as (3.5) but now where each block entry is a k ℓ × k m matrix. On the fermionic zero-modes M αA and λ αA (also matrices) the orbifold projection enforces the conditions\n\nM αA = R(g) A B γ(g)M αB γ(g) -1 , λ αA = γ(g)λ αB γ(g) -1 R(g) B A (3.6) 4\n\nThere is another Euclidean brane which preserves two supercharges, namely the Euclidean (anti) D3-branes orthogonal to the 4 dimensions of space-time. We will be considering here only the Dinstantons, leaving the complete analysis of the other effects to future work. In this context, note that the extended brane instantons would have an infinite action (and thus a vanishing contribution) in the strict non-compact set up we are using here.\n\nwhere R(g) is the orbifold action of Table 1 in the spinor representation which can be chosen as R(g 1 ) = -Γ 6789 , R(g 2 ) = -Γ 4589 . (3.7) It is easy to find an explicit representation of the Dirac matrices such that M αA and λ αA for A = 1, 2, 3 also have the structure of (3.5) while for A = 4 they are block diagonal.\n\nEquivalently, one could write the spinor indices in the internal space in terms of the three SO(2) charges associated to the embedding SO(2) × SO(2) × SO(2) ⊂ SO(6) ≃ SU(4)\n\nM α-++ = M α1 , M α+-+ = M α2 , M α++-= M α3 , M α---= M α4 , λ α+--= λ α1 , λ α-+-= λ α2 , λ α--+ = λ α3 , λ α+++ = λ α4 . (3.8)\n\nThe most notable difference between the neutral sector and the gauge theory on the D3-branes is that, whereas in the four-dimensional theory the U(1) gauge factors are rendered massive by a generalization of the Green-Schwarz mechanism and do not appear in the low energy action, for the instanton they are in fact present and enter crucially into the dynamics. Let us finally turn to the charged sector, describing strings going from the instantons to the D3-branes. The analysis of the spectrum and the action of the orbifold group on the Chan-Paton factors show, in particular, that the bosonic zero-modes are diagonal in the gauge factors. There are four block diagonal matrices of bosonic zeromodes ω α, ω α with entries N ℓ ×k ℓ and k ℓ ×N ℓ respectively and eight fermionic matrices µ A , μA with entries N ℓ × k m and k m × N ℓ , that again display the same structure as above -same as (3.5) for A = 1, 2, 3 and diagonal for A = 4."
},
{
"section_type": "OTHER",
"section_title": "Recovery of the ADS superpotential",
"text": "The measure on the moduli space of the instantons and the ADHM constraints are simply obtained by inserting the above expressions into the moduli integral (2.7). If one chooses some of the N ℓ or k ℓ to vanish one can deduce immediately from the structure of the projection which modes will survive and which will not.\n\nAs a consistency check, one can try to reproduce the ADS correction to the superpotential [27, 28] for the theory with two nodes. Take fractional branes corresponding to a rank assignment (N c , N f , 0, 0), and consider the effect of a ED1 corresponding to instanton numbers (1, 0, 0, 0) .\n\nThe only chiral fields present are the two components of Φ 1 connecting the first 11 and second node\n\nΦ 1 = 0 Q 0 0 Q 0 0 0 0 0 0 0 0 0 0 0 . (3.9)\n\nSince the instanton is sitting only at one node, all off diagonal neutral modes are absent, as they connect instantons at two distinct nodes. Thus, the only massless modes present in the neutral sector are four bosons x µ , denoting the upper-left component of a µ , two fermions θ α denoting the upper-left component of M α4 and two more fermions λ α denoting the upper-left component of λ α4 . We have identified the non zero entries of a µ and M α4 with the super-coordinates x µ and θ α since they precisely correspond to the Goldstone modes of the super-translation symmetries broken by the instanton and do not appear in S 1 + S 2 (cfr. (2.1) and (2.3)). Their integration produces the integral over space-time and half of Grassmann space which precedes the superpotential term to which the instanton contributes. On the contrary, λ α appears in S 1 and when it is integrated it yields the fermionic ADHM constraint.\n\nIn the charged sector, we have bosonic zero-modes ω u α and ω αu , with u an index in the fundamental or anti-fundamental of SU(N c ). In addition, there are fermionic zero-modes µ u and μu with indices in SU(N c ), together with additional fermionic zeromodes µ ′f and μ′ f where the index f is now in the fundamental or anti-fundamental of SU(N f ). 5 Note that the µ zero-modes carry an SU(4) index 4 (being on the diagonal) while the µ ′ zero-modes carry an SU(4) index 1, since they are of the same form as Φ 1 .\n\nAll this can be conveniently summarized in a generalized quiver diagram as represented in Fig. 2 , which accounts for both the brane configuration and the instanton zero-modes.\n\nFor a single instanton, the action (2.1) greatly simplifies since many fields are vanishing as well as all commutators and one gets\n\nS 1 = i (μ u ω u α + ω αu µ u ) λ α -iD c ω α u (τ c ) β αω u β . (3.10)\n\nSimilarly, the coupling of the charged modes to the chiral superfield can be expressed by writing eq. (2.3) as\n\nS 2 = 1 2 ω αu Q u f Q †f v + Q †u f Qf v ω αv - i 2 μu Q †u f µ ′f + i 2 μ′ f Q †f u µ u . (3.11)\n\nNote that it is the anti-holomorphic superfields that enter in the couplings with the fermionic zero-modes, as is clear by comparing with (2.6). The above action is exactly the same which appears in the ADHM construction as reviewed in [18] .\n\n5 Recall that the bosonic zero-modes are diagonal in the gauge factors; therefore there are no ω f α and ω αf zero-modes.\n\n12 c 1 E ω Q Q µ µ λ ~µ µ SU(N ) SU(N ) f _ ' _ ' ω _\n\nFigure 2: Quiver diagram describing an ordinary instanton in a SU(N c ) × SU(N f ) theory. Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1 is wrapped on the same cycle as the color branes. All zero-modes are included except the θ's and the x µ 's, which only contribute to the measure for the integral over chiral superspace.\n\nWe are now ready to perform the integral (2.7) over all the existing zero-modes. Writing (3.12) we see that the instanton induced superpotential is\n\nZ = dx 4 dθ 2 W ,\n\nW = C d{λ, D, ω, ω, µ, μ} e -S 1 -S 2 . (3.13)\n\nThe integrals over D and λ enforce the bosonic and fermionic ADHM constraints, respectively. Thus\n\nW = C d{ω, ω, µ, μ} δ(μ u ω u α + ω αu µ u ) δ(ω α u (τ c ) β αω u β ) e -S 2 . (3.14)\n\nWe essentially arrive at the point of having to evaluate an integral over a set of zeromodes which is exactly the same as the one discussed in detail in the literature, e.g. [18] . We thus quickly go to the result referring the reader to the above review for further details. First of all, it is easy to see that, due to the presence of extra µ modes in the integrand from the fermionic delta function, only when N f = N c -1 we obtain a non-vanishing result. After having integrated over the µ and µ ′ , we are left with a (constrained) gaussian integration that can be performed e.g. by going to a region 13 of the moduli space where the chiral fields are diagonal, up to a row/column of zeroes. Furthermore, the D-terms in the gauge sector constrain the quark superfields to obey QQ † = Q † Q, so that the bosonic integration brings the square of a simple determinant in the denominator. The last fermionic integration conspires to cancel the anti-holomorphic contributions and gives\n\nW ADS = Λ 2Nc+1 det( QQ) , (3.15)\n\nwhich is just the expected ADS superpotential for N f = N c -1, the only case where such non-perturbative contribution is generated by a genuine one-instanton effect and not by gaugino condensation. In (3.15) Λ is the SQCD strong coupling scale that is reconstructed by the combination of e -8π 2 /g 2 coming from the instanton action with various dimensional factors coming from the normalization of the instanton measure [18]."
},
{
"section_type": "OTHER",
"section_title": "Absence of exotic contributions",
"text": "Until now, we have reproduced from stringy considerations the effect that is supposed to be generated also by instantons in the gauge theory. Considering a slightly different set up, we would like to study the possibility of generating other terms.\n\nLet us consider a system with rank assignment (N c , N f , 0, 0), as before, but fractional instanton numbers (0, 0, 1, 0) . In other words, we study the effect of a single fractional instanton sitting on an unoccupied node of the gauge theory. The quiver diagram, with the relevant zero-modes structure, is given in Fig. 3 .\n\nThe neutral zero-modes of the instanton sector are the same as before. This is because the quantization of this sector does not know the whereabouts of the D3branes and thus all nodes are equivalent, in this respect. In the mixed sector, we have no bosonic zero-modes now, since the ω and ω are diagonal. Note that, although we always have four mixed (ND) boundary conditions, due to the quiver structure induced by the orbifold, here we effectively realize the same situation one has when there are eight ND directions, namely that the bosonic sector of the charged moduli is empty.\n\nOn the other hand, there are fermionic zero-modes µ u , μu , µ ′f and μ′ f , as in the previous case. Note that despite having the same name, these zero-modes correspond actually to different Chan-Paton matrix elements with respect to the previous ones, the difference being in the instanton index that is not written explicitly. In particular we can think of µ and µ ′ as carrying an SU(4) index 2 and 3 respectively.\n\nBecause of the absence of bosonic charged modes, the action (2.1) is identically 14 c E 1 λ Q Q μ µ µ µ SU(N ) f SU(N ) _ ' _ ' Figure 3: Quiver diagram describing an exotic instanton in a SU(N c ) × SU(N f ) theory. Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1 is wrapped on a different cycle with respect to both sets of quiver branes.\n\nzero and the action (2.3) contains only the last term:\n\nS 1 = 0 S 2 = i 2 μu Q u f µ ′f - i 2 μ′ f Qf u µ u . (3.16)\n\nNote that in this case it is the holomorphic superfields which appear above, as is clear from (2.6) and from noticing that the diagonal fermionic zero-mode µ 4 is not present. We are thus led to consider W = C d{λ, D, µ, μ} e -S 2 . (3.17) One notices right away that the integral over the charged modes is non vanishing (only) for the case N f = N c and gives a tantalizing contribution proportional to B B, where B = det Q and B = det Q are the baryon fields of the theory. However, we must carefully analyze the integration over the remaining zero-modes of the neutral sector. Now neither D nor λ appear in the integrand. The integral over D does not raise any concern: it is, after all, an auxiliary field and its disappearance from the integrand is due to the peculiarities of the ADHM limit. Before taking this limit, D appeared quadratically in the action and could be integrated out, leaving an overall normalization constant. The integral over λ is another issue. In this case, λ is absent from the integrand even before taking the ADHM limit and its integration multiplies 15 the above result by zero, making the overall contribution of such instantons to the superpotential vanishing. Of course, the presence of such extra zero-modes should not come as a surprise since they correspond to the two extra broken supersymmetries of an instanton on a CY. Therefore we see that the neutral zero-modes contribution, in the exotic instanton case, plays a dramatic role and conspires to make everything vanishing (as opposite to the ADS case analyzed before). A natural question is to see whether these zeromodes get lifted by some effect we have not taken into account, yet. For one thing, supersymmetry arguments would make one think that taking into account the backreaction of the D3-branes might change things. However, in the following subsection we show that this seems not to be the case."
},
{
"section_type": "OTHER",
"section_title": "Study of the back-reaction",
"text": "Let us stick to the case N f = N c , which is the only one where the integral (3.17) might give a non-vanishing contribution. In this case the fractional brane system is nothing but a stack of (N c ) N = 2 fractional branes. These branes couple to only one of the 3 closed string twisted sectors [24] . More specifically, they source the metric h µν , the R-R four-form potential C µνρσ and two twisted scalars b and c from the NS-NS and R-R sector respectively. This means that the disk one-point function of their vertex operators [31, 32] is non vanishing when the disk boundary is attached to such D3-branes. (Indeed in this way or, equivalently, by using the boundary-state formalism [33, 34] , one can derive the profile for these fields.) If the back-reaction of these fields on the instanton lifted the extra zero-modes λ's, this should be visible when computing the one point function of the corresponding closed string vertex operators on a disk with insertions on this boundary of the vertex operators for such moduli. To see whether such coupling is there, we first need to write down the vertex operators for the λ's in the (±1/2) superghost pictures. The vertex in the (-1/2) picture is found e.g. in [6] and reads V -1/2 λ\n\n(z) = λ αA S α(z)S A (z)e -φ(z)/2 , (3.18)\n\nwhere S α(z) and S A (z) are the spin-fields in the first four and last six directions respectively. For our argument we need to focus on the S A (z) dependence. Since the modulus that survives the orbifold projection is, with our conventions, λ α4 = λ α+++ , we write the corresponding spin-field as S +++ (z) = e iH 1 (z)/2 e iH 2 (z)/2 e iH 3 (z)/2 , (3.19) where H i (z) is the free boson used to bosonize the fermionic sector in the i-th complex direction: ψ i (z) = e iH i (z) . The vertex operator in the +1/2 picture can be obtained by 16 applying the picture-changing operator to (3.18) V 1/2 λ (z) = [Q BRST , ξV -1/2 λ (z)] . (3.20)\n\nThe crucial part in Q BRST is [31]\n\nQ BRST = dz 2πi η e φ ψ µ ∂X µ + ψi ∂Z i + ψ i ∂ Zi + . . . (3.21)\n\nBecause of the nature of the supercurrent, we see that (3.21) flips at most one sign in (3.19), hence the product V -1/2 λ V 1/2 λ will always carry an unbalanced charge in some of the three internal SO(2) groups. On the other hand, the vertex operators for the fields sourced by the fractional D3's cannot compensate such an unbalance. Hence, their correlation function on the D-instanton with the insertion of V -1/2 λ V 1/2 λ carries a charge unbalance and therefore vanishes. Therefore, at least within the above perturbative approach, the neutral zero-modes seem not to get lifted by the back-reaction of the D3-branes.\n\nOne might consider some additional ingredients which could provide the lifting. A natural guess would be moving in the CY moduli space or adding suitable background fluxes [35, 36] . There are indeed non-vanishing background fields at the orbifold point, i.e. the b fields of the twisted sectors which the N = 2 fractional branes do not couple to. These fields, however, being not associated to geometric deformations of the internal space should be described by a CFT vertex operator uncharged under the SO(2)'s, simply because of Lorentz invariance in the internal space. Therefore, the only way to get an effective mass term for the zero-modes λ would be to move out of the orbifold point in the CY moduli space. Indeed, the other moduli of the NS-NS twisted sector, being associated to geometric blow-ups of the singularity, are charged under (some of) the internal SO(2)'s and can have a non vanishing coupling with the λ's. More generically, complicated closed string background fluxes might be suitable. This is an interesting option which however we do not pursue here, since we want to stick to situations where a CFT description is available.\n\nA more radical thing to do is to remove the zero-modes from the very start, for instance by means of an orientifold projection [37, 38] . This is the option we are going to consider in the remainder of this work.\n\nN = 1 Z 2 × Z 2 orientifold\n\nIn this section we supplement our orbifold background by an O3 orientifold and show that in this case exotic instanton contributions do arise and provide new terms in the 17 superpotential. We refer to e.g. [39, 40, 41] for a comprehensive discussion of N = 1 and N = 2 orientifolds.\n\nThe first ingredient we need is the action of the O3-plane on the various fields. Denote by Ω the generator of the orientifold. The action of Ω on the vertex operators for the various fields (ignoring for the time being the Chan-Paton factors) is well known. The vertex operators for the bosonic fields on the D3-brane contain, in the 0 picture, the following terms:\n\nA µ ∼ ∂ τ x µ and Φ i ∼ ∂ σ\n\nzi . They both change sign under Ω, the first because of the derivative ∂ τ and the second because the orientifold action for the O3-plane is always accompanied by a simultaneous reflection of all the transverse coordinates z i .\n\nThe action of the orientifold on the Chan-Paton factors is realized by means of a matrix γ(Ω) which in presence of an orbifold must satisfy the following consistency condition [39] γ(g)γ(Ω)γ(g) T = + γ(Ω) (4.1) for all orbifold generators g. This amounts to require that the orientifold projection commutes with the orbifold projection. The matrix γ(Ω) can be either symmetric or anti-symmetric. We choose to perform an anti-symmetric orientifold projection on the D3 branes and denote the corresponding matrix by γ -(Ω). This requires having an even number N ℓ of D3 branes on each node of the quiver so that we can write\n\nγ -(Ω) = ǫ 1 0 0 0 0 ǫ 2 0 0 0 0 ǫ 3 0 0 0 0 ǫ 4 (4.2)\n\nwhere the ǫ ℓ 's are N ℓ × N ℓ antisymmetric matrices obeying ǫ 2 ℓ = -1. Using (3.3) and (4.2) it is straightforward to verify that the consistency condition (4.1) is verified.\n\nThe field content of the stacks of fractional D3-branes in this orientifold model is obtained by supplementing the orbifold conditions (3.4) with the orientifold ones\n\nA µ = -γ -(Ω)A T µ γ -(Ω) -1 , Φ l = -γ -(Ω)Φ lT γ -(Ω) -1 . (4.3) This implies that A µ = diag (A 1 µ , A 2 µ , A 3 µ , A 4 µ ) with A ℓ µ = ǫ ℓ A iT µ ǫ ℓ .\n\nThus, the resulting gauge theory is a USp(N\n\n1 ) × USp(N 2 ) × USp(N 3 ) × USp(N 4 ) theory.\n\nThe chiral superfields, which after the orbifold have the structure (3.5), are such that the Φ ℓm component joining the nodes ℓ and m of the quiver, must obey the orientifold condition Φ ℓm = ǫ ℓ Φ T mℓ ǫ m . In the following, we will take N 3 = N 4 = 0 so that we are left with only two gauge groups and no tree level superpotential."
},
{
"section_type": "OTHER",
"section_title": "Instanton sector",
"text": "Let us now consider the instanton sector, starting by analyzing the zero-mode content in the neutral sector. There are two basic changes to the previous story. The first is that the vertex operator for a µ is now proportional to ∂ σ x µ , not to ∂ τ x µ and it remains invariant under Ω (the vertex operator for χ a still changes sign). The second is that the crucial consistency condition discussed in [38] requires that we now represent the action of Ω on the Chan-Paton factors of the neutral modes by a symmetric matrix which can be taken to be\n\nγ + (Ω) = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , (4.4)\n\nwhere the 1's are k ℓ × k ℓ unit matrices. The matrix a µ will be 4 × 4 block diagonal, e.g.\n\na µ = diag (a 1 µ , a 2 µ , a 3 µ , a 4 µ ), but now a ℓ µ = a ℓT µ .\n\nThe most generic situation is to have a configuration with instanton numbers (k 1 , k 2 , k 3 , k 4 ). By considering a configuration with k 3 = 1 and k 1 = k 2 = k 4 = 0, we can project out all bosonic zero-modes except for the four components a 3 µ that we denote by x µ . The scalars χ 4 . . . χ 9 are off-diagonal and we shall not consider them further.\n\nThe nice surprise comes when considering the orientifold action on the fermionic neutral zero-modes M αA and λ αA . The orbifold part of the group acts on the spinor indices as in (3.7), while the orientifold projection acts as the reflection in the transverse space, namely R(Ω) = -i Γ 456789 (4.5) Putting together the orbifold projections (3.6) with the orientifold ones\n\nM αA = R A B (Ω)γ + (Ω)(M αB ) T γ + (Ω) -1 , λ αA = γ + (Ω)(λ αB ) T γ + (Ω) -1 R B A (Ω) (4.6)\n\nwe can find the spectrum of surviving fermionic zero-modes. Using (4.4) and (4.5), it is easy to see that (4.6) implies\n\nM αA = (M αA ) T , λ αA = -(λ αA ) T . (4.7)\n\nThus, for the simple case where k 3 = 1 and k 1 = k 2 = k 4 = 0, all λ's are projected out and only two chiral M zero-modes remain: M α---, to be identified with the N = 1 chiral superspace coordinates θ α . Also the charged zero-modes are easy to discuss in this simple scenario. There are no bosonic modes since the D-instanton and the D3-branes sit at different nodes while 19 the bosonic modes are necessarily diagonal. Most of the fermionic zero-modes µ A and μA are also projected out by the orbifold condition µ A = R(g) A B γ(g)µ B γ(g) -1 , μA = R(g) A B γ(g)μ B γ(g) -1 . (4.8) Finally, the orientifold condition relates this time the fields in the conjugate sectors, allowing one to express μ as a linear combination of the µ\n\nμA = R(Ω) A B γ + (Ω)(µ B ) T γ -(Ω) -1 . (4.9)\n\nThe only charged modes surviving these projections can be expressed, in block 4 × 4 notation, as\n\nµ 2 = 0 0 µ 13 0 0 0 0 0 0 0 0 0 0 0 0 0 , μ2 = 0 0 0 0 0 0 0 0 μ31 0 0 0 0 0 0 0 , µ 3 = 0 0 0 0 0 0 µ 23 0 0 0 0 0 0 0 0 0 , μ3 = 0 0 0 0 0 0 0 0 0 μ32 0 0 0 0 0 0 , (4.10)\n\nwhere the entries, to be thought of as column/row vectors in the fundamental/antifundamental of SU(N ℓ ) depending on their position, are such that μ31 = -µ T 13 ǫ 1 and μ32 = -µ T 23 ǫ 2 . Thus, in the case where we have fractional D3 branes (N 1 , N 2 , 0, 0) and an exotic instanton (0, 0, 1, 0) , the only surviving chiral field is Φ 12 ≡ ǫ 1 Φ T 21 ǫ 2 , the orientifold projection eliminates the offending λ's and we are left with just the neutral zero-modes x µ and θ α and the charged ones µ 13 and µ 23 . This is summarized in the generalized quiver of Fig. 4 .\n\nIn this case the instanton partition function is\n\nZ = dx 4 dθ 2 W (4.11)\n\nwhere the superpotential W is\n\nW = C dµ e -S 1 -S 2 = C dµ 13 dµ 23 e iµ T 13 ǫ 1 Φ 12 µ 23 . (4.12)\n\nThis integral clearly vanishes unless N 1 = N 2 , in which case we have\n\nW ∝ det(Φ 12 ) (4.13) 20 E 1 µ µ USP(N ) USP(N ) 1 2 Φ Φ 12 21 _ 32 23 13 µ 31 µ _\n\nFigure 4: The generalized Z 2 × Z 2 orientifold quiver and the exotic instanton contribution.\n\nWe thus see that exotic instanton corrections are possible in this simple model. 6\n\nIt is interesting to note that the above correction is present in the same case (N 1 = N 2 ≡ N) where the usual ADS superpotential for USp(N) is generated [42]\n\nW ADS = Λ 2N +3 det(Φ 12 ) (4.14)\n\nand its presence stabilizes the runaway behavior and gives a theory with a non-trivial moduli space of supersymmetric vacua given by det(Φ 12 ) = const. Of course, the ADS superpotential for this case can also be constructed along the same lines as section 3.2, see e.g. [18] . In fact, this derivation is somewhat simpler than the one for the SU(N) gauge group since there are no ADHM constraints at all in the one instanton case. We think the above situation is not specific to the background we have been considering, but is in fact quite generic. As soon as the λ zero-modes are consistently lifted, we expect the exotic instantons to contribute new superpotential terms. As a further example, in the next section we will consider a N = 2 model, where exotic instantons will turn out to contribute to the prepotential."
},
{
"section_type": "OTHER",
"section_title": "An N = 2 example: the Z 3 orientifold",
"text": "Let us now consider the quiver gauge theory obtained by placing an orientifold O3-plane at a C × C 2 /Z 3 orbifold singularity. In what follows we will use N = 1 superspace notation. We first briefly repeat the steps that led to the constructions of such a quiver 6 The gauge invariant quantity above can be rewritten as the Pfaffian of a suitably defined mesonic matrix.\n\n21 theory in the seminal paper [39] . Define ξ = e 2πi/3 and let the generator of the orbifold group act on the first two complex coordinates as g : z 1 z 2 → ξ 0 0 ξ -1 z 1 z 2 , (5.1)\n\nwhile leaving the third one invariant. This preserves N = 2 SUSY. The action of the generator g on the Chan-Paton factors is given by the matrix\n\nγ(g) = 1 0 0 0 ξ 0 0 0 ξ 2 . (5.2)\n\nThe N = 2 theory obtained this way, summarized in Fig. 5 , is a three node quiver gauge theory with gauge groups SU(N 1 ) × SU(N 2 ) × SU(N 3 ), supplemented by a cubic superpotential which is nothing but the orbifold projection of the N = 4 superpotential (its precise form is not relevant for the present purposes).\n\nSU(N ) SU(N ) SU(N ) 1 3 2 Figure 5: The Z 3 (un-orientifolded) theory. The lines with both ends on a single node represent adjoint chiral multiplets which, together with the vector multiplets at each node constitute the N = 2 vector multiplets. Similarly, lines between nodes represent chiral multiplets which pair up into hyper-multiplets, in N = 2 language.\n\nAs for the action of Ω on the Chan-Paton factors, we choose again to perform the symplectic projection on the D3-branes. To do so, we must take N 1 to be even and N 2 = N 3 , so that we can write\n\nγ -(Ω) = ǫ 0 0 0 0 1 0 -1 0 , (5.3) 22\n\nwhere ǫ is a N 1 × N 1 antisymmetric matrix obeying ǫ 2 = -1 and the 1's denote N 2 × N 2 identity matrices. The matrices γ(g) and γ -(Ω) satisfy the usual consistency condition [38, 39] as in (4.1). The field content on the fractional D3-branes at the singularity will be given by implementing the conditions\n\nA µ = γ(g)A µ γ(g) -1 , Φ i = ξ -i γ(g)Φ i γ(g) -1 , A µ = -γ -(Ω)A T µ γ -(Ω) -1 , Φ i = -γ -(Ω)Φ iT γ -(Ω) -1 . (5.4)\n\nThe orbifold part of these conditions forces A µ and Φ 3 to be 3 × 3 block diagonal matrices, e.g. A µ = diag (A 1 µ , A 2 µ , A 3 µ ), while the orientifold imposes that A 1 µ = ǫA 1T µ ǫ and A 2 µ = -A 3T µ . The resulting gauge theory is thus a USp(N 1 ) × SU(N 2 ) theory. It is convenient, however, to still denote A 2 µ and A 3 µ diagramatically as belonging to different nodes with the understanding that these should be identified in the above sense.\n\nThe projection on the chiral fields can be done similarly and we obtain, denoting by Φ ℓm the non-zero entries of the fields Φ 1 and Φ 2 (only one can be non-zero for each pair ℓm)\n\nΦ 12 = -ǫΦ T 31 , Φ 13 = +ǫΦ T 21 , Φ 23 = Φ T 23 , Φ 32 = Φ T 32 . (5.5)\n\nThe field content is summarized in Table 2 .\n\nUSp(N 1 ) SU(N 2 ) Φ 12 Φ 21 Φ 13 Φ 31 Φ 23 • Φ 32 •\n\nTable 2: Chiral fields making up the quiver gauge theory.\n\nThe theory we want to focus on in the following has rank assignment (N 1 , N 2 ) = (0, N). This yields an N = 2 SU(N) gauge theory with an hyper-multiplet in the symmetric/(conjugate)symmetric representation. We denote the N = 2 vector multiplet by A whose field content in the block 3 × 3 notation is thus\n\n = 0 0 0 0 A 0 0 0 -A T . (5.6) 23\n\nIn what follows we will be interested in studying corrections to the prepotential F coming from exotic instantons associated to the first node (the one that is not populated by D3-branes). Let us then analyze the structure of the stringy instanton sector of the present model, first."
},
{
"section_type": "OTHER",
"section_title": "Instanton sector",
"text": "The most generic situation is to have a configuration with instanton numbers (k 1 , k 2 ) (later we will be mainly concerned with a configuration with instanton numbers (1, 0)).\n\nLet us start analyzing the zero-modes content in neutral sector. The story is pretty similar to the one discussed in the previous section. The vertex operator for a µ is proportional to ∂ σ x µ and so it remains invariant under Ω. The action on the Chan-Paton factors of these D-instantons must now be represented by a symmetric matrix which we take to be\n\nγ + (Ω) = 1 ′ 0 0 0 0 1 0 1 0 (5.7)\n\nwhere 1 ′ is a k 1 × k 1 unit matrix and the 1's are k 2 × k 2 unit matrices. Because of the different orientifold projection, the matrices of bosonic zero-modes behave slightly differently. The matrices a µ , χ 8 and χ 9 will still be 3 × 3 block diagonal, e.g. a µ = diag (a 1 µ , a 2 µ , a 3 µ ), but now a 1 µ = a 1T µ and a 2 µ = a 3T µ whereas the same relations for χ 8 and χ 9 will have an additional minus sign. The remaining fields χ 4...7 are off diagonal and we shall not consider them further since we will consider only the case of one type of instanton. By considering a configuration with k 1 = 1 and k 2 = 0, we can project out all bosonic zero-modes except for the four components a 1 µ that we denote by x µ .\n\nLet us now consider the orientifold action on the fermionic neutral zero-modes M αA and λ αA . The orbifold part of the group acts on the internal spinor indices as a rotation\n\nR(g) = e π 3 Γ 45 e -π 3 Γ 67 , (5.8)\n\nwhile the orientifold acts through the matrix R(Ω) given in (4.5). The orbifold and orientifold projections thus require\n\nM αA = R(g) A B γ(g)M αB γ(g) -1 , λ αA = γ(g)λ αB γ(g) -1 R(g) B A , (5.9) M αA = R(Ω) A B γ + (Ω)(M αB ) T γ + (Ω) -1 , λ αA = γ + (Ω)(λ αB ) T γ + (Ω) -1 R(Ω) B A .\n\nUsing the explicit expressions for the various matrices, we see that, for the simple case where k 1 = 1 and k 2 = 0, all λ's are projected out and only four chiral M zeromodes remain: M α---and M α++-to be identified with the N = 2 chiral superspace 24 coordinates θ 1 α and θ 2 α . Hence, also in this case the orientifold projection has cured the problem encountered in section 3 (albeit in a N = 2 context now) and we can rest assured that the integration over the charged modes will yield a contribution to the prepotential.\n\nLet us now move to the charged zero-modes sector. Just as in the previous model, there are no bosonic modes since the D-instanton and the D3-branes sit at different nodes while the bosonic modes are necessarily diagonal. Most of the fermionic zeromodes µ A and μA are projected out by the orbifold condition which is formally the same as in (4.8), while the orientifold condition relates the fields in the conjugate sectors, giving μ as a linear combination of the µ's according to\n\nμA = R(Ω) A B γ + (Ω)(µ B ) T γ -(Ω) -1 . (5.10)\n\nTo summarize, the only charged modes surviving the projection can be expressed, in block 3 × 3 notation as\n\nµ 1 = 0 0 0 0 0 0 µ 0 0 , μ1 = 0 µ T 0 0 0 0 0 0 0 , µ 2 = 0 0 0 µ ′ 0 0 0 0 0 , μ2 = 0 0 -µ ′T 0 0 0 0 0 0 (5.11)\n\nwhere the entries are to be thought of as column/row vectors in the fundamental/antifundamental of SU(N) depending on their position. As anticipated, the configuration we want to consider is a (0, N) fractional D3branes system together with an exotic (1, 0) instanton. The quiver structure, including the relevant moduli, is depicted in Fig. 6 . It is now easy to see that inserting the expressions (5.6) and (5.11) into Eqs. (2.1), (2.3) and (2.7) we finally obtain\n\nZ = dx 4 dθ 4 F with F = C dµdµ ′ e iµ T Aµ ′ ∝ det A .\n\n(5.12) It would be interesting to study the potential implications of this result in the gauge theory. There are many other simple models that could be analyzed along these lines."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusions",
"text": "In this paper we have presented some simple examples of what seem to be rather generic phenomena in the context of string instanton physics. We paid particular attention to 25 E 1 µ Φ Φ 32 23 _ µ U(N) U(N) -1,T 1 2 1 µ 2 µ _\n\nFigure 6: The extended Z 3 orientifold theory with (0, N ) fractional D3-branes and (1, 0) instanton number. The upper node (which would represent the USp(N 1 ) gauge group and disappears when we set N 1 = 0 as in the case under consideration) is where the instanton sits. The lower nodes denote only one gauge group. The charged fermionic zero-modes follow Eq. (5.11). For simplicity we have not drawn the lines denoting the adjoint.\n\nthe study of the fermionic zero-modes and their effects on the holomorphic quantities of the theory. We have seen both examples where the instanton contributions vanish due to the presence of extra zero-modes and where they do not. In the second case, as explicitly shown in a N = 1 example, exotic instantons can have a stabilizing effect on the theory.\n\nAlthough we have only considered some simple examples, we would like to stress that these results are quite generic and can be carried over to all orbifold gauge theories. A future direction would be to try to be more systematic and analyze the various possibilities encountered in more complex N = 2 and N = 1 models. In a similar spirit, one should analyze the multi-instanton contributions as well, since the total correction to the holomorphic quantities will be the sum of all such terms. The study of the zero-modes is expected to be even more relevant in this case as it will probably make many contributions vanish. With an eye to string phenomenology, one should also incorporate these models into globally consistent compactifications and study the effects of these terms there.\n\nLastly, it would be interesting to study the dynamical implications of some of the terms generated. We briefly touched upon this at the end of section 4 when we mentioned the stabilizing effect of the exotic instanton on the USp(N) theory. Although from the strict field theory point of view these terms are thought of as ordinary polyno-26 mial terms in the holomorphic quantities, 7 they are \"special\" when seen from the point of view of string theory and they might therefore induce a particular type of dynamics."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgements",
"text": "We would like to thank many people for discussions and email exchanges at various stages of this work that helped us sharpen the focus of the presentation: M. Bianchi, M. Billò, P. Di Vecchia, S. Franco, M. Frau, F. Fucito, S. Kachru, R. Marotta, L. Martucci, F. Morales, B. E. W. Nilsson, D. Persson, I. Pesando, D. Robles-Llana, R. Russo, A. Tanzini, A. Tomasiello, A. Uranga, T. Weigand and N. Wyllard. R.A., M.B. and A.L. are partially supported by the European Commission FP6 Programme MRTN-CT-2004-005104, in which R.A is associated to V.U. Brussel, M.B. to University of Padova and A.L. to University of Torino. R.A. is a Research Associate of the Fonds National de la Recherche Scientifique (Belgium). The research of R.A. is also supported by IISN -Belgium (convention 4.4505.86) and by the \"Interuniversity Attraction Poles Programme -Belgian Science Policy\". M.B. is also supported by Italian MIUR under contract PRIN-2005023102 and by a MIUR fellowship within the program \"Rientro dei Cervelli\". The research of G.F. is supported by the Swedish Research Council (Vetenskapsrådet) contracts 622-2003-1124 and 621-2002-3884. A.L. thanks the Galileo Galilei Institute for the hospitality and support during the completion of this work.\n\n[8] L. E. Ibanez and A. M. Uranga, arXiv:hep-th/0609213. [9] B. Florea, S. Kachru, J. McGreevy and N. Saulina, arXiv:hep-th/0610003. [10] S. A. Abel and M. D. Goodsell, arXiv:hep-th/0612110. [11] N. Akerblom, R. Blumenhagen, D. Lust, E. Plauschinn and M. Schmidt-Sommerfeld, arXiv:hep-th/0612132. [12] M. Bianchi and E. Kiritsis, arXiv:hep-th/0702015. [13] M. Cvetic, R. Richter and T. Weigand, arXiv:hep-th/0703028. [14] M. Bianchi, F. Fucito, J.F. Morales, arXiv:0704.0784 [hep-th]. [15] K. Intriligator and N. Seiberg, JHEP 0602 (2006) 031 [arXiv:hep-th/0512347]. [16] R. Argurio, M. Bertolini, C. Closset and S. Cremonesi, JHEP 0609, 030 (2006) [arXiv:hep-th/0606175]. [17] R. Argurio, M. Bertolini, S. Franco and S. Kachru, arXiv:hep-th/0703236. [18] N. Dorey, T. J. Hollowood, V. V. Khoze and M. P. Mattis, Phys. Rept. 371 (2002) 231 [arXiv:hep-th/0206063]. [19] M. Bianchi, S. Kovacs and G. Rossi, arXiv:hep-th/0703142. [20] M. Billo, M. Frau, S. Sciuto, G. Vallone, and A. Lerda, JHEP 0603 (2006) 023 [arXiv:hep-th/0511036]. [21] M. Billo, M. Frau, I. Pesando and A. Lerda, JHEP 0405 (2004) 023 [arXiv:hepth/0402160]; M. Billo, M. Frau, F. Lonegro and A. Lerda, JHEP 0505 (2005) 047 [arXiv:hep-th/0502084]; M. Billo, M. Frau, F. Fucito and A. Lerda, JHEP 0611 (2006) 012 [arXiv:hep-th/0606013]. [22] E. Witten, Nucl. Phys. B 460 (1996) 335 [arXiv:hep-th/9510135]. [23] M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Yu. I. Manin, Phys. Lett. A 65, 185 (1978). [24] M. Bertolini, P. Di Vecchia, G. Ferretti and R. Marotta, Nucl. Phys. B 630 (2002) 222 [arXiv:hep-th/0112187]. [25] M. Bertolini, Int. J. Mod. Phys. A 18 (2003) 5647 [arXiv:hep-th/0303160]. [26] S. Franco, A. Hanany, F. Saad and A. M. Uranga, JHEP 0601 (2006) 011 [arXiv:hepth/0505040].\n\n[27] T. R. Taylor, G. Veneziano and S. Yankielowicz, Nucl. Phys. B 218 (1983) 493. [28] I. Affleck, M. Dine and N. Seiberg, Phys. B 241 (1984) 493. [29] D. Berenstein, C. P. Herzog, P. Ouyang and S. Pinansky, JHEP 0509 (2005) 084 [arXiv:hep-th/0505029]. [30] M. Bertolini, F. Bigazzi and A. L. Cotrone, Phys. Rev. D 72 (2005) 061902 [arXiv:hepth/0505055]. [31] D. Friedan, E. J. Martinec and S. H. Shenker, Nucl. Phys. B 271 (1986) 93.\n\n[32] L. J. Dixon, D. Friedan, E. J. Martinec and S. H. Shenker, Nucl. Phys. B 282 (1987) 13.\n\n[33] P. Di Vecchia, M. Frau, I. Pesando, S. Sciuto, A. Lerda and R. Russo, Nucl. Phys. B 507 (1997) 259 [arXiv:hep-th/9707068]; M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda, R. Marotta and I. Pesando, JHEP 0102 (2001) 014 [arXiv:hep-th/0011077]; M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda and R. Marotta, Nucl. Phys. B 621 (2002) 157 [arXiv:hep-th/0107057]. [34] P. Di Vecchia and A. Liccardo, NATO Adv. Study Inst. Ser. C. Math. Phys. Sci. 556 (2000) 1 [arXiv:hep-th/9912161]. [35] L. Martucci, J. Rosseel, D. Van den Bleeken and A. Van Proeyen, Class. Quant. Grav. 22, 2745 (2005) [arXiv:hep-th/0504041]. [36] E. Bergshoeff, R. Kallosh, A. K. Kashani-Poor, D. Sorokin and A. Tomasiello, JHEP 0510, 102 (2005) [arXiv:hep-th/0507069]. [37] G. Pradisi and A. Sagnotti, Phys. Lett. B 216 (1989) 59. [38] E. G. Gimon and J. Polchinski, Phys. Rev. D 54 (1996) 1667 [arXiv:hep-th/9601038]. [39] M. R. Douglas and G. W. Moore, arXiv:hep-th/9603167. [40] M. Berkooz and R. G. Leigh, Nucl. Phys. B 483 (1997) 187 [arXiv:hep-th/9605049]. [41] G. Zwart, Nucl. Phys. B 526 (1998) 378 [arXiv:hep-th/9708040]. [42] K. A. Intriligator and P. Pouliot, Phys. Lett. B 353 (1995) 471 [arXiv:hep-th/9505006]."
}
] | |
arxiv:0704.0264 | 0704.0264 | 1 | 0ab569bfe706c57e3dc4b25b50ee03583e2c1873f6ddb49382cb12928fa8323e | Gluon Radiation of an Expanding Color Skyrmion in the Quark-Gluon Plasma | The density of states and energy spectrum of the gluon radiation are
calculated for the color current of an expanding hydrodynamic skyrmion in the
quark gluon plasma with a semiclassical method. Results are compared with those
in literatures. | [
"Jian Dai"
] | [
"hep-ph",
"hep-th",
"nucl-th"
] | hep-ph | [] | 2007-04-02 | 2026-02-26 | 1 E-mail: jdai@sci.ccny.cuny.edu
In this letter, we address the issue of gluon radiation during the hydrodynamic stage in the evolution of the deconfined hot QCD matter or quark gluon plasma (QGP) [1] (for review see for example [2] ).
The medium induced gluon radiation has been thoroughly explored in the context of final state partonic energy loss or "jet quenching" [3] . The spatially extended nuclear matter affects the processes of fragmentation and hadronization of the hard partons produced in the relativistic heavy ion collisions. Essentially all high p ⊥ hadronic observables are affected at collider energies and the degree of the medium modification can give a characterization of the hot QCD matter in the deconfined phase. In principle, the medium induced radiation effect emerges from thermal QCD per se. However, in practice, different approximation schemes are applied giving consistent results [4, 5] . On the other hand, gluon radiation has also been considered in the context of gluon density saturation in the initial stage, where a strongly interacting gluonic atmosphere is crucial for the rapid local thermalization for the deconfined QCD matter [6] .
The time evolution of the RHIC "fireball" can influence the observable particle production spectra. Given a strong initial interaction, the resulting state of matter is usually modeled as a relativistic fluid undergoing a hydrodynamic flow. Generalized fluid mechanics that characterizes the long-distance physics of the transport of color charges has been developed for this purpose [7] (for review see [8] ). Recently, we discovered a type of single skyrmion solutions in color fluid [9] . Moreover, we found an interesting case in which the time-dependent skyrmion expands in time, which is in accordance with the expanding nature of the fireball generated in RHIC experiments [10] . The pattern of gluon radiation pertaining to the color current of these non-static configurations is an important character of this color skyrmion. So in this letter we calculate this radiation spectrum in a semiclassical approach. The main results from our calculation are the following. There is a fast fall-off in the UV side of the spectrum but a smooth peak dominates the intermediate energy. And in IR, a long tail is the characteristic feature.
The organization of this paper is the following. In Sect. 2, after a brief review of the nonabelian fluid mechanics, we calculate the nonabelian current corresponding to the soliton solution. In Sect. 3, semiclassical gluon radiation is calculated. In Sect. 4, comparison of the radiation spectrum in our hydrodynamic approach and in other approaches is carried out.
Given the thermalization of hot QCD matter above the deconfinement transition temperature, the transport of the color charges in the volume of the nuclear size can be modeled by a nonlinear sigma model in a first-order formalism
L = j µ ω µ -F (n) -g ef f J aµ A a µ . ( 1
)
This nonlinear sigma model describes an ideal fluid system. The configuration of this fluid is described by a group element field U , which shows up in the velocity field ω µ
ω µ = - i 2 T r(σ 3 U † ∂ µ U ). ( 2
)
Conjugate to the velocity is the abelian charge current j µ . It is easy to see that the first term in the lagrangian density (1) gives rise to the canonical structure of the fluid system. The fact that we will consider only one abelian charge current means that U takes value in an SU (2) group. The information about the equation of state (EOS) of the fluid is contained in the second term, which is essentially the free energy density of the fluid. In fact, energy and pressure densities are given by the ideal fluid formula
ǫ = F, p = nF ′ -F. ( 3
)
Here n is the invariant length of j µ , n 2 = j µ j µ . The third term is the gauge coupling of the fluid with an external gluon field A a µ with an effective coupling g ef f . J aµ is the nonabelian charge current which is related to the abelian current by the Eckart factorization J aµ = Q a j µ where Q a is the nonabelian charge density of the fluid configuration
Q a = 1 2 T r(σ 3 U † σ a U ). ( 4
)
For SU (2) group, a = 1, 2, 3.
When the temperature is relatively high, we approximate the EOS by
ǫ = 3p ( 5
)
which is known in relativistic fluid mechanics to describe radiation. As a result, the free energy density can be obtained by integrating Eq. (3),
F = β 4/3 n 4/3 ( 6
)
where β is a dimensionless constant of integration. In this case, and without an external gluon field, the fluid system in (1) possesses a class of expanding soliton solutions which can be studied via variational and collective coordinate methods [10] .
U = U x R(t) , R(t) ≈ R 0 ( t τ + 1) 4/3 θ(t) ( 7
) 2
where R 0 and τ are the spacial and temporal characterizations of the variational soliton and θ(t) the usual step function in time direction. Physically, it is certainly very interesting to understand the origin of these two scales from a fundamental level. The approximation in (7) is valid provided τ ≪ R 0 . This condition enables us to define a small parameter
λ = τ R 0 . ( 8
)
For our purpose, we calculate the nonabelian current in (1) corresponding to the soliton solution in Eq. ( 7 ). To do so, the hedgehog ansatz is specified for the solution (7)
U = cos φ + iσ • x sin φ ( 9
)
where x is the unit vector and φ is given by the stereographic map
sin φ = 2s 1 + s 2 , cos φ = ± 1 -s 2 1 + s 2 . ( 10
)
We write s as the dimensionless coordinate x/R(t). The sign in the expression of cos φ signifies a topological charge which is the skyrmion number. The negative sign gives the skyrmion number +1 or a skyrmion and the positive sign the skyrmion number is -1 or an anti-skyrmion. We will take the positive sign in the following. By expressing the abelian current j µ in terms of the velocity ω µ through the equation of motion, we derive the following expression for the nonabelian current
d 3 xJ aµ = 2 β 3 • d 3 s (1 + s 2 ) 6 • (ŝ 2 3 s 2 Ṙ2 -1) • δ a 3 (1 -6s 2 + s 4 ) + 4ǫ a3b ŝb s(1 -s 2 ) + 8ŝ 3 ŝa s 2 • -ŝ 3 s(1 + s 2 ) Ṙ 2ŝ 1 ŝ3 s 2 -2ŝ 2 s 2ŝ 2 ŝ3 s 2 + 2ŝ 1 s 2ŝ 2 3 s 2 -s 2 + 1 .( 11
)
The current in (11) has a natural form of a multipole expansion due to the skyrmion orientation in the color space. In this letter we only consider the effect of the lowest mode and the effects of higher polarization will be considered elsewhere. The spherically symmetric part in the current is contained only in the third component
d 3 xJ a3 0 = -δ a 3 2 β 3 d 3 s (1 + s 2 ) 6 P 6 (s) ( 12
)
where P 6 (s) = 1 -7s 2 + 7s 4s 6 .
Now we consider the interaction between the expanding color skyrmion and the hard partons. Since the transfer momentum between hard partons is in high order to that between hard 3 parton and soliton, we expect a hierarchy between the partonic coupling g Y M and the effective coupling g ef f . Accordingly, gluon self-interaction in terms like F a µν F aµν can be omitted so we can work with a free parton picture. Then the gauge coupling in (1) becomes the coupling between a classical current and a free quantum field for gluon. In this approximation, the lowest order semiclassical amplitude is given by
iM = g ef f 1| d 4 xJ aµ Âa µ |0 . ( 13
)
|0 and |1 are gluonic Fock vacuum and one-gluon state. The gluon factor in (13) is given by the wave function
1| Âa µ (x)|0 = ϕ a ε µ e ik•x √ 2ω ( 14
)
where the color and helicity parts ϕ, ε will be summed over eventually. Putting the current in, we have
iM = A(k) dte iωt d 3 s (1 + s 2 ) 6 e -iR(t)k•s P 6 (s) ( 15
)
where A(k) = -(2/β) 3 g ef f ϕ 3 ε 3 / √ 2ω. The spatial Fourier transformation can be completed analytically
iM = B(k) dte iωt-R(t)k Q 4 (R(t)k) ( 16
)
where B(k) = π 2 A(k)/120 and Q 4 (x) = 5x 2 -5x 3 + x 4 . To go further, we need to specify R(t) in this equation to the form given in (7) . This gives iM = B(k)e -iωτ η ω ∞ ωτ η
dte iηt-t 4/3 Q 4 (t 4/3 ) ( 17
)
where η = ωτ /(kR 0 ) 3/4 . With onshell condition ω = k, η = λκ 1/4 where κ is defined to be R 0 k. Accordingly,
iM = - π 2 15 √ 2 g ef f β 3 λR 3/2 0 ϕ 3 ε 3 e -iωτ i M λ (κ) κ 5/4 ( 18
)
where
i M λ (κ) = ∞ κ 3/4 dte iλκ 1/4 t-t 4/3 Q 4 (t 4/3 ) ( 19
)
The radiation spectrum is given by dE = kdN . E(k) is the total energy radiated over the entire time of expansion as a function of k. The number distribution is
dN = c,h |M| 2 d 3 k ( 20
) 4
where the summation is over colors and helicities of the gluon. In a spherically symmetric setting, dN = ndk where n is the density of states
n = 4πk 2 c,h |M| 2 . ( 21
)
By straightforward calculation,
n = αR 0 λ 2 κ -1/2 | M λ (κ)| 2 , (22) dE dk = αλ 2 κ 1/2 | M λ (κ)| 2 . ( 23
)
where α ≡ (2π 5 /225)(g 2 ef f /β 6 ). The numerical results for λ = 1/15, 2/15, 1/5 are given in Fig. 1. 0.2 0.5 1 2 5 10 Κ 0 0.05 0.1 0.15 0.2 n ΑR 0 0.2 0.5 1 2 5 10 Κ 0 0.05 0.1 0.15 0.2 0.25 0.3 dE Αdk Figure 1: Density of states and energy spectrum for λ = 1/5 (Black), 2/15 (Deep Gray) and 1/15 (Light Gray).
Understanding the pattern of gluon radiation in relativistic heavy ion collision processes is important for making an accurate determination of the physical mechanisms from the measurement of its decay products.
In [6], the authors extracted the asymptotic behavior of the number density in small k is of the 1/k form. In our case, the asymptotic of the number density in small k is ∼ 1/ √ k. (See Fig. 2 .) The difference comes from the fact that the medium size is taken to be infinitely large in [6] while in our case the medium size is characterized by the soliton size R 0 . So the IR behavior in our case is softer.
For the case of jet quenching, the radiation energy lost is due to scattering off the hard quarks. A popular approach is to model the medium as a collection of colored static scattering 5 0.02 0.05 0.1 0.2 0.5 1 Κ 0.11 0.115 0.12 0.125 0.13 0.135 0.14 n Κ ΑR 0 Figure 2: n/(1/ √ κ) in small k for λ = .2 centers [11] . This approach can be extended to the expanding medium [4] though the gluon radiation by the expanding medium itself is not included. In fact, the medium induced gluon radiation is characterized by the frequency
ω C = 1 2 qL 2 ( 24
)
where q is the quenching parameter, estimated to be .04 ∼ .16GeV 2 /f m, and L is the inmedium path length of a hard parton [12] . In general ω C is significantly larger than the characteristic momentum in our case 1/R 0 . So there is a hierarchy between the medium induced gluon radiation spectrum and the gluon radiation spectrum by the medium.
Our hydrodynamical approach opens up another interesting possibility to address the eccentricity of the elliptic flow either intrinsically by considering the nonabelian color current or exogenously by considering the gluon radiation patterns. This will be the topic of the follow-up to this work.
Acknowledgment. This work was supported by a CUNY Collaborative Research Incentive grant. The author has greatly benefited from the mentoring by V. P. Nair. 6 References [1] PHENIX Collaboration, K. Adcox, et al, Nucl. Phys. A757 (2005) 184-283, nuclex/0410003; I. Arsene et al. BRAHMS collaboration, Nucl. Phys. A757 (2005) 1-27, nucl-ex/0410020; B. B. Back et al (PHOBOS), Nucl. Phys. A757 (2005) 28-101, nuclex/0410022; STAR Collaboration: J. Adams, et al, Nucl. Phys. A757 (2005) 102-183, nucl-ex/0501009. [2] Berndt Muller, James L. Nagle, nucl-th/0602029. [3] Alexander Kovner, Urs A. Wiedemann, "Gluon Radiation and Parton Energy Loss", in Quark Gluon Plasma 3 Editors: R. C. Hwa and X. Wang World Scientific Singapore, hep-ph/0304151. [4] Carlos A. Salgado, Urs Achim Wiedemann, Phys. Rev. D68 (2003) 014008, hepph/0302184. [5] Urs A. Wiedemann, Nucl. Phys. B588 (2000) 303, hep-ph/0005129. [6] Yuri V. Kovchegov, Dirk H. Rischke, Phys. Rev. C56 (1997) 1084, hep-ph/9704201. [7] R. Jackiw, V.P. Nair, So-Young Pi, Phys. Rev. D62 (2000) 085018, hep-th/0004084; B. Bistrovic, R. Jackiw, H. Li, V.P. Nair, S.-Y. Pi, Phys. Rev. D67 (2003) 025013, hep-th/0210143. [8] R. Jackiw, V.P. Nair, S.-Y. Pi, A.P. Polychronakos, J. Phys.A. Math. Gen. 37 (2004) R327. [9] Jian Dai, V.P. Nair, Phys. Rev. D74 (2006) 085014, hep-ph/0605090. [10] Jian Dai, "Stability and Evolution of Color Skyrmions in the Quark-Gluon Plasma", hep-ph/0612260. [11] M. Gyulassy, X. Wang, Nucl. Phys. B420 (1994) 583. [12] Miklos Gyulassy, Ivan Vitev, Xin-Nian Wang, Ben-Wei Zhang, "Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter", in Quark Gluon Plasma 3 Editors: R. C. Hwa and X. Wang World Scientific Singapore, nucl-th/0302077. 7 | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "1 E-mail: jdai@sci.ccny.cuny.edu"
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "In this letter, we address the issue of gluon radiation during the hydrodynamic stage in the evolution of the deconfined hot QCD matter or quark gluon plasma (QGP) [1] (for review see for example [2] ).\n\nThe medium induced gluon radiation has been thoroughly explored in the context of final state partonic energy loss or \"jet quenching\" [3] . The spatially extended nuclear matter affects the processes of fragmentation and hadronization of the hard partons produced in the relativistic heavy ion collisions. Essentially all high p ⊥ hadronic observables are affected at collider energies and the degree of the medium modification can give a characterization of the hot QCD matter in the deconfined phase. In principle, the medium induced radiation effect emerges from thermal QCD per se. However, in practice, different approximation schemes are applied giving consistent results [4, 5] . On the other hand, gluon radiation has also been considered in the context of gluon density saturation in the initial stage, where a strongly interacting gluonic atmosphere is crucial for the rapid local thermalization for the deconfined QCD matter [6] .\n\nThe time evolution of the RHIC \"fireball\" can influence the observable particle production spectra. Given a strong initial interaction, the resulting state of matter is usually modeled as a relativistic fluid undergoing a hydrodynamic flow. Generalized fluid mechanics that characterizes the long-distance physics of the transport of color charges has been developed for this purpose [7] (for review see [8] ). Recently, we discovered a type of single skyrmion solutions in color fluid [9] . Moreover, we found an interesting case in which the time-dependent skyrmion expands in time, which is in accordance with the expanding nature of the fireball generated in RHIC experiments [10] . The pattern of gluon radiation pertaining to the color current of these non-static configurations is an important character of this color skyrmion. So in this letter we calculate this radiation spectrum in a semiclassical approach. The main results from our calculation are the following. There is a fast fall-off in the UV side of the spectrum but a smooth peak dominates the intermediate energy. And in IR, a long tail is the characteristic feature.\n\nThe organization of this paper is the following. In Sect. 2, after a brief review of the nonabelian fluid mechanics, we calculate the nonabelian current corresponding to the soliton solution. In Sect. 3, semiclassical gluon radiation is calculated. In Sect. 4, comparison of the radiation spectrum in our hydrodynamic approach and in other approaches is carried out."
},
{
"section_type": "OTHER",
"section_title": "2 Color current of an expanding soliton",
"text": "Given the thermalization of hot QCD matter above the deconfinement transition temperature, the transport of the color charges in the volume of the nuclear size can be modeled by a nonlinear sigma model in a first-order formalism\n\nL = j µ ω µ -F (n) -g ef f J aµ A a µ . ( 1\n\n)\n\nThis nonlinear sigma model describes an ideal fluid system. The configuration of this fluid is described by a group element field U , which shows up in the velocity field ω µ\n\nω µ = - i 2 T r(σ 3 U † ∂ µ U ). ( 2\n\n)\n\nConjugate to the velocity is the abelian charge current j µ . It is easy to see that the first term in the lagrangian density (1) gives rise to the canonical structure of the fluid system. The fact that we will consider only one abelian charge current means that U takes value in an SU (2) group. The information about the equation of state (EOS) of the fluid is contained in the second term, which is essentially the free energy density of the fluid. In fact, energy and pressure densities are given by the ideal fluid formula\n\nǫ = F, p = nF ′ -F. ( 3\n\n)\n\nHere n is the invariant length of j µ , n 2 = j µ j µ . The third term is the gauge coupling of the fluid with an external gluon field A a µ with an effective coupling g ef f . J aµ is the nonabelian charge current which is related to the abelian current by the Eckart factorization J aµ = Q a j µ where Q a is the nonabelian charge density of the fluid configuration\n\nQ a = 1 2 T r(σ 3 U † σ a U ). ( 4\n\n)\n\nFor SU (2) group, a = 1, 2, 3.\n\nWhen the temperature is relatively high, we approximate the EOS by\n\nǫ = 3p ( 5\n\n)\n\nwhich is known in relativistic fluid mechanics to describe radiation. As a result, the free energy density can be obtained by integrating Eq. (3),\n\nF = β 4/3 n 4/3 ( 6\n\n)\n\nwhere β is a dimensionless constant of integration. In this case, and without an external gluon field, the fluid system in (1) possesses a class of expanding soliton solutions which can be studied via variational and collective coordinate methods [10] .\n\nU = U x R(t) , R(t) ≈ R 0 ( t τ + 1) 4/3 θ(t) ( 7\n\n) 2\n\nwhere R 0 and τ are the spacial and temporal characterizations of the variational soliton and θ(t) the usual step function in time direction. Physically, it is certainly very interesting to understand the origin of these two scales from a fundamental level. The approximation in (7) is valid provided τ ≪ R 0 . This condition enables us to define a small parameter\n\nλ = τ R 0 . ( 8\n\n)\n\nFor our purpose, we calculate the nonabelian current in (1) corresponding to the soliton solution in Eq. ( 7 ). To do so, the hedgehog ansatz is specified for the solution (7)\n\nU = cos φ + iσ • x sin φ ( 9\n\n)\n\nwhere x is the unit vector and φ is given by the stereographic map\n\nsin φ = 2s 1 + s 2 , cos φ = ± 1 -s 2 1 + s 2 . ( 10\n\n)\n\nWe write s as the dimensionless coordinate x/R(t). The sign in the expression of cos φ signifies a topological charge which is the skyrmion number. The negative sign gives the skyrmion number +1 or a skyrmion and the positive sign the skyrmion number is -1 or an anti-skyrmion. We will take the positive sign in the following. By expressing the abelian current j µ in terms of the velocity ω µ through the equation of motion, we derive the following expression for the nonabelian current\n\nd 3 xJ aµ = 2 β 3 • d 3 s (1 + s 2 ) 6 • (ŝ 2 3 s 2 Ṙ2 -1) • δ a 3 (1 -6s 2 + s 4 ) + 4ǫ a3b ŝb s(1 -s 2 ) + 8ŝ 3 ŝa s 2 • -ŝ 3 s(1 + s 2 ) Ṙ 2ŝ 1 ŝ3 s 2 -2ŝ 2 s 2ŝ 2 ŝ3 s 2 + 2ŝ 1 s 2ŝ 2 3 s 2 -s 2 + 1 .( 11\n\n)\n\nThe current in (11) has a natural form of a multipole expansion due to the skyrmion orientation in the color space. In this letter we only consider the effect of the lowest mode and the effects of higher polarization will be considered elsewhere. The spherically symmetric part in the current is contained only in the third component\n\nd 3 xJ a3 0 = -δ a 3 2 β 3 d 3 s (1 + s 2 ) 6 P 6 (s) ( 12\n\n)\n\nwhere P 6 (s) = 1 -7s 2 + 7s 4s 6 ."
},
{
"section_type": "OTHER",
"section_title": "Semiclassical gluon radiation",
"text": "Now we consider the interaction between the expanding color skyrmion and the hard partons. Since the transfer momentum between hard partons is in high order to that between hard 3 parton and soliton, we expect a hierarchy between the partonic coupling g Y M and the effective coupling g ef f . Accordingly, gluon self-interaction in terms like F a µν F aµν can be omitted so we can work with a free parton picture. Then the gauge coupling in (1) becomes the coupling between a classical current and a free quantum field for gluon. In this approximation, the lowest order semiclassical amplitude is given by\n\niM = g ef f 1| d 4 xJ aµ Âa µ |0 . ( 13\n\n)\n\n|0 and |1 are gluonic Fock vacuum and one-gluon state. The gluon factor in (13) is given by the wave function\n\n1| Âa µ (x)|0 = ϕ a ε µ e ik•x √ 2ω ( 14\n\n)\n\nwhere the color and helicity parts ϕ, ε will be summed over eventually. Putting the current in, we have\n\niM = A(k) dte iωt d 3 s (1 + s 2 ) 6 e -iR(t)k•s P 6 (s) ( 15\n\n)\n\nwhere A(k) = -(2/β) 3 g ef f ϕ 3 ε 3 / √ 2ω. The spatial Fourier transformation can be completed analytically\n\niM = B(k) dte iωt-R(t)k Q 4 (R(t)k) ( 16\n\n)\n\nwhere B(k) = π 2 A(k)/120 and Q 4 (x) = 5x 2 -5x 3 + x 4 . To go further, we need to specify R(t) in this equation to the form given in (7) . This gives iM = B(k)e -iωτ η ω ∞ ωτ η\n\ndte iηt-t 4/3 Q 4 (t 4/3 ) ( 17\n\n)\n\nwhere η = ωτ /(kR 0 ) 3/4 . With onshell condition ω = k, η = λκ 1/4 where κ is defined to be R 0 k. Accordingly,\n\niM = - π 2 15 √ 2 g ef f β 3 λR 3/2 0 ϕ 3 ε 3 e -iωτ i M λ (κ) κ 5/4 ( 18\n\n)\n\nwhere\n\ni M λ (κ) = ∞ κ 3/4 dte iλκ 1/4 t-t 4/3 Q 4 (t 4/3 ) ( 19\n\n)\n\nThe radiation spectrum is given by dE = kdN . E(k) is the total energy radiated over the entire time of expansion as a function of k. The number distribution is\n\ndN = c,h |M| 2 d 3 k ( 20\n\n) 4\n\nwhere the summation is over colors and helicities of the gluon. In a spherically symmetric setting, dN = ndk where n is the density of states\n\nn = 4πk 2 c,h |M| 2 . ( 21\n\n)\n\nBy straightforward calculation,\n\nn = αR 0 λ 2 κ -1/2 | M λ (κ)| 2 , (22) dE dk = αλ 2 κ 1/2 | M λ (κ)| 2 . ( 23\n\n)\n\nwhere α ≡ (2π 5 /225)(g 2 ef f /β 6 ). The numerical results for λ = 1/15, 2/15, 1/5 are given in Fig. 1. 0.2 0.5 1 2 5 10 Κ 0 0.05 0.1 0.15 0.2 n ΑR 0 0.2 0.5 1 2 5 10 Κ 0 0.05 0.1 0.15 0.2 0.25 0.3 dE Αdk Figure 1: Density of states and energy spectrum for λ = 1/5 (Black), 2/15 (Deep Gray) and 1/15 (Light Gray)."
},
{
"section_type": "DISCUSSION",
"section_title": "Comparison and discussion",
"text": "Understanding the pattern of gluon radiation in relativistic heavy ion collision processes is important for making an accurate determination of the physical mechanisms from the measurement of its decay products.\n\nIn [6], the authors extracted the asymptotic behavior of the number density in small k is of the 1/k form. In our case, the asymptotic of the number density in small k is ∼ 1/ √ k. (See Fig. 2 .) The difference comes from the fact that the medium size is taken to be infinitely large in [6] while in our case the medium size is characterized by the soliton size R 0 . So the IR behavior in our case is softer.\n\nFor the case of jet quenching, the radiation energy lost is due to scattering off the hard quarks. A popular approach is to model the medium as a collection of colored static scattering 5 0.02 0.05 0.1 0.2 0.5 1 Κ 0.11 0.115 0.12 0.125 0.13 0.135 0.14 n Κ ΑR 0 Figure 2: n/(1/ √ κ) in small k for λ = .2 centers [11] . This approach can be extended to the expanding medium [4] though the gluon radiation by the expanding medium itself is not included. In fact, the medium induced gluon radiation is characterized by the frequency\n\nω C = 1 2 qL 2 ( 24\n\n)\n\nwhere q is the quenching parameter, estimated to be .04 ∼ .16GeV 2 /f m, and L is the inmedium path length of a hard parton [12] . In general ω C is significantly larger than the characteristic momentum in our case 1/R 0 . So there is a hierarchy between the medium induced gluon radiation spectrum and the gluon radiation spectrum by the medium.\n\nOur hydrodynamical approach opens up another interesting possibility to address the eccentricity of the elliptic flow either intrinsically by considering the nonabelian color current or exogenously by considering the gluon radiation patterns. This will be the topic of the follow-up to this work.\n\nAcknowledgment. This work was supported by a CUNY Collaborative Research Incentive grant. The author has greatly benefited from the mentoring by V. P. Nair. 6 References [1] PHENIX Collaboration, K. Adcox, et al, Nucl. Phys. A757 (2005) 184-283, nuclex/0410003; I. Arsene et al. BRAHMS collaboration, Nucl. Phys. A757 (2005) 1-27, nucl-ex/0410020; B. B. Back et al (PHOBOS), Nucl. Phys. A757 (2005) 28-101, nuclex/0410022; STAR Collaboration: J. Adams, et al, Nucl. Phys. A757 (2005) 102-183, nucl-ex/0501009. [2] Berndt Muller, James L. Nagle, nucl-th/0602029. [3] Alexander Kovner, Urs A. Wiedemann, \"Gluon Radiation and Parton Energy Loss\", in Quark Gluon Plasma 3 Editors: R. C. Hwa and X. Wang World Scientific Singapore, hep-ph/0304151. [4] Carlos A. Salgado, Urs Achim Wiedemann, Phys. Rev. D68 (2003) 014008, hepph/0302184. [5] Urs A. Wiedemann, Nucl. Phys. B588 (2000) 303, hep-ph/0005129. [6] Yuri V. Kovchegov, Dirk H. Rischke, Phys. Rev. C56 (1997) 1084, hep-ph/9704201. [7] R. Jackiw, V.P. Nair, So-Young Pi, Phys. Rev. D62 (2000) 085018, hep-th/0004084; B. Bistrovic, R. Jackiw, H. Li, V.P. Nair, S.-Y. Pi, Phys. Rev. D67 (2003) 025013, hep-th/0210143. [8] R. Jackiw, V.P. Nair, S.-Y. Pi, A.P. Polychronakos, J. Phys.A. Math. Gen. 37 (2004) R327. [9] Jian Dai, V.P. Nair, Phys. Rev. D74 (2006) 085014, hep-ph/0605090. [10] Jian Dai, \"Stability and Evolution of Color Skyrmions in the Quark-Gluon Plasma\", hep-ph/0612260. [11] M. Gyulassy, X. Wang, Nucl. Phys. B420 (1994) 583. [12] Miklos Gyulassy, Ivan Vitev, Xin-Nian Wang, Ben-Wei Zhang, \"Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter\", in Quark Gluon Plasma 3 Editors: R. C. Hwa and X. Wang World Scientific Singapore, nucl-th/0302077. 7"
}
] | ||
arxiv:0704.0268 | 0704.0268 | 1 | 18e9df294e9c7831bbe67f56a5262e1041a2ed0cc8162203d75ad72592c837f1 | Automated Generation of Layout and Control for Quantum Circuits | We present a computer-aided design flow for quantum circuits, complete with
automatic layout and control logic extraction. To motivate automated layout for
quantum circuits, we investigate grid-based layouts and show a performance
variance of four times as we vary grid structure and initial qubit placement.
We then propose two polynomial-time design heuristics: a greedy algorithm
suitable for small, congestion-free quantum circuits and a dataflow-based
analysis approach to placement and routing with implicit initial placement of
qubits. Finally, we show that our dataflow-based heuristic generates better
layouts than the state-of-the-art automated grid-based layout and scheduling
mechanism in terms of latency and potential pipelinability, but at the cost of
some area. | [
"Mark Whitney",
"Nemanja Isailovic",
"Yatish Patel",
"John Kubiatowicz"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | We present a computer-aided design flow for quantum circuits, complete with automatic layout and control logic extraction. To motivate automated layout for quantum circuits, we investigate grid-based layouts and show a performance variance of four times as we vary grid structure and initial qubit placement. We then propose two polynomial-time design heuristics: a greedy algorithm suitable for small, congestionfree quantum circuits and a dataflow-based analysis approach to placement and routing with implicit initial placement of qubits. Finally, we show that our dataflow-based heuristic generates better layouts than the state-of-the-art automated grid-based layout and scheduling mechanism in terms of latency and potential pipelinability, but at the cost of some area.
Quantum computing offers us the opportunity to solve certain problems thought to be intractable on a classical machine. For example, the following classically hard problems benefit from quantum algorithms: factorization [19] , unsorted database search [6] , and simulation of quantum mechanical systems [26] .
In addition to significant work on quantum algorithms and underlying physics, there have been several studies exploring architectural trade-offs for quantum computers. Most such research [3, 16] has focused on simulating quantum algorithms on a fixed layout rather than on techniques for quantum circuit synthesis and layout generation. These studies tend
Classical Control: HDL Format (plus annotations for scheduling) Quantum Layout (including initial qubit placement) Basic Blocks Custom Modules Quantum Circuit Specification CAD Flow for Quantum Circuits New Custom Module The goal of our CAD flow is to automate the laying out of a quantum circuit to generate a physical layout, an intelligent initial placement of qubits, the associated classical control logic and annotations to help the online scheduler better use the layout optimizations as they were intended. This flow may then be used recursively to design larger blocks using previously created modules.
to use hand-generated and hand-optimized layouts on which efficient scheduling is then performed. While this approach is quite informative in a new field, it quickly becomes intractable as the size of the circuit grows.
Our goal is to automate most of the tasks involved in generating a physical layout and its associated control logic from a high-level quantum circuit specification (Figure 1 ). Our computer-aided design (CAD) flow should process a quantum circuit specification and produce the following:
• a physical layout in the desired technology
• an intelligent initial qubit placement in the layout
• classical control circuitry specified in some hardware description language (HDL), which may then be run through a classical CAD flow
• a set of annotations or "hints" for the online scheduler, allowing a tighter coupling of layout optimizations to actual runtime operation Much like a classical CAD flow, this quantum CAD flow is intended to be used hierarchically. We begin with a set of technology-specific basic blocks (some ion trap technology examples are given in Section 2). We then lay out some simple quantum circuits with the CAD flow, thus creating custom modules. The CAD flow may then be used recursively to create ever larger designs. This approach allows us to develop, evaluate and reuse design heuristics and avoids both the uncertainty and time-intensive nature of handgenerated layouts.
Quantum circuits that are large enough to be "interesting" require the orchestration of hundreds of thousands of physical components. In approaching such problems, it is important to build upon prior work in classical CAD flows. Although the specifics of quantum technologies (such as are discussed in Section 2) are different from classical CMOS technologies, prior work in CAD research can give us insight into how to approach the automated layout of quantum gates and channels. Further, quantum circuits exhibit some interesting properties that lend themselves to automatic synthesis and computer-aided design techniques:
Quantum ECC Quantum data is extremely fragile and consequently must remain encoded at all times -while being stored, moved, and computed upon. The encoded version of a circuit is often two or three orders of magnitude larger than the unencoded version. Further, the appropriate level of encoding may need to be selected as part of the layout process in order to achieve an appropriate "threshold" of error-free execution. Rather than burdening the designer with the complexities of adding fault-tolerance to a circuit, computer-aided synthesis, design and verification can perform such tasks automatically.
Ancillae Quantum computations use many helper qubits known as ancillae. Ancillae consist of bits that are constructed, utilized and recycled as part of a computation. Sometimes, ancillae are explicit in a designer's view of the circuit.
Often, however, they should be added automatically in the process of circuit synthesis, such as during the construction of fault-tolerant circuits from high-level circuit descriptions. An automatic design flow can insert appropriate circuits to generate and recycle ancillae without involving the designer.
Teleportation Quantum circuits present two possibilities for data transport: ballistic movement and teleportation. Ballistic movement is relatively simple over short distances in technologies such as ion traps (Section 2). Teleportation is an alternative that utilizes a higher-overhead distribution network of entangled quantum bits to distribute information with lower error over longer distances [9] . The choice to employ teleportation is ideally done after an initial layout has determined long communication paths. Consequently, it is a natural target for a computer-aided design flow.
In this paper, we make the following contributions:
• We propose a CAD flow for automated design of quantum circuits and detail the necessary components of the flow.
• We describe a technique for automatic synthesis of the classical control circuitry for a given layout.
• We show that different grid-based architectures, which have been the focus of most prior work in this field, exhibit vastly varying performance for the same circuit.
• We present heuristics for the placement and routing of quantum circuits in ion trap technology.
• We lay out some quantum error correction circuits and evaluate the effectiveness of the heuristics in terms of circuit area and latency. Each macroblock has a specific number of ports (shown as P0-P3) along with a set of electrodes used for ion movement and trapping. Some macroblocks contain a trap region where gates may be performed (black square).
The rest of this paper is organized as follows. We introduce our chosen technology in Section 2, followed by an overview of prior work in the field in Section 3. In Section 4, we detail our proposed CAD flow and our evaluation metrics. In Section 5, we describe the control circuitry interface and scheduling protocol that we use in the following sections. Section 6 contains a study of grid-based layouts, which have been the basis of most prior work on this subject. In Section 7, we present a greedy approach to laying out quantum circuits, followed in Section 8 by a much more scalable dataflow analysis-based approach to layout. Section 9 contains our experimental results for all three approaches to layout generation, and we conclude in Section 10.
For our initial study, we choose trapped ions [4, 17] as our substrate technology. Trapped ions have shown good potential for scalability [10] . In this technology, a physical qubit is an ion, and a gate is a location where a trapped ion may be operated upon by a modulated laser.
The ion is both trapped and ballistically moved by applying pulse sequences to discrete electrodes which line the edges of ion traps. Figure 3a shows an experimentally-demonstrated layout for a three-way intersection [7] . A qubit may be held in place at any trap region, or it may be ballistically moved between them using the gray electrodes lining the paths.
Rather than using ion traps as basic blocks, we define a library of macroblocks consisting of multiple traps for two reasons. First, macroblocks abstract out some of the low-level details, insulating our analyses from variations in the technology implementations of ion traps. Details such as which ion species is used, specific electrode sizing and geometry (clearly variable in the layout in Figure 3a ) and exact voltage levels necessary for trapping and movement are all encapsulated within the macroblock. Second, ballistic movement along a channel requires carefully timed application of pulse sequences to electrodes in nonadjacent traps. By defining basic blocks consisting of a few ion traps, we gain the benefit that crossing an interface between basic blocks requires communication only between the two blocks involved.
We use the library of macroblocks shown in Figure 2 , each of which consists of a 3x3 grid of trap regions and electrodes, with ports to allow qubit movement between macroblocks. The black squares are gate locations, which may not be performed at intersections or turns in ion trap technology. Each of these macroblocks may be rotated in a layout. This library is by no means exhaustive, however it does provide the major pieces necessary to construct many physical circuits. The macroblocks we present are abstractions of experimentally-demonstrated ion trap technology [7, 18] . In Figure 3 , we show how one can map a demonstrated layout (Figure 3a ) to our macroblock abstractions (Figure 3b ). We model this layout as a set of StraightChannel and ThreeWayIntersection macroblocks. Above the ion trap plane is an array of MEMS mirrors which routes laser pulses to the gate locations in order to apply quantum gates [11] , as shown in Figure 3c .
Some key differences between this quantum circuit technology and classical CMOS are as follows:
• "Wires" in ion traps consist of rectangular channels, lined with electrodes, with atomic ions suspended above the channel regions and moved ballistically [13] . Ballistic movement of qubits requires synchronized application of voltages on channel electrodes to move data around. Thus each wire requires movement control circuitry to handle any qubit communication.
• A by-product of the synchronous nature of the qubit wire channels is that these circuits can be used in a synchronous manner with no additional overhead. This enables some convenient pipelining options which will be discussed in Section 8.1.
Figure 3 : a) Experimentally demonstrated physical layout of a T-junction (three-way intersection). b) Abstraction of the circuit in (a), built using the StraightChannel and ThreeWayIntersection macroblocks shown in Figure 2 . c) The ion traps are laid out on a plane, above which is an array of MEMS mirrors used to route and split the laser beams that apply quantum gates.
• Each gate location will likely have the ability to perform any operation available in ion trap technology. This enables the reuse gate locations within a quantum circuit.
• Scalable ion trap systems will almost certainly be two-dimensional due to the difficulty of fabricating and controlling ion traps in a third dimension [8] . This means that all ion crossings must be intersections.
• Any routing channel may be shared by multiple ions as long as control circuits prevent multiion occupancy. Consequently, our circuit model resembles a general network, although scheduling the movement in a general networking model adds substantial complexity to our circuit.
• Movement latency of ions is not only dependent on Manhattan distance but also on the geometry of the wire channel. Experimentally, it has been shown that a right angle turn takes substantially longer than a straight channel over the same distance [18, 7] .
Prior research has laid the groundwork for our quantum circuit CAD flow. Svore et al [22, 23] proposed a design flow capable of pushing a quantum program down to physical operations. Their work outlined various file formats and provided initial implementations of some of the necessary tools. Similarly, Balensiefer et al [2, 3] proposed a design flow and compilation techniques to address fault-tolerance and provided some tools to evaluate simple layouts. While our CAD flow builds upon some of these ideas, we concentrate on automatic layout generation and control circuitry extraction. Additionally, initial hand-optimized layouts have been proposed in the literature. Metodi et al [15] proposed a uniform Quantum Logic Array architecture, which was later extended and improved in [24] . Their work concentrated on architectural research and did not delve into details of physical layout or scheduling. Finally, Metodi et al [16] created a tool to automatically generate a physical operations schedule given a quantum circuit and a fixed grid-based layout structure. We extend and improve upon their work by adding new scheduling heuristics capable of running on grid-based and non-grid-based layouts.
Maslov et al [14] have recently proposed heuristics for the mapping of quantum circuits onto molecules used in liquid state NMR quantum computing technology. Their algorithm starts with a molecule to be used for computation, modeled as a weighted graph with edges representing atomic couplings within the molecule. The dataflow graph of the circuit is mapped onto the molecule graph with an effort to minimize overall circuit runtime. Our techniques focus on circuit placement and routing in an ion trap technology and do not use a predefined physical substrate topology as in the NMR case. A new ion trap geometry is instead generated by our toolset for each circuit.
The ultimate goal of a quantum CAD flow is identical to that of a standard classical CAD flow: to automate
High-Level Description Synthesizer Tech-Independent Netlist Tech Mapping (Tech-Specific Gates, Encoding, Fault Tolerance) Tech Parameter File (Basic Blocks) Custom Modules Tech-Dependent Netlist Interconnect (if necessary) Placement and Routing Classical Control Synthesis for Qubit Movement Geometry-Aware Netlist the synthesis and laying out of a circuit. For a quantum CAD flow, the output circuit consists of both the quantum portion and the associated classical control logic.
The quantum CAD flow we present elaborates on the design flows described in prior works [3, 22, 23] . Unlike prior work, our CAD flow addresses the need to integrate automatic generation of classical control into the flow. Figure 4 shows an overview of our CAD toolset. Rectangles are tools, while ovals represent intermediate file formats. Our toolset is built to be as similar to classical CAD flows as possible, while still accounting for the differences between classical and quantum computing described in Section 1.1.
At the top, we begin with a high-level description of the desired quantum circuit. At present this specification consists of a sequence of quantum assembly language (QASM [3] ) instructions implementing the desired circuit, since this is a convenient format already being used by various third-party tools. We are currently investigating extension of this high-level description to other formats, such as schematic entry, mathematical formulae or a more general high-level language.
The synthesizer parses the QASM file and generates a technology-independent netlist stored in XML format. From this point onward (downward in the figure), all file formats are XML. Additionally, information may be modified or added but generally not removed. As we move down the flow, we add more and more low level details, but we also keep highlevel information such as encoded qubit groupings, nested layout modules, distinction between ancillae and data, etc. This allows low-level tools to make more intelligent decisions concerning qubit placement and channel needs based on high-level circuit structure. It likewise allows logical level modification at the lowest levels without having to attempt to deduce qubit groupings.
A technology parameter file specifies the complete set of basic blocks available for the layout (see examples in Figure 2 ), as well as design rules for connecting them. A basic block specification contains the following:
• the geometry of the block in enough detail to allow fabrication
• control logic for each operation possible within the block (including both movement and gates)
• control logic for handling each operation possible at each interface
The most basic function of the technology mapping tool is to take a technology-independent netlist and map it onto allowed basic blocks to create the technology-dependent netlist. This may be more or less complicated depending upon the complexity of the basic blocks. In addition, it may need to translate to technology-specific gates (in case the QASM file uses gates not available in this technology), encode the qubits used in the circuit (perhaps also automatically adding the ancilla and operation sequences necessary for error correction) and add fault tolerance to the final physical circuit.
In the initial technology-dependent netlist, all qubits are physical qubits, meaning that encoding levels have been set (though they may still be modified later). At this point, any technology-specific optimizations may optionally be applied to the physical circuit encapsulated in this netlist. Additionally, if the circuit is complex enough to warrant the inclusion of a teleportation-based interconnection network [9] , it is added to the netlist here using the higher level qubit grouping information in the netlist.
Once the designer is happy with the netlist, a placement and routing tool lays out the netlist and adds any further channels needed for communication. This geometry-aware netlist may be iterated upon as necessary to refine the layout. Once the layout is finalized, the classical control synthesis tool combines the control logic of the various components of the design, integrates interface control mechanisms to function properly and generates the unified control structure for the entire layout. Our control synthesis tool generates a Verilog file, which may then be run through a classical CAD flow for implementation.
The layout specification along with the control logic file together comprise the geometry-aware netlist, which is the end result for the quantum circuit initially specified in the high-level description. In order to allow hierarchical design of larger quantum circuits, we may now add this geometry-aware netlist to our set of custom modules. Future technology mappings may use both the basic blocks specified in the technology parameter file and any custom modules we create (or acquire).
The gray area in Figure 4 identifies the portions we shall be focusing on for the rest of this paper. We currently process the high-level description (a QASM file) directly into a technology-dependent netlist for ion traps using the macroblocks shown in Figure 2 . Thus we perform a tech mapping, but no automatic encoding, interconnect or addition of gates for fault tolerance. In this paper, we focus on laying out lowlevel circuits, such as those for encoded ancilla generation and error correction. The classical control synthesis box of the CAD flow is discussed in Section 5, while placement and routing are analyzed and compared in Sections 6, 7, 8 and 9.
We use two main metrics to evaluate the performance of our CAD flow: area and latency. For area, we consider the bounding box around the layout, so irregularly-shaped layouts are penalized (since they have wasted space). To determine latency of circuit execution, we use the scheduling heuristic described in Section 5.2 and extended in Section 8.3. A third metric of interest is fault-tolerance. For small layouts and circuits, we can use third-party tools to determine whether a given layout and schedule is faulttolerant [5] , but we do not currently use the fault-tolerance metric in our iterative design flow. We use area and latency because, to a first approximation, lower area and lower latency are likely to decrease decoherence. Previous algorithms to accurately determine the error tolerance of a quantum circuit have involved very computationally-intensive analyses that would be inappropriate for circuits with more than a few dozen gates [1] . However, we are looking into ways to incorporate fault tolerance as a metric.
The classical control system is responsible for executing the quantum circuit, including deciding where and when gate operations occur and tracking and managing every qubit in the system. It is composed of the following major components: instruction issue logic, gate control logic and macroblock control logic. Instruction issue logic handles all instruction scheduling and determines qubit movement paths. Gate control logic oversees laser resource arbitration, deciding which requested gate operations may occur at any given time. The macroblock control logic, which consists of an individual logic block for each macroblock in the system, handles all the internals of the macroblock, including details of gate operation for each gate possible within the macroblock, qubit movement within the macroblock and qubit movement into and out of the ports.
The first step in the control flow involves processing the quantum circuit's high-level description (the QASM file). The instruction issue logic accepts this stream of instructions as input and creates a series of qubit control messages. Using these qubit control messages, macroblock control logic blocks can determine where to move qubits and when to execute a gate operation. Qubit control messages are simple bit streams composed of a qubit ID, along with a sequence of commands, as shown in Figure 5 . When a qubit needs to perform an action, the instruction issue logic sends to it an appropriate control message which travels with the qubit as it traverses the layout. Once a macroblock receives a qubit and its corresponding control message, it uses the first command in the sequence to determine the operation it must perform. The macroblock then removes the com-Figure 5 : Example of how a qubit control message is constructed to move a qubit through a series of macroblocks. The qubit enters M0 and travels through M1 and M2, arriving at M3 where it is instructed to perform a CNOT. mand bits used and passes on the remaining control message to the next macroblock into which the qubit travels. In this manner, the instruction issue logic can create a multi-command qubit control message that specifies the path a qubit will traverse through consecutive macroblocks, along with where gate operations take place. The instruction issue logic only has to transmit this control message to the source macroblock, relying on the inter-macroblock communication interface to handle the rest.
Communication between the instruction issue logic and the macroblocks takes place using a shared control message bus in order to minimize the number of wire connections required by the instruction issue logic. Each macroblock listens to the control message bus for messages addressed to it and only processes messages with a destination ID that match the macroblock's ID. A macroblock is only responsible for monitoring the control message bus if it contains a qubit that has no remaining command bits. This condition generally occurs after a gate operation, when the instruction issue logic is deciding what action the qubit should take next. Once the instruction issue logic sends a new control message for the qubit, the macroblock resumes operation.
Macroblocks communicate with each other via control signals associated with each quantum port in the macroblock. Each port has signals to control qubit movement into the macroblock and signals to control movement out of the macroblock via that port. These signals are connected to the corresponding signals of the neighboring macroblocks. The macroblocks as-sert a request signal to a destination macroblock when a qubit command indicates the qubit should cross into the next macroblock. If an available signal response is received, the qubit, along with its control message, can move across into the neighboring macroblock; if not, the qubit must wait until the available signal is present.
The macroblock interface enables the instruction issue logic to schedule qubit movement as a path through a sequence of macroblocks, without concerning itself with the low level details of qubit movement. This modular system allows macroblocks to be replaced with any other macroblock that implements the defined interface, without modifying the instruction issue logic.
Additionally, macroblocks have an interface to the laser control logic. Whenever a macroblock is instructed to perform a gate operation, it must request a laser resource through the laser control logic. The laser controller is responsible for aggregating requests from all the macroblocks in the system, and deciding when and where to send laser pulses. The laser controller also attempts to parallelize as many operations as possible. Once the laser pulses have completed, the laser controller notifies the macroblocks, indicating that the gate operation is complete.
The instruction issue logic is responsible for determining the runtime execution order of the instructions in the quantum circuit, which involves both preprocessing and online scheduling. The instruction sequence is first preprocessed to assign priorities that will help during scheduling. The sequence is traversed from end to beginning, scheduling instructions as late as dependencies allow, using realistic gate latencies but ignoring movement. Essentially, each instruction is labeled with the length of its critical path to the end of the program. This is similar to the method used in [16] , but we use critical path with gate times rather than the size of the dependent subtree.
The instruction preprocessing generates an optimal schedule assuming infinite gates and zero movement cost. However, we wish to evaluate a layout with more realistic characteristics. Our scheduler is designed to schedule on an arbitrary graph, but the layouts provided to it by the place and route tool are in fact planar layouts using only right angles. In ad-dition, the scheduler requires that the qubit initial positions be provided as well.
Our scheduler implements a greedy scheduling technique. It keeps the set of instructions which have had all their dependencies fulfilled (and thus are ready to be executed). It attempts to schedule them in priority order. So the highest priority ready instruction (according to critical path) is attempted first and is thus more likely to get access to the resources it needs. These contested resources include both gates and channels/intersections. Once all possible instructions have been scheduled, time advances until one or more resources is freed and more instructions may be scheduled. This scheduling and stalling cycle continues until the full sequence has been executed or until deadlock occurs, in which case it is detected and the highest priority unscheduled instruction at the time of deadlock is reported.
Since we are interested in evaluating layouts rather than in designing an efficient online scheduler, we use very thorough searches over the graph in both gate assignment and pathfinding. This causes the scheduler to take longer but takes much of the uncertainty concerning schedule quality out of our tests. In addition, the scheduler reports stalling information which may be used for iterating upon the layout.
Armed with well defined component interfaces and a method to execute the quantum instructions, all that remains to create the control system for a given quantum circuit is putting the pieces together. The quantum datapath is composed of an arbitrary number of macroblocks pulled from the component library. Each macroblock in our component library has associated with it classical control logic. The control logic handles all the internals of the macroblock including details of ion movement, ion trapping and gate operation. In our library, the macroblock control logic is specified using behavioral Verilog modules.
When the layout stage of the CAD flow creates a physical layout of macroblocks, we extract the corresponding control logic blocks and assemble them together in a top-level Verilog module for the full control system, stitching together all necessary macroblock interfaces. This module instantiates all the appropriate macroblock control modules, along with the instruction issue logic and laser controller unit. Combined, these modules are assembled into a sin- gle Verilog module which implements the full classical control system for the quantum circuit and which may be input to a classical CAD flow for synthesis.
We begin our exploration of placement and routing heuristics by considering grid-based layouts. A majority of the work done in the field has concentrated on these types of layouts. In all of these works, a layout is constructed by first designing a primitive cell and then tiling this cell into a larger physical layout. For example, the authors of [15, 16] manually design a single cell, and for any given quantum circuit, they use that cell to construct an appropriately sized layout. In [23] , the authors automate the generation of an H-Tree based layout constructed from a single cell pattern. Similarly, [3] uses a cell such as in [23] but also provides some tools to evaluate the performance of a circuit when the number of communication channels and gate locations within the cell is varied. We use a combination of these methods to implement a tool that automatically creates a grid-based physical layout for a given quantum circuit.
The grid-based physical layouts generated by our tools are constructed by first creating a primitive cell out of the macroblocks mentioned in Section 2 and then tiling the cell to fill up the desired area. For example, Figure 6 shows how a 2 × 2 sized cell can be tiled to create the layout used in [16] (referred to henceforth as the QPOS grid). These types of simple structures are easy to automatically generate given only the number of qubits and gate operations in the quantum circuit. Furthermore, grid-based structures are very appealing to consider because, apart from selecting the number of cells in the layout and the initial qubit placement, no other customization is required in order to map a quantum circuit onto the layout. The regular pattern also makes it easy to determine how qubits move through the system, as simple schemes such as dimension-ordered routing can be used.
The approach we use to generate the grid-based layout for a given quantum circuit is as follows:
1. Given the cell size, create a valid cell structure out of macroblocks.
2. Create a layout by tiling the cell to fill up the desired area.
3. Assign initial qubit locations.
4. Simulate the quantum circuit on the layout to determine the execution time.
The first step finds a valid cell structure. A cell is valid if all the macroblocks that open to the perimeter of the cell have an open macroblock to connect to when the cell is tiled. Also, a cell cannot have an isolated macroblock within it that is unreachable. Once we tile this valid cell to create a larger layout, we must decide on how to assign initial qubit locations. The two methods we utilize are: a systematic left to right, one qubit per cell approach, and a randomized placement. The systematic placement allows us to fairly compare different layouts. However, since the initial placement of the qubits can affect the performance of the circuit, the tool also tries a number of random placements in an effort to determine if the systematic placement unfairly handicapped the circuit.
This layout generation and evaluation procedure is iterated upon until all valid cell configurations of the given size are searched. We then repeat this process for different cell sizes. The cell structure that results in the minimum simulated time for the circuit is used to create the final layout.
an example, Figure 7 shows the results of searching for the best layout composed of 3 × 2 sized cells targeting the [ [23, 1, 7] ] Golay encode circuit [21] , one of our benchmarks shown in Table 1 . More than 900 valid cell configurations were tested. For each cell configuration, we try multiple initial qubit placements (as mentioned earlier) resulting in a range of runtimes for each cell configuration. Differences in the runtime of the circuit are not limited to just variations on the cell configuration but are in fact also highly dependent on the initial qubit placement.
Figure 8 shows the best cell structure found by conducting a search of all 2 × 2, 2 × 3, and 3 × 2 sized cells for two different circuits. The main result of this search is that the best cell structure used to create the grid-based layout is dependent on what circuit will be run upon it. By varying the location of gates and communication channels, we tailor the structure of the layout to match the circuit requirements.
While this type of exhaustive search of physical layouts is capable of finding an optimal layout for a quantum circuit, it suffers from a number of drawbacks. Namely, as the size of the cell increases, the number of possible cell configurations grows exponentially. Searching for a good layout for anything but the smallest cell sizes is not a realistic option. Furthermore, while small circuits may be able to take advantage of primitive cell based grids, larger circuits will require a less homogeneous layout. One approach to doing this is to construct a large layout out of smaller grid-based pieces, all with different cell configurations. While this approach is interesting, we feel a more promising approach is one that resembles a classical CAD flow, where information extracted from the circuit is used to construct the layout.
One problem we observed in the regular grid layout design was that the high amount of channel congestion due to limited bandwidth causes densely-packed (occupied) gates. Additionally, we found that a number of gate locations and channels in many of the grids were not even used by the scheduler to perform the circuit.
We present a new heuristic that attempts to solve some of these problems. The heuristic is a simple greedy algorithm that starts with only as many gate locations as qubits (because we assume that qubits only rest in storage/gate locations) and no channels connecting the gates. It iterates with the circuit scheduler, moving and connecting gate locations until the qubits can communicate sufficiently to perform the specified circuit. The current layout is fed into the circuit scheduler which tries to schedule until it finds qubits in gate locations that cannot communicate to perform a gate. The place and router then connects the problematic gate locations and tries scheduling on the layout again. The iteration finishes once the circuit can be successfully completed. Our algorithm bears some similarity to the iterative procedure in adaptive cluster growth placement [12] in classical CAD. Gate locations are placed from the center outward as the circuit grows to fit a rectilinear boundary.
The placer can move gate locations that have to be connected if they are not already connected to something else. The router connects gate locations by making a direct path in the x and y directions between them and placing a new channel, shifting existing channels out of the way. Since channels are allowed to overlap, intersections are inserted where the new channels cut across existing ones.
This technique has the advantage that, since the circuit scheduler prioritizes gates based on gate delay critical path, potentially critical gates are mapped to gate locations and connected early in the process. Thus critical gates tend to be initially placed close together to shorten the circuit critical path. Additionally, gate locations that need to communicate can be connected directly instead of using a general shared grid channel network, where congestion can occur and cause qubits to be routed along unnecessarily long paths.
A disadvantage of this heuristic is that gate placement is done to optimize critical path, not to minimize channel intersections. This means that the layout could end up having many 4-way channel intersections and turns, both of which have more delay than 2-way straight channels. Additionally, even though critical gates are mapped and placed near each other, the channel routing algorithm tends to spread these gate locations apart as more channels cut through the center of the circuit. We discuss our experimental evaluation of this heuristic in Section 9.
As described in Section 6, a systematic row by row initial placement for qubits allows us to make somewhat accurate comparisons between different gridbased layouts, while a random initial qubit placement allows us to test a single grid's dependence on qubit starting positions. However, in laying out a quantum circuit, we would like to have a more intelligent and natural means of determining initial qubit placement. For this, we turn to the dataflow graph representation of the circuit.
Figure 9a shows a QASM instruction sequence consisting of Hadamard gates (H) and controlled bitflips (CX) operating on qubits Q0, Q1, Q2 and Q3, with each instruction labeled by a letter. Figure 9b shows the equivalent sequence of operations in standard quantum circuit format. Either of these may
A) H Q0 B) H Q1 C) H Q2 D) H Q3 E) CX Q0,Q1 F) CX Q2,Q3 G) CX Q1,Q2 H) CX Q2,Q3 I) CX Q0,Q2 H H H H Q0 Q1 Q2 Q3 A E G D F B C I H (b) (c) (a)
Figure 9 : a) A QASM instruction sequence. b) A quantum circuit equivalent to the instruction sequence in (a). c) A dataflow graph equivalent to the instruction sequence in (a). Each node represents an instruction, as labeled in (a). Each arc represents a qubit dependency.
A B C D E F I A B C D F A B C D E F G H I NG1 NG9 NG8 NG7 NG6 NG5 NG4 NG3 NG2 (b) (c) (a)
8 1 1 4 5 1 2 6 4 1 NG1 NG9 NG10 NG6 NG5 NG4 NG3 NG2 8 1 1 4 1 2 6 1 6? G H NG1 NG10 NG6 NG4 NG3 NG2 1 1 4 1 6? 6 1 6? G H E I NG11
Figure 10 : a) Each node (instruction) is initialized in its own node group (NG, outlined by the dotted lines), which corresponds to a physical gate location in a layout. Once placed, we extract physical distances between the nodes (the edge labels). b) We find the longest edge weight on the longest critical path (the length 5 edge on the path C-F-G-H-I; solid bold arrows) and merge its two node groups to eliminate that latency. c) We recompute the critical path (A-E-I; dashed bold arrows) and merge its node groups, and so on. be translated into the dataflow graph shown in Fig- ure 9c , where each node represents a QASM instruction (as labeled in Figure 9a ) and each arc represents a qubit dependency. With this dataflow graph, we may perform some analyses to help us place and route a layout for our quantum circuit.
The general idea is that we shall create node groups in the dataflow graph which correspond to distinct gate locations that may then be placed and routed on a layout. All instructions within a single node group are guaranteed to be executed at a single gate location, as elaborated upon in Section 8.3. To begin with, we create a node group for each instruction, giving us a dataflow group graph, as shown in Figure 10a . If we lay out this group graph with a distinct designated gate for each instruction (using heuristics discussed in Section 8.2), we get a layout in which the starting location of each qubit is specified implicitly by its first gate location, so no additional initial placement heuristic is needed.
From this layout we can extract movement latency between nodes and label the edges with weights (as in Figure 10a ). We now find the longest critical path by qubit. The critical path A-E-I of qubit Q0 has length 14 (the dashed bold arrows), while the critical path C-F-G-H-I of qubit Q2 has length 15 (the solid bold arrows). We select the longest edge on the longest critical path, which is the edge G-H with weight 5. We merge these two node groups to eliminate this latency, in effect specifying that these two instructions should occur at the same gate location (Figure 10b ). We then update the layout and recompute distances. Assuming we merged these two node groups to the location of H (NG8), then the weight of edge F-G changes to 1 (to match the weight of edge F-H) and the weight of edge E-G probably changes to 6 (former E-G plus former G-H), but the exact change really depends on layout decisions. The new critical path is now A-E-I, so if we do this again, we merge node groups NG5 and NG9 to eliminate the edge of weight 8, and we get the group graph in Figure 10c . In merging nodes, there is the possibility that two qubit starting locations get merged, complicating the assignment of initial placement. For this reason, we add a dummy input node for each qubit before its first instruction. The merging heuristic doesn't allow more than one input node in any single node group, so we maintain the benefit of having an intelligent initial qubit placement without extra work.
There is an important trade-off to consider when taking this merging approach. A tiled grid layout provides plenty of gate location reuse but is unlikely to provide any pipelinability without great effort. A layout of the group graph in Figure 10a (with each instruction assigned to a distinct gate location) provides no gate location reuse at all but high potential pipelinability. This raises the question of whether we wish to minimize area and time (for critical data qubits), maximize throughput of a pipeline (for ancilla generation), or compromise at some middle ground where small sets of nearby nodes are merged in order to exploit locality while still retaining some pipelinability. We intend to further explore this topic in the future.
Taking the group graph from the dataflow analysis heuristic, the placement algorithm takes advantage of the fanout-limited gate output imposed by the No-Cloning Theorem [25] to lay out the dataflow-ordered gate locations in a roughly rectangular block. We adopt a gate array-style design, where gate locations are laid out in columns according to the graph, with space left between each pair of columns for necessary channels. This can lead to wasted space due to a linear layout of uneven column sizes, so we may also perform a folding operation, wherein a short column may be folded in (joined) with the previous column, thus filling out the rectangular bounding box of the layout as much as possible and decreasing area. The columns are then sorted to position gate locations that need to be connected roughly horizontal to one another. This further minimizes channel distance between connected gate locations and reduces the number of high-latency turns.
Once gate locations are placed, we use a grid-based model in which we first route local wire channels between gate locations that are in adjacent or the same columns. These channels tend to be only a few macroblocks long each. A separate global channel is then inserted between each pair of rows and between each
Technology-Dependent Netlist Dataflow Analysis & Gate Combination Sorted Dataflow Placement Local/Global Channel Router Geometry-Aware Netlist Gate Scheduler/ Simulator Placement and Routing Figure 11: The placement and routing portion of our CAD flow (shown in Figure 4) takes a technologydependent netlist and translates it into a geometryaware netlist through an iterative process involving dataflow analysis and placement and routing techniques. pair of columns of gate locations. These global channels stretch the full length of the layout. There are no real routing constraints in our simple model since channels are allowed to overlap and turn into 3-or 4-way intersections. We depend on the dataflow column sorting in the placement phase to reduce the number of intersections and shared local channels. While local channels could technically be used for global routing and vice versa, we've found that this division in routing tends to divide the traffic and separate local from long-distance congestion. With these basic placement and routing schemes, we may now iterate upon the layout, as shown in Figure 11. The technology-dependent netlist is translated into a dataflow group graph with a separate gate location for each instruction (Figure 10a ). This group graph is then placed, routed and scheduled to get latency and identify the runtime critical path (as opposed to the critical path in the group graph, which fails to take congestion into account). The longest latency move on the runtime critical path (between two node groups) is merged into one node group, thus eliminating the move since a node group represents a single gate location. This new group graph is then placed, routed and scheduled again to find the next pair of node groups to merge.
Once this process has iterated enough times, we reach a point where congestion at some heavily merged node group is actually hurting the latency with each further merge. We alleviate this congestion by adding storage nodes (essentially gate locations that don't actually perform gates) near the congested node group. This increases the area slightly but maintains the locality exploited by the merging heuristic. If congestion persists, we halt the algorithm, back up a few merging steps and output the geometry-aware netlist.
The scheduling heuristic described in Section 5.2 schedules an arbitrary QASM instruction sequence on an arbitrary layout. However, once we have assigned instructions in a dataflow graph to node groups (as described in Section 8.1), we wish those instructions to be executed at their proper location on any layout placed and routed from the group graph. To this end, we annotate each instruction in the instruction sequence with the name of the gate location where it must be executed. Additionally, since we have the gate locations in advance, we can incorporate movement in the back-prioritization of the instruction sequence. Thus, the priority assigned to each qubit now incorporates both gate latencies and movement through an uncongested layout, which gives us a better approximation of each qubit's critical path. We use this extended scheduler in our dataflow-based experiments presented in Section 9.
We now present our simulation results for the heuristics described in earlier sections.
Relatively high error rates of operations in a quantum computer necessitate heavy encodings of qubits. As such, we focus on encoding circuits (useful for both data and ancillae) and error correction circuits to experiment with circuit layout techniques. We lay out a number of error correction and encoding circuits to Qubit Gate Circuit name count count [ [7, 1, 3] ] L1 encode [20] 7 21 [ [23, 1, 7] ] L1 encode [21] 23 116 [ [7, 1, 3] ] L1 correction [1] 21 136 [ [7, 1, 3] ] L2 encode [20] 49 245
Table 1 : List of our QECC benchmarks, with quantum gate count and number of qubits processed in the circuit.
evaluate the effectiveness of the heuristics used in our CAD flow in terms of circuit area and latency, as determined by our scheduler. Our circuit benchmarks are shown in Table 1 . We use two level 1 (L1) encoding circuits, a level 2 (L2) recursive encoding circuit and a fault-tolerant level 1 correction circuit. The idea of the encoding circuits is that they will provide a constant stream of encoded ancillae to interact with encoded data qubit blocks. Thus, for these circuits, throughput is a more important measure than latency, implying that they would benefit greatly from pipelining. Nonetheless, a high latency circuit could introduce non-trivial error due to increased qubit idle time. On the other hand, correction circuits are much more latency dependent, since they are on the critical path for the processing of data qubit blocks.
We have evaluated a variety of layout design heuristics on the four benchmarks shown in Table 1 . The results are in Table 2 . "QPOS Grid" refers to the best scheduled layout from the literature [16] (see Section 6) . "Optimal Grid" refers to the best grid with an area matching the QPOS Grid used that was found by the exhaustive search described in Section 6. "Greedy" refers to the heuristic described in Section 7. "DF" refers to the dataflowbased approach from Section 8. "Non-folded" means the dataflow graph is laid out with varying column widths; "folded" means the layout has been made more rectangular by stacking columns. The number of global channels is between each pair of rows and columns of gate locations. "Critical combining" refers to our dataflow group graph merging heuristic.
The exhaustive search over grids yields the best latency for all benchmarks, which is not surprising. This kind of search becomes intractable quickly as circuit size grows, and additionally, it is based on the unproven assumption that a regular layout pattern is the best approach. We include this data point as something to keep in mind as a target latency. Among the polynomial-time heuristics, we first note that no single heuristic is optimal for all four benchmarks and that, in fact, no single heuristic optimizes both latency and area for any single circuit. Dataflow-based place and route techniques in general produce the lowest latency circuits. We find that the optimal global channel count per column (1 or 2) depends on the circuit being laid out. This is an artifact of the lack of maturity in our routing methodology. We intend to explore more adaptive routing optimization in our ongoing work.
The dataflow approach and the QPOS Grid tend to trade off between latency and area. However, we expect that the dataflow approach will show greater potential for pipelining, thus allowing us to target circuits such as an encoded ancilla generation factory, for which throughput is of greater importance than latency. We also observe that non-folded dataflow layouts are likely to have even greater pipelinability than folded ones, but at the likely cost of greater area. Although, we should note that the area estimates for the non-folded DF-based layouts are in fact overestimates due to our use of a liberal bounding box for these calculations.
We find that the greedy heuristic tends to find the best design area-wise, but the latency penalty increases with circuit complexity. This is expected, as greedy is unable to handle congestion problems, so it works best for small circuits where congestion is not an issue. It is for the opposite reason that the DF heuristics fail on the [ [7, 1, 3] ] L1 encode. They insert too much complexity into an otherwise simple problem.
We presented a computer-aided design flow for the layout, scheduling and control of ion trap-based quantum circuits. We focused on physical quantum circuits, that is, ones for which all ancillae, encodings and interconnect are explicitly specified. We explored several mechanisms for generating optimal layouts and schedules for our benchmark circuits.
Prior work has tended to assume a specific regular grid structure and to schedule operations within this structure. We investigated a variety of grid structures and showed a performance variance of a factor of four as we varied grid structure and initial qubit placement. Since exhaustive search is clearly impractical for large circuits, we also explored two polynomialtime heuristics for automated layout design. Our greedy algorithm produces good results for very simple circuits, but quickly begins to be suboptimal as circuit size grows. For larger circuits, we investigated a dataflow-based analysis of the quantum circuit to a place and route mechanism which leverages from classical algorithms. We found that our our dataflow approach generally offers the best latency, often at the cost of area. However, we expect that a layout based on the dataflow graph analysis also offers better potential for pipelining than a grid-based approach, and we intend to investigate this further in the future. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We present a computer-aided design flow for quantum circuits, complete with automatic layout and control logic extraction. To motivate automated layout for quantum circuits, we investigate grid-based layouts and show a performance variance of four times as we vary grid structure and initial qubit placement. We then propose two polynomial-time design heuristics: a greedy algorithm suitable for small, congestionfree quantum circuits and a dataflow-based analysis approach to placement and routing with implicit initial placement of qubits. Finally, we show that our dataflow-based heuristic generates better layouts than the state-of-the-art automated grid-based layout and scheduling mechanism in terms of latency and potential pipelinability, but at the cost of some area."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Quantum computing offers us the opportunity to solve certain problems thought to be intractable on a classical machine. For example, the following classically hard problems benefit from quantum algorithms: factorization [19] , unsorted database search [6] , and simulation of quantum mechanical systems [26] .\n\nIn addition to significant work on quantum algorithms and underlying physics, there have been several studies exploring architectural trade-offs for quantum computers. Most such research [3, 16] has focused on simulating quantum algorithms on a fixed layout rather than on techniques for quantum circuit synthesis and layout generation. These studies tend\n\nClassical Control: HDL Format (plus annotations for scheduling) Quantum Layout (including initial qubit placement) Basic Blocks Custom Modules Quantum Circuit Specification CAD Flow for Quantum Circuits New Custom Module The goal of our CAD flow is to automate the laying out of a quantum circuit to generate a physical layout, an intelligent initial placement of qubits, the associated classical control logic and annotations to help the online scheduler better use the layout optimizations as they were intended. This flow may then be used recursively to design larger blocks using previously created modules.\n\nto use hand-generated and hand-optimized layouts on which efficient scheduling is then performed. While this approach is quite informative in a new field, it quickly becomes intractable as the size of the circuit grows.\n\nOur goal is to automate most of the tasks involved in generating a physical layout and its associated control logic from a high-level quantum circuit specification (Figure 1 ). Our computer-aided design (CAD) flow should process a quantum circuit specification and produce the following:\n\n• a physical layout in the desired technology\n\n• an intelligent initial qubit placement in the layout\n\n• classical control circuitry specified in some hardware description language (HDL), which may then be run through a classical CAD flow\n\n• a set of annotations or \"hints\" for the online scheduler, allowing a tighter coupling of layout optimizations to actual runtime operation Much like a classical CAD flow, this quantum CAD flow is intended to be used hierarchically. We begin with a set of technology-specific basic blocks (some ion trap technology examples are given in Section 2). We then lay out some simple quantum circuits with the CAD flow, thus creating custom modules. The CAD flow may then be used recursively to create ever larger designs. This approach allows us to develop, evaluate and reuse design heuristics and avoids both the uncertainty and time-intensive nature of handgenerated layouts."
},
{
"section_type": "OTHER",
"section_title": "Motivation for a Quantum CAD Flow",
"text": "Quantum circuits that are large enough to be \"interesting\" require the orchestration of hundreds of thousands of physical components. In approaching such problems, it is important to build upon prior work in classical CAD flows. Although the specifics of quantum technologies (such as are discussed in Section 2) are different from classical CMOS technologies, prior work in CAD research can give us insight into how to approach the automated layout of quantum gates and channels. Further, quantum circuits exhibit some interesting properties that lend themselves to automatic synthesis and computer-aided design techniques:\n\nQuantum ECC Quantum data is extremely fragile and consequently must remain encoded at all times -while being stored, moved, and computed upon. The encoded version of a circuit is often two or three orders of magnitude larger than the unencoded version. Further, the appropriate level of encoding may need to be selected as part of the layout process in order to achieve an appropriate \"threshold\" of error-free execution. Rather than burdening the designer with the complexities of adding fault-tolerance to a circuit, computer-aided synthesis, design and verification can perform such tasks automatically.\n\nAncillae Quantum computations use many helper qubits known as ancillae. Ancillae consist of bits that are constructed, utilized and recycled as part of a computation. Sometimes, ancillae are explicit in a designer's view of the circuit.\n\nOften, however, they should be added automatically in the process of circuit synthesis, such as during the construction of fault-tolerant circuits from high-level circuit descriptions. An automatic design flow can insert appropriate circuits to generate and recycle ancillae without involving the designer.\n\nTeleportation Quantum circuits present two possibilities for data transport: ballistic movement and teleportation. Ballistic movement is relatively simple over short distances in technologies such as ion traps (Section 2). Teleportation is an alternative that utilizes a higher-overhead distribution network of entangled quantum bits to distribute information with lower error over longer distances [9] . The choice to employ teleportation is ideally done after an initial layout has determined long communication paths. Consequently, it is a natural target for a computer-aided design flow."
},
{
"section_type": "OTHER",
"section_title": "Contributions",
"text": "In this paper, we make the following contributions:\n\n• We propose a CAD flow for automated design of quantum circuits and detail the necessary components of the flow.\n\n• We describe a technique for automatic synthesis of the classical control circuitry for a given layout.\n\n• We show that different grid-based architectures, which have been the focus of most prior work in this field, exhibit vastly varying performance for the same circuit.\n\n• We present heuristics for the placement and routing of quantum circuits in ion trap technology.\n\n• We lay out some quantum error correction circuits and evaluate the effectiveness of the heuristics in terms of circuit area and latency. Each macroblock has a specific number of ports (shown as P0-P3) along with a set of electrodes used for ion movement and trapping. Some macroblocks contain a trap region where gates may be performed (black square)."
},
{
"section_type": "OTHER",
"section_title": "Paper Organization",
"text": "The rest of this paper is organized as follows. We introduce our chosen technology in Section 2, followed by an overview of prior work in the field in Section 3. In Section 4, we detail our proposed CAD flow and our evaluation metrics. In Section 5, we describe the control circuitry interface and scheduling protocol that we use in the following sections. Section 6 contains a study of grid-based layouts, which have been the basis of most prior work on this subject. In Section 7, we present a greedy approach to laying out quantum circuits, followed in Section 8 by a much more scalable dataflow analysis-based approach to layout. Section 9 contains our experimental results for all three approaches to layout generation, and we conclude in Section 10."
},
{
"section_type": "OTHER",
"section_title": "Ion Traps",
"text": "For our initial study, we choose trapped ions [4, 17] as our substrate technology. Trapped ions have shown good potential for scalability [10] . In this technology, a physical qubit is an ion, and a gate is a location where a trapped ion may be operated upon by a modulated laser.\n\nThe ion is both trapped and ballistically moved by applying pulse sequences to discrete electrodes which line the edges of ion traps. Figure 3a shows an experimentally-demonstrated layout for a three-way intersection [7] . A qubit may be held in place at any trap region, or it may be ballistically moved between them using the gray electrodes lining the paths.\n\nRather than using ion traps as basic blocks, we define a library of macroblocks consisting of multiple traps for two reasons. First, macroblocks abstract out some of the low-level details, insulating our analyses from variations in the technology implementations of ion traps. Details such as which ion species is used, specific electrode sizing and geometry (clearly variable in the layout in Figure 3a ) and exact voltage levels necessary for trapping and movement are all encapsulated within the macroblock. Second, ballistic movement along a channel requires carefully timed application of pulse sequences to electrodes in nonadjacent traps. By defining basic blocks consisting of a few ion traps, we gain the benefit that crossing an interface between basic blocks requires communication only between the two blocks involved.\n\nWe use the library of macroblocks shown in Figure 2 , each of which consists of a 3x3 grid of trap regions and electrodes, with ports to allow qubit movement between macroblocks. The black squares are gate locations, which may not be performed at intersections or turns in ion trap technology. Each of these macroblocks may be rotated in a layout. This library is by no means exhaustive, however it does provide the major pieces necessary to construct many physical circuits. The macroblocks we present are abstractions of experimentally-demonstrated ion trap technology [7, 18] . In Figure 3 , we show how one can map a demonstrated layout (Figure 3a ) to our macroblock abstractions (Figure 3b ). We model this layout as a set of StraightChannel and ThreeWayIntersection macroblocks. Above the ion trap plane is an array of MEMS mirrors which routes laser pulses to the gate locations in order to apply quantum gates [11] , as shown in Figure 3c .\n\nSome key differences between this quantum circuit technology and classical CMOS are as follows:\n\n• \"Wires\" in ion traps consist of rectangular channels, lined with electrodes, with atomic ions suspended above the channel regions and moved ballistically [13] . Ballistic movement of qubits requires synchronized application of voltages on channel electrodes to move data around. Thus each wire requires movement control circuitry to handle any qubit communication.\n\n• A by-product of the synchronous nature of the qubit wire channels is that these circuits can be used in a synchronous manner with no additional overhead. This enables some convenient pipelining options which will be discussed in Section 8.1.\n\nFigure 3 : a) Experimentally demonstrated physical layout of a T-junction (three-way intersection). b) Abstraction of the circuit in (a), built using the StraightChannel and ThreeWayIntersection macroblocks shown in Figure 2 . c) The ion traps are laid out on a plane, above which is an array of MEMS mirrors used to route and split the laser beams that apply quantum gates.\n\n• Each gate location will likely have the ability to perform any operation available in ion trap technology. This enables the reuse gate locations within a quantum circuit.\n\n• Scalable ion trap systems will almost certainly be two-dimensional due to the difficulty of fabricating and controlling ion traps in a third dimension [8] . This means that all ion crossings must be intersections.\n\n• Any routing channel may be shared by multiple ions as long as control circuits prevent multiion occupancy. Consequently, our circuit model resembles a general network, although scheduling the movement in a general networking model adds substantial complexity to our circuit.\n\n• Movement latency of ions is not only dependent on Manhattan distance but also on the geometry of the wire channel. Experimentally, it has been shown that a right angle turn takes substantially longer than a straight channel over the same distance [18, 7] ."
},
{
"section_type": "BACKGROUND",
"section_title": "Related Work",
"text": "Prior research has laid the groundwork for our quantum circuit CAD flow. Svore et al [22, 23] proposed a design flow capable of pushing a quantum program down to physical operations. Their work outlined various file formats and provided initial implementations of some of the necessary tools. Similarly, Balensiefer et al [2, 3] proposed a design flow and compilation techniques to address fault-tolerance and provided some tools to evaluate simple layouts. While our CAD flow builds upon some of these ideas, we concentrate on automatic layout generation and control circuitry extraction. Additionally, initial hand-optimized layouts have been proposed in the literature. Metodi et al [15] proposed a uniform Quantum Logic Array architecture, which was later extended and improved in [24] . Their work concentrated on architectural research and did not delve into details of physical layout or scheduling. Finally, Metodi et al [16] created a tool to automatically generate a physical operations schedule given a quantum circuit and a fixed grid-based layout structure. We extend and improve upon their work by adding new scheduling heuristics capable of running on grid-based and non-grid-based layouts.\n\nMaslov et al [14] have recently proposed heuristics for the mapping of quantum circuits onto molecules used in liquid state NMR quantum computing technology. Their algorithm starts with a molecule to be used for computation, modeled as a weighted graph with edges representing atomic couplings within the molecule. The dataflow graph of the circuit is mapped onto the molecule graph with an effort to minimize overall circuit runtime. Our techniques focus on circuit placement and routing in an ion trap technology and do not use a predefined physical substrate topology as in the NMR case. A new ion trap geometry is instead generated by our toolset for each circuit."
},
{
"section_type": "OTHER",
"section_title": "Quantum CAD Flow",
"text": "The ultimate goal of a quantum CAD flow is identical to that of a standard classical CAD flow: to automate\n\nHigh-Level Description Synthesizer Tech-Independent Netlist Tech Mapping (Tech-Specific Gates, Encoding, Fault Tolerance) Tech Parameter File (Basic Blocks) Custom Modules Tech-Dependent Netlist Interconnect (if necessary) Placement and Routing Classical Control Synthesis for Qubit Movement Geometry-Aware Netlist the synthesis and laying out of a circuit. For a quantum CAD flow, the output circuit consists of both the quantum portion and the associated classical control logic.\n\nThe quantum CAD flow we present elaborates on the design flows described in prior works [3, 22, 23] . Unlike prior work, our CAD flow addresses the need to integrate automatic generation of classical control into the flow. Figure 4 shows an overview of our CAD toolset. Rectangles are tools, while ovals represent intermediate file formats. Our toolset is built to be as similar to classical CAD flows as possible, while still accounting for the differences between classical and quantum computing described in Section 1.1.\n\nAt the top, we begin with a high-level description of the desired quantum circuit. At present this specification consists of a sequence of quantum assembly language (QASM [3] ) instructions implementing the desired circuit, since this is a convenient format already being used by various third-party tools. We are currently investigating extension of this high-level description to other formats, such as schematic entry, mathematical formulae or a more general high-level language.\n\nThe synthesizer parses the QASM file and generates a technology-independent netlist stored in XML format. From this point onward (downward in the figure), all file formats are XML. Additionally, information may be modified or added but generally not removed. As we move down the flow, we add more and more low level details, but we also keep highlevel information such as encoded qubit groupings, nested layout modules, distinction between ancillae and data, etc. This allows low-level tools to make more intelligent decisions concerning qubit placement and channel needs based on high-level circuit structure. It likewise allows logical level modification at the lowest levels without having to attempt to deduce qubit groupings.\n\nA technology parameter file specifies the complete set of basic blocks available for the layout (see examples in Figure 2 ), as well as design rules for connecting them. A basic block specification contains the following:\n\n• the geometry of the block in enough detail to allow fabrication\n\n• control logic for each operation possible within the block (including both movement and gates)\n\n• control logic for handling each operation possible at each interface\n\nThe most basic function of the technology mapping tool is to take a technology-independent netlist and map it onto allowed basic blocks to create the technology-dependent netlist. This may be more or less complicated depending upon the complexity of the basic blocks. In addition, it may need to translate to technology-specific gates (in case the QASM file uses gates not available in this technology), encode the qubits used in the circuit (perhaps also automatically adding the ancilla and operation sequences necessary for error correction) and add fault tolerance to the final physical circuit.\n\nIn the initial technology-dependent netlist, all qubits are physical qubits, meaning that encoding levels have been set (though they may still be modified later). At this point, any technology-specific optimizations may optionally be applied to the physical circuit encapsulated in this netlist. Additionally, if the circuit is complex enough to warrant the inclusion of a teleportation-based interconnection network [9] , it is added to the netlist here using the higher level qubit grouping information in the netlist.\n\nOnce the designer is happy with the netlist, a placement and routing tool lays out the netlist and adds any further channels needed for communication. This geometry-aware netlist may be iterated upon as necessary to refine the layout. Once the layout is finalized, the classical control synthesis tool combines the control logic of the various components of the design, integrates interface control mechanisms to function properly and generates the unified control structure for the entire layout. Our control synthesis tool generates a Verilog file, which may then be run through a classical CAD flow for implementation.\n\nThe layout specification along with the control logic file together comprise the geometry-aware netlist, which is the end result for the quantum circuit initially specified in the high-level description. In order to allow hierarchical design of larger quantum circuits, we may now add this geometry-aware netlist to our set of custom modules. Future technology mappings may use both the basic blocks specified in the technology parameter file and any custom modules we create (or acquire).\n\nThe gray area in Figure 4 identifies the portions we shall be focusing on for the rest of this paper. We currently process the high-level description (a QASM file) directly into a technology-dependent netlist for ion traps using the macroblocks shown in Figure 2 . Thus we perform a tech mapping, but no automatic encoding, interconnect or addition of gates for fault tolerance. In this paper, we focus on laying out lowlevel circuits, such as those for encoded ancilla generation and error correction. The classical control synthesis box of the CAD flow is discussed in Section 5, while placement and routing are analyzed and compared in Sections 6, 7, 8 and 9.\n\nWe use two main metrics to evaluate the performance of our CAD flow: area and latency. For area, we consider the bounding box around the layout, so irregularly-shaped layouts are penalized (since they have wasted space). To determine latency of circuit execution, we use the scheduling heuristic described in Section 5.2 and extended in Section 8.3. A third metric of interest is fault-tolerance. For small layouts and circuits, we can use third-party tools to determine whether a given layout and schedule is faulttolerant [5] , but we do not currently use the fault-tolerance metric in our iterative design flow. We use area and latency because, to a first approximation, lower area and lower latency are likely to decrease decoherence. Previous algorithms to accurately determine the error tolerance of a quantum circuit have involved very computationally-intensive analyses that would be inappropriate for circuits with more than a few dozen gates [1] . However, we are looking into ways to incorporate fault tolerance as a metric."
},
{
"section_type": "OTHER",
"section_title": "Control",
"text": "The classical control system is responsible for executing the quantum circuit, including deciding where and when gate operations occur and tracking and managing every qubit in the system. It is composed of the following major components: instruction issue logic, gate control logic and macroblock control logic. Instruction issue logic handles all instruction scheduling and determines qubit movement paths. Gate control logic oversees laser resource arbitration, deciding which requested gate operations may occur at any given time. The macroblock control logic, which consists of an individual logic block for each macroblock in the system, handles all the internals of the macroblock, including details of gate operation for each gate possible within the macroblock, qubit movement within the macroblock and qubit movement into and out of the ports."
},
{
"section_type": "OTHER",
"section_title": "Control Interfaces",
"text": "The first step in the control flow involves processing the quantum circuit's high-level description (the QASM file). The instruction issue logic accepts this stream of instructions as input and creates a series of qubit control messages. Using these qubit control messages, macroblock control logic blocks can determine where to move qubits and when to execute a gate operation. Qubit control messages are simple bit streams composed of a qubit ID, along with a sequence of commands, as shown in Figure 5 . When a qubit needs to perform an action, the instruction issue logic sends to it an appropriate control message which travels with the qubit as it traverses the layout. Once a macroblock receives a qubit and its corresponding control message, it uses the first command in the sequence to determine the operation it must perform. The macroblock then removes the com-Figure 5 : Example of how a qubit control message is constructed to move a qubit through a series of macroblocks. The qubit enters M0 and travels through M1 and M2, arriving at M3 where it is instructed to perform a CNOT. mand bits used and passes on the remaining control message to the next macroblock into which the qubit travels. In this manner, the instruction issue logic can create a multi-command qubit control message that specifies the path a qubit will traverse through consecutive macroblocks, along with where gate operations take place. The instruction issue logic only has to transmit this control message to the source macroblock, relying on the inter-macroblock communication interface to handle the rest.\n\nCommunication between the instruction issue logic and the macroblocks takes place using a shared control message bus in order to minimize the number of wire connections required by the instruction issue logic. Each macroblock listens to the control message bus for messages addressed to it and only processes messages with a destination ID that match the macroblock's ID. A macroblock is only responsible for monitoring the control message bus if it contains a qubit that has no remaining command bits. This condition generally occurs after a gate operation, when the instruction issue logic is deciding what action the qubit should take next. Once the instruction issue logic sends a new control message for the qubit, the macroblock resumes operation.\n\nMacroblocks communicate with each other via control signals associated with each quantum port in the macroblock. Each port has signals to control qubit movement into the macroblock and signals to control movement out of the macroblock via that port. These signals are connected to the corresponding signals of the neighboring macroblocks. The macroblocks as-sert a request signal to a destination macroblock when a qubit command indicates the qubit should cross into the next macroblock. If an available signal response is received, the qubit, along with its control message, can move across into the neighboring macroblock; if not, the qubit must wait until the available signal is present.\n\nThe macroblock interface enables the instruction issue logic to schedule qubit movement as a path through a sequence of macroblocks, without concerning itself with the low level details of qubit movement. This modular system allows macroblocks to be replaced with any other macroblock that implements the defined interface, without modifying the instruction issue logic.\n\nAdditionally, macroblocks have an interface to the laser control logic. Whenever a macroblock is instructed to perform a gate operation, it must request a laser resource through the laser control logic. The laser controller is responsible for aggregating requests from all the macroblocks in the system, and deciding when and where to send laser pulses. The laser controller also attempts to parallelize as many operations as possible. Once the laser pulses have completed, the laser controller notifies the macroblocks, indicating that the gate operation is complete."
},
{
"section_type": "OTHER",
"section_title": "Instruction Scheduling",
"text": "The instruction issue logic is responsible for determining the runtime execution order of the instructions in the quantum circuit, which involves both preprocessing and online scheduling. The instruction sequence is first preprocessed to assign priorities that will help during scheduling. The sequence is traversed from end to beginning, scheduling instructions as late as dependencies allow, using realistic gate latencies but ignoring movement. Essentially, each instruction is labeled with the length of its critical path to the end of the program. This is similar to the method used in [16] , but we use critical path with gate times rather than the size of the dependent subtree.\n\nThe instruction preprocessing generates an optimal schedule assuming infinite gates and zero movement cost. However, we wish to evaluate a layout with more realistic characteristics. Our scheduler is designed to schedule on an arbitrary graph, but the layouts provided to it by the place and route tool are in fact planar layouts using only right angles. In ad-dition, the scheduler requires that the qubit initial positions be provided as well.\n\nOur scheduler implements a greedy scheduling technique. It keeps the set of instructions which have had all their dependencies fulfilled (and thus are ready to be executed). It attempts to schedule them in priority order. So the highest priority ready instruction (according to critical path) is attempted first and is thus more likely to get access to the resources it needs. These contested resources include both gates and channels/intersections. Once all possible instructions have been scheduled, time advances until one or more resources is freed and more instructions may be scheduled. This scheduling and stalling cycle continues until the full sequence has been executed or until deadlock occurs, in which case it is detected and the highest priority unscheduled instruction at the time of deadlock is reported.\n\nSince we are interested in evaluating layouts rather than in designing an efficient online scheduler, we use very thorough searches over the graph in both gate assignment and pathfinding. This causes the scheduler to take longer but takes much of the uncertainty concerning schedule quality out of our tests. In addition, the scheduler reports stalling information which may be used for iterating upon the layout."
},
{
"section_type": "OTHER",
"section_title": "Control Extraction",
"text": "Armed with well defined component interfaces and a method to execute the quantum instructions, all that remains to create the control system for a given quantum circuit is putting the pieces together. The quantum datapath is composed of an arbitrary number of macroblocks pulled from the component library. Each macroblock in our component library has associated with it classical control logic. The control logic handles all the internals of the macroblock including details of ion movement, ion trapping and gate operation. In our library, the macroblock control logic is specified using behavioral Verilog modules.\n\nWhen the layout stage of the CAD flow creates a physical layout of macroblocks, we extract the corresponding control logic blocks and assemble them together in a top-level Verilog module for the full control system, stitching together all necessary macroblock interfaces. This module instantiates all the appropriate macroblock control modules, along with the instruction issue logic and laser controller unit. Combined, these modules are assembled into a sin- gle Verilog module which implements the full classical control system for the quantum circuit and which may be input to a classical CAD flow for synthesis."
},
{
"section_type": "OTHER",
"section_title": "Grid-based Layouts",
"text": "We begin our exploration of placement and routing heuristics by considering grid-based layouts. A majority of the work done in the field has concentrated on these types of layouts. In all of these works, a layout is constructed by first designing a primitive cell and then tiling this cell into a larger physical layout. For example, the authors of [15, 16] manually design a single cell, and for any given quantum circuit, they use that cell to construct an appropriately sized layout. In [23] , the authors automate the generation of an H-Tree based layout constructed from a single cell pattern. Similarly, [3] uses a cell such as in [23] but also provides some tools to evaluate the performance of a circuit when the number of communication channels and gate locations within the cell is varied. We use a combination of these methods to implement a tool that automatically creates a grid-based physical layout for a given quantum circuit.\n\nThe grid-based physical layouts generated by our tools are constructed by first creating a primitive cell out of the macroblocks mentioned in Section 2 and then tiling the cell to fill up the desired area. For example, Figure 6 shows how a 2 × 2 sized cell can be tiled to create the layout used in [16] (referred to henceforth as the QPOS grid). These types of simple structures are easy to automatically generate given only the number of qubits and gate operations in the quantum circuit. Furthermore, grid-based structures are very appealing to consider because, apart from selecting the number of cells in the layout and the initial qubit placement, no other customization is required in order to map a quantum circuit onto the layout. The regular pattern also makes it easy to determine how qubits move through the system, as simple schemes such as dimension-ordered routing can be used.\n\nThe approach we use to generate the grid-based layout for a given quantum circuit is as follows:\n\n1. Given the cell size, create a valid cell structure out of macroblocks.\n\n2. Create a layout by tiling the cell to fill up the desired area.\n\n3. Assign initial qubit locations.\n\n4. Simulate the quantum circuit on the layout to determine the execution time.\n\nThe first step finds a valid cell structure. A cell is valid if all the macroblocks that open to the perimeter of the cell have an open macroblock to connect to when the cell is tiled. Also, a cell cannot have an isolated macroblock within it that is unreachable. Once we tile this valid cell to create a larger layout, we must decide on how to assign initial qubit locations. The two methods we utilize are: a systematic left to right, one qubit per cell approach, and a randomized placement. The systematic placement allows us to fairly compare different layouts. However, since the initial placement of the qubits can affect the performance of the circuit, the tool also tries a number of random placements in an effort to determine if the systematic placement unfairly handicapped the circuit.\n\nThis layout generation and evaluation procedure is iterated upon until all valid cell configurations of the given size are searched. We then repeat this process for different cell sizes. The cell structure that results in the minimum simulated time for the circuit is used to create the final layout.\n\nan example, Figure 7 shows the results of searching for the best layout composed of 3 × 2 sized cells targeting the [ [23, 1, 7] ] Golay encode circuit [21] , one of our benchmarks shown in Table 1 . More than 900 valid cell configurations were tested. For each cell configuration, we try multiple initial qubit placements (as mentioned earlier) resulting in a range of runtimes for each cell configuration. Differences in the runtime of the circuit are not limited to just variations on the cell configuration but are in fact also highly dependent on the initial qubit placement.\n\nFigure 8 shows the best cell structure found by conducting a search of all 2 × 2, 2 × 3, and 3 × 2 sized cells for two different circuits. The main result of this search is that the best cell structure used to create the grid-based layout is dependent on what circuit will be run upon it. By varying the location of gates and communication channels, we tailor the structure of the layout to match the circuit requirements.\n\nWhile this type of exhaustive search of physical layouts is capable of finding an optimal layout for a quantum circuit, it suffers from a number of drawbacks. Namely, as the size of the cell increases, the number of possible cell configurations grows exponentially. Searching for a good layout for anything but the smallest cell sizes is not a realistic option. Furthermore, while small circuits may be able to take advantage of primitive cell based grids, larger circuits will require a less homogeneous layout. One approach to doing this is to construct a large layout out of smaller grid-based pieces, all with different cell configurations. While this approach is interesting, we feel a more promising approach is one that resembles a classical CAD flow, where information extracted from the circuit is used to construct the layout."
},
{
"section_type": "OTHER",
"section_title": "Greedy Place and Route",
"text": "One problem we observed in the regular grid layout design was that the high amount of channel congestion due to limited bandwidth causes densely-packed (occupied) gates. Additionally, we found that a number of gate locations and channels in many of the grids were not even used by the scheduler to perform the circuit.\n\nWe present a new heuristic that attempts to solve some of these problems. The heuristic is a simple greedy algorithm that starts with only as many gate locations as qubits (because we assume that qubits only rest in storage/gate locations) and no channels connecting the gates. It iterates with the circuit scheduler, moving and connecting gate locations until the qubits can communicate sufficiently to perform the specified circuit. The current layout is fed into the circuit scheduler which tries to schedule until it finds qubits in gate locations that cannot communicate to perform a gate. The place and router then connects the problematic gate locations and tries scheduling on the layout again. The iteration finishes once the circuit can be successfully completed. Our algorithm bears some similarity to the iterative procedure in adaptive cluster growth placement [12] in classical CAD. Gate locations are placed from the center outward as the circuit grows to fit a rectilinear boundary.\n\nThe placer can move gate locations that have to be connected if they are not already connected to something else. The router connects gate locations by making a direct path in the x and y directions between them and placing a new channel, shifting existing channels out of the way. Since channels are allowed to overlap, intersections are inserted where the new channels cut across existing ones.\n\nThis technique has the advantage that, since the circuit scheduler prioritizes gates based on gate delay critical path, potentially critical gates are mapped to gate locations and connected early in the process. Thus critical gates tend to be initially placed close together to shorten the circuit critical path. Additionally, gate locations that need to communicate can be connected directly instead of using a general shared grid channel network, where congestion can occur and cause qubits to be routed along unnecessarily long paths.\n\nA disadvantage of this heuristic is that gate placement is done to optimize critical path, not to minimize channel intersections. This means that the layout could end up having many 4-way channel intersections and turns, both of which have more delay than 2-way straight channels. Additionally, even though critical gates are mapped and placed near each other, the channel routing algorithm tends to spread these gate locations apart as more channels cut through the center of the circuit. We discuss our experimental evaluation of this heuristic in Section 9."
},
{
"section_type": "OTHER",
"section_title": "Dataflow-Based Layouts",
"text": "As described in Section 6, a systematic row by row initial placement for qubits allows us to make somewhat accurate comparisons between different gridbased layouts, while a random initial qubit placement allows us to test a single grid's dependence on qubit starting positions. However, in laying out a quantum circuit, we would like to have a more intelligent and natural means of determining initial qubit placement. For this, we turn to the dataflow graph representation of the circuit."
},
{
"section_type": "OTHER",
"section_title": "Dataflow Graph Analysis",
"text": "Figure 9a shows a QASM instruction sequence consisting of Hadamard gates (H) and controlled bitflips (CX) operating on qubits Q0, Q1, Q2 and Q3, with each instruction labeled by a letter. Figure 9b shows the equivalent sequence of operations in standard quantum circuit format. Either of these may\n\nA) H Q0 B) H Q1 C) H Q2 D) H Q3 E) CX Q0,Q1 F) CX Q2,Q3 G) CX Q1,Q2 H) CX Q2,Q3 I) CX Q0,Q2 H H H H Q0 Q1 Q2 Q3 A E G D F B C I H (b) (c) (a)\n\nFigure 9 : a) A QASM instruction sequence. b) A quantum circuit equivalent to the instruction sequence in (a). c) A dataflow graph equivalent to the instruction sequence in (a). Each node represents an instruction, as labeled in (a). Each arc represents a qubit dependency.\n\nA B C D E F I A B C D F A B C D E F G H I NG1 NG9 NG8 NG7 NG6 NG5 NG4 NG3 NG2 (b) (c) (a)\n\n8 1 1 4 5 1 2 6 4 1 NG1 NG9 NG10 NG6 NG5 NG4 NG3 NG2 8 1 1 4 1 2 6 1 6? G H NG1 NG10 NG6 NG4 NG3 NG2 1 1 4 1 6? 6 1 6? G H E I NG11\n\nFigure 10 : a) Each node (instruction) is initialized in its own node group (NG, outlined by the dotted lines), which corresponds to a physical gate location in a layout. Once placed, we extract physical distances between the nodes (the edge labels). b) We find the longest edge weight on the longest critical path (the length 5 edge on the path C-F-G-H-I; solid bold arrows) and merge its two node groups to eliminate that latency. c) We recompute the critical path (A-E-I; dashed bold arrows) and merge its node groups, and so on. be translated into the dataflow graph shown in Fig- ure 9c , where each node represents a QASM instruction (as labeled in Figure 9a ) and each arc represents a qubit dependency. With this dataflow graph, we may perform some analyses to help us place and route a layout for our quantum circuit.\n\nThe general idea is that we shall create node groups in the dataflow graph which correspond to distinct gate locations that may then be placed and routed on a layout. All instructions within a single node group are guaranteed to be executed at a single gate location, as elaborated upon in Section 8.3. To begin with, we create a node group for each instruction, giving us a dataflow group graph, as shown in Figure 10a . If we lay out this group graph with a distinct designated gate for each instruction (using heuristics discussed in Section 8.2), we get a layout in which the starting location of each qubit is specified implicitly by its first gate location, so no additional initial placement heuristic is needed.\n\nFrom this layout we can extract movement latency between nodes and label the edges with weights (as in Figure 10a ). We now find the longest critical path by qubit. The critical path A-E-I of qubit Q0 has length 14 (the dashed bold arrows), while the critical path C-F-G-H-I of qubit Q2 has length 15 (the solid bold arrows). We select the longest edge on the longest critical path, which is the edge G-H with weight 5. We merge these two node groups to eliminate this latency, in effect specifying that these two instructions should occur at the same gate location (Figure 10b ). We then update the layout and recompute distances. Assuming we merged these two node groups to the location of H (NG8), then the weight of edge F-G changes to 1 (to match the weight of edge F-H) and the weight of edge E-G probably changes to 6 (former E-G plus former G-H), but the exact change really depends on layout decisions. The new critical path is now A-E-I, so if we do this again, we merge node groups NG5 and NG9 to eliminate the edge of weight 8, and we get the group graph in Figure 10c . In merging nodes, there is the possibility that two qubit starting locations get merged, complicating the assignment of initial placement. For this reason, we add a dummy input node for each qubit before its first instruction. The merging heuristic doesn't allow more than one input node in any single node group, so we maintain the benefit of having an intelligent initial qubit placement without extra work.\n\nThere is an important trade-off to consider when taking this merging approach. A tiled grid layout provides plenty of gate location reuse but is unlikely to provide any pipelinability without great effort. A layout of the group graph in Figure 10a (with each instruction assigned to a distinct gate location) provides no gate location reuse at all but high potential pipelinability. This raises the question of whether we wish to minimize area and time (for critical data qubits), maximize throughput of a pipeline (for ancilla generation), or compromise at some middle ground where small sets of nearby nodes are merged in order to exploit locality while still retaining some pipelinability. We intend to further explore this topic in the future."
},
{
"section_type": "OTHER",
"section_title": "Placement and Routing",
"text": "Taking the group graph from the dataflow analysis heuristic, the placement algorithm takes advantage of the fanout-limited gate output imposed by the No-Cloning Theorem [25] to lay out the dataflow-ordered gate locations in a roughly rectangular block. We adopt a gate array-style design, where gate locations are laid out in columns according to the graph, with space left between each pair of columns for necessary channels. This can lead to wasted space due to a linear layout of uneven column sizes, so we may also perform a folding operation, wherein a short column may be folded in (joined) with the previous column, thus filling out the rectangular bounding box of the layout as much as possible and decreasing area. The columns are then sorted to position gate locations that need to be connected roughly horizontal to one another. This further minimizes channel distance between connected gate locations and reduces the number of high-latency turns.\n\nOnce gate locations are placed, we use a grid-based model in which we first route local wire channels between gate locations that are in adjacent or the same columns. These channels tend to be only a few macroblocks long each. A separate global channel is then inserted between each pair of rows and between each\n\nTechnology-Dependent Netlist Dataflow Analysis & Gate Combination Sorted Dataflow Placement Local/Global Channel Router Geometry-Aware Netlist Gate Scheduler/ Simulator Placement and Routing Figure 11: The placement and routing portion of our CAD flow (shown in Figure 4) takes a technologydependent netlist and translates it into a geometryaware netlist through an iterative process involving dataflow analysis and placement and routing techniques. pair of columns of gate locations. These global channels stretch the full length of the layout. There are no real routing constraints in our simple model since channels are allowed to overlap and turn into 3-or 4-way intersections. We depend on the dataflow column sorting in the placement phase to reduce the number of intersections and shared local channels. While local channels could technically be used for global routing and vice versa, we've found that this division in routing tends to divide the traffic and separate local from long-distance congestion. With these basic placement and routing schemes, we may now iterate upon the layout, as shown in Figure 11. The technology-dependent netlist is translated into a dataflow group graph with a separate gate location for each instruction (Figure 10a ). This group graph is then placed, routed and scheduled to get latency and identify the runtime critical path (as opposed to the critical path in the group graph, which fails to take congestion into account). The longest latency move on the runtime critical path (between two node groups) is merged into one node group, thus eliminating the move since a node group represents a single gate location. This new group graph is then placed, routed and scheduled again to find the next pair of node groups to merge.\n\nOnce this process has iterated enough times, we reach a point where congestion at some heavily merged node group is actually hurting the latency with each further merge. We alleviate this congestion by adding storage nodes (essentially gate locations that don't actually perform gates) near the congested node group. This increases the area slightly but maintains the locality exploited by the merging heuristic. If congestion persists, we halt the algorithm, back up a few merging steps and output the geometry-aware netlist."
},
{
"section_type": "RESULTS",
"section_title": "Annotated Scheduling - Results",
"text": "The scheduling heuristic described in Section 5.2 schedules an arbitrary QASM instruction sequence on an arbitrary layout. However, once we have assigned instructions in a dataflow graph to node groups (as described in Section 8.1), we wish those instructions to be executed at their proper location on any layout placed and routed from the group graph. To this end, we annotate each instruction in the instruction sequence with the name of the gate location where it must be executed. Additionally, since we have the gate locations in advance, we can incorporate movement in the back-prioritization of the instruction sequence. Thus, the priority assigned to each qubit now incorporates both gate latencies and movement through an uncongested layout, which gives us a better approximation of each qubit's critical path. We use this extended scheduler in our dataflow-based experiments presented in Section 9.\n\nWe now present our simulation results for the heuristics described in earlier sections."
},
{
"section_type": "OTHER",
"section_title": "Benchmarks",
"text": "Relatively high error rates of operations in a quantum computer necessitate heavy encodings of qubits. As such, we focus on encoding circuits (useful for both data and ancillae) and error correction circuits to experiment with circuit layout techniques. We lay out a number of error correction and encoding circuits to Qubit Gate Circuit name count count [ [7, 1, 3] ] L1 encode [20] 7 21 [ [23, 1, 7] ] L1 encode [21] 23 116 [ [7, 1, 3] ] L1 correction [1] 21 136 [ [7, 1, 3] ] L2 encode [20] 49 245\n\nTable 1 : List of our QECC benchmarks, with quantum gate count and number of qubits processed in the circuit.\n\nevaluate the effectiveness of the heuristics used in our CAD flow in terms of circuit area and latency, as determined by our scheduler. Our circuit benchmarks are shown in Table 1 . We use two level 1 (L1) encoding circuits, a level 2 (L2) recursive encoding circuit and a fault-tolerant level 1 correction circuit. The idea of the encoding circuits is that they will provide a constant stream of encoded ancillae to interact with encoded data qubit blocks. Thus, for these circuits, throughput is a more important measure than latency, implying that they would benefit greatly from pipelining. Nonetheless, a high latency circuit could introduce non-trivial error due to increased qubit idle time. On the other hand, correction circuits are much more latency dependent, since they are on the critical path for the processing of data qubit blocks."
},
{
"section_type": "OTHER",
"section_title": "Evaluation",
"text": "We have evaluated a variety of layout design heuristics on the four benchmarks shown in Table 1 . The results are in Table 2 . \"QPOS Grid\" refers to the best scheduled layout from the literature [16] (see Section 6) . \"Optimal Grid\" refers to the best grid with an area matching the QPOS Grid used that was found by the exhaustive search described in Section 6. \"Greedy\" refers to the heuristic described in Section 7. \"DF\" refers to the dataflowbased approach from Section 8. \"Non-folded\" means the dataflow graph is laid out with varying column widths; \"folded\" means the layout has been made more rectangular by stacking columns. The number of global channels is between each pair of rows and columns of gate locations. \"Critical combining\" refers to our dataflow group graph merging heuristic.\n\nThe exhaustive search over grids yields the best latency for all benchmarks, which is not surprising. This kind of search becomes intractable quickly as circuit size grows, and additionally, it is based on the unproven assumption that a regular layout pattern is the best approach. We include this data point as something to keep in mind as a target latency. Among the polynomial-time heuristics, we first note that no single heuristic is optimal for all four benchmarks and that, in fact, no single heuristic optimizes both latency and area for any single circuit. Dataflow-based place and route techniques in general produce the lowest latency circuits. We find that the optimal global channel count per column (1 or 2) depends on the circuit being laid out. This is an artifact of the lack of maturity in our routing methodology. We intend to explore more adaptive routing optimization in our ongoing work.\n\nThe dataflow approach and the QPOS Grid tend to trade off between latency and area. However, we expect that the dataflow approach will show greater potential for pipelining, thus allowing us to target circuits such as an encoded ancilla generation factory, for which throughput is of greater importance than latency. We also observe that non-folded dataflow layouts are likely to have even greater pipelinability than folded ones, but at the likely cost of greater area. Although, we should note that the area estimates for the non-folded DF-based layouts are in fact overestimates due to our use of a liberal bounding box for these calculations.\n\nWe find that the greedy heuristic tends to find the best design area-wise, but the latency penalty increases with circuit complexity. This is expected, as greedy is unable to handle congestion problems, so it works best for small circuits where congestion is not an issue. It is for the opposite reason that the DF heuristics fail on the [ [7, 1, 3] ] L1 encode. They insert too much complexity into an otherwise simple problem."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusion",
"text": "We presented a computer-aided design flow for the layout, scheduling and control of ion trap-based quantum circuits. We focused on physical quantum circuits, that is, ones for which all ancillae, encodings and interconnect are explicitly specified. We explored several mechanisms for generating optimal layouts and schedules for our benchmark circuits.\n\nPrior work has tended to assume a specific regular grid structure and to schedule operations within this structure. We investigated a variety of grid structures and showed a performance variance of a factor of four as we varied grid structure and initial qubit placement. Since exhaustive search is clearly impractical for large circuits, we also explored two polynomialtime heuristics for automated layout design. Our greedy algorithm produces good results for very simple circuits, but quickly begins to be suboptimal as circuit size grows. For larger circuits, we investigated a dataflow-based analysis of the quantum circuit to a place and route mechanism which leverages from classical algorithms. We found that our our dataflow approach generally offers the best latency, often at the cost of area. However, we expect that a layout based on the dataflow graph analysis also offers better potential for pipelining than a grid-based approach, and we intend to investigate this further in the future."
}
] | ||
arxiv:0704.0278 | 0704.0278 | 1 | 10.1088/0264-9381/24/13/009 | fed9bb61e28a64d27ff273594f1325a8e6ee8168c1801dc30de7af1494379996 | q-Deformed spin foam models of quantum gravity | We numerically study Barrett-Crane models of Riemannian quantum gravity. We
have extended the existing numerical techniques to handle q-deformed models and
arbitrary space-time triangulations. We present and interpret expectation
values of a few selected observables for each model, including a spin-spin
correlation function which gives insight into the behaviour of the models. We
find the surprising result that, as the deformation parameter q goes to 1
through roots of unity, the limit is discontinuous. | [
"Igor Khavkine",
"J. Daniel Christensen"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | Spin foam models were first introduced as a space-time alternative to the spin network description of states in loop quantum gravity [3] . The most studied spin foam models are due to Barrett and Crane [8, 9] . A spin foam is a discretization of space-time where the fundamental degrees of freedom are the areas labelling its 2-dimensional faces.
An important goal in the investigation of spin foam models is to obtain predictions that can be compared to the large scale, classical, or semiclassical behavior of gravity. This work continues the numerical investigation of the physical properties of spin foam models of Riemannian quantum gravity begun in [5] [6] [7] 13] . In this paper, we extend the computations to the q-deformed Barrett-Crane model and to larger space-time triangulations.
The main applications of q-deformation are two-fold. On the one hand, it can act as a regulator for divergent models, as is apparent in the link between the Ponzano-Regge models. On the other hand, Smolin [30] has argued that q-deformation is necessary to account for a positive cosmological constant. Both of these aspects are explored in more detail in Section 2.2. A surprising result of our work is evidence that the limit, as the cosmological constant is taken to zero through positive values, is discontinuous.
Large triangulations are necessary to approximate semiclassical space-times. The possibility of obtaining numerical results from larger triangulations takes us one step closer to that goal and increases the number of facets from which the physical properties of a spin foam model may be examined. As an example, we are able to study how the spin-spin correlation varies with the distance between faces in the triangulation. 1 This paper is structured as follows. We begin in Section 2 by reviewing the basics of q-deformation and discussing in detail its aforementioned applications. Section 3 reviews the details of the Barrett-Crane model, summarizes the necessary changes for its q-deformation, and defines several observables associated to spin foams. In Section 4, we review the existing numerical simulation techniques and how they need to be generalized to handle q-deformation and larger triangulations. Section 5 presents the results of our numerical simulations. In Section 6, we give our conclusions and list some avenues for future research. The Appendix briefly summarizes our notational conventions and useful formulas.
2 Deformation of su(2)
In this section, we describe the q-deformation of the Lie algebra su(2) into the algebra su q (2) (also denoted U q (su(2))), the representations of su q (2), and the applications of q-deformation. The deformations of spin(4) are then obtained through the isomorphism spin(4) ∼ = su(2) ⊕ su (2) .
The following is part of the general subject of quantum groups [21] . Here we shall concentrate solely on the su(2) and spin(4) cases.
The Lie algebra su(2) is generated by the well known Pauli matrices σ i , which obey the commutation relations
[σ + , σ -] = 4σ 3 , [σ 3 , σ + ] = 2σ + , [σ 3 , σ -] = -2σ -, ( 1
)
where σ ± = σ 1 ± iσ 2 . The universal enveloping algebra of su(2) is the associative algebra generated by σ ± and σ 3 subject to the above identities, with the Lie bracket being interpreted as [A, B] = AB -BA.
The q-deformed algebra su q (2) is constructed by replacing σ 3 with another generator. Formally, it is thought of as Σ = q 1 2 σ3 , where q ∈ C with the exceptions q = 0, 1, -1. The Lie bracket relations are replaced by the identities
[σ + , σ -] = 4 Σ 2 -Σ -2 q -q -1 , Σσ + = qσ + Σ, Σσ -= -qσ -Σ. ( 2
)
We can rewrite q = 1 + 2ε and think of ε as a small complex number. Then, formally at leading order in ε, the substitution Σ = q 1 2 σ3 = 1 + εσ 3 + O(ε 2 ) reduces the deformed identities (2) to the standard Lie algebra relations (1). The associative algebra generated by σ ± and σ 3 subject to the deformed identities (2) is the algebra su q (2) .
For generic q, that is, when q is not a root of unity, the finite-dimensional irreducible representations of su q (2) are classified by a half-integer, j = 0, 1/2, 1, 3/2, . . . , referred to as the spin, in direct analogy with the representations of su(2) and the theory of angular momentum. The dimension of the representation j is 2j + 1. When q = exp(iπ/r) is a 2rth root of unity (ROU), where r > 2 is an integer called the ROU parameter, the representations j are still defined, but become reducible for j > (r -2)/2. They decompose into a sum of representations with spin at most (r -2)/2 and so-called trace 0 ones, whose nature will be explained below.
For the purposes of this paper we are concerned only with intertwiners between representations of su q (2), i.e., linear maps commuting with the action of the algebra, and their (quantum) traces 1 .
Any such intertwiner can be constructed from a small set of generators and elementary operations on them. These constructions, as well as traces, can be represented graphically. Such graphs are called (abstract) spin networks. Their calculus is well developed and is described in [18] , whose conventions we follow throughout the paper with one exception: we use spins (half-integers) instead of twice-spins (integers). A brief review of our notation and conventions can be found in the Appendix.
Trace 0 representations of su q (2) are so called because the trace of an intertwiner from such a representation to itself is always zero. Thus, they can be freely discarded, as they do not contribute to the evaluation of q-deformed spin networks.
Deformation, especially with q = exp(iπ/r) a 2rth primitive ROU, is important for spin foam models for at least two reasons. Replacing q = 1 by some ROU can act as a regulator for a model whose partition function and observable values are otherwise divergent. Also, su q (2) spin networks 2 naturally appear when considering a positive cosmological constant in loop quantum gravity.
The original Ponzano-Regge model [27] attempts to express the path integral for 3-dimensional Riemannian general relativity as a sum over labelled triangulations of a 3-manifold. The edges of the triangulation are labelled by discrete lengths, identified with spin labels of irreducible SU (2) representations. Each tetrahedron contributes a 6j-symbol factor to the summand, normalized to ensure invariance of the overall sum under change of triangulation. Unfortunately, the Ponzano-Regge model turned out to be divergent. Motivated by the construction of 3-manifold invariants, Turaev and Viro were able to regularize the Ponzano-Regge model [1, 31] by replacing the SU (2) 6j-symbols with their q-deformed analogs at a ROU q. The key feature of the regularization is the truncation of the summation to only the irreducible representations of su q (2) of non-zero trace, which leaves only a finite number of terms in the model's partition function.
A version of the Barrett-Crane model, derived from a group field theory by De Pietri, Freidel, Krasnov and Rovelli [16] (DFKR for short), was also found to be divergent. A q-deformed version of the same model at a ROU q is similarly regularized (see Section 3.2). Some numerical results for the regularized version of this model are given in Section 5.2.
The argument linking q-deformation to the presence of a positive cosmological constant is due to Smolin [29] and is given in more refined form in [30] . It is briefly summarized as follows. Loop quantum gravity begins by writing the degrees of freedom of general relativity in terms of an SU (2) connection on a spatial slice and the slice's extrinsic curvature. A state in the Schrödinger picture, a wave function on the space of connections, can be constructed by integrating the Chern-Simons 3-form over the spatial slice. This state, known as the Kodama state, simultaneously satisfies all the canonical constraints of the theory and semiclassically approximates de Sit-1 When q = 1, this notion of trace reduces up to sign to the usual trace of a linear map, but is slightly different otherwise, cf. [10, Chapter 4] .
2 These are graphs embedded in a 3-manifold, labelled by representations of suq(2). They are similar to but distinct from the abstract spin networks referred to above. See [4] for the distinction.
ter spacetime, which is a solution of the vacuum Einstein equations with a positive cosmological constant. The requirement that the Kodama state also be invariant under large gauge transformations implies discretization of the cosmological constant, Λ ∼ 1/r, with r a positive integer. The coefficients of the Kodama state in the spin network basis are obtained by evaluating the labelled graph, associated to a basis state, as an abstract su q (2) spin network. Here the deformation parameter q is a ROU, q = exp(iπ/r), where the ROU parameter r is identified with the discretization parameter of the cosmological constant.
Given the heuristic link [4] between spin networks of loop quantum gravity and spin foams, it is natural to q-deform a spin foam model as an attempt to account for a positive cosmological constant. With this aim, Noui and Roche [23] have given a q-deformed version of the Lorentzian Barrett-Crane model. The possibility of qdeformation has been with the Riemannian Barrett-Crane model since its inception [8] and all the necessary ingredients have been present in the literature for some time. In the next section these details are collected in a form ready for numerical investigation.
Consider a triangulated 4-manifold. Let ∆ n denote the set of n-dimensional simplices of the triangulation. The dual 2-skeleton is formed by associating a dual vertex, edge and polygonal face to each 4-simplex, tetrahedron, and triangle of the triangulation, respectively. A spin foam is an assignment of labels, usually called spins, to the dual faces of the dual 2-skeleton. Each dual edge has 4 spins incident on it, while each dual vertex has 10. A spin foam model assigns amplitudes A F , A E and A V , that depend on all the incident spins, to each dual face, edge and vertex, respectively. The amplitude Z(F ) assigned to a spin foam F is the product of the amplitudes for individual cells of the 2-complex, while the total amplitude Z tot assigned to a triangulation is obtained by summing over all spin foams based on the triangulation:
Z(F ) = f ∈∆2 A F (f ) e∈∆3 A E (e) v∈∆4 A V (v), Z tot = F Z(F ). ( 3
)
Some models, such as those based on group field theory [16, 17, 24] , also include a sum over triangulations in the definition of the total partition function.
The Riemannian Barrett-Crane model was first proposed in [8] . Its relation to the Crane-Yetter [15] spin foam model is analogous to the relation of the Plebanski [26] formulation of general relativity (GR) to 4-dimensional BF theory with Spin(4) as the structure group. Both BF theory and the Crane-Yetter model are topological and the latter is considered a quantization of the former [2] . In the Plebanski formulation, GR is a constrained version of BF theory. Similarly, the Barrett-Crane model restricts the spin labels summed over in the Crane-Yetter model. With this restriction, Barrett and Crane hoped to produce a discrete model of quantum (Riemannian) GR.
All amplitudes are defined in terms of spin(4) spin networks. However, given the isomorphism spin(4) ∼ = su(2) ⊕ su(2), all irreducible representations of spin(4) can be 4 written as tensor products of irreducible representations of su(2). The Barrett-Crane model specifically limits itself to balanced representations, which are of the form j ⊗ j, where j is the irreducible representation of su(2) of spin j. Since the tensor product corresponds to a juxtaposition of edges in a spin network, any spin(4) spin network may be written as an su(2) spin network where an edge labelled j ⊗ j is replaced by two parallel edges, each labelled j. To avoid redundancy of notation, we use a single j instead of j ⊗ j to label spin(4) spin network edges. We then distinguish them from su(2) networks by placing a bold dot at every vertex. The Barrett-Crane vertex is an intertwiner between four balanced representations:
b c a d = e j d e a c e b b a d c e ⊗ b a d c e
. ( 4
)
The graphs on the right hand side of the definition are su(2) spin networks and the sum runs over all admissible labels e. The graphical notation and the conditions for admissibility are defined in the Appendix. The above expression defines the Barrett-Crane vertex in a way that breaks rotational symmetry. However, it can be shown that the vertex is in fact rotationally symmetric. Up to normalization, this property makes the Barrett-Crane vertex unique [28] . The above formula defines a vertical splitting of the vertex. A ninety degree rotation will define an analogous horizontal splitting. Both possibilities are important in the derivation of the algorithm presented in Section 4.1.
Given a 4-simplex v of a triangulation, the corresponding vertex of the dual 2complex is assigned the amplitude
A V (v) = 0 1 2 3 4 j 1,0 j 1,1 j 1,4 j 1,2 j 1,3 j 2,0 j 2,1 j 2,4 j 2,2 j 2,3 . ( 5
)
This spin network is called the 10j-symbol. The 4-simplex v is bounded by five tetrahedra, which correspond to the vertices of the 10j graph. The four edges incident on a vertex correspond to the four faces of the corresponding tetrahedron; the spin labels are assigned accordingly. The edge joining two vertices corresponds to the face shared by corresponding tetrahedra. Evaluation of the 10j-symbol is discussed in Section 4.1. While the crossing structure depicted above is immaterial in the undeformed case, it is essential at nontrivial values of q. It is given here for reference. 5
The original paper of Barrett and Crane did not specify dual edge and face amplitudes. Three different dual edge and face amplitude assignments were considered in a previous paper [7] . We concentrate on the same possibilities. For the Perez-Rovelli model [25], we have
A F (f ) = j , A E (e) = j 4 j 3 j 2 j 1 j 1 j 2 j 3 j 4 . ( 6
)
For the DFKR model [16], we have
A F (f ) = j , A E (e) = 1 j 4 j 3 j 2 j 1 . ( 7
)
For the Baez-Christensen model [7], we have
A F (f ) = 1, A E (e) = 1 j 4 j 3 j 2 j 1 . ( 8
)
The bubble diagram, when translated into su(2) spin networks, corresponds to two bubbles (see Appendix)
j = j 2 . ( 9
)
and evaluates to (2j + 1) 2 . The so-called eye diagram simply counts the dimension of the space of 4-valent intertwiners, which is also the number of admissible e-edges summed over in Equation (4) . In symmetric form, it is given by
j 4 j 3 j 2 j 1 = 1 + min{2j, s -2J} if positive and s is integral, 0 otherwise, ( 10
) where s = k j k , j = min k j k , and J = max k j k . 6
Thanks to graphical notation, the q-deformation of the spin foam amplitudes described above is straightforward, with only a few subtleties. The main distinction is that q-deformed graphs are actually ribbon (framed) graphs with braiding. Thus, any undeformed spin network has to be supplemented with information about twists and crossings before evaluation.
In [32] , Yetter generalized the Barrett-Crane 4-vertex for a q-deformed version of spin(4). Since spin(4) ∼ = su(2) ⊕ su(2), there is a two parameter family of possible deformations of the Lie algebra, 2) . Yetter singles out the one parameter family q ′ = q -1 , restricted to balanced representations, since it preserves the invariance of the Barrett-Crane vertex under rotations. This family also has especially simple curl and twist identities:
spin q,q ′ (4) ∼ = su q (2) ⊕ su q ′ (
j = j and c a b = c b a , ( 11
)
where the left factor of j ⊗ j corresponds to su q (2) and the right one to su q -1 (2), and the 3-vertex is the obvious juxtaposition of two su q (2) and su q -1 (2) 3-vertices. Once this deformation is adopted, the ribbon structure can be ignored [32] , so one only needs to specify the crossing structure for a given spin(4) spin network to obtain a well-defined q-evaluation.
There are three basic graphs needed to define the Barrett-Crane simplex amplitudes: the bubble, the eye, and the 10j-symbol. The evaluation of the bubble graph, Equation (9), is [2j + 1] 2 , where the quantum integer [2j + 1] is defined in the Appendix. Remarkably, the value of the eye diagram turns out not to depend on q and its value is still given by Equation (10) . The only exception is when q is a ROU with parameter r. Then, the dimension of the space of 4-valent intertwiners changes to
j 4 j 3 j 2 j 1 = min 1 + min{2j, s -2J} r -1 -max{2J, s -2j}
if positive and s is integral,
0 otherwise, ( 12
) where again s = k j k , j = min k j k , and J = max k j k .
The 10j-symbol is the only network with a non-planar graph. Originally, it was defined in terms of the 15j-symbol from the Crane-Yetter model. This 15j-symbol was defined with q-deformation in mind, so its crossing and ribbon structure was fully specified [14, Section 3]. Adapted to the 10j-graph, it can be summarized as follows: Consider a 4-simplex. The dual 1-skeleton of the boundary has five dual vertices and ten dual edges, and is the complete graph K 5 on these five dual vertices. If we remove one of the (non-dual) vertices from the boundary of the 4-simplex, what remains is homeomorphic to R 3 . For any such homeomorphism, the embedding of K 5 into R 3 can be projected onto a 2-dimensional plane. The crossing structure of the 10j graph is defined by such a projection. It is illustrated in Equation (5) . Although, with crossings, the 10j graph is no longer manifestly invariant under permutations of its vertices, it can be shown to be so. 7
The definition of observables in a spin foam model of quantum gravity is still open to interpretation (see Section 6 of [7] for a brief discussion). For a fixed spin foam, the half-integer spin labels of its faces are the fundamental variables of the model. Practically speaking, any observable of a spin foam model should be an expectation value of some function O(F ) of the spin labels of a spin foam F , averaged over all spin foams with amplitudes specified by Equation (3):
O = F O(F )Z(F ) Z tot . ( 13
)
In this paper we choose to concentrate on a few observables representative of the kind of quantities computable in a spin foam model. As before, fix a triangulation of a 4-manifold, let ∆ 2 represent the set of its faces and let j : ∆ 2 → {0, 1/2, 1, . . .} be the spin labelling. We define:
J(F ) = 1 |∆ 2 | f ∈∆2 ⌊j(f )⌉ , ( 14
) (δJ) 2 (F ) = 1 |∆ 2 | f ∈∆2 (⌊j(f )⌉ -J ) 2 , ( 15
) A(F ) = 1 |∆ 2 | f ∈∆2 ⌊j(f )⌉ ⌊j(f ) + 1⌉, ( 16
)
C d (F ) = 1 N d f,f ′ ∈∆2 dist(f,f ′ )=d ⌊j(f )⌉ ⌊j(f ′ )⌉ -J 2 (δJ) 2 . ( 17
) where ⌊n⌉ denotes a quantum half-integer (see Appendix), | • | denotes cardinality, dist(f, f ′
) denotes the distance between faces, and N d is a normalization factor (see below for the definition of distance and N d ). These observables represent average spin per face, variance of spin per face, average area per face, and spin-spin correlation as a function of d. The choice of observables given above is somewhat arbitrary. For instance, there are several subtly distinct choices for the expression for (δJ) 2 . Fortunately, they all yield expectation values that are nearly identical. The expression given above has the technical advantage of falling into the class of so-called single spin observables. These are observables whose expectation value can be directly obtained from the knowledge of probability with which spin j occurs on any face of a spin foam. All of J, (δJ) 2 , and A are single spin observables, while C d is not.
Note that on a fixed triangulation with no other background geometry, there is no physical notion of distance. We can, instead, define a combinatorial analog. For any two faces f and f ′ of a given triangulation, let dist(f, f ′ ) be the smallest number of face-sharing tetrahedra that connect f to f ′ . Given the discrete structure of our spacetime model, it is conceivable that this combinatorial distance, multiplied by a fundamental unit of length, approximates some notion of distance derived from the dynamical geometry of the spin foam model.
The correlation function C d may be thought of as analogous to a normalized 2point function of quantum field theory. The d-degree of face f is the number of faces 8 f ′ such that dist(f, f ′ ) = d. If the d-degree of every face is the same, the normalization factor N d can be taken to be the number of terms in the sum (17) , that is, the number of face pairs separated by distance d. This choice ensures the inequality |C d | ≤ 1. If not all faces have the same d-degree, then the normalization factor has to be modified to
N d = |∆ 2 |D d , ( 18
)
where D d is the maximum d-degree of a face, which reduces to the simpler definition in the case of uniform d-degree.
The choice of the q-dependent expression ⌊j⌉, instead of simply using the halfinteger j, is motivated in Section 5.1. For some q, the argument of the square root in A(F ) may be negative or even complex. In that case, a branch choice will have to be made. Luckily, if q = 1, q is a ROU, or q is real, the expression under the square root is always non-negative.
The key development that made possible numerical simulation of variations of the (undeformed) Barrett-Crane model [6, 7] is the development by Christensen and Egan of a fast algorithm for evaluating 10j-symbols [13] . In this section, we show how this algorithm generalizes to the q-deformed case and discuss numerical evaluation of observables for the previously described spin foam models.
The derivation of the Christensen-Egan algorithm given in [13] is contingent on the possibility of splitting the and on the recoupling identity, Equation (43) of the Appendix. Both identities still hold in the q-deformed case. The validity of the 4-vertex splitting was proved by Yetter [32] and the recoupling identity is a standard part of su q (2) representation theory.
The only remaining detail of the algorithm's generalization is the crossing structure of the 10j graph, which was established in Section 3.2. However, its only consequence is an extra factor from the twist implicit in the bubble diagram of Section 4 of [13], cf. Equation (50) of the Appendix. We will not reproduce the derivation of the algorithm here. However, the way in which the twist arises is schematically illustrated in Figure 1 . Note that the triviality of the twist for Yetter's balanced representations, Equation (11) , does not apply here since the twist occurs separately in distinct su q (2) networks.
The algorithm itself can be summarized in the following form:
{10j} = (-) 2S m1,m2 φ tr[M 4 M 3 M 2 M 1 M 0 ]. ( 19
)
The 10j-symbol depends on the ten spins j i,k , (i = 1, 2, k = 0, . . . , 4) specified in Equation (5) . The overall prefactor depends on the total spin S = i,k j i,k and the per-term prefactor is
φ = (-) m1-m2 [2m 1 + 1][2m 2 + 1]q m1(m1+1)-m2(m2+1) . ( 20
) 9
(a) (b) (c) (d) Figure 1: In reference to [13] , (a) corresponds to Equation (1), (b) corresponds to Equation ( 2), while (c) and (d) correspond to the "ladder" and "bubble" diagrams of Section 4, respectively. The illustrated twist introduces the explicitly q-dependent factor into Equation (20) .
The exponents of (-) and q are always integers. The M k are matrices (not all of the same size) of dimensions compatible with the five-fold product and trace. Their matrix elements are
(M k ) l k+1 l k = [2l k + 1](T 1 ) l k+1 l k (T 2 ) l k+1 l k θ(j 2,k-1 , l k+1 , j 1,k ) θ(j 2,k+1 , l k+1 , j 1,k+1 ) , ( 21
) (T i ) l k+1 l k = Tet l k j 2,k m i l k+1 j 2,k-1 j 1,k θ(j 2,k , l k+1 , m i ) . ( 22
)
The quantum integers [n], as well as the theta θ(a, b, c) and tetrahedral Tet[• • •] su q (2) spin networks are defined in the Appendix. The quantities l k and m i are spin labels (half-integers). They are constrained by admissibility conditions (parity conditions and triangle inequalities). The parity of each index is determined by the conditions
l k ≡ j 1,k + j 2,k ≡ j 1,k-1 + j 2,k-2 , ( 23
) m i ≡ l k + j 2,k-1 , ( 24
)
for i = 1, 2 and k = 0, . . . , 4, where ≡ denotes equivalence mod 1 and the second subscript of j is taken mod 5. Summation bounds are determined by the triangle inequalities, which must be checked for each trivalent vertex introduced in the derivation 10 of the algorithm. They boil down to
lb 3 (j 1,k , j 2,k , j 2,k-1 ) ≤ m i ≤ j 1,k + j 2,k + j 2,k-1 , (25) |j 1,k-1 -j 2,k-2 | ≤ l k ≤ j 1,k-1 + j 2,k-2 , ( 26
) |j 1,k -j 2,k | ≤ l k ≤ j 1,k + j 2,k , (27) |m i -j 2,k-1 | ≤ l k ≤ m i + j 2,k-1 , ( 28
)
for i = 1, 2 and k = 0, . . . , 4, where we have used the notation lb 3 (a, b, c) = 2 max{a, b, c} -(a + b + c). (29) When q = exp(iπ/r) is a ROU, extra inequalities must be taken into account to exclude summation over reducible representations. These are
m i ≥ j 1,k + j 2,k + j 2,k-1 -(r -2), (30) m i ≤ ub 3 (j 1,k , j 2,k , j 2,k-1 ) + (r -2), ( 31
)
l k ≤ (r -2) -(j 1,k + 2,k ), (32) l k ≤ (r -2) -(j 1,k-1 + j 2,k-2 ), ( 33
) l k ≤ (r -2) -(m + j 2,k-1 ), ( 34
)
where now ub 3 (a, b, c) = 2 min{a, b, c} -(a + b + c).
If any of the parity constraints or inequalities cannot be satisfied, the 10j-symbol evaluates to zero. algorithm has been implemented and tested in the q = 1 and ROU cases, for both j and r up to several hundreds. Unfortunately, for generic q, when Q = max{|q|, |q| -1 } > 1, the quantum integers grow exponentially as |[n]| ∼ Q n . Such a rapid growth makes the sums involved in this algorithm numerically unstable. It is still possible to use this algorithm with Q close to 1 or symbolically, using rational functions of q instead of limited precision floating point numbers. Symbolic computation is, however, significantly slower (by up to a factor of 10 6 ) than its floating point counterpart. The software library spinnet which implements these and other spin network evaluations is available from the authors and will be described in a future publication.
The sums involved in evaluating expectation values of observables, as in Equation (13) , are very high-dimensional. For instance, a minimal triangulation of the 4-sphere (seen as the boundary of a 5-simplex) contains 20 faces. Hence, any brute force evaluation of an expectation value, even on such a small lattice, involves a sum over the 20dimensional space of half-integer spin labels.
Fortunately, in the undeformed case, the total amplitude Z(F ) for a closed spin foam is never negative 3 [5]. The proof for the q = 1 case generalizes to the ROU case. One need only realize two facts. The first is that, in the ROU case, quantum integers are non-negative. The second is that, for q a ROU, an su q -1 (2) spin network evaluates to the complex conjugate of the corresponding su q (2) spin network. The 3 We expect the same thing to hold in Lorentzian signature [5, 12].
disjoint union of any two such spin networks evaluates to their product, the absolute value squared of either of them, and hence is non-negative. Then, the same positivity result follows as from Equation (1) of [5] . This positivity allows us to treat Z(F ) as a statistical distribution and use Monte Carlo methods to extract expectation values with much greater efficiency than brute force summation.
The main tool for evaluating expectation values is the Metropolis algorithm [20, 22] . The algorithm consists of a walk on the space of spin labellings. Each step is randomly picked from a set of elementary moves and is either accepted or rejected based on the relative amplitudes of spin foam configurations before and after the move. An expectation value is extracted as the average of the observable over the configurations constituting the walk. Elementary moves for spin foam simulations are discussed in the next section.
A Metropolis-like algorithm is possible even if individual spin foam amplitudes Z(F ) are negative or even complex. However, if the total partition function Z tot sums to zero, then the expectation values in Equation (13) become ill defined. Moreover, in numerical simulations, if Z tot is even close to zero, expectation value estimates may exhibit great loss of precision and slow convergence. In the path-integral Monte Carlo literature, this situation is known as the sign problem [11] . Still, the sign problem need not occur or, depending on the severity of the problem, there may be ways of effectively dealing with it.
Independent Metropolis runs can be thought of as providing independent estimates of a given expectation value. Thus, the error in the computed value of an observable can be estimated through the standard deviation of the results of many independent simulation runs [19].
The choice of elementary moves for spin foam simulations must satisfy several criteria. Theoretically, the most important one is ergodicity. That is, any spin foam must be able to transform into any other one through a sequence of elementary moves which avoid configurations with zero amplitude. Practically, it is important that these moves usually preserve admissibility. A spin foam F is called admissible if the associated amplitude Z(F ) is non-zero. If, starting with an admissible spin foam, most elementary moves produce an inadmissible spin foam, the simulation will spend a lot of time rejecting such moves without any practical benefit.
As before, consider a fixed triangulation of a compact 4-manifold. The parity conditions (23) imposed on the j i,k ,
j 1,k + j 2,k ≡ j 1,k-1 + j 2,k-2 , 0 ≤ k ≤ 4,
when taken together with the total spin foam amplitude (3), provide strong constraints on admissible spin foams. One can show that a move that changes spin labels by ±1/2 (mod 1) on each face of a closed surface in the dual 2-skeleton preserves the parity constraint. We take as the elementary moves the moves that change the spin labels by ±1/2 on the boundaries of the dual 3-cells of the dual 3-complex; the dual 3-cells correspond to the edges of the triangulation. If the manifold has non-trivial mod 2 homology in dimension 2, additional moves would be necessary, but for the examples we consider the moves above suffice. From a practical point of view, extra moves might improve the simulation's equilibration time. For instance, in the ROU case, parity preserving moves that change the spins from 0 to (r -2)/2 or (r -3)/2 were 12 introduced, since spins close to either admissible extreme may have large amplitudes. This property of the Perez-Rovelli and Baez-Christensen models is illustrated in the following section.
Unfortunately, the inequalities constraining spin labels do not have a similar geometric interpretation and cannot be used to easily restrict the set of elementary moves in advance.
Using methods described in the previous section, we ran simulations of the three variations of the Barrett-Crane model described in Section 3 and obtained expectation values for observables listed in Section 3.3. While previous work [7] performed simulations only on the minimal triangulation of the 4-sphere, which we will refer to simply as the minimal triangulation, we have extended the same techniques to arbitrary triangulations of closed manifolds.
The most striking result we can report is a discontinuity in the transition to the limit r → ∞, where r, a positive integer, is the ROU parameter with q = exp(iπ/r). As r → ∞, the deformation parameter q tends to its classical value 1. If we interpret the cosmological constant as inversely proportional to r, Λ ∼ 1/r, this limit also corresponds to Λ → 0, through positive values. For a fixed spin foam, the amplitudes and observables we study tend continuously to their undeformed values as r → ∞.
However, we find that observable expectation values do not tend to their undeformed values in the same limit, that is, O r O q=1 as r → ∞. The discontinuity is most simply illustrated with the single spin distribution, that is the probability of finding spin j at any spin foam face. This probability can be estimated from the histogram of all spin labels that have occurred during a Monte Carlo simulation. The points in Figure 2 (a) show the single spin distributions for the Baez-Christensen model with r = 50 and q = 1. The curves show the corresponding single bubble amplitude. It is the amplitude Z(F j ) of a spin foam F j with all spin labels zero, except for the boundary of an elementary dual 3-cell, whose faces are all labelled with spin j. The amplitudes and distributions are normalized as probability distributions so their sums over j yield 1. The similarity between the points and the continuous curves is consistent with the hypothesis that spin foams with isolated bubbles dominate the partition function sum. The behavior of the single spin distribution for the Perez-Rovelli model is very similar, except that its peaks are much more pronounced.
Note that the undeformed single spin distribution has a single peak at j = 0, while the r = 50 case has two peaks, one at j = 0 and the other at j = (r -2)/2, the largest non-trace 0 irreducible representation. The bimodal nature of the single spin distribution has an important impact on the large r behavior of observable expectation values, as is most easily seen with single spin observables (Section 3.3). For instance, if we consider the average, j, of the half-integers j, the large j peak would dominate the expectation value and j would diverge linearly in r, as r → ∞. On the other hand, since J is the average of the quantum half-integers ⌊j⌉, J at least approaches a constant in the same limit. This is illustrated in Figure 2 (b). 13 Figure 2: (a) Single spin distribution and single bubble amplitude for the Baez-Christensen model. The distribution was obtained from 10 9 steps of Metropolis simulation on a triangulation with 202 faces (cf. Section 5.3). (b) Some single spin observables as functions of j, with r = 50.
However, as shown in Figure 3 , this limit is not the same as the undeformed expectation value. At the same time, as can be seen from the plot of the Perez-Rovelli average area in the same figure, there are some observables whose large r limits are at least very close to the undeformed values. The area observable summand A j = ⌊j⌉ ⌊j + 1⌉ is exactly zero at both j = 0 and j = (r -2)/2, while the spin observable summand J j = ⌊j⌉ is zero at j = 0 but still positive at j = (r -2)/2, Figure 2 (b). The large j peak of the Perez-Rovelli model is very narrow and thus the expectation value of a single spin observable is strongly influenced by its value at j = (r -2)/2. The data for larger triangulations is qualitatively similar.
As expected, the ROU deformation of the DFKR model yields a finite partition function and finite expectation values. For instance, its single spin distribution for r = 40 is illustrated in Figure 4 . The divergence of the amplitude for large spins in the undeformed, q = 1, case makes numerical simulation impossible without an artificial spin cutoff. Thus, we do not have an undeformed analog of the single spin distribution. For the minimal triangulation, the ROU spin distribution deviates slightly from the single bubble amplitude close to the boundaries of admissible j. For the larger triangulation, the deviation is much more pronounced and is not restricted to the edges. This suggests that there are other significant contributions to the partition function besides single bubble spin foams. Note the large weight associated with spins around j = r/4. Around this value of j, both the area A j = ⌊j⌉ ⌊j + 1⌉ and the spin J j = ⌊j⌉ attain their maximal values and are proportional to r. Thus, it is natural to expect their expectation values to grow linearly in r, which is consistent with the divergent nature of the undeformed DFKR model. This is precisely the behavior shown in Figure 5 . On the minimal triangulation, the best linear fits for the average spin expectation value and for the 14 Figure 3: Observables for the Baez-Christensen (BCh) and Perez-Rovelli (PR) models as functions of the ROU parameter r. For large r, observables do not in general tend to their undeformed, q = 1, values; arrows show the deviation. Some observables were scaled to fit on the graph. Data is from Metropolis simulations on the minimal triangulation.
square root of the average spin variance are
J r = 0.146 r -0.064, ( 35
) (δJ) 2 1/2 r = 0.014 r + 0.187. ( 36
)
For larger triangulations, the dependence of these observables is also approximately linear in r, with only slight variation in the effective slope.
The ability to work with larger lattices allows us to explore a broader range of observables. One of them is the spin-spin correlation function C d defined in Section 3.3. In general C 0 = 1 and C d → 0 for large d. The decay of the correlation shows how quickly the spin labels on different spin foam faces become independent. A positive value of C d indicates that, on average, any two faces distance d apart both have spins above (or both below) the mean J . On the other hand, a negative value of C d indicates that, on average, any two faces distance d apart have one above and one below the mean J . A small triangulation limits the maximum distance between faces. For example, the minimal triangulation has maximum distance d = 3. Larger triangulations of the 4-sphere were obtained by refining the minimal one by applying Pachner moves randomly and uniformly over the whole triangulation. We restricted the Pachner moves to those that did not decrease the number of simplices.
Figure 4: Single spin distributions and single bubble amplitudes for the DFKR model. The distributions were obtained from 10 9 steps of Metropolis simulation on the minimal triangulation and on a triangulation with 202 faces (cf. Section 5.3).
The largest triangulation we have used has maximum distance d = 6. Its correlations for different models are shown in Figure 6 along with those from the minimal triangulation. Correlation functions for different values of ROU parameter r (including the q = 1 case) and other triangulations are qualitatively similar.
Notice the small negative dip for small values of d for the Perez-Rovelli and Baez-Christensen models. As discussed in previous sections, the partition functions of these models are dominated by spin foams with isolated bubbles. The correlation data is consistent with this hypothesis. The values of the spins assigned to faces of the bubble will be strongly correlated, while the values of the spins on two faces, one of which lies on the bubble and the other does not, should be strongly anti-correlated. Since a given face usually has fewer nearest neighbors that lie on the same bubble than that do not, on average, the short distance correlation is expected to be negative. At slightly larger distances, the correlation function turns positive again. This indicates that on a larger triangulations, spin foams with several isolated bubbles contribute strongly to the partition function. Although, with so few data points, it is difficult to extrapolate the behavior of the correlation function to larger triangulations and distances, its features are qualitatively similar to that of a condensed fluid, where the density-density correlation function exhibits oscillations on the scale of the molecular dimensions.
Note that the behavior of the DFKR correlation function is significantly different from the other two. This is also consistent with the already observed fact that its partition function has strong contributions from other than single or isolated bubble spin foams.
Figure 5: Observables for the DFKR model: area A , average spin J , spin standard deviation (δJ) 2 . Metropolis simulation, minimal triangulation. Error bars are smaller than the data points.
We have numerically investigated the behavior of physical observables for the Perez-Rovelli, DFKR, and Baez-Christensen versions of the Barrett-Crane spin foam model. Each version assigns different dual edge and face amplitudes to a spin foam, and these choices greatly affect the behavior of the resulting model. The behavior of the models was also greatly affected by q-deformation.
The limiting behavior of observables was found to be discontinuous in the limit of large ROU parameter r, i.e., q = exp(iπ/r) close to its undeformed value of 1. This result is at odds with the physical interpretation of the relation Λ ∼ 1/r between the cosmological constant Λ and the ROU parameter. Finally, the behavior of the examined physical observables, especially of the spin-spin correlation function, indicates the dominance of isolated bubble spin foams in the Perez-Rovelli and Baez-Christensen partition functions, while less so for the the DFKR one. Some questions raised by these results deserve attention. For instance, it is not known whether the same q → 1 limit behavior will be observed when q is taken through non-ROU values. While calculations with max{|q|, |q| -1 } > 1 are numerically unstable, they should still be possible for |q| ∼ 1.
Another important project is to perform a more extensive study of the effects of triangulation size in order to better understand the semi-classical limit.
Finally, all of this work should also be carried out for the Lorentzian models, which are physically much more interesting but computationally much more difficult.
These and other questions will be the subject of future investigations.
17 Figure 6: Spin-spin correlation functions for the Baez-Christensen (BCh), Perez-Rovelli (PR) and DFKR models, on the minimal triangulation (6 vertices, 15 edges, 20 faces, 15 tetrahedra, and 6 4-simplices) as well as a larger triangulation (23 vertices, 103 edges, 202 faces, 200 tetrahedra, and 80 4-simplices). ROU parameter r = 10.
The authors would like to thank Wade Cherrington for helpful discussions. The first author was supported by NSERC and FQRNT postgraduate scholarships and the second author by an NSERC grant. Computational resources for this project were provided by SHARCNET.
A Spin network notation and conventions
Quantum integers are a q-deformation of integers. For an integer n, the corresponding quantum integer is denoted by [n] and is given by
[n] = q n -q -n q -q -1 . ( 37
)
In the limit q → 1, we recover the regular integers, [n] → n. Note that [n] is invariant under the transformation q → q -1 . When q = exp(iπ/r) is a root of unity (ROU), for some integer r > 1, an equivalent definition is [n] = sin(nπ/r) sin(π/r) . (
) 38
This expression is non-negative in the range 0 ≤ n ≤ r. Quantum factorials are defined as
[n]! = [1][2] • • • [n]. ( 39
)
18 In many cases, q-deformed spin network evaluations can be obtained from their undeformed counterparts by simply replacing factorials with quantum factorials. For convenience, when dealing with half-integral spins, we also define quantum half-integers as ⌊j⌉ = [2j] 2 (40) when j is a half-integer. Abstract su q (2) spin networks can be approached from two different directions. They can represent contractions and compositions of su q (2)-invariant tensors and intertwiners [10]. At the same time, they can represent traces of tangles evaluated according to the rules of the Kauffman bracket [18] . Either way, the computations turn out to be the same. We present here formulas for the evaluation of a few spin networks of interest.
The single bubble network evaluates to what is sometimes called the superdimension of the spin-j representation:
j = (-) 2j [2j + 1]. ( 41
)
(As in the rest of the paper, the spin labels are half-integers.) Up to a constant, there is a unique 3-valent vertex (corresponding to the Clebsch-Gordan intertwiner) whose normalization is fixed up to sign by the value of the θnetwork :
θ(a, b, c) = b c a = (-) s [s + 1]![s -2a]![s -2b]![s -2c]! [2a]![2b]![2c]! , ( 42
)
where s = a + b + c. The θ-network is non-vanishing, together with the three-vertex itself, if and only if s is an integer and the triangle inequalities are satisfied: a ≤ b + c, b ≤ c + a, and c ≤ a + b. In addition, when q is a ROU, one extra inequality must be satisfied: s ≤ r -2. The triple (a, b, c) of spin labels is called admissible if θ(a, b, c) is non-zero.
The recoupling identity gives the transformation between different bases for the linear space of 4-valent tangles (or intertwiners):
b c a d f = e (-) 2e [2e + 1] Tet a b e c d f θ(a, d, e) θ(c, b, e) b a d c e
, ( 43
)
where the sum is over all admissible labels e and the value of the tetrahedral network is Tet a b e c d f = b a c d f e
= I! E! m≤S≤M (-) S [S + 1]! i [S -a i ]! j [b j -S]! , ( 44
) 19
where
I! = i,j [b j -a i ]! E! = [2A]![2B]![2C]![2D]![2E]![2F ]! ( 45
) a 1 = (a + d + e) b 1 = (b + d + e + f ) ( 46
)
a 2 = (b + c + e) b 2 = (a + c + e + f ) (47) a 3 = (a + b + f ) b 3 = (a + b + c + d) (
) 48
a 4 = (c + d + f ) m = max{a i } M = min{b j }. ( 49
)
Due to parity constraints, the a i , b j , m, M , and S are all integers. Since the three-vertex is unique up to scale, its composition with with a braiding applied to two incoming legs yields a multiplicative factor: c a b = (-) a+b-c q a(a+1)+b(b+1)-c(c+1) c b a .
(50) Note that the above braiding factor is not invariant under the transformation q → q -1 , while the bubble, tetrahedral and θ-networks are all invariant under this transformation, by virtue of their expressions in terms of quantum integers. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Spin foam models were first introduced as a space-time alternative to the spin network description of states in loop quantum gravity [3] . The most studied spin foam models are due to Barrett and Crane [8, 9] . A spin foam is a discretization of space-time where the fundamental degrees of freedom are the areas labelling its 2-dimensional faces.\n\nAn important goal in the investigation of spin foam models is to obtain predictions that can be compared to the large scale, classical, or semiclassical behavior of gravity. This work continues the numerical investigation of the physical properties of spin foam models of Riemannian quantum gravity begun in [5] [6] [7] 13] . In this paper, we extend the computations to the q-deformed Barrett-Crane model and to larger space-time triangulations.\n\nThe main applications of q-deformation are two-fold. On the one hand, it can act as a regulator for divergent models, as is apparent in the link between the Ponzano-Regge models. On the other hand, Smolin [30] has argued that q-deformation is necessary to account for a positive cosmological constant. Both of these aspects are explored in more detail in Section 2.2. A surprising result of our work is evidence that the limit, as the cosmological constant is taken to zero through positive values, is discontinuous.\n\nLarge triangulations are necessary to approximate semiclassical space-times. The possibility of obtaining numerical results from larger triangulations takes us one step closer to that goal and increases the number of facets from which the physical properties of a spin foam model may be examined. As an example, we are able to study how the spin-spin correlation varies with the distance between faces in the triangulation. 1 This paper is structured as follows. We begin in Section 2 by reviewing the basics of q-deformation and discussing in detail its aforementioned applications. Section 3 reviews the details of the Barrett-Crane model, summarizes the necessary changes for its q-deformation, and defines several observables associated to spin foams. In Section 4, we review the existing numerical simulation techniques and how they need to be generalized to handle q-deformation and larger triangulations. Section 5 presents the results of our numerical simulations. In Section 6, we give our conclusions and list some avenues for future research. The Appendix briefly summarizes our notational conventions and useful formulas.\n\n2 Deformation of su(2)\n\nIn this section, we describe the q-deformation of the Lie algebra su(2) into the algebra su q (2) (also denoted U q (su(2))), the representations of su q (2), and the applications of q-deformation. The deformations of spin(4) are then obtained through the isomorphism spin(4) ∼ = su(2) ⊕ su (2) .\n\nThe following is part of the general subject of quantum groups [21] . Here we shall concentrate solely on the su(2) and spin(4) cases."
},
{
"section_type": "OTHER",
"section_title": "The algebra su q (2) and its representations",
"text": "The Lie algebra su(2) is generated by the well known Pauli matrices σ i , which obey the commutation relations\n\n[σ + , σ -] = 4σ 3 , [σ 3 , σ + ] = 2σ + , [σ 3 , σ -] = -2σ -, ( 1\n\n)\n\nwhere σ ± = σ 1 ± iσ 2 . The universal enveloping algebra of su(2) is the associative algebra generated by σ ± and σ 3 subject to the above identities, with the Lie bracket being interpreted as [A, B] = AB -BA.\n\nThe q-deformed algebra su q (2) is constructed by replacing σ 3 with another generator. Formally, it is thought of as Σ = q 1 2 σ3 , where q ∈ C with the exceptions q = 0, 1, -1. The Lie bracket relations are replaced by the identities\n\n[σ + , σ -] = 4 Σ 2 -Σ -2 q -q -1 , Σσ + = qσ + Σ, Σσ -= -qσ -Σ. ( 2\n\n)\n\nWe can rewrite q = 1 + 2ε and think of ε as a small complex number. Then, formally at leading order in ε, the substitution Σ = q 1 2 σ3 = 1 + εσ 3 + O(ε 2 ) reduces the deformed identities (2) to the standard Lie algebra relations (1). The associative algebra generated by σ ± and σ 3 subject to the deformed identities (2) is the algebra su q (2) .\n\nFor generic q, that is, when q is not a root of unity, the finite-dimensional irreducible representations of su q (2) are classified by a half-integer, j = 0, 1/2, 1, 3/2, . . . , referred to as the spin, in direct analogy with the representations of su(2) and the theory of angular momentum. The dimension of the representation j is 2j + 1. When q = exp(iπ/r) is a 2rth root of unity (ROU), where r > 2 is an integer called the ROU parameter, the representations j are still defined, but become reducible for j > (r -2)/2. They decompose into a sum of representations with spin at most (r -2)/2 and so-called trace 0 ones, whose nature will be explained below.\n\nFor the purposes of this paper we are concerned only with intertwiners between representations of su q (2), i.e., linear maps commuting with the action of the algebra, and their (quantum) traces 1 .\n\nAny such intertwiner can be constructed from a small set of generators and elementary operations on them. These constructions, as well as traces, can be represented graphically. Such graphs are called (abstract) spin networks. Their calculus is well developed and is described in [18] , whose conventions we follow throughout the paper with one exception: we use spins (half-integers) instead of twice-spins (integers). A brief review of our notation and conventions can be found in the Appendix.\n\nTrace 0 representations of su q (2) are so called because the trace of an intertwiner from such a representation to itself is always zero. Thus, they can be freely discarded, as they do not contribute to the evaluation of q-deformed spin networks."
},
{
"section_type": "OTHER",
"section_title": "Applications of q-deformation",
"text": "Deformation, especially with q = exp(iπ/r) a 2rth primitive ROU, is important for spin foam models for at least two reasons. Replacing q = 1 by some ROU can act as a regulator for a model whose partition function and observable values are otherwise divergent. Also, su q (2) spin networks 2 naturally appear when considering a positive cosmological constant in loop quantum gravity.\n\nThe original Ponzano-Regge model [27] attempts to express the path integral for 3-dimensional Riemannian general relativity as a sum over labelled triangulations of a 3-manifold. The edges of the triangulation are labelled by discrete lengths, identified with spin labels of irreducible SU (2) representations. Each tetrahedron contributes a 6j-symbol factor to the summand, normalized to ensure invariance of the overall sum under change of triangulation. Unfortunately, the Ponzano-Regge model turned out to be divergent. Motivated by the construction of 3-manifold invariants, Turaev and Viro were able to regularize the Ponzano-Regge model [1, 31] by replacing the SU (2) 6j-symbols with their q-deformed analogs at a ROU q. The key feature of the regularization is the truncation of the summation to only the irreducible representations of su q (2) of non-zero trace, which leaves only a finite number of terms in the model's partition function.\n\nA version of the Barrett-Crane model, derived from a group field theory by De Pietri, Freidel, Krasnov and Rovelli [16] (DFKR for short), was also found to be divergent. A q-deformed version of the same model at a ROU q is similarly regularized (see Section 3.2). Some numerical results for the regularized version of this model are given in Section 5.2.\n\nThe argument linking q-deformation to the presence of a positive cosmological constant is due to Smolin [29] and is given in more refined form in [30] . It is briefly summarized as follows. Loop quantum gravity begins by writing the degrees of freedom of general relativity in terms of an SU (2) connection on a spatial slice and the slice's extrinsic curvature. A state in the Schrödinger picture, a wave function on the space of connections, can be constructed by integrating the Chern-Simons 3-form over the spatial slice. This state, known as the Kodama state, simultaneously satisfies all the canonical constraints of the theory and semiclassically approximates de Sit-1 When q = 1, this notion of trace reduces up to sign to the usual trace of a linear map, but is slightly different otherwise, cf. [10, Chapter 4] .\n\n2 These are graphs embedded in a 3-manifold, labelled by representations of suq(2). They are similar to but distinct from the abstract spin networks referred to above. See [4] for the distinction.\n\nter spacetime, which is a solution of the vacuum Einstein equations with a positive cosmological constant. The requirement that the Kodama state also be invariant under large gauge transformations implies discretization of the cosmological constant, Λ ∼ 1/r, with r a positive integer. The coefficients of the Kodama state in the spin network basis are obtained by evaluating the labelled graph, associated to a basis state, as an abstract su q (2) spin network. Here the deformation parameter q is a ROU, q = exp(iπ/r), where the ROU parameter r is identified with the discretization parameter of the cosmological constant.\n\nGiven the heuristic link [4] between spin networks of loop quantum gravity and spin foams, it is natural to q-deform a spin foam model as an attempt to account for a positive cosmological constant. With this aim, Noui and Roche [23] have given a q-deformed version of the Lorentzian Barrett-Crane model. The possibility of qdeformation has been with the Riemannian Barrett-Crane model since its inception [8] and all the necessary ingredients have been present in the literature for some time. In the next section these details are collected in a form ready for numerical investigation."
},
{
"section_type": "OTHER",
"section_title": "Deformation of the Barrett-Crane model",
"text": "Consider a triangulated 4-manifold. Let ∆ n denote the set of n-dimensional simplices of the triangulation. The dual 2-skeleton is formed by associating a dual vertex, edge and polygonal face to each 4-simplex, tetrahedron, and triangle of the triangulation, respectively. A spin foam is an assignment of labels, usually called spins, to the dual faces of the dual 2-skeleton. Each dual edge has 4 spins incident on it, while each dual vertex has 10. A spin foam model assigns amplitudes A F , A E and A V , that depend on all the incident spins, to each dual face, edge and vertex, respectively. The amplitude Z(F ) assigned to a spin foam F is the product of the amplitudes for individual cells of the 2-complex, while the total amplitude Z tot assigned to a triangulation is obtained by summing over all spin foams based on the triangulation:\n\nZ(F ) = f ∈∆2 A F (f ) e∈∆3 A E (e) v∈∆4 A V (v), Z tot = F Z(F ). ( 3\n\n)\n\nSome models, such as those based on group field theory [16, 17, 24] , also include a sum over triangulations in the definition of the total partition function."
},
{
"section_type": "OTHER",
"section_title": "Review of the undeformed model",
"text": "The Riemannian Barrett-Crane model was first proposed in [8] . Its relation to the Crane-Yetter [15] spin foam model is analogous to the relation of the Plebanski [26] formulation of general relativity (GR) to 4-dimensional BF theory with Spin(4) as the structure group. Both BF theory and the Crane-Yetter model are topological and the latter is considered a quantization of the former [2] . In the Plebanski formulation, GR is a constrained version of BF theory. Similarly, the Barrett-Crane model restricts the spin labels summed over in the Crane-Yetter model. With this restriction, Barrett and Crane hoped to produce a discrete model of quantum (Riemannian) GR."
},
{
"section_type": "OTHER",
"section_title": "Dual vertex amplitude",
"text": "All amplitudes are defined in terms of spin(4) spin networks. However, given the isomorphism spin(4) ∼ = su(2) ⊕ su(2), all irreducible representations of spin(4) can be 4 written as tensor products of irreducible representations of su(2). The Barrett-Crane model specifically limits itself to balanced representations, which are of the form j ⊗ j, where j is the irreducible representation of su(2) of spin j. Since the tensor product corresponds to a juxtaposition of edges in a spin network, any spin(4) spin network may be written as an su(2) spin network where an edge labelled j ⊗ j is replaced by two parallel edges, each labelled j. To avoid redundancy of notation, we use a single j instead of j ⊗ j to label spin(4) spin network edges. We then distinguish them from su(2) networks by placing a bold dot at every vertex. The Barrett-Crane vertex is an intertwiner between four balanced representations:\n\nb c a d = e j d e a c e b b a d c e ⊗ b a d c e\n\n. ( 4\n\n)\n\nThe graphs on the right hand side of the definition are su(2) spin networks and the sum runs over all admissible labels e. The graphical notation and the conditions for admissibility are defined in the Appendix. The above expression defines the Barrett-Crane vertex in a way that breaks rotational symmetry. However, it can be shown that the vertex is in fact rotationally symmetric. Up to normalization, this property makes the Barrett-Crane vertex unique [28] . The above formula defines a vertical splitting of the vertex. A ninety degree rotation will define an analogous horizontal splitting. Both possibilities are important in the derivation of the algorithm presented in Section 4.1.\n\nGiven a 4-simplex v of a triangulation, the corresponding vertex of the dual 2complex is assigned the amplitude\n\nA V (v) = 0 1 2 3 4 j 1,0 j 1,1 j 1,4 j 1,2 j 1,3 j 2,0 j 2,1 j 2,4 j 2,2 j 2,3 . ( 5\n\n)\n\nThis spin network is called the 10j-symbol. The 4-simplex v is bounded by five tetrahedra, which correspond to the vertices of the 10j graph. The four edges incident on a vertex correspond to the four faces of the corresponding tetrahedron; the spin labels are assigned accordingly. The edge joining two vertices corresponds to the face shared by corresponding tetrahedra. Evaluation of the 10j-symbol is discussed in Section 4.1. While the crossing structure depicted above is immaterial in the undeformed case, it is essential at nontrivial values of q. It is given here for reference. 5"
},
{
"section_type": "OTHER",
"section_title": "Dual edge and face amplitudes",
"text": "The original paper of Barrett and Crane did not specify dual edge and face amplitudes. Three different dual edge and face amplitude assignments were considered in a previous paper [7] . We concentrate on the same possibilities. For the Perez-Rovelli model [25], we have\n\nA F (f ) = j , A E (e) = j 4 j 3 j 2 j 1 j 1 j 2 j 3 j 4 . ( 6\n\n)\n\nFor the DFKR model [16], we have\n\nA F (f ) = j , A E (e) = 1 j 4 j 3 j 2 j 1 . ( 7\n\n)\n\nFor the Baez-Christensen model [7], we have\n\nA F (f ) = 1, A E (e) = 1 j 4 j 3 j 2 j 1 . ( 8\n\n)\n\nThe bubble diagram, when translated into su(2) spin networks, corresponds to two bubbles (see Appendix)\n\nj = j 2 . ( 9\n\n)\n\nand evaluates to (2j + 1) 2 . The so-called eye diagram simply counts the dimension of the space of 4-valent intertwiners, which is also the number of admissible e-edges summed over in Equation (4) . In symmetric form, it is given by\n\nj 4 j 3 j 2 j 1 = 1 + min{2j, s -2J} if positive and s is integral, 0 otherwise, ( 10\n\n) where s = k j k , j = min k j k , and J = max k j k . 6"
},
{
"section_type": "OTHER",
"section_title": "The q-deformed model",
"text": "Thanks to graphical notation, the q-deformation of the spin foam amplitudes described above is straightforward, with only a few subtleties. The main distinction is that q-deformed graphs are actually ribbon (framed) graphs with braiding. Thus, any undeformed spin network has to be supplemented with information about twists and crossings before evaluation.\n\nIn [32] , Yetter generalized the Barrett-Crane 4-vertex for a q-deformed version of spin(4). Since spin(4) ∼ = su(2) ⊕ su(2), there is a two parameter family of possible deformations of the Lie algebra, 2) . Yetter singles out the one parameter family q ′ = q -1 , restricted to balanced representations, since it preserves the invariance of the Barrett-Crane vertex under rotations. This family also has especially simple curl and twist identities:\n\nspin q,q ′ (4) ∼ = su q (2) ⊕ su q ′ (\n\nj = j and c a b = c b a , ( 11\n\n)\n\nwhere the left factor of j ⊗ j corresponds to su q (2) and the right one to su q -1 (2), and the 3-vertex is the obvious juxtaposition of two su q (2) and su q -1 (2) 3-vertices. Once this deformation is adopted, the ribbon structure can be ignored [32] , so one only needs to specify the crossing structure for a given spin(4) spin network to obtain a well-defined q-evaluation.\n\nThere are three basic graphs needed to define the Barrett-Crane simplex amplitudes: the bubble, the eye, and the 10j-symbol. The evaluation of the bubble graph, Equation (9), is [2j + 1] 2 , where the quantum integer [2j + 1] is defined in the Appendix. Remarkably, the value of the eye diagram turns out not to depend on q and its value is still given by Equation (10) . The only exception is when q is a ROU with parameter r. Then, the dimension of the space of 4-valent intertwiners changes to\n\nj 4 j 3 j 2 j 1 = min 1 + min{2j, s -2J} r -1 -max{2J, s -2j}\n\nif positive and s is integral,\n\n0 otherwise, ( 12\n\n) where again s = k j k , j = min k j k , and J = max k j k .\n\nThe 10j-symbol is the only network with a non-planar graph. Originally, it was defined in terms of the 15j-symbol from the Crane-Yetter model. This 15j-symbol was defined with q-deformation in mind, so its crossing and ribbon structure was fully specified [14, Section 3]. Adapted to the 10j-graph, it can be summarized as follows: Consider a 4-simplex. The dual 1-skeleton of the boundary has five dual vertices and ten dual edges, and is the complete graph K 5 on these five dual vertices. If we remove one of the (non-dual) vertices from the boundary of the 4-simplex, what remains is homeomorphic to R 3 . For any such homeomorphism, the embedding of K 5 into R 3 can be projected onto a 2-dimensional plane. The crossing structure of the 10j graph is defined by such a projection. It is illustrated in Equation (5) . Although, with crossings, the 10j graph is no longer manifestly invariant under permutations of its vertices, it can be shown to be so. 7"
},
{
"section_type": "OTHER",
"section_title": "Observables",
"text": "The definition of observables in a spin foam model of quantum gravity is still open to interpretation (see Section 6 of [7] for a brief discussion). For a fixed spin foam, the half-integer spin labels of its faces are the fundamental variables of the model. Practically speaking, any observable of a spin foam model should be an expectation value of some function O(F ) of the spin labels of a spin foam F , averaged over all spin foams with amplitudes specified by Equation (3):\n\nO = F O(F )Z(F ) Z tot . ( 13\n\n)\n\nIn this paper we choose to concentrate on a few observables representative of the kind of quantities computable in a spin foam model. As before, fix a triangulation of a 4-manifold, let ∆ 2 represent the set of its faces and let j : ∆ 2 → {0, 1/2, 1, . . .} be the spin labelling. We define:\n\nJ(F ) = 1 |∆ 2 | f ∈∆2 ⌊j(f )⌉ , ( 14\n\n) (δJ) 2 (F ) = 1 |∆ 2 | f ∈∆2 (⌊j(f )⌉ -J ) 2 , ( 15\n\n) A(F ) = 1 |∆ 2 | f ∈∆2 ⌊j(f )⌉ ⌊j(f ) + 1⌉, ( 16\n\n)\n\nC d (F ) = 1 N d f,f ′ ∈∆2 dist(f,f ′ )=d ⌊j(f )⌉ ⌊j(f ′ )⌉ -J 2 (δJ) 2 . ( 17\n\n) where ⌊n⌉ denotes a quantum half-integer (see Appendix), | • | denotes cardinality, dist(f, f ′\n\n) denotes the distance between faces, and N d is a normalization factor (see below for the definition of distance and N d ). These observables represent average spin per face, variance of spin per face, average area per face, and spin-spin correlation as a function of d. The choice of observables given above is somewhat arbitrary. For instance, there are several subtly distinct choices for the expression for (δJ) 2 . Fortunately, they all yield expectation values that are nearly identical. The expression given above has the technical advantage of falling into the class of so-called single spin observables. These are observables whose expectation value can be directly obtained from the knowledge of probability with which spin j occurs on any face of a spin foam. All of J, (δJ) 2 , and A are single spin observables, while C d is not.\n\nNote that on a fixed triangulation with no other background geometry, there is no physical notion of distance. We can, instead, define a combinatorial analog. For any two faces f and f ′ of a given triangulation, let dist(f, f ′ ) be the smallest number of face-sharing tetrahedra that connect f to f ′ . Given the discrete structure of our spacetime model, it is conceivable that this combinatorial distance, multiplied by a fundamental unit of length, approximates some notion of distance derived from the dynamical geometry of the spin foam model.\n\nThe correlation function C d may be thought of as analogous to a normalized 2point function of quantum field theory. The d-degree of face f is the number of faces 8 f ′ such that dist(f, f ′ ) = d. If the d-degree of every face is the same, the normalization factor N d can be taken to be the number of terms in the sum (17) , that is, the number of face pairs separated by distance d. This choice ensures the inequality |C d | ≤ 1. If not all faces have the same d-degree, then the normalization factor has to be modified to\n\nN d = |∆ 2 |D d , ( 18\n\n)\n\nwhere D d is the maximum d-degree of a face, which reduces to the simpler definition in the case of uniform d-degree.\n\nThe choice of the q-dependent expression ⌊j⌉, instead of simply using the halfinteger j, is motivated in Section 5.1. For some q, the argument of the square root in A(F ) may be negative or even complex. In that case, a branch choice will have to be made. Luckily, if q = 1, q is a ROU, or q is real, the expression under the square root is always non-negative."
},
{
"section_type": "OTHER",
"section_title": "Numerical simulation",
"text": "The key development that made possible numerical simulation of variations of the (undeformed) Barrett-Crane model [6, 7] is the development by Christensen and Egan of a fast algorithm for evaluating 10j-symbols [13] . In this section, we show how this algorithm generalizes to the q-deformed case and discuss numerical evaluation of observables for the previously described spin foam models."
},
{
"section_type": "OTHER",
"section_title": "The q-deformation of the fast 10j algorithm",
"text": "The derivation of the Christensen-Egan algorithm given in [13] is contingent on the possibility of splitting the and on the recoupling identity, Equation (43) of the Appendix. Both identities still hold in the q-deformed case. The validity of the 4-vertex splitting was proved by Yetter [32] and the recoupling identity is a standard part of su q (2) representation theory.\n\nThe only remaining detail of the algorithm's generalization is the crossing structure of the 10j graph, which was established in Section 3.2. However, its only consequence is an extra factor from the twist implicit in the bubble diagram of Section 4 of [13], cf. Equation (50) of the Appendix. We will not reproduce the derivation of the algorithm here. However, the way in which the twist arises is schematically illustrated in Figure 1 . Note that the triviality of the twist for Yetter's balanced representations, Equation (11) , does not apply here since the twist occurs separately in distinct su q (2) networks.\n\nThe algorithm itself can be summarized in the following form:\n\n{10j} = (-) 2S m1,m2 φ tr[M 4 M 3 M 2 M 1 M 0 ]. ( 19\n\n)\n\nThe 10j-symbol depends on the ten spins j i,k , (i = 1, 2, k = 0, . . . , 4) specified in Equation (5) . The overall prefactor depends on the total spin S = i,k j i,k and the per-term prefactor is\n\nφ = (-) m1-m2 [2m 1 + 1][2m 2 + 1]q m1(m1+1)-m2(m2+1) . ( 20\n\n) 9\n\n(a) (b) (c) (d) Figure 1: In reference to [13] , (a) corresponds to Equation (1), (b) corresponds to Equation ( 2), while (c) and (d) correspond to the \"ladder\" and \"bubble\" diagrams of Section 4, respectively. The illustrated twist introduces the explicitly q-dependent factor into Equation (20) .\n\nThe exponents of (-) and q are always integers. The M k are matrices (not all of the same size) of dimensions compatible with the five-fold product and trace. Their matrix elements are\n\n(M k ) l k+1 l k = [2l k + 1](T 1 ) l k+1 l k (T 2 ) l k+1 l k θ(j 2,k-1 , l k+1 , j 1,k ) θ(j 2,k+1 , l k+1 , j 1,k+1 ) , ( 21\n\n) (T i ) l k+1 l k = Tet l k j 2,k m i l k+1 j 2,k-1 j 1,k θ(j 2,k , l k+1 , m i ) . ( 22\n\n)\n\nThe quantum integers [n], as well as the theta θ(a, b, c) and tetrahedral Tet[• • •] su q (2) spin networks are defined in the Appendix. The quantities l k and m i are spin labels (half-integers). They are constrained by admissibility conditions (parity conditions and triangle inequalities). The parity of each index is determined by the conditions\n\nl k ≡ j 1,k + j 2,k ≡ j 1,k-1 + j 2,k-2 , ( 23\n\n) m i ≡ l k + j 2,k-1 , ( 24\n\n)\n\nfor i = 1, 2 and k = 0, . . . , 4, where ≡ denotes equivalence mod 1 and the second subscript of j is taken mod 5. Summation bounds are determined by the triangle inequalities, which must be checked for each trivalent vertex introduced in the derivation 10 of the algorithm. They boil down to\n\nlb 3 (j 1,k , j 2,k , j 2,k-1 ) ≤ m i ≤ j 1,k + j 2,k + j 2,k-1 , (25) |j 1,k-1 -j 2,k-2 | ≤ l k ≤ j 1,k-1 + j 2,k-2 , ( 26\n\n) |j 1,k -j 2,k | ≤ l k ≤ j 1,k + j 2,k , (27) |m i -j 2,k-1 | ≤ l k ≤ m i + j 2,k-1 , ( 28\n\n)\n\nfor i = 1, 2 and k = 0, . . . , 4, where we have used the notation lb 3 (a, b, c) = 2 max{a, b, c} -(a + b + c). (29) When q = exp(iπ/r) is a ROU, extra inequalities must be taken into account to exclude summation over reducible representations. These are\n\nm i ≥ j 1,k + j 2,k + j 2,k-1 -(r -2), (30) m i ≤ ub 3 (j 1,k , j 2,k , j 2,k-1 ) + (r -2), ( 31\n\n)\n\nl k ≤ (r -2) -(j 1,k + 2,k ), (32) l k ≤ (r -2) -(j 1,k-1 + j 2,k-2 ), ( 33\n\n) l k ≤ (r -2) -(m + j 2,k-1 ), ( 34\n\n)\n\nwhere now ub 3 (a, b, c) = 2 min{a, b, c} -(a + b + c).\n\nIf any of the parity constraints or inequalities cannot be satisfied, the 10j-symbol evaluates to zero. algorithm has been implemented and tested in the q = 1 and ROU cases, for both j and r up to several hundreds. Unfortunately, for generic q, when Q = max{|q|, |q| -1 } > 1, the quantum integers grow exponentially as |[n]| ∼ Q n . Such a rapid growth makes the sums involved in this algorithm numerically unstable. It is still possible to use this algorithm with Q close to 1 or symbolically, using rational functions of q instead of limited precision floating point numbers. Symbolic computation is, however, significantly slower (by up to a factor of 10 6 ) than its floating point counterpart. The software library spinnet which implements these and other spin network evaluations is available from the authors and will be described in a future publication."
},
{
"section_type": "METHOD",
"section_title": "Positivity and statistical methods",
"text": "The sums involved in evaluating expectation values of observables, as in Equation (13) , are very high-dimensional. For instance, a minimal triangulation of the 4-sphere (seen as the boundary of a 5-simplex) contains 20 faces. Hence, any brute force evaluation of an expectation value, even on such a small lattice, involves a sum over the 20dimensional space of half-integer spin labels.\n\nFortunately, in the undeformed case, the total amplitude Z(F ) for a closed spin foam is never negative 3 [5]. The proof for the q = 1 case generalizes to the ROU case. One need only realize two facts. The first is that, in the ROU case, quantum integers are non-negative. The second is that, for q a ROU, an su q -1 (2) spin network evaluates to the complex conjugate of the corresponding su q (2) spin network. The 3 We expect the same thing to hold in Lorentzian signature [5, 12].\n\ndisjoint union of any two such spin networks evaluates to their product, the absolute value squared of either of them, and hence is non-negative. Then, the same positivity result follows as from Equation (1) of [5] . This positivity allows us to treat Z(F ) as a statistical distribution and use Monte Carlo methods to extract expectation values with much greater efficiency than brute force summation.\n\nThe main tool for evaluating expectation values is the Metropolis algorithm [20, 22] . The algorithm consists of a walk on the space of spin labellings. Each step is randomly picked from a set of elementary moves and is either accepted or rejected based on the relative amplitudes of spin foam configurations before and after the move. An expectation value is extracted as the average of the observable over the configurations constituting the walk. Elementary moves for spin foam simulations are discussed in the next section.\n\nA Metropolis-like algorithm is possible even if individual spin foam amplitudes Z(F ) are negative or even complex. However, if the total partition function Z tot sums to zero, then the expectation values in Equation (13) become ill defined. Moreover, in numerical simulations, if Z tot is even close to zero, expectation value estimates may exhibit great loss of precision and slow convergence. In the path-integral Monte Carlo literature, this situation is known as the sign problem [11] . Still, the sign problem need not occur or, depending on the severity of the problem, there may be ways of effectively dealing with it.\n\nIndependent Metropolis runs can be thought of as providing independent estimates of a given expectation value. Thus, the error in the computed value of an observable can be estimated through the standard deviation of the results of many independent simulation runs [19]."
},
{
"section_type": "OTHER",
"section_title": "Elementary moves for spin foams",
"text": "The choice of elementary moves for spin foam simulations must satisfy several criteria. Theoretically, the most important one is ergodicity. That is, any spin foam must be able to transform into any other one through a sequence of elementary moves which avoid configurations with zero amplitude. Practically, it is important that these moves usually preserve admissibility. A spin foam F is called admissible if the associated amplitude Z(F ) is non-zero. If, starting with an admissible spin foam, most elementary moves produce an inadmissible spin foam, the simulation will spend a lot of time rejecting such moves without any practical benefit.\n\nAs before, consider a fixed triangulation of a compact 4-manifold. The parity conditions (23) imposed on the j i,k ,\n\nj 1,k + j 2,k ≡ j 1,k-1 + j 2,k-2 , 0 ≤ k ≤ 4,\n\nwhen taken together with the total spin foam amplitude (3), provide strong constraints on admissible spin foams. One can show that a move that changes spin labels by ±1/2 (mod 1) on each face of a closed surface in the dual 2-skeleton preserves the parity constraint. We take as the elementary moves the moves that change the spin labels by ±1/2 on the boundaries of the dual 3-cells of the dual 3-complex; the dual 3-cells correspond to the edges of the triangulation. If the manifold has non-trivial mod 2 homology in dimension 2, additional moves would be necessary, but for the examples we consider the moves above suffice. From a practical point of view, extra moves might improve the simulation's equilibration time. For instance, in the ROU case, parity preserving moves that change the spins from 0 to (r -2)/2 or (r -3)/2 were 12 introduced, since spins close to either admissible extreme may have large amplitudes. This property of the Perez-Rovelli and Baez-Christensen models is illustrated in the following section.\n\nUnfortunately, the inequalities constraining spin labels do not have a similar geometric interpretation and cannot be used to easily restrict the set of elementary moves in advance."
},
{
"section_type": "RESULTS",
"section_title": "Results",
"text": "Using methods described in the previous section, we ran simulations of the three variations of the Barrett-Crane model described in Section 3 and obtained expectation values for observables listed in Section 3.3. While previous work [7] performed simulations only on the minimal triangulation of the 4-sphere, which we will refer to simply as the minimal triangulation, we have extended the same techniques to arbitrary triangulations of closed manifolds."
},
{
"section_type": "OTHER",
"section_title": "Discontinuity of the r → ∞ limit",
"text": "The most striking result we can report is a discontinuity in the transition to the limit r → ∞, where r, a positive integer, is the ROU parameter with q = exp(iπ/r). As r → ∞, the deformation parameter q tends to its classical value 1. If we interpret the cosmological constant as inversely proportional to r, Λ ∼ 1/r, this limit also corresponds to Λ → 0, through positive values. For a fixed spin foam, the amplitudes and observables we study tend continuously to their undeformed values as r → ∞.\n\nHowever, we find that observable expectation values do not tend to their undeformed values in the same limit, that is, O r O q=1 as r → ∞. The discontinuity is most simply illustrated with the single spin distribution, that is the probability of finding spin j at any spin foam face. This probability can be estimated from the histogram of all spin labels that have occurred during a Monte Carlo simulation. The points in Figure 2 (a) show the single spin distributions for the Baez-Christensen model with r = 50 and q = 1. The curves show the corresponding single bubble amplitude. It is the amplitude Z(F j ) of a spin foam F j with all spin labels zero, except for the boundary of an elementary dual 3-cell, whose faces are all labelled with spin j. The amplitudes and distributions are normalized as probability distributions so their sums over j yield 1. The similarity between the points and the continuous curves is consistent with the hypothesis that spin foams with isolated bubbles dominate the partition function sum. The behavior of the single spin distribution for the Perez-Rovelli model is very similar, except that its peaks are much more pronounced.\n\nNote that the undeformed single spin distribution has a single peak at j = 0, while the r = 50 case has two peaks, one at j = 0 and the other at j = (r -2)/2, the largest non-trace 0 irreducible representation. The bimodal nature of the single spin distribution has an important impact on the large r behavior of observable expectation values, as is most easily seen with single spin observables (Section 3.3). For instance, if we consider the average, j, of the half-integers j, the large j peak would dominate the expectation value and j would diverge linearly in r, as r → ∞. On the other hand, since J is the average of the quantum half-integers ⌊j⌉, J at least approaches a constant in the same limit. This is illustrated in Figure 2 (b). 13 Figure 2: (a) Single spin distribution and single bubble amplitude for the Baez-Christensen model. The distribution was obtained from 10 9 steps of Metropolis simulation on a triangulation with 202 faces (cf. Section 5.3). (b) Some single spin observables as functions of j, with r = 50.\n\nHowever, as shown in Figure 3 , this limit is not the same as the undeformed expectation value. At the same time, as can be seen from the plot of the Perez-Rovelli average area in the same figure, there are some observables whose large r limits are at least very close to the undeformed values. The area observable summand A j = ⌊j⌉ ⌊j + 1⌉ is exactly zero at both j = 0 and j = (r -2)/2, while the spin observable summand J j = ⌊j⌉ is zero at j = 0 but still positive at j = (r -2)/2, Figure 2 (b). The large j peak of the Perez-Rovelli model is very narrow and thus the expectation value of a single spin observable is strongly influenced by its value at j = (r -2)/2. The data for larger triangulations is qualitatively similar."
},
{
"section_type": "OTHER",
"section_title": "Regularization of the DFKR model",
"text": "As expected, the ROU deformation of the DFKR model yields a finite partition function and finite expectation values. For instance, its single spin distribution for r = 40 is illustrated in Figure 4 . The divergence of the amplitude for large spins in the undeformed, q = 1, case makes numerical simulation impossible without an artificial spin cutoff. Thus, we do not have an undeformed analog of the single spin distribution. For the minimal triangulation, the ROU spin distribution deviates slightly from the single bubble amplitude close to the boundaries of admissible j. For the larger triangulation, the deviation is much more pronounced and is not restricted to the edges. This suggests that there are other significant contributions to the partition function besides single bubble spin foams. Note the large weight associated with spins around j = r/4. Around this value of j, both the area A j = ⌊j⌉ ⌊j + 1⌉ and the spin J j = ⌊j⌉ attain their maximal values and are proportional to r. Thus, it is natural to expect their expectation values to grow linearly in r, which is consistent with the divergent nature of the undeformed DFKR model. This is precisely the behavior shown in Figure 5 . On the minimal triangulation, the best linear fits for the average spin expectation value and for the 14 Figure 3: Observables for the Baez-Christensen (BCh) and Perez-Rovelli (PR) models as functions of the ROU parameter r. For large r, observables do not in general tend to their undeformed, q = 1, values; arrows show the deviation. Some observables were scaled to fit on the graph. Data is from Metropolis simulations on the minimal triangulation.\n\nsquare root of the average spin variance are\n\nJ r = 0.146 r -0.064, ( 35\n\n) (δJ) 2 1/2 r = 0.014 r + 0.187. ( 36\n\n)\n\nFor larger triangulations, the dependence of these observables is also approximately linear in r, with only slight variation in the effective slope."
},
{
"section_type": "OTHER",
"section_title": "Spin-spin correlation",
"text": "The ability to work with larger lattices allows us to explore a broader range of observables. One of them is the spin-spin correlation function C d defined in Section 3.3. In general C 0 = 1 and C d → 0 for large d. The decay of the correlation shows how quickly the spin labels on different spin foam faces become independent. A positive value of C d indicates that, on average, any two faces distance d apart both have spins above (or both below) the mean J . On the other hand, a negative value of C d indicates that, on average, any two faces distance d apart have one above and one below the mean J . A small triangulation limits the maximum distance between faces. For example, the minimal triangulation has maximum distance d = 3. Larger triangulations of the 4-sphere were obtained by refining the minimal one by applying Pachner moves randomly and uniformly over the whole triangulation. We restricted the Pachner moves to those that did not decrease the number of simplices.\n\nFigure 4: Single spin distributions and single bubble amplitudes for the DFKR model. The distributions were obtained from 10 9 steps of Metropolis simulation on the minimal triangulation and on a triangulation with 202 faces (cf. Section 5.3).\n\nThe largest triangulation we have used has maximum distance d = 6. Its correlations for different models are shown in Figure 6 along with those from the minimal triangulation. Correlation functions for different values of ROU parameter r (including the q = 1 case) and other triangulations are qualitatively similar.\n\nNotice the small negative dip for small values of d for the Perez-Rovelli and Baez-Christensen models. As discussed in previous sections, the partition functions of these models are dominated by spin foams with isolated bubbles. The correlation data is consistent with this hypothesis. The values of the spins assigned to faces of the bubble will be strongly correlated, while the values of the spins on two faces, one of which lies on the bubble and the other does not, should be strongly anti-correlated. Since a given face usually has fewer nearest neighbors that lie on the same bubble than that do not, on average, the short distance correlation is expected to be negative. At slightly larger distances, the correlation function turns positive again. This indicates that on a larger triangulations, spin foams with several isolated bubbles contribute strongly to the partition function. Although, with so few data points, it is difficult to extrapolate the behavior of the correlation function to larger triangulations and distances, its features are qualitatively similar to that of a condensed fluid, where the density-density correlation function exhibits oscillations on the scale of the molecular dimensions.\n\nNote that the behavior of the DFKR correlation function is significantly different from the other two. This is also consistent with the already observed fact that its partition function has strong contributions from other than single or isolated bubble spin foams.\n\nFigure 5: Observables for the DFKR model: area A , average spin J , spin standard deviation (δJ) 2 . Metropolis simulation, minimal triangulation. Error bars are smaller than the data points."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusion",
"text": "We have numerically investigated the behavior of physical observables for the Perez-Rovelli, DFKR, and Baez-Christensen versions of the Barrett-Crane spin foam model. Each version assigns different dual edge and face amplitudes to a spin foam, and these choices greatly affect the behavior of the resulting model. The behavior of the models was also greatly affected by q-deformation.\n\nThe limiting behavior of observables was found to be discontinuous in the limit of large ROU parameter r, i.e., q = exp(iπ/r) close to its undeformed value of 1. This result is at odds with the physical interpretation of the relation Λ ∼ 1/r between the cosmological constant Λ and the ROU parameter. Finally, the behavior of the examined physical observables, especially of the spin-spin correlation function, indicates the dominance of isolated bubble spin foams in the Perez-Rovelli and Baez-Christensen partition functions, while less so for the the DFKR one. Some questions raised by these results deserve attention. For instance, it is not known whether the same q → 1 limit behavior will be observed when q is taken through non-ROU values. While calculations with max{|q|, |q| -1 } > 1 are numerically unstable, they should still be possible for |q| ∼ 1.\n\nAnother important project is to perform a more extensive study of the effects of triangulation size in order to better understand the semi-classical limit.\n\nFinally, all of this work should also be carried out for the Lorentzian models, which are physically much more interesting but computationally much more difficult.\n\nThese and other questions will be the subject of future investigations.\n\n17 Figure 6: Spin-spin correlation functions for the Baez-Christensen (BCh), Perez-Rovelli (PR) and DFKR models, on the minimal triangulation (6 vertices, 15 edges, 20 faces, 15 tetrahedra, and 6 4-simplices) as well as a larger triangulation (23 vertices, 103 edges, 202 faces, 200 tetrahedra, and 80 4-simplices). ROU parameter r = 10."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgements",
"text": "The authors would like to thank Wade Cherrington for helpful discussions. The first author was supported by NSERC and FQRNT postgraduate scholarships and the second author by an NSERC grant. Computational resources for this project were provided by SHARCNET.\n\nA Spin network notation and conventions\n\nQuantum integers are a q-deformation of integers. For an integer n, the corresponding quantum integer is denoted by [n] and is given by\n\n[n] = q n -q -n q -q -1 . ( 37\n\n)\n\nIn the limit q → 1, we recover the regular integers, [n] → n. Note that [n] is invariant under the transformation q → q -1 . When q = exp(iπ/r) is a root of unity (ROU), for some integer r > 1, an equivalent definition is [n] = sin(nπ/r) sin(π/r) . (\n\n) 38\n\nThis expression is non-negative in the range 0 ≤ n ≤ r. Quantum factorials are defined as\n\n[n]! = [1][2] • • • [n]. ( 39\n\n)\n\n18 In many cases, q-deformed spin network evaluations can be obtained from their undeformed counterparts by simply replacing factorials with quantum factorials. For convenience, when dealing with half-integral spins, we also define quantum half-integers as ⌊j⌉ = [2j] 2 (40) when j is a half-integer. Abstract su q (2) spin networks can be approached from two different directions. They can represent contractions and compositions of su q (2)-invariant tensors and intertwiners [10]. At the same time, they can represent traces of tangles evaluated according to the rules of the Kauffman bracket [18] . Either way, the computations turn out to be the same. We present here formulas for the evaluation of a few spin networks of interest.\n\nThe single bubble network evaluates to what is sometimes called the superdimension of the spin-j representation:\n\nj = (-) 2j [2j + 1]. ( 41\n\n)\n\n(As in the rest of the paper, the spin labels are half-integers.) Up to a constant, there is a unique 3-valent vertex (corresponding to the Clebsch-Gordan intertwiner) whose normalization is fixed up to sign by the value of the θnetwork :\n\nθ(a, b, c) = b c a = (-) s [s + 1]![s -2a]![s -2b]![s -2c]! [2a]![2b]![2c]! , ( 42\n\n)\n\nwhere s = a + b + c. The θ-network is non-vanishing, together with the three-vertex itself, if and only if s is an integer and the triangle inequalities are satisfied: a ≤ b + c, b ≤ c + a, and c ≤ a + b. In addition, when q is a ROU, one extra inequality must be satisfied: s ≤ r -2. The triple (a, b, c) of spin labels is called admissible if θ(a, b, c) is non-zero.\n\nThe recoupling identity gives the transformation between different bases for the linear space of 4-valent tangles (or intertwiners):\n\nb c a d f = e (-) 2e [2e + 1] Tet a b e c d f θ(a, d, e) θ(c, b, e) b a d c e\n\n, ( 43\n\n)\n\nwhere the sum is over all admissible labels e and the value of the tetrahedral network is Tet a b e c d f = b a c d f e\n\n= I! E! m≤S≤M (-) S [S + 1]! i [S -a i ]! j [b j -S]! , ( 44\n\n) 19\n\nwhere\n\nI! = i,j [b j -a i ]! E! = [2A]![2B]![2C]![2D]![2E]![2F ]! ( 45\n\n) a 1 = (a + d + e) b 1 = (b + d + e + f ) ( 46\n\n)\n\na 2 = (b + c + e) b 2 = (a + c + e + f ) (47) a 3 = (a + b + f ) b 3 = (a + b + c + d) (\n\n) 48\n\na 4 = (c + d + f ) m = max{a i } M = min{b j }. ( 49\n\n)\n\nDue to parity constraints, the a i , b j , m, M , and S are all integers. Since the three-vertex is unique up to scale, its composition with with a braiding applied to two incoming legs yields a multiplicative factor: c a b = (-) a+b-c q a(a+1)+b(b+1)-c(c+1) c b a .\n\n(50) Note that the above braiding factor is not invariant under the transformation q → q -1 , while the bubble, tetrahedral and θ-networks are all invariant under this transformation, by virtue of their expressions in terms of quantum integers."
}
] | |
arxiv:0704.0284 | 0704.0284 | 1 | 10.1103/PhysRevD.75.124008 | 050e5d6a0fc0de02938b89b75e31009b008c3dde3207e3d2b4c1e1bcc6ef1b80 | Second Order Perturbative Calculation of Quasinormal Modes of
Schwarzschild Black Holes | We analytically calculate to second order the correction to the asymptotic
form of quasinormal frequencies of four dimensional Schwarzschild black holes
based on the monodromy analysis proposed by Motl and Neitzke. Our results are
in good agreement with those obtained from numerical calculation. | [
"Hsien-chung Kao"
] | [
"hep-th",
"gr-qc"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | Quasinormal modes (QNMs) were originally observed in considering the scattering or emission of gravitational waves by Schwarzschild black holes [1] . It was found that a characteristic damped oscillation, which only depends on the black hole mass, dominated the time evolution in a certain period of time. Since then QNMs have been investigated extensively both analytically and numerically. For a general review and classification, see Refs. [2, 3] . From numerical studies, an asymptotic formula for quasinormal frequencies of Schwarzschild black holes was obtained [4]:
2GM ω n ≈ 0.0874247 + 1 2 n - 1 2 i + O[n -1/2 ]. ( 1
)
The real part in the above formula was later postulated to be 1 4π ln 3 [5] based on a discrete area spectrum of quantum black holes proposed in Ref. [6] . This was confirmed later by Motl and Neitzke [7] . The recent surge of interest in the QNMs derived from its possible application in determining the Immirzi parameter in loop quantum gravity [8] . The numerical value ln 3 in the real part of the asymptotic quasinormal frequencies in Schwarzschild black holes was at first taken as a hint that the relevant gauge group in loop quantum gravity is SO(3) instead of the commonly believed SU (2). However, as shown in Ref. [7] , the value ln 3 is not universal and one should take the argument with a grain of salt.
Another interesting application of QNMs was pointed out by Horowitz and Hubeny in their study of a scalar field in the background of a Schwarzschild anti-de Sitter black hole [9] . According to AdS/CFT correspondence, a large black hole in AdS spacetime corresponds to a thermal state in CFT [10] . They argued the decay of the scalar field corresponds to the decay of a perturbation of this state. In the BTZ black hole, a one-to-one correspondence was found between the QNMs in the bulk and the poles of the retarded correlation function in the dual conformal field theory on the boundary [11] . The idea of dS/CFT correspondence has also been proposed and formulated [15] . Since there is a cosmological horizon in de Sitter spacetime, QNMs may also be defined in principle. Similar studies of QNMs have also been carried out in de Sitter spacetime trying to lent support for such correspondence [16] . However, the situation there is more subtle and it seems QNMs only exist in odd dimensions [3]. Therefore, it is not clear whether such correspondence makes sense in even dimensions, and further study is necessary.
In Ref. [12] , the author calculated the first order correction to the asymptotic form of quasinormal frequencies of a Schwarzschild black hole using a WKB analysis. The result was extended to include the scalar field case using the monodromy analysis developed by Motl and Neitzke [13] . The agreement with numerical results is excellent. We will begin with a brief review of their method which made systematic expansion more accessible. In a background spacetime described by a metric g µν , a massless scalar Φ satisfies the following Klein-Gordon equation: 1 √ -g ∂ µ g µν √ -g∂ ν Φ = 0. (2) 2 For four dimensional Schwarzschild black holes, the metric is given by ds 2 = -f (r)dt 2 + f (r) -1 dr 2 + r 2 dΩ 2 , with f (r) = (1 -r0 r ) and r 0 = 2GM. Let Φ(r, t, Ω) = r φ(r)Y lm (Ω) e iωt . (3) φ(r) now satisfies the following equation:
-f (r) d dr f (r) dφ dr + V (r)φ = ω 2 φ, (4) with V (r) = (1 -r 0 r ) l(l + 1) r 2 + r 0 r 3 .
By a simple modification in the potential V (r) [2], V (r) = (1 -r 0 r ) l(l + 1) r 2 + (1 -j 2 )r 0 r 3 , (5) the previous equation can also describes linearized perturbation of the metric or an electromagnetic test fields. Here, j = 0, 1, 2 which is the spin of the relevant field. They can also be classified as the tensor, vector, and scalar types of perturbation to the background Schwarzschild metric using the master equations derived by Ishibashi and Kodama [14] . Introducing the tortoise coordinate: x(r) = r + r 0 ln(r/r 0 -1), one obtain a Schrodinger-like equation
- d 2 dx 2 + V [r(x)] φ = ω 2 φ. ( 6
)
Because of our convention in eq (3), QNMs are defined through the following out-going wave boundary condition:
φ(x) ∼ e iωx as x → -∞ (horizon), e -iωx as x → ∞ (spatial infinity), ( 7
)
assuming Re ω > 0. Define
f (x) = e iωx φ ∼ e 2iωx as x → -∞, 1 as x → ∞. ( 8
)
According to Ref. [7] , the boundary condition at the horizon translates to the monodromy of f (x) around it M(r 0 ) = e 4πωr0 . (9)
The same monodromy can also be accounted for by those around r = 0 and r = ∞, and it has been shown that only the former one is non-trivial. To find the monodromy around r = 0, one need to introduce the complex coordinate variable
z = ω(x -iπr 0 ) = ω[r + r 0 ln(1 -r/r 0 )], ( 10
)
3 which is vanishing at the black hole singularity r = 0. In the limit |r/r 0 | ≪ 1, the potential can be expanded as a series in z/(ωr 0 ):
V (z) = - ω 2 (1 -j 2 ) 4z 2 + 3l(l + 1) + 1 -j 2 6 √ 2(-ωr 0 ) 1/2 z 3/2 - 3l(l + 1) + 1 -j 2 36 √ 2(-ωr 0 ) 3/2 z 1/2 + . . . . ( 11
)
Note that the third term in the above expression is of order (-ωr 0 ) -3/2 and would not contribute until we consider third order perturbation. To second order in perturbation theory, the wavefunction can be expanded as
φ = φ (0) + 1 √ -ωr 0 φ (1) + 1 -ωr 0 φ (2) + O(ω -3/2 ). ( 12
)
The zeroth, first and second order equations are given by
dφ (0) dz 2 + 1 -j 2 4z 2 + 1 φ (0) = 0; ( 13
) dφ (1) dz 2 + 1 -j 2 4z 2 + 1 φ (1) = √ -ωr 0 δV (z) φ (0) ; ( 14
) dφ (2) dz 2 + 1 -j 2 4z 2 + 1 φ (2) = √ -ωr 0 δV (z) φ (1) , ( 15
)
respectively. Here, z) to be the two linearly independent solutions to the zeroth order equation
δV (z) = 3l(l + 1) + 1 -j 2 6 √ 2(-ωr 0 ) 1/2 z 3/2 . ( 16
) Define φ ( 0
) ± (
φ ( 0
) ± (z) = πz 2 J ±j/2 (z). ( 17
)
In the asymptotic region z ≫ 1 φ
(0) ± (z) ≈ cos[z -π(1 ± j)/4]. ( 18
)
It has been shown by Musiri and Siopsis that φ (1) ± can be expressed in terms of φ
(0) ± φ (1) + (z) = Cφ (0) + (z) z 0 dz 1 δV (z 1 ) φ ( 0
) -(z 1 ) φ (0) + (z 1 ) -Cφ ( 0
) -(z) z 0 dz 1 δV (z 1 ) φ (0) + (z 1 ) φ (0) + (z 1 ); ( 19
) φ ( 1
) -(z) = Cφ (0) + (z) z 0 dz 1 δV (z 1 ) φ ( 0
) -(z 1 ) φ ( 0
) -(z 1 ) -Cφ ( 0
) -(z) z 0 dz 1 δV (z 1 ) φ (0) + (z 1 ) φ ( 0
) -(z 1 ). ( 20
)
where C = √ -ωr 0 / sin(πj/2) [13] . Similarly, φ (2) ± can in turn be expressed in terms of φ
(1) ± φ (2) + (z) = Cφ ( 0
) + (z) z 0 dz 2 δV (z 2 ) φ ( 0
) -(z 2 ) φ (1) + (z 2 ) -Cφ ( 0
) -(z) z 0 dz 2 δV (z 2 ) φ (0) + (z 2 ) φ (1) + (z 2 ); ( 21
) φ ( 2
) -(z) = Cφ ( 0
) + (z) z 0 dz 2 δV (z 2 ) φ ( 0
) -(z 2 ) φ ( 1
) -(z 2 ) -Cφ ( 0
) -(z) z 0 dz 2 δV (z 2 ) φ (0) + (z 2 ) φ ( 1
) -(z 2 ). ( 22
)
In the limit, z → ∞,
φ ( 1
) ± (z) = c -± φ ( 0
) + (z) -c +± φ ( 0
) -(z); ( 23
) φ ( 2
) ± (z) = d -± φ ( 0
) + (z) -d +± φ ( 0
) -(z). ( 24
)
4 Here, 17) are in fact linearly dependent to each other when j is an even integer. As a result, each of these coefficients is divergent by itself in these cases. It is reassuring to see that all the divergent pieces cancel among themselves so that physically interested quantities do have a smooth limit when j is an even integer. In zeroth order, the combination
c ±± = C ∞ 0 dz 1 δV (z 1 ) φ ( 0
) ± (z 1 ) φ ( 0
) ± (z 1 ); ( 25
) d ±± = C 2 ∞ 0 dz 2 z1 0 dz 1 δV (z 2 ) δV (z 1 ) φ ( 0
) ± (z 2 ) φ (0) + (z 2 )φ ( 0
) -(z 1 ) -φ ( 0
) -(z 2 )φ ( 0
) + (z 1 ) φ ( 0
) ± (z 1 ). ( 26
) Notice that φ ( 0
) ± defined in eq (
φ (0) (z) = φ (0) + (z) -e -iπ(j/2) φ (0) -(z) ∼ e -iz ( 27
)
in the asymptotic region z ≫ 1. This can be extended to second order φ(z) = φ
(0) + (z) + 1 √ -ωr 0 φ (1) + (z) + 1 -ωr 0 φ (2) + (z) -e -iπ(j/2) 1 - ξ √ -ωr 0 - ζ -ωr 0 φ (0) -(z) + 1 √ -ωr 0 φ (1) -(z) + 1 -ωr 0 φ (2) -(z) , ( 28
)
by introducing two parameters ξ and ζ. Naturally, they are determined by the condition that the coefficient of the e iz term is vanishing when z → ∞:
ξ = ξ + + ξ -; ( 29
) ζ = -ξξ -+ d ++ e iπj/2 -d +-+ d --e -iπj/2 -d -+ , ( 30
)
where
ξ + = c ++ e iπj/2 -c +-, ξ -= c --e -iπj/2 -c -+ . ( 31
)
Substitute the above result back to eq (28), we have
φ(z) = i e iπ(1-j)/4 sin(πj/2) e -iz 1 - ξ - √ -ωr 0 + ξ(ξ m + c +-) -d --e -iπj/2 + d -+ -ωr 0 , ( 32
)
where the identity c -+ = c +-has been used to simplify the expression. When going around the black hole singularity by 3π, φ (1) ± and φ (2) ± both pick up an extra phase:
φ ( 1
) ± ( e 3iπ z) = e 3iπ(2±j)/2 φ ( 1
) ± (-z); ( 33
) φ ( 2
) ± ( e 3iπ z) = e 3iπ(3±j)/2 φ ( 2
) ± (-z). ( 34
) Consequently, φ( e 3iπ z) = e 3iπ(1+j)/2 φ ( 0
) + (-z) -i 1 √ -ωr 0 φ (1) + (-z) - 1 -ωr 0 φ (2) + (-z) -e -iπ(j/2) 1 - ξ √ -ωr 0 - ζ -ωr 0 e 3iπ(1-j)/2 φ (0) -(-z) -i 1 √ -ωr 0 φ (1) -(-z) - 1 -ωr 0 φ (2) -(-z) . ( 35
) 5
To second order, φ( e 3iπ z) = -i e iπ(1-j)/4 sin(3πj/2) e -iz
1 + (1 + ie 3iπj )ξ + + (1 + i)ξ - √ -ωr 0 (-1 + e i3πj ) + -(1 + i)ξ ξ -+ [(1 + e i3πj )(d ++ e iπj/2 -d -+ ) + 2d --e -iπj/2 -2d +-] -ωr 0 (-1 + e i3πj ) + . . . , ( 36
)
where the term e iz is not relevant for our calculation and has been neglected. Taking the ratio between the coefficients of the term e -iz in eqs (36) and (32) , we obtain the monodromy to second order:
M(r 0 ) = -[1 + 2 cos(jπ)] 1 + ∆ 1 √ -ωr 0 + ∆ 2c + ∆ 2d -ωr 0 . ( 37
)
Here,
∆ 1 = (1 + ie 3iπj )ξ + + (i + e 3iπj )ξ - (-1 + e i3πj ) ; ( 38
) ∆ 2c = -(1 -i)ξ + ξ --ξc +-; ( 39
) ∆ 2d = (1 + e i3πj )(d ++ e iπj/2 + d --e -iπj/2 ) -2d +--2d -+ e i3πj (-1 + e i3πj ) . ( 40
)
The terms ∆ 2c and ∆ 2d depend on coefficients c µν and d µν , respectively. Although our expression for ∆ 1 here is different from that in Ref. [13] by a phase factor, our final result is identical to their. Making use of the formula
I 1 (µ, ν) ≡ ∞ 0 dz z -1/2 J µ (z)J ν (z) = π/2Γ( 1+2µ+2ν 4 ) Γ( 3-2µ-2ν 4 )Γ( 3+2µ-2ν 4 )Γ( 3-2µ+2ν 4 ) , ( 41
)
one can obtain explicitly
c ++ = 3l 2 + 3l + 1 -j 2 Γ 2 ( 1 4 ) Γ( 1-2j 4 ) Γ( 1+2j 4 ) sin[ π(1-2j) 4 ] 48 π 3 2 sin( j π 2 ) ; ( 42
) c --= 3l 2 + 3l + 1 -j 2 Γ 2 ( 1 4 ) Γ( 1-2j 4 ) Γ( 1+2j 4 ) sin[ π(1+2j) 4 ] 48 π 3 2 sin( j π 2 ) ; ( 43
) c +-= 3l 2 + 3l + 1 -j 2 Γ 2 ( 1 4 ) Γ( 1-2j 4 ) Γ( 1+2j 4 ) sin[ π(1-2j) 4 ] sin[ π(1+2j) 4 ] 24 √ 2 π 3 2 sin( j π 2 ) . ( 44
) Note that c --= -c ++ (j → -j); c +-= -c -+ (j → -j). ( 45
)
These relation are also obeyed by d µν 's, which can be used to reduce our work. With the above results, we are ready to find ∆ 1 and ∆ 2c in eq (40) :
∆ 1 = - i(3l 2 + 3l + 1 -j 2 ) Γ 2 ( 1 4 ) Γ( 1-2j 4 ) Γ( 1+2j 4 ) cos( jπ 2 ) cos(jπ) 6 √ 2π 3/2 [1 + 2 cos(jπ)] ; ( 46
) ∆ 2c = - (3l 2 + 3l + 1 -j 2 ) 2 Γ 4 ( 1 4 ) Γ 2 ( 1-2j 4 ) Γ 2 ( 1+2j 4 ) cos(jπ) 1152π 3 . ( 47
)
6 The double integral
I 2 (µ 2 , ν 2 ; µ 1 , ν 1 ) ≡ ∞ 0 dz 2 z2 0 dz 1 z -1/2 2 z -1/2 1 J µ2 (z 2 )J ν2 (z 2 )J µ1 (z 1 )J ν1 (z 1 ) ( 48
)
can be expressed in terms of the generalized hypergeometric functions, but the general formula is quite complicated and not particularly illuminating. Therefore, we will just give the final result explicitly for the coefficients d ++ and d +-:
d ++ = - π 2 3l 2 + 3l + 1 -j 2 2 cot( jπ 2 )Γ( 1 4 ) 5 G 4 ( 1 4 , 1 2 , 1 2 , 1+j 2 , 1-j 2 ; 5 4 , 1, 2+j 2 , 2-j 2 ; 1) 576 sin 2 ( j π 2 ) + √ π 3l 2 + 3l + 1 -j 2 2 cot( j π 2 ) cot(j π) Γ( 1+2j 4 ) Γ( 1+2j 2 ) Γ 2 ( 1+j 2 ) 288 sin( jπ 2 ) 5 G 4 ( 1 2 , 1 + 2j 4 , 1 + j 2 , 1 + j 2 , 1 + 2j 2 ; 2 + j 2 , 2 + j 2 , 5 + 2j 4 , 1 + j; 1) + 3l 2 + 3l + 1 -j 2 2 Γ 4 ( 1 4 ) Γ 2 ( 1-2j 4 ) Γ 2 ( 1+2j 4 ) sin 2 [ π(1-2j) 4 ] sin[ π(1+2j) 4 ] 1152 √ 2 π 3 sin 2 ( j π 2 ) ; ( 49
) d +-= - π 2 3l 2 + 3l + 1 -j 2 2 Γ( 1 4 ) 5 G 4 ( 1 4 , 1 2 , 1 2 , 1+j 2 , 1-j 2 ; 5 4 , 1, 2+j 2 , 2-j 2 ; 1) 1152 sin 3 ( j π 2 ) + √ π 3l 2 + 3l + 1 -j 2 2 cot 2 ( j π 2 ) cot(j π) Γ( 1-2j 4 ) Γ( 1-2j 2 ) Γ 2 ( 1-j 2 ) 576 5 G 4 ( 1 2 , 1 -2j 4 , 1 -j 2 , 1 -j 2 , 1 -2j 2 ; 2 -j 2 , 2 -j 2 , 5 -2j 4 , 1 -j; 1) - 3l 2 + 3l + 1 -j 2 2 Γ 4 ( 1 4 ) Γ 2 ( 1-2j 4 ) Γ 2 ( 1+2j 4 ) sin 2 [ π(1-2j) 4 ] sin 2 [ π(1+2j) 4 ] 2304 π 3 sin 2 ( j π 2 ) . ( 50
)
Here, we have used the regularized generalized hypergeometric function
5 G 4 (a 1 , a 2 , a 3 , a 4 , a 5 ; b 1 , b 2 , b 3 , b 4 ; z)
so that the pole structure of each term in these expressions are more explicit. It is related to the usual generalized hypergeometric function by
5 G 4 (a 1 , a 2 , a 3 , a 4 , a 5 ; b 1 , b 2 , b 3 , b 4 ; z) = 5 F 4 (a 1 , a 2 , a 3 , a 4 , a 5 ; b 1 , b 2 , b 3 , b 4 ; z) Γ(b 1 )Γ(b 2 )Γ(b 3 )Γ(b 4 ) . ( 51
)
The other two coefficients can be obtained by relations analogous to those in eq (45)
d --= -d ++ (j → -j); d -+ = -d +-(j → -j). ( 52
)
On the face of it, each of the d µν 's has a third order pole coming from terms involving the generalized hypergeometric function when j is an even integer. On closer look, we see there are some cancelation among the divergences and in the end all they have are just simple poles in such limit similar to the c µν 's. Another possible divergence arises in d --when j = 1, which will again be canceled when we calculate the monodromy. 7 It is now straightforward to obtain ∆ 2d by making use of the following two identities
-4π 7/2 cos 2 ( jπ 2 ) cot(jπ)Γ( 1 + 2j 4 ) Γ( 1 + 2j 2 ) Γ 2 ( 1 + j 2 ) 5 G 4 ( 1 2 , 1 + 2j 4 , 1 + j 2 , 1 + j 2 , 1 + 2j 2 ; 2 + j 2 , 2 + j 2 , 5 + 2j 4 , 1 + j; 1) +4π 7/2 cos 2 ( jπ 2 ) cot(jπ)Γ( 1 -2j 4 ) Γ( 1 -2j 2 ) Γ 2 ( 1 -j 2 ) 5 G 4 ( 1 2 , 1 -2j 4 , 1 -j 2 , 1 -j 2 , 1 -2j 2 ; 2 -j 2 , 2 -j 2 , 5 -2j 4 , 1 -j; 1) -Γ 4 ( 1 4 ) Γ 2 ( 1 -2j 4 ) Γ 2 ( 1 + 2j 4 ) sin[ π(1 -2j) 4 ] sin[ π(1 + 2j) 4 ] = 0; ( 53
) -4π 7/2 cos( jπ 2 ) Γ( 1 + 2j 4 ) Γ( 1 + 2j 2 ) Γ 2 ( 1 + j 2 ) 5 G 4 ( 1 2 , 1 + 2j 4 , 1 + j 2 , 1 + j 2 , 1 + 2j 2 ; 2 + j 2 , 2 + j 2 , 5 + 2j 4 , 1 + j; 1) -4π 7/2 cos( jπ 2 ) Γ( 1 -2j 4 ) Γ( 1 -2j 2 ) Γ 2 ( 1 -j 2 ) 5 G 4 ( 1 2 , 1 -2j 4 , 1 -j 2 , 1 -j 2 , 1 -2j 2 ; 2 -j 2 , 2 -j 2 , 5 -2j 4 , 1 -j; 1); +8π 5 Γ( 1 4 ) 5 G 4 ( 1 4 , 1 2 , 1 2 , 1 + j 2 , 1 -j 2 ; 5 4 , 1, 2 + j 2 , 2 -j 2 ; 1) -Γ 4 ( 1 4 ) Γ 2 ( 1 -2j 4 ) Γ 2 ( 1 + 2j 4 ) cos( jπ 2 )[1 -cos(jπ)] = 0. ( 54
)
Eventually, we achieve the following nice result
∆ 2d = (3l 2 + 3l + 1 -j 2 ) 2 Γ 4 ( 1 4 ) Γ 2 ( 1-2j 4 ) Γ 2 ( 1+2j 4 ) cos(jπ) 1152π 3 [1 + 2 cos(jπ)] , ( 55
)
where all divergences have been canceled out.
Together with the result from eq (47) , the asymptotic form of quasinormal frequencies of a four dimensional Schwarzschild black hole is found to be
4πωr 0 = (2n + 1)πi + ln[1 + 2 cos(jπ)] - i(3l 2 + 3l + 1 -j 2 ) Γ 2 ( 1 4 ) Γ( 1-2j 4 ) Γ( 1+2j 4 ) cos( jπ 2 ) cos(jπ) 6 √ 2π 3/2 [1 + 2 cos(jπ)] √ -ωr 0 + (3l 2 + 3l + 1 -j 2 ) 2 Γ 4 ( 1 4 ) Γ 2 ( 1-2j 4 ) Γ 2 ( 1+2j 4 ) cos 2 (jπ) 576π 3 [1 + 2 cos(jπ)] 2 (-ωr 0 ) + O[(-ωr 0 ) -3/2 ]. ( 56
)
The physically interested cases are
ω n T H ≈ (2n + 1)πi + ln 3 + 1 -i √ n (l 2 + l -1)Γ 4 (1/4) 18 √ 2π 3/2 + i n (l 2 + l -1) 2 Γ 8 (1/4) 2592π 3 , for j = 2; ( 57
) ω n T H ≈ (2n + 1)πi + ln 3 + 1 -i √ n (l 2 + l + 1/3)Γ 4 (1/4) 6 √ 2π 3/2 + i n (l 2 + l + 1/3) 2 Γ 8 (1/4) 288π 3 , for j = 0; ( 58
) ω n T H ≈ 2nπi + i2π(l 2 + l) 2 n , for j = 1. ( 59
)
A few comments are in order. First, all the second order corrections are purely imaginary. In particular, when j = 2 (gravitational perturbation) the numerical coefficients of the i/n term (after divided by 8 4π) are 0.739, 3.58, 49.7 for l = 2, 3, 6, respectively. They are in good agreement with the known numerical studies [4]. As for the real part, our result predicts vanishing correction. For j = 2, this is again consistent with the numerical results in Ref. [4] for l = 2, 3. For l = 6 the numerical result is 0.263, which seems to be contradictory to ours. However, the numerical value for l = 6 has opposite sign relative to those of l = 2, 3. This is peculiar, since in all other cases a given type of corrections are always of the same sign irrespective of the specific value of angular momentum. Therefore, we believe more study is needed to clarify whether there is really a discrepancy. As for the j = 1 case, the numerical study in Ref. [17] suggests the leading correction is of the form b n 3/2 . However, this does not necessarily mean the two results are inconsistent. In fact, one can only extract the behavior of the leading correction to the real part from their Fig. 2 and further numerical study is needed to confirm or refute our prediction.
In sum, we have calculated to second order the correction to the asymptotic form of quasinormal frequencies for Schwarzschild black holes in four dimensions. Most of our results are consistent with the numerical ones when available. In cases where there seem to be contradiction, we think further numerical studies are needed to clarify the situation. It would also be helpful if more detailed numerical studies can be carried out for the j = 0 case so that more thorough comparisons are possible. It would be interesting to generalize the method to other spacetime backgrounds [18] . Extension to higher order is also desirable. It might enable us to find a quantitative prediction for the "algebraically special" frequencies in Schwarzschild black holes, where the quasinormal frequency is purely imaginary and it increases with the fourth power of l [19, 4] .
The author thanks Chong-Sun Chu for helpful discussions. The work is supported in part by the National Science Council and the National Center for Theoretical Sciences, Taiwan.
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10 [12] A. Maassen van den Brink, "WKB analysis of the Regge-Wheeler Equation Down in the Frequency Plane," J. Math. Phys. 45, 327 (2004) [gr-qc/0303095]. [13] S. Musiri and G. Siopsis, Perturbative calculation of quasi-normal modes of Schwarzschild black holes, Class. Quant. Grav. 20 L285, (2003) [hep-th/0308168]. [14] A. Ishibashi and H. Kodama, "A Master Equation for Gravitational Perturbations of Maximally Symmetric Black Holes in Higher Dimensions," Prog. Theor. Phys. 110, 701 (2003) [hep-th/0305147]; A. Ishibashi and H. Kodama, "Stability of Higher Dimensional Schwarzschild Black Holes," Prog. Theor. Phys. 110, 901 (2003) [hep-th/0305185]; A. Ishibashi and H. Kodama, "Master Equations for Perturbations of Generalized Static Black Holes with Charge in Higher Dimensions," Prog. Theor. Phys. 111, 29 (2004) [hep-th/0308128]. [15] E. Witten, "Quantum Gravity In De Sitter Space," [hep-th/0106109]; A. Strominger, "The dS/CFT Correspondence," JHEP 0110, 034 (2001) [hep-th/0106113]; D. Klemm, "Some Aspects Of The De Sitter/CFT Correspondence," Nucl. Phys. B 625, 295 (2002) [hep-th/0106247]. [16] E. Abdalla, B. Wang, A. Lima-Santos, and W. G. Qiu, "Support Of dS/CFT Correspondence From Perturbations Of Three-Dimensional Space-Time," Phys. Lett. B 538, 435 (2002) [hep-th/0204030]; E. Abdalla, K. H. C. Castello-Branco, and A. Lima-Santos, "Support of dS/CFT Correspondence from Space-time Perturbations," Phys. Rev. D 66, 104018 (2002) [hep-th/0208065]; Y. S. Myung and N. J. Kim, "Difference between AdS and dS Spaces: Wave Equation Approach," Class. Quant. Grav. 21, 63 (2004) [hep-th/0304231]; T. R. Choudhury and T. Padmanabhan, "Quasinormal Modes in Schwarzschild-de Sitter Space-time: A Simple Derivation of the Level Spacing of the Frequencies," Phys. Rev. D 69, 064033 (2004) [gr-qc/0311064]. [17] V. Cardoso, J. P. S. Lemos and S. Yoshida, Quasinormal Modes of Schwarzschild Black Holes in Four and Higher Dimensions, Phys. Rev. D69, 044004 (2004) [gr-qc/0309112]. [18] S. Musiri, S. Ness and G. Siopsis, "Perturbative Calculation of Quasi-normal Modes of AdS Schwarzschild Black Holes, Phys. Rev. D 73, 064001 (2006), [hep-th/0511113]; F.-W. Shu 11 and Y.-G. Shen, " it Perturbative Calculation of Quasinormal Modes of d-Dimensional Black Holes," JHEP 0608, 087 (2006) [hep-th/0605128]. [19] S. Chandrasekhar, "On Algebraically Special Perturbations of Black Holes," Proc. R. Soc. London A392, 1 (1984). 12 | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Quasinormal modes (QNMs) were originally observed in considering the scattering or emission of gravitational waves by Schwarzschild black holes [1] . It was found that a characteristic damped oscillation, which only depends on the black hole mass, dominated the time evolution in a certain period of time. Since then QNMs have been investigated extensively both analytically and numerically. For a general review and classification, see Refs. [2, 3] . From numerical studies, an asymptotic formula for quasinormal frequencies of Schwarzschild black holes was obtained [4]:\n\n2GM ω n ≈ 0.0874247 + 1 2 n - 1 2 i + O[n -1/2 ]. ( 1\n\n)\n\nThe real part in the above formula was later postulated to be 1 4π ln 3 [5] based on a discrete area spectrum of quantum black holes proposed in Ref. [6] . This was confirmed later by Motl and Neitzke [7] . The recent surge of interest in the QNMs derived from its possible application in determining the Immirzi parameter in loop quantum gravity [8] . The numerical value ln 3 in the real part of the asymptotic quasinormal frequencies in Schwarzschild black holes was at first taken as a hint that the relevant gauge group in loop quantum gravity is SO(3) instead of the commonly believed SU (2). However, as shown in Ref. [7] , the value ln 3 is not universal and one should take the argument with a grain of salt.\n\nAnother interesting application of QNMs was pointed out by Horowitz and Hubeny in their study of a scalar field in the background of a Schwarzschild anti-de Sitter black hole [9] . According to AdS/CFT correspondence, a large black hole in AdS spacetime corresponds to a thermal state in CFT [10] . They argued the decay of the scalar field corresponds to the decay of a perturbation of this state. In the BTZ black hole, a one-to-one correspondence was found between the QNMs in the bulk and the poles of the retarded correlation function in the dual conformal field theory on the boundary [11] . The idea of dS/CFT correspondence has also been proposed and formulated [15] . Since there is a cosmological horizon in de Sitter spacetime, QNMs may also be defined in principle. Similar studies of QNMs have also been carried out in de Sitter spacetime trying to lent support for such correspondence [16] . However, the situation there is more subtle and it seems QNMs only exist in odd dimensions [3]. Therefore, it is not clear whether such correspondence makes sense in even dimensions, and further study is necessary."
},
{
"section_type": "OTHER",
"section_title": "Perturbative calculation of the asymptotic form of quasinormal frequencies",
"text": "In Ref. [12] , the author calculated the first order correction to the asymptotic form of quasinormal frequencies of a Schwarzschild black hole using a WKB analysis. The result was extended to include the scalar field case using the monodromy analysis developed by Motl and Neitzke [13] . The agreement with numerical results is excellent. We will begin with a brief review of their method which made systematic expansion more accessible. In a background spacetime described by a metric g µν , a massless scalar Φ satisfies the following Klein-Gordon equation: 1 √ -g ∂ µ g µν √ -g∂ ν Φ = 0. (2) 2 For four dimensional Schwarzschild black holes, the metric is given by ds 2 = -f (r)dt 2 + f (r) -1 dr 2 + r 2 dΩ 2 , with f (r) = (1 -r0 r ) and r 0 = 2GM. Let Φ(r, t, Ω) = r φ(r)Y lm (Ω) e iωt . (3) φ(r) now satisfies the following equation:\n\n-f (r) d dr f (r) dφ dr + V (r)φ = ω 2 φ, (4) with V (r) = (1 -r 0 r ) l(l + 1) r 2 + r 0 r 3 .\n\nBy a simple modification in the potential V (r) [2], V (r) = (1 -r 0 r ) l(l + 1) r 2 + (1 -j 2 )r 0 r 3 , (5) the previous equation can also describes linearized perturbation of the metric or an electromagnetic test fields. Here, j = 0, 1, 2 which is the spin of the relevant field. They can also be classified as the tensor, vector, and scalar types of perturbation to the background Schwarzschild metric using the master equations derived by Ishibashi and Kodama [14] . Introducing the tortoise coordinate: x(r) = r + r 0 ln(r/r 0 -1), one obtain a Schrodinger-like equation\n\n- d 2 dx 2 + V [r(x)] φ = ω 2 φ. ( 6\n\n)\n\nBecause of our convention in eq (3), QNMs are defined through the following out-going wave boundary condition:\n\nφ(x) ∼ e iωx as x → -∞ (horizon), e -iωx as x → ∞ (spatial infinity), ( 7\n\n)\n\nassuming Re ω > 0. Define\n\nf (x) = e iωx φ ∼ e 2iωx as x → -∞, 1 as x → ∞. ( 8\n\n)\n\nAccording to Ref. [7] , the boundary condition at the horizon translates to the monodromy of f (x) around it M(r 0 ) = e 4πωr0 . (9)\n\nThe same monodromy can also be accounted for by those around r = 0 and r = ∞, and it has been shown that only the former one is non-trivial. To find the monodromy around r = 0, one need to introduce the complex coordinate variable\n\nz = ω(x -iπr 0 ) = ω[r + r 0 ln(1 -r/r 0 )], ( 10\n\n)\n\n3 which is vanishing at the black hole singularity r = 0. In the limit |r/r 0 | ≪ 1, the potential can be expanded as a series in z/(ωr 0 ):\n\nV (z) = - ω 2 (1 -j 2 ) 4z 2 + 3l(l + 1) + 1 -j 2 6 √ 2(-ωr 0 ) 1/2 z 3/2 - 3l(l + 1) + 1 -j 2 36 √ 2(-ωr 0 ) 3/2 z 1/2 + . . . . ( 11\n\n)\n\nNote that the third term in the above expression is of order (-ωr 0 ) -3/2 and would not contribute until we consider third order perturbation. To second order in perturbation theory, the wavefunction can be expanded as\n\nφ = φ (0) + 1 √ -ωr 0 φ (1) + 1 -ωr 0 φ (2) + O(ω -3/2 ). ( 12\n\n)\n\nThe zeroth, first and second order equations are given by\n\ndφ (0) dz 2 + 1 -j 2 4z 2 + 1 φ (0) = 0; ( 13\n\n) dφ (1) dz 2 + 1 -j 2 4z 2 + 1 φ (1) = √ -ωr 0 δV (z) φ (0) ; ( 14\n\n) dφ (2) dz 2 + 1 -j 2 4z 2 + 1 φ (2) = √ -ωr 0 δV (z) φ (1) , ( 15\n\n)\n\nrespectively. Here, z) to be the two linearly independent solutions to the zeroth order equation\n\nδV (z) = 3l(l + 1) + 1 -j 2 6 √ 2(-ωr 0 ) 1/2 z 3/2 . ( 16\n\n) Define φ ( 0\n\n) ± (\n\nφ ( 0\n\n) ± (z) = πz 2 J ±j/2 (z). ( 17\n\n)\n\nIn the asymptotic region z ≫ 1 φ\n\n(0) ± (z) ≈ cos[z -π(1 ± j)/4]. ( 18\n\n)\n\nIt has been shown by Musiri and Siopsis that φ (1) ± can be expressed in terms of φ\n\n(0) ± φ (1) + (z) = Cφ (0) + (z) z 0 dz 1 δV (z 1 ) φ ( 0\n\n) -(z 1 ) φ (0) + (z 1 ) -Cφ ( 0\n\n) -(z) z 0 dz 1 δV (z 1 ) φ (0) + (z 1 ) φ (0) + (z 1 ); ( 19\n\n) φ ( 1\n\n) -(z) = Cφ (0) + (z) z 0 dz 1 δV (z 1 ) φ ( 0\n\n) -(z 1 ) φ ( 0\n\n) -(z 1 ) -Cφ ( 0\n\n) -(z) z 0 dz 1 δV (z 1 ) φ (0) + (z 1 ) φ ( 0\n\n) -(z 1 ). ( 20\n\n)\n\nwhere C = √ -ωr 0 / sin(πj/2) [13] . Similarly, φ (2) ± can in turn be expressed in terms of φ\n\n(1) ± φ (2) + (z) = Cφ ( 0\n\n) + (z) z 0 dz 2 δV (z 2 ) φ ( 0\n\n) -(z 2 ) φ (1) + (z 2 ) -Cφ ( 0\n\n) -(z) z 0 dz 2 δV (z 2 ) φ (0) + (z 2 ) φ (1) + (z 2 ); ( 21\n\n) φ ( 2\n\n) -(z) = Cφ ( 0\n\n) + (z) z 0 dz 2 δV (z 2 ) φ ( 0\n\n) -(z 2 ) φ ( 1\n\n) -(z 2 ) -Cφ ( 0\n\n) -(z) z 0 dz 2 δV (z 2 ) φ (0) + (z 2 ) φ ( 1\n\n) -(z 2 ). ( 22\n\n)\n\nIn the limit, z → ∞,\n\nφ ( 1\n\n) ± (z) = c -± φ ( 0\n\n) + (z) -c +± φ ( 0\n\n) -(z); ( 23\n\n) φ ( 2\n\n) ± (z) = d -± φ ( 0\n\n) + (z) -d +± φ ( 0\n\n) -(z). ( 24\n\n)\n\n4 Here, 17) are in fact linearly dependent to each other when j is an even integer. As a result, each of these coefficients is divergent by itself in these cases. It is reassuring to see that all the divergent pieces cancel among themselves so that physically interested quantities do have a smooth limit when j is an even integer. In zeroth order, the combination\n\nc ±± = C ∞ 0 dz 1 δV (z 1 ) φ ( 0\n\n) ± (z 1 ) φ ( 0\n\n) ± (z 1 ); ( 25\n\n) d ±± = C 2 ∞ 0 dz 2 z1 0 dz 1 δV (z 2 ) δV (z 1 ) φ ( 0\n\n) ± (z 2 ) φ (0) + (z 2 )φ ( 0\n\n) -(z 1 ) -φ ( 0\n\n) -(z 2 )φ ( 0\n\n) + (z 1 ) φ ( 0\n\n) ± (z 1 ). ( 26\n\n) Notice that φ ( 0\n\n) ± defined in eq (\n\nφ (0) (z) = φ (0) + (z) -e -iπ(j/2) φ (0) -(z) ∼ e -iz ( 27\n\n)\n\nin the asymptotic region z ≫ 1. This can be extended to second order φ(z) = φ\n\n(0) + (z) + 1 √ -ωr 0 φ (1) + (z) + 1 -ωr 0 φ (2) + (z) -e -iπ(j/2) 1 - ξ √ -ωr 0 - ζ -ωr 0 φ (0) -(z) + 1 √ -ωr 0 φ (1) -(z) + 1 -ωr 0 φ (2) -(z) , ( 28\n\n)\n\nby introducing two parameters ξ and ζ. Naturally, they are determined by the condition that the coefficient of the e iz term is vanishing when z → ∞:\n\nξ = ξ + + ξ -; ( 29\n\n) ζ = -ξξ -+ d ++ e iπj/2 -d +-+ d --e -iπj/2 -d -+ , ( 30\n\n)\n\nwhere\n\nξ + = c ++ e iπj/2 -c +-, ξ -= c --e -iπj/2 -c -+ . ( 31\n\n)\n\nSubstitute the above result back to eq (28), we have\n\nφ(z) = i e iπ(1-j)/4 sin(πj/2) e -iz 1 - ξ - √ -ωr 0 + ξ(ξ m + c +-) -d --e -iπj/2 + d -+ -ωr 0 , ( 32\n\n)\n\nwhere the identity c -+ = c +-has been used to simplify the expression. When going around the black hole singularity by 3π, φ (1) ± and φ (2) ± both pick up an extra phase:\n\nφ ( 1\n\n) ± ( e 3iπ z) = e 3iπ(2±j)/2 φ ( 1\n\n) ± (-z); ( 33\n\n) φ ( 2\n\n) ± ( e 3iπ z) = e 3iπ(3±j)/2 φ ( 2\n\n) ± (-z). ( 34\n\n) Consequently, φ( e 3iπ z) = e 3iπ(1+j)/2 φ ( 0\n\n) + (-z) -i 1 √ -ωr 0 φ (1) + (-z) - 1 -ωr 0 φ (2) + (-z) -e -iπ(j/2) 1 - ξ √ -ωr 0 - ζ -ωr 0 e 3iπ(1-j)/2 φ (0) -(-z) -i 1 √ -ωr 0 φ (1) -(-z) - 1 -ωr 0 φ (2) -(-z) . ( 35\n\n) 5\n\nTo second order, φ( e 3iπ z) = -i e iπ(1-j)/4 sin(3πj/2) e -iz\n\n1 + (1 + ie 3iπj )ξ + + (1 + i)ξ - √ -ωr 0 (-1 + e i3πj ) + -(1 + i)ξ ξ -+ [(1 + e i3πj )(d ++ e iπj/2 -d -+ ) + 2d --e -iπj/2 -2d +-] -ωr 0 (-1 + e i3πj ) + . . . , ( 36\n\n)\n\nwhere the term e iz is not relevant for our calculation and has been neglected. Taking the ratio between the coefficients of the term e -iz in eqs (36) and (32) , we obtain the monodromy to second order:\n\nM(r 0 ) = -[1 + 2 cos(jπ)] 1 + ∆ 1 √ -ωr 0 + ∆ 2c + ∆ 2d -ωr 0 . ( 37\n\n)\n\nHere,\n\n∆ 1 = (1 + ie 3iπj )ξ + + (i + e 3iπj )ξ - (-1 + e i3πj ) ; ( 38\n\n) ∆ 2c = -(1 -i)ξ + ξ --ξc +-; ( 39\n\n) ∆ 2d = (1 + e i3πj )(d ++ e iπj/2 + d --e -iπj/2 ) -2d +--2d -+ e i3πj (-1 + e i3πj ) . ( 40\n\n)\n\nThe terms ∆ 2c and ∆ 2d depend on coefficients c µν and d µν , respectively. Although our expression for ∆ 1 here is different from that in Ref. [13] by a phase factor, our final result is identical to their. Making use of the formula\n\nI 1 (µ, ν) ≡ ∞ 0 dz z -1/2 J µ (z)J ν (z) = π/2Γ( 1+2µ+2ν 4 ) Γ( 3-2µ-2ν 4 )Γ( 3+2µ-2ν 4 )Γ( 3-2µ+2ν 4 ) , ( 41\n\n)\n\none can obtain explicitly\n\nc ++ = 3l 2 + 3l + 1 -j 2 Γ 2 ( 1 4 ) Γ( 1-2j 4 ) Γ( 1+2j 4 ) sin[ π(1-2j) 4 ] 48 π 3 2 sin( j π 2 ) ; ( 42\n\n) c --= 3l 2 + 3l + 1 -j 2 Γ 2 ( 1 4 ) Γ( 1-2j 4 ) Γ( 1+2j 4 ) sin[ π(1+2j) 4 ] 48 π 3 2 sin( j π 2 ) ; ( 43\n\n) c +-= 3l 2 + 3l + 1 -j 2 Γ 2 ( 1 4 ) Γ( 1-2j 4 ) Γ( 1+2j 4 ) sin[ π(1-2j) 4 ] sin[ π(1+2j) 4 ] 24 √ 2 π 3 2 sin( j π 2 ) . ( 44\n\n) Note that c --= -c ++ (j → -j); c +-= -c -+ (j → -j). ( 45\n\n)\n\nThese relation are also obeyed by d µν 's, which can be used to reduce our work. With the above results, we are ready to find ∆ 1 and ∆ 2c in eq (40) :\n\n∆ 1 = - i(3l 2 + 3l + 1 -j 2 ) Γ 2 ( 1 4 ) Γ( 1-2j 4 ) Γ( 1+2j 4 ) cos( jπ 2 ) cos(jπ) 6 √ 2π 3/2 [1 + 2 cos(jπ)] ; ( 46\n\n) ∆ 2c = - (3l 2 + 3l + 1 -j 2 ) 2 Γ 4 ( 1 4 ) Γ 2 ( 1-2j 4 ) Γ 2 ( 1+2j 4 ) cos(jπ) 1152π 3 . ( 47\n\n)\n\n6 The double integral\n\nI 2 (µ 2 , ν 2 ; µ 1 , ν 1 ) ≡ ∞ 0 dz 2 z2 0 dz 1 z -1/2 2 z -1/2 1 J µ2 (z 2 )J ν2 (z 2 )J µ1 (z 1 )J ν1 (z 1 ) ( 48\n\n)\n\ncan be expressed in terms of the generalized hypergeometric functions, but the general formula is quite complicated and not particularly illuminating. Therefore, we will just give the final result explicitly for the coefficients d ++ and d +-:\n\nd ++ = - π 2 3l 2 + 3l + 1 -j 2 2 cot( jπ 2 )Γ( 1 4 ) 5 G 4 ( 1 4 , 1 2 , 1 2 , 1+j 2 , 1-j 2 ; 5 4 , 1, 2+j 2 , 2-j 2 ; 1) 576 sin 2 ( j π 2 ) + √ π 3l 2 + 3l + 1 -j 2 2 cot( j π 2 ) cot(j π) Γ( 1+2j 4 ) Γ( 1+2j 2 ) Γ 2 ( 1+j 2 ) 288 sin( jπ 2 ) 5 G 4 ( 1 2 , 1 + 2j 4 , 1 + j 2 , 1 + j 2 , 1 + 2j 2 ; 2 + j 2 , 2 + j 2 , 5 + 2j 4 , 1 + j; 1) + 3l 2 + 3l + 1 -j 2 2 Γ 4 ( 1 4 ) Γ 2 ( 1-2j 4 ) Γ 2 ( 1+2j 4 ) sin 2 [ π(1-2j) 4 ] sin[ π(1+2j) 4 ] 1152 √ 2 π 3 sin 2 ( j π 2 ) ; ( 49\n\n) d +-= - π 2 3l 2 + 3l + 1 -j 2 2 Γ( 1 4 ) 5 G 4 ( 1 4 , 1 2 , 1 2 , 1+j 2 , 1-j 2 ; 5 4 , 1, 2+j 2 , 2-j 2 ; 1) 1152 sin 3 ( j π 2 ) + √ π 3l 2 + 3l + 1 -j 2 2 cot 2 ( j π 2 ) cot(j π) Γ( 1-2j 4 ) Γ( 1-2j 2 ) Γ 2 ( 1-j 2 ) 576 5 G 4 ( 1 2 , 1 -2j 4 , 1 -j 2 , 1 -j 2 , 1 -2j 2 ; 2 -j 2 , 2 -j 2 , 5 -2j 4 , 1 -j; 1) - 3l 2 + 3l + 1 -j 2 2 Γ 4 ( 1 4 ) Γ 2 ( 1-2j 4 ) Γ 2 ( 1+2j 4 ) sin 2 [ π(1-2j) 4 ] sin 2 [ π(1+2j) 4 ] 2304 π 3 sin 2 ( j π 2 ) . ( 50\n\n)\n\nHere, we have used the regularized generalized hypergeometric function\n\n5 G 4 (a 1 , a 2 , a 3 , a 4 , a 5 ; b 1 , b 2 , b 3 , b 4 ; z)\n\nso that the pole structure of each term in these expressions are more explicit. It is related to the usual generalized hypergeometric function by\n\n5 G 4 (a 1 , a 2 , a 3 , a 4 , a 5 ; b 1 , b 2 , b 3 , b 4 ; z) = 5 F 4 (a 1 , a 2 , a 3 , a 4 , a 5 ; b 1 , b 2 , b 3 , b 4 ; z) Γ(b 1 )Γ(b 2 )Γ(b 3 )Γ(b 4 ) . ( 51\n\n)\n\nThe other two coefficients can be obtained by relations analogous to those in eq (45)\n\nd --= -d ++ (j → -j); d -+ = -d +-(j → -j). ( 52\n\n)\n\nOn the face of it, each of the d µν 's has a third order pole coming from terms involving the generalized hypergeometric function when j is an even integer. On closer look, we see there are some cancelation among the divergences and in the end all they have are just simple poles in such limit similar to the c µν 's. Another possible divergence arises in d --when j = 1, which will again be canceled when we calculate the monodromy. 7 It is now straightforward to obtain ∆ 2d by making use of the following two identities\n\n-4π 7/2 cos 2 ( jπ 2 ) cot(jπ)Γ( 1 + 2j 4 ) Γ( 1 + 2j 2 ) Γ 2 ( 1 + j 2 ) 5 G 4 ( 1 2 , 1 + 2j 4 , 1 + j 2 , 1 + j 2 , 1 + 2j 2 ; 2 + j 2 , 2 + j 2 , 5 + 2j 4 , 1 + j; 1) +4π 7/2 cos 2 ( jπ 2 ) cot(jπ)Γ( 1 -2j 4 ) Γ( 1 -2j 2 ) Γ 2 ( 1 -j 2 ) 5 G 4 ( 1 2 , 1 -2j 4 , 1 -j 2 , 1 -j 2 , 1 -2j 2 ; 2 -j 2 , 2 -j 2 , 5 -2j 4 , 1 -j; 1) -Γ 4 ( 1 4 ) Γ 2 ( 1 -2j 4 ) Γ 2 ( 1 + 2j 4 ) sin[ π(1 -2j) 4 ] sin[ π(1 + 2j) 4 ] = 0; ( 53\n\n) -4π 7/2 cos( jπ 2 ) Γ( 1 + 2j 4 ) Γ( 1 + 2j 2 ) Γ 2 ( 1 + j 2 ) 5 G 4 ( 1 2 , 1 + 2j 4 , 1 + j 2 , 1 + j 2 , 1 + 2j 2 ; 2 + j 2 , 2 + j 2 , 5 + 2j 4 , 1 + j; 1) -4π 7/2 cos( jπ 2 ) Γ( 1 -2j 4 ) Γ( 1 -2j 2 ) Γ 2 ( 1 -j 2 ) 5 G 4 ( 1 2 , 1 -2j 4 , 1 -j 2 , 1 -j 2 , 1 -2j 2 ; 2 -j 2 , 2 -j 2 , 5 -2j 4 , 1 -j; 1); +8π 5 Γ( 1 4 ) 5 G 4 ( 1 4 , 1 2 , 1 2 , 1 + j 2 , 1 -j 2 ; 5 4 , 1, 2 + j 2 , 2 -j 2 ; 1) -Γ 4 ( 1 4 ) Γ 2 ( 1 -2j 4 ) Γ 2 ( 1 + 2j 4 ) cos( jπ 2 )[1 -cos(jπ)] = 0. ( 54\n\n)\n\nEventually, we achieve the following nice result\n\n∆ 2d = (3l 2 + 3l + 1 -j 2 ) 2 Γ 4 ( 1 4 ) Γ 2 ( 1-2j 4 ) Γ 2 ( 1+2j 4 ) cos(jπ) 1152π 3 [1 + 2 cos(jπ)] , ( 55\n\n)\n\nwhere all divergences have been canceled out.\n\nTogether with the result from eq (47) , the asymptotic form of quasinormal frequencies of a four dimensional Schwarzschild black hole is found to be\n\n4πωr 0 = (2n + 1)πi + ln[1 + 2 cos(jπ)] - i(3l 2 + 3l + 1 -j 2 ) Γ 2 ( 1 4 ) Γ( 1-2j 4 ) Γ( 1+2j 4 ) cos( jπ 2 ) cos(jπ) 6 √ 2π 3/2 [1 + 2 cos(jπ)] √ -ωr 0 + (3l 2 + 3l + 1 -j 2 ) 2 Γ 4 ( 1 4 ) Γ 2 ( 1-2j 4 ) Γ 2 ( 1+2j 4 ) cos 2 (jπ) 576π 3 [1 + 2 cos(jπ)] 2 (-ωr 0 ) + O[(-ωr 0 ) -3/2 ]. ( 56\n\n)\n\nThe physically interested cases are\n\nω n T H ≈ (2n + 1)πi + ln 3 + 1 -i √ n (l 2 + l -1)Γ 4 (1/4) 18 √ 2π 3/2 + i n (l 2 + l -1) 2 Γ 8 (1/4) 2592π 3 , for j = 2; ( 57\n\n) ω n T H ≈ (2n + 1)πi + ln 3 + 1 -i √ n (l 2 + l + 1/3)Γ 4 (1/4) 6 √ 2π 3/2 + i n (l 2 + l + 1/3) 2 Γ 8 (1/4) 288π 3 , for j = 0; ( 58\n\n) ω n T H ≈ 2nπi + i2π(l 2 + l) 2 n , for j = 1. ( 59\n\n)\n\nA few comments are in order. First, all the second order corrections are purely imaginary. In particular, when j = 2 (gravitational perturbation) the numerical coefficients of the i/n term (after divided by 8 4π) are 0.739, 3.58, 49.7 for l = 2, 3, 6, respectively. They are in good agreement with the known numerical studies [4]. As for the real part, our result predicts vanishing correction. For j = 2, this is again consistent with the numerical results in Ref. [4] for l = 2, 3. For l = 6 the numerical result is 0.263, which seems to be contradictory to ours. However, the numerical value for l = 6 has opposite sign relative to those of l = 2, 3. This is peculiar, since in all other cases a given type of corrections are always of the same sign irrespective of the specific value of angular momentum. Therefore, we believe more study is needed to clarify whether there is really a discrepancy. As for the j = 1 case, the numerical study in Ref. [17] suggests the leading correction is of the form b n 3/2 . However, this does not necessarily mean the two results are inconsistent. In fact, one can only extract the behavior of the leading correction to the real part from their Fig. 2 and further numerical study is needed to confirm or refute our prediction."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusion",
"text": "In sum, we have calculated to second order the correction to the asymptotic form of quasinormal frequencies for Schwarzschild black holes in four dimensions. Most of our results are consistent with the numerical ones when available. In cases where there seem to be contradiction, we think further numerical studies are needed to clarify the situation. It would also be helpful if more detailed numerical studies can be carried out for the j = 0 case so that more thorough comparisons are possible. It would be interesting to generalize the method to other spacetime backgrounds [18] . Extension to higher order is also desirable. It might enable us to find a quantitative prediction for the \"algebraically special\" frequencies in Schwarzschild black holes, where the quasinormal frequency is purely imaginary and it increases with the fourth power of l [19, 4] ."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgment",
"text": "The author thanks Chong-Sun Chu for helpful discussions. The work is supported in part by the National Science Council and the National Center for Theoretical Sciences, Taiwan.\n\n[2] H. P. Nollert, \"Quasinormal modes: the characteristic 'sound' of black holes and neutron stars,\" Class Quantum Grav. 16, R159 (1999); K. D. Kokkotas and B. G. Schmidt, \"Quasi-normal modes of stars and black holes,\" Living Rev. Relativ. 2, 2 (1999) [gr-qc/9909058]. 9 [3] J. Noatario and R. Schiappa, \"On the classification of asymptotic quasinormal frequencies for d-dimensional black hole and quantum gravity,\" Adv. Theor. Math. Phys. 8, 1001 (2004) [hep-th/0411267].\n\n[4] H. P. Nollert, \"Quasinormal Modes of Schwarzschild Black Holes: the Determination of Quasinormal Frequencies of Very Large Imaginary Parts,\" Phys. Rev. D 47, 5253 (1993); N. Andersson, \"On the Asymptotic Distribution of Quasinormal-mode Frequencies of Schwarzschild Black Holes, \" Class Quantum Grav. 10, L61 (1993). [5] S. Hod, \"Bohr's Correspondence Principle and the Area Spectrum of Quantum Black Holes,\" Phys. Rev. Lett. 81, 4293 (1998) [gr-qc/9812002]. [6] J. D. Bekenstein and V. F. Mukhanov, \"Spectroscopy of the Quantum Black Hole,\" Phys. Lett. B360, 7 (1995) [gr-qc/9505012]. [7] L. Motl, \"An Analytic Computation of Asymptotic Schwarzschild Quasinormal Frequencies, \" Adv. Theor. Math. Phys. 6, 1135 (2003) [gr-qc/0212096]; L. Motl, and A. Neitzke, \"Asymptotic Black Hole Quasinormal Frequencies,\" Adv. Theor. Math. Phys. 7, 307 (2003) [hep-th/0301173]. [8] O. Dreyer, \"Quasinormal Modes, the Area Spectrum, and Black Hole Entropy,\" Phys. Rev. Lett. 90, 081301 (2003) [gr-qc/0211076]. [9] G. T. Horowitz and V. E. Hubeny, \"Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium,\" Phys. Rev. D62, 024027 (2000) [hep-th/9909056]. [10] J. Maldacena, \"The Large N Limit Of Superconformal Field Theories And Supergravity,\" Adv. Theor. Math. Phys. 2, 231 (1998) [hep-th/9711200]; E. Witten, \"Anti-De Sitter Space And Holography,\" Adv. Theor. Math. Phys. 2, 253 (1998) [hep-th/9802150]; S. Gubser, I. Klebanov, and A. Polyakov\"Gauge Theory Correlators From Noncritical String Theory,\" Phys. Lett. B 428, 105 (1998) [hep-th/9802109]. [11] D. Birmingham, I. Sachs, and S. N. Solodukhin, \"Conformal Field Theory Interpretation of Black Hole Quasinormal Modes,\" Phys. Rev. Lett. 88, 151301 (2002) [hep-th/0112055].\n\n10 [12] A. Maassen van den Brink, \"WKB analysis of the Regge-Wheeler Equation Down in the Frequency Plane,\" J. Math. Phys. 45, 327 (2004) [gr-qc/0303095]. [13] S. Musiri and G. Siopsis, Perturbative calculation of quasi-normal modes of Schwarzschild black holes, Class. Quant. Grav. 20 L285, (2003) [hep-th/0308168]. [14] A. Ishibashi and H. Kodama, \"A Master Equation for Gravitational Perturbations of Maximally Symmetric Black Holes in Higher Dimensions,\" Prog. Theor. Phys. 110, 701 (2003) [hep-th/0305147]; A. Ishibashi and H. Kodama, \"Stability of Higher Dimensional Schwarzschild Black Holes,\" Prog. Theor. Phys. 110, 901 (2003) [hep-th/0305185]; A. Ishibashi and H. Kodama, \"Master Equations for Perturbations of Generalized Static Black Holes with Charge in Higher Dimensions,\" Prog. Theor. Phys. 111, 29 (2004) [hep-th/0308128]. [15] E. Witten, \"Quantum Gravity In De Sitter Space,\" [hep-th/0106109]; A. Strominger, \"The dS/CFT Correspondence,\" JHEP 0110, 034 (2001) [hep-th/0106113]; D. Klemm, \"Some Aspects Of The De Sitter/CFT Correspondence,\" Nucl. Phys. B 625, 295 (2002) [hep-th/0106247]. [16] E. Abdalla, B. Wang, A. Lima-Santos, and W. G. Qiu, \"Support Of dS/CFT Correspondence From Perturbations Of Three-Dimensional Space-Time,\" Phys. Lett. B 538, 435 (2002) [hep-th/0204030]; E. Abdalla, K. H. C. Castello-Branco, and A. Lima-Santos, \"Support of dS/CFT Correspondence from Space-time Perturbations,\" Phys. Rev. D 66, 104018 (2002) [hep-th/0208065]; Y. S. Myung and N. J. Kim, \"Difference between AdS and dS Spaces: Wave Equation Approach,\" Class. Quant. Grav. 21, 63 (2004) [hep-th/0304231]; T. R. Choudhury and T. Padmanabhan, \"Quasinormal Modes in Schwarzschild-de Sitter Space-time: A Simple Derivation of the Level Spacing of the Frequencies,\" Phys. Rev. D 69, 064033 (2004) [gr-qc/0311064]. [17] V. Cardoso, J. P. S. Lemos and S. Yoshida, Quasinormal Modes of Schwarzschild Black Holes in Four and Higher Dimensions, Phys. Rev. D69, 044004 (2004) [gr-qc/0309112]. [18] S. Musiri, S. Ness and G. Siopsis, \"Perturbative Calculation of Quasi-normal Modes of AdS Schwarzschild Black Holes, Phys. Rev. D 73, 064001 (2006), [hep-th/0511113]; F.-W. Shu 11 and Y.-G. Shen, \" it Perturbative Calculation of Quasinormal Modes of d-Dimensional Black Holes,\" JHEP 0608, 087 (2006) [hep-th/0605128]. [19] S. Chandrasekhar, \"On Algebraically Special Perturbations of Black Holes,\" Proc. R. Soc. London A392, 1 (1984). 12"
}
] | |
arxiv:0704.0296 | 0704.0296 | 1 | 10.1103/PhysRevD.75.104015 | ef23e0c2953d4682cf77d3e5947190e55072c520e3502c8369ae74ec26c34517 | Generalized Twistor Transform And Dualities, With A New Description of
Particles With Spin, Beyond Free and Massless | A generalized twistor transform for spinning particles in 3+1 dimensions is
constructed that beautifully unifies many types of spinning systems by mapping
them to the same twistor, thus predicting an infinite set of duality relations
among spinning systems with different Hamiltonians. Usual 1T-physics is not
equipped to explain the duality relationships and unification between these
systems. We use 2T-physics in 4+2 dimensions to uncover new properties of
twistors, and expect that our approach will prove to be useful for practical
applications as well as for a deeper understanding of fundamental physics.
Unexpected structures for a new description of spinning particles emerge. A
unifying symmetry SU(2,3) that includes conformal symmetry SU(2,2)=SO(4,2) in
the massless case, turns out to be a fundamental property underlying the
dualities of a large set of spinning systems, including those that occur in
high spin theories. This may lead to new forms of string theory backgrounds as
well as to new methods for studying various corners of M theory. In this paper
we present the main concepts, and in a companion paper we give other details. | [
"Itzhak Bars and Bora Orcal"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | A generalized twistor transform for spinning particles in 3+1 dimensions is constructed that beautifully unifies many types of spinning systems by mapping them to the same twistor , thus predicting an infinite set of duality relations among spinning systems with different Hamiltonians. Usual 1T-physics is not equipped to explain the duality relationships
and unification between these systems. We use 2T-physics in 4+2 dimensions to uncover new properties of twistors, and expect that our approach will prove to be useful for practical applications as well as for a deeper understanding of fundamental physics. Unexpected structures for a new description of spinning particles emerge. A unifying symmetry SU (2, 3) that includes conformal symmetry SU(2, 2) =SO (4, 2) in the massless case, turns out to be a fundamental property underlying the dualities of a large set of spinning systems, including those that occur in high spin theories. This may lead to new forms of string theory backgrounds as well as to new methods for studying various corners of M theory. In this paper we present the main concepts, and in a companion paper we give other details [1] .
The Penrose twistor transform [2] - [5] brings to the foreground the conformal symmetry SO (4, 2) in the dynamics of massless relativistic particles of any spin in 3 + 1 dimensions.
The transform relates the phase space and spin degrees of freedom x µ , p µ , s µν to a twistor
Z A = µ α λα
and reformulates the dynamics in terms of twistors instead of phase space. The twistor Z A is made up of a pair of SL(2, C) spinors µ α, λ α , α, α = 1, 2, and is regarded as the 4 components A = 1, 2, 3, 4 of the Weyl spinor of SO(4, 2) =SU (2, 2) .
The well known twistor transform for a spinning massless particle is [5]
µ α = -i (x + iȳ) αβ λ β , λ α λ β = p α β , (1.1)
where (x + iȳ) αβ = 1 √ 2 (x µ + iy µ ) (σ µ ) αβ , and p α β = 1 √ 2 p µ (σ µ ) α β , while σ µ = (1, σ) , σµ = (-1, σ) are Pauli matrices. x µ + iy µ is a complexification of spacetime [2] . The helicity h of the particle is determined by p • y = h. The spin tensor is given by s µν = ε µνρσ y ρ p σ , and it leads to 1 2 s µν s µν = h 2 . The Pauli-Lubanski vector is proportional to the momentum W µ = 1 2 ε µνρσ s νρ p σ = (y • p) p µ -p 2 y µ = hp µ , appropriate for a massless particle of helicity h. The reformulation of the dynamics in terms of twistors is manifestly SU(2, 2) covariant.
It was believed that twistors and the SO(4, 2)=SU(2, 2) symmetry, interpreted as conformal symmetry, govern the dynamics of massless particles only, since the momentum p µ of the form p α β = λ α λ β automatically satisfies p µ p µ = 0. However, recent work has shown that the same twistor Z A = µ α λα that describes massless spinless particles (h = 0) also describes an assortment of other spinless particle dynamical systems [6] [7] . These include massive and interacting particles. The mechanism that avoids p µ p µ = 0 [6] [7] is explained following Eq.(6.9) below. The list of systems includes the following examples worked out explicitly in previous publications and in unpublished notes.
The massless relativistic particle in d = 4 flat Minkowski space.
The massive relativistic particle in d = 4 flat Minkowski space. The nonrelativistic free massive particle in 3 space dimensions. The nonrelativistic hydrogen atom (i.e. 1/r potential) in 3 space dimensions.
The harmonic oscillator in 2 space dimensions, with its mass ⇔ an extra dimension.
The particle on AdS 4 , or on dS 4 .
The particle on AdS 3 ×S 1 or on R × S 3 .
The particle on AdS 2 ×S 2 .
The particle on the Robertson-Walker spacetime.
The particle on any maximally symmetric space of positive or negative curvature.
The particle on any of the above spaces modified by any conformal factor.
A related family of other particle systems, including some black hole backgrounds. In this paper we will discuss these for the case of d = 4 with spin (h = 0). It must be emphasized that while the phase spaces (and therefore dynamics, Hamiltonian, etc.) in these systems are different, the twistors µ α, λ α are the same. For example, the massive particle phase space (x µ , p µ ) massive and the one for the massless particle (x µ , p µ ) massless are not the same (x µ , p µ ) , rather they can be obtained from one another by a non-linear transformation for any value of the mass parameter m [6] , and similarly, for all the other spaces mentioned above. However, under such "duality" transformations from one system to another, the twistors for all the cases are the same up to an overall phase transformation
µ α, λ α massive = µ α, λ α massless = • • • = µ α, λ α . (1.2)
This unification also shows that all of these systems share the same SO(4, 2)=SU (2, 2) global symmetry of the twistors. This SU(2, 2) is interpreted as conformal symmetry for the massless particle phase space, but has other meanings as a hidden symmetry of all the other systems in their own phase spaces. Furthermore, in the quantum physical Hilbert space, the symmetry is realized in the same unitary representation of SU (2, 2) , with the same Casimir eigenvalues (see (7.16,7 .17) below), for all the systems listed above.
The underlying reason for such fantastic looking properties cannot be found in one-time physics (1T-physics) in 3+1 dimensions, but is explained in two-time physics (2T-physics) [8] as being due to a local Sp(2, R) symmetry. The Sp(2, R) symmetry which acts in phase space makes position and momentum indistinguishable at any instant and requires one extra space and one extra time dimensions to implement it, thus showing that the unification relies on an underlying spacetime in 4+2 dimensions. It was realized sometime ago that in 2Tphysics twistors emerge as a gauge choice [9] , while the other systems are also gauge choices of the same theory in 4+2 dimensions. The 4+2 phase space can be gauge fixed to many 3+1 phase spaces that are distinguishable from the point of view of 1T-physics, without any Kaluza-Klein remnants, and this accounts for the different Hamiltonians that have a duality relationship with one another. We will take advantage of the properties of 2T-physics to build the general twistor transform that relates these systems including spin.
Given that the field theoretic formulation of 2T-physics in 4+2 dimensions yields the Standard Model of Particles and Forces in 3+1 dimensions as a gauge choice [10] , including spacetime supersymmetry [11] , and given that twistors have simplified QCD computations [12] [13], we expect that our twistor methods will find useful applications.
The Penrose twistor description of massless spinning particles requires that the pairs µ α, i λ α or their complex conjugates (λ α , iμ α ) be canonical conjugates and satisfy the helicity constraint given by ZA Z A = λ αµ α + μα λ α = 2h.
(2.1) Indeed, Eq.(1.1) satisfies this property provided y • p = h. Here we have defined the 4 of SU(2, 2) as the contravariant twistor
ZA ≡ Z † η 2,2 A = λ α μα , η 2,2 = 0 1 1 0 = SU (2, 2) metric. (2.
2)
The canonical structure, along with the constraint ZA Z A = 2h follows from the following worldline action for twistors
S h = dτ i ZA D τ Z A -2h à , D τ Z A ≡ ∂Z A ∂τ -i ÃZ A . (2.3)
In the case of h = 0 it was shown that this action emerges as a gauge choice of a more general action in 2T-physics [6][7] . Later in the paper, in Eq.( 4 .1) we give the h = 0 2T-physics action from which (2.3) is derived as a gauge choice. The derivative part of this action gives the canonical structure S 0 = dτ i ZA ∂ τ Z A = i dτ λ α∂ τ µ α + μα ∂ τ λ α that requires µ α, i λ α or their complex conjugates (λ α , iμ α ) to be canonical conjugates.
The 1-form Ãdτ is a U(1) gauge field on the worldline, D τ Z A is the U(1) gauge covariant derivative that satisfies
δ ε D τ Z A = iε D τ Z A for δ ε Ã = ∂ε/∂τ and δ ε Z A = iεZ A .
The term 2h à is gauge invariant since it transforms as a total derivative under the infinitesimal gauge transformation. 2h à was introduced in [6] [7] as being an integral part of the twistor formulation of the spinning particle action.
Our aim is to show that this action describes not only massless spinning particles, but also all of the other particle systems listed above with spin. This will be done by constructing the twistor transform from Z A to the phase space and spin degrees of freedom of these systems, and claiming the unification of dynamics via the generalized twistor transform.
This generalizes the work of [6][7] which was done for the h = 0 case of the action in (2.3).
We will use 2T-physics as a tool to construct the general twistor transform, so this unification is equivalent to the unification achieved in 2T-physics.
In our quest for the general twistor transform with spin, we first discuss an alternative to the well known twistor transform of Eq.(1.1). Instead of the y µ (τ ) that appears in the complexified spacetime x µ + iy µ we introduce an SL(2, C) bosonic 2 spinor v α (τ ) and its complex conjugate vα (τ ) , and write the general vector y µ in the matrix form as y αβ = hv α vβ + ωp αβ , where ω (τ ) is an arbitrary gauge freedom that drops out. Then the helicity condition y • p = h takes the form vpv = 1. Furthermore, we can write λ α = p α β v β since this automatically satisfies λ α λ β = p α β when p 2 = (vpv -1) = 0 are true. With this choice of variables, the Penrose transform of Eq.(1.1) takes the new form
λ α = (pv) α , µ α = [(-ixp + h) v] α , p 2 = (vpv -1) = 0, (3.1)
where the last equation is a set of constraints on the degrees of freedom x µ , p µ , v α, vα .
If we insert the twistor transform (3.1) into the action (2.3), the twistor action turns into the action for the phase space and spin degrees of freedom x µ , p µ , v α, vα
S h = dτ ẋµ p µ - e 2 p 2 + ih (vp) D τ v -D τ v (pv) -2h à . (3.2)
where D τ v = v -i Ãv is the U(1) gauge covariant derivative and we have included the
Lagrange multiplier e to impose p 2 = 0 when we don't refer to twistors. The equation of motion for à imposes the second constraint vpv -1 = 0 that implies U(1) gauge invariance 3 .
From the global Lorentz symmetry of (3.2), the Lorentz generator is computed via Noether's theorem J µν = x µ p ν -x ν p µ + s µν , with s µν = i 2 hv (pσ µν + σ µν p) v. The helic-2 This is similar to the fermionic case in [5] . The bosonic spinor v can describe any spin h.
3 If this action is taken without the U(1) constraint à = 0 , then the excitations in the v sector describe an infinite tower of massless states with all helicities from zero to infinity (here we rescale
√ 2hv → v) S all spins = dτ ẋµ p µ - e 2 p 2 + i 2 vp v -vpv (3.3)
The spectrum coincides with the spectrum of the infinite slope limit of string theory with all helicities 1 2 vpv. This action has a hidden SU(2, 3) symmetry that includes SU(2, 2) conformal symmetry. This is explained in the rest of the paper by the fact that this action is a gauge fixed version of a 2T-physics master action (4.1,5.4) in 4+2 dimensions with manifest SU(2, 3) symmetry. A related approach has been pursued also in [15] - [18] in 3+1 dimensions in the context of only massless particles. Along with the manifestly SU(2, 3) symmetric 2T-physics actions, we are proposing here a unified 2T-physics setting for discussing high spin theories [14] including all the dual versions of the high spin theories related to the spinning physical systems listed in section (I).
ity is determined by computing the Pauli-Lubanski vector W µ = 1 2 ε µνλσ s νλ p σ = (hvpv) p µ . The helicity operator hvpv reduces to the constant h in the U(1) gauge invariant sector.
The action (3.2) gives a description of a massless particle with any helicity h in terms of the SL(2, C) bosonic spinors v, v. We note its similarity to the standard superparticle action [20][21] written in the first order formalism. The difference with the superparticle is that the fermionic spacetime spinor θ α of the superparticle is replaced with the bosonic spacetime spinor v α, and the gauge field à imposes the U(1) gauge symmetry constraint vpv -1 = 0 that restricts the system to a single, but arbitrary helicity state given by h.
Just like the superparticle case, our action has a local kappa symmetry with a bosonic local spinor parameter κ α (τ ), namely
δ κ v α = p αβ κ β , δ κ x µ = ih √ 2 ((δ κ v) σ µ v -vσ µ (δ κ v)) , (3.4)
δ κ p µ = 0, δ κ e = -ih κ (D τ v) -D τ v κ , δ κ Ã = 0. (3.5)
These kappa transformations mix the phase space degrees of freedom (x, p) with the spin degrees of freedom v, v. The transformations δ κ x µ , δ κ e are non-linear.
Let us count physical degrees of freedom. By using the kappa and the τ -reparametrization symmetries one can choose the lightcone gauge. From phase space x µ , p µ there remains 3 positions and 3 momentum degrees of freedom. One of the two complex components of v α is set to zero by using the kappa symmetry, so v α = v 0 . The phase of the remaining component is eliminated by choosing the U(1) gauge, and finally its magnitude is fixed by solving the constraint vpv -1 = 0 to obtain v α = (p + ) -1/2 1 0 . Therefore, there are no independent physical degrees of freedom in v. The remaining degrees of freedom for the particle of any spin are just the three positions and momenta, and the constant h that appears in s µν . This is as it should be, as seen also by counting the physical degrees of freedom from the twistor point of view. When we consider the other systems listed in the first section, we should expect that they too are described by the same number of degrees of freedom since they will be obtained from the same twistor, although they obey different dynamics (different Hamiltonians) in their respective phase spaces.
The lightcone quantization of the the massless particle systems described by the actions (3.2,3.3) is performed after identifying the physical degrees of freedom as discussed above.
The lightcone quantum spectrum and wavefunction are the expected ones for spinning massless particles, and agree with their covariant quantization given in [15] - [19] .
IV. 2T-PHYSICS WITH SP(2, R) , SU(2, 3) AND KAPPA SYMMETRIES The similarity of (3.2) to the action of the superparticle provides the hint for how to lift it to the 2T-physics formalism, as was done for the superparticle [22] [9] and the twistor superstring [23][24] . This requires lifting 3+1 phase space (x µ , p µ ) to 4+2 phase space X M , P M and lifting the SL(2, C) spinors v, v to the SU(2, 2) spinors V A , V A . The larger set of degrees of freedom X M , P M , V A , V A that are covariant under the global symmetry SU(2, 2) =SO(4, 2) , include gauge degrees of freedom, and are subject to gauge symmetries and constraints that follow from them as described below.
The point is that the SU(2, 2) invariant constraints on X M , P M , V A , V A have a wider set of solutions than just the 3+1 system of Eq.(3.2) we started from. This is because 3+1 dimensional spin & phase space has many different embeddings in 4+2 dimensions, and those are distinguishable from the point of view of 1T-physics because target space "time" and corresponding "Hamiltonian" are different in different embeddings, thus producing the different dynamical systems listed in section (I). The various 1T-physics solutions are reached by simply making gauge choices. One of the gauge choices for the action we give below in Eq.(4.1) is the twistor action of Eq.(2.3). Another gauge choice is the 4+2 spin & phase space action in terms of the lifted spin & phase space X M , P M , V A , V A as given in Eq. (5.4) .
The latter can be further gauge fixed to produce all of the systems listed in section (I) including the action (3.2) for the massless spinning particle with any spin. All solutions still remember that there is a hidden global symmetry SU(2, 2) =SO(4, 2) , so all systems listed in section (I) are realizations of the same unitary representation of SU(2, 2) whose Casimir eigenvalues will be given below.
For the 4 + 2 version of the superparticle [22] that is similar to the action in (5.4) , this program was taken to a higher level in [9] by embedding the fermionic supercoordinates in the coset of the supergroup SU(2, 2|1) /SU(2, 2) ×U(1). We will follow the same route here, and embed the bosonic SU(2, 2) spinors V A , V A in the left coset SU(2, 3) /SU(2, 2) ×U(1) . This coset will be regarded as the gauging of the group SU(2, 3) under the subgroup [SU (2, 2) × U (1)] L from the left side. Thus the most powerful version of the action that reveals the global and gauge symmetries is obtained when it is organized in terms of the X M i (τ ), g (τ ) and à (τ ) degrees of freedom described as 4+2 phase space X M (τ )
P M (τ ) ≡ X M i (τ ) , i = 1, 2, doublets of Sp (2, R) gauge symmetry, group element g (τ ) ⊂ SU (2, 3) subject to [SU (2, 2) × U (1) ] L × U (1) L+R gauge symmetry.
We should mention that the h = 0 version of this theory, and the corresponding twistor property, was discussed in [6] , by taking g (τ ) ⊂SU(2, 2) and dropping all of the U(1)'s. So, the generalized theory that includes spin has the new features that involves SU(2, 2) →SU (2, 3) and the U(1) structures. The action has the following form
S h = dτ 1 2 ε ij D τ X M i X N j η M N + T r (iD τ g) g -1 L 0 0 0 -2h à , (4.1)
where
ε ij = 0 -1 1 0 ij is the antisymmetric Sp(2, R) metric, and D τ X M i = ∂ τ X M i -A j i X M j is the Sp(2, R) gauge covariant derivative, with the 3 gauge potentials A ij = ε ik A j k = A C C B . For SU(2, 3) the group element is pseudo-unitary, g -1 = (η 2,3 ) g † (η 2,3 ) -1 , where η 2,3 is the SU(2, 3) metric η 2,3 = η 2,2 0 0 -1 . The covariant derivative D τ g is given by D τ g = ∂ τ g -i à [q, g] , q = 1 5 1 4×4 0 0 -4 (4.2)
where the generator of U(1) L+R is proportional to the 5×5 traceless matrix q ∈ u(1) ∈ su(2, 3) L+R . The last term of the action -2h Ã, which is also the last term of the action (2.3), is invariant under the U(1) L+R since it transforms to a total derivative.
Finally, the 4 × 4 traceless matrix (L) B A ∈su(2, 2) ∈su(2, 3) that appears on the left side of g (or right side of g -1 ) is
(L) B A ≡ 1 4i Γ M N B A L M N , L M N = ε ij X M i X N j = X M P N -X N P M . (4.3)
where Γ M N = 1 2 Γ M ΓN -Γ N ΓM are the 4×4 gamma-matrix representation of the 15 generators of SU (2, 2) . A detailed description of these gamma matrices is given in [11] .
The symmetries of actions of this type for any group or supergroup g were discussed in [9] [23] [24] [7] . The only modification of that discussion here is due to the inclusion of the U(1) gauge field Ã. In the absence of the à coupling the global symmetry is given by the transformation of g (τ ) from the right side g (τ ) → g (τ ) g R where g R ⊂SU(2, 3) R .
However, in our case, the presence of the coupling with the U(1) L+R charge q breaks the global symmetry down to the (SU(2, 2)×U( 1 )) R subgroup that acts on the right side of g.
: g (τ ) → g (τ ) h R , h R ∈ [SU(2, 2) × U (1)] R ⊂ SU(2, 3) R . (4.4)
Using Noether's theorem we deduce the conserved global charges as the [SU(2, 2)×U( 1 )] R components of the the following SU(2, 3) R Lie algebra valued matrix J (2,3)
J (2,3) = g -1 L 0 0 0 g = J + 1 4 J 0 - j j -J 0 , J 2,3 = η 2,3 (J 2,3 ) † (η 2,3 ) -1 , (4.5)
The traceless 4 × 4 matrix (J
) B A = 1 4i Γ M N J M N is the conserved SU(2, 2) =SO(4,
2) charge and J 0 is the conserved U(1) charge. Namely, by using the equations of motion one can verify ∂ τ (J ) B A = 0 and ∂ τ J 0 = 0. The spinor charges j A , jA are not conserved foot_1 due to the coupling of Ã. As we will find out later in Eq.(6.8), j A is proportional to the twistor
j A = J 0 Z A , (4.6)
up to an irrelevant gauge transformation. It is important to note that J and J 0 are invariant on shell under the gauge symmetries discussed below. Therefore they generate physical symmetries [SU(2, 2)×U( 1 )] R under which all gauge invariant physical states are classified.
The local symmetries of this action are summarized as
Sp (2, R) × SU (2, 2) 3 4 kappa 3 4 kappa U (1) lef t (4.7)
The Sp(2, R) is manifest in (4.1). The rest corresponds to making local SU(2, 3) transformations on g (τ ) from the left side g (τ ) → g L (τ ) g (τ ) , as well as transforming X M i = X M , P M as vectors with the local subgroup SU(2, 2) L =SO(4, 2) , and A ij under the kappa. The 3/4 kappa symmetry which is harder to see will be discussed in more detail below. These symmetries coincide with those given in previous discussions in [9] [23][24] [7] despite the presence of Ã. The reason is that the U(1) L+R covariant derivative D τ g in Eq.(4.2) can be replaced by a purely U(1) R covariant derivative D τ g = ∂ τ g + igq à because the difference drops out in the trace in the action (4.1). Hence the symmetries on left side of g (τ ) → g L (τ ) g (τ ) remain the same despite the coupling of Ã.
We outline the roles of each of these local symmetries. The Sp(2, R) gauge symmetry can reduce X M , P M to any of the phase spaces in 3+1 dimensions listed in section (I). This is the same as the h = 0 case discussed in [6] . The [SU(2, 2) ×U( 1 In terms of counting, there remains only 3 position and 3 momentum physical degrees of freedom, plus the constant h, in agreement with the counting of physical degrees of freedom of the twistors.
It is possible to gauge fix the symmetries (4.7) partially to exhibit some intermediate covariant forms. For example, to reach the SL(2, C) covariant massless particle described by the action (3.2) from the 2T-physics action above, we take the massless particle gauge by using two out of the three Sp(2, R) gauge parameters to rotate the M = + ′ doublet to the form X + ′ P + ′ (τ ) = 1 0 , and solving explicitly two of the Sp(2, R) constraints X 2 = X • P = 0
X M = ( + ′ 1 , -′ x 2 2 , µ x µ (τ )), P M = ( + ′ 0 , -′ x • p , µ p µ (τ )). (4.8)
This is the same as the h = 0 massless case in [6] . There is a tau reparametrization gauge symmetry as a remnant of Sp(2, R) . Next, the [SU(2, 2) ×U(1)] L gauge symmetry reduces g (τ ) → t (V ) written in terms of V A , V A as given in Eq.(5.3), and the 3/4 kappa symmetry
reduces the SU(2, 2) spinor V A → v α 0
to the two components SL(2, C) doublet v α, with a leftover kappa symmetry as discussed in Eqs. (3.4-3.5) . The gauge fixed form of g is then
g = exp 0 0 √ 2hv α 0 0 0 0 √ 2hv α 0 = 1 hv α vβ √ 2hv α 0 1 0 0 √ 2hv α 1 ∈ SU (2, 3) . (4.9)
The inverse g -1 = (η 2,3 ) g † (η 2,3 ) -1 is given by replacing v, v by (-v) , (-v) . Inserting the gauge fixed forms of X, P, g (4.8,4.9) into the action (4.1) reduces it to the massless spinning particle action (3.2). Furthermore, inserting these X, P, g into the expression for the current in (4.5) gives the conserved SU(2, 2) charges J (see Eqs.(5.9,5.20)) which have the significance of the hidden conformal symmetry of the gauge fixed action (3.2). This hidden symmetry is far from obvious in the form (3.2), but it is straightforward to derive from the 2T-physics action as we have just outlined.
Partial or full gauge fixings of (4.1) similar to (4.8,4.9) produce the actions, the hidden SU(2, 2) symmetry, and the twistor transforms with spin of all the systems listed in section (I). These were discussed for h = 0 in [6] , and we have now shown how they generalize to any spin h = 0, with further details below. It is revealing, for example, to realize that the massive spinning particle has a hidden SU(2, 2) "mass-deformed conformal symmetry", including spin, not known before, and that its action can be reached by gauge fixing the action (4.1), or by a twistor transform from (2.3). The same remarks applied to all the other systems listed in section (I) are equally revealing. For more information see our related paper [1] .
Through the gauge (4.8,4.9), the twistor transform (3.1), and the massless particle action (3.2), we have constructed a bridge between the manifestly SU(2, 2) invariant twistor action (2.3) for any spin and the 2T-physics action (4.1) for any spin. This bridge will be made much more transparent in the following sections by building the general twistor transform.
V. 2T-PHYSICS ACTION WITH X M , P M , V A , V A IN 4+2 DIMENSIONS
We have hinted above that there is an intimate relation between the 2T-physics action
, R) constraints X i • X j = X 2 = P 2 = X • P = 0 [6][7] X M = ( + ′ 1 , -′ 0 , + 0, - 0 , i 0), P M = ( + ′ 0 , -′ 0 , + 1, - 0 , i 0). (5.1)
This completely eliminates all phase space degrees of freedom. We are left with the gauge
fixed action S h = dτ T r 1 2 (D τ g) g -1 Γ -′ - 0 0 0 -2h à , where (iL) → 1 2 Γ -′ -L + ′ -′ , and L + ′ -′ = 1.
Due to the many zero entries in the 4×4 matrix Γ -′ -[6], only one column from g in the form Z A Z 5 and one row from g -1 in the form ZA , -Z5 can contribute in the trace, and therefore the action becomes
S h = dτ i ZA ŻA -i Z5 Ż5 + Ã Z5 Z 5 -2h .
Here Z5 Ż5 drops out as a total derivative since the magnitude of the complex number Z 5 is a constant Z5 Z 5 = 2h. Furthermore, we must take into account ZA Z A -Z5 Z 5 = 0 which is an off-diagonal entry in the matrix equation g -1 g = 1. Then we see that the 2T-physics action (4.1) reduces to the twistor action (2.3) with the gauge choice (5.1) 5 .
Next let us gauge fix the 2T-physics action (4.1) to a manifestly SU(2, 2) =SO (4, 2) invariant version in flat 4+2 dimensions, in terms of the phase space & spin degrees of freedom X M , P M , V A , V A . For this we use the [SU(2, 2)×U( 1 )] lef t symmetry to gauge fix g gauge fix:
g → t (V ) ∈ SU(2, 3) [SU(2, 2) × U(1)] lef t (5.2)
The coset element t (V ) is parameterized by the SU(2, 2) spinor V and its conjugate V = V † η 2,2 and given by the 5×5 SU(2, 3) matrix foot_2
t (V ) = 1 -2hV V -1/2 0 0 1 -2h V V -1/2 1 √ 2hV √ 2h V 1 . (5.3)
The factor 2h is inserted for a convenient normalization of V. Note that the first matrix commutes with the second one, so it can be written in either order. The inverse of the group
element is t -1 (V ) = (η 2,3 ) t † (η 2,3 ) -1 = t (-V ) , as can be checked explicitly t (V ) t (-V ) = 1.
Inserting this gauge in (4.1) the action becomes
S h = dτ Ẋ • P - 1 2 A ij X i • X j - 1 2 Ω M N L M N -2h à V LV 1 -2h V V -1 (5.4) = dτ 1 2 ε ij Dτ X M i X N j η M N -2h à V LV 1 -2h V V -1 (5.5)
where Dτ
X M i = ∂ τ X M i -A j i X M j -Ω M N X iN (5.6)
is a covariant derivative for local Sp(2, R) as well as local SU(2, 2) =SO(4, 2) but with a composite SO(4, 2) connection Ω M N (V (τ )) given conveniently in the following forms
1 2 Ω M N Γ M N = (i∂ τ t) t -1 SU (2,2) , or 1 2 Ω M N L M N = -T r (i∂ τ t) t -1 L 0 0 0 . (5.7)
Thus, Ω is the SU(2, 2) projection of the SU(2, 3) Cartan connection and given explicitly as The action (5.4,5.5) is manifestly invariant under global SU(2, 2) =SO(4, 2) rotations, and under local U (1) phase transformations applied on V A , V A . The conserved global symmetry currents J and J 0 can be derived either directly from (5.4) by using Noether's theorem, or by inserting the gauge fixed form of g → t (V ) into Eq.(4.5
1 2 Ω M N Γ M N = 2h V -V V V V V V -V V - V V V V V √ 1 -2h V V 1 + √ 1 -2h V V + h V V V V - 1 4 V V -V V 1 -2h V V (5.
) 7 J (2,3) = t -1 L 0 0 0 t J = 1 √ 1 -2hV V L 1 √ 1 -2hV V - 1 4 J 0 , J 0 = 2h V LV 1 -2h V V
(5.9)
j A = √ 2h 1 √ 1 -2hV V LV 1 √ 1 -2h V V (5.10)
According to the equation of motion for à that follows from the action (5.4) we must have the following constraint (this means U(1) gauge invariant physical sector)
V LV 1 -2h V V = 1. (5.11)
Therefore, in the physical sector the conserved [SU(2, 2)×U( 1 )] right charges take the form physical sector:
J 0 = 2h, J = 1 √ 1 -2hV V L 1 √ 1 -2hV V - h 2 . ( 5
A K A = X i • Γκ i (τ ) A = X M Γ M κ 1 A + P M Γ M κ 2 A , (5.13)
with κ iA (τ ) two arbitrary local spinors 8 . Now that g has been gauge fixed g → t (V ), the kappa transformation must be taken as the naive kappa transformation on g followed by a
[SU(2, 2) ×U(1)] lef t gauge transformation which restores the gauge fixed form of t (V )
t (V ) → t (V ′ ) = exp -ω 0 0 T r (ω) exp 0 K K 0 t (V ) (5.14)
The SU(2, 2) part of the restoring gauge transformation must also be applied on X M , P M .
Performing these steps we find the infinitesimal version of this transformation [22]
δ κ V = 1 √ 1 -2hV V K 1 √ 1 -2h V V , δ κ X M i = ω M N X iN , δ κ A ij = see below, ( 5
.15) 7 In the high spin version ( Ã = 0) the conserved charges include j A as part of SU(2, 3) R global symmetry.
It is then also convenient to rescale √ 2hV → V in Eqs.(5.3-5.10) to eliminate an irrelevant constant. 8 In this special form only 3 out of the 4 components of K A are effectively independent gauge parameters.
This can be seen easily in the special frame for X M , P M given in Eq.(5.1).
where ω M N (K, V ) has the same form as Ω M N in Eq.(5.8) but with V replaced by the δ κ V given above. The covariant derivative Dτ X M i in Eq.(5.6) is covariant under the local SU(2, 2) transformation with parameter ω M N (K, V ) (this is best seen from the projected Cartan connection form Ω = [(i∂ τ t) t -1 ] SU (2,2) ). Therefore, the kappa transformations (5.15) inserted in (5.5) give
δ κ S h = dτ - 1 2 δ κ A ij X i • X j + iT r (D τ t) t -1 0 -KL LK 0 .
(5.16)
In computing the second term the derivative terms that contain ∂ τ K have dropped out in the trace. Using Eq.( 5 .13) we see that
LK = 1 4i ε li X M l X N i X L j Γ M N Γ L κ j (5.17) = 1 4i ε li X M l X N i X L j (Γ M N L + η N L Γ M -η M L Γ N ) κ l (5.18) = 1 2i ε li X i • X j X l • Γκ j (5.19)
The completely antisymmetric X M i X N j X L l Γ M N L term in the second line vanishes since i, j, l can only take two values. The crucial observation is that the remaining term in LK is proportional to the dot products X i • X j . Therefore the second term in (5. 16 ) is cancelled by the first term by choosing the appropriate δ κ A ij in Eq.(5.16), thus establishing the kappa symmetry.
The local kappa transformations (5.15) are also a symmetry of the global SU(2, 3) R charges δ κ J = δ κ J 0 = δ κ j A = 0 provided the constraints X i • X j = 0 are used. Hence these charges are kappa invariant in the physical sector.
We have established the global SO(4, 2) and local Sp(2, R) × (3/4 kappa)×U(1) symmetries of the phase space action (5.4) in 4+2 dimensions. From it we can derive all of the phase space actions of the systems listed in section (I) by making various gauge choices for the local Sp(2, R) × (3/4 kappa)×U(1) symmetries. This was demonstrated for the spinless case h = 0 in [6] . The gauge choices for X M , P M discussed in [6] now need to be supplemented with gauge choices for V A , V A by using the kappa×U(1) local symmetries.
Here we demonstrate the gauge fixing described above for the massless particle of any spin h. The kappa symmetry effectively has 3 complex gauge parameters as explained in footnote (8) . If the kappa gauge is fixed by using two of its parameters we reach the following forms For other examples of gauge fixing that generates some of the systems in the list of section (I) see our related paper [1] .
V A → v α 0 , V A → (0 vα ) , V V → 0, 1 -2hV V -1/2 → 1 0 hvv 1 . ( 5
The various formulations of spinning particles described above all contain gauge degrees of freedom of various kinds. However, they all have the global symmetry SU(2, 2)=SO(4, 2)
whose conserved charges J B A are gauge invariant in all the formulations. The most symmetric 2T-physics version gave the J B A as embedded in SU(2, 3) R in the SU(2, 2) projected form in Eq.(4.5)
J = g -1 L 0 0 0 g SU (2,2) . (6.1)
Since this is gauge invariant, when gauge fixed, it must agree with the Noether charges computed in any version of the theory. So we can equate the general phase space version of Eq.(5.9) with the twistor version that follows from the Noether currents of (2.3) as follows
J = Z (h) Z(h) - 1 4 T r Z (h) Z(h) = 1 √ 1 -2hV V L 1 √ 1 -2hV V - 1 4 J 0 (6.2)
The trace corresponds to the U(1) charge J 0 = T r Z (h) Z(h) , so
J + 1 4 J 0 = Z (h) Z(h) = 1 √ 1 -2hV V L 1 √ 1 -2hV V . (6.3)
In the case of h = 0 this becomes
Z (0) Z(0) = L. (6.4)
Therefore the equality (6.3) is solved up to an irrelevant phase by
Z (h) = 1 √ 1 -2hV V Z (0) . (6.5)
By inserting (6.4) into the constraint (5.11) we learn a new form of the constraints
V Z (0) = 1 -2h V V , V Z (h) = 1. (6.6)
In turn, this implies
Z (0) = LV √ 1 -2h V V (6.7)
which is consistent 9 with Z (0) Z(0) = L , and its vanishing trace Z(0) Z (0) = 0 since LL = 0 (due to X 2 = P 2 = X • P = 0). Putting it all together we then have
Z (h) = 1 √ 1 -2hV V LV 1 √ 1 -2h V V = J + 1 4 J 0 V. (6.8)
We note that this Z (h) is proportional to the non-conserved coset part of the SU(2, 3) charges
J 2,3 , that is j A = √ J 0 Z (h)
given in Eqs.(4.5,4.6) or (5.10), when g and L are replaced by their gauge fixed forms, and use the constraint 10 J 0 = 2h.
The key for the general twistor transform for any spin is Eq.(6.5), or equivalently (6.8).
The general twistor transform between Z (0) and X M , P M which satisfies Z (0) Z(0) = L is already given in [6] as
Z (0) = µ (0) λ (0) , µ (0) α = -i X µ X + ′ σµ λ (0) α , λ (0) α λ(0) β = X + P µ -X µ P + (σ µ ) α β . (6.9)
Note that (X + P µ -X µ P + ) is compatible with the requirement that any SL(2, C) vector constructed as λ (0) α λ(0) β must be lightlike. This property is satisfied thanks to the Sp(2, R) constraints X 2 = P 2 = X • P = 0 in 4+2 dimensions, thus allowing a particle of any mass in the 3 + 1 subspace (since P µ P µ is not restricted to be lightlike). Besides satisfying Z (0) Z(0) = L, this Z (0) also satisfies Z(0) Z (0) = 0, as well as the canonical properties of twistors. Namely, Z (0) has the property [6] dτ Z(0) ∂ τ Z (0) = dτ ẊM P M .
(6.10)
From here, by gauge fixing the Sp(2, R) gauge symmetry, we obtain the twistor transforms for all the systems listed in section (I) for h = 0 directly from Eq.(6.9), as demonstrated in [6] . All of that is now generalized at once to any spin h through Eq.(6.5). Hence (6.5) together with (6.9) tell us how to construct explicitly the general twistor Z (h)
A in terms 9 To see this, we note that Eqs.(6.4,6.6) lead to 10 For the high spin version ( Ã = 0) we don't use the constraint. Instead, we use
LV V L 1-2h V V = Z (0) Z(0) V V Z (0) Z(0) 1-2h V V = Z (0) Z(0) = L.
Z (h) = 1 √ 1-2hV
V Z (0) only in its form (6.5), and note that, after using Eq.(6.4), the j A in Eq.(5.10) takes the form
j A = √ J 0 Z (h) with √ J 0 = Z0 V √ 2h √ 1-2h V V
, and it is possible to rescale h away everywhere √ 2hV → V.
of spin & phase space degrees of freedom X M , P M , V A , V A . Then the Sp(2, R) and kappa gauge symmetries that act on X M , P M , V A , V A can be gauge fixed for any spin h, to give the specific twistor transform for any of the systems under consideration.
We have already seen in Eq.( 6 .2) that the twistor transform (6.5) relates the conserved SU(2, 2) charges in twistor and phase space versions. Let us now verify that (6.5) provides the transformation between the twistor action (2.3) and the spin & phase space action (5.4).
We compute the canonical structure as follows
dτ Z(h) ∂ τ Z (h) = dτ Z(0) 1 √ 1 -2hV V ∂ τ 1 √ 1 -2hV V Z (0) (6.11) = dτ Z(0) 1 √ 1-2hV V ∂ τ 1 √ 1-2hV V Z (0) + Z(0) 1 1-2hV V ∂ τ Z (0) (6.12) = dτ Ẋ • P + T r (i∂ τ t) t -1 L 0 0 0 (6.13)
The last form is the canonical structure of spin & phase space as given in (5.4). To prove this result we used Eq.(6.10), footnote (6) , and the other properties of Z (0) including Eqs.(6.4-6.7), as well as the constraints X 2 = P 2 = X • P = 0, and dropped some total derivatives. This proves that the canonical properties of Z (h) determine the canonical properties of spin & phase space degrees of freedom and vice versa.
Then, including the terms that impose the constraints, the twistor action (2.3) and the phase space action (5.4) are equivalent. Of course, this is expected since they are both gauge fixed versions of the master action (4.1), but is useful to establish it also directly via the general twistor transform given in Eq.(6.5).
In this section we derive the quantum algebra of the gauge invariant observables J B A and J 0 which are the conserved charges of [SU(2, 2) ×U(1)] R . Since these are gauge invariant symmetry currents they govern the system in any of its gauge fixed versions, including in any of its versions listed in section (I). From the quantum algebra we deduce the constraints among the physical observables J B A ,J 0 and quantize the theory covariantly. Among other things, we compute the Casimir eigenvalues of the unitary irreducible representation of SU(2, 2) which classifies the physical states in any of the gauge fixed version of the theory (with the different 1T-physics interpretations listed in section (I)).
The simplest way to quantize the theory is to use the twistor variables, and from them compute the gauge invariant properties that apply in any gauge fixed version. We will apply the covariant quantization approach, which means that the constraint due to the U(1) gauge symmetry will be applied on states. Since the quantum variables will generally not satisfy the constraints, we will call the quantum twistors in this section Z A , ZA to distinguish them from the classical Z (h) A , Z(h)A of the previous sections that were constrained at the classical level. So the formalism in this section can also be applied to the high spin theories (discussed in several footnotes up to this point in the paper) by ignoring the constraint on the states.
According to the twistor action (2.3) Z A and i ZA (or equivalently λ α and iμ α ) are canonical conjugates. Therefore the quantum rules (equivalent to spin & phase space quantum rules) are The operator Ĵ0 has non-trivial commutation relations with Z A , Z A which follow from the basic commutation rules above A at the quantum level as follows
Z A , ZB = δ B A . ( 7
Ĵ0 , Z A = -Z A , Ĵ0 , Z A = Z A . ( 7
J B A = Z A ZB - 1 4 T r Z Z δ B A = Z Z - Ĵ0 + 2 4 B A . (7.5)
In this expression the order of the quantum operators matters and gives rise to the shift J 0 → Ĵ0 + 2 in contrast to the corresponding classical expression. The commutation rules among the generators J B A and the Z A , ZA are computed from the basic commutators (7.1),
J B A , Z C = -δ B C Z A + 1 4 Z C δ B A , J B A , ZD = δ D A ZB - 1 4 ZD δ B A (7.6) J B A , J D C = δ D A J B C -δ B C J D A , Ĵ0 , J B A = 0. (7.7)
We see from these that the gauge invariant observables J B A satisfy the SU(2, 2) Lie algebra, while the Z A , ZA transform like the quartets 4, 4 of SU (2, 2) . Note that the operator Ĵ0 commutes with the generators J B A , therefore J B A is U(1) gauge invariant, and furthermore Ĵ0 must be a function of the Casimir operators of SU (2, 2) . When Ĵ0 takes the value 2h on physical states, then the Casimir operators also will have eigenvalues on physical states which determine the SU(2, 2) representation in the physical sector.
From the quantum rules (7.3), it is evident that the U(1) generator Ĵ0 can only have integer eigenvalues since it acts like a number of operator. More directly, through Eq. (7.4) it is related to the number operator ZZ. Therefore the theory is consistent at the quantum level (7.2) provided 2h is an integer.
Let us now compute the square of the matrix J B A . By using the form (7.5) we have
(J J ) = Z Z -Ĵ0 +2 4 Z Z -Ĵ0 +2 4 = Z ZZ Z -2 Ĵ0 +2 4 Z Z + Ĵ0 +2 4 2
where we have used
Ĵ0 , Z A ZB = 0. Now we elaborate Z ZZ Z B A = Z A Ĵ0 -2 ZB = Ĵ0 -1 Z A ZB where
we first used (7.4) and then (7.3). Finally we note from (7.5) that
Z A ZB = J B A + Ĵ0 +2 4 δ B A .
Putting these observations together we can rewrite the right hand side of (J J ) in terms of J and Ĵ0 as follows 11
(J J ) = Ĵ0 2 -2 J + 3 16 Ĵ2 0 -4 . ( 7
11 A similar structure at the classical level can be easily computed by squaring the expression for J in Eq.( 6 .2)
and applying the classical constraint J 0 = ZA Z A = 2h. This yields the classical version
J C A J B C = J0 2 J B A + 3 16 J 2 0 δ B A = hJ B A + 3 4 h 2 δ B A
, which is different than the quantum equation (7.8) . Thus, the quadratic Casimir at the classical level is computed as C 2 = 3 4 J 2 0 = 3h 2 which is different than the quantum value in (7.16 ).
A , Ĵ0 which are gauge invariant physical observables. It is a correct equation for all the states in the theory, including those that do not satisfy the U(1) constraint (7.2). We call this the quantum master equation because it will determine completely all the SU(2, 2) properties of the physical states for all the systems listed in section (I) for any spin.
By multiplying the master equation with J and using (7.8) again we can compute J J J .
Using this process repeatedly we find all the powers of the matrix J
(J ) n = α n J + β n , (7.9)
where
α n ( Ĵ0 ) = 1 Ĵ0 -1 3 4 Ĵ0 -2 n - -1 4 Ĵ0 + 2 n , (7.10)
β n ( Ĵ0 ) = 3 16 Ĵ2 0 -4 α n-1 ( Ĵ0 ). (7.11)
Remarkably, these formulae apply to all powers, including negative powers of the matrix J .
Using this result, any function of the matrix J constructed as a Taylor series takes the form
f (J ) = α Ĵ0 J +β Ĵ0 (7.12)
where
α Ĵ0 = 1 Ĵ0 -1 f 3 4 Ĵ0 -2 -f -1 4 Ĵ0 + 2 , (7.13)
β Ĵ0 = 1 Ĵ0 -1 Ĵ0 + 2 4 f 3 4 Ĵ0 -2 + 3 Ĵ0 -2 4 f -1 4 Ĵ0 + 2 . (7.14)
We can compute all the Casimir operators by taking the trace of J n in Eq.(7.9), so we
find 12 C n ( Ĵ0 ) ≡ T r (J ) n = 4β n ( Ĵ0 ) = 3 4 Ĵ2 0 -4 α n-1 ( Ĵ0 ). (7.15)
In particular the quadratic, cubic and quartic Casimir operators of SU(2, 2) =SO(6, 2) are computed at the quantum level as
C 2 ( Ĵ0 ) = 3 4 Ĵ2 0 -4 , C 3 ( Ĵ0 ) = 3 8 Ĵ2 0 -4 Ĵ0 -4 , (7.16)
C 4 ( Ĵ0 ) = 3 64 Ĵ2 0 -4 7 Ĵ2 0 -32 Ĵ0 + 52 . (7.17)
all have the same Casimir eigenvalues C 2 = -3, C 3 = 6, C 4 = -39 4 at the quantum level. Much more elaborate tests of the dualities can be performed both at the classical and quantum levels by computing any function of the gauge invariant J B
A and checking that it has the same value when computed in terms of the spin & phase space of any of the systems listed in section (I). At the quantum level all of these systems have the same Casimir eigenvalues of the C n for a given h. So their spectra must correspond to the same unitary irreducible representation of SU(2, 2) as seen above. But the rest of the labels of the representation correspond to simultaneously commuting operators that include the Hamiltonian.
The Hamiltonian of each system is some operator constructed from the observables J B A , and so are the other simultaneously diagonalizable observables. Therefore, the different systems are related to one another by unitary transformations that sends one Hamiltonian to another, but staying within the same representation. These unitary transformations are the quantum versions of the gauge transformations of Eq.(4.7), and so they are the duality transformations at the quantum level. In particular the twistor transform applied to any of the systems is one of those duality transformations. By applying the twistor transforms we can map the Hilbert space of one system to another, and then compute any function of the gauge invariant J B A between dually related states of different systems. The prediction is that all such computations within different systems must give the same result.
A is expressed in terms of rather different phase space and spin degrees of freedom in each dynamical system with a different Hamiltonian, this predicted duality is remarkable. 1T-physics simply is not equipped to explain why or for which systems there are such dualities, although it can be used to check it. The origin as well as the proof of the duality is the unification of the systems in the form of the 2T-physics master action of Eq.(4.1) in 4+2 dimensions. The existence of the dualities, which can laboriously be checked using 1T-physics, is the evidence that the underlying spacetime is more beneficially understood as being a spacetime in 4+2 dimensions.
We have established a master equation for physical observables J at the quantum level. Now, we also want to establish the twistor transform at the quantum level expressed as much as possible in terms of the gauge invariant physical quantum observables J . To this end we write the master equation (7.8) in the form
J - 3 4 Ĵ0 -2 J + 1 4 Ĵ0 + 2 = 0. (8.1)
Recall the quantum equation (7.5) J + Ĵ0 +2 4 = Z Z, so the equation above is equivalent to
J - 3 4 Ĵ0 -2 Z = 0. (8.2)
This is a 4 × 4 matrix eigenvalue equation with operator entries. The general solution is
Z = J + 1 4 Ĵ0 + 2 V (8.3)
where VA is any spinor up to a normalization. This is verified by using the master equation (8.1) which gives J -
3 4 Ĵ0 -2 Z = J -3 4 Ĵ0 -2 J + 1 4 Ĵ0 + 2 V = 0. Not-
ing that the solution (8.3) has the same form as the classical version of the twistor transform in Eq.(6.8), except for the quantum shift J 0 → Ĵ0 + 2, we conclude that the VA introduced above is the quantum version of the V A discussed earlier (up to a possible renormalization 13 ), as belonging to the coset SU(2, 3)
/[SU(2, 2) ×U(1)].
Now VA is a quantum operator whose commutation rules must be compatible with those of Z A , ZA , Ĵ0 and J B A . Its commutation rules with J B A , Ĵ0 are straightforward and fixed uniquely by the SU(2, 2) ×U(1) covariance
Ĵ0 , VA = -VA , Ĵ0 , V A = V A , (8.4)
J B A , VC = -δ B C VA + 1 4 VC δ B A , J B A , V D = δ D A V B - 1 4 V D δ B A . (8.5)
Other quantum properties of VA follow from imposing the quantum property ZZ = Ĵ0 -2 in (7.4). Inserting Z of the form (8.3), using the master equation, and observing the commutation rules (8.4), we obtain
V J + Ĵ0 + 2 4 V = 1. (8.6)
13 The quantum version of V is valid in the whole Hilbert space, not only in the subspace that satisfies the U(1) constraint Ĵ0 → 2h. In particular, in the high spin version, already at the classical level we must take V = V ( √ 2h/ √ J 0 ) and then rescale it V √ 2h → V as described in previous footnotes. So in the full quantum Hilbert space we must take V = √ 2hV ( Ĵ0 + γ) -1/2 (or the rescaled version V √ 2h → V ) with the possibly quantum shifted operator ( Ĵ0 + γ) -1/2 . This is related to (5.11) if we take (5.9) into account by including the quantum shift J 0 → Ĵ0 + 2. Considering (8.3) this equation may also be written as
V Z = Z V = 1. (8.7)
Next we impose Z A , ZB = δ B A to deduce the quantum rules for [ VA , V B ]. After some algebra we learn that the most general form compatible with
Z A , ZB = δ B A is VA , V B = - V V Ĵ0 -1 δ B A + M(J -3 Ĵ0 -2 4 ) + (J -3 Ĵ0 -2 4 ) M B A , (8.8)
where M B A is some complex matrix and M = (η 2,2 ) M † (η 2,2 ) -1 . The matrix M B A could not be determined uniquely because of the 3/4 kappa gauge freedom in the choice of VA itself.
A maximally gauge fixed version of VA corresponds to eliminating 3 of its components V2,3,4 = 0 by using the 3/4 kappa symmetry, leaving only A ≡ V1 = 0. Then we find V 1,2,4 = 0 and V 3 = A † . Let us analyze the quantum properties of this gauge in the context of the formalism above. From Eq.(8.6) we determine A = (J 1 3 ) -1/2 e -iφ , where φ is a phase, and then from Eq.( 8 .3) we find Z A .
Z A = J 1 A + Ĵ0 + 2 4 δ 1 A J 1 3 -1/2 e -iφ , ZA = e iφ J 1 3 -1/2 J A 3 + Ĵ0 + 2 4 δ A 3 . (8.9)
We see that, except for the overall phase, Z A is completely determined in terms of the gauge invariant J B A . We use a set of gamma matrices Γ M given in ( [6] , [11] ) to write
J B A = 1 4i J M N (Γ M N ) B
A as an explicit matrix so that Z A can be written in terms of the 15 SO(4, 2) =SU(2, 2) generators J M N . We find
Z A = 1 2 J 12 + 1 2i J +-+ 1 2i J + ′ -′ + Ĵ0 +2 4 i √ 2 (J +1 + iJ +2 ) J + ′ + i √ 2 J + ′ 1 + iJ + ′ 2 e -iφ √ J + ′ + , (8.10)
and ZA = Z † η 2,2 A . The orders of the operators here are important. The basis M = ± ′ , ±, i with i = 1, 2 corresponds to using the lightcone combinations
X ± ′ = 1 √ 2 X 0 ′ ± X 1 ′ , X ± = 1 √ 2 (X 0 ± X 1 ).
From our setup above, the Z A , ZA in (8.10) are guaranteed to satisfy the twistor commutation rules Z A , ZB = δ B A provided we insure that the VA , V B have the quantum properties given in Eqs. (8.4,8.5,8.8) . These are satisfied provided we take the following non-trivial commutation rules for φ φ, Ĵ0 = i, [φ, J 12 ] = i 2 ⇒ Ĵ0 , e ±iφ = ±e ±iφ , [J 12 , e ±iφ ] = ± 1 2 e ±iφ (8.11) while all other commutators between φ and J M N vanish. Then (8.8) becomes [ VA ,
V B ] = 0, so M B A vanishes in this gauge. Indeed one can check directly that only by using the Lie algebra for the J M N , Ĵ0 and the commutation rules for φ in (8.11) , we obtain Z A , ZB = δ B A , which a remarkable form of the twistor transform at the quantum level.
The expression (8.10) for the twistor is not SU(2, 2) covariant. Of course, this is because we chose a non-covariant gauge for VA . However, the global symmetry SU(2, 2) is still intact since the correct commutation rules between the twistors and J M N or the J B A as given in (7.6,7.7) are built in, and are automatically satisfied. Therefore, despite the lack of manifest covariance, the expression for Z A in (8.10) transforms covariantly as the spinor of SU (2, 2) .
It is now evident that one has many choices of gauges for VA . Once a gauge is picked the procedure outlined above will automatically produce the quantum twistor transform in that gauge, and it will have the correct commutation rules and SU (2, 2) properties at the quantum level. For example, in the SL(2, C) covariant gauge of Eq.(5.20), the quantum twistor transform in terms of J M N is
µ α = 1 4i J µν (σ µν ) αβ v β + 1 2i J + ′ -′ v α, λ α = 1 √ 2 J + ′ µ (σ µ ) α β v β . (8.12)
with the constraint 1 √ 2 vσ µ vJ + ′ µ = 1. (8.13) This gauge for VM covers several of the systems listed in section (I). The spinless case was discussed at the classical level in ( [6] ). The quantum properties of this gauge are discussed in more detail in ( [1] ).
The result for Z A in (8.10) is a quantum twistor transform that relies only on the gauge invariants J B A or equivalently J M N . It generalizes a similar result in [6] that was given at the classical level. In the present case it is quantum and with spin. All the information on spin is included in the generators J M N = L M N + S M N . There are other ways of describing spinning particles. For example, one can start with a 2T-physics action that uses fermions ψ M (τ ) [27] instead of our bosonic variables V A (τ ) . Since we only use the gauge invariant J M N , our quantum twistor transform (8.3) applies to all such descriptions of spinning particles, with an appropriate relation between V and the new spin degrees of freedom. In particular in the gauge fixed form of V that yields (8.10) there is no need to seek a relation between V and the other spin degrees of freedom. Therefore, in the form (8.10), if the J M N are produced with the correct quantum algebra SU(2, 2) =SO(4, 2) in any theory, (for example bosonic spinors, or fermions ψ M , or the list of systems in section (I), or any other) then our formula (8.3) gives the twistor transform for the corresponding degrees of freedom of that theory.
Those degrees of freedom appear as the building blocks of J M N . So, the machinery proposed in this section contains some very powerful tools.
The 2T-physics action (4.1) offered the group SU(2, 3) as the most symmetric unifying property of the spinning particles for all the systems listed in section (I), including twistors.
Here we discuss how this fundamental underlying structure governs and simplifies the quantum theory.
We examine the SU(2, 3) charges J B A , Ĵ0 , j A , jA given in (4.5,5.9,5.10). Since these are gauge invariant under all the gauge symmetries (4.7) they are physical quantities that should have the properties of the Lie algebra 14 of SU (2, 3) in all the systems listed in section (I).
Using covariant quantization we construct the quantum version of all these charges in terms of twistors. By using the general quantum twistor transform of the previous section, these charges can also be written in terms of the quantized spin and phase space degrees of freedom of any of the relevant systems.
The twistor expressions for Ĵ0 , J B A are already given in Eqs.(7.2,7.5)
Ĵ0 = 1 2 Z A ZA + ZA Z A , J B A = Z A ZB - Ĵ0 + 2 4 δ B A . (9.1)
We have seen that at the classical level (j A ) classical = √ J 0 Z A and now we must figure out 14 Even when j A is not a conserved charge when the U(1) constraint is imposed, its commutation rules are still the same in the covariant quantization approach, independently than the constraint.
the quantum version j A = Ĵ0 + αZ A that gives the correct SU(2, 3) closure property
j A , jB = J B A + 5 4 Ĵ0 δ B A . (9.2)
The coefficient 5 4 is determined by consistency with the Jacobi identity j A , jB , j C + jB , j C , j A + [j C , j A , ] , jB = 0, and the requirement that the commutators of j A with J B A , Ĵ0 be just like those of Z A given in Eqs.(7.6,7.7), as part of the SU(2, 3) Lie algebra. So we carry out the computation in Eq.(9.2) as follows
j A , jB = Ĵ0 + αZ A ZB Ĵ0 + α -ZB Ĵ0 + α Ĵ0 + αZ A (9.3) = Ĵ0 + α Z A ZB -Ĵ0 + α -1 ZB Z A (9.4) = Ĵ0 + α -1 Z A , ZB + Z A ZB (9.5) = δ B A Ĵ0 + α -1 + Ĵ0 + 2 4 + J B A (9.6)
To get (9.4) we have used the properties
Z A f Ĵ0 = f Ĵ0 + 1 Z A and ZB f Ĵ0 = f Ĵ0 -1 ZB for any function f Ĵ0 . These follow from the commutator Ĵ0 , Z A = -Z A
written in the form Z A Ĵ0 = Ĵ0 + 1 Z A which is used repeatedly, and similarly for ZB . To get (9.6) we have used Z A , ZB = δ B A and then used the definitions (9.1). By comparing (9.6) and (9.2) we fix α = 1/2. Hence the correct quantum version of j A is
j A = Ĵ0 + 1 2 Z A = Z A Ĵ0 - 1 2 . (9.7)
The second form is obtained by using
Z A f Ĵ0 = f Ĵ0 + 1 Z A .
Note the following properties of the j A , jA
jA j A = Ĵ0 - 1 2 ZZ Ĵ0 - 1 2 = Ĵ0 - 1 2 Ĵ0 -2 (9.8) j A jB = Ĵ0 + 1 2 Z A ZB Ĵ0 + 1 2 = Ĵ0 + 1 2 J + 1 4 Ĵ0 + 2 (9.9)
which will be used below.
With the above arguments we have now constructed the quantum version of the SU(2, 3) charges written as a 5 × 5 traceless matrix
Ĵ2,3 = g -1 L 0 0 0 g quantum = J + 1 4 Ĵ0 - j j -Ĵ0 (9.10) = Z A ZB -1 2 δ B A Ĵ0 + 1 2 Z A -ZB Ĵ0 + 1 2 -Ĵ0 , (9.11)
with Ĵ0 , J given in Eq.(9.1).
At the classical level, the square of the matrix J 2,3 vanishes since L 2 = 0 as follows
(J 2,3 ) 2 classical = g -1 L 0 0 0 g g -1 L 0 0 0 g = g -1 L 2 0 0 0 g = 0. (9.12)
At the quantum level we find the following non-zero result which is SU(2, 3) covariant Ĵ2,3
2 = Z Z -1 2 Ĵ0 + 1 2 Z -Z Ĵ0 + 1 2 -Ĵ0 2 (9.13) = - 5 2 Ĵ2,3 -1. ( 9
Written out in terms of the charges, Eq.(9.14) becomes
J + 1 4 Ĵ0 - j j -Ĵ0 2 = - 5 2 J + 1 4 Ĵ0 - j j -Ĵ0 -1. (9.16)
Collecting terms in each block we obtain the following relations among the gauge invariant charges J , Ĵ0 , j, j
J + 1 4 Ĵ0 2 -j j + 5 2 J + 1 4 Ĵ0 + 1 = 0, (9.17)
J + 1 4 Ĵ0 j -j Ĵ0 + 5 2 j = 0, (9.18)
-jj + Ĵ0 2 - 5 2 Ĵ0 + 1 = 0. (9.19)
Combined with the information in Eq.(9.9) the first equation is equivalent to the master quantum equation (7.8). After using j Ĵ0 = Ĵ0 j + j, the second equation is equivalent to the eigenvalue equation (8.2) whose solution is the quantum twistor transform (8.3). The third equation is identical to (9.8).
Hence the SU(2, 3) quantum property Ĵ2,3 This is a remarkable simple unifying description of a diverse set of spinning systems, that shows the existence of the sophisticated higher structure SU(2, 3) for which there was no clue whatsoever from the point of view of 1T-physics.
One can consider several paths that generalizes our discussion, including the following.
• It is straightforward to generalize our theory by replacing SU(2, 3) with the supergroup SU(2, (2 + n) |N) . This generalizes the spinor V A to V a A where a labels the fundamental representation of the supergroup SU(n|N) . The case of N = 0 and n = 1 is what we discussed in this paper. The case of n = 0 and any N relates to the superparticle with N supersymmetries (and all its duals) discussed in [22] and in [6] [7] . The massless particle gauge is investigated in [17] , but the other cases listed metries insure that the theory has no negative norm states. In the massless particle gauge, this model corresponds to supersymmetrizing spinning particles rather than supersymmetrizing the zero spin particle. The usual R-symmetry group in SUSY is replaced here by SU(n|N) ×U (1) . For all these cases with non-zero n, N, the 2Tphysics and twistor formalisms unify a large class of new 1T-physics systems and establishes dualities among them.
• One can generalize our discussion in 4+2 dimensions, including the previous paragraph, to higher dimensions. The starting point in 4+2 dimensions was SU(2, 2) =SO (4, 2) embedded in g =SU(2, 3) . For higher dimensions we start from SO(d, 2) and seek a group or supergroup that contains SO(d, 2) in the spinor representation. For example for 6+2 dimensions, the starting point is the 8×8 spinor version of SO(8 * ) =SO (6, 2) embedded in g =SO(9 * ) =SO (6, 3) or g =SO(10 * ) =SO (6, 4) . The spinor variables in 6+2 dimensions V A will then be the spinor of SO(8 * ) =SO(6, 2) parametrizing the coset SO(9 * ) /SO(8 * ) (real spinor) or SO(10 * ) /SO(8 * ) ×SO(2) (complex spinor). This can be supersymmetrized. The pure superparticle version of this program for various dimensions is discussed in [6] [7] , where all the relevant supergroups are classified.
That discussion can now be taken further by including bosonic variables embedded in a supergroup as just outlined in the previous item. As explained before [6] [7], it must be mentioned that when d + 2 exceeds 6 + 2 it seems that we need to include also brane degrees of freedom in addition to particle degrees of freedom. Also, even in lower dimensions, if the group element g belongs to a group larger than the minimal one [6] [7] , extra degrees of freedom will appear.
• The methods in this paper overlap with those in [28] where a similar master quantum equation technique for the supergroup SU(2, 2|4) was used to describe the spectrum of type-IIB supergravity compactified on AdS 5 ×S 5 . So our methods have a direct bearing on M theory. In the case of [28] the matrix insertion L 0 0 0 in the 2T-physics action was generalized to L (4,2) 0 0 L (6,0) to describe a theory in 10+2 dimensions. This approach to higher dimensions can avoid the brane degrees of freedom and concentrate only on the particle limit. Similar generalizations can be used with our present better develped methods and richer set of groups mentioned above to explore various corners of M theory.
• One of the projects in 2T-physics is to take advantage of its flexible gauge fixing mechanisms in the context of 2T-physics field theory. Applying this concept to the 2T-physics version of the Standard Model [10] will generate duals to the Standard Model in 3+1 dimensions. The study of the duals could provide some non-perturbative or other physical information on the usual Standard Model. This program is about to be launched in the near future [29] . Applying the twistor techniques developed here to 2T-physics field theory should shed light on how to connect the Standard Model with a twistor version. This could lead to further insight and to new computational techniques for the types of twistor computations that proved to be useful in QCD [12] [13] .
• Our new models and methods can also be applied to the study of high spin theories by generalizing the techniques in [14] which are closely related to 2T-physics. The high spin version of our model has been discussed in many of the footnotes, and can be supersymmetrized and written in higher dimensions as outlined above in this section. The new ingredient from the 2T point of view is the bosonic spinor V A and the higher symmetry, such as SU(2, 3) and its generalizations in higher dimensions or with supersymmetry. The massless particle gauge of our theory in 3+1 dimensions coincides with the high spin studies in [15] - [18] . Our theory of course applies broadly to all the spinning systems that emerge in the other gauges, not only to massless particles.
The last three sections on the quantum theory discussed in this paper would apply also in the high spin version of our theory. The more direct 4+2 higher dimensional quantization of high spin theories including the spinor V A (or its generalizations V a A ) is obtained from our SU(2, 3) quantum formalism in the last section.
• One can consider applying the bosonic spinor that worked well in the particle case to strings and branes. This may provide new string backgrounds with spin degrees of freedom other than the familiar Neveu-Schwarz or Green-Schwarz formulations that involve fermions.
More details and applications of our theory will be presented in a companion paper [1] .
We gratefully acknowledge discussions with S-H. Chen, Y-C. Kuo, and G. Quelin. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "A generalized twistor transform for spinning particles in 3+1 dimensions is constructed that beautifully unifies many types of spinning systems by mapping them to the same twistor , thus predicting an infinite set of duality relations among spinning systems with different Hamiltonians. Usual 1T-physics is not equipped to explain the duality relationships"
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "and unification between these systems. We use 2T-physics in 4+2 dimensions to uncover new properties of twistors, and expect that our approach will prove to be useful for practical applications as well as for a deeper understanding of fundamental physics. Unexpected structures for a new description of spinning particles emerge. A unifying symmetry SU (2, 3) that includes conformal symmetry SU(2, 2) =SO (4, 2) in the massless case, turns out to be a fundamental property underlying the dualities of a large set of spinning systems, including those that occur in high spin theories. This may lead to new forms of string theory backgrounds as well as to new methods for studying various corners of M theory. In this paper we present the main concepts, and in a companion paper we give other details [1] ."
},
{
"section_type": "OTHER",
"section_title": "I. SPINNING PARTICLES IN 3+1 -BEYOND FREE AND MASSLESS",
"text": "The Penrose twistor transform [2] - [5] brings to the foreground the conformal symmetry SO (4, 2) in the dynamics of massless relativistic particles of any spin in 3 + 1 dimensions.\n\nThe transform relates the phase space and spin degrees of freedom x µ , p µ , s µν to a twistor\n\nZ A = µ α λα\n\nand reformulates the dynamics in terms of twistors instead of phase space. The twistor Z A is made up of a pair of SL(2, C) spinors µ α, λ α , α, α = 1, 2, and is regarded as the 4 components A = 1, 2, 3, 4 of the Weyl spinor of SO(4, 2) =SU (2, 2) .\n\nThe well known twistor transform for a spinning massless particle is [5]\n\nµ α = -i (x + iȳ) αβ λ β , λ α λ β = p α β , (1.1)\n\nwhere (x + iȳ) αβ = 1 √ 2 (x µ + iy µ ) (σ µ ) αβ , and p α β = 1 √ 2 p µ (σ µ ) α β , while σ µ = (1, σ) , σµ = (-1, σ) are Pauli matrices. x µ + iy µ is a complexification of spacetime [2] . The helicity h of the particle is determined by p • y = h. The spin tensor is given by s µν = ε µνρσ y ρ p σ , and it leads to 1 2 s µν s µν = h 2 . The Pauli-Lubanski vector is proportional to the momentum W µ = 1 2 ε µνρσ s νρ p σ = (y • p) p µ -p 2 y µ = hp µ , appropriate for a massless particle of helicity h. The reformulation of the dynamics in terms of twistors is manifestly SU(2, 2) covariant.\n\nIt was believed that twistors and the SO(4, 2)=SU(2, 2) symmetry, interpreted as conformal symmetry, govern the dynamics of massless particles only, since the momentum p µ of the form p α β = λ α λ β automatically satisfies p µ p µ = 0. However, recent work has shown that the same twistor Z A = µ α λα that describes massless spinless particles (h = 0) also describes an assortment of other spinless particle dynamical systems [6] [7] . These include massive and interacting particles. The mechanism that avoids p µ p µ = 0 [6] [7] is explained following Eq.(6.9) below. The list of systems includes the following examples worked out explicitly in previous publications and in unpublished notes.\n\nThe massless relativistic particle in d = 4 flat Minkowski space.\n\nThe massive relativistic particle in d = 4 flat Minkowski space. The nonrelativistic free massive particle in 3 space dimensions. The nonrelativistic hydrogen atom (i.e. 1/r potential) in 3 space dimensions.\n\nThe harmonic oscillator in 2 space dimensions, with its mass ⇔ an extra dimension.\n\nThe particle on AdS 4 , or on dS 4 .\n\nThe particle on AdS 3 ×S 1 or on R × S 3 .\n\nThe particle on AdS 2 ×S 2 .\n\nThe particle on the Robertson-Walker spacetime.\n\nThe particle on any maximally symmetric space of positive or negative curvature.\n\nThe particle on any of the above spaces modified by any conformal factor.\n\nA related family of other particle systems, including some black hole backgrounds. In this paper we will discuss these for the case of d = 4 with spin (h = 0). It must be emphasized that while the phase spaces (and therefore dynamics, Hamiltonian, etc.) in these systems are different, the twistors µ α, λ α are the same. For example, the massive particle phase space (x µ , p µ ) massive and the one for the massless particle (x µ , p µ ) massless are not the same (x µ , p µ ) , rather they can be obtained from one another by a non-linear transformation for any value of the mass parameter m [6] , and similarly, for all the other spaces mentioned above. However, under such \"duality\" transformations from one system to another, the twistors for all the cases are the same up to an overall phase transformation\n\nµ α, λ α massive = µ α, λ α massless = • • • = µ α, λ α . (1.2)\n\nThis unification also shows that all of these systems share the same SO(4, 2)=SU (2, 2) global symmetry of the twistors. This SU(2, 2) is interpreted as conformal symmetry for the massless particle phase space, but has other meanings as a hidden symmetry of all the other systems in their own phase spaces. Furthermore, in the quantum physical Hilbert space, the symmetry is realized in the same unitary representation of SU (2, 2) , with the same Casimir eigenvalues (see (7.16,7 .17) below), for all the systems listed above.\n\nThe underlying reason for such fantastic looking properties cannot be found in one-time physics (1T-physics) in 3+1 dimensions, but is explained in two-time physics (2T-physics) [8] as being due to a local Sp(2, R) symmetry. The Sp(2, R) symmetry which acts in phase space makes position and momentum indistinguishable at any instant and requires one extra space and one extra time dimensions to implement it, thus showing that the unification relies on an underlying spacetime in 4+2 dimensions. It was realized sometime ago that in 2Tphysics twistors emerge as a gauge choice [9] , while the other systems are also gauge choices of the same theory in 4+2 dimensions. The 4+2 phase space can be gauge fixed to many 3+1 phase spaces that are distinguishable from the point of view of 1T-physics, without any Kaluza-Klein remnants, and this accounts for the different Hamiltonians that have a duality relationship with one another. We will take advantage of the properties of 2T-physics to build the general twistor transform that relates these systems including spin.\n\nGiven that the field theoretic formulation of 2T-physics in 4+2 dimensions yields the Standard Model of Particles and Forces in 3+1 dimensions as a gauge choice [10] , including spacetime supersymmetry [11] , and given that twistors have simplified QCD computations [12] [13], we expect that our twistor methods will find useful applications."
},
{
"section_type": "OTHER",
"section_title": "II. TWISTOR LAGRANGIAN",
"text": "The Penrose twistor description of massless spinning particles requires that the pairs µ α, i λ α or their complex conjugates (λ α , iμ α ) be canonical conjugates and satisfy the helicity constraint given by ZA Z A = λ αµ α + μα λ α = 2h.\n\n(2.1) Indeed, Eq.(1.1) satisfies this property provided y • p = h. Here we have defined the 4 of SU(2, 2) as the contravariant twistor\n\nZA ≡ Z † η 2,2 A = λ α μα , η 2,2 = 0 1 1 0 = SU (2, 2) metric. (2.\n\n2)\n\nThe canonical structure, along with the constraint ZA Z A = 2h follows from the following worldline action for twistors\n\nS h = dτ i ZA D τ Z A -2h à , D τ Z A ≡ ∂Z A ∂τ -i ÃZ A . (2.3)\n\nIn the case of h = 0 it was shown that this action emerges as a gauge choice of a more general action in 2T-physics [6][7] . Later in the paper, in Eq.( 4 .1) we give the h = 0 2T-physics action from which (2.3) is derived as a gauge choice. The derivative part of this action gives the canonical structure S 0 = dτ i ZA ∂ τ Z A = i dτ λ α∂ τ µ α + μα ∂ τ λ α that requires µ α, i λ α or their complex conjugates (λ α , iμ α ) to be canonical conjugates.\n\nThe 1-form Ãdτ is a U(1) gauge field on the worldline, D τ Z A is the U(1) gauge covariant derivative that satisfies\n\nδ ε D τ Z A = iε D τ Z A for δ ε à = ∂ε/∂τ and δ ε Z A = iεZ A .\n\nThe term 2h à is gauge invariant since it transforms as a total derivative under the infinitesimal gauge transformation. 2h à was introduced in [6] [7] as being an integral part of the twistor formulation of the spinning particle action.\n\nOur aim is to show that this action describes not only massless spinning particles, but also all of the other particle systems listed above with spin. This will be done by constructing the twistor transform from Z A to the phase space and spin degrees of freedom of these systems, and claiming the unification of dynamics via the generalized twistor transform.\n\nThis generalizes the work of [6][7] which was done for the h = 0 case of the action in (2.3).\n\nWe will use 2T-physics as a tool to construct the general twistor transform, so this unification is equivalent to the unification achieved in 2T-physics."
},
{
"section_type": "OTHER",
"section_title": "III. MASSLESS PARTICLE WITH ANY SPIN IN 3+1 DIMENSIONS",
"text": "In our quest for the general twistor transform with spin, we first discuss an alternative to the well known twistor transform of Eq.(1.1). Instead of the y µ (τ ) that appears in the complexified spacetime x µ + iy µ we introduce an SL(2, C) bosonic 2 spinor v α (τ ) and its complex conjugate vα (τ ) , and write the general vector y µ in the matrix form as y αβ = hv α vβ + ωp αβ , where ω (τ ) is an arbitrary gauge freedom that drops out. Then the helicity condition y • p = h takes the form vpv = 1. Furthermore, we can write λ α = p α β v β since this automatically satisfies λ α λ β = p α β when p 2 = (vpv -1) = 0 are true. With this choice of variables, the Penrose transform of Eq.(1.1) takes the new form\n\nλ α = (pv) α , µ α = [(-ixp + h) v] α , p 2 = (vpv -1) = 0, (3.1)\n\nwhere the last equation is a set of constraints on the degrees of freedom x µ , p µ , v α, vα .\n\nIf we insert the twistor transform (3.1) into the action (2.3), the twistor action turns into the action for the phase space and spin degrees of freedom x µ , p µ , v α, vα\n\nS h = dτ ẋµ p µ - e 2 p 2 + ih (vp) D τ v -D τ v (pv) -2h à . (3.2)\n\nwhere D τ v = v -i Ãv is the U(1) gauge covariant derivative and we have included the\n\nLagrange multiplier e to impose p 2 = 0 when we don't refer to twistors. The equation of motion for à imposes the second constraint vpv -1 = 0 that implies U(1) gauge invariance 3 .\n\nFrom the global Lorentz symmetry of (3.2), the Lorentz generator is computed via Noether's theorem J µν = x µ p ν -x ν p µ + s µν , with s µν = i 2 hv (pσ µν + σ µν p) v. The helic-2 This is similar to the fermionic case in [5] . The bosonic spinor v can describe any spin h.\n\n3 If this action is taken without the U(1) constraint à = 0 , then the excitations in the v sector describe an infinite tower of massless states with all helicities from zero to infinity (here we rescale\n\n√ 2hv → v) S all spins = dτ ẋµ p µ - e 2 p 2 + i 2 vp v -vpv (3.3)\n\nThe spectrum coincides with the spectrum of the infinite slope limit of string theory with all helicities 1 2 vpv. This action has a hidden SU(2, 3) symmetry that includes SU(2, 2) conformal symmetry. This is explained in the rest of the paper by the fact that this action is a gauge fixed version of a 2T-physics master action (4.1,5.4) in 4+2 dimensions with manifest SU(2, 3) symmetry. A related approach has been pursued also in [15] - [18] in 3+1 dimensions in the context of only massless particles. Along with the manifestly SU(2, 3) symmetric 2T-physics actions, we are proposing here a unified 2T-physics setting for discussing high spin theories [14] including all the dual versions of the high spin theories related to the spinning physical systems listed in section (I).\n\nity is determined by computing the Pauli-Lubanski vector W µ = 1 2 ε µνλσ s νλ p σ = (hvpv) p µ . The helicity operator hvpv reduces to the constant h in the U(1) gauge invariant sector.\n\nThe action (3.2) gives a description of a massless particle with any helicity h in terms of the SL(2, C) bosonic spinors v, v. We note its similarity to the standard superparticle action [20][21] written in the first order formalism. The difference with the superparticle is that the fermionic spacetime spinor θ α of the superparticle is replaced with the bosonic spacetime spinor v α, and the gauge field à imposes the U(1) gauge symmetry constraint vpv -1 = 0 that restricts the system to a single, but arbitrary helicity state given by h.\n\nJust like the superparticle case, our action has a local kappa symmetry with a bosonic local spinor parameter κ α (τ ), namely\n\nδ κ v α = p αβ κ β , δ κ x µ = ih √ 2 ((δ κ v) σ µ v -vσ µ (δ κ v)) , (3.4)\n\nδ κ p µ = 0, δ κ e = -ih κ (D τ v) -D τ v κ , δ κ à = 0. (3.5)\n\nThese kappa transformations mix the phase space degrees of freedom (x, p) with the spin degrees of freedom v, v. The transformations δ κ x µ , δ κ e are non-linear.\n\nLet us count physical degrees of freedom. By using the kappa and the τ -reparametrization symmetries one can choose the lightcone gauge. From phase space x µ , p µ there remains 3 positions and 3 momentum degrees of freedom. One of the two complex components of v α is set to zero by using the kappa symmetry, so v α = v 0 . The phase of the remaining component is eliminated by choosing the U(1) gauge, and finally its magnitude is fixed by solving the constraint vpv -1 = 0 to obtain v α = (p + ) -1/2 1 0 . Therefore, there are no independent physical degrees of freedom in v. The remaining degrees of freedom for the particle of any spin are just the three positions and momenta, and the constant h that appears in s µν . This is as it should be, as seen also by counting the physical degrees of freedom from the twistor point of view. When we consider the other systems listed in the first section, we should expect that they too are described by the same number of degrees of freedom since they will be obtained from the same twistor, although they obey different dynamics (different Hamiltonians) in their respective phase spaces.\n\nThe lightcone quantization of the the massless particle systems described by the actions (3.2,3.3) is performed after identifying the physical degrees of freedom as discussed above.\n\nThe lightcone quantum spectrum and wavefunction are the expected ones for spinning massless particles, and agree with their covariant quantization given in [15] - [19] .\n\nIV. 2T-PHYSICS WITH SP(2, R) , SU(2, 3) AND KAPPA SYMMETRIES The similarity of (3.2) to the action of the superparticle provides the hint for how to lift it to the 2T-physics formalism, as was done for the superparticle [22] [9] and the twistor superstring [23][24] . This requires lifting 3+1 phase space (x µ , p µ ) to 4+2 phase space X M , P M and lifting the SL(2, C) spinors v, v to the SU(2, 2) spinors V A , V A . The larger set of degrees of freedom X M , P M , V A , V A that are covariant under the global symmetry SU(2, 2) =SO(4, 2) , include gauge degrees of freedom, and are subject to gauge symmetries and constraints that follow from them as described below.\n\nThe point is that the SU(2, 2) invariant constraints on X M , P M , V A , V A have a wider set of solutions than just the 3+1 system of Eq.(3.2) we started from. This is because 3+1 dimensional spin & phase space has many different embeddings in 4+2 dimensions, and those are distinguishable from the point of view of 1T-physics because target space \"time\" and corresponding \"Hamiltonian\" are different in different embeddings, thus producing the different dynamical systems listed in section (I). The various 1T-physics solutions are reached by simply making gauge choices. One of the gauge choices for the action we give below in Eq.(4.1) is the twistor action of Eq.(2.3). Another gauge choice is the 4+2 spin & phase space action in terms of the lifted spin & phase space X M , P M , V A , V A as given in Eq. (5.4) .\n\nThe latter can be further gauge fixed to produce all of the systems listed in section (I) including the action (3.2) for the massless spinning particle with any spin. All solutions still remember that there is a hidden global symmetry SU(2, 2) =SO(4, 2) , so all systems listed in section (I) are realizations of the same unitary representation of SU(2, 2) whose Casimir eigenvalues will be given below.\n\nFor the 4 + 2 version of the superparticle [22] that is similar to the action in (5.4) , this program was taken to a higher level in [9] by embedding the fermionic supercoordinates in the coset of the supergroup SU(2, 2|1) /SU(2, 2) ×U(1). We will follow the same route here, and embed the bosonic SU(2, 2) spinors V A , V A in the left coset SU(2, 3) /SU(2, 2) ×U(1) . This coset will be regarded as the gauging of the group SU(2, 3) under the subgroup [SU (2, 2) × U (1)] L from the left side. Thus the most powerful version of the action that reveals the global and gauge symmetries is obtained when it is organized in terms of the X M i (τ ), g (τ ) and à (τ ) degrees of freedom described as 4+2 phase space X M (τ )\n\nP M (τ ) ≡ X M i (τ ) , i = 1, 2, doublets of Sp (2, R) gauge symmetry, group element g (τ ) ⊂ SU (2, 3) subject to [SU (2, 2) × U (1) ] L × U (1) L+R gauge symmetry.\n\nWe should mention that the h = 0 version of this theory, and the corresponding twistor property, was discussed in [6] , by taking g (τ ) ⊂SU(2, 2) and dropping all of the U(1)'s. So, the generalized theory that includes spin has the new features that involves SU(2, 2) →SU (2, 3) and the U(1) structures. The action has the following form\n\nS h = dτ 1 2 ε ij D τ X M i X N j η M N + T r (iD τ g) g -1 L 0 0 0 -2h à , (4.1)\n\nwhere\n\nε ij = 0 -1 1 0 ij is the antisymmetric Sp(2, R) metric, and D τ X M i = ∂ τ X M i -A j i X M j is the Sp(2, R) gauge covariant derivative, with the 3 gauge potentials A ij = ε ik A j k = A C C B . For SU(2, 3) the group element is pseudo-unitary, g -1 = (η 2,3 ) g † (η 2,3 ) -1 , where η 2,3 is the SU(2, 3) metric η 2,3 = η 2,2 0 0 -1 . The covariant derivative D τ g is given by D τ g = ∂ τ g -i à [q, g] , q = 1 5 1 4×4 0 0 -4 (4.2)\n\nwhere the generator of U(1) L+R is proportional to the 5×5 traceless matrix q ∈ u(1) ∈ su(2, 3) L+R . The last term of the action -2h Ã, which is also the last term of the action (2.3), is invariant under the U(1) L+R since it transforms to a total derivative.\n\nFinally, the 4 × 4 traceless matrix (L) B A ∈su(2, 2) ∈su(2, 3) that appears on the left side of g (or right side of g -1 ) is\n\n(L) B A ≡ 1 4i Γ M N B A L M N , L M N = ε ij X M i X N j = X M P N -X N P M . (4.3)\n\nwhere Γ M N = 1 2 Γ M ΓN -Γ N ΓM are the 4×4 gamma-matrix representation of the 15 generators of SU (2, 2) . A detailed description of these gamma matrices is given in [11] .\n\nThe symmetries of actions of this type for any group or supergroup g were discussed in [9] [23] [24] [7] . The only modification of that discussion here is due to the inclusion of the U(1) gauge field Ã. In the absence of the à coupling the global symmetry is given by the transformation of g (τ ) from the right side g (τ ) → g (τ ) g R where g R ⊂SU(2, 3) R .\n\nHowever, in our case, the presence of the coupling with the U(1) L+R charge q breaks the global symmetry down to the (SU(2, 2)×U( 1 )) R subgroup that acts on the right side of g."
},
{
"section_type": "OTHER",
"section_title": "So the global symmetry is given by global",
"text": ": g (τ ) → g (τ ) h R , h R ∈ [SU(2, 2) × U (1)] R ⊂ SU(2, 3) R . (4.4)\n\nUsing Noether's theorem we deduce the conserved global charges as the [SU(2, 2)×U( 1 )] R components of the the following SU(2, 3) R Lie algebra valued matrix J (2,3)\n\nJ (2,3) = g -1 L 0 0 0 g = J + 1 4 J 0 - j j -J 0 , J 2,3 = η 2,3 (J 2,3 ) † (η 2,3 ) -1 , (4.5)\n\nThe traceless 4 × 4 matrix (J\n\n) B A = 1 4i Γ M N J M N is the conserved SU(2, 2) =SO(4,\n\n2) charge and J 0 is the conserved U(1) charge. Namely, by using the equations of motion one can verify ∂ τ (J ) B A = 0 and ∂ τ J 0 = 0. The spinor charges j A , jA are not conserved foot_1 due to the coupling of Ã. As we will find out later in Eq.(6.8), j A is proportional to the twistor\n\nj A = J 0 Z A , (4.6)\n\nup to an irrelevant gauge transformation. It is important to note that J and J 0 are invariant on shell under the gauge symmetries discussed below. Therefore they generate physical symmetries [SU(2, 2)×U( 1 )] R under which all gauge invariant physical states are classified.\n\nThe local symmetries of this action are summarized as\n\nSp (2, R) × SU (2, 2) 3 4 kappa 3 4 kappa U (1) lef t (4.7)\n\nThe Sp(2, R) is manifest in (4.1). The rest corresponds to making local SU(2, 3) transformations on g (τ ) from the left side g (τ ) → g L (τ ) g (τ ) , as well as transforming X M i = X M , P M as vectors with the local subgroup SU(2, 2) L =SO(4, 2) , and A ij under the kappa. The 3/4 kappa symmetry which is harder to see will be discussed in more detail below. These symmetries coincide with those given in previous discussions in [9] [23][24] [7] despite the presence of Ã. The reason is that the U(1) L+R covariant derivative D τ g in Eq.(4.2) can be replaced by a purely U(1) R covariant derivative D τ g = ∂ τ g + igq à because the difference drops out in the trace in the action (4.1). Hence the symmetries on left side of g (τ ) → g L (τ ) g (τ ) remain the same despite the coupling of Ã.\n\nWe outline the roles of each of these local symmetries. The Sp(2, R) gauge symmetry can reduce X M , P M to any of the phase spaces in 3+1 dimensions listed in section (I). This is the same as the h = 0 case discussed in [6] . The [SU(2, 2) ×U( 1 In terms of counting, there remains only 3 position and 3 momentum physical degrees of freedom, plus the constant h, in agreement with the counting of physical degrees of freedom of the twistors.\n\nIt is possible to gauge fix the symmetries (4.7) partially to exhibit some intermediate covariant forms. For example, to reach the SL(2, C) covariant massless particle described by the action (3.2) from the 2T-physics action above, we take the massless particle gauge by using two out of the three Sp(2, R) gauge parameters to rotate the M = + ′ doublet to the form X + ′ P + ′ (τ ) = 1 0 , and solving explicitly two of the Sp(2, R) constraints X 2 = X • P = 0\n\nX M = ( + ′ 1 , -′ x 2 2 , µ x µ (τ )), P M = ( + ′ 0 , -′ x • p , µ p µ (τ )). (4.8)\n\nThis is the same as the h = 0 massless case in [6] . There is a tau reparametrization gauge symmetry as a remnant of Sp(2, R) . Next, the [SU(2, 2) ×U(1)] L gauge symmetry reduces g (τ ) → t (V ) written in terms of V A , V A as given in Eq.(5.3), and the 3/4 kappa symmetry\n\nreduces the SU(2, 2) spinor V A → v α 0\n\nto the two components SL(2, C) doublet v α, with a leftover kappa symmetry as discussed in Eqs. (3.4-3.5) . The gauge fixed form of g is then\n\ng = exp 0 0 √ 2hv α 0 0 0 0 √ 2hv α 0 = 1 hv α vβ √ 2hv α 0 1 0 0 √ 2hv α 1 ∈ SU (2, 3) . (4.9)\n\nThe inverse g -1 = (η 2,3 ) g † (η 2,3 ) -1 is given by replacing v, v by (-v) , (-v) . Inserting the gauge fixed forms of X, P, g (4.8,4.9) into the action (4.1) reduces it to the massless spinning particle action (3.2). Furthermore, inserting these X, P, g into the expression for the current in (4.5) gives the conserved SU(2, 2) charges J (see Eqs.(5.9,5.20)) which have the significance of the hidden conformal symmetry of the gauge fixed action (3.2). This hidden symmetry is far from obvious in the form (3.2), but it is straightforward to derive from the 2T-physics action as we have just outlined.\n\nPartial or full gauge fixings of (4.1) similar to (4.8,4.9) produce the actions, the hidden SU(2, 2) symmetry, and the twistor transforms with spin of all the systems listed in section (I). These were discussed for h = 0 in [6] , and we have now shown how they generalize to any spin h = 0, with further details below. It is revealing, for example, to realize that the massive spinning particle has a hidden SU(2, 2) \"mass-deformed conformal symmetry\", including spin, not known before, and that its action can be reached by gauge fixing the action (4.1), or by a twistor transform from (2.3). The same remarks applied to all the other systems listed in section (I) are equally revealing. For more information see our related paper [1] .\n\nThrough the gauge (4.8,4.9), the twistor transform (3.1), and the massless particle action (3.2), we have constructed a bridge between the manifestly SU(2, 2) invariant twistor action (2.3) for any spin and the 2T-physics action (4.1) for any spin. This bridge will be made much more transparent in the following sections by building the general twistor transform.\n\nV. 2T-PHYSICS ACTION WITH X M , P M , V A , V A IN 4+2 DIMENSIONS\n\nWe have hinted above that there is an intimate relation between the 2T-physics action\n\n, R) constraints X i • X j = X 2 = P 2 = X • P = 0 [6][7] X M = ( + ′ 1 , -′ 0 , + 0, - 0 , i 0), P M = ( + ′ 0 , -′ 0 , + 1, - 0 , i 0). (5.1)\n\nThis completely eliminates all phase space degrees of freedom. We are left with the gauge\n\nfixed action S h = dτ T r 1 2 (D τ g) g -1 Γ -′ - 0 0 0 -2h à , where (iL) → 1 2 Γ -′ -L + ′ -′ , and L + ′ -′ = 1.\n\nDue to the many zero entries in the 4×4 matrix Γ -′ -[6], only one column from g in the form Z A Z 5 and one row from g -1 in the form ZA , -Z5 can contribute in the trace, and therefore the action becomes\n\nS h = dτ i ZA ŻA -i Z5 Ż5 + à Z5 Z 5 -2h .\n\nHere Z5 Ż5 drops out as a total derivative since the magnitude of the complex number Z 5 is a constant Z5 Z 5 = 2h. Furthermore, we must take into account ZA Z A -Z5 Z 5 = 0 which is an off-diagonal entry in the matrix equation g -1 g = 1. Then we see that the 2T-physics action (4.1) reduces to the twistor action (2.3) with the gauge choice (5.1) 5 .\n\nNext let us gauge fix the 2T-physics action (4.1) to a manifestly SU(2, 2) =SO (4, 2) invariant version in flat 4+2 dimensions, in terms of the phase space & spin degrees of freedom X M , P M , V A , V A . For this we use the [SU(2, 2)×U( 1 )] lef t symmetry to gauge fix g gauge fix:\n\ng → t (V ) ∈ SU(2, 3) [SU(2, 2) × U(1)] lef t (5.2)\n\nThe coset element t (V ) is parameterized by the SU(2, 2) spinor V and its conjugate V = V † η 2,2 and given by the 5×5 SU(2, 3) matrix foot_2\n\nt (V ) = 1 -2hV V -1/2 0 0 1 -2h V V -1/2 1 √ 2hV √ 2h V 1 . (5.3)\n\nThe factor 2h is inserted for a convenient normalization of V. Note that the first matrix commutes with the second one, so it can be written in either order. The inverse of the group\n\nelement is t -1 (V ) = (η 2,3 ) t † (η 2,3 ) -1 = t (-V ) , as can be checked explicitly t (V ) t (-V ) = 1.\n\nInserting this gauge in (4.1) the action becomes\n\nS h = dτ Ẋ • P - 1 2 A ij X i • X j - 1 2 Ω M N L M N -2h à V LV 1 -2h V V -1 (5.4) = dτ 1 2 ε ij Dτ X M i X N j η M N -2h à V LV 1 -2h V V -1 (5.5)\n\nwhere Dτ\n\nX M i = ∂ τ X M i -A j i X M j -Ω M N X iN (5.6)\n\nis a covariant derivative for local Sp(2, R) as well as local SU(2, 2) =SO(4, 2) but with a composite SO(4, 2) connection Ω M N (V (τ )) given conveniently in the following forms\n\n1 2 Ω M N Γ M N = (i∂ τ t) t -1 SU (2,2) , or 1 2 Ω M N L M N = -T r (i∂ τ t) t -1 L 0 0 0 . (5.7)\n\nThus, Ω is the SU(2, 2) projection of the SU(2, 3) Cartan connection and given explicitly as The action (5.4,5.5) is manifestly invariant under global SU(2, 2) =SO(4, 2) rotations, and under local U (1) phase transformations applied on V A , V A . The conserved global symmetry currents J and J 0 can be derived either directly from (5.4) by using Noether's theorem, or by inserting the gauge fixed form of g → t (V ) into Eq.(4.5\n\n1 2 Ω M N Γ M N = 2h V -V V V V V V -V V - V V V V V √ 1 -2h V V 1 + √ 1 -2h V V + h V V V V - 1 4 V V -V V 1 -2h V V (5.\n\n) 7 J (2,3) = t -1 L 0 0 0 t J = 1 √ 1 -2hV V L 1 √ 1 -2hV V - 1 4 J 0 , J 0 = 2h V LV 1 -2h V V\n\n(5.9)\n\nj A = √ 2h 1 √ 1 -2hV V LV 1 √ 1 -2h V V (5.10)\n\nAccording to the equation of motion for à that follows from the action (5.4) we must have the following constraint (this means U(1) gauge invariant physical sector)\n\nV LV 1 -2h V V = 1. (5.11)\n\nTherefore, in the physical sector the conserved [SU(2, 2)×U( 1 )] right charges take the form physical sector:\n\nJ 0 = 2h, J = 1 √ 1 -2hV V L 1 √ 1 -2hV V - h 2 . ( 5\n\nA K A = X i • Γκ i (τ ) A = X M Γ M κ 1 A + P M Γ M κ 2 A , (5.13)\n\nwith κ iA (τ ) two arbitrary local spinors 8 . Now that g has been gauge fixed g → t (V ), the kappa transformation must be taken as the naive kappa transformation on g followed by a\n\n[SU(2, 2) ×U(1)] lef t gauge transformation which restores the gauge fixed form of t (V )\n\nt (V ) → t (V ′ ) = exp -ω 0 0 T r (ω) exp 0 K K 0 t (V ) (5.14)\n\nThe SU(2, 2) part of the restoring gauge transformation must also be applied on X M , P M .\n\nPerforming these steps we find the infinitesimal version of this transformation [22]\n\nδ κ V = 1 √ 1 -2hV V K 1 √ 1 -2h V V , δ κ X M i = ω M N X iN , δ κ A ij = see below, ( 5\n\n.15) 7 In the high spin version ( à = 0) the conserved charges include j A as part of SU(2, 3) R global symmetry.\n\nIt is then also convenient to rescale √ 2hV → V in Eqs.(5.3-5.10) to eliminate an irrelevant constant. 8 In this special form only 3 out of the 4 components of K A are effectively independent gauge parameters.\n\nThis can be seen easily in the special frame for X M , P M given in Eq.(5.1).\n\nwhere ω M N (K, V ) has the same form as Ω M N in Eq.(5.8) but with V replaced by the δ κ V given above. The covariant derivative Dτ X M i in Eq.(5.6) is covariant under the local SU(2, 2) transformation with parameter ω M N (K, V ) (this is best seen from the projected Cartan connection form Ω = [(i∂ τ t) t -1 ] SU (2,2) ). Therefore, the kappa transformations (5.15) inserted in (5.5) give\n\nδ κ S h = dτ - 1 2 δ κ A ij X i • X j + iT r (D τ t) t -1 0 -KL LK 0 .\n\n(5.16)\n\nIn computing the second term the derivative terms that contain ∂ τ K have dropped out in the trace. Using Eq.( 5 .13) we see that\n\nLK = 1 4i ε li X M l X N i X L j Γ M N Γ L κ j (5.17) = 1 4i ε li X M l X N i X L j (Γ M N L + η N L Γ M -η M L Γ N ) κ l (5.18) = 1 2i ε li X i • X j X l • Γκ j (5.19)\n\nThe completely antisymmetric X M i X N j X L l Γ M N L term in the second line vanishes since i, j, l can only take two values. The crucial observation is that the remaining term in LK is proportional to the dot products X i • X j . Therefore the second term in (5. 16 ) is cancelled by the first term by choosing the appropriate δ κ A ij in Eq.(5.16), thus establishing the kappa symmetry.\n\nThe local kappa transformations (5.15) are also a symmetry of the global SU(2, 3) R charges δ κ J = δ κ J 0 = δ κ j A = 0 provided the constraints X i • X j = 0 are used. Hence these charges are kappa invariant in the physical sector.\n\nWe have established the global SO(4, 2) and local Sp(2, R) × (3/4 kappa)×U(1) symmetries of the phase space action (5.4) in 4+2 dimensions. From it we can derive all of the phase space actions of the systems listed in section (I) by making various gauge choices for the local Sp(2, R) × (3/4 kappa)×U(1) symmetries. This was demonstrated for the spinless case h = 0 in [6] . The gauge choices for X M , P M discussed in [6] now need to be supplemented with gauge choices for V A , V A by using the kappa×U(1) local symmetries.\n\nHere we demonstrate the gauge fixing described above for the massless particle of any spin h. The kappa symmetry effectively has 3 complex gauge parameters as explained in footnote (8) . If the kappa gauge is fixed by using two of its parameters we reach the following forms For other examples of gauge fixing that generates some of the systems in the list of section (I) see our related paper [1] .\n\nV A → v α 0 , V A → (0 vα ) , V V → 0, 1 -2hV V -1/2 → 1 0 hvv 1 . ( 5"
},
{
"section_type": "OTHER",
"section_title": "VI. GENERAL TWISTOR TRANSFORM (CLASSICAL)",
"text": "The various formulations of spinning particles described above all contain gauge degrees of freedom of various kinds. However, they all have the global symmetry SU(2, 2)=SO(4, 2)\n\nwhose conserved charges J B A are gauge invariant in all the formulations. The most symmetric 2T-physics version gave the J B A as embedded in SU(2, 3) R in the SU(2, 2) projected form in Eq.(4.5)\n\nJ = g -1 L 0 0 0 g SU (2,2) . (6.1)\n\nSince this is gauge invariant, when gauge fixed, it must agree with the Noether charges computed in any version of the theory. So we can equate the general phase space version of Eq.(5.9) with the twistor version that follows from the Noether currents of (2.3) as follows\n\nJ = Z (h) Z(h) - 1 4 T r Z (h) Z(h) = 1 √ 1 -2hV V L 1 √ 1 -2hV V - 1 4 J 0 (6.2)\n\nThe trace corresponds to the U(1) charge J 0 = T r Z (h) Z(h) , so\n\nJ + 1 4 J 0 = Z (h) Z(h) = 1 √ 1 -2hV V L 1 √ 1 -2hV V . (6.3)\n\nIn the case of h = 0 this becomes\n\nZ (0) Z(0) = L. (6.4)\n\nTherefore the equality (6.3) is solved up to an irrelevant phase by\n\nZ (h) = 1 √ 1 -2hV V Z (0) . (6.5)\n\nBy inserting (6.4) into the constraint (5.11) we learn a new form of the constraints\n\nV Z (0) = 1 -2h V V , V Z (h) = 1. (6.6)\n\nIn turn, this implies\n\nZ (0) = LV √ 1 -2h V V (6.7)\n\nwhich is consistent 9 with Z (0) Z(0) = L , and its vanishing trace Z(0) Z (0) = 0 since LL = 0 (due to X 2 = P 2 = X • P = 0). Putting it all together we then have\n\nZ (h) = 1 √ 1 -2hV V LV 1 √ 1 -2h V V = J + 1 4 J 0 V. (6.8)\n\nWe note that this Z (h) is proportional to the non-conserved coset part of the SU(2, 3) charges\n\nJ 2,3 , that is j A = √ J 0 Z (h)\n\ngiven in Eqs.(4.5,4.6) or (5.10), when g and L are replaced by their gauge fixed forms, and use the constraint 10 J 0 = 2h.\n\nThe key for the general twistor transform for any spin is Eq.(6.5), or equivalently (6.8).\n\nThe general twistor transform between Z (0) and X M , P M which satisfies Z (0) Z(0) = L is already given in [6] as\n\nZ (0) = µ (0) λ (0) , µ (0) α = -i X µ X + ′ σµ λ (0) α , λ (0) α λ(0) β = X + P µ -X µ P + (σ µ ) α β . (6.9)\n\nNote that (X + P µ -X µ P + ) is compatible with the requirement that any SL(2, C) vector constructed as λ (0) α λ(0) β must be lightlike. This property is satisfied thanks to the Sp(2, R) constraints X 2 = P 2 = X • P = 0 in 4+2 dimensions, thus allowing a particle of any mass in the 3 + 1 subspace (since P µ P µ is not restricted to be lightlike). Besides satisfying Z (0) Z(0) = L, this Z (0) also satisfies Z(0) Z (0) = 0, as well as the canonical properties of twistors. Namely, Z (0) has the property [6] dτ Z(0) ∂ τ Z (0) = dτ ẊM P M .\n\n(6.10)\n\nFrom here, by gauge fixing the Sp(2, R) gauge symmetry, we obtain the twistor transforms for all the systems listed in section (I) for h = 0 directly from Eq.(6.9), as demonstrated in [6] . All of that is now generalized at once to any spin h through Eq.(6.5). Hence (6.5) together with (6.9) tell us how to construct explicitly the general twistor Z (h)\n\nA in terms 9 To see this, we note that Eqs.(6.4,6.6) lead to 10 For the high spin version ( Ã = 0) we don't use the constraint. Instead, we use\n\nLV V L 1-2h V V = Z (0) Z(0) V V Z (0) Z(0) 1-2h V V = Z (0) Z(0) = L.\n\nZ (h) = 1 √ 1-2hV\n\nV Z (0) only in its form (6.5), and note that, after using Eq.(6.4), the j A in Eq.(5.10) takes the form\n\nj A = √ J 0 Z (h) with √ J 0 = Z0 V √ 2h √ 1-2h V V\n\n, and it is possible to rescale h away everywhere √ 2hV → V.\n\nof spin & phase space degrees of freedom X M , P M , V A , V A . Then the Sp(2, R) and kappa gauge symmetries that act on X M , P M , V A , V A can be gauge fixed for any spin h, to give the specific twistor transform for any of the systems under consideration.\n\nWe have already seen in Eq.( 6 .2) that the twistor transform (6.5) relates the conserved SU(2, 2) charges in twistor and phase space versions. Let us now verify that (6.5) provides the transformation between the twistor action (2.3) and the spin & phase space action (5.4).\n\nWe compute the canonical structure as follows\n\ndτ Z(h) ∂ τ Z (h) = dτ Z(0) 1 √ 1 -2hV V ∂ τ 1 √ 1 -2hV V Z (0) (6.11) = dτ Z(0) 1 √ 1-2hV V ∂ τ 1 √ 1-2hV V Z (0) + Z(0) 1 1-2hV V ∂ τ Z (0) (6.12) = dτ Ẋ • P + T r (i∂ τ t) t -1 L 0 0 0 (6.13)\n\nThe last form is the canonical structure of spin & phase space as given in (5.4). To prove this result we used Eq.(6.10), footnote (6) , and the other properties of Z (0) including Eqs.(6.4-6.7), as well as the constraints X 2 = P 2 = X • P = 0, and dropped some total derivatives. This proves that the canonical properties of Z (h) determine the canonical properties of spin & phase space degrees of freedom and vice versa.\n\nThen, including the terms that impose the constraints, the twistor action (2.3) and the phase space action (5.4) are equivalent. Of course, this is expected since they are both gauge fixed versions of the master action (4.1), but is useful to establish it also directly via the general twistor transform given in Eq.(6.5)."
},
{
"section_type": "OTHER",
"section_title": "VII. QUANTUM MASTER EQUATION, SPECTRUM, AND DUALITIES",
"text": "In this section we derive the quantum algebra of the gauge invariant observables J B A and J 0 which are the conserved charges of [SU(2, 2) ×U(1)] R . Since these are gauge invariant symmetry currents they govern the system in any of its gauge fixed versions, including in any of its versions listed in section (I). From the quantum algebra we deduce the constraints among the physical observables J B A ,J 0 and quantize the theory covariantly. Among other things, we compute the Casimir eigenvalues of the unitary irreducible representation of SU(2, 2) which classifies the physical states in any of the gauge fixed version of the theory (with the different 1T-physics interpretations listed in section (I)).\n\nThe simplest way to quantize the theory is to use the twistor variables, and from them compute the gauge invariant properties that apply in any gauge fixed version. We will apply the covariant quantization approach, which means that the constraint due to the U(1) gauge symmetry will be applied on states. Since the quantum variables will generally not satisfy the constraints, we will call the quantum twistors in this section Z A , ZA to distinguish them from the classical Z (h) A , Z(h)A of the previous sections that were constrained at the classical level. So the formalism in this section can also be applied to the high spin theories (discussed in several footnotes up to this point in the paper) by ignoring the constraint on the states.\n\nAccording to the twistor action (2.3) Z A and i ZA (or equivalently λ α and iμ α ) are canonical conjugates. Therefore the quantum rules (equivalent to spin & phase space quantum rules) are The operator Ĵ0 has non-trivial commutation relations with Z A , Z A which follow from the basic commutation rules above A at the quantum level as follows\n\nZ A , ZB = δ B A . ( 7\n\nĴ0 , Z A = -Z A , Ĵ0 , Z A = Z A . ( 7\n\nJ B A = Z A ZB - 1 4 T r Z Z δ B A = Z Z - Ĵ0 + 2 4 B A . (7.5)\n\nIn this expression the order of the quantum operators matters and gives rise to the shift J 0 → Ĵ0 + 2 in contrast to the corresponding classical expression. The commutation rules among the generators J B A and the Z A , ZA are computed from the basic commutators (7.1),\n\nJ B A , Z C = -δ B C Z A + 1 4 Z C δ B A , J B A , ZD = δ D A ZB - 1 4 ZD δ B A (7.6) J B A , J D C = δ D A J B C -δ B C J D A , Ĵ0 , J B A = 0. (7.7)\n\nWe see from these that the gauge invariant observables J B A satisfy the SU(2, 2) Lie algebra, while the Z A , ZA transform like the quartets 4, 4 of SU (2, 2) . Note that the operator Ĵ0 commutes with the generators J B A , therefore J B A is U(1) gauge invariant, and furthermore Ĵ0 must be a function of the Casimir operators of SU (2, 2) . When Ĵ0 takes the value 2h on physical states, then the Casimir operators also will have eigenvalues on physical states which determine the SU(2, 2) representation in the physical sector.\n\nFrom the quantum rules (7.3), it is evident that the U(1) generator Ĵ0 can only have integer eigenvalues since it acts like a number of operator. More directly, through Eq. (7.4) it is related to the number operator ZZ. Therefore the theory is consistent at the quantum level (7.2) provided 2h is an integer.\n\nLet us now compute the square of the matrix J B A . By using the form (7.5) we have\n\n(J J ) = Z Z -Ĵ0 +2 4 Z Z -Ĵ0 +2 4 = Z ZZ Z -2 Ĵ0 +2 4 Z Z + Ĵ0 +2 4 2\n\nwhere we have used\n\nĴ0 , Z A ZB = 0. Now we elaborate Z ZZ Z B A = Z A Ĵ0 -2 ZB = Ĵ0 -1 Z A ZB where\n\nwe first used (7.4) and then (7.3). Finally we note from (7.5) that\n\nZ A ZB = J B A + Ĵ0 +2 4 δ B A .\n\nPutting these observations together we can rewrite the right hand side of (J J ) in terms of J and Ĵ0 as follows 11\n\n(J J ) = Ĵ0 2 -2 J + 3 16 Ĵ2 0 -4 . ( 7\n\n11 A similar structure at the classical level can be easily computed by squaring the expression for J in Eq.( 6 .2)\n\nand applying the classical constraint J 0 = ZA Z A = 2h. This yields the classical version\n\nJ C A J B C = J0 2 J B A + 3 16 J 2 0 δ B A = hJ B A + 3 4 h 2 δ B A\n\n, which is different than the quantum equation (7.8) . Thus, the quadratic Casimir at the classical level is computed as C 2 = 3 4 J 2 0 = 3h 2 which is different than the quantum value in (7.16 )."
},
{
"section_type": "OTHER",
"section_title": "This equation is a constraint satisfied by the global [SU(2, 2) ×U(1)] R charges J B",
"text": "A , Ĵ0 which are gauge invariant physical observables. It is a correct equation for all the states in the theory, including those that do not satisfy the U(1) constraint (7.2). We call this the quantum master equation because it will determine completely all the SU(2, 2) properties of the physical states for all the systems listed in section (I) for any spin.\n\nBy multiplying the master equation with J and using (7.8) again we can compute J J J .\n\nUsing this process repeatedly we find all the powers of the matrix J\n\n(J ) n = α n J + β n , (7.9)\n\nwhere\n\nα n ( Ĵ0 ) = 1 Ĵ0 -1 3 4 Ĵ0 -2 n - -1 4 Ĵ0 + 2 n , (7.10)\n\nβ n ( Ĵ0 ) = 3 16 Ĵ2 0 -4 α n-1 ( Ĵ0 ). (7.11)\n\nRemarkably, these formulae apply to all powers, including negative powers of the matrix J .\n\nUsing this result, any function of the matrix J constructed as a Taylor series takes the form\n\nf (J ) = α Ĵ0 J +β Ĵ0 (7.12)\n\nwhere\n\nα Ĵ0 = 1 Ĵ0 -1 f 3 4 Ĵ0 -2 -f -1 4 Ĵ0 + 2 , (7.13)\n\nβ Ĵ0 = 1 Ĵ0 -1 Ĵ0 + 2 4 f 3 4 Ĵ0 -2 + 3 Ĵ0 -2 4 f -1 4 Ĵ0 + 2 . (7.14)\n\nWe can compute all the Casimir operators by taking the trace of J n in Eq.(7.9), so we\n\nfind 12 C n ( Ĵ0 ) ≡ T r (J ) n = 4β n ( Ĵ0 ) = 3 4 Ĵ2 0 -4 α n-1 ( Ĵ0 ). (7.15)\n\nIn particular the quadratic, cubic and quartic Casimir operators of SU(2, 2) =SO(6, 2) are computed at the quantum level as\n\nC 2 ( Ĵ0 ) = 3 4 Ĵ2 0 -4 , C 3 ( Ĵ0 ) = 3 8 Ĵ2 0 -4 Ĵ0 -4 , (7.16)\n\nC 4 ( Ĵ0 ) = 3 64 Ĵ2 0 -4 7 Ĵ2 0 -32 Ĵ0 + 52 . (7.17)\n\nall have the same Casimir eigenvalues C 2 = -3, C 3 = 6, C 4 = -39 4 at the quantum level. Much more elaborate tests of the dualities can be performed both at the classical and quantum levels by computing any function of the gauge invariant J B\n\nA and checking that it has the same value when computed in terms of the spin & phase space of any of the systems listed in section (I). At the quantum level all of these systems have the same Casimir eigenvalues of the C n for a given h. So their spectra must correspond to the same unitary irreducible representation of SU(2, 2) as seen above. But the rest of the labels of the representation correspond to simultaneously commuting operators that include the Hamiltonian.\n\nThe Hamiltonian of each system is some operator constructed from the observables J B A , and so are the other simultaneously diagonalizable observables. Therefore, the different systems are related to one another by unitary transformations that sends one Hamiltonian to another, but staying within the same representation. These unitary transformations are the quantum versions of the gauge transformations of Eq.(4.7), and so they are the duality transformations at the quantum level. In particular the twistor transform applied to any of the systems is one of those duality transformations. By applying the twistor transforms we can map the Hilbert space of one system to another, and then compute any function of the gauge invariant J B A between dually related states of different systems. The prediction is that all such computations within different systems must give the same result."
},
{
"section_type": "OTHER",
"section_title": "Given that J B",
"text": "A is expressed in terms of rather different phase space and spin degrees of freedom in each dynamical system with a different Hamiltonian, this predicted duality is remarkable. 1T-physics simply is not equipped to explain why or for which systems there are such dualities, although it can be used to check it. The origin as well as the proof of the duality is the unification of the systems in the form of the 2T-physics master action of Eq.(4.1) in 4+2 dimensions. The existence of the dualities, which can laboriously be checked using 1T-physics, is the evidence that the underlying spacetime is more beneficially understood as being a spacetime in 4+2 dimensions."
},
{
"section_type": "OTHER",
"section_title": "VIII. QUANTUM TWISTOR TRANSFORM",
"text": "We have established a master equation for physical observables J at the quantum level. Now, we also want to establish the twistor transform at the quantum level expressed as much as possible in terms of the gauge invariant physical quantum observables J . To this end we write the master equation (7.8) in the form\n\nJ - 3 4 Ĵ0 -2 J + 1 4 Ĵ0 + 2 = 0. (8.1)\n\nRecall the quantum equation (7.5) J + Ĵ0 +2 4 = Z Z, so the equation above is equivalent to\n\nJ - 3 4 Ĵ0 -2 Z = 0. (8.2)\n\nThis is a 4 × 4 matrix eigenvalue equation with operator entries. The general solution is\n\nZ = J + 1 4 Ĵ0 + 2 V (8.3)\n\nwhere VA is any spinor up to a normalization. This is verified by using the master equation (8.1) which gives J -\n\n3 4 Ĵ0 -2 Z = J -3 4 Ĵ0 -2 J + 1 4 Ĵ0 + 2 V = 0. Not-\n\ning that the solution (8.3) has the same form as the classical version of the twistor transform in Eq.(6.8), except for the quantum shift J 0 → Ĵ0 + 2, we conclude that the VA introduced above is the quantum version of the V A discussed earlier (up to a possible renormalization 13 ), as belonging to the coset SU(2, 3)\n\n/[SU(2, 2) ×U(1)].\n\nNow VA is a quantum operator whose commutation rules must be compatible with those of Z A , ZA , Ĵ0 and J B A . Its commutation rules with J B A , Ĵ0 are straightforward and fixed uniquely by the SU(2, 2) ×U(1) covariance\n\nĴ0 , VA = -VA , Ĵ0 , V A = V A , (8.4)\n\nJ B A , VC = -δ B C VA + 1 4 VC δ B A , J B A , V D = δ D A V B - 1 4 V D δ B A . (8.5)\n\nOther quantum properties of VA follow from imposing the quantum property ZZ = Ĵ0 -2 in (7.4). Inserting Z of the form (8.3), using the master equation, and observing the commutation rules (8.4), we obtain\n\nV J + Ĵ0 + 2 4 V = 1. (8.6)\n\n13 The quantum version of V is valid in the whole Hilbert space, not only in the subspace that satisfies the U(1) constraint Ĵ0 → 2h. In particular, in the high spin version, already at the classical level we must take V = V ( √ 2h/ √ J 0 ) and then rescale it V √ 2h → V as described in previous footnotes. So in the full quantum Hilbert space we must take V = √ 2hV ( Ĵ0 + γ) -1/2 (or the rescaled version V √ 2h → V ) with the possibly quantum shifted operator ( Ĵ0 + γ) -1/2 . This is related to (5.11) if we take (5.9) into account by including the quantum shift J 0 → Ĵ0 + 2. Considering (8.3) this equation may also be written as\n\nV Z = Z V = 1. (8.7)\n\nNext we impose Z A , ZB = δ B A to deduce the quantum rules for [ VA , V B ]. After some algebra we learn that the most general form compatible with\n\nZ A , ZB = δ B A is VA , V B = - V V Ĵ0 -1 δ B A + M(J -3 Ĵ0 -2 4 ) + (J -3 Ĵ0 -2 4 ) M B A , (8.8)\n\nwhere M B A is some complex matrix and M = (η 2,2 ) M † (η 2,2 ) -1 . The matrix M B A could not be determined uniquely because of the 3/4 kappa gauge freedom in the choice of VA itself.\n\nA maximally gauge fixed version of VA corresponds to eliminating 3 of its components V2,3,4 = 0 by using the 3/4 kappa symmetry, leaving only A ≡ V1 = 0. Then we find V 1,2,4 = 0 and V 3 = A † . Let us analyze the quantum properties of this gauge in the context of the formalism above. From Eq.(8.6) we determine A = (J 1 3 ) -1/2 e -iφ , where φ is a phase, and then from Eq.( 8 .3) we find Z A .\n\nZ A = J 1 A + Ĵ0 + 2 4 δ 1 A J 1 3 -1/2 e -iφ , ZA = e iφ J 1 3 -1/2 J A 3 + Ĵ0 + 2 4 δ A 3 . (8.9)\n\nWe see that, except for the overall phase, Z A is completely determined in terms of the gauge invariant J B A . We use a set of gamma matrices Γ M given in ( [6] , [11] ) to write\n\nJ B A = 1 4i J M N (Γ M N ) B\n\nA as an explicit matrix so that Z A can be written in terms of the 15 SO(4, 2) =SU(2, 2) generators J M N . We find\n\nZ A = 1 2 J 12 + 1 2i J +-+ 1 2i J + ′ -′ + Ĵ0 +2 4 i √ 2 (J +1 + iJ +2 ) J + ′ + i √ 2 J + ′ 1 + iJ + ′ 2 e -iφ √ J + ′ + , (8.10)\n\nand ZA = Z † η 2,2 A . The orders of the operators here are important. The basis M = ± ′ , ±, i with i = 1, 2 corresponds to using the lightcone combinations\n\nX ± ′ = 1 √ 2 X 0 ′ ± X 1 ′ , X ± = 1 √ 2 (X 0 ± X 1 ).\n\nFrom our setup above, the Z A , ZA in (8.10) are guaranteed to satisfy the twistor commutation rules Z A , ZB = δ B A provided we insure that the VA , V B have the quantum properties given in Eqs. (8.4,8.5,8.8) . These are satisfied provided we take the following non-trivial commutation rules for φ φ, Ĵ0 = i, [φ, J 12 ] = i 2 ⇒ Ĵ0 , e ±iφ = ±e ±iφ , [J 12 , e ±iφ ] = ± 1 2 e ±iφ (8.11) while all other commutators between φ and J M N vanish. Then (8.8) becomes [ VA ,\n\nV B ] = 0, so M B A vanishes in this gauge. Indeed one can check directly that only by using the Lie algebra for the J M N , Ĵ0 and the commutation rules for φ in (8.11) , we obtain Z A , ZB = δ B A , which a remarkable form of the twistor transform at the quantum level.\n\nThe expression (8.10) for the twistor is not SU(2, 2) covariant. Of course, this is because we chose a non-covariant gauge for VA . However, the global symmetry SU(2, 2) is still intact since the correct commutation rules between the twistors and J M N or the J B A as given in (7.6,7.7) are built in, and are automatically satisfied. Therefore, despite the lack of manifest covariance, the expression for Z A in (8.10) transforms covariantly as the spinor of SU (2, 2) .\n\nIt is now evident that one has many choices of gauges for VA . Once a gauge is picked the procedure outlined above will automatically produce the quantum twistor transform in that gauge, and it will have the correct commutation rules and SU (2, 2) properties at the quantum level. For example, in the SL(2, C) covariant gauge of Eq.(5.20), the quantum twistor transform in terms of J M N is\n\nµ α = 1 4i J µν (σ µν ) αβ v β + 1 2i J + ′ -′ v α, λ α = 1 √ 2 J + ′ µ (σ µ ) α β v β . (8.12)\n\nwith the constraint 1 √ 2 vσ µ vJ + ′ µ = 1. (8.13) This gauge for VM covers several of the systems listed in section (I). The spinless case was discussed at the classical level in ( [6] ). The quantum properties of this gauge are discussed in more detail in ( [1] ).\n\nThe result for Z A in (8.10) is a quantum twistor transform that relies only on the gauge invariants J B A or equivalently J M N . It generalizes a similar result in [6] that was given at the classical level. In the present case it is quantum and with spin. All the information on spin is included in the generators J M N = L M N + S M N . There are other ways of describing spinning particles. For example, one can start with a 2T-physics action that uses fermions ψ M (τ ) [27] instead of our bosonic variables V A (τ ) . Since we only use the gauge invariant J M N , our quantum twistor transform (8.3) applies to all such descriptions of spinning particles, with an appropriate relation between V and the new spin degrees of freedom. In particular in the gauge fixed form of V that yields (8.10) there is no need to seek a relation between V and the other spin degrees of freedom. Therefore, in the form (8.10), if the J M N are produced with the correct quantum algebra SU(2, 2) =SO(4, 2) in any theory, (for example bosonic spinors, or fermions ψ M , or the list of systems in section (I), or any other) then our formula (8.3) gives the twistor transform for the corresponding degrees of freedom of that theory.\n\nThose degrees of freedom appear as the building blocks of J M N . So, the machinery proposed in this section contains some very powerful tools."
},
{
"section_type": "OTHER",
"section_title": "IX. THE UNIFYING SU(2, 3) LIE ALGEBRA",
"text": "The 2T-physics action (4.1) offered the group SU(2, 3) as the most symmetric unifying property of the spinning particles for all the systems listed in section (I), including twistors.\n\nHere we discuss how this fundamental underlying structure governs and simplifies the quantum theory.\n\nWe examine the SU(2, 3) charges J B A , Ĵ0 , j A , jA given in (4.5,5.9,5.10). Since these are gauge invariant under all the gauge symmetries (4.7) they are physical quantities that should have the properties of the Lie algebra 14 of SU (2, 3) in all the systems listed in section (I).\n\nUsing covariant quantization we construct the quantum version of all these charges in terms of twistors. By using the general quantum twistor transform of the previous section, these charges can also be written in terms of the quantized spin and phase space degrees of freedom of any of the relevant systems.\n\nThe twistor expressions for Ĵ0 , J B A are already given in Eqs.(7.2,7.5)\n\nĴ0 = 1 2 Z A ZA + ZA Z A , J B A = Z A ZB - Ĵ0 + 2 4 δ B A . (9.1)\n\nWe have seen that at the classical level (j A ) classical = √ J 0 Z A and now we must figure out 14 Even when j A is not a conserved charge when the U(1) constraint is imposed, its commutation rules are still the same in the covariant quantization approach, independently than the constraint.\n\nthe quantum version j A = Ĵ0 + αZ A that gives the correct SU(2, 3) closure property\n\nj A , jB = J B A + 5 4 Ĵ0 δ B A . (9.2)\n\nThe coefficient 5 4 is determined by consistency with the Jacobi identity j A , jB , j C + jB , j C , j A + [j C , j A , ] , jB = 0, and the requirement that the commutators of j A with J B A , Ĵ0 be just like those of Z A given in Eqs.(7.6,7.7), as part of the SU(2, 3) Lie algebra. So we carry out the computation in Eq.(9.2) as follows\n\nj A , jB = Ĵ0 + αZ A ZB Ĵ0 + α -ZB Ĵ0 + α Ĵ0 + αZ A (9.3) = Ĵ0 + α Z A ZB -Ĵ0 + α -1 ZB Z A (9.4) = Ĵ0 + α -1 Z A , ZB + Z A ZB (9.5) = δ B A Ĵ0 + α -1 + Ĵ0 + 2 4 + J B A (9.6)\n\nTo get (9.4) we have used the properties\n\nZ A f Ĵ0 = f Ĵ0 + 1 Z A and ZB f Ĵ0 = f Ĵ0 -1 ZB for any function f Ĵ0 . These follow from the commutator Ĵ0 , Z A = -Z A\n\nwritten in the form Z A Ĵ0 = Ĵ0 + 1 Z A which is used repeatedly, and similarly for ZB . To get (9.6) we have used Z A , ZB = δ B A and then used the definitions (9.1). By comparing (9.6) and (9.2) we fix α = 1/2. Hence the correct quantum version of j A is\n\nj A = Ĵ0 + 1 2 Z A = Z A Ĵ0 - 1 2 . (9.7)\n\nThe second form is obtained by using\n\nZ A f Ĵ0 = f Ĵ0 + 1 Z A .\n\nNote the following properties of the j A , jA\n\njA j A = Ĵ0 - 1 2 ZZ Ĵ0 - 1 2 = Ĵ0 - 1 2 Ĵ0 -2 (9.8) j A jB = Ĵ0 + 1 2 Z A ZB Ĵ0 + 1 2 = Ĵ0 + 1 2 J + 1 4 Ĵ0 + 2 (9.9)\n\nwhich will be used below.\n\nWith the above arguments we have now constructed the quantum version of the SU(2, 3) charges written as a 5 × 5 traceless matrix\n\nĴ2,3 = g -1 L 0 0 0 g quantum = J + 1 4 Ĵ0 - j j -Ĵ0 (9.10) = Z A ZB -1 2 δ B A Ĵ0 + 1 2 Z A -ZB Ĵ0 + 1 2 -Ĵ0 , (9.11)\n\nwith Ĵ0 , J given in Eq.(9.1).\n\nAt the classical level, the square of the matrix J 2,3 vanishes since L 2 = 0 as follows\n\n(J 2,3 ) 2 classical = g -1 L 0 0 0 g g -1 L 0 0 0 g = g -1 L 2 0 0 0 g = 0. (9.12)\n\nAt the quantum level we find the following non-zero result which is SU(2, 3) covariant Ĵ2,3\n\n2 = Z Z -1 2 Ĵ0 + 1 2 Z -Z Ĵ0 + 1 2 -Ĵ0 2 (9.13) = - 5 2 Ĵ2,3 -1. ( 9\n\nWritten out in terms of the charges, Eq.(9.14) becomes\n\nJ + 1 4 Ĵ0 - j j -Ĵ0 2 = - 5 2 J + 1 4 Ĵ0 - j j -Ĵ0 -1. (9.16)\n\nCollecting terms in each block we obtain the following relations among the gauge invariant charges J , Ĵ0 , j, j\n\nJ + 1 4 Ĵ0 2 -j j + 5 2 J + 1 4 Ĵ0 + 1 = 0, (9.17)\n\nJ + 1 4 Ĵ0 j -j Ĵ0 + 5 2 j = 0, (9.18)\n\n-jj + Ĵ0 2 - 5 2 Ĵ0 + 1 = 0. (9.19)\n\nCombined with the information in Eq.(9.9) the first equation is equivalent to the master quantum equation (7.8). After using j Ĵ0 = Ĵ0 j + j, the second equation is equivalent to the eigenvalue equation (8.2) whose solution is the quantum twistor transform (8.3). The third equation is identical to (9.8).\n\nHence the SU(2, 3) quantum property Ĵ2,3 This is a remarkable simple unifying description of a diverse set of spinning systems, that shows the existence of the sophisticated higher structure SU(2, 3) for which there was no clue whatsoever from the point of view of 1T-physics."
},
{
"section_type": "OTHER",
"section_title": "X. FUTURE DIRECTIONS",
"text": "One can consider several paths that generalizes our discussion, including the following.\n\n• It is straightforward to generalize our theory by replacing SU(2, 3) with the supergroup SU(2, (2 + n) |N) . This generalizes the spinor V A to V a A where a labels the fundamental representation of the supergroup SU(n|N) . The case of N = 0 and n = 1 is what we discussed in this paper. The case of n = 0 and any N relates to the superparticle with N supersymmetries (and all its duals) discussed in [22] and in [6] [7] . The massless particle gauge is investigated in [17] , but the other cases listed metries insure that the theory has no negative norm states. In the massless particle gauge, this model corresponds to supersymmetrizing spinning particles rather than supersymmetrizing the zero spin particle. The usual R-symmetry group in SUSY is replaced here by SU(n|N) ×U (1) . For all these cases with non-zero n, N, the 2Tphysics and twistor formalisms unify a large class of new 1T-physics systems and establishes dualities among them.\n\n• One can generalize our discussion in 4+2 dimensions, including the previous paragraph, to higher dimensions. The starting point in 4+2 dimensions was SU(2, 2) =SO (4, 2) embedded in g =SU(2, 3) . For higher dimensions we start from SO(d, 2) and seek a group or supergroup that contains SO(d, 2) in the spinor representation. For example for 6+2 dimensions, the starting point is the 8×8 spinor version of SO(8 * ) =SO (6, 2) embedded in g =SO(9 * ) =SO (6, 3) or g =SO(10 * ) =SO (6, 4) . The spinor variables in 6+2 dimensions V A will then be the spinor of SO(8 * ) =SO(6, 2) parametrizing the coset SO(9 * ) /SO(8 * ) (real spinor) or SO(10 * ) /SO(8 * ) ×SO(2) (complex spinor). This can be supersymmetrized. The pure superparticle version of this program for various dimensions is discussed in [6] [7] , where all the relevant supergroups are classified.\n\nThat discussion can now be taken further by including bosonic variables embedded in a supergroup as just outlined in the previous item. As explained before [6] [7], it must be mentioned that when d + 2 exceeds 6 + 2 it seems that we need to include also brane degrees of freedom in addition to particle degrees of freedom. Also, even in lower dimensions, if the group element g belongs to a group larger than the minimal one [6] [7] , extra degrees of freedom will appear.\n\n• The methods in this paper overlap with those in [28] where a similar master quantum equation technique for the supergroup SU(2, 2|4) was used to describe the spectrum of type-IIB supergravity compactified on AdS 5 ×S 5 . So our methods have a direct bearing on M theory. In the case of [28] the matrix insertion L 0 0 0 in the 2T-physics action was generalized to L (4,2) 0 0 L (6,0) to describe a theory in 10+2 dimensions. This approach to higher dimensions can avoid the brane degrees of freedom and concentrate only on the particle limit. Similar generalizations can be used with our present better develped methods and richer set of groups mentioned above to explore various corners of M theory.\n\n• One of the projects in 2T-physics is to take advantage of its flexible gauge fixing mechanisms in the context of 2T-physics field theory. Applying this concept to the 2T-physics version of the Standard Model [10] will generate duals to the Standard Model in 3+1 dimensions. The study of the duals could provide some non-perturbative or other physical information on the usual Standard Model. This program is about to be launched in the near future [29] . Applying the twistor techniques developed here to 2T-physics field theory should shed light on how to connect the Standard Model with a twistor version. This could lead to further insight and to new computational techniques for the types of twistor computations that proved to be useful in QCD [12] [13] .\n\n• Our new models and methods can also be applied to the study of high spin theories by generalizing the techniques in [14] which are closely related to 2T-physics. The high spin version of our model has been discussed in many of the footnotes, and can be supersymmetrized and written in higher dimensions as outlined above in this section. The new ingredient from the 2T point of view is the bosonic spinor V A and the higher symmetry, such as SU(2, 3) and its generalizations in higher dimensions or with supersymmetry. The massless particle gauge of our theory in 3+1 dimensions coincides with the high spin studies in [15] - [18] . Our theory of course applies broadly to all the spinning systems that emerge in the other gauges, not only to massless particles.\n\nThe last three sections on the quantum theory discussed in this paper would apply also in the high spin version of our theory. The more direct 4+2 higher dimensional quantization of high spin theories including the spinor V A (or its generalizations V a A ) is obtained from our SU(2, 3) quantum formalism in the last section.\n\n• One can consider applying the bosonic spinor that worked well in the particle case to strings and branes. This may provide new string backgrounds with spin degrees of freedom other than the familiar Neveu-Schwarz or Green-Schwarz formulations that involve fermions.\n\nMore details and applications of our theory will be presented in a companion paper [1] .\n\nWe gratefully acknowledge discussions with S-H. Chen, Y-C. Kuo, and G. Quelin."
}
] | |
arxiv:0704.0299 | 0704.0299 | 1 | 10.1103/PhysRevD.75.124022 | ea0edaa46fecaf2fdba24d6d9faf816a86beb0298436e6e342bffaadcce1f210 | Parametrized Post-Newtonian Expansion of Chern-Simons Gravity | We investigate the weak-field, post-Newtonian expansion to the solution of
the field equations in Chern-Simons gravity with a perfect fluid source. In
particular, we study the mapping of this solution to the parameterized
post-Newtonian formalism to 1 PN order in the metric. We find that the PPN
parameters of Chern-Simons gravity are identical to those of general
relativity, with the exception of the inclusion of a new term that is
proportional to the Chern-Simons coupling parameter and the curl of the PPN
vector potentials. We also find that the new term is naturally enhanced by the
non-linearity of spacetime and we provide a physical interpretation for it. By
mapping this correction to the gravito-electro-magnetic framework, we study the
corrections that this new term introduces to the acceleration of point
particles and the frame-dragging effect in gyroscopic precession. We find that
the Chern-Simons correction to these classical predictions could be used by
current and future experiments to place bounds on intrinsic parameters of
Chern-Simons gravity and, thus, string theory. | [
"Stephon Alexander and Nicolas Yunes"
] | [
"hep-th",
"astro-ph",
"gr-qc"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | We investigate the weak-field, post-Newtonian expansion to the solution of the field equations in Chern-Simons gravity with a perfect fluid source. In particular, we study the mapping of this solution to the parameterized post-Newtonian formalism to 1 PN order in the metric. We find that the PPN parameters of Chern-Simons gravity are identical to those of general relativity, with the exception of the inclusion of a new term that is proportional to the Chern-Simons coupling parameter and the curl of the PPN vector potentials. We also find that the new term is naturally enhanced by the non-linearity of spacetime and we provide a physical interpretation for it. By mapping this correction to the gravito-electro-magnetic framework, we study the corrections that this new term introduces to the acceleration of point particles and the frame-dragging effect in gyroscopic precession. We find that the Chern-Simons correction to these classical predictions could be used by current and future experiments to place bounds on intrinsic parameters of Chern-Simons gravity and, thus, string theory.
Tests of alternative theories of gravity that modify general relativity (GR) at a fundamental level are essential to the advancement of physics. One formalism that has had incredible success in this task is the parameterized post-Newtonian (PPN) framework [1, 2, 3, 4, 5, 6] . In this formalism, the metric of the alternative theory is solved for in the weak-field limit and its deviations from GR are expressed in terms of PPN parameters. Once a metric has been obtained, one can calculate predictions of the alternative theory, such as light deflection and the perihelion shift of Mercury, which shall depend on these PPN parameters. Therefore, experimental measurements of such physical effects directly lead to constraints on the parameters of the alternative theory. This framework, together with the relevant experiments, have already been successfully employed to constrain scalar-tensor theories (Brans-Dicke, Bekenstein) [7] , vector-tensor theories (Will-Nordtvedt [8] , Hellings-Nordtvedt [9] ), bimetric theories (Rosen [10, 11] ) and stratified theories (Ni [12] ) (see [13] for definitions and an updated review.)
Only recently has this framework been used to study quantum gravitational and string-theoretical inspired ideas. On the string theoretical side, Kalyana [14] investigated the PPN parameters associated with the gravitondilaton system in low-energy string theory. More recently, Ivashchuk, et. al. [15] studied PPN parameters in the context of general black holes and p-brane spherically symmetric solutions, while Bezerra, et. al. [16] considered domain wall spacetimes for low energy effective string theories and derived the corresponding PPN parameters for the metric of a wall. On the quantum gravitational side, Gleiser and Kozameh [17] and more recently Fan, et. al. [18] studied the possibility of testing gravitational birefringence induced by quantum gravity, which was proposed by Amelino-Camelia, el. al. [19] and Gambini and Pullin [20] . Other non-PPN proposals have been also put forth to test quantum gravity, for example through gravitational waves [21, 22, 23, 24, 25, 26, 27, 28 ], but we shall not discuss those tests here.
Chern-Simons (CS) gravity [29, 30] is one such extension of GR, where the gravitational action is modified by the addition of a parity-violating term. This extension is promising because it is required by all 4dimensional compactifications of string theory [31] for mathematical consistency because it cancels the Green-Schwarz anomaly [32] . CS gravity, however, is not unique to string theory and in fact has its roots in the standard model, where it arises as a gravitational anomaly provided that there are more flavours of left handed leptons than right handed ones. Moreover the CS extension to GR can arise via the embedding of the three dimensional Chern-Simons topological current into a 4D space-time manifold, decsribed by Jackiw and Pi [30] Chern-Simons gravity has been recently studied in the cosmological context. In particular, this framework was used to shed light on the anisotropies of the cosmic microwave background (CMB) [33, 34, 35] and the leptogenesis problem [34, 36, 37] . Parity violation has also been shown to produce birefringent gravitational waves [28, 29] , where different polarizations modes acquire varying amplitudes. These modes obey different propagation equations because the imaginary sector of the classical dispersion relation is CS corrected. Different from [20] , in CS birefringence the velocity of the gravitational wave remains that of light.
In this paper we study CS gravity in the PPN framework, extending the analysis of [38] and providing some missing details. In particular, we shall consider the effect of the CS correction to the gravitational field of, for instance, a pulsar, a binary system or a star in the weak-field limit. These corrections are obtained by solv-ing the modified field equations in the weak-field limit for post-Newtonian (PN) sources, defined as those that are weakly-gravitating and slowly-moving [39] . Such an expansion requires the calculation of the Ricci and Cotton tensors to second order in the metric perturbation. We then find that CS gravity leads to the same gravitational field as that of classical GR and, thus, the same PPN parameters, except for the inclusion of a new term in the vectorial sector of the metric, namely
g (CS) 0i = 2 ḟ (∇ × V ) i , (1)
where ḟ acts as a coupling parameter of CS theory and V i is a PPN potential. We also show that this solution can be alternatively obtained by finding a formal solution to the modified field equations and performing a PN expansion, as is done in PN theory. The full solution is further shown to satisfy the additional CS constraint, which leads to equations of motion given only by the divergence of the stress-energy tensor.
The CS correction to the metric found here leads to an interesting interpretation of CS gravity and forces us to consider a new type of coupling. The interpretation consists of thinking of the field that sources the CS correction as a fluid that permeates all of spacetime. Then the CS correction in the metric is due to the "dragging" of such a fluid by the motion of the source. Until now, couplings of the CS correction to the angular momentum of the source had been neglected by the string theory community. Similarly, curl-type terms had also been considered unnecessary in the traditional PPN framework, since previous alternative gravity theories had not required it. As we shall show, in CS gravity and thus in string theory, such a coupling is naturally occurring. Therefore, a proper PPN mapping requires the introduction of a new curl-type term with a corresponding new PPN parameter of the type of Eq. (1) .
A modification to the gravitational field leads naturally to corrections of the standard predictions of GR. In order to illustrate such a correction, we consider the CS term in the gravito-electro-magnetic analogy [40, 41] , where we find that the CS correction accounts for a modification of gravitomagnetism. Furthermore, we calculate the modification to the acceleration of point particles and the frame dragging effect in the precession of gyroscopes. We find that these corrections are given by
δa i = - 3 2 ḟ r G m c 2 r 2 v c • n v c × n i , δΩ i = - ḟ r G m c 3 r 2 3 v c • n n i - v i c , (2)
where m and v are the mass and velocities of the source, while r is the distance to the source and n i = x i /r is a unit vector, with • and × the flat-space scalar and cross products. Both corrections are found to be naturally enhanced in regions of high spacetime curvature. We then conclude that experiments that measure the gravitomagnetic sector of the metric either in the weak-field (such as Gravity Probe B [42] ) and particularly in the non-linear regime, will lead to a direct constraint on the CS coupling parameter ḟ . In this paper we develop the details of how to calculate these corrections, while the specifics of how to actually impose a constraint, which depend on the experimental setup, are beyond the scope of this paper.
The remainder of this paper deals with the details of the calculations discussed in the previous paragraphs. We have divided the paper as follows: Sec. II describes the basics of the PPN framework; Sec. III discusses CS modified gravity, the modified field equations and computes a formal solution; Sec. IV expands the field equations to second order in the metric perturbation; Sec. V iteratively solves the field equations in the PN approximation and finds the PPN parameters of CS gravity; Sec. VI discusses the correction to the acceleration of point particles and the frame dragging effect; Sec. VII concludes and points to future research.
The conventions that we use throughout this work are the following: Greek letters represent spacetime indices, while Latin letters stand for spatial indices only; semicolons stand for covariant derivatives, while colons stand for partial derivatives; overhead dots stand for derivatives with respects to time. We denote uncontrolled remainders with the symbol O(A), which stands for terms of order A. We also use the Einstein summation convention unless otherwise specified. Finally, we use geometrized units, where G = c = 1, and the metric signature (-, +, +, +).
In this section we summarize the basics of the PPN framework, following [6] . This framework was first developed by Eddington, Robertson and Schiff [1, 6] , but it came to maturity through the seminal papers of Nordtvedt and Will [2, 3, 4, 5] . In this section, we describe the latter formulation, since it is the most widely used in experimental tests of gravitational theories.
The goal of the PPN formalism is to allow for comparisons of different metric theories of gravity with each other and with experiment. Such comparisons become manageable through a slow-motion, weak-field expansion of the metric and the equations of motion, the so-called PN expansion. When such an expansion is carried out to sufficiently high but finite order, the resultant solution is an accurate approximation to the exact solution in most of the spacetime. This approximation, however, does break down for systems that are not slowly-moving, such as merging binary systems, or weakly gravitating, such as near the apparent horizons of black hole binaries. Nonetheless, as far as solar system tests are concerned, the PN expansion is not only valid but also highly accurate.
The PPN framework employs an order countingscheme that is similar to that used in multiple-scale anal-ysis [43, 44, 45, 46] . The symbol O(A) stands for terms of order ǫ A , where ǫ ≪ 1 is a PN expansion parameter. For convenience, it is customary to associate this parameter with the orbital velocity of the system v/c = O(1), which embodies the slow-motion approximation. By the Virial theorem, this velocity is related to the Newtonian potential U via U ∼ v 2 , which then implies that U = O(2) and embodies the weak-gravity approximation. These expansions can be thought of as two independent series: one in inverse powers of the speed of light c and the other in positive powers of Newton's gravitational constant.
Other quantities, such as matter densities and derivatives, can and should also be classified within this ordercounting scheme. Matter density ρ, pressure p and specific energy density Π, however, are slightly more complicated to classify because they are not dimensionless. Dimensionlessness can be obtained by comparing the pressure and the energy density to the matter density, which we assume is the largest component of the stressenergy tensor, namely p/ρ ∼ Π/ρ = O(2). Derivatives can also be classified in this fashion, where we find that ∂ t /∂ x = O(1). Such a relation can be derived by noting that ∂ t ∼ v i ∇ i , which comes from the Euler equations of hydrodynamics to Newtonian order.
With such an order-counting scheme developed, it is instructive to study the action of a single neutral particle. The Lagrangian of this system is given by
L = (g µν u µ u ν ) 1/2 , = -g 00 -2g 0i v i -g ij v i v j 1/2 (3)
where u µ = dx µ /dt = (1, v i ) is the 4-velocity of the particle and v i is its 3-velocity. From Eq. (3), note that knowledge of L to O(A) implies knowledge of g 00 to O(A), g 0i to O(A -1) and g ij to O(A -2). Therefore, since the Lagrangian is already known to O(2) (the Newtonian solution), the first PN correction to the equations of motion requires g 00 to O(4), g 0i to O(3) and g ij to O(2). Such order counting is the reason for calculating different sectors of the metric perturbation to different PN orders. A PPN analysis is usually performed in a particular background, which defines a particular coordinate system, and in an specific gauge, called the standard PPN gauge. The background is usually taken to be Minkowski because for solar system experiments deviations due to cosmological effects are negligible and can, in principle, be treated as adiabatic corrections. Moreover, one usually chooses a standard PPN frame, whose outer regions are at rest with respect to the rest frame of the universe. Such a frame, for example, forces the spatial sector of the metric to be diagonal and isotropic [6] . The gauge employed is very similar to the PN expansion of the Lorentz gauge of linearized gravitational wave theory. The differences between the standard PPN and Lorentz gauge are of O(3) and they allow for the presence of certain PPN potentials in the vectorial sector of the metric perturbation.
The last ingredient in the PPN recipe is the choice of a stress-energy tensor. The standard choice is that of a perfect fluid, given by
T µν = (ρ + ρΠ + p) u µ u ν + pg µν . (4)
Such a stress-energy density suffices to obtain the PN expansion of the gravitational field outside a fluid body, like the Sun, or of compact binary system. One can show that the internal structure of the fluid bodies can be neglected to 1 PN order by the effacement principle [39] in GR. Such effacement principle might actually not hold in modified field theories, but we shall study this subject elsewhere [47] .
With all these machinery, on can write down a supermetric [6] , namely
g 00 = -1 + 2U -2βU 2 -2ξΦ W + (2γ + 2 + α 3 + ζ 1 -2ξ) Φ 1 + 2 (3γ -2β + 1 + ζ 2 + ξ) Φ 2 + 2 (1 + ζ 3 ) Φ 3 + 2 (3γ + 3ζ 4 -2ξ) Φ 4 -(ζ 1 -2ξ) A, g 0i = - 1 2 (4γ + 3 + α 1 -α 2 + ζ 1 -2ξ) V i - 1 2 (1 + α 2 -ζ 1 + 2ξ) W i , g ij = (1 + 2γU ) δ ij , (5)
where δ ij is the Kronecker delta and where the PPN potentials
(U, Φ W , Φ 1 , Φ 2 , Φ 3 , Φ 4 , A, V i , W i )
are defined in Appendix A. Equation (5) describes a super-metric theory of gravity, because it reduces to different metric theories, such as GR or other alternative theories [6], through the appropriate choice of PPN parameters (γ, β, ξ, α 1 , α 2 , α 3 , ζ 1 , ζ 2 , ζ 3 , ζ 4
). One could obtain a more general form of the PPN metric by performing a post-Galilean transformation on Eq. ( 5 ), but such a procedure shall not be necessary in this paper.
The super-metric of Eq. ( 5 ) is parameterized in terms of a specific number of PPN potentials, where one usually employs certain criteria to narrow the space of possible potentials to consider. Some of these restriction include the following: the potentials tend to zero as an inverse power of the distance to the source; the origin of the coordinate system is chosen to coincide with the source, such that the metric does not contain constant terms; and the metric perturbations h 00 , h 0i and h ij transform as a scalar, vector and tensor. The above restrictions are reasonable, but, in general, an additional subjective condition is usually imposed that is based purely on simplicity: the metric perturbations are not generated by gradients or curls of velocity vectors or other generalized vector functions. As of yet, no reason had arisen for relaxing such a condition, but as we shall see in this paper, such terms are indeed needed for CS modified theories.
What is the physical meaning of all these parameters? One can understand what these parameters mean by calculating the generalized geodesic equations of motion and conservation laws [6] . For example, the parameter γ measures how much space-curvature is produced by a unit rest mass, while the parameter β determines how much "non-linearity" is there in the superposition law of gravity. Similarly, the parameter ξ determines whether there are preferred-location effects, while α i represent preferred-frame effects. Finally, the parameters ζ i measure the amount of violation of conservation of total momentum. In terms of conservation laws, one can interpret these parameters as measuring whether a theory is fully conservative, with linear and angular momentum conserved (ζ i and α i vanish), semi-conservative, with linear momentum conserved (ζ i and α 3 vanish), or nonconservative, where only the energy is conserved through lowest Newtonian order. One can verify that in GR, γ = β = 1 and all other parameters vanish, which implies that there are no preferred-location or frame effects and that the theory is fully conservative.
A PPN analysis of an alternative theory of gravity then reduces to mapping its solutions to Eq. ( 5 ) and then determining the PPN parameters in terms of intrinsic parameters of the theory. The procedure is simply as follows: expand the modified field equations in the metric perturbation and in the PN approximation; iteratively solve for the metric perturbation to O(4) in h 00 , to O(3) in h 0i and to O(2) in h ij ; compare the solution to the PPN metric of Eq. ( 5 ) and read off the PPN parameters of the alternative theory. We shall employ this procedure in Sec. V to obtain the PPN parameters of CS gravity.
In this section, we describe the basics of CS gravity, following mainly [29, 30] . In the standard CS formalism, GR is modified by adding a new term to the gravitational action. This term is given by [30]
S CS = m 2 pl 64π d 4 xf ( ⋆ R R) , (6)
where m pl is the Planck mass, f is a prescribed external quantity with units of squared mass (or squared length in geometrized units), R is the Ricci scalar and the star stands the dual operation, such that
R ⋆ R = 1 2 R αβγδ ǫ αβµν R γδ µν , (7)
with ǫ µνδγ the totally-antisymmetric Levi-Civita tensor and R µνδγ the Riemann tensor. Such a correction to the gravitational action is interesting because of the unavoidable parity violation that is introduced. Such parity violation is inspired from CP violation in the standard model, where such corrections act as anomaly-canceling terms. A similar scenario occurs in string theory, where the Green-Schwarz anomaly is canceled by precisely such a CS correction [32] , although CS gravity is not exclusively tied to string theory. Parity violation in CS gravity inexorably leads to birefringence in gravitational propagation, where here we mean that different polarization modes obey different propagation equations but travel at the same speed, that of light [29, 30, 36, 47] . If CS gravity were to lead to polarization modes that travel at different speeds, then one could use recently proposed experiments [17] to test this effect, but such is not the case in CS gravity. Birefringent gravitational waves, and thus CS gravity, have been proposed as possible explanations to the cosmic-microwave-background (CMB) anisotropies [36] , as well as the baryogenesis problem during the inflationary epoch [33] .
The magnitude of the CS correction is controlled by the externally-prescribed quantity f , which depends on the specific theory under consideration. When we consider CS gravity as an effective quantum theory, then the correction is suppressed by some mass scale M , which could be the electro-weak scale or some other scale, since it is unconstrained. In the context of string theory, the quantity f has been calculated only in conservative scenarios, where it was found to be suppressed by the Planck mass. In other scenarios, however, enhancements have been proposed, such as in cosmologies where the string coupling vanishes at late times [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58] , or where the field that generates f couples to spacetime regions with large curvature [59, 60] or stress-energy density [28, 47] . For simplicity, we here assume that this quantity is spatially homogeneous and its magnitude is small but non-negligible, so that we work to first order in the string-theoretical correction. Therefore, we treat ḟ as an independent perturbation parameter, [70] unrelated to ǫ, the PN perturbation parameter.
The field equations of CS modified gravity can be obtained by varying the action with respect to the metric. Doing so, one obtains
G µν + C µν = 8πT µν , (8)
where G µν is the Einstein tensor, T µν is a stress-energy tensor and C µν is the Cotton tensor. The latter tensor is defined via
C µν = - 1 √ -g f ,σ ǫ σαβ (µ D α R ν)β + (D σ f ,τ ) ⋆ R τ (µ σ ν) , (9)
where parenthesis stand for symmetrization, g is the determinant of the metric, D a stands for covariant differentiation and colon subscripts stand for partial differentiation.
Formally, the introduction of such a modification to the field equations leads to a new constraint, which is compensated by the introduction of the new scalar field degree of freedom f . This constraint originates by requiring that the divergence of the field equations vanish, namely
D µ C µν = 1 8 √ -g D ν f ( ⋆ RR) = 0, ( 10
)
where the divergence of the Einstein tensor vanished by the Bianchi identities. If this constraint is satisfied, then the equations of motion for the stress-energy D µ T µν are unaffected by CS gravity. A common source of confusion is that Eq. ( 10 ) is sometimes interpreted as requiring that R ⋆ R also vanish, which would then force the correction to the action to vanish. However, this is not the case because, in general, f is an exact form (d 2 f = 0) and, thus, Eq. ( 10 ) only implies an additional constraint that forces all solutions to the field equations to have a vanishing R ⋆ R.
The previous success of CS gravity in proposing plausible explanations to important cosmological problems prompts us to consider this extension of GR in the weakfield regime. For this purpose, it is convenient to rewrite the field equations in trace-reversed form, since this form is most amenable to a PN expansion. Doing so, we find,
R µν + C µν = 8π T µν - 1 2 g µν T , (11)
where the trace of the Cotton tensor vanishes identically and T = g µν T µν is the four dimensional trace of the stress-energy tensor. To linear order, the Ricci and Cotton tensors are given by [30]
R µν = - 1 2 h µν + O(h) 2 , C µν = - ḟ 2 ǫ0αβ (µ η h ν)β,α + O(h) 2 , ( 12
)
where ǫαβγδ is the Levi-Civita symbol, with convention ǫ0123 = +1, and η = -∂ 2 t + η ij ∂ i ∂ j is the flat space D'Alambertian, with η µν the Minkowski metric. In Eq. ( 12 ), we have employed the Lorentz gauge condition h µα, α = h ,µ /2, where h = g µν h µν is the four dimensional trace of the metric perturbation.
The Cotton tensor changes the characteristic behavior of the Einstein equations by forcing them to become third order instead of second order. Third-order partial differential equations are common in boundary layer theory [43] . However, in CS gravity, the third-order contributions are multiplied by a factor of f and we shall treat this function as a small independent expansion parameter. Therefore, the change in characteristics in the modified field equations can also be treated perturbatively, which is justified because eventhough ḟ might be enhanced by standard model currents, extra dimensions or a vanishing string coupling, it must still carry some type of mass suppression.
The trace-reversed form of the field equations is useful because it allows us to immediately find a formal solution. Inverting the D'Alambertian operator we obtain
H µν = -16π -1 η T µν - 1 2 g µν T + O(h) 2 , ( 13
)
where we have defined an effective metric perturbation as
H µν ≡ h µν + ḟ ǫ0αβ (µ h ν)β,α . (14)
Note that this formal solution is identical to the formal PN solution to the field equations in the limit ḟ → 0.
Also note that the second term in Eq. ( 14 ) is in essence a curl operator acting on the metric. This antisymmetric operator naturally forces the trace of the CS correction to vanish, as well as the 00 component and the symmetric spatial part.
From the formal solution to the modified field equations, we immediately identify the only two possible nonzero CS contributions: a coupling to the vector component of the metric h 0i ; and coupling to the transversetraceless part of the spatial metric h T T ij . The latter has already been studied in the gravitational wave context [29, 30, 47] and it vanishes identically if we require the spatial sector of the metric perturbation to be conformally flat. The former coupling is a new curl-type contribution to the metric perturbation that, to our knowledge, had so far been neglected both by the string theory and PPN communities. In fact, as we shall see in later sections, terms of this type will force us to introduce a new PPN parameter that is proportional to the curl of certain PPN potentials.
Let us conclude this section by pushing the formal solution to the modified field equations further to obtain a formal solution in terms of the actual metric perturbation h µν . Combining Eqs. ( 13 ) and ( 14 ) we arrive at the differential equation
h µν + ḟ ǫ0αβ (µ h ν)β,α = -16π -1 η T µν - 1 2 g µν T +O(h) 2 .
(15) Since we are searching for perturbations about the general relativistic solution, we shall make the ansatz
h µν = h (GR) µν + ḟ ζ µν + O(h) 2 , ( 16
)
where h
(GR) µν is the solution predicted by general relativity
h (GR) µν ≡ -16π -1 η T µν - 1 2 g µν T , (17)
and where ζ µν is an unknown function we are solving for. Inserting this ansatz into Eq. ( 15 ) we obtain
ζ µν + ḟ ǫ0αβ (µ ζ ν)β,α = 16πǫ 0αβ (µ ∂ α -1 η T ν)β - 1 2 g ν)β T . ( 18
)
We shall neglect the second term on the left-hand side because it would produce a second order correction. Such conclusion was also reached when studying parity violation in GR to explain certain features of the CMB [35] . We thus obtain the formal solution
ζ µν = 16πǫ 0αβ (µ ∂ α -1 η T ν)β - 1 2 g ν)β T (19)
and the actual metric perturbation to linear order becomes
h µν = -16π -1 η T µν - 1 2 η µν T (20)
+ 16π ḟ ǫkℓi -1
η δ i(µ T ν)ℓ,k - 1 2 δ i(µ η ν)ℓ T ,k + O(h) 2 ,
where we have used some properties of the Levi-Civita symbol to simplify this expression. The procedure presented here is general enough that it can be directly applied to study CS gravity in the PPN framework, as well as possibly find PN solutions to CS gravity.
In this section, we perform a PN expansion of the field equations and obtain a solution in the form of a PN se-ries. This solution then allows us to read off the PPN parameters by comparing it to the standard PPN supermetric [Eq. ( 5 )]. In this section we shall follow closely the methods of [6] and [61] and indices shall be manipulated with the Minkowski metric, unless otherwise specified.
Let us begin by expanding the field equations to second order in the metric perturbation. Doing so we find that the Ricci and Cotton tensors are given to second order by
R µν = - 1 2 η h µν -2h σ(µ,ν) σ + h ,µν - 1 2 h ρ λ 2h ρ(µ,ν)λ -h µν,ρλ -h ρλ,µν - 1 2 h ρλ ,µ h ρλ,ν + h λ µ,ρ h ρ ν,λ (21)
-h ρ µ,λ h ρν, λ + 1 2 h ,λ -2h λρ ,ρ h µν,λ -2h λ(µ,ν) + O(h) 3 , C µν = - ḟ 2 ǫ0αβ (µ η h ν)β,α -h σβ,αν) σ - ḟ 2 ǫ0αβ (µ h η h ν)β,α -h σβ,αν) σ + 1 2 2h ν)(λ,α) -h λα,ν) × η h λ β -2h σ (λ ,β) σ + h ,β λ -2 QR ν)β,α - ḟ 4 ǫσαβ (µ 2h 0 (σ,τ ) -h στ, 0 h τ [β,α]ν) -h ν)[β,α] τ - ḟ 2 h µλ ǫ0αβ(λ η h ν)β,α -h σβ,αν) σ - ḟ 2 ǫ0αβ(µ η h λ) β,α -h σβ,α σλ) h νλ + O(h) 3 . ( 22
)
where index contraction is carried out with the Minkowski metric and where we have not assumed any gauge condition. The operator Q(•) takes the quadratic part of its operand [of O(h) 2 ] and it is explained in more detail in Appendix B, where the derivation of the expansion of the Cotton tensor is presented in more detail. In this derivation, we have used the definition of the Levi-Civita tensor
ǫ αβγδ = (-g) 1/2 ǫαβγδ = 1 - 1 2 h ǫαβγδ + O(h) 2 , ( 23
)
ǫ αβγδ = -(-g) -1/2 ǫαβγδ = -1 + 1 2 h ǫαβγδ + O(h) 2 .
Note that the PN expanded version of the linearized Ricci tensor of Eq. ( 21 ) agrees with previous results [6] . Also note that if the Lorentz condition is enforced, several terms in both expressions vanish identically and the Cotton tensor to first order reduces to Eq. ( 12 ), which agrees with previous results [30] .
Let us now specialize the analysis to the standard PPN gauge. For this purpose, we shall impose the following gauge conditions
h jk, k - 1 2 h ,j = O(4), h 0k, k - 1 2 h k k,0 = O(5), (24)
where h k k is the spatial trace of the metric perturbation. Note that the first equation is the PN expansion of one of the Lorentz gauge conditions, while the second equation is not. This is the reason why the previous equations where not expanded in the Lorentz gauge. Nonetheless, such a gauge condition does not uniquely fix the coordinate system, since we can still perform an infinitesimal gauge transformation that leaves the modified field equations invariant. One can show that the Lorentz and PPN gauge are related to each other by such a gauge transformation. In the PPN gauge, then, the Ricci tensor takes the usual form
R 00 = - 1 2 ∇ 2 h 00 - 1 2 h 00,i h 00, i + 1 2 h ij h 00,ij + O(6), R 0i = - 1 2 ∇ 2 h 0i - 1 4 h 00,0i + O(5), R ij = - 1 2 ∇ 2 h ij + O(4), (25)
which agrees with previous results [6] , while the Cotton tensor reduces to
C 00 = O(6), C 0i = - 1 4 ḟ ǫ0kl i ∇ 2 h 0l,k + O(5), C ij = - 1 2 ḟ ǫ0kl (i ∇ 2 h j)l,k + O(4), (26)
where ∇ = η ij ∂ i ∂ j is the Laplacian of flat space [see Appendix B for the derivation of Eq. ( 26 ).] Note again the explicit appearance of two coupling terms of the Cotton tensor to the metric perturbation: one to the transversetraceless part of the spatial metric and the other to the vector metric perturbation. The PN expansions of the linearized Ricci and Cotton tensor then allow us to solve the modified field equations in the PPN framework.
In this section we shall proceed to systematically solve the modified field equation following the standard PPN iterative procedure [6] . We shall begin with the 00 and ij components of the metric to O(2), and then proceed with the 0i components to O(3) and the 00 component to O(4). Once all these components have been solved for in terms of PPN potentials, we shall be able to read off the PPN parameters adequate to CS gravity.
Let us begin with the modified field equations for the scalar sector of the metric perturbation. These equations are given to O(2) by
∇ 2 h 00 = -8πρ, (27)
because T = -ρ. Eq. ( 27 ) is the Poisson equation, whose solution in terms of PPN potentials is
h 00 = 2U + O(4). (28)
Let us now proceed with the solution to the field equation for the spatial sector of the metric perturbation. This equation to O(2) is given by
∇ 2 h ij + ḟ ǫ0kl (i ∇ 2 h j)l,k = -8πρδ ij , (29)
where note that this is the first appearance of a Cotton tensor contribution. Since the Levi-Civita symbol is a constant and ḟ is only time-dependent, we can factor out the Laplacian and rewrite this equation in terms of the effective metric H ij as
∇ 2 H ij = -8πρδ ij , (30)
where, as defined in Sec. III,
H ij = h ij + ḟ ǫ0kl (i h j)l,k . (31)
The solution of Eq. ( 30 ) can be immediately found in terms of PPN potentials as
H ij = 2U δ ij + O(4), (32)
which is nothing but Eq. ( 13 ). Recall, however, that in Sec. III we explicitly used the Lorentz gauge to simplify the field equations, whereas here we are using the PPN gauge. The reason why the solutions are the same is that the PPN and Lorentz gauge are indistinguishable to this order.
Once the effective metric has been solved for, we can obtain the actual metric perturbation following the procedure described in Sec. III. Combining Eq. ( 31 ) with Eq. ( 32 ), we arrive at the following differential equation
h ij + ḟ ǫ0kl (i h j)l,k = 2U δ ij . (33)
We look for solutions whose zeroth-order result is that predicted by GR and the CS term is a perturbative correction, namely
h ij = 2U δ ij + ḟ ζ ij , (34)
where ζ is assumed to be of O( ḟ ) 0 . Inserting this ansatz into Eq. ( 33 ) we arrive at
ζ ij + ḟ ǫ0kl (i ζ j)l,k = 0, ( 35
)
where the contraction of the Levi-Civita symbol and the Kronecker delta vanished. As in Sec. III, note that the second term on the left hand side is a second order correction and can thus be neglected to discover that ζ ij vanishes to this order. The spatial metric perturbation to O( 2 ) is then simply given by the GR prediction without any CS correction, namely
h ij = 2U δ ij + O(4). (36)
Physically, the reason why the spatial metric is unaffected by the CS correction is related to the use of a perfect fluid stress-energy tensor, which, together with the PPN gauge condition, forces the metric to be spatially conformally flat. In fact, if the spatial metric were not flat, then the spatial sector of the metric perturbation would be corrected by the CS term. Such would be the case if we had pursued a solution to 2 PN order, where the Landau-Lifshitz pseudo-tensor sources a non-conformal correction to the spatial metric [39] , or if we had searched for gravitational wave solutions, whose stress-energy tensor vanishes [29, 36] . In fact, one can check that, in such a scenario, Eq. ( 30 ) reduces to that found by [29, 30, 36, 47] as ρ → 0. We have then found that the weak-field expansion of the gravitational field outside a fluid body, like the Sun or a compact binary, is unaffected by the CS correction to O(2).
Let us now look for solutions to the field equations for the vector sector of the metric perturbation. The field equations to O(3) become
∇ 2 h 0i + 1 2 h 00,0i + 1 2 ḟ ǫ0kl i ∇ 2 h 0l,k = 16πρv i , (37)
where we have used that T 0i = -T 0i . Using the lower order solutions and the effective metric, as in Sec. III, we obtain
∇ 2 H 0i + U ,0i = 16πρv i , (38)
where the vectorial sector of the effective metric is
H 0i = h 0i + 1 2 ḟ ǫ0kl i h 0l,k . (39)
We recognize Eq. ( 38 ) as the standard GR field equation to O(3), except that the dependent function is the effective metric instead of the metric perturbation. We can thus solve this equation in terms of PPN potentials to obtain
H 0i = - 7 2 V i - 1 2 W i , (40)
where we have used that the superpotential X satisfies X ,0j = V j -W j (see Appendix A for the definitions.) Combining Eq. ( 39 ) with Eq. ( 40 ) we arrive at a differential equation for the metric perturbation, namely
h 0i + 1 2 ḟ ǫ0kl i h 0l,k = - 7 2 V i - 1 2 W i . ( 41
)
Once more, let us look for solutions that are perturbation about the GR prediction, namely
h 0i = - 7 2 V i - 1 2 W i + ḟ ζ i , (42)
where we again assume that ζ i is of O( ḟ ) 0 . The field equation becomes
ζ i + 1 2 ḟ (∇ × ζ) i = 1 2 7 2 (∇ × V ) i + 1 2 (∇ × W ) i , (43)
where (∇ × A) i = ǫ ijk ∂ j A k is the standard curl operator of flat space. As in Sec. III, note once more that the second term on the left-hand side is again a second order correction and we shall thus neglect it. Also note that the curl of the V i potential happens to be equal to the curl of the W i potential. The solution for the vectorial sector of the actual gravitational field then simplifies to
h 0i = - 7 2 V i - 1 2 W i + 2 ḟ (∇ × V ) i + O(5). (44)
We have arrived at the first contribution of CS modified gravity to the metric for a perfect fluid source. Chern-Simons gravity was previously seen to couple to the transverse-traceless sector of the metric perturbation for gravitational wave solutions [29, 30, 36, 47] . The CS correction is also believed to couple to Noether vector currents, such as neutron currents, which partially fueled the idea that this correction could be enhanced. However, to our knowledge, this correction was never thought to couple to vector metric perturbations. From the analysis presented here, we see that in fact CS gravity does couple to such terms, even if the matter source is neutrally charged. The only requirement for such couplings is that the source is not static, ie. that the object is either moving or spinning relative to the PPN rest frame so that the PPN vector potential does not vanish. The latter is suppressed by a relative O(1) because in the far field the velocity of a compact object produces a term of O(3) in V i , while the spin produces a term of O(4). In a later section, we shall discuss some of the physical and observational implications of such a modification to the metric.
A full analysis of the PPN structure of a modified theory of gravity requires that we solve for the 00 component of the metric perturbation to O(4). The field equations to this order are
- 1 2 ∇ 2 h 00 - 1 2 h 00,i h 00,i + 1 2 h ij h 00,ij = 4πρ [1 + 2 v 2 -U + 1 2 Π + 3 2 p ρ , (45)
where the CS correction does not contribute at this order (see Appendix B.) Note that the h 0i sector of the metric perturbation to O(3) does not feed back into the field equations at this order either. The terms that do come into play are the h 00 and h ij sectors of the metric, which are not modified to lowest order by the CS correction. The field equation, thus, reduce to the standard one of GR, whose solution in terms of PPN potentials is
h 00 = 2U -2U 2 + 4Φ 1 + 4Φ 2 + 2Φ 3 + 6Φ 4 + O(6). ( 46
)
We have thus solved for all components of the metric perturbation to 1 PN order beyond the Newtonian answer, namely g 00 to O(4), g 0i to O(3) and g ij to O(2).
We now have all the necessary ingredients to read off the PPN parameters of CS modified gravity. Let us begin by writing the full metric with the solutions found in the previous subsections:
g 00 = -1 + 2U -2U 2 + 4Φ 1 + 4Φ 2 + 2Φ 3 + 6Φ 4 + O(6), g 0i = - 7 2 V i - 1 2 W i + 2 ḟ (∇ × V ) i + O(5)
,
g ij = (1 + 2U ) δ ij + O(4). ( 47
)
can verify that this metric is indeed a solution of Eqs. ( 27 ), ( 29 ), (37) and (45) to the appropriate PN order and to first order in the CS coupling parameter. Also note that the solution of Eq. ( 47 ) automatically satisfies the constraint ⋆ RR = 0 to linear order because the contraction of the Levi-Civita symbol with two partial derivatives vanishes. Such a solution is then allowed in CS gravity, just as other classical solutions are [62] , and the equations of motion for the fluid can be obtained directly from the covariant derivative of the stress-energy tensor.
We can now read off the PPN parameters of the CS modified theory by comparing Eq. ( 5 ) to Eq. (47) . A visual inspection reveals that the CS solution is identical to the classical GR one, which implies that γ = β = 1, ζ = 0 and α 1 = α 2 = α 3 = ξ 1 = ξ 2 = ξ 3 = ξ 4 = 0 and there are no preferred frame effects. However, Eq. ( 5 ) contains an extra term that cannot be modeled by the standard PPN metric of Eq. ( 5 ), namely the curl contribution to g 0i . We then see that the PPN metric must be enhanced by the addition of a curl-type term to the 0i components of the metric, namely
g 0i ≡ - 1 2 (4γ + 3 + α 1 -α 2 + ζ 1 -2ξ) V i - 1 2 (1 + α 2 -ζ 1 + 2ξ) W i + χ (r∇ × V ) i , ( 48
)
where χ is a new PPN parameter and where we have multiplied the curl operator by the radial distance to the source, r, in order to make χ a proper dimensionless parameter. Note that there is no need to introduce any additional PPN parameters because the curl of W i equals the curl of V i . In fact, we could have equally parameterized the new contribution to the PPN metric in terms of the curl of W i , but we chose not to because V i appears more frequently in PN theory. For the case of CS modified gravity, the new χ parameter is simply
χ = 2 ḟ r , (49)
which is dimensionless since ḟ has units of length. If an experiment could measure or place bounds on the value of χ, then ḟ could also be bounded, thus placing a constraint on the CS coupling parameter.
In this section we shall propose a physical interpretation to the CS modification to the PPN metric and we shall calculate some GR predictions that are modified by this correction. This section, however, is by no means a complete study of all the possible consequences of the CS correction, which is beyond the scope of this paper.
Let us begin by considering a system of A nearly spherical bodies, for which the gravitational vector potentials are simply [6]
V i = A m A r A v i A + 1 2 A J A r 2 A × n A i , (50)
W i = A m A r A (v A • n A ) n i A + 1 2 A J A r 2 A × n A i ,
where m A is the mass of the Ath body, r A is the field point distance to the Ath body, n i A = x i A /r A is a unit vector pointing to the Ath body, v A is the velocity of the Ath body and J i A is the spin-angular momentum of the Ath For example, the spin angular momentum for a Kerr spacetime is given by J = m 2 a i , where a is the dimensionless Kerr spin parameter. Note that if A = 2 then the system being modeled could be a binary of spinning compact objects, while if A = 1 it could represent the field of the sun or that of a rapidly spinning neutron star or pulsar. In obtaining Eq. ( 50 ), we have implicitly assumed a point-particle approximation, which in classical GR is justified by the effacement principle. This principle postulates that the internal structure of bodies contributes to the solution of the field equations to higher PN order. One can verify that this is indeed the case in classical GR, where internal structure contributions appear at 5 PN order. In CS gravity, however, it is a priori unclear whether an analogous effacement principle holds because the CS term is expected to couple with matter current via standard model-like interactions. If such is the case, it is possible that a "mountain" on the surface of a neutron star [63] or an r-mode instability [64, 65, 66] enhances the CS contribution. In this paper, however, we shall neglect these interactions, and relegate such possibilities to future work [47] .
With such a vector potential, we can calculate the CS correction to the metric. For this purpose, we define the correction δg 0i ≡ g 0i -g
(GR) 0i
, where g
(GR) 0i
is the GR prediction without CS gravity. We then find that the CS corrections is given by
δg 0i = 2 A ḟ r A m A r A (v A × n A ) i - J i A 2r 2 A + 3 2 (J A • n A ) r 2 A n i A , (51)
where the • operator is the flat-space inner product and where we have used the identities ǫijk ǫklm = δ il δ jmδ im δ jl and ǫilk ǫjlm = 2δ ij . Note that the first term of Eq. ( 51 ) is of O(3), while the second and third terms are of O(4) as previously anticipated. Also note that ḟ couples both to the spin and orbital angular momentum. Therefore, whether the system under consideration is the Solar system (v i of the Sun is zero while J i is small), the Crab pulsar (v i is again zero but J i is large) or a binary system of compact objects (neither v i nor J i vanish), there will in general be a non-vanishing coupling between the CS correction and the vector potential of the system.
From the above analysis, it is also clear that the CS correction increases with the non-linearity of the spacetime. In other words, the CS term is larger not only for systems with large velocities and spins, but also in regions near the source. For a binary system, this fact implies that the CS correction is naturally enhanced in the last stages of inspiral and during merger. Note that this enhancement is different from all previous enhancements proposed, since it does not require the presence of charge [28, 47] , a fifth dimension with warped compactifications [59, 60] , or a vanishing string coupling [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58] . Unfortunately, the end of the inspiral stage coincides with the edge of the PN region of validity and, thus, a com-plete analysis of such a natural enhancement will have to be carried out through numerical simulations.
In the presence of a source with the vector potentials of Eq. ( 50 ), we can write the vectorial sector of the metric perturbation in a suggestive way, namely
g 0i = A - 7 2 m A r A v i A - A m A 6r 2 A v A -v (ef f ) A i (52) - 1 2 A n i A m A r A v (ef f ) A • n A -2 A J (ef f ) A r 2 A × n A i ,
where we have defined an effective velocity and angular momentum vector via
v i A(ef f ) = v i A -6 ḟ J i A m A r 2 A , J i A(ef f ) = J i A -ḟ m A v i A , (53)
or in terms of the Newtonian orbital angular momentum
L i A(N ) = r A × p A and linear momentum p i A(N ) = m A v i A L i A(ef f ) = L i A(N ) -6 ḟ (n A × J A ) i , J i A(ef f ) = J i A -ḟ p i A . (54)
From this analysis, it is clear that the CS corrections seems to couple to both a quantity that resembles the orbital and the spin angular momentum vector. Note that when the spin angular momentum vanishes the vectorial metric perturbation is identical to that of a spinning moving fluid, but where the spin is induced by the coupling of the orbital angular momentum to the CS term. The presence of an effective CS spin angular momentum in non-spinning sources leads to an interesting physical interpretation. Let us model the field that sources ḟ as a fluid that permeates all of spacetime. This field could be, for example, a model-independent axion, inspired by the quantity introduced in the standard model to resolve the strong CP problem [67] . In this scenario, then the fluid is naturally "dragged" by the motion of any source and the CS modification to the metric is nothing but such dragging. This analogy is inspired by the ergosphere of the Kerr solution, where inertial frames are dragged with the rotation of the black hole. In fact, one could push this analogy further and try to construct the shear and bulk viscosity of such a fluid, but we shall not attempt this here. Of course, this interpretation is to be understood only qualitatively, since its purpose is only to allow the reader to picture the CS modification to the metric in physical terms.
An alternative interpretation can be given to the CS modification in terms of the gravito-electro-magnetic (GED) analogy [40, 41] , which shall allow us to easily construct the predictions of the modified theory. In this analogy, one realizes that the PN solution to the linearized field equations can be written in terms of a potential and vector potential, namely
ds 2 = -(1 -2Φ) dt 2 -4 (A • dx) dt + (1 + 2Φ) δ ij dx i dx j , (55)
where Φ reduces to the Newtonian potential U in the Newtonian limit [41] and A i is a vector potential related to the metric via A i = -g 0i /4. One can then construct GED fields in analogy to Maxwell's electromagnetic theory via
E i = -(∇Φ) i -∂ t 1 2 A i , B i = (∇ × A) i , (56)
which in terms of the vectorial sector of the metric perturbation becomes
E i = -(∇Φ) i + 1 8 ġi , B i = - 1 4 (∇ × g) i , (57)
where we have defined the vector g i = g 0i . The geodesic equations for a test particle then reduce to the Lorentz force law, namely
F i = -mE i -2m (v × B) i . ( 58
)
We can now work out the effect of the CS correction on the GED fields and equations of motion. First note that the CS correction only affects g. We can then write the CS modification to the Lorentz force law by defining δa i = a i -a i (GR) , where a i (GR) is the acceleration vector predicted by GR, to obtain,
δa i = 1 8 δ ġi + 1 2 (v × δΩ) i , (59)
where we have defined the angular velocity
δΩ i = (∇ × δg) i . ( 60
)
The time derivative of the vector g i is of O( 5 ) and can thus be neglected, but the angular velocity cannot and it is given by
δΩ i = - A ḟ m A r 3 A 3 (v A • n A ) n i A -v i A , (61)
which is clearly of O(3). Note that although the first term between square brackets cancels for circular orbits because n i A is perpendicular to v i A to Newtonian order, the second term does not. The angular velocity adds a correction to the acceleration of O(4), namely
δa i = - 3 2 A ḟ m A r 3 A (v A • n A ) (v A × n A ) i , (62)
which for a system in circular orbit vanishes to Newtonian order. One could use this formalism to find the perturbations in the motion of moving objects by integrating Eq. ( 62 ) twice. However, for systems in a circular orbit, such as the Earth-Moon system or compact binaries, this correction vanishes to leading order. Therefore, lunar ranging experiments [68] might not be able to constraint ḟ .
Another correction to the predictions of GR is that of the precession of gyroscopes by the so-called Lense-Thirring or frame-dragging effect. In this process, the spin angular momentum of a source twists spacetime in such a way that gyroscopes are dragged with it. The precession angular velocity depends on the vector sector of the metric perturbation via Eq. ( 61 ). Thus, the full Lense-Thirring term in the precession angular velocity of precessing gyroscopes is
Ω i LT = - 1 r 3 A A J i A(ef f ) -3n i A J A(ef f ) • n A i . ( 63
)
Note that this angular velocity is identical to the GR prediction, except for the replacement J i A → J i A(ef f ) . In CS modified gravity, then, the Lense-Thirring effect is not only produced by the spin angular momentum of the gyroscope but also by the orbital angular momentum that couples to the CS correction. Therefore, if an experiment were to measure the precession of gyroscopes by the curvature of spacetime (see, for example, Gravity Probe B [42] ) one could constraint ḟ and thus some intrinsic parameters of string theory. Note, however, that the CS correction depends on the velocity of the bodies with respect to the inertial PPN rest-frame. In order to relate these predictions to the quantities that are actually measured in the experiment, one would have to transform to the experiment's frame, or perhaps to a basis aligned with the direction of distant stars [6] .
Are there other experiments that could be performed to measure such a deviation from GR? Any experiment that samples the vectorial sector of the metric would in effect be measuring such a deviation. In this paper, we have only discussed modifications to the frame-dragging effect and the acceleration of bodies through the GED analogy, but this need not be the only corrections to classical GR predictions. In fact, any predictions that depends on g 0i indirectly, for example via Christoffel symbols, will probably also be modified unless the corrections is fortuitously canceled. In this paper, we have laid the theoretical foundations of the weak-field correction to the metric due to CS gravity and studied some possible corrections to classical predictions. A detailed study of other corrections is beyond the scope of this paper.
We have studied the weak-field expansion of the solution to the CS modified field equations in the presence of a perfect fluid PN source in the point particle limit. Such an expansion required that we linearize the Ricci and Cotton tensor to second order in the metric perturbation without any gauge assumption. An iterative PPN formalism was then employed to solve for the metric perturbation in this modified theory of gravity. We have found that CS gravity possesses the same PPN parameters as those of GR, but it also requires the introduction of a new term and PPN parameter that is proportional to the curl of the PPN vector potentials. Such a term is enhanced in non-linear scenarios without requiring the presence of standard model currents, large extra dimensions or a vanishing string coupling.
We have proposed an interpretation for the new term in the metric produced by CS gravity and studied some of the possible consequences it might have on GR predictions. The interpretation consists of picturing the field that sources the CS term as a fluid that permeates all of spacetime. In this scenario, the CS term is nothing but the "dragging" of the fluid by the motion of the source. Irrespective of the validity of such an interpretation, the inclusion of a new term to the weak-field expansion of the metric naturally leads to corrections to the standard GR predictions. We have studied the acceleration of point particles and the Lense-Thirring contribution to the precession of gyroscopes. We have found that both corrections are proportional to the CS coupling parameter and, therefore, experimental measurement of these effects might be used to constraint CS and, possibly, string theory.
Future work could concentrate on studying further the non-linear enhancement of the CS correction and the modifications to the predictions of GR. The PPN analysis performed here breaks down very close to the source due to the use of a point particle approximation in the stress energy tensor. One possible research route could consists of studying the CS correction in a perturbed Kerr background [69] . Another possible route could be to analyze other predictions of the theory, such as the perihelion shift of Mercury or the Nordtvedt effect. Furthermore, in light of the imminent highly-accurate measurement of the Lense-Thirring effect by Gravity Probe B, it might be useful to revisit this correction in a frame better-adapted to the experimental setup. Finally, the CS modification to the weak-field metric might lead to non-conservative effects and the breaking of the effacement principle [47] , which could be studied through the evaluation of the gravitational pseudo stress-energy tensor. Ultimately, it will be experiments that will determine the viability of CS modified gravity and string theory.
APPENDIX A: PPN POTENTIALS
In this appendix, we present explicit expressions for the PPN potentials used to parameterize the metric in Eq. ( 5 ). These potentials are the following:
U ≡ ρ |x -x ′ | d 3 x ′ , V i ≡ ρ ′ v ′ i |x -x ′ | d 3 x ′ , W i ≡ ρ ′ v ′ j (x -x ′ ) j (x -x ′ ) i |x -x ′ | 3 d 3 x ′ , Φ W ≡ ρ ′ ρ ′′ (x -x ′ ) i |x -x ′ | 3 (x ′ -x ′′ ) i |x -x ′′ | - (x -x ′′ ) i |x ′ -x ′′ | d 3 x ′ d 3 x ′′ , Φ 1 ≡ ρ ′ v ′2 |x -x ′ | d 3 x ′ , Φ 2 ≡ ρ ′ U ′ |x -x ′ | d 3 x ′ , Φ 3 ≡ ρ ′ Π ′ |x -x ′ | d 3 x ′ , Φ 4 ≡ p ′ |x -x ′ | d 3 x ′ , A ≡ ρ ′ v ′ i (x -x ′ ) i 2 |x -x ′ | d 3 x ′ , X ≡ ρ ′ |x -x ′ |d 3 x ′ . (A1)
These potentials satisfy the following relations
∇ 2 U = -4πρ, ∇ 2 V i = -4πρv i , ∇ 2 Φ 1 = -4πρv 2 , ∇ 2 Φ 2 = -4πρU, ∇ 2 Φ 3 = -4πρΠ, ∇ 2 Φ 4 = -4πp, ∇ 2 X = -2U (A2)
The potential X is sometimes referred to as the superpotential because it acts as a potential for the Newtonian potential.
In this appendix, we present some more details on the derivation of the linearized Cotton tensor to second order. We begin with the definition of the Cotton tensor [30] in terms of the symmetrization operator, namely
C µν = - 1 √ -g (D σ f ) ǫ σαβ(µ D α R ν) β + (D στ f ) ⋆ R τ (µ|σ|ν) . (B1)
Using the symmetries of the Levi-Civita and Riemann tensor, as well as the fact that f depends only on time, we can simplify the Cotton tensor to
C µν = (-g) -1 ḟ ǫ0αβ(µ R ν) β,α + ǫ0αβ(µ Γ ν) λα R λ β + 1 2 Γ 0 στ ǫσαβ(µ R ν)τ αβ . (B2)
Noting that the determinant of the metric is simply g = -1 + h, so that (-g) -1 = 1 + h, we can identify four terms in the Cotton tensor
C µν A = ḟ ǫ0αβ(µ LR ν) β,α , C µν B = ḟ ǫ0αβ(µ h ρρ LR ν) β,α , C µν C = ḟ ǫ0αβ(µ LΓ ν) λα LR λ β , C µν D = ḟ 2 ǫσαβ(µ LΓ 0 στ LR ν)τ αβ , C µν E = ḟ ǫ0αβ(µ QR ν) β,α , (B3)
where the L operator stands for the linear part of its operand, while the Q operator isolates the quadratic part of its operand. For example, if we act L and Q on (1+h) n , where n is some integer, we obtain
L(1 + h) n = nh, Q(1 + h) n = n(n -1) 2 h 2 . ( B4
)
Let us now compute each of these terms separately. The first four terms are given by
C µν A = -
ḟ 2 ǫ0αβ(µ η h ν) β,α -h σβ, ν ασ , C µν B = -ḟ 2 hǫ 0αβ(µ η h ν) β,α -h σβ, ν ασ , C µν C = -ḟ 4 ǫ0αβ(µ h ν) λ,α + h ν α,λ -h λα, ν) × η h λ β -h σ λ ,β σ -h σβ, λσ + h , λ β , C µν D = ḟ 4 ǫσαβ(µ 2h 0 (σ,τ ) -h στ, 0 h τ [β,α] ν -h ν [β,α] τ .
(B5)
The last term of the Cotton tensor is simply the derivative of the Ricci tensor which we already calculated to second order in Eq. (21) . In order to avoid notation clutter, we shall not present it again here, but instead we combine all the Cotton tensor pieces to obtain
C µν = - ḟ 2
ǫ0αβ(µ η h ν) β,α -h σβ,α σν) -ḟ 2 ǫ0αβ(µ h η h ν) β,α -h σβ,α σν) + 1 2 2h ν) (λ,α) -h λα, ν) (B6) × η h λ β -2h σ (λ ,β) σ + h ,β λ -2 QR ν) β,α + ḟ 4 ǫσαβ(µ 2h 0 (σ,τ ) -h στ, 0 h τ [β,α] ν) -h ν) [β,α] τ + O(h) 3 where its covariant form is C µν = -ḟ 2 ǫ0αβ (µ η h ν)β,α -h σβ,αν) σḟ 2 ǫ0αβ (µ h η h ν)β,α -h σβ,αν) σ + 1 2 2h ν)(λ,α) -h λα,ν) × η h λ β -2h σ (λ ,β) σ + h ,β λ -2 QR ν)β,α + h νλ η h λ) β,α -h σβ,α σλ) + ḟ 4 ǫσαβ (µ 2h 0 (σ,τ ) -h στ, 0 h τ [β,α]ν) -h ν)[β,α] τḟ 2 h µλ ǫ0αβ(λ η h ν)β,α -h σβ,αν) σ + O(h) 3 . (B7) For the PPN mapping of CS modified gravity, only the 00 component of the metric is needed to second order, which implies we only need C 00 to O(h) 2 . This component is given by C 00 = ḟ 4 ǫijk 0 2h 0 (i,ℓ) -h iℓ, 0 h ℓ [k,j]0 -h 0[k,j] ℓ ḟ 2 h 0ℓ ǫ0jk(ℓ η h 0k,j -h ik,j0 i + O(h) 3 , (B8)
where in fact the last term vanishes due to the PPN gauge condition. Note that this term is automatically of O( 6 ), which is well beyond the required order we need in h 00 . | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We investigate the weak-field, post-Newtonian expansion to the solution of the field equations in Chern-Simons gravity with a perfect fluid source. In particular, we study the mapping of this solution to the parameterized post-Newtonian formalism to 1 PN order in the metric. We find that the PPN parameters of Chern-Simons gravity are identical to those of general relativity, with the exception of the inclusion of a new term that is proportional to the Chern-Simons coupling parameter and the curl of the PPN vector potentials. We also find that the new term is naturally enhanced by the non-linearity of spacetime and we provide a physical interpretation for it. By mapping this correction to the gravito-electro-magnetic framework, we study the corrections that this new term introduces to the acceleration of point particles and the frame-dragging effect in gyroscopic precession. We find that the Chern-Simons correction to these classical predictions could be used by current and future experiments to place bounds on intrinsic parameters of Chern-Simons gravity and, thus, string theory."
},
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Tests of alternative theories of gravity that modify general relativity (GR) at a fundamental level are essential to the advancement of physics. One formalism that has had incredible success in this task is the parameterized post-Newtonian (PPN) framework [1, 2, 3, 4, 5, 6] . In this formalism, the metric of the alternative theory is solved for in the weak-field limit and its deviations from GR are expressed in terms of PPN parameters. Once a metric has been obtained, one can calculate predictions of the alternative theory, such as light deflection and the perihelion shift of Mercury, which shall depend on these PPN parameters. Therefore, experimental measurements of such physical effects directly lead to constraints on the parameters of the alternative theory. This framework, together with the relevant experiments, have already been successfully employed to constrain scalar-tensor theories (Brans-Dicke, Bekenstein) [7] , vector-tensor theories (Will-Nordtvedt [8] , Hellings-Nordtvedt [9] ), bimetric theories (Rosen [10, 11] ) and stratified theories (Ni [12] ) (see [13] for definitions and an updated review.)\n\nOnly recently has this framework been used to study quantum gravitational and string-theoretical inspired ideas. On the string theoretical side, Kalyana [14] investigated the PPN parameters associated with the gravitondilaton system in low-energy string theory. More recently, Ivashchuk, et. al. [15] studied PPN parameters in the context of general black holes and p-brane spherically symmetric solutions, while Bezerra, et. al. [16] considered domain wall spacetimes for low energy effective string theories and derived the corresponding PPN parameters for the metric of a wall. On the quantum gravitational side, Gleiser and Kozameh [17] and more recently Fan, et. al. [18] studied the possibility of testing gravitational birefringence induced by quantum gravity, which was proposed by Amelino-Camelia, el. al. [19] and Gambini and Pullin [20] . Other non-PPN proposals have been also put forth to test quantum gravity, for example through gravitational waves [21, 22, 23, 24, 25, 26, 27, 28 ], but we shall not discuss those tests here.\n\nChern-Simons (CS) gravity [29, 30] is one such extension of GR, where the gravitational action is modified by the addition of a parity-violating term. This extension is promising because it is required by all 4dimensional compactifications of string theory [31] for mathematical consistency because it cancels the Green-Schwarz anomaly [32] . CS gravity, however, is not unique to string theory and in fact has its roots in the standard model, where it arises as a gravitational anomaly provided that there are more flavours of left handed leptons than right handed ones. Moreover the CS extension to GR can arise via the embedding of the three dimensional Chern-Simons topological current into a 4D space-time manifold, decsribed by Jackiw and Pi [30] Chern-Simons gravity has been recently studied in the cosmological context. In particular, this framework was used to shed light on the anisotropies of the cosmic microwave background (CMB) [33, 34, 35] and the leptogenesis problem [34, 36, 37] . Parity violation has also been shown to produce birefringent gravitational waves [28, 29] , where different polarizations modes acquire varying amplitudes. These modes obey different propagation equations because the imaginary sector of the classical dispersion relation is CS corrected. Different from [20] , in CS birefringence the velocity of the gravitational wave remains that of light.\n\nIn this paper we study CS gravity in the PPN framework, extending the analysis of [38] and providing some missing details. In particular, we shall consider the effect of the CS correction to the gravitational field of, for instance, a pulsar, a binary system or a star in the weak-field limit. These corrections are obtained by solv-ing the modified field equations in the weak-field limit for post-Newtonian (PN) sources, defined as those that are weakly-gravitating and slowly-moving [39] . Such an expansion requires the calculation of the Ricci and Cotton tensors to second order in the metric perturbation. We then find that CS gravity leads to the same gravitational field as that of classical GR and, thus, the same PPN parameters, except for the inclusion of a new term in the vectorial sector of the metric, namely\n\ng (CS) 0i = 2 ḟ (∇ × V ) i , (1)\n\nwhere ḟ acts as a coupling parameter of CS theory and V i is a PPN potential. We also show that this solution can be alternatively obtained by finding a formal solution to the modified field equations and performing a PN expansion, as is done in PN theory. The full solution is further shown to satisfy the additional CS constraint, which leads to equations of motion given only by the divergence of the stress-energy tensor.\n\nThe CS correction to the metric found here leads to an interesting interpretation of CS gravity and forces us to consider a new type of coupling. The interpretation consists of thinking of the field that sources the CS correction as a fluid that permeates all of spacetime. Then the CS correction in the metric is due to the \"dragging\" of such a fluid by the motion of the source. Until now, couplings of the CS correction to the angular momentum of the source had been neglected by the string theory community. Similarly, curl-type terms had also been considered unnecessary in the traditional PPN framework, since previous alternative gravity theories had not required it. As we shall show, in CS gravity and thus in string theory, such a coupling is naturally occurring. Therefore, a proper PPN mapping requires the introduction of a new curl-type term with a corresponding new PPN parameter of the type of Eq. (1) .\n\nA modification to the gravitational field leads naturally to corrections of the standard predictions of GR. In order to illustrate such a correction, we consider the CS term in the gravito-electro-magnetic analogy [40, 41] , where we find that the CS correction accounts for a modification of gravitomagnetism. Furthermore, we calculate the modification to the acceleration of point particles and the frame dragging effect in the precession of gyroscopes. We find that these corrections are given by\n\nδa i = - 3 2 ḟ r G m c 2 r 2 v c • n v c × n i , δΩ i = - ḟ r G m c 3 r 2 3 v c • n n i - v i c , (2)\n\nwhere m and v are the mass and velocities of the source, while r is the distance to the source and n i = x i /r is a unit vector, with • and × the flat-space scalar and cross products. Both corrections are found to be naturally enhanced in regions of high spacetime curvature. We then conclude that experiments that measure the gravitomagnetic sector of the metric either in the weak-field (such as Gravity Probe B [42] ) and particularly in the non-linear regime, will lead to a direct constraint on the CS coupling parameter ḟ . In this paper we develop the details of how to calculate these corrections, while the specifics of how to actually impose a constraint, which depend on the experimental setup, are beyond the scope of this paper.\n\nThe remainder of this paper deals with the details of the calculations discussed in the previous paragraphs. We have divided the paper as follows: Sec. II describes the basics of the PPN framework; Sec. III discusses CS modified gravity, the modified field equations and computes a formal solution; Sec. IV expands the field equations to second order in the metric perturbation; Sec. V iteratively solves the field equations in the PN approximation and finds the PPN parameters of CS gravity; Sec. VI discusses the correction to the acceleration of point particles and the frame dragging effect; Sec. VII concludes and points to future research.\n\nThe conventions that we use throughout this work are the following: Greek letters represent spacetime indices, while Latin letters stand for spatial indices only; semicolons stand for covariant derivatives, while colons stand for partial derivatives; overhead dots stand for derivatives with respects to time. We denote uncontrolled remainders with the symbol O(A), which stands for terms of order A. We also use the Einstein summation convention unless otherwise specified. Finally, we use geometrized units, where G = c = 1, and the metric signature (-, +, +, +)."
},
{
"section_type": "OTHER",
"section_title": "II. THE ABC OF PPN",
"text": "In this section we summarize the basics of the PPN framework, following [6] . This framework was first developed by Eddington, Robertson and Schiff [1, 6] , but it came to maturity through the seminal papers of Nordtvedt and Will [2, 3, 4, 5] . In this section, we describe the latter formulation, since it is the most widely used in experimental tests of gravitational theories.\n\nThe goal of the PPN formalism is to allow for comparisons of different metric theories of gravity with each other and with experiment. Such comparisons become manageable through a slow-motion, weak-field expansion of the metric and the equations of motion, the so-called PN expansion. When such an expansion is carried out to sufficiently high but finite order, the resultant solution is an accurate approximation to the exact solution in most of the spacetime. This approximation, however, does break down for systems that are not slowly-moving, such as merging binary systems, or weakly gravitating, such as near the apparent horizons of black hole binaries. Nonetheless, as far as solar system tests are concerned, the PN expansion is not only valid but also highly accurate.\n\nThe PPN framework employs an order countingscheme that is similar to that used in multiple-scale anal-ysis [43, 44, 45, 46] . The symbol O(A) stands for terms of order ǫ A , where ǫ ≪ 1 is a PN expansion parameter. For convenience, it is customary to associate this parameter with the orbital velocity of the system v/c = O(1), which embodies the slow-motion approximation. By the Virial theorem, this velocity is related to the Newtonian potential U via U ∼ v 2 , which then implies that U = O(2) and embodies the weak-gravity approximation. These expansions can be thought of as two independent series: one in inverse powers of the speed of light c and the other in positive powers of Newton's gravitational constant.\n\nOther quantities, such as matter densities and derivatives, can and should also be classified within this ordercounting scheme. Matter density ρ, pressure p and specific energy density Π, however, are slightly more complicated to classify because they are not dimensionless. Dimensionlessness can be obtained by comparing the pressure and the energy density to the matter density, which we assume is the largest component of the stressenergy tensor, namely p/ρ ∼ Π/ρ = O(2). Derivatives can also be classified in this fashion, where we find that ∂ t /∂ x = O(1). Such a relation can be derived by noting that ∂ t ∼ v i ∇ i , which comes from the Euler equations of hydrodynamics to Newtonian order.\n\nWith such an order-counting scheme developed, it is instructive to study the action of a single neutral particle. The Lagrangian of this system is given by\n\nL = (g µν u µ u ν ) 1/2 , = -g 00 -2g 0i v i -g ij v i v j 1/2 (3)\n\nwhere u µ = dx µ /dt = (1, v i ) is the 4-velocity of the particle and v i is its 3-velocity. From Eq. (3), note that knowledge of L to O(A) implies knowledge of g 00 to O(A), g 0i to O(A -1) and g ij to O(A -2). Therefore, since the Lagrangian is already known to O(2) (the Newtonian solution), the first PN correction to the equations of motion requires g 00 to O(4), g 0i to O(3) and g ij to O(2). Such order counting is the reason for calculating different sectors of the metric perturbation to different PN orders. A PPN analysis is usually performed in a particular background, which defines a particular coordinate system, and in an specific gauge, called the standard PPN gauge. The background is usually taken to be Minkowski because for solar system experiments deviations due to cosmological effects are negligible and can, in principle, be treated as adiabatic corrections. Moreover, one usually chooses a standard PPN frame, whose outer regions are at rest with respect to the rest frame of the universe. Such a frame, for example, forces the spatial sector of the metric to be diagonal and isotropic [6] . The gauge employed is very similar to the PN expansion of the Lorentz gauge of linearized gravitational wave theory. The differences between the standard PPN and Lorentz gauge are of O(3) and they allow for the presence of certain PPN potentials in the vectorial sector of the metric perturbation.\n\nThe last ingredient in the PPN recipe is the choice of a stress-energy tensor. The standard choice is that of a perfect fluid, given by\n\nT µν = (ρ + ρΠ + p) u µ u ν + pg µν . (4)\n\nSuch a stress-energy density suffices to obtain the PN expansion of the gravitational field outside a fluid body, like the Sun, or of compact binary system. One can show that the internal structure of the fluid bodies can be neglected to 1 PN order by the effacement principle [39] in GR. Such effacement principle might actually not hold in modified field theories, but we shall study this subject elsewhere [47] .\n\nWith all these machinery, on can write down a supermetric [6] , namely\n\ng 00 = -1 + 2U -2βU 2 -2ξΦ W + (2γ + 2 + α 3 + ζ 1 -2ξ) Φ 1 + 2 (3γ -2β + 1 + ζ 2 + ξ) Φ 2 + 2 (1 + ζ 3 ) Φ 3 + 2 (3γ + 3ζ 4 -2ξ) Φ 4 -(ζ 1 -2ξ) A, g 0i = - 1 2 (4γ + 3 + α 1 -α 2 + ζ 1 -2ξ) V i - 1 2 (1 + α 2 -ζ 1 + 2ξ) W i , g ij = (1 + 2γU ) δ ij , (5)\n\nwhere δ ij is the Kronecker delta and where the PPN potentials\n\n(U, Φ W , Φ 1 , Φ 2 , Φ 3 , Φ 4 , A, V i , W i )\n\nare defined in Appendix A. Equation (5) describes a super-metric theory of gravity, because it reduces to different metric theories, such as GR or other alternative theories [6], through the appropriate choice of PPN parameters (γ, β, ξ, α 1 , α 2 , α 3 , ζ 1 , ζ 2 , ζ 3 , ζ 4\n\n). One could obtain a more general form of the PPN metric by performing a post-Galilean transformation on Eq. ( 5 ), but such a procedure shall not be necessary in this paper.\n\nThe super-metric of Eq. ( 5 ) is parameterized in terms of a specific number of PPN potentials, where one usually employs certain criteria to narrow the space of possible potentials to consider. Some of these restriction include the following: the potentials tend to zero as an inverse power of the distance to the source; the origin of the coordinate system is chosen to coincide with the source, such that the metric does not contain constant terms; and the metric perturbations h 00 , h 0i and h ij transform as a scalar, vector and tensor. The above restrictions are reasonable, but, in general, an additional subjective condition is usually imposed that is based purely on simplicity: the metric perturbations are not generated by gradients or curls of velocity vectors or other generalized vector functions. As of yet, no reason had arisen for relaxing such a condition, but as we shall see in this paper, such terms are indeed needed for CS modified theories.\n\nWhat is the physical meaning of all these parameters? One can understand what these parameters mean by calculating the generalized geodesic equations of motion and conservation laws [6] . For example, the parameter γ measures how much space-curvature is produced by a unit rest mass, while the parameter β determines how much \"non-linearity\" is there in the superposition law of gravity. Similarly, the parameter ξ determines whether there are preferred-location effects, while α i represent preferred-frame effects. Finally, the parameters ζ i measure the amount of violation of conservation of total momentum. In terms of conservation laws, one can interpret these parameters as measuring whether a theory is fully conservative, with linear and angular momentum conserved (ζ i and α i vanish), semi-conservative, with linear momentum conserved (ζ i and α 3 vanish), or nonconservative, where only the energy is conserved through lowest Newtonian order. One can verify that in GR, γ = β = 1 and all other parameters vanish, which implies that there are no preferred-location or frame effects and that the theory is fully conservative.\n\nA PPN analysis of an alternative theory of gravity then reduces to mapping its solutions to Eq. ( 5 ) and then determining the PPN parameters in terms of intrinsic parameters of the theory. The procedure is simply as follows: expand the modified field equations in the metric perturbation and in the PN approximation; iteratively solve for the metric perturbation to O(4) in h 00 , to O(3) in h 0i and to O(2) in h ij ; compare the solution to the PPN metric of Eq. ( 5 ) and read off the PPN parameters of the alternative theory. We shall employ this procedure in Sec. V to obtain the PPN parameters of CS gravity."
},
{
"section_type": "OTHER",
"section_title": "III. CS GRAVITY IN A NUTSHELL",
"text": "In this section, we describe the basics of CS gravity, following mainly [29, 30] . In the standard CS formalism, GR is modified by adding a new term to the gravitational action. This term is given by [30]\n\nS CS = m 2 pl 64π d 4 xf ( ⋆ R R) , (6)\n\nwhere m pl is the Planck mass, f is a prescribed external quantity with units of squared mass (or squared length in geometrized units), R is the Ricci scalar and the star stands the dual operation, such that\n\nR ⋆ R = 1 2 R αβγδ ǫ αβµν R γδ µν , (7)\n\nwith ǫ µνδγ the totally-antisymmetric Levi-Civita tensor and R µνδγ the Riemann tensor. Such a correction to the gravitational action is interesting because of the unavoidable parity violation that is introduced. Such parity violation is inspired from CP violation in the standard model, where such corrections act as anomaly-canceling terms. A similar scenario occurs in string theory, where the Green-Schwarz anomaly is canceled by precisely such a CS correction [32] , although CS gravity is not exclusively tied to string theory. Parity violation in CS gravity inexorably leads to birefringence in gravitational propagation, where here we mean that different polarization modes obey different propagation equations but travel at the same speed, that of light [29, 30, 36, 47] . If CS gravity were to lead to polarization modes that travel at different speeds, then one could use recently proposed experiments [17] to test this effect, but such is not the case in CS gravity. Birefringent gravitational waves, and thus CS gravity, have been proposed as possible explanations to the cosmic-microwave-background (CMB) anisotropies [36] , as well as the baryogenesis problem during the inflationary epoch [33] .\n\nThe magnitude of the CS correction is controlled by the externally-prescribed quantity f , which depends on the specific theory under consideration. When we consider CS gravity as an effective quantum theory, then the correction is suppressed by some mass scale M , which could be the electro-weak scale or some other scale, since it is unconstrained. In the context of string theory, the quantity f has been calculated only in conservative scenarios, where it was found to be suppressed by the Planck mass. In other scenarios, however, enhancements have been proposed, such as in cosmologies where the string coupling vanishes at late times [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58] , or where the field that generates f couples to spacetime regions with large curvature [59, 60] or stress-energy density [28, 47] . For simplicity, we here assume that this quantity is spatially homogeneous and its magnitude is small but non-negligible, so that we work to first order in the string-theoretical correction. Therefore, we treat ḟ as an independent perturbation parameter, [70] unrelated to ǫ, the PN perturbation parameter.\n\nThe field equations of CS modified gravity can be obtained by varying the action with respect to the metric. Doing so, one obtains\n\nG µν + C µν = 8πT µν , (8)\n\nwhere G µν is the Einstein tensor, T µν is a stress-energy tensor and C µν is the Cotton tensor. The latter tensor is defined via\n\nC µν = - 1 √ -g f ,σ ǫ σαβ (µ D α R ν)β + (D σ f ,τ ) ⋆ R τ (µ σ ν) , (9)\n\nwhere parenthesis stand for symmetrization, g is the determinant of the metric, D a stands for covariant differentiation and colon subscripts stand for partial differentiation.\n\nFormally, the introduction of such a modification to the field equations leads to a new constraint, which is compensated by the introduction of the new scalar field degree of freedom f . This constraint originates by requiring that the divergence of the field equations vanish, namely\n\nD µ C µν = 1 8 √ -g D ν f ( ⋆ RR) = 0, ( 10\n\n)\n\nwhere the divergence of the Einstein tensor vanished by the Bianchi identities. If this constraint is satisfied, then the equations of motion for the stress-energy D µ T µν are unaffected by CS gravity. A common source of confusion is that Eq. ( 10 ) is sometimes interpreted as requiring that R ⋆ R also vanish, which would then force the correction to the action to vanish. However, this is not the case because, in general, f is an exact form (d 2 f = 0) and, thus, Eq. ( 10 ) only implies an additional constraint that forces all solutions to the field equations to have a vanishing R ⋆ R.\n\nThe previous success of CS gravity in proposing plausible explanations to important cosmological problems prompts us to consider this extension of GR in the weakfield regime. For this purpose, it is convenient to rewrite the field equations in trace-reversed form, since this form is most amenable to a PN expansion. Doing so, we find,\n\nR µν + C µν = 8π T µν - 1 2 g µν T , (11)\n\nwhere the trace of the Cotton tensor vanishes identically and T = g µν T µν is the four dimensional trace of the stress-energy tensor. To linear order, the Ricci and Cotton tensors are given by [30]\n\nR µν = - 1 2 h µν + O(h) 2 , C µν = - ḟ 2 ǫ0αβ (µ η h ν)β,α + O(h) 2 , ( 12\n\n)\n\nwhere ǫαβγδ is the Levi-Civita symbol, with convention ǫ0123 = +1, and η = -∂ 2 t + η ij ∂ i ∂ j is the flat space D'Alambertian, with η µν the Minkowski metric. In Eq. ( 12 ), we have employed the Lorentz gauge condition h µα, α = h ,µ /2, where h = g µν h µν is the four dimensional trace of the metric perturbation.\n\nThe Cotton tensor changes the characteristic behavior of the Einstein equations by forcing them to become third order instead of second order. Third-order partial differential equations are common in boundary layer theory [43] . However, in CS gravity, the third-order contributions are multiplied by a factor of f and we shall treat this function as a small independent expansion parameter. Therefore, the change in characteristics in the modified field equations can also be treated perturbatively, which is justified because eventhough ḟ might be enhanced by standard model currents, extra dimensions or a vanishing string coupling, it must still carry some type of mass suppression.\n\nThe trace-reversed form of the field equations is useful because it allows us to immediately find a formal solution. Inverting the D'Alambertian operator we obtain\n\nH µν = -16π -1 η T µν - 1 2 g µν T + O(h) 2 , ( 13\n\n)\n\nwhere we have defined an effective metric perturbation as\n\nH µν ≡ h µν + ḟ ǫ0αβ (µ h ν)β,α . (14)\n\nNote that this formal solution is identical to the formal PN solution to the field equations in the limit ḟ → 0.\n\nAlso note that the second term in Eq. ( 14 ) is in essence a curl operator acting on the metric. This antisymmetric operator naturally forces the trace of the CS correction to vanish, as well as the 00 component and the symmetric spatial part.\n\nFrom the formal solution to the modified field equations, we immediately identify the only two possible nonzero CS contributions: a coupling to the vector component of the metric h 0i ; and coupling to the transversetraceless part of the spatial metric h T T ij . The latter has already been studied in the gravitational wave context [29, 30, 47] and it vanishes identically if we require the spatial sector of the metric perturbation to be conformally flat. The former coupling is a new curl-type contribution to the metric perturbation that, to our knowledge, had so far been neglected both by the string theory and PPN communities. In fact, as we shall see in later sections, terms of this type will force us to introduce a new PPN parameter that is proportional to the curl of certain PPN potentials.\n\nLet us conclude this section by pushing the formal solution to the modified field equations further to obtain a formal solution in terms of the actual metric perturbation h µν . Combining Eqs. ( 13 ) and ( 14 ) we arrive at the differential equation\n\nh µν + ḟ ǫ0αβ (µ h ν)β,α = -16π -1 η T µν - 1 2 g µν T +O(h) 2 .\n\n(15) Since we are searching for perturbations about the general relativistic solution, we shall make the ansatz\n\nh µν = h (GR) µν + ḟ ζ µν + O(h) 2 , ( 16\n\n)\n\nwhere h\n\n(GR) µν is the solution predicted by general relativity\n\nh (GR) µν ≡ -16π -1 η T µν - 1 2 g µν T , (17)\n\nand where ζ µν is an unknown function we are solving for. Inserting this ansatz into Eq. ( 15 ) we obtain\n\nζ µν + ḟ ǫ0αβ (µ ζ ν)β,α = 16πǫ 0αβ (µ ∂ α -1 η T ν)β - 1 2 g ν)β T . ( 18\n\n)\n\nWe shall neglect the second term on the left-hand side because it would produce a second order correction. Such conclusion was also reached when studying parity violation in GR to explain certain features of the CMB [35] . We thus obtain the formal solution\n\nζ µν = 16πǫ 0αβ (µ ∂ α -1 η T ν)β - 1 2 g ν)β T (19)\n\nand the actual metric perturbation to linear order becomes\n\nh µν = -16π -1 η T µν - 1 2 η µν T (20)\n\n+ 16π ḟ ǫkℓi -1\n\nη δ i(µ T ν)ℓ,k - 1 2 δ i(µ η ν)ℓ T ,k + O(h) 2 ,\n\nwhere we have used some properties of the Levi-Civita symbol to simplify this expression. The procedure presented here is general enough that it can be directly applied to study CS gravity in the PPN framework, as well as possibly find PN solutions to CS gravity."
},
{
"section_type": "OTHER",
"section_title": "IV. PN EXPANSION OF CS GRAVITY",
"text": "In this section, we perform a PN expansion of the field equations and obtain a solution in the form of a PN se-ries. This solution then allows us to read off the PPN parameters by comparing it to the standard PPN supermetric [Eq. ( 5 )]. In this section we shall follow closely the methods of [6] and [61] and indices shall be manipulated with the Minkowski metric, unless otherwise specified.\n\nLet us begin by expanding the field equations to second order in the metric perturbation. Doing so we find that the Ricci and Cotton tensors are given to second order by\n\nR µν = - 1 2 η h µν -2h σ(µ,ν) σ + h ,µν - 1 2 h ρ λ 2h ρ(µ,ν)λ -h µν,ρλ -h ρλ,µν - 1 2 h ρλ ,µ h ρλ,ν + h λ µ,ρ h ρ ν,λ (21)\n\n-h ρ µ,λ h ρν, λ + 1 2 h ,λ -2h λρ ,ρ h µν,λ -2h λ(µ,ν) + O(h) 3 , C µν = - ḟ 2 ǫ0αβ (µ η h ν)β,α -h σβ,αν) σ - ḟ 2 ǫ0αβ (µ h η h ν)β,α -h σβ,αν) σ + 1 2 2h ν)(λ,α) -h λα,ν) × η h λ β -2h σ (λ ,β) σ + h ,β λ -2 QR ν)β,α - ḟ 4 ǫσαβ (µ 2h 0 (σ,τ ) -h στ, 0 h τ [β,α]ν) -h ν)[β,α] τ - ḟ 2 h µλ ǫ0αβ(λ η h ν)β,α -h σβ,αν) σ - ḟ 2 ǫ0αβ(µ η h λ) β,α -h σβ,α σλ) h νλ + O(h) 3 . ( 22\n\n)\n\nwhere index contraction is carried out with the Minkowski metric and where we have not assumed any gauge condition. The operator Q(•) takes the quadratic part of its operand [of O(h) 2 ] and it is explained in more detail in Appendix B, where the derivation of the expansion of the Cotton tensor is presented in more detail. In this derivation, we have used the definition of the Levi-Civita tensor\n\nǫ αβγδ = (-g) 1/2 ǫαβγδ = 1 - 1 2 h ǫαβγδ + O(h) 2 , ( 23\n\n)\n\nǫ αβγδ = -(-g) -1/2 ǫαβγδ = -1 + 1 2 h ǫαβγδ + O(h) 2 .\n\nNote that the PN expanded version of the linearized Ricci tensor of Eq. ( 21 ) agrees with previous results [6] . Also note that if the Lorentz condition is enforced, several terms in both expressions vanish identically and the Cotton tensor to first order reduces to Eq. ( 12 ), which agrees with previous results [30] .\n\nLet us now specialize the analysis to the standard PPN gauge. For this purpose, we shall impose the following gauge conditions\n\nh jk, k - 1 2 h ,j = O(4), h 0k, k - 1 2 h k k,0 = O(5), (24)\n\nwhere h k k is the spatial trace of the metric perturbation. Note that the first equation is the PN expansion of one of the Lorentz gauge conditions, while the second equation is not. This is the reason why the previous equations where not expanded in the Lorentz gauge. Nonetheless, such a gauge condition does not uniquely fix the coordinate system, since we can still perform an infinitesimal gauge transformation that leaves the modified field equations invariant. One can show that the Lorentz and PPN gauge are related to each other by such a gauge transformation. In the PPN gauge, then, the Ricci tensor takes the usual form\n\nR 00 = - 1 2 ∇ 2 h 00 - 1 2 h 00,i h 00, i + 1 2 h ij h 00,ij + O(6), R 0i = - 1 2 ∇ 2 h 0i - 1 4 h 00,0i + O(5), R ij = - 1 2 ∇ 2 h ij + O(4), (25)\n\nwhich agrees with previous results [6] , while the Cotton tensor reduces to\n\nC 00 = O(6), C 0i = - 1 4 ḟ ǫ0kl i ∇ 2 h 0l,k + O(5), C ij = - 1 2 ḟ ǫ0kl (i ∇ 2 h j)l,k + O(4), (26)\n\nwhere ∇ = η ij ∂ i ∂ j is the Laplacian of flat space [see Appendix B for the derivation of Eq. ( 26 ).] Note again the explicit appearance of two coupling terms of the Cotton tensor to the metric perturbation: one to the transversetraceless part of the spatial metric and the other to the vector metric perturbation. The PN expansions of the linearized Ricci and Cotton tensor then allow us to solve the modified field equations in the PPN framework."
},
{
"section_type": "OTHER",
"section_title": "V. PPN SOLUTION OF CS GRAVITY",
"text": "In this section we shall proceed to systematically solve the modified field equation following the standard PPN iterative procedure [6] . We shall begin with the 00 and ij components of the metric to O(2), and then proceed with the 0i components to O(3) and the 00 component to O(4). Once all these components have been solved for in terms of PPN potentials, we shall be able to read off the PPN parameters adequate to CS gravity."
},
{
"section_type": "OTHER",
"section_title": "A. h00 and hij to O(2)",
"text": "Let us begin with the modified field equations for the scalar sector of the metric perturbation. These equations are given to O(2) by\n\n∇ 2 h 00 = -8πρ, (27)\n\nbecause T = -ρ. Eq. ( 27 ) is the Poisson equation, whose solution in terms of PPN potentials is\n\nh 00 = 2U + O(4). (28)\n\nLet us now proceed with the solution to the field equation for the spatial sector of the metric perturbation. This equation to O(2) is given by\n\n∇ 2 h ij + ḟ ǫ0kl (i ∇ 2 h j)l,k = -8πρδ ij , (29)\n\nwhere note that this is the first appearance of a Cotton tensor contribution. Since the Levi-Civita symbol is a constant and ḟ is only time-dependent, we can factor out the Laplacian and rewrite this equation in terms of the effective metric H ij as\n\n∇ 2 H ij = -8πρδ ij , (30)\n\nwhere, as defined in Sec. III,\n\nH ij = h ij + ḟ ǫ0kl (i h j)l,k . (31)\n\nThe solution of Eq. ( 30 ) can be immediately found in terms of PPN potentials as\n\nH ij = 2U δ ij + O(4), (32)\n\nwhich is nothing but Eq. ( 13 ). Recall, however, that in Sec. III we explicitly used the Lorentz gauge to simplify the field equations, whereas here we are using the PPN gauge. The reason why the solutions are the same is that the PPN and Lorentz gauge are indistinguishable to this order.\n\nOnce the effective metric has been solved for, we can obtain the actual metric perturbation following the procedure described in Sec. III. Combining Eq. ( 31 ) with Eq. ( 32 ), we arrive at the following differential equation\n\nh ij + ḟ ǫ0kl (i h j)l,k = 2U δ ij . (33)\n\nWe look for solutions whose zeroth-order result is that predicted by GR and the CS term is a perturbative correction, namely\n\nh ij = 2U δ ij + ḟ ζ ij , (34)\n\nwhere ζ is assumed to be of O( ḟ ) 0 . Inserting this ansatz into Eq. ( 33 ) we arrive at\n\nζ ij + ḟ ǫ0kl (i ζ j)l,k = 0, ( 35\n\n)\n\nwhere the contraction of the Levi-Civita symbol and the Kronecker delta vanished. As in Sec. III, note that the second term on the left hand side is a second order correction and can thus be neglected to discover that ζ ij vanishes to this order. The spatial metric perturbation to O( 2 ) is then simply given by the GR prediction without any CS correction, namely\n\nh ij = 2U δ ij + O(4). (36)\n\nPhysically, the reason why the spatial metric is unaffected by the CS correction is related to the use of a perfect fluid stress-energy tensor, which, together with the PPN gauge condition, forces the metric to be spatially conformally flat. In fact, if the spatial metric were not flat, then the spatial sector of the metric perturbation would be corrected by the CS term. Such would be the case if we had pursued a solution to 2 PN order, where the Landau-Lifshitz pseudo-tensor sources a non-conformal correction to the spatial metric [39] , or if we had searched for gravitational wave solutions, whose stress-energy tensor vanishes [29, 36] . In fact, one can check that, in such a scenario, Eq. ( 30 ) reduces to that found by [29, 30, 36, 47] as ρ → 0. We have then found that the weak-field expansion of the gravitational field outside a fluid body, like the Sun or a compact binary, is unaffected by the CS correction to O(2)."
},
{
"section_type": "OTHER",
"section_title": "B. h0i to O(3)",
"text": "Let us now look for solutions to the field equations for the vector sector of the metric perturbation. The field equations to O(3) become\n\n∇ 2 h 0i + 1 2 h 00,0i + 1 2 ḟ ǫ0kl i ∇ 2 h 0l,k = 16πρv i , (37)\n\nwhere we have used that T 0i = -T 0i . Using the lower order solutions and the effective metric, as in Sec. III, we obtain\n\n∇ 2 H 0i + U ,0i = 16πρv i , (38)\n\nwhere the vectorial sector of the effective metric is\n\nH 0i = h 0i + 1 2 ḟ ǫ0kl i h 0l,k . (39)\n\nWe recognize Eq. ( 38 ) as the standard GR field equation to O(3), except that the dependent function is the effective metric instead of the metric perturbation. We can thus solve this equation in terms of PPN potentials to obtain\n\nH 0i = - 7 2 V i - 1 2 W i , (40)\n\nwhere we have used that the superpotential X satisfies X ,0j = V j -W j (see Appendix A for the definitions.) Combining Eq. ( 39 ) with Eq. ( 40 ) we arrive at a differential equation for the metric perturbation, namely\n\nh 0i + 1 2 ḟ ǫ0kl i h 0l,k = - 7 2 V i - 1 2 W i . ( 41\n\n)\n\nOnce more, let us look for solutions that are perturbation about the GR prediction, namely\n\nh 0i = - 7 2 V i - 1 2 W i + ḟ ζ i , (42)\n\nwhere we again assume that ζ i is of O( ḟ ) 0 . The field equation becomes\n\nζ i + 1 2 ḟ (∇ × ζ) i = 1 2 7 2 (∇ × V ) i + 1 2 (∇ × W ) i , (43)\n\nwhere (∇ × A) i = ǫ ijk ∂ j A k is the standard curl operator of flat space. As in Sec. III, note once more that the second term on the left-hand side is again a second order correction and we shall thus neglect it. Also note that the curl of the V i potential happens to be equal to the curl of the W i potential. The solution for the vectorial sector of the actual gravitational field then simplifies to\n\nh 0i = - 7 2 V i - 1 2 W i + 2 ḟ (∇ × V ) i + O(5). (44)\n\nWe have arrived at the first contribution of CS modified gravity to the metric for a perfect fluid source. Chern-Simons gravity was previously seen to couple to the transverse-traceless sector of the metric perturbation for gravitational wave solutions [29, 30, 36, 47] . The CS correction is also believed to couple to Noether vector currents, such as neutron currents, which partially fueled the idea that this correction could be enhanced. However, to our knowledge, this correction was never thought to couple to vector metric perturbations. From the analysis presented here, we see that in fact CS gravity does couple to such terms, even if the matter source is neutrally charged. The only requirement for such couplings is that the source is not static, ie. that the object is either moving or spinning relative to the PPN rest frame so that the PPN vector potential does not vanish. The latter is suppressed by a relative O(1) because in the far field the velocity of a compact object produces a term of O(3) in V i , while the spin produces a term of O(4). In a later section, we shall discuss some of the physical and observational implications of such a modification to the metric."
},
{
"section_type": "OTHER",
"section_title": "C. h00 to O(4)",
"text": "A full analysis of the PPN structure of a modified theory of gravity requires that we solve for the 00 component of the metric perturbation to O(4). The field equations to this order are\n\n- 1 2 ∇ 2 h 00 - 1 2 h 00,i h 00,i + 1 2 h ij h 00,ij = 4πρ [1 + 2 v 2 -U + 1 2 Π + 3 2 p ρ , (45)\n\nwhere the CS correction does not contribute at this order (see Appendix B.) Note that the h 0i sector of the metric perturbation to O(3) does not feed back into the field equations at this order either. The terms that do come into play are the h 00 and h ij sectors of the metric, which are not modified to lowest order by the CS correction. The field equation, thus, reduce to the standard one of GR, whose solution in terms of PPN potentials is\n\nh 00 = 2U -2U 2 + 4Φ 1 + 4Φ 2 + 2Φ 3 + 6Φ 4 + O(6). ( 46\n\n)\n\nWe have thus solved for all components of the metric perturbation to 1 PN order beyond the Newtonian answer, namely g 00 to O(4), g 0i to O(3) and g ij to O(2)."
},
{
"section_type": "OTHER",
"section_title": "D. PPN Parameters for CS Gravity",
"text": "We now have all the necessary ingredients to read off the PPN parameters of CS modified gravity. Let us begin by writing the full metric with the solutions found in the previous subsections:\n\ng 00 = -1 + 2U -2U 2 + 4Φ 1 + 4Φ 2 + 2Φ 3 + 6Φ 4 + O(6), g 0i = - 7 2 V i - 1 2 W i + 2 ḟ (∇ × V ) i + O(5)\n\n,\n\ng ij = (1 + 2U ) δ ij + O(4). ( 47\n\n)\n\ncan verify that this metric is indeed a solution of Eqs. ( 27 ), ( 29 ), (37) and (45) to the appropriate PN order and to first order in the CS coupling parameter. Also note that the solution of Eq. ( 47 ) automatically satisfies the constraint ⋆ RR = 0 to linear order because the contraction of the Levi-Civita symbol with two partial derivatives vanishes. Such a solution is then allowed in CS gravity, just as other classical solutions are [62] , and the equations of motion for the fluid can be obtained directly from the covariant derivative of the stress-energy tensor.\n\nWe can now read off the PPN parameters of the CS modified theory by comparing Eq. ( 5 ) to Eq. (47) . A visual inspection reveals that the CS solution is identical to the classical GR one, which implies that γ = β = 1, ζ = 0 and α 1 = α 2 = α 3 = ξ 1 = ξ 2 = ξ 3 = ξ 4 = 0 and there are no preferred frame effects. However, Eq. ( 5 ) contains an extra term that cannot be modeled by the standard PPN metric of Eq. ( 5 ), namely the curl contribution to g 0i . We then see that the PPN metric must be enhanced by the addition of a curl-type term to the 0i components of the metric, namely\n\ng 0i ≡ - 1 2 (4γ + 3 + α 1 -α 2 + ζ 1 -2ξ) V i - 1 2 (1 + α 2 -ζ 1 + 2ξ) W i + χ (r∇ × V ) i , ( 48\n\n)\n\nwhere χ is a new PPN parameter and where we have multiplied the curl operator by the radial distance to the source, r, in order to make χ a proper dimensionless parameter. Note that there is no need to introduce any additional PPN parameters because the curl of W i equals the curl of V i . In fact, we could have equally parameterized the new contribution to the PPN metric in terms of the curl of W i , but we chose not to because V i appears more frequently in PN theory. For the case of CS modified gravity, the new χ parameter is simply\n\nχ = 2 ḟ r , (49)\n\nwhich is dimensionless since ḟ has units of length. If an experiment could measure or place bounds on the value of χ, then ḟ could also be bounded, thus placing a constraint on the CS coupling parameter."
},
{
"section_type": "OTHER",
"section_title": "VI. ASTROPHYSICAL IMPLICATIONS",
"text": "In this section we shall propose a physical interpretation to the CS modification to the PPN metric and we shall calculate some GR predictions that are modified by this correction. This section, however, is by no means a complete study of all the possible consequences of the CS correction, which is beyond the scope of this paper.\n\nLet us begin by considering a system of A nearly spherical bodies, for which the gravitational vector potentials are simply [6]\n\nV i = A m A r A v i A + 1 2 A J A r 2 A × n A i , (50)\n\nW i = A m A r A (v A • n A ) n i A + 1 2 A J A r 2 A × n A i ,\n\nwhere m A is the mass of the Ath body, r A is the field point distance to the Ath body, n i A = x i A /r A is a unit vector pointing to the Ath body, v A is the velocity of the Ath body and J i A is the spin-angular momentum of the Ath For example, the spin angular momentum for a Kerr spacetime is given by J = m 2 a i , where a is the dimensionless Kerr spin parameter. Note that if A = 2 then the system being modeled could be a binary of spinning compact objects, while if A = 1 it could represent the field of the sun or that of a rapidly spinning neutron star or pulsar. In obtaining Eq. ( 50 ), we have implicitly assumed a point-particle approximation, which in classical GR is justified by the effacement principle. This principle postulates that the internal structure of bodies contributes to the solution of the field equations to higher PN order. One can verify that this is indeed the case in classical GR, where internal structure contributions appear at 5 PN order. In CS gravity, however, it is a priori unclear whether an analogous effacement principle holds because the CS term is expected to couple with matter current via standard model-like interactions. If such is the case, it is possible that a \"mountain\" on the surface of a neutron star [63] or an r-mode instability [64, 65, 66] enhances the CS contribution. In this paper, however, we shall neglect these interactions, and relegate such possibilities to future work [47] .\n\nWith such a vector potential, we can calculate the CS correction to the metric. For this purpose, we define the correction δg 0i ≡ g 0i -g\n\n(GR) 0i\n\n, where g\n\n(GR) 0i\n\nis the GR prediction without CS gravity. We then find that the CS corrections is given by\n\nδg 0i = 2 A ḟ r A m A r A (v A × n A ) i - J i A 2r 2 A + 3 2 (J A • n A ) r 2 A n i A , (51)\n\nwhere the • operator is the flat-space inner product and where we have used the identities ǫijk ǫklm = δ il δ jmδ im δ jl and ǫilk ǫjlm = 2δ ij . Note that the first term of Eq. ( 51 ) is of O(3), while the second and third terms are of O(4) as previously anticipated. Also note that ḟ couples both to the spin and orbital angular momentum. Therefore, whether the system under consideration is the Solar system (v i of the Sun is zero while J i is small), the Crab pulsar (v i is again zero but J i is large) or a binary system of compact objects (neither v i nor J i vanish), there will in general be a non-vanishing coupling between the CS correction and the vector potential of the system.\n\nFrom the above analysis, it is also clear that the CS correction increases with the non-linearity of the spacetime. In other words, the CS term is larger not only for systems with large velocities and spins, but also in regions near the source. For a binary system, this fact implies that the CS correction is naturally enhanced in the last stages of inspiral and during merger. Note that this enhancement is different from all previous enhancements proposed, since it does not require the presence of charge [28, 47] , a fifth dimension with warped compactifications [59, 60] , or a vanishing string coupling [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58] . Unfortunately, the end of the inspiral stage coincides with the edge of the PN region of validity and, thus, a com-plete analysis of such a natural enhancement will have to be carried out through numerical simulations.\n\nIn the presence of a source with the vector potentials of Eq. ( 50 ), we can write the vectorial sector of the metric perturbation in a suggestive way, namely\n\ng 0i = A - 7 2 m A r A v i A - A m A 6r 2 A v A -v (ef f ) A i (52) - 1 2 A n i A m A r A v (ef f ) A • n A -2 A J (ef f ) A r 2 A × n A i ,\n\nwhere we have defined an effective velocity and angular momentum vector via\n\nv i A(ef f ) = v i A -6 ḟ J i A m A r 2 A , J i A(ef f ) = J i A -ḟ m A v i A , (53)\n\nor in terms of the Newtonian orbital angular momentum\n\nL i A(N ) = r A × p A and linear momentum p i A(N ) = m A v i A L i A(ef f ) = L i A(N ) -6 ḟ (n A × J A ) i , J i A(ef f ) = J i A -ḟ p i A . (54)\n\nFrom this analysis, it is clear that the CS corrections seems to couple to both a quantity that resembles the orbital and the spin angular momentum vector. Note that when the spin angular momentum vanishes the vectorial metric perturbation is identical to that of a spinning moving fluid, but where the spin is induced by the coupling of the orbital angular momentum to the CS term. The presence of an effective CS spin angular momentum in non-spinning sources leads to an interesting physical interpretation. Let us model the field that sources ḟ as a fluid that permeates all of spacetime. This field could be, for example, a model-independent axion, inspired by the quantity introduced in the standard model to resolve the strong CP problem [67] . In this scenario, then the fluid is naturally \"dragged\" by the motion of any source and the CS modification to the metric is nothing but such dragging. This analogy is inspired by the ergosphere of the Kerr solution, where inertial frames are dragged with the rotation of the black hole. In fact, one could push this analogy further and try to construct the shear and bulk viscosity of such a fluid, but we shall not attempt this here. Of course, this interpretation is to be understood only qualitatively, since its purpose is only to allow the reader to picture the CS modification to the metric in physical terms.\n\nAn alternative interpretation can be given to the CS modification in terms of the gravito-electro-magnetic (GED) analogy [40, 41] , which shall allow us to easily construct the predictions of the modified theory. In this analogy, one realizes that the PN solution to the linearized field equations can be written in terms of a potential and vector potential, namely\n\nds 2 = -(1 -2Φ) dt 2 -4 (A • dx) dt + (1 + 2Φ) δ ij dx i dx j , (55)\n\nwhere Φ reduces to the Newtonian potential U in the Newtonian limit [41] and A i is a vector potential related to the metric via A i = -g 0i /4. One can then construct GED fields in analogy to Maxwell's electromagnetic theory via\n\nE i = -(∇Φ) i -∂ t 1 2 A i , B i = (∇ × A) i , (56)\n\nwhich in terms of the vectorial sector of the metric perturbation becomes\n\nE i = -(∇Φ) i + 1 8 ġi , B i = - 1 4 (∇ × g) i , (57)\n\nwhere we have defined the vector g i = g 0i . The geodesic equations for a test particle then reduce to the Lorentz force law, namely\n\nF i = -mE i -2m (v × B) i . ( 58\n\n)\n\nWe can now work out the effect of the CS correction on the GED fields and equations of motion. First note that the CS correction only affects g. We can then write the CS modification to the Lorentz force law by defining δa i = a i -a i (GR) , where a i (GR) is the acceleration vector predicted by GR, to obtain,\n\nδa i = 1 8 δ ġi + 1 2 (v × δΩ) i , (59)\n\nwhere we have defined the angular velocity\n\nδΩ i = (∇ × δg) i . ( 60\n\n)\n\nThe time derivative of the vector g i is of O( 5 ) and can thus be neglected, but the angular velocity cannot and it is given by\n\nδΩ i = - A ḟ m A r 3 A 3 (v A • n A ) n i A -v i A , (61)\n\nwhich is clearly of O(3). Note that although the first term between square brackets cancels for circular orbits because n i A is perpendicular to v i A to Newtonian order, the second term does not. The angular velocity adds a correction to the acceleration of O(4), namely\n\nδa i = - 3 2 A ḟ m A r 3 A (v A • n A ) (v A × n A ) i , (62)\n\nwhich for a system in circular orbit vanishes to Newtonian order. One could use this formalism to find the perturbations in the motion of moving objects by integrating Eq. ( 62 ) twice. However, for systems in a circular orbit, such as the Earth-Moon system or compact binaries, this correction vanishes to leading order. Therefore, lunar ranging experiments [68] might not be able to constraint ḟ .\n\nAnother correction to the predictions of GR is that of the precession of gyroscopes by the so-called Lense-Thirring or frame-dragging effect. In this process, the spin angular momentum of a source twists spacetime in such a way that gyroscopes are dragged with it. The precession angular velocity depends on the vector sector of the metric perturbation via Eq. ( 61 ). Thus, the full Lense-Thirring term in the precession angular velocity of precessing gyroscopes is\n\nΩ i LT = - 1 r 3 A A J i A(ef f ) -3n i A J A(ef f ) • n A i . ( 63\n\n)\n\nNote that this angular velocity is identical to the GR prediction, except for the replacement J i A → J i A(ef f ) . In CS modified gravity, then, the Lense-Thirring effect is not only produced by the spin angular momentum of the gyroscope but also by the orbital angular momentum that couples to the CS correction. Therefore, if an experiment were to measure the precession of gyroscopes by the curvature of spacetime (see, for example, Gravity Probe B [42] ) one could constraint ḟ and thus some intrinsic parameters of string theory. Note, however, that the CS correction depends on the velocity of the bodies with respect to the inertial PPN rest-frame. In order to relate these predictions to the quantities that are actually measured in the experiment, one would have to transform to the experiment's frame, or perhaps to a basis aligned with the direction of distant stars [6] .\n\nAre there other experiments that could be performed to measure such a deviation from GR? Any experiment that samples the vectorial sector of the metric would in effect be measuring such a deviation. In this paper, we have only discussed modifications to the frame-dragging effect and the acceleration of bodies through the GED analogy, but this need not be the only corrections to classical GR predictions. In fact, any predictions that depends on g 0i indirectly, for example via Christoffel symbols, will probably also be modified unless the corrections is fortuitously canceled. In this paper, we have laid the theoretical foundations of the weak-field correction to the metric due to CS gravity and studied some possible corrections to classical predictions. A detailed study of other corrections is beyond the scope of this paper."
},
{
"section_type": "CONCLUSION",
"section_title": "VII. CONCLUSION",
"text": "We have studied the weak-field expansion of the solution to the CS modified field equations in the presence of a perfect fluid PN source in the point particle limit. Such an expansion required that we linearize the Ricci and Cotton tensor to second order in the metric perturbation without any gauge assumption. An iterative PPN formalism was then employed to solve for the metric perturbation in this modified theory of gravity. We have found that CS gravity possesses the same PPN parameters as those of GR, but it also requires the introduction of a new term and PPN parameter that is proportional to the curl of the PPN vector potentials. Such a term is enhanced in non-linear scenarios without requiring the presence of standard model currents, large extra dimensions or a vanishing string coupling.\n\nWe have proposed an interpretation for the new term in the metric produced by CS gravity and studied some of the possible consequences it might have on GR predictions. The interpretation consists of picturing the field that sources the CS term as a fluid that permeates all of spacetime. In this scenario, the CS term is nothing but the \"dragging\" of the fluid by the motion of the source. Irrespective of the validity of such an interpretation, the inclusion of a new term to the weak-field expansion of the metric naturally leads to corrections to the standard GR predictions. We have studied the acceleration of point particles and the Lense-Thirring contribution to the precession of gyroscopes. We have found that both corrections are proportional to the CS coupling parameter and, therefore, experimental measurement of these effects might be used to constraint CS and, possibly, string theory.\n\nFuture work could concentrate on studying further the non-linear enhancement of the CS correction and the modifications to the predictions of GR. The PPN analysis performed here breaks down very close to the source due to the use of a point particle approximation in the stress energy tensor. One possible research route could consists of studying the CS correction in a perturbed Kerr background [69] . Another possible route could be to analyze other predictions of the theory, such as the perihelion shift of Mercury or the Nordtvedt effect. Furthermore, in light of the imminent highly-accurate measurement of the Lense-Thirring effect by Gravity Probe B, it might be useful to revisit this correction in a frame better-adapted to the experimental setup. Finally, the CS modification to the weak-field metric might lead to non-conservative effects and the breaking of the effacement principle [47] , which could be studied through the evaluation of the gravitational pseudo stress-energy tensor. Ultimately, it will be experiments that will determine the viability of CS modified gravity and string theory.\n\nAPPENDIX A: PPN POTENTIALS\n\nIn this appendix, we present explicit expressions for the PPN potentials used to parameterize the metric in Eq. ( 5 ). These potentials are the following:\n\nU ≡ ρ |x -x ′ | d 3 x ′ , V i ≡ ρ ′ v ′ i |x -x ′ | d 3 x ′ , W i ≡ ρ ′ v ′ j (x -x ′ ) j (x -x ′ ) i |x -x ′ | 3 d 3 x ′ , Φ W ≡ ρ ′ ρ ′′ (x -x ′ ) i |x -x ′ | 3 (x ′ -x ′′ ) i |x -x ′′ | - (x -x ′′ ) i |x ′ -x ′′ | d 3 x ′ d 3 x ′′ , Φ 1 ≡ ρ ′ v ′2 |x -x ′ | d 3 x ′ , Φ 2 ≡ ρ ′ U ′ |x -x ′ | d 3 x ′ , Φ 3 ≡ ρ ′ Π ′ |x -x ′ | d 3 x ′ , Φ 4 ≡ p ′ |x -x ′ | d 3 x ′ , A ≡ ρ ′ v ′ i (x -x ′ ) i 2 |x -x ′ | d 3 x ′ , X ≡ ρ ′ |x -x ′ |d 3 x ′ . (A1)\n\nThese potentials satisfy the following relations\n\n∇ 2 U = -4πρ, ∇ 2 V i = -4πρv i , ∇ 2 Φ 1 = -4πρv 2 , ∇ 2 Φ 2 = -4πρU, ∇ 2 Φ 3 = -4πρΠ, ∇ 2 Φ 4 = -4πp, ∇ 2 X = -2U (A2)\n\nThe potential X is sometimes referred to as the superpotential because it acts as a potential for the Newtonian potential."
},
{
"section_type": "OTHER",
"section_title": "APPENDIX B: LINEARIZATION OF THE COTTON TENSOR",
"text": "In this appendix, we present some more details on the derivation of the linearized Cotton tensor to second order. We begin with the definition of the Cotton tensor [30] in terms of the symmetrization operator, namely\n\nC µν = - 1 √ -g (D σ f ) ǫ σαβ(µ D α R ν) β + (D στ f ) ⋆ R τ (µ|σ|ν) . (B1)\n\nUsing the symmetries of the Levi-Civita and Riemann tensor, as well as the fact that f depends only on time, we can simplify the Cotton tensor to\n\nC µν = (-g) -1 ḟ ǫ0αβ(µ R ν) β,α + ǫ0αβ(µ Γ ν) λα R λ β + 1 2 Γ 0 στ ǫσαβ(µ R ν)τ αβ . (B2)\n\nNoting that the determinant of the metric is simply g = -1 + h, so that (-g) -1 = 1 + h, we can identify four terms in the Cotton tensor\n\nC µν A = ḟ ǫ0αβ(µ LR ν) β,α , C µν B = ḟ ǫ0αβ(µ h ρρ LR ν) β,α , C µν C = ḟ ǫ0αβ(µ LΓ ν) λα LR λ β , C µν D = ḟ 2 ǫσαβ(µ LΓ 0 στ LR ν)τ αβ , C µν E = ḟ ǫ0αβ(µ QR ν) β,α , (B3)\n\nwhere the L operator stands for the linear part of its operand, while the Q operator isolates the quadratic part of its operand. For example, if we act L and Q on (1+h) n , where n is some integer, we obtain\n\nL(1 + h) n = nh, Q(1 + h) n = n(n -1) 2 h 2 . ( B4\n\n)\n\nLet us now compute each of these terms separately. The first four terms are given by\n\nC µν A = -\n\nḟ 2 ǫ0αβ(µ η h ν) β,α -h σβ, ν ασ , C µν B = -ḟ 2 hǫ 0αβ(µ η h ν) β,α -h σβ, ν ασ , C µν C = -ḟ 4 ǫ0αβ(µ h ν) λ,α + h ν α,λ -h λα, ν) × η h λ β -h σ λ ,β σ -h σβ, λσ + h , λ β , C µν D = ḟ 4 ǫσαβ(µ 2h 0 (σ,τ ) -h στ, 0 h τ [β,α] ν -h ν [β,α] τ .\n\n(B5)\n\nThe last term of the Cotton tensor is simply the derivative of the Ricci tensor which we already calculated to second order in Eq. (21) . In order to avoid notation clutter, we shall not present it again here, but instead we combine all the Cotton tensor pieces to obtain\n\nC µν = - ḟ 2\n\nǫ0αβ(µ η h ν) β,α -h σβ,α σν) -ḟ 2 ǫ0αβ(µ h η h ν) β,α -h σβ,α σν) + 1 2 2h ν) (λ,α) -h λα, ν) (B6) × η h λ β -2h σ (λ ,β) σ + h ,β λ -2 QR ν) β,α + ḟ 4 ǫσαβ(µ 2h 0 (σ,τ ) -h στ, 0 h τ [β,α] ν) -h ν) [β,α] τ + O(h) 3 where its covariant form is C µν = -ḟ 2 ǫ0αβ (µ η h ν)β,α -h σβ,αν) σḟ 2 ǫ0αβ (µ h η h ν)β,α -h σβ,αν) σ + 1 2 2h ν)(λ,α) -h λα,ν) × η h λ β -2h σ (λ ,β) σ + h ,β λ -2 QR ν)β,α + h νλ η h λ) β,α -h σβ,α σλ) + ḟ 4 ǫσαβ (µ 2h 0 (σ,τ ) -h στ, 0 h τ [β,α]ν) -h ν)[β,α] τḟ 2 h µλ ǫ0αβ(λ η h ν)β,α -h σβ,αν) σ + O(h) 3 . (B7) For the PPN mapping of CS modified gravity, only the 00 component of the metric is needed to second order, which implies we only need C 00 to O(h) 2 . This component is given by C 00 = ḟ 4 ǫijk 0 2h 0 (i,ℓ) -h iℓ, 0 h ℓ [k,j]0 -h 0[k,j] ℓ ḟ 2 h 0ℓ ǫ0jk(ℓ η h 0k,j -h ik,j0 i + O(h) 3 , (B8)\n\nwhere in fact the last term vanishes due to the PPN gauge condition. Note that this term is automatically of O( 6 ), which is well beyond the required order we need in h 00 ."
}
] | |
arxiv:0704.0314 | 0704.0314 | 1 | 10.1103/PhysRevD.76.027502 | 1b38d11269562682f9f9bac6c878b017939e7d85962ba39ba8426a667a4df365 | Extra dimensions and Lorentz invariance violation | We consider effective model where photons interact with scalar field
corresponding to conformal excitations of the internal space (geometrical
moduli/gravexcitons). We demonstrate that this interaction results in a
modified dispersion relation for photons, and consequently, the photon group
velocity depends on the energy implying the propagation time delay effect. We
suggest to use the experimental bounds of the time delay of gamma ray bursts
(GRBs) photons propagation as an additional constrain for the gravexciton
parameters. | [
"Viktor Baukh",
"Alexander Zhuk",
"Tina Kahniashvili"
] | [
"hep-ph",
"astro-ph",
"hep-th"
] | hep-ph | [] | 2007-04-03 | 2026-02-26 | We consider effective model where photons interact with scalar field corresponding to conformal excitations of the internal space (geometrical moduli/gravexcitons). We demonstrate that this interaction results in a modified dispersion relation for photons, and consequently, the photon group velocity depends on the energy implying the propagation time delay effect. We suggest to use the experimental bounds of the time delay of gamma ray bursts (GRBs) photons propagation as an additional constrain for the gravexciton parameters.
Lorentz invariance (LI) of physical laws is one of the corner stone of modern physics. There is a number of experiments confirming this symmetry at energies we can approach now. For example, on a classical level, the rotation invariance has been tested in Michelson-Morley experiments, and the boost invariance has been tested in Kennedy-Torhndike experiments [1] . Although, up to now, LI is well established experimentally, we cannot say surely that at higher energies it is still valid. Moreover, modern astrophysical and cosmological data (e.g. UHECR, dark matter, dark energy, etc) indicate for a possible LI violation (LV). To resolve these challenges, there are number of attempts to create new physical models, such as M/string theory, Kaluza-Klein models, brane-world models, etc. [1] .
In this paper we investigate LV test related to photon dispersion measure (PhDM). This test is based on the LV effect of a phenomenological energy-dependent speed of photon [2, 3, 4, 5, 6, 7, 8] , for recent studies see Ref. [9] and references therein.
The formalism that we use is based on the analogy with electromagnetic waves propagation in a magnetized medium, and extends previous works [8, 10, 11] . In our model, instead of propagation in a magnetized medium, the electromagnetic waves are propagating in vacuum filled with a scalar field ψ. LV occurs because of an interaction term f(ψ)F 2 where F is an amplitude of the electromagnetic field. Such an interaction might have different origins. In the string theory ψ could be a dilaton field [12, 13] . The field ψ could be associated with geometrical moduli. In brane-world models the similar term describes an interaction between the bulk dilaton and the Standard Model fields on the brane [14] . In Ref. [15] , such an interaction was obtained in N = 4 * Electronic address: bauch˙vGR@ukr.net † Electronic address: zhuk@paco.net ‡ Electronic address: tinatin@phys.ksu.edu super-gravity in four dimensions. In Kaluza-Klein models the term f(ψ)F 2 has the pure geometrical origin, and it appears in the effective, dimensionally reduced, four dimensional action (see e.g. [16, 17] ). In particular, in reduced Einstein-Yang-Mills theories, the function f (ψ) coincides (up to a numerical prefactor) with the volume of the internal space. Phenomenological (exactly solvable) models with spherical symmetries were considered in Refs. [18] . To be more specific, we consider the model which is based on the reduced Einstein-Yang-Mills theory [17] , where the term ∝ ψF 2 describes the interaction between the conformal excitations of the internal space (gravexcitons) and photons. It is clear that the similar LV effect exists for all types of interactions of the form f(ψ)F 2 mentioned above.
Obviously, the interaction term f(ψ)F 2 modifies the Maxwell equations, and, consequently, results in a modified dispersion relation for photons. We show that this modification has rather specific form. For example, we demonstrate that refractive indices for the left and right circularly polarized waves coincide with each other. Thus, rotational invariance is preserved. However, the speed of the electromagnetic wave's propagation in vacuum differs from the speed of light c. This difference implies the time delay effect which can be measured via high-energy GRB photons propagation over cosmological distances (see e.g. Ref. [9] ). It is clear that gravexcitons should not overclose the Universe and should not result in variations of the fine structure constant. These demands lead to a certain constrains for gravexcitons (see Refs. [17, 19] ). We use the time delay effect, caused by the interaction between photons and gravexcitons, to get additional bounds on the parameters of gravexcitons.
The starting point of our investigation is the Abelian part of D-dimensional action of the Einstein-Yang-Mills theory:
S EM = - 1 2 M d D x |g| F MN F MN , (1)
where the D-dimensional metric, g = g MN (X)dX M ⊗ dX N = g (0) (x) µν dx µ ⊗ dx ν + a 2 1 (x)g (1) , is defined on the product manifold M = M 0 × M 1 . Here, M 0 is the (D 0 = d 0 + 1)-dimensional external space. The d 1dimensional internal space M 1 has a constant curvature with the scale factor a 1 (x) ≡ L P l exp β 1 (x). Dimensional reduction of the action (1) results in the following effective D 0 -dimensional action [17]
SEM = - 1 2 M0 d D0 x |g (0) | [(1 -Dκ 0 ψ) F µν F µν ] , (2)
which is written in the Einstein frame with the D 0dimensional metric, g(0
) µν = (exp d 1 β1 ) -2/(D0-2) g (0)
µν . Here,
κ 0 ψ ≡ -β1 (D 0 -2)/d 1 (D -2) ≪ 1 and β1 ≡ β 1 -β 1
0 are small fluctuations of the internal space scale factor over the stable background β 1 0 (0 subscript denotes the present day value). These internal space scalefactor small fluctuations/oscillations have the form of a scalar field (so called gravexciton [20] ) with a mass m ψ defined by the curvature of the effective potential (see for detail [20] ). Action ( 2 ) is defined under the approximation κ 0 ψ < 1 that obviously holds for the condition
1 ψ < M P l . κ 2 0 = 8π/M 2 P l is four dimen- sional gravitational constant, M P l is the Plank mass, D = 2 d 1 /[(D 0 -1)(D -1)
] is a model dependent constant. The Lagrangian density for the scalar field ψ reads:
L ψ = |g (0) |(-g µν ψ ,µ ψ ,ν -m 2 ψ ψψ)/2.
For simplicity we assume that g0 is the flat Friedman-Lemaitre-Robertson-Walker (FLRW) metric with the scale factor a(t).
Let's consider Eq. ( 2 ). It is worth of noting that the D 0 -dimensional field strength tensor, F µν , is gauge invariant. 2 Secondly, action (2) is conformally invariant in the case when D 0 = 4. The transform to the Einstein frame does not break gauge invariance of the action (2), and the electromagnetic field is antisymmetric as usual, (2) with respect to the electromagnetic vector potential,
F µν = ∂ µ A ν -∂ ν A µ . Varying
∂ ν √ -g (1 -Dκ 0 ψ) F µν = 0. (3)
The second term in the round brackets Dκ 0 ψF µν reflects the interaction between photons and the scalar field ψ, and as we show below, it is responsible for LV. In particular, coupling between photons and the scalar field ψ makes the speed of photons different from the standard speed of light. Eq. ( 3 ) together with Bianchi identity (which is preserved in the considered model due to gaugeinvariance of the tensor, F µν [17] ) defines a complete set 1 In the brane-world model the prefactor κ 0 in the expression for κ 0 ψ is replaced by the parameter proportional to M -1 EW [14] . Thus, the smallness condition holds for ψ < M EW . 2 Eq. ( 2 ) can be rewritten in the more familiar form SEM = [17] . The field strength tensor Fµν is not gauge invariant here.
-(1/2) R M 0 d D 0 x q |g (0) | Fµν F µν
of the generalized Maxwell equations. As we noted, action ( 2 ) is conformally invariant in the 4D dimensional space-time. So, it is convenient to present the flat FLRW metric g0 in the conformally flat form: g0 µν = a 2 η µν , where η µν is the Minkowski metric.
Using the standard definition of the electromagnetic field tensor, F µν , we obtain the complete set of the Maxwell equations in vacuum,
∇ • B = 0 , (4)
∇ • E = Dκ 0 1 -Dκ 0 ψ (∇ψ • E) , (5)
∇ × B = ∂E ∂η - Dκ 0 ψ 1 -Dκ 0 ψ E + Dκ 0 1 -Dκ 0 ψ [∇ψ × B] , (6)
∇ × E = - ∂B ∂η , (7)
where all operations are performed in the Minkowski space-time, η denotes conformal time related to physical time t as dt = a(η)dη, and an overdot represents a derivative with respect to conformal time η.
Eqs. ( 4 ) and ( 7 ) correspond to Bianchi identity, and since it is preserved, Eqs. ( 4 ) and (7) keep their usual forms. Eqs. ( 5 ) and ( 6 ) are modified due to interactions between photons and gravexcitons (∝ κ 0 ψ). These modifications have simple physical meaning: the interaction between photons and the scalar field ψ acts as an effective electric charge e ef f . This effective charge is proportional to the scalar product of the ψ field gradient and the E field, and it vanishes for an homogeneous ψ field. The modification of Eq. ( 6 ) corresponds to an effective current J ef f , which depends on both electric and magnetic fields. This effective current is determined by variations of the ψ field over the time ( ψ) and space (∇ψ). For the case of a homogeneous ψ field the effective current is still present and LV takes place. The modified Maxwell equations are conformally invariant. To account for the expansion of the Universe we rescale the field components asB, E → B, E a 2 [21] .
To obtain a dispersion relation for photons, we use the Fourier transform between position and wavenumber spaces as,
F(k, ω) = dη d 3 x e -i(ωη-k•x) F(x, η) , F(x, η) = 1 (2π) 4 dω d 3 ke i(ωη-k•x) F(k, ω) . (8)
Here, F is a vector function describing either the electric or the magnetic field, ω is the angular frequency of the electro-magnetic wave measured today, and k is the wave-vector. We assume that the field ψ is an oscillatory field with the frequency ω ψ and the momentum q, so ψ(x, η) = Ce i(ω ψ η-q•x) , C = const . Eq. ( 4 ) implies B ⊥ k. Without loosing of generality, and for simplicity of description we assume that the wave-vector k is oriented along the z axis. Using Eq. ( 7 ) we get E ⊥ B.
A linearly polarized wave can be expressed as a superposition of left (L, -) and right (R, +) circularly polarized (LCP and RCP) waves. Using the polarization basis of Sec. 1.1.3 of Ref. [22] , we derive E ± = (E x ± iE y )/ √ 2. Rewriting Eqs. ( 4 ) -( 7 ) in the components, 3 for LCP and RCP waves we get,
(1 -n 2 + )E + = 0, (1 -n 2 (-) )E -= 0 , (9)
where n + and n -are refractive indices for RCP and LCP electromagnetic waves
n 2 + = k 2 [1 -Dκ 0 ψ(1 + q z /k)] ω 2 [1 -Dκ 0 ψ(1 + ω ψ /ω)] = n 2 -. (10)
In the case when LI is preserved the electromagnetic waves propagating in vacuum have n
+ = n -= n = k/ω ≡ 1.
For the electromagnetic waves propagating in the magnetized plasma, k/ω = 1, and the difference between the LCP and RCP refractive indices describes the Faraday rotation effect, α ∝ ω(n + -n -) [23] . In the considered model, since n + = n -the rotation effect is absent, but the speed of electromagnetic waves propagation in vacuum differs from the speed of light c (see also Ref. [24] for LV induced by electromagnetic field coupling to other generic field). This difference implies the propagation time delay effect, ∆t = ∆l(1-∂k/∂ω) (∆l is a propagation distance), ∆t is the difference between the photon travel time and that for a "photon" which travels at the speed of light c. Here, t is physical synchronous time. This formula does not take into account the evolution of the Universe. However, it is easy to show that the effect of the Universe expansion is negligibly small. Solving the dispersion relation as a square equation, we obtain
∂k ∂ω ≃ ± 1 + 1 2 ω 2 ψ -q 2 z 4ω 2 (Dκ 0 ψ) 2 , (11)
where ± signs correspond to photons forward and backward directions respectively. The modified inverse group velocity (11) shows that the LV effect can be measured if we know the gravexciton frequency ω ψ , z-component of the momentum q z and its amplitude ψ. For our estimates, we assume that ψ is the oscillatory field, satisfying (in local Lorentz frame) the dispersion relation, ω 2 ψ = m 2 ψ + q 2 , where m ψ is the mass of gravexcitons 4 . Unfortunately, we do not have any information concerning parameters of gravexcitons (some estimates can be found in [17, 19] ). Thus, we intend to use possible LV effects (supposing it is caused by interaction between photons and gravexcitons) to set limits on gravexciton parameters. For example, we can easily get the following estimate for the upper limit of the amplitude of gravexciton oscillations:
|ψ| ≈ 1 √ π D ∆t ∆l ω m ψ M P l , (12)
where for ω and m ψ we can use their physical values.
In the case of GRB with ω ∼ 10 21 ÷ 10 22 Hz ∼ 10 -4 ÷ 10 -3 GeV and ∆l ∼ 3 ÷ 5 × 10 9 y ∼ 10 17 sec the typical upper limit for the time delay is ∆t ∼ 10 -4 sec [9] . For these values the upper limit on gravexciton amplitude of oscillations is foot_2
|κ 0 ψ| ≈ 10 -13 GeV m ψ . (13)
This estimate shows that our approximation κ 0 ψ < 1 works for gravexciton masses m ψ > 10 -13 GeV. Future measurements of the time-delay effect for GRBs at frequencies ω ∼ 1 -10GeV would increase significantly the limit up to m ψ > 10 -9 GeV. On the other hand, Cavendish-type experiments [26, 27] ) exclude fifth force particles with masses m ψ 1/(10 -2 cm) ∼ 10 -12 GeV which is rather close to our lower bound for ψ field masses. Respectively we slightly shift the considered mass lower limit to be m ψ ≥ 10 -12 GeV. These masses considerably higher than the mass corresponding to the equality between the energy densities of the matter and radiation (matter/radiation equality), m eq ∼ H eq ∼ 10 -37 GeV, where H eq is the Hubble "constant" at matter/radiation equality. It means that such ψ-particles start to oscillate during the radiation dominated epoch (see appendix). Another bound on the ψ-particles masses comes from the condition of their stability. With respect to decay ψ → γγ the life-time of ψ-particles is τ ∼ (M P l /m ψ ) 3 t P l [17] , and the stability conditions requires that the decay time should be greater than the age of the Universe. According this we consider light gravexcitons with masses m ψ ≤ 10 -21 M P l ∼ 10 -2 GeV ∼ 20m e (where m e is the electron mass).
As an additional restriction arises from the condition that such cosmological gravexcitons should not overclose the observable Universe. This reads m ψ m eq (M P l /ψ in ) 4 which implies the following restriction for the amplitude of the initial oscillations: ψ in (m eq /m ψ )
1/4 M P L << M P l [19] . Thus, for the range of masses 10 -12 GeV ≤ m ψ ≤ 10 -2 GeV, we obtain respectively ψ in 10 - foot_3 M P l and ψ in 10 -9 M P l . According to Eq. (A.3), we can also get the estimate for the amplitude of oscillations of the considered gravexciton at the present time. Together with the non-overcloseness condition, we obtain from this expression that |κ 0 ψ| ∼ 10 -43 for m ψ ∼ 10 -12 GeV and ψ in ∼ 10 -6 M P l and |κ 0 ψ| ∼ 10 -53 for m ψ ∼ 10 -2 GeV and ψ in ∼ 10 -9 M P l . Obviously, it is much less than the upper limit (13) . Note, as we mentioned above, gravexcitons with masses m ψ 10 -2 GeV can start to decay at the present epoch. However, taking into account the estimate |κ 0 ψ| ∼ 10 -53 , we can easily get that their energy density ρ ψ ∼ (|κ 0 ψ| 2 /8π)M 2 P l m 2 ψ ∼ 10 -55 g/cm 3 is much less than the present energy density of the radiation ρ γ ∼ 10 -34 g/cm 3 . Thus, ρ ψ contributes negligibly in ρ γ . Otherwise, the gravexcitons with masses m ψ 10 -2 GeV should be observed at the present time, which, obviously, is not the case.
Additionally, it follows from Eq. (42) in Ref. [17] that to avoid the problem of the fine structure constant variation, the amplitude of the initial oscillations should satisfy the condition: ψ in 10 -5 M P l which, obviously, completely agrees with our upper bound ψ in 10 -6 GeV.
Summarizing we shown that LV effects can give additional restrictions on parameters of gravexcitons. First, we found that gravexcitons should not be lighter than 10 -13 GeV. It is very close to the limit following from the fifth-force experiment. Moreover, experiments for GRB at frequencies ω > 1GeV can result in significant shift of this lower limit making it much stronger than the fifthforce estimates. Together with the non-overcloseness condition, this estimate leads to the upper limit on the amplitude of the gravexciton initial oscillations. It should not exceed ψ in 10 -6 GeV. Thus, the bound on the initial amplitude obtained from the fine structure constant variation is one magnitude weaker than our one even for the limiting case of the gravexciton masses. Increasing the mass of gravexcitons makes our limit stronger. Our estimates for the present day amplitude of the gravexciton oscillations, following from the obtained above limitations, show that we cannot use the LV effect for the direct detections of the gravexcitons. Nevertheless, the obtained bounds can be useful for astrophysical and cosmological applications. For example, let us suppose that gravexcitons with masses m ψ > 10 -2 GeV are produced during late stages of the Universe expansion in some regions and GRB photons travel to us through these regions. Then, Eq. (A.3) is not valid for such gravexcitons having astrophysical origin and the only upper limit on the amplitude of their oscillations (in these regions) follows from Eq. ( 13 ). In the case of TeV masses we get |κ 0 ψ| ∼ 10 -16 . If GRB photons have frequencies up to 1 TeV, ω ∼ 1TeV, then this estimate is increased by 6 orders of magnitude. dominated and matter dominated stages, correspondingly. We are interested in the gravexciton oscillations at the present time t = t univ . In this case s = 2/3 and for B(t univ ) we obtain: B(t univ ) ∼ t -1 univ ≈ 10 -61 M P l . Thus, the amplitude of the light gravexciton oscillations at the present time reads: | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We consider effective model where photons interact with scalar field corresponding to conformal excitations of the internal space (geometrical moduli/gravexcitons). We demonstrate that this interaction results in a modified dispersion relation for photons, and consequently, the photon group velocity depends on the energy implying the propagation time delay effect. We suggest to use the experimental bounds of the time delay of gamma ray bursts (GRBs) photons propagation as an additional constrain for the gravexciton parameters."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Lorentz invariance (LI) of physical laws is one of the corner stone of modern physics. There is a number of experiments confirming this symmetry at energies we can approach now. For example, on a classical level, the rotation invariance has been tested in Michelson-Morley experiments, and the boost invariance has been tested in Kennedy-Torhndike experiments [1] . Although, up to now, LI is well established experimentally, we cannot say surely that at higher energies it is still valid. Moreover, modern astrophysical and cosmological data (e.g. UHECR, dark matter, dark energy, etc) indicate for a possible LI violation (LV). To resolve these challenges, there are number of attempts to create new physical models, such as M/string theory, Kaluza-Klein models, brane-world models, etc. [1] .\n\nIn this paper we investigate LV test related to photon dispersion measure (PhDM). This test is based on the LV effect of a phenomenological energy-dependent speed of photon [2, 3, 4, 5, 6, 7, 8] , for recent studies see Ref. [9] and references therein.\n\nThe formalism that we use is based on the analogy with electromagnetic waves propagation in a magnetized medium, and extends previous works [8, 10, 11] . In our model, instead of propagation in a magnetized medium, the electromagnetic waves are propagating in vacuum filled with a scalar field ψ. LV occurs because of an interaction term f(ψ)F 2 where F is an amplitude of the electromagnetic field. Such an interaction might have different origins. In the string theory ψ could be a dilaton field [12, 13] . The field ψ could be associated with geometrical moduli. In brane-world models the similar term describes an interaction between the bulk dilaton and the Standard Model fields on the brane [14] . In Ref. [15] , such an interaction was obtained in N = 4 * Electronic address: bauch˙vGR@ukr.net † Electronic address: zhuk@paco.net ‡ Electronic address: tinatin@phys.ksu.edu super-gravity in four dimensions. In Kaluza-Klein models the term f(ψ)F 2 has the pure geometrical origin, and it appears in the effective, dimensionally reduced, four dimensional action (see e.g. [16, 17] ). In particular, in reduced Einstein-Yang-Mills theories, the function f (ψ) coincides (up to a numerical prefactor) with the volume of the internal space. Phenomenological (exactly solvable) models with spherical symmetries were considered in Refs. [18] . To be more specific, we consider the model which is based on the reduced Einstein-Yang-Mills theory [17] , where the term ∝ ψF 2 describes the interaction between the conformal excitations of the internal space (gravexcitons) and photons. It is clear that the similar LV effect exists for all types of interactions of the form f(ψ)F 2 mentioned above.\n\nObviously, the interaction term f(ψ)F 2 modifies the Maxwell equations, and, consequently, results in a modified dispersion relation for photons. We show that this modification has rather specific form. For example, we demonstrate that refractive indices for the left and right circularly polarized waves coincide with each other. Thus, rotational invariance is preserved. However, the speed of the electromagnetic wave's propagation in vacuum differs from the speed of light c. This difference implies the time delay effect which can be measured via high-energy GRB photons propagation over cosmological distances (see e.g. Ref. [9] ). It is clear that gravexcitons should not overclose the Universe and should not result in variations of the fine structure constant. These demands lead to a certain constrains for gravexcitons (see Refs. [17, 19] ). We use the time delay effect, caused by the interaction between photons and gravexcitons, to get additional bounds on the parameters of gravexcitons.\n\nThe starting point of our investigation is the Abelian part of D-dimensional action of the Einstein-Yang-Mills theory:\n\nS EM = - 1 2 M d D x |g| F MN F MN , (1)\n\nwhere the D-dimensional metric, g = g MN (X)dX M ⊗ dX N = g (0) (x) µν dx µ ⊗ dx ν + a 2 1 (x)g (1) , is defined on the product manifold M = M 0 × M 1 . Here, M 0 is the (D 0 = d 0 + 1)-dimensional external space. The d 1dimensional internal space M 1 has a constant curvature with the scale factor a 1 (x) ≡ L P l exp β 1 (x). Dimensional reduction of the action (1) results in the following effective D 0 -dimensional action [17]\n\nSEM = - 1 2 M0 d D0 x |g (0) | [(1 -Dκ 0 ψ) F µν F µν ] , (2)\n\nwhich is written in the Einstein frame with the D 0dimensional metric, g(0\n\n) µν = (exp d 1 β1 ) -2/(D0-2) g (0)\n\nµν . Here,\n\nκ 0 ψ ≡ -β1 (D 0 -2)/d 1 (D -2) ≪ 1 and β1 ≡ β 1 -β 1\n\n0 are small fluctuations of the internal space scale factor over the stable background β 1 0 (0 subscript denotes the present day value). These internal space scalefactor small fluctuations/oscillations have the form of a scalar field (so called gravexciton [20] ) with a mass m ψ defined by the curvature of the effective potential (see for detail [20] ). Action ( 2 ) is defined under the approximation κ 0 ψ < 1 that obviously holds for the condition\n\n1 ψ < M P l . κ 2 0 = 8π/M 2 P l is four dimen- sional gravitational constant, M P l is the Plank mass, D = 2 d 1 /[(D 0 -1)(D -1)\n\n] is a model dependent constant. The Lagrangian density for the scalar field ψ reads:\n\nL ψ = |g (0) |(-g µν ψ ,µ ψ ,ν -m 2 ψ ψψ)/2.\n\nFor simplicity we assume that g0 is the flat Friedman-Lemaitre-Robertson-Walker (FLRW) metric with the scale factor a(t).\n\nLet's consider Eq. ( 2 ). It is worth of noting that the D 0 -dimensional field strength tensor, F µν , is gauge invariant. 2 Secondly, action (2) is conformally invariant in the case when D 0 = 4. The transform to the Einstein frame does not break gauge invariance of the action (2), and the electromagnetic field is antisymmetric as usual, (2) with respect to the electromagnetic vector potential,\n\nF µν = ∂ µ A ν -∂ ν A µ . Varying\n\n∂ ν √ -g (1 -Dκ 0 ψ) F µν = 0. (3)\n\nThe second term in the round brackets Dκ 0 ψF µν reflects the interaction between photons and the scalar field ψ, and as we show below, it is responsible for LV. In particular, coupling between photons and the scalar field ψ makes the speed of photons different from the standard speed of light. Eq. ( 3 ) together with Bianchi identity (which is preserved in the considered model due to gaugeinvariance of the tensor, F µν [17] ) defines a complete set 1 In the brane-world model the prefactor κ 0 in the expression for κ 0 ψ is replaced by the parameter proportional to M -1 EW [14] . Thus, the smallness condition holds for ψ < M EW . 2 Eq. ( 2 ) can be rewritten in the more familiar form SEM = [17] . The field strength tensor Fµν is not gauge invariant here.\n\n-(1/2) R M 0 d D 0 x q |g (0) | Fµν F µν\n\nof the generalized Maxwell equations. As we noted, action ( 2 ) is conformally invariant in the 4D dimensional space-time. So, it is convenient to present the flat FLRW metric g0 in the conformally flat form: g0 µν = a 2 η µν , where η µν is the Minkowski metric.\n\nUsing the standard definition of the electromagnetic field tensor, F µν , we obtain the complete set of the Maxwell equations in vacuum,\n\n∇ • B = 0 , (4)\n\n∇ • E = Dκ 0 1 -Dκ 0 ψ (∇ψ • E) , (5)\n\n∇ × B = ∂E ∂η - Dκ 0 ψ 1 -Dκ 0 ψ E + Dκ 0 1 -Dκ 0 ψ [∇ψ × B] , (6)\n\n∇ × E = - ∂B ∂η , (7)\n\nwhere all operations are performed in the Minkowski space-time, η denotes conformal time related to physical time t as dt = a(η)dη, and an overdot represents a derivative with respect to conformal time η.\n\nEqs. ( 4 ) and ( 7 ) correspond to Bianchi identity, and since it is preserved, Eqs. ( 4 ) and (7) keep their usual forms. Eqs. ( 5 ) and ( 6 ) are modified due to interactions between photons and gravexcitons (∝ κ 0 ψ). These modifications have simple physical meaning: the interaction between photons and the scalar field ψ acts as an effective electric charge e ef f . This effective charge is proportional to the scalar product of the ψ field gradient and the E field, and it vanishes for an homogeneous ψ field. The modification of Eq. ( 6 ) corresponds to an effective current J ef f , which depends on both electric and magnetic fields. This effective current is determined by variations of the ψ field over the time ( ψ) and space (∇ψ). For the case of a homogeneous ψ field the effective current is still present and LV takes place. The modified Maxwell equations are conformally invariant. To account for the expansion of the Universe we rescale the field components asB, E → B, E a 2 [21] .\n\nTo obtain a dispersion relation for photons, we use the Fourier transform between position and wavenumber spaces as,\n\nF(k, ω) = dη d 3 x e -i(ωη-k•x) F(x, η) , F(x, η) = 1 (2π) 4 dω d 3 ke i(ωη-k•x) F(k, ω) . (8)\n\nHere, F is a vector function describing either the electric or the magnetic field, ω is the angular frequency of the electro-magnetic wave measured today, and k is the wave-vector. We assume that the field ψ is an oscillatory field with the frequency ω ψ and the momentum q, so ψ(x, η) = Ce i(ω ψ η-q•x) , C = const . Eq. ( 4 ) implies B ⊥ k. Without loosing of generality, and for simplicity of description we assume that the wave-vector k is oriented along the z axis. Using Eq. ( 7 ) we get E ⊥ B.\n\nA linearly polarized wave can be expressed as a superposition of left (L, -) and right (R, +) circularly polarized (LCP and RCP) waves. Using the polarization basis of Sec. 1.1.3 of Ref. [22] , we derive E ± = (E x ± iE y )/ √ 2. Rewriting Eqs. ( 4 ) -( 7 ) in the components, 3 for LCP and RCP waves we get,\n\n(1 -n 2 + )E + = 0, (1 -n 2 (-) )E -= 0 , (9)\n\nwhere n + and n -are refractive indices for RCP and LCP electromagnetic waves\n\nn 2 + = k 2 [1 -Dκ 0 ψ(1 + q z /k)] ω 2 [1 -Dκ 0 ψ(1 + ω ψ /ω)] = n 2 -. (10)\n\nIn the case when LI is preserved the electromagnetic waves propagating in vacuum have n\n\n+ = n -= n = k/ω ≡ 1.\n\nFor the electromagnetic waves propagating in the magnetized plasma, k/ω = 1, and the difference between the LCP and RCP refractive indices describes the Faraday rotation effect, α ∝ ω(n + -n -) [23] . In the considered model, since n + = n -the rotation effect is absent, but the speed of electromagnetic waves propagation in vacuum differs from the speed of light c (see also Ref. [24] for LV induced by electromagnetic field coupling to other generic field). This difference implies the propagation time delay effect, ∆t = ∆l(1-∂k/∂ω) (∆l is a propagation distance), ∆t is the difference between the photon travel time and that for a \"photon\" which travels at the speed of light c. Here, t is physical synchronous time. This formula does not take into account the evolution of the Universe. However, it is easy to show that the effect of the Universe expansion is negligibly small. Solving the dispersion relation as a square equation, we obtain\n\n∂k ∂ω ≃ ± 1 + 1 2 ω 2 ψ -q 2 z 4ω 2 (Dκ 0 ψ) 2 , (11)\n\nwhere ± signs correspond to photons forward and backward directions respectively. The modified inverse group velocity (11) shows that the LV effect can be measured if we know the gravexciton frequency ω ψ , z-component of the momentum q z and its amplitude ψ. For our estimates, we assume that ψ is the oscillatory field, satisfying (in local Lorentz frame) the dispersion relation, ω 2 ψ = m 2 ψ + q 2 , where m ψ is the mass of gravexcitons 4 . Unfortunately, we do not have any information concerning parameters of gravexcitons (some estimates can be found in [17, 19] ). Thus, we intend to use possible LV effects (supposing it is caused by interaction between photons and gravexcitons) to set limits on gravexciton parameters. For example, we can easily get the following estimate for the upper limit of the amplitude of gravexciton oscillations:\n\n|ψ| ≈ 1 √ π D ∆t ∆l ω m ψ M P l , (12)\n\nwhere for ω and m ψ we can use their physical values.\n\nIn the case of GRB with ω ∼ 10 21 ÷ 10 22 Hz ∼ 10 -4 ÷ 10 -3 GeV and ∆l ∼ 3 ÷ 5 × 10 9 y ∼ 10 17 sec the typical upper limit for the time delay is ∆t ∼ 10 -4 sec [9] . For these values the upper limit on gravexciton amplitude of oscillations is foot_2\n\n|κ 0 ψ| ≈ 10 -13 GeV m ψ . (13)\n\nThis estimate shows that our approximation κ 0 ψ < 1 works for gravexciton masses m ψ > 10 -13 GeV. Future measurements of the time-delay effect for GRBs at frequencies ω ∼ 1 -10GeV would increase significantly the limit up to m ψ > 10 -9 GeV. On the other hand, Cavendish-type experiments [26, 27] ) exclude fifth force particles with masses m ψ 1/(10 -2 cm) ∼ 10 -12 GeV which is rather close to our lower bound for ψ field masses. Respectively we slightly shift the considered mass lower limit to be m ψ ≥ 10 -12 GeV. These masses considerably higher than the mass corresponding to the equality between the energy densities of the matter and radiation (matter/radiation equality), m eq ∼ H eq ∼ 10 -37 GeV, where H eq is the Hubble \"constant\" at matter/radiation equality. It means that such ψ-particles start to oscillate during the radiation dominated epoch (see appendix). Another bound on the ψ-particles masses comes from the condition of their stability. With respect to decay ψ → γγ the life-time of ψ-particles is τ ∼ (M P l /m ψ ) 3 t P l [17] , and the stability conditions requires that the decay time should be greater than the age of the Universe. According this we consider light gravexcitons with masses m ψ ≤ 10 -21 M P l ∼ 10 -2 GeV ∼ 20m e (where m e is the electron mass).\n\nAs an additional restriction arises from the condition that such cosmological gravexcitons should not overclose the observable Universe. This reads m ψ m eq (M P l /ψ in ) 4 which implies the following restriction for the amplitude of the initial oscillations: ψ in (m eq /m ψ )\n\n1/4 M P L << M P l [19] . Thus, for the range of masses 10 -12 GeV ≤ m ψ ≤ 10 -2 GeV, we obtain respectively ψ in 10 - foot_3 M P l and ψ in 10 -9 M P l . According to Eq. (A.3), we can also get the estimate for the amplitude of oscillations of the considered gravexciton at the present time. Together with the non-overcloseness condition, we obtain from this expression that |κ 0 ψ| ∼ 10 -43 for m ψ ∼ 10 -12 GeV and ψ in ∼ 10 -6 M P l and |κ 0 ψ| ∼ 10 -53 for m ψ ∼ 10 -2 GeV and ψ in ∼ 10 -9 M P l . Obviously, it is much less than the upper limit (13) . Note, as we mentioned above, gravexcitons with masses m ψ 10 -2 GeV can start to decay at the present epoch. However, taking into account the estimate |κ 0 ψ| ∼ 10 -53 , we can easily get that their energy density ρ ψ ∼ (|κ 0 ψ| 2 /8π)M 2 P l m 2 ψ ∼ 10 -55 g/cm 3 is much less than the present energy density of the radiation ρ γ ∼ 10 -34 g/cm 3 . Thus, ρ ψ contributes negligibly in ρ γ . Otherwise, the gravexcitons with masses m ψ 10 -2 GeV should be observed at the present time, which, obviously, is not the case.\n\nAdditionally, it follows from Eq. (42) in Ref. [17] that to avoid the problem of the fine structure constant variation, the amplitude of the initial oscillations should satisfy the condition: ψ in 10 -5 M P l which, obviously, completely agrees with our upper bound ψ in 10 -6 GeV.\n\nSummarizing we shown that LV effects can give additional restrictions on parameters of gravexcitons. First, we found that gravexcitons should not be lighter than 10 -13 GeV. It is very close to the limit following from the fifth-force experiment. Moreover, experiments for GRB at frequencies ω > 1GeV can result in significant shift of this lower limit making it much stronger than the fifthforce estimates. Together with the non-overcloseness condition, this estimate leads to the upper limit on the amplitude of the gravexciton initial oscillations. It should not exceed ψ in 10 -6 GeV. Thus, the bound on the initial amplitude obtained from the fine structure constant variation is one magnitude weaker than our one even for the limiting case of the gravexciton masses. Increasing the mass of gravexcitons makes our limit stronger. Our estimates for the present day amplitude of the gravexciton oscillations, following from the obtained above limitations, show that we cannot use the LV effect for the direct detections of the gravexcitons. Nevertheless, the obtained bounds can be useful for astrophysical and cosmological applications. For example, let us suppose that gravexcitons with masses m ψ > 10 -2 GeV are produced during late stages of the Universe expansion in some regions and GRB photons travel to us through these regions. Then, Eq. (A.3) is not valid for such gravexcitons having astrophysical origin and the only upper limit on the amplitude of their oscillations (in these regions) follows from Eq. ( 13 ). In the case of TeV masses we get |κ 0 ψ| ∼ 10 -16 . If GRB photons have frequencies up to 1 TeV, ω ∼ 1TeV, then this estimate is increased by 6 orders of magnitude. dominated and matter dominated stages, correspondingly. We are interested in the gravexciton oscillations at the present time t = t univ . In this case s = 2/3 and for B(t univ ) we obtain: B(t univ ) ∼ t -1 univ ≈ 10 -61 M P l . Thus, the amplitude of the light gravexciton oscillations at the present time reads:"
}
] | |
arxiv:0704.0323 | 0704.0323 | 1 | 10.1088/1751-8113/41/15/155303 | a843256b0bab16e58310af032903a5199db900cbb46a890de67a9f4205ca8ae9 | General sequential quantum cloning | Some multipartite quantum states can be generated in a sequential manner
which may be implemented by various physical setups like microwave and optical
cavity QED, trapped ions, and quantum dots etc. We analyze the general N to M
qubits Universal Quantum Cloning Machine (UQCM) within a sequential generation
scheme. We show that the N to M sequential UQCM is available. The case of
d-level quantum states sequential cloning is also presented. | [
"Gui-Fang Dang",
"Heng Fan"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Some multipartite quantum states can be generated in a sequential manner which may be implemented by various physical setups like microwave and optical cavity QED, trapped ions, and quantum dots etc. We analyze the general N to M (N ≤ M ) qubits Universal Quantum Cloning Machine (UQCM) within a sequential generation scheme. We show that the N to M sequential UQCM is available. The case of d-level quantum states sequential cloning is also presented.
Quantum entanglement plays a key role in quantum computation and quantum information [1] . Multipartite entangled states arise as a resource for quantum information processing tasks such as the well known quantum teleportation [2] , quantum communication [3, 4] , clock synchronization [5] etc. In general it is extremely difficult to generate experimentally multipartite entangled states through single global unitary operations. In this sense, the sequential generation of the entangled states appears to be promising. Actually most of the quantum computation networks are designed to implement quantum logic gates through a sequential procedure [6] . Recently sequential implementing of quantum information processing tasks has been attracting much attention. It is pointed out that photonic multiqubit states can be generated by letting a source emit photonic qubits in a sequential manner [7] . The general sequential generation of entangled multiqubit states in the realm of cavity QED was systematically studied in Refs. [8, 9] . It is also shown that the class of sequentially generated states is identical to the matrix-product-state (MPS) which is very useful in study of spin chains of condensed matter physics [10] .
On the other hand, much progress has already been made in the past years in studying quantum cloning machines, for reviews see, for example, Refs. [11, 12, 13] . And various quantum cloning machines have been implemented experimently by polarization of photons [14, 15, 16, 17, 18] ,nuclear spins in Nuclear Magnetic Resonance [19, 20] , etc. However, these experiments are for 1 to 2 (one qubit input and two-qubit output) or 1 to 3 cloning machines. The more general case will be much difficult. There are some schemes proposed for the general quantum cloning machines which are not in a sequential manner, see for example, [21, 22] . Recently a 1 to M sequential universal quantum cloning is proposed [23] by using the cloning transformation presented in Ref. [24] . Since it is in a sequential procedure, potentially it reduces the difficult in implementing this quantum cloning machine. However, as is well known the collective quantum cloning machine (the N identical input states are cloned collectively to M copies) is better than the quantum cloning machine which can only deal with the individual input(only one input is copied to several copies each time). We know that the general N to M cloning transformation is also available in Refs. [24, 25] . Then a natural question arise is that whether the general N to M sequential cloning machine is possible. In this Letter, we will present the general sequential universal quantum cloning machine.
The 1 to M cloning transformations used in Ref. [23] was proposed by Gisin and Massar in Ref. [24] . And the N to M UQCM was also presented in Ref. [24] . However, to use the method proposed in Refs. [8, 23] to find the sequential cloning machine, the input state |Φ ⊗N should be expanded in computational basis {|0 , |1 }. The explicit quantum cloning transformations with this kind of input were proposed by Fan et al in Ref. [25] . In this Letter, based on the result of Ref. [25] , the general sequential UQCM will be presented.
As presented in Refs. [8, 23] , the sequential generation of a multiqubit state is like the following. Let H A be a D-dimensional Hilbert space which acts as the ancillary system, and a single qubit (e.g., a time-bin qubit) is in a two-dimensional Hilbert space H B . In every step of the sequential generation of a multiqubit state, a unitary time evolution will be acting on the joint system H A ⊗ H B . We assume that each qubit is initially in the state |0 which is like a blank or an empty state and will not be written out in the formulas. So the unitary time evolution is written in the form of an isometry V :
H A → H A ⊗H B , where V = i,α,β V i α,β |α, i β|, each V i is a D × D matrix,
and the isometry condition takes the form
1 i=0 V i † V i = 1.
By applying successively n operations of V (not necessarily the same) on an initial ancillary state |φ I ∈ H A , we obtain |Ψ = V [n] ...V [2] V [1] |φ I . The generated n qubits are in general an entangled state, but the last step qubit-ancilla interaction can be chosen so as to decouple the final multiqubit entangled state from the auxiliary system, so the sequentially generated state is
|ψ = 1 i1...in=0 φ F |V [n]in ...V [1]i1 |φ I |i n , ..., i 1 , (1)
where |φ F is the final state of the ancilla. This is the MPS. It was proven that any MPS can be sequentially generated [8] .
Suppose there are N identical pure quantum states
|Φ ⊗N = (x 0 |0 +x 1 |1 ) ⊗N need to be cloned to M copies, where |x 0 | 2 + |x 1 | 2 = 1.
We know that the input state can be represented by a basis in symmetric subspace.
|Φ ⊗N = N m=0 x N -m 0 x m 1 C m N |(N -m)0, m1 , (2)
where |(N -m)0, m1 denotes the symmetric and normalized state with (N -m) qubits in the state |0 and m qubits in the state |1 , and we have C m N = N !/(N -m)!m! in standard notation. So if we find the quantum cloning transformations for all states in symmetric subspace, we can clone N pure states to M copies. The UQCM with input in symmetric subspace can be written as [25] ,
|(N -m)0, m1 → |Φ m M , (3)
where
|Φ m M = M-N j=0 β mj |(M -m -j)0, (m + j)1 ⊗ R j , (4)
β mj = C M-N -j M-m-j C j (m+j) /C N +1 M+1 , (5)
where R j are the ancillary states of the cloning machine and are orthogonal with each other for different j. For a sequential quantum cloning machine in this Letter, we choose a realization R j ≡ |(M -N -j)1, j0 for the ancilla states. This UQCM is optimal in the sense that the fidelity between single qubit output state reduced density operator ρ out reduced and the single input |Φ is optimal. The optimal fidelity is F = Φ|ρ out reduced |Φ = (M N + M + N )/M (N + 2), see Refs. [11, 12, 13] for reviews and the references therein. A realization of this UQCM with photon stimulated emission can be found in Ref. [22] which is not in a sequential manner. We next show that this general N to M UQCM can be generated through a sequential procedure.
The basic idea is to show that the final state of the cloning, |Φ m M in (4), can be expressed in its MPS form. As shown in Ref. [8] , any MPS can be sequentially generated. We shall follow the method, for example, as in Refs. [23, 26] . By Schmidt decomposition, we first express the quantum state |Φ m M as a bi-partite state across 1 : 2... cut,
|Φ m M = λ [1] 1 |0 |φ [2...(2M-N )] 1 + λ [1] 2 |1 |φ [2...(2M-N )] 2 = α1i1 Γ [1]i1 α1 λ [1] α1 |i 1 |φ [2...(2M-N )] α1 , (6)
where Γ
[1]0
α1 = δ α1,1 , Γ [1]1 α1 = δ α1,2
, and λ [1] α1 are eigenvalues of the first qubit reduced density operator, and we find λ
[1] 1 = M-m-1 k=-m β 2 mk C m+k M-1 /C m+k M , λ [1] 2 = M-m-1 k=-m β 2 mk+1 C m+k M-1 /C m+k+1 M .
To correspond with the MPS in (1), we can define
V [1]i1 α1 = Γ [1]i1 α1 λ [1]
α1 . Successively by Schmidt decomposition, the quantum state |Φ m M in ( 4 ) is divided into a bi-partite state with the first n qubits as one part, and the rest as another part, where 1 < n ≤ M -1. We find
|Φ m M = n ′ j=0 λ [n] j+1 |(n -j)0, j1 |φ [(n+1)...(2M-N )] j+1 , (7)
when 1 < n ≤ M -N +m, n ′ = n; when M -N +m < n ≤ M -1, n ′ = M -N + m, λ [n]
j+1 are eigenvalues of the first n qubits reduced density operator of |Φ m M . According to the results in Eqs. (4, 5) , we can obtain,
λ [n] j+1 = C j n M-m-n k=-m β 2 m(j+k) C m+k M-n C m+j+k M . ( 8
)
And we also have
|φ [(n+1)...(2M-N )] j+1 = C j n λ [n] j+1 M-m-n k=-m β 2 m(j+k) × × C (m+k) M-n C m+j+k M |(M -n -m -k)0, (m + k)1 ⊗ R j+k .
By induction and a concise formula, we have
|Φ n...(2M-N )] j+1 = αn,in Γ [n]in (j+1)αn λ [n] αn |i n |φ [(n+1)...(2M-N )] αn = C j n-1 λ [n-1] j+1 |0 |φ [(n+1)...(2M-N )] j+1 λ [n] j+1 C j n +|1 |φ [(n+1)...(2M-N )] j+2 λ [n] j+2 √ C j+1 n , (9)
where we denote
Γ [n]0 (j+1)αn = δ (j+1)αn C j n-1 /(λ [n-1] j+1 C j n ), (10)
Γ [n]1 (j+1)αn = δ (j+2)αn C j n-1 /(λ [n-1] j+1 C j+1 n ). ( 11
)
Still we define that
V [n]in αnαn-1 = Γ [n]in αn-1αn λ [n] αn . (12)
It is thus in the MPS representation. We can further consider other cases including the ancilla state of the cloning machine represented as R j (Note it is not the ancilla state in the MPS representation). We can find that the output state of the general UQCM can be expressed as MPS as in form (1) . So it can be created sequentially. The explicit results are summarized in the appendix.
We have shown that the output states of the general UQCM in (4,5) are MPS's and thus can be generated sequentially. The sequential matrices V [n] of course depend on the input |(N -m)0, m1 which are W-like states and are generally multiqubit entangled. For later convenience, we denote V (m) to express that it depends on input state for different m. By a straightforward method, the sequential cloning operation, i.e., the isometrices, depending on different input may take the form m |(N -m)0, m (N -m)0, m1| ⊗ V (m). However, this operation may need a single global unitary operator which involves N -qubit entangled states except for m = 0, m = N . This contradicts with our aim that each operation should be divided into sequential unitary operators in a quDit (quantum state in D-dimensional space) times qubit system. Here we can use a scheme like the following: the ancillary state interacts with each qubit according to the (N + 1)
× D-dimensional isometrices as m C m N |0 0| ⊗N -m ⊗ |1 1| ⊗m ⊗ V (m)
sequentially, here a whole normalization factor is omitted. We know that the operation |0 0| ⊗N -m ⊗ |1 1| ⊗m acts on each qubit individually. Thus this scheme reduces the complexity of the operation. This finishes our general sequential UQCM for the case of qubit. In case N = 1, we recover the result of Ref. [23] for 1 to M cloning. We should remark that similar as the case of sequential 1 to M UQCM in Ref. [23] , for the general sequential UQCM, the minimal dimension D of the ancillary state grows linearly at most with M -N/2 + 1 for even N or M -(N -1)/2 for odd N .
Next we will consider a more general case that the sequential cloning machine is about the quantum state in ddimensional Hilbert space. We will use the d-dimensional UQCM proposed by Fan et al in Ref. [25] . This UQCM is a generalization of the cloning machine proposed in Ref. [24] and we can use this UQCM to study its sequential form for d-dimensional case.
An arbitrary d-dimensional pure state takes the form
|Φ = d-1 i=0 x i |i with d-1 i=0 |x i | 2 = 1. N identical pure states can be expanded in terms of state in symmet- ric subspace |Φ ⊗N = N m=0 N ! m1!...m d ! x m1 0 ...x m d d-1 | m , where | m ≡ |m 1 , ...,
| m → |Φ m M = M-N j=0 β m j | m + j ⊗ | j , (13)
β m j = d i=1 C ji mi+ji C M-N M+d-1 (14)
where j should satisfy i j i = M -N . This cloning machine is optimal and the corresponding fidelity of a single quantum state between input and output is
F = (N (d + M ) + M -N ) /(d + N )M .
As for qubit system, we next show that the output states for all symmetric states input can be expressed as the sequential form. We consider the case 1 < n ≤ M -1, and the state |Φ m M is a bipartite state across 1...n : (n + 1)... cut,
|Φ m M = M j=0 M-n k=0 λ [n] j | j |φ [(n+1)...(M+1)] j (15)
where
λ [n] j = M-n k=0 β 2 m( j-m+ k) d i=1 C ji ji+ki C n M , (16)
|φ
[(n+1)...(M+1)] j = M-n k=0 β m( j-m+ k) d i=1 C ji ji +ki C n M | k | j -m + k /λ [n] j . (17)
By the same procedure as that of qubit case, we can obtain the following
|φ [n...(M+1)] j = αnin Γ [n]in jαn λ [n] αn |i n |φ (n+1)...(M+1)] αn . (18)
Then we have Γ
[n]in jαn = δ αn( j+ ei n +1)
j in+1 + 1 n /λ [n-1] j . ( 19
)
Still we can define V
[n]in
αnαn-1 = Γ [n]in αn-1αn λ [n]
αn , and thus we can find that each state |Φ m M is a MPS and thus can be sequentially generated. The detailed result of this part will be presented elsewhere [27] .
In conclusion, we show that the general N to M universal quantum cloning machine can be implemented by a sequential manner. Since the sequential generation of multipartite state can be implemented in various physical setups such as microwave and optical cavity QED, trapped ions and quantum dots etc. This general sequential quantum cloning machine may be implemented much easier than the single global implementation scheme. This reduces dramatically the complexity in implementing the general UQCM. We also show that for d-dimensional quantum state, the sequential UQCM is also available. Besides the universal cloning machine, the 1 to M phasecovariant quantum cloning machine can also be sequentially implemented. It will be interesting to consider similarly the general N to M phase-covariant cloning and the economic phase-covariant cloning. The sequential asymmetric quantum cloning machine may also be an interesting topic.
Acknowledgements: HF was supported by "Bairen" program, NSFC and "973" program (2006CB921107).
Appendix.-The explicit form of matrices V are presented as:
V [n]0 αnαn-1 = δ αnαn-1 × × M-m-n k=-m X C m+k M -n C m+α n-1 -1+k M M-m-n+1 k=-m X C m+k M -n+1 C m+α n-1 -1+k M 1/2 , V [n]0 αnαn-1 = δ αnαn-1+1 × × M-m-n k=-m X ′ C m+k M -n C m+α n-1 +k M M-m-n+1 k=-m X C m+k M -n+1 C m+α n-1 -1+k M 1/2
, where notations
X = β 2 m(αn-1-1+k) , X ′ = β 2 m(αn-1+k)
are used. For case 1 < n ≤ M -N + m, α n-1 = 1, ..., n; α n = 1, ..., (n+ 1), and for case M -N + m < n ≤ M -1, α n-1 , α n = 1, ..., (M -N + m + 1). We can check that the above defined V satisfies the isometry condition For case concerning about ancilla state of the UQCM, assume 1 ≤ l ≤ M -N , we have (2) For (m + 1) ≤ α M+l , α M+l-1 ≤ (M -N + m -l + 1),
in V [n]in † V [n]in = 1. Similarly we have V [M]0 αM αM-1 = δ αM αM-1 × × β 2 m(α M -1 -1-m) C α M -1 M β 2 m(α M -1 -1-m) C α M -1 -1 M + β 2 m(α M -1 -m) C α M -1 M 1/2 , V [
V [M+l]0 α M +l α M +l-1 = δ α M +l (α M +l-1 -1) × × α M+l-1 -m -1 M -N -l + 1 1/2 , V [M+l]1 α M +l α M +l-1 = δ α M +l α M +l-1 × × M -N -l -α M+l-1 + m + 1 M -N -l + 1
V [M+l]1 α M +l α M +l-1 = δ α M +l α M +l-1 M-N -l-α M +l-1 +m+2 M-N -l+1
.
For α M+l = (M -N + m -l + 2), 1 ≤ α M+l-1 ≤ (M -N + m + 1), V
[M+l]0 α M +l α M +l-1 = 0.
Otherwise
V [M+l]0 α M +l α M +l-1 = δ α M +l α M +l-1 1 √ 2 . | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Some multipartite quantum states can be generated in a sequential manner which may be implemented by various physical setups like microwave and optical cavity QED, trapped ions, and quantum dots etc. We analyze the general N to M (N ≤ M ) qubits Universal Quantum Cloning Machine (UQCM) within a sequential generation scheme. We show that the N to M sequential UQCM is available. The case of d-level quantum states sequential cloning is also presented."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Quantum entanglement plays a key role in quantum computation and quantum information [1] . Multipartite entangled states arise as a resource for quantum information processing tasks such as the well known quantum teleportation [2] , quantum communication [3, 4] , clock synchronization [5] etc. In general it is extremely difficult to generate experimentally multipartite entangled states through single global unitary operations. In this sense, the sequential generation of the entangled states appears to be promising. Actually most of the quantum computation networks are designed to implement quantum logic gates through a sequential procedure [6] . Recently sequential implementing of quantum information processing tasks has been attracting much attention. It is pointed out that photonic multiqubit states can be generated by letting a source emit photonic qubits in a sequential manner [7] . The general sequential generation of entangled multiqubit states in the realm of cavity QED was systematically studied in Refs. [8, 9] . It is also shown that the class of sequentially generated states is identical to the matrix-product-state (MPS) which is very useful in study of spin chains of condensed matter physics [10] .\n\nOn the other hand, much progress has already been made in the past years in studying quantum cloning machines, for reviews see, for example, Refs. [11, 12, 13] . And various quantum cloning machines have been implemented experimently by polarization of photons [14, 15, 16, 17, 18] ,nuclear spins in Nuclear Magnetic Resonance [19, 20] , etc. However, these experiments are for 1 to 2 (one qubit input and two-qubit output) or 1 to 3 cloning machines. The more general case will be much difficult. There are some schemes proposed for the general quantum cloning machines which are not in a sequential manner, see for example, [21, 22] . Recently a 1 to M sequential universal quantum cloning is proposed [23] by using the cloning transformation presented in Ref. [24] . Since it is in a sequential procedure, potentially it reduces the difficult in implementing this quantum cloning machine. However, as is well known the collective quantum cloning machine (the N identical input states are cloned collectively to M copies) is better than the quantum cloning machine which can only deal with the individual input(only one input is copied to several copies each time). We know that the general N to M cloning transformation is also available in Refs. [24, 25] . Then a natural question arise is that whether the general N to M sequential cloning machine is possible. In this Letter, we will present the general sequential universal quantum cloning machine.\n\nThe 1 to M cloning transformations used in Ref. [23] was proposed by Gisin and Massar in Ref. [24] . And the N to M UQCM was also presented in Ref. [24] . However, to use the method proposed in Refs. [8, 23] to find the sequential cloning machine, the input state |Φ ⊗N should be expanded in computational basis {|0 , |1 }. The explicit quantum cloning transformations with this kind of input were proposed by Fan et al in Ref. [25] . In this Letter, based on the result of Ref. [25] , the general sequential UQCM will be presented.\n\nAs presented in Refs. [8, 23] , the sequential generation of a multiqubit state is like the following. Let H A be a D-dimensional Hilbert space which acts as the ancillary system, and a single qubit (e.g., a time-bin qubit) is in a two-dimensional Hilbert space H B . In every step of the sequential generation of a multiqubit state, a unitary time evolution will be acting on the joint system H A ⊗ H B . We assume that each qubit is initially in the state |0 which is like a blank or an empty state and will not be written out in the formulas. So the unitary time evolution is written in the form of an isometry V :\n\nH A → H A ⊗H B , where V = i,α,β V i α,β |α, i β|, each V i is a D × D matrix,\n\nand the isometry condition takes the form\n\n1 i=0 V i † V i = 1.\n\nBy applying successively n operations of V (not necessarily the same) on an initial ancillary state |φ I ∈ H A , we obtain |Ψ = V [n] ...V [2] V [1] |φ I . The generated n qubits are in general an entangled state, but the last step qubit-ancilla interaction can be chosen so as to decouple the final multiqubit entangled state from the auxiliary system, so the sequentially generated state is\n\n|ψ = 1 i1...in=0 φ F |V [n]in ...V [1]i1 |φ I |i n , ..., i 1 , (1)\n\nwhere |φ F is the final state of the ancilla. This is the MPS. It was proven that any MPS can be sequentially generated [8] .\n\nSuppose there are N identical pure quantum states\n\n|Φ ⊗N = (x 0 |0 +x 1 |1 ) ⊗N need to be cloned to M copies, where |x 0 | 2 + |x 1 | 2 = 1.\n\nWe know that the input state can be represented by a basis in symmetric subspace.\n\n|Φ ⊗N = N m=0 x N -m 0 x m 1 C m N |(N -m)0, m1 , (2)\n\nwhere |(N -m)0, m1 denotes the symmetric and normalized state with (N -m) qubits in the state |0 and m qubits in the state |1 , and we have C m N = N !/(N -m)!m! in standard notation. So if we find the quantum cloning transformations for all states in symmetric subspace, we can clone N pure states to M copies. The UQCM with input in symmetric subspace can be written as [25] ,\n\n|(N -m)0, m1 → |Φ m M , (3)\n\nwhere\n\n|Φ m M = M-N j=0 β mj |(M -m -j)0, (m + j)1 ⊗ R j , (4)\n\nβ mj = C M-N -j M-m-j C j (m+j) /C N +1 M+1 , (5)\n\nwhere R j are the ancillary states of the cloning machine and are orthogonal with each other for different j. For a sequential quantum cloning machine in this Letter, we choose a realization R j ≡ |(M -N -j)1, j0 for the ancilla states. This UQCM is optimal in the sense that the fidelity between single qubit output state reduced density operator ρ out reduced and the single input |Φ is optimal. The optimal fidelity is F = Φ|ρ out reduced |Φ = (M N + M + N )/M (N + 2), see Refs. [11, 12, 13] for reviews and the references therein. A realization of this UQCM with photon stimulated emission can be found in Ref. [22] which is not in a sequential manner. We next show that this general N to M UQCM can be generated through a sequential procedure.\n\nThe basic idea is to show that the final state of the cloning, |Φ m M in (4), can be expressed in its MPS form. As shown in Ref. [8] , any MPS can be sequentially generated. We shall follow the method, for example, as in Refs. [23, 26] . By Schmidt decomposition, we first express the quantum state |Φ m M as a bi-partite state across 1 : 2... cut,\n\n|Φ m M = λ [1] 1 |0 |φ [2...(2M-N )] 1 + λ [1] 2 |1 |φ [2...(2M-N )] 2 = α1i1 Γ [1]i1 α1 λ [1] α1 |i 1 |φ [2...(2M-N )] α1 , (6)\n\nwhere Γ\n\n[1]0\n\nα1 = δ α1,1 , Γ [1]1 α1 = δ α1,2\n\n, and λ [1] α1 are eigenvalues of the first qubit reduced density operator, and we find λ\n\n[1] 1 = M-m-1 k=-m β 2 mk C m+k M-1 /C m+k M , λ [1] 2 = M-m-1 k=-m β 2 mk+1 C m+k M-1 /C m+k+1 M .\n\nTo correspond with the MPS in (1), we can define\n\nV [1]i1 α1 = Γ [1]i1 α1 λ [1]\n\nα1 . Successively by Schmidt decomposition, the quantum state |Φ m M in ( 4 ) is divided into a bi-partite state with the first n qubits as one part, and the rest as another part, where 1 < n ≤ M -1. We find\n\n|Φ m M = n ′ j=0 λ [n] j+1 |(n -j)0, j1 |φ [(n+1)...(2M-N )] j+1 , (7)\n\nwhen 1 < n ≤ M -N +m, n ′ = n; when M -N +m < n ≤ M -1, n ′ = M -N + m, λ [n]\n\nj+1 are eigenvalues of the first n qubits reduced density operator of |Φ m M . According to the results in Eqs. (4, 5) , we can obtain,\n\nλ [n] j+1 = C j n M-m-n k=-m β 2 m(j+k) C m+k M-n C m+j+k M . ( 8\n\n)\n\nAnd we also have\n\n|φ [(n+1)...(2M-N )] j+1 = C j n λ [n] j+1 M-m-n k=-m β 2 m(j+k) × × C (m+k) M-n C m+j+k M |(M -n -m -k)0, (m + k)1 ⊗ R j+k .\n\nBy induction and a concise formula, we have\n\n|Φ n...(2M-N )] j+1 = αn,in Γ [n]in (j+1)αn λ [n] αn |i n |φ [(n+1)...(2M-N )] αn = C j n-1 λ [n-1] j+1 |0 |φ [(n+1)...(2M-N )] j+1 λ [n] j+1 C j n +|1 |φ [(n+1)...(2M-N )] j+2 λ [n] j+2 √ C j+1 n , (9)\n\nwhere we denote\n\nΓ [n]0 (j+1)αn = δ (j+1)αn C j n-1 /(λ [n-1] j+1 C j n ), (10)\n\nΓ [n]1 (j+1)αn = δ (j+2)αn C j n-1 /(λ [n-1] j+1 C j+1 n ). ( 11\n\n)\n\nStill we define that\n\nV [n]in αnαn-1 = Γ [n]in αn-1αn λ [n] αn . (12)\n\nIt is thus in the MPS representation. We can further consider other cases including the ancilla state of the cloning machine represented as R j (Note it is not the ancilla state in the MPS representation). We can find that the output state of the general UQCM can be expressed as MPS as in form (1) . So it can be created sequentially. The explicit results are summarized in the appendix.\n\nWe have shown that the output states of the general UQCM in (4,5) are MPS's and thus can be generated sequentially. The sequential matrices V [n] of course depend on the input |(N -m)0, m1 which are W-like states and are generally multiqubit entangled. For later convenience, we denote V (m) to express that it depends on input state for different m. By a straightforward method, the sequential cloning operation, i.e., the isometrices, depending on different input may take the form m |(N -m)0, m (N -m)0, m1| ⊗ V (m). However, this operation may need a single global unitary operator which involves N -qubit entangled states except for m = 0, m = N . This contradicts with our aim that each operation should be divided into sequential unitary operators in a quDit (quantum state in D-dimensional space) times qubit system. Here we can use a scheme like the following: the ancillary state interacts with each qubit according to the (N + 1)\n\n× D-dimensional isometrices as m C m N |0 0| ⊗N -m ⊗ |1 1| ⊗m ⊗ V (m)\n\nsequentially, here a whole normalization factor is omitted. We know that the operation |0 0| ⊗N -m ⊗ |1 1| ⊗m acts on each qubit individually. Thus this scheme reduces the complexity of the operation. This finishes our general sequential UQCM for the case of qubit. In case N = 1, we recover the result of Ref. [23] for 1 to M cloning. We should remark that similar as the case of sequential 1 to M UQCM in Ref. [23] , for the general sequential UQCM, the minimal dimension D of the ancillary state grows linearly at most with M -N/2 + 1 for even N or M -(N -1)/2 for odd N .\n\nNext we will consider a more general case that the sequential cloning machine is about the quantum state in ddimensional Hilbert space. We will use the d-dimensional UQCM proposed by Fan et al in Ref. [25] . This UQCM is a generalization of the cloning machine proposed in Ref. [24] and we can use this UQCM to study its sequential form for d-dimensional case.\n\nAn arbitrary d-dimensional pure state takes the form\n\n|Φ = d-1 i=0 x i |i with d-1 i=0 |x i | 2 = 1. N identical pure states can be expanded in terms of state in symmet- ric subspace |Φ ⊗N = N m=0 N ! m1!...m d ! x m1 0 ...x m d d-1 | m , where | m ≡ |m 1 , ...,\n\n| m → |Φ m M = M-N j=0 β m j | m + j ⊗ | j , (13)\n\nβ m j = d i=1 C ji mi+ji C M-N M+d-1 (14)\n\nwhere j should satisfy i j i = M -N . This cloning machine is optimal and the corresponding fidelity of a single quantum state between input and output is\n\nF = (N (d + M ) + M -N ) /(d + N )M .\n\nAs for qubit system, we next show that the output states for all symmetric states input can be expressed as the sequential form. We consider the case 1 < n ≤ M -1, and the state |Φ m M is a bipartite state across 1...n : (n + 1)... cut,\n\n|Φ m M = M j=0 M-n k=0 λ [n] j | j |φ [(n+1)...(M+1)] j (15)\n\nwhere\n\nλ [n] j = M-n k=0 β 2 m( j-m+ k) d i=1 C ji ji+ki C n M , (16)\n\n|φ\n\n[(n+1)...(M+1)] j = M-n k=0 β m( j-m+ k) d i=1 C ji ji +ki C n M | k | j -m + k /λ [n] j . (17)\n\nBy the same procedure as that of qubit case, we can obtain the following\n\n|φ [n...(M+1)] j = αnin Γ [n]in jαn λ [n] αn |i n |φ (n+1)...(M+1)] αn . (18)\n\nThen we have Γ\n\n[n]in jαn = δ αn( j+ ei n +1)\n\nj in+1 + 1 n /λ [n-1] j . ( 19\n\n)\n\nStill we can define V\n\n[n]in\n\nαnαn-1 = Γ [n]in αn-1αn λ [n]\n\nαn , and thus we can find that each state |Φ m M is a MPS and thus can be sequentially generated. The detailed result of this part will be presented elsewhere [27] .\n\nIn conclusion, we show that the general N to M universal quantum cloning machine can be implemented by a sequential manner. Since the sequential generation of multipartite state can be implemented in various physical setups such as microwave and optical cavity QED, trapped ions and quantum dots etc. This general sequential quantum cloning machine may be implemented much easier than the single global implementation scheme. This reduces dramatically the complexity in implementing the general UQCM. We also show that for d-dimensional quantum state, the sequential UQCM is also available. Besides the universal cloning machine, the 1 to M phasecovariant quantum cloning machine can also be sequentially implemented. It will be interesting to consider similarly the general N to M phase-covariant cloning and the economic phase-covariant cloning. The sequential asymmetric quantum cloning machine may also be an interesting topic.\n\nAcknowledgements: HF was supported by \"Bairen\" program, NSFC and \"973\" program (2006CB921107).\n\nAppendix.-The explicit form of matrices V are presented as:\n\nV [n]0 αnαn-1 = δ αnαn-1 × × M-m-n k=-m X C m+k M -n C m+α n-1 -1+k M M-m-n+1 k=-m X C m+k M -n+1 C m+α n-1 -1+k M 1/2 , V [n]0 αnαn-1 = δ αnαn-1+1 × × M-m-n k=-m X ′ C m+k M -n C m+α n-1 +k M M-m-n+1 k=-m X C m+k M -n+1 C m+α n-1 -1+k M 1/2\n\n, where notations\n\nX = β 2 m(αn-1-1+k) , X ′ = β 2 m(αn-1+k)\n\nare used. For case 1 < n ≤ M -N + m, α n-1 = 1, ..., n; α n = 1, ..., (n+ 1), and for case M -N + m < n ≤ M -1, α n-1 , α n = 1, ..., (M -N + m + 1). We can check that the above defined V satisfies the isometry condition For case concerning about ancilla state of the UQCM, assume 1 ≤ l ≤ M -N , we have (2) For (m + 1) ≤ α M+l , α M+l-1 ≤ (M -N + m -l + 1),\n\nin V [n]in † V [n]in = 1. Similarly we have V [M]0 αM αM-1 = δ αM αM-1 × × β 2 m(α M -1 -1-m) C α M -1 M β 2 m(α M -1 -1-m) C α M -1 -1 M + β 2 m(α M -1 -m) C α M -1 M 1/2 , V [\n\nV [M+l]0 α M +l α M +l-1 = δ α M +l (α M +l-1 -1) × × α M+l-1 -m -1 M -N -l + 1 1/2 , V [M+l]1 α M +l α M +l-1 = δ α M +l α M +l-1 × × M -N -l -α M+l-1 + m + 1 M -N -l + 1\n\nV [M+l]1 α M +l α M +l-1 = δ α M +l α M +l-1 M-N -l-α M +l-1 +m+2 M-N -l+1\n\n.\n\nFor α M+l = (M -N + m -l + 2), 1 ≤ α M+l-1 ≤ (M -N + m + 1), V\n\n[M+l]0 α M +l α M +l-1 = 0.\n\nOtherwise\n\nV [M+l]0 α M +l α M +l-1 = δ α M +l α M +l-1 1 √ 2 ."
}
] | |
arxiv:0704.0332 | 0704.0332 | 1 | ab1b535c10971747aea9297bf8482333825895fbc83801d4337ff9487811ca54 | The effect of a fifth large-scale space-time dimension on the
conservation of energy in a four dimensional Universe | The effect of introducing a fifth large-scale space-time dimension to the
equations of orbital dynamics was analysed in an earlier paper by the authors.
The results showed good agreement with the observed flat rotation curves of
galaxies and the Pioneer Anomaly. This analysis did not require the
modification of Newtonian dynamics, but rather only their restatement in a five
dimensional framework. The same analysis derived a acceleration parameter ar,
which plays an important role in the restated equations of orbital dynamics,
and suggested a value for ar. In this companion paper, the principle of
conservation of energy is restated within the same five-dimensional framework.
The resulting analysis provides an alternative route to estimating the value of
ar, without reference to the equations of orbital dynamics, and based solely on
key cosmological constants and parameters, including the gravitational
constant, G. The same analysis suggests that: (i) the inverse square law of
gravity may itself be due to the conservation of energy at the boundary between
a four-dimensional universe and a fifth large-scale space-time dimension; and
(ii) there is a limiting case for the Tulley-Fisher relationship linking the
speed of light to the mass of the Universe. | [
"M.B. Gerrard and T.J. Sumner"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | Conservation of energy with 5 dimensions 2 1. Introduction In an earlier paper [1] we introduced a fifth large-scale space-time dimension, r to Newton's Second Law, as applied to systems with angular velocity. The resulting analysis of the orbital motion of galaxies, which considered only the role of baryonic matter, is consistent with their observed rotation curves and the Tulley-Fisher relationship. The dimension r, is orthogonal to the three space dimensions s(x, y, z) and the time dimension, t of a four-dimensional universe, but does not represent a degree of freedom of motion in this analysis. For a closed isotropic universe, r is the radius of curvature of (four-dimensional) space-time and has a value, r u remote from gravitating matter that is estimated to be ∼ 7.5 × 10 26 m. The parameter a r is derived from the relationship a r = c 2 /r. In the case of r being equal to r u , a r has a value of 1.2 × 10 -10 ms -2 , which is the same as the MOND parameter a 0 derived by Milgrom [2] from observing the rotation curves of more than eighty galaxies.
Using the same five-dimensional analytical framework, this paper examines the relationships between a r , the principle of conservation of energy and gravity. The resulting derivation of a r is, therefore, unrelated to orbital dynamics and Newton's Second Law and instead relies on key cosmological constants, such as the gravitational constant, G and parameters, such as the mass density of the universe.
The large-scale distribution of matter across the universe creates a background gravitational acceleration, g b which is isotropic if matter itself is evenly distributed on this scale. The mutual attraction of each particle of matter towards all other matter, as represented by g b , is similar in concept to a three dimensional "surface tension" stretching across the universe.
If space is assumed to be flat and open and matter is assumed to be evenly distributed on this large scale, with (baryonic) mass density ρ, then the background gravitational acceleration, g b , can be derived as follows:
g b = πGρH H ( 1
)
where G is the gravitational constant (6.67 × 10 -11 m 3 Kg -1 s -2 ), ρ for baryonic matter has a currently estimated value ρ u = 3 × 10 -28 Kg m -3 and H H is the Hubble Horizon given by H H = c/H with H being Hubble's Constant (71 Km s -1 Mpc -1 ). Substitution in equation (1) gives a current value for g b of 8.2 × 10 -12 ms -2 which is noted to be two orders of magnitude less than the value of a 0 . The accuracy of equation (1) depends on three potential sources of uncertainty, namely: the value of ρ, the method of calculation of the volume of the universe within the Hubble Horizon and the value of H itself. These will be discussed later.
Conservation of energy with 5 dimensions 3 ~M σ r u g x a r r ~x Figure 1. Locus of points r(x) at which there is balance between the two accelerations g x and a r .
In section 3.2 of the earlier paper [1] an expression was derived for the locus of points r(x) adjacent to a gravitating mass, M which defined the balance condition between gravitational acceleration g x and the acceleration a r acting everywhere in the universe in the direction of r.
r (x) = r u 1 - GM c 2 x ( 2
)
where r u is the radius of curvature of four-dimensional space-time remote from gravitating matter M and x is the distance away from M as shown in figure 1 .
The effect which matter has on the local radius of curvature of space-time, r is cumulative and can be found by the superposition (∆r/r u = Σ ∆r i /r u , where ∆r i = (r u -r i )) from all individual masses, M i . Applying equation (2) to all baryonic matter contained within the Hubble Horizon (again assumed to be evenly distributed across space with density ρ and lying within a spherical volume defined by 4/3 (πs 3 ) where s here is H H ) it is possible to calculate an overall background value of r(x). This value will inevitably be somewhat less than r u given that no point is, in practice, completely remote from all matter. This background value of r is referred to as r b and is derived by integrating the contributions from matter lying within concentric spherical shells of space to give:
r b = r u 1 - 2πGρH 2 H c 2 ( 3
)
Substituting for known parameters and constants in equation (3), including the current value of the mass density ρ u , gives a value for r b equal to 0.98 × r u . Substituting either value for r into the key relationship a r = c 2 /r gives the same value for a r to within one decimal place, namely 1.2 × 10 -10 ms -2 .
The average mass density of the universe, ρ, decreases over time in an expanding universe. For a Euclidean (although expanding) universe, the volume of space within
Conservation of energy with 5 dimensions 4 the Hubble Horizon is given by (4/3) πH 3 H ≃ (4/3) π(ct) 3 . Given that (to a first order) the total mass lying within the Hubble Horizon is constant, it follows that we can derive an expression for the average mass density ρ(t) of the universe at any time t, in terms of the average mass density observed for the current era ρ u (i.e. ∼ 3 × 10 -28 Kg m -3 ) and the current estimated age of the universe t u (i.e. 13.7 Bn years).
ρ ≃ ρ u t 3 u t 3 ( 4
)
Given that this equation is derived (in part) from the approximation H H ≃ ct, it is assumed only to be applicable in the current analysis for perturbations of time about the current era. Substituting for ρ from equation (4) into equation (3) provides an expression for the local time-dependency of the background radius of curvature of space-time r b in equation (5) , which is similarly limited in its range of extrapolation.
r b = r u 1 - 2πGρ u t 3 u t ( 5
)
In section 3.1 in the earlier paper [1] a r was described as a universal acceleration of expansion acting at all points in space in the direction of r. To maintain conservation of energy within four-dimensional space-time, it follows that for any mass m at a point in space P there must be an acceleration equal and opposite to a r which prevents energy being transferred from within the four-dimensional universe to the fifth dimension r, as shown in figure 2. Accordingly, this principle may be written as:
a r + d 2 r b dt 2 = 0 ( 6
)
The second term of this equation (r b ) is identified as the acceleration acting on a mass in the direction of the dimension r (decreasing) by virtue of the expansion of the universe in the dimension r which causes r b the background value of r to increase over time (but at a decelerating rate -see equation (5)). In other words, given that the universal acceleration a r is acting everywhere along the boundary between the fourdimensional space-time and the fifth dimension r, energy can only be conserved (within four dimensional space-time) if the background radius of curvature of space-time r b varies in time so as to satisfy equation (6) . This conservation of energy at the boundary between the four dimensional universe and the fifth dimension r is, of course, the reason why the dimension r is not itself directly observable. As referred above, for the current era a r is 1.2 × 10 -10 ms -2 .
Assuming only r b varies with time equation (5) gives:
d 2 r b dt 2 = -4πρGr u ( 7
)
Substituting values for known parameters and constants on the right-hand side (including the current mass density of the universe, ρ u , provides the result: rb = Conservation of energy with 5 dimensions
5 a r d 2 r dt 2 s(x,y,z) ~P m r
Figure 2. Conservation of energy requires the two accelerations a r and rb to be equal and opposite.
-1.9 × 10 -10 ms -2 . Given the approximations used to derive equation (7) , this value for rb appears to be in reasonably good agreement with the value expected from equation (6), namely: -1.2 × 10 -10 ms -2 ). The substitution for r u in equation (7) using the relationship a r = c 2 /r (section (1) above), but with the identification of a r = a o for r = r u for the current era, and the combining of equations (6) and (7) allows an expression for a o as:
a o = 4πρ u Gc 2 1/2 ( 8
)
which has the value of 1.5 × 10 -10 ms -2 for current era. The level of agreement between a r and rb , calculated from equation (7) , can only be properly assessed by considering the uncertainty in the three key components to equation (7): the value of the Hubble Horizon, the average mass density of the universe and the estimated volume of the universe. Consistency between rb from equation (7) and equation (6) ) lies within the uncertainty ranges of ±12% in each of these three components. However, the principal source of uncertainty in rb is expected to be the method used to calculate the volume of the universe lying within the Hubble Horizon.
The form of universe that underpins the derivation of a r is closed (i.e. curved) and isotropic Section 3.2 in [1] and yet, so far in this paper, we have used the Euclidean derivation of a three dimension spherical volume 4/3 (πs 3 ), where s is the radius of the volume -i.e. a derivation appropriate to a flat and open universe. A closed isotropic three dimensional space is the "surface area" of a 4-dimensional hyper-sphere, the 3dimensional volume of which is given not by 4/3 (πs 3 ) but by the expression 2π 2 R 3 , where R is the radius of curvature of the hyper-sphere. The relevant feature of this 3dimensional "surface area" is that at increasing distances s from a point P, the volumes of concentric spherical shells of space centred on P become progressively smaller than those derived from the (Euclidean) expression 4πs 2 ds.
Accordingly, failure to take account of this effect will have led to an over-estimation of the volume of the universe lying within the Hubble Horizon and so to an overestimation of rb in equation (7) . The value of g b in equation (1) will have, likewise, been overestimated for this reason.
Conservation of energy with 5 dimensions 6 There are two important aspects of the application of ρ in the calculation of rb and g b that also need to be highlighted: the first in relation to a closed universe; and the second in relation to an expanding universe.
The application of a single average value for ρ to a closed universe, defined by the 3-dimensional "surface area" of a 4-dimensional hyper-sphere, means that the contributions of matter lying within ever more distant volumes of space ‡ to the measured values of rb and g b , will ultimately diminish with distance. Consequently, inaccuracies in the value of H and, thereby, the Hubble Horizon should not be primary sources of error in rb and g b . Moreover, recent observations that indicate lower values for H at the furthest distances should not, for the same reason, undermine the validity of using a single value for H in the derivation of equations (1) or (7).
The nature of expansion of the universe (whether open or closed) that is assumed here, is one in which mass density is determined by a fixed amount of matter lying within the Hubble Horizon assumed to be receding at the speed of light. To a first order it is not affected by mass flows across either the Hubble Horizon, or across regions of space lying within the Hubble Horizon, nor by the inter-change between matter and energy. Accordingly, a profile of steadily increasing mass densities at further distances from a point P , due to these further distances being observations of the universe's past, should not affect the determination of rb and g b , to the extent that greater mass densities (in the past) are off-set by reductions in the volume of space (in the past).
If the same adjustment for space being closed as would be needed to bring to rb into equality with a r in equation (6) is also applied to the derivation of g b in equation (1), g b reduces by circa 25% to 6.0 × 10 -12 ms -2 . Having made the same correction for volume, the relationship between the background value for the radius of curvature of space-time r b and r u also remain unchanged (to one decimal place), namely r b = 0.98 × r u . Hence, the corrected calculation of the volume of space lying within the Hubble Horizon does not affect the calculated value for a r , which remains 1.2 × 10 -10 ms -2 (i.e. the same as a 0 ).
Hence, if account is taken of a closed and isotropic nature of space in applying the current value for the mass density of the universe ρ u , then the principle of conservation of energy appears to offer an alternative approach to the valuation of a r and, moreover, an approach that is based on key cosmological parameters and the gravitational constant G and that is independent of orbital dynamics and Newton's Second Law used in the earlier paper.
‡ i.e. the volume of concentric shells of space centred on point P and lying at distance s from P depart increasingly from 4πs 2 ds as s increases
Conservation of energy with 5 dimensions 7 5. Discussion A number of simplifying assumptions have been made in this paper. These include assumptions about the Hubble Horizon, the mass density of the universe and the calculation of volumes of space over large distances. Nonetheless, the value for a r derived from the principle of conservation of energy is in good agreement with that expected from MOND observations [2] and from the derivation based on the Hubble Constant [1] .
The relative dominance of proximate matter over very distant matter in the determination of the background universal gravitational acceleration g b and in the background value for the radius of curvature of space-time r b (assuming matter is evenly distributed on a very-large scale and the universe is closed), should make the calculations used in this paper reasonably robust to inaccuracies in the estimation of the Hubble Horizon and of volumes of space at greater distances.
The time dependencies of r b evident in equation (5) (i.e. increasing with age of the universe) and of |r b | evident in equation (7) (i.e. decreasing with age of the universe) imply that we should modify the central equation for a r proposed in the earlier paper and write it as:
a r = c 2 r b ( 9
)
For a value of r b = 0.98 × r u , the value of a r derived from equation (9) remains the same as a 0 (the MOND parameter) to one decimal place (i.e. 1.2 × 10 -10 ms -2 ) for the current era. The substitution of r b for r u in the equation for a r and the principle of conservation of energy (i.e. equation (6)) are consistent with higher values for ρ, |r b | and a r in earlier ages of the universe. The observations of rotation curves of galaxies which support the MOND parameter a 0 proposed by Milgrom have, so far, mostly covered galaxies out as far as ∼ 100 Mpc from earth. To one decimal place, there is no change to r b from equation (5) over these distances and so no corresponding departure from the MOND value for a 0 would be expected from equation (9) .
The analysis in sections 3 and 4 can, of course, be reversed and the principle of conservation of energy as expressed by equation (6) can be used as the starting point to derive the underlying relationship between matter and the radius of curvature of 4dimensional space-time in an expanding universe, namely equation (2) . If this approach is adopted, then the inverse square law of gravity (which is a derivative of equation (2) §) may be inferred as a consequence of the conservation of energy at the boundary between a (closed) expanding four-dimensional universe and a fifth large-scale dimension of spacetime. Accordingly, a description of gravity based upon this principle of conservation of energy would appear to offer a derivation based on thermodynamics for the key dimensionless term of General Relativity (GM/c 2 x). Furthermore, equations (7) and (9) may be substituted in equation (6) to provide an expression for the gravitational § For the relationship between equation (2) and the inverse square law of gravity, see section 3.2 [1]
Conservation of energy with 5 dimensions 8 constant (G), of the following form:
G = kc 2 r u M universe ( 10
)
where M universe is the mass of the universe and k is a dimensionless constant determined by the correct approach to calculating the volume of the universe. This equation suggests a link between G and the key fifth dimensional parameter r u , which is identified in this and the earlier paper as the radius of curvature of space-time remote from gravitating matter; albeit with the same limitations as equation (7) from which it is derived. All the terms on the right-hand side of equation (10) are, as expected, constant.
Finally, equation (10) can itself be restated in terms of the parameter a o rather than r u by substituting the expression a o = c 2 /r u :
c 4 = a o GM universe k -1 ( 11
)
which is of the form of the Tulley-Fisher relationship (see equation (25) in [1]). The equation suggests a limiting case for this relationship and, moreover, one which is derived from the principle of conservation of energy at a universal level and without reference to the orbital dynamics of individual galaxies or the universe as a whole. | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Conservation of energy with 5 dimensions 2 1. Introduction In an earlier paper [1] we introduced a fifth large-scale space-time dimension, r to Newton's Second Law, as applied to systems with angular velocity. The resulting analysis of the orbital motion of galaxies, which considered only the role of baryonic matter, is consistent with their observed rotation curves and the Tulley-Fisher relationship. The dimension r, is orthogonal to the three space dimensions s(x, y, z) and the time dimension, t of a four-dimensional universe, but does not represent a degree of freedom of motion in this analysis. For a closed isotropic universe, r is the radius of curvature of (four-dimensional) space-time and has a value, r u remote from gravitating matter that is estimated to be ∼ 7.5 × 10 26 m. The parameter a r is derived from the relationship a r = c 2 /r. In the case of r being equal to r u , a r has a value of 1.2 × 10 -10 ms -2 , which is the same as the MOND parameter a 0 derived by Milgrom [2] from observing the rotation curves of more than eighty galaxies.\n\nUsing the same five-dimensional analytical framework, this paper examines the relationships between a r , the principle of conservation of energy and gravity. The resulting derivation of a r is, therefore, unrelated to orbital dynamics and Newton's Second Law and instead relies on key cosmological constants, such as the gravitational constant, G and parameters, such as the mass density of the universe."
},
{
"section_type": "BACKGROUND",
"section_title": "Background Gravitational Acceleration in the Universe",
"text": "The large-scale distribution of matter across the universe creates a background gravitational acceleration, g b which is isotropic if matter itself is evenly distributed on this scale. The mutual attraction of each particle of matter towards all other matter, as represented by g b , is similar in concept to a three dimensional \"surface tension\" stretching across the universe.\n\nIf space is assumed to be flat and open and matter is assumed to be evenly distributed on this large scale, with (baryonic) mass density ρ, then the background gravitational acceleration, g b , can be derived as follows:\n\ng b = πGρH H ( 1\n\n)\n\nwhere G is the gravitational constant (6.67 × 10 -11 m 3 Kg -1 s -2 ), ρ for baryonic matter has a currently estimated value ρ u = 3 × 10 -28 Kg m -3 and H H is the Hubble Horizon given by H H = c/H with H being Hubble's Constant (71 Km s -1 Mpc -1 ). Substitution in equation (1) gives a current value for g b of 8.2 × 10 -12 ms -2 which is noted to be two orders of magnitude less than the value of a 0 . The accuracy of equation (1) depends on three potential sources of uncertainty, namely: the value of ρ, the method of calculation of the volume of the universe within the Hubble Horizon and the value of H itself. These will be discussed later.\n\nConservation of energy with 5 dimensions 3 ~M σ r u g x a r r ~x Figure 1. Locus of points r(x) at which there is balance between the two accelerations g x and a r ."
},
{
"section_type": "BACKGROUND",
"section_title": "Background Radius of Curvature of the Universe",
"text": "In section 3.2 of the earlier paper [1] an expression was derived for the locus of points r(x) adjacent to a gravitating mass, M which defined the balance condition between gravitational acceleration g x and the acceleration a r acting everywhere in the universe in the direction of r.\n\nr (x) = r u 1 - GM c 2 x ( 2\n\n)\n\nwhere r u is the radius of curvature of four-dimensional space-time remote from gravitating matter M and x is the distance away from M as shown in figure 1 .\n\nThe effect which matter has on the local radius of curvature of space-time, r is cumulative and can be found by the superposition (∆r/r u = Σ ∆r i /r u , where ∆r i = (r u -r i )) from all individual masses, M i . Applying equation (2) to all baryonic matter contained within the Hubble Horizon (again assumed to be evenly distributed across space with density ρ and lying within a spherical volume defined by 4/3 (πs 3 ) where s here is H H ) it is possible to calculate an overall background value of r(x). This value will inevitably be somewhat less than r u given that no point is, in practice, completely remote from all matter. This background value of r is referred to as r b and is derived by integrating the contributions from matter lying within concentric spherical shells of space to give:\n\nr b = r u 1 - 2πGρH 2 H c 2 ( 3\n\n)\n\nSubstituting for known parameters and constants in equation (3), including the current value of the mass density ρ u , gives a value for r b equal to 0.98 × r u . Substituting either value for r into the key relationship a r = c 2 /r gives the same value for a r to within one decimal place, namely 1.2 × 10 -10 ms -2 .\n\nThe average mass density of the universe, ρ, decreases over time in an expanding universe. For a Euclidean (although expanding) universe, the volume of space within\n\nConservation of energy with 5 dimensions 4 the Hubble Horizon is given by (4/3) πH 3 H ≃ (4/3) π(ct) 3 . Given that (to a first order) the total mass lying within the Hubble Horizon is constant, it follows that we can derive an expression for the average mass density ρ(t) of the universe at any time t, in terms of the average mass density observed for the current era ρ u (i.e. ∼ 3 × 10 -28 Kg m -3 ) and the current estimated age of the universe t u (i.e. 13.7 Bn years).\n\nρ ≃ ρ u t 3 u t 3 ( 4\n\n)\n\nGiven that this equation is derived (in part) from the approximation H H ≃ ct, it is assumed only to be applicable in the current analysis for perturbations of time about the current era. Substituting for ρ from equation (4) into equation (3) provides an expression for the local time-dependency of the background radius of curvature of space-time r b in equation (5) , which is similarly limited in its range of extrapolation.\n\nr b = r u 1 - 2πGρ u t 3 u t ( 5\n\n)"
},
{
"section_type": "OTHER",
"section_title": "Conservation of Energy",
"text": "In section 3.1 in the earlier paper [1] a r was described as a universal acceleration of expansion acting at all points in space in the direction of r. To maintain conservation of energy within four-dimensional space-time, it follows that for any mass m at a point in space P there must be an acceleration equal and opposite to a r which prevents energy being transferred from within the four-dimensional universe to the fifth dimension r, as shown in figure 2. Accordingly, this principle may be written as:\n\na r + d 2 r b dt 2 = 0 ( 6\n\n)\n\nThe second term of this equation (r b ) is identified as the acceleration acting on a mass in the direction of the dimension r (decreasing) by virtue of the expansion of the universe in the dimension r which causes r b the background value of r to increase over time (but at a decelerating rate -see equation (5)). In other words, given that the universal acceleration a r is acting everywhere along the boundary between the fourdimensional space-time and the fifth dimension r, energy can only be conserved (within four dimensional space-time) if the background radius of curvature of space-time r b varies in time so as to satisfy equation (6) . This conservation of energy at the boundary between the four dimensional universe and the fifth dimension r is, of course, the reason why the dimension r is not itself directly observable. As referred above, for the current era a r is 1.2 × 10 -10 ms -2 .\n\nAssuming only r b varies with time equation (5) gives:\n\nd 2 r b dt 2 = -4πρGr u ( 7\n\n)\n\nSubstituting values for known parameters and constants on the right-hand side (including the current mass density of the universe, ρ u , provides the result: rb = Conservation of energy with 5 dimensions\n\n5 a r d 2 r dt 2 s(x,y,z) ~P m r\n\nFigure 2. Conservation of energy requires the two accelerations a r and rb to be equal and opposite.\n\n-1.9 × 10 -10 ms -2 . Given the approximations used to derive equation (7) , this value for rb appears to be in reasonably good agreement with the value expected from equation (6), namely: -1.2 × 10 -10 ms -2 ). The substitution for r u in equation (7) using the relationship a r = c 2 /r (section (1) above), but with the identification of a r = a o for r = r u for the current era, and the combining of equations (6) and (7) allows an expression for a o as:\n\na o = 4πρ u Gc 2 1/2 ( 8\n\n)\n\nwhich has the value of 1.5 × 10 -10 ms -2 for current era. The level of agreement between a r and rb , calculated from equation (7) , can only be properly assessed by considering the uncertainty in the three key components to equation (7): the value of the Hubble Horizon, the average mass density of the universe and the estimated volume of the universe. Consistency between rb from equation (7) and equation (6) ) lies within the uncertainty ranges of ±12% in each of these three components. However, the principal source of uncertainty in rb is expected to be the method used to calculate the volume of the universe lying within the Hubble Horizon.\n\nThe form of universe that underpins the derivation of a r is closed (i.e. curved) and isotropic Section 3.2 in [1] and yet, so far in this paper, we have used the Euclidean derivation of a three dimension spherical volume 4/3 (πs 3 ), where s is the radius of the volume -i.e. a derivation appropriate to a flat and open universe. A closed isotropic three dimensional space is the \"surface area\" of a 4-dimensional hyper-sphere, the 3dimensional volume of which is given not by 4/3 (πs 3 ) but by the expression 2π 2 R 3 , where R is the radius of curvature of the hyper-sphere. The relevant feature of this 3dimensional \"surface area\" is that at increasing distances s from a point P, the volumes of concentric spherical shells of space centred on P become progressively smaller than those derived from the (Euclidean) expression 4πs 2 ds.\n\nAccordingly, failure to take account of this effect will have led to an over-estimation of the volume of the universe lying within the Hubble Horizon and so to an overestimation of rb in equation (7) . The value of g b in equation (1) will have, likewise, been overestimated for this reason.\n\nConservation of energy with 5 dimensions 6 There are two important aspects of the application of ρ in the calculation of rb and g b that also need to be highlighted: the first in relation to a closed universe; and the second in relation to an expanding universe."
},
{
"section_type": "OTHER",
"section_title": "A closed universe",
"text": "The application of a single average value for ρ to a closed universe, defined by the 3-dimensional \"surface area\" of a 4-dimensional hyper-sphere, means that the contributions of matter lying within ever more distant volumes of space ‡ to the measured values of rb and g b , will ultimately diminish with distance. Consequently, inaccuracies in the value of H and, thereby, the Hubble Horizon should not be primary sources of error in rb and g b . Moreover, recent observations that indicate lower values for H at the furthest distances should not, for the same reason, undermine the validity of using a single value for H in the derivation of equations (1) or (7)."
},
{
"section_type": "OTHER",
"section_title": "An expanding universe",
"text": "The nature of expansion of the universe (whether open or closed) that is assumed here, is one in which mass density is determined by a fixed amount of matter lying within the Hubble Horizon assumed to be receding at the speed of light. To a first order it is not affected by mass flows across either the Hubble Horizon, or across regions of space lying within the Hubble Horizon, nor by the inter-change between matter and energy. Accordingly, a profile of steadily increasing mass densities at further distances from a point P , due to these further distances being observations of the universe's past, should not affect the determination of rb and g b , to the extent that greater mass densities (in the past) are off-set by reductions in the volume of space (in the past).\n\nIf the same adjustment for space being closed as would be needed to bring to rb into equality with a r in equation (6) is also applied to the derivation of g b in equation (1), g b reduces by circa 25% to 6.0 × 10 -12 ms -2 . Having made the same correction for volume, the relationship between the background value for the radius of curvature of space-time r b and r u also remain unchanged (to one decimal place), namely r b = 0.98 × r u . Hence, the corrected calculation of the volume of space lying within the Hubble Horizon does not affect the calculated value for a r , which remains 1.2 × 10 -10 ms -2 (i.e. the same as a 0 ).\n\nHence, if account is taken of a closed and isotropic nature of space in applying the current value for the mass density of the universe ρ u , then the principle of conservation of energy appears to offer an alternative approach to the valuation of a r and, moreover, an approach that is based on key cosmological parameters and the gravitational constant G and that is independent of orbital dynamics and Newton's Second Law used in the earlier paper.\n\n‡ i.e. the volume of concentric shells of space centred on point P and lying at distance s from P depart increasingly from 4πs 2 ds as s increases\n\nConservation of energy with 5 dimensions 7 5. Discussion A number of simplifying assumptions have been made in this paper. These include assumptions about the Hubble Horizon, the mass density of the universe and the calculation of volumes of space over large distances. Nonetheless, the value for a r derived from the principle of conservation of energy is in good agreement with that expected from MOND observations [2] and from the derivation based on the Hubble Constant [1] .\n\nThe relative dominance of proximate matter over very distant matter in the determination of the background universal gravitational acceleration g b and in the background value for the radius of curvature of space-time r b (assuming matter is evenly distributed on a very-large scale and the universe is closed), should make the calculations used in this paper reasonably robust to inaccuracies in the estimation of the Hubble Horizon and of volumes of space at greater distances.\n\nThe time dependencies of r b evident in equation (5) (i.e. increasing with age of the universe) and of |r b | evident in equation (7) (i.e. decreasing with age of the universe) imply that we should modify the central equation for a r proposed in the earlier paper and write it as:\n\na r = c 2 r b ( 9\n\n)\n\nFor a value of r b = 0.98 × r u , the value of a r derived from equation (9) remains the same as a 0 (the MOND parameter) to one decimal place (i.e. 1.2 × 10 -10 ms -2 ) for the current era. The substitution of r b for r u in the equation for a r and the principle of conservation of energy (i.e. equation (6)) are consistent with higher values for ρ, |r b | and a r in earlier ages of the universe. The observations of rotation curves of galaxies which support the MOND parameter a 0 proposed by Milgrom have, so far, mostly covered galaxies out as far as ∼ 100 Mpc from earth. To one decimal place, there is no change to r b from equation (5) over these distances and so no corresponding departure from the MOND value for a 0 would be expected from equation (9) .\n\nThe analysis in sections 3 and 4 can, of course, be reversed and the principle of conservation of energy as expressed by equation (6) can be used as the starting point to derive the underlying relationship between matter and the radius of curvature of 4dimensional space-time in an expanding universe, namely equation (2) . If this approach is adopted, then the inverse square law of gravity (which is a derivative of equation (2) §) may be inferred as a consequence of the conservation of energy at the boundary between a (closed) expanding four-dimensional universe and a fifth large-scale dimension of spacetime. Accordingly, a description of gravity based upon this principle of conservation of energy would appear to offer a derivation based on thermodynamics for the key dimensionless term of General Relativity (GM/c 2 x). Furthermore, equations (7) and (9) may be substituted in equation (6) to provide an expression for the gravitational § For the relationship between equation (2) and the inverse square law of gravity, see section 3.2 [1]\n\nConservation of energy with 5 dimensions 8 constant (G), of the following form:\n\nG = kc 2 r u M universe ( 10\n\n)\n\nwhere M universe is the mass of the universe and k is a dimensionless constant determined by the correct approach to calculating the volume of the universe. This equation suggests a link between G and the key fifth dimensional parameter r u , which is identified in this and the earlier paper as the radius of curvature of space-time remote from gravitating matter; albeit with the same limitations as equation (7) from which it is derived. All the terms on the right-hand side of equation (10) are, as expected, constant.\n\nFinally, equation (10) can itself be restated in terms of the parameter a o rather than r u by substituting the expression a o = c 2 /r u :\n\nc 4 = a o GM universe k -1 ( 11\n\n)\n\nwhich is of the form of the Tulley-Fisher relationship (see equation (25) in [1]). The equation suggests a limiting case for this relationship and, moreover, one which is derived from the principle of conservation of energy at a universal level and without reference to the orbital dynamics of individual galaxies or the universe as a whole."
}
] | ||
arxiv:0704.0340 | 0704.0340 | 1 | cd1ccc64371eab104adafb8415e5f5fbdd051067c946e8a0113e4f759380cc2f | Phonon-mediated decay of an atom in a surface-induced potential | We study phonon-mediated transitions between translational levels of an atom
in a surface-induced potential. We present a general master equation governing
the dynamics of the translational states of the atom. In the framework of the
Debye model, we derive compact expressions for the rates for both upward and
downward transitions. Numerical calculations for the transition rates are
performed for a deep silica-induced potential allowing for a large number of
bound levels as well as free states of a cesium atom. The total absorption rate
is shown to be determined mainly by the bound-to-bound transitions for deep
bound levels and by bound-to-free transitions for shallow bound levels.
Moreover, the phonon emission and absorption processes can be orders of
magnitude larger for deep bound levels as compared to the shallow bound ones.
We also study various types of transitions from free states. We show that, for
thermal atomic cesium with temperature in the range from 100 $\mu$K to 400
$\mu$K in the vicinity of a silica surface with temperature of 300 K, the
adsorption (free-to-bound decay) rate is about two times larger than the
heating (free-to-free upward decay) rate, while the cooling (free-to-free
downward decay) rate is negligible. | [
"Fam Le Kien",
"S. Dutta Gupta",
"K. Hakuta"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Over the past few years, tight confinement of cold atoms has drawn considerable attention. The interest in this area is motivated not only by the fundamental nature of the problem, but also by its potential applications in atom optics and quantum information. A method for microscopic trapping and guiding of individual atoms along a nanofiber has been proposed [1] . Surface-atom quantum electrodynamic effects have constituted another interesting area, where a great deal of work has been carried out. Modification of spontaneous emission of an atom [2] and radiative exchange between two distant atoms [3] mediated by a nanofiber have been investigated. Surface-induced deep potentials have played a major role and have received due attention in recent years. Oria et al. have studied various theoretical schemes to load atoms into such potentials [4, 5] . A rigorous theory of spontaneous decay of an atom in a surface-induced potential invoking the density-matrix formalism has been developed [6] . The role of interference between the emitted and reflected fields and also the role of transmission into the evanescent modes were identified. Further calculations on the excitation spectrum have been carried out [7] . Bound-to-bound transitions were shown to lead to significant effects like a large red tail of the excitation spectrum as compared to the weak consequences of free-to-bound transitions. A crucial step in this direction was the experimental observation of the excitation spectrum and the channeling of the fluorescent photons along the nanofiber [8] , opening up avenues for novel quantum information devices.
In most of the problems involving surface-atom interaction, the macroscopic surface is usually kept at room temperature. Thus the pertinent question that can be asked is what would be the effect of heating on the cold atoms. It is understood that transfer of heat to the trapped atoms will lead to a change in the occupation probability of the vibrational levels as well as their coherence. Phonon-induced changes in the populations of the vibrational levels have been studied by several groups [5, 9, 10] . In a nice and compact treatment based on the dyadic Green function and the Fermi golden rule, Henkel et al. showed that the effects can be very different depending on the nature of the atomic/molecular species [9] . The time scales for various species were estimated. It should be stressed that the trap considered by Henkel et al. was not necessarily a surface trap and misses out on many of the aspects of the surface-atom interaction [9] . Based on the assumption that the surface-atom interaction can be represented by a Morse potential, the phonon-mediated decay was estimated by Oria et al. [5] . Their estimate was based on the formalism developed by Gortel et al. [10] . However, all the previous theories focus on only the transition rates and thus are not general enough. In this paper, we present a general densitymatrix formalism to calculate the phonon-mediated decay of populations as well as the changes in coherence. We derive the relevant master equation for the density matrix of the atom. We emphasize that our densitymatrix equation describes the full dynamics of the coupling between trapped atoms and phonons and does not assume any particular form of the trapping potential. Under the Debye approximation, we derive compact expressions for the phonon-mediated decay rates. Numerical calculations are carried out assuming the potential model considered in [4] . In contrast to the previous work, we include a large number of vibrational levels due to the deep surface-atom potential. We show that there can be significant differences in the decay rates when the initial level is chosen as one of the shallow or deep bound levels. We also calculate and analyze the decay rates for various 2 types of transitions from free states. The paper is organized as follows. In Sec. II we describe the model. In Sec. III we derive the basic dynamical equations for the phonon-mediated decay processes. In Sec. IV we present the results of numerical calculations. Our conclusions are given in Sec. V.
We assume the whole space to be divided into two regions, namely, the half-space x < 0, occupied by a nondispersive nonabsorbing dielectric medium (medium 1), and the half-space x > 0, occupied by vacuum (medium 2). We examine a single atom moving in the empty halfspace x > 0. We assume that the atom is in a fixed internal state |i with energy hω i . Without loss of generality, we assume that the energy of the internal state |i is zero, i.e. ω i = 0. We describe the interaction between the atom and the surface. We first consider the surface-induced interaction potential and then add the atom-phonon interaction.
In this subsection, we describe the interaction between the atom and the surface in the case where thermal vibrations of the surface are absent. The potential energy of the surface-atom interaction is a combination of a long-range van der Waals attraction and a short-range repulsion [11] . Despite a large volume of research on the surface-atom interaction, due to the complexity of surface physics and the lack of data, the actual form of the potential is yet to be ascertained [11] . For the purpose of numerical demonstration of our formalism, we choose the following model for the potential [4, 11] :
U (x) = Ae -αx - C 3 x 3 . ( 1
)
Here, C 3 is the van der Waals coefficient, while A and α determine the height and range, respectively, of the surface repulsion. The potential parameters C 3 , A, and α depend on the nature of the dielectric and the atom.
In numerical calculations, we use the parameters of fused silica, for the dielectric, and the parameters of groundstate atomic cesium, for the atom. The parameters for the interaction between silica and ground-state atomic cesium are theoretically estimated to be C 3 = 1.56 kHz µm 3 , A = 1.6 × 10 18 Hz, and α = 53 nm -1 [6]. We introduce the notation ϕ ν (x) for the eigenfunctions of the center-of-mass motion of the atom in the potential U (x). They are determined by the stationary Schrödinger equation -h2 2m
d 2 dx 2 + U (x) ϕ ν (x) = E ν ϕ ν (x). ( 2
)
Here m is the mass of the atom. In the numerical example with atomic cesium, we have m = 132.9 a.u. = 2.21 × 10 -25 kg. The eigenvalues E ν are the centerof-mass energies of the translational levels of the atom. These eigenvalues are the shifts of the energies of the translational levels from the energy of the internal state |i . Without loss of generality, we assume that the center-of-mass eigenfunctions ϕ ν (x) are real functions, i.e. ϕ * ν (x) = ϕ ν (x). In Fig. 1 , we show the potential U (x) and the wave functions ϕ ν (x) of a number of bound levels with energies in the range from -1 GHz to -5 MHz. We also plot the wave function of a free state with energy of about 4.25 MHz. In order to have some estimate about the spatial extent of a wave function ϕ ν (x), we define a crossing point x cross , which corresponds to the rightmost solution of the equation U (x) = E ν . Note that, for shallow levels, the wave function generally peaks close to the point x cross . We plot the eigenvalue modulus |E ν | and the crossing point x cross in Figs. 2(a) and 2(b), respectively. It is clear from the figure that, for ν in the range from 0 to 300, the eigenvalue varies dramatically from about 158 THz to about 322 kHz, while the wave function extends only up to 170 nm.
FIG. 1: Energies and wave functions of the center-of-mass motion of an atom in a surface-induced potential. The parameters of the potential are C3 = 1.56 kHz µm 3 , A = 1.6 × 10 18 Hz, and α = 53 nm -1 . The mass of the atom is m = 2.21 × 10 -25 kg. We plot bound levels with energies in the range from -1 GHz to -5 MHz and also a free state with energy of about 4.25 MHz. FIG. 2: Eigenvalue modulus |Eν| (a) and crossing point xcross (b) as functions of the vibrational quantum number ν. The parameters used are as in Fig. 1.
We introduce the notation |ν = |ϕ ν and ω ν = E ν /h for the state vectors and frequencies of translational levels. Then, the Hamiltonian of the atom in the surfaceinduced potential can be represented in the diagonal form
H A = ν hω ν σ νν . ( 3
)
Here, σ νν = |ν ν| is the population operator for the translational level ν. We emphasize that the summation over ν includes both the discrete (E ν < 0) and continuous (E ν > 0) spectra. The levels ν with E ν < 0 are called the bound (or vibrational) levels. In such a state, the atom is bound to the surface. It is vibrating, or more exactly, moving back and forth between the walls formed by the van der Waals part and the repulsive part of the potential. The levels ν with E ν > 0 are called the free (or continuum) levels. The center-of-mass wave functions of 3 the bound states are normalized to unity. The center-ofmass wave functions of the free states are normalized to the delta function of energy.
In this subsection, we incorporate the thermal vibrations of the solid into the model. Due to the thermal effects, the surface of the dielectric vibrates. The surfaceinduced potential for the atom is then U (xx s ), where x s is the displacement of the surface from the mean position x s = 0. We approximate the vibrating potential U (xx s ) by expanding it to the first order in x s ,
U (x -x s ) = U (x) -U ′ (x)x s . ( 4
)
The first term, U (x), when combined with the kinetic energy p 2 /2m, yields the Hamiltonian H A [see Eq. ( 3 )], which leads to the formation of translational levels of the atom. The second term, -U ′ (x)x s , accounts for the thermal effects in the interaction of the atom with the solid. Note that the quantity F = -U ′ (x) is the force of the surface upon the atom. Hence, the force of the atom upon the surface is -F = U ′ (x) and, consequently, U ′ (x)x s is the work required to displace the surface for a small distance x s . It is well known that, for a smooth surface, the gas atom interacts only with the phonons polarized along the x direction [10] . In the harmonic approximation, we have
x s = q h 2M N ω q 1/2 (b q e iqR + b † q e -iqR ). ( 5
)
Here, M is the mass of a particle of the solid, N is the particle number density, ω q and q are the frequency and wave vector of the x-polarized acoustic phonons, respectively, R = (0, y, z) is the lateral component of the position vector (x, y, z) of the atom, and b q and b † q are the annihilation and creation phonon operators, respectively. Without loss of generality, we choose R = 0. Meanwhile, the operator U ′ can be decomposed as
U ′ = νν ′ σ νν ′ ν|U ′ |ν ′ , where σ νν ′ = |ν ν ′ | is
the operator for the translational transition ν ↔ ν ′ . Hence, the energy term -U ′ (x)x s leads to the atom-phonon interaction Hamiltonian [10]
H I = h q 1 √ ω q S(b q + b † q ), ( 6
) with S = νν ′ g νν ′ σ νν ′ . ( 7
)
Here we have introduced the atom-phonon coupling coefficients
g νν ′ = F νν ′ √ 2M N h , ( 8
)
with
F νν ′ = - ∞ -∞ ϕ ν (x)U ′ (x)ϕ ν ′ (x)dx ( 9
)
being the matrix elements for the force of the surface upon the atom. We note that F νν ′ = -mω 2 νν ′ x νν ′ , where x νν ′ = ν|x|ν ′ and ω νν ′ = ω νω ν ′ are the surface-atom dipole matrix element and the translational transition frequency, respectively. Hence, the coupling coefficient g νν ′ depends on the dipole matrix element x νν ′ and the transition frequency ω νν ′ . Since ω νν = 0, we have g νν = 0.
We note that the Hamiltonian of the x-polarized acoustic phonons is given by
H B = q hω q b † q b q . ( 10
)
The total Hamiltonian of the atom-phonon system is
H = H A + H I + H B . ( 11
)
We use the above Hamiltonian to study the phononmediated decay of the atom.
In this section, we present the basic equations for the phonon-mediated decay processes. We derive a general master equation for the reduced density operator of the atom in subsection III A, obtain analytical expressions for the relaxation rates and frequency shifts in subsection III B, and calculate the rates and the shifts in the framework of the Debye model in subsection III C.
In the Heisenberg picture, the equation for the phonon operator b q (t) is
ḃq (t) = -iω q b q (t) - i √ ω q S(t), ( 12
)
which has a solution of the form
b q (t) = b q (t 0 )e -iωq(t-t0) -iW q (t). ( 13
)
Here, t 0 is the initial time and W q is given by
W q (t) = 1 √ ω q t t0 e -iωq(t-τ ) S(τ ) dτ. ( 14
)
Consider an arbitrary atomic operator O which acts only on the atomic states but not on the phonon states. The time evolution of this operator is governed by the Heisenberg equation
∂O(t) ∂t = i h [H A (t) + H I (t), O(t)], ( 15
)
4 which, with account of Eqs. (6) and (13), yields ∂O(t) ∂t = i h [H A (t), O(t)] + q i √ ω q [S(t), O(t)][b q (t 0 )e -iωq(t-t0) -iW q (t)]
- q i √ ω q [b † q (t 0 )e iωq(t-t0) + iW † q (t)][O(t), S(t)]. ( 16
)
We assume the initial density of the atom-phonon system to be the direct product state
ρ Σ (t 0 ) = ρ(t 0 )ρ B (t 0 ), ( 17
)
with the atom in an arbitrary state ρ(t 0 ) and the phonons in a thermal state
ρ B (t 0 ) = Z -1 exp[-H B (t 0 )/k B T ]. ( 18
)
Here, Z is the normalization constant and T is the temperature of the phonon bath. For the initial condition (17), the Bogolubov's lemma [12] , applied to an arbitrary operator Θ(t), asserts the following:
Θ(t)b q (t 0 ) = nq [b q (t 0 ), Θ(t)] , ( 19
)
where the mean number of phonons in the mode q is given by
nq = 1 exp(hω q /k B T ) -1 . ( 20
)
Let Θ be an atomic operator. We then have the commutation relation [b q (t), Θ(t)] = 0, which yields [b q (t 0 ), Θ(t)] = ie iωq(t-t0) [W q (t), Θ(t)]. (21) Combining Eq. (19) with Eq. (21) leads to
Θ(t)b q (t 0 ) = ie iωq(t-t0) nq [W q (t), Θ(t)] . ( 22
)
We perform the quantum mechanical averaging for expression (16) and use Eq. ( 22 ) to eliminate the phonon operators b q (t 0 ) and b † q (t 0 ). The resulting equation can be written as
∂ O(t) ∂t = i h [H A (t), O(t)] + q nq + 1 √ ω q [S(t), O(t)]W q (t) + W † q (t)[O(t), S(t)] + q nq √ ω q W q (t)[O(t), S(t)] + [S(t), O(t)]W † q (t) . ( 23
)
We note that Eq. (23) is exact. It does not contain phonon operators explicitly. The dependence on the phonon operators is hidden in the time shift of the operator S(τ ) in expression (14) for the operator W q (t).
We now show how the dependence of the operator W q (t) on the phonon operators can be approximately eliminated. We assume that the atom-phonon coupling coefficients g νν ′ are small. The use of the zeroth-order approximation σ νν ′ (τ ) = σ νν ′ (t)e iω νν ′ (τ -t) in the expression for S(τ ) [see Eq. ( 7 )] yields
S(τ ) = νν ′ g νν ′ σ νν ′ (t)e iω νν ′ (τ -t) , ( 24
)
which is accurate to first order in the coupling coefficients. Inserting Eq. (24) into Eq. ( 14 ) gives
W q (t) = 2π √ ω q νν ′ g νν ′ σ νν ′ (t)δ -(ω ν ′ ν -ω q ), ( 25
)
where
δ -(ω) = lim ǫ→0 1 2π 0 -∞ e -i(ω+iǫ)τ dτ = i 2π P ω + 1 2 δ(ω). ( 26
)
Here, in order to take into account the effect of adiabatic turn-on of interaction, we have added a small positive parameter ǫ to the integral and have used the limit t 0 → -∞. Introducing the notation
K q = W q √ ω q = 2π ω q νν ′ g νν ′ σ νν ′ δ -(ω ν ′ ν -ω q ), ( 27
)
we can rewrite Eq. ( 23 ) in the form
∂ O(t) ∂t = i h [H A (t), O(t)] + q (n q + 1) [S(t), O(t)]K q (t) + K † q (t)[O(t), S(t)] + q nq K q (t)[O(t), S(t)] + [S(t), O(t)]K † q (t) . ( 28
)
In order to examine the time evolution of the reduced density operator ρ(t) of the atom in the Schrödinger picture, we use the relation O(t) = Tr[O(t)ρ(0)] = Tr[O(0)ρ(t)], transform to arrange the operator O(0) at the first position in each operator product, and eliminate O(0). Then, we obtain the Liouville master equation
∂ρ(t) ∂t = - i h [H A , ρ(t)] + q (n q + 1){[K q ρ(t), S] + [S, ρ(t)K † q ]} + q nq {[S, ρ(t)K q ] + [K † q ρ(t), S]}. ( 29
)
Equations (28) and (29) are valid to second order in the coupling coefficients. These equations allow us to study the time evolution and dynamical characteristics of the atom interacting with the thermal phonon bath. We note that Eq. (29) is a particular form of the Zwanzig's generalized master equation, which can be obtained by the projection operator method [13]. 5 B. Relaxation rates and frequency shifts We use Eq. (29) to derive an equation for the matrix elements ρ jj ′ ≡ j|ρ|j ′ of the reduced density operator of the atom. The result is
∂ρ jj ′ ∂t = -iω jj ′ ρ jj ′ + νν ′ (γ e jj ′ νν ′ + γ a jj ′ νν ′ )ρ νν ′ - ν [(γ e jν + γ a jν )ρ νj ′ + (γ e * j ′ ν + γ a * j ′ ν )ρ jν ], ( 30
)
where the coefficients
γ e jj ′ νν ′ = 2π q nq + 1 ω q g jν g j ′ ν ′ [δ -(ω νj -ω q ) + δ + (ω ν ′ j ′ -ω q )], γ e jν = 2π qµ nq + 1 ω q g jµ g νµ δ -(ω νµ -ω q ) ( 31
) and γ a jj ′ νν ′ = 2π q nq ω q g jν g j ′ ν ′ [δ -(ω j ′ ν ′ -ω q ) + δ + (ω jν -ω q )], γ a jν = 2π qµ nq ω q g jµ g νµ δ + (ω µν -ω q ) ( 32
)
are the decay parameters associated with the phonon emission and absorption, respectively. Here, the notation δ + (ω) = δ * -(ω) has been used. Equation (30) describes phonon-induced variations in the populations and coherences of the translational levels of the atom. We analyze the characteristics of the relaxation processes. For simplicity of mathematical treatment, we first consider only transitions from discrete levels. The equation for the diagonal matrix element ρ jj for a discrete level j can be written in the form
∂ρ jj ∂t = ν (γ e jjνν + γ a jjνν )ρ νν -(γ e jj + γ a jj + c.c.)ρ jj + off-diagonal terms. ( 33
)
When the off-diagonal terms are neglected, Eq. (33) reduces to a simple rate equation. It is clear from Eq. ( 33 ) that the rate for the downward transition from an upper level l to a lower level k (k < l) is
R e kl = γ e kkll = 2π q nq + 1 ω q g 2 lk δ(ω lk -ω q ), ( 34
)
while the rate for the upward transition from a lower level k to an upper level l (l > k) is
R a lk = γ a llkk = 2π q nq ω q g 2 lk δ(ω lk -ω q ). ( 35
)
Equations (34) and (35) are in agreement with the results of Gortel et al. [10], obtained by using the Fermi golden rule. We note that R e kl and R a lk with l ≤ k are mathematically equal to zero because they have no physical meaning. For convenience, we introduce the notation R lk = R e lk , R a lk , or 0 for l < k, l > k, or l = k, respectively. It is clear that the off-diagonal coefficients R lk with l = k are the rates of transitions. However, the diagonal coefficients R kk have no physical meaning and are mathematically equal to zero.
As seen from Eq. ( 33 ), the phonon-mediated depletion rate of a level k is Γ kk = 2Re(γ e kk + γ a kk ). The explicit expression for this rate is
Γ kk = 2π qµ nq + 1 ω q g 2 kµ δ(ω kµ -ω q ) + 2π qµ nq ω q g 2 µk δ(ω µk -ω q ). ( 36
)
We note that Γ kk = µ (R e µk + R a µk ) = µ R µk . We can write Γ kk = Γ e kk + Γ a kk , where
Γ e kk = µ<k R e µk ( 37
) and Γ a kk = µ>k R a µk ( 38
)
are the contributions due to downward transitions (phonon emission) and upward transitions (phonon absorption), respectively. In the above equations, the summation over µ can be extended to cover not only the discrete levels but also the continuum levels. Meanwhile, the equation for the off-diagonal matrix element ρ lk for a pair of discrete levels l and k can be written in the form
∂ρ lk /∂t = -(iω lk + γ e ll + γ a ll + γ e * kk + γ a * kk )ρ lk + . . . , or, equivalently, ∂ρ lk ∂t = -i(ω lk + ∆ lk -iΓ lk )ρ lk + . . . . ( 39
)
Here the frequency shift ∆ lk is given by
∆ lk = qµ nq + 1 ω q g 2 lµ ω lµ -ω q + g 2 µk ω µk + ω q + qµ nq ω q g 2 lµ ω lµ + ω q + g 2 µk ω µk -ω q , ( 40
)
while the coherence decay rate Γ lk is expressed as
Γ lk = π qµ nq + 1 ω q g 2 lµ δ(ω lµ -ω q ) + g 2 kµ δ(ω kµ -ω q ) + π qµ nq ω q g 2 µl δ(ω µl -ω q ) + g 2 µk δ(ω µk -ω q ) . ( 41
)
6 When we set l = k in Eq. (40), we find ∆ kk = 0. When we set l = k in Eq. (41), we recover Eq. (36). We note that Γ lk = µ (R e µl + R e µk + R a µl + R a µk )/2 = µ (R µl + R µk )/2. Comparison between Eqs. (41) and (36) yields the relation Γ lk = (Γ ll + Γ kk )/2. We can also write Γ lk = Γ e lk + Γ a lk , where Γ e lk = µ (R e µl + R e µk )/2 and Γ a lk = µ (R a µl + R a µk )/2 are the contributions due to downward transitions (phonon emission) and upward transitions (phonon absorption), respectively. In the above equations, the summation over µ can be extended to cover not only the discrete levels but also the continuum levels.
We now discuss phonon-mediated transitions from continuum (free) levels. We start by considering free-tobound transitions. For a continuum level f with energy E f > 0, the center-of-mass wave function ϕ f (x) is normalized per unit energy. In this case, the quantity R νf becomes the density of the transition rate. A free level f can be approximated by a level of a quasicontinuum [14] . A discretization of the continuum can be realized by using a large box of length L with reflecting boundary conditions [15] . We label E n the energies of the eigenstates in the box and φ n (x) the corresponding wave functions. Note that such states are standing-wave states [14, 15] . The relation between a quasicontinuum-state wave function φ n f (x), normalized to unity in the box, and the corresponding continuum-state wave function ϕ f (x), normalized per unit energy, with equal energies
E n f = E f , is [15] ϕ f (x) ∼ = ∂E n f ∂n f -1/2 φ n f (x) ∼ = L πh 1/2 m 2E n f 1/4 φ n f (x). ( 42
)
Consequently, for a single atom initially prepared in the quasicontinuum standing-wave state |n f = |φ n f , the rate for the transition to an arbitrary bound state |ν is approximately given by
G νf = πh L v f R νf , ( 43
)
where v f = (2E f /m) 1/2 is the velocity of the atom in the initial standing-wave state |f . The phonon-mediated free-to-bound decay rate (adsorption rate) is then given by
G f = ν G νf , ( 44
)
where the summation includes only bound levels. It is clear from Eq. ( 43 ) that, in the continuum limit L → ∞, the rate G νf tends to zero. This is because a free atom can be anywhere in free space and therefore the effect of phonons on a single free atom is negligible. In order to get deeper insight into the free-to-bound transition rate density R νf , we consider a macroscopic atomic ensemble in the thermodynamic limit [14] . Suppose that there are N 0 atoms in a volume with a large length L and a transverse cross section area S 0 . Assume all the atoms are in the same quasicontinuum state |n f and interact with the dielectric independently. The rate for the transitions of the atoms from the quasicontinuum state |n f to an arbitrary bound state |ν , defined as the time derivative of the number of atoms in the state |ν , is D νf = N 0 G νf . In order to get the rate for the continuum state |f , we need to take the thermodynamical limit, where L → ∞ and N 0 → ∞ but N 0 /L remains constant. Then, the rate for the transitions of the atoms from the continuum state |f to an arbitrary bound state |ν is given by
D νf = πhρ 0 S 0 v f R νf = 2πhN f R νf .
Here, ρ 0 = N 0 /LS 0 is the atomic number density and
N f = ρ 0 S 0 v f /2
is the number of atoms incident into the dielectric surface per unit time. It is clear that the transition rate D νf is proportional to the incidence rate N f as well as the transition rate density R νf . We emphasize that D νf is a characteristics for a macroscopic atomic ensemble in the thermodynamic limit while G νf is a measure for a single atom. When the length of the box, L, and the number of atoms, N 0 , are finite, the dynamics of the atoms cannot be described by the free-to-bound rate D νf directly. Instead, we must use the transition rate per atom G νf = D νf /N 0 , which depends on the length L of the box that contains the free atoms [see Eq. (43)].
In a thermal gas, the atoms have different velocities and, therefore, different energies. For a thermal Maxwell-Boltzmann gas with temperature T 0 , the distribution of the kinetic energy E f of the atomic center-of-mass motion along the x direction is
P (E f ) = 1 √ πk B T 0 e -E f /kB T0 E f . ( 45
)
The transition rate to an arbitrary bound state |ν is then given by
G νT0 = ∞ 0 G νf P (E f ) dE f , i.e. G νT0 = λ D L ∞ 0 e -E f /kB T0 R νf dE f , ( 46
)
where λ D = (2πh 2 /mk B T 0 ) 1/2 the thermal de Broglie wavelength. The phonon-mediated free-to-bound decay rate (adsorption rate) is given by
G T0 = ν G νT0 = ∞ 0 G f P (E f ) dE f . ( 47
)
In the above equation, the summation over ν includes only bound levels. Note that Eq. ( 46 ) is in qualitative agreement with the results of Refs. [5, 14] .
It is easy to extend the above results to the case of free-to-free transitions. Indeed, it can be shown that the density of the rate for the transition from a quasicontinuum state |n f , which corresponds to a free state |f , to a different free state |f ′ is given by
Q f ′ f = πh L v f R f ′ f . ( 48
)
7 For convenience, we introduce the notation
Q e f ′ f = Q f ′ f or 0 for E f ′ < E f or E f ′ ≥ E f , respectively, and Q a f ′ f = Q f ′ f or 0 for E f ′ > E f or E f ′ ≤ E f , respectively. Then, we have Q f ′ f = Q e f ′ f , 0, or Q a f ′ f for E f ′ < E f , E f ′ = E f , or E f ′ > E f , respectively.
The downward (phonon-emission) and upward (phonon-absorption) free-to-free decay rates for the free state |f are given by
Q e f = E f 0 Q e f ′ f dE f ′ ( 49
)
and
Q a f = ∞ E f Q a f ′ f dE f ′ , ( 50
)
respectively. The total free-to-free decay rate for the free state |f is
Q f = Q e f + Q a f = ∞ 0 Q f ′ f dE f ′ .
For a thermal gas, we need to replace the transition rate density Q f ′ f and the decay rate
Q f by Q f ′ T0 = ∞ 0 Q f ′ f P (E f ) dE f and Q T0 = ∞ 0 Q f P (E f ) dE f , respec-
tively, which are the averages of Q ′ f and Q f , respectively, with respect to the energy distribution P (E f ) of the initial state. Like in the other cases, we have
Q f ′ T0 = Q e f ′ T0 + Q a f ′ T0 and Q T0 = Q e T0 + Q a T0 , where Q e f ′ T0 = ∞ E f ′ Q e f ′ f P (E f ) dE f , Q a f ′ T0 = E f ′ 0 Q a f ′ f P (E f ) dE f ( 51
)
are the downward and upward transition rate densities and
Q e T0 = ∞ 0 Q e f P (E f ) dE f , Q a T0 = ∞ 0 Q a f P (E f ) dE f ( 52
)
are the downward and upward decay rates. The thermal decay rates Q e T0 and Q a T0 describe the cooling and heating processes, respectively. It can be easily shown that
Q e T0 < Q a T0 , Q e T0 > Q a T0 , and Q e T0 = Q a T0 when T 0 < T , T 0 > T , and T 0 = T , respectively. The relation Q e T0 < Q a T0 (Q e T0 > Q a T0 ), obtained for T 0 < T (T 0 > T ),
indicates the dominance of heating (cooling) of free atoms by the surface.
In order to get insight into the relaxation rates and frequency shifts, we approximate them using the Debye model for phonons. In this model, the phonon frequency ω q is related to the phonon wave number q as ω q = vq, where v is the sound velocity. Furthermore, the summation over the first Brillouin zone is replaced by an integral over a sphere of radius q D = (6π 2 N/V ) 1/3 , where V is the volume of the solid. The Debye frequency and the Debye temperature are given by ω D = vq D and T D = hω D /k B , respectively. For fused silica, we have v = 5.96 km/s, N M/V = 2.2 g/cm 3 , and M = 9.98 × 10 -26 kg [16]. Using these parameters, we find q D = 109.29 × 10 6 cm -1 , ω D = 10.4 THz, and T D = 498 K. In order to perform the summation over phonon states in the framework of the Debye model, we invoke the thermodynamic limit, i.e., replace
q • • • = V 8π 3 |q|≤qD . . . dq = 3N ω 3 D ωD 0 . . . ω 2 q dω q . ( 53
)
Then, for transitions between an upper level l and a lower level k, where 0 < ω lk < ω D , Eqs. (34) and (35) yield
R e kl = 3π M hω 3 D (n lk + 1)ω lk F 2 lk ( 54
) and R a lk = 3π M hω 3 D nlk ω lk F 2 lk . ( 55
)
Here, nlk is given by Eq. ( 20 ) with ω q replaced by ω lk . We emphasize that, according to Eqs. (54) and (55), the phonon-emission rate R e kl and the phonon-absorption rate R a lk depend not only on the matrix element F lk of the force but also on the translational transition frequency ω lk . The frequency dependences of the transition rates are comprised of the frequency dependences of the mean phonon number nlk , the phonon mode density 3N ω 2 lk /ω 3 D , and the matrix element F lk = -U ′ lk = -mω 2 lk x lk of the force. An additional factor comes from the presence of the phonon frequency in Eq. (5) for the surface displacement and, consequently, in the atomphonon interaction Hamiltonian (6). It is clear that an increase in the phonon frequency leads to a decrease in the mean phonon number and an increase in the phonon mode density. The matrix element of the force usually first increases and then decreases with increasing phonon frequency. Due to the existence of several competing factors, the frequency dependences of the transition rates are rather complicated. They usually first increase and then decrease with increasing phonon frequency. We note that, for transitions with ω lk > ω D , we have R e kl = R a lk = 0. We conclude this section by noting that the use of Eq. (53) in Eq. (40) yields the frequency shift
∆ lk = ∆ ( 0
) lk + ∆ (T ) lk , ( 56
) where ∆ ( 0
) lk = 3 2M hω 3 D µ ωD 0 F 2 lµ ω lµ -ω + F 2 µk ω µk + ω ωdω ( 57
)
8 and ∆ (T ) lk = 3 M hω 3 D µ ωD 0 ω lµ F 2 lµ ω 2 lµω 2 + ω µk F 2 µk ω 2 µkω 2 nω ωdω (58) are the zero-and finite-temperature contributions, respectively. In Eq. (58), nω is given by Eq. (20) with ω q replaced by ω.
In this section, we present the numerical results based on the analytical expressions derived in the previous section for the phonon-mediated relaxation rates of the translational levels of the atom. In particular, we use Eqs. (54) and (55) , obtained in the framework of the Debye model, for our numerical calculations. We consider transitions from bound states as well as free states. The transitions from bound states to other translational levels occur in the case where the atom is initially already adsorbed or trapped near the surface. The transitions from free states to other translational levels occur in the processes of adsorbing, heating, and cooling of free atoms by the surface. Due to the difference in physics of the initial situations, we study the transitions from bound and free states separately.
A. Transitions from bound states FIG. 3: Phonon-emission rates R e ν ′ ν from the vibrational levels (a) ν = 280 and (b) ν = 120 to other levels ν ′ as functions of the lower-level energy E ν ′ . The arrows mark the initial states. The parameters of the solid are M = 9.98 × 10 -26 kg and ωD = 10.4 THz. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 1. FIG. 4: Phonon-absorption rates R a ν ′ ν from the vibrational levels (a) ν = 280 and (b) ν = 120 to other levels ν ′ as functions of the upper-level energy E ν ′ . The left (right) panel in each row corresponds to bound-to-bound (bound-to-free) transitions. The arrows mark the initial states. The parameters used are as in Fig. 3 . The temperature of the phonon bath is T = 300 K.
We start from a given bound level and calculate the rates of phonon-mediated atomic transitions, both downward and upward. The profiles of the phonon-emission (downward-transition) rate R e ν ′ ν [see Eq. ( 54 )] and the phonon-absorption (upward-transition) rate R a ν ′ ν [see Eq. (55)] are shown in Figs. 3 and 4, respectively. The upper (lower) part of each of these figures corresponds to the case of the initial level ν = 280 (ν = 120), with energy E ν = -156 MHz (E ν = -8.4 THz). The left (right) panel of Fig. 4 corresponds to bound-to-bound (bound-to-free) upward transitions. The temperature of the surface is assumed to be T = 300 K. As seen from Figs. 3 and 4, the transition rates have pronounced localized profiles. Due to the competing effects of the mean phonon number, the phonon mode density, and the matrix element of the force, the transition rates usually first increase and then decrease with increasing phonon frequency. It is clear from a comparison of Figs. 3(a) and 3(b) and also a comparison of Figs. 4(a) and 4(b) that transitions from shallow levels have probabilities orders of magnitude lower than those from deeper levels. The main reason is that the wave functions of the shallow states are spread further away from the surface than those for the deep states. Due to this difference, the effects of the surface vibrations are weaker for the shallow levels than for the deep levels. Another pertinent feature that should be noted from the figure is the following: Since transition frequencies involved are large, they may overshoot the Debye frequency ω D = 10.4 THz, leading to a cutoff on the lower (higher) side of the frequency axis for the emission (absorption) curve.
In order to see the overall effect of the individual transition rates shown above, we add them up. First we examine the phonon-absorption rates of bound levels. The total phonon-absorption rate Γ a νν of a bound level ν is the sum of the individual absorption rates R a µν over all the upper levels µ, both bound and free [see Eq. (38)]. We plot in Fig. 5 the contributions to Γ a νν from two types of transitions, bound-to-bound and bound-to-free (desorption) transitions. The solid curve of the figure shows that the bound-to-bound phonon-absorption rate is large (above 10 10 s -1 ) for deep and intermediate levels. However, it reduces dramatically with increasing ν in the region of large ν and becomes very small (below 10 -5 s -1 ) for shallow levels. Meanwhile, the dashed curve of Fig. 5 shows that the bound-to-free phonon-absorption rate (i.e., the desorption rate) is zero for deep levels, since the energy required for the transition is greater than the Debye energy [5] . However, the desorption rate is substantial (above 10 5 s -1 ) for intermediate and shallow levels. Thus, the total phonon-absorption rate Γ a νν is mainly determined by the bound-to-bound transitions in the case of deep levels and by the bound-to-free transitions in the case of shallow levels. One of the reasons for the dramatic reduction of the bound-to-bound phonon-absorption rate in the region of shallow levels is that the number of upper bound levels µ becomes small. The second reason is that the frequency of each individual transition becomes small, leading to a decrease of the phonon mode density. The third reason is that the center-of-mass wave functions of shallow levels are spread far away from the surface, leading to a reduction of the effect of phonons on the atom.
Unlike the bound-to-bound phonon-absorption rate, 9 the bound-to-free phonon-absorption rate is substantial in the region of shallow levels. This is because the freestate spectrum is continuous and the range of the boundto-free transition frequency can be large (up to the Debye frequency ω D = 10.4 THz). The gradual reduction of the bound-to-free phonon-absorption rate in the region of shallow levels is mainly due to the reduction of the time that the atom spends in the proximity of the surface.
FIG. 5: Contributions of bound-to-bound (solid curve) and bound-to-free (dashed curve) transitions to the total phononabsorption rate Γ a νν versus the vibrational quantum number ν of the initial level. The parameters used are as in Fig. 3 . The temperature of the phonon bath is T = 300 K.
The total phonon-emission rate Γ e νν [see Eq. (37)] and the total phonon-absorption rate Γ a νν [see Eq. ( 38 )] are shown in Fig. 6 by the solid and dashed curves, respectively. It is clear from the figure that emission is comparable to but slightly stronger than absorption. Such a dominance is due to the fact that phonon emission moves the atom to a center-of-mass state closer to the surface while phonon absorption changes the atomic state in the opposite direction (see Figs. 1 and 2). Our results for the rates are in good qualitative agreement with the results of Oria et al., albeit with the Morse potential [5] . We stress that we include a large number of vibrational levels as a consequence of the deep silica-cesium potential. Note that the earlier work on this theme involved much fewer levels [5].
FIG. 6: Phonon-emission decay rate Γ e νν (solid lines) and phonon-absorption decay rate Γ a νν (dashed lines) of a bound level as functions of the vibrational quantum number ν. The inset shows the rates in the linear scale to highlight the differences in the dissociation limit. The parameters used are as in Fig. 3 . The temperature of the phonon bath is T = 300 K.
FIG. 7: Same as in Fig. 6 except that T = 30 K.
We next study the effect of temperature on the decay rates. The results for the phonon-mediated decay rates for T = 30 K are shown in Fig. 7 . In contrast to Fig. 6 , the absorption rate is now much smaller than the corresponding emission rate for both shallow and deep levels. Thus, while it is difficult to distinguish the two log-scale curves for deep and shallow levels at room temperature (see Fig. 6 ), they are well resolved at low temperature.
We now calculate the rates for transitions from free states to other levels. We first examine free-to-bound transitions, which correspond to the adsorption process. According to Eq. (43), the free-to-bound (more exactly, quasicontinuum-to-bound) transition rate G νf depends not only on the continuum-to-bound transition rate density R νf but also on the length L of the free-atom quantization box. To be specific, we use in our numerical calculations the value L = 1 mm, which is a typical size of atomic clouds in magneto-optical traps [17].
FIG. 8: Free-to-bound transition rates G νf for transitions from the free plane-wave states with energies (a) E f = 2 MHz and (b) E f = 3.1 THz to bound levels ν as functions of the bound-level energy Eν. The arrows mark the energies of the initial free states. The insets show G νf on the log scale versus Eν in the range from -200 MHz to -0.2 MHz to highlight the rates to shallow bound levels. The length of the freeatom quantization box is L = 1 mm. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 3.
We plot in Fig. 8 the free-to-bound transition rate G νf [see Eq. (43)] as a function of the vibrational quantum number ν. The upper (lower) part of the figure corresponds to the case of the initial-state energy E f = 2 MHz (E f = 3.1 THz), which is close to the average kinetic energy per atom in an ideal gas with temperature T 0 = 200 µK (T 0 = 300 K). We observe that the free-tobound transition rate first increases and then decreases with increasing transition frequency ω f ν = (E f -E ν )/h. Such behavior results from the competing effects of the mean phonon number, the phonon mode density, and the matrix element of the force, like in the case of boundto-bound transitions (see Fig. 3 ). We also see a cutoff of the transition frequency, which is associated with the Debye frequency. Comparison of Figs. 8(a) and 8(b) shows that the transitions from low-energy free states have probabilities orders of magnitude smaller than those from high-energy free states. One of the reasons is that the transition rate G νf is proportional to the velocity v f = (2E f /m) 1/2 [see Eq. (43)]. The dependence of the transition rate density R νf on the transition frequency ω f ν plays an important role. Because of this, the rates for the transitions from low-energy free states to shallow bound levels are very small [see the inset of Fig. 8(a)].
FIG. 9: Free-to-bound decay rate G f as a function of the free-state energy E f . The inset highlights the magnitude and profile of the decay rate for E f in the range from 0 to 20 MHz. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8
We show in Fig. 9 the free-to-bound decay rate G f [see Eq. (44)], which is a characteristic of the adsorption process, as a function of the free-state energy E f . We see that G f first increases and then decreases with 10 increasing E f . The increase of G f with increasing E f in the region of small E f (see the inset) is mainly due to the increase in the atomic incidence velocity v f . In this region, we have G f ∝ v f ∝ E f [see Eqs. (43) and (44)]. For E f in the range from 0 to 20 MHz, which is typical for atoms in magneto-optical traps, the maximum value of G f is on the order of 10 4 s -1 (see the inset of Fig. 9 ). Such free-to-bound (adsorption) rates are several orders of magnitude smaller than the bound-to-free (desorption) rates (see the dashed curve in Fig. 5 ). The decrease of G f with increasing E f in the region of large E f is mainly due to the reduction of the atom-phonon coupling coefficients.
FIG. 10: Free-to-bound transition rates GνT 0 for transitions from the thermal states with temperatures (a) T0 = 200 µK and (b) T0 = 300 K to bound levels ν as functions of the bound-level energy Eν. The insets show GνT 0 on the log scale versus Eν in the range from -200 MHz to -0.2 MHz to highlight the rates to shallow bound levels. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. FIG. 11: Free-to-bound decay rate GT 0 as a function of the atomic temperature T0 in the ranges (a) from 100 µK to 400 µK and (b) from 50 K to 350 K. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8.
In a thermal gas, the adsorption process is characterized by the transition rate G νT0 [see Eq. (46)] and the decay rate G T0 [see Eq. ( 47 )], which are the averages of the free-to-bound transition rate G νf and the free-to-bound decay rate G f , respectively, over the freestate energy distribution (45) . We plot the free-to-bound transition rate G νT0 and the free-to-bound decay rate G T0 in Figs. 10 and 11, respectively. Comparison between Figs. 10(a) and 9(a) shows that the transition rates from low-temperature thermal states and low-energy free states look quite similar to each other. The reason is that the spread of the energy distribution is not substantial in the case of low temperatures. The spread of the energy distribution is however substantial in the case of high temperatures, leading to the softening of the cutoff frequency effect [compare Fig. 10(b ) with Fig. 9(b)] . Figure 11 shows that the free-to-bound decay rate G T0 first increases and then reduces with increasing atomic temperature T 0 . For T 0 in the range from 100 µK to 400 µK, which is typical for atoms in magneto-optical traps, the maximum value of G T0 is on the order of 10 4 s -1 [see Fig. 11(a) ]. Such free-to-bound (adsorption) rates are several orders of magnitude smaller than the bound-tofree (desorption) rates (see the dashed curve in Fig. 5 ).
Figure 11( a) shows that, in the region of low atomic temperature T 0 , one has G T0 ∝ √ T 0 , in agreement with the asymptotic behavior of Eqs. (46) and (47) .
FIG. 12: Free-to-free transition rate densities Q f ′ f for the upward (solid lines) and downward (dashed lines) transitions from the free states |f with energies (a) E f = 2 MHz and (b) E f = 3.1 THz to other free states |f ′ as functions of the finallevel energy E f ′ . The arrows mark the energies of the initial free states. The inset in part (a) shows Q f ′ f versus E f ′ in the range from 0 to 4 MHz to highlight the small magnitude of the rate density for downward transitions (dashed line). The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8 .
We now examine free-to-free transitions, both upward and downward, which corresponding to the heating and cooling processes of free atoms by the surface. We plot in Fig. 12 the free-to-free transition rate density Q f ′ f [see Eq. ( 48 )] as a function of the final-level energy E f ′ . The upper (lower) part of the figure corresponds to the case of the initial-state energy E f = 2 MHz (E f = 3.1 THz), which is close to the average kinetic energy per atom in an ideal gas with temperature T 0 = 200 µK (T 0 = 300 K). The rate densities are shown for the upward (phononabsorption) and downward (phonon-emission) transitions by the solid and dashed lines, respectively. The figure shows that the free-to-free transition rate density increases or decreases with increasing transition frequency if the latter is not too large or is large enough, respectively. We also observe a signature of the Debye cutoff of the phonon frequency. Comparison of Figs. 12(a) and 12(b) shows that transitions from low-energy free states have probabilities orders of magnitude smaller than those from high-energy free states. Figure 12 (a) and its inset show that, when the energy of the free state is low, the free-to-free downward (cooling) transition rate is very small as compared to the free-to-free upward (heating) transition rate.
FIG. 13: Free-to-free upward and downward decay rates Q a f (solid lines) and Q e f (dashed lines) as functions of the energy E f of the initial free state. The insets highlight the magnitudes and profiles of the decay rates for E f in the range from 0 to 20 MHz. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. We show in Fig. 13 the free-to-free upward (phononabsorption) and downward (phonon-emission) decay rates Q a f [see Eq. ( 50 )] and Q e f [see Eq. ( 49 )] as functions of the free-state energy E f . We observe that Q a f and Q e f increase with increasing E f in the range from 0 to 8 THz. The increase of Q a f with increasing E f in the region of small E f (see the left inset) is mainly due to the increase in the atomic incidence velocity v f . In this region, we have Q a f ∝ v f ∝ E f [see Eqs. (48) and (50)]. The increase of Q e f with increasing E f in the region of small E f (see the right inset) is due to not only the increase in the atomic incidence velocity v f [see Eq. (48)] but also 11 the increase of the transition rate density Q e f ′ f and the increase of the integration interval (0, E f ) [see Eq. (49)]. In this region, the dependence of Q e f on the energy E f is of higher order than E 3/2 f . The left inset of Fig. 13 shows that, for E f in the range from 0 to 20 MHz, the maximum value of Q a f is on the order of 10 4 s -1 . Such free-to-free upward (heating) decay rates are comparable to but about two times smaller than the corresponding free-to-bound (adsorption) decay rates (see the inset of Fig. 9 ). Meanwhile, the right inset of Fig. 13 shows that, in the region of small E f , the free-to-free downward (cooling) decay rate Q e f is very small.
FIG. 14: Free-to-free transition rate densities Q a f T 0 for upward transitions (solid lines) and Q e f T 0 for downward transitions (dashed lines) from the thermal states with temperatures (a) T0 = 200 µK and (b) T0 = 300 K to free levels f as functions of the free-level energy E f . The inset in part (a) shows the rate densities versus E f in the range from 0 to 8 MHz to highlight the small magnitude of Q e f T 0 (dashed line). The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. FIG. 15: Free-to-free decay rates Q a T 0 (solid lines) and Q e T 0 (dashed lines) for upward and downward transitions, respectively, as functions of the atomic temperature T0 in the ranges (a) from 100 µK to 400 µK and (b) from 50 K to 350 K. For comparison, the free-to-bound decay rate GT 0 is replotted from Fig. 11 by the dotted lines. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8.
In the case of a thermal gas, the phonon-mediated heat transfer between the gas and the surface is characterized by the free-to-free transition rate densities Q a f T0 and Q e f T0
[see Eqs. (51)] and the free-to-free decay rates Q a T0 and Q e T0 [see Eqs. (52)]. We plot the free-to-free transition rate densities Q a f T0 and Q e f T0 in Fig. 14 . Comparison between Figs. 14(a) and 12(a) shows that the transition rate densities from low-temperature thermal states and low-energy free states are quite similar to each other. The spread of the initial-state energy distribution is not substantial in this case. However, the energy spread of the initial state is substantial in the case of high temperatures, concealing the cutoff frequency effect [compare Fig. 14(b ) with Fig. 12(b)] . We display the freeto-free decay rates Q a T0 and Q e T0 in Fig. 15 . The solid and dashed lines correspond to the upward (heating) and downward (cooling) transitions, respectively. For comparison, the free-to-bound decay rate (adsorption rate) G T0 is re-plotted from Fig. 11 by the dotted lines. We observe that, for T 0 in the range from 100 µK to 400 µK [see Fig. 15(a) ], the adsorption rate G T0 (dotted line) is about two times larger than the heating rate Q a T0 (solid line), while the cooling rate Q e T0 (dashed line) is negligible. Figure 15 (a) shows that, in the region of low atomic temperatures, one has Q T0 ∼ = Q a T0 ∝ √ T 0 , in agreement with the asymptotic behavior of expressions (52) . The figure also shows that Q e T0 quickly increases with increasing atomic temperature T 0 . The relation Q e T0 < Q a T0 , obtained for T 0 < T , indicates the dominance of heating of cold free atoms by the surface. The substantial magnitude of the free-to-bound transition rate G T0 (dotted line) indicates that a significant number of atoms can be adsorbed by the surface. According to Fig. 15(b) , the free-to-free downward transition rate Q e T0 (dashed line) crosses the upward transition rate Q a T0 (solid line) when T 0 = T = 300 K, and then becomes the dominant decay rate. The relation Q e T0 > Q a T0 , obtained for T 0 > T , indicates the dominance of cooling of hot free atoms by the surface.
In conclusion, we have studied the phonon-mediated transitions of an atom in a surface-induced potential. We developed a general formalism, which is applicable for any surface-atom potential. A systematic derivation of the corresponding density-matrix equation enables us to investigate the dynamics of both diagonal and offdiagonal elements. We included a large number of vibrational levels originating from the deep silica-cesium potential. We calculated the transition and decay rates from both bound and free levels. We found that the rates of phonon-mediated transitions between translational levels depend on the mean phonon number, the phonon mode density, and the matrix element of the force from the surface upon the atom. Due to the effects of these competing factors, the transition rates usually first increase and then reduce with increasing transition frequency. We focused on the transitions from bound states. Two specific examples, namely, when the initial level is a shallow level also when it can be one of the deep levels have been worked out. We have shown that there can be marked differences in the absorption and emission behavior in the two cases. For example, both the absorption and emission rates from the deep bound levels can be several orders (in our case, six orders) of magnitude larger than the corresponding rates from the shallow bound levels. We also analyzed various types of transitions from free states. We have shown that, for thermal atomic cesium with temperature in the range from 100 µK to 400 µK in the vicinity of a silica surface with temperature of 300 K, the adsorption (free-to-bound decay) rate is about two times larger than the heating (free-to-free upward decay) rate, while the cooling (free-to-free downward decay) rate is negligible. 12 Acknowledgments We thank M. Chevrollier for fruitful discussions. This work was carried out under the 21st Century COE pro-gram on "Coherent Optical Science." [ * ] Also at Institute of Physics and Electronics, Vietnamese Academy of Science and Technology, Hanoi, Vietnam. [1] V. I. Balykin, K. Hakuta, Fam Le Kien, J. Q. Liang, and M. Morinaga, Phys. Rev. A 70, 011401(R) (2004); Fam Le Kien, V. I. Balykin, and K. Hakuta, Phys. Rev. A 70, 063403 (2004). [2] Fam Le Kien, S. Dutta Gupta, V. I. Balykin, and K.
Hakuta, Phys. Rev. A 72, 032509 (2005). [3] Fam Le Kien, S. Dutta Gupta, K. P. Nayak, and K.
Hakuta, Phys. Rev. A 72, 063815 (2005). [4] E. G. Lima, M. Chevrollier, O. Di Lorenzo, P. C. Segundo, and M. Oriá, Phys. Rev. A 62, 013410 (2000). [5] T. Passerat de Silans, B. Farias, M. Oriá, and M.
Chevrollier, Appl. Phys. B 82, 367 (2006). [6] Fam Le Kien and K. Hakuta, Phys. Rev. A 75, 013423 (2007). [7] Fam Le Kien, S. Dutta Gupta, and K. Hakuta, e-print quant-ph/0610067. [8] K. P. Nayak, P. N. Melentiev, M. Morinaga, Fam Le Kien, V. I. Balykin, and K. Hakuta, e-print quant-ph/0610136. [9] C. Henkel and M. Wilkens, Europhys. Lett. 47, 414 (1999). [10] Z. W. Gortel, H. J. Kreuzer, and R. Teshima, Phys. Rev. B 22, 5655 (1980). [11] H. Hoinkes, Rev. Mod. Phys. 52, 933 (1980). [12] N. N. Bogolubov, Commun. of JINR, E17-11822, Dubna (1978); N. N. Bogolubov and N. N. Bogolubov Jr., Elementary Particles and Nuclei (USSR) 11, 245 (1980). [13] R. Zwanzig, Lectures in Theoretical Physics, eds. W. E. Brittin, B. W. Downs, and J. Downs (Interscience, New York, 1961) Vol. 3, p. 106; G. S. Agarwal, Progress in Optics, ed. E. Wolf (North-Holland, Amsterdam, 1973) Vol. 11, p. 3; L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995) p. 880. [14] J. Javanainen and M. Mackie, Phys. Rev. A 58, R789 (1998); M. Mackie and J. Javanainen, ibid. 60, 3174 (1999). [15] E. Luc-Koenig, M. Vatasescu, and F. Masnou-Seeuws, Eur. Phys. J. D 31, 239 (2004). [16] See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 2001). [17] H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, New York, 1999). | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Over the past few years, tight confinement of cold atoms has drawn considerable attention. The interest in this area is motivated not only by the fundamental nature of the problem, but also by its potential applications in atom optics and quantum information. A method for microscopic trapping and guiding of individual atoms along a nanofiber has been proposed [1] . Surface-atom quantum electrodynamic effects have constituted another interesting area, where a great deal of work has been carried out. Modification of spontaneous emission of an atom [2] and radiative exchange between two distant atoms [3] mediated by a nanofiber have been investigated. Surface-induced deep potentials have played a major role and have received due attention in recent years. Oria et al. have studied various theoretical schemes to load atoms into such potentials [4, 5] . A rigorous theory of spontaneous decay of an atom in a surface-induced potential invoking the density-matrix formalism has been developed [6] . The role of interference between the emitted and reflected fields and also the role of transmission into the evanescent modes were identified. Further calculations on the excitation spectrum have been carried out [7] . Bound-to-bound transitions were shown to lead to significant effects like a large red tail of the excitation spectrum as compared to the weak consequences of free-to-bound transitions. A crucial step in this direction was the experimental observation of the excitation spectrum and the channeling of the fluorescent photons along the nanofiber [8] , opening up avenues for novel quantum information devices.\n\nIn most of the problems involving surface-atom interaction, the macroscopic surface is usually kept at room temperature. Thus the pertinent question that can be asked is what would be the effect of heating on the cold atoms. It is understood that transfer of heat to the trapped atoms will lead to a change in the occupation probability of the vibrational levels as well as their coherence. Phonon-induced changes in the populations of the vibrational levels have been studied by several groups [5, 9, 10] . In a nice and compact treatment based on the dyadic Green function and the Fermi golden rule, Henkel et al. showed that the effects can be very different depending on the nature of the atomic/molecular species [9] . The time scales for various species were estimated. It should be stressed that the trap considered by Henkel et al. was not necessarily a surface trap and misses out on many of the aspects of the surface-atom interaction [9] . Based on the assumption that the surface-atom interaction can be represented by a Morse potential, the phonon-mediated decay was estimated by Oria et al. [5] . Their estimate was based on the formalism developed by Gortel et al. [10] . However, all the previous theories focus on only the transition rates and thus are not general enough. In this paper, we present a general densitymatrix formalism to calculate the phonon-mediated decay of populations as well as the changes in coherence. We derive the relevant master equation for the density matrix of the atom. We emphasize that our densitymatrix equation describes the full dynamics of the coupling between trapped atoms and phonons and does not assume any particular form of the trapping potential. Under the Debye approximation, we derive compact expressions for the phonon-mediated decay rates. Numerical calculations are carried out assuming the potential model considered in [4] . In contrast to the previous work, we include a large number of vibrational levels due to the deep surface-atom potential. We show that there can be significant differences in the decay rates when the initial level is chosen as one of the shallow or deep bound levels. We also calculate and analyze the decay rates for various 2 types of transitions from free states. The paper is organized as follows. In Sec. II we describe the model. In Sec. III we derive the basic dynamical equations for the phonon-mediated decay processes. In Sec. IV we present the results of numerical calculations. Our conclusions are given in Sec. V."
},
{
"section_type": "OTHER",
"section_title": "II. DESCRIPTION OF THE MODEL SYSTEM",
"text": "We assume the whole space to be divided into two regions, namely, the half-space x < 0, occupied by a nondispersive nonabsorbing dielectric medium (medium 1), and the half-space x > 0, occupied by vacuum (medium 2). We examine a single atom moving in the empty halfspace x > 0. We assume that the atom is in a fixed internal state |i with energy hω i . Without loss of generality, we assume that the energy of the internal state |i is zero, i.e. ω i = 0. We describe the interaction between the atom and the surface. We first consider the surface-induced interaction potential and then add the atom-phonon interaction."
},
{
"section_type": "OTHER",
"section_title": "A. Surface-induced interaction potential",
"text": "In this subsection, we describe the interaction between the atom and the surface in the case where thermal vibrations of the surface are absent. The potential energy of the surface-atom interaction is a combination of a long-range van der Waals attraction and a short-range repulsion [11] . Despite a large volume of research on the surface-atom interaction, due to the complexity of surface physics and the lack of data, the actual form of the potential is yet to be ascertained [11] . For the purpose of numerical demonstration of our formalism, we choose the following model for the potential [4, 11] :\n\nU (x) = Ae -αx - C 3 x 3 . ( 1\n\n)\n\nHere, C 3 is the van der Waals coefficient, while A and α determine the height and range, respectively, of the surface repulsion. The potential parameters C 3 , A, and α depend on the nature of the dielectric and the atom.\n\nIn numerical calculations, we use the parameters of fused silica, for the dielectric, and the parameters of groundstate atomic cesium, for the atom. The parameters for the interaction between silica and ground-state atomic cesium are theoretically estimated to be C 3 = 1.56 kHz µm 3 , A = 1.6 × 10 18 Hz, and α = 53 nm -1 [6]. We introduce the notation ϕ ν (x) for the eigenfunctions of the center-of-mass motion of the atom in the potential U (x). They are determined by the stationary Schrödinger equation -h2 2m\n\nd 2 dx 2 + U (x) ϕ ν (x) = E ν ϕ ν (x). ( 2\n\n)\n\nHere m is the mass of the atom. In the numerical example with atomic cesium, we have m = 132.9 a.u. = 2.21 × 10 -25 kg. The eigenvalues E ν are the centerof-mass energies of the translational levels of the atom. These eigenvalues are the shifts of the energies of the translational levels from the energy of the internal state |i . Without loss of generality, we assume that the center-of-mass eigenfunctions ϕ ν (x) are real functions, i.e. ϕ * ν (x) = ϕ ν (x). In Fig. 1 , we show the potential U (x) and the wave functions ϕ ν (x) of a number of bound levels with energies in the range from -1 GHz to -5 MHz. We also plot the wave function of a free state with energy of about 4.25 MHz. In order to have some estimate about the spatial extent of a wave function ϕ ν (x), we define a crossing point x cross , which corresponds to the rightmost solution of the equation U (x) = E ν . Note that, for shallow levels, the wave function generally peaks close to the point x cross . We plot the eigenvalue modulus |E ν | and the crossing point x cross in Figs. 2(a) and 2(b), respectively. It is clear from the figure that, for ν in the range from 0 to 300, the eigenvalue varies dramatically from about 158 THz to about 322 kHz, while the wave function extends only up to 170 nm.\n\nFIG. 1: Energies and wave functions of the center-of-mass motion of an atom in a surface-induced potential. The parameters of the potential are C3 = 1.56 kHz µm 3 , A = 1.6 × 10 18 Hz, and α = 53 nm -1 . The mass of the atom is m = 2.21 × 10 -25 kg. We plot bound levels with energies in the range from -1 GHz to -5 MHz and also a free state with energy of about 4.25 MHz. FIG. 2: Eigenvalue modulus |Eν| (a) and crossing point xcross (b) as functions of the vibrational quantum number ν. The parameters used are as in Fig. 1.\n\nWe introduce the notation |ν = |ϕ ν and ω ν = E ν /h for the state vectors and frequencies of translational levels. Then, the Hamiltonian of the atom in the surfaceinduced potential can be represented in the diagonal form\n\nH A = ν hω ν σ νν . ( 3\n\n)\n\nHere, σ νν = |ν ν| is the population operator for the translational level ν. We emphasize that the summation over ν includes both the discrete (E ν < 0) and continuous (E ν > 0) spectra. The levels ν with E ν < 0 are called the bound (or vibrational) levels. In such a state, the atom is bound to the surface. It is vibrating, or more exactly, moving back and forth between the walls formed by the van der Waals part and the repulsive part of the potential. The levels ν with E ν > 0 are called the free (or continuum) levels. The center-of-mass wave functions of 3 the bound states are normalized to unity. The center-ofmass wave functions of the free states are normalized to the delta function of energy."
},
{
"section_type": "OTHER",
"section_title": "B. Atom-phonon interaction",
"text": "In this subsection, we incorporate the thermal vibrations of the solid into the model. Due to the thermal effects, the surface of the dielectric vibrates. The surfaceinduced potential for the atom is then U (xx s ), where x s is the displacement of the surface from the mean position x s = 0. We approximate the vibrating potential U (xx s ) by expanding it to the first order in x s ,\n\nU (x -x s ) = U (x) -U ′ (x)x s . ( 4\n\n)\n\nThe first term, U (x), when combined with the kinetic energy p 2 /2m, yields the Hamiltonian H A [see Eq. ( 3 )], which leads to the formation of translational levels of the atom. The second term, -U ′ (x)x s , accounts for the thermal effects in the interaction of the atom with the solid. Note that the quantity F = -U ′ (x) is the force of the surface upon the atom. Hence, the force of the atom upon the surface is -F = U ′ (x) and, consequently, U ′ (x)x s is the work required to displace the surface for a small distance x s . It is well known that, for a smooth surface, the gas atom interacts only with the phonons polarized along the x direction [10] . In the harmonic approximation, we have\n\nx s = q h 2M N ω q 1/2 (b q e iqR + b † q e -iqR ). ( 5\n\n)\n\nHere, M is the mass of a particle of the solid, N is the particle number density, ω q and q are the frequency and wave vector of the x-polarized acoustic phonons, respectively, R = (0, y, z) is the lateral component of the position vector (x, y, z) of the atom, and b q and b † q are the annihilation and creation phonon operators, respectively. Without loss of generality, we choose R = 0. Meanwhile, the operator U ′ can be decomposed as\n\nU ′ = νν ′ σ νν ′ ν|U ′ |ν ′ , where σ νν ′ = |ν ν ′ | is\n\nthe operator for the translational transition ν ↔ ν ′ . Hence, the energy term -U ′ (x)x s leads to the atom-phonon interaction Hamiltonian [10]\n\nH I = h q 1 √ ω q S(b q + b † q ), ( 6\n\n) with S = νν ′ g νν ′ σ νν ′ . ( 7\n\n)\n\nHere we have introduced the atom-phonon coupling coefficients\n\ng νν ′ = F νν ′ √ 2M N h , ( 8\n\n)\n\nwith\n\nF νν ′ = - ∞ -∞ ϕ ν (x)U ′ (x)ϕ ν ′ (x)dx ( 9\n\n)\n\nbeing the matrix elements for the force of the surface upon the atom. We note that F νν ′ = -mω 2 νν ′ x νν ′ , where x νν ′ = ν|x|ν ′ and ω νν ′ = ω νω ν ′ are the surface-atom dipole matrix element and the translational transition frequency, respectively. Hence, the coupling coefficient g νν ′ depends on the dipole matrix element x νν ′ and the transition frequency ω νν ′ . Since ω νν = 0, we have g νν = 0.\n\nWe note that the Hamiltonian of the x-polarized acoustic phonons is given by\n\nH B = q hω q b † q b q . ( 10\n\n)\n\nThe total Hamiltonian of the atom-phonon system is\n\nH = H A + H I + H B . ( 11\n\n)\n\nWe use the above Hamiltonian to study the phononmediated decay of the atom."
},
{
"section_type": "OTHER",
"section_title": "III. DYNAMICS OF THE ATOM",
"text": "In this section, we present the basic equations for the phonon-mediated decay processes. We derive a general master equation for the reduced density operator of the atom in subsection III A, obtain analytical expressions for the relaxation rates and frequency shifts in subsection III B, and calculate the rates and the shifts in the framework of the Debye model in subsection III C."
},
{
"section_type": "OTHER",
"section_title": "A. Master equation",
"text": "In the Heisenberg picture, the equation for the phonon operator b q (t) is\n\nḃq (t) = -iω q b q (t) - i √ ω q S(t), ( 12\n\n)\n\nwhich has a solution of the form\n\nb q (t) = b q (t 0 )e -iωq(t-t0) -iW q (t). ( 13\n\n)\n\nHere, t 0 is the initial time and W q is given by\n\nW q (t) = 1 √ ω q t t0 e -iωq(t-τ ) S(τ ) dτ. ( 14\n\n)\n\nConsider an arbitrary atomic operator O which acts only on the atomic states but not on the phonon states. The time evolution of this operator is governed by the Heisenberg equation\n\n∂O(t) ∂t = i h [H A (t) + H I (t), O(t)], ( 15\n\n)\n\n4 which, with account of Eqs. (6) and (13), yields ∂O(t) ∂t = i h [H A (t), O(t)] + q i √ ω q [S(t), O(t)][b q (t 0 )e -iωq(t-t0) -iW q (t)]\n\n- q i √ ω q [b † q (t 0 )e iωq(t-t0) + iW † q (t)][O(t), S(t)]. ( 16\n\n)\n\nWe assume the initial density of the atom-phonon system to be the direct product state\n\nρ Σ (t 0 ) = ρ(t 0 )ρ B (t 0 ), ( 17\n\n)\n\nwith the atom in an arbitrary state ρ(t 0 ) and the phonons in a thermal state\n\nρ B (t 0 ) = Z -1 exp[-H B (t 0 )/k B T ]. ( 18\n\n)\n\nHere, Z is the normalization constant and T is the temperature of the phonon bath. For the initial condition (17), the Bogolubov's lemma [12] , applied to an arbitrary operator Θ(t), asserts the following:\n\nΘ(t)b q (t 0 ) = nq [b q (t 0 ), Θ(t)] , ( 19\n\n)\n\nwhere the mean number of phonons in the mode q is given by\n\nnq = 1 exp(hω q /k B T ) -1 . ( 20\n\n)\n\nLet Θ be an atomic operator. We then have the commutation relation [b q (t), Θ(t)] = 0, which yields [b q (t 0 ), Θ(t)] = ie iωq(t-t0) [W q (t), Θ(t)]. (21) Combining Eq. (19) with Eq. (21) leads to\n\nΘ(t)b q (t 0 ) = ie iωq(t-t0) nq [W q (t), Θ(t)] . ( 22\n\n)\n\nWe perform the quantum mechanical averaging for expression (16) and use Eq. ( 22 ) to eliminate the phonon operators b q (t 0 ) and b † q (t 0 ). The resulting equation can be written as\n\n∂ O(t) ∂t = i h [H A (t), O(t)] + q nq + 1 √ ω q [S(t), O(t)]W q (t) + W † q (t)[O(t), S(t)] + q nq √ ω q W q (t)[O(t), S(t)] + [S(t), O(t)]W † q (t) . ( 23\n\n)\n\nWe note that Eq. (23) is exact. It does not contain phonon operators explicitly. The dependence on the phonon operators is hidden in the time shift of the operator S(τ ) in expression (14) for the operator W q (t).\n\nWe now show how the dependence of the operator W q (t) on the phonon operators can be approximately eliminated. We assume that the atom-phonon coupling coefficients g νν ′ are small. The use of the zeroth-order approximation σ νν ′ (τ ) = σ νν ′ (t)e iω νν ′ (τ -t) in the expression for S(τ ) [see Eq. ( 7 )] yields\n\nS(τ ) = νν ′ g νν ′ σ νν ′ (t)e iω νν ′ (τ -t) , ( 24\n\n)\n\nwhich is accurate to first order in the coupling coefficients. Inserting Eq. (24) into Eq. ( 14 ) gives\n\nW q (t) = 2π √ ω q νν ′ g νν ′ σ νν ′ (t)δ -(ω ν ′ ν -ω q ), ( 25\n\n)\n\nwhere\n\nδ -(ω) = lim ǫ→0 1 2π 0 -∞ e -i(ω+iǫ)τ dτ = i 2π P ω + 1 2 δ(ω). ( 26\n\n)\n\nHere, in order to take into account the effect of adiabatic turn-on of interaction, we have added a small positive parameter ǫ to the integral and have used the limit t 0 → -∞. Introducing the notation\n\nK q = W q √ ω q = 2π ω q νν ′ g νν ′ σ νν ′ δ -(ω ν ′ ν -ω q ), ( 27\n\n)\n\nwe can rewrite Eq. ( 23 ) in the form\n\n∂ O(t) ∂t = i h [H A (t), O(t)] + q (n q + 1) [S(t), O(t)]K q (t) + K † q (t)[O(t), S(t)] + q nq K q (t)[O(t), S(t)] + [S(t), O(t)]K † q (t) . ( 28\n\n)\n\nIn order to examine the time evolution of the reduced density operator ρ(t) of the atom in the Schrödinger picture, we use the relation O(t) = Tr[O(t)ρ(0)] = Tr[O(0)ρ(t)], transform to arrange the operator O(0) at the first position in each operator product, and eliminate O(0). Then, we obtain the Liouville master equation\n\n∂ρ(t) ∂t = - i h [H A , ρ(t)] + q (n q + 1){[K q ρ(t), S] + [S, ρ(t)K † q ]} + q nq {[S, ρ(t)K q ] + [K † q ρ(t), S]}. ( 29\n\n)\n\nEquations (28) and (29) are valid to second order in the coupling coefficients. These equations allow us to study the time evolution and dynamical characteristics of the atom interacting with the thermal phonon bath. We note that Eq. (29) is a particular form of the Zwanzig's generalized master equation, which can be obtained by the projection operator method [13]. 5 B. Relaxation rates and frequency shifts We use Eq. (29) to derive an equation for the matrix elements ρ jj ′ ≡ j|ρ|j ′ of the reduced density operator of the atom. The result is\n\n∂ρ jj ′ ∂t = -iω jj ′ ρ jj ′ + νν ′ (γ e jj ′ νν ′ + γ a jj ′ νν ′ )ρ νν ′ - ν [(γ e jν + γ a jν )ρ νj ′ + (γ e * j ′ ν + γ a * j ′ ν )ρ jν ], ( 30\n\n)\n\nwhere the coefficients\n\nγ e jj ′ νν ′ = 2π q nq + 1 ω q g jν g j ′ ν ′ [δ -(ω νj -ω q ) + δ + (ω ν ′ j ′ -ω q )], γ e jν = 2π qµ nq + 1 ω q g jµ g νµ δ -(ω νµ -ω q ) ( 31\n\n) and γ a jj ′ νν ′ = 2π q nq ω q g jν g j ′ ν ′ [δ -(ω j ′ ν ′ -ω q ) + δ + (ω jν -ω q )], γ a jν = 2π qµ nq ω q g jµ g νµ δ + (ω µν -ω q ) ( 32\n\n)\n\nare the decay parameters associated with the phonon emission and absorption, respectively. Here, the notation δ + (ω) = δ * -(ω) has been used. Equation (30) describes phonon-induced variations in the populations and coherences of the translational levels of the atom. We analyze the characteristics of the relaxation processes. For simplicity of mathematical treatment, we first consider only transitions from discrete levels. The equation for the diagonal matrix element ρ jj for a discrete level j can be written in the form\n\n∂ρ jj ∂t = ν (γ e jjνν + γ a jjνν )ρ νν -(γ e jj + γ a jj + c.c.)ρ jj + off-diagonal terms. ( 33\n\n)\n\nWhen the off-diagonal terms are neglected, Eq. (33) reduces to a simple rate equation. It is clear from Eq. ( 33 ) that the rate for the downward transition from an upper level l to a lower level k (k < l) is\n\nR e kl = γ e kkll = 2π q nq + 1 ω q g 2 lk δ(ω lk -ω q ), ( 34\n\n)\n\nwhile the rate for the upward transition from a lower level k to an upper level l (l > k) is\n\nR a lk = γ a llkk = 2π q nq ω q g 2 lk δ(ω lk -ω q ). ( 35\n\n)\n\nEquations (34) and (35) are in agreement with the results of Gortel et al. [10], obtained by using the Fermi golden rule. We note that R e kl and R a lk with l ≤ k are mathematically equal to zero because they have no physical meaning. For convenience, we introduce the notation R lk = R e lk , R a lk , or 0 for l < k, l > k, or l = k, respectively. It is clear that the off-diagonal coefficients R lk with l = k are the rates of transitions. However, the diagonal coefficients R kk have no physical meaning and are mathematically equal to zero.\n\nAs seen from Eq. ( 33 ), the phonon-mediated depletion rate of a level k is Γ kk = 2Re(γ e kk + γ a kk ). The explicit expression for this rate is\n\nΓ kk = 2π qµ nq + 1 ω q g 2 kµ δ(ω kµ -ω q ) + 2π qµ nq ω q g 2 µk δ(ω µk -ω q ). ( 36\n\n)\n\nWe note that Γ kk = µ (R e µk + R a µk ) = µ R µk . We can write Γ kk = Γ e kk + Γ a kk , where\n\nΓ e kk = µ<k R e µk ( 37\n\n) and Γ a kk = µ>k R a µk ( 38\n\n)\n\nare the contributions due to downward transitions (phonon emission) and upward transitions (phonon absorption), respectively. In the above equations, the summation over µ can be extended to cover not only the discrete levels but also the continuum levels. Meanwhile, the equation for the off-diagonal matrix element ρ lk for a pair of discrete levels l and k can be written in the form\n\n∂ρ lk /∂t = -(iω lk + γ e ll + γ a ll + γ e * kk + γ a * kk )ρ lk + . . . , or, equivalently, ∂ρ lk ∂t = -i(ω lk + ∆ lk -iΓ lk )ρ lk + . . . . ( 39\n\n)\n\nHere the frequency shift ∆ lk is given by\n\n∆ lk = qµ nq + 1 ω q g 2 lµ ω lµ -ω q + g 2 µk ω µk + ω q + qµ nq ω q g 2 lµ ω lµ + ω q + g 2 µk ω µk -ω q , ( 40\n\n)\n\nwhile the coherence decay rate Γ lk is expressed as\n\nΓ lk = π qµ nq + 1 ω q g 2 lµ δ(ω lµ -ω q ) + g 2 kµ δ(ω kµ -ω q ) + π qµ nq ω q g 2 µl δ(ω µl -ω q ) + g 2 µk δ(ω µk -ω q ) . ( 41\n\n)\n\n6 When we set l = k in Eq. (40), we find ∆ kk = 0. When we set l = k in Eq. (41), we recover Eq. (36). We note that Γ lk = µ (R e µl + R e µk + R a µl + R a µk )/2 = µ (R µl + R µk )/2. Comparison between Eqs. (41) and (36) yields the relation Γ lk = (Γ ll + Γ kk )/2. We can also write Γ lk = Γ e lk + Γ a lk , where Γ e lk = µ (R e µl + R e µk )/2 and Γ a lk = µ (R a µl + R a µk )/2 are the contributions due to downward transitions (phonon emission) and upward transitions (phonon absorption), respectively. In the above equations, the summation over µ can be extended to cover not only the discrete levels but also the continuum levels.\n\nWe now discuss phonon-mediated transitions from continuum (free) levels. We start by considering free-tobound transitions. For a continuum level f with energy E f > 0, the center-of-mass wave function ϕ f (x) is normalized per unit energy. In this case, the quantity R νf becomes the density of the transition rate. A free level f can be approximated by a level of a quasicontinuum [14] . A discretization of the continuum can be realized by using a large box of length L with reflecting boundary conditions [15] . We label E n the energies of the eigenstates in the box and φ n (x) the corresponding wave functions. Note that such states are standing-wave states [14, 15] . The relation between a quasicontinuum-state wave function φ n f (x), normalized to unity in the box, and the corresponding continuum-state wave function ϕ f (x), normalized per unit energy, with equal energies\n\nE n f = E f , is [15] ϕ f (x) ∼ = ∂E n f ∂n f -1/2 φ n f (x) ∼ = L πh 1/2 m 2E n f 1/4 φ n f (x). ( 42\n\n)\n\nConsequently, for a single atom initially prepared in the quasicontinuum standing-wave state |n f = |φ n f , the rate for the transition to an arbitrary bound state |ν is approximately given by\n\nG νf = πh L v f R νf , ( 43\n\n)\n\nwhere v f = (2E f /m) 1/2 is the velocity of the atom in the initial standing-wave state |f . The phonon-mediated free-to-bound decay rate (adsorption rate) is then given by\n\nG f = ν G νf , ( 44\n\n)\n\nwhere the summation includes only bound levels. It is clear from Eq. ( 43 ) that, in the continuum limit L → ∞, the rate G νf tends to zero. This is because a free atom can be anywhere in free space and therefore the effect of phonons on a single free atom is negligible. In order to get deeper insight into the free-to-bound transition rate density R νf , we consider a macroscopic atomic ensemble in the thermodynamic limit [14] . Suppose that there are N 0 atoms in a volume with a large length L and a transverse cross section area S 0 . Assume all the atoms are in the same quasicontinuum state |n f and interact with the dielectric independently. The rate for the transitions of the atoms from the quasicontinuum state |n f to an arbitrary bound state |ν , defined as the time derivative of the number of atoms in the state |ν , is D νf = N 0 G νf . In order to get the rate for the continuum state |f , we need to take the thermodynamical limit, where L → ∞ and N 0 → ∞ but N 0 /L remains constant. Then, the rate for the transitions of the atoms from the continuum state |f to an arbitrary bound state |ν is given by\n\nD νf = πhρ 0 S 0 v f R νf = 2πhN f R νf .\n\nHere, ρ 0 = N 0 /LS 0 is the atomic number density and\n\nN f = ρ 0 S 0 v f /2\n\nis the number of atoms incident into the dielectric surface per unit time. It is clear that the transition rate D νf is proportional to the incidence rate N f as well as the transition rate density R νf . We emphasize that D νf is a characteristics for a macroscopic atomic ensemble in the thermodynamic limit while G νf is a measure for a single atom. When the length of the box, L, and the number of atoms, N 0 , are finite, the dynamics of the atoms cannot be described by the free-to-bound rate D νf directly. Instead, we must use the transition rate per atom G νf = D νf /N 0 , which depends on the length L of the box that contains the free atoms [see Eq. (43)].\n\nIn a thermal gas, the atoms have different velocities and, therefore, different energies. For a thermal Maxwell-Boltzmann gas with temperature T 0 , the distribution of the kinetic energy E f of the atomic center-of-mass motion along the x direction is\n\nP (E f ) = 1 √ πk B T 0 e -E f /kB T0 E f . ( 45\n\n)\n\nThe transition rate to an arbitrary bound state |ν is then given by\n\nG νT0 = ∞ 0 G νf P (E f ) dE f , i.e. G νT0 = λ D L ∞ 0 e -E f /kB T0 R νf dE f , ( 46\n\n)\n\nwhere λ D = (2πh 2 /mk B T 0 ) 1/2 the thermal de Broglie wavelength. The phonon-mediated free-to-bound decay rate (adsorption rate) is given by\n\nG T0 = ν G νT0 = ∞ 0 G f P (E f ) dE f . ( 47\n\n)\n\nIn the above equation, the summation over ν includes only bound levels. Note that Eq. ( 46 ) is in qualitative agreement with the results of Refs. [5, 14] .\n\nIt is easy to extend the above results to the case of free-to-free transitions. Indeed, it can be shown that the density of the rate for the transition from a quasicontinuum state |n f , which corresponds to a free state |f , to a different free state |f ′ is given by\n\nQ f ′ f = πh L v f R f ′ f . ( 48\n\n)\n\n7 For convenience, we introduce the notation\n\nQ e f ′ f = Q f ′ f or 0 for E f ′ < E f or E f ′ ≥ E f , respectively, and Q a f ′ f = Q f ′ f or 0 for E f ′ > E f or E f ′ ≤ E f , respectively. Then, we have Q f ′ f = Q e f ′ f , 0, or Q a f ′ f for E f ′ < E f , E f ′ = E f , or E f ′ > E f , respectively.\n\nThe downward (phonon-emission) and upward (phonon-absorption) free-to-free decay rates for the free state |f are given by\n\nQ e f = E f 0 Q e f ′ f dE f ′ ( 49\n\n)\n\nand\n\nQ a f = ∞ E f Q a f ′ f dE f ′ , ( 50\n\n)\n\nrespectively. The total free-to-free decay rate for the free state |f is\n\nQ f = Q e f + Q a f = ∞ 0 Q f ′ f dE f ′ .\n\nFor a thermal gas, we need to replace the transition rate density Q f ′ f and the decay rate\n\nQ f by Q f ′ T0 = ∞ 0 Q f ′ f P (E f ) dE f and Q T0 = ∞ 0 Q f P (E f ) dE f , respec-\n\ntively, which are the averages of Q ′ f and Q f , respectively, with respect to the energy distribution P (E f ) of the initial state. Like in the other cases, we have\n\nQ f ′ T0 = Q e f ′ T0 + Q a f ′ T0 and Q T0 = Q e T0 + Q a T0 , where Q e f ′ T0 = ∞ E f ′ Q e f ′ f P (E f ) dE f , Q a f ′ T0 = E f ′ 0 Q a f ′ f P (E f ) dE f ( 51\n\n)\n\nare the downward and upward transition rate densities and\n\nQ e T0 = ∞ 0 Q e f P (E f ) dE f , Q a T0 = ∞ 0 Q a f P (E f ) dE f ( 52\n\n)\n\nare the downward and upward decay rates. The thermal decay rates Q e T0 and Q a T0 describe the cooling and heating processes, respectively. It can be easily shown that\n\nQ e T0 < Q a T0 , Q e T0 > Q a T0 , and Q e T0 = Q a T0 when T 0 < T , T 0 > T , and T 0 = T , respectively. The relation Q e T0 < Q a T0 (Q e T0 > Q a T0 ), obtained for T 0 < T (T 0 > T ),\n\nindicates the dominance of heating (cooling) of free atoms by the surface."
},
{
"section_type": "OTHER",
"section_title": "C. Relaxation rates and frequency shifts in the framework of the Debye model",
"text": "In order to get insight into the relaxation rates and frequency shifts, we approximate them using the Debye model for phonons. In this model, the phonon frequency ω q is related to the phonon wave number q as ω q = vq, where v is the sound velocity. Furthermore, the summation over the first Brillouin zone is replaced by an integral over a sphere of radius q D = (6π 2 N/V ) 1/3 , where V is the volume of the solid. The Debye frequency and the Debye temperature are given by ω D = vq D and T D = hω D /k B , respectively. For fused silica, we have v = 5.96 km/s, N M/V = 2.2 g/cm 3 , and M = 9.98 × 10 -26 kg [16]. Using these parameters, we find q D = 109.29 × 10 6 cm -1 , ω D = 10.4 THz, and T D = 498 K. In order to perform the summation over phonon states in the framework of the Debye model, we invoke the thermodynamic limit, i.e., replace\n\nq • • • = V 8π 3 |q|≤qD . . . dq = 3N ω 3 D ωD 0 . . . ω 2 q dω q . ( 53\n\n)\n\nThen, for transitions between an upper level l and a lower level k, where 0 < ω lk < ω D , Eqs. (34) and (35) yield\n\nR e kl = 3π M hω 3 D (n lk + 1)ω lk F 2 lk ( 54\n\n) and R a lk = 3π M hω 3 D nlk ω lk F 2 lk . ( 55\n\n)\n\nHere, nlk is given by Eq. ( 20 ) with ω q replaced by ω lk . We emphasize that, according to Eqs. (54) and (55), the phonon-emission rate R e kl and the phonon-absorption rate R a lk depend not only on the matrix element F lk of the force but also on the translational transition frequency ω lk . The frequency dependences of the transition rates are comprised of the frequency dependences of the mean phonon number nlk , the phonon mode density 3N ω 2 lk /ω 3 D , and the matrix element F lk = -U ′ lk = -mω 2 lk x lk of the force. An additional factor comes from the presence of the phonon frequency in Eq. (5) for the surface displacement and, consequently, in the atomphonon interaction Hamiltonian (6). It is clear that an increase in the phonon frequency leads to a decrease in the mean phonon number and an increase in the phonon mode density. The matrix element of the force usually first increases and then decreases with increasing phonon frequency. Due to the existence of several competing factors, the frequency dependences of the transition rates are rather complicated. They usually first increase and then decrease with increasing phonon frequency. We note that, for transitions with ω lk > ω D , we have R e kl = R a lk = 0. We conclude this section by noting that the use of Eq. (53) in Eq. (40) yields the frequency shift\n\n∆ lk = ∆ ( 0\n\n) lk + ∆ (T ) lk , ( 56\n\n) where ∆ ( 0\n\n) lk = 3 2M hω 3 D µ ωD 0 F 2 lµ ω lµ -ω + F 2 µk ω µk + ω ωdω ( 57\n\n)\n\n8 and ∆ (T ) lk = 3 M hω 3 D µ ωD 0 ω lµ F 2 lµ ω 2 lµω 2 + ω µk F 2 µk ω 2 µkω 2 nω ωdω (58) are the zero-and finite-temperature contributions, respectively. In Eq. (58), nω is given by Eq. (20) with ω q replaced by ω."
},
{
"section_type": "RESULTS",
"section_title": "IV. NUMERICAL RESULTS AND DISCUSSIONS",
"text": "In this section, we present the numerical results based on the analytical expressions derived in the previous section for the phonon-mediated relaxation rates of the translational levels of the atom. In particular, we use Eqs. (54) and (55) , obtained in the framework of the Debye model, for our numerical calculations. We consider transitions from bound states as well as free states. The transitions from bound states to other translational levels occur in the case where the atom is initially already adsorbed or trapped near the surface. The transitions from free states to other translational levels occur in the processes of adsorbing, heating, and cooling of free atoms by the surface. Due to the difference in physics of the initial situations, we study the transitions from bound and free states separately.\n\nA. Transitions from bound states FIG. 3: Phonon-emission rates R e ν ′ ν from the vibrational levels (a) ν = 280 and (b) ν = 120 to other levels ν ′ as functions of the lower-level energy E ν ′ . The arrows mark the initial states. The parameters of the solid are M = 9.98 × 10 -26 kg and ωD = 10.4 THz. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 1. FIG. 4: Phonon-absorption rates R a ν ′ ν from the vibrational levels (a) ν = 280 and (b) ν = 120 to other levels ν ′ as functions of the upper-level energy E ν ′ . The left (right) panel in each row corresponds to bound-to-bound (bound-to-free) transitions. The arrows mark the initial states. The parameters used are as in Fig. 3 . The temperature of the phonon bath is T = 300 K.\n\nWe start from a given bound level and calculate the rates of phonon-mediated atomic transitions, both downward and upward. The profiles of the phonon-emission (downward-transition) rate R e ν ′ ν [see Eq. ( 54 )] and the phonon-absorption (upward-transition) rate R a ν ′ ν [see Eq. (55)] are shown in Figs. 3 and 4, respectively. The upper (lower) part of each of these figures corresponds to the case of the initial level ν = 280 (ν = 120), with energy E ν = -156 MHz (E ν = -8.4 THz). The left (right) panel of Fig. 4 corresponds to bound-to-bound (bound-to-free) upward transitions. The temperature of the surface is assumed to be T = 300 K. As seen from Figs. 3 and 4, the transition rates have pronounced localized profiles. Due to the competing effects of the mean phonon number, the phonon mode density, and the matrix element of the force, the transition rates usually first increase and then decrease with increasing phonon frequency. It is clear from a comparison of Figs. 3(a) and 3(b) and also a comparison of Figs. 4(a) and 4(b) that transitions from shallow levels have probabilities orders of magnitude lower than those from deeper levels. The main reason is that the wave functions of the shallow states are spread further away from the surface than those for the deep states. Due to this difference, the effects of the surface vibrations are weaker for the shallow levels than for the deep levels. Another pertinent feature that should be noted from the figure is the following: Since transition frequencies involved are large, they may overshoot the Debye frequency ω D = 10.4 THz, leading to a cutoff on the lower (higher) side of the frequency axis for the emission (absorption) curve.\n\nIn order to see the overall effect of the individual transition rates shown above, we add them up. First we examine the phonon-absorption rates of bound levels. The total phonon-absorption rate Γ a νν of a bound level ν is the sum of the individual absorption rates R a µν over all the upper levels µ, both bound and free [see Eq. (38)]. We plot in Fig. 5 the contributions to Γ a νν from two types of transitions, bound-to-bound and bound-to-free (desorption) transitions. The solid curve of the figure shows that the bound-to-bound phonon-absorption rate is large (above 10 10 s -1 ) for deep and intermediate levels. However, it reduces dramatically with increasing ν in the region of large ν and becomes very small (below 10 -5 s -1 ) for shallow levels. Meanwhile, the dashed curve of Fig. 5 shows that the bound-to-free phonon-absorption rate (i.e., the desorption rate) is zero for deep levels, since the energy required for the transition is greater than the Debye energy [5] . However, the desorption rate is substantial (above 10 5 s -1 ) for intermediate and shallow levels. Thus, the total phonon-absorption rate Γ a νν is mainly determined by the bound-to-bound transitions in the case of deep levels and by the bound-to-free transitions in the case of shallow levels. One of the reasons for the dramatic reduction of the bound-to-bound phonon-absorption rate in the region of shallow levels is that the number of upper bound levels µ becomes small. The second reason is that the frequency of each individual transition becomes small, leading to a decrease of the phonon mode density. The third reason is that the center-of-mass wave functions of shallow levels are spread far away from the surface, leading to a reduction of the effect of phonons on the atom.\n\nUnlike the bound-to-bound phonon-absorption rate, 9 the bound-to-free phonon-absorption rate is substantial in the region of shallow levels. This is because the freestate spectrum is continuous and the range of the boundto-free transition frequency can be large (up to the Debye frequency ω D = 10.4 THz). The gradual reduction of the bound-to-free phonon-absorption rate in the region of shallow levels is mainly due to the reduction of the time that the atom spends in the proximity of the surface.\n\nFIG. 5: Contributions of bound-to-bound (solid curve) and bound-to-free (dashed curve) transitions to the total phononabsorption rate Γ a νν versus the vibrational quantum number ν of the initial level. The parameters used are as in Fig. 3 . The temperature of the phonon bath is T = 300 K.\n\nThe total phonon-emission rate Γ e νν [see Eq. (37)] and the total phonon-absorption rate Γ a νν [see Eq. ( 38 )] are shown in Fig. 6 by the solid and dashed curves, respectively. It is clear from the figure that emission is comparable to but slightly stronger than absorption. Such a dominance is due to the fact that phonon emission moves the atom to a center-of-mass state closer to the surface while phonon absorption changes the atomic state in the opposite direction (see Figs. 1 and 2). Our results for the rates are in good qualitative agreement with the results of Oria et al., albeit with the Morse potential [5] . We stress that we include a large number of vibrational levels as a consequence of the deep silica-cesium potential. Note that the earlier work on this theme involved much fewer levels [5].\n\nFIG. 6: Phonon-emission decay rate Γ e νν (solid lines) and phonon-absorption decay rate Γ a νν (dashed lines) of a bound level as functions of the vibrational quantum number ν. The inset shows the rates in the linear scale to highlight the differences in the dissociation limit. The parameters used are as in Fig. 3 . The temperature of the phonon bath is T = 300 K.\n\nFIG. 7: Same as in Fig. 6 except that T = 30 K.\n\nWe next study the effect of temperature on the decay rates. The results for the phonon-mediated decay rates for T = 30 K are shown in Fig. 7 . In contrast to Fig. 6 , the absorption rate is now much smaller than the corresponding emission rate for both shallow and deep levels. Thus, while it is difficult to distinguish the two log-scale curves for deep and shallow levels at room temperature (see Fig. 6 ), they are well resolved at low temperature."
},
{
"section_type": "OTHER",
"section_title": "B. Transitions from free states",
"text": "We now calculate the rates for transitions from free states to other levels. We first examine free-to-bound transitions, which correspond to the adsorption process. According to Eq. (43), the free-to-bound (more exactly, quasicontinuum-to-bound) transition rate G νf depends not only on the continuum-to-bound transition rate density R νf but also on the length L of the free-atom quantization box. To be specific, we use in our numerical calculations the value L = 1 mm, which is a typical size of atomic clouds in magneto-optical traps [17].\n\nFIG. 8: Free-to-bound transition rates G νf for transitions from the free plane-wave states with energies (a) E f = 2 MHz and (b) E f = 3.1 THz to bound levels ν as functions of the bound-level energy Eν. The arrows mark the energies of the initial free states. The insets show G νf on the log scale versus Eν in the range from -200 MHz to -0.2 MHz to highlight the rates to shallow bound levels. The length of the freeatom quantization box is L = 1 mm. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 3.\n\nWe plot in Fig. 8 the free-to-bound transition rate G νf [see Eq. (43)] as a function of the vibrational quantum number ν. The upper (lower) part of the figure corresponds to the case of the initial-state energy E f = 2 MHz (E f = 3.1 THz), which is close to the average kinetic energy per atom in an ideal gas with temperature T 0 = 200 µK (T 0 = 300 K). We observe that the free-tobound transition rate first increases and then decreases with increasing transition frequency ω f ν = (E f -E ν )/h. Such behavior results from the competing effects of the mean phonon number, the phonon mode density, and the matrix element of the force, like in the case of boundto-bound transitions (see Fig. 3 ). We also see a cutoff of the transition frequency, which is associated with the Debye frequency. Comparison of Figs. 8(a) and 8(b) shows that the transitions from low-energy free states have probabilities orders of magnitude smaller than those from high-energy free states. One of the reasons is that the transition rate G νf is proportional to the velocity v f = (2E f /m) 1/2 [see Eq. (43)]. The dependence of the transition rate density R νf on the transition frequency ω f ν plays an important role. Because of this, the rates for the transitions from low-energy free states to shallow bound levels are very small [see the inset of Fig. 8(a)].\n\nFIG. 9: Free-to-bound decay rate G f as a function of the free-state energy E f . The inset highlights the magnitude and profile of the decay rate for E f in the range from 0 to 20 MHz. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8\n\nWe show in Fig. 9 the free-to-bound decay rate G f [see Eq. (44)], which is a characteristic of the adsorption process, as a function of the free-state energy E f . We see that G f first increases and then decreases with 10 increasing E f . The increase of G f with increasing E f in the region of small E f (see the inset) is mainly due to the increase in the atomic incidence velocity v f . In this region, we have G f ∝ v f ∝ E f [see Eqs. (43) and (44)]. For E f in the range from 0 to 20 MHz, which is typical for atoms in magneto-optical traps, the maximum value of G f is on the order of 10 4 s -1 (see the inset of Fig. 9 ). Such free-to-bound (adsorption) rates are several orders of magnitude smaller than the bound-to-free (desorption) rates (see the dashed curve in Fig. 5 ). The decrease of G f with increasing E f in the region of large E f is mainly due to the reduction of the atom-phonon coupling coefficients.\n\nFIG. 10: Free-to-bound transition rates GνT 0 for transitions from the thermal states with temperatures (a) T0 = 200 µK and (b) T0 = 300 K to bound levels ν as functions of the bound-level energy Eν. The insets show GνT 0 on the log scale versus Eν in the range from -200 MHz to -0.2 MHz to highlight the rates to shallow bound levels. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. FIG. 11: Free-to-bound decay rate GT 0 as a function of the atomic temperature T0 in the ranges (a) from 100 µK to 400 µK and (b) from 50 K to 350 K. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8.\n\nIn a thermal gas, the adsorption process is characterized by the transition rate G νT0 [see Eq. (46)] and the decay rate G T0 [see Eq. ( 47 )], which are the averages of the free-to-bound transition rate G νf and the free-to-bound decay rate G f , respectively, over the freestate energy distribution (45) . We plot the free-to-bound transition rate G νT0 and the free-to-bound decay rate G T0 in Figs. 10 and 11, respectively. Comparison between Figs. 10(a) and 9(a) shows that the transition rates from low-temperature thermal states and low-energy free states look quite similar to each other. The reason is that the spread of the energy distribution is not substantial in the case of low temperatures. The spread of the energy distribution is however substantial in the case of high temperatures, leading to the softening of the cutoff frequency effect [compare Fig. 10(b ) with Fig. 9(b)] . Figure 11 shows that the free-to-bound decay rate G T0 first increases and then reduces with increasing atomic temperature T 0 . For T 0 in the range from 100 µK to 400 µK, which is typical for atoms in magneto-optical traps, the maximum value of G T0 is on the order of 10 4 s -1 [see Fig. 11(a) ]. Such free-to-bound (adsorption) rates are several orders of magnitude smaller than the bound-tofree (desorption) rates (see the dashed curve in Fig. 5 ).\n\nFigure 11( a) shows that, in the region of low atomic temperature T 0 , one has G T0 ∝ √ T 0 , in agreement with the asymptotic behavior of Eqs. (46) and (47) .\n\nFIG. 12: Free-to-free transition rate densities Q f ′ f for the upward (solid lines) and downward (dashed lines) transitions from the free states |f with energies (a) E f = 2 MHz and (b) E f = 3.1 THz to other free states |f ′ as functions of the finallevel energy E f ′ . The arrows mark the energies of the initial free states. The inset in part (a) shows Q f ′ f versus E f ′ in the range from 0 to 4 MHz to highlight the small magnitude of the rate density for downward transitions (dashed line). The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8 .\n\nWe now examine free-to-free transitions, both upward and downward, which corresponding to the heating and cooling processes of free atoms by the surface. We plot in Fig. 12 the free-to-free transition rate density Q f ′ f [see Eq. ( 48 )] as a function of the final-level energy E f ′ . The upper (lower) part of the figure corresponds to the case of the initial-state energy E f = 2 MHz (E f = 3.1 THz), which is close to the average kinetic energy per atom in an ideal gas with temperature T 0 = 200 µK (T 0 = 300 K). The rate densities are shown for the upward (phononabsorption) and downward (phonon-emission) transitions by the solid and dashed lines, respectively. The figure shows that the free-to-free transition rate density increases or decreases with increasing transition frequency if the latter is not too large or is large enough, respectively. We also observe a signature of the Debye cutoff of the phonon frequency. Comparison of Figs. 12(a) and 12(b) shows that transitions from low-energy free states have probabilities orders of magnitude smaller than those from high-energy free states. Figure 12 (a) and its inset show that, when the energy of the free state is low, the free-to-free downward (cooling) transition rate is very small as compared to the free-to-free upward (heating) transition rate.\n\nFIG. 13: Free-to-free upward and downward decay rates Q a f (solid lines) and Q e f (dashed lines) as functions of the energy E f of the initial free state. The insets highlight the magnitudes and profiles of the decay rates for E f in the range from 0 to 20 MHz. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. We show in Fig. 13 the free-to-free upward (phononabsorption) and downward (phonon-emission) decay rates Q a f [see Eq. ( 50 )] and Q e f [see Eq. ( 49 )] as functions of the free-state energy E f . We observe that Q a f and Q e f increase with increasing E f in the range from 0 to 8 THz. The increase of Q a f with increasing E f in the region of small E f (see the left inset) is mainly due to the increase in the atomic incidence velocity v f . In this region, we have Q a f ∝ v f ∝ E f [see Eqs. (48) and (50)]. The increase of Q e f with increasing E f in the region of small E f (see the right inset) is due to not only the increase in the atomic incidence velocity v f [see Eq. (48)] but also 11 the increase of the transition rate density Q e f ′ f and the increase of the integration interval (0, E f ) [see Eq. (49)]. In this region, the dependence of Q e f on the energy E f is of higher order than E 3/2 f . The left inset of Fig. 13 shows that, for E f in the range from 0 to 20 MHz, the maximum value of Q a f is on the order of 10 4 s -1 . Such free-to-free upward (heating) decay rates are comparable to but about two times smaller than the corresponding free-to-bound (adsorption) decay rates (see the inset of Fig. 9 ). Meanwhile, the right inset of Fig. 13 shows that, in the region of small E f , the free-to-free downward (cooling) decay rate Q e f is very small.\n\nFIG. 14: Free-to-free transition rate densities Q a f T 0 for upward transitions (solid lines) and Q e f T 0 for downward transitions (dashed lines) from the thermal states with temperatures (a) T0 = 200 µK and (b) T0 = 300 K to free levels f as functions of the free-level energy E f . The inset in part (a) shows the rate densities versus E f in the range from 0 to 8 MHz to highlight the small magnitude of Q e f T 0 (dashed line). The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8. FIG. 15: Free-to-free decay rates Q a T 0 (solid lines) and Q e T 0 (dashed lines) for upward and downward transitions, respectively, as functions of the atomic temperature T0 in the ranges (a) from 100 µK to 400 µK and (b) from 50 K to 350 K. For comparison, the free-to-bound decay rate GT 0 is replotted from Fig. 11 by the dotted lines. The temperature of the phonon bath is T = 300 K. Other parameters are as in Fig. 8.\n\nIn the case of a thermal gas, the phonon-mediated heat transfer between the gas and the surface is characterized by the free-to-free transition rate densities Q a f T0 and Q e f T0\n\n[see Eqs. (51)] and the free-to-free decay rates Q a T0 and Q e T0 [see Eqs. (52)]. We plot the free-to-free transition rate densities Q a f T0 and Q e f T0 in Fig. 14 . Comparison between Figs. 14(a) and 12(a) shows that the transition rate densities from low-temperature thermal states and low-energy free states are quite similar to each other. The spread of the initial-state energy distribution is not substantial in this case. However, the energy spread of the initial state is substantial in the case of high temperatures, concealing the cutoff frequency effect [compare Fig. 14(b ) with Fig. 12(b)] . We display the freeto-free decay rates Q a T0 and Q e T0 in Fig. 15 . The solid and dashed lines correspond to the upward (heating) and downward (cooling) transitions, respectively. For comparison, the free-to-bound decay rate (adsorption rate) G T0 is re-plotted from Fig. 11 by the dotted lines. We observe that, for T 0 in the range from 100 µK to 400 µK [see Fig. 15(a) ], the adsorption rate G T0 (dotted line) is about two times larger than the heating rate Q a T0 (solid line), while the cooling rate Q e T0 (dashed line) is negligible. Figure 15 (a) shows that, in the region of low atomic temperatures, one has Q T0 ∼ = Q a T0 ∝ √ T 0 , in agreement with the asymptotic behavior of expressions (52) . The figure also shows that Q e T0 quickly increases with increasing atomic temperature T 0 . The relation Q e T0 < Q a T0 , obtained for T 0 < T , indicates the dominance of heating of cold free atoms by the surface. The substantial magnitude of the free-to-bound transition rate G T0 (dotted line) indicates that a significant number of atoms can be adsorbed by the surface. According to Fig. 15(b) , the free-to-free downward transition rate Q e T0 (dashed line) crosses the upward transition rate Q a T0 (solid line) when T 0 = T = 300 K, and then becomes the dominant decay rate. The relation Q e T0 > Q a T0 , obtained for T 0 > T , indicates the dominance of cooling of hot free atoms by the surface."
},
{
"section_type": "CONCLUSION",
"section_title": "V. CONCLUSIONS",
"text": "In conclusion, we have studied the phonon-mediated transitions of an atom in a surface-induced potential. We developed a general formalism, which is applicable for any surface-atom potential. A systematic derivation of the corresponding density-matrix equation enables us to investigate the dynamics of both diagonal and offdiagonal elements. We included a large number of vibrational levels originating from the deep silica-cesium potential. We calculated the transition and decay rates from both bound and free levels. We found that the rates of phonon-mediated transitions between translational levels depend on the mean phonon number, the phonon mode density, and the matrix element of the force from the surface upon the atom. Due to the effects of these competing factors, the transition rates usually first increase and then reduce with increasing transition frequency. We focused on the transitions from bound states. Two specific examples, namely, when the initial level is a shallow level also when it can be one of the deep levels have been worked out. We have shown that there can be marked differences in the absorption and emission behavior in the two cases. For example, both the absorption and emission rates from the deep bound levels can be several orders (in our case, six orders) of magnitude larger than the corresponding rates from the shallow bound levels. We also analyzed various types of transitions from free states. We have shown that, for thermal atomic cesium with temperature in the range from 100 µK to 400 µK in the vicinity of a silica surface with temperature of 300 K, the adsorption (free-to-bound decay) rate is about two times larger than the heating (free-to-free upward decay) rate, while the cooling (free-to-free downward decay) rate is negligible. 12 Acknowledgments We thank M. Chevrollier for fruitful discussions. This work was carried out under the 21st Century COE pro-gram on \"Coherent Optical Science.\" [ * ] Also at Institute of Physics and Electronics, Vietnamese Academy of Science and Technology, Hanoi, Vietnam. [1] V. I. Balykin, K. Hakuta, Fam Le Kien, J. Q. Liang, and M. Morinaga, Phys. Rev. A 70, 011401(R) (2004); Fam Le Kien, V. I. Balykin, and K. Hakuta, Phys. Rev. A 70, 063403 (2004). [2] Fam Le Kien, S. Dutta Gupta, V. I. Balykin, and K.\n\nHakuta, Phys. Rev. A 72, 032509 (2005). [3] Fam Le Kien, S. Dutta Gupta, K. P. Nayak, and K.\n\nHakuta, Phys. Rev. A 72, 063815 (2005). [4] E. G. Lima, M. Chevrollier, O. Di Lorenzo, P. C. Segundo, and M. Oriá, Phys. Rev. A 62, 013410 (2000). [5] T. Passerat de Silans, B. Farias, M. Oriá, and M.\n\nChevrollier, Appl. Phys. B 82, 367 (2006). [6] Fam Le Kien and K. Hakuta, Phys. Rev. A 75, 013423 (2007). [7] Fam Le Kien, S. Dutta Gupta, and K. Hakuta, e-print quant-ph/0610067. [8] K. P. Nayak, P. N. Melentiev, M. Morinaga, Fam Le Kien, V. I. Balykin, and K. Hakuta, e-print quant-ph/0610136. [9] C. Henkel and M. Wilkens, Europhys. Lett. 47, 414 (1999). [10] Z. W. Gortel, H. J. Kreuzer, and R. Teshima, Phys. Rev. B 22, 5655 (1980). [11] H. Hoinkes, Rev. Mod. Phys. 52, 933 (1980). [12] N. N. Bogolubov, Commun. of JINR, E17-11822, Dubna (1978); N. N. Bogolubov and N. N. Bogolubov Jr., Elementary Particles and Nuclei (USSR) 11, 245 (1980). [13] R. Zwanzig, Lectures in Theoretical Physics, eds. W. E. Brittin, B. W. Downs, and J. Downs (Interscience, New York, 1961) Vol. 3, p. 106; G. S. Agarwal, Progress in Optics, ed. E. Wolf (North-Holland, Amsterdam, 1973) Vol. 11, p. 3; L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995) p. 880. [14] J. Javanainen and M. Mackie, Phys. Rev. A 58, R789 (1998); M. Mackie and J. Javanainen, ibid. 60, 3174 (1999). [15] E. Luc-Koenig, M. Vatasescu, and F. Masnou-Seeuws, Eur. Phys. J. D 31, 239 (2004). [16] See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 2001). [17] H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, New York, 1999)."
}
] | ||
arxiv:0704.0353 | 0704.0353 | 1 | 10.1103/PhysRevC.75.047303 | 28deb216dbe5c3af9126240a6e9997c54fd309608bb5238cec5b8dbbafdf9e2a | Spin and pseudospin symmetries and the equivalent spectra of
relativistic spin-1/2 and spin-0 particles | We show that the conditions which originate the spin and pseudospin
symmetries in the Dirac equation are the same that produce equivalent energy
spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector
and scalar potentials. The conclusions do not depend on the particular shapes
of the potentials and can be important in different fields of physics. When
both scalar and vector potentials are spherical, these conditions for
isospectrality imply that the spin-orbit and Darwin terms of either the upper
component or the lower component of the Dirac spinor vanish, making it
equivalent, as far as energy is concerned, to a spin-0 state. In this case,
besides energy, a scalar particle will also have the same orbital angular
momentum as the (conserved) orbital angular momentum of either the upper or
lower component of the corresponding spin-1/2 particle. We point out a few
possible applications of this result. | [
"P. Alberto",
"A. S. de Castro",
"M. Malheiro"
] | [
"nucl-th",
"quant-ph"
] | nucl-th | [] | 2007-04-03 | 2026-02-26 | We show that the conditions which originate the spin and pseudospin symmetries in the Dirac equation are the same that produce equivalent energy spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector and scalar potentials. The conclusions do not depend on the particular shapes of the potentials and can be important in different fields of physics. When both scalar and vector potentials are spherical, these conditions for isospectrality imply that the spinorbit and Darwin terms of either the upper component or the lower component of the Dirac spinor vanish, making it equivalent, as far as energy is concerned, to a spin-0 state. In this case, besides energy, a scalar particle will also have the same orbital angular momentum as the (conserved) orbital angular momentum of either the upper or lower component of the corresponding spin-1/2 particle. We point out a few possible applications of this result.
When describing some strong interacting systems it is often useful, because of simplicity, to approximate the behavior of relativistic spin-1/2 particles by scalar spin-0 particles obeying the Klein-Gordon equation. An example is the case of relativistic quark models used for studying quark-hadron duality because of the added complexity of structure functions of Dirac particles as compared to scalar ones. It turns out that some results (e.g., the onset of scaling in some structure functions) almost do not depend on the spin structure of the particle [1] . In this work we will give another example of an observable, the energy, whose value may not depend on the spinor structure of the particle, i.e., whether one has a spin-1/2 or a spin-0 particle. We will show that when a Dirac particle is subjected to scalar and vector potentials of equal magnitude, it will have exactly the same energy spectrum as a scalar particle of the same mass under the same potentials. As we will see, this happens because the spin-orbit and Darwin terms in the second-order equation for either the upper or lower spinor component vanish when the scalar and vector potentials have equal magnitude. It is not uncommon to find physical systems in which strong interacting relativistic particles are subject to Lorentz scalar potentials (or position-dependent effective masses) that are of the same order of magnitude of potentials which couple to the energy (time components of Lorentz four-vectors). For instance, the scalar and vector (hereafter meaning time-component of a four-vector potential) nuclear mean-field potentials have opposite signs but similar magnitudes, whereas relativistic models of mesons with a heavy and a light quark, like Dor B-mesons, explain the observed small spin-orbit splitting by having vector and scalar potentials with the same sign and similar strengths [2] .
It is well-known that all the components of the free Dirac spinor, i.e., the solution of the free Dirac equation, satisfy the free Klein-Gordon equation. Indeed, from the free Dirac equation
(i γ µ ∂ µ -mc)Ψ = 0 (1)
one gets
(-i γ ν ∂ ν -mc)(i γ µ ∂ µ -mc)Ψ = ( 2 ∂ µ ∂ µ + m 2 c 2 )Ψ = 0 , (2)
where use has been made of the relation
γ µ γ ν ∂ µ ∂ ν = ∂ µ ∂ µ .
In a similar way, for the time-independent free Dirac equation we would have
(c α • p + βmc 2 )ψ = (-i c α • ∇ + βmc 2 )ψ = Eψ , (3)
where, as usual, ψ(r) = Ψ(r, t) exp (i E t/ ), α = γ 0 γ and β = γ 0 . Then, by left multiplying Eq. ( 3 ) by cα• p + βmc 2 , one gets the time-independent free Klein-Gordon equation
(c 2 p 2 + m 2 c 4 )ψ = (-2 c 2 ∇ 2 + m 2 c 2 )ψ = E 2 ψ , (4)
where the relation {β, α} = 0 was used. This all means that the free four-component Dirac spinor, and of course all of its components, satisfy the Klein-Gordon equation. This is not surprising, because, after all, both free spin-1/2 and spin-0 particles obey the same relativistic dispersion relation, E 2 = p 2 c 2 + m 2 c 4 , in spite of having different spinor structures and thus different wave functions. Since there is no spin-dependent interaction, one expects both to have the same energy spectrum.
We consider now the case of a spin-1/2 particle subject to a Lorentz scalar potential V s plus a vector potential V v . The time-independent Dirac equation is given by
[c α • p + β(mc 2 + V s )]ψ = (E -V v )ψ (5)
It is convenient to define the four-spinors
ψ ± = P ± ψ = [(I ± β)/2] ψ such that ψ + = φ 0 ψ -= 0 χ , (6)
where φ and χ are respectively the upper and lower two-component spinors. Using the properties and anticommutation relations of the matrices β and α we can apply the projectors P ± to the Dirac equation ( 5 ) and decompose it into two coupled equations for ψ + and ψ -:
c α • p ψ -+ (mc 2 + V s )ψ + = (E -V v )ψ + (7) c α • p ψ + -(mc 2 + V s )ψ -= (E -V v )ψ -. (8)
Applying the operator c α • p on the left of these equations and using them to write ψ + and ψ -in terms of α • p ψ - and α • p ψ + respectively, we finally get second-order equations for ψ + and ψ -:
c 2 p 2 ψ + + c 2 [α • p ∆]α • p ψ + E -∆ + mc 2 = (E -∆ + mc 2 )(E -Σ -mc 2 )ψ + (9) c 2 p 2 ψ -+ c 2 [α • p Σ]α • p ψ - E -Σ -mc 2 = (E -∆ + mc 2 )(E -Σ -mc 2 )ψ - (10)
where the square brackets [ ] mean that the operator α • p only acts on the potential in front of it and we defined Σ = V v + V s and ∆ = V v -V s . The second term in these equations can be further elaborated noting that the Dirac α i matrices satisfy the relation α i α j = δ ij + 2 iǫ ijk S k where S k , k = 1, 2, 3, are the spin operator components. The second-order equations read now
c 2 p 2 ψ + + c 2 [p ∆] • p ψ + + 2i [p ∆] × p • S ψ + E -∆ + mc 2 = (E -∆ + mc 2 )(E -Σ -mc 2 )ψ + (11) c 2 p 2 ψ -+ c 2 [p Σ] • p ψ -+ 2i [p Σ] × p • S ψ - E -Σ -mc 2 = (E -∆ + mc 2 )(E -Σ -mc 2 )ψ -. (12)
Now, if p ∆ = 0, meaning that ∆ is constant or zero (if ∆ goes to zero at infinity, the two conditions are equivalent), then the second term in eq. ( 11 ) disappears and we have
c 2 p 2 ψ + = (E -∆ + mc 2 )(E -Σ -mc 2 )ψ + = [(E -V v ) 2 -(mc 2 + V s ) 2 ]ψ + , (13)
which is precisely the time-independent Klein-Gordon equation for a scalar potential V s plus a vector potential V v [14] .
Since the second-order equation determines the eigenvalues for the spin-1/2 particle, this means that when p ∆ = 0, a spin-1/2 and a spin-0 particle with the same mass and subject to the same potentials V s and V v will have the same energy spectrum, including both bound and scattering states. This last sufficient condition for isospectrality can be relaxed to demand that just the combination mc 2 + V s be the same for both particles, allowing them to have different masses. This is so because this weaker condition does not change the gradient of ∆ and Σ and therefore the condition p ∆ = 0 will still hold. On the other hand, if the scalar and vector potentials are such that p Σ = 0, we would obtain a Klein-Gordon equation for ψ -, and again the spectrum for spin-0 and spin-1/2 particles would be the same, provided they are subjected to the same vector potential and mc 2 + V s is the same for both particles. If both V s and V v are central potentials, i.e., only depend on the radial coordinate, then the numerators of the second terms in equations ( 11 ) and ( 12 ) read
[p ∆] • p ψ + + 2i [p ∆] × p • S ψ + = -2 ∆ ′ ∂ψ + ∂r + 2 r ∆ ′ L • S ψ + (14) [p Σ] • p ψ -+ 2i [p Σ] × p • S ψ -= -2 Σ ′ ∂ψ - ∂r + 2 r Σ ′ L • S ψ -, (15)
where ∆ ′ and Σ ′ are the derivatives with respect to r of the radial potentials ∆(r) and Σ(r), and L = r × p is the orbital angular momentum operator. From these equations ones sees that these terms, which set apart the Dirac second-order equations for the upper and lower components of the Dirac spinor from the Klein-Gordon equation and thus are the origin of the different spectra for spin-1/2 and spin-0 particles, are composed of a derivative term, related to the Darwin term which appears in the Foldy-Wouthuysen expansion, and a L • S spin-orbit term. If ∆ ′ = 0 (Σ ′ = 0), then there is no spin-orbit term for the upper (lower) component of the Dirac spinor. In turn, since the second-order equation determines the energy eigenvalues, this means that the orbital angular momentum of the respective component is a good quantum number of the Dirac spinor. This can be a bit surprising, since one knows that in general the orbital quantum number is not a good quantum number for a Dirac particle, since L 2 does not commute with a Dirac Hamiltonian with radial potentials. The reason why this does not happen in these cases was reported in Refs. [3, 4] , and we now review it in a slight different fashion. Let us consider in more detail the case of spherical potentials such that ∆ ′ = 0. One knows that a spinor that is a solution of a Dirac equation with spherically symmetric potentials can be generally written as
ψ jm (r) = i g j l (r) r Y j l m (r) f j l(r) r Y j l m (r) . ( 16
)
where Y j l m are the spinor spherical harmonics. These result from the coupling of spherical harmonics and twodimensional Pauli spinors χ ms , Y j l m = ms m l l m l ; 1/2 m s | j m Y l m l χ ms , where l m l ; 1/2 m s | j m is a Clebsch-Gordan coefficient and l = l ± 1, the plus and minus signs being related to whether one has aligned or anti-aligned spin, i.e., j = l ± 1/2. The spinor spherical harmonics for the lower component satisfy the relation
Y j l m = -σ • r Y j l m .
The fact that the upper and lower components have different orbital angular momenta is related to the fact, mentioned before, that L 2 does not commute with the Dirac Hamiltonian
H = c α • p + β(V s + mc 2 ) + V v = c α • p + βmc 2 + ΣP + + ∆P -, (17)
where P ± are the projectors defined above. However, when ∆ ′ = 0, there is an extra SU(2) symmetry of H (so-called "spin symmetry") as first shown by Bell and Ruegg [5] . When we have spherical potentials, Ginocchio showed that there is an additional SU(2) symmetry (for a recent review see [4] ). The generators of this last symmetry are
L = LP + + 1 p 2 α • p L α • p P -= L 0 0 U p L U p , (18)
where U p = σ • p/( p 2 ) is the helicity operator. One can check that L commutes with the Dirac Hamiltonian,
[H, L] = [c α • p, LP + + 1 p 2 α • p L α • p P -] + [∆, 1 p 2 α • p L α • p] + [Σ, L] = [∆, 1 p 2 α • p L α • p ] = 0 , (19)
where the last equality comes from the fact that ∆ ′ = 0. The Casimir L 2 operator is given by L 2 = L 2 P + + 1 p 2 α • p L 2 α • p P -. Applying this operator to the spinor ψ jm (16), we get
L 2 ψ jm = L 2 ψ + jm + 1 p 2 α • p L 2 α • p ψ - jm = 2 l(l + 1)ψ + jm + α • p cL 2 ψ + jm E -∆ + mc 2 = 2 l(l + 1)ψ + jm + 2 l(l + 1)ψ - jm = 2 l(l + 1)ψ jm , (20)
where ψ ± jm = P ± ψ jm and we used the relation, valid when ∆ ′ = 0,
ψ + jm = (E -∆ + mc 2 ) α • p cp 2 ψ - jm .
From (20) we see that ψ jm is indeed an eigenstate of L 2 . Thus the orbital quantum number of the upper component l is a good quantum number of the system when the spherical potentials V s (r) and V v (r) are such that V v (r) = V s (r) + C ∆ , where C ∆ is an arbitrary constant. Also, according to we have said before, there is a state of a spin-0 particle subjected to these same spherical potentials (or, at least, with a scalar potential such that the sum V s + mc 2 is the same) that has the same energy and the same orbital angular momentum as ψ jm . In addition, the wave function of this scalar particle would be proportional to the spatial part of the wave function of the upper component.
Note that the generator of the "spin symmetry" S is given by a similar expression as (18) just replacing L by /2 σ [4, 5], meaning that S 2 ≡ S 2 = 3/4 2 I so that spin is also a good quantum number, as would be expected. Actually, one can show that the total angular momentum operator J can be written as L + S, so that l, m l (eigenvalue of L z ), s = 1/2, m s (eigenvalue of S z ) are good quantum numbers. Then, of course, j and m = m l + m s are also good quantum numbers, but only in a trivial way, because there is no longer spin-orbit coupling. Therefore, in the spinor (16) one could just replace the spinor spherical harmonic Y j l m by Y l m l χ ms and Y j l m by -σ • r Y l m l χ ms . Note that if ∆ is a nonrelativistic potential, ∆ ≪ mc 2 and ∆ ′ ≪ m 2 c 4 /( c), i.e., it is slowly varying over a Compton wavelength. In this case, the spin-orbit term will also get suppressed. In fact, the derivative of the ∆ potential is the origin of the well-known relativistic spin-orbit effect which appears as a relativistic correction term in atomic physics or in the v/c Foldy-Wouthuysen expansion (only the derivative of V v appears because usually no Lorentz scalar potential V s is considered, and therefore ∆ = V v ).
When Σ ′ = 0, or V v (r) = -V s (r) + C Σ , with C Σ an arbitrary constant, there is again a SU (2) symmetry, usually called pseudospin symmetry ( [5, 6] ) which is relevant for describing the single-particle level structure of several nuclei. This symmetry has a dynamical character and cannot be fully realized in nuclei because in Relativistic Mean-field Theories the Σ potential is the only binding potential for nucleons [7, 8] . For harmonic oscillator potentials this is no longer the case, since ∆, acting as an effective mass going to infinity, can bind Dirac particles [9, 10] , even when Σ = 0. As before, in the special case of spherical potentials, there is another SU(2) symmetry whose generators are
L = 1 p 2 α • p L α • p P + + LP -= U p L U p 0 0 L . (21)
In the same way as before, applying L2 to ψ jm , we would find that L2 ψ jm = 2l ( l + 1)ψ jm , that is, this time it is the orbital quantum number of the lower component l which is a good quantum number of the system and can be used to classify energy levels. Again, provided the vector and scalar potentials are adequately related, there would be a corresponding state of a spin-0 particle with the same energy and same orbital angular momentum l, and, furthermore, its wave function would be proportional to the spatial part of the wave function of the lower component. As before, the pseudospin symmetry generator S can be obtained from L by replacing L by /2 σ. The good quantum numbers of the system would be, besides l, m l, s ≡ s = 1/2 and m s. Again, J = L + S. It is interesting that, as has been noted by Ginocchio [9] , the generators of spin and pseudospin symmetries are related through a γ 5 transformation since S = γ 5 Sγ 5 and L = γ 5 Lγ 5 . This property was used in a recent work to relate spin symmetric and pseudospin symmetric spectra of harmonic oscillator potentials [11] . There it was shown that for massless particles (or ultrarelativistic particles) the spin-and pseudo-spin spectra of Dirac particles are the same. In addition, this means that spin-symmetric massless eigenstates of γ 5 would be also pseudo-spin symmetric and vice-versa. Since in this case ∆ = Σ = 0, or V v = V s = 0, this is, of course, just another way of stating the well-known fact that free massless Dirac particles have good chirality.
Naturally, for free spin-1/2 particles described by spherical waves, both l and l are good quantum numbers, which just reflects the fact that one can have free spherical waves with any orbital angular momentum for the upper or lower component and still have the same energy, as long as their linear momentum magnitude is the same, or, put in another way, the energy of a free spin-1/2 particle cannot depend on its direction of motion.
In summary, we showed that when a relativistic spin-1/2 particle is subject to vector and scalar potentials such that V v = ±V s + C ± , where C ± are constants, its energy spectrum does not depend on their spinorial structure, being identical to the spectrum of a spin-0 particle which has no spinorial structure. This amounts to say that if the potentials have these configurations there is no spin-orbit coupling and Darwin term. If the scalar and vector potentials are spherical, one can classify the energy levels according to the orbital angular momentum quantum number of either the upper or the lower component of the Dirac spinor. This would then correspond to having a spin-0 particle with orbital angular momentum l or l, respectively. This spectral identity can of course happen only with potentials which do not involve the spinorial structure of the Dirac equation in an intrinsic way. For instance, a tensor potential of the form iβσ µν (∂ µ A ν -∂ ν A µ ) does not have an analog in the Klein-Gordon equation, so that one could not have a spin-0 particle with the same spectrum as a spin-1/2 particle with such a potential. This is the case of the so-called Dirac oscillator [12] (see [10] for a complete reference list), in which the Dirac equation contains a potential of the form iβσ 0i mωr i = imωβα • r. Another important potential, the electromagnetic vector potential A, which is the spatial part of the electromagnetic four-vector potential, can be added via the minimal coupling scheme to both the Dirac and the Klein-Gordon equations. Since α • (p -eA)α • (p -eA) = (p -eA) 2 + 2e ∇ × A • S, the spectra of spin-0 and spin-1/2 particles cannot be identical as long as there is a magnetic field present, even though the condition V v = ±V s + C ± is fulfilled. It is important also to remark that, since for an electromagnetic interaction V v is the time-component of the electromagnetic four-vector potential, this last condition is gauge invariant in the present case, in which we are dealing with stationary states, i.e, time-independent potentials. So, in the absence of a external magnetic field (allowing, for instance, an electromagnetic vector potential A which is constant or a gradient of a scalar function), a spin-0 and spin-1/2 particle subject to the same electromagnetic potential V v and a Lorentz scalar potential fulfilling the above relation would have the same spectrum.
The remark made above about the similarity of spin-0 and spin-1/2 wave functions can be relevant for calculations in which the observables do not depend on the spin structure of the particle, like some structure functions. One such calculation was made by Paris [13] in a massless confined Dirac particle, in which V v = V s . It would be interesting to see how a Klein-Gordon particle would behave under the same potentials. More generally, this spectral identity can also have experimental implications in different fields of physics, since, should such an identity be found, it would signal the presence of a Lorentz scalar field having a similar magnitude as that of a time-component of a Lorentz vector field, or at least differing just by a constant. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We show that the conditions which originate the spin and pseudospin symmetries in the Dirac equation are the same that produce equivalent energy spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector and scalar potentials. The conclusions do not depend on the particular shapes of the potentials and can be important in different fields of physics. When both scalar and vector potentials are spherical, these conditions for isospectrality imply that the spinorbit and Darwin terms of either the upper component or the lower component of the Dirac spinor vanish, making it equivalent, as far as energy is concerned, to a spin-0 state. In this case, besides energy, a scalar particle will also have the same orbital angular momentum as the (conserved) orbital angular momentum of either the upper or lower component of the corresponding spin-1/2 particle. We point out a few possible applications of this result."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "When describing some strong interacting systems it is often useful, because of simplicity, to approximate the behavior of relativistic spin-1/2 particles by scalar spin-0 particles obeying the Klein-Gordon equation. An example is the case of relativistic quark models used for studying quark-hadron duality because of the added complexity of structure functions of Dirac particles as compared to scalar ones. It turns out that some results (e.g., the onset of scaling in some structure functions) almost do not depend on the spin structure of the particle [1] . In this work we will give another example of an observable, the energy, whose value may not depend on the spinor structure of the particle, i.e., whether one has a spin-1/2 or a spin-0 particle. We will show that when a Dirac particle is subjected to scalar and vector potentials of equal magnitude, it will have exactly the same energy spectrum as a scalar particle of the same mass under the same potentials. As we will see, this happens because the spin-orbit and Darwin terms in the second-order equation for either the upper or lower spinor component vanish when the scalar and vector potentials have equal magnitude. It is not uncommon to find physical systems in which strong interacting relativistic particles are subject to Lorentz scalar potentials (or position-dependent effective masses) that are of the same order of magnitude of potentials which couple to the energy (time components of Lorentz four-vectors). For instance, the scalar and vector (hereafter meaning time-component of a four-vector potential) nuclear mean-field potentials have opposite signs but similar magnitudes, whereas relativistic models of mesons with a heavy and a light quark, like Dor B-mesons, explain the observed small spin-orbit splitting by having vector and scalar potentials with the same sign and similar strengths [2] .\n\nIt is well-known that all the components of the free Dirac spinor, i.e., the solution of the free Dirac equation, satisfy the free Klein-Gordon equation. Indeed, from the free Dirac equation\n\n(i γ µ ∂ µ -mc)Ψ = 0 (1)\n\none gets\n\n(-i γ ν ∂ ν -mc)(i γ µ ∂ µ -mc)Ψ = ( 2 ∂ µ ∂ µ + m 2 c 2 )Ψ = 0 , (2)\n\nwhere use has been made of the relation\n\nγ µ γ ν ∂ µ ∂ ν = ∂ µ ∂ µ .\n\nIn a similar way, for the time-independent free Dirac equation we would have\n\n(c α • p + βmc 2 )ψ = (-i c α • ∇ + βmc 2 )ψ = Eψ , (3)\n\nwhere, as usual, ψ(r) = Ψ(r, t) exp (i E t/ ), α = γ 0 γ and β = γ 0 . Then, by left multiplying Eq. ( 3 ) by cα• p + βmc 2 , one gets the time-independent free Klein-Gordon equation\n\n(c 2 p 2 + m 2 c 4 )ψ = (-2 c 2 ∇ 2 + m 2 c 2 )ψ = E 2 ψ , (4)\n\nwhere the relation {β, α} = 0 was used. This all means that the free four-component Dirac spinor, and of course all of its components, satisfy the Klein-Gordon equation. This is not surprising, because, after all, both free spin-1/2 and spin-0 particles obey the same relativistic dispersion relation, E 2 = p 2 c 2 + m 2 c 4 , in spite of having different spinor structures and thus different wave functions. Since there is no spin-dependent interaction, one expects both to have the same energy spectrum.\n\nWe consider now the case of a spin-1/2 particle subject to a Lorentz scalar potential V s plus a vector potential V v . The time-independent Dirac equation is given by\n\n[c α • p + β(mc 2 + V s )]ψ = (E -V v )ψ (5)\n\nIt is convenient to define the four-spinors\n\nψ ± = P ± ψ = [(I ± β)/2] ψ such that ψ + = φ 0 ψ -= 0 χ , (6)\n\nwhere φ and χ are respectively the upper and lower two-component spinors. Using the properties and anticommutation relations of the matrices β and α we can apply the projectors P ± to the Dirac equation ( 5 ) and decompose it into two coupled equations for ψ + and ψ -:\n\nc α • p ψ -+ (mc 2 + V s )ψ + = (E -V v )ψ + (7) c α • p ψ + -(mc 2 + V s )ψ -= (E -V v )ψ -. (8)\n\nApplying the operator c α • p on the left of these equations and using them to write ψ + and ψ -in terms of α • p ψ - and α • p ψ + respectively, we finally get second-order equations for ψ + and ψ -:\n\nc 2 p 2 ψ + + c 2 [α • p ∆]α • p ψ + E -∆ + mc 2 = (E -∆ + mc 2 )(E -Σ -mc 2 )ψ + (9) c 2 p 2 ψ -+ c 2 [α • p Σ]α • p ψ - E -Σ -mc 2 = (E -∆ + mc 2 )(E -Σ -mc 2 )ψ - (10)\n\nwhere the square brackets [ ] mean that the operator α • p only acts on the potential in front of it and we defined Σ = V v + V s and ∆ = V v -V s . The second term in these equations can be further elaborated noting that the Dirac α i matrices satisfy the relation α i α j = δ ij + 2 iǫ ijk S k where S k , k = 1, 2, 3, are the spin operator components. The second-order equations read now\n\nc 2 p 2 ψ + + c 2 [p ∆] • p ψ + + 2i [p ∆] × p • S ψ + E -∆ + mc 2 = (E -∆ + mc 2 )(E -Σ -mc 2 )ψ + (11) c 2 p 2 ψ -+ c 2 [p Σ] • p ψ -+ 2i [p Σ] × p • S ψ - E -Σ -mc 2 = (E -∆ + mc 2 )(E -Σ -mc 2 )ψ -. (12)\n\nNow, if p ∆ = 0, meaning that ∆ is constant or zero (if ∆ goes to zero at infinity, the two conditions are equivalent), then the second term in eq. ( 11 ) disappears and we have\n\nc 2 p 2 ψ + = (E -∆ + mc 2 )(E -Σ -mc 2 )ψ + = [(E -V v ) 2 -(mc 2 + V s ) 2 ]ψ + , (13)\n\nwhich is precisely the time-independent Klein-Gordon equation for a scalar potential V s plus a vector potential V v [14] .\n\nSince the second-order equation determines the eigenvalues for the spin-1/2 particle, this means that when p ∆ = 0, a spin-1/2 and a spin-0 particle with the same mass and subject to the same potentials V s and V v will have the same energy spectrum, including both bound and scattering states. This last sufficient condition for isospectrality can be relaxed to demand that just the combination mc 2 + V s be the same for both particles, allowing them to have different masses. This is so because this weaker condition does not change the gradient of ∆ and Σ and therefore the condition p ∆ = 0 will still hold. On the other hand, if the scalar and vector potentials are such that p Σ = 0, we would obtain a Klein-Gordon equation for ψ -, and again the spectrum for spin-0 and spin-1/2 particles would be the same, provided they are subjected to the same vector potential and mc 2 + V s is the same for both particles. If both V s and V v are central potentials, i.e., only depend on the radial coordinate, then the numerators of the second terms in equations ( 11 ) and ( 12 ) read\n\n[p ∆] • p ψ + + 2i [p ∆] × p • S ψ + = -2 ∆ ′ ∂ψ + ∂r + 2 r ∆ ′ L • S ψ + (14) [p Σ] • p ψ -+ 2i [p Σ] × p • S ψ -= -2 Σ ′ ∂ψ - ∂r + 2 r Σ ′ L • S ψ -, (15)\n\nwhere ∆ ′ and Σ ′ are the derivatives with respect to r of the radial potentials ∆(r) and Σ(r), and L = r × p is the orbital angular momentum operator. From these equations ones sees that these terms, which set apart the Dirac second-order equations for the upper and lower components of the Dirac spinor from the Klein-Gordon equation and thus are the origin of the different spectra for spin-1/2 and spin-0 particles, are composed of a derivative term, related to the Darwin term which appears in the Foldy-Wouthuysen expansion, and a L • S spin-orbit term. If ∆ ′ = 0 (Σ ′ = 0), then there is no spin-orbit term for the upper (lower) component of the Dirac spinor. In turn, since the second-order equation determines the energy eigenvalues, this means that the orbital angular momentum of the respective component is a good quantum number of the Dirac spinor. This can be a bit surprising, since one knows that in general the orbital quantum number is not a good quantum number for a Dirac particle, since L 2 does not commute with a Dirac Hamiltonian with radial potentials. The reason why this does not happen in these cases was reported in Refs. [3, 4] , and we now review it in a slight different fashion. Let us consider in more detail the case of spherical potentials such that ∆ ′ = 0. One knows that a spinor that is a solution of a Dirac equation with spherically symmetric potentials can be generally written as\n\nψ jm (r) = i g j l (r) r Y j l m (r) f j l(r) r Y j l m (r) . ( 16\n\n)\n\nwhere Y j l m are the spinor spherical harmonics. These result from the coupling of spherical harmonics and twodimensional Pauli spinors χ ms , Y j l m = ms m l l m l ; 1/2 m s | j m Y l m l χ ms , where l m l ; 1/2 m s | j m is a Clebsch-Gordan coefficient and l = l ± 1, the plus and minus signs being related to whether one has aligned or anti-aligned spin, i.e., j = l ± 1/2. The spinor spherical harmonics for the lower component satisfy the relation\n\nY j l m = -σ • r Y j l m .\n\nThe fact that the upper and lower components have different orbital angular momenta is related to the fact, mentioned before, that L 2 does not commute with the Dirac Hamiltonian\n\nH = c α • p + β(V s + mc 2 ) + V v = c α • p + βmc 2 + ΣP + + ∆P -, (17)\n\nwhere P ± are the projectors defined above. However, when ∆ ′ = 0, there is an extra SU(2) symmetry of H (so-called \"spin symmetry\") as first shown by Bell and Ruegg [5] . When we have spherical potentials, Ginocchio showed that there is an additional SU(2) symmetry (for a recent review see [4] ). The generators of this last symmetry are\n\nL = LP + + 1 p 2 α • p L α • p P -= L 0 0 U p L U p , (18)\n\nwhere U p = σ • p/( p 2 ) is the helicity operator. One can check that L commutes with the Dirac Hamiltonian,\n\n[H, L] = [c α • p, LP + + 1 p 2 α • p L α • p P -] + [∆, 1 p 2 α • p L α • p] + [Σ, L] = [∆, 1 p 2 α • p L α • p ] = 0 , (19)\n\nwhere the last equality comes from the fact that ∆ ′ = 0. The Casimir L 2 operator is given by L 2 = L 2 P + + 1 p 2 α • p L 2 α • p P -. Applying this operator to the spinor ψ jm (16), we get\n\nL 2 ψ jm = L 2 ψ + jm + 1 p 2 α • p L 2 α • p ψ - jm = 2 l(l + 1)ψ + jm + α • p cL 2 ψ + jm E -∆ + mc 2 = 2 l(l + 1)ψ + jm + 2 l(l + 1)ψ - jm = 2 l(l + 1)ψ jm , (20)\n\nwhere ψ ± jm = P ± ψ jm and we used the relation, valid when ∆ ′ = 0,\n\nψ + jm = (E -∆ + mc 2 ) α • p cp 2 ψ - jm .\n\nFrom (20) we see that ψ jm is indeed an eigenstate of L 2 . Thus the orbital quantum number of the upper component l is a good quantum number of the system when the spherical potentials V s (r) and V v (r) are such that V v (r) = V s (r) + C ∆ , where C ∆ is an arbitrary constant. Also, according to we have said before, there is a state of a spin-0 particle subjected to these same spherical potentials (or, at least, with a scalar potential such that the sum V s + mc 2 is the same) that has the same energy and the same orbital angular momentum as ψ jm . In addition, the wave function of this scalar particle would be proportional to the spatial part of the wave function of the upper component.\n\nNote that the generator of the \"spin symmetry\" S is given by a similar expression as (18) just replacing L by /2 σ [4, 5], meaning that S 2 ≡ S 2 = 3/4 2 I so that spin is also a good quantum number, as would be expected. Actually, one can show that the total angular momentum operator J can be written as L + S, so that l, m l (eigenvalue of L z ), s = 1/2, m s (eigenvalue of S z ) are good quantum numbers. Then, of course, j and m = m l + m s are also good quantum numbers, but only in a trivial way, because there is no longer spin-orbit coupling. Therefore, in the spinor (16) one could just replace the spinor spherical harmonic Y j l m by Y l m l χ ms and Y j l m by -σ • r Y l m l χ ms . Note that if ∆ is a nonrelativistic potential, ∆ ≪ mc 2 and ∆ ′ ≪ m 2 c 4 /( c), i.e., it is slowly varying over a Compton wavelength. In this case, the spin-orbit term will also get suppressed. In fact, the derivative of the ∆ potential is the origin of the well-known relativistic spin-orbit effect which appears as a relativistic correction term in atomic physics or in the v/c Foldy-Wouthuysen expansion (only the derivative of V v appears because usually no Lorentz scalar potential V s is considered, and therefore ∆ = V v ).\n\nWhen Σ ′ = 0, or V v (r) = -V s (r) + C Σ , with C Σ an arbitrary constant, there is again a SU (2) symmetry, usually called pseudospin symmetry ( [5, 6] ) which is relevant for describing the single-particle level structure of several nuclei. This symmetry has a dynamical character and cannot be fully realized in nuclei because in Relativistic Mean-field Theories the Σ potential is the only binding potential for nucleons [7, 8] . For harmonic oscillator potentials this is no longer the case, since ∆, acting as an effective mass going to infinity, can bind Dirac particles [9, 10] , even when Σ = 0. As before, in the special case of spherical potentials, there is another SU(2) symmetry whose generators are\n\nL = 1 p 2 α • p L α • p P + + LP -= U p L U p 0 0 L . (21)\n\nIn the same way as before, applying L2 to ψ jm , we would find that L2 ψ jm = 2l ( l + 1)ψ jm , that is, this time it is the orbital quantum number of the lower component l which is a good quantum number of the system and can be used to classify energy levels. Again, provided the vector and scalar potentials are adequately related, there would be a corresponding state of a spin-0 particle with the same energy and same orbital angular momentum l, and, furthermore, its wave function would be proportional to the spatial part of the wave function of the lower component. As before, the pseudospin symmetry generator S can be obtained from L by replacing L by /2 σ. The good quantum numbers of the system would be, besides l, m l, s ≡ s = 1/2 and m s. Again, J = L + S. It is interesting that, as has been noted by Ginocchio [9] , the generators of spin and pseudospin symmetries are related through a γ 5 transformation since S = γ 5 Sγ 5 and L = γ 5 Lγ 5 . This property was used in a recent work to relate spin symmetric and pseudospin symmetric spectra of harmonic oscillator potentials [11] . There it was shown that for massless particles (or ultrarelativistic particles) the spin-and pseudo-spin spectra of Dirac particles are the same. In addition, this means that spin-symmetric massless eigenstates of γ 5 would be also pseudo-spin symmetric and vice-versa. Since in this case ∆ = Σ = 0, or V v = V s = 0, this is, of course, just another way of stating the well-known fact that free massless Dirac particles have good chirality.\n\nNaturally, for free spin-1/2 particles described by spherical waves, both l and l are good quantum numbers, which just reflects the fact that one can have free spherical waves with any orbital angular momentum for the upper or lower component and still have the same energy, as long as their linear momentum magnitude is the same, or, put in another way, the energy of a free spin-1/2 particle cannot depend on its direction of motion.\n\nIn summary, we showed that when a relativistic spin-1/2 particle is subject to vector and scalar potentials such that V v = ±V s + C ± , where C ± are constants, its energy spectrum does not depend on their spinorial structure, being identical to the spectrum of a spin-0 particle which has no spinorial structure. This amounts to say that if the potentials have these configurations there is no spin-orbit coupling and Darwin term. If the scalar and vector potentials are spherical, one can classify the energy levels according to the orbital angular momentum quantum number of either the upper or the lower component of the Dirac spinor. This would then correspond to having a spin-0 particle with orbital angular momentum l or l, respectively. This spectral identity can of course happen only with potentials which do not involve the spinorial structure of the Dirac equation in an intrinsic way. For instance, a tensor potential of the form iβσ µν (∂ µ A ν -∂ ν A µ ) does not have an analog in the Klein-Gordon equation, so that one could not have a spin-0 particle with the same spectrum as a spin-1/2 particle with such a potential. This is the case of the so-called Dirac oscillator [12] (see [10] for a complete reference list), in which the Dirac equation contains a potential of the form iβσ 0i mωr i = imωβα • r. Another important potential, the electromagnetic vector potential A, which is the spatial part of the electromagnetic four-vector potential, can be added via the minimal coupling scheme to both the Dirac and the Klein-Gordon equations. Since α • (p -eA)α • (p -eA) = (p -eA) 2 + 2e ∇ × A • S, the spectra of spin-0 and spin-1/2 particles cannot be identical as long as there is a magnetic field present, even though the condition V v = ±V s + C ± is fulfilled. It is important also to remark that, since for an electromagnetic interaction V v is the time-component of the electromagnetic four-vector potential, this last condition is gauge invariant in the present case, in which we are dealing with stationary states, i.e, time-independent potentials. So, in the absence of a external magnetic field (allowing, for instance, an electromagnetic vector potential A which is constant or a gradient of a scalar function), a spin-0 and spin-1/2 particle subject to the same electromagnetic potential V v and a Lorentz scalar potential fulfilling the above relation would have the same spectrum.\n\nThe remark made above about the similarity of spin-0 and spin-1/2 wave functions can be relevant for calculations in which the observables do not depend on the spin structure of the particle, like some structure functions. One such calculation was made by Paris [13] in a massless confined Dirac particle, in which V v = V s . It would be interesting to see how a Klein-Gordon particle would behave under the same potentials. More generally, this spectral identity can also have experimental implications in different fields of physics, since, should such an identity be found, it would signal the presence of a Lorentz scalar field having a similar magnitude as that of a time-component of a Lorentz vector field, or at least differing just by a constant."
}
] | |
arxiv:0704.0362 | 0704.0362 | 1 | 0d2f7b5e699452837dc706c96b5b46bce67719bb41f8c12fe46dfcc9004589f4 | The Arctic Circle Revisited | The problem of limit shapes in the six-vertex model with domain wall boundary
conditions is addressed by considering a specially tailored bulk correlation
function, the emptiness formation probability. A closed expression of this
correlation function is given, both in terms of certain determinant and
multiple integral, which allows for a systematic treatment of the limit shapes
of the model for full range of values of vertex weights. Specifically, we show
that for vertex weights corresponding to the free-fermion line on the phase
diagram, the emptiness formation probability is related to a one-matrix model
with a triple logarithmic singularity, or Triple Penner model. The saddle-point
analysis of this model leads to the Arctic Circle Theorem, and its
generalization to the Arctic Ellipses, known previously from domino tilings. | [
"F. Colomo and A.G. Pronko"
] | [
"math-ph",
"hep-th",
"math.MP"
] | math-ph | [] | 2007-04-03 | 2026-02-26 | The Arctic Circle has first appeared in the study of domino tilings of large Aztec diamonds [EKLP, JPS]. The name originates from the fact that in most configurations the dominoes are 'frozen' outside the circle inscribed into the diamond, while the interior of the circle is a disordered, or 'temperate', zone. Further investigations of the domino tilings of Aztec diamonds, such as details of statistics near the circle, can be found in [CEP, J1, J2]. Here we mention that the Arctic Circle is a particular example of a limit shape in dimer models, in the sense that it describes the shape of a spatial phase separation of order and disorder. Apart from domino tilings, many more examples have been discussed recently, see, among others, papers [CKP, CLP, KO, KOS, OR].
As long as only dimer models are considered, this amounts to restrict to discrete free-fermionic models, although with nontrivial boundary conditions. Indeed, many of them can be viewed as a six-vertex model at its Free Fermion point (the correspondence being however usually not bijective), with suitably chosen fixed boundary conditions. In particular, this is the case of domino tilings of Aztec diamonds [EKLP], and the corresponding boundary conditions of the six-vertex model are the so-called Domain Wall Boundary Conditions (DWBC). Hence the problem of limit shapes extends to the six-vertex model with generic weights, and with fixed boundary conditions, among which the case of DWBC is the most interesting.
Historically, the six-vertex model with DWBC was first considered in paper [K] within the framework of Quantum Inverse Scattering Method [KBI] to prove the Gaudin hypothesis for norms of Bethe states. The model was subsequently solved 2000 Mathematics Subject Classification. 15A52, 82B05, 82B20, 82B23. 1 2 F. COLOMO AND A.G. PRONKO in paper [I] where a determinant formula for the partition function was given; see also [ICK] for a detailed exposition. Quite independently, the model was later found, under certain restrictions on the vertex weights, to be deeply related with enumerations of alternating sign matrices (see, e.g., [Br] for a review) and, as already mentioned, to domino tilings of Aztec diamonds [EKLP].
Concerning the problem of limit shapes for the six-vertex model with DWBC, as far as the Free Fermion point is considered, the relation with domino tilings provided apparently an indirect proof of the corresponding Arctic Circle. The nonbijective nature of the correspondence between the two models asked for more direct results, purposely for the free-fermion six-vertex model, see [Zi1, FS, KP]. Out of the Free Fermion point, however, only very few analytical results are available, such as exact expressions for boundary one-point [BPZ] and two-point [FP, CP1] correlation functions. The present knowledge on the subject is based mainly on numerics [E, SZ, AR]; some steps towards finding the limit shapes of the model have been done recently in [PR] .
In the present note we propose a rather direct strategy to address the problem: after briefly reviewing the six-vertex model with DWBC, we define a bulk correlation function, the Emptiness Formation Probability (EFP), which discriminates the ordered and disordered phase regions. We give for this correlation function two equivalent representations, in terms of a determinant and of a multiple integral. The core derivation of EFP is heavily based on the Quantum Inverse Scattering Method [KBI], along the lines of papers [BPZ, CP1]; it is out of the scope of the present paper, corresponding details being given in a separate publication [CP4]. Here our aim is to demonstrate how the limit shapes for the considered model can be extracted from EFP in a suitable scaling limit, by making use of ideas and techniques of Random Matrix Models.
To be more specific, and to establish a contact with previous results, we specialize here our further discussion to the case of free-fermion six-vertex model. We show that the asymptotic analysis of multiple integral formula for EFP in the scaling limit reduces to a saddle-point problem for a one-matrix model with a triple logarithmic singularity, or triple Penner model. We argue that the limit shape corresponds to condensation of all saddle-point solutions to a single point. This allows us to recover the known Arctic Circle and Ellipses.
As a comment to our approach, it is to be stressed that it is directly tailored on the six-vertex model, rather than domino tilings. For this reason it is not restricted to the free-fermion models, even if, of course, further significant efforts might be necessary, essentially from the point of view of Random Matrix Model reformulation, for application to more general situations. On the basis of our previous results in [CP2], however, the application of the method to the particular case of the so-called Ice Point of the model should be straightforward. This would provide the limit shape of alternating sign matrices.
2. The model 2.1. The six-vertex model. The six-vertex model (for reviews, see [LW, Ba] ) is formulated on a square lattice with arrows lying on edges, and obeying the so-called 'ice-rule', namely, the only admitted configurations are such that there are always two arrows pointing away from, and two arrows pointing into, each lattice vertex. An equivalent and graphically simpler description of the configurations of THE ARCTIC CIRCLE REVISITED 3 w 1 w 2 w 3 w 4 w 5 w 6 Figure 1. The six allowed types of vertices in terms of arrows and lines, and their Boltzmann weights.
Figure 2. A possible configuration of the six-vertex model with DWBC at N = 4, in terms of arrows and lines.
the model can be given in terms of lines flowing through the vertices: for each arrow pointing downward or to the left, draw a thick line on the corresponding edge. This line picture implements the 'ice-rule' in an automated way. The six possible vertex states and the Boltzmann weights w 1 , w 2 , . . . , w 6 assigned to each vertex according to its state are shown in Figure 1.
Conditions. The Domain Wall Boundary Conditions (DWBC) are imposed on the N ×N square lattice by fixing the direction of all arrows on the boundaries in a specific way. Namely, the vertical arrows on the top and bottom of the lattice point inward, while the horizontal arrows on the left and right sides point outward. Equivalently, a generic configuration of the model with DWBC can be depicted by N lines flowing from the upper boundary to the left one. A possible state of the model both in terms of arrows and of lines is shown in Figure 2.
The partition function is defined, as usual, as a sum over all possible arrow configurations, compatible with the imposed DWBC, each configuration being assigned its Boltzmann weight, given as the product of all the corresponding vertex weights, Z N = arrow configurations with DWBC w n1 1 w n2 2 . . . w n6 6 .
Here n 1 , n 2 , . . . , n 6 denote the numbers of vertices with weights w 1 , w 2 , . . . , w 6 , respectively, in each arrow configuration (n
1 + n 2 + • • • + n 6 = N 2 ).
2.4. Anisotropy parameter and phases of the model. The six-vertex model with DWBC can be considered, with no loss of generality, with its weights invariant under the simultaneous reversal of all arrows, w 1 = w 2 =: a , w 3 = w 4 =: b , w 5 = w 6 =: c . 4 F. COLOMO AND A.G. PRONKO
Under different choices of Boltzmann weights the six-vertex model exhibits different behaviours, according to the value of the parameter ∆, defined as ∆ = a 2 + b 2 -c 2 2ab .
It is well known that there are three physical regions or phases for the six-vertex model: the ferroelectric phase, ∆ > 1; the anti-ferroelectric phase, ∆ < -1; the disordered phase, -1 < ∆ < 1. Here we restrict ourselves to the disordered phase, where the Boltzmann weights are conveniently parameterized as a = sin(λ + η) , b = sin(λ -η) , c = sin 2η . (2.1) With this choice one has ∆ = cos 2η. The parameter λ is the so-called spectral parameter and η is the crossing parameter. The physical requirement of positive Boltzmann weights, in the disordered regime, restricts the values of the crossing and spectral parameters to 0 < η < π/2 and η < λ < π -η.
The special case η = π/4 (or ∆ = 0) is related to free fermions on a lattice, and there is a well-known correspondence with dimers and domino tilings. In particular, at λ = π/2, the ∆ = 0 six-vertex model with DWBC is related to the domino tilings of Aztec diamond. For arbitrary λ ∈ [π/4, 3π/4], we shall refer to the ∆ = 0 case as the Free Fermion line.
The case η = π/6 (i.e. ∆ = 1/2) and λ = π/2, where all weights are equal, a = b = c, is known as the Ice Point; all configurations are given the same weight. In this case there is a one to one correspondence between configurations of the model with DWBC and N × N alternating sign matrices.
2.5. Phase separation and limit shapes. The six-vertex model exhibits spatial separation of phases for a wide choice of fixed boundary conditions, and, in particular, in the case of DWBC. Roughly speaking, the effect is related to the fact that ordered configurations on the boundary can induce, through the ice-rule, a macroscopic order inside the lattice.
The notion of phase separation acquires a precise meaning in the scaling limit, that is the thermodynamic/continuum limit, performed by sending the number of sites N to infinity and the lattice spacing to zero, while keeping the total size of the lattice fixed, e.g., to 1. On a finite lattice, several macroscopic regions may appear, which in the scaling limit are expected to be sharply separated by some curves, the so-called Arctic curves.
For the six-vertex model with DWBC the shape of the Artic curve, or limit shape, has been found rigorously only on the Free Fermion line, and for the closely related domino tilings of Aztec diamond [JPS, CEP, Zi1, FS, KP]. For generic values of weights the limit shapes are not known, but the whole picture is strongly supported both numerically [E, SZ, AR] and analytically [KZ, Zi2, BF, PR].
3. Emptiness Formation Probability 3.1. Definition. We shall use the following coordinates on the lattice: r = 1, . . . , N labels the vertical lines from right to left; s = 1, . . . , N labels the horizontal lines from top to bottom. We may now introduce the correlation function F N (r, s), measuring the probability for the first s horizontal edges between the r-th and r + 1-th line to be all 'full' (i.e. thick in the line picture, or with a left arrow in the
THE ARCTIC CIRCLE REVISITED 5 r 1 s 1 Figure 3. Emptiness Formation Probability. The sum in (3.1) is performed over all configurations compatible with the drawn arrows.
standard picture of the six-vertex model):
F N (r, s) = 1 Z N 'constrained' arrow configurations with DWBC w n1 1 w n2 2 . . . w n6 6 . (3.1)
Here the sum is performed over all arrow configurations on the N × N lattice, subjected to the restriction of DWBC, and to the condition that all arrows on the first s edges between the r-th and r + 1-th line should point left, see Figure 3 . Although this correlation function may appear rather sophisticated, it is computable in some closed form by means of the Quantum Inverse Scattering Method, on which DWBC are indeed tailored. It is the natural adaptation of the Emptiness Formation Probability of quantum spin chains to the present model. For this reason, and to link to the common practice in the quantum integrable models community, even if F N (r, s) actually describes 'fullness' formation probability, we shall call it Emptiness Formation Probability (EFP). r, s) . Let us restrict ourselves to the disordered regime, -1 < ∆ < 1, for definiteness. From previous analytical and numerical work, in the large N limit the emergence of a limit shape, in the form of a continuous closed curve touching once each of the four sides of the lattice, is expected. It follows that five regions emerge in the lattice: a central region, enclosed by the curve, and four corner regions, lying outside the closed curve and delimited by the sides of the lattice. The central region is disordered, while the four corners are frozen, with mainly vertices of type 1, 3, 2, 4 (see Figure 1 ) appearing in the top-left, top-right, bottom-right and bottom-left corner, respectively.
F N (
By construction, EFP is expected to be almost one in frozen regions of type 1, or 3, bordering the top side of the lattice, and to be rather small otherwise. DWBC exclude a region of type 3 to emerge in the upper part of the lattice. Hence F N (r, s) describes, at a given value of r, as s increases, a transition from a frozen region of vertices of type 1, where F N (r, s) ∼ 1, to a generic region where F N (r, s) ∼ 0. It follows that F N (r, s) can describe only the upper left portion of the closed curve, between its top and left contact points. Nevertheless, it should be mentioned that the full curve can be built from the knowledge of its top left portion, just 6 F. COLOMO AND A.G. PRONKO
exploiting the crossing symmetry of the six-vertex model. Hence EFP, F N (r, s), is well suited to describe limit shapes.
3.3. Some notations. For a given choice of parameters λ, η we define ϕ := c ab = sin 2η sin(λ + η) sin(λ -η) ,
and the integration measure on the real line µ(x) := e x(λ-π/2) sinh(ηx) sinh(πx/2) , related to ϕ as follows:
ϕ = ∞ -∞ µ(x) dx .
Let us introduce the complete set of monic orthogonal polynomial {P n (x)} n=0,1,... associated to the integration measure µ(x), with the orthogonality relation
∞ -∞ P n (x)P m (x)µ(x) dx = h n δ nm .
The square norms h n are completely determined by the measure µ(x), and may be expressed, in principle, in terms of its moments. In the following we shall be interested in the complete set of orthogonal polynomials {K n (x)} n=0,1,... defined as
K n (x) = n! ϕ n+1 1 h n P n (x) .
We moreover define ω(ǫ) := a b sin(ǫ) sin(ǫ -2η) , ω(ǫ) := b a sin(ǫ) sin(ǫ + 2η) .
Note that the following relation holds
a 2 ω -2∆ab ωω + b 2 ω = 0 , (3.2)
allowing to express ω in terms of ω.
For EFP in the six-vertex model with DWBC, the following representation holds:
F N (r, s) = (-1) s det 1≤j,k≤s K N -k (∂ ǫj ) s j=1 [ω(ǫ j )] N -r [ω(ǫ j ) -1] N × 1≤j<k≤s [ω(ǫ j ) -1] [ω(ǫ k ) -1] ω(ǫ j )ω(ǫ k ) -1 ǫ1=0,...,ǫs=0
. (3.3) This representation has been obtained in the framework of the Quantum Inverse Scattering Method [KBI], along the lines of analogous derivations worked out for one-point and two-point boundary correlation functions of the model [BPZ, CP1].
The details of the derivation can be found in [CP4].
THE ARCTIC CIRCLE REVISITED 7 3.5. The boundary correlation function. If we consider expression (3.3) when s = 1, we recover the boundary polarization, introduced and computed in [BPZ]. It is convenient to consider the closely related boundary correlation function
H N (r) := F N (r, 1) -F N (r -1, 1) .
As shown in [BPZ, CP1], the following representation holds:
H N (r) = K N -1 (∂ ǫ ) [ω(ǫ)] N -r [ω(ǫ) -1] N -1 ǫ=0 .
We define the corresponding generating function
h N (z) := N r=1 H N (r) z r-1 .
(3.4) Noticing that ω(ǫ) → 0 as ǫ → 0, it can be shown that, given any arbitrary function f (z) regular in a neighbourhood of the origin, the following inverse representation holds
K N -1 (∂ ǫ )f (ω(ǫ)) ǫ=0 = 1 2πi C0 (z -1) N -1 z N h N (z)f (z) dz . (3.5)
Here C 0 is a closed counterclockwise contour in the complex plane, enclosing the origin, and no other singularity of the integrand.
3.6. Multiple integral representation. Plugging (3.5) into representation (3.3), we readily obtain the following multiple integral representation for EFP:
F N (r, s) = - 1 2πi s C0 • • • C0 d s ω det 1≤j,k≤s h N -k+1 (ω j ) ω j -1 ω j N -k × s j=1 ω N -r-1 j (ω j -1) N 1≤j<k≤s (ω j -1)(ω k -1) ωj ω k -1 . (3.6)
Here ωj 's should be expressed in terms of ω j 's through (3.2). Indeed, due to (3.5), relation (3.2) for functions ω(ǫ), ω(ǫ), translates directly into the same relation between ω j and ωj , j = 1, . . . , s. Representation (3.6), and all results in this Section hold for any choice of parameters λ and η within the disordered regime. Moreover, by analytical continuation in parameters λ and η, these results can be easily extended to all other regimes.
The determinant in expression (3.6) is a particular representation of the partition function of the six-vertex model with DWBC, when the homogeneous limit is performed only on a subset of the spectral parameters [CP3]. The structure of the previous multiple integral representation therefore closely recalls analogous ones for the Heisenberg XXZ quantum spin chain correlation functions [JM, KMT].
For generic values of λ and η, the orthogonal polynomials K n (x), or the generating function h N (z), are known only in terms of rather implicit representations. Fortunately, there are three notable exceptions [CP2]: the Free Fermion line (η = π/4, -π/4 < λ < π/4, ∆ = 0), the Ice Point (η = π/6, λ = π/2, ∆ = 1/2), and the Dual Ice Point (η = π/3, λ = π/2, ∆ = -1/2). In these three cases, the K n (x) turn out to be classical orthogonal polynomials, namely Meixner-Pollaczek, Continuous Hahn and Continuous Dual Hahn polynomial, respectively. Correspondingly, the 8 F. COLOMO AND A.G. PRONKO
generating function can be represented explicitly in terms of terminating hypergeometric functions that may simplify considerably further evaluation of EFP. In the next Section we shall focus on the case of Free Fermion line.
4. Multiple integral representation at ∆ = 0 4.1. Specialization to η = π/4. We shall now restrict ourselves to the case η = π/4. We have ∆ = 0, and the six-vertex model reduces to a model of free fermions on the lattice. The parameter λ can still assume any value in the interval (-π/4, π/4). It is convenient to trade λ for the new parameter
τ = tan 2 (λ -π/4) , 0 < τ < ∞ .
The symmetric point (related to the domino tiling of Aztec Diamond) corresponds now to τ = 1. For generic values of τ we have: ω = -τ ω .
The generating function (3.4) is known explicitely (see [CP2] for details):
h N (z) = 1 + τ z 1 + τ N -1 .
Plugging this expression into (3.6), we get
F N (r, s) = - 1 2πi s C0 • • • C0 d s ω det 1≤j,k≤s (1 + τ ω j )(ω j -1) (1 + τ )ω j N -k × s j=1 ω N -r-1 j (ω j -1) N 1≤j<k≤s (1 + τ ω j )(ω k -1) 1 + τ ω j ω k . (4.1)
4.2. Symmetrization. After extracting a common factor s j=1
(1 + τ ω j )(ω j -1) (1 + τ )ω j N -s
from the determinant in (4.1), we recognize it to be of Vandermonde type. We can therefore collect from the integrand of (4.1) the double product
1≤j<k≤s (1 + τ ω j )(ω j -1) (1 + τ )ω j - (1 + τ ω k )(ω k -1) (1 + τ )ω k (1 + τ ω j )(ω k -1) 1 + τ ω j ω k .
Noticing that the integration and the remaining of integrand are fully symmetric under permutation of variables ω 1 , . . . , ω j , we can perform total symmetrization of the previous double product over all its variables, with the result
1 s! (-1) s(s-1)/2 s j=1 1 ω s-1 j 1≤j<k≤s (ω j -ω k ) 2 .
THE ARCTIC CIRCLE REVISITED 9 Hence, we finally obtain the following representation for EFP on the Free Fermion line:
F N (r, s) = (-1) s(s+1)/2 s!(1 + τ ) s(N -s) (2πi) s × C0 • • • C0 d s ω 1≤j<k≤s (ω j -ω k ) 2 s j=1 (1 + τ ω j ) N -s (ω j -1) s ω r j . (4.2)
The appearance of a squared Vandermonde determinant in this expression naturally recalls the partition functions of s × s Random Matrix Models.
Triple Penner model and Arctic Ellipses 5.1. Scaling limit. We shall now address the asymptotic behaviour of expression (4.2) for EFP in the ∆ = 0 case. We are interested in the limit N, r, s → ∞, while keeping the ratios r/N = x , s/N = y , fixed. In this limit, x, y ∈ [0, 1] will parameterize the unit square to which the lattice is rescaled. Correspondingly EFP is expected to approach a limit function
F (x, y) := lim N →∞ F N (xN, yN ) , x, y ∈ [0, 1] .
We shall exploit the standard approach developed for instance in the investigation of asymptotic behaviour for Random Matrix Models. Before this let us however point out some facts which holds already for any finite value of s. 4.2 ) only in the integration contours. Here C 1 is a closed, clockwise oriented contour (note the change in orientation) in the complex plane enclosing point ω = 1, and no other singularity of the integrand. We have the identity I N (r, s) = 1 (5.1) for any integer r, s = 1, . . . , N . The simplest way to prove the previous identity is by shifting ω j → ω j + 1, and rewriting I N (r, s) as an Hankel determinant; indeed we have
I N (r, s) := (-1) s(s+1)/2 s!(1 + τ ) s(N -s) (2πi) s × C1 • • • C1 d s ω 1≤j<k≤s (ω j -ω k ) 2 s j=1 (1 + τ ω j ) N -s (ω j -1) s ω r j , which differs from (
I N (r, s) = (-1) s(s-1)/2 (1 + τ ) s(N -s) det 1≤j,k≤s 1 2πi C0 ω j+k-2-s (1 + τ + τ ω) N -s (1 + ω) r dω .
The entries of the Hankel matrix vanish for j +k > s+1, and hence the determinant is simply given by the product of the antidiagonal entries, j + k = s + 1 (modulo a sign (-1) s(s-1)/2 emerging from the permutation of all columns). Identity (5.1) follows immediately.
10 F. COLOMO AND A.G. PRONKO 5.3. Saddle-point evaluation for large N and finite s. When using the saddle-point method in variables ω 1 , . . . , ω s to evaluate the behaviour of F N (r, s) for large N and r, and finite s, it is rather easy to see that the saddle-point equations decouple at leading order, and that each saddle-point will be on the real axis, contributing with a factor e -N Sj with S j positive.
If a given saddle-point is smaller than 1, the contour C 0 can be deformed through the saddle-point without encountering any pole, and its contribution will vanish as e -N Sj in the large N limit. If however the saddle-point, still on the real axis, happens to be larger than 1, the deformation of the contour C 0 through the saddle-point will pick up the contribution of the pole at ω = 1 (with a reversed orientation of the contour), and the j-th integral will behave as 1 + e -N Sj . Hence, in the large N limit (at fixed s) the quantity F N (r, s) will vanish unless all the saddle-points are greater than 1, in which case F N (r, s) ∼ I N (r, s) = 1. Note that in the present situation the s saddle-points coincide. A detailed analysis shows that in this case the position of the s saddle-points depends on the value x = r/N as ω 0 = x τ (1-x) . In correspondence to the value x 0 = τ 1+τ , for which these saddlepoints are exactly 1, the function F (x, 0) has a step discontinuity. More precisely, it is easy to show that for x ∈ [0, 1], F (x, 0) = θ(x -x 0 ), where θ(x) is Heaviside step function. From a physical point of view x 0 is the contact point between the limit shape and the boundary. What have been discussed here can easily be verified in the case s = 1. The extension to finite s > 1 is rather direct as well.
Having in mind the analogy with s × s Random Matrix Models, and the scaling limit specified in Section 5.1, we rewrite our expression for F N (r, s) at ∆ = 0 as follows:
F N (r, s) = (-1) s(s+1)/2 s!(1 + τ ) s 2 (1/y-1) (2πi) s C0 • • • C0 d s ω exp s j,k=1 j =k ln |ω j -ω k | + s s j=1 1 y -1 ln(τ ω j + 1) -ln(ω j -1) - x y ln(ω j ) . (5.2)
Both sums in the exponent are O(s 2 ). The corresponding (coupled) saddle-point equations read
1 ω j -1 + x/y ω j - (1/y -1)τ τ ω j + 1 = 2 s s k=1 k =j 1 ω j -ω k . (5.3)
A standard physical picture reinterprets the saddle-point equations as the equilibrium condition for the positions of s charged particle confined to the real axis, with logarithmic electrostatic repulsion, in an external potential. In the present case the latter can be seen as generated by three external charges, 1, x/y, and -(1/y -1) at positions 1, 0, and -1/τ , respectively. It is natural to refer to this model as the triple Penner model. Although the simple Penner [P] matrix model has been widely investigated, not so much is known about the much more complicate double Penner model [M, PW]. We have not been able to trace any previous study concerning the triple Penner model.
THE ARCTIC CIRCLE REVISITED 11 5.5. The exact Green function at finite s. To investigate the structure of solutions of the saddle-point equations (5.3) for large s we first introduce the Green function
G s (z) = 1 s s j=1 1 z -ω j ,
which, if the ω j 's solves (5.3), has to satisfy the differential equation:
z(z -1)(τ z + 1) sG ′ s (z) + s 2 G 2 s (z) -s(αz 2 + βz + γ)sG s (z) = τ s(s -1) -αs 2 z + (1 -τ )s(s -1) -βs 2 + Ω 2τ s(s -1) -αs 2 . (5.4)
The coefficients α, β and γ are readily obtained as the coefficients of the second order polynomial appearing in the numerator, when setting to common denominator the left hand side of (5.3). We give them explicitly for later convenience:
α = τ 2 - 1 -x y , β = τ y + (1 -τ ) 1 + x y , γ = - x y .
The derivation of the differential equation is very standard (see, e.g., [SD]). The left hand side is built by suitably combining the explicit definition of the Green function and its derivative. The result has to be a polynomial of the first degree in z, whose coefficients are constructed by matching the leading and first subleading behaviour of the left hand side as |z| → ∞.
5.6. The first moment Ω. The quantity Ω appearing in (5.4) is defined as the first moment of the solutions of the saddle-point equations: Ω := 1 s s j=1 ω j .
It is related in a obvious way to the first subleading coefficient of G s (z); indeed, from the definition of the Green function, it is evident that
G s (z) = 1 z + 1 s s j=1 ω j 1 z 2 + O(z -3 ) , |z| → ∞ .
It is worth to emphasize that Ω is still unknown, and that in principle its value should be determined self consistently by first working out the explicit solution of G s (z) (which will depend implicitly on Ω), from (5.4) and then demanding that 1 s s j=1 ω j evaluated from this solution coincides with Ω. The appearance of the undetermined parameter Ω is a manifestation of the 'two-cuts' nature of the Random Matrix Model related to (5.2), see, e.g., par. 6.7 of [D1].
5.7. The asymptotic Green function. We are now in condition to perform the large s (and large N , r) limit at fixed x, y. In the limit, we can neglect terms of order O(s) in the differential equation (5.4), which therefore reduces to an algebraic equation for the limiting Green function G(z):
z(z-1)(τ z+1)[G(z)] 2 -(αz 2 +βz+γ)G(z) = (τ -α)z+(1-τ -β)+Ω(2τ -α) . (5.5)
The previous algebraic equation has to be supplemented by the normalization condition
G(z) ∼ 1 z , |z| → ∞ .
(5.6) 12 F. COLOMO AND A.G. PRONKO
Hence the Green function describing the large s asymptotic distribution of solutions for the saddle equation (5.3) reads:
G(z) = 1 2z(z -1)(τ z + 1) (αz 2 + βz + γ) + (αz 2 + βz + γ) 2 + 4z(z -1)(τ z + 1)[(τ -α)z + Ω(2τ -α) + 1 -τ -β] . (5.7)
We have selected the positive branch of the square root, to satisfy the normalization condition (note that the coefficient of z 4 under the square root is (α -2τ ) 2 , and α -2τ is negative for any x, y ∈ [0, 1]). However, the expression for G(z) is not completely specified yet, because Ω is still undetermined.
5.8. Limit shape and condensation of roots. The polynomial under the square root is of fourth order, hence G(z) will have in general two cuts in the complex plane. The emergence of a two-cut problem was already expected from the appearance of the undetermined first moment Ω in (5.4). The discontinuity of G(z) across these cuts defines, when positive, the density of solutions of the saddlepoint equations (5.3) when s → ∞. The problem of explicitly finding this density, for arbitrary α, β, γ (or x, y), is a formidable one, not to mention the evaluation of the corresponding 'free energy', and of the saddle-point contribution to the integral in (5.2). But our aim is much more modest, since we are presently interested only in the expression of the limit shape, i.e. in the curve in the square x, y ∈ [0, 1], delimiting regions where F (x, y) = 0 from regions where F (x, y) = 1. Of course we are here somehow assuming that the transition of F (x, y) from 0 to 1 is stepwise in the scaling limit, but this is supported both by the physical interpretation of EFP (in the disordered region, by definition, the number of 'thin' lines is macroscopic, and the probability of finding no 'thin' horizontal edges immediately vanishes in the scaling limit) and by the discussion of Section 5.3.
As explained in the discussion of the double Penner model in paper [PW], the logarithmic wells in the potential can behave as condensation germs for the saddlepoint solutions. In our case, this can role can be played only by the 'charge' at ω = 1 in the Penner potential since the charge at ω = -1/τ is always repulsive, while the one at ω = 0 is larger than 1, at least in the region of interest. [PW] have shown that condensation can occur only for charges less than or equal to 1, since this will be the fraction of condensed solutions. This consideration, together with the expected stepwise behaviour and the discussion in Section 5.3, suggest the following picture for the evolution of saddle-point solution density from the disordered region, F (x, y) ∼ 0, to the upper left frozen region, F (x, y) ∼ 1: in the disordered region there is a macroscopic fraction of solutions which are real and smaller than 1, while in the upper left frozen region this fraction vanish. On the basis of the discussion here and in Sections 3.2 and 5.3, we shall assume that at the transition between the two regions all saddle-point solutions have condensed at ω = 1.
We claim that the Arctic curve in the square x, y ∈ [0, 1] separating the disordered phase from the upper left frozen phase is defined by the condition that all solutions of the saddle-point equation lies at ω = 1.
In the derivation of the limit shape, this is indeed the only assumption to which we are unable to provide a proof. There is in fact no guarantee, at this level, for
THE ARCTIC CIRCLE REVISITED 13 this possibility to occur, and limit shapes could in principle emerge from a different condition. But if for some values of x, y ∈ [0, 1] we have all solutions of the saddlepoint equation condensing at ω = 1, then this provides a transition mechanism from 0 to 1 for F (x, y), and this might correspondingly define some limit shape.
If all saddle-point solutions condensate at ω = 1, then we obviously have: Ω = 1 .
Moreover, the complicate expression (5.7) for G(z) should simply reduce to
G(z) = 1 z -1 , (5.8)
since we expect to have no cuts, and only one pole at z = 1 with unit residue.
Consider the quartic polynomial under the square root in (5.7). It is convenient to rewrite it in terms of α := 2τ -α = τ 1 -x y ,
β := 2 -β = τ x + y -1 y + y -x y , γ := -γ = x y . (5.9)
Note that α and γ are always positive for x, y ∈ [0, 1]. When Ω = 1, our quartic polynomial reads α2 z 4 + 2 α βz 3 + ( β2 + 2 αγ)z 2 + 2 βγz + γ2 , which may be equivalently rewritten as
(αz 2 + βz + γ) 2 .
We see that the quartic polynomial reduces to a perfect square, and hence, when Ω = 1, the two cuts of G(z) disappear, as expected. Now, when Ω = 1, in our new notations, the Green function reads:
G(z) = [(2τ -α)z 2 + (2 -β)z -γ] + (αz 2 + βz + γ) 2 2z(τ z + 1)(z -1) . (5.10)
We now require the coefficients α, β, γ to be such that the polynomial under the square root combines with the first part of the numerator in (5.10) to give 2z(τ z + 1) and simplify the Green function according to (5.8). Once we have chosen a given branch of the square root (the positive one, in order to satisfy normalization condition (5.6)), it is obvious that the required simplification can occur for any z in the complex plane only if the second order polynomial αz 2 + βz + γ does not change its sign, i.e. only if its two roots coincide, implying: β2 -4 αγ = 0 .
Rewriting the last relation in terms of x, y, through (5.9), we readily get
(1 + τ ) 2 x 2 + (1 + τ ) 2 y 2 -2(1 -τ 2 )xy -2τ (1 + τ )x -2τ (1 + τ )y + τ 2 = 0 .
We have therefore recovered the limit shape, which in this Free Fermion case is the well-known Arctic Ellipse (Arctic Circle for τ = 1) [JPS, CEP]. We recall that, as discussed in Section 3.2, F (x, y) is non-vanishing only in the upper left region 14 F. COLOMO AND A.G. PRONKO of the unit square. Therefore, concerning EFP, only the upper left portion of the Arctic curve, between the two contact points at ( τ 1+τ , 0) and (1, 1 1+τ ), is relevant.
Our starting point has been the definition of a relatively simple but relevant correlation function for the six-vertex model with DWBC, the Emptiness Formation Probability. We have provided both a determinant representation and a multiple integral representation for the proposed correlation function. This is the first example in literature of a bulk (as opposed to boundary) correlation function for the considered model, for generic weights.
The multiple integral representation, specialized to the Free Fermion case, has been studied in the scaling limit. In the standard picture of Random Matrix Models, we recognize the emergence of a triple Penner model. Assuming condensation of the roots of saddle point equations in correspondence to a limit shape, we recover the well-known Arctic Circle and Ellipse. It would be interesting to investigate whether universality considerations of Random Matrix Models (see, e.g., [D2]) can be extended to the Penner model in the neighbourhood of its logarithmic singularities. This would imply directly the results of [CEP, J1, J2] on the Tracy-Widom distribution and the Airy process, emerging in a suitably rescaled neighbourhood of the Arctic Ellipse.
It is worth to stress that the multiple integral representation for EFP presented in Section 3 can be studied beyond the usual Free Fermion situation. We expect that condensation of roots of the saddle point equation in correspondence of the limit shape is a general phenomenon. We believe that this assumption could be of importance in addressing the problem of limit shapes in the six-vertex model with DWBC.
Our derivation of the limit shape in the Free Fermion case uses the explicit knowledge of function h N (z), standing in the multiple integral representation (3.6). It is worth mentioning that function h N (z) is also known explicitly at Ice Point, (∆ = 1/2), and Dual Ice Point, (∆ = -1/2), being expressible in terms of (polynomial) Gauss hypergeometric function [Ze, CP2]. For instance, at Ice Point the triple Penner model discussed above generalizes to a two-matrix Penner model. This model can be studied along the lines presented here, thus providing a solution to the longstanding problem of limit shape for Alternating Sign Matrices.
We thank Nicolai Reshetikhin for useful discussion, and for giving us a draft of [PR] before completion. FC is grateful to Percy Deift, and Courant Institute of Mathematical Science, for warm hospitality. AGP thanks INFN, Sezione di Firenze, where part of this work was done. We acknowledge financial support from MIUR PRIN program (SINTESI 2004). One of us (AGP) is also supported in part by Civilian Research and Development Foundation (grant RUM1-2622-ST-04), by Russian Foundation for Basic Research (grant 04-01-00825), and by the program Mathematical Methods in Nonlinear Dynamics of Russian Academy of Sciences. This work is partially done within the European Community network EUCLID (HPRN-CT-2002-00325), and the European Science Foundation program INSTANS.
THE ARCTIC CIRCLE REVISITED 15 References [AR] D. Allison and N. Reshetikhin, Numerical study of the 6-vertex model with domain wall boundary conditions, Ann. Inst. Fourier (Grenoble) 55 (2005) 1847-1869. [Ba] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic press, San Diego, 1982. [Br] D. M. Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Cambridge University Press, Cambridge, 1999. [BF] P. Bleher and V. Fokin, Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase, preprint (2005) arXiv:math-ph/0510033. [BPZ] N.M. Bogoliubov, A.G. Pronko, and M.B. Zvonarev, Boundary correlation functions of the six-vertex model, J. Phys. A: Math. Gen. 35 (2002) 5525-5541. [CEP] H. Cohn, N. Elkies and J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J. 85 (1996) 117-166. [CKP] H. Cohn, R. Kenyon and J. Propp, A variational principle for domino tilings, J. Amer.
Math. Soc. 14 (2001) 297-346 [CLP] H. Cohn, M. Larsen and J. Propp, The shape of a typical boxed plane partition, New York J. Math. 4 (1998) 137-165. [CP1] F. Colomo and A.G. Pronko, On two-point boundary correlations in the six-vertex model with DWBC, J. Stat. Mech.: Theor. Exp. JSTAT(2005)P05010, arXiv:math-ph/0503049. [CP2] F. Colomo and A.G. Pronko, Square ice, alternating sign matrices, and classical orthogonal polynomials, J. Stat. Mech.: Theor. Exp. JSTAT(2005)P01005, arXiv:math-ph/0411076. [CP3] F. Colomo and A.G. Pronko, The role of orthogonal polynomials in the six-vertex model and its combinatorial applications, J. Phys. A: Math. Gen. 39 (2006) 9015-9033. [CP4] F. Colomo and A.G. Pronko, Emptiness Formation Probability in the Domain Wall sixvertex marodel, in preparation. [D1] P. Deift, Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, Amer. Math. Soc., Providence, RI, 2000. [D2] P. Deift, Universality for mathematical and physical systems, preprint (2006) arXiv: math-ph/0603038. [E] K. Eloranta, Diamond Ice, J. Statist. Phys. 96 (1999) 1091-1109. [EKLP] N. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating sign matrices and domino tilings , J. Algebraic Combin. 1 (1992) 111-132; 219-234. [FP] O. Foda and I. Preston, On the correlation functions of the domain wall six-vertex model, J. Stat. Mech.: Theor. Exp. JSTAT(2004)P11001. [FS] P.L. Ferrari and H. Spohn, Domino tilings and the six-vertex model at its free fermion point, J. Phys. A: Math. Gen. 39 (2006) 10297-10306. [I] A.G. Izergin, Partition function of the six-vertex model in the finite volume, Sov. Phys. Dokl. 32 (1987) 878. [ICK] A.G. Izergin, D.A. Coker and V.E. Korepin, Determinant formula for the six-vertex model, J. Phys. A: Math. Gen. 25 (1992) 4315-4334. [J1] K. Johansson, Non-intersecting paths, random tilings and random matrices, Probab. Theory Related Fields 123 (2002) 225-280. [J2] K. Johansson, The arctic circle boundary and the Airy process, Annals of Probability 33 (2005) 1-30. [JM] M. Jimbo and T. Miwa, Algebraic analysis of solvable lattice models, CBMS Lecture Notes Series, vol. 85, Amer. Math. Soc., Providence, RI (1995). [JPS] W. Jockush, J. Propp and P. Shor, Random domino tilings and the arctic circle theorem, preprint (1995) arXiv:math.CO/9801068. [K] V.E. Korepin, Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982) 391-418. [KBI] V.E. Korepin, N.M. Bogoliubov, and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge, 1993. [KMT] N. Kitanine, J. M. Maillet and V. Terras, Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field, Nucl. Phys. B 567 (2000) 554-582. [KO] R. Kenyon and A. Okounkov, Limit shapes and the complex Burgers equation, preprint (2005) arXiv:math-ph/0507007. 16 F. COLOMO AND A.G. PRONKO [KOS] R. Kenyon, A. Okounkov and S. Sheffield, Dimers and Amoebae, Ann. of Math. (2) 163 (2006) 1019-1056. [KP] V. Kapitonov and A. Pronko, On the arctic ellipse phenomenon in the six-vertex model, in preparation. [KZ] V. Korepin, P. Zinn-Justin, Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions, J. Phys. A 33 (2000) 7053-7066. [LW] E.H. Lieb and F.Y. Wu, Two-dimensional ferroelectric models, in Phase Transitions and Critical Phenomena, Vol. 1, edited by C. Domb and M.S. Green, Academic Press, London, 1972, pp. 321-490. [M] Yu. Makeenko, Critical Scaling and Continuum Limits in the D > 1 Kazakov-Migdal Model, Int.J.Mod.Phys. A10 (1995) 2615-2660. [OR] A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003) 581-603, [P] R.C. Penner, Perturbative series and the moduli space of Riemann surfaces, J. Diff. Geom. 27 (1988) 35-53. [PR] K. Palamarchuk and N. Reshetikhin, The six-vertex model with fixed boundary conditions, in preparation. [PW] L. Paniak and N. Weiss, Kazakov-Migdal Model with Logarithmic potential and the Double Penner Matrix Model, J. Math. Phys. 36 (1995) 2512-2530. [SD] B. Sriram Shastry and A. Dhar, Solution of a generalized Stieltjes problem J. Phys. A: Math.
Gen. 34 6197-6208. [SZ] O.F. Syljuasen and M.B. Zvonarev, Directed-loop Monte Carlo simulations of Vertex models,
Phys. Rev. E 70 (2004) 016118. [Ze] D. Zeilberger, Proof of the refined alternating sign matrix conjecture, New York J. Math. 2 (1996) 59-68. [Zi1] P. Zinn-Justin, The influence of boundary conditions in the six-vertex model, preprint (2002) arXiv:cond-mat/0205192. [Zi2] P. Zinn-Justin, Six-Vertex Model with Domain Wall Boundary Conditions and One-Matrix Model, Phys. Rev. E 62 (2000), 3411-3418. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The Arctic Circle has first appeared in the study of domino tilings of large Aztec diamonds [EKLP, JPS]. The name originates from the fact that in most configurations the dominoes are 'frozen' outside the circle inscribed into the diamond, while the interior of the circle is a disordered, or 'temperate', zone. Further investigations of the domino tilings of Aztec diamonds, such as details of statistics near the circle, can be found in [CEP, J1, J2]. Here we mention that the Arctic Circle is a particular example of a limit shape in dimer models, in the sense that it describes the shape of a spatial phase separation of order and disorder. Apart from domino tilings, many more examples have been discussed recently, see, among others, papers [CKP, CLP, KO, KOS, OR].\n\nAs long as only dimer models are considered, this amounts to restrict to discrete free-fermionic models, although with nontrivial boundary conditions. Indeed, many of them can be viewed as a six-vertex model at its Free Fermion point (the correspondence being however usually not bijective), with suitably chosen fixed boundary conditions. In particular, this is the case of domino tilings of Aztec diamonds [EKLP], and the corresponding boundary conditions of the six-vertex model are the so-called Domain Wall Boundary Conditions (DWBC). Hence the problem of limit shapes extends to the six-vertex model with generic weights, and with fixed boundary conditions, among which the case of DWBC is the most interesting.\n\nHistorically, the six-vertex model with DWBC was first considered in paper [K] within the framework of Quantum Inverse Scattering Method [KBI] to prove the Gaudin hypothesis for norms of Bethe states. The model was subsequently solved 2000 Mathematics Subject Classification. 15A52, 82B05, 82B20, 82B23. 1 2 F. COLOMO AND A.G. PRONKO in paper [I] where a determinant formula for the partition function was given; see also [ICK] for a detailed exposition. Quite independently, the model was later found, under certain restrictions on the vertex weights, to be deeply related with enumerations of alternating sign matrices (see, e.g., [Br] for a review) and, as already mentioned, to domino tilings of Aztec diamonds [EKLP].\n\nConcerning the problem of limit shapes for the six-vertex model with DWBC, as far as the Free Fermion point is considered, the relation with domino tilings provided apparently an indirect proof of the corresponding Arctic Circle. The nonbijective nature of the correspondence between the two models asked for more direct results, purposely for the free-fermion six-vertex model, see [Zi1, FS, KP]. Out of the Free Fermion point, however, only very few analytical results are available, such as exact expressions for boundary one-point [BPZ] and two-point [FP, CP1] correlation functions. The present knowledge on the subject is based mainly on numerics [E, SZ, AR]; some steps towards finding the limit shapes of the model have been done recently in [PR] .\n\nIn the present note we propose a rather direct strategy to address the problem: after briefly reviewing the six-vertex model with DWBC, we define a bulk correlation function, the Emptiness Formation Probability (EFP), which discriminates the ordered and disordered phase regions. We give for this correlation function two equivalent representations, in terms of a determinant and of a multiple integral. The core derivation of EFP is heavily based on the Quantum Inverse Scattering Method [KBI], along the lines of papers [BPZ, CP1]; it is out of the scope of the present paper, corresponding details being given in a separate publication [CP4]. Here our aim is to demonstrate how the limit shapes for the considered model can be extracted from EFP in a suitable scaling limit, by making use of ideas and techniques of Random Matrix Models.\n\nTo be more specific, and to establish a contact with previous results, we specialize here our further discussion to the case of free-fermion six-vertex model. We show that the asymptotic analysis of multiple integral formula for EFP in the scaling limit reduces to a saddle-point problem for a one-matrix model with a triple logarithmic singularity, or triple Penner model. We argue that the limit shape corresponds to condensation of all saddle-point solutions to a single point. This allows us to recover the known Arctic Circle and Ellipses.\n\nAs a comment to our approach, it is to be stressed that it is directly tailored on the six-vertex model, rather than domino tilings. For this reason it is not restricted to the free-fermion models, even if, of course, further significant efforts might be necessary, essentially from the point of view of Random Matrix Model reformulation, for application to more general situations. On the basis of our previous results in [CP2], however, the application of the method to the particular case of the so-called Ice Point of the model should be straightforward. This would provide the limit shape of alternating sign matrices.\n\n2. The model 2.1. The six-vertex model. The six-vertex model (for reviews, see [LW, Ba] ) is formulated on a square lattice with arrows lying on edges, and obeying the so-called 'ice-rule', namely, the only admitted configurations are such that there are always two arrows pointing away from, and two arrows pointing into, each lattice vertex. An equivalent and graphically simpler description of the configurations of THE ARCTIC CIRCLE REVISITED 3 w 1 w 2 w 3 w 4 w 5 w 6 Figure 1. The six allowed types of vertices in terms of arrows and lines, and their Boltzmann weights.\n\nFigure 2. A possible configuration of the six-vertex model with DWBC at N = 4, in terms of arrows and lines.\n\nthe model can be given in terms of lines flowing through the vertices: for each arrow pointing downward or to the left, draw a thick line on the corresponding edge. This line picture implements the 'ice-rule' in an automated way. The six possible vertex states and the Boltzmann weights w 1 , w 2 , . . . , w 6 assigned to each vertex according to its state are shown in Figure 1."
},
{
"section_type": "OTHER",
"section_title": "Domain Wall Boundary",
"text": "Conditions. The Domain Wall Boundary Conditions (DWBC) are imposed on the N ×N square lattice by fixing the direction of all arrows on the boundaries in a specific way. Namely, the vertical arrows on the top and bottom of the lattice point inward, while the horizontal arrows on the left and right sides point outward. Equivalently, a generic configuration of the model with DWBC can be depicted by N lines flowing from the upper boundary to the left one. A possible state of the model both in terms of arrows and of lines is shown in Figure 2."
},
{
"section_type": "OTHER",
"section_title": "Partition function.",
"text": "The partition function is defined, as usual, as a sum over all possible arrow configurations, compatible with the imposed DWBC, each configuration being assigned its Boltzmann weight, given as the product of all the corresponding vertex weights, Z N = arrow configurations with DWBC w n1 1 w n2 2 . . . w n6 6 .\n\nHere n 1 , n 2 , . . . , n 6 denote the numbers of vertices with weights w 1 , w 2 , . . . , w 6 , respectively, in each arrow configuration (n\n\n1 + n 2 + • • • + n 6 = N 2 ).\n\n2.4. Anisotropy parameter and phases of the model. The six-vertex model with DWBC can be considered, with no loss of generality, with its weights invariant under the simultaneous reversal of all arrows, w 1 = w 2 =: a , w 3 = w 4 =: b , w 5 = w 6 =: c . 4 F. COLOMO AND A.G. PRONKO\n\nUnder different choices of Boltzmann weights the six-vertex model exhibits different behaviours, according to the value of the parameter ∆, defined as ∆ = a 2 + b 2 -c 2 2ab .\n\nIt is well known that there are three physical regions or phases for the six-vertex model: the ferroelectric phase, ∆ > 1; the anti-ferroelectric phase, ∆ < -1; the disordered phase, -1 < ∆ < 1. Here we restrict ourselves to the disordered phase, where the Boltzmann weights are conveniently parameterized as a = sin(λ + η) , b = sin(λ -η) , c = sin 2η . (2.1) With this choice one has ∆ = cos 2η. The parameter λ is the so-called spectral parameter and η is the crossing parameter. The physical requirement of positive Boltzmann weights, in the disordered regime, restricts the values of the crossing and spectral parameters to 0 < η < π/2 and η < λ < π -η.\n\nThe special case η = π/4 (or ∆ = 0) is related to free fermions on a lattice, and there is a well-known correspondence with dimers and domino tilings. In particular, at λ = π/2, the ∆ = 0 six-vertex model with DWBC is related to the domino tilings of Aztec diamond. For arbitrary λ ∈ [π/4, 3π/4], we shall refer to the ∆ = 0 case as the Free Fermion line.\n\nThe case η = π/6 (i.e. ∆ = 1/2) and λ = π/2, where all weights are equal, a = b = c, is known as the Ice Point; all configurations are given the same weight. In this case there is a one to one correspondence between configurations of the model with DWBC and N × N alternating sign matrices.\n\n2.5. Phase separation and limit shapes. The six-vertex model exhibits spatial separation of phases for a wide choice of fixed boundary conditions, and, in particular, in the case of DWBC. Roughly speaking, the effect is related to the fact that ordered configurations on the boundary can induce, through the ice-rule, a macroscopic order inside the lattice.\n\nThe notion of phase separation acquires a precise meaning in the scaling limit, that is the thermodynamic/continuum limit, performed by sending the number of sites N to infinity and the lattice spacing to zero, while keeping the total size of the lattice fixed, e.g., to 1. On a finite lattice, several macroscopic regions may appear, which in the scaling limit are expected to be sharply separated by some curves, the so-called Arctic curves.\n\nFor the six-vertex model with DWBC the shape of the Artic curve, or limit shape, has been found rigorously only on the Free Fermion line, and for the closely related domino tilings of Aztec diamond [JPS, CEP, Zi1, FS, KP]. For generic values of weights the limit shapes are not known, but the whole picture is strongly supported both numerically [E, SZ, AR] and analytically [KZ, Zi2, BF, PR].\n\n3. Emptiness Formation Probability 3.1. Definition. We shall use the following coordinates on the lattice: r = 1, . . . , N labels the vertical lines from right to left; s = 1, . . . , N labels the horizontal lines from top to bottom. We may now introduce the correlation function F N (r, s), measuring the probability for the first s horizontal edges between the r-th and r + 1-th line to be all 'full' (i.e. thick in the line picture, or with a left arrow in the\n\nTHE ARCTIC CIRCLE REVISITED 5 r 1 s 1 Figure 3. Emptiness Formation Probability. The sum in (3.1) is performed over all configurations compatible with the drawn arrows.\n\nstandard picture of the six-vertex model):\n\nF N (r, s) = 1 Z N 'constrained' arrow configurations with DWBC w n1 1 w n2 2 . . . w n6 6 . (3.1)\n\nHere the sum is performed over all arrow configurations on the N × N lattice, subjected to the restriction of DWBC, and to the condition that all arrows on the first s edges between the r-th and r + 1-th line should point left, see Figure 3 . Although this correlation function may appear rather sophisticated, it is computable in some closed form by means of the Quantum Inverse Scattering Method, on which DWBC are indeed tailored. It is the natural adaptation of the Emptiness Formation Probability of quantum spin chains to the present model. For this reason, and to link to the common practice in the quantum integrable models community, even if F N (r, s) actually describes 'fullness' formation probability, we shall call it Emptiness Formation Probability (EFP). r, s) . Let us restrict ourselves to the disordered regime, -1 < ∆ < 1, for definiteness. From previous analytical and numerical work, in the large N limit the emergence of a limit shape, in the form of a continuous closed curve touching once each of the four sides of the lattice, is expected. It follows that five regions emerge in the lattice: a central region, enclosed by the curve, and four corner regions, lying outside the closed curve and delimited by the sides of the lattice. The central region is disordered, while the four corners are frozen, with mainly vertices of type 1, 3, 2, 4 (see Figure 1 ) appearing in the top-left, top-right, bottom-right and bottom-left corner, respectively."
},
{
"section_type": "DISCUSSION",
"section_title": "Qualitative discussion of",
"text": "F N (\n\nBy construction, EFP is expected to be almost one in frozen regions of type 1, or 3, bordering the top side of the lattice, and to be rather small otherwise. DWBC exclude a region of type 3 to emerge in the upper part of the lattice. Hence F N (r, s) describes, at a given value of r, as s increases, a transition from a frozen region of vertices of type 1, where F N (r, s) ∼ 1, to a generic region where F N (r, s) ∼ 0. It follows that F N (r, s) can describe only the upper left portion of the closed curve, between its top and left contact points. Nevertheless, it should be mentioned that the full curve can be built from the knowledge of its top left portion, just 6 F. COLOMO AND A.G. PRONKO\n\nexploiting the crossing symmetry of the six-vertex model. Hence EFP, F N (r, s), is well suited to describe limit shapes.\n\n3.3. Some notations. For a given choice of parameters λ, η we define ϕ := c ab = sin 2η sin(λ + η) sin(λ -η) ,\n\nand the integration measure on the real line µ(x) := e x(λ-π/2) sinh(ηx) sinh(πx/2) , related to ϕ as follows:\n\nϕ = ∞ -∞ µ(x) dx .\n\nLet us introduce the complete set of monic orthogonal polynomial {P n (x)} n=0,1,... associated to the integration measure µ(x), with the orthogonality relation\n\n∞ -∞ P n (x)P m (x)µ(x) dx = h n δ nm .\n\nThe square norms h n are completely determined by the measure µ(x), and may be expressed, in principle, in terms of its moments. In the following we shall be interested in the complete set of orthogonal polynomials {K n (x)} n=0,1,... defined as\n\nK n (x) = n! ϕ n+1 1 h n P n (x) .\n\nWe moreover define ω(ǫ) := a b sin(ǫ) sin(ǫ -2η) , ω(ǫ) := b a sin(ǫ) sin(ǫ + 2η) .\n\nNote that the following relation holds\n\na 2 ω -2∆ab ωω + b 2 ω = 0 , (3.2)\n\nallowing to express ω in terms of ω."
},
{
"section_type": "OTHER",
"section_title": "Determinant representation.",
"text": "For EFP in the six-vertex model with DWBC, the following representation holds:\n\nF N (r, s) = (-1) s det 1≤j,k≤s K N -k (∂ ǫj ) s j=1 [ω(ǫ j )] N -r [ω(ǫ j ) -1] N × 1≤j<k≤s [ω(ǫ j ) -1] [ω(ǫ k ) -1] ω(ǫ j )ω(ǫ k ) -1 ǫ1=0,...,ǫs=0\n\n. (3.3) This representation has been obtained in the framework of the Quantum Inverse Scattering Method [KBI], along the lines of analogous derivations worked out for one-point and two-point boundary correlation functions of the model [BPZ, CP1].\n\nThe details of the derivation can be found in [CP4].\n\nTHE ARCTIC CIRCLE REVISITED 7 3.5. The boundary correlation function. If we consider expression (3.3) when s = 1, we recover the boundary polarization, introduced and computed in [BPZ]. It is convenient to consider the closely related boundary correlation function\n\nH N (r) := F N (r, 1) -F N (r -1, 1) .\n\nAs shown in [BPZ, CP1], the following representation holds:\n\nH N (r) = K N -1 (∂ ǫ ) [ω(ǫ)] N -r [ω(ǫ) -1] N -1 ǫ=0 .\n\nWe define the corresponding generating function\n\nh N (z) := N r=1 H N (r) z r-1 .\n\n(3.4) Noticing that ω(ǫ) → 0 as ǫ → 0, it can be shown that, given any arbitrary function f (z) regular in a neighbourhood of the origin, the following inverse representation holds\n\nK N -1 (∂ ǫ )f (ω(ǫ)) ǫ=0 = 1 2πi C0 (z -1) N -1 z N h N (z)f (z) dz . (3.5)\n\nHere C 0 is a closed counterclockwise contour in the complex plane, enclosing the origin, and no other singularity of the integrand.\n\n3.6. Multiple integral representation. Plugging (3.5) into representation (3.3), we readily obtain the following multiple integral representation for EFP:\n\nF N (r, s) = - 1 2πi s C0 • • • C0 d s ω det 1≤j,k≤s h N -k+1 (ω j ) ω j -1 ω j N -k × s j=1 ω N -r-1 j (ω j -1) N 1≤j<k≤s (ω j -1)(ω k -1) ωj ω k -1 . (3.6)\n\nHere ωj 's should be expressed in terms of ω j 's through (3.2). Indeed, due to (3.5), relation (3.2) for functions ω(ǫ), ω(ǫ), translates directly into the same relation between ω j and ωj , j = 1, . . . , s. Representation (3.6), and all results in this Section hold for any choice of parameters λ and η within the disordered regime. Moreover, by analytical continuation in parameters λ and η, these results can be easily extended to all other regimes.\n\nThe determinant in expression (3.6) is a particular representation of the partition function of the six-vertex model with DWBC, when the homogeneous limit is performed only on a subset of the spectral parameters [CP3]. The structure of the previous multiple integral representation therefore closely recalls analogous ones for the Heisenberg XXZ quantum spin chain correlation functions [JM, KMT].\n\nFor generic values of λ and η, the orthogonal polynomials K n (x), or the generating function h N (z), are known only in terms of rather implicit representations. Fortunately, there are three notable exceptions [CP2]: the Free Fermion line (η = π/4, -π/4 < λ < π/4, ∆ = 0), the Ice Point (η = π/6, λ = π/2, ∆ = 1/2), and the Dual Ice Point (η = π/3, λ = π/2, ∆ = -1/2). In these three cases, the K n (x) turn out to be classical orthogonal polynomials, namely Meixner-Pollaczek, Continuous Hahn and Continuous Dual Hahn polynomial, respectively. Correspondingly, the 8 F. COLOMO AND A.G. PRONKO\n\ngenerating function can be represented explicitly in terms of terminating hypergeometric functions that may simplify considerably further evaluation of EFP. In the next Section we shall focus on the case of Free Fermion line.\n\n4. Multiple integral representation at ∆ = 0 4.1. Specialization to η = π/4. We shall now restrict ourselves to the case η = π/4. We have ∆ = 0, and the six-vertex model reduces to a model of free fermions on the lattice. The parameter λ can still assume any value in the interval (-π/4, π/4). It is convenient to trade λ for the new parameter\n\nτ = tan 2 (λ -π/4) , 0 < τ < ∞ .\n\nThe symmetric point (related to the domino tiling of Aztec Diamond) corresponds now to τ = 1. For generic values of τ we have: ω = -τ ω .\n\nThe generating function (3.4) is known explicitely (see [CP2] for details):\n\nh N (z) = 1 + τ z 1 + τ N -1 .\n\nPlugging this expression into (3.6), we get\n\nF N (r, s) = - 1 2πi s C0 • • • C0 d s ω det 1≤j,k≤s (1 + τ ω j )(ω j -1) (1 + τ )ω j N -k × s j=1 ω N -r-1 j (ω j -1) N 1≤j<k≤s (1 + τ ω j )(ω k -1) 1 + τ ω j ω k . (4.1)\n\n4.2. Symmetrization. After extracting a common factor s j=1\n\n(1 + τ ω j )(ω j -1) (1 + τ )ω j N -s\n\nfrom the determinant in (4.1), we recognize it to be of Vandermonde type. We can therefore collect from the integrand of (4.1) the double product\n\n1≤j<k≤s (1 + τ ω j )(ω j -1) (1 + τ )ω j - (1 + τ ω k )(ω k -1) (1 + τ )ω k (1 + τ ω j )(ω k -1) 1 + τ ω j ω k .\n\nNoticing that the integration and the remaining of integrand are fully symmetric under permutation of variables ω 1 , . . . , ω j , we can perform total symmetrization of the previous double product over all its variables, with the result\n\n1 s! (-1) s(s-1)/2 s j=1 1 ω s-1 j 1≤j<k≤s (ω j -ω k ) 2 .\n\nTHE ARCTIC CIRCLE REVISITED 9 Hence, we finally obtain the following representation for EFP on the Free Fermion line:\n\nF N (r, s) = (-1) s(s+1)/2 s!(1 + τ ) s(N -s) (2πi) s × C0 • • • C0 d s ω 1≤j<k≤s (ω j -ω k ) 2 s j=1 (1 + τ ω j ) N -s (ω j -1) s ω r j . (4.2)\n\nThe appearance of a squared Vandermonde determinant in this expression naturally recalls the partition functions of s × s Random Matrix Models."
},
{
"section_type": "OTHER",
"section_title": "5. Triple Penner model and Arctic Ellipses 5.1.",
"text": "Triple Penner model and Arctic Ellipses 5.1. Scaling limit. We shall now address the asymptotic behaviour of expression (4.2) for EFP in the ∆ = 0 case. We are interested in the limit N, r, s → ∞, while keeping the ratios r/N = x , s/N = y , fixed. In this limit, x, y ∈ [0, 1] will parameterize the unit square to which the lattice is rescaled. Correspondingly EFP is expected to approach a limit function\n\nF (x, y) := lim N →∞ F N (xN, yN ) , x, y ∈ [0, 1] .\n\nWe shall exploit the standard approach developed for instance in the investigation of asymptotic behaviour for Random Matrix Models. Before this let us however point out some facts which holds already for any finite value of s. 4.2 ) only in the integration contours. Here C 1 is a closed, clockwise oriented contour (note the change in orientation) in the complex plane enclosing point ω = 1, and no other singularity of the integrand. We have the identity I N (r, s) = 1 (5.1) for any integer r, s = 1, . . . , N . The simplest way to prove the previous identity is by shifting ω j → ω j + 1, and rewriting I N (r, s) as an Hankel determinant; indeed we have"
},
{
"section_type": "OTHER",
"section_title": "A useful identity. Consider the quantity",
"text": "I N (r, s) := (-1) s(s+1)/2 s!(1 + τ ) s(N -s) (2πi) s × C1 • • • C1 d s ω 1≤j<k≤s (ω j -ω k ) 2 s j=1 (1 + τ ω j ) N -s (ω j -1) s ω r j , which differs from (\n\nI N (r, s) = (-1) s(s-1)/2 (1 + τ ) s(N -s) det 1≤j,k≤s 1 2πi C0 ω j+k-2-s (1 + τ + τ ω) N -s (1 + ω) r dω .\n\nThe entries of the Hankel matrix vanish for j +k > s+1, and hence the determinant is simply given by the product of the antidiagonal entries, j + k = s + 1 (modulo a sign (-1) s(s-1)/2 emerging from the permutation of all columns). Identity (5.1) follows immediately.\n\n10 F. COLOMO AND A.G. PRONKO 5.3. Saddle-point evaluation for large N and finite s. When using the saddle-point method in variables ω 1 , . . . , ω s to evaluate the behaviour of F N (r, s) for large N and r, and finite s, it is rather easy to see that the saddle-point equations decouple at leading order, and that each saddle-point will be on the real axis, contributing with a factor e -N Sj with S j positive.\n\nIf a given saddle-point is smaller than 1, the contour C 0 can be deformed through the saddle-point without encountering any pole, and its contribution will vanish as e -N Sj in the large N limit. If however the saddle-point, still on the real axis, happens to be larger than 1, the deformation of the contour C 0 through the saddle-point will pick up the contribution of the pole at ω = 1 (with a reversed orientation of the contour), and the j-th integral will behave as 1 + e -N Sj . Hence, in the large N limit (at fixed s) the quantity F N (r, s) will vanish unless all the saddle-points are greater than 1, in which case F N (r, s) ∼ I N (r, s) = 1. Note that in the present situation the s saddle-points coincide. A detailed analysis shows that in this case the position of the s saddle-points depends on the value x = r/N as ω 0 = x τ (1-x) . In correspondence to the value x 0 = τ 1+τ , for which these saddlepoints are exactly 1, the function F (x, 0) has a step discontinuity. More precisely, it is easy to show that for x ∈ [0, 1], F (x, 0) = θ(x -x 0 ), where θ(x) is Heaviside step function. From a physical point of view x 0 is the contact point between the limit shape and the boundary. What have been discussed here can easily be verified in the case s = 1. The extension to finite s > 1 is rather direct as well."
},
{
"section_type": "OTHER",
"section_title": "Saddle-point equation.",
"text": "Having in mind the analogy with s × s Random Matrix Models, and the scaling limit specified in Section 5.1, we rewrite our expression for F N (r, s) at ∆ = 0 as follows:\n\nF N (r, s) = (-1) s(s+1)/2 s!(1 + τ ) s 2 (1/y-1) (2πi) s C0 • • • C0 d s ω exp s j,k=1 j =k ln |ω j -ω k | + s s j=1 1 y -1 ln(τ ω j + 1) -ln(ω j -1) - x y ln(ω j ) . (5.2)\n\nBoth sums in the exponent are O(s 2 ). The corresponding (coupled) saddle-point equations read\n\n1 ω j -1 + x/y ω j - (1/y -1)τ τ ω j + 1 = 2 s s k=1 k =j 1 ω j -ω k . (5.3)\n\nA standard physical picture reinterprets the saddle-point equations as the equilibrium condition for the positions of s charged particle confined to the real axis, with logarithmic electrostatic repulsion, in an external potential. In the present case the latter can be seen as generated by three external charges, 1, x/y, and -(1/y -1) at positions 1, 0, and -1/τ , respectively. It is natural to refer to this model as the triple Penner model. Although the simple Penner [P] matrix model has been widely investigated, not so much is known about the much more complicate double Penner model [M, PW]. We have not been able to trace any previous study concerning the triple Penner model.\n\nTHE ARCTIC CIRCLE REVISITED 11 5.5. The exact Green function at finite s. To investigate the structure of solutions of the saddle-point equations (5.3) for large s we first introduce the Green function\n\nG s (z) = 1 s s j=1 1 z -ω j ,\n\nwhich, if the ω j 's solves (5.3), has to satisfy the differential equation:\n\nz(z -1)(τ z + 1) sG ′ s (z) + s 2 G 2 s (z) -s(αz 2 + βz + γ)sG s (z) = τ s(s -1) -αs 2 z + (1 -τ )s(s -1) -βs 2 + Ω 2τ s(s -1) -αs 2 . (5.4)\n\nThe coefficients α, β and γ are readily obtained as the coefficients of the second order polynomial appearing in the numerator, when setting to common denominator the left hand side of (5.3). We give them explicitly for later convenience:\n\nα = τ 2 - 1 -x y , β = τ y + (1 -τ ) 1 + x y , γ = - x y .\n\nThe derivation of the differential equation is very standard (see, e.g., [SD]). The left hand side is built by suitably combining the explicit definition of the Green function and its derivative. The result has to be a polynomial of the first degree in z, whose coefficients are constructed by matching the leading and first subleading behaviour of the left hand side as |z| → ∞.\n\n5.6. The first moment Ω. The quantity Ω appearing in (5.4) is defined as the first moment of the solutions of the saddle-point equations: Ω := 1 s s j=1 ω j .\n\nIt is related in a obvious way to the first subleading coefficient of G s (z); indeed, from the definition of the Green function, it is evident that\n\nG s (z) = 1 z + 1 s s j=1 ω j 1 z 2 + O(z -3 ) , |z| → ∞ .\n\nIt is worth to emphasize that Ω is still unknown, and that in principle its value should be determined self consistently by first working out the explicit solution of G s (z) (which will depend implicitly on Ω), from (5.4) and then demanding that 1 s s j=1 ω j evaluated from this solution coincides with Ω. The appearance of the undetermined parameter Ω is a manifestation of the 'two-cuts' nature of the Random Matrix Model related to (5.2), see, e.g., par. 6.7 of [D1].\n\n5.7. The asymptotic Green function. We are now in condition to perform the large s (and large N , r) limit at fixed x, y. In the limit, we can neglect terms of order O(s) in the differential equation (5.4), which therefore reduces to an algebraic equation for the limiting Green function G(z):\n\nz(z-1)(τ z+1)[G(z)] 2 -(αz 2 +βz+γ)G(z) = (τ -α)z+(1-τ -β)+Ω(2τ -α) . (5.5)\n\nThe previous algebraic equation has to be supplemented by the normalization condition\n\nG(z) ∼ 1 z , |z| → ∞ .\n\n(5.6) 12 F. COLOMO AND A.G. PRONKO\n\nHence the Green function describing the large s asymptotic distribution of solutions for the saddle equation (5.3) reads:\n\nG(z) = 1 2z(z -1)(τ z + 1) (αz 2 + βz + γ) + (αz 2 + βz + γ) 2 + 4z(z -1)(τ z + 1)[(τ -α)z + Ω(2τ -α) + 1 -τ -β] . (5.7)\n\nWe have selected the positive branch of the square root, to satisfy the normalization condition (note that the coefficient of z 4 under the square root is (α -2τ ) 2 , and α -2τ is negative for any x, y ∈ [0, 1]). However, the expression for G(z) is not completely specified yet, because Ω is still undetermined.\n\n5.8. Limit shape and condensation of roots. The polynomial under the square root is of fourth order, hence G(z) will have in general two cuts in the complex plane. The emergence of a two-cut problem was already expected from the appearance of the undetermined first moment Ω in (5.4). The discontinuity of G(z) across these cuts defines, when positive, the density of solutions of the saddlepoint equations (5.3) when s → ∞. The problem of explicitly finding this density, for arbitrary α, β, γ (or x, y), is a formidable one, not to mention the evaluation of the corresponding 'free energy', and of the saddle-point contribution to the integral in (5.2). But our aim is much more modest, since we are presently interested only in the expression of the limit shape, i.e. in the curve in the square x, y ∈ [0, 1], delimiting regions where F (x, y) = 0 from regions where F (x, y) = 1. Of course we are here somehow assuming that the transition of F (x, y) from 0 to 1 is stepwise in the scaling limit, but this is supported both by the physical interpretation of EFP (in the disordered region, by definition, the number of 'thin' lines is macroscopic, and the probability of finding no 'thin' horizontal edges immediately vanishes in the scaling limit) and by the discussion of Section 5.3.\n\nAs explained in the discussion of the double Penner model in paper [PW], the logarithmic wells in the potential can behave as condensation germs for the saddlepoint solutions. In our case, this can role can be played only by the 'charge' at ω = 1 in the Penner potential since the charge at ω = -1/τ is always repulsive, while the one at ω = 0 is larger than 1, at least in the region of interest. [PW] have shown that condensation can occur only for charges less than or equal to 1, since this will be the fraction of condensed solutions. This consideration, together with the expected stepwise behaviour and the discussion in Section 5.3, suggest the following picture for the evolution of saddle-point solution density from the disordered region, F (x, y) ∼ 0, to the upper left frozen region, F (x, y) ∼ 1: in the disordered region there is a macroscopic fraction of solutions which are real and smaller than 1, while in the upper left frozen region this fraction vanish. On the basis of the discussion here and in Sections 3.2 and 5.3, we shall assume that at the transition between the two regions all saddle-point solutions have condensed at ω = 1."
},
{
"section_type": "OTHER",
"section_title": "Main assumption.",
"text": "We claim that the Arctic curve in the square x, y ∈ [0, 1] separating the disordered phase from the upper left frozen phase is defined by the condition that all solutions of the saddle-point equation lies at ω = 1.\n\nIn the derivation of the limit shape, this is indeed the only assumption to which we are unable to provide a proof. There is in fact no guarantee, at this level, for\n\nTHE ARCTIC CIRCLE REVISITED 13 this possibility to occur, and limit shapes could in principle emerge from a different condition. But if for some values of x, y ∈ [0, 1] we have all solutions of the saddlepoint equation condensing at ω = 1, then this provides a transition mechanism from 0 to 1 for F (x, y), and this might correspondingly define some limit shape.\n\nIf all saddle-point solutions condensate at ω = 1, then we obviously have: Ω = 1 .\n\nMoreover, the complicate expression (5.7) for G(z) should simply reduce to\n\nG(z) = 1 z -1 , (5.8)\n\nsince we expect to have no cuts, and only one pole at z = 1 with unit residue."
},
{
"section_type": "OTHER",
"section_title": "Arctic Ellipses.",
"text": "Consider the quartic polynomial under the square root in (5.7). It is convenient to rewrite it in terms of α := 2τ -α = τ 1 -x y ,\n\nβ := 2 -β = τ x + y -1 y + y -x y , γ := -γ = x y . (5.9)\n\nNote that α and γ are always positive for x, y ∈ [0, 1]. When Ω = 1, our quartic polynomial reads α2 z 4 + 2 α βz 3 + ( β2 + 2 αγ)z 2 + 2 βγz + γ2 , which may be equivalently rewritten as\n\n(αz 2 + βz + γ) 2 .\n\nWe see that the quartic polynomial reduces to a perfect square, and hence, when Ω = 1, the two cuts of G(z) disappear, as expected. Now, when Ω = 1, in our new notations, the Green function reads:\n\nG(z) = [(2τ -α)z 2 + (2 -β)z -γ] + (αz 2 + βz + γ) 2 2z(τ z + 1)(z -1) . (5.10)\n\nWe now require the coefficients α, β, γ to be such that the polynomial under the square root combines with the first part of the numerator in (5.10) to give 2z(τ z + 1) and simplify the Green function according to (5.8). Once we have chosen a given branch of the square root (the positive one, in order to satisfy normalization condition (5.6)), it is obvious that the required simplification can occur for any z in the complex plane only if the second order polynomial αz 2 + βz + γ does not change its sign, i.e. only if its two roots coincide, implying: β2 -4 αγ = 0 .\n\nRewriting the last relation in terms of x, y, through (5.9), we readily get\n\n(1 + τ ) 2 x 2 + (1 + τ ) 2 y 2 -2(1 -τ 2 )xy -2τ (1 + τ )x -2τ (1 + τ )y + τ 2 = 0 .\n\nWe have therefore recovered the limit shape, which in this Free Fermion case is the well-known Arctic Ellipse (Arctic Circle for τ = 1) [JPS, CEP]. We recall that, as discussed in Section 3.2, F (x, y) is non-vanishing only in the upper left region 14 F. COLOMO AND A.G. PRONKO of the unit square. Therefore, concerning EFP, only the upper left portion of the Arctic curve, between the two contact points at ( τ 1+τ , 0) and (1, 1 1+τ ), is relevant."
},
{
"section_type": "OTHER",
"section_title": "Concluding remarks",
"text": "Our starting point has been the definition of a relatively simple but relevant correlation function for the six-vertex model with DWBC, the Emptiness Formation Probability. We have provided both a determinant representation and a multiple integral representation for the proposed correlation function. This is the first example in literature of a bulk (as opposed to boundary) correlation function for the considered model, for generic weights.\n\nThe multiple integral representation, specialized to the Free Fermion case, has been studied in the scaling limit. In the standard picture of Random Matrix Models, we recognize the emergence of a triple Penner model. Assuming condensation of the roots of saddle point equations in correspondence to a limit shape, we recover the well-known Arctic Circle and Ellipse. It would be interesting to investigate whether universality considerations of Random Matrix Models (see, e.g., [D2]) can be extended to the Penner model in the neighbourhood of its logarithmic singularities. This would imply directly the results of [CEP, J1, J2] on the Tracy-Widom distribution and the Airy process, emerging in a suitably rescaled neighbourhood of the Arctic Ellipse.\n\nIt is worth to stress that the multiple integral representation for EFP presented in Section 3 can be studied beyond the usual Free Fermion situation. We expect that condensation of roots of the saddle point equation in correspondence of the limit shape is a general phenomenon. We believe that this assumption could be of importance in addressing the problem of limit shapes in the six-vertex model with DWBC.\n\nOur derivation of the limit shape in the Free Fermion case uses the explicit knowledge of function h N (z), standing in the multiple integral representation (3.6). It is worth mentioning that function h N (z) is also known explicitly at Ice Point, (∆ = 1/2), and Dual Ice Point, (∆ = -1/2), being expressible in terms of (polynomial) Gauss hypergeometric function [Ze, CP2]. For instance, at Ice Point the triple Penner model discussed above generalizes to a two-matrix Penner model. This model can be studied along the lines presented here, thus providing a solution to the longstanding problem of limit shape for Alternating Sign Matrices."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgements",
"text": "We thank Nicolai Reshetikhin for useful discussion, and for giving us a draft of [PR] before completion. FC is grateful to Percy Deift, and Courant Institute of Mathematical Science, for warm hospitality. AGP thanks INFN, Sezione di Firenze, where part of this work was done. We acknowledge financial support from MIUR PRIN program (SINTESI 2004). One of us (AGP) is also supported in part by Civilian Research and Development Foundation (grant RUM1-2622-ST-04), by Russian Foundation for Basic Research (grant 04-01-00825), and by the program Mathematical Methods in Nonlinear Dynamics of Russian Academy of Sciences. This work is partially done within the European Community network EUCLID (HPRN-CT-2002-00325), and the European Science Foundation program INSTANS.\n\nTHE ARCTIC CIRCLE REVISITED 15 References [AR] D. Allison and N. Reshetikhin, Numerical study of the 6-vertex model with domain wall boundary conditions, Ann. Inst. Fourier (Grenoble) 55 (2005) 1847-1869. [Ba] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic press, San Diego, 1982. [Br] D. M. Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Cambridge University Press, Cambridge, 1999. [BF] P. Bleher and V. Fokin, Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase, preprint (2005) arXiv:math-ph/0510033. [BPZ] N.M. Bogoliubov, A.G. Pronko, and M.B. Zvonarev, Boundary correlation functions of the six-vertex model, J. Phys. A: Math. Gen. 35 (2002) 5525-5541. [CEP] H. Cohn, N. Elkies and J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J. 85 (1996) 117-166. [CKP] H. Cohn, R. Kenyon and J. Propp, A variational principle for domino tilings, J. Amer.\n\nMath. Soc. 14 (2001) 297-346 [CLP] H. Cohn, M. Larsen and J. Propp, The shape of a typical boxed plane partition, New York J. Math. 4 (1998) 137-165. [CP1] F. Colomo and A.G. Pronko, On two-point boundary correlations in the six-vertex model with DWBC, J. Stat. Mech.: Theor. Exp. JSTAT(2005)P05010, arXiv:math-ph/0503049. [CP2] F. Colomo and A.G. Pronko, Square ice, alternating sign matrices, and classical orthogonal polynomials, J. Stat. Mech.: Theor. Exp. JSTAT(2005)P01005, arXiv:math-ph/0411076. [CP3] F. Colomo and A.G. Pronko, The role of orthogonal polynomials in the six-vertex model and its combinatorial applications, J. Phys. A: Math. Gen. 39 (2006) 9015-9033. [CP4] F. Colomo and A.G. Pronko, Emptiness Formation Probability in the Domain Wall sixvertex marodel, in preparation. [D1] P. Deift, Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, Amer. Math. Soc., Providence, RI, 2000. [D2] P. Deift, Universality for mathematical and physical systems, preprint (2006) arXiv: math-ph/0603038. [E] K. Eloranta, Diamond Ice, J. Statist. Phys. 96 (1999) 1091-1109. [EKLP] N. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating sign matrices and domino tilings , J. Algebraic Combin. 1 (1992) 111-132; 219-234. [FP] O. Foda and I. Preston, On the correlation functions of the domain wall six-vertex model, J. Stat. Mech.: Theor. Exp. JSTAT(2004)P11001. [FS] P.L. Ferrari and H. Spohn, Domino tilings and the six-vertex model at its free fermion point, J. Phys. A: Math. Gen. 39 (2006) 10297-10306. [I] A.G. Izergin, Partition function of the six-vertex model in the finite volume, Sov. Phys. Dokl. 32 (1987) 878. [ICK] A.G. Izergin, D.A. Coker and V.E. Korepin, Determinant formula for the six-vertex model, J. Phys. A: Math. Gen. 25 (1992) 4315-4334. [J1] K. Johansson, Non-intersecting paths, random tilings and random matrices, Probab. Theory Related Fields 123 (2002) 225-280. [J2] K. Johansson, The arctic circle boundary and the Airy process, Annals of Probability 33 (2005) 1-30. [JM] M. Jimbo and T. Miwa, Algebraic analysis of solvable lattice models, CBMS Lecture Notes Series, vol. 85, Amer. Math. Soc., Providence, RI (1995). [JPS] W. Jockush, J. Propp and P. Shor, Random domino tilings and the arctic circle theorem, preprint (1995) arXiv:math.CO/9801068. [K] V.E. Korepin, Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982) 391-418. [KBI] V.E. Korepin, N.M. Bogoliubov, and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge, 1993. [KMT] N. Kitanine, J. M. Maillet and V. Terras, Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field, Nucl. Phys. B 567 (2000) 554-582. [KO] R. Kenyon and A. Okounkov, Limit shapes and the complex Burgers equation, preprint (2005) arXiv:math-ph/0507007. 16 F. COLOMO AND A.G. PRONKO [KOS] R. Kenyon, A. Okounkov and S. Sheffield, Dimers and Amoebae, Ann. of Math. (2) 163 (2006) 1019-1056. [KP] V. Kapitonov and A. Pronko, On the arctic ellipse phenomenon in the six-vertex model, in preparation. [KZ] V. Korepin, P. Zinn-Justin, Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions, J. Phys. A 33 (2000) 7053-7066. [LW] E.H. Lieb and F.Y. Wu, Two-dimensional ferroelectric models, in Phase Transitions and Critical Phenomena, Vol. 1, edited by C. Domb and M.S. Green, Academic Press, London, 1972, pp. 321-490. [M] Yu. Makeenko, Critical Scaling and Continuum Limits in the D > 1 Kazakov-Migdal Model, Int.J.Mod.Phys. A10 (1995) 2615-2660. [OR] A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003) 581-603, [P] R.C. Penner, Perturbative series and the moduli space of Riemann surfaces, J. Diff. Geom. 27 (1988) 35-53. [PR] K. Palamarchuk and N. Reshetikhin, The six-vertex model with fixed boundary conditions, in preparation. [PW] L. Paniak and N. Weiss, Kazakov-Migdal Model with Logarithmic potential and the Double Penner Matrix Model, J. Math. Phys. 36 (1995) 2512-2530. [SD] B. Sriram Shastry and A. Dhar, Solution of a generalized Stieltjes problem J. Phys. A: Math.\n\nGen. 34 6197-6208. [SZ] O.F. Syljuasen and M.B. Zvonarev, Directed-loop Monte Carlo simulations of Vertex models,\n\nPhys. Rev. E 70 (2004) 016118. [Ze] D. Zeilberger, Proof of the refined alternating sign matrix conjecture, New York J. Math. 2 (1996) 59-68. [Zi1] P. Zinn-Justin, The influence of boundary conditions in the six-vertex model, preprint (2002) arXiv:cond-mat/0205192. [Zi2] P. Zinn-Justin, Six-Vertex Model with Domain Wall Boundary Conditions and One-Matrix Model, Phys. Rev. E 62 (2000), 3411-3418."
}
] | ||
arxiv:0704.0366 | 0704.0366 | 1 | 10.1103/PhysRevD.76.064033 | 5157bc79c20f18512af060924b37c7f790332765bba987a088039c64f82c758e | Generalized Nariai Solutions for Yang-type Monopoles | A detailed study of the geometries that emerge by a gravitating generalized
Yang monopole in even dimensions is carried out. In particular, those which
present black hole and cosmological horizons. This two-horizon system is
thermally unstable. The process of thermalization will drive both horizons to
coalesce. This limit is what is profusely studied in this paper. It is shown
that eventhough coordinate distance shrinks to zero, physical distance does
not. So, there is some remaining space which geometry has been computed and
identified as a generalized Nariai solution. The thermal properties of this new
spacetime are then calculated. Topics, as the elliptical relation between radii
of spheres in the geometry or a discussion about whether a mass-type term
should be present in the line element or not, are also included. | [
"Pablo Diaz",
"Antonio Segui"
] | [
"gr-qc",
"hep-th"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | 1 Introduction 2 2 The gravitational coupling. Some geometrical features 3 3 The horizon coalecence geometry 4 3.1 Case m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Case m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Conclusions 11 A Proof of the finite nonzero physical distance 12 B Horizon coalescence as a flow on the line 13 1 Introduction
Monopoles have been subject of deep study and controversy all over the last century. This is so because, although no experimental evidence of their existence has been found, many theoretical issues make them almost unavoidable. They already appeared as solutions of Maxwell equations as long as the null B-divergence condition was relaxed, that is, ∇•B = 0. It was Dirac [1] in the early thirties who first proposed the theoretical possibility of creating an experiment to actually produce a "fake" monopole, in a way that its fakeness, say, the Dirac string, was undetectable. As a consequence, the product of the electric and the magnetic charges was quantized. Many years later, in 1959, the quantization requirement was confirmed by the celebrated Aharonov-Bohm experiment [2] . Since 1954, owing to the papers by Yang and Mills [3] and by Utiyama [4] , gauge theories of a group of symmetry larger than U(1), in particular non abelian symmetry groups SU(2) and SU(3) (which eventually would conform the Standard Model of particle physics) where gradually developped. In 1969, Lubkin [5] realized that monopoles can be classified by the homotopy group of the gauge symmetry group of the theory, so that the magnetic charge is replaced by the topological charge of the field configuration. In the case of the Dirac monopole, the homotopy group π 1 of U(1) is exactly Z. However, it was not till 1975 that Yang [6] generalized the abelian monopole to the case of an SU(2)-invariant gauge theory in six dimensions, see also [7] . Modern approaches use the formalism of fiber bundles for a suitable description of monopoles. It generalizes the traditional classification in terms of the homotopic group of the gauge theory. In this way, magnetic monopoles are identified with the different instanton configurations which come up basically as non trivial maps of the gauge group, usually SU(N), onto S d , where d is the spatial dimension. That is, magnetic monopoles are all those non trivial principal bundles with group structure SU(N) that can be realized on the hypersurface S d . The classification coincides, as said before, with the different classes of homotopy groups. The genalization of Yang monopoles to an arbitrary even dimension was carried out in [8] . Using slightly different methods similar analysis have recently been done [9] . The reader can find good reviews on the subject in [10] , [11] and the references therein.
As every existing object in nature, monopoles couple to gravity via their energymomentum tensor. The resulting geometry is obtained by solving the Yang-Mills-Einstein equations, which get greatly simplified by imposing spherical symmetry (as expected from a magnetic monopole field configuration). This geometry is fully specified by choosing a point in the space of parameters {µ, m, Λ, k}, the meaning of which will be explained in detail later on. For a given range of parameters, it is easy to prove that the geometry presents both a cosmological and an event horizon. A full analogy with the Schwarzschild-de Sitter solution reveals that, in these cases, the geometry is dynamically driven through the parameter space into a thermally stable point where both horizons coalesce [12] , the final line element being the analogue of Nariai's spacetime in four dimensions.
This paper is organized as follows: the next section sets a general framework and fixes the notation used later. The main body of the article concerns the analysis of the coalescence solutions. This is achieved in two subsections corresponding to the massless and massive cases respectively. An explicit computation of the resulting geometry is carried out in each case. A final section includes some conclusions and comments. Two appendixes have been added to the article. They are topics which lie somehow out of the main line of the paper, either for being technical aspects of a computation (Appendix A) or for presenting a new idea the exposition of which would need a new section, as in Appendix B. the absence of B, in turn, would not have prevented the reader from a full understanding of the paper.
The gravitational effects of these monopoles have been recently studied [9] . It was done, as usual, by minimally coupling the Yang-Mills energy-momentum tensor to gravity. Variations of the Einstein-Hilbert action
S = dx d -det g 1 16πG (R -2Λ) - 1 2γ 2 T r|F | 2 ( 1
)
with respect to the metric tensor leads to
G mn = 8πGT mn -g mn Λ, ( 2
)
where
T mn = γ -2 tr(F p m F np ) - 1 4 g mn tr(F pq F pq ) ( 3
)
is the energy momentum tensor of the YM strength field. The traces are taken in the colour index and γ is the YM coupling constant. Finding general solutions for (2) is a highly complicated problem. However, imposing spherical symmetry simplifies the task enormously. According to this, the ansatz will be a spatially spherically symmetric (2k + 2)-dimensional metric whose line element reads
ds 2 = -∆dt 2 + ∆ -1 dr 2 + r 2 dΩ 2 2k . ( 4
)
The last equation is consistent with (2) and (3) when [9] ∆(r) = 1 -2Gm r 2k-1 -µ 2 r 2 -
r 2 R 2 , ( 5
)
where R = k(2k+1) Λ is the de Sitter radius, µ 2 is proportional to 1 2k-3 and measures the magnetic charge of the monopole, m comes up as a constant of integration with dimensions of mass and G is the Newton constant in 2k + 2 spacetime dimension. At first sight, (4) with (5) look like a Schwarzschild-de Sitter geometry in 2k + 1 spatial dimensions with an extra term, the one involving µ, which seems to be independent of the dimension of spacetime. It seems reasonable to think of this term as a contribution of the magnetic monopole. This simple image, even if not exact 1 , is helpful and, unless we face the vanishing limits, it may be kept in mind in the following.
The next step (and the next temptation) is to analyze how the causal structure of this spacetime depends on given values of the parameters. The main body of this work concerns a deep analysis of the solution in the case when parameters µ, Λ, m and k allow the existence of two horizons. Then, inspired by the Schwarzschild-de Sitter unstable solution, it is claimed that the system gets dynamically driven to a value of the parameters where both horizons coalesce. Eventhough coordinate distance shrinks to zero, physical distance does not. A generalized Nariai geometry "between" the horizons is then explicitly obtained. The Nariai line element [13] is a nonsingular solution of the Einstein's vacuum equations with a positive cosmological constant, R µν = Λg µν . It was first found by Kasner [14] and its electrically charged generalization dates of 1959 [15]. However, the important fact that it emerges as an extremal limit of Schwarzschild-de Sitter black holes was not noticed until 1983 [12] .
Nariai spacetime in four dimensions is the direct product dS 2 × S 2 , dS 2 being no more than the hyperbolic version of S 2 as we change t → iτ . In 2k + 2 dimensions, the solution gets generalized to dS 2 × S 2k . Again, it is the direct product of two constant curvature spaces and admits a 3 + k(2k + 1) group of isometries SO(2, 1) × SO(2k + 1). The space is homogeneous since the group acts transitively and is locally static, given that a global dS-type spacetime cannot be described by merely one static coordinate chart. In four dimensions, radii of curvature of the two product spaces are equal if the black hole is neutral, and different in the charged case. If the black hole is electrically charged, the respective radii a and b are different and related by the equation
a -2 + b -2 = 2Λ ( 6
)
as shown in [15] . This relation will be generalized in the magnetic case, the object of our study. A short but instructive recent work on the four dimensional geometry can be found in [16] .
Studying the horizons of a geometry like (4) is equivalent to searching the divergencies of g rr for finite values of the coordinates. This leads us to analyze the zeroes of function ∆(r), where the horizons will be located. For a certain range of values of {µ, Λ, k, m} there will be two horizons. Finding this region in the parameter space will be the first 1 The resulting geometry is, of course, not just the sum of terms of different geometries, but it casually coincides. Differences are bound to exist on the limit of vanishing of a given contribution. For instance, let us suppose that, given a set of parameters, say {m, µ, Λ, k }, we can switch off µ (by neglecting it with respect to the others). The resulting geometry is topologically different to the one obtained by not assuming any monopole at all at the beginning, that is, the limit does not coincide. However, in the cases studied here, this is no more than an enough-to-be-aware-of subtlety.
task. After that, attention will be focused on the coalescence point of the horizons 2 . The analysis consists of two steps, first, the parameterization of the coordinate separation of the horizons (ǫ) and the calculation of the physical distance between them when coalescence takes place (ǫ → 0). Then, following the strategy in [12] , the computation of the line element of the remaining geometry. This program is carried out on two cases: m = 0 and m = 0, which are treated in the next subsections, respectively. The massless case must be seen as a toy model of the massive one. This distinction is made not merely for simplicity but also because, as will be explained, the mass parameter comes out naturally for dynamical requirements.
In the massless case, ∆(r) gets reduced to
∆(r) = 1 - µ 2 r 2 - r 2 R 2 . ( 7
)
Solving ∆ = 0 is equivalent to finding the zeroes of a biquadratic equation as long as r = 0 is not considered. We perform the change z ≡ r 2 and solve a second order ordinary equation. The horizons are found to be at
z + = R 2 2 1 -1 - 4µ 2 R 2 ( 8
) z ++ = R 2 2 1 + 1 - 4µ 2 R 2 . ( 9
)
R > 2µ guarantees the existence of two positive solutions and, therefore, four solutions for the quartic equation. Two of them, r + = + √ z + and r ++ = + √ z ++ , correspond to the radial coordinate of the inner (black hole) and outer (cosmological) horizon respectively. If R = 2µ, both solutions coincide, which means that the horizons coalesce. As said before, this does not mean that the geometry vanishes as a naive observation (given a wrong choice of coordinates) would make one think. Physical distance between the horizons, on the contrary, remains finite at the limit. In order to prove this, let us compute it. For fixed time and angular coordinates, the physical distance is
D(µ, R) = r ++ r + rR [-r 4 + R 2 r 2 -µ 2 R 2 ] 1/2 dr = = z ++ z + R 2 R 4 4 -µ 2 R 2 -z -R 2 2 2 1/2 dz ( 10
)
The requirement R > 2µ implies R 4 4 -µ 2 R 2 > 0 so the above integral is exactly solved as an cos -1 -type. The result is
D(R) = 1 2 πR. ( 11
)
Surprisingly, the physical distance does not depend on µ. It means that, given a cosmological constant Λ, one could "switch on" the monopole and go on till the horizons coalesce but the distance would remain unchanged. However, because of quantization 2 Coalescence as seen in Schwarzschild coordinates.
requirements, monopole charge µ cannot be tuned, but needs to have, instead, a fixed value upto a sign. On the other hand, the cosmological constant, Λ, should be chosen when writing the lagrangian. It means that changing its value does not drive us from one model to another but implies an essential change in the theory [17] . Therefore, we are not free to adjust any parameter arbitrarily as done with the mass of the black hole in the Schwarzschild-de Sitter case. Then, eventhough physical reasons would lead the horizons to coalesce, the absence of any free parameter in our model makes it impossible. In the next section, m will come to our help as a free parameter for the model. Despite the last remark, one could wonder about the kind of geometry that remains when the horizons coalesce. This task, even if seems just a curious exercise now, will be useful for the next section. Applying a technique similar to the one Gingspar and Perry [12] used to study the geometry of Nariai's solution, we proceed by, first, parametrizing the separation of horizons as
R = 2µ(1 + ǫ 2 ), ( 12
)
in a way that coalescence corresponds to taking ǫ = 0. Then, we define a "wise" change of coordinates
χ = cos -1 -2 R 2 A (r 2 -r 2 0 ) τ = ǫ 2it R , ( 13
)
where A = 1 -4µ 2 R 2 and r 2 0 = R 2 2 , and the angular coordinates remain unchanged. The new coordinates (13) might seem randomly chosen at first sight. However, there are some reasons that justify such a functional dependence. For instance, χ is nothing but the physical distance between r + and r. The timelike coordinate t is multiplied by i in order to work in the Euclidean region 3 and by ǫ because ∆/ǫ 2 is expected to have a finite limit when ǫ → 0. Now, we apply (12) and (13) and expand ∆(r(χ))dτ 2 , ∆ -1 (r(χ))dχ 2 and r 2 (χ) up to first order in ǫ. The line element (4) reads
ds 2 = µ 2 dχ 2 + µ 2 sin 2 (χ) 1 + ǫ √ 2 cos(χ) dτ 2 + + 2µ 2 1 - √ 2 cos(χ)ǫ dΩ 2 2k . ( 14
)
We take limit ǫ → 0 to obtain
ds 2 = µ 2 dχ 2 + sin 2 (χ)dτ 2 + 2µ 2 dΩ 2 2k . ( 15
)
As seen in ( 15 ), the 2k-sphere decouples from the rest. The resulting geometry is S 2 ×S 2k for k ≥ 2. Notice the parallelism between this geometry and Nariai's solution, which is S 2 × S 2 . The "classical" relation between radii (6) gets also generalized to
a -2 + b -2 = C 0 Λ, ( 16
)
where C 0 = 6 k(2k+1) . The geometry (15) can be viewed as a "degenerate" black hole, in which the two horizons have the same (maximum) size and are in thermal equilibrium. This could be interpreted by an observer as a bath of radiation coming from both horizons 3 τ will be periodic at both horizons, although different in each case. Equality will hold at the coalescence point, when thermal stability is reached.
at a precise temperature [19] . The temperature can be calculated by means of surface gravity κ, as computed in the new coordinates (13) T = 1 2π √ 2µ = 1 π k(2k + 1) 2Λ 1/2
. ( 17
)
The entropy can also be computed as a quarter of the sum of the two horizons [18], so
S = 1 4 A T = 1 2 A H = 1 2 ω 2k k(2k + 1) 2Λ 2 , ( 18
)
where ω 2k is the area of the 2k-dimension unit sphere.
In the massive case we recover the full expression (5) for ∆. Since the singular point r = 0 is not to be considered, we better analyze the function r 2k-1 ∆(r)
∆ ≡ r 2k-1 ∆ = - r 2k+1 R 2 + r 2k-1 -µ 2 r 2k-3 -2Gm. ( 19
)
It is known that a polynomial equation with powers equal to or higher than five is not generally solvable in a symbolical way. This happens for k ≥ 2. So, the purpose of doing a study for the massive case analogous to that achieved in the first section is ruined. Nevertheless, some information can be extracted from (19) . We should first remember the sign of the parameters: R 2 > 0 (de Sitter), µ 2 > 0 for k ≥ 2, and m will be free in principle. Derivating (19) and equating to zero leads to a biquadratic equation of the form
- 1 R 2 (2k + 1)r 4 + (2k -1)r 2 -(2k -3)µ 2 = 0, ( 20
)
which, as long as
Λµ 2 ≤ k 4 (2k -1) 2 2k -3 , ( 21
)
has two positive (and two negative) roots, r min and r max ≡ r c . In terms of the cosmological constant
r 2 c ≡ k(2k -1) 2Λ 1 + 1 - 4(2k -3)Λµ 2 k(2k -1) 2 , ( 22
)
r min is obtained from (22) by swapping the sign of the square root. A quick look at (19) shows that the smallest root is a minimum and the largest is a maximum of function ∆. Now, let us plug r c into (19): 1. If m > 0, then (see fig. 1 ) a) ∆(r c ) ≥ 0 implies that there are two event horizons, the black hole and the cosmological horizon. The inequality gets saturated at the coalescence point.
b) ∆(r c ) < 0 means that no horizon is found.
2. If m < 0, then (see fig. 2 ) a) ∆(r min ) < 0 together with ∆(r c ) < 0 implies that there is just one Cauchy horizon. b) ∆(r min ) < 0 together with ∆(r c ) > 0 assures the existence of a Cauchy horizon and both black hole and cosmological horizon. c) ∆(r min ) > 0 leaves us with the cosmological horizon only.
The case we will study is ∆(r min ) < 0 and ∆(r c ) > 0 which, independently of the sign of m, assures 4 the existence of black hole and cosmological horizons. This corresponds to values of m within range (see fig. 3) 1.a 1.b
Figure 1: Case m > 0. The curve represents function ∆(r). Figure 1 .a has two roots which correspond to the black hole (r + ) and cosmological horizon (r ++ ) respectively. Figure 1.b shows the absence of horizons.
2.a 2.b 2.c
Figure 2: Case m < 0. This time ∆(r) permits the existence of one (Cauchy) horizon as in Figure 2.a, three horizons (Cauchy, black hole and cosmological) as in 2.b, or just the cosmological horizon as shown in 2.c.
m -< m < m + , ( 23
)
where
Gm c ≡ Gm + = 1 1 + 2k r 2k-3 c (r 2 c -2µ 2 ). ( 24
)
The value of Gm -is obtained by replacing r c → r min . In terms of Λ and µ we get
Gm ± = (2Λ) -k+1/2 1 + 2k -k + 2k 2 ± k 2 (1 -2k) 2 -4Λµ 2 (2k -3)k k-3/2 -k + 2k 2 -4Λµ 2 ± k 2 (1 -2k) 2 -4Λµ 2 (2k -3)k . ( 25
)
4 The value of m can be negative. That is because m should not be thought of as an entity with physical meaning but as a geometrical parameter. Short calculation in (25) shows that m gets negative values for Λµ 2 ≥ k 4 (1 + 2k).
r r,m r,m r c Figure 3: This figure shows the range of "masses" which are consistent with the existence of both black hole and cosmological horizons. The curve ∆(r) "moves down" in the process of coalescence.
The crucial point is that both horizons coalesce when r c is a root of (19) which happens at m = m c (k, Λµ 2 , Λ). Two relations have been imposed so far: d e ∆(r;m) dr 20) , which defines r c , and ∆(r c ; m c ) = 0 which leads to m c . In order for m to be real, the bound which must be impossed on Λµ 2 coincides with (21) which, in turn, is nothing but the condition for the existence of two horizons. So, if a given a value for Λµ 2 is low enough to produce two horizons, there always exists a real value of m which makes them coalesce. Again, as in the Schwarzschild-de Sitter example, the system is unstable and the equilibrium point is reached at m = m c . Unlike the massless case, plugging m gives us enough room for maneuvre to drive the system to equilibrium. At this point, we would like to remark that the procedure of horizon coalescence, as studied in detail below, may be seen as a flow in a line which undergoes a Pitchfork bifurcation at the coalescence point. Parameter m, moved by thermal instability, drives the system to the critical situation. For concreteness see Appendix B. Let us focus on the near coalescence point. This can be parameterized by
| rc = 0, that is, (
r = r c + δr = r c (1 + ǫ cos χ) (26) m = m c -δm = m c (1 + bǫ 2 ).
Parameterization of r also involves a change of coordinates r → χ and should be taken as imposed at the moment although it will be justified later. The horizons will be symmetrically located at: r + = r c (1 -ǫ) and r ++ = r c (1 + ǫ) which correspond to χ + = π and χ ++ = 2π, respectively 5 . The value of b as well as the absence of a linear term in ǫ of the parameterization of m may be explained as follows. Near the coalescence point one should Taylor expand ∆ around r c and have in mind that, for δm mc ≪ 1, ∆ is aproximately parabolic, so that second order expansion is enough. By definition ∆(r + ) = ∆(r ++ ) = 0 5 For a small enough ǫ, it is expected that the parabolic approach holds and, then, both horizons are symmetrically located with respect to r c .
and ∆ reaches a maximum at r c . So,
0 = ∆(r ++ ) = ∆(r c , m) + ∆ ′ (r c , m)(r c ǫ) + 1 2 ∆ ′′ (r c , m)(r c ǫ) 2 = = 2Gδm r 2k-1 c + 1 2 ∆ ′′ (r c ; m c )r 2 c ǫ 2 , ( 27
)
which means that
b = ∆ ′′ (r c ; m c )r 2k+1 c 4Gm c . ( 28
)
Calculating the physical distance near the coalescence point would, again, imply solving the integration
D(ǫ) = r ++ r + dr ∆ 1/2 (r) , ( 29
)
where r ++ = r + + 2r c ǫ. Although the exact result is not computed, an explicit proof of its finite nonzero value is given in Appendix A. The procedure of calculating the physical distance also brings us some light on which is the change of coordinates that should be made in order to understand the resulting geometry. It turns out to be
χ = cos -1 1 ǫr c (r -r c ) τ = ǫi Br c t, ( 30
) where B = k k -2k 2 + 2Λr 2 c ( 31
)
is a dimensionless factor.
The coalescence of horizons takes place at ǫ = 0. In order to study the geometry at the limit we proceed by calculating -∆dt 2 , ∆ -1 dr 2 and r 2 in the new coordinates (30) and expand in ǫ around ǫ = 0. The new line element gets determined by taking the zero order of the expansion. The relations for r and m in (26) are in accordance with (30) , where b takes the value of (28) , by virtue of the parabolic approach. From (30), it is straightforward to see that r 2 takes a constant value r 2 c . Surprisingly, as in the massless and Schwarzschild-de Sitter cases, the geometry splits in two disconnected parts which lead to a product manifold S 2 × S 2k . The line element reads
ds 2 = Br 2 c dχ 2 + sin 2 (χ)dτ 2 + r 2 c dΩ 2 2k , ( 32
)
where χ ∈ [π, 2π] and τ is periodic 6 . As seen in (32) , S 2 has radius a 2 = Br 2 c , and S 2k has radius b 2 = r 2 c . Now, the generalized Bertotti relation (6) is
a -2 + b -2 = 2(1 -k) r 2 c + 2Λ k = CΛ, ( 33
)
where C(k, Λµ 2 ) is obtained by inserting (22) in (33) . Note that C k, Λµ 2 = k(2k + 1)/4 = C 0 , and then (33) turns into (16) , that is, into the massless case. This is no 6 τ is periodic on both horizon surfaces all over the process in order to avoid the conical singularity at the horizons. At the coalescence point, however, both periods equal.
surprising since Λµ 2 = k(2k + 1)/4 is the condition for coalescence in the massless case (equivalent to R = 2µ), and, at the same time, it makes m c = 0. So, the massive geometry is a consistent extension of the massless one. Now, fixing Λ does not determine uniquely the geometry. Another dimensionless variable Λµ 2 is required.
As in the last section, the geometry (32) can be viewed as a "degenerate" black hole, in which the two horizons have the same (maximum) size and are in thermal equilibrium.
In the present case the temperature is given in terms of the surface gravity κ by
T = κ 2π = 1 2πr c √ B . ( 34
)
In Planck units,the entropy associated with this solution may be calculated (given that it is not extreme 7 ) by means of the total area of the horizons as
S = 1 4 A T = 1 2 A H = 1 2 ω 2k r 4 c . ( 35
)
The spherically symmetric solution of gravity due to a magnetic monopole in arbitrary dimension has been studied, in particular, when the set of parameters {Λ, µ, m, k} allows the existence of two horizons. In these cases, thermal instabilities drive a process of horizon coalescence. Even though coordinate separation between the horizons shrinks to zero, it has been proven in both the massless and the massive case that the physical distance does not. The geometry of the remaining space between the horizons has been calculated in both cases. They turned out to be Nariai-type solutions, that is, the product of a 2-sphere and a 2k-sphere for a (2k + 2)-dimensional spacetime. In each solution, the radii of the spheres are not independent. They are related by an elliptical equation which should be understood as the generalization of the relation found by Bertotti. The unique generalized equation involving these radii for both the massless and the massive case has been given. After computing the line element in each case, the thermodynamical properties (Hawking temperature and entropy) due to the existence of horizons have been calculated. The Yang monopole corresponds to the six dimensional case, where k = 2. The geometry obtained after coalescence is S 2 × S 4 as can be explicitly read in (32) . This case is especially interesting since it may be described in String Theory (a realization of the Yang monopole in Heterotic String Theory has recently been done [21] as well as another complementary picture in Type-IIA String Theory [20] ). In the same context, it looks possible to find results (18) and (35) for the entropy by application of some attractor mechanism [22, 23] . We believe that this would be an interesting topic to be addressed in future research.
7 A charge black hole is said to be extreme when it has the minimum mass. Then, as it cannot release any energy without losing charge, it is supposed not to emit, and its associated Hawking temperature is 0. The black hole we are dealing with in this paper is extreme in the sense of carrying the "maximum mass" allowed by the cosmological constant Λ. Obviously, the temperature will not be zero.
A Proof of the finite nonzero physical distance
Computing the physical distance is equivalent to performing the integration
D = r ++ r + dr ∆ 1/2 (r) , ( 36
)
where, for small ǫ, r ++ = r + + 2r c ǫ. Divergencies might appear at the points where ∆ → 0. The case we have been considering all along section (3.2) concerns the existence of two horizons which coalesce, that is, two single roots r + and r ++ of ∆ which join to form a double one. Function ∆ can always be expressed as ∆ = (r -r + )(r ++ -r)g(r), where g(r) is a polynomial function of powers of degree 2k -1 and no zeroes within the range [r + , r ++ ] are to be found by construction. Explicitly, equation (36) is
D(ǫ) = r + +2rcǫ r + dr (r -r + ) 1/2 (r + + 2r c ǫ -r) 1/2 r k-1/2 g 1/2 (r) h(r) . ( 37
)
Now, h(r) is a continuous divergenceless strictly positive function in the compact [r + , r ++ ], which means that it will reach a positive maximum and minimum for certain r ′ s. Let us call h max and h min the values of the function h in these points 8 . Then h min r + +2rcǫ
r + dr (r -r + ) 1/2 (r + + 2r c ǫ -r) 1/2 ≤ D(ǫ) ≤ ≤ h max r + +2rcǫ r + dr (r -r + ) 1/2 (r + + 2r c ǫ -r) 1/2 . ( 38
)
The integration can be performed:
r + +2rcǫ r + dr (r -r + ) 1/2 (r + + 2r c ǫ -r) 1/2 = π. ( 39
) Now D(ǫ → 0) = πh(r c ), ( 40
)
where the value of r c is given in (22) . Integrations of form (39) are solved exactly by a cos -1 type function, and a nonzero finite result is obtained. It is remarkable that the same can be said for any ∆ we would choose, as long as no more than two single roots were to join to form a double one. The key point is that (39) , which could be problematic, is independent of ǫ and therefore the distance is finite in the limit, when ǫ → 0. So, eventhough (39) was neither exactly the physical distance in the massive case nor in Schwarzschild-de Sitter solution (however, it was in the massless case as we have already seen in the first section), it is closely related to it. This fact gives us a hint or, at least, justifies the change of coordinates we were performing once and again to study the geometry at the limit ǫ → 0.
8 These, in principle, depend on ǫ but coincide when ǫ → 0:
h min = h max ≡ h(r c ).
B Horizon coalescence as a flow on the line
The main phenomenon that concerns this paper, as said before, can be described in terms of the dynamics of a vector field on the line. The coalescence point, in this picture, is no more than a supercritical Pitchfork bifurcation. Let us remember some general features of the dynamics of a one-dimensional flow. The equation of a general vector field on the line can be expressed as: ẋ = f (x, α) (41)
where f is any real function with real support, the dot means differentiation with respect to t and α is a parameter of the model. Fixed points of (41) require ẋ = 0, which must be obtained by finding the roots of f , that is f (x * , α) = 0. (42) Equation (42) is solved by an n-collection of fixed points x * i for a given value of α. Let us suppose that f has three roots if α = α 0 . Fixed points come closer as α moves and get "condensed" in a "fat" fixed point (bifurcation point) at α = α c . A paradigmatic example of a Pitchfork bifurcation is shown by function
f (x) = x(α -x 2 ). ( 43
)
One question arises naturally now about the role the horizons play in this picture. Let us claim that horizons are fixed points and the role of α is played by m. We will justify this identification by constructing the vector flow. Constructing a flow in a manifold (in our case it will be a line) is equivalent to giving a family of curves r(t) which covers the manifold or part of it. Each of the curves gets specified by the initial condition, say, r(t = 0). Now, let us consider geodesic motions. Without loss of generality, the angular coordinates of our geometry will be frozen, θ and φ are constants, and only radial curves r(t) are to be regarded. Static coordinate system will serve us to describe the movement for any r ∈ (r + , r ++ ). Let us invoke intuition at this point. If r(0) is near the cosmological or the black hole horizon it is clear that a test particle will move out of the region by approaching each horizon respectively. Then, there is a point r = r g where the test particle will not "feel" any force and, consequently, it will not move 9 . This is the first (unstable) fixed point.
Let us move the origin by defining r ′ = r -r g , after this, primes will be dropped out to simplify notation. The flow at each point will be determined by the physical velocity ṙ(r) (as measured by an observer placed at r = 0) that a test particle would adquire at r if it is dropped with ṙ = 0 at around r = 0 (as close as possible). It is not hard to see that the velocity of the test particle, as seen by the static geodesic observer, is bound to be zero at both horizons. So, horizons are fixed points. Now, our system can be treated as a vector flow ṙ(r) which covers the region between the horizons. The vector flow has three fixed points: {r + , r ++ , 0} where the first two are stable. As m runs towards m c , the system shrinks into a Pitchfork bifurcation. Near the bifurcation point the flow can be approximated by
ṙ = βr(r -r + )(r ++ -r), ( 44
)
9 r g in our geometry, plays the role the asymptotic infinity does in Schwarzschild solution,that is, the point where the time-like Killing vector should be normalized in order to define the horizon temperature. Note that r g ≡ r c at the coalescence point, that is, when ǫ = 0.
where β is a positive constant which depends on µ, k and Λ. On the one hand, in the coordinate system {χ, τ }, and using (30), we have dr dt = dr dχ dχ dτ dτ dt -→ ṙ = iǫ 2 B dχ dτ . (45) On the other hand, equation (44), expressed in the new coordinate system, reads ṙ = ǫ 3 βr 3 c cos χ sin 2 χ, and so dχ dτ = -iǫβr 3 c B cos χ sin 2 χ. (46)
As expected, in the new coordinate system, every point converts into a fixed point as horizons coalesce (ǫ → 0). Since the flux lines were identified with geodesics of test particles, this can be understood as the abscence of forces at the end of the process.
We thank P. K. Townsend and Adil Belhaj for helpful discussions and Jean Nuyts for critical reading of the manuscript. This work has been supported by MCYT ( Spain) under grant FPA 2003-02948.
References [1] P. A. M. Dirac, Quantised singularities in the electromagnetic field, Proc. Roy. Soc. Lond. A 133, 60 (1931). [2] Y. Aharonov and D. Bohm, Significance of Electromagnetic Potentials in the Quantum Theory, Phys. Rev. 115, 485 (1959). [3] C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96, 191 (1954). [4] Ryoyu Utiyama, Invariant Theoretical Interpretation of Interaction, Phys. Rev. 101, 1597 (1956). [5] E. Lubkin, Geometric Definition of Gauge Invariance, Ann. Phys. 23, 233 (1963). [6] C. N. Yang, Generalization of Dirac's monopole to SU2 gauge fields, J. Math. Phys. 19, 320 (1978).
[7] P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories And Magnetic Charge, Nucl. Phys. B 125, 1 (1977). [8] Zalan Horvath, Laszlo Palla, Spontaneous Compactification And 'Monopoles' In Higher Dimensions., Nucl. Phys. B 142, 327 (1978). [9] G.W. Gibbons and P.K. Townsend, Self-graviting Yang monopoles in all dimensions, Class. Quantum Grav. 23, 4873 (2006).
[10] S. Coleman, The magnetic monopole fifty years later, in The Unity of the Fundamental Interactions, ed. A. Zichichi (Plenum, New York, 1983).
[11] E. J. Weinberg and P. Yi, Magnetic Monopole Dynamics, supersymmetry, and Duality, Phys. Rept. 43, 65 (2007). hep-th/0609055. [12] Paul Ginsparg and Malcom J. Perry, Semiclassical perdurance of de Sitter space, Nuclear Physics B 222, 245 (1983).
[13] H. Nariai, Sci. Rep. Tohoku Univ., Ser. 1 35, 62 (1951). [14] E. Kasner, Trans. Am. Math. Soc. 27, 101 (1925).
[15] B. Bertotti, Uniform Magnetic Field in the Theory of General Relativity, Phys. Rev. 116, 1331 (1959).
[16] Marcello Ortaggio, Impulsive waves in the Nariai Universe, Phys. Rev. D 65, 084046 (2002). [17] G. W. Gibbons and S. W. Hawking, Cosmological Event Horizons, Thermodynamics, And Particle Creation, Phys. Rev. D 15, 2738 (1977). [18] S. W. Hawking and Simon F. Ross, Duality between electric and magnetic black holes, Phys. Rev. D 52, 5865 (1995) [19] R. Bousso and S. W. Hawking, Pair creation of black holes during inflation, Phys. Rev. D 54, 6312 (1996). [20] A. Belhaj, P. Diaz, A. Segui, On the Superstring Realization of the Yang Monopole, (2007). hep-th/0703255. [21] E. A. Bergshoeff, G. W. Gibbons and P. K. Townsend, Open M5-branes, Phys. Rev. Lett. 97, 231601 (2006). hep-th/0607193. [22] S. Ferrara, R. Kallosh, A. Strominger, N=2 extremal black holes, Phys. Rev. D 52, 5412 (1995). [23] S. Ferrara, R. Kallosh, Supersymmetry and Attractors, Phys. Rev. D 54, 1514 (1996). | [
{
"section_type": "OTHER",
"section_title": "Contents",
"text": "1 Introduction 2 2 The gravitational coupling. Some geometrical features 3 3 The horizon coalecence geometry 4 3.1 Case m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Case m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Conclusions 11 A Proof of the finite nonzero physical distance 12 B Horizon coalescence as a flow on the line 13 1 Introduction\n\nMonopoles have been subject of deep study and controversy all over the last century. This is so because, although no experimental evidence of their existence has been found, many theoretical issues make them almost unavoidable. They already appeared as solutions of Maxwell equations as long as the null B-divergence condition was relaxed, that is, ∇•B = 0. It was Dirac [1] in the early thirties who first proposed the theoretical possibility of creating an experiment to actually produce a \"fake\" monopole, in a way that its fakeness, say, the Dirac string, was undetectable. As a consequence, the product of the electric and the magnetic charges was quantized. Many years later, in 1959, the quantization requirement was confirmed by the celebrated Aharonov-Bohm experiment [2] . Since 1954, owing to the papers by Yang and Mills [3] and by Utiyama [4] , gauge theories of a group of symmetry larger than U(1), in particular non abelian symmetry groups SU(2) and SU(3) (which eventually would conform the Standard Model of particle physics) where gradually developped. In 1969, Lubkin [5] realized that monopoles can be classified by the homotopy group of the gauge symmetry group of the theory, so that the magnetic charge is replaced by the topological charge of the field configuration. In the case of the Dirac monopole, the homotopy group π 1 of U(1) is exactly Z. However, it was not till 1975 that Yang [6] generalized the abelian monopole to the case of an SU(2)-invariant gauge theory in six dimensions, see also [7] . Modern approaches use the formalism of fiber bundles for a suitable description of monopoles. It generalizes the traditional classification in terms of the homotopic group of the gauge theory. In this way, magnetic monopoles are identified with the different instanton configurations which come up basically as non trivial maps of the gauge group, usually SU(N), onto S d , where d is the spatial dimension. That is, magnetic monopoles are all those non trivial principal bundles with group structure SU(N) that can be realized on the hypersurface S d . The classification coincides, as said before, with the different classes of homotopy groups. The genalization of Yang monopoles to an arbitrary even dimension was carried out in [8] . Using slightly different methods similar analysis have recently been done [9] . The reader can find good reviews on the subject in [10] , [11] and the references therein.\n\nAs every existing object in nature, monopoles couple to gravity via their energymomentum tensor. The resulting geometry is obtained by solving the Yang-Mills-Einstein equations, which get greatly simplified by imposing spherical symmetry (as expected from a magnetic monopole field configuration). This geometry is fully specified by choosing a point in the space of parameters {µ, m, Λ, k}, the meaning of which will be explained in detail later on. For a given range of parameters, it is easy to prove that the geometry presents both a cosmological and an event horizon. A full analogy with the Schwarzschild-de Sitter solution reveals that, in these cases, the geometry is dynamically driven through the parameter space into a thermally stable point where both horizons coalesce [12] , the final line element being the analogue of Nariai's spacetime in four dimensions.\n\nThis paper is organized as follows: the next section sets a general framework and fixes the notation used later. The main body of the article concerns the analysis of the coalescence solutions. This is achieved in two subsections corresponding to the massless and massive cases respectively. An explicit computation of the resulting geometry is carried out in each case. A final section includes some conclusions and comments. Two appendixes have been added to the article. They are topics which lie somehow out of the main line of the paper, either for being technical aspects of a computation (Appendix A) or for presenting a new idea the exposition of which would need a new section, as in Appendix B. the absence of B, in turn, would not have prevented the reader from a full understanding of the paper."
},
{
"section_type": "OTHER",
"section_title": "The gravitational coupling. Some geometrical features",
"text": "The gravitational effects of these monopoles have been recently studied [9] . It was done, as usual, by minimally coupling the Yang-Mills energy-momentum tensor to gravity. Variations of the Einstein-Hilbert action\n\nS = dx d -det g 1 16πG (R -2Λ) - 1 2γ 2 T r|F | 2 ( 1\n\n)\n\nwith respect to the metric tensor leads to\n\nG mn = 8πGT mn -g mn Λ, ( 2\n\n)\n\nwhere\n\nT mn = γ -2 tr(F p m F np ) - 1 4 g mn tr(F pq F pq ) ( 3\n\n)\n\nis the energy momentum tensor of the YM strength field. The traces are taken in the colour index and γ is the YM coupling constant. Finding general solutions for (2) is a highly complicated problem. However, imposing spherical symmetry simplifies the task enormously. According to this, the ansatz will be a spatially spherically symmetric (2k + 2)-dimensional metric whose line element reads\n\nds 2 = -∆dt 2 + ∆ -1 dr 2 + r 2 dΩ 2 2k . ( 4\n\n)\n\nThe last equation is consistent with (2) and (3) when [9] ∆(r) = 1 -2Gm r 2k-1 -µ 2 r 2 -\n\nr 2 R 2 , ( 5\n\n)\n\nwhere R = k(2k+1) Λ is the de Sitter radius, µ 2 is proportional to 1 2k-3 and measures the magnetic charge of the monopole, m comes up as a constant of integration with dimensions of mass and G is the Newton constant in 2k + 2 spacetime dimension. At first sight, (4) with (5) look like a Schwarzschild-de Sitter geometry in 2k + 1 spatial dimensions with an extra term, the one involving µ, which seems to be independent of the dimension of spacetime. It seems reasonable to think of this term as a contribution of the magnetic monopole. This simple image, even if not exact 1 , is helpful and, unless we face the vanishing limits, it may be kept in mind in the following.\n\nThe next step (and the next temptation) is to analyze how the causal structure of this spacetime depends on given values of the parameters. The main body of this work concerns a deep analysis of the solution in the case when parameters µ, Λ, m and k allow the existence of two horizons. Then, inspired by the Schwarzschild-de Sitter unstable solution, it is claimed that the system gets dynamically driven to a value of the parameters where both horizons coalesce. Eventhough coordinate distance shrinks to zero, physical distance does not. A generalized Nariai geometry \"between\" the horizons is then explicitly obtained. The Nariai line element [13] is a nonsingular solution of the Einstein's vacuum equations with a positive cosmological constant, R µν = Λg µν . It was first found by Kasner [14] and its electrically charged generalization dates of 1959 [15]. However, the important fact that it emerges as an extremal limit of Schwarzschild-de Sitter black holes was not noticed until 1983 [12] .\n\nNariai spacetime in four dimensions is the direct product dS 2 × S 2 , dS 2 being no more than the hyperbolic version of S 2 as we change t → iτ . In 2k + 2 dimensions, the solution gets generalized to dS 2 × S 2k . Again, it is the direct product of two constant curvature spaces and admits a 3 + k(2k + 1) group of isometries SO(2, 1) × SO(2k + 1). The space is homogeneous since the group acts transitively and is locally static, given that a global dS-type spacetime cannot be described by merely one static coordinate chart. In four dimensions, radii of curvature of the two product spaces are equal if the black hole is neutral, and different in the charged case. If the black hole is electrically charged, the respective radii a and b are different and related by the equation\n\na -2 + b -2 = 2Λ ( 6\n\n)\n\nas shown in [15] . This relation will be generalized in the magnetic case, the object of our study. A short but instructive recent work on the four dimensional geometry can be found in [16] ."
},
{
"section_type": "OTHER",
"section_title": "The horizon coalecence geometry",
"text": "Studying the horizons of a geometry like (4) is equivalent to searching the divergencies of g rr for finite values of the coordinates. This leads us to analyze the zeroes of function ∆(r), where the horizons will be located. For a certain range of values of {µ, Λ, k, m} there will be two horizons. Finding this region in the parameter space will be the first 1 The resulting geometry is, of course, not just the sum of terms of different geometries, but it casually coincides. Differences are bound to exist on the limit of vanishing of a given contribution. For instance, let us suppose that, given a set of parameters, say {m, µ, Λ, k }, we can switch off µ (by neglecting it with respect to the others). The resulting geometry is topologically different to the one obtained by not assuming any monopole at all at the beginning, that is, the limit does not coincide. However, in the cases studied here, this is no more than an enough-to-be-aware-of subtlety.\n\ntask. After that, attention will be focused on the coalescence point of the horizons 2 . The analysis consists of two steps, first, the parameterization of the coordinate separation of the horizons (ǫ) and the calculation of the physical distance between them when coalescence takes place (ǫ → 0). Then, following the strategy in [12] , the computation of the line element of the remaining geometry. This program is carried out on two cases: m = 0 and m = 0, which are treated in the next subsections, respectively. The massless case must be seen as a toy model of the massive one. This distinction is made not merely for simplicity but also because, as will be explained, the mass parameter comes out naturally for dynamical requirements."
},
{
"section_type": "OTHER",
"section_title": "Case m = 0",
"text": "In the massless case, ∆(r) gets reduced to\n\n∆(r) = 1 - µ 2 r 2 - r 2 R 2 . ( 7\n\n)\n\nSolving ∆ = 0 is equivalent to finding the zeroes of a biquadratic equation as long as r = 0 is not considered. We perform the change z ≡ r 2 and solve a second order ordinary equation. The horizons are found to be at\n\nz + = R 2 2 1 -1 - 4µ 2 R 2 ( 8\n\n) z ++ = R 2 2 1 + 1 - 4µ 2 R 2 . ( 9\n\n)\n\nR > 2µ guarantees the existence of two positive solutions and, therefore, four solutions for the quartic equation. Two of them, r + = + √ z + and r ++ = + √ z ++ , correspond to the radial coordinate of the inner (black hole) and outer (cosmological) horizon respectively. If R = 2µ, both solutions coincide, which means that the horizons coalesce. As said before, this does not mean that the geometry vanishes as a naive observation (given a wrong choice of coordinates) would make one think. Physical distance between the horizons, on the contrary, remains finite at the limit. In order to prove this, let us compute it. For fixed time and angular coordinates, the physical distance is\n\nD(µ, R) = r ++ r + rR [-r 4 + R 2 r 2 -µ 2 R 2 ] 1/2 dr = = z ++ z + R 2 R 4 4 -µ 2 R 2 -z -R 2 2 2 1/2 dz ( 10\n\n)\n\nThe requirement R > 2µ implies R 4 4 -µ 2 R 2 > 0 so the above integral is exactly solved as an cos -1 -type. The result is\n\nD(R) = 1 2 πR. ( 11\n\n)\n\nSurprisingly, the physical distance does not depend on µ. It means that, given a cosmological constant Λ, one could \"switch on\" the monopole and go on till the horizons coalesce but the distance would remain unchanged. However, because of quantization 2 Coalescence as seen in Schwarzschild coordinates.\n\nrequirements, monopole charge µ cannot be tuned, but needs to have, instead, a fixed value upto a sign. On the other hand, the cosmological constant, Λ, should be chosen when writing the lagrangian. It means that changing its value does not drive us from one model to another but implies an essential change in the theory [17] . Therefore, we are not free to adjust any parameter arbitrarily as done with the mass of the black hole in the Schwarzschild-de Sitter case. Then, eventhough physical reasons would lead the horizons to coalesce, the absence of any free parameter in our model makes it impossible. In the next section, m will come to our help as a free parameter for the model. Despite the last remark, one could wonder about the kind of geometry that remains when the horizons coalesce. This task, even if seems just a curious exercise now, will be useful for the next section. Applying a technique similar to the one Gingspar and Perry [12] used to study the geometry of Nariai's solution, we proceed by, first, parametrizing the separation of horizons as\n\nR = 2µ(1 + ǫ 2 ), ( 12\n\n)\n\nin a way that coalescence corresponds to taking ǫ = 0. Then, we define a \"wise\" change of coordinates\n\nχ = cos -1 -2 R 2 A (r 2 -r 2 0 ) τ = ǫ 2it R , ( 13\n\n)\n\nwhere A = 1 -4µ 2 R 2 and r 2 0 = R 2 2 , and the angular coordinates remain unchanged. The new coordinates (13) might seem randomly chosen at first sight. However, there are some reasons that justify such a functional dependence. For instance, χ is nothing but the physical distance between r + and r. The timelike coordinate t is multiplied by i in order to work in the Euclidean region 3 and by ǫ because ∆/ǫ 2 is expected to have a finite limit when ǫ → 0. Now, we apply (12) and (13) and expand ∆(r(χ))dτ 2 , ∆ -1 (r(χ))dχ 2 and r 2 (χ) up to first order in ǫ. The line element (4) reads\n\nds 2 = µ 2 dχ 2 + µ 2 sin 2 (χ) 1 + ǫ √ 2 cos(χ) dτ 2 + + 2µ 2 1 - √ 2 cos(χ)ǫ dΩ 2 2k . ( 14\n\n)\n\nWe take limit ǫ → 0 to obtain\n\nds 2 = µ 2 dχ 2 + sin 2 (χ)dτ 2 + 2µ 2 dΩ 2 2k . ( 15\n\n)\n\nAs seen in ( 15 ), the 2k-sphere decouples from the rest. The resulting geometry is S 2 ×S 2k for k ≥ 2. Notice the parallelism between this geometry and Nariai's solution, which is S 2 × S 2 . The \"classical\" relation between radii (6) gets also generalized to\n\na -2 + b -2 = C 0 Λ, ( 16\n\n)\n\nwhere C 0 = 6 k(2k+1) . The geometry (15) can be viewed as a \"degenerate\" black hole, in which the two horizons have the same (maximum) size and are in thermal equilibrium. This could be interpreted by an observer as a bath of radiation coming from both horizons 3 τ will be periodic at both horizons, although different in each case. Equality will hold at the coalescence point, when thermal stability is reached.\n\nat a precise temperature [19] . The temperature can be calculated by means of surface gravity κ, as computed in the new coordinates (13) T = 1 2π √ 2µ = 1 π k(2k + 1) 2Λ 1/2\n\n. ( 17\n\n)\n\nThe entropy can also be computed as a quarter of the sum of the two horizons [18], so\n\nS = 1 4 A T = 1 2 A H = 1 2 ω 2k k(2k + 1) 2Λ 2 , ( 18\n\n)\n\nwhere ω 2k is the area of the 2k-dimension unit sphere."
},
{
"section_type": "OTHER",
"section_title": "Case m = 0",
"text": "In the massive case we recover the full expression (5) for ∆. Since the singular point r = 0 is not to be considered, we better analyze the function r 2k-1 ∆(r)\n\n∆ ≡ r 2k-1 ∆ = - r 2k+1 R 2 + r 2k-1 -µ 2 r 2k-3 -2Gm. ( 19\n\n)\n\nIt is known that a polynomial equation with powers equal to or higher than five is not generally solvable in a symbolical way. This happens for k ≥ 2. So, the purpose of doing a study for the massive case analogous to that achieved in the first section is ruined. Nevertheless, some information can be extracted from (19) . We should first remember the sign of the parameters: R 2 > 0 (de Sitter), µ 2 > 0 for k ≥ 2, and m will be free in principle. Derivating (19) and equating to zero leads to a biquadratic equation of the form\n\n- 1 R 2 (2k + 1)r 4 + (2k -1)r 2 -(2k -3)µ 2 = 0, ( 20\n\n)\n\nwhich, as long as\n\nΛµ 2 ≤ k 4 (2k -1) 2 2k -3 , ( 21\n\n)\n\nhas two positive (and two negative) roots, r min and r max ≡ r c . In terms of the cosmological constant\n\nr 2 c ≡ k(2k -1) 2Λ 1 + 1 - 4(2k -3)Λµ 2 k(2k -1) 2 , ( 22\n\n)\n\nr min is obtained from (22) by swapping the sign of the square root. A quick look at (19) shows that the smallest root is a minimum and the largest is a maximum of function ∆. Now, let us plug r c into (19): 1. If m > 0, then (see fig. 1 ) a) ∆(r c ) ≥ 0 implies that there are two event horizons, the black hole and the cosmological horizon. The inequality gets saturated at the coalescence point.\n\nb) ∆(r c ) < 0 means that no horizon is found.\n\n2. If m < 0, then (see fig. 2 ) a) ∆(r min ) < 0 together with ∆(r c ) < 0 implies that there is just one Cauchy horizon. b) ∆(r min ) < 0 together with ∆(r c ) > 0 assures the existence of a Cauchy horizon and both black hole and cosmological horizon. c) ∆(r min ) > 0 leaves us with the cosmological horizon only.\n\nThe case we will study is ∆(r min ) < 0 and ∆(r c ) > 0 which, independently of the sign of m, assures 4 the existence of black hole and cosmological horizons. This corresponds to values of m within range (see fig. 3) 1.a 1.b\n\nFigure 1: Case m > 0. The curve represents function ∆(r). Figure 1 .a has two roots which correspond to the black hole (r + ) and cosmological horizon (r ++ ) respectively. Figure 1.b shows the absence of horizons.\n\n2.a 2.b 2.c\n\nFigure 2: Case m < 0. This time ∆(r) permits the existence of one (Cauchy) horizon as in Figure 2.a, three horizons (Cauchy, black hole and cosmological) as in 2.b, or just the cosmological horizon as shown in 2.c.\n\nm -< m < m + , ( 23\n\n)\n\nwhere\n\nGm c ≡ Gm + = 1 1 + 2k r 2k-3 c (r 2 c -2µ 2 ). ( 24\n\n)\n\nThe value of Gm -is obtained by replacing r c → r min . In terms of Λ and µ we get\n\nGm ± = (2Λ) -k+1/2 1 + 2k -k + 2k 2 ± k 2 (1 -2k) 2 -4Λµ 2 (2k -3)k k-3/2 -k + 2k 2 -4Λµ 2 ± k 2 (1 -2k) 2 -4Λµ 2 (2k -3)k . ( 25\n\n)\n\n4 The value of m can be negative. That is because m should not be thought of as an entity with physical meaning but as a geometrical parameter. Short calculation in (25) shows that m gets negative values for Λµ 2 ≥ k 4 (1 + 2k).\n\nr r,m r,m r c Figure 3: This figure shows the range of \"masses\" which are consistent with the existence of both black hole and cosmological horizons. The curve ∆(r) \"moves down\" in the process of coalescence.\n\nThe crucial point is that both horizons coalesce when r c is a root of (19) which happens at m = m c (k, Λµ 2 , Λ). Two relations have been imposed so far: d e ∆(r;m) dr 20) , which defines r c , and ∆(r c ; m c ) = 0 which leads to m c . In order for m to be real, the bound which must be impossed on Λµ 2 coincides with (21) which, in turn, is nothing but the condition for the existence of two horizons. So, if a given a value for Λµ 2 is low enough to produce two horizons, there always exists a real value of m which makes them coalesce. Again, as in the Schwarzschild-de Sitter example, the system is unstable and the equilibrium point is reached at m = m c . Unlike the massless case, plugging m gives us enough room for maneuvre to drive the system to equilibrium. At this point, we would like to remark that the procedure of horizon coalescence, as studied in detail below, may be seen as a flow in a line which undergoes a Pitchfork bifurcation at the coalescence point. Parameter m, moved by thermal instability, drives the system to the critical situation. For concreteness see Appendix B. Let us focus on the near coalescence point. This can be parameterized by\n\n| rc = 0, that is, (\n\nr = r c + δr = r c (1 + ǫ cos χ) (26) m = m c -δm = m c (1 + bǫ 2 ).\n\nParameterization of r also involves a change of coordinates r → χ and should be taken as imposed at the moment although it will be justified later. The horizons will be symmetrically located at: r + = r c (1 -ǫ) and r ++ = r c (1 + ǫ) which correspond to χ + = π and χ ++ = 2π, respectively 5 . The value of b as well as the absence of a linear term in ǫ of the parameterization of m may be explained as follows. Near the coalescence point one should Taylor expand ∆ around r c and have in mind that, for δm mc ≪ 1, ∆ is aproximately parabolic, so that second order expansion is enough. By definition ∆(r + ) = ∆(r ++ ) = 0 5 For a small enough ǫ, it is expected that the parabolic approach holds and, then, both horizons are symmetrically located with respect to r c .\n\nand ∆ reaches a maximum at r c . So,\n\n0 = ∆(r ++ ) = ∆(r c , m) + ∆ ′ (r c , m)(r c ǫ) + 1 2 ∆ ′′ (r c , m)(r c ǫ) 2 = = 2Gδm r 2k-1 c + 1 2 ∆ ′′ (r c ; m c )r 2 c ǫ 2 , ( 27\n\n)\n\nwhich means that\n\nb = ∆ ′′ (r c ; m c )r 2k+1 c 4Gm c . ( 28\n\n)\n\nCalculating the physical distance near the coalescence point would, again, imply solving the integration\n\nD(ǫ) = r ++ r + dr ∆ 1/2 (r) , ( 29\n\n)\n\nwhere r ++ = r + + 2r c ǫ. Although the exact result is not computed, an explicit proof of its finite nonzero value is given in Appendix A. The procedure of calculating the physical distance also brings us some light on which is the change of coordinates that should be made in order to understand the resulting geometry. It turns out to be\n\nχ = cos -1 1 ǫr c (r -r c ) τ = ǫi Br c t, ( 30\n\n) where B = k k -2k 2 + 2Λr 2 c ( 31\n\n)\n\nis a dimensionless factor.\n\nThe coalescence of horizons takes place at ǫ = 0. In order to study the geometry at the limit we proceed by calculating -∆dt 2 , ∆ -1 dr 2 and r 2 in the new coordinates (30) and expand in ǫ around ǫ = 0. The new line element gets determined by taking the zero order of the expansion. The relations for r and m in (26) are in accordance with (30) , where b takes the value of (28) , by virtue of the parabolic approach. From (30), it is straightforward to see that r 2 takes a constant value r 2 c . Surprisingly, as in the massless and Schwarzschild-de Sitter cases, the geometry splits in two disconnected parts which lead to a product manifold S 2 × S 2k . The line element reads\n\nds 2 = Br 2 c dχ 2 + sin 2 (χ)dτ 2 + r 2 c dΩ 2 2k , ( 32\n\n)\n\nwhere χ ∈ [π, 2π] and τ is periodic 6 . As seen in (32) , S 2 has radius a 2 = Br 2 c , and S 2k has radius b 2 = r 2 c . Now, the generalized Bertotti relation (6) is\n\na -2 + b -2 = 2(1 -k) r 2 c + 2Λ k = CΛ, ( 33\n\n)\n\nwhere C(k, Λµ 2 ) is obtained by inserting (22) in (33) . Note that C k, Λµ 2 = k(2k + 1)/4 = C 0 , and then (33) turns into (16) , that is, into the massless case. This is no 6 τ is periodic on both horizon surfaces all over the process in order to avoid the conical singularity at the horizons. At the coalescence point, however, both periods equal.\n\nsurprising since Λµ 2 = k(2k + 1)/4 is the condition for coalescence in the massless case (equivalent to R = 2µ), and, at the same time, it makes m c = 0. So, the massive geometry is a consistent extension of the massless one. Now, fixing Λ does not determine uniquely the geometry. Another dimensionless variable Λµ 2 is required.\n\nAs in the last section, the geometry (32) can be viewed as a \"degenerate\" black hole, in which the two horizons have the same (maximum) size and are in thermal equilibrium.\n\nIn the present case the temperature is given in terms of the surface gravity κ by\n\nT = κ 2π = 1 2πr c √ B . ( 34\n\n)\n\nIn Planck units,the entropy associated with this solution may be calculated (given that it is not extreme 7 ) by means of the total area of the horizons as\n\nS = 1 4 A T = 1 2 A H = 1 2 ω 2k r 4 c . ( 35\n\n)"
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusions",
"text": "The spherically symmetric solution of gravity due to a magnetic monopole in arbitrary dimension has been studied, in particular, when the set of parameters {Λ, µ, m, k} allows the existence of two horizons. In these cases, thermal instabilities drive a process of horizon coalescence. Even though coordinate separation between the horizons shrinks to zero, it has been proven in both the massless and the massive case that the physical distance does not. The geometry of the remaining space between the horizons has been calculated in both cases. They turned out to be Nariai-type solutions, that is, the product of a 2-sphere and a 2k-sphere for a (2k + 2)-dimensional spacetime. In each solution, the radii of the spheres are not independent. They are related by an elliptical equation which should be understood as the generalization of the relation found by Bertotti. The unique generalized equation involving these radii for both the massless and the massive case has been given. After computing the line element in each case, the thermodynamical properties (Hawking temperature and entropy) due to the existence of horizons have been calculated. The Yang monopole corresponds to the six dimensional case, where k = 2. The geometry obtained after coalescence is S 2 × S 4 as can be explicitly read in (32) . This case is especially interesting since it may be described in String Theory (a realization of the Yang monopole in Heterotic String Theory has recently been done [21] as well as another complementary picture in Type-IIA String Theory [20] ). In the same context, it looks possible to find results (18) and (35) for the entropy by application of some attractor mechanism [22, 23] . We believe that this would be an interesting topic to be addressed in future research.\n\n7 A charge black hole is said to be extreme when it has the minimum mass. Then, as it cannot release any energy without losing charge, it is supposed not to emit, and its associated Hawking temperature is 0. The black hole we are dealing with in this paper is extreme in the sense of carrying the \"maximum mass\" allowed by the cosmological constant Λ. Obviously, the temperature will not be zero.\n\nA Proof of the finite nonzero physical distance\n\nComputing the physical distance is equivalent to performing the integration\n\nD = r ++ r + dr ∆ 1/2 (r) , ( 36\n\n)\n\nwhere, for small ǫ, r ++ = r + + 2r c ǫ. Divergencies might appear at the points where ∆ → 0. The case we have been considering all along section (3.2) concerns the existence of two horizons which coalesce, that is, two single roots r + and r ++ of ∆ which join to form a double one. Function ∆ can always be expressed as ∆ = (r -r + )(r ++ -r)g(r), where g(r) is a polynomial function of powers of degree 2k -1 and no zeroes within the range [r + , r ++ ] are to be found by construction. Explicitly, equation (36) is\n\nD(ǫ) = r + +2rcǫ r + dr (r -r + ) 1/2 (r + + 2r c ǫ -r) 1/2 r k-1/2 g 1/2 (r) h(r) . ( 37\n\n)\n\nNow, h(r) is a continuous divergenceless strictly positive function in the compact [r + , r ++ ], which means that it will reach a positive maximum and minimum for certain r ′ s. Let us call h max and h min the values of the function h in these points 8 . Then h min r + +2rcǫ\n\nr + dr (r -r + ) 1/2 (r + + 2r c ǫ -r) 1/2 ≤ D(ǫ) ≤ ≤ h max r + +2rcǫ r + dr (r -r + ) 1/2 (r + + 2r c ǫ -r) 1/2 . ( 38\n\n)\n\nThe integration can be performed:\n\nr + +2rcǫ r + dr (r -r + ) 1/2 (r + + 2r c ǫ -r) 1/2 = π. ( 39\n\n) Now D(ǫ → 0) = πh(r c ), ( 40\n\n)\n\nwhere the value of r c is given in (22) . Integrations of form (39) are solved exactly by a cos -1 type function, and a nonzero finite result is obtained. It is remarkable that the same can be said for any ∆ we would choose, as long as no more than two single roots were to join to form a double one. The key point is that (39) , which could be problematic, is independent of ǫ and therefore the distance is finite in the limit, when ǫ → 0. So, eventhough (39) was neither exactly the physical distance in the massive case nor in Schwarzschild-de Sitter solution (however, it was in the massless case as we have already seen in the first section), it is closely related to it. This fact gives us a hint or, at least, justifies the change of coordinates we were performing once and again to study the geometry at the limit ǫ → 0.\n\n8 These, in principle, depend on ǫ but coincide when ǫ → 0:\n\nh min = h max ≡ h(r c ).\n\nB Horizon coalescence as a flow on the line\n\nThe main phenomenon that concerns this paper, as said before, can be described in terms of the dynamics of a vector field on the line. The coalescence point, in this picture, is no more than a supercritical Pitchfork bifurcation. Let us remember some general features of the dynamics of a one-dimensional flow. The equation of a general vector field on the line can be expressed as: ẋ = f (x, α) (41)\n\nwhere f is any real function with real support, the dot means differentiation with respect to t and α is a parameter of the model. Fixed points of (41) require ẋ = 0, which must be obtained by finding the roots of f , that is f (x * , α) = 0. (42) Equation (42) is solved by an n-collection of fixed points x * i for a given value of α. Let us suppose that f has three roots if α = α 0 . Fixed points come closer as α moves and get \"condensed\" in a \"fat\" fixed point (bifurcation point) at α = α c . A paradigmatic example of a Pitchfork bifurcation is shown by function\n\nf (x) = x(α -x 2 ). ( 43\n\n)\n\nOne question arises naturally now about the role the horizons play in this picture. Let us claim that horizons are fixed points and the role of α is played by m. We will justify this identification by constructing the vector flow. Constructing a flow in a manifold (in our case it will be a line) is equivalent to giving a family of curves r(t) which covers the manifold or part of it. Each of the curves gets specified by the initial condition, say, r(t = 0). Now, let us consider geodesic motions. Without loss of generality, the angular coordinates of our geometry will be frozen, θ and φ are constants, and only radial curves r(t) are to be regarded. Static coordinate system will serve us to describe the movement for any r ∈ (r + , r ++ ). Let us invoke intuition at this point. If r(0) is near the cosmological or the black hole horizon it is clear that a test particle will move out of the region by approaching each horizon respectively. Then, there is a point r = r g where the test particle will not \"feel\" any force and, consequently, it will not move 9 . This is the first (unstable) fixed point.\n\nLet us move the origin by defining r ′ = r -r g , after this, primes will be dropped out to simplify notation. The flow at each point will be determined by the physical velocity ṙ(r) (as measured by an observer placed at r = 0) that a test particle would adquire at r if it is dropped with ṙ = 0 at around r = 0 (as close as possible). It is not hard to see that the velocity of the test particle, as seen by the static geodesic observer, is bound to be zero at both horizons. So, horizons are fixed points. Now, our system can be treated as a vector flow ṙ(r) which covers the region between the horizons. The vector flow has three fixed points: {r + , r ++ , 0} where the first two are stable. As m runs towards m c , the system shrinks into a Pitchfork bifurcation. Near the bifurcation point the flow can be approximated by\n\nṙ = βr(r -r + )(r ++ -r), ( 44\n\n)\n\n9 r g in our geometry, plays the role the asymptotic infinity does in Schwarzschild solution,that is, the point where the time-like Killing vector should be normalized in order to define the horizon temperature. Note that r g ≡ r c at the coalescence point, that is, when ǫ = 0.\n\nwhere β is a positive constant which depends on µ, k and Λ. On the one hand, in the coordinate system {χ, τ }, and using (30), we have dr dt = dr dχ dχ dτ dτ dt -→ ṙ = iǫ 2 B dχ dτ . (45) On the other hand, equation (44), expressed in the new coordinate system, reads ṙ = ǫ 3 βr 3 c cos χ sin 2 χ, and so dχ dτ = -iǫβr 3 c B cos χ sin 2 χ. (46)\n\nAs expected, in the new coordinate system, every point converts into a fixed point as horizons coalesce (ǫ → 0). Since the flux lines were identified with geodesics of test particles, this can be understood as the abscence of forces at the end of the process."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgment",
"text": "We thank P. K. Townsend and Adil Belhaj for helpful discussions and Jean Nuyts for critical reading of the manuscript. This work has been supported by MCYT ( Spain) under grant FPA 2003-02948.\n\nReferences [1] P. A. M. Dirac, Quantised singularities in the electromagnetic field, Proc. Roy. Soc. Lond. A 133, 60 (1931). [2] Y. Aharonov and D. Bohm, Significance of Electromagnetic Potentials in the Quantum Theory, Phys. Rev. 115, 485 (1959). [3] C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96, 191 (1954). [4] Ryoyu Utiyama, Invariant Theoretical Interpretation of Interaction, Phys. Rev. 101, 1597 (1956). [5] E. Lubkin, Geometric Definition of Gauge Invariance, Ann. Phys. 23, 233 (1963). [6] C. N. Yang, Generalization of Dirac's monopole to SU2 gauge fields, J. Math. Phys. 19, 320 (1978).\n\n[7] P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories And Magnetic Charge, Nucl. Phys. B 125, 1 (1977). [8] Zalan Horvath, Laszlo Palla, Spontaneous Compactification And 'Monopoles' In Higher Dimensions., Nucl. Phys. B 142, 327 (1978). [9] G.W. Gibbons and P.K. Townsend, Self-graviting Yang monopoles in all dimensions, Class. Quantum Grav. 23, 4873 (2006).\n\n[10] S. Coleman, The magnetic monopole fifty years later, in The Unity of the Fundamental Interactions, ed. A. Zichichi (Plenum, New York, 1983).\n\n[11] E. J. Weinberg and P. Yi, Magnetic Monopole Dynamics, supersymmetry, and Duality, Phys. Rept. 43, 65 (2007). hep-th/0609055. [12] Paul Ginsparg and Malcom J. Perry, Semiclassical perdurance of de Sitter space, Nuclear Physics B 222, 245 (1983).\n\n[13] H. Nariai, Sci. Rep. Tohoku Univ., Ser. 1 35, 62 (1951). [14] E. Kasner, Trans. Am. Math. Soc. 27, 101 (1925).\n\n[15] B. Bertotti, Uniform Magnetic Field in the Theory of General Relativity, Phys. Rev. 116, 1331 (1959).\n\n[16] Marcello Ortaggio, Impulsive waves in the Nariai Universe, Phys. Rev. D 65, 084046 (2002). [17] G. W. Gibbons and S. W. Hawking, Cosmological Event Horizons, Thermodynamics, And Particle Creation, Phys. Rev. D 15, 2738 (1977). [18] S. W. Hawking and Simon F. Ross, Duality between electric and magnetic black holes, Phys. Rev. D 52, 5865 (1995) [19] R. Bousso and S. W. Hawking, Pair creation of black holes during inflation, Phys. Rev. D 54, 6312 (1996). [20] A. Belhaj, P. Diaz, A. Segui, On the Superstring Realization of the Yang Monopole, (2007). hep-th/0703255. [21] E. A. Bergshoeff, G. W. Gibbons and P. K. Townsend, Open M5-branes, Phys. Rev. Lett. 97, 231601 (2006). hep-th/0607193. [22] S. Ferrara, R. Kallosh, A. Strominger, N=2 extremal black holes, Phys. Rev. D 52, 5412 (1995). [23] S. Ferrara, R. Kallosh, Supersymmetry and Attractors, Phys. Rev. D 54, 1514 (1996)."
}
] | |
arxiv:0704.0367 | 0704.0367 | 1 | d1ea2d5ebc8bc890820988399fa841935bb6bc94bd400b7160c0b0b2e3602f9e | Instanton representation of Plebanski gravity. Consistency of the
initital value constraints under time evolution | The instanton representation of Plebanski gravity provides as equations of
motion a Hodge self-duality condition and a set of `generalized' Maxwell's
equations, subject to gravitational degrees of freedom encoded in the initial
value constraints of general relativity. The main result of the present paper
will be to prove that this constraint surface is preserved under time
evolution. We carry this out not using the usual Dirac procedure, but rather
the Lagrangian equations of motion themsleves. Finally, we provide a comparison
with the Ashtekar formulation to place these results into overall context. | [
"Eyo Eyo Ita III"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | In [1] a new formulation of general relativity was presented, named the instanton representation of Plebanski gravity. The basic dynamical variables are an SO(3, C) gauge connection A a µ and a matrix Ψ ae taking its values in two copies of SO(3, C). 1 The consequences of the associated action I Inst were determined via its equations of motion, which hinge crucially on weak equalities implied by the the initial value constraints. For these consequences to be self-consistent, the constraint surface must be preserved for all time by the evolution equations. The present paper will demonstrate that this is indeed the case. We will not use the usual Hamiltonian formulation for totally constrained systems [2] , since we will not make use of any canonical structure implied by I Inst . Rather, we will deduce the time evolution of the dynamical variables directly from the equations of motion of I Inst .
Sections 2 and 3 of this paper present the instanton representation action and derive the time evolution of the basic variables. Sections 4, 5 and 6 demonstrate that the nondynamical equations, referred to as the diffeomorphism, Gauss' law and Hamiltonian constraints, evolve into combinations of the same constraint set. The result is that the time derivatives of these constraints are weakly equal to zero with no additional constraints generated on the system. While we do not use the usual Dirac method in this paper, the result is still that the instanton representation is in a sense Dirac consistent. We will make this inference clearer by comparison with the Ashtekar variables in the discussion section. On a final note, the terms 'diffeomorphism' and 'Gauss' law' constraints are used loosely in this paper, in that we have not specified what transformations of the basic variables these constraints generate. The use of these terms is mainly for notational purposes, due to their counterparts which appear in the Ashtekar variables.
The starting action for the instanton representation of Plebanski gravity is given by [1]
I Inst = dt Σ d 3 xΨ ae B k e F a 0i + ǫ kjm B j a N m -iN (detB) 1/2 √ detΨ Λ + trΨ -1 , ( 1
)
1 Index labelling conventions for this paper are that symbols a, b, . . . from the beginiing of the Latin alphabet denote internal SO(3, C) indices while those from the middle i, j, k, . . . denote spatial indices. Both of these sets of indices take takes 1, 2 and 3. The Greek symbols µ, ν, . . . refer to spacetime indices which take values 0, 1, 2, 3.
where N µ = (N, N i ) are the lapse function and shift vector from metric general relativity, and Λ is the cosmological constant. The basic fields are Ψ ae and A a i , and we action (1) is defined only on configurations restricted to (detB) = 0 and (detΨ) = 0. 2 In the Dirac procedure one refers to N µ as nondynamical fields, since their velocities do not appear in the action. While the velocity Ψae also does not appear, we will distinguish this field from N µ since the action (1), unlike for the latter, is nonlinear in Ψ ae .
The equation of motion for the shift vector N i , the analogue of the Hamilton equation for its conjugate momentum Π N , is given by δI Inst δN i = ǫ mjk B j a B k e Ψ ae = (detB)(B -1
) d i ψ d ∼ 0, ( 2
)
where ψ d = ǫ dae Ψ ae is the antisymmetric part of Ψ ae . This is equivalent to the diffeomorphism constraint H i owing to the nondegeneracy of B i a , and we will often use H i and ψ d interchangeably in this paper. The equation of motion for the lapse function N , the analogue of the Hamilton equation for its conjugte momentum Π N , is given by δI Inst δN = (detB) 1/2 √ detΨ Λ + trΨ -1 = 0. (3) Nondegeneracy of Ψ ae and the magnetic field B i e implies that on-shell, the following relation must be satisfied Λ + trΨ -1 = 0, (4) which we will similarly take as synonymous with the Hamiltonian constraint. The equation of motion for Ψ ae is
δI Inst δΨ ae = B k e F a 0k + ǫ kjm B k e B j a N m + iN √ detB √ detΨ(Ψ -1 Ψ -1 ) ea ∼ 0, ( 5
)
up to a term proportional to (4) which we have set weakly equal to zero. One could attempt to define a momentum conjugate to Ψ ae , for which (5) would be the associated Hamilton's equation of motion. But since Ψ ae forms part of the canonical structure of (1), then our interpretation is that this is not technically correct. 3 The equation of motion for the connection A a µ is given by
δI Inst δA a µ ∼ ǫ µσνρ D σ (Ψ ae F e νρ ) - i 2 δ µ i D ij da 4ǫ mjk N m B k e Ψ [de] +N (B -1 ) d j √ detB √ detΨ Λ + trΨ -1 , ( 6
) 2
The latter case limits the application of our results to spacetimes of Petroc Types I, D and O (See e.g. [3] and [4] .
3 This is because (5) contains a velocity Ȧa k within F a 0k and will therefore be regarded as an evolution equation rather than a constraint. This is in stark contrast with (2) and (3), which are genuine constraint equations due to the absence of any velocities.
where we have defined D ji ea (x, y) ≡ δ δA a i (x) B j e (y) = ǫ jki -δ ae ∂ k + f eda A d k δ (3) (x, y); D
0i ea ≡ 0. ( 7
)
The terms in large round brackets in (6) vanish weakly, since they are proportional to the constraints (2) and (4) and their spatial derivatives. For the purposes of this paper we will regard (6) as synonymous with ǫ µσνρ D σ (Ψ ae F e νρ ) ∼ 0. (
) 8
In an abuse of notation, we will treat (5) and (8) as strong equalities in this paper. This will be justified once we have completed the demonstration that the constraint surface defined collectively by (2), (3) and the Gauss' constraint from (8) is indeed preserved under time evolution. As a note prior to proceeding we will often make the identification
N (detB) 1/2 √ detΨ ≡ √ -g ( 9
)
as a shorthand notation, to avoid cluttering many of the derivations which follow in this paper.
Prior to embarking upon the issue of consistency of time evolution of the initial value constraints, we will check for internal consistency of I Inst , which entails probing of the physical content implied by (8) and (5). First, equation (8) can be decomposed into its spatial and temporal parts as
D i (Ψ bf B i f ) = 0; D 0 (Ψ bf B i f ) = ǫ ijk D j (Ψ bf F f 0k ). ( 10
)
The first equation of (10) is the Gauss' law constraint of a SO(3) Yang-Mills theory, when one makes the identification of Ψ bf B i f ∼ E i b with the Yang-Mills electric field. The Maxwell equations for U (1) gauge theory with sources (ρ, J ), in units where c = 1, are given by
∇ • B = 0; Ḃ = -∇ × E = 0; ∇ • E = ρ; ˙ E = -J + ∇ × B. ( 11
)
Equations (10) can be seen as a generalization of the first two equations of (11) to SO(3) nonabelian gauge theory in flat space when one: (i) identifies F f 0k ≡ E f k with the SO(3) generalization of the electric field E, and (ii) one chooses Ψ ae = kδ ae for some numerical constant k.
When ρ = 0 and J = 0, then one has the vacuum theory and equations (11) are invariant under the transformation
( E, B) -→ (-B, E). ( 12
)
3 Then the second pair of equations of (11) become implied by the first pair. This is the condition that the Abelian curvature F µν , where F 0i = E i and ǫ ijk F jk = B i , is Hodge self-dual with respect to the metric of a conformally flat spacetime. But equations (10) for more general Ψ ae encode gravitational degrees of freedom, which as shown in [1] generalizes the concept of selfduality to more general spacetimes solving the Einstein equations. Let us first attempt to derive the analogue for (10) of the second pair of (11) in the vacuum case. Acting on the first equation of (10) with D 0 yields
D 0 D i (Ψ bf B i f ) = D i D 0 (Ψ bf B i f ) + [D 0 , D i ](Ψ bf B i f ) = 0. ( 13
)
Substituting the second equation of (10) into the first term on the right hand side of (13) and using the definition of temporal curvature as the commutator of covariant derivatives on the second term we have
D i (ǫ ijk D j (Ψ bf F f 0k )) + f bcd F c 0i Ψ df B i f = f bcd B k c F f 0k + B k f F c 0k Ψ df = 0 ( 14
)
where we have also used the spatial part of the commutator
ǫ ijk D i D j v a = f abc B k b v c .
Note that the term in brackets in (14) is symmetric in f and c, and also forms the symmetric part of the left hand side of (5)
B i f F b 0i + i √ -g(Ψ -1 Ψ -1 ) f b + ǫ ijk B i f B j b N k = 0, ( 15
)
re-written here for completeness. To make progress from (14), we will substitute (15) into (14). This causes the last term of (15) to drop out due to antisymmetry, which leaves us with
-i √ -gf bcd Ψ df (Ψ -1 Ψ -1 ) f c + Ψ df (Ψ -1 Ψ -1 ) f c = -2i √ -gf bcd Ψ -1 dc . ( 16
)
The equations are consistent only if (16) vanishes, which is the requirement that Ψ ae = Ψ ea be symmetric. This of course is the requirement that the diffeomorphism constraint (2) be satisfied. So the analogue of the second pair of (11) in the vacuum case must be encoded in the requirement that Ψ ae = Ψ ea be symmetric.
We must now verify that the initial value constraints are preserved under time evolution defined by the equations of motion (5) and (6). These equations are respectively the Hodge duality condition
B k f F b 0k + i √ -g(Ψ -1 Ψ -1 ) f b + ǫ ijk N i B j b B k f = 0, ( 17
)
4 and one of the Bianchi identity-like equations
ǫ ijk D j (Ψ ae F e ok ) = D 0 (Ψ ae B i e ). ( 18
)
Since the initial value constraints were used to obtain the second line of (17) from (1), then we must verify that these constraints are preserved under time evolution as a requirement of consistency.
Using F b 0i = Ȧb i -D i A b 0 and defining √ -g(B -1 ) f i (Ψ -1 Ψ -1 ) f b + ǫ mnk N m B n b ≡ iH b k , ( 19
)
Then equation (17) can be written as a time evolution equation for the connection, which is not the same as a constraint equation as noted earlier
F b 0i = -iH b i -→ Ȧb i = D i A b 0 -iH b i . ( 20
)
From equation (20) we can obtain the following equation governing time evolution equation for the magnetic field
Ḃi e = ǫ ijk D j Ȧe k = ǫ ijk D j D k A e 0 -iH e k = f ebc B i b A c 0 -iǫ ijk D j H e k = -δ θ B i e -iǫ ijk D j H e k , ( 21
)
which will be useful. On the first term on the right hand side of (21) we have used the definition of the curvature as the commutator of covariant derivatives. The notation δ θ in (21) suggests that that B i e transforms as a SO(3, C) vector under gauge transformations parametrized by θ b ≡ A b 0 . 4 Since we have not specified anything about the canonical structure of I Inst , then δ θ as used in (21) and in (24) should at this stage simply be regarded as a definition useful for shorthand notation.
We will now apply the Liebnitz rule in conjunction with the definition of the temporal covariant derivatives to (18) to determine the equation governing the time evolution of Ψ ae . This is given by
D 0 (Ψ ae B i e ) = B i e Ψae + Ψ ae Ḃi e + f abc A b 0 (Ψ ce B i e ) = ǫ ijk D j (Ψ ae F e 0k ). ( 22
)
Substituting (21) and (20) into the left and right hand sides of (22), we have
B i e Ψae + Ψ ae f ebc B i b A c 0 -iǫ ijk D j H e k + f abc A b 0 (Ψ ce B i e ) = -iǫ ijk D j (Ψ ae H e k ).( 23
)
In what follows, it will be convenient to use the following transformation properties for Ψ ae as A a i under SO(3, C) gauge transformations
δ θ Ψ ae = f abc Ψ ce + f ebc Ψ ac A b 0 ; δ θ A a i = -D i A a 0 ; δ θ B i e = -f ebc B i b A c 0 . ( 24
)
4 We will make the identification with SO(3, C) gauge transformations later in this paper when we bring in the relation of IInst with the Ashtekar variables.
Then using (24), the time evolution equations for the phase space variables Ω Inst can be written in the following compact form
Ȧb i = -δ θ A b i -iH b i ; Ψae = -δ θ Ψ ae -iǫ ijk (B -1 ) e i (D j Ψ af )H f k . ( 25
)
We have found evolution equations for Ψ ae and A a i from the covariant equations of A a µ and the Hodge-duality condition We have obtained these without using Poisson brackets, and by assuming that the Hamiltonian and diffeomorphism constraints are satisfied. Therefore the first order of business is then to check for the preservation of the initial value constraints under the time evolution generated by (25) . This means that we must check that the time evolution of the diffeomorphism, Gauss' law and Hamiltonian constraints are combinations of terms proportional to the same constraints and their spatial derivatives, and terms which vanish when the constraints hold. 5 These constraints are given by
w e {Ψ ae } = 0; (detB)(B -1 ) d i ψ d = 0; (detB) 1/2 √ detΨ Λ + trΨ -1 = 0(26)
where (detB) = 0 and (detΨ) = 0. We will occasionally make the identification
N (detB) 1/2 (detΨ) 1/2 ≡ √ -g ( 27
)
for a shorthand notation. Additionally, the following definitions are provided for the vector fields appearing in the Gauss' constraint
w e = B i e D i ; v e = B i e ∂ i ( 28
)
where D i is the SO(3, C) covariant derivative with respect to the connection A a i . Equations (26) are the equations of motion for the auxilliary fields A a 0 , N i and N .
The diffeomorphism constraint is directly proportional to ψ d = ǫ dae Ψ ae , the antisymmetric part of Ψ ae . So to establish the consistency condition for this constraint, it suffices to show that the antisymmetric part of the second equation of (25) weakly vanishes. This is given by
ǫ dae Ψae = -δ θ (ǫ dae Ψ ae ) -iǫ dae ǫ ijk (B -1 ) e i (D j Ψ af )H f k , ( 29
)
5 This includes any nonlinear function of linear order or higher in the constraints, a situation which involves the diffeomorphism constraint.
which splits into two terms. Using (24), one finds that the first term of (29) is given by
-ǫ dae δ θ Ψ ae = -ǫ dae f abc Ψ ce + Ψ ac f ebc A b 0 = -δ eb δ dc -δ ec δ bd Ψ ce + δ db δ ac -δ dc δ ab Ψ ac A b 0 = -Ψ db -δ bd trΨ + δ db trΨ -Ψ bd A b 0 = 2Ψ [bd] A b 0 = -ǫ dbh A b 0 ψ h , ( 30
)
which is proportional to the diffeomorphism constraint. The second term of (30) has two contributions due to H f k as defined in (19) . The first contribution reduces to
-iǫ dae ǫ ijk (B -1 ) e i (D j Ψ af )(H (1) ) f k = -iǫ dae ǫ ijk (B -1 ) e i (D j Ψ af ) √ -g(B -1 ) g k (Ψ -1 Ψ -1 ) gf = iǫ dae (detB) -1 ǫ egh (Ψ -1 Ψ -1 ) gf B j h D j Ψ af = i(detB) -1 (Ψ -1 Ψ -1 ) gf δ g d δ h a -δ g a δ h d v v {Ψ af } = i(detB) -1 (Ψ -1 Ψ -1 ) gf δ g d v a {Ψ af } -v d {Ψ gf } = i(detB) -1 (Ψ -1 Ψ -1 ) df G f + v d {Λ + trΨ -1 } . ( 31
)
The first term on the final right hand side of (31) is the Gauss' constraint and the second term is the derivative of a term direction proportional to the Hamiltonian constraint. 6 The second contribution to the second term of (29) is given by
ǫ dae ǫ ijk (B -1 ) e i (D j Ψ af )(H (2) ) f k = ǫ dae ǫ ijk (B -1 ) e i (D j Ψ af )ǫ mnk N m B n f = ǫ dac δ i m δ j n -δ i n δ j m (B -1 ) e i (D j Ψ af )N m B n f = ǫ dae N i (B -1 ) e i v f {Ψ af } -N j D j (ǫ dae Ψ ae ) = ǫ dae N i (B -1 ) e f G a -N j D j ψ d .( 32
)
The result is that the time evolution of the diffeomorphism constraint is directly proportional to
ψd = i(detB) -1 (Ψ -1 Ψ -1 ) da + ǫ dae N i (B -1 ) e i G a + A b 0 ǫ bdh -δ dh N j D j ψ h + i(detB) -1 v d {(-g) -1/2 H}, ( 33
)
which is a linear combination of terms proportional to the constraints (26) and their spatial derivatives. The result is that the diffeomorphism constraint H i = 0 is consistent with respect to the Hamiltonian evolution generated by the equations (25). So it remains to verify consistency of Gauss' law and the Hamiltonian constraints G a and H.
6 We have added in a term Λ, which can be regarded as a constant of integration with respect to the spatial derivatives from v d .
Having verified the consistency of the diffeomorphism constraint under time evolution, we now move on to the Gauss' constraint. Application of the Liebnitz rule to the first equation of (26) yields
Ġa = Ḃi e D i Ψ ae + B i e D i Ψae + B i e f abf Ψ f e + f ebg Ψ ag Ȧa i . ( 34
)
Upon substituion of (21) and (25) into (34), we have
Ġa = -δ θ B i e -iǫ ijk D j H e k D i Ψ ae + B m e D m -δ θ Ψ ae -iǫ ijk (B -1 ) e i (D j Ψ af )H f k +B i e f abf Ψ f e + f ebg Ψ ag -δ θ A b i -iH b i .( 35
)
Using the Liebniz rule to combine the δ θ terms of (35), we have
Ġa = -δ θ G a -iǫ ijk (D j H e k )D i Ψ ae + B m e D m ((B -1 ) e i (D j Ψ af )H f k ) -i f abf Ψ f e + f ebg Ψ ag B i e H b i . ( 36
)
The requirement of consistency is that we must show that the right hand side of (36) vanishes weakly. First, we will show that the third term on the right hand side of (36) vanishes up to terms of linear order and higher in the diffeomorphism constraint. This term, up to an insignificant numerical factor, has two contributions. The first contribution is
f abf Ψ f e + f ebg Ψ ag B i e (H (1) ) b i = √ -g f abf Ψ f e + f ebg Ψ ag (Ψ -1 Ψ -1 ) eb = √ -g f abf (Ψ -1 ) f b + f ebg (Ψ -1 Ψ -1 ) eb Ψ ag ∼ δ (1) a ( ψ) ∼ 0, ( 37
)
which is directly proportional to a nonlinear function of first order in ψ d which is proportional to the diffeomorphism constraint. The second contribution to the third term on the right hand side of (36) is
f abf Ψ f e + f ebg Ψ ag B i e (H (2) ) b i = f abf Ψ f e + f ebg Ψ ag ǫ kmn N k B m e B n b = f abf Ψ f e + f ebg Ψ ag (detB)N k (B -1 ) d k ǫ deb = (detB)N k (B -1 ) d k δ f d δ ae -δ f e δ ad Ψ f e + 2δ dg Ψ ag = (detB)N k (B -1 ) d k Ψ da -δ ad trΨ + 2Ψ ad ≡ δ (2) a ( N ) ( 38
)
which does not vanish, and neither is it expressible as a constraint. For the Gauss' law constraint to be consistent under time evolution, a necessary condition is that this δ (2) a ( N ) term must be exactly cancelled by another term arising from the variation.
Let us expand the terms in square brackets in (36) . This is given, using the Liebniz rule on the second term, by
ǫ ijk (D j H e k )(D i Ψ ae ) + ǫ ijk B m e D m ((B -1 ) e i (D j Ψ ae )H f k ) = ǫ ijk (D j H e k )(D i Ψ ae ) -ǫ ijk B m e (B -1 ) e n (D m B n g )(B -1 ) g i (D j Ψ af )H f k +ǫ mjk (D m D j Ψ af )H f k + ǫ mjk (D j Ψ af )(D m H f k ). ( 39
)
The first and last terms on the right hand side of (39) cancel, which can be seen by relabelling of indices. Upon application of the definition of curvature as the commutator of covariant derivatives to the third term, then (39) reduces to
-ǫ ijk (D n B n g )(B -1 ) g i (D j Ψ af )H f k + H f k B k b f abc Ψ cf + f f bc Ψ ac . ( 40
)
The first term of (40) vanishes on account of the Bianchi identity and the second term contains two contributions which we must evaluate. The first contribution is given by 37) . So putting the results of (39), (40) and (41) into (36), we have
(H (2) ) f k B k a f abc Ψ cf + f f bc Ψ ac = (detB)N k (B -1 ) d k ǫ dbf f abc Ψ cf + f f bc Ψ ac = (detB)N k (B -1 ) d k δ da δ f c -δ dc δ f a Ψ cf -2δ dc Ψ ac = (detB)N k (B -1 ) d k δ da trΨ -Ψ da -2Ψ ad = -δ (2) a ( N ),( 41
) with δ ( 2
) a ( N ) as given in (
Ġa = -δ θ G a + δ (2) a ( N ) + δ (1) a ( ψ) + δ (1) a ( ψ) -δ (2) a ( N ) = -δ θ G a + 2δ (1) ( ψ).( 42
)
The velocity of the Gauss' law constraint is a linear combination of the Gauss' constraint with terms of the diffeomorphism constraint of linear order and higher. Hence the time evolution of the Gauss' law constraint is consistent in the sense that we have defined, since δ (1) ( ψ) vanishes for ψ d = 0.
6 Consistency of the Hamiltonian constraint under time evolution
The time derivative of the Hamiltonian constraint, the third equation of (26) , is given by
Ḣ = d dt ((detB) 1/2 (detΨ) 1/2 (Λ + trΨ -1 ) + √ -g N d dt (Λ + trΨ -1 ) ( 43
)
which has split up into two terms. The first term is directly proportional to the Hamiltonian constraint, therefore it is already consistent. We will 9 nevertheless expand it using (21) and (25)
1 2 (B -1 ) d i Ḃi d + (Ψ -1 ) ae Ψae (detB) 1/2 (detΨ) 1/2 (Λ + trΨ -1 ) = 1 2 (B -1 ) d i -δ θ B i d -iǫ ijk D j H d k +(Ψ -1 ) ae -δ θ Ψ ae -iǫ ijk (B -1 ) e i (D j Ψ af )H f k H. ( 44
)
We will be content to compute the δ θ terms of (44). These are
(B -1 ) d i δ θ B i d = (B -1 ) d i f dbf B i b A f 0 = δ db f dbf A f 0 = 0 ( 45
)
on account of antisymmetry of the structure constants, and
(Ψ -1 ) ea δ θ Ψ ae = (Ψ -1 ) ea f abf Ψ f e + f ebg Ψ ag = 0, ( 46
)
also due to antisymmetry of the structure constants. We have shown that the first term on the right hand side of (43) is consistent with respect to time evolution. To verify consistency of the Hamiltonian constraint under time evolution, it remains to show that the second term is weakly equal to zero. It suffices to show this just for the second term, in brackets, of (43)
d dt (Λ + trΨ -1 ) = -(Ψ -1 Ψ -1 ) f e Ψef = (Ψ -1 Ψ -1 ) ef δ θ Ψ ae -iǫ ijk (B -1 ) e i (D j Ψ af )H f k , ( 47
)
where we have used (25). Equation (47) has split up into two terms, of which the first term is
(Ψ -1 Ψ -1 ) ea δ θ Ψ ae = (Ψ -1 Ψ -1 ) ea f Abf Ψ f e + f ebg Ψ ag A b 0 f abf (Ψ -1 ) f a + f ebg (Ψ -1 ) eg A b 0 = m( ψ) ∼ 0 ( 48
)
which vanishes weakly since it is a nonlinear function of at least linear order in ψ d . The second term of (47) splits into two terms which we must evaluate. The first contribution is proportional to
(Ψ -1 Ψ -1 ) ea ǫ ijk (B -1 ) e i (D j Ψ af )(H (1) ) f k = √ -g(Ψ -1 Ψ -1 ) ea ǫ ijk (B -1 ) e i (D j Ψ af )(B -1 ) d k (Ψ -1 Ψ -1 ) df = - √ -g(Ψ -1 Ψ -1 ) ea (Ψ -1 Ψ -1 ) df (detB) -1 ǫ edg B j g D j Ψ af = - √ -g(detB) -1 ǫ edg (Ψ -1 Ψ -1 ) ea (Ψ -1 Ψ -1 ) df v g {Ψ af } ≡ v{ ψ} ( 49
)
for some vector field v. We have used the fact that the term in (49) quartic in Ψ -1 in antisymmetric in a and f due to the epsilon symbol. Hence Ψ af as acted upon by v g can only appear in an antisymmetric combination, and is therefore proportional to the diffeomorphism constraint ψ d whose spatial 10 derivatives weakly vanish. Hence (49) presents a consistent contribution to the time evolution of H, which leaves remaining the second contribution to the second term of (47). This term is proportional to
(Ψ -1 Ψ -1 ) ea ǫ ijk (B -1 ) e i (D j Ψ af )(H (2) ) f k = (Ψ -1 Ψ -1 ) ea ǫ ijk (B -1 ) e i (D j Ψ af )ǫ mnk N m B n f = δ i m δ j n -δ i n δ j m (B -1 ) e i B n f (Ψ -1 Ψ -1 ) ea (D j Ψ af ) = N i (B -1 ) e i B j f -δ ef N j (Ψ -1 Ψ -1 ) ea (D j Ψ af ) = (-g) -1/2 N i H a i v f {Ψ af } -(Ψ -1 Ψ -1 ) f a (N j D j Ψ af ) = (-1) -1/2 N i H a i G a -N j D j (Λ + trΨ -1 ). ( 50
)
The first term on the final right hand side of (50) is proportional to the Gauss' law constraint, and the second term is proportional to the derivative of the Hamiltonian constraint. To obtain this second term we have added in Λ as a constant of differentiation with respect to ∂ j . Substituting (48), (49) and (50) into (47), then we have
Ḣ =∼ Ô( ψ) + (-g) -1/2 N i H a i G a + T ((-g) -1/2 H), ( 51
)
where Ô and T are operators consisting of spatial derivatives acting to the right and c numbers. The time derivative of the Hamiltonian constraint is a linear combination of the Gauss' law and Hamiltonian constraints and its spatial derivatives, plus terms of linear order and higher in the diffeomorphism constraint and its spatial derivatives. Hence the Hamiltonian constraint is consistent under time evolution.
The final equations governing the time evolution of the initial value constraints are given weakly by
ψd = i(detB) -1 (Ψ -1 Ψ -1 ) da + ǫ dae N i (B -1 ) e i G a + A b 0 ǫ bdh -δ dh N j D j ψ h + i(detB) -1 v d {Λ + trΨ -1 }; Ġa = -f abc A b 0 G c + δ (1) a ( ψ); Ḣ = - i 2 ǫ ijk (B -1 ) d i (D j H d k ) + ǫ ijk (B -1 ) e i (Ψ -1 ) ae (D j Ψ af )H f k -N j ∂ j (Λ + trΨ -1 ) +(-g) -1/2 N i H a i G a - √ -g(detB) -1 ǫ edg (Ψ -2 Ψ -1 ) ea (Ψ -1 Ψ -1 ) df v g {ǫ af h ψ h } + m( ψ).( 52
)
Equations (52) show that all constraints derivable from the the action (1) are preserved under time evolution, since their time derivatives yield linear combinations of the same set of constraints and their spatial derivatives.
There are no additional constraints generated which implies that the action (1) is consistent in the Dirac sense. On the other hand, we have not defined the canonical structure of (52) or any Poisson brackets. Equations (52) can be written schematically in the following form
˙ H ∼ H + G + H; ˙ G ∼ G + Φ( H); Ḣ ∼ H + G + Φ( H), ( 53
)
where Φ is some nonlinear function of the diffeomorphism constraint H, which is of at least first order in H. In the Hamiltonian formulation of a theory, one identifies time derivatives of a variable f with via ḟ = {f, H} the Poisson brackets of the variable with the Hamiltonian H. So while we have not specified Poisson brackets, equation (53) implies the existence of Poisson brackets associated to some Hamiltonian H Inst for the action (1), with
{ H, H Inst } ∼ H + G + H; { G, H Inst } ∼ G + Φ( H); {H, H Inst } ∼ H + Φ( H) + G. ( 54
)
So the main result of this paper has been to demonstrate that the instanton representation of Plebanski gravity forms a consistent system, in the sense that the constraint surface is preserved under time evolution. As a direction of future research we will compute the algebra of constraints for (1) directly from its canonical structure. Nevertheless it will be useful for the present paper to think of equations (52) in the Dirac context, mainly for comparison with other formulations of general relativity. This will bring us to the Ashtekar variables.
8 Discussion: Relation of the instanton representation to the Ashtekar variables
We will now provide the rationale for not following the Dirac procedure for constrained systems [2] with respect to (1), by comparison with the Ashekar formulation of GR. The action for the instanton representation (1) can be written in the following 3+1 decomposed form
I Inst = dt Σ d 3 x Ψ ae B i e Ȧa i + A a 0 w e {Ψ ae } -ǫ ijk N i B j a B k e Ψ ae -iN (detB) 1/2 (detΨ) 1/2 Λ + trΨ -1 , ( 55
)
which regards Ψ ae and A a i as phase space variables. But the phase space of (55) is noncanonical since its symplectic two form
δθ Inst = δ Σ d 3 xΨ ae B i e δA a i = Σ d 3 xB i e δΨ ae ∧ δA a i + Σ d 3 xΨ ae ǫ ijk D j (δA e k ) ∧ δA a i , ( 56
)
12 is not closed owing to the presence of the second term on the right hand side. The initial stages of the Dirac procedure applied to (55) state that the momentum conjugate to A a i yields the primary constraint
Π i a = δI Inst δ Ȧa i = Ψ ae B i e . ( 57
)
Then making the identification σ i a = Π i a and upon substitution into (57) and into (55), one obtains the action
I Ash = dt Σ d 3 x σ i a Ȧa i + A a 0 G a -N i H i - i 2 N H , ( 58
)
which is the action for the Ashtekar complex formalism of general relativity [5], [6] , with σ i a being the densitized triad. This is a totally constrained system with (A a 0 , N i , N ), respectively the SO(3, C) rotation angle A a 0 , the shift vector N i and the densitized lapse function N = N (det σ) -1/2 as auxilliary fields. The constraints in (58) smearing the auxilliary fields are the Gauss' law, vector and Hamiltonian constraints
G a = D i σ i a ; H i = ǫ ijk σ j a B k a ; H = ǫ ijk ǫ abc σ i a σ j b Λ 3 σ k c + B k c . ( 59
)
From (58) one reads off the symplectic two form Ω Ash given by
Ω Ash = Σ d 3 xδ σ i a ∧ δA a i = δ Σ d 3 x σ i a δA a i = δθ Ash , ( 60
)
which is the exact functional variation of the canonical one form θ Ash . The actions (55) and (58) are transformable into each other only under the condition (detB) = 0 and (detΨ) = 0. In (58) it is clear that σ i a and A a i form a canonically conjugate pair, which suggests that (55) is a noncanonical version of (58). The constraints algebra for (59) is
{ H[ N ], H[ M ]} = H k N i ∂ k M i -M i ∂ k N i ; { H[N ], G a [θ a ]} = G a [N i ∂ i θ a ]; {G a [θ a ], G b [λ b ]} = G a f a bc θ b λ c ; {H(N ), H[ N ]} = H[N i ∂ i N {H(N ), G a (θ a )} = 0; H(N ), H(M ) = H i [ N ∂ j M -M ∂ j N H ij ], ( 61
)
which is first class due to closure of the algebra, and is therefore consistent in the Dirac sense. Let us consider (61) for each constraint with the total Hamiltonian H Ash and compare with (54). This is given schematically by
{ H, H Ash } ∼ H + G + H; { G, H Ash } ∼ G + H; {H, H Ash } ∼ H + H. ( 62
) 13
Comparison of (62) with (54) shows an essentially similar structure for the top two lines involving H and G. 7 But there is a marked dissimilarity with respect to the Hamiltonian constraint H. Note that there is a Gauss' law constraint appearing in the right hand side of the last line of (54) whereas there is no such constraint on the corresponding right hand side of (62). This means that while the Hamiltonian constraint is gauge-invariant under SO(3, C) gauge-transformations as implied by (61) and (62), this is not the case in (54) . This means that the action (1), which as shown in [1] describes general relativity for Petrov Types I, D and O, has a different role for the Gauss' law and Hamiltonian constraints than the action (58), which also describes general relativity. Therefore I Inst and I Ash at some level correspond to genuinely different descriptions of GR, a feature which would have been missed had we applied the step-by-step Dirac procedure.
9 Appendix: Commutation relations for I Inst
We will now infer the Poisson brackets for (55) by inference from the corresponding canonical Ashtekar Poisson brackets
{A a i (x), σ j b (y)} = δ a b δ j i δ (3) (x, y) ( 63
)
along with the vanishing brackets
{A a i (x), A b j (y)} = { σ i a (x), σ j b (x)} = 0. ( 64
)
To find the analogue of (63) and (64) for (55), we will use the tranformation equation
σ i a = Ψ ae B i e , ( 65
)
which corresponds to a noncanonical transformation. Substitution of (65) into (63) yields
{A a i (x), Ψ bf (y)B i f (y)} = δ j i δ a b δ (3) (x, y) {A a i (x), Ψ bf (y)}B j f (y) + Ψ bf (x){A a i (x), B j f (y)}. ( 66
)
The second term on the right hand side of (66) vanishes on account of the first relations of (64), and upon multiplying (66) by the inverse magnetic field (B -1 ) e i , assumed to be nondegenerate, we obtain
{A a i (x), Ψ bf (y)} = δ a b (B -1 (y)) f i δ (3) (x, y). ( 67
)
7 The linearly versus nonlinearly of the diffeomorphism constraints on the right hand side is just a minor difference.
This gives us the Poisson brackets {A, A} ∼ 0 and {A, Ψ} ∼ B -1 , which leaves remaining the brackets {Ψ, Ψ}. To obtain these, we substitute (65) into the second equation of (64), yielding { σ i a (x), σ b j (y)} = {Ψ ae (x)B i e (x), Ψ bf (y)B j f (y)} = Ψ ae (x){B i e (x), Ψ bf (y)}B j f (y) + {Ψ ae (x), Ψ bf (y)}B i e (x)B j f (y) +Ψ bf (x)Ψ ae (x){B i e (x), B j f (y)} + Ψ bf (y){Ψ ae (x), B j f (y)}B i e (x) = 0. (68) Noting that the third term vanishes on account of the first equation of (64), equation (68) reduces to {Ψ ae (x), Ψ bf (y)}B i e (x)B j f (y) +Ψ ae (x){B i e (x), Ψ bf (y)}B j f (y) -Ψ bf (y){B j f (y), Ψ ae (x)}B i e (x) = 0. (69)
The bottom two terms of (69) can be computed using (67) {B i e (x), Ψ bf (y)} = ǫ imn D x m {A e n (x), Ψ bf (y)} = ǫ imn D x m (δ e b (B -1 (y)) f n δ (3) (x, y)).(70) Substituting (70) into (69) and cancelling a pair of magnetic fields, then we have that {Ψ ae (x), Ψ bf (y)}B i e (x)B j f (y) = ǫ ijm Ψ ae (x)D x m + Ψ ba (y)D y m δ (3) (x, y). (71) Left and right multiplying (71) by the inverse of the magnetic fields, we have {Ψ ae (x), Ψ bf (y)} = ǫ ijm (B -1 (y)) f j D x m Ψ ab (x)(B -1 (x)) e i
+(B -1 (x)) e i D y m Ψ ba (y)(B -1 (y)) f j δ (3) (x, y). ( 72
)
One sees that the internal components of Ψ ae have nontrivial commutation relations with themselves. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "In [1] a new formulation of general relativity was presented, named the instanton representation of Plebanski gravity. The basic dynamical variables are an SO(3, C) gauge connection A a µ and a matrix Ψ ae taking its values in two copies of SO(3, C). 1 The consequences of the associated action I Inst were determined via its equations of motion, which hinge crucially on weak equalities implied by the the initial value constraints. For these consequences to be self-consistent, the constraint surface must be preserved for all time by the evolution equations. The present paper will demonstrate that this is indeed the case. We will not use the usual Hamiltonian formulation for totally constrained systems [2] , since we will not make use of any canonical structure implied by I Inst . Rather, we will deduce the time evolution of the dynamical variables directly from the equations of motion of I Inst .\n\nSections 2 and 3 of this paper present the instanton representation action and derive the time evolution of the basic variables. Sections 4, 5 and 6 demonstrate that the nondynamical equations, referred to as the diffeomorphism, Gauss' law and Hamiltonian constraints, evolve into combinations of the same constraint set. The result is that the time derivatives of these constraints are weakly equal to zero with no additional constraints generated on the system. While we do not use the usual Dirac method in this paper, the result is still that the instanton representation is in a sense Dirac consistent. We will make this inference clearer by comparison with the Ashtekar variables in the discussion section. On a final note, the terms 'diffeomorphism' and 'Gauss' law' constraints are used loosely in this paper, in that we have not specified what transformations of the basic variables these constraints generate. The use of these terms is mainly for notational purposes, due to their counterparts which appear in the Ashtekar variables."
},
{
"section_type": "OTHER",
"section_title": "Instanton representation of Plebanski gravity",
"text": "The starting action for the instanton representation of Plebanski gravity is given by [1]\n\nI Inst = dt Σ d 3 xΨ ae B k e F a 0i + ǫ kjm B j a N m -iN (detB) 1/2 √ detΨ Λ + trΨ -1 , ( 1\n\n)\n\n1 Index labelling conventions for this paper are that symbols a, b, . . . from the beginiing of the Latin alphabet denote internal SO(3, C) indices while those from the middle i, j, k, . . . denote spatial indices. Both of these sets of indices take takes 1, 2 and 3. The Greek symbols µ, ν, . . . refer to spacetime indices which take values 0, 1, 2, 3.\n\nwhere N µ = (N, N i ) are the lapse function and shift vector from metric general relativity, and Λ is the cosmological constant. The basic fields are Ψ ae and A a i , and we action (1) is defined only on configurations restricted to (detB) = 0 and (detΨ) = 0. 2 In the Dirac procedure one refers to N µ as nondynamical fields, since their velocities do not appear in the action. While the velocity Ψae also does not appear, we will distinguish this field from N µ since the action (1), unlike for the latter, is nonlinear in Ψ ae .\n\nThe equation of motion for the shift vector N i , the analogue of the Hamilton equation for its conjugate momentum Π N , is given by δI Inst δN i = ǫ mjk B j a B k e Ψ ae = (detB)(B -1\n\n) d i ψ d ∼ 0, ( 2\n\n)\n\nwhere ψ d = ǫ dae Ψ ae is the antisymmetric part of Ψ ae . This is equivalent to the diffeomorphism constraint H i owing to the nondegeneracy of B i a , and we will often use H i and ψ d interchangeably in this paper. The equation of motion for the lapse function N , the analogue of the Hamilton equation for its conjugte momentum Π N , is given by δI Inst δN = (detB) 1/2 √ detΨ Λ + trΨ -1 = 0. (3) Nondegeneracy of Ψ ae and the magnetic field B i e implies that on-shell, the following relation must be satisfied Λ + trΨ -1 = 0, (4) which we will similarly take as synonymous with the Hamiltonian constraint. The equation of motion for Ψ ae is\n\nδI Inst δΨ ae = B k e F a 0k + ǫ kjm B k e B j a N m + iN √ detB √ detΨ(Ψ -1 Ψ -1 ) ea ∼ 0, ( 5\n\n)\n\nup to a term proportional to (4) which we have set weakly equal to zero. One could attempt to define a momentum conjugate to Ψ ae , for which (5) would be the associated Hamilton's equation of motion. But since Ψ ae forms part of the canonical structure of (1), then our interpretation is that this is not technically correct. 3 The equation of motion for the connection A a µ is given by\n\nδI Inst δA a µ ∼ ǫ µσνρ D σ (Ψ ae F e νρ ) - i 2 δ µ i D ij da 4ǫ mjk N m B k e Ψ [de] +N (B -1 ) d j √ detB √ detΨ Λ + trΨ -1 , ( 6\n\n) 2\n\nThe latter case limits the application of our results to spacetimes of Petroc Types I, D and O (See e.g. [3] and [4] .\n\n3 This is because (5) contains a velocity Ȧa k within F a 0k and will therefore be regarded as an evolution equation rather than a constraint. This is in stark contrast with (2) and (3), which are genuine constraint equations due to the absence of any velocities.\n\nwhere we have defined D ji ea (x, y) ≡ δ δA a i (x) B j e (y) = ǫ jki -δ ae ∂ k + f eda A d k δ (3) (x, y); D\n\n0i ea ≡ 0. ( 7\n\n)\n\nThe terms in large round brackets in (6) vanish weakly, since they are proportional to the constraints (2) and (4) and their spatial derivatives. For the purposes of this paper we will regard (6) as synonymous with ǫ µσνρ D σ (Ψ ae F e νρ ) ∼ 0. (\n\n) 8\n\nIn an abuse of notation, we will treat (5) and (8) as strong equalities in this paper. This will be justified once we have completed the demonstration that the constraint surface defined collectively by (2), (3) and the Gauss' constraint from (8) is indeed preserved under time evolution. As a note prior to proceeding we will often make the identification\n\nN (detB) 1/2 √ detΨ ≡ √ -g ( 9\n\n)\n\nas a shorthand notation, to avoid cluttering many of the derivations which follow in this paper."
},
{
"section_type": "OTHER",
"section_title": "Internal consistency of the equations of motion",
"text": "Prior to embarking upon the issue of consistency of time evolution of the initial value constraints, we will check for internal consistency of I Inst , which entails probing of the physical content implied by (8) and (5). First, equation (8) can be decomposed into its spatial and temporal parts as\n\nD i (Ψ bf B i f ) = 0; D 0 (Ψ bf B i f ) = ǫ ijk D j (Ψ bf F f 0k ). ( 10\n\n)\n\nThe first equation of (10) is the Gauss' law constraint of a SO(3) Yang-Mills theory, when one makes the identification of Ψ bf B i f ∼ E i b with the Yang-Mills electric field. The Maxwell equations for U (1) gauge theory with sources (ρ, J ), in units where c = 1, are given by\n\n∇ • B = 0; Ḃ = -∇ × E = 0; ∇ • E = ρ; ˙ E = -J + ∇ × B. ( 11\n\n)\n\nEquations (10) can be seen as a generalization of the first two equations of (11) to SO(3) nonabelian gauge theory in flat space when one: (i) identifies F f 0k ≡ E f k with the SO(3) generalization of the electric field E, and (ii) one chooses Ψ ae = kδ ae for some numerical constant k.\n\nWhen ρ = 0 and J = 0, then one has the vacuum theory and equations (11) are invariant under the transformation\n\n( E, B) -→ (-B, E). ( 12\n\n)\n\n3 Then the second pair of equations of (11) become implied by the first pair. This is the condition that the Abelian curvature F µν , where F 0i = E i and ǫ ijk F jk = B i , is Hodge self-dual with respect to the metric of a conformally flat spacetime. But equations (10) for more general Ψ ae encode gravitational degrees of freedom, which as shown in [1] generalizes the concept of selfduality to more general spacetimes solving the Einstein equations. Let us first attempt to derive the analogue for (10) of the second pair of (11) in the vacuum case. Acting on the first equation of (10) with D 0 yields\n\nD 0 D i (Ψ bf B i f ) = D i D 0 (Ψ bf B i f ) + [D 0 , D i ](Ψ bf B i f ) = 0. ( 13\n\n)\n\nSubstituting the second equation of (10) into the first term on the right hand side of (13) and using the definition of temporal curvature as the commutator of covariant derivatives on the second term we have\n\nD i (ǫ ijk D j (Ψ bf F f 0k )) + f bcd F c 0i Ψ df B i f = f bcd B k c F f 0k + B k f F c 0k Ψ df = 0 ( 14\n\n)\n\nwhere we have also used the spatial part of the commutator\n\nǫ ijk D i D j v a = f abc B k b v c .\n\nNote that the term in brackets in (14) is symmetric in f and c, and also forms the symmetric part of the left hand side of (5)\n\nB i f F b 0i + i √ -g(Ψ -1 Ψ -1 ) f b + ǫ ijk B i f B j b N k = 0, ( 15\n\n)\n\nre-written here for completeness. To make progress from (14), we will substitute (15) into (14). This causes the last term of (15) to drop out due to antisymmetry, which leaves us with\n\n-i √ -gf bcd Ψ df (Ψ -1 Ψ -1 ) f c + Ψ df (Ψ -1 Ψ -1 ) f c = -2i √ -gf bcd Ψ -1 dc . ( 16\n\n)\n\nThe equations are consistent only if (16) vanishes, which is the requirement that Ψ ae = Ψ ea be symmetric. This of course is the requirement that the diffeomorphism constraint (2) be satisfied. So the analogue of the second pair of (11) in the vacuum case must be encoded in the requirement that Ψ ae = Ψ ea be symmetric."
},
{
"section_type": "OTHER",
"section_title": "The time evolution equations",
"text": "We must now verify that the initial value constraints are preserved under time evolution defined by the equations of motion (5) and (6). These equations are respectively the Hodge duality condition\n\nB k f F b 0k + i √ -g(Ψ -1 Ψ -1 ) f b + ǫ ijk N i B j b B k f = 0, ( 17\n\n)\n\n4 and one of the Bianchi identity-like equations\n\nǫ ijk D j (Ψ ae F e ok ) = D 0 (Ψ ae B i e ). ( 18\n\n)\n\nSince the initial value constraints were used to obtain the second line of (17) from (1), then we must verify that these constraints are preserved under time evolution as a requirement of consistency.\n\nUsing F b 0i = Ȧb i -D i A b 0 and defining √ -g(B -1 ) f i (Ψ -1 Ψ -1 ) f b + ǫ mnk N m B n b ≡ iH b k , ( 19\n\n)\n\nThen equation (17) can be written as a time evolution equation for the connection, which is not the same as a constraint equation as noted earlier\n\nF b 0i = -iH b i -→ Ȧb i = D i A b 0 -iH b i . ( 20\n\n)\n\nFrom equation (20) we can obtain the following equation governing time evolution equation for the magnetic field\n\nḂi e = ǫ ijk D j Ȧe k = ǫ ijk D j D k A e 0 -iH e k = f ebc B i b A c 0 -iǫ ijk D j H e k = -δ θ B i e -iǫ ijk D j H e k , ( 21\n\n)\n\nwhich will be useful. On the first term on the right hand side of (21) we have used the definition of the curvature as the commutator of covariant derivatives. The notation δ θ in (21) suggests that that B i e transforms as a SO(3, C) vector under gauge transformations parametrized by θ b ≡ A b 0 . 4 Since we have not specified anything about the canonical structure of I Inst , then δ θ as used in (21) and in (24) should at this stage simply be regarded as a definition useful for shorthand notation.\n\nWe will now apply the Liebnitz rule in conjunction with the definition of the temporal covariant derivatives to (18) to determine the equation governing the time evolution of Ψ ae . This is given by\n\nD 0 (Ψ ae B i e ) = B i e Ψae + Ψ ae Ḃi e + f abc A b 0 (Ψ ce B i e ) = ǫ ijk D j (Ψ ae F e 0k ). ( 22\n\n)\n\nSubstituting (21) and (20) into the left and right hand sides of (22), we have\n\nB i e Ψae + Ψ ae f ebc B i b A c 0 -iǫ ijk D j H e k + f abc A b 0 (Ψ ce B i e ) = -iǫ ijk D j (Ψ ae H e k ).( 23\n\n)\n\nIn what follows, it will be convenient to use the following transformation properties for Ψ ae as A a i under SO(3, C) gauge transformations\n\nδ θ Ψ ae = f abc Ψ ce + f ebc Ψ ac A b 0 ; δ θ A a i = -D i A a 0 ; δ θ B i e = -f ebc B i b A c 0 . ( 24\n\n)\n\n4 We will make the identification with SO(3, C) gauge transformations later in this paper when we bring in the relation of IInst with the Ashtekar variables.\n\nThen using (24), the time evolution equations for the phase space variables Ω Inst can be written in the following compact form\n\nȦb i = -δ θ A b i -iH b i ; Ψae = -δ θ Ψ ae -iǫ ijk (B -1 ) e i (D j Ψ af )H f k . ( 25\n\n)\n\nWe have found evolution equations for Ψ ae and A a i from the covariant equations of A a µ and the Hodge-duality condition We have obtained these without using Poisson brackets, and by assuming that the Hamiltonian and diffeomorphism constraints are satisfied. Therefore the first order of business is then to check for the preservation of the initial value constraints under the time evolution generated by (25) . This means that we must check that the time evolution of the diffeomorphism, Gauss' law and Hamiltonian constraints are combinations of terms proportional to the same constraints and their spatial derivatives, and terms which vanish when the constraints hold. 5 These constraints are given by\n\nw e {Ψ ae } = 0; (detB)(B -1 ) d i ψ d = 0; (detB) 1/2 √ detΨ Λ + trΨ -1 = 0(26)\n\nwhere (detB) = 0 and (detΨ) = 0. We will occasionally make the identification\n\nN (detB) 1/2 (detΨ) 1/2 ≡ √ -g ( 27\n\n)\n\nfor a shorthand notation. Additionally, the following definitions are provided for the vector fields appearing in the Gauss' constraint\n\nw e = B i e D i ; v e = B i e ∂ i ( 28\n\n)\n\nwhere D i is the SO(3, C) covariant derivative with respect to the connection A a i . Equations (26) are the equations of motion for the auxilliary fields A a 0 , N i and N ."
},
{
"section_type": "OTHER",
"section_title": "Consistency of the diffeomorphism constraint under time evolution",
"text": "The diffeomorphism constraint is directly proportional to ψ d = ǫ dae Ψ ae , the antisymmetric part of Ψ ae . So to establish the consistency condition for this constraint, it suffices to show that the antisymmetric part of the second equation of (25) weakly vanishes. This is given by\n\nǫ dae Ψae = -δ θ (ǫ dae Ψ ae ) -iǫ dae ǫ ijk (B -1 ) e i (D j Ψ af )H f k , ( 29\n\n)\n\n5 This includes any nonlinear function of linear order or higher in the constraints, a situation which involves the diffeomorphism constraint.\n\nwhich splits into two terms. Using (24), one finds that the first term of (29) is given by\n\n-ǫ dae δ θ Ψ ae = -ǫ dae f abc Ψ ce + Ψ ac f ebc A b 0 = -δ eb δ dc -δ ec δ bd Ψ ce + δ db δ ac -δ dc δ ab Ψ ac A b 0 = -Ψ db -δ bd trΨ + δ db trΨ -Ψ bd A b 0 = 2Ψ [bd] A b 0 = -ǫ dbh A b 0 ψ h , ( 30\n\n)\n\nwhich is proportional to the diffeomorphism constraint. The second term of (30) has two contributions due to H f k as defined in (19) . The first contribution reduces to\n\n-iǫ dae ǫ ijk (B -1 ) e i (D j Ψ af )(H (1) ) f k = -iǫ dae ǫ ijk (B -1 ) e i (D j Ψ af ) √ -g(B -1 ) g k (Ψ -1 Ψ -1 ) gf = iǫ dae (detB) -1 ǫ egh (Ψ -1 Ψ -1 ) gf B j h D j Ψ af = i(detB) -1 (Ψ -1 Ψ -1 ) gf δ g d δ h a -δ g a δ h d v v {Ψ af } = i(detB) -1 (Ψ -1 Ψ -1 ) gf δ g d v a {Ψ af } -v d {Ψ gf } = i(detB) -1 (Ψ -1 Ψ -1 ) df G f + v d {Λ + trΨ -1 } . ( 31\n\n)\n\nThe first term on the final right hand side of (31) is the Gauss' constraint and the second term is the derivative of a term direction proportional to the Hamiltonian constraint. 6 The second contribution to the second term of (29) is given by\n\nǫ dae ǫ ijk (B -1 ) e i (D j Ψ af )(H (2) ) f k = ǫ dae ǫ ijk (B -1 ) e i (D j Ψ af )ǫ mnk N m B n f = ǫ dac δ i m δ j n -δ i n δ j m (B -1 ) e i (D j Ψ af )N m B n f = ǫ dae N i (B -1 ) e i v f {Ψ af } -N j D j (ǫ dae Ψ ae ) = ǫ dae N i (B -1 ) e f G a -N j D j ψ d .( 32\n\n)\n\nThe result is that the time evolution of the diffeomorphism constraint is directly proportional to\n\nψd = i(detB) -1 (Ψ -1 Ψ -1 ) da + ǫ dae N i (B -1 ) e i G a + A b 0 ǫ bdh -δ dh N j D j ψ h + i(detB) -1 v d {(-g) -1/2 H}, ( 33\n\n)\n\nwhich is a linear combination of terms proportional to the constraints (26) and their spatial derivatives. The result is that the diffeomorphism constraint H i = 0 is consistent with respect to the Hamiltonian evolution generated by the equations (25). So it remains to verify consistency of Gauss' law and the Hamiltonian constraints G a and H.\n\n6 We have added in a term Λ, which can be regarded as a constant of integration with respect to the spatial derivatives from v d ."
},
{
"section_type": "OTHER",
"section_title": "5 Consistency of the Gauss' constraint under time evolution",
"text": "Having verified the consistency of the diffeomorphism constraint under time evolution, we now move on to the Gauss' constraint. Application of the Liebnitz rule to the first equation of (26) yields\n\nĠa = Ḃi e D i Ψ ae + B i e D i Ψae + B i e f abf Ψ f e + f ebg Ψ ag Ȧa i . ( 34\n\n)\n\nUpon substituion of (21) and (25) into (34), we have\n\nĠa = -δ θ B i e -iǫ ijk D j H e k D i Ψ ae + B m e D m -δ θ Ψ ae -iǫ ijk (B -1 ) e i (D j Ψ af )H f k +B i e f abf Ψ f e + f ebg Ψ ag -δ θ A b i -iH b i .( 35\n\n)\n\nUsing the Liebniz rule to combine the δ θ terms of (35), we have\n\nĠa = -δ θ G a -iǫ ijk (D j H e k )D i Ψ ae + B m e D m ((B -1 ) e i (D j Ψ af )H f k ) -i f abf Ψ f e + f ebg Ψ ag B i e H b i . ( 36\n\n)\n\nThe requirement of consistency is that we must show that the right hand side of (36) vanishes weakly. First, we will show that the third term on the right hand side of (36) vanishes up to terms of linear order and higher in the diffeomorphism constraint. This term, up to an insignificant numerical factor, has two contributions. The first contribution is\n\nf abf Ψ f e + f ebg Ψ ag B i e (H (1) ) b i = √ -g f abf Ψ f e + f ebg Ψ ag (Ψ -1 Ψ -1 ) eb = √ -g f abf (Ψ -1 ) f b + f ebg (Ψ -1 Ψ -1 ) eb Ψ ag ∼ δ (1) a ( ψ) ∼ 0, ( 37\n\n)\n\nwhich is directly proportional to a nonlinear function of first order in ψ d which is proportional to the diffeomorphism constraint. The second contribution to the third term on the right hand side of (36) is\n\nf abf Ψ f e + f ebg Ψ ag B i e (H (2) ) b i = f abf Ψ f e + f ebg Ψ ag ǫ kmn N k B m e B n b = f abf Ψ f e + f ebg Ψ ag (detB)N k (B -1 ) d k ǫ deb = (detB)N k (B -1 ) d k δ f d δ ae -δ f e δ ad Ψ f e + 2δ dg Ψ ag = (detB)N k (B -1 ) d k Ψ da -δ ad trΨ + 2Ψ ad ≡ δ (2) a ( N ) ( 38\n\n)\n\nwhich does not vanish, and neither is it expressible as a constraint. For the Gauss' law constraint to be consistent under time evolution, a necessary condition is that this δ (2) a ( N ) term must be exactly cancelled by another term arising from the variation.\n\nLet us expand the terms in square brackets in (36) . This is given, using the Liebniz rule on the second term, by\n\nǫ ijk (D j H e k )(D i Ψ ae ) + ǫ ijk B m e D m ((B -1 ) e i (D j Ψ ae )H f k ) = ǫ ijk (D j H e k )(D i Ψ ae ) -ǫ ijk B m e (B -1 ) e n (D m B n g )(B -1 ) g i (D j Ψ af )H f k +ǫ mjk (D m D j Ψ af )H f k + ǫ mjk (D j Ψ af )(D m H f k ). ( 39\n\n)\n\nThe first and last terms on the right hand side of (39) cancel, which can be seen by relabelling of indices. Upon application of the definition of curvature as the commutator of covariant derivatives to the third term, then (39) reduces to\n\n-ǫ ijk (D n B n g )(B -1 ) g i (D j Ψ af )H f k + H f k B k b f abc Ψ cf + f f bc Ψ ac . ( 40\n\n)\n\nThe first term of (40) vanishes on account of the Bianchi identity and the second term contains two contributions which we must evaluate. The first contribution is given by 37) . So putting the results of (39), (40) and (41) into (36), we have\n\n(H (2) ) f k B k a f abc Ψ cf + f f bc Ψ ac = (detB)N k (B -1 ) d k ǫ dbf f abc Ψ cf + f f bc Ψ ac = (detB)N k (B -1 ) d k δ da δ f c -δ dc δ f a Ψ cf -2δ dc Ψ ac = (detB)N k (B -1 ) d k δ da trΨ -Ψ da -2Ψ ad = -δ (2) a ( N ),( 41\n\n) with δ ( 2\n\n) a ( N ) as given in (\n\nĠa = -δ θ G a + δ (2) a ( N ) + δ (1) a ( ψ) + δ (1) a ( ψ) -δ (2) a ( N ) = -δ θ G a + 2δ (1) ( ψ).( 42\n\n)\n\nThe velocity of the Gauss' law constraint is a linear combination of the Gauss' constraint with terms of the diffeomorphism constraint of linear order and higher. Hence the time evolution of the Gauss' law constraint is consistent in the sense that we have defined, since δ (1) ( ψ) vanishes for ψ d = 0.\n\n6 Consistency of the Hamiltonian constraint under time evolution\n\nThe time derivative of the Hamiltonian constraint, the third equation of (26) , is given by\n\nḢ = d dt ((detB) 1/2 (detΨ) 1/2 (Λ + trΨ -1 ) + √ -g N d dt (Λ + trΨ -1 ) ( 43\n\n)\n\nwhich has split up into two terms. The first term is directly proportional to the Hamiltonian constraint, therefore it is already consistent. We will 9 nevertheless expand it using (21) and (25)\n\n1 2 (B -1 ) d i Ḃi d + (Ψ -1 ) ae Ψae (detB) 1/2 (detΨ) 1/2 (Λ + trΨ -1 ) = 1 2 (B -1 ) d i -δ θ B i d -iǫ ijk D j H d k +(Ψ -1 ) ae -δ θ Ψ ae -iǫ ijk (B -1 ) e i (D j Ψ af )H f k H. ( 44\n\n)\n\nWe will be content to compute the δ θ terms of (44). These are\n\n(B -1 ) d i δ θ B i d = (B -1 ) d i f dbf B i b A f 0 = δ db f dbf A f 0 = 0 ( 45\n\n)\n\non account of antisymmetry of the structure constants, and\n\n(Ψ -1 ) ea δ θ Ψ ae = (Ψ -1 ) ea f abf Ψ f e + f ebg Ψ ag = 0, ( 46\n\n)\n\nalso due to antisymmetry of the structure constants. We have shown that the first term on the right hand side of (43) is consistent with respect to time evolution. To verify consistency of the Hamiltonian constraint under time evolution, it remains to show that the second term is weakly equal to zero. It suffices to show this just for the second term, in brackets, of (43)\n\nd dt (Λ + trΨ -1 ) = -(Ψ -1 Ψ -1 ) f e Ψef = (Ψ -1 Ψ -1 ) ef δ θ Ψ ae -iǫ ijk (B -1 ) e i (D j Ψ af )H f k , ( 47\n\n)\n\nwhere we have used (25). Equation (47) has split up into two terms, of which the first term is\n\n(Ψ -1 Ψ -1 ) ea δ θ Ψ ae = (Ψ -1 Ψ -1 ) ea f Abf Ψ f e + f ebg Ψ ag A b 0 f abf (Ψ -1 ) f a + f ebg (Ψ -1 ) eg A b 0 = m( ψ) ∼ 0 ( 48\n\n)\n\nwhich vanishes weakly since it is a nonlinear function of at least linear order in ψ d . The second term of (47) splits into two terms which we must evaluate. The first contribution is proportional to\n\n(Ψ -1 Ψ -1 ) ea ǫ ijk (B -1 ) e i (D j Ψ af )(H (1) ) f k = √ -g(Ψ -1 Ψ -1 ) ea ǫ ijk (B -1 ) e i (D j Ψ af )(B -1 ) d k (Ψ -1 Ψ -1 ) df = - √ -g(Ψ -1 Ψ -1 ) ea (Ψ -1 Ψ -1 ) df (detB) -1 ǫ edg B j g D j Ψ af = - √ -g(detB) -1 ǫ edg (Ψ -1 Ψ -1 ) ea (Ψ -1 Ψ -1 ) df v g {Ψ af } ≡ v{ ψ} ( 49\n\n)\n\nfor some vector field v. We have used the fact that the term in (49) quartic in Ψ -1 in antisymmetric in a and f due to the epsilon symbol. Hence Ψ af as acted upon by v g can only appear in an antisymmetric combination, and is therefore proportional to the diffeomorphism constraint ψ d whose spatial 10 derivatives weakly vanish. Hence (49) presents a consistent contribution to the time evolution of H, which leaves remaining the second contribution to the second term of (47). This term is proportional to\n\n(Ψ -1 Ψ -1 ) ea ǫ ijk (B -1 ) e i (D j Ψ af )(H (2) ) f k = (Ψ -1 Ψ -1 ) ea ǫ ijk (B -1 ) e i (D j Ψ af )ǫ mnk N m B n f = δ i m δ j n -δ i n δ j m (B -1 ) e i B n f (Ψ -1 Ψ -1 ) ea (D j Ψ af ) = N i (B -1 ) e i B j f -δ ef N j (Ψ -1 Ψ -1 ) ea (D j Ψ af ) = (-g) -1/2 N i H a i v f {Ψ af } -(Ψ -1 Ψ -1 ) f a (N j D j Ψ af ) = (-1) -1/2 N i H a i G a -N j D j (Λ + trΨ -1 ). ( 50\n\n)\n\nThe first term on the final right hand side of (50) is proportional to the Gauss' law constraint, and the second term is proportional to the derivative of the Hamiltonian constraint. To obtain this second term we have added in Λ as a constant of differentiation with respect to ∂ j . Substituting (48), (49) and (50) into (47), then we have\n\nḢ =∼ Ô( ψ) + (-g) -1/2 N i H a i G a + T ((-g) -1/2 H), ( 51\n\n)\n\nwhere Ô and T are operators consisting of spatial derivatives acting to the right and c numbers. The time derivative of the Hamiltonian constraint is a linear combination of the Gauss' law and Hamiltonian constraints and its spatial derivatives, plus terms of linear order and higher in the diffeomorphism constraint and its spatial derivatives. Hence the Hamiltonian constraint is consistent under time evolution."
},
{
"section_type": "OTHER",
"section_title": "Recapitulation",
"text": "The final equations governing the time evolution of the initial value constraints are given weakly by\n\nψd = i(detB) -1 (Ψ -1 Ψ -1 ) da + ǫ dae N i (B -1 ) e i G a + A b 0 ǫ bdh -δ dh N j D j ψ h + i(detB) -1 v d {Λ + trΨ -1 }; Ġa = -f abc A b 0 G c + δ (1) a ( ψ); Ḣ = - i 2 ǫ ijk (B -1 ) d i (D j H d k ) + ǫ ijk (B -1 ) e i (Ψ -1 ) ae (D j Ψ af )H f k -N j ∂ j (Λ + trΨ -1 ) +(-g) -1/2 N i H a i G a - √ -g(detB) -1 ǫ edg (Ψ -2 Ψ -1 ) ea (Ψ -1 Ψ -1 ) df v g {ǫ af h ψ h } + m( ψ).( 52\n\n)\n\nEquations (52) show that all constraints derivable from the the action (1) are preserved under time evolution, since their time derivatives yield linear combinations of the same set of constraints and their spatial derivatives.\n\nThere are no additional constraints generated which implies that the action (1) is consistent in the Dirac sense. On the other hand, we have not defined the canonical structure of (52) or any Poisson brackets. Equations (52) can be written schematically in the following form\n\n˙ H ∼ H + G + H; ˙ G ∼ G + Φ( H); Ḣ ∼ H + G + Φ( H), ( 53\n\n)\n\nwhere Φ is some nonlinear function of the diffeomorphism constraint H, which is of at least first order in H. In the Hamiltonian formulation of a theory, one identifies time derivatives of a variable f with via ḟ = {f, H} the Poisson brackets of the variable with the Hamiltonian H. So while we have not specified Poisson brackets, equation (53) implies the existence of Poisson brackets associated to some Hamiltonian H Inst for the action (1), with\n\n{ H, H Inst } ∼ H + G + H; { G, H Inst } ∼ G + Φ( H); {H, H Inst } ∼ H + Φ( H) + G. ( 54\n\n)\n\nSo the main result of this paper has been to demonstrate that the instanton representation of Plebanski gravity forms a consistent system, in the sense that the constraint surface is preserved under time evolution. As a direction of future research we will compute the algebra of constraints for (1) directly from its canonical structure. Nevertheless it will be useful for the present paper to think of equations (52) in the Dirac context, mainly for comparison with other formulations of general relativity. This will bring us to the Ashtekar variables.\n\n8 Discussion: Relation of the instanton representation to the Ashtekar variables\n\nWe will now provide the rationale for not following the Dirac procedure for constrained systems [2] with respect to (1), by comparison with the Ashekar formulation of GR. The action for the instanton representation (1) can be written in the following 3+1 decomposed form\n\nI Inst = dt Σ d 3 x Ψ ae B i e Ȧa i + A a 0 w e {Ψ ae } -ǫ ijk N i B j a B k e Ψ ae -iN (detB) 1/2 (detΨ) 1/2 Λ + trΨ -1 , ( 55\n\n)\n\nwhich regards Ψ ae and A a i as phase space variables. But the phase space of (55) is noncanonical since its symplectic two form\n\nδθ Inst = δ Σ d 3 xΨ ae B i e δA a i = Σ d 3 xB i e δΨ ae ∧ δA a i + Σ d 3 xΨ ae ǫ ijk D j (δA e k ) ∧ δA a i , ( 56\n\n)\n\n12 is not closed owing to the presence of the second term on the right hand side. The initial stages of the Dirac procedure applied to (55) state that the momentum conjugate to A a i yields the primary constraint\n\nΠ i a = δI Inst δ Ȧa i = Ψ ae B i e . ( 57\n\n)\n\nThen making the identification σ i a = Π i a and upon substitution into (57) and into (55), one obtains the action\n\nI Ash = dt Σ d 3 x σ i a Ȧa i + A a 0 G a -N i H i - i 2 N H , ( 58\n\n)\n\nwhich is the action for the Ashtekar complex formalism of general relativity [5], [6] , with σ i a being the densitized triad. This is a totally constrained system with (A a 0 , N i , N ), respectively the SO(3, C) rotation angle A a 0 , the shift vector N i and the densitized lapse function N = N (det σ) -1/2 as auxilliary fields. The constraints in (58) smearing the auxilliary fields are the Gauss' law, vector and Hamiltonian constraints\n\nG a = D i σ i a ; H i = ǫ ijk σ j a B k a ; H = ǫ ijk ǫ abc σ i a σ j b Λ 3 σ k c + B k c . ( 59\n\n)\n\nFrom (58) one reads off the symplectic two form Ω Ash given by\n\nΩ Ash = Σ d 3 xδ σ i a ∧ δA a i = δ Σ d 3 x σ i a δA a i = δθ Ash , ( 60\n\n)\n\nwhich is the exact functional variation of the canonical one form θ Ash . The actions (55) and (58) are transformable into each other only under the condition (detB) = 0 and (detΨ) = 0. In (58) it is clear that σ i a and A a i form a canonically conjugate pair, which suggests that (55) is a noncanonical version of (58). The constraints algebra for (59) is\n\n{ H[ N ], H[ M ]} = H k N i ∂ k M i -M i ∂ k N i ; { H[N ], G a [θ a ]} = G a [N i ∂ i θ a ]; {G a [θ a ], G b [λ b ]} = G a f a bc θ b λ c ; {H(N ), H[ N ]} = H[N i ∂ i N {H(N ), G a (θ a )} = 0; H(N ), H(M ) = H i [ N ∂ j M -M ∂ j N H ij ], ( 61\n\n)\n\nwhich is first class due to closure of the algebra, and is therefore consistent in the Dirac sense. Let us consider (61) for each constraint with the total Hamiltonian H Ash and compare with (54). This is given schematically by\n\n{ H, H Ash } ∼ H + G + H; { G, H Ash } ∼ G + H; {H, H Ash } ∼ H + H. ( 62\n\n) 13\n\nComparison of (62) with (54) shows an essentially similar structure for the top two lines involving H and G. 7 But there is a marked dissimilarity with respect to the Hamiltonian constraint H. Note that there is a Gauss' law constraint appearing in the right hand side of the last line of (54) whereas there is no such constraint on the corresponding right hand side of (62). This means that while the Hamiltonian constraint is gauge-invariant under SO(3, C) gauge-transformations as implied by (61) and (62), this is not the case in (54) . This means that the action (1), which as shown in [1] describes general relativity for Petrov Types I, D and O, has a different role for the Gauss' law and Hamiltonian constraints than the action (58), which also describes general relativity. Therefore I Inst and I Ash at some level correspond to genuinely different descriptions of GR, a feature which would have been missed had we applied the step-by-step Dirac procedure.\n\n9 Appendix: Commutation relations for I Inst\n\nWe will now infer the Poisson brackets for (55) by inference from the corresponding canonical Ashtekar Poisson brackets\n\n{A a i (x), σ j b (y)} = δ a b δ j i δ (3) (x, y) ( 63\n\n)\n\nalong with the vanishing brackets\n\n{A a i (x), A b j (y)} = { σ i a (x), σ j b (x)} = 0. ( 64\n\n)\n\nTo find the analogue of (63) and (64) for (55), we will use the tranformation equation\n\nσ i a = Ψ ae B i e , ( 65\n\n)\n\nwhich corresponds to a noncanonical transformation. Substitution of (65) into (63) yields\n\n{A a i (x), Ψ bf (y)B i f (y)} = δ j i δ a b δ (3) (x, y) {A a i (x), Ψ bf (y)}B j f (y) + Ψ bf (x){A a i (x), B j f (y)}. ( 66\n\n)\n\nThe second term on the right hand side of (66) vanishes on account of the first relations of (64), and upon multiplying (66) by the inverse magnetic field (B -1 ) e i , assumed to be nondegenerate, we obtain\n\n{A a i (x), Ψ bf (y)} = δ a b (B -1 (y)) f i δ (3) (x, y). ( 67\n\n)\n\n7 The linearly versus nonlinearly of the diffeomorphism constraints on the right hand side is just a minor difference.\n\nThis gives us the Poisson brackets {A, A} ∼ 0 and {A, Ψ} ∼ B -1 , which leaves remaining the brackets {Ψ, Ψ}. To obtain these, we substitute (65) into the second equation of (64), yielding { σ i a (x), σ b j (y)} = {Ψ ae (x)B i e (x), Ψ bf (y)B j f (y)} = Ψ ae (x){B i e (x), Ψ bf (y)}B j f (y) + {Ψ ae (x), Ψ bf (y)}B i e (x)B j f (y) +Ψ bf (x)Ψ ae (x){B i e (x), B j f (y)} + Ψ bf (y){Ψ ae (x), B j f (y)}B i e (x) = 0. (68) Noting that the third term vanishes on account of the first equation of (64), equation (68) reduces to {Ψ ae (x), Ψ bf (y)}B i e (x)B j f (y) +Ψ ae (x){B i e (x), Ψ bf (y)}B j f (y) -Ψ bf (y){B j f (y), Ψ ae (x)}B i e (x) = 0. (69)\n\nThe bottom two terms of (69) can be computed using (67) {B i e (x), Ψ bf (y)} = ǫ imn D x m {A e n (x), Ψ bf (y)} = ǫ imn D x m (δ e b (B -1 (y)) f n δ (3) (x, y)).(70) Substituting (70) into (69) and cancelling a pair of magnetic fields, then we have that {Ψ ae (x), Ψ bf (y)}B i e (x)B j f (y) = ǫ ijm Ψ ae (x)D x m + Ψ ba (y)D y m δ (3) (x, y). (71) Left and right multiplying (71) by the inverse of the magnetic fields, we have {Ψ ae (x), Ψ bf (y)} = ǫ ijm (B -1 (y)) f j D x m Ψ ab (x)(B -1 (x)) e i\n\n+(B -1 (x)) e i D y m Ψ ba (y)(B -1 (y)) f j δ (3) (x, y). ( 72\n\n)\n\nOne sees that the internal components of Ψ ae have nontrivial commutation relations with themselves."
}
] | ||
arxiv:0704.0371 | 0704.0371 | 1 | 10.1007/s10714-007-0428-0 | dd1f8cc3e5015dd43a6b13bb28a43a7df9d6d3060cda78a0bc035ea5a9ff314c | Dark energy interacting with neutrinos and dark matter: a
phenomenological theory | A model for a flat homogeneous and isotropic Universe composed of dark
energy, dark matter, neutrinos, radiation and baryons is analyzed. The fields
of dark matter and neutrinos are supposed to interact with the dark energy. The
dark energy is considered to obey either the van der Waals or the Chaplygin
equations of state. The ratio between the pressure and the energy density of
the neutrinos varies with the red-shift simulating massive and non-relativistic
neutrinos at small red-shifts and non-massive relativistic neutrinos at high
red-shifts. The model can reproduce the expected red-shift behaviors of the
deceleration parameter and of the density parameters of each constituent. | [
"G. M. Kremer"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | A model for a flat homogeneous and isotropic Universe composed of dark energy, dark matter, neutrinos, radiation and baryons is analyzed. The fields of dark matter and neutrinos are supposed to interact with the dark energy. The dark energy is considered to obey either the van der Waals or the Chaplygin equations of state. The ratio between the pressure and the energy density of the neutrinos varies with the red-shift simulating massive and non-relativistic neutrinos at small red-shifts and non-massive relativistic neutrinos at high red-shifts. The model can reproduce the expected red-shift behaviors of the deceleration parameter and of the density parameters of each constituent.
The recent astronomical measurements of type-IA supernovae [1, 2, 3, 4] and the analysis of the power spectrum of the CMBR [5, 6, 7, 8, 9] provided strong evidence for a present accelerated expansion of the Universe [3, 10, 11, 12, 13, 14] ; the nature of the responsible entity, called dark energy, still remains unknown. Furthermore, the measurements of the rotation curves of spiral galaxies [15] as well as other astronomical experiments suggest that the luminous matter represents only a small amount of the massive particles of the Universe, and that the more significant amount is related to dark matter. That offered a new setting for cosmological models with dark energy and dark matter and in these contexts many interesting phenomenological models appear in the literature analyzing the interaction of neutrinos [16, 17, 18] and dark matter [19, 20, 21, 22, 23, 24] with dark energy. With respect to dark energy some exotic equations of state were proposed in the literature and among others we quote the van der Waals [25, 26, 27, 28, 29] and the Chaplygin [30, 31, 32, 33] equations of state.
In the present work a very simple cosmological model -for a homogeneous, isotropic and flat Universe composed by dark matter, dark energy, baryons, radiation and neutrinos -is investigated where the dark energy is modeled either by the van der Waals or the Chaplygin equations of state and interact with neutrinos and dark matter. Units have been chosen so that 8πG/3 = c = 1, whereas the metric tensor has signature (+, -, -, -).
Let a homogeneous, isotropic and spatially flat Universe be characterized by the Robertson Walker metric ds 2 = dt 2 -a(t) 2 δ ij dx i dx j , where a(t) denotes the cosmic scale factor. The sources of the gravitational field are related to a mixture of five constituents described by the fields of dark energy, dark matter, baryons, neutrinos and radiation. The components of the energy-momentum tensor of the sources is written as
(T µ ν ) = diag(ρ, -p, -p, -p), (1)
where ρ and p denote the total energy density and pressure of the sources, respectively. In terms of the energy densities and pressures of the constituents it follows
ρ = ρ b + ρ dm + ρ r + ρ ν + ρ de , p = p b + p dm + p r + p ν + p de . (2)
Above the indexes (b, dm, r, ν, de) refer to the baryons, dark matter, radiation, neutrinos and dark energy, respectively. The conservation law of the energy-momentum tensor T µν ;ν = 0 leads to the evolution equation for the total energy density of the sources, namely
ρ + 3 ȧ a (ρ + p) = 0, (3)
where the dot refers to a differentiation with respect to time.
The baryons and radiation are considered as non-interacting fields so that the evolution equations for their energy densities read
ρb + 3 ȧ a ρ b = 0, ρr + 4 ȧ a ρ r = 0, (4)
once the baryons represent a pressureless fluid, i.e., p b = 0, and the radiation pressure is given in terms of its energy density by p r = ρ r /3.
According to a model proposed by Wetterich [19] the evolution equation for the energy density of a pressureless (p dm = 0) dark matter field which interacts with a scalar field φ is given by ρdm + 3 ȧ a ρ dm = βρ dm φ.
(
) 5
Here the scalar field plays the role of the dark energy and β is a constant which couples the fields of dark matter and dark energy.
For interacting neutrinos with dark energy it is supposed that the evolution equation of the energy density is given by (see [17, 18] )
ρν + 3 ȧ a (ρ ν + p ν ) = α(ρ ν -3p ν ) φ. ( 6
)
The coefficient α is connected with the mass of the neutrinos and for more details one is referred to [17, 18] and to the references therein. Here α will be consider a phenomenological coefficient that couples the dark energy field with the neutrinos. Note that if p ν = ρ ν /3, there is no coupling between the fields of dark energy and neutrinos. Moreover, it is also important to note that the neutrinos in the past must behave as massless particles where the relationship between the pressure and the energy density is p ν = ρ ν /3. Due to the coupling of the neutrinos with the scalar field they become massive and non-relativistic. For these reasons a barotropic equation of state for the neutrinos is proposed where the ratio between the pressure and the energy density w ν = p ν /ρ ν , given in terms of the red-shift z, reads
w ν = 1 z 2 + 5 z K 3 (1/z) K 2 (1/z) - 1 z 2 K 3 (1/z) K 2 (1/z) 2 -1 -1 . (7)
Above K 2 (1/z) and K 3 (1/z) are modified Bessel functions of second kind. For small values of z, w ν tends to the non-relativistic limit equal to 2/3, whereas for large values of z, w ν tends to the relativistic limit equal to 1/3. It is noteworthy that for red-shifts z ≈ 10 this ratio reaches the value w ν ≈ 1/3 and the coupling between the neutrinos and the dark energy is negligible. The expression given in (7) is motivated by the equation of the specific heat of a relativistic gas (see e.g. [34] ). The evolution equation for the energy density of the dark energy field is obtained from equations (2) through (6) ,
yielding ρde + 3 ȧ a (ρ de + p de ) = -α φ(ρ ν -3p ν ) -βρ dm φ. ( 8
)
The energy density and pressure of the dark energy are connected with the scalar field by φ = √ ρ de + p de . Since the purpose of this work is to develop a phenomenological theory, it is assumed that the dark energy field behaves either as a van der Waals or a Chaplygin fluid with an equation of state given by [28, 29, 30, 31, 32, 33 ]
p vw = 8w vw ρ vw 3 -ρ vw -3ρ 2 vw , p ch = - A ρ ch , (9)
where w vw and A are positive free parameters in the van der Waals and Chaplygin equations of state, respectively. For the determination of the time evolution of the energy densities one has to close the system of differential equations by introducing the Friedmann equation
ȧ a 2 = ρ. ( 10
)
From now on the red-shift will be used as a variable instead of time thanks to the following relationships
z = 1 a -1, d dt = - √ ρ(1 + z) d dz . (11)
Equations ( 4 ) can be easily integrated leading to the well-known dependence of the energy densities of the baryons and radiation with the red-shift
ρ r (z) = ρ r (0)(1 + z) 4 , ρ b (z) = ρ b (0)(1 + z) 3 , (12)
whereas equations ( 5 ), ( 6 ) and (8) become a system of coupled differential equations for the energy densities ρ dm , ρ ν and ρ de , namely,
(1 + z)ρ ′ dm -3ρ dm (ρ de + p de )/ρ = -βρ dm , (13)
(1 + z)ρ ′ ν -3(ρ ν + p ν ) (ρ de + p de )/ρ = -α(ρ ν -3p ν ), (14)
(1 + z)ρ ′ de -3(ρ de + p de ) (ρ de + p de )/ρ = βρ dm + α(ρ ν -3p ν ). ( 15
)
In the above equations the prime refers to a differentiation with respect to the red-shift.
In order to solve the coupled system of differential equations ( 13 ) -( 15 ) one has to specify initial values for the energy densities at z = 0. The following initial values for the density parameters Ω i (z) = ρ i (z)/ρ(z) taken from the literature (see [35] for a review) were chosen: Ω de (0) = 0.72,
0 500 1000 1500 2000 2500 3000 z 0 0.2 0.4 0.6 0.8 1 Ω dm (0) = 0.229916, Ω b (0) = 5 × 10 -2 , Ω r (0) = 5 × 10 -5 , Ω ν (0) = 3.4 × 10 -5 . Moreover, one has to specify values for the coupling parameters α and β and for the parameters w vw and A which appear in the van der Waals and Chaplygin equations of state (9) . One way to fix the two last parameters is through the use of the value of the deceleration parameter q = 1/2 + 3p/2ρ at z = 0. Indeed, by considering q(0) = -0.55 it follows w vw = 0.33851 and A = 0.50403. For the coupling parameters two sets of values were chosen, namely, (a) α = 5 × 10 -5 and β = -5 × 10 -5 for the van der Waals equation of state and (b) α = 10 -1 and β = -10 -2 for the Chaplygin equation of state. Its is also important to note that by increasing the value of the coupling parameter α (and/ or β) the transfer of energy between the dark energy and neutrinos (and/or dark matter) becomes more efficient. In Fig. 1 the density parameters are plotted as functions of the red-shift for values in the range 0 ≤ z ≤ 10. The straight lines refer to the case where the van der Waals equation of state is used to describe the dark energy field whereas the dashed lines correspond to the Chaplygin equation of state. The two density parameters that represent the dark energy field are denoted by Ω vw and Ω ch . One can infer from this figure that the dark energy density parameter tends to zero for high red-shifts when the van der Waals equation of state is used, whereas it tends to a constant value for the Chaplygin equation of state. While for high red-shifts the van der Waals equation of state simulates a cosmological constant with p vw = -ρ vw , the pressure of the Chaplygin fluid vanishes indicating that it becomes another component of the dark matter field (see also the behavior of the pressures indicated in Fig. 4 ). It is also important to note that the density parameters of the baryons and of the dark matter increase more with the red-shift for the van der Waals equation of state, since there is an accentuated decrease in the density parameter of the dark energy for this case. Note that the density parameters of the radiation and neutrinos are very small in this range of the red-shift and are not represented in this figure .
The behavior of the density parameters for the cases of the van der Waals and Chaplygin equations of state are shown in Figs. 2 and 3 , respectively, for red-shifts in the range 0 ≤ z ≤ 3000. One can conclude from these figures, as expected, that the density parameters of the neutrinos and radiation increase with the red-shift whereas those of the baryons and dark matter decrease. Furthermore, the equality between the "matter" and "radiation" fields occurs when z ≈ 3000 for the case where the dark matter field is modeled as a van der Waals fluid and z ≈ 4200 for the case of a Chaplygin fluid. This can be easily understood, since in the latter case the dark energy becomes dark matter for high red-shifts contributing for the density parameter of the "matter" field.
Ω dm + Ω b Ω r + Ω ν Ω dm Ω r Ω b Ω ν
In Fig. 4 are plotted the deceleration parameter and the ratio between the pressure and the energy density for both cases, the large frame corresponding to the van de Waals fluid whereas the small frame to the Chaplygin fluid. For both cases the deceleration parameter at z = 0 is equal to q(0) = -0.55, since this value was fixed in order to find the parameters w vw and A in the equations of state (9) . The transition from the decelerated to the accelerated phase of the Universe occurs at z T = 0.73 and z T = 0.53 for the van der Waals and Chaplygin equations of state, respectively. It
0 500 1000 1500 2000 2500 3000 z 0 0.2 0.4 0.6 0.8 1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 1.5 2 z -1 -0.5 0 0.5 1 1.5 -0.5 0 0.5 1 1.5 1.5 2 z -1 -0.5 0 0.5 p vw / ρ vw q q p ch / ρ ch is interesting to note that while the Chaplygin equation of state simulates a cosmological constant with p ch = -ρ ch for negative red-shifts which implies an accelerated phase of the Universe in the future, the van der Waals equation of state leads to a positive pressure and brings the Universe to another decelerated phase in the future. It is noteworthy to call attention that for positive values of the red-shift, the solution of the coupled differential equations ( 13 ) through (15) predicts that the van der Waals fluid behaves close to a cosmological constant with p vw ≈ -ρ vw . This behavior does not lead to a new transition from a decelerated to an accelerated phase in the very early Universe, since the energy density of the radiation field increases so that the radiation pressure becomes larger than that of the van der Waals fluid. For high red-shifts the Universe first becomes dominated by the baryon and dark matter fields and for higher red-shifts by the radiation field. This model does not attempt to model the inflationary period, where the inflaton field dominates a short rapid evolution of the Universe.
Ω dm + Ω b + Ω ch Ω dm Ω r + Ω ν Ω r Ω b Ω ν Ω ch
As final remarks we call attention to the fact that one expects that the coupling between dark energy, dark matter and neutrinos should be weak so that the parameters α and β are restricted to small values. The difference between the parameters adopted for the van der Waals and Chaplygin equations of state is due to stability conditions of the non-linear coupled system of differential equations ( 13 ) - (15) , the van der Waals equation of state being more unstable for large values of these parameters than the Chaplygin equation of state. In Fig. 5 we have plotted the density parameters as functions of the red-shift for the case where a Chaplygin equation of state is used as dark energy. One can infer from this figure that the decay of the dark energy density parameter and the increase of the dark matter density parameter with the red-shift are more pronounced when there exists a coupling between the fields. The density parameter of the baryons remains unchanged since the baryons are uncoupled.
As final comment it is important to note that even without couplings between the fields of dark energy, dark matter and neutrinos, this phenomenological model -with the equations of state of van der Waals and Chaplyging as dark energy -can describe satisfactorily the evolution of a Universe whose constituents are dark energy, dark matter, baryons, neutrinos and radiation. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "A model for a flat homogeneous and isotropic Universe composed of dark energy, dark matter, neutrinos, radiation and baryons is analyzed. The fields of dark matter and neutrinos are supposed to interact with the dark energy. The dark energy is considered to obey either the van der Waals or the Chaplygin equations of state. The ratio between the pressure and the energy density of the neutrinos varies with the red-shift simulating massive and non-relativistic neutrinos at small red-shifts and non-massive relativistic neutrinos at high red-shifts. The model can reproduce the expected red-shift behaviors of the deceleration parameter and of the density parameters of each constituent."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "The recent astronomical measurements of type-IA supernovae [1, 2, 3, 4] and the analysis of the power spectrum of the CMBR [5, 6, 7, 8, 9] provided strong evidence for a present accelerated expansion of the Universe [3, 10, 11, 12, 13, 14] ; the nature of the responsible entity, called dark energy, still remains unknown. Furthermore, the measurements of the rotation curves of spiral galaxies [15] as well as other astronomical experiments suggest that the luminous matter represents only a small amount of the massive particles of the Universe, and that the more significant amount is related to dark matter. That offered a new setting for cosmological models with dark energy and dark matter and in these contexts many interesting phenomenological models appear in the literature analyzing the interaction of neutrinos [16, 17, 18] and dark matter [19, 20, 21, 22, 23, 24] with dark energy. With respect to dark energy some exotic equations of state were proposed in the literature and among others we quote the van der Waals [25, 26, 27, 28, 29] and the Chaplygin [30, 31, 32, 33] equations of state.\n\nIn the present work a very simple cosmological model -for a homogeneous, isotropic and flat Universe composed by dark matter, dark energy, baryons, radiation and neutrinos -is investigated where the dark energy is modeled either by the van der Waals or the Chaplygin equations of state and interact with neutrinos and dark matter. Units have been chosen so that 8πG/3 = c = 1, whereas the metric tensor has signature (+, -, -, -).\n\nLet a homogeneous, isotropic and spatially flat Universe be characterized by the Robertson Walker metric ds 2 = dt 2 -a(t) 2 δ ij dx i dx j , where a(t) denotes the cosmic scale factor. The sources of the gravitational field are related to a mixture of five constituents described by the fields of dark energy, dark matter, baryons, neutrinos and radiation. The components of the energy-momentum tensor of the sources is written as\n\n(T µ ν ) = diag(ρ, -p, -p, -p), (1)\n\nwhere ρ and p denote the total energy density and pressure of the sources, respectively. In terms of the energy densities and pressures of the constituents it follows\n\nρ = ρ b + ρ dm + ρ r + ρ ν + ρ de , p = p b + p dm + p r + p ν + p de . (2)\n\nAbove the indexes (b, dm, r, ν, de) refer to the baryons, dark matter, radiation, neutrinos and dark energy, respectively. The conservation law of the energy-momentum tensor T µν ;ν = 0 leads to the evolution equation for the total energy density of the sources, namely\n\nρ + 3 ȧ a (ρ + p) = 0, (3)\n\nwhere the dot refers to a differentiation with respect to time.\n\nThe baryons and radiation are considered as non-interacting fields so that the evolution equations for their energy densities read\n\nρb + 3 ȧ a ρ b = 0, ρr + 4 ȧ a ρ r = 0, (4)\n\nonce the baryons represent a pressureless fluid, i.e., p b = 0, and the radiation pressure is given in terms of its energy density by p r = ρ r /3.\n\nAccording to a model proposed by Wetterich [19] the evolution equation for the energy density of a pressureless (p dm = 0) dark matter field which interacts with a scalar field φ is given by ρdm + 3 ȧ a ρ dm = βρ dm φ.\n\n(\n\n) 5\n\nHere the scalar field plays the role of the dark energy and β is a constant which couples the fields of dark matter and dark energy.\n\nFor interacting neutrinos with dark energy it is supposed that the evolution equation of the energy density is given by (see [17, 18] )\n\nρν + 3 ȧ a (ρ ν + p ν ) = α(ρ ν -3p ν ) φ. ( 6\n\n)\n\nThe coefficient α is connected with the mass of the neutrinos and for more details one is referred to [17, 18] and to the references therein. Here α will be consider a phenomenological coefficient that couples the dark energy field with the neutrinos. Note that if p ν = ρ ν /3, there is no coupling between the fields of dark energy and neutrinos. Moreover, it is also important to note that the neutrinos in the past must behave as massless particles where the relationship between the pressure and the energy density is p ν = ρ ν /3. Due to the coupling of the neutrinos with the scalar field they become massive and non-relativistic. For these reasons a barotropic equation of state for the neutrinos is proposed where the ratio between the pressure and the energy density w ν = p ν /ρ ν , given in terms of the red-shift z, reads\n\nw ν = 1 z 2 + 5 z K 3 (1/z) K 2 (1/z) - 1 z 2 K 3 (1/z) K 2 (1/z) 2 -1 -1 . (7)\n\nAbove K 2 (1/z) and K 3 (1/z) are modified Bessel functions of second kind. For small values of z, w ν tends to the non-relativistic limit equal to 2/3, whereas for large values of z, w ν tends to the relativistic limit equal to 1/3. It is noteworthy that for red-shifts z ≈ 10 this ratio reaches the value w ν ≈ 1/3 and the coupling between the neutrinos and the dark energy is negligible. The expression given in (7) is motivated by the equation of the specific heat of a relativistic gas (see e.g. [34] ). The evolution equation for the energy density of the dark energy field is obtained from equations (2) through (6) ,\n\nyielding ρde + 3 ȧ a (ρ de + p de ) = -α φ(ρ ν -3p ν ) -βρ dm φ. ( 8\n\n)\n\nThe energy density and pressure of the dark energy are connected with the scalar field by φ = √ ρ de + p de . Since the purpose of this work is to develop a phenomenological theory, it is assumed that the dark energy field behaves either as a van der Waals or a Chaplygin fluid with an equation of state given by [28, 29, 30, 31, 32, 33 ]\n\np vw = 8w vw ρ vw 3 -ρ vw -3ρ 2 vw , p ch = - A ρ ch , (9)\n\nwhere w vw and A are positive free parameters in the van der Waals and Chaplygin equations of state, respectively. For the determination of the time evolution of the energy densities one has to close the system of differential equations by introducing the Friedmann equation\n\nȧ a 2 = ρ. ( 10\n\n)\n\nFrom now on the red-shift will be used as a variable instead of time thanks to the following relationships\n\nz = 1 a -1, d dt = - √ ρ(1 + z) d dz . (11)\n\nEquations ( 4 ) can be easily integrated leading to the well-known dependence of the energy densities of the baryons and radiation with the red-shift\n\nρ r (z) = ρ r (0)(1 + z) 4 , ρ b (z) = ρ b (0)(1 + z) 3 , (12)\n\nwhereas equations ( 5 ), ( 6 ) and (8) become a system of coupled differential equations for the energy densities ρ dm , ρ ν and ρ de , namely,\n\n(1 + z)ρ ′ dm -3ρ dm (ρ de + p de )/ρ = -βρ dm , (13)\n\n(1 + z)ρ ′ ν -3(ρ ν + p ν ) (ρ de + p de )/ρ = -α(ρ ν -3p ν ), (14)\n\n(1 + z)ρ ′ de -3(ρ de + p de ) (ρ de + p de )/ρ = βρ dm + α(ρ ν -3p ν ). ( 15\n\n)\n\nIn the above equations the prime refers to a differentiation with respect to the red-shift.\n\nIn order to solve the coupled system of differential equations ( 13 ) -( 15 ) one has to specify initial values for the energy densities at z = 0. The following initial values for the density parameters Ω i (z) = ρ i (z)/ρ(z) taken from the literature (see [35] for a review) were chosen: Ω de (0) = 0.72,\n\n0 500 1000 1500 2000 2500 3000 z 0 0.2 0.4 0.6 0.8 1 Ω dm (0) = 0.229916, Ω b (0) = 5 × 10 -2 , Ω r (0) = 5 × 10 -5 , Ω ν (0) = 3.4 × 10 -5 . Moreover, one has to specify values for the coupling parameters α and β and for the parameters w vw and A which appear in the van der Waals and Chaplygin equations of state (9) . One way to fix the two last parameters is through the use of the value of the deceleration parameter q = 1/2 + 3p/2ρ at z = 0. Indeed, by considering q(0) = -0.55 it follows w vw = 0.33851 and A = 0.50403. For the coupling parameters two sets of values were chosen, namely, (a) α = 5 × 10 -5 and β = -5 × 10 -5 for the van der Waals equation of state and (b) α = 10 -1 and β = -10 -2 for the Chaplygin equation of state. Its is also important to note that by increasing the value of the coupling parameter α (and/ or β) the transfer of energy between the dark energy and neutrinos (and/or dark matter) becomes more efficient. In Fig. 1 the density parameters are plotted as functions of the red-shift for values in the range 0 ≤ z ≤ 10. The straight lines refer to the case where the van der Waals equation of state is used to describe the dark energy field whereas the dashed lines correspond to the Chaplygin equation of state. The two density parameters that represent the dark energy field are denoted by Ω vw and Ω ch . One can infer from this figure that the dark energy density parameter tends to zero for high red-shifts when the van der Waals equation of state is used, whereas it tends to a constant value for the Chaplygin equation of state. While for high red-shifts the van der Waals equation of state simulates a cosmological constant with p vw = -ρ vw , the pressure of the Chaplygin fluid vanishes indicating that it becomes another component of the dark matter field (see also the behavior of the pressures indicated in Fig. 4 ). It is also important to note that the density parameters of the baryons and of the dark matter increase more with the red-shift for the van der Waals equation of state, since there is an accentuated decrease in the density parameter of the dark energy for this case. Note that the density parameters of the radiation and neutrinos are very small in this range of the red-shift and are not represented in this figure . \nThe behavior of the density parameters for the cases of the van der Waals and Chaplygin equations of state are shown in Figs. 2 and 3 , respectively, for red-shifts in the range 0 ≤ z ≤ 3000. One can conclude from these figures, as expected, that the density parameters of the neutrinos and radiation increase with the red-shift whereas those of the baryons and dark matter decrease. Furthermore, the equality between the \"matter\" and \"radiation\" fields occurs when z ≈ 3000 for the case where the dark matter field is modeled as a van der Waals fluid and z ≈ 4200 for the case of a Chaplygin fluid. This can be easily understood, since in the latter case the dark energy becomes dark matter for high red-shifts contributing for the density parameter of the \"matter\" field.\n\nΩ dm + Ω b Ω r + Ω ν Ω dm Ω r Ω b Ω ν\n\nIn Fig. 4 are plotted the deceleration parameter and the ratio between the pressure and the energy density for both cases, the large frame corresponding to the van de Waals fluid whereas the small frame to the Chaplygin fluid. For both cases the deceleration parameter at z = 0 is equal to q(0) = -0.55, since this value was fixed in order to find the parameters w vw and A in the equations of state (9) . The transition from the decelerated to the accelerated phase of the Universe occurs at z T = 0.73 and z T = 0.53 for the van der Waals and Chaplygin equations of state, respectively. It\n\n0 500 1000 1500 2000 2500 3000 z 0 0.2 0.4 0.6 0.8 1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 1.5 2 z -1 -0.5 0 0.5 1 1.5 -0.5 0 0.5 1 1.5 1.5 2 z -1 -0.5 0 0.5 p vw / ρ vw q q p ch / ρ ch is interesting to note that while the Chaplygin equation of state simulates a cosmological constant with p ch = -ρ ch for negative red-shifts which implies an accelerated phase of the Universe in the future, the van der Waals equation of state leads to a positive pressure and brings the Universe to another decelerated phase in the future. It is noteworthy to call attention that for positive values of the red-shift, the solution of the coupled differential equations ( 13 ) through (15) predicts that the van der Waals fluid behaves close to a cosmological constant with p vw ≈ -ρ vw . This behavior does not lead to a new transition from a decelerated to an accelerated phase in the very early Universe, since the energy density of the radiation field increases so that the radiation pressure becomes larger than that of the van der Waals fluid. For high red-shifts the Universe first becomes dominated by the baryon and dark matter fields and for higher red-shifts by the radiation field. This model does not attempt to model the inflationary period, where the inflaton field dominates a short rapid evolution of the Universe.\n\nΩ dm + Ω b + Ω ch Ω dm Ω r + Ω ν Ω r Ω b Ω ν Ω ch\n\nAs final remarks we call attention to the fact that one expects that the coupling between dark energy, dark matter and neutrinos should be weak so that the parameters α and β are restricted to small values. The difference between the parameters adopted for the van der Waals and Chaplygin equations of state is due to stability conditions of the non-linear coupled system of differential equations ( 13 ) - (15) , the van der Waals equation of state being more unstable for large values of these parameters than the Chaplygin equation of state. In Fig. 5 we have plotted the density parameters as functions of the red-shift for the case where a Chaplygin equation of state is used as dark energy. One can infer from this figure that the decay of the dark energy density parameter and the increase of the dark matter density parameter with the red-shift are more pronounced when there exists a coupling between the fields. The density parameter of the baryons remains unchanged since the baryons are uncoupled.\n\nAs final comment it is important to note that even without couplings between the fields of dark energy, dark matter and neutrinos, this phenomenological model -with the equations of state of van der Waals and Chaplyging as dark energy -can describe satisfactorily the evolution of a Universe whose constituents are dark energy, dark matter, baryons, neutrinos and radiation."
}
] | |
arxiv:0704.0372 | 0704.0372 | 1 | 10.1088/1751-8113/40/11/013 | 0e304d6a4369de20e39f410749114d5070c26242ff3a56ae76d8c25256102174 | Levy-Lieb constrained-search formulation as a minimization of the
correlation functional | The constrained-search formulation of Levy and Lieb, which formally defines
the exact Hohenberg-Kohn functional for any N-representable electron density,
is here shown to be equivalent to the minimization of the correlation
functional with respect to the N-1 conditional probability density, where N is
number of electrons of the system. The consequences and implications of such a
result are here analyzed and discussed via a practical example. | [
"L.Delle Site"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | The constrained-search formulation of Levy and Lieb, which formally defines the exact Hohenberg-Kohn functional for any N -representable electron density, is here shown to be equivalent to the minimization of the correlation functional with respect to the N -1 conditional probability density, where N is number of electrons of the system. The consequences and implications of such a result are here analyzed and discussed via a practical example.
The Hohenberg-Kohn (HK) theorem [1] has opened new perspectives to the calculations of electronic-based properties of condensed matter [2] , and, an aspect often disregarded, has given profound new insights into the general understanding of quantum mechanics. In fact the 3N-dimensional Schrödinger problem for the ground state of an electronic system:
H N ψ(r 1 , ...r 2 ) = E 0 ψ(r 1 , ...r 2 ); H N = i=1,N (- 1 2 ∇ 2 i ) + i=1,N v(r i ) + i<j 1 r ij (1)
where v(r i ) is the external potential, i<j 1 r ij is the electron-electron Coulomb term, E 0 is the energy of the ground state and ψ(r 1 , ...r N ) is the 3N-dimensional antisymmetric ground state wavefunction [3] , is transformed into a "manageable" variational problem in three dimensions where the central role is played by the electron density: ρ(r) = N Ω N-1 ψ * (r, r 2 , .....r N )ψ(r, r 2 , .....r N )dr 2 ....dr N , where Ω N -1 is the N -1 spatial domain.
In explicit terms the variational problems is written as:
E 0 = Min ρ E[ρ] (2)
where Ω ρ(r)dr = N (Ω being the spatial domain of definition) and
E[ρ] = T [ρ] + V ee [ρ] + V ext [ρ]
is the energy functional composed respectively by the kinetic, electron-electron potential and the external potential functional. However in its original formulation the HK theorem and the related variational problem have got a restricted field of applicability; it is valid only if the electron density ρ(r) is v-representable, that is if ρ(r) is the density corresponding to an antisymmetric wavefunction of the ground-state of an Hamiltonian of the form of Eq.1. It follows that the correct formulation of the variational problem becomes:
E 0 = Min ρ E v [ρ] (3)
where v refers to the v-representability of ρ(r). As discussed in Ref. [2] , there are no general conditions for a density to be v-representable and this makes the use of the HK theorem and its associated variational principle not practical. A generalization of the HK theorem which does not require ρ(r) to be v-representable was found, in parallel, by M.Levy [4] and E.Lieb [5] and it is usually known as the Levy constrained-search formulation or Levy-Lieb constrained-search formulation [6] ; in this paper we adopt the latter terminology. We also notice that recently P.Ayers [7] has further clarified this concept and developed an axiomatic treatment of the Hohenberg-Kohn functional. In the following we briefly describe the crucial aspects of the abovementioned approach which are relevant for the current work.
The starting point of the theory is the distinction between the ground state wavefunction, ψ, and a wavefunction ψ λ that also integrates to the ground state electron density ρ(r). Since ψ is the ground state wavefunction, we have:
ψ λ |H N | ψ λ ≥ ψ |H N | ψ = E 0 . (4)
Taking into account that V ext [ρ] is a functional of ρ only, Eq.4 can be written as:
ψ λ |T + V ee | ψ λ ≥ ψ |T + V ee | ψ (5)
where T and V ee are respectively the kinetic and Coulomb electron-electron operator as defined in Eq.1. The meaning of Eq.5 is that ψ is the wavefunction that minimize the kinetic plus the electron-electron repulsion energy and integrates to ρ. It follows that the initial variational problem of Eq.2 can be transformed in a double hierarchical minimization procedure which formally allows for searching among all the ρ's which are N-representable,
i.e. it can be obtained from some antisymmetric wavefunction; this is a condition which is much weaker and more controllable than the v-representability. In explicit terms such a formulation is written as:
E 0 = Min ρ Min ψ λ →ρ ψ λ |T + V ee | ψ λ + v(r)ρ(r)dr . (6)
The inner minimization is restricted to all wave functions ψ λ leading to ρ(r), while the outer minimization searches over all the ρ's which integrate to N. The original HK formulation can then be seen as a part of this new one once its universal functional,
F [ρ] = ψ |T + V ee | ψ
is written as:
F [ρ] = Min ψ→ρ ψ |T + V ee | ψ . (7)
The purpose of this work is to show that F [ρ] can be determined solely by a minimization with respect to the N -1 conditional probability density of the electron correlation functional. This latter will be shown to be composed by the non local Fisher information functional [8] and the electron-electron two-particle Coulomb term. The advantage of this representation is manifold; it further clarifies the connection of electronic properties to the Fisher theory and shows that the knowledge of such a functional is the crucial ingredient in density functional based approaches; it also identifies the Weizsacker kinetic term,
|∇ρ(r)| 2
ρ(r) dr, as necessary component of the universal functional F [ρ] and, in practical terms, offers an objective criterion of evaluation of "approximate" exchange and correlation functional, i.e. among two functionals, the physically better founded is the "smaller" one. In order to show the practical aspects of our idea we illustrate a potential application.
Before writing the functional in the conditional probability density formalism, we need to define such a quantity. Let us consider a generic fermionic wavefunction ψ(r 1 , ....r N ), for simplicity we consider a real wavefunction, but the extension to a complex one can be also done [9] ; we do not consider the spin dependence explicitly, however this will not influence the main conclusions. Then the N-particle probability density is [10, 11] :
Nψ * (r 1 , ....r N )ψ(r 1 , ....r N ) = Θ 2 (r 1 , ...., r N ) (8)
and this can formally decomposed as [10, 11] :
Θ 2 (r 1 , ...., r N ) = ρ(r 1 )f (r 2 , ......., r N /r 1 ) (9)
where ρ(r 1 ) is the one particle probability density (normalized to N) and f (r 2 , ......., r N /r 1 )
is the N -1 electron conditional (w.r.t. r 1 ) probability density, i.e. the probability density of finding an N -1 electron configuration, C(r 2 , ......., r N ), for a given fixed value of r 1 . The function f satisfies the following properties:
(i )
Ω N-1 f (r 2 , ........, r N /r 1 )dr 2 .......dr N = 1∀r 1 (ii ) f (r 1 , ..r i ...r j-1 , r j+1 ..., r N /r j ) = 0; f or i = j; ∀i, j = 1, N (iii ) f (r 1 , .....r i .., r j ...r k-1 , r k+1 .., r N /r k ) = 0; f or i = j; ∀i, j = k (10)
The property (iii ) of Eq.10 assures us that f reflects the fermionic character of an electronic wavefunction. In fact it says that if any two particles are in the same 'state' "r" the probability of that specific global configuration is zero. In principle, together with condition (ii), this is a way to mimic the antisymmetric character of the fermionic wavefunction since for fermions |ψ(r 1 , ...r i , ...r j , ...r N )| 2 = 0; f or i = j, ∀i, j.It must be noticed that condition (iii) is complementary to (ii). With this formalism the term ψ |T + V ee | ψ can be written as (see Refs. [9, 10, 12] ):
ψ |T + V ee | ψ = 1 8 |∇ρ(r)| 2 ρ(r) dr + 1 8 ρ(r) Ω N-1 |∇ r f (r ′ , ...., r N /r)| 2 f (r ′ , ....., r N /r) dr ′ ....dr N dr + (N -1) ρ(r) Ω N-1 f (r ′ , ....., r N /r) |r -r ′ | dr ′ ....dr N dr (11)
where we have identified r 1 with r and made use of the property of electron indistinguishability, thus r could be identified with any of the r i (and the same for r ′ identified here with r 2 ) without changing the results; a further consequence is that the Coulomb expression (last term on the r.h.s.) is written as the sum of N -1 identical terms for the generic r and r ′ particles. Using Eq.11 the Levy-Lieb constrained-search formulation can then be written as:
E 0 = Min ρ Min f (Γ[f, ρ]) + 1 8 |∇ρ(r)| 2 ρ(r) dr + v(r)ρ(r)dr (12)
where
Γ[f, ρ] = 1 8 ρ(r) Ω N-1 |∇ r f (r ′ , ...., r N /r)| 2 f (r ′ , ....., r N /r) dr ′ ....dr N dr + (N -1) ρ(r) Ω N-1 f (r ′ , ....., r N /r) |r -r ′ | dr ′ ....dr N dr. (13)
In this way we have transferred the problem from from ψ to f which means that the focus is now on Γ[f, ρ], i.e., as discussed in Ref. [12] , the correlation functional.
In our previous work [12] , we have proposed an approximation for f based on a twoparticle factorization:
f = Π N i=2 h i (E H (r, r i )) = Π N i=2 e (N -1)E(r) e -E H (r,r i ) (14)
where
e -E(r) = ω e -E H (r,r i ) dr i . ( 15
)
here E H (r, r i ) = ρ(r)ρ(r i ) |r-r i | , N is the number of particle, and ω the volume corresponding to one particle. Such an approximation, due to its simplicity, allows us to write an analytic expression of the Fisher functional which can be used in a straightforward way in numerical calculations. However it does not satisfy the condition (iii ) of Eq.10, and, for this reason, in order to use it into the Levy-Lieb constrained-search scheme it must be extended. The expression we propose here is the following:
f (r 2 , ...r N /r) = Π n=2,N e E(r)-γE H (r,rn) × Π i>j =1 e -βE H (r i ,r j ) (16)
with:
e -E(r) = Π n=2,N Π i>j =1 e -γE H (r,rn)-βE H (r i ,r j ) dr 2 .....dr N ( 17
)
Here γ and β are two free parameters. As it can be easily verified this expression of f satisfies all the requirements of Eq.10. The meaning of f as expressed in Eq.16 is that the probability of finding a certain configuration for the N -1 particles, having fixed particle r 1 = r, depends not only on the fixed particle and its interaction with the N -1 other particles as before, but also on the mutual arrangements of the N -1 particles (it has also to be kept in mind that using the particle indistinguishability the formalism can be applied to any r i as a fixed particle). The parameters γ and β express how important the N -1 mutual interactions are with respect to the interactions with r. Being now f a biparametric function, one can use the Levy-Lieb constrained search in our formulation and find the optimal values for γ and β. This practical example shows two different aspects of our formulation; basically we have shown that indeed it is possible to build a function f and actually it can be chosen in a way that its optimal expression can be determined via the constrained-search formulation. It must be noticed that this form of f is still rather simple since the spins are not explicitly considered when constructing the function and thus one cannot distinguish between the exchange and the correlation part of the electron-electron interaction as it is done in standard Density Functional Theory; as a consequence one should expect only an overall average description of these two terms which are here incorporated into the global correlation. However the construction of a more complete expression of f , which takes care of the effects of the spins, is the subject of current investigation. This emphasizes once more the merit of the general procedure shown here, that is different expressions of f , with different degrees of complexity, can be proposed and their relative validity checked by the constrained-search procedure.
As anticipated in the introduction, the consequences of Eqs.12,13 are rather interesting.
The Levy-Lieb variational principle can be reformulated as: The universal functional
F [ρ]
is the one with the minimum correlation functional with respect to the electron conditional probability density. This new interpretation of the HK universal functional tells us that only an accurate description of the correlation effects, considering the Weizsacker term as a necessary term, leads to an accurate description of the whole energy functional; such a criterion is necessary and sufficient. It is obvious that it is necessary; without knowing
Γ[f, ρ], F [ρ] cannot be known; it is sufficient because once Γ[f, ρ] or better f (r 2 , ...r N /r 1 )
is (in principle) known than the whole energy functional is known explicitely. Clearly, the "true" f (r 2 , ...r N /r 1 ) is very difficult if not impossible to obtain [13] , however it can be sufficiently well described on the basis of mathematical requirements and physical intuition as done for example in Ref. [12] and as shown in the previous section. From this point of view, Eqs.12,13, can be seen as an objective criterion to design, on the basis of physical intuition and fundamental mathematical requirements, valid energy functionals. In fact, as done in Ref. [12] and in the previous section, one can construct well-founded expressions for f keeping in mind the physical meaning of the electron correlation effects and the necessary related mathematical prescriptions of Eq.10. Next one can make use of Eqs.12,13 and choose among different functional forms of f , the one giving the "smaller" Γ. It must be noticed that in this work we do not claim that finding a functional form of f is easier or more rigorous than to find an exchange-correlation functional in standard Density Functional Theory; it represents an alternative or complementary approach to the latter.
However, the approach based on f allows one to express in a more direct way, via the choice of different forms of f , the physical principles related to the electron correlation effects and to have an explicit form of the correlation term for the kinetic functional which is of great advantage for kinetic functional based methods (see e.g. Refs. [14, 15] ).An important aspect linked to the statement above is that the term: is available (see e.g. [10, 16, 17] and references therein); this term is very often linked to the electron correlation functional and electronic properties(see Refs. [18, 19] ), our work further clarifies this connection, suggesting that the results known from the analysis of the Fisher functional could be employed in this context. In conclusion we have shown an alternative view of the Levy-Lieb constrained search approach and provided an example which clarifies the practical advantage of our idea; in this sense the present work it is not merely a marginal new formal contribution to a rather well-known method, but gives a new powerful insight into the field of applicability for realistic systems. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The constrained-search formulation of Levy and Lieb, which formally defines the exact Hohenberg-Kohn functional for any N -representable electron density, is here shown to be equivalent to the minimization of the correlation functional with respect to the N -1 conditional probability density, where N is number of electrons of the system. The consequences and implications of such a result are here analyzed and discussed via a practical example."
},
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The Hohenberg-Kohn (HK) theorem [1] has opened new perspectives to the calculations of electronic-based properties of condensed matter [2] , and, an aspect often disregarded, has given profound new insights into the general understanding of quantum mechanics. In fact the 3N-dimensional Schrödinger problem for the ground state of an electronic system:\n\nH N ψ(r 1 , ...r 2 ) = E 0 ψ(r 1 , ...r 2 ); H N = i=1,N (- 1 2 ∇ 2 i ) + i=1,N v(r i ) + i<j 1 r ij (1)\n\nwhere v(r i ) is the external potential, i<j 1 r ij is the electron-electron Coulomb term, E 0 is the energy of the ground state and ψ(r 1 , ...r N ) is the 3N-dimensional antisymmetric ground state wavefunction [3] , is transformed into a \"manageable\" variational problem in three dimensions where the central role is played by the electron density: ρ(r) = N Ω N-1 ψ * (r, r 2 , .....r N )ψ(r, r 2 , .....r N )dr 2 ....dr N , where Ω N -1 is the N -1 spatial domain.\n\nIn explicit terms the variational problems is written as:\n\nE 0 = Min ρ E[ρ] (2)\n\nwhere Ω ρ(r)dr = N (Ω being the spatial domain of definition) and\n\nE[ρ] = T [ρ] + V ee [ρ] + V ext [ρ]\n\nis the energy functional composed respectively by the kinetic, electron-electron potential and the external potential functional. However in its original formulation the HK theorem and the related variational problem have got a restricted field of applicability; it is valid only if the electron density ρ(r) is v-representable, that is if ρ(r) is the density corresponding to an antisymmetric wavefunction of the ground-state of an Hamiltonian of the form of Eq.1. It follows that the correct formulation of the variational problem becomes:\n\nE 0 = Min ρ E v [ρ] (3)\n\nwhere v refers to the v-representability of ρ(r). As discussed in Ref. [2] , there are no general conditions for a density to be v-representable and this makes the use of the HK theorem and its associated variational principle not practical. A generalization of the HK theorem which does not require ρ(r) to be v-representable was found, in parallel, by M.Levy [4] and E.Lieb [5] and it is usually known as the Levy constrained-search formulation or Levy-Lieb constrained-search formulation [6] ; in this paper we adopt the latter terminology. We also notice that recently P.Ayers [7] has further clarified this concept and developed an axiomatic treatment of the Hohenberg-Kohn functional. In the following we briefly describe the crucial aspects of the abovementioned approach which are relevant for the current work.\n\nThe starting point of the theory is the distinction between the ground state wavefunction, ψ, and a wavefunction ψ λ that also integrates to the ground state electron density ρ(r). Since ψ is the ground state wavefunction, we have:\n\nψ λ |H N | ψ λ ≥ ψ |H N | ψ = E 0 . (4)\n\nTaking into account that V ext [ρ] is a functional of ρ only, Eq.4 can be written as:\n\nψ λ |T + V ee | ψ λ ≥ ψ |T + V ee | ψ (5)\n\nwhere T and V ee are respectively the kinetic and Coulomb electron-electron operator as defined in Eq.1. The meaning of Eq.5 is that ψ is the wavefunction that minimize the kinetic plus the electron-electron repulsion energy and integrates to ρ. It follows that the initial variational problem of Eq.2 can be transformed in a double hierarchical minimization procedure which formally allows for searching among all the ρ's which are N-representable,\n\ni.e. it can be obtained from some antisymmetric wavefunction; this is a condition which is much weaker and more controllable than the v-representability. In explicit terms such a formulation is written as:\n\nE 0 = Min ρ Min ψ λ →ρ ψ λ |T + V ee | ψ λ + v(r)ρ(r)dr . (6)\n\nThe inner minimization is restricted to all wave functions ψ λ leading to ρ(r), while the outer minimization searches over all the ρ's which integrate to N. The original HK formulation can then be seen as a part of this new one once its universal functional,\n\nF [ρ] = ψ |T + V ee | ψ\n\nis written as:\n\nF [ρ] = Min ψ→ρ ψ |T + V ee | ψ . (7)\n\nThe purpose of this work is to show that F [ρ] can be determined solely by a minimization with respect to the N -1 conditional probability density of the electron correlation functional. This latter will be shown to be composed by the non local Fisher information functional [8] and the electron-electron two-particle Coulomb term. The advantage of this representation is manifold; it further clarifies the connection of electronic properties to the Fisher theory and shows that the knowledge of such a functional is the crucial ingredient in density functional based approaches; it also identifies the Weizsacker kinetic term,\n\n|∇ρ(r)| 2\n\nρ(r) dr, as necessary component of the universal functional F [ρ] and, in practical terms, offers an objective criterion of evaluation of \"approximate\" exchange and correlation functional, i.e. among two functionals, the physically better founded is the \"smaller\" one. In order to show the practical aspects of our idea we illustrate a potential application."
},
{
"section_type": "OTHER",
"section_title": "II. THE NEW REPRESENTATION",
"text": "Before writing the functional in the conditional probability density formalism, we need to define such a quantity. Let us consider a generic fermionic wavefunction ψ(r 1 , ....r N ), for simplicity we consider a real wavefunction, but the extension to a complex one can be also done [9] ; we do not consider the spin dependence explicitly, however this will not influence the main conclusions. Then the N-particle probability density is [10, 11] :\n\nNψ * (r 1 , ....r N )ψ(r 1 , ....r N ) = Θ 2 (r 1 , ...., r N ) (8)\n\nand this can formally decomposed as [10, 11] :\n\nΘ 2 (r 1 , ...., r N ) = ρ(r 1 )f (r 2 , ......., r N /r 1 ) (9)\n\nwhere ρ(r 1 ) is the one particle probability density (normalized to N) and f (r 2 , ......., r N /r 1 )\n\nis the N -1 electron conditional (w.r.t. r 1 ) probability density, i.e. the probability density of finding an N -1 electron configuration, C(r 2 , ......., r N ), for a given fixed value of r 1 . The function f satisfies the following properties:\n\n(i )\n\nΩ N-1 f (r 2 , ........, r N /r 1 )dr 2 .......dr N = 1∀r 1 (ii ) f (r 1 , ..r i ...r j-1 , r j+1 ..., r N /r j ) = 0; f or i = j; ∀i, j = 1, N (iii ) f (r 1 , .....r i .., r j ...r k-1 , r k+1 .., r N /r k ) = 0; f or i = j; ∀i, j = k (10)\n\nThe property (iii ) of Eq.10 assures us that f reflects the fermionic character of an electronic wavefunction. In fact it says that if any two particles are in the same 'state' \"r\" the probability of that specific global configuration is zero. In principle, together with condition (ii), this is a way to mimic the antisymmetric character of the fermionic wavefunction since for fermions |ψ(r 1 , ...r i , ...r j , ...r N )| 2 = 0; f or i = j, ∀i, j.It must be noticed that condition (iii) is complementary to (ii). With this formalism the term ψ |T + V ee | ψ can be written as (see Refs. [9, 10, 12] ):\n\nψ |T + V ee | ψ = 1 8 |∇ρ(r)| 2 ρ(r) dr + 1 8 ρ(r) Ω N-1 |∇ r f (r ′ , ...., r N /r)| 2 f (r ′ , ....., r N /r) dr ′ ....dr N dr + (N -1) ρ(r) Ω N-1 f (r ′ , ....., r N /r) |r -r ′ | dr ′ ....dr N dr (11)\n\nwhere we have identified r 1 with r and made use of the property of electron indistinguishability, thus r could be identified with any of the r i (and the same for r ′ identified here with r 2 ) without changing the results; a further consequence is that the Coulomb expression (last term on the r.h.s.) is written as the sum of N -1 identical terms for the generic r and r ′ particles. Using Eq.11 the Levy-Lieb constrained-search formulation can then be written as:\n\nE 0 = Min ρ Min f (Γ[f, ρ]) + 1 8 |∇ρ(r)| 2 ρ(r) dr + v(r)ρ(r)dr (12)\n\nwhere\n\nΓ[f, ρ] = 1 8 ρ(r) Ω N-1 |∇ r f (r ′ , ...., r N /r)| 2 f (r ′ , ....., r N /r) dr ′ ....dr N dr + (N -1) ρ(r) Ω N-1 f (r ′ , ....., r N /r) |r -r ′ | dr ′ ....dr N dr. (13)\n\nIn this way we have transferred the problem from from ψ to f which means that the focus is now on Γ[f, ρ], i.e., as discussed in Ref. [12] , the correlation functional."
},
{
"section_type": "OTHER",
"section_title": "III. A PRACTICAL EXAMPLE: THE PARAMETRIC EXPONENTIAL FORM OF f",
"text": "In our previous work [12] , we have proposed an approximation for f based on a twoparticle factorization:\n\nf = Π N i=2 h i (E H (r, r i )) = Π N i=2 e (N -1)E(r) e -E H (r,r i ) (14)\n\nwhere\n\ne -E(r) = ω e -E H (r,r i ) dr i . ( 15\n\n)\n\nhere E H (r, r i ) = ρ(r)ρ(r i ) |r-r i | , N is the number of particle, and ω the volume corresponding to one particle. Such an approximation, due to its simplicity, allows us to write an analytic expression of the Fisher functional which can be used in a straightforward way in numerical calculations. However it does not satisfy the condition (iii ) of Eq.10, and, for this reason, in order to use it into the Levy-Lieb constrained-search scheme it must be extended. The expression we propose here is the following:\n\nf (r 2 , ...r N /r) = Π n=2,N e E(r)-γE H (r,rn) × Π i>j =1 e -βE H (r i ,r j ) (16)\n\nwith:\n\ne -E(r) = Π n=2,N Π i>j =1 e -γE H (r,rn)-βE H (r i ,r j ) dr 2 .....dr N ( 17\n\n)\n\nHere γ and β are two free parameters. As it can be easily verified this expression of f satisfies all the requirements of Eq.10. The meaning of f as expressed in Eq.16 is that the probability of finding a certain configuration for the N -1 particles, having fixed particle r 1 = r, depends not only on the fixed particle and its interaction with the N -1 other particles as before, but also on the mutual arrangements of the N -1 particles (it has also to be kept in mind that using the particle indistinguishability the formalism can be applied to any r i as a fixed particle). The parameters γ and β express how important the N -1 mutual interactions are with respect to the interactions with r. Being now f a biparametric function, one can use the Levy-Lieb constrained search in our formulation and find the optimal values for γ and β. This practical example shows two different aspects of our formulation; basically we have shown that indeed it is possible to build a function f and actually it can be chosen in a way that its optimal expression can be determined via the constrained-search formulation. It must be noticed that this form of f is still rather simple since the spins are not explicitly considered when constructing the function and thus one cannot distinguish between the exchange and the correlation part of the electron-electron interaction as it is done in standard Density Functional Theory; as a consequence one should expect only an overall average description of these two terms which are here incorporated into the global correlation. However the construction of a more complete expression of f , which takes care of the effects of the spins, is the subject of current investigation. This emphasizes once more the merit of the general procedure shown here, that is different expressions of f , with different degrees of complexity, can be proposed and their relative validity checked by the constrained-search procedure."
},
{
"section_type": "DISCUSSION",
"section_title": "IV. DISCUSSION AND CONCLUSIONS",
"text": "As anticipated in the introduction, the consequences of Eqs.12,13 are rather interesting.\n\nThe Levy-Lieb variational principle can be reformulated as: The universal functional\n\nF [ρ]\n\nis the one with the minimum correlation functional with respect to the electron conditional probability density. This new interpretation of the HK universal functional tells us that only an accurate description of the correlation effects, considering the Weizsacker term as a necessary term, leads to an accurate description of the whole energy functional; such a criterion is necessary and sufficient. It is obvious that it is necessary; without knowing\n\nΓ[f, ρ], F [ρ] cannot be known; it is sufficient because once Γ[f, ρ] or better f (r 2 , ...r N /r 1 )\n\nis (in principle) known than the whole energy functional is known explicitely. Clearly, the \"true\" f (r 2 , ...r N /r 1 ) is very difficult if not impossible to obtain [13] , however it can be sufficiently well described on the basis of mathematical requirements and physical intuition as done for example in Ref. [12] and as shown in the previous section. From this point of view, Eqs.12,13, can be seen as an objective criterion to design, on the basis of physical intuition and fundamental mathematical requirements, valid energy functionals. In fact, as done in Ref. [12] and in the previous section, one can construct well-founded expressions for f keeping in mind the physical meaning of the electron correlation effects and the necessary related mathematical prescriptions of Eq.10. Next one can make use of Eqs.12,13 and choose among different functional forms of f , the one giving the \"smaller\" Γ. It must be noticed that in this work we do not claim that finding a functional form of f is easier or more rigorous than to find an exchange-correlation functional in standard Density Functional Theory; it represents an alternative or complementary approach to the latter.\n\nHowever, the approach based on f allows one to express in a more direct way, via the choice of different forms of f , the physical principles related to the electron correlation effects and to have an explicit form of the correlation term for the kinetic functional which is of great advantage for kinetic functional based methods (see e.g. Refs. [14, 15] ).An important aspect linked to the statement above is that the term: is available (see e.g. [10, 16, 17] and references therein); this term is very often linked to the electron correlation functional and electronic properties(see Refs. [18, 19] ), our work further clarifies this connection, suggesting that the results known from the analysis of the Fisher functional could be employed in this context. In conclusion we have shown an alternative view of the Levy-Lieb constrained search approach and provided an example which clarifies the practical advantage of our idea; in this sense the present work it is not merely a marginal new formal contribution to a rather well-known method, but gives a new powerful insight into the field of applicability for realistic systems."
}
] | |
arxiv:0704.0373 | 0704.0373 | 1 | 73b49bc09e35b75ceb60fe75a9c900fcfb3c49ccef47c52d7651e63b64d3f27e | Reality of linear and angular momentum expectation values in bound
states | In quantum mechanics textbooks the momentum operator is defined in the
Cartesian coordinates and rarely the form of the momentum operator in spherical
polar coordinates is discussed. Consequently one always generalizes the
Cartesian prescription to other coordinates and falls in a trap. In this work
we introduce the difficulties one faces when the question of the momentum
operator in spherical polar coordinate comes. We have tried to point out most
of the elementary quantum mechanical results, related to the momentum operator,
which has coordinate dependence. We explicitly calculate the momentum
expectation values in various bound states and show that the expectation value
really turns out to be zero, a consequence of the fact that the momentum
expectation value is real. We comment briefly on the status of the angular
variables in quantum mechanics and the problems related in interpreting them as
dynamical variables. At the end, we calculate the Heisenberg's equation of
motion for the radial component of the momentum for the Hydrogen atom. | [
"Utpal Roy",
"Suranjana Ghosh",
"T. Shreecharan",
"Kaushik Bhattacharya"
] | [
"quant-ph",
"hep-th"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Quantum mechanics is a treasure house of peculiar and interesting things. Elementary textbooks of quantum mechanics [1, 2, 3] generally start with the postulates which are required to define the nature of the dynamical variables in the theory and their commutation relations. The choice of the dynamical variables is not that clear, as the coordinates in Cartesian system are all elevated to the status of operators where as time remains a parameter. More over in spherical polar coordinates only the radial component can be represented as an operator while the angles still remain as a problem. The difficulty of giving different status to the spatial coordinates and time is bypassed in quantum field theories where all the coordinates and time become parameters of the theory. But the problem with angles still remain a puzzle which requires to be understood in future.
When we start to learn quantum mechanics, most of the time we begin with elementary calculations relating to the particle in a one dimensional infinite well, particle in a finite potential well, linear harmonic oscillator and so on. The main aim of these calculations is to solve the Schrödinger equation in the specific cases and find out the bound state energies and the energy eigenfunctions in coordinate space representation. While solving these problems we overlook the subtleties of other quantum mechanical objects as the definition of the momentum operator in various coordinates, the reality of its expectation value, etc.. In the last one or two decades there has been a number of studies regarding the self-adjointness of various operators [4] . The aim of these studies has been to analyze the self-adjointness of various operators like momentum, Hamiltonian etc. and find out whether these operators are really self-adjoint in some interval of space where the theory is defined, if not then can there be any mathematical method by which we can make these operators to be self-adjoint in the specified intervals ?
In the present work we deal with a much elementary concept in quantum mechanics related to the reality of the expectation values of the momentum operator, be it linear or angular. We do not analyze the self-adjointness of the operators which requires different mathematical techniques. To test the self-adjointness of an operator we have to see whether the operator is symmetric in a specific spatial interval and the functional domain of the operator and its adjoint are the same. In the article we always keep in touch with the recent findings from modern research about self-adjoint extensions but loosely we assume that the operators which we are dealing with are Hermitian. If something is in contrary we point it out in the main text. A considerable portion of our article deals with the analysis 2 of the fact that the expectation value of the momentum operator in various bound states are zero, a result which most of the textbooks only quote but never show. In the simpler cases the result can be shown by one or two lines of calculation, but in nontrivial potentials as the Morse potential, the Coulomb potential the result is established by using various properties of the special functions such as the associated Legendre and the associated Laguerre.
The presentation of various materials in our article is done in the following way. Next section deals with the definition of the momentum operator and its properties. Section III deals with the intricacies of the definition of the momentum operator in spherical polar coordinates and the problems we face when we try to mechanically implement the quantization condition, which is invariably always written in Cartesian coordinates in most of the textbooks on quantum mechanics. In section IV we explicitly calculate the momentum expectation values in various potentials and show that in bound states we always get the expectation value of the linear momentum to be zero. Section V gives a brief discussion on the Ehrenfest theorem when we are using it to find out the time derivative of the expectation value of the radial component of momentum in the case of the Hydrogen atom. We end with the concluding section which summarizes the findings in our article.
Before going into the main discussion we would like to mention about the convention. We have deliberately put a hat over various symbols to show that they are operators in quantum mechanics. Some times this convention becomes tricky when we are dealing with angular variables as there the status of these variables is in question. The other symbols have their conventional meaning. As we are always using the coordinate representation sometimes we may omit the hat over the position operator as in this representation the position operator and its eigenvalues can be trivially interchanged.
From the Poisson bracket formalism of classical mechanics we can infer: [x i , pj ] = ih δ i j , (2.1) where δ i j = 1 when i = j and zero for all other cases, and i, j = 1, 2, 3. In the above equation xi is the position operator and pj is the linear momentum operator in Cartesian coordinates. From the above equation we can also find the form of the momentum operator in position representation, which is: pi = -ih ∂ ∂x i . (2.2) It is interesting to note that the above expression of the momentum operator also gives us the form of the generator of translations. This is because of the property: [p x , F (x)] = -ih dF (x) dx , (2.3) where F (x) is an arbitrary well defined function of x. The above equation ensures that the momentum operator generates translations along the x direction. Particularly in one-dimension the expression of the momentum operator becomes px = -ih ∂ ∂x . We know that the expectation value of the momentum operator must be real. If we focus on one-dimensional systems to start with, where the system is specified by the wave-function ψ(x, t), the expectation value of any operator Ô is defined by: Ô ≡ ∞ -∞ ψ * (x, t) Ô ψ(x, t) dx , (2.4) where ψ * signifies complex conjugation of ψ and the extent of the system is taken as -∞ < x < ∞. From the above equation we can write, Ô * = ∞ -∞ ψ(x, t) Ô * ψ(x, t) * dx . (2.5) If Ô = Ô * then the condition of the reality of the expectation value becomes: ∞ -∞ ψ * (x, t) Ô ψ(x, t) dx = ∞ -∞ ψ(x, t) Ô * ψ(x, t) * dx . (2.6) 3 For a three-dimensional system the above condition becomes, ∞ -∞ ψ * (x, t) Ô ψ(x, t) d 3 x = ∞ -∞ ψ(x, t) Ô * ψ(x, t) * d 3 x . (2.7) Now if we take the specific case of the momentum operator in one-dimension we can explicitly show that its expectation value is real if the extent of the system is infinite and the wave-function vanishes at infinity. The proof is as follows.
The expectation value of the momentum operator is: ∞ -∞ ψ * (x, t) px ψ(x, t) dx = -ih ∞ -∞ ψ * (x, t) ∂ψ(x, t) ∂x dx = -ih ψ * (x, t)ψ(x, t)| ∞ -∞ -∞ -∞ ψ(x, t) ∂ψ(x, t) * ∂x dx , (2.8) If the wave functions vanish at infinity then the first term on the second line on the right-hand side of the above equation drops and we have, ∞ -∞ ψ * (x, t) px ψ(x, t) dx = -ih ∞ -∞ ψ * (x, t) ∂ψ(x, t) ∂x dx = ih ∞ -∞ ψ(x, t) ∂ψ(x, t) * ∂x dx , = ∞ -∞ ψ(x, t) p * x ψ(x, t) * dx . (2.9) A similar proof holds for the three-dimensional case where it is assumed that the wave-function vanishes at the boundary surface at infinity.
In non-relativistic version of quantum mechanics we know that if we have a particle of mass m which is present in a time-independent potential we can separate the Schrödinger equation:
ih ∂ψ(x, t) ∂t = -h2 2m ∇ 2 + V (x) ψ(x, t) , (3.1) into two equations, one is the time-dependent one which gives the trivial solution e -iEt h where E is the total energy of the particle, and the other equation is the time-independent Schrödinger equation: ∇ 2 u(x) + 2m h2 (E -V (x))u(x) = 0 , (3.2) where u(x) is the solution of the time-independent Schrödinger equation and the complete solution of the Eq. (3.1) is: ψ(x, t) = u(x)e -iEt h . (3.3) In the case of the free-particle, where V (x) = 0, we have u(x, t) = e ik•x where E = k 2 h2 2m and k = |k|. The freeparticle solution is an eigenfunction of the momentum operator with eigen value hk. Although if we try to find out the expectation value of the momentum operator as is done in the last section we will be in trouble as these wave-functions do not vanish at infinity, a typical property of free-particle solutions. But this problem is not related to the Hermiticity property of the momentum operator, it is related with the de-localized nature of the free-particle solution.
In physics many times we require to solve a problem using curvilinear coordinate systems. The choice of our coordinate system depends upon the specific symmetry which we have at hand. Suppose we are working in spherical polar coordinates and the solution of Eq. (3.2) can be separated into well behaved functions of r, θ and φ as,
u(x) = u(r, θ, φ) = R(r) Θ(θ) Φ(φ) . (3.4) 4
If we try to follow the proof of the Hermiticity of the linear momenta components, as done in the last section, in spherical polar coordinates, then we should write: p = -ih τ u * (r, θ, φ)∇u(r, θ, φ) dτ , = -ih R * (r)Θ * (θ)Φ * (φ) e r ∂ ∂r + e θ r ∂ ∂θ + e φ r sin θ ∂ ∂φ R(r)Θ(θ)Φ(φ)r 2 drdΩ , (3.5)
where in the above equation e r , e θ , e φ respectively are the unit vectors along r, θ and φ, and dΩ = sin θ dθ dφ. τ is the volume over which we integrate the expression in the above equation. From the last equation we can write:
pr = -ih Ω |Θ(θ)| 2 |Φ(φ)| 2 dΩ ∞ 0 r 2 R * (r) dR(r) dr dr , (3.6)
As Θ(θ) and Φ(φ) are normalized, the integration:
Ω |Θ(θ)| 2 |Φ(φ)| 2
dΩ = 1 and we can proceed as in Eq. (2.9) as:
pr = -ih ∞ 0 r 2 R * (r) dR(r) dr dr , = -ih r 2 R * (r)R(r) ∞ 0 - ∞ 0 2rR * (r) + r 2 dR * (r) dr R(r) dr . (3.7)
If R(r) vanishes at infinity then the above equation reduces to,
pr = ih ∞ 0 r 2 R(r) dR * (r) dr dr + 2ih ∞ 0 r|R(r)| 2 dr , = pr * + 2ih ∞ 0 r|R(r)| 2 dr . (3.8)
The above equation implies that pr is not real in spherical polar coordinates. The solution of the above problem lies in redefining pr as is evident from Eq. (3.8), and it was given by Dirac [5, 6] . The redefined linear momentum operator along r can be:
pr ≡ -ih ∂ ∂r + 1 r = -ih 1 r ∂ ∂r r . (3.9)
This definition of the pr is suitable because in this form it satisfies the commutation relation as given in Eq. (2.1) where now the operator conjugate to r is pr . The form of pr in Eq. (3.9) shows that for any arbitrary function of r as F (r) we must still have Eq. (2.3) satisfied. This implies that the modified form of pr is still a generator of translations along the r direction. Up to this point we were following what was said by Dirac regarding the status of the radial momentum operator. Still everything is not that smooth with the redefined operator as we can see that it turns out to be singular around r = 0, more over, although the radial momentum acts like a translation generator along r but near r = 0 it cannot generate a translation towards the left as the interval ends there. In this regard we can state that the issue of the reality of the radial component of the momentum in spherical polar coordinates is a topic of modern research in theoretical physics [7, 8] . It has been shown that the operator -ih ∂ ∂r is not Hermitian and more over it can be shown [4] that such an operator cannot be self-adjoint in the interval [0, ∞].
In some recent work [8] the author claims that there can be an unitary operator which connects -ih ∂ ∂r to -ih 1 r ∂ ∂r r, and as the former operator does not have a self-adjoint extension in the semi-infinite interval so the latter is also not self-adjoint in the same interval.
If we further try to find out whether pθ and pφ are real, then we will face difficulties. Working out naively if we claim that pφ = 1 rsinθ ∂ ∂φ as suggested by the φ component of Eq. (3.5) we will notice that φ pφ does not have the dimension of action. This means pφ or pθ is not conjugate to φ or θ. This is a direct representation of the special coordinate dependence of the quantization condition. Only in Cartesian coordinates the variables conjugate to x, y and z are p x , p y and p z . Taking the clue from classical mechanics we know the proper dynamical variables conjugate to φ and θ are the angular momentum operators, namely Lθ and Lφ . In general Lφ is given by: Lφ = -ih ∂ ∂φ , (3.10) 5 which can be shown to posses real expectation values by following a similar proof as is done in Eq. (2.8) and Eq. (2.9), if we assume Φ(0) = Φ(2π). In this form it is tempting to say that we can have a relation of the form, [ φ, Lφ ] = ih , (3.11) which looks algebraically correct. But the difficulty in writing such an equation is in the interpretation of φ which has been elevated from an angular variable to a dynamical operator. In spherical polar coordinates both θ and φ are compact variables and consequently have their own subtleties. Much work is being done in trying to understand the status of angular variables and phases [9, 10] , in this work we only present one example showing the difficulty of accepting φ as an operator. From the solution of the time-independent Schrödinger equation for an isotropic potential we will always have:
Φ(φ) = 1 √ 2π e iMφ , (3.12)
where M = 0, ±1, ±2, •, •. Now if φ is an operator we can find its expectation value, and it turns out to be: φ = 1 2π 2π 0 φe iMφ e -iMφ dφ , = π , (3.13) and the expectation value of φ2 is:
φ2 = 1 2π 2π 0 φ 2 e iMφ e -iMφ dφ , = 4 3 π 2 . (3.14) Consequently ∆φ = φ2 -φ 2 = π √ 3
. Similarly calculating Lφ we get: Lφ = M h 2π 2π 0 e -iMφ e iMφ dφ , = M h , (3.15) as expected, and L2 φ = M 2 h2 . This implies ∆L φ = L2 φ -Lφ 2 = 0. So we can immediately see that the Heisenberg uncertainty relation between φ and Lφ , ∆φ∆L φ ≥ h/2 breaks down. This fact makes life difficult and we have no means to eradicate this problem.
Taking the clue from the φ part we can propose that Lθ is also of the form -ih ∂ ∂θ . With this definition of Lθ let us try to prove its Hermitian nature as done in Eq. (3.7). Taking R(r) and Φ(φ) in Eq. (3.4) separately normalized, we can write:
Lθ = -ih π 0 Θ * (θ) dΘ(θ) dθ sin θdθ = -ih sin θ Θ * (θ)Θ(θ)| π 0 - π 0 cos θ Θ * (θ) + sin θ dΘ * (θ) dθ Θ(θ) dθ , = ih π 0 sin θ Θ(θ) dΘ * (θ) dθ dθ + ih π 0 cos θ |Θ(θ)| 2 dθ , = Lθ * + ih π 0 cos θ |Θ(θ)| 2 dθ . (3.16)
The above equation shows that Lθ is not real. The rest is similar to the analysis following Eq. (3.8) where now we have to redefine the angular momentum operator conjugate to θ as [11]: Lθ ≡ -ih ∂ ∂θ + 1 2 cot θ . (3.17) 6 Unlike the φ case, Θ(θ) are not eigenfunctions of Lθ . But the difficulties of establishing θ as an operator still persists and in general θ is not taken to be a dynamical operator in quantum mechanics. It is known that both θ and φ are compact variables, i.e. they have a finite extent. But there is a difference between them. In spherical polar coordinates the range of φ and θ are not the same, 0 ≤ φ < 2π and 0 ≤ θ ≤ π. This difference can have physical effects. As φ runs over the whole angular range so the wave-function corresponding to it Φ(φ) is periodic in nature while due to the range of θ, Θ(θ) need not be periodic. Consequently there can be a net angular momentum along the φ direction while there cannot be any net angular momentum along θ direction. And this can be easily shown to be true. As the time-independent Schrödinger equation for an isotropic potential yields Φ(φ) as given in Eq. (3.12) similarly it is known that in such a potential the form of Θ(θ) is given by:
Θ(θ) = N θ P L M (cos θ) , (3.18)
where N θ is a normalization constant depending on L, M and P L M (cos θ) is the associated Legendre function, which is real. In the above equation L and M are integers where L = 0, 1, 2, 3, •, • and M = 0, ±1, ±2, ±3, •, •. The quantum number M appearing in Eq. (3.12) and in Eq. (3.18) are the same. This becomes evident when we solve the time-independent Schrödinger equation in spherical polar coordinates by the method of separation of variables. A requirement of the solution is -L ≤ M ≤ L. Now we can calculate the expectation value of Lθ using the above wave-function and it is:
Lθ = -ihN 2 θ π 0 P L M (cos θ) dP L M (cos θ) dθ + 1 2 cot θP L M (cos θ) sin θdθ = -ihN 2 θ π 0 P L M (cos θ) dP L M (cos θ) dθ sin θdθ + 1 2 π 0 P L M (cos θ)P L M (cos θ) cos θ dθ . (3.19)
To evaluate the integrals on the right hand side of the above equation we can take x = cos θ and then the expectation value becomes:
Lθ = -ihN 2 θ -1 1 P L M (x) dP M L (x) dx (1 -x 2 ) 1 2 dx - 1 2 -1 1 P L M(x) P L M(x) x √ 1 -x 2 dx . (3.20)
The second term in the right hand side of the above equation vanishes as the integrand is an odd function in the integration range. For the first integral we use the following recurrence relation [12]:
(x 2 -1) dP L M (x) dx = M xP L M (x) -(L + M )P L M-1 (x) , (3.21)
the last integral can be written as, (3.23) we can see immediately that both the integrands in the right hand side of the above equation is odd and consequently Lθ = 0 as expected. A similar analysis gives Lφ = M h. It must be noted that the form of Lθ still permits it to be the generator of rotations along the θ direction. As the motion along φ is closed so there can be a net flow of angular momentum along that direction but because the motion along θ is not so, a net momentum along θ direction will not conserve probability and consequently for probability conservation we must have expectation value of angular momentum along such a direction to be zero. In elementary quantum mechanics text books it is often loosely written that the solution of the time-independent Schrödinger equation is real when we are solving it for a real potential. But this statement is not correct. The reality 7 of the solution also depends upon the coordinate system used. Specially for compact periodic coordinates we can always have complex functions as solutions without breaking any laws of physics. Before leaving the discussion on angular variables in spherical polar coordinates we want to point out one simple thing which is interesting. In Cartesian coordinates when we deal with angular momentum we know that: [ Li , Lj ] = iǫ ijk Lk , (3.24) where Li stands for Lx , Ly or Lz . For this reason there cannot be any state which can be labelled by the quantum numbers of any two of the above angular momenta. But from the expressions of Lφ and Lθ we see that, [ Lφ , Lθ ] = 0 , (3.25) and consequently in spherical polar coordinates we can have wave-function solutions of the Schrödinger equation which are simultaneous eigenfunctions of both Lφ and Lθ as P L M (θ). For real V (x), we expect the solution of the time-independent Schrödinger equation u(x) to be real, when we are solving the problem in Cartesian coordinates. In all these cases the expectation value of the linear momentum operators must vanish. The reason is simple and can be understood in one-dimensional cases where with real u(x) we directly see that the integral ∞ -∞ u * (x) ∂u(x) ∂x dx is real and so ∞ -∞ u * (x) px u(x) dx becomes imaginary as px contains i, as is evident from the first line in Eq. (2.9). So if the expectation value of the momentum operator has to be real then the only outcome can be that for all those cases where we have a time-independent solution in a bounded region of space, with a real potential and working in Cartesian coordinates, the expectation value of the momentum operator must vanish. The above statement is true in curvilinear coordinates also, but in those cases the definition of the momentum operators have to be modified. This fact becomes clear when we write the relationship between the probability flux and the expectation value of the momentum operator. The probability flux for a particle of mass m is:
Lθ = ihN 2 θ M -1 1 x(1 -x 2 ) -1 2 P L M (x)P L M (x) dx -(L + M ) -1 1 (1 -x 2 ) -1 2 P L M (x)P L M-1 (x) dx . (3.22) As, P L M (x) = (-1) L+M P L M (-x) ,
j(x, t) = - ih 2m [ψ * (x, t)∇ψ(x, t) -(∇ψ * (x, t))ψ(x, t)] , = h m Im (ψ * (x, t)∇ψ(x, t)) , (3.26)
where 'Im' implies the imaginary part of some quantity. Most of the elementary quantum mechanics books then proceeds to show that:
d 3 x j(x, t) = p m , (3.27)
which is obtained from Eq. (3.26) by integrating both sides of it over the whole volume. From Eq. (3.26) we immediately see that if the solution of the time-independent Schrödinger equation is real we will have j(x, t) = 0 and consequently from Eq. (3.27), p = 0. But this statement is also coordinate dependent, which is rarely said in elementary textbooks of quantum mechanics. Eq. (3.26) evidently does not hold in spherical polar coordinates. If we take Eq. (3.4) as the solution in a general isotropic central potential and use the general form of ∇ in spherical polar coordinates then it can be seen that j r (r, θ, φ, t) = 0 for a real potential. But then Eq. (3.27) does not hold as here pr is simply the radial component of ∇ and not as given in Eq. (3.9), and we know d dr is not zero. The reason why Eq. (3.26) is not suitable in spherical polar coordinates is related to the fact that in deriving Eq. (3.26) one assumes that the probability density of finding the quantum state within position x and x + dx at time t is |ψ(x, t)| 2 . But this statement is only true in Cartesian coordinates, in spherical polar coordinates the probability density of the system to be within a region r and r + dr, θ and θ + dθ, φ and φ
+ dφ is not |ψ(r, θ, φ)| 2 but |ψ(r, θ, φ)| 2 r 2 sin θ
and consequently the steps which follow leading to Eq. (3.26) in Cartesian coordinates are not valid in spherical polar coordinates. In general Eq. (3.26) will not be valid in any curvilinear coordinate system.
The next section contains the actual calculations of the expectation values of the momentum operator in various cases where we have bound state solutions. In all the relevant cases discussed in this article it is seen that although px = 0 but p2 x is not zero as it is related to the Hamiltonian operator. In all the cases we must have,
(p x ) s = 0 , s = odd integer . (3.28)
The above equation can be guessed from the reality of the expectation value of the momentum operator.
8
In this section we will calculate the momentum expectation values in various bound states with stiff or slowly varying potentials.
A. Particle in one-dimensional stiff potential wells
In this case we consider a particle to be confined in region -L 2 to L 2 along the x-axis where the potential is specified by,
V (x) = ∞ , |x| ≥ L 2 , = 0 , |x| < L 2 . (4.1)
In this case the solution of the time-independent Schrödinger equation, Eq. (3.2), satisfies the boundary condition,
u - L 2 = u L 2 = 0 , (4.2)
and as the potential has parity symmetry about x = 0 we have two sets of solutions, the odd solutions:
u (o) n (x) = 2 L sin 2nπx L , (4.3)
and the even solutions:
u (e) n (x) = 2 L cos (2n -1)πx L . (4.4)
In the above equations n is a positive integer. Both of these functions, u (o) n (x) for the odd case and u (e) n (x) for the even case, are real and are not momentum eigenstates. But the momentum expectation values can be found out from the above solutions. For the odd solutions we have:
px = -ih L 2 -L 2 u (o) n (x) du (o) n (x) dx dx , = - 4inπh L 2 L 2 -L 2 sin 2nπx L cos 2nπx L , = 0 , (4.5)
as expected. Similarly for the even solutions it is also easy to show that the expectation value of the momentum operator vanishes.
In this case,
V (x) = 0 , |x| ≥ a , = -V 0 , |x| < a , (V 0 > 0) . (4.6)
If we are not interested in the normalization constant of the bound state solution then the solution of the timeindependent Schrödinger equation in this case is:
u(x) ∼ e -κ|x| , |x| > a , ∼ cos(kx) , |x| < a , (even parity) ∼ sin(kx) , |x| < a , (odd parity) , (4.7) 9
where,
k 2 = 2m(-|E| + V 0 ) h2 , (4.8) κ 2 = 2m|E| h2 . (4.9)
In this case the expectation value of the momentum operator is:
px ∼ -ih +∞ -∞ u(x) du(x) dx dx , ∼ -ih κ -a -∞ e 2κx dx - ∞ a e -2κx dx + k a -a sin(kx) cos(kx) dx , = 0 , (4.10)
where the first two lines of the above equation holds up to a constant arising from the normalization of the wavefunction. In deriving the last equation we have taken the odd parity solution, but the result remains unaffected if we take the even parity solution as well.
In this case the potential is:
V (x) = -V 0 δ(x) , (V 0 > 0) . (4.11)
In this case there can be one bound state solution which is obtained after solving the Eq. (3.2). Demanding that the solution u(x) satisfies the boundary conditions: (4.13) where ǫ is an infinitesimal quantity tending to zero, we get the form of the solution which is:
u(x = -ǫ) = u(x = +ǫ) , (4.12) du dx x=+ǫ - du dx x=-ǫ = - 2mV 0 h2 u(x = 0) ,
u(x) = √ κ e κx , x ≤ 0 , (4.14) = √ κ e κx , x ≥ 0 , (4.15)
where κ = mV0 h2 and the energy of the bound state is E = -mV 2 0 2h 2 . The expectation value of the momentum operator in this case is:
px = -ih ∞ -∞ u(x) du(x) dx dx , = -ihκ 0 -∞ e 2κx dx - ∞ 0 e -2κx dx , = 0 . (4.16)
In this case, also from Hermiticity of the momentum operator we see that Eq. (3.28) holds true.
B. Particle in one-dimensional slowly varying potentials
In the case of the linear harmonic oscillator we have:
V (x) = 1 2 mω 2 x2 , (4.17)
10 where ω is the angular frequency of the oscillator. The solution of Eq. (3.2) in this case, using the series solution method, yields:
u n (q) = N n e -q 2 2 H n (q) , (4.18)
where n = 0, 1, 2, •, • and q = √ αx where α = mω h2 . H n (q) are Hermite polynomials of order n and N n is the normalization constant given by,
N n = 1 √ π n! 2 n 1 2 . (4.19)
The momentum expectation value in this case turns out to be, px = -ih √ α ∞ -∞
u n (q) du n (q) dq dq , = -ih √ αN 2 n ∞ -∞ e -q 2 H n (q) dH n (q) dq dq - ∞ -∞ q e -q 2 H 2 n (q) dq , = 0 . (4.20)
The first integral on the right side of the second line of the last equation vanishes because, dHn(q) dq = 2nH n-1 (q) and consequently the integral transforms into the orthogonality condition of the Hermite polynomials. The second integral on the second line of the right side of the above equation vanishes because the integrand is an odd function of q.
The linear harmonic oscillator (LHO) has some very interesting properties. To unravel them we have to digress a bit from the wave-mechanics approach which we have been following and follow the Dirac notation of bra and kets. The Hamiltonian of the LHO in one-dimension is: Ĥ = p2 x 2m + 1 2 mω 2 x2 , (4.21) which can also be written as: Ĥ = hω â † â + 1 2 , (4.22) where â and â † are the annihilation and the creation operators given by:
â ≡ mω 2h x + ip x mω , â † ≡ mω 2h x - ip x mω . (4.23)
It can be seen clearly from the above definitions that â is not an Hermitian operator. More over from the definition of the operators we see that, [â, â † ] = 1 . (4.24) Conventionally the number operator is defined as: N ≡ â † â , (4.25) and its eigen-basis are the number states |n such that,
N |n = n|n . (4.26)
The Hamiltonian of the LHO can be written in terms of the number operator and consequently the number states are energy eigenstates. In this basis the action of the annihilation and creation operators are as:
â|n = √ n |n -1 , (4.27) â † |n = √ n + 1 |n + 1 . (4.28)
From the definitions of the annihilation and creation operators we can write the momentum operator as: px = -i mhω 2 (â -â † ) . (4.29) 11 From Eq. (4.27), Eq. (4.28) and the above equation we can write the matrix elements of the momentum operator as:
n ′ |p x |n = i mhω 2 - √ n δ n ′ , n-1 + √ n + 1 δ n ′ , n+1 . (4.30)
The above equation shows that the momentum operator can connect two different energy eigenstates. In the case of LHO, except the number operator states, we can have another state which is an eigenstate of the annihilation operator â. This state is conventionally called the coherent state and it is given as:
|α = e -|α| 2 2 ∞ n=0 α n √ n! |n , (4.31)
where α is an arbitrary complex number. Now from Eq. (4.29) we can find the momentum expectation value of the coherent state and it is,
px α ≡ α|p x |α = mhω 2 Im(α) . (4.32)
From the above equation we can see that although the expectation value of the momentum operator is zero in the energy eigen-basis but it is not so when we compute the momentum expectation value in the coherent state basis, which is essentially a superposition of energy eigenstates. It must be noted that the momentum expectation value is non zero only when the parameter α has an imaginary part.
Among the potentials belonging to the hypergeometric class the Pöschl-Teller potentials have been the most extensively studied and used. This class of potentials consist of trigonometric as well as the hyperbolic type. The trigonometric versions have found applications in molecular and solid state physics and the hyperbolic variants have been used in various studies related to black hole perturbations.
In the present work we use the trigonometric, symmetric Pöschl-Teller potential given by: (4.33) where V 0 can be parameterized as: (4.34) with for some positive number λ > 1 and a is some scaling factor. The energy eigenvalues of the bound state solutions are:
V (x) = V 0 tan 2 (ax) ,
V 0 = h2 a 2 2m λ(λ -1) ,
E n = - h2 a 2 2m (n 2 + 2nλ + λ) , (4.35)
and the solution of the time-independent Schrödinger equation is,
u n (x) = N n cos(ax) P 1/2-λ n+λ-1/2 (sin(ax)) , (4.36)
where,
N n = a(n + λ)Γ(n + 2λ) Γ(n + 1) 1/2 , (4.37)
is the normalization constant and P µ ν (x) is the associated Legendre function. At this point it is fair to point out that P µ ν (x) is not the Legendre polynomial P L M (x) appearing in Eq. (3.18), as µ and ν need not be integers as L and M . P µ ν (x) is not a polynomial but the function appearing in the right hand side of Eq. (4.36) is a polynomial. Now as claimed in the text let us show that the momentum expectation value is indeed zero. Before we proceed let us simplify the notation a bit by calling µ = 1/2 -λ and ν = n + λ -1/2. Substituting z = ax we can write the momentum expectation value as:
px = -ihN 2 n π/2 -π/2 dz cos(z) P µ ν (sin(z)) d dz cos(z) P µ ν (sin(z)) . (4.38)
12 Note the limits of the integration range from π/2 to -π/2 since at this value the potential becomes infinity hence we need not consider the integration range to be the whole real line. For the sake of convenience let us make a change of variable; letting y = sin(z) the above integral becomes: px = -ihN 2 n +1 -1
dy (1 -y 2 ) 1/4 P µ ν (y) d dy (1 -y 2 ) 1/4 P µ ν (y) . (4.39)
Taking the derivative inside the integral we get: px = -ihN 2 n +1 -1
dy (1 -y 2 ) 1/2 P µ ν (y) dP µ ν (y) dy - y(1 -y 2 ) -1/2 2 P µ ν (y)P µ ν (y) . (4.40)
It is known that for associated Legendre functions [13],
P µ ν (-x) = cos[(µ + ν)π] P µ ν (x) - 2 π sin[(µ + ν)π] Q µ ν (x) , (4.41)
where Q µ ν (x) is the other linearly independent solution of the associated Legendre differential equation. As in our case µ + ν = n so P µ ν (x) will have definite parity. As P µ ν (x) has definite parity so the contribution of the second term in the above integral vanishes since the total integrand is an odd function. The first integral is similar to the one in Eq. (3.20) and, due to the typical parity property of P µ ν (x) as shown in Eq. (4.41), it also vanishes. Consequently we have px = 0 as expected.
Diatomic molecule is an exactly solvable system, if one neglects the molecular rotation. The most convenient model to describe the system, is the Morse potential [14]:
V (x) = D(e -2β x -2e -β x) , (4.42)
where x = r/r 0 -1, which is the distance from the equilibrium position scaled by the equilibrium value of the inter-nuclear distance r 0 . D is the depth of the potential, called dissociation energy of the molecule and β being a parameter which controls the width of the potential.
In terms of the above scaled variable x, the time-independent Schrödinger equation becomes:
- h2 2µr 0 d 2 u(x) dx 2 + D(e -2βx -2e -βx
)u(x) = Eu(x) . (4.43) Here µ is the reduced mass of the molecule and the corresponding bound state eigen function comes out to be:
u λ n (ξ) = N e -ξ/2 ξ s/2 L s n (ξ) , (4.44)
where the variables are described as, ξ = 2λe -y ; y = βx; 0 < ξ < ∞ , (4.45) and n = 0, 1, ..., [λ -1/2] , (4.46) which is nothing but the quantum number of the vibrational bound states. Here [ρ] denotes the largest integer smaller than ρ, thus total number of bound states is [λ -1/2] + 1. The parameters,
λ = 2µDr 2 0 β 2 h2 and s = - 8µr 2 0 β 2 h2 E , (4.47)
satisfy the constraint condition s + 2n = 2λ -1. We note that the parameter λ is potential dependent and s is related to energy E. In Eq. (4.44), L s n (y) is the associated Laguerre polynomial and N is the normalization constant [15]:
N = β(2λ -2n -1)Γ(n + 1) Γ(2λ -n)r 0 1/2 . (4.48) 13
We are looking for the expectation value of linear momentum for a vibrating diatomic molecule, and its expression is: px = -ih ∞ -∞ u * n (ξ) d dx u n (ξ)dx . (4.49) In terms of the changed variable ξ = 2λe -βx the integration limit changes to ∞ to 0 and the expectation value becomes:
px = -ih 0 ∞ u * n (ξ) d dξ u n (ξ)dξ = ihN 2 - 1 2 ∞ 0 e -ξ ξ s (L s n (ξ)) 2 dξ + s 2 ∞ 0 e -ξ ξ s-1 (L s n (ξ)) 2 dξ + ∞ 0 e -ξ ξ s L s n (ξ) d dξ L s n (ξ)dξ = ihN 2 - 1 2 I 1 + s 2 I 2 + I 3 . (4.50)
Integral I 1 is the orthogonality relation of the associated Laguerre polynomials, which is:
∞ 0 e -ξ ξ s L s n (ξ)L s m (ξ)dξ = Γ(s + n + 1) Γ(n + 1) δ m,n . (4.51)
To evaluate the second integral one uses the normalization integral of Morse eigenstates. The normalization relation is:
∞ -∞ u * (ξ)u(ξ)dr = |N | 2 r 0 β ∞ 0 e -ξ ξ s-1 (L s n (ξ)) 2 dξ = 1 . (4.52)
The above integral involving ξ, is explicitly I 2 . N , being the normalization constant as given in Eq. 4.48. Thus it is very straight forward to evaluate I 2 from the above relation as,
I 2 = Γ(n + s + 1) s Γ(n + 1) . (4.53)
The last integrand I 3 includes a differentiation which can be written as [16]:
d dξ L s n (ξ) = -L s+1 n-1 (ξ) . (4.54)
Writing the right hand side of the above equation as a summation [17]:
L s+1 n = n m=0 L s m , (4.55)
and substituting the derivative term in integral I 3 we obtain:
I 3 = - n-1 m=0 ∞ 0 e -ξ ξ s L s n (ξ)L s m (ξ)dξ . (4.56)
In the above integral m = n because m can go only upto (n -1). Thus the integral vanishes. Now let us see what is the expectation value of momentum observable, after evaluating the three integrals above. Substituting the non-zero values I 1 and I 2 in Eq. 4.50, it is clear that the expectation value of momentum is zero as has been expected.
After a thorough discussion about the momentum expectation values for various solvable one-dimensional potentials, it is worth spending some time discussing about the average position of the particle inside the bound states. Among 14 all the above examples, in each case we had V (x) = V (-x) except the Morse potential as Morse potential is not an example of a symmetric potential: V (x) = V (-x).
In deriving the expectation values of momentum for above symmetric cases, we often considered that the integrals of odd functions over the symmetric limits vanishes. This result does not hold true for the asymmetric Morse potential. Already we have shown that the momentum expectation value: < p >= 0 for all the above potentials. When it comes to the expectation values of position, one can easily see that < x >= 0 for symmetric potentials whose centers are at the origin. On the other hand if this is not the case, suppose the infinite square well is defined in the range 0 ≤ x ≤ L also then the expectation value of position does not vanish. It becomes L/2. Thus, more accurately the average position of the particle is dependent on the symmetry of the potential where as the average momentum is solely guided by the reality of it's eigenvalues and consequently it is zero always.
Below we will briefly discuss how the asymmetry of the potential affects the expectation value of x in the case of the Morse potential. The expectation value of the position operator is:
x = ∞ -∞ u λ * n (ξ) x u λ n (ξ)dx. (4.57)
The eigen function and the variables are respectively substituted from Eq. (4.44) and Eq. (4.45). We obtain
x = N 2 β 2 ln( 2λ
) ∞ 0 e -ξ ξ s-1 (L s n (ξ)) 2 dξ + ∞ 0 e -ξ ξ s-1 (L s n (ξ)) 2 ln(ξ)dξ . (4.58)
The first integral is already been obtained in Eq. (4.53). This result is independent of the quantum number n. The second integral (say I) is not that straight forward, because it contains associated Laguerre polynomial, logarithm, exponential and monomial functions. Here at best we can evaluate the integral atleast for some specific n as, n = 0 or n = 1, when the Laguerre polynomial is respectively replaced by 1 and (-ξ + s + 1). For the ground state wave function (n = 0), I would be
I n=0 = ∞ 0 e -ξ ξ s-1 ln(ξ)dξ, (4.59)
which can be written in terms of Ψ(s) and Γ function [18]:
I n=0 = Γ(s)Ψ(s), (4.60)
where, Ψ(s) is the logarithmic factorial function, defined as d(ln(s)! ds = (s!) ′ s! = Ψ(s). For n = 0, first integral reduces to Γ(s) from Eq. (4.53). Above two evaluations gives the ground state expectation value:
x n=0 = 1 r 0 β
[ln(s + 1) -Ψ(s)] . (4.61) For n = 1, one can proceed in the same way
x n=1 = N 2 β 2 ∞ 0 e -ξ ξ s+1 ln(ξ)dξ + (s + 1) 2 ∞ 0 e -ξ ξ s-1 ln(ξ)dξ2(s + 1) ∞ 0 e -ξ ξ s ln(ξ)dξ (4.62) = N 2 β 2 Γ(s + 2)Ψ(s + 2) + (s + 1) 2 Γ(s)Ψ(s) -2(s + 1)Γ(s + 1)Ψ(s + 1) ,
which simplifies to give the expectation value corresponding to the second eigen state:
x n=1 = 1 r 0 β ln(s + 3) -Ψ(s + 2) + 3 (s + 2) ) . (4.63)
Other expectation values for n > 1 can also be obtained in a similar fashion. The important point which is to be noted here is, though the average momentum vanishes, the average position is non-zero for Morse potential and remain so, irrespective of the choice of coordinate origin. This result is also true for all eigen states of the same Hamiltonian.
In three dimensions, for a spherically symmetric potential the solution of the Schrödinger equation is given in Eq. (3.4). Here we have assumed that the variables can be separated. The expectation values of Lθ and Lφ have been evaluated in section III. In this section we take the case of the Hydrogen atom and calculate the expectation value of the radial component of the linear momentum. 15 1. The Hydrogen atom In this case, V (r) = -e 2 r . (4.64)
where e is the electronic charge and r = x 2 + y 2 + z 2 . Now we have to write Eq. (3.2) in spherical polar coordinates and the solution of the time-independent Schrödinger equation is:
u n L M (r, θ, φ) = N r R n L (r) Y L M (θ, φ) , = N r e -r/na0 2r na 0 L L 2L+1 n-L-1 2r na 0 Y L M (θ, φ) , (4.65)
where a 0 = h2 me 2 is the Bohr radius and m is the reduced mass of the system comprising of the proton and the electron. n is the principal quantum number which is a positive integer, L 2L+1 n-L-1 (x) are the associated Laguerre polynomials, Y L M (θ, φ) are the spherical-harmonics, and N r is the normalization arising from the radial part of the eigenfunction. The values which L and M can take is discussed in section III. The radial normalization constant is given by:
N r = 2 na 0 3 (n -L -1)! (n + L)!2n 1/2 . (4.66)
The spherical-harmonics are given by,
Y L M (θ, φ) = 2 L + 1 4π (L -M )! (L + M )! 1/2 P L M (cos θ)e iMφ , (4.67)
where P M L (cos θ) are the associated Legendre functions. It is noted that although the Coulomb potential is a real potential but the solution in spherical polar coordinates is not real, e iMφ , is complex. The spherical-harmonics are ortho-normalized according to the relation,
π θ=0 2π φ=0 dθ dφ sin θ Y L M (θ, φ)Y L M (θ, φ) = δ L L δ M M .
(4.68)
Let us write the eigenfunctions in terms of dimensionless quantity: ρ = 2r/na 0 ≡ αr. Also we define k ≡ (2L + 1) and n r ≡ (n -L -1) for the sake of convenience. With this amount of notational machinery the eigenfunctions can be written as:
u n L M (r, θ, φ) = N r R nL (ρ) Y L M (θ, φ) . (4.69)
The radial momentum expectation value in this case is not given by -ih ∂ ∂ρ , its form is (already discussed in section III):
pρ = -ih Ñ 2 ∞ 0 dρ ρ 2 R * nL (ρ) ∂ ∂ρ + 1 ρ R nL (ρ) dΩ [Y L M (θ, φ)] 2 . (4.70)
Where Ñ 2 = N 2 r /α 2 . The integral for the spherical harmonics yields identity. The radial expectation value then becomes,
pρ = -ih Ñ 2 ∞ 0 dρ - 1 2 e -ρ ρ k+1 [L k nr (ρ)] 2 + (L + 1) e -ρ ρ k [L k nr (ρ)] 2 + e -ρ ρ k+1 L k nr (ρ) d dρ [L k nr (ρ)] . (4.71)
Using the recurrence relation [16]:
d dρ L k nr (ρ) = ρ -1 n r L k nr (ρ) -(n r + k) L k nr -1 (ρ) , (4.72)
the expectation value integral acquires the form:
pρ = -ih Ñ 2 ∞ 0 dρ - 1 2 e -ρ ρ k+1 [L k nr (ρ)] 2 + (n r + L + 1) e -ρ ρ k [L k nr (ρ)] 2 + e -ρ ρ k L k nr (ρ)L k nr -1 (ρ) . (4.73) 16
The third contribution of the becomes zero from the orthogonality property of the associated Laguerre polynomials as given in Eq. (4.51). The contribution from the second term can also be found similarly. To find the share of the first term we make use of [19]: ∞ 0 dρ e -ρ ρ k+1 [L k nr (ρ)] 2 = (n r + k)! n r ! (2n r + k + 1). (4.74)
Collecting all the contributions we get the radial expectation value to be zero as expected.
The time evolution of any operator Ô in the Heisenberg picture is given by:
d Ô dt = 1 ih [ Ô, Ĥ] , (5.1)
where Ĥ is the Hamiltonian of the system. The Hamiltonian of a quantum system comprising of a particle of mass m is given by:
Ĥ = p2 2m + V (x) . (5.2)
From the above two equations we can write the time evolution of the momentum operator in one dimension, in Cartesian coordinates as:
dp x dt = 1 ih [p x , Ĥ] = - d dx V (x) , (5.3)
which is the operator version of Newton's second law in a time independent potential. Now if we take the expectation values of both sides of Eq. (5.3) in any basis we get:
d px dt = - d dx V (x) , (5.4)
and historically the above equation is called the Ehrenfest theorem, which was deduced in a different way by P.
Ehrenfest. Using the Ehrenfest theorem we can deduce that the rate of change of the expectation value of the momentum operator is zero in the case of the linear harmonic oscillator. In the case of the linear harmonic oscillator we have:
d dx V (x) = mω 2 x , (5.5)
and it can be trivially shown that x = 0. This directly implies that,
d px dt = 0 , (5.6)
for the linear harmonic oscillator. The above equation shows that the expectation value of the momentum along x direction is constant, and this constant is zero is known from other sources. Next we focus on the Hydrogen atom. The Hamiltonian of the Hydrogen atom is:
Ĥ = - h2 2m 1 r ∂ 2 ∂r 2 r + 1 2mr 2 L2 - e 2 r , (5.7) where, L2 = -h 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂φ 2 , (5.8)
whose eigenvalues are of the form h2 L(L + 1) in the basis Y L M (θ, φ). In the expression of the Hamiltonian m is the reduced mass of the system comprising of the proton and electron. Next we try to apply Heisenberg's equation to the 17 radial momentum operator. Noting that the first term of the Hamiltonian is nothing but p2 r the Heisenberg equation is: dp r dt = -L2 2m 1 r ∂ ∂r r , 1 r 2 + e 2 1 r ∂ ∂r r , 1 r , = L2 mr 3 -e 2 r 2 . (5.9)
The above equation is the operator form of Newton's second law in spherical polar coordinates. Next we evaluate the expectation value of both the sides of the above equation using the wave-functions given in Eq. (4.65). We know,
1 r 2 = 1 n 3 a 2 0 (L + 1 2 ) , (5.10) 1 r 3 = 1 a 3 0 n 3 L(L + 1 2 )(L + 1) . (5.11)
Using the above expectation values in Eq. (5.9) and noting that L2 = h2 L(L + 1) we see that the time derivative of the expectation value of the radial momentum operator of the Hydrogen atom vanishes. The above analysis shows that the form of the Ehrenfest theorem as given in Eq. (5.4) is only valid in Cartesian coordinates. In the case of the Hydrogen atom if we used Eq. (5.4) we should have never got the correct result.
In the present work we have emphasized on the reality of the momentum expectation value and using the reality of the expectation value as a bench mark we did find out the form of the momentum operator in spherical polar coordinate system. We found that most of the concepts which define the momentum operator in Cartesian coordinates do not hold good in spherical polar coordinates and in general in any other coordinate system. The reason being that whenever we do an integration in curvilinear coordinates the Jacobian of the coordinate transformation matrix comes inside the picture and the Cartesian results start to falter if we do not change the rules appropriately. The forms of the momentum along the radial direction and the form of the angular momentum operators are derived in section III. The status of the angular variables was briefly discussed in the same section. We explicitly calculated the expectation values of the momentum operator in various important cases and showed that the expectation value of the momentum operator do really come out to be zero as expected. Although the expectation value of the momentum operator vanishes in most of the bound states, with a real potential, the expectation value of the position is not required to vanish. The expectation value of the position operator is directly related with the parity property of the potential which was briefly discussed in subsection IV C. At the end we calculated the Heisenberg equation of motion for the radial momentum operator for the Hydrogen atom and showed its formal semblance with Newton's second law. It was also shown that if we properly write the Heisenberg equation of motion in spherical polar coordinates then Ehrenfest's theorem follows naturally.
In short we conclude by saying: 1. the forms of the various momentum operators, in most of the coordinate systems, in quantum mechanics can be obtained by imposing the condition of the reality of their eigenvalues. The form of the probability conservation equation and Ehrenfest theorem must be modified in curvilinear coordinates to yield meaningful results. 2. There are obvious problems in elevating the status of angular variables to dynamical variables in quantum mechanics.
3. For compact variables, if the variable is periodic the expectation value of the angular momentum conjugate to it is non-zero. If the compact variable is not periodic then the angular momentum conjugate to it must vanish.
4. The momentum expectation values in cases of bound state motions vanish, whereas the position expectation values in those cases depends on the symmetry of the potential.
The authors thank Professors D. P. Dewangan, S. Rindani, J. Banerji, P. Panigrahi and Ms. Suratna Das for stimulating discussions and constant encouragements.
[1] J. J. Sakurai, "Modern quantum mechanics", International student edition, Addison-Wesley, 1999. [2] L I. Schiff, "Quantum mechanics", McGraw-Hill International Editions, third edition. [3] R. Shankar, "Principles of quantum mechanics", Plenum Press, New York 1994, second edition. [4] G. Bonneau, J. Faraut, G. Valent, Am. J. Phys. 69 322, (2001). [5] P. A. M Dirac. "The principles of quantum mechanics", Fourth edition, Oxford university press, 1958. [6] S. Flügge. "Practical quantum mechanics I", Springer-Verlag Berlin Heidelberg 1971. [7] G. Paz, Euro. J. Phys. 22 337, (2001). [8] G. Paz, J. Phys. A: Math. Gen. 35 3727, (2002). [9] P. Carruthers, M. M. Nieto, Rev. Mod. Phys. 40 411, (1968). [10] D. T. Pegg, S. M. Barnett, Phys. Rev. A 39 1665, (1989). [11] H. Essén, Am. J. Phys. 46 983, (1978). [12] I. S. Gradshteyn, I. M. Ryzhik. "Table of integrals, series, and products". Academic Press, Harcourt India, sixth edition, page 955, 8.733 1 [13] I. S. Gradshteyn, I. M. Ryzhik. "Table of integrals, series, and products". Academic Press, Harcourt India, sixth edition, page 956, 8.737 2. [14] P. M. Morse, Phys. Rev. 34 57, (1929). [15] S. Ghosh, A. Chiruvelli, J. Banerji and P. K. Panigrahi, Phys. Rev. A 73, 013411, (2006). [16] I. S. Gradshteyn, I. M. Ryzhik. "Table of integrals, series, and products". Academic Press, Harcourt India, sixth edition, page 991, 8.971 2. [17] I. S. Gradshteyn, I. M. Ryzhik. "Table of integrals, series, and products". Academic Press, Harcourt India, sixth edition, page 992, 8.974 3. [18] I. S. Gradshteyn, I. M. Ryzhik. "Table of integrals, series, and products", sixth edition, (Academic Press, Harcourt India). [19] The specific integration result and other related expressions can be found in the web page: http://mathworld.wolfram.com/LaguerrePolynomial.html , Equation 24 . | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Quantum mechanics is a treasure house of peculiar and interesting things. Elementary textbooks of quantum mechanics [1, 2, 3] generally start with the postulates which are required to define the nature of the dynamical variables in the theory and their commutation relations. The choice of the dynamical variables is not that clear, as the coordinates in Cartesian system are all elevated to the status of operators where as time remains a parameter. More over in spherical polar coordinates only the radial component can be represented as an operator while the angles still remain as a problem. The difficulty of giving different status to the spatial coordinates and time is bypassed in quantum field theories where all the coordinates and time become parameters of the theory. But the problem with angles still remain a puzzle which requires to be understood in future.\n\nWhen we start to learn quantum mechanics, most of the time we begin with elementary calculations relating to the particle in a one dimensional infinite well, particle in a finite potential well, linear harmonic oscillator and so on. The main aim of these calculations is to solve the Schrödinger equation in the specific cases and find out the bound state energies and the energy eigenfunctions in coordinate space representation. While solving these problems we overlook the subtleties of other quantum mechanical objects as the definition of the momentum operator in various coordinates, the reality of its expectation value, etc.. In the last one or two decades there has been a number of studies regarding the self-adjointness of various operators [4] . The aim of these studies has been to analyze the self-adjointness of various operators like momentum, Hamiltonian etc. and find out whether these operators are really self-adjoint in some interval of space where the theory is defined, if not then can there be any mathematical method by which we can make these operators to be self-adjoint in the specified intervals ?\n\nIn the present work we deal with a much elementary concept in quantum mechanics related to the reality of the expectation values of the momentum operator, be it linear or angular. We do not analyze the self-adjointness of the operators which requires different mathematical techniques. To test the self-adjointness of an operator we have to see whether the operator is symmetric in a specific spatial interval and the functional domain of the operator and its adjoint are the same. In the article we always keep in touch with the recent findings from modern research about self-adjoint extensions but loosely we assume that the operators which we are dealing with are Hermitian. If something is in contrary we point it out in the main text. A considerable portion of our article deals with the analysis 2 of the fact that the expectation value of the momentum operator in various bound states are zero, a result which most of the textbooks only quote but never show. In the simpler cases the result can be shown by one or two lines of calculation, but in nontrivial potentials as the Morse potential, the Coulomb potential the result is established by using various properties of the special functions such as the associated Legendre and the associated Laguerre.\n\nThe presentation of various materials in our article is done in the following way. Next section deals with the definition of the momentum operator and its properties. Section III deals with the intricacies of the definition of the momentum operator in spherical polar coordinates and the problems we face when we try to mechanically implement the quantization condition, which is invariably always written in Cartesian coordinates in most of the textbooks on quantum mechanics. In section IV we explicitly calculate the momentum expectation values in various potentials and show that in bound states we always get the expectation value of the linear momentum to be zero. Section V gives a brief discussion on the Ehrenfest theorem when we are using it to find out the time derivative of the expectation value of the radial component of momentum in the case of the Hydrogen atom. We end with the concluding section which summarizes the findings in our article.\n\nBefore going into the main discussion we would like to mention about the convention. We have deliberately put a hat over various symbols to show that they are operators in quantum mechanics. Some times this convention becomes tricky when we are dealing with angular variables as there the status of these variables is in question. The other symbols have their conventional meaning. As we are always using the coordinate representation sometimes we may omit the hat over the position operator as in this representation the position operator and its eigenvalues can be trivially interchanged."
},
{
"section_type": "OTHER",
"section_title": "II. DEFINITION OF THE MOMENTUM OPERATOR AND THE REALITY OF ITS EXPECTATION VALUE",
"text": "From the Poisson bracket formalism of classical mechanics we can infer: [x i , pj ] = ih δ i j , (2.1) where δ i j = 1 when i = j and zero for all other cases, and i, j = 1, 2, 3. In the above equation xi is the position operator and pj is the linear momentum operator in Cartesian coordinates. From the above equation we can also find the form of the momentum operator in position representation, which is: pi = -ih ∂ ∂x i . (2.2) It is interesting to note that the above expression of the momentum operator also gives us the form of the generator of translations. This is because of the property: [p x , F (x)] = -ih dF (x) dx , (2.3) where F (x) is an arbitrary well defined function of x. The above equation ensures that the momentum operator generates translations along the x direction. Particularly in one-dimension the expression of the momentum operator becomes px = -ih ∂ ∂x . We know that the expectation value of the momentum operator must be real. If we focus on one-dimensional systems to start with, where the system is specified by the wave-function ψ(x, t), the expectation value of any operator Ô is defined by: Ô ≡ ∞ -∞ ψ * (x, t) Ô ψ(x, t) dx , (2.4) where ψ * signifies complex conjugation of ψ and the extent of the system is taken as -∞ < x < ∞. From the above equation we can write, Ô * = ∞ -∞ ψ(x, t) Ô * ψ(x, t) * dx . (2.5) If Ô = Ô * then the condition of the reality of the expectation value becomes: ∞ -∞ ψ * (x, t) Ô ψ(x, t) dx = ∞ -∞ ψ(x, t) Ô * ψ(x, t) * dx . (2.6) 3 For a three-dimensional system the above condition becomes, ∞ -∞ ψ * (x, t) Ô ψ(x, t) d 3 x = ∞ -∞ ψ(x, t) Ô * ψ(x, t) * d 3 x . (2.7) Now if we take the specific case of the momentum operator in one-dimension we can explicitly show that its expectation value is real if the extent of the system is infinite and the wave-function vanishes at infinity. The proof is as follows.\n\nThe expectation value of the momentum operator is: ∞ -∞ ψ * (x, t) px ψ(x, t) dx = -ih ∞ -∞ ψ * (x, t) ∂ψ(x, t) ∂x dx = -ih ψ * (x, t)ψ(x, t)| ∞ -∞ -∞ -∞ ψ(x, t) ∂ψ(x, t) * ∂x dx , (2.8) If the wave functions vanish at infinity then the first term on the second line on the right-hand side of the above equation drops and we have, ∞ -∞ ψ * (x, t) px ψ(x, t) dx = -ih ∞ -∞ ψ * (x, t) ∂ψ(x, t) ∂x dx = ih ∞ -∞ ψ(x, t) ∂ψ(x, t) * ∂x dx , = ∞ -∞ ψ(x, t) p * x ψ(x, t) * dx . (2.9) A similar proof holds for the three-dimensional case where it is assumed that the wave-function vanishes at the boundary surface at infinity."
},
{
"section_type": "OTHER",
"section_title": "III. THE EXPECTATION VALUE OF THE MOMENTUM OPERATOR IN CARTESIAN AND SPHERICAL POLAR COORDINATES",
"text": "In non-relativistic version of quantum mechanics we know that if we have a particle of mass m which is present in a time-independent potential we can separate the Schrödinger equation:\n\nih ∂ψ(x, t) ∂t = -h2 2m ∇ 2 + V (x) ψ(x, t) , (3.1) into two equations, one is the time-dependent one which gives the trivial solution e -iEt h where E is the total energy of the particle, and the other equation is the time-independent Schrödinger equation: ∇ 2 u(x) + 2m h2 (E -V (x))u(x) = 0 , (3.2) where u(x) is the solution of the time-independent Schrödinger equation and the complete solution of the Eq. (3.1) is: ψ(x, t) = u(x)e -iEt h . (3.3) In the case of the free-particle, where V (x) = 0, we have u(x, t) = e ik•x where E = k 2 h2 2m and k = |k|. The freeparticle solution is an eigenfunction of the momentum operator with eigen value hk. Although if we try to find out the expectation value of the momentum operator as is done in the last section we will be in trouble as these wave-functions do not vanish at infinity, a typical property of free-particle solutions. But this problem is not related to the Hermiticity property of the momentum operator, it is related with the de-localized nature of the free-particle solution.\n\nIn physics many times we require to solve a problem using curvilinear coordinate systems. The choice of our coordinate system depends upon the specific symmetry which we have at hand. Suppose we are working in spherical polar coordinates and the solution of Eq. (3.2) can be separated into well behaved functions of r, θ and φ as,\n\nu(x) = u(r, θ, φ) = R(r) Θ(θ) Φ(φ) . (3.4) 4\n\nIf we try to follow the proof of the Hermiticity of the linear momenta components, as done in the last section, in spherical polar coordinates, then we should write: p = -ih τ u * (r, θ, φ)∇u(r, θ, φ) dτ , = -ih R * (r)Θ * (θ)Φ * (φ) e r ∂ ∂r + e θ r ∂ ∂θ + e φ r sin θ ∂ ∂φ R(r)Θ(θ)Φ(φ)r 2 drdΩ , (3.5)\n\nwhere in the above equation e r , e θ , e φ respectively are the unit vectors along r, θ and φ, and dΩ = sin θ dθ dφ. τ is the volume over which we integrate the expression in the above equation. From the last equation we can write:\n\npr = -ih Ω |Θ(θ)| 2 |Φ(φ)| 2 dΩ ∞ 0 r 2 R * (r) dR(r) dr dr , (3.6)\n\nAs Θ(θ) and Φ(φ) are normalized, the integration:\n\nΩ |Θ(θ)| 2 |Φ(φ)| 2\n\ndΩ = 1 and we can proceed as in Eq. (2.9) as:\n\npr = -ih ∞ 0 r 2 R * (r) dR(r) dr dr , = -ih r 2 R * (r)R(r) ∞ 0 - ∞ 0 2rR * (r) + r 2 dR * (r) dr R(r) dr . (3.7)\n\nIf R(r) vanishes at infinity then the above equation reduces to,\n\npr = ih ∞ 0 r 2 R(r) dR * (r) dr dr + 2ih ∞ 0 r|R(r)| 2 dr , = pr * + 2ih ∞ 0 r|R(r)| 2 dr . (3.8)\n\nThe above equation implies that pr is not real in spherical polar coordinates. The solution of the above problem lies in redefining pr as is evident from Eq. (3.8), and it was given by Dirac [5, 6] . The redefined linear momentum operator along r can be:\n\npr ≡ -ih ∂ ∂r + 1 r = -ih 1 r ∂ ∂r r . (3.9)\n\nThis definition of the pr is suitable because in this form it satisfies the commutation relation as given in Eq. (2.1) where now the operator conjugate to r is pr . The form of pr in Eq. (3.9) shows that for any arbitrary function of r as F (r) we must still have Eq. (2.3) satisfied. This implies that the modified form of pr is still a generator of translations along the r direction. Up to this point we were following what was said by Dirac regarding the status of the radial momentum operator. Still everything is not that smooth with the redefined operator as we can see that it turns out to be singular around r = 0, more over, although the radial momentum acts like a translation generator along r but near r = 0 it cannot generate a translation towards the left as the interval ends there. In this regard we can state that the issue of the reality of the radial component of the momentum in spherical polar coordinates is a topic of modern research in theoretical physics [7, 8] . It has been shown that the operator -ih ∂ ∂r is not Hermitian and more over it can be shown [4] that such an operator cannot be self-adjoint in the interval [0, ∞].\n\nIn some recent work [8] the author claims that there can be an unitary operator which connects -ih ∂ ∂r to -ih 1 r ∂ ∂r r, and as the former operator does not have a self-adjoint extension in the semi-infinite interval so the latter is also not self-adjoint in the same interval.\n\nIf we further try to find out whether pθ and pφ are real, then we will face difficulties. Working out naively if we claim that pφ = 1 rsinθ ∂ ∂φ as suggested by the φ component of Eq. (3.5) we will notice that φ pφ does not have the dimension of action. This means pφ or pθ is not conjugate to φ or θ. This is a direct representation of the special coordinate dependence of the quantization condition. Only in Cartesian coordinates the variables conjugate to x, y and z are p x , p y and p z . Taking the clue from classical mechanics we know the proper dynamical variables conjugate to φ and θ are the angular momentum operators, namely Lθ and Lφ . In general Lφ is given by: Lφ = -ih ∂ ∂φ , (3.10) 5 which can be shown to posses real expectation values by following a similar proof as is done in Eq. (2.8) and Eq. (2.9), if we assume Φ(0) = Φ(2π). In this form it is tempting to say that we can have a relation of the form, [ φ, Lφ ] = ih , (3.11) which looks algebraically correct. But the difficulty in writing such an equation is in the interpretation of φ which has been elevated from an angular variable to a dynamical operator. In spherical polar coordinates both θ and φ are compact variables and consequently have their own subtleties. Much work is being done in trying to understand the status of angular variables and phases [9, 10] , in this work we only present one example showing the difficulty of accepting φ as an operator. From the solution of the time-independent Schrödinger equation for an isotropic potential we will always have:\n\nΦ(φ) = 1 √ 2π e iMφ , (3.12)\n\nwhere M = 0, ±1, ±2, •, •. Now if φ is an operator we can find its expectation value, and it turns out to be: φ = 1 2π 2π 0 φe iMφ e -iMφ dφ , = π , (3.13) and the expectation value of φ2 is:\n\nφ2 = 1 2π 2π 0 φ 2 e iMφ e -iMφ dφ , = 4 3 π 2 . (3.14) Consequently ∆φ = φ2 -φ 2 = π √ 3\n\n. Similarly calculating Lφ we get: Lφ = M h 2π 2π 0 e -iMφ e iMφ dφ , = M h , (3.15) as expected, and L2 φ = M 2 h2 . This implies ∆L φ = L2 φ -Lφ 2 = 0. So we can immediately see that the Heisenberg uncertainty relation between φ and Lφ , ∆φ∆L φ ≥ h/2 breaks down. This fact makes life difficult and we have no means to eradicate this problem.\n\nTaking the clue from the φ part we can propose that Lθ is also of the form -ih ∂ ∂θ . With this definition of Lθ let us try to prove its Hermitian nature as done in Eq. (3.7). Taking R(r) and Φ(φ) in Eq. (3.4) separately normalized, we can write:\n\nLθ = -ih π 0 Θ * (θ) dΘ(θ) dθ sin θdθ = -ih sin θ Θ * (θ)Θ(θ)| π 0 - π 0 cos θ Θ * (θ) + sin θ dΘ * (θ) dθ Θ(θ) dθ , = ih π 0 sin θ Θ(θ) dΘ * (θ) dθ dθ + ih π 0 cos θ |Θ(θ)| 2 dθ , = Lθ * + ih π 0 cos θ |Θ(θ)| 2 dθ . (3.16)\n\nThe above equation shows that Lθ is not real. The rest is similar to the analysis following Eq. (3.8) where now we have to redefine the angular momentum operator conjugate to θ as [11]: Lθ ≡ -ih ∂ ∂θ + 1 2 cot θ . (3.17) 6 Unlike the φ case, Θ(θ) are not eigenfunctions of Lθ . But the difficulties of establishing θ as an operator still persists and in general θ is not taken to be a dynamical operator in quantum mechanics. It is known that both θ and φ are compact variables, i.e. they have a finite extent. But there is a difference between them. In spherical polar coordinates the range of φ and θ are not the same, 0 ≤ φ < 2π and 0 ≤ θ ≤ π. This difference can have physical effects. As φ runs over the whole angular range so the wave-function corresponding to it Φ(φ) is periodic in nature while due to the range of θ, Θ(θ) need not be periodic. Consequently there can be a net angular momentum along the φ direction while there cannot be any net angular momentum along θ direction. And this can be easily shown to be true. As the time-independent Schrödinger equation for an isotropic potential yields Φ(φ) as given in Eq. (3.12) similarly it is known that in such a potential the form of Θ(θ) is given by:\n\nΘ(θ) = N θ P L M (cos θ) , (3.18)\n\nwhere N θ is a normalization constant depending on L, M and P L M (cos θ) is the associated Legendre function, which is real. In the above equation L and M are integers where L = 0, 1, 2, 3, •, • and M = 0, ±1, ±2, ±3, •, •. The quantum number M appearing in Eq. (3.12) and in Eq. (3.18) are the same. This becomes evident when we solve the time-independent Schrödinger equation in spherical polar coordinates by the method of separation of variables. A requirement of the solution is -L ≤ M ≤ L. Now we can calculate the expectation value of Lθ using the above wave-function and it is:\n\nLθ = -ihN 2 θ π 0 P L M (cos θ) dP L M (cos θ) dθ + 1 2 cot θP L M (cos θ) sin θdθ = -ihN 2 θ π 0 P L M (cos θ) dP L M (cos θ) dθ sin θdθ + 1 2 π 0 P L M (cos θ)P L M (cos θ) cos θ dθ . (3.19)\n\nTo evaluate the integrals on the right hand side of the above equation we can take x = cos θ and then the expectation value becomes:\n\nLθ = -ihN 2 θ -1 1 P L M (x) dP M L (x) dx (1 -x 2 ) 1 2 dx - 1 2 -1 1 P L M(x) P L M(x) x √ 1 -x 2 dx . (3.20)\n\nThe second term in the right hand side of the above equation vanishes as the integrand is an odd function in the integration range. For the first integral we use the following recurrence relation [12]:\n\n(x 2 -1) dP L M (x) dx = M xP L M (x) -(L + M )P L M-1 (x) , (3.21)\n\nthe last integral can be written as, (3.23) we can see immediately that both the integrands in the right hand side of the above equation is odd and consequently Lθ = 0 as expected. A similar analysis gives Lφ = M h. It must be noted that the form of Lθ still permits it to be the generator of rotations along the θ direction. As the motion along φ is closed so there can be a net flow of angular momentum along that direction but because the motion along θ is not so, a net momentum along θ direction will not conserve probability and consequently for probability conservation we must have expectation value of angular momentum along such a direction to be zero. In elementary quantum mechanics text books it is often loosely written that the solution of the time-independent Schrödinger equation is real when we are solving it for a real potential. But this statement is not correct. The reality 7 of the solution also depends upon the coordinate system used. Specially for compact periodic coordinates we can always have complex functions as solutions without breaking any laws of physics. Before leaving the discussion on angular variables in spherical polar coordinates we want to point out one simple thing which is interesting. In Cartesian coordinates when we deal with angular momentum we know that: [ Li , Lj ] = iǫ ijk Lk , (3.24) where Li stands for Lx , Ly or Lz . For this reason there cannot be any state which can be labelled by the quantum numbers of any two of the above angular momenta. But from the expressions of Lφ and Lθ we see that, [ Lφ , Lθ ] = 0 , (3.25) and consequently in spherical polar coordinates we can have wave-function solutions of the Schrödinger equation which are simultaneous eigenfunctions of both Lφ and Lθ as P L M (θ). For real V (x), we expect the solution of the time-independent Schrödinger equation u(x) to be real, when we are solving the problem in Cartesian coordinates. In all these cases the expectation value of the linear momentum operators must vanish. The reason is simple and can be understood in one-dimensional cases where with real u(x) we directly see that the integral ∞ -∞ u * (x) ∂u(x) ∂x dx is real and so ∞ -∞ u * (x) px u(x) dx becomes imaginary as px contains i, as is evident from the first line in Eq. (2.9). So if the expectation value of the momentum operator has to be real then the only outcome can be that for all those cases where we have a time-independent solution in a bounded region of space, with a real potential and working in Cartesian coordinates, the expectation value of the momentum operator must vanish. The above statement is true in curvilinear coordinates also, but in those cases the definition of the momentum operators have to be modified. This fact becomes clear when we write the relationship between the probability flux and the expectation value of the momentum operator. The probability flux for a particle of mass m is:\n\nLθ = ihN 2 θ M -1 1 x(1 -x 2 ) -1 2 P L M (x)P L M (x) dx -(L + M ) -1 1 (1 -x 2 ) -1 2 P L M (x)P L M-1 (x) dx . (3.22) As, P L M (x) = (-1) L+M P L M (-x) ,\n\nj(x, t) = - ih 2m [ψ * (x, t)∇ψ(x, t) -(∇ψ * (x, t))ψ(x, t)] , = h m Im (ψ * (x, t)∇ψ(x, t)) , (3.26)\n\nwhere 'Im' implies the imaginary part of some quantity. Most of the elementary quantum mechanics books then proceeds to show that:\n\nd 3 x j(x, t) = p m , (3.27)\n\nwhich is obtained from Eq. (3.26) by integrating both sides of it over the whole volume. From Eq. (3.26) we immediately see that if the solution of the time-independent Schrödinger equation is real we will have j(x, t) = 0 and consequently from Eq. (3.27), p = 0. But this statement is also coordinate dependent, which is rarely said in elementary textbooks of quantum mechanics. Eq. (3.26) evidently does not hold in spherical polar coordinates. If we take Eq. (3.4) as the solution in a general isotropic central potential and use the general form of ∇ in spherical polar coordinates then it can be seen that j r (r, θ, φ, t) = 0 for a real potential. But then Eq. (3.27) does not hold as here pr is simply the radial component of ∇ and not as given in Eq. (3.9), and we know d dr is not zero. The reason why Eq. (3.26) is not suitable in spherical polar coordinates is related to the fact that in deriving Eq. (3.26) one assumes that the probability density of finding the quantum state within position x and x + dx at time t is |ψ(x, t)| 2 . But this statement is only true in Cartesian coordinates, in spherical polar coordinates the probability density of the system to be within a region r and r + dr, θ and θ + dθ, φ and φ\n\n+ dφ is not |ψ(r, θ, φ)| 2 but |ψ(r, θ, φ)| 2 r 2 sin θ\n\nand consequently the steps which follow leading to Eq. (3.26) in Cartesian coordinates are not valid in spherical polar coordinates. In general Eq. (3.26) will not be valid in any curvilinear coordinate system.\n\nThe next section contains the actual calculations of the expectation values of the momentum operator in various cases where we have bound state solutions. In all the relevant cases discussed in this article it is seen that although px = 0 but p2 x is not zero as it is related to the Hamiltonian operator. In all the cases we must have,\n\n(p x ) s = 0 , s = odd integer . (3.28)\n\nThe above equation can be guessed from the reality of the expectation value of the momentum operator.\n\n8"
},
{
"section_type": "OTHER",
"section_title": "IV. MOMENTUM EXPECTATION VALUES IN VARIOUS BOUND STATES",
"text": "In this section we will calculate the momentum expectation values in various bound states with stiff or slowly varying potentials.\n\nA. Particle in one-dimensional stiff potential wells"
},
{
"section_type": "OTHER",
"section_title": "Infinite square well potential",
"text": "In this case we consider a particle to be confined in region -L 2 to L 2 along the x-axis where the potential is specified by,\n\nV (x) = ∞ , |x| ≥ L 2 , = 0 , |x| < L 2 . (4.1)\n\nIn this case the solution of the time-independent Schrödinger equation, Eq. (3.2), satisfies the boundary condition,\n\nu - L 2 = u L 2 = 0 , (4.2)\n\nand as the potential has parity symmetry about x = 0 we have two sets of solutions, the odd solutions:\n\nu (o) n (x) = 2 L sin 2nπx L , (4.3)\n\nand the even solutions:\n\nu (e) n (x) = 2 L cos (2n -1)πx L . (4.4)\n\nIn the above equations n is a positive integer. Both of these functions, u (o) n (x) for the odd case and u (e) n (x) for the even case, are real and are not momentum eigenstates. But the momentum expectation values can be found out from the above solutions. For the odd solutions we have:\n\npx = -ih L 2 -L 2 u (o) n (x) du (o) n (x) dx dx , = - 4inπh L 2 L 2 -L 2 sin 2nπx L cos 2nπx L , = 0 , (4.5)\n\nas expected. Similarly for the even solutions it is also easy to show that the expectation value of the momentum operator vanishes."
},
{
"section_type": "OTHER",
"section_title": "Finite square well potential",
"text": "In this case,\n\nV (x) = 0 , |x| ≥ a , = -V 0 , |x| < a , (V 0 > 0) . (4.6)\n\nIf we are not interested in the normalization constant of the bound state solution then the solution of the timeindependent Schrödinger equation in this case is:\n\nu(x) ∼ e -κ|x| , |x| > a , ∼ cos(kx) , |x| < a , (even parity) ∼ sin(kx) , |x| < a , (odd parity) , (4.7) 9\n\nwhere,\n\nk 2 = 2m(-|E| + V 0 ) h2 , (4.8) κ 2 = 2m|E| h2 . (4.9)\n\nIn this case the expectation value of the momentum operator is:\n\npx ∼ -ih +∞ -∞ u(x) du(x) dx dx , ∼ -ih κ -a -∞ e 2κx dx - ∞ a e -2κx dx + k a -a sin(kx) cos(kx) dx , = 0 , (4.10)\n\nwhere the first two lines of the above equation holds up to a constant arising from the normalization of the wavefunction. In deriving the last equation we have taken the odd parity solution, but the result remains unaffected if we take the even parity solution as well."
},
{
"section_type": "OTHER",
"section_title": "Dirac-delta potential",
"text": "In this case the potential is:\n\nV (x) = -V 0 δ(x) , (V 0 > 0) . (4.11)\n\nIn this case there can be one bound state solution which is obtained after solving the Eq. (3.2). Demanding that the solution u(x) satisfies the boundary conditions: (4.13) where ǫ is an infinitesimal quantity tending to zero, we get the form of the solution which is:\n\nu(x = -ǫ) = u(x = +ǫ) , (4.12) du dx x=+ǫ - du dx x=-ǫ = - 2mV 0 h2 u(x = 0) ,\n\nu(x) = √ κ e κx , x ≤ 0 , (4.14) = √ κ e κx , x ≥ 0 , (4.15)\n\nwhere κ = mV0 h2 and the energy of the bound state is E = -mV 2 0 2h 2 . The expectation value of the momentum operator in this case is:\n\npx = -ih ∞ -∞ u(x) du(x) dx dx , = -ihκ 0 -∞ e 2κx dx - ∞ 0 e -2κx dx , = 0 . (4.16)\n\nIn this case, also from Hermiticity of the momentum operator we see that Eq. (3.28) holds true.\n\nB. Particle in one-dimensional slowly varying potentials"
},
{
"section_type": "OTHER",
"section_title": "Linear harmonic oscillator potential",
"text": "In the case of the linear harmonic oscillator we have:\n\nV (x) = 1 2 mω 2 x2 , (4.17)\n\n10 where ω is the angular frequency of the oscillator. The solution of Eq. (3.2) in this case, using the series solution method, yields:\n\nu n (q) = N n e -q 2 2 H n (q) , (4.18)\n\nwhere n = 0, 1, 2, •, • and q = √ αx where α = mω h2 . H n (q) are Hermite polynomials of order n and N n is the normalization constant given by,\n\nN n = 1 √ π n! 2 n 1 2 . (4.19)\n\nThe momentum expectation value in this case turns out to be, px = -ih √ α ∞ -∞\n\nu n (q) du n (q) dq dq , = -ih √ αN 2 n ∞ -∞ e -q 2 H n (q) dH n (q) dq dq - ∞ -∞ q e -q 2 H 2 n (q) dq , = 0 . (4.20)\n\nThe first integral on the right side of the second line of the last equation vanishes because, dHn(q) dq = 2nH n-1 (q) and consequently the integral transforms into the orthogonality condition of the Hermite polynomials. The second integral on the second line of the right side of the above equation vanishes because the integrand is an odd function of q.\n\nThe linear harmonic oscillator (LHO) has some very interesting properties. To unravel them we have to digress a bit from the wave-mechanics approach which we have been following and follow the Dirac notation of bra and kets. The Hamiltonian of the LHO in one-dimension is: Ĥ = p2 x 2m + 1 2 mω 2 x2 , (4.21) which can also be written as: Ĥ = hω â † â + 1 2 , (4.22) where â and â † are the annihilation and the creation operators given by:\n\nâ ≡ mω 2h x + ip x mω , â † ≡ mω 2h x - ip x mω . (4.23)\n\nIt can be seen clearly from the above definitions that â is not an Hermitian operator. More over from the definition of the operators we see that, [â, â † ] = 1 . (4.24) Conventionally the number operator is defined as: N ≡ â † â , (4.25) and its eigen-basis are the number states |n such that,\n\nN |n = n|n . (4.26)\n\nThe Hamiltonian of the LHO can be written in terms of the number operator and consequently the number states are energy eigenstates. In this basis the action of the annihilation and creation operators are as:\n\nâ|n = √ n |n -1 , (4.27) â † |n = √ n + 1 |n + 1 . (4.28)\n\nFrom the definitions of the annihilation and creation operators we can write the momentum operator as: px = -i mhω 2 (â -â † ) . (4.29) 11 From Eq. (4.27), Eq. (4.28) and the above equation we can write the matrix elements of the momentum operator as:\n\nn ′ |p x |n = i mhω 2 - √ n δ n ′ , n-1 + √ n + 1 δ n ′ , n+1 . (4.30)\n\nThe above equation shows that the momentum operator can connect two different energy eigenstates. In the case of LHO, except the number operator states, we can have another state which is an eigenstate of the annihilation operator â. This state is conventionally called the coherent state and it is given as:\n\n|α = e -|α| 2 2 ∞ n=0 α n √ n! |n , (4.31)\n\nwhere α is an arbitrary complex number. Now from Eq. (4.29) we can find the momentum expectation value of the coherent state and it is,\n\npx α ≡ α|p x |α = mhω 2 Im(α) . (4.32)\n\nFrom the above equation we can see that although the expectation value of the momentum operator is zero in the energy eigen-basis but it is not so when we compute the momentum expectation value in the coherent state basis, which is essentially a superposition of energy eigenstates. It must be noted that the momentum expectation value is non zero only when the parameter α has an imaginary part."
},
{
"section_type": "OTHER",
"section_title": "Pöschl-Teller potential",
"text": "Among the potentials belonging to the hypergeometric class the Pöschl-Teller potentials have been the most extensively studied and used. This class of potentials consist of trigonometric as well as the hyperbolic type. The trigonometric versions have found applications in molecular and solid state physics and the hyperbolic variants have been used in various studies related to black hole perturbations.\n\nIn the present work we use the trigonometric, symmetric Pöschl-Teller potential given by: (4.33) where V 0 can be parameterized as: (4.34) with for some positive number λ > 1 and a is some scaling factor. The energy eigenvalues of the bound state solutions are:\n\nV (x) = V 0 tan 2 (ax) ,\n\nV 0 = h2 a 2 2m λ(λ -1) ,\n\nE n = - h2 a 2 2m (n 2 + 2nλ + λ) , (4.35)\n\nand the solution of the time-independent Schrödinger equation is,\n\nu n (x) = N n cos(ax) P 1/2-λ n+λ-1/2 (sin(ax)) , (4.36)\n\nwhere,\n\nN n = a(n + λ)Γ(n + 2λ) Γ(n + 1) 1/2 , (4.37)\n\nis the normalization constant and P µ ν (x) is the associated Legendre function. At this point it is fair to point out that P µ ν (x) is not the Legendre polynomial P L M (x) appearing in Eq. (3.18), as µ and ν need not be integers as L and M . P µ ν (x) is not a polynomial but the function appearing in the right hand side of Eq. (4.36) is a polynomial. Now as claimed in the text let us show that the momentum expectation value is indeed zero. Before we proceed let us simplify the notation a bit by calling µ = 1/2 -λ and ν = n + λ -1/2. Substituting z = ax we can write the momentum expectation value as:\n\npx = -ihN 2 n π/2 -π/2 dz cos(z) P µ ν (sin(z)) d dz cos(z) P µ ν (sin(z)) . (4.38)\n\n12 Note the limits of the integration range from π/2 to -π/2 since at this value the potential becomes infinity hence we need not consider the integration range to be the whole real line. For the sake of convenience let us make a change of variable; letting y = sin(z) the above integral becomes: px = -ihN 2 n +1 -1\n\ndy (1 -y 2 ) 1/4 P µ ν (y) d dy (1 -y 2 ) 1/4 P µ ν (y) . (4.39)\n\nTaking the derivative inside the integral we get: px = -ihN 2 n +1 -1\n\ndy (1 -y 2 ) 1/2 P µ ν (y) dP µ ν (y) dy - y(1 -y 2 ) -1/2 2 P µ ν (y)P µ ν (y) . (4.40)\n\nIt is known that for associated Legendre functions [13],\n\nP µ ν (-x) = cos[(µ + ν)π] P µ ν (x) - 2 π sin[(µ + ν)π] Q µ ν (x) , (4.41)\n\nwhere Q µ ν (x) is the other linearly independent solution of the associated Legendre differential equation. As in our case µ + ν = n so P µ ν (x) will have definite parity. As P µ ν (x) has definite parity so the contribution of the second term in the above integral vanishes since the total integrand is an odd function. The first integral is similar to the one in Eq. (3.20) and, due to the typical parity property of P µ ν (x) as shown in Eq. (4.41), it also vanishes. Consequently we have px = 0 as expected."
},
{
"section_type": "OTHER",
"section_title": "Morse potential",
"text": "Diatomic molecule is an exactly solvable system, if one neglects the molecular rotation. The most convenient model to describe the system, is the Morse potential [14]:\n\nV (x) = D(e -2β x -2e -β x) , (4.42)\n\nwhere x = r/r 0 -1, which is the distance from the equilibrium position scaled by the equilibrium value of the inter-nuclear distance r 0 . D is the depth of the potential, called dissociation energy of the molecule and β being a parameter which controls the width of the potential.\n\nIn terms of the above scaled variable x, the time-independent Schrödinger equation becomes:\n\n- h2 2µr 0 d 2 u(x) dx 2 + D(e -2βx -2e -βx\n\n)u(x) = Eu(x) . (4.43) Here µ is the reduced mass of the molecule and the corresponding bound state eigen function comes out to be:\n\nu λ n (ξ) = N e -ξ/2 ξ s/2 L s n (ξ) , (4.44)\n\nwhere the variables are described as, ξ = 2λe -y ; y = βx; 0 < ξ < ∞ , (4.45) and n = 0, 1, ..., [λ -1/2] , (4.46) which is nothing but the quantum number of the vibrational bound states. Here [ρ] denotes the largest integer smaller than ρ, thus total number of bound states is [λ -1/2] + 1. The parameters,\n\nλ = 2µDr 2 0 β 2 h2 and s = - 8µr 2 0 β 2 h2 E , (4.47)\n\nsatisfy the constraint condition s + 2n = 2λ -1. We note that the parameter λ is potential dependent and s is related to energy E. In Eq. (4.44), L s n (y) is the associated Laguerre polynomial and N is the normalization constant [15]:\n\nN = β(2λ -2n -1)Γ(n + 1) Γ(2λ -n)r 0 1/2 . (4.48) 13\n\nWe are looking for the expectation value of linear momentum for a vibrating diatomic molecule, and its expression is: px = -ih ∞ -∞ u * n (ξ) d dx u n (ξ)dx . (4.49) In terms of the changed variable ξ = 2λe -βx the integration limit changes to ∞ to 0 and the expectation value becomes:\n\npx = -ih 0 ∞ u * n (ξ) d dξ u n (ξ)dξ = ihN 2 - 1 2 ∞ 0 e -ξ ξ s (L s n (ξ)) 2 dξ + s 2 ∞ 0 e -ξ ξ s-1 (L s n (ξ)) 2 dξ + ∞ 0 e -ξ ξ s L s n (ξ) d dξ L s n (ξ)dξ = ihN 2 - 1 2 I 1 + s 2 I 2 + I 3 . (4.50)\n\nIntegral I 1 is the orthogonality relation of the associated Laguerre polynomials, which is:\n\n∞ 0 e -ξ ξ s L s n (ξ)L s m (ξ)dξ = Γ(s + n + 1) Γ(n + 1) δ m,n . (4.51)\n\nTo evaluate the second integral one uses the normalization integral of Morse eigenstates. The normalization relation is:\n\n∞ -∞ u * (ξ)u(ξ)dr = |N | 2 r 0 β ∞ 0 e -ξ ξ s-1 (L s n (ξ)) 2 dξ = 1 . (4.52)\n\nThe above integral involving ξ, is explicitly I 2 . N , being the normalization constant as given in Eq. 4.48. Thus it is very straight forward to evaluate I 2 from the above relation as,\n\nI 2 = Γ(n + s + 1) s Γ(n + 1) . (4.53)\n\nThe last integrand I 3 includes a differentiation which can be written as [16]:\n\nd dξ L s n (ξ) = -L s+1 n-1 (ξ) . (4.54)\n\nWriting the right hand side of the above equation as a summation [17]:\n\nL s+1 n = n m=0 L s m , (4.55)\n\nand substituting the derivative term in integral I 3 we obtain:\n\nI 3 = - n-1 m=0 ∞ 0 e -ξ ξ s L s n (ξ)L s m (ξ)dξ . (4.56)\n\nIn the above integral m = n because m can go only upto (n -1). Thus the integral vanishes. Now let us see what is the expectation value of momentum observable, after evaluating the three integrals above. Substituting the non-zero values I 1 and I 2 in Eq. 4.50, it is clear that the expectation value of momentum is zero as has been expected."
},
{
"section_type": "OTHER",
"section_title": "C. Position expectation values for various potentials",
"text": "After a thorough discussion about the momentum expectation values for various solvable one-dimensional potentials, it is worth spending some time discussing about the average position of the particle inside the bound states. Among 14 all the above examples, in each case we had V (x) = V (-x) except the Morse potential as Morse potential is not an example of a symmetric potential: V (x) = V (-x).\n\nIn deriving the expectation values of momentum for above symmetric cases, we often considered that the integrals of odd functions over the symmetric limits vanishes. This result does not hold true for the asymmetric Morse potential. Already we have shown that the momentum expectation value: < p >= 0 for all the above potentials. When it comes to the expectation values of position, one can easily see that < x >= 0 for symmetric potentials whose centers are at the origin. On the other hand if this is not the case, suppose the infinite square well is defined in the range 0 ≤ x ≤ L also then the expectation value of position does not vanish. It becomes L/2. Thus, more accurately the average position of the particle is dependent on the symmetry of the potential where as the average momentum is solely guided by the reality of it's eigenvalues and consequently it is zero always.\n\nBelow we will briefly discuss how the asymmetry of the potential affects the expectation value of x in the case of the Morse potential. The expectation value of the position operator is:\n\nx = ∞ -∞ u λ * n (ξ) x u λ n (ξ)dx. (4.57)\n\nThe eigen function and the variables are respectively substituted from Eq. (4.44) and Eq. (4.45). We obtain\n\nx = N 2 β 2 ln( 2λ\n\n) ∞ 0 e -ξ ξ s-1 (L s n (ξ)) 2 dξ + ∞ 0 e -ξ ξ s-1 (L s n (ξ)) 2 ln(ξ)dξ . (4.58)\n\nThe first integral is already been obtained in Eq. (4.53). This result is independent of the quantum number n. The second integral (say I) is not that straight forward, because it contains associated Laguerre polynomial, logarithm, exponential and monomial functions. Here at best we can evaluate the integral atleast for some specific n as, n = 0 or n = 1, when the Laguerre polynomial is respectively replaced by 1 and (-ξ + s + 1). For the ground state wave function (n = 0), I would be\n\nI n=0 = ∞ 0 e -ξ ξ s-1 ln(ξ)dξ, (4.59)\n\nwhich can be written in terms of Ψ(s) and Γ function [18]:\n\nI n=0 = Γ(s)Ψ(s), (4.60)\n\nwhere, Ψ(s) is the logarithmic factorial function, defined as d(ln(s)! ds = (s!) ′ s! = Ψ(s). For n = 0, first integral reduces to Γ(s) from Eq. (4.53). Above two evaluations gives the ground state expectation value:\n\nx n=0 = 1 r 0 β\n\n[ln(s + 1) -Ψ(s)] . (4.61) For n = 1, one can proceed in the same way\n\nx n=1 = N 2 β 2 ∞ 0 e -ξ ξ s+1 ln(ξ)dξ + (s + 1) 2 ∞ 0 e -ξ ξ s-1 ln(ξ)dξ2(s + 1) ∞ 0 e -ξ ξ s ln(ξ)dξ (4.62) = N 2 β 2 Γ(s + 2)Ψ(s + 2) + (s + 1) 2 Γ(s)Ψ(s) -2(s + 1)Γ(s + 1)Ψ(s + 1) ,\n\nwhich simplifies to give the expectation value corresponding to the second eigen state:\n\nx n=1 = 1 r 0 β ln(s + 3) -Ψ(s + 2) + 3 (s + 2) ) . (4.63)\n\nOther expectation values for n > 1 can also be obtained in a similar fashion. The important point which is to be noted here is, though the average momentum vanishes, the average position is non-zero for Morse potential and remain so, irrespective of the choice of coordinate origin. This result is also true for all eigen states of the same Hamiltonian."
},
{
"section_type": "OTHER",
"section_title": "D. Momentum expectation value for a three-dimensional slowly varying spherically symmetric potential",
"text": "In three dimensions, for a spherically symmetric potential the solution of the Schrödinger equation is given in Eq. (3.4). Here we have assumed that the variables can be separated. The expectation values of Lθ and Lφ have been evaluated in section III. In this section we take the case of the Hydrogen atom and calculate the expectation value of the radial component of the linear momentum. 15 1. The Hydrogen atom In this case, V (r) = -e 2 r . (4.64)\n\nwhere e is the electronic charge and r = x 2 + y 2 + z 2 . Now we have to write Eq. (3.2) in spherical polar coordinates and the solution of the time-independent Schrödinger equation is:\n\nu n L M (r, θ, φ) = N r R n L (r) Y L M (θ, φ) , = N r e -r/na0 2r na 0 L L 2L+1 n-L-1 2r na 0 Y L M (θ, φ) , (4.65)\n\nwhere a 0 = h2 me 2 is the Bohr radius and m is the reduced mass of the system comprising of the proton and the electron. n is the principal quantum number which is a positive integer, L 2L+1 n-L-1 (x) are the associated Laguerre polynomials, Y L M (θ, φ) are the spherical-harmonics, and N r is the normalization arising from the radial part of the eigenfunction. The values which L and M can take is discussed in section III. The radial normalization constant is given by:\n\nN r = 2 na 0 3 (n -L -1)! (n + L)!2n 1/2 . (4.66)\n\nThe spherical-harmonics are given by,\n\nY L M (θ, φ) = 2 L + 1 4π (L -M )! (L + M )! 1/2 P L M (cos θ)e iMφ , (4.67)\n\nwhere P M L (cos θ) are the associated Legendre functions. It is noted that although the Coulomb potential is a real potential but the solution in spherical polar coordinates is not real, e iMφ , is complex. The spherical-harmonics are ortho-normalized according to the relation,\n\nπ θ=0 2π φ=0 dθ dφ sin θ Y L M (θ, φ)Y L M (θ, φ) = δ L L δ M M .\n\n(4.68)\n\nLet us write the eigenfunctions in terms of dimensionless quantity: ρ = 2r/na 0 ≡ αr. Also we define k ≡ (2L + 1) and n r ≡ (n -L -1) for the sake of convenience. With this amount of notational machinery the eigenfunctions can be written as:\n\nu n L M (r, θ, φ) = N r R nL (ρ) Y L M (θ, φ) . (4.69)\n\nThe radial momentum expectation value in this case is not given by -ih ∂ ∂ρ , its form is (already discussed in section III):\n\npρ = -ih Ñ 2 ∞ 0 dρ ρ 2 R * nL (ρ) ∂ ∂ρ + 1 ρ R nL (ρ) dΩ [Y L M (θ, φ)] 2 . (4.70)\n\nWhere Ñ 2 = N 2 r /α 2 . The integral for the spherical harmonics yields identity. The radial expectation value then becomes,\n\npρ = -ih Ñ 2 ∞ 0 dρ - 1 2 e -ρ ρ k+1 [L k nr (ρ)] 2 + (L + 1) e -ρ ρ k [L k nr (ρ)] 2 + e -ρ ρ k+1 L k nr (ρ) d dρ [L k nr (ρ)] . (4.71)\n\nUsing the recurrence relation [16]:\n\nd dρ L k nr (ρ) = ρ -1 n r L k nr (ρ) -(n r + k) L k nr -1 (ρ) , (4.72)\n\nthe expectation value integral acquires the form:\n\npρ = -ih Ñ 2 ∞ 0 dρ - 1 2 e -ρ ρ k+1 [L k nr (ρ)] 2 + (n r + L + 1) e -ρ ρ k [L k nr (ρ)] 2 + e -ρ ρ k L k nr (ρ)L k nr -1 (ρ) . (4.73) 16\n\nThe third contribution of the becomes zero from the orthogonality property of the associated Laguerre polynomials as given in Eq. (4.51). The contribution from the second term can also be found similarly. To find the share of the first term we make use of [19]: ∞ 0 dρ e -ρ ρ k+1 [L k nr (ρ)] 2 = (n r + k)! n r ! (2n r + k + 1). (4.74)\n\nCollecting all the contributions we get the radial expectation value to be zero as expected."
},
{
"section_type": "DISCUSSION",
"section_title": "V. A DISCUSSION ON HEISENBERG'S EQUATION OF MOTION AND EHRENFEST THEOREM",
"text": "The time evolution of any operator Ô in the Heisenberg picture is given by:\n\nd Ô dt = 1 ih [ Ô, Ĥ] , (5.1)\n\nwhere Ĥ is the Hamiltonian of the system. The Hamiltonian of a quantum system comprising of a particle of mass m is given by:\n\nĤ = p2 2m + V (x) . (5.2)\n\nFrom the above two equations we can write the time evolution of the momentum operator in one dimension, in Cartesian coordinates as:\n\ndp x dt = 1 ih [p x , Ĥ] = - d dx V (x) , (5.3)\n\nwhich is the operator version of Newton's second law in a time independent potential. Now if we take the expectation values of both sides of Eq. (5.3) in any basis we get:\n\nd px dt = - d dx V (x) , (5.4)\n\nand historically the above equation is called the Ehrenfest theorem, which was deduced in a different way by P.\n\nEhrenfest. Using the Ehrenfest theorem we can deduce that the rate of change of the expectation value of the momentum operator is zero in the case of the linear harmonic oscillator. In the case of the linear harmonic oscillator we have:\n\nd dx V (x) = mω 2 x , (5.5)\n\nand it can be trivially shown that x = 0. This directly implies that,\n\nd px dt = 0 , (5.6)\n\nfor the linear harmonic oscillator. The above equation shows that the expectation value of the momentum along x direction is constant, and this constant is zero is known from other sources. Next we focus on the Hydrogen atom. The Hamiltonian of the Hydrogen atom is:\n\nĤ = - h2 2m 1 r ∂ 2 ∂r 2 r + 1 2mr 2 L2 - e 2 r , (5.7) where, L2 = -h 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂φ 2 , (5.8)\n\nwhose eigenvalues are of the form h2 L(L + 1) in the basis Y L M (θ, φ). In the expression of the Hamiltonian m is the reduced mass of the system comprising of the proton and electron. Next we try to apply Heisenberg's equation to the 17 radial momentum operator. Noting that the first term of the Hamiltonian is nothing but p2 r the Heisenberg equation is: dp r dt = -L2 2m 1 r ∂ ∂r r , 1 r 2 + e 2 1 r ∂ ∂r r , 1 r , = L2 mr 3 -e 2 r 2 . (5.9)\n\nThe above equation is the operator form of Newton's second law in spherical polar coordinates. Next we evaluate the expectation value of both the sides of the above equation using the wave-functions given in Eq. (4.65). We know,\n\n1 r 2 = 1 n 3 a 2 0 (L + 1 2 ) , (5.10) 1 r 3 = 1 a 3 0 n 3 L(L + 1 2 )(L + 1) . (5.11)\n\nUsing the above expectation values in Eq. (5.9) and noting that L2 = h2 L(L + 1) we see that the time derivative of the expectation value of the radial momentum operator of the Hydrogen atom vanishes. The above analysis shows that the form of the Ehrenfest theorem as given in Eq. (5.4) is only valid in Cartesian coordinates. In the case of the Hydrogen atom if we used Eq. (5.4) we should have never got the correct result."
},
{
"section_type": "CONCLUSION",
"section_title": "VI. CONCLUSION",
"text": "In the present work we have emphasized on the reality of the momentum expectation value and using the reality of the expectation value as a bench mark we did find out the form of the momentum operator in spherical polar coordinate system. We found that most of the concepts which define the momentum operator in Cartesian coordinates do not hold good in spherical polar coordinates and in general in any other coordinate system. The reason being that whenever we do an integration in curvilinear coordinates the Jacobian of the coordinate transformation matrix comes inside the picture and the Cartesian results start to falter if we do not change the rules appropriately. The forms of the momentum along the radial direction and the form of the angular momentum operators are derived in section III. The status of the angular variables was briefly discussed in the same section. We explicitly calculated the expectation values of the momentum operator in various important cases and showed that the expectation value of the momentum operator do really come out to be zero as expected. Although the expectation value of the momentum operator vanishes in most of the bound states, with a real potential, the expectation value of the position is not required to vanish. The expectation value of the position operator is directly related with the parity property of the potential which was briefly discussed in subsection IV C. At the end we calculated the Heisenberg equation of motion for the radial momentum operator for the Hydrogen atom and showed its formal semblance with Newton's second law. It was also shown that if we properly write the Heisenberg equation of motion in spherical polar coordinates then Ehrenfest's theorem follows naturally.\n\nIn short we conclude by saying: 1. the forms of the various momentum operators, in most of the coordinate systems, in quantum mechanics can be obtained by imposing the condition of the reality of their eigenvalues. The form of the probability conservation equation and Ehrenfest theorem must be modified in curvilinear coordinates to yield meaningful results. 2. There are obvious problems in elevating the status of angular variables to dynamical variables in quantum mechanics.\n\n3. For compact variables, if the variable is periodic the expectation value of the angular momentum conjugate to it is non-zero. If the compact variable is not periodic then the angular momentum conjugate to it must vanish.\n\n4. The momentum expectation values in cases of bound state motions vanish, whereas the position expectation values in those cases depends on the symmetry of the potential."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgements 18",
"text": "The authors thank Professors D. P. Dewangan, S. Rindani, J. Banerji, P. Panigrahi and Ms. Suratna Das for stimulating discussions and constant encouragements.\n\n[1] J. J. Sakurai, \"Modern quantum mechanics\", International student edition, Addison-Wesley, 1999. [2] L I. Schiff, \"Quantum mechanics\", McGraw-Hill International Editions, third edition. [3] R. Shankar, \"Principles of quantum mechanics\", Plenum Press, New York 1994, second edition. [4] G. Bonneau, J. Faraut, G. Valent, Am. J. Phys. 69 322, (2001). [5] P. A. M Dirac. \"The principles of quantum mechanics\", Fourth edition, Oxford university press, 1958. [6] S. Flügge. \"Practical quantum mechanics I\", Springer-Verlag Berlin Heidelberg 1971. [7] G. Paz, Euro. J. Phys. 22 337, (2001). [8] G. Paz, J. Phys. A: Math. Gen. 35 3727, (2002). [9] P. Carruthers, M. M. Nieto, Rev. Mod. Phys. 40 411, (1968). [10] D. T. Pegg, S. M. Barnett, Phys. Rev. A 39 1665, (1989). [11] H. Essén, Am. J. Phys. 46 983, (1978). [12] I. S. Gradshteyn, I. M. Ryzhik. \"Table of integrals, series, and products\". Academic Press, Harcourt India, sixth edition, page 955, 8.733 1 [13] I. S. Gradshteyn, I. M. Ryzhik. \"Table of integrals, series, and products\". Academic Press, Harcourt India, sixth edition, page 956, 8.737 2. [14] P. M. Morse, Phys. Rev. 34 57, (1929). [15] S. Ghosh, A. Chiruvelli, J. Banerji and P. K. Panigrahi, Phys. Rev. A 73, 013411, (2006). [16] I. S. Gradshteyn, I. M. Ryzhik. \"Table of integrals, series, and products\". Academic Press, Harcourt India, sixth edition, page 991, 8.971 2. [17] I. S. Gradshteyn, I. M. Ryzhik. \"Table of integrals, series, and products\". Academic Press, Harcourt India, sixth edition, page 992, 8.974 3. [18] I. S. Gradshteyn, I. M. Ryzhik. \"Table of integrals, series, and products\", sixth edition, (Academic Press, Harcourt India). [19] The specific integration result and other related expressions can be found in the web page: http://mathworld.wolfram.com/LaguerrePolynomial.html , Equation 24 ."
}
] | ||
arxiv:0704.0376 | 0704.0376 | 1 | 10.1103/PhysRevA.76.012337 | 2b23c349f79ec23643ab7e201b461b290f8bcb9bf9938ef0f1cc65aefa49574b | Environmental noise reduction for holonomic quantum gates | We study the performance of holonomic quantum gates, driven by lasers, under
the effect of a dissipative environment modeled as a thermal bath of
oscillators. We show how to enhance the performance of the gates by suitable
choice of the loop in the manifold of the controllable parameters of the laser.
For a simplified, albeit realistic model, we find the surprising result that
for a long time evolution the performance of the gate (properly estimated in
terms of average fidelity) increases. On the basis of this result, we compare
holonomic gates with the so-called stimulated Raman adiabatic passage (STIRAP)
gates. | [
"Daniele Parodi",
"Maura Sassetti",
"Paolo Solinas",
"Nino Zangh\\`i"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | The major challenge for quantum computation is posed by the fact that generically quantum states are very delicate objects quite difficult to control with the required accuracy-typically, by means of external driving fields, e.g., a laser. The interaction with the many degrees of freedom of the environment causes decoherence; moreover, errors in processing the information may lead to a wrong output state.
Among the approaches aiming at overcoming these difficulties are those for which the quantum gate depends very weakly on the details of the dynamics, in particular, the holonomic quantum computation (HQC) [1] and the so-called Stimulated Raman adiabatic passage (STI-RAP) [2, 3, 4] . In the latter, the gate operator is obtained acting on the phase difference of the driving lasers during the evolution, while in the former the same goal is achieved by exploiting the non-commutative analogue of the Berry phase collected by a quantum state during a cyclic evolution. Concrete proposals have been put forward, for both Abelian [5, 6] and non-Abelian holonomies [7, 8, 9, 10, 11, 12] . The main advantage of the HQC is the robustness against noise deriving from a imperfect control of the driving fields [13, 14, 15, 16, 17, 18, 19, 20] .
In a recent paper [21] we have shown that the disturbance of the environment on holonomic gates can be suppressed and the performance of the gate optimized for particular environments (purely superohmic thermal bath). In the present paper we consider a different sort of optimization, which is independent of the particular nature of the environment.
By exploiting the full geometrical structure of HQC, we show how the performance of a holonomic gate can be enhanced by a suitable choice of the loop in the manifold of the parameters of the external driving field: by choosing the optimal loop which minimizes the "error" (properly estimated in terms of average fidelity loss). Our result is based on the observation that there are different loops in the parameter manifold producing the same gate and, since decoherence and dissipation crucially depend on the dynamics, it is possible to drive the system over trajectories which are less perturbed by the noise. For a simplified, albeit realistic model, we find the surprising result that the error decreases linearly as the gating time increases. Thus the disturbance of the environment can be drastically reduced. On the basis of this result, we compare holonomic gates with the STIRAP gates.
In Sec. II the model is introduced and the explicit expression of the error is derived. In Sec. III we find the optimal loop, calculate the error, make a comparison with other approaches, and briefly sketch how to treat a different coupling with the environment.
The physical model is given by three degenerate (or quasidegenerate) states, |+ , |-, and |0 , optically connected to another state |G . The system is driven by lasers with different frequencies and polarizations, acting selectively on the degenerate states. This model describes various quantum systems interacting with a laser radiation, ranging from semiconductor quantum dots, such as excitons [12] and spin-degenerate electron states [3], to trapped ions [8] or neutral atoms [7] .
The (approximate) Hamiltonian modeling the effect of the laser on the system is (for simplicity, = 1) [8, 12] H 0 (t) = j=+,-,0 ǫ|j j| + (e -iǫt Ω j (t)|j G| + H.c) , (1) where Ω j (t) are the timedependent Rabi frequencies de-2 pending on controllable parameters, such as the phase and intensity of the lasers, and ǫ is the energy of the degenerate electron states. The Rabi frequencies are modulated within the adiabatic time t ad , (which coincides with the gating time), to produce a loop in the parameter space and thereby realize the periodic condition H 0 (t ad ) = H 0 (0).
The Hamiltonian (1) has four time dependent eigenstates: two eigenstates |E i (t) , i = 1, 2, called bright states, and two eigenstates |E i (t) , i = 3, 4, called dark states. The two dark states have degenerate eigenvalue ǫ and the two bright states have timedependent energies λ ± (t) = [ǫ ± ǫ 2 + 4Ω 2 (t)]/2 with Ω 2 (t) = i=±,0 |Ω i (t)| 2 [22] . The evolution of the state is generated by U t = T e -i R t
0 dt ′ H0(t ′ ) , ( 2
)
where T is the time-ordered operator. In the adiabatic approximation, the evolution of the state takes place in the degenerate subspace generated by |+ , |-, and |0 . This approximation allows to separate the dynamic contribution and the geometric contribution from the evolution operator. Expanding U t in the basis of instantaneous eigenstates of H 0 (t) (the bright and dark states), in the adiabatic approximation, we have
U t ∼ = j e -i R t 0 Ej (t ′ )dt ′ |E j (t) E j (t)| U t , ( 3
)
where
U t = T e R t 0 dτ V (τ ) , ( 4
)
here V is the operator with matrix elements t) . The unitary operator U t plays the role of timedependent holonomic operator and is the fundamental ingredient for realizing complex geometric transformation whereas j e -i
V ij (t) = E i (t)|∂ t |E j (
R t 0 Ej (t ′ )dt ′ |E j (t) E j (t)| is the dynamic contribution.
Consider U t for a closed loop, i.e., for t = t ad ,
U = U t ad . ( 5
)
If the initial state |ψ 0 is a superposition of |+ and |-, then U|ψ 0 is still a superposition of the same vectors (in general, with different coefficients) [12] . Thus the space spanned by |+ and |-can be regarded as the "logical space" on which the "logical operator" U acts as a "quantum gate" operator. Note that for t < t ad , U t |ψ 0 has, in general, also a component along |0 . However, as it is easy to show [22] , at any instant t < t ad , U t |ψ 0 can be expanded in the twodimensional space spanned by the dark states |E 3 (t) and |E 4 (t) . It is important to observe that U depends only on global geometric features of the path in the parameter manifold and not on the details of the dynamical evolution [1, 12] .
To construct a complete set of holonomic quantum gates, it is sufficient to restrict the Rabi frequencies Ω j (t) in such a way that the norm Ω of the vector Ω = [Ω 0 (t), Ω + (t), Ω -(t)] is time independent and the vector lies on a real three dimensional sphere [8, 12] . We parametrize the evolution on this sphere as Ω + (t) = sin θ(t) cos φ(t), Ω -(t) = sin θ(t) sin φ(t) and Ω 0 (t) = cos θ(t) with fixed initial (and final) point in θ(0) = 0, the north pole By straightforward calculation we obtain the analytical expression for V (t) in eq. (4), V (t) = iσ y cos[θ(t)] φ(t), where σ y is the usual Pauli matrix written in the basis of dark states. Thus, the operator (4) becomes U t = cos[a(t)] -iσ y sin[a(t)], here a(t) = t 0 dτ φ(τ ) cos θ(τ ). Accordingly, the logical operator U (5) is
U = cos a -iσ y sin a, ( 6
)
where
a = a(t ad ) = t ad 0 dτ φ(τ ) cos θ(τ ) ( 7
)
is the solid angle spanned on the sphere during the evolution. Note that the are many paths on the sphere which generate the same logical operator U, and span the same solid angle a.
In a previous work we have studied how interaction with the environment disturbs the logical operator U [21] . The goal of the present paper is to analyze whether and how such a disturbance can be minimized for a given U. To this end, we model the environment as a thermal bath of harmonic oscillators with linear coupling between system and environment [23] . The total Hamiltonian is then
H = H 0 (t) + N α=1 ( p 2 α 2m α + 1 2 m α ω 2 α x 2 α + c α x α A), ( 8
)
where A is the system interaction operator called, from now on, noise operator. We now consider the time evolution of the reduced density matrix of the system, determined by the Hamiltonian (8). We rely on the standard methods of the "master equation approach," with the environment treated in the Born approximation and assumed to be at each time in its own thermal equilibrium state at temperature T . This allows to include the effect of the environment in the correlation function (k B = 1)
g(τ ) = ∞ 0 J(ω) coth ω 2T cos(ωτ ) -i sin(ωτ ) dω.
(9) Here the spectral density is
J(ω) = π 2 N α=1 c 2 α m α ω α δ(ω -ω α ), ( 10
)
at the low frequencies regimes, is proportional to ω s , with s ≥ 0, i.e., s = 1 describes a Ohmic environments, typical of baths of conduction electrons, s = 3 describes 3 a super-Ohmic environment, typical of baths of phonons [21, 24] . The asymptotic decay of the real part of g(τ ) defines the characteristic memory time of the environment. Denoting with ρ(t) the time evolution of the reduced density matrix of the system in the interaction picture, e.g., ρ(t) = U † t ρU t , one has [24]
ρ(t ad ) = ρ(0) + -i
Z t ad 0 dt Z t 0 dτ {g(τ )[ Ã Ã′ ρ(t -τ ) -Ã′ ρ(t -τ ) Ã] + g(-τ )[ρ(t -τ ) Ã′ Ã -Ãρ(t -τ ) Ã′ ]. ( 11
)
Here à and Ã′ stand for Ã(t) and Ã(t -τ ), with the tilde denoting the time evolution in the interaction picture.
In quantum information the quality of a gate is usually evaluated by the fidelity F , which measures the closeness between the unperturbed state and the final state,
F = ψ 0 (0)|U † ρ(t ad )U|ψ 0 (0) , ( 12
)
where |ψ 0 (0) is the initial state, and ρ(t ad ) = U ρ(t ad )U † is the reduced density matrix in the Schrödinger picture starting from the initial condition ρ(0
) = |ψ 0 (0) ψ 0 (0)|.
The average error is defined as the average fidelity loss, i.e.,
δ =< 1 -F >= 1-< ψ 0 (0)|ρ(t ad )|ψ 0 (0) >, ( 13
)
where < • • • > denotes averaging with respect to the uniform distribution over the initial state |ψ 0 (0) . The right-handside of Eq. ( 13 ) can be computed by the following steps: (1) solving Eq. (11) in strictly second order approximation; this approximation corresponds to replace ρ(t -τ ) with ρ(0); (2) using the adiabatic approximation U (t -τ, t) ≈ exp(iτ H 0 (t)); (3) expanding the scalar product in Eq. (13) with respect to a complete orthonormal basis {|ϕ n (t) }, n = 1, 2, 3, orthogonal to |ψ 0 (t) . In this way, one obtains
δ = 3 n=1 t ad 0 dt G(t)| ψ 0 (t)|A|ϕ n (t) | 2 , ( 14
)
where
G(t) = t 0 dτ Re[g(τ )] cos(ω 0n t) + Im[g(τ )] sin(ω 0n t)) . (15) Here, ω 0n = ω 0 -ω n are the energy differences associated to the transition ψ 0 ↔ φ n , with ω 0 = ǫ, ω 1 = λ + , ω 2 = λ -, and ω 3 = ǫ.
The interaction between system and environment is expressed by the noise operator A in Eq. ( 8 ). We shall now make the assumption that A = diag{0, 0, 0, 1} in the |G , |± , and |0 basis. In this case the transition between degenerate states are forbidden, however the noise breaks their degeneracy, shifting one of them. In spite of its simple form, this A is nevertheless a realistic noise operator for physical semiconductor systems [4].
The problem can be stated in the following way: given the noise operator A and the logical operator U, find a path on the parameter space (the surface of the sphere, described above) which minimizes the error δ.
The total error δ, given by Eq. ( 14 ), can be decomposed as
δ = δ tr + δ pd , ( 16
)
where the transition error, δ tr , is the contribution to the sum of the nondegenerate states (ω 0n = 0) and the pure dephasing error δ pd is the contribution of the degenerate states (ω 0n = 0). Thus
δ pd = π 8 t ad 0 dt ∞ 0 dω J(ω) ω coth ω 2T sin(ωt) 1 + 1 2 sin 2 2a(t) sin 4 θ(t) ( 17
)
and
δ tr = n=+,- 1 8 1 + [(λ n -ǫ)/Ω] 2 Γ 0n t ad 0 sin 2 2θ(t)dt, ( 18
) where Γ 0n = J(|ω 0n |) coth |ω 0n | 2T -sgn(ω 0n ) ( 19
)
correspond to the transition rates calculated by standard Fermi golden rules, supposing, as usual, G(t) ≈ G(∞) for g(τ ) strongly peaked around τ = 0. In the following we define for simplicity
K = n=+,- 1 8 1 + [(λ n -ǫ)/Ω] 2 Γ 0n .
Since we are interested at long time evolution, we start discussing the transition error which dominates in this regime [4, 25] .
As explained in Sec. II, the holonomic paths are closed curves on the surface of the sphere which start from the north pole. It turns out that the curve minimizing δ tr can be found among the loops which are composed by a simple sequence of three paths (see the Appendix): evolution along a meridian (φ = const), evolution along a
4 Π 4 Π 2 3 Π 4 Π 1 2 3 4 θ M vδtr K
FIG. 1: The error δtr versus θM for two different a values: a = π/2 (dashed line) and a = π/4 (full line) correspond to NOT and Hadamard gate, respectively.
parallel (θ = const) and a final evolution along a meridian to come back to the north pole.
The error δ tr in (18) , depends on a given by Eq. ( 7 ), θ M (the maximum angle spanned during the evolution along the meridian), ∆φ (the angle spanned along the parallel), and angular velocity v. We allow ∆φ ≥ 2π which corresponds to cover more than one loop along the parallel. The velocity along the parallel is v(t) = φ(t) sin θ and that along the meridian is v(t) = θ(t). In the following we assume that v is constant, and it cannot exceed the maximal value of v max , fixed by adiabatic condition v max ≪ Ω.
The parameters a, θ M , and ∆φ are connected by the relation a = ∆φ(1 -cos θ M ). The error δ tr is then
δ tr = δ M tr + δ P tr , ( 20
)
where
δ M tr = K v θ M - 1 4 sin 4θ M ( 21
)
is the contribution along the meridian and
δ P tr = K a v sin θ M sin 2 2θ M 1 -cos θ M ( 22
)
is the contribution along the parallel. In Fig. 1 δ tr is plotted for a = π/2 and a = π/4 (corresponding to NOT and Hadamard gate, respectively) as a function of θ M . One can see that δ tr has a local minimum for θ M = π/2 and a global minimum for θ M = 0 where the error vanishes. This suggests that the best choice is to take θ M as small as possible.
It is interesting to consider the dependence of δ tr also on the evolution time t ad . For simplicity, we set the velocity v = v max . In this case, changing θ M (and then ∆φ) corresponds to a change in the evolution time. We obtain
θ M = arccos 1 - a 2πm , ( 23
)
5 10 15 20 25 30 1 2 3 4 v max t ad δtr K FIG. 2: The error δtr versus vmaxt ad for two different a values: a = π/2 (dashed line) and a = π/4 (full line) correspond to NOT and Hadamard gate, respectively. The dotted-dashed line shows the value of the error at θ = π/2. The circles show the critical value of vmaxt ad above which the best loop is the one with the minimal θM .
where
m = 1 4πa (v max t ad ) 2 + a 2 . ( 24
)
Using these relations, δ M tr and δ P tr , given by (21) and (22) become functions of t ad , v max , and a. Note that m measures the space covered along the parallel, in fact ∆φ = 2πm.
In Fig. 2 we see the behavior of δ tr as a function of v max t ad . The first minimum for both curves corresponds to θ M = π/2, then the curves for long t ad decrease asymptotically to zero corresponding to the region in which θ M → 0. In this regime we have δ tr ∝ 1/t ad which is drastically different from the results obtained with other methods where δ tr ∝ t ad , (see Refs [4, 25] and below Sec. III C). It should be observed that this surprising results is a merit of holonomic approach which allows to choose the loop in the parameter space, without changing the logical operation as long as it subtends the same solid angle. Observe that small θ M and long t ad mean large value of m, i.e., multiple loops around the north pole.
Figure 2 shows that, for a given gate, there is a critical value k c of v max t ad which discriminate between the choice of θ M (e.g., k = 6 for the Hadamard gate and k = 25 for the NOT gate). For v max t ad < k c the best choice for the loop is θ M = π/2; For v max t ad > k c the best choice is the value of θ M determined by eq. (23) and (24) .
Note that the region v max t ad > k c is accessible with physical realistic parameters [12] . For example, if we choose the laser intensity Ω = 20 meV and v max = Ω/50 (for which values the nonadiabatic transitions are forbidden), the critical parameter corresponds to the critical time of 15 ps for the Hadamard gate and 42 ps for the NOT gate. 5 B. Pure Dephasing Until now we have ignored the pure dephasing effect because we have assumed that it is negligible in comparison with the transition error for long evolution time. Now, we check that the pure dephasing error contribution can indeed be neglected. We can write the pure dephasing error using Eq. (17) and splitting to parallel and meridian part as
δ P pd = t ad 0 dt ∞ 0 dω J(ω) ω coth ω 2T Q[a(t)] sin ωt sin 4 θ M ( 25
)
and
δ M pd = θ M v max 0 dt ∞ 0 dω J(ω) ω coth ω 2T Q[a(t)] sin ωt sin 4 (v max t) + sin 4 θ M 1 - v max t θ M , ( 26
)
where Q[a(t)] = 1 + 1/2 sin 2 [2a(t)].
To estimate δ pd we assume that t ad is longer with respect to the characteristic time of the bath. Remembering that J(ω) ∝ ω s , the pure dephasing error behavior along the parallel part at the temperature T is
δ P pd ∝ 1 t ad s+3 , T ≪ 1/t ad T 1 t ad s+2 , T ≫ 1/t ad ( 27
)
while the along meridian is
δ M pd ∝ 1 t ad 3 . ( 28
)
Then, we can conclude that the pure dephasing can always be neglected for long time evolution because it decreases faster than the transition error.
We make a comparison between holonomic quantum computation (HQC) and the STIRAP procedure which is an analogous approach to process quantum information. The STIRAP procedure ([2, 4] ) is, in its basic points, very similar to the holonomic information manipulation. The level spectrum, the information encoding, the evolution produced by adiabatic evolving laser are exactly the same. The fundamental difference is that in STIRAP the dynamical evolution is fixed (we must pass through a precise sequence of states) and then the corresponding loop in the parameter space is fixed. In particular, we go from the north pole to the south pole and back to the north pole along meridians. Since the loop, as in our model, is a sequence of meridian-parallel-meridian path, we can calculate the error and make a direct comparison. In this case, the transition error results proportional to δ tr ∝ t ad and grows linearly in time while for HQC δ tr ∝ 1/t ad . Therefore, the HQC is fundamentally the favorite for long application times with respect to the STIRAP ones.
Moreover, we can show that the freedom in the choice of the loop allows us to construct HQC which perform better than the best STIRAP gates. In Ref. [4] the minimum error (not depending on the evolution time) for STI-RAP was obtained reaching a compromise between the necessity to minimize the transition, pure dephasing error and the constraint of adiabatic evolution. With realistic physical parameters [21] (J(ω) = kω 3 e (-ω/ωc) 2 , Ω = 10 meV, ǫ = 1eV, v max = Ω/50, k = 10 -2 (meV) -2 , ω c = 0.5 meV and for low temperature), the total minimum error in Ref. [4] is δ stirap = 10 -3 . With the same parameters, we still have the possibility to increase the evolution time in order to reduce the environmental error. However, for evolution time t ad = 50 ps we obtain a total error δ = 1.5 × 10 -4 for the NOT gate and δ = 4 × 10 -5 for the Hadamard gate, respectively. As can be seen, the logical gate performance is greatly increased.
Until now we have discussed the possibility to minimize the environmental error by choosing a particular loop in the parameter sphere but the structure of the error functional clearly depends on the system-environment interaction. Then one might wonder if the same approach can be used for a different noise environment.
For this reason, we now briefly analyze the case of noise matrix in the form A = diag{0, 1, 0, -1}. Again, for long evolution we can neglect the contribution of the pure dephasing and focus on the transition error. In this case the interesting part of the error functional takes the form
δ tr = K[( 1 2 sin 2θ cos 2θ) 2 + (sin θ sin 2φ) 2 ]. ( 29
)
Even if the analysis in this case is much more complicated, it can be seen that δ tr has an absolute minimum for θ M = 0. The long time behavior is the same (δ tr ∝ 1/t ad ) such that the results are qualitatively analogous to the above ones: for small θ M loops (or long evolution at fixed velocity) the holonomic quantum gate presents a decreasing error. Then even in this case it is possible to minimize the environmental error.
In summary, we have analyzed the performance of holonomic quantum gates in the presence of environmen-6 tal noise by focusing on the possibility to have small errors choosing different loops in the parameter manifold. Due to the geometric dependence, we can implement the same logical gate with different loops. Since different loops correspond to different dynamical evolutions, we have used this freedom to construct an evolution through "protected" or "weakly influenced" states leading to good holonomic quantum gates performances. This allows to select (once that the physical parameter are fixed) the best loop which minimizes the environmental effect. (Note that this optimization procedure is rather independent of the details of the simple model we have considered and arguably, it could be extended to more complicated systems without any substantial modification.) We have shown that for long time evolutions the noise decreases as 1/t ad while in the other cases it increases linearly with adiabatic time. We also have shown that the same features can be found with different kinds of noise suggesting the possibility to find a way to minimize the environmental effect in the presence of any noise. These results open a new possibility for implementation of holonomic quantum gates to quantum computation because they seem robust against both control error and environmental noise.
The autors thank E. De Vito for useful discussions. One of the authors (P. S.) acknowledges support from INFN. Financial support by the italian MIUR via PRIN05 and INFN is acknowledged.
APPENDIX A: MINIMIZING THEOREM
Let us consider the family C n composed of the closed curves generated by a sequence of n paths along a parallel (θ = const) alternated with paths along a meridian (φ = const). We call C n a generic curve in this family. For example, the family C 1 contains all the closed curves composed by the sequence of path meridian-parallel-meridian while the family C 2 contains the curves meridian-parallelmeridian-parallel-meridian.
We argue that the closed curve minimizing the error in Eq. ( 18 ) can be found in the C 1 family. First, we show that any closed curve in C 2 spanning a solid angle a on the sphere can be replaced by a closed curve in C 1 spanning the same angle and producing a smaller error. In analogous way any closed curve in C 3 can be replaced by a closed curve in C 2 with smaller error and so on. By induction we obtain that any closed curve in C n can be replaced by a curve in C 1 spanning the same solid angle but producing smaller error. Since the curve belonging to C n can approximate any closed curve on the sphere, the best curve can be found in C 1 .
The crucial point is to show that any curve in C 2 can be replaced by a curve in C 1 . Let us consider a generic curve C 2 in C 2 spanning a solid angle a: composed by a segment of a meridian (with θ going from 0 to θ 1 ), a parallel (spanning a ∆φ 1 angle), meridian (with θ : θ 1 → θ 2 ), a parallel (spanning a ∆φ 2 angle), and finally a segment to the north pole along a meridian. Let us consider two closed curves C 1 1 and C 2 1 in C 1 subtending the same solid angle a with, respectively, θ 1 and θ 2 as maximum angle spanned during the evolution along the meridian. First we analyze (20) along the meridian. Without losing generality, we can take θ 1 < θ 2 ; it is clear from Eq. (21) that the value of δ tr along the meridian for C 1 1 is smaller that for C 2 1 : δ M C 1 1 < δ M C 2 1 . We note from the Eq. ( 21 ), suitable extended to C 2 , that the two paths along the meridians depends only on θ 2 and then produce the same error of C 2 1 ,
δ M C 1 1 < δ M C 2 1 = δ M C2 . ( A1
)
The difference between the contribution along the parallel is
δ P C2 -δ P C 2 1 = ∆φ 1 sin θ 1 sin 2 2θ 1 - 1 -cos θ 1 1 -cos θ 2 sin θ 2 sin 2 2θ 2 (A2) and δ P C2 -δ P C 1 1 = ∆φ 2 sin θ 2 sin 2 2θ 2 - 1 -cos θ 2 1 -cos θ 1 sin θ 1 sin 2 2θ 1 .
(A3) Analysis of the positivity of the quantities given by Eqs. (A2) and (A3) shows that δ P C2 cannot be at the same time smaller than δ P C 2 1 and δ P C 2 2 . In fact, there are two possibilities: If δ P C2 > δ P C 1 1 , from Eq. ( A1 ) and (A3),
δ C2 = δ M C2 + δ P C2 > δ M C 1 1 + δ P C 1 1 = δ C 1 1 , ( A4
)
and the best closed curve is C 1 1 . If δ P C2 > δ P C 2 1 , from Eqs. (A1) and (A2),
δ C2 = δ M C2 + δ P C2 > δ M C 2 1 + δ P C 2 1 = δ C 2 1 , ( A5
)
and the best closed curve is C 2 1 . In the same way it can be shown that any closed curve in C 3 can be replaced by a closed curve in C 2 with smaller error.
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Rev. B 67, 121307(R) (2003). [13] A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, Phys. Rev. Lett. 90, 160402 (2003). [14] G. De Chiara and G. M. Palma, Phys. Rev. Lett. 91, 090404 (2003). [15] A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, Phys. Rev. Lett. 92, 020402 (2004).
[16] V.I. Kuvshinov and A.V. Kuzmin, Phys. Lett. A, 316, 391 (2003). [17] P. Solinas, P. Zanardi, and N. Zanghì, Phys. Rev. A 70, 042316 (2004). [18] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301(R) (2005). [19] G. Florio, P. Facchi, R. Fazio, V. Giovannetti, and S. Pascazio, Phys. Rev. A 73, 022327 (2006). [20] I. Fuentes-Guridi, F. Girelli, and E. Livine, Phys. Rev. Lett. 94, 020503 (2005) [21] D. Parodi, M. Sassetti, P. Solinas, P. Zanardi, and N. Zanghì, Phys. Rev. A 73, 052304 (2006). [22] The explicit expression for the bright states is |E1 = 1 √ 2Ω (Ω|e + P i Ωi|i ) and |E2 = 1 √ 2Ω (-Ω|e + P i Ωi|i ); for the dark states is |E3 = 1/(Ω [23] A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981). [24] U. Weiss, Quantum dissipative systems (World Scientific, Singapore, 1999). [25] R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki, L. Jacak, and P. Machnikowski, Phys. Rev. A 70, 010501(R) (2004).
p |Ω+| 2 + |Ω-| 2 )[Ω0(Ω+|+ + Ω-|-) - (Ω 2 -|Ω0| 2 )|0 ]) and |E4 = 1/ p |Ω+| 2 + |Ω-| 2 [Ω-|+ - Ω+|-]. | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The major challenge for quantum computation is posed by the fact that generically quantum states are very delicate objects quite difficult to control with the required accuracy-typically, by means of external driving fields, e.g., a laser. The interaction with the many degrees of freedom of the environment causes decoherence; moreover, errors in processing the information may lead to a wrong output state.\n\nAmong the approaches aiming at overcoming these difficulties are those for which the quantum gate depends very weakly on the details of the dynamics, in particular, the holonomic quantum computation (HQC) [1] and the so-called Stimulated Raman adiabatic passage (STI-RAP) [2, 3, 4] . In the latter, the gate operator is obtained acting on the phase difference of the driving lasers during the evolution, while in the former the same goal is achieved by exploiting the non-commutative analogue of the Berry phase collected by a quantum state during a cyclic evolution. Concrete proposals have been put forward, for both Abelian [5, 6] and non-Abelian holonomies [7, 8, 9, 10, 11, 12] . The main advantage of the HQC is the robustness against noise deriving from a imperfect control of the driving fields [13, 14, 15, 16, 17, 18, 19, 20] .\n\nIn a recent paper [21] we have shown that the disturbance of the environment on holonomic gates can be suppressed and the performance of the gate optimized for particular environments (purely superohmic thermal bath). In the present paper we consider a different sort of optimization, which is independent of the particular nature of the environment.\n\nBy exploiting the full geometrical structure of HQC, we show how the performance of a holonomic gate can be enhanced by a suitable choice of the loop in the manifold of the parameters of the external driving field: by choosing the optimal loop which minimizes the \"error\" (properly estimated in terms of average fidelity loss). Our result is based on the observation that there are different loops in the parameter manifold producing the same gate and, since decoherence and dissipation crucially depend on the dynamics, it is possible to drive the system over trajectories which are less perturbed by the noise. For a simplified, albeit realistic model, we find the surprising result that the error decreases linearly as the gating time increases. Thus the disturbance of the environment can be drastically reduced. On the basis of this result, we compare holonomic gates with the STIRAP gates.\n\nIn Sec. II the model is introduced and the explicit expression of the error is derived. In Sec. III we find the optimal loop, calculate the error, make a comparison with other approaches, and briefly sketch how to treat a different coupling with the environment."
},
{
"section_type": "OTHER",
"section_title": "II. MODEL",
"text": "The physical model is given by three degenerate (or quasidegenerate) states, |+ , |-, and |0 , optically connected to another state |G . The system is driven by lasers with different frequencies and polarizations, acting selectively on the degenerate states. This model describes various quantum systems interacting with a laser radiation, ranging from semiconductor quantum dots, such as excitons [12] and spin-degenerate electron states [3], to trapped ions [8] or neutral atoms [7] .\n\nThe (approximate) Hamiltonian modeling the effect of the laser on the system is (for simplicity, = 1) [8, 12] H 0 (t) = j=+,-,0 ǫ|j j| + (e -iǫt Ω j (t)|j G| + H.c) , (1) where Ω j (t) are the timedependent Rabi frequencies de-2 pending on controllable parameters, such as the phase and intensity of the lasers, and ǫ is the energy of the degenerate electron states. The Rabi frequencies are modulated within the adiabatic time t ad , (which coincides with the gating time), to produce a loop in the parameter space and thereby realize the periodic condition H 0 (t ad ) = H 0 (0).\n\nThe Hamiltonian (1) has four time dependent eigenstates: two eigenstates |E i (t) , i = 1, 2, called bright states, and two eigenstates |E i (t) , i = 3, 4, called dark states. The two dark states have degenerate eigenvalue ǫ and the two bright states have timedependent energies λ ± (t) = [ǫ ± ǫ 2 + 4Ω 2 (t)]/2 with Ω 2 (t) = i=±,0 |Ω i (t)| 2 [22] . The evolution of the state is generated by U t = T e -i R t\n\n0 dt ′ H0(t ′ ) , ( 2\n\n)\n\nwhere T is the time-ordered operator. In the adiabatic approximation, the evolution of the state takes place in the degenerate subspace generated by |+ , |-, and |0 . This approximation allows to separate the dynamic contribution and the geometric contribution from the evolution operator. Expanding U t in the basis of instantaneous eigenstates of H 0 (t) (the bright and dark states), in the adiabatic approximation, we have\n\nU t ∼ = j e -i R t 0 Ej (t ′ )dt ′ |E j (t) E j (t)| U t , ( 3\n\n)\n\nwhere\n\nU t = T e R t 0 dτ V (τ ) , ( 4\n\n)\n\nhere V is the operator with matrix elements t) . The unitary operator U t plays the role of timedependent holonomic operator and is the fundamental ingredient for realizing complex geometric transformation whereas j e -i\n\nV ij (t) = E i (t)|∂ t |E j (\n\nR t 0 Ej (t ′ )dt ′ |E j (t) E j (t)| is the dynamic contribution.\n\nConsider U t for a closed loop, i.e., for t = t ad ,\n\nU = U t ad . ( 5\n\n)\n\nIf the initial state |ψ 0 is a superposition of |+ and |-, then U|ψ 0 is still a superposition of the same vectors (in general, with different coefficients) [12] . Thus the space spanned by |+ and |-can be regarded as the \"logical space\" on which the \"logical operator\" U acts as a \"quantum gate\" operator. Note that for t < t ad , U t |ψ 0 has, in general, also a component along |0 . However, as it is easy to show [22] , at any instant t < t ad , U t |ψ 0 can be expanded in the twodimensional space spanned by the dark states |E 3 (t) and |E 4 (t) . It is important to observe that U depends only on global geometric features of the path in the parameter manifold and not on the details of the dynamical evolution [1, 12] .\n\nTo construct a complete set of holonomic quantum gates, it is sufficient to restrict the Rabi frequencies Ω j (t) in such a way that the norm Ω of the vector Ω = [Ω 0 (t), Ω + (t), Ω -(t)] is time independent and the vector lies on a real three dimensional sphere [8, 12] . We parametrize the evolution on this sphere as Ω + (t) = sin θ(t) cos φ(t), Ω -(t) = sin θ(t) sin φ(t) and Ω 0 (t) = cos θ(t) with fixed initial (and final) point in θ(0) = 0, the north pole By straightforward calculation we obtain the analytical expression for V (t) in eq. (4), V (t) = iσ y cos[θ(t)] φ(t), where σ y is the usual Pauli matrix written in the basis of dark states. Thus, the operator (4) becomes U t = cos[a(t)] -iσ y sin[a(t)], here a(t) = t 0 dτ φ(τ ) cos θ(τ ). Accordingly, the logical operator U (5) is\n\nU = cos a -iσ y sin a, ( 6\n\n)\n\nwhere\n\na = a(t ad ) = t ad 0 dτ φ(τ ) cos θ(τ ) ( 7\n\n)\n\nis the solid angle spanned on the sphere during the evolution. Note that the are many paths on the sphere which generate the same logical operator U, and span the same solid angle a.\n\nIn a previous work we have studied how interaction with the environment disturbs the logical operator U [21] . The goal of the present paper is to analyze whether and how such a disturbance can be minimized for a given U. To this end, we model the environment as a thermal bath of harmonic oscillators with linear coupling between system and environment [23] . The total Hamiltonian is then\n\nH = H 0 (t) + N α=1 ( p 2 α 2m α + 1 2 m α ω 2 α x 2 α + c α x α A), ( 8\n\n)\n\nwhere A is the system interaction operator called, from now on, noise operator. We now consider the time evolution of the reduced density matrix of the system, determined by the Hamiltonian (8). We rely on the standard methods of the \"master equation approach,\" with the environment treated in the Born approximation and assumed to be at each time in its own thermal equilibrium state at temperature T . This allows to include the effect of the environment in the correlation function (k B = 1)\n\ng(τ ) = ∞ 0 J(ω) coth ω 2T cos(ωτ ) -i sin(ωτ ) dω.\n\n(9) Here the spectral density is\n\nJ(ω) = π 2 N α=1 c 2 α m α ω α δ(ω -ω α ), ( 10\n\n)\n\nat the low frequencies regimes, is proportional to ω s , with s ≥ 0, i.e., s = 1 describes a Ohmic environments, typical of baths of conduction electrons, s = 3 describes 3 a super-Ohmic environment, typical of baths of phonons [21, 24] . The asymptotic decay of the real part of g(τ ) defines the characteristic memory time of the environment. Denoting with ρ(t) the time evolution of the reduced density matrix of the system in the interaction picture, e.g., ρ(t) = U † t ρU t , one has [24]\n\nρ(t ad ) = ρ(0) + -i\n\nZ t ad 0 dt Z t 0 dτ {g(τ )[ à Ã′ ρ(t -τ ) -Ã′ ρ(t -τ ) Ã] + g(-τ )[ρ(t -τ ) Ã′ à -Ãρ(t -τ ) Ã′ ]. ( 11\n\n)\n\nHere à and Ã′ stand for Ã(t) and Ã(t -τ ), with the tilde denoting the time evolution in the interaction picture.\n\nIn quantum information the quality of a gate is usually evaluated by the fidelity F , which measures the closeness between the unperturbed state and the final state,\n\nF = ψ 0 (0)|U † ρ(t ad )U|ψ 0 (0) , ( 12\n\n)\n\nwhere |ψ 0 (0) is the initial state, and ρ(t ad ) = U ρ(t ad )U † is the reduced density matrix in the Schrödinger picture starting from the initial condition ρ(0\n\n) = |ψ 0 (0) ψ 0 (0)|.\n\nThe average error is defined as the average fidelity loss, i.e.,\n\nδ =< 1 -F >= 1-< ψ 0 (0)|ρ(t ad )|ψ 0 (0) >, ( 13\n\n)\n\nwhere < • • • > denotes averaging with respect to the uniform distribution over the initial state |ψ 0 (0) . The right-handside of Eq. ( 13 ) can be computed by the following steps: (1) solving Eq. (11) in strictly second order approximation; this approximation corresponds to replace ρ(t -τ ) with ρ(0); (2) using the adiabatic approximation U (t -τ, t) ≈ exp(iτ H 0 (t)); (3) expanding the scalar product in Eq. (13) with respect to a complete orthonormal basis {|ϕ n (t) }, n = 1, 2, 3, orthogonal to |ψ 0 (t) . In this way, one obtains\n\nδ = 3 n=1 t ad 0 dt G(t)| ψ 0 (t)|A|ϕ n (t) | 2 , ( 14\n\n)\n\nwhere\n\nG(t) = t 0 dτ Re[g(τ )] cos(ω 0n t) + Im[g(τ )] sin(ω 0n t)) . (15) Here, ω 0n = ω 0 -ω n are the energy differences associated to the transition ψ 0 ↔ φ n , with ω 0 = ǫ, ω 1 = λ + , ω 2 = λ -, and ω 3 = ǫ.\n\nThe interaction between system and environment is expressed by the noise operator A in Eq. ( 8 ). We shall now make the assumption that A = diag{0, 0, 0, 1} in the |G , |± , and |0 basis. In this case the transition between degenerate states are forbidden, however the noise breaks their degeneracy, shifting one of them. In spite of its simple form, this A is nevertheless a realistic noise operator for physical semiconductor systems [4]."
},
{
"section_type": "OTHER",
"section_title": "III. MINIMIZING THE ERROR",
"text": "The problem can be stated in the following way: given the noise operator A and the logical operator U, find a path on the parameter space (the surface of the sphere, described above) which minimizes the error δ.\n\nThe total error δ, given by Eq. ( 14 ), can be decomposed as\n\nδ = δ tr + δ pd , ( 16\n\n)\n\nwhere the transition error, δ tr , is the contribution to the sum of the nondegenerate states (ω 0n = 0) and the pure dephasing error δ pd is the contribution of the degenerate states (ω 0n = 0). Thus\n\nδ pd = π 8 t ad 0 dt ∞ 0 dω J(ω) ω coth ω 2T sin(ωt) 1 + 1 2 sin 2 2a(t) sin 4 θ(t) ( 17\n\n)\n\nand\n\nδ tr = n=+,- 1 8 1 + [(λ n -ǫ)/Ω] 2 Γ 0n t ad 0 sin 2 2θ(t)dt, ( 18\n\n) where Γ 0n = J(|ω 0n |) coth |ω 0n | 2T -sgn(ω 0n ) ( 19\n\n)\n\ncorrespond to the transition rates calculated by standard Fermi golden rules, supposing, as usual, G(t) ≈ G(∞) for g(τ ) strongly peaked around τ = 0. In the following we define for simplicity\n\nK = n=+,- 1 8 1 + [(λ n -ǫ)/Ω] 2 Γ 0n .\n\nSince we are interested at long time evolution, we start discussing the transition error which dominates in this regime [4, 25] ."
},
{
"section_type": "OTHER",
"section_title": "A. Transition rate",
"text": "As explained in Sec. II, the holonomic paths are closed curves on the surface of the sphere which start from the north pole. It turns out that the curve minimizing δ tr can be found among the loops which are composed by a simple sequence of three paths (see the Appendix): evolution along a meridian (φ = const), evolution along a\n\n4 Π 4 Π 2 3 Π 4 Π 1 2 3 4 θ M vδtr K\n\nFIG. 1: The error δtr versus θM for two different a values: a = π/2 (dashed line) and a = π/4 (full line) correspond to NOT and Hadamard gate, respectively.\n\nparallel (θ = const) and a final evolution along a meridian to come back to the north pole.\n\nThe error δ tr in (18) , depends on a given by Eq. ( 7 ), θ M (the maximum angle spanned during the evolution along the meridian), ∆φ (the angle spanned along the parallel), and angular velocity v. We allow ∆φ ≥ 2π which corresponds to cover more than one loop along the parallel. The velocity along the parallel is v(t) = φ(t) sin θ and that along the meridian is v(t) = θ(t). In the following we assume that v is constant, and it cannot exceed the maximal value of v max , fixed by adiabatic condition v max ≪ Ω.\n\nThe parameters a, θ M , and ∆φ are connected by the relation a = ∆φ(1 -cos θ M ). The error δ tr is then\n\nδ tr = δ M tr + δ P tr , ( 20\n\n)\n\nwhere\n\nδ M tr = K v θ M - 1 4 sin 4θ M ( 21\n\n)\n\nis the contribution along the meridian and\n\nδ P tr = K a v sin θ M sin 2 2θ M 1 -cos θ M ( 22\n\n)\n\nis the contribution along the parallel. In Fig. 1 δ tr is plotted for a = π/2 and a = π/4 (corresponding to NOT and Hadamard gate, respectively) as a function of θ M . One can see that δ tr has a local minimum for θ M = π/2 and a global minimum for θ M = 0 where the error vanishes. This suggests that the best choice is to take θ M as small as possible.\n\nIt is interesting to consider the dependence of δ tr also on the evolution time t ad . For simplicity, we set the velocity v = v max . In this case, changing θ M (and then ∆φ) corresponds to a change in the evolution time. We obtain\n\nθ M = arccos 1 - a 2πm , ( 23\n\n)\n\n5 10 15 20 25 30 1 2 3 4 v max t ad δtr K FIG. 2: The error δtr versus vmaxt ad for two different a values: a = π/2 (dashed line) and a = π/4 (full line) correspond to NOT and Hadamard gate, respectively. The dotted-dashed line shows the value of the error at θ = π/2. The circles show the critical value of vmaxt ad above which the best loop is the one with the minimal θM .\n\nwhere\n\nm = 1 4πa (v max t ad ) 2 + a 2 . ( 24\n\n)\n\nUsing these relations, δ M tr and δ P tr , given by (21) and (22) become functions of t ad , v max , and a. Note that m measures the space covered along the parallel, in fact ∆φ = 2πm.\n\nIn Fig. 2 we see the behavior of δ tr as a function of v max t ad . The first minimum for both curves corresponds to θ M = π/2, then the curves for long t ad decrease asymptotically to zero corresponding to the region in which θ M → 0. In this regime we have δ tr ∝ 1/t ad which is drastically different from the results obtained with other methods where δ tr ∝ t ad , (see Refs [4, 25] and below Sec. III C). It should be observed that this surprising results is a merit of holonomic approach which allows to choose the loop in the parameter space, without changing the logical operation as long as it subtends the same solid angle. Observe that small θ M and long t ad mean large value of m, i.e., multiple loops around the north pole.\n\nFigure 2 shows that, for a given gate, there is a critical value k c of v max t ad which discriminate between the choice of θ M (e.g., k = 6 for the Hadamard gate and k = 25 for the NOT gate). For v max t ad < k c the best choice for the loop is θ M = π/2; For v max t ad > k c the best choice is the value of θ M determined by eq. (23) and (24) .\n\nNote that the region v max t ad > k c is accessible with physical realistic parameters [12] . For example, if we choose the laser intensity Ω = 20 meV and v max = Ω/50 (for which values the nonadiabatic transitions are forbidden), the critical parameter corresponds to the critical time of 15 ps for the Hadamard gate and 42 ps for the NOT gate. 5 B. Pure Dephasing Until now we have ignored the pure dephasing effect because we have assumed that it is negligible in comparison with the transition error for long evolution time. Now, we check that the pure dephasing error contribution can indeed be neglected. We can write the pure dephasing error using Eq. (17) and splitting to parallel and meridian part as\n\nδ P pd = t ad 0 dt ∞ 0 dω J(ω) ω coth ω 2T Q[a(t)] sin ωt sin 4 θ M ( 25\n\n)\n\nand\n\nδ M pd = θ M v max 0 dt ∞ 0 dω J(ω) ω coth ω 2T Q[a(t)] sin ωt sin 4 (v max t) + sin 4 θ M 1 - v max t θ M , ( 26\n\n)\n\nwhere Q[a(t)] = 1 + 1/2 sin 2 [2a(t)].\n\nTo estimate δ pd we assume that t ad is longer with respect to the characteristic time of the bath. Remembering that J(ω) ∝ ω s , the pure dephasing error behavior along the parallel part at the temperature T is\n\nδ P pd ∝ 1 t ad s+3 , T ≪ 1/t ad T 1 t ad s+2 , T ≫ 1/t ad ( 27\n\n)\n\nwhile the along meridian is\n\nδ M pd ∝ 1 t ad 3 . ( 28\n\n)\n\nThen, we can conclude that the pure dephasing can always be neglected for long time evolution because it decreases faster than the transition error."
},
{
"section_type": "OTHER",
"section_title": "C. Comparison between HQC and STIRAP",
"text": "We make a comparison between holonomic quantum computation (HQC) and the STIRAP procedure which is an analogous approach to process quantum information. The STIRAP procedure ([2, 4] ) is, in its basic points, very similar to the holonomic information manipulation. The level spectrum, the information encoding, the evolution produced by adiabatic evolving laser are exactly the same. The fundamental difference is that in STIRAP the dynamical evolution is fixed (we must pass through a precise sequence of states) and then the corresponding loop in the parameter space is fixed. In particular, we go from the north pole to the south pole and back to the north pole along meridians. Since the loop, as in our model, is a sequence of meridian-parallel-meridian path, we can calculate the error and make a direct comparison. In this case, the transition error results proportional to δ tr ∝ t ad and grows linearly in time while for HQC δ tr ∝ 1/t ad . Therefore, the HQC is fundamentally the favorite for long application times with respect to the STIRAP ones.\n\nMoreover, we can show that the freedom in the choice of the loop allows us to construct HQC which perform better than the best STIRAP gates. In Ref. [4] the minimum error (not depending on the evolution time) for STI-RAP was obtained reaching a compromise between the necessity to minimize the transition, pure dephasing error and the constraint of adiabatic evolution. With realistic physical parameters [21] (J(ω) = kω 3 e (-ω/ωc) 2 , Ω = 10 meV, ǫ = 1eV, v max = Ω/50, k = 10 -2 (meV) -2 , ω c = 0.5 meV and for low temperature), the total minimum error in Ref. [4] is δ stirap = 10 -3 . With the same parameters, we still have the possibility to increase the evolution time in order to reduce the environmental error. However, for evolution time t ad = 50 ps we obtain a total error δ = 1.5 × 10 -4 for the NOT gate and δ = 4 × 10 -5 for the Hadamard gate, respectively. As can be seen, the logical gate performance is greatly increased."
},
{
"section_type": "OTHER",
"section_title": "D. More general noise",
"text": "Until now we have discussed the possibility to minimize the environmental error by choosing a particular loop in the parameter sphere but the structure of the error functional clearly depends on the system-environment interaction. Then one might wonder if the same approach can be used for a different noise environment.\n\nFor this reason, we now briefly analyze the case of noise matrix in the form A = diag{0, 1, 0, -1}. Again, for long evolution we can neglect the contribution of the pure dephasing and focus on the transition error. In this case the interesting part of the error functional takes the form\n\nδ tr = K[( 1 2 sin 2θ cos 2θ) 2 + (sin θ sin 2φ) 2 ]. ( 29\n\n)\n\nEven if the analysis in this case is much more complicated, it can be seen that δ tr has an absolute minimum for θ M = 0. The long time behavior is the same (δ tr ∝ 1/t ad ) such that the results are qualitatively analogous to the above ones: for small θ M loops (or long evolution at fixed velocity) the holonomic quantum gate presents a decreasing error. Then even in this case it is possible to minimize the environmental error."
},
{
"section_type": "CONCLUSION",
"section_title": "IV. CONCLUSIONS",
"text": "In summary, we have analyzed the performance of holonomic quantum gates in the presence of environmen-6 tal noise by focusing on the possibility to have small errors choosing different loops in the parameter manifold. Due to the geometric dependence, we can implement the same logical gate with different loops. Since different loops correspond to different dynamical evolutions, we have used this freedom to construct an evolution through \"protected\" or \"weakly influenced\" states leading to good holonomic quantum gates performances. This allows to select (once that the physical parameter are fixed) the best loop which minimizes the environmental effect. (Note that this optimization procedure is rather independent of the details of the simple model we have considered and arguably, it could be extended to more complicated systems without any substantial modification.) We have shown that for long time evolutions the noise decreases as 1/t ad while in the other cases it increases linearly with adiabatic time. We also have shown that the same features can be found with different kinds of noise suggesting the possibility to find a way to minimize the environmental effect in the presence of any noise. These results open a new possibility for implementation of holonomic quantum gates to quantum computation because they seem robust against both control error and environmental noise."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgment",
"text": "The autors thank E. De Vito for useful discussions. One of the authors (P. S.) acknowledges support from INFN. Financial support by the italian MIUR via PRIN05 and INFN is acknowledged.\n\nAPPENDIX A: MINIMIZING THEOREM\n\nLet us consider the family C n composed of the closed curves generated by a sequence of n paths along a parallel (θ = const) alternated with paths along a meridian (φ = const). We call C n a generic curve in this family. For example, the family C 1 contains all the closed curves composed by the sequence of path meridian-parallel-meridian while the family C 2 contains the curves meridian-parallelmeridian-parallel-meridian.\n\nWe argue that the closed curve minimizing the error in Eq. ( 18 ) can be found in the C 1 family. First, we show that any closed curve in C 2 spanning a solid angle a on the sphere can be replaced by a closed curve in C 1 spanning the same angle and producing a smaller error. In analogous way any closed curve in C 3 can be replaced by a closed curve in C 2 with smaller error and so on. By induction we obtain that any closed curve in C n can be replaced by a curve in C 1 spanning the same solid angle but producing smaller error. Since the curve belonging to C n can approximate any closed curve on the sphere, the best curve can be found in C 1 .\n\nThe crucial point is to show that any curve in C 2 can be replaced by a curve in C 1 . Let us consider a generic curve C 2 in C 2 spanning a solid angle a: composed by a segment of a meridian (with θ going from 0 to θ 1 ), a parallel (spanning a ∆φ 1 angle), meridian (with θ : θ 1 → θ 2 ), a parallel (spanning a ∆φ 2 angle), and finally a segment to the north pole along a meridian. Let us consider two closed curves C 1 1 and C 2 1 in C 1 subtending the same solid angle a with, respectively, θ 1 and θ 2 as maximum angle spanned during the evolution along the meridian. First we analyze (20) along the meridian. Without losing generality, we can take θ 1 < θ 2 ; it is clear from Eq. (21) that the value of δ tr along the meridian for C 1 1 is smaller that for C 2 1 : δ M C 1 1 < δ M C 2 1 . We note from the Eq. ( 21 ), suitable extended to C 2 , that the two paths along the meridians depends only on θ 2 and then produce the same error of C 2 1 ,\n\nδ M C 1 1 < δ M C 2 1 = δ M C2 . ( A1\n\n)\n\nThe difference between the contribution along the parallel is\n\nδ P C2 -δ P C 2 1 = ∆φ 1 sin θ 1 sin 2 2θ 1 - 1 -cos θ 1 1 -cos θ 2 sin θ 2 sin 2 2θ 2 (A2) and δ P C2 -δ P C 1 1 = ∆φ 2 sin θ 2 sin 2 2θ 2 - 1 -cos θ 2 1 -cos θ 1 sin θ 1 sin 2 2θ 1 .\n\n(A3) Analysis of the positivity of the quantities given by Eqs. (A2) and (A3) shows that δ P C2 cannot be at the same time smaller than δ P C 2 1 and δ P C 2 2 . In fact, there are two possibilities: If δ P C2 > δ P C 1 1 , from Eq. ( A1 ) and (A3),\n\nδ C2 = δ M C2 + δ P C2 > δ M C 1 1 + δ P C 1 1 = δ C 1 1 , ( A4\n\n)\n\nand the best closed curve is C 1 1 . If δ P C2 > δ P C 2 1 , from Eqs. (A1) and (A2),\n\nδ C2 = δ M C2 + δ P C2 > δ M C 2 1 + δ P C 2 1 = δ C 2 1 , ( A5\n\n)\n\nand the best closed curve is C 2 1 . In the same way it can be shown that any closed curve in C 3 can be replaced by a closed curve in C 2 with smaller error.\n\n[1] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [2] Z. Kis and F. Renzoni, Phys. Rev. A 65, 032318 (2002).\n\n7 [3] F. Troiani, E. Molinari, and U. Hohenester, Phys. Rev. Lett. 90, 206802 (2003). [4] K. Roszak, A. Grodecka, P. Machnikowski, and T. Kuhn, Phys. Rev. B 71, 195333 (2005). [5] J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Nature (London) 403, 869 (2000). [6] G. Falci, R. Fazio, G. M. Palma, J. Siewert, and V. Vedral, Nature (London) 407, 355 (2000). [7] R. G. Unanyan, B.W. Shore, and K. Bergmann, Phys.\n\nRev. A 59, 2910 (1999). [8] L.-M. Duan, J.I. Cirac, and P. Zoller, Science 292, 1695 (2001). [9] L. Faoro, J. Siewert, and R. Fazio, Phys. Rev. Lett. 90, 028301 (2003). [10] I. Fuentes-Guridi, J. Pachos, S. Bose, V. Vedral, and S.\n\nChoi, Phys. Rev. A 66, 022102 (2002). [11] A. Recati, T. Calarco, P. Zanardi, J. I. Cirac, and P.\n\nZoller, Phys. Rev. A 66, 032309 (2002). [12] P. Solinas, P. Zanardi, N. Zanghì, and F. Rossi, Phys.\n\nRev. B 67, 121307(R) (2003). [13] A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, Phys. Rev. Lett. 90, 160402 (2003). [14] G. De Chiara and G. M. Palma, Phys. Rev. Lett. 91, 090404 (2003). [15] A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, Phys. Rev. Lett. 92, 020402 (2004).\n\n[16] V.I. Kuvshinov and A.V. Kuzmin, Phys. Lett. A, 316, 391 (2003). [17] P. Solinas, P. Zanardi, and N. Zanghì, Phys. Rev. A 70, 042316 (2004). [18] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301(R) (2005). [19] G. Florio, P. Facchi, R. Fazio, V. Giovannetti, and S. Pascazio, Phys. Rev. A 73, 022327 (2006). [20] I. Fuentes-Guridi, F. Girelli, and E. Livine, Phys. Rev. Lett. 94, 020503 (2005) [21] D. Parodi, M. Sassetti, P. Solinas, P. Zanardi, and N. Zanghì, Phys. Rev. A 73, 052304 (2006). [22] The explicit expression for the bright states is |E1 = 1 √ 2Ω (Ω|e + P i Ωi|i ) and |E2 = 1 √ 2Ω (-Ω|e + P i Ωi|i ); for the dark states is |E3 = 1/(Ω [23] A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981). [24] U. Weiss, Quantum dissipative systems (World Scientific, Singapore, 1999). [25] R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki, L. Jacak, and P. Machnikowski, Phys. Rev. A 70, 010501(R) (2004).\n\np |Ω+| 2 + |Ω-| 2 )[Ω0(Ω+|+ + Ω-|-) - (Ω 2 -|Ω0| 2 )|0 ]) and |E4 = 1/ p |Ω+| 2 + |Ω-| 2 [Ω-|+ - Ω+|-]."
}
] | |
arxiv:0704.0377 | 0704.0377 | 1 | 10.1088/0954-3899/36/4/045001 | c9b267f312751f7fb7b5e1c66f7f4a0de81e94dad51fcf4ab1923542757529fc | The lifetime of unstable particles in electromagnetic fields | We show that the electromagnetic moments of unstable particles (resonances)
have an absorptive contribution which quantifies the change of the particle's
lifetime in an external electromagnetic field. To give an example we compute
here the imaginary part of the magnetic moment for the cases of the muon and
the neutron at leading order in the electroweak coupling. We also consider an
analogous effect for the strongly-decaying $\Delta$(1232) resonance. The result
for the muon is Im$ \mu = e G_F^2 m^3/768 \pi^3$, with $e$ the charge and $m$
the mass of the muon, $G_F$ the Fermi constant, which in an external magnetic
field of $B$ Tesla give rise to the relative change in the muon lifetime of
$3\times 10^{-15} B$. For neutron the effect is of a similar magnitude. We
speculate on the observable implications of this effect. | [
"Daniele Binosi (ECT*",
"Trento)",
"Vladimir Pascalutsa (ECT*",
"Trento &\n Mainz U.)"
] | [
"hep-ph",
"hep-th",
"nucl-th"
] | hep-ph | [] | 2007-04-03 | 2026-02-26 | The electromagnetic (e.m.) moments of a particle are among the few fundamental quantities which describe the particle properties and as such have thoroughly been studied. The most renowned examples are the magnetic moments of the electron and the muon which have been measured to unprecendented accuracy and yielded a number of physical insights, see [1] for recent reviews. What is far lesser known is that the e.m. moments of unstable particles are complex numbers in general [2, 3] . Their imaginary part reflects, of course, the unstable nature of the particle, however, the precise interpretation has been missing. In this paper we work out the relation, suggested first by Holstein [4] , which should exist between the imaginary part of the magnetic moment and the effect of an external magnetic field on particle's lifetime.
The argument for such a relation is very simple. The (self-)energy of the particle with a lifetime τ has an absorptive part, which has an interpretation of the width Γ = 1/τ . The particle's magnetic moment µ in the presence of magnetic field B induces the change in the energy: -µ • B. The latter contribution can then also change the width, provided the magnetic moment has an absorptive part (Im µ = 0).
The decay properties of unstable particles, such as muon or neutron are extremely well studied and are widely used for the precise determination of the Standard Model parameters [5, 6] . There are also a plethora of studies of how these particles behave in e.m. fields. A well-known example is the search for the neutron's electric dipole moment [7] .
In view of these studies it is compelling to investigate how the decay properties of unstable particles may be affected by e.m. fields.
The lifetime of unstable quantum-mechanical systems is known to be affected by an e.m. field. Positronium provides a textbook example[8], where the effect arises due to the admixture of para-(S = 0) and ortho-(S = 1) positronium states with orbital momentum l = 0 by the magnetic field interacting with the magnetic moments of the constituents. As the result, already in the field of B = 0.2 Tesla, the lifetime of ortho-positronium decreases by almost a factor of 2.
It is far from obvious how the same kind of an effect can arise for an elementary unstable particle, e.g., the muon. The above-mentioned relation between the imaginary part of the magnetic moment and the lifetime change may, therefore, provide us with both an interpre-2 µ ν µ µ W W ν e e FIG. 1: The muon self-energy contributing to its decay width.
tation for the imaginary part of the magnetic moment and the means to compute the effect of the lifetime change.
In the following we examine in detail the case of the muon, compute the leading contribution to Im µ and the corresponding effect on the lifetime. Then we will briefly discuss the cases of the neutron and of the ∆-resonance.
II. MUON DECAY (µ → e ν e ν µ )
The leading contribution to the muon decay width arises at two-loop level, see Fig. 1. For our purposes, the W propagators in this graph can safely be assumed to be static -Fermi theory. We also neglect the mass of the electron in the loops, since it leads to an underpercent correction of O(m e /m); here and in what follows, m is the muon mass. The graphs with other Standard Model fermions (e.g., quarks) in the loops need not to be considered here, because they cannot give any contribution to the muon width.
Using dimensional regularization, we compute this graph in d = 4 -2ǫ dimensions (in
the limit ǫ → 0 + ),[14] Σ (p /) = g 2 8M 4 W i d d k ( 2π
) d 2γ µ (1 -γ 5 ) (p / -k /) γ ν (p -k) 2 + iε Π µν (k). ( 1
)
where M W is the W -boson mass, g = |e|/ sin θ W is the electroweak coupling related to the Fermi constant by G F / √ 2 = g 2 /8M 2 W , e is the charge, θ W is the Weinberg angle, and
Π µν (k) = g 2 8 d (d -2) (4π) d/2 (d -1) Γ (ǫ) Γ (1 -ǫ) 2 Γ (2 -2ǫ) × (-k 2 ) -ǫ k 2 g µν -k µ k ν ( 2
)
is the one-loop correction to the polarization tensor of the W boson. The decay width can then be found as Γ = -2 ImΣ (p / = m). A brief calculation shows that the self-energy has the following form:
Σ (p /) = v(s) p / (1 -γ 5 ) , ( 3
)
3 with s = p 2 and the scalar function v given by:
v(s) = - G 2 F s 2 3(4π) 4 1 ǫ + 21 4 -2γ E -2 ln -s 4π + O(ǫ) , ( 4
)
where γ E = -Γ ′ (1) is the Euler's constant. The absorptive part of this function stems from the logarithm [ln(-s -iε) = ln s -iπ, for s > 0]:
Im v(s) = - G 2 F s 2 384π 3 . ( 5
)
Terefore, the width is Γ = -2m Im v(m 2 ), and the muon lifetime:
τ = 192π 3 /(G 2 F m 5 ) ≃ 2.187 × 10 -6 sec, ( 6
)
This result is of course long-known due to the seminal work of Feynman and Gell-Mann on Fermi theory [9] . It is in a percent agreement with the experimental value[5]:
τ (exp) = (2.19703 ± 0.00004) 10 -6 sec, ( 7
)
The discrepancy is due the neglect of the electron mass and some radiative corrections, c.f. [10] . We now investigate the influence of the e.m. field on the leading contribution given by Eq. ( 6 ).
Let us denote by Σ (x, y; A µ ) the self-energy in the presence of an external e.m. field A µ .
It is obtained by minimal substitution (∂ µ → ∂ µ -ieA µ ) of the derivatives of all charged fields into the self-energy of Fig. 1. Expanding in the e.m. coupling, we obtain:
Σ [x, y; A µ ] = Σ (i∂ / x ) δ 4 (x -y) + dz Λ µ (x, y; z) A µ (z) + O(e 2 A 2 ), ( 8
)
where Σ (i∂ /) is the already computed self-energy in the vacuum, while Λ is the e.m. vertex correction of Fig. 2 , with static W 's.
Denoting p (p ′ ) the 4-momentum of the initial (final) muon and assuming the on-shell
situation (p 2 = p ′ 2 = p • p ′ = m 2 ),
the vertex correction has in the momentum space the following general form:
Λ µ (p ′ , p) = e F γ µ + G (p + p ′ ) µ 2m + F A γ µ γ 5 , ( 9
)
where F , G and F A are complex numbers. Note that eF/2m is the correction to the magnetic moment, and eF + eG is the correction to the electric charge. The Ward-Takahashi (WT) identity:
(p ′ -p) • Λ(p ′ , p) = e [Σ (p /) -Σ (p / ′ )] ( 10
) 4 γ µ µ ν µ W W ν e e e
FIG. 2: Electromagnetic correction to the muon decay.
with the self-energy in Eq. ( 3 ) leads to the following conditions:
F + G = -v(m 2 ) -2m 2 v ′ (m 2 ), F A = v(m 2 ) . ( 11
)
Therefore, the term F A is in fact necessary by the e.m. gauge invariance. The γ 5 terms, in both self-energy and the vertex, are shown to vanish when summing over all the fermions in Standard Model [11] . However, this does not happen for the imaginary part because the heavier fermions do not contribute.
The expression for the graph in Fig. 2 is (in Fermi theory) given by
Λ µ (p ′ , p) = - eg 4 64M 4 W d d k 1 (2π) d d d k 2 (2π) d 2γ α k / 2 k 2 2 γ β (1 -γ 5 ) × Tr 2γ α (1 -γ 5 ) (p / ′ -k / 1 ) γ µ (p / -k / 1 ) γ β (k / 1 -k / 2 ) (k 1 -k 2 ) 2 (p -k 1 ) 2 (p ′ -k 1 ) 2 . ( 12
)
After a lengthy calculation we obtain the following result:
Im F = G 2 F m 4 384π 3 , Im G = G 2 F m 4 96π 3 , Im F A = - G 2 F m 4 384π 3 , ( 13
)
hence satisfying the gauge-invariance conditions Eq. ( 11 ), for Im v given by Eq. ( 5 ).
We would like to emphasize here that, of course, not only the magnetic moment, but also the charge operator receives an imaginary contribution, equal to e Im(F + G). However, through the WT identity, this contribution is completely fixed by the momentum dependence of the self-energy, and therefore is not independent. The same holds for F A . We thus discuss only the effect of the absorptive part of the magnetic moment, here given by Im µ = e Im F/2m = eG 2 F m 3 /768π 3 . The energy of the magnetic moment interacting with the magnetic field is equal to -µB z , with B z being the projection of the field along the muon spin. Then the total energy, in the 5 muon rest-frame, is given by: m -(i/2)Γ -µB z . We thus deduce that the absorptive part yields the following change in the muon width: ∆Γ = 2 Im µ B z = e 2m
G 2 F m 4 192π 3 B z , ( 14
)
while the change in the lifetime is ∆τ = -(∆Γ/Γ) τ , for ∆Γ/Γ ≪ 1.
Given this result, we conclude that the positively-charged muons live shorter (longer) in a uniform magnetic field if their spin is aligned along (against) the field. For the relative change in the width we find:
|∆Γ| Γ = |eB z | 2m 2 < ∼ 3 × 10 -15 B T -1 , ( 15
)
where B is the strength of the field in Tesla. Therefore, in moderate magnetic fields the change in the muon lifetime is tiny, well beyond the present experimental accuracy (which is at the ppm level). We will dwell on this more in the concluding part of the paper, but for now we turn to a more technical issue.
It is interesting to observe that the result of Eq. ( 13 ), can simply be obtained by the minimal substitution into Eq. ( 3 ), rather than into the electron propagator in Eq. ( 1 ). To show this we go to coordinate space and hence write the self-energy as Σ (x, y) = Σ (i∂ / ) δ(x -y). The minimal substitution to the first order in e leads to the following vertex correction:
Λµ (x, y; z) = -δ/δA µ (z) Σ (i∂ / + eA / ) δ(x -y) | A=0 . ( 16
)
Note that in general this is different from the vertex function in Eq. ( 8 ), since in the latter the minimal substitution is performed also in the internal lines. The general form of Eq. (9), of course, applies here as well, but now the scalar functions are completely specified by the self-energy:
F = -v(m 2 ), G = -2m 2 v ′ (m 2 ), FA = v(m 2 ) . ( 17
)
Substituting the explicit form of Im v, we see that this method unambiguously leads to exactly the same result [Eq. ( 13 )] as the full calculation. We emphasize though, that this method cannot always work (see, e.g., Ref. [12] ), as will also be clear from the following examples. Nevertheless, it is worthwhile to investigate this method further, since knowing whether it is applicable a priori can enormously facilitate the calculations.
We consider now the neutron β-decay. Assuming exact V -A interaction (g A = 1) and neglecting the electron mass (but not the proton mass, m p ), the corresponding two-loop self-energy can still be written in the form of Eq. ( 3 ). We introduce δ = (s -m 2 p )/2s and treat it as a small parameter, since in the physical case (where s = m 2 n ), δ ≃ 1.293 × 10 -3 . A simple calculation then yields:
Im v(s) = - G 2 F |V ud | 2 30π 3 s 2 δ 5 , ( 18
)
where V ud is the relevant quark-mixing (CKM) matrix element. We note in passing that this result leads to the lifetime of τ n ≈ 622 sec, to be compared with the experimental value of 886 sec. This 30 % disagreement is largely due to the fact that in reality the axial coupling g A deviates from 1. However, for our order-of-magnitude estimate this discrepancy is unimportant.
What is important is that the derivative of the self-energy is enhanced by one power of δ:
Im v ′ (s) = - (G F |V ud |) 2 12π 3 s δ 4 . ( 19
)
and this opens the possibility for the enhancement of the effect in the lifetime. Namely, the relative change in the neutron width then goes as
|∆Γ n | Γ n ∼ µ N |B z | m n -m p < ∼ 3 × 10 -14 B T -1 , ( 20
)
where µ N ≃ 3.15 × 10 -14 MeV T -1 is the nuclear magneton. A more precise analysis of this effect for the neutron is beyond the scope of this paper. We focus instead on the example of the ∆-resonance, where such an enhancement will be shown to be even more dramatic, at least qualitatively.
The ∆ resonance decays strongly into the pion and nucleon, ∆ → πN, and the corresponding self-energy, to leading order in chiral effective-field theory, yields the following result for the absorptive part[3]:
Im Σ ∆ (p /) = -2 3 πλ 3 C 2 (α p / + m N ) , ( 21
)
where the isospin symmetry is assumed, e.g.,
m p = m n = m N . The constant C = h A m ∆ /8πf π ≃ 1.5,
where h A represents the πN∆ coupling and is fitted to the empiracal width of the ∆, f π ≃ 93 MeV is the pion-decay constant, and m ∆ = 1232 MeV is the ∆ 7 γ ∆ ∆ γ ∆ ∆ π N π N N π (b) (a)
FIG. 3: The leading chiral-loop correction to the magnetic moment of the ∆.
mass. For simplicity we neglect the pion mass (i.e., take the chiral limit). Then, in Eq. (21),
λ = (s -m 2 N )/2s, α = 1 -λ. For s = m ∆ , λ ≈ (m ∆ -m N )/m N ∼ 1
/3 is a small parameter in the chiral effective-field theory with ∆'s (see Ref. [13] for a recent review), and will so be treated here too.
The absorptive part of the magnetic dipole moment of the ∆ arises at this order from graphs in Fig. 3 . These graphs, computed in Ref.[3], in the chiral limit yield the following result (upto λ 4 terms):
Im F (a) = 4πC 2 (λ -3λ 2 + 43 12 λ 3 ) , Im G (a) = 4πC 2 (-λ + 4λ 2 -71 12 λ 3 ) , Im F (b) = 4πC 2 (λ 2 + 1 3 λ 3 ) , ( 22
) Im G (b) = -32 3 πC 2 λ 3 ,
where F and G correspond with the decomposition in Eq. ( 9 ), with the superscript referring to the corresponding graphs in Fig. 3 ; F A is absent in this case, of course.
First of all, we observe that this result satisfies the WT conditions, Eq. ( 11 ), for each of the four charge states of the ∆,
∆ ++ : Im [F (a) + G (a) + F (b) + G (b) ] = -2 Im Σ ′ ∆ , ∆ + : Im [ 1 3 (F (a) + G (a) ) + 2 3 (F (b) + G (b) )] = -Im Σ ′ ∆ , ∆ 0 : Im [-1 3 (F (a) + G (a) ) + 1 3 (F (b) + G (b) )] = 0 , ( 23
) ∆ -: -Im [F (a) + G (a) ] = Im Σ ′ ∆ , where Σ ′ ∆ = ∂/∂p / Σ ∆ (p /)| p /=m ∆ , and hence Im Σ ′ ∆ = 4πC 2 (-λ 2 + 7 3 λ 3
). At the same time, the 'naive' minimal-substitution procedure [Eq. ( 16 )], that happens to work for the muon, fails here miserably. It would predict that the magnetic moment 8 contribution would go with the same power as the self-energy [Eq. (17)], which for the absorptive part means Im µ ∼ Im Σ (m ∆ ) ∼ λ 3 . In reality it goes as λ. E.g., for the ∆ + :
Im µ ∆ + = (e/2m ∆ ) Im[ 1 3 F (a) + 2 3 F (b) ] = 4 3 π µ N C 2 λ + O(λ 2 ). ( 24
)
The fact that the self-energy goes as λ 3 , while Im µ as λ has as a consequence the enhancement of the lifetime change in the magnetic field by two powers of λ.
Quantitatively such enhancements of the lifetime change over the lifetime by the phasespace volume do not make much difference in the above examples. However, it shows that it might be useful to look for manifestations of the lifetime change in the medium where the phase-space volume can be varied.
We have examined her a concept of the 'absorptive magnetic moment' -an intrinsic property of an unstable particle, together with the width or the lifetime. It manifests itself in the change of the particle's lefetime in an external magnetic field, see Eq. (25) below.
We have computed this quantity for the examples of muon, neutron and ∆-resonance to leading order in couplings. In all the three cosidered cases the effect on the lifetime is tiny for normal magnetic fields: in a uniform field of 1 Tesla the change in the lifetime is of order of 10 -13 percent, at most.
In the case of the muon we have computed this effect to the leading order in the electroweak coupling; the change in the lifetime is
∆τ = -2 Im µ B z τ 2 = -96π 3 eB z /(G 2 F m 7 ) , ( 25
) or, numerically, |∆τ | < ∼ 6 × 10 -21
(B/T) sec. A direct measurement of this effect is therefore beyond the present experimental precision. Nevertheless, it is worthwhile to investigate the effect of the magnetic field on the differential decay rates, with the hope that some asymmetries could show a significantly bigger sensitivity.
A notable feature of this effect is that the relative change of the lifetime is inversely proportional to the phase space. It goes as (m n -m p ) -1 in the neutron case, and as (m ∆m N ) -2 in the ∆-resonance case. (The difference in power is apparently because the neutron 9 decays solely into fermions while the ∆ has a boson in the decay product.) One can expect that in the conditions where the phase-space is significantly reduced, e.g. for the neutron in nuclear medium, the effect of the lifetime change may become measurable.
Especially interesting would be to evaluate the manifestations of this effect in neutron star formations. Not only the phase-space of the neutron decay is shrinking, the protons decay too, and all that occurs in magnetic fields as large as 10 1 0 Tesla. Even larger fields can be achieved in atomic or nuclear systems. Finally, it is worthwhile to point out that in lattice QCD studies strong magnetic fields are standardly used to compute the electromagnetic properties of hadrons. Combined with the lattice techniques of extracting the width, the relation between the absorprive part and the lifetime change may allow to compute the former on the lattice for unstable hadrons.
We thank Barry Holstein and Marc Vanderhaeghen for a number of insightful discussions. The work of V.P. is partially supported by the European Community-Research Infrastructure Activity under the FP6 "Structuring the European Research Area" programme (HadronPhysics, contract RII3-CT-2004-506078).
[1] M. Passera, J. Phys. G 31, R75 (2005); J. P. Miller, E. de Rafael and B. L. Roberts, Rept.
Prog. Phys. 70, 795 (2007). [2] L. V. Avdeev and M. Y. Kalmykov, Phys. Lett. B 436, 132 (1998). [3] V. Pascalutsa and M. Vanderhaeghen, Phys. Rev. Lett. 94, 102003 (2005); Phys. Rev. D 77, 014027 (2008). [4] B. R. Holstein, unpublished. [5] W. M. Yao et al. [Particle Data Group],"Review of particle physics,"J. Phys. G 33, 1 (2006). [6] D. Tomono [RIKEN RAL R77 Collaboration], AIP Conf. Proc. 842, 906 (2006) ; K. R. Lynch, AIP Conf. Proc. 870, 333 (2006); J. S. Nico, AIP Conf. Proc. 870, 132 (2006); A. P. Serebrov et al., arXiv:nucl-ex/0702009. [7] P. G. Harris et al., Phys. Rev. Lett. 82, 904 (1999); C. A. Baker et al., Phys. Rev. Lett. 97, 10 131801 (2006). [8] J. L. Basdevant and J. Dalibard, "Quantum Mechanics Solver," (Springer, Berlin, 2005). [9] R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958). [10] T. van Ritbergen and R. G. Stuart, Phys. Rev. Lett. 82, 488 (1999). [11] A. Czarnecki and B. Krause, Nucl. Phys. Proc. Suppl. 51C, 148 (1996). [12] J. H. Koch, V. Pascalutsa and S. Scherer, Phys. Rev. C 65, 045202 (2002). [13] V. Pascalutsa, M. Vanderhaeghen and S. N. Yang, Phys. Rept. 437, 125 (2007). [14] Our conventions are: metric (+, -, -, -), ε 0123 = +1, γ 5 = iγ 0 γ 1 γ 2 γ 3 , γ's stand for Dirac matrices and their totally-antisymmetric products: γ µν = 1 2 [γ µ , γ ν ], γ µνα = 1 2 {γ µν , γ α }, γ µναβ = 1 2 [γ µνα , γ β ]. | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The electromagnetic (e.m.) moments of a particle are among the few fundamental quantities which describe the particle properties and as such have thoroughly been studied. The most renowned examples are the magnetic moments of the electron and the muon which have been measured to unprecendented accuracy and yielded a number of physical insights, see [1] for recent reviews. What is far lesser known is that the e.m. moments of unstable particles are complex numbers in general [2, 3] . Their imaginary part reflects, of course, the unstable nature of the particle, however, the precise interpretation has been missing. In this paper we work out the relation, suggested first by Holstein [4] , which should exist between the imaginary part of the magnetic moment and the effect of an external magnetic field on particle's lifetime.\n\nThe argument for such a relation is very simple. The (self-)energy of the particle with a lifetime τ has an absorptive part, which has an interpretation of the width Γ = 1/τ . The particle's magnetic moment µ in the presence of magnetic field B induces the change in the energy: -µ • B. The latter contribution can then also change the width, provided the magnetic moment has an absorptive part (Im µ = 0).\n\nThe decay properties of unstable particles, such as muon or neutron are extremely well studied and are widely used for the precise determination of the Standard Model parameters [5, 6] . There are also a plethora of studies of how these particles behave in e.m. fields. A well-known example is the search for the neutron's electric dipole moment [7] .\n\nIn view of these studies it is compelling to investigate how the decay properties of unstable particles may be affected by e.m. fields.\n\nThe lifetime of unstable quantum-mechanical systems is known to be affected by an e.m. field. Positronium provides a textbook example[8], where the effect arises due to the admixture of para-(S = 0) and ortho-(S = 1) positronium states with orbital momentum l = 0 by the magnetic field interacting with the magnetic moments of the constituents. As the result, already in the field of B = 0.2 Tesla, the lifetime of ortho-positronium decreases by almost a factor of 2.\n\nIt is far from obvious how the same kind of an effect can arise for an elementary unstable particle, e.g., the muon. The above-mentioned relation between the imaginary part of the magnetic moment and the lifetime change may, therefore, provide us with both an interpre-2 µ ν µ µ W W ν e e FIG. 1: The muon self-energy contributing to its decay width.\n\ntation for the imaginary part of the magnetic moment and the means to compute the effect of the lifetime change.\n\nIn the following we examine in detail the case of the muon, compute the leading contribution to Im µ and the corresponding effect on the lifetime. Then we will briefly discuss the cases of the neutron and of the ∆-resonance.\n\nII. MUON DECAY (µ → e ν e ν µ )\n\nThe leading contribution to the muon decay width arises at two-loop level, see Fig. 1. For our purposes, the W propagators in this graph can safely be assumed to be static -Fermi theory. We also neglect the mass of the electron in the loops, since it leads to an underpercent correction of O(m e /m); here and in what follows, m is the muon mass. The graphs with other Standard Model fermions (e.g., quarks) in the loops need not to be considered here, because they cannot give any contribution to the muon width.\n\nUsing dimensional regularization, we compute this graph in d = 4 -2ǫ dimensions (in\n\nthe limit ǫ → 0 + ),[14] Σ (p /) = g 2 8M 4 W i d d k ( 2π\n\n) d 2γ µ (1 -γ 5 ) (p / -k /) γ ν (p -k) 2 + iε Π µν (k). ( 1\n\n)\n\nwhere M W is the W -boson mass, g = |e|/ sin θ W is the electroweak coupling related to the Fermi constant by G F / √ 2 = g 2 /8M 2 W , e is the charge, θ W is the Weinberg angle, and\n\nΠ µν (k) = g 2 8 d (d -2) (4π) d/2 (d -1) Γ (ǫ) Γ (1 -ǫ) 2 Γ (2 -2ǫ) × (-k 2 ) -ǫ k 2 g µν -k µ k ν ( 2\n\n)\n\nis the one-loop correction to the polarization tensor of the W boson. The decay width can then be found as Γ = -2 ImΣ (p / = m). A brief calculation shows that the self-energy has the following form:\n\nΣ (p /) = v(s) p / (1 -γ 5 ) , ( 3\n\n)\n\n3 with s = p 2 and the scalar function v given by:\n\nv(s) = - G 2 F s 2 3(4π) 4 1 ǫ + 21 4 -2γ E -2 ln -s 4π + O(ǫ) , ( 4\n\n)\n\nwhere γ E = -Γ ′ (1) is the Euler's constant. The absorptive part of this function stems from the logarithm [ln(-s -iε) = ln s -iπ, for s > 0]:\n\nIm v(s) = - G 2 F s 2 384π 3 . ( 5\n\n)\n\nTerefore, the width is Γ = -2m Im v(m 2 ), and the muon lifetime:\n\nτ = 192π 3 /(G 2 F m 5 ) ≃ 2.187 × 10 -6 sec, ( 6\n\n)\n\nThis result is of course long-known due to the seminal work of Feynman and Gell-Mann on Fermi theory [9] . It is in a percent agreement with the experimental value[5]:\n\nτ (exp) = (2.19703 ± 0.00004) 10 -6 sec, ( 7\n\n)\n\nThe discrepancy is due the neglect of the electron mass and some radiative corrections, c.f. [10] . We now investigate the influence of the e.m. field on the leading contribution given by Eq. ( 6 ).\n\nLet us denote by Σ (x, y; A µ ) the self-energy in the presence of an external e.m. field A µ .\n\nIt is obtained by minimal substitution (∂ µ → ∂ µ -ieA µ ) of the derivatives of all charged fields into the self-energy of Fig. 1. Expanding in the e.m. coupling, we obtain:\n\nΣ [x, y; A µ ] = Σ (i∂ / x ) δ 4 (x -y) + dz Λ µ (x, y; z) A µ (z) + O(e 2 A 2 ), ( 8\n\n)\n\nwhere Σ (i∂ /) is the already computed self-energy in the vacuum, while Λ is the e.m. vertex correction of Fig. 2 , with static W 's.\n\nDenoting p (p ′ ) the 4-momentum of the initial (final) muon and assuming the on-shell\n\nsituation (p 2 = p ′ 2 = p • p ′ = m 2 ),\n\nthe vertex correction has in the momentum space the following general form:\n\nΛ µ (p ′ , p) = e F γ µ + G (p + p ′ ) µ 2m + F A γ µ γ 5 , ( 9\n\n)\n\nwhere F , G and F A are complex numbers. Note that eF/2m is the correction to the magnetic moment, and eF + eG is the correction to the electric charge. The Ward-Takahashi (WT) identity:\n\n(p ′ -p) • Λ(p ′ , p) = e [Σ (p /) -Σ (p / ′ )] ( 10\n\n) 4 γ µ µ ν µ W W ν e e e\n\nFIG. 2: Electromagnetic correction to the muon decay.\n\nwith the self-energy in Eq. ( 3 ) leads to the following conditions:\n\nF + G = -v(m 2 ) -2m 2 v ′ (m 2 ), F A = v(m 2 ) . ( 11\n\n)\n\nTherefore, the term F A is in fact necessary by the e.m. gauge invariance. The γ 5 terms, in both self-energy and the vertex, are shown to vanish when summing over all the fermions in Standard Model [11] . However, this does not happen for the imaginary part because the heavier fermions do not contribute.\n\nThe expression for the graph in Fig. 2 is (in Fermi theory) given by\n\nΛ µ (p ′ , p) = - eg 4 64M 4 W d d k 1 (2π) d d d k 2 (2π) d 2γ α k / 2 k 2 2 γ β (1 -γ 5 ) × Tr 2γ α (1 -γ 5 ) (p / ′ -k / 1 ) γ µ (p / -k / 1 ) γ β (k / 1 -k / 2 ) (k 1 -k 2 ) 2 (p -k 1 ) 2 (p ′ -k 1 ) 2 . ( 12\n\n)\n\nAfter a lengthy calculation we obtain the following result:\n\nIm F = G 2 F m 4 384π 3 , Im G = G 2 F m 4 96π 3 , Im F A = - G 2 F m 4 384π 3 , ( 13\n\n)\n\nhence satisfying the gauge-invariance conditions Eq. ( 11 ), for Im v given by Eq. ( 5 ).\n\nWe would like to emphasize here that, of course, not only the magnetic moment, but also the charge operator receives an imaginary contribution, equal to e Im(F + G). However, through the WT identity, this contribution is completely fixed by the momentum dependence of the self-energy, and therefore is not independent. The same holds for F A . We thus discuss only the effect of the absorptive part of the magnetic moment, here given by Im µ = e Im F/2m = eG 2 F m 3 /768π 3 . The energy of the magnetic moment interacting with the magnetic field is equal to -µB z , with B z being the projection of the field along the muon spin. Then the total energy, in the 5 muon rest-frame, is given by: m -(i/2)Γ -µB z . We thus deduce that the absorptive part yields the following change in the muon width: ∆Γ = 2 Im µ B z = e 2m\n\nG 2 F m 4 192π 3 B z , ( 14\n\n)\n\nwhile the change in the lifetime is ∆τ = -(∆Γ/Γ) τ , for ∆Γ/Γ ≪ 1.\n\nGiven this result, we conclude that the positively-charged muons live shorter (longer) in a uniform magnetic field if their spin is aligned along (against) the field. For the relative change in the width we find:\n\n|∆Γ| Γ = |eB z | 2m 2 < ∼ 3 × 10 -15 B T -1 , ( 15\n\n)\n\nwhere B is the strength of the field in Tesla. Therefore, in moderate magnetic fields the change in the muon lifetime is tiny, well beyond the present experimental accuracy (which is at the ppm level). We will dwell on this more in the concluding part of the paper, but for now we turn to a more technical issue.\n\nIt is interesting to observe that the result of Eq. ( 13 ), can simply be obtained by the minimal substitution into Eq. ( 3 ), rather than into the electron propagator in Eq. ( 1 ). To show this we go to coordinate space and hence write the self-energy as Σ (x, y) = Σ (i∂ / ) δ(x -y). The minimal substitution to the first order in e leads to the following vertex correction:\n\nΛµ (x, y; z) = -δ/δA µ (z) Σ (i∂ / + eA / ) δ(x -y) | A=0 . ( 16\n\n)\n\nNote that in general this is different from the vertex function in Eq. ( 8 ), since in the latter the minimal substitution is performed also in the internal lines. The general form of Eq. (9), of course, applies here as well, but now the scalar functions are completely specified by the self-energy:\n\nF = -v(m 2 ), G = -2m 2 v ′ (m 2 ), FA = v(m 2 ) . ( 17\n\n)\n\nSubstituting the explicit form of Im v, we see that this method unambiguously leads to exactly the same result [Eq. ( 13 )] as the full calculation. We emphasize though, that this method cannot always work (see, e.g., Ref. [12] ), as will also be clear from the following examples. Nevertheless, it is worthwhile to investigate this method further, since knowing whether it is applicable a priori can enormously facilitate the calculations."
},
{
"section_type": "OTHER",
"section_title": "III. NEUTRON DECAY AND THE ∆-RESONANCE",
"text": "We consider now the neutron β-decay. Assuming exact V -A interaction (g A = 1) and neglecting the electron mass (but not the proton mass, m p ), the corresponding two-loop self-energy can still be written in the form of Eq. ( 3 ). We introduce δ = (s -m 2 p )/2s and treat it as a small parameter, since in the physical case (where s = m 2 n ), δ ≃ 1.293 × 10 -3 . A simple calculation then yields:\n\nIm v(s) = - G 2 F |V ud | 2 30π 3 s 2 δ 5 , ( 18\n\n)\n\nwhere V ud is the relevant quark-mixing (CKM) matrix element. We note in passing that this result leads to the lifetime of τ n ≈ 622 sec, to be compared with the experimental value of 886 sec. This 30 % disagreement is largely due to the fact that in reality the axial coupling g A deviates from 1. However, for our order-of-magnitude estimate this discrepancy is unimportant.\n\nWhat is important is that the derivative of the self-energy is enhanced by one power of δ:\n\nIm v ′ (s) = - (G F |V ud |) 2 12π 3 s δ 4 . ( 19\n\n)\n\nand this opens the possibility for the enhancement of the effect in the lifetime. Namely, the relative change in the neutron width then goes as\n\n|∆Γ n | Γ n ∼ µ N |B z | m n -m p < ∼ 3 × 10 -14 B T -1 , ( 20\n\n)\n\nwhere µ N ≃ 3.15 × 10 -14 MeV T -1 is the nuclear magneton. A more precise analysis of this effect for the neutron is beyond the scope of this paper. We focus instead on the example of the ∆-resonance, where such an enhancement will be shown to be even more dramatic, at least qualitatively.\n\nThe ∆ resonance decays strongly into the pion and nucleon, ∆ → πN, and the corresponding self-energy, to leading order in chiral effective-field theory, yields the following result for the absorptive part[3]:\n\nIm Σ ∆ (p /) = -2 3 πλ 3 C 2 (α p / + m N ) , ( 21\n\n)\n\nwhere the isospin symmetry is assumed, e.g.,\n\nm p = m n = m N . The constant C = h A m ∆ /8πf π ≃ 1.5,\n\nwhere h A represents the πN∆ coupling and is fitted to the empiracal width of the ∆, f π ≃ 93 MeV is the pion-decay constant, and m ∆ = 1232 MeV is the ∆ 7 γ ∆ ∆ γ ∆ ∆ π N π N N π (b) (a)\n\nFIG. 3: The leading chiral-loop correction to the magnetic moment of the ∆.\n\nmass. For simplicity we neglect the pion mass (i.e., take the chiral limit). Then, in Eq. (21),\n\nλ = (s -m 2 N )/2s, α = 1 -λ. For s = m ∆ , λ ≈ (m ∆ -m N )/m N ∼ 1\n\n/3 is a small parameter in the chiral effective-field theory with ∆'s (see Ref. [13] for a recent review), and will so be treated here too.\n\nThe absorptive part of the magnetic dipole moment of the ∆ arises at this order from graphs in Fig. 3 . These graphs, computed in Ref.[3], in the chiral limit yield the following result (upto λ 4 terms):\n\nIm F (a) = 4πC 2 (λ -3λ 2 + 43 12 λ 3 ) , Im G (a) = 4πC 2 (-λ + 4λ 2 -71 12 λ 3 ) , Im F (b) = 4πC 2 (λ 2 + 1 3 λ 3 ) , ( 22\n\n) Im G (b) = -32 3 πC 2 λ 3 ,\n\nwhere F and G correspond with the decomposition in Eq. ( 9 ), with the superscript referring to the corresponding graphs in Fig. 3 ; F A is absent in this case, of course.\n\nFirst of all, we observe that this result satisfies the WT conditions, Eq. ( 11 ), for each of the four charge states of the ∆,\n\n∆ ++ : Im [F (a) + G (a) + F (b) + G (b) ] = -2 Im Σ ′ ∆ , ∆ + : Im [ 1 3 (F (a) + G (a) ) + 2 3 (F (b) + G (b) )] = -Im Σ ′ ∆ , ∆ 0 : Im [-1 3 (F (a) + G (a) ) + 1 3 (F (b) + G (b) )] = 0 , ( 23\n\n) ∆ -: -Im [F (a) + G (a) ] = Im Σ ′ ∆ , where Σ ′ ∆ = ∂/∂p / Σ ∆ (p /)| p /=m ∆ , and hence Im Σ ′ ∆ = 4πC 2 (-λ 2 + 7 3 λ 3\n\n). At the same time, the 'naive' minimal-substitution procedure [Eq. ( 16 )], that happens to work for the muon, fails here miserably. It would predict that the magnetic moment 8 contribution would go with the same power as the self-energy [Eq. (17)], which for the absorptive part means Im µ ∼ Im Σ (m ∆ ) ∼ λ 3 . In reality it goes as λ. E.g., for the ∆ + :\n\nIm µ ∆ + = (e/2m ∆ ) Im[ 1 3 F (a) + 2 3 F (b) ] = 4 3 π µ N C 2 λ + O(λ 2 ). ( 24\n\n)\n\nThe fact that the self-energy goes as λ 3 , while Im µ as λ has as a consequence the enhancement of the lifetime change in the magnetic field by two powers of λ.\n\nQuantitatively such enhancements of the lifetime change over the lifetime by the phasespace volume do not make much difference in the above examples. However, it shows that it might be useful to look for manifestations of the lifetime change in the medium where the phase-space volume can be varied."
},
{
"section_type": "CONCLUSION",
"section_title": "IV. CONCLUSIONS AND OUTLOOK",
"text": "We have examined her a concept of the 'absorptive magnetic moment' -an intrinsic property of an unstable particle, together with the width or the lifetime. It manifests itself in the change of the particle's lefetime in an external magnetic field, see Eq. (25) below.\n\nWe have computed this quantity for the examples of muon, neutron and ∆-resonance to leading order in couplings. In all the three cosidered cases the effect on the lifetime is tiny for normal magnetic fields: in a uniform field of 1 Tesla the change in the lifetime is of order of 10 -13 percent, at most.\n\nIn the case of the muon we have computed this effect to the leading order in the electroweak coupling; the change in the lifetime is\n\n∆τ = -2 Im µ B z τ 2 = -96π 3 eB z /(G 2 F m 7 ) , ( 25\n\n) or, numerically, |∆τ | < ∼ 6 × 10 -21\n\n(B/T) sec. A direct measurement of this effect is therefore beyond the present experimental precision. Nevertheless, it is worthwhile to investigate the effect of the magnetic field on the differential decay rates, with the hope that some asymmetries could show a significantly bigger sensitivity.\n\nA notable feature of this effect is that the relative change of the lifetime is inversely proportional to the phase space. It goes as (m n -m p ) -1 in the neutron case, and as (m ∆m N ) -2 in the ∆-resonance case. (The difference in power is apparently because the neutron 9 decays solely into fermions while the ∆ has a boson in the decay product.) One can expect that in the conditions where the phase-space is significantly reduced, e.g. for the neutron in nuclear medium, the effect of the lifetime change may become measurable.\n\nEspecially interesting would be to evaluate the manifestations of this effect in neutron star formations. Not only the phase-space of the neutron decay is shrinking, the protons decay too, and all that occurs in magnetic fields as large as 10 1 0 Tesla. Even larger fields can be achieved in atomic or nuclear systems. Finally, it is worthwhile to point out that in lattice QCD studies strong magnetic fields are standardly used to compute the electromagnetic properties of hadrons. Combined with the lattice techniques of extracting the width, the relation between the absorprive part and the lifetime change may allow to compute the former on the lattice for unstable hadrons."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "We thank Barry Holstein and Marc Vanderhaeghen for a number of insightful discussions. The work of V.P. is partially supported by the European Community-Research Infrastructure Activity under the FP6 \"Structuring the European Research Area\" programme (HadronPhysics, contract RII3-CT-2004-506078).\n\n[1] M. Passera, J. Phys. G 31, R75 (2005); J. P. Miller, E. de Rafael and B. L. Roberts, Rept.\n\nProg. Phys. 70, 795 (2007). [2] L. V. Avdeev and M. Y. Kalmykov, Phys. Lett. B 436, 132 (1998). [3] V. Pascalutsa and M. Vanderhaeghen, Phys. Rev. Lett. 94, 102003 (2005); Phys. Rev. D 77, 014027 (2008). [4] B. R. Holstein, unpublished. [5] W. M. Yao et al. [Particle Data Group],\"Review of particle physics,\"J. Phys. G 33, 1 (2006). [6] D. Tomono [RIKEN RAL R77 Collaboration], AIP Conf. Proc. 842, 906 (2006) ; K. R. Lynch, AIP Conf. Proc. 870, 333 (2006); J. S. Nico, AIP Conf. Proc. 870, 132 (2006); A. P. Serebrov et al., arXiv:nucl-ex/0702009. [7] P. G. Harris et al., Phys. Rev. Lett. 82, 904 (1999); C. A. Baker et al., Phys. Rev. Lett. 97, 10 131801 (2006). [8] J. L. Basdevant and J. Dalibard, \"Quantum Mechanics Solver,\" (Springer, Berlin, 2005). [9] R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958). [10] T. van Ritbergen and R. G. Stuart, Phys. Rev. Lett. 82, 488 (1999). [11] A. Czarnecki and B. Krause, Nucl. Phys. Proc. Suppl. 51C, 148 (1996). [12] J. H. Koch, V. Pascalutsa and S. Scherer, Phys. Rev. C 65, 045202 (2002). [13] V. Pascalutsa, M. Vanderhaeghen and S. N. Yang, Phys. Rept. 437, 125 (2007). [14] Our conventions are: metric (+, -, -, -), ε 0123 = +1, γ 5 = iγ 0 γ 1 γ 2 γ 3 , γ's stand for Dirac matrices and their totally-antisymmetric products: γ µν = 1 2 [γ µ , γ ν ], γ µνα = 1 2 {γ µν , γ α }, γ µναβ = 1 2 [γ µνα , γ β ]."
}
] | |
arxiv:0704.0386 | 0704.0386 | 1 | 10.1103/PhysRevLett.99.150401 | 4cdf211d453d201045c8a8892026ddb0e2d51231bc554fbbf685f8b762731c33 | Quantum non-local effects with Bose-Einstein condensates | We study theoretically the properties of two Bose-Einstein condensates in
different spin states, represented by a double Fock state. Individual
measurements of the spins of the particles are performed in transverse
directions, giving access to the relative phase of the condensates. Initially,
this phase is completely undefined, and the first measurements provide random
results. But a fixed value of this phase rapidly emerges under the effect of
the successive quantum measurements, giving rise to a quasi-classical situation
where all spins have parallel transverse orientations. If the number of
measurements reaches its maximum (the number of particles), quantum effects
show up again, giving rise to violations of Bell type inequalities. The
violation of BCHSH inequalities with an arbitrarily large number of spins may
be comparable (or even equal) to that obtained with two spins. | [
"Franck Lalo\\\"e (LKB - Lhomond)",
"William J. Mullin (UMASS)"
] | [
"quant-ph",
"cond-mat.other"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | We study theoretically the properties of two Bose-Einstein condensates in different spin states, represented by a double Fock state. Individual measurements of the spins of the particles are performed in transverse directions, giving access to the relative phase of the condensates. Initially, this phase is completely undefined, and the first measurements provide random results. But a fixed value of this phase rapidly emerges under the effect of the successive quantum measurements, giving rise to a quasi-classical situation where all spins have parallel transverse orientations. If the number of measurements reaches its maximum (the number of particles), quantum effects show up again, giving rise to violations of Bell type inequalities. The violation of BCHSH inequalities with an arbitrarily large number of spins may be comparable (or even equal) to that obtained with two spins.
The notion of non-locality in quantum mechanics (QM) takes its roots in a chain of two theorems, the EPR (Einstein Podolsky Rosen) theorem [1] and its logical continuation, the Bell theorem. The EPR theorem starts from three assumptions (Einstein realism, locality, the predictions of quantum mechanics concerning some perfect correlations are correct) and proves that QM is incomplete: additional quantities, traditionally named λ, are necessary to complete the description of physical reality. The Bell theorem [2, 3] then proves that, if λ exists, the predictions of QM concerning other imperfect correlations cannot always be correct. The ensemble of the three assumptions: Einstein realism, locality, all predictions of QM are correct, is therefore self-contradictory; if Einstein realism is valid, QM is non-local. Bohr [4] rejected Einstein realism because, in his view, the notion of physical reality could not correctly be applied to microscopic quantum systems, defined independently of the measurement apparatuses. Indeed, since EPR consider a system of two microscopic particles, which can be "seen" only with the help of measurement apparatuses, the notion of their independent physical reality is open to discussion.
Nevertheless, it has been pointed out recently [5, 6] that the EPR theorem also applies to macroscopic systems, namely Bose-Einstein (BE) condensates in two different internal states. The λ introduced by EPR then corresponds to the relative phase of the condensates, i.e. to macroscopic transverse spin orientations, physical quantities at a human scale; it then seems more difficult to deny the existence of their reality, even in the absence of measurement devices. This gives even more strength to the EPR argument and weakens Bohr's refutation. It is then natural to ask whether the Bell theorem can be transposed to this stronger case.
The purpose of this article is to show that it can. We consider an ensemble of N + particles in a state defined by an orbital state u and a spin state +, and N -particles in the same state with spin orientation -. The whole system is described quantum mechanically by a double Fock state, that is, a "double BE condensate":
| Φ > = (a u,+ ) † N+ (a u,-) † N- | vac. > (1)
where a u,+ and a u,-are the destruction operators associated with the two populated single-particle states and |vac. > is the vacuum state. We introduce a sequence of transverse spin measurements that leads to quantum predictions violating the so called BCHSH [7, 8] Bell inequality. This is reminiscent of the work of Mermin [9] , who finds exponential violations of local realist inequalities with N -particle spin states that are maximally entangled. By contrast, here we consider the simplest way in which many bosons can be put in two different internal levels, with a N -particle state containing only the minimal possible correlations, those due to statistics. We find violations of inequalities that are the same order of magnitude as with the usual singlet spin state and may actually saturate the Cirel'son bound [10] .
Double Fock states are experimentally more accessible and much less sensitive to dissipation and decoherence than maximally entangled states [11] . Considering a system in a double Fock state, we assume that a series of rapid spin measurements can be performed and described by the usual QM postulate of measurement, without worrying about decoherence between the measurements, thermal effects, etc.
The operators associated with the local density of particles and spins can be expressed as functions of the two fields operators Ψ ± (r) associated with the two internal states ± as:
n(r) = Ψ † + (r)Ψ + (r) + Ψ † -(r)Ψ -(r), σ z (r) = Ψ † + (r)Ψ + (r)-Ψ † -(r)Ψ -(r)
, while the spin component in the direction of plane xOy making an angle ϕ with Ox is: σ ϕ (r) = e -iϕ Ψ † + (r)Ψ -(r) + e iϕ Ψ † -(r)Ψ + (r). Consider now a measurement of this component performed at point r and providing result η = ±1. The corresponding projector is:
P η=±1 (r, ϕ) = 1 2 [n(r) + η σ ϕ (r)] (2)
and, because the measurements are supposed to be performed at different points (ensuring that these projectors all commute) the probability P(η 1 , η 2 , ...η N ) for a series of results η i ± 1 for spin measurements at points r i along directions ϕ i can be written as:
< Φ | P η1 (r 1 , ϕ 1 ) × P η2 (r 2 , ϕ 2 ) × ....P ηN (r N , ϕ N ) | Φ > (3
) We now substitute the expression of σ ϕ (r) into ( 2 ) and ( 3 ), exactly as in the calculation of ref. [5] , but with one difference: here we do not assume that the number of measurements is much smaller than N ± , but equal to its maximum value N = N + + N -. In the product of projectors appearing in (3) , because all r's are different, commutation allows us to push all the field operators to the right, all their conjugates to the left; one can then easily see that each Ψ ± (r) acting on | Φ > can be replaced by u(r) × a u,± , and similarly for the Hermitian conjugates. With our initial state, a non-zero result can be obtained only if exactly N + operators a u,+ appear in the term considered, and N -operators a u,-; a similar condition exists for the Hermitian conjugate operators. To express these conditions, we introduce two additional variables. As in [5] , the first variable λ ensures an equal number of creation and destruction operators in the internal states ± through the mathematical identity:
π -π dλ 2π e inλ = δ n,0 (4)
The second variable Λ expresses in a similar way that the difference between the number of destruction operators in states + and -is exactly N + -N -, through the integral:
π -π dΛ 2π e -inΛ e i(N+-N-)Λ = δ n,N+-N- (5)
The introduction of the corresponding exponentials into the product of projectors ( 2 ) in ( 3 ) provides the expression (c.c. means complex conjugate):
N j=1 |u(r j )| 2 1 2
e iΛ + e -iΛ + η j e i(λ-ϕj +Λ) + c.c.
(6) where, after integration over λ and Λ, the only surviving terms are all associated with the same matrix element in state | Φ > (that of the product of N + operators a † u,+ and N -operators a † u,-followed by the same sequence of destruction operators, providing the constant result N + !N -!). We can thus write the probability as:
P(η 1 , η 2 , ...η N ) ∼ π -π dλ 2π +π -π dΛ 2π e i(N+-N-)Λ N j=1 |u(r j )| 2 1 2
e iΛ + e -iΛ + η j e i(λ-ϕj +Λ) + c.c. (7) or, by using Λ parity and changing one integration variable (λ ′ = λ + Λ), as:
P(η 1 , η 2 , ...η N ) = 1 2 N C N +π -π dΛ 2π cos [(N + -N -)Λ] +π -π dλ ′ 2π N j=1 {cos (Λ) + η j cos (λ ′ -ϕ j )} (8)
The normalization coefficient C N is readily obtained by writing that the sum of probabilities of all possible sequences of η's is 1 (this step requires discussion; we come back to this point at the end of this article):
C N = +π -π dΛ 2π cos [(N + -N -)Λ] [cos (Λ)] N (9)
Finally, we generalize (8) to any number of measurements M < N . A sequence of M measurements can always be completed by additional N -M measurements, leading to probability (8) . We can therefore take the sum of ( 8 ) over all possible results of the additional N -M measurements to obtain the probability for any M as:
P(η 1 , η 2 , ...η M ) = 1 2 M C N +π -π dΛ 2π cos [(N + -N -)Λ] [cos Λ] N -M +π -π dλ ′ 2π M j=1 {cos (Λ) + η j cos (λ ′ -ϕ j )} (10)
The Λ integral can be replaced by twice the integral between ±π/2 (a change of Λ into π -Λ multiplies the function by (-1) N+-N-+N -M+M = 1). If M ≪ N , the large power of cos Λ in the first integral concentrates its contribution around Λ ≃ 0, so that a good approximation is Λ = 0. We then recover the results of refs [5, 6] , with a single integral over λ defining the relative phase of the condensates (Anderson phase), initially completely undetermined, so that the first spin measurement provides a completely random result. But the phase rapidly emerges under the effect of a few measurements, and remains constant [12, 13, 14] ; it takes a different value for each realization of the experiment, as if it was revealing the pre-existing value of a classical quantity. Moreover, when cos Λ is replaced by 1, each factor of the product over j remains positive (or zero), leading to a result similar to that of stochastic local realist theories; the Bell inequalities can then be obtained. However, when N -M is small or even vanishes, cos Λ can take values that are smaller than 1 and the factors may become negative, opening the possibility of violations. In a sense, the additional variable Λ controls the amount of quantum effects in the series of measurements. We now discuss when these standard QM predictions violate Bell inequalities. We need the value of the quantum average of the product of results, that is the sum of η 1 , η 2 , ...η M × P(η 1 , η 2 , ...η M ) over all possible values of the η's, which according to (10) is given by:
E(ϕ 1 , ϕ 2 , ..ϕ M ) = 1 C N +π -π dΛ 2π cos [(N + -N -)Λ] [cos Λ] N -M +π -π dλ ′ 2π M j=1 cos λ ′ -ϕ j (11)
Consider a thought experiment where two condensates in different spin states (two eigenstates of the Oz spin component) overlap in two remote regions of space A and B , with two experimentalists Alice and Bob; they measure the spins of the particles in arbitrary transverse directions (perpendicular to Oz) at points of space where the orbital wave functions of the two condensates are equal. All measurements performed by Alice are made along a single direction ϕ a , which plays here the usual role of the "setting" a, while all those performed by Bob are made along angle ϕ b . We assume that Alice retains just the product A of all her measurements, while Bob retains only the product B of his; A and B are both ±1.
We now assume two possible orientations ϕ a and ϕ ′ a for Alice, two possible orientations ϕ b and ϕ ′ b for Bob. Within deterministic local realism, for each realization of the experiment, it is possible to define two numbers A, A ′ , both equal to ±1, associated with the two possible products of results η that Alice will observe, depending of her choice of orientation; the same is obviously true for Bob, introducing B and B ′ . Within stochastic local realism [8, 15] , A and A ′ are the difference of probabilities associated with Alice observing +1 or -1, i.e. numbers that have values between +1 and -1. In both cases, the following inequalities (BCHSH) are obeyed:
-2 ≤ AB + AB ′ ± (A ′ B -A ′ B ′ ) ≤ 2 (12)
In standard quantum mechanics, of course, "unperformed experiments have no results" [16] , and several of the numbers appearing in (12) are undefined; only two of them can be defined after the experiment has been performed with a given choice of the orientations. Thus, while one can calculate from (11) the quantum average value < Q > of the sum of products of results appearing in (12) , there is no special reason why < Q > should be limited between +2 and -2. Situations where the limit is exceeded are called "quantum non-local".
We have seen that the most interesting situations occur when the cosines do not introduce their peaking effect around Λ = 0, i.e. when N + = N -and M has its maximum value N . Then, for a given N , the only remaining choice is how the number of measurements is shared between N a measurements for Alice and N b for Bob.
Assume first that N a = 1 (Alice makes one measurement) and therefore N b = N -1 (Bob makes all the others). Since we assume that N + = N -and M = N , the Λ integral in (11) disappears, and the λ integral contains only the product of cos (λ ′ -ϕ a ) by the (N -1)th power of cos (λ ′ -ϕ b ), which is straightforward and provides cos (ϕ a -ϕ b ) times the normalization integral C N . The quantum average associated with the product AB is thus merely equal to cos (ϕ a -ϕ b ), exactly as the usual case of two spins in a singlet state. Then it is well-known that, when the angles form a "fan" [17] spaced by χ = π/4, a strong violation of (12) occurs, by a factor √ 2, saturating the Cirel'son bound [10] . A similar calculation can be performed when Alice makes 2 measurements and Bob N -2, and shows that the quantum average is now equal to
1 2 1 + 1 N -1 + (1 -1 N -1 ) cos 2 (ϕ a -ϕ b ) , no longer independent of N. If N = 4, the maximum of < Q > is 2.28 < 2 √ 2
, and rises to 2.41 as N → ∞. An expression for the generalization of the quantum average to any number P and N -P of measurements by Alice and Bob, respectively, is (with χ = ϕ a -ϕ b ):
E(χ) = N 2 ! N ! {P/2} k=0 P !(N -2k)! k!(P -2k)!( N 2 -k)! sin 2k χ cos P -2k χ (13)
where {P/2} is the integer part of P/2. The maximum of < Q > can then be found using a numerical Mathematica routine. Results are shown for several values of P in Fig. 1 . The angles maximizing the quantum Bell quantity always occur in the fan shape, although the basic angle χ changes with P and N. All of the curves where P is held fixed have a finite < Q > limit with increasing N , and the optimum values of the angles approach constants. For the curve P = N/2, the limit is 2.32 when N → ∞, and the fan opening decreases as 1/ √ N .
2.8 2.6 2.4 2.2 2.0 <Q> Max 100 80 60 40 20 N P 1 2 4 6 N/2
FIG. 1: The maximum of the quantum average < Q > for Alice doing P experiments and Bob N -P , as a function of the total number of particles N . The usual Bell situation is obtained for N = 2, P = 1. Local realist theories predict an upper limit of 2; large violations of this limit are obtained, even with macroscopic systems (N → ∞). If P = 1, the violation saturates the Cirel'son limit for any N .
We can also study cases where the number of measurements is M < N : if Bob makes all his measurements, but ignores one or two of them (independently of the order of the measurements), when he correlates his results with Alice, the BCHSH inequality is never violated. All measurements have to be taken into account to obtain violations. Furthermore, if the number of particles in the two condensates are not equal, no violation occurs either. Finally, it is possible to consider cases where we generalize the angles considered: experimenter Carole makes measurements at ϕ c and ϕ ′ c , and David at ϕ d and ϕ ′ d . We then find that a maximization of < Q > reduces to the cases already studied, where the new angles collapse to the previous angles ϕ a , • • • , ϕ ′ b . For the sake of simplicity, we have not yet discussed some important issues that underlie our calculations. One is related to the so called "sample bias loophole" (or "detection/efficiency loophole") and to the normalization condition (9) , which assumes that one spin is detected at each point of measurement. A more detailed study (see second ref. [5] ) should include the integration of each r in a small detection volume and the possibility that no particle is detected in it. This is a well-known difficulty, which already appears in the usual two-photon experiments [8] , where most photons are missed by the detectors. If this loophole still raises a real experimental challenge, the difficulty can be resolved in theory by assuming the presence of additional spin-independent detectors [2, 8] , which ensure the detection of one particle in each detector and create appropriate initial conditions (see for instance [18] for a description of an experiment with veto detectors). We postpone this discussion to another article [19] . A second issue deals with the definition of the local realist quantities A, B, etc. For two condensates, we have a slightly different situation than in the usual EPR situation: the local realist reasoning leads to the existence of a well-defined phase λ between the con- [5] , not necessarily to deterministic properties of the individual particles. Fortunately, Bell inequalities can also be derived within stochastic local realist theories [3, 8] (see also for instance [9] or appendix I of [15] ), and this difference is not a problem [19] .
In conclusion, strong violations of local realism may occur for large quantum systems, even if the state is a simple double Fock state with equal populations; within present experimental techniques, this seems reachable with N ∼ 10 or 20. We have assumed that the measured quantity is the product of many microscopic measurements, not their sum, which would be macroscopic; a product of results remains sensitive to the last measurement, even after a long sequence of others. Curiously, for very few measurements only the results are quantum, for many measurements they can be interpreted in terms of a classical phase, but become again strongly quantum when the maximum number of measurements is reached, a sort of revival of quantum-ness of the system.
Laboratoire Kastler Brossel is "UMR 8552 du CNRS, de l'ENS, et de l'Université Pierre et Marie Curie". | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We study theoretically the properties of two Bose-Einstein condensates in different spin states, represented by a double Fock state. Individual measurements of the spins of the particles are performed in transverse directions, giving access to the relative phase of the condensates. Initially, this phase is completely undefined, and the first measurements provide random results. But a fixed value of this phase rapidly emerges under the effect of the successive quantum measurements, giving rise to a quasi-classical situation where all spins have parallel transverse orientations. If the number of measurements reaches its maximum (the number of particles), quantum effects show up again, giving rise to violations of Bell type inequalities. The violation of BCHSH inequalities with an arbitrarily large number of spins may be comparable (or even equal) to that obtained with two spins."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "The notion of non-locality in quantum mechanics (QM) takes its roots in a chain of two theorems, the EPR (Einstein Podolsky Rosen) theorem [1] and its logical continuation, the Bell theorem. The EPR theorem starts from three assumptions (Einstein realism, locality, the predictions of quantum mechanics concerning some perfect correlations are correct) and proves that QM is incomplete: additional quantities, traditionally named λ, are necessary to complete the description of physical reality. The Bell theorem [2, 3] then proves that, if λ exists, the predictions of QM concerning other imperfect correlations cannot always be correct. The ensemble of the three assumptions: Einstein realism, locality, all predictions of QM are correct, is therefore self-contradictory; if Einstein realism is valid, QM is non-local. Bohr [4] rejected Einstein realism because, in his view, the notion of physical reality could not correctly be applied to microscopic quantum systems, defined independently of the measurement apparatuses. Indeed, since EPR consider a system of two microscopic particles, which can be \"seen\" only with the help of measurement apparatuses, the notion of their independent physical reality is open to discussion.\n\nNevertheless, it has been pointed out recently [5, 6] that the EPR theorem also applies to macroscopic systems, namely Bose-Einstein (BE) condensates in two different internal states. The λ introduced by EPR then corresponds to the relative phase of the condensates, i.e. to macroscopic transverse spin orientations, physical quantities at a human scale; it then seems more difficult to deny the existence of their reality, even in the absence of measurement devices. This gives even more strength to the EPR argument and weakens Bohr's refutation. It is then natural to ask whether the Bell theorem can be transposed to this stronger case.\n\nThe purpose of this article is to show that it can. We consider an ensemble of N + particles in a state defined by an orbital state u and a spin state +, and N -particles in the same state with spin orientation -. The whole system is described quantum mechanically by a double Fock state, that is, a \"double BE condensate\":\n\n| Φ > = (a u,+ ) † N+ (a u,-) † N- | vac. > (1)\n\nwhere a u,+ and a u,-are the destruction operators associated with the two populated single-particle states and |vac. > is the vacuum state. We introduce a sequence of transverse spin measurements that leads to quantum predictions violating the so called BCHSH [7, 8] Bell inequality. This is reminiscent of the work of Mermin [9] , who finds exponential violations of local realist inequalities with N -particle spin states that are maximally entangled. By contrast, here we consider the simplest way in which many bosons can be put in two different internal levels, with a N -particle state containing only the minimal possible correlations, those due to statistics. We find violations of inequalities that are the same order of magnitude as with the usual singlet spin state and may actually saturate the Cirel'son bound [10] .\n\nDouble Fock states are experimentally more accessible and much less sensitive to dissipation and decoherence than maximally entangled states [11] . Considering a system in a double Fock state, we assume that a series of rapid spin measurements can be performed and described by the usual QM postulate of measurement, without worrying about decoherence between the measurements, thermal effects, etc.\n\nThe operators associated with the local density of particles and spins can be expressed as functions of the two fields operators Ψ ± (r) associated with the two internal states ± as:\n\nn(r) = Ψ † + (r)Ψ + (r) + Ψ † -(r)Ψ -(r), σ z (r) = Ψ † + (r)Ψ + (r)-Ψ † -(r)Ψ -(r)\n\n, while the spin component in the direction of plane xOy making an angle ϕ with Ox is: σ ϕ (r) = e -iϕ Ψ † + (r)Ψ -(r) + e iϕ Ψ † -(r)Ψ + (r). Consider now a measurement of this component performed at point r and providing result η = ±1. The corresponding projector is:\n\nP η=±1 (r, ϕ) = 1 2 [n(r) + η σ ϕ (r)] (2)\n\nand, because the measurements are supposed to be performed at different points (ensuring that these projectors all commute) the probability P(η 1 , η 2 , ...η N ) for a series of results η i ± 1 for spin measurements at points r i along directions ϕ i can be written as:\n\n< Φ | P η1 (r 1 , ϕ 1 ) × P η2 (r 2 , ϕ 2 ) × ....P ηN (r N , ϕ N ) | Φ > (3\n\n) We now substitute the expression of σ ϕ (r) into ( 2 ) and ( 3 ), exactly as in the calculation of ref. [5] , but with one difference: here we do not assume that the number of measurements is much smaller than N ± , but equal to its maximum value N = N + + N -. In the product of projectors appearing in (3) , because all r's are different, commutation allows us to push all the field operators to the right, all their conjugates to the left; one can then easily see that each Ψ ± (r) acting on | Φ > can be replaced by u(r) × a u,± , and similarly for the Hermitian conjugates. With our initial state, a non-zero result can be obtained only if exactly N + operators a u,+ appear in the term considered, and N -operators a u,-; a similar condition exists for the Hermitian conjugate operators. To express these conditions, we introduce two additional variables. As in [5] , the first variable λ ensures an equal number of creation and destruction operators in the internal states ± through the mathematical identity:\n\nπ -π dλ 2π e inλ = δ n,0 (4)\n\nThe second variable Λ expresses in a similar way that the difference between the number of destruction operators in states + and -is exactly N + -N -, through the integral:\n\nπ -π dΛ 2π e -inΛ e i(N+-N-)Λ = δ n,N+-N- (5)\n\nThe introduction of the corresponding exponentials into the product of projectors ( 2 ) in ( 3 ) provides the expression (c.c. means complex conjugate):\n\nN j=1 |u(r j )| 2 1 2\n\ne iΛ + e -iΛ + η j e i(λ-ϕj +Λ) + c.c.\n\n(6) where, after integration over λ and Λ, the only surviving terms are all associated with the same matrix element in state | Φ > (that of the product of N + operators a † u,+ and N -operators a † u,-followed by the same sequence of destruction operators, providing the constant result N + !N -!). We can thus write the probability as:\n\nP(η 1 , η 2 , ...η N ) ∼ π -π dλ 2π +π -π dΛ 2π e i(N+-N-)Λ N j=1 |u(r j )| 2 1 2\n\ne iΛ + e -iΛ + η j e i(λ-ϕj +Λ) + c.c. (7) or, by using Λ parity and changing one integration variable (λ ′ = λ + Λ), as:\n\nP(η 1 , η 2 , ...η N ) = 1 2 N C N +π -π dΛ 2π cos [(N + -N -)Λ] +π -π dλ ′ 2π N j=1 {cos (Λ) + η j cos (λ ′ -ϕ j )} (8)\n\nThe normalization coefficient C N is readily obtained by writing that the sum of probabilities of all possible sequences of η's is 1 (this step requires discussion; we come back to this point at the end of this article):\n\nC N = +π -π dΛ 2π cos [(N + -N -)Λ] [cos (Λ)] N (9)\n\nFinally, we generalize (8) to any number of measurements M < N . A sequence of M measurements can always be completed by additional N -M measurements, leading to probability (8) . We can therefore take the sum of ( 8 ) over all possible results of the additional N -M measurements to obtain the probability for any M as:\n\nP(η 1 , η 2 , ...η M ) = 1 2 M C N +π -π dΛ 2π cos [(N + -N -)Λ] [cos Λ] N -M +π -π dλ ′ 2π M j=1 {cos (Λ) + η j cos (λ ′ -ϕ j )} (10)\n\nThe Λ integral can be replaced by twice the integral between ±π/2 (a change of Λ into π -Λ multiplies the function by (-1) N+-N-+N -M+M = 1). If M ≪ N , the large power of cos Λ in the first integral concentrates its contribution around Λ ≃ 0, so that a good approximation is Λ = 0. We then recover the results of refs [5, 6] , with a single integral over λ defining the relative phase of the condensates (Anderson phase), initially completely undetermined, so that the first spin measurement provides a completely random result. But the phase rapidly emerges under the effect of a few measurements, and remains constant [12, 13, 14] ; it takes a different value for each realization of the experiment, as if it was revealing the pre-existing value of a classical quantity. Moreover, when cos Λ is replaced by 1, each factor of the product over j remains positive (or zero), leading to a result similar to that of stochastic local realist theories; the Bell inequalities can then be obtained. However, when N -M is small or even vanishes, cos Λ can take values that are smaller than 1 and the factors may become negative, opening the possibility of violations. In a sense, the additional variable Λ controls the amount of quantum effects in the series of measurements. We now discuss when these standard QM predictions violate Bell inequalities. We need the value of the quantum average of the product of results, that is the sum of η 1 , η 2 , ...η M × P(η 1 , η 2 , ...η M ) over all possible values of the η's, which according to (10) is given by:\n\nE(ϕ 1 , ϕ 2 , ..ϕ M ) = 1 C N +π -π dΛ 2π cos [(N + -N -)Λ] [cos Λ] N -M +π -π dλ ′ 2π M j=1 cos λ ′ -ϕ j (11)\n\nConsider a thought experiment where two condensates in different spin states (two eigenstates of the Oz spin component) overlap in two remote regions of space A and B , with two experimentalists Alice and Bob; they measure the spins of the particles in arbitrary transverse directions (perpendicular to Oz) at points of space where the orbital wave functions of the two condensates are equal. All measurements performed by Alice are made along a single direction ϕ a , which plays here the usual role of the \"setting\" a, while all those performed by Bob are made along angle ϕ b . We assume that Alice retains just the product A of all her measurements, while Bob retains only the product B of his; A and B are both ±1.\n\nWe now assume two possible orientations ϕ a and ϕ ′ a for Alice, two possible orientations ϕ b and ϕ ′ b for Bob. Within deterministic local realism, for each realization of the experiment, it is possible to define two numbers A, A ′ , both equal to ±1, associated with the two possible products of results η that Alice will observe, depending of her choice of orientation; the same is obviously true for Bob, introducing B and B ′ . Within stochastic local realism [8, 15] , A and A ′ are the difference of probabilities associated with Alice observing +1 or -1, i.e. numbers that have values between +1 and -1. In both cases, the following inequalities (BCHSH) are obeyed:\n\n-2 ≤ AB + AB ′ ± (A ′ B -A ′ B ′ ) ≤ 2 (12)\n\nIn standard quantum mechanics, of course, \"unperformed experiments have no results\" [16] , and several of the numbers appearing in (12) are undefined; only two of them can be defined after the experiment has been performed with a given choice of the orientations. Thus, while one can calculate from (11) the quantum average value < Q > of the sum of products of results appearing in (12) , there is no special reason why < Q > should be limited between +2 and -2. Situations where the limit is exceeded are called \"quantum non-local\".\n\nWe have seen that the most interesting situations occur when the cosines do not introduce their peaking effect around Λ = 0, i.e. when N + = N -and M has its maximum value N . Then, for a given N , the only remaining choice is how the number of measurements is shared between N a measurements for Alice and N b for Bob.\n\nAssume first that N a = 1 (Alice makes one measurement) and therefore N b = N -1 (Bob makes all the others). Since we assume that N + = N -and M = N , the Λ integral in (11) disappears, and the λ integral contains only the product of cos (λ ′ -ϕ a ) by the (N -1)th power of cos (λ ′ -ϕ b ), which is straightforward and provides cos (ϕ a -ϕ b ) times the normalization integral C N . The quantum average associated with the product AB is thus merely equal to cos (ϕ a -ϕ b ), exactly as the usual case of two spins in a singlet state. Then it is well-known that, when the angles form a \"fan\" [17] spaced by χ = π/4, a strong violation of (12) occurs, by a factor √ 2, saturating the Cirel'son bound [10] . A similar calculation can be performed when Alice makes 2 measurements and Bob N -2, and shows that the quantum average is now equal to\n\n1 2 1 + 1 N -1 + (1 -1 N -1 ) cos 2 (ϕ a -ϕ b ) , no longer independent of N. If N = 4, the maximum of < Q > is 2.28 < 2 √ 2\n\n, and rises to 2.41 as N → ∞. An expression for the generalization of the quantum average to any number P and N -P of measurements by Alice and Bob, respectively, is (with χ = ϕ a -ϕ b ):\n\nE(χ) = N 2 ! N ! {P/2} k=0 P !(N -2k)! k!(P -2k)!( N 2 -k)! sin 2k χ cos P -2k χ (13)\n\nwhere {P/2} is the integer part of P/2. The maximum of < Q > can then be found using a numerical Mathematica routine. Results are shown for several values of P in Fig. 1 . The angles maximizing the quantum Bell quantity always occur in the fan shape, although the basic angle χ changes with P and N. All of the curves where P is held fixed have a finite < Q > limit with increasing N , and the optimum values of the angles approach constants. For the curve P = N/2, the limit is 2.32 when N → ∞, and the fan opening decreases as 1/ √ N .\n\n2.8 2.6 2.4 2.2 2.0 <Q> Max 100 80 60 40 20 N P 1 2 4 6 N/2\n\nFIG. 1: The maximum of the quantum average < Q > for Alice doing P experiments and Bob N -P , as a function of the total number of particles N . The usual Bell situation is obtained for N = 2, P = 1. Local realist theories predict an upper limit of 2; large violations of this limit are obtained, even with macroscopic systems (N → ∞). If P = 1, the violation saturates the Cirel'son limit for any N .\n\nWe can also study cases where the number of measurements is M < N : if Bob makes all his measurements, but ignores one or two of them (independently of the order of the measurements), when he correlates his results with Alice, the BCHSH inequality is never violated. All measurements have to be taken into account to obtain violations. Furthermore, if the number of particles in the two condensates are not equal, no violation occurs either. Finally, it is possible to consider cases where we generalize the angles considered: experimenter Carole makes measurements at ϕ c and ϕ ′ c , and David at ϕ d and ϕ ′ d . We then find that a maximization of < Q > reduces to the cases already studied, where the new angles collapse to the previous angles ϕ a , • • • , ϕ ′ b . For the sake of simplicity, we have not yet discussed some important issues that underlie our calculations. One is related to the so called \"sample bias loophole\" (or \"detection/efficiency loophole\") and to the normalization condition (9) , which assumes that one spin is detected at each point of measurement. A more detailed study (see second ref. [5] ) should include the integration of each r in a small detection volume and the possibility that no particle is detected in it. This is a well-known difficulty, which already appears in the usual two-photon experiments [8] , where most photons are missed by the detectors. If this loophole still raises a real experimental challenge, the difficulty can be resolved in theory by assuming the presence of additional spin-independent detectors [2, 8] , which ensure the detection of one particle in each detector and create appropriate initial conditions (see for instance [18] for a description of an experiment with veto detectors). We postpone this discussion to another article [19] . A second issue deals with the definition of the local realist quantities A, B, etc. For two condensates, we have a slightly different situation than in the usual EPR situation: the local realist reasoning leads to the existence of a well-defined phase λ between the con- [5] , not necessarily to deterministic properties of the individual particles. Fortunately, Bell inequalities can also be derived within stochastic local realist theories [3, 8] (see also for instance [9] or appendix I of [15] ), and this difference is not a problem [19] .\n\nIn conclusion, strong violations of local realism may occur for large quantum systems, even if the state is a simple double Fock state with equal populations; within present experimental techniques, this seems reachable with N ∼ 10 or 20. We have assumed that the measured quantity is the product of many microscopic measurements, not their sum, which would be macroscopic; a product of results remains sensitive to the last measurement, even after a long sequence of others. Curiously, for very few measurements only the results are quantum, for many measurements they can be interpreted in terms of a classical phase, but become again strongly quantum when the maximum number of measurements is reached, a sort of revival of quantum-ness of the system.\n\nLaboratoire Kastler Brossel is \"UMR 8552 du CNRS, de l'ENS, et de l'Université Pierre et Marie Curie\"."
}
] | |
arxiv:0704.0389 | 0704.0389 | 1 | 10.1103/PhysRevD.75.124007 10.1103/PhysRevD.82.029901
10.1103/PhysRevD.82.129903 | 8bbc0a5c277ac9087fae73f2351273d64758f1d12c30292eb64f6a3c3c9216cc | Evolution of the Carter constant for inspirals into a black hole: effect
of the black hole quadrupole | We analyze the effect of gravitational radiation reaction on generic orbits
around a body with an axisymmetric mass quadrupole moment Q to linear order in
Q, to the leading post-Newtonian order, and to linear order in the mass ratio.
This system admits three constants of the motion in absence of radiation
reaction: energy, angular momentum, and a third constant analogous to the
Carter constant. We compute instantaneous and time-averaged rates of change of
these three constants. For a point particle orbiting a black hole, Ryan has
computed the leading order evolution of the orbit's Carter constant, which is
linear in the spin. Our result, when combined with an interaction quadratic in
the spin (the coupling of the black hole's spin to its own radiation reaction
field), gives the next to leading order evolution. The effect of the
quadrupole, like that of the linear spin term, is to circularize eccentric
orbits and to drive the orbital plane towards antialignment with the symmetry
axis. In addition we consider a system of two point masses where one body has a
single mass multipole or current multipole. To linear order in the mass ratio,
to linear order in the multipole, and to the leading post-Newtonian order, we
show that there does not exist an analog of the Carter constant for such a
system (except for the cases of spin and mass quadrupole). With mild additional
assumptions, this result falsifies the conjecture that all vacuum, axisymmetric
spacetimes posess a third constant of geodesic motion. | [
"Eanna E. Flanagan and Tanja Hinderer"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | We analyze the effect of gravitational radiation reaction on generic orbits around a body with an axisymmetric mass quadrupole moment Q to linear order in Q, to the leading post-Newtonian order, and to linear order in the mass ratio. This system admits three constants of the motion in absence of radiation reaction: energy, angular momentum along the symmetry axis, and a third constant analogous to the Carter constant. We compute instantaneous and time-averaged rates of change of these three constants. For a point particle orbiting a black hole, Ryan [15] has computed the leading order evolution of the orbit's Carter constant, which is linear in the spin. Our result, when combined with an interaction quadratic in the spin (the coupling of the black hole's spin to its own radiation reaction field), gives the next to leading order evolution. The effect of the quadrupole, like that of the linear spin term, is to circularize eccentric orbits and to drive the orbital plane towards antialignment with the symmetry axis. In addition we consider a system of two point masses where one body has a single mass multipole or current multipole of order l. To linear order in the mass ratio, to linear order in the multipole, and to the leading post-Newtonian order, we show that there does not exist an analog of the Carter constant for such a system (except for the cases of an l = 1 current moment and an l = 2 mass moment). Thus, the existence of the Carter constant in Kerr depends on interaction effects between the different multipoles. With mild additional assumptions, this result falsifies the conjecture that all vacuum, axisymmetric spacetimes posess a third constant of the motion for geodesic motion.
Erratum: Evolution of the Carter constant for inspirals into a black hole: Effect of the black hole quadrupole [Phys. Rev. D 75, 124007 (2007) ] Éanna É. Flanagan, Tanja Hinderer In Eqs. (3.16 ), (3.17), (3.18) , (3.24) , (3.25) and (3.26) of this paper, the variable r should be replaced everywhere by the variable r, and the variable θ should be replaced everywhere by the variable θ. The definitions of r and θ are given in Eq. (2.11) . These replacements do not affect the any of the subsequent results in the paper.
Also, the right hand side of Eq. ( B3 ) is missing a term -4SL z r and Eq. (2.24) is missing a factor of dϕ/d t in front of Q.
Some terms are missing in Eqs. (3.18) , (3.26 ) and (3.30) - (3.33) . The additional terms in Eqs. (3.18 ) and (3.26) are -8Q 15r 7 -75K 2 + 2K r(51rE + 50) + 8r 2 (rE + 1)(3rE + 5) , and 8Q 15p 2 r7 25p 3 (3p -4r) + p 2 r2 11 -51e 2 + 32pr 3 1 -e 2 + 6r 4 1 -e 2 2 , respectively. These result in additional fractional corrections to Eq. (3.30) given by -Q p 2 1 2 + 73 48 e 2 + 37 192 e 4 , and the full expression replacing the O(Q) terms in Eq. (3.30) is then K = -64 5 (1 -e 2 ) 3/2 p 3 1 + 7e 2 8 -Q p 2 1 + 8 3 e 2 + 11 12 e 4 + 13 4 + 841 96 e 2 + 449 192 e 4 cos(2ι) + O(S), O(S 2 ) -terms. Equations (3.31), (3.32) and (3.33) contain typos in the O(S) and O(Q) terms, the corrected expressions are given below. We thank P. Komorowski for pointing this out. Equation (3.31) should be replaced by ṗ = -64 5 (1 -e 2 ) 3/2 p 3 1 + 7e 2 8 -S cos(ι) 96p 3/2 1064 + 1516e 2 + 475e 4 -Q 8p 2 14 + 149e 2 12 + 19e 4 48 + 50 + 469e 2 12 + 227e 4 24 cos(2ι) + S 2 64p 2 1 3 + e 2 + e 4 8 [13 -cos(2ι)] , (0.1) Equation (3.32) should be replaced by ė = -304 15 e(1 -e 2 ) 3/2 p 4 1 + 121e 2 304 + Se(1 -e 2 ) 3/2 cos(ι) 5p 11/2 1172 + 932e 2 + 1313e 4 6 + Q(1 -e 2 ) 3/2 ep 6 32 + 785e 2 3 -219e 4 2 + 13e 6 + 32 + 2195e 2 3 + 251e 4 + 218e 6 3 cos(2ι) -S 2 e(1 -e 2 ) 3/2 8p 6 2 + 3e 2 + e 4 4 [13 -cos(2ι)] , (0.2) and the corrected Eq. (3.33) is ι = S sin(ι)(1 -e 2 ) 3/2 p 11/2 244 15 + 252 5 e 2 + 19 2 e 4 -1 -e 2 3/2 S 2 sin(2ι) 240p 6 8 + 3e 2 8 + e 2 + Q cot(ι)(1 -e 2 ) 3/2 60p 6 312 + 736e 2 -83e 4 -408 + 1268e 2 + 599e 4 cos(2ι) . (0.3)
The inspiral of stellar mass compact objects with masses µ in the range µ ∼ 1 -100M ⊙ into massive black holes with masses M ∼ 10 5 -10 7 M ⊙ is one of the most important sources for the future space-based gravitational wave detector LISA. Observing such events will provide a variety of information: (i) the masses and spins of black holes can be measured to high accuracy (∼ 10 -4 ); which can constrain the black hole's growth history [1] ; (ii) the observations will give a precise test of general relativity in the strong field regime and unambiguously identify whether the central object is a black hole [2] ; and (iii) the measured event rate will give insight into the complex stellar dynamics in galactic nuclei [1] . Analogous inspirals may also be interesting for the advanced stages of ground-based detectors: it has been estimated that advanced LIGO could detect up to ∼ 10 -30 inspirals per year of stellar mass compact objects into intermediate mass black holes with masses M ∼ 10 2 -10 4 M ⊙ in globular clusters [3] . Detecting these inspirals and extracting information from the datastream will require accurate models of the gravitational waveform as templates for matched filtering. For computing templates, we therefore need a detailed understanding of the how radiation reaction influences the evolution of bound orbits around Kerr black holes [4] [5] [6] [7] .
There are three dimensionless parameters characterizing inspirals of bodies into black holes:
• the dimensionless spin parameter a = |S|/M 2 of the black hole, where S is the spin.
• the strength of the interaction potential ǫ 2 = GM/rc 2 , i.e. the expansion parameter used in post-Newtonian (PN) theory.
• the mass ratio µ/M .
For LISA data analysis we will need waveforms that are accurate to all orders in a and ǫ 2 , and to leading order in µ/M . However, it is useful to have analytic results in the regimes a ≪ 1 and/or ǫ 2 ≪ 1. Such approximate results can be useful as a check of numerical schemes that compute more accurate waveforms, for scoping out LISA's data analysis requirements [1, 6] , and for assessing the accuracy of the leading order in µ/M or adiabatic approximation [8] [9] [10] . There is substantial literature on such approximate analytic results, and in this paper we will extend some of these results to higher order. A long standing difficulty in computing the evolution of generic orbits has been the evolution of the orbit's "Carter constant", a constant of motion which governs the orbital shape and inclination. A theoretical prescription now exists for computing Carter constant evolution to all orders in ǫ and a in the adiabatic limit µ ≪ M [9, [11] [12] [13] , but it has not yet been implemented numerically. In this paper we focus on computing analytically the evolution of the Carter constant in the regime a ≪ 1, ǫ ≪ 1, µ/M ≪ 1, extending earlier results by Ryan [14, 15] .
We next review existing analytical work on the effects of multipole moments on inspiral waveforms. For non-spinning point masses, the phase of the l = 2 piece of the waveform is known to O(ǫ 7 ) beyond leading order [16] , while spin corrections are not known to such high order. To study the leading order effects of the central body's multipole moments on the inspiral waveform, in the test mass limit µ ≪ M , one has to correct both the conservative and dissipative pieces of the forces on the bodies. For the conservative pieces, it suffices to use the Newtonian action for a binary with an additional multipole interaction potential. For the dissipative pieces, the multipole corrections to the fluxes at infinity of the conserved quantities can simply be added to the known PN point mass results. The lowest order spin-orbit coupling effects on the gravitational radiation were first derived by Kidder [17] , then extended by Ryan [14, 15] , Gergely [18] , and Will [19] . Recently, the corrections of O(ǫ 2 ) beyond the leading order to the spin-orbit effects on the fluxes were derived [20, 21] . Corrections to the waveform due to the quadrupole -mass monopole interaction were first considered by Poisson [22] , who derived the effect on the time averaged energy flux for circular equatorial orbits. Gergely [23] extended this work to generic orbits and computed the radiative instantaneous and time averaged rates of change of energy E, magnitude of angular momentum |L|, and the angle κ = cos -1 (S • L) between the spin S and orbital angular momentum L. Instead of the Carter constant, Gergely identified the angular average of the magnitude of the orbital angular momentum, L, as a constant of motion. The fact that to post-2-Newtonian (2PN) order there is no time averaged secular evolution of the spin allowed Gergely to obtain expressions for L and κ from the quadrupole formula for the evolution of the total angular momentum J = L+ S. In a different paper, Gergely [18] showed that in addition to the quadrupole, self-interaction spin effects also contribute at 2PN order, which was seen previously in the black hole perturbation calculations of Shibata et al. [24] . Gergely calculated the effect of this interaction on the instantaneous and time-averaged fluxes of E and |L| but did not derive the evolution of the third constant of motion.
In this paper, we will re-examine the effects of the quadrupole moment of the black hole and of the leading order spin self interaction. For a black hole, our analysis will thus contain all effects that are quadratic in spin to the leading order in ǫ 2 and in µ/M . Our work will extend earlier work by
• Considering generic orbits.
• Using a natural generalization of the Carter-type constant that can be defined for two point particles when one of them has a quadrupole. This facilitates applying our analysis to Kerr inspirals.
• Computing instantaneous as well as time-averaged fluxes for all three constants of motion: energy E, z-component of angular momentum L z , and Carter-type constant K. For most purposes, only time-averaged fluxes are needed as only they are gauge invariant and physically relevant. However, there is one effect for which the time-averaged fluxes are insufficient, namely transient resonances that occur during an inspiral in Kerr in the vicinity of geodesics for which the radial and azimuthal frequencies are commensurate [10, 25] . The instantaneous fluxes derived in this paper will be used in [10] for studying the effect of these resonances on the gravitational wave phasing.
We will analyze the effect of gravitational radiation reaction on orbits around a body with an axisymmetric mass quadrupole moment Q to leading order in Q, to the leading post-Newtonian order, and to leading order in the mass ratio. With these approximations the adiabatic approximation holds: gravitational radiation reaction takes place over a timescale much longer than the orbital period, so the orbit looks geodesic on short timescales. We follow Ryan's method of computation [14] : First, we calculate the orbital motion in the absence of radiation reaction and the associated constants of motion. Next, we use the leading order radiation reaction accelerations that act on the particle (given by the Burke-Thorne formula [26] augmented by the relevant spin corrections [14] ) to compute the evolution of the constants of motion. In the adiabatic limit, the time-averaged rates of change of the constants of motion can be used to infer the secular orbital evolution. Our results show that a mass quadrupole has the same qualitative effect on the evolution as spin: it tends to circularize eccentric orbits and drive the orbital plane towards antialignment with the symmetry axis of the quadrupole.
The relevance of our result to point particles inspiralling into black holes is as follows. The vacuum spacetime geometry around any stationary body is completely characterized by the body's mass multipole moments I L = I a1,a2...a l and current multipole moments S L = S a1,a2...a l [27] . These moments are defined as coefficients in a power series expansion of the metric in the body's local asymptotic rest frame [28] . For nearly Newtonian sources, they are given by integrals over the source as
I L ≡ I a1,...a l = ρx <a1 . . . x a l > d 3 x, (1.1)
S L ≡ S a1,...a l = ρx p v q ǫ pq<a1 x a2 . . . x a l > d 3 x.(1.2)
Here ρ is the mass density and v q is the velocity, and "< • • • >" means "symmetrize and remove all traces". For axisymmetric situations, the tensor multipole moments I L (S L ) contain only a single independent component, conventionally denoted by I l (S l ) [27] . For a Kerr black hole of mass M and spin S, these moments are given by [27]
I l + iS l = M l+1 (ia) l , (1.3)
where a is the dimensionless spin parameter defined by a = |S|/M 2 . Note that S l = 0 for even l and I l = 0 for odd l.
Consider now inspirals into an axisymmetric body which has some arbitrary mass and current multipoles I l and S l . Then we can consider effects that are linear in I l and S l for each l, effects that are quadratic in the multipoles proportional to I l I l ′ , I l S l ′ , S l S l ′ , effects that are cubic, etc. For a general body, all these effects can be separated using their scalings, but for a black hole, I l ∝ a l for even l and S l ∝ a l for odd l [see Eq.(1.3)], so the effects cannot be separated. For example, a physical effect that scales as O(a 2 ) could be an effect that is quadratic in the spin or linear in the quadrupole; an analysis in Kerr cannot distinguish these two possibilities. For this reason, it is useful to analyze spacetimes that are more general than Kerr, characterized by arbitrary I l and S l , as we do in this paper. For recent work on computing exact metrics characterized by sets of moments I l and S l , see Refs. [29, 30] and references therein.
The leading order effect of the black hole's multipoles on the inspiral is the O(a) effect computed by Ryan [15] . This O(a) effect depends linearly on the spin S 1 and is independent of the higher multipoles S l and I l since these all scale as O(a 2 ) or smaller. In this paper we compute the O(a 2 ) effect on the inspiral, which includes the leading order linear effect of the black hole's quadrupole (linear in I 2 ≡ Q) and the leading order spin self-interaction (quadratic in S 1 ).
We next discuss how these O(a 2 ) effects scale with the post-Newtonian expansion parameter ǫ. Consider first the conservative orbital dynamics. Here it is easy to see that fractional corrections that are linear in I 2 scale as O(a 2 ǫ 4 ), while those quadratic in S 1 scale as O(a 2 ǫ 6 ). Thus, the two types of terms cleanly separate. We compute only the leading order, O(a 2 ǫ 4 ), term. For the dissipative contributions to the orbital motion, however, the scalings are different. There are corrections to the radiation reaction acceleration whose fractional magnitudes are O(a 2 ǫ 4 ) from both types of effects linear in I 2 and quadratic in S 1 . The effects quadratic in S 1 are due to the backscattering of the radiation off the piece of spacetime curvature due to the black hole's spin. This effect was first pointed out by Shibata et al. [24] , who computed the time-averaged energy flux for circular orbits and small inclination angles based on a PN expansion of black hole perturbations. Later, Gergely [18] analyzed this effect on the instantaneous and time-averaged fluxes of energy and magnitude of orbital angular momentum within the PN framework.
The organization of this paper is as follows. In Sec. II, we study the conservative orbital dynamics of two point particles when one particle is endowed with an axisymmetric quadrupole, in the weak field regime, and to leading order in the mass ratio. In Sec. III, we compute the radiation reaction accelerations and the instantaneous and time-averaged fluxes. In order to have all the contributions at O(a 2 ǫ 4 ) for a black hole, we include in our computations of radiation reaction acceleration the interaction that is quadratic in the spin S 1 . The application to black holes in Sec. IV briefly discusses the qualitative predictions of our results and also compares with previous results.
The methods used in this paper can be applied only to the black hole spin (as analyzed by Ryan [14] ) and the black hole quadrupole (as analyzed here). We show in Sec. V that for the higher order mass and current multipole moments taken individually, an analog of the Carter constant cannot be defined to the order of our approximations. We then show that under mild assumptions, this non-existence result can be extended to exact spacetimes, thus falsifying the conjecture that all vacuum axisymmetric spacetimes possess a third constant of geodesic motion.
Consider two point particles m 1 and m 2 interacting in Newtonian gravity, where m 2 ≪ m 1 and where the mass m 1 has a quadrupole moment Q ij which is axisymmetric:
Q ij = d 3 xρ(r) x i x j - 1 3 r 2 δ ij (2.1) = Q n i n j - 1 3 δ ij . (2.2)
For a Kerr black hole of mass M and dimensionless spin parameter a with spin axis along n, the quadrupole scalar is Q = -M 3 a 2 . The action describing this system, to leading order in m 2 /m 1 , is
S = dt 1 2 µv 2 -µΦ(r) , (2.3)
where v = ṙ is the velocity, the potential is
Φ(r) = - M r - 3 2r 5 x i x j Q ij , (2.4)
µ is the reduced mass and M the total mass of the binary, and we are using units with G = c = 1. We work to linear order in Q, to linear order in m 2 /m 1 , and to leading order in M/r. In this regime, the action (2.3) also describes the conservative effect of the black hole's mass quadrupole on bound test particles in Kerr, as discussed in the introduction. We shall assume that the quadrupole Q ij is constant in time. In reality, the quadrupole will evolve due to torques that act to change the orientation of the central body. An estimate based on treating m 1 as a rigid body in the Newtonian field of m 2 gives the scaling of the timescale for the quadrupole to evolve compared to the radiation reaction time as (see Appendix I for details)
T evol T rr ∼ m 1 m 2 M r S Q ∼ M µ M r 1 a .
(2.5)
Here, we have denoted the dimensionless spin and quadrupole of the body by S and Q respectively, and the last relation applies for a Kerr black hole. Since µ/M ≪ 1, the first factor in Eq. (2.5) will be large, and since 1/a ≥ 1 and for the relativistic regime M/r ∼ 1, the evolution time is long compared to the radiation reaction time. Therefore we can neglect the evolution of the quadrupole at leading order. This system admits three conserved quantities, the energy
E = 1 2 µv 2 + µΦ(r), (2.6)
the z-component of angular momentum
L z = e z • (µr × v), (2.7)
and the Carter-type constant
K = µ 2 (r × v) 2 - 2Qµ 2 r 3 (n • r) 2 + Qµ 2 M (n • v) 2 - 1 2 v 2 + M r . (2.8)
(See below for a derivation of this expression for K).
A. Conservative orbital dynamics in a Boyer-Lindquist-like coordinate system
We next specialize to units where M = 1. We also define the rescaled conserved quantities by Ẽ = E/µ, Lz = L z /µ, K = K/µ 2 , and drop the tildes. These specializations and definitions have the effect of eliminating all factors of µ and M from the analysis. In spherical polar coordinates (r, θ, ϕ) the constants of motion E and
L z become E = 1 2 ( ṙ2 + r 2 θ2 + r 2 sin 2 θ φ2 ) - 1 r + Q 2r 3 (1 -3 cos 2 θ), (2.9)
L z = r 2 sin 2 θ φ. (2.10)
In these coordinates, the Hamilton-Jacobi equation is not separable, so a separation constant K cannot readily be derived. For this reason we switch to a different coordinate system (r, θ, ϕ) defined by
r cos θ = r cos θ 1 + Q 4r 2 , r sin θ = r sin θ 1 - Q 4r 2 . (2.11)
We also define a new time variable t by
dt = 1 - Q 2r 2 cos(2 θ) d t.
(2.12)
The action (2.3) in terms of the new variables to linear order in Q is
S = d t 1 2 dr d t 2 + 1 2 r2 d θ d t 2 + 1 2 r2 sin 2 θ dϕ d t 2 1 - Q r2 sin 2 θ + 1 r + Q 4r 3 . (2.13)
However, a difficulty is that the action (2.13) does not give the same dynamics as the original action (2.3). The reason is that for solutions of the equations of motion for the action (2.3), the variation of the action vanishes for paths with fixed endpoints for which the time interval ∆t is fixed. Similarly, for solutions of the equations of motion for the action (2.13), the variation of the action vanishes for paths with fixed endpoints for which the time interval ∆ t is fixed. The two sets of varied paths are not the same, since ∆t = ∆ t in general. Therefore, solutions of the Euler-Lagrange equations for the action (2.3) do not correspond to solutions of the Euler-Lagrange equations for the action (2.13). However, in the special case of zeroenergy motions, the extra terms in the variation of the action vanish. Thus, a way around this difficulty is to modify the original action to be
Ŝ = dt 1 2 µv 2 -µΦ(r) + E . (2.14)
This action has the same extrema as the action (2.3), and for motion with physical energy E, the energy computed with this action is zero. Transforming to the new variables yields, to linear order in Q:
Ŝ = d t 1 2
dr d t 2 + 1 2 r2 d θ d t 2 + 1 2 r2 sin 2 θ dϕ d t 2 1 - Q r2 sin 2 θ + 1 r + Q 4r 3 + E - QE 2r 2 cos(2 θ) . (2.15)
The zero-energy motions for this action coincide with the zero energy motions for the action (2.14). We use this action (2.15) as the foundation for the remainder of our analysis in this section. The z-component of angular momentum in terms of the new variables (r, θ, ϕ, t) is
L z = r2 sin 2 θ dϕ d t 1 - Q r2 sin 2 θ . (2.16)
We now transform to the Hamiltonian:
Ĥ = 1 2 p 2 r - 1 r -E - Q 4r 3 + QL 2 z 2r 4 + 1 2r 2 p 2 θ + L 2 z sin 2 θ + QE cos(2 θ) (2.17)
and solve the Hamiltonian Jacobi equation. Denoting the separation constant by K we obtain the following two equations for the r and θ motions:
dr d t 2 = 2E + 2 r - K r2 + Q 2 1 r3 - 2L 2 z r4 , (2.18)
and r4 d θ
d t 2 = K - L 2 z sin 2 θ -QE cos(2 θ). ( 2
.19) Note that the equations of motion (2.18) and (2.19) have the same structure as the equations of motion for Kerr geodesic motion. Using Eqs. (2.18), (2.19) and (2.16) together with the inverse of the transformation (2.11) to linear order in Q, we obtain the expression for K in spherical polar coordinates:
K = r 4 ( θ2 + sin 2 θ φ2 ) + Q( ṙ cos θ -r θ sin θ) 2 + Q r - Q 2 ( ṙ2 + r 2 θ2 + r 2 sin 2 θ φ2 ) - 2Q r cos 2 θ. (2.20)
This is equivalent to the formula (2.8) quoted earlier.
To include the linear in spin effects, we repeat Ryan's analysis [14, 15] (he only gives the final, time averaged fluxes; we will also give the instantaneous fluxes). We can simply add these linear in spin terms to our results because any terms of order O(SQ) will be higher than the order a 2 to which we are working. The correction to the action (2.3) due to spin-orbit coupling is
S spin-orbit = dt - 2µSn i ǫ ijk x j ẋk r 3 . (2.21)
We will restrict our analysis to the case when the unit vectors n i corresponding to the axisymmetric quadrupole Q ij and to the spin S i coincide, as they do in Kerr.
Including the spin-orbit term in the action (2.3) results in the following modified expressions for L z and K:
L z = n • (µr × v) - 2S r 3 [r 2 -(n • r) 2 ], (2.22)
and
K = (r × v) 2 - 4S r n • (r × v) - 2Q r 3 (n • r) 2 +Q (n • v) 2 - 1 2 v 2 + 1 r . (2.23)
In terms of the Boyer-Lindquist like coordinates, the conserved quantities with the linear in spin terms included are:
L z = r2 sin 2 θ dϕ d t - 2S r sin 2 θ -Q sin 4 θ dϕ d t , (2.24)
K = r 4 ( θ2 + sin 2 θ φ2 ) -4Sr sin 2 θ φ - 2Q r cos 2 θ + Q( ṙ cos θ -r θ sin θ) 2 + QM r - Q 2 ( ṙ2 + r 2 θ2 + r 2 sin 2 θ φ2 ). (2.25)
The equations of motion are
dr d t 2 = 2E + 2 r - K r2 - 4SL z r3 + Q 2 1 r3 - 2L 2 z r4 , (2.26) and r4 d θ d t 2 = K - L 2 z sin 2 θ -QE cos(2 θ). ( 2
.27) III. EFFECTS LINEAR IN QUADRUPOLE AND QUADRATIC IN SPIN ON THE EVOLUTION OF THE CONSTANTS OF MOTION A. Evaluation of the radiation reaction force
The relative acceleration of the two bodies can be written as
a = -∇Φ(r) + a rr , (3.1)
where a rr is the radiation-reaction acceleration. Combining this with Eqs. (2.6), (2.22) and (2.23) for E, L z and K gives the following formulae for the time derivatives of the conserved quantities:
Ė = v • a rr , (3.2) Lz = n • (r × a rr ), (3.3)
K = 2(r × v) • (r × a rr ) - 4S r n • (r × a rr ) +2Q(n • v) (n • a rr ) -Qv • a rr . (3.4)
The standard expression for the leading order radiation reaction acceleration acting on one of the bodies is [31] :
a j rr = - 2 5 I
(5)
jk x k + 16 45 ǫ jpq S (6) pk x k x q + 32 45 ǫ jpq S (5)
pk x k v q + 32 45 ǫ pq[j S (5) k]p x q v k . (3.5)
Here the superscripts in parentheses indicate the number of time derivatives and square brackets on the indices denote antisymmetrization. The multipole moments I jk (t) and S jk (t) in Eq. (3.5) are the total multipole moments of the spacetime, i.e. approximately those of the black hole plus those due to the orbital motion. The expression (3.5) is formulated in asymptotically Cartesian mass centered (ACMC) coordinates of the system, which are displaced from the coordinates used in Sec. II by an amount [28]
δr(t) = - µ M r(t). (3.6)
This displacement contributes to the radiation reaction acceleration in the following ways:
1. The black hole multipole moments I l and S l , which are time-independent in the coordinates used in Sec. II, will be displaced by δr and thus will contribute to the (l + 1)th ACMC radiative multipole [28] .
2. The constants of motion are defined in terms of the black hole centered coordinates used in Sec. II, so the acceleration a rr we need in Eqs. (3.2) -(3.4) is the relative acceleration. This requires calculating the acceleration of both the black hole and the point mass in the ACMC coordinates using (3.5), and then subtracting to find a rr = a µ rra M rr [14] . To leading order in µ, the only effect of the acceleration of the black hole is via a backreaction of the radiation field: the lth black hole moments couple to the (l + 1)th radiative moments, thus producing an additional contribution to the acceleration.
For our calculations at O(S 1 ǫ 3 ), O(I 2 ǫ 4 ), O(S 2 1 ǫ 4 ), we can make the following simplifications:
• quadrupole corrections: The fractional corrections linear in I 2 = Q that scale as O(a 2 ǫ 4 ) require only the effect of I 2 on the conservative orbital dynamics as computed in Sec. IIA and the Burke-Thorne formula for the radiation reaction acceleration [given by the first term in Eq. (3.5)].
• spin-spin corrections: As discussed in the introduction, the fractional corrections quadratic in S 1 to the conservative dynamics scale as O(a 2 ǫ 6 ) and are subleading order effects which we neglect. At O(a 2 ǫ 4 ), the only effect quadratic in S 1 is the backscattering of the radiation off the spacetime curvature due to the spin. As discussed in item 1. above, the black hole's current dipole S i = S 1 δ i3 (taking the z-axis to be the symmetry axis) will contribute to the radiative current quadrupole an amount
S spin ij = - 3 2 µ M S 1 x i δ j3 . (3.7)
The black hole's current dipole S i will couple to the gravitomagnetic radiation field due to S ij as discussed in item 2. above, and contribute to the relative acceleration as [14] : x k x q + 32 45 ǫ jpq S
a j spin rr = 8 15 S 1 δ i3 S (
(5) spin pk
x k v q + 32 45 ǫ pq[j S (5) spin k]p x q v k + 8 15 S 1 δ i3 S (5) orbit ij + S (5) spin ij . (3.9)
To justify these approximations, consider the scaling of the contribution of black hole's acceleration to the orbital dynamics. The mass and current multipoles of the black hole contribute terms to the Hamiltonian that scale with ǫ as
∆H ∼ S l ǫ 2l+3 & I l ǫ 2l+2 .
(3.10)
Since the Newtonian energy scales as ǫ 2 , the fractional correction to the orbital dynamics scale as
∆H/E ∼ S l ǫ 2l+1 & I l ǫ 2l . (3.11)
To O(ǫ 4 ), the only radiative multipole moments that contribute to the acceleration (3.5) are the mass quadrupole I 2 , the mass octupole I 3 , and the current quadrupole S 2 (cf. [17] ). Since we are focusing only on the leading order terms quadratic in spin (these can simply be added to the known 2PN point particle and 1.5PN linear in spin results), the only terms in Eq. (3.5) relevant for our purposes are those given in Eq. (3.9). The results from a computation of the fully relativistic metric perturbation for black hole inspirals [24] show that quadratic in spin corrections to the l = 2 piece compared to the flat space Burke-Thorne formula first appear at O(a 2 ǫ 4 ), which is consistent with the above arguments.
We evaluate the radiation reaction force as follows. The total mass and current quadrupole moment of the system are
Q T ij = Q ij + µx i x j , (3.12
)
S T ij = S spin ij + x i ǫ jkm x k ẋm , (3.13)
where from Eq. (2.11)
x i = r sin θ 1 - Q 4r 2 cos ϕ, r sin θ 1 - Q 4r 2 sin ϕ, r cos θ 1 + Q 4r 2 . (3.14)
Only the second term in Eq. (3.12) contributes to the time derivative of the quadrupole. We differentiate five times by using
d dt = 1 + Q 2r 2 cos(2 θ) d d t , (3.15)
to the order we are working as discussed above. After each differentiation, we eliminate any occurrences of dϕ/d t using Eq. ( 2 .24), and we eliminate any occurrences of the second order time derivatives d 2 r/d t2 and d 2 θ/d t2 in favor of first order time derivatives using (the time derivatives of) Eqs. (2.26) and (2.27). For computing the terms linear and quadratic in S 1 , we set the quadrupole Q to zero in all the formulae. We insert the resulting expression into the formula (3.9) for the self-acceleration, and then into Eqs. (3.2) -(3.4). We eliminate (dr/d t) 2 , (d θ/d t) 2 , and (dϕ/d t) in favor of E, L z , and K using Eqs. (2.24) -(2.27). In the final expressions for the instantaneous fluxes, we keep only terms that are of O(S), O(Q) and O(S 2 ) and obtain the following results: Ė = 160K 3r 6 + 64 3r 5 + 512E 15r 4 -40K 2 r7 + 272KE 5r 5 + 64E 2 5r 3
+ SL z r9 196K 2 + 952 3 r2 - 3668 5 K r -352KE r2 + 1024 3 E r3 + 128 5 E 2 r4 + 2Q r9 -49K 2 -169KL 2 z + r 532 5 K + 3307 15 L 2 z + 2r 2 - 20 3 + 47KE + 548 5 L 2 z E - 152 5 r3 E -16r 4 E 2 + Q r9 -562K 2 + 2998 3 K r - 320 3 r2 + 5072 5 KE r2 - 4048 15 r3 E -160r 4 E 2 cos(2 θ) + Q r6 sin(2 θ) 439K - 926 3 r - 1528 5 r2 E θ ṙ + S 2 r9 -K 2 + 22 3 K r - 28
3 r2 + 32 5 KE r2 -236 15 r3 E -32 5 r4 E 2 cos(2 θ) -r3 sin(2 θ) K + 2 3 r + 8 5 r2 E θ ṙ + S 2 r9 -49K 2 + 6KL 2 z + 2r 63K -16 3 L 2 z -98 3 + r2 112KE -48 5 L 2 z E -1652 15 r3 E -224 5 r4 E 2 , (3.16) Lz = 32L z r4 + 144L z E 5r 3 -24KL z r5 + S r7 -50K 2 + 240KL 2 z + 62 5 K r -7376 15 L 2 z r + 316 3 r2 + 56KE r2 -1824 5 EL 2 z r2 + 624 5 E r3 + 128 5 E 2 r4 + S r7 50K 2 -62 5 K r -316 3 r2 -56KE r2 -624 5 E r3 -128 5 E 2 r4 cos(2 θ) + S r4 -104K + 64r + 64E r2 sin(2 θ) ṙ θ + QL z 5r 7 660E r2 + 753r -360L 2 z -435K + 1601r + 1512r 2 E -1185K cos 2 θ + 174QL z r4 sin(2 θ) ṙ θ + 2S 2 L z r7 72 5 E r2 + 16r -9K , (3.17) and K = 16K 5r 5 20r + 18r 2 E -15K + SL z r7 280K 2 -14008 15 K r + 1264 3 r2 + 2496 5 E r3 -2528 5 KE r2 + 512 5 E 2 r4 + 12Q 5r 7 -45K 2 + rL 2 z (83 + 80rE) -115KL 2 z + 14K r(6 + 5rE) + 4Q 15r 7 cos(2 θ) -2175K 2 + 2975K r + 80r 2 + 3012KE r2 -112E r3 -168E 2 r4 + 2Q 15r 4 3075K -20r -192E r2 sin(2 θ) θ ṙ + 2S 2 r7 7K -2L 2 z -3K + 16 3 r + 24 5 E r2 + K cos(2 θ) 3K -16 3 r -24 5 E r2 + 2S 2 r4 sin(2 θ) -4K + 14 3 r + 16 5 E r2 θ ṙ. (3.18) C. Alternative set of constants of the motion
A body in a generic bound orbit in Kerr traces an open ellipse precessing about the hole's spin axis. For stable orbits the motion is confined to a toroidal region whose shape is determined by E, L z , K. The motion can equivalently be characterized by the set of constants inclination angle ι, eccentricity e, and semi-latus rectum p defined by Hughes [32] . The constants ι, p and e are defined by cos ι = L z / √ K, and by r± = p/(1 ± e), where r± are the turning points of the radial motion, and r is the Boyer-Lindquist radial coordinate. This parameterization has a simple physical interpretation: in the Newtonian limit of large p, the orbit of the particle is an ellipse of eccentricity e and semilatus rectum p on a plane whose inclination angle to the hole's equatorial plane is ι. In the relativistic regime p ∼ M , this interpretation of the constants e, p, and ι is no longer valid because the orbit is not an ellipse and ι is not the angle at which the object crosses the equatorial plane (see Ryan [14] for a discussion).
We adopt here analogous definitions of constants of motion ι, e and p, namely
cos(ι) = L z / √ K, (3.19) p 1 ± e = r± . (3.20)
Here K is the conserved quantity (2.23) or (2.25), and r± are the turning points of the radial motion using the r coordinate defined by Eq. (2.11), given by the vanishing of the right-hand side of Eq. (2.26). We now rewrite our results in terms of the new constants of the motion e, p and ι. We can use Eq. (2.26) together with the equations (3. 19 ) and (3.20) to write E, L z and K as functions of p, e and ι. To leading order in Q and S we obtain
K = p 1 - 2S cos ι p 3/2 3 + e 2 -1 + e 2 2Q cos 2 ι p 2 + 3 + e 2 Q 4p 2 , (3.21)
E = - (1 -e 2 ) 2p 1 + 2S cos ι p 3/2 1 -e 2 + 1 -e 2 Q p 2 cos 2 ι - 1 4 , (3.22)
L z = √ p cos ι 1 - S cos ι p 3/2 (3 + e 2 ) -1 + e 2 Q cos 2 ι p 2 + 3 + e 2 Q 8p 2 . (3.23)
As discussed in the introduction, the effects quadratic in S on the conservative dynamics scale as O(a 2 ǫ 6 ) and thus are not included in this analysis to O(a 2 ǫ 4 ). Inserting these relations into the expressions (3.16)-(3.18) gives, dropping terms of O(QS), O(Q 2 ) and O(QS 2 ): Ė = -8 15p 2 r7 75p 4 -100p 3 r + p 2 r2 11 -51e 2 + 32pr 3 1 -e 2 ) -6r 4 1 -e 2 2
4S cos ι 15p 7/2 r9 735p 6 -2751p 5 r + 10p 4 r2 (365 -6e 2 ) -128pr 5 (1 -e 2 ) 2 -48r 6 (e 2 -1) 3 + 64S cos ι 15p 3/2 r6 5p(-23 + 3e 2 ) -3r(-9 + e 2 + 8e 4 ) -Q 15p 4 r9 4005p 6 -6499p 5 r + 2p 4 r2 1577 -1977e 2 -24r 6 1 -e 2 3 -32p 3 r3 8 -33e 2 + 64pr 5
1 -2e 2 + e 4 - Q 15p 4 r9 24p 2 r4 5 -27e 2 + 22e 4 -pr 3 sin(2 θ) 6585p 2 -4630pr + 2292r 2 (1 -e 2 ) θ ṙ - Q 15p 4 r9 2p 2 cos(2 θ) 4215p 4 -7495p 3 r + 4p 2 r2 (1151 -951e 2 ) -1012pr 3 (1 -e 2 ) + 300r 4 (1 -2e 2 + e 4 ) - Q 15p 4 r9 cos(2ι) 2535p 6 -3307p 5 r + 12p 4 r2 (37 -237e 2 ) -48r 6 (1 -e 2 ) 3 + 800p 3 r3 (1 + e 2 ) + 128pr 5 (1 -2e 2 + e 4 ) + 204Q 15p 2 r5 cos(2ι) 1 + 2e 2 -3e 4 - 2S 2 15p 2 r9 84r 4 (1 -e 2 ) 2 (1 + e 2 ) 2 + 345p 4 -905p 3 r -413pr 3 (1 -e 2 ) + 2p 2 r2 (446 -201e 2 ) - S 2 15p 2 r9 cos(2 θ) 15p 4 -110p 3 r + 4p 2 r2 (47 -12e 2 ) -118pr 3 (1 -e 2 ) + 24r 4 (1 -e 2 ) 2 (1 + e 2 ) 2 + S 2 15r 9 cos(2ι) 45p 2 -80pr + 36r 2 (1 -e 2 ) - S 2 15pr 6 sin(2 θ) ṙ θ 15p 2 + 10pr -12r 2 (1 -e 2 ) , (3.24)
Lz = -
8 cos ι 5 √ pr 5 15p 2 -20pr + 9r 2 (1 -e 2 ) + 2S 15p 2 r7 525p 4 -1751p 3 r + 34p 2 r2 (61 -6e 2 ) + 12pr 3 (-69 + 29e 2 ) + 6r 4 (17 + 2e 2 -19e 4 ) + 2S 15p 2 r7 375p 4 -93p 3 r + 468pr 3 (1 -e 2 ) -10p 2 r2 (58 + 21e 2 ) -48r 4 (1 -2e 2 + e 4 ) cos(2 θ) + 4S 15p 2 r7 450p 4 -922p 3 r -60pr 3 (3 + e 2 ) -9p 2 r2 (-83 + 23e 2 ) + 27r 4 (1 + 2e 2 -3e 4 ) cos(2ι) - 8S pr 4 13p 2 -8pr + 4r 2 (1 -e 2 ) sin(2 θ) ṙ θ - Q cos ι 5p 5/2 r7 615p 4 -753p 3 r + 15p 2 r2 19 -31e 2 + 20pr 3 1 + 3e 2 + 9r 4 1 -6e 2 + 5e 4 - Q cos ι 5p 1/2 r7 cos(2 θ) 1185p 2 -1601pr + 756r 2 (1 -e 2 )
-2Q cos ι 5p 5/2 r7 2 cos(2ι) 45p
4 -18r 4 e 2 (1 -e 2 ) -45p 2 r2 (1 + e 2 ) + 20pr 3 (1 + e 2 ) -435p 3 r3 sin(2 θ) θ ṙ -2S 2 cos ι p 1/2 r7 9p 2 -16pr + 36 5 r2 (1 -e 2 ) , (3.25) and K = 16 5r 5 20pr -15p 2 -9r 2 (1 -e 2 ) + 8S cos ι 15p 3/2 r7 525p 4 -1751p 3 r + 2p 2 r2 (1172 -57e 2 ) + 12pr 3 (-99 + 19e 2 ) -24r 4 (-11 + 4e 2 + 7e 4 ) + 2Q 5p 2 r7 -615p 4 + 753p 3 r + 30p 2 r2 (17e 2 -9) + 72r 4 e 2 (1 -e 2 ) -40pr 3 (1 + 3e 2 ) + 2Q 5p 2 r7 cos(2ι) -345p 4 + 249p 3 r -160pr 3 (1 + e 2 ) + 120p 2 r2 (1 + 3e 2 ) + 36r 4 (1 + 2e 2 -3e 4 ) + 2Q 15p 2 r7 2 cos(2 θ) 2175p 4 -2975p 3 r -56pr 3 (1 -e 2 ) + 2p 2 r2 (713 -753e 2 ) + 42r 4 (1 -2e 2 + e 4 ) + 2Q 15pr 4 sin(2 θ) 3075p 2 -20pr + 96r 2 (1 -e 2 ) ṙ θ + 2S 2 r7 2 -9p 2 + 16pr -36 5 r2 (1 -e 2 ) + cos(2 θ) + cos(2ι) 3p 2 -16 3 pr + 12 5 r2 (1 -e 2 ) + 4S 2 pr 4 sin(2 θ) ṙ θ -2p 2 + 7 3 pr -4 5 r2 (1 -e 2 ) . (3.26)
In this section we will compute the infinite timeaverages Ė , Lz and K of the fluxes. These averages are defined by
Ė ≡ lim T →∞ 1 T T /2 -T /2 Ė(t)dt.
(3.27)
These time-averaged fluxes are sufficient to evolve orbits in the adiabatic regime (except for the effect of res-onances) [12, 25] . In Appendix II, we present two different ways of computing the time averages. The first approach is based on decoupling the r and θ motion using the analog of the Mino time parameter for geodesic motion in Kerr [12] . The second approach uses the explicit Newtonian parameterization of the orbital motion.
Both averaging methods give the following results:
Ė = - 32 5 (1 -e 2 )
3/2 p 5 1 + 73 24 e 2 + 37 96 e 4 -S p 3/2 73 12 + 823 24 e 2 + 949 32 e 4 + 491 192 e 6 cos(ι) -Q p 2 1 2 + 85 32 e 2 + 349 128 e 4 + 107 384 e 6 + 11 4 + 273 16 e 2 + 847 64 e 4 + 179 192 e 6 cos(2ι) + S 2 p 2 13 192 + 247 384 e 2 + 299 512 e 4 + 39 1024 e 6 -1 192 + 19 384 e 2 + 23 512 e 4 + 3 1024 e 6 cos(2ι) ,(3.28) Lz = -32 5 (1 -e 2 ) 3/2 p 7/2 cos ι 1 + 7 8 e 2 -S 2p 3/2 cos ι 61 24 + 7e 2 + 271 64 e 4 + 61 8 + 91 4 e 2 + 461 64 e 4 cos(2ι) -Q 16p 2 -3 -45 4 e 2 + 19 8 e 4 + 45 + 148e 2 + 331 8 e 4 cos(2ι) + S 2 16p 2 1 + 3e 2 + 3 8 e 4 , (3.29) K = -64 5 (1 -e 2 ) 3/2 p 3 1 + 7 8 e 2 -S 2p 3/2 97 6 + 37e 2 + 211 16 e 4 cos(ι) -Q p 2 1 + 8 3 e 2 + 11 12 e 4 + 13 4 + 841 96 e 2 + 449 192 e 4 cos(2ι) + S 2 p 2 13 192 + 13 64 e 2 + 13 512 e 4 -1 192 + 1 64 e 2 + 1 512 e 4 cos(2ι) . (3.30) Using Eqs. (3.21) and (3.23), we obtain from (3.28) -(3.30) the following time averaged rates of change of the orbital elements e, p, ι: ṗ = -64 5 (1 -e 2 ) 3/2 p 3 1 + 7e 2 8 -S cos(ι) 96p 3/2 1064 + 1516e 2 + 475e 4 -Q 8p 2 14 + 149e 2 12 + 19e 4 48 + 50 + 469e 2 12 + 227e 4 24 cos(2ι) + S 2 64p 2 1 3 + e 2 + e 4 8 [13 -cos(2ι)] , (3.31) ė = -304 15 e(1 -e 2 ) 3/2 p 4 1 + 121e 2 304 + Se(1 -e 2 ) 3/2 cos(ι) 5p 11/2 1172 + 932e 2 + 1313e 4 6 + Q(1 -e 2 ) 3/2 ep 6 32 + 785e 2 3 -219e 4 2 + 13e 6 + 32 + 2195e 2 3 + 251e 4 + 218e 6 3 cos(2ι) -S 2 e(1 -e 2 ) 3/2 8p 6 2 + 3e 2 + e 4 4 [13 -cos(2ι)] , (3.32) ι = S sin(ι)(1 -e 2 ) 3/2 p 11/2 244 15 + 252 5 e 2 + 19 2 e 4 -1 -e 2 3/2 S 2 sin(2ι) 240p 6 8 + 3e 2 8 + e 2 + Q cot(ι)(1 -e 2 ) 3/2 60p 6 312 + 736e 2 -83e 4 -408 + 1268e 2 + 599e 4 cos(2ι) . (3.33) IV. APPLICATION TO BLACK HOLES A. Qualitative discussion of results
The above results for the fluxes, Eqs. (3.31), (3.32) and (3.33) show that the correction terms at O(a 2 ǫ 4 ) due to the quadrupole have the same type of effect on the evolution as the linear spin correction computed by Ryan: they tend to circularize eccentric orbits and change the angle ι such as to become antialigned with the symmetry axis of the quadrupole.
The effects of the terms quadratic in spin are qualitatively different. In the expression (3.28) for Ė , the coefficient of cos(2ι) due to the spin self-interaction has the same sign as the quadrupole term, while the terms not involving ι have the opposite sign. The terms involving cos(2ι) in Eq. (3.30) for K of O(Q) and O(S 2 ) terms have the same sign, while the terms not involving ι have the opposite sign. The fractional spin-spin correction to Lz , Eq. (3.29), has no ι-dependence, and in expression (3.33) for ι , the dependence on ι of the two effects O(Q) and O(S 2 ) is different, too. This is not surprising as the O(Q) effects included here are corrections to the conservative orbital dynamics, while the effects of O(S 2 ) that we included are due to radiation reaction.
The terms linear in the spin in our results for the time averaged fluxes, Eqs. (3.28) - (3.33) , agree with those computed by Ryan, Eqs. (14a) -(15c) of [15] , and with those given in Eqs. (2.5) -(2.7) of Ref. [33] , when we use the transformations to the variables used by Ryan given in Eqs. (2.3) -(2.4) in [33] .
Equation (3.28) for the time averaged energy flux agrees with Eq. (3.10) of Gergely [23] and Eq. (4.15) of [18] when we use the following transformations:
K = L2 1 - Q 2 L4 Ā2 sin 2 κ cos δ -(1 -Ā2 ) cos 2 κ = L2 1 - Q L4 E cos 2 κ - Q 2 L4 (1 + 2 L2 ) sin 2 κ cos δ , (4.1
)
cos ι = cos κ 1 + Q 2 L4 E cos 2 κ + Q 2 L4 (1 + 2 L2 ) sin 2 κ cos δ , (4.2)
ξ 0 = 1 2 (δ + κ), (4.3
)
ξ 0 = (ψ 0 -ψ i ) + π 2 , (4.4)
where Ā, L, κ, δ, ψ 0 and ψ i are the quantities used by Gergely. The first relation here is obtained from the turning points of the radial motion as follows. We compute r± in terms of E and K and map these expressions back to r using Eqs. (2.11) . The result can then be compared with the turning points in Gergely's variables, Eq.
(2.19) of [23] , using the fact that E is the same in both cases. Instead of the evolution of the constants of motion K and L z , Gergely computes the rates of change of the magnitude L of the orbital angular momentum and of the angle κ defined by cos κ = (L • S)/L. Using the transformations (4.1) -( 4 .4) and the definition of κ we verify that our Eq. (3.29) agrees with the Lz computed using Gergely's Eqs. (3.23) and (3.35) in [23] and Eq. (4.30) of [18] .
In the limit of the circular equatorial orbits analyzed by Poisson [22] , our Eq. (3.28) agrees with Poisson's Eq. ( 22 ) when we use the transformations and specializations: The main improvement of our analysis over Gergely's is that we express the results in terms of the Carter-type constant K, which facilitates comparing our results with other analyses of black hole inspirals. Our computations also include the spin curvature scattering effects for all three constants of motion; Gergely [18] only considers these effects for two of them: the energy and magnitude of angular momentum, not for the third conserved quantity.
p = 1 v 2 1 - Q 4 v 4 , (4.5
When we expand Eq. (3.28) for small inclination angles and specialize to circular orbits, then after converting p to the parameter v using Eq. (4.5), we obtain Ė = -32 5p 5 1 -
1 p 2 2Q + S 2 16 + ι 2 2p 2 11Q - S 2 48 = - 32 5p 5 1 - a 2 v 4 16 33 - 527 6 ι 2 . (4.9)
This result agrees with the terms at O(a 2 v 4 ) of Eq. (3.13) of Shibata et al. [24] , whose calculations were based on the fully relativistic expressions. This agreement is a check that we have taken into account all the contributions at O(a 2 ǫ 4 ). The analysis in Ref. [24] could not distinguish between effects due to the quadrupole and those due curvature scattering, but we can see from Eq. (4.9) that those two interactions have the opposite dependence on ι. Comparing (4.9) with Eq. (3.7) of [24] (which gives the fluxes into the different modes (l = 2, m, n), where m and n are the multiples of the ϕ and θ frequencies), we see that the terms in the (2, ±2, 0) and the (2, ±1, ±1) modes are entirely due to the quadrupole, while the spin-spin interaction effects are fully contained in the (2, ±1, 0) and (2, 0, ±1) modes.
In this section, we show that for a single axisymmetric multipole interaction, it is not possible to find an analog of the Carter constant (a conserved quantity which does not correspond to a symmetry of the Lagrangian), except for the cases of spin (treated by Ryan [15] ) and mass quadrupole moment (treated in this paper). Our proof is valid only in the approximations in which we work -expanding to linear order in the mass ratio, to the leading post-Newtonian order, and to linear order in the multipole. However we will show below that with very mild additional smoothness assumptions, our nonexistence result extends to exact geodesic motion in exact vacuum spacetimes.
We start in Sec. V A by showing that there is no coordinate system in which the Hamilton-Jacobi equation is separable. Now separability of the Hamilton-Jacobi equation is a sufficient but not a necessary condition for the existence of a additional conserved quantity. Hence, this result does not yield information about the existence or non-existence of an additional constant. Nevertheless we find it to be a suggestive result. Our actual derivation of the non-existence is based on Poisson bracket computations, and is given in Sec. V B.
Consider a binary of two point masses m 1 and m 2 , where the mass m 1 is endowed with a single axisymmetric current multipole moment S l or axisymmetric mass multipole moment I l . In this section, we show that the Hamilton-Jacobi equation for this motion, to linear order in the multipoles, to linear order in the mass ratio and to the leading post-Newtonian order, is separable only for the cases S 1 and I 2 .
We choose the symmetry axis to be the z-axis and write the action for a general multipole as
S = dt 1 2 ṙ2 + r 2 θ2 + r 2 sin 2 θ φ2 + 1 r + f (r, θ) + g(r, θ) φ + E] . (5.1)
For mass moments, g(r, θ) = 0, while for current moments f (r, θ) = 0. For an axisymmetric multipole of order l, the functions f and g will be of the form
f (r, θ) = c l I l P l (cos θ) r l+1 , g(r, θ) = d l S l sin θ∂ θ P l (cos θ) r l , (5.2
) where P l (cos θ) are the Legendre polynomials and c l and d l are constants. We will work to linear order in f and g. In Eq. (5.1), we have added the energy term needed when doing a change of time variables, cf. the discussion before Eq. (2.14) in section III. Since ϕ is a cyclic coordinate, p ϕ = L z is a constant of motion and the system has effectively only two degrees of freedom. Note that in the case of a current moment, there will be correction term in L z :
L z = r 2 sin 2 θ φ + g(r, θ).
(
Next, we switch to a different coordinate system (r, θ, ϕ) defined by
r = r + α(r, θ, L z ), (5.4)
θ = θ + β(r, θ, L z ), (5.5)
where the functions α and β are yet undetermined. We also define a new time variable t by
dt = 1 + γ(r, θ, L z ) d t. (5.6)
Since we work to linear order in f and g, we can work to linear order in α, β, and γ. We then compute the action in the new coordinates and drop the tildes. The Hamiltonian is given by
H = 1 2 p 2 r (1 + γ -2α ,r ) + p 2 θ 2r 2 (1 - 2α r -2β ,θ + γ) + p r p θ r 2 (-α ,θ -r 2 β ,r ) -E(1 + γ) + L 2 z 2r 2 sin 2 θ (1 + γ - 2α r -2β cot θ) - 1 r (1 - α r + γ) -f - gL z r 2 sin 2 θ (5.7)
and the corresponding Hamilton-Jacobi equation is
0 = ∂W ∂r 2 Ĉ1 + ∂W ∂θ 2 Ĉ2 r 2 +2 ∂W ∂r ∂W ∂θ Ĉ3 r 2 + 2 V , (5.8)
where we have denoted
Ĉ1 = J(r, θ) [1 + γ -2α ,r ] = 1 + γ -2α ,r + j, (5.9) Ĉ2 = J(r, θ) 1 - 2α r -2β ,θ + γ = 1 - 2α r -2β ,θ + γ + j, (5.10)
Ĉ3 = J(r, θ) -α ,θ -r 2 β ,r = -α ,θ -r 2 β ,r , (5.11)
V = J(r, θ) L 2 z 2r 2 sin 2 θ (1 + γ - 2α r -2β cot θ) - 1 r (1 - α r + γ) -E(1 + γ) -f - gL z r 2 sin 2 θ = L 2 z 2r 2 sin 2 θ (1 + γ - 2α r -2β cot θ + j) -E(1 + γ + j) - 1 r (1 - α r + γ + j) -f - gL z r 2 sin 2 θ . (5.12)
The unperturbed problem is separable, so make the perturbed problem separable, we have multiplied the Hamilton-Jacobi equation by an arbitrary function J(r, θ), which can be expanded as J(r, θ) = 1 + j(r, θ), where j(r, θ) is a small perturbation.
To find a solution of the form W = W r (r) + W θ (θ), we first specialize to the case where Ĉ3 = 0:
-Ĉ3 = β ,r r 2 + α ,θ = 0.
(5.13)
We differentiate Eq. (5.8) with respect to θ, using Eq.
(5.8) to write (dW r /dr) 2 in terms of (dW θ /dθ) 2 and then differentiate the result with respect to r to obtain
0 = dW θ dθ 2 ∂ r ∂ θ Ĉ2 Ĉ2 - ∂ θ Ĉ1 Ĉ1 +2∂ r r 2 ∂ θ V Ĉ2 - r 2 V ∂ θ Ĉ1 Ĉ1 Ĉ2 . (5.14)
Expanding Eq. (5.14) to linear order in the small quantities then yields the two conditions for the kinetic and the potential part of the Hamiltonian to be separable:
0 = ∂ r ∂ θ 2α ,r - 2α r -2β ,θ , (5.15) 0
= L 2 z sin 2 θ 2β ,r cot 2 θ -3β ,rθ cot θ + β ,r csc 2 θ + L 2 z sin 2 θ ∂ r - α ,θ r + α ,rθ -∂ r ∂ θ c l I l r l-1 P l (cos θ) + d l S l L z r l sin θ ∂ θ P l (cos θ) -∂ r r 2α ,rθ - α ,θ r + 2Er 2 α ,rθ , (5.16)
The unperturbed motion for a bound orbit is in a plane, so we can switch from spherical to plane polar coordinates (r, ψ). In terms of these coordinates, we have H 0 = p 2 r /2+p 2 ψ /2, K 0 = p 2 ψ , and cos θ = sin ι sin(ψ+ψ 0 ), with cos ι = L z / √ K and the constant ψ 0 denoting the angle between the direction of the periastron and the intersection between the orbital and equatorial plane. Then Eq. (5.32) becomes
d dt δK = η(t), (5.33)
η(t) = - 2p ψ d l S l L z sin ι r l+2 (t) ∂ ψ ∂ ψ P l (sin ι sin(ψ(t) + ψ 0 )) cos(ψ(t) + ψ 0 ) + 2p ψ c l I l r l+1 (t)
∂ ψ P l (sin ι sin(ψ(t) + ψ 0 )).
(5.34)
For unbound orbits, one can always integrate Eq. (5.33) to determine δK. However, for bound periodic orbits there is a possible obstruction: the solution for the conserved quantity K 0 + δK will be single valued if and only if the integral of the source over the closed orbit vanishes, T orb 0 η(t)dt = 0.
(5.35)
Here, T orb is the orbital period. In other words, the partial differential equation (5.32) has a solution δK if and only if the condition (5.35) is satisfied. This is the same condition as obtained by the Poincare-Mel'nikov-Arnold method, a technique for showing the non-integrability and existence of chaos in certain classes of perturbed dynamical systems [35] . Thus, it suffices to show that the condition (5.35) is violated for all multipoles other than the spin and mass quadrupole. To perform the integral in Eq. (5.35), we use the parameterization for the unperturbed motion, r = K/(1 + e cos ψ) and dt/dψ = K 3/2 /(1 + e cos ψ) 2 , so that the condition for the existence of a conserved quantity A lnjk e j (sin ι) l-2n (sin ψ 0 ) k (cos ψ 0 ) j-k 2π 0 dχ (sin χ) j-k+1 (cos χ) k+l-2n-1
K 0 + δK becomes 2π 0 dψ c l I l (1 + e cos ψ) l-1 ∂ ψ P l (sin ι sin(ψ + ψ 0 )) - d l S l L z K sin ι (1 + e cos ψ) l ∂ ψ ∂ ψ P l (sin ι sin(ψ + ψ 0 )) cos(ψ + ψ 0 ) = 0.
+ d l S l L z K N n=0 l j=0
B lnjk e j (sin ι) l-2n-1 (sin ψ 0 ) k (cos ψ 0 ) j-k 2π 0 dχ (sin χ) j-k+1 (cos χ) k+l-2n-2 .
(5.38)
The coefficients A lnkj and B lnkj are
A lnkj = (-1) n+k+1 (l -1)!(2l -2n)! 2 l n!(l -1 -j)!k!(j -k)!(l -n)!(l -2n -1)! , B lnkj = (-1) n+k l!(2l -2n)! 2 l n!(l -j)!k!(j -k)!(l -n)!(l -2n -2)! . (5.39)
The only non-vanishing contribution to the integrals in Eq. (5.38) will come from terms with even powers of both cos χ and sin χ. These can be evaluated as multiples of the beta function:
0 = c l I l N n=0 l-1 j=0
C lnjk e j (sin ι) l-2n (sin ψ 0 ) k (cos ψ 0 ) j-k δ (j-k+1),even δ (l+k-1),even
+ d l S l L z K N n=0 l j=0
D lnjk e j (sin ι) l-2n-1 (sin ψ 0 ) k (cos ψ 0 ) j-k δ (j-k+1),even δ (l+k),even .
(5.40)
Here, the coefficients are
C lnjk = 2Γ( j 2 -k 2 + 1)Γ( k 2 + l 2 -n) Γ( j 2 + l 2 -n + 1) A lnkj , D lnjk = 2Γ( j 2 -k 2 + 1)Γ( k 2 + l 2 -n -1 2 ) Γ( j 2 + l 2 -n + 3 2 )
B lnkj (5.41)
Eq. (5.40) shows that for even l, terms with j =even (odd) and k =odd (even) give a non-vanishing contribution for the case of a mass (current) multipole, and hence K 0 +δK is not a conserved quantity for the perturbed motion. Note that terms with j =even and k =odd for even l occur only for l > 3, so for l = 2 the mass quadrupole term in Eq. ( 5 .40) vanishes and therefore there exists an analog of the Carter constant, which is consistent with our results of Sec. II and our separability analysis. For odd l, terms with j =odd (even) and k =even (odd) are finite for I l (S l ). Note that for the case l = 1 of the spin, the derivatives with respect to χ in Eq. (5.37) evaluate to zero, so in this case there also exists a Carter-type constant. These results show that for a general multipole other than I 2 and S 1 , there will not be a Carter-type constant for such a system.
Our result on the non-existence of a Carter-type constant can be extended, with mild smoothness assumptions, to falsify the conjecture that all exact, axisymmetric vacuum spacetimes posess a third constant of the motion for geodesic motion. Specifically, we fix a multipole order l, and we assume:
• There exists a one parameter family (M, g ab (λ)) of spacetimes, which is smooth in the parameter λ, such that λ = 0 is Schwarzschild, and each spacetime g ab (λ) is stationary and axisymmetric with commuting Killing fields ∂/∂t and ∂/∂φ, and such that all the mass and current multipole moments of the spacetime vanish except for the one of order l. On physical grounds, one expects a one parameter family of metrics with these properties to exist.
• We denote by H(λ) the Hamiltonian on the tangent bundle over M for geodesic motion in the metric g ab (λ). By hypothesis, there exists for each λ a conserved quantity M (λ) which is functionally independent of the conserved energy and angular momentum. Our second assumption is that M (λ) is differentiable in λ at λ = 0. One would expect this to be true on physical grounds.
• We assume that the conserved quantity M (λ) is invariant under the symmetries of the system:
L ξ M (λ) = L η M (λ) = 0,
where ξ and η are the natural extensions to the 8 dimensional phase space of the Killing vectors ∂/∂t and ∂/∂φ. This is a very natural assumption.
These assumptions, when combined with our result of the previous section, lead to a contradiction, showing that the conjecture is false under our assumptions.
To prove this, we start by noting that M (0) is a conserved quantity for geodesic motion in Schwarzschild, so it must be possible to express it as some function f of the three independent conserved quantities:
M (0) = f (E, L z , K 0 ).
(5.42)
Here E is the energy, L z is the angular momentum, and K 0 is the Carter constant. Differentiating the exact relation {H(λ), M (λ)} = 0 and evaluating at λ = 0 gives
{H 0 , M 1 } = ∂f ∂E {E, H 1 }+ ∂f ∂L z {L z , H 1 }+ ∂f ∂K 0 {K 0 , H 1 }, (5.43
) where H 0 = H(0), H 1 = H ′ (0), and M 1 = M ′ (0). As before, we can regard this is a partial differential equation that determines M 1 , and a necessary condition for solutions to exist and be single valued is that the integral of the right hand side over any closed orbit must vanish:
∂f ∂E {E, H 1 } + ∂f ∂L z {L z , H 1 } + ∂f ∂K 0 {K 0 , H 1 } = 0.
(5.44) Now strictly speaking, there are no closed orbits in the eight dimensional phase space. However, the argument of the previous section applies to orbits which are closed in the four dimensional space with coordinates (r, θ, p r , p θ ), since by the third assumption above everything is independent of t and φ, and p t and p φ are conserved. Here (t, r, θ, φ) are Schwarzschild coordinates and (p t , p r , p θ , p φ ) are the corresponding conjugate momenta.
Next, we can pull the partial derivatives ∂f /∂E etc. outside of the integral. It is then easy to see that the first two terms vanish, since there do exist a conserved energy and a conserved z-component of angular momentum for the perturbed system. Thus, Eq. (5.44) reduces to ∂f ∂K 0 {K 0 , H 1 } = 0.
(5.45)
Since M (0) is functionally independent of E and L z , the prefactor ∂f /∂K 0 must be nonzero, so we obtain {K 0 , H 1 } = 0.
(5.46)
The result (5.46) applies to fully relativistic orbits in Schwarzschild. We need to take the Newtonian limit of this result in order to use the result we derived in the previous section. However, the Newtonian limit is a little subtle since Newtonian orbits are closed and generic relativistic orbits are not closed. We now discuss how the limit is taken.
The integral (5.46) is taken over any closed orbit in the four dimensional phase space (r, θ, p r , p θ ) which corresponds to a geodesic in Schwarzschild. Such orbits are non generic; they are the orbits for which the ratio between the radial and angular frequencies ω r and ω θ is a rational number. We denote by q r and q θ the angle variables corresponding to the r and θ motions [36] . These variables evolve with proper time τ according to q r = q r,0 + ω r τ, (5.47a)
q θ = q θ,0 + ω θ τ, (5.47b)
where q r,0 and q θ,0 are the initial values. We denote the integrand in Eq. (5.46) by I(q r , q θ , a, ε, ι),
where I is some function, and a, ε and ι are the parameters of the geodesic defined by Hughes [32] (functions of E, L z and K 0 ). The result (5.46) can be written as
1 T T /2 -T /2 dτ I[q r (τ ), q θ (τ ), a, ε, ι] = 0, (5.48)
where T = T (a, ε, ι) is the period of the r, θ motion. Since the variables q r and q θ are periodic with period 2π, we can express the function I as a Fourier series I(q r , q θ , a, ε, ι) =
= ∞ n,m=-∞ I nm (a, ε, ι)e inqr,0+imq θ,0 ×Si [(nω r + mω θ )T /2] , (5.50)
where Si(x) = sin(x)/x. Since the initial conditions q r,0 and q θ,0 are arbitrary, it follows that
I nm (a, ε, ι)Si [(nω r + mω θ )T /2] = 0 (5.51)
for all n, m.
Next, for closed orbits the ratio of the frequencies must be a rational number, so
w r w θ = p q , (5.52)
where p and q are integers with no factor in common. These integers depend on a, ε and ι. The period T is given by 2π/T = qω r = pω θ . The second factor in Eq.
(
.51) now simplifies to Si (np + mq)π pq , (5.53) which vanishes if and only if n = nq, m = mp, n + m = 0, (5.54) for integers n, m. It follows that I nm (a, ε, ι) = 0 (5.55)
for all n, m except for values of n, m which satisfy the condition (5.54) Consider now the Newtonian limit, which is the limit a → ∞ while keeping fixed ε and ι and the mass of the black hole. We denote by I N (q r , q θ , a, ε, ι) the Newtonian limit of the function I(q r , q θ , a, ε, ι). The integral (5.48) in the Newtonian limit is given by the above computation with p = q = 1, since ω r = ω θ in this limit. This gives
1 T dτ I N = ∞ n=-∞
I N n,-n (a, ε, ι) e in(qr,0-q θ,0 ) , (5.56)
where I N nm are the Fourier components of I N . In the previous subsection, we showed that this function is nonzero, which implies that there exists a value k of n for which I N k,-k = 0. Now as a → ∞, we have ω r /ω θ → 1, and hence from Eq. (5.52) there exists a critical value a c of a such that the values of p and q exceed k for all closed orbits with a > a c . (We are keeping fixed the values of ε and ι). as a → ∞. This completes the proof. Hence, if the three assumptions listed at the start of this subsection are satisfied, then the conjecture that all vacuum, axisymmetric spacetimes possess a third constant of the motion is false.
Finally, it is sometimes claimed in the classical dynamics literature that perturbation theory is not a sufficiently powerful tool to assess whether the integrability of a system is preserved under deformations. An example that is often quoted is the Toda lattice Hamiltonian [38, 39] . This system is integrable and admits a full set of constants of motion in involution. However, if one approximates the Hamiltonian by Taylor expanding the potential about the origin to third order, one obtains a system which is not integrable. This would seem to indicate that perturbation theory can indicate a non-integrability, while the exact system is still integrable.
In fact, the Toda lattice example does not invalidate the method of proof we use here. If we write the Toda lattice Hamiltonian as H(q, p), then the situation is that H(λq, p) is integrable for λ = 1, but it is not integrable for 0 < λ < 1. Expanding H(λq, p) to third order in λ gives a non-integrable Hamiltonian. Thus, the perturbative result is not in disagreement with the exact result for 0 < λ < 1, it only disagrees with the exact result for λ = 1. In other words, the example shows that perturbation theory can fail to yield the correct result for finite values of λ, but there is no indication that it fails in arbitrarily small neighborhoods of λ = 0. Our application is qualitatively different from the Toda lattice example since we have a one parameter family of Hamiltonians H(λ) which by assumption are integrable for all values of λ.
We have examined the effect of an axisymmetric quadrupole moment Q of a central body on test particle inspirals, to linear order in Q, to the leading post-Newtonian order, and to linear order in the mass ratio. Our analysis shows that a natural generalization of the Carter constant can be defined for the quadrupole interaction. We have also analyzed the leading order spin selfinteraction effect due to the scattering of the radiation off the spacetime curvature due to the spin. Combining the effects of the quadrupole and the leading order effects linear and quadratic in the spin, we have obtained expressions for the instantaneous as well as time-averaged evolution of the constants of motion for generic orbits under gravitational radiation reaction, complete at O(a 2 ǫ 4 ). We have also shown that for a single multipole interaction other than Q or spin, in our approximations, a Cartertype constant does not exist. With mild additional assumptions, this result can be extended to exact spacetimes and falsifies the conjecture that all axisymmetric vacuum spacetimes possess a third constant of motion for geodesic motion.
This research was partially supported by NSF grant PHY-0457200. We thank Jeandrew Brink for useful correspondence.
Appendix A: Time variation of quadrupole: order of magnitude estimates
In this appendix, we give an estimate of the timescale T evol for the quadrupole to change. The analysis in the body of this paper is valid only when T evol ≫ T rr , where T rr is the radiation reaction time, since we have neglected the time evolution of the quadrupole. We distinguish between two cases: (i) when the central body is exactly nonspinning but has a quadrupole, and (ii) when the central body has finite spin in addition to the quadrupole.
For the purpose of a crude estimate, the relevant interaction is the tidal interaction with energy
Q ij E ij ∼ - m 2 r 3 QI cos 2 θ, (A1)
where E ij is the tidal field, θ is the angle between the symmetry axis and the normal to the orbital plane of m 2 , and we have written the quadrupole as Q ∼ QI, where Q is dimensionless and I is the moment of inertia.
For small deviations from equilibrium, the relevant piece of the Lagrangian is schematically
L ∼ I ψ2 + QI m 2 r 3 ψ 2 . (A2)
We define the evolution timescale T evol to be the time it takes for the angle to change by an amount of order unity, and since the amplitude of the oscillation scales roughly as ∼ m 2 /m 1 , the evolution time scales as
T -2 evol ∼ m 2 2 m 2 1 Q m 2 M ω 2 orbit , (A3)
where ω 2 orbit = M/r 3 . Thus, the ratio of the evolution timescale compared to the radiation reaction timescale scales as
T evol /T rr ∼ 1/ Q m 1 m 2 µ M 1/2 M r 5/2
. (A4)
When the body is spinning the effect of the tidal coupling is to cause a precession. For the purpose of this estimate, we calculate the torque on m 1 due to the companion's Newtonian field. The torque N scales as
N i ∼ ǫ imj Q mk E jk . ( A5
)
We assume that the precession is slow, i.e.
ω prec ≪ S/m 1 m 2 M , ( A6
)
where ω prec is the precession frequency and S = S/m 2 1 is the dimensionless spin. This gives the approximate scaling of the precession timescale as (cf. [37] ) )
T prec /T rr ∼ S Q M r . ( A7 | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We analyze the effect of gravitational radiation reaction on generic orbits around a body with an axisymmetric mass quadrupole moment Q to linear order in Q, to the leading post-Newtonian order, and to linear order in the mass ratio. This system admits three constants of the motion in absence of radiation reaction: energy, angular momentum along the symmetry axis, and a third constant analogous to the Carter constant. We compute instantaneous and time-averaged rates of change of these three constants. For a point particle orbiting a black hole, Ryan [15] has computed the leading order evolution of the orbit's Carter constant, which is linear in the spin. Our result, when combined with an interaction quadratic in the spin (the coupling of the black hole's spin to its own radiation reaction field), gives the next to leading order evolution. The effect of the quadrupole, like that of the linear spin term, is to circularize eccentric orbits and to drive the orbital plane towards antialignment with the symmetry axis. In addition we consider a system of two point masses where one body has a single mass multipole or current multipole of order l. To linear order in the mass ratio, to linear order in the multipole, and to the leading post-Newtonian order, we show that there does not exist an analog of the Carter constant for such a system (except for the cases of an l = 1 current moment and an l = 2 mass moment). Thus, the existence of the Carter constant in Kerr depends on interaction effects between the different multipoles. With mild additional assumptions, this result falsifies the conjecture that all vacuum, axisymmetric spacetimes posess a third constant of the motion for geodesic motion."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Erratum: Evolution of the Carter constant for inspirals into a black hole: Effect of the black hole quadrupole [Phys. Rev. D 75, 124007 (2007) ] Éanna É. Flanagan, Tanja Hinderer In Eqs. (3.16 ), (3.17), (3.18) , (3.24) , (3.25) and (3.26) of this paper, the variable r should be replaced everywhere by the variable r, and the variable θ should be replaced everywhere by the variable θ. The definitions of r and θ are given in Eq. (2.11) . These replacements do not affect the any of the subsequent results in the paper.\n\nAlso, the right hand side of Eq. ( B3 ) is missing a term -4SL z r and Eq. (2.24) is missing a factor of dϕ/d t in front of Q.\n\nSome terms are missing in Eqs. (3.18) , (3.26 ) and (3.30) - (3.33) . The additional terms in Eqs. (3.18 ) and (3.26) are -8Q 15r 7 -75K 2 + 2K r(51rE + 50) + 8r 2 (rE + 1)(3rE + 5) , and 8Q 15p 2 r7 25p 3 (3p -4r) + p 2 r2 11 -51e 2 + 32pr 3 1 -e 2 + 6r 4 1 -e 2 2 , respectively. These result in additional fractional corrections to Eq. (3.30) given by -Q p 2 1 2 + 73 48 e 2 + 37 192 e 4 , and the full expression replacing the O(Q) terms in Eq. (3.30) is then K = -64 5 (1 -e 2 ) 3/2 p 3 1 + 7e 2 8 -Q p 2 1 + 8 3 e 2 + 11 12 e 4 + 13 4 + 841 96 e 2 + 449 192 e 4 cos(2ι) + O(S), O(S 2 ) -terms. Equations (3.31), (3.32) and (3.33) contain typos in the O(S) and O(Q) terms, the corrected expressions are given below. We thank P. Komorowski for pointing this out. Equation (3.31) should be replaced by ṗ = -64 5 (1 -e 2 ) 3/2 p 3 1 + 7e 2 8 -S cos(ι) 96p 3/2 1064 + 1516e 2 + 475e 4 -Q 8p 2 14 + 149e 2 12 + 19e 4 48 + 50 + 469e 2 12 + 227e 4 24 cos(2ι) + S 2 64p 2 1 3 + e 2 + e 4 8 [13 -cos(2ι)] , (0.1) Equation (3.32) should be replaced by ė = -304 15 e(1 -e 2 ) 3/2 p 4 1 + 121e 2 304 + Se(1 -e 2 ) 3/2 cos(ι) 5p 11/2 1172 + 932e 2 + 1313e 4 6 + Q(1 -e 2 ) 3/2 ep 6 32 + 785e 2 3 -219e 4 2 + 13e 6 + 32 + 2195e 2 3 + 251e 4 + 218e 6 3 cos(2ι) -S 2 e(1 -e 2 ) 3/2 8p 6 2 + 3e 2 + e 4 4 [13 -cos(2ι)] , (0.2) and the corrected Eq. (3.33) is ι = S sin(ι)(1 -e 2 ) 3/2 p 11/2 244 15 + 252 5 e 2 + 19 2 e 4 -1 -e 2 3/2 S 2 sin(2ι) 240p 6 8 + 3e 2 8 + e 2 + Q cot(ι)(1 -e 2 ) 3/2 60p 6 312 + 736e 2 -83e 4 -408 + 1268e 2 + 599e 4 cos(2ι) . (0.3)"
},
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION AND SUMMARY",
"text": "The inspiral of stellar mass compact objects with masses µ in the range µ ∼ 1 -100M ⊙ into massive black holes with masses M ∼ 10 5 -10 7 M ⊙ is one of the most important sources for the future space-based gravitational wave detector LISA. Observing such events will provide a variety of information: (i) the masses and spins of black holes can be measured to high accuracy (∼ 10 -4 ); which can constrain the black hole's growth history [1] ; (ii) the observations will give a precise test of general relativity in the strong field regime and unambiguously identify whether the central object is a black hole [2] ; and (iii) the measured event rate will give insight into the complex stellar dynamics in galactic nuclei [1] . Analogous inspirals may also be interesting for the advanced stages of ground-based detectors: it has been estimated that advanced LIGO could detect up to ∼ 10 -30 inspirals per year of stellar mass compact objects into intermediate mass black holes with masses M ∼ 10 2 -10 4 M ⊙ in globular clusters [3] . Detecting these inspirals and extracting information from the datastream will require accurate models of the gravitational waveform as templates for matched filtering. For computing templates, we therefore need a detailed understanding of the how radiation reaction influences the evolution of bound orbits around Kerr black holes [4] [5] [6] [7] .\n\nThere are three dimensionless parameters characterizing inspirals of bodies into black holes:\n\n• the dimensionless spin parameter a = |S|/M 2 of the black hole, where S is the spin.\n\n• the strength of the interaction potential ǫ 2 = GM/rc 2 , i.e. the expansion parameter used in post-Newtonian (PN) theory.\n\n• the mass ratio µ/M .\n\nFor LISA data analysis we will need waveforms that are accurate to all orders in a and ǫ 2 , and to leading order in µ/M . However, it is useful to have analytic results in the regimes a ≪ 1 and/or ǫ 2 ≪ 1. Such approximate results can be useful as a check of numerical schemes that compute more accurate waveforms, for scoping out LISA's data analysis requirements [1, 6] , and for assessing the accuracy of the leading order in µ/M or adiabatic approximation [8] [9] [10] . There is substantial literature on such approximate analytic results, and in this paper we will extend some of these results to higher order. A long standing difficulty in computing the evolution of generic orbits has been the evolution of the orbit's \"Carter constant\", a constant of motion which governs the orbital shape and inclination. A theoretical prescription now exists for computing Carter constant evolution to all orders in ǫ and a in the adiabatic limit µ ≪ M [9, [11] [12] [13] , but it has not yet been implemented numerically. In this paper we focus on computing analytically the evolution of the Carter constant in the regime a ≪ 1, ǫ ≪ 1, µ/M ≪ 1, extending earlier results by Ryan [14, 15] .\n\nWe next review existing analytical work on the effects of multipole moments on inspiral waveforms. For non-spinning point masses, the phase of the l = 2 piece of the waveform is known to O(ǫ 7 ) beyond leading order [16] , while spin corrections are not known to such high order. To study the leading order effects of the central body's multipole moments on the inspiral waveform, in the test mass limit µ ≪ M , one has to correct both the conservative and dissipative pieces of the forces on the bodies. For the conservative pieces, it suffices to use the Newtonian action for a binary with an additional multipole interaction potential. For the dissipative pieces, the multipole corrections to the fluxes at infinity of the conserved quantities can simply be added to the known PN point mass results. The lowest order spin-orbit coupling effects on the gravitational radiation were first derived by Kidder [17] , then extended by Ryan [14, 15] , Gergely [18] , and Will [19] . Recently, the corrections of O(ǫ 2 ) beyond the leading order to the spin-orbit effects on the fluxes were derived [20, 21] . Corrections to the waveform due to the quadrupole -mass monopole interaction were first considered by Poisson [22] , who derived the effect on the time averaged energy flux for circular equatorial orbits. Gergely [23] extended this work to generic orbits and computed the radiative instantaneous and time averaged rates of change of energy E, magnitude of angular momentum |L|, and the angle κ = cos -1 (S • L) between the spin S and orbital angular momentum L. Instead of the Carter constant, Gergely identified the angular average of the magnitude of the orbital angular momentum, L, as a constant of motion. The fact that to post-2-Newtonian (2PN) order there is no time averaged secular evolution of the spin allowed Gergely to obtain expressions for L and κ from the quadrupole formula for the evolution of the total angular momentum J = L+ S. In a different paper, Gergely [18] showed that in addition to the quadrupole, self-interaction spin effects also contribute at 2PN order, which was seen previously in the black hole perturbation calculations of Shibata et al. [24] . Gergely calculated the effect of this interaction on the instantaneous and time-averaged fluxes of E and |L| but did not derive the evolution of the third constant of motion.\n\nIn this paper, we will re-examine the effects of the quadrupole moment of the black hole and of the leading order spin self interaction. For a black hole, our analysis will thus contain all effects that are quadratic in spin to the leading order in ǫ 2 and in µ/M . Our work will extend earlier work by\n\n• Considering generic orbits.\n\n• Using a natural generalization of the Carter-type constant that can be defined for two point particles when one of them has a quadrupole. This facilitates applying our analysis to Kerr inspirals.\n\n• Computing instantaneous as well as time-averaged fluxes for all three constants of motion: energy E, z-component of angular momentum L z , and Carter-type constant K. For most purposes, only time-averaged fluxes are needed as only they are gauge invariant and physically relevant. However, there is one effect for which the time-averaged fluxes are insufficient, namely transient resonances that occur during an inspiral in Kerr in the vicinity of geodesics for which the radial and azimuthal frequencies are commensurate [10, 25] . The instantaneous fluxes derived in this paper will be used in [10] for studying the effect of these resonances on the gravitational wave phasing.\n\nWe will analyze the effect of gravitational radiation reaction on orbits around a body with an axisymmetric mass quadrupole moment Q to leading order in Q, to the leading post-Newtonian order, and to leading order in the mass ratio. With these approximations the adiabatic approximation holds: gravitational radiation reaction takes place over a timescale much longer than the orbital period, so the orbit looks geodesic on short timescales. We follow Ryan's method of computation [14] : First, we calculate the orbital motion in the absence of radiation reaction and the associated constants of motion. Next, we use the leading order radiation reaction accelerations that act on the particle (given by the Burke-Thorne formula [26] augmented by the relevant spin corrections [14] ) to compute the evolution of the constants of motion. In the adiabatic limit, the time-averaged rates of change of the constants of motion can be used to infer the secular orbital evolution. Our results show that a mass quadrupole has the same qualitative effect on the evolution as spin: it tends to circularize eccentric orbits and drive the orbital plane towards antialignment with the symmetry axis of the quadrupole.\n\nThe relevance of our result to point particles inspiralling into black holes is as follows. The vacuum spacetime geometry around any stationary body is completely characterized by the body's mass multipole moments I L = I a1,a2...a l and current multipole moments S L = S a1,a2...a l [27] . These moments are defined as coefficients in a power series expansion of the metric in the body's local asymptotic rest frame [28] . For nearly Newtonian sources, they are given by integrals over the source as\n\nI L ≡ I a1,...a l = ρx <a1 . . . x a l > d 3 x, (1.1)\n\nS L ≡ S a1,...a l = ρx p v q ǫ pq<a1 x a2 . . . x a l > d 3 x.(1.2)\n\nHere ρ is the mass density and v q is the velocity, and \"< • • • >\" means \"symmetrize and remove all traces\". For axisymmetric situations, the tensor multipole moments I L (S L ) contain only a single independent component, conventionally denoted by I l (S l ) [27] . For a Kerr black hole of mass M and spin S, these moments are given by [27]\n\nI l + iS l = M l+1 (ia) l , (1.3)\n\nwhere a is the dimensionless spin parameter defined by a = |S|/M 2 . Note that S l = 0 for even l and I l = 0 for odd l.\n\nConsider now inspirals into an axisymmetric body which has some arbitrary mass and current multipoles I l and S l . Then we can consider effects that are linear in I l and S l for each l, effects that are quadratic in the multipoles proportional to I l I l ′ , I l S l ′ , S l S l ′ , effects that are cubic, etc. For a general body, all these effects can be separated using their scalings, but for a black hole, I l ∝ a l for even l and S l ∝ a l for odd l [see Eq.(1.3)], so the effects cannot be separated. For example, a physical effect that scales as O(a 2 ) could be an effect that is quadratic in the spin or linear in the quadrupole; an analysis in Kerr cannot distinguish these two possibilities. For this reason, it is useful to analyze spacetimes that are more general than Kerr, characterized by arbitrary I l and S l , as we do in this paper. For recent work on computing exact metrics characterized by sets of moments I l and S l , see Refs. [29, 30] and references therein.\n\nThe leading order effect of the black hole's multipoles on the inspiral is the O(a) effect computed by Ryan [15] . This O(a) effect depends linearly on the spin S 1 and is independent of the higher multipoles S l and I l since these all scale as O(a 2 ) or smaller. In this paper we compute the O(a 2 ) effect on the inspiral, which includes the leading order linear effect of the black hole's quadrupole (linear in I 2 ≡ Q) and the leading order spin self-interaction (quadratic in S 1 ).\n\nWe next discuss how these O(a 2 ) effects scale with the post-Newtonian expansion parameter ǫ. Consider first the conservative orbital dynamics. Here it is easy to see that fractional corrections that are linear in I 2 scale as O(a 2 ǫ 4 ), while those quadratic in S 1 scale as O(a 2 ǫ 6 ). Thus, the two types of terms cleanly separate. We compute only the leading order, O(a 2 ǫ 4 ), term. For the dissipative contributions to the orbital motion, however, the scalings are different. There are corrections to the radiation reaction acceleration whose fractional magnitudes are O(a 2 ǫ 4 ) from both types of effects linear in I 2 and quadratic in S 1 . The effects quadratic in S 1 are due to the backscattering of the radiation off the piece of spacetime curvature due to the black hole's spin. This effect was first pointed out by Shibata et al. [24] , who computed the time-averaged energy flux for circular orbits and small inclination angles based on a PN expansion of black hole perturbations. Later, Gergely [18] analyzed this effect on the instantaneous and time-averaged fluxes of energy and magnitude of orbital angular momentum within the PN framework.\n\nThe organization of this paper is as follows. In Sec. II, we study the conservative orbital dynamics of two point particles when one particle is endowed with an axisymmetric quadrupole, in the weak field regime, and to leading order in the mass ratio. In Sec. III, we compute the radiation reaction accelerations and the instantaneous and time-averaged fluxes. In order to have all the contributions at O(a 2 ǫ 4 ) for a black hole, we include in our computations of radiation reaction acceleration the interaction that is quadratic in the spin S 1 . The application to black holes in Sec. IV briefly discusses the qualitative predictions of our results and also compares with previous results.\n\nThe methods used in this paper can be applied only to the black hole spin (as analyzed by Ryan [14] ) and the black hole quadrupole (as analyzed here). We show in Sec. V that for the higher order mass and current multipole moments taken individually, an analog of the Carter constant cannot be defined to the order of our approximations. We then show that under mild assumptions, this non-existence result can be extended to exact spacetimes, thus falsifying the conjecture that all vacuum axisymmetric spacetimes possess a third constant of geodesic motion."
},
{
"section_type": "OTHER",
"section_title": "II. EFFECT OF AN AXISYMMETRIC MASS QUADRUPOLE ON THE CONSERVATIVE ORBITAL DYNAMICS",
"text": "Consider two point particles m 1 and m 2 interacting in Newtonian gravity, where m 2 ≪ m 1 and where the mass m 1 has a quadrupole moment Q ij which is axisymmetric:\n\nQ ij = d 3 xρ(r) x i x j - 1 3 r 2 δ ij (2.1) = Q n i n j - 1 3 δ ij . (2.2)\n\nFor a Kerr black hole of mass M and dimensionless spin parameter a with spin axis along n, the quadrupole scalar is Q = -M 3 a 2 . The action describing this system, to leading order in m 2 /m 1 , is\n\nS = dt 1 2 µv 2 -µΦ(r) , (2.3)\n\nwhere v = ṙ is the velocity, the potential is\n\nΦ(r) = - M r - 3 2r 5 x i x j Q ij , (2.4)\n\nµ is the reduced mass and M the total mass of the binary, and we are using units with G = c = 1. We work to linear order in Q, to linear order in m 2 /m 1 , and to leading order in M/r. In this regime, the action (2.3) also describes the conservative effect of the black hole's mass quadrupole on bound test particles in Kerr, as discussed in the introduction. We shall assume that the quadrupole Q ij is constant in time. In reality, the quadrupole will evolve due to torques that act to change the orientation of the central body. An estimate based on treating m 1 as a rigid body in the Newtonian field of m 2 gives the scaling of the timescale for the quadrupole to evolve compared to the radiation reaction time as (see Appendix I for details)\n\nT evol T rr ∼ m 1 m 2 M r S Q ∼ M µ M r 1 a .\n\n(2.5)\n\nHere, we have denoted the dimensionless spin and quadrupole of the body by S and Q respectively, and the last relation applies for a Kerr black hole. Since µ/M ≪ 1, the first factor in Eq. (2.5) will be large, and since 1/a ≥ 1 and for the relativistic regime M/r ∼ 1, the evolution time is long compared to the radiation reaction time. Therefore we can neglect the evolution of the quadrupole at leading order. This system admits three conserved quantities, the energy\n\nE = 1 2 µv 2 + µΦ(r), (2.6)\n\nthe z-component of angular momentum\n\nL z = e z • (µr × v), (2.7)\n\nand the Carter-type constant\n\nK = µ 2 (r × v) 2 - 2Qµ 2 r 3 (n • r) 2 + Qµ 2 M (n • v) 2 - 1 2 v 2 + M r . (2.8)\n\n(See below for a derivation of this expression for K).\n\nA. Conservative orbital dynamics in a Boyer-Lindquist-like coordinate system\n\nWe next specialize to units where M = 1. We also define the rescaled conserved quantities by Ẽ = E/µ, Lz = L z /µ, K = K/µ 2 , and drop the tildes. These specializations and definitions have the effect of eliminating all factors of µ and M from the analysis. In spherical polar coordinates (r, θ, ϕ) the constants of motion E and\n\nL z become E = 1 2 ( ṙ2 + r 2 θ2 + r 2 sin 2 θ φ2 ) - 1 r + Q 2r 3 (1 -3 cos 2 θ), (2.9)\n\nL z = r 2 sin 2 θ φ. (2.10)\n\nIn these coordinates, the Hamilton-Jacobi equation is not separable, so a separation constant K cannot readily be derived. For this reason we switch to a different coordinate system (r, θ, ϕ) defined by\n\nr cos θ = r cos θ 1 + Q 4r 2 , r sin θ = r sin θ 1 - Q 4r 2 . (2.11)\n\nWe also define a new time variable t by\n\ndt = 1 - Q 2r 2 cos(2 θ) d t.\n\n(2.12)\n\nThe action (2.3) in terms of the new variables to linear order in Q is\n\nS = d t 1 2 dr d t 2 + 1 2 r2 d θ d t 2 + 1 2 r2 sin 2 θ dϕ d t 2 1 - Q r2 sin 2 θ + 1 r + Q 4r 3 . (2.13)\n\nHowever, a difficulty is that the action (2.13) does not give the same dynamics as the original action (2.3). The reason is that for solutions of the equations of motion for the action (2.3), the variation of the action vanishes for paths with fixed endpoints for which the time interval ∆t is fixed. Similarly, for solutions of the equations of motion for the action (2.13), the variation of the action vanishes for paths with fixed endpoints for which the time interval ∆ t is fixed. The two sets of varied paths are not the same, since ∆t = ∆ t in general. Therefore, solutions of the Euler-Lagrange equations for the action (2.3) do not correspond to solutions of the Euler-Lagrange equations for the action (2.13). However, in the special case of zeroenergy motions, the extra terms in the variation of the action vanish. Thus, a way around this difficulty is to modify the original action to be\n\nŜ = dt 1 2 µv 2 -µΦ(r) + E . (2.14)\n\nThis action has the same extrema as the action (2.3), and for motion with physical energy E, the energy computed with this action is zero. Transforming to the new variables yields, to linear order in Q:\n\nŜ = d t 1 2\n\ndr d t 2 + 1 2 r2 d θ d t 2 + 1 2 r2 sin 2 θ dϕ d t 2 1 - Q r2 sin 2 θ + 1 r + Q 4r 3 + E - QE 2r 2 cos(2 θ) . (2.15)\n\nThe zero-energy motions for this action coincide with the zero energy motions for the action (2.14). We use this action (2.15) as the foundation for the remainder of our analysis in this section. The z-component of angular momentum in terms of the new variables (r, θ, ϕ, t) is\n\nL z = r2 sin 2 θ dϕ d t 1 - Q r2 sin 2 θ . (2.16)\n\nWe now transform to the Hamiltonian:\n\nĤ = 1 2 p 2 r - 1 r -E - Q 4r 3 + QL 2 z 2r 4 + 1 2r 2 p 2 θ + L 2 z sin 2 θ + QE cos(2 θ) (2.17)\n\nand solve the Hamiltonian Jacobi equation. Denoting the separation constant by K we obtain the following two equations for the r and θ motions:\n\ndr d t 2 = 2E + 2 r - K r2 + Q 2 1 r3 - 2L 2 z r4 , (2.18)\n\nand r4 d θ\n\nd t 2 = K - L 2 z sin 2 θ -QE cos(2 θ). ( 2\n\n.19) Note that the equations of motion (2.18) and (2.19) have the same structure as the equations of motion for Kerr geodesic motion. Using Eqs. (2.18), (2.19) and (2.16) together with the inverse of the transformation (2.11) to linear order in Q, we obtain the expression for K in spherical polar coordinates:\n\nK = r 4 ( θ2 + sin 2 θ φ2 ) + Q( ṙ cos θ -r θ sin θ) 2 + Q r - Q 2 ( ṙ2 + r 2 θ2 + r 2 sin 2 θ φ2 ) - 2Q r cos 2 θ. (2.20)\n\nThis is equivalent to the formula (2.8) quoted earlier."
},
{
"section_type": "OTHER",
"section_title": "B. Effects linear in spin on the conservative orbital dynamics",
"text": "To include the linear in spin effects, we repeat Ryan's analysis [14, 15] (he only gives the final, time averaged fluxes; we will also give the instantaneous fluxes). We can simply add these linear in spin terms to our results because any terms of order O(SQ) will be higher than the order a 2 to which we are working. The correction to the action (2.3) due to spin-orbit coupling is\n\nS spin-orbit = dt - 2µSn i ǫ ijk x j ẋk r 3 . (2.21)\n\nWe will restrict our analysis to the case when the unit vectors n i corresponding to the axisymmetric quadrupole Q ij and to the spin S i coincide, as they do in Kerr.\n\nIncluding the spin-orbit term in the action (2.3) results in the following modified expressions for L z and K:\n\nL z = n • (µr × v) - 2S r 3 [r 2 -(n • r) 2 ], (2.22)\n\nand\n\nK = (r × v) 2 - 4S r n • (r × v) - 2Q r 3 (n • r) 2 +Q (n • v) 2 - 1 2 v 2 + 1 r . (2.23)\n\nIn terms of the Boyer-Lindquist like coordinates, the conserved quantities with the linear in spin terms included are:\n\nL z = r2 sin 2 θ dϕ d t - 2S r sin 2 θ -Q sin 4 θ dϕ d t , (2.24)\n\nK = r 4 ( θ2 + sin 2 θ φ2 ) -4Sr sin 2 θ φ - 2Q r cos 2 θ + Q( ṙ cos θ -r θ sin θ) 2 + QM r - Q 2 ( ṙ2 + r 2 θ2 + r 2 sin 2 θ φ2 ). (2.25)\n\nThe equations of motion are\n\ndr d t 2 = 2E + 2 r - K r2 - 4SL z r3 + Q 2 1 r3 - 2L 2 z r4 , (2.26) and r4 d θ d t 2 = K - L 2 z sin 2 θ -QE cos(2 θ). ( 2\n\n.27) III. EFFECTS LINEAR IN QUADRUPOLE AND QUADRATIC IN SPIN ON THE EVOLUTION OF THE CONSTANTS OF MOTION A. Evaluation of the radiation reaction force\n\nThe relative acceleration of the two bodies can be written as\n\na = -∇Φ(r) + a rr , (3.1)\n\nwhere a rr is the radiation-reaction acceleration. Combining this with Eqs. (2.6), (2.22) and (2.23) for E, L z and K gives the following formulae for the time derivatives of the conserved quantities:\n\nĖ = v • a rr , (3.2) Lz = n • (r × a rr ), (3.3)\n\nK = 2(r × v) • (r × a rr ) - 4S r n • (r × a rr ) +2Q(n • v) (n • a rr ) -Qv • a rr . (3.4)\n\nThe standard expression for the leading order radiation reaction acceleration acting on one of the bodies is [31] :\n\na j rr = - 2 5 I\n\n(5)\n\njk x k + 16 45 ǫ jpq S (6) pk x k x q + 32 45 ǫ jpq S (5)\n\npk x k v q + 32 45 ǫ pq[j S (5) k]p x q v k . (3.5)\n\nHere the superscripts in parentheses indicate the number of time derivatives and square brackets on the indices denote antisymmetrization. The multipole moments I jk (t) and S jk (t) in Eq. (3.5) are the total multipole moments of the spacetime, i.e. approximately those of the black hole plus those due to the orbital motion. The expression (3.5) is formulated in asymptotically Cartesian mass centered (ACMC) coordinates of the system, which are displaced from the coordinates used in Sec. II by an amount [28]\n\nδr(t) = - µ M r(t). (3.6)\n\nThis displacement contributes to the radiation reaction acceleration in the following ways:\n\n1. The black hole multipole moments I l and S l , which are time-independent in the coordinates used in Sec. II, will be displaced by δr and thus will contribute to the (l + 1)th ACMC radiative multipole [28] .\n\n2. The constants of motion are defined in terms of the black hole centered coordinates used in Sec. II, so the acceleration a rr we need in Eqs. (3.2) -(3.4) is the relative acceleration. This requires calculating the acceleration of both the black hole and the point mass in the ACMC coordinates using (3.5), and then subtracting to find a rr = a µ rra M rr [14] . To leading order in µ, the only effect of the acceleration of the black hole is via a backreaction of the radiation field: the lth black hole moments couple to the (l + 1)th radiative moments, thus producing an additional contribution to the acceleration.\n\nFor our calculations at O(S 1 ǫ 3 ), O(I 2 ǫ 4 ), O(S 2 1 ǫ 4 ), we can make the following simplifications:\n\n• quadrupole corrections: The fractional corrections linear in I 2 = Q that scale as O(a 2 ǫ 4 ) require only the effect of I 2 on the conservative orbital dynamics as computed in Sec. IIA and the Burke-Thorne formula for the radiation reaction acceleration [given by the first term in Eq. (3.5)].\n\n• spin-spin corrections: As discussed in the introduction, the fractional corrections quadratic in S 1 to the conservative dynamics scale as O(a 2 ǫ 6 ) and are subleading order effects which we neglect. At O(a 2 ǫ 4 ), the only effect quadratic in S 1 is the backscattering of the radiation off the spacetime curvature due to the spin. As discussed in item 1. above, the black hole's current dipole S i = S 1 δ i3 (taking the z-axis to be the symmetry axis) will contribute to the radiative current quadrupole an amount\n\nS spin ij = - 3 2 µ M S 1 x i δ j3 . (3.7)\n\nThe black hole's current dipole S i will couple to the gravitomagnetic radiation field due to S ij as discussed in item 2. above, and contribute to the relative acceleration as [14] : x k x q + 32 45 ǫ jpq S\n\na j spin rr = 8 15 S 1 δ i3 S (\n\n(5) spin pk\n\nx k v q + 32 45 ǫ pq[j S (5) spin k]p x q v k + 8 15 S 1 δ i3 S (5) orbit ij + S (5) spin ij . (3.9)\n\nTo justify these approximations, consider the scaling of the contribution of black hole's acceleration to the orbital dynamics. The mass and current multipoles of the black hole contribute terms to the Hamiltonian that scale with ǫ as\n\n∆H ∼ S l ǫ 2l+3 & I l ǫ 2l+2 .\n\n(3.10)\n\nSince the Newtonian energy scales as ǫ 2 , the fractional correction to the orbital dynamics scale as\n\n∆H/E ∼ S l ǫ 2l+1 & I l ǫ 2l . (3.11)\n\nTo O(ǫ 4 ), the only radiative multipole moments that contribute to the acceleration (3.5) are the mass quadrupole I 2 , the mass octupole I 3 , and the current quadrupole S 2 (cf. [17] ). Since we are focusing only on the leading order terms quadratic in spin (these can simply be added to the known 2PN point particle and 1.5PN linear in spin results), the only terms in Eq. (3.5) relevant for our purposes are those given in Eq. (3.9). The results from a computation of the fully relativistic metric perturbation for black hole inspirals [24] show that quadratic in spin corrections to the l = 2 piece compared to the flat space Burke-Thorne formula first appear at O(a 2 ǫ 4 ), which is consistent with the above arguments."
},
{
"section_type": "OTHER",
"section_title": "B. Instantaneous fluxes",
"text": "We evaluate the radiation reaction force as follows. The total mass and current quadrupole moment of the system are\n\nQ T ij = Q ij + µx i x j , (3.12\n\n)\n\nS T ij = S spin ij + x i ǫ jkm x k ẋm , (3.13)\n\nwhere from Eq. (2.11)\n\nx i = r sin θ 1 - Q 4r 2 cos ϕ, r sin θ 1 - Q 4r 2 sin ϕ, r cos θ 1 + Q 4r 2 . (3.14)\n\nOnly the second term in Eq. (3.12) contributes to the time derivative of the quadrupole. We differentiate five times by using\n\nd dt = 1 + Q 2r 2 cos(2 θ) d d t , (3.15)\n\nto the order we are working as discussed above. After each differentiation, we eliminate any occurrences of dϕ/d t using Eq. ( 2 .24), and we eliminate any occurrences of the second order time derivatives d 2 r/d t2 and d 2 θ/d t2 in favor of first order time derivatives using (the time derivatives of) Eqs. (2.26) and (2.27). For computing the terms linear and quadratic in S 1 , we set the quadrupole Q to zero in all the formulae. We insert the resulting expression into the formula (3.9) for the self-acceleration, and then into Eqs. (3.2) -(3.4). We eliminate (dr/d t) 2 , (d θ/d t) 2 , and (dϕ/d t) in favor of E, L z , and K using Eqs. (2.24) -(2.27). In the final expressions for the instantaneous fluxes, we keep only terms that are of O(S), O(Q) and O(S 2 ) and obtain the following results: Ė = 160K 3r 6 + 64 3r 5 + 512E 15r 4 -40K 2 r7 + 272KE 5r 5 + 64E 2 5r 3\n\n+ SL z r9 196K 2 + 952 3 r2 - 3668 5 K r -352KE r2 + 1024 3 E r3 + 128 5 E 2 r4 + 2Q r9 -49K 2 -169KL 2 z + r 532 5 K + 3307 15 L 2 z + 2r 2 - 20 3 + 47KE + 548 5 L 2 z E - 152 5 r3 E -16r 4 E 2 + Q r9 -562K 2 + 2998 3 K r - 320 3 r2 + 5072 5 KE r2 - 4048 15 r3 E -160r 4 E 2 cos(2 θ) + Q r6 sin(2 θ) 439K - 926 3 r - 1528 5 r2 E θ ṙ + S 2 r9 -K 2 + 22 3 K r - 28\n\n3 r2 + 32 5 KE r2 -236 15 r3 E -32 5 r4 E 2 cos(2 θ) -r3 sin(2 θ) K + 2 3 r + 8 5 r2 E θ ṙ + S 2 r9 -49K 2 + 6KL 2 z + 2r 63K -16 3 L 2 z -98 3 + r2 112KE -48 5 L 2 z E -1652 15 r3 E -224 5 r4 E 2 , (3.16) Lz = 32L z r4 + 144L z E 5r 3 -24KL z r5 + S r7 -50K 2 + 240KL 2 z + 62 5 K r -7376 15 L 2 z r + 316 3 r2 + 56KE r2 -1824 5 EL 2 z r2 + 624 5 E r3 + 128 5 E 2 r4 + S r7 50K 2 -62 5 K r -316 3 r2 -56KE r2 -624 5 E r3 -128 5 E 2 r4 cos(2 θ) + S r4 -104K + 64r + 64E r2 sin(2 θ) ṙ θ + QL z 5r 7 660E r2 + 753r -360L 2 z -435K + 1601r + 1512r 2 E -1185K cos 2 θ + 174QL z r4 sin(2 θ) ṙ θ + 2S 2 L z r7 72 5 E r2 + 16r -9K , (3.17) and K = 16K 5r 5 20r + 18r 2 E -15K + SL z r7 280K 2 -14008 15 K r + 1264 3 r2 + 2496 5 E r3 -2528 5 KE r2 + 512 5 E 2 r4 + 12Q 5r 7 -45K 2 + rL 2 z (83 + 80rE) -115KL 2 z + 14K r(6 + 5rE) + 4Q 15r 7 cos(2 θ) -2175K 2 + 2975K r + 80r 2 + 3012KE r2 -112E r3 -168E 2 r4 + 2Q 15r 4 3075K -20r -192E r2 sin(2 θ) θ ṙ + 2S 2 r7 7K -2L 2 z -3K + 16 3 r + 24 5 E r2 + K cos(2 θ) 3K -16 3 r -24 5 E r2 + 2S 2 r4 sin(2 θ) -4K + 14 3 r + 16 5 E r2 θ ṙ. (3.18) C. Alternative set of constants of the motion\n\nA body in a generic bound orbit in Kerr traces an open ellipse precessing about the hole's spin axis. For stable orbits the motion is confined to a toroidal region whose shape is determined by E, L z , K. The motion can equivalently be characterized by the set of constants inclination angle ι, eccentricity e, and semi-latus rectum p defined by Hughes [32] . The constants ι, p and e are defined by cos ι = L z / √ K, and by r± = p/(1 ± e), where r± are the turning points of the radial motion, and r is the Boyer-Lindquist radial coordinate. This parameterization has a simple physical interpretation: in the Newtonian limit of large p, the orbit of the particle is an ellipse of eccentricity e and semilatus rectum p on a plane whose inclination angle to the hole's equatorial plane is ι. In the relativistic regime p ∼ M , this interpretation of the constants e, p, and ι is no longer valid because the orbit is not an ellipse and ι is not the angle at which the object crosses the equatorial plane (see Ryan [14] for a discussion).\n\nWe adopt here analogous definitions of constants of motion ι, e and p, namely\n\ncos(ι) = L z / √ K, (3.19) p 1 ± e = r± . (3.20)\n\nHere K is the conserved quantity (2.23) or (2.25), and r± are the turning points of the radial motion using the r coordinate defined by Eq. (2.11), given by the vanishing of the right-hand side of Eq. (2.26). We now rewrite our results in terms of the new constants of the motion e, p and ι. We can use Eq. (2.26) together with the equations (3. 19 ) and (3.20) to write E, L z and K as functions of p, e and ι. To leading order in Q and S we obtain\n\nK = p 1 - 2S cos ι p 3/2 3 + e 2 -1 + e 2 2Q cos 2 ι p 2 + 3 + e 2 Q 4p 2 , (3.21)\n\nE = - (1 -e 2 ) 2p 1 + 2S cos ι p 3/2 1 -e 2 + 1 -e 2 Q p 2 cos 2 ι - 1 4 , (3.22)\n\nL z = √ p cos ι 1 - S cos ι p 3/2 (3 + e 2 ) -1 + e 2 Q cos 2 ι p 2 + 3 + e 2 Q 8p 2 . (3.23)\n\nAs discussed in the introduction, the effects quadratic in S on the conservative dynamics scale as O(a 2 ǫ 6 ) and thus are not included in this analysis to O(a 2 ǫ 4 ). Inserting these relations into the expressions (3.16)-(3.18) gives, dropping terms of O(QS), O(Q 2 ) and O(QS 2 ): Ė = -8 15p 2 r7 75p 4 -100p 3 r + p 2 r2 11 -51e 2 + 32pr 3 1 -e 2 ) -6r 4 1 -e 2 2\n\n4S cos ι 15p 7/2 r9 735p 6 -2751p 5 r + 10p 4 r2 (365 -6e 2 ) -128pr 5 (1 -e 2 ) 2 -48r 6 (e 2 -1) 3 + 64S cos ι 15p 3/2 r6 5p(-23 + 3e 2 ) -3r(-9 + e 2 + 8e 4 ) -Q 15p 4 r9 4005p 6 -6499p 5 r + 2p 4 r2 1577 -1977e 2 -24r 6 1 -e 2 3 -32p 3 r3 8 -33e 2 + 64pr 5\n\n1 -2e 2 + e 4 - Q 15p 4 r9 24p 2 r4 5 -27e 2 + 22e 4 -pr 3 sin(2 θ) 6585p 2 -4630pr + 2292r 2 (1 -e 2 ) θ ṙ - Q 15p 4 r9 2p 2 cos(2 θ) 4215p 4 -7495p 3 r + 4p 2 r2 (1151 -951e 2 ) -1012pr 3 (1 -e 2 ) + 300r 4 (1 -2e 2 + e 4 ) - Q 15p 4 r9 cos(2ι) 2535p 6 -3307p 5 r + 12p 4 r2 (37 -237e 2 ) -48r 6 (1 -e 2 ) 3 + 800p 3 r3 (1 + e 2 ) + 128pr 5 (1 -2e 2 + e 4 ) + 204Q 15p 2 r5 cos(2ι) 1 + 2e 2 -3e 4 - 2S 2 15p 2 r9 84r 4 (1 -e 2 ) 2 (1 + e 2 ) 2 + 345p 4 -905p 3 r -413pr 3 (1 -e 2 ) + 2p 2 r2 (446 -201e 2 ) - S 2 15p 2 r9 cos(2 θ) 15p 4 -110p 3 r + 4p 2 r2 (47 -12e 2 ) -118pr 3 (1 -e 2 ) + 24r 4 (1 -e 2 ) 2 (1 + e 2 ) 2 + S 2 15r 9 cos(2ι) 45p 2 -80pr + 36r 2 (1 -e 2 ) - S 2 15pr 6 sin(2 θ) ṙ θ 15p 2 + 10pr -12r 2 (1 -e 2 ) , (3.24)\n\nLz = -\n\n8 cos ι 5 √ pr 5 15p 2 -20pr + 9r 2 (1 -e 2 ) + 2S 15p 2 r7 525p 4 -1751p 3 r + 34p 2 r2 (61 -6e 2 ) + 12pr 3 (-69 + 29e 2 ) + 6r 4 (17 + 2e 2 -19e 4 ) + 2S 15p 2 r7 375p 4 -93p 3 r + 468pr 3 (1 -e 2 ) -10p 2 r2 (58 + 21e 2 ) -48r 4 (1 -2e 2 + e 4 ) cos(2 θ) + 4S 15p 2 r7 450p 4 -922p 3 r -60pr 3 (3 + e 2 ) -9p 2 r2 (-83 + 23e 2 ) + 27r 4 (1 + 2e 2 -3e 4 ) cos(2ι) - 8S pr 4 13p 2 -8pr + 4r 2 (1 -e 2 ) sin(2 θ) ṙ θ - Q cos ι 5p 5/2 r7 615p 4 -753p 3 r + 15p 2 r2 19 -31e 2 + 20pr 3 1 + 3e 2 + 9r 4 1 -6e 2 + 5e 4 - Q cos ι 5p 1/2 r7 cos(2 θ) 1185p 2 -1601pr + 756r 2 (1 -e 2 )\n\n-2Q cos ι 5p 5/2 r7 2 cos(2ι) 45p\n\n4 -18r 4 e 2 (1 -e 2 ) -45p 2 r2 (1 + e 2 ) + 20pr 3 (1 + e 2 ) -435p 3 r3 sin(2 θ) θ ṙ -2S 2 cos ι p 1/2 r7 9p 2 -16pr + 36 5 r2 (1 -e 2 ) , (3.25) and K = 16 5r 5 20pr -15p 2 -9r 2 (1 -e 2 ) + 8S cos ι 15p 3/2 r7 525p 4 -1751p 3 r + 2p 2 r2 (1172 -57e 2 ) + 12pr 3 (-99 + 19e 2 ) -24r 4 (-11 + 4e 2 + 7e 4 ) + 2Q 5p 2 r7 -615p 4 + 753p 3 r + 30p 2 r2 (17e 2 -9) + 72r 4 e 2 (1 -e 2 ) -40pr 3 (1 + 3e 2 ) + 2Q 5p 2 r7 cos(2ι) -345p 4 + 249p 3 r -160pr 3 (1 + e 2 ) + 120p 2 r2 (1 + 3e 2 ) + 36r 4 (1 + 2e 2 -3e 4 ) + 2Q 15p 2 r7 2 cos(2 θ) 2175p 4 -2975p 3 r -56pr 3 (1 -e 2 ) + 2p 2 r2 (713 -753e 2 ) + 42r 4 (1 -2e 2 + e 4 ) + 2Q 15pr 4 sin(2 θ) 3075p 2 -20pr + 96r 2 (1 -e 2 ) ṙ θ + 2S 2 r7 2 -9p 2 + 16pr -36 5 r2 (1 -e 2 ) + cos(2 θ) + cos(2ι) 3p 2 -16 3 pr + 12 5 r2 (1 -e 2 ) + 4S 2 pr 4 sin(2 θ) ṙ θ -2p 2 + 7 3 pr -4 5 r2 (1 -e 2 ) . (3.26)"
},
{
"section_type": "OTHER",
"section_title": "D. Time averaged fluxes",
"text": "In this section we will compute the infinite timeaverages Ė , Lz and K of the fluxes. These averages are defined by\n\nĖ ≡ lim T →∞ 1 T T /2 -T /2 Ė(t)dt.\n\n(3.27)\n\nThese time-averaged fluxes are sufficient to evolve orbits in the adiabatic regime (except for the effect of res-onances) [12, 25] . In Appendix II, we present two different ways of computing the time averages. The first approach is based on decoupling the r and θ motion using the analog of the Mino time parameter for geodesic motion in Kerr [12] . The second approach uses the explicit Newtonian parameterization of the orbital motion.\n\nBoth averaging methods give the following results:\n\nĖ = - 32 5 (1 -e 2 )\n\n3/2 p 5 1 + 73 24 e 2 + 37 96 e 4 -S p 3/2 73 12 + 823 24 e 2 + 949 32 e 4 + 491 192 e 6 cos(ι) -Q p 2 1 2 + 85 32 e 2 + 349 128 e 4 + 107 384 e 6 + 11 4 + 273 16 e 2 + 847 64 e 4 + 179 192 e 6 cos(2ι) + S 2 p 2 13 192 + 247 384 e 2 + 299 512 e 4 + 39 1024 e 6 -1 192 + 19 384 e 2 + 23 512 e 4 + 3 1024 e 6 cos(2ι) ,(3.28) Lz = -32 5 (1 -e 2 ) 3/2 p 7/2 cos ι 1 + 7 8 e 2 -S 2p 3/2 cos ι 61 24 + 7e 2 + 271 64 e 4 + 61 8 + 91 4 e 2 + 461 64 e 4 cos(2ι) -Q 16p 2 -3 -45 4 e 2 + 19 8 e 4 + 45 + 148e 2 + 331 8 e 4 cos(2ι) + S 2 16p 2 1 + 3e 2 + 3 8 e 4 , (3.29) K = -64 5 (1 -e 2 ) 3/2 p 3 1 + 7 8 e 2 -S 2p 3/2 97 6 + 37e 2 + 211 16 e 4 cos(ι) -Q p 2 1 + 8 3 e 2 + 11 12 e 4 + 13 4 + 841 96 e 2 + 449 192 e 4 cos(2ι) + S 2 p 2 13 192 + 13 64 e 2 + 13 512 e 4 -1 192 + 1 64 e 2 + 1 512 e 4 cos(2ι) . (3.30) Using Eqs. (3.21) and (3.23), we obtain from (3.28) -(3.30) the following time averaged rates of change of the orbital elements e, p, ι: ṗ = -64 5 (1 -e 2 ) 3/2 p 3 1 + 7e 2 8 -S cos(ι) 96p 3/2 1064 + 1516e 2 + 475e 4 -Q 8p 2 14 + 149e 2 12 + 19e 4 48 + 50 + 469e 2 12 + 227e 4 24 cos(2ι) + S 2 64p 2 1 3 + e 2 + e 4 8 [13 -cos(2ι)] , (3.31) ė = -304 15 e(1 -e 2 ) 3/2 p 4 1 + 121e 2 304 + Se(1 -e 2 ) 3/2 cos(ι) 5p 11/2 1172 + 932e 2 + 1313e 4 6 + Q(1 -e 2 ) 3/2 ep 6 32 + 785e 2 3 -219e 4 2 + 13e 6 + 32 + 2195e 2 3 + 251e 4 + 218e 6 3 cos(2ι) -S 2 e(1 -e 2 ) 3/2 8p 6 2 + 3e 2 + e 4 4 [13 -cos(2ι)] , (3.32) ι = S sin(ι)(1 -e 2 ) 3/2 p 11/2 244 15 + 252 5 e 2 + 19 2 e 4 -1 -e 2 3/2 S 2 sin(2ι) 240p 6 8 + 3e 2 8 + e 2 + Q cot(ι)(1 -e 2 ) 3/2 60p 6 312 + 736e 2 -83e 4 -408 + 1268e 2 + 599e 4 cos(2ι) . (3.33) IV. APPLICATION TO BLACK HOLES A. Qualitative discussion of results\n\nThe above results for the fluxes, Eqs. (3.31), (3.32) and (3.33) show that the correction terms at O(a 2 ǫ 4 ) due to the quadrupole have the same type of effect on the evolution as the linear spin correction computed by Ryan: they tend to circularize eccentric orbits and change the angle ι such as to become antialigned with the symmetry axis of the quadrupole.\n\nThe effects of the terms quadratic in spin are qualitatively different. In the expression (3.28) for Ė , the coefficient of cos(2ι) due to the spin self-interaction has the same sign as the quadrupole term, while the terms not involving ι have the opposite sign. The terms involving cos(2ι) in Eq. (3.30) for K of O(Q) and O(S 2 ) terms have the same sign, while the terms not involving ι have the opposite sign. The fractional spin-spin correction to Lz , Eq. (3.29), has no ι-dependence, and in expression (3.33) for ι , the dependence on ι of the two effects O(Q) and O(S 2 ) is different, too. This is not surprising as the O(Q) effects included here are corrections to the conservative orbital dynamics, while the effects of O(S 2 ) that we included are due to radiation reaction."
},
{
"section_type": "RESULTS",
"section_title": "B. Comparison with previous results",
"text": "The terms linear in the spin in our results for the time averaged fluxes, Eqs. (3.28) - (3.33) , agree with those computed by Ryan, Eqs. (14a) -(15c) of [15] , and with those given in Eqs. (2.5) -(2.7) of Ref. [33] , when we use the transformations to the variables used by Ryan given in Eqs. (2.3) -(2.4) in [33] .\n\nEquation (3.28) for the time averaged energy flux agrees with Eq. (3.10) of Gergely [23] and Eq. (4.15) of [18] when we use the following transformations:\n\nK = L2 1 - Q 2 L4 Ā2 sin 2 κ cos δ -(1 -Ā2 ) cos 2 κ = L2 1 - Q L4 E cos 2 κ - Q 2 L4 (1 + 2 L2 ) sin 2 κ cos δ , (4.1\n\n)\n\ncos ι = cos κ 1 + Q 2 L4 E cos 2 κ + Q 2 L4 (1 + 2 L2 ) sin 2 κ cos δ , (4.2)\n\nξ 0 = 1 2 (δ + κ), (4.3\n\n)\n\nξ 0 = (ψ 0 -ψ i ) + π 2 , (4.4)\n\nwhere Ā, L, κ, δ, ψ 0 and ψ i are the quantities used by Gergely. The first relation here is obtained from the turning points of the radial motion as follows. We compute r± in terms of E and K and map these expressions back to r using Eqs. (2.11) . The result can then be compared with the turning points in Gergely's variables, Eq.\n\n(2.19) of [23] , using the fact that E is the same in both cases. Instead of the evolution of the constants of motion K and L z , Gergely computes the rates of change of the magnitude L of the orbital angular momentum and of the angle κ defined by cos κ = (L • S)/L. Using the transformations (4.1) -( 4 .4) and the definition of κ we verify that our Eq. (3.29) agrees with the Lz computed using Gergely's Eqs. (3.23) and (3.35) in [23] and Eq. (4.30) of [18] .\n\nIn the limit of the circular equatorial orbits analyzed by Poisson [22] , our Eq. (3.28) agrees with Poisson's Eq. ( 22 ) when we use the transformations and specializations: The main improvement of our analysis over Gergely's is that we express the results in terms of the Carter-type constant K, which facilitates comparing our results with other analyses of black hole inspirals. Our computations also include the spin curvature scattering effects for all three constants of motion; Gergely [18] only considers these effects for two of them: the energy and magnitude of angular momentum, not for the third conserved quantity.\n\np = 1 v 2 1 - Q 4 v 4 , (4.5\n\nWhen we expand Eq. (3.28) for small inclination angles and specialize to circular orbits, then after converting p to the parameter v using Eq. (4.5), we obtain Ė = -32 5p 5 1 -\n\n1 p 2 2Q + S 2 16 + ι 2 2p 2 11Q - S 2 48 = - 32 5p 5 1 - a 2 v 4 16 33 - 527 6 ι 2 . (4.9)\n\nThis result agrees with the terms at O(a 2 v 4 ) of Eq. (3.13) of Shibata et al. [24] , whose calculations were based on the fully relativistic expressions. This agreement is a check that we have taken into account all the contributions at O(a 2 ǫ 4 ). The analysis in Ref. [24] could not distinguish between effects due to the quadrupole and those due curvature scattering, but we can see from Eq. (4.9) that those two interactions have the opposite dependence on ι. Comparing (4.9) with Eq. (3.7) of [24] (which gives the fluxes into the different modes (l = 2, m, n), where m and n are the multiples of the ϕ and θ frequencies), we see that the terms in the (2, ±2, 0) and the (2, ±1, ±1) modes are entirely due to the quadrupole, while the spin-spin interaction effects are fully contained in the (2, ±1, 0) and (2, 0, ±1) modes."
},
{
"section_type": "OTHER",
"section_title": "V. NON-EXISTENCE OF A CARTER-TYPE CONSTANT FOR HIGHER MULTIPOLES",
"text": "In this section, we show that for a single axisymmetric multipole interaction, it is not possible to find an analog of the Carter constant (a conserved quantity which does not correspond to a symmetry of the Lagrangian), except for the cases of spin (treated by Ryan [15] ) and mass quadrupole moment (treated in this paper). Our proof is valid only in the approximations in which we work -expanding to linear order in the mass ratio, to the leading post-Newtonian order, and to linear order in the multipole. However we will show below that with very mild additional smoothness assumptions, our nonexistence result extends to exact geodesic motion in exact vacuum spacetimes.\n\nWe start in Sec. V A by showing that there is no coordinate system in which the Hamilton-Jacobi equation is separable. Now separability of the Hamilton-Jacobi equation is a sufficient but not a necessary condition for the existence of a additional conserved quantity. Hence, this result does not yield information about the existence or non-existence of an additional constant. Nevertheless we find it to be a suggestive result. Our actual derivation of the non-existence is based on Poisson bracket computations, and is given in Sec. V B."
},
{
"section_type": "OTHER",
"section_title": "A. Separability analysis",
"text": "Consider a binary of two point masses m 1 and m 2 , where the mass m 1 is endowed with a single axisymmetric current multipole moment S l or axisymmetric mass multipole moment I l . In this section, we show that the Hamilton-Jacobi equation for this motion, to linear order in the multipoles, to linear order in the mass ratio and to the leading post-Newtonian order, is separable only for the cases S 1 and I 2 .\n\nWe choose the symmetry axis to be the z-axis and write the action for a general multipole as\n\nS = dt 1 2 ṙ2 + r 2 θ2 + r 2 sin 2 θ φ2 + 1 r + f (r, θ) + g(r, θ) φ + E] . (5.1)\n\nFor mass moments, g(r, θ) = 0, while for current moments f (r, θ) = 0. For an axisymmetric multipole of order l, the functions f and g will be of the form\n\nf (r, θ) = c l I l P l (cos θ) r l+1 , g(r, θ) = d l S l sin θ∂ θ P l (cos θ) r l , (5.2\n\n) where P l (cos θ) are the Legendre polynomials and c l and d l are constants. We will work to linear order in f and g. In Eq. (5.1), we have added the energy term needed when doing a change of time variables, cf. the discussion before Eq. (2.14) in section III. Since ϕ is a cyclic coordinate, p ϕ = L z is a constant of motion and the system has effectively only two degrees of freedom. Note that in the case of a current moment, there will be correction term in L z :\n\nL z = r 2 sin 2 θ φ + g(r, θ).\n\n(\n\nNext, we switch to a different coordinate system (r, θ, ϕ) defined by\n\nr = r + α(r, θ, L z ), (5.4)\n\nθ = θ + β(r, θ, L z ), (5.5)\n\nwhere the functions α and β are yet undetermined. We also define a new time variable t by\n\ndt = 1 + γ(r, θ, L z ) d t. (5.6)\n\nSince we work to linear order in f and g, we can work to linear order in α, β, and γ. We then compute the action in the new coordinates and drop the tildes. The Hamiltonian is given by\n\nH = 1 2 p 2 r (1 + γ -2α ,r ) + p 2 θ 2r 2 (1 - 2α r -2β ,θ + γ) + p r p θ r 2 (-α ,θ -r 2 β ,r ) -E(1 + γ) + L 2 z 2r 2 sin 2 θ (1 + γ - 2α r -2β cot θ) - 1 r (1 - α r + γ) -f - gL z r 2 sin 2 θ (5.7)\n\nand the corresponding Hamilton-Jacobi equation is\n\n0 = ∂W ∂r 2 Ĉ1 + ∂W ∂θ 2 Ĉ2 r 2 +2 ∂W ∂r ∂W ∂θ Ĉ3 r 2 + 2 V , (5.8)\n\nwhere we have denoted\n\nĈ1 = J(r, θ) [1 + γ -2α ,r ] = 1 + γ -2α ,r + j, (5.9) Ĉ2 = J(r, θ) 1 - 2α r -2β ,θ + γ = 1 - 2α r -2β ,θ + γ + j, (5.10)\n\nĈ3 = J(r, θ) -α ,θ -r 2 β ,r = -α ,θ -r 2 β ,r , (5.11)\n\nV = J(r, θ) L 2 z 2r 2 sin 2 θ (1 + γ - 2α r -2β cot θ) - 1 r (1 - α r + γ) -E(1 + γ) -f - gL z r 2 sin 2 θ = L 2 z 2r 2 sin 2 θ (1 + γ - 2α r -2β cot θ + j) -E(1 + γ + j) - 1 r (1 - α r + γ + j) -f - gL z r 2 sin 2 θ . (5.12)\n\nThe unperturbed problem is separable, so make the perturbed problem separable, we have multiplied the Hamilton-Jacobi equation by an arbitrary function J(r, θ), which can be expanded as J(r, θ) = 1 + j(r, θ), where j(r, θ) is a small perturbation.\n\nTo find a solution of the form W = W r (r) + W θ (θ), we first specialize to the case where Ĉ3 = 0:\n\n-Ĉ3 = β ,r r 2 + α ,θ = 0.\n\n(5.13)\n\nWe differentiate Eq. (5.8) with respect to θ, using Eq.\n\n(5.8) to write (dW r /dr) 2 in terms of (dW θ /dθ) 2 and then differentiate the result with respect to r to obtain\n\n0 = dW θ dθ 2 ∂ r ∂ θ Ĉ2 Ĉ2 - ∂ θ Ĉ1 Ĉ1 +2∂ r r 2 ∂ θ V Ĉ2 - r 2 V ∂ θ Ĉ1 Ĉ1 Ĉ2 . (5.14)\n\nExpanding Eq. (5.14) to linear order in the small quantities then yields the two conditions for the kinetic and the potential part of the Hamiltonian to be separable:\n\n0 = ∂ r ∂ θ 2α ,r - 2α r -2β ,θ , (5.15) 0\n\n= L 2 z sin 2 θ 2β ,r cot 2 θ -3β ,rθ cot θ + β ,r csc 2 θ + L 2 z sin 2 θ ∂ r - α ,θ r + α ,rθ -∂ r ∂ θ c l I l r l-1 P l (cos θ) + d l S l L z r l sin θ ∂ θ P l (cos θ) -∂ r r 2α ,rθ - α ,θ r + 2Er 2 α ,rθ , (5.16)\n\nThe unperturbed motion for a bound orbit is in a plane, so we can switch from spherical to plane polar coordinates (r, ψ). In terms of these coordinates, we have H 0 = p 2 r /2+p 2 ψ /2, K 0 = p 2 ψ , and cos θ = sin ι sin(ψ+ψ 0 ), with cos ι = L z / √ K and the constant ψ 0 denoting the angle between the direction of the periastron and the intersection between the orbital and equatorial plane. Then Eq. (5.32) becomes\n\nd dt δK = η(t), (5.33)\n\nη(t) = - 2p ψ d l S l L z sin ι r l+2 (t) ∂ ψ ∂ ψ P l (sin ι sin(ψ(t) + ψ 0 )) cos(ψ(t) + ψ 0 ) + 2p ψ c l I l r l+1 (t)\n\n∂ ψ P l (sin ι sin(ψ(t) + ψ 0 )).\n\n(5.34)\n\nFor unbound orbits, one can always integrate Eq. (5.33) to determine δK. However, for bound periodic orbits there is a possible obstruction: the solution for the conserved quantity K 0 + δK will be single valued if and only if the integral of the source over the closed orbit vanishes, T orb 0 η(t)dt = 0.\n\n(5.35)\n\nHere, T orb is the orbital period. In other words, the partial differential equation (5.32) has a solution δK if and only if the condition (5.35) is satisfied. This is the same condition as obtained by the Poincare-Mel'nikov-Arnold method, a technique for showing the non-integrability and existence of chaos in certain classes of perturbed dynamical systems [35] . Thus, it suffices to show that the condition (5.35) is violated for all multipoles other than the spin and mass quadrupole. To perform the integral in Eq. (5.35), we use the parameterization for the unperturbed motion, r = K/(1 + e cos ψ) and dt/dψ = K 3/2 /(1 + e cos ψ) 2 , so that the condition for the existence of a conserved quantity A lnjk e j (sin ι) l-2n (sin ψ 0 ) k (cos ψ 0 ) j-k 2π 0 dχ (sin χ) j-k+1 (cos χ) k+l-2n-1\n\nK 0 + δK becomes 2π 0 dψ c l I l (1 + e cos ψ) l-1 ∂ ψ P l (sin ι sin(ψ + ψ 0 )) - d l S l L z K sin ι (1 + e cos ψ) l ∂ ψ ∂ ψ P l (sin ι sin(ψ + ψ 0 )) cos(ψ + ψ 0 ) = 0.\n\n+ d l S l L z K N n=0 l j=0\n\nB lnjk e j (sin ι) l-2n-1 (sin ψ 0 ) k (cos ψ 0 ) j-k 2π 0 dχ (sin χ) j-k+1 (cos χ) k+l-2n-2 .\n\n(5.38)\n\nThe coefficients A lnkj and B lnkj are\n\nA lnkj = (-1) n+k+1 (l -1)!(2l -2n)! 2 l n!(l -1 -j)!k!(j -k)!(l -n)!(l -2n -1)! , B lnkj = (-1) n+k l!(2l -2n)! 2 l n!(l -j)!k!(j -k)!(l -n)!(l -2n -2)! . (5.39)\n\nThe only non-vanishing contribution to the integrals in Eq. (5.38) will come from terms with even powers of both cos χ and sin χ. These can be evaluated as multiples of the beta function:\n\n0 = c l I l N n=0 l-1 j=0\n\nC lnjk e j (sin ι) l-2n (sin ψ 0 ) k (cos ψ 0 ) j-k δ (j-k+1),even δ (l+k-1),even\n\n+ d l S l L z K N n=0 l j=0\n\nD lnjk e j (sin ι) l-2n-1 (sin ψ 0 ) k (cos ψ 0 ) j-k δ (j-k+1),even δ (l+k),even .\n\n(5.40)\n\nHere, the coefficients are\n\nC lnjk = 2Γ( j 2 -k 2 + 1)Γ( k 2 + l 2 -n) Γ( j 2 + l 2 -n + 1) A lnkj , D lnjk = 2Γ( j 2 -k 2 + 1)Γ( k 2 + l 2 -n -1 2 ) Γ( j 2 + l 2 -n + 3 2 )\n\nB lnkj (5.41)\n\nEq. (5.40) shows that for even l, terms with j =even (odd) and k =odd (even) give a non-vanishing contribution for the case of a mass (current) multipole, and hence K 0 +δK is not a conserved quantity for the perturbed motion. Note that terms with j =even and k =odd for even l occur only for l > 3, so for l = 2 the mass quadrupole term in Eq. ( 5 .40) vanishes and therefore there exists an analog of the Carter constant, which is consistent with our results of Sec. II and our separability analysis. For odd l, terms with j =odd (even) and k =even (odd) are finite for I l (S l ). Note that for the case l = 1 of the spin, the derivatives with respect to χ in Eq. (5.37) evaluate to zero, so in this case there also exists a Carter-type constant. These results show that for a general multipole other than I 2 and S 1 , there will not be a Carter-type constant for such a system."
},
{
"section_type": "OTHER",
"section_title": "Exact vacuum spacetimes",
"text": "Our result on the non-existence of a Carter-type constant can be extended, with mild smoothness assumptions, to falsify the conjecture that all exact, axisymmetric vacuum spacetimes posess a third constant of the motion for geodesic motion. Specifically, we fix a multipole order l, and we assume:\n\n• There exists a one parameter family (M, g ab (λ)) of spacetimes, which is smooth in the parameter λ, such that λ = 0 is Schwarzschild, and each spacetime g ab (λ) is stationary and axisymmetric with commuting Killing fields ∂/∂t and ∂/∂φ, and such that all the mass and current multipole moments of the spacetime vanish except for the one of order l. On physical grounds, one expects a one parameter family of metrics with these properties to exist.\n\n• We denote by H(λ) the Hamiltonian on the tangent bundle over M for geodesic motion in the metric g ab (λ). By hypothesis, there exists for each λ a conserved quantity M (λ) which is functionally independent of the conserved energy and angular momentum. Our second assumption is that M (λ) is differentiable in λ at λ = 0. One would expect this to be true on physical grounds.\n\n• We assume that the conserved quantity M (λ) is invariant under the symmetries of the system:\n\nL ξ M (λ) = L η M (λ) = 0,\n\nwhere ξ and η are the natural extensions to the 8 dimensional phase space of the Killing vectors ∂/∂t and ∂/∂φ. This is a very natural assumption.\n\nThese assumptions, when combined with our result of the previous section, lead to a contradiction, showing that the conjecture is false under our assumptions.\n\nTo prove this, we start by noting that M (0) is a conserved quantity for geodesic motion in Schwarzschild, so it must be possible to express it as some function f of the three independent conserved quantities:\n\nM (0) = f (E, L z , K 0 ).\n\n(5.42)\n\nHere E is the energy, L z is the angular momentum, and K 0 is the Carter constant. Differentiating the exact relation {H(λ), M (λ)} = 0 and evaluating at λ = 0 gives\n\n{H 0 , M 1 } = ∂f ∂E {E, H 1 }+ ∂f ∂L z {L z , H 1 }+ ∂f ∂K 0 {K 0 , H 1 }, (5.43\n\n) where H 0 = H(0), H 1 = H ′ (0), and M 1 = M ′ (0). As before, we can regard this is a partial differential equation that determines M 1 , and a necessary condition for solutions to exist and be single valued is that the integral of the right hand side over any closed orbit must vanish:\n\n∂f ∂E {E, H 1 } + ∂f ∂L z {L z , H 1 } + ∂f ∂K 0 {K 0 , H 1 } = 0.\n\n(5.44) Now strictly speaking, there are no closed orbits in the eight dimensional phase space. However, the argument of the previous section applies to orbits which are closed in the four dimensional space with coordinates (r, θ, p r , p θ ), since by the third assumption above everything is independent of t and φ, and p t and p φ are conserved. Here (t, r, θ, φ) are Schwarzschild coordinates and (p t , p r , p θ , p φ ) are the corresponding conjugate momenta.\n\nNext, we can pull the partial derivatives ∂f /∂E etc. outside of the integral. It is then easy to see that the first two terms vanish, since there do exist a conserved energy and a conserved z-component of angular momentum for the perturbed system. Thus, Eq. (5.44) reduces to ∂f ∂K 0 {K 0 , H 1 } = 0.\n\n(5.45)\n\nSince M (0) is functionally independent of E and L z , the prefactor ∂f /∂K 0 must be nonzero, so we obtain {K 0 , H 1 } = 0.\n\n(5.46)\n\nThe result (5.46) applies to fully relativistic orbits in Schwarzschild. We need to take the Newtonian limit of this result in order to use the result we derived in the previous section. However, the Newtonian limit is a little subtle since Newtonian orbits are closed and generic relativistic orbits are not closed. We now discuss how the limit is taken.\n\nThe integral (5.46) is taken over any closed orbit in the four dimensional phase space (r, θ, p r , p θ ) which corresponds to a geodesic in Schwarzschild. Such orbits are non generic; they are the orbits for which the ratio between the radial and angular frequencies ω r and ω θ is a rational number. We denote by q r and q θ the angle variables corresponding to the r and θ motions [36] . These variables evolve with proper time τ according to q r = q r,0 + ω r τ, (5.47a)\n\nq θ = q θ,0 + ω θ τ, (5.47b)\n\nwhere q r,0 and q θ,0 are the initial values. We denote the integrand in Eq. (5.46) by I(q r , q θ , a, ε, ι),\n\nwhere I is some function, and a, ε and ι are the parameters of the geodesic defined by Hughes [32] (functions of E, L z and K 0 ). The result (5.46) can be written as\n\n1 T T /2 -T /2 dτ I[q r (τ ), q θ (τ ), a, ε, ι] = 0, (5.48)\n\nwhere T = T (a, ε, ι) is the period of the r, θ motion. Since the variables q r and q θ are periodic with period 2π, we can express the function I as a Fourier series I(q r , q θ , a, ε, ι) =\n\n= ∞ n,m=-∞ I nm (a, ε, ι)e inqr,0+imq θ,0 ×Si [(nω r + mω θ )T /2] , (5.50)\n\nwhere Si(x) = sin(x)/x. Since the initial conditions q r,0 and q θ,0 are arbitrary, it follows that\n\nI nm (a, ε, ι)Si [(nω r + mω θ )T /2] = 0 (5.51)\n\nfor all n, m.\n\nNext, for closed orbits the ratio of the frequencies must be a rational number, so\n\nw r w θ = p q , (5.52)\n\nwhere p and q are integers with no factor in common. These integers depend on a, ε and ι. The period T is given by 2π/T = qω r = pω θ . The second factor in Eq.\n\n(\n\n.51) now simplifies to Si (np + mq)π pq , (5.53) which vanishes if and only if n = nq, m = mp, n + m = 0, (5.54) for integers n, m. It follows that I nm (a, ε, ι) = 0 (5.55)\n\nfor all n, m except for values of n, m which satisfy the condition (5.54) Consider now the Newtonian limit, which is the limit a → ∞ while keeping fixed ε and ι and the mass of the black hole. We denote by I N (q r , q θ , a, ε, ι) the Newtonian limit of the function I(q r , q θ , a, ε, ι). The integral (5.48) in the Newtonian limit is given by the above computation with p = q = 1, since ω r = ω θ in this limit. This gives\n\n1 T dτ I N = ∞ n=-∞\n\nI N n,-n (a, ε, ι) e in(qr,0-q θ,0 ) , (5.56)\n\nwhere I N nm are the Fourier components of I N . In the previous subsection, we showed that this function is nonzero, which implies that there exists a value k of n for which I N k,-k = 0. Now as a → ∞, we have ω r /ω θ → 1, and hence from Eq. (5.52) there exists a critical value a c of a such that the values of p and q exceed k for all closed orbits with a > a c . (We are keeping fixed the values of ε and ι). as a → ∞. This completes the proof. Hence, if the three assumptions listed at the start of this subsection are satisfied, then the conjecture that all vacuum, axisymmetric spacetimes possess a third constant of the motion is false.\n\nFinally, it is sometimes claimed in the classical dynamics literature that perturbation theory is not a sufficiently powerful tool to assess whether the integrability of a system is preserved under deformations. An example that is often quoted is the Toda lattice Hamiltonian [38, 39] . This system is integrable and admits a full set of constants of motion in involution. However, if one approximates the Hamiltonian by Taylor expanding the potential about the origin to third order, one obtains a system which is not integrable. This would seem to indicate that perturbation theory can indicate a non-integrability, while the exact system is still integrable.\n\nIn fact, the Toda lattice example does not invalidate the method of proof we use here. If we write the Toda lattice Hamiltonian as H(q, p), then the situation is that H(λq, p) is integrable for λ = 1, but it is not integrable for 0 < λ < 1. Expanding H(λq, p) to third order in λ gives a non-integrable Hamiltonian. Thus, the perturbative result is not in disagreement with the exact result for 0 < λ < 1, it only disagrees with the exact result for λ = 1. In other words, the example shows that perturbation theory can fail to yield the correct result for finite values of λ, but there is no indication that it fails in arbitrarily small neighborhoods of λ = 0. Our application is qualitatively different from the Toda lattice example since we have a one parameter family of Hamiltonians H(λ) which by assumption are integrable for all values of λ."
},
{
"section_type": "CONCLUSION",
"section_title": "VI. CONCLUSION",
"text": "We have examined the effect of an axisymmetric quadrupole moment Q of a central body on test particle inspirals, to linear order in Q, to the leading post-Newtonian order, and to linear order in the mass ratio. Our analysis shows that a natural generalization of the Carter constant can be defined for the quadrupole interaction. We have also analyzed the leading order spin selfinteraction effect due to the scattering of the radiation off the spacetime curvature due to the spin. Combining the effects of the quadrupole and the leading order effects linear and quadratic in the spin, we have obtained expressions for the instantaneous as well as time-averaged evolution of the constants of motion for generic orbits under gravitational radiation reaction, complete at O(a 2 ǫ 4 ). We have also shown that for a single multipole interaction other than Q or spin, in our approximations, a Cartertype constant does not exist. With mild additional assumptions, this result can be extended to exact spacetimes and falsifies the conjecture that all axisymmetric vacuum spacetimes possess a third constant of motion for geodesic motion."
},
{
"section_type": "OTHER",
"section_title": "VII. ACKNOWLEDGMENTS",
"text": "This research was partially supported by NSF grant PHY-0457200. We thank Jeandrew Brink for useful correspondence.\n\nAppendix A: Time variation of quadrupole: order of magnitude estimates\n\nIn this appendix, we give an estimate of the timescale T evol for the quadrupole to change. The analysis in the body of this paper is valid only when T evol ≫ T rr , where T rr is the radiation reaction time, since we have neglected the time evolution of the quadrupole. We distinguish between two cases: (i) when the central body is exactly nonspinning but has a quadrupole, and (ii) when the central body has finite spin in addition to the quadrupole."
},
{
"section_type": "OTHER",
"section_title": "Estimate of the scaling for the nonspinning case",
"text": "For the purpose of a crude estimate, the relevant interaction is the tidal interaction with energy\n\nQ ij E ij ∼ - m 2 r 3 QI cos 2 θ, (A1)\n\nwhere E ij is the tidal field, θ is the angle between the symmetry axis and the normal to the orbital plane of m 2 , and we have written the quadrupole as Q ∼ QI, where Q is dimensionless and I is the moment of inertia.\n\nFor small deviations from equilibrium, the relevant piece of the Lagrangian is schematically\n\nL ∼ I ψ2 + QI m 2 r 3 ψ 2 . (A2)\n\nWe define the evolution timescale T evol to be the time it takes for the angle to change by an amount of order unity, and since the amplitude of the oscillation scales roughly as ∼ m 2 /m 1 , the evolution time scales as\n\nT -2 evol ∼ m 2 2 m 2 1 Q m 2 M ω 2 orbit , (A3)\n\nwhere ω 2 orbit = M/r 3 . Thus, the ratio of the evolution timescale compared to the radiation reaction timescale scales as\n\nT evol /T rr ∼ 1/ Q m 1 m 2 µ M 1/2 M r 5/2\n\n. (A4)"
},
{
"section_type": "OTHER",
"section_title": "Estimate of the scaling for the spinning case",
"text": "When the body is spinning the effect of the tidal coupling is to cause a precession. For the purpose of this estimate, we calculate the torque on m 1 due to the companion's Newtonian field. The torque N scales as\n\nN i ∼ ǫ imj Q mk E jk . ( A5\n\n)\n\nWe assume that the precession is slow, i.e.\n\nω prec ≪ S/m 1 m 2 M , ( A6\n\n)\n\nwhere ω prec is the precession frequency and S = S/m 2 1 is the dimensionless spin. This gives the approximate scaling of the precession timescale as (cf. [37] ) )\n\nT prec /T rr ∼ S Q M r . ( A7"
}
] | |
arxiv:0704.0397 | 0704.0397 | 1 | 10.1103/PhysRevA.75.063803 | ee238a6a21e36cb923522ce06c9ab6af9261e7ac7d5d8cbd300f92fb188bc89f | Conditional generation of path-entangled optical NOON states | We propose a measurement protocol to generate path-entangled NOON states
conditionally from two pulsed type II optical parametric oscillators. We
calculate the fidelity of the produced states and the success probability of
the protocol. The trigger detectors are assumed to have finite dead time, and
for short pulse trigger fields they are modeled as on/off detectors with finite
efficiency. Continuous-wave operation of the parametric oscillators is also
considered. | [
"Anne E. B. Nielsen and Klaus Molmer"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Nonclassical states of light have many applications, and a number of different protocols exist for the generation of various classes of states. The two-mode maximally entangled N -photon states
|NOON = 1 √ 2 |N, 0 + e iφ |0, N , ( 1
)
the so-called NOON states, are particularly interesting because a single-photon phase shift of χ induced in one of the two components changes the relative phase of the two terms by N χ. This special property of NOON states may be utilized to enhance spatial resolution in (quantum) microscopy and lithography [1] , and in interferometry it has been shown that a certain measurement strategy, using NOON states, leads to a phase estimation error scaling as L -1/4 N -3/4 T if the phase to be estimated is known to lie within an interval from -π/L to π/L, where N T is the total number of photons used in the measurements [2] . This is better than the classical shot noise limit of N -1/2 T , and NOON states are thus useful to perform accurate measurements and may be a valuable field resource in sensors. NOON states are also a source of entanglement with applications in quantum information protocols and in fundamental studies such as tests of Bells inequality [3] .
It is thus of great interest to be able to produce NOON states, and various NOON state generation schemes have been suggested theoretically [4, 5, 6, 7, 8, 9] and studied in experiments [7, 8, 10, 11] . The N = 1 and N = 2 NOON states may be generated by combining either a single photon and a vacuum state or two single-photon states on a 50 : 50 beam splitter, but this simple approach is not directly extendable to N > 2, and we shall thus mainly be concerned with generation of N = 3 NOON states in the present paper, even though the suggested protocol is, in principle, applicable for all N . Mitchell, Lundeen, and Steinberg have generated NOON states with N = 3 from a pair of down converted photons and a local oscillator photon using certain polarization transforming components and post-selection [10] . In this experiment, however, the successful generation of the NOON state is witnessed by a destructive detection of the state. In the present paper we propose and analyze in detail a nondestructive generation protocol, which conditions the successful generation of the N -photon NOON state on the registration of N photo detection events in other field modes, and which uses as resource only linear optics and the output from two optical parametric oscillators (OPOs). The protocol does not rely on efficient photo detection. The analysis is carried out in terms of Wigner function formalism, and effects of finite detector efficiency and finite detector dead time are considered.
Conditional generation of nonclassical states occupying a single mode has been investigated both experimentally and theoretically [12, 13, 14, 15, 16, 17, 18, 19, 20] . With the correlated output from a single nondegenerate OPO it is, for instance, possible to generate n-photon Fock states of light in the signal beam conditioned on n photo detections in the idler (trigger) beam [16, 19, 20] , and in principle the entanglement of a highly squeezed twomode field from an OPO makes it possible to prepare any state in the signal beam that can be either measured as an eigenstate of a suitable observable of the idler beam or produced as the final state of a generalized measurement. The basic idea of the protocol proposed in the present paper is to mix the output from two OPOs and employ the entanglement to prepare a two-mode state in two of the output beams by detection of the desired output state in the remaining beams.
In Sec. II we explain the NOON state generation protocol in detail. In Sec. III we analyze the performance of the protocol quantitatively for pulsed OPO sources. We provide the fidelity of the generated states and the success probability. In Sec. IV we consider production of NOON states from continuous-wave OPO sources, and Sec. V concludes the paper.
The experimental setup is illustrated in Fig. 1 . Two pulses of two-mode squeezed states are generated by two identical OPOs via type II parametric down conversion. The field mode operators of the modes generated by the 2 first OPO are denoted â+ and â-, respectively, while the field mode operators of the modes generated by the second OPO are denoted b+ and b-, respectively. For definiteness, we assume that the plus modes are vertically polarized and that the minus modes are horizontally polarized. The modes are separated spatially by the first two polarizing beam splitters, and the third polarizing beam splitter combines the â-and b+ modes, which are subsequently subjected to the NOON state measurement proposed in [21] and illustrated for N = 3 in Fig. 1 . The idea behind this measurement is to apply the highly nonlinear operator ÂN = âN --( b+ e iθ ) N to the state. The result is only nonzero if either the â-mode or the b+ mode contains at least N photons. On the other hand, if the squeezing is sufficiently small, it is unlikely to have more than a total of N photons in the two trigger modes, and by conditioning on the successful application of ÂN , we select the pulses of the system where N photon pairs are generated in one OPO and zero photon pairs in the other. It is equally probable that the photons originate from the first OPO or from the second OPO, and, as we shall see in detail below, the result is that a NOON state is generated conditionally in the output modes â+ and b-.
As stated in [21] , ÂN can be rewritten as a simple product of single photon annihilation operators âN --b+ e iθ N = N n=1 â--b+ e iθ e i2πn/N , (2) and it is thus possible to implement ÂN by means of beam splitters and photo detectors. We first consider odd values of N . Beam splitters are used to divide the input into N distinct spatial modes labeled by n = 1, . . . , N . The beam splitter reflectivities are chosen to obtain the same expectation value of the intensity in each of the modes. The vertically polarized modes are then phase shifted by the factor e i2πn/N +iπ relative to the horizontally polarized modes, i.e., b+ → -b+ e i2πn/N , and finally polarizing beam splitters with principal planes oriented at 45 • relative to the horizontal polarization transform â-and -b+ e i2πn/N into (â --b+ e i2πn/N )/ √ 2 (the transmitted mode) and (â -+ b+ e i2πn/N )/ √ 2 (the reflected mode) [22] . The annihilation of a photon in each of the modes transmitted by the beam splitters witnesses the overall application of the operator ÂN . If one observes both reflected and transmitted modes simultaneously, one conditions on detection events in all the transmitted modes and no detection events in all the reflected modes. If detection events are instead observed in all the reflected modes and in none of the transmitted modes, an operator of the form (2) is also obtained, but θ is effectively transformed into θ + π due to the phase shift at the polarizing beam splitter, and the value of φ of the generated NOON states is changed by N π (see below). The success probability is thus increased by a factor of two if both outcomes are accepted.
FIG. 1: Experimental setup for NOON state generation. OPO, optical parametric oscillator; PBS, polarizing beam splitter; PS, phase shifter; and APD, avalanche photo diode. The part of the setup enclosed in the dashed box performs the NOON state measurement, and here it is shown for N = 3. Note that the polarizing beam splitters inside the box are oriented at 45 • . The numbers denote beam splitter reflectivities of 1/3 and 1/2, and the three phase shifters transform b+ into -b+e 2πin/3 , where n = 1, 2, 3, respectively. See text for details.
For even values of N a similar measurement scheme is applicable, but it is sufficient to divide the field into N/2 spatial modes initially, and in this case the NOON state generation is conditioned on detection events in both transmitted and reflected modes (see [21]).
After this presentation of the basic idea and the physical setup we now consider the actual outcome of the detection process. For short pulse OPO output the dead time of the photo detectors may typically be longer than the pulse duration, and we shall thus assume that it is impossible to obtain more than a single detection event per detector per pulse, i.e., if the detector efficiency is unity, the detectors are only able to distinguish between vacuum and states different from the vacuum state. Such detectors are denoted on/off detectors, and they are discussed in detail in Ref. [23] . The finite dead time of the detectors is not severe to the measurement procedure described in [21] because the on/off detector model and the conventional photo detector model, represented by the annihilation operator, lead to identical signal states if the total number of photons in the idler modes is guaranteed to be less than or equal to the number of conditioning detection events, i.e., N .
We analyze the performance of the setup using Gaussian Wigner function formalism [15, 19, 20] , which is applicable because the squeezed states generated by the OPOs and the vacuum states coupled into the system via the beam splitters are all Gaussian. In general, the Wigner function of an n-mode Gaussian state with zero mean field amplitude takes the form
W V (x 1 , p 1 , . . . , x n , p n ) = 1 π n det(V ) e -y T V -1 y , ( 3
)
3 where y ≡ (x 1 , p 1 , . . . , x n , p n ) T and V is the 2n × 2n covariance matrix. If ĉi denotes the field mode annihilation operator of mode i, the elements of V are given in terms of the real and imaginary parts of the expectation values ĉ † i ĉj and ĉi ĉj . Note that for a multi-mode Gaussian state we are free to include only the modes of interest in (3) because the partial trace operation over unobserved modes is equivalent to integration over the corresponding quadrature variables. A unit efficiency 'on' detection in mode i projects mode i on the subspace of Hilbert space that is orthogonal to the vacuum state, i.e., the Wigner function is multiplied by (1
-2πW 0 (x i , p i )), where W 0 (x, p) = exp(-x 2 -p 2
)/π is the Wigner function of the vacuum state, the variables x i and p i are integrated out, and the state is renormalized. Since the Gaussian nature of a state is preserved under linear transformations, and since a detector with single-photon efficiency η is equivalent to a beam splitter with transmission η followed by a unit efficiency detector [23] , effects of non-unit detector efficiency are easily included in the covariance matrix.
To calculate ĉ † i ĉj and ĉi ĉj explicitly we note that the state generated by the OPOs is [24]
|ψ i = (1 -r 2 ) ∞ n=0 ∞ m=0 n+m |n, n, m, m , ( 4
)
where r is the squeezing parameter and the modes are listed in the order: â+ , â-, b+ , b-. We assume that N is odd and consider the transmitted trigger modes (which we number from 1 to N ), the â+ mode (mode N + 1), and the bmode (mode N + 2). By expressing the field operators of the trigger modes (those observed by the unit efficiency detectors) in terms of â-, b+ , and field operators representing vacuum states we find
ĉ † j ĉk = ψ i | η 2N (â † --e -2πij/N b † + e -iθ ) η 2N (â --e 2πik/N b+ e iθ )|ψ i = 1 + e 2πi(k-j)/N λ 2N , ( 5
)
where j ∈ {1, 2, . . . , N }, k ∈ {1, 2, . . . , N }, λ ≡ ηr 2 /(1r 2 ), and we allow of a constant phase shift θ of b+ relative to â-. Furthermore
ĉ † N +1 ĉN+1 = ψ i |â † + â+ |ψ i = r 2 /(1 -r 2 ), ( 6
) ĉ † N +2 ĉN+2 = ψ i | b † -b-|ψ i = r 2 /(1 -r 2 ), ( 7
) ĉk ĉN+1 = ψ i | η 2N (â --e 2πik/N +iθ b+ )â + |ψ i = η 2N r 1 -r 2 , ( 8
) ĉk ĉN+2 = ψ i | η 2N (â --e 2πik/N +iθ b+ ) b-|ψ i = - η 2N r 1 -r 2 e 2πik/N +iθ , ( 9
)
and ĉj ĉk = ĉN+1 ĉN+1 = ĉN+2 ĉN+2 = ĉN+1 ĉN+2 =
ĉ † N +1 ĉN+2 = ĉ † k ĉN+1 = ĉ † k ĉN+2 = 0. ( 10
)
For even values of N the factors η/(2N ) are replaced by η/N . Note that loss in the signal beam may be taken into account by performing the transformations â+ → √ η s â+ and b-→ √ η s bin the above expressions, where 1 -η s is the loss. The NOON state fidelity F N of the signal state conditioned on N photo detection events in the transmitted trigger modes is
F N = 4π 2 P N W NOON (x N +1 , p N +1 , x N +2 , p N +2 ) N i=1 (1 -2πW 0 (x i , p i )) W V (x 1 , p 1 , . . . , x N +2 , p N +2 ) N +2 i=1 dx i dp i , ( 11
)
where W NOON is the Wigner function of the NOON state (1), and
P N = N i=1 (1 -2πW 0 (x i , p i )) W V (x 1 , p 1 , . . . , x N +2 , p N +2 ) N +2 i=1 dx i dp i , ( 12
)
is the success probability, i.e., the probability to obtain the conditioning detection events and produce the NOON state in a given pulse of the OPO system. We expand the product
N i=1 (1 -2πW 0 (x i , p i )) = d N i=1 (-2πW 0 (x i , p i )) di , (13) where the sum is over all d ≡ (d 1 , d 2 , . . . , d N ) with d i ∈ {0, 1}
, and define the diagonal matrix J d = 4 diag(d 1 , d 1 , d 2 , d 2 , . . . , d N , d N ) and the n× n identity matrix I n . Furthermore, we divide the covariance matrix into four parts
V = V tt V ts V T ts V ss , ( 14
)
where V tt is the 2N × 2N covariance matrix of the trigger modes, V ss is the 4 × 4 covariance matrix of the signal modes, while V ts contains the correlations between the trigger and the signal modes, and we define the vector
y s = (x N +1 , p N +1 , x N +2 , p N +2 ) T
and the matrix
U d = V ss -V T ts J d (J d V tt J d + I 2N ) -1 J d V ts . ( 15
)
This allows us to write Eqs. (11) and (12) in the following compact forms [25]
F N = 4π 2 P N d (-2) P N i=1 di det(I 2N + J d V tt ) W NOON (y s )W U d (y s )dy s , ( 16
)
and
P N = d (-2) P N i=1 di det(I 2N + J d V tt ) . ( 17
)
Since W NOON is a product of a polynomial and a Gaussian the integral in Eq. ( 16 ) may be evaluated analytically and for N = 3 and η = 1 we find
F η=1 3 = (1 -r 2 ) 2 (2 -r 2 ) 2 (3 -2r 2 )(6 -5r 2 ) 18(4 -3r 2 ) , ( 18
)
where the optimal value φ = N θ + π + 2πn, n ∈ Z, is assumed. Expressions for P N are given in table I for N = 1, 2, 3, and 4, and F N and P N are plotted for N = 3 in Figs. 2 and 3, respectively. We observe that high probabilities are only found in the parameter regime, where the fidelity is low. If, for instance, we want a NOON state fidelity of at least 0.9, we choose r = 0.14, and if η = 0.25, P 3 is of order 10 -8 . With a repetition rate of order 10 6 s -1 (see [16] ) one state is produced every second minute on average. The production rate is very dependent on detector efficiency, and if η is increased to unity, the rate is increased by approximately a factor of 60. For odd values of N we may observe both reflected and transmitted trigger modes and condition on detection events in all the transmitted trigger modes and no detection events in all the reflected trigger modes, or, vice versa. In this case we also include the reflected trigger modes in the covariance matrix, which we now denote by V + . By a similar analysis as above we obtain the success probability
P + N = 2 d 2 N (-2) P N i=1 di det(I 4N + J + d V + tt ) , ( 19
)
N PN P + N 1 λ λ+1 2λ (λ+1) 2 2 λ 2 (λ+1) 2 -3 λ 3 (λ+4) (λ+2) 2 (λ+3)(λ+6) 2λ 3 (3λ+4) (λ+1) 2 (λ+2) 2 (2λ+3)(5λ+6) 4 λ 4 (λ 2 +6λ+6) (λ+1) 2 (λ+2) 2 (λ 2 +8λ+8) -TABLE I: Success probabilities calculated from Eqs. (17) and (19). λ ≡ ηr 2 /(1 -r 2 ). 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r F 3 and F + 3 FIG. 2: NOON state fidelity F3 (solid lines) and F + 3 (dashed lines) as a function of squeezing parameter r for η = 1 (upper lines), η = 0.25 (middle lines), and η → 0 (lower lines). Note that in the latter case F3 = F + 3 .
where J + d ≡ diag(d 1 , d 1 , . . . , d N , d N , 1, 1, . . . , 1, 1), while the NOON state fidelity F + N is given by Eq. (11) with V replaced by the matrix
V -(V + R ) T (V + RR + I 2N ) -1 V + R , ( 20
)
where V + RR is the covariance matrix of the reflected trigger modes and V + R consists of the correlations between the reflected trigger modes and the signal and transmitted trigger modes. Explicit results for P + N are given in table I for N = 1 and 3. F + 3 and P + 3 are compared to F 3 and P 3 in Figs. 2 and 3, and it is observed that F + 3 and P + 3 are both larger than F 3 and P 3 if r is not large (and η > 0). For r → 1, P + 3 → 0 because in this limit it is very unlikely to obtain no detection events in all the reflected or in all the transmitted trigger modes.
In the limit of very small detector efficiency a simple expression for the NOON state fidelity for the case of N trigger detectors is easily derived without using Wigner function formalism. In general, if the state of interest is expressed in the photon number basis, the mathematical operation corresponding to an 'on' detection is to multiply each term by 1 -(1 -η) n , where n is the number of photons in the mode observed by the nonunit efficiency detector, trace out the detected mode, and renormalize. If nη ≪ 1 for all contributing terms, 1 -(1 -η) n ≈ √ nη ∝ √ n, and the on/off detector 5 0 0.2 0.4 0.6 0.8 1 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 r P 3 and P 3 + FIG. 3: Success probability P3 (solid lines) and P + 3 (dashed lines) as a function of squeezing for η = 1 (upper lines) and η = 0.25 (lower lines). The dotted lines represent the approximate expression (23).
model becomes equivalent to the photo detector model. In this case the density operator of the output state is obtained as
ρ = M ∞ p=0 ∞ q=0 p| q|(â N --( b+ e iθ ) N )|ψ i ψ i |((â † -) N -( b † + e -iθ ) N )|q |p = (1 -r 2 ) N +2 2N !r 2N ∞ n=N ∞ m=0 (r 2 ) n+m n! (n -N )! |n, m n, m| -e -iN θ n!(m + N )! (n -N )!m! |n, m n -N, m + N | + ∞ n=0 ∞ m=N (r 2 ) n+m m! (m -N )! |n, m n, m| -e iN θ (n + N )!m! n!(m -N )! |n, m n + N, m -N | , ( 21
)
where M is a normalization constant and the traces are over the â-and b+ modes. This leads to the NOON state fidelity
F η≪1 N = NOON|ρ|NOON = (1 -r 2 ) N +2 , ( 22
)
where again φ = N θ + π + 2πn, n ∈ Z, is assumed. It is interesting to compare this result with the fidelity (1 -r 2 ) N +1 obtained for production of N -photon states from a single two-mode squeezed state by conditioning on N detection events in the idler beam and using detectors with very small efficiency. If a single-photon state is produced by this method and transformed into an N = 1 NOON state as explained in the Introduction, the NOON state fidelity is F 1,s = (1-r 2 ) 2 , and the success probability is P 1,s = λ/(λ + 1). Choosing squeezing parameters such that F 1,s = F 1 , we find that P + 1 = (4/3)P 1,s in the high fidelity limit. It is thus possible to achieve a higher success probability using the scheme with two OPOs, but the price to pay is a more technically involved setup, and NOON states with two different values of φ are produced. For N = 2 the present protocol and combination of two single-photon states on a 50 : 50 beam splitter, each produced conditionally from a single OPO, lead to identical fidelities and success probabilities. Finally we note that the photo detector model underestimates
F N for η > 0 because 1 -(1 -η) n = η n-1 i=0 (1 -η) i < nη
for n = 2, 3, . . . while 1 -(1 -η) n = nη for n = 0, 1, i.e., the 'wrong' terms containing more than N photons are given a too large weight. This is also what we observe in Fig. 2 .
In the limit of small r and for odd values of N the success probability is given approximately by the simple expression
P N ≈ η 2N N ψ i |((â † -) N -( b † + e -iθ ) N ) (â N --( b+ e iθ ) N )|ψ i = 2N ! (2N ) N λ N (N odd). (23)
Again η/(2N ) must be replaced by η/N to obtain P N for even values of N . The approximation to P 3 is shown in Fig. 3 .
Our protocol is not limited to pulsed fields, and for completeness we now briefly consider NOON state generation from continuously driven OPOs. We assume N = 3. For continuous-wave fields each of the three detected trigger beams and the two signal beams are described by time dependent field operators ĉi (t). The trigger detections take place in particular modes localized around the three detection times t c1 , t c2 , and t c3 , and we want to determine the NOON state fidelity of an output state occupying one temporal mode in each signal beam. Following the general multimode formalism in Refs. [15, 20] , the five relevant modes are specified by the mode functions f i (t), and the corresponding five single mode operators are ĉi = f i (t)ĉ i (t)dt. In general, we are free to choose the two output mode functions at will, and in the present case it is natural to choose the mode function which gives rise to the largest three-photon state fidelity when three-photon states are generated from a single type II continuous-wave OPO. Since we are mainly interested in the parameter region where the squeezing is small and the NOON state fidelity is large, we use the optimal three-photon state mode function derived for very low beam intensity in [20]
, i.e., f 4 (t) = f 5 (t) = 3 k=1 c k γ/2 exp(-γ|t -t ck |/2)
, where 6 0 0.05 0.1 0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ε/γ F 3 FIG. 4: NOON state fidelity as a function of ǫ/γ for states generated from a pair of continuous-wave OPO sources when conditioning on three simultaneous trigger detection events tc1 = tc2 = tc3. 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (t c3 -t c1 )γ F 3 FIG. 5: Fidelity of NOON states from continuous-wave OPO sources as a function of separation between trigger detection events (tc3 -tc1)γ for N = 3, tc3 -tc2 = tc2 -tc1, and ǫ/γ = 0.01.
the coefficients c k are functions of the intervals between the detection times and γ is the leakage rate of the OPO output mirror. We furthermore assume that the trigger mode functions are nonzero only in an infinitesimal time interval centered at the detection time, which is valid if the trigger detections take place on a time scale much shorter than γ -1 . Since we consider a low intensity continuous beam, and since we formally assume that the trigger detectors only register the light field in infinitesimal time intervals around the detection times, the annihilation operator detector model is perfectly valid in this case and detector dead time is insignificant.
We may now proceed as above and eliminate all the irrelevant modes from the analysis by writing down the Gaussian Wigner function of the five interesting modes.
The only difference is that this time ĉ † i ĉj and ĉi ĉj are expressed in terms of the two time correlation functions of the OPO output. Also, the operators applied to the Wigner function to take conditioning into account are different because the annihilation detector model is used. The reader is referred to Refs. [15, 20] for details.
The resulting fidelity is shown as a function of ǫ/γ in Fig. 4 , where ǫ is the nonlinear gain in the OPO, and as a function of the temporal distance between the conditioning detection events in Fig. 5 . As in the pulsed case the fidelity decreases when the degree of squeezing increases. The fidelity also decreases when the temporal distance between the conditioning detection events increases from zero, but it is permissible to have a small time interval between the trigger detection events. We note that the curves represent a lower limit to the theoretically achievable fidelity since a better fidelity may be obtained for another choice of output state mode functions.
In conclusion we have analyzed a method to generate path entangled NOON states from the output from two optical parametric oscillators. The method relies on the joint detection of photons in a number of trigger beams, and we presented a theoretical analysis of the role of detector efficiency and dead time for the fidelity of the states obtained and the success probability of the protocol. Our specific NOON state protocol applies for general photon numbers of the states, but in practice it is not realistic to go beyond the case of N = 3, studied here. This is due to the reduction of the fidelity due to unwanted contributions from higher number states, when the OPO output power gets too high, combined with the severe reduction of the probability to obtain the number of conditioning detection events needed when the OPO output power is too low. The N = 3 NOON states, which can be produced at 90% fidelity at the rate of one state produced every 10 -100 seconds, seem to be at the limit of realistic experiments of the proposed kind. Finally, we also determined the NOON state fidelity for continuouswave fields, where the best NOON state occupies a pair of suitably selected temporal mode functions, and where we find high fidelities as long as the trigger events occur within a short time window compared to the lifetime of the OPO cavity field.
We presented this analysis for the production of optical NOON states, but we note that recent theoretical proposals and experiments with four wave mixing of matter waves [26] , engineered quadratic interactions among trapped ions [27], and entanglement between field and atomic degrees of freedom [28, 29] bring promise for similar conditional generation of atomic and interspecies atom-field NOON states.
This work was supported by the European Integrated project SCALA. 7 [1] A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000). [2] L. Pezzé and A. Smerzi, quant-ph/0508158. [3] M. D'Angelo, A. Zavatta, V. Parigi, and M. Bellini, Phys.
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40, 1573 (1993). [23] P. P. Rohde and T. C. Ralph, J. Mod. Opt. 53, 1589 (2006). [24] A. Ekert and P. Knight, Am. J. Phys. 63, 415 (1995). [25] J. Eisert and M. B. Plenio, Int. J. Quantum Inf. 1, 479 (2003). [26] G. K. Campbell, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, Phys. Rev. Lett. 96, 020406 (2006) [27] D. Porras and J. I. Cirac, Phys. Rev. Lett. 93, 263602 (2004). [28] B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Monroe, Nature, 428, 153 (2004). [29] J. Volz, M. Weber, D. Schlenk, W. Rosenfeld, J. Vrana, K. Saucke, C. Kurtsiefer, and H. Weinfurter, Phys. Rev. Lett. 96, 030404 (2006). | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Nonclassical states of light have many applications, and a number of different protocols exist for the generation of various classes of states. The two-mode maximally entangled N -photon states\n\n|NOON = 1 √ 2 |N, 0 + e iφ |0, N , ( 1\n\n)\n\nthe so-called NOON states, are particularly interesting because a single-photon phase shift of χ induced in one of the two components changes the relative phase of the two terms by N χ. This special property of NOON states may be utilized to enhance spatial resolution in (quantum) microscopy and lithography [1] , and in interferometry it has been shown that a certain measurement strategy, using NOON states, leads to a phase estimation error scaling as L -1/4 N -3/4 T if the phase to be estimated is known to lie within an interval from -π/L to π/L, where N T is the total number of photons used in the measurements [2] . This is better than the classical shot noise limit of N -1/2 T , and NOON states are thus useful to perform accurate measurements and may be a valuable field resource in sensors. NOON states are also a source of entanglement with applications in quantum information protocols and in fundamental studies such as tests of Bells inequality [3] .\n\nIt is thus of great interest to be able to produce NOON states, and various NOON state generation schemes have been suggested theoretically [4, 5, 6, 7, 8, 9] and studied in experiments [7, 8, 10, 11] . The N = 1 and N = 2 NOON states may be generated by combining either a single photon and a vacuum state or two single-photon states on a 50 : 50 beam splitter, but this simple approach is not directly extendable to N > 2, and we shall thus mainly be concerned with generation of N = 3 NOON states in the present paper, even though the suggested protocol is, in principle, applicable for all N . Mitchell, Lundeen, and Steinberg have generated NOON states with N = 3 from a pair of down converted photons and a local oscillator photon using certain polarization transforming components and post-selection [10] . In this experiment, however, the successful generation of the NOON state is witnessed by a destructive detection of the state. In the present paper we propose and analyze in detail a nondestructive generation protocol, which conditions the successful generation of the N -photon NOON state on the registration of N photo detection events in other field modes, and which uses as resource only linear optics and the output from two optical parametric oscillators (OPOs). The protocol does not rely on efficient photo detection. The analysis is carried out in terms of Wigner function formalism, and effects of finite detector efficiency and finite detector dead time are considered.\n\nConditional generation of nonclassical states occupying a single mode has been investigated both experimentally and theoretically [12, 13, 14, 15, 16, 17, 18, 19, 20] . With the correlated output from a single nondegenerate OPO it is, for instance, possible to generate n-photon Fock states of light in the signal beam conditioned on n photo detections in the idler (trigger) beam [16, 19, 20] , and in principle the entanglement of a highly squeezed twomode field from an OPO makes it possible to prepare any state in the signal beam that can be either measured as an eigenstate of a suitable observable of the idler beam or produced as the final state of a generalized measurement. The basic idea of the protocol proposed in the present paper is to mix the output from two OPOs and employ the entanglement to prepare a two-mode state in two of the output beams by detection of the desired output state in the remaining beams.\n\nIn Sec. II we explain the NOON state generation protocol in detail. In Sec. III we analyze the performance of the protocol quantitatively for pulsed OPO sources. We provide the fidelity of the generated states and the success probability. In Sec. IV we consider production of NOON states from continuous-wave OPO sources, and Sec. V concludes the paper."
},
{
"section_type": "METHOD",
"section_title": "II. EXPERIMENTAL SETUP FOR NOON STATE GENERATION",
"text": "The experimental setup is illustrated in Fig. 1 . Two pulses of two-mode squeezed states are generated by two identical OPOs via type II parametric down conversion. The field mode operators of the modes generated by the 2 first OPO are denoted â+ and â-, respectively, while the field mode operators of the modes generated by the second OPO are denoted b+ and b-, respectively. For definiteness, we assume that the plus modes are vertically polarized and that the minus modes are horizontally polarized. The modes are separated spatially by the first two polarizing beam splitters, and the third polarizing beam splitter combines the â-and b+ modes, which are subsequently subjected to the NOON state measurement proposed in [21] and illustrated for N = 3 in Fig. 1 . The idea behind this measurement is to apply the highly nonlinear operator ÂN = âN --( b+ e iθ ) N to the state. The result is only nonzero if either the â-mode or the b+ mode contains at least N photons. On the other hand, if the squeezing is sufficiently small, it is unlikely to have more than a total of N photons in the two trigger modes, and by conditioning on the successful application of ÂN , we select the pulses of the system where N photon pairs are generated in one OPO and zero photon pairs in the other. It is equally probable that the photons originate from the first OPO or from the second OPO, and, as we shall see in detail below, the result is that a NOON state is generated conditionally in the output modes â+ and b-.\n\nAs stated in [21] , ÂN can be rewritten as a simple product of single photon annihilation operators âN --b+ e iθ N = N n=1 â--b+ e iθ e i2πn/N , (2) and it is thus possible to implement ÂN by means of beam splitters and photo detectors. We first consider odd values of N . Beam splitters are used to divide the input into N distinct spatial modes labeled by n = 1, . . . , N . The beam splitter reflectivities are chosen to obtain the same expectation value of the intensity in each of the modes. The vertically polarized modes are then phase shifted by the factor e i2πn/N +iπ relative to the horizontally polarized modes, i.e., b+ → -b+ e i2πn/N , and finally polarizing beam splitters with principal planes oriented at 45 • relative to the horizontal polarization transform â-and -b+ e i2πn/N into (â --b+ e i2πn/N )/ √ 2 (the transmitted mode) and (â -+ b+ e i2πn/N )/ √ 2 (the reflected mode) [22] . The annihilation of a photon in each of the modes transmitted by the beam splitters witnesses the overall application of the operator ÂN . If one observes both reflected and transmitted modes simultaneously, one conditions on detection events in all the transmitted modes and no detection events in all the reflected modes. If detection events are instead observed in all the reflected modes and in none of the transmitted modes, an operator of the form (2) is also obtained, but θ is effectively transformed into θ + π due to the phase shift at the polarizing beam splitter, and the value of φ of the generated NOON states is changed by N π (see below). The success probability is thus increased by a factor of two if both outcomes are accepted.\n\nFIG. 1: Experimental setup for NOON state generation. OPO, optical parametric oscillator; PBS, polarizing beam splitter; PS, phase shifter; and APD, avalanche photo diode. The part of the setup enclosed in the dashed box performs the NOON state measurement, and here it is shown for N = 3. Note that the polarizing beam splitters inside the box are oriented at 45 • . The numbers denote beam splitter reflectivities of 1/3 and 1/2, and the three phase shifters transform b+ into -b+e 2πin/3 , where n = 1, 2, 3, respectively. See text for details.\n\nFor even values of N a similar measurement scheme is applicable, but it is sufficient to divide the field into N/2 spatial modes initially, and in this case the NOON state generation is conditioned on detection events in both transmitted and reflected modes (see [21])."
},
{
"section_type": "OTHER",
"section_title": "III. PERFORMANCE OF THE PROTOCOL",
"text": "After this presentation of the basic idea and the physical setup we now consider the actual outcome of the detection process. For short pulse OPO output the dead time of the photo detectors may typically be longer than the pulse duration, and we shall thus assume that it is impossible to obtain more than a single detection event per detector per pulse, i.e., if the detector efficiency is unity, the detectors are only able to distinguish between vacuum and states different from the vacuum state. Such detectors are denoted on/off detectors, and they are discussed in detail in Ref. [23] . The finite dead time of the detectors is not severe to the measurement procedure described in [21] because the on/off detector model and the conventional photo detector model, represented by the annihilation operator, lead to identical signal states if the total number of photons in the idler modes is guaranteed to be less than or equal to the number of conditioning detection events, i.e., N .\n\nWe analyze the performance of the setup using Gaussian Wigner function formalism [15, 19, 20] , which is applicable because the squeezed states generated by the OPOs and the vacuum states coupled into the system via the beam splitters are all Gaussian. In general, the Wigner function of an n-mode Gaussian state with zero mean field amplitude takes the form\n\nW V (x 1 , p 1 , . . . , x n , p n ) = 1 π n det(V ) e -y T V -1 y , ( 3\n\n)\n\n3 where y ≡ (x 1 , p 1 , . . . , x n , p n ) T and V is the 2n × 2n covariance matrix. If ĉi denotes the field mode annihilation operator of mode i, the elements of V are given in terms of the real and imaginary parts of the expectation values ĉ † i ĉj and ĉi ĉj . Note that for a multi-mode Gaussian state we are free to include only the modes of interest in (3) because the partial trace operation over unobserved modes is equivalent to integration over the corresponding quadrature variables. A unit efficiency 'on' detection in mode i projects mode i on the subspace of Hilbert space that is orthogonal to the vacuum state, i.e., the Wigner function is multiplied by (1\n\n-2πW 0 (x i , p i )), where W 0 (x, p) = exp(-x 2 -p 2\n\n)/π is the Wigner function of the vacuum state, the variables x i and p i are integrated out, and the state is renormalized. Since the Gaussian nature of a state is preserved under linear transformations, and since a detector with single-photon efficiency η is equivalent to a beam splitter with transmission η followed by a unit efficiency detector [23] , effects of non-unit detector efficiency are easily included in the covariance matrix.\n\nTo calculate ĉ † i ĉj and ĉi ĉj explicitly we note that the state generated by the OPOs is [24]\n\n|ψ i = (1 -r 2 ) ∞ n=0 ∞ m=0 n+m |n, n, m, m , ( 4\n\n)\n\nwhere r is the squeezing parameter and the modes are listed in the order: â+ , â-, b+ , b-. We assume that N is odd and consider the transmitted trigger modes (which we number from 1 to N ), the â+ mode (mode N + 1), and the bmode (mode N + 2). By expressing the field operators of the trigger modes (those observed by the unit efficiency detectors) in terms of â-, b+ , and field operators representing vacuum states we find\n\nĉ † j ĉk = ψ i | η 2N (â † --e -2πij/N b † + e -iθ ) η 2N (â --e 2πik/N b+ e iθ )|ψ i = 1 + e 2πi(k-j)/N λ 2N , ( 5\n\n)\n\nwhere j ∈ {1, 2, . . . , N }, k ∈ {1, 2, . . . , N }, λ ≡ ηr 2 /(1r 2 ), and we allow of a constant phase shift θ of b+ relative to â-. Furthermore\n\nĉ † N +1 ĉN+1 = ψ i |â † + â+ |ψ i = r 2 /(1 -r 2 ), ( 6\n\n) ĉ † N +2 ĉN+2 = ψ i | b † -b-|ψ i = r 2 /(1 -r 2 ), ( 7\n\n) ĉk ĉN+1 = ψ i | η 2N (â --e 2πik/N +iθ b+ )â + |ψ i = η 2N r 1 -r 2 , ( 8\n\n) ĉk ĉN+2 = ψ i | η 2N (â --e 2πik/N +iθ b+ ) b-|ψ i = - η 2N r 1 -r 2 e 2πik/N +iθ , ( 9\n\n)\n\nand ĉj ĉk = ĉN+1 ĉN+1 = ĉN+2 ĉN+2 = ĉN+1 ĉN+2 =\n\nĉ † N +1 ĉN+2 = ĉ † k ĉN+1 = ĉ † k ĉN+2 = 0. ( 10\n\n)\n\nFor even values of N the factors η/(2N ) are replaced by η/N . Note that loss in the signal beam may be taken into account by performing the transformations â+ → √ η s â+ and b-→ √ η s bin the above expressions, where 1 -η s is the loss. The NOON state fidelity F N of the signal state conditioned on N photo detection events in the transmitted trigger modes is\n\nF N = 4π 2 P N W NOON (x N +1 , p N +1 , x N +2 , p N +2 ) N i=1 (1 -2πW 0 (x i , p i )) W V (x 1 , p 1 , . . . , x N +2 , p N +2 ) N +2 i=1 dx i dp i , ( 11\n\n)\n\nwhere W NOON is the Wigner function of the NOON state (1), and\n\nP N = N i=1 (1 -2πW 0 (x i , p i )) W V (x 1 , p 1 , . . . , x N +2 , p N +2 ) N +2 i=1 dx i dp i , ( 12\n\n)\n\nis the success probability, i.e., the probability to obtain the conditioning detection events and produce the NOON state in a given pulse of the OPO system. We expand the product\n\nN i=1 (1 -2πW 0 (x i , p i )) = d N i=1 (-2πW 0 (x i , p i )) di , (13) where the sum is over all d ≡ (d 1 , d 2 , . . . , d N ) with d i ∈ {0, 1}\n\n, and define the diagonal matrix J d = 4 diag(d 1 , d 1 , d 2 , d 2 , . . . , d N , d N ) and the n× n identity matrix I n . Furthermore, we divide the covariance matrix into four parts\n\nV = V tt V ts V T ts V ss , ( 14\n\n)\n\nwhere V tt is the 2N × 2N covariance matrix of the trigger modes, V ss is the 4 × 4 covariance matrix of the signal modes, while V ts contains the correlations between the trigger and the signal modes, and we define the vector\n\ny s = (x N +1 , p N +1 , x N +2 , p N +2 ) T\n\nand the matrix\n\nU d = V ss -V T ts J d (J d V tt J d + I 2N ) -1 J d V ts . ( 15\n\n)\n\nThis allows us to write Eqs. (11) and (12) in the following compact forms [25]\n\nF N = 4π 2 P N d (-2) P N i=1 di det(I 2N + J d V tt ) W NOON (y s )W U d (y s )dy s , ( 16\n\n)\n\nand\n\nP N = d (-2) P N i=1 di det(I 2N + J d V tt ) . ( 17\n\n)\n\nSince W NOON is a product of a polynomial and a Gaussian the integral in Eq. ( 16 ) may be evaluated analytically and for N = 3 and η = 1 we find\n\nF η=1 3 = (1 -r 2 ) 2 (2 -r 2 ) 2 (3 -2r 2 )(6 -5r 2 ) 18(4 -3r 2 ) , ( 18\n\n)\n\nwhere the optimal value φ = N θ + π + 2πn, n ∈ Z, is assumed. Expressions for P N are given in table I for N = 1, 2, 3, and 4, and F N and P N are plotted for N = 3 in Figs. 2 and 3, respectively. We observe that high probabilities are only found in the parameter regime, where the fidelity is low. If, for instance, we want a NOON state fidelity of at least 0.9, we choose r = 0.14, and if η = 0.25, P 3 is of order 10 -8 . With a repetition rate of order 10 6 s -1 (see [16] ) one state is produced every second minute on average. The production rate is very dependent on detector efficiency, and if η is increased to unity, the rate is increased by approximately a factor of 60. For odd values of N we may observe both reflected and transmitted trigger modes and condition on detection events in all the transmitted trigger modes and no detection events in all the reflected trigger modes, or, vice versa. In this case we also include the reflected trigger modes in the covariance matrix, which we now denote by V + . By a similar analysis as above we obtain the success probability\n\nP + N = 2 d 2 N (-2) P N i=1 di det(I 4N + J + d V + tt ) , ( 19\n\n)\n\nN PN P + N 1 λ λ+1 2λ (λ+1) 2 2 λ 2 (λ+1) 2 -3 λ 3 (λ+4) (λ+2) 2 (λ+3)(λ+6) 2λ 3 (3λ+4) (λ+1) 2 (λ+2) 2 (2λ+3)(5λ+6) 4 λ 4 (λ 2 +6λ+6) (λ+1) 2 (λ+2) 2 (λ 2 +8λ+8) -TABLE I: Success probabilities calculated from Eqs. (17) and (19). λ ≡ ηr 2 /(1 -r 2 ). 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r F 3 and F + 3 FIG. 2: NOON state fidelity F3 (solid lines) and F + 3 (dashed lines) as a function of squeezing parameter r for η = 1 (upper lines), η = 0.25 (middle lines), and η → 0 (lower lines). Note that in the latter case F3 = F + 3 .\n\nwhere J + d ≡ diag(d 1 , d 1 , . . . , d N , d N , 1, 1, . . . , 1, 1), while the NOON state fidelity F + N is given by Eq. (11) with V replaced by the matrix\n\nV -(V + R ) T (V + RR + I 2N ) -1 V + R , ( 20\n\n)\n\nwhere V + RR is the covariance matrix of the reflected trigger modes and V + R consists of the correlations between the reflected trigger modes and the signal and transmitted trigger modes. Explicit results for P + N are given in table I for N = 1 and 3. F + 3 and P + 3 are compared to F 3 and P 3 in Figs. 2 and 3, and it is observed that F + 3 and P + 3 are both larger than F 3 and P 3 if r is not large (and η > 0). For r → 1, P + 3 → 0 because in this limit it is very unlikely to obtain no detection events in all the reflected or in all the transmitted trigger modes.\n\nIn the limit of very small detector efficiency a simple expression for the NOON state fidelity for the case of N trigger detectors is easily derived without using Wigner function formalism. In general, if the state of interest is expressed in the photon number basis, the mathematical operation corresponding to an 'on' detection is to multiply each term by 1 -(1 -η) n , where n is the number of photons in the mode observed by the nonunit efficiency detector, trace out the detected mode, and renormalize. If nη ≪ 1 for all contributing terms, 1 -(1 -η) n ≈ √ nη ∝ √ n, and the on/off detector 5 0 0.2 0.4 0.6 0.8 1 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 r P 3 and P 3 + FIG. 3: Success probability P3 (solid lines) and P + 3 (dashed lines) as a function of squeezing for η = 1 (upper lines) and η = 0.25 (lower lines). The dotted lines represent the approximate expression (23).\n\nmodel becomes equivalent to the photo detector model. In this case the density operator of the output state is obtained as\n\nρ = M ∞ p=0 ∞ q=0 p| q|(â N --( b+ e iθ ) N )|ψ i ψ i |((â † -) N -( b † + e -iθ ) N )|q |p = (1 -r 2 ) N +2 2N !r 2N ∞ n=N ∞ m=0 (r 2 ) n+m n! (n -N )! |n, m n, m| -e -iN θ n!(m + N )! (n -N )!m! |n, m n -N, m + N | + ∞ n=0 ∞ m=N (r 2 ) n+m m! (m -N )! |n, m n, m| -e iN θ (n + N )!m! n!(m -N )! |n, m n + N, m -N | , ( 21\n\n)\n\nwhere M is a normalization constant and the traces are over the â-and b+ modes. This leads to the NOON state fidelity\n\nF η≪1 N = NOON|ρ|NOON = (1 -r 2 ) N +2 , ( 22\n\n)\n\nwhere again φ = N θ + π + 2πn, n ∈ Z, is assumed. It is interesting to compare this result with the fidelity (1 -r 2 ) N +1 obtained for production of N -photon states from a single two-mode squeezed state by conditioning on N detection events in the idler beam and using detectors with very small efficiency. If a single-photon state is produced by this method and transformed into an N = 1 NOON state as explained in the Introduction, the NOON state fidelity is F 1,s = (1-r 2 ) 2 , and the success probability is P 1,s = λ/(λ + 1). Choosing squeezing parameters such that F 1,s = F 1 , we find that P + 1 = (4/3)P 1,s in the high fidelity limit. It is thus possible to achieve a higher success probability using the scheme with two OPOs, but the price to pay is a more technically involved setup, and NOON states with two different values of φ are produced. For N = 2 the present protocol and combination of two single-photon states on a 50 : 50 beam splitter, each produced conditionally from a single OPO, lead to identical fidelities and success probabilities. Finally we note that the photo detector model underestimates\n\nF N for η > 0 because 1 -(1 -η) n = η n-1 i=0 (1 -η) i < nη\n\nfor n = 2, 3, . . . while 1 -(1 -η) n = nη for n = 0, 1, i.e., the 'wrong' terms containing more than N photons are given a too large weight. This is also what we observe in Fig. 2 .\n\nIn the limit of small r and for odd values of N the success probability is given approximately by the simple expression\n\nP N ≈ η 2N N ψ i |((â † -) N -( b † + e -iθ ) N ) (â N --( b+ e iθ ) N )|ψ i = 2N ! (2N ) N λ N (N odd). (23)\n\nAgain η/(2N ) must be replaced by η/N to obtain P N for even values of N . The approximation to P 3 is shown in Fig. 3 ."
},
{
"section_type": "OTHER",
"section_title": "IV. NOON STATES FROM CONTINUOUS-WAVE OPO SOURCES",
"text": "Our protocol is not limited to pulsed fields, and for completeness we now briefly consider NOON state generation from continuously driven OPOs. We assume N = 3. For continuous-wave fields each of the three detected trigger beams and the two signal beams are described by time dependent field operators ĉi (t). The trigger detections take place in particular modes localized around the three detection times t c1 , t c2 , and t c3 , and we want to determine the NOON state fidelity of an output state occupying one temporal mode in each signal beam. Following the general multimode formalism in Refs. [15, 20] , the five relevant modes are specified by the mode functions f i (t), and the corresponding five single mode operators are ĉi = f i (t)ĉ i (t)dt. In general, we are free to choose the two output mode functions at will, and in the present case it is natural to choose the mode function which gives rise to the largest three-photon state fidelity when three-photon states are generated from a single type II continuous-wave OPO. Since we are mainly interested in the parameter region where the squeezing is small and the NOON state fidelity is large, we use the optimal three-photon state mode function derived for very low beam intensity in [20]\n\n, i.e., f 4 (t) = f 5 (t) = 3 k=1 c k γ/2 exp(-γ|t -t ck |/2)\n\n, where 6 0 0.05 0.1 0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ε/γ F 3 FIG. 4: NOON state fidelity as a function of ǫ/γ for states generated from a pair of continuous-wave OPO sources when conditioning on three simultaneous trigger detection events tc1 = tc2 = tc3. 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (t c3 -t c1 )γ F 3 FIG. 5: Fidelity of NOON states from continuous-wave OPO sources as a function of separation between trigger detection events (tc3 -tc1)γ for N = 3, tc3 -tc2 = tc2 -tc1, and ǫ/γ = 0.01.\n\nthe coefficients c k are functions of the intervals between the detection times and γ is the leakage rate of the OPO output mirror. We furthermore assume that the trigger mode functions are nonzero only in an infinitesimal time interval centered at the detection time, which is valid if the trigger detections take place on a time scale much shorter than γ -1 . Since we consider a low intensity continuous beam, and since we formally assume that the trigger detectors only register the light field in infinitesimal time intervals around the detection times, the annihilation operator detector model is perfectly valid in this case and detector dead time is insignificant.\n\nWe may now proceed as above and eliminate all the irrelevant modes from the analysis by writing down the Gaussian Wigner function of the five interesting modes.\n\nThe only difference is that this time ĉ † i ĉj and ĉi ĉj are expressed in terms of the two time correlation functions of the OPO output. Also, the operators applied to the Wigner function to take conditioning into account are different because the annihilation detector model is used. The reader is referred to Refs. [15, 20] for details.\n\nThe resulting fidelity is shown as a function of ǫ/γ in Fig. 4 , where ǫ is the nonlinear gain in the OPO, and as a function of the temporal distance between the conditioning detection events in Fig. 5 . As in the pulsed case the fidelity decreases when the degree of squeezing increases. The fidelity also decreases when the temporal distance between the conditioning detection events increases from zero, but it is permissible to have a small time interval between the trigger detection events. We note that the curves represent a lower limit to the theoretically achievable fidelity since a better fidelity may be obtained for another choice of output state mode functions."
},
{
"section_type": "CONCLUSION",
"section_title": "V. CONCLUSION",
"text": "In conclusion we have analyzed a method to generate path entangled NOON states from the output from two optical parametric oscillators. The method relies on the joint detection of photons in a number of trigger beams, and we presented a theoretical analysis of the role of detector efficiency and dead time for the fidelity of the states obtained and the success probability of the protocol. Our specific NOON state protocol applies for general photon numbers of the states, but in practice it is not realistic to go beyond the case of N = 3, studied here. This is due to the reduction of the fidelity due to unwanted contributions from higher number states, when the OPO output power gets too high, combined with the severe reduction of the probability to obtain the number of conditioning detection events needed when the OPO output power is too low. The N = 3 NOON states, which can be produced at 90% fidelity at the rate of one state produced every 10 -100 seconds, seem to be at the limit of realistic experiments of the proposed kind. Finally, we also determined the NOON state fidelity for continuouswave fields, where the best NOON state occupies a pair of suitably selected temporal mode functions, and where we find high fidelities as long as the trigger events occur within a short time window compared to the lifetime of the OPO cavity field.\n\nWe presented this analysis for the production of optical NOON states, but we note that recent theoretical proposals and experiments with four wave mixing of matter waves [26] , engineered quadratic interactions among trapped ions [27], and entanglement between field and atomic degrees of freedom [28, 29] bring promise for similar conditional generation of atomic and interspecies atom-field NOON states.\n\nThis work was supported by the European Integrated project SCALA. 7 [1] A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000). [2] L. Pezzé and A. Smerzi, quant-ph/0508158. [3] M. D'Angelo, A. Zavatta, V. Parigi, and M. Bellini, Phys.\n\nRev. A 74, 052114. [4] P. Kok, H. Lee, and J. P. Dowling, Phys. Rev. A 65, 052104 (2002). [5] J. Fiurasek, Phys. Rev. A 65, 053818 (2002). [6] H. F. Hofmann, Phys. Rev. A 70, 023812 (2004). [7] P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, Nature (London) 429, 158 (2004). [8] H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, Phys. Rev. Lett. 94, 090502 (2005). [9] N. M. VanMeter, P. Lougovski, D. B. Uskov, K. Kieling, J. Eisert, and J. P. Dowling, quant-ph/0612154. [10] M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, Nature (London) 429, 161 (2004). [11] A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P.\n\nGrangier, Phys. Rev. Lett. 98, 030502 (2007). [12] M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G.\n\nWelsch, Phys. Rev. A 55, 3184 (1997). [13] A. B. URen, C. Silberhorn, J. L. Ball, K. Banaszek, and I. A. Walmsley, Phys. Rev. A 72, 021802(R) (2005). [14] A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P.\n\nGrangier, Science 312, 83 (2006). [15] K. Mølmer, Phys. Rev. A 73, 063804 (2006). [16] A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 96, 213601 (2006). [17] J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, Phys. Rev. Lett. 97, 083604 (2006). [18] K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, Opt. Express 15, 3568 (2007). [19] A. E. B. Nielsen and K. Mølmer, Phys. Rev. A 75, 023806 (2007). [20] A. E. B. Nielsen and K. Mølmer, Phys. Rev. A 75, 043801 (2007). [21] F. W. Sun, Z. Y. Ou, and G. C. Guo, Phys. Rev. A 73, 023808 (2006). [22] P. Hariharan, N. Brown, and B. C. Sanders, J. Mod. Opt.\n\n40, 1573 (1993). [23] P. P. Rohde and T. C. Ralph, J. Mod. Opt. 53, 1589 (2006). [24] A. Ekert and P. Knight, Am. J. Phys. 63, 415 (1995). [25] J. Eisert and M. B. Plenio, Int. J. Quantum Inf. 1, 479 (2003). [26] G. K. Campbell, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, Phys. Rev. Lett. 96, 020406 (2006) [27] D. Porras and J. I. Cirac, Phys. Rev. Lett. 93, 263602 (2004). [28] B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Monroe, Nature, 428, 153 (2004). [29] J. Volz, M. Weber, D. Schlenk, W. Rosenfeld, J. Vrana, K. Saucke, C. Kurtsiefer, and H. Weinfurter, Phys. Rev. Lett. 96, 030404 (2006)."
}
] | |
arxiv:0704.0399 | 0704.0399 | 1 | 10.1016/j.physletb.2007.05.052 | 4afd0ebd0871067862405df8d67c23ae9c96c7ba7cb0d4a02d6c7877a8d4ae34 | Hawking radiation of linear dilaton black holes | We compute exactly the semi-classical radiation spectrum for a class of
non-asymptotically flat charged dilaton black holes, the so-called linear
dilaton black holes. In the high frequency regime, the temperature for these
black holes generically agrees with the surface gravity result. In the special
case where the black hole is massless, we show that, although the surface
gravity remains finite, there is no radiation, in agreement with the fact that
massless objects cannot radiate. | [
"G. Clement",
"J.C. Fabris and G.T. Marques"
] | [
"gr-qc",
"hep-th"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | We compute exactly the semi-classical radiation spectrum for a class of non-asymptotically flat charged dilaton black holes, the socalled linear dilaton black holes. In the high frequency regime, the temperature for these black holes generically agrees with the surface gravity result. In the special case where the black hole is massless, we show that, although the surface gravity remains finite, there is no radiation, in agreement with the fact that massless objects cannot radiate.
Quantum field theory in curved spacetime predicts new phenomena such as particle emission by a black hole [1] . This is due to the fact that the vacuum for a quantum field near the horizon is different from the observer's vacuum at spatial infinity. A distant observer thus receives from a black hole a steady flux of particles exhibiting, in the high frequency regime, a black body spectrum with a temperature proportional to the surface gravity [2] . Although Hawking's original derivation of this black hole evaporation dealt with realistic collapsing black holes, Unruh [3] showed that the same results are obtained when the collapse is replaced by appropriate boundary conditions on the horizon of an eternal black hole. In the semi-classical approximation, the black hole radiation spectrum may be evaluated by computing the Bogoliubov coefficients relating the two vacua. An equivalent procedure is to compute the reflection and absorption coefficients of a wave by the black hole. Usually, the wave equation cannot be solved exactly, and one must resort to match solutions in an overlap region between the near-horizon and asymptotic regions [4, 5] . In the special case of the (2+1)dimensional BTZ black hole [6] , an exact solution of the wave equation is available, which allows for an exact computation of the radiation spectrum, leading to the Hawking temperature [7, 8, 9] .
In this Letter, we discuss another case of black holes also allowing for an exact semi-classical computation of their radiation spectrum, that of linear dilaton black hole solutions to Einstein-Maxwell dilaton (EMD) theory in four dimensions. Linear dilaton black holes are a special case of the more general class of non-asymptotically flat black hole solutions to EMD [10, 11] , which we first briefly present. We discuss the evaporation of these non-asymptotically flat black holes and show that they either collapse to a naked singularity in a finite time, or evaporate in an infinite time. We then specialize to linear dilaton black holes, and outline the analytical computation of their radiation spectrum. For massive black holes, this computation leads, in the high frequency regime, to the same temperature which is obtained from the surface gravity. However in the case of massless extreme black holes, we find that, although the surface gravity remains finite, there is no radiation, in agreement with the fact that a massless object cannot radiate.
EMD is defined by the following action
S = 1 16π dx 4 √ -g R -2∂ µ φ∂ µ φ -e -2αφ F µν F µν , (1)
where F µν is the electromagnetic field, and φ is the dilatonic field, with coupling constant α. This theory admits static spherically symmetric solutions representing black holes. Among these black hole solutions there are asymptotically flat ones [12, 13] as well as non-asymptotically flat configurations [10, 11] . In the present work, we are interested in the non-asymptotically flat black hole solutions
ds 2 = r γ (r -b) r γ+1 0 dt 2 - r γ+1 0 r γ (r -b) dr 2 + r(r -b)dΩ 2 , (2)
F = 1 + γ 2 ν r 0 dr ∧ dt , e 2αφ = ν 2 r r 0 1-γ . ( 3
) with γ = 1 -α 2 1 + α 2 . ( 4
)
The constants b and r 0 are related to the mass and to the electric charge of the black hole through
M = (1 -γ)b/4 , Q = 1 + γ 2 r 0 ν . (5)
The solutions ( 2 ),(3) interpolate between the Schwarzschild solution for γ = -1 (α 2 → ∞) and the Bertotti-Robinson solution for γ = +1 (α 2 = 0). For b > 0 the horizon at r = b hides the singularity at r = 0, while in the extreme black hole case b = 0 the horizon coincides with the singularity. This is a curious case, with vanishing mass but a finite electric charge. For -1 < γ < 0 (α 2 > 1) the central singularity is timelike and clearly naked [11] . On the other hand, for 0 ≤ γ < 1 (0 < α 2 ≤ 1), the central singularity is null and marginally trapped [14] , so that signals coming from the centre never reach external observers. Thus in this case, extreme black holes can be still considered as black holes indeed.
The statistical Hawking temperature of the black holes (2), computed as usual by dividing the surface gravity by 2π is given by
T H = 1 4π b γ r 1+γ 0 . (6)
It is finite for all γ if b = 0. For b = 0 and -1 < γ < 0 (naked singularity). the temperature is infinite, while for b = 0 and 0 < γ < 1 (extreme black hole), the temperature vanishes.
The case b = γ = 0 is intriguing. Although this an extreme black hole, the situation is different from that of asymptotically flat extreme black holes. The near-horizon Euclidean extreme Reissner-Nordström geometry is cylindrical, rather than conical, so that its statistical temperature is arbitrary, contrary to the zero value derived from surface gravity [15] . In the present case the two-dimensional Euclidean continuation of the metric (2) with γ = 0 clearly has a conical singularity at r = b for all values of b, including b = 0, leading for this particular extreme black hole to the finite temperature T H = 1/4πr 0 , in agreement with the value (6) . However this result is questionable. A black hole with pointlike horizon and zero mass clearly cannot radiate, so one should rather expect its temperature to be zero. We will return to this question presently.
As black holes (2) radiate, they loose mass according to Stefan's law
dM dt = -σA h T 4 H , (7)
where σ is Stefan's constant, and A h = 4πr 1+γ 0 b 1-γ is the horizon area. Assuming that only electrically neutral quanta are radiated, (7) implies that the horizon area decreases according to
db dt = - 4σ (4π) 3 (1 -γ) r -3(1+γ) 0 b 1+3γ , (8)
which is solved by
b(t) = r 0 γc 1 -γ t -t 0 r 3 0 -1/3γ (γ = 0) , b(t) = r 0 exp - c 3 t -t 0 r 3 0 (γ = 0) , (9)
where c = 3σ/16π 3 , and t 0 is an integration constant. The outcome depends on the sign of γ. For γ < 0, the Hawking temperature increases with decreasing mass and the black hole collapses to a naked singularity (or evaporates away altogether in the Schwarzschild case γ = -1) in a finite time according to b ∼ (t 0 -t) 1/3|γ| . On the other hand, for γ ≥ 0, the Hawking temperature decreases (or is constant for γ = 0) with decreasing mass, and the black hole evaporates in an infinite time, reaching the extreme black hole state b = 0 only asymptotically. We now proceed to a more precise evaluation of the temperature of nonasymptotically flat black holes from the study of wave scattering in these spacetimes. The wave equation
∇ 2 φ = 0 ( 10
)
does not generically allow for an exact solution in the spacetimes (2). However, it can be solved analytically [16] in the case of linear dilaton black holes with γ = 0 and b = 0, with the metric
ds 2 = r -b r 0 dt 2 - r 0 r -b dr 2 + r(r -b)dΩ 2 , (11)
Considering the harmonic eigenmodes
φ(x) = ψ(r, t)Y lm (θ, ϕ) , ψ(r, t) = R(r)e -iωt , (12)
we obtain the following radial equation:
∂ r r(r -b)∂ r R + ω2 r r -b -l(l + 1) R = 0 ( 13
) (ω 2 ≡ ω 2 r 2 0 ). Putting y = b -r b , R = y iω f , (14)
reduces (13) to the equation
y(1 -y)∂ 2 y f + 1 + 2iω -2(1 + iω)y ∂ y f + ω2 -iω -λ2 -1/4 f = 0 , ( 15
)
with λ2 = ω2 -(l + 1/2) 2 . ( 16
)
This is a hypergeometric equation
y(1 -y)∂ 2 y f + c -(a + b + 1)y ∂ y f -abf = 0 , (17)
with a = 1 2 + i(ω + λ) , b = 1 2 + i(ω -λ) , c = 1 + 2iω . ( 18
)
It follows that the general solution of equation ( 13 ) is
R = C 1 r -b b iω F 1 2 + i(ω + λ), 1 2 + i(ω -λ), 1 + 2iω; b -r b + C 2 r -b b -iω F 1 2 -i(ω + λ), 1 2 -i(ω -λ), 1 -2iω; b -r b .(19) Putting r -b b = e x/r 0 , (20)
the partial wave near the horizon (x → -∞) is thus
ψ ≃ C 1 e iω(x-t) + C 2 e -iω(x+t) . ( 21
)
To obtain the behavior of the partial wave near spatial infinity, we must expand the solutions of ( 15 ) in hypergeometric functions of argument 1/y. The relevant transformation is
F (a, b, c; y) = Γ(c)Γ(b -a) Γ(b)Γ(c -a) (-y) -a F (a, a + 1 -c, a + 1 -b; 1/y) + Γ(c)Γ(a -b) Γ(a)Γ(c -b) (-y) -b F (b, b + 1 -c, b + 1 -a; 1/y) . ( 22
)
This leads to the asymptotic behavior
ψ ≃ r b -1/2 B 1 e i(λx-ωt) + B 2 e -i(λx+ωt) (23)
(λ = λ/r 0 ), where the amplitudes of the asymptotic outgoing and ingoing waves B 1 and B 2 are related to the amplitudes of the near-horizon outgoing and ingoing waves C 1 and C 2 by
B 1 = Γ(2i λ) Γ(1 + 2iω) Γ(1/2 + i(ω + λ)) 2 C 1 + Γ(1 -2iω) Γ(1/2 -i(ω -λ)) 2 C 2 , B 2 = Γ(-2i λ) Γ(1 + 2iω) Γ(1/2 + i(ω -λ)) 2 C 1 + Γ(1 -2iω) Γ(1/2 -i(ω + λ)) 2 C 2 . (24)
Hawking radiation can be considered as the inverse process of scattering by the black hole, with the asymptotic boundary condition B 1 = 0 (the outgoing mode is absent). The coefficient for reflection by the black hole is then given by
R = |C 1 | 2 |C 2 | 2 B 1 =0 = |Γ(1/2 + i(ω + λ)) 2 | 2 |Γ(1/2 + i(ω -λ)) 2 | 2 = cosh 2 π(ω -λ) cosh 2 π(ω + λ) . ( 25
)
The resulting radiation spectrum is
N = R 1 -R = (e ω/T H -1) -1 . ( 26
)
For high frequencies, λ ≃ ω = ω/r 0 , and we recover from (25) the Hawking temperature as computed from the surface gravity,
T H = 1 4πr 0 . ( 27
)
The above computation fails in the linear dilaton vacuum case b = 0. The question of assigning a temperature to such massless black holes might be evacuated by arguing that they cannot be formed, either through central collapse of matter, or (as we have seen above) through evaporation of massive black holes. Nevertheless, as a matter of principle one should consider the possibility of primordial massless black holes. From the general temperature law (6) these should have a finite temperature. On the other hand, being massless they cannot radiate energy away, so their temperature should vanish.
The question can be settled by solving the massless Klein-Gordon equation in the metric (11) with b = 0,
ds 2 = r r 0 dt 2 - r 0 r dr 2 -r 0 rdΩ 2 . ( 28
)
This metric can be rewritten as
ds 2 = Σ 2 dτ 2 -dx 2 -dΩ 2 , (29)
with x = ln(r/r 0 ) , τ = t/r 0 , Σ = r 0 e x/2 , (30)
showing that the linear dilaton vacuum metric is conformal to the product M 2 × S 2 of a two-dimensional Minkowski spacetime with the two-sphere.
Performing also the redefinition
φ = Σ -1 ψ , (31)
the Klein-Gordon equation ( 10 ) is reduced to
∇ 2 φ = Σ -3 ∂ τ τ -∂ xx -∇ 2 Ω + 1 4 ψ = 0 , (32)
where ∇ 2 Ω is the Laplacian operator on the two-sphere. For a given spherical harmonic with orbital quantum number l, the reduced Klein-Gordon equation is thus
∇ 2 2 ψ l + (l + 1/2) 2 ψ l = 0 , (33)
with ∇ 2 2 the Dalembertian operator on M 2 . Also, for a given spherical harmonic the four-dimensional Klein-Gordon norm reduces to the M 2 norm:
φ 2 = 1 2i d 3 x |g|g 0µ φ * ↔ ∂ µ φ = 2π i dx ψ * l ↔ ∂ τ ψ l . (34)
Thus, the problem of wave propagation in the linear dilaton vacuum reduces to the propagation of eigenmodes of a free Klein-Gordon field in two dimensions, with effective mass µ = l + 1/2. Clearly there is no reflection, so that the linear dilaton vacuum does not radiate and hence its Hawking temperature vanishes, contrary to the naive surface gravity value (6) . A similar reasoning holds in 2+1 dimensions for the BTZ vacuum [6] (M = L = 0), which is conformal to M 2 × S 1 .
We have shown that a complete analytical computation of the radiation spectrum is possible for linear dilaton black hole solutions of EMD. For massive black holes, this leads in the high frequency regime to a Planckian distribution with a temperature independent of the black hole mass, in accordance with the surface gravity value. On the other hand, we find that extreme, massless black holes do not radiate, thereby solving the paradox presented by apparently hot (if the surface gravity temperature is taken seriously) yet massless black holes. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We compute exactly the semi-classical radiation spectrum for a class of non-asymptotically flat charged dilaton black holes, the socalled linear dilaton black holes. In the high frequency regime, the temperature for these black holes generically agrees with the surface gravity result. In the special case where the black hole is massless, we show that, although the surface gravity remains finite, there is no radiation, in agreement with the fact that massless objects cannot radiate."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Quantum field theory in curved spacetime predicts new phenomena such as particle emission by a black hole [1] . This is due to the fact that the vacuum for a quantum field near the horizon is different from the observer's vacuum at spatial infinity. A distant observer thus receives from a black hole a steady flux of particles exhibiting, in the high frequency regime, a black body spectrum with a temperature proportional to the surface gravity [2] . Although Hawking's original derivation of this black hole evaporation dealt with realistic collapsing black holes, Unruh [3] showed that the same results are obtained when the collapse is replaced by appropriate boundary conditions on the horizon of an eternal black hole. In the semi-classical approximation, the black hole radiation spectrum may be evaluated by computing the Bogoliubov coefficients relating the two vacua. An equivalent procedure is to compute the reflection and absorption coefficients of a wave by the black hole. Usually, the wave equation cannot be solved exactly, and one must resort to match solutions in an overlap region between the near-horizon and asymptotic regions [4, 5] . In the special case of the (2+1)dimensional BTZ black hole [6] , an exact solution of the wave equation is available, which allows for an exact computation of the radiation spectrum, leading to the Hawking temperature [7, 8, 9] .\n\nIn this Letter, we discuss another case of black holes also allowing for an exact semi-classical computation of their radiation spectrum, that of linear dilaton black hole solutions to Einstein-Maxwell dilaton (EMD) theory in four dimensions. Linear dilaton black holes are a special case of the more general class of non-asymptotically flat black hole solutions to EMD [10, 11] , which we first briefly present. We discuss the evaporation of these non-asymptotically flat black holes and show that they either collapse to a naked singularity in a finite time, or evaporate in an infinite time. We then specialize to linear dilaton black holes, and outline the analytical computation of their radiation spectrum. For massive black holes, this computation leads, in the high frequency regime, to the same temperature which is obtained from the surface gravity. However in the case of massless extreme black holes, we find that, although the surface gravity remains finite, there is no radiation, in agreement with the fact that a massless object cannot radiate.\n\nEMD is defined by the following action\n\nS = 1 16π dx 4 √ -g R -2∂ µ φ∂ µ φ -e -2αφ F µν F µν , (1)\n\nwhere F µν is the electromagnetic field, and φ is the dilatonic field, with coupling constant α. This theory admits static spherically symmetric solutions representing black holes. Among these black hole solutions there are asymptotically flat ones [12, 13] as well as non-asymptotically flat configurations [10, 11] . In the present work, we are interested in the non-asymptotically flat black hole solutions\n\nds 2 = r γ (r -b) r γ+1 0 dt 2 - r γ+1 0 r γ (r -b) dr 2 + r(r -b)dΩ 2 , (2)\n\nF = 1 + γ 2 ν r 0 dr ∧ dt , e 2αφ = ν 2 r r 0 1-γ . ( 3\n\n) with γ = 1 -α 2 1 + α 2 . ( 4\n\n)\n\nThe constants b and r 0 are related to the mass and to the electric charge of the black hole through\n\nM = (1 -γ)b/4 , Q = 1 + γ 2 r 0 ν . (5)\n\nThe solutions ( 2 ),(3) interpolate between the Schwarzschild solution for γ = -1 (α 2 → ∞) and the Bertotti-Robinson solution for γ = +1 (α 2 = 0). For b > 0 the horizon at r = b hides the singularity at r = 0, while in the extreme black hole case b = 0 the horizon coincides with the singularity. This is a curious case, with vanishing mass but a finite electric charge. For -1 < γ < 0 (α 2 > 1) the central singularity is timelike and clearly naked [11] . On the other hand, for 0 ≤ γ < 1 (0 < α 2 ≤ 1), the central singularity is null and marginally trapped [14] , so that signals coming from the centre never reach external observers. Thus in this case, extreme black holes can be still considered as black holes indeed.\n\nThe statistical Hawking temperature of the black holes (2), computed as usual by dividing the surface gravity by 2π is given by\n\nT H = 1 4π b γ r 1+γ 0 . (6)\n\nIt is finite for all γ if b = 0. For b = 0 and -1 < γ < 0 (naked singularity). the temperature is infinite, while for b = 0 and 0 < γ < 1 (extreme black hole), the temperature vanishes.\n\nThe case b = γ = 0 is intriguing. Although this an extreme black hole, the situation is different from that of asymptotically flat extreme black holes. The near-horizon Euclidean extreme Reissner-Nordström geometry is cylindrical, rather than conical, so that its statistical temperature is arbitrary, contrary to the zero value derived from surface gravity [15] . In the present case the two-dimensional Euclidean continuation of the metric (2) with γ = 0 clearly has a conical singularity at r = b for all values of b, including b = 0, leading for this particular extreme black hole to the finite temperature T H = 1/4πr 0 , in agreement with the value (6) . However this result is questionable. A black hole with pointlike horizon and zero mass clearly cannot radiate, so one should rather expect its temperature to be zero. We will return to this question presently.\n\nAs black holes (2) radiate, they loose mass according to Stefan's law\n\ndM dt = -σA h T 4 H , (7)\n\nwhere σ is Stefan's constant, and A h = 4πr 1+γ 0 b 1-γ is the horizon area. Assuming that only electrically neutral quanta are radiated, (7) implies that the horizon area decreases according to\n\ndb dt = - 4σ (4π) 3 (1 -γ) r -3(1+γ) 0 b 1+3γ , (8)\n\nwhich is solved by\n\nb(t) = r 0 γc 1 -γ t -t 0 r 3 0 -1/3γ (γ = 0) , b(t) = r 0 exp - c 3 t -t 0 r 3 0 (γ = 0) , (9)\n\nwhere c = 3σ/16π 3 , and t 0 is an integration constant. The outcome depends on the sign of γ. For γ < 0, the Hawking temperature increases with decreasing mass and the black hole collapses to a naked singularity (or evaporates away altogether in the Schwarzschild case γ = -1) in a finite time according to b ∼ (t 0 -t) 1/3|γ| . On the other hand, for γ ≥ 0, the Hawking temperature decreases (or is constant for γ = 0) with decreasing mass, and the black hole evaporates in an infinite time, reaching the extreme black hole state b = 0 only asymptotically. We now proceed to a more precise evaluation of the temperature of nonasymptotically flat black holes from the study of wave scattering in these spacetimes. The wave equation\n\n∇ 2 φ = 0 ( 10\n\n)\n\ndoes not generically allow for an exact solution in the spacetimes (2). However, it can be solved analytically [16] in the case of linear dilaton black holes with γ = 0 and b = 0, with the metric\n\nds 2 = r -b r 0 dt 2 - r 0 r -b dr 2 + r(r -b)dΩ 2 , (11)\n\nConsidering the harmonic eigenmodes\n\nφ(x) = ψ(r, t)Y lm (θ, ϕ) , ψ(r, t) = R(r)e -iωt , (12)\n\nwe obtain the following radial equation:\n\n∂ r r(r -b)∂ r R + ω2 r r -b -l(l + 1) R = 0 ( 13\n\n) (ω 2 ≡ ω 2 r 2 0 ). Putting y = b -r b , R = y iω f , (14)\n\nreduces (13) to the equation\n\ny(1 -y)∂ 2 y f + 1 + 2iω -2(1 + iω)y ∂ y f + ω2 -iω -λ2 -1/4 f = 0 , ( 15\n\n)\n\nwith λ2 = ω2 -(l + 1/2) 2 . ( 16\n\n)\n\nThis is a hypergeometric equation\n\ny(1 -y)∂ 2 y f + c -(a + b + 1)y ∂ y f -abf = 0 , (17)\n\nwith a = 1 2 + i(ω + λ) , b = 1 2 + i(ω -λ) , c = 1 + 2iω . ( 18\n\n)\n\nIt follows that the general solution of equation ( 13 ) is\n\nR = C 1 r -b b iω F 1 2 + i(ω + λ), 1 2 + i(ω -λ), 1 + 2iω; b -r b + C 2 r -b b -iω F 1 2 -i(ω + λ), 1 2 -i(ω -λ), 1 -2iω; b -r b .(19) Putting r -b b = e x/r 0 , (20)\n\nthe partial wave near the horizon (x → -∞) is thus\n\nψ ≃ C 1 e iω(x-t) + C 2 e -iω(x+t) . ( 21\n\n)\n\nTo obtain the behavior of the partial wave near spatial infinity, we must expand the solutions of ( 15 ) in hypergeometric functions of argument 1/y. The relevant transformation is\n\nF (a, b, c; y) = Γ(c)Γ(b -a) Γ(b)Γ(c -a) (-y) -a F (a, a + 1 -c, a + 1 -b; 1/y) + Γ(c)Γ(a -b) Γ(a)Γ(c -b) (-y) -b F (b, b + 1 -c, b + 1 -a; 1/y) . ( 22\n\n)\n\nThis leads to the asymptotic behavior\n\nψ ≃ r b -1/2 B 1 e i(λx-ωt) + B 2 e -i(λx+ωt) (23)\n\n(λ = λ/r 0 ), where the amplitudes of the asymptotic outgoing and ingoing waves B 1 and B 2 are related to the amplitudes of the near-horizon outgoing and ingoing waves C 1 and C 2 by\n\nB 1 = Γ(2i λ) Γ(1 + 2iω) Γ(1/2 + i(ω + λ)) 2 C 1 + Γ(1 -2iω) Γ(1/2 -i(ω -λ)) 2 C 2 , B 2 = Γ(-2i λ) Γ(1 + 2iω) Γ(1/2 + i(ω -λ)) 2 C 1 + Γ(1 -2iω) Γ(1/2 -i(ω + λ)) 2 C 2 . (24)\n\nHawking radiation can be considered as the inverse process of scattering by the black hole, with the asymptotic boundary condition B 1 = 0 (the outgoing mode is absent). The coefficient for reflection by the black hole is then given by\n\nR = |C 1 | 2 |C 2 | 2 B 1 =0 = |Γ(1/2 + i(ω + λ)) 2 | 2 |Γ(1/2 + i(ω -λ)) 2 | 2 = cosh 2 π(ω -λ) cosh 2 π(ω + λ) . ( 25\n\n)\n\nThe resulting radiation spectrum is\n\nN = R 1 -R = (e ω/T H -1) -1 . ( 26\n\n)\n\nFor high frequencies, λ ≃ ω = ω/r 0 , and we recover from (25) the Hawking temperature as computed from the surface gravity,\n\nT H = 1 4πr 0 . ( 27\n\n)\n\nThe above computation fails in the linear dilaton vacuum case b = 0. The question of assigning a temperature to such massless black holes might be evacuated by arguing that they cannot be formed, either through central collapse of matter, or (as we have seen above) through evaporation of massive black holes. Nevertheless, as a matter of principle one should consider the possibility of primordial massless black holes. From the general temperature law (6) these should have a finite temperature. On the other hand, being massless they cannot radiate energy away, so their temperature should vanish.\n\nThe question can be settled by solving the massless Klein-Gordon equation in the metric (11) with b = 0,\n\nds 2 = r r 0 dt 2 - r 0 r dr 2 -r 0 rdΩ 2 . ( 28\n\n)\n\nThis metric can be rewritten as\n\nds 2 = Σ 2 dτ 2 -dx 2 -dΩ 2 , (29)\n\nwith x = ln(r/r 0 ) , τ = t/r 0 , Σ = r 0 e x/2 , (30)\n\nshowing that the linear dilaton vacuum metric is conformal to the product M 2 × S 2 of a two-dimensional Minkowski spacetime with the two-sphere.\n\nPerforming also the redefinition\n\nφ = Σ -1 ψ , (31)\n\nthe Klein-Gordon equation ( 10 ) is reduced to\n\n∇ 2 φ = Σ -3 ∂ τ τ -∂ xx -∇ 2 Ω + 1 4 ψ = 0 , (32)\n\nwhere ∇ 2 Ω is the Laplacian operator on the two-sphere. For a given spherical harmonic with orbital quantum number l, the reduced Klein-Gordon equation is thus\n\n∇ 2 2 ψ l + (l + 1/2) 2 ψ l = 0 , (33)\n\nwith ∇ 2 2 the Dalembertian operator on M 2 . Also, for a given spherical harmonic the four-dimensional Klein-Gordon norm reduces to the M 2 norm:\n\nφ 2 = 1 2i d 3 x |g|g 0µ φ * ↔ ∂ µ φ = 2π i dx ψ * l ↔ ∂ τ ψ l . (34)\n\nThus, the problem of wave propagation in the linear dilaton vacuum reduces to the propagation of eigenmodes of a free Klein-Gordon field in two dimensions, with effective mass µ = l + 1/2. Clearly there is no reflection, so that the linear dilaton vacuum does not radiate and hence its Hawking temperature vanishes, contrary to the naive surface gravity value (6) . A similar reasoning holds in 2+1 dimensions for the BTZ vacuum [6] (M = L = 0), which is conformal to M 2 × S 1 .\n\nWe have shown that a complete analytical computation of the radiation spectrum is possible for linear dilaton black hole solutions of EMD. For massive black holes, this leads in the high frequency regime to a Planckian distribution with a temperature independent of the black hole mass, in accordance with the surface gravity value. On the other hand, we find that extreme, massless black holes do not radiate, thereby solving the paradox presented by apparently hot (if the surface gravity temperature is taken seriously) yet massless black holes."
}
] | |
arxiv:0704.0400 | 0704.0400 | 1 | 78c2458a58debec64016d66dde5a09da5009f228daa68bafc495157d156633b4 | The S-Matrix of AdS/CFT and Yangian Symmetry | We review the algebraic construction of the S-matrix of AdS/CFT. We also
present its symmetry algebra which turns out to be a Yangian of the centrally
extended su(2|2) superalgebra. | [
"Niklas Beisert"
] | [
"nlin.SI",
"cond-mat.stat-mech",
"hep-th"
] | nlin.SI | [] | 2007-04-03 | 2026-02-26 | Bethe's ansatz [1] for solving a one-dimensional integrable model was and remains a powerful tool in contemporary theoretical physics: 75 years ago it solved one of the first models of quantum mechanics, the Heisenberg spin chain [2] ; today it provides exact solutions for the spectra of certain gauge and string theories and thus helps us understand their duality [3] better. Since the discovery of integrable structures in planar N = 4 supersymmetric gauge theory [4] and in planar IIB string theory on AdS 5 × S 5 [5] the tools for computing and comparing the spectra of both models have evolved rapidly. We now have complete asymptotic Bethe equations [6, 7] which interpolate smoothly between the perturbative regimes in gauge and string theory and which agree with all available data.
In this note we will focus on the S-matrix [8] in the excitation picture above a ferromagnetic ground state. We start by reviewing the algebraic construction of the S-matrix in Sec. 2. In Sec. 3 we subsequently show that this S-matrix has indeed a larger symmetry algebra: a Yangian. 1 2 The Universal Enveloping Algebra U(su(2|2) ⋉ R 2 )
In this section the results on the S-matrix of AdS/CFT shall be reviewed from an algebraic point of view. The applicable symmetry is a central extension h of the Lie superalgebra su(2|2) which we consider first. We continue by presenting the Hopf algebra structure of its universal enveloping algebra and its fundamental representation. Finally, we comment on the S-matrix and its dressing phase factor. Lie Superalgebra. The symmetry in the excitation picture for light cone string theory on AdS 5 × S 5 and for single-trace local operators in N = 4 supersymmetric gauge theory is given by two copies of the Lie superalgebra [9, 10] h := su(2|2) ⋉ R 2 = psu(2|2) ⋉ R 3 . (2.1) It is a central extension of the standard Lie superalgebras su(2|2) or psu(2|2), see [11] . It is generated by the su(2) × su(2) generators R a b , L α β , the supercharges Q α b , S a β and the central charges C, P, K. The Lie brackets of the su(2) generators take the standard form
[R a b , R c d ] = δ c b R a d -δ a d R c b , [L α β , L γ δ ] = δ γ β L α δ -δ α δ L γ β , [R a b , Q γ d ] = -δ a d Q γ b + 1 2 δ a b Q γ d , [L α β , Q γ d ] = +δ γ β Q α d -1 2 δ α β Q γ d , [R a b
, S c δ ] = +δ c b S a δ -1 2 δ a b S c δ , [L α β , S c δ ] = -δ α δ S c β + 1 2 δ α β S c δ . (2.2)
The Lie brackets of two supercharges yield {Q α b , S c δ } = δ c b L α δ + δ α δ R c b + δ c b δ α δ C, {Q α b , Q γ d } = ε αγ ε bd P, {S a β , S c δ } = ε ac ε βδ K. (2.3) The remaining Lie brackets vanish. Where appropriate, we shall use the collective symbol J A for the generators. The Lie brackets then take the standard form [J A , J B ] = f AB C J C . (2.4) For simplicity of notation, we shall pretend that all generators are bosonic; the generalisation to fermionic generators by insertion of suitable signs and graded commutators is straightforward.
Hopf Algebra. Next we consider the universal enveloping algebra U(h) of h. The construction of the product is standard, and one identifies the Lie brackets (2.4) with graded commutators. For the coproduct one can introduce a non-trivial braiding [12, 13] ∆J A = J A ⊗ 1 + U [A] ⊗ J A (2.5) 2 ∆R a b = R a b ⊗ 1 + 1 ⊗ R a b , ∆L α β = L α β ⊗ 1 + 1 ⊗ L α β , ∆Q α b = Q α b ⊗ 1 + U +1 ⊗ Q α b , ∆S a β = S a β ⊗ 1 + U -1 ⊗ S a β , ∆C = C ⊗ 1 + 1 ⊗ C, ∆P = P ⊗ 1 + U +2 ⊗ P, ∆K = K ⊗ 1 + U -2 ⊗ K, ∆U = U ⊗ U.
Table 1: The coproduct of the braided universal enveloping algebra U(h).
with some abelian 1 generator U (a priori unrelated to the algebra) and the grading [R] = [L] = [C] = 0, [Q] = +1, [S] = -1, [P] = +2, [K] = -2.
(2.6)
The coproduct is spelt out in Tab. 1 for the individual generators. The above grading is derived from the Cartan charge of the sl(2) automorphism [11] of the algebra h and therefore the coproduct is compatible with the algebra relations. We should define the remaining structures of the Hopf algebra: the antipode S and the counit ε [12, 13] . The antipode is an anti-homomorphism which acts on the generators as S(1) = 1, S(U) = U -1 , S(J A ) = -U -[A] J A . (2.7)
The counit acts non-trivially only on 1 and U ε(1) = ε(U) = 1, ε(J A ) = 0. (2.8) Cocommutativity. This coproduct is in general not quasi-cocommutative as can easily be seen by considering the central charges P, K in Tab. 1. To make it quasi-cocommutative we have to satisfy the constraints [12] P ⊗ 1 -U +2 = 1 -U +2 ⊗ P, K ⊗ 1 -U -2 = 1 -U -2 ⊗ K. (2.9)
They are solved by identifying the central charges P, K with the braiding factor U as follows [13]
P = gα 1 -U +2 , K = gα -1 1 -U -2 . (2.10)
This leads to the following quadratic constraint PKgα -1 P -gαK = 0. (2.11) It was furthermore shown in [14] that the coproduct is quasi-triangular, at least at the level of central charges, see also [15] .
1 Curiously, we can include the supersymmetric grading (-1) F in the generator U to manually impose the correct statistics. This is helpful for an implementation within a computer algebra system. In this case U would anticommute with fermionic generators.
Fundamental Representation. The algebra h has a four-dimensional representation [10] which we will call fundamental. The corresponding multiplet has two bosonic states |φ a and two fermionic states |ψ α . The action of the two sets of su(2) generators has to be canonical
R a b |φ c = δ c b |φ a -1 2 δ a b |φ c , L α β |ψ γ = δ γ β |ψ α -1 2 δ α β |ψ γ . (2.12)
The supersymmetry generators must also act in a manifestly su(2) × su(2) covariant way
Q α a |φ b = a δ b a |ψ α , Q α a |ψ β = b ε αβ ε ab |φ b , S a α |φ b = c ε ab ε αβ |ψ β , S a α |ψ β = d δ β α |φ a . (2.13)
We can write the four parameters a, b, c, d using the parameters x ± , γ and the constants g, α as
a = √ g γ, b = √ g α γ 1 - x + x -, c = √ g iγ αx + , d = √ g x + iγ 1 - x - x + . (2.14)
The parameters x ± (together with γ) label the representation and they must obey the constraint
x + + 1 x + -x -- 1 x -= i g . (2.15)
The three central charges C, P, K and U are represented by the values C, P, K and U which read
C = 1 2 1 + 1/x + x - 1 -1/x + x -, P = gα 1 - x + x -, K = g α 1 - x - x + , U = x + x -. (2.16)
They furthermore obey the quadratic relation C 2 -P K = 1 4 . Note that the corresponding quadratic combination of central charges C 2 -PK is singled out by being invariant under the sl(2) external automorphism.
Fundamental S-Matrix. In [10, 14] an S-matrix acting on the tensor product of two fundamental representations was derived. It was constructed by imposing invariance under the algebra h [∆J A , S] = 0. (2.17) We will not reproduce the result here, it is given in [14] . Note that we have to fix the parameters ξ = U = x + /x -in order to make the action of the generators compatible with the coproduct (2.5). 2 2 This identification removes all braiding factors from the S-matrix in [14] which will thus satisfy the standard Yang-Baxter (matrix) equation, see also [10, 16, 17] .
This S-matrix has several interesting properties. Firstly, it is not of difference form; it cannot be written as a function of the difference of some spectral parameters. Secondly, the S-matrix could be determined uniquely up to one overall function merely by imposing a Lie-type symmetry (2.17) [10] . This unusual fact is related to an unusual feature of representation theory of the algebra h: The tensor product of two fundamental representations is irreducible in almost all cases [14] .
Intriguingly this S-matrix is equivalent to Shastry's R-matrix [18] of the one-dimensional Hubbard model [19] . Furthermore the Bethe equations [10] contain two copies of the Lieb-Wu equations for the Hubbard model [20] . These observations of [14] establish a link between an important model of condensed matter physics and string theory (complementary to the one in [21] ).
Finally, let us note that one can derive (asymptotic) Bethe equations from the Smatrix and thus confirm the conjecture in [6] . So far this step has been performed in two different ways: by means of the nested coordinate [10] and the algebraic [17] Bethe ansatz.
Phase Factor. The remaining overall phase factor of the S-matrix clearly cannot be determined by demanding invariance under h. The phase factor was computed to some approximation from gauge theory [22] and from string theory [23] . The problem of an algebraically undetermined phase factor is in fact generic. Usually one imposes a further crossing symmetry relation to obtain a constraint on it. Indeed the known string phase factor is consistent with crossing symmetry [24] as was shown in [25] . By substituting a suitable ansatz [26] for the phase factor into the crossing symmetry relation a conjecture for the all-orders phase factor at strong coupling was made in [27] .
A corresponding all-orders expansion at weak coupling was presented in [7] . The latter conjecture was obtained by a sort of analytic continuation in the perturbative order of the series. Let us illustrate this principle by means of a very simple example: Consider the rational function f (x) = 1/(1x). It has the following expansions at x = 0 and at x = ∞
f (x) x→0 = ∞ n=0 a n x n , f (x) x→∞ = ∞ n=1 b n x -n (2.18)
with a n = 1 and b n = -1. When we consider a n and b n as analytic functions of the index, we can make the observation ("reciprocity") a n = -b -n . (2.19) Of course there are various ways in which the two functions +1 and -1 could be related, but the choice (2.19) appears to work for a surprisingly large class of functions! 3 It was proved in [30] that it does apply for the conjectured expansion of the phase factor. Very useful integral expressions for the phase have recently appeared in [31] . The analytic expression of the dressing phase can formally be obtained from the psu(2, 2|4) Bethe 3 Among other physical examples, we have identified circular Maldacena-Wilson loops [28] and noncritical string theory [29] where this reciprocity can be applied. Furthermore, summation by the Euler-MacLaurin formula (also known as zeta-function regularisation) is consistent with it. I thank Curt Callan, Marcos Mariño and Tristan McLoughlin for discussions of this principle.
equations [32] (see however appendix D in [33] ) in analogy to the covariant approach of [34, 21, 35] . While this proposal may seem to be encouraging in general, it is at the same time strange from the Hopf algebra point of view to use an S-matrix which does not obey the crossing relation [32] . This calls for further investigations.
Several tests of the phase have recently appeared, they are based on four-loop unitary scattering methods [36] , numerical evaluation [37, 38] , analytic methods [37, 30, 39] and on taking a certain highly non-trivial limit [40].
3 The Yangian Y(su(2|2) ⋉ R 2 )
In the section we investigate Yangian symmetry [41, 42] for the above S-matrix. We will start with a very brief review of Yangian symmetry for generic S-matrices (see [43] for more extensive reviews), and then we apply the framework to the S-matrix discussed above.
Yangians and S-Matrices. Typically the symmetries of rational S-matrices are of Yangian type. The Yangian Y(g) of a Lie algebra g is a deformation of the universal enveloping algebra of half the affine extension of g.
More plainly, it is generated by the g-generators J A and the Yangian generators J A . Their commutators take the generic form
[J A , J B ] = f AB C J C , [J A , J B ] = f AB C J C , (3.1)
and they should obey the Jacobi and Serre relations
J [A , [J B , J C] ] = 0, J [A , [J B , J C] ] = 0, J [A , [ J B , J C] ] = 1 4 2 f AG D f BH E f CK F f GHK J {D J E J F } . (3.2) The symbol f ABC = g AD g BE f DE C
represents the structure constants f AD C with two indices lowered by means of the inverse of the Cartan-Killing forms g AD and g BE . The brackets { } and [ ] at the level of indices imply total symmetrisation and anti-symmetrisation, respectively. Finally, is a scale parameter whose value plays no physical role. The first two relations lead to a constraint on the structure constants f AB C . The third relation 4 is a deformation of the Serre relation for affine extensions of Lie algebras.
The Yangian is a Hopf algebra and the coproduct takes the standard form
∆J A = J A ⊗ 1 + 1 ⊗ J A , ∆ J A = J A ⊗ 1 + 1 ⊗ J A + 1 2 f A BC J B ⊗ J C . (3.3) where f A BC = g BD f AD C . The antipode S is defined by S(J A ) = -J A , S( J A ) = -J A + 1 4 f A BC f BC D J D , (3.4)
4 For g = su(2) it has to be replaced by a quartic relation.
and the counit ε takes the standard form ε(1) = 1, ε(J A ) = ε( J A ) = 0. (3.5) For the study of integrable systems, the evaluation representations of the Yangian are of special interest. For these the action of the Yangian generators J A is proportional to the Lie generators
J A |u = uJ A |u . (3.6)
Here |u is some state of the evaluation module with spectral parameter u. This Yangian representation is finite-dimensional if the g-representation is. One merely has to ensure that the Serre relation (3.2) is satisfied. This is indeed not the case for all representations of all Lie algebras. The power of the Yangian symmetry lies in the fact that tensor products of evaluation representations are typically irreducible (except for special values of their spectral parameters). This allows for simple proofs (e.g. for the Yang-Baxter relation) by representation theory arguments. Let us finally consider the connection to the S-matrix. The S-matrix is a permutation operator; it acts by interchanging two modules of the algebra
S : V 1 ⊗ V 2 → V 2 ⊗ V 1 . (3.7)
In particular, for the tensor product of two evaluation modules one has
S|u 1 , u 2 ∼ |u 2 , u 1 . (3.8)
Invariance of the S-matrix under the Yangian means [∆J A , S] = [∆ J A , S] = 0 (3.9) for all generators J A , J A . The existence of such an S-matrix is equivalent to quasicocommutativity of Y(g). Note that only the difference of spectral parameters appears in the invariance condition: We can write the action of the coproduct of Yangian generators on the evaluation module |u 1 , u 2 as
∆ J A ≃ (u 1 -u 2 )J A ⊗ 1 + u 2 ∆J A + f A BC J B ⊗ J C . (3.10)
Here the first equation in (3.9) ensures that the term proportional to u 2 drops out from the second equation. Therefore the S-matrix typically depends on the difference u 1u 2 of spectral parameters only.
Yangians in AdS/CFT. Yangian symmetries for planar AdS/CFT have been investigated in [44] , both for classical string theory and for gauge theory at leading order, see also [45] Yangian symmetry also persists to higher perturbative orders in both models [22, 46] and it is likely that it also exists at finite coupling. This Yangian can be understood as a symmetry of the Hamiltonian on an infinite world sheet or as an expansion of the full monodromy matrix. The Lie symmetry in this picture is psu(2, 2|4) and the Yangian would be Y(psu(2, 2|4)). Here we consider a different picture of well-separated excitations on a ferromagnetic ground state and of their scattering matrix. In this picture the Lie symmetry reduces to two copies of h and the corresponding Yangian would be Y(h). Our Yangian should arise as a subalgebra of the full Yangian Y(psu(2, 2|4)) when acting on asymptotic excitation states. 7 Hopf Algebra. Let us now consider Y(h). We have already studied the universal enveloping algebra U(h). All we still need to do is to introduce one generator J A for each J A obeying the relations (3.1,3.2), and define its coproduct, antipode as well as counit.
In (2.5) we have defined a braided coproduct for the universal enveloping algebra. For consistency with the Serre relations, we also have to apply an analogous braiding to the standard Yangian coproduct
∆ J A = J A ⊗ 1 + U [A] ⊗ J A + f A BC J B U [C] ⊗ J C . (3.11)
Note that lowering an index requires to use the inverse Cartan-Killing form of the algebra.
In the case of h the Cartan-Killing form is degenerate and we need to extend the algebra by the sl(2) outer automorphism, see [14] . Effectively, lowering an index leads to an interchange of the automorphism generators with the central charges. We refrain from spelling out the Cartan-Killing form or the modified structure constants. Instead we present the complete set of coproducts of Yangian generators in Tab. 2, where we also fix the value of .
For the sake of completeness we state the antipode 5 and the counit
S( J A ) = -U -[A] J A , ε( J A ) = 0. (3.12)
Cocommutativity. An important question is if this coproduct can be quasi-cocommutative. 6 A first step is to consider the central generators C, P, K. For that purpose it is favourable to choose suitable combinations (3.13) for whom the coproduct almost trivialises
C ′ = C + 1 2 gα -1 P -1 2 gαK, P ′ = P + C P -2gα , K ′ = K -C K -2gα -1 ,
∆ C ′ = C ′ ⊗ 1 + 1 ⊗ C ′ , ∆ P ′ = P ′ ⊗ 1 + U +2 ⊗ P ′ , ∆ K ′ = K ′ ⊗ 1 + U -2 ⊗ K ′ . (3.14)
The combination C ′ is already cocommutative, and in order to make the generators P ′ , K ′ cocommutative we have to set as above in (2.9,2.10)
P ′ = igu P P, K ′ = igu K K (3.15)
with two universal constants u P and u K . With this choice, C, P, K also become cocommutative because they differ from C ′ , P ′ , K ′ only by central elements.
5 Note that f A BC f BC D = 0 here, so there is no contribution from the Lie generators. 6 The braiding factors in (3.11) turn out to be very important for the Yangian. It can easily be seen that without them the coproduct cannot be quasi-cocommutative. This is in contradistinction to the universal enveloping algebra where the braided as well as the unbraided coproduct are quasicocommutative.
∆ R a b = R a b ⊗ 1 + 1 ⊗ R a b + 1 2 R a c ⊗ R c b -1 2 R c b ⊗ R a c -1 2 S a γ U +1 ⊗ Q γ b -1 2 Q γ b U -1 ⊗ S a γ + 1 4 δ a b S d γ U +1 ⊗ Q γ d + 1 4 δ a b Q γ d U -1 ⊗ S d γ , ∆ L α β = L α β ⊗ 1 + 1 ⊗ L α β -1 2 L α γ ⊗ L γ β + 1 2 L γ β ⊗ L α γ + 1 2 Q α c U -1 ⊗ S c β + 1 2 S c β U +1 ⊗ Q α c -1 4 δ α β Q δ c U -1 ⊗ S c δ -1 4 δ α β S c δ U +1 ⊗ Q δ c , ∆ Q α b = Q α b ⊗ 1 + U +1 ⊗ Q α b -1 2 L α γ U +1 ⊗ Q γ b + 1 2 Q γ b ⊗ L α γ -1 2 R c b U +1 ⊗ Q α c + 1 2 Q α c ⊗ R c b -1 2 CU +1 ⊗ Q α b + 1 2 Q α b ⊗ C + 1 2 ε αγ ε bd PU -1 ⊗ S d γ -1 2 ε αγ ε bd S d γ U +2 ⊗ P, ∆ S a β = S a β ⊗ 1 + U -1 ⊗ S a β + 1 2 R a c U -1 ⊗ S c β -1 2 S c β ⊗ R a c + 1 2 L γ β U -1 ⊗ S a γ -1 2 S a γ ⊗ L γ β + 1 2 CU -1 ⊗ S a β -1 2 S a β ⊗ C -1 2 ε ac ε βδ KU +1 ⊗ Q δ c + 1 2 ε ac ε βδ Q δ c U -2 ⊗ K, ∆ C = C ⊗ 1 + 1 ⊗ C + 1 2 PU -2 ⊗ K -1 2 KU +2 ⊗ P, ∆ P = P ⊗ 1 + U +2 ⊗ P -CU +2 ⊗ P + P ⊗ C, ∆ K = K ⊗ 1 + U -2 ⊗ K + CU -2 ⊗ K -K ⊗ C.
Table 2: The coproduct of the Yangian generators in Y(h).
Fundamental Evaluation Representation. For the fundamental evaluation representation we make the ansatz 7
J A |X = ig(u + u 0 )J A |X .
(3.16) By comparison with (3.13,3.15) we can infer that u has to be related to the parameters of the fundamental representation by
u = x + + 1 x + - i 2g = x -+ 1 x -+ i 2g = 1 2 (x + + x -)(1 + 1/x + x -) . (3.17)
Furthermore u P and u K in (3.15) have to both coincide with the universal constant
u 0 = u P = u K . 8
As an aside we state the eigenvalue of the quadratic combination
C C -1 2 P K -1 2 K P = 1 4 ig(u + u 0 ). (3.18)
Fundamental S-Matrix. Using the coproducts in Tab. 2 we have confirmed that the S-matrix is also invariant under all of the Yangian generators [∆ J A , S] = 0. (3.19) We have used a computer algebra system to evaluate the action of the Yangian generators and the S-matrix. 9 To show invariance requires heavy use of the identity (2.15). Superficially it is very surprising to find all these additional symmetries of the S-matrix. The deeper reason however should be that the coproduct is quasi-cocommutative. We have thus proved quasi-cocommutativity when acting on fundamental representations. It is interesting to see that the S-matrix is based on standard evaluation representations of the Yangian. Nevertheless, it is not a function of the difference of spectral parameters. This unusual property traces back to the link between the spectral parameter u and the h-representation parameters x ± in (3.17). The latter is again related to the braiding in the coproduct (3.11).
As our S-matrix is equivalent [14] to Shastry's R-matrix, our Yangian is presumably an extension of the su(2) × su(2) Yangian symmetry of the Hubbard model found in [47] .
In this note we have reviewed the construction of the S-matrix with centrally extended su(2|2) symmetry that appears in the context of the planar AdS/CFT correspondence and the one-dimensional Hubbard model. We have furthermore shown that the S-matrix has an additional Yangian symmetry whose Hopf algebra structure we have presented. This Yangian is not quite a standard Yangian, but its coproduct needs to be braided in order to be quasi-cocommutative. This fact is intimately related to the existence of a 7 We believe, but we have not verified that this is compatible with the Serre relations (3.2). 8 It is conceivable that a further consistency requirement fixes the value of u 0 , presumably to zero. 9 We have also confirmed the invariance of the singlet state found in [10].
10 triplet of central charges with non-trivial coproduct and leads to the wealth of unusual features of the S-matrix.
In connection to the Yangian there are many points left to be clarified. Most importantly the representation theory needs to be understood. Which representations of h lift to evaluation representations of Y(h)? At what values of the spectral parameters do their tensor products become reducible? This information could be used to prove that the coproduct is quasi-cocommutative. Also the Yang-Baxter equation for the S-matrix should follow straightforwardly. It might also give some further understanding of bound states [48] .
Then it would be highly desirable to construct a universal R-matrix for this Yangian and show that it is quasi-triangular. This would put large parts of the integrable structure for arbitrary representations of this algebra on solid ground much like for the case of generic simple Lie algebras. Some further interesting questions include: Is this Yangian the unique quasi-cocommutative Hopf algebra based on h? Does the double Yangian [42] exist and what is its structure? Can the sl(2) automorphism of the algebra be included at the Yangian level such that the coproduct is quasi-cocommutative? What would the representations be in this case?
Acknowledgements. I am grateful to C. Callan, D. Erkal, A. Kleinschmidt, P. Koroteev, N. MacKay, M. Mariño, T. McLoughlin, J. Plefka, F. Spill and B. Zwiebel for interesting discussions.
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{
"section_type": "BACKGROUND",
"section_title": "Introduction and Overview",
"text": "Bethe's ansatz [1] for solving a one-dimensional integrable model was and remains a powerful tool in contemporary theoretical physics: 75 years ago it solved one of the first models of quantum mechanics, the Heisenberg spin chain [2] ; today it provides exact solutions for the spectra of certain gauge and string theories and thus helps us understand their duality [3] better. Since the discovery of integrable structures in planar N = 4 supersymmetric gauge theory [4] and in planar IIB string theory on AdS 5 × S 5 [5] the tools for computing and comparing the spectra of both models have evolved rapidly. We now have complete asymptotic Bethe equations [6, 7] which interpolate smoothly between the perturbative regimes in gauge and string theory and which agree with all available data.\n\nIn this note we will focus on the S-matrix [8] in the excitation picture above a ferromagnetic ground state. We start by reviewing the algebraic construction of the S-matrix in Sec. 2. In Sec. 3 we subsequently show that this S-matrix has indeed a larger symmetry algebra: a Yangian. 1 2 The Universal Enveloping Algebra U(su(2|2) ⋉ R 2 )\n\nIn this section the results on the S-matrix of AdS/CFT shall be reviewed from an algebraic point of view. The applicable symmetry is a central extension h of the Lie superalgebra su(2|2) which we consider first. We continue by presenting the Hopf algebra structure of its universal enveloping algebra and its fundamental representation. Finally, we comment on the S-matrix and its dressing phase factor. Lie Superalgebra. The symmetry in the excitation picture for light cone string theory on AdS 5 × S 5 and for single-trace local operators in N = 4 supersymmetric gauge theory is given by two copies of the Lie superalgebra [9, 10] h := su(2|2) ⋉ R 2 = psu(2|2) ⋉ R 3 . (2.1) It is a central extension of the standard Lie superalgebras su(2|2) or psu(2|2), see [11] . It is generated by the su(2) × su(2) generators R a b , L α β , the supercharges Q α b , S a β and the central charges C, P, K. The Lie brackets of the su(2) generators take the standard form\n\n[R a b , R c d ] = δ c b R a d -δ a d R c b , [L α β , L γ δ ] = δ γ β L α δ -δ α δ L γ β , [R a b , Q γ d ] = -δ a d Q γ b + 1 2 δ a b Q γ d , [L α β , Q γ d ] = +δ γ β Q α d -1 2 δ α β Q γ d , [R a b\n\n, S c δ ] = +δ c b S a δ -1 2 δ a b S c δ , [L α β , S c δ ] = -δ α δ S c β + 1 2 δ α β S c δ . (2.2)\n\nThe Lie brackets of two supercharges yield {Q α b , S c δ } = δ c b L α δ + δ α δ R c b + δ c b δ α δ C, {Q α b , Q γ d } = ε αγ ε bd P, {S a β , S c δ } = ε ac ε βδ K. (2.3) The remaining Lie brackets vanish. Where appropriate, we shall use the collective symbol J A for the generators. The Lie brackets then take the standard form [J A , J B ] = f AB C J C . (2.4) For simplicity of notation, we shall pretend that all generators are bosonic; the generalisation to fermionic generators by insertion of suitable signs and graded commutators is straightforward.\n\nHopf Algebra. Next we consider the universal enveloping algebra U(h) of h. The construction of the product is standard, and one identifies the Lie brackets (2.4) with graded commutators. For the coproduct one can introduce a non-trivial braiding [12, 13] ∆J A = J A ⊗ 1 + U [A] ⊗ J A (2.5) 2 ∆R a b = R a b ⊗ 1 + 1 ⊗ R a b , ∆L α β = L α β ⊗ 1 + 1 ⊗ L α β , ∆Q α b = Q α b ⊗ 1 + U +1 ⊗ Q α b , ∆S a β = S a β ⊗ 1 + U -1 ⊗ S a β , ∆C = C ⊗ 1 + 1 ⊗ C, ∆P = P ⊗ 1 + U +2 ⊗ P, ∆K = K ⊗ 1 + U -2 ⊗ K, ∆U = U ⊗ U.\n\nTable 1: The coproduct of the braided universal enveloping algebra U(h).\n\nwith some abelian 1 generator U (a priori unrelated to the algebra) and the grading [R] = [L] = [C] = 0, [Q] = +1, [S] = -1, [P] = +2, [K] = -2.\n\n(2.6)\n\nThe coproduct is spelt out in Tab. 1 for the individual generators. The above grading is derived from the Cartan charge of the sl(2) automorphism [11] of the algebra h and therefore the coproduct is compatible with the algebra relations. We should define the remaining structures of the Hopf algebra: the antipode S and the counit ε [12, 13] . The antipode is an anti-homomorphism which acts on the generators as S(1) = 1, S(U) = U -1 , S(J A ) = -U -[A] J A . (2.7)\n\nThe counit acts non-trivially only on 1 and U ε(1) = ε(U) = 1, ε(J A ) = 0. (2.8) Cocommutativity. This coproduct is in general not quasi-cocommutative as can easily be seen by considering the central charges P, K in Tab. 1. To make it quasi-cocommutative we have to satisfy the constraints [12] P ⊗ 1 -U +2 = 1 -U +2 ⊗ P, K ⊗ 1 -U -2 = 1 -U -2 ⊗ K. (2.9)\n\nThey are solved by identifying the central charges P, K with the braiding factor U as follows [13]\n\nP = gα 1 -U +2 , K = gα -1 1 -U -2 . (2.10)\n\nThis leads to the following quadratic constraint PKgα -1 P -gαK = 0. (2.11) It was furthermore shown in [14] that the coproduct is quasi-triangular, at least at the level of central charges, see also [15] .\n\n1 Curiously, we can include the supersymmetric grading (-1) F in the generator U to manually impose the correct statistics. This is helpful for an implementation within a computer algebra system. In this case U would anticommute with fermionic generators.\n\nFundamental Representation. The algebra h has a four-dimensional representation [10] which we will call fundamental. The corresponding multiplet has two bosonic states |φ a and two fermionic states |ψ α . The action of the two sets of su(2) generators has to be canonical\n\nR a b |φ c = δ c b |φ a -1 2 δ a b |φ c , L α β |ψ γ = δ γ β |ψ α -1 2 δ α β |ψ γ . (2.12)\n\nThe supersymmetry generators must also act in a manifestly su(2) × su(2) covariant way\n\nQ α a |φ b = a δ b a |ψ α , Q α a |ψ β = b ε αβ ε ab |φ b , S a α |φ b = c ε ab ε αβ |ψ β , S a α |ψ β = d δ β α |φ a . (2.13)\n\nWe can write the four parameters a, b, c, d using the parameters x ± , γ and the constants g, α as\n\na = √ g γ, b = √ g α γ 1 - x + x -, c = √ g iγ αx + , d = √ g x + iγ 1 - x - x + . (2.14)\n\nThe parameters x ± (together with γ) label the representation and they must obey the constraint\n\nx + + 1 x + -x -- 1 x -= i g . (2.15)\n\nThe three central charges C, P, K and U are represented by the values C, P, K and U which read\n\nC = 1 2 1 + 1/x + x - 1 -1/x + x -, P = gα 1 - x + x -, K = g α 1 - x - x + , U = x + x -. (2.16)\n\nThey furthermore obey the quadratic relation C 2 -P K = 1 4 . Note that the corresponding quadratic combination of central charges C 2 -PK is singled out by being invariant under the sl(2) external automorphism.\n\nFundamental S-Matrix. In [10, 14] an S-matrix acting on the tensor product of two fundamental representations was derived. It was constructed by imposing invariance under the algebra h [∆J A , S] = 0. (2.17) We will not reproduce the result here, it is given in [14] . Note that we have to fix the parameters ξ = U = x + /x -in order to make the action of the generators compatible with the coproduct (2.5). 2 2 This identification removes all braiding factors from the S-matrix in [14] which will thus satisfy the standard Yang-Baxter (matrix) equation, see also [10, 16, 17] .\n\nThis S-matrix has several interesting properties. Firstly, it is not of difference form; it cannot be written as a function of the difference of some spectral parameters. Secondly, the S-matrix could be determined uniquely up to one overall function merely by imposing a Lie-type symmetry (2.17) [10] . This unusual fact is related to an unusual feature of representation theory of the algebra h: The tensor product of two fundamental representations is irreducible in almost all cases [14] .\n\nIntriguingly this S-matrix is equivalent to Shastry's R-matrix [18] of the one-dimensional Hubbard model [19] . Furthermore the Bethe equations [10] contain two copies of the Lieb-Wu equations for the Hubbard model [20] . These observations of [14] establish a link between an important model of condensed matter physics and string theory (complementary to the one in [21] ).\n\nFinally, let us note that one can derive (asymptotic) Bethe equations from the Smatrix and thus confirm the conjecture in [6] . So far this step has been performed in two different ways: by means of the nested coordinate [10] and the algebraic [17] Bethe ansatz.\n\nPhase Factor. The remaining overall phase factor of the S-matrix clearly cannot be determined by demanding invariance under h. The phase factor was computed to some approximation from gauge theory [22] and from string theory [23] . The problem of an algebraically undetermined phase factor is in fact generic. Usually one imposes a further crossing symmetry relation to obtain a constraint on it. Indeed the known string phase factor is consistent with crossing symmetry [24] as was shown in [25] . By substituting a suitable ansatz [26] for the phase factor into the crossing symmetry relation a conjecture for the all-orders phase factor at strong coupling was made in [27] .\n\nA corresponding all-orders expansion at weak coupling was presented in [7] . The latter conjecture was obtained by a sort of analytic continuation in the perturbative order of the series. Let us illustrate this principle by means of a very simple example: Consider the rational function f (x) = 1/(1x). It has the following expansions at x = 0 and at x = ∞\n\nf (x) x→0 = ∞ n=0 a n x n , f (x) x→∞ = ∞ n=1 b n x -n (2.18)\n\nwith a n = 1 and b n = -1. When we consider a n and b n as analytic functions of the index, we can make the observation (\"reciprocity\") a n = -b -n . (2.19) Of course there are various ways in which the two functions +1 and -1 could be related, but the choice (2.19) appears to work for a surprisingly large class of functions! 3 It was proved in [30] that it does apply for the conjectured expansion of the phase factor. Very useful integral expressions for the phase have recently appeared in [31] . The analytic expression of the dressing phase can formally be obtained from the psu(2, 2|4) Bethe 3 Among other physical examples, we have identified circular Maldacena-Wilson loops [28] and noncritical string theory [29] where this reciprocity can be applied. Furthermore, summation by the Euler-MacLaurin formula (also known as zeta-function regularisation) is consistent with it. I thank Curt Callan, Marcos Mariño and Tristan McLoughlin for discussions of this principle.\n\nequations [32] (see however appendix D in [33] ) in analogy to the covariant approach of [34, 21, 35] . While this proposal may seem to be encouraging in general, it is at the same time strange from the Hopf algebra point of view to use an S-matrix which does not obey the crossing relation [32] . This calls for further investigations.\n\nSeveral tests of the phase have recently appeared, they are based on four-loop unitary scattering methods [36] , numerical evaluation [37, 38] , analytic methods [37, 30, 39] and on taking a certain highly non-trivial limit [40].\n\n3 The Yangian Y(su(2|2) ⋉ R 2 )\n\nIn the section we investigate Yangian symmetry [41, 42] for the above S-matrix. We will start with a very brief review of Yangian symmetry for generic S-matrices (see [43] for more extensive reviews), and then we apply the framework to the S-matrix discussed above.\n\nYangians and S-Matrices. Typically the symmetries of rational S-matrices are of Yangian type. The Yangian Y(g) of a Lie algebra g is a deformation of the universal enveloping algebra of half the affine extension of g.\n\nMore plainly, it is generated by the g-generators J A and the Yangian generators J A . Their commutators take the generic form\n\n[J A , J B ] = f AB C J C , [J A , J B ] = f AB C J C , (3.1)\n\nand they should obey the Jacobi and Serre relations\n\nJ [A , [J B , J C] ] = 0, J [A , [J B , J C] ] = 0, J [A , [ J B , J C] ] = 1 4 2 f AG D f BH E f CK F f GHK J {D J E J F } . (3.2) The symbol f ABC = g AD g BE f DE C\n\nrepresents the structure constants f AD C with two indices lowered by means of the inverse of the Cartan-Killing forms g AD and g BE . The brackets { } and [ ] at the level of indices imply total symmetrisation and anti-symmetrisation, respectively. Finally, is a scale parameter whose value plays no physical role. The first two relations lead to a constraint on the structure constants f AB C . The third relation 4 is a deformation of the Serre relation for affine extensions of Lie algebras.\n\nThe Yangian is a Hopf algebra and the coproduct takes the standard form\n\n∆J A = J A ⊗ 1 + 1 ⊗ J A , ∆ J A = J A ⊗ 1 + 1 ⊗ J A + 1 2 f A BC J B ⊗ J C . (3.3) where f A BC = g BD f AD C . The antipode S is defined by S(J A ) = -J A , S( J A ) = -J A + 1 4 f A BC f BC D J D , (3.4)\n\n4 For g = su(2) it has to be replaced by a quartic relation.\n\nand the counit ε takes the standard form ε(1) = 1, ε(J A ) = ε( J A ) = 0. (3.5) For the study of integrable systems, the evaluation representations of the Yangian are of special interest. For these the action of the Yangian generators J A is proportional to the Lie generators\n\nJ A |u = uJ A |u . (3.6)\n\nHere |u is some state of the evaluation module with spectral parameter u. This Yangian representation is finite-dimensional if the g-representation is. One merely has to ensure that the Serre relation (3.2) is satisfied. This is indeed not the case for all representations of all Lie algebras. The power of the Yangian symmetry lies in the fact that tensor products of evaluation representations are typically irreducible (except for special values of their spectral parameters). This allows for simple proofs (e.g. for the Yang-Baxter relation) by representation theory arguments. Let us finally consider the connection to the S-matrix. The S-matrix is a permutation operator; it acts by interchanging two modules of the algebra\n\nS : V 1 ⊗ V 2 → V 2 ⊗ V 1 . (3.7)\n\nIn particular, for the tensor product of two evaluation modules one has\n\nS|u 1 , u 2 ∼ |u 2 , u 1 . (3.8)\n\nInvariance of the S-matrix under the Yangian means [∆J A , S] = [∆ J A , S] = 0 (3.9) for all generators J A , J A . The existence of such an S-matrix is equivalent to quasicocommutativity of Y(g). Note that only the difference of spectral parameters appears in the invariance condition: We can write the action of the coproduct of Yangian generators on the evaluation module |u 1 , u 2 as\n\n∆ J A ≃ (u 1 -u 2 )J A ⊗ 1 + u 2 ∆J A + f A BC J B ⊗ J C . (3.10)\n\nHere the first equation in (3.9) ensures that the term proportional to u 2 drops out from the second equation. Therefore the S-matrix typically depends on the difference u 1u 2 of spectral parameters only.\n\nYangians in AdS/CFT. Yangian symmetries for planar AdS/CFT have been investigated in [44] , both for classical string theory and for gauge theory at leading order, see also [45] Yangian symmetry also persists to higher perturbative orders in both models [22, 46] and it is likely that it also exists at finite coupling. This Yangian can be understood as a symmetry of the Hamiltonian on an infinite world sheet or as an expansion of the full monodromy matrix. The Lie symmetry in this picture is psu(2, 2|4) and the Yangian would be Y(psu(2, 2|4)). Here we consider a different picture of well-separated excitations on a ferromagnetic ground state and of their scattering matrix. In this picture the Lie symmetry reduces to two copies of h and the corresponding Yangian would be Y(h). Our Yangian should arise as a subalgebra of the full Yangian Y(psu(2, 2|4)) when acting on asymptotic excitation states. 7 Hopf Algebra. Let us now consider Y(h). We have already studied the universal enveloping algebra U(h). All we still need to do is to introduce one generator J A for each J A obeying the relations (3.1,3.2), and define its coproduct, antipode as well as counit.\n\nIn (2.5) we have defined a braided coproduct for the universal enveloping algebra. For consistency with the Serre relations, we also have to apply an analogous braiding to the standard Yangian coproduct\n\n∆ J A = J A ⊗ 1 + U [A] ⊗ J A + f A BC J B U [C] ⊗ J C . (3.11)\n\nNote that lowering an index requires to use the inverse Cartan-Killing form of the algebra.\n\nIn the case of h the Cartan-Killing form is degenerate and we need to extend the algebra by the sl(2) outer automorphism, see [14] . Effectively, lowering an index leads to an interchange of the automorphism generators with the central charges. We refrain from spelling out the Cartan-Killing form or the modified structure constants. Instead we present the complete set of coproducts of Yangian generators in Tab. 2, where we also fix the value of .\n\nFor the sake of completeness we state the antipode 5 and the counit\n\nS( J A ) = -U -[A] J A , ε( J A ) = 0. (3.12)\n\nCocommutativity. An important question is if this coproduct can be quasi-cocommutative. 6 A first step is to consider the central generators C, P, K. For that purpose it is favourable to choose suitable combinations (3.13) for whom the coproduct almost trivialises\n\nC ′ = C + 1 2 gα -1 P -1 2 gαK, P ′ = P + C P -2gα , K ′ = K -C K -2gα -1 ,\n\n∆ C ′ = C ′ ⊗ 1 + 1 ⊗ C ′ , ∆ P ′ = P ′ ⊗ 1 + U +2 ⊗ P ′ , ∆ K ′ = K ′ ⊗ 1 + U -2 ⊗ K ′ . (3.14)\n\nThe combination C ′ is already cocommutative, and in order to make the generators P ′ , K ′ cocommutative we have to set as above in (2.9,2.10)\n\nP ′ = igu P P, K ′ = igu K K (3.15)\n\nwith two universal constants u P and u K . With this choice, C, P, K also become cocommutative because they differ from C ′ , P ′ , K ′ only by central elements.\n\n5 Note that f A BC f BC D = 0 here, so there is no contribution from the Lie generators. 6 The braiding factors in (3.11) turn out to be very important for the Yangian. It can easily be seen that without them the coproduct cannot be quasi-cocommutative. This is in contradistinction to the universal enveloping algebra where the braided as well as the unbraided coproduct are quasicocommutative.\n\n∆ R a b = R a b ⊗ 1 + 1 ⊗ R a b + 1 2 R a c ⊗ R c b -1 2 R c b ⊗ R a c -1 2 S a γ U +1 ⊗ Q γ b -1 2 Q γ b U -1 ⊗ S a γ + 1 4 δ a b S d γ U +1 ⊗ Q γ d + 1 4 δ a b Q γ d U -1 ⊗ S d γ , ∆ L α β = L α β ⊗ 1 + 1 ⊗ L α β -1 2 L α γ ⊗ L γ β + 1 2 L γ β ⊗ L α γ + 1 2 Q α c U -1 ⊗ S c β + 1 2 S c β U +1 ⊗ Q α c -1 4 δ α β Q δ c U -1 ⊗ S c δ -1 4 δ α β S c δ U +1 ⊗ Q δ c , ∆ Q α b = Q α b ⊗ 1 + U +1 ⊗ Q α b -1 2 L α γ U +1 ⊗ Q γ b + 1 2 Q γ b ⊗ L α γ -1 2 R c b U +1 ⊗ Q α c + 1 2 Q α c ⊗ R c b -1 2 CU +1 ⊗ Q α b + 1 2 Q α b ⊗ C + 1 2 ε αγ ε bd PU -1 ⊗ S d γ -1 2 ε αγ ε bd S d γ U +2 ⊗ P, ∆ S a β = S a β ⊗ 1 + U -1 ⊗ S a β + 1 2 R a c U -1 ⊗ S c β -1 2 S c β ⊗ R a c + 1 2 L γ β U -1 ⊗ S a γ -1 2 S a γ ⊗ L γ β + 1 2 CU -1 ⊗ S a β -1 2 S a β ⊗ C -1 2 ε ac ε βδ KU +1 ⊗ Q δ c + 1 2 ε ac ε βδ Q δ c U -2 ⊗ K, ∆ C = C ⊗ 1 + 1 ⊗ C + 1 2 PU -2 ⊗ K -1 2 KU +2 ⊗ P, ∆ P = P ⊗ 1 + U +2 ⊗ P -CU +2 ⊗ P + P ⊗ C, ∆ K = K ⊗ 1 + U -2 ⊗ K + CU -2 ⊗ K -K ⊗ C.\n\nTable 2: The coproduct of the Yangian generators in Y(h).\n\nFundamental Evaluation Representation. For the fundamental evaluation representation we make the ansatz 7\n\nJ A |X = ig(u + u 0 )J A |X .\n\n(3.16) By comparison with (3.13,3.15) we can infer that u has to be related to the parameters of the fundamental representation by\n\nu = x + + 1 x + - i 2g = x -+ 1 x -+ i 2g = 1 2 (x + + x -)(1 + 1/x + x -) . (3.17)\n\nFurthermore u P and u K in (3.15) have to both coincide with the universal constant\n\nu 0 = u P = u K . 8\n\nAs an aside we state the eigenvalue of the quadratic combination\n\nC C -1 2 P K -1 2 K P = 1 4 ig(u + u 0 ). (3.18)\n\nFundamental S-Matrix. Using the coproducts in Tab. 2 we have confirmed that the S-matrix is also invariant under all of the Yangian generators [∆ J A , S] = 0. (3.19) We have used a computer algebra system to evaluate the action of the Yangian generators and the S-matrix. 9 To show invariance requires heavy use of the identity (2.15). Superficially it is very surprising to find all these additional symmetries of the S-matrix. The deeper reason however should be that the coproduct is quasi-cocommutative. We have thus proved quasi-cocommutativity when acting on fundamental representations. It is interesting to see that the S-matrix is based on standard evaluation representations of the Yangian. Nevertheless, it is not a function of the difference of spectral parameters. This unusual property traces back to the link between the spectral parameter u and the h-representation parameters x ± in (3.17). The latter is again related to the braiding in the coproduct (3.11).\n\nAs our S-matrix is equivalent [14] to Shastry's R-matrix, our Yangian is presumably an extension of the su(2) × su(2) Yangian symmetry of the Hubbard model found in [47] ."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusions and Outlook",
"text": "In this note we have reviewed the construction of the S-matrix with centrally extended su(2|2) symmetry that appears in the context of the planar AdS/CFT correspondence and the one-dimensional Hubbard model. We have furthermore shown that the S-matrix has an additional Yangian symmetry whose Hopf algebra structure we have presented. This Yangian is not quite a standard Yangian, but its coproduct needs to be braided in order to be quasi-cocommutative. This fact is intimately related to the existence of a 7 We believe, but we have not verified that this is compatible with the Serre relations (3.2). 8 It is conceivable that a further consistency requirement fixes the value of u 0 , presumably to zero. 9 We have also confirmed the invariance of the singlet state found in [10].\n\n10 triplet of central charges with non-trivial coproduct and leads to the wealth of unusual features of the S-matrix.\n\nIn connection to the Yangian there are many points left to be clarified. Most importantly the representation theory needs to be understood. Which representations of h lift to evaluation representations of Y(h)? At what values of the spectral parameters do their tensor products become reducible? This information could be used to prove that the coproduct is quasi-cocommutative. Also the Yang-Baxter equation for the S-matrix should follow straightforwardly. It might also give some further understanding of bound states [48] .\n\nThen it would be highly desirable to construct a universal R-matrix for this Yangian and show that it is quasi-triangular. This would put large parts of the integrable structure for arbitrary representations of this algebra on solid ground much like for the case of generic simple Lie algebras. Some further interesting questions include: Is this Yangian the unique quasi-cocommutative Hopf algebra based on h? Does the double Yangian [42] exist and what is its structure? Can the sl(2) automorphism of the algebra be included at the Yangian level such that the coproduct is quasi-cocommutative? What would the representations be in this case?\n\nAcknowledgements. I am grateful to C. Callan, D. Erkal, A. Kleinschmidt, P. Koroteev, N. MacKay, M. Mariño, T. McLoughlin, J. Plefka, F. Spill and B. Zwiebel for interesting discussions.\n\n[21] A. Rej, D. Serban and M. Staudacher, JHEP 0603, 018 (2006), hep-th/0512077. [22] D. Serban and M. Staudacher, JHEP 0406, 001 (2004), hep-th/0401057. [23] G. Arutyunov, S. Frolov and M. Staudacher, JHEP 0410, 016 (2004), hep-th/0406256. N. Beisert and A. A. Tseytlin, Phys. Lett. B629, 102 (2005), hep-th/0509084. R. Hernández and E. López, JHEP 0607, 004 (2006), hep-th/0603204. N. Gromov and P. Vieira, Nucl. Phys. B789, 175 (2008), hep-th/0703191. [24] R. A. Janik, Phys. Rev. D73, 086006 (2006), hep-th/0603038. [25] G. Arutyunov and S. Frolov, Phys. Lett. B639, 378 (2006), hep-th/0604043. [26] N. Beisert and T. Klose, J. Stat. Mech. 06, P07006 (2006), hep-th/0510124. [27] N. Beisert, R. Hernández and E. López, JHEP 0611, 070 (2006), hep-th/0609044. [28] J. K. Erickson, G. W. Semenoff and K. Zarembo, Nucl. Phys. B582, 155 (2000), hep-th/0003055. [29] D. J. Gross and N. Miljkovic, Phys. Lett. B238, 217 (1990). [30] A. V. Kotikov and L. N. Lipatov, Nucl. Phys. B769, 217 (2007), hep-th/0611204. [31] A. V. Belitsky, Phys. Lett. B650, 72 (2007), hep-th/0703058. N. Dorey, D. M. Hofman and J. Maldacena, Phys. Rev. D76, 025011 (2007), hep-th/0703104. [32] K. Sakai and Y. Satoh, Phys. Lett. B661, 216 (2008), hep-th/0703177. [33] A. Rej, M. Staudacher and S. Zieme, J. Stat. Mech. 0708, P08006 (2007), hep-th/0702151v2. [34] N. Mann and J. Polchinski, Phys. Rev. D72, 086002 (2005), hep-th/0508232. [35] N. Gromov, V. Kazakov, K. Sakai and P. Vieira, Nucl. Phys. B764, 15 (2007), hep-th/0603043. [36] Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower and V. A. Smirnov, Phys. Rev. D75, 085010 (2007), hep-th/0610248. [37] M. K. Benna, S. Benvenuti, I. R. Klebanov and A. Scardicchio, Phys. Rev. Lett. 98, 131603 (2007), hep-th/0611135. [38] M. Beccaria, G. F. De Angelis and V. Forini, JHEP 0704, 066 (2007), hep-th/0703131. [39] L. F. Alday, G. Arutyunov, M. K. Benna, B. Eden and I. R. Klebanov, JHEP 0704, 082 (2007), hep-th/0702028. I. Kostov, D. Serban and D. Volin, Nucl. Phys. B789, 413 (2008), hep-th/0703031. [40] J. Maldacena and I. Swanson, Phys. Rev. D76, 026002 (2007), hep-th/0612079. [41] V. G. Drinfel'd, Sov. Math. Dokl. 32, 254 (1985). [42] V. G. Drinfel'd, J. Math. Sci. 41, 898 (1988). [43] D. Bernard, Int. J. Mod. Phys. B7, 3517 (1993), hep-th/9211133. N. J. MacKay, Int. J. Mod. Phys. A20, 7189 (2005), hep-th/0409183. [44] L. Dolan, C. R. Nappi and E. Witten, JHEP 0310, 017 (2003), hep-th/0308089. [45] L. Dolan, C. R. Nappi and E. Witten, hep-th/0401243, in: \"Quantum theory and symmetries\", ed.: P. C. Argyres et al., World Scientific (2004), Singapore. M. Hatsuda and K. Yoshida, Adv. Theor. Math. Phys. 9, 703 (2005), hep-th/0407044. L. Dolan and C. R. Nappi, Nucl. Phys. B717, 361 (2005), hep-th/0411020. [46] A. Agarwal and S. G. Rajeev, Int. J. Mod. Phys. A20, 5453 (2005), hep-th/0409180. N. Berkovits, JHEP 0503, 041 (2005), hep-th/0411170. B. I. Zwiebel, J. Phys. A40, 1141 (2007), hep-th/0610283. N. Beisert and D. Erkal, J. Stat. Mech. 0803, P03001 (2008), arxiv:0711.4813. [47] D. B. Uglov and V. E. Korepin, Phys. Lett. A190, 238 (1994), hep-th/9310158. [48] N. Dorey, J. Phys. A39, 13119 (2006), hep-th/0604175. H.-Y. Chen, N. Dorey and K. Okamura, JHEP 0611, 035 (2006), hep-th/0608047."
}
] | ||
arxiv:0704.0403 | 0704.0403 | 1 | 10.1038/nphoton.2007.46 | 3379021d8f8fb881332ac988f54b65201d8a992a55c2cb7511ddb58b81618ee8 | Review: Semiconductor Quantum Light Sources | Lasers and LEDs display a statistical distribution in the number of photons
emitted in a given time interval. New applications exploiting the quantum
properties of light require sources for which either individual photons, or
pairs, are generated in a regulated stream. Here we review recent research on
single-photon sources based on the emission of a single semiconductor quantum
dot. In just a few years remarkable progress has been made in generating
indistinguishable single-photons and entangled photon pairs using such
structures. It suggests it may be possible to realise compact, robust, LED-like
semiconductor devices for quantum light generation. | [
"Andrew J Shields"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Applying quantum light states to photonic applications allows functionalities that are not possible using 'ordinary' classical light. For example, carrying information with single-photons provides a means to test the secrecy of optical communications, which could soon be applied to the problem of sharing digital cryptographic keys. 1 2 Although secure quantum key distribution systems based on weak laser pulses have already been realised for simple point-to-point links, true single-photon sources would improve their performance. 3 Furthermore, quantum light sources are important for future quantum communication protocols such as quantum teleportation. 4 Here quantum networks sharing entanglement could be used to distribute keys over longer distance or through more complex topologies. 5 A natural progression would be to use photons for quantum information processing, as well as communication. In this regard it is relatively straightforward to encode and manipulate quantum information on a photon. On the other hand, single-photons do not interact strongly with one-another, a prerequisite for a simple photon logic gate. In linear optics quantum computing 67 (LOQC) this problem is solved using projective measurements to induce an effective interaction between the photons. Here triggered sources of single-photons and entangled pairs are required as both the qubit carriers, as well as auxiliary sources to test the successful operation of the gates. Although the component requirements for LOQC are challenging, they have recently been relaxed significantly by new theoretical schemes. 7 Quantum light states are also likely to become increasingly important for various types of precision optical measurement. 8 For these applications we would ideally like light sources which generate pure single-photon states "on demand" in response to an external trigger signal. Key performance measures for such a source are the efficiency, defined as the fraction of photons collected into the experiment or application per trigger, and the second order correlation function at zero delay, see text box. The latter is essentially a measure of the two-photon rate compared to a classical source with random emission times of the same average intensity. In order to construct applications involving more than one photon, it is also important that photons emitted from the source (at different times), as well as those from different sources, are otherwise indistinguishable.
In the absence of a convenient triggered single-photon source, most experiments in quantum optics rely on non-linear optical processes for generating quantum light states. Optically pumping a crystal with a χ(2) non-linearity has a finite probability of generating a pair of lower energy photons via parametric down conversion. This may be used to prepare photon pairs with time-bin entanglement, 9 entangled polarisations, 1011 or alternatively single-photon states 'heralded' by the second photon in the pair. 12 A χ(3) non-linearity in a semiconductor has also been used to generate entangled pairs. 13 As these non-linear processes occur randomly, there is always a finite probability of generating two pairs that increases with pump power. As double pairs degrade the fidelity of quantum optical gates, the pump laser power must be restricted to reduce the rate of double pairs to an acceptable level, which has a detrimental effect upon the efficiency of the source. 14 This means that although down-conversion sources continue to be highly successful in demonstrating few photon quantum optical gates, scaling to large numbers may be problematic. Solutions have been proposed based on switching multiple sources, 15 or storing photons in a switched fibre loop. 16 Ideally we would like a quantum light source that generates exactly one single-photon, or entangled-pair, per excitation trigger pulse. This may be achieved using the emission of a single quantum system. After relaxation, a quantum system is by definition no longer excited and therefore unable to re-emit. Photon anti-bunching, the tendency of a quantum source to emit photons separated in time, was first demonstrated in the resonance fluorescence of a low density vapour of Na atoms, 17 and subsequently for a single ion. 18 2 Quantum dots are often referred to as "artificial atoms", as their electron motion is quantised in all three spatial directions, resulting in a discrete energy level spectrum, like that of an atom. They provide a quantum system which can be grown within robust, monolithic semiconductor devices and can be engineered to have a wide range of desired properties. In the following we review recent progress towards the realisation of a semiconductor technology for quantum photonics. An excellent account of the early work can be found in Ref. 19 . Space restrictions limit discussion of work on other quantised systems. For this we refer the reader to the comprehensive review in Ref 20 .
Nano-scale quantum dots with good optical properties can be fabricated using a natural growth mode of strained layer semiconductors. 21 When InAs is deposited on GaAs it initially grows as a strained two-dimensional sheet, but beyond some critical thickness, tiny islands like those shown in Fig. 1a form in order to minimize the surface strain.
Overgrowth of the islands leads to the coherent incorporation of In x Ga 1-x As dots into the crystal structure of the device, as can be seen in the cross-sectional image of Fig. 1c . The most intensively studied are small InAs dots on GaAs emitting around 900-950nm at low temperatures, which can be conveniently measured with low noise Si single photon detectors.
A less desirable feature of the self-organising technique is that the dots form at random positions on the growth surface. However, recently considerable progress has been made on controlling the dot position (Fig. 1b ) within the device structure by patterning nanometer sized pits on the growth surface. 2223 As InGaAs has a lower energy bandgap than GaAs, the quantum dot forms a potential trap for electrons and holes. If sufficiently small, the dot contains just a few quantised levels in the conduction and valence bands, each of which holds two electrons or holes of opposite spin. Illumination by a picosecond laser pulse excites electrons and holes which rapidly relax to the lowest lying energy states either side of the bandgap. A quantum dot can thus capture two electrons and two holes to form the biexciton state, which decays by a radiative cascade, as shown schematically in Fig. 2a . One of the trapped electrons recombines with one of the holes and generates a first photon (called the biexciton photon, X 2 ). This leaves a single electron-hole pair in the dot (the exciton state), which subsequently also recombines to generate a second (exciton, X) photon. The biexciton and exciton photons have distinct energies, as can be seen in the low temperature photoluminescence spectrum of Fig. 2a , due to the different Coulomb energies of their initial and final states. Often a number of other weaker lines can also be seen due to recombination of charged excitons which form intermittently when the dot captures an excess electron or hole. 24 Larger quantum dots, with several confined electron and hole levels, have a richer optical signature due to the large number of exciton complexes that can be confined.
High resolution spectroscopy reveals that the X 2 and X transitions of a dot are in fact both doublets with linearly polarised components parallel to the [110] and [1-10] axes of the semiconductor crystal, labelled here H and V, respectively. 2526 The origin of this polarisation is an asymmetry in the electron-hole exchange interaction of the dot which produces a splitting of the exciton spin states. The asymmetry derives from an elongation of the dot along one crystal axis and in-built strain in the crystal. It mixes the exciton eigenstates of a symmetric dot with total z-spin J z = +1 and -1 into symmetric and anti-symmetric combinations, which couple to two H or two V polarised photons, respectively, as shown in Fig. 2 .
The exciton state of the dot has a typical lifetime of ~1ns, which is due purely to radiative decay. As this is much longer than the duration of the exciting laser pulse, or the lifetime of the photo-excited carrier population in the surrounding semiconductor, only one X photon can be emitted per laser pulse. This can be proven, as first reported 27 by Peter Michler, Atac Imamoglu and their colleagues in Santa Barbara, by measuring the second order correlation function, g (2) (τ) of the exciton photoluminescence, 2829 see text box. In fact each of the exciton complexes of the dot generates at most one photon per excitation cycle, which allows single-photon emission from also the biexciton or charged exciton transitions. 30 Cross-correlation measurements 313233 between the X and X 2 photons confirm the time correlation expected for the cascade in Fig. 2a , ie the X photon follows the X 2 one. Indeed the shape of the cross-correlation function for both CW and pulsed excitation can be accurately described with a simple rate equation model and the experimentally measured X and X 2 decay rates. 34
A major advantage of using self-assembled quantum dots for single-photon generation is that they can be easily incorporated into cavities using standard semiconductor growth and processing techniques. Cavity effects are useful for 3 directing the emission from the dot into an experiment or application, as well as for modifying the photon emission dynamics. 3536 Purcell 37 predicted enhanced spontaneous emission from a source in a cavity when its energy coincides with that of the cavity mode, due to the greater density of optical states to emit into. For an ideal cavity, in which the emitter is located at the maximum of the electric field with its dipole aligned with the local electric field, the enhancement in decay rate is given by F p = (3/4π 2 ) (λ/n) 3 Q/V, where Q is the quality factor, a measure of the time a photon is trapped in the cavity, and V is the effective mode volume. Thus high photon collection efficiency, and simultaneously fast radiative decay, requires small cavities with highly reflecting mirrors and a high degree of structural perfection. However, without controlling the location of the dot in the cavity, as discussed below, it may be difficult to achieve the full enhancement predicted by the Purcell formula.
Figure 3 shows images of some of the single quantum dot cavity structures that have proven most successful. Pillar microcavities, formed by etching cylindrical pillars into semiconductor Bragg mirrors placed either side of the dot layer, have shown large Purcell enhancements and have a highly directional emission profile, thus making good single-photon sources. 38394041 Purcell factors of around 6 have been measured directly, 4041 through the rate of cavity-enhanced radiative decay compared to that of a dot without cavity, implying a coupling to the cavity mode of β=F p /(1+F p )>85%, if we assume the leaky modes are unaffected by the cavity. However, the experimentally determined photon collection efficiency, which is a more pertinent parameter for applications, is typically ~10%, due the fact that not all the cavity mode can be coupled into an experiment and scattering of the mode by the rough pillar edges. We can expect that the photon collection efficiency will increase with improvements to the processing technology or new designs of microcavity.
Another means of forming a cavity is to etch a series of holes in a suspended slab of semiconductor, so as to form a lateral variation in the refractive index which creates a forbidden energy gap for photonic modes in which light cannot propagate. 42 Photons can then be trapped in a central irregularity in this structure: usually an unetched portion of the slab. Such photonic bandgap defect cavities have been fabricated in Si with Q values approaching 10 6 . 4344 High quality active cavities have also been demonstrated in GaAs containing InAs quantum dots. 45464748 A radiative lifetime of 86 ps, corresponding to a Purcell factor of F p ~12, has been reported. 47 Very recently a lifetime of 60ps was measured for a cavity in the strong coupling regeme. 48 If the Q-value is sufficiently large, the system enters the strong coupling regime where the excitation oscillates coherently between an exciton in the dot and a photon in the cavity. The spectral signature of strong coupling, an anticrossing between the dot line and the cavity mode, has been observed for quantum dots in pillar microcavities, 49 photonic bandgap defect cavities, 50 microdisks 51 and microspheres. 52 It has been demonstrated for atom cavities that strong coupling allows the deterministic generation of single-photons. 5354 Single-photon sources in the strong coupling regime can be expected to have very high extraction efficiencies and be time-bandwidth limited. 55 Encouragingly single-photon emission has been reported recently for a dot in a strongly coupled pillar microcavity. 56 Another interesting recent development is the ability to locate a single quantum dot within the cavity, as this ensures the largest possible coupling and removes background emission, as well as other undesirable effects, due to other dots in the cavity. Above we discussed techniques to control the dot position on the growth surface. The other way is to position the cavity around the dot. One technique combines micro-photoluminescence spectroscopy to locate the dot position, with in-situ laser photolithography to pattern markers on the wafer surface. 57 An alternative involves growing a vertical stack of dots so that their location can be revealed by scanning the wafer surface, 58 as shown in Fig. 3 . Recently this technique has allowed larger coupling energies for a single dot in a photonic bandgap defect cavity. 48
Cavity effects are important for rendering different photons from the source indistinguishable, which is essential for many applications in quantum information. When two identical photons are incident simultaneously on the opposite input ports of a 50/50 beamsplitter, they will always exit via the same output port, 59 as shown schematically in Fig. 4a . This occurs because of a destructive interference in the probability amplitude of the final state in which one photon exits through each output port. The amplitude of the case where both photons are reflected exactly cancels with that where both are transmitted, due to the π/2 phase change upon reflection, provided the two photons are entirely identical.
Two-photon interference of two single-photons emitted successively from a quantum dot in a weakly-coupled pillar microcavity was first reported by the Stanford group. 60 Fig. 4b shows a schematic of their experiment. Notice the reduction of the co-incidence count rate measured between detectors in either output port, when the two photons are injected simultaneously (Fig. 4c ). The dip does not extend completely to zero, indicating that the two photons sometime exit the beamsplitter in opposite ports. The measured reduction in co-incidence rate at zero delay of 69%, implies an overlap for the single-photon wavepackets of 0.81, after correcting for the imperfect single-photon visibility of the 4 interferometer. Two-photon interference dips of 66% and 75% have been reported by Bennett et al 61 and Vauroutsis et al. 62 Similar results have been obtained for a single dot in a photonic bandgap defect cavity. 63 This two-photon interference visibility is limited by the finite coherence time of the photons emitted by the quantum dot, 64 which renders them distinguishable. The depth of the dip in Fig. 4c depends upon the ratio of radiative decay time to the coherence time of the dot, ie R=2τ decay /τ coh . When unity, the coherence time is limited by radiative decay and the source will display perfect 2-photon interference. The most successful approach thus far has been to extend τ coh by resonant optical excitation of the dot and reduce τ decay using the Purcell effect in a pillar microcavity, to values R~1.5. In the future higher visibilities may be achieved with a larger Purcell enhancement, using a single dot cavity in the strongcoupling regime or with electrical gating described in the next section.
A source of indistinguishable single-photons was used by Fattal et al to generate entanglement between post-selected pairs. 65 66 This involves simply rotating the polarisation of one of the photons incident on the final beamsplitter in Fig. 4a by 90 o . By post-selecting the results where the two photons arrive at the beamsplitter at the same time and where there is one photon in each output arm (labelled 1 and 2), the measured pairs should correspond to the Bell state
ψ -= 1/√2 (¦H 1 V 2 > -¦V 1 H 2 >)
Eq.1 Note that only if the two photons are indistinguishable and thus the entanglement is only in the photon polarisation, are the two terms in Eq1 able to interfere. Analysis of the density matrix published by Fattal et al 65 reveals a fidelity of the post-selected pairs to the state in Eq.1 of 0.69, beyond the classical limit of 0.5. This source of entangled pairs has an importance difference to that based on the biexciton cascade described below. Post-selection implies that the photons are destroyed when this scheme succeeds. This is a problem for some quantum information applications such as LOQC, but could be usefully applied to quantum key distribution. 65
An early proposal for an electrical single-photon source by Kim et al 67 was based upon etching a semiconductor heterostructure displaying Coulomb blockade. However, the light emission from this etched structure was too weak to allow the second-order correlation function to be studied. Recently encouraging progress has been made towards the realisation of a single-photon source based on quantising a lateral electrical injection current. 6869 However the most successful approach so far has been to integrate self-assembled quantum dots into conventional p-i-n doped junctions.
In the first report of electrically-driven single-photon emission by Yuan et al, 70 the electroluminescence of a single dot was isolated by forming a micron-diameter emission aperture in the opaque top contact of the p-i-n diode. Fig. 5a shows an improved emission aperture single-photon LED after Bennett et al, 71 which incorporates an optical cavity formed between a high reflectivity Bragg mirror and the semiconductor/air interface in the aperture. This structure forms a weak cavity, which enhances the measured collection efficiency 10-fold compared to devices without a cavity. 72 Single-photon pulses are generated by exciting the diode with a train of short voltage pulses. The second order correlation function g (2) (τ) of either the X or X 2 electroluminescence (Fig. 5c ) shows the suppression of the zero delay peak indicative of single-photon emission. 71 The finite rate of multi-photon pulses is due mostly to background emission from layers other than the dot, which is also seen for non-resonant optical excitation. Electrical contacts also allow the temporal characteristics of the single-photon source to be tailored. By applying a negative bias to the diode between the electrical injection pulses, Bennett et al 73 reduced the jitter in the photon emission time <100ps. This allowed the repetition rate of the single-photon source to be increased to 1.07GHz (Fig. 5d ) while retaining good singlephoton emission characteristics (Fig. 5e ). Electrical gating could provide a technique for producing time-bandwidthlimited single-photons from quantum dots. Another promising approach is to aperture the current flowing through the device. 7475 This is achieved by growing a thin AlAs layer within the intrinsic region of the p-i-n junction and later exposing the mesa to wet oxidation in a furnace, converting the AlAs layer around the outer edge of the mesa to insulating Aluminium oxide. By careful control of the oxidation time, a µm-diameter conducting aperture can be formed within the insulating ring of AlOx. Such structures have the advantage of exciting just a single dot within the structure, thereby reducing the amount of background emission. The oxide annulus also confines the optical mode laterally within the structure, potentially allowing high photon extraction efficiency.
Altering the nanostructure or materials that comprise the quantum dot allows considerable control over the emission wavelength and other characteristics. Most of the experimental work done so far has concentrated on small InAs quantum dots emitting around 900-950nm, as these have well understood optical properties and can be detected with 5 low noise Si single-photon detectors. On the other hand the shallow confinement potentials of this system means they emit only at low temperatures. At shorter wavelengths optically-pumped single-photon emission has been demonstrated at ~350nm using GaN/AlGaN, 76 500nm using CdSe/ZnSSe 77 and 682nm InP/GaInP 78 quantum dot. The former two systems have been shown to operate at 200K. It is very important for quantum communications to develop sources at longer wavelengths in the fibre optic transmission bands at 1.3 and 1.55µm. This may be achieved using InAs/GaAs heterostructures by depositing more InAs to form larger quantum dots. These larger dots offer deeper confinement potentials than those at 900nm and thus often display room temperature emission. 79 Optically pumped single-photon emission at telecom wavelengths has been achieved using a number of techniques to prepare low densities of longer wavelength dots, including a bimodal growth mode in MBE to form low densities of large dots, 80 ultra-low growth rate MBE 81 and MOCVD. 82 Recently, the first electrically-driven single-photon source at a telecom wavelength has been demonstrated. 83
By collecting both the X 2 and X photons emitted by the biexciton cascade, a single quantum dot may also be used as a source of photon pairs. Polarisation correlation measurements on these pairs discovered that the two photons were classically-correlated with the same linear polarisation. 848586 This occurs because the cascade can proceed via one of two intermediate exciton spin states, as described above and shown in Fig. 2a , one of which couples to two H-and the other two V-polarised photons. The emission is thus a statistical mixture of |H X2 H X > and |V X2 V X >, although exciton spin scattering during the cascade (discussed below) ensures there are also some cross-polarised pairs.
The spin splitting 87,88 of the exciton state of the dot distinguishes the H and V polarised pairs and prevents the emission of entangled pairs predicted by Benson et al. 89 If this splitting could be removed, the H and V components would interfere in appropriately designed experiments. The emitted 2-photon state should then be written as a superposition of HH and VV, which can be recast in either the diagonal (spanned by D, A) or circular (σ + , σ -) polarisation bases, ie
Φ + = 1/√2 (¦H X2 H X > + ¦V X2 V X >) = 1/√2 (¦D X2 D X > + ¦Α X2 Α X >) = 1/√2 (¦σ + X2 σ - X > + ¦σ + X2 σ - X >) Eq.2.
Equal weighting of the HH and VV terms assumes the source to be unpolarised, as indicated by experimental measurements.
Eq.2 suggests that, for zero exciton spin splitting, the biexciton cascade generates entangled photon pairs, similar to those seen for atoms. 90 Entanglement of the X or X 2 photons was recently observed experimentally for the first time by Stevenson, Young and co-workers, 9192 using two different schemes to cancel the exciton spin splitting. An alternative approach by Akopian et al, 93 using dots with finite exciton splitting, post-selects photons emitted in a narrow spectral band where the two polarisation lines overlap.
The exciton spin splitting depends on the exciton emission energy, tending to zero for InAs dots emitting close to 1.4eV and then inverting for higher emission energy. 94 95 These correspond to shallow quantum dots for which the carrier wavefunctions extend into the barrier material reducing the electron-hole exchange. Zero splitting can be achieved by either careful control of the growth conditions to achieve dots emitting close to the desired energy, or by annealing samples emitting at lower energy. 94 The exciton spin splitting may be continuously tuned by applying a magnetic field in the plane of the dot. 96 It has been observed that the signatures of entanglement then appear only when the exciton splitting is close to zero. 91 Other promising schemes to tune the exciton splitting are now emerging, including application of strain 97 and electric field. 9899 Figure 6a plots polarisation correlations reported by Young et al 92 for a dot with zero exciton splitting (by control of the growth conditions). Pairs emitted in the same cascade (ie zero delay) shows a very striking positive correlation (copolarisation) measuring in either, rectilinear or diagonal bases and anti-correlation (cross-polarisation) when measuring in circular basis. This is exactly the behaviour expected for the entangled state of Eq.2. In contrast, a dot with finite splitting shows polarisation correlation for the rectilinear basis only, with no correlation for diagonal or circular measurements, see Figure 6b . The strong correlations seen for all three bases in Fig. 6a could not be produced by any classical light source or mixture of classical sources and is proof that the source generates entangled photons. The measured 92 two-photon density matrix (Fig. 6c ) projects onto the expected
1/√2 (¦H X2 H X > + ¦V X2 V X >) state
with fidelity (ie probability) 0.702 ± 0.022, exceeding the classical limit (0.5) by 9 standard deviations.
Two processes contribute to the 'wrongly' correlated pairs which impair the fidelity of the entangled photon source. The first of these is due to background emission from layers in the sample other than the dot. This background emission, which is unpolarised and dilutes the entangled photons from the dot, limited the fidelity observed in the first report 91 of triggered entangled photon pairs from a quantum dot and has been subsequently reduced with better sample design. 92 The second mechanism, which is an intrinsic feature of the dot, is exciton spin scattering during the biexciton cascade. It is interesting that this process does not seem to depend strongly upon the exciton spin splitting. It may be reduced by suppressing the scattering using resonant excitation or alternatively using cavity effects to reduce the time required for the radiative cascade.
The past several years have seen remarkable progress in quantum light generation using semiconductor devices. However, despite considerable progress many challenges still remain. The structural integrity of cavities must continue to improve, thereby enhancing quality factors. This, combined with the ability to reliably position single dots within the cavity, will further enhance photon collection efficiencies and the Rabi energy in the strong coupling regime. It is also important to realise all the benefits of these cavity effects in more practical electrically-driven sources. Meanwhile bandstructure engineering of the quantum dots will allow a wider range of wavelengths to be accessed for both single and entangled photon sources, as well as structures that can operate at higher temperatures. Techniques for fine tuning the characteristics of individual emitters will also be important.
One of the most interesting aspects of semiconductor quantum optics is that we may be able to use quantum dots not only as quantum light emitters, but also as the logic and memory elements which are required in quantum information processing. Although LOQC is scalable theoretically, quantum computing with photons would be much easier with a useful single-photon non-linearity. Such non-linearity may be achieved with a quantum dot in a cavity in the strong coupling regime. Encouragingly strong coupling of a single quantum dot with various type of cavity has already been observed in the spectral domain. Eventually it may even be possible to integrate photon emission, logic, memory and detection elements into single semiconductor chips to form a photonic integrated circuit for quantum information processing.
The author would like to thank Mark Stevenson, Robert Young, Anthony Bennett, Martin Ward and Andy Hudson for their useful comments during the preparation of the manuscript and the UK DTI "Optical Systems for Digital Age", EPSRC and EC Future and Emerging Technologies programmes for supporting research on quantum light sources.
7
The photon statistics of light can be studied via the second order correlation function, g (2) (τ), which describes the correlation between the intensity of the light field with that after a delay τ and is given by 100 2 ) 2 ( ) ( ) ( ) ( ) ( > < > + < = t I t I t I g τ τ
This function can be measured directly using the Hanbury-Brown and Twiss 101 interferometer, comprising a 50/50 beamsplitter and two single-photon detectors, shown in the figure . For delays much less than the average time between detection events (ie for low intensities), the distribution in the delays between clicks in each of the two detectors is proportional to g (2) (τ).
For a continuous light source with random emission times, such as an ideal laser or LED, g (2) (τ)=1. It shows there is no correlation in the emission time of any two photons from the source. A source for which g (2) (τ=0)>1 is described as 'bunched' since there is an enhanced probability of two photons being emitted within a short time interval. Photons emitted by quantum light sources are typically 'anti-bunched', (g (2) (τ=0)<1) and tend to be separated in time.
In communication and computing systems, we are interested in pulsed light sources, for which the emission occurs at times defined by an external clock. In this case g (2) (τ) consists of a series of peaks separated by a clock period. For an ideal single-photon source, the peak at zero time delay is absent, g (2) (τ=0)=0; as the source cannot produce more than one photon per excitation period, clearly the two detectors cannot fire simultaneously.
The figure shows g (2) (τ) recorded for resonant pulsed optical excitation of the X emission of a single quantum dot in a pillar microcavity. Notice the almost complete absence of the peak at zero delay: the definitive signature of a singlephoton source. The weak peak seen at τ=0 demonstrates that the rate of two-photon emission is 50 times less than that of an ideal laser with the same average intensity. The bunching behaviour observed for the finite delay peaks is explained by intermittent trapping of a charge carrier in the dot. 102 This trace was taken for quasi-resonant laser excitation of the dot which avoids creating carriers in the surrounding semiconductor. For higher energy laser excitation, the suppression in g (2) (0) is typically reduced indicating occasional 2-photon pulses due to emission from the layers surrounding the dot, but can be minimised with careful sample design.
Figure textbox: (a) Schematic of the set-up used for photon correlation measurements, (b) second order correlation function of the exciton emission of a single dot in a pillar microcavity.
Figure 1: Self assembled quantum dots (a) Image of a layer of InAs/GaAs self assembled quantum dots recorded in an Atomic Force Microscope (AFM). Each yellow blob corresponds to a dot with typical lateral diameters of 20-30nm and a height of 4-8nm. (b) AFM image 23 of a layer of InAs quantum dots whose locations have been seeded by a matrix of nanometer sized pits patterned onto the wafer surface. Under optimal conditions up to 60% of the etch pits contain a single dot (Courtesy of P Atkinson & D A Ritchie, Cambridge). (c) Cross-sectional STM image of an InAs dot inside a GaAs device (Courtesy of P. Koenraad, Eindhoven).
Figure 2: Optical spectrum of a quantum dot. (a) Schematic of the biexciton cascade of a quantum dot. (b) Typical photoluminescence spectrum of a single quantum dot showing sharp line emission due to the biexciton X 2 and exciton X photon emitted by the cascade. The inset shows the polarisation splitting of the transitions originating from the spin splitting of the exciton level.
Figure 3: SEM images of semiconductor cavities, including pillar microcavities (a) 56 and (b), microdisk (c) 51 and photonic bandgap defect cavities (d) 47 , (e) and (f). 48 (Structures fabricated at Univ Wuerzburg (a), CNRS-LPN (UPR-20), Marcoussis (b, c, e), Univ Cambridge (d), UCSB/ETHZ Zurich (f)) Figure 4: Two Photon Interference. (a) If the two photons are indistinguishable, the two outcomes resulting in one photon in either arm interfere destructively. This results in the two photons always exiting the beamsplitter together. (b) Schematic of an experiment using two photons emitted successively from a quantum dot, (c) experimental data showing suppression of the co-incidence rate in (b) when the delay between input photons is zero due to two-photon interference. 60 (Courtesy of Y Yamamoto, Stanford Univ.) Figure 5: Electrically driven single-photon emission. (a) Schematic of a single-photon LED. (b) Electroluminescence spectra of the device. Notice the spectra are dominated by the exciton X and biexciton X 2 lines, which have linear and quadratic dependence on drive current, respectively. Other weak lines are due to charged excitons. (c) second order correlation function recorded for the exciton (i) and biexciton (ii) emission lines, (d) time-resolved electroluminescence from a device operate with a 1.07GHz repetition rate, (e) measured (i) and modelled (ii) second order correlation function of the biexciton electroluminescence at 1.07GHz. (adapted from Refs. 71and 73) Figure 6: Generation of entangled photons by a quantum dot. (a) Degree of correlation measured for a dot with exciton polarisation splitting S=0 µeV in linear (i), diagonal (ii) and circular (iii) polarisation bases as a function of the delay between the X and X 2 photons (in units of the repetition cycle). The correlation is defined as the rate of co-polarised pairs minus the rate of cross-polarised pairs divided by the total rate. Notice that the values at finite delay show no correlation, as expected for pairs emitted in different laser excitation cycles. More interesting are the peaks close to zero time delay, corresponding to X and X 2 photon emitted from the same cascade. The presence of strong correlations for all three types of measurement for the dot with zero exciton splitting can only be explained if the X and X 2 polarisations are entangled. (b) Degree of correlations measured for the dot in (a) subject to in-plane magnetic field so as to produce an exciton polarisation splitting of S=25 µeV. Notice that the correlation in diagonal and circular bases have vanished, indicating only classical correlations at finite splitting. (c) Two-photon density matrix of the device emission in (a). The strong off-diagonal terms appear due to entanglement. (adapted from Ref 92) 9 References 1 Gisin, N., Ribordy, G., Tittel, W. & Zbinden, H. Quantum cryptography. Rev. Mod Physics 74, 145-195 (2001). 2 Dusek, M., Lutkenhaus, N. & Hendrych, M. Quantum Cryptography. Progress in Optics 49, Edt. E. Wolf (Elsevier 2006). 3 Waks, E., Inoue, K., Santori, C., Fattal, D., Vucković, J., Solomono, G. S. & Yamamoto, Y. Secure communication: Quantum cryptography with a photon turnstile. Nature 420, 762-762 (2002). 4 Bouwmeester, D., Pan, J. W., Mattle, K., Eibl, M., Weinfurter, H. & Zeilinger, A. Experimental quantum teleportation. Nature 390, 575-579 (1997). 5 Briegel, H.-J., Dür, W., Cirac, J. I. & Zoller, P. Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication. Phys. Rev. Lett. 81, 5932-5935 (1998). 6 Knill, E., Laflamme, R. & Milburn, G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 46-52 (2001). 7 Kok, P., Munro, W. J., Nemoto, K., Ralph, T. C., Dowling, J. P. & Milburn, G. J. Linear optical quantum computing. Quantph/0512071 (2005). 8 Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: Beating the standard quantum limit. Science 306, 1330-1336 (2004). 9 Brendel J, Gisin N, Tittel W and Zbinden H, Pulsed Energy-Time Entangled Twin-Photon Source for Quantum Communication Phys. Rev. Lett. 82, 2594 (1999).
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Lett. 88, 131101 (2006). 48 Hennessy et al, quant-ph/0610034; to be published in Nature on 22 Feb doi:10.1038/nature05586 49 Reithmaier, J. P., Sek, G., Löffler, A., Hofmann, C., Kuhn, S., Reitzenstein, S., Keldysh, L. V., Kulakovskii, V. D., Reinecke, T. L. & Forchel, A. Strong coupling in a single quantum dot-semiconductor microcavity system. Nature 432, 197-200 (2004). 50 Yoshie, T., Scherer, A., Hendrickson, J., Khitrova, G., Gibbs, H. M., Rupper, G, Ell, C., Shchekin, O. B. & Deppeet, D. G. Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity. Nature 432, 200-203 (2004). 51 Peter, E., Senellart, P., Martrou, D., Lemaître, A., Hours, J., Gérard, J. M. & Bloch, J. Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity. Phys. Rev. Lett. 95, 067401 (2005). 52 LeThomas N., Woggon U., Schöps O., Artemyev M.V., Kazes M., Banin U., Nano Lett. 6, 557 (2006). 53 Kuhn A., Hennrich M., Rempe G., Determinsitic single photon source for distributed quantum networking, Phys. Rev. Lett. 89, 067901 (202) 54 McKeever J., Boca A., Boozer A.D., Miller R., Buck J.R., Kuzmich A., Kimble H.J., Determinsitic generation of single photons from one atom trapped in a cavity, Science 303, 1992 (2004). 55 Cui G and Raymer M.G., Quantum efficiency of single photon sources in cavity-QED strong-coupling regime, Optics Express 13, 9660 (2005). 56 Press, D., Goetzinger, S., Reitzenstein, S., Hofmann, C., Loeffler, A., Kamp, M., Forchel, A. & Yamamoto, Y. Photon antibunching from a single quantum dot-microcavity system in the strong coupling regime. Quant-ph/0609193 (2006). 57 Lee K. H. et al, Registration of single quantum dots using cryogenic laser photolithography, Appl. Phys. Lett. 88, 193106 (2006) 58 Badolato A. et al, Deterministic Coupling of single quantum dots to single nanocavity modes, Science 308, 1158 (2005). 59 Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044-2046 (1987). 60 Santori, C., Fattal, D., Vucković, J., Solomon, G. S. & Yamamoto, Y. Indistinguishable photons from a single-photon device. Nature 419, 594-597 (2002). 61 Bennett, A. J., Unitt, D., Atkinson, P., Ritchie, D. A. & Shields, A. J. Influence of exciton dynamics on the interference of two photons from a microcavity single-photon source. Opt. Express 13, 7772-7778 (2005). 62 Varoutsis, S., Laurent, S., Kramper, P., Lemaître, A., Sagnes, I., Robert-Philip, I. & Abram, I. Restoration of photon indistinguishability in the emission of a semiconductor quantum dot. Phys. Rev. B 72, 041303 (2005). 63 Laurent, S., Varoutsis, S., Le Gratier, L., Lemaître, A., Sagnes, I., Raineri, F., Levenson, A., Robert-Philip, I. & Abram, I. Indistinguishable single photons from a single quantum dot in two-dimensional photonic crystal cavity. Appl. Phys. Lett. 87, 163107 (2005). 64 Kammerer. C., Cassabois, G., Voisin, C., Perrin, M., Delalande, C., Roussignol, Ph. & Gérard, J. M. Interferometric correlation spectroscopy in single quantum dots. Appl. Phys. Lett. 81, 2737-2739 (2002).. 65 Fattal, D., Diamanti, E., Inoue, K. & Yamamoto, Y. Quantum teleportation with a quantum dot single-photon source. Phys. Rev. Lett. 92, 037904 (2004). 66 Fattal, D., Inoue, K., Vucković, J., Santori, C., Solomon, G. S. & Yamamoto, Y. Entanglement formation and violation of Bell's inequality with a semiconductor single-photon source. Phys. Rev. Lett. 92, 037903 (2004). 67 Imamoglu, A. & Yamamoto, Y. Turnstile device for heralded single photons: Coulomb blockade of electron and hole tunneling in quantum confined p-i-n heterojunctions. Phys. Rev. Lett. 72, 210 (1994). 68 Cecchini, M., De Simoni, G., Piazza, V., Beltram, Berre, H. E. & Ritchie, D. A. Surface acoustic wave-driven planar light-emitting device. Appl. Phys. Lett. 85, 3020-3022 (2005). 69 Gell, J. R. et al, Surface acoustic wave driven luminescence from a lateral p-n junction. Appl. Phys. Lett. accepted (2006). 70 Yuan, Z., Kardynal, B. E., Stevenson, R. M., Shields, A. J., Lobo, C. J., Cooper, K., Beattie, N. S., Ritchie, D. A. & Pepper, M. Electrically driven single-photon source. Science 295, 102-105 (2002). 71 Bennett, A. J., Unitt, D. C., See, P., Shields, A. J., Atkinson, P, Cooper, K. & Ritchie, D. A. A microcavity single-photon emitting diode. Appl. Phys. Lett. 86, 181102 (2005). 72 Abram, I., Robert, I. & Kuszelewicz, R. Spontaneous emission control in semiconductor microcavities with metallic or Bragg mirrors. IEEE. J. Of Quant. Elect. 34, 71-76 (1998) 11 73 Bennett, A. J., Unitt, D. C., See, P., Shields, A. J., Atkinson, P., Cooper, K. & Ritchie, D. A. Electrical control of the uncertainty in the time of single-photon emission events. Phys Rev B 72, 033316 (2005) 74 Ellis, D., Bennett, A. J., Shields, A. J., Atkinson, P. & Ritchie, D. A. Electrically addressing a single self-assembled quantum dot. Appl. Phys. Lett. 88, 133509 (2006). 75 Lochman, A., Stock, E., Schulz, O, Hopfer, F., Bimberg, D., Haisler, V. A., Toropov, A. I., Bakarov, A. K. & Kalagin, A. K. Electrically driven single quantum dot polarised single photon emitter. Electron. Lett. 42, 774-775 (2006). 76 Kako, S., Santori, C., Hoshino, K., Götzinger, S., Yamamoto, Y. & Arakawa, Y. A gallium nitride single-photon source operating at 200 K. Nature Materials 5, 887-892 (2006). 77 Sebald, K., Michler, P., Passow, T., Hommel, D., Bacher, G. & Forchel, A. Single-photon emission of CdSe quantum dots at temperatures up to 200 K. Appl. Phys. Lett. 81, 2920-2922 (2002). 78 Aichele, T., Zwiller, V. & Benson O. Visible single-photon generation from semiconductor quantum dots. New J. Phys. 6, 90 (2004). 79 Le Ru, E.C., Fack, J. & Murray, R. Temperature and excitation density dependence of the photoluminescence from annealed InAs/GaAs quantum dots. Phys. Rev. B 67, 245318 (2003). 80 Ward, M. B., Karimov, O. Z., Unitt, D. C., Yuan, Z. L., See, P., Gevaux, D. G., Shields, A. J., Atkinson, P. & Ritchie, D. A. Ondemand single-photon source for 1.3 µm telecom fiber. Appl. Phys. Lett. 86, 201111 (2005). 81 Zinoni, C., Alloing, B., Monat, C., Zwiller, V., Li, L. H., Fiore, A., Lunghi, L., Gerardino, A., de Riedmatten, H., Zbinden, H. & Gisin, N. Time-resolved and antibunching experiments on single quantum dots at 1300nm. Appl. Phys. Lett. 88, 131102 (2006). 82 Miyazawa T., Takemoto, K., Sakuma, Y., Hirose, S., Usuki, T., Yokoyama, N., Miyazawa, T., Takatsu, M. & Arakawa, Y., Single-Photon Generation in the 1.55-um Optical-Fiber Band from an InAs/InP Quantum Dot, Jpn. J. Appl. Phys., vol 44, no 20, pp. L620-622 (2005) 83 Ward M B, Farrow T, See P, Yuan Z L, Karimov O Z, Bennett A J, Shields A J, Atkinson P, Cooper K, Ritchie D A,, Electrically driven telecommunication wavelength single-photon source Appl. Phys. Lett. 90, 063512 (2007) 84 Stevenson, R. M., Thompson, R. M., Shields, A. J., Farrer, I., Kardynal, B. E., Ritchie, D. A. & Pepper, M. Quantum dots as a photon source for passive quantum key encoding. Phys. Rev. B 66, 081302 (2002). 85 Santori, C., Fattal, D., Pelton, M., Solomon, G. S. & Yamamoto, Y. Polarization-correlated photon pairs from a single quantum dot. Phys. Rev. B 66, 045308 (2002). 86 Ulrich, S. M., Strauf, S., Michler, P., Bacher, G. & Forchel, A. Triggered polarization-correlated photon pairs from a single CdSe quantum dot. Appl. Phys. Lett. 83, 1848-1850 (2003). 87 van Kesteren, H. W., Cosman, E. C., van der Poel, W. A. J. A. & Foxon C. T. Fine structure of excitons in type-II GaAs/AlAs quantum wells. Phys. Rev. B 41, 5283-5292 (1990) 88 Blackwood, E., Snelling, M. J., Harley, R. T., Andrews, S. R. & Foxon C. T. B. Exchange interaction of excitons in GaAs Heterostructures. Phys. Rev. B 50, 14246-14254 (1994) 89 Benson, O., Santori, C., Pelton, M. & Yamamoto, Y. Regulated and entangled photons from a single quantum dot. Phys. Rev. Lett. 2513-2516 (2000). 90 Aspect, A., Grangier, P. & Roger, G. Experimental tests of realistic local theories via Bell's theorem. Phys. Rev. Lett. 47, 460-463 (1981). 91 Stevenson, R. M., Young, R. J., Atkinson, P., Cooper, K., Ritchie, D. A. & Shields, A. J. A semiconductor source of triggered entangled photon pairs. Nature 439, 179 (2006). 92 Young, R. J., Stevenson, R. M., Atkinson, P., Cooper, K., Ritchie, D. A. & Shields, A. J. Improved fidelity of triggered entangled photons from single quantum dots. New J. Phys. 8, 29 (2006). 93 Akopian, N., Lindner, N. H., Poem, E., Berlatzky, Y., Avron, J. & Gershoni, D. Entangled photon pairs from semiconductor quantum dots. Phys. Rev. Lett. 96, 130501 (2006). 94 Young, R. J., Stevenson, R. M., Shields, A. J., Atkinson, P., Cooper, K., Ritchie, D. A., Groom, K. M., Tartakovskii, A. I. & Skolnick, M. S. Inversion of exciton level splitting in quantum dots. Phys. Rev. B 72, 113305 (2005). 95 Seguin R., Schliwa A., Rodt S., Poetschke K., Pojl U.W. Bimberg D., Size-Dependent Fine-Structure Splitting in Self-Organized InAs/GaAs Quantum Dots, Phys. Rev. Lett. 95, 257402 (2005). 96 Stevenson, R. M., Young, R. J., See, P., Gevaux, D. G., Cooper, K., Atkinson, P., Farrer, I., Ritchie, D. A. & Shields, A. J. Magnetic-field-induced reduction of the exciton polarisation splitting in InAs quantum dots. Phys. Rev. B 73, 033306 (2006). 97 Seidl, S., Kroner, M., Högele, A. & Karrai, K. Effect of uniaxial stress on excitons in a self-assembled quantum dot. Appl. Phys. Lett. 88, 203113 (2006). 98 Geradot, B. D., Seidl, S., Dalgarno, P. A., Warburton, R. J., Granados, D., Garcia, J. M., Kowalik, K., Krebs, O., Karrai, K., Badolato, A. & Petroff, P. M. Manipulating exciton fine-structure in quantum dot with a lateral electric field. Cond-mat/0608711 (2006). 99 Kowalik, K., Lemaître, A., Laurent, S., Senellart, P., Voisin, P. & Gaj, J. A. Influence of an in-plane electric field on exciton fine structure in InAs-GaAs self-assembled quantum dots. Appl. Phys. Lett. 86, 041907 (2005). 100 Walls, D. F. & Milburn, G. J. Quantum Optics (Springer, Berlin, 1994). 101 Hanbury Brown, R. & Twiss, R. Q. A New Type of Interferometer for Use in Radio Astronomy. Phil. Mag. 45, 663 (1954). 102 Santori, C., Fattal, D., Vucković, J, Solomon, G. S., Waks, E. & Yamamoto, Y. Submicrosecond correlations in photoluminescence from InAs quantum dots. Phys. Rev. B 69, 205324 (2004).
(c) 10nm (a) 500 nm 500 nm (b) Fig. 1 1375 1380 1385 Detection polarisation: Vertical Horizontal X 2 X X 2 X PL Intensity Photon Energy (meV) 1378.0 1378.5 1380.0 1380.5 (b) (a) + -X X 2 -+ + -H H V V Fig. 2 S ground state S S 500 nm 500nm (d) (c) (a) (b) (e) (f) Fig. 3 or (a) (b) (c) (c) Fig. 4 substrate/buffer n-ohmic contact InAs QD insulator Al p-ohmic contact emission n+ Bragg mirror cavity layer contact metal p+ GaAs Semicon/air interface (a) 905 910 915 X X - X 2 x100 0.11 µA 12.0 µA electroluminescence wavelength (nm) 95.1 µ A X + (b) -40 -20 0 20 40 Time (ns) X 2 g (2) (τ) -40 -20 0 20 40 delay (ns)
(ii) X (c) (i) -10 -5 5 10 delay (ns) 0 (i) calculated (ii) measured (e) g (2) (τ) (d) -2 2 time (ns) 0 EL Fig. 5 HH HV VH VV HH HV VH VV HH HV VH VV HH HV VH VV HH HV VH VV HH HV VH VV -0.05 0.05 0.15 0.25 0.35 0.45 (c) Real Part Imaginary Part Errors (magnitude) S = 0µeV delay period (/12.5ns) degree of correlation C (a) (i) (ii) (iii) -15 0 15 S = 25µeV (b) (i) (ii) (iii) -15 0 15 0 0 0.6 0 0.6 -0.6 Fig. 6 detector beamsplitter detector device emission -40 -20 0 20 40 g (2) (τ) delay, τ [ns] 1 0 (a) (b) Fig. textbox | [
{
"section_type": "OTHER",
"section_title": "Applications of Quantum Photonics",
"text": "Applying quantum light states to photonic applications allows functionalities that are not possible using 'ordinary' classical light. For example, carrying information with single-photons provides a means to test the secrecy of optical communications, which could soon be applied to the problem of sharing digital cryptographic keys. 1 2 Although secure quantum key distribution systems based on weak laser pulses have already been realised for simple point-to-point links, true single-photon sources would improve their performance. 3 Furthermore, quantum light sources are important for future quantum communication protocols such as quantum teleportation. 4 Here quantum networks sharing entanglement could be used to distribute keys over longer distance or through more complex topologies. 5 A natural progression would be to use photons for quantum information processing, as well as communication. In this regard it is relatively straightforward to encode and manipulate quantum information on a photon. On the other hand, single-photons do not interact strongly with one-another, a prerequisite for a simple photon logic gate. In linear optics quantum computing 67 (LOQC) this problem is solved using projective measurements to induce an effective interaction between the photons. Here triggered sources of single-photons and entangled pairs are required as both the qubit carriers, as well as auxiliary sources to test the successful operation of the gates. Although the component requirements for LOQC are challenging, they have recently been relaxed significantly by new theoretical schemes. 7 Quantum light states are also likely to become increasingly important for various types of precision optical measurement. 8 For these applications we would ideally like light sources which generate pure single-photon states \"on demand\" in response to an external trigger signal. Key performance measures for such a source are the efficiency, defined as the fraction of photons collected into the experiment or application per trigger, and the second order correlation function at zero delay, see text box. The latter is essentially a measure of the two-photon rate compared to a classical source with random emission times of the same average intensity. In order to construct applications involving more than one photon, it is also important that photons emitted from the source (at different times), as well as those from different sources, are otherwise indistinguishable.\n\nIn the absence of a convenient triggered single-photon source, most experiments in quantum optics rely on non-linear optical processes for generating quantum light states. Optically pumping a crystal with a χ(2) non-linearity has a finite probability of generating a pair of lower energy photons via parametric down conversion. This may be used to prepare photon pairs with time-bin entanglement, 9 entangled polarisations, 1011 or alternatively single-photon states 'heralded' by the second photon in the pair. 12 A χ(3) non-linearity in a semiconductor has also been used to generate entangled pairs. 13 As these non-linear processes occur randomly, there is always a finite probability of generating two pairs that increases with pump power. As double pairs degrade the fidelity of quantum optical gates, the pump laser power must be restricted to reduce the rate of double pairs to an acceptable level, which has a detrimental effect upon the efficiency of the source. 14 This means that although down-conversion sources continue to be highly successful in demonstrating few photon quantum optical gates, scaling to large numbers may be problematic. Solutions have been proposed based on switching multiple sources, 15 or storing photons in a switched fibre loop. 16 Ideally we would like a quantum light source that generates exactly one single-photon, or entangled-pair, per excitation trigger pulse. This may be achieved using the emission of a single quantum system. After relaxation, a quantum system is by definition no longer excited and therefore unable to re-emit. Photon anti-bunching, the tendency of a quantum source to emit photons separated in time, was first demonstrated in the resonance fluorescence of a low density vapour of Na atoms, 17 and subsequently for a single ion. 18 2 Quantum dots are often referred to as \"artificial atoms\", as their electron motion is quantised in all three spatial directions, resulting in a discrete energy level spectrum, like that of an atom. They provide a quantum system which can be grown within robust, monolithic semiconductor devices and can be engineered to have a wide range of desired properties. In the following we review recent progress towards the realisation of a semiconductor technology for quantum photonics. An excellent account of the early work can be found in Ref. 19 . Space restrictions limit discussion of work on other quantised systems. For this we refer the reader to the comprehensive review in Ref 20 ."
},
{
"section_type": "OTHER",
"section_title": "Optical Properties of Single Quantum Dots",
"text": "Nano-scale quantum dots with good optical properties can be fabricated using a natural growth mode of strained layer semiconductors. 21 When InAs is deposited on GaAs it initially grows as a strained two-dimensional sheet, but beyond some critical thickness, tiny islands like those shown in Fig. 1a form in order to minimize the surface strain.\n\nOvergrowth of the islands leads to the coherent incorporation of In x Ga 1-x As dots into the crystal structure of the device, as can be seen in the cross-sectional image of Fig. 1c . The most intensively studied are small InAs dots on GaAs emitting around 900-950nm at low temperatures, which can be conveniently measured with low noise Si single photon detectors.\n\nA less desirable feature of the self-organising technique is that the dots form at random positions on the growth surface. However, recently considerable progress has been made on controlling the dot position (Fig. 1b ) within the device structure by patterning nanometer sized pits on the growth surface. 2223 As InGaAs has a lower energy bandgap than GaAs, the quantum dot forms a potential trap for electrons and holes. If sufficiently small, the dot contains just a few quantised levels in the conduction and valence bands, each of which holds two electrons or holes of opposite spin. Illumination by a picosecond laser pulse excites electrons and holes which rapidly relax to the lowest lying energy states either side of the bandgap. A quantum dot can thus capture two electrons and two holes to form the biexciton state, which decays by a radiative cascade, as shown schematically in Fig. 2a . One of the trapped electrons recombines with one of the holes and generates a first photon (called the biexciton photon, X 2 ). This leaves a single electron-hole pair in the dot (the exciton state), which subsequently also recombines to generate a second (exciton, X) photon. The biexciton and exciton photons have distinct energies, as can be seen in the low temperature photoluminescence spectrum of Fig. 2a , due to the different Coulomb energies of their initial and final states. Often a number of other weaker lines can also be seen due to recombination of charged excitons which form intermittently when the dot captures an excess electron or hole. 24 Larger quantum dots, with several confined electron and hole levels, have a richer optical signature due to the large number of exciton complexes that can be confined.\n\nHigh resolution spectroscopy reveals that the X 2 and X transitions of a dot are in fact both doublets with linearly polarised components parallel to the [110] and [1-10] axes of the semiconductor crystal, labelled here H and V, respectively. 2526 The origin of this polarisation is an asymmetry in the electron-hole exchange interaction of the dot which produces a splitting of the exciton spin states. The asymmetry derives from an elongation of the dot along one crystal axis and in-built strain in the crystal. It mixes the exciton eigenstates of a symmetric dot with total z-spin J z = +1 and -1 into symmetric and anti-symmetric combinations, which couple to two H or two V polarised photons, respectively, as shown in Fig. 2 .\n\nThe exciton state of the dot has a typical lifetime of ~1ns, which is due purely to radiative decay. As this is much longer than the duration of the exciting laser pulse, or the lifetime of the photo-excited carrier population in the surrounding semiconductor, only one X photon can be emitted per laser pulse. This can be proven, as first reported 27 by Peter Michler, Atac Imamoglu and their colleagues in Santa Barbara, by measuring the second order correlation function, g (2) (τ) of the exciton photoluminescence, 2829 see text box. In fact each of the exciton complexes of the dot generates at most one photon per excitation cycle, which allows single-photon emission from also the biexciton or charged exciton transitions. 30 Cross-correlation measurements 313233 between the X and X 2 photons confirm the time correlation expected for the cascade in Fig. 2a , ie the X photon follows the X 2 one. Indeed the shape of the cross-correlation function for both CW and pulsed excitation can be accurately described with a simple rate equation model and the experimentally measured X and X 2 decay rates. 34"
},
{
"section_type": "OTHER",
"section_title": "Semiconductor Microcavities",
"text": "A major advantage of using self-assembled quantum dots for single-photon generation is that they can be easily incorporated into cavities using standard semiconductor growth and processing techniques. Cavity effects are useful for 3 directing the emission from the dot into an experiment or application, as well as for modifying the photon emission dynamics. 3536 Purcell 37 predicted enhanced spontaneous emission from a source in a cavity when its energy coincides with that of the cavity mode, due to the greater density of optical states to emit into. For an ideal cavity, in which the emitter is located at the maximum of the electric field with its dipole aligned with the local electric field, the enhancement in decay rate is given by F p = (3/4π 2 ) (λ/n) 3 Q/V, where Q is the quality factor, a measure of the time a photon is trapped in the cavity, and V is the effective mode volume. Thus high photon collection efficiency, and simultaneously fast radiative decay, requires small cavities with highly reflecting mirrors and a high degree of structural perfection. However, without controlling the location of the dot in the cavity, as discussed below, it may be difficult to achieve the full enhancement predicted by the Purcell formula.\n\nFigure 3 shows images of some of the single quantum dot cavity structures that have proven most successful. Pillar microcavities, formed by etching cylindrical pillars into semiconductor Bragg mirrors placed either side of the dot layer, have shown large Purcell enhancements and have a highly directional emission profile, thus making good single-photon sources. 38394041 Purcell factors of around 6 have been measured directly, 4041 through the rate of cavity-enhanced radiative decay compared to that of a dot without cavity, implying a coupling to the cavity mode of β=F p /(1+F p )>85%, if we assume the leaky modes are unaffected by the cavity. However, the experimentally determined photon collection efficiency, which is a more pertinent parameter for applications, is typically ~10%, due the fact that not all the cavity mode can be coupled into an experiment and scattering of the mode by the rough pillar edges. We can expect that the photon collection efficiency will increase with improvements to the processing technology or new designs of microcavity.\n\nAnother means of forming a cavity is to etch a series of holes in a suspended slab of semiconductor, so as to form a lateral variation in the refractive index which creates a forbidden energy gap for photonic modes in which light cannot propagate. 42 Photons can then be trapped in a central irregularity in this structure: usually an unetched portion of the slab. Such photonic bandgap defect cavities have been fabricated in Si with Q values approaching 10 6 . 4344 High quality active cavities have also been demonstrated in GaAs containing InAs quantum dots. 45464748 A radiative lifetime of 86 ps, corresponding to a Purcell factor of F p ~12, has been reported. 47 Very recently a lifetime of 60ps was measured for a cavity in the strong coupling regeme. 48 If the Q-value is sufficiently large, the system enters the strong coupling regime where the excitation oscillates coherently between an exciton in the dot and a photon in the cavity. The spectral signature of strong coupling, an anticrossing between the dot line and the cavity mode, has been observed for quantum dots in pillar microcavities, 49 photonic bandgap defect cavities, 50 microdisks 51 and microspheres. 52 It has been demonstrated for atom cavities that strong coupling allows the deterministic generation of single-photons. 5354 Single-photon sources in the strong coupling regime can be expected to have very high extraction efficiencies and be time-bandwidth limited. 55 Encouragingly single-photon emission has been reported recently for a dot in a strongly coupled pillar microcavity. 56 Another interesting recent development is the ability to locate a single quantum dot within the cavity, as this ensures the largest possible coupling and removes background emission, as well as other undesirable effects, due to other dots in the cavity. Above we discussed techniques to control the dot position on the growth surface. The other way is to position the cavity around the dot. One technique combines micro-photoluminescence spectroscopy to locate the dot position, with in-situ laser photolithography to pattern markers on the wafer surface. 57 An alternative involves growing a vertical stack of dots so that their location can be revealed by scanning the wafer surface, 58 as shown in Fig. 3 . Recently this technique has allowed larger coupling energies for a single dot in a photonic bandgap defect cavity. 48"
},
{
"section_type": "OTHER",
"section_title": "Photon Indistinguishability",
"text": "Cavity effects are important for rendering different photons from the source indistinguishable, which is essential for many applications in quantum information. When two identical photons are incident simultaneously on the opposite input ports of a 50/50 beamsplitter, they will always exit via the same output port, 59 as shown schematically in Fig. 4a . This occurs because of a destructive interference in the probability amplitude of the final state in which one photon exits through each output port. The amplitude of the case where both photons are reflected exactly cancels with that where both are transmitted, due to the π/2 phase change upon reflection, provided the two photons are entirely identical.\n\nTwo-photon interference of two single-photons emitted successively from a quantum dot in a weakly-coupled pillar microcavity was first reported by the Stanford group. 60 Fig. 4b shows a schematic of their experiment. Notice the reduction of the co-incidence count rate measured between detectors in either output port, when the two photons are injected simultaneously (Fig. 4c ). The dip does not extend completely to zero, indicating that the two photons sometime exit the beamsplitter in opposite ports. The measured reduction in co-incidence rate at zero delay of 69%, implies an overlap for the single-photon wavepackets of 0.81, after correcting for the imperfect single-photon visibility of the 4 interferometer. Two-photon interference dips of 66% and 75% have been reported by Bennett et al 61 and Vauroutsis et al. 62 Similar results have been obtained for a single dot in a photonic bandgap defect cavity. 63 This two-photon interference visibility is limited by the finite coherence time of the photons emitted by the quantum dot, 64 which renders them distinguishable. The depth of the dip in Fig. 4c depends upon the ratio of radiative decay time to the coherence time of the dot, ie R=2τ decay /τ coh . When unity, the coherence time is limited by radiative decay and the source will display perfect 2-photon interference. The most successful approach thus far has been to extend τ coh by resonant optical excitation of the dot and reduce τ decay using the Purcell effect in a pillar microcavity, to values R~1.5. In the future higher visibilities may be achieved with a larger Purcell enhancement, using a single dot cavity in the strongcoupling regime or with electrical gating described in the next section.\n\nA source of indistinguishable single-photons was used by Fattal et al to generate entanglement between post-selected pairs. 65 66 This involves simply rotating the polarisation of one of the photons incident on the final beamsplitter in Fig. 4a by 90 o . By post-selecting the results where the two photons arrive at the beamsplitter at the same time and where there is one photon in each output arm (labelled 1 and 2), the measured pairs should correspond to the Bell state\n\nψ -= 1/√2 (¦H 1 V 2 > -¦V 1 H 2 >)\n\nEq.1 Note that only if the two photons are indistinguishable and thus the entanglement is only in the photon polarisation, are the two terms in Eq1 able to interfere. Analysis of the density matrix published by Fattal et al 65 reveals a fidelity of the post-selected pairs to the state in Eq.1 of 0.69, beyond the classical limit of 0.5. This source of entangled pairs has an importance difference to that based on the biexciton cascade described below. Post-selection implies that the photons are destroyed when this scheme succeeds. This is a problem for some quantum information applications such as LOQC, but could be usefully applied to quantum key distribution. 65"
},
{
"section_type": "OTHER",
"section_title": "Single-Photon LEDs",
"text": "An early proposal for an electrical single-photon source by Kim et al 67 was based upon etching a semiconductor heterostructure displaying Coulomb blockade. However, the light emission from this etched structure was too weak to allow the second-order correlation function to be studied. Recently encouraging progress has been made towards the realisation of a single-photon source based on quantising a lateral electrical injection current. 6869 However the most successful approach so far has been to integrate self-assembled quantum dots into conventional p-i-n doped junctions.\n\nIn the first report of electrically-driven single-photon emission by Yuan et al, 70 the electroluminescence of a single dot was isolated by forming a micron-diameter emission aperture in the opaque top contact of the p-i-n diode. Fig. 5a shows an improved emission aperture single-photon LED after Bennett et al, 71 which incorporates an optical cavity formed between a high reflectivity Bragg mirror and the semiconductor/air interface in the aperture. This structure forms a weak cavity, which enhances the measured collection efficiency 10-fold compared to devices without a cavity. 72 Single-photon pulses are generated by exciting the diode with a train of short voltage pulses. The second order correlation function g (2) (τ) of either the X or X 2 electroluminescence (Fig. 5c ) shows the suppression of the zero delay peak indicative of single-photon emission. 71 The finite rate of multi-photon pulses is due mostly to background emission from layers other than the dot, which is also seen for non-resonant optical excitation. Electrical contacts also allow the temporal characteristics of the single-photon source to be tailored. By applying a negative bias to the diode between the electrical injection pulses, Bennett et al 73 reduced the jitter in the photon emission time <100ps. This allowed the repetition rate of the single-photon source to be increased to 1.07GHz (Fig. 5d ) while retaining good singlephoton emission characteristics (Fig. 5e ). Electrical gating could provide a technique for producing time-bandwidthlimited single-photons from quantum dots. Another promising approach is to aperture the current flowing through the device. 7475 This is achieved by growing a thin AlAs layer within the intrinsic region of the p-i-n junction and later exposing the mesa to wet oxidation in a furnace, converting the AlAs layer around the outer edge of the mesa to insulating Aluminium oxide. By careful control of the oxidation time, a µm-diameter conducting aperture can be formed within the insulating ring of AlOx. Such structures have the advantage of exciting just a single dot within the structure, thereby reducing the amount of background emission. The oxide annulus also confines the optical mode laterally within the structure, potentially allowing high photon extraction efficiency.\n\nAltering the nanostructure or materials that comprise the quantum dot allows considerable control over the emission wavelength and other characteristics. Most of the experimental work done so far has concentrated on small InAs quantum dots emitting around 900-950nm, as these have well understood optical properties and can be detected with 5 low noise Si single-photon detectors. On the other hand the shallow confinement potentials of this system means they emit only at low temperatures. At shorter wavelengths optically-pumped single-photon emission has been demonstrated at ~350nm using GaN/AlGaN, 76 500nm using CdSe/ZnSSe 77 and 682nm InP/GaInP 78 quantum dot. The former two systems have been shown to operate at 200K. It is very important for quantum communications to develop sources at longer wavelengths in the fibre optic transmission bands at 1.3 and 1.55µm. This may be achieved using InAs/GaAs heterostructures by depositing more InAs to form larger quantum dots. These larger dots offer deeper confinement potentials than those at 900nm and thus often display room temperature emission. 79 Optically pumped single-photon emission at telecom wavelengths has been achieved using a number of techniques to prepare low densities of longer wavelength dots, including a bimodal growth mode in MBE to form low densities of large dots, 80 ultra-low growth rate MBE 81 and MOCVD. 82 Recently, the first electrically-driven single-photon source at a telecom wavelength has been demonstrated. 83"
},
{
"section_type": "OTHER",
"section_title": "Generation of Entangled Photons",
"text": "By collecting both the X 2 and X photons emitted by the biexciton cascade, a single quantum dot may also be used as a source of photon pairs. Polarisation correlation measurements on these pairs discovered that the two photons were classically-correlated with the same linear polarisation. 848586 This occurs because the cascade can proceed via one of two intermediate exciton spin states, as described above and shown in Fig. 2a , one of which couples to two H-and the other two V-polarised photons. The emission is thus a statistical mixture of |H X2 H X > and |V X2 V X >, although exciton spin scattering during the cascade (discussed below) ensures there are also some cross-polarised pairs.\n\nThe spin splitting 87,88 of the exciton state of the dot distinguishes the H and V polarised pairs and prevents the emission of entangled pairs predicted by Benson et al. 89 If this splitting could be removed, the H and V components would interfere in appropriately designed experiments. The emitted 2-photon state should then be written as a superposition of HH and VV, which can be recast in either the diagonal (spanned by D, A) or circular (σ + , σ -) polarisation bases, ie\n\nΦ + = 1/√2 (¦H X2 H X > + ¦V X2 V X >) = 1/√2 (¦D X2 D X > + ¦Α X2 Α X >) = 1/√2 (¦σ + X2 σ - X > + ¦σ + X2 σ - X >) Eq.2.\n\nEqual weighting of the HH and VV terms assumes the source to be unpolarised, as indicated by experimental measurements.\n\nEq.2 suggests that, for zero exciton spin splitting, the biexciton cascade generates entangled photon pairs, similar to those seen for atoms. 90 Entanglement of the X or X 2 photons was recently observed experimentally for the first time by Stevenson, Young and co-workers, 9192 using two different schemes to cancel the exciton spin splitting. An alternative approach by Akopian et al, 93 using dots with finite exciton splitting, post-selects photons emitted in a narrow spectral band where the two polarisation lines overlap.\n\nThe exciton spin splitting depends on the exciton emission energy, tending to zero for InAs dots emitting close to 1.4eV and then inverting for higher emission energy. 94 95 These correspond to shallow quantum dots for which the carrier wavefunctions extend into the barrier material reducing the electron-hole exchange. Zero splitting can be achieved by either careful control of the growth conditions to achieve dots emitting close to the desired energy, or by annealing samples emitting at lower energy. 94 The exciton spin splitting may be continuously tuned by applying a magnetic field in the plane of the dot. 96 It has been observed that the signatures of entanglement then appear only when the exciton splitting is close to zero. 91 Other promising schemes to tune the exciton splitting are now emerging, including application of strain 97 and electric field. 9899 Figure 6a plots polarisation correlations reported by Young et al 92 for a dot with zero exciton splitting (by control of the growth conditions). Pairs emitted in the same cascade (ie zero delay) shows a very striking positive correlation (copolarisation) measuring in either, rectilinear or diagonal bases and anti-correlation (cross-polarisation) when measuring in circular basis. This is exactly the behaviour expected for the entangled state of Eq.2. In contrast, a dot with finite splitting shows polarisation correlation for the rectilinear basis only, with no correlation for diagonal or circular measurements, see Figure 6b . The strong correlations seen for all three bases in Fig. 6a could not be produced by any classical light source or mixture of classical sources and is proof that the source generates entangled photons. The measured 92 two-photon density matrix (Fig. 6c ) projects onto the expected\n\n1/√2 (¦H X2 H X > + ¦V X2 V X >) state\n\nwith fidelity (ie probability) 0.702 ± 0.022, exceeding the classical limit (0.5) by 9 standard deviations.\n\nTwo processes contribute to the 'wrongly' correlated pairs which impair the fidelity of the entangled photon source. The first of these is due to background emission from layers in the sample other than the dot. This background emission, which is unpolarised and dilutes the entangled photons from the dot, limited the fidelity observed in the first report 91 of triggered entangled photon pairs from a quantum dot and has been subsequently reduced with better sample design. 92 The second mechanism, which is an intrinsic feature of the dot, is exciton spin scattering during the biexciton cascade. It is interesting that this process does not seem to depend strongly upon the exciton spin splitting. It may be reduced by suppressing the scattering using resonant excitation or alternatively using cavity effects to reduce the time required for the radiative cascade."
},
{
"section_type": "OTHER",
"section_title": "Outlook",
"text": "The past several years have seen remarkable progress in quantum light generation using semiconductor devices. However, despite considerable progress many challenges still remain. The structural integrity of cavities must continue to improve, thereby enhancing quality factors. This, combined with the ability to reliably position single dots within the cavity, will further enhance photon collection efficiencies and the Rabi energy in the strong coupling regime. It is also important to realise all the benefits of these cavity effects in more practical electrically-driven sources. Meanwhile bandstructure engineering of the quantum dots will allow a wider range of wavelengths to be accessed for both single and entangled photon sources, as well as structures that can operate at higher temperatures. Techniques for fine tuning the characteristics of individual emitters will also be important.\n\nOne of the most interesting aspects of semiconductor quantum optics is that we may be able to use quantum dots not only as quantum light emitters, but also as the logic and memory elements which are required in quantum information processing. Although LOQC is scalable theoretically, quantum computing with photons would be much easier with a useful single-photon non-linearity. Such non-linearity may be achieved with a quantum dot in a cavity in the strong coupling regime. Encouragingly strong coupling of a single quantum dot with various type of cavity has already been observed in the spectral domain. Eventually it may even be possible to integrate photon emission, logic, memory and detection elements into single semiconductor chips to form a photonic integrated circuit for quantum information processing.\n\nThe author would like to thank Mark Stevenson, Robert Young, Anthony Bennett, Martin Ward and Andy Hudson for their useful comments during the preparation of the manuscript and the UK DTI \"Optical Systems for Digital Age\", EPSRC and EC Future and Emerging Technologies programmes for supporting research on quantum light sources.\n\n7"
},
{
"section_type": "OTHER",
"section_title": "TextBox : Photon Correlation Measurements",
"text": "The photon statistics of light can be studied via the second order correlation function, g (2) (τ), which describes the correlation between the intensity of the light field with that after a delay τ and is given by 100 2 ) 2 ( ) ( ) ( ) ( ) ( > < > + < = t I t I t I g τ τ\n\nThis function can be measured directly using the Hanbury-Brown and Twiss 101 interferometer, comprising a 50/50 beamsplitter and two single-photon detectors, shown in the figure . For delays much less than the average time between detection events (ie for low intensities), the distribution in the delays between clicks in each of the two detectors is proportional to g (2) (τ).\n\nFor a continuous light source with random emission times, such as an ideal laser or LED, g (2) (τ)=1. It shows there is no correlation in the emission time of any two photons from the source. A source for which g (2) (τ=0)>1 is described as 'bunched' since there is an enhanced probability of two photons being emitted within a short time interval. Photons emitted by quantum light sources are typically 'anti-bunched', (g (2) (τ=0)<1) and tend to be separated in time.\n\nIn communication and computing systems, we are interested in pulsed light sources, for which the emission occurs at times defined by an external clock. In this case g (2) (τ) consists of a series of peaks separated by a clock period. For an ideal single-photon source, the peak at zero time delay is absent, g (2) (τ=0)=0; as the source cannot produce more than one photon per excitation period, clearly the two detectors cannot fire simultaneously.\n\nThe figure shows g (2) (τ) recorded for resonant pulsed optical excitation of the X emission of a single quantum dot in a pillar microcavity. Notice the almost complete absence of the peak at zero delay: the definitive signature of a singlephoton source. The weak peak seen at τ=0 demonstrates that the rate of two-photon emission is 50 times less than that of an ideal laser with the same average intensity. The bunching behaviour observed for the finite delay peaks is explained by intermittent trapping of a charge carrier in the dot. 102 This trace was taken for quasi-resonant laser excitation of the dot which avoids creating carriers in the surrounding semiconductor. For higher energy laser excitation, the suppression in g (2) (0) is typically reduced indicating occasional 2-photon pulses due to emission from the layers surrounding the dot, but can be minimised with careful sample design.\n\nFigure textbox: (a) Schematic of the set-up used for photon correlation measurements, (b) second order correlation function of the exciton emission of a single dot in a pillar microcavity."
},
{
"section_type": "OTHER",
"section_title": "Figure Captions",
"text": "Figure 1: Self assembled quantum dots (a) Image of a layer of InAs/GaAs self assembled quantum dots recorded in an Atomic Force Microscope (AFM). Each yellow blob corresponds to a dot with typical lateral diameters of 20-30nm and a height of 4-8nm. (b) AFM image 23 of a layer of InAs quantum dots whose locations have been seeded by a matrix of nanometer sized pits patterned onto the wafer surface. Under optimal conditions up to 60% of the etch pits contain a single dot (Courtesy of P Atkinson & D A Ritchie, Cambridge). (c) Cross-sectional STM image of an InAs dot inside a GaAs device (Courtesy of P. Koenraad, Eindhoven).\n\nFigure 2: Optical spectrum of a quantum dot. (a) Schematic of the biexciton cascade of a quantum dot. (b) Typical photoluminescence spectrum of a single quantum dot showing sharp line emission due to the biexciton X 2 and exciton X photon emitted by the cascade. The inset shows the polarisation splitting of the transitions originating from the spin splitting of the exciton level.\n\nFigure 3: SEM images of semiconductor cavities, including pillar microcavities (a) 56 and (b), microdisk (c) 51 and photonic bandgap defect cavities (d) 47 , (e) and (f). 48 (Structures fabricated at Univ Wuerzburg (a), CNRS-LPN (UPR-20), Marcoussis (b, c, e), Univ Cambridge (d), UCSB/ETHZ Zurich (f)) Figure 4: Two Photon Interference. (a) If the two photons are indistinguishable, the two outcomes resulting in one photon in either arm interfere destructively. This results in the two photons always exiting the beamsplitter together. (b) Schematic of an experiment using two photons emitted successively from a quantum dot, (c) experimental data showing suppression of the co-incidence rate in (b) when the delay between input photons is zero due to two-photon interference. 60 (Courtesy of Y Yamamoto, Stanford Univ.) Figure 5: Electrically driven single-photon emission. (a) Schematic of a single-photon LED. (b) Electroluminescence spectra of the device. Notice the spectra are dominated by the exciton X and biexciton X 2 lines, which have linear and quadratic dependence on drive current, respectively. Other weak lines are due to charged excitons. (c) second order correlation function recorded for the exciton (i) and biexciton (ii) emission lines, (d) time-resolved electroluminescence from a device operate with a 1.07GHz repetition rate, (e) measured (i) and modelled (ii) second order correlation function of the biexciton electroluminescence at 1.07GHz. (adapted from Refs. 71and 73) Figure 6: Generation of entangled photons by a quantum dot. (a) Degree of correlation measured for a dot with exciton polarisation splitting S=0 µeV in linear (i), diagonal (ii) and circular (iii) polarisation bases as a function of the delay between the X and X 2 photons (in units of the repetition cycle). The correlation is defined as the rate of co-polarised pairs minus the rate of cross-polarised pairs divided by the total rate. Notice that the values at finite delay show no correlation, as expected for pairs emitted in different laser excitation cycles. More interesting are the peaks close to zero time delay, corresponding to X and X 2 photon emitted from the same cascade. The presence of strong correlations for all three types of measurement for the dot with zero exciton splitting can only be explained if the X and X 2 polarisations are entangled. (b) Degree of correlations measured for the dot in (a) subject to in-plane magnetic field so as to produce an exciton polarisation splitting of S=25 µeV. Notice that the correlation in diagonal and circular bases have vanished, indicating only classical correlations at finite splitting. (c) Two-photon density matrix of the device emission in (a). The strong off-diagonal terms appear due to entanglement. 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Submicrosecond correlations in photoluminescence from InAs quantum dots. Phys. Rev. B 69, 205324 (2004).\n\n(c) 10nm (a) 500 nm 500 nm (b) Fig. 1 1375 1380 1385 Detection polarisation: Vertical Horizontal X 2 X X 2 X PL Intensity Photon Energy (meV) 1378.0 1378.5 1380.0 1380.5 (b) (a) + -X X 2 -+ + -H H V V Fig. 2 S ground state S S 500 nm 500nm (d) (c) (a) (b) (e) (f) Fig. 3 or (a) (b) (c) (c) Fig. 4 substrate/buffer n-ohmic contact InAs QD insulator Al p-ohmic contact emission n+ Bragg mirror cavity layer contact metal p+ GaAs Semicon/air interface (a) 905 910 915 X X - X 2 x100 0.11 µA 12.0 µA electroluminescence wavelength (nm) 95.1 µ A X + (b) -40 -20 0 20 40 Time (ns) X 2 g (2) (τ) -40 -20 0 20 40 delay (ns)\n\n(ii) X (c) (i) -10 -5 5 10 delay (ns) 0 (i) calculated (ii) measured (e) g (2) (τ) (d) -2 2 time (ns) 0 EL Fig. 5 HH HV VH VV HH HV VH VV HH HV VH VV HH HV VH VV HH HV VH VV HH HV VH VV -0.05 0.05 0.15 0.25 0.35 0.45 (c) Real Part Imaginary Part Errors (magnitude) S = 0µeV delay period (/12.5ns) degree of correlation C (a) (i) (ii) (iii) -15 0 15 S = 25µeV (b) (i) (ii) (iii) -15 0 15 0 0 0.6 0 0.6 -0.6 Fig. 6 detector beamsplitter detector device emission -40 -20 0 20 40 g (2) (τ) delay, τ [ns] 1 0 (a) (b) Fig. textbox"
}
] | |
arxiv:0704.0409 | 0704.0409 | 1 | 10.1103/PhysRevA.76.032114 | 1272817a2a40683f12f237ad18c903107c8e6e73ef9bf6f87210091b7dfac177 | On the over-barrier reflection in quantum mechanics with multiple
degrees of freedom | We present an analytic example of two dimensional quantum mechanical system,
where the exponential suppression of the probability of over-barrier reflection
changes non-monotonically with energy. The suppression is minimal at certain
"optimal" energies where reflection occurs with exponentially larger
probability than at other energies. | [
"D.G. Levkov",
"A.G. Panin",
"S.M. Sibiryakov"
] | [
"quant-ph",
"hep-th",
"nlin.CD",
"physics.atom-ph",
"physics.chem-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Tunneling and over-barrier reflection are the characteristic non-perturbative phenomena in quantum mechanics. They typically occur with exponentially small probabilities,
P ∝ e -F/ , ( 1
)
where F is the suppression exponent; still, the above phenomena are indispensable in understanding a wide variety of physical situations, from the generation of baryon number asymmetry in the early Universe [1] to chemical reactions [2] and atom ionization processes [3]. During the last decades extensive investigations of tunneling processes in systems with many degrees of freedom have been performed [2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] . These studies 1 levkov@ms2.inr.ac.ru 2 panin@ms2.inr.ac.ru 3 Sergey.Sibiryakov@cern.ch, sibir@ms2.inr.ac.ru 1 revealed a rich variety of features of multidimensional tunneling which are in striking contrast to the properties of one-dimensional tunneling and over-barrier reflection. In particular, the following phenomenon has been observed: the probability of tunneling may depend nonmonotonically on the total energy of the system and exhibit resonance-like peaks. One can envisage three physically different mechanisms of this phenomenon. The first mechanism, present already in one-dimensional case, is tunneling via creation of a metastable state. In this case the tunneling probability at the maximum of the resonance is exponentially higher than at other energies. On the other hand, the resonance width ∆E is exponentially suppressed; so, after averaging with an energy distribution of a finite width the effect of the resonance is washed out in the semiclassical limit → 0. The second possible mechanism of non-monotonic behavior of P(E) is quantum interference [7, 13] (see also [14] ). In this case the peak value of the tunneling probability is only by a factor of order one higher than the average value, while the width of the resonance scales as ∆E ∝ . Again, the resonances become indiscernible in the semiclassical limit. In both these cases the resonances can be attributed to the subleading semiclassical corrections, i.e. non-monotonic behavior of the pre-exponential factor omitted in Eq. ( 1 ). The third possibility is that the suppression exponent F (E) is non-monotonic. In this case the existence of the "resonances" is the leading semiclassical effect: the optimal tunneling probability at the maximum of the resonance is exponentially higher than the probability at other energies. At the same time the resonance width scales as 4 ∆E ∝ √ . This last possibility of "optimal tunneling" is definitely of interest; yet, it did not receive much attention in literature. We are aware of only a few works mentioning non-monotonic dependence of the suppression exponent on energy [15, 16, 14] .
It is worthwhile studying this phenomenon in detail; this can provide a new insight into the dynamics of multidimensional tunneling.
In this paper we consider the process of over-barrier reflection in a simple model with two degrees of freedom. Our setup is interesting in two respects. First, the model under study is essentially non-linear and the variables cannot be separated; still, over-barrier reflections in this model can be described analytically within the semiclassical framework. Thus, this model can serve as an analytic laboratory for the study of multidimensional tunneling. Second, the suppression exponent F of the reflection process behaves non-monotonically as the 4 This follows from the representation
P(E) ∝ exp - F (E o ) - F ′′ (E o )(E -E o ) 2 2
of the tunneling probability in the vicinity of the maximum.
2 total energy E changes. We demonstrate that the function F (E) possesses a number of local minima E = E o , where reflection is optimal. We stress that the process we study is exponentially preferable at "optimal" energies as compared to other energies. Our model describes the motion of a quantum particle in the two dimensional harmonic waveguide (see Refs. [8, 10, 14] for similar models). The Hamiltonian is
H = p 2 x 2m + p 2 y 2m + mω 2 2 w 2 (x, y) ,
where x, y are the Cartesian coordinates and m is the mass of the particle. The function U = mω 2 w 2 /2 represents the waveguide potential in two dimensions: a particle with small energy is bound to move along the line w(x, y) ≈ 0. We do not introduce a potential barrier across the waveguide and consider the case when the line w = 0 stretches all the way from x → -∞ to x → +∞. We also assume that the function w(x, y) is linear in the initial asymptotic region, w(x, y) → y as x → -∞ .
In the present paper we consider two particular cases of the function w(x, y) describing waveguides with one and two sharp turns 5 , see Fig. 1. The motion of the particle at x → -∞ is a superposition of free translatory motion in x direction and oscillations of frequency ω along y coordinate; the state of such a particle is fully characterized by two quantum numbers, the total energy E and y-oscillator excitation number N. The particle sent into the waveguide from the asymptotic region x → -∞ with given E, N may either continue to move towards x → +∞, or reflect back into the region x → -∞. We are interested in the probability P(E, N) of reflection.
Let us discuss reflections at the classical level. [Note that the classical counterpart of N is the energy of transverse oscillations.] Consider first the waveguide with one sharp turn (Fig. 1a ). One observes that the outcome of the classical evolution, i.e. whether or not the particle reflects from the turn, depends not only on the total energy E, but on other dynamical quantities as well. In particular, the direction of the momentum of the particle in the vicinity of the turn (point C on the graph) is important. This means that the entire dynamics in the waveguide should be taken into account in order to determine the possibility of classical reflection. This is in sharp contrast with the situation in one-dimensional case, where reflection from the potential barrier (or transition through it) is ensured by the value of the conserved energy of the particle.
The explicit expressions for the waveguide functions w(x, y) will be presented in the subsequent sections.
(a) C U=0 U=E U=E x y (b) L C C' U=E U=E U = 0 Figure 1: The equipotential contour U = E for the waveguides with (a) one and (b) two sharp turns. An example of classical trajectory is shown in the case (b).
Now, consider the waveguide with two turns. The model is characterized by the angles of the turns and the distance L between them (see Fig. 1b ). Suppose the particle starts moving classically from x → -∞ with N = 0 along the valley w = 0. Then, the transverse oscillations get excited only after the particle crosses the first turn, point C ′ on the plot, so that at the time of arrival to the second turn (point C) approximately ωτ /2π oscillations are made, where τ ∼ L m/2E is the time of motion between the two turns. The state of the particle (coordinates and momenta) at which it comes across the second turn depends periodically on the phase of transverse oscillations ωτ . Hence, one expects that the regime of motion of the classical particle can change from transmission to reflection and back as the energy grows (τ decreases); the energies where it happens can be roughly estimated as
E n ∼ mω 2 L 2 2(2πn) 2 . ( 2
)
We will see that this is indeed the case for the waveguides with certain angles of the turns. At some values of E, N the reflection process cannot proceed classically. Then, at the quantum mechanical level its probability is exponentially suppressed, F (E, N) > 0. It is natural to call such a process "over-barrier reflection" 6 . The central quantity to be studied below is the suppression exponent F (E, N) of this process. The above discussion suggests that F (E, N), being determined by the entire dynamics in the waveguide, may be a highly non-trivial function. For the particular case of the waveguide with alternating regimes of 6 By this term we want to emphasize that the process is classically forbidden. Recall, however, that there is no actual potential barrier across the waveguide in our setup.
4 classical reflections and transmissions F should oscillate: F = 0 at the energies where the classical reflections are allowed, and F > 0 at the energies where the reflections are classically forbidden. One can expect that the similar oscillatory behavior of the suppression exponent persists for other two-turn models as well. Now, instead of reaching zero, F may possess a number of local positive minima implying that the reflection at the "optimal" energies is still a tunneling process.
Let us emphasize the difference of the "optimal tunneling" from quantum interference and resonance phenomena in our two-turn model. The interference of the de Broglie waves reflected from the two turns can, in principle, lead to oscillations in the reflection probability P(E). One can estimate the positions of the interference peaks by equating the De Broglie wavelength of the particle to an integer fraction of the distance between the turns, 2π / √ 2mE ∼ L/n. This yields the energies of the interference peaks, E int n ∼ (2πn) 2 2 2mL 2 .
This formula is completely different from Eq. ( 2 ) for the peaks due to "optimal tunneling". In particular, the distance between the adjacent inteference peaks, ∆E int ∼ 2π L 2E m , scales proportional to . Thus, these peaks should be averaged over in the semiclassical limit. Besides, the amplitude of the interference peaks is at most of order one and does not affect the suppression exponent. Indeed, the exponential increase of the scattering amplitude can arise due to quantum interference only in the presence of a resonant state with exponentially long life-time. This state should be supported somewhere in between the turns and should be classically stable. In Sec. 4.2 we show that such states are absent in our system. One concludes that the peak-like structure of the probability P(E) of "optimal tunneling" is caused by completely different physical reasons as compared to the case of resonance scattering in quantum theory.
It is worth noting that the phenomenon of "optimal tunneling" has an important implementation in field theory. Recently it was argued [17] (see also Ref. [16] ) that the probability of tunneling induced by particle collisions [18, 19] reaches its maximum at a certain "optimal" energy and stays constant 7 at higher energies. This result, if generic, provides the 7 As opposed to the quantum mechanical case, the tunneling probability does not decrease at energies higher than the "optimal" one. This is due to the possibility, specific to the field theoretical setup, to emit the excess of energy into a few hard particles, so that tunneling effectively occurs at the "optimal" energy. [20] about the high-energy behavior of the probability of collision-induced nonperturbative transitions in field theory. The quantum mechanical model presented here supports the generic nature of the phenomenon of "optimal tunneling"; the simplicity of our model enables one to get an intuitive insight into the nature of this phenomenon.
The paper is organized as follows. In Sec. 2 we review the semiclassical method of complex trajectories, which is exploited in the rest of the paper. Reflections in the waveguides with one and two turns are considered in Secs. 3 and 4 respectively. We discuss our results in Sec. 5. In appendix we analyze the validity of some assumptions made in the main body of the paper.
We start by describing the semiclassical method 8 of complex trajectories which will be used in the study of over-barrier reflections. We concentrate on the derivation of the formula for the suppression exponent F (E, N) (see Refs. [2, 8, 9] for the details of the method and Ref. [19] for the field theory formulation). In what follows we use the system of units
= m = ω = 1 ,
where the Hamiltonian takes the form,
H = 1 2 p 2 x + p 2 y + w 2 (x, y) . ( 3
)
One starts with the amplitude of reflection into the state with definite coordinates
x f < 0 , y f , A = x f , y f |e -i Ĥ(t f -t i ) |E, N . ( 4
)
Here |E, N is the initial state of the particle moving in the asymptotic region x i → -∞ with fixed translatory momentum p 0 = 2(E -N) and the oscillator excitation number N. Semiclassically,
x i , y i |E, N = e ip 0 x i cos y i √ 2N p y (y ′ )dy ′ + π/4 , ( 5
)
8 Note that the method has been confirmed by the explicit comparison with the exact quantum mechanical results in Refs. [8, 9, 14] ; specifically, the recent check [14] deals with the case when the dependence of the suppression exponent on energy is not monotonic.
where x i , y i denote initial coordinates,
p y (y ′ ) = 2N -y ′2 , ( 6
)
and we omitted the pre-exponential factor which is irrelevant for our purposes. Using Eq. ( 5 ), one rewrites the amplitude (4) as a path integral,
A = dx i dy i [dx][dy] x f , y f x i , y i e iS+ip 0 x i cos y i √ 2N p y (y ′ )dy ′ + π/4 , ( 7
)
where S is the classical action of the model (3).
In the semiclassical case the integral (7) is dominated by the (generically complex) saddle point. Note that, as we continue the integrand in Eq. ( 7 ) into the plane of complex coordinates, one of the exponents constituting the initial oscillator wave function grows, while the other becomes negligibly small. Within the validity of our approximation, we omit the decaying exponent by writing
cos y i √ 2N p y (y ′ )dy ′ + π/4 → exp i y i √ 2N p y (y ′ )dy ′ , ( 8
)
with the standard choice 9 of the branch of the square root in Eq. ( 6 ). One proceeds by finding the saddle point for the integral (7) with the substitution (8). Extremization with respect to x(t), y(t) leads to the classical equations of motion,
ẍ = -ww x , ÿ = -ww y . ( 9
)
Differentiating with respect to
x i ≡ x(t i ), y i ≡ y(t i ), one obtains, ẋi = p 0 = 2(E -N) , ẏi = p y (y i ) = 2N -y 2 i .
The latter equations are equivalent to fixing the total energy E and initial oscillator energy N of the complex trajectory,
E = 1 2 ẋ2 i + N , ( 10a
) N = 1 2 ẏ2 i + y 2 i . ( 10b
)
9 The correct branch is fixed by drawing a cut between the oscillator turning points y = ± √ 2N , and choosing Im p y > 0 at y ∈ R, y > √ 2N , see, e.g., Refs. [21].
Substituting the saddle-point configuration 10 into Eq. ( 7 ), one obtains the amplitude of the process with exponential accuracy, A ∝ e iS+iB(x i , y i ) ,
where the term
B(x i , y i ) = p 0 x i + y i √ 2N p y (y ′ )dy ′ ( 11
)
is the initial-state contribution. For the inclusive reflection probability one writes,
P = dx f dy f |A| 2 ∝ dx f dy f
e iS-iS * +iB-iB * .
The integral over the final states can also be evaluated by the saddle point technique; extremization with respect to x f ≡ x(t f ), y f ≡ y(t f ) fixes the boundary conditions in the asymptotic future,
Im ẋf = Im x f = 0 , Im ẏf = Im y f = 0 . ( 12
)
In this way one obtains the expression (1) for the reflection probability, where the suppression exponent F is given by the value of the functional
F (E, N) = 2 Im S + 2 Im B(x i , y i )
evaluated on the saddle-point configuration -a complex trajectory satisfying the boundary value problem (9), (10), (12). The contribution B(x i , y i ) of the initial state is simplified after one uses the asymptotic form of the solution at t → -∞ (x i → -∞),
x = p 0 t + x 0 , y = ae -it + āe it . ( 13
)
Equations (10) guarantee that the quantities p 0 = 2(E -N) and 2aā = N are real, since E, N ∈ R. Therefore, one may introduce two real parameters T , θ as follows,
2 Im x 0 = -p 0 T , ā = a * e T +θ . ( 14
)
One finds for the initial term (11),
2 Im B(x i , y i ) = Im 2p 0 x i -2Narccos(y i / √ 2N) + y i 2N -y 2 i = -p 2 0 T -N(T + θ) + Im(y i ẏi ) ,
10 For simplicity we assume that the saddle-point configuration is unique. Otherwise, one should take the saddle point corresponding to the weakest exponential suppression.
8 and thus
F = 2 Im S -ET -Nθ , ( 15
)
where S is the classical action of the system (3) integrated by parts, S = -1 2
t f t i dt xẍ + y ÿ + w 2 (x, y) . ( 16
)
Let us comment on the physical meaning of the parameters T , θ. Consider two trajectories which are solutions to the boundary value problem (9), (10), (12) at neighbouring values of E, N. The differential of the quantity 2 Im S as one deforms one trajectory into the other is
d (2 Im S) = d Im(2S + x i ẋi + y i ẏi ) = Im(x i d ẋi -ẋi dx i + y i d ẏi -ẏi dy i ) = EdT + Ndθ ,
where in the last equality we used the asymptotic form (13), (14) of the solution. Then, from Eq. (15) one finds, dF (E, N) = -T dE -θdN . (17) Thus, the parameters T and θ are (up to sign) the derivatives of the suppression exponent with respect to energy E and initial oscillator excitation number N respectively. Our final remark is that the boundary value problem (9), (10), (12) is invariant with respect to the trivial time translation symmetry,
t → t + δt , δt ∈ R , ( 18
)
which can be fixed in any convenient way.
3 The model with one turn
To warm up, we consider the simplest model, where the waveguide has one sharp turn,
w = y θ(-x + y tg β) + cos β (x sin β + y cos β) θ(x -y tg β) . ( 19
)
Here θ(x) is the step function. It is convenient to use the rotated coordinate system, ξ η = cos β -sin β sin β cos β x y .
The waveguide function takes the form,
w = η cos β -ξ sin β θ(-ξ) . ( 20
) 9 x y ξ η A B C β β x y ξ η A B C β β x y ξ η A B C β β
Figure 2: The equipotential contour w 2 (x, y) = 2N for the waveguide (20) and the trajectory of the critical solution with energy N/ cos 2 β.
The equipotential contour w 2 (ξ, η) = const is shown in Fig. 2 . One observes that the motion of the particle in two regions, ξ < 0 and ξ > 0, decomposes into the translatory motion and oscillations in the coordinates x, y and ξ, η respectively (see. Eqs. (19) and (20)); the frequency of η-oscillations in the latter case is cos β. Due to the presence of the step function, the first derivatives of the potential (20) are discontinuous 11 at ξ = 0. Strictly speaking, the semiclassical method is not applicable in this situation [21] . Thus, the formula (20) should be regarded as an approximation to some waveguide function with smooth turn. Generically the width of the smoothened turn is characterized by a parameter b; the sharp-turn approximation (20) corresponds to b → 0. An example of smoothening is provided by the following substitution in Eq. (20),
θ(ξ) → θ b (ξ) = 1 1 + e -ξ/b . ( 21
)
The semiclassical description can be used as long as the de Broglie wavelength of the particle is small compared to the linear size of the potential 12 , 1/ √ E ≪ b. We conclude that the sharp-turn and semiclassical approximations are valid simultaneously for smooth waveguides with
1 ≫ b ≫ 1/ √ E . ( 22
)
11 Note that the potential itself is continuous. 12 Another semiclassical condition is that the energy is sufficient to excite a lot of oscillator levels, E ≫ 1.
It is satisfied provided Eq. (22) holds.
10 An important property of the model (20) is invariance of the classical equations of motion (9) under the rescaling of the coordinates,
x → Λx , y → Λy . ( 23
)
Using the transformation (23), one may express a solution x(t), y(t) with energy E in terms of the "normalized" one, x = x√ E , y = ỹ√ E ,
where the solution x(t), ỹ(t) has unit energy; its initial oscillator excitation number is ν = N/E .
The suppression exponent (15) takes the form,
F (E, N) = Ef β (ν) , ( 24
)
where f β (ν) is the exponent for the "normalized" solution. Substituting the expression (24) into Eq. ( 17 ), one obtains,
f β (ν) = -T -θν . ( 25
)
We will exploit Eq. ( 25 ) in the end of this section. Now, we proceed to finding the "normalized" trajectories. At certain initial data ν > ν cr the particle can reflect from the turn classically, so that
f β (ν > ν cr ) = 0 .
Let us find the value of ν cr . In the region ξ < 0 the classical solution takes the form,
x(t) = p 0 t + x 0 , ( 26a
) y(t) = A 0 sin(t + ϕ) . ( 26b
)
Having crossed the line ξ = 0 (line AB in Fig. 2 ), the classical particle can never return back into the region ξ < 0. Indeed, in this case it moves at ξ > 0 with constant momentum p ξ > 0. Thus, the particle can reflect classically only if its trajectory touches the line ξ = 0. The potential of our model has ill-defined derivatives at ξ = 0, and the fate of the particle moving along the line AB depends on the particular choice of the smoothening of the potential. In appendix we consider the motion of the classical particle in the case when nonzero smoothening of width b is switched on. For a class of smoothenings we show that 11 in the small vicinity (δξ ∼ b) of any trajectory touching the line ξ = 0 there exists some "smoothened" trajectory, which reflects classically from the turn. Consequently, below we associate the trajectories touching the line ξ = 0 with the classical reflected solutions. One notices that the inclination of the trajectory (26) is bounded from above
dy dx ≤ A 0 p 0 ;
therefore, the classical trajectory of the particle can touch the line ξ = 0, that is, y/x = ctg β only at
A 0 /p 0 ≥ ctg β . ( 27
)
From Eqs. (27), (26), (10) one extracts the condition for the particle to reflect classically from the turn,
ν ≥ ν cr = cos 2 β . ( 28
)
The critical classical solution at ν = ν cr touches the line ξ = 0 at η = 0 (point C in Fig. 2 ), where its trajectory
x cr (t) = √ 2t sin β , ( 29
) y cr (t) = √ 2 sin t cos β .
has the largest inclination. We now turn to the classically forbidden reflections at ν < ν cr , which are described by the boundary value problem (9), (10), (12). One makes the following important observation. The waveguide function (20) has the form of two analytic functions glued together at ξ = 0. Hence, the equations of motion (9) can be continued analytically to the complex values of coordinates in two different ways, starting from the regions ξ < 0 and ξ > 0 respectively. In this way one obtains two complex solutions, ξ -(t), η -(t) and ξ + (t), η + (t). These solutions and their first derivatives should be matched at some moment of time t 1 , ξ(t 1 ) = 0. [Note that the matching time t 1 does not need to be real.] Below we conventionally refer to these solutions as the ones belonging to the regions ξ < 0 and ξ > 0. By the same reasoning as above we find that once the particle arrives into the region ξ > 0, it never reflects back to ξ < 0, unless p ξ = 0. So, in the region ξ > 0 one writes,
ξ + (t) = 0 , ( 30a
) η + (t) = √ 2 cos β sin(t cos β + ϕ η ) , ( 30b
) 12
where the "normalization" condition E = 1 has been used explicitly. Due to the conditions in the asymptotic future, Eqs. (12), the parameter ϕ η is real. We use the translational invariance (18) to set ϕ η = 0. Note that we again associate the trajectory going along the line ξ = 0 with the reflected one.
The physical picture of over-barrier reflection that comes to mind matches with the new mechanism of multidimensional tunneling proposed recently in Refs. [9, 11] . The process proceeds in two steps. The first step, which is exponentially suppressed, is formation of the periodic classical orbit (30) oscillating along the line ξ = 0. This orbit is unstable. At the second step of the process the unstable orbit decays classically forming a trajectory going back to x → -∞ at t → +∞. Clearly, the second step does not affect the suppression exponent of the whole process, and we do not consider it explicitly. In what follows we concentrate on the determination of the tunneling trajectory describing the first step of the process.
One should find the solution at ξ < 0 and impose the boundary conditions (10). Note, however, that the energy of our solution is fixed already. As for the initial oscillator excitation number ν, it does not change during the evolution in the region ξ < 0. Thus, one may fix it at the matching time t = t 1 . One writes,
ν = 1 2 ( ẏ2 + y 2 ) t=t 1 = cos 2 β + sin 2 β sin 2 (t 1 cos β) .
This complex equation allows one to express t 1 as sin(t
1 cos β) = -i √ ν cr -ν sin β , ( 31
)
where the choice of the sign is dictated by the condition in footnote 9. It is convenient to introduce notation t 1 = iT 1 , T 1 ∈ R.
In order to find the suppression exponent f β (ν), one needs to evaluate the parameters T (ν), θ(ν). At ξ < 0 the solution has the form,
x -(t) = p 0 (t -iT /2) + x ′ 0 , ( 32a
) y -(t) = ae -it + a * e T +θ+it , ( 32b
)
where the definitions (13), (14) have been taken into account explicitly, so that p 0 , x ′ 0 ∈ R. One evaluates p 0 , x ′ 0 , a, T , θ by matching the coordinates x ± , y ± and their first derivatives 13 0.2 0.15 0.1 0.05 0 ν cr 0.2 0.15 0.1 0.05 0 ν f β Figure 3: The suppression exponent f β (ν) for the waveguide (20); β = π/3. ẋ± , ẏ± at t = iT 1 ; this yields
x ′ 0 = 0 , p 0 = 2(1 -ν) , a = i ν 2 e -T +θ 2 , T 1 - T 2 = - 1 -ν/ cos 2 β 1 -ν , sh T 1 - T + θ 2 = - cos 2 β -ν sin β √ ν
The last two equations, together with Eq.( 25 ), define the function f β (ν),
f β (ν) = 2 cos β arcsh √ ν cr -ν sin β -ν cos β arcsh √ ν cr -ν sin β √ ν -(ν cr -ν)(1 -ν) ;
this finction is plotted in Fig. 3 . One observes that at ν → ν cr the quantities T 1 , T, θ, f β tend to zero, and the complex trajectory tends to the classically allowed critical solution, cf. Eqs. (29),
p 0 → √ 2 sin β , a → i √ 2 cos β .
At ν = 0 one has,
f β (0) = -2 + 2 cos β arcth (cos β) . ( 33
)
To summarize, we obtained the suppression exponent for the reflection of a particle in the simplest waveguide with one sharp turn.
14 L x' x y' y ξ η A B A' B' C C' β α α L x' x y' y ξ η A B A' B' C C' β α α L x' x y' y ξ η A B A' B' C C' β α α L x' x y' y ξ η A B A' B' C C' β α α
Figure 4: The equipotential contour w 2 (x, y) = 2N ′ for the waveguide (35) and the trajectory of the critical solution with energy N ′ / cos 2 β > E B . The matching points C, C ′ are shown by the thick black dots.
4 The model with two turns
In the model of the previous section the suppression exponent was proportional to energy because of the coordinate rescaling symmetry (23) . Now, we are going to demonstrate that small violation of this symmetry results in highly non-trivial graph for F (E). One introduces a second turn into the waveguide, see Fig. 4 . We want to consider this turn as a small perturbation, so, we assume its angle α to be smaller than β. It is convenient to introduce two additional coordinate systems, x ′ , y ′ and ξ, η, bound to the central and rightmost parts of the waveguide respectively. They are related to the original coordinate system x, y as follows,
x ′ y ′ = cos α sin α -sin α cos α x y , ξ η = cos β -sin β sin β cos β x ′ -L y ′ .
(34) Note that the origin of the coordinate system ξ, η is shifted by the distance L. The waveguide 15 function is
w = θ(-x ′ )θ(-ξ)y + θ(-ξ)θ(x ′ )y ′ cos α + θ(ξ)η cos α cos β ; ( 35
)
it consists of three pieces glued together continuously at x ′ = 0 and ξ = 0 (lines A ′ B ′ and AB in Fig. 4 respectively). At t → -∞ the particle comes flying from the asymptotic region
x ′ < 0, where w = y.
In the intermediate region x ′ > 0, ξ < 0 the particle moves in the x ′ direction oscillating along the y ′ coordinate with the frequency cos α. Finally, in the region ξ > 0 its motion is free in the coordinates ξ, η; the frequency of η-oscillations is cos α cos β.
The model (35) no longer possesses the symmetry (23): rescaling of coordinates changes the length L of the central part of the waveguide. In what follows it is convenient to work in terms of the rescaled dynamical variables, x = x/L , ỹ = y/L .
In new terms the parameter L disappears from the classical equations of motion, entering the theory through the overall coefficient L 2 in front of the action. The initial-state quantum numbers are also proportional to
L 2 , E = L 2 Ẽ , N = L 2 Ñ . ( 36
)
Thus, the conditions (22) for the validity of the semiclassical approximation are satisfied in the limit L → ∞ , Ẽ, Ñ = fixed .
The suppression exponent takes the form
F (E, N) = L 2 F ( Ẽ, Ñ) . ( 37
)
To simplify notations, we omit tildes over the rescaled quantities in the rest of this section. Rescaling back to the physical units can be easily performed in the final formulae by implementing Eqs. (36), (37).
Let us begin this subsection by demonstrating that there are no stable classical solutions localized in the region between the turns. This is important for the determination of the tunneling probability, since such stable solutions could lead to exponential resonances in the tunneling amplitude. The argument proceeds as follows. Any trajectory which is localized in 16 the intermediate region should reflect from the line AB infinitely many times. Each reflection involves touching the unstable orbit living at the line AB. This implies that the trajectory itself is unstable. We proceed by determining the region of initial data E, N, which correspond to the classical reflections. [For brevity we will refer to this region as the "classically allowed region", as opposed to the "classically forbidden region" where reflections occur only at the quantum mechanical level. We stress that these are the regions in the plane of quantum numbers E, N.] Let us search for the critical classical solutions which correspond to the smallest initial oscillator number N = N cr (E) at given energy E. As in the previous section, one finds that the particle must get stuck at the line 13 AB for some time in order to reflect back. Let us first make an assumption inspired by the study of the one-turn model that the critical solutions touch the line AB at their maximum inclination point (point C in Fig. 4 ).
We will see shortly that this is true only at energies above a certain value E B , see Eq. (50). Still, the analysis based on the above assumption enables one to catch the qualitative features of the critical line N = N cr (E). Besides, the analysis is considerably simplified in this case;
we postpone the accurate study until the end of this subsection. Keeping in mind the above remarks, one writes for the solution in the intermediate region,
x ′ cr (t) = t √ 2E sin β + 1 , ( 38a
) y ′ cr (t) = √ 2E cos β cos α sin(t cos α) . ( 38b
)
Before entering the intermediate region, the particle crosses the line A ′ B ′ (point C ′ in Fig. 4 ).
The initial oscillator number N is most conveniently calculated at the moment t = t 0 ≡ -1 √ 2E sin β of crossing. Using the relations (34) one obtains,
ẋcr (t 0 ) = √ 2E sin β cos α -cos β sin α cos cos α √ 2E sin β , ( 39
)
and thus
N cr (E) = E - 1 2 ẋ2 cr (t 0 ) = E -E sin β cos α -cos β sin α cos cos α √ 2E sin β 2 , E > E B . ( 40
) 17 0 0.05 0.1 0.15 0.2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.1 0
N/α 2 E/α 2 E B E 2 E 3 E 4
Figure 5: The boundary N = N cr (E) of the classically allowed region at E > E B for the waveguide model (35); β = π/3, α = π/30. The region of the classically allowed initial data lies above this boundary. The empty circles correspond to the energies E = E n , where the curve N = N cr (E) touches its lower envelope N = E cos 2 (β + α).
As an example, we show in Fig. 5 the region of the classically allowed initial data for β = π/3, α = π/30. One observes that the function N cr (E) oscillates between two linear envelopes, E cos 2 (β + α) and E cos 2 (β -α); the period of oscillations decreases as E → 0. Moreover, the curve N cr (E) has a number of minima at the points E = E cr n . This means that the energies E = E cr n are optimal for reflection: in the vicinity of any point
E = E cr n , N = N cr (E cr n
) reflections become exponentially suppressed independently of whether the energy gets increased or decreased. This feature is particularly pronounced in the case α + β = π/2, when the lower envelope coincides with the line N = 0. Then, the classical reflections (i.e.
reflections with the probability of order 1) at N = 0 are possible only in the vicinities of the points
E = 1 8π 2 (n -1/2) 2 .
This is the case we used in Introduction to illustrate the effect.
The minima E = E cr n exist at other values of the parameters as well. For instance, let 13 We do not consider reflections from the line A ′ B ′ . They disappear at larger values of N than reflections from the line AB if α is small enough.
18 us find the positions of these minima in the case α ≪ 1. One differentiates Eq. (40) with respect to energy and obtains,
E cr n = E n 1 - 1 π(n -1/2) arcsin ctg β 2πα(n -1/2) + O(α 2 ) , ( 41
)
where
E n = 1 8π 2 (n -1/2) 2 sin 2 β ( 42
)
are the points where the curve N = N cr (E) touches its lower envelope. The argument of arcsine in Eq. (41) should be smaller than one, so, the minima E cr n exist only at large enough n,
n ≥ n 0 ≡ ctg β 2πα + 1 2 + 1 , ( 43
)
where [•] stands for the integer part. Let us make several comments. First, note that n 0 ∼ O(1/α), consequently, all the optimal points E cr n lie in the region of small energies E ∼ 1/n 2 0 ∼ O(α 2 ). Second, as we pointed out before, the formula (40) for the function N cr (E) holds at E > E B . Comparing the expressions (42), (43) and (50), one observes that E n 0 > E B if tg β > 1. So, there does exist a range of energies where the non-monotonic behavior of the function N cr (E) can be inferred from the formula (40). In fact, the conclusion about the existence of the local minima of N cr (E), as well as the expressions (41), (42), (43) determining their positions, remain valid also at E < E B . This follows from the rigorous analysis of the boundary of the classically allowed region to which we turn now. The reader who is more interested in the tunneling processes may skip this part and proceed directly to subsection 4.3. Ansatz (38) . Instead, we start with the general solution in the intermediate region,
x ′ = p ′ 0 (t -t 0 ) , ( 44a
) y ′ = A ′ 0 sin [(t -t 0 ) cos α + ϕ ′ ] . ( 44b
)
It is convenient to parametrize it by the total energy E = p ′2 0 /2 + cos 2 αA ′2 0 /2 and the "inclination" γ defined by the relation
p ′ 0 /A ′ 0 = tg γ cos α .
Expressions (44) take the following form,
x ′ = √ 2E (t -t 0 ) sin γ , ( 45a
)
y ′ = √ 2E cos γ cos α sin [(t -t 0 ) cos α + ϕ ′ ] . ( 45b
) 19
The constants t 0 and ϕ ′ are fixed by demanding the trajectory (45) to reflect classically from the second turn, i.e. touch the line ξ = 0 at t = 0,
(x ′ -1) cos β -y ′ sin β t=0 = 0 , dy ′ dx ′ t=0 = ctg β .
These conditions imply,
t 0 = - 1 √ 2E sin γ + 1 cos α tg 2 β tg 2 γ -1 , ( 46a
)
ϕ ′ = - cos α √ 2E sin γ + tg 2 β tg 2 γ -1 -arccos tg γ tg β . ( 46b
)
One sees that the classical reflections are possible only at γ ∈ [0; β]; the boundary value γ = β reproduces the solution (38). In order to find N cr (E), one should minimize the value of the incoming oscillator excitation number with respect to γ at fixed E. At t = t 0 , when the particle crosses the first turn,
p 0 ≡ ẋ(t 0 ) = √ 2E(cos α sin γ -sin α cos γ cos ϕ ′ ) . ( 47
)
Since N = E -p 2 0 /2, one can maximize the value of the translatory momentum p 0 instead of minimizing N(γ). Formula (39) represents the value γ = β lying at the boundary of the accessible γ-domain; this value should be compared to p 0 (γ) taken at local maxima.
Let us consider the case α ≪ 1. At large enough energies, E ∼ 1, Eq. (47) is dominated by the first term, which grows with γ, so that the maximum of p 0 (γ) is indeed achieved at γ = β. At small energies, however, the second term in Eq. (47) becomes essential because of the quickly oscillating cos ϕ ′ multiplier: the frequency of cos ϕ ′ oscillations grows as E → 0, and at E ∼ α 2 , in spite of the small magnitude proportional to sin α, the second term produces the sequence of local maxima of the function p 0 (γ).
One expects the parameters of the trajectory at small α not to be very different from the ones at α = 0 (the latter case was considered in Sec. 3). So, we write,
γ = β -δγ ,
where 0 < δγ ≪ 1. Expanding the expressions (46), (47) and taking into account that E ∼ α 2 one obtains,
ϕ ′ = - 1 √ 2E sin β (1 + δγ ctg β) , ( 48a
) p 0 = √ 2E(sin β -δγ cos β -α cos β cos ϕ ′ ) . ( 48b
)
20 Now, the local maxima of the initial translatory momentum can be obtained explicitly by differentiating Eqs. (48) with respect to δγ. One finds the sequence of them,
δγ n = -tg β + √ 2E sin 2 β cos β 2πn -π -arcsin √ 2E sin 2 β α cos β . ( 49
)
Only the maxima with δγ n > 0 should be taken into account. The local maxima exist when
E ≤ E B ≡ α 2 cos 2 β 2 sin 4 β . ( 50
)
Substituting Eq. (49) into the expressions (48), one evaluates the values of p 0 at the local maxima,
p 0,n (E) =2 √ 2E sin β -2E sin 2 β 2πn -π -arcsin √ 2E sin 2 β α cos β + α √ 2E cos β 1 - 2E sin 4 β α 2 cos 2 β .
The graphs N n (E) = E -p 2 0,n (E)/2 are shown in Fig. 6 for the case β = π/3, α = π/30. Each graph is plotted for the energy range E > E An restricted by the condition δγ n > 0. They are presented together with the curve given by the formula (40). By definition, the critical solution corresponds to the lowest of these graphs. Clearly, for each "local" curve representing the n-th local minimum of N(γ) there is a range of energies E An < E < E Bn where it lies lower than the "global" curve (40). This means that the parameter γ of the critical solution changes discontinuously across the points E = E Bn . Correspondingly, the curve N cr (E) has a break at these points. On the other hand, the function N cr (E) is smooth at the points A n as the "local" graphs end up exactly at δγ = 0, where the parameters of the n-th "local" solution coincide with the ones of the "global" solution.
To summarize, we have observed that the boundary of the classically allowed region is given by a collection of many branches of classical solutions, each branch being relevant in its own energy interval. We will see that a similar branch structure is present in the complex trajectories describing over-barrier reflections in the classically forbidden region of E, N.
In this subsection we demonstrate that the suppression exponent F (E, N) viewed as a function of energy at fixed N exhibits oscillations deep inside the classically forbidden region 21 0 0.02 0.04 0.06 0 0.05 0.1 0.15 0.2 N/α 2 E/α 2 4 5 6 4 5 6 A 4 B 4 A 5 B 5 A 6 B 6 Figure 6: The graphs N n (E) corresponding to the local minima of the function N(γ) (dashed lines) plotted together with the "global" curve, Eq. (40) (solid line); β = π/3, α = π/30. The critical curve N = N cr (E) is obtained by taking the minimum among all the graphs. of initial data. This result comes without surprise if one takes into account the non-monotonic behavior of the boundary N cr (E) of the classically allowed region. Indeed, the curve N = N cr (E) coincides with the line F (E, N) = 0. One has,
dN cr dE = - ∂ E F ∂ N F N =Ncr(E) , so that ∂F ∂E (E cr n , N cr n ) = 0 .
We conclude that the points E = E cr n are the local minima of the function F (E) at fixed N = N cr n . It is natural to expect that such local minima of F (E) exist at other values of N as well. To illustrate this fact explicitly, we study the complex trajectories, solutions to Eqs. (9), (10), (12) .
Following the tactics of the previous section, we find solutions in three separate regions: initial region x ′ < 0, final region ξ > 0, and the intermediate region x ′ > 0, ξ < 0. These solutions, together with their first derivatives, should be glued at t = t 0 , when the complex trajectory crosses the line x ′ = 0, and at t = t 1 , when ξ = 0. Besides, we are looking for the 22 tunneling solution which ends up oscillating along the line AB, see Fig. 4. As discussed in Sec. 3 this assumes existence of the second step of the process: classical decay of the unstable orbit living at ξ = 0; the latter decay is described by a real trajectory 14 going to x → -∞ at t → +∞.
The solution in the final region ξ > 0 is (cf. Eqs. (30)),
ξ + (t) = 0 , ( 51a
) η + (t) = √ 2E cos α cos β sin(t cos α cos β) , ( 51b
)
where we used the time translation invariance (18) to fix the final oscillator phase ϕ η = 0. In the intermediate region x ′ > 0, ξ < 0 one writes,
x ′ (t) = p ′ 0 t + x ′ 0 , ( 52a
) y ′ (t) = a ′ e -it cos α + ā′ e it cos α . ( 52b
)
Note that the final solution (51) does not contain free parameters; thus, the matching of x ′ , ẋ′ , y ′ , ẏ′ at t = t 1 enables one to express all the parameters in Eqs. (52) in terms of one complex variable t 1 ,
p ′ 0 = √ 2E sin β cos φ 1 , ( 53a
) x ′ 0 = 1 + √ 2E tg β cos α [sin φ 1 -φ 1 cos φ 1 ] , ( 53b
)
a ′ = E/2 cos α e iφ 1 / cos β [sin φ 1 + i cos β cos φ 1 ] , ( 53c
) ā′ = E/2 cos α e -iφ 1 / cos β [sin φ 1 -i cos β cos φ 1 ] , ( 53d
)
where we introduced φ 1 = t 1 cos α cos β.
As the energy of the solution has been fixed already, the only remaining initial condition involves initial oscillator excitation number at x ′ < 0, see Eqs. (10) . It is convenient to impose this condition at the matching point t = t 0 . One recalls the definition of the matching time
t 0 , p ′ 0 t 0 + x ′ 0 = 0 ,
14 One wonders why this trajectory does not reflect from the turn A ′ B ′ on its way back. This concern is removed by the observation that the trajectory produced in the decay of the unstable orbit is not unique: in appendix we show that the decay can occur at any point of the segment AC giving rise to a whole bunch of potential decay trajectories. Most of these trajectories pass through the turn A ′ B ′ without reflection.
which, after taking into account the expressions (53a), (53b), leads to the following equation,
cos α √ 2E sin β + sin φ 1 cos β -cos φ 1 ∆φ = 0 , ( 54
)
where ∆φ = cos α(t 1 -t 0 ). At t = t 0 one has,
ẋ(t 0 ) = p ′ 0 cos α -ẏ′ (t 0 ) sin α = 2(E -N) ,
and thus √ 1 -ν sin α = ctg α sin β cos φ 1 -sin φ 1 sin ∆φ -cos β cos φ 1 cos ∆φ . (55)
As before, ν = N/E.
Two complex equations (54), (55) determine the matching times t 0 , t 1 , and, consequently, the complex trajectory. Although these equations cannot be solved explicitly, they can be simplified in the case α ≪ 1, which we consider from now on. For concreteness, we study reflections at = 0. It is important to keep in mind that in the region of interest E ∼ E cr n ∼ O(α 2 ); thus, one should regard all the momenta p and oscillator amplitudes a, ā, as the quantities of order O(α). At the same time, for the distances along the waveguide one has x ∼ O(1), so that the real parts of time intervals may be parametrically large, Re t ∼ x/p ∼ O(1/α).
Further on, it will be convenient to work in terms of real variables, so, we represent φ 1 and ∆φ as φ 1 = cos α cos β(τ 1 + iT 1 ) , ∆φ = cos α(τ + i∆T ) .
Note that τ and ∆T are the real and imaginary parts of the time interval t 1 -t 0 which the particle spends in the intermediate region. Now, equation (54) enables one to express
τ = 1 √ 2E sin βch(T 1 cos β) + O(α) , ( 56
) τ 1 = - 1 τ cos β 1 cos β -∆T cth(T 1 cos β) + O(α 3 ) . ( 57
)
Note that τ 1 ∼ O(α), τ ∼ O(1/α). Then, the real part of Eq. (55) implies that ch(T
1 cos β) = 1 sin β 1 + α ctgβ cos τ e ∆T + O(α 2 ) . ( 58
)
While deriving this formula we imposed T 1 < 0 which follows from the requirement that in the limit α → 0 equation (31) should be recovered; besides, we assumed e ∆T ∼ O(1).
24 Substituting Eq. (58) into Eq. ( 56 ) and the imaginary part of Eq. ( 55 ), we obtain the final set of equations,
1 -τ √ 2E = α ctgβ cos τ e ∆T + O(α 2 ) , ( 59a
) (1 + ∆T )e -∆T = α ctgβτ sin τ + O(α) . ( 59b
)
These two nonlinear equations, still, cannot be solved explicitly. Nevertheless, one can get a pretty accurate idea about the structure of their solutions. Before proceeding to the analysis of the above equations, let us derive a convenient expression for the suppression exponent F 0 (E) ≡ F (E, N = 0). Note that on general grounds one expects to obtain an expression of the form,
F 0 (E) = E(f β (0) + O(α)) ,
where f β (0) is given by Eq. (33) . We are interested in the O(α) correction in this expression, so, one must be careful to keep track of the subleading terms during the derivation.
Making use of the equations of motion, one obtains for the incomplete action (16) of the system, 2 Im S = Im p ′ 0 = √ 2E sin β Im(cos φ 1 ) .
Substitution of Eqs. (56), (57), (58) into this formula yields 2 Im S = 2E -1 -∆T -α ctg β cos τ e ∆T 1 + 1 cos 2 β + 2∆T + O(α 2 ) .
For the parameter T one has (see Eqs. (14)),
T = - 2 Im x 0 p 0 = - 2 Im(x(t 0 ) -p 0 t 0 ) p 0 = 2(T 1 -∆T ) + 2 E sin α Im y ′ (t 0 ) , ( 60
)
where in the last equality we used Eqs. (34) and x ′ (t 0 ) = 0. The quantity Im y ′ (t 0 ) is evaluated by using Eqs. (52b), (53) and (58); one finds,
Im y ′ (t 0 ) = - √ 2E ctg β cos τ e ∆T + O(α) .
Substituting everything into the formula (15), we obtain,
F 0 (E) = E f β (0) -4α ctg β cos τ ∆T e ∆T + O(α 2 ) . ( 61
)
This expression implies that determination of the O(α) correction to the suppression exponent involves finding τ , ∆T with O(1)-accuracy. This is precisely the level of accuracy of 25 Eqs. (59). Below we will also need the following formulae, which can be easily obtained by using T = -F . E . and Eq. ( 60 ),
dF 0 dE = f β (0) + 2(∆T + 1) + O(α) , ( 62
) d dE F 0 E = 2(∆T + 1 + O(α)) E . ( 63
)
Note that, though the suppression exponent differs from that in the one-turn case only by O(α) correction, its derivative gets modified in the zeroth order in α.
Now, we are ready to analyze Eqs. (59). One begins by solving Eq. (59b) graphically, see Fig. 7. The important property of this equation is as follows. One notices that the l.h.s. of Eq. (59b) is always smaller than 1, the maximum being achieved at ∆T = 0. Therefore, the solutions to this equation are confined to the bands τ sin τ < tgβ α .
This corresponds to
τ ∈ [0; 2π(n 1 -1) + δτ n 1 ] or τ ∈ [2πn -π -δτ n ; 2πn + δτ n ] , n ≥ n 1 ( 64
)
where
δτ n = arcsin tgβ 2πα(n -1/2) + O(α) , n 1 = tgβ 2πα + 1 2 + 1 , ( 65
)
with [•] in the last formula standing for the integer part. The forbidden bands, where τ sin τ > tgβ/α, are marked in Fig. 7 by yellow shading. The property (64) introduces a topological classification of the solutions τ , ∆T to Eqs. (59). Namely, these solutions fall into a set of continuous branches: the "local" branches τ n (E), ∆T n (E) living inside the
strips τ ∈ [2πn -π -δτ n ; 2πn + δτ n ], n ≥ n 1 , and the "global" branch τ g (E), ∆T g (E)
inhabiting the very first band τ ∈ [0; 2π(n 1 -1) + δτ n 1 ]. As follows from the definition of τ , the topological number n counts the number of y ′ -oscillations during the evolution in the intermediate region.
Let us consider the "global" branch. From Eqs. (59) one has,
τ g → 2π(n 1 -1) + O(α ln α) , ∆T g → ln(tg β/α) , E → 0 , τ g → 0 , ∆T g → -1 , E → +∞ . 26 -2 -1 0 1 2 3 4 10π 9π 8π 7π 6π 5π 4π 3π 2π π 0 ∆T τ g 4 5 -2 -1 0 1 2 3 4 10π 9π 8π 7π 6π 5π 4π 3π 2π π 0 ∆T τ g 4 5
Figure 7: Curves representing solutions to Eq. ( 59b ); β = π/3, α = π/30.
By inspection of Fig. 7 one can work out the qualitative behavior of the functions τ g (E), ∆T g (E). Alternatively, these functions can be found numerically. They are plotted in Fig. 8 for the case β = π/3, α = π/30 (the curves marked with "g"). One observes that at high enough energies the function ∆T g (E) exhibits oscillations around the line ∆T = -1.
According to the formula (63) this means that the function F 0 (E)/E is non-monotonic, it attains local minima at the points
E ′ n = 1 8π 2 (n -1/2) 2 1 + 2αe -1 ctgβ + O(α 2 ) . ( 66
) Moreover, if n ≥ n ′ 0 ≡ tg β 4πα f β (0) exp 1 + f β (0) 2 + 1 2 + 1 ( 67
) there exist E o n = E ′ n (1 + O(α)), such that ∆T (E o n ) = -1 -f β (0)/2.
Then, according to Eq. (62) the points E o n are the "optimal" energies corresponding to the local minima of the suppression exponent F 0 (E).
At low energies the function ∆T g (E) ceases to oscillate and becomes large and positive.
According to Eq. (62) this means that the suppression exponent F 0,g (E) of the "global" solution becomes negative at low energies 15 , see Fig. 9. This is a clear signal that the 15 It is worth mentioning that Eqs. (59) and the expression (61) for the suppression exponent become 27 0 0.1 0.2 0.3 τ E/α 2 4π 5π 6π 7π 8π 9π 10π 0 0.1 0.2 0.3 τ E/α 2 4π 5π 6π 7π 8π 9π 10π g g 4 4 5 5
E' 3 E' 4 E' 5 -2 -1 0 1 2 3 4 0 0.1 0.2 0.3 ∆T E/α 2 g 4 5 g 4 5 E' 3 E' 4 E' 5
Figure 8: Several first branches of solutions to Eqs. (59): "global" branch ("g") and two "local" branches ("4", "5"); β = π/3, α = π/30. 28 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.2 0.4 0.6 F 0 /α 2 E/α 2 E' 2 E' 3 g g 4 4 0 0.02 0.04 0.12 0.11 0.10 0.09 A B Figure 9: The suppression exponent F 0 (E) for the "global" and first "local" (n = 4) branches; β = π/3, α = π/30. The vicinity of intersection of the graphs is enlarged in the upper right corner. "global" solution becomes unphysical at these energies and its contribution to the reflection probability should be discarded: negative suppression exponent contradicts the unitarity requirement 16 , P < 1. One is forced to conclude that at low energies reflection is described by the "local" solutions. Let us study them in detail.
For the n-th branch one obtains,
τ n → 2πn + O(α ln α) , ∆T n → ln(tg β/α) , E → 0 , τ n → 2πn -π , ∆T g → +∞ , E → +∞ .
From Fig. 7 one learns that the n-th solution passes through the points
∆T n = -1 , τ = 2πn or τ = 2πn -π . ( 68
)
inapplicable at large ∆T : the assumption e ∆T ∼ O(1) which was used in the derivation of these equations gets violated. Nevertheless, by analyzing the full equations (54), (55) one can show that dF 0,g /dE = -T g is large and positive at E → 0. This is sufficient for concluding that F 0,g (E) is negative in the low-energy domain. 16 Another indication that the "global" solution is unphysical at small E is that the function τ g (E) is bounded from above. Indeed, τ is the time interval the particle spends in the intermediate part of the waveguide, one expects it to tend to infinity as E → 0 for a physically relevant solution.
Thus, each curve ∆T n (E) has one sharp dip, its minimum is smaller than -1, see Fig. 8 . As in the case with the "global" branch, the points (68) represent the extrema of the functions F 0,n (E)/E; the positions of the local minima are again given by Eq. (66). Making use of Eq. ( 61 ), we find that the suppressions F 0,n (E) of the "local" branches are large and positive at high energies. Hence, these solutions give subdominant contributions to the reflection probability at such E as compared to the "global" solution. As energy decreases, F 0,n (E) also decreases, then makes one oscillation and drops to negative values at small E. The latter property means that each "local" branch becomes unphysical at small enough energies. The suppression exponent of the first "local" branch (corresponding to n = 4 in the case β = π/3, α = π/30) is presented in Fig. 9 . An alert reader may have already guessed that we have met here the typical Stokes phenomenon [21] . In fact, the Stokes phenomenon is specific to the situations where some integral (e.g., the path integral (7) in our case) is evaluated by the saddle-point method. Essentially, it means the following: as one gradually changes the parameters of the integral in question, a given saddle point may become non-contributing after the values of these parameters cross a certain curve drawn in the parameter space, the Stokes line. Since the result of the computation should be continuous, this phenomenon occurs only for subdominant saddle points (saddle-point trajectories in our case). Unfortunately, apart from several heuristic conjectures [21, 12] , sometimes rather suggestive [13] , there is presently no general method of dealing with the Stokes phenomenon in the semiclassical calculations. However, in the situation encountered above it suffices to use the simplest logic lying at the heart of all other approaches 17 .
When gathering the final result for the suppression exponent, we follow two guidelines. First, it is clear that, as energy decreases, each branch becomes unphysical before F 0,n (E) crosses zero. On the other hand, at high energies one should pick up the branch corresponding to the smallest value of the suppression exponent. Looking at Fig. 9 , one notes that the curves F 0,g (E), F 0,4 (E) have two intersections, A and B. At E > E B one chooses the "global" branch. In the region E A < E < E B we switch to the first "local" branch, because in this region F 0,4 (E) < F 0,g (E). Naively, at E = E A one should jump back to the "global" branch; however, in order to preserve unitarity at small energies, we suppose that somewhere in between the points B and A the "global" branch becomes non-contributing, so that one should stay at the "local" branch at E < E A . Similarly, the adjacent "local" branches have 17 The simplification in the present case is related to the fact that we concentrate on the dominant semiclassical contribution, leaving aside the subdominant ones.
two intersections; as the energy decreases, we switch from n-th branch to n + 1-th at the first intersection, and stay there until the intersection with the n + 2-th branch. Overall, one obtains the graph for the suppression exponent plotted in Fig. 10 . The suppression exponent 0 0.02 0.04 0.06 0.08 0 0.05 0.1 0.15 0.2 0.25
F 0 /α 2 E/α 2 E' 3 E' 4 E' 5 g 4 5
Figure 10: The final result for the suppression exponent F 0 (E) in the region of small energies; β = π/3, α = π/30. The points where different branches merge are shown with thick black dots. oscillates between two linear envelopes, F = E(f β (0) ± 4e -1 α ctg β); oscillations pile up in the region of low energies. The reflection process is optimal in the vicinities of the minima of the function F 0 (E).
By considering a class of two-dimensional waveguide models, we have demonstrated explicitly that the probability of over-barrier reflection can be non-monotonic function of energy. The origin of the effect lies in the classical dynamics: the parameters of the complex trajectory describing over-barrier reflection change quasi-periodically as the energy gets decreased.
This results in the oscillatory behavior of the suppression exponent. Reflection occurs with 31 exponentially larger probability in the vicinities of "optimal" energies (local minima of the suppression exponent) while being highly suppressed in between.
Our results are obtained for a fairly specific class of waveguides, namely, the ones with very sharp turns. However, the qualitative features observed in this paper should be valid for quite general waveguide models: a classical particle with high energy feels any large-scale turn of the waveguide as a sharp one 18 ; if two turns are separated by a long interval of free motion, one arrives to the model (35) . We remark that the phenomenon of optimal tunneling has been observed also in numerical investigation of a smooth waveguide, see Ref. [14] .
The branch structure of solutions observed in the region of small energies is interesting from the mathematical point of view. We have shown that there exists an infinite sequence of complex trajectories marked by the topological number n. Each branch produces physically consistent result for the suppression exponent in some energy interval; outside of this interval the n-th branch would correspond either to highly suppressed transitions (high energies) or to violation of unitarity (low energies). We collected the final graph for the suppression exponent basing on the empirical considerations, which hardly may be acknowledged as satisfactory. Our study clearly shows that the method of complex trajectories should be equipped with a convenient rule to pick up the physical trajectory among the discrete set of solutions to the boundary value problem (9), (10), (12) (in other words, the method to deal with the Stokes phenomenon). Presently, such a rule is absent.
We note that the described physical phenomenon of optimal tunneling is present independently of the way the branches of solutions are glued together. The result at relatively high energies is given by the "global" branch, which displays a large number of local minima 65) . This is the case for the illustrative example considered throughout this paper, see Fig. 9 .
if n ′ 0 > n 1 , see Eqs. (67), (
As a final remark, we point out some open issues. We have calculated the suppression exponent of reflection using the sharp-turn approximation. It would be instructive to extend our analysis by finding corrections due to the finite turn widths. The motivation is twofold. First, the analysis performed in appendix implies existence of a rich variety of distinctive semiclassical solutions contributing almost equally into the reflection probability. This feature might be a manifestation of chaos [7] which is present in our system but hidden by the sharp-turn approximation. [Note that chaos is inherent in a very similar waveguide model
18 More precisely, one should compare the width b of the turn to the quantity 2π ω p0 m , where p 0 is the translatory momentum of the particle and ω stands for the frequency of transverse oscillations; if b ≪ 2πp0 ωm , one is in the class of models with sharp turns.
with smooth potential, see Ref. [14] .] Clearly, the structure of solutions in the vicinities of the turns is worth further investigation.
Second, it was proposed recently in Refs. [9, 11] that the process of dynamical tunneling in quantum systems with multiple degrees of freedom (including field theoretical models, see Refs. [19] ) can proceed differently from the ordinary case of one-dimensional tunneling. Namely, classically unstable state can be created during the process; this state decays subsequently into the final asymptotic region. The analysis performed in the present paper naturally conforms with this tunneling mechanism: all our complex trajectories are matched with the unstable orbit living at the turn. Still, the sharp-turn approximation does not allow to distinguish between the truly unstable trajectories staying at the turn forever and those which reflect from the turn in a finite time. To decide whether the tunneling mechanism of Refs. [9, 11] is indeed realized in our model one needs to go beyond the sharp-turn approximation. Then, the candidate for the "mediator" unstable state is the "excited sphaleron", the solution considered in the appendix. Presumably, in our model one can answer analytically to the question of whether or not the "excited sphaleron" acts as an intermediate state of the tunneling process. This study is quite beyond the scope of the present paper and we leave it for future investigations.
Acknowledgments. We are indebted to F.L. Bezrukov and V.A. Rubakov for the encouraging interest and helpful suggestions. This work is supported in part by the Russian Foundation for Basic Research, grant 05-02-17363-a; Grants of the President of Russian Federation NS-7293.2006.2 (government contract 02.445.11.7370), MK-2563.2006.2 (D.L.), MK-2205.2005.2 (S.S.); Grants of the Russian Science Support Foundation (D.L. and S.S.); the personal fellowship of the "Dynasty" foundation (awarded by the Scientific board of ICFPM) (A.P.) and INTAS grant YS 03-55-2362 (D.L.). D.L. is grateful to Universite Libre de Bruxelles and EPFL (Lausanne) for hospitality during his visits.
In this appendix we analyze the motion of the particle near the sharp turn of the waveguide (20) at nonzero smoothening of the turn, see, e.g., Eq. (21) . We suppose that in the small vicinity of the turn the function w(ξ, η) can be represented in the form w(ξ, η) = cos β (η -bv(ξ/b)) , (69) 33 where v(ψ) does not depend explicitly on b. Moreover, we consider the case when v(ψ) has a maximum 19 , v ′ (ψ 0 ) = 0 . (70) Due to the property (70) one immediately obtains the exact periodic solution to the equations of motion (9), which we call "excited sphaleron" [9],
ξ sp = bψ 0 , η sp = A η sin(t cos β + ϕ η ) + bv(ψ 0 ) . ( 71
)
We are going to show that this solution is unstable: a small perturbation above it grows with time and the particle flies away to either end of the waveguide. In particular, there are solutions that describe the decay of the sphaleron to ξ → -∞ both at t → ±∞. Clearly, such solutions correspond to reflections from the turn. In the vicinity of the sphaleron the trajectory of the particle can be represented in the form, ξ = bψ(t) , η = η sp (t) + bρ(t) , (72)
where ψ, ρ ∼ O(1). Writing down the classical equations of motion (9) in the leading order in b, one obtains,
d 2 ψ ds 2 = 4 b A η sin(2s)v ′ (ψ) , ( 73
) d 2 ρ ds 2 + 4ρ = 4[v(ψ) -v(ψ 0 )] , ( 74
)
where s = (t cos β + ϕ η )/2. It is worth noting that the right hand side of Eqs. (73), (74) are of different order in b. We will see that due to this difference ρ = 0 in the leading order in b.
Let us first consider the linear perturbations above the excited sphaleron,
ψ = ψ 0 + δψ , δψ ≪ 1 .
Equation (73) can be linearized with respect to δψ leading to the Mathieu equation
d 2 ds 2 δψ + 2q sin(2s)δψ = 0 ,
with canonical parameter q = -2v ′′ 0 A η /b > 0. As q ∼ O(1/b) ≫ 1, one can apply the WKB formula,
δψ = A cos W dW/ds , ( 75
)
19 For the smoothening (21), the properties (69), (70) hold with v(ψ) = ψtgβ 1+e ψ , ψ 0 ≈ 1.28.
where |A| ≪ 1, and W = 2q s π/4 ds ′ sin(2s ′ ) .
Note that we have chosen the solution symmetric with respect to time reflections,
δψ(π/2 -s) = δψ(s) . (76) At s ∈ [0; π/2] the exponent W is real and the particle gets stuck at ψ ≈ ψ 0 , oscillating around this point with high frequency dW/ds ∼ O(b -1/2 ). At s < 0 the solution (75) grows exponentially, meaning that the particle flies away from the excited sphaleron,
δψ(s < 0) = A cos(W (0) -π/4) |dW/ds| e |W (s)-W (0)| .
In what follows, we choose A cos(W (0) -π/4) < 0, so that δψ < 0 at s < 0. Let us denote by s 1 < 0 the point where δψ becomes formally equal to -1,
A cos(W (0) -π/4) |dW/ds| e |W (s 1 )-W (0)| = -1 .
In what follows we suppose that s 1 ∼ O(1), hence, A is exponentially small. Then, in the vicinity of this point, |s -s 1 | ≪ 1, one has,
δψ = -exp -2q sin(2s 1 )(s 1 -s) = -exp 4v ′′ 0 A η sin(2s 1 ) (s 1 -s) √ b . ( 77
)
We notice that δψ evolves from exponentially small values to δψ ∼ O(1) during the characteristic time |s -s 1 | ∼ O( √ b). When δψ ∼ O(1) the linear approximation breaks down and one has to solve the nonlinear equation (73). Using s = s 1 + O( √ b) one writes
d 2 ψ ds 2 = 4 b A η sin(2s 1 )v ′ (ψ) . ( 78
)
This equation permits to draw a useful analogy with one-dimensional particle moving in the effective potential V ef f (ψ) = -4b -1 A η sin(2s 1 )v(ψ) (see Fig. 11 ). This auxiliary particle starts in the region near the maximum of the potential at (s -s 1 )/ √ b → +∞ with energy E ≈ V max and rolls down toward ψ → -∞ at (s-s 1 )/ √ b → -∞. In this limit v(ψ) → ψ tg β and the solution takes the form
ψ = C 1 + C 2 (s -s 1 ) + 2b -1 A η sin(2s 1 ) tg β (s -s 1 ) 2 . 35 ψ 0 V max ψ V eff
Figure 11: The effective potential for Eq. (78).
Note that the coefficients C 1 , C 2 here are not independent: they are determined by the parameter s 1 through matching of the solution with Eq. (77) at (s -s 1 )/ √ b → +∞. We do not need their explicit form, however.
Let us argue that the function ρ remains small during the whole evolution of the particle in the vicinity of the sphaleron. Indeed, in the linear regime one has δψ ≪ 1 and the r.h.s. of Eq. (74) is small. So, ρ does not get excited. On the other hand, the nonlinear evolution of ψ proceeds in a short time interval ∆s = O( √ b); so, again, ρ is suppressed by some power of b. The trajectory (72) found in the vicinity of the sphaleron should be matched at
1 ≫ |s -s 1 | ≫ √ b
with the free solution in the asymptotic region ξ < 0, see Eqs. (26) . It is straightforward to check that matching can be performed up to the second order in (t -t 1 ), which is consistent with our approximations. In this way one determines the free asymptotic solution which, up to corrections of order O(b), coincides with the sinusoid coming from ξ → -∞ at t → -∞ and touching the line ξ = 0 at t = t 1 . Now we recall that, by construction, the obtained solution is symmetric with respect to time reflections, ξ(s) = ξ(π/2 -s) , η(s) = η(π/2 -s) .
This means that it satisfies ξ → -∞ at t → ±∞. This solution describes reflection of the particle from the turn.
The reasoning presented in this appendix puts considerations of the main body of this paper on the firm ground: we have found the "smoothened" solutions which reflect classically from the turn, and in the limit b → 0 coincide with the free solutions of Sec. 3 touching the line ξ = 0.
It is worth mentioning that, apart from the reflected solution we have found, in the vicinity of any trajectory touching the line ξ = 0 there exists a rich variety of qualitatively different motions. First of all, one may successfully search for solutions which are odd with respect to time reflections (Eq. (76) with minus sign). Such solutions, though close to the reflected ones at t < 0, describe transmissions of the particle through the sharp turn into the asymptotic region ξ → +∞. Relaxing the time reflection symmetry, one can find solutions leaving the vicinity of the turn at any point η < 0, which is different, in general, from the starting point η = η(s 1 ). Yet another types of solutions are obtained in the case when the amplitude A of δψ-oscillations at s ∈ [0; π/2] is so small that δψ does not reach the values of order one during the time period s ∈ [-π/2; 0]. If the particle is still in the vicinity of the point ψ 0 at s = -π/2, it remains for sure in this vicinity at s ∈ [-π; -π/2], because the r.h.s. of Eq. (73) is positive again. In this way one obtains solutions, which spend two, three, etc. sphaleron periods at ψ ≈ ψ 0 before escaping into the asymptotic regions ψ → ±∞. In the leading order in b all these solutions correspond to the identical initial state, and (in the case of classically forbidden transitions) to the same value of the suppression exponent. However, an accurate study of the dynamics in the vicinity of the the sphaleron is generically required to obtain the correct value of the suppression exponent in the case b ∼ 1, cf. Ref. [14] . | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Tunneling and over-barrier reflection are the characteristic non-perturbative phenomena in quantum mechanics. They typically occur with exponentially small probabilities,\n\nP ∝ e -F/ , ( 1\n\n)\n\nwhere F is the suppression exponent; still, the above phenomena are indispensable in understanding a wide variety of physical situations, from the generation of baryon number asymmetry in the early Universe [1] to chemical reactions [2] and atom ionization processes [3]. During the last decades extensive investigations of tunneling processes in systems with many degrees of freedom have been performed [2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] . These studies 1 levkov@ms2.inr.ac.ru 2 panin@ms2.inr.ac.ru 3 Sergey.Sibiryakov@cern.ch, sibir@ms2.inr.ac.ru 1 revealed a rich variety of features of multidimensional tunneling which are in striking contrast to the properties of one-dimensional tunneling and over-barrier reflection. In particular, the following phenomenon has been observed: the probability of tunneling may depend nonmonotonically on the total energy of the system and exhibit resonance-like peaks. One can envisage three physically different mechanisms of this phenomenon. The first mechanism, present already in one-dimensional case, is tunneling via creation of a metastable state. In this case the tunneling probability at the maximum of the resonance is exponentially higher than at other energies. On the other hand, the resonance width ∆E is exponentially suppressed; so, after averaging with an energy distribution of a finite width the effect of the resonance is washed out in the semiclassical limit → 0. The second possible mechanism of non-monotonic behavior of P(E) is quantum interference [7, 13] (see also [14] ). In this case the peak value of the tunneling probability is only by a factor of order one higher than the average value, while the width of the resonance scales as ∆E ∝ . Again, the resonances become indiscernible in the semiclassical limit. In both these cases the resonances can be attributed to the subleading semiclassical corrections, i.e. non-monotonic behavior of the pre-exponential factor omitted in Eq. ( 1 ). The third possibility is that the suppression exponent F (E) is non-monotonic. In this case the existence of the \"resonances\" is the leading semiclassical effect: the optimal tunneling probability at the maximum of the resonance is exponentially higher than the probability at other energies. At the same time the resonance width scales as 4 ∆E ∝ √ . This last possibility of \"optimal tunneling\" is definitely of interest; yet, it did not receive much attention in literature. We are aware of only a few works mentioning non-monotonic dependence of the suppression exponent on energy [15, 16, 14] .\n\nIt is worthwhile studying this phenomenon in detail; this can provide a new insight into the dynamics of multidimensional tunneling.\n\nIn this paper we consider the process of over-barrier reflection in a simple model with two degrees of freedom. Our setup is interesting in two respects. First, the model under study is essentially non-linear and the variables cannot be separated; still, over-barrier reflections in this model can be described analytically within the semiclassical framework. Thus, this model can serve as an analytic laboratory for the study of multidimensional tunneling. Second, the suppression exponent F of the reflection process behaves non-monotonically as the 4 This follows from the representation\n\nP(E) ∝ exp - F (E o ) - F ′′ (E o )(E -E o ) 2 2\n\nof the tunneling probability in the vicinity of the maximum.\n\n2 total energy E changes. We demonstrate that the function F (E) possesses a number of local minima E = E o , where reflection is optimal. We stress that the process we study is exponentially preferable at \"optimal\" energies as compared to other energies. Our model describes the motion of a quantum particle in the two dimensional harmonic waveguide (see Refs. [8, 10, 14] for similar models). The Hamiltonian is\n\nH = p 2 x 2m + p 2 y 2m + mω 2 2 w 2 (x, y) ,\n\nwhere x, y are the Cartesian coordinates and m is the mass of the particle. The function U = mω 2 w 2 /2 represents the waveguide potential in two dimensions: a particle with small energy is bound to move along the line w(x, y) ≈ 0. We do not introduce a potential barrier across the waveguide and consider the case when the line w = 0 stretches all the way from x → -∞ to x → +∞. We also assume that the function w(x, y) is linear in the initial asymptotic region, w(x, y) → y as x → -∞ .\n\nIn the present paper we consider two particular cases of the function w(x, y) describing waveguides with one and two sharp turns 5 , see Fig. 1. The motion of the particle at x → -∞ is a superposition of free translatory motion in x direction and oscillations of frequency ω along y coordinate; the state of such a particle is fully characterized by two quantum numbers, the total energy E and y-oscillator excitation number N. The particle sent into the waveguide from the asymptotic region x → -∞ with given E, N may either continue to move towards x → +∞, or reflect back into the region x → -∞. We are interested in the probability P(E, N) of reflection.\n\nLet us discuss reflections at the classical level. [Note that the classical counterpart of N is the energy of transverse oscillations.] Consider first the waveguide with one sharp turn (Fig. 1a ). One observes that the outcome of the classical evolution, i.e. whether or not the particle reflects from the turn, depends not only on the total energy E, but on other dynamical quantities as well. In particular, the direction of the momentum of the particle in the vicinity of the turn (point C on the graph) is important. This means that the entire dynamics in the waveguide should be taken into account in order to determine the possibility of classical reflection. This is in sharp contrast with the situation in one-dimensional case, where reflection from the potential barrier (or transition through it) is ensured by the value of the conserved energy of the particle.\n\nThe explicit expressions for the waveguide functions w(x, y) will be presented in the subsequent sections.\n\n(a) C U=0 U=E U=E x y (b) L C C' U=E U=E U = 0 Figure 1: The equipotential contour U = E for the waveguides with (a) one and (b) two sharp turns. An example of classical trajectory is shown in the case (b).\n\nNow, consider the waveguide with two turns. The model is characterized by the angles of the turns and the distance L between them (see Fig. 1b ). Suppose the particle starts moving classically from x → -∞ with N = 0 along the valley w = 0. Then, the transverse oscillations get excited only after the particle crosses the first turn, point C ′ on the plot, so that at the time of arrival to the second turn (point C) approximately ωτ /2π oscillations are made, where τ ∼ L m/2E is the time of motion between the two turns. The state of the particle (coordinates and momenta) at which it comes across the second turn depends periodically on the phase of transverse oscillations ωτ . Hence, one expects that the regime of motion of the classical particle can change from transmission to reflection and back as the energy grows (τ decreases); the energies where it happens can be roughly estimated as\n\nE n ∼ mω 2 L 2 2(2πn) 2 . ( 2\n\n)\n\nWe will see that this is indeed the case for the waveguides with certain angles of the turns. At some values of E, N the reflection process cannot proceed classically. Then, at the quantum mechanical level its probability is exponentially suppressed, F (E, N) > 0. It is natural to call such a process \"over-barrier reflection\" 6 . The central quantity to be studied below is the suppression exponent F (E, N) of this process. The above discussion suggests that F (E, N), being determined by the entire dynamics in the waveguide, may be a highly non-trivial function. For the particular case of the waveguide with alternating regimes of 6 By this term we want to emphasize that the process is classically forbidden. Recall, however, that there is no actual potential barrier across the waveguide in our setup.\n\n4 classical reflections and transmissions F should oscillate: F = 0 at the energies where the classical reflections are allowed, and F > 0 at the energies where the reflections are classically forbidden. One can expect that the similar oscillatory behavior of the suppression exponent persists for other two-turn models as well. Now, instead of reaching zero, F may possess a number of local positive minima implying that the reflection at the \"optimal\" energies is still a tunneling process.\n\nLet us emphasize the difference of the \"optimal tunneling\" from quantum interference and resonance phenomena in our two-turn model. The interference of the de Broglie waves reflected from the two turns can, in principle, lead to oscillations in the reflection probability P(E). One can estimate the positions of the interference peaks by equating the De Broglie wavelength of the particle to an integer fraction of the distance between the turns, 2π / √ 2mE ∼ L/n. This yields the energies of the interference peaks, E int n ∼ (2πn) 2 2 2mL 2 .\n\nThis formula is completely different from Eq. ( 2 ) for the peaks due to \"optimal tunneling\". In particular, the distance between the adjacent inteference peaks, ∆E int ∼ 2π L 2E m , scales proportional to . Thus, these peaks should be averaged over in the semiclassical limit. Besides, the amplitude of the interference peaks is at most of order one and does not affect the suppression exponent. Indeed, the exponential increase of the scattering amplitude can arise due to quantum interference only in the presence of a resonant state with exponentially long life-time. This state should be supported somewhere in between the turns and should be classically stable. In Sec. 4.2 we show that such states are absent in our system. One concludes that the peak-like structure of the probability P(E) of \"optimal tunneling\" is caused by completely different physical reasons as compared to the case of resonance scattering in quantum theory.\n\nIt is worth noting that the phenomenon of \"optimal tunneling\" has an important implementation in field theory. Recently it was argued [17] (see also Ref. [16] ) that the probability of tunneling induced by particle collisions [18, 19] reaches its maximum at a certain \"optimal\" energy and stays constant 7 at higher energies. This result, if generic, provides the 7 As opposed to the quantum mechanical case, the tunneling probability does not decrease at energies higher than the \"optimal\" one. This is due to the possibility, specific to the field theoretical setup, to emit the excess of energy into a few hard particles, so that tunneling effectively occurs at the \"optimal\" energy. [20] about the high-energy behavior of the probability of collision-induced nonperturbative transitions in field theory. The quantum mechanical model presented here supports the generic nature of the phenomenon of \"optimal tunneling\"; the simplicity of our model enables one to get an intuitive insight into the nature of this phenomenon."
},
{
"section_type": "OTHER",
"section_title": "answer to the long-standing question",
"text": "The paper is organized as follows. In Sec. 2 we review the semiclassical method of complex trajectories, which is exploited in the rest of the paper. Reflections in the waveguides with one and two turns are considered in Secs. 3 and 4 respectively. We discuss our results in Sec. 5. In appendix we analyze the validity of some assumptions made in the main body of the paper."
},
{
"section_type": "METHOD",
"section_title": "The semiclassical method",
"text": "We start by describing the semiclassical method 8 of complex trajectories which will be used in the study of over-barrier reflections. We concentrate on the derivation of the formula for the suppression exponent F (E, N) (see Refs. [2, 8, 9] for the details of the method and Ref. [19] for the field theory formulation). In what follows we use the system of units\n\n= m = ω = 1 ,\n\nwhere the Hamiltonian takes the form,\n\nH = 1 2 p 2 x + p 2 y + w 2 (x, y) . ( 3\n\n)\n\nOne starts with the amplitude of reflection into the state with definite coordinates\n\nx f < 0 , y f , A = x f , y f |e -i Ĥ(t f -t i ) |E, N . ( 4\n\n)\n\nHere |E, N is the initial state of the particle moving in the asymptotic region x i → -∞ with fixed translatory momentum p 0 = 2(E -N) and the oscillator excitation number N. Semiclassically,\n\nx i , y i |E, N = e ip 0 x i cos y i √ 2N p y (y ′ )dy ′ + π/4 , ( 5\n\n)\n\n8 Note that the method has been confirmed by the explicit comparison with the exact quantum mechanical results in Refs. [8, 9, 14] ; specifically, the recent check [14] deals with the case when the dependence of the suppression exponent on energy is not monotonic.\n\nwhere x i , y i denote initial coordinates,\n\np y (y ′ ) = 2N -y ′2 , ( 6\n\n)\n\nand we omitted the pre-exponential factor which is irrelevant for our purposes. Using Eq. ( 5 ), one rewrites the amplitude (4) as a path integral,\n\nA = dx i dy i [dx][dy] x f , y f x i , y i e iS+ip 0 x i cos y i √ 2N p y (y ′ )dy ′ + π/4 , ( 7\n\n)\n\nwhere S is the classical action of the model (3).\n\nIn the semiclassical case the integral (7) is dominated by the (generically complex) saddle point. Note that, as we continue the integrand in Eq. ( 7 ) into the plane of complex coordinates, one of the exponents constituting the initial oscillator wave function grows, while the other becomes negligibly small. Within the validity of our approximation, we omit the decaying exponent by writing\n\ncos y i √ 2N p y (y ′ )dy ′ + π/4 → exp i y i √ 2N p y (y ′ )dy ′ , ( 8\n\n)\n\nwith the standard choice 9 of the branch of the square root in Eq. ( 6 ). One proceeds by finding the saddle point for the integral (7) with the substitution (8). Extremization with respect to x(t), y(t) leads to the classical equations of motion,\n\nẍ = -ww x , ÿ = -ww y . ( 9\n\n)\n\nDifferentiating with respect to\n\nx i ≡ x(t i ), y i ≡ y(t i ), one obtains, ẋi = p 0 = 2(E -N) , ẏi = p y (y i ) = 2N -y 2 i .\n\nThe latter equations are equivalent to fixing the total energy E and initial oscillator energy N of the complex trajectory,\n\nE = 1 2 ẋ2 i + N , ( 10a\n\n) N = 1 2 ẏ2 i + y 2 i . ( 10b\n\n)\n\n9 The correct branch is fixed by drawing a cut between the oscillator turning points y = ± √ 2N , and choosing Im p y > 0 at y ∈ R, y > √ 2N , see, e.g., Refs. [21].\n\nSubstituting the saddle-point configuration 10 into Eq. ( 7 ), one obtains the amplitude of the process with exponential accuracy, A ∝ e iS+iB(x i , y i ) ,\n\nwhere the term\n\nB(x i , y i ) = p 0 x i + y i √ 2N p y (y ′ )dy ′ ( 11\n\n)\n\nis the initial-state contribution. For the inclusive reflection probability one writes,\n\nP = dx f dy f |A| 2 ∝ dx f dy f\n\ne iS-iS * +iB-iB * .\n\nThe integral over the final states can also be evaluated by the saddle point technique; extremization with respect to x f ≡ x(t f ), y f ≡ y(t f ) fixes the boundary conditions in the asymptotic future,\n\nIm ẋf = Im x f = 0 , Im ẏf = Im y f = 0 . ( 12\n\n)\n\nIn this way one obtains the expression (1) for the reflection probability, where the suppression exponent F is given by the value of the functional\n\nF (E, N) = 2 Im S + 2 Im B(x i , y i )\n\nevaluated on the saddle-point configuration -a complex trajectory satisfying the boundary value problem (9), (10), (12). The contribution B(x i , y i ) of the initial state is simplified after one uses the asymptotic form of the solution at t → -∞ (x i → -∞),\n\nx = p 0 t + x 0 , y = ae -it + āe it . ( 13\n\n)\n\nEquations (10) guarantee that the quantities p 0 = 2(E -N) and 2aā = N are real, since E, N ∈ R. Therefore, one may introduce two real parameters T , θ as follows,\n\n2 Im x 0 = -p 0 T , ā = a * e T +θ . ( 14\n\n)\n\nOne finds for the initial term (11),\n\n2 Im B(x i , y i ) = Im 2p 0 x i -2Narccos(y i / √ 2N) + y i 2N -y 2 i = -p 2 0 T -N(T + θ) + Im(y i ẏi ) ,\n\n10 For simplicity we assume that the saddle-point configuration is unique. Otherwise, one should take the saddle point corresponding to the weakest exponential suppression.\n\n8 and thus\n\nF = 2 Im S -ET -Nθ , ( 15\n\n)\n\nwhere S is the classical action of the system (3) integrated by parts, S = -1 2\n\nt f t i dt xẍ + y ÿ + w 2 (x, y) . ( 16\n\n)\n\nLet us comment on the physical meaning of the parameters T , θ. Consider two trajectories which are solutions to the boundary value problem (9), (10), (12) at neighbouring values of E, N. The differential of the quantity 2 Im S as one deforms one trajectory into the other is\n\nd (2 Im S) = d Im(2S + x i ẋi + y i ẏi ) = Im(x i d ẋi -ẋi dx i + y i d ẏi -ẏi dy i ) = EdT + Ndθ ,\n\nwhere in the last equality we used the asymptotic form (13), (14) of the solution. Then, from Eq. (15) one finds, dF (E, N) = -T dE -θdN . (17) Thus, the parameters T and θ are (up to sign) the derivatives of the suppression exponent with respect to energy E and initial oscillator excitation number N respectively. Our final remark is that the boundary value problem (9), (10), (12) is invariant with respect to the trivial time translation symmetry,\n\nt → t + δt , δt ∈ R , ( 18\n\n)\n\nwhich can be fixed in any convenient way.\n\n3 The model with one turn\n\nTo warm up, we consider the simplest model, where the waveguide has one sharp turn,\n\nw = y θ(-x + y tg β) + cos β (x sin β + y cos β) θ(x -y tg β) . ( 19\n\n)\n\nHere θ(x) is the step function. It is convenient to use the rotated coordinate system, ξ η = cos β -sin β sin β cos β x y .\n\nThe waveguide function takes the form,\n\nw = η cos β -ξ sin β θ(-ξ) . ( 20\n\n) 9 x y ξ η A B C β β x y ξ η A B C β β x y ξ η A B C β β\n\nFigure 2: The equipotential contour w 2 (x, y) = 2N for the waveguide (20) and the trajectory of the critical solution with energy N/ cos 2 β.\n\nThe equipotential contour w 2 (ξ, η) = const is shown in Fig. 2 . One observes that the motion of the particle in two regions, ξ < 0 and ξ > 0, decomposes into the translatory motion and oscillations in the coordinates x, y and ξ, η respectively (see. Eqs. (19) and (20)); the frequency of η-oscillations in the latter case is cos β. Due to the presence of the step function, the first derivatives of the potential (20) are discontinuous 11 at ξ = 0. Strictly speaking, the semiclassical method is not applicable in this situation [21] . Thus, the formula (20) should be regarded as an approximation to some waveguide function with smooth turn. Generically the width of the smoothened turn is characterized by a parameter b; the sharp-turn approximation (20) corresponds to b → 0. An example of smoothening is provided by the following substitution in Eq. (20),\n\nθ(ξ) → θ b (ξ) = 1 1 + e -ξ/b . ( 21\n\n)\n\nThe semiclassical description can be used as long as the de Broglie wavelength of the particle is small compared to the linear size of the potential 12 , 1/ √ E ≪ b. We conclude that the sharp-turn and semiclassical approximations are valid simultaneously for smooth waveguides with\n\n1 ≫ b ≫ 1/ √ E . ( 22\n\n)\n\n11 Note that the potential itself is continuous. 12 Another semiclassical condition is that the energy is sufficient to excite a lot of oscillator levels, E ≫ 1.\n\nIt is satisfied provided Eq. (22) holds.\n\n10 An important property of the model (20) is invariance of the classical equations of motion (9) under the rescaling of the coordinates,\n\nx → Λx , y → Λy . ( 23\n\n)\n\nUsing the transformation (23), one may express a solution x(t), y(t) with energy E in terms of the \"normalized\" one, x = x√ E , y = ỹ√ E ,\n\nwhere the solution x(t), ỹ(t) has unit energy; its initial oscillator excitation number is ν = N/E .\n\nThe suppression exponent (15) takes the form,\n\nF (E, N) = Ef β (ν) , ( 24\n\n)\n\nwhere f β (ν) is the exponent for the \"normalized\" solution. Substituting the expression (24) into Eq. ( 17 ), one obtains,\n\nf β (ν) = -T -θν . ( 25\n\n)\n\nWe will exploit Eq. ( 25 ) in the end of this section. Now, we proceed to finding the \"normalized\" trajectories. At certain initial data ν > ν cr the particle can reflect from the turn classically, so that\n\nf β (ν > ν cr ) = 0 .\n\nLet us find the value of ν cr . In the region ξ < 0 the classical solution takes the form,\n\nx(t) = p 0 t + x 0 , ( 26a\n\n) y(t) = A 0 sin(t + ϕ) . ( 26b\n\n)\n\nHaving crossed the line ξ = 0 (line AB in Fig. 2 ), the classical particle can never return back into the region ξ < 0. Indeed, in this case it moves at ξ > 0 with constant momentum p ξ > 0. Thus, the particle can reflect classically only if its trajectory touches the line ξ = 0. The potential of our model has ill-defined derivatives at ξ = 0, and the fate of the particle moving along the line AB depends on the particular choice of the smoothening of the potential. In appendix we consider the motion of the classical particle in the case when nonzero smoothening of width b is switched on. For a class of smoothenings we show that 11 in the small vicinity (δξ ∼ b) of any trajectory touching the line ξ = 0 there exists some \"smoothened\" trajectory, which reflects classically from the turn. Consequently, below we associate the trajectories touching the line ξ = 0 with the classical reflected solutions. One notices that the inclination of the trajectory (26) is bounded from above\n\ndy dx ≤ A 0 p 0 ;\n\ntherefore, the classical trajectory of the particle can touch the line ξ = 0, that is, y/x = ctg β only at\n\nA 0 /p 0 ≥ ctg β . ( 27\n\n)\n\nFrom Eqs. (27), (26), (10) one extracts the condition for the particle to reflect classically from the turn,\n\nν ≥ ν cr = cos 2 β . ( 28\n\n)\n\nThe critical classical solution at ν = ν cr touches the line ξ = 0 at η = 0 (point C in Fig. 2 ), where its trajectory\n\nx cr (t) = √ 2t sin β , ( 29\n\n) y cr (t) = √ 2 sin t cos β .\n\nhas the largest inclination. We now turn to the classically forbidden reflections at ν < ν cr , which are described by the boundary value problem (9), (10), (12). One makes the following important observation. The waveguide function (20) has the form of two analytic functions glued together at ξ = 0. Hence, the equations of motion (9) can be continued analytically to the complex values of coordinates in two different ways, starting from the regions ξ < 0 and ξ > 0 respectively. In this way one obtains two complex solutions, ξ -(t), η -(t) and ξ + (t), η + (t). These solutions and their first derivatives should be matched at some moment of time t 1 , ξ(t 1 ) = 0. [Note that the matching time t 1 does not need to be real.] Below we conventionally refer to these solutions as the ones belonging to the regions ξ < 0 and ξ > 0. By the same reasoning as above we find that once the particle arrives into the region ξ > 0, it never reflects back to ξ < 0, unless p ξ = 0. So, in the region ξ > 0 one writes,\n\nξ + (t) = 0 , ( 30a\n\n) η + (t) = √ 2 cos β sin(t cos β + ϕ η ) , ( 30b\n\n) 12\n\nwhere the \"normalization\" condition E = 1 has been used explicitly. Due to the conditions in the asymptotic future, Eqs. (12), the parameter ϕ η is real. We use the translational invariance (18) to set ϕ η = 0. Note that we again associate the trajectory going along the line ξ = 0 with the reflected one.\n\nThe physical picture of over-barrier reflection that comes to mind matches with the new mechanism of multidimensional tunneling proposed recently in Refs. [9, 11] . The process proceeds in two steps. The first step, which is exponentially suppressed, is formation of the periodic classical orbit (30) oscillating along the line ξ = 0. This orbit is unstable. At the second step of the process the unstable orbit decays classically forming a trajectory going back to x → -∞ at t → +∞. Clearly, the second step does not affect the suppression exponent of the whole process, and we do not consider it explicitly. In what follows we concentrate on the determination of the tunneling trajectory describing the first step of the process.\n\nOne should find the solution at ξ < 0 and impose the boundary conditions (10). Note, however, that the energy of our solution is fixed already. As for the initial oscillator excitation number ν, it does not change during the evolution in the region ξ < 0. Thus, one may fix it at the matching time t = t 1 . One writes,\n\nν = 1 2 ( ẏ2 + y 2 ) t=t 1 = cos 2 β + sin 2 β sin 2 (t 1 cos β) .\n\nThis complex equation allows one to express t 1 as sin(t\n\n1 cos β) = -i √ ν cr -ν sin β , ( 31\n\n)\n\nwhere the choice of the sign is dictated by the condition in footnote 9. It is convenient to introduce notation t 1 = iT 1 , T 1 ∈ R.\n\nIn order to find the suppression exponent f β (ν), one needs to evaluate the parameters T (ν), θ(ν). At ξ < 0 the solution has the form,\n\nx -(t) = p 0 (t -iT /2) + x ′ 0 , ( 32a\n\n) y -(t) = ae -it + a * e T +θ+it , ( 32b\n\n)\n\nwhere the definitions (13), (14) have been taken into account explicitly, so that p 0 , x ′ 0 ∈ R. One evaluates p 0 , x ′ 0 , a, T , θ by matching the coordinates x ± , y ± and their first derivatives 13 0.2 0.15 0.1 0.05 0 ν cr 0.2 0.15 0.1 0.05 0 ν f β Figure 3: The suppression exponent f β (ν) for the waveguide (20); β = π/3. ẋ± , ẏ± at t = iT 1 ; this yields\n\nx ′ 0 = 0 , p 0 = 2(1 -ν) , a = i ν 2 e -T +θ 2 , T 1 - T 2 = - 1 -ν/ cos 2 β 1 -ν , sh T 1 - T + θ 2 = - cos 2 β -ν sin β √ ν\n\nThe last two equations, together with Eq.( 25 ), define the function f β (ν),\n\nf β (ν) = 2 cos β arcsh √ ν cr -ν sin β -ν cos β arcsh √ ν cr -ν sin β √ ν -(ν cr -ν)(1 -ν) ;\n\nthis finction is plotted in Fig. 3 . One observes that at ν → ν cr the quantities T 1 , T, θ, f β tend to zero, and the complex trajectory tends to the classically allowed critical solution, cf. Eqs. (29),\n\np 0 → √ 2 sin β , a → i √ 2 cos β .\n\nAt ν = 0 one has,\n\nf β (0) = -2 + 2 cos β arcth (cos β) . ( 33\n\n)\n\nTo summarize, we obtained the suppression exponent for the reflection of a particle in the simplest waveguide with one sharp turn.\n\n14 L x' x y' y ξ η A B A' B' C C' β α α L x' x y' y ξ η A B A' B' C C' β α α L x' x y' y ξ η A B A' B' C C' β α α L x' x y' y ξ η A B A' B' C C' β α α\n\nFigure 4: The equipotential contour w 2 (x, y) = 2N ′ for the waveguide (35) and the trajectory of the critical solution with energy N ′ / cos 2 β > E B . The matching points C, C ′ are shown by the thick black dots.\n\n4 The model with two turns"
},
{
"section_type": "OTHER",
"section_title": "Introducing the system",
"text": "In the model of the previous section the suppression exponent was proportional to energy because of the coordinate rescaling symmetry (23) . Now, we are going to demonstrate that small violation of this symmetry results in highly non-trivial graph for F (E). One introduces a second turn into the waveguide, see Fig. 4 . We want to consider this turn as a small perturbation, so, we assume its angle α to be smaller than β. It is convenient to introduce two additional coordinate systems, x ′ , y ′ and ξ, η, bound to the central and rightmost parts of the waveguide respectively. They are related to the original coordinate system x, y as follows,\n\nx ′ y ′ = cos α sin α -sin α cos α x y , ξ η = cos β -sin β sin β cos β x ′ -L y ′ .\n\n(34) Note that the origin of the coordinate system ξ, η is shifted by the distance L. The waveguide 15 function is\n\nw = θ(-x ′ )θ(-ξ)y + θ(-ξ)θ(x ′ )y ′ cos α + θ(ξ)η cos α cos β ; ( 35\n\n)\n\nit consists of three pieces glued together continuously at x ′ = 0 and ξ = 0 (lines A ′ B ′ and AB in Fig. 4 respectively). At t → -∞ the particle comes flying from the asymptotic region\n\nx ′ < 0, where w = y.\n\nIn the intermediate region x ′ > 0, ξ < 0 the particle moves in the x ′ direction oscillating along the y ′ coordinate with the frequency cos α. Finally, in the region ξ > 0 its motion is free in the coordinates ξ, η; the frequency of η-oscillations is cos α cos β.\n\nThe model (35) no longer possesses the symmetry (23): rescaling of coordinates changes the length L of the central part of the waveguide. In what follows it is convenient to work in terms of the rescaled dynamical variables, x = x/L , ỹ = y/L .\n\nIn new terms the parameter L disappears from the classical equations of motion, entering the theory through the overall coefficient L 2 in front of the action. The initial-state quantum numbers are also proportional to\n\nL 2 , E = L 2 Ẽ , N = L 2 Ñ . ( 36\n\n)\n\nThus, the conditions (22) for the validity of the semiclassical approximation are satisfied in the limit L → ∞ , Ẽ, Ñ = fixed .\n\nThe suppression exponent takes the form\n\nF (E, N) = L 2 F ( Ẽ, Ñ) . ( 37\n\n)\n\nTo simplify notations, we omit tildes over the rescaled quantities in the rest of this section. Rescaling back to the physical units can be easily performed in the final formulae by implementing Eqs. (36), (37)."
},
{
"section_type": "OTHER",
"section_title": "Classical evolution",
"text": "Let us begin this subsection by demonstrating that there are no stable classical solutions localized in the region between the turns. This is important for the determination of the tunneling probability, since such stable solutions could lead to exponential resonances in the tunneling amplitude. The argument proceeds as follows. Any trajectory which is localized in 16 the intermediate region should reflect from the line AB infinitely many times. Each reflection involves touching the unstable orbit living at the line AB. This implies that the trajectory itself is unstable. We proceed by determining the region of initial data E, N, which correspond to the classical reflections. [For brevity we will refer to this region as the \"classically allowed region\", as opposed to the \"classically forbidden region\" where reflections occur only at the quantum mechanical level. We stress that these are the regions in the plane of quantum numbers E, N.] Let us search for the critical classical solutions which correspond to the smallest initial oscillator number N = N cr (E) at given energy E. As in the previous section, one finds that the particle must get stuck at the line 13 AB for some time in order to reflect back. Let us first make an assumption inspired by the study of the one-turn model that the critical solutions touch the line AB at their maximum inclination point (point C in Fig. 4 ).\n\nWe will see shortly that this is true only at energies above a certain value E B , see Eq. (50). Still, the analysis based on the above assumption enables one to catch the qualitative features of the critical line N = N cr (E). Besides, the analysis is considerably simplified in this case;\n\nwe postpone the accurate study until the end of this subsection. Keeping in mind the above remarks, one writes for the solution in the intermediate region,\n\nx ′ cr (t) = t √ 2E sin β + 1 , ( 38a\n\n) y ′ cr (t) = √ 2E cos β cos α sin(t cos α) . ( 38b\n\n)\n\nBefore entering the intermediate region, the particle crosses the line A ′ B ′ (point C ′ in Fig. 4 ).\n\nThe initial oscillator number N is most conveniently calculated at the moment t = t 0 ≡ -1 √ 2E sin β of crossing. Using the relations (34) one obtains,\n\nẋcr (t 0 ) = √ 2E sin β cos α -cos β sin α cos cos α √ 2E sin β , ( 39\n\n)\n\nand thus\n\nN cr (E) = E - 1 2 ẋ2 cr (t 0 ) = E -E sin β cos α -cos β sin α cos cos α √ 2E sin β 2 , E > E B . ( 40\n\n) 17 0 0.05 0.1 0.15 0.2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.1 0\n\nN/α 2 E/α 2 E B E 2 E 3 E 4\n\nFigure 5: The boundary N = N cr (E) of the classically allowed region at E > E B for the waveguide model (35); β = π/3, α = π/30. The region of the classically allowed initial data lies above this boundary. The empty circles correspond to the energies E = E n , where the curve N = N cr (E) touches its lower envelope N = E cos 2 (β + α).\n\nAs an example, we show in Fig. 5 the region of the classically allowed initial data for β = π/3, α = π/30. One observes that the function N cr (E) oscillates between two linear envelopes, E cos 2 (β + α) and E cos 2 (β -α); the period of oscillations decreases as E → 0. Moreover, the curve N cr (E) has a number of minima at the points E = E cr n . This means that the energies E = E cr n are optimal for reflection: in the vicinity of any point\n\nE = E cr n , N = N cr (E cr n\n\n) reflections become exponentially suppressed independently of whether the energy gets increased or decreased. This feature is particularly pronounced in the case α + β = π/2, when the lower envelope coincides with the line N = 0. Then, the classical reflections (i.e.\n\nreflections with the probability of order 1) at N = 0 are possible only in the vicinities of the points\n\nE = 1 8π 2 (n -1/2) 2 .\n\nThis is the case we used in Introduction to illustrate the effect.\n\nThe minima E = E cr n exist at other values of the parameters as well. For instance, let 13 We do not consider reflections from the line A ′ B ′ . They disappear at larger values of N than reflections from the line AB if α is small enough.\n\n18 us find the positions of these minima in the case α ≪ 1. One differentiates Eq. (40) with respect to energy and obtains,\n\nE cr n = E n 1 - 1 π(n -1/2) arcsin ctg β 2πα(n -1/2) + O(α 2 ) , ( 41\n\n)\n\nwhere\n\nE n = 1 8π 2 (n -1/2) 2 sin 2 β ( 42\n\n)\n\nare the points where the curve N = N cr (E) touches its lower envelope. The argument of arcsine in Eq. (41) should be smaller than one, so, the minima E cr n exist only at large enough n,\n\nn ≥ n 0 ≡ ctg β 2πα + 1 2 + 1 , ( 43\n\n)\n\nwhere [•] stands for the integer part. Let us make several comments. First, note that n 0 ∼ O(1/α), consequently, all the optimal points E cr n lie in the region of small energies E ∼ 1/n 2 0 ∼ O(α 2 ). Second, as we pointed out before, the formula (40) for the function N cr (E) holds at E > E B . Comparing the expressions (42), (43) and (50), one observes that E n 0 > E B if tg β > 1. So, there does exist a range of energies where the non-monotonic behavior of the function N cr (E) can be inferred from the formula (40). In fact, the conclusion about the existence of the local minima of N cr (E), as well as the expressions (41), (42), (43) determining their positions, remain valid also at E < E B . This follows from the rigorous analysis of the boundary of the classically allowed region to which we turn now. The reader who is more interested in the tunneling processes may skip this part and proceed directly to subsection 4.3. Ansatz (38) . Instead, we start with the general solution in the intermediate region,"
},
{
"section_type": "OTHER",
"section_title": "Now, we do not appeal to the",
"text": "x ′ = p ′ 0 (t -t 0 ) , ( 44a\n\n) y ′ = A ′ 0 sin [(t -t 0 ) cos α + ϕ ′ ] . ( 44b\n\n)\n\nIt is convenient to parametrize it by the total energy E = p ′2 0 /2 + cos 2 αA ′2 0 /2 and the \"inclination\" γ defined by the relation\n\np ′ 0 /A ′ 0 = tg γ cos α .\n\nExpressions (44) take the following form,\n\nx ′ = √ 2E (t -t 0 ) sin γ , ( 45a\n\n)\n\ny ′ = √ 2E cos γ cos α sin [(t -t 0 ) cos α + ϕ ′ ] . ( 45b\n\n) 19\n\nThe constants t 0 and ϕ ′ are fixed by demanding the trajectory (45) to reflect classically from the second turn, i.e. touch the line ξ = 0 at t = 0,\n\n(x ′ -1) cos β -y ′ sin β t=0 = 0 , dy ′ dx ′ t=0 = ctg β .\n\nThese conditions imply,\n\nt 0 = - 1 √ 2E sin γ + 1 cos α tg 2 β tg 2 γ -1 , ( 46a\n\n)\n\nϕ ′ = - cos α √ 2E sin γ + tg 2 β tg 2 γ -1 -arccos tg γ tg β . ( 46b\n\n)\n\nOne sees that the classical reflections are possible only at γ ∈ [0; β]; the boundary value γ = β reproduces the solution (38). In order to find N cr (E), one should minimize the value of the incoming oscillator excitation number with respect to γ at fixed E. At t = t 0 , when the particle crosses the first turn,\n\np 0 ≡ ẋ(t 0 ) = √ 2E(cos α sin γ -sin α cos γ cos ϕ ′ ) . ( 47\n\n)\n\nSince N = E -p 2 0 /2, one can maximize the value of the translatory momentum p 0 instead of minimizing N(γ). Formula (39) represents the value γ = β lying at the boundary of the accessible γ-domain; this value should be compared to p 0 (γ) taken at local maxima.\n\nLet us consider the case α ≪ 1. At large enough energies, E ∼ 1, Eq. (47) is dominated by the first term, which grows with γ, so that the maximum of p 0 (γ) is indeed achieved at γ = β. At small energies, however, the second term in Eq. (47) becomes essential because of the quickly oscillating cos ϕ ′ multiplier: the frequency of cos ϕ ′ oscillations grows as E → 0, and at E ∼ α 2 , in spite of the small magnitude proportional to sin α, the second term produces the sequence of local maxima of the function p 0 (γ).\n\nOne expects the parameters of the trajectory at small α not to be very different from the ones at α = 0 (the latter case was considered in Sec. 3). So, we write,\n\nγ = β -δγ ,\n\nwhere 0 < δγ ≪ 1. Expanding the expressions (46), (47) and taking into account that E ∼ α 2 one obtains,\n\nϕ ′ = - 1 √ 2E sin β (1 + δγ ctg β) , ( 48a\n\n) p 0 = √ 2E(sin β -δγ cos β -α cos β cos ϕ ′ ) . ( 48b\n\n)\n\n20 Now, the local maxima of the initial translatory momentum can be obtained explicitly by differentiating Eqs. (48) with respect to δγ. One finds the sequence of them,\n\nδγ n = -tg β + √ 2E sin 2 β cos β 2πn -π -arcsin √ 2E sin 2 β α cos β . ( 49\n\n)\n\nOnly the maxima with δγ n > 0 should be taken into account. The local maxima exist when\n\nE ≤ E B ≡ α 2 cos 2 β 2 sin 4 β . ( 50\n\n)\n\nSubstituting Eq. (49) into the expressions (48), one evaluates the values of p 0 at the local maxima,\n\np 0,n (E) =2 √ 2E sin β -2E sin 2 β 2πn -π -arcsin √ 2E sin 2 β α cos β + α √ 2E cos β 1 - 2E sin 4 β α 2 cos 2 β .\n\nThe graphs N n (E) = E -p 2 0,n (E)/2 are shown in Fig. 6 for the case β = π/3, α = π/30. Each graph is plotted for the energy range E > E An restricted by the condition δγ n > 0. They are presented together with the curve given by the formula (40). By definition, the critical solution corresponds to the lowest of these graphs. Clearly, for each \"local\" curve representing the n-th local minimum of N(γ) there is a range of energies E An < E < E Bn where it lies lower than the \"global\" curve (40). This means that the parameter γ of the critical solution changes discontinuously across the points E = E Bn . Correspondingly, the curve N cr (E) has a break at these points. On the other hand, the function N cr (E) is smooth at the points A n as the \"local\" graphs end up exactly at δγ = 0, where the parameters of the n-th \"local\" solution coincide with the ones of the \"global\" solution.\n\nTo summarize, we have observed that the boundary of the classically allowed region is given by a collection of many branches of classical solutions, each branch being relevant in its own energy interval. We will see that a similar branch structure is present in the complex trajectories describing over-barrier reflections in the classically forbidden region of E, N."
},
{
"section_type": "OTHER",
"section_title": "Classically forbidden reflections",
"text": "In this subsection we demonstrate that the suppression exponent F (E, N) viewed as a function of energy at fixed N exhibits oscillations deep inside the classically forbidden region 21 0 0.02 0.04 0.06 0 0.05 0.1 0.15 0.2 N/α 2 E/α 2 4 5 6 4 5 6 A 4 B 4 A 5 B 5 A 6 B 6 Figure 6: The graphs N n (E) corresponding to the local minima of the function N(γ) (dashed lines) plotted together with the \"global\" curve, Eq. (40) (solid line); β = π/3, α = π/30. The critical curve N = N cr (E) is obtained by taking the minimum among all the graphs. of initial data. This result comes without surprise if one takes into account the non-monotonic behavior of the boundary N cr (E) of the classically allowed region. Indeed, the curve N = N cr (E) coincides with the line F (E, N) = 0. One has,\n\ndN cr dE = - ∂ E F ∂ N F N =Ncr(E) , so that ∂F ∂E (E cr n , N cr n ) = 0 .\n\nWe conclude that the points E = E cr n are the local minima of the function F (E) at fixed N = N cr n . It is natural to expect that such local minima of F (E) exist at other values of N as well. To illustrate this fact explicitly, we study the complex trajectories, solutions to Eqs. (9), (10), (12) .\n\nFollowing the tactics of the previous section, we find solutions in three separate regions: initial region x ′ < 0, final region ξ > 0, and the intermediate region x ′ > 0, ξ < 0. These solutions, together with their first derivatives, should be glued at t = t 0 , when the complex trajectory crosses the line x ′ = 0, and at t = t 1 , when ξ = 0. Besides, we are looking for the 22 tunneling solution which ends up oscillating along the line AB, see Fig. 4. As discussed in Sec. 3 this assumes existence of the second step of the process: classical decay of the unstable orbit living at ξ = 0; the latter decay is described by a real trajectory 14 going to x → -∞ at t → +∞.\n\nThe solution in the final region ξ > 0 is (cf. Eqs. (30)),\n\nξ + (t) = 0 , ( 51a\n\n) η + (t) = √ 2E cos α cos β sin(t cos α cos β) , ( 51b\n\n)\n\nwhere we used the time translation invariance (18) to fix the final oscillator phase ϕ η = 0. In the intermediate region x ′ > 0, ξ < 0 one writes,\n\nx ′ (t) = p ′ 0 t + x ′ 0 , ( 52a\n\n) y ′ (t) = a ′ e -it cos α + ā′ e it cos α . ( 52b\n\n)\n\nNote that the final solution (51) does not contain free parameters; thus, the matching of x ′ , ẋ′ , y ′ , ẏ′ at t = t 1 enables one to express all the parameters in Eqs. (52) in terms of one complex variable t 1 ,\n\np ′ 0 = √ 2E sin β cos φ 1 , ( 53a\n\n) x ′ 0 = 1 + √ 2E tg β cos α [sin φ 1 -φ 1 cos φ 1 ] , ( 53b\n\n)\n\na ′ = E/2 cos α e iφ 1 / cos β [sin φ 1 + i cos β cos φ 1 ] , ( 53c\n\n) ā′ = E/2 cos α e -iφ 1 / cos β [sin φ 1 -i cos β cos φ 1 ] , ( 53d\n\n)\n\nwhere we introduced φ 1 = t 1 cos α cos β.\n\nAs the energy of the solution has been fixed already, the only remaining initial condition involves initial oscillator excitation number at x ′ < 0, see Eqs. (10) . It is convenient to impose this condition at the matching point t = t 0 . One recalls the definition of the matching time\n\nt 0 , p ′ 0 t 0 + x ′ 0 = 0 ,\n\n14 One wonders why this trajectory does not reflect from the turn A ′ B ′ on its way back. This concern is removed by the observation that the trajectory produced in the decay of the unstable orbit is not unique: in appendix we show that the decay can occur at any point of the segment AC giving rise to a whole bunch of potential decay trajectories. Most of these trajectories pass through the turn A ′ B ′ without reflection.\n\nwhich, after taking into account the expressions (53a), (53b), leads to the following equation,\n\ncos α √ 2E sin β + sin φ 1 cos β -cos φ 1 ∆φ = 0 , ( 54\n\n)\n\nwhere ∆φ = cos α(t 1 -t 0 ). At t = t 0 one has,\n\nẋ(t 0 ) = p ′ 0 cos α -ẏ′ (t 0 ) sin α = 2(E -N) ,\n\nand thus √ 1 -ν sin α = ctg α sin β cos φ 1 -sin φ 1 sin ∆φ -cos β cos φ 1 cos ∆φ . (55)\n\nAs before, ν = N/E.\n\nTwo complex equations (54), (55) determine the matching times t 0 , t 1 , and, consequently, the complex trajectory. Although these equations cannot be solved explicitly, they can be simplified in the case α ≪ 1, which we consider from now on. For concreteness, we study reflections at = 0. It is important to keep in mind that in the region of interest E ∼ E cr n ∼ O(α 2 ); thus, one should regard all the momenta p and oscillator amplitudes a, ā, as the quantities of order O(α). At the same time, for the distances along the waveguide one has x ∼ O(1), so that the real parts of time intervals may be parametrically large, Re t ∼ x/p ∼ O(1/α).\n\nFurther on, it will be convenient to work in terms of real variables, so, we represent φ 1 and ∆φ as φ 1 = cos α cos β(τ 1 + iT 1 ) , ∆φ = cos α(τ + i∆T ) .\n\nNote that τ and ∆T are the real and imaginary parts of the time interval t 1 -t 0 which the particle spends in the intermediate region. Now, equation (54) enables one to express\n\nτ = 1 √ 2E sin βch(T 1 cos β) + O(α) , ( 56\n\n) τ 1 = - 1 τ cos β 1 cos β -∆T cth(T 1 cos β) + O(α 3 ) . ( 57\n\n)\n\nNote that τ 1 ∼ O(α), τ ∼ O(1/α). Then, the real part of Eq. (55) implies that ch(T\n\n1 cos β) = 1 sin β 1 + α ctgβ cos τ e ∆T + O(α 2 ) . ( 58\n\n)\n\nWhile deriving this formula we imposed T 1 < 0 which follows from the requirement that in the limit α → 0 equation (31) should be recovered; besides, we assumed e ∆T ∼ O(1).\n\n24 Substituting Eq. (58) into Eq. ( 56 ) and the imaginary part of Eq. ( 55 ), we obtain the final set of equations,\n\n1 -τ √ 2E = α ctgβ cos τ e ∆T + O(α 2 ) , ( 59a\n\n) (1 + ∆T )e -∆T = α ctgβτ sin τ + O(α) . ( 59b\n\n)\n\nThese two nonlinear equations, still, cannot be solved explicitly. Nevertheless, one can get a pretty accurate idea about the structure of their solutions. Before proceeding to the analysis of the above equations, let us derive a convenient expression for the suppression exponent F 0 (E) ≡ F (E, N = 0). Note that on general grounds one expects to obtain an expression of the form,\n\nF 0 (E) = E(f β (0) + O(α)) ,\n\nwhere f β (0) is given by Eq. (33) . We are interested in the O(α) correction in this expression, so, one must be careful to keep track of the subleading terms during the derivation.\n\nMaking use of the equations of motion, one obtains for the incomplete action (16) of the system, 2 Im S = Im p ′ 0 = √ 2E sin β Im(cos φ 1 ) .\n\nSubstitution of Eqs. (56), (57), (58) into this formula yields 2 Im S = 2E -1 -∆T -α ctg β cos τ e ∆T 1 + 1 cos 2 β + 2∆T + O(α 2 ) .\n\nFor the parameter T one has (see Eqs. (14)),\n\nT = - 2 Im x 0 p 0 = - 2 Im(x(t 0 ) -p 0 t 0 ) p 0 = 2(T 1 -∆T ) + 2 E sin α Im y ′ (t 0 ) , ( 60\n\n)\n\nwhere in the last equality we used Eqs. (34) and x ′ (t 0 ) = 0. The quantity Im y ′ (t 0 ) is evaluated by using Eqs. (52b), (53) and (58); one finds,\n\nIm y ′ (t 0 ) = - √ 2E ctg β cos τ e ∆T + O(α) .\n\nSubstituting everything into the formula (15), we obtain,\n\nF 0 (E) = E f β (0) -4α ctg β cos τ ∆T e ∆T + O(α 2 ) . ( 61\n\n)\n\nThis expression implies that determination of the O(α) correction to the suppression exponent involves finding τ , ∆T with O(1)-accuracy. This is precisely the level of accuracy of 25 Eqs. (59). Below we will also need the following formulae, which can be easily obtained by using T = -F . E . and Eq. ( 60 ),\n\ndF 0 dE = f β (0) + 2(∆T + 1) + O(α) , ( 62\n\n) d dE F 0 E = 2(∆T + 1 + O(α)) E . ( 63\n\n)\n\nNote that, though the suppression exponent differs from that in the one-turn case only by O(α) correction, its derivative gets modified in the zeroth order in α.\n\nNow, we are ready to analyze Eqs. (59). One begins by solving Eq. (59b) graphically, see Fig. 7. The important property of this equation is as follows. One notices that the l.h.s. of Eq. (59b) is always smaller than 1, the maximum being achieved at ∆T = 0. Therefore, the solutions to this equation are confined to the bands τ sin τ < tgβ α .\n\nThis corresponds to\n\nτ ∈ [0; 2π(n 1 -1) + δτ n 1 ] or τ ∈ [2πn -π -δτ n ; 2πn + δτ n ] , n ≥ n 1 ( 64\n\n)\n\nwhere\n\nδτ n = arcsin tgβ 2πα(n -1/2) + O(α) , n 1 = tgβ 2πα + 1 2 + 1 , ( 65\n\n)\n\nwith [•] in the last formula standing for the integer part. The forbidden bands, where τ sin τ > tgβ/α, are marked in Fig. 7 by yellow shading. The property (64) introduces a topological classification of the solutions τ , ∆T to Eqs. (59). Namely, these solutions fall into a set of continuous branches: the \"local\" branches τ n (E), ∆T n (E) living inside the\n\nstrips τ ∈ [2πn -π -δτ n ; 2πn + δτ n ], n ≥ n 1 , and the \"global\" branch τ g (E), ∆T g (E)\n\ninhabiting the very first band τ ∈ [0; 2π(n 1 -1) + δτ n 1 ]. As follows from the definition of τ , the topological number n counts the number of y ′ -oscillations during the evolution in the intermediate region.\n\nLet us consider the \"global\" branch. From Eqs. (59) one has,\n\nτ g → 2π(n 1 -1) + O(α ln α) , ∆T g → ln(tg β/α) , E → 0 , τ g → 0 , ∆T g → -1 , E → +∞ . 26 -2 -1 0 1 2 3 4 10π 9π 8π 7π 6π 5π 4π 3π 2π π 0 ∆T τ g 4 5 -2 -1 0 1 2 3 4 10π 9π 8π 7π 6π 5π 4π 3π 2π π 0 ∆T τ g 4 5\n\nFigure 7: Curves representing solutions to Eq. ( 59b ); β = π/3, α = π/30.\n\nBy inspection of Fig. 7 one can work out the qualitative behavior of the functions τ g (E), ∆T g (E). Alternatively, these functions can be found numerically. They are plotted in Fig. 8 for the case β = π/3, α = π/30 (the curves marked with \"g\"). One observes that at high enough energies the function ∆T g (E) exhibits oscillations around the line ∆T = -1.\n\nAccording to the formula (63) this means that the function F 0 (E)/E is non-monotonic, it attains local minima at the points\n\nE ′ n = 1 8π 2 (n -1/2) 2 1 + 2αe -1 ctgβ + O(α 2 ) . ( 66\n\n) Moreover, if n ≥ n ′ 0 ≡ tg β 4πα f β (0) exp 1 + f β (0) 2 + 1 2 + 1 ( 67\n\n) there exist E o n = E ′ n (1 + O(α)), such that ∆T (E o n ) = -1 -f β (0)/2.\n\nThen, according to Eq. (62) the points E o n are the \"optimal\" energies corresponding to the local minima of the suppression exponent F 0 (E).\n\nAt low energies the function ∆T g (E) ceases to oscillate and becomes large and positive.\n\nAccording to Eq. (62) this means that the suppression exponent F 0,g (E) of the \"global\" solution becomes negative at low energies 15 , see Fig. 9. This is a clear signal that the 15 It is worth mentioning that Eqs. (59) and the expression (61) for the suppression exponent become 27 0 0.1 0.2 0.3 τ E/α 2 4π 5π 6π 7π 8π 9π 10π 0 0.1 0.2 0.3 τ E/α 2 4π 5π 6π 7π 8π 9π 10π g g 4 4 5 5\n\nE' 3 E' 4 E' 5 -2 -1 0 1 2 3 4 0 0.1 0.2 0.3 ∆T E/α 2 g 4 5 g 4 5 E' 3 E' 4 E' 5\n\nFigure 8: Several first branches of solutions to Eqs. (59): \"global\" branch (\"g\") and two \"local\" branches (\"4\", \"5\"); β = π/3, α = π/30. 28 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.2 0.4 0.6 F 0 /α 2 E/α 2 E' 2 E' 3 g g 4 4 0 0.02 0.04 0.12 0.11 0.10 0.09 A B Figure 9: The suppression exponent F 0 (E) for the \"global\" and first \"local\" (n = 4) branches; β = π/3, α = π/30. The vicinity of intersection of the graphs is enlarged in the upper right corner. \"global\" solution becomes unphysical at these energies and its contribution to the reflection probability should be discarded: negative suppression exponent contradicts the unitarity requirement 16 , P < 1. One is forced to conclude that at low energies reflection is described by the \"local\" solutions. Let us study them in detail.\n\nFor the n-th branch one obtains,\n\nτ n → 2πn + O(α ln α) , ∆T n → ln(tg β/α) , E → 0 , τ n → 2πn -π , ∆T g → +∞ , E → +∞ .\n\nFrom Fig. 7 one learns that the n-th solution passes through the points\n\n∆T n = -1 , τ = 2πn or τ = 2πn -π . ( 68\n\n)\n\ninapplicable at large ∆T : the assumption e ∆T ∼ O(1) which was used in the derivation of these equations gets violated. Nevertheless, by analyzing the full equations (54), (55) one can show that dF 0,g /dE = -T g is large and positive at E → 0. This is sufficient for concluding that F 0,g (E) is negative in the low-energy domain. 16 Another indication that the \"global\" solution is unphysical at small E is that the function τ g (E) is bounded from above. Indeed, τ is the time interval the particle spends in the intermediate part of the waveguide, one expects it to tend to infinity as E → 0 for a physically relevant solution.\n\nThus, each curve ∆T n (E) has one sharp dip, its minimum is smaller than -1, see Fig. 8 . As in the case with the \"global\" branch, the points (68) represent the extrema of the functions F 0,n (E)/E; the positions of the local minima are again given by Eq. (66). Making use of Eq. ( 61 ), we find that the suppressions F 0,n (E) of the \"local\" branches are large and positive at high energies. Hence, these solutions give subdominant contributions to the reflection probability at such E as compared to the \"global\" solution. As energy decreases, F 0,n (E) also decreases, then makes one oscillation and drops to negative values at small E. The latter property means that each \"local\" branch becomes unphysical at small enough energies. The suppression exponent of the first \"local\" branch (corresponding to n = 4 in the case β = π/3, α = π/30) is presented in Fig. 9 . An alert reader may have already guessed that we have met here the typical Stokes phenomenon [21] . In fact, the Stokes phenomenon is specific to the situations where some integral (e.g., the path integral (7) in our case) is evaluated by the saddle-point method. Essentially, it means the following: as one gradually changes the parameters of the integral in question, a given saddle point may become non-contributing after the values of these parameters cross a certain curve drawn in the parameter space, the Stokes line. Since the result of the computation should be continuous, this phenomenon occurs only for subdominant saddle points (saddle-point trajectories in our case). Unfortunately, apart from several heuristic conjectures [21, 12] , sometimes rather suggestive [13] , there is presently no general method of dealing with the Stokes phenomenon in the semiclassical calculations. However, in the situation encountered above it suffices to use the simplest logic lying at the heart of all other approaches 17 .\n\nWhen gathering the final result for the suppression exponent, we follow two guidelines. First, it is clear that, as energy decreases, each branch becomes unphysical before F 0,n (E) crosses zero. On the other hand, at high energies one should pick up the branch corresponding to the smallest value of the suppression exponent. Looking at Fig. 9 , one notes that the curves F 0,g (E), F 0,4 (E) have two intersections, A and B. At E > E B one chooses the \"global\" branch. In the region E A < E < E B we switch to the first \"local\" branch, because in this region F 0,4 (E) < F 0,g (E). Naively, at E = E A one should jump back to the \"global\" branch; however, in order to preserve unitarity at small energies, we suppose that somewhere in between the points B and A the \"global\" branch becomes non-contributing, so that one should stay at the \"local\" branch at E < E A . Similarly, the adjacent \"local\" branches have 17 The simplification in the present case is related to the fact that we concentrate on the dominant semiclassical contribution, leaving aside the subdominant ones.\n\ntwo intersections; as the energy decreases, we switch from n-th branch to n + 1-th at the first intersection, and stay there until the intersection with the n + 2-th branch. Overall, one obtains the graph for the suppression exponent plotted in Fig. 10 . The suppression exponent 0 0.02 0.04 0.06 0.08 0 0.05 0.1 0.15 0.2 0.25\n\nF 0 /α 2 E/α 2 E' 3 E' 4 E' 5 g 4 5\n\nFigure 10: The final result for the suppression exponent F 0 (E) in the region of small energies; β = π/3, α = π/30. The points where different branches merge are shown with thick black dots. oscillates between two linear envelopes, F = E(f β (0) ± 4e -1 α ctg β); oscillations pile up in the region of low energies. The reflection process is optimal in the vicinities of the minima of the function F 0 (E)."
},
{
"section_type": "DISCUSSION",
"section_title": "Discussion",
"text": "By considering a class of two-dimensional waveguide models, we have demonstrated explicitly that the probability of over-barrier reflection can be non-monotonic function of energy. The origin of the effect lies in the classical dynamics: the parameters of the complex trajectory describing over-barrier reflection change quasi-periodically as the energy gets decreased.\n\nThis results in the oscillatory behavior of the suppression exponent. Reflection occurs with 31 exponentially larger probability in the vicinities of \"optimal\" energies (local minima of the suppression exponent) while being highly suppressed in between.\n\nOur results are obtained for a fairly specific class of waveguides, namely, the ones with very sharp turns. However, the qualitative features observed in this paper should be valid for quite general waveguide models: a classical particle with high energy feels any large-scale turn of the waveguide as a sharp one 18 ; if two turns are separated by a long interval of free motion, one arrives to the model (35) . We remark that the phenomenon of optimal tunneling has been observed also in numerical investigation of a smooth waveguide, see Ref. [14] .\n\nThe branch structure of solutions observed in the region of small energies is interesting from the mathematical point of view. We have shown that there exists an infinite sequence of complex trajectories marked by the topological number n. Each branch produces physically consistent result for the suppression exponent in some energy interval; outside of this interval the n-th branch would correspond either to highly suppressed transitions (high energies) or to violation of unitarity (low energies). We collected the final graph for the suppression exponent basing on the empirical considerations, which hardly may be acknowledged as satisfactory. Our study clearly shows that the method of complex trajectories should be equipped with a convenient rule to pick up the physical trajectory among the discrete set of solutions to the boundary value problem (9), (10), (12) (in other words, the method to deal with the Stokes phenomenon). Presently, such a rule is absent.\n\nWe note that the described physical phenomenon of optimal tunneling is present independently of the way the branches of solutions are glued together. The result at relatively high energies is given by the \"global\" branch, which displays a large number of local minima 65) . This is the case for the illustrative example considered throughout this paper, see Fig. 9 .\n\nif n ′ 0 > n 1 , see Eqs. (67), (\n\nAs a final remark, we point out some open issues. We have calculated the suppression exponent of reflection using the sharp-turn approximation. It would be instructive to extend our analysis by finding corrections due to the finite turn widths. The motivation is twofold. First, the analysis performed in appendix implies existence of a rich variety of distinctive semiclassical solutions contributing almost equally into the reflection probability. This feature might be a manifestation of chaos [7] which is present in our system but hidden by the sharp-turn approximation. [Note that chaos is inherent in a very similar waveguide model\n\n18 More precisely, one should compare the width b of the turn to the quantity 2π ω p0 m , where p 0 is the translatory momentum of the particle and ω stands for the frequency of transverse oscillations; if b ≪ 2πp0 ωm , one is in the class of models with sharp turns.\n\nwith smooth potential, see Ref. [14] .] Clearly, the structure of solutions in the vicinities of the turns is worth further investigation.\n\nSecond, it was proposed recently in Refs. [9, 11] that the process of dynamical tunneling in quantum systems with multiple degrees of freedom (including field theoretical models, see Refs. [19] ) can proceed differently from the ordinary case of one-dimensional tunneling. Namely, classically unstable state can be created during the process; this state decays subsequently into the final asymptotic region. The analysis performed in the present paper naturally conforms with this tunneling mechanism: all our complex trajectories are matched with the unstable orbit living at the turn. Still, the sharp-turn approximation does not allow to distinguish between the truly unstable trajectories staying at the turn forever and those which reflect from the turn in a finite time. To decide whether the tunneling mechanism of Refs. [9, 11] is indeed realized in our model one needs to go beyond the sharp-turn approximation. Then, the candidate for the \"mediator\" unstable state is the \"excited sphaleron\", the solution considered in the appendix. Presumably, in our model one can answer analytically to the question of whether or not the \"excited sphaleron\" acts as an intermediate state of the tunneling process. This study is quite beyond the scope of the present paper and we leave it for future investigations.\n\nAcknowledgments. We are indebted to F.L. Bezrukov and V.A. Rubakov for the encouraging interest and helpful suggestions. This work is supported in part by the Russian Foundation for Basic Research, grant 05-02-17363-a; Grants of the President of Russian Federation NS-7293.2006.2 (government contract 02.445.11.7370), MK-2563.2006.2 (D.L.), MK-2205.2005.2 (S.S.); Grants of the Russian Science Support Foundation (D.L. and S.S.); the personal fellowship of the \"Dynasty\" foundation (awarded by the Scientific board of ICFPM) (A.P.) and INTAS grant YS 03-55-2362 (D.L.). D.L. is grateful to Universite Libre de Bruxelles and EPFL (Lausanne) for hospitality during his visits."
},
{
"section_type": "OTHER",
"section_title": "A Classical motion near the turn",
"text": "In this appendix we analyze the motion of the particle near the sharp turn of the waveguide (20) at nonzero smoothening of the turn, see, e.g., Eq. (21) . We suppose that in the small vicinity of the turn the function w(ξ, η) can be represented in the form w(ξ, η) = cos β (η -bv(ξ/b)) , (69) 33 where v(ψ) does not depend explicitly on b. Moreover, we consider the case when v(ψ) has a maximum 19 , v ′ (ψ 0 ) = 0 . (70) Due to the property (70) one immediately obtains the exact periodic solution to the equations of motion (9), which we call \"excited sphaleron\" [9],\n\nξ sp = bψ 0 , η sp = A η sin(t cos β + ϕ η ) + bv(ψ 0 ) . ( 71\n\n)\n\nWe are going to show that this solution is unstable: a small perturbation above it grows with time and the particle flies away to either end of the waveguide. In particular, there are solutions that describe the decay of the sphaleron to ξ → -∞ both at t → ±∞. Clearly, such solutions correspond to reflections from the turn. In the vicinity of the sphaleron the trajectory of the particle can be represented in the form, ξ = bψ(t) , η = η sp (t) + bρ(t) , (72)\n\nwhere ψ, ρ ∼ O(1). Writing down the classical equations of motion (9) in the leading order in b, one obtains,\n\nd 2 ψ ds 2 = 4 b A η sin(2s)v ′ (ψ) , ( 73\n\n) d 2 ρ ds 2 + 4ρ = 4[v(ψ) -v(ψ 0 )] , ( 74\n\n)\n\nwhere s = (t cos β + ϕ η )/2. It is worth noting that the right hand side of Eqs. (73), (74) are of different order in b. We will see that due to this difference ρ = 0 in the leading order in b.\n\nLet us first consider the linear perturbations above the excited sphaleron,\n\nψ = ψ 0 + δψ , δψ ≪ 1 .\n\nEquation (73) can be linearized with respect to δψ leading to the Mathieu equation\n\nd 2 ds 2 δψ + 2q sin(2s)δψ = 0 ,\n\nwith canonical parameter q = -2v ′′ 0 A η /b > 0. As q ∼ O(1/b) ≫ 1, one can apply the WKB formula,\n\nδψ = A cos W dW/ds , ( 75\n\n)\n\n19 For the smoothening (21), the properties (69), (70) hold with v(ψ) = ψtgβ 1+e ψ , ψ 0 ≈ 1.28.\n\nwhere |A| ≪ 1, and W = 2q s π/4 ds ′ sin(2s ′ ) .\n\nNote that we have chosen the solution symmetric with respect to time reflections,\n\nδψ(π/2 -s) = δψ(s) . (76) At s ∈ [0; π/2] the exponent W is real and the particle gets stuck at ψ ≈ ψ 0 , oscillating around this point with high frequency dW/ds ∼ O(b -1/2 ). At s < 0 the solution (75) grows exponentially, meaning that the particle flies away from the excited sphaleron,\n\nδψ(s < 0) = A cos(W (0) -π/4) |dW/ds| e |W (s)-W (0)| .\n\nIn what follows, we choose A cos(W (0) -π/4) < 0, so that δψ < 0 at s < 0. Let us denote by s 1 < 0 the point where δψ becomes formally equal to -1,\n\nA cos(W (0) -π/4) |dW/ds| e |W (s 1 )-W (0)| = -1 .\n\nIn what follows we suppose that s 1 ∼ O(1), hence, A is exponentially small. Then, in the vicinity of this point, |s -s 1 | ≪ 1, one has,\n\nδψ = -exp -2q sin(2s 1 )(s 1 -s) = -exp 4v ′′ 0 A η sin(2s 1 ) (s 1 -s) √ b . ( 77\n\n)\n\nWe notice that δψ evolves from exponentially small values to δψ ∼ O(1) during the characteristic time |s -s 1 | ∼ O( √ b). When δψ ∼ O(1) the linear approximation breaks down and one has to solve the nonlinear equation (73). Using s = s 1 + O( √ b) one writes\n\nd 2 ψ ds 2 = 4 b A η sin(2s 1 )v ′ (ψ) . ( 78\n\n)\n\nThis equation permits to draw a useful analogy with one-dimensional particle moving in the effective potential V ef f (ψ) = -4b -1 A η sin(2s 1 )v(ψ) (see Fig. 11 ). This auxiliary particle starts in the region near the maximum of the potential at (s -s 1 )/ √ b → +∞ with energy E ≈ V max and rolls down toward ψ → -∞ at (s-s 1 )/ √ b → -∞. In this limit v(ψ) → ψ tg β and the solution takes the form\n\nψ = C 1 + C 2 (s -s 1 ) + 2b -1 A η sin(2s 1 ) tg β (s -s 1 ) 2 . 35 ψ 0 V max ψ V eff\n\nFigure 11: The effective potential for Eq. (78).\n\nNote that the coefficients C 1 , C 2 here are not independent: they are determined by the parameter s 1 through matching of the solution with Eq. (77) at (s -s 1 )/ √ b → +∞. We do not need their explicit form, however.\n\nLet us argue that the function ρ remains small during the whole evolution of the particle in the vicinity of the sphaleron. Indeed, in the linear regime one has δψ ≪ 1 and the r.h.s. of Eq. (74) is small. So, ρ does not get excited. On the other hand, the nonlinear evolution of ψ proceeds in a short time interval ∆s = O( √ b); so, again, ρ is suppressed by some power of b. The trajectory (72) found in the vicinity of the sphaleron should be matched at\n\n1 ≫ |s -s 1 | ≫ √ b\n\nwith the free solution in the asymptotic region ξ < 0, see Eqs. (26) . It is straightforward to check that matching can be performed up to the second order in (t -t 1 ), which is consistent with our approximations. In this way one determines the free asymptotic solution which, up to corrections of order O(b), coincides with the sinusoid coming from ξ → -∞ at t → -∞ and touching the line ξ = 0 at t = t 1 . Now we recall that, by construction, the obtained solution is symmetric with respect to time reflections, ξ(s) = ξ(π/2 -s) , η(s) = η(π/2 -s) .\n\nThis means that it satisfies ξ → -∞ at t → ±∞. This solution describes reflection of the particle from the turn.\n\nThe reasoning presented in this appendix puts considerations of the main body of this paper on the firm ground: we have found the \"smoothened\" solutions which reflect classically from the turn, and in the limit b → 0 coincide with the free solutions of Sec. 3 touching the line ξ = 0.\n\nIt is worth mentioning that, apart from the reflected solution we have found, in the vicinity of any trajectory touching the line ξ = 0 there exists a rich variety of qualitatively different motions. First of all, one may successfully search for solutions which are odd with respect to time reflections (Eq. (76) with minus sign). Such solutions, though close to the reflected ones at t < 0, describe transmissions of the particle through the sharp turn into the asymptotic region ξ → +∞. Relaxing the time reflection symmetry, one can find solutions leaving the vicinity of the turn at any point η < 0, which is different, in general, from the starting point η = η(s 1 ). Yet another types of solutions are obtained in the case when the amplitude A of δψ-oscillations at s ∈ [0; π/2] is so small that δψ does not reach the values of order one during the time period s ∈ [-π/2; 0]. If the particle is still in the vicinity of the point ψ 0 at s = -π/2, it remains for sure in this vicinity at s ∈ [-π; -π/2], because the r.h.s. of Eq. (73) is positive again. In this way one obtains solutions, which spend two, three, etc. sphaleron periods at ψ ≈ ψ 0 before escaping into the asymptotic regions ψ → ±∞. In the leading order in b all these solutions correspond to the identical initial state, and (in the case of classically forbidden transitions) to the same value of the suppression exponent. However, an accurate study of the dynamics in the vicinity of the the sphaleron is generically required to obtain the correct value of the suppression exponent in the case b ∼ 1, cf. Ref. [14] ."
}
] | |
arxiv:0704.0418 | 0704.0418 | 1 | 10.1103/PhysRevLett.99.147202 | 4841c2d087beb058c39c1425bfc13ff2ca5d9f6c1afcf6090f8283292d79ce39 | Entanglement entropy at infinite randomness fixed points in higher
dimensions | The entanglement entropy of the two-dimensional random transverse Ising model
is studied with a numerical implementation of the strong disorder
renormalization group. The asymptotic behavior of the entropy per surface area
diverges at, and only at, the quantum phase transition that is governed by an
infinite randomness fixed point. Here we identify a double-logarithmic
multiplicative correction to the area law for the entanglement entropy. This
contrasts with the pure area law valid at the infinite randomness fixed point
in the diluted transverse Ising model in higher dimensions. | [
"Yu-Cheng Lin",
"Ferenc Igloi and Heiko Rieger"
] | [
"cond-mat.dis-nn",
"quant-ph"
] | cond-mat.dis-nn | [] | 2007-04-03 | 2026-02-26 | The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite randomness fixed point in the diluted transverse Ising model in higher dimensions.
Extensive studies have been devoted recently to understand ground state entanglement in quantum many-body systems [1] . In particular, the behavior of various entanglement measures at/near quantum phase transitions has been of special interest. One of the widely used entanglement measures is the von Neumann entropy, which quantifies entanglement of a pure quantum state in a bipartite system. Critical ground states in one dimension (1D) are known to have entanglement entropy that diverges logarithmically in the subsystem size with a universal coefficient determined by the central charge of the associated conformal field theory [2] . Away from the critical point, the entanglement entropy saturates to a finite value, which is related to the finite correlation length.
In higher dimensions, the scaling behavior of the entanglement entropy is far less clear. A standard expectation is that non-critical entanglement entropy scales as the area of the boundary between the subsystems, known as the "area law" [3, 4] . This area-relationship is known to be violated for gapless fermionic systems [5] in which a logarithmic multiplicative correction is found. One might suspect that whether the area law holds or not depends on whether the correlation length is finite or diverges. However, it has turned out that the situation is more complex: numerical findings [7] and a recent analytical study [8] have shown that the area law holds even for critical bosonic systems, despite a divergent correlation length. This indicates that the length scale associated with entanglement may differ from the correlation length. Another ongoing research activity for entanglement in higher spatial dimensions is to understand topological contributions to the entanglement entropy [9] .
The nature of quantum phase transitions with quenched randomness is in many systems quite different from the pure case. For instance, in a class of systems the critical behavior is governed by a so-called infiniterandomness fixed point (IRFP), at which the energy scale ǫ and the length scale L are related as: ln ǫ ∼ L ψ with 0 < ψ < 1. In these systems the off-critical regions are also gapless and the excitation energies in these so-called Griffiths phases scale as ǫ ∼ L -z with a nonuniversal dynamical exponent z < ∞. Even so, certain random critical points in 1D are shown to have logarithmic divergences of entanglement entropy with universal coefficients, as in the pure case; these include infiniterandomness fixed points in the random-singlet universality class [12, 13, 14, 15, 16] and a class of aperiodic singlet phases [17] .
In this paper we consider the random quantum Ising model in two dimensions (2D), and examine the disorderaveraged entanglement entropy. The critical behavior of this system is governed by an IRFP [10, 11] implying that the disorder strength grows without limit as the system is coarse grained in the renormalization group (RG) sense. In our study, the ground state of the system and the entanglement entropy are numerically calculated using a strong-disorder RG method [18, 19] , which yields asymptotically exact results at an IRFP. To our knowledge this is the first study of entanglement in higher dimensional interacting quantum systems with disorder.
The random transverse Ising model is defined by the Hamiltonian
H = - i,j J ij σ z i σ z j - i h i σ x i . (1)
Here the {σ α i } are spin-1/2 Pauli matrices at site i of an L × L square lattice with periodic boundary conditions. The nearest neighbor bonds J ij (≥ 0) are independent random variables, while the transverse fields h i (≥ 0) are random or constant. For a given realization of randomness we consider a square block A of linear size ℓ, and calculate the entanglement between A and the rest of the system B, which is quantified by the von Neumann entropy of the reduced density matrix for either subsystems:
S = -Tr(ρ A log 2 ρ A ) = -Tr(ρ B log 2 ρ B ).
(
) 2
The basic idea of the strong disorder RG (SDRG) approach is as follows [18, 19] : The ground state of the system is calculated by successively eliminating the largest local terms in the Hamiltonian and by generating a new effective Hamiltonian in the frame of the perturbation theory. If the strongest bond is J ij , the two spins at i and j are combined into a ferromagnetic cluster with an effective transverse field h (ij) = hihj Jij . If, on the other hand, the largest term is the field h i , the spin at i is decimated and an effective bond is generated between its neighboring sites, say j and k, with strength J jk = Jij J ik hi . After decimating all degrees of freedom, we obtain the ground state of the system, consisting of a collection of independent ferromagnetic clusters of various sizes; each cluster of n spins is frozen in an entangled state of the form:
1 √ 2 (| ↑↑ • • • ↑ n times + | ↓↓ • • • ↓ n times ). (3)
In this representation, the entanglement entropy of a block is given by the number of clusters that connect sites inside to sites outside the block [Fig. 1 ]. We note that correlations between remote sites also contribute to the entropy due to long-range effective bonds generated under renormalization.
In 1D the RG calculation can be carried out analytically and the disorder-averaged entropy S ℓ of a segment of length ℓ has been obtained as S ℓ = ln 2 6 log 2 ℓ [12] . In higher dimensions the RG method can only be implemented numerically. The major complication in this case is that the model is not self-dual and thus the location of the critical point is not exactly known. To locate the crit-
-1.5 -1 -0.5 10 -2 10 -1 10 0 -1.5 -1 -0.5 0 10 -2 10 -1 10 0 -0.15 -0.1 -0.05 0 10 -1 10 0 10 1 L = 16 L = 32 L = 64 -1.5 -1 -0.5 10 -2 10 -1 10 0 -1.5 -1 -0.5 0 10 -2 10 -1 10 0 (c) (a) (b) (d) (e) h 0 =1.175 h 0 =1.175 h 0 =1.175
h 0 =1.18 h 0 =1.17 PSfrag replacements ln h ∞ ln h ∞ ln J ∞ ln J ∞ ln h ∞ /L 0.55 P (ln h ∞ ) P (ln h ∞ ) P (ln J ∞ ) P (ln J ∞ ) P (ln h ∞ /L 0.55
) FIG. 2: (color online). The distribution of the last decimated effective log-fields ln e h∞, and the distribution of the last decimated effective log-bonds ln e J∞ in the RG calculations. At h0 = 1.175, the distributions, shown in (a) and (b), get broader with increasing system sizes, indicating the RG flow towards infinite randomness, i.e. the system is critical. A scaling plot of the data in (a) using energy-length scaling ln e h∞ ∼ L ψ with ψ = 0.55 is presented in (c). The solid line is just a guide to the eye. The subfigures (d) and (e) show the log-field distribution at h0 = 1.18 and the log-bond distribution at h0 = 1.17, respectively; the distributions show a power-law decaying tail in the low energy region, which is clear evidence that the system is in the Griffiths phases.
ical point, we can make use of the fact that the excitation energy of the system has the scaling behavior ln ǫ ∼ L ψ at criticality, while it follows ǫ ∼ L -z in the off-critical
In the numerical implementation of the SDRG method, the low-energy excitations of a given sample can be identified with the effective transverse field h∞ of the last decimated spin cluster, or with the effective coupling J∞ of the last decimated cluster-pair.
In our implementation we set for convenience the transverse fields to be a constant h 0 and the random bond variables were taken from a rectangular distribution centered at J = 1 with a width ∆ = 0.5. The critical point was approached by varying the single control parameter h 0 . Although this initial disorder appears to be The block entropy per surface area vs. ln ℓ on a log-scale for different system sizes L at the critical point. The data show a straight line (guided by the dashed line), corresponding to the scaling obeying the area law with a double-logarithmic correction, as given in Eq. ( 4 ).
weak, the renormalized field and bond distributions become extremely broad even on a logarithmic scale [Fig. 2 ] at the critical point h 0 = h c = 1.175. This indicates the RG flow towards infinite randomness. Slightly away from the critical point, both in the disordered Griffiths-phase with h 0 = 1.18 and in the ordered Griffiths-phase with h 0 = 1.17, the distributions have a finite width and obey quantum-Griffiths scaling h ∞ ∼ L -z . At the critical point one has IRFP scaling ln h ∞ ∼ L ψ and we estimate the scaling exponent as ψ = 0.55, quite close to the value ψ = 0.5 for the 1D case [18] . Now we consider the entanglement entropy near the infinite randomness critical point. To obtain the disorderaveraged entanglement entropy S ℓ of a square block of size ℓ, we averaged the entropies over blocks in different positions of the whole system for a given disorder realization and then averaged over a few thousand samples. In Fig. 3 we show the entropy per surface unit S ℓ /ℓ = s ℓ for different values of h 0 . This average entropy density is found to be saturated outside the critical point, which corresponds to the area law. At the critical point s ℓ increases monotonously with ℓ, and the numerical data are consistent with a log-log dependence:
S ℓ ∼ ℓ log 2 log 2 ℓ (4)
as illustrated in Fig. 3 . In this way we have identified an alternative route to locate the infinite randomness critical point: it is given by the field h 0 for which the average block entropy at ℓ = L/2 is maximal. Indeed the numerical results in Fig. 3 predict the same value of h c as obtained from the scaling of the gaps. We note that the same quantity, the position of the maxima of the average entropy, can be used for the random quantum Ising chain to locate finite-size transition points [21] .
The log-log size dependence of the average entropy in Eq.( 4 ) at criticality is completely new; it differs from the scaling behavior observed in 2D pure systems, like the area law, S ℓ ∼ ℓ, for critical bosonic systems [7, 8] , or a logarithmic multiplicative correction to the area law, S ℓ ∼ ℓ log 2 ℓ, as found in free fermions [5, 6, 7, 8] . This double-logarithmic correction can be understood via a SDRG argument: In the 1D case a characteristic length scale r at a given RG step is identified with the average length of the effective bonds, i.e. the average size of the effective clusters. At the scale r(< ℓ) the fraction of the total number of spins, n r , that have not been decimated is given by n r ∼ 1/r [18] ; these active (i.e., undecimated) spins have a finite probability to form a cluster across the boundary of the block (a segment ℓ in the 1D case) and thus to give contributions to the entanglement entropy. Repeating the renormalization until the scale r ∼ ℓ, the contributions to the entropy are summed up: S ℓ ∼ ℓ r0 dr n r ∼ ln ℓ, leading to the logarithmic dependence of the 1D model [12] . For the 2D case with the same type of RG transformation with a length scale r < ℓ, the fraction of active spins in the renormalized surface layer of the block is n r ∼ ℓ/r. Here we have to consider the situation in which some of these active surface spins would form clusters within the surface layer and thus contribute zero entanglement entropy; the number of the active spins that are already engaged in clusters on the surface at RG scale r is proportional to ln r, as known from the 1D case, and only O(1) of the active surface spins would form clusters connecting the block with the rest of the system. Consequently, the entropy contribution in 2D can be estimated as: S ℓ ∼ ℓ r0 dr n r / ln r ∼ ℓ ln ln ℓ, i.e. a double-logarithmic ℓdependence, as reflected by the numerical data in Fig. 3 .
Based on the SDRG argument described above, the double-logarithmic correction to the area law appears to be applicable for a broad class of critical points in 2D with infinite randomness. For instance, the critical points of quantum Ising spin glasses are believed to belong to the same universality class as ferromagnets since the frustration becomes irrelevant under RG transformation, and the same type of cluster formations as observed in our numerics for the ferromagnet is expected to be generated during the action of the RG. The entanglement entropy at the IRFP is completely determined by the cluster geometries occurring during the SDRG. Another type of IRFP in higher dimensions occurs in the bond-diluted quantum Ising ferromagnet: The Hamiltonian is again given by ( 1 ), but now J ij = 0 with probability p and J ij = J > 0 with probability 1 -p. At percolation threshold p = p c there is a quantum critical line along small nonzero transverse fields, which is con- trolled by the classical percolation fixed point, and the energy scaling across this transition line obeys ln ǫ ∼ L ψ , implying an IRFP [20] . The ground state of the system is given by a set of ordered clusters in the same geometry as in the classical percolation model -only nearest neighboring sites are combined into a cluster. In this cluster structure, the block entropy, determined by the number of the clusters connecting the block and the rest of the system, is bounded by the area of the block, i.e. S ℓ ∼ ℓ d-1 with d being the dimensionality of the system. To examine this, we determined the entanglement entropy by analyzing the cluster geometry of the bonddiluted transverse Ising model. Fig. 4 shows our results for the square lattice, which follow a pure area-law with an additive constant: S ℓ = aℓ + b + O(1/ℓ).
To summarize, we have found that the entanglement properties at quantum phase transitions of disordered systems in dimensions larger than one can behave quite differently. Generalizing our arguments for the 2D case, we expect for the random bond transverse Ising systems a multiplicative d-fold logarithmic correction to the area law in d dimensions at the critical point, whereas for diluted Ising model at small transverse fields the area law will hold in any dimension d > 1 at the percolation threshold. Although both critical points are described by infinite randomness fixed points, the structure of the strongly coupled clusters in both cases is fundamentally different, reflecting the different degrees of quantum mechanical entanglement in the ground state of the two systems. This behavior appears to be in contrast to onedimensional systems governed by IRFPs [12] .
Other disordered quantum systems in higher dimensions might also display interesting entanglement prop-erties: For instance, the numerical SDRG has also been applied to higher dimensional random Heisenberg antiferromagnets which do not display an IRFP [22] . The ground states involve both singlet spins and clusters with larger moments; therefore, we expect the correction to the area law to be weaker than a multiplicative logarithm and different from the valence bond entanglement entropy in the Néel Phase [23] .
Useful discussions with Cécile Monthus are gratefully acknowledged. This work has been supported by the National Office of Research and Technology under Grant No. ASEP1111, by a German-Hungarian exchange program (DAAD-M ÖB), by the Hungarian National Research Fund under grant No OTKA TO48721, K62588, MO45596. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite randomness fixed point in the diluted transverse Ising model in higher dimensions."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Extensive studies have been devoted recently to understand ground state entanglement in quantum many-body systems [1] . In particular, the behavior of various entanglement measures at/near quantum phase transitions has been of special interest. One of the widely used entanglement measures is the von Neumann entropy, which quantifies entanglement of a pure quantum state in a bipartite system. Critical ground states in one dimension (1D) are known to have entanglement entropy that diverges logarithmically in the subsystem size with a universal coefficient determined by the central charge of the associated conformal field theory [2] . Away from the critical point, the entanglement entropy saturates to a finite value, which is related to the finite correlation length.\n\nIn higher dimensions, the scaling behavior of the entanglement entropy is far less clear. A standard expectation is that non-critical entanglement entropy scales as the area of the boundary between the subsystems, known as the \"area law\" [3, 4] . This area-relationship is known to be violated for gapless fermionic systems [5] in which a logarithmic multiplicative correction is found. One might suspect that whether the area law holds or not depends on whether the correlation length is finite or diverges. However, it has turned out that the situation is more complex: numerical findings [7] and a recent analytical study [8] have shown that the area law holds even for critical bosonic systems, despite a divergent correlation length. This indicates that the length scale associated with entanglement may differ from the correlation length. Another ongoing research activity for entanglement in higher spatial dimensions is to understand topological contributions to the entanglement entropy [9] .\n\nThe nature of quantum phase transitions with quenched randomness is in many systems quite different from the pure case. For instance, in a class of systems the critical behavior is governed by a so-called infiniterandomness fixed point (IRFP), at which the energy scale ǫ and the length scale L are related as: ln ǫ ∼ L ψ with 0 < ψ < 1. In these systems the off-critical regions are also gapless and the excitation energies in these so-called Griffiths phases scale as ǫ ∼ L -z with a nonuniversal dynamical exponent z < ∞. Even so, certain random critical points in 1D are shown to have logarithmic divergences of entanglement entropy with universal coefficients, as in the pure case; these include infiniterandomness fixed points in the random-singlet universality class [12, 13, 14, 15, 16] and a class of aperiodic singlet phases [17] .\n\nIn this paper we consider the random quantum Ising model in two dimensions (2D), and examine the disorderaveraged entanglement entropy. The critical behavior of this system is governed by an IRFP [10, 11] implying that the disorder strength grows without limit as the system is coarse grained in the renormalization group (RG) sense. In our study, the ground state of the system and the entanglement entropy are numerically calculated using a strong-disorder RG method [18, 19] , which yields asymptotically exact results at an IRFP. To our knowledge this is the first study of entanglement in higher dimensional interacting quantum systems with disorder.\n\nThe random transverse Ising model is defined by the Hamiltonian\n\nH = - i,j J ij σ z i σ z j - i h i σ x i . (1)\n\nHere the {σ α i } are spin-1/2 Pauli matrices at site i of an L × L square lattice with periodic boundary conditions. The nearest neighbor bonds J ij (≥ 0) are independent random variables, while the transverse fields h i (≥ 0) are random or constant. For a given realization of randomness we consider a square block A of linear size ℓ, and calculate the entanglement between A and the rest of the system B, which is quantified by the von Neumann entropy of the reduced density matrix for either subsystems:\n\nS = -Tr(ρ A log 2 ρ A ) = -Tr(ρ B log 2 ρ B ).\n\n(\n\n) 2\n\nThe basic idea of the strong disorder RG (SDRG) approach is as follows [18, 19] : The ground state of the system is calculated by successively eliminating the largest local terms in the Hamiltonian and by generating a new effective Hamiltonian in the frame of the perturbation theory. If the strongest bond is J ij , the two spins at i and j are combined into a ferromagnetic cluster with an effective transverse field h (ij) = hihj Jij . If, on the other hand, the largest term is the field h i , the spin at i is decimated and an effective bond is generated between its neighboring sites, say j and k, with strength J jk = Jij J ik hi . After decimating all degrees of freedom, we obtain the ground state of the system, consisting of a collection of independent ferromagnetic clusters of various sizes; each cluster of n spins is frozen in an entangled state of the form:\n\n1 √ 2 (| ↑↑ • • • ↑ n times + | ↓↓ • • • ↓ n times ). (3)\n\nIn this representation, the entanglement entropy of a block is given by the number of clusters that connect sites inside to sites outside the block [Fig. 1 ]. We note that correlations between remote sites also contribute to the entropy due to long-range effective bonds generated under renormalization.\n\nIn 1D the RG calculation can be carried out analytically and the disorder-averaged entropy S ℓ of a segment of length ℓ has been obtained as S ℓ = ln 2 6 log 2 ℓ [12] . In higher dimensions the RG method can only be implemented numerically. The major complication in this case is that the model is not self-dual and thus the location of the critical point is not exactly known. To locate the crit-\n\n-1.5 -1 -0.5 10 -2 10 -1 10 0 -1.5 -1 -0.5 0 10 -2 10 -1 10 0 -0.15 -0.1 -0.05 0 10 -1 10 0 10 1 L = 16 L = 32 L = 64 -1.5 -1 -0.5 10 -2 10 -1 10 0 -1.5 -1 -0.5 0 10 -2 10 -1 10 0 (c) (a) (b) (d) (e) h 0 =1.175 h 0 =1.175 h 0 =1.175\n\nh 0 =1.18 h 0 =1.17 PSfrag replacements ln h ∞ ln h ∞ ln J ∞ ln J ∞ ln h ∞ /L 0.55 P (ln h ∞ ) P (ln h ∞ ) P (ln J ∞ ) P (ln J ∞ ) P (ln h ∞ /L 0.55\n\n) FIG. 2: (color online). The distribution of the last decimated effective log-fields ln e h∞, and the distribution of the last decimated effective log-bonds ln e J∞ in the RG calculations. At h0 = 1.175, the distributions, shown in (a) and (b), get broader with increasing system sizes, indicating the RG flow towards infinite randomness, i.e. the system is critical. A scaling plot of the data in (a) using energy-length scaling ln e h∞ ∼ L ψ with ψ = 0.55 is presented in (c). The solid line is just a guide to the eye. The subfigures (d) and (e) show the log-field distribution at h0 = 1.18 and the log-bond distribution at h0 = 1.17, respectively; the distributions show a power-law decaying tail in the low energy region, which is clear evidence that the system is in the Griffiths phases.\n\nical point, we can make use of the fact that the excitation energy of the system has the scaling behavior ln ǫ ∼ L ψ at criticality, while it follows ǫ ∼ L -z in the off-critical\n\nIn the numerical implementation of the SDRG method, the low-energy excitations of a given sample can be identified with the effective transverse field h∞ of the last decimated spin cluster, or with the effective coupling J∞ of the last decimated cluster-pair.\n\nIn our implementation we set for convenience the transverse fields to be a constant h 0 and the random bond variables were taken from a rectangular distribution centered at J = 1 with a width ∆ = 0.5. The critical point was approached by varying the single control parameter h 0 . Although this initial disorder appears to be The block entropy per surface area vs. ln ℓ on a log-scale for different system sizes L at the critical point. The data show a straight line (guided by the dashed line), corresponding to the scaling obeying the area law with a double-logarithmic correction, as given in Eq. ( 4 ).\n\nweak, the renormalized field and bond distributions become extremely broad even on a logarithmic scale [Fig. 2 ] at the critical point h 0 = h c = 1.175. This indicates the RG flow towards infinite randomness. Slightly away from the critical point, both in the disordered Griffiths-phase with h 0 = 1.18 and in the ordered Griffiths-phase with h 0 = 1.17, the distributions have a finite width and obey quantum-Griffiths scaling h ∞ ∼ L -z . At the critical point one has IRFP scaling ln h ∞ ∼ L ψ and we estimate the scaling exponent as ψ = 0.55, quite close to the value ψ = 0.5 for the 1D case [18] . Now we consider the entanglement entropy near the infinite randomness critical point. To obtain the disorderaveraged entanglement entropy S ℓ of a square block of size ℓ, we averaged the entropies over blocks in different positions of the whole system for a given disorder realization and then averaged over a few thousand samples. In Fig. 3 we show the entropy per surface unit S ℓ /ℓ = s ℓ for different values of h 0 . This average entropy density is found to be saturated outside the critical point, which corresponds to the area law. At the critical point s ℓ increases monotonously with ℓ, and the numerical data are consistent with a log-log dependence:\n\nS ℓ ∼ ℓ log 2 log 2 ℓ (4)\n\nas illustrated in Fig. 3 . In this way we have identified an alternative route to locate the infinite randomness critical point: it is given by the field h 0 for which the average block entropy at ℓ = L/2 is maximal. Indeed the numerical results in Fig. 3 predict the same value of h c as obtained from the scaling of the gaps. We note that the same quantity, the position of the maxima of the average entropy, can be used for the random quantum Ising chain to locate finite-size transition points [21] .\n\nThe log-log size dependence of the average entropy in Eq.( 4 ) at criticality is completely new; it differs from the scaling behavior observed in 2D pure systems, like the area law, S ℓ ∼ ℓ, for critical bosonic systems [7, 8] , or a logarithmic multiplicative correction to the area law, S ℓ ∼ ℓ log 2 ℓ, as found in free fermions [5, 6, 7, 8] . This double-logarithmic correction can be understood via a SDRG argument: In the 1D case a characteristic length scale r at a given RG step is identified with the average length of the effective bonds, i.e. the average size of the effective clusters. At the scale r(< ℓ) the fraction of the total number of spins, n r , that have not been decimated is given by n r ∼ 1/r [18] ; these active (i.e., undecimated) spins have a finite probability to form a cluster across the boundary of the block (a segment ℓ in the 1D case) and thus to give contributions to the entanglement entropy. Repeating the renormalization until the scale r ∼ ℓ, the contributions to the entropy are summed up: S ℓ ∼ ℓ r0 dr n r ∼ ln ℓ, leading to the logarithmic dependence of the 1D model [12] . For the 2D case with the same type of RG transformation with a length scale r < ℓ, the fraction of active spins in the renormalized surface layer of the block is n r ∼ ℓ/r. Here we have to consider the situation in which some of these active surface spins would form clusters within the surface layer and thus contribute zero entanglement entropy; the number of the active spins that are already engaged in clusters on the surface at RG scale r is proportional to ln r, as known from the 1D case, and only O(1) of the active surface spins would form clusters connecting the block with the rest of the system. Consequently, the entropy contribution in 2D can be estimated as: S ℓ ∼ ℓ r0 dr n r / ln r ∼ ℓ ln ln ℓ, i.e. a double-logarithmic ℓdependence, as reflected by the numerical data in Fig. 3 .\n\nBased on the SDRG argument described above, the double-logarithmic correction to the area law appears to be applicable for a broad class of critical points in 2D with infinite randomness. For instance, the critical points of quantum Ising spin glasses are believed to belong to the same universality class as ferromagnets since the frustration becomes irrelevant under RG transformation, and the same type of cluster formations as observed in our numerics for the ferromagnet is expected to be generated during the action of the RG. The entanglement entropy at the IRFP is completely determined by the cluster geometries occurring during the SDRG. Another type of IRFP in higher dimensions occurs in the bond-diluted quantum Ising ferromagnet: The Hamiltonian is again given by ( 1 ), but now J ij = 0 with probability p and J ij = J > 0 with probability 1 -p. At percolation threshold p = p c there is a quantum critical line along small nonzero transverse fields, which is con- trolled by the classical percolation fixed point, and the energy scaling across this transition line obeys ln ǫ ∼ L ψ , implying an IRFP [20] . The ground state of the system is given by a set of ordered clusters in the same geometry as in the classical percolation model -only nearest neighboring sites are combined into a cluster. In this cluster structure, the block entropy, determined by the number of the clusters connecting the block and the rest of the system, is bounded by the area of the block, i.e. S ℓ ∼ ℓ d-1 with d being the dimensionality of the system. To examine this, we determined the entanglement entropy by analyzing the cluster geometry of the bonddiluted transverse Ising model. Fig. 4 shows our results for the square lattice, which follow a pure area-law with an additive constant: S ℓ = aℓ + b + O(1/ℓ).\n\nTo summarize, we have found that the entanglement properties at quantum phase transitions of disordered systems in dimensions larger than one can behave quite differently. Generalizing our arguments for the 2D case, we expect for the random bond transverse Ising systems a multiplicative d-fold logarithmic correction to the area law in d dimensions at the critical point, whereas for diluted Ising model at small transverse fields the area law will hold in any dimension d > 1 at the percolation threshold. Although both critical points are described by infinite randomness fixed points, the structure of the strongly coupled clusters in both cases is fundamentally different, reflecting the different degrees of quantum mechanical entanglement in the ground state of the two systems. This behavior appears to be in contrast to onedimensional systems governed by IRFPs [12] .\n\nOther disordered quantum systems in higher dimensions might also display interesting entanglement prop-erties: For instance, the numerical SDRG has also been applied to higher dimensional random Heisenberg antiferromagnets which do not display an IRFP [22] . The ground states involve both singlet spins and clusters with larger moments; therefore, we expect the correction to the area law to be weaker than a multiplicative logarithm and different from the valence bond entanglement entropy in the Néel Phase [23] .\n\nUseful discussions with Cécile Monthus are gratefully acknowledged. This work has been supported by the National Office of Research and Technology under Grant No. ASEP1111, by a German-Hungarian exchange program (DAAD-M ÖB), by the Hungarian National Research Fund under grant No OTKA TO48721, K62588, MO45596."
}
] | |
arxiv:0704.0420 | 0704.0420 | 1 | bdacf2eacc8ee140d5826823b0c6aa04e4cbe11cafca16852f4ed47d78546a24 | The Hourglass - Consequences of Pure Hamiltonian Evolution of a
Radiating System | Hourglass is the name given here to a formal isolated quantum system that can
radiate. Starting from a time when it defines the system it represents clearly
and no radiation is present, it is given straightforward Hamiltonian evolution.
The question of what significance hourglasses have is raised, and this question
is proposed to be more consequential than the measurement problem. | [
"Donald McCartor"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Hourglass is the name given here to a formal isolated quantum system that can radiate. Starting from a time when it defines the system it represents clearly and no radiation is present, it is given straightforward Hamiltonian evolution. The question of what significance hourglasses have is raised, and this question is proposed to be more consequential than the measurement problem. 1 Hourglasses 2 Physics without true histories 3 But histories are sometimes good 4 Phlogiston and oxygen 5 A closer look at quantum engineering 6 Conclusion
But I want to know the particular go of it -the plea of James Clerk Maxwell as a young child concerning, among many things, the bell-wires that ring the bells that summon servants. [Mahon] 1 Hourglasses Suppose that theory develops in such a way that quantum fields can be handled like nonrelativistic quantum mechanics. Then if we are interested in something, perhaps gooseberry bushes, we can model one as we would conceive it to be at some instant and then follow its development through time. And not only the atoms and molecules would be modeled, but also the radiation. This is a scheme for the imagination. The gooseberry bush, though not isolated, would grow within a suitable environment that would be an isolated system, complete in itself. We do learn well from isolated systems, both real ones in the laboratory and those envisaged in our theoretical musings.
We will provide the bush with air, earth, and water. And there can be life-giving sunlight shining on it. As for the light that had been reflected or emitted from the bush before the present time, we will leave that out. Such light goes off and away, so it could only matter as information about what the bush had been doing. We will take the bush just as it is now.
The gooseberry bush is then developed forward in time. Lagrange or Hamilton would have recognized what we are doing, for we are doing physics the classical way. We have an initial condition and we are finding out what will happen next.
As we move toward the future, light shoots out from the bush, as we expect. But, disconcertingly, the bush starts to lose definition. Its parts lose their precise places. Within a few weeks it is a scarcely recognizable mess. Let's go back in time, then. This is terrifyingly worse. The bush has been the subject of a vast conspiracy. Light has been streaming in on it from the entire universe. The bush swallows it up. Then at the present time this suddenly all stops. Time symmetry of the Hamiltonian makes it happen like that. This is the hourglass. It is really more like a cone, with the light streaming in before the set-up time forming one nappe and the light streaming out after it the other. But hourglass is a more colorful name.
What to do? We will try to bring quantum mechanics to the rescue. We will make what is conventionally called a measurement, but cautiously. A place is chosen well outside the gooseberry bush, and a time chosen that is later than when we set the state of the bush up. A check is made of whether there are at this place and time any photons coming from the direction of the bush. By doing things this way, we won't disturb the bush at all, and we don't care if we disturb the escaping light. We get from this, of course, a probability distribution over various possibilities for photons at this place and time. Encouraged by this small success, we choose another place and time and do the same. And this is what is nice: the two measurements are compatible. Thus we get correlations between them, too. Emboldened by this opportunity, we do millions of them, which all formally combine into a single measurement with a single set of possible results. Each possible result of the single, combined measurement is a combination of results of all the individual measurements of light made at the various times and places. Thus each combined result constitutes a kind of movie of the gooseberry bush.
What will the most probable of these results be like? This is the problem of the hourglass. To begin with, however, it may be that there is no hourglass. The deepest quantum theory might not provide a system with a state and its evolution. Or if it does, it could still be objected that the Hamiltonian evolution should not have been allowed to run on unchecked. There should have been many quantum jumps. By leaving them out, quantum mechanics has been misused, and what results is no matter.
But Lagrange and Hamilton and would have been best pleased if these objections did not hold. And surely we would then hope to see in each of the most probable results something like a movie of a bush producing gooseberries: physics working right. The bushes in these movies would look much alike at the start but then gradually differ, as chance has it. We would learn something about how gooseberry bushes grow gooseberries! Certainly Lagrange and Hamilton would have thought the problem of the hourglass a leading one, if they had known of quantum mechanics. Indeed, every physicist might like to take a stab at guessing its solution, just to orient themselves in their science. Does the hourglass fail, and if so, where and why? Or if it does produce movies true to our world, but not from a developing quantum state that might be the true history of a gooseberry bush, rather from a "history" that does at one time represent a gooseberry bush well, but soon is unlike anything that ever did exist, then how can this be?
Here is what I think about it. But before we go into that, see if you don't agree that the hourglass question has gravity, and this regardless of the ideas that I or anyone might have for its answer. Now my guess is that quantum mechanics will give us movies of ripening gooseberries, produced by hourglasses through the means described or something rather like that. And I think that to understand hourglasses, not to solve the measurement problem, is the central question for the understanding of quantum mechanics.
For the measurement problem begs a question, which makes it futile. It assumes that we learn from physics simply because physics describes well those things that exist. Like this example from classical physics. There exist in a gas a multitude of zipping molecules. At any given moment, each particle has its particular position and momentum, and over time this forms their history. Physics has told us what a gas is-precisely what exists there. This is what lets us learn about gases. Undoubtedly this is how Boltzmann saw it.
But when we look at the statistical mechanics he produced, and even more at that of Gibbs, a person will acquire deep qualms about this viewpoint. Boltzmann's analysis of the collision of molecules seems like straightforward common sense. He is looking at what they are likely to do. But when Loschmidt's reversibility objection is brought forward, the lucidity vanishes.
Gibbs's more abstract statistical mechanics made the problem even starker. Gibbs found beautiful mathematical form in Boltzmann's (and Maxwell's) work, which he generalized. He held that thermodynamic systems should be represented as being in states that have the form of certain probability distributions over classical states. Gibbs could not well understand what these probabilities were about, but he saw that his theory was good nevertheless. To keep this lack of clear comprehension from poisoning work with the theory, he devised a work-around. The axioms of probability theory are reflected in the axioms of finite set theory. One can effectively solve problems of probability by thinking about finite sets. So Gibbs suggested that we simply think about these probabilities in terms of sets. The word he used was ensembles.
Gibbs described his intent in these words: "The application of this principle is not limited to cases in which there is a formal and explicit reference to an ensemble of systems. Yet the conception of such an ensemble may serve to give precision to notions of probability. It is in fact customary in the discussion of probabilities to describe anything which is imperfectly known as something taken at random from a great number of things which are completely described." [Gibbs] But physicists have never been able to accept gracefully that they don't understand the elements of their science. So they have been moved to think that they do understand Gibbs's probabilities somehow, and this has led to two missteps.
One has been to regard the probabilities in Gibbs's theory as being the result of our ignorance of the detailed state of the system we are considering. But when a probability distribution is useful, this is a very great step up in order from chaos. Ignorance cannot create order. If water always boils at the same temperature, it is not our fault. Rather than being so explained, for it is not, Gibbs's theory shows that there is something deeply wrong with classical mechanics. Classical statistical mechanics is not really a form of classical mechanics. It is quantum mechanics being born.
The following words of Gibbs seem to show that Gibbs himself took the view just scotched. "The states of the bodies which we handle are certainly not known to us exactly. What we know about a body can generally be described most accurately and most simply by saying that it is one taken at random from a great number (ensemble) of bodies which are completely described." [Gibbs] The impression that I get, though, is that Gibbs is cautiously hedging. He is not saying plainly, as he might have, that a body we handle will be in some completely described state, so that if we describe it with an ensemble, the probabilities in the ensemble simply represent our partial ignorance about that state. He does say plainly that his method seems to work.
The other misstep has come about because quantum theory is a mirror of Gibbs's statistical mechanics in the sense that it is based on what are probabilities in form (in other words, sets of non-negative real numbers that add up to one) and we don't know what they mean in general. It is true that we can make good sense of them as real probabilities in various special cases. For instance, when quantum mechanics is applied to the Stern-Gerlach experiment, to see the detector react is like seeing a coin tossed. But in the general case no such kind of experience is directly implied by these probability forms. There are, for example, canonical distributions in quantum mechanics too, and we don't ever expect to see a detector pick a pure state out of a hot cup of coffee.
We then sometimes think about these formal probabilities in terms of ensembles, just as Gibbs did, and for the same reason. Where the formal probabilities are highest and the members of the ensemble most numerous, there the greatest significance will lie, whatever it may be. This is fine. But quite often physicists say that ensembles (that is to say, Gibbs's work-around) provide the means to understand quantum theory. This is clearly wrong.
But to get back to the measurement problem. As you well know, but for explicitness I will say it anyway, to see a problem in measurement is to suppose that quantum mechanics can describe the equipment in the lab as it exists at the start of an experiment, but when the representation is continued, the equipment becomes entangled with the microscopic systems it is examining and gets smeared. Then quantum mechanics has stopped describing what we know exists in the lab and needs to be corrected so that it will continue to describe what exists.
But it isn't so that quantum mechanics, if it is to show us some predictability in nature, must provide us directly with histories of the existence of things, as by a developing wave packet. As evidence, I offer the hourglass.
If physics does not work simply because it describes what exists, and if, rather, the way of the hourglass is right, then a corollary is that how we learn about nature necessarily becomes more indirect. We are given such information as radiation provides about something, not directly told what exists there. And for the purpose of inferring useful rules of nature's behavior, what we deal with are imagined situations that we think typical of what we want to learn about, not faithful descriptions of actual things. No real radiating system is like an hourglass, except momentarily near the hourglass's neck.
But quantum engineering may temper the truth of that judgment just a bit. For there is also an engineering use of quantum mechanics where, somewhat as classical mechanics does it, for a time we can use a wave packet to represent the development of an actual situation we are dealing with. But this is rather more special, for we must take care to set things up so that this will work. The vacuum must be excellent, etc. Isolation is important.
A simple example of quantum engineering is an ion that alternately blinks for a spell and remains dark for a spell while sitting in an ion trap that is irradiated by lasers. You can picture the ion well enough by thinking of Schrödinger evolution of a wave packet with occasional quantum jumps interspersed. You might then be tempted to think that everything can be handled effectively in the same way, at least in principle. We have just not been clever enough to find New York City's wave packet and its measurement collapses. This is trouble. The worst of it is that you will be led to ignore hourglasses and what they imply, since clearly hourglasses cannot represent the history of things in the same manner that you have advantageously represented the history of the blinking ion.
On the other hand, imagine that decades ago physicists had taken hourglasses to their hearts, as well I think they might have. Then they could have been tempted to look upon representing an ion in a trap by Schrödinger evolution of a wave packet with quantum jumps as 'following the wrong philosophy' (by trying to represent the actual histories of things with wave packets), and might have disdained to do so. There is a lesson here. Don't take your philosophical ideas too seriously, we're not good enough for that.
I believe, though, that from hourglasses you would be able to infer that Schrödinger evolution with jumps is a simple and effective (not perfect) way to regard a blinking ion in a trap. The hourglasses would then be in this sense the more fundamental theory.
But what is a quantum jump? Here is where I think the community of physicists has been careless in the use of words, perhaps mixed with real misunderstanding. Two principles of quantum physics have been formulated. The first principle (promoted by Dirac and von Neumann) is that when a measurement is made on a system, an immediately following measurement will give the same result. Therefore, right after any measurement the system must be in the eigenstate corresponding to the value found.
The second principle is that if the probabilities of the possible results of all the measurements that may be made on a system are defined, then there will be a (unique) quantum state that the system may be said to be in that will yield these probabilities. Add to this that sometimes two measurements may be made on a system without interfering with each other. Then when one of the two measurements has a certain result this will define a conditional probability for any result of the other measurement (simply divide the probability that both results occur by the probability that this result of the first measurement occurs). According to the second principle, then, there will be a quantum state that yields these probabilities (for the possible results of any measurement that may be made without interfering with, or suffering interference from, a given measurement that has had a certain result).
Please notice that the argument above assumes that the set of all the measurements compatible with a given measurement effectively constitutes 'all the measurements that may be made on a system' as needed by the second principle. Now consider a system A in the state α. It is composed of two subsystems, B and C, in reduced states β and γ respectively. A measurement is made on subsystem B and it has a result. By the first principle, there is a quantum state β ′ that will yield the probabilities of the possible results of any immediately following measurement that might be made on subsystem B. And by the second principle, there is a quantum state γ ′ that will yield the probabilities of the possible results of any compatible measurement made on subsystem C.
For the supplanting in one's considerations of β by β ′ there is the historical name 'collapse of the wave packet'. For the supplanting in one's considerations of α by γ ′ most physicists use the same phrase (or any of its several synonyms). It would easier to think about these things if different names were used for the two. 'Collapse of the wave packet' might be retained for the first and, say, 'conditioning of the wave packet' adopted for the second. This is all the more important because the first principle is an out and out mistake by Dirac and von Neumann, whereas the second is an inalienable part of quantum mechanics. To those two mathematically minded, and so logically minded, people, the dignity of quantum mechanics required that there be measurements, so that quantum mechanics might be real physics. And since quantum mechanics did not say that a system had to have, before the measurement, the value found in the measurement, the dignity of measurement required that it at least have that value afterward, or what sort of measurement was this anyway?
Tacked on to this was the fact that so distressed Schrödinger: wave packets spread interminably. If a developing wave packet were to represent the history of a system, which they assumed to be necessary, then the spreading had to be checked, and an occasional quantum jump such as their measurement theory presupposed might do that.
And experiment lent some support. Above all, if an electron went splat somewhere on a screen, which they regarded as a measurement by the experimenter of the electron's position, then conservation of charge suggested strongly that the electron could be found subsequently thereabout. This was the origin of the phrase 'collapse of the wave packet'. Too, the famous Stern-Gerlach experiment allows a following measurement of spin, which will give the same result as the first if the first measurement's detection has been delicate enough.
But the idea of a quickly following measurement is just not well-defined in general. And there are cases where the principle must prove false under any reasonable definition of a following measurement. For example, a particle might lose most of its energy in those collisions that measured its energy. Or if the momentum of a charged particle were measured by the curvature of its path in a magnetic field, the particle might end up going in the wrong direction, although this is, to be sure, correctable. Those events called "measurements" are what they are, and if they fall short of truly being measurements of properties, so be it! If the first principle is an error, then that leaves us with only one principle, the second, and people might then be inclined to continue to use the traditional phrase 'collapse of the wave packet', but now meaning the replacements the second principle defines. This would result in the transfer, in the course of history, of the meaning of the phrase from the first principle to the second. I think that this would have the same unhappy effect as if Lavoisier, not wishing to burden the world with a neologism, had instead given to the word phlogiston a new sense.
The second principle has a very different flavor from the first. For it leads to conditional probabilities, and these lend themselves to imaginative thinking. In this mind-set you are free to take up points of view according to what you wish to learn. The first principle, however, leads to probabilities that are thought to be the properties of real events, such as an actual toss of a coin. You are now in a reality mind-set. That probability is as much a part of the coin toss as is the silver of the coin, and you must deal with it. You have no choice. But I don't mean to say that this is an absolute difference between the two principles. Rather, they tend to lead us into these respective modes of thought, and vice versa. Bearing this in mind, let us look at the hourglass and quantum engineering.
First consider the hourglass that represents a gooseberry bush. By choosing one among the more probable of the results of the course of observation of light, we will select what is in effect a likely movie of such a bush. We can look at the movie, and the marvelous algorithms of our brains will construct an idea of a gooseberry bush and follow it through its history. We have gotten something good out of this, and we have made no use of the conditional probabilities offered by the second principle at all. However, if we are not limited to one movie then we can use conditional probabilities as they are normally used, to explore various interesting possibilities while taking into account how likely they are when we are supplied with certain information.
Notice that we have been thinking imaginatively. No one would suppose that we have directly grasped the reality of a gooseberry bush in our garden in this way, particularly because real gooseberry bushes do not start to exist at a special time. Now consider quantum engineering. By means of careful construction of the equipment a clearly defined situation can be set up where the power of wave packets to give understanding will be enhanced. On the other hand, here there can be significant entanglement. The power of our minds to achieve understanding through their everyday methods will be set at nought.
Then for quantum engineering, a history formed by wave packet development with occasional saltations may be a quite good route to understanding. We would take up this idea of what exists simply because it is good enough to help us with the job at hand. And for this case, where we find it fit to think that we are dealing with an actual system that is an evolving wave packet, and with saltations that we regard as actual events, but in a way so different from that intended by Dirac and von Neumann, then perhaps a third term, say, 'change of the wave packet', would be appropriate for the saltations.
These changes of the wave packet would differ from collapses of the wave packet because, although they would be thought of as real events just as collapses have been, they would be derived from conditioning of wave packets, in the following manner. When a system sends out radiation (or anything else) that will not return, in one way you can consider the system of interest to be the whole, including the radiation, and in another way you can consider it to be the reduced system that does not include the radiation. Upon observation of the radiation you will derive from the result and from the wave packet of the whole system a wave packet for the system less the radiation, and this we have called conditioning of the wave packet. But if before the observation your interest had been focussed on the system less the radiation, and thus on its reduced wave packet, then you will have gone from one wave packet to another wave packet for the system less the radiation. And since you are reckoning these wave packets as being portions of the system's history, this looks like a quantum jump. This is what is meant by a change of the wave packet. There is no need to define any such change of the wave packet precisely, of course. No more is there need to suppose that it can be defined with precision.
If hourglasses cannot be true histories, how can it be that we can learn from them? What lets them tell us how gooseberry bushes grow, when they are only momentarily like a gooseberry bush? I haven't said a word about this yet.
First of all, there is an assumption hidden behind this puzzlement of ours. The assumption is that we have no reason to be perplexed that we can learn from things that can be true histories. For if it did not seem so perfectly natural to us that we learn from true histories, then it would not appear unnatural to learn from what clearly cannot be a true history. But I think this assumption of ours is thoughtless, and I will try to explain why.
We make judgments about when we are better informed and when less so. The ideas we hold true when thought to be better informed are compared with those that we held when not so well informed. In this way, through the device of taking the ideas that we presently have most confidence in as trustworthy, we try to gather how successfully our ideas tend to stack up against reality. It is not quite so simple, however, since we know from sad experience that the ideas we now trust may fail us. But we have the conviction, or hope, that if such happens we can land on our feet again. We will search for still better ideas until we find something that works.
We are apt to give to this situation a logical cast. Namely, by positing that there is a best of all possible ideas in whose direction we are headed. This posit can be helpful. It can give us greater confidence in our search for better ideas. If we guess that this best idea will have a certain form, and we guess well, it can guide our search. But there is no necessity for this posit; all we really know is what was said above.
Another thing we like to do is to find where things are and when. Our vision, touch, and hearing do this automatically all the time, and we often give them some conscious help, say by turning the head. When we are a teenager it is likely to occur to us that there must be a best of all possible such ideas, a complete map of where everything is, and has been, and perhaps will be too. A further thought may cross one's mind. Maybe this is all that our world is. For instance, if one person likes another, this should show up in that person's actions, which the map will completely define. Maybe the liking simply is those actions. Now I will propose some physics, the red dust theory. According to this theory the world is made up of an exceedingly large number of very fine specks of a scarlet dust. Because of its ruddiness, the dust is extremely beautiful, if only we could see it, but we will not be concerned with that. The red dust theory differs from most physics in that the flight of the particles does not have to satisfy a differential equation, it is merely continuous.
The interpretation of the theory is quite simple. Where we find things there will be a crowd of these specks, and where we find vacancy they will be much sparser. But can our world be as this theory says? Surely it can. There will be among its solutions one that maps the entire history of our universe with extraordinary precision. The collisions of galaxies, the evolution of whales, the experiments in laboratories, all will be there and rightly shown. Now you may think that the red dust theory is hopelessly bad physics and should be ignored. It may be hopelessly bad, but it should not be ignored. It is a benchmark. If another physics theory is proposed, is it better than the red dust theory, and if so, just why? This is especially pertinent if the other theory intends, as does the red dust theory, to give a precise description of all that exists. Bohmian quantum mechanics is an example.
But what I intend to put up against the benchmark is classical mechanics. Everyone will agree that classical mechanics is far better than the red dust theory. You can do things with classical mechanics; you can't do anything with the red dust theory. For instance, you can pull a pendulum to the side and let it go. It will swing. Classical mechanics can give you the history of that swing ahead of time. The red dust theory has so many solutions compatible with the way things are at the start that it won't tell you anything useful about how things will go.
Our experience with classical mechanics is that it is practical, but why is this so? The most natural idea is that the world must at bottom be classical mechanical. Since we understood the pendulum by assigning a classical mechanical state to it and evolving the state, there must then be an evolving classical mechanical state that the whole world is in, and that would explain why classical mechanics is so useful.
When we look at the history of our universe, however, and particularly at the evolution of life over billions of years, and when we consider the resources that it is likely that classical mechanics has to offer in its solutions, it doesn't really seem possible that there is any classical mechanical history that would match our universe's history, no matter how exquisitely the initial conditions are chosen. For the more detailed structures of the classical representation must in time dissolve into lasting chaos, and I would think rather quickly.
Still, this does depend on a point I don't actually know the answer to. For in order to make the universe behave as you wish, that is to say, give a good account of continents rifting and hummingbirds feeding, it might be that to obtain each additional second of the desired history it is always sufficient to correctly calculate another, say, thousand decimal places for the positions and momenta of the molecules in the initial state. Or to the contrary, the first thousand decimal places might give you one second, the next thousand only a further half second, then a fourth of a second, and so on.
Yet even if I am wrong in this, we would just go from Scylla to Charybdis. For in that case classical mechanics must be like the red dust theory, where, from our point of view, anything is possible, or too close to anything. In either case the classical solution set would imply no structure such as we experience in life. No sculpted dunes, no ants carting morsels, no shower of hail would pop out of it. Nor can one imagine any reason why the solution set would show a preference for depicting creatures learning classical mechanics, or if so doing benefiting by it. In short, there is a total disconnect between the fact that classical mechanics is useful and the hypothesis that the universe as a whole is a classical mechanical system.
That leaves us with an unsolved mystery: why does classical mechanics work for us? And classical mechanics is the archetype of the kind of physics where we learn from what can be true histories of things.
To my mind, the hourglass with observation of its emitted light is deeply conservative physics. It makes quantum mechanics as seamless a continuation of the physics of the previous centuries as is at all possible. This is because of the mathematical form of the hourglass, which is a continuous development from initial conditions, as well as the form of the observations, which impinge as little as can be. And when this leads to our being given movies rather than direct histories, then I am surprised (and amused) by this, but accept it for the sake of the qualities mentioned, which I consider to be virtues that promise. Nature is teaching us another lesson.
Bohr's old quantum theory was based on quantum jumps, and I think this was a wonderful piece of exploration in the dark. When Heisenberg's new quantum mechanics came along, quantum jumps were kept. The jumps would allow direct histories to be retained as the foundation of our physics, though at the expense of the continuous Hamiltonian evolution of the wave packets (and at the expense of clear definition, for no one has ever been able to specify just when and where and what the quantum jumps are). Like Schrödinger, I am jarred by this. If we are given the choice of preserving philosophical principle or mathematical form, I think we should prefer mathematical form.
A final thought: If learning from the movies provided by hourglasses is how we do physics, then to know why quantum mechanics works would be to know why all the inferences we might make from the movies will fit together with sufficient coherence. But to know this would require that we know all the things we might ever think of. It's hopeless. Though we might nibble at the problem, by showing that the hourglasses have some needed characteristics. So I think the hourglasses will leave us with an essentially unfathomable mystery. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Hourglass is the name given here to a formal isolated quantum system that can radiate. Starting from a time when it defines the system it represents clearly and no radiation is present, it is given straightforward Hamiltonian evolution. The question of what significance hourglasses have is raised, and this question is proposed to be more consequential than the measurement problem. 1 Hourglasses 2 Physics without true histories 3 But histories are sometimes good 4 Phlogiston and oxygen 5 A closer look at quantum engineering 6 Conclusion"
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "But I want to know the particular go of it -the plea of James Clerk Maxwell as a young child concerning, among many things, the bell-wires that ring the bells that summon servants. [Mahon] 1 Hourglasses Suppose that theory develops in such a way that quantum fields can be handled like nonrelativistic quantum mechanics. Then if we are interested in something, perhaps gooseberry bushes, we can model one as we would conceive it to be at some instant and then follow its development through time. And not only the atoms and molecules would be modeled, but also the radiation. This is a scheme for the imagination. The gooseberry bush, though not isolated, would grow within a suitable environment that would be an isolated system, complete in itself. We do learn well from isolated systems, both real ones in the laboratory and those envisaged in our theoretical musings.\n\nWe will provide the bush with air, earth, and water. And there can be life-giving sunlight shining on it. As for the light that had been reflected or emitted from the bush before the present time, we will leave that out. Such light goes off and away, so it could only matter as information about what the bush had been doing. We will take the bush just as it is now.\n\nThe gooseberry bush is then developed forward in time. Lagrange or Hamilton would have recognized what we are doing, for we are doing physics the classical way. We have an initial condition and we are finding out what will happen next.\n\nAs we move toward the future, light shoots out from the bush, as we expect. But, disconcertingly, the bush starts to lose definition. Its parts lose their precise places. Within a few weeks it is a scarcely recognizable mess. Let's go back in time, then. This is terrifyingly worse. The bush has been the subject of a vast conspiracy. Light has been streaming in on it from the entire universe. The bush swallows it up. Then at the present time this suddenly all stops. Time symmetry of the Hamiltonian makes it happen like that. This is the hourglass. It is really more like a cone, with the light streaming in before the set-up time forming one nappe and the light streaming out after it the other. But hourglass is a more colorful name.\n\nWhat to do? We will try to bring quantum mechanics to the rescue. We will make what is conventionally called a measurement, but cautiously. A place is chosen well outside the gooseberry bush, and a time chosen that is later than when we set the state of the bush up. A check is made of whether there are at this place and time any photons coming from the direction of the bush. By doing things this way, we won't disturb the bush at all, and we don't care if we disturb the escaping light. We get from this, of course, a probability distribution over various possibilities for photons at this place and time. Encouraged by this small success, we choose another place and time and do the same. And this is what is nice: the two measurements are compatible. Thus we get correlations between them, too. Emboldened by this opportunity, we do millions of them, which all formally combine into a single measurement with a single set of possible results. Each possible result of the single, combined measurement is a combination of results of all the individual measurements of light made at the various times and places. Thus each combined result constitutes a kind of movie of the gooseberry bush.\n\nWhat will the most probable of these results be like? This is the problem of the hourglass. To begin with, however, it may be that there is no hourglass. The deepest quantum theory might not provide a system with a state and its evolution. Or if it does, it could still be objected that the Hamiltonian evolution should not have been allowed to run on unchecked. There should have been many quantum jumps. By leaving them out, quantum mechanics has been misused, and what results is no matter.\n\nBut Lagrange and Hamilton and would have been best pleased if these objections did not hold. And surely we would then hope to see in each of the most probable results something like a movie of a bush producing gooseberries: physics working right. The bushes in these movies would look much alike at the start but then gradually differ, as chance has it. We would learn something about how gooseberry bushes grow gooseberries! Certainly Lagrange and Hamilton would have thought the problem of the hourglass a leading one, if they had known of quantum mechanics. Indeed, every physicist might like to take a stab at guessing its solution, just to orient themselves in their science. Does the hourglass fail, and if so, where and why? Or if it does produce movies true to our world, but not from a developing quantum state that might be the true history of a gooseberry bush, rather from a \"history\" that does at one time represent a gooseberry bush well, but soon is unlike anything that ever did exist, then how can this be?"
},
{
"section_type": "OTHER",
"section_title": "Physics without true histories",
"text": "Here is what I think about it. But before we go into that, see if you don't agree that the hourglass question has gravity, and this regardless of the ideas that I or anyone might have for its answer. Now my guess is that quantum mechanics will give us movies of ripening gooseberries, produced by hourglasses through the means described or something rather like that. And I think that to understand hourglasses, not to solve the measurement problem, is the central question for the understanding of quantum mechanics.\n\nFor the measurement problem begs a question, which makes it futile. It assumes that we learn from physics simply because physics describes well those things that exist. Like this example from classical physics. There exist in a gas a multitude of zipping molecules. At any given moment, each particle has its particular position and momentum, and over time this forms their history. Physics has told us what a gas is-precisely what exists there. This is what lets us learn about gases. Undoubtedly this is how Boltzmann saw it.\n\nBut when we look at the statistical mechanics he produced, and even more at that of Gibbs, a person will acquire deep qualms about this viewpoint. Boltzmann's analysis of the collision of molecules seems like straightforward common sense. He is looking at what they are likely to do. But when Loschmidt's reversibility objection is brought forward, the lucidity vanishes.\n\nGibbs's more abstract statistical mechanics made the problem even starker. Gibbs found beautiful mathematical form in Boltzmann's (and Maxwell's) work, which he generalized. He held that thermodynamic systems should be represented as being in states that have the form of certain probability distributions over classical states. Gibbs could not well understand what these probabilities were about, but he saw that his theory was good nevertheless. To keep this lack of clear comprehension from poisoning work with the theory, he devised a work-around. The axioms of probability theory are reflected in the axioms of finite set theory. One can effectively solve problems of probability by thinking about finite sets. So Gibbs suggested that we simply think about these probabilities in terms of sets. The word he used was ensembles.\n\nGibbs described his intent in these words: \"The application of this principle is not limited to cases in which there is a formal and explicit reference to an ensemble of systems. Yet the conception of such an ensemble may serve to give precision to notions of probability. It is in fact customary in the discussion of probabilities to describe anything which is imperfectly known as something taken at random from a great number of things which are completely described.\" [Gibbs] But physicists have never been able to accept gracefully that they don't understand the elements of their science. So they have been moved to think that they do understand Gibbs's probabilities somehow, and this has led to two missteps.\n\nOne has been to regard the probabilities in Gibbs's theory as being the result of our ignorance of the detailed state of the system we are considering. But when a probability distribution is useful, this is a very great step up in order from chaos. Ignorance cannot create order. If water always boils at the same temperature, it is not our fault. Rather than being so explained, for it is not, Gibbs's theory shows that there is something deeply wrong with classical mechanics. Classical statistical mechanics is not really a form of classical mechanics. It is quantum mechanics being born.\n\nThe following words of Gibbs seem to show that Gibbs himself took the view just scotched. \"The states of the bodies which we handle are certainly not known to us exactly. What we know about a body can generally be described most accurately and most simply by saying that it is one taken at random from a great number (ensemble) of bodies which are completely described.\" [Gibbs] The impression that I get, though, is that Gibbs is cautiously hedging. He is not saying plainly, as he might have, that a body we handle will be in some completely described state, so that if we describe it with an ensemble, the probabilities in the ensemble simply represent our partial ignorance about that state. He does say plainly that his method seems to work.\n\nThe other misstep has come about because quantum theory is a mirror of Gibbs's statistical mechanics in the sense that it is based on what are probabilities in form (in other words, sets of non-negative real numbers that add up to one) and we don't know what they mean in general. It is true that we can make good sense of them as real probabilities in various special cases. For instance, when quantum mechanics is applied to the Stern-Gerlach experiment, to see the detector react is like seeing a coin tossed. But in the general case no such kind of experience is directly implied by these probability forms. There are, for example, canonical distributions in quantum mechanics too, and we don't ever expect to see a detector pick a pure state out of a hot cup of coffee.\n\nWe then sometimes think about these formal probabilities in terms of ensembles, just as Gibbs did, and for the same reason. Where the formal probabilities are highest and the members of the ensemble most numerous, there the greatest significance will lie, whatever it may be. This is fine. But quite often physicists say that ensembles (that is to say, Gibbs's work-around) provide the means to understand quantum theory. This is clearly wrong.\n\nBut to get back to the measurement problem. As you well know, but for explicitness I will say it anyway, to see a problem in measurement is to suppose that quantum mechanics can describe the equipment in the lab as it exists at the start of an experiment, but when the representation is continued, the equipment becomes entangled with the microscopic systems it is examining and gets smeared. Then quantum mechanics has stopped describing what we know exists in the lab and needs to be corrected so that it will continue to describe what exists.\n\nBut it isn't so that quantum mechanics, if it is to show us some predictability in nature, must provide us directly with histories of the existence of things, as by a developing wave packet. As evidence, I offer the hourglass."
},
{
"section_type": "OTHER",
"section_title": "But histories are sometimes good",
"text": "If physics does not work simply because it describes what exists, and if, rather, the way of the hourglass is right, then a corollary is that how we learn about nature necessarily becomes more indirect. We are given such information as radiation provides about something, not directly told what exists there. And for the purpose of inferring useful rules of nature's behavior, what we deal with are imagined situations that we think typical of what we want to learn about, not faithful descriptions of actual things. No real radiating system is like an hourglass, except momentarily near the hourglass's neck.\n\nBut quantum engineering may temper the truth of that judgment just a bit. For there is also an engineering use of quantum mechanics where, somewhat as classical mechanics does it, for a time we can use a wave packet to represent the development of an actual situation we are dealing with. But this is rather more special, for we must take care to set things up so that this will work. The vacuum must be excellent, etc. Isolation is important.\n\nA simple example of quantum engineering is an ion that alternately blinks for a spell and remains dark for a spell while sitting in an ion trap that is irradiated by lasers. You can picture the ion well enough by thinking of Schrödinger evolution of a wave packet with occasional quantum jumps interspersed. You might then be tempted to think that everything can be handled effectively in the same way, at least in principle. We have just not been clever enough to find New York City's wave packet and its measurement collapses. This is trouble. The worst of it is that you will be led to ignore hourglasses and what they imply, since clearly hourglasses cannot represent the history of things in the same manner that you have advantageously represented the history of the blinking ion.\n\nOn the other hand, imagine that decades ago physicists had taken hourglasses to their hearts, as well I think they might have. Then they could have been tempted to look upon representing an ion in a trap by Schrödinger evolution of a wave packet with quantum jumps as 'following the wrong philosophy' (by trying to represent the actual histories of things with wave packets), and might have disdained to do so. There is a lesson here. Don't take your philosophical ideas too seriously, we're not good enough for that.\n\nI believe, though, that from hourglasses you would be able to infer that Schrödinger evolution with jumps is a simple and effective (not perfect) way to regard a blinking ion in a trap. The hourglasses would then be in this sense the more fundamental theory."
},
{
"section_type": "OTHER",
"section_title": "Phlogiston and oxygen",
"text": "But what is a quantum jump? Here is where I think the community of physicists has been careless in the use of words, perhaps mixed with real misunderstanding. Two principles of quantum physics have been formulated. The first principle (promoted by Dirac and von Neumann) is that when a measurement is made on a system, an immediately following measurement will give the same result. Therefore, right after any measurement the system must be in the eigenstate corresponding to the value found.\n\nThe second principle is that if the probabilities of the possible results of all the measurements that may be made on a system are defined, then there will be a (unique) quantum state that the system may be said to be in that will yield these probabilities. Add to this that sometimes two measurements may be made on a system without interfering with each other. Then when one of the two measurements has a certain result this will define a conditional probability for any result of the other measurement (simply divide the probability that both results occur by the probability that this result of the first measurement occurs). According to the second principle, then, there will be a quantum state that yields these probabilities (for the possible results of any measurement that may be made without interfering with, or suffering interference from, a given measurement that has had a certain result).\n\nPlease notice that the argument above assumes that the set of all the measurements compatible with a given measurement effectively constitutes 'all the measurements that may be made on a system' as needed by the second principle. Now consider a system A in the state α. It is composed of two subsystems, B and C, in reduced states β and γ respectively. A measurement is made on subsystem B and it has a result. By the first principle, there is a quantum state β ′ that will yield the probabilities of the possible results of any immediately following measurement that might be made on subsystem B. And by the second principle, there is a quantum state γ ′ that will yield the probabilities of the possible results of any compatible measurement made on subsystem C.\n\nFor the supplanting in one's considerations of β by β ′ there is the historical name 'collapse of the wave packet'. For the supplanting in one's considerations of α by γ ′ most physicists use the same phrase (or any of its several synonyms). It would easier to think about these things if different names were used for the two. 'Collapse of the wave packet' might be retained for the first and, say, 'conditioning of the wave packet' adopted for the second. This is all the more important because the first principle is an out and out mistake by Dirac and von Neumann, whereas the second is an inalienable part of quantum mechanics. To those two mathematically minded, and so logically minded, people, the dignity of quantum mechanics required that there be measurements, so that quantum mechanics might be real physics. And since quantum mechanics did not say that a system had to have, before the measurement, the value found in the measurement, the dignity of measurement required that it at least have that value afterward, or what sort of measurement was this anyway?\n\nTacked on to this was the fact that so distressed Schrödinger: wave packets spread interminably. If a developing wave packet were to represent the history of a system, which they assumed to be necessary, then the spreading had to be checked, and an occasional quantum jump such as their measurement theory presupposed might do that.\n\nAnd experiment lent some support. Above all, if an electron went splat somewhere on a screen, which they regarded as a measurement by the experimenter of the electron's position, then conservation of charge suggested strongly that the electron could be found subsequently thereabout. This was the origin of the phrase 'collapse of the wave packet'. Too, the famous Stern-Gerlach experiment allows a following measurement of spin, which will give the same result as the first if the first measurement's detection has been delicate enough.\n\nBut the idea of a quickly following measurement is just not well-defined in general. And there are cases where the principle must prove false under any reasonable definition of a following measurement. For example, a particle might lose most of its energy in those collisions that measured its energy. Or if the momentum of a charged particle were measured by the curvature of its path in a magnetic field, the particle might end up going in the wrong direction, although this is, to be sure, correctable. Those events called \"measurements\" are what they are, and if they fall short of truly being measurements of properties, so be it! If the first principle is an error, then that leaves us with only one principle, the second, and people might then be inclined to continue to use the traditional phrase 'collapse of the wave packet', but now meaning the replacements the second principle defines. This would result in the transfer, in the course of history, of the meaning of the phrase from the first principle to the second. I think that this would have the same unhappy effect as if Lavoisier, not wishing to burden the world with a neologism, had instead given to the word phlogiston a new sense."
},
{
"section_type": "OTHER",
"section_title": "A closer look at quantum engineering",
"text": "The second principle has a very different flavor from the first. For it leads to conditional probabilities, and these lend themselves to imaginative thinking. In this mind-set you are free to take up points of view according to what you wish to learn. The first principle, however, leads to probabilities that are thought to be the properties of real events, such as an actual toss of a coin. You are now in a reality mind-set. That probability is as much a part of the coin toss as is the silver of the coin, and you must deal with it. You have no choice. But I don't mean to say that this is an absolute difference between the two principles. Rather, they tend to lead us into these respective modes of thought, and vice versa. Bearing this in mind, let us look at the hourglass and quantum engineering.\n\nFirst consider the hourglass that represents a gooseberry bush. By choosing one among the more probable of the results of the course of observation of light, we will select what is in effect a likely movie of such a bush. We can look at the movie, and the marvelous algorithms of our brains will construct an idea of a gooseberry bush and follow it through its history. We have gotten something good out of this, and we have made no use of the conditional probabilities offered by the second principle at all. However, if we are not limited to one movie then we can use conditional probabilities as they are normally used, to explore various interesting possibilities while taking into account how likely they are when we are supplied with certain information.\n\nNotice that we have been thinking imaginatively. No one would suppose that we have directly grasped the reality of a gooseberry bush in our garden in this way, particularly because real gooseberry bushes do not start to exist at a special time. Now consider quantum engineering. By means of careful construction of the equipment a clearly defined situation can be set up where the power of wave packets to give understanding will be enhanced. On the other hand, here there can be significant entanglement. The power of our minds to achieve understanding through their everyday methods will be set at nought.\n\nThen for quantum engineering, a history formed by wave packet development with occasional saltations may be a quite good route to understanding. We would take up this idea of what exists simply because it is good enough to help us with the job at hand. And for this case, where we find it fit to think that we are dealing with an actual system that is an evolving wave packet, and with saltations that we regard as actual events, but in a way so different from that intended by Dirac and von Neumann, then perhaps a third term, say, 'change of the wave packet', would be appropriate for the saltations.\n\nThese changes of the wave packet would differ from collapses of the wave packet because, although they would be thought of as real events just as collapses have been, they would be derived from conditioning of wave packets, in the following manner. When a system sends out radiation (or anything else) that will not return, in one way you can consider the system of interest to be the whole, including the radiation, and in another way you can consider it to be the reduced system that does not include the radiation. Upon observation of the radiation you will derive from the result and from the wave packet of the whole system a wave packet for the system less the radiation, and this we have called conditioning of the wave packet. But if before the observation your interest had been focussed on the system less the radiation, and thus on its reduced wave packet, then you will have gone from one wave packet to another wave packet for the system less the radiation. And since you are reckoning these wave packets as being portions of the system's history, this looks like a quantum jump. This is what is meant by a change of the wave packet. There is no need to define any such change of the wave packet precisely, of course. No more is there need to suppose that it can be defined with precision."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusion",
"text": "If hourglasses cannot be true histories, how can it be that we can learn from them? What lets them tell us how gooseberry bushes grow, when they are only momentarily like a gooseberry bush? I haven't said a word about this yet.\n\nFirst of all, there is an assumption hidden behind this puzzlement of ours. The assumption is that we have no reason to be perplexed that we can learn from things that can be true histories. For if it did not seem so perfectly natural to us that we learn from true histories, then it would not appear unnatural to learn from what clearly cannot be a true history. But I think this assumption of ours is thoughtless, and I will try to explain why.\n\nWe make judgments about when we are better informed and when less so. The ideas we hold true when thought to be better informed are compared with those that we held when not so well informed. In this way, through the device of taking the ideas that we presently have most confidence in as trustworthy, we try to gather how successfully our ideas tend to stack up against reality. It is not quite so simple, however, since we know from sad experience that the ideas we now trust may fail us. But we have the conviction, or hope, that if such happens we can land on our feet again. We will search for still better ideas until we find something that works.\n\nWe are apt to give to this situation a logical cast. Namely, by positing that there is a best of all possible ideas in whose direction we are headed. This posit can be helpful. It can give us greater confidence in our search for better ideas. If we guess that this best idea will have a certain form, and we guess well, it can guide our search. But there is no necessity for this posit; all we really know is what was said above.\n\nAnother thing we like to do is to find where things are and when. Our vision, touch, and hearing do this automatically all the time, and we often give them some conscious help, say by turning the head. When we are a teenager it is likely to occur to us that there must be a best of all possible such ideas, a complete map of where everything is, and has been, and perhaps will be too. A further thought may cross one's mind. Maybe this is all that our world is. For instance, if one person likes another, this should show up in that person's actions, which the map will completely define. Maybe the liking simply is those actions. Now I will propose some physics, the red dust theory. According to this theory the world is made up of an exceedingly large number of very fine specks of a scarlet dust. Because of its ruddiness, the dust is extremely beautiful, if only we could see it, but we will not be concerned with that. The red dust theory differs from most physics in that the flight of the particles does not have to satisfy a differential equation, it is merely continuous.\n\nThe interpretation of the theory is quite simple. Where we find things there will be a crowd of these specks, and where we find vacancy they will be much sparser. But can our world be as this theory says? Surely it can. There will be among its solutions one that maps the entire history of our universe with extraordinary precision. The collisions of galaxies, the evolution of whales, the experiments in laboratories, all will be there and rightly shown. Now you may think that the red dust theory is hopelessly bad physics and should be ignored. It may be hopelessly bad, but it should not be ignored. It is a benchmark. If another physics theory is proposed, is it better than the red dust theory, and if so, just why? This is especially pertinent if the other theory intends, as does the red dust theory, to give a precise description of all that exists. Bohmian quantum mechanics is an example.\n\nBut what I intend to put up against the benchmark is classical mechanics. Everyone will agree that classical mechanics is far better than the red dust theory. You can do things with classical mechanics; you can't do anything with the red dust theory. For instance, you can pull a pendulum to the side and let it go. It will swing. Classical mechanics can give you the history of that swing ahead of time. The red dust theory has so many solutions compatible with the way things are at the start that it won't tell you anything useful about how things will go.\n\nOur experience with classical mechanics is that it is practical, but why is this so? The most natural idea is that the world must at bottom be classical mechanical. Since we understood the pendulum by assigning a classical mechanical state to it and evolving the state, there must then be an evolving classical mechanical state that the whole world is in, and that would explain why classical mechanics is so useful.\n\nWhen we look at the history of our universe, however, and particularly at the evolution of life over billions of years, and when we consider the resources that it is likely that classical mechanics has to offer in its solutions, it doesn't really seem possible that there is any classical mechanical history that would match our universe's history, no matter how exquisitely the initial conditions are chosen. For the more detailed structures of the classical representation must in time dissolve into lasting chaos, and I would think rather quickly.\n\nStill, this does depend on a point I don't actually know the answer to. For in order to make the universe behave as you wish, that is to say, give a good account of continents rifting and hummingbirds feeding, it might be that to obtain each additional second of the desired history it is always sufficient to correctly calculate another, say, thousand decimal places for the positions and momenta of the molecules in the initial state. Or to the contrary, the first thousand decimal places might give you one second, the next thousand only a further half second, then a fourth of a second, and so on.\n\nYet even if I am wrong in this, we would just go from Scylla to Charybdis. For in that case classical mechanics must be like the red dust theory, where, from our point of view, anything is possible, or too close to anything. In either case the classical solution set would imply no structure such as we experience in life. No sculpted dunes, no ants carting morsels, no shower of hail would pop out of it. Nor can one imagine any reason why the solution set would show a preference for depicting creatures learning classical mechanics, or if so doing benefiting by it. In short, there is a total disconnect between the fact that classical mechanics is useful and the hypothesis that the universe as a whole is a classical mechanical system.\n\nThat leaves us with an unsolved mystery: why does classical mechanics work for us? And classical mechanics is the archetype of the kind of physics where we learn from what can be true histories of things.\n\nTo my mind, the hourglass with observation of its emitted light is deeply conservative physics. It makes quantum mechanics as seamless a continuation of the physics of the previous centuries as is at all possible. This is because of the mathematical form of the hourglass, which is a continuous development from initial conditions, as well as the form of the observations, which impinge as little as can be. And when this leads to our being given movies rather than direct histories, then I am surprised (and amused) by this, but accept it for the sake of the qualities mentioned, which I consider to be virtues that promise. Nature is teaching us another lesson.\n\nBohr's old quantum theory was based on quantum jumps, and I think this was a wonderful piece of exploration in the dark. When Heisenberg's new quantum mechanics came along, quantum jumps were kept. The jumps would allow direct histories to be retained as the foundation of our physics, though at the expense of the continuous Hamiltonian evolution of the wave packets (and at the expense of clear definition, for no one has ever been able to specify just when and where and what the quantum jumps are). Like Schrödinger, I am jarred by this. If we are given the choice of preserving philosophical principle or mathematical form, I think we should prefer mathematical form."
},
{
"section_type": "OTHER",
"section_title": "Isn't this what Copernicus did?",
"text": "A final thought: If learning from the movies provided by hourglasses is how we do physics, then to know why quantum mechanics works would be to know why all the inferences we might make from the movies will fit together with sufficient coherence. But to know this would require that we know all the things we might ever think of. It's hopeless. Though we might nibble at the problem, by showing that the hourglasses have some needed characteristics. So I think the hourglasses will leave us with an essentially unfathomable mystery."
}
] | ||
arxiv:0704.0425 | 0704.0425 | 1 | ef888e2c75769b2020de3d3b4cdef51d358f0413bb8eb3c4e2e378f7e6ec387c | QED for fields obeying a square root operator equation | Instead of using local field equations - like the Dirac equation for spin-1/2
and the Klein-Gordon equation for spin-0 particles - one could try to use
non-local field equations in order to describe scattering processes. The latter
equations can be obtained by means of the relativistic energy together with the
correspondence principle, resulting in equations with a square root operator.
By coupling them to an electromagnetic field and expanding the square root (and
taking into account terms of quadratic order in the electromagnetic coupling
constant e), it is possible to calculate scattering matrix elements within the
framework of quantum electrodynamics, e.g. like those for Compton scattering or
for the scattering of two identical particles. This will be done here for the
scalar case. These results are then compared with the corresponding ones based
on the Klein-Gordon equation. A proposal of how to transfer these reflections
to the spin-1/2 case is also presented. | [
"Tobias Gleim"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | Instead of using local field equations -like the Dirac equation for spin-1/2 and the Klein-Gordon equation for spin-0 particles -one could try to use non-local field equations in order to describe scattering processes. The latter equations can be obtained by means of the relativistic energy together with the correspondence principle, resulting in equations with a square root operator. By coupling them to an electromagnetic field and expanding the square root (and taking into account terms of quadratic order in the electromagnetic coupling constant e), it is possible to calculate scattering matrix elements within the framework of quantum electrodynamics, e.g. like those for Compton scattering or for the scattering of two identical particles. This will be done here for the scalar case. These results are then compared with the corresponding ones based on the Klein-Gordon equation. A proposal of how to transfer these reflections to the spin-1/2 case is also presented.
Free scalar particles are usually described by means of the well-known Klein-Gordon equation (see e.g. [4, 5, 6, 10] ):
( ) ( ) 0 , ˆ2 2 2 = + + ∂ t x m p t r r φ , (1)
where we have used the momentum operator in configuration space ∇ -= r r i p ˆ (and set the velocity of light as well as Planck's constant h to one). ( 1 ) can be regarded as an iteration of the following square root operator equation (see e.g. [1, 2, 4, 5] ):
( ) ( )
t x p m t x i t , , 2 2 r r r + = ∂ φ φ . ( 2
)
Introducing an electromagnetic field with a 4-vector potential ( ) ( )
[ ] ( )
x x A x A e x A x A ie x m p t φ φ µ µ µ µ µ µ 2 2 2 2 ˆ+ ∂ + ∂ - = + + ∂ r , (3)
where we have used the 4-vector notation ( ) ( ) ( )
x t x x x r r
, , 0 = = µ and Einstein's summation convention.
Here, the coupling terms on the right hand side of (3) could easily be separated from the term with the free particle Hamiltonian on the left hand side of (3). This is unfortunately no longer possible, if one couples the non-local equation (2) to the electromagnetic field:
( ) ( ) ( ) ( ) ( ) ( ) x x eA x x A e p m x i t φ φ φ 0 2 2 ˆ+ - + = ∂ r r , (4)
because the vector potential A r appears under the square root. But in a perturbation analysis of scattering processes, this property is useful, since such an analysis is based on the assumption that the coupling terms make only small contributions to the free particle solution due to the small value of the coupling constant e. By rewriting the Hamiltonian in (4) , ( ) ( ) ( ) ( ) 2 2 2 2 0 2 2 0 ˆA e p A e A p e p m eA x A e p m eA H r r r r r r r r + ⋅ -⋅ -+ + + = -+ + = ′ , (5) one can split off a factor with the free Hamiltonian 2 2 0 p m H + = ′ , (6) which yields ( )( ) ( ) 2 1 2 2 1 2 2 2 2 0 1 ˆp m p m A e p A e A p e eA H r r r r r r r + + + ⋅ -⋅ -+ + = ′ -.
With the above-mentioned assumption, it is now very tempting to expand the first square root factor. A very similar approach has already been proposed by [3] . We would like to restrict ourselves to a series expansion of the kind ...
1 1 2 8 1 2 1 + -+ ≈ + y y y (8) containing only constant, linear and quadratic terms, where y ˆ denotes ( )( ) 1 2 2 2 2 ˆ-+ + ⋅ -⋅ -= p m A e p A e A p e y r r r r r r .
Hamiltonian ( 7 ) is therefore approximated by
2 1 0 Ĥ H H H ′ + ′ + ′ ≈ ′ (10) with ( )( )
+ + ⋅ + ⋅ - = ′ - 0 2 2 2 1 1 2 1 ˆA p m p A A p e H r r r r r , ( 11
) ( ) ( )( ) ( )( ) + ⋅ + ⋅ + ⋅ + ⋅ - + = ′ - - - 2 1 2 1 2 2 1 2 2 8 1 2 2 2 2 1 2 2 p m p A A p p m p A A p p m A e H r r r r r r r r r r r r , (12)
where we have reordered the terms of expansion (9) , retaining only terms to (and including the) quadratic order in e and recollected powers of ( )
2 2 p m r +
. What we have won by (10) is a separation of the free Hamiltonian (6) from the coupling terms in ( 5 ) that approximately result in the sum of 1 Ĥ ′ and 2 Ĥ ′ (i.e. ( 11 ) and ( 12 ) respectively). For we are only interested in corrections to the free Hamiltonian anyway, this approximation might not hurt very much. But however, this separation seems not to be a true one, because of the multiple factors of powers of ( )
2 2 p m r +
in (11) and (12) . That is, we need an interpretation of these operators. To this end, it is useful to know that for the free square root operator equation ( 2 ), an integral representation can be given (see [1, 2] ):
( ) ( ) ( ) ( )( ) t x t x x x x d t x i t , : , , 3 r r r r r φ φ φ Ω = ′ ′ - Ω ′ = ∂ ∫ , ( 13
)
where Ω denotes an energy distribution
( ) ( ) ( ) ∫ ′ - ⋅ - = ′ - Ω x x p i p e p d x x
r r r r r ω π 3 3 2 (14) with 2 2 p m p p r r + = = ω ω .
(15) (13) results from the fact, that one would expect to obtain the following momentum space representation of (2):
( ) ( )
t
p t p i p t , , ~r r φ ω φ = ∂ with ( ) t p, ~r φ denoting the Fourier-transformed ( ) t x, r φ . If the operator 2 2 p m r + corresponds to
( ) ∫ ′ - Ω ′ x x x d r r 3 , the operator ( ) 2 1 2 2 ˆ- + p m r must correspond to ( ) ∫ ′ - Ω ′ - x x x d r r 1 3 with ( ) ( ) ( ) ∫ ′ - ⋅ - - - = ′ - Ω x x p i p e p d x x r r r r r 1 3 3 1 2 ω π , (16) because
( ) ( ) ( ) ( ) ( ) x x x x x x x d x x x x x d ′ - = ′ - ′ ′ Ω ′ ′ - Ω ′ ′ = ′ - ′ ′ Ω ′ ′ - Ω ′ ′ - - ∫ ∫ r r r r r r r r r r 3 1 3 1 3 δ (17) with the Dirac distribution ( ) ( ) ( ) ∫ ′ - ⋅ - = ′ - x x p i e p d x x r r r r r 3 3 3 2π δ . ( 18
)
(17) should be an integral representation of the "symbolic equation"
( ) ( ) ( ) ( ) 1 ˆ2 1 2 1 2 1 2 1 2 2 2 2 2 2 2 2 = + + = + + - - p m p m p m p m r r r r .
Accordingly, terms with the nth power of ( )
2 1 2 2 p m r + , ( ) 2 2 2 ˆn p m r + , (19)
correspond to integrals over "the nth power of Ω ":
( ) ( ) ( ) ∫ ′ - ⋅ - = ′ - Ω x x p i n p n e p d x x r r r r r ω π 3 3 2 . ( 20
)
By replacing the operators of type (19) by integrals over "powers of Ω " as given in (20), 1 Ĥ ′ and 2 Ĥ ′ (see (11) and (12) , respectively) can now be given a configuration space representation, too.
With these preparations, we can now address to the quantisation of the scalar field with the aim to be able to calculate scattering matrix elements.
Starting with Hamiltonian (10) , it is now possible to describe scattering processes within the framework of quantum field theory. For free scalar particles, a quantum field theoretic ansatz is described e.g. in [2] and [4] , using (2) and ( 13 ), respectively, as equations for a field operator ( )
x φ .
The latter one can be formulated with the help of creation and annihilation operators + must be a commutator (for fermions we would use here an anti-commutator instead, cf. e.g. [4] ). Equations ( 21 ) to (24) are identical to those that one would postulate within a non-relativistic quantum field theory for bosons. For the density of a Hamiltonian, we make the usual ansatz (see e.g. [7] ):
( ) ( ) x H x H φ φ ′ = + ˆ ( 25
)
which one can retrieve from a density of a Lagrangian (see [2] ):
( ) ( ) ( ) φ φ φ φ φ φ φ φ + + + + Ω - Ω - ∂ - ∂ = 2 1 2 1 2 t t i L . ( 25 a)
Substituting (10) into (25), we get
2 1 0 Ĥ H H H + + ≈ (26) with ( ) ( ) x H x H φ φ 0 0 ˆ′ = + , ( 27
) ( ) ( ) x H x H φ φ 1 1 ˆ′ = + , ( 28
) ( ) ( ) x H x H φ φ 2 2 ˆ′ = + . ( 29
) (25) is (among other things) motivated by the fact that p p p a a p d H x d r r 3 0 3 + ∫ ∫ = ω (30) reproduces the relativistic analogue of the free non-relativistic Hamiltonian: p p a a m p p d r r r 2 2 3 + ∫ . (31) (28) together with (29) are the densities of the Hamiltonian to (and including the) quadratic order in e. (21) to (24) are valid for free particles, but can also be used for interacting ones, if Dirac's representation is used instead of the so far applied Heisenberg representation. Then, with (28) and (29) combined to a Hamiltonian density 2 1 Ĥ H H I + =
(32) for the interaction of scalar bosons with photons, we can now start to calculate scattering matrix elements. To this purpose, we need the serial expansion of the S-operator (see e.g. [7] ) to the order of 2 e : ( ) ( ) ...
1 ˆ2 1 + + + = S S S (33) with ( ) ( ) ( )
∫ - = x H T x d i S I 4 1 , ( 34
) ( ) ( ) ( ) ( ) ( ) ∫ ∫ - = 2 1 2 4 1 4 2 2 1 2 ˆx H x H T x d x d i S I I , ( 35
)
where we have introduced a time ordering operator
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 0 1 0 2 2 1 0 2 0 1 2 1 ˆx H x H T x H x H x x x H x H x x x H x H T I I I I I I I I = - + - = θ θ (36) with ( ) < ≥ = 0 , 0 0 , 1 t t t θ . (37) ( ) 1
Ŝ does not only contribute to the expansion (33) with terms of order e, but also to order 2 e .
Therefore, we can split off ( )
1 Ŝ into a term 1 Ŝ containing only terms in e, ( )
( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ∫ + - + + Ω ⋅ + ⋅ - - = x x A x x p x A x A p x T x d ie S φ φ φ φ 0 1 2 1 4 1 r r r r (38)
with
( )( ) ( ) ( ) ∫ - Ω = Ω - - t x x x x d x , 1 1 1 1 3 1 r r r φ φ (39)
and a part containing only terms in 2 e :
( )
( ) ( )( )( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ) ∫ ∫ - - + - + Ω ⋅ + ⋅ - Ω ⋅ + ⋅ - Ω - = 1 2 1 2 1 2 1 2 2 2 1 2 2 3 1 1 1 1 1 8 1 1 1 1 2 1 2 1 1 4 2 12 , , , ˆt x p t x A t x A p x x x d p x A x A p x x x A x T x d ie S r r r r r r r r r r r r r r φ φ φ φ (40)
where the momentum operator 1 p r contains a gradient acting on 1 x r and 2 p r acting on 2 x r . In (38) and
(40), we have already substituted (32) into (34) and replaced powers of ( )
2 1 2 2 p m r +
by integrals over "powers of Ω " (see (20)) in (11) . Thus we can rewrite (34) as
( ) 12 1 1 Ŝ S S + = . ( 41
)
The time ordering operator appearing in ( )
1
Ŝ can be left out, because it contains only one time. In ( )
2 Ŝ
we only want to retain terms of order 2 e , therefore I H ˆ can be approximated by 1 Ĥ :
( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) 2 2 0 2 1 1 0 1 2 1 2 2 2 2 2 1 1 1 1 1 1 1 4 1 2 2 0 2 1 1 1 1 1 1 1 2 1 2 4 1 4 2 1 2 2 x x A x x x A x x p x A x A p x x p x A x A p x x x A x x p x A x A p x T x d x d ie S φ φ φ φ φ φ φ φ φ φ φ φ + + - + - + + - + + Ω ⋅ + ⋅ Ω ⋅ + ⋅ + Ω ⋅ + ⋅ - - = ∫ ∫ r r r r r r r r r r r r . (42)
Here, the time ordering operator must be taken into account, because there are two times: 1 t and 2 t . We have already quantised the field of scalar bosons, but must do now the corresponding with the electromagnetic field. Since the scalar field in (38), ( 40 ) and ( 42 ) couple in a different way to the vector potential A r than to the scalar potential 0 A , the choice of a Coulomb gauge seems to be appropriate:
0 = ⋅ ∇ A r r . ( 43
)
Then the field equations take on the form ( )
j A A t t r r r r = ∇ ∂ + ∇ - ∂ 0 2 2 , ( 44 a) ρ -
= ∇ 0 2 A r ( 44 b)
with charge and current densities ρ and j r , respectively. From (44 b) we can see that in this gauge the scalar potential is just a c-number:
( ) ( ) ∫ ′ - ′ ′ = x x t x x d t x A r r r r , 4 1 , 3 0 ρ π , ( 45
)
whereas the vector potential A r becomes an operator when being quantised. For a free field, A r can be chosen like (see e.g. [4, 7, 10] )
( ) ( ) ( ) ( ) λ ε ω π λ λ λ , 2 2 ˆ21 3 3 k e c e c k d x A x ik k x ik k k r r r r r ∫ ∑ = ⋅ + ⋅ - + = (46)
with the usual photon frequency
2 ~k k r = ω and with creation and annihilation operators λ k c r ˆ and + λ k c r ˆ, respectively, for photons:
] ( ) k k c c k k r r r r - ′ = ′ + ′ ′ 3 , ˆδ δ λ λ λ λ , [ ] [ ] 0 , , ˆ= = + + ′ ′ ′ ′ λ λ λ λ k k k k c c c c r r r r , ( 47
)
and 0
A would even vanish. The polarisation vectors ( )
λ ε , k r r
fulfil the relation (see e.g. [4, 7] ):
( ) ( ) 2 2 1 , , k k k k k j i ij j i r r r - = ∑ = δ λ ε λ ε λ . ( 48
)
Due to the Coulomb gauge condition (43),
( ) 0 , = ⋅ λ ε k k r r r (49)
is valid, too.
In the following sections, we are going to calculate ( )
1 Ŝ and ( ) 2 Ŝ by substituting the field operators ( ) x φ and ( ) x A r from ( 21 ) and ( 46 ) as well as the distributions (20). Since we are considering electromagnetic interactions between (charged) spin-0 bosons, we have to take 0 A in (45) into account, too. Therefore, we first have to find out what the density of charge in (45) will be in this case. This can be done by coupling the Lagrangian density for free spin-0 bosons (25 a) to an electromagnetic field with the aid of the minimal coupling scheme With these results, one can then continue to calculate scattering matrix elements.
The first term we calculate is
( ) ( ) ( ) ( )( )( ) ∫ - + Ω ⋅ + ⋅ = x p x A x A p x x d I φ φ 1 4 1 : ˆr r r r (50)
appearing in ( ) 1 Ŝ (see ( 38 )). To this end, it is useful to recognise that by means of (16) we get
( )( ) ( ) ∫ ⋅ - - - = Ω p x ip p a e p d x r 2 1 3 1 2 3 ω π φ . ( 51
)
With (51) and integration by parts, (50) yields:
( ) ( ) ( ) ( ) ( ) + + - - + + - + ⋅ = ∫ ∑ λ λ λ δ δ ω λ ε ω π k k p p p k c k p p c k p p a a p p k p d p d k d I r r r r r r r r , 2 2 ˆ2 1 4 2 1 4 2 1 2 3 1 3 3 1 1 2 1 , (52) where ( ) (
) ( )
k p p k p p k p p r r r ± - ± - = ± - 2 1 3 2 2 1 4 1 δ ω ω ω δ δ (53)
and, of course, the a operators commute with the c operators. In 12 Ŝ (40), the first integral can be expressed in a similar way:
( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ) 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 , , 1 2 2 2 : ˆ2 1 2 1 4 2 1 2 1 4 2 1 2 1 4 2 1 2 1 4 , 2 2 1 1 1 2 3 1 3 2 3 1 3 1 2 4 2 λ λ λ λ λ λ λ λ λ λ δ δ δ δ λ ε λ ε ω ω ω π φ φ k k k k k k k k p p p k k c c p p k k c c p p k k c c p p k k c c p p k k k k a a p d p d k d k d x x A x x d I r r r r r r r r r r r r r r r - + - - + - + + - + - + - + - + + ⋅ ⋅ = Ω = + + + + + - + ∫ ∑ ∫ (54)
After a quite lengthy but straightforward calculation, the second integral in (40) yields (see (42)) which is not just like the product of two operators 1 Î due to the time ordering operator as defined in (36). Unfortunately, we cannot use the famous Wick theorem, because the scalar field operator (21) contains only contributions to positive energy solutions: it does not consist of a sum of both positive and negative energy solutions as it would be the case for the field operator of the Klein-Gordon equation. Due to the symmetry of the time ordering operator (36) in its arguments, we may conclude
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) = Ω ⋅ + ⋅ ⋅ - Ω ⋅ + ⋅ = - - + ∫ ∫ 1 2 1 2 1 2 1 2 2 2 1 2 1 1 1 1 1 2 3 1 4 3 , , , ˆ: ˆt x p t x A t x A p x x p x A x A p x x d x d I r r r r r r r r r r r r r φ φ (55) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) - + - - + - + + - - ⋅ + - + - + - + + - + ⋅ ⋅ + ⋅ ∑ ∑ ∑ ∫ + + + - + + + 1 1 1 2 2 1 1 2 2 1 1 1 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 1 2 , 2 , , 1 2 2 2 2 1 2 1 4 2 1 2 1 4 2 1 1 1 1 2 1 2 1 4 2 1 2 1 4 2 1 1 1 1 2 1 2 2 2 2 3 1 3 2 3 1 3 λ λ λ λ λ λ λ λ λ λ λ δ δ ω λ ε δ δ ω λ ε λ ε ω ω ω π k k k k k p k k k k k p p p p k k c c p p k k c c p p k k k p k c c p p k k c c p p k k k p k p p k a a p d p d k d k d r r r r
( ) ( ) ( ) ( ) ( ) ( ) 2 1 0 2 0 1 2 4 1 4 2 1 2 4 1 4 2 ˆx H x H x x x d x d x H x H T x d x d I I I I ∫ ∫ ∫ ∫ - = θ . ( 56
)
With this property, the calculation of the second term in (42) can be simplified a bit:
( ) ( ) ( ) ( )( )( ) ( ( ) ( ) ( ) ( )( ) ( ) = Ω ⋅ + ⋅ ⋅ Ω ⋅ + ⋅ = - + - + ∫ ∫ 2 1 2 2 2 2 2 1 1 1 1 1 1 1 2 4 1 4 4 : ˆx p x A x A p x x p x A x A p x T x d x d I φ φ φ φ r r r r r r r r (57 a) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) - - - - - - - ⋅ - - ′ - ′ - ′ + + - - - - - - ⋅ + - ′ - ′ - ′ + - - - + - - - ⋅ - - ′ + ′ - ′ + + - - + - - - ⋅ + - ′ + ′ - ′ ⋅ + ⋅ ′ + ′ ⋅ ′ ′ ′ ′ ′ ∫ ∫ ∫ ∫ ∑ ∑ ∫ ′ ′ ′ + + ′ ′ ′ ′ ′ + ′ ′ ′ ′ ′ + ′ ′ ′ ′ ′ ′ ′ ′ ′ + ′ + ′ ′ 2 1 2 1 2 2 1 2 1 3 2 1 3 2 1 2 1 2 2 1 2 1 3 2 1 3 2 1 2 1 2 2 1 2 1 3 2 1 3 2 1 2 1 2 2 1 2 1 3 2 1 3 2 1 2 1 2 3 1 3 2 3 1 3 3 3 ẽxp ẽxp 2 ˆẽxp ẽxp 2 ˆẽxp ẽxp 2 ˆẽxp ẽxp 2 ˆ, , 2 2 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 2 1 2 t i t i t t
dt dt k p p k p p c c t i t i t t dt dt k p p k p p c c t i t i t t dt dt k p p k p p c c t i t i t t dt dt k p p k p p c c p p k p p k a a a a p d p d p d p d k d k d k p p k p p k k k p p k p p k k k p p k p p k k k p p k p p k k p p p p p p k k
ω ω ω ω ω ω θ π δ δ ω ω ω ω ω ω θ π δ δ ω ω ω ω ω ω θ π δ δ ω ω ω ω ω ω θ π δ δ ω λ ε ω λ ε ω ω π λ λ λ λ λ λ λ λ λ λ
r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r (57) contains four terms of the subsequent type that can be simplified with the help of ( 58 ) and (59): (
τ ω ω τ θ π τ ω ω δ ω ω π τ ω ω τ θ π τ ω ω θ π 2 1 2 2 1 2 1 2 2 1 2 2 1 2 2 1 1 2 1 2 2 1 exp 2 exp 2 exp 2 exp 2 - - + = + - - - = - - - ∫ ∫ ∫ ∫ i i i d T dT d t i t i xp e t t dt dt (60 a) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
The function θ can be expressed by an integral in the complex plane (see e.g. [6, 8] ), ( )
∫ ∞ ∞ - - ± ± = ± ε π τ θ τ i p e dp i ip 0 0 1 2 0 (61)
with an ε approaching zero. Substituting this into (60 a), we get:
( ) ( ) ( ) ( ) ( ) 2 1 2 2 1 2 2 1 1 2 exp 2 ω ω δ ε ω π τ ω ω τ θ π τ ω ω δ + + = - - + ∫ i i d i . (60 b)
With this result, (57) becomes:
( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
+ - - ⋅ - - ′ - ′ - ′ - - + - - + + + - ⋅ + - ′ - ′ - ′ + - + - - + + - - ⋅ - - ′ + ′ - ′ - - + + - + + + - ⋅ + - ′ + ′ - ′ + - + + - ⋅ + ⋅ ′ + ′ ⋅ ′ ′ ′ ′ ′ = ′ ′ ′ + + ′ ′ ′ ′ ′ + ′ ′ ′ ′ ′ + ′ ′ ′ ′ ′ ′ ′ ′ ′ + ′ + ′ ′ ∑ ∑ ∫ ε ω ω ω δ δ ω ω ω ω ω ω δ ε ω ω ω δ δ ω ω ω ω ω ω δ ε ω ω ω δ δ ω ω ω ω ω ω δ ε ω ω ω δ δ ω ω ω ω ω ω δ ω λ ε ω λ ε ω ω π λ λ λ λ λ λ λ λ λ λ i k p p k p p c c i k p p k p p c c i k p p k p p c c i k p p k p p c c p p k p p k a a a a p d p d p d p d k d k d i I k p p k p p k p p k k k p p k p p k p p k k k p p k p p k p p k k k p p k p p k p p k k p p p p p p k k 1 ~1 ~1 ~1 ~, , 2 2 2 2 ˆ2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 2 1 2 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 2 1 2
- ′ - - ′ + ′ ′ ′ - = + ′ + ′ ∫ δ π . ( 62 b)
(42) contains two terms in 0 A . The first term consists of a combination of (50) and (62 a), but taken at different times and therefore joined via the time ordering operator. That is why we have to use (60) as well as (45) and (63) again:
( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( ) ( ) (
)
+ - - - - + - - + - - + + - - + - - + - - + - - + + + - + - + - - + + - + + - - + + - + - - + + - ⋅ - ′ - - ′ + ′ + ⋅ ′ ′ = Ω ⋅ + ⋅ = + + ′ + ′ ′ ′ + + ′ + ′ ′ ′ ′ ′ + ′ + ′ ′ ′ + ′ + ′ ′ ′ ′ ′ = + - + ∑ ∫ ∫ λ λ λ λ λ ε ω ω ω ω ω ω ω ω ω ω δ δ ε ω ω ω ω ω ω ω ω ω ω ω δ δ ε ω ω ω ω ω ω ω ω ω ω δ δ ε ω ω ω ω ω ω ω ω ω ω ω δ δ δ λ ε ω ω π φ φ φ φ 1 1 1 2 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 2 1 1 1 1 2 1 1 1 2 3 1 ~1 ~1 ~1 ~, ~1 2 2 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 2 1 2 3 2 1 1 3 3 3 3 1 3 2 3 1 3 7 2 2 0 2 1 1 1 1 1 1 1
2 4 1 4 6 k p p k k p k p p k p p p k p k k p k k p p p k p k k p p p k p k k p p k k p k p p k p p p k p k k p k k p p p k p k k p p p k p k p k c a a a a a i k p p c a a a a a i k p p c a a a a a i k p p c a a a a a i k p p k k k p k p p p k k d k d p d p d p d p d k d i x x A x x p x A x A p x T x d x d I
r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r (64)
The second term of (42) containing a scalar potential is even quadratic in 0 A :
( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ + + = 2 2 0 2 1 1 0 1 2 4 1 4 7 ˆx x A x x x A x T x d x d I φ φ φ φ . ( 65 a)
(65 a) contains a factor of two integrands of the kind of (62 a), but taken at two different times. Thus the time ordering operator must be taken into account. With the same substitutions as in ( 62 ) and (64), we obtain the following result:
( ) ( ) ( ) ( ) ( ) ( ) ε ω ω ω ω δ δ ω ω ω ω ω ω ω ω δ π i k k k k k k p p k k p p a a a a a a a a p d k d k d p d p d k d k d p d i I k p k p k p k p k p k p p k k p p k k p + - - + - ′ - ′ ⋅ - ′ + - ′ - ′ + - ′ - - + + - - + ⋅ ′ ′ ′ ′ = ′ ′ ′ ′ ′ ′ + ′ + ′ + ′ + ′ ∫ 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 2 2 1 ˆ2 2 2 2 1 1 2 2 2 2 3 1 1 1 1 3 5 3 2
3 2 3 2 3 2 3 1 3 1 3 1 3 1 3 7 r r r r r r r r r r r r r r r r r r r r (65 b)
The results ( 52 ), ( 54 ), ( 55 ), ( 57 ), ( 62 ), ( 64 ) and (65) substituted into (38) to (42) now enable us to evaluate scattering matrix elements for scattering processes to (and including) the order 2 e . As two examples, we turn first to the scalar analogue of Compton scattering in order to address then to the scattering of two identical scalar bosons. These two scattering processes can be compared easily with the corresponding results of the well-known scalar QED dealing with the Klein-Gordon equation (3) .
For Compton scattering, we need one scalar boson and one photon each in the input and output channel. This means, we have to evaluate the element
0 0 : ˆ+ + ′ ′ ′ = µ µ h q q h c a S a c S r r r r , ( 66
)
with the S ˆ-operator (33). Firstly, we realise that the terms based on 1 Î (see (52)) as well as 6 Î (see (64)) must vanish, because the subsequent two elements in the photon operators become zero:
0 0 0 0 0 = = ′ ′ + ′ ′ µ λµ µ λ µ δ δ h h k h k h c c c c r r r r r r , ( 67
) 0 0 0 = + + ′ ′ µ λ µ h k h c c c r r r
. There, we have used the commutation relations (47), the properties of creation and annihilation operators corresponding to those of ( 22 ) and ( 23 ) as well as abbreviated the delta functional by ( )
h k h k r r r r - = 3 δ δ . ( 68
)
The terms with 2 Î and 3 Î need
q p p q q p p q a a a a r r r r r r r r 2 1 2 1 0 0 δ δ ′ + + ′ = , ( 69
)
whereas a term q p p p p q q p p p p q a a a a a a r r r r r r r r r r r r
1 2 1 2 1 2 1 2 0 0 δ δ δ ′ ′ ′ + + ′ + ′ ′ = belongs to 4
Î . Here we have used again ( 22 ) to (24).
For 2 Î , 3 Î and 4 Î the following equations are necessary, too:
0 0 0 = + ′ ′ ′ ′ µ λ λ µ h k k h c c c c r r r r , 0 0 0 = + + + ′ ′ ′ ′ µ λ λ µ h k k h c c c c r r r r , ( 70
) µ λ λ µ µ λ λ µ δ δ δ δ ′ ′ ′ ′ + + ′ ′ ′ ′ = k h k h h k k h c c c c r r r r r r r r 0 0 , λµ λ µ µ λ λ µ δ δ δ δ ′ ′ ′ ′ + + ′ ′ ′ ′ = k h k h h k k h c c c c r r r r r r r r 0 0
, where we have commutated the creation operators successively to the left and the annihilation operators to the right. Furthermore, we see that the term based on 5 Î from (62) becomes zero, due to
0 0 0 = + + ′ + ′ ′ q p k k p q a a a a a a r r r r r r . ( 71
)
A similar result holds true for the term based on 7 Î from (65).
0 0 0 2 2 2 2 1 1 1 1 = + + ′ + ′ + ′ + ′ ′ q p k k p p k k p q a a a a a a a a a a
I I i I i ie I I ie S + - + - - + + - - ≈ - . ( 73
)
For Compton scattering, 1 Î , 2 Î , 3 Î , 4 Î , 5 Î , 6 Î and 7 Î can be evaluated explicitly:
0 ˆ7 6 5 1 = = = = I I I I , ( 74
) ( ) ( ) ( ) µ ε µ ε ω ω ω π δ ′ ′ ⋅ ′ - ′ - + = ′ , , 2 ˆ4 2 h h h q h q I q h h r r r r , ( 75
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ′ - ⋅ ′ ′ + ′ - ⋅ + + ⋅ ′ - + ⋅ ′ ′ ′ - ′ - + = ′ - + ′ h q h h h q h h q h h h q h h q h q I h q h q q h h r r r r r r r r r r r r r r r r r r r r r r 2 , 2 , 1 2 , 2 , 1 2 2 ˆ2 2 2 4 3 µ ε µ ε ω µ ε µ ε ω ω ω ω π δ (76) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + - + + ⋅ ′ + ′ ⋅ ′ ′ + + - - ′ - ⋅ ′ ′ - ′ ⋅ ′ - ′ - + = + + ′ - ′ ′ - ′ ε ω ω ω µ ε µ ε ω ε ω ω ω µ ε µ ε ω ω ω ω π δ i h q h h q h i h q h h q h h q h q I h q h q h q h q h q h q q h h r r r r r r r r r r r r r r r r r r r r r r r r r r r r ~2 , 2 , 1 ~2 , 2 , 1 2 ˆ2 4 4 (77)
The two terms 2 Î and 4 Î in (73) resemble the three terms in the corresponding formula of Compton scattering for scalar bosons, but this time being based on the Klein-Gordon equation (3) (see e.g. [6, 8] ):
(
ε µ ε µ ε µ ε µ ε µ ε δ ω ω ω ω π ′ ′ ⋅ - ′ - ⋅ ′ ′ - ′ - - ′ ⋅ + ′ + ′ ⋅ ′ ′ - + + ⋅ ⋅ ′ - ′ - + - = - ′ ′ , , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )] µ
2 2 2 2 2 1 ˆ2 2 2 2 4 2 2 h h i h q h m h q i h q h h q h m h q i h q h h q h q ie S q q h h (78) 2 2 , 2 , 2 , 2 ,
where ε is the 4-dimensional generalisation of the three dimensional polarisation vector ε r used so far. In (78), the first two terms correspond to 4 Î . We realise that these terms in (78) look very much like those in (77), but also that especially the propagators of both theories are a bit different. (
resembles the third term in (78). On the other hand, 3 Î in (76) could also be regarded as the analogue of the first two terms in (78) -at least after having used the Coulomb gauge condition (49) in (76).
The first two terms in (78) vanish, if we choose the incoming scalar boson to be at rest, ( ) ( )
0 , : , 0 r r m q q q = = , (79)
and want to have transversally polarised photons in this laboratory system, ( ) ( )
0 , , = ⋅ ′ ′ = ⋅ q h q h µ ε µ ε , (80)
and use the Lorentz gauge condition
( ) ( ) 0 , , = ′ ⋅ ′ ′ = ⋅ h h h h µ ε µ ε . ( 81
)
Accordingly, 3 Î and 4 Î in (73) vanish too, if we adopt (79) again and use the analogue of (80),
( ) ( ) 0 , , = ⋅ ′ ′ = ⋅ q h q h r r r r r r µ ε µ ε , ( 82
)
as well as apply the Coulomb gauge condition (49). That is, under these conditions in both versions of scalar QED, only one term remains (i.e. (75) in (73) and, accordingly, the third term in (78)) which is a relativistically generalised version of the matrix element from which the well-known Thomson scattering cross section can be evaluated. Even though the propagators in both theories are rather different, for this example of scattering process, the results do not seem to differ very much from each other in the laboratory system chosen above. Therefore the question arises, whether this is also the case for further scattering processes. To this end, we are going to investigate what happens, if two identical scalar bosons interact with each other.
If we want to calculate matrix elements of the S-operator (33) for the scattering of two identical scalar bosons, we need two scalar bosons in the input channel and two in the output channel:
0 0 : ˆ1 2 1 2 + + ′ ′ = q q q q a a S a a S r r r r . ( 83
)
We can reuse (73), but have to evaluate 1 Î , 2 Î , 3 Î , 4 Î , 5 Î , 6 Î and 7 Î again. Firstly, we recognise that due to (52) and the analogue of ( 22 ) and ( 23 ) for the photon operators
0 ˆ1 = I , (84)
is valid. Furthermore, we need
2 1 2 1 2 2 1 1 0 0 λ λ λ λ δ δ k k k k c c r r r r = + (85)
for calculating 2 Î with the help of ( 54 ), whereas all the other photon operator terms therein vanish.
The same holds true for 3 Î with (55) and 4 Î with (57).
As far as the scalar boson operators are concerned, for the evaluation of 2 Î and 3 Î 0 0 0
1 2 1 2 1 2 = + + + ′ ′ q q p p q q a a a a a a r r r r r r .
Hence, these two terms,
0 ˆ2 = I , (86) 0
ˆ3 = I ,
are zero, too. For the evaluation of the non-vanishing term 4 Î , the following result is useful:
2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 1 2 1 2 1 2 1 2
0 0 p q p q p q p q p q p q p q p q p q p q p q p q p q p q p q p q q q p p p p q q a a a a a a a a
′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ + + + ′ + ′ ′ ′ + + + =
r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ . (87) With (85), (87) and the invariance of (48) under the transformation k k r r -→ , we get
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 2 1 2 1 2 1 1 2 1 1 1 1 1 1 2 2 2 2 1 2 1 4 2 2 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 ~, , 1 1 ~, , 2 8 1
ˆA in terms i i q q q q q q q q i i q q q q q q q q q q q q ie S q q q q q q q q q q q q q q q q q q q q q q
+ - + - - + - - ⋅ ⋅ ′ + ⋅ ′ - ′ - ⋅ ′ + + - + - - + - - ⋅ ⋅ ′ + ⋅ ′ - ′ - ⋅ ′ + ′ - ′ - + - = - ′ - ′ ′ - ′ ′ - ′ - ′ ′ - ′ ′ - ∑ ∑ ε ω ω ω ε ω ω ω ω λ ε λ ε ε ω ω ω ε ω ω ω ω λ ε λ ε ω ω π δ λ λ r r
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ′ - ′ + ⋅ ′ + + ′ - ′ + ⋅ ′ + ′ - ′ - + = - ′ ′ 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 2 1 2 1 4 2 2 1 2 1 2 4
1 ˆq q q q q q q q q q q q q q q q ie S q q q q ω ω ω ω π 1 ˆA in terms q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ie S q q
+ ′ - ′ - ′ + ⋅ ′ - ′ - ⋅ ′ + - ′ - ′ - ′ + ⋅ ′ - ′ - ⋅ ′ + - ′ - ′ + ⋅ ′ + + ′ - ′ + ⋅ ′ + ′ - ′ - + - = - v r r r v r v r r r v r r r v r v r r r r r r r r r v r ω ω π δ (88 b)
The first two terms in (88 b) correspond to (89) which contains the photon propagator in the Feynman gauge. But since we use a Coulomb gauge, only the space-like components of the linear 4-momenta appear in the numerators of the first two terms in (88 b). Moreover, in the second line of (88 b) two additional terms are present which can be reformulated by means of the delta distribution in (88 b):
( )( ) ( ) ( ) ( )( ) ( ) ( ) 2 2 1 2 2 1 2 2 2 1 2 2 2 1 2 1 1 2 1 1 2 1 2 1 2 2 2 2
q q q q q q q q q q q q q q q q v r r r r r v r r r r r
′ - ′ - ′ - - ′ - ′ - ′ - ′ - ′ - ′ - . ( 90
)
The structure of (88 b) is the same as that of (89). We have two terms: in the second term, the momenta of the two scalar bosons in the output channel have been exchanged compared to the first term.
Now we address to the terms in the scalar potential 0 A in (88). The Coulomb term 5 Î contains a term p q k q q p q k k q p q q p q k p q k q q k q p k q p q q k q p q q p k k p q q a a a a a a a a
′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ + + + ′ + ′ ′ ′ - - - =
2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 0 0 δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ . Therefore, 5 Î yields ( ) ( ) ( ) ( ) - ′ - - ′ - - ′ + ′ = 2 1 2 2 1 1 2 1 2 1 4 2 5 1 1 2 1 ˆq q q q q q q q I r r r r δ π . ( 91
)
The term 6 Î vanishes, since the annihilation operators for photons in (64) act directly on the vacuum states.
On the other hand, 7 Î becomes a rather lengthy because of (
( )
1 1 1 1 1 1 1 1 1 2 1 1 2 1 2 1 1 2 2 2 2 2 2 1 1 2 1 1 2 1 1 2 2 2 2 2 1 2 1 1 2 1 2 1 1 2 2 2 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 2 2 1 1 1 1 1 2 0 0 k p k p k p k p a a a a a a a a a a a a k q p q k p p k q p q k q p k k k q p q p p q p q k q p k q p q k p p k q k q p k q p q k k q k p p q p q q p k k p p k k p q q ′ ↔ ′ - + ′ ↔ ′ - - ′ ↔ ′ - - ′ ↔ ′ - = ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ + + + ′ + ′ + ′ + ′ ′ ′
r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ where ( ) 1 1 k p ′ ↔ ′ r r denotes the same term as the immediately preceding one, but with exchanged momenta p′ r and 1 k ′ r , respectively. Thus, 7 Î gives
( ) ( ) ( ) + - - ⋅ - - + - - ′ + ′ = - ′ - - ′ - + ′ - + ′ - - + ∫ 2 2 2 2 2 2 2 2 3 3 2 1 2 1 4 2 7 1 2 1 2 2 2 1 2 1 2 1 1 1 1 1 1 2 2 8
ˆq p q p q p q p q p q p q p q p p q q p q q p d q q q q i I
r r r r r r r r r r r r r r r r r r r r r r ω ω ω ω ω ω ω ω ω ω ω ω π δ π .
Hence in place of the time-like components (i.e. the energy terms) of the linear 4-momenta in the numerators of (89) derived from the Klein-Gordon equation, several terms arise: (90), ( 91 ) and (92). But (91) and (the non relativistic limit of) (92) would also have appeared, if we had started from the non-relativistic Schrödinger equation. Thus, these terms state the fact that the equation used for obtaining (88) (together with (90), ( 91 ) and ( 92 )) is Schrödinger equation like.
For scalar bosons, we could see that it is possible to describe scattering processes by means of a square root operator equation being coupled to an electromagnetic field. We achieved this by splitting off a factor in the shape of the free square root operator from the equation and by a series expansion of a remaining square root factor containing terms in the electromagnetic vector potential and powers of the (inverse) free square root operator. The latter ones could be given an integral representation.
Having quantised the fields involved, we could evaluate the scattering matrix elements for Compton scattering and for the scattering of two identical bosons (to -and including -the quadratic order of e) which, on the one hand, resembled the results derived with the help of the corresponding Klein-Gordon equation and, on the other hand, the results one would have obtained with a non-relativistic Schrödinger equation. Of course, now several questions arise, e.g.:
• Can we formulate Feynman rules at all for our non-local scalar QED?
• Do divergent terms appear and, if yes, can a renormalisation procedure be found?
• Can the results of this non-local QED be confirmed (or refuted) by experiments?
To the first question: if one tries to formulate Feynman rules, one must be aware that in each step of the approximation procedure, we have to expand the first square root factor (in the second term of) (7) to the desired order in e. Thus, the Hamiltonian we are using within that procedure must be adapted in each order of e of the approximation. Therefore we conclude that even if it were possible to formulate Feynman rules, they would be much more complicated than those for the Klein-Gordon theory. This is the price we have to pay for non-locality. But without Feynman rules, it is not very easy to analyse the renormalisability of that non-local scalar QED either.
The answer to the last question listed above is negative due to a lack of elementary spinless bosons in nature. But that question would be sensible, if we had a corresponding non-local theory for spin-1/2 particles. We could even make a guess, how this theory would look like: for free spin-1/2 particles, the wave functions in (2) should be 2-spinors. It would also be possible to couple this equation to an electromagnetic field, but the usual minimal coupling scheme does not work. Instead we would have to postulate an equation:
φ φ m H i t ˆ′ = ∂ , (93)
with the Hamilton operator ( ) ( ) 0 2 2 ˆeA E i B e A e p m H + ⋅ --+ = ′ r m r r r r m σ ,
which contains an additional term with the Pauli matrices σ r and the magnetic field B r as well as the electric field E r under the square root. For this equation, it has already been shown that it can reproduce the gyromagnetic factor of 2 for the electron as well as, when being applied to a hydrogen atom, that it can reproduce correct binding energies of the electron at least to (and including) the quadratic order in 2 e (see [12] ). We can apply the same approximation procedure to this equation as we have presented here for the scalar case. But of course, additional difficulties emerge: the Hamiltonian corresponding to the one shown in (25) should be Hermitian. And at least for the free spin-1/2 case, that Hamiltonian should be relativistically invariant. This means, that for the free case,
+ ± ′ = ′ H H m . ( 97
)
The results of a QED based on (95) or (96) could then be compared with the ones based on a corresponding Dirac equation. The author does not know, whether such an approach has already been performed for the spin-1/2 case or for the here presented spinless case. He does not know either, if an application for it can be found, where e.g. non-local properties are indispensable. But it seems to him that from the technical point of view, square root operator equations coupled to an electromagnetic field like (4) (or maybe even like (93) together with (94)) can be handled within the framework of a quantum field theory. Such equations were given up quite early in the history of quantum mechanics for several good reasons, e.g. due to their lack of relativistic invariance (see [13] ), their non-local character accompanied by the difficulty of finding an appropriate mathematical interpretation and description, respectively (see e.g. [4, 5, 6] ). While the first reason mentioned remains still valid, from today's perspective, those non-local properties might not be refused as vehemently as in the past (e.g. when one looks out for approximations of the so called Bethe-Salpeter equation [11] ). The author hopes to have shown, that at least answers to the question of possible descriptions of such non-local square root operator equations can be found. | [
{
"section_type": "OTHER",
"section_title": "Tobias Gleim",
"text": "Instead of using local field equations -like the Dirac equation for spin-1/2 and the Klein-Gordon equation for spin-0 particles -one could try to use non-local field equations in order to describe scattering processes. The latter equations can be obtained by means of the relativistic energy together with the correspondence principle, resulting in equations with a square root operator. By coupling them to an electromagnetic field and expanding the square root (and taking into account terms of quadratic order in the electromagnetic coupling constant e), it is possible to calculate scattering matrix elements within the framework of quantum electrodynamics, e.g. like those for Compton scattering or for the scattering of two identical particles. This will be done here for the scalar case. These results are then compared with the corresponding ones based on the Klein-Gordon equation. A proposal of how to transfer these reflections to the spin-1/2 case is also presented.\n\nFree scalar particles are usually described by means of the well-known Klein-Gordon equation (see e.g. [4, 5, 6, 10] ):\n\n( ) ( ) 0 , ˆ2 2 2 = + + ∂ t x m p t r r φ , (1)\n\nwhere we have used the momentum operator in configuration space ∇ -= r r i p ˆ (and set the velocity of light as well as Planck's constant h to one). ( 1 ) can be regarded as an iteration of the following square root operator equation (see e.g. [1, 2, 4, 5] ):\n\n( ) ( )\n\nt x p m t x i t , , 2 2 r r r + = ∂ φ φ . ( 2\n\n)\n\nIntroducing an electromagnetic field with a 4-vector potential ( ) ( )\n\n[ ] ( )\n\nx x A x A e x A x A ie x m p t φ φ µ µ µ µ µ µ 2 2 2 2 ˆ+ ∂ + ∂ - = + + ∂ r , (3)\n\nwhere we have used the 4-vector notation ( ) ( ) ( )\n\nx t x x x r r\n\n, , 0 = = µ and Einstein's summation convention.\n\nHere, the coupling terms on the right hand side of (3) could easily be separated from the term with the free particle Hamiltonian on the left hand side of (3). This is unfortunately no longer possible, if one couples the non-local equation (2) to the electromagnetic field:\n\n( ) ( ) ( ) ( ) ( ) ( ) x x eA x x A e p m x i t φ φ φ 0 2 2 ˆ+ - + = ∂ r r , (4)\n\nbecause the vector potential A r appears under the square root. But in a perturbation analysis of scattering processes, this property is useful, since such an analysis is based on the assumption that the coupling terms make only small contributions to the free particle solution due to the small value of the coupling constant e. By rewriting the Hamiltonian in (4) , ( ) ( ) ( ) ( ) 2 2 2 2 0 2 2 0 ˆA e p A e A p e p m eA x A e p m eA H r r r r r r r r + ⋅ -⋅ -+ + + = -+ + = ′ , (5) one can split off a factor with the free Hamiltonian 2 2 0 p m H + = ′ , (6) which yields ( )( ) ( ) 2 1 2 2 1 2 2 2 2 0 1 ˆp m p m A e p A e A p e eA H r r r r r r r + + + ⋅ -⋅ -+ + = ′ -.\n\nWith the above-mentioned assumption, it is now very tempting to expand the first square root factor. A very similar approach has already been proposed by [3] . We would like to restrict ourselves to a series expansion of the kind ...\n\n1 1 2 8 1 2 1 + -+ ≈ + y y y (8) containing only constant, linear and quadratic terms, where y ˆ denotes ( )( ) 1 2 2 2 2 ˆ-+ + ⋅ -⋅ -= p m A e p A e A p e y r r r r r r .\n\nHamiltonian ( 7 ) is therefore approximated by\n\n2 1 0 Ĥ H H H ′ + ′ + ′ ≈ ′ (10) with ( )( )\n\n + + ⋅ + ⋅ - = ′ - 0 2 2 2 1 1 2 1 ˆA p m p A A p e H r r r r r , ( 11\n\n) ( ) ( )( ) ( )( ) + ⋅ + ⋅ + ⋅ + ⋅ - + = ′ - - - 2 1 2 1 2 2 1 2 2 8 1 2 2 2 2 1 2 2 p m p A A p p m p A A p p m A e H r r r r r r r r r r r r , (12)\n\nwhere we have reordered the terms of expansion (9) , retaining only terms to (and including the) quadratic order in e and recollected powers of ( )\n\n2 2 p m r +\n\n. What we have won by (10) is a separation of the free Hamiltonian (6) from the coupling terms in ( 5 ) that approximately result in the sum of 1 Ĥ ′ and 2 Ĥ ′ (i.e. ( 11 ) and ( 12 ) respectively). For we are only interested in corrections to the free Hamiltonian anyway, this approximation might not hurt very much. But however, this separation seems not to be a true one, because of the multiple factors of powers of ( )\n\n2 2 p m r +\n\nin (11) and (12) . That is, we need an interpretation of these operators. To this end, it is useful to know that for the free square root operator equation ( 2 ), an integral representation can be given (see [1, 2] ):\n\n( ) ( ) ( ) ( )( ) t x t x x x x d t x i t , : , , 3 r r r r r φ φ φ Ω = ′ ′ - Ω ′ = ∂ ∫ , ( 13\n\n)\n\nwhere Ω denotes an energy distribution\n\n( ) ( ) ( ) ∫ ′ - ⋅ - = ′ - Ω x x p i p e p d x x\n\nr r r r r ω π 3 3 2 (14) with 2 2 p m p p r r + = = ω ω .\n\n(15) (13) results from the fact, that one would expect to obtain the following momentum space representation of (2):\n\n( ) ( )\n\nt\n\np t p i p t , , ~r r φ ω φ = ∂ with ( ) t p, ~r φ denoting the Fourier-transformed ( ) t x, r φ . If the operator 2 2 p m r + corresponds to\n\n( ) ∫ ′ - Ω ′ x x x d r r 3 , the operator ( ) 2 1 2 2 ˆ- + p m r must correspond to ( ) ∫ ′ - Ω ′ - x x x d r r 1 3 with ( ) ( ) ( ) ∫ ′ - ⋅ - - - = ′ - Ω x x p i p e p d x x r r r r r 1 3 3 1 2 ω π , (16) because\n\n( ) ( ) ( ) ( ) ( ) x x x x x x x d x x x x x d ′ - = ′ - ′ ′ Ω ′ ′ - Ω ′ ′ = ′ - ′ ′ Ω ′ ′ - Ω ′ ′ - - ∫ ∫ r r r r r r r r r r 3 1 3 1 3 δ (17) with the Dirac distribution ( ) ( ) ( ) ∫ ′ - ⋅ - = ′ - x x p i e p d x x r r r r r 3 3 3 2π δ . ( 18\n\n)\n\n(17) should be an integral representation of the \"symbolic equation\"\n\n( ) ( ) ( ) ( ) 1 ˆ2 1 2 1 2 1 2 1 2 2 2 2 2 2 2 2 = + + = + + - - p m p m p m p m r r r r .\n\nAccordingly, terms with the nth power of ( )\n\n2 1 2 2 p m r + , ( ) 2 2 2 ˆn p m r + , (19)\n\ncorrespond to integrals over \"the nth power of Ω \":\n\n( ) ( ) ( ) ∫ ′ - ⋅ - = ′ - Ω x x p i n p n e p d x x r r r r r ω π 3 3 2 . ( 20\n\n)\n\nBy replacing the operators of type (19) by integrals over \"powers of Ω \" as given in (20), 1 Ĥ ′ and 2 Ĥ ′ (see (11) and (12) , respectively) can now be given a configuration space representation, too.\n\nWith these preparations, we can now address to the quantisation of the scalar field with the aim to be able to calculate scattering matrix elements."
},
{
"section_type": "OTHER",
"section_title": "Quantisation of the scalar field and the description of scattering processes",
"text": "Starting with Hamiltonian (10) , it is now possible to describe scattering processes within the framework of quantum field theory. For free scalar particles, a quantum field theoretic ansatz is described e.g. in [2] and [4] , using (2) and ( 13 ), respectively, as equations for a field operator ( )\n\nx φ .\n\nThe latter one can be formulated with the help of creation and annihilation operators + must be a commutator (for fermions we would use here an anti-commutator instead, cf. e.g. [4] ). Equations ( 21 ) to (24) are identical to those that one would postulate within a non-relativistic quantum field theory for bosons. For the density of a Hamiltonian, we make the usual ansatz (see e.g. [7] ):\n\n( ) ( ) x H x H φ φ ′ = + ˆ ( 25\n\n)\n\nwhich one can retrieve from a density of a Lagrangian (see [2] ):\n\n( ) ( ) ( ) φ φ φ φ φ φ φ φ + + + + Ω - Ω - ∂ - ∂ = 2 1 2 1 2 t t i L . ( 25 a)\n\nSubstituting (10) into (25), we get\n\n2 1 0 Ĥ H H H + + ≈ (26) with ( ) ( ) x H x H φ φ 0 0 ˆ′ = + , ( 27\n\n) ( ) ( ) x H x H φ φ 1 1 ˆ′ = + , ( 28\n\n) ( ) ( ) x H x H φ φ 2 2 ˆ′ = + . ( 29\n\n) (25) is (among other things) motivated by the fact that p p p a a p d H x d r r 3 0 3 + ∫ ∫ = ω (30) reproduces the relativistic analogue of the free non-relativistic Hamiltonian: p p a a m p p d r r r 2 2 3 + ∫ . (31) (28) together with (29) are the densities of the Hamiltonian to (and including the) quadratic order in e. (21) to (24) are valid for free particles, but can also be used for interacting ones, if Dirac's representation is used instead of the so far applied Heisenberg representation. Then, with (28) and (29) combined to a Hamiltonian density 2 1 Ĥ H H I + =\n\n(32) for the interaction of scalar bosons with photons, we can now start to calculate scattering matrix elements. To this purpose, we need the serial expansion of the S-operator (see e.g. [7] ) to the order of 2 e : ( ) ( ) ...\n\n1 ˆ2 1 + + + = S S S (33) with ( ) ( ) ( )\n\n∫ - = x H T x d i S I 4 1 , ( 34\n\n) ( ) ( ) ( ) ( ) ( ) ∫ ∫ - = 2 1 2 4 1 4 2 2 1 2 ˆx H x H T x d x d i S I I , ( 35\n\n)\n\nwhere we have introduced a time ordering operator\n\n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 0 1 0 2 2 1 0 2 0 1 2 1 ˆx H x H T x H x H x x x H x H x x x H x H T I I I I I I I I = - + - = θ θ (36) with ( ) < ≥ = 0 , 0 0 , 1 t t t θ . (37) ( ) 1\n\nŜ does not only contribute to the expansion (33) with terms of order e, but also to order 2 e .\n\nTherefore, we can split off ( )\n\n1 Ŝ into a term 1 Ŝ containing only terms in e, ( )\n\n( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ∫ + - + + Ω ⋅ + ⋅ - - = x x A x x p x A x A p x T x d ie S φ φ φ φ 0 1 2 1 4 1 r r r r (38)\n\nwith\n\n( )( ) ( ) ( ) ∫ - Ω = Ω - - t x x x x d x , 1 1 1 1 3 1 r r r φ φ (39)\n\nand a part containing only terms in 2 e :\n\n( )\n\n( ) ( )( )( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ) ∫ ∫ - - + - + Ω ⋅ + ⋅ - Ω ⋅ + ⋅ - Ω - = 1 2 1 2 1 2 1 2 2 2 1 2 2 3 1 1 1 1 1 8 1 1 1 1 2 1 2 1 1 4 2 12 , , , ˆt x p t x A t x A p x x x d p x A x A p x x x A x T x d ie S r r r r r r r r r r r r r r φ φ φ φ (40)\n\nwhere the momentum operator 1 p r contains a gradient acting on 1 x r and 2 p r acting on 2 x r . In (38) and\n\n(40), we have already substituted (32) into (34) and replaced powers of ( )\n\n2 1 2 2 p m r +\n\nby integrals over \"powers of Ω \" (see (20)) in (11) . Thus we can rewrite (34) as\n\n( ) 12 1 1 Ŝ S S + = . ( 41\n\n)\n\nThe time ordering operator appearing in ( )\n\n1\n\nŜ can be left out, because it contains only one time. In ( )\n\n2 Ŝ\n\nwe only want to retain terms of order 2 e , therefore I H ˆ can be approximated by 1 Ĥ :\n\n( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) 2 2 0 2 1 1 0 1 2 1 2 2 2 2 2 1 1 1 1 1 1 1 4 1 2 2 0 2 1 1 1 1 1 1 1 2 1 2 4 1 4 2 1 2 2 x x A x x x A x x p x A x A p x x p x A x A p x x x A x x p x A x A p x T x d x d ie S φ φ φ φ φ φ φ φ φ φ φ φ + + - + - + + - + + Ω ⋅ + ⋅ Ω ⋅ + ⋅ + Ω ⋅ + ⋅ - - = ∫ ∫ r r r r r r r r r r r r . (42)\n\nHere, the time ordering operator must be taken into account, because there are two times: 1 t and 2 t . We have already quantised the field of scalar bosons, but must do now the corresponding with the electromagnetic field. Since the scalar field in (38), ( 40 ) and ( 42 ) couple in a different way to the vector potential A r than to the scalar potential 0 A , the choice of a Coulomb gauge seems to be appropriate:\n\n0 = ⋅ ∇ A r r . ( 43\n\n)\n\nThen the field equations take on the form ( )\n\nj A A t t r r r r = ∇ ∂ + ∇ - ∂ 0 2 2 , ( 44 a) ρ -\n\n= ∇ 0 2 A r ( 44 b)\n\nwith charge and current densities ρ and j r , respectively. From (44 b) we can see that in this gauge the scalar potential is just a c-number:\n\n( ) ( ) ∫ ′ - ′ ′ = x x t x x d t x A r r r r , 4 1 , 3 0 ρ π , ( 45\n\n)\n\nwhereas the vector potential A r becomes an operator when being quantised. For a free field, A r can be chosen like (see e.g. [4, 7, 10] )\n\n( ) ( ) ( ) ( ) λ ε ω π λ λ λ , 2 2 ˆ21 3 3 k e c e c k d x A x ik k x ik k k r r r r r ∫ ∑ = ⋅ + ⋅ - + = (46)\n\nwith the usual photon frequency\n\n2 ~k k r = ω and with creation and annihilation operators λ k c r ˆ and + λ k c r ˆ, respectively, for photons:\n\n] ( ) k k c c k k r r r r - ′ = ′ + ′ ′ 3 , ˆδ δ λ λ λ λ , [ ] [ ] 0 , , ˆ= = + + ′ ′ ′ ′ λ λ λ λ k k k k c c c c r r r r , ( 47\n\n)\n\nand 0\n\nA would even vanish. The polarisation vectors ( )\n\nλ ε , k r r\n\nfulfil the relation (see e.g. [4, 7] ):\n\n( ) ( ) 2 2 1 , , k k k k k j i ij j i r r r - = ∑ = δ λ ε λ ε λ . ( 48\n\n)\n\nDue to the Coulomb gauge condition (43),\n\n( ) 0 , = ⋅ λ ε k k r r r (49)\n\nis valid, too.\n\nIn the following sections, we are going to calculate ( )\n\n1 Ŝ and ( ) 2 Ŝ by substituting the field operators ( ) x φ and ( ) x A r from ( 21 ) and ( 46 ) as well as the distributions (20). Since we are considering electromagnetic interactions between (charged) spin-0 bosons, we have to take 0 A in (45) into account, too. Therefore, we first have to find out what the density of charge in (45) will be in this case. This can be done by coupling the Lagrangian density for free spin-0 bosons (25 a) to an electromagnetic field with the aid of the minimal coupling scheme With these results, one can then continue to calculate scattering matrix elements.\n\nThe first term we calculate is\n\n( ) ( ) ( ) ( )( )( ) ∫ - + Ω ⋅ + ⋅ = x p x A x A p x x d I φ φ 1 4 1 : ˆr r r r (50)\n\nappearing in ( ) 1 Ŝ (see ( 38 )). To this end, it is useful to recognise that by means of (16) we get\n\n( )( ) ( ) ∫ ⋅ - - - = Ω p x ip p a e p d x r 2 1 3 1 2 3 ω π φ . ( 51\n\n)\n\nWith (51) and integration by parts, (50) yields:\n\n( ) ( ) ( ) ( ) ( ) + + - - + + - + ⋅ = ∫ ∑ λ λ λ δ δ ω λ ε ω π k k p p p k c k p p c k p p a a p p k p d p d k d I r r r r r r r r , 2 2 ˆ2 1 4 2 1 4 2 1 2 3 1 3 3 1 1 2 1 , (52) where ( ) (\n\n) ( )\n\nk p p k p p k p p r r r ± - ± - = ± - 2 1 3 2 2 1 4 1 δ ω ω ω δ δ (53)\n\nand, of course, the a operators commute with the c operators. In 12 Ŝ (40), the first integral can be expressed in a similar way:\n\n( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ) 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 , , 1 2 2 2 : ˆ2 1 2 1 4 2 1 2 1 4 2 1 2 1 4 2 1 2 1 4 , 2 2 1 1 1 2 3 1 3 2 3 1 3 1 2 4 2 λ λ λ λ λ λ λ λ λ λ δ δ δ δ λ ε λ ε ω ω ω π φ φ k k k k k k k k p p p k k c c p p k k c c p p k k c c p p k k c c p p k k k k a a p d p d k d k d x x A x x d I r r r r r r r r r r r r r r r - + - - + - + + - + - + - + - + + ⋅ ⋅ = Ω = + + + + + - + ∫ ∑ ∫ (54)\n\nAfter a quite lengthy but straightforward calculation, the second integral in (40) yields (see (42)) which is not just like the product of two operators 1 Î due to the time ordering operator as defined in (36). Unfortunately, we cannot use the famous Wick theorem, because the scalar field operator (21) contains only contributions to positive energy solutions: it does not consist of a sum of both positive and negative energy solutions as it would be the case for the field operator of the Klein-Gordon equation. Due to the symmetry of the time ordering operator (36) in its arguments, we may conclude\n\n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) = Ω ⋅ + ⋅ ⋅ - Ω ⋅ + ⋅ = - - + ∫ ∫ 1 2 1 2 1 2 1 2 2 2 1 2 1 1 1 1 1 2 3 1 4 3 , , , ˆ: ˆt x p t x A t x A p x x p x A x A p x x d x d I r r r r r r r r r r r r r φ φ (55) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) - + - - + - + + - - ⋅ + - + - + - + + - + ⋅ ⋅ + ⋅ ∑ ∑ ∑ ∫ + + + - + + + 1 1 1 2 2 1 1 2 2 1 1 1 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 1 2 , 2 , , 1 2 2 2 2 1 2 1 4 2 1 2 1 4 2 1 1 1 1 2 1 2 1 4 2 1 2 1 4 2 1 1 1 1 2 1 2 2 2 2 3 1 3 2 3 1 3 λ λ λ λ λ λ λ λ λ λ λ δ δ ω λ ε δ δ ω λ ε λ ε ω ω ω π k k k k k p k k k k k p p p p k k c c p p k k c c p p k k k p k c c p p k k c c p p k k k p k p p k a a p d p d k d k d r r r r\n\n( ) ( ) ( ) ( ) ( ) ( ) 2 1 0 2 0 1 2 4 1 4 2 1 2 4 1 4 2 ˆx H x H x x x d x d x H x H T x d x d I I I I ∫ ∫ ∫ ∫ - = θ . ( 56\n\n)\n\nWith this property, the calculation of the second term in (42) can be simplified a bit:\n\n( ) ( ) ( ) ( )( )( ) ( ( ) ( ) ( ) ( )( ) ( ) = Ω ⋅ + ⋅ ⋅ Ω ⋅ + ⋅ = - + - + ∫ ∫ 2 1 2 2 2 2 2 1 1 1 1 1 1 1 2 4 1 4 4 : ˆx p x A x A p x x p x A x A p x T x d x d I φ φ φ φ r r r r r r r r (57 a) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )\n\n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) - - - - - - - ⋅ - - ′ - ′ - ′ + + - - - - - - ⋅ + - ′ - ′ - ′ + - - - + - - - ⋅ - - ′ + ′ - ′ + + - - + - - - ⋅ + - ′ + ′ - ′ ⋅ + ⋅ ′ + ′ ⋅ ′ ′ ′ ′ ′ ∫ ∫ ∫ ∫ ∑ ∑ ∫ ′ ′ ′ + + ′ ′ ′ ′ ′ + ′ ′ ′ ′ ′ + ′ ′ ′ ′ ′ ′ ′ ′ ′ + ′ + ′ ′ 2 1 2 1 2 2 1 2 1 3 2 1 3 2 1 2 1 2 2 1 2 1 3 2 1 3 2 1 2 1 2 2 1 2 1 3 2 1 3 2 1 2 1 2 2 1 2 1 3 2 1 3 2 1 2 1 2 3 1 3 2 3 1 3 3 3 ẽxp ẽxp 2 ˆẽxp ẽxp 2 ˆẽxp ẽxp 2 ˆẽxp ẽxp 2 ˆ, , 2 2 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 2 1 2 t i t i t t\n\ndt dt k p p k p p c c t i t i t t dt dt k p p k p p c c t i t i t t dt dt k p p k p p c c t i t i t t dt dt k p p k p p c c p p k p p k a a a a p d p d p d p d k d k d k p p k p p k k k p p k p p k k k p p k p p k k k p p k p p k k p p p p p p k k\n\nω ω ω ω ω ω θ π δ δ ω ω ω ω ω ω θ π δ δ ω ω ω ω ω ω θ π δ δ ω ω ω ω ω ω θ π δ δ ω λ ε ω λ ε ω ω π λ λ λ λ λ λ λ λ λ λ\n\nr r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r (57) contains four terms of the subsequent type that can be simplified with the help of ( 58 ) and (59): (\n\nτ ω ω τ θ π τ ω ω δ ω ω π τ ω ω τ θ π τ ω ω θ π 2 1 2 2 1 2 1 2 2 1 2 2 1 2 2 1 1 2 1 2 2 1 exp 2 exp 2 exp 2 exp 2 - - + = + - - - = - - - ∫ ∫ ∫ ∫ i i i d T dT d t i t i xp e t t dt dt (60 a) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )\n\nThe function θ can be expressed by an integral in the complex plane (see e.g. [6, 8] ), ( )\n\n∫ ∞ ∞ - - ± ± = ± ε π τ θ τ i p e dp i ip 0 0 1 2 0 (61)\n\nwith an ε approaching zero. Substituting this into (60 a), we get:\n\n( ) ( ) ( ) ( ) ( ) 2 1 2 2 1 2 2 1 1 2 exp 2 ω ω δ ε ω π τ ω ω τ θ π τ ω ω δ + + = - - + ∫ i i d i . (60 b)\n\nWith this result, (57) becomes:\n\n( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )\n\n + - - ⋅ - - ′ - ′ - ′ - - + - - + + + - ⋅ + - ′ - ′ - ′ + - + - - + + - - ⋅ - - ′ + ′ - ′ - - + + - + + + - ⋅ + - ′ + ′ - ′ + - + + - ⋅ + ⋅ ′ + ′ ⋅ ′ ′ ′ ′ ′ = ′ ′ ′ + + ′ ′ ′ ′ ′ + ′ ′ ′ ′ ′ + ′ ′ ′ ′ ′ ′ ′ ′ ′ + ′ + ′ ′ ∑ ∑ ∫ ε ω ω ω δ δ ω ω ω ω ω ω δ ε ω ω ω δ δ ω ω ω ω ω ω δ ε ω ω ω δ δ ω ω ω ω ω ω δ ε ω ω ω δ δ ω ω ω ω ω ω δ ω λ ε ω λ ε ω ω π λ λ λ λ λ λ λ λ λ λ i k p p k p p c c i k p p k p p c c i k p p k p p c c i k p p k p p c c p p k p p k a a a a p d p d p d p d k d k d i I k p p k p p k p p k k k p p k p p k p p k k k p p k p p k p p k k k p p k p p k p p k k p p p p p p k k 1 ~1 ~1 ~1 ~, , 2 2 2 2 ˆ2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 2 1 2 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 2 1 2\n\n- ′ - - ′ + ′ ′ ′ - = + ′ + ′ ∫ δ π . ( 62 b)\n\n(42) contains two terms in 0 A . The first term consists of a combination of (50) and (62 a), but taken at different times and therefore joined via the time ordering operator. That is why we have to use (60) as well as (45) and (63) again:\n\n( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( ) ( ) (\n\n)\n\n + - - - - + - - + - - + + - - + - - + - - + - - + + + - + - + - - + + - + + - - + + - + - - + + - ⋅ - ′ - - ′ + ′ + ⋅ ′ ′ = Ω ⋅ + ⋅ = + + ′ + ′ ′ ′ + + ′ + ′ ′ ′ ′ ′ + ′ + ′ ′ ′ + ′ + ′ ′ ′ ′ ′ = + - + ∑ ∫ ∫ λ λ λ λ λ ε ω ω ω ω ω ω ω ω ω ω δ δ ε ω ω ω ω ω ω ω ω ω ω ω δ δ ε ω ω ω ω ω ω ω ω ω ω δ δ ε ω ω ω ω ω ω ω ω ω ω ω δ δ δ λ ε ω ω π φ φ φ φ 1 1 1 2 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 2 1 1 1 1 2 1 1 1 2 3 1 ~1 ~1 ~1 ~, ~1 2 2 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 2 1 2 3 2 1 1 3 3 3 3 1 3 2 3 1 3 7 2 2 0 2 1 1 1 1 1 1 1\n\n2 4 1 4 6 k p p k k p k p p k p p p k p k k p k k p p p k p k k p p p k p k k p p k k p k p p k p p p k p k k p k k p p p k p k k p p p k p k p k c a a a a a i k p p c a a a a a i k p p c a a a a a i k p p c a a a a a i k p p k k k p k p p p k k d k d p d p d p d p d k d i x x A x x p x A x A p x T x d x d I\n\nr r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r (64)\n\nThe second term of (42) containing a scalar potential is even quadratic in 0 A :\n\n( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ + + = 2 2 0 2 1 1 0 1 2 4 1 4 7 ˆx x A x x x A x T x d x d I φ φ φ φ . ( 65 a)\n\n(65 a) contains a factor of two integrands of the kind of (62 a), but taken at two different times. Thus the time ordering operator must be taken into account. With the same substitutions as in ( 62 ) and (64), we obtain the following result:\n\n( ) ( ) ( ) ( ) ( ) ( ) ε ω ω ω ω δ δ ω ω ω ω ω ω ω ω δ π i k k k k k k p p k k p p a a a a a a a a p d k d k d p d p d k d k d p d i I k p k p k p k p k p k p p k k p p k k p + - - + - ′ - ′ ⋅ - ′ + - ′ - ′ + - ′ - - + + - - + ⋅ ′ ′ ′ ′ = ′ ′ ′ ′ ′ ′ + ′ + ′ + ′ + ′ ∫ 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 2 2 1 ˆ2 2 2 2 1 1 2 2 2 2 3 1 1 1 1 3 5 3 2\n\n3 2 3 2 3 2 3 1 3 1 3 1 3 1 3 7 r r r r r r r r r r r r r r r r r r r r (65 b)\n\nThe results ( 52 ), ( 54 ), ( 55 ), ( 57 ), ( 62 ), ( 64 ) and (65) substituted into (38) to (42) now enable us to evaluate scattering matrix elements for scattering processes to (and including) the order 2 e . As two examples, we turn first to the scalar analogue of Compton scattering in order to address then to the scattering of two identical scalar bosons. These two scattering processes can be compared easily with the corresponding results of the well-known scalar QED dealing with the Klein-Gordon equation (3) ."
},
{
"section_type": "OTHER",
"section_title": "Compton scattering",
"text": "For Compton scattering, we need one scalar boson and one photon each in the input and output channel. This means, we have to evaluate the element\n\n0 0 : ˆ+ + ′ ′ ′ = µ µ h q q h c a S a c S r r r r , ( 66\n\n)\n\nwith the S ˆ-operator (33). Firstly, we realise that the terms based on 1 Î (see (52)) as well as 6 Î (see (64)) must vanish, because the subsequent two elements in the photon operators become zero:\n\n0 0 0 0 0 = = ′ ′ + ′ ′ µ λµ µ λ µ δ δ h h k h k h c c c c r r r r r r , ( 67\n\n) 0 0 0 = + + ′ ′ µ λ µ h k h c c c r r r\n\n. There, we have used the commutation relations (47), the properties of creation and annihilation operators corresponding to those of ( 22 ) and ( 23 ) as well as abbreviated the delta functional by ( )\n\nh k h k r r r r - = 3 δ δ . ( 68\n\n)\n\nThe terms with 2 Î and 3 Î need\n\nq p p q q p p q a a a a r r r r r r r r 2 1 2 1 0 0 δ δ ′ + + ′ = , ( 69\n\n)\n\nwhereas a term q p p p p q q p p p p q a a a a a a r r r r r r r r r r r r\n\n1 2 1 2 1 2 1 2 0 0 δ δ δ ′ ′ ′ + + ′ + ′ ′ = belongs to 4\n\nÎ . Here we have used again ( 22 ) to (24).\n\nFor 2 Î , 3 Î and 4 Î the following equations are necessary, too:\n\n0 0 0 = + ′ ′ ′ ′ µ λ λ µ h k k h c c c c r r r r , 0 0 0 = + + + ′ ′ ′ ′ µ λ λ µ h k k h c c c c r r r r , ( 70\n\n) µ λ λ µ µ λ λ µ δ δ δ δ ′ ′ ′ ′ + + ′ ′ ′ ′ = k h k h h k k h c c c c r r r r r r r r 0 0 , λµ λ µ µ λ λ µ δ δ δ δ ′ ′ ′ ′ + + ′ ′ ′ ′ = k h k h h k k h c c c c r r r r r r r r 0 0\n\n, where we have commutated the creation operators successively to the left and the annihilation operators to the right. Furthermore, we see that the term based on 5 Î from (62) becomes zero, due to\n\n0 0 0 = + + ′ + ′ ′ q p k k p q a a a a a a r r r r r r . ( 71\n\n)\n\nA similar result holds true for the term based on 7 Î from (65).\n\n0 0 0 2 2 2 2 1 1 1 1 = + + ′ + ′ + ′ + ′ ′ q p k k p p k k p q a a a a a a a a a a\n\nI I i I i ie I I ie S + - + - - + + - - ≈ - . ( 73\n\n)\n\nFor Compton scattering, 1 Î , 2 Î , 3 Î , 4 Î , 5 Î , 6 Î and 7 Î can be evaluated explicitly:\n\n0 ˆ7 6 5 1 = = = = I I I I , ( 74\n\n) ( ) ( ) ( ) µ ε µ ε ω ω ω π δ ′ ′ ⋅ ′ - ′ - + = ′ , , 2 ˆ4 2 h h h q h q I q h h r r r r , ( 75\n\n) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ′ - ⋅ ′ ′ + ′ - ⋅ + + ⋅ ′ - + ⋅ ′ ′ ′ - ′ - + = ′ - + ′ h q h h h q h h q h h h q h h q h q I h q h q q h h r r r r r r r r r r r r r r r r r r r r r r 2 , 2 , 1 2 , 2 , 1 2 2 ˆ2 2 2 4 3 µ ε µ ε ω µ ε µ ε ω ω ω ω π δ (76) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + - + + ⋅ ′ + ′ ⋅ ′ ′ + + - - ′ - ⋅ ′ ′ - ′ ⋅ ′ - ′ - + = + + ′ - ′ ′ - ′ ε ω ω ω µ ε µ ε ω ε ω ω ω µ ε µ ε ω ω ω ω π δ i h q h h q h i h q h h q h h q h q I h q h q h q h q h q h q q h h r r r r r r r r r r r r r r r r r r r r r r r r r r r r ~2 , 2 , 1 ~2 , 2 , 1 2 ˆ2 4 4 (77)\n\nThe two terms 2 Î and 4 Î in (73) resemble the three terms in the corresponding formula of Compton scattering for scalar bosons, but this time being based on the Klein-Gordon equation (3) (see e.g. [6, 8] ):\n\n(\n\nε µ ε µ ε µ ε µ ε µ ε δ ω ω ω ω π ′ ′ ⋅ - ′ - ⋅ ′ ′ - ′ - - ′ ⋅ + ′ + ′ ⋅ ′ ′ - + + ⋅ ⋅ ′ - ′ - + - = - ′ ′ , , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )] µ\n\n2 2 2 2 2 1 ˆ2 2 2 2 4 2 2 h h i h q h m h q i h q h h q h m h q i h q h h q h q ie S q q h h (78) 2 2 , 2 , 2 , 2 ,\n\nwhere ε is the 4-dimensional generalisation of the three dimensional polarisation vector ε r used so far. In (78), the first two terms correspond to 4 Î . We realise that these terms in (78) look very much like those in (77), but also that especially the propagators of both theories are a bit different. (\n\nresembles the third term in (78). On the other hand, 3 Î in (76) could also be regarded as the analogue of the first two terms in (78) -at least after having used the Coulomb gauge condition (49) in (76).\n\nThe first two terms in (78) vanish, if we choose the incoming scalar boson to be at rest, ( ) ( )\n\n0 , : , 0 r r m q q q = = , (79)\n\nand want to have transversally polarised photons in this laboratory system, ( ) ( )\n\n0 , , = ⋅ ′ ′ = ⋅ q h q h µ ε µ ε , (80)\n\nand use the Lorentz gauge condition\n\n( ) ( ) 0 , , = ′ ⋅ ′ ′ = ⋅ h h h h µ ε µ ε . ( 81\n\n)\n\nAccordingly, 3 Î and 4 Î in (73) vanish too, if we adopt (79) again and use the analogue of (80),\n\n( ) ( ) 0 , , = ⋅ ′ ′ = ⋅ q h q h r r r r r r µ ε µ ε , ( 82\n\n)\n\nas well as apply the Coulomb gauge condition (49). That is, under these conditions in both versions of scalar QED, only one term remains (i.e. (75) in (73) and, accordingly, the third term in (78)) which is a relativistically generalised version of the matrix element from which the well-known Thomson scattering cross section can be evaluated. Even though the propagators in both theories are rather different, for this example of scattering process, the results do not seem to differ very much from each other in the laboratory system chosen above. Therefore the question arises, whether this is also the case for further scattering processes. To this end, we are going to investigate what happens, if two identical scalar bosons interact with each other."
},
{
"section_type": "OTHER",
"section_title": "Scattering of two identical scalar bosons",
"text": "If we want to calculate matrix elements of the S-operator (33) for the scattering of two identical scalar bosons, we need two scalar bosons in the input channel and two in the output channel:\n\n0 0 : ˆ1 2 1 2 + + ′ ′ = q q q q a a S a a S r r r r . ( 83\n\n)\n\nWe can reuse (73), but have to evaluate 1 Î , 2 Î , 3 Î , 4 Î , 5 Î , 6 Î and 7 Î again. Firstly, we recognise that due to (52) and the analogue of ( 22 ) and ( 23 ) for the photon operators\n\n0 ˆ1 = I , (84)\n\nis valid. Furthermore, we need\n\n2 1 2 1 2 2 1 1 0 0 λ λ λ λ δ δ k k k k c c r r r r = + (85)\n\nfor calculating 2 Î with the help of ( 54 ), whereas all the other photon operator terms therein vanish.\n\nThe same holds true for 3 Î with (55) and 4 Î with (57).\n\nAs far as the scalar boson operators are concerned, for the evaluation of 2 Î and 3 Î 0 0 0\n\n1 2 1 2 1 2 = + + + ′ ′ q q p p q q a a a a a a r r r r r r .\n\nHence, these two terms,\n\n0 ˆ2 = I , (86) 0\n\nˆ3 = I ,\n\nare zero, too. For the evaluation of the non-vanishing term 4 Î , the following result is useful:\n\n2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 1 2 1 2 1 2 1 2\n\n0 0 p q p q p q p q p q p q p q p q p q p q p q p q p q p q p q p q q q p p p p q q a a a a a a a a\n\n′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ + + + ′ + ′ ′ ′ + + + =\n\nr r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ . (87) With (85), (87) and the invariance of (48) under the transformation k k r r -→ , we get\n\n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 2 1 2 1 2 1 1 2 1 1 1 1 1 1 2 2 2 2 1 2 1 4 2 2 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 ~, , 1 1 ~, , 2 8 1\n\nˆA in terms i i q q q q q q q q i i q q q q q q q q q q q q ie S q q q q q q q q q q q q q q q q q q q q q q\n\n+ - + - - + - - ⋅ ⋅ ′ + ⋅ ′ - ′ - ⋅ ′ + + - + - - + - - ⋅ ⋅ ′ + ⋅ ′ - ′ - ⋅ ′ + ′ - ′ - + - = - ′ - ′ ′ - ′ ′ - ′ - ′ ′ - ′ ′ - ∑ ∑ ε ω ω ω ε ω ω ω ω λ ε λ ε ε ω ω ω ε ω ω ω ω λ ε λ ε ω ω π δ λ λ r r\n\n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ′ - ′ + ⋅ ′ + + ′ - ′ + ⋅ ′ + ′ - ′ - + = - ′ ′ 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 2 1 2 1 4 2 2 1 2 1 2 4\n\n1 ˆq q q q q q q q q q q q q q q q ie S q q q q ω ω ω ω π 1 ˆA in terms q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ie S q q\n\n+ ′ - ′ - ′ + ⋅ ′ - ′ - ⋅ ′ + - ′ - ′ - ′ + ⋅ ′ - ′ - ⋅ ′ + - ′ - ′ + ⋅ ′ + + ′ - ′ + ⋅ ′ + ′ - ′ - + - = - v r r r v r v r r r v r r r v r v r r r r r r r r r v r ω ω π δ (88 b)\n\nThe first two terms in (88 b) correspond to (89) which contains the photon propagator in the Feynman gauge. But since we use a Coulomb gauge, only the space-like components of the linear 4-momenta appear in the numerators of the first two terms in (88 b). Moreover, in the second line of (88 b) two additional terms are present which can be reformulated by means of the delta distribution in (88 b):\n\n( )( ) ( ) ( ) ( )( ) ( ) ( ) 2 2 1 2 2 1 2 2 2 1 2 2 2 1 2 1 1 2 1 1 2 1 2 1 2 2 2 2\n\nq q q q q q q q q q q q q q q q v r r r r r v r r r r r\n\n′ - ′ - ′ - - ′ - ′ - ′ - ′ - ′ - ′ - . ( 90\n\n)\n\nThe structure of (88 b) is the same as that of (89). We have two terms: in the second term, the momenta of the two scalar bosons in the output channel have been exchanged compared to the first term.\n\nNow we address to the terms in the scalar potential 0 A in (88). The Coulomb term 5 Î contains a term p q k q q p q k k q p q q p q k p q k q q k q p k q p q q k q p q q p k k p q q a a a a a a a a\n\n′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ + + + ′ + ′ ′ ′ - - - =\n\n2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 1 2 0 0 δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ . Therefore, 5 Î yields ( ) ( ) ( ) ( ) - ′ - - ′ - - ′ + ′ = 2 1 2 2 1 1 2 1 2 1 4 2 5 1 1 2 1 ˆq q q q q q q q I r r r r δ π . ( 91\n\n)\n\nThe term 6 Î vanishes, since the annihilation operators for photons in (64) act directly on the vacuum states.\n\nOn the other hand, 7 Î becomes a rather lengthy because of (\n\n( )\n\n1 1 1 1 1 1 1 1 1 2 1 1 2 1 2 1 1 2 2 2 2 2 2 1 1 2 1 1 2 1 1 2 2 2 2 2 1 2 1 1 2 1 2 1 1 2 2 2 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 2 2 1 1 1 1 1 2 0 0 k p k p k p k p a a a a a a a a a a a a k q p q k p p k q p q k q p k k k q p q p p q p q k q p k q p q k p p k q k q p k q p q k k q k p p q p q q p k k p p k k p q q ′ ↔ ′ - + ′ ↔ ′ - - ′ ↔ ′ - - ′ ↔ ′ - = ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ + + + ′ + ′ + ′ + ′ ′ ′\n\nr r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ where ( ) 1 1 k p ′ ↔ ′ r r denotes the same term as the immediately preceding one, but with exchanged momenta p′ r and 1 k ′ r , respectively. Thus, 7 Î gives\n\n( ) ( ) ( ) + - - ⋅ - - + - - ′ + ′ = - ′ - - ′ - + ′ - + ′ - - + ∫ 2 2 2 2 2 2 2 2 3 3 2 1 2 1 4 2 7 1 2 1 2 2 2 1 2 1 2 1 1 1 1 1 1 2 2 8\n\nˆq p q p q p q p q p q p q p q p p q q p q q p d q q q q i I\n\nr r r r r r r r r r r r r r r r r r r r r r ω ω ω ω ω ω ω ω ω ω ω ω π δ π .\n\nHence in place of the time-like components (i.e. the energy terms) of the linear 4-momenta in the numerators of (89) derived from the Klein-Gordon equation, several terms arise: (90), ( 91 ) and (92). But (91) and (the non relativistic limit of) (92) would also have appeared, if we had started from the non-relativistic Schrödinger equation. Thus, these terms state the fact that the equation used for obtaining (88) (together with (90), ( 91 ) and ( 92 )) is Schrödinger equation like."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusions and outlook",
"text": "For scalar bosons, we could see that it is possible to describe scattering processes by means of a square root operator equation being coupled to an electromagnetic field. We achieved this by splitting off a factor in the shape of the free square root operator from the equation and by a series expansion of a remaining square root factor containing terms in the electromagnetic vector potential and powers of the (inverse) free square root operator. The latter ones could be given an integral representation.\n\nHaving quantised the fields involved, we could evaluate the scattering matrix elements for Compton scattering and for the scattering of two identical bosons (to -and including -the quadratic order of e) which, on the one hand, resembled the results derived with the help of the corresponding Klein-Gordon equation and, on the other hand, the results one would have obtained with a non-relativistic Schrödinger equation. Of course, now several questions arise, e.g.:\n\n• Can we formulate Feynman rules at all for our non-local scalar QED?\n\n• Do divergent terms appear and, if yes, can a renormalisation procedure be found?\n\n• Can the results of this non-local QED be confirmed (or refuted) by experiments?\n\nTo the first question: if one tries to formulate Feynman rules, one must be aware that in each step of the approximation procedure, we have to expand the first square root factor (in the second term of) (7) to the desired order in e. Thus, the Hamiltonian we are using within that procedure must be adapted in each order of e of the approximation. Therefore we conclude that even if it were possible to formulate Feynman rules, they would be much more complicated than those for the Klein-Gordon theory. This is the price we have to pay for non-locality. But without Feynman rules, it is not very easy to analyse the renormalisability of that non-local scalar QED either.\n\nThe answer to the last question listed above is negative due to a lack of elementary spinless bosons in nature. But that question would be sensible, if we had a corresponding non-local theory for spin-1/2 particles. We could even make a guess, how this theory would look like: for free spin-1/2 particles, the wave functions in (2) should be 2-spinors. It would also be possible to couple this equation to an electromagnetic field, but the usual minimal coupling scheme does not work. Instead we would have to postulate an equation:\n\nφ φ m H i t ˆ′ = ∂ , (93)\n\nwith the Hamilton operator ( ) ( ) 0 2 2 ˆeA E i B e A e p m H + ⋅ --+ = ′ r m r r r r m σ ,\n\nwhich contains an additional term with the Pauli matrices σ r and the magnetic field B r as well as the electric field E r under the square root. For this equation, it has already been shown that it can reproduce the gyromagnetic factor of 2 for the electron as well as, when being applied to a hydrogen atom, that it can reproduce correct binding energies of the electron at least to (and including) the quadratic order in 2 e (see [12] ). We can apply the same approximation procedure to this equation as we have presented here for the scalar case. But of course, additional difficulties emerge: the Hamiltonian corresponding to the one shown in (25) should be Hermitian. And at least for the free spin-1/2 case, that Hamiltonian should be relativistically invariant. This means, that for the free case,\n\n+ ± ′ = ′ H H m . ( 97\n\n)\n\nThe results of a QED based on (95) or (96) could then be compared with the ones based on a corresponding Dirac equation. The author does not know, whether such an approach has already been performed for the spin-1/2 case or for the here presented spinless case. He does not know either, if an application for it can be found, where e.g. non-local properties are indispensable. But it seems to him that from the technical point of view, square root operator equations coupled to an electromagnetic field like (4) (or maybe even like (93) together with (94)) can be handled within the framework of a quantum field theory. Such equations were given up quite early in the history of quantum mechanics for several good reasons, e.g. due to their lack of relativistic invariance (see [13] ), their non-local character accompanied by the difficulty of finding an appropriate mathematical interpretation and description, respectively (see e.g. [4, 5, 6] ). While the first reason mentioned remains still valid, from today's perspective, those non-local properties might not be refused as vehemently as in the past (e.g. when one looks out for approximations of the so called Bethe-Salpeter equation [11] ). The author hopes to have shown, that at least answers to the question of possible descriptions of such non-local square root operator equations can be found."
}
] | ||
arxiv:0704.0431 | 0704.0431 | 1 | 10.1142/9789812834300_0199 | 4eb59b844cbe83ff4aab47b61143f1b80337823676b9269d898b997200243f54 | Fragmentation of general relativistic quasi-toroidal polytropes | We investigate the role of rotational instabilities in the context of black
hole formation in relativistic stars. In addition to the standard scenario - an
axially symmetric dynamical instability forming a horizon at the star's center
- the recently found low-$T/|W|$ instabilities are shown to lead to
fragmentation and off-center horizon formation in differentially rotating
stars. This process might be an alternative pathway to produce SMBHs from
supermassive stars with inefficient angular momentum transport. | [
"Burkhard Zink",
"Nikolaos Stergioulas",
"Ian Hawke",
"Christian D. Ott",
"Erik\n Schnetter",
"Ewald Mueller"
] | [
"gr-qc",
"astro-ph.HE"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | How do black holes form from relativistic stars? This question is of great fundamental and practical importance in gravitational physics and general relativistic astrophysics. On the fundamental level, black holes are genuinely relativistic objects, and thus the study of their production involves questions about horizon dynamics, global structure of spacetimes, and the nature of the singularities predicted as a consequence of the occurrence of trapped surfaces. On the level of astrophysical applications, systems involving black holes are possible engines for highly energetic phenomena like AGNs or gamma-ray bursts, and also likely a comparatively strong source of gravitational radiation.
The most simple model of black hole formation from, say, cold neutron stars, is a fluid in spherically symmetric polytropic equilibrium moving on a sequence of increasing mass due to accretion [1] . This assumes that (i) the stellar structure and dynamics are represented reasonably by the ideal fluid equation of state and the polytropic stratification, (ii) accretion processes are slow compared to the dynamical timescales of the star, and (iii) rotation is negligible. Our focus has been to study the effects of relaxing the third assumption.
In spherical symmetry, the sequence of equilibrium polytropes has a maximum in the mass function M (ρ c ), where ρ c denotes the central rest-mass density of the polytrope. This maximum is connected to a change in the stability of the fundamental mode of oscillation [1] , and thus collapse sets in via a dynamical instability to radial deformations. During the subsequent evolution, a trapped tube forms at the center which traverses the stellar material entirely [2] .
How much of this behaviour is preserved when rotation is taken into account? Rotation is known to change the equilibrium structure of the star, and, in consequence, its modes of oscillation and set of unstable perturbations. The collapse might also lead to the formation of a massive disk around the new-born black hole, and finally only systems without spherical symmetry can be a source of gravitational radiation.
Numerical simulations have been used to study the collapse and black hole formation of general relativistic rotating polytropic stars [3] . For the uniformly and moderately differentially rotating models investigated in those studies, the dynamical process is described by the instability of a quasi-radial mode and subsequent collapse of the star up to the formation of an accreting Kerr black hole at the star's center.
Will strong differential rotation modify this picture? Even before our study, there was evidence that this should be the case. (i) Strong differential rotation can deform the high- density regions of a star into a toroidal shape, thus changing the equilibrium structure considerably. (ii) It admits stars of high normalized rotational energy T /|W | [1] which are stable to axisymmetric perturbations. (iii) It admits nonaxisymmetric instabilities, for example by the occurrence of corotation points [4] , at low values of T /|W | [5] . (iv) A barmode instability of the type found in Maclaurin spheroids [6] would likely express itself by the formation of two orbiting fragments if the initial high-density region has toroidal shape.
This last property has motivated us to ask this question: Can a bar deformation transform a strongly differentially rotating star into a binary black hole merger with a massive accretion disk? If so, this process might occur in supermassive stars if the timescales associated with angular momentum transport are too large to enforce uniform rotation.
We have investigated black hole formation in strongly differentially rotating, quasi-toroidal models of supermassive stars [7, 8] , and found that a non-axisymmetric instability can lead to the off-center formation of a trapped surface (see fig- ure ). An extensive parameter space study of this fragmentation instability [8] reveals that many quasi-toroidal stars of this kind are dynamically unstable in this manner, even for low values of T /|W |, and we have found evidence that the corotation mechanism observed by Watts et al. [4] might be active in these models. Since, on a sequence of increasing T /|W |, one of the low order m = 1 modes becomes dynamically un-stable before m = 2 and higher order modes, one would not expect a binary black hole system to form in many situations (although this may depend on the rotation law and details of the pre-collapse evolution as well). Rather, the off-center production of a single black hole with a massive accretion disk appears more likely.
Since the normalized angular momentum J/M 2 of the initial model is greater than unity, there is another interesting consequence of this formation process: the resulting black hole, unless it is ejected from its shell, may very well be rapidly rotating, spun up by accretion of the material remaining outside the initial location of the trapped surface. Investigating the late time behaviour of this accretion process, estimating possible kick velocities of the resulting black hole, and finding the mass of the final accretion disk is, however, beyond our present-day capabilities and subject of future study. | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "How do black holes form from relativistic stars? This question is of great fundamental and practical importance in gravitational physics and general relativistic astrophysics. On the fundamental level, black holes are genuinely relativistic objects, and thus the study of their production involves questions about horizon dynamics, global structure of spacetimes, and the nature of the singularities predicted as a consequence of the occurrence of trapped surfaces. On the level of astrophysical applications, systems involving black holes are possible engines for highly energetic phenomena like AGNs or gamma-ray bursts, and also likely a comparatively strong source of gravitational radiation.\n\nThe most simple model of black hole formation from, say, cold neutron stars, is a fluid in spherically symmetric polytropic equilibrium moving on a sequence of increasing mass due to accretion [1] . This assumes that (i) the stellar structure and dynamics are represented reasonably by the ideal fluid equation of state and the polytropic stratification, (ii) accretion processes are slow compared to the dynamical timescales of the star, and (iii) rotation is negligible. Our focus has been to study the effects of relaxing the third assumption.\n\nIn spherical symmetry, the sequence of equilibrium polytropes has a maximum in the mass function M (ρ c ), where ρ c denotes the central rest-mass density of the polytrope. This maximum is connected to a change in the stability of the fundamental mode of oscillation [1] , and thus collapse sets in via a dynamical instability to radial deformations. During the subsequent evolution, a trapped tube forms at the center which traverses the stellar material entirely [2] .\n\nHow much of this behaviour is preserved when rotation is taken into account? Rotation is known to change the equilibrium structure of the star, and, in consequence, its modes of oscillation and set of unstable perturbations. The collapse might also lead to the formation of a massive disk around the new-born black hole, and finally only systems without spherical symmetry can be a source of gravitational radiation.\n\nNumerical simulations have been used to study the collapse and black hole formation of general relativistic rotating polytropic stars [3] . For the uniformly and moderately differentially rotating models investigated in those studies, the dynamical process is described by the instability of a quasi-radial mode and subsequent collapse of the star up to the formation of an accreting Kerr black hole at the star's center.\n\nWill strong differential rotation modify this picture? Even before our study, there was evidence that this should be the case. (i) Strong differential rotation can deform the high- density regions of a star into a toroidal shape, thus changing the equilibrium structure considerably. (ii) It admits stars of high normalized rotational energy T /|W | [1] which are stable to axisymmetric perturbations. (iii) It admits nonaxisymmetric instabilities, for example by the occurrence of corotation points [4] , at low values of T /|W | [5] . (iv) A barmode instability of the type found in Maclaurin spheroids [6] would likely express itself by the formation of two orbiting fragments if the initial high-density region has toroidal shape.\n\nThis last property has motivated us to ask this question: Can a bar deformation transform a strongly differentially rotating star into a binary black hole merger with a massive accretion disk? If so, this process might occur in supermassive stars if the timescales associated with angular momentum transport are too large to enforce uniform rotation.\n\nWe have investigated black hole formation in strongly differentially rotating, quasi-toroidal models of supermassive stars [7, 8] , and found that a non-axisymmetric instability can lead to the off-center formation of a trapped surface (see fig- ure ). An extensive parameter space study of this fragmentation instability [8] reveals that many quasi-toroidal stars of this kind are dynamically unstable in this manner, even for low values of T /|W |, and we have found evidence that the corotation mechanism observed by Watts et al. [4] might be active in these models. Since, on a sequence of increasing T /|W |, one of the low order m = 1 modes becomes dynamically un-stable before m = 2 and higher order modes, one would not expect a binary black hole system to form in many situations (although this may depend on the rotation law and details of the pre-collapse evolution as well). Rather, the off-center production of a single black hole with a massive accretion disk appears more likely.\n\nSince the normalized angular momentum J/M 2 of the initial model is greater than unity, there is another interesting consequence of this formation process: the resulting black hole, unless it is ejected from its shell, may very well be rapidly rotating, spun up by accretion of the material remaining outside the initial location of the trapped surface. Investigating the late time behaviour of this accretion process, estimating possible kick velocities of the resulting black hole, and finding the mass of the final accretion disk is, however, beyond our present-day capabilities and subject of future study."
}
] | |
arxiv:0704.0435 | 0704.0435 | 1 | 10.1088/0264-9381/24/17/009 | 89e4dcf26ccedb9a759319a45998444094d94075b23c34890556082636a15cdf | Type D Einstein spacetimes in higher dimensions | We show that all static spacetimes in higher dimensions are of Weyl types G,
I_i, D or O. This applies also to stationary spacetimes if additional
conditions are fulfilled, as for most known black hole/ring solutions. (The
conclusions change when the Killing generator becomes null, such as at Killing
horizons.) Next we demonstrate that the same Weyl types characterize warped
product spacetimes with a one-dimensional Lorentzian (timelike) factor, whereas
warped spacetimes with a two-dimensional Lorentzian factor are restricted to
the types D or O. By exploring the Bianchi identities, we then analyze the
simplest non-trivial case from the above classes - type D vacuum spacetimes,
possibly with a cosmological constant, dropping, however, the assumptions that
the spacetime is static, stationary or warped. It is shown that for ``generic''
type D vacuum spacetimes the corresponding principal null directions are
geodetic in any dimension (this applies also to type II spacetimes). For n>=5,
however, there may exist particular cases of type D spacetimes which admit
non-geodetic multiple principal null directions and we present such examples in
any n>=7. Further studies are restricted to five dimensions, where the type D
Weyl tensor is described by a 3x3 matrix \Phi_{ij}. In the case with
``twistfree'' (A_{ij}=0) principal null geodesics we show that in a ``generic''
case \Phi_{ij} is symmetric and eigenvectors of \Phi_{ij} coincide with those
of the expansion matrix S_{ij}, providing us with three preferred spacelike
directions of the spacetime. Similar results are also obtained when relaxing
the twistfree condition and assuming instead that \Phi_{ij} is symmetric. The
n=5 Myers-Perry black hole and Kerr-NUT-AdS metrics in arbitrary dimension are
briefly studied as specific examples of type D vacuum spacetime. | [
"Vojtech Pravda",
"Alena Pravdova",
"Marcello Ortaggio"
] | [
"gr-qc",
"hep-th"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | We show that all static spacetimes in higher dimensions n > 4 are necessarily of Weyl types G, I i , D or O. This applies also to stationary spacetimes provided additional conditions are fulfilled, as for most known black hole/ring solutions. (The conclusions change when the Killing generator becomes null, such as at Killing horizons, on which we briefly comment.) Next we demonstrate that the same Weyl types characterize warped product spacetimes with a onedimensional Lorentzian (timelike) factor, whereas warped spacetimes with a twodimensional Lorentzian factor are restricted to the types D or O. By exploring algebraic consequences of the Bianchi identities, we then analyze the simplest non-trivial case from the above classes -type D vacuum spacetimes, possibly with a cosmological constant, dropping, however, the assumptions that the spacetime is static, stationary or warped. It is shown that for "generic" type D vacuum spacetimes (as defined in the text) the corresponding principal null directions are geodetic in arbitrary dimension (this in fact applies also to type II spacetimes). For n ≥ 5, however, there may exist particular cases of type D vacuum spacetimes which admit non-geodetic multiple principal null directions and we explicitly present such examples in any n ≥ 7. Further studies are restricted to five dimensions, where the type D Weyl tensor is fully described by a 3 × 3 real matrix Φ ij . In the case with "twistfree" (A ij = 0) principal null geodesics we show that in a "generic" case Φ ij is symmetric and eigenvectors of Φ ij coincide with eigenvectors of the expansion matrix S ij providing us thus in general with three preferred spacelike directions of the spacetime. Similar results are also obtained when relaxing the twistfree condition and assuming instead that Φ ij is symmetric. The five dimensional Myers-Perry black hole and Kerr-NUT-AdS metrics in arbitrary dimension are also briefly studied as specific illustrative examples of type D vacuum spacetimes.
Algebraically special spacetimes play an essential role in the field of exact solutions of Einstein's equations and many known exact solutions in four dimensions are indeed algebraically special [1] . Recently a generalization of the Petrov classification to higher dimensions was developed in [2, 3] and it turned out that many higher-dimensional solutions of Einstein's equations are algebraically special as well (see e.g. [4] ), in fact ‡ Now at: Departament de Física Fonamental, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain so far there is only one known solution identified [5] as algebraically general -the static charged black ring [6] .
There is, however, one important difference between four dimensional and n > 4 dimensional cases -the Goldberg-Sachs theorem does not hold in higher dimensions. Recall that for n = 4 the Goldberg-Sachs theorem implies that principal null directions of an algebraically special vacuum spacetime are necessarily geodetic and shearfree. It was stressed already in [7, 8] that the Goldberg-Sachs theorem cannot be straightforwardly extended to higher dimensions. Namely in [7] it was pointed out that principal null directions (or Weyl aligned null directions -WANDs [2] ) of the n = 5 Myers-Perry black holes [9] are shearing though the spacetime is of type D. In [8] it was shown that in fact all vacuum, n > 4, type N and III expanding spacetimes are shearing. In [10] it was also shown that for n > 4, n odd, all geodetic WANDs with non-vanishing twist are again shearing.
In this paper we study various properties of algebraically special vacuum spacetimes, such as geodeticity of multiple WANDs (not guaranteed in higher dimensions -another "violation" of the Goldberg-Sachs theorem) and relationships between optical matrices S ij and A ij and the Weyl tensor. Before approaching these problems, we study in the first part of the paper (sections 3 and 4) constraints on Weyl types of the spacetime following from various assumptions on the geometry.
In section 3 we show that in arbitrary dimension (i.e., hereafter, n ≥ 4) the only Weyl types compatible with static spacetimes (and expanding stationary spacetimes with appropriate reflection symmetry) are types G, I i , D and O.
In section 4 we study direct or warped product spacetimes. It turns out that warped spacetimes with a one-dimensional Lorentzian factor are again of types G, I i , D and O and that warped spacetimes with a two-dimensional Lorentzian factor are necessarily of type D or O. This also implies that spherically symmetric spacetimes are of type D or O.
It follows that type D spacetimes play an important role as the simplest non-trivial case compatible with the above mentioned assumptions. Therefore, in the second part of the paper (sections 5 and 6) we focus on studying properties of type D Einstein spacetimes (i.e., vacuum with an arbitrary cosmological constant), dropping, however, the assumptions that the spacetime is static, stationary or warped.
In section 5 we study type D spacetimes in arbitrary dimension and analyze geodeticity of WANDs. It turns out that in a "generic" case in vacuum the multiple WANDs are geodetic. Let us also point out that negative boost weight Weyl components do not enter relevant equations and thus the same results also hold for multiple WANDs in type II Einstein spacetimes. Surprisingly, it also turns out that explicit examples of special vacuum type D spacetimes not belonging to our "generic" class and admitting non-geodetic multiple WANDs can easily be constructed. Such examples for arbitrary dimension n ≥ 7 are given in section 5. 4 . This shows that there exist even more striking "violations" of the Goldberg-Sachs theorem in higher dimensions than the examples with non-zero shear discussed above. In section 5 we also study various properties of shearfree type D vacuum spacetimes.
Perhaps not surprisingly, the situation in five dimensions is considerably simpler than for n > 5. In fact it turns out that for n = 5 the Weyl tensor of type D is fully determined by a 3 × 3 real matrix Φ ij . At the same time, five dimensional gravity is already an interesting arena where qualitatively new phenomena appear. We thus devote section 6 to five dimensional vacuum type D spacetimes. We study relationships between the Weyl tensor represented by Φ ij and optical matrices S ij and A ij . One of the results is that for "generic" spacetimes with non-twisting WANDs (A ij = 0) the antisymmetric part of Φ ij , Φ A ij , vanishes and the symmetric part Φ S ij is aligned with S ij (in the sense that the matrices Φ S ij and S ij can be diagonalized together). Similarly, in the "generic" case the condition Φ A ij = 0 implies vanishing of A ij . Again, there exist particular cases for which the "generic" proof does not hold, see section 6 for details. In this section a simple explicit example of a five-dimensional vacuum type D spacetime, the Myers-Perry metric, is also presented and S ij , A ij , Φ S ij , and Φ A ij are explicitly given.
Finally in section 7 we concisely summarize main results and in the Appendix we briefly study geometric optics of type D Kerr-NUT-AdS metrics in arbitrary dimension.
Let us first briefly summarize our notation, further details can be found in [8] . In an n-dimensional spacetime let us introduce a frame of n real vectors m (a) (a, b . . . = 0, . . . , n -1): two null vectors
m (0) = m (1) = n, m (1) = m (0) = ℓ and n -2 orthonormal spacelike vectors m (i) = m (i) (i, j . . . = 2, . . . , n -1) satisfying ℓ a ℓ a = n a n a = ℓ a m (i) a = n a m (i) a = 0, ℓ a n a = 1, m (i)a m (j) a = δ ij . (1)
The metric now reads
g ab = 2ℓ (a n b) + δ ij m (i) a m (j) b . (2)
We will use the following decomposition of the covariant derivative of the vector ℓ and the covariant derivative in the direction of ℓ
ℓ a;b = L cd m (c) a m (d) b , D ≡ ℓ a ∇ a . (3)
Note that ℓ is geodetic iff L i0 = 0 and for an affine parameterization also L 10 = 0. We will often use the symmetric and antisymmetric parts of
L ij , S ij ≡ L (ij) (its trace S ≡ S ii ), A ij ≡ L [ij]
. In case of geodetic ℓ, the trace of S ij represents expansion θ ≡ 1 n-2 S, the tracefree part of S ij is shear σ ij ≡ S ij -θδ ij and the antisymmetric matrix A ij is twist. § Optical scalars can be expressed in terms of ℓ (when
L i0 = 0 = L 10 ) σ 2 ≡ σ ij σ ji = ℓ (a;b) ℓ (a;b) -1 n-2 ℓ a ;a 2 , θ = 1 n-2 ℓ a ;a , ω 2 ≡ A ij A ij = ℓ [a;b] ℓ a;b . (4)
The decomposition of the Weyl tensor in the frame (1) in full generality is given by [8]
C abcd = 4C 0i0j n {a m (i) b n c m (j) d} + 8C 010i n {a ℓ b n c m (i) d} + 4C 0ijk n {a m (i) b m (j) c m (k) d} + 4C 0101 n {a ℓ b n c ℓ d} + 4C 01ij n {a ℓ b m (i) c m (j) d} + 8C 0i1j n {a m (i) b ℓ c m (j) d} + C ijkl m (i) {a m (j) b m (k) c m (l) d} + 8C 101i ℓ {a n b ℓ c m (i) d} + 4C 1ijk ℓ {a m (i) b m (j) c m (k) d} + 4C 1i1j ℓ {a m (i) b ℓ c m (j)
b} , where the operation { } is defined as
w {a x b y c z d} ≡ 1 2 (w [a x b] y [c z d] + w [c x d] y [a z b] ).
§ For the sake of brevity, throughout the paper we shall refer to the corresponding quantities for nongeodetic congruences as "expansion", "shear", and "twist" (in inverted commas), bearing in mind that in that case expressions (4) do not hold.
In the second part of this paper we will focus on type D spacetimes, possessing (in an adapted frame) only boost order zero components (see [8] ) C 0101 , C 01ij , C 0i1j , C ijkl . For simplicity let us define the (n -2) × (n -2) real matrix
Φ ij ≡ C 0i1j , (5)
with Φ S ij , Φ A ij , and Φ ≡ Φ ii being the symmetric and antisymmetric parts of Φ ij and its trace, respectively. Let us observe that for static spacetimes and for a large class of warped geometries one has Φ A ij = 0 (see section 4). Note also that the above mentioned boost order zero components of the Weyl tensor are not completely independent. In fact from the symmetries and the tracelessness of the Weyl tensor (cf. eqs. ( 7 ) and ( 9 ) in [8] ) it follows that
C 01ij = 2C 0[i|1|j] = 2Φ A ij , C 0(i|1|j) = Φ S ij = -1 2 C ikjk , C 0101 = -1 2 C ijij = Φ. (6)
The type D Weyl tensor is thus completely determined by m(m-1)
2 independent components of Φ A ij and m 2 (m 2 -1)
12 independent components of C ijkl , where n = m -2.
Algebraically special spacetimes in higher dimensions are characterized by the existence of preferred null directions -Weyl aligned null directions (WANDs). A necessary and sufficient condition for a null vector ℓ being WAND in arbitrary dimension is [3, 11]
ℓ b ℓ c ℓ [e C a]bc[d ℓ f ] = 0, (7)
where C abcd is the Weyl tensor. Let us now assume that a spacetime of interest is algebraically special and thus the equation ( 7 ) possesses a null solution ℓ = (ℓ t , ℓ A ), A = 1 . . . n -1 (note that necessarily ℓ t = 0 and at least one of the remaining components is also non-zero).
For static spacetimes the metric does not depend on the direction of time and consequently the form of the metric and of the Weyl tensor remains unchanged under the transformation t = -t. Therefore, in these new coordinates equation (7) has the same form as in the original coordinates and admits a second solution ñ = (ℓ t , ℓ A ). In the original coordinates n = (-ℓ t , ℓ A ). Thus for static spacetimes the existence of a WAND ℓ implies the existence of a distinct WAND n which in fact has the same order of alignment. The only Weyl types compatible with this property are types G, I i and D (or, trivially, O, i.e. conformally flat spacetimes). Therefore Proposition 1 All static spacetimes in arbitrary dimension are of Weyl types G, I i or D, unless conformally flat.
In fact explicit examples of static spacetimes of these Weyl types are knowncharged static black ring (type G - [5] ), vacuum static black ring (type I i - [11] ), the Schwarzschild-Tangherlini black hole (type D - [8] ) and the Einstein universe R×S n-1 (type O -cf. the results summarized in section 4). Cf. also the static examples given in [4] .
In the standard n = 4 (i.e., m = 2) case these are essentially the imaginary and real part of Ψ 2 . More specifically, with the conventions of [1] , one has Φ Note that in four dimensions there is no type G and type I is equivalent to type I i [2, 3] . Thus for n = 4 only types I, D and O are compatible with static spacetimes. This was discussed already in [12] in the case of static, n = 4, vacuum spacetimes (see also additional comments in [13] and in section 6.2 of [1] ).
S ij = 1 2 Φδ ij with Φ = -2Re(Ψ 2 ), Φ A 23 = Φ 23 = -Im(Ψ 2 )
One can use the same arguments as above for stationary spacetimes with the metric remaining unchanged under reflection symmetry involving time and some other coordinates. E.g. in Boyer-Lindquist coordinates the Kerr metric remains unchanged under t = -t, φ = -φ and n = 5 Myers-Perry under t = -t, φ = -φ, ψ = -ψ or, for general dimension, Myers-Perry under t = -t, φi = -φ i . Note, however, that in contrast to the static case, in some special stationary cases one could in principle get from the original WAND ℓ a "new" WAND n = -ℓ which represents the same null direction. In order to deal with these special cases we note that the "divergence scalar" (or, loosely speaking, "expansion", since it does coincide with the standard expansion scalar in the case of geodetic, affinely parameterized null directions) of both WANDs n and ℓ related by reflection symmetry is the same (as well as all the other optical scalars and the geodeticity parameters -this also applies to the static case), i.e. ℓ a ;a = n a ;a while the "expansion" of -ℓ is equal to -ℓ a ;a . Therefore for all "expanding" spacetimes n = -ℓ. Thus Proposition 2 In arbitrary dimension, all stationary spacetimes with non-vanishing divergence scalar ("expansion") and invariant under appropriate reflection symmetry are of Weyl types G, I i or D, unless conformally flat.
Note also that it is shown in [14] that Kerr-Schild spacetimes with the assumption R 00 = 0 are of type II (or more special) in arbitrary dimension with the Kerr-Schild vector being the multiple WAND. Therefore all Kerr-Schild spacetimes that are either static or belong to the above mentioned class of stationary spacetimes are necessarily of type D. In particular, the Myers-Perry metric in arbitrary dimension is thus of type D. ¶
In addition to the rotating Myers-Perry black holes for n ≥ 4, of type D, we can mention a number of physically relevant solutions as explicit examples of spacetimes subject to Proposition 2. + First, rotating vacuum black rings [17] , of type I i [11] . To our knowledge, no stationary (non-static) type G solution has been so far explicitly identified. It is, however, plausible to expect that a rotating charged black ring (so far unknown in the standard Einstein-Maxwell theory) will be of type G as its static counterparts. Further interesting examples fulfilling our assumptions are expanding ¶ This was already known in the case n = 5 [4, 8] . Furthermore, it has been demonstrated recently in [15] by explicit computation of the full curvature tensor that the family [16] of higher dimensional rotating black holes with a cosmological constant and NUT parameter is of type D for any n. We observe in addition that, using the connection 1-forms given in [15] , it is also straightforward to show (see the Appendix) that the mutiple WANDs (which are related by reflection symmetry) of all such solutions are twisting, expanding and shearing (except that the shear vanishes for n = 4). The fact that the WANDs found in [15] are complex is only due to the analytical continuation trick used in [16] to cast the line element in a nicely symmetric form -the WANDs of the associated "physical" spacetimes are thus real after Wick-rotating back one of the coordinates. + It is straightforward to verify the "reflexion symmetry" of the metric we mention in this context. The "expansion" condition, instead, has not been verified explicitly in all cases. However, it is plausible that these spacetimes are indeed "expanding" since they contain as special limits or subcases solutions with expansion, e.g. Myers-Perry black holes (cf. section 6.4, [8] and the preceding footnote).
stationary axisymmetric spacetimes with n -2 commuting Killing vector fields [18] , which contain, apart from the (n = 5) black holes/rings mentioned above, also e.g. the recently obtained "black saturn" [19] , doubly spinning black rings [20] and black di-rings [21] . In any dimension also rotating uniform black strings/branes satisfy the assumptions of Proposition 2 (see section 4), and so does the ansatz recently used in [22] for the numerical construction of corresponding n = 6 non-uniform solutions. Other examples are all the stationary solutions discussed in [4] and various black ring solutions reviewed in [23] .
First, it is worth observing that we have not used any field equations for the gravitational field in the considerations presented above and the results are thus purely geometrical.
Note that one can not relax the assumption ℓ a ;a = 0 in the case of stationary spacetimes. For example, the special pp -wave metric ds 2 = g ij dx i dx j -2dudv-2Hdu 2 such that H ,u = 0 (note that it is always H ,v = 0 by the definition of pp -waves) and ∂ u • ∂ u = -2H < 0 represents stationary spacetimes (cf., e.g., [24] for the n = 4 vacuum case) that are invariant under reflection symmetry (ũ = -u, ṽ = -v) and yet of type N [25] . In fact, the geodetic multiple WAND ℓ = ∂ v is non-expanding (and n = -ℓ is not a new WAND).
Furthermore, if we assume a null Killing vector field k instead of a timelike one we are led to different conclusions. Namely, it is easy to show that k must be geodetic, shearfree and non-expanding, which for R ab k a k b = 0 implies that k is a twistfree WAND [10] . We thus end up with a subfamily of the Kundt class, for which (under the alignment requirement R ab k a ∝ k b , obeyed e.g. in vacuum) the algebraic type is II or more special [10] (cf. section 24.4 of [1] for n = 4). In particular, a similar argument applies locally at Killing horizons, where the type must thus be again II or more special (provided R ab k a ∝ k b ). * This is in agreement with the result of [26] for generic isolated horizons. As an explicit example, vacuum black rings (which are of type I i in the stationary region) become locally of type II on the horizon [11] .
Finally, spacelike Killing vectors do not impose any constraint on the algebraic type of the Weyl tensor, in general, and all types are in fact possible. For example charged static black rings are of type G, vacuum black rings of type I i , vacuum black holes of type D, and they all admit at least one spacelike Killing vector; Kundt spacetimes can be constructed that admit axial symmetry with all types II, D, III and N being possible (see, e.g., [1] for n = 4).
In this section we show that the algebraic types discussed above also characterize certain classes of direct/warped product geometries of physical relevance. In addition we discuss some optical properties of these spacetimes.
* The proof is a bit more tricky in this case since the Killing vector is null only at the horizon.
Still, one can adapt techniques used in [26, 27] for related investigations. Note that the horizon of higher dimensional stationary black holes is indeed a Killing horizon (at least in the non-degenerate case) [27] .
Let us consider two (pseudo-)Riemannian spaces (M 1 , g (1) ) and (M 2 , g (2) ) of dimension n 1 and n 2 (n 1 , n 2 ≥ 1 and n 1 + n 2 ≥ 4), parameterized by coordinates x A (A, B = 0, . . . , n 1 -1) and x I (I, J = n 1 , . . . , n 1 + n 2 -1), respectively. Using adapted coordinates x µ (µ, ν = 0, . . . , n 1 + n 2 -1) constructed from the coordinates x A of M 1 and x I of M 2 , we define the direct product (M, g) to be the product manifold
M = M 1 × M 2 , of dimension n = n 1 + n 2 , equipped with the metric tensor g(x µ ) = g (1) (x A ) ⊕ g (2) (x I ) defined (locally) by g AB = g (1)AB , g IJ = g (2)IJ , g AI = 0.
For the sake of definiteness, we shall assume hereafter that (M 1 , g 1 ) is Lorentzian and (M 2 , g 2 ) is Riemannian.
In general, any geometric quantity which can be split like the product metric (i.e., with no mixed components and with the A[I] components depending only on the x A [x I ] coordinates) is called a "product object" (or "decomposable"). Various interesting geometrical properties then follow [28] and, in particular, the Riemann and Ricci tensors and the Ricci scalar are all decomposable. As a consequence, a product space is an Einstein space iff each factor is an Einstein space and their Ricci scalars satisfy R (1) /n 1 = R (2) /n 2 [28] .
Using the above coordinates it follows from the standard definition that the mixed components of the Weyl tensor are given by
C ABCI = C ABIJ = C AIJK = 0, (8)
C AIBJ = - 1 n -2 g (1)AB R (2)IJ + g (2)IJ R (1)AB + R (1) + R (2) (n -1)(n -2) g (1)AB g (2)IJ , (9)
where
R (1)AB [R (2)IJ ] is the Ricci tensor of (M 1 , g 1 ) [(M 2 , g 2 )]
. For the non-mixed components one has to distinguish the special cases n 1 = 1, 2 (and the "symmetric" cases n 2 = 1, 2, which we omit for brevity). If n 1 = 1 there are of course no non-mixed components C ABCD since now the x A span a one-dimensional space. If n 1 = 2 there is only one independent component, i.e. C 0101 (notice that here, exceptionally, 0 and 1 are not frame indices but refer to the coordinates x 0 and x 1 in the factor space M 1 ).
For n 1 ≥ 3,
C ABCD = C (1)ABCD + 2n 2 (n -2)(n 1 -2) g (1)A[C R (1)D]B -g (1)B[C R (1)D]A + 2 (n -1)(n -2) R (2) -R (1) n 2 (n 2 + 2n 1 -3) (n 1 -1)(n 1 -2) g (1)A[C g (1)D]B (n 1 ≥ 3), ( 10
)
where C (1)ABCD is the Weyl tensor of (M 1 , g 1 ), whereas the remaining non-mixed components are given for any n 1 ≥ 1 by
C IJKL = C (2)IJKL + 2n 1 (n -2)(n 2 -2) g (2)I[K R (2)L]J -g (2)J[K R (2)L]I + 2 (n -1)(n -2) R (1) -R (2) n 1 (n 1 + 2n 2 -3) (n 2 -1)(n 2 -2) g (2)I[K g (2)L]J (n 2 ≥ 3), ( 11
)
where C (2)IJKL is the Weyl tensor of (M 2 , g 2 ). It is thus obvious that the Weyl tensor is not decomposable, in general. It turns out that the Weyl tensor is decomposable iff both product spaces are Einstein spaces and n 2 (n 2 -1)R (1) + n 1 (n 1 -1)R (2) = 0 (the latter condition is identically satisfied whenever n 1 = 1 or n 2 = 1, while for
n 1 = 2 [n 2 = 2] it implies that (M 1 , g 1 ) [(M 2 , g 2 )] must be of constant curvature).
When the Weyl tensor is decomposable the only non-vanishing components take the simple form C ABCD = C (1)ABCD , C IJKL = C (2)IJKL . Therefore, in particular, the product space is conformally flat iff both product spaces are of constant curvature and
n 2 (n 2 -1)R (1) + n 1 (n 1 -1)R (2) = 0.
Determining the possible algebraic types of the Weyl tensor requires considering various possible choices for the dimension n 1 of the Lorentzian factor.
If n 1 = 1, the full metric can always be cast in the special static form ds 2 = -dt 2 + g IJ dx I dx J . Recalling the result of section 3, the Weyl tensor can thus only be of type G, I i , D or O. In particular, one can show that C 0i1j = C 0j1i , so that for direct product spacetimes with n 1 = 1 one has Φ A ij = 0 identically. If n 1 ≥ 2, it is convenient to adapt the null frame (1) to the natural product structure, so that
g ab = 2ℓ (a n b) + δ Â B m ( Â) a m ( B) b + δ Î Ĵ m ( Î) a m ( Ĵ) b
(where Â, B = 2, . . . , n 1 -1, Î, Ĵ = n 1 , . . . , n -1 are now frame indices, and the frame vectors do not have mixed coordinate components, e.g. ℓ I = 0 = n I etc.). From (10) and (11) it thus follows that C ABCD and C IJKL do not give rise to mixed frame components, and from (9) that C AIBJ does not give rise to non-mixed frame components. Hence the only non-vanishing mixed components are (ordered by boost weight)
C 0 Î0 Ĵ = - 1 n -2 R (1)00 δ Î Ĵ , C 0 Î Â Ĵ = - 1 n -2 R (1)0 Âδ Î Ĵ , C 0 Î1 Ĵ = - 1 n -2 R (2) Î Ĵ + R (1)01 δ Î Ĵ + R (1) + R (2) (n -1)(n -2) δ Î Ĵ , C Â Î B Ĵ = - 1 n -2 R (2) Î Ĵ δ Â B + R (1) Â B δ Î Ĵ + R (1) + R (2) (n -1)(n -2) δ Â B δ Î Ĵ , (12)
C 1 Î Â Ĵ = - 1 n -2 R (1)1 Âδ Î Ĵ , C 1 Î1 Ĵ = - 1 n -2 R (1)11 δ Î Ĵ .
The non-mixed frame components are given for n 1 = 2 by
C 0101 = - 1 2(n 2 + 1) (n 2 -1)R (1) + 2R (2) n 2 (n 1 = 2), (13)
and for n 1 ≥ 3 by
C 0 Â0 B = C (1)0 Â0 B + n 2 (n -2)(n 1 -2) R (1)00 δ Â B , C 010 Â = C (1)010 Â - n 2 (n -2)(n 1 -2) R (1)0 Â, C 0 Â B Ĉ = C (1)0 Â B Ĉ - 2n 2 (n -2)(n 1 -2) R (1)0[ Ĉ δ B] Â, C 0101 = C (1)0101 - 2n 2 (n -2)(n 1 -2) R (1)01 - 1 (n -1)(n -2) R (2) -R (1) n 2 (n 2 + 2n 1 -3) (n 1 -1)(n 1 -2) , C 01 Â B = C (1)01 Â B (n 1 ≥ 3), ( 14
)
C 0 Â1 B = C (1)0 Â1 B + n 2 (n -2)(n 1 -2) R (1) Â B + R (1)01 δ Â B + 1 (n -1)(n -2) R (2) -R (1) n 2 (n 2 + 2n 1 -3) (n 1 -1)(n 1 -2) δ Â B , C Â B Ĉ D = C (1) Â B Ĉ D + 2n 2 (n -2)(n 1 -2) R (1) B[ Dδ Ĉ] Â -R (1) Â[ Dδ Ĉ] B + 2 (n -1)(n -2) R (2) -R (1) n 2 (n 2 + 2n 1 -3) (n 1 -1)(n 1 -2) δ B[ Dδ Ĉ] Â, C Î Ĵ K L = C (2) Î Ĵ K L + 2n 1 (n -2)(n 2 -2) δ Î[ K R (2) L] Ĵ -δ Ĵ[ K R (2) L] Î + 2 (n -1)(n -2) R (1) -R (2) n 1 (n 1 + 2n 2 -3) (n 2 -1)(n 2 -2) δ Î[ K δ L] Ĵ , C 1 Â B Ĉ = C (1)1 Â B Ĉ - 2n 2 (n -2)(n 1 -2) R (1)1[ Ĉ δ B] Â, C 101 Â = C (1)101 Â - n 2 (n -2)(n 1 -2) R (1)1 Â, C 1 Â1 B = C (1)1 Â1 B + n 2 (n -2)(n 1 -2) R (1)11 δ Â B .
(The expression for C Î Ĵ K L holds only when n 2 ≥ 3, while for n 2 = 2 one gets only one component C 2323 similar to (13).)
For n 1 = 2 the Weyl tensor of (M 1 , g 1 ) of course vanishes, and in addition we have R (1)00 = 0 = R (1)11 identically (any 2-space satisfies 2R (1)AB = R (1) g (1)AB ). Therefore among the above components ( 12 ) and ( 13 ) only the boost weight zero components C 0 Î1 Ĵ and C 0101 survive, so that the corresponding spacetime can be only of type D (or conformally flat), and ℓ and n, as chosen above, are multiple WANDs. Note also that Φ ij reduces to Φ Î Ĵ = C 0 Î1 Ĵ = C 0 Ĵ 1 Î in this case, therefore Φ A ij = 0. As an example, the higher dimensional electric Bertotti-Robinson solutions fall in this class, cf., e.g, [29, 30] .
For n 1 = 3, again the Weyl tensor of (M 1 , g 1 ) vanishes. With the additional assumption that (M 1 , g 1 ) is Einstein, we get R (1)00 = R (1)11 = R (1)0  = R (1)1  = 0 (here  = 2 only), and as above the Weyl tensor is of type D with Φ A ij = 0. Similarly, for any n 1 > 3, if (M 1 , g 1 ) is an Einstein space the only non-zero mixed Weyl components (12) will have boost weight zero, and the non-mixed components (14) simplify considerably. As a particular consequence, if (M 1 , g 1 ) is an Einstein space of type D, (M, g) will also be of type D (but now Φ A ij = 0, in general) -this is the case, for example, of uniform black strings/branes (either static or rotating, see also the discussion concluding this section). If (M 1 , g 1 ) is of constant curvature, (M, g) will be of type D with Φ A ij = 0 (or O) -this includes the higher dimensional magnetic Bertotti-Robinson solutions [29] . One can consider other special cases using similar simple arguments.
A spacetime conformal to a direct product spacetime is called a warped product spacetime if the conformal factor depends only on one of the two coordinate sets x A , x I (see e.g. [1] ). Obviously, the algebraic type of two conformal spaces is the same.♯ Some of the results presented above can thus be straightforwardly generalized to the more general case of warped products. For example, Proposition 3 In arbitrary dimension, a warped spacetime with a one-dimensional Lorentzian (timelike) factor can be only of type G, I i , D (with Φ A ij = 0) or O. This case includes, in particular, the conclusion of section 3 for static spacetimes. As warped non-static/non-stationary examples we can mention the de Sitter universe (in global coordinates) and FRW cosmologies. For n = 4 Proposition 3 reduces to a result of [32] . Furthermore, Proposition 4 In arbitrary dimension, a warped spacetime with a two-dimensional Lorentzian factor can be only of type D (with Φ A ij = 0) or O. Cf. again [32] for n = 4. Notice that in this case the line element can always be cast in one of the two (conformally related) forms ds 2 = 2A(u, v)dudv + f (u, v)h IJ (x)dx I dx J or ds 2 = 2 f (x)A(u, v)dudv + g IJ (x)dx I dx J (so that multiple WANDs are given by ∂ u and ∂ v ), which include a number of known spacetimes. For example, the first possibility includes all spherically symmetric spacetimes, hence as a special case of Proposition 4 we have Proposition 5 In arbitrary dimension, a spherically symmetric spacetime is of type D (with Φ A ij = 0) or O. For n = 4 this has been known for a long time (see e.g. [33] and sections 15.2, 15.3 of [1] ), and in this case Φ A ij = 0 means that Ψ 2 is real (see the footnote on p. 4). For n > 4 this result has been proven in [34] in the static case.
Other properties of decomposable Weyl tensors were discussed in [2] .
Let us define an n-dimensional spacetime (M, g) as the warped product of an n 1dimensional Lorentzian space (M 1 , g (1) ) and an n 2 -dimensional Riemannian space (M 2 , g (2) ), with n = n 1 + n 2 as in the preceding subsection. Hereafter we shall assume n 1 ≥ 2. Using the adapted coordinates defined above, the metric can take one of the following two forms
ds 2 = g AB dx A dx B + f (x A )h IJ dx I dx J , (15)
ds 2 = f (x I )h AB dx A dx B + g IJ dx I dx J , ( 16
)
where g AB , h AB = g (1)AB depend only on the x A coordinates and g IJ , h IJ = g (2)IJ only on the x I coordinates. Given a null vector ℓ (1) = ℓ A (1) ∂ A of M 1 , this can be "lifted" to define a null vector ℓ of M with covariant components ℓ A = ℓ (1)A (functions of the x A only) and ℓ I = 0. From equations ( 15 ), ( 16 ) it follows that if ℓ (1) is geodetic (and affinely parameterized) in M 1 then ℓ is automatically geodetic (and affinely parameterized) in M . We can thus "compare" the optical scalars of ℓ (1) in M 1 with those of ℓ in M . For the warped metric (15) , with the definitions (4) one finds
σ 2 = σ 2 (1) + (n 1 -2)n 2 n 1 + n 2 -2 θ (1) - 1 2 (ln f ) ,A ℓ A 2 , θ = 1 n 1 + n 2 -2 (n 1 -2)θ (1) + n 2 2 (ln f ) ,A ℓ A , ( 17
)
ω 2 = ω 2 (1)
, where σ 2
(1) , θ (1) and ω 2 (1) are the optical scalars of ℓ (1) in (M 1 , g (1) ). For the warped metric ( 16 ) one has
σ 2 = f -2 σ 2 (1) + (n 1 -2)n 2 n 1 + n 2 -2 θ 2 (1) , θ = n 1 -2 n 1 + n 2 -2 f -1 θ (1) , (18)
ω 2 = f -2 ω 2 (1)
.
The special case of direct products is recovered for f, f = const. (which can be rescaled to 1), in which case the shear of the full spacetime originates in the shear and expansion of the Lorentzian factor (while expansion and twist are essentially the same as in (M 1 , g (1) )).
Note that for n 1 = 2 the definitions (4) for σ 2 (1) and θ (1) become formally singular because of the normalization, but for a Lorentzian 2-space (e.g., ds 2 = 2A(u, v)dudv with the geodetic null vector ℓ = A -1 ∂ v ) one has ℓ (a;b) ℓ (a;b) = ℓ a ;a = ℓ [a;b] ℓ a;b = 0, so that we can essentially take σ 2
(1) = θ (1) = ω (1) = 0 and formulae ( 17 ), ( 18 ) still hold. The results of this section can be applied to several known solutions. For example, static [rotating] black strings and branes (i.e, direct products of Schwarzschild [Kerr] cross a flat space) are type D vacuum spacetimes with two shearing, expanding, twistfree [twisting] multiple WANDs. As such, they clearly "violate" the Golberg-Sachs theorem. In addition, spherically symmetric solutions in any dimensions (which necessarily take the metric form (15) with n 1 = 2) are type D spacetimes with two shearfree, expanding, twistfree multiple WANDs (independently of any specific field equations; in the "exceptional case" (ln f ) ,A ℓ A = 0 the vector ℓ is non-expanding, e.g. for Bertotti-Robinson/Nariai geometries, or for null generators of horizons).
From the results of the previous sections it follows that type D spacetimes are the simplest non-trivial examples of static/stationary ("expanding" and with an appropriate reflection)/warped spacetimes. Therefore we will focus on type D spacetimes in general (without assuming staticity etc.).
Recall that the quantities/symbols used below (e.g. Φ ij , L ij , D) are defined in section 2.
Various contractions of Bianchi identities R abcd;e + R abde;c + R abec;d = 0 (19) lead to a set of first-order PDEs for frame components of the Riemann tensor given in Appendix B of [8] . In the following we shall concentrate on Einstein spaces (defined by R ab = R n g ab ), for which the same set of equations holds unchanged also for components of the Weyl tensor. In case of algebraically special spacetimes, some of these differential equations reduce to algebraical equations due to the vanishing of some components of the Weyl tensor. Here we derive algebraic conditions following from the Bianchi equations for type D Einstein spacetimes. These conditions will be employed in subsequent sections.
In particular, by contracting (19) with m (i) , ℓ, m (j) , m (k) and ℓ (equation (B.8) in [8] ) and assuming to have a type D Einstein space we get the first algebraic condition
Φ ij L k -Φ ik L j + 2Φ A kj L i -C isjk L s = 0, (20)
where we denoted L i0 by L i . We will also denote L i L i by L.
The second algebraic equation follows from equation (B.15, [8] )
0 = 2 Φ A jk L im + Φ A mj L ik + Φ A km L ij + Φ ij A mk + Φ ik A jm + Φ im A kj + C isjk L sm + C ismj L sk + C iskm L sj (21)
and contraction of k with i leads to
0 = SΦ A mj + ΦA jm -(Φ S mi + Φ A mi )S ij + (Φ S ji + Φ A ji )S im + 2(Φ A im A ij -Φ A ij A im ) + 1 2 C ismj A si . (22)
By contracting m with j in equation (B.12) from [8] we get
2DΦ S ik = 4Φ A ij A kj + Φ kj L ij + Φ ji L jk -Φ ki S -ΦL ik -2Φ S is L sk -2Φ S sk s M i0 -2Φ S is s M k0 +C ijks L sj , (23)
where we employed
C iskj s M j0 +C ijks s M j0 = 0 ( s M j0 + j M s0 = 0, cf. [8]
). The symmetric part of equation (B.5, [8] ) and equation (B.3) (that is equivalent to the antisymmetric part of (B.5)) give, respectively,
2DΦ S ik = -2ΦS ik + (-2Φ is + Φ si )L sk + (-2Φ ks + Φ sk )L si -2Φ S sk s M i0 -2Φ S is s M k0 , (24)
2DΦ A ik = -2ΦA ik +(-2Φ is +Φ si )L sk -(-2Φ ks +Φ sk )L si -2Φ A sk s M i0 +2Φ A si s M k0 .( 25
)
By subtracting ( 24 ) from ( 23 ) we finally obtain the third algebraic equation 26 ) Its antisymmetric part is, thanks to C ikjm A mj = 2C ijks A sj , equal to equation (22) and its symmetric part reads
0 = -Φ ki S + ΦL ki + Φ kj L ij + 4Φ A ij A kj + (2Φ kj -Φ jk )L ji + 2Φ A ij L jk + C ijks L sj .(
0 = -SΦ S ik + ΦS ik + Φ S ij S jk + Φ S kj S ij + 3(Φ A ij S jk + Φ A kj S ji ) + C ijks S sj . (27)
Equations ( 20 ), ( 22 ) and ( 27 ) will be extensively used in the following sections.
In passing, let us observe here in what sense the n = 4 case is unique. Recalling the footnote on p. 4, from (20) we get L i = 0 (geodetic property) unless Φ ij = 0 (trivial case of zero Weyl tensor); equation ( 22 ) is identically satisfied (noting that necessarily Φ A ij ∝ A ij when n = 4); equation ( 27 ) implies S ij ∝ δ ij (vanishing shear) again unless Φ ij = 0. Thus for n = 4 we correctly recover the standard Goldberg-Sachs result (here restricted to type D spacetimes) that multiple WANDs (PNDs) are geodetic and shearfree in vacuum (and Einstein) spaces [1] . The situation in higher dimensions, which is qualitatively different from the n = 4 case, is studied in the following sections.
In this section we study equation (20) in order to determine under which circumstances the multiple WAND ℓ is geodetic.
By contracting i with k in (20) and using (6) we get
3Φ A ij -Φ S ij L i = ΦL j (28)
and after multiplying (28) by L j we obtain
Φ S ij L i L j = -ΦL. (29)
By multiplying (20) by L i L j and using (29) we get
L 3Φ A ik L i + Φ S ik L i + ΦL k = 0. ( 30
)
Thus either L = 0 or
-3Φ A ij -Φ S ij L i = ΦL j . (31)
By adding and subtracting ( 28 ) and ( 31 ) we get
Φ S ij L i = -ΦL j , Φ A ij L i = 0. (32)
Finally multiplying (20) by L i and using (32) we get
LΦ A ij = 0. (33)
This implies that for a type D vacuum spacetime with non-vanishing Φ A ij in arbitrary dimension corresponding WANDs are geodetic.
In the case with vanishing Φ A ij , let us choose a frame in which
Φ S ij is diagonal Φ S ij = diag{p (2) , p (3)
, . . . , p (n-1) }. Then from the first equation (32
) it follows (p (i) + Φ)L i = 0, (34)
where (from now on) we do not sum over indices in brackets. If p (i) = -Φ, ∀i, then L i = 0, ∀i, i.e. ℓ is geodetic. Note that so far we have employed only equation (20) , which corresponds to equation (B.8) in [8] and which does not contain Weyl tensor components with negative boost order. Consequently, the same conclusions hold also for type II Einstein spacetimes.
Proposition 6 In arbitrary dimension, multiple WANDs of type II and D Einstein spacetimes are geodetic if at least one of the following conditions is satisfied: i) Φ A ij is non-vanishing; ii) for all eigenvalues of Φ S ij : p (i) = -Φ. Note that the above argument can not be extended to more special algebraic classes of spacetimes since it relies on the fact that some Weyl components with boost weight zero are non-vanishing. However, it was already shown in [8] that multiple WANDs in type N and III vacuum spacetimes are geodetic (in that case with no need of extra assumptions). Therefore we can conclude that under most "generic" conditions multiple WANDs are geodetic. Note, however, that certain special type-D vacuum solutions with Φ A ij = 0 and p (i) = -Φ (for some i) admit non-geodetic multiple WANDs. Explicit example of such spacetime is given in section 5.4.
The algebraic equations ( 22 ) and ( 27 ) are quite complicated in general dimension and thus here we will limit ourselves to the "shearfree" case. This is of interest since it includes, for instance, the Robinson-Trautman solutions containing static black holes [35] .
With the "shearfree" condition
S ij = S n-2 δ ij , (35)
equation ( 27 ) leads for S = 0 to
Φ S ij = Φ n-2 δ ij (S = 0), ( 36
)
whereas it is identically satisfied for S = 0. In the rest of this subsection we thus consider only the "expanding" case S = 0. For Φ S ij in the form (36) with Φ = 0 the condition ii) of Proposition 6 is satisfied and thus the WAND ℓ is geodetic.
Proposition 7 In arbitrary dimension, multiple "shearfree" and "expanding" WAND in a type D Einstein spacetime is geodetic whenever Φ ij = 0. the only non-vanishing components of the Weyl tensor have boost weight zero and are given by [35]
C ijkl = r 2 (R ijkl -2Kh i[k h l]j ), (40)
where R ijkl is the Riemann tensor associated to h ij . This implies that the spacetime (38) is of type D, with Φ ij = 0, and that both ℓ and n are multiple WANDs. Now, the vector ℓ is geodetic, shearfree and twistfree by construction [35] .
Next, one can easily show that
∇ n n = -H ,r n + H ,i dx i , (41)
where, by (38), H ,i = -r(ln P ) ,ui . Therefore n is geodetic if and only if (ln P ) ,ui = 0 ⇔ P = p 1 (u)p 2 (x 2 , x 3 , . . .). For a general (non-factorized) function P the multiple WAND n is thus non-geodetic (one can also easily check that it "shearfree", "twistfree" and "expanding"). A simple explicit example of such spacetimes is obtained by extending to any n ≥ 7 the n = 7 dimensional solution discussed in [36] , i.e. by taking in eq. ( 38 )
K = -1, P = f (u, z) -1/2 ρ n-5 (det η αβ ) 1/2 1/(2-n) , h ij dx i dx j = f (u, z) dz 2 + V (ρ)dτ 2 + 1 V (ρ) dρ 2 + ρ 2 η αβ dx α dx β , (42)
f (u, z) = 4b(u)e 2z/l l 2 [e 2z/l -b(u)] 2 , V (ρ) = 1 - µ ρ n-6 - ρ 2 l 2 ,
where z ≡ x 2 , τ ≡ x 3 , ρ ≡ x 4 , η αβ = η αβ (x 5 , x 6 , . . .) is the metric of an (n -5)dimensional unit sphere (α, β = 5, . . . , n -1), µ and l are constants and b(u) > 0 is an arbitrary function. The multiple WAND n is non-geodetic as long as db/du = 0. Note that there is not contradiction with the results of the previous subsections precisely because Φ ij = 0 here.
Let us now study the five-dimensional case. Note that the algebraic relation (6) between -2Φ S ij and C ijkl is equivalent to the relation between the Ricci and the Riemann tensor of a m -2 dimensional space. Therefore in five dimensions C ijkl is equivalent to Φ S ij and thus a type D Weyl tensor in five dimesions is fully determined by Φ ij . In fact, for n = 5 it is possible to solve the second constraint from (6) for
C ijkl : C ijkl (n=5) = 2 δ il Φ S jk -δ ik Φ S jl -δ jl Φ S ik + δ jk Φ S il -Φ (δ il δ jk -δ ik δ jl ) . (43)
Thus in the five dimensional case the algebraic equations we consider, ( 20 ), ( 21 ), ( 22 ), (27) , can be expressed in terms of Φ ij , L i , and L ij . Plugging (43) into (20), recalling equation ( 32 ) and contracting with L k one finds the equation
LΦ S ij + 2ΦL i L j -ΦLδ ij = 0. ( 44
)
For n = 5 equation ( 21 ) takes the form
0 = Φ A jk L im + (Φ A im + 3Φ S im )A kj + Φ A mj L ik + (Φ A ik + 3Φ S ik )A jm + Φ A km L ij +(Φ A ij + 3Φ S ij )A mk + δ ij (Φ S ms L sk -Φ S ks L sm ) + δ ik (Φ S js L sm -Φ S ms L sj ) +δ im (Φ S ks L sj -Φ S js L sk ) + Φ[δ ij A km + δ ik A mj + δ im A jk ]. (45)
Subtracting ( 61 ) and (62) we obtain (p (m) + p (j) )A mj = 0 and thus in the "generic" case p (m) + p (j) = 0, ∀m, j, A mj = 0. (68)
Proposition 11 In five dimensions, the multiple WAND ℓ in a "generic" (p (i) + p (j) = 0, ∀i, j) type D spacetime with Φ A ik = 0 and Φ S ik = 0, is geodetic and nontwisting (A ij = 0) and Φ S ik and S ij can be diagonalized together. There are some special cases to be treated: -Case a) one p (i) = 0 and Φ = 0: without loss of generality we choose p (2) = 0, then from ( 61
p (m) + p (j) )A mj = 0, 2ΦA mj = (p (j) -p (m) )S mj (69)
and thus if Φ = 0, A 34 = -p
Φ S 34 . If Φ = 0, then S 34 = 0 and S ij is diagonal and A 23 is arbitrary. -Case d) two pairs satisfy p (m) + p (j) = 0: without loss of generality we choose p (2) = p (3) = -p (4) = Φ. From (64) it follows that the diagonal components of S ij , s (2) and s (3) , vanish and s (4) is arbitrary. Equation (63) implies that S 24 and S 34 are arbitrary and from equation (69) we get A 23 = 0, A 24 = -S 24 , A 34 = -S 34 . This case is the non-geodetic case (48) from section 6.1.
As an illustrative example we give S ij , A ij , Φ S ij and Φ A ij for the five-dimensional Myers-Perry black hole [9]
ds 2 = ρ 2 4∆ dx 2 + ρ 2 dθ 2 -dt 2 + (x + a 2 ) sin 2 θdφ 2 + (x + b 2 ) cos 2 θdψ 2 + r 0 2 ρ 2 (dt + a sin 2 θdφ + b cos 2 θdψ) 2 , where ρ 2 = x + a 2 cos 2 θ + b 2 sin 2 θ, ∆ = (x + a 2 )(x + b 2 ) -r 0 2 x.
Two (multiple, geodetic) WANDs (related by reflection symmetry) are given by [7] ℓ
= (x + a 2 )(x + b 2 ) ∆ ∂ t - a x + a 2 ∂ φ - b x + b 2 ∂ ψ + 2 √ x∂ x , (70)
n = α (x + a 2 )(x + b 2 ) ∆ ∂ t - a x + a 2 ∂ φ - b x + b 2 ∂ ψ -2 √ x∂ x , (71)
where we chose α = -∆/2ρ 2 x in order to satisfy the normalization condition ℓ • n = 1.
As a basis of spacelike vectors we choose three eigenvectors of S ij m
(2) = 1 ρ ∂ θ , m (3) = 1 √ xχ (-ab∂ t + b∂ φ + a∂ ψ ) , (72)
m (4) = 1 ρχ (a 2 -b 2 ) sin θ cos θ∂ t -a tan -1 θ∂ φ + b tan θ∂ ψ ,
with χ = a 2 cos 2 θ + b 2 sin 2 θ. In this frame
S ij = √ x ρ 2 0 0 0 1 √ x 0 0 0 √ x ρ 2 , A ij = χ ρ 2 0 0 -1 0 0 0 1 0 0 , (73) and Φ
S ij = r 0 2 ρ 4 ρ 2 -2x ρ 2 0 0 0 -1 0 0 0 ρ 2 -2x ρ 2 , Φ A ij = 2r 0 2 χ √ x ρ 6 0 0 1 0 0 0 -1 0 0 .(74)
Notice that in the static (Schwarzschild) limit (a = 0 = b so that ρ 2 = x) one has
S ij = δ ij / √
x and σ ij = 0 = A ij , and indeed for Φ ij we recover the form discussed in subsection 5.3 in the shearfree expanding case and in subsection 6.2 in the "generic" non-twisting case (with p (2) = p (3) = p (4) ).
Let us finally outline main results presented in the paper.
In the first part of the paper (Sections 3 and 4) we study constraints on Weyl types of a spacetime following from various assumptions on geometry. It turns out that: -Static spacetimes are of types G, I i , D or conformally flat (Proposition 1).
-"Expanding" stationary spacetimes with appropriate reflection symmetry belong to these types as well (Proposition 2).
-Warped spacetimes with one-dimensional Lorentzian factor are again of types G, I i , D and O (Proposition 3).
-Warped spacetimes with two-dimensional Lorentzian factor are necessarily of types D or O (Proposition 4), in particular this also applies to spherically symmetric spacetimes (Proposition 5).
These results may have useful practical applications in determining the algebraic type of specific spacetimes (or at least in ruling out some types) just by "inspecting" the given metric and without performing any calculations. This is particularly important in higher dimensions, where it is more difficult to determine the algebraic class of a given metric.
In the second part of the paper (sections 5 and 6) we study properties of type D vacuum spacetimes in general (without assuming that the spacetime is static, stationary or warped). In five dimensions a type D Weyl tensor is determined by a 3×3 matrix Φ ij with symmetric and antisymmetric parts being Φ A ij and Φ S ij , respectively.
In general in the non-twisting case Φ ij is symmetric while in the twisting case antisymmetric part Φ A ij appears. In higher dimensions n > 5 the (n-2)×(n-2) matrix Φ ij does not contain complete information about the Weyl tensor, but it still plays an important role. The matrix Φ ij can also be used for further classification of type D or II spacetimes, for example according to possible degeneracy of eigendirections of Φ ij . Special classes are also cases with Φ ij being symmetric or vanishing (such examples for n ≥ 7 are given in section 5.4) etc.
First we focused on the geodeticity of multiple WANDs in type D vacuum spacetimes (these are always geodetic for n = 4). It was shown that: -The multiple WAND in a vacuum spacetime is geodetic in the "generic" case, i.e. if Φ A ij = 0 or if all eigenvalues of Φ S ij are distinct from minus the trace of Φ ij (Proposition 6).
-It is also geodetic in the type D, shearfree case whenever Φ ij = 0 (Proposition 7). -However, explicit examples of vacuum type D spacetimes with non-geodetic multiple WAND in n ≥ 7 dimensions are given in section 5.4. This provides us with the first examples of spacetimes "violating" the geodetic part of the Goldberg-Sachs theorem.
-In five dimensions multiple WANDs are also geodetic when Φ A ij = 0 and Φ S ij = 0 has a "generic" form (Proposition 11), special cases are discussed in section 6.3.
Properties of the matrix Φ ij , as well as the expansion and twist matrices S ij and A ij have been also studied: -For warped spacetimes with a one/two-dimensional Lorentzian factor (thus also for static spacetimes) the antisymmetric part of Φ ij , Φ A ij , vanishes. -In vacuum type D spacetimes admitting a shearfree expanding WAND, Φ S ij is proportional to δ ij and if A ij = 0 (this always holds in odd dimensions [10] ) then Φ A ij = 0 and in the case with Φ S ij = 0 also vice versa (Proposition 8). -In five dimensions in a "generic" Einstein type D non-twisting spacetime, Φ A ij vanishes and eigendirections of Φ ij coincide with those of S ij (Proposition 10).
-In five dimensions in a "generic" vacuum type D spacetime with symmetric Φ ij , the multiple WAND ℓ is non-twisting and eigendirections of Φ ij and S ij coincide (Proposition 11).
These results provide interesting connections between geometric properties of principal null congruences and Weyl curvature. Hopefully, they can be also used for constructing exact type D solutions with particular properties. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We show that all static spacetimes in higher dimensions n > 4 are necessarily of Weyl types G, I i , D or O. This applies also to stationary spacetimes provided additional conditions are fulfilled, as for most known black hole/ring solutions. (The conclusions change when the Killing generator becomes null, such as at Killing horizons, on which we briefly comment.) Next we demonstrate that the same Weyl types characterize warped product spacetimes with a onedimensional Lorentzian (timelike) factor, whereas warped spacetimes with a twodimensional Lorentzian factor are restricted to the types D or O. By exploring algebraic consequences of the Bianchi identities, we then analyze the simplest non-trivial case from the above classes -type D vacuum spacetimes, possibly with a cosmological constant, dropping, however, the assumptions that the spacetime is static, stationary or warped. It is shown that for \"generic\" type D vacuum spacetimes (as defined in the text) the corresponding principal null directions are geodetic in arbitrary dimension (this in fact applies also to type II spacetimes). For n ≥ 5, however, there may exist particular cases of type D vacuum spacetimes which admit non-geodetic multiple principal null directions and we explicitly present such examples in any n ≥ 7. Further studies are restricted to five dimensions, where the type D Weyl tensor is fully described by a 3 × 3 real matrix Φ ij . In the case with \"twistfree\" (A ij = 0) principal null geodesics we show that in a \"generic\" case Φ ij is symmetric and eigenvectors of Φ ij coincide with eigenvectors of the expansion matrix S ij providing us thus in general with three preferred spacelike directions of the spacetime. Similar results are also obtained when relaxing the twistfree condition and assuming instead that Φ ij is symmetric. The five dimensional Myers-Perry black hole and Kerr-NUT-AdS metrics in arbitrary dimension are also briefly studied as specific illustrative examples of type D vacuum spacetimes."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Algebraically special spacetimes play an essential role in the field of exact solutions of Einstein's equations and many known exact solutions in four dimensions are indeed algebraically special [1] . Recently a generalization of the Petrov classification to higher dimensions was developed in [2, 3] and it turned out that many higher-dimensional solutions of Einstein's equations are algebraically special as well (see e.g. [4] ), in fact ‡ Now at: Departament de Física Fonamental, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain so far there is only one known solution identified [5] as algebraically general -the static charged black ring [6] .\n\nThere is, however, one important difference between four dimensional and n > 4 dimensional cases -the Goldberg-Sachs theorem does not hold in higher dimensions. Recall that for n = 4 the Goldberg-Sachs theorem implies that principal null directions of an algebraically special vacuum spacetime are necessarily geodetic and shearfree. It was stressed already in [7, 8] that the Goldberg-Sachs theorem cannot be straightforwardly extended to higher dimensions. Namely in [7] it was pointed out that principal null directions (or Weyl aligned null directions -WANDs [2] ) of the n = 5 Myers-Perry black holes [9] are shearing though the spacetime is of type D. In [8] it was shown that in fact all vacuum, n > 4, type N and III expanding spacetimes are shearing. In [10] it was also shown that for n > 4, n odd, all geodetic WANDs with non-vanishing twist are again shearing.\n\nIn this paper we study various properties of algebraically special vacuum spacetimes, such as geodeticity of multiple WANDs (not guaranteed in higher dimensions -another \"violation\" of the Goldberg-Sachs theorem) and relationships between optical matrices S ij and A ij and the Weyl tensor. Before approaching these problems, we study in the first part of the paper (sections 3 and 4) constraints on Weyl types of the spacetime following from various assumptions on the geometry.\n\nIn section 3 we show that in arbitrary dimension (i.e., hereafter, n ≥ 4) the only Weyl types compatible with static spacetimes (and expanding stationary spacetimes with appropriate reflection symmetry) are types G, I i , D and O.\n\nIn section 4 we study direct or warped product spacetimes. It turns out that warped spacetimes with a one-dimensional Lorentzian factor are again of types G, I i , D and O and that warped spacetimes with a two-dimensional Lorentzian factor are necessarily of type D or O. This also implies that spherically symmetric spacetimes are of type D or O.\n\nIt follows that type D spacetimes play an important role as the simplest non-trivial case compatible with the above mentioned assumptions. Therefore, in the second part of the paper (sections 5 and 6) we focus on studying properties of type D Einstein spacetimes (i.e., vacuum with an arbitrary cosmological constant), dropping, however, the assumptions that the spacetime is static, stationary or warped.\n\nIn section 5 we study type D spacetimes in arbitrary dimension and analyze geodeticity of WANDs. It turns out that in a \"generic\" case in vacuum the multiple WANDs are geodetic. Let us also point out that negative boost weight Weyl components do not enter relevant equations and thus the same results also hold for multiple WANDs in type II Einstein spacetimes. Surprisingly, it also turns out that explicit examples of special vacuum type D spacetimes not belonging to our \"generic\" class and admitting non-geodetic multiple WANDs can easily be constructed. Such examples for arbitrary dimension n ≥ 7 are given in section 5. 4 . This shows that there exist even more striking \"violations\" of the Goldberg-Sachs theorem in higher dimensions than the examples with non-zero shear discussed above. In section 5 we also study various properties of shearfree type D vacuum spacetimes.\n\nPerhaps not surprisingly, the situation in five dimensions is considerably simpler than for n > 5. In fact it turns out that for n = 5 the Weyl tensor of type D is fully determined by a 3 × 3 real matrix Φ ij . At the same time, five dimensional gravity is already an interesting arena where qualitatively new phenomena appear. We thus devote section 6 to five dimensional vacuum type D spacetimes. We study relationships between the Weyl tensor represented by Φ ij and optical matrices S ij and A ij . One of the results is that for \"generic\" spacetimes with non-twisting WANDs (A ij = 0) the antisymmetric part of Φ ij , Φ A ij , vanishes and the symmetric part Φ S ij is aligned with S ij (in the sense that the matrices Φ S ij and S ij can be diagonalized together). Similarly, in the \"generic\" case the condition Φ A ij = 0 implies vanishing of A ij . Again, there exist particular cases for which the \"generic\" proof does not hold, see section 6 for details. In this section a simple explicit example of a five-dimensional vacuum type D spacetime, the Myers-Perry metric, is also presented and S ij , A ij , Φ S ij , and Φ A ij are explicitly given.\n\nFinally in section 7 we concisely summarize main results and in the Appendix we briefly study geometric optics of type D Kerr-NUT-AdS metrics in arbitrary dimension."
},
{
"section_type": "OTHER",
"section_title": "Preliminaries",
"text": "Let us first briefly summarize our notation, further details can be found in [8] . In an n-dimensional spacetime let us introduce a frame of n real vectors m (a) (a, b . . . = 0, . . . , n -1): two null vectors\n\nm (0) = m (1) = n, m (1) = m (0) = ℓ and n -2 orthonormal spacelike vectors m (i) = m (i) (i, j . . . = 2, . . . , n -1) satisfying ℓ a ℓ a = n a n a = ℓ a m (i) a = n a m (i) a = 0, ℓ a n a = 1, m (i)a m (j) a = δ ij . (1)\n\nThe metric now reads\n\ng ab = 2ℓ (a n b) + δ ij m (i) a m (j) b . (2)\n\nWe will use the following decomposition of the covariant derivative of the vector ℓ and the covariant derivative in the direction of ℓ\n\nℓ a;b = L cd m (c) a m (d) b , D ≡ ℓ a ∇ a . (3)\n\nNote that ℓ is geodetic iff L i0 = 0 and for an affine parameterization also L 10 = 0. We will often use the symmetric and antisymmetric parts of\n\nL ij , S ij ≡ L (ij) (its trace S ≡ S ii ), A ij ≡ L [ij]\n\n. In case of geodetic ℓ, the trace of S ij represents expansion θ ≡ 1 n-2 S, the tracefree part of S ij is shear σ ij ≡ S ij -θδ ij and the antisymmetric matrix A ij is twist. § Optical scalars can be expressed in terms of ℓ (when\n\nL i0 = 0 = L 10 ) σ 2 ≡ σ ij σ ji = ℓ (a;b) ℓ (a;b) -1 n-2 ℓ a ;a 2 , θ = 1 n-2 ℓ a ;a , ω 2 ≡ A ij A ij = ℓ [a;b] ℓ a;b . (4)\n\nThe decomposition of the Weyl tensor in the frame (1) in full generality is given by [8]\n\nC abcd = 4C 0i0j n {a m (i) b n c m (j) d} + 8C 010i n {a ℓ b n c m (i) d} + 4C 0ijk n {a m (i) b m (j) c m (k) d} + 4C 0101 n {a ℓ b n c ℓ d} + 4C 01ij n {a ℓ b m (i) c m (j) d} + 8C 0i1j n {a m (i) b ℓ c m (j) d} + C ijkl m (i) {a m (j) b m (k) c m (l) d} + 8C 101i ℓ {a n b ℓ c m (i) d} + 4C 1ijk ℓ {a m (i) b m (j) c m (k) d} + 4C 1i1j ℓ {a m (i) b ℓ c m (j)\n\nb} , where the operation { } is defined as\n\nw {a x b y c z d} ≡ 1 2 (w [a x b] y [c z d] + w [c x d] y [a z b] ).\n\n§ For the sake of brevity, throughout the paper we shall refer to the corresponding quantities for nongeodetic congruences as \"expansion\", \"shear\", and \"twist\" (in inverted commas), bearing in mind that in that case expressions (4) do not hold.\n\nIn the second part of this paper we will focus on type D spacetimes, possessing (in an adapted frame) only boost order zero components (see [8] ) C 0101 , C 01ij , C 0i1j , C ijkl . For simplicity let us define the (n -2) × (n -2) real matrix\n\nΦ ij ≡ C 0i1j , (5)\n\nwith Φ S ij , Φ A ij , and Φ ≡ Φ ii being the symmetric and antisymmetric parts of Φ ij and its trace, respectively. Let us observe that for static spacetimes and for a large class of warped geometries one has Φ A ij = 0 (see section 4). Note also that the above mentioned boost order zero components of the Weyl tensor are not completely independent. In fact from the symmetries and the tracelessness of the Weyl tensor (cf. eqs. ( 7 ) and ( 9 ) in [8] ) it follows that\n\nC 01ij = 2C 0[i|1|j] = 2Φ A ij , C 0(i|1|j) = Φ S ij = -1 2 C ikjk , C 0101 = -1 2 C ijij = Φ. (6)\n\nThe type D Weyl tensor is thus completely determined by m(m-1)\n\n2 independent components of Φ A ij and m 2 (m 2 -1)\n\n12 independent components of C ijkl , where n = m -2."
},
{
"section_type": "OTHER",
"section_title": "Static spacetimes",
"text": "Algebraically special spacetimes in higher dimensions are characterized by the existence of preferred null directions -Weyl aligned null directions (WANDs). A necessary and sufficient condition for a null vector ℓ being WAND in arbitrary dimension is [3, 11]\n\nℓ b ℓ c ℓ [e C a]bc[d ℓ f ] = 0, (7)\n\nwhere C abcd is the Weyl tensor. Let us now assume that a spacetime of interest is algebraically special and thus the equation ( 7 ) possesses a null solution ℓ = (ℓ t , ℓ A ), A = 1 . . . n -1 (note that necessarily ℓ t = 0 and at least one of the remaining components is also non-zero).\n\nFor static spacetimes the metric does not depend on the direction of time and consequently the form of the metric and of the Weyl tensor remains unchanged under the transformation t = -t. Therefore, in these new coordinates equation (7) has the same form as in the original coordinates and admits a second solution ñ = (ℓ t , ℓ A ). In the original coordinates n = (-ℓ t , ℓ A ). Thus for static spacetimes the existence of a WAND ℓ implies the existence of a distinct WAND n which in fact has the same order of alignment. The only Weyl types compatible with this property are types G, I i and D (or, trivially, O, i.e. conformally flat spacetimes). Therefore Proposition 1 All static spacetimes in arbitrary dimension are of Weyl types G, I i or D, unless conformally flat.\n\nIn fact explicit examples of static spacetimes of these Weyl types are knowncharged static black ring (type G - [5] ), vacuum static black ring (type I i - [11] ), the Schwarzschild-Tangherlini black hole (type D - [8] ) and the Einstein universe R×S n-1 (type O -cf. the results summarized in section 4). Cf. also the static examples given in [4] .\n\nIn the standard n = 4 (i.e., m = 2) case these are essentially the imaginary and real part of Ψ 2 . More specifically, with the conventions of [1] , one has Φ Note that in four dimensions there is no type G and type I is equivalent to type I i [2, 3] . Thus for n = 4 only types I, D and O are compatible with static spacetimes. This was discussed already in [12] in the case of static, n = 4, vacuum spacetimes (see also additional comments in [13] and in section 6.2 of [1] ).\n\nS ij = 1 2 Φδ ij with Φ = -2Re(Ψ 2 ), Φ A 23 = Φ 23 = -Im(Ψ 2 )"
},
{
"section_type": "OTHER",
"section_title": "Stationary spacetimes",
"text": "One can use the same arguments as above for stationary spacetimes with the metric remaining unchanged under reflection symmetry involving time and some other coordinates. E.g. in Boyer-Lindquist coordinates the Kerr metric remains unchanged under t = -t, φ = -φ and n = 5 Myers-Perry under t = -t, φ = -φ, ψ = -ψ or, for general dimension, Myers-Perry under t = -t, φi = -φ i . Note, however, that in contrast to the static case, in some special stationary cases one could in principle get from the original WAND ℓ a \"new\" WAND n = -ℓ which represents the same null direction. In order to deal with these special cases we note that the \"divergence scalar\" (or, loosely speaking, \"expansion\", since it does coincide with the standard expansion scalar in the case of geodetic, affinely parameterized null directions) of both WANDs n and ℓ related by reflection symmetry is the same (as well as all the other optical scalars and the geodeticity parameters -this also applies to the static case), i.e. ℓ a ;a = n a ;a while the \"expansion\" of -ℓ is equal to -ℓ a ;a . Therefore for all \"expanding\" spacetimes n = -ℓ. Thus Proposition 2 In arbitrary dimension, all stationary spacetimes with non-vanishing divergence scalar (\"expansion\") and invariant under appropriate reflection symmetry are of Weyl types G, I i or D, unless conformally flat.\n\nNote also that it is shown in [14] that Kerr-Schild spacetimes with the assumption R 00 = 0 are of type II (or more special) in arbitrary dimension with the Kerr-Schild vector being the multiple WAND. Therefore all Kerr-Schild spacetimes that are either static or belong to the above mentioned class of stationary spacetimes are necessarily of type D. In particular, the Myers-Perry metric in arbitrary dimension is thus of type D. ¶\n\nIn addition to the rotating Myers-Perry black holes for n ≥ 4, of type D, we can mention a number of physically relevant solutions as explicit examples of spacetimes subject to Proposition 2. + First, rotating vacuum black rings [17] , of type I i [11] . To our knowledge, no stationary (non-static) type G solution has been so far explicitly identified. It is, however, plausible to expect that a rotating charged black ring (so far unknown in the standard Einstein-Maxwell theory) will be of type G as its static counterparts. Further interesting examples fulfilling our assumptions are expanding ¶ This was already known in the case n = 5 [4, 8] . Furthermore, it has been demonstrated recently in [15] by explicit computation of the full curvature tensor that the family [16] of higher dimensional rotating black holes with a cosmological constant and NUT parameter is of type D for any n. We observe in addition that, using the connection 1-forms given in [15] , it is also straightforward to show (see the Appendix) that the mutiple WANDs (which are related by reflection symmetry) of all such solutions are twisting, expanding and shearing (except that the shear vanishes for n = 4). The fact that the WANDs found in [15] are complex is only due to the analytical continuation trick used in [16] to cast the line element in a nicely symmetric form -the WANDs of the associated \"physical\" spacetimes are thus real after Wick-rotating back one of the coordinates. + It is straightforward to verify the \"reflexion symmetry\" of the metric we mention in this context. The \"expansion\" condition, instead, has not been verified explicitly in all cases. However, it is plausible that these spacetimes are indeed \"expanding\" since they contain as special limits or subcases solutions with expansion, e.g. Myers-Perry black holes (cf. section 6.4, [8] and the preceding footnote).\n\nstationary axisymmetric spacetimes with n -2 commuting Killing vector fields [18] , which contain, apart from the (n = 5) black holes/rings mentioned above, also e.g. the recently obtained \"black saturn\" [19] , doubly spinning black rings [20] and black di-rings [21] . In any dimension also rotating uniform black strings/branes satisfy the assumptions of Proposition 2 (see section 4), and so does the ansatz recently used in [22] for the numerical construction of corresponding n = 6 non-uniform solutions. Other examples are all the stationary solutions discussed in [4] and various black ring solutions reviewed in [23] ."
},
{
"section_type": "RESULTS",
"section_title": "Remarks and \"limitations\" of the results",
"text": "First, it is worth observing that we have not used any field equations for the gravitational field in the considerations presented above and the results are thus purely geometrical.\n\nNote that one can not relax the assumption ℓ a ;a = 0 in the case of stationary spacetimes. For example, the special pp -wave metric ds 2 = g ij dx i dx j -2dudv-2Hdu 2 such that H ,u = 0 (note that it is always H ,v = 0 by the definition of pp -waves) and ∂ u • ∂ u = -2H < 0 represents stationary spacetimes (cf., e.g., [24] for the n = 4 vacuum case) that are invariant under reflection symmetry (ũ = -u, ṽ = -v) and yet of type N [25] . In fact, the geodetic multiple WAND ℓ = ∂ v is non-expanding (and n = -ℓ is not a new WAND).\n\nFurthermore, if we assume a null Killing vector field k instead of a timelike one we are led to different conclusions. Namely, it is easy to show that k must be geodetic, shearfree and non-expanding, which for R ab k a k b = 0 implies that k is a twistfree WAND [10] . We thus end up with a subfamily of the Kundt class, for which (under the alignment requirement R ab k a ∝ k b , obeyed e.g. in vacuum) the algebraic type is II or more special [10] (cf. section 24.4 of [1] for n = 4). In particular, a similar argument applies locally at Killing horizons, where the type must thus be again II or more special (provided R ab k a ∝ k b ). * This is in agreement with the result of [26] for generic isolated horizons. As an explicit example, vacuum black rings (which are of type I i in the stationary region) become locally of type II on the horizon [11] .\n\nFinally, spacelike Killing vectors do not impose any constraint on the algebraic type of the Weyl tensor, in general, and all types are in fact possible. For example charged static black rings are of type G, vacuum black rings of type I i , vacuum black holes of type D, and they all admit at least one spacelike Killing vector; Kundt spacetimes can be constructed that admit axial symmetry with all types II, D, III and N being possible (see, e.g., [1] for n = 4)."
},
{
"section_type": "OTHER",
"section_title": "Direct/warped product spacetimes",
"text": "In this section we show that the algebraic types discussed above also characterize certain classes of direct/warped product geometries of physical relevance. In addition we discuss some optical properties of these spacetimes.\n\n* The proof is a bit more tricky in this case since the Killing vector is null only at the horizon.\n\nStill, one can adapt techniques used in [26, 27] for related investigations. Note that the horizon of higher dimensional stationary black holes is indeed a Killing horizon (at least in the non-degenerate case) [27] ."
},
{
"section_type": "OTHER",
"section_title": "Weyl tensor",
"text": "Let us consider two (pseudo-)Riemannian spaces (M 1 , g (1) ) and (M 2 , g (2) ) of dimension n 1 and n 2 (n 1 , n 2 ≥ 1 and n 1 + n 2 ≥ 4), parameterized by coordinates x A (A, B = 0, . . . , n 1 -1) and x I (I, J = n 1 , . . . , n 1 + n 2 -1), respectively. Using adapted coordinates x µ (µ, ν = 0, . . . , n 1 + n 2 -1) constructed from the coordinates x A of M 1 and x I of M 2 , we define the direct product (M, g) to be the product manifold\n\nM = M 1 × M 2 , of dimension n = n 1 + n 2 , equipped with the metric tensor g(x µ ) = g (1) (x A ) ⊕ g (2) (x I ) defined (locally) by g AB = g (1)AB , g IJ = g (2)IJ , g AI = 0.\n\nFor the sake of definiteness, we shall assume hereafter that (M 1 , g 1 ) is Lorentzian and (M 2 , g 2 ) is Riemannian.\n\nIn general, any geometric quantity which can be split like the product metric (i.e., with no mixed components and with the A[I] components depending only on the x A [x I ] coordinates) is called a \"product object\" (or \"decomposable\"). Various interesting geometrical properties then follow [28] and, in particular, the Riemann and Ricci tensors and the Ricci scalar are all decomposable. As a consequence, a product space is an Einstein space iff each factor is an Einstein space and their Ricci scalars satisfy R (1) /n 1 = R (2) /n 2 [28] .\n\nUsing the above coordinates it follows from the standard definition that the mixed components of the Weyl tensor are given by\n\nC ABCI = C ABIJ = C AIJK = 0, (8)\n\nC AIBJ = - 1 n -2 g (1)AB R (2)IJ + g (2)IJ R (1)AB + R (1) + R (2) (n -1)(n -2) g (1)AB g (2)IJ , (9)\n\nwhere\n\nR (1)AB [R (2)IJ ] is the Ricci tensor of (M 1 , g 1 ) [(M 2 , g 2 )]\n\n. For the non-mixed components one has to distinguish the special cases n 1 = 1, 2 (and the \"symmetric\" cases n 2 = 1, 2, which we omit for brevity). If n 1 = 1 there are of course no non-mixed components C ABCD since now the x A span a one-dimensional space. If n 1 = 2 there is only one independent component, i.e. C 0101 (notice that here, exceptionally, 0 and 1 are not frame indices but refer to the coordinates x 0 and x 1 in the factor space M 1 ).\n\nFor n 1 ≥ 3,\n\nC ABCD = C (1)ABCD + 2n 2 (n -2)(n 1 -2) g (1)A[C R (1)D]B -g (1)B[C R (1)D]A + 2 (n -1)(n -2) R (2) -R (1) n 2 (n 2 + 2n 1 -3) (n 1 -1)(n 1 -2) g (1)A[C g (1)D]B (n 1 ≥ 3), ( 10\n\n)\n\nwhere C (1)ABCD is the Weyl tensor of (M 1 , g 1 ), whereas the remaining non-mixed components are given for any n 1 ≥ 1 by\n\nC IJKL = C (2)IJKL + 2n 1 (n -2)(n 2 -2) g (2)I[K R (2)L]J -g (2)J[K R (2)L]I + 2 (n -1)(n -2) R (1) -R (2) n 1 (n 1 + 2n 2 -3) (n 2 -1)(n 2 -2) g (2)I[K g (2)L]J (n 2 ≥ 3), ( 11\n\n)\n\nwhere C (2)IJKL is the Weyl tensor of (M 2 , g 2 ). It is thus obvious that the Weyl tensor is not decomposable, in general. It turns out that the Weyl tensor is decomposable iff both product spaces are Einstein spaces and n 2 (n 2 -1)R (1) + n 1 (n 1 -1)R (2) = 0 (the latter condition is identically satisfied whenever n 1 = 1 or n 2 = 1, while for\n\nn 1 = 2 [n 2 = 2] it implies that (M 1 , g 1 ) [(M 2 , g 2 )] must be of constant curvature).\n\nWhen the Weyl tensor is decomposable the only non-vanishing components take the simple form C ABCD = C (1)ABCD , C IJKL = C (2)IJKL . Therefore, in particular, the product space is conformally flat iff both product spaces are of constant curvature and\n\nn 2 (n 2 -1)R (1) + n 1 (n 1 -1)R (2) = 0.\n\nDetermining the possible algebraic types of the Weyl tensor requires considering various possible choices for the dimension n 1 of the Lorentzian factor.\n\nIf n 1 = 1, the full metric can always be cast in the special static form ds 2 = -dt 2 + g IJ dx I dx J . Recalling the result of section 3, the Weyl tensor can thus only be of type G, I i , D or O. In particular, one can show that C 0i1j = C 0j1i , so that for direct product spacetimes with n 1 = 1 one has Φ A ij = 0 identically. If n 1 ≥ 2, it is convenient to adapt the null frame (1) to the natural product structure, so that\n\ng ab = 2ℓ (a n b) + δ  B m ( Â) a m ( B) b + δ Î Ĵ m ( Î) a m ( Ĵ) b\n\n(where Â, B = 2, . . . , n 1 -1, Î, Ĵ = n 1 , . . . , n -1 are now frame indices, and the frame vectors do not have mixed coordinate components, e.g. ℓ I = 0 = n I etc.). From (10) and (11) it thus follows that C ABCD and C IJKL do not give rise to mixed frame components, and from (9) that C AIBJ does not give rise to non-mixed frame components. Hence the only non-vanishing mixed components are (ordered by boost weight)\n\nC 0 Î0 Ĵ = - 1 n -2 R (1)00 δ Î Ĵ , C 0 Î Â Ĵ = - 1 n -2 R (1)0 Âδ Î Ĵ , C 0 Î1 Ĵ = - 1 n -2 R (2) Î Ĵ + R (1)01 δ Î Ĵ + R (1) + R (2) (n -1)(n -2) δ Î Ĵ , C  ΠB Ĵ = - 1 n -2 R (2) Î Ĵ δ  B + R (1)  B δ Î Ĵ + R (1) + R (2) (n -1)(n -2) δ  B δ Î Ĵ , (12)\n\nC 1 Î Â Ĵ = - 1 n -2 R (1)1 Âδ Î Ĵ , C 1 Î1 Ĵ = - 1 n -2 R (1)11 δ Î Ĵ .\n\nThe non-mixed frame components are given for n 1 = 2 by\n\nC 0101 = - 1 2(n 2 + 1) (n 2 -1)R (1) + 2R (2) n 2 (n 1 = 2), (13)\n\nand for n 1 ≥ 3 by\n\nC 0 Â0 B = C (1)0 Â0 B + n 2 (n -2)(n 1 -2) R (1)00 δ  B , C 010  = C (1)010  - n 2 (n -2)(n 1 -2) R (1)0 Â, C 0  B Ĉ = C (1)0  B Ĉ - 2n 2 (n -2)(n 1 -2) R (1)0[ Ĉ δ B] Â, C 0101 = C (1)0101 - 2n 2 (n -2)(n 1 -2) R (1)01 - 1 (n -1)(n -2) R (2) -R (1) n 2 (n 2 + 2n 1 -3) (n 1 -1)(n 1 -2) , C 01  B = C (1)01  B (n 1 ≥ 3), ( 14\n\n)\n\nC 0 Â1 B = C (1)0 Â1 B + n 2 (n -2)(n 1 -2) R (1)  B + R (1)01 δ  B + 1 (n -1)(n -2) R (2) -R (1) n 2 (n 2 + 2n 1 -3) (n 1 -1)(n 1 -2) δ  B , C  B Ĉ D = C (1)  B Ĉ D + 2n 2 (n -2)(n 1 -2) R (1) B[ Dδ Ĉ]  -R (1) Â[ Dδ Ĉ] B + 2 (n -1)(n -2) R (2) -R (1) n 2 (n 2 + 2n 1 -3) (n 1 -1)(n 1 -2) δ B[ Dδ Ĉ] Â, C Î Ĵ K L = C (2) Î Ĵ K L + 2n 1 (n -2)(n 2 -2) δ Î[ K R (2) L] Ĵ -δ Ĵ[ K R (2) L] Î + 2 (n -1)(n -2) R (1) -R (2) n 1 (n 1 + 2n 2 -3) (n 2 -1)(n 2 -2) δ Î[ K δ L] Ĵ , C 1  B Ĉ = C (1)1  B Ĉ - 2n 2 (n -2)(n 1 -2) R (1)1[ Ĉ δ B] Â, C 101  = C (1)101  - n 2 (n -2)(n 1 -2) R (1)1 Â, C 1 Â1 B = C (1)1 Â1 B + n 2 (n -2)(n 1 -2) R (1)11 δ  B .\n\n(The expression for C Î Ĵ K L holds only when n 2 ≥ 3, while for n 2 = 2 one gets only one component C 2323 similar to (13).)\n\nFor n 1 = 2 the Weyl tensor of (M 1 , g 1 ) of course vanishes, and in addition we have R (1)00 = 0 = R (1)11 identically (any 2-space satisfies 2R (1)AB = R (1) g (1)AB ). Therefore among the above components ( 12 ) and ( 13 ) only the boost weight zero components C 0 Î1 Ĵ and C 0101 survive, so that the corresponding spacetime can be only of type D (or conformally flat), and ℓ and n, as chosen above, are multiple WANDs. Note also that Φ ij reduces to Φ Î Ĵ = C 0 Î1 Ĵ = C 0 Ĵ 1 Î in this case, therefore Φ A ij = 0. As an example, the higher dimensional electric Bertotti-Robinson solutions fall in this class, cf., e.g, [29, 30] .\n\nFor n 1 = 3, again the Weyl tensor of (M 1 , g 1 ) vanishes. With the additional assumption that (M 1 , g 1 ) is Einstein, we get R (1)00 = R (1)11 = R (1)0  = R (1)1  = 0 (here  = 2 only), and as above the Weyl tensor is of type D with Φ A ij = 0. Similarly, for any n 1 > 3, if (M 1 , g 1 ) is an Einstein space the only non-zero mixed Weyl components (12) will have boost weight zero, and the non-mixed components (14) simplify considerably. As a particular consequence, if (M 1 , g 1 ) is an Einstein space of type D, (M, g) will also be of type D (but now Φ A ij = 0, in general) -this is the case, for example, of uniform black strings/branes (either static or rotating, see also the discussion concluding this section). If (M 1 , g 1 ) is of constant curvature, (M, g) will be of type D with Φ A ij = 0 (or O) -this includes the higher dimensional magnetic Bertotti-Robinson solutions [29] . One can consider other special cases using similar simple arguments.\n\nA spacetime conformal to a direct product spacetime is called a warped product spacetime if the conformal factor depends only on one of the two coordinate sets x A , x I (see e.g. [1] ). Obviously, the algebraic type of two conformal spaces is the same.♯ Some of the results presented above can thus be straightforwardly generalized to the more general case of warped products. For example, Proposition 3 In arbitrary dimension, a warped spacetime with a one-dimensional Lorentzian (timelike) factor can be only of type G, I i , D (with Φ A ij = 0) or O. This case includes, in particular, the conclusion of section 3 for static spacetimes. As warped non-static/non-stationary examples we can mention the de Sitter universe (in global coordinates) and FRW cosmologies. For n = 4 Proposition 3 reduces to a result of [32] . Furthermore, Proposition 4 In arbitrary dimension, a warped spacetime with a two-dimensional Lorentzian factor can be only of type D (with Φ A ij = 0) or O. Cf. again [32] for n = 4. Notice that in this case the line element can always be cast in one of the two (conformally related) forms ds 2 = 2A(u, v)dudv + f (u, v)h IJ (x)dx I dx J or ds 2 = 2 f (x)A(u, v)dudv + g IJ (x)dx I dx J (so that multiple WANDs are given by ∂ u and ∂ v ), which include a number of known spacetimes. For example, the first possibility includes all spherically symmetric spacetimes, hence as a special case of Proposition 4 we have Proposition 5 In arbitrary dimension, a spherically symmetric spacetime is of type D (with Φ A ij = 0) or O. For n = 4 this has been known for a long time (see e.g. [33] and sections 15.2, 15.3 of [1] ), and in this case Φ A ij = 0 means that Ψ 2 is real (see the footnote on p. 4). For n > 4 this result has been proven in [34] in the static case.\n\nOther properties of decomposable Weyl tensors were discussed in [2] ."
},
{
"section_type": "OTHER",
"section_title": "\"Factorized\" geodetic null vector fields",
"text": "Let us define an n-dimensional spacetime (M, g) as the warped product of an n 1dimensional Lorentzian space (M 1 , g (1) ) and an n 2 -dimensional Riemannian space (M 2 , g (2) ), with n = n 1 + n 2 as in the preceding subsection. Hereafter we shall assume n 1 ≥ 2. Using the adapted coordinates defined above, the metric can take one of the following two forms\n\nds 2 = g AB dx A dx B + f (x A )h IJ dx I dx J , (15)\n\nds 2 = f (x I )h AB dx A dx B + g IJ dx I dx J , ( 16\n\n)\n\nwhere g AB , h AB = g (1)AB depend only on the x A coordinates and g IJ , h IJ = g (2)IJ only on the x I coordinates. Given a null vector ℓ (1) = ℓ A (1) ∂ A of M 1 , this can be \"lifted\" to define a null vector ℓ of M with covariant components ℓ A = ℓ (1)A (functions of the x A only) and ℓ I = 0. From equations ( 15 ), ( 16 ) it follows that if ℓ (1) is geodetic (and affinely parameterized) in M 1 then ℓ is automatically geodetic (and affinely parameterized) in M . We can thus \"compare\" the optical scalars of ℓ (1) in M 1 with those of ℓ in M . For the warped metric (15) , with the definitions (4) one finds\n\nσ 2 = σ 2 (1) + (n 1 -2)n 2 n 1 + n 2 -2 θ (1) - 1 2 (ln f ) ,A ℓ A 2 , θ = 1 n 1 + n 2 -2 (n 1 -2)θ (1) + n 2 2 (ln f ) ,A ℓ A , ( 17\n\n)\n\nω 2 = ω 2 (1)\n\n, where σ 2\n\n(1) , θ (1) and ω 2 (1) are the optical scalars of ℓ (1) in (M 1 , g (1) ). For the warped metric ( 16 ) one has\n\nσ 2 = f -2 σ 2 (1) + (n 1 -2)n 2 n 1 + n 2 -2 θ 2 (1) , θ = n 1 -2 n 1 + n 2 -2 f -1 θ (1) , (18)\n\nω 2 = f -2 ω 2 (1)\n\n.\n\nThe special case of direct products is recovered for f, f = const. (which can be rescaled to 1), in which case the shear of the full spacetime originates in the shear and expansion of the Lorentzian factor (while expansion and twist are essentially the same as in (M 1 , g (1) )).\n\nNote that for n 1 = 2 the definitions (4) for σ 2 (1) and θ (1) become formally singular because of the normalization, but for a Lorentzian 2-space (e.g., ds 2 = 2A(u, v)dudv with the geodetic null vector ℓ = A -1 ∂ v ) one has ℓ (a;b) ℓ (a;b) = ℓ a ;a = ℓ [a;b] ℓ a;b = 0, so that we can essentially take σ 2\n\n(1) = θ (1) = ω (1) = 0 and formulae ( 17 ), ( 18 ) still hold. The results of this section can be applied to several known solutions. For example, static [rotating] black strings and branes (i.e, direct products of Schwarzschild [Kerr] cross a flat space) are type D vacuum spacetimes with two shearing, expanding, twistfree [twisting] multiple WANDs. As such, they clearly \"violate\" the Golberg-Sachs theorem. In addition, spherically symmetric solutions in any dimensions (which necessarily take the metric form (15) with n 1 = 2) are type D spacetimes with two shearfree, expanding, twistfree multiple WANDs (independently of any specific field equations; in the \"exceptional case\" (ln f ) ,A ℓ A = 0 the vector ℓ is non-expanding, e.g. for Bertotti-Robinson/Nariai geometries, or for null generators of horizons)."
},
{
"section_type": "OTHER",
"section_title": "Type D Einstein spacetimes in higher dimensions",
"text": "From the results of the previous sections it follows that type D spacetimes are the simplest non-trivial examples of static/stationary (\"expanding\" and with an appropriate reflection)/warped spacetimes. Therefore we will focus on type D spacetimes in general (without assuming staticity etc.).\n\nRecall that the quantities/symbols used below (e.g. Φ ij , L ij , D) are defined in section 2."
},
{
"section_type": "OTHER",
"section_title": "Algebraic conditions following from the Bianchi equations",
"text": "Various contractions of Bianchi identities R abcd;e + R abde;c + R abec;d = 0 (19) lead to a set of first-order PDEs for frame components of the Riemann tensor given in Appendix B of [8] . In the following we shall concentrate on Einstein spaces (defined by R ab = R n g ab ), for which the same set of equations holds unchanged also for components of the Weyl tensor. In case of algebraically special spacetimes, some of these differential equations reduce to algebraical equations due to the vanishing of some components of the Weyl tensor. Here we derive algebraic conditions following from the Bianchi equations for type D Einstein spacetimes. These conditions will be employed in subsequent sections.\n\nIn particular, by contracting (19) with m (i) , ℓ, m (j) , m (k) and ℓ (equation (B.8) in [8] ) and assuming to have a type D Einstein space we get the first algebraic condition\n\nΦ ij L k -Φ ik L j + 2Φ A kj L i -C isjk L s = 0, (20)\n\nwhere we denoted L i0 by L i . We will also denote L i L i by L.\n\nThe second algebraic equation follows from equation (B.15, [8] )\n\n0 = 2 Φ A jk L im + Φ A mj L ik + Φ A km L ij + Φ ij A mk + Φ ik A jm + Φ im A kj + C isjk L sm + C ismj L sk + C iskm L sj (21)\n\nand contraction of k with i leads to\n\n0 = SΦ A mj + ΦA jm -(Φ S mi + Φ A mi )S ij + (Φ S ji + Φ A ji )S im + 2(Φ A im A ij -Φ A ij A im ) + 1 2 C ismj A si . (22)\n\nBy contracting m with j in equation (B.12) from [8] we get\n\n2DΦ S ik = 4Φ A ij A kj + Φ kj L ij + Φ ji L jk -Φ ki S -ΦL ik -2Φ S is L sk -2Φ S sk s M i0 -2Φ S is s M k0 +C ijks L sj , (23)\n\nwhere we employed\n\nC iskj s M j0 +C ijks s M j0 = 0 ( s M j0 + j M s0 = 0, cf. [8]\n\n). The symmetric part of equation (B.5, [8] ) and equation (B.3) (that is equivalent to the antisymmetric part of (B.5)) give, respectively,\n\n2DΦ S ik = -2ΦS ik + (-2Φ is + Φ si )L sk + (-2Φ ks + Φ sk )L si -2Φ S sk s M i0 -2Φ S is s M k0 , (24)\n\n2DΦ A ik = -2ΦA ik +(-2Φ is +Φ si )L sk -(-2Φ ks +Φ sk )L si -2Φ A sk s M i0 +2Φ A si s M k0 .( 25\n\n)\n\nBy subtracting ( 24 ) from ( 23 ) we finally obtain the third algebraic equation 26 ) Its antisymmetric part is, thanks to C ikjm A mj = 2C ijks A sj , equal to equation (22) and its symmetric part reads\n\n0 = -Φ ki S + ΦL ki + Φ kj L ij + 4Φ A ij A kj + (2Φ kj -Φ jk )L ji + 2Φ A ij L jk + C ijks L sj .(\n\n0 = -SΦ S ik + ΦS ik + Φ S ij S jk + Φ S kj S ij + 3(Φ A ij S jk + Φ A kj S ji ) + C ijks S sj . (27)\n\nEquations ( 20 ), ( 22 ) and ( 27 ) will be extensively used in the following sections.\n\nIn passing, let us observe here in what sense the n = 4 case is unique. Recalling the footnote on p. 4, from (20) we get L i = 0 (geodetic property) unless Φ ij = 0 (trivial case of zero Weyl tensor); equation ( 22 ) is identically satisfied (noting that necessarily Φ A ij ∝ A ij when n = 4); equation ( 27 ) implies S ij ∝ δ ij (vanishing shear) again unless Φ ij = 0. Thus for n = 4 we correctly recover the standard Goldberg-Sachs result (here restricted to type D spacetimes) that multiple WANDs (PNDs) are geodetic and shearfree in vacuum (and Einstein) spaces [1] . The situation in higher dimensions, which is qualitatively different from the n = 4 case, is studied in the following sections."
},
{
"section_type": "OTHER",
"section_title": "WANDs in \"generic\" vacuum type D and II spacetimes in arbitrary dimension are geodetic",
"text": "In this section we study equation (20) in order to determine under which circumstances the multiple WAND ℓ is geodetic.\n\nBy contracting i with k in (20) and using (6) we get\n\n3Φ A ij -Φ S ij L i = ΦL j (28)\n\nand after multiplying (28) by L j we obtain\n\nΦ S ij L i L j = -ΦL. (29)\n\nBy multiplying (20) by L i L j and using (29) we get\n\nL 3Φ A ik L i + Φ S ik L i + ΦL k = 0. ( 30\n\n)\n\nThus either L = 0 or\n\n-3Φ A ij -Φ S ij L i = ΦL j . (31)\n\nBy adding and subtracting ( 28 ) and ( 31 ) we get\n\nΦ S ij L i = -ΦL j , Φ A ij L i = 0. (32)\n\nFinally multiplying (20) by L i and using (32) we get\n\nLΦ A ij = 0. (33)\n\nThis implies that for a type D vacuum spacetime with non-vanishing Φ A ij in arbitrary dimension corresponding WANDs are geodetic.\n\nIn the case with vanishing Φ A ij , let us choose a frame in which\n\nΦ S ij is diagonal Φ S ij = diag{p (2) , p (3)\n\n, . . . , p (n-1) }. Then from the first equation (32\n\n) it follows (p (i) + Φ)L i = 0, (34)\n\nwhere (from now on) we do not sum over indices in brackets. If p (i) = -Φ, ∀i, then L i = 0, ∀i, i.e. ℓ is geodetic. Note that so far we have employed only equation (20) , which corresponds to equation (B.8) in [8] and which does not contain Weyl tensor components with negative boost order. Consequently, the same conclusions hold also for type II Einstein spacetimes.\n\nProposition 6 In arbitrary dimension, multiple WANDs of type II and D Einstein spacetimes are geodetic if at least one of the following conditions is satisfied: i) Φ A ij is non-vanishing; ii) for all eigenvalues of Φ S ij : p (i) = -Φ. Note that the above argument can not be extended to more special algebraic classes of spacetimes since it relies on the fact that some Weyl components with boost weight zero are non-vanishing. However, it was already shown in [8] that multiple WANDs in type N and III vacuum spacetimes are geodetic (in that case with no need of extra assumptions). Therefore we can conclude that under most \"generic\" conditions multiple WANDs are geodetic. Note, however, that certain special type-D vacuum solutions with Φ A ij = 0 and p (i) = -Φ (for some i) admit non-geodetic multiple WANDs. Explicit example of such spacetime is given in section 5.4."
},
{
"section_type": "OTHER",
"section_title": "Vacuum type D spacetimes with a \"shearfree\" WAND",
"text": "The algebraic equations ( 22 ) and ( 27 ) are quite complicated in general dimension and thus here we will limit ourselves to the \"shearfree\" case. This is of interest since it includes, for instance, the Robinson-Trautman solutions containing static black holes [35] .\n\nWith the \"shearfree\" condition\n\nS ij = S n-2 δ ij , (35)\n\nequation ( 27 ) leads for S = 0 to\n\nΦ S ij = Φ n-2 δ ij (S = 0), ( 36\n\n)\n\nwhereas it is identically satisfied for S = 0. In the rest of this subsection we thus consider only the \"expanding\" case S = 0. For Φ S ij in the form (36) with Φ = 0 the condition ii) of Proposition 6 is satisfied and thus the WAND ℓ is geodetic.\n\nProposition 7 In arbitrary dimension, multiple \"shearfree\" and \"expanding\" WAND in a type D Einstein spacetime is geodetic whenever Φ ij = 0. the only non-vanishing components of the Weyl tensor have boost weight zero and are given by [35]\n\nC ijkl = r 2 (R ijkl -2Kh i[k h l]j ), (40)\n\nwhere R ijkl is the Riemann tensor associated to h ij . This implies that the spacetime (38) is of type D, with Φ ij = 0, and that both ℓ and n are multiple WANDs. Now, the vector ℓ is geodetic, shearfree and twistfree by construction [35] .\n\nNext, one can easily show that\n\n∇ n n = -H ,r n + H ,i dx i , (41)\n\nwhere, by (38), H ,i = -r(ln P ) ,ui . Therefore n is geodetic if and only if (ln P ) ,ui = 0 ⇔ P = p 1 (u)p 2 (x 2 , x 3 , . . .). For a general (non-factorized) function P the multiple WAND n is thus non-geodetic (one can also easily check that it \"shearfree\", \"twistfree\" and \"expanding\"). A simple explicit example of such spacetimes is obtained by extending to any n ≥ 7 the n = 7 dimensional solution discussed in [36] , i.e. by taking in eq. ( 38 )\n\nK = -1, P = f (u, z) -1/2 ρ n-5 (det η αβ ) 1/2 1/(2-n) , h ij dx i dx j = f (u, z) dz 2 + V (ρ)dτ 2 + 1 V (ρ) dρ 2 + ρ 2 η αβ dx α dx β , (42)\n\nf (u, z) = 4b(u)e 2z/l l 2 [e 2z/l -b(u)] 2 , V (ρ) = 1 - µ ρ n-6 - ρ 2 l 2 ,\n\nwhere z ≡ x 2 , τ ≡ x 3 , ρ ≡ x 4 , η αβ = η αβ (x 5 , x 6 , . . .) is the metric of an (n -5)dimensional unit sphere (α, β = 5, . . . , n -1), µ and l are constants and b(u) > 0 is an arbitrary function. The multiple WAND n is non-geodetic as long as db/du = 0. Note that there is not contradiction with the results of the previous subsections precisely because Φ ij = 0 here."
},
{
"section_type": "OTHER",
"section_title": "Type D vacuum spacetimes in five dimensions",
"text": "Let us now study the five-dimensional case. Note that the algebraic relation (6) between -2Φ S ij and C ijkl is equivalent to the relation between the Ricci and the Riemann tensor of a m -2 dimensional space. Therefore in five dimensions C ijkl is equivalent to Φ S ij and thus a type D Weyl tensor in five dimesions is fully determined by Φ ij . In fact, for n = 5 it is possible to solve the second constraint from (6) for\n\nC ijkl : C ijkl (n=5) = 2 δ il Φ S jk -δ ik Φ S jl -δ jl Φ S ik + δ jk Φ S il -Φ (δ il δ jk -δ ik δ jl ) . (43)\n\nThus in the five dimensional case the algebraic equations we consider, ( 20 ), ( 21 ), ( 22 ), (27) , can be expressed in terms of Φ ij , L i , and L ij . Plugging (43) into (20), recalling equation ( 32 ) and contracting with L k one finds the equation\n\nLΦ S ij + 2ΦL i L j -ΦLδ ij = 0. ( 44\n\n)\n\nFor n = 5 equation ( 21 ) takes the form\n\n0 = Φ A jk L im + (Φ A im + 3Φ S im )A kj + Φ A mj L ik + (Φ A ik + 3Φ S ik )A jm + Φ A km L ij +(Φ A ij + 3Φ S ij )A mk + δ ij (Φ S ms L sk -Φ S ks L sm ) + δ ik (Φ S js L sm -Φ S ms L sj ) +δ im (Φ S ks L sj -Φ S js L sk ) + Φ[δ ij A km + δ ik A mj + δ im A jk ]. (45)\n\nSubtracting ( 61 ) and (62) we obtain (p (m) + p (j) )A mj = 0 and thus in the \"generic\" case p (m) + p (j) = 0, ∀m, j, A mj = 0. (68)\n\nProposition 11 In five dimensions, the multiple WAND ℓ in a \"generic\" (p (i) + p (j) = 0, ∀i, j) type D spacetime with Φ A ik = 0 and Φ S ik = 0, is geodetic and nontwisting (A ij = 0) and Φ S ik and S ij can be diagonalized together. There are some special cases to be treated: -Case a) one p (i) = 0 and Φ = 0: without loss of generality we choose p (2) = 0, then from ( 61\n\np (m) + p (j) )A mj = 0, 2ΦA mj = (p (j) -p (m) )S mj (69)\n\nand thus if Φ = 0, A 34 = -p\n\nΦ S 34 . If Φ = 0, then S 34 = 0 and S ij is diagonal and A 23 is arbitrary. -Case d) two pairs satisfy p (m) + p (j) = 0: without loss of generality we choose p (2) = p (3) = -p (4) = Φ. From (64) it follows that the diagonal components of S ij , s (2) and s (3) , vanish and s (4) is arbitrary. Equation (63) implies that S 24 and S 34 are arbitrary and from equation (69) we get A 23 = 0, A 24 = -S 24 , A 34 = -S 34 . This case is the non-geodetic case (48) from section 6.1."
},
{
"section_type": "OTHER",
"section_title": "An example -Myers-Perry black hole",
"text": "As an illustrative example we give S ij , A ij , Φ S ij and Φ A ij for the five-dimensional Myers-Perry black hole [9]\n\nds 2 = ρ 2 4∆ dx 2 + ρ 2 dθ 2 -dt 2 + (x + a 2 ) sin 2 θdφ 2 + (x + b 2 ) cos 2 θdψ 2 + r 0 2 ρ 2 (dt + a sin 2 θdφ + b cos 2 θdψ) 2 , where ρ 2 = x + a 2 cos 2 θ + b 2 sin 2 θ, ∆ = (x + a 2 )(x + b 2 ) -r 0 2 x.\n\nTwo (multiple, geodetic) WANDs (related by reflection symmetry) are given by [7] ℓ\n\n= (x + a 2 )(x + b 2 ) ∆ ∂ t - a x + a 2 ∂ φ - b x + b 2 ∂ ψ + 2 √ x∂ x , (70)\n\nn = α (x + a 2 )(x + b 2 ) ∆ ∂ t - a x + a 2 ∂ φ - b x + b 2 ∂ ψ -2 √ x∂ x , (71)\n\nwhere we chose α = -∆/2ρ 2 x in order to satisfy the normalization condition ℓ • n = 1.\n\nAs a basis of spacelike vectors we choose three eigenvectors of S ij m\n\n(2) = 1 ρ ∂ θ , m (3) = 1 √ xχ (-ab∂ t + b∂ φ + a∂ ψ ) , (72)\n\nm (4) = 1 ρχ (a 2 -b 2 ) sin θ cos θ∂ t -a tan -1 θ∂ φ + b tan θ∂ ψ ,\n\nwith χ = a 2 cos 2 θ + b 2 sin 2 θ. In this frame\n\nS ij = √ x ρ 2 0 0 0 1 √ x 0 0 0 √ x ρ 2 , A ij = χ ρ 2 0 0 -1 0 0 0 1 0 0 , (73) and Φ\n\nS ij = r 0 2 ρ 4 ρ 2 -2x ρ 2 0 0 0 -1 0 0 0 ρ 2 -2x ρ 2 , Φ A ij = 2r 0 2 χ √ x ρ 6 0 0 1 0 0 0 -1 0 0 .(74)\n\nNotice that in the static (Schwarzschild) limit (a = 0 = b so that ρ 2 = x) one has\n\nS ij = δ ij / √\n\nx and σ ij = 0 = A ij , and indeed for Φ ij we recover the form discussed in subsection 5.3 in the shearfree expanding case and in subsection 6.2 in the \"generic\" non-twisting case (with p (2) = p (3) = p (4) )."
},
{
"section_type": "DISCUSSION",
"section_title": "Discussion",
"text": "Let us finally outline main results presented in the paper.\n\nIn the first part of the paper (Sections 3 and 4) we study constraints on Weyl types of a spacetime following from various assumptions on geometry. It turns out that: -Static spacetimes are of types G, I i , D or conformally flat (Proposition 1).\n\n-\"Expanding\" stationary spacetimes with appropriate reflection symmetry belong to these types as well (Proposition 2).\n\n-Warped spacetimes with one-dimensional Lorentzian factor are again of types G, I i , D and O (Proposition 3).\n\n-Warped spacetimes with two-dimensional Lorentzian factor are necessarily of types D or O (Proposition 4), in particular this also applies to spherically symmetric spacetimes (Proposition 5).\n\nThese results may have useful practical applications in determining the algebraic type of specific spacetimes (or at least in ruling out some types) just by \"inspecting\" the given metric and without performing any calculations. This is particularly important in higher dimensions, where it is more difficult to determine the algebraic class of a given metric.\n\nIn the second part of the paper (sections 5 and 6) we study properties of type D vacuum spacetimes in general (without assuming that the spacetime is static, stationary or warped). In five dimensions a type D Weyl tensor is determined by a 3×3 matrix Φ ij with symmetric and antisymmetric parts being Φ A ij and Φ S ij , respectively.\n\nIn general in the non-twisting case Φ ij is symmetric while in the twisting case antisymmetric part Φ A ij appears. In higher dimensions n > 5 the (n-2)×(n-2) matrix Φ ij does not contain complete information about the Weyl tensor, but it still plays an important role. The matrix Φ ij can also be used for further classification of type D or II spacetimes, for example according to possible degeneracy of eigendirections of Φ ij . Special classes are also cases with Φ ij being symmetric or vanishing (such examples for n ≥ 7 are given in section 5.4) etc.\n\nFirst we focused on the geodeticity of multiple WANDs in type D vacuum spacetimes (these are always geodetic for n = 4). It was shown that: -The multiple WAND in a vacuum spacetime is geodetic in the \"generic\" case, i.e. if Φ A ij = 0 or if all eigenvalues of Φ S ij are distinct from minus the trace of Φ ij (Proposition 6).\n\n-It is also geodetic in the type D, shearfree case whenever Φ ij = 0 (Proposition 7). -However, explicit examples of vacuum type D spacetimes with non-geodetic multiple WAND in n ≥ 7 dimensions are given in section 5.4. This provides us with the first examples of spacetimes \"violating\" the geodetic part of the Goldberg-Sachs theorem.\n\n-In five dimensions multiple WANDs are also geodetic when Φ A ij = 0 and Φ S ij = 0 has a \"generic\" form (Proposition 11), special cases are discussed in section 6.3.\n\nProperties of the matrix Φ ij , as well as the expansion and twist matrices S ij and A ij have been also studied: -For warped spacetimes with a one/two-dimensional Lorentzian factor (thus also for static spacetimes) the antisymmetric part of Φ ij , Φ A ij , vanishes. -In vacuum type D spacetimes admitting a shearfree expanding WAND, Φ S ij is proportional to δ ij and if A ij = 0 (this always holds in odd dimensions [10] ) then Φ A ij = 0 and in the case with Φ S ij = 0 also vice versa (Proposition 8). -In five dimensions in a \"generic\" Einstein type D non-twisting spacetime, Φ A ij vanishes and eigendirections of Φ ij coincide with those of S ij (Proposition 10).\n\n-In five dimensions in a \"generic\" vacuum type D spacetime with symmetric Φ ij , the multiple WAND ℓ is non-twisting and eigendirections of Φ ij and S ij coincide (Proposition 11).\n\nThese results provide interesting connections between geometric properties of principal null congruences and Weyl curvature. Hopefully, they can be also used for constructing exact type D solutions with particular properties."
}
] | |
arxiv:0704.0440 | 0704.0440 | 1 | 10.1103/PhysRevLett.99.130402 | 242e9f1047522986c26fd027b61b1fd1b1be0f7092c6f585d9fa2765a3a83eea | Dynamics of a quantum phase transition in a ferromagnetic Bose-Einstein
condensate | We discuss dynamics of a slow quantum phase transition in a spin-1
Bose-Einstein condensate. We determine analytically the scaling properties of
the system magnetization and verify them with numerical simulations in a one
dimensional model. | [
"Bogdan Damski and Wojciech H. Zurek"
] | [
"cond-mat.other",
"hep-th",
"quant-ph"
] | cond-mat.other | [] | 2007-04-03 | 2026-02-26 | We discuss dynamics of a slow quantum phase transition in a spin-1 Bose-Einstein condensate. We determine analytically the scaling properties of the system magnetization and verify them with numerical simulations in a one dimensional model.
Studies of phase transitions have traditionally focused on equilibrium scalings of various properties near the critical point. Dynamics of the phase transition presents new challenges and there is a strong motivation for analyzing it. Nonequilibrium phase transitions may play a role in the evolution of the early Universe [1] . Their analogues can be studied in the condensed matter experiments. The latter observation led to development of the theory based on the universality of critical behavior [2] , which in turn resulted in a series of beautiful experiments [3] . The recent progress in the cold atom experiments allows for time dependent realizations of different models undergoing a quantum phase transition (QPT) [4, 5] . These experimental developments are only a proverbial tip of the iceberg, but they call for an in-depth theoretical understanding of the QPT dynamics.
A QPT is a fundamental change in ground state (GS) of the system as a result of small variations of an external parameter, e.g., a magnetic field [6] . It takes place ideally at zero absolute temperature, which is in striking contrast to thermodynamical phase transitions. The most complete description of the QPT dynamics has been obtained so far in spin models [7, 8] that are exactly solvable. In these systems the gap in the excitation spectrum goes to zero at the critical point, which precludes the adiabatic evolution across the phase boundary. It leads to creation of excitations whose density and scaling with a quench rate follow from a quantum version [7, 9] of the Kibble-Zurek (KZ) theory [1, 2] .
We study dynamics of a ferromagnetic condensate of spin-1 particles [10]. For simplicity, we consider 1D homogeneous (untrapped) clouds: atoms in a box as in the experiment [11] with spinless bosons. We adopt the parameters for our 1D model such that the length and time scales are comparable to experimental ones [12] . Assuming that the system is placed in a magnetic field B aligned in the z direction, one gets the following dimensionless mean-field energy functional [12]
E[Ψ] = dz 1 2 dΨ † dz dΨ dz + c 0 2 Ψ † Ψ 2 + Q Ψ|F 2 z |Ψ + c 1 2 α Ψ|F α |Ψ 2 (1)
where Ψ T = (ψ 1 , ψ 0 , ψ -1 ) describes the m = 0, ±1 condensate components, dz m |ψ m | 2 = 1, and F x,y,z are spin-1 matrices [13] . The first term in (1) is the kinetic energy, the second and the fourth term describe spinindependent and spin-dependent atom interactions respectively, the third term is a quadratic Zeeman shift coming from atom interactions with a magnetic field.
For 87 Rb atoms considered here c 1 < 0, which results in an interesting phase diagram due to the competition between the last two terms in (1) . Restricting analysis to zero longitudinal magnetization case, and introducing
q = Q/(n|c 1 |), n = Ψ † Ψ
one finds a polar phase for q > 2, described by Ψ T P ∼ (0, 1, 0), and the broken-symmetry phase where
Ψ T B ∼ ( 4 -2qe iχ1 , 8 + 4qe i(χ1+χ-1)/2 , 4 -2qe iχ-1 )
for 0 ≤ q < 2. The freedom of choosing the χ ±1 results in rotational symmetry of the transverse magnetization on the (x, y) plane. The transition between these phases can be driven by the change of the magnetic field B imposed on the atom cloud, q ∼ Q ∼ B 2 [14] , which was experimentally done in [5] .
The dynamics of a QPT depends on the rate of quench driving the system across the phase boundary. For very fast "impulse" transition, the system has no time to adjust to the changes of the Hamiltonian and arrives in a region where a new phase is expected with the "old" wavefunction untouched during the evolution. Slow transitions are different: the system has time to "probe" various broken symmetry "vacua" in the neighborhood of the critical point where it gets excited. We are interested in evolutions that are fast enough to produce macroscopic excitations of the system, but slow enough to reflect scalings of the critical region. By comparing analytical findings to numerical simulations for experimentally relevant parameters we provide the first complete description of QPT dynamics in a ferromagnetic condensate.
Fast transitions were realized in the Berkeley experiment [5] . The 3D numerical simulations closely following this experiment were reported in [14] . Analytical studies of the evolution after "impulse" quench were presented in [15, 16] . The paper of Lamacraft [15] also discusses dynamics of non instantaneous transitions in 2D spinor condensates focusing on analytical predictions on the growth of the transverse magnetization correlation functions.
We start with a qualitative discussion. Considering small perturbations around the GS of the brokensymmetry phase one finds three Bogolubov modes as in [13] where quantum fluctuations are studied. In the long wavelength limit (important for slow transitions) there is only one nonzero eigenvalue mode: the gapped mode having eigenenergy ∆ ∼ 4 -q 2 . Suppose now that we drive the system from polar to broken-symmetry phase. The system reaction time to Hamiltonian changes in the broken-symmetry phase is given by the inverse of the gap: τ ∼ 1 ∆ [7, 9] . For example, when τ is small enough the evolution becomes adiabatic so the system adjusts fast to parameter changes. Right after entering the brokensymmetry phase, the reaction time is large with respect to the transition time, ∆/ d∆ dt , and so the system undergoes the "impulse" evolution where its state is "frozen". The gapped mode starts to be populated around the instant t after entering the broken-symmetry phase: the system leaves the "impulse" regime to catch up with instantaneous GS solution. This happens when the two time scales become comparable: 1/∆( t) ∼ ∆/ d∆ dt | t= t. We consider here transitions driven by
q(t) = 2 -t/τ Q , (2)
where τ Q is the quench time inversely proportional to the speed of driving the system through the phase transition.
For slow transitions of interest here,
τ Q ≫ 1, we obtain t ∼ τ 1/3 Q . (3)
In the following we analyze dynamics induced by a linear decrease of q(t) (2). The evolution starts from t < 0, i.e., in the polar phase, and ends at t = 2τ Q (q = 0). Such q(t) dependence is achieved by ramping down the magnetic field as ∼ 2 -t/τ Q . The initial state is chosen as a slightly perturbed GS in the polar phase, Ψ T ∼ (δψ 1 , 1/ √ L+δψ 0 , δψ -1 ), where |δψ m | ≪ 1/ √ L are random. We generate the real and imaginary part of δΨ m at different grid points with the probability distribution p(x) = exp(-x 2 /2σ 2 )/ √ 2πσ. We take σ = 10 -4 to start evolution closely to the polar phase GS.
To find the full numerical solution within the meanfield approximation, we integrate three coupled nonlinear Schrödinger equations for the ψ m condensates that can be easily obtained by the variation of (1). During evolution we look at the magnetization of the sample
f α = Ψ|F α |Ψ , α = x, y, z.
The transverse magnetization. A total transverse (to the magnetic field in the z direction) magnetization reads
M T (t) = dz[f 2 x (z, t) + f 2 y (z, t)] = dz m T , (4)
and is experimentally measurable. It disappears in the GS of the polar phase and equals (1 -q 2 /4)/L in the broken-symmetry GS. Its typical evolution is depicted in Fig. 1 . We see there that nothing happens in the polar phase. The system starts nontrivial evolution in the 8 10 -9 70 30 (a) 10 2 -13 75 45 (b)
FIG. 2: The vectors represent (fx(z), fy(z)) × 10 3 . Plot (a): snapshot at q(t = 2.81) = 1.72, i.e., at the first peak in MT L (see Fig. 1 ). Plot (b): snapshot by the end of time evolution: q(t = 20) = 0. The results come from the same numerical simulation as in Fig. 1 (see [12] for units).
broken-symmetry phase at a distance t/τ Q after the critical point was passed. The magnetization grows fast from that point until it exceeds the static prediction and starts oscillations with the amplitude decreasing in time. We consider slow transitions. Therefore, by the end of time evolution, when q = 0, the system is in the slightly perturbed ferromagnetic GS: globally M T L ≈ 1 (Fig. 1 ) and locally L 2 m T (z) ≈ 1 (Fig. 3 ). We can now ask: Does the scaling (3) hold? To find out we define arbitrarily t as the instant when M T L intersects 1%. A fit to numerics for τ Q ≥ 10 yields ln t = (0.056 ± 0.01) + (0.332 ± 0.002) lnτ Q which confirms prediction (3). This fit is presented in Fig. 1a , where the gradual departure of the numerical data for τ Q < 10 from t ∼ τ 1/3 Q indicates that τ Q ≫ 1 or 37ms has to be taken for the observation of 1/3 exponent: quench has to be slow enough to reflect the critical dynamics. In the GS configuration of the broken-symmetry phase the vector (f x , f y ) can have arbitrary orientation, so in the dynamical problem considered here it is interesting to find out how is this rotational symmetry broken. When unstable evolution starts, spatial correlations in magnetization appear (Fig. 2a ). In the subsequent evolution these correlations evolve such that the correlation length increases: see Fig. 2b obtained by the end of time evolution. This is a generic picture though the details depend on the quench time τ Q and initial state of the system. This behavior suggests creation of spin textures [17, 18] . In our case, topological textures are spin configurations where the magnetization direction varies in space so that the kinetic energy term in (1) is not minimized, but magnetization magnitude follows closely a GS result. Such structures appear in 1D when the first homotopy group of the vacuum manifold M is nontrivial, which happens here: π 1 (M) = Z [19] . These textures are characterized by the winding number, 1 2π dz d dz Arg(f x + if y ), which is not conserved. Indeed, it reads +1 in Fig. 2a , while by the end of that evolution (Fig. 2b ) it equals 0.
Are different stages of this evolution experimentally observable? Let's look at τ Q = 10 case presented in Figs. 1 2 3 . The evolution from the phase boundary to the first peak in magnetization M T (the q = 0 point) takes 2.81 × 37ms ∼ = 104ms (2τ Q = 740ms). Both these time scales are well within the reach of the experiment [5] .
The longitudinal magnetization. Initially, f z (z) ≈ 0 so that dzf z ≈ 0. The conservation of the latter allows only for creation of a network of magnetic domains (nontopological structures with fixed f z sign) having opposite polarizations. The domains appear by the time when the system enters unstable evolution and the maxima of |f z | tend to move towards the minima of m T (Fig. 3 ). More quantitatively, we performed N r evolutions starting from different initial conditions, but fixed σ. As in the experiment [5] , we average over these runs to wash out shotto-shot fluctuations. In Fig. 4 we plot the mean domain size: ξ = i ξ z (i)/N r , where i = 1, ..., N r and ξ z (i) is the mean domain size in the i-th run. As shown in Fig. 4a , for t t we observe ξ ≈ f (t/τ 1/3 Q ) as for M T (t). The domains are formed on a time scale of ∼ t. A simple analysis based on KZ theory [1, 2] suggests that their characteristic post-transition size, ξ, should be roughly given by ∼ t 0 dtv s (t), where v s (t) is a sound velocity. There are two sound modes in the broken-symmetry phase that propagate both spin and density fluctuations [13] : the faster (slower) one has velocity ∼ √ c 0 (∼ q|c 1 |). Putting any of these as v s into the integral, and assuming τ Q ≫ 1 for the slower mode, we get
ξ ∼ τ 1/3 Q . (5)
This result correctly predicts the scaling property of the size of post-transition "defects" as is evident from the overlap of different curves in Fig. 4 , which shows up for τ Q ≥ 25 or 0.9s. Quantitatively, we define ξ as the value of ξ averaged over q ∈ [1/2, 1] to wash out post-transition fluctuations. A fit got us ln ξ = (-0.38 ± 0.03) + (0.30 ± 0.01) ln τ Q , in good agreement with (5) . The fit was done to τ Q ≥ 30 data and is presented in Fig. 4b which illustrates that smaller τ Q data gradually departs from 1/3 scaling law. Now we focus on the analytical calculations providing predictions about early stages of time-evolution. We assume that the wave-function stays close to the polar phase GS, Ψ T = (δψ
1 (t), 1/ √ L + δψ 0 (t), δψ -1 (t)) exp(-iµt), where the chemical potential is µ = c 0 /L, |δψ m | ≪ 1/
√ L, and dz(δΨ 0 + δΨ * 0 ) ≡ 0 to keep dzΨ † Ψ = 1 + O(δΨ 2 ). Linearizing the coupled nonlinear-Schrödinger equations that describe the system we get f χ = ReG χ , where χ = x, y, G x = √ 2(δΨ
1 + δΨ -1 )/ √ L, G y = i √ 2(δΨ 1 -δΨ -1 )/ √ L, and
i d dt G χ = - 1 2 d 2 dz 2 G χ + α 2 qG χ - α 2 (G χ + G * χ ),
where α = 2|c 1 |/L. To solve this equation we go to momentum space, a χ (k) = dzf χ exp(ikz) and b χ (k) = dzImG χ exp(ikz), getting
d dt a χ b χ = 1 2 0 k 2 + αq 2α -k 2 -αq 0 a χ b χ . (6)
Diagonalizing the matrix from Eq. ( 6 ) we see that there is instability for k 2 /α < 2 -q as in the Bogolubov spectrum of this model [13] . Thus, the system is stable in the polar phase, and so small initial perturbations do not grow during the evolution towards broken-symmetry phase. The instability for q < 2 is responsible for the magnetization jump in Fig. 1 and the subsequent breakdown of the linear approach (Fig. 1b ). To solve Eq. ( 6 ) with q(t) given by (2) we derive the equation for d 2 a χ (t)/dt 2 , keep leading order terms in the slow transition (τ Q ≫ 1) and long-wavelength (k 2 /α ≪ 2) limits, and get that where κ = (α 2 /2) 1/3 , α kχ and β kχ are constants given by initial conditions, while Ai and Bi are Airy functions. From (7) we see that the instability arises from unbounded increase of the Bi(s) function happening for s > 0, i.e., k 2 /α < 2-q(t), which is a dynamical manifestation of the static result for unstable modes. This solution works till t ∼ t ∼ τ 1/3 Q when a significant increase of f χ invalidates the linearized theory: this calculation rigorously derives scaling (3). Additionally, the solution ( 7 ) can be reliably used as long as τ Q ≫ 1 or 37ms, which is also supported by numerics (Fig. 1a ). The quench time scale in the experiment [5] is much smaller than this bound. Finally, these results hold for any initial state spread over the k modes.
a χ (k, t) = α kχ Ai(s)+β kχ Bi(s), s κ = t τ 1/3 Q - k 2 τ 2/3 Q α , (7)
The (re)scalings t/τ 1/3
Q and ξ ∼ t ∼ τ 1/3
Q derived above in a 1D system were also found by different means in a 2D spinor condensate [15] . A trivial extension of our meanfield analytical calculations to 2D and 3D systems shows that they hold for any number of spatial dimensions.
To summarize, we have developed a theory of the dynamics of symmetry-breaking in the quantum phase transition inspired by the experiment [5] , but for the range of quench rates that are sufficiently slow so that the critical scalings can determine phase transition dynamics. This regime should be accessible by a "slower" version of the quench [5] . Our analysis points to a Kibble-Zurek-like scenario, where the state of the system departs from the old symmetric vacuum with a delay ∼ t after the critical point was crossed. This sets up an initial post-transition state with a characteristic length scale ξ (5) . This scale should determine the initial density of topological features. In our 1D simulations textures appear, but we predict that in real 3D experiments other topological defects are created (as they were in [5] ), and the distance between them should be initially ∼ ξ. Such topological defects are more stable than textures so measurement of their density should be possible and would be a good test of the theory we have presented. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We discuss dynamics of a slow quantum phase transition in a spin-1 Bose-Einstein condensate. We determine analytically the scaling properties of the system magnetization and verify them with numerical simulations in a one dimensional model."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Studies of phase transitions have traditionally focused on equilibrium scalings of various properties near the critical point. Dynamics of the phase transition presents new challenges and there is a strong motivation for analyzing it. Nonequilibrium phase transitions may play a role in the evolution of the early Universe [1] . Their analogues can be studied in the condensed matter experiments. The latter observation led to development of the theory based on the universality of critical behavior [2] , which in turn resulted in a series of beautiful experiments [3] . The recent progress in the cold atom experiments allows for time dependent realizations of different models undergoing a quantum phase transition (QPT) [4, 5] . These experimental developments are only a proverbial tip of the iceberg, but they call for an in-depth theoretical understanding of the QPT dynamics.\n\nA QPT is a fundamental change in ground state (GS) of the system as a result of small variations of an external parameter, e.g., a magnetic field [6] . It takes place ideally at zero absolute temperature, which is in striking contrast to thermodynamical phase transitions. The most complete description of the QPT dynamics has been obtained so far in spin models [7, 8] that are exactly solvable. In these systems the gap in the excitation spectrum goes to zero at the critical point, which precludes the adiabatic evolution across the phase boundary. It leads to creation of excitations whose density and scaling with a quench rate follow from a quantum version [7, 9] of the Kibble-Zurek (KZ) theory [1, 2] .\n\nWe study dynamics of a ferromagnetic condensate of spin-1 particles [10]. For simplicity, we consider 1D homogeneous (untrapped) clouds: atoms in a box as in the experiment [11] with spinless bosons. We adopt the parameters for our 1D model such that the length and time scales are comparable to experimental ones [12] . Assuming that the system is placed in a magnetic field B aligned in the z direction, one gets the following dimensionless mean-field energy functional [12]\n\nE[Ψ] = dz 1 2 dΨ † dz dΨ dz + c 0 2 Ψ † Ψ 2 + Q Ψ|F 2 z |Ψ + c 1 2 α Ψ|F α |Ψ 2 (1)\n\nwhere Ψ T = (ψ 1 , ψ 0 , ψ -1 ) describes the m = 0, ±1 condensate components, dz m |ψ m | 2 = 1, and F x,y,z are spin-1 matrices [13] . The first term in (1) is the kinetic energy, the second and the fourth term describe spinindependent and spin-dependent atom interactions respectively, the third term is a quadratic Zeeman shift coming from atom interactions with a magnetic field.\n\nFor 87 Rb atoms considered here c 1 < 0, which results in an interesting phase diagram due to the competition between the last two terms in (1) . Restricting analysis to zero longitudinal magnetization case, and introducing\n\nq = Q/(n|c 1 |), n = Ψ † Ψ\n\none finds a polar phase for q > 2, described by Ψ T P ∼ (0, 1, 0), and the broken-symmetry phase where\n\nΨ T B ∼ ( 4 -2qe iχ1 , 8 + 4qe i(χ1+χ-1)/2 , 4 -2qe iχ-1 )\n\nfor 0 ≤ q < 2. The freedom of choosing the χ ±1 results in rotational symmetry of the transverse magnetization on the (x, y) plane. The transition between these phases can be driven by the change of the magnetic field B imposed on the atom cloud, q ∼ Q ∼ B 2 [14] , which was experimentally done in [5] .\n\nThe dynamics of a QPT depends on the rate of quench driving the system across the phase boundary. For very fast \"impulse\" transition, the system has no time to adjust to the changes of the Hamiltonian and arrives in a region where a new phase is expected with the \"old\" wavefunction untouched during the evolution. Slow transitions are different: the system has time to \"probe\" various broken symmetry \"vacua\" in the neighborhood of the critical point where it gets excited. We are interested in evolutions that are fast enough to produce macroscopic excitations of the system, but slow enough to reflect scalings of the critical region. By comparing analytical findings to numerical simulations for experimentally relevant parameters we provide the first complete description of QPT dynamics in a ferromagnetic condensate.\n\nFast transitions were realized in the Berkeley experiment [5] . The 3D numerical simulations closely following this experiment were reported in [14] . Analytical studies of the evolution after \"impulse\" quench were presented in [15, 16] . The paper of Lamacraft [15] also discusses dynamics of non instantaneous transitions in 2D spinor condensates focusing on analytical predictions on the growth of the transverse magnetization correlation functions.\n\nWe start with a qualitative discussion. Considering small perturbations around the GS of the brokensymmetry phase one finds three Bogolubov modes as in [13] where quantum fluctuations are studied. In the long wavelength limit (important for slow transitions) there is only one nonzero eigenvalue mode: the gapped mode having eigenenergy ∆ ∼ 4 -q 2 . Suppose now that we drive the system from polar to broken-symmetry phase. The system reaction time to Hamiltonian changes in the broken-symmetry phase is given by the inverse of the gap: τ ∼ 1 ∆ [7, 9] . For example, when τ is small enough the evolution becomes adiabatic so the system adjusts fast to parameter changes. Right after entering the brokensymmetry phase, the reaction time is large with respect to the transition time, ∆/ d∆ dt , and so the system undergoes the \"impulse\" evolution where its state is \"frozen\". The gapped mode starts to be populated around the instant t after entering the broken-symmetry phase: the system leaves the \"impulse\" regime to catch up with instantaneous GS solution. This happens when the two time scales become comparable: 1/∆( t) ∼ ∆/ d∆ dt | t= t. We consider here transitions driven by\n\nq(t) = 2 -t/τ Q , (2)\n\nwhere τ Q is the quench time inversely proportional to the speed of driving the system through the phase transition.\n\nFor slow transitions of interest here,\n\nτ Q ≫ 1, we obtain t ∼ τ 1/3 Q . (3)\n\nIn the following we analyze dynamics induced by a linear decrease of q(t) (2). The evolution starts from t < 0, i.e., in the polar phase, and ends at t = 2τ Q (q = 0). Such q(t) dependence is achieved by ramping down the magnetic field as ∼ 2 -t/τ Q . The initial state is chosen as a slightly perturbed GS in the polar phase, Ψ T ∼ (δψ 1 , 1/ √ L+δψ 0 , δψ -1 ), where |δψ m | ≪ 1/ √ L are random. We generate the real and imaginary part of δΨ m at different grid points with the probability distribution p(x) = exp(-x 2 /2σ 2 )/ √ 2πσ. We take σ = 10 -4 to start evolution closely to the polar phase GS.\n\nTo find the full numerical solution within the meanfield approximation, we integrate three coupled nonlinear Schrödinger equations for the ψ m condensates that can be easily obtained by the variation of (1). During evolution we look at the magnetization of the sample\n\nf α = Ψ|F α |Ψ , α = x, y, z.\n\nThe transverse magnetization. A total transverse (to the magnetic field in the z direction) magnetization reads\n\nM T (t) = dz[f 2 x (z, t) + f 2 y (z, t)] = dz m T , (4)\n\nand is experimentally measurable. It disappears in the GS of the polar phase and equals (1 -q 2 /4)/L in the broken-symmetry GS. Its typical evolution is depicted in Fig. 1 . We see there that nothing happens in the polar phase. The system starts nontrivial evolution in the 8 10 -9 70 30 (a) 10 2 -13 75 45 (b)\n\nFIG. 2: The vectors represent (fx(z), fy(z)) × 10 3 . Plot (a): snapshot at q(t = 2.81) = 1.72, i.e., at the first peak in MT L (see Fig. 1 ). Plot (b): snapshot by the end of time evolution: q(t = 20) = 0. The results come from the same numerical simulation as in Fig. 1 (see [12] for units).\n\nbroken-symmetry phase at a distance t/τ Q after the critical point was passed. The magnetization grows fast from that point until it exceeds the static prediction and starts oscillations with the amplitude decreasing in time. We consider slow transitions. Therefore, by the end of time evolution, when q = 0, the system is in the slightly perturbed ferromagnetic GS: globally M T L ≈ 1 (Fig. 1 ) and locally L 2 m T (z) ≈ 1 (Fig. 3 ). We can now ask: Does the scaling (3) hold? To find out we define arbitrarily t as the instant when M T L intersects 1%. A fit to numerics for τ Q ≥ 10 yields ln t = (0.056 ± 0.01) + (0.332 ± 0.002) lnτ Q which confirms prediction (3). This fit is presented in Fig. 1a , where the gradual departure of the numerical data for τ Q < 10 from t ∼ τ 1/3 Q indicates that τ Q ≫ 1 or 37ms has to be taken for the observation of 1/3 exponent: quench has to be slow enough to reflect the critical dynamics. In the GS configuration of the broken-symmetry phase the vector (f x , f y ) can have arbitrary orientation, so in the dynamical problem considered here it is interesting to find out how is this rotational symmetry broken. When unstable evolution starts, spatial correlations in magnetization appear (Fig. 2a ). In the subsequent evolution these correlations evolve such that the correlation length increases: see Fig. 2b obtained by the end of time evolution. This is a generic picture though the details depend on the quench time τ Q and initial state of the system. This behavior suggests creation of spin textures [17, 18] . In our case, topological textures are spin configurations where the magnetization direction varies in space so that the kinetic energy term in (1) is not minimized, but magnetization magnitude follows closely a GS result. Such structures appear in 1D when the first homotopy group of the vacuum manifold M is nontrivial, which happens here: π 1 (M) = Z [19] . These textures are characterized by the winding number, 1 2π dz d dz Arg(f x + if y ), which is not conserved. Indeed, it reads +1 in Fig. 2a , while by the end of that evolution (Fig. 2b ) it equals 0.\n\nAre different stages of this evolution experimentally observable? Let's look at τ Q = 10 case presented in Figs. 1 2 3 . The evolution from the phase boundary to the first peak in magnetization M T (the q = 0 point) takes 2.81 × 37ms ∼ = 104ms (2τ Q = 740ms). Both these time scales are well within the reach of the experiment [5] .\n\nThe longitudinal magnetization. Initially, f z (z) ≈ 0 so that dzf z ≈ 0. The conservation of the latter allows only for creation of a network of magnetic domains (nontopological structures with fixed f z sign) having opposite polarizations. The domains appear by the time when the system enters unstable evolution and the maxima of |f z | tend to move towards the minima of m T (Fig. 3 ). More quantitatively, we performed N r evolutions starting from different initial conditions, but fixed σ. As in the experiment [5] , we average over these runs to wash out shotto-shot fluctuations. In Fig. 4 we plot the mean domain size: ξ = i ξ z (i)/N r , where i = 1, ..., N r and ξ z (i) is the mean domain size in the i-th run. As shown in Fig. 4a , for t t we observe ξ ≈ f (t/τ 1/3 Q ) as for M T (t). The domains are formed on a time scale of ∼ t. A simple analysis based on KZ theory [1, 2] suggests that their characteristic post-transition size, ξ, should be roughly given by ∼ t 0 dtv s (t), where v s (t) is a sound velocity. There are two sound modes in the broken-symmetry phase that propagate both spin and density fluctuations [13] : the faster (slower) one has velocity ∼ √ c 0 (∼ q|c 1 |). Putting any of these as v s into the integral, and assuming τ Q ≫ 1 for the slower mode, we get\n\nξ ∼ τ 1/3 Q . (5)\n\nThis result correctly predicts the scaling property of the size of post-transition \"defects\" as is evident from the overlap of different curves in Fig. 4 , which shows up for τ Q ≥ 25 or 0.9s. Quantitatively, we define ξ as the value of ξ averaged over q ∈ [1/2, 1] to wash out post-transition fluctuations. A fit got us ln ξ = (-0.38 ± 0.03) + (0.30 ± 0.01) ln τ Q , in good agreement with (5) . The fit was done to τ Q ≥ 30 data and is presented in Fig. 4b which illustrates that smaller τ Q data gradually departs from 1/3 scaling law. Now we focus on the analytical calculations providing predictions about early stages of time-evolution. We assume that the wave-function stays close to the polar phase GS, Ψ T = (δψ\n\n1 (t), 1/ √ L + δψ 0 (t), δψ -1 (t)) exp(-iµt), where the chemical potential is µ = c 0 /L, |δψ m | ≪ 1/\n\n√ L, and dz(δΨ 0 + δΨ * 0 ) ≡ 0 to keep dzΨ † Ψ = 1 + O(δΨ 2 ). Linearizing the coupled nonlinear-Schrödinger equations that describe the system we get f χ = ReG χ , where χ = x, y, G x = √ 2(δΨ\n\n1 + δΨ -1 )/ √ L, G y = i √ 2(δΨ 1 -δΨ -1 )/ √ L, and\n\ni d dt G χ = - 1 2 d 2 dz 2 G χ + α 2 qG χ - α 2 (G χ + G * χ ),\n\nwhere α = 2|c 1 |/L. To solve this equation we go to momentum space, a χ (k) = dzf χ exp(ikz) and b χ (k) = dzImG χ exp(ikz), getting\n\nd dt a χ b χ = 1 2 0 k 2 + αq 2α -k 2 -αq 0 a χ b χ . (6)\n\nDiagonalizing the matrix from Eq. ( 6 ) we see that there is instability for k 2 /α < 2 -q as in the Bogolubov spectrum of this model [13] . Thus, the system is stable in the polar phase, and so small initial perturbations do not grow during the evolution towards broken-symmetry phase. The instability for q < 2 is responsible for the magnetization jump in Fig. 1 and the subsequent breakdown of the linear approach (Fig. 1b ). To solve Eq. ( 6 ) with q(t) given by (2) we derive the equation for d 2 a χ (t)/dt 2 , keep leading order terms in the slow transition (τ Q ≫ 1) and long-wavelength (k 2 /α ≪ 2) limits, and get that where κ = (α 2 /2) 1/3 , α kχ and β kχ are constants given by initial conditions, while Ai and Bi are Airy functions. From (7) we see that the instability arises from unbounded increase of the Bi(s) function happening for s > 0, i.e., k 2 /α < 2-q(t), which is a dynamical manifestation of the static result for unstable modes. This solution works till t ∼ t ∼ τ 1/3 Q when a significant increase of f χ invalidates the linearized theory: this calculation rigorously derives scaling (3). Additionally, the solution ( 7 ) can be reliably used as long as τ Q ≫ 1 or 37ms, which is also supported by numerics (Fig. 1a ). The quench time scale in the experiment [5] is much smaller than this bound. Finally, these results hold for any initial state spread over the k modes.\n\na χ (k, t) = α kχ Ai(s)+β kχ Bi(s), s κ = t τ 1/3 Q - k 2 τ 2/3 Q α , (7)\n\nThe (re)scalings t/τ 1/3\n\nQ and ξ ∼ t ∼ τ 1/3\n\nQ derived above in a 1D system were also found by different means in a 2D spinor condensate [15] . A trivial extension of our meanfield analytical calculations to 2D and 3D systems shows that they hold for any number of spatial dimensions.\n\nTo summarize, we have developed a theory of the dynamics of symmetry-breaking in the quantum phase transition inspired by the experiment [5] , but for the range of quench rates that are sufficiently slow so that the critical scalings can determine phase transition dynamics. This regime should be accessible by a \"slower\" version of the quench [5] . Our analysis points to a Kibble-Zurek-like scenario, where the state of the system departs from the old symmetric vacuum with a delay ∼ t after the critical point was crossed. This sets up an initial post-transition state with a characteristic length scale ξ (5) . This scale should determine the initial density of topological features. In our 1D simulations textures appear, but we predict that in real 3D experiments other topological defects are created (as they were in [5] ), and the distance between them should be initially ∼ ξ. Such topological defects are more stable than textures so measurement of their density should be possible and would be a good test of the theory we have presented."
}
] | |
arxiv:0704.0442 | 0704.0442 | 1 | 10.1364/JOSAB.24.002195 | 552e0178ea613901c96919ff19108b49213ee274a88c9ee7976deec1049e13b7 | Quantum electromagnetic X-waves | We show that two distinct quantum states of the electromagnetic field can be
associated to a classical vector X wave or a propagation-invariant solution of
Maxwell equations. The difference between the two states is of pure quantum
mechanical origin since they are internally entangled and disentangled,
respectively and can be generated by different linear or nonlinear processes.
Detection and generation of Schr\"odinger-cat states comprising two entangled
X-waves and their possible applications are discussed. | [
"Alessandro Ciattoni",
"Claudio Conti"
] | [
"physics.optics",
"physics.gen-ph",
"quant-ph"
] | physics.optics | [] | 2007-04-03 | 2026-02-26 | We show that two distinct quantum states of the electromagnetic field can be associated to a classical vector X wave or a propagation-invariant solution of Maxwell equations. The difference between the two states is of pure quantum mechanical origin since they are internally entangled and disentangled, respectively and can be generated by different linear or nonlinear processes. Detection and generation of Schrödinger-cat states comprising two entangled X-waves and their possible applications are discussed.
Electromagnetic X waves and, more generally, "localized waves" [1] are propagation-invariant solutions of classical Maxwell equations in vacuum [2, 3, 4] , dielectric media [5] , plasmas [6] , optically nonlinear materials [7, 8, 9, 10] and periodic structures [11, 12, 13, 14] . In recent years these waves have attracted a considerable interest since, despite at a first glance they are non-physical (for example some of them apparently travel superluminally and carry an infinite amount of energy), propagation-invariant fields support several applications like image transmission, reconstruction and telecommunications (for a recent review see Ref. [1] ).
Since the first observation of localized optical X waves [15] , the fundamental implications of rigidly travelling spatiotemporal correlations have been recognized. Classically, these waves can be generated by means of linear devices (like axicons) [15] and, recently, their experimental feasibility has been also proved through nonlinear optical processes [16, 17] . Within the quantum framework, analytical techniques borrowed from the subject of localized waves have allowed some authors to consider highly localized states for the photon-wavefunction [18] , whereas the role of quantum X waves in nonlinear optical process (limited to the paraxial regime) were analyzed in [19] .
In this paper we consider the issue of "constructing a quantum state of the electromagnetic field whose classical counterpart is an X-wave satisfying the full set of Maxwell equations in vacuum". More precisely, we investigate the possibility of introducing various quantum states on which the mean value of the electric field operator coincides with a prescribed classical vector X wave. We show that the solution of this problem is not-trivial by finding two different quantum states characterized by the above property. Even if from a classical perspective they describe the same entity, these two states essentially differ in the degree of entanglement of their internal modal structure so that both a "disentangled" and an "entangled" quantum description can be provided of an arbitrary classical X wave. In this respect one can investigate a method for discerning between the two possibilities and, correspondingly, for recognizing which kind of quantum X wave is produced by a given mechanism (e.g. axicons or nonlinear processes). In addition, we show that some X wave states can be considered as multi-mode, multi-dimensional Schrödinger cats, exhibiting the relevant feature of being propagation invariant; the corresponding macroscopic quantum content is controlled by the velocity of the X waves and can be maintained for long distances.
Within a classical formulation, the x-component of the electric field of a radially symmetric linearly polarized X wave can be written as (extension to more general X waves solutions [1] is trivial)
E x (r, t) = ℜ ∞ 0 dkf (k)J 0 (sin θ 0 kr ⊥ )e ik cos θ0 z-c cos θ 0 t (1)
where r ⊥ = x 2 + y 2 and ℜ stands for the real part, whereas the longitudinal component is simply expressed in terms of the transversal components. (see e.g. [4] ) This kind of classical waves propagates along the z-axis without distortion and it is fully characterized by its spectrum f (k) and its velocity c/ cos θ 0 . Physically, such an X wave arises as the superposition of various plane-waves at different frequencies whose wave-vectors are all inclined at the same angle θ 0 with the propagation direction z. In order to give a quantum description of X waves, we construct a quantum state of the electromagnetic field through the requirement that the quantum mean value of the electric field operator on this state yields a propagation invariant wave of the kind of Eq.( 1 ) and we show that this can be done in two different manners. In order to fix the notation, we start from the standard expression of the electric field operator in Heisenberg representation, Ê(r, t) = k,s hc|k| 2ε0L 3 iâ k,s e i(k•r-c|k|t) e k,s + h.c., where L is the edge of the quantization cubic box (see e.g. [20, 21] ), k = (2π/L)(n x e x + n y e y + n z e z ) (where n x , n y and n z are integers), e k,s are the pair (s = 1, 2) of polarization unit vectors associated to the mode k and âk,s are the standard photon annihilation operators. Bearing in mind that the mean value of the electric field operator on a coherent state is a plane wave, let us consider the quantum states
|X = k,s Dk,s [∆ θ0 (k)β(k, s)]|0 (2)
where Dk,s (α) is the displacement operator which, acting on the vacuum state |0 , produces the coherent state |α k,s (associated to the mode k, s) according to the relation Dk,s (α)|0 = |α k,s [20] . Here β(k, s) is a complex weight function to be determined below and, in order to deal with the discreteness of the k vectors, we have defined ∆ θ0 (k) = 1 if k ∈ C(θ 0 , δθ) and ∆ θ0 (k) = 0 elsewhere, where C(θ 0 , δθ) is the portion of space comprised between the two coaxial cones of aperture angles θ 0δθ/2 and θ 0 + δθ/2 and whose common axis coincides with the z-axis. The presence of the function ∆ θ0 (k) assures that the state |X of Eq.( 2 ) is the tensor product of coherent states whose modes have wave vectors globally lying, for δθ ≪ θ 0 , on a cone with aperture angle θ 0 so that the relation k z = η|k ⊥ | with η = 1/ tan θ 0 holds for each wave vector. The mean value of the electric field operator on the state |X is given by
X| Ê|X = ℜ k,s ∆ θ0 (k) 2hc|k| ε 0 L 3 iβ(k, s) e i(k•r-c|k|t) e k,s (3)
from which we envisage that, for L → ∞ and δθ ≪ θ 0 , X| Ê(r, t)|X → E(x, y, z-V t), which is a genuine propagationinvariant vector field travelling along the z-axis. In order to prove this assertion we note that the limit is different from zero only if β is dependent on δθ and L and it is given by
β(k, s) = (2π) 3 iδθ ε 0 2hc|k|L 3 |k| |k ⊥ | f s (k), (4)
where k ⊥ = k x e x + k y e y , f s (k) are two arbitrary function of k and the various coefficients are chosen for later convenience. Substituting Eq.( 4 ) into Eq.( 3 ) and performing the limits L → +∞ and δθ → 0 by means of the rule
1 L 3 k → 1 (2π) 3 d 3 k and the relation lim δθ→0 ∆ θ0 (k)/δθ = δ(θ -θ 0 ) we obtain X| Ê|X = ℜ ∞ 0 dkk 2 2π 0 dφ [f 1 (k)e k1 + f 2 (k)e k2 ] e i(k•r-ckt) (5)
where, after defining k = k[sin θ 0 (cos φe x + sin φe y ) + cos θ 0 e z ], polar coordinates has been introduced for the k integration. Note that, inside the integral of Eq.( 5 ), the relation k•r-ckt = k[(x cos φ sin θ 0 +y sin φ sin θ 0 )+cos θ 0 (zct/ cos θ 0 )] holds, so that the field in Eq.( 5 ) describes a wave travelling undistorted along the z-axis with velocity V = c/ cos θ 0 , as expected. Note that, strictly speaking, this rigorous nondiffracting behavior is a consequence of the limit δθ → 0 which is an idealization of the realistic condition δθ ≪ θ 0 , δθ representing the experimental uncertainty of the wave vectors smeared around the selected conical surface. Since δθ can be experimentally chosen much smaller than θ 0 , the actual fields are nearly nondiffracting objects, their departure from ideal fields being experimentally tunable.
A convenient choice for the modes polarization unit vectors is given by e k1 = cos θ 0 (cos φe x + sin φe y )sin θ 0 e z and e k2 =sin φe x + cos φe y so that a linearly polarized wave along the x-direction is obtained by letting
f 2 = -f 1 tan φ cos θ 0 with f 1 = (cos φ/ cos θ 0 )[f (k)/k 2 ] from which we get f 1 (k)e k1 + f 2 (k)e k2 = f (k) (e x -
cos φ tan θ 0 e z ) /k 2 where the factor k -2 has been added for later convenience. Inserting this expression into Eq.( 5 ), we obtain
X| Ê|X = ℜ ∞ 0 dkf (k) J 0 (sin θ 0 kr ⊥ ) e x -
x tan θ 0 r ⊥ J 1 (sin θ 0 kr ⊥ ) e z e ik cos θ0(z-V t)
where J n (ξ) is the Bessel function of the first kind of order n and the relation 2π 0 dφe ih(x cos φ+y sin φ) = 2πJ 0 (hr ⊥ ) has been exploited. The expectation value of the electric field in Eqs.(6) describes a linearly polarized X-waves (see Eq.( 1 )) which is an exact solution of Maxwell equations in vacuum (see Ref. [4] ). Note that the longitudinal component E z is due to the full non-paraxial character of the exact approach we are considering and it is negligible for small aperture angles θ 0 .
Let us consider the state of the electromagnetic field given by
|X E = k,s ∆ θ0 (k)Φ(k, s) Dk,s [α(k, s)] |0 (7)
where α(k, s) and Φ(k, s) are arbitrary complex function, the second one being constrained by the normalization conditions X E |X E = 1. Note that the very presence of the function ∆ θ0 (k) in Eq.( 7 ) implies that modes are collected in such a way that their wave vectors almost lies, in the limit δθ ≪ θ 0 , on the surface of the cone characterizing the spectrum of an arbitrary classical X wave and, therefore |X E is expected to describe a quantum X wave. In order to prove this assertion we note that
X E |â ks |X E = ∆ θ0 (k)α(k, s)Q(k, s) (8)
where
Q(k, s) = |Φ(k, s)| 2 + Φ(k, s)e -1 2 |α(k,s)| 2 k ′ =k,s ′ =s ∆ θ0 (k ′ )Φ * (k ′ , s ′ )e -1 2 |α(k ′ ,s ′ )| 2 so
that the mean value of the electric field on |X E turns out to be
X E | Ê|X E = ℜ k,s ∆ θ0 (k)G(k, s)e i(k•r-c|k|t) e k,s (9)
where G(k, s) = i 2hc|k| ε0L 3 α(k, s)Q(k, s). Comparing Eq.( 9 ) with Eq.( 3 ) and noting that their structures are identical, we deduce that, in the limit δθ → 0 and L → ∞, the state |X E describes a quantum X wave. It is worth noting that this state is completely different from the |X of Eq.( 2 ) and it is obtained as a linear combination of the states Dk,s [α(k, s)]|0 each corresponding to a coherent state in the mode (k, s), all other modes being in the vacuum state. This implies that the state |X E arises from an entangled superposition of modes in coherent states, where with "entanglement" we mean that the state cannot be expressed as a product of kets containing different modes. We therefore conclude that a prescribed classical X wave can be represented by two different quantum states |X and |X E exhibiting an internal disentangled and entangled modal structure, respectively whose quantum difference is expected to play a fundamental role during the measurement process. As a matter of fact, a realistic measurement device sensible to a bounded set S of wave vectors k affects the states |X and |X E in a dramatically different way after the measurement since a measurement carried out on |X leaves the modes k / ∈ S unaltered whereas the same measurement on |X E profoundly changes the structure of the modes k / ∈ S causing, as usual, an irreversible loss of information and notably of the propagation invariance.
The same difference between the two proposed quantum X waves also raises the problem of discerning which one (i.e. disentangled or entangled) is obtained from a prescribed mechanism capable of generating a classical X wave. Since the disentangled X wave is just the product of coherent states with different frequencies but with the same axicon angle, |X is well expected to be produced by a simple linear device, namely an axicon (as in [15] ), exposed to a coherent non-monochromatic source like a mode-locked laser. Conversely the entangled X wave is expected when considering nonlinear processes since it is well known that frequency-conversion or parametric processes are accompanied by tight phase-matching condition able to provide the spectral structure characterizing a classical X wave, as addressed by various authors with reference to various kind of nonlinearities (as in [22, 23, 24, 25] ).
A viable scheme for detecting entanglement on a generated X-waves is offered by homodyne detection that select a specific mode, eventually including a prism or a grating spatially separating the various angular frequencies. This correspond to consider the projection of the state |X E onto a specific mode (at frequency ω 1 and within a normalization constant)
|X E projected = |α 1 + Φ E |0 (10)
where Φ E will in general be dependent on the specific X-wave or on its axicon angle. Using homodyne detection (see [26] ) and tuning the local field phase θ it is possible to unveil fringes in the probability distribution of the detected current. As an example, for θ =arg(α 1 ) + π/2, the probability of detecting the value x of the output current | x|α | 2 is an oscillating function:
| x|α | 2 = e -x 2 √ π 1 + |Φ E | 2 + 2|Φ E | cos( √ 2|α 1 |x -arg Φ E ) . (11)
Note that the same reasoning with the state |X yields Φ E = 0 so that oscillations in | x|α | 2 do not appear if the original state is not entangled. More general results can be obtained by considering beam-splitters and generic phases θ, but their mathematical description is cumbersome and will be reported elsewhere.
In addition to the considered X waves, we discuss here the interesting possibility of turning a disentangled X wave into an entangled one by letting the former (appropriately generated by an axicon) to pass through a nonlinear Kerr medium. In order to avoid mathematical complications, we consider the typical spectrum of a broad band mode-locked laser containing M harmonics ω m = mω 0 (m = 1, 2, ..., M ) and we focus on the case where each mode is in a coherent state corresponding to a plane wave propagating with the conical angle θ 0 . The disentangled X wave generated by the axicon is formed by the modes whose wave vectors are such that k m = mω 0 /c with k z = k m /(1 + η 2 ) and reads |X = |α 1 1 |α 2 2 ...|α M M where α m denotes the complex parameter of the coherent state with angular frequency ω m . When travelling through a Kerr medium the various modes induce cross and self-phase modulation and the relevant and well-known interaction Hamiltonian corresponds to that of a nonlinear oscillator (see [26] ) and reads ĤI = χ(h/2) m,p nm np , where nm = â † m âm is the photon number operator of the mode m. If t is the interaction time, the output state from the crystal |X out = e -i h ĤI t |X is given by
|X out = e -1 2 M r=0 |αr | 2 ∞ n1,n2,...,nM =0 e -i 2 χt M m,p=0 nmnp q=1,2,...,M α nq q n q ! |n q . (12)
Note that, for t = 4π/χ the output state coincides with the incoming one whereas for t = 2π/χ the state |X out is obtained from |X after the replacement α m → -α m and the resulting disentangled X wave state can be denoted as | -X . More interesting is the situation for t = π/χ since, defining N = m n m and exploiting the relation e -i(π/2)N 2 = [e -iπ/4 + (-1) N e iπ/4 ]/ √ 2, we obtain obtain
|X out = 1 √ 2 e -iπ/4 |X + e iπ/4 | -X . (13)
which is an entangled superposition of two disentangled quantum X waves travelling at the same velocity. The key point is here that the interaction time t coincides with the time spent by the classical X wave to pass through the nonlinear medium or t = D cos θ 0 /c, D being the nonlinear medium length. As for coherent states, the states periodically evolve with temporal period 4π/χ and with spatial period 4πc/(cos θ 0 χ). As a consequence, by simply acting on the velocity of the X-wave (e.g. by varying the axicon angle) it is possible to fine-tune the degree of entanglement of the two macroscopic classical X-waves, and even to switching from classical states to Schrödinger-cat states. This is a not-trivial outcome of the interplay between the classical interference process supporting X waves and the entanglement of coherent states. In addition, X waves travel at a velocity which is different from the other linear waves so that, by using coincidences measurements, it is straightforward to distinguish these states from a statistical mixture, and hence to point out their purely quantum properties as described above.
In conclusion we have shown that classical electromagnetic X-waves may hide different degrees of quantum entanglement, depending on their generation mechanism (exploiting a linear, a nonlinear medium or both of them). In this respect their velocity (i.e. the axicon angle) plays a prominent role. Other interesting consequences of the X-waves structure arise when dealing with interferometric setup, as for example that considered in [27] . If a nonlinear medium is placed in one or both arms of an interferometer, the output state is a "progressive undistorted squeezed vacuum" or entangled superposition of classical X waves so that any detection scheme is affected by the axicon angle, and tuning among various quantum states can be attained, as it will be detailed elsewhere. The quantum properties of X-waves can hence be exploited for free-space quantum communications, highly sensible interferometers or quantum computing. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We show that two distinct quantum states of the electromagnetic field can be associated to a classical vector X wave or a propagation-invariant solution of Maxwell equations. The difference between the two states is of pure quantum mechanical origin since they are internally entangled and disentangled, respectively and can be generated by different linear or nonlinear processes. Detection and generation of Schrödinger-cat states comprising two entangled X-waves and their possible applications are discussed."
},
{
"section_type": "BACKGROUND",
"section_title": "INTRODUCTION",
"text": "Electromagnetic X waves and, more generally, \"localized waves\" [1] are propagation-invariant solutions of classical Maxwell equations in vacuum [2, 3, 4] , dielectric media [5] , plasmas [6] , optically nonlinear materials [7, 8, 9, 10] and periodic structures [11, 12, 13, 14] . In recent years these waves have attracted a considerable interest since, despite at a first glance they are non-physical (for example some of them apparently travel superluminally and carry an infinite amount of energy), propagation-invariant fields support several applications like image transmission, reconstruction and telecommunications (for a recent review see Ref. [1] ).\n\nSince the first observation of localized optical X waves [15] , the fundamental implications of rigidly travelling spatiotemporal correlations have been recognized. Classically, these waves can be generated by means of linear devices (like axicons) [15] and, recently, their experimental feasibility has been also proved through nonlinear optical processes [16, 17] . Within the quantum framework, analytical techniques borrowed from the subject of localized waves have allowed some authors to consider highly localized states for the photon-wavefunction [18] , whereas the role of quantum X waves in nonlinear optical process (limited to the paraxial regime) were analyzed in [19] .\n\nIn this paper we consider the issue of \"constructing a quantum state of the electromagnetic field whose classical counterpart is an X-wave satisfying the full set of Maxwell equations in vacuum\". More precisely, we investigate the possibility of introducing various quantum states on which the mean value of the electric field operator coincides with a prescribed classical vector X wave. We show that the solution of this problem is not-trivial by finding two different quantum states characterized by the above property. Even if from a classical perspective they describe the same entity, these two states essentially differ in the degree of entanglement of their internal modal structure so that both a \"disentangled\" and an \"entangled\" quantum description can be provided of an arbitrary classical X wave. In this respect one can investigate a method for discerning between the two possibilities and, correspondingly, for recognizing which kind of quantum X wave is produced by a given mechanism (e.g. axicons or nonlinear processes). In addition, we show that some X wave states can be considered as multi-mode, multi-dimensional Schrödinger cats, exhibiting the relevant feature of being propagation invariant; the corresponding macroscopic quantum content is controlled by the velocity of the X waves and can be maintained for long distances."
},
{
"section_type": "OTHER",
"section_title": "QUANTUM DESCRIPTION OF X WAVES",
"text": "Within a classical formulation, the x-component of the electric field of a radially symmetric linearly polarized X wave can be written as (extension to more general X waves solutions [1] is trivial)\n\nE x (r, t) = ℜ ∞ 0 dkf (k)J 0 (sin θ 0 kr ⊥ )e ik cos θ0 z-c cos θ 0 t (1)\n\nwhere r ⊥ = x 2 + y 2 and ℜ stands for the real part, whereas the longitudinal component is simply expressed in terms of the transversal components. (see e.g. [4] ) This kind of classical waves propagates along the z-axis without distortion and it is fully characterized by its spectrum f (k) and its velocity c/ cos θ 0 . Physically, such an X wave arises as the superposition of various plane-waves at different frequencies whose wave-vectors are all inclined at the same angle θ 0 with the propagation direction z. In order to give a quantum description of X waves, we construct a quantum state of the electromagnetic field through the requirement that the quantum mean value of the electric field operator on this state yields a propagation invariant wave of the kind of Eq.( 1 ) and we show that this can be done in two different manners. In order to fix the notation, we start from the standard expression of the electric field operator in Heisenberg representation, Ê(r, t) = k,s hc|k| 2ε0L 3 iâ k,s e i(k•r-c|k|t) e k,s + h.c., where L is the edge of the quantization cubic box (see e.g. [20, 21] ), k = (2π/L)(n x e x + n y e y + n z e z ) (where n x , n y and n z are integers), e k,s are the pair (s = 1, 2) of polarization unit vectors associated to the mode k and âk,s are the standard photon annihilation operators. Bearing in mind that the mean value of the electric field operator on a coherent state is a plane wave, let us consider the quantum states\n\n|X = k,s Dk,s [∆ θ0 (k)β(k, s)]|0 (2)\n\nwhere Dk,s (α) is the displacement operator which, acting on the vacuum state |0 , produces the coherent state |α k,s (associated to the mode k, s) according to the relation Dk,s (α)|0 = |α k,s [20] . Here β(k, s) is a complex weight function to be determined below and, in order to deal with the discreteness of the k vectors, we have defined ∆ θ0 (k) = 1 if k ∈ C(θ 0 , δθ) and ∆ θ0 (k) = 0 elsewhere, where C(θ 0 , δθ) is the portion of space comprised between the two coaxial cones of aperture angles θ 0δθ/2 and θ 0 + δθ/2 and whose common axis coincides with the z-axis. The presence of the function ∆ θ0 (k) assures that the state |X of Eq.( 2 ) is the tensor product of coherent states whose modes have wave vectors globally lying, for δθ ≪ θ 0 , on a cone with aperture angle θ 0 so that the relation k z = η|k ⊥ | with η = 1/ tan θ 0 holds for each wave vector. The mean value of the electric field operator on the state |X is given by\n\nX| Ê|X = ℜ k,s ∆ θ0 (k) 2hc|k| ε 0 L 3 iβ(k, s) e i(k•r-c|k|t) e k,s (3)\n\nfrom which we envisage that, for L → ∞ and δθ ≪ θ 0 , X| Ê(r, t)|X → E(x, y, z-V t), which is a genuine propagationinvariant vector field travelling along the z-axis. In order to prove this assertion we note that the limit is different from zero only if β is dependent on δθ and L and it is given by\n\nβ(k, s) = (2π) 3 iδθ ε 0 2hc|k|L 3 |k| |k ⊥ | f s (k), (4)\n\nwhere k ⊥ = k x e x + k y e y , f s (k) are two arbitrary function of k and the various coefficients are chosen for later convenience. Substituting Eq.( 4 ) into Eq.( 3 ) and performing the limits L → +∞ and δθ → 0 by means of the rule\n\n1 L 3 k → 1 (2π) 3 d 3 k and the relation lim δθ→0 ∆ θ0 (k)/δθ = δ(θ -θ 0 ) we obtain X| Ê|X = ℜ ∞ 0 dkk 2 2π 0 dφ [f 1 (k)e k1 + f 2 (k)e k2 ] e i(k•r-ckt) (5)\n\nwhere, after defining k = k[sin θ 0 (cos φe x + sin φe y ) + cos θ 0 e z ], polar coordinates has been introduced for the k integration. Note that, inside the integral of Eq.( 5 ), the relation k•r-ckt = k[(x cos φ sin θ 0 +y sin φ sin θ 0 )+cos θ 0 (zct/ cos θ 0 )] holds, so that the field in Eq.( 5 ) describes a wave travelling undistorted along the z-axis with velocity V = c/ cos θ 0 , as expected. Note that, strictly speaking, this rigorous nondiffracting behavior is a consequence of the limit δθ → 0 which is an idealization of the realistic condition δθ ≪ θ 0 , δθ representing the experimental uncertainty of the wave vectors smeared around the selected conical surface. Since δθ can be experimentally chosen much smaller than θ 0 , the actual fields are nearly nondiffracting objects, their departure from ideal fields being experimentally tunable.\n\nA convenient choice for the modes polarization unit vectors is given by e k1 = cos θ 0 (cos φe x + sin φe y )sin θ 0 e z and e k2 =sin φe x + cos φe y so that a linearly polarized wave along the x-direction is obtained by letting\n\nf 2 = -f 1 tan φ cos θ 0 with f 1 = (cos φ/ cos θ 0 )[f (k)/k 2 ] from which we get f 1 (k)e k1 + f 2 (k)e k2 = f (k) (e x -\n\ncos φ tan θ 0 e z ) /k 2 where the factor k -2 has been added for later convenience. Inserting this expression into Eq.( 5 ), we obtain\n\nX| Ê|X = ℜ ∞ 0 dkf (k) J 0 (sin θ 0 kr ⊥ ) e x -\n\nx tan θ 0 r ⊥ J 1 (sin θ 0 kr ⊥ ) e z e ik cos θ0(z-V t)\n\nwhere J n (ξ) is the Bessel function of the first kind of order n and the relation 2π 0 dφe ih(x cos φ+y sin φ) = 2πJ 0 (hr ⊥ ) has been exploited. The expectation value of the electric field in Eqs.(6) describes a linearly polarized X-waves (see Eq.( 1 )) which is an exact solution of Maxwell equations in vacuum (see Ref. [4] ). Note that the longitudinal component E z is due to the full non-paraxial character of the exact approach we are considering and it is negligible for small aperture angles θ 0 .\n\nLet us consider the state of the electromagnetic field given by\n\n|X E = k,s ∆ θ0 (k)Φ(k, s) Dk,s [α(k, s)] |0 (7)\n\nwhere α(k, s) and Φ(k, s) are arbitrary complex function, the second one being constrained by the normalization conditions X E |X E = 1. Note that the very presence of the function ∆ θ0 (k) in Eq.( 7 ) implies that modes are collected in such a way that their wave vectors almost lies, in the limit δθ ≪ θ 0 , on the surface of the cone characterizing the spectrum of an arbitrary classical X wave and, therefore |X E is expected to describe a quantum X wave. In order to prove this assertion we note that\n\nX E |â ks |X E = ∆ θ0 (k)α(k, s)Q(k, s) (8)\n\nwhere\n\nQ(k, s) = |Φ(k, s)| 2 + Φ(k, s)e -1 2 |α(k,s)| 2 k ′ =k,s ′ =s ∆ θ0 (k ′ )Φ * (k ′ , s ′ )e -1 2 |α(k ′ ,s ′ )| 2 so\n\nthat the mean value of the electric field on |X E turns out to be\n\nX E | Ê|X E = ℜ k,s ∆ θ0 (k)G(k, s)e i(k•r-c|k|t) e k,s (9)\n\nwhere G(k, s) = i 2hc|k| ε0L 3 α(k, s)Q(k, s). Comparing Eq.( 9 ) with Eq.( 3 ) and noting that their structures are identical, we deduce that, in the limit δθ → 0 and L → ∞, the state |X E describes a quantum X wave. It is worth noting that this state is completely different from the |X of Eq.( 2 ) and it is obtained as a linear combination of the states Dk,s [α(k, s)]|0 each corresponding to a coherent state in the mode (k, s), all other modes being in the vacuum state. This implies that the state |X E arises from an entangled superposition of modes in coherent states, where with \"entanglement\" we mean that the state cannot be expressed as a product of kets containing different modes. We therefore conclude that a prescribed classical X wave can be represented by two different quantum states |X and |X E exhibiting an internal disentangled and entangled modal structure, respectively whose quantum difference is expected to play a fundamental role during the measurement process. As a matter of fact, a realistic measurement device sensible to a bounded set S of wave vectors k affects the states |X and |X E in a dramatically different way after the measurement since a measurement carried out on |X leaves the modes k / ∈ S unaltered whereas the same measurement on |X E profoundly changes the structure of the modes k / ∈ S causing, as usual, an irreversible loss of information and notably of the propagation invariance."
},
{
"section_type": "OTHER",
"section_title": "GENERATION AND DETECTION OF QUANTUM X WAVES",
"text": "The same difference between the two proposed quantum X waves also raises the problem of discerning which one (i.e. disentangled or entangled) is obtained from a prescribed mechanism capable of generating a classical X wave. Since the disentangled X wave is just the product of coherent states with different frequencies but with the same axicon angle, |X is well expected to be produced by a simple linear device, namely an axicon (as in [15] ), exposed to a coherent non-monochromatic source like a mode-locked laser. Conversely the entangled X wave is expected when considering nonlinear processes since it is well known that frequency-conversion or parametric processes are accompanied by tight phase-matching condition able to provide the spectral structure characterizing a classical X wave, as addressed by various authors with reference to various kind of nonlinearities (as in [22, 23, 24, 25] ).\n\nA viable scheme for detecting entanglement on a generated X-waves is offered by homodyne detection that select a specific mode, eventually including a prism or a grating spatially separating the various angular frequencies. This correspond to consider the projection of the state |X E onto a specific mode (at frequency ω 1 and within a normalization constant)\n\n|X E projected = |α 1 + Φ E |0 (10)\n\nwhere Φ E will in general be dependent on the specific X-wave or on its axicon angle. Using homodyne detection (see [26] ) and tuning the local field phase θ it is possible to unveil fringes in the probability distribution of the detected current. As an example, for θ =arg(α 1 ) + π/2, the probability of detecting the value x of the output current | x|α | 2 is an oscillating function:\n\n| x|α | 2 = e -x 2 √ π 1 + |Φ E | 2 + 2|Φ E | cos( √ 2|α 1 |x -arg Φ E ) . (11)\n\nNote that the same reasoning with the state |X yields Φ E = 0 so that oscillations in | x|α | 2 do not appear if the original state is not entangled. More general results can be obtained by considering beam-splitters and generic phases θ, but their mathematical description is cumbersome and will be reported elsewhere.\n\nIn addition to the considered X waves, we discuss here the interesting possibility of turning a disentangled X wave into an entangled one by letting the former (appropriately generated by an axicon) to pass through a nonlinear Kerr medium. In order to avoid mathematical complications, we consider the typical spectrum of a broad band mode-locked laser containing M harmonics ω m = mω 0 (m = 1, 2, ..., M ) and we focus on the case where each mode is in a coherent state corresponding to a plane wave propagating with the conical angle θ 0 . The disentangled X wave generated by the axicon is formed by the modes whose wave vectors are such that k m = mω 0 /c with k z = k m /(1 + η 2 ) and reads |X = |α 1 1 |α 2 2 ...|α M M where α m denotes the complex parameter of the coherent state with angular frequency ω m . When travelling through a Kerr medium the various modes induce cross and self-phase modulation and the relevant and well-known interaction Hamiltonian corresponds to that of a nonlinear oscillator (see [26] ) and reads ĤI = χ(h/2) m,p nm np , where nm = â † m âm is the photon number operator of the mode m. If t is the interaction time, the output state from the crystal |X out = e -i h ĤI t |X is given by\n\n|X out = e -1 2 M r=0 |αr | 2 ∞ n1,n2,...,nM =0 e -i 2 χt M m,p=0 nmnp q=1,2,...,M α nq q n q ! |n q . (12)\n\nNote that, for t = 4π/χ the output state coincides with the incoming one whereas for t = 2π/χ the state |X out is obtained from |X after the replacement α m → -α m and the resulting disentangled X wave state can be denoted as | -X . More interesting is the situation for t = π/χ since, defining N = m n m and exploiting the relation e -i(π/2)N 2 = [e -iπ/4 + (-1) N e iπ/4 ]/ √ 2, we obtain obtain\n\n|X out = 1 √ 2 e -iπ/4 |X + e iπ/4 | -X . (13)\n\nwhich is an entangled superposition of two disentangled quantum X waves travelling at the same velocity. The key point is here that the interaction time t coincides with the time spent by the classical X wave to pass through the nonlinear medium or t = D cos θ 0 /c, D being the nonlinear medium length. As for coherent states, the states periodically evolve with temporal period 4π/χ and with spatial period 4πc/(cos θ 0 χ). As a consequence, by simply acting on the velocity of the X-wave (e.g. by varying the axicon angle) it is possible to fine-tune the degree of entanglement of the two macroscopic classical X-waves, and even to switching from classical states to Schrödinger-cat states. This is a not-trivial outcome of the interplay between the classical interference process supporting X waves and the entanglement of coherent states. In addition, X waves travel at a velocity which is different from the other linear waves so that, by using coincidences measurements, it is straightforward to distinguish these states from a statistical mixture, and hence to point out their purely quantum properties as described above."
},
{
"section_type": "CONCLUSION",
"section_title": "CONCLUSIONS",
"text": "In conclusion we have shown that classical electromagnetic X-waves may hide different degrees of quantum entanglement, depending on their generation mechanism (exploiting a linear, a nonlinear medium or both of them). In this respect their velocity (i.e. the axicon angle) plays a prominent role. Other interesting consequences of the X-waves structure arise when dealing with interferometric setup, as for example that considered in [27] . If a nonlinear medium is placed in one or both arms of an interferometer, the output state is a \"progressive undistorted squeezed vacuum\" or entangled superposition of classical X waves so that any detection scheme is affected by the axicon angle, and tuning among various quantum states can be attained, as it will be detailed elsewhere. The quantum properties of X-waves can hence be exploited for free-space quantum communications, highly sensible interferometers or quantum computing."
}
] | |
arxiv:0704.0444 | 0704.0444 | 1 | 10.1103/PhysRevD.79.046005 | d1636fe8181ec4bdd12a40cdd36e0aa6bed81159e7b6428fc691bd17a1bc8fa3 | Multiple Unfoldings of Orbifold Singularities: Engineering Geometric
Analogies to Unification | Katz and Vafa showed how charged matter can arise geometrically by the
deformation of ADE-type orbifold singularities in type IIa, M-theory, and
F-theory compactifications. In this paper we use those same basic ingredients,
used there to geometrically engineer specific matter representations, here to
deform the compactification manifold itself in a way which naturally
compliments many features of unified model building. We realize this idea
explicitly by deforming a manifold engineered to give rise to an $SU_5$ grand
unified model into a one giving rise to the Standard Model. In this framework,
the relative local positions of the singularities giving rise to Standard Model
fields are specified in terms of the values of a small number of complex
structure moduli which deform the original manifold, greatly reducing the
arbitrariness of their relative positions. | [
"Jacob L. Bourjaily"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | One of the ways in which a gauge theory with massless charged matter can arise in type IIa, Mtheory, and F-theory is known as geometrical engineering. In this framework, gauge theory at low energy arises from co-dimension four singular surfaces in the compactification manifold [2] and charged matter arises as isolated points (curves in F-theory) on these surfaces over which the singularity is enhanced. Katz and Vafa [1] constructed explicit examples of local geometry which would give rise to different representations of various gauge groups. Their work was presented explicitly in the language of type IIa or F-theory, but the general results have been shown to apply much more broadly to M-theory as well [3, 4, 5, 6, 7] .
The picture of matter and gauge theory arising from pure geometry via singular structures has been used very fruitfully in much of the progress of Mtheory phenomenology. In [8] Witten engineered an interesting phenomenological model in M-theory which could possibly solve the Higgs doublet-triplet splitting problem; this model was explored in great detail together with Friedmann in [9] . There, the explicit topology of the ADE-singular surface and the relative locations of all the isolated conical singularities was motivated by phenomenology-the description of the geometry of the singularities themselves was taken for granted.
Unlike model building with D-branes, for example, geometrical engineering as it has been understood provides little information about the number, type, and relative locations of the many different singularities needed for any phenomenological model. This information must either come a posteriori from phenomenological success or via duality to a concrete string model. But recent successes in M-theory model building (for example, [10, 11] ) motivate a new look at how to describe the relative structure of singularities-at least locally-within the framework of geometrical engineering itself.
In this paper, we reduce the apparent arbitrariness of the number and relative positions of the singularities required by low-energy phenomenology by showing how they can be obtained from deforming a smaller number of singularities in a more unified model. In section II we review the ingredients of geometrical engineering as described in [1] . The basic framework is then interpreted in a novel way in section III to relate manifolds with matter singularities to those with more or less symmetry. The idea is used explicitly to deform an SU 5 grand unified model into the Standard model.
To be clear, as in [1] our results apply only strictly to N = 2 models from type IIa compactifications or N = 1 models from F-theory compactifications 1 ; but we suspect that this framework has an M-theory analogue in the spirit of [7].
In the framework of geometrical engineering the compactification manifold is described as a fibration of (singular) K3 surfaces over a base space of appropriate dimension. The collection of point-like (codimension four) singularities of the K3 fibres would then be a co-dimension four surface in the compactified manifold, giving rise to gauge theory of type corresponding to the singularities on each K3 fibre. Table I lists polynomials in C 3 whose solutions can be (locally) taken to be the fibres for each corresponding gauge group.
One of the strengths of this framework is its generality: the local geometry is specified in terms of 2 TABLE I: Hypersurfaces in C 3 giving rise to the desired orbifold singularities. Gauge group Polynomial SUn (≡ An-1) xy = z n SO2n (≡ Dn) x 2 + y 2 z = z n-1 E6 x 2 = y 3 + z 4 E7 x 2 + y 3 = yz 3 E8 x 2 + y 3 = z 5 the K3 fibres, so that the description applies equally well to compactifications in type IIa, M-theory, and F-theory-the difference being the dimension of the space over which the surfaces in Table I are fibred.
To obtain massless charged matter, however, additional structure is necessary. Specifically, at isolated points (in type IIa or M-theory) on the co-dimension four singular surface, the type of singularity of the K3 fibres must be enhanced by one rank. Mathematically, this requires that one can describe how the various polynomials in Table I can be deformed into each other; and the possible two-dimensional deformations have been classified [12] .
For example, to describe the embedding of a massless 5 of SU 5 in type IIa, you would need to start with a K3-fibred Calabi-Yau where each of the fibres are of the type giving rise to SU 5 gauge theory. From Table I we see that these four-dimensional fibres are locally the set of solutions to the equation
xy = z 5 , ( 1
)
in C 3 . Now, to obtain matter in the 5 representation, there would need to be an isolated point somewhere on the two-dimensional base space where the fibre is enhanced to SU 6 , [1]. A description of the local geometry can be given by
xy = (z + 5t)(z -t) 5 , ( 2
)
where t is a complex coordinate on the base over which the K3's are fibred. Notice that when t = 0 the equation describes precisely the fibre which would have given rise to SU 6 gauge theory if it were fibred over the entire base manifold. However, because it is the fibre only over the origin in the complex t-plane, there is no SU 6 gauge theory. Equation ( 2) is said to describe the 'resolution' SU 6 → SU 5 , which is found to give rise to SU 5 gauge theory at low energy with a single massless 5 at t = 0. This and many other explicit examples of such resolutions and the matter representations obtained are given in [1] . One subtlety which makes the description above not automatically apply to M-theory constructions, however, is that in equation (2) the complex parameter t is two-dimensional: taken as a coordinate over which the K3 surfaces are fibred, it gives rise to a six-dimensional compactification manifold. In M-theory, co-dimension four singularities are three-dimensional and chiral matter would live at isolated points on these three dimensional orbifold singularities. So in M-theory the resolution SU 6 → SU 5 would need a three-dimensional deformation. Morally, the structure is identical to that described in equation (2), but the parameter t must be upgraded to describe three-dimensional deformations. This can be done in terms of hyper-Kähler quotients. We suspect that all the resolutions described explicitly for type IIa here and in [1] can be upgraded to three-dimensional deformations needed in M-theory, and in many cases these generalizations have already been given [4, 5, 7] .
The main result of this paper is that distinct conical singularities on a surface with some gauge symmetry can be deformed into each other in ways analogous to unification; and conversely, that a description of a single matter field in a unified theory can be 'unfolded' into distinct matter fields in a theory of lower gauge symmetry. Because the tools used to perform these unification-like deformations are precisely the same as those used to describe the singularities themselves, some care must be taken to avoid unnecessary confusion.
We will start by reinterpreting the tools used above to engineer charged matter, and then we will use both interpretations simultaneously to construct explicit examples of the geometric analogue to unified model building.
Consider again the resolution SU 6 → SU 5 described by
xy = (z + 5s)(z -s) 5 , ( 3
)
where we have replaced t → s from equation (2) to make a interpretative distinction that will soon become clear. We propose to momentarily discuss pure gauge theory and ignore any description of matter. With this in mind, take a fixed (real) twodimensional neighborhood over which every point is fibred by the solutions to equation (3) for any fixed value of s. Because the fibres are the same everywhere on the manifold, there is no matter: for any s the geometry would give rise to pure gauge theory at low energy. For s = 0 solutions to equation (3) are SU 5 fibres and so the compactification manifold would give rise to pure SU 5 . However, when s = 0 the fibres are all SU 6 and so the low-energy theory would be pure SU 6 . Therefore s is a 'global' parameter which deforms the gauge content of the theory such that for arbitrary values of s = 0 the theory is pure SU 5 and for s = 0 it is pure SU 6 . That this deformation is 'smooth' is apparent at least when s = 0. An obvious question to ask is how this framework applies when conical singularities are present. We
3 (1, 2) (3, 1) SU 6 SU 5 SU 3 ×SU 3 SU 4 ×SU 2 t s
FIG. 1: The plane describing the deformation of a theory with a single of SU into one of SU SU gauge theory with one ( and one ( as a function of as described by equation (4). For a fixed value of , the base space over which solutions to (4) are fibred are indicated by the black line. Notice that the relative positions of the two isolated (conical) singulari ties are fixed by will show that when the ADE-surface singularity changes because of some complex structure modulus such as above, the conical singularities giving rise to charged matter (often) behave as one would expect from unified model building intuition. This is best demonstrated with explicit examples.
Suppose that the singular 3 surfaces are fibred over a two-dimensional base space with local complex coordinate . And say the four-dimensional fibre over the point is given by the solutions to
xy = ( + 5 )( + 3 ( 4
)
for a given value of , which is now to be interpreted as a complex structure modulus deforming the entire local geometry near = 0. When = 0 the geometry is of course identical to our previous description of SU SU and so the theory would be SU with a single massless located at = 0. Consider now to be fixed at some non-zero value. The gauge theory is then SU SU : for generic values of , the fibres given by equation (4) have two singular points, at + 3 = 0 and = 0, and so the union of these points over the base manifold coordinatized by will be two distinct, two-dimensional singular surfaces: one giving rise to SU and the other SU . These surfaces become coincident as a single SU surface when = 0.
Along the complex -plane, there are two isolated points over which the singularities are enhanced: at s/2 the fibre is visibly SU SU and at s/3 the fibre is SU SU . Therefore the theory has two two charged, massless fields, in the ( and ( representations of SU SU at s/2 and s/3, respectively. Figure 1 indicates the singularity structure as a function of Notice how this description parallels unified model building: the = 0 theory of one of SU deforms smoothly into one with ( of SU SU Similarly, we may ask how a 10 of SU would deform into distinct singularities supporting Standard Model matter fields. The fibre structure giving rise to a massless 10 of SU is given as follows. Let be a local coordinate on the base space over which fibres are given by solutions to + 2yt 10 ; (5) at = 0, equation (5) describes an SO 10 fibre, while for = 0 the fibres are SU -although in this case the result is harder to read off. This resolution, SO 10 SU , gives rise to a 10 of SU [1]. Following the same idea as before, the deformation of this singularity into SU SU is given by
+ 2 + ( + ( ( 6
)
where is again interpreted as a complex structure modulus deforming the geometry near the singularity. Notice as before that = 0 describes an SU theory with one massless 10 located at = 0. However, for = 0 there are again two orbifold singularities corresponding to SU SU gauge theory. At three distinct points on the complex plane the rank of the fibre is enhanced: = 0, and give rise to matter in the ( , ( , and representations of SU SU , respectively. The structure of the deformation achieved by varying is shown in Figure 2 .
Again, our intuition from unified model building is realized naturally in this framework.
FIG. 1: The t-s plane describing the deformation of a theory with a single 5 of SU5 into one of SU3 × SU2 × U1 gauge theory with one (3, 1) 1/3 and one (1, 2) -1/2 as a function of s as described by equation (4). For a fixed value of s, the base space over which solutions to (4) are fibred are indicated by the black line. Notice that the relative positions of the two isolated (conical) singularities are fixed by s.
will show that when the ADE-surface singularity changes because of some complex structure modulus such as s above, the conical singularities giving rise to charged matter (often) behave as one would expect from unified model building intuition. This is best demonstrated with explicit examples.
Suppose that the singular K3 surfaces are fibred over a two-dimensional base space with local complex coordinate t. And say the four-dimensional fibre over the point t is given by the solutions to 4) for a given value of s, which is now to be interpreted as a complex structure modulus deforming the entire local geometry near t = 0. When s = 0 the geometry is of course identical to our previous description of SU 6 → SU 5 and so the theory would be SU 5 with a single massless 5 located at t = 0. Consider now s to be fixed at some non-zero value. The gauge theory is then SU 3 ×SU 2 ×U 1 : for generic values of t, the fibres given by equation (4) have two singular points, at x = y = zt + 3s = 0 and x = y = zt -2s = 0, and so the union of these points over the base manifold coordinatized by t will two distinct, two-dimensional singular surfaces: one giving rise to SU 2 and the other SU 3 . These surfaces become coincident as a single SU 5 surface when s = 0.
xy = (z + 5t)(z -t + 3s) 2 (z -t -2s) 3 , (
Along the complex t-plane, there are two isolated points over which the singularities are enhanced: at t = s/2 the fibre is visibly SU 3 × SU 3 , and at t = -s/3 the fibre is SU 4 × SU 2 . Therefore the theory has two two charged, massless fields, in the (1, 2) -1/2 and (3, 1) 1/3 representations of SU 3 × SU 2 × U 1 at t = s/2 and t = -s/3, respectively. Figure 1 indicates the singularity structure as a function of s.
Notice how this description parallels unified model building: the s = 0 theory of one 5 of SU 5 deforms smoothly into one with (3,
1) 1/3 ⊕ (1, 2) -1/2 of SU 3 × SU 2 × U 1 .
Similarly, we may ask how a 10 of SU 5 would deform into distinct singularities supporting Standard Model matter fields. The fibre structure giving rise to a massless 10 of SU 5 is given as follows. Let t be a local coordinate on the base space over which fibres are given by solutions to
x 2 + y 2 z + 2yt 5 = 1 z z + t 2 5 -t 10 ; ( 5
)
at t = 0, equation (5) describes an SO 10 fibre, while for t = 0 the fibres are SU 5 -although in this case the result is harder to read off. This resolution, SO 10 → SU 5 , gives rise to a 10 of SU 5 [1]. Following the same idea as before, the deformation of this singularity into SU 3 × SU 2 is given by
x 2 + y 2 z + 2y(t + s) 3 (t -s) 2 = 1 z z + (t -s) 2 2 z + (t + s) 2 3 -(t -s) 4 (t + s) 6 , ( 6
)
where s is again interpreted as a complex structure modulus deforming the geometry near the singularity. Notice as before that s = 0 describes an SU 5 theory with one massless 10 located at t = 0. However, for s = 0 there are again two orbifold singularities corresponding to SU 3 × SU 2 × U 1 gauge theory. At three distinct points on the complex t plane the rank of the fibre is enhanced: t = -s, t = 0, and t = s give rise to matter in the (3, 1) -2/3 , (3, 2) 1/6 , and
(1, 1) 1 representations of SU 3 × SU 2 × U 1 , respec- tively.
The structure of the deformation achieved by varying s is shown in Figure 2 . Again, our intuition from unified model building is realized naturally in this framework.
4 (3, 2) (3, 1) (1, 1) SO 10 SU 5 SU 5 SU 3 ×SU 2 ×SU 2 SU 4 ×SU 2 t s
FIG. 2: The plane describing the deformation of a theory with a single 10 of SU into one of SU SU gauge theory, with matter content ( , as a function of as described by equation (6). For a fixed value of , the base space over which solutions to (6) are fibred are indicated by the black line. Notice that the relative positions of the three isolated (conical) singularities are fixed by
One of the primary reasons why geometrical engineering had not been more widely used phenomenologically is because the number, type, and relative locations of the singularities giving rise to various matter fields were explicitly ad hoc: the inherent local framework prevented relationships between distinct singularities from being discussed. In this paper, we have shown a framework in which these questions can be addressed concretely, systematically reducing the arbitrariness of these models.
Of course, the local nature of geometrical engineering is still inherent in this framework, and continues to prevent us from addressing questions about the global structure such as stability, quantum gravity, and the quantization of seemingly continuous parameters like . However, in the spirit of [13] , we think that local engineering is a good step toward realistic string phenomenology, and may perhaps offer new insights.
In this paper we explicitly illustrated the geometric unfolding of the matter content of an SU grand unified model into the Standard Model. But the procedure can easily be generalized. It is not difficult to see how this will work for a more unified theory. For example, one can envision how an entire family could unfold out of a single SO 10 resolution (which starts as a 16 of SO 10 ), or how all three families of the Standard Mode could be unfolded out of a single SO 10 SU or SU resolution. However, these examples require more sophisticated tools of analysis, and so we have chosen to describe these in a forthcoming work.
This work originated from discussions with Malcolm Perry whose insights drove this work forward in its earliest steps. The author also appreciates helpful discussions, comments, and suggestions from Herman Verlinde, Sergei Gukov, Gordon Kane, Edward Witten, Paul Langacker, Bobby Acharya, Dmitry Malyshev, Matthew Buican, Piyush Kumar, and Konstantin Bobkov.
This research was supported in part by the Michigan Center for Theoretical Physics and a Graduate Research Fellowship from the National Science Foundation.
[1] S. Katz and C. Vafa, "Matter from Geometry," Nucl. Phys., vol. B497, pp. 146-154, 1997, hepth/9606086. [2] A. Klemm, W. Lerche, and P. Mayr, "K3 Fibrations and Heterotic Type II String Duality," Phys. Lett. vol. B357, pp. 313-322, 1995, hep-th/9506112. [3] M. Atiyah and E. Witten, "M-theory Dynamics on a Manifold of Holonomy," Adv. Theor. Math. Phys., vol. 6, pp. 1-106, 2003, hep-th/0107177. [4] E. Witten, "Anomaly Cancellation on Manifolds," 2001, hep-th/0108165. [5] B. Acharya and E. Witten, "Chiral Fermions from Manifolds of Holonomy," 2001, hep-th/0109152. [6] B. S. Acharya and S. Gukov, "M-theory and Singularities of Exceptional Holonomy Manifolds," Phys. Rept., vol. 392, pp. 121-189, 2004, hep-th/0409191. [7] P. Berglund and A. Brandhuber, "Matter from Manifolds," Nucl. Phys., vol. B641, pp. 351-375, 2002, hep-th/0205184. [8] E. Witten, "Deconstruction, Holonomy, and Doublet-Triplet Splitting," 2001, hep-ph/0201018. [9] T. Friedmann and E. Witten, "Unification Scale, Proton Decay, and Manifolds of G(2) Holonomy," Adv. Theor. Math. Phys., vol. 7, pp. 577-617, 2003, hep-th/0211269. [10] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and D. Vaman, "An M Theory Solution to the Hierarchy Problem," Phys. Rev. Lett., vol. 97, p. 191601, 2006, hep-th/0606262. [11] B. S. Acharya, K. Bobkov, G. L. Kane, P. Kumar, and J. Shao, "Explaining the Electroweak Scale and Stabilizing Moduli in M-Theory," 2007, hepth/0701034. [12] S. Katz and D. Morrison, "Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups," J. Algebraic Geometry vol. 1, pp. 449-530, 1992. [13] H. Verlinde and M. Wijnholt, "Building the Standard Model on a D3-Brane," JHEP, vol. 01, p. 106, 2007, hep-th/0508089.
FIG. 2: The t-s plane describing the deformation of a theory with a single 10 of SU5 into one of SU3 ×SU2 ×U1 gauge theory, with matter content (3, 1) -2/3 ⊕(3, 2) 1/6 ⊕ (1, 1)1, as a function of s as described by equation (6). For a fixed value of s, the base space over which solutions to (6) are fibred are indicated by the black line. Notice that the relative positions of the three isolated (conical) singularities are fixed by s.
One of the primary reasons why geometrical engineering had not been more widely used phenomenologically is because the number, type, and relative locations of the singularities giving rise to various matter fields were explicitly ad hoc: the inherent local framework prevented relationships between distinct singularities from being discussed. In this paper, we have shown a framework in which these questions can be addressed concretely, systematically reduc-ing the arbitrariness of these models.
Of course, the local nature of geometrical engineering is still inherent in this framework, and continues to prevent us from addressing questions about the global structure such as stability, quantum gravity, and the quantization of seemingly continuous parameters like s. However, in the spirit of [13] , we think that local engineering is a good step toward realistic string phenomenology, and may perhaps offer new insights.
In this paper we explicitly illustrated the geometric unfolding of the matter content of an SU 5 grand unified model into the Standard Model. But the procedure can easily be generalized. It is not difficult to see how this will work for a more unified theory. For example, one can envision how an entire family could unfold out of a single E 6 → SO 10 resolution (which starts as a 16 of SO 10 ), or how all three families of the Standard Mode could be unfolded out of a single
E 8 → SO 10 × SU 3 or E 8 → E 6 × SU 2 resolution.
However, these examples require more sophisticated tools of analysis, and so we have chosen to describe these in a forthcoming work.
This work originated from discussions with Malcolm Perry whose insights drove this work forward in its earliest steps. The author also appreciates helpful discussions, comments, and suggestions from Herman Verlinde, Sergei Gukov, Gordon Kane, Edward Witten, Paul Langacker, Bobby Acharya, Dmitry Malyshev, Matthew Buican, Piyush Kumar, and Konstantin Bobkov.
This research was supported in part by the Michigan Center for Theoretical Physics and a Graduate Research Fellowship from the National Science Foundation.
[1] S. Katz and C. Vafa, "Matter from Geometry," Nucl. Phys., vol. B497, pp. 146-154, 1997, hep-th/9606086. [2] A. Klemm, W. Lerche, and P. Mayr, "K3 Fibrations and Heterotic Type II String Duality," Phys. Lett., vol. B357, pp. 313-322, 1995, hep-th/9506112. [3] M. Atiyah and E. Witten, "M-theory Dynamics on a Manifold of G2 Holonomy," Adv. Theor. Math. Phys., vol. 6, pp. 1-106, 2003, hep-th/0107177. [4] E. Witten, "Anomaly Cancellation on G2 Manifolds," 2001, hep-th/0108165. [5] B. Acharya and E. Witten, "Chiral Fermions from Manifolds of G2 Holonomy," 2001, hep-th/0109152. [6] B. S. Acharya and S. Gukov, "M-theory and Singularities of Exceptional Holonomy Manifolds," Phys. Rept., vol. 392, pp. 121-189, 2004, hep-th/0409191. [7] P. Berglund and A. Brandhuber, "Matter from G2 Manifolds," Nucl. Phys., vol. B641, pp. 351-375, 2002, hep-th/0205184. [8] E. Witten, "Deconstruction, G2 Holonomy, and Doublet-Triplet Splitting," 2001, hep-ph/0201018. [9] T. Friedmann and E. Witten, "Unification Scale, Proton Decay, and Manifolds of G(2) Holonomy," Adv. Theor. Math. Phys., vol. 7, pp. 577-617, 2003, hep-th/0211269. [10] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and D. Vaman, "An M Theory Solution to the Hierarchy Problem," Phys. Rev. Lett., vol. 97, p. 191601, 2006, hep-th/0606262. [11] B. S. Acharya, K. Bobkov, G. L. Kane, P. Kumar, and J. Shao, "Explaining the Electroweak Scale and Stabilizing Moduli in M-Theory," 2007, hep-th/0701034. [12] S. Katz and D. Morrison, "Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups," J. Algebraic Geometry, vol. 1, pp. 449-530, 1992. [13] H. Verlinde and M. Wijnholt, "Building the Standard Model on a D3-Brane," JHEP, vol. 01, p. 106, 2007, hep-th/0508089. | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "One of the ways in which a gauge theory with massless charged matter can arise in type IIa, Mtheory, and F-theory is known as geometrical engineering. In this framework, gauge theory at low energy arises from co-dimension four singular surfaces in the compactification manifold [2] and charged matter arises as isolated points (curves in F-theory) on these surfaces over which the singularity is enhanced. Katz and Vafa [1] constructed explicit examples of local geometry which would give rise to different representations of various gauge groups. Their work was presented explicitly in the language of type IIa or F-theory, but the general results have been shown to apply much more broadly to M-theory as well [3, 4, 5, 6, 7] .\n\nThe picture of matter and gauge theory arising from pure geometry via singular structures has been used very fruitfully in much of the progress of Mtheory phenomenology. In [8] Witten engineered an interesting phenomenological model in M-theory which could possibly solve the Higgs doublet-triplet splitting problem; this model was explored in great detail together with Friedmann in [9] . There, the explicit topology of the ADE-singular surface and the relative locations of all the isolated conical singularities was motivated by phenomenology-the description of the geometry of the singularities themselves was taken for granted.\n\nUnlike model building with D-branes, for example, geometrical engineering as it has been understood provides little information about the number, type, and relative locations of the many different singularities needed for any phenomenological model. This information must either come a posteriori from phenomenological success or via duality to a concrete string model. But recent successes in M-theory model building (for example, [10, 11] ) motivate a new look at how to describe the relative structure of singularities-at least locally-within the framework of geometrical engineering itself.\n\nIn this paper, we reduce the apparent arbitrariness of the number and relative positions of the singularities required by low-energy phenomenology by showing how they can be obtained from deforming a smaller number of singularities in a more unified model. In section II we review the ingredients of geometrical engineering as described in [1] . The basic framework is then interpreted in a novel way in section III to relate manifolds with matter singularities to those with more or less symmetry. The idea is used explicitly to deform an SU 5 grand unified model into the Standard model.\n\nTo be clear, as in [1] our results apply only strictly to N = 2 models from type IIa compactifications or N = 1 models from F-theory compactifications 1 ; but we suspect that this framework has an M-theory analogue in the spirit of [7]."
},
{
"section_type": "OTHER",
"section_title": "II. GEOMETRICAL ENGINEERING",
"text": "In the framework of geometrical engineering the compactification manifold is described as a fibration of (singular) K3 surfaces over a base space of appropriate dimension. The collection of point-like (codimension four) singularities of the K3 fibres would then be a co-dimension four surface in the compactified manifold, giving rise to gauge theory of type corresponding to the singularities on each K3 fibre. Table I lists polynomials in C 3 whose solutions can be (locally) taken to be the fibres for each corresponding gauge group.\n\nOne of the strengths of this framework is its generality: the local geometry is specified in terms of 2 TABLE I: Hypersurfaces in C 3 giving rise to the desired orbifold singularities. Gauge group Polynomial SUn (≡ An-1) xy = z n SO2n (≡ Dn) x 2 + y 2 z = z n-1 E6 x 2 = y 3 + z 4 E7 x 2 + y 3 = yz 3 E8 x 2 + y 3 = z 5 the K3 fibres, so that the description applies equally well to compactifications in type IIa, M-theory, and F-theory-the difference being the dimension of the space over which the surfaces in Table I are fibred.\n\nTo obtain massless charged matter, however, additional structure is necessary. Specifically, at isolated points (in type IIa or M-theory) on the co-dimension four singular surface, the type of singularity of the K3 fibres must be enhanced by one rank. Mathematically, this requires that one can describe how the various polynomials in Table I can be deformed into each other; and the possible two-dimensional deformations have been classified [12] .\n\nFor example, to describe the embedding of a massless 5 of SU 5 in type IIa, you would need to start with a K3-fibred Calabi-Yau where each of the fibres are of the type giving rise to SU 5 gauge theory. From Table I we see that these four-dimensional fibres are locally the set of solutions to the equation\n\nxy = z 5 , ( 1\n\n)\n\nin C 3 . Now, to obtain matter in the 5 representation, there would need to be an isolated point somewhere on the two-dimensional base space where the fibre is enhanced to SU 6 , [1]. A description of the local geometry can be given by\n\nxy = (z + 5t)(z -t) 5 , ( 2\n\n)\n\nwhere t is a complex coordinate on the base over which the K3's are fibred. Notice that when t = 0 the equation describes precisely the fibre which would have given rise to SU 6 gauge theory if it were fibred over the entire base manifold. However, because it is the fibre only over the origin in the complex t-plane, there is no SU 6 gauge theory. Equation ( 2) is said to describe the 'resolution' SU 6 → SU 5 , which is found to give rise to SU 5 gauge theory at low energy with a single massless 5 at t = 0. This and many other explicit examples of such resolutions and the matter representations obtained are given in [1] . One subtlety which makes the description above not automatically apply to M-theory constructions, however, is that in equation (2) the complex parameter t is two-dimensional: taken as a coordinate over which the K3 surfaces are fibred, it gives rise to a six-dimensional compactification manifold. In M-theory, co-dimension four singularities are three-dimensional and chiral matter would live at isolated points on these three dimensional orbifold singularities. So in M-theory the resolution SU 6 → SU 5 would need a three-dimensional deformation. Morally, the structure is identical to that described in equation (2), but the parameter t must be upgraded to describe three-dimensional deformations. This can be done in terms of hyper-Kähler quotients. We suspect that all the resolutions described explicitly for type IIa here and in [1] can be upgraded to three-dimensional deformations needed in M-theory, and in many cases these generalizations have already been given [4, 5, 7] ."
},
{
"section_type": "OTHER",
"section_title": "III. ENGINEERING GEOMETRIC ANALOGIES TO UNIFICATION",
"text": "The main result of this paper is that distinct conical singularities on a surface with some gauge symmetry can be deformed into each other in ways analogous to unification; and conversely, that a description of a single matter field in a unified theory can be 'unfolded' into distinct matter fields in a theory of lower gauge symmetry. Because the tools used to perform these unification-like deformations are precisely the same as those used to describe the singularities themselves, some care must be taken to avoid unnecessary confusion.\n\nWe will start by reinterpreting the tools used above to engineer charged matter, and then we will use both interpretations simultaneously to construct explicit examples of the geometric analogue to unified model building.\n\nConsider again the resolution SU 6 → SU 5 described by\n\nxy = (z + 5s)(z -s) 5 , ( 3\n\n)\n\nwhere we have replaced t → s from equation (2) to make a interpretative distinction that will soon become clear. We propose to momentarily discuss pure gauge theory and ignore any description of matter. With this in mind, take a fixed (real) twodimensional neighborhood over which every point is fibred by the solutions to equation (3) for any fixed value of s. Because the fibres are the same everywhere on the manifold, there is no matter: for any s the geometry would give rise to pure gauge theory at low energy. For s = 0 solutions to equation (3) are SU 5 fibres and so the compactification manifold would give rise to pure SU 5 . However, when s = 0 the fibres are all SU 6 and so the low-energy theory would be pure SU 6 . Therefore s is a 'global' parameter which deforms the gauge content of the theory such that for arbitrary values of s = 0 the theory is pure SU 5 and for s = 0 it is pure SU 6 . That this deformation is 'smooth' is apparent at least when s = 0. An obvious question to ask is how this framework applies when conical singularities are present. We\n\n3 (1, 2) (3, 1) SU 6 SU 5 SU 3 ×SU 3 SU 4 ×SU 2 t s\n\nFIG. 1: The plane describing the deformation of a theory with a single of SU into one of SU SU gauge theory with one ( and one ( as a function of as described by equation (4). For a fixed value of , the base space over which solutions to (4) are fibred are indicated by the black line. Notice that the relative positions of the two isolated (conical) singulari ties are fixed by will show that when the ADE-surface singularity changes because of some complex structure modulus such as above, the conical singularities giving rise to charged matter (often) behave as one would expect from unified model building intuition. This is best demonstrated with explicit examples.\n\nSuppose that the singular 3 surfaces are fibred over a two-dimensional base space with local complex coordinate . And say the four-dimensional fibre over the point is given by the solutions to\n\nxy = ( + 5 )( + 3 ( 4\n\n)\n\nfor a given value of , which is now to be interpreted as a complex structure modulus deforming the entire local geometry near = 0. When = 0 the geometry is of course identical to our previous description of SU SU and so the theory would be SU with a single massless located at = 0. Consider now to be fixed at some non-zero value. The gauge theory is then SU SU : for generic values of , the fibres given by equation (4) have two singular points, at + 3 = 0 and = 0, and so the union of these points over the base manifold coordinatized by will be two distinct, two-dimensional singular surfaces: one giving rise to SU and the other SU . These surfaces become coincident as a single SU surface when = 0.\n\nAlong the complex -plane, there are two isolated points over which the singularities are enhanced: at s/2 the fibre is visibly SU SU and at s/3 the fibre is SU SU . Therefore the theory has two two charged, massless fields, in the ( and ( representations of SU SU at s/2 and s/3, respectively. Figure 1 indicates the singularity structure as a function of Notice how this description parallels unified model building: the = 0 theory of one of SU deforms smoothly into one with ( of SU SU Similarly, we may ask how a 10 of SU would deform into distinct singularities supporting Standard Model matter fields. The fibre structure giving rise to a massless 10 of SU is given as follows. Let be a local coordinate on the base space over which fibres are given by solutions to + 2yt 10 ; (5) at = 0, equation (5) describes an SO 10 fibre, while for = 0 the fibres are SU -although in this case the result is harder to read off. This resolution, SO 10 SU , gives rise to a 10 of SU [1]. Following the same idea as before, the deformation of this singularity into SU SU is given by\n\n+ 2 + ( + ( ( 6\n\n)\n\nwhere is again interpreted as a complex structure modulus deforming the geometry near the singularity. Notice as before that = 0 describes an SU theory with one massless 10 located at = 0. However, for = 0 there are again two orbifold singularities corresponding to SU SU gauge theory. At three distinct points on the complex plane the rank of the fibre is enhanced: = 0, and give rise to matter in the ( , ( , and representations of SU SU , respectively. The structure of the deformation achieved by varying is shown in Figure 2 .\n\nAgain, our intuition from unified model building is realized naturally in this framework.\n\nFIG. 1: The t-s plane describing the deformation of a theory with a single 5 of SU5 into one of SU3 × SU2 × U1 gauge theory with one (3, 1) 1/3 and one (1, 2) -1/2 as a function of s as described by equation (4). For a fixed value of s, the base space over which solutions to (4) are fibred are indicated by the black line. Notice that the relative positions of the two isolated (conical) singularities are fixed by s.\n\nwill show that when the ADE-surface singularity changes because of some complex structure modulus such as s above, the conical singularities giving rise to charged matter (often) behave as one would expect from unified model building intuition. This is best demonstrated with explicit examples.\n\nSuppose that the singular K3 surfaces are fibred over a two-dimensional base space with local complex coordinate t. And say the four-dimensional fibre over the point t is given by the solutions to 4) for a given value of s, which is now to be interpreted as a complex structure modulus deforming the entire local geometry near t = 0. When s = 0 the geometry is of course identical to our previous description of SU 6 → SU 5 and so the theory would be SU 5 with a single massless 5 located at t = 0. Consider now s to be fixed at some non-zero value. The gauge theory is then SU 3 ×SU 2 ×U 1 : for generic values of t, the fibres given by equation (4) have two singular points, at x = y = zt + 3s = 0 and x = y = zt -2s = 0, and so the union of these points over the base manifold coordinatized by t will two distinct, two-dimensional singular surfaces: one giving rise to SU 2 and the other SU 3 . These surfaces become coincident as a single SU 5 surface when s = 0.\n\nxy = (z + 5t)(z -t + 3s) 2 (z -t -2s) 3 , (\n\nAlong the complex t-plane, there are two isolated points over which the singularities are enhanced: at t = s/2 the fibre is visibly SU 3 × SU 3 , and at t = -s/3 the fibre is SU 4 × SU 2 . Therefore the theory has two two charged, massless fields, in the (1, 2) -1/2 and (3, 1) 1/3 representations of SU 3 × SU 2 × U 1 at t = s/2 and t = -s/3, respectively. Figure 1 indicates the singularity structure as a function of s.\n\nNotice how this description parallels unified model building: the s = 0 theory of one 5 of SU 5 deforms smoothly into one with (3,\n\n1) 1/3 ⊕ (1, 2) -1/2 of SU 3 × SU 2 × U 1 .\n\nSimilarly, we may ask how a 10 of SU 5 would deform into distinct singularities supporting Standard Model matter fields. The fibre structure giving rise to a massless 10 of SU 5 is given as follows. Let t be a local coordinate on the base space over which fibres are given by solutions to\n\nx 2 + y 2 z + 2yt 5 = 1 z z + t 2 5 -t 10 ; ( 5\n\n)\n\nat t = 0, equation (5) describes an SO 10 fibre, while for t = 0 the fibres are SU 5 -although in this case the result is harder to read off. This resolution, SO 10 → SU 5 , gives rise to a 10 of SU 5 [1]. Following the same idea as before, the deformation of this singularity into SU 3 × SU 2 is given by\n\nx 2 + y 2 z + 2y(t + s) 3 (t -s) 2 = 1 z z + (t -s) 2 2 z + (t + s) 2 3 -(t -s) 4 (t + s) 6 , ( 6\n\n)\n\nwhere s is again interpreted as a complex structure modulus deforming the geometry near the singularity. Notice as before that s = 0 describes an SU 5 theory with one massless 10 located at t = 0. However, for s = 0 there are again two orbifold singularities corresponding to SU 3 × SU 2 × U 1 gauge theory. At three distinct points on the complex t plane the rank of the fibre is enhanced: t = -s, t = 0, and t = s give rise to matter in the (3, 1) -2/3 , (3, 2) 1/6 , and\n\n(1, 1) 1 representations of SU 3 × SU 2 × U 1 , respec- tively.\n\nThe structure of the deformation achieved by varying s is shown in Figure 2 . Again, our intuition from unified model building is realized naturally in this framework.\n\n4 (3, 2) (3, 1) (1, 1) SO 10 SU 5 SU 5 SU 3 ×SU 2 ×SU 2 SU 4 ×SU 2 t s\n\nFIG. 2: The plane describing the deformation of a theory with a single 10 of SU into one of SU SU gauge theory, with matter content ( , as a function of as described by equation (6). For a fixed value of , the base space over which solutions to (6) are fibred are indicated by the black line. Notice that the relative positions of the three isolated (conical) singularities are fixed by"
},
{
"section_type": "DISCUSSION",
"section_title": "IV. DISCUSSION",
"text": "One of the primary reasons why geometrical engineering had not been more widely used phenomenologically is because the number, type, and relative locations of the singularities giving rise to various matter fields were explicitly ad hoc: the inherent local framework prevented relationships between distinct singularities from being discussed. In this paper, we have shown a framework in which these questions can be addressed concretely, systematically reducing the arbitrariness of these models.\n\nOf course, the local nature of geometrical engineering is still inherent in this framework, and continues to prevent us from addressing questions about the global structure such as stability, quantum gravity, and the quantization of seemingly continuous parameters like . However, in the spirit of [13] , we think that local engineering is a good step toward realistic string phenomenology, and may perhaps offer new insights.\n\nIn this paper we explicitly illustrated the geometric unfolding of the matter content of an SU grand unified model into the Standard Model. But the procedure can easily be generalized. It is not difficult to see how this will work for a more unified theory. For example, one can envision how an entire family could unfold out of a single SO 10 resolution (which starts as a 16 of SO 10 ), or how all three families of the Standard Mode could be unfolded out of a single SO 10 SU or SU resolution. However, these examples require more sophisticated tools of analysis, and so we have chosen to describe these in a forthcoming work."
},
{
"section_type": "OTHER",
"section_title": "V. ACKNOWLEDGEMENTS",
"text": "This work originated from discussions with Malcolm Perry whose insights drove this work forward in its earliest steps. The author also appreciates helpful discussions, comments, and suggestions from Herman Verlinde, Sergei Gukov, Gordon Kane, Edward Witten, Paul Langacker, Bobby Acharya, Dmitry Malyshev, Matthew Buican, Piyush Kumar, and Konstantin Bobkov.\n\nThis research was supported in part by the Michigan Center for Theoretical Physics and a Graduate Research Fellowship from the National Science Foundation.\n\n[1] S. Katz and C. Vafa, \"Matter from Geometry,\" Nucl. Phys., vol. B497, pp. 146-154, 1997, hepth/9606086. [2] A. Klemm, W. Lerche, and P. Mayr, \"K3 Fibrations and Heterotic Type II String Duality,\" Phys. Lett. vol. B357, pp. 313-322, 1995, hep-th/9506112. [3] M. Atiyah and E. Witten, \"M-theory Dynamics on a Manifold of Holonomy,\" Adv. Theor. Math. Phys., vol. 6, pp. 1-106, 2003, hep-th/0107177. [4] E. Witten, \"Anomaly Cancellation on Manifolds,\" 2001, hep-th/0108165. [5] B. Acharya and E. Witten, \"Chiral Fermions from Manifolds of Holonomy,\" 2001, hep-th/0109152. [6] B. S. Acharya and S. Gukov, \"M-theory and Singularities of Exceptional Holonomy Manifolds,\" Phys. Rept., vol. 392, pp. 121-189, 2004, hep-th/0409191. [7] P. Berglund and A. Brandhuber, \"Matter from Manifolds,\" Nucl. Phys., vol. B641, pp. 351-375, 2002, hep-th/0205184. [8] E. Witten, \"Deconstruction, Holonomy, and Doublet-Triplet Splitting,\" 2001, hep-ph/0201018. [9] T. Friedmann and E. Witten, \"Unification Scale, Proton Decay, and Manifolds of G(2) Holonomy,\" Adv. Theor. Math. Phys., vol. 7, pp. 577-617, 2003, hep-th/0211269. [10] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and D. Vaman, \"An M Theory Solution to the Hierarchy Problem,\" Phys. Rev. Lett., vol. 97, p. 191601, 2006, hep-th/0606262. [11] B. S. Acharya, K. Bobkov, G. L. Kane, P. Kumar, and J. Shao, \"Explaining the Electroweak Scale and Stabilizing Moduli in M-Theory,\" 2007, hepth/0701034. [12] S. Katz and D. Morrison, \"Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups,\" J. Algebraic Geometry vol. 1, pp. 449-530, 1992. [13] H. Verlinde and M. Wijnholt, \"Building the Standard Model on a D3-Brane,\" JHEP, vol. 01, p. 106, 2007, hep-th/0508089.\n\nFIG. 2: The t-s plane describing the deformation of a theory with a single 10 of SU5 into one of SU3 ×SU2 ×U1 gauge theory, with matter content (3, 1) -2/3 ⊕(3, 2) 1/6 ⊕ (1, 1)1, as a function of s as described by equation (6). For a fixed value of s, the base space over which solutions to (6) are fibred are indicated by the black line. Notice that the relative positions of the three isolated (conical) singularities are fixed by s."
},
{
"section_type": "DISCUSSION",
"section_title": "IV. DISCUSSION",
"text": "One of the primary reasons why geometrical engineering had not been more widely used phenomenologically is because the number, type, and relative locations of the singularities giving rise to various matter fields were explicitly ad hoc: the inherent local framework prevented relationships between distinct singularities from being discussed. In this paper, we have shown a framework in which these questions can be addressed concretely, systematically reduc-ing the arbitrariness of these models.\n\nOf course, the local nature of geometrical engineering is still inherent in this framework, and continues to prevent us from addressing questions about the global structure such as stability, quantum gravity, and the quantization of seemingly continuous parameters like s. However, in the spirit of [13] , we think that local engineering is a good step toward realistic string phenomenology, and may perhaps offer new insights.\n\nIn this paper we explicitly illustrated the geometric unfolding of the matter content of an SU 5 grand unified model into the Standard Model. But the procedure can easily be generalized. It is not difficult to see how this will work for a more unified theory. For example, one can envision how an entire family could unfold out of a single E 6 → SO 10 resolution (which starts as a 16 of SO 10 ), or how all three families of the Standard Mode could be unfolded out of a single\n\nE 8 → SO 10 × SU 3 or E 8 → E 6 × SU 2 resolution.\n\nHowever, these examples require more sophisticated tools of analysis, and so we have chosen to describe these in a forthcoming work."
},
{
"section_type": "OTHER",
"section_title": "V. ACKNOWLEDGEMENTS",
"text": "This work originated from discussions with Malcolm Perry whose insights drove this work forward in its earliest steps. The author also appreciates helpful discussions, comments, and suggestions from Herman Verlinde, Sergei Gukov, Gordon Kane, Edward Witten, Paul Langacker, Bobby Acharya, Dmitry Malyshev, Matthew Buican, Piyush Kumar, and Konstantin Bobkov.\n\nThis research was supported in part by the Michigan Center for Theoretical Physics and a Graduate Research Fellowship from the National Science Foundation.\n\n[1] S. Katz and C. Vafa, \"Matter from Geometry,\" Nucl. Phys., vol. B497, pp. 146-154, 1997, hep-th/9606086. [2] A. Klemm, W. Lerche, and P. Mayr, \"K3 Fibrations and Heterotic Type II String Duality,\" Phys. Lett., vol. B357, pp. 313-322, 1995, hep-th/9506112. [3] M. Atiyah and E. Witten, \"M-theory Dynamics on a Manifold of G2 Holonomy,\" Adv. Theor. Math. Phys., vol. 6, pp. 1-106, 2003, hep-th/0107177. [4] E. Witten, \"Anomaly Cancellation on G2 Manifolds,\" 2001, hep-th/0108165. [5] B. Acharya and E. Witten, \"Chiral Fermions from Manifolds of G2 Holonomy,\" 2001, hep-th/0109152. [6] B. S. Acharya and S. Gukov, \"M-theory and Singularities of Exceptional Holonomy Manifolds,\" Phys. Rept., vol. 392, pp. 121-189, 2004, hep-th/0409191. [7] P. Berglund and A. Brandhuber, \"Matter from G2 Manifolds,\" Nucl. Phys., vol. B641, pp. 351-375, 2002, hep-th/0205184. [8] E. Witten, \"Deconstruction, G2 Holonomy, and Doublet-Triplet Splitting,\" 2001, hep-ph/0201018. [9] T. Friedmann and E. Witten, \"Unification Scale, Proton Decay, and Manifolds of G(2) Holonomy,\" Adv. Theor. Math. Phys., vol. 7, pp. 577-617, 2003, hep-th/0211269. [10] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and D. Vaman, \"An M Theory Solution to the Hierarchy Problem,\" Phys. Rev. Lett., vol. 97, p. 191601, 2006, hep-th/0606262. [11] B. S. Acharya, K. Bobkov, G. L. Kane, P. Kumar, and J. Shao, \"Explaining the Electroweak Scale and Stabilizing Moduli in M-Theory,\" 2007, hep-th/0701034. [12] S. Katz and D. Morrison, \"Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups,\" J. Algebraic Geometry, vol. 1, pp. 449-530, 1992. [13] H. Verlinde and M. Wijnholt, \"Building the Standard Model on a D3-Brane,\" JHEP, vol. 01, p. 106, 2007, hep-th/0508089."
}
] | |
arxiv:0704.0445 | 0704.0445 | 1 | 10.1103/PhysRevD.76.046004 | 4da8123da9f581af73bb6ab2b9302999a3c6bb3aaa17b17d9163f3be237d2596 | Geometrically Engineering the Standard Model: Locally Unfolding Three
Families out of E8 | This paper extends and builds upon the results of an earlier paper, in which
we described how to use the tools of geometrical engineering to deform
geometrically-engineered grand unified models into ones with lower symmetry.
This top-down unfolding has the advantage that the relative positions of
singularities giving rise to the many `low energy' matter fields are related by
only a few parameters which deform the geometry of the unified model. And
because the relative positions of singularities are necessary to compute the
superpotential, for example, this is a framework in which the arbitrariness of
geometrically engineered models can be greatly reduced.
In our earlier paper, this picture was made concrete for the case of
deforming the representations of an SU(5) model into their Standard Model
content. In this paper we continue that discussion to show how a geometrically
engineered 16 of SO(10) can be unfolded into the Standard Model, and how the
three families of the Standard Model uniquely emerge from the unfolding of a
single, isolated E8 singularity. | [
"Jacob L. Bourjaily"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | In [2] , Katz and Vafa showed how to geometrically engineer matter representations in terms of the local singularity structure of type IIa, M-theory, and Ftheory compactifications. In that framework matter and gauge theory both have purely geometrical origins: SU n , SO 2n and E n gauge theories arise from the existence of co-dimension four singular curves of certain types in the compactification manifold [3]; and massless matter representations arise from isolated points (in type IIa or M-theory) or curves (in F-theory) along the singular surface over which the type of singularity is enhanced by one rank.
Despite the extraordinary generality of this framework, it has not been widely used phenomenologically. This is largely because the description of the isolated enhancements of singularities giving rise to various matter representations is inherently local: although the geometry near any particular enhancement could be described concretely, the framework had nothing to say about numbers, types, and relative locations of different matter fields. This global data was either to be determined by duality to a concrete, global string theory model 1 , or suggested via the a posteriori success of a given set of relative positions (as in e.g. [5, 6] ). Another way to relate the number and relative positions of (enhanced singularities giving rise to) matter fields was given in [1] : in that paper, we described for example how a local description of the geometry giving rise to a massless 5 of SU 5 could be smoothly deformed into a local description of a (3, 1) and a (1, 2) of SU 3 ×SU 2 which live at distinct In this paper we describe pedagogically how to extend that idea to engineer analogies to SO 10 and E 6 × SU 2 grand unified models 2 . Although in [1] we were able to analyze explicit unfoldings of SO 10 and SU 6 singularities sufficiently well by sight, this will not be possible for our present examples. All of the examples in this paper involve the unfolding of isolated E n singularities; and although algebraic descriptions of these are known and classified [7] , it would be unnecessarily cumbersome and unenlightening to analyze them explicitly as we did in [1] . Therefore, in section II we describe a much more powerful and elegant language in which to study these resolutions.
In section III we describe in detail how the unfolding of a 16 of SO 10 into the Standard Model is derived in the language of section II. This is achieved in two stages: in the first stage, we unfold the 16 into 10 ⊕ 5 ⊕ 1 of SU 5 ; we then unfold the resulting SU 5 model into a single 'family' of the Standard Model. At the end, all the relative positions of the singularities of the family are set by the non-zero values of two complex structure moduli, thereby greatly reducing the arbitrariness of their relative positions. The next most obvious example would be a description of how a 27 of E 6 geometrically unfolds into the Standard Model. However, there are two reasons to leave this example to the reader: first, it is a most natural extension of the results of section III; secondly, it is a consequence of the E 6 × SU 2 grand unified model which we describe in section IV. Al-2 As described in section IV, the resolution E 8 → E 6 × SU 2 naturally starts as a theory with three 27's of E 6 related by an SU 2 family symmetry.
FIG. 1: A cartoon of the geometric deformation of a 5 and 10 of SU5 into the Standard Model as described in [1] . The surface over which the singularities are enhanced is coordinatized by a complex parameter t and the geometry is deformed by changing the value of a complex parameter s. The relative locations of the 'resolved' singularities are given in terms of s.
though not given as an example in [2] , it is not hard to see 3 that a single isolated E 8 singularity at the intersection of a co-dimension four surfaces of types E 6 and SU 2 gives rise to matter in the representation (27, 2) ⊕ (27, 1) ⊕ (1, 2). It is easy to see how this would unfold into the matter content of three families-one coming from each of the 27's. That three families emerge from E 8 is a general consequence of group theory and can be understood from the fact that E 6 × SU 3 is a maximal subgroup of E 8 into which the adjoint of E 8 partially branches into an SU 3 triplet of 27's.
As in the preceding paper [1], this work is presented concretely in the language of Calabi-Yau compactifications of type IIa string theory, which can also be naturally extended to F-theory models. Here, we engineer the explicit local geometry of (non-compact) Calabi-Yau three-folds which are K3-fibrations over C 1 . If type IIa string theory is compactified on this three-fold, a four-dimensional N = 2 theory with various massless hypermultiplets will result. But if, for example, the C 1 base of this three-fold were fibred as an O(-2) bundle over CP 1 , the resulting total space would be a Calabi-Yau four-fold 4 upon which F-theory would compactify to an N = 1 theory with chiral multiplets. However, because the manifold over which the singular K3's are fibred in M-theory is a real, three-dimensional space, our fibrations over C 1 do not have a direct application to M-theory.
It would of course be desirable to have a similar description of geometric unfolding explicitly in the language of G 2 -manifolds so that this picture could be realized concretely in M-theory as well. This is particularly important in light of the recent advances in M-theory phenomenology (e.g. [8, 9] ). By exten-3 This is described in section II.
4 This is just one example of the ways in which these Calabi-Yau three-folds could be fibred over CP 1 to result in a Calabi-Yau four-fold.
sion of the work of Berglund and Brandhuber in [10] , such a generalization should be relatively straightforward, but we will not attempt to do this here.
Recall that a gauge theory in type IIa string theory can arise from compactification to six dimensions over a singular K3 surface (similar statements apply to M-theory and F-theory) [3]. The complex structures of the singular compactification manifolds giving rise to SU n (≡ A n-1 ), SO 2n (≡ D n ), and E n gauge theory are given in Table I -where the surfaces are labelled conveniently by the name of the resulting gauge theory 5 .
We can generalize this discussion by considering a complex, one-dimensional space B over which a smooth family of singular K3 surfaces are fibred. If almost everywhere over B the K3-fibres have singularities of a single type, then compactification of type IIa string theory over the total space will give rise
Gauge group Polynomial SUn (≡ An-1) xy = z n SO2n (≡ Dn) x 2 + y 2 z = z n-1 E6 x 2 = y 3 + z 4 E7 x 2 + y 3 = 16yz 3 E8 x 2 + y 3 = z 5
TABLE I: Hypersurfaces in C 3 giving rise to the desired orbifold singularities.
5 It is of curious historical interest that the equations listed in Table I were first identified by Fleix Klein in 1884 [11] .
The reader may also be amused that the full resolutions of these surfaces were almost completely classified-up to a few computational errors-by Bramble in 1918 [12].
3 to gauge theory in four-dimensions of the type corresponding to the typical fibre. Massless charged matter will arise if over isolated points in B the type of fibre is enhanced by one rank. The geometry about a single such isolated point where the singularity is enhanced was described in detail by Katz and Vafa in [2] . The representation of matter living at these 'moresingular' points was also given in [2] : suppose that G ⊃ H × U 1 and that the rank of H is one less than G; then, if there is an isolated G-type singularity over a surface of H-type singularities, the resulting massless representation is given by those parts of the decomposition of the adjoint of G into H × U 1 which are charged under the U 1 .
Because the question of how to (smoothly) deform the surfaces of Table I into ones of lower rank has intrinsic mathematical interest, it is not too surprising that all possible two-dimensional deformations have been classified. Our discussion below will make use of the notation and results presented in [7] .
In our present work, we are interested in deformations of E n singularities into ones of lower rank. Unlike SU n singularities, the resolutions of which are easy enough to read off by sight, the algebraic complexity of E n singularities is formidable. To appreciate what is meant by this, consider the resolution of E 7 . From Table I we know that an E 7 singularity is locally isomorphic to the surface x 2 + y 3 = 16yz 3 in C 3 . Its full resolution in terms of the seven deformation parameters t = (t 1 , t 2 , . . . , t 7 ) is given by
-x 2 -y 3 + 16yz 3 + ǫ 2 ( t)y 2 z + ǫ 6 ( t)y 2 + ǫ 8 ( t)yz + ǫ 10 ( t)z 2 + ǫ 12 ( t)y + ǫ 14 ( t)z + ǫ 18 ( t) = 0, ( 1
)
where the ǫ n ( t) are n th order symmetric polynomials in the components of t which are tabulated over several pages of the appendix of [7].
A naïve way to determine the type of singularity found by resolving E 7 "in the direction t" would be to expand equation (1) completely using the explicit functions ǫ n ( t), find each of its singular points, and expand locally about each until an isomorphism with a singularity of lower rank in Table I was clear. This is the way, for example, that [2] demonstrated that the resolution of E 7 in the direction (0, 0, 0, 0, 0, t, 0) gives rise to E 6 for t = 0. All of the results in this paper could be verified in this way. Luckily, however, Katz and Morrison described a much more powerful and direct way to analyze the deformations of E n singularities [7].
We would like a pragmatic answer to the following question: what is the type of fibre found by resolving an E n singularity in the direction t? That there is an easy answer to this question makes our work much simpler. Although an adequate treatment would take us well beyond the scope of our present discussion, the answer given in [7] is at least very easy to make use of 6 : for each of the equations in Table II satisfied by the components of t, the singularity has the corresponding root. Given the list of roots, it is then a straight-forward exercise to construct the Dynkin diagram corresponding to the singularity 7 .
used. As stated before, we are working with the conventions of [7]. 7 Misusing the notation of [7] in a way applicable only to
In an admittedly bad notation, we consider each of the n deformation parameters t i (t) to be functions of t, the local coordinate on the base space B. A (non-Abelian) gauge theory will be present if there are roots implied by Table II which are preserved for generic values of t. And charged massless matter will exist if at isolated points {t * } an additional root is added-or, in terms of Dynkin diagrams, if an additional node is added. At each isolated point we can therefore identify the resolution G → H and thereby determine the resulting representation.
Root ti -tj = 0 =⇒ ei -ej ti + tj + t k = 0 =⇒ e0 -ei -ej -e k P 6 j=1 ti j = 0 =⇒ 2e0 - P 6 j=1 ei j 2ti 1 + P 7 j=2 ti j = 0 =⇒ 3e0 -2ei 1 - P 7 j=2 ei j
TABLE II: The roots of the singularity resulting from the resolution of En in the direction t. This is a reproduction of Table 4 of Ref. [7] .
SUn, SO 2n and En, one can think of the vectors e i as an orthonormal basis in Minkowski space which is equipped with a mostly-plus metric. Then roots are vectors in this space of norm +2. Each (positive) root gives rise to a node in the resulting Dynkin diagram, and two nodes are connected by a line if their inner product is -1 and disconnected if they are orthogonal.
III. GEOMETRIC ANALOGUE OF SO10 GRAND UNIFICATION A. The Description of a 16 of SO10
A necessary starting point to describe the unfolding of a 16 of SO 10 into the Standard Model is a description of the initial geometry as was done in [2] . We will briefly review that construction in the language described above before we unfold it, first into an SU 5 model, and later all the way into SU 3 × SU 2 .
Let t be a local complex coordinate on the space B over which is fibred the resolution of E 6 parameterized by t = (t, t, t, t, t, -2t). To be clear, for each value of t, the vector t describes an explicit surface in C 3 given in reference [7] analogous to that of equation (1) above.
Considering the rules of Table II , we see that for an arbitrary value of t = 0 the root lattice of the fibre is
(e 0 -e 1 -e 2 -e 6 ) (e 1 -e 2 ) (e 2 -e 3 ) (e 3 -e 4 ) (e 4 -e 5 ) , (2)
where we have displayed the roots suggestively so as to reproduce the SO 10 Dynkin diagram. At t = 0, however, E 6 is restored. So we have an isolated E 6 fibre over the point t = 0, while for any t = 0 the fibre is SO 10 . This gives rise to SO 10 gauge theory with a single massless 16 located at the origin in the t-plane.
We would like to unfold the manifold described above into one with SU 5 gauge theory. It is not hard to guess in what 'directions' t we may deform the the geometry so that the fibre over a generic point is SU 5 . Let a denote a parameter independent of t which adjusts the whole geometry over the region which is coordinatized by t. Then let the fibre over t be given by the resolution of E 6 in the direction (t, t, t, t, t + a, -2ta). Obviously when a = 0 the situation is the same as above and results in a single massless 16 of SO 10 . However, when a = 0 the situation is different: for generic values of t it is easy to see that the simple roots are (3) which means that the generic fibre over t is just SU 5 -and so the resulting gauge theory is SU 5 .
(e 0 -e 1 -e 5 -e 6 ) (e 1 -e 2 ) (e 2 -e 3 ) (e 3 -e 4 ) ,
To find what matter representations exist, we must determine over which locations t the rank of the fibre is enhanced. This means we are seeking special values of t (determined by a) at which an additional equation in Table II is satisfied. For each of these points, we can draw the resulting Dynkin dia-
TABLE III: The locations on the complex t-plane over which the singularity of the fibre is enhanced, and the representations of SU5 × U1 that result. Location Fibre Representation of SU5 × U1 3t + 2a = 0 SU5 × SU2 1-5 3t + a = 0 SO10 10-1 t = 0 SU6 53 gram to determine the fibre over that point, thereby determining the representation which arises there. It is not hard to exhaustively find all these 'more singular' points. They are give in Table III . Notice that we have included the U 1 -charge assignments that result; these are normalized as in the appendix of [13].
To complete our task and unfold the 16 of SO 10 all the way to the Standard Model, we must deform the fibres by another 'global' parameter, which we will denote b. It is not hard to guess a direction over which the generic fibre will be SU 3 × SU 2 : try for example (t, t, t, t + b, t + a, -2tab). Again, we notice that for a general location t and generic fixed values a, b = 0, the singularity has the root structure
(e 1 -e 2 ) (e 2 -e 3 ) ⊗ (e 0 -e 4 -e 5 -e 6 ) , (4) which is visibly SU 3 × SU 2 .
Like above, it is a straight-forward exercise to determine all the locations over which the singularity is enhanced, and the resulting representation which arises. These points including their resulting representations (with U 1 -charges as normalized in [13] ) are listed in Table IV . The entire unfolding is reproduced graphically in Figure 2 .
Representation
of SU 3 ×SU 2 ×U 1 Name 3t + 2a + b = 0 SU3 × SU2 × SU2 (1, 1)0 ν c L 3t + a + 2b = 0 SU3 × SU2 × SU2 (1, 1)6 e c L 3t + a + b = 0 SU5 (3, 2)1 Q 3t + a = 0 SU4 × SU2 (3, 1)-4 u c L 3t + b = 0 SU4 × SU2 (3, 1)2 d c L t = 0 SU3 × SU3 (1, 2)-3 L
TABLE IV: The locations on the complex t-plane over which the singularity of the fibre is enhanced and the representations of SU3 × SU2 × U1 that result.
FIG. 2: An illustration of the resolution of a geometrically engineered 16 of SO10 into the Standard Model as a function of two complex structure moduli a and b as described in section III. The coordinate along the base space is t and runs vertically in the diagram. When a = b = 0, along the left-hand side, there is just one isolated E6 singularity at t = 0. When b = 0 but a is allowed to vary, this single singularity splits into three, and any a =constant slice will have three isolated singularities in the complex t-plane as shown above. Moving rightward in the diagram, at the dashed line a is held fixed and b is allowed to grow, causing the three enhancements of SU5 to break apart into six total isolated singularities over SU3 × SU2, which is shown on the right-hand-side. Also shown are the (appropriately normalized) U1 charges of fields obtained via this multiple unfolding.
After having completed the unfolding of a 16 of SO 10 into the Standard Model, it is natural to ask if this idea can be extended to relate all the singularities of the Standard Model as perhaps the unfolding of a single isolated singularity of higher-rank. The answer is in fact yes-and there is a sense in which precisely three families arise if the notion of 'geometric unification' is saturated.
Because a 16 of SO 10 arises from the resolution E 6 → SO 10 , it can only be unfolded out an exceptional singularity. Clearly the highest level of unification one can achieve along this line would
be to start with a resolution E 8 → H where H is a rank-seven subgroup of E 8 which contains SO 10 .
The possible 'top-level' gauge groups are then
E 7 , E 6 × SU 2 , and SO 10 × SU 3 . We choose to study E 8 → E 6 × SU 2
as our example because it will naturally include a description of the unfolding of 27 of E 6 into the Standard Model, which is interesting in its own right, and because it follows quite directly from our work in section III.
The initial geometry which we will deform into the Standard Model is given as follows. Let t be a complex coordinate on the base space B over which is fibred the resolution (t, t, 0, 0, 0, 0, 0, 0) of E 8 . Clearly, when t = 0 we recover E 8 ; when t = 0 we see that the roots of the fibre are
(e 0 -e 3 -e 4 -e 5 ) (e 3 -e 4 ) (e 4 -e 5 ) (e 5 -e 6 ) (e 6 -e 7 ) (e 7 -e 8 ) ⊗ (e 1 -e 2 ) , ( 5
) which is visibly E 6 × SU 2 .
Following the general rule to determine the representation resulting from a given resolution [2] , we find that at t = 0 lives massless matter charged in the (27, 2)
1 ⊕(27, 1) -2 ⊕ (1, 2) 3 representation of E 6 × SU 2 × U 1ϕ .
To avoid pedantic redundancy, in Figure 3 we have summarized in great detail the entire unfolding into
SU 3 ×SU 2 ×U 1Y ×U 1χ ×U 1Ψ ×U 1 ζ ×U 1ϕ .
An outline of the steps involved in deriving this unfolding is given presently.
First, the unfolding of the E 6 × SU 2 gauge theory into E 6 gauge theory is obtained by defining the fibre over t to be given by the resolution of E 8 in the direction (t + a, ta, 0, 0, 0, 0, 0, 0) for some a = 0. This clearly kills the SU 2 node of the fibre in equation (5) . There are five locations at which the singularity is enhanced by one rank, giving rise to three 27's and two singlets as shown in the leftmost section of Figure 3. 6 FIG. 3: An illustration of the resolution of a single isolated E8 into the Standard Model in terms of four deformation parameters a, b, c, d. Along the left hand side, for a = b = c = d = 0, the generic E6 × SU2 fibre is enhanced to E8 at t = 0. Moving from left to right, a, b, c, d are sequentially allowed to grow to some non-zero value-and between dashed lines all but one of the moduli are held fixed. Solid lines indicate the locations of enhanced singularities relative to the plane for as functions of a, b, c, d. The complete list of isolated singularities, their locations, and charge assignments are given on the right hand side of the diagram.
The rest of the unfolding is a natural application of the work in section III. Let us now set the fibre over t to be given by the resolution (t + a, ta, b, b, b, b, b, -2b) of E 8 for arbitrary complex deformation parameters a, b = 0. From section III we see immediately that the generic fibre is SO 10 . A thorough scanning for possible solutions to equations in Table II shows that there are 11 isolated points on the complex t-plane over which the singularity is enhanced. These correspond to the 'breaking' of each 27 of E 6 into 16 ⊕ 10 ⊕ 1 of SO 10 , while the singlets remain singlets. This is seen in the second vertical strip (from the left) in Figure 3 .
Again, following our discussion above, it is easy to guess possibilities for the next two resolution directions. First, we set the fibre over t to be given by (t + a, ta, b, b, b, b, b + c, -2bc) which will result SU 5 gauge theory with matter content corresponding to the 'canonical' decomposition of three 27's of E 6 with two singlets. And finally, the full resolution of the E 6 × SU 2 grand unified model into SU 3 × SU 2 can be given by letting the fibre over t be given by the (t+a, t-a, t, t, t, t+d, t+c, -2b-c-d) resolution of E 8 for (generic) arbitrary fixed complex structure moduli a, b, c, d = 0.
Let us clarify what we have done. For a given set of fixed, nonzero complex structure moduli, the resolution given above describes the explicit, local geometry of a non-compact Calabi-Yau three-fold, which is a K3-fibration over C 1 . If type IIa string theory is compactified on this three-fold, the resulting fourdimensional theory will have SU 3 × SU 2 gauge theory with hypermultiplets at isolated points as given in Figure 3 which reproduce the spectrum of three families of the Standard Model with an extended Higgs sector and some exotics. Alternatively, if one takes this (non-compact) Calabi-Yau three-fold and fibres it over CP 1 as described in section I so that the total space is Calabi-Yau, then F-theory on this space will give rise to N = 1 supersymmetry with SU 3 × SU 2 gauge theory and chiral multiplets in the representations given in Figure 3 . And although it does not follow directly from our construction above, considering the close similarities between two-and three-dimensional resolutions of the singular K3 surfaces we have every reason to suspect an analogous geometry can be engineered for M-theory in terms of hyper-Kähler quotients by extension of the results in [10, 14, 15] . We are currently working on building this geometry in M-theory, and we expect to report on this work soon.
Given these four complex structure moduli, all the relative positions of the 35 disparate singularities giving rise to all three families of the (extended) Standard Model are then known 8 . Beyond the usual three families of the Standard Model, the manifold also gives rise to two Higgs doublets for each family, six Higgs colour triplets, three right-handed neutrinos and five other Standard Model singlets. We should point out that this matter content (and their U 1 -charge assignments) is a consequence of group theory and algebraic geometry alone-it is simply what is found when unfolding E 8 all the way to the Standard Model.
And given the relative positions and local geometry of the singularities together with the U 1structure, one can in principle compute the full superpotential coming from instantons wrapping different singularities. Because these are fixed by the values of the complex structure moduli, there is a (complex) four-dimensional landscape 9 of different, explicit SU 3 × SU 2 embeddings at the compactification scale. Although this large landscape may appear to have too much freedom, we remind the reader that in the traditional understanding of geometrical engineering there would be hundreds of parameters describing the (independent) relative locations of each of the isolated singularities.
There are a few things to notice about the form of the superpotential that will emerge. First, because of the U 1 -charge assignments, each term in the superpotential must combine exactly one term arising from each of the 27's. This greatly limits the form of the superpotential. And in particular, it implies that neither mass nor flavour eigenstates will arise from any single 27-that is, the 'families' in the colloquial sense are necessarily linear combinations of fields resulting from different 27's.
Also notice that in general the terms in the superpotential will be proportional to e - R dVol where dVol is the volume form of some cycle wrapping singularities (the details of which depends on whether we are talking about type IIa, M-theory, or F-theory realizations), and are in principle calculable in terms of the deformation moldui. And because these coefficients are exponentially related to the volumes of cycles, we expect the high-scale Lagrangian will be generically hierarchical. This structure could be important for solving problems in phenomenology-for example the µ problem in the Higgs potential, the Higgs doublet-triplet splitting problem, or avoiding 8 A subtlety, however, is that because our language has been explicitly that of N = 2 theory from type IIa, we are unable to distinguish the 5 from the 5 in the splitting of the 10's of SO 10 . In Figure 3 , a consistent choice was made-and although we do not justify this claim here, it is the choice that will be correct for the M-theory generalization of this work. 9 That it is continuous is a consequence of the fact that we are engineering non-compact Calabi-Yaus. If one matched this local geometry to a compact global structure, the landscape would of course be discrete.
8 proton decay. We are in the process of studying the phenomenology of models on this landscape. At first glance, the U 1 -structure combined with high-scale hierarchies could possibly be complex enough to be able to avoid some of the typical problems of E 6 -like grand unified models. We should point out that if there were no high-scale hierarchies, however, then the allowed terms in the superpotential would generically give rise to low-energy lepton and baryon number violation, similar to any 'generic' E 6 model-i.e. one which includes all types of terms allowed by the E 6mandated U 1 -structure [16] . We could always impose additional symmetries and add fields by hand to solve these problems, but this would not be very compelling. However, if viable models already exist in the landscape which do not require additional fields or symmetries, these would be compelling even if we do not yet understand how they are selected.
One of the most important phenomenological questions about these models is the fate of the additional U 1 symmetries. Although we suspect that one can determine which of the U 1 symmetries are dynamical below the compactification scale by studying the normalizability of their corresponding vector multiplets, we do not presently have have a complete understanding of this situation. Of course, if any additional U 1 's survive to low energy they could have very interesting-or damningphenomenological consequences.
An important point to bear in mind when considering geometrically engineered models is that there generically exist 10 moduli which can deform the geometry into one which gives rise to a theory with less gauge symmetry. For example, if you are given a geometrically-engineered SO 10 grand unified model, then our results show explicitly that the model can be locally deformed into an SU 5 model, and this can be deformed further into the Standard Model; the original SO 10 theory is seen to be a single point in a (complex) two-dimensional landscape of SU 3 × SU 2 theories. And because larger symmetries always lie in lower dimensional surfaces of moduli space, it is very relevant to ask what physics prevents this unfolding from taking place. Indeed, this question applies to the Standard Model as wellour analysis could easily go further to unfold away SU 3 ×SU 2 . We are not presently able to answer why this does not happen 11 ; although this observation 10 There could be global obstructions which prevent such a deformation from taking place. But these are invisible to the non-compact, local constructions considered here. 11 Although, perhaps the unfolding of SU 2 may provide an alternative to tuning in the usual Higgs sector [17] . It would be interesting to understand in greater detail the relation-
suggests that perhaps theories with less symmetry, like SU 3 × SU 2 , could be much more natural than grand unified theories. More generally, it is not presently understood what physics controls the values of the geometric moduli which deform the manifold-the parameters which deform the E 8 → E 6 ×SU 2 complex structure, for example. We do not yet have a general mechanism which would fix these parameters; we simply observe that any non-zero values of the moduli will give rise to a geometrically engineered manifold with SU 3 × SU 2 gauge theory 'peppered' with all the necessary singularities of the three families of the Standard Model together with the usual E 6 -like exotics. And importantly, for any point in the complex fourdimensional 'landscape,' the relative locations of all the relevant singularities are known-and hence in principle so is the superpotential.
This relationship between moduli-fixing and gauge symmetry breaking could be a novel feature of geometrically-unfolded models. It may allow one to apply the results in [8] , for example, to single out theories on the landscape. However, a prerequisite to this type of analysis would be an identification of which moduli should be identified with the ones which deform the geometry as described here.
Although the motivation in this paper and in [1] appears to be a top-down realization of grand unification, there is a sense in which we are really engineering from the bottom-up. Specifically, because the local geometry we have described is non-compact, the resulting theory is decoupled from quantum gravity, and the parameters along the landscape of deformations are continuous. This is not unlike the situation in [18] . But what we lose in global constraints we perhaps gain by concrete local structure. Not only do we have a framework which naturally predicts three families with a rather detailed phenomenological structure, but we have done so in a way that preserves all the information about the local geometry. And because this framework realizes the 'physics from pure geometry' paradigm in a potentially powerful way, it could prove important to concrete phenomenological constructions in M-theory, for example.
Of course we envision these local geometries to be embedded within compact Calabi-Yau manifolds. It is an assumption of the framework that the precise global topology of the compactification manifold can be ignored at least as a good first approximation. One may ask the extent to which these constructions can be glued into compact manifolds. Concretely: under what circumstances can a noncompact Calabi-Yau three-fold which is a fibration of K3 surfaces with asymptotically uniform ADE-type ship between unfolding and the Higgs mechanism.
9 singularities be compactified? This is an important question for mathematicians, the answer to which would likely lead to important physical insight-e.g. quantization of the moduli space of deformations.
A possible objection to this framework is that our constructions appear to depend on several seemingly arbitrary choices (the specific chain from E 8 to the Standard Model, which roots were eliminated at each step, etc.). However, it is likely that the particle content, for example, which results is completely independent of these choices. Furthermore, we suspect that different realizations of the unfolding merely result in different parameterizations of the landscape, and do not reflect true additional arbitrariness. But this is still an area that deserves attention.
Lastly, because in this picture the Standard Model is seen to unfold at the compactification scale, one may ask what has become of gauge coupling unification. Because the gauge coupling constants are functions of the volumes of their corresponding codimension four singular surfaces 12 which depend on the deformation moduli, the traditional meaning of grand unification is more subtle here-as is typical in string phenomenology. For example, although we chose to unfold the Standard Model sequentially as a series of less unified models, there is no reason to suspect that that order has any physical importance. Surely, if as we parameterized the unfolding in section IV, setting d → 0 (or c → 0) would result in an SU 5 grand unified theory; but setting a → 0 instead would result in a restoration of family symmetry. The four complex structure moduli tune different types of unification separately-and should simultaneously be at play in the question of gauge coupling unification.
It is interesting to note, however, that if one were to simultaneously scale the values of all the moduli to be very small, the spectrum would be more and more unified: the relative distances between singularities shrink, unifying the coefficients in the superpotential; and the volumes of the co-dimension four singularities (if realized in a compact manifold) would approach one another, resulting in a unification of their gauge couplings. What this may mean phenomenologically remains to be understood.
In this paper we have described a local, purely geometric framework in which gauge symmetry 'breaking' can be re-cast as a problem of moduli fixingand in which the same moduli which describe this geometric 'unfolding' also determine the physics of massless matter. And although we still do not understand the mechanisms by which these moduli are fixed, the landscape of possibilities is already enormously reduced: what would have been the hundreds of parameters describing the relative positions on the compactification manifold of the Standard Model's three families worth of matter fields, we specify them all in terms of only four complex structure moduli which describe the unfolding of an isolated E 8 singularity. And the fact that three families emerges is group-theoretic and not added by hand.
It is a pleasure to thank helpful discussions with and insightful comments of Herman Verlinde, Sergei Gukov, Gordy Kane, Paul Langacker, Edward Witten, Cumrun Vafa, Brent Nelson, Malcolm Perry, Dmitry Malyshev, Matthew Buican, Piyush Kumar, and Konstantin Bobkov.
This work was funded in part by a Graduate Research Fellowship from the National Science Foundation.
12 Of course, this can only be discussed concretely when the compact manifold is known.
[1] J. L. Bourjaily, "Multiple Unfoldings of Orbifold Singularities: Engineering Geometric Analogies to Unification," arXiv:0704.0444. [2] S. Katz and C. Vafa, "Matter from Geometry," Nucl. Phys., vol. B497, pp. 146-154, 1997, hep-th/9606086. [3] A. Klemm, W. Lerche, and P. Mayr, "K3 Fibrations and Heterotic Type II String Duality," Phys. Lett., vol. B357, pp. 313-322, 1995, hep-th/9506112. [4] B. S. Acharya and S. Gukov, "M-theory and Singularities of Exceptional Holonomy Manifolds," Phys. Rept., vol. 392, pp. 121-189, 2004, hep-th/0409191. [5] E. Witten, "Deconstruction, G2 Holonomy, and Doublet-Triplet Splitting," 2001, hep-ph/0201018. [6] T. Friedmann and E. Witten, "Unification Scale, Proton Decay, and Manifolds of G(2) Holonomy," Adv. Theor. Math. Phys., vol. 7, pp. 577-617, 2003, hep-th/0211269. [7] S. Katz and D. Morrison, "Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups," J. Algebraic Geometry, vol. 1, pp. 449-530, 1992. [8] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and D. Vaman, "An M Theory Solution to the Hierarchy Problem," Phys. Rev. Lett., vol. 97, p. 191601, 2006, hep-th/0606262. [9] B. S. Acharya, K. Bobkov, G. L. Kane, P. Kumar, and J. Shao, "Explaining the Electroweak Scale and Stabilizing Moduli in M-Theory," 2007, hep-th/0701034. [10] P. Berglund and A. Brandhuber, "Matter from G2 Manifolds," Nucl. Phys., vol. B641, pp. 351-375, 2002, hep-th/0205184. [11] F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom funften Grade. Leipzig: Teubner, 1884. [12] C. C. Bramble, "A Collineation Group Isomorphic with the Group of Double Tangents of the Plane 10 Quartic," Amer. J. Math., vol. 40, pp. 351-365, 1918. [13] R. Slansky, "Group Theory for Unified Model Building," Phys. Rept., vol. 79, pp. 1-128, 1981. [14] M. Atiyah and E. Witten, "M-theory Dynamics on a Manifold of G2 Holonomy," Adv. Theor. Math. Phys., vol. 6, pp. 1-106, 2003, hep-th/0107177. [15] B. Acharya and E. Witten, "Chiral Fermions from Manifolds of G2 Holonomy," 2001, hep-th/0109152. [16] J. L. Hewett and T. G. Rizzo, "Low-Energy Phenomenology of Superstring Inspired E(6) Models," Phys. Rept., vol. 183, p. 193, 1989. [17] This was pointed out by Paul Langacker in a private correspondance. [18] H. Verlinde and M. Wijnholt, "Building the Standard Model on a D3-Brane," JHEP, vol. 01, p. 106, 2007, hep-th/0508089. | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In [2] , Katz and Vafa showed how to geometrically engineer matter representations in terms of the local singularity structure of type IIa, M-theory, and Ftheory compactifications. In that framework matter and gauge theory both have purely geometrical origins: SU n , SO 2n and E n gauge theories arise from the existence of co-dimension four singular curves of certain types in the compactification manifold [3]; and massless matter representations arise from isolated points (in type IIa or M-theory) or curves (in F-theory) along the singular surface over which the type of singularity is enhanced by one rank.\n\nDespite the extraordinary generality of this framework, it has not been widely used phenomenologically. This is largely because the description of the isolated enhancements of singularities giving rise to various matter representations is inherently local: although the geometry near any particular enhancement could be described concretely, the framework had nothing to say about numbers, types, and relative locations of different matter fields. This global data was either to be determined by duality to a concrete, global string theory model 1 , or suggested via the a posteriori success of a given set of relative positions (as in e.g. [5, 6] ). Another way to relate the number and relative positions of (enhanced singularities giving rise to) matter fields was given in [1] : in that paper, we described for example how a local description of the geometry giving rise to a massless 5 of SU 5 could be smoothly deformed into a local description of a (3, 1) and a (1, 2) of SU 3 ×SU 2 which live at distinct In this paper we describe pedagogically how to extend that idea to engineer analogies to SO 10 and E 6 × SU 2 grand unified models 2 . Although in [1] we were able to analyze explicit unfoldings of SO 10 and SU 6 singularities sufficiently well by sight, this will not be possible for our present examples. All of the examples in this paper involve the unfolding of isolated E n singularities; and although algebraic descriptions of these are known and classified [7] , it would be unnecessarily cumbersome and unenlightening to analyze them explicitly as we did in [1] . Therefore, in section II we describe a much more powerful and elegant language in which to study these resolutions.\n\nIn section III we describe in detail how the unfolding of a 16 of SO 10 into the Standard Model is derived in the language of section II. This is achieved in two stages: in the first stage, we unfold the 16 into 10 ⊕ 5 ⊕ 1 of SU 5 ; we then unfold the resulting SU 5 model into a single 'family' of the Standard Model. At the end, all the relative positions of the singularities of the family are set by the non-zero values of two complex structure moduli, thereby greatly reducing the arbitrariness of their relative positions. The next most obvious example would be a description of how a 27 of E 6 geometrically unfolds into the Standard Model. However, there are two reasons to leave this example to the reader: first, it is a most natural extension of the results of section III; secondly, it is a consequence of the E 6 × SU 2 grand unified model which we describe in section IV. Al-2 As described in section IV, the resolution E 8 → E 6 × SU 2 naturally starts as a theory with three 27's of E 6 related by an SU 2 family symmetry.\n\nFIG. 1: A cartoon of the geometric deformation of a 5 and 10 of SU5 into the Standard Model as described in [1] . The surface over which the singularities are enhanced is coordinatized by a complex parameter t and the geometry is deformed by changing the value of a complex parameter s. The relative locations of the 'resolved' singularities are given in terms of s.\n\nthough not given as an example in [2] , it is not hard to see 3 that a single isolated E 8 singularity at the intersection of a co-dimension four surfaces of types E 6 and SU 2 gives rise to matter in the representation (27, 2) ⊕ (27, 1) ⊕ (1, 2). It is easy to see how this would unfold into the matter content of three families-one coming from each of the 27's. That three families emerge from E 8 is a general consequence of group theory and can be understood from the fact that E 6 × SU 3 is a maximal subgroup of E 8 into which the adjoint of E 8 partially branches into an SU 3 triplet of 27's.\n\nAs in the preceding paper [1], this work is presented concretely in the language of Calabi-Yau compactifications of type IIa string theory, which can also be naturally extended to F-theory models. Here, we engineer the explicit local geometry of (non-compact) Calabi-Yau three-folds which are K3-fibrations over C 1 . If type IIa string theory is compactified on this three-fold, a four-dimensional N = 2 theory with various massless hypermultiplets will result. But if, for example, the C 1 base of this three-fold were fibred as an O(-2) bundle over CP 1 , the resulting total space would be a Calabi-Yau four-fold 4 upon which F-theory would compactify to an N = 1 theory with chiral multiplets. However, because the manifold over which the singular K3's are fibred in M-theory is a real, three-dimensional space, our fibrations over C 1 do not have a direct application to M-theory.\n\nIt would of course be desirable to have a similar description of geometric unfolding explicitly in the language of G 2 -manifolds so that this picture could be realized concretely in M-theory as well. This is particularly important in light of the recent advances in M-theory phenomenology (e.g. [8, 9] ). By exten-3 This is described in section II.\n\n4 This is just one example of the ways in which these Calabi-Yau three-folds could be fibred over CP 1 to result in a Calabi-Yau four-fold.\n\nsion of the work of Berglund and Brandhuber in [10] , such a generalization should be relatively straightforward, but we will not attempt to do this here."
},
{
"section_type": "OTHER",
"section_title": "II. RESOLVING En-TYPE SINGULARITIES",
"text": "Recall that a gauge theory in type IIa string theory can arise from compactification to six dimensions over a singular K3 surface (similar statements apply to M-theory and F-theory) [3]. The complex structures of the singular compactification manifolds giving rise to SU n (≡ A n-1 ), SO 2n (≡ D n ), and E n gauge theory are given in Table I -where the surfaces are labelled conveniently by the name of the resulting gauge theory 5 .\n\nWe can generalize this discussion by considering a complex, one-dimensional space B over which a smooth family of singular K3 surfaces are fibred. If almost everywhere over B the K3-fibres have singularities of a single type, then compactification of type IIa string theory over the total space will give rise\n\nGauge group Polynomial SUn (≡ An-1) xy = z n SO2n (≡ Dn) x 2 + y 2 z = z n-1 E6 x 2 = y 3 + z 4 E7 x 2 + y 3 = 16yz 3 E8 x 2 + y 3 = z 5\n\nTABLE I: Hypersurfaces in C 3 giving rise to the desired orbifold singularities.\n\n5 It is of curious historical interest that the equations listed in Table I were first identified by Fleix Klein in 1884 [11] .\n\nThe reader may also be amused that the full resolutions of these surfaces were almost completely classified-up to a few computational errors-by Bramble in 1918 [12].\n\n3 to gauge theory in four-dimensions of the type corresponding to the typical fibre. Massless charged matter will arise if over isolated points in B the type of fibre is enhanced by one rank. The geometry about a single such isolated point where the singularity is enhanced was described in detail by Katz and Vafa in [2] . The representation of matter living at these 'moresingular' points was also given in [2] : suppose that G ⊃ H × U 1 and that the rank of H is one less than G; then, if there is an isolated G-type singularity over a surface of H-type singularities, the resulting massless representation is given by those parts of the decomposition of the adjoint of G into H × U 1 which are charged under the U 1 .\n\nBecause the question of how to (smoothly) deform the surfaces of Table I into ones of lower rank has intrinsic mathematical interest, it is not too surprising that all possible two-dimensional deformations have been classified. Our discussion below will make use of the notation and results presented in [7] .\n\nIn our present work, we are interested in deformations of E n singularities into ones of lower rank. Unlike SU n singularities, the resolutions of which are easy enough to read off by sight, the algebraic complexity of E n singularities is formidable. To appreciate what is meant by this, consider the resolution of E 7 . From Table I we know that an E 7 singularity is locally isomorphic to the surface x 2 + y 3 = 16yz 3 in C 3 . Its full resolution in terms of the seven deformation parameters t = (t 1 , t 2 , . . . , t 7 ) is given by\n\n-x 2 -y 3 + 16yz 3 + ǫ 2 ( t)y 2 z + ǫ 6 ( t)y 2 + ǫ 8 ( t)yz + ǫ 10 ( t)z 2 + ǫ 12 ( t)y + ǫ 14 ( t)z + ǫ 18 ( t) = 0, ( 1\n\n)\n\nwhere the ǫ n ( t) are n th order symmetric polynomials in the components of t which are tabulated over several pages of the appendix of [7].\n\nA naïve way to determine the type of singularity found by resolving E 7 \"in the direction t\" would be to expand equation (1) completely using the explicit functions ǫ n ( t), find each of its singular points, and expand locally about each until an isomorphism with a singularity of lower rank in Table I was clear. This is the way, for example, that [2] demonstrated that the resolution of E 7 in the direction (0, 0, 0, 0, 0, t, 0) gives rise to E 6 for t = 0. All of the results in this paper could be verified in this way. Luckily, however, Katz and Morrison described a much more powerful and direct way to analyze the deformations of E n singularities [7].\n\nWe would like a pragmatic answer to the following question: what is the type of fibre found by resolving an E n singularity in the direction t? That there is an easy answer to this question makes our work much simpler. Although an adequate treatment would take us well beyond the scope of our present discussion, the answer given in [7] is at least very easy to make use of 6 : for each of the equations in Table II satisfied by the components of t, the singularity has the corresponding root. Given the list of roots, it is then a straight-forward exercise to construct the Dynkin diagram corresponding to the singularity 7 ."
},
{
"section_type": "OTHER",
"section_title": "Of course, this answer does depend on the parameterization",
"text": "used. As stated before, we are working with the conventions of [7]. 7 Misusing the notation of [7] in a way applicable only to\n\nIn an admittedly bad notation, we consider each of the n deformation parameters t i (t) to be functions of t, the local coordinate on the base space B. A (non-Abelian) gauge theory will be present if there are roots implied by Table II which are preserved for generic values of t. And charged massless matter will exist if at isolated points {t * } an additional root is added-or, in terms of Dynkin diagrams, if an additional node is added. At each isolated point we can therefore identify the resolution G → H and thereby determine the resulting representation."
},
{
"section_type": "OTHER",
"section_title": "Equation",
"text": "Root ti -tj = 0 =⇒ ei -ej ti + tj + t k = 0 =⇒ e0 -ei -ej -e k P 6 j=1 ti j = 0 =⇒ 2e0 - P 6 j=1 ei j 2ti 1 + P 7 j=2 ti j = 0 =⇒ 3e0 -2ei 1 - P 7 j=2 ei j\n\nTABLE II: The roots of the singularity resulting from the resolution of En in the direction t. This is a reproduction of Table 4 of Ref. [7] .\n\nSUn, SO 2n and En, one can think of the vectors e i as an orthonormal basis in Minkowski space which is equipped with a mostly-plus metric. Then roots are vectors in this space of norm +2. Each (positive) root gives rise to a node in the resulting Dynkin diagram, and two nodes are connected by a line if their inner product is -1 and disconnected if they are orthogonal.\n\nIII. GEOMETRIC ANALOGUE OF SO10 GRAND UNIFICATION A. The Description of a 16 of SO10\n\nA necessary starting point to describe the unfolding of a 16 of SO 10 into the Standard Model is a description of the initial geometry as was done in [2] . We will briefly review that construction in the language described above before we unfold it, first into an SU 5 model, and later all the way into SU 3 × SU 2 .\n\nLet t be a local complex coordinate on the space B over which is fibred the resolution of E 6 parameterized by t = (t, t, t, t, t, -2t). To be clear, for each value of t, the vector t describes an explicit surface in C 3 given in reference [7] analogous to that of equation (1) above.\n\nConsidering the rules of Table II , we see that for an arbitrary value of t = 0 the root lattice of the fibre is\n\n(e 0 -e 1 -e 2 -e 6 ) (e 1 -e 2 ) (e 2 -e 3 ) (e 3 -e 4 ) (e 4 -e 5 ) , (2)\n\nwhere we have displayed the roots suggestively so as to reproduce the SO 10 Dynkin diagram. At t = 0, however, E 6 is restored. So we have an isolated E 6 fibre over the point t = 0, while for any t = 0 the fibre is SO 10 . This gives rise to SO 10 gauge theory with a single massless 16 located at the origin in the t-plane."
},
{
"section_type": "OTHER",
"section_title": "B. Unfolding the 16 of SO10 into SU5",
"text": "We would like to unfold the manifold described above into one with SU 5 gauge theory. It is not hard to guess in what 'directions' t we may deform the the geometry so that the fibre over a generic point is SU 5 . Let a denote a parameter independent of t which adjusts the whole geometry over the region which is coordinatized by t. Then let the fibre over t be given by the resolution of E 6 in the direction (t, t, t, t, t + a, -2ta). Obviously when a = 0 the situation is the same as above and results in a single massless 16 of SO 10 . However, when a = 0 the situation is different: for generic values of t it is easy to see that the simple roots are (3) which means that the generic fibre over t is just SU 5 -and so the resulting gauge theory is SU 5 .\n\n(e 0 -e 1 -e 5 -e 6 ) (e 1 -e 2 ) (e 2 -e 3 ) (e 3 -e 4 ) ,\n\nTo find what matter representations exist, we must determine over which locations t the rank of the fibre is enhanced. This means we are seeking special values of t (determined by a) at which an additional equation in Table II is satisfied. For each of these points, we can draw the resulting Dynkin dia-\n\nTABLE III: The locations on the complex t-plane over which the singularity of the fibre is enhanced, and the representations of SU5 × U1 that result. Location Fibre Representation of SU5 × U1 3t + 2a = 0 SU5 × SU2 1-5 3t + a = 0 SO10 10-1 t = 0 SU6 53 gram to determine the fibre over that point, thereby determining the representation which arises there. It is not hard to exhaustively find all these 'more singular' points. They are give in Table III . Notice that we have included the U 1 -charge assignments that result; these are normalized as in the appendix of [13]."
},
{
"section_type": "OTHER",
"section_title": "C. Unfolding a 16 of SO10 into the Standard Model",
"text": "To complete our task and unfold the 16 of SO 10 all the way to the Standard Model, we must deform the fibres by another 'global' parameter, which we will denote b. It is not hard to guess a direction over which the generic fibre will be SU 3 × SU 2 : try for example (t, t, t, t + b, t + a, -2tab). Again, we notice that for a general location t and generic fixed values a, b = 0, the singularity has the root structure\n\n(e 1 -e 2 ) (e 2 -e 3 ) ⊗ (e 0 -e 4 -e 5 -e 6 ) , (4) which is visibly SU 3 × SU 2 .\n\nLike above, it is a straight-forward exercise to determine all the locations over which the singularity is enhanced, and the resulting representation which arises. These points including their resulting representations (with U 1 -charges as normalized in [13] ) are listed in Table IV . The entire unfolding is reproduced graphically in Figure 2 ."
},
{
"section_type": "OTHER",
"section_title": "Location Fibre",
"text": "Representation\n\nof SU 3 ×SU 2 ×U 1 Name 3t + 2a + b = 0 SU3 × SU2 × SU2 (1, 1)0 ν c L 3t + a + 2b = 0 SU3 × SU2 × SU2 (1, 1)6 e c L 3t + a + b = 0 SU5 (3, 2)1 Q 3t + a = 0 SU4 × SU2 (3, 1)-4 u c L 3t + b = 0 SU4 × SU2 (3, 1)2 d c L t = 0 SU3 × SU3 (1, 2)-3 L\n\nTABLE IV: The locations on the complex t-plane over which the singularity of the fibre is enhanced and the representations of SU3 × SU2 × U1 that result.\n\nFIG. 2: An illustration of the resolution of a geometrically engineered 16 of SO10 into the Standard Model as a function of two complex structure moduli a and b as described in section III. The coordinate along the base space is t and runs vertically in the diagram. When a = b = 0, along the left-hand side, there is just one isolated E6 singularity at t = 0. When b = 0 but a is allowed to vary, this single singularity splits into three, and any a =constant slice will have three isolated singularities in the complex t-plane as shown above. Moving rightward in the diagram, at the dashed line a is held fixed and b is allowed to grow, causing the three enhancements of SU5 to break apart into six total isolated singularities over SU3 × SU2, which is shown on the right-hand-side. Also shown are the (appropriately normalized) U1 charges of fields obtained via this multiple unfolding."
},
{
"section_type": "OTHER",
"section_title": "IV. GEOMETRIC ANALOGUE OF E6 × SU2 GRAND UNIFICATION",
"text": "After having completed the unfolding of a 16 of SO 10 into the Standard Model, it is natural to ask if this idea can be extended to relate all the singularities of the Standard Model as perhaps the unfolding of a single isolated singularity of higher-rank. The answer is in fact yes-and there is a sense in which precisely three families arise if the notion of 'geometric unification' is saturated.\n\nBecause a 16 of SO 10 arises from the resolution E 6 → SO 10 , it can only be unfolded out an exceptional singularity. Clearly the highest level of unification one can achieve along this line would\n\nbe to start with a resolution E 8 → H where H is a rank-seven subgroup of E 8 which contains SO 10 .\n\nThe possible 'top-level' gauge groups are then\n\nE 7 , E 6 × SU 2 , and SO 10 × SU 3 . We choose to study E 8 → E 6 × SU 2\n\nas our example because it will naturally include a description of the unfolding of 27 of E 6 into the Standard Model, which is interesting in its own right, and because it follows quite directly from our work in section III.\n\nThe initial geometry which we will deform into the Standard Model is given as follows. Let t be a complex coordinate on the base space B over which is fibred the resolution (t, t, 0, 0, 0, 0, 0, 0) of E 8 . Clearly, when t = 0 we recover E 8 ; when t = 0 we see that the roots of the fibre are\n\n(e 0 -e 3 -e 4 -e 5 ) (e 3 -e 4 ) (e 4 -e 5 ) (e 5 -e 6 ) (e 6 -e 7 ) (e 7 -e 8 ) ⊗ (e 1 -e 2 ) , ( 5\n\n) which is visibly E 6 × SU 2 .\n\nFollowing the general rule to determine the representation resulting from a given resolution [2] , we find that at t = 0 lives massless matter charged in the (27, 2)\n\n1 ⊕(27, 1) -2 ⊕ (1, 2) 3 representation of E 6 × SU 2 × U 1ϕ .\n\nTo avoid pedantic redundancy, in Figure 3 we have summarized in great detail the entire unfolding into\n\nSU 3 ×SU 2 ×U 1Y ×U 1χ ×U 1Ψ ×U 1 ζ ×U 1ϕ .\n\nAn outline of the steps involved in deriving this unfolding is given presently.\n\nFirst, the unfolding of the E 6 × SU 2 gauge theory into E 6 gauge theory is obtained by defining the fibre over t to be given by the resolution of E 8 in the direction (t + a, ta, 0, 0, 0, 0, 0, 0) for some a = 0. This clearly kills the SU 2 node of the fibre in equation (5) . There are five locations at which the singularity is enhanced by one rank, giving rise to three 27's and two singlets as shown in the leftmost section of Figure 3. 6 FIG. 3: An illustration of the resolution of a single isolated E8 into the Standard Model in terms of four deformation parameters a, b, c, d. Along the left hand side, for a = b = c = d = 0, the generic E6 × SU2 fibre is enhanced to E8 at t = 0. Moving from left to right, a, b, c, d are sequentially allowed to grow to some non-zero value-and between dashed lines all but one of the moduli are held fixed. Solid lines indicate the locations of enhanced singularities relative to the plane for as functions of a, b, c, d. The complete list of isolated singularities, their locations, and charge assignments are given on the right hand side of the diagram.\n\nThe rest of the unfolding is a natural application of the work in section III. Let us now set the fibre over t to be given by the resolution (t + a, ta, b, b, b, b, b, -2b) of E 8 for arbitrary complex deformation parameters a, b = 0. From section III we see immediately that the generic fibre is SO 10 . A thorough scanning for possible solutions to equations in Table II shows that there are 11 isolated points on the complex t-plane over which the singularity is enhanced. These correspond to the 'breaking' of each 27 of E 6 into 16 ⊕ 10 ⊕ 1 of SO 10 , while the singlets remain singlets. This is seen in the second vertical strip (from the left) in Figure 3 .\n\nAgain, following our discussion above, it is easy to guess possibilities for the next two resolution directions. First, we set the fibre over t to be given by (t + a, ta, b, b, b, b, b + c, -2bc) which will result SU 5 gauge theory with matter content corresponding to the 'canonical' decomposition of three 27's of E 6 with two singlets. And finally, the full resolution of the E 6 × SU 2 grand unified model into SU 3 × SU 2 can be given by letting the fibre over t be given by the (t+a, t-a, t, t, t, t+d, t+c, -2b-c-d) resolution of E 8 for (generic) arbitrary fixed complex structure moduli a, b, c, d = 0."
},
{
"section_type": "OTHER",
"section_title": "V. IMPLICATIONS",
"text": "Let us clarify what we have done. For a given set of fixed, nonzero complex structure moduli, the resolution given above describes the explicit, local geometry of a non-compact Calabi-Yau three-fold, which is a K3-fibration over C 1 . If type IIa string theory is compactified on this three-fold, the resulting fourdimensional theory will have SU 3 × SU 2 gauge theory with hypermultiplets at isolated points as given in Figure 3 which reproduce the spectrum of three families of the Standard Model with an extended Higgs sector and some exotics. Alternatively, if one takes this (non-compact) Calabi-Yau three-fold and fibres it over CP 1 as described in section I so that the total space is Calabi-Yau, then F-theory on this space will give rise to N = 1 supersymmetry with SU 3 × SU 2 gauge theory and chiral multiplets in the representations given in Figure 3 . And although it does not follow directly from our construction above, considering the close similarities between two-and three-dimensional resolutions of the singular K3 surfaces we have every reason to suspect an analogous geometry can be engineered for M-theory in terms of hyper-Kähler quotients by extension of the results in [10, 14, 15] . We are currently working on building this geometry in M-theory, and we expect to report on this work soon.\n\nGiven these four complex structure moduli, all the relative positions of the 35 disparate singularities giving rise to all three families of the (extended) Standard Model are then known 8 . Beyond the usual three families of the Standard Model, the manifold also gives rise to two Higgs doublets for each family, six Higgs colour triplets, three right-handed neutrinos and five other Standard Model singlets. We should point out that this matter content (and their U 1 -charge assignments) is a consequence of group theory and algebraic geometry alone-it is simply what is found when unfolding E 8 all the way to the Standard Model.\n\nAnd given the relative positions and local geometry of the singularities together with the U 1structure, one can in principle compute the full superpotential coming from instantons wrapping different singularities. Because these are fixed by the values of the complex structure moduli, there is a (complex) four-dimensional landscape 9 of different, explicit SU 3 × SU 2 embeddings at the compactification scale. Although this large landscape may appear to have too much freedom, we remind the reader that in the traditional understanding of geometrical engineering there would be hundreds of parameters describing the (independent) relative locations of each of the isolated singularities.\n\nThere are a few things to notice about the form of the superpotential that will emerge. First, because of the U 1 -charge assignments, each term in the superpotential must combine exactly one term arising from each of the 27's. This greatly limits the form of the superpotential. And in particular, it implies that neither mass nor flavour eigenstates will arise from any single 27-that is, the 'families' in the colloquial sense are necessarily linear combinations of fields resulting from different 27's.\n\nAlso notice that in general the terms in the superpotential will be proportional to e - R dVol where dVol is the volume form of some cycle wrapping singularities (the details of which depends on whether we are talking about type IIa, M-theory, or F-theory realizations), and are in principle calculable in terms of the deformation moldui. And because these coefficients are exponentially related to the volumes of cycles, we expect the high-scale Lagrangian will be generically hierarchical. This structure could be important for solving problems in phenomenology-for example the µ problem in the Higgs potential, the Higgs doublet-triplet splitting problem, or avoiding 8 A subtlety, however, is that because our language has been explicitly that of N = 2 theory from type IIa, we are unable to distinguish the 5 from the 5 in the splitting of the 10's of SO 10 . In Figure 3 , a consistent choice was made-and although we do not justify this claim here, it is the choice that will be correct for the M-theory generalization of this work. 9 That it is continuous is a consequence of the fact that we are engineering non-compact Calabi-Yaus. If one matched this local geometry to a compact global structure, the landscape would of course be discrete.\n\n8 proton decay. We are in the process of studying the phenomenology of models on this landscape. At first glance, the U 1 -structure combined with high-scale hierarchies could possibly be complex enough to be able to avoid some of the typical problems of E 6 -like grand unified models. We should point out that if there were no high-scale hierarchies, however, then the allowed terms in the superpotential would generically give rise to low-energy lepton and baryon number violation, similar to any 'generic' E 6 model-i.e. one which includes all types of terms allowed by the E 6mandated U 1 -structure [16] . We could always impose additional symmetries and add fields by hand to solve these problems, but this would not be very compelling. However, if viable models already exist in the landscape which do not require additional fields or symmetries, these would be compelling even if we do not yet understand how they are selected.\n\nOne of the most important phenomenological questions about these models is the fate of the additional U 1 symmetries. Although we suspect that one can determine which of the U 1 symmetries are dynamical below the compactification scale by studying the normalizability of their corresponding vector multiplets, we do not presently have have a complete understanding of this situation. Of course, if any additional U 1 's survive to low energy they could have very interesting-or damningphenomenological consequences."
},
{
"section_type": "DISCUSSION",
"section_title": "VI. DISCUSSION",
"text": "An important point to bear in mind when considering geometrically engineered models is that there generically exist 10 moduli which can deform the geometry into one which gives rise to a theory with less gauge symmetry. For example, if you are given a geometrically-engineered SO 10 grand unified model, then our results show explicitly that the model can be locally deformed into an SU 5 model, and this can be deformed further into the Standard Model; the original SO 10 theory is seen to be a single point in a (complex) two-dimensional landscape of SU 3 × SU 2 theories. And because larger symmetries always lie in lower dimensional surfaces of moduli space, it is very relevant to ask what physics prevents this unfolding from taking place. Indeed, this question applies to the Standard Model as wellour analysis could easily go further to unfold away SU 3 ×SU 2 . We are not presently able to answer why this does not happen 11 ; although this observation 10 There could be global obstructions which prevent such a deformation from taking place. But these are invisible to the non-compact, local constructions considered here. 11 Although, perhaps the unfolding of SU 2 may provide an alternative to tuning in the usual Higgs sector [17] . It would be interesting to understand in greater detail the relation-\n\nsuggests that perhaps theories with less symmetry, like SU 3 × SU 2 , could be much more natural than grand unified theories. More generally, it is not presently understood what physics controls the values of the geometric moduli which deform the manifold-the parameters which deform the E 8 → E 6 ×SU 2 complex structure, for example. We do not yet have a general mechanism which would fix these parameters; we simply observe that any non-zero values of the moduli will give rise to a geometrically engineered manifold with SU 3 × SU 2 gauge theory 'peppered' with all the necessary singularities of the three families of the Standard Model together with the usual E 6 -like exotics. And importantly, for any point in the complex fourdimensional 'landscape,' the relative locations of all the relevant singularities are known-and hence in principle so is the superpotential.\n\nThis relationship between moduli-fixing and gauge symmetry breaking could be a novel feature of geometrically-unfolded models. It may allow one to apply the results in [8] , for example, to single out theories on the landscape. However, a prerequisite to this type of analysis would be an identification of which moduli should be identified with the ones which deform the geometry as described here.\n\nAlthough the motivation in this paper and in [1] appears to be a top-down realization of grand unification, there is a sense in which we are really engineering from the bottom-up. Specifically, because the local geometry we have described is non-compact, the resulting theory is decoupled from quantum gravity, and the parameters along the landscape of deformations are continuous. This is not unlike the situation in [18] . But what we lose in global constraints we perhaps gain by concrete local structure. Not only do we have a framework which naturally predicts three families with a rather detailed phenomenological structure, but we have done so in a way that preserves all the information about the local geometry. And because this framework realizes the 'physics from pure geometry' paradigm in a potentially powerful way, it could prove important to concrete phenomenological constructions in M-theory, for example.\n\nOf course we envision these local geometries to be embedded within compact Calabi-Yau manifolds. It is an assumption of the framework that the precise global topology of the compactification manifold can be ignored at least as a good first approximation. One may ask the extent to which these constructions can be glued into compact manifolds. Concretely: under what circumstances can a noncompact Calabi-Yau three-fold which is a fibration of K3 surfaces with asymptotically uniform ADE-type ship between unfolding and the Higgs mechanism.\n\n9 singularities be compactified? This is an important question for mathematicians, the answer to which would likely lead to important physical insight-e.g. quantization of the moduli space of deformations.\n\nA possible objection to this framework is that our constructions appear to depend on several seemingly arbitrary choices (the specific chain from E 8 to the Standard Model, which roots were eliminated at each step, etc.). However, it is likely that the particle content, for example, which results is completely independent of these choices. Furthermore, we suspect that different realizations of the unfolding merely result in different parameterizations of the landscape, and do not reflect true additional arbitrariness. But this is still an area that deserves attention.\n\nLastly, because in this picture the Standard Model is seen to unfold at the compactification scale, one may ask what has become of gauge coupling unification. Because the gauge coupling constants are functions of the volumes of their corresponding codimension four singular surfaces 12 which depend on the deformation moduli, the traditional meaning of grand unification is more subtle here-as is typical in string phenomenology. For example, although we chose to unfold the Standard Model sequentially as a series of less unified models, there is no reason to suspect that that order has any physical importance. Surely, if as we parameterized the unfolding in section IV, setting d → 0 (or c → 0) would result in an SU 5 grand unified theory; but setting a → 0 instead would result in a restoration of family symmetry. The four complex structure moduli tune different types of unification separately-and should simultaneously be at play in the question of gauge coupling unification.\n\nIt is interesting to note, however, that if one were to simultaneously scale the values of all the moduli to be very small, the spectrum would be more and more unified: the relative distances between singularities shrink, unifying the coefficients in the superpotential; and the volumes of the co-dimension four singularities (if realized in a compact manifold) would approach one another, resulting in a unification of their gauge couplings. What this may mean phenomenologically remains to be understood.\n\nIn this paper we have described a local, purely geometric framework in which gauge symmetry 'breaking' can be re-cast as a problem of moduli fixingand in which the same moduli which describe this geometric 'unfolding' also determine the physics of massless matter. And although we still do not understand the mechanisms by which these moduli are fixed, the landscape of possibilities is already enormously reduced: what would have been the hundreds of parameters describing the relative positions on the compactification manifold of the Standard Model's three families worth of matter fields, we specify them all in terms of only four complex structure moduli which describe the unfolding of an isolated E 8 singularity. And the fact that three families emerges is group-theoretic and not added by hand."
},
{
"section_type": "OTHER",
"section_title": "VII. ACKNOWLEDGEMENTS",
"text": "It is a pleasure to thank helpful discussions with and insightful comments of Herman Verlinde, Sergei Gukov, Gordy Kane, Paul Langacker, Edward Witten, Cumrun Vafa, Brent Nelson, Malcolm Perry, Dmitry Malyshev, Matthew Buican, Piyush Kumar, and Konstantin Bobkov.\n\nThis work was funded in part by a Graduate Research Fellowship from the National Science Foundation.\n\n12 Of course, this can only be discussed concretely when the compact manifold is known.\n\n[1] J. L. Bourjaily, \"Multiple Unfoldings of Orbifold Singularities: Engineering Geometric Analogies to Unification,\" arXiv:0704.0444. [2] S. Katz and C. Vafa, \"Matter from Geometry,\" Nucl. Phys., vol. B497, pp. 146-154, 1997, hep-th/9606086. [3] A. Klemm, W. Lerche, and P. Mayr, \"K3 Fibrations and Heterotic Type II String Duality,\" Phys. Lett., vol. B357, pp. 313-322, 1995, hep-th/9506112. [4] B. S. Acharya and S. Gukov, \"M-theory and Singularities of Exceptional Holonomy Manifolds,\" Phys. Rept., vol. 392, pp. 121-189, 2004, hep-th/0409191. [5] E. Witten, \"Deconstruction, G2 Holonomy, and Doublet-Triplet Splitting,\" 2001, hep-ph/0201018. [6] T. Friedmann and E. Witten, \"Unification Scale, Proton Decay, and Manifolds of G(2) Holonomy,\" Adv. Theor. Math. Phys., vol. 7, pp. 577-617, 2003, hep-th/0211269. [7] S. Katz and D. Morrison, \"Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups,\" J. Algebraic Geometry, vol. 1, pp. 449-530, 1992. [8] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and D. Vaman, \"An M Theory Solution to the Hierarchy Problem,\" Phys. Rev. Lett., vol. 97, p. 191601, 2006, hep-th/0606262. [9] B. S. Acharya, K. Bobkov, G. L. Kane, P. Kumar, and J. Shao, \"Explaining the Electroweak Scale and Stabilizing Moduli in M-Theory,\" 2007, hep-th/0701034. [10] P. Berglund and A. Brandhuber, \"Matter from G2 Manifolds,\" Nucl. Phys., vol. B641, pp. 351-375, 2002, hep-th/0205184. [11] F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom funften Grade. Leipzig: Teubner, 1884. [12] C. C. Bramble, \"A Collineation Group Isomorphic with the Group of Double Tangents of the Plane 10 Quartic,\" Amer. J. Math., vol. 40, pp. 351-365, 1918. [13] R. Slansky, \"Group Theory for Unified Model Building,\" Phys. Rept., vol. 79, pp. 1-128, 1981. [14] M. Atiyah and E. Witten, \"M-theory Dynamics on a Manifold of G2 Holonomy,\" Adv. Theor. Math. Phys., vol. 6, pp. 1-106, 2003, hep-th/0107177. [15] B. Acharya and E. Witten, \"Chiral Fermions from Manifolds of G2 Holonomy,\" 2001, hep-th/0109152. [16] J. L. Hewett and T. G. Rizzo, \"Low-Energy Phenomenology of Superstring Inspired E(6) Models,\" Phys. Rept., vol. 183, p. 193, 1989. [17] This was pointed out by Paul Langacker in a private correspondance. [18] H. Verlinde and M. Wijnholt, \"Building the Standard Model on a D3-Brane,\" JHEP, vol. 01, p. 106, 2007, hep-th/0508089."
}
] | |
arxiv:0704.0449 | 0704.0449 | 1 | 10.1088/1126-6708/2007/10/023 | 58e5b0bbc995ed67138e9dcf8a2726c11c1bccc96b640d73f50fa59508d16626 | Worldsheet Instantons and Torsion Curves, Part B: Mirror Symmetry | We apply mirror symmetry to the problem of counting holomorphic rational
curves in a Calabi-Yau threefold X with Z_3 x Z_3 Wilson lines. As we found in
Part A [hep-th/0703182], the integral homology group H_2(X,Z)=Z^3 + Z_3 + Z_3
contains torsion curves. Using the B-model on the mirror of X as well as its
covering spaces, we compute the instanton numbers. We observe that X is
self-mirror even at the quantum level. Using the self-mirror property, we
derive the complete prepotential on X, going beyond the results of Part A. In
particular, this yields the first example where the instanton number depends on
the torsion part of its homology class. Another consequence is that the
threefold X provides a non-toric example for the conjectured exchange of
torsion subgroups in mirror manifolds. | [
"Volker Braun",
"Maximilian Kreuzer",
"Burt A. Ovrut",
"Emanuel Scheidegger"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z 3 ⊕ Z 3 Wilson lines. As we found in Part A [1], the integral homology group H 2 (X, Z) = Z 3 ⊕ Z 3 ⊕ Z 3 contains torsion curves. Using the B-model on the mirror of X as well as its covering spaces, we compute the instanton numbers. We observe that X is self-mirror even at the quantum level. Using the selfmirror property, we derive the complete prepotential on X, going beyond the results of Part A. In particular, this yields the first example where the instanton number depends on the torsion part of its homology class. Another consequence is that the threefold X provides a non-toric example for the conjectured exchange of torsion subgroups in mirror manifolds.
was essentially solved by mirror symmetry several years ago [2] . The purpose of this paper is to take into account an important subtlety that does not appear in very simple Calabi-Yau manifolds like hypersurfaces in smooth toric varieties. This subtlety is the appearance of torsion curve classes. That is, the homology foot_0 group
H 2 X, Z = Z 3 ⊕ Z 3 ⊕ Z 3
(1) contains the torsion 2 subgroup Z 3 ⊕ Z 3 . Here, the manifold of interest X is a quotient of one of Schoen's Calabi-Yau manifolds [3, 4] by a freely acting symmetry group.
There are already a few known examples of such Calabi-Yau manifolds with torsion curves [5, 6, 7, 8, 9] , but the proper instanton counting has never been done before.
The prime motivation for studying these curves is that one would like to compute the superpotential for the vector bundle moduli [10, 11, 12, 13, 14, 15, 16] in a heterotic MSSM [17, 18, 19, 20, 21, 22, 23, 24, 25] . Our main result will be that there exist smooth rigid rational curves in X that are alone in their homology class. This proves that, in general, no cancellation between contributions to the superpotential W from instantons in the same homology class can occur.
Therefore we would like to count rational curves on X. In physical terms, we need to find the instanton correction F np X,0 to the genus zero prepotential of the (A-model) topological string on X. This is usually written as a (convergent) power series in h 11 variables q a = e 2πit a . Each summand is the contribution of an instanton, and the (integer) coefficients are the multiplicities of instantons in each homology class. According to [26, 27, 1] the novel feature of the 3-torsion curves on X is that for each 3-torsion generator we need an additional variable b j such that b 3 j = 1. The Fourier series of the prepotential on X becomes
F np X,0 (p, q, r, b 1 , b 2 ) = n 1 ,n 2 ,n 3 ∈Z m 1 ,m 2 ∈Z 3 n (n 1 ,n 2 ,n 3 ,m 1 ,m 2 ) Li 3 p n 1 q n 2 r n 3 b m 1 1 b m 2 2 , (2)
where n (n 1 ,n 2 ,n 3 ,m 1 ,m 2 ) is the instanton number in the curve class (n 1 , n 2 , n 3 , m 1 , m 2 ).
For the purpose of computing the prepotential, we can either use directly the Amodel or start with the B-model and apply mirror symmetry. The A-model calculation was carried out in the companion paper [1] , entitled Part A. The results were:
• A set of powerful techniques to compute the torsion subgroups in the integral homology and cohomology groups of X. They are spectral sequences starting with the so-called group (co)homology of the group action on the universal cover X.
• A closed formula for the genus zero prepotential
F np X,0 (p, q, r, b 1 , b 2 ) = 2 i,j=0
pb i 1 b j 2 P (q) 4 P (r) 4 +O(p 2 ) = 2 i,j=0
Li
3 (pb i 1 b j 2 )+• • • (3)
to linear order in p, extending the one computed in [28] for the universal cover X.
Here, if p(k) is the number of partitions of k ∈ Z ≥ , then P (q) is the generating function for partitions,
P (q) def = ∞ i=0 p(i)q i = q 1 24 η( 1 2πi ln q) . (4)
• Expanding eq. ( 3 ) as an instanton series we find that the number of rational curves of degree (1, 0, 0, m 1 , m 2 ) is:
n (1,0,0,m 1 ,m 2 ) = 1, ∀ m 1 , m 2 ∈ Z 3 . (5)
Furthermore, these curves have normal bundle O P 1 (-1)⊕O P 1 (-1). Hence, there are indeed 9 smooth rigid rational curves which are alone in their homology class.
Alternatively, one can start with the B-model topological string and apply mirror symmetry, which is what we will do in this paper, entitled Part B. This will allow us to obtain the higher order terms in p. The order in p up to which one wants to compute the instanton numbers is only limited by computer power. We will again find a closed formula at every order in p, however, this time by guessing it from the instanton calculation, and hence only up to the order given by this limitation. The way to arrive at this result is as follows:
• The universal cover X admits a simple realization as a complete intersection in a toric variety. In this situation, mirror symmetry boils down to an algorithm to compute instanton numbers. Unfortunately, there are many non-toric divisors which cannot be treated this way. It turns out that, after descending to X, precisely the torsion information is lost. In this approach one can only compute F np X,0 (q 1 , q 2 , q 3 , 1, 1).
• As a pleasant surprise we find strong evidence that the manifold X is self-mirror.
In particular, we attempt to compute the instanton numbers on the mirror X * by descending from the covering space X * . The embedding of X * into a toric variety is such that all 19 divisors are toric. In principle, this allows for a complete analysis including the full Z 3 ⊕ Z 3 torsion information, but this is too demanding in view of current computer power.
• Although the full quotient X = X/(Z 3 × Z 3 ) is not toric, it turns out that a certain partial quotient X = X/Z 3 can be realized as a complete intersection in a toric variety. That way, one only has to deal with h 11 (X) = 7 parameters, which is manageable on a desktop computer. Assuming the self-mirror property, we work with the mirror X * , for which again all divisors are toric, and we can compute the expansion of F np X,0 (p, q, r, 1, b 2 ) to any desired degree. A symmetry argument then allows one to recover the b 1 dependence as well. Finally, we can extract the instanton numbers n (n 1 ,n 2 ,n 3 ,m 1 ,m 2 ) including the torsion information.
• As can be seen from the A-model result eq. ( 3 ), we observe that the prepotential F np X,0 at order p factors into 2 i,j=0 b i 1 b j 2 times a function of p, q, r only. This means that the instanton number does not depend on the torsion part of its homology class. We will explain the underlying reason for this factorization and show that it breaks down at order p 3 . This fits nicely with the B-model computation at order p 3 , where the instanton numbers do depend on the torsion part.
• Another consequence of the self-mirror property is that X is a non-toric example for the conjecture of [6] . By this conjecture, certain torsion subgroups of the integral homology groups are exchanged under mirror symmetry.
An easily readable overview and a discussion of the physical consequences of our findings for superpotentials and moduli stabilization of heterotic models was presented in [27] . The present Part B is self-contained and can be read independently of Part A [1] . All necessary results from Part A are reproduced in this part.
As a guide through this paper, we start in Section 2 with a brief overview of the topology of the various spaces involved as determined in Part A [1]. This is followed by a review of the Batyrev-Borisov construction of mirror pairs of complete intersections in toric varieties in Section 3. We illustrate this construction by means of the covering spaces X and X * of our example. The review includes the techniques to compute the B-model prepotential and the mirror map. These are applied in Section 4 to the partial quotients X and X * yielding the main results stated above. This assumes that X as well as X are self-mirror, and evidence for this property is recapitulated in Section 5. Moreover, we show how the torsion subgroups are exchanged. Section 6 contains an explanation for the breakdown of the factorization alluded to above. Putting all the information together we try to guess a closed form for the prepotential in Section 7. Finally, we present our conclusions in Section 8. In the course of this work we will notice that a certain flop of X is very natural from the toric point of view, and we will present it in Appendix B.
2 Calabi-Yau Threefolds 2.1 The Calabi-Yau Threefold X
The Calabi-Yau manifold X of interest is constructed as a free G def = Z 3 × Z 3 quotient of its universal covering space X. The latter is one of Schoen's Calabi-Yau threefolds [3] . It is simply connected and hence easier to study. Among its various descriptions are the fiber product of two dP 9 surfaces, a resolution of a certain T 6 orbifold [29] , or a complete intersection in a toric variety. In the present Part B, we will mostly use the latter viewpoint. The simplest way is to introduce the toric ambient variety P 2 × P 1 × P 2 with homogeneous coordinates [x 0 : x 1 : x 2 ], [t 0 : t 1 ], [y 0 : y 1 :
y 2 ] ∈ P 2 × P 1 × P 2 . ( 6
)
The embedded Calabi-Yau threefold X is then obtained as the complete intersection of a degree (0, 1, 3) and a degree (3, 1, 0) hypersurface in P 2 × P 1 × P 2 . We restrict the coefficients of their defining equations F i = 0 to a particular set of three complex parameters λ 1 , λ 2 , λ 3 , such that the polynomials F i read
t 0 x 3 0 + x 3 1 + x 3 2 + t 1 x 0 x 1 x 2 def = F 1 (7a)
λ 1 t 0 + t 1 y 3 0 + y 3 1 + y 3 2 + λ 2 t 0 + λ 3 t 1 y 0 y 1 y 2 def = F 2 . ( 7b
)
For the special complex structure parametrized by λ 1 , λ 2 , λ 3 the complete intersection is invariant under the G = Z 3 × Z 3 action generated by (ζ def = e 2πi 3 )
g 1 : [x 0 : x 1 : x 2 ] → [x 0 : ζx 1 : ζ 2 x 2 ] [t 0 : t 1 ] → [t 0 : t 1 ] (no action) [y 0 : y 1 : y 2 ] → [y 0 : ζy 1 : ζ 2 y 2 ] (8a)
and
g 2 : [x 0 : x 1 : x 2 ] → [x 1 : x 2 : x 0 ] [t 0 : t 1 ] → [t 0 : t 1 ] (no action) [y 0 : y 1 : y 2 ] → [y 1 : y 2 : y 0 ] (8b)
One can show that the fixed points of this group action in P 2 × P 1 × P 2 do not satisfy eqns. (7a) and (7b), hence the action on X is free.
The partial quotient
X def = X G 1 (9)
will be of particular interest in this paper because this quotient is generated by phase symmetries, see eq. (8a), and hence is toric. In particular, we will need a basis of Kähler classes. As usual, we will not distinguish degree-2 cohomology and degree-4 homology classes but identify them via Poincaré duality. Part A [1] ?? shows that foot_1
H 2 X, Z = H 2 ( X, Z) G 1 ⊕ Z 3 = span Z φ, τ 1 , υ 1 , ψ 1 , τ 2 , υ 2 , ψ 2 ⊕ Z 3 . (10)
Hence, by abuse of notation, we can identify the free generators on X with the G 1invariant generators on X, see Part A eq. (??), via the pull back by the quotient map.
The triple intersection numbers on X = X/Z 3 are one-third of the corresponding intersection numbers on X listed in Part A eq. (??). Hence, the intersection numbers on X are
φτ 1 τ 2 = 3 φτ 1 υ 2 = 3 φτ 1 ψ 2 = 6 φυ 1 τ 2 = 3 φυ 1 υ 2 = 3 φυ 1 ψ 2 = 6 φψ 1 τ 2 = 6 φψ 1 υ 2 = 6 φψ 1 ψ 2 = 12 τ 2 1 τ 2 = 1 τ 2 1 υ 2 = 1 τ 2 1 ψ 2 = 2 τ 1 υ 1 τ 2 = 3 τ 1 υ 1 υ 2 = 3 τ 1 υ 1 ψ 2 = 6 τ 1 ψ 1 τ 2 = 3 τ 1 ψ 1 υ 2 = 3 τ 1 ψ 1 ψ 2 = 6 τ 1 τ 2 2 = 1 τ 1 τ 2 υ 2 = 3 τ 1 τ 2 ψ 2 = 3 τ 1 υ 2 2 = 3 τ 1 υ 2 ψ 2 = 6 τ 1 ψ 2 2 = 6 υ 2 1 τ 2 = 3 υ 2 1 υ 2 = 3 υ 2 1 ψ 2 = 6 υ 1 ψ 1 τ 2 = 6 υ 1 ψ 1 υ 2 = 6 υ 1 ψ 1 ψ 2 = 12 υ 1 τ 2 2 = 1 υ 1 τ 2 υ 2 = 3 υ 1 τ 2 ψ 2 = 3 υ 1 υ 2 2 = 3 υ 1 υ 2 ψ 2 = 6 υ 1 ψ 2 2 = 6 ψ 2 1 τ 2 = 6 ψ 2 1 υ 2 = 6 ψ 2 1 ψ 2 = 12 ψ 1 τ 2 2 = 2 ψ 1 τ 2 υ 2 = 6 ψ 1 τ 2 ψ 2 = 6 ψ 1 υ 2 2 = 6 ψ 1 υ 2 ψ 2 = 12 ψ 1 ψ 2 2 = 12. (11)
Clearly, G 2 acts on the partial quotient X. From Part A eq. (??) it follows that, of the 7 non-torsion divisors above, only 3 are G 2 -invariant. This invariant part is particularly manageable and will be important in the following. We find
H 2 X, Z G free = H 2 X, Z G 2 free = span Z φ, τ 1 , τ 2 (12)
with products 3τ 2 i = τ i φ. In particular, the triple intersection numbers on X are
τ 2 1 τ 2 = 1, τ 1 φτ 2 = 3, τ 1 τ 2 2 = 1, (13)
and 0 otherwise. Finally, the second Chern class of
X is c 2 (X) = 12(τ 2 1 + τ 2 2 ). There- fore, c 2 X • τ 1 = 12, c 2 X • φ = 0, c 2 X • τ 2 = 12. ( 14
)
As we discussed in Part A ??, the instanton-generated superpotential should be thought of as a series with one variable for each generator in H 2 . In particular, we Calabi-Yau threefold H 2 -, Z Free generators Torsion generators X Z 19 p 0 , q 0 , . . . , q 8 , r 0 , . . . , r 8 ∅
X = X/G 1 Z 7 ⊕ Z 3 P, Q 1 , Q 2 , Q 3 , R 1 , R 2 , R 3 b 1 X = X/G Z 3 ⊕ Z 3 ⊕ Z 3 p, q, r b 1 , b 2
Table 1: The different Calabi-Yau threefolds, curve classes, and variables used to expand the prepotential.
will be interested in the Calabi-Yau threefolds X, X, and X. For these, the degree-2 integral homology and the variables used (see Part A [1] for precise definitions) are in summarized Table 1 . Pushing down the curves by the respective quotients lets us express the prepotential on the quotient in terms of the prepotential on the covering space. We found in Part A that
F np X,0 P, Q 1 , Q 2 , Q 3 , R 1 , R 2 , R 3 , b 1 ) = 1 |G 1 | F np e X,0 P Q 5 1 Q 6 2 R 5 1 R 6 2 , Q 5 1 Q 6 2 , Q -2 1 Q -2 2 Q -3 3 b 1 , Q -1 1 Q -1 2 , Q 3 3 , Q 2 3 b 1 , Q 3 , 1, b 1 , Q 1 Q 3 3 , R 5 1 R 6 2 , R -2 1 R -2 2 R -3 3 b 2 1 , R -1 1 R -1 2 , R 3 3 , R 2 3 b 2 1 , R 3 , 1, b 2 1 , R 1 R 3 3 (15)
and
F np X,0 p, q, r, b 1 , b 2 ) = 1 |G 2 | F np X,0 p, q, b 2 , b 2 , r, b 2 2 , b 2 2 , b 1 . ( 16
)
3 Toric Geometry and Mirror Symmetry
In this section we review mirror symmetry and the construction of the B-model for the mirror of the covering space X. Since X is a complete intersection in a toric variety, we can use the standard constructions. Because we expect the model to be self-mirror, we will analyze the B-model for X and its mirror X * . The toric geometry for X is much simpler 4 than for X * , but contains less information. In this section we will start with the simpler model in order to review the Batyrev-Borisov construction for the mirror of a complete intersection in a toric variety. Then we will apply this construction to the more complicated model, now without going into too many details. We will see that, on the simpler side, not all parameters are toric and no torsion is visible. However, on the more complicated mirror side, all parameters are toric which will allow us, in principle, to perform the B-model computation of the complete prepotential. As X ∼ = X * is expected to be self-mirror, this determines the complete prepotential F np e X,0 = F np e X * ,0 as well. In practice, however, the analysis is computationally too involved.
Fortunately, the space X = X/G 1 and its mirror will turn out to be both tractable with toric methods and sufficiently informative. This quotient will be the subject of Section 4. Finally, this is also the starting point for arguing in Section 5 that the self-mirror property persists at the level of instanton corrections.
Recall that, in Subsection 2.1 we defined our Calabi-Yau manifold as the complete intersection
X def = F 1 = 0, F 2 = 0 ⊂ P 2 × P 1 × P 2 (17)
with the two polynomials F 1 , F 2 as in eqns. (7a) and (7b), respectively. In order to construct the mirror manifold following Batyrev and Borisov, we need to reformulate this definition in terms of toric geometry. We review here some essential ingredients of toric geometry, for details we refer to [30, 31] and references therein. We will give the abstract definitions and concepts step by step, and at each step illustrate them with the example X and its mirror manifold X * .
Given a lattice N of dimension d, a toric variety V Σ is defined in terms of a fan Σ which is a collection of rational polyhedral foot_3 cones σ ⊂ N such that it contains all faces and intersections of its elements. V Σ is compact if the support of Σ covers all of the real extension N R of the lattice N. The resulting d-dimensional variety V Σ is smooth if all cones are simplicial and if all maximal cones are generated by a lattice basis.
Let Σ (1) denote the set of one-dimensional cones (rays) with primitive generators ρ i , i = 1, . . . , n. The simplest description of V Σ introduces n homogeneous coordinates z i corresponding to the generators ρ i of the rays in Σ (1) . These homogeneous coordinates are then subjected to weighted projective identifications
z 1 : • • • : z n = λ q (a) 1 z 1 : • • • : λ q (a) n z n a = 1, . . . , h (18)
for any nonzero complex number λ ∈ C × , where the integer n-vectors q (a) i are generators of the linear relations q (a)
i ρ i = 0 among the primitive lattice vectors 6 ρ i . In order to obtain a well-behaved quotient, we must exclude an exceptional set Z(Σ) ⊂ C n that is defined in terms of the fan, as will be explained below. Hence, the quotient is
V Σ = C n -Z(Σ) C × h × Γ , (19)
where Γ ≃ N/ span{ρ i } is a finite abelian group. There are h = n -d independent C × identifications, therefore the complex dimension of V Σ equals the rank d of the lattice N. The identifications by Γ are only non-trivial if the ρ i do not span the lattice N.
Refinements of the lattice N with fixed ρ i can hence be used to construct quotients of toric varieties V Σ by discrete phase symmetries such as Z 3 . Such quotients will be discussed in Section 4. Note that the rays ρ i are in 1-to-1 correspondence with the (C × )-invariant divisors D i on V Σ , which are defined as
D i = z i = 0 ⊂ V Σ . (20)
Conversely, the homogeneous coordinate z i is a section of the line bundle O(D i ).
For example, consider the simplest compact toric variety, the projective space P d . Its fan Σ = Σ(∆) is generated by the n = d + 1 vectors
ρ 1 = e 1 , ρ 2 = e 2 , . . . , ρ n-1 = e d , ρ n = - d i=1 e i (21)
of a d-dimensional simplex ∆. They satisfy a single linear relation, n i=1 ρ i = 0. Therefore q i = 1 for all i, and the homogeneous coordinates in eq. ( 18 ) are the usual homogeneous coordinates on P d .
For products of toric varieties we simply extend the relations for any single factor by zeros and take the union of them. Hence, the fan of the polyhedron ∆ * describing the 5-dimensional toric variety P 2 ×P 1 ×P 2 in eq. ( 17 ) is generated by the n = 5+3 = 8 vectors ρ 1 = e 1 , ρ 2 = e 2 , ρ 3 = -e 1 -e 2 , ρ 4 = e 3 , ρ 5 = -e 3 , ρ 6 = e 4 , ρ 7 = e 5 , ρ 8 = -e 4 -e 5 (22) satisfying the linear relations
3 i=1 ρ i = 5 i=4 ρ i = 8 i=6 ρ i = 0. ( 23
)
Except for the origin, there are no other lattice points in the interior of ∆ * . The corresponding homogeneous coordinates will be denoted by
z 1 = x 0 , z 2 = x 1 , z 3 = x 2 , z 4 = t 0 , z 5 = t 1 , z 6 = y 0 , z 7 = y 1 , z 8 = y 2 . ( 24
)
In more general situations, given a polytope ∆ * ⊂ N we will denote the resulting toric variety by P ∆ * = V Σ(∆ * ) .
Batyrev showed that a generic section of K -1 P ∆ * , the anticanonical bundle of P ∆ * , defines a Calabi-Yau hypersurface if ∆ * is reflexive, which means, by definition, that ∆ * and its dual
∆ = x ∈ M R (x, y) ≥ -1 ∀y ∈ ∆ * (25)
are both lattice polytopes. Here, M = Hom(N, Z) is the lattice dual to N and M R is its real extension. Mirror symmetry corresponds to the exchange of ∆ and ∆ * [32] .
The generalization of this construction to complete intersections of codimension r > 1 is due to Batyrev and Borisov [33, 34] . For that purpose, they introduced the notion of a nef partition. Consider a dual pair of
d-dimensional reflexive polytopes ∆ ⊂ M R , ∆ * ⊂ N R . In that context, a partition E = E 1 ∪ • • • ∪ E r of
the set of vertices of ∆ * into disjoint subsets E 1 , . . . , E r is called a nef-partition if there exist r integral upper convex Σ(∆ * )-piecewise linear support functions φ l
: N R → R, l = 1, . . . , r such that φ l (ρ) = 1 if ρ ∈ E l , 0 otherwise. ( 26
)
Each φ l corresponds to a divisor
D 0,l = ρ∈E l D ρ (27)
on P ∆ * , and their intersection
Y = D 0,1 ∩ • • • ∩ D 0,r (28)
defines a family Y of Calabi-Yau complete intersections of codimension r. Moreover, each φ l corresponds to a lattice polyhedron ∆ l defined as
∆ l = x ∈ M R (x, y) ≥ -φ l (y) ∀y ∈ N R . (29)
The lattice points m ∈ ∆ l correspond to monomials
z m = n i=1 z m,ρ i i ∈ Γ (P ∆ * , O(D 0,l )) . ( 30
)
One can show that the sum of the functions φ l is equal to the support function of K -1
P ∆ *
and, therefore, the corresponding Minkowski sum is ∆ 1 + • • • + ∆ r = ∆. Moreover, the knowledge of the decomposition E = E 1 ∪ • • • ∪ E r is equivalent to that of the set of supporting polyhedra Π(∆) = {∆ 1 , . . . , ∆ r }, and therefore this data is often also called a nef partition.
It can be shown that given a nef partition Π(∆) the polytopes foot_5
∇ l = {0} ∪ E l ⊂ N R (31)
define again a nef partition Π * (∇) = {∇ 1 , . . . , ∇ r } such that the Minkowski sum ∇ = ∇ 1 + • • • + ∇ r is a reflexive polytope. This is the combinatorial manifestation of mirror symmetry in terms of dual pairs of nef partitions of ∆ * and ∇ * , which we summarize in the diagram
∆ = ∆ 1 + . . . + ∆ r ∆ * = ∇ 1 , . . . , ∇ r 5 5
Mirror Symmetry j j j j j j j j j j j j j j u u j j j j j j j j j j j j j j
M R N R ∇ * = ∆ 1 , . . . , ∆ r (∆ l , ∇ l ′ ) ≥ -δ l l ′ ∇ = ∇ 1 + . . . + ∇ r . ( 32
)
In the horizontal direction, we have the duality between the lattices M and N and mirror symmetry goes from the upper right to the lower left. The other diagonal has also a meaning in terms of mirror symmetry as we will explain below. The complete intersections Y ⊂ P ∆ * and Y * ⊂ P ∇ * associated to the dual nef partitions are then mirror Calabi-Yau varieties.
Let us now apply the Batyrev-Borisov construction to the complete intersection eq. ( 17 ), hence r = 2. There exist several nef-partitions of ∆ * . The one which has the correct degrees (3, 1, 0) and (0, 1, 3) is, up to exchange of t 0 and t 1 , E 1 = {ρ i |i = 1, . . . , 4} and E 2 = {ρ i |i = 5, . . . , 8}. Adding the origin and taking the convex hull yields the polytopes
∇ 1 = ρ 1 , . . . , ρ 4 , 0 , ∇ 2 = ρ 5 , . . . , ρ 8 , 0 , (33)
where the ρ i are defined in eq. (22) . The two divisors cutting out the Calabi-Yau threefold are, according to eq. ( 27 ),
D 0,1 = 4 i=1 D i , D 0,2 = 8 i=5 D i ⇒ X = D 0,1 ∩ D 0,2 ⊂ P ∆ * (34)
Note that, while ∆ * has no further lattice points, its dual ∆ has 18 vertices and 300 lattice points. Using the computer package PALP [35] , we determine the associated polytopes ∆ 1 and ∆ 2 of the global sections of O(D 0,1 ) and O(D 0,2 ), respectively. In an appropriate lattice basis there is, up to symmetry, a unique nef partition consisting of
∆ 1 = ν 1 , . . . , ν 6 , 0 , ∆ 2 = ν 7 , . . . , ν 12 , 0 , (35)
where
ν 1 = 2e 1 -e 2 , ν 2 = -e 1 + 2e 2 , ν 3 = -e 1 -e 2 , ν 4 = 2e 1 -e 2 -
Among these 12 vectors there are the 7 independent linear relations
3ν 3 + ν 4 + ν 5 -2ν 6 = 0, 3ν 9 + ν 10 + ν 11 -2ν 12 = 0, ν 1 -ν 3 -ν 4 + ν 6 = 0, -ν 1 + ν 2 + ν 4 -ν 5 = 0, ν 7 -ν 9 -ν 10 + ν 12 = 0, -ν 7 + ν 8 + ν 10 -ν 11 = 0, -ν 2 + ν 5 -ν 8 + ν 11 = 0. ( 37
)
The convex hull ∇ * = ∆ 1 , ∆ 2 yields the fan Σ(∇ * ) and, consequently, the toric variety P ∇ * . Let D * i , i = 1, . . . , 12 be the divisors associated to the vertices ν i . Then, by eq. ( 27 ), the nef partition eq. ( 35 ) defines the divisors
D * 0,1 = 6 i=1 D * i , D * 0,2 = 12 i=7 D * i , ⇒ X * = D * 0,1 ∩ D * 0,2 ⊂ P ∇ * (38)
cutting out the mirror complete intersection X * . In contrast to ∆ * , the polytope ∇ * contains extra integral points. We find that it contains, in addition to the origin and the vertices in eq. ( 36 ), the 26 points
ν 13 = 1 3 (ν 4 + ν 5 + ν 6 ) = -e 3 , ν 12+6k+i+j = 1 3 (ν 3k+i + 2ν 3k+j ), ν 14 = 1 3 (ν 10 + ν 11 + ν 12 ) = e 3 , ν 15+6k+i+j = 1 3 (ν 3k+j + 2ν 3k+i ) ∀ k ∈ {0, . . . , 3}, (i, j) ∈ (1, 2), (1, 3), (2, 3) . (39)
For completeness, note that the dual polytope ∇ has 15 vertices and 24 lattice points. Running PALP to compute the Hodge numbers using the formula of [36] , we obtain
h 1,1 X = h 1,2 X = h 1,1 X * = h 1,2 X * = 19, (40)
in agreement with Part A [1], eq. (??). So far, we have mainly focused on the information contained in the reflexive polytopes ∆ * and ∇ * and ignored their duals. We have already mentioned that in the reflexive case a generic section of K -1 P ∆ * defines a Calabi-Yau manifold, and that such sections are provided by the lattice points of ∆. In other words, ∆ and ∇ are the Newton polytopes of Y and Y * , respectively. That is, the complete intersection Y (Y * ) is defined by r polynomial equations, and the exponents of the monomials in each equation are the lattice points in ∆ (∇). More precisely, the Minkowski sum for, say, ∆ = ∆ 1 + • • • + ∆ r defines r homogeneous polynomials
F l (z) = m∈ ∆ l ∩M a l,m r l ′ =1 ρ i ∈ ∇ l ′ ∩N z m,ρ i +δ l l ′ i , l = 1, . . . , r (41)
with coefficients a l,m ∈ C. The simultaneous vanishing of F 1 , . . . , F r then defines the complete intersection Calabi-Yau manifold Y ⊂ P ∆ * . Exchanging ∆ l and ∇ l ′ in eq. ( 41 ) yields the equations F * l defining the mirror manifold Y * . It is in this sense that the map from the upper left to the lower right in eq. ( 32 ) is also a manifestation of mirror symmetry. Since we will not need the actual polynomials for X and X * , we refrain from writing them explicitly. Instead, we refer the reader to Section 4, where we determine the equations in a simpler situation.
Up to now we have only considered one of the ingredients in the fan Σ, namely, the generators ρ ∈ Σ (1) which defined the C × action in eq. (19) . The second ingredient is the exceptional set Z(Σ). It corresponds to fixed loci of a continuous subgroup of (C × ) h for which the quotient eq. ( 19 ) is not well defined. Therefore, these loci have to be removed. In terms of the homogeneous coordinates z i , this happens precisely when a subset {z i |i ∈ I}, I ⊆ {1, . . . , n}, of the coordinates vanishes simultaneously such that there is no cone σ ∈ Σ containing all of the ρ i ⊆ σ, i ∈ I. Hence, the set Z(Σ) is the union of the sets
Z I = {[z 1 : • • • : z n ] |z i = 0 ∀i ∈ I}.
Minimal index sets I with this property are called primitive collections [37] . In order to determine the index sets I we need a coherent 8 triangulation T = T (∆ * ) of the polytope ∆ * for which all simplices contain the origin. Different triangulations will yield different exceptional sets and, hence, different toric varieties. However, for simplicity, we will mostly suppress the choice of a triangulation in the notation. In the case of complete intersections, only those triangulations of ∆ * are compatible with a given nef partition that can be lifted to a triangulation of the corresponding Gorenstein cone, see [38] .
The polytope defining projective space P d admits a unique triangulation with the required properties, and this triangulation consists of n = d + 1 simplices. The only primitive collection is I = {1, . . . , n}. This is well-known from the definition of projective space, where we have to remove the origin z 1 = • • • = z d+1 = 0 from C d+1 . Similarly, the polyhedron ∆ * for the ambient space P ∆ * of X admits a unique triangulation, and the primitive collections are those of its factors, that is,
I 1 = {1, 2, 3}, I 2 = {4, 5}, I 3 = {6, 7, 8}. ( 42
)
8 Coherent triangulations, sometimes also called regular triangulations, satisfy some technical property that is equivalent to the associated toric quotient being Kähler.
The mirror polyhedron ∇ * , on the other hand, admits a huge number of triangulations. We will discuss particularly interesting triangulations of the mirror polyhedron at the end of Appendix A.
The primitive collections determine the cohomology ring of toric varieties and, together with the nef partition, complete intersections. Recall that if the collection ρ i 1 , . . . , ρ i k of rays is not contained in at least one cone, then the corresponding homogeneous coordinates z i l are not allowed to vanish simultaneously. Therefore, the corresponding divisors D i l have no common intersection. Hence, we obtain non-linear relations
R I = D i 1 • . . . • D i k = 0 in
the intersection ring. It can be shown that all such relations are generated by the primitive collections I = {i 1 , . . . , i k } defined above. The ideal generated by these R I is called Stanley-Reisner ideal
I SR = R I , I primitive collection ⊂ Z[D 1 , . . . , D n ], (43)
and Z[D 1 , . . . , D n ]/I SR is the Stanley-Reisner ring. The intersection ring of a nonsingular compact toric variety P Σ is [39]
H * P Σ , Z = Z [D 1 , . . . , D n ] I SR , i (m, ρ i )D i . (44)
In other words, the intersection ring can be obtained from the Stanley-Reisner ring by adding the linear relations i (m, ρ i )D i = 0, where it is sufficient to take a set of basis vectors for m ∈ M. In particular, the intersection number of the divisors spanning a maximal-dimensional simplicial cone σ = span R≥ {ρ i 1 , . . . , ρ i d } is
D i 1 • . . . • D i d = 1 Vol(σ) , (45)
where Vol(σ) is the lattice-volume, that is, the geometric volume divided by the volume 1 d! of a basic simplex. For practical purposes it is sufficient to compute one of these volumes, the remaining intersections can be obtained using the linear and non-linear relations.
Having found the intersection ring of the ambient toric variety, we now turn to the complete intersection Y ⊂ P ∆ * . The toric part of its even-degree intersection ring is [40] H
ev toric Y, Q = Q [D 1 , . . . , D n ] I Y , (46)
where I Y is the ideal quotient
I Y = I SR , i (m, ρ i )D i : r l=1 D 0,l . (47)
Note that it can happen that some of the D i appear as generators of I Y . This means that they can be set to zero in the intersection ring. Geometrically, this means that these divisors do not intersect a generic complete intersection Y . While the intersection ring depends on the triangulation T (∆ * ) through the primitive collections defining the Stanley-Reisner ideal, we conjecture that the divisors D i not intersecting Y are independent of the choice of triangulation. This conjecture is proven for r = 1 and supported by a large amount of empirical evidence for r > 1. We conclude that the dimension dim H 2 toric (Y ) is in general smaller than h 1,1 (Y ) for the following two reasons: Only h = n -d = dim H 2 (P ∆ * , Z) divisors are realized in the ambient toric variety P ∆ * , and some of them may not descend to the complete intersection Y . Using the adjunction formula we can compute the the Chern classes of Y by expanding
c(Y ) = n i=1 (1 + D i ) r l=1 (1 + D 0,l ) . ( 48
)
The intersection ring together with the second Chern class determine the diffeomorphism type of a simply-connected Calabi-Yau manifold [41] . If we consider the cohomology with integral coefficients there can be torsion and, in fact, this is what this paper is all about. Unfortunately, a combinatorial formula in terms of the fan Σ(∆) for the torsion in the integral cohomology of a toric Calabi-Yau manifold is only known in the hypersurface case [6] . We now illustrate these concepts in the example of the complete intersection X ⊂ P ∆ * = P 2 × P 1 × P 2 and its mirror manifold X * . In eq. ( 42 ) we already determined the primitive collections, hence the corresponding Stanley-Reisner ideal is
I SR = D 1 D 2 D 3 , D 4 D 5 , D 6 D 7 D 8 . (49)
The linear equivalences are
D 1 = D 2 , D 1 = D 3 , D 4 = D 5 , D 6 = D 7 , D 6 = D 8 and, hence, we can choose K 1 = D 4 , K 2 = D 1 , K 3 = D 6
as a basis for H 2 (P ∆ * ). In terms of this basis, we obtain D 0,1 = K 1 +3K 2 and D 0,2 = K 1 +3K 3 , see eq. ( 27 ). Therefore, the ideal I e X in eq. ( 34 ) is
I e X = K 3 2 K 2 -K 2 2 K 3 , K 1 K 2 -3 K 2 2 , K 1 K 3 -3 K 3 2 , K 1 2 , K 2 3 , K 3 3 . ( 50
)
Next, we define the restriction of the K i to X to be the divisors
Ji = K i • X = K i (K 1 + 3K 2 )(K 1 + 3K 3 ). ( 51
)
We need to compute one of the intersection numbers directly from the volume of a cone, say,
J1 J2 J3 = K 1 K 2 K 3 (K 1 + 3K 2 )(K 1 + 3K 3 ) = 9K 1 K 2 2 K 2 3
, where we made use of the relations in I e X . Using eq. ( 45 ), this intersection can be evaluated to be
9K 1 K 2 2 K 2 3 = 9D 1 D 2 D 4 D 6 D 7 = 9/ Vol ρ 1 , ρ 2 , ρ 4 , ρ 6 , ρ 7 = 9/ Vol e 1 ,
Then, again using eq. ( 50 ), we see that the only non-vanishing intersection numbers and the second Chern class are
J2 2 J3 = 3, J1 J2 J3 = 9, J2 J2 3 = 3, c 2 X • J1 = 0, c 2 X • J2 = 36, c 2 X • J3 = 36. ( 53
)
Note that only h 1,1 toric ( X) = 3 of the h 1,1 ( X) = 19 parameters are realized torically. Comparing the triple intersection numbers with eq. ( 13 ), it is clear that these 3 toric divisors are precisely the G-invariant divisors on X.
A similar, though much more complicated, calculation can be done for X * ⊂ P ∇ * . Using the results of Appendix A one can show that, among the points in eq. ( 39 ), the 14 divisors D * 13 , D * 14 , D * 12+6k+i+j , D * 15+6k+i+j , k = 0, 2 appear as generators of eq. ( 47 ) and, therefore, do not intersect X * . Subtracting from the remaining 24 divisors in eqns. ( 36 ) and ( 39 ) the remaining 5 linear relations in eq. ( 37 ), we find that all h 1,1 toric ( X * ) = h 1,1 ( X * ) = 19 moduli are realized torically.
As we have just seen, the cohomology classes
D i span H 2 (P Σ , R) = H 1,1 (P Σ ).
The Kähler classes of a smooth projective toric variety P Σ form an open cone in H 1,1 (P Σ ) called the Kähler cone K(P Σ ). This cone has a combinatorial description in terms of the fan Σ, which we now review. First, define a support function to be a continuous function
ψ : N R → R given on each cone σ ∈ Σ by an m σ ∈ M R via ψ(ρ) = (m σ , ρ) ∀ρ ∈ σ ⊂ N R . (54)
A support function determines a divisor D = i ψ(ρ i )D i . We say that D is convex if ψ is a convex function on N R . The convex classes form a non-empty strongly convex polyhedral cone in H 1,1 (P Σ ) whose interior is the Kähler cone K(P Σ ). Such a support function is strictly convex if and only if
ψ(ρ i 1 + • • • + ρ i k ) > ψ(ρ i 1 ) + • • • + ψ(ρ i k ) (55)
for every primitive collection I = {i 1 , . . . , i k } [40] . The dual of the Kähler cone K(P Σ ) is called the Mori cone or the cone of numerically effective curves NE(P Σ ). Its generators can be described by vectors l (a) of the corresponding linear relations i l (a)
i ρ i = 0. Each face of the Kähler cone K(P Σ ) is dual to an edge of NE(P Σ ). These edges are generated by curves c (a) , and the entries of the vector l (a) are
l (a) i = c (a) • D i . (56)
A practical algorithm to find the generators for l (a) in terms of the triangulation T (∆ * ) is described in [42] . Of course, we are not interested in the ambient space but in a complete intersection Y ⊂ P ∆ * . The restriction of a Kähler class on the ambient space yields a Kähler class on Y , but not every Kähler class on Y arises that way. We define the toric part of the Kähler cone on Y as the restriction [43] K(Y
) toric = K(P Σ ) Y ⊂ K(Y ). ( 57
)
In the simplicial case, we can always take the basis J i of H 2 toric (Y, Q) to be edges of the Kähler cone. The dual of the toric Kähler cone of Y is the (toric) Mori cone NE(Y ) toric . This is sufficient for mirror symmetry purposes, however, it can be larger than the actual cone of effective curves. Once the generators l (a) of NE(P ∆ * ) are determined, we need to add the information about the nef partition. For this purpose, we define
l (a) 0,m def = -D 0,m • c (a) m = 1, . . . , r. (58)
Finally, it is customary to write the generators of the Mori cone NE(Y ) toric as
l (a) = l (a) 0,1 , . . . , l (a) 0,r ; l (a) 1 , . . . , l (a) n , (59)
which are, by abuse of notation, again denoted by l (a) . The knowledge of the (toric) Mori cone is important for several reasons. It defines the local coordinates on the complex structure moduli space of the mirror Y * near the point of maximal unipotent monodromy. Moreover, the generators enter the coefficients of the fundamental period which is a solution of the Picard-Fuchs equations as we will review in Subsection 3.5. For example, using the unique primitive collections in eq. ( 42 ), the Mori cone for P ∆ * is generated 9 by l (1) =(0, 0, 0, 1, 1, 0, 0, 0)
l (2) =(1, 1, 1, 0, 0, 0, 0, 0) l (3) =(0, 0, 0, 0, 0, 1, 1, 1). ( 60
)
Recalling the nef partition
D 0,1 = D 1 + • • • + D 4 , D 0,2 = D 5 + • • • + D 8 , we prepend (-D 0,1 • c (a) , -D 0,2 • c (a)
) = (-3, 0), (-1, -1), (0, -3), a = 1, 2, 3, to obtain the generators l (1) =(-1,-1; 0, 0, 0, 1, 1, 0, 0, 0)
l (2) =(-3, 0; 1, 1, 1, 0, 0, 0, 0, 0) l (3) =( 0,-3; 0, 0, 0, 0, 0, 1, 1, 1) (61)
of the Mori cone NE( X) toric . Due to the large number of toric moduli, the calculation for the Mori cone NE(P ∇ * ) of the ambient toric variety of the mirror X * is much more complex.
Mirror symmetry identifies the quantum corrected Kähler moduli space of Y with the classical complex structure moduli space of Y * , see the excellent treatise in [43] for details. The deformations of the complex structure of Y * are encoded in the periods ̟ = γ Ω and the latter can be computed from the equations F * l that cut out Y * ⊂ P ∇ * . Given the Mori cone eq. ( 59 ) and the classical intersections numbers [44, 45, 38, 43] to write down a local expansion of the periods, convergent near the large complex structure point, which is characterized by its maximal unipotent monodromy. In the following, we will review just the bare essentials.
κ abc = J a • J b • J c we follow
The coefficients a i in the polynomial constraints F * l of the complete intersection Y * , see eq. ( 41 ), define the complex structure of Y * . A particular set of local coordinates u a on the complex structure moduli space on Y * is defined by
u b = r m=1 a l (b) 0,m m,0 n i=1 a l (b) i i b = 1, . . . , h (62)
where h def = h 1,1 toric (Y ) and a m,0 is the coefficient in (41) corresponding to the origin in ∇ l . In these coordinates, the point of maximal unipotent monodromy is at u b = 0. We define the cohomology-valued period
̟(u, J) = {na≥=0} r m=1 1 - h a=1 l (a) 0,m J a - P h a=1 l (a) 0,m na n i=1 1 + h a=1 l (a) i J a P h a=1 l (a) i na h a=1 u na+Ja a . ( 63
)
where (x) n = Γ(x + n)/Γ(x) is the Pochhammer symbol. Note that the choice of triangulation is implicit in the generators l (a) of the Mori cone. Expanding ̟(u, J) by cohomology degree yields
̟(u, J) = ̟ (0) (u) + h a=1 ̟ (1) a (u)J a + h a=1 ̟ (2) a (u)κ abc J b J c -̟ (3) (u) dVol, ( 64
)
where dVol is the volume form. The coefficients in eq. ( 64 ) are the fundamental period ̟ (0) (u), that is, the unique solution to the Picard-Fuchs equations holomorphic at u a = 0, and
̟ (1) a (u) = ∂ Ja ̟(u, J)| J=0 , ̟ (2) a (u) = 1 2 κ abc ∂ J b ∂ Jc ̟(u, J)| J=0 , ̟ (3) (u) = - 1 6 κ abc ∂ Ja ∂ J b ∂ Jc ̟(u, J)| J=0 . (65)
These coefficients coincide with the basis of solutions of the Picard-Fuchs equations obtained from the Frobenius method in [46, 31] . The B-model prepotential
F B Y * ,0 is F B Y * ,0 (u) = 1 2̟ (0) (u) 2 ̟ (0) (u)̟ (3) (u) + h a=1 ̟ (1) a (u)̟ (2) a (u) . ( 66
)
At the large complex structure point the mirror map defines natural flat coordinates on the Kähler moduli space of the original manifold Y , which are
t i = ̟ (1) i (u) ̟ 0 (u) , i = 1, . . . , h. (67)
We also define q j = e 2πit j = u j + O(u 2 ). One way to obtain the prepotential is to compute its third derivatives
C * abc = D a D b D c F B Y * ,0 = Y * Ω ∧ ∂ a ∂ b ∂ c Ω, (68)
and apply the Picard-Fuchs operators. This leads to linear differential equations, which determine C * abc up to a common constant, see again [46, 43] for details. The quantum corrected three point function C ijk (q) on Y follows from C * abc (u) using the inverse mirror map eq. ( 67 ) u = u(t), and one obtains
C ijk (q) = 1 ̟ (0) (u(q)) 2 ∂u a ∂t i ∂u b ∂t j ∂u c ∂t k C * abc (u(q)). ( 69
)
In practice, we use the formula
C ijk (q) = ∂ t i ∂ t j ̟ (2) k (u(q)) ̟ (0) (u(q)) . ( 70
)
Integrating three times with respect to t i yields the prepotential F B Y * ,0 (t) up to a polynomial of degree three in t i which can be determined partially by the topological data of Y .
Mirror symmetry then ensures that the B-model prepotential, eq. ( 66 ), is equal to the A-model prepotential. That is,
F Y,0 (q) = F B Y * ,0 (u(q)). ( 71
)
This allows us to compute the instanton numbers n d . For the case of interest,
X ∈ P ∆ * = P 2 × P 1 × P 2 , (72)
we refer to [28] where this program been carried out in detail. The same calculation can in principle be done on the mirror X * , but the large number of toric moduli again makes it highly extensive. Instead, we refer to the next section where a suitable quotient of X * will be treated in detail for which the computations are reasonably simple.
In this section we consider the quotient X = X/G in terms of toric geometry and study the mirror of X in this context. In order to achieve this, we first analyze the partial quotient X = X/G 1 . Using the techniques introduced in Section 3, we construct the mirror X * . Using their toric realization, we perform the B-model computation for the non-perturbative prepotentials F np X,0 and F np X * ,0 , respectively. Finally, we explain how one can implement the quotient by G 2 on both sides in order to obtain X and X * .
We start with a review of the general discussion of free quotients of complete intersections in toric geometry in [31] . Consider a fan Σ ⊂ N R and pick a lattice refinement N such that Γ = N/N is a finite abelian group. Such a lattice refinement consists of a finite sequence of lattice refinements of the form N → N + w p Z which are described by a vector w p = 1 kp α pi ρ i with α pi ∈ Z. The group Γ is then isomorphic to p Z kp . Let Σ be the fan obtained from Σ by relating everything to the lattice N. In this context, we make some additional identifications in the toric quotient eq. ( 19 ) [47] . One finds that VΣ = V Σ /Γ is the quotient of V Σ by the finite abelian group Γ. Its action on the homogeneous coordinates is by multiplication by phases
z 1 : • • • : z n → ξ α 1 z 1 : • • • : ξ αn z n , ξ = e 2πi k , (73)
for every cyclic subgroup of order k. We will denote such group actions by Z k : (α 1 , . . . , α n ). If V Σ is a compact toric variety, then the quotient VΣ is never free [39] . However, a hypersurface or complete intersection in V Σ need not intersect the set of fixed points, and in that case we get a smooth quotient manifold with nontrivial fundamental group. We now apply this to P ∆ * = P 2 × P 1 × P 2 defined in eq. ( 22 ). The first step in performing the quotient of P ∆ * by G 1 thus amounts to a refinement N = w Z + N of the lattice N with index |G 1 | = 3. From the definition eq. (8a) of the action of G 1 on P ∆ * and eq. ( 24 ) we read off that the refinement is by a vector
w ∈ 1 3 ρ 2 + 2ρ 3 + ρ 7 + 2ρ 8 + Z 5 . ( 74
)
The resulting polytope ∆ * admits the same nef partition as ∆ * in eq. ( 33 ), ∇1 = ρ1 , . . . , ρ4 , 0 , ∇2 = ρ5 , . . . , ρ8 , 0 .
where we express the generators ρ in terms of ρ as ρi = ρ i , i = 1, . . . , 6 , ρ7 = ρ 7 + e 1 + 2e 2 + e 4 + 2e 5 , ρ8 = ρ 8 -e 1 -2e 2 -e 4 -2e 5 .
(76)
It is easy to check that the ρi satisfy the same linear relations eq. ( 23 ) as the ρ i , and that w = 1 3 (ρ 1 -ρ2 + ρ6 -ρ7 ) = -e 2 -e 5 . The ρi together with w therefore indeed generate the lattice N . Note that, while all 8 non-zero lattice points of ∆ * are vertices, the dual polytope ∆ has 18 vertices and 102 points. Using PALP [35] again, we compute the lattice points of the polytope ∇ * = ∆1 , ∆2 ⊂ M R , which will describe the ambient space of the mirror X * of X. We find ∆1 = ν1 , . . . , ν6 , 0 , ∆2 = ν7 , . . . , ν12 , 0 ,
where we express the vertices νi in terms of the vertices ν i of ∇ * as ν3k+1 = ν 3k+1 , ν3k+2 = ν 3k+2 -e 5 , ν3k+3 = ν 3k+3 + e 5 , k = 0, . . . , 3.
Again, it is easy to check that the νi satisfy the same linear relations eq. ( 37 ) as the ν i . It turns out that the lattice points of ∇ * generate a sublattice M of index 3 in M, and the lattice refinement is generated by
w * = 1 3 ν1 + 2ν 2 + 2ν 7 + ν8 = e 2 + e 4 -e 5 . (79)
Among the points of ∇ * listed in eq. ( 39 ) only ν 13 and ν 14 are also lattice points of the sublattice M. In fact, we have ν13 = ν 13 and ν14 = ν 14 . Hence, ∇ * has 12 vertices and 15 lattice points; its dual ∇ = ∇1 + ∇2 has 42 lattice points among which 15 are vertices 10 . Once we have the polytopes ∆ * and ∇ * , we can construct X and X * as complete intersections entirely analogous to X and X * , see Section 3. That is, using eq. ( 27 ), we define
X = D0,1 ∩ D0,2 , X * = D * 0,1 ∩ D * 0,2 (81)
in terms of the nef partitions eq. ( 75 ) and (77), respectively. Here, Di and D * i denote the divisors associated to the generators ρi and νi , respectively. The absence of fixed points of the G 1 action on the complete intersection X is guaranteed by the fact that the resulting polytope ∆ * ⊂ NR has no additional lattice points [31] . Hence, X = X/G 1 has a non-trivial fundamental group π 1 (X) = Z 3 . Surprisingly, it turns out that the mirror X * is a free quotient as well. To see this recall that, as noticed above, the lattice points of ∇ * generate a sublattice M of index 3 in M. Furthermore, ∇ * also has no additional lattice points with respect to ∇ * . Therefore, there is a 10 Note that all of our polytopes differ from the non-free Z 3 × Z 3 quotient of ∆ * defined in [28] , Proposition 7.1. In the notation of [31] their quotient is
∇ * = P
1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1
3 0 0 3 1 1 Z 3 : 0 1 2 0 0 0 0 0 Z 3 : 0 0 0 0 1 2 0 0 ( 80
)
and has 21 points and 8 vertices in the lattice N .
group G * 1 ≃ Z 3 acting torically on P ∇ * . On the homogeneous coordinates this action is
g * 1 : z 1 : • • • : z 12 → ζz 1 : ζ 2 z 2 : z 3 : • • • : z 6 : ζ 2 z 7 : ζz 8 : z 9 : • • • : z 12 . (82)
Hence, X * = X * /G * 1 also has a non-trivial fundamental group π 1 (X * ) = Z 3 . Note that this never happens for hypersurfaces in toric varieties [6] . Having the toric representation of X and X * , we can now compute their Hodge numbers. It turns out
that h 1,1 X = h 1,2 X = h 1,1 X * = h 1,2 X * = 7, (83)
in agreement with Part A [1], eq. (??).
We now turn to the G 2 action, which does not act torically. Hence, we cannot, in principle, find a toric variety containing X = X/G 2 as we did for the G 1 quotient above. However, at least we have to ensure that X and X * are G 2 -symmetric. This can be achieved via suitable symmetries in the toric data.
The easy part of the toric data for X is the polytope ∆ * . The G 2 action on the ambient space permutes the homogeneous coordinates, see eq. (8b). In terms of toric geometry, this means that it permutes the corresponding points of the polytope. That is 11 ,
g 2 : ρi → ρ1+(i mod 3) ∀i ∈ {1, 2, 3}, g 2 : ρ4 → ρ4 , ρ5 → ρ5 , g 2 : ρ5+i → ρ6+(i mod 3) ∀i ∈ {1, 2, 3}. ( 84
)
It induces a mirror group action G * 2 on X * which is geometrical, rather than a quantum symmetry as discussed in [48] . The action of G * 2 is obviously the dual group action on the dual lattice M, which again must be a symmetry of the relevant polytope ∇ * . We find that
g * 2 : ν3k+i → ν3k+1+(i mod 3) ∀k = 0, . . . , 3, i ∈ {1, 2, 3}. ( 85
)
As a check on the mirror group action, note that the matrix of scalar products, see eq. (87) below, is invariant. That is,
g 2 (ρ l ), g * 2 (ν l ′ ) = ρl , νl ′ ∀ l, l ′ . ( 86
)
By abuse of notation, we denote the corresponding cyclic permutation of homogeneous coordinates by g * 2 as well. Using this action, we define the mirror of X to be X * = X * /G * 2 . This idea has already been used for the construction of mirrors of orbifolds of the quintic [49] soon after the discovery of the first mirror construction by Greene and Plesser.
Following eq. ( 41 ), the equations for the Calabi-Yau complete intersections X and X * are defined by evaluating the matrix of scalar products ρi , νj + δ l l ′ , which are , + δ l l ′ ν1 ν2 ν3 ν4 ν5 ν6 ν13 ν7 ν8 ν9 ν10 ν11 ν12 ν14 ρ1 3 0 0 3 0 0 1 0 0 0 0 0 0 0 ρ2 0 3 0 0 3 0 1 0 0 0 0 0 0 0 ρ3 0 0 3 0 0 3 1 0 0 0 0 0 0 0 ρ4 1 1 1 0 0 0 0 0 0 0 1 1 1 1 ρ5 0 0 0 1 1 1 1 1 1 1 0 0 0 0 ρ6 0 0 0 0 0 0 0 3 0 0 3 0 0 1 ρ7 0 0 0 0 0 0 0 0 3 0 0 3 0 1 ρ8 0 0 0 0 0 0 0 0 0 3 0 0 3 1 (87)
The equations of X can now be read off from the columns of eq. ( 87 ), and one finds
F 1 = (λ 5 t 0 + λ 6 t 1 )(x 3 0 + x 3 1 + x 3 2 ) + (λ 7 t 0 + λ 8 t 1 )x 0 x 1 x 2 , (88a)
F 2 = (λ 1 t 0 + λ 4 t 1 )(y 3 0 + y 3 1 + y 3 2 ) + (λ 2 t 0 + λ 3 t 1 )y 0 y 1 y 2 , (88b)
where the G 2 -symmetry has been imposed. Note that the last monomial in each equation corresponds to the vector 0 ∈ ∆l , l = 1, 2. Two of the eight coefficients λ m can be fixed by normalizing the equations, say λ 4 = λ 5 = 1, and three correspond to the symmetries of P 1 , that is, SL(2) transformations of [t 0 : t 1 ]. Hence, we can, for example, set λ 6 = λ 7 = λ 8 = 0. This leaves us with 3 complex structure deformations λ 1 , λ 2 , and λ 3 , see eqns. (7a) and (7b). The equations defining X * correspond to the rows of eq. ( 87 ), that is,
F * 1 = a 1 (z 3 1 z 3 4 + z 3 2 z 3 5 + z 3 3 z 3
6 )z 13 + (a 2 z 10 z 11 z 12 z 14 + a 3 z 4 z 5 z 6 z 13 )z 1 z 2 z 3 , (89a) F * 2 = a 4 (z 3 7 z 3 10 + z 3 8 z 3 11 + z 3 9 z 3 12 )z 14 + (a 5 z 4 z 5 z 6 z 13 + a 6 z 10 z 11 z 12 z 14 )z 7 z 8 z 9 ,
where, again, invariance under G * 2 has been imposed and the last monomial of each equation comes from the lattice point 0 ∈ ∇l , , l = 1, 2. Both equations are homogeneous with respect to all seven scaling degrees that follow from the linear relations eq. ( 37 ). Among the twelve scalings of the coordinates z i , six are compatible with the cyclic permutations g * 2 , see eq. (85). Subtracting the three G 2 symmetric independent scalings among the relations eq. ( 37 ), there remains one torus action that acts effectively on the parameters plus two normalizations of the equations. As expected, the six parameters a m of the equations of X * thus become the 3 complex structure moduli.
So far, we only considered the polytopes ∆ * and ∇ * . However, this is only part of the toric data defining the manifolds X and X * , respectively. In addition, we need the triangulations and the corresponding exceptional sets. A change in the triangulation corresponds to a flop of the toric variety. The very real danger is that not all, and perhaps none, of the flopped Calabi-Yau manifolds are G 2 -symmetric. For X ⊂ P ∆ * this turns out to be unproblematic, but for X * ⊂ P ∇ * we will find a condition for the choice of a triangulation.
We now return to the discussion of the triangulations and the intersection ring of X.
The analogous, but technically much more involved discussion of X * will be presented in Subsection 4.5. For X everything is straightforward since the G 1 -quotient did not introduce additional lattice points in the associated polytope ∆ * . Therefore, just like for the polytope ∆ * of the covering space X, there exists a unique triangulation. In particular the primitive collections, the Stanley-Reisner ideal, and the ideal I X are identical to the ones in eqns. ( 42 ), (49) , and (50) since they are derived from the same triangulation. Moreover, one can easily see that this triangulation is G 2 -invariant and, hence,
X is G 2 symmetric.
The only change is in the normalization of the intersection ring in eq. ( 52 ), since the total volume has to be divided by 3 = |G 1 |. This can also be seen in eq. ( 76 ), where the volume of the cone is now 3 instead of 1. Hence, on X the intersection ring and the second Chern class are
J2 2 J3 = 1, J1 J2 J3 = 3, J2 J2 3 = 1, c 2 X • J1 = 0, c 2 X • J2 = 12, c 2 X • J3 = 12. ( 90
)
Comparing these intersection numbers with eq. ( 13 ), it is clear that the toric divisors should be identified with the G 1 -invariant divisors on X as
J1 = φ, J2 = τ 1 , J3 = τ 2 . ( 91
)
The curves spanning the Mori cone on the cover turn out to be G 1 -invariant as well. Therefore, the Mori cones NE(P ∆ * ) and NE(X) toric are identical to those in eqns. ( 60 ) and (61), respectively. Following the steps given in Section 3 we now want to compute the B-model prepotential F B X * ,0 , plug in the mirror map, and obtain the prepotential on X
F np X,0 (P, Q 1 , Q 2 , Q 3 , R 1 , R 2 , R 3 , b 1 ). ( 92
)
We immediately realize the following two caveats:
• We do not know how to incorporate the torsion curves H 2 (X, Z) tors = Z 3 into the toric mirror symmetry calculation.
• Of the 7 Kähler classes on X, only 3 are toric.
This means that only 3 out of the 7 + 1 variables in the prepotential are accessible, and the remaining ones are set to one. Looking at the intersection numbers eq. ( 90 ), it is clear that the 3 divisors are precisely the G 2 -invariant divisors on X, see eq. ( 13 ). Therefore, these 3 variables must be those that map to the variables p, q, and r on X. By comparing with eq. ( 16 ), we see that the corresponding variables on X are P , Q 1 , and R 1 . Hence, we actually only compute
F np X,0 (P, Q 1 , 1, 1, R 1 , 1, 1, 1) = n 1 ,n 2 ,n 3 n X (n 1 ,n 2 ,n 3 ) Li 3 P n 1 Q n 2 1 R n 3 1 . (93)
In effect, this means that the resulting instanton numbers are not just the instantons in a single integral homology class, but the instanton numbers in a whole set of integral homology classes. The instanton numbers sum over all curve classes that cannot be distinguished by P, Q 1 , R 1 ∈ Hom H 2 (X, Z), C × . Up to total degree 4 and the symmetry n
X (n 1 ,n 2 ,n 3 ) = n X (n 1 ,n 3 ,n 2 ) , (94)
the resulting instanton numbers are
n X (1,0,0) = 27 n X (1,0,1) = 108 n X (1,0,2) = 378 n X (1,0,3) = 1080 n X (1,1,1) = 432 n X (1,1,2) = 1512 n X (2,0,1) = -54 n X (2,0,2) = -756 n X
(2,1,1) = 864 n X (3,0,1) = 9.
(95)
Knowing the prepotential on X, we now want to divide out the free G 2 action and arrive at the prepotential on X. Since we do not know the complete expansion but only eq. ( 93 ), we have to set b 1 = b 2 = 1 in the descent equation ( 16 ). This yields
F np X,0 p, q, r, 1, 1) = 1 3 F np X,0 p, q, 1, 1, r, 1, 1, 1 = n 1 ,n 2 ,n 3 n X (n 1 ,n 2 ,n 3 ) Li 3 p n 1 q n 2 r n 3 . (96)
Up to the symmetry n X (n 1 ,n 2 ,n 3 ) = n X (n 1 ,n 3 ,n 2 ) , the non-vanishing instanton numbers for X up to total degree 5 are
n X (1,0,0) = 9 n X (1,0,1) = 36 n X (1,0,2) = 126 n X (1,0,3) = 360 n X (1,0,4) = 945 n X (1,1,1) = 144 n X (1,1,2) = 504 n X (1,1,3) = 1440 n X (1,2,2) = 1764 n X (2,0,1) = -18 n X (2,0,2) = -252 n X (2,0,3) = -1728 n X (2,1,1) = 288 n X (2,1,2) = 3960 n X (3,0,1) = 3 n X (3,0,2) = 252 n X (3,1,1) = 756, (97)
Unfortunately, this direct calculation misses the torsion information and only yields the expansion F np X,0 (p, q, r, 1, 1). The b 1 dependence was lost because the toric methods do not yield this part, and the b 2 dependence was lost because the relevant divisor on X was not toric. Comparing with the full expansion of the prepotential
F np X,0 p, q, r, b 1 , b 2 ) = n 1 ,n 2 ,n 3 m 1 ,m 2 n X (n 1 ,n 2 ,n 3 ,m 1 ,m 2 ) Li 3 p n 1 q n 2 r n 3 b m 1 1 b m 2 2 , (98)
see Part A eq. (??), this means we only obtain the sum of the instanton numbers over all torsion classes
n X (n 1 ,n 2 ,n 3 ) = 2 m 1 ,m 2 =0 n X (n 1 ,n 2 ,n 3 ,m 1 ,m 2 ) . (99)
Clearly, this destroys the torsion information, that is, the instanton numbers n X
(n 1 ,n 2 ,n 3 )
do not depend on the torsion part of the integral homology. For comparison purposes, we list the instanton numbers n X (n 1 ,n 2 ,n 3 ) for 0 ≤ n 1 , n 2 , n 3 ≤ 5 in Table 2 .
* We now study the mirror X * , which sits in a more complicated ambient toric variety.
Consequently, the analysis is more involved. The big advantage, however, will turn out to be that all h 11 (X * ) = 7 Kähler moduli are toric, which will enable us to obtain the full instanton expansion. Since the polytope ∇ * in eq. ( 78 ) is not simplicial, we have to specify a resolution of the singularities, that is, a triangulation T ( ∇ * ). Moreover, not any triangulation will do, but we have to make sure that it is compatible with the action of the permutation group G * 2 . While a tedious technicality, the existence of such a resolution has to be shown in order to establish the existence of a geometrical mirror family of X.
In particular, we show in Appendix A that there is no projective resolution of the ambient space among the 720 coherent star triangulations of ∇ * that respects the permutation symmetry eq. ( 85 ). In other words, if one demands G * 2 symmetry then the ambient toric variety cannot be chosen to be Kähler, but only a complex manifold. Clearly, in that case there is no Kähler cone and the usual toric mirror symmetry algorithm does not work. What comes to the rescue is that there are two classes of non-symmetric projective resolutions for which the symmetry-violating exceptional sets do not intersect X * . Hence the complete intersection is G 2 -symmetric, even though the ambient space is not. We conclude that the extended Kähler moduli space of X * contains two symmetric phases. We will denote these two classes of triangulations by T ± = T ± ( ∇ * ), see Appendix A. In fact, the two phases are topologically distinct, and only the triangulation T + describes the threefold X * that we are interested in. In Appendix B, we will investigate the other triangulation T -which describes a flop of X * .
n X (0,n 2 ,n 3 ) n X (3,n 2 ,n 3 ) ❅ ❅ ❅
n 2 n 3 0 1 2 3 4 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 0 4 0 0 0 0 0 0 5 0 0 0 0 0 0
❅ ❅ ❅ n 2 n 3 0 1 2 3
4 5 0 0 3 252 4158 40173 287415 1 3 756 15390 164280 1259685 7763364 2 252 15390 426708 5427684 46537092 310465062 3 4158 164280 5427684 73971360 657552966 4487097816 4 40173 1259685 46537092 657552966 5948103483 41016575313 5 287415 7763364 310465062 4487097816 41016575313 284581389204 n X (1,n 2 ,n 3 ) n X (4,n 2 ,n 3 ) ❅ ❅ ❅ n 2 n 3 0 1 2 3 4 5 0 9 36 126 360 945 2268 1 36 144 504 1440 3780 9072 2 126 504 1764 5040 13230 31752 3 360 1440 5040 14400 37800 90720 4 945 3780 13230 37800 99225 238140 5 2268 9072 31752 90720 238140 571536 ❅ ❅ ❅ n 2 n 3 0 1 2 3 4 5 0 0 0 -144 -6048 -107280 -1235520 1 0 -306 -12348 -207000 -2273400 -19066500 2 -144 -12348 348480 14609520 235219680 2505155400 3 -6048 -207000 14609520 520226784 8245864800 87989812560 4 -107280 -2273400 235219680 8245864800 131759049600 1417949658000 5 -1235520 -19066500 2505155400 87989812560 1417949658000 15365394415800 n X (2,n 2 ,n 3 ) n X (5,n 2 ,n 3 ) ❅ ❅ ❅ n 2 n 3 0 1 2 3 4 5 0 0 -18 -252 -1728 -9000 -38808 1 -18 288 3960 27648 143748 620928 2 -252 3960 54432 380160 1976472 8537760 3 -1728 27648 380160 2654208 13799808 59609088 4 -9000 143748 1976472 13799808 71748000 309920688 5 -38808 620928 8537760 59609088 309920688 1338720768 ❅ ❅ ❅ n 2 n 3 0 1 2 3 4 5 0 0 0 45 5670 189990 3508920 1 0 36 13140 474840 8793648 111499020 2 45 13140 1112886 38961252 777759975 10723515300 3 5670 474840 38961252 1952428464 47357606430 732897531720
4 189990 8793648 777759975 47357606430 1237373786439 19911043749420 5 3508920 111499020 10723515300 732897531720 19911043749420 327006066948660
Table 2: Summed instanton numbers n X (n 1 ,n 2 ,n 3 ) = m 1 ,m 2 n X (n 1 ,n 2 ,n 3 ,m 1 ,m 2 )
(hence not distinguishing torsion) computed by mirror symmetry. The table contains all non-vanishing instanton numbers for 0 ≤ n 1 , n 2 , n 3 ≤ 6. where we dropped the superscript * on D for ease of notation. From this, in turn, we obtain the generators l(a) + of the Mori cone NE(P ∇ * ):
l(1) + =( 0, 0, 0, 0, 0, 0, 1, 0, 0,-1, 0, 0, 0, 1) l(2) + =( 1, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) l(3) + =(-1, 1, 0, 1,-1, 0, 0, 0, 0, 0, 0, 0, 0, 0) l(4) + =( 0,-1, 1, 0, 1,-1, 0, 0, 0, 0, 0, 0, 0, 0) l( 5 ) + =( 0, 0,-1, 0, 0, 1, 0,-1, 0, 0, 1, 0, 0, 0) l( 6 ) + =( 0, 0, 0, 0, 0, 0,-1, 0, 1, 1, 0,-1, 0, 0) l( 7 ) + =( 0, 0, 0, 0, 0, 0, 0, 1,-1, 0,-1, 1, 0, 0) l(8) + =( 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0,-3, 0) l(9) + =( 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0,-3).
(101)
A dual basis for the generators of the Kähler cone K(P ∇ * ) is
K1 = D13 + 2 D1 -D2 -D3 + D9 + D7 + D8 + 3 D4 , K2 = 3 D1 + D13 + 3 D4 , K3 = D13 + 2 D1 + 3 D4 , K4 = D13 + 2 D1 -D2 + 3 D4 , K5 = D13 + 2 D1 -D2 -D3 + 3 D4 , K6 = D13 + 2 D1 -D2 -D3 + D9 + D8 + 3 D4 , K7 = D8 + D13 + 2 D1 -D2 -D3 + 3 D4 , K8 = D4 + D1 , K9 = D10 + D7 . ( 102
)
The Calabi-Yau complete intersection X * is then defined by X * = K1 K2 . It turns out that the divisors D13 , D14 do not intersect X * . Therefore, all
h 1,1 toric X * = h 1,1 X * = 7 ( 103
)
Kähler moduli are realized torically. Since there are two divisors that do not intersect, finding the Mori cone is somewhat subtle. First, we have to restrict the lattice of linear relations to the sublattice orthogonal to these two directions. For the generators of the toric Mori cone NE(X * ) toric , this means that l(1)
+ → 3 l(1) + + l(9) + , l(2) + → 3 l(2) + + l (8)
+ and that we drop l(8) + , l(9) + as well as the entries corresponding to intersections with D13 , D14 . In addition, we prepend the intersection numbers with D0,1 and D0,2 . This yields l(1) + =(-3, 0; 0, 0, 0, 0, 0, 0, 3, 0, 0,-2, 1, 1) l( 2 ) + =( 0,-3; 3, 0, 0,-2, 1, 1, 0, 0, 0, 0, 0, 0) l(3) + =( 0, 0;-1, 1, 0, 1,-1, 0, 0, 0, 0, 0, 0, 0) l(4) + =( 0, 0; 0,-1, 1, 0, 1,-1, 0, 0, 0, 0, 0, 0) l( 5 ) + =( 0, 0; 0, 0,-1, 0, 0, 1, 0,-1, 0, 0, 1, 0) l( 6 ) + =( 0, 0; 0, 0, 0, 0, 0, 0,-1, 0, 1, 1, 0,-1) l( 7 ) + =( 0, 0; 0, 0, 0, 0, 0, 0, 0, 1,-1, 0,-1, 1).
(
) 104
The dual basis of divisors is J *
1 = 1 3 K2 1 K2 , J * 2 = 1 3 K1 K2 2 , J * 5 = K1 K2 K5 , J * 3 = K1 K2 K3 , J * 4 = K1 K2 K4 , J * 6 = K1 K2 K6 , J * 7 = K1 K2 K7 . (105)
We now try to identify this basis J * 1 , . . . , J * 7 of divisors on X * with the basis {φ, τ 1 , υ 1 , ψ 1 , τ 2 , υ 2 , ψ 2 } of divisors on X in eq. (10) . It turns out that there is more than one way to identify the bases if one only wants to preserve the triple intersection numbers. To obtain a unique answer, we also need to identify the actions by G * 2 and G 2 as well. First, the G * 2 action on H 2 (P ∇ * , Z) is defined by eq. ( 85 ). Using the linear equivalence relations
2 D1 -D2 -D3 + 2 D4 -D5 -D6 = 0 -D1 + 2 D2 -D3 -D4 + 2 D5 -D6 = 0 2 D7 -D8 -D9 + 2 D10 -D11 -D12 = 0 -D2 + D3 -D5 + D6 -D8 + D9 -D11 + D12 = 0 -D4 -D5 -D6 + D10 + D11 + D12 -D13 + D14 = 0 (106)
and the definition eq. ( 105 ), one can compute the induced group action on H 2 (X * , Z).
We find
g * 2 J * 1 J * 2 J * 3 J * 4 J * 5 J * 6 J * 7 =
1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 -1 1 0 0 0 0 3 -1 0 1 0 0 0 0 0 0 1 0 0 3 0 0 0 1 0 -1
0 0 0 0 1 1 -1 J * 1 J * 2 J * 3 J * 4 J * 5 J * 6 J * 7 . ( 107
)
Second, recall that the G 2 action on the divisors of X * is
g 2 φ τ 1 υ 1 ψ 1 τ 2 υ 2 ψ 2 =
1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 3 0 -1 0 0 0 0 3 1 -1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 3 0 -1
0 0 0 0 3 1 -1 φ τ 1 υ 1 ψ 1 τ 2 υ 2 ψ 2 , ( 108
)
see also Part A eq. (??). The instanton numbers on X * are the expansion coefficients
F np X * ,0 (P * , Q * 1 , Q * 2 , Q * 3 , R * 1 , R * 2 , R * 3 , 1) = n 1 ,...,n 7 n X * (n 1 ,n 2 ,n 3 ,n 4 ,n 5 ,n 6 ,n 7 ) Li 3 P * n 1 Q * n 2 1 Q * n 3 2 Q * n 4 3 R * n 5 1 R * n 6 2 R * n 7 3 . (113)
We see that we almost get the complete instanton expansion eq. ( 15 ), we only miss the expansion in the b * 1 variable which is not computed by the toric mirror symmetry algorithm. Up to total degree 5, the instanton numbers are
n X * (1,0,0,0,0,0,0) =3 n X * (1,0,0,0,0,1,0) =3 n X * (1,0,0,0,0,1,1) =3 n X * (1,0,1,0,0,0,0) =3 n X * (1,0,1,0,0,1,0) =3 n X * (1,0,1,0,0,1,1) =3 n X * (1,0,1,1,0,0,0) =3 n X * (1,0,1,1,0,1,0) =3 n X * (1,0,1,1,0,1,1) =3 n X * (1,1,0,0,0,1,2) =9 n X * (1,0,1,2,1,0,0) =9 n X * (1,0,1,1,1,1,0) =3 n X * (1,1,1,0,0,1,1) =3 n X * (1,1,0,0,0,1,1) =3 n X * (1,0,1,1,1,0,0) =3. ( 114
)
Finally, let us take a look at the G * 2 action, see eq. ( 85 ). Of the 7 generators of the toric Mori cone, eq. ( 104 ), only the 3 generators l(1)
+ , l (2)
+ and l( 5 )
+ are invariant. Not surprisingly, the dual G * 2 -invariant divisors J * 5 = φ, J * 1 = τ 1 , J * 2 = τ 2 (115)
were identified with the G 2 -invariant divisors on X in eq. ( 109 ). Therefore, only 3 Kähler parameters survive to the quotient X * = X * /G * 2 , and we have
h 1,1 X = h 1,2 X = h 1,1 X * = h 1,2 X * = 3. ( 116
)
Now that we have the expression eq. ( 113 ) for the prepotential on X * , we can again apply a suitable variable substitution
P * , Q * 1 , Q * 2 , Q * 3 , R * 1 , R * 2 , R * 3 , b * 1 , b * 2 -→ p * , q * , r * , b * 1 , b * 2 (117)
and obtain the prepotential on the quotient
X * = X * /G * 2 .
The correct way to replace the variables is determined by the group action on the homology and cohomology as we explained in Part A. Having computed the G * 2 -action in eq. ( 107 ), we determine the descent equation for the prepotential to be 13
F np X * ,0 p * , q * , r * , b * 1 , b * 2 ) = 1 |G * 2 | F np X * ,0 p * , q * , b * 2 , b * 2 , r * , b * 2 2 , b * 2 2 , b * 1 . ( 118
)
13 Interestingly, eq. ( 118 ) turns out to be exactly analogous to eq. ( 16 ), even though the identification of divisors on X * and X is not just a relabeling of divisors.
Using the series expansion of the prepotential for b * 1 = 1 on X * from Subsection 4.5, we now find that
F np X * ,0 (p * , q * , r * , 1, b * 2 ) = 2 j=0 3 × Li 3 (p * b * j 2 ) + 4 Li 3 (p * q * b * j 2 ) + 4 Li 3 (p * r * b * j 2 ) + 14 Li 3 (p * q * 2 b * j 2 ) + 16 Li 3 (p * q * r * b * j 2 ) + 14 Li 3 (p * r * 2 b * j 2 ) + 40 Li 3 (p * q * 3 b * j 2 ) + 56 Li 3 (p * q * 2 rb * j 2 ) + 56 Li 3 (p * q * r 2 b * j 2 ) + 40 Li 3 (p * r * 3 b * j 2 ) + 105 Li 3 (p * q * 4 b * j 2 ) + 160 Li 3 (p * q * 3 r * b * j 2 ) -2 Li 3 (p * 2 q * b * j 2 ) -2 Li 3 (p * 2 r * b * j 2 ) -28 Li 3 (p * 2 q * 2 b * j 2 ) + 32 Li 3 (p * 2 q * r * b * j 2 ) -28 Li 3 (p * 2 r * 2 b * j 2 ) + 3 Li 3 (p * 3 q * ) + 3 Li 3 (p * 3 r * ) + total p * , q * , r * -degree ≥ 5 . (119)
The corresponding instanton numbers
F np X * ,0 (p * , q * , r * , 1, b * 2 ) = n 1 ,n 2 ,n 3 ,m 2 n X * (n 1 ,n 2 ,n 3 ,m 2 ) Li 3 p * n 1 q * n 2 r * n 3 b * m 2 2 (120)
are listed in Table 3 . For comparison purposes, we list the summed instanton numbers
(n 1 , n 2 , n 3 ) n X * (n 1 ,n 2 ,n 3 ,0) n X * (n 1 ,n 2 ,n 3 ,1) n X * (n 1 ,n 2 ,n 3 ,2) n X (n 1 ,n 2 ,n 3 )
(1, 0, 0)
3 3 3 9 (1, 0, 1) 12 12 12 36 (1, 0, 2) 42 42 42 126 (1, 0, 3) 120 120 120 360 (1, 1, 1) 48 48 48 144 (1, 1, 2) 168 168 168 504 (2, 0, 1) -6 -6 -6 -18 (2, 0, 2) -84 -84 -84 -252 (2, 1, 1) 96 96 96 288 (3, 0, 1) 3 0 0 3
Table
3: Instanton numbers n X * (n 1 ,n 2 ,n 3 ,m 2 )
computed by toric mirror symmetry. They are invariant under the exchange n 2 ↔ n 3 , so we only display them for n 2 ≤ n 3 . on X as well, see eq. ( 99 ). One observes that the sum over the more refined instanton numbers on X * equals the summed instanton number on X, another clue towards X being self-mirror.
So far, we have alluded to X being possibly self-mirror, but not actually made use of this property. Now we are going to assume the self-mirror property and, hence, obtain the prepotential on X as
F np X,0 (p, q, r, b 1 , b 2 ) = F np X * ,0 (p, q, r, b 1 , b 2 ). (121)
Note that at linear and quadratic order in p we can actually recover the b 1 , b 2 expansion from the summed instanton numbers in Subsection 4.4 and the factorization which we will prove in Section 6.
In contrast, for the prepotential terms at order p 3 we have to use the X * prepotential to obtain the b 2 expansion from eq. ( 119 ). Since this is based on a toric computation on X * , we do not directly obtain the b 1 expansion. However, note that the fact that g 1 acted torically, eq. (8a), and g 2 non-torically, eq. (8b), is just a consequence of the choice of coordinate system on P 2 × P 1 × P 2 . By a suitable coordinate choice, we could have made any one of the four Z 3 subgroups of G = Z 3 × Z 3 act torically. Therefore, any combination of b 1 , b 2 other than 1 = b 0 1 b 0 2 has to occur in the same way in the complete series expansion of the prepotential. We conclude that the prepotential can only depend on b 1 and b 2 through the combinations 1,
2 i,j=0 b i 1 b j 2 . (122)
This observation lets us recover the full b 1 , b 2 expansion of the prepotential. To
n X (1,n 2 ,n 3 ,0,0) n X (1,n 2 ,n 3 ,m 1 ,m 2 ) , (m 1 , m 2 ) = (0, 0) ❅ ❅ ❅ n 2 n 3 0 1 2 3
4 0 1 4 14 40 105 1 4 16 56 160 2 14 56 196 3 40 160 4 105 ❅ ❅ ❅ n 2 n 3 0 1 2 3 4 0 1 4 14 40 105 1 4 16 56 160 2 14 56 196 3 40 160 4 105
n X (2,n 2 ,n 3 ,0,0) n X (2,n 2 ,n 3 ,m 1 ,m 2 ) , (m 1 , m 2 ) = (0, 0) ❅ ❅ ❅ n 2 n 3 0 1 2 3 0 0 -2 -28 -192 1 -2 32 440 2 -28 440 3 -192 ❅ ❅ ❅ n 2 n 3 0 1 2 3 0 0 -2 -28 -192 1 -2 440 2 -28 440 3 -192 n X (3,n 2 ,n 3 ,0,0) n X (3,n 2 ,n 3 ,m 1 ,m 2 ) , (m 1 , m 2 ) = (0, 0) ❅ ❅ ❅ n 2 n 3 0 1 2 0 0 3 36
1 3 108
2 36 ❅ ❅ ❅ n 2 n 3 1 2 0 0 27
1 81 2 27 Table 4: Instanton numbers n X (n 1 ,n 2 ,n 3 ,m 1 ,m 2 )
computed by mirror symmetry. The table contains all non-vanishing instanton numbers for n 1 + n 2 + n 3 ≤ 5. The entries marked in bold depend non-trivially on the torsion part of their respective homology class.
an integral lattice structure and form a ring, and therefore have a product. Because of Poincaré duality, that is,
H 2 (Y ) = H 4 (Y ) ∨ , it is sufficient to look at H 2 (Y ). There is a product H 2 (Y ) × H 2 (Y ) → H 2 (Y )
whose structure constants κ ijk are the triple intersection numbers. These intersection numbers are finer invariants than just the dimensions of the cohomology groups, and a self-mirror Calabi-Yau threefold should satisfy
κ ijk (Y ) = κ ijk (Y * ). (125)
For simply connected threefolds with torsion-free homology a theorem of Wall [41] states that the cohomology groups with the intersection product κ ijk (Y ) together with the second Chern class c 2 (Y ) determine the diffeomorphism type of Y .
If, however, Y and Y * have non-trivial fundamental groups then we cannot conclude that easily that they are diffeomorphic. But the non-trivial fundamental group is often reflected in torsion in homology (for example if π 1 (Y ) is Abelian). In that case, the conjecture of [6] says that for any Calabi-Yau threefold Z
H 3 Z, Z tors ≃ H 2 Z * , Z tors , H 2 Z, Z tors ≃ H 3 Z * , Z tors . (126)
Therefore, a self-mirror manifold Y = Y * is expected to satisfy
H 2 Y, Z tors ≃ H 3 Y, Z tors . (127)
Of the many spaces Y satisfying eq. ( 124 ) there are only a few which also satisfy eq. ( 125 ). So far we only considered classical topology, but we know that the ring H 2 (Y ) experiences quantum corrections when going far away from the large volume limit. At small volume the intersection numbers are replaced by the three-point functions C ijk (q) of (topological) conformal field theory in eq. ( 69 ). In the large volume limit q goes to zero and the C ijk (q) go to κ ijk , as expected. The C ijk (q) are characterized by the genus zero instanton numbers n
d (Y ) = n (0) d (Y * ). ( 128
)
One can go even further and couple the topological conformal field theory to topological gravity and define higher genus instanton numbers n
(g) d , where now n (g)
d (Y ) = n (g) d (Y * ), g > 0 (129)
has to hold. These invariants are very difficult to compute, however see [52, 53] for recent progress. We do not know whether they contain more information about the symplectic structure than the genus zero invariants. In other words, there are presently no examples known whose n (g) d agree for g = 0 but differ for g > 0. Now, one can start with any Y and use some method to construct the mirror Y * . Among these are the Greene-Plesser construction in conformal field theory, or its geometric generalizations by Batyrev and Borisov for complete intersections in toric varieties. Then, to show that Y is self-mirror one proceeds to compute the various invariants. The simplest condition, eq. ( 124 ), can directly be checked in terms of the toric data. This concretely means that one starts with a mirror pair Y and Y * satisfying eq. ( 124 ) and checks whether eqns. ( 125 ), ( 127 ), (128), and (129) are satisfied. In fact, in Section 4 we collected a large amount of evidence in favor of the claim that X and its Batyrev-Borisov mirror threefold X * are the same. Indeed, eqns. ( 40 ), ( 83 ) and (116) show that X, X, and X satisfy by construction the constraint eq. ( 124 ) on the Euler number. More interestingly, by the identifications found in eqns. ( 109 ) and ( 115 ) we observed that the condition on the intersection ring, eq. ( 125 ), is satisfied for X and X, respectively. Next, eq. ( 97 ) and Table 3 show that X also fulfils the requirement eq. ( 128 ) on the genus zero instanton numbers. It would be very interesting to see whether also the condition eq. ( 129 ) for higher genus curves can be met.
Finally, we consider the torsion in cohomology. In Part A ?? we have shown that
H 3 X, Z tors ≃ H 2 X, Z tors ≃ Z 3 ⊕ Z 3 , (130)
as we expect from a self-mirror threefold. Moreover, we can actually compute the fundamental group of the Batyrev-Borisov mirror independently. For that, first notice that the quotient X * = X * /G * 1 is fixed-point free, see Subsection 4.2. The mirror permutation G * 2 on X * acts freely as well. Therefore, both X and X * are free quotients by a group isomorphic to Z 3 ⊕ Z 3 , thus their fundamental groups are
π 1 X ≃ π 1 X * ≃ Z 3 ⊕ Z 3 . (131)
Moreover, on can easily show that on a proper foot_8 Calabi-Yau threefold Z one has H 2 (Z, Z) tors = π 1 (Z) ab , the Abelianization of the fundamental group. Hence, we see that
H 3 (X, Z) tors ≃ Z 3 ⊕ Z 3 ≃ H 2 (X * , Z) tors (132)
and the first of eq. ( 126 ) is true. This provides the first evidence for the conjecture of [6] in a context other than toric hypersurfaces. Another point of view is that there is a geometrical or rather combinatorial reason for the self-mirror property in this case. From eqns. (36) and (39) one can easily see that the lattice points ν i , ν 6+i , ν 13 , ν 14 , i = 1, . . . , 3, span a sub-polytope of ∇ * satisfying the same linear relations as all the lattice points ρ i of ∆ * in eq. (23) . Hence, this sub-polytope is isomorphic to ∆ * . The same is true for the polytopes ∇ * and ∆ * . The toric variety P ∇ * which is the ambient space of X * can therefore be regarded as a blow-up of a quotient of P ∆ * , the ambient space of X. Actually, this blow-up makes all 7 divisors of X * toric. Similarly, P ∇ * can be regarded as a blow-up of a quotient of P ∆ * . As shown in Subsection 3.3 this entails that all 19 Kähler moduli of X * are realized torically. Note that it is possible that the mirror polytopes ∆ * and ∇ * are actually isomorphic. In fact, for toric hypersurfaces there are 41, 710 self-dual polytopes [54] . The novel feature in our case is that non-isomorphic polytopes lead to self-mirror complete intersections, consistent with the nef partitions.
6 Factorization vs. The (3, 1, 0, 0, 0) Curve
One interesting observation is that the prepotential F np X,0 at order p, see eq. ( 123 ) in this paper and eq. (??) in Part A [1], factors into 2 i,j=0 b i 1 b j 2 times a function of p, q, r only. This means that the instanton number for any pseudo-section (curve contributing at order p) does not depend on the torsion part of its homology class. In other words, for any pseudo-section there are 8 other pseudo-sections with the same class in H 2 (X, Z) free and together filling up all of H 2 (X, Z) tors = Z 3 ⊕ Z 3 . In contrast, this factorization does not hold at order p 3 . For example,
F np X,0 (p, q, r, b 1 , b 2 ) = • • • + 3p 3 q + 0 b 1 + b 2 1 + b 2 + b 1 b 2 + b 2 1 b 2 + b 2 2 + b 1 b 2 2 + b 2 1 b 2 2 p 3 q + • • • . (133)
The purpose of this subsection is to understand this behavior.
First, the factorization of the prepotential at any order of p not divisible by 3 follows from an extra symmetry that we have not utilized so far. The covering space X is, in addition to eqns. (8a) and (8b), also invariant under another Ĝ = Z 3 × Z 3 action generated by (ζ def = e 2πi 3 ) ĝ1 :
[x 0 : x 1 : x 2 ] → [x 0 : ζx 1 : ζ 2 x 2 ] [t 0 : t 1 ] → [t 0 : t 1 ] (no action) [y 0 : y 1 : y 2 ] → [y 0 : y 1 : y 2 ] (no action) (134a) and ĝ2 : [x 0 : x 1 : x 2 ] → [x 1 : x 2 : x 0 ] [t 0 : t 1 ] → [t 0 : t 1 ] (no action) [y 0 : y 1 : y 2 ] → [y 0 : y 1 : y 2 ] (no action) (134b)
This symmetry has fixed points and, therefore, cannot be used if one is looking for a smooth quotient of X. However, it commutes with G and hence descends to a Ĝ = Z 3 × Z 3 symmetry of X (with fixed points). Clearly, the instanton sum must observe this additional geometric symmetry. To make use of this symmetry, we have to express its action on the variables in F np X,0 (p, q, r, b 1 , b 2 ). We can do so by first noting that the basic 81 curves
s 1 ×s 2 ⊂ X, s 1 ∈ MW (B 1 ), s 2 ∈ MW (B 2 ) ( 135
)
are really one orbit under G × Ĝ. Recall that, after dividing out G, these curves became the 9 sections in MW (X) = Z 3 ⊕ Z 3 , see Part A ??. We now observe that MW (X) = {s ij } is one Ĝ-orbit; since each of these sections contributes pb i 1 b j 2 , i, j = 0, . . . , 2 the induced Ĝ action on the prepotential must be ĝ1 :
F np X,0 (p, q, r, b 1 , b 2 ) → F np X,0 (b 1 p, q, r, b 1 , b 2 ), ĝ2 : F np X,0 (p, q, r, b 1 , b 2 ) → F np X,0 (b 2 p, q, r, b 1 , b 2 ). (136)
Clearly, the prepotential must be invariant under the ĝ1 , ĝ2 action. While imposing no constraint on the p 3n terms in the prepotential, all other powers of p must appear in the combination
p n 2 i,j=0 b i 1 b j 2 , n ≡ 0 mod 3. (137)
This proves the factorization observed at the beginning of this subsection. Second, we would like to understand the p 3 q terms in eq. ( 133 ). These are the curves in the homology classes 15 (3, 1, 0, * , * ) ∈
Z 3 ⊕ Z 3 ⊕ Z 3 = H 2 X, Z . (138)
We will show that the rational curves in this class come in a single family, that is, the moduli space of genus 0 curves on X in these homology classes
M 0 X, (3, 1, 0, * , * ) (139)
is connected. In particular, all such curves have the same homology class (3, 1, 0, 0, 0) and only contribute to p 3 q in the prepotential eq. ( 133 ). As discussed in Part A ??, any such map C X : P 1 → X factors
P 1 C X / / C
e X ? ? ? ? ? ? ? ? X X q ? ?
.
(140) 15 Recall that the exponent of p is the degree along the base P 1 . This is why we pick a basis in H 2 (X, Z) free such that a curve in (n 1 , n 2 , n 3 , m 1 , m 2 ) contributes at order p n1 q n2 r n3 b m1 1 b m2 2 in the prepotential.
The map C e X can be written in terms of homogeneous coordinates as a function
C e X : P 1 [z 0 :z 1 ] → P 2 [x 0 :x 1 :x 2 ] × P 1 [t 0 :t 1 ] × P 2 [y 0 :y 1 :y 2 ] (141)
satisfying the equations (7a) and (7b) defining X,
F 1 • C e X [z 0 : z 1 ] = 0 = F 2 • C e X [z 0 : z 1 ] ∀[z 0 : z 1 ] ∈ P 1 . ( 142
)
The curve C X ends up in the homology class (3, 1, 0, * , * ) if and only if the defining equation ( 141 ) is of degree (3, 1, 0) in P 2 × P 1 × P 2 . Hence, eq. ( 141 ) is defined by complex constants α ij , β ij , γ i (up to rescaling) such that
x i = α i0 z 0 + α i1 z 1 i = 0, 1, 2 t i = β i0 z 3 0 + β i1 z 2 0 z 1 + β i2 z 0 z 2 1 + β i3 z 3 1 i = 0, 1 y i = γ i i = 0, 1, 2. ( 143
)
These constants have to be picked such that the resulting curve lies on the complete intersection X, that is, they have to satisfy eq. ( 142 ). Inserting eq. ( 143 ), we find that
F 1 • C e X [z 0 : z 1 ] is a homogeneous degree 6 polynomial in [z 0 : z 1 ]. Since the coefficients of z k 0 z 6-k 1
must vanish individually, this yields 7 constraints for the parameters α ij , β ij . What makes this system of constraint equations tractable is the fact that they are all linear in β ij ,
F 1 • C e X = 0 ⇔ A 1 0 0 0 A 5 0 0 0 A 2 A 1 0 0 A 6 A 5 0 0 A 3 A 2 A 1 0 A 7 A 6 A 5 0 A 4 A 3 A 2 A 1 A 8 A 7 A 6 A 5 0 A 4 A 3 A 2 0 A 8 A 7 A 6 0 0 A 4 A 3 0 0 A 8 A 7 0 0 0 A 4 0 0 0 A 8 β 00 β 01 β 02 β 03 β 10 β 11 β 12 β 13 = 0 (144)
where (145) Thinking of this as 7 linear equations for the 8 parameters β ij , there is always a nonzero solution. The solution is generically unique up to an overall factor, and turns into an P n for special values of the α ij . Moreover, the parameter space of the α ij is connected (essentially, the moduli space of lines in P 2 ). Since we just identified the parameter space of the (α ij , β ij ) as a blow-up thereof, it is therefore connected as well.
A 1 def = α 3 00 + α 3 10 + α 3
It remains to satisfy F 2 • C e X = 0. One can easily see that the only way is to pick the γ i to be simultaneous solutions of
γ 3 0 + γ 3 1 + γ 3 2 = 0 = γ 1 γ 2 γ 3 . (146)
Since two cubics intersect in 9 points, there are 9 such solutions, permuted by G. Therefore, the parameter space of (α ij , β ij , γ i ) has 9 connected components, permuted by the G-action. The moduli space of curves C X on X is the G-quotient of the moduli space of curves C e X on X, and therefore has only a single connected component. By continuity, every curve C X in this connected family has the same homology class, explaining the piece of the prepotential given in eq. ( 133 ).
Putting all the information together we found out about the prepotential on X, one can try to divine a closed form for the prepotential. We guess that the order p n terms have the closed form
F np X,0 (p, q, r, b 1 , b 2 ) p n = p n 8 n-1 i,j∈Z 3 b i 1 b j 2 P (q) 4 P (r) 4 n M 2n-2 (q, r) (147)
if n is not a multiple of 3 and, slightly weaker, that F np X,0 (p, q, r, 1, 1)
p n = 9p n 8 n-1 P (q) 4 P (r) 4 n M 2n-2 (q, r) (148)
if n is a multiple of 3. Here,
• P (q) is the usual generating function of partitions eq. ( 4 ).
• The M 2n-2 are polynomials in the Eisenstein series E 2 (q), E 4 (q), E 6 (q) and E 2 (r), E 4 (r), E 6 (r), starting with
M -2 (q, r) = 0 M 0 (q, r) = 1 M 2 (q, r) = E 2 (q)E 2 (r) M 4 (q, r) = 13 108 E 4 (q)E 4 (r) + 1 4 E 4 (q)E 2 (r) 2 + E 2 (q) 2 E 4 (r) + 7 4 E 2 (q) 2 E 2 (r) 2 M 6 (q, r) = 1 27 E 6 (q)E 6 (r) + 13 54 E 6 (q)E 4 (r)E 2 (r) + E 4 (q)E 2 (q)E 6 (r) + 1 6 E 6 (q)E 2 (r) 3 + E 2 (q) 3 E 6 (r) + 79 108 E 4 (q)E 2 (q)E 4 (r)E 2 (r) + 5 4 E 2 (q) 3 E 4 (r)E 2 (r) + E 4 (q)E 2 (q)E 2 (r) 3 + 47 12 E 2 (q) 3 E 2 (r) 3 M 8 (q, r) = 2 3 E 6 (q)E 2 (q)E 6 (r)E 2 (r) + 1309 6750 E 4 (q)E 4 (q)E 4 (r)E 4 (r) + 25 108 E 6 (q)E 2 (q)E 4 (r)E 4 (r) + E 4 (q)E 4 (q)E 6 (r)E 2 (r) + 85 54 E 6 (q)E 2 (q)E 4 (r)E 2 (r) 2 + E 4 (q)E 2 (q) 2 E 6 (r)E 2 (r) + 13 12 E 6 (q)E 2 (q)E 2 (r) 4 + E 2 (q) 4 E 6 (r)E 2 (r) + 137 216 E 4 (q)E 4 (q)E 4 (r)E 2 (r) 2 + E 4 (q)E 2 (q) 2 E 4 (r)E 4 (r) + 3 8 E 4 (q)E 4 (q)E 2 (r) 4 + E 2 (q) 4 E 4 (r)E 4 (r) + 34 9 E 4 (q)E 2 (q) 2 E 4 (r)E 2 (r) 2 + 121 12 E 2 (q) 4 E 2 (r) 4 + 41 8 E 4 (q)E 2 (q) 2 E 2 (r) 4 + E 2 (q) 4 E 4 (r)E 2 (r) 2 . ( 149
)
They are symmetric under the exchange q ↔ r and of weight 2n in q and r separately. But, for example, M 4 above does not factor into a function of q and a function of r. So the M 2n-2 are not the products of the polynomials appearing in the dP 9 prepotential. However, by setting q = 0 or r = 0 one recovers the corresponding polynomials in the dP 9 prepotential [55] .
• The E 2i are the usual Eisenstein series
E 2 (q) = 1 -24q -72q 2 -96q 3 -168q 4 -144q 5 -288q 6 + O(q 7 ) E 4 (q) = 1 + 240q + 2160q 2 + 6720q 3 + 17520q 4 + 30240q 5 + O(q 6 ) E 6 (q) = 1 -504q -16632q 2 -122976q 3 -532728q 4 + O(q 5 ). (150)
Note that the naive Taylor series coefficients of the prepotential are fractional, but when expanding in terms of Li 3 's (which account for the multicover contributions) one finds integral instanton numbers.
These expressions for the prepotential agree with all instanton numbers computed in this paper. Unfortunately, we have not been able to guess a closed formula that includes the b 1 and b 2 dependence of the prepotential F np X,0 (p, q, r, b 1 , b 2 )| p n if n is divisible by 3. We expect that these involve extra functions beyond the Eisenstein series.
In the initial paper Part A [1], we analyzed the topology of the Calabi-Yau manifold of interest and found that
H 2 X, Z = Z 3 ⊕ Z 3 ⊕ Z 3 . (151)
Although the presence of torsion curve classes complicates the counting of rational curves, we managed to derive the A-model prepotential to linear order in p. The goal of this paper is to go beyond the results of Part A using mirror symmetry. By carefully adapting methods designed for complete intersections in toric varieties, we can apply mirror symmetry to compute the instanton numbers on X, even though X is not toric. Using that X is self-mirror, we completely solve this problem and are able to calculate the complete A-model prepotential to any desired precision (and for arbitrary degrees in p), limited only by computer power. Carrying out this computation, we find the first examples of instanton numbers that do depend on the torsion part of their integral homology class, see Table 4 on Page 35.
Since the self-mirror property of X is important, we investigate it in detail. In doing so, we go far beyond just checking that the Hodge numbers are self-mirror. In particular, we find that the intersection rings are identical and that torsion in homology obeys the conjectured mirror relation [6] . Finally, going beyond classical geometry, we independently calculate certain instanton numbers on X and its Batyrev-Borisov mirror X * . Again, we find that X and X * are indistinguishable, providing strong evidence for X being self-mirror. Both of these results extend those found in Part A [1] .
Using these results, we are able to guess certain closed expressions for the prepotential of X in terms of modular forms. In certain limits it specializes to the dP 9 prepotential of [55] . There it is known that the coefficients in p of the dP 9 prepotential satisfy a recursion relation. Moreover, there is a gap condition, that is, a certain number of subsequent terms in a series expansion is absent. This condition provides sufficient data to determine the integration constants for the recursion and allows to determine the prepotential completely, even at higher genus. We expect a similar story to be valid for the prepotential of X.
Furthermore, we introduce the notation:
a i = νi , b i = ν3+i , c i = ν6+i , d i = ν12+i , i = 1, 2, 3, e = ν13 , f = ν14 . ( 154
)
Among these 14 vectors in eq. ( 154 ) there are 9 independent linear relations, see eq. ( 37 ),
a 1 + a 2 + a 3 = 0, c 1 + c 2 + c 3 = 0, e + f = 0, b i = a i + e, d l = c l + f, (155)
which imply others like a i + b j = a j + b i and a
i + c l = b i + d l or e = 1 3 (b 1 + b 2 + b 3 ) and f = 1 3 (d 1 + d 2 + d 3 )
. Lemma 1. ∇ * has 15 facets, 6 of which are simplicial:
a i a j b i b j c l c m d l d m ] i<j l<m , a i a j d 1 d 2 d 3 i<j , b 1 b 2 b 3 c l c m l<m . (156)
The nine non-simplicial facets form an orbit under the permutation symmetries Z ab 3 × Z cd 3 generated by g ab : (
a i b i ) → a i+1 b i+1
and g cd :
1 -1 -1 1 0 0 0 0 1 0 -1 0 1 0 -1 0 0 0 0 0 1 -1 -1 1 , (157)
which is the coefficient matrix of the basis a i -a j -b i + b j = 0, a i -b i + c l -d l = 0 and c l -c m -d l + d m = 0 of linear relations. The coherent triangulations are in one-to-one correspondence to chambers that are seperated by the facets of the cones generated by all linear bases µ = {v 1 , v 2 , v 3 } with v i selected among the 8 column vectors of the Gale transform [57, 58] . In the present case the cones over the faces of the parallelepiped in Figure 1 are subdivided into 24 chambers, which are indicated by dashed lines. The triangulations, which we can label by the facet containing and the edge adjoining the chamber, are obtained as the sets of complements of those bases µ that span a cone containing the respective chamber. Hence, each non-simplicial facet has 24 coherent triangulations, which can be characterized by the triangulations of its 2 pure and of its 4 mixed circuits: Calling the triangulation a i c l |b i d l positive and the triangulation a i c l | b i d l negative, and arranging the cyclic permutations g ab and g cd in the horizontal and vertical direction, respectively, we can assign one of 16 different types ± ± ± ± to each triangulation, where the signs indicate the induced triangulations of the mixed circuits. The constraints that reduce the a priori 32 = 2 6 combinations to 24 all derive from the following rules:
a i a j b i b j c l a i c l | b i d l ∧ a j c l |b j d l ⇒ a i b j | a j b i a i c l |b i d l ∧ a j c l | b j d l ⇒ a i b j |a j b i (158)
i.e. a triangular prism can be triangulated in 6 different ways, which correlates the a priori 8 combinations of the triangulations of the 3 squares (with analogous constraints for the two "horizontal" prisms [a
i b i c l c m d l d m ] contained in the facet [a i a j b i b j c l c m d l d m ]).
Putting the pieces together we obtain Lemma 2. The 24 triangulations of the non-simplicial facets can be assorted as follows:
• For + + + + , -- --the pure circuits are unconstrained, yielding 2 • 2 2 = 8 triangulations.
• For + + --, -- + + the pure ab-circuit is unconstrained; with the transposed types + -+ -, -+ -+ this accounts for another 8 triangulations.
• The final 8 triangulations come from the 8 types with an odd number of positive signs, for which the triangulation of the pure circuits is unique.
• The two types + - -+ and -+ + -cannot occur because of contradictory implications for the triangulations of the pure circuits.
The secondary fan and the induced triangulations for the codimension-two faces at which the non-simplical facets intersect can be obtained from Figure 1 by projection along the dropped vertices. The secondary fan of the prism of eq. (158), for example, which is shown in Figure 2 , is obtained from Figure 1 c m d m . The wall crossings between the six cones in Figure 2 are labeled by the circuits whose flops relate the adjoining triangulations [57] .
For the construction of the complete star triangulation we now observe that the non-simplicial intersections of the 9 non-simplicial facets Table 5: The 824 = 2 (64 + 36 + 72 + 96 + 36 + 72 + 36) star triangulations of ∇ * , including the 720 = 2 (36 + 36 + 72 + 72 + 36 + 72 + 36) coherent triangulations.
striction on the compatible signs is due to the absence of the inconsistent types + - -+ and -+ + -as subgraphs on the torus. The multiplicities µ • 2 n come from the number n of unconstrained pure circuits and from the order µ of the effective part of the symmetry group generated by transposition and permutations of lines and columns. We thus find a total of 824 triangulations. The cyclic permutation symmetry that we want to keep on the Calabi-Yau manifold X * amounts to a diagonal shift, i.e. its induced action on the graph is generated by g ab g cd . We are hence left with the 5 , which have unbroken horizontal symmetry and for which the multiplicity is reduced from 12 • 8 to 12 • 6. This poses a problem for the eight Z 3 -symmetric triangulations, which are all non-coherent. Coherence of the remaining 720 triangulations can be established by checking that their Mori cones are all strictly convex [59] .
What comes to our rescue is that, even if all projective ambient spaces break the diagonal Z 3 permutation symmetry, it may be preserved on X * if the obstructing exceptional sets do not overlap with the complete intersection. In the present case these are the blow-ups of the singularities coming from the pure circuits, i.e. codimension two sets of the form a i • b j or c l • d m , where we use, for simplicity, the symbol of the vertex νj for the corresponding divisor D j . Recall from eq. ( 77 ) that X * is given by the product D * The polytope ∇ * of the mirror X * of the universal cover has 39 lattice points, with the same 12 vertices as ∇ * but living on the finer lattice M . The 24 additional lattice points, see eq. ( 39 ), are
a ij = 1 3 (a i + 2a j ), b ij = 1 3 (b i + 2b j ), (166)
c ij = 1 3 (c i + 2c j ), d ij = 1 3 (d i + 2d j ), (167)
where i = j. These additional points are all located on edges of ∇ * . It is natural to consider triangulations that are refinements of the ones that we just discussed. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z 3 ⊕ Z 3 Wilson lines. As we found in Part A [1], the integral homology group H 2 (X, Z) = Z 3 ⊕ Z 3 ⊕ Z 3 contains torsion curves. Using the B-model on the mirror of X as well as its covering spaces, we compute the instanton numbers. We observe that X is self-mirror even at the quantum level. Using the selfmirror property, we derive the complete prepotential on X, going beyond the results of Part A. In particular, this yields the first example where the instanton number depends on the torsion part of its homology class. Another consequence is that the threefold X provides a non-toric example for the conjectured exchange of torsion subgroups in mirror manifolds."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "was essentially solved by mirror symmetry several years ago [2] . The purpose of this paper is to take into account an important subtlety that does not appear in very simple Calabi-Yau manifolds like hypersurfaces in smooth toric varieties. This subtlety is the appearance of torsion curve classes. That is, the homology foot_0 group\n\nH 2 X, Z = Z 3 ⊕ Z 3 ⊕ Z 3\n\n(1) contains the torsion 2 subgroup Z 3 ⊕ Z 3 . Here, the manifold of interest X is a quotient of one of Schoen's Calabi-Yau manifolds [3, 4] by a freely acting symmetry group.\n\nThere are already a few known examples of such Calabi-Yau manifolds with torsion curves [5, 6, 7, 8, 9] , but the proper instanton counting has never been done before.\n\nThe prime motivation for studying these curves is that one would like to compute the superpotential for the vector bundle moduli [10, 11, 12, 13, 14, 15, 16] in a heterotic MSSM [17, 18, 19, 20, 21, 22, 23, 24, 25] . Our main result will be that there exist smooth rigid rational curves in X that are alone in their homology class. This proves that, in general, no cancellation between contributions to the superpotential W from instantons in the same homology class can occur.\n\nTherefore we would like to count rational curves on X. In physical terms, we need to find the instanton correction F np X,0 to the genus zero prepotential of the (A-model) topological string on X. This is usually written as a (convergent) power series in h 11 variables q a = e 2πit a . Each summand is the contribution of an instanton, and the (integer) coefficients are the multiplicities of instantons in each homology class. According to [26, 27, 1] the novel feature of the 3-torsion curves on X is that for each 3-torsion generator we need an additional variable b j such that b 3 j = 1. The Fourier series of the prepotential on X becomes\n\nF np X,0 (p, q, r, b 1 , b 2 ) = n 1 ,n 2 ,n 3 ∈Z m 1 ,m 2 ∈Z 3 n (n 1 ,n 2 ,n 3 ,m 1 ,m 2 ) Li 3 p n 1 q n 2 r n 3 b m 1 1 b m 2 2 , (2)\n\nwhere n (n 1 ,n 2 ,n 3 ,m 1 ,m 2 ) is the instanton number in the curve class (n 1 , n 2 , n 3 , m 1 , m 2 ).\n\nFor the purpose of computing the prepotential, we can either use directly the Amodel or start with the B-model and apply mirror symmetry. The A-model calculation was carried out in the companion paper [1] , entitled Part A. The results were:\n\n• A set of powerful techniques to compute the torsion subgroups in the integral homology and cohomology groups of X. They are spectral sequences starting with the so-called group (co)homology of the group action on the universal cover X.\n\n• A closed formula for the genus zero prepotential\n\nF np X,0 (p, q, r, b 1 , b 2 ) = 2 i,j=0\n\npb i 1 b j 2 P (q) 4 P (r) 4 +O(p 2 ) = 2 i,j=0\n\nLi\n\n3 (pb i 1 b j 2 )+• • • (3)\n\nto linear order in p, extending the one computed in [28] for the universal cover X.\n\nHere, if p(k) is the number of partitions of k ∈ Z ≥ , then P (q) is the generating function for partitions,\n\nP (q) def = ∞ i=0 p(i)q i = q 1 24 η( 1 2πi ln q) . (4)\n\n• Expanding eq. ( 3 ) as an instanton series we find that the number of rational curves of degree (1, 0, 0, m 1 , m 2 ) is:\n\nn (1,0,0,m 1 ,m 2 ) = 1, ∀ m 1 , m 2 ∈ Z 3 . (5)\n\nFurthermore, these curves have normal bundle O P 1 (-1)⊕O P 1 (-1). Hence, there are indeed 9 smooth rigid rational curves which are alone in their homology class.\n\nAlternatively, one can start with the B-model topological string and apply mirror symmetry, which is what we will do in this paper, entitled Part B. This will allow us to obtain the higher order terms in p. The order in p up to which one wants to compute the instanton numbers is only limited by computer power. We will again find a closed formula at every order in p, however, this time by guessing it from the instanton calculation, and hence only up to the order given by this limitation. The way to arrive at this result is as follows:\n\n• The universal cover X admits a simple realization as a complete intersection in a toric variety. In this situation, mirror symmetry boils down to an algorithm to compute instanton numbers. Unfortunately, there are many non-toric divisors which cannot be treated this way. It turns out that, after descending to X, precisely the torsion information is lost. In this approach one can only compute F np X,0 (q 1 , q 2 , q 3 , 1, 1).\n\n• As a pleasant surprise we find strong evidence that the manifold X is self-mirror.\n\nIn particular, we attempt to compute the instanton numbers on the mirror X * by descending from the covering space X * . The embedding of X * into a toric variety is such that all 19 divisors are toric. In principle, this allows for a complete analysis including the full Z 3 ⊕ Z 3 torsion information, but this is too demanding in view of current computer power.\n\n• Although the full quotient X = X/(Z 3 × Z 3 ) is not toric, it turns out that a certain partial quotient X = X/Z 3 can be realized as a complete intersection in a toric variety. That way, one only has to deal with h 11 (X) = 7 parameters, which is manageable on a desktop computer. Assuming the self-mirror property, we work with the mirror X * , for which again all divisors are toric, and we can compute the expansion of F np X,0 (p, q, r, 1, b 2 ) to any desired degree. A symmetry argument then allows one to recover the b 1 dependence as well. Finally, we can extract the instanton numbers n (n 1 ,n 2 ,n 3 ,m 1 ,m 2 ) including the torsion information.\n\n• As can be seen from the A-model result eq. ( 3 ), we observe that the prepotential F np X,0 at order p factors into 2 i,j=0 b i 1 b j 2 times a function of p, q, r only. This means that the instanton number does not depend on the torsion part of its homology class. We will explain the underlying reason for this factorization and show that it breaks down at order p 3 . This fits nicely with the B-model computation at order p 3 , where the instanton numbers do depend on the torsion part.\n\n• Another consequence of the self-mirror property is that X is a non-toric example for the conjecture of [6] . By this conjecture, certain torsion subgroups of the integral homology groups are exchanged under mirror symmetry.\n\nAn easily readable overview and a discussion of the physical consequences of our findings for superpotentials and moduli stabilization of heterotic models was presented in [27] . The present Part B is self-contained and can be read independently of Part A [1] . All necessary results from Part A are reproduced in this part.\n\nAs a guide through this paper, we start in Section 2 with a brief overview of the topology of the various spaces involved as determined in Part A [1]. This is followed by a review of the Batyrev-Borisov construction of mirror pairs of complete intersections in toric varieties in Section 3. We illustrate this construction by means of the covering spaces X and X * of our example. The review includes the techniques to compute the B-model prepotential and the mirror map. These are applied in Section 4 to the partial quotients X and X * yielding the main results stated above. This assumes that X as well as X are self-mirror, and evidence for this property is recapitulated in Section 5. Moreover, we show how the torsion subgroups are exchanged. Section 6 contains an explanation for the breakdown of the factorization alluded to above. Putting all the information together we try to guess a closed form for the prepotential in Section 7. Finally, we present our conclusions in Section 8. In the course of this work we will notice that a certain flop of X is very natural from the toric point of view, and we will present it in Appendix B.\n\n2 Calabi-Yau Threefolds 2.1 The Calabi-Yau Threefold X\n\nThe Calabi-Yau manifold X of interest is constructed as a free G def = Z 3 × Z 3 quotient of its universal covering space X. The latter is one of Schoen's Calabi-Yau threefolds [3] . It is simply connected and hence easier to study. Among its various descriptions are the fiber product of two dP 9 surfaces, a resolution of a certain T 6 orbifold [29] , or a complete intersection in a toric variety. In the present Part B, we will mostly use the latter viewpoint. The simplest way is to introduce the toric ambient variety P 2 × P 1 × P 2 with homogeneous coordinates [x 0 : x 1 : x 2 ], [t 0 : t 1 ], [y 0 : y 1 :\n\ny 2 ] ∈ P 2 × P 1 × P 2 . ( 6\n\n)\n\nThe embedded Calabi-Yau threefold X is then obtained as the complete intersection of a degree (0, 1, 3) and a degree (3, 1, 0) hypersurface in P 2 × P 1 × P 2 . We restrict the coefficients of their defining equations F i = 0 to a particular set of three complex parameters λ 1 , λ 2 , λ 3 , such that the polynomials F i read\n\nt 0 x 3 0 + x 3 1 + x 3 2 + t 1 x 0 x 1 x 2 def = F 1 (7a)\n\nλ 1 t 0 + t 1 y 3 0 + y 3 1 + y 3 2 + λ 2 t 0 + λ 3 t 1 y 0 y 1 y 2 def = F 2 . ( 7b\n\n)\n\nFor the special complex structure parametrized by λ 1 , λ 2 , λ 3 the complete intersection is invariant under the G = Z 3 × Z 3 action generated by (ζ def = e 2πi 3 )\n\ng 1 : [x 0 : x 1 : x 2 ] → [x 0 : ζx 1 : ζ 2 x 2 ] [t 0 : t 1 ] → [t 0 : t 1 ] (no action) [y 0 : y 1 : y 2 ] → [y 0 : ζy 1 : ζ 2 y 2 ] (8a)\n\nand\n\ng 2 : [x 0 : x 1 : x 2 ] → [x 1 : x 2 : x 0 ] [t 0 : t 1 ] → [t 0 : t 1 ] (no action) [y 0 : y 1 : y 2 ] → [y 1 : y 2 : y 0 ] (8b)\n\nOne can show that the fixed points of this group action in P 2 × P 1 × P 2 do not satisfy eqns. (7a) and (7b), hence the action on X is free."
},
{
"section_type": "OTHER",
"section_title": "The Intermediate Quotient X",
"text": "The partial quotient\n\nX def = X G 1 (9)\n\nwill be of particular interest in this paper because this quotient is generated by phase symmetries, see eq. (8a), and hence is toric. In particular, we will need a basis of Kähler classes. As usual, we will not distinguish degree-2 cohomology and degree-4 homology classes but identify them via Poincaré duality. Part A [1] ?? shows that foot_1\n\nH 2 X, Z = H 2 ( X, Z) G 1 ⊕ Z 3 = span Z φ, τ 1 , υ 1 , ψ 1 , τ 2 , υ 2 , ψ 2 ⊕ Z 3 . (10)\n\nHence, by abuse of notation, we can identify the free generators on X with the G 1invariant generators on X, see Part A eq. (??), via the pull back by the quotient map.\n\nThe triple intersection numbers on X = X/Z 3 are one-third of the corresponding intersection numbers on X listed in Part A eq. (??). Hence, the intersection numbers on X are\n\nφτ 1 τ 2 = 3 φτ 1 υ 2 = 3 φτ 1 ψ 2 = 6 φυ 1 τ 2 = 3 φυ 1 υ 2 = 3 φυ 1 ψ 2 = 6 φψ 1 τ 2 = 6 φψ 1 υ 2 = 6 φψ 1 ψ 2 = 12 τ 2 1 τ 2 = 1 τ 2 1 υ 2 = 1 τ 2 1 ψ 2 = 2 τ 1 υ 1 τ 2 = 3 τ 1 υ 1 υ 2 = 3 τ 1 υ 1 ψ 2 = 6 τ 1 ψ 1 τ 2 = 3 τ 1 ψ 1 υ 2 = 3 τ 1 ψ 1 ψ 2 = 6 τ 1 τ 2 2 = 1 τ 1 τ 2 υ 2 = 3 τ 1 τ 2 ψ 2 = 3 τ 1 υ 2 2 = 3 τ 1 υ 2 ψ 2 = 6 τ 1 ψ 2 2 = 6 υ 2 1 τ 2 = 3 υ 2 1 υ 2 = 3 υ 2 1 ψ 2 = 6 υ 1 ψ 1 τ 2 = 6 υ 1 ψ 1 υ 2 = 6 υ 1 ψ 1 ψ 2 = 12 υ 1 τ 2 2 = 1 υ 1 τ 2 υ 2 = 3 υ 1 τ 2 ψ 2 = 3 υ 1 υ 2 2 = 3 υ 1 υ 2 ψ 2 = 6 υ 1 ψ 2 2 = 6 ψ 2 1 τ 2 = 6 ψ 2 1 υ 2 = 6 ψ 2 1 ψ 2 = 12 ψ 1 τ 2 2 = 2 ψ 1 τ 2 υ 2 = 6 ψ 1 τ 2 ψ 2 = 6 ψ 1 υ 2 2 = 6 ψ 1 υ 2 ψ 2 = 12 ψ 1 ψ 2 2 = 12. (11)\n\nClearly, G 2 acts on the partial quotient X. From Part A eq. (??) it follows that, of the 7 non-torsion divisors above, only 3 are G 2 -invariant. This invariant part is particularly manageable and will be important in the following. We find\n\nH 2 X, Z G free = H 2 X, Z G 2 free = span Z φ, τ 1 , τ 2 (12)\n\nwith products 3τ 2 i = τ i φ. In particular, the triple intersection numbers on X are\n\nτ 2 1 τ 2 = 1, τ 1 φτ 2 = 3, τ 1 τ 2 2 = 1, (13)\n\nand 0 otherwise. Finally, the second Chern class of\n\nX is c 2 (X) = 12(τ 2 1 + τ 2 2 ). There- fore, c 2 X • τ 1 = 12, c 2 X • φ = 0, c 2 X • τ 2 = 12. ( 14\n\n)"
},
{
"section_type": "OTHER",
"section_title": "Variables",
"text": "As we discussed in Part A ??, the instanton-generated superpotential should be thought of as a series with one variable for each generator in H 2 . In particular, we Calabi-Yau threefold H 2 -, Z Free generators Torsion generators X Z 19 p 0 , q 0 , . . . , q 8 , r 0 , . . . , r 8 ∅\n\nX = X/G 1 Z 7 ⊕ Z 3 P, Q 1 , Q 2 , Q 3 , R 1 , R 2 , R 3 b 1 X = X/G Z 3 ⊕ Z 3 ⊕ Z 3 p, q, r b 1 , b 2\n\nTable 1: The different Calabi-Yau threefolds, curve classes, and variables used to expand the prepotential.\n\nwill be interested in the Calabi-Yau threefolds X, X, and X. For these, the degree-2 integral homology and the variables used (see Part A [1] for precise definitions) are in summarized Table 1 . Pushing down the curves by the respective quotients lets us express the prepotential on the quotient in terms of the prepotential on the covering space. We found in Part A that\n\nF np X,0 P, Q 1 , Q 2 , Q 3 , R 1 , R 2 , R 3 , b 1 ) = 1 |G 1 | F np e X,0 P Q 5 1 Q 6 2 R 5 1 R 6 2 , Q 5 1 Q 6 2 , Q -2 1 Q -2 2 Q -3 3 b 1 , Q -1 1 Q -1 2 , Q 3 3 , Q 2 3 b 1 , Q 3 , 1, b 1 , Q 1 Q 3 3 , R 5 1 R 6 2 , R -2 1 R -2 2 R -3 3 b 2 1 , R -1 1 R -1 2 , R 3 3 , R 2 3 b 2 1 , R 3 , 1, b 2 1 , R 1 R 3 3 (15)\n\nand\n\nF np X,0 p, q, r, b 1 , b 2 ) = 1 |G 2 | F np X,0 p, q, b 2 , b 2 , r, b 2 2 , b 2 2 , b 1 . ( 16\n\n)\n\n3 Toric Geometry and Mirror Symmetry\n\nIn this section we review mirror symmetry and the construction of the B-model for the mirror of the covering space X. Since X is a complete intersection in a toric variety, we can use the standard constructions. Because we expect the model to be self-mirror, we will analyze the B-model for X and its mirror X * . The toric geometry for X is much simpler 4 than for X * , but contains less information. In this section we will start with the simpler model in order to review the Batyrev-Borisov construction for the mirror of a complete intersection in a toric variety. Then we will apply this construction to the more complicated model, now without going into too many details. We will see that, on the simpler side, not all parameters are toric and no torsion is visible. However, on the more complicated mirror side, all parameters are toric which will allow us, in principle, to perform the B-model computation of the complete prepotential. As X ∼ = X * is expected to be self-mirror, this determines the complete prepotential F np e X,0 = F np e X * ,0 as well. In practice, however, the analysis is computationally too involved.\n\nFortunately, the space X = X/G 1 and its mirror will turn out to be both tractable with toric methods and sufficiently informative. This quotient will be the subject of Section 4. Finally, this is also the starting point for arguing in Section 5 that the self-mirror property persists at the level of instanton corrections.\n\nRecall that, in Subsection 2.1 we defined our Calabi-Yau manifold as the complete intersection\n\nX def = F 1 = 0, F 2 = 0 ⊂ P 2 × P 1 × P 2 (17)\n\nwith the two polynomials F 1 , F 2 as in eqns. (7a) and (7b), respectively. In order to construct the mirror manifold following Batyrev and Borisov, we need to reformulate this definition in terms of toric geometry. We review here some essential ingredients of toric geometry, for details we refer to [30, 31] and references therein. We will give the abstract definitions and concepts step by step, and at each step illustrate them with the example X and its mirror manifold X * ."
},
{
"section_type": "OTHER",
"section_title": "Toric Varieties",
"text": "Given a lattice N of dimension d, a toric variety V Σ is defined in terms of a fan Σ which is a collection of rational polyhedral foot_3 cones σ ⊂ N such that it contains all faces and intersections of its elements. V Σ is compact if the support of Σ covers all of the real extension N R of the lattice N. The resulting d-dimensional variety V Σ is smooth if all cones are simplicial and if all maximal cones are generated by a lattice basis.\n\nLet Σ (1) denote the set of one-dimensional cones (rays) with primitive generators ρ i , i = 1, . . . , n. The simplest description of V Σ introduces n homogeneous coordinates z i corresponding to the generators ρ i of the rays in Σ (1) . These homogeneous coordinates are then subjected to weighted projective identifications\n\nz 1 : • • • : z n = λ q (a) 1 z 1 : • • • : λ q (a) n z n a = 1, . . . , h (18)\n\nfor any nonzero complex number λ ∈ C × , where the integer n-vectors q (a) i are generators of the linear relations q (a)\n\ni ρ i = 0 among the primitive lattice vectors 6 ρ i . In order to obtain a well-behaved quotient, we must exclude an exceptional set Z(Σ) ⊂ C n that is defined in terms of the fan, as will be explained below. Hence, the quotient is\n\nV Σ = C n -Z(Σ) C × h × Γ , (19)\n\nwhere Γ ≃ N/ span{ρ i } is a finite abelian group. There are h = n -d independent C × identifications, therefore the complex dimension of V Σ equals the rank d of the lattice N. The identifications by Γ are only non-trivial if the ρ i do not span the lattice N.\n\nRefinements of the lattice N with fixed ρ i can hence be used to construct quotients of toric varieties V Σ by discrete phase symmetries such as Z 3 . Such quotients will be discussed in Section 4. Note that the rays ρ i are in 1-to-1 correspondence with the (C × )-invariant divisors D i on V Σ , which are defined as\n\nD i = z i = 0 ⊂ V Σ . (20)\n\nConversely, the homogeneous coordinate z i is a section of the line bundle O(D i ).\n\nFor example, consider the simplest compact toric variety, the projective space P d . Its fan Σ = Σ(∆) is generated by the n = d + 1 vectors\n\nρ 1 = e 1 , ρ 2 = e 2 , . . . , ρ n-1 = e d , ρ n = - d i=1 e i (21)\n\nof a d-dimensional simplex ∆. They satisfy a single linear relation, n i=1 ρ i = 0. Therefore q i = 1 for all i, and the homogeneous coordinates in eq. ( 18 ) are the usual homogeneous coordinates on P d .\n\nFor products of toric varieties we simply extend the relations for any single factor by zeros and take the union of them. Hence, the fan of the polyhedron ∆ * describing the 5-dimensional toric variety P 2 ×P 1 ×P 2 in eq. ( 17 ) is generated by the n = 5+3 = 8 vectors ρ 1 = e 1 , ρ 2 = e 2 , ρ 3 = -e 1 -e 2 , ρ 4 = e 3 , ρ 5 = -e 3 , ρ 6 = e 4 , ρ 7 = e 5 , ρ 8 = -e 4 -e 5 (22) satisfying the linear relations\n\n3 i=1 ρ i = 5 i=4 ρ i = 8 i=6 ρ i = 0. ( 23\n\n)\n\nExcept for the origin, there are no other lattice points in the interior of ∆ * . The corresponding homogeneous coordinates will be denoted by\n\nz 1 = x 0 , z 2 = x 1 , z 3 = x 2 , z 4 = t 0 , z 5 = t 1 , z 6 = y 0 , z 7 = y 1 , z 8 = y 2 . ( 24\n\n)\n\nIn more general situations, given a polytope ∆ * ⊂ N we will denote the resulting toric variety by P ∆ * = V Σ(∆ * ) ."
},
{
"section_type": "OTHER",
"section_title": "The Batyrev-Borisov Construction",
"text": "Batyrev showed that a generic section of K -1 P ∆ * , the anticanonical bundle of P ∆ * , defines a Calabi-Yau hypersurface if ∆ * is reflexive, which means, by definition, that ∆ * and its dual\n\n∆ = x ∈ M R (x, y) ≥ -1 ∀y ∈ ∆ * (25)\n\nare both lattice polytopes. Here, M = Hom(N, Z) is the lattice dual to N and M R is its real extension. Mirror symmetry corresponds to the exchange of ∆ and ∆ * [32] .\n\nThe generalization of this construction to complete intersections of codimension r > 1 is due to Batyrev and Borisov [33, 34] . For that purpose, they introduced the notion of a nef partition. Consider a dual pair of\n\nd-dimensional reflexive polytopes ∆ ⊂ M R , ∆ * ⊂ N R . In that context, a partition E = E 1 ∪ • • • ∪ E r of\n\nthe set of vertices of ∆ * into disjoint subsets E 1 , . . . , E r is called a nef-partition if there exist r integral upper convex Σ(∆ * )-piecewise linear support functions φ l\n\n: N R → R, l = 1, . . . , r such that φ l (ρ) = 1 if ρ ∈ E l , 0 otherwise. ( 26\n\n)\n\nEach φ l corresponds to a divisor\n\nD 0,l = ρ∈E l D ρ (27)\n\non P ∆ * , and their intersection\n\nY = D 0,1 ∩ • • • ∩ D 0,r (28)\n\ndefines a family Y of Calabi-Yau complete intersections of codimension r. Moreover, each φ l corresponds to a lattice polyhedron ∆ l defined as\n\n∆ l = x ∈ M R (x, y) ≥ -φ l (y) ∀y ∈ N R . (29)\n\nThe lattice points m ∈ ∆ l correspond to monomials\n\nz m = n i=1 z m,ρ i i ∈ Γ (P ∆ * , O(D 0,l )) . ( 30\n\n)\n\nOne can show that the sum of the functions φ l is equal to the support function of K -1\n\nP ∆ *\n\nand, therefore, the corresponding Minkowski sum is ∆ 1 + • • • + ∆ r = ∆. Moreover, the knowledge of the decomposition E = E 1 ∪ • • • ∪ E r is equivalent to that of the set of supporting polyhedra Π(∆) = {∆ 1 , . . . , ∆ r }, and therefore this data is often also called a nef partition.\n\nIt can be shown that given a nef partition Π(∆) the polytopes foot_5\n\n∇ l = {0} ∪ E l ⊂ N R (31)\n\ndefine again a nef partition Π * (∇) = {∇ 1 , . . . , ∇ r } such that the Minkowski sum ∇ = ∇ 1 + • • • + ∇ r is a reflexive polytope. This is the combinatorial manifestation of mirror symmetry in terms of dual pairs of nef partitions of ∆ * and ∇ * , which we summarize in the diagram\n\n∆ = ∆ 1 + . . . + ∆ r ∆ * = ∇ 1 , . . . , ∇ r 5 5\n\nMirror Symmetry j j j j j j j j j j j j j j u u j j j j j j j j j j j j j j\n\nM R N R ∇ * = ∆ 1 , . . . , ∆ r (∆ l , ∇ l ′ ) ≥ -δ l l ′ ∇ = ∇ 1 + . . . + ∇ r . ( 32\n\n)\n\nIn the horizontal direction, we have the duality between the lattices M and N and mirror symmetry goes from the upper right to the lower left. The other diagonal has also a meaning in terms of mirror symmetry as we will explain below. The complete intersections Y ⊂ P ∆ * and Y * ⊂ P ∇ * associated to the dual nef partitions are then mirror Calabi-Yau varieties.\n\nLet us now apply the Batyrev-Borisov construction to the complete intersection eq. ( 17 ), hence r = 2. There exist several nef-partitions of ∆ * . The one which has the correct degrees (3, 1, 0) and (0, 1, 3) is, up to exchange of t 0 and t 1 , E 1 = {ρ i |i = 1, . . . , 4} and E 2 = {ρ i |i = 5, . . . , 8}. Adding the origin and taking the convex hull yields the polytopes\n\n∇ 1 = ρ 1 , . . . , ρ 4 , 0 , ∇ 2 = ρ 5 , . . . , ρ 8 , 0 , (33)\n\nwhere the ρ i are defined in eq. (22) . The two divisors cutting out the Calabi-Yau threefold are, according to eq. ( 27 ),\n\nD 0,1 = 4 i=1 D i , D 0,2 = 8 i=5 D i ⇒ X = D 0,1 ∩ D 0,2 ⊂ P ∆ * (34)\n\nNote that, while ∆ * has no further lattice points, its dual ∆ has 18 vertices and 300 lattice points. Using the computer package PALP [35] , we determine the associated polytopes ∆ 1 and ∆ 2 of the global sections of O(D 0,1 ) and O(D 0,2 ), respectively. In an appropriate lattice basis there is, up to symmetry, a unique nef partition consisting of\n\n∆ 1 = ν 1 , . . . , ν 6 , 0 , ∆ 2 = ν 7 , . . . , ν 12 , 0 , (35)\n\nwhere\n\nν 1 = 2e 1 -e 2 , ν 2 = -e 1 + 2e 2 , ν 3 = -e 1 -e 2 , ν 4 = 2e 1 -e 2 -\n\nAmong these 12 vectors there are the 7 independent linear relations\n\n3ν 3 + ν 4 + ν 5 -2ν 6 = 0, 3ν 9 + ν 10 + ν 11 -2ν 12 = 0, ν 1 -ν 3 -ν 4 + ν 6 = 0, -ν 1 + ν 2 + ν 4 -ν 5 = 0, ν 7 -ν 9 -ν 10 + ν 12 = 0, -ν 7 + ν 8 + ν 10 -ν 11 = 0, -ν 2 + ν 5 -ν 8 + ν 11 = 0. ( 37\n\n)\n\nThe convex hull ∇ * = ∆ 1 , ∆ 2 yields the fan Σ(∇ * ) and, consequently, the toric variety P ∇ * . Let D * i , i = 1, . . . , 12 be the divisors associated to the vertices ν i . Then, by eq. ( 27 ), the nef partition eq. ( 35 ) defines the divisors\n\nD * 0,1 = 6 i=1 D * i , D * 0,2 = 12 i=7 D * i , ⇒ X * = D * 0,1 ∩ D * 0,2 ⊂ P ∇ * (38)\n\ncutting out the mirror complete intersection X * . In contrast to ∆ * , the polytope ∇ * contains extra integral points. We find that it contains, in addition to the origin and the vertices in eq. ( 36 ), the 26 points\n\nν 13 = 1 3 (ν 4 + ν 5 + ν 6 ) = -e 3 , ν 12+6k+i+j = 1 3 (ν 3k+i + 2ν 3k+j ), ν 14 = 1 3 (ν 10 + ν 11 + ν 12 ) = e 3 , ν 15+6k+i+j = 1 3 (ν 3k+j + 2ν 3k+i ) ∀ k ∈ {0, . . . , 3}, (i, j) ∈ (1, 2), (1, 3), (2, 3) . (39)\n\nFor completeness, note that the dual polytope ∇ has 15 vertices and 24 lattice points. Running PALP to compute the Hodge numbers using the formula of [36] , we obtain\n\nh 1,1 X = h 1,2 X = h 1,1 X * = h 1,2 X * = 19, (40)\n\nin agreement with Part A [1], eq. (??). So far, we have mainly focused on the information contained in the reflexive polytopes ∆ * and ∇ * and ignored their duals. We have already mentioned that in the reflexive case a generic section of K -1 P ∆ * defines a Calabi-Yau manifold, and that such sections are provided by the lattice points of ∆. In other words, ∆ and ∇ are the Newton polytopes of Y and Y * , respectively. That is, the complete intersection Y (Y * ) is defined by r polynomial equations, and the exponents of the monomials in each equation are the lattice points in ∆ (∇). More precisely, the Minkowski sum for, say, ∆ = ∆ 1 + • • • + ∆ r defines r homogeneous polynomials\n\nF l (z) = m∈ ∆ l ∩M a l,m r l ′ =1 ρ i ∈ ∇ l ′ ∩N z m,ρ i +δ l l ′ i , l = 1, . . . , r (41)\n\nwith coefficients a l,m ∈ C. The simultaneous vanishing of F 1 , . . . , F r then defines the complete intersection Calabi-Yau manifold Y ⊂ P ∆ * . Exchanging ∆ l and ∇ l ′ in eq. ( 41 ) yields the equations F * l defining the mirror manifold Y * . It is in this sense that the map from the upper left to the lower right in eq. ( 32 ) is also a manifestation of mirror symmetry. Since we will not need the actual polynomials for X and X * , we refrain from writing them explicitly. Instead, we refer the reader to Section 4, where we determine the equations in a simpler situation."
},
{
"section_type": "OTHER",
"section_title": "Toric Intersection Ring",
"text": "Up to now we have only considered one of the ingredients in the fan Σ, namely, the generators ρ ∈ Σ (1) which defined the C × action in eq. (19) . The second ingredient is the exceptional set Z(Σ). It corresponds to fixed loci of a continuous subgroup of (C × ) h for which the quotient eq. ( 19 ) is not well defined. Therefore, these loci have to be removed. In terms of the homogeneous coordinates z i , this happens precisely when a subset {z i |i ∈ I}, I ⊆ {1, . . . , n}, of the coordinates vanishes simultaneously such that there is no cone σ ∈ Σ containing all of the ρ i ⊆ σ, i ∈ I. Hence, the set Z(Σ) is the union of the sets\n\nZ I = {[z 1 : • • • : z n ] |z i = 0 ∀i ∈ I}.\n\nMinimal index sets I with this property are called primitive collections [37] . In order to determine the index sets I we need a coherent 8 triangulation T = T (∆ * ) of the polytope ∆ * for which all simplices contain the origin. Different triangulations will yield different exceptional sets and, hence, different toric varieties. However, for simplicity, we will mostly suppress the choice of a triangulation in the notation. In the case of complete intersections, only those triangulations of ∆ * are compatible with a given nef partition that can be lifted to a triangulation of the corresponding Gorenstein cone, see [38] .\n\nThe polytope defining projective space P d admits a unique triangulation with the required properties, and this triangulation consists of n = d + 1 simplices. The only primitive collection is I = {1, . . . , n}. This is well-known from the definition of projective space, where we have to remove the origin z 1 = • • • = z d+1 = 0 from C d+1 . Similarly, the polyhedron ∆ * for the ambient space P ∆ * of X admits a unique triangulation, and the primitive collections are those of its factors, that is,\n\nI 1 = {1, 2, 3}, I 2 = {4, 5}, I 3 = {6, 7, 8}. ( 42\n\n)\n\n8 Coherent triangulations, sometimes also called regular triangulations, satisfy some technical property that is equivalent to the associated toric quotient being Kähler.\n\nThe mirror polyhedron ∇ * , on the other hand, admits a huge number of triangulations. We will discuss particularly interesting triangulations of the mirror polyhedron at the end of Appendix A.\n\nThe primitive collections determine the cohomology ring of toric varieties and, together with the nef partition, complete intersections. Recall that if the collection ρ i 1 , . . . , ρ i k of rays is not contained in at least one cone, then the corresponding homogeneous coordinates z i l are not allowed to vanish simultaneously. Therefore, the corresponding divisors D i l have no common intersection. Hence, we obtain non-linear relations\n\nR I = D i 1 • . . . • D i k = 0 in\n\nthe intersection ring. It can be shown that all such relations are generated by the primitive collections I = {i 1 , . . . , i k } defined above. The ideal generated by these R I is called Stanley-Reisner ideal\n\nI SR = R I , I primitive collection ⊂ Z[D 1 , . . . , D n ], (43)\n\nand Z[D 1 , . . . , D n ]/I SR is the Stanley-Reisner ring. The intersection ring of a nonsingular compact toric variety P Σ is [39]\n\nH * P Σ , Z = Z [D 1 , . . . , D n ] I SR , i (m, ρ i )D i . (44)\n\nIn other words, the intersection ring can be obtained from the Stanley-Reisner ring by adding the linear relations i (m, ρ i )D i = 0, where it is sufficient to take a set of basis vectors for m ∈ M. In particular, the intersection number of the divisors spanning a maximal-dimensional simplicial cone σ = span R≥ {ρ i 1 , . . . , ρ i d } is\n\nD i 1 • . . . • D i d = 1 Vol(σ) , (45)\n\nwhere Vol(σ) is the lattice-volume, that is, the geometric volume divided by the volume 1 d! of a basic simplex. For practical purposes it is sufficient to compute one of these volumes, the remaining intersections can be obtained using the linear and non-linear relations.\n\nHaving found the intersection ring of the ambient toric variety, we now turn to the complete intersection Y ⊂ P ∆ * . The toric part of its even-degree intersection ring is [40] H\n\nev toric Y, Q = Q [D 1 , . . . , D n ] I Y , (46)\n\nwhere I Y is the ideal quotient\n\nI Y = I SR , i (m, ρ i )D i : r l=1 D 0,l . (47)\n\nNote that it can happen that some of the D i appear as generators of I Y . This means that they can be set to zero in the intersection ring. Geometrically, this means that these divisors do not intersect a generic complete intersection Y . While the intersection ring depends on the triangulation T (∆ * ) through the primitive collections defining the Stanley-Reisner ideal, we conjecture that the divisors D i not intersecting Y are independent of the choice of triangulation. This conjecture is proven for r = 1 and supported by a large amount of empirical evidence for r > 1. We conclude that the dimension dim H 2 toric (Y ) is in general smaller than h 1,1 (Y ) for the following two reasons: Only h = n -d = dim H 2 (P ∆ * , Z) divisors are realized in the ambient toric variety P ∆ * , and some of them may not descend to the complete intersection Y . Using the adjunction formula we can compute the the Chern classes of Y by expanding\n\nc(Y ) = n i=1 (1 + D i ) r l=1 (1 + D 0,l ) . ( 48\n\n)\n\nThe intersection ring together with the second Chern class determine the diffeomorphism type of a simply-connected Calabi-Yau manifold [41] . If we consider the cohomology with integral coefficients there can be torsion and, in fact, this is what this paper is all about. Unfortunately, a combinatorial formula in terms of the fan Σ(∆) for the torsion in the integral cohomology of a toric Calabi-Yau manifold is only known in the hypersurface case [6] . We now illustrate these concepts in the example of the complete intersection X ⊂ P ∆ * = P 2 × P 1 × P 2 and its mirror manifold X * . In eq. ( 42 ) we already determined the primitive collections, hence the corresponding Stanley-Reisner ideal is\n\nI SR = D 1 D 2 D 3 , D 4 D 5 , D 6 D 7 D 8 . (49)\n\nThe linear equivalences are\n\nD 1 = D 2 , D 1 = D 3 , D 4 = D 5 , D 6 = D 7 , D 6 = D 8 and, hence, we can choose K 1 = D 4 , K 2 = D 1 , K 3 = D 6\n\nas a basis for H 2 (P ∆ * ). In terms of this basis, we obtain D 0,1 = K 1 +3K 2 and D 0,2 = K 1 +3K 3 , see eq. ( 27 ). Therefore, the ideal I e X in eq. ( 34 ) is\n\nI e X = K 3 2 K 2 -K 2 2 K 3 , K 1 K 2 -3 K 2 2 , K 1 K 3 -3 K 3 2 , K 1 2 , K 2 3 , K 3 3 . ( 50\n\n)\n\nNext, we define the restriction of the K i to X to be the divisors\n\nJi = K i • X = K i (K 1 + 3K 2 )(K 1 + 3K 3 ). ( 51\n\n)\n\nWe need to compute one of the intersection numbers directly from the volume of a cone, say,\n\nJ1 J2 J3 = K 1 K 2 K 3 (K 1 + 3K 2 )(K 1 + 3K 3 ) = 9K 1 K 2 2 K 2 3\n\n, where we made use of the relations in I e X . Using eq. ( 45 ), this intersection can be evaluated to be\n\n9K 1 K 2 2 K 2 3 = 9D 1 D 2 D 4 D 6 D 7 = 9/ Vol ρ 1 , ρ 2 , ρ 4 , ρ 6 , ρ 7 = 9/ Vol e 1 ,\n\nThen, again using eq. ( 50 ), we see that the only non-vanishing intersection numbers and the second Chern class are\n\nJ2 2 J3 = 3, J1 J2 J3 = 9, J2 J2 3 = 3, c 2 X • J1 = 0, c 2 X • J2 = 36, c 2 X • J3 = 36. ( 53\n\n)\n\nNote that only h 1,1 toric ( X) = 3 of the h 1,1 ( X) = 19 parameters are realized torically. Comparing the triple intersection numbers with eq. ( 13 ), it is clear that these 3 toric divisors are precisely the G-invariant divisors on X.\n\nA similar, though much more complicated, calculation can be done for X * ⊂ P ∇ * . Using the results of Appendix A one can show that, among the points in eq. ( 39 ), the 14 divisors D * 13 , D * 14 , D * 12+6k+i+j , D * 15+6k+i+j , k = 0, 2 appear as generators of eq. ( 47 ) and, therefore, do not intersect X * . Subtracting from the remaining 24 divisors in eqns. ( 36 ) and ( 39 ) the remaining 5 linear relations in eq. ( 37 ), we find that all h 1,1 toric ( X * ) = h 1,1 ( X * ) = 19 moduli are realized torically."
},
{
"section_type": "OTHER",
"section_title": "Mori Cone",
"text": "As we have just seen, the cohomology classes\n\nD i span H 2 (P Σ , R) = H 1,1 (P Σ ).\n\nThe Kähler classes of a smooth projective toric variety P Σ form an open cone in H 1,1 (P Σ ) called the Kähler cone K(P Σ ). This cone has a combinatorial description in terms of the fan Σ, which we now review. First, define a support function to be a continuous function\n\nψ : N R → R given on each cone σ ∈ Σ by an m σ ∈ M R via ψ(ρ) = (m σ , ρ) ∀ρ ∈ σ ⊂ N R . (54)\n\nA support function determines a divisor D = i ψ(ρ i )D i . We say that D is convex if ψ is a convex function on N R . The convex classes form a non-empty strongly convex polyhedral cone in H 1,1 (P Σ ) whose interior is the Kähler cone K(P Σ ). Such a support function is strictly convex if and only if\n\nψ(ρ i 1 + • • • + ρ i k ) > ψ(ρ i 1 ) + • • • + ψ(ρ i k ) (55)\n\nfor every primitive collection I = {i 1 , . . . , i k } [40] . The dual of the Kähler cone K(P Σ ) is called the Mori cone or the cone of numerically effective curves NE(P Σ ). Its generators can be described by vectors l (a) of the corresponding linear relations i l (a)\n\ni ρ i = 0. Each face of the Kähler cone K(P Σ ) is dual to an edge of NE(P Σ ). These edges are generated by curves c (a) , and the entries of the vector l (a) are\n\nl (a) i = c (a) • D i . (56)\n\nA practical algorithm to find the generators for l (a) in terms of the triangulation T (∆ * ) is described in [42] . Of course, we are not interested in the ambient space but in a complete intersection Y ⊂ P ∆ * . The restriction of a Kähler class on the ambient space yields a Kähler class on Y , but not every Kähler class on Y arises that way. We define the toric part of the Kähler cone on Y as the restriction [43] K(Y\n\n) toric = K(P Σ ) Y ⊂ K(Y ). ( 57\n\n)\n\nIn the simplicial case, we can always take the basis J i of H 2 toric (Y, Q) to be edges of the Kähler cone. The dual of the toric Kähler cone of Y is the (toric) Mori cone NE(Y ) toric . This is sufficient for mirror symmetry purposes, however, it can be larger than the actual cone of effective curves. Once the generators l (a) of NE(P ∆ * ) are determined, we need to add the information about the nef partition. For this purpose, we define\n\nl (a) 0,m def = -D 0,m • c (a) m = 1, . . . , r. (58)\n\nFinally, it is customary to write the generators of the Mori cone NE(Y ) toric as\n\nl (a) = l (a) 0,1 , . . . , l (a) 0,r ; l (a) 1 , . . . , l (a) n , (59)\n\nwhich are, by abuse of notation, again denoted by l (a) . The knowledge of the (toric) Mori cone is important for several reasons. It defines the local coordinates on the complex structure moduli space of the mirror Y * near the point of maximal unipotent monodromy. Moreover, the generators enter the coefficients of the fundamental period which is a solution of the Picard-Fuchs equations as we will review in Subsection 3.5. For example, using the unique primitive collections in eq. ( 42 ), the Mori cone for P ∆ * is generated 9 by l (1) =(0, 0, 0, 1, 1, 0, 0, 0)\n\nl (2) =(1, 1, 1, 0, 0, 0, 0, 0) l (3) =(0, 0, 0, 0, 0, 1, 1, 1). ( 60\n\n)\n\nRecalling the nef partition\n\nD 0,1 = D 1 + • • • + D 4 , D 0,2 = D 5 + • • • + D 8 , we prepend (-D 0,1 • c (a) , -D 0,2 • c (a)\n\n) = (-3, 0), (-1, -1), (0, -3), a = 1, 2, 3, to obtain the generators l (1) =(-1,-1; 0, 0, 0, 1, 1, 0, 0, 0)\n\nl (2) =(-3, 0; 1, 1, 1, 0, 0, 0, 0, 0) l (3) =( 0,-3; 0, 0, 0, 0, 0, 1, 1, 1) (61)\n\nof the Mori cone NE( X) toric . Due to the large number of toric moduli, the calculation for the Mori cone NE(P ∇ * ) of the ambient toric variety of the mirror X * is much more complex."
},
{
"section_type": "OTHER",
"section_title": "B-Model Prepotential",
"text": "Mirror symmetry identifies the quantum corrected Kähler moduli space of Y with the classical complex structure moduli space of Y * , see the excellent treatise in [43] for details. The deformations of the complex structure of Y * are encoded in the periods ̟ = γ Ω and the latter can be computed from the equations F * l that cut out Y * ⊂ P ∇ * . Given the Mori cone eq. ( 59 ) and the classical intersections numbers [44, 45, 38, 43] to write down a local expansion of the periods, convergent near the large complex structure point, which is characterized by its maximal unipotent monodromy. In the following, we will review just the bare essentials.\n\nκ abc = J a • J b • J c we follow\n\nThe coefficients a i in the polynomial constraints F * l of the complete intersection Y * , see eq. ( 41 ), define the complex structure of Y * . A particular set of local coordinates u a on the complex structure moduli space on Y * is defined by\n\nu b = r m=1 a l (b) 0,m m,0 n i=1 a l (b) i i b = 1, . . . , h (62)\n\nwhere h def = h 1,1 toric (Y ) and a m,0 is the coefficient in (41) corresponding to the origin in ∇ l . In these coordinates, the point of maximal unipotent monodromy is at u b = 0. We define the cohomology-valued period\n\n̟(u, J) = {na≥=0} r m=1 1 - h a=1 l (a) 0,m J a - P h a=1 l (a) 0,m na n i=1 1 + h a=1 l (a) i J a P h a=1 l (a) i na h a=1 u na+Ja a . ( 63\n\n)\n\nwhere (x) n = Γ(x + n)/Γ(x) is the Pochhammer symbol. Note that the choice of triangulation is implicit in the generators l (a) of the Mori cone. Expanding ̟(u, J) by cohomology degree yields\n\n̟(u, J) = ̟ (0) (u) + h a=1 ̟ (1) a (u)J a + h a=1 ̟ (2) a (u)κ abc J b J c -̟ (3) (u) dVol, ( 64\n\n)\n\nwhere dVol is the volume form. The coefficients in eq. ( 64 ) are the fundamental period ̟ (0) (u), that is, the unique solution to the Picard-Fuchs equations holomorphic at u a = 0, and\n\n̟ (1) a (u) = ∂ Ja ̟(u, J)| J=0 , ̟ (2) a (u) = 1 2 κ abc ∂ J b ∂ Jc ̟(u, J)| J=0 , ̟ (3) (u) = - 1 6 κ abc ∂ Ja ∂ J b ∂ Jc ̟(u, J)| J=0 . (65)\n\nThese coefficients coincide with the basis of solutions of the Picard-Fuchs equations obtained from the Frobenius method in [46, 31] . The B-model prepotential\n\nF B Y * ,0 is F B Y * ,0 (u) = 1 2̟ (0) (u) 2 ̟ (0) (u)̟ (3) (u) + h a=1 ̟ (1) a (u)̟ (2) a (u) . ( 66\n\n)\n\nAt the large complex structure point the mirror map defines natural flat coordinates on the Kähler moduli space of the original manifold Y , which are\n\nt i = ̟ (1) i (u) ̟ 0 (u) , i = 1, . . . , h. (67)\n\nWe also define q j = e 2πit j = u j + O(u 2 ). One way to obtain the prepotential is to compute its third derivatives\n\nC * abc = D a D b D c F B Y * ,0 = Y * Ω ∧ ∂ a ∂ b ∂ c Ω, (68)\n\nand apply the Picard-Fuchs operators. This leads to linear differential equations, which determine C * abc up to a common constant, see again [46, 43] for details. The quantum corrected three point function C ijk (q) on Y follows from C * abc (u) using the inverse mirror map eq. ( 67 ) u = u(t), and one obtains\n\nC ijk (q) = 1 ̟ (0) (u(q)) 2 ∂u a ∂t i ∂u b ∂t j ∂u c ∂t k C * abc (u(q)). ( 69\n\n)\n\nIn practice, we use the formula\n\nC ijk (q) = ∂ t i ∂ t j ̟ (2) k (u(q)) ̟ (0) (u(q)) . ( 70\n\n)\n\nIntegrating three times with respect to t i yields the prepotential F B Y * ,0 (t) up to a polynomial of degree three in t i which can be determined partially by the topological data of Y .\n\nMirror symmetry then ensures that the B-model prepotential, eq. ( 66 ), is equal to the A-model prepotential. That is,\n\nF Y,0 (q) = F B Y * ,0 (u(q)). ( 71\n\n)\n\nThis allows us to compute the instanton numbers n d . For the case of interest,\n\nX ∈ P ∆ * = P 2 × P 1 × P 2 , (72)\n\nwe refer to [28] where this program been carried out in detail. The same calculation can in principle be done on the mirror X * , but the large number of toric moduli again makes it highly extensive. Instead, we refer to the next section where a suitable quotient of X * will be treated in detail for which the computations are reasonably simple."
},
{
"section_type": "OTHER",
"section_title": "Quotienting the B-Model",
"text": "In this section we consider the quotient X = X/G in terms of toric geometry and study the mirror of X in this context. In order to achieve this, we first analyze the partial quotient X = X/G 1 . Using the techniques introduced in Section 3, we construct the mirror X * . Using their toric realization, we perform the B-model computation for the non-perturbative prepotentials F np X,0 and F np X * ,0 , respectively. Finally, we explain how one can implement the quotient by G 2 on both sides in order to obtain X and X * ."
},
{
"section_type": "OTHER",
"section_title": "The Quotient by G 1",
"text": "We start with a review of the general discussion of free quotients of complete intersections in toric geometry in [31] . Consider a fan Σ ⊂ N R and pick a lattice refinement N such that Γ = N/N is a finite abelian group. Such a lattice refinement consists of a finite sequence of lattice refinements of the form N → N + w p Z which are described by a vector w p = 1 kp α pi ρ i with α pi ∈ Z. The group Γ is then isomorphic to p Z kp . Let Σ be the fan obtained from Σ by relating everything to the lattice N. In this context, we make some additional identifications in the toric quotient eq. ( 19 ) [47] . One finds that VΣ = V Σ /Γ is the quotient of V Σ by the finite abelian group Γ. Its action on the homogeneous coordinates is by multiplication by phases\n\nz 1 : • • • : z n → ξ α 1 z 1 : • • • : ξ αn z n , ξ = e 2πi k , (73)\n\nfor every cyclic subgroup of order k. We will denote such group actions by Z k : (α 1 , . . . , α n ). If V Σ is a compact toric variety, then the quotient VΣ is never free [39] . However, a hypersurface or complete intersection in V Σ need not intersect the set of fixed points, and in that case we get a smooth quotient manifold with nontrivial fundamental group. We now apply this to P ∆ * = P 2 × P 1 × P 2 defined in eq. ( 22 ). The first step in performing the quotient of P ∆ * by G 1 thus amounts to a refinement N = w Z + N of the lattice N with index |G 1 | = 3. From the definition eq. (8a) of the action of G 1 on P ∆ * and eq. ( 24 ) we read off that the refinement is by a vector\n\nw ∈ 1 3 ρ 2 + 2ρ 3 + ρ 7 + 2ρ 8 + Z 5 . ( 74\n\n)\n\nThe resulting polytope ∆ * admits the same nef partition as ∆ * in eq. ( 33 ), ∇1 = ρ1 , . . . , ρ4 , 0 , ∇2 = ρ5 , . . . , ρ8 , 0 .\n\nwhere we express the generators ρ in terms of ρ as ρi = ρ i , i = 1, . . . , 6 , ρ7 = ρ 7 + e 1 + 2e 2 + e 4 + 2e 5 , ρ8 = ρ 8 -e 1 -2e 2 -e 4 -2e 5 .\n\n(76)\n\nIt is easy to check that the ρi satisfy the same linear relations eq. ( 23 ) as the ρ i , and that w = 1 3 (ρ 1 -ρ2 + ρ6 -ρ7 ) = -e 2 -e 5 . The ρi together with w therefore indeed generate the lattice N . Note that, while all 8 non-zero lattice points of ∆ * are vertices, the dual polytope ∆ has 18 vertices and 102 points. Using PALP [35] again, we compute the lattice points of the polytope ∇ * = ∆1 , ∆2 ⊂ M R , which will describe the ambient space of the mirror X * of X. We find ∆1 = ν1 , . . . , ν6 , 0 , ∆2 = ν7 , . . . , ν12 , 0 ,\n\nwhere we express the vertices νi in terms of the vertices ν i of ∇ * as ν3k+1 = ν 3k+1 , ν3k+2 = ν 3k+2 -e 5 , ν3k+3 = ν 3k+3 + e 5 , k = 0, . . . , 3.\n\nAgain, it is easy to check that the νi satisfy the same linear relations eq. ( 37 ) as the ν i . It turns out that the lattice points of ∇ * generate a sublattice M of index 3 in M, and the lattice refinement is generated by\n\nw * = 1 3 ν1 + 2ν 2 + 2ν 7 + ν8 = e 2 + e 4 -e 5 . (79)\n\nAmong the points of ∇ * listed in eq. ( 39 ) only ν 13 and ν 14 are also lattice points of the sublattice M. In fact, we have ν13 = ν 13 and ν14 = ν 14 . Hence, ∇ * has 12 vertices and 15 lattice points; its dual ∇ = ∇1 + ∇2 has 42 lattice points among which 15 are vertices 10 . Once we have the polytopes ∆ * and ∇ * , we can construct X and X * as complete intersections entirely analogous to X and X * , see Section 3. That is, using eq. ( 27 ), we define\n\nX = D0,1 ∩ D0,2 , X * = D * 0,1 ∩ D * 0,2 (81)\n\nin terms of the nef partitions eq. ( 75 ) and (77), respectively. Here, Di and D * i denote the divisors associated to the generators ρi and νi , respectively. The absence of fixed points of the G 1 action on the complete intersection X is guaranteed by the fact that the resulting polytope ∆ * ⊂ NR has no additional lattice points [31] . Hence, X = X/G 1 has a non-trivial fundamental group π 1 (X) = Z 3 . Surprisingly, it turns out that the mirror X * is a free quotient as well. To see this recall that, as noticed above, the lattice points of ∇ * generate a sublattice M of index 3 in M. Furthermore, ∇ * also has no additional lattice points with respect to ∇ * . Therefore, there is a 10 Note that all of our polytopes differ from the non-free Z 3 × Z 3 quotient of ∆ * defined in [28] , Proposition 7.1. In the notation of [31] their quotient is\n\n∇ * = P \n\n1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1\n\n 3 0 0 3 1 1 Z 3 : 0 1 2 0 0 0 0 0 Z 3 : 0 0 0 0 1 2 0 0 ( 80\n\n)\n\nand has 21 points and 8 vertices in the lattice N .\n\ngroup G * 1 ≃ Z 3 acting torically on P ∇ * . On the homogeneous coordinates this action is\n\ng * 1 : z 1 : • • • : z 12 → ζz 1 : ζ 2 z 2 : z 3 : • • • : z 6 : ζ 2 z 7 : ζz 8 : z 9 : • • • : z 12 . (82)\n\nHence, X * = X * /G * 1 also has a non-trivial fundamental group π 1 (X * ) = Z 3 . Note that this never happens for hypersurfaces in toric varieties [6] . Having the toric representation of X and X * , we can now compute their Hodge numbers. It turns out\n\nthat h 1,1 X = h 1,2 X = h 1,1 X * = h 1,2 X * = 7, (83)\n\nin agreement with Part A [1], eq. (??)."
},
{
"section_type": "OTHER",
"section_title": "The Quotient by G 2",
"text": "We now turn to the G 2 action, which does not act torically. Hence, we cannot, in principle, find a toric variety containing X = X/G 2 as we did for the G 1 quotient above. However, at least we have to ensure that X and X * are G 2 -symmetric. This can be achieved via suitable symmetries in the toric data.\n\nThe easy part of the toric data for X is the polytope ∆ * . The G 2 action on the ambient space permutes the homogeneous coordinates, see eq. (8b). In terms of toric geometry, this means that it permutes the corresponding points of the polytope. That is 11 ,\n\ng 2 : ρi → ρ1+(i mod 3) ∀i ∈ {1, 2, 3}, g 2 : ρ4 → ρ4 , ρ5 → ρ5 , g 2 : ρ5+i → ρ6+(i mod 3) ∀i ∈ {1, 2, 3}. ( 84\n\n)\n\nIt induces a mirror group action G * 2 on X * which is geometrical, rather than a quantum symmetry as discussed in [48] . The action of G * 2 is obviously the dual group action on the dual lattice M, which again must be a symmetry of the relevant polytope ∇ * . We find that\n\ng * 2 : ν3k+i → ν3k+1+(i mod 3) ∀k = 0, . . . , 3, i ∈ {1, 2, 3}. ( 85\n\n)\n\nAs a check on the mirror group action, note that the matrix of scalar products, see eq. (87) below, is invariant. That is,\n\ng 2 (ρ l ), g * 2 (ν l ′ ) = ρl , νl ′ ∀ l, l ′ . ( 86\n\n)\n\nBy abuse of notation, we denote the corresponding cyclic permutation of homogeneous coordinates by g * 2 as well. Using this action, we define the mirror of X to be X * = X * /G * 2 . This idea has already been used for the construction of mirrors of orbifolds of the quintic [49] soon after the discovery of the first mirror construction by Greene and Plesser.\n\nFollowing eq. ( 41 ), the equations for the Calabi-Yau complete intersections X and X * are defined by evaluating the matrix of scalar products ρi , νj + δ l l ′ , which are , + δ l l ′ ν1 ν2 ν3 ν4 ν5 ν6 ν13 ν7 ν8 ν9 ν10 ν11 ν12 ν14 ρ1 3 0 0 3 0 0 1 0 0 0 0 0 0 0 ρ2 0 3 0 0 3 0 1 0 0 0 0 0 0 0 ρ3 0 0 3 0 0 3 1 0 0 0 0 0 0 0 ρ4 1 1 1 0 0 0 0 0 0 0 1 1 1 1 ρ5 0 0 0 1 1 1 1 1 1 1 0 0 0 0 ρ6 0 0 0 0 0 0 0 3 0 0 3 0 0 1 ρ7 0 0 0 0 0 0 0 0 3 0 0 3 0 1 ρ8 0 0 0 0 0 0 0 0 0 3 0 0 3 1 (87)\n\nThe equations of X can now be read off from the columns of eq. ( 87 ), and one finds\n\nF 1 = (λ 5 t 0 + λ 6 t 1 )(x 3 0 + x 3 1 + x 3 2 ) + (λ 7 t 0 + λ 8 t 1 )x 0 x 1 x 2 , (88a)\n\nF 2 = (λ 1 t 0 + λ 4 t 1 )(y 3 0 + y 3 1 + y 3 2 ) + (λ 2 t 0 + λ 3 t 1 )y 0 y 1 y 2 , (88b)\n\nwhere the G 2 -symmetry has been imposed. Note that the last monomial in each equation corresponds to the vector 0 ∈ ∆l , l = 1, 2. Two of the eight coefficients λ m can be fixed by normalizing the equations, say λ 4 = λ 5 = 1, and three correspond to the symmetries of P 1 , that is, SL(2) transformations of [t 0 : t 1 ]. Hence, we can, for example, set λ 6 = λ 7 = λ 8 = 0. This leaves us with 3 complex structure deformations λ 1 , λ 2 , and λ 3 , see eqns. (7a) and (7b). The equations defining X * correspond to the rows of eq. ( 87 ), that is,\n\nF * 1 = a 1 (z 3 1 z 3 4 + z 3 2 z 3 5 + z 3 3 z 3\n\n6 )z 13 + (a 2 z 10 z 11 z 12 z 14 + a 3 z 4 z 5 z 6 z 13 )z 1 z 2 z 3 , (89a) F * 2 = a 4 (z 3 7 z 3 10 + z 3 8 z 3 11 + z 3 9 z 3 12 )z 14 + (a 5 z 4 z 5 z 6 z 13 + a 6 z 10 z 11 z 12 z 14 )z 7 z 8 z 9 ,\n\nwhere, again, invariance under G * 2 has been imposed and the last monomial of each equation comes from the lattice point 0 ∈ ∇l , , l = 1, 2. Both equations are homogeneous with respect to all seven scaling degrees that follow from the linear relations eq. ( 37 ). Among the twelve scalings of the coordinates z i , six are compatible with the cyclic permutations g * 2 , see eq. (85). Subtracting the three G 2 symmetric independent scalings among the relations eq. ( 37 ), there remains one torus action that acts effectively on the parameters plus two normalizations of the equations. As expected, the six parameters a m of the equations of X * thus become the 3 complex structure moduli.\n\nSo far, we only considered the polytopes ∆ * and ∇ * . However, this is only part of the toric data defining the manifolds X and X * , respectively. In addition, we need the triangulations and the corresponding exceptional sets. A change in the triangulation corresponds to a flop of the toric variety. The very real danger is that not all, and perhaps none, of the flopped Calabi-Yau manifolds are G 2 -symmetric. For X ⊂ P ∆ * this turns out to be unproblematic, but for X * ⊂ P ∇ * we will find a condition for the choice of a triangulation."
},
{
"section_type": "OTHER",
"section_title": "B-Model on X",
"text": "We now return to the discussion of the triangulations and the intersection ring of X.\n\nThe analogous, but technically much more involved discussion of X * will be presented in Subsection 4.5. For X everything is straightforward since the G 1 -quotient did not introduce additional lattice points in the associated polytope ∆ * . Therefore, just like for the polytope ∆ * of the covering space X, there exists a unique triangulation. In particular the primitive collections, the Stanley-Reisner ideal, and the ideal I X are identical to the ones in eqns. ( 42 ), (49) , and (50) since they are derived from the same triangulation. Moreover, one can easily see that this triangulation is G 2 -invariant and, hence,\n\nX is G 2 symmetric.\n\nThe only change is in the normalization of the intersection ring in eq. ( 52 ), since the total volume has to be divided by 3 = |G 1 |. This can also be seen in eq. ( 76 ), where the volume of the cone is now 3 instead of 1. Hence, on X the intersection ring and the second Chern class are\n\nJ2 2 J3 = 1, J1 J2 J3 = 3, J2 J2 3 = 1, c 2 X • J1 = 0, c 2 X • J2 = 12, c 2 X • J3 = 12. ( 90\n\n)\n\nComparing these intersection numbers with eq. ( 13 ), it is clear that the toric divisors should be identified with the G 1 -invariant divisors on X as\n\nJ1 = φ, J2 = τ 1 , J3 = τ 2 . ( 91\n\n)\n\nThe curves spanning the Mori cone on the cover turn out to be G 1 -invariant as well. Therefore, the Mori cones NE(P ∆ * ) and NE(X) toric are identical to those in eqns. ( 60 ) and (61), respectively. Following the steps given in Section 3 we now want to compute the B-model prepotential F B X * ,0 , plug in the mirror map, and obtain the prepotential on X\n\nF np X,0 (P, Q 1 , Q 2 , Q 3 , R 1 , R 2 , R 3 , b 1 ). ( 92\n\n)\n\nWe immediately realize the following two caveats:\n\n• We do not know how to incorporate the torsion curves H 2 (X, Z) tors = Z 3 into the toric mirror symmetry calculation.\n\n• Of the 7 Kähler classes on X, only 3 are toric.\n\nThis means that only 3 out of the 7 + 1 variables in the prepotential are accessible, and the remaining ones are set to one. Looking at the intersection numbers eq. ( 90 ), it is clear that the 3 divisors are precisely the G 2 -invariant divisors on X, see eq. ( 13 ). Therefore, these 3 variables must be those that map to the variables p, q, and r on X. By comparing with eq. ( 16 ), we see that the corresponding variables on X are P , Q 1 , and R 1 . Hence, we actually only compute\n\nF np X,0 (P, Q 1 , 1, 1, R 1 , 1, 1, 1) = n 1 ,n 2 ,n 3 n X (n 1 ,n 2 ,n 3 ) Li 3 P n 1 Q n 2 1 R n 3 1 . (93)\n\nIn effect, this means that the resulting instanton numbers are not just the instantons in a single integral homology class, but the instanton numbers in a whole set of integral homology classes. The instanton numbers sum over all curve classes that cannot be distinguished by P, Q 1 , R 1 ∈ Hom H 2 (X, Z), C × . Up to total degree 4 and the symmetry n\n\nX (n 1 ,n 2 ,n 3 ) = n X (n 1 ,n 3 ,n 2 ) , (94)\n\nthe resulting instanton numbers are\n\nn X (1,0,0) = 27 n X (1,0,1) = 108 n X (1,0,2) = 378 n X (1,0,3) = 1080 n X (1,1,1) = 432 n X (1,1,2) = 1512 n X (2,0,1) = -54 n X (2,0,2) = -756 n X\n\n(2,1,1) = 864 n X (3,0,1) = 9.\n\n(95)"
},
{
"section_type": "OTHER",
"section_title": "Instanton Numbers of X",
"text": "Knowing the prepotential on X, we now want to divide out the free G 2 action and arrive at the prepotential on X. Since we do not know the complete expansion but only eq. ( 93 ), we have to set b 1 = b 2 = 1 in the descent equation ( 16 ). This yields\n\nF np X,0 p, q, r, 1, 1) = 1 3 F np X,0 p, q, 1, 1, r, 1, 1, 1 = n 1 ,n 2 ,n 3 n X (n 1 ,n 2 ,n 3 ) Li 3 p n 1 q n 2 r n 3 . (96)\n\nUp to the symmetry n X (n 1 ,n 2 ,n 3 ) = n X (n 1 ,n 3 ,n 2 ) , the non-vanishing instanton numbers for X up to total degree 5 are\n\nn X (1,0,0) = 9 n X (1,0,1) = 36 n X (1,0,2) = 126 n X (1,0,3) = 360 n X (1,0,4) = 945 n X (1,1,1) = 144 n X (1,1,2) = 504 n X (1,1,3) = 1440 n X (1,2,2) = 1764 n X (2,0,1) = -18 n X (2,0,2) = -252 n X (2,0,3) = -1728 n X (2,1,1) = 288 n X (2,1,2) = 3960 n X (3,0,1) = 3 n X (3,0,2) = 252 n X (3,1,1) = 756, (97)\n\nUnfortunately, this direct calculation misses the torsion information and only yields the expansion F np X,0 (p, q, r, 1, 1). The b 1 dependence was lost because the toric methods do not yield this part, and the b 2 dependence was lost because the relevant divisor on X was not toric. Comparing with the full expansion of the prepotential\n\nF np X,0 p, q, r, b 1 , b 2 ) = n 1 ,n 2 ,n 3 m 1 ,m 2 n X (n 1 ,n 2 ,n 3 ,m 1 ,m 2 ) Li 3 p n 1 q n 2 r n 3 b m 1 1 b m 2 2 , (98)\n\nsee Part A eq. (??), this means we only obtain the sum of the instanton numbers over all torsion classes\n\nn X (n 1 ,n 2 ,n 3 ) = 2 m 1 ,m 2 =0 n X (n 1 ,n 2 ,n 3 ,m 1 ,m 2 ) . (99)\n\nClearly, this destroys the torsion information, that is, the instanton numbers n X\n\n(n 1 ,n 2 ,n 3 )\n\ndo not depend on the torsion part of the integral homology. For comparison purposes, we list the instanton numbers n X (n 1 ,n 2 ,n 3 ) for 0 ≤ n 1 , n 2 , n 3 ≤ 5 in Table 2 ."
},
{
"section_type": "OTHER",
"section_title": "B-Model on X",
"text": "* We now study the mirror X * , which sits in a more complicated ambient toric variety.\n\nConsequently, the analysis is more involved. The big advantage, however, will turn out to be that all h 11 (X * ) = 7 Kähler moduli are toric, which will enable us to obtain the full instanton expansion. Since the polytope ∇ * in eq. ( 78 ) is not simplicial, we have to specify a resolution of the singularities, that is, a triangulation T ( ∇ * ). Moreover, not any triangulation will do, but we have to make sure that it is compatible with the action of the permutation group G * 2 . While a tedious technicality, the existence of such a resolution has to be shown in order to establish the existence of a geometrical mirror family of X.\n\nIn particular, we show in Appendix A that there is no projective resolution of the ambient space among the 720 coherent star triangulations of ∇ * that respects the permutation symmetry eq. ( 85 ). In other words, if one demands G * 2 symmetry then the ambient toric variety cannot be chosen to be Kähler, but only a complex manifold. Clearly, in that case there is no Kähler cone and the usual toric mirror symmetry algorithm does not work. What comes to the rescue is that there are two classes of non-symmetric projective resolutions for which the symmetry-violating exceptional sets do not intersect X * . Hence the complete intersection is G 2 -symmetric, even though the ambient space is not. We conclude that the extended Kähler moduli space of X * contains two symmetric phases. We will denote these two classes of triangulations by T ± = T ± ( ∇ * ), see Appendix A. In fact, the two phases are topologically distinct, and only the triangulation T + describes the threefold X * that we are interested in. In Appendix B, we will investigate the other triangulation T -which describes a flop of X * .\n\nn X (0,n 2 ,n 3 ) n X (3,n 2 ,n 3 ) ❅ ❅ ❅\n\nn 2 n 3 0 1 2 3 4 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 0 4 0 0 0 0 0 0 5 0 0 0 0 0 0\n\n❅ ❅ ❅ n 2 n 3 0 1 2 3\n\n4 5 0 0 3 252 4158 40173 287415 1 3 756 15390 164280 1259685 7763364 2 252 15390 426708 5427684 46537092 310465062 3 4158 164280 5427684 73971360 657552966 4487097816 4 40173 1259685 46537092 657552966 5948103483 41016575313 5 287415 7763364 310465062 4487097816 41016575313 284581389204 n X (1,n 2 ,n 3 ) n X (4,n 2 ,n 3 ) ❅ ❅ ❅ n 2 n 3 0 1 2 3 4 5 0 9 36 126 360 945 2268 1 36 144 504 1440 3780 9072 2 126 504 1764 5040 13230 31752 3 360 1440 5040 14400 37800 90720 4 945 3780 13230 37800 99225 238140 5 2268 9072 31752 90720 238140 571536 ❅ ❅ ❅ n 2 n 3 0 1 2 3 4 5 0 0 0 -144 -6048 -107280 -1235520 1 0 -306 -12348 -207000 -2273400 -19066500 2 -144 -12348 348480 14609520 235219680 2505155400 3 -6048 -207000 14609520 520226784 8245864800 87989812560 4 -107280 -2273400 235219680 8245864800 131759049600 1417949658000 5 -1235520 -19066500 2505155400 87989812560 1417949658000 15365394415800 n X (2,n 2 ,n 3 ) n X (5,n 2 ,n 3 ) ❅ ❅ ❅ n 2 n 3 0 1 2 3 4 5 0 0 -18 -252 -1728 -9000 -38808 1 -18 288 3960 27648 143748 620928 2 -252 3960 54432 380160 1976472 8537760 3 -1728 27648 380160 2654208 13799808 59609088 4 -9000 143748 1976472 13799808 71748000 309920688 5 -38808 620928 8537760 59609088 309920688 1338720768 ❅ ❅ ❅ n 2 n 3 0 1 2 3 4 5 0 0 0 45 5670 189990 3508920 1 0 36 13140 474840 8793648 111499020 2 45 13140 1112886 38961252 777759975 10723515300 3 5670 474840 38961252 1952428464 47357606430 732897531720\n\n4 189990 8793648 777759975 47357606430 1237373786439 19911043749420 5 3508920 111499020 10723515300 732897531720 19911043749420 327006066948660\n\nTable 2: Summed instanton numbers n X (n 1 ,n 2 ,n 3 ) = m 1 ,m 2 n X (n 1 ,n 2 ,n 3 ,m 1 ,m 2 )\n\n(hence not distinguishing torsion) computed by mirror symmetry. The table contains all non-vanishing instanton numbers for 0 ≤ n 1 , n 2 , n 3 ≤ 6. where we dropped the superscript * on D for ease of notation. From this, in turn, we obtain the generators l(a) + of the Mori cone NE(P ∇ * ):\n\nl(1) + =( 0, 0, 0, 0, 0, 0, 1, 0, 0,-1, 0, 0, 0, 1) l(2) + =( 1, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) l(3) + =(-1, 1, 0, 1,-1, 0, 0, 0, 0, 0, 0, 0, 0, 0) l(4) + =( 0,-1, 1, 0, 1,-1, 0, 0, 0, 0, 0, 0, 0, 0) l( 5 ) + =( 0, 0,-1, 0, 0, 1, 0,-1, 0, 0, 1, 0, 0, 0) l( 6 ) + =( 0, 0, 0, 0, 0, 0,-1, 0, 1, 1, 0,-1, 0, 0) l( 7 ) + =( 0, 0, 0, 0, 0, 0, 0, 1,-1, 0,-1, 1, 0, 0) l(8) + =( 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0,-3, 0) l(9) + =( 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0,-3).\n\n(101)\n\nA dual basis for the generators of the Kähler cone K(P ∇ * ) is\n\nK1 = D13 + 2 D1 -D2 -D3 + D9 + D7 + D8 + 3 D4 , K2 = 3 D1 + D13 + 3 D4 , K3 = D13 + 2 D1 + 3 D4 , K4 = D13 + 2 D1 -D2 + 3 D4 , K5 = D13 + 2 D1 -D2 -D3 + 3 D4 , K6 = D13 + 2 D1 -D2 -D3 + D9 + D8 + 3 D4 , K7 = D8 + D13 + 2 D1 -D2 -D3 + 3 D4 , K8 = D4 + D1 , K9 = D10 + D7 . ( 102\n\n)\n\nThe Calabi-Yau complete intersection X * is then defined by X * = K1 K2 . It turns out that the divisors D13 , D14 do not intersect X * . Therefore, all\n\nh 1,1 toric X * = h 1,1 X * = 7 ( 103\n\n)\n\nKähler moduli are realized torically. Since there are two divisors that do not intersect, finding the Mori cone is somewhat subtle. First, we have to restrict the lattice of linear relations to the sublattice orthogonal to these two directions. For the generators of the toric Mori cone NE(X * ) toric , this means that l(1)\n\n+ → 3 l(1) + + l(9) + , l(2) + → 3 l(2) + + l (8)\n\n+ and that we drop l(8) + , l(9) + as well as the entries corresponding to intersections with D13 , D14 . In addition, we prepend the intersection numbers with D0,1 and D0,2 . This yields l(1) + =(-3, 0; 0, 0, 0, 0, 0, 0, 3, 0, 0,-2, 1, 1) l( 2 ) + =( 0,-3; 3, 0, 0,-2, 1, 1, 0, 0, 0, 0, 0, 0) l(3) + =( 0, 0;-1, 1, 0, 1,-1, 0, 0, 0, 0, 0, 0, 0) l(4) + =( 0, 0; 0,-1, 1, 0, 1,-1, 0, 0, 0, 0, 0, 0) l( 5 ) + =( 0, 0; 0, 0,-1, 0, 0, 1, 0,-1, 0, 0, 1, 0) l( 6 ) + =( 0, 0; 0, 0, 0, 0, 0, 0,-1, 0, 1, 1, 0,-1) l( 7 ) + =( 0, 0; 0, 0, 0, 0, 0, 0, 0, 1,-1, 0,-1, 1).\n\n(\n\n) 104\n\nThe dual basis of divisors is J *\n\n1 = 1 3 K2 1 K2 , J * 2 = 1 3 K1 K2 2 , J * 5 = K1 K2 K5 , J * 3 = K1 K2 K3 , J * 4 = K1 K2 K4 , J * 6 = K1 K2 K6 , J * 7 = K1 K2 K7 . (105)\n\nWe now try to identify this basis J * 1 , . . . , J * 7 of divisors on X * with the basis {φ, τ 1 , υ 1 , ψ 1 , τ 2 , υ 2 , ψ 2 } of divisors on X in eq. (10) . It turns out that there is more than one way to identify the bases if one only wants to preserve the triple intersection numbers. To obtain a unique answer, we also need to identify the actions by G * 2 and G 2 as well. First, the G * 2 action on H 2 (P ∇ * , Z) is defined by eq. ( 85 ). Using the linear equivalence relations\n\n2 D1 -D2 -D3 + 2 D4 -D5 -D6 = 0 -D1 + 2 D2 -D3 -D4 + 2 D5 -D6 = 0 2 D7 -D8 -D9 + 2 D10 -D11 -D12 = 0 -D2 + D3 -D5 + D6 -D8 + D9 -D11 + D12 = 0 -D4 -D5 -D6 + D10 + D11 + D12 -D13 + D14 = 0 (106)\n\nand the definition eq. ( 105 ), one can compute the induced group action on H 2 (X * , Z).\n\nWe find\n\ng * 2 J * 1 J * 2 J * 3 J * 4 J * 5 J * 6 J * 7 = \n\n1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 -1 1 0 0 0 0 3 -1 0 1 0 0 0 0 0 0 1 0 0 3 0 0 0 1 0 -1\n\n0 0 0 0 1 1 -1 J * 1 J * 2 J * 3 J * 4 J * 5 J * 6 J * 7 . ( 107\n\n)\n\nSecond, recall that the G 2 action on the divisors of X * is\n\ng 2 φ τ 1 υ 1 ψ 1 τ 2 υ 2 ψ 2 = \n\n1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 3 0 -1 0 0 0 0 3 1 -1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 3 0 -1\n\n0 0 0 0 3 1 -1 φ τ 1 υ 1 ψ 1 τ 2 υ 2 ψ 2 , ( 108\n\n)\n\nsee also Part A eq. (??). The instanton numbers on X * are the expansion coefficients\n\nF np X * ,0 (P * , Q * 1 , Q * 2 , Q * 3 , R * 1 , R * 2 , R * 3 , 1) = n 1 ,...,n 7 n X * (n 1 ,n 2 ,n 3 ,n 4 ,n 5 ,n 6 ,n 7 ) Li 3 P * n 1 Q * n 2 1 Q * n 3 2 Q * n 4 3 R * n 5 1 R * n 6 2 R * n 7 3 . (113)\n\nWe see that we almost get the complete instanton expansion eq. ( 15 ), we only miss the expansion in the b * 1 variable which is not computed by the toric mirror symmetry algorithm. Up to total degree 5, the instanton numbers are\n\nn X * (1,0,0,0,0,0,0) =3 n X * (1,0,0,0,0,1,0) =3 n X * (1,0,0,0,0,1,1) =3 n X * (1,0,1,0,0,0,0) =3 n X * (1,0,1,0,0,1,0) =3 n X * (1,0,1,0,0,1,1) =3 n X * (1,0,1,1,0,0,0) =3 n X * (1,0,1,1,0,1,0) =3 n X * (1,0,1,1,0,1,1) =3 n X * (1,1,0,0,0,1,2) =9 n X * (1,0,1,2,1,0,0) =9 n X * (1,0,1,1,1,1,0) =3 n X * (1,1,1,0,0,1,1) =3 n X * (1,1,0,0,0,1,1) =3 n X * (1,0,1,1,1,0,0) =3. ( 114\n\n)\n\nFinally, let us take a look at the G * 2 action, see eq. ( 85 ). Of the 7 generators of the toric Mori cone, eq. ( 104 ), only the 3 generators l(1)\n\n+ , l (2)\n\n+ and l( 5 )\n\n+ are invariant. Not surprisingly, the dual G * 2 -invariant divisors J * 5 = φ, J * 1 = τ 1 , J * 2 = τ 2 (115)\n\nwere identified with the G 2 -invariant divisors on X in eq. ( 109 ). Therefore, only 3 Kähler parameters survive to the quotient X * = X * /G * 2 , and we have\n\nh 1,1 X = h 1,2 X = h 1,1 X * = h 1,2 X * = 3. ( 116\n\n)"
},
{
"section_type": "OTHER",
"section_title": "Instanton Numbers of X *",
"text": "Now that we have the expression eq. ( 113 ) for the prepotential on X * , we can again apply a suitable variable substitution\n\nP * , Q * 1 , Q * 2 , Q * 3 , R * 1 , R * 2 , R * 3 , b * 1 , b * 2 -→ p * , q * , r * , b * 1 , b * 2 (117)\n\nand obtain the prepotential on the quotient\n\nX * = X * /G * 2 .\n\nThe correct way to replace the variables is determined by the group action on the homology and cohomology as we explained in Part A. Having computed the G * 2 -action in eq. ( 107 ), we determine the descent equation for the prepotential to be 13\n\nF np X * ,0 p * , q * , r * , b * 1 , b * 2 ) = 1 |G * 2 | F np X * ,0 p * , q * , b * 2 , b * 2 , r * , b * 2 2 , b * 2 2 , b * 1 . ( 118\n\n)\n\n13 Interestingly, eq. ( 118 ) turns out to be exactly analogous to eq. ( 16 ), even though the identification of divisors on X * and X is not just a relabeling of divisors.\n\nUsing the series expansion of the prepotential for b * 1 = 1 on X * from Subsection 4.5, we now find that\n\nF np X * ,0 (p * , q * , r * , 1, b * 2 ) = 2 j=0 3 × Li 3 (p * b * j 2 ) + 4 Li 3 (p * q * b * j 2 ) + 4 Li 3 (p * r * b * j 2 ) + 14 Li 3 (p * q * 2 b * j 2 ) + 16 Li 3 (p * q * r * b * j 2 ) + 14 Li 3 (p * r * 2 b * j 2 ) + 40 Li 3 (p * q * 3 b * j 2 ) + 56 Li 3 (p * q * 2 rb * j 2 ) + 56 Li 3 (p * q * r 2 b * j 2 ) + 40 Li 3 (p * r * 3 b * j 2 ) + 105 Li 3 (p * q * 4 b * j 2 ) + 160 Li 3 (p * q * 3 r * b * j 2 ) -2 Li 3 (p * 2 q * b * j 2 ) -2 Li 3 (p * 2 r * b * j 2 ) -28 Li 3 (p * 2 q * 2 b * j 2 ) + 32 Li 3 (p * 2 q * r * b * j 2 ) -28 Li 3 (p * 2 r * 2 b * j 2 ) + 3 Li 3 (p * 3 q * ) + 3 Li 3 (p * 3 r * ) + total p * , q * , r * -degree ≥ 5 . (119)\n\nThe corresponding instanton numbers\n\nF np X * ,0 (p * , q * , r * , 1, b * 2 ) = n 1 ,n 2 ,n 3 ,m 2 n X * (n 1 ,n 2 ,n 3 ,m 2 ) Li 3 p * n 1 q * n 2 r * n 3 b * m 2 2 (120)\n\nare listed in Table 3 . For comparison purposes, we list the summed instanton numbers\n\n(n 1 , n 2 , n 3 ) n X * (n 1 ,n 2 ,n 3 ,0) n X * (n 1 ,n 2 ,n 3 ,1) n X * (n 1 ,n 2 ,n 3 ,2) n X (n 1 ,n 2 ,n 3 )\n\n(1, 0, 0)\n\n3 3 3 9 (1, 0, 1) 12 12 12 36 (1, 0, 2) 42 42 42 126 (1, 0, 3) 120 120 120 360 (1, 1, 1) 48 48 48 144 (1, 1, 2) 168 168 168 504 (2, 0, 1) -6 -6 -6 -18 (2, 0, 2) -84 -84 -84 -252 (2, 1, 1) 96 96 96 288 (3, 0, 1) 3 0 0 3\n\nTable\n\n3: Instanton numbers n X * (n 1 ,n 2 ,n 3 ,m 2 )\n\ncomputed by toric mirror symmetry. They are invariant under the exchange n 2 ↔ n 3 , so we only display them for n 2 ≤ n 3 . on X as well, see eq. ( 99 ). One observes that the sum over the more refined instanton numbers on X * equals the summed instanton number on X, another clue towards X being self-mirror."
},
{
"section_type": "OTHER",
"section_title": "Instanton Numbers Assuming The Self-Mirror Property",
"text": "So far, we have alluded to X being possibly self-mirror, but not actually made use of this property. Now we are going to assume the self-mirror property and, hence, obtain the prepotential on X as\n\nF np X,0 (p, q, r, b 1 , b 2 ) = F np X * ,0 (p, q, r, b 1 , b 2 ). (121)\n\nNote that at linear and quadratic order in p we can actually recover the b 1 , b 2 expansion from the summed instanton numbers in Subsection 4.4 and the factorization which we will prove in Section 6.\n\nIn contrast, for the prepotential terms at order p 3 we have to use the X * prepotential to obtain the b 2 expansion from eq. ( 119 ). Since this is based on a toric computation on X * , we do not directly obtain the b 1 expansion. However, note that the fact that g 1 acted torically, eq. (8a), and g 2 non-torically, eq. (8b), is just a consequence of the choice of coordinate system on P 2 × P 1 × P 2 . By a suitable coordinate choice, we could have made any one of the four Z 3 subgroups of G = Z 3 × Z 3 act torically. Therefore, any combination of b 1 , b 2 other than 1 = b 0 1 b 0 2 has to occur in the same way in the complete series expansion of the prepotential. We conclude that the prepotential can only depend on b 1 and b 2 through the combinations 1,\n\n2 i,j=0 b i 1 b j 2 . (122)\n\nThis observation lets us recover the full b 1 , b 2 expansion of the prepotential. To\n\nn X (1,n 2 ,n 3 ,0,0) n X (1,n 2 ,n 3 ,m 1 ,m 2 ) , (m 1 , m 2 ) = (0, 0) ❅ ❅ ❅ n 2 n 3 0 1 2 3\n\n4 0 1 4 14 40 105 1 4 16 56 160 2 14 56 196 3 40 160 4 105 ❅ ❅ ❅ n 2 n 3 0 1 2 3 4 0 1 4 14 40 105 1 4 16 56 160 2 14 56 196 3 40 160 4 105\n\nn X (2,n 2 ,n 3 ,0,0) n X (2,n 2 ,n 3 ,m 1 ,m 2 ) , (m 1 , m 2 ) = (0, 0) ❅ ❅ ❅ n 2 n 3 0 1 2 3 0 0 -2 -28 -192 1 -2 32 440 2 -28 440 3 -192 ❅ ❅ ❅ n 2 n 3 0 1 2 3 0 0 -2 -28 -192 1 -2 440 2 -28 440 3 -192 n X (3,n 2 ,n 3 ,0,0) n X (3,n 2 ,n 3 ,m 1 ,m 2 ) , (m 1 , m 2 ) = (0, 0) ❅ ❅ ❅ n 2 n 3 0 1 2 0 0 3 36\n\n1 3 108\n\n2 36 ❅ ❅ ❅ n 2 n 3 1 2 0 0 27\n\n1 81 2 27 Table 4: Instanton numbers n X (n 1 ,n 2 ,n 3 ,m 1 ,m 2 )\n\ncomputed by mirror symmetry. The table contains all non-vanishing instanton numbers for n 1 + n 2 + n 3 ≤ 5. The entries marked in bold depend non-trivially on the torsion part of their respective homology class.\n\nan integral lattice structure and form a ring, and therefore have a product. Because of Poincaré duality, that is,\n\nH 2 (Y ) = H 4 (Y ) ∨ , it is sufficient to look at H 2 (Y ). There is a product H 2 (Y ) × H 2 (Y ) → H 2 (Y )\n\nwhose structure constants κ ijk are the triple intersection numbers. These intersection numbers are finer invariants than just the dimensions of the cohomology groups, and a self-mirror Calabi-Yau threefold should satisfy\n\nκ ijk (Y ) = κ ijk (Y * ). (125)\n\nFor simply connected threefolds with torsion-free homology a theorem of Wall [41] states that the cohomology groups with the intersection product κ ijk (Y ) together with the second Chern class c 2 (Y ) determine the diffeomorphism type of Y .\n\nIf, however, Y and Y * have non-trivial fundamental groups then we cannot conclude that easily that they are diffeomorphic. But the non-trivial fundamental group is often reflected in torsion in homology (for example if π 1 (Y ) is Abelian). In that case, the conjecture of [6] says that for any Calabi-Yau threefold Z\n\nH 3 Z, Z tors ≃ H 2 Z * , Z tors , H 2 Z, Z tors ≃ H 3 Z * , Z tors . (126)\n\nTherefore, a self-mirror manifold Y = Y * is expected to satisfy\n\nH 2 Y, Z tors ≃ H 3 Y, Z tors . (127)\n\nOf the many spaces Y satisfying eq. ( 124 ) there are only a few which also satisfy eq. ( 125 ). So far we only considered classical topology, but we know that the ring H 2 (Y ) experiences quantum corrections when going far away from the large volume limit. At small volume the intersection numbers are replaced by the three-point functions C ijk (q) of (topological) conformal field theory in eq. ( 69 ). In the large volume limit q goes to zero and the C ijk (q) go to κ ijk , as expected. The C ijk (q) are characterized by the genus zero instanton numbers n\n\nd (Y ) = n (0) d (Y * ). ( 128\n\n)\n\nOne can go even further and couple the topological conformal field theory to topological gravity and define higher genus instanton numbers n\n\n(g) d , where now n (g)\n\nd (Y ) = n (g) d (Y * ), g > 0 (129)\n\nhas to hold. These invariants are very difficult to compute, however see [52, 53] for recent progress. We do not know whether they contain more information about the symplectic structure than the genus zero invariants. In other words, there are presently no examples known whose n (g) d agree for g = 0 but differ for g > 0. Now, one can start with any Y and use some method to construct the mirror Y * . Among these are the Greene-Plesser construction in conformal field theory, or its geometric generalizations by Batyrev and Borisov for complete intersections in toric varieties. Then, to show that Y is self-mirror one proceeds to compute the various invariants. The simplest condition, eq. ( 124 ), can directly be checked in terms of the toric data. This concretely means that one starts with a mirror pair Y and Y * satisfying eq. ( 124 ) and checks whether eqns. ( 125 ), ( 127 ), (128), and (129) are satisfied. In fact, in Section 4 we collected a large amount of evidence in favor of the claim that X and its Batyrev-Borisov mirror threefold X * are the same. Indeed, eqns. ( 40 ), ( 83 ) and (116) show that X, X, and X satisfy by construction the constraint eq. ( 124 ) on the Euler number. More interestingly, by the identifications found in eqns. ( 109 ) and ( 115 ) we observed that the condition on the intersection ring, eq. ( 125 ), is satisfied for X and X, respectively. Next, eq. ( 97 ) and Table 3 show that X also fulfils the requirement eq. ( 128 ) on the genus zero instanton numbers. It would be very interesting to see whether also the condition eq. ( 129 ) for higher genus curves can be met.\n\nFinally, we consider the torsion in cohomology. In Part A ?? we have shown that\n\nH 3 X, Z tors ≃ H 2 X, Z tors ≃ Z 3 ⊕ Z 3 , (130)\n\nas we expect from a self-mirror threefold. Moreover, we can actually compute the fundamental group of the Batyrev-Borisov mirror independently. For that, first notice that the quotient X * = X * /G * 1 is fixed-point free, see Subsection 4.2. The mirror permutation G * 2 on X * acts freely as well. Therefore, both X and X * are free quotients by a group isomorphic to Z 3 ⊕ Z 3 , thus their fundamental groups are\n\nπ 1 X ≃ π 1 X * ≃ Z 3 ⊕ Z 3 . (131)\n\nMoreover, on can easily show that on a proper foot_8 Calabi-Yau threefold Z one has H 2 (Z, Z) tors = π 1 (Z) ab , the Abelianization of the fundamental group. Hence, we see that\n\nH 3 (X, Z) tors ≃ Z 3 ⊕ Z 3 ≃ H 2 (X * , Z) tors (132)\n\nand the first of eq. ( 126 ) is true. This provides the first evidence for the conjecture of [6] in a context other than toric hypersurfaces. Another point of view is that there is a geometrical or rather combinatorial reason for the self-mirror property in this case. From eqns. (36) and (39) one can easily see that the lattice points ν i , ν 6+i , ν 13 , ν 14 , i = 1, . . . , 3, span a sub-polytope of ∇ * satisfying the same linear relations as all the lattice points ρ i of ∆ * in eq. (23) . Hence, this sub-polytope is isomorphic to ∆ * . The same is true for the polytopes ∇ * and ∆ * . The toric variety P ∇ * which is the ambient space of X * can therefore be regarded as a blow-up of a quotient of P ∆ * , the ambient space of X. Actually, this blow-up makes all 7 divisors of X * toric. Similarly, P ∇ * can be regarded as a blow-up of a quotient of P ∆ * . As shown in Subsection 3.3 this entails that all 19 Kähler moduli of X * are realized torically. Note that it is possible that the mirror polytopes ∆ * and ∇ * are actually isomorphic. In fact, for toric hypersurfaces there are 41, 710 self-dual polytopes [54] . The novel feature in our case is that non-isomorphic polytopes lead to self-mirror complete intersections, consistent with the nef partitions.\n\n6 Factorization vs. The (3, 1, 0, 0, 0) Curve\n\nOne interesting observation is that the prepotential F np X,0 at order p, see eq. ( 123 ) in this paper and eq. (??) in Part A [1], factors into 2 i,j=0 b i 1 b j 2 times a function of p, q, r only. This means that the instanton number for any pseudo-section (curve contributing at order p) does not depend on the torsion part of its homology class. In other words, for any pseudo-section there are 8 other pseudo-sections with the same class in H 2 (X, Z) free and together filling up all of H 2 (X, Z) tors = Z 3 ⊕ Z 3 . In contrast, this factorization does not hold at order p 3 . For example,\n\nF np X,0 (p, q, r, b 1 , b 2 ) = • • • + 3p 3 q + 0 b 1 + b 2 1 + b 2 + b 1 b 2 + b 2 1 b 2 + b 2 2 + b 1 b 2 2 + b 2 1 b 2 2 p 3 q + • • • . (133)\n\nThe purpose of this subsection is to understand this behavior.\n\nFirst, the factorization of the prepotential at any order of p not divisible by 3 follows from an extra symmetry that we have not utilized so far. The covering space X is, in addition to eqns. (8a) and (8b), also invariant under another Ĝ = Z 3 × Z 3 action generated by (ζ def = e 2πi 3 ) ĝ1 :\n\n [x 0 : x 1 : x 2 ] → [x 0 : ζx 1 : ζ 2 x 2 ] [t 0 : t 1 ] → [t 0 : t 1 ] (no action) [y 0 : y 1 : y 2 ] → [y 0 : y 1 : y 2 ] (no action) (134a) and ĝ2 : [x 0 : x 1 : x 2 ] → [x 1 : x 2 : x 0 ] [t 0 : t 1 ] → [t 0 : t 1 ] (no action) [y 0 : y 1 : y 2 ] → [y 0 : y 1 : y 2 ] (no action) (134b)\n\nThis symmetry has fixed points and, therefore, cannot be used if one is looking for a smooth quotient of X. However, it commutes with G and hence descends to a Ĝ = Z 3 × Z 3 symmetry of X (with fixed points). Clearly, the instanton sum must observe this additional geometric symmetry. To make use of this symmetry, we have to express its action on the variables in F np X,0 (p, q, r, b 1 , b 2 ). We can do so by first noting that the basic 81 curves\n\ns 1 ×s 2 ⊂ X, s 1 ∈ MW (B 1 ), s 2 ∈ MW (B 2 ) ( 135\n\n)\n\nare really one orbit under G × Ĝ. Recall that, after dividing out G, these curves became the 9 sections in MW (X) = Z 3 ⊕ Z 3 , see Part A ??. We now observe that MW (X) = {s ij } is one Ĝ-orbit; since each of these sections contributes pb i 1 b j 2 , i, j = 0, . . . , 2 the induced Ĝ action on the prepotential must be ĝ1 :\n\nF np X,0 (p, q, r, b 1 , b 2 ) → F np X,0 (b 1 p, q, r, b 1 , b 2 ), ĝ2 : F np X,0 (p, q, r, b 1 , b 2 ) → F np X,0 (b 2 p, q, r, b 1 , b 2 ). (136)\n\nClearly, the prepotential must be invariant under the ĝ1 , ĝ2 action. While imposing no constraint on the p 3n terms in the prepotential, all other powers of p must appear in the combination\n\np n 2 i,j=0 b i 1 b j 2 , n ≡ 0 mod 3. (137)\n\nThis proves the factorization observed at the beginning of this subsection. Second, we would like to understand the p 3 q terms in eq. ( 133 ). These are the curves in the homology classes 15 (3, 1, 0, * , * ) ∈\n\nZ 3 ⊕ Z 3 ⊕ Z 3 = H 2 X, Z . (138)\n\nWe will show that the rational curves in this class come in a single family, that is, the moduli space of genus 0 curves on X in these homology classes\n\nM 0 X, (3, 1, 0, * , * ) (139)\n\nis connected. In particular, all such curves have the same homology class (3, 1, 0, 0, 0) and only contribute to p 3 q in the prepotential eq. ( 133 ). As discussed in Part A ??, any such map C X : P 1 → X factors\n\nP 1 C X / / C\n\ne X ? ? ? ? ? ? ? ? X X q ? ?\n\n.\n\n(140) 15 Recall that the exponent of p is the degree along the base P 1 . This is why we pick a basis in H 2 (X, Z) free such that a curve in (n 1 , n 2 , n 3 , m 1 , m 2 ) contributes at order p n1 q n2 r n3 b m1 1 b m2 2 in the prepotential.\n\nThe map C e X can be written in terms of homogeneous coordinates as a function\n\nC e X : P 1 [z 0 :z 1 ] → P 2 [x 0 :x 1 :x 2 ] × P 1 [t 0 :t 1 ] × P 2 [y 0 :y 1 :y 2 ] (141)\n\nsatisfying the equations (7a) and (7b) defining X,\n\nF 1 • C e X [z 0 : z 1 ] = 0 = F 2 • C e X [z 0 : z 1 ] ∀[z 0 : z 1 ] ∈ P 1 . ( 142\n\n)\n\nThe curve C X ends up in the homology class (3, 1, 0, * , * ) if and only if the defining equation ( 141 ) is of degree (3, 1, 0) in P 2 × P 1 × P 2 . Hence, eq. ( 141 ) is defined by complex constants α ij , β ij , γ i (up to rescaling) such that\n\nx i = α i0 z 0 + α i1 z 1 i = 0, 1, 2 t i = β i0 z 3 0 + β i1 z 2 0 z 1 + β i2 z 0 z 2 1 + β i3 z 3 1 i = 0, 1 y i = γ i i = 0, 1, 2. ( 143\n\n)\n\nThese constants have to be picked such that the resulting curve lies on the complete intersection X, that is, they have to satisfy eq. ( 142 ). Inserting eq. ( 143 ), we find that\n\nF 1 • C e X [z 0 : z 1 ] is a homogeneous degree 6 polynomial in [z 0 : z 1 ]. Since the coefficients of z k 0 z 6-k 1\n\nmust vanish individually, this yields 7 constraints for the parameters α ij , β ij . What makes this system of constraint equations tractable is the fact that they are all linear in β ij ,\n\nF 1 • C e X = 0 ⇔ A 1 0 0 0 A 5 0 0 0 A 2 A 1 0 0 A 6 A 5 0 0 A 3 A 2 A 1 0 A 7 A 6 A 5 0 A 4 A 3 A 2 A 1 A 8 A 7 A 6 A 5 0 A 4 A 3 A 2 0 A 8 A 7 A 6 0 0 A 4 A 3 0 0 A 8 A 7 0 0 0 A 4 0 0 0 A 8 β 00 β 01 β 02 β 03 β 10 β 11 β 12 β 13 = 0 (144)\n\nwhere (145) Thinking of this as 7 linear equations for the 8 parameters β ij , there is always a nonzero solution. The solution is generically unique up to an overall factor, and turns into an P n for special values of the α ij . Moreover, the parameter space of the α ij is connected (essentially, the moduli space of lines in P 2 ). Since we just identified the parameter space of the (α ij , β ij ) as a blow-up thereof, it is therefore connected as well.\n\nA 1 def = α 3 00 + α 3 10 + α 3\n\nIt remains to satisfy F 2 • C e X = 0. One can easily see that the only way is to pick the γ i to be simultaneous solutions of\n\nγ 3 0 + γ 3 1 + γ 3 2 = 0 = γ 1 γ 2 γ 3 . (146)\n\nSince two cubics intersect in 9 points, there are 9 such solutions, permuted by G. Therefore, the parameter space of (α ij , β ij , γ i ) has 9 connected components, permuted by the G-action. The moduli space of curves C X on X is the G-quotient of the moduli space of curves C e X on X, and therefore has only a single connected component. By continuity, every curve C X in this connected family has the same homology class, explaining the piece of the prepotential given in eq. ( 133 )."
},
{
"section_type": "OTHER",
"section_title": "Towards a Closed Formula",
"text": "Putting all the information together we found out about the prepotential on X, one can try to divine a closed form for the prepotential. We guess that the order p n terms have the closed form\n\nF np X,0 (p, q, r, b 1 , b 2 ) p n = p n 8 n-1 i,j∈Z 3 b i 1 b j 2 P (q) 4 P (r) 4 n M 2n-2 (q, r) (147)\n\nif n is not a multiple of 3 and, slightly weaker, that F np X,0 (p, q, r, 1, 1)\n\np n = 9p n 8 n-1 P (q) 4 P (r) 4 n M 2n-2 (q, r) (148)\n\nif n is a multiple of 3. Here,\n\n• P (q) is the usual generating function of partitions eq. ( 4 ).\n\n• The M 2n-2 are polynomials in the Eisenstein series E 2 (q), E 4 (q), E 6 (q) and E 2 (r), E 4 (r), E 6 (r), starting with\n\nM -2 (q, r) = 0 M 0 (q, r) = 1 M 2 (q, r) = E 2 (q)E 2 (r) M 4 (q, r) = 13 108 E 4 (q)E 4 (r) + 1 4 E 4 (q)E 2 (r) 2 + E 2 (q) 2 E 4 (r) + 7 4 E 2 (q) 2 E 2 (r) 2 M 6 (q, r) = 1 27 E 6 (q)E 6 (r) + 13 54 E 6 (q)E 4 (r)E 2 (r) + E 4 (q)E 2 (q)E 6 (r) + 1 6 E 6 (q)E 2 (r) 3 + E 2 (q) 3 E 6 (r) + 79 108 E 4 (q)E 2 (q)E 4 (r)E 2 (r) + 5 4 E 2 (q) 3 E 4 (r)E 2 (r) + E 4 (q)E 2 (q)E 2 (r) 3 + 47 12 E 2 (q) 3 E 2 (r) 3 M 8 (q, r) = 2 3 E 6 (q)E 2 (q)E 6 (r)E 2 (r) + 1309 6750 E 4 (q)E 4 (q)E 4 (r)E 4 (r) + 25 108 E 6 (q)E 2 (q)E 4 (r)E 4 (r) + E 4 (q)E 4 (q)E 6 (r)E 2 (r) + 85 54 E 6 (q)E 2 (q)E 4 (r)E 2 (r) 2 + E 4 (q)E 2 (q) 2 E 6 (r)E 2 (r) + 13 12 E 6 (q)E 2 (q)E 2 (r) 4 + E 2 (q) 4 E 6 (r)E 2 (r) + 137 216 E 4 (q)E 4 (q)E 4 (r)E 2 (r) 2 + E 4 (q)E 2 (q) 2 E 4 (r)E 4 (r) + 3 8 E 4 (q)E 4 (q)E 2 (r) 4 + E 2 (q) 4 E 4 (r)E 4 (r) + 34 9 E 4 (q)E 2 (q) 2 E 4 (r)E 2 (r) 2 + 121 12 E 2 (q) 4 E 2 (r) 4 + 41 8 E 4 (q)E 2 (q) 2 E 2 (r) 4 + E 2 (q) 4 E 4 (r)E 2 (r) 2 . ( 149\n\n)\n\nThey are symmetric under the exchange q ↔ r and of weight 2n in q and r separately. But, for example, M 4 above does not factor into a function of q and a function of r. So the M 2n-2 are not the products of the polynomials appearing in the dP 9 prepotential. However, by setting q = 0 or r = 0 one recovers the corresponding polynomials in the dP 9 prepotential [55] .\n\n• The E 2i are the usual Eisenstein series\n\nE 2 (q) = 1 -24q -72q 2 -96q 3 -168q 4 -144q 5 -288q 6 + O(q 7 ) E 4 (q) = 1 + 240q + 2160q 2 + 6720q 3 + 17520q 4 + 30240q 5 + O(q 6 ) E 6 (q) = 1 -504q -16632q 2 -122976q 3 -532728q 4 + O(q 5 ). (150)\n\nNote that the naive Taylor series coefficients of the prepotential are fractional, but when expanding in terms of Li 3 's (which account for the multicover contributions) one finds integral instanton numbers.\n\nThese expressions for the prepotential agree with all instanton numbers computed in this paper. Unfortunately, we have not been able to guess a closed formula that includes the b 1 and b 2 dependence of the prepotential F np X,0 (p, q, r, b 1 , b 2 )| p n if n is divisible by 3. We expect that these involve extra functions beyond the Eisenstein series."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusion",
"text": "In the initial paper Part A [1], we analyzed the topology of the Calabi-Yau manifold of interest and found that\n\nH 2 X, Z = Z 3 ⊕ Z 3 ⊕ Z 3 . (151)\n\nAlthough the presence of torsion curve classes complicates the counting of rational curves, we managed to derive the A-model prepotential to linear order in p. The goal of this paper is to go beyond the results of Part A using mirror symmetry. By carefully adapting methods designed for complete intersections in toric varieties, we can apply mirror symmetry to compute the instanton numbers on X, even though X is not toric. Using that X is self-mirror, we completely solve this problem and are able to calculate the complete A-model prepotential to any desired precision (and for arbitrary degrees in p), limited only by computer power. Carrying out this computation, we find the first examples of instanton numbers that do depend on the torsion part of their integral homology class, see Table 4 on Page 35.\n\nSince the self-mirror property of X is important, we investigate it in detail. In doing so, we go far beyond just checking that the Hodge numbers are self-mirror. In particular, we find that the intersection rings are identical and that torsion in homology obeys the conjectured mirror relation [6] . Finally, going beyond classical geometry, we independently calculate certain instanton numbers on X and its Batyrev-Borisov mirror X * . Again, we find that X and X * are indistinguishable, providing strong evidence for X being self-mirror. Both of these results extend those found in Part A [1] .\n\nUsing these results, we are able to guess certain closed expressions for the prepotential of X in terms of modular forms. In certain limits it specializes to the dP 9 prepotential of [55] . There it is known that the coefficients in p of the dP 9 prepotential satisfy a recursion relation. Moreover, there is a gap condition, that is, a certain number of subsequent terms in a series expansion is absent. This condition provides sufficient data to determine the integration constants for the recursion and allows to determine the prepotential completely, even at higher genus. We expect a similar story to be valid for the prepotential of X.\n\nFurthermore, we introduce the notation:\n\na i = νi , b i = ν3+i , c i = ν6+i , d i = ν12+i , i = 1, 2, 3, e = ν13 , f = ν14 . ( 154\n\n)\n\nAmong these 14 vectors in eq. ( 154 ) there are 9 independent linear relations, see eq. ( 37 ),\n\na 1 + a 2 + a 3 = 0, c 1 + c 2 + c 3 = 0, e + f = 0, b i = a i + e, d l = c l + f, (155)\n\nwhich imply others like a i + b j = a j + b i and a\n\ni + c l = b i + d l or e = 1 3 (b 1 + b 2 + b 3 ) and f = 1 3 (d 1 + d 2 + d 3 )\n\n. Lemma 1. ∇ * has 15 facets, 6 of which are simplicial:\n\na i a j b i b j c l c m d l d m ] i<j l<m , a i a j d 1 d 2 d 3 i<j , b 1 b 2 b 3 c l c m l<m . (156)\n\nThe nine non-simplicial facets form an orbit under the permutation symmetries Z ab 3 × Z cd 3 generated by g ab : (\n\na i b i ) → a i+1 b i+1\n\nand g cd :\n\n 1 -1 -1 1 0 0 0 0 1 0 -1 0 1 0 -1 0 0 0 0 0 1 -1 -1 1 , (157)\n\nwhich is the coefficient matrix of the basis a i -a j -b i + b j = 0, a i -b i + c l -d l = 0 and c l -c m -d l + d m = 0 of linear relations. The coherent triangulations are in one-to-one correspondence to chambers that are seperated by the facets of the cones generated by all linear bases µ = {v 1 , v 2 , v 3 } with v i selected among the 8 column vectors of the Gale transform [57, 58] . In the present case the cones over the faces of the parallelepiped in Figure 1 are subdivided into 24 chambers, which are indicated by dashed lines. The triangulations, which we can label by the facet containing and the edge adjoining the chamber, are obtained as the sets of complements of those bases µ that span a cone containing the respective chamber. Hence, each non-simplicial facet has 24 coherent triangulations, which can be characterized by the triangulations of its 2 pure and of its 4 mixed circuits: Calling the triangulation a i c l |b i d l positive and the triangulation a i c l | b i d l negative, and arranging the cyclic permutations g ab and g cd in the horizontal and vertical direction, respectively, we can assign one of 16 different types ± ± ± ± to each triangulation, where the signs indicate the induced triangulations of the mixed circuits. The constraints that reduce the a priori 32 = 2 6 combinations to 24 all derive from the following rules:\n\na i a j b i b j c l a i c l | b i d l ∧ a j c l |b j d l ⇒ a i b j | a j b i a i c l |b i d l ∧ a j c l | b j d l ⇒ a i b j |a j b i (158)\n\ni.e. a triangular prism can be triangulated in 6 different ways, which correlates the a priori 8 combinations of the triangulations of the 3 squares (with analogous constraints for the two \"horizontal\" prisms [a\n\ni b i c l c m d l d m ] contained in the facet [a i a j b i b j c l c m d l d m ]).\n\nPutting the pieces together we obtain Lemma 2. The 24 triangulations of the non-simplicial facets can be assorted as follows:\n\n• For + + + + , -- --the pure circuits are unconstrained, yielding 2 • 2 2 = 8 triangulations.\n\n• For + + --, -- + + the pure ab-circuit is unconstrained; with the transposed types + -+ -, -+ -+ this accounts for another 8 triangulations.\n\n• The final 8 triangulations come from the 8 types with an odd number of positive signs, for which the triangulation of the pure circuits is unique.\n\n• The two types + - -+ and -+ + -cannot occur because of contradictory implications for the triangulations of the pure circuits.\n\nThe secondary fan and the induced triangulations for the codimension-two faces at which the non-simplical facets intersect can be obtained from Figure 1 by projection along the dropped vertices. The secondary fan of the prism of eq. (158), for example, which is shown in Figure 2 , is obtained from Figure 1 c m d m . The wall crossings between the six cones in Figure 2 are labeled by the circuits whose flops relate the adjoining triangulations [57] .\n\nFor the construction of the complete star triangulation we now observe that the non-simplicial intersections of the 9 non-simplicial facets Table 5: The 824 = 2 (64 + 36 + 72 + 96 + 36 + 72 + 36) star triangulations of ∇ * , including the 720 = 2 (36 + 36 + 72 + 72 + 36 + 72 + 36) coherent triangulations.\n\nstriction on the compatible signs is due to the absence of the inconsistent types + - -+ and -+ + -as subgraphs on the torus. The multiplicities µ • 2 n come from the number n of unconstrained pure circuits and from the order µ of the effective part of the symmetry group generated by transposition and permutations of lines and columns. We thus find a total of 824 triangulations. The cyclic permutation symmetry that we want to keep on the Calabi-Yau manifold X * amounts to a diagonal shift, i.e. its induced action on the graph is generated by g ab g cd . We are hence left with the 5 , which have unbroken horizontal symmetry and for which the multiplicity is reduced from 12 • 8 to 12 • 6. This poses a problem for the eight Z 3 -symmetric triangulations, which are all non-coherent. Coherence of the remaining 720 triangulations can be established by checking that their Mori cones are all strictly convex [59] .\n\nWhat comes to our rescue is that, even if all projective ambient spaces break the diagonal Z 3 permutation symmetry, it may be preserved on X * if the obstructing exceptional sets do not overlap with the complete intersection. In the present case these are the blow-ups of the singularities coming from the pure circuits, i.e. codimension two sets of the form a i • b j or c l • d m , where we use, for simplicity, the symbol of the vertex νj for the corresponding divisor D j . Recall from eq. ( 77 ) that X * is given by the product D * The polytope ∇ * of the mirror X * of the universal cover has 39 lattice points, with the same 12 vertices as ∇ * but living on the finer lattice M . The 24 additional lattice points, see eq. ( 39 ), are\n\na ij = 1 3 (a i + 2a j ), b ij = 1 3 (b i + 2b j ), (166)\n\nc ij = 1 3 (c i + 2c j ), d ij = 1 3 (d i + 2d j ), (167)\n\nwhere i = j. These additional points are all located on edges of ∇ * . It is natural to consider triangulations that are refinements of the ones that we just discussed."
}
] | |
arxiv:0704.0456 | 0704.0456 | 1 | 10.1088/1751-8113/40/25/S37 | 515eb130c1ea7f3255b1b6a9f4dae76d33c4ab7fd315a3c6938003a8c23328da | Particle propagation in cosmological backgrounds | We study the quantum propagation of particles in cosmological backgrounds, by
considering a doublet of massive scalar fields propagating in an expanding
universe, possibly filled with radiation. We focus on the dissipative effects
related to the expansion rate. At first order, we recover the expected result
that the decay rate is determined by the local temperature. Beyond linear
order, the decay rate has an additional contribution governed by the expansion
parameter. This latter contribution is present even for stable particles in the
vacuum. Finally, we analyze the long time behaviour of the propagator and
briefly discuss applications to the trans-Planckian question. | [
"Daniel Arteaga"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | We study the quantum propagation of particles in cosmological backgrounds, by considering a doublet of massive scalar fields propagating in an expanding universe, possibly filled with radiation. We focus on the dissipative effects related to the expansion rate. At first order, we recover the expected result that the decay rate is determined by the local temperature. Beyond linear order, the decay rate has an additional contribution governed by the expansion parameter. This latter contribution is present even for stable particles in the vacuum. Finally, we analyze the long time behaviour of the propagator and briefly discuss applications to the trans-Planckian question.
In this contribution we study the quantum propagation of particles in a cosmological background. We are particularly interested in understanding the dissipative phenomena related to the time dependence of the metric. To this end, we analyze the propagator of a massive particle which interacts with a massless radiation field in an expanding universe. Several points must be considered.
First, we are dealing with an interacting field theory in a curved spacetime. In this situation, the asymptotic in and out vacua generally do not coincide. Being interested in expectation values, rather than in in-out matrix elements, we adopt the Keldysh-Schwinger formalism, or Closed Time Path (CTP) method [1] [2] [3] in curved spacetime [4] [5] [6] [7] .
Second, as it is well known, in a curved spacetime there is no single definition for the vacuum nor for the concept of particle. We face this issue by working within the adiabatic approximation [8] : the massive particles will have their Compton wavelengths much smaller than the typical curvature radius of the universe (in our case, the Hubble radius).
Third, as explained in [9] , in theories such as QED or perturbative quantum gravity, dissipative effects appear only at two loops, because the one-loop diagrams which could lead to dissipation vanish on the mass shell. Here, in order to keep the calculations simple, we have chosen a simple, yet physically meaningful, model which exhibits dissipation at one loop. We expect the behaviour of QED or perturbative quantum gravity to be similar at two loops.
In this contribution we compute the retarded self-energy of the lightest field in a massive doublet which propagates in a thermal bath of massless particles in an expanding universe, and from it we extract the decay rate. Notice that in the Minkowski vacuum the excitations of the lighter field are stable, hence the decay rate is zero. We concentrate on the physical insights and summarize the main results. A more detailed account will be given in separate publications [10, 11] The contribution is organized as follows. In section 2 we introduce the model and motivate the use of the adiabatic approximation for the massive fields. In section 3 we present the results for the imaginary part of the self-energy and the decay rate. In section 4 we study the time evolution of the interacting propagator. Finally, in section 5 we summarize the main points of the contribution and discuss its relevance to the trans-Planckian question. We use a system of natural units with = c = 1, and the metric has the signature (-, +, +, +).
We consider spatially isotropic and homogeneous Friedmann-Lemaître-Robertson-Walker models with flat spatial sections:
ds 2 = -dt 2 + a 2 (t)dx 2 .
(
) 1
The particle model is the following: two massive fields φ m , and φ M , interacting with a massless field, χ, via a trilinear coupling. The total action is S = S m + S M + S χ + S int , where each term is given by
S m = 1 2 dt d 3 x a 3 (t) (∂ t φ m ) 2 - 1 a 2 (t) (∂ x φ m ) 2 -m 2 φ 2 m , (2a)
S M = 1 2 dt d 3 x a 3 (t) (∂ t φ M ) 2 - 1 a 2 (t) (∂ x φ M ) 2 -M 2 φ 2 M , (2b)
S χ = 1 2 dt d 3 x a 3 (t) (∂ t χ) 2 - 1 a 2 (t) (∂ x χ) 2 -ξR(t)χ 2 , (2c)
S int = gM dt d 3 x a 3 (t)φ m φ M χ, (2d)
with R(t) being the Ricci scalar. We assume that the massless field is conformally coupled to gravity, so that ξ = 1/6. It is useful to work with rescaled massive fields defined by φ(t,
x) := [-g(t, x)] 1/4 φ(t, x) = a 3/2 (t)φ(t, x).
We consider the two massive fields having large masses but with a small mass difference ∆m := M -m ≪ M. As shown in [12] , the model can be interpreted as a field-theory description of a relativistic two-level atom (of mass m and energy gap ∆m) interacting with a scalar radiation field χ. The radiation field χ is assumed to be at some conformal temperature θ (which can eventually be zero). The corresponding physical temperature, as well as the Hubble rate H(t) := ȧ(t)/a(t), are chosen to be much smaller than the masses of the fields. These restrictions ensure that the number of massive particles is strictly conserved. The non-trivial dynamics concerns the transitions between the two massive fields accompanied by emission and absorption of massless quanta.
In a curved spacetimes it is not a trivial task to compute even the free field vacuum propagators. For massless conformally coupled fields there is a natural vacuum state, the conformal vacuum. Propagators in this vacuum, when expressed in conformal time, essentially correspond to the flat spacetime propagators. For the massive fields, rather than attempting to find the exact free propagator, we will exploit the fact that their Compton wavelengths is much smaller than the Hubble length H -1 . In this regime, the adiabatic (WKB) approximation is valid and explicit expressions for the free propagators can be computed [8, 10] -see for instance (19) .
In this section we consider the interacting retarded Green function
G R (t, t ′ ; p) := θ(t -t ′ ) [ φmp (t), φmp (t ′ )] (3)
within the adiabatic approximation. It is related to the retarded self-energy Σ R via [12] G R (t,
t ′ ; p) = G (0) R (t, t ′ ; p) -i ds ds ′ -g(s) -g(s ′ )G (0) R (t, s; p)Σ R (s, s ′ ; p)G R (s ′ , t ′ ; p) (4)
where G (0) R (t, t ′ ; p) is the free retarded propagator. In terms of the rescaled fields, φ(t; p) = a 3/2 φ(t; p), the above relation becomes
ḠR (t, t ′ ; p) = Ḡ(0) R (t, t ′ ; p) -i ds ds ′ Ḡ(0) R (t, s; p) ΣR (s, s ′ ; p) ḠR (s ′ , t ′ ; p). (5)
We assume that the massive fields are in the adiabatic vacuum, and that the massless field χ is in a thermal state, characterized by a fixed conformal temperature θ. We will compute the imaginary part of the one-loop self energy to order g 2 in the adiabatic approximation, evaluated at the mass shell. It will be evaluated in a a frequency representation around the average time coordinate T = (t 1 +t 2 )/2, by Fouriertransforming with respect to the difference coordinate ∆ = t 1 -t 2 , which amounts to a local frequency representation (it is further analyzed in next section). As for the spatial part, we work in the momentum representation to exploit conservation of the conformal momentum.
As a first step, we approximate the evolution of the scale factor by a linear expansion:
a(t) ≈ a(T )[1 + H(T )(t -T )]. (6)
This approximation for the scale factor is appropriate when considering physical temperatures which are much larger than the expansion rate (but still much smaller than the fields masses). On the mass shell, thermal corrections will dominate over curvature corrections, since the thermal energy scale is much larger than the curvature energy scale. Therefore, we expect the on-shell self-energy to be governed by the thermal bath at the instantaneous physical temperature at each moment of the expansion, θ/a(T ). The explicit calculation [10] confirms that the imaginary part of the on-shell selfenergy is given by that of a thermal bath in Minkowski at a physical temperature θ/a(T ). In the limit in which the atoms are at rest this result is [10, 12] Im ΣR (m, T ; 0) = -
g 2 8π M∆m n θ/a(T ) (∆m), (7)
where n θ/a(T ) (∆m) is the Bose-Einstein function:
n θ/a(T ) (∆m) := 1 e ∆m a(T )/θ -1 . (8)
As in Minkowski spacetime, the self-energy corresponds to a decay rate,
Γ = - 1 m Im ΣR (m, T ; 0) = g 2 8π ∆m n θ/a(T ) (∆m), (9)
which amounts to the probability per unit time for the lightest state to absorb a massless particle from the thermal bath.
When the expansion rate of the universe is of the order of the temperature or larger, vacuum effects become relevant. Energy conservation does not hold for energy scales of the order of the expansion rate, and therefore we expect new channels for the particle decay which will contribute to the imaginary part of the self-energy.
In order to study the vacuum effects we need to choose a explicit model for the evolution of the scale factor. For instance, in the case of de Sitter,
a(t) = a(T ) e H(t-T ) , (10)
the vacuum contribution to the imaginary part of the retarded self-energy given by [11] Im ΣR (m,
T ; 0) = - g 2 8π M∆m n H/(2π) (∆m), ( 11
)
which coicides with the self-energy in a Minkowski thermal bath at a temperature H/(2π). The result is not unexpected since the effective de Sitter temperature [13] is recovered. The corresponding decay rate
Γ = - 1 m Im ΣR (m, T ; 0) = g 2 8π ∆m n H/(2π) (∆m), (12)
amounts for the probability per unit time for the lightest field to emit a massless particle. Energy conservation forbids this process in Minkowski spacetime, but this restriction does not apply in an expanding universe.
In expanding universes the propagators are no longer time-translation invariant. We can nevertheless always express the propagator in a frequency representation, ḠR (ω, T ; p) := d∆ e iω∆ ḠR (T + ∆/2, T -∆/2; p) .
For short time differences as compared to the inverse expansion rate, i.e., |t-t ′ | ≪ H -1 , (5) can be diagonalized:
ḠR (ω, T ; p) = -i [-i Ḡ(0) (ω, T ; p)] -1 + ΣR (ω, T ; p) . (14)
Fourier-transforming again we get the short-time behavior:
ḠR (t, t ′ ; p) = -i R p (T ) sin [R p (T )(t -t ′ )] e -Γp(T )(t-t ′ )/2 θ(t -t ′ ). ( 15
)
with
R 2 p (T ) := E 2 p (T ) + Re ΣR (E p , T ; p) := m 2 + p 2 a 2 (T ) + Re ΣR (E p , T ; p) (16)
and
Γ p (T ) := - 1 R p (T ) Im ΣR (E p , T ; p). (17)
Therefore one recovers the usual interpretation, in which the real part of the self-energy corresponds to the energy shift, and in which the imaginary part corresponds to the decay rate. Notice that both quantities depend in general on time. One may also be interested in considering large time lapses, and in this case the frequency representation of the propagator around the average time does not make sense. Lifting the short-time requirement, and only imposing the adiabatic approximation, the following expression for the evolution of the retarded propagator is found [10] :
ḠR (t 1 , t 2 ; p) = -i R p (t 1 )R p (t 2 ) sin t 1 t 2 dt ′ R p (t ′ ) e - R t 1 t 2 dt ′ Γ k (t ′ )/2 θ(t 1 -t 2 ). ( 18
)
Notice that the long-time evolution of the propagator can be expressed in terms of integrals of quantities evaluated in the local frequency representation. Two time scales are clearely separated: the interaction timescale, in which the interaction process take place and in which the self-energy is evaluated, and the evolution timescale, which can be much longer and during which the propagators deviate significantly from the corresponding Minkowski expression. Equation ( 18 ) can be derived in a very similar way as the well-known adiabatic approximation for the free retarded propagator [8] :
Ḡ(0) R (t 1 , t 2 ; p) = -i E p (t 1 )E p (t 2 ) sin t 1 t 2 dt ′ E p (t ′ ) θ(t 1 -t 2 ). (19)
The goal of this contribution is to analyze the quantum effects in the propagation of interacting fields in a cosmological background. This issue may play an important role in justifying the non-trivial dispersion relations which have been used when addressing the trans-Planckian question in the context of black holes [14] [15] [16] [17] and cosmology [18] [19] [20] [21] .
Interactions could indeed significantly modify the field propagation when approaching the event horizon of a black hole [22] [23] [24] [25] or at primordial stages of inflation [9] . In our model, the masses of the fields were assumed to be much larger than the expansion rate of the universe. This was a key assumption, because it allowed to introduce the adiabatic (WKB) approximation, which not only makes the problem solvable, but also allows having a well-defined particle concept even in absence of asymptotic regimes. Within this approximation, the time-evolution of the interacting propagators can be computed from the integral of the retarded self-energy, evaluated on-shell in a frequency representation around the mid time.
The imaginary part of the self-energy determines the decay of the retarded propagator, and hence it is an expression of the dissipative properties. For temperatures higher than the expansion parameter the decay of the propagator is determined by the local temperature at each moment of expansion. For lower temperatures, the decay of the propagator is driven by the expansion rate of the universe. This second contribution, which is present even in the vacuum, can be interpreted as being a consequence of the absence of energy conservation at those energy scales comparable to the expansion rate.
The decay rate, derived from the imaginary part of the self-energy, has a secular character. Even small decay rates could thus give an important effect when integrated over large periods of time. The exact significance of the generically dissipative properties of the propagator will be further analyzed elsewhere [11] . | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We study the quantum propagation of particles in cosmological backgrounds, by considering a doublet of massive scalar fields propagating in an expanding universe, possibly filled with radiation. We focus on the dissipative effects related to the expansion rate. At first order, we recover the expected result that the decay rate is determined by the local temperature. Beyond linear order, the decay rate has an additional contribution governed by the expansion parameter. This latter contribution is present even for stable particles in the vacuum. Finally, we analyze the long time behaviour of the propagator and briefly discuss applications to the trans-Planckian question."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "In this contribution we study the quantum propagation of particles in a cosmological background. We are particularly interested in understanding the dissipative phenomena related to the time dependence of the metric. To this end, we analyze the propagator of a massive particle which interacts with a massless radiation field in an expanding universe. Several points must be considered.\n\nFirst, we are dealing with an interacting field theory in a curved spacetime. In this situation, the asymptotic in and out vacua generally do not coincide. Being interested in expectation values, rather than in in-out matrix elements, we adopt the Keldysh-Schwinger formalism, or Closed Time Path (CTP) method [1] [2] [3] in curved spacetime [4] [5] [6] [7] .\n\nSecond, as it is well known, in a curved spacetime there is no single definition for the vacuum nor for the concept of particle. We face this issue by working within the adiabatic approximation [8] : the massive particles will have their Compton wavelengths much smaller than the typical curvature radius of the universe (in our case, the Hubble radius).\n\nThird, as explained in [9] , in theories such as QED or perturbative quantum gravity, dissipative effects appear only at two loops, because the one-loop diagrams which could lead to dissipation vanish on the mass shell. Here, in order to keep the calculations simple, we have chosen a simple, yet physically meaningful, model which exhibits dissipation at one loop. We expect the behaviour of QED or perturbative quantum gravity to be similar at two loops.\n\nIn this contribution we compute the retarded self-energy of the lightest field in a massive doublet which propagates in a thermal bath of massless particles in an expanding universe, and from it we extract the decay rate. Notice that in the Minkowski vacuum the excitations of the lighter field are stable, hence the decay rate is zero. We concentrate on the physical insights and summarize the main results. A more detailed account will be given in separate publications [10, 11] The contribution is organized as follows. In section 2 we introduce the model and motivate the use of the adiabatic approximation for the massive fields. In section 3 we present the results for the imaginary part of the self-energy and the decay rate. In section 4 we study the time evolution of the interacting propagator. Finally, in section 5 we summarize the main points of the contribution and discuss its relevance to the trans-Planckian question. We use a system of natural units with = c = 1, and the metric has the signature (-, +, +, +)."
},
{
"section_type": "OTHER",
"section_title": "The model",
"text": "We consider spatially isotropic and homogeneous Friedmann-Lemaître-Robertson-Walker models with flat spatial sections:\n\nds 2 = -dt 2 + a 2 (t)dx 2 .\n\n(\n\n) 1\n\nThe particle model is the following: two massive fields φ m , and φ M , interacting with a massless field, χ, via a trilinear coupling. The total action is S = S m + S M + S χ + S int , where each term is given by\n\nS m = 1 2 dt d 3 x a 3 (t) (∂ t φ m ) 2 - 1 a 2 (t) (∂ x φ m ) 2 -m 2 φ 2 m , (2a)\n\nS M = 1 2 dt d 3 x a 3 (t) (∂ t φ M ) 2 - 1 a 2 (t) (∂ x φ M ) 2 -M 2 φ 2 M , (2b)\n\nS χ = 1 2 dt d 3 x a 3 (t) (∂ t χ) 2 - 1 a 2 (t) (∂ x χ) 2 -ξR(t)χ 2 , (2c)\n\nS int = gM dt d 3 x a 3 (t)φ m φ M χ, (2d)\n\nwith R(t) being the Ricci scalar. We assume that the massless field is conformally coupled to gravity, so that ξ = 1/6. It is useful to work with rescaled massive fields defined by φ(t,\n\nx) := [-g(t, x)] 1/4 φ(t, x) = a 3/2 (t)φ(t, x).\n\nWe consider the two massive fields having large masses but with a small mass difference ∆m := M -m ≪ M. As shown in [12] , the model can be interpreted as a field-theory description of a relativistic two-level atom (of mass m and energy gap ∆m) interacting with a scalar radiation field χ. The radiation field χ is assumed to be at some conformal temperature θ (which can eventually be zero). The corresponding physical temperature, as well as the Hubble rate H(t) := ȧ(t)/a(t), are chosen to be much smaller than the masses of the fields. These restrictions ensure that the number of massive particles is strictly conserved. The non-trivial dynamics concerns the transitions between the two massive fields accompanied by emission and absorption of massless quanta.\n\nIn a curved spacetimes it is not a trivial task to compute even the free field vacuum propagators. For massless conformally coupled fields there is a natural vacuum state, the conformal vacuum. Propagators in this vacuum, when expressed in conformal time, essentially correspond to the flat spacetime propagators. For the massive fields, rather than attempting to find the exact free propagator, we will exploit the fact that their Compton wavelengths is much smaller than the Hubble length H -1 . In this regime, the adiabatic (WKB) approximation is valid and explicit expressions for the free propagators can be computed [8, 10] -see for instance (19) ."
},
{
"section_type": "OTHER",
"section_title": "The self-energy and decay rates",
"text": "In this section we consider the interacting retarded Green function\n\nG R (t, t ′ ; p) := θ(t -t ′ ) [ φmp (t), φmp (t ′ )] (3)\n\nwithin the adiabatic approximation. It is related to the retarded self-energy Σ R via [12] G R (t,\n\nt ′ ; p) = G (0) R (t, t ′ ; p) -i ds ds ′ -g(s) -g(s ′ )G (0) R (t, s; p)Σ R (s, s ′ ; p)G R (s ′ , t ′ ; p) (4)\n\nwhere G (0) R (t, t ′ ; p) is the free retarded propagator. In terms of the rescaled fields, φ(t; p) = a 3/2 φ(t; p), the above relation becomes\n\nḠR (t, t ′ ; p) = Ḡ(0) R (t, t ′ ; p) -i ds ds ′ Ḡ(0) R (t, s; p) ΣR (s, s ′ ; p) ḠR (s ′ , t ′ ; p). (5)\n\nWe assume that the massive fields are in the adiabatic vacuum, and that the massless field χ is in a thermal state, characterized by a fixed conformal temperature θ. We will compute the imaginary part of the one-loop self energy to order g 2 in the adiabatic approximation, evaluated at the mass shell. It will be evaluated in a a frequency representation around the average time coordinate T = (t 1 +t 2 )/2, by Fouriertransforming with respect to the difference coordinate ∆ = t 1 -t 2 , which amounts to a local frequency representation (it is further analyzed in next section). As for the spatial part, we work in the momentum representation to exploit conservation of the conformal momentum."
},
{
"section_type": "OTHER",
"section_title": "Linear approximation to the scale factor",
"text": "As a first step, we approximate the evolution of the scale factor by a linear expansion:\n\na(t) ≈ a(T )[1 + H(T )(t -T )]. (6)\n\nThis approximation for the scale factor is appropriate when considering physical temperatures which are much larger than the expansion rate (but still much smaller than the fields masses). On the mass shell, thermal corrections will dominate over curvature corrections, since the thermal energy scale is much larger than the curvature energy scale. Therefore, we expect the on-shell self-energy to be governed by the thermal bath at the instantaneous physical temperature at each moment of the expansion, θ/a(T ). The explicit calculation [10] confirms that the imaginary part of the on-shell selfenergy is given by that of a thermal bath in Minkowski at a physical temperature θ/a(T ). In the limit in which the atoms are at rest this result is [10, 12] Im ΣR (m, T ; 0) = -\n\ng 2 8π M∆m n θ/a(T ) (∆m), (7)\n\nwhere n θ/a(T ) (∆m) is the Bose-Einstein function:\n\nn θ/a(T ) (∆m) := 1 e ∆m a(T )/θ -1 . (8)\n\nAs in Minkowski spacetime, the self-energy corresponds to a decay rate,\n\nΓ = - 1 m Im ΣR (m, T ; 0) = g 2 8π ∆m n θ/a(T ) (∆m), (9)\n\nwhich amounts to the probability per unit time for the lightest state to absorb a massless particle from the thermal bath."
},
{
"section_type": "OTHER",
"section_title": "Beyond linear order: vacuum effects",
"text": "When the expansion rate of the universe is of the order of the temperature or larger, vacuum effects become relevant. Energy conservation does not hold for energy scales of the order of the expansion rate, and therefore we expect new channels for the particle decay which will contribute to the imaginary part of the self-energy.\n\nIn order to study the vacuum effects we need to choose a explicit model for the evolution of the scale factor. For instance, in the case of de Sitter,\n\na(t) = a(T ) e H(t-T ) , (10)\n\nthe vacuum contribution to the imaginary part of the retarded self-energy given by [11] Im ΣR (m,\n\nT ; 0) = - g 2 8π M∆m n H/(2π) (∆m), ( 11\n\n)\n\nwhich coicides with the self-energy in a Minkowski thermal bath at a temperature H/(2π). The result is not unexpected since the effective de Sitter temperature [13] is recovered. The corresponding decay rate\n\nΓ = - 1 m Im ΣR (m, T ; 0) = g 2 8π ∆m n H/(2π) (∆m), (12)\n\namounts for the probability per unit time for the lightest field to emit a massless particle. Energy conservation forbids this process in Minkowski spacetime, but this restriction does not apply in an expanding universe."
},
{
"section_type": "OTHER",
"section_title": "Retarded propagator and self-energy in cosmology",
"text": "In expanding universes the propagators are no longer time-translation invariant. We can nevertheless always express the propagator in a frequency representation, ḠR (ω, T ; p) := d∆ e iω∆ ḠR (T + ∆/2, T -∆/2; p) .\n\nFor short time differences as compared to the inverse expansion rate, i.e., |t-t ′ | ≪ H -1 , (5) can be diagonalized:\n\nḠR (ω, T ; p) = -i [-i Ḡ(0) (ω, T ; p)] -1 + ΣR (ω, T ; p) . (14)\n\nFourier-transforming again we get the short-time behavior:\n\nḠR (t, t ′ ; p) = -i R p (T ) sin [R p (T )(t -t ′ )] e -Γp(T )(t-t ′ )/2 θ(t -t ′ ). ( 15\n\n)\n\nwith\n\nR 2 p (T ) := E 2 p (T ) + Re ΣR (E p , T ; p) := m 2 + p 2 a 2 (T ) + Re ΣR (E p , T ; p) (16)\n\nand\n\nΓ p (T ) := - 1 R p (T ) Im ΣR (E p , T ; p). (17)\n\nTherefore one recovers the usual interpretation, in which the real part of the self-energy corresponds to the energy shift, and in which the imaginary part corresponds to the decay rate. Notice that both quantities depend in general on time. One may also be interested in considering large time lapses, and in this case the frequency representation of the propagator around the average time does not make sense. Lifting the short-time requirement, and only imposing the adiabatic approximation, the following expression for the evolution of the retarded propagator is found [10] :\n\nḠR (t 1 , t 2 ; p) = -i R p (t 1 )R p (t 2 ) sin t 1 t 2 dt ′ R p (t ′ ) e - R t 1 t 2 dt ′ Γ k (t ′ )/2 θ(t 1 -t 2 ). ( 18\n\n)\n\nNotice that the long-time evolution of the propagator can be expressed in terms of integrals of quantities evaluated in the local frequency representation. Two time scales are clearely separated: the interaction timescale, in which the interaction process take place and in which the self-energy is evaluated, and the evolution timescale, which can be much longer and during which the propagators deviate significantly from the corresponding Minkowski expression. Equation ( 18 ) can be derived in a very similar way as the well-known adiabatic approximation for the free retarded propagator [8] :\n\nḠ(0) R (t 1 , t 2 ; p) = -i E p (t 1 )E p (t 2 ) sin t 1 t 2 dt ′ E p (t ′ ) θ(t 1 -t 2 ). (19)"
},
{
"section_type": "DISCUSSION",
"section_title": "Summary and discussion",
"text": "The goal of this contribution is to analyze the quantum effects in the propagation of interacting fields in a cosmological background. This issue may play an important role in justifying the non-trivial dispersion relations which have been used when addressing the trans-Planckian question in the context of black holes [14] [15] [16] [17] and cosmology [18] [19] [20] [21] .\n\nInteractions could indeed significantly modify the field propagation when approaching the event horizon of a black hole [22] [23] [24] [25] or at primordial stages of inflation [9] . In our model, the masses of the fields were assumed to be much larger than the expansion rate of the universe. This was a key assumption, because it allowed to introduce the adiabatic (WKB) approximation, which not only makes the problem solvable, but also allows having a well-defined particle concept even in absence of asymptotic regimes. Within this approximation, the time-evolution of the interacting propagators can be computed from the integral of the retarded self-energy, evaluated on-shell in a frequency representation around the mid time.\n\nThe imaginary part of the self-energy determines the decay of the retarded propagator, and hence it is an expression of the dissipative properties. For temperatures higher than the expansion parameter the decay of the propagator is determined by the local temperature at each moment of expansion. For lower temperatures, the decay of the propagator is driven by the expansion rate of the universe. This second contribution, which is present even in the vacuum, can be interpreted as being a consequence of the absence of energy conservation at those energy scales comparable to the expansion rate.\n\nThe decay rate, derived from the imaginary part of the self-energy, has a secular character. Even small decay rates could thus give an important effect when integrated over large periods of time. The exact significance of the generically dissipative properties of the propagator will be further analyzed elsewhere [11] ."
}
] | |
arxiv:0704.0457 | 0704.0457 | 1 | 10.1103/PhysRevB.76.014511 | 4e8ebebd01ba06ac30445d5b77bde13649dba9ea0e7dbcd07646f21bc4448016 | Quantum analysis of a linear DC SQUID mechanical displacement detector | We provide a quantum analysis of a DC SQUID mechanical displacement detector
within the sub-critical Josephson current regime. A segment of the SQUID loop
forms the mechanical resonator and motion of the latter is transduced
inductively through changes in the flux threading the loop. Expressions are
derived for the detector signal response and noise, which are used to evaluate
the position and force detection sensitivity. We also investigate cooling of
the mechanical resonator due to back reaction noise from the detector. | [
"M. P. Blencowe",
"E. Buks"
] | [
"cond-mat.supr-con",
"cond-mat.mes-hall",
"quant-ph"
] | cond-mat.supr-con | [] | 2007-04-03 | 2026-02-26 | In a series of recent experiments 1, 2, 3 and related theoretical work, 4, 5, 6, 7, 8, 9 it was demonstrated that a displacement detector based on either a normal or superconducting single electronic transistor (SSET) can resolve the motion of a micron-scale mechanical resonator close to the quantum limit as set by Heisenberg's Uncertainty Principle. 10, 11, 12 The displacement transduction was achieved by capacitively coupling the gated mechanical resonator to the SSET metallic island. When the resonator is voltage biased, motion of the latter changes the island charging energy and hence the Cooper pair tunnel rates. The resulting modulation in the source-drain tunnel current through the SSET is then read out as a signature of the mechanical motion. Given the success of this capacitive-based transduction method in approaching the quantum limit, it is natural to consider complementary, inductive-based transduction methods in which, for example, a superconducting quantum interference device (SQUID) is similarly used as an intermediate quantum-limited stage between the micron-scale mechanical resonator and secondary amplification stages. 13, 14, 15, 16 Unavoidable, fundamental noise sources and how they affect the SSET and SQUID devices are not necessarily the same. Furthermore, achievable coupling strengths between each type of device and a micron-scale mechanical resonator may be different. Therefore, it would be interesting to address the merits of the SQUID in comparison with the established SSET for approaching the quantum limit of displacement detection.
In the present paper, we analyze a DC SQUID-based displacement detector. The SQUID is integrated with a mechanical resonator in the form of a doubly-clamped beam, shown schematically in Fig. 1 . Motion of the beam changes the magnetic flux Φ threading the SQUID loop, hence modulating the current circulating the loop. We shall address the operation of the SQUID displacement detector in the regime for which the loop current is smaller than the Josephson junction critical current I c and at temperatures well below the superconducting critical temperature. We thus assume that resistive (normal) current flow through the junctions and accompanying current noise can be neglected. (See for example Ref. 17 for a quantum noise analysis of resistively shunted Josephson junctions and Ref. 18 for a related analysis of the DC SQUID.) Such an assumption cannot be made with the usual mode of operation for the SSET devices, where the tunnel current unavoidably involves the 2 quasiparticle decay of Cooper pairs, resulting in shot noise.
As noise source, we will consider the quantum electromagnetic fluctuations within the pump/probe feedline and also transmission line resonator that is connected to the SQUID. This noise is a consequence of the necessary dissipative coupling to the outside world and affects the mechanical signal output in two ways. First, the noise is added directly to the output in the probe line and, second, the noise acts back on the mechanical resonator via the SQUID, affecting the resonator's motion.
With the Josephson junction plasma frequencies assumed to be much larger than the other resonant modes of relevance for the device, the SQUID can be modeled to a good approximation as an effective inductance that depends on the external current I entering and exiting the loop, as well as on the applied flux. In this first of two papers, we shall make the further approximation of neglecting the I-dependence of the SQUID effective inductance, which requires the condition I ≪ I c . In the sequel, 19 we will relax this condition somewhat by including the next to leading O(I 2 ) term in the inductance and address the consequences of this non-linear correction for quantum-limited displacement detection.
Modeling the SQUID approximately as a passive inductance element, the transmission line resonator-mechanical resonator effective Hamiltonian is given by Eq. (24). This Hamiltonian describes many other detector-oscillator systems that are modeled as two coupled harmonic oscillators, including the examples of an LC resonator capacitively coupled to a mechanical resonator 20, 21 and an optical cavity coupled to a mechanically compliant mirror via radiation pressure; 22, 23, 24, 25, 26 the various systems are distinguished only by the dependences of the coupling strengths on the parameters particular to each system. Thus, many of the results of this paper are of more general relevance.
The central results of the paper are Eqs. (69) and (70), giving the detector response to a mechanical resonator undergoing quantum Brownian motion and also subject to a classical driving force. In the derivation of these expressions, we do not approximate the response as a perturbation series in the coupling between the SQUID and mechanical resonator as is conventionally done, but rather find it more natural to base our approximations instead on assumed weak coupling between the mechanical resonator and its external heat bath and weak classical driving force. Thus, in the context of the linear response paradigm, our detector should properly be viewed as including the mechanical resonator degrees of freedom as well, with the weak perturbative signal instead consisting of the heat bath force noise 3 and classical drive force acting on the mechanical resonator. Since the quality factors of actual, micron-scale mechanical resonators can be very large at sub-Kelvin temperatures (E.g., Q ∼ 10 5 in the experiments of Refs. 2,3), quantum electromagnetic noise in the transmission line part of the detector can have strong back reaction effects on the motion of the mechanical resonator, even when the coupling between the resonator and the SQUID is very weak. One consequence that we shall consider is cooling of the mechanical resonator fundamental mode, which requires strong back reaction damping combined with low noise.
Nevertheless, as we will also show, one can still analyze the quantum-limited detector linear response to the mechanical resonator's position signal using general expressions (69) and (70), under the appropriate conditions of small pump drive and weak coupling between the SQUID and mechanical resonator such that back reaction effects are small.
The outline of the paper is as follows. In Sec. II, we write down the SQUID-mechanical resonator equations of motion corresponding to the circuit scheme shown in Fig. 1 and then derive the Heisenberg equations for the various mode raising and lowering operators, subject to the above-mentioned approximations. In Sec. III, we solve the equations within the linear response approximation to derive the detector signal response and noise. In Sec. IV, we analyze both the position and force detection sensitivity, and address also back reaction cooling of the mechanical resonator. Sec. V provides concluding remarks.
Fig. 1 shows the displacement detector scheme. The device consists of a stripline resonator (transmission line T ) made of two sections, each of length l/2, connected via a DC SQUID (see Refs. 27,28,29,30 for related, qubit detection schemes). The transmission line inductance and capacitance per unit length are L T and C T respectively. The Josephson junctions in each arm of the SQUID are assumed to have identical critical currents I c and capacitances C J . A length l osc segment of the SQUID loop is free to vibrate as a doubly-clamped bar resonator and the fundamental flexural mode of interest (in the plane of the loop) is treated as a harmonic oscillator with mass m, frequency ω m and displacement coordinate y. The total external magnetic flux applied perpendicular to the SQUID loop is given by Φ ext +λB ext l osc y,
where Φ ext is the flux corresponding to the case y = 0, B ext is the normal component of the magnetic field at the location of the vibrating loop segment (oscillator), and the dimensionless parameter λ < 1 is a geometrical correction factor accounting for the nonuniform displacement of the doubly-clamped resonator in the fundamental flexural mode. C J C J I c I c y p T T m,ω m γ pT γ eT γ bm Φ FIG. 1: Scheme for the displacement detector showing the pump/probe line 'p', transmission line resonator 'T ', and DC SQUID with mechanically compliant loop segment having effective mass m and fundamental frequency ω m . Note that the scale of the DC SQUID is exaggerated relative to that of the stripline for clarity.
The transmission line is weakly coupled to a pump/probe feedline (p), with inductance and capacitance per unit length L p and C p respectively, employed for delivering the input and output RF signals; the coupling can be characterized by a transmission line mode amplitude damping rate γ pT (see section II B below). Other possible damping mechanisms in the transmission line may be taken into account by adding a fictitious semi-infinite stripline environment (e), weakly coupled to the transmission line characterized by mode amplitude damping rate γ eT . 31 While γ eT can be made much smaller than γ pT with suitable transmission line resonator design, we shall nevertheless include both sources of damping in our analysis so as to eventually be able to gauge their relative effects on the detector displacement sensitivity [see Eq. (90)]. The SQUID, on the other hand, is assumed to be dissipationless. The 5 mechanical oscillator is also assumed to be coupled to an external heat bath (b), characterized by mode amplitude damping rate γ bm .
A convenient choice of dynamical coordinates for the SQUID are γ ± = (φ 1 ± φ 2 ) /2, where φ 1 and φ 2 are the gauge invariant phases across each of the two Josephson junctions. 32 For the transmission line, we similarly use its phase field coordinate φ(x, t), 30, 33 where x describes the longitudinal location along the transmission line: -l/2 < x < l/2, with the SQUID located at x = 0. In terms of φ, the transmission line current and voltage are I T (x, t) = -Φ 0 2πL T ∂φ(x, t) ∂x (1) and V T (x, t) = Φ 0 2π ∂φ(x, t) ∂t , (
) 2
where Φ 0 = h/(2e) is the flux quantum. Neglecting for now the couplings to the feedline, stripline and mechanical oscillator environments, the equations of motion for the closed system comprising the superconducting transmission line-SQUID-mechanical oscillator are as follows (see, e.g., Ref. 14 for a derivation of related equations of motion for a mechanical rf-SQUID): ∂ 2 φ ∂t 2 = (L T C T ) -1 ∂ 2 φ ∂x 2 , (3) ω -2 J γ-+ cos(γ + ) sin(γ -) + 2β -1 L γ --π n + (Φ ext + λB ext l osc y) Φ 0 = 0, (4) ω -2 J γ+ + sin(γ + ) cos(γ -) -
I T 2I c = 0, ( 5
)
and
mÿ + mω 2 m y - Φ 0 πL λB ext l osc γ -= 0, ( 6
)
where ω J = 2πI c /(C J Φ 0 ) is the plasma frequency of the SQUID Josephson junctions, the dimensionless parameter β L = 2πLI c /Φ 0 , L is the self inductance of the SQUID, n is an integer arising from the single-valuedness condition for the phase 2γ -around the loop, and I T is shorthand for I T (x = 0, t). Eq. ( 3 ) is simply the wave equation for the phase field coordinate φ(x, t) of the transmission line. Eq. ( 4 ) describes the current circulating the loop, which depends on the external flux threading the loop. Eq. ( 5 ) describes the average current threading the loop, which from current conservation is equal to one-half the transmission line current at x = 0. With the circulating SQUID current given by Φ 0 γ -/(πL) (up to a 6 Φ ext dependent term), we recognize in Eq. ( 6 ) the Lorentz force acting on the mechanical oscillator.
In addition to the equations of motion, we have the following current and voltage boundary conditions:
I T (x = ±l/2, t) = 0 ( 7
) and ∂ (L eff [Φ ext (y), I T ]I T ) ∂t = V T (0 -, t) -V T (0 + , t), ( 8
)
where the external flux and current-dependent, effective inductance L eff [Φ ext (y), I T ] of the SQUID as 'seen' by the transmission line is
L eff [Φ ext (y), I T ] = Φ 0 γ + 2πI T + L 4 , ( 9
)
with Φ ext (y) = Φ ext + λB ext l osc y. Note that we have set n = 0, since observable quantities do not depend on n.
We now make the following assumptions and consequent approximations: (a) ω J ≫ ω T ≫ ω m (where ω T is the relevant resonant mode of the transmission line); neglect the SQUID inertia terms ω -2 J γ± . (b) β L ≪ 1; solve for γ ± as series expansions to first order in β L . (c) |B ext l osc y| /Φ 0 ≪ 1; series expand the equations of motion to first order in y(t).
(d) |I T /I c | = Φ 0 2πL T Ic ∂φ(0,t) ∂x
≪ 1; series expand the equations of motion to second order in I T .
With ω J 's typically in the tens of GHz, assumption (a) is reasonable. From Eq. ( 4 ), we see that a small β L value prevents the γ -coordinate from getting trapped in its various potential minima, causing unwanted hysteresis. With the γ + expansion in I T consisting of only odd powers, approximations (a) and (d) amount to describing the SQUID simply as a current independent, Φ ext -tunable passive inductance element L eff [Φ ext (y)] that also depends on the mechanical oscillator position coordinate y. Including the next-to-leading, I 3 T term in the γ + expansion gives an I 2 T -dependent, nonlinear correction to the SQUID effective inductance. The consequences of including this nonlinear correction term for the quantum-limited displacement detection sensitivity will be considered in a forthcoming paper. 19 Solving for γ + to order I T and substituting in Eq. ( 9 ), we obtain:
L eff [Φ ext (y)] ≈ Φ 0 4πI c sec πΦ ext (y) Φ 0 , ( 10
)
where the self inductance L contribution has been neglected since it is of order β L ≪ 1.
Solving for γ -to order I 2 T and substituting into Eq. ( 6 ), we obtain for the mechanical 7 oscillator equation of motion:
mÿ + mω 2 m y - πλB ext l osc I 2 T 8I c tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) = 0, ( 11
)
where from (c), we have set y = 0 in the solution for γ -and have dropped an overall constant term. Since the γ -expansion in I T consists only of even powers, we must go to second order in I T so as to have a non-trivial transmission line-oscillator effective coupling.
Thus, the SQUID phase coordinates γ ± have been completely eliminated from the equations of motion, a consequence of approximation (a); the SQUID mediates the interaction between the mechanical oscillator coordinate y and transmission line coordinate φ without retardation effects. From Eq. ( 11 ), it might appear that the force on the mechanical oscillator due to the transmission line can be made arbitrarily large by tuning Φ ext close to Φ 0 /2. Note, however, that the proper conditions for the validity of the I T and β L expansions are:
I T I c sec (πΦ ext /Φ 0 ) ≪ 1 ( 12
) and |β L sec (πΦ ext /Φ 0 )| ≪ 1. ( 13
)
We now restrict ourselves to a single transmission line mode and derive approximate equations of motion for the mode amplitude. Suppose that the mechanical oscillator position coordinate is held fixed at y = 0. The following phase field satisfies the current boundary conditions (7):
φ(x, t) = -φ(t) cos [k 0 (x + l/2)] ; x < 0 +φ(t) cos [k 0 (x -l/2)] ; x > 0 , ( 14
)
with the wavenumber k 0 determined by the voltage boundary condition (8):
k 0 l 2 tan k 0 l 2 = - L T l L eff (Φ ext ) . ( 15
)
The wave equation (3) gives for the transmission mode frequency: ω T = k 0 / √ L T C T . Substituting the phase field (14) into the I T part of the oscillator equation of motion (11) furthermore gives the transmission line force acting on the oscillator with fixed coordinate y = 0. Now release the mechanical oscillator coordinate and suppose that for small [condition (c)] , slow [condition (a)] displacements, the force is the same to a good approximation.Then the 8 oscillator equation of motion becomes mÿ(t) + mω 2 m y(t) +
1 4 C T l Φ 0 2π 2 sin 2 (k 0 l/2) × - λB ext l osc (Φ 0 /2π) • Φ 0 4πL T lI c tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) ω 2 T φ 2 (t) = 0, ( 16
)
From Eq. ( 16 ), we can determine the mechanical sector of the Lagrangian, along with the interaction potential involving y and the mode amplitude φ. The remaining transmission line sector follows from the wave equation (3) and we thus have for the total Lagrangian:
L φ, y, φ, ẏ = 1 2 m ẏ2 - 1 2 mω 2 m y 2 + 1 2 C T l Φ 0 2π 2 sin 2 (k 0 l/2) × 1 2 φ2 - 1 2 1 - λB ext l osc y (Φ 0 /2π) • Φ 0 4πL T lI c tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) ω 2 T φ 2 . ( 17
)
From Eq. ( 17 ), we see that for motion occuring on the much longer timescale ω -1 m ≫ ω -1 T , the mechanical oscillator has the effect of modulating the frequency of the transmission line mode.
The associated Hamiltonian is
H (φ, y, p φ , p y ) = 2 C T l Φ 0 2π 2 sin 2 (k 0 l/2) 1 2 p 2 φ + 1 2 C T l Φ 0 2π 2 sin 2 (k 0 l/2) × 1 - λB ext l osc y (Φ 0 /2π) • Φ 0 4πL T lI c tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) 1 2 ω 2 T φ 2 + p 2 y 2m + 1 2 mω 2 m y 2 . ( 18
)
Let us now quantize. For the transmission line mode coordinate, the raising(lowering) operator is defined as:
â± T = 1 2 ω T 1 2 C T l (Φ 0 /2π) 2 sin 2 (k 0 l/2) 1 2 C T l Φ 0 2π 2 sin 2 (k 0 l/2) ω T φ ∓ ip φ ( 19
)
and for the mechanical oscillator
â± m = 1 √ 2mω (mω ŷ ∓ ip y ) . ( 20
)
In terms of these operators, the Hamiltionian (18) becomes (for notational convenience we omit from now on the 'hats' on the operators and also the 'minus' superscript on the lowering operator):
H = ω T a + T a T + ω m a + m a m + 1 2 ω T K T m a T + a + T 2 a m + a + m , ( 21
) 9
where the dimensionless coupling parameter between the mechanical oscillator and transmission line mode is
K T m = - λB ext l osc ∆x zp (Φ 0 /2π) Φ 0 4πL T lI c tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) , ( 22
)
with ∆x zp = /(2mω m ) the zero-point uncertainty of the mechanical oscillator. From expression (10) for the effective inductance, another way to express the coupling parameter is as follows:
K T m = - λB ext l osc ∆x zp (Φ 0 /2π) Φ 0 π dL eff /dΦ ext L T l . ( 23
)
From Eq. ( 23 ), we see that in order to increase the coupling between the mechanical oscillator and transmission line, the SQUID effective inductance-to-transmission line inductance ratio must be increased. The advantage of using a SQUID over an ordinary, geometrical mutual inductance between a transmission line and micron-sized mechanical oscillator is that the former can give a much larger effective inductance. As we shall see in Sec. IV, just requiring that the inductances be matched such that Φ 0 π dL eff /dΦext L T l ∼ 1 is sufficient for strong back reaction effects with modest drive powers, even though the other term in K T m describing the flux induced for a zero-point displacement is typically very small.
Assuming then that K T m ≪ 1 and making the rotating wave approximation (RWA) for the 'T ' part of the interaction term in the system Hamiltonian (21), i.e., neglecting the terms (a T ) 2 and (a + T ) 2 , we have (up to an unimportant additive constant):
H = ω T a + T a T + ω m a + m a m + ω T K T m a + T a T a m + a + m . ( 24
)
Many other systems are modeled by this form of Hamiltonian, a notable example being the single mode of an optical cavity interacting via radiation pressure with a mechanically compliant mirror. 22, 23, 24, 25, 26 Thus, much of the subsequent analysis will be relevant to a broad class of coupled resonator devices-not to just the transmission line-SQUID-mechanical resonator system.
So far, we have treated the transmission line and mechanical resonator as a closed system with SQUID-induced effective coupling . Of course, a real transmission line mode will experience damping and accompanying fluctuations, not least because it must be coupled to the 10 outside world in order for its state to be measured. Furthermore, the mechanical resonator mode will of course be damped even when decoupled from the SQUID. It is straightforward to incorporate the various baths and pump/probe feedline in terms of raising/lowering operators. Assuming weak system-bath couplings, which again justify the RWA, we have for the full Hamiltonian:
H = ω T a + T a T + ω m a + m a m + ω T K T m a + T a T a m + a + m + dωωa + p (ω)a p (ω) + dωωa + e (ω)a e (ω) + dωωa + b (ω)a b (ω) + dω K * pT a + p (ω)a T + K pT a + T a p (ω) + dω K * eT a + e (ω)a T + K pT a + T a e (ω) + dω K * bm a + b (ω)a m + K bm a + m a b (ω) -2mω m (a m + a + m )F ext (t), ( 25
)
where a p denotes the pump/probe (p) feed line operator, a e the transmission line bath ('e' for 'environment') operator, and a b the mechanical resonator bath (b) operator. These operators satisfy the usual canonical commutation relations:
a i (ω), a + j (ω ′ ) = δ ij δ(ω -ω ′ ). ( 26
)
The couplings between these baths and the transmission line and mechanical resonator systems are denoted as K pT , K eT , and K bm . Note we have also included for generality a classical driving force F ext (t) acting on the mechanical resonator. This allows us the opportunity to later on analyze quantum limits on force detection in addition to displacement detection.
Within the RWA, it is straightforward to solve the Heisenberg equations for the bath operators and substitute these solutions into the Heisenberg equations for the transmission line and mechanical oscillator to give
da m dt = -iω m a m + i 2mω m F ext (t) -iω T K T m a + T a T -dω |K T m | 2 t t 0 dt ′ e -iω(t-t ′ ) a m (t ′ ) -i dωK bm e -iω(t-t 0 ) a b (ω, t 0 ) ( 27
)
and
da T dt = -iω T a T -iω T K T m a T a m + a + m -dω |K pT | 2 t t 0 dt ′ e -iω(t-t ′ ) a T (t ′ ) -i dωK pT e -iω(t-t 0 ) a p (ω, t 0 ) 11 -dω |K eT | 2 t t 0 dt ′ e -iω(t-t ′ ) a T (t ′ ) -i dωK eT e -iω(t-t 0 ) a e (ω, t 0 ). ( 28
)
We now make the so-called 'first Markov approximation', 34, 35 in which the frequency dependences of the couplings to the baths are neglected:
K pT (ω) = γ pT π e iφ pT K eT (ω) = γ eT π e iφ eT K bm (ω) = γ bm π e iφ bm , ( 29
)
where the γ's and φ's are independent of ω as stated. The Heisenberg equations of motion (27) and (28) then simplify to
da m dt = -iω m a m + i 2mω m F ext (t) -iω T K T m a + T a T -γ bm a m (t) -i 2γ bm e iφ bm a in b (t) ( 30
) and da T dt = -iω T a T -iω T K T m a T a m + a + m -γ pT a T (t) -i 2γ pT e iφ pT a in p (t) -γ eT a T (t) -i 2γ eT e iφ eT a in e (t), ( 31
)
where the γ i 's are the various mode amplitude damping rates (assumed much smaller than their associated mode frequencies) and the 'in' operators 10, 31, 34, 35 are defined as
a in i (t) = 1 √ 2π dωe -iω(t-t 0 ) a i (ω, t 0 ), ( 32
)
with t > t 0 . The time t 0 can be taken to be an instant in the distant past before the measurement commences and when the initial conditions are specified (see below). We can similarly define 'out' operators:
a out i (t) = 1 √ 2π dωe -iω(t-t 1 ) a i (ω, t 1 ), ( 33
)
with t 1 > t. The time t 1 can be taken to be an instant in the distant future after the measurement has finished. From the Heisenberg equations for the bath operators and the definitions of the 'in' and 'out' operators, we obtain the following identities between them: 34,35
a out p (t) -a in p (t) = -i 2γ pT e -iφ pT a T ( t
) 12 a out b (t) -a in b (t) = -i 2γ bm e -iφ bm a m (t) a out e (t) -a in e (t) = -i 2γ eT e -iφ eT a T (t). (34)
In outline, the method of solution runs in principle as follows: 31, 34, 35, 36 (1) specify the 'in' operators. (2) Solve for the system operators a m (t) and a T (t) in terms of the 'in' operators.
(3) Use the relevant identity (34) to determine the 'out' operator a out p (t), which yields the desired probe signal. It is more convenient to solve the Heisenberg equations in the frequency domain with the Fourier transformed operators O(t) = 1 √ 2π ∞ -∞ dωe -iωt O(ω). The equations for the system operators then become
a m (ω) = 1 ω -ω m + iγ bm 2γ bm e iφ bm a in b (ω) - 1 √ 2m ω m F ext (ω) + ω T K T m 2 √ 2π ∞ -∞ dω ′ a T (ω ′ )a + T (ω ′ -ω) + a + T (ω ′ )a T (ω + ω ′ ) ( 35
) and a T (ω) = 1 ω -ω T + i(γ pT + γ eT ) 2γ pT e iφ pT a in p (ω) + 2γ eT e iφ eT a in e (ω) + ω T K T m √ 2π ∞ -∞ dω ′ a T (ω ′ ) a m (ω -ω ′ ) + a + m (ω ′ -ω) , ( 36
)
while the relevant 'in/out' operator identity becomes
a out p (ω) = -i 2γ pT e -iφ pT a T (ω) + a in p (ω). ( 37
)
C. Observables and 'in' states
Before proceeding with the solution to Eqs. (35) and (36), let us first devote some time to deriving expressions for observables that we actually measure in terms of a out p (ω). Model the pump/probe feedline as a semi-infinite transmission line -∞ < x < 0. Solving the wave equation for the decoupled transmission line and then using the expressions (1), (2) relating the current/voltage to the phase coordinate, we obtain
I out (x, t) = - ∞ -∞ dω ω πZ p sin (ωx/v p ) e -iωt a out p (ω) + e iωt a out+ p (ω) ( 38
)
and
V out (x, t) = i ∞ -∞ dω Z p ω π cos (ωx/v p ) e -iωt a out p (ω) -e iωt a out+ p (ω) , ( 39
) 13
where the sinusoidal x dependence in the current expression follows from the vanishing of the current boundary condition at x = 0, the feedline impedance is Z p = L p /C p and the wave propagation velocity is v p = 1/ L p C p . Suppose the current/volt meter is at x → -∞, so that the actual observables correspond to measuring the left-propagating component of the current/voltage. Then decomposing the x-dependent trig terms into their real and imaginary parts, we can identify the left propagating current/voltage operators as
I out (x, t) = -i 4πZ p ∞ 0 dω √ ω e -iω(x/vp+t) a out p (ω) -a out+ p (-ω) +e iω(x/vp+t) a out p (-ω) -a out+ p (ω) ( 40
)
and
V out (x, t) = i Z p 4π ∞ 0 dω √ ω e -iω(x/vp+t) a out p (ω) -a out+ p (-ω) +e iω(x/vp+t) a out p (-ω) -a out+ p (ω) . ( 41
)
The output signal of interest due to the mechanical oscillator signal input will lie within some bandwidth δω centered at ω s , the 'signal' frequency, and so we define the filtered output current I out (x, t|ω s , δω) and voltage V out (x, t|ω s , δω) to be the same as the above, left-moving operators, but with the integration range instead restricted to the interval [ω s -δω/2, ω s + δω/2].
Since the motion of the mechanical resonator modulates the transmission line frequency, one way to transduce displacements is to measure the relative phase shift between the 'in' pump current and 'out' probe current using the homodyne detection procedure. 35 Another common way is to measure the 'out' power relative to the 'in' power, or equivalently the mean-squared current/voltage (all three quantities differ by trivial factors of Z p ). We will discuss the latter method of transduction; the former, homodyne method can be straightforwardly addressed using similar techniques to those presented here. Thus, we consider the following expectation value:
δI out (x, t|ω s , δω) 2 = I out (x, t|ω s , δω) 2 -I out (x, t|ω s , δω) 2 , ( 42
)
where the angle brackets denote an ensemble average with respect to the 'in' states of the various baths and feedline (see below). If the mechanical oscillator is being driven by a classical external force whose fluctuations are invariant under time translations, i.e., 14 F ext (t)F ext (t ′ ) = C(t -t ′ ), then the above, mean-squared current will be time-independent.
Alternatively, if F ext (t) is, e.g., some deterministic, AC drive, then we must also time-average so as to get a time-independent measure of the detector response:
[δI out (x, t|ω s , δω)] 2 = 1 T M T M /2 -T M /2 dt I out (x, t|ω s , δω) 2 , ( 43
)
where T M is duration of the measurement, assumed much larger than all other timescales associated with the detector dynamics. We have also assumed that the time-averaged current vanishes in the signal bandwidth of interest: I out (ω s , δω) = 0. Substituting in the expression (40) for I out (x, t|ω s , δω) in terms of the a out p operators, we obtain after some algebra:
[δI out (ω s , δω)] 2 = 1 Z p ωs+δω/2 ωs-δω/2 dω 1 dω 2 2π ω 1 2 (ω 1 -ω 2 ) T M sin [(ω 1 -ω 2 )T M /2] × 1 2 a out p (ω 1 )a out+ p (ω 2 ) + a out+ p (ω 2 )a out p (ω 1 ) . ( 44
)
As 'in' states, we suppose k B T ≪ ω T , such that the relevant transmission line 'in' bath modes (ω e ∼ ω T ) are assumed to be approximately in the vacuum state. On the other hand, with the mechanical mode typically at a much lower frequency ω m ≪ ω T , we assume that its relevant 'in' bath modes (ω b ∼ ω m ) are in the proper, non-zero temperature thermal state.
For the pump/probe feedline, we consider the following coherent state:
30 |{α(ω)} p = exp dωα(ω) a in+ p (ω) -a in p (ω) |0 p , ( 45
)
where |0 p is the vacuum state and
α(ω) = -I 0 Z p T 2 M 2 e -(ω-ωp) 2 T 2 M /2 √ ω , ( 46
)
normalized such that the amplitude of the expectation value of I in [the right propagating version of (40) with a out p replaced by a in p ] with respect to this state is just I 0 . Again, we suppose k B T ≪ ω p , so that thermal fluctuations of the feedline are neglected. The frequency width of this pump drive is assumed to be the inverse lifetime of the measurement.
Below we shall see that the output mechanical signal will appear as two 'satellite' peaks on either side of the central peak at ω p due to the pump signal, i.e, the mechanical signal can be extracted by centering the filter at either of ω s = ω p ± ω m (up to a renormalization of the mechanical oscillator frequency), corresponding to the anti-Stokes and Stokes bands. 15 Note that we do not have to specify the initial t 0 states of the mechanical resonator and transmission line systems; a T (t 0 ) and a m (t 0 )-dependent initial transients have been dropped in the above equations for a T (ω) and a m (ω), since they give a negligible contribution to the long-time, steady-state behavior of interest.
We are now ready to solve for [δI out ] 2 . Introduce the following shorthand notation:
S T (ω) = 2γ pT e iφ pT a in p (ω) + 2γ eT e iφ eT a in e (ω) S m (ω) = 2γ bm e iφ bm a in b (ω) - 1 √ 2m ω m F ext (ω) K = ω T K T m √ 2π , ( 47
)
and γ T = γ pT +γ eT , the net transmission line mode amplitude dissipation rate due to loss via the probe line and the transmission line bath. Substituting Eq. (35) for a m (ω) into Eq. (36) for a T (ω) yields the following, single equation in terms of a T (ω) only:
a T (ω) = ∞ -∞ dω ′ a T (ω -ω ′ )A(ω, ω ′ ) + ∞ -∞ dω ′ B(ω, ω ′ )a T (ω -ω ′ ) × ∞ -∞ dω ′′ a T (ω ′′ )a + T (ω ′′ -ω ′ ) + a + T (ω ′′ )a T (ω ′′ + ω ′ ) + C(ω), ( 48
)
where, for the convenience of subsequent calculations, we have made this equation as concise as possible with the following definitions:
A(ω, ω ′ ) = K ω -ω T + iγ T S m (ω ′ ) ω ′ -ω m + iγ bm + S + m (-ω ′ ) -ω ′ -ω m -iγ bm , B(ω, ω ′ ) = K 2 /2 ω -ω T + iγ T 1 ω ′ -ω m + iγ bm + 1 -ω ′ -ω m -iγ bm , C(ω) = S T (ω) ω -ω T + iγ T . ( 49
)
We expand Eq. (48) for a T (ω) to first order in the mechanical oscillator bath operator a in b (ω) and external driving force
F ext (ω) [equivalently expand in A(ω, ω ′ )]: a T (ω) ≈ a ( 0
) T (ω)+ a (1) T (ω), where a (0) T (ω) = ∞ -∞ dω ′ B(ω, ω ′ )a (0) T (ω -ω ′ ) 16 × ∞ -∞ dω ′′ a (0) T (ω ′′ )a (0)+ T (ω ′′ -ω ′ ) + a (0)+ T (ω ′′ )a ( 0
) T (ω ′′ + ω ′ ) + C(ω) (50) and a (1) T (ω) = ∞ -∞ dω ′ a (0) T (ω -ω ′ )A(ω, ω ′ ) + ∞ -∞ dω ′ B(ω, ω ′ )a ( 1
) T (ω -ω ′ ) × ∞ -∞ dω ′′ a (0) T (ω ′′ )a (0)+ T (ω ′′ -ω ′ ) + a (0)+ T (ω ′′ )a (0) T (ω ′′ + ω ′ ) + ∞ -∞ dω ′ B(ω, ω ′ )a ( 0
) T (ω -ω ′ ) ∞ -∞ dω ′′ a (0) T (ω ′′ )a (1)+ T (ω ′′ -ω ′ ) +a (1)+ T (ω ′′ )a ( 0
) T (ω ′′ + ω ′ ) + a (1) T (ω ′′ )a (0)+ T (ω ′′ -ω ′ ) +a (0)+ T (ω ′′ )a ( 1
) T (ω ′′ + ω ′ ) . ( 51
)
Eq. (50) then yields the detector noise, while (51) yields the detector response to the signal within the linear response approximation. Thus, our approach here is to treat the mechanical oscillator as part of the detector degrees of freedom, with the signal defined as the thermal bath fluctuations and classical external force acting on the oscillator. This is the appropriate viewpoint for force detection. On the other hand, if the focus is on measuring the quantum state of the mechanical oscillator itself, then the oscillator should not be included as part of the detector degrees of freedom. Nevertheless, as we shall later see, the latter viewpoint can be straightforwardly extracted from the former under not too strong coupling K T m and pump drive current amplitude I 0 conditions.
The sequence of solution steps to Eqs. (50) and (51) are in principle as follows: (1) Solve first equation (50) for a (0) T (ω) in terms of B(ω, ω ′ ) and C(ω); (2) Substitute the solution for a (0) T (ω) into Eq. (51) for a (1) T (ω) and invert this Eq. (which is linear in a (1) T (ω)) to obtain the solution for a (1) T (ω) in terms of A(ω, ω ′ ), B(ω, ω ′ ), and C(ω). It is not clear how to carry out these steps in practice, however, since the equations involve products of noncommuting operators. Thus, we must find some way to solve by further approximation.
The key observation is that the feedline is in a coherent state, which is classical-like for sufficiently large current amplitude I 0 so as to ensure signal amplification. We therefore decompose a (0) T (ω) into a classical, expectation-valued part and quantum, operator-valued fluctuation part, a (0) ω), and subtitute into Eq. (50) for a (0) T (ω), 17 linearizing with respect to the quantum fluctuation δa (0) T (ω). This gives two equations, one for the expectation value a (0)
T (ω) = a (0) T (ω) + δa (0) T (
T (ω) = ∞ -∞ dω ′ B(ω, ω ′ ) a (0) T (ω -ω ′ ) ∞ -∞ dω ′′ a (0) T (ω ′′ ) a (0)+ T (ω ′′ -ω ′ ) + a (0)+ T (ω ′′ ) a (0) T (ω ′′ + ω ′ ) + C(ω) ( 52
)
and the other for the quantum fluctuation:
δa ( 0
) T (ω) = ∞ -∞ dω ′ B(ω, ω ′ )δa ( 0
) T (ω -ω ′ ) ∞ -∞ dω ′′ a ( 0
) T (ω ′′ ) a (0)+ T (ω ′′ -ω ′ ) + a (0)+ T (ω ′′ ) a ( 0
) T (ω ′′ + ω ′ ) + ∞ -∞ dω ′ B(ω, ω ′ ) a ( 0
) T (ω -ω ′ ) × ∞ -∞ dω ′′ δa ( 0
) T (ω ′′ ) a (0)+ T (ω ′′ -ω ′ ) + a ( 0
) T (ω ′′ ) δa (0)+ T (ω ′′ -ω ′ ) +δa (0)+ T (ω ′′ ) a ( 0
) T (ω ′′ + ω ′ ) + a (0)+ T (ω ′′ ) δa ( 0
) T (ω ′′ + ω ′ ) + δC(ω). ( 53
)
Eq. (51) for a (1) T (ω) is approximated by replacing a (0) T (ω) with its expectation value a (0) T (ω) , i.e., we drop the quantum fluctuation part δa (0) T (ω). This is because Eq. (51) already depends linearly on the quantum fluctuating signal term A(ω, ω ′ ), which we of course want to keep. Dropping the δa (0) T (ω) contribution to Eq. (51) amounts to neglecting multiplicative detector noise, which is reasonable given that we are concerned with large signal amplification.
The sequence of solutions steps are therefore in practice as follows: (1) Solve Eq. (52) first for a (0) T (ω) ; (2) Substitute this solution into Eq. (51) for a (1) T (ω) and invert; (3) Substitute the solution for a (0) T (ω) into the Eq. (53) for δa (0) T (ω) and invert; (4) Use these solutions for a (1) T (ω) and δa (0) T (ω) to determine the detector signal and noise terms, respectively. Beginning with step (1), we have
C(ω) = - i 2γ pT e iφ pT γ T -i∆ω a in p (ω) = i 2γ pT e iφ pT γ T -i∆ω • I 0 Z p T 2 M 2 ω e -(ω-ωp) 2 T 2 M /2 , ( 54
)
where ∆ω = ω p -ω T is the detuning frequency (not to be confused with the bandwidth δω) and note a in e (ω) = 0 (recall, we assume the transmission line resonant frequency ω T mode is in the vacuum state). Given that T M is the longest timescale in the system dynamics, C(ω) is sharply peaked about the frequency ω p and we will therefore approximate the 18 exponential with a delta function: C(ω) = cδ(ω -ω p ), where
c = i √ 2πe iφ pT γ T -i∆ω I 2 0 Z p γ pT ω p . ( 55
)
Considering for the moment an iterative solution to Eq. (52) for a (0) T (ω) , we see that a (0) T (ω) must also have the form of a delta function peaked at ω p : a (0) T (ω) = χδ(ω -ω p ). Substituting this ansatz into Eq. (52), we obtain the following equation for χ:
χ = 2χ |χ| 2 B(ω p , 0) + c. ( 56
)
This equation has a rather involved analytical solution. For sufficiently large |c| 2 |B(ω p , 0)| the response can become bistable (i.e., two locally stable solutions for χ). This region will not be discussed in the present paper, however. When we consider actual device parameters later in Sec. IV, we will assume sufficiently small drive such that χ ≈ c, allowing much simpler analytical expressions to be written down for the detector response.
Proceeding now to step (2), we substitute the expectation value a (0) T (ω) = χδ(ω -ω p ) for the operator a (0) T (ω) into Eq. (51) for a (1) T (ω). Carrying out the integrals, we obtain
1 -2 |χ| 2 [B(ω, 0) + B(ω, ω -ω p )] a (1) T (ω) -2χ 2 B(ω, ω -ω p )a (1)+ T (2ω p -ω) = χA(ω, ω -ω p ). ( 57
)
Before we can invert to obtain a (1) T (ω), we require a second linearly independent equation also involving a (1)+ T (2ω p -ω) and a (1) T (ω). This equation can be obtained by replacing ω with 2ω p -ω in Eq. ( 57 ) and then taking the adjoint:
1 + 2 |χ| 2 [B(ω -2∆ω, 0) + B(ω -2∆ω, ω -ω p )] a (1)+ T (2ω p -ω) +2χ * 2 B(ω -2∆ω, ω -ω p )a (1) T (ω) = -χ * A(ω -2∆ω, ω -ω p ), ( 58
)
where we have used the identities 2∆ω, 0). Inverting, we obtain a (1)
A + (2ω p -ω, ω p -ω) = -A(ω -∆ω, ω -ω p ), B * (2ω p - ω, ω p -ω) = -B(ω -2∆ω, ω -ω p ), and B * (2ω p -ω, 0) = -B(ω -
T (ω) = α 1 (ω)A(ω, ω -ω p ) + α 2 (ω)A(ω -2∆ω, ω -ω p ), ( 59
) where α 1 (ω) = D(ω) -1 1 + 2 |χ| 2 [B(ω -2∆ω, 0) + B(ω -2∆ω, ω -ω p )] χ ( 60
)
19 and α 2 (ω) = -2D(ω) -1 |χ| 2 B(ω, ω -ω p )χ, (61) with determinant
D(ω) = 1 -2 |χ| 2 [B(ω, 0) + B(ω, ω -ω p )] × 1 + 2 |χ| 2 [B(ω -2∆ω, 0) + B(ω -2∆ω, ω -ω p )] +4 |χ| 4 B(ω, ω -ω p )B(ω -2∆ω, ω -ω p ). ( 62
)
Moving on now to step (3), we substitute the expectation value a (0) T (ω) = χδ(ω -ω p ) into Eq. (53) for δa (0) T (ω) and carry out the integrals to obtain:
1 -2 |χ| 2 [B(ω, 0) + B(ω, ω -ω p )] δa (0) T (ω) -2χ 2 B(ω, ω -ω p )δa (0)+ T (2ω p -ω) = δC(ω). ( 63
)
Replacing ω with 2ω p -ω in Eq. ( 63 ) and then taking the adjoint:
1 + 2 |χ| 2 [B(ω -2∆ω, 0) + B(ω -2∆ω, ω -ω p )] δa (0)+ T (2ω p -ω) +2χ * 2 B(ω -2∆ω, ω -ω p )δa ( 0
) T (ω) = δC + (2ω p -ω). ( 64
)
Inverting Eqs. (63) and (64), we obtain
δa ( 0
) T (ω) = β 1 (ω)δC(ω) + β 2 (ω)δC + (2ω p -ω), ( 65
) where β 1 (ω) = D(ω) -1 1 + 2 |χ| 2 [B(ω -2∆ω, 0) + B(ω -2∆ω, ω -ω p )] ( 66
) and β 2 (ω) = 2D(ω) -1 χ 2 B(ω, ω -ω p ) ( 67
)
We are now ready to carry out step (4). To obtain the detector response, we substitute into expression (44) for [δI out ] 2 the linear response approximation to the 'out' probe operator [see Eq. (37)]:
a out p (ω) = -i 2γ pT e -iφ pT a ( 1
) T (ω) + -i 2γ pT e -iφ pT δa ( 0
) T (ω) + δa in p (ω) . ( 68
)
The first square-bracketed term will give the signal contribution to the detector response, while the second bracketed term gives the noise contribution. Note that the average values 20 a (0) T (ω) and a in p (ω) are not required in the noise term since they give negligible contribution in the signal bandwidths of interest centered at ω s = ω p ± ω m . Substituting in the signal part of a out p (ω), we obtain after some algebra:
[δI out (ω s , δω)] 2 signal = I 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 ωs+δω/2 ωs-δω/2 dω 2π ω ω p γ 2 T (ω -ω p + ∆ω) 2 + γ 2 T × α 1 (ω) c + α 2 (ω) c ω -ω p + ∆ω + iγ T ω -ω p -∆ω + iγ T 2 × 2γ bm (ω -ω p -ω m ) 2 + γ 2 bm [2n(ω -ω p ) + 1] + 2γ bm (ω p -ω -ω m ) 2 + γ 2 bm [2n(ω p -ω) + 1] + I 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 1 2m ω m γ bm ωs+δω/2 ωs-δω/2 dωdω ′ 2π ω ω p γ 2 T (ω -ω p + ∆ω) 2 + γ 2 T × α 1 (ω) c + α 2 (ω) c ω -ω p + ∆ω + iγ T ω -ω p -∆ω + iγ T 2 × sin [(ω -ω ′ )T M /2] (ω -ω ′ ) T M /2 2γ bm (ω -ω p -ω m ) 2 + γ 2 bm F ext (ω -ω p )F * ext (ω ′ -ω p ) + 2γ bm (ω p -ω -ω m ) 2 + γ 2 bm F ext (ω p -ω)F * ext (ω p -ω ′ ) , ( 69
)
where n(ω) = e ω/k B T -1 -1 is the Bose-Einstein thermal occupation number average for bath mode ω. The signal part of the detector response comprises a thermal component and a classical force component. In the limit of weak coupling K T m → 0 and or small drive current amplitude I 0 → 0, we have α 1 (ω)/c → 1, α 2 (ω)/c → 0 and we note that the frequency resolved detector response has the form of two Lorentzians centered at ω p ± ω m .
The resulting expression for the detector response coincides with an O(K 2 T m ) perturbative solution to the detector response (44) via the linear response Eqs. (50) and (51) (but no semiclassical approximation). However, as shall be described in Sec. IV, when the current drive is not small and or coupling is not weak, then the α i terms will modify this simple form, at the next level of approximation renormalizing the Lorentzians, i.e., shifting their location and changing their width.
Substituting in the noise part of a out p (ω), we obtain after some algebra:
[δI out (ω s , δω)] 2 noise = Z -1 p ωs+δω/2 ωs-δω/2 dω 2π ω 2γ T γ pT (ω -ω p + ∆ω) 2 + γ 2 T × |β 1 (ω)| 2 + (ω -ω p + ∆ω) 2 + γ 2 T (ω -ω p -∆ω) 2 + γ 2 T |β 2 (ω)| 2 -Re [β 1 (ω)] + (ω -ω p + ∆ω) γ T Im [β 1 (ω)] +Z -1 p ω s 2 δω 2π . ( 70
)
21 The noise part of the detector response comprises a back reaction component (the integral term) where transmission line noise drives the mechanical oscillator via the SQUID coupling, and a component that is added at the output due to zero-point fluctuations in the probe line. While not as obvious given the form of Eq. (70), one may again verify (see Sec. IV) that the detector back reaction on the mechanical oscillator takes the form of two Lorentzians centered at ω p ± ω m in the weak coupling and or weak current drive limit, coinciding with an O(K 2 T m ) perturbative calculation. Eqs. (69) and (70) are the main results of the paper, their sum giving the net output mean-squared current.
As articulated by Caves, 10 the fact that the 'in' and 'out' operators satisfy canonical commutation relations places a lower, quantum limit on the noise contribution to the detector response, Eq. (70). We now derive this quantum limit. First write the 'out' operator (68) as
a out p (ω) = -i 2γ pT e -iφ pT a ( 1
) T (ω) + N(ω), ( 71
)
where N(ω) = -i 2γ pT e -iφ pT δa (0) T (ω) + δa in p (ω) is the noise part. Taking commutators, we have the following identity relating the noise and signal operator terms:
N(ω), N + (ω ′ ) = δ(ω -ω ′ ) -2γ pT a (1) T (ω), a (1)+ T (ω ′ ) . ( 72
)
Now, from the Heisenberg Uncertainty Principle, one can derive the following general inequality:
N[f ]N + [f ] + N + [f ]N[f ] ≥ N[f ], N + [f ] , ( 73
)
where N[f ] = ∞ 0 dωf (ω)N(ω) and f (ω) is an arbitrary function. Inserting the commutator identity (72), Eq. (73) becomes
N[f ]N + [f ] + N + [f ]N[f ] ≥ ∞ 0 dω |f (ω)| 2 -2γ pT a (1) T [f ], a (1)+ T [f ] . ( 74
)
Choosing the 'filter' function f (ω) = ωΘ(ω -ω s + δω/2)Θ(ω s + δω/2 -ω) and evaluating the commutator, we obtain the following lower bound on the detector noise:
[δI out (ω s , δω)] 2 noise ≥ 22 Z -1 p ω s 2 δω 2π - I 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 ωs+δω/2 ωs-δω/2 dω 2π ω ω p γ 2 T (ω -ω p + ∆ω) 2 + γ 2 T × α 1 (ω) c + α 2 (ω) c ω -ω p + ∆ω + iγ T ω -ω p -∆ω + iγ T 2 × 2γ bm (ω -ω p -ω m ) 2 + γ 2 bm - 2γ bm (ω p -ω -ω m ) 2 + γ 2 bm . ( 75
)
In the next section we will address the extent to which the detector noise can approach the quantum bound on the right hand side of Eq. (75), depending on the current drive amplitude I 0 and other detector parameters.
To gain a better understanding of the detector response, we now provide analytical approximations to Eqs. (69) and (70) that are valid under the condition 56 )], i.e., the expectation value a (0) T (ω) for the transmission line depends approximately only on the pump/probe feedline state and not on the mechanical oscillator state. Explicitly, this condition reads:
|c| 2 |B(ω p , 0)| ≪ 1 such that χ ≈ c [see Eq. (
2I 2 0 Z p K 2 T m ω T γ pT ω m (γ 2 T + ∆ω 2 ) 3/2 ≪ 1, ( 76
)
placing an upper limit on I 0 and K T m for the validity of this approximation. We also assume that the mechanical and transmission line mode frequencies are widely separated:
ω m ≪ ω T , and with small damping rates: γ bm ≪ ω m , γ T ≪ ω T .
We do not restrict the relative magnitudes of ω m and γ T , however. A simple picture emerges in which the detector back reaction 'renormalizes' the mechanical oscillator frequency and damping rate:
ω m → R ω ω m and γ bm → R γ γ bm , where R ω ω m = ω m + ∆ω + |c| 2 ω 2 T K 2 T m πω m |c| 2 ω 2 T K 2 T m [γ 2 T + ∆ω 2 -ω 2 m ] π γ 2 T + (∆ω + ω m ) 2 γ 2 T + (∆ω -ω m ) 2 ( 77
)
and
R γ γ bm = γ bm -∆ω + |c| 2 ω 2 T K 2 T m πω m 2|c| 2 ω 2 T K 2 T m ω m γ T π γ 2 T + (∆ω + ω m ) 2 γ 2 T + (∆ω -ω m ) 2 , ( 78
) 23
where c is defined in Eq. ( 55 ). With the measurement filter bandwidth centered at either of ω s = ω p ± R ω ω m , the approximation to Eq. (69) for the signal response is (with the classical force term omitted):
[δI out (ω s = ω p ± R ω ω m , δω)] 2 signal = I 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 γ 2 T γ 2 T + (∆ω ± ω m ) 2 × ωs+δω/2 ωs-δω/2 dω 2π 2γ bm (ω -ω p ∓ R ω ω m ) 2 + (R γ γ bm ) 2 [2n(R ω ω m ) + 1] . ( 79
)
When there is a classical force acting on the mechanical oscillator, we must add to Eq. (79) the term
I 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 1 2m ω m γ bm ωs+δω/2 ωs-δω/2 dωdω ′ 2π γ 2 T (ω -ω p + ∆ω) 2 + γ 2 T × sin [(ω -ω ′ )T M /2] (ω -ω ′ ) T M /2 2γ bm (ω -ω p -R ω ω m ) 2 + (R γ γ bm ) 2 F ext (ω -ω p )F * ext (ω ′ -ω p ) + 2γ bm (ω p -ω -R ω ω m ) 2 + (R γ γ bm ) 2 F ext (ω p -ω)F * ext (ω p -ω ′ ) . ( 80
)
The approximation to Eq. (70) for the detector noise is
[δI out (ω s = ω p ± R ω ω m , δω)] 2 noise = I 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 γ 2 T γ 2 T + (∆ω ± ω m ) 2 × ωs+δω/2 ωs-δω/2 dω 2π 2γ bm (ω -ω p ∓ R ω ω m ) 2 + (R γ γ bm ) 2 N ± + Z -1 p ω s 2 δω 2π , ( 81
)
where the back reaction noise parameter is
N ± = |c| 2 K 2 T m ω 2 T γ T πγ bm γ 2 T + (∆ω ∓ ω m ) 2 ∓ 1 = 2I 2 0 Z p K 2 T m ω T γ T γ pT γ bm [γ 2 T + ∆ω 2 ] γ 2 T + (∆ω ∓ ω m ) 2 ∓ 1. ( 82
)
The ∓1 term in the back reaction noise parameter depends on whether the filter is centered
at ω s = ω p + ω m or ω s = ω p -ω m
and corresponds respectively to 'phase preserving' or 'phase conjugating' detection as discussed in Caves. 10 In the limit I 0 → 0 and or K T m → 0, we see from Eqs. (79), (81), and (82) that the back reaction noise amounts to doubling the oscillator quantum zero-point motion signal in the phase conjugating case, while the back reaction noise exactly cancels the quantum zero-point motion signal in the phase preserving case. In both cases, the noise coincides with the lower quantum bound (75). However, in this small drive/coupling limit, we do not have a detector or amplifier but rather an attenuator, which is of only academic interest to us.
Comparing the detector response (79) and back reaction part of Eq. ( 81 ), we see that the mechanical oscillator behaves in the steady state as if in contact with a thermal bath. 8, 9, 12, 26, 37, 38, 39 The back reaction of the detector on the mechanical oscillator is effectively that of a thermal bath with damping rate γ back = γ bm (R γ -1) and effective thermal average occupation number n back defined as follows:
γ back (2n ± back + 1) = γ bm N ± . ( 83
)
Thus,
n ± back = (R γ -1) -1 1 2 N ± - 1 2 . ( 84
)
The failure to approach the lower quantum bound (75) when N ± ≫ 1 then translates into having (2n ± back + 1)γ back /γ bm ≫ 1. Thus, to get close to the bound, we necessarily require γ back ≪ γ bm ; 12 the back reaction occupation number n ± back does not have to be small. With the mechanical oscillator also in thermal contact with its external bath, the net damping rate of the oscillator is γ net = γ bm + γ back = R γ γ bm and the net, effective thermal average occupation number n net of the oscillator is defined as follows:
γ net 2n ± net + 1 = γ bm [2n(R ω ω m ) + 1] + γ back 2n ± back + 1 . ( 85
)
Thus,
n ± net = R -1 γ n(R ω ω m ) + 1 2 + 1 2 N ± - 1 2 . ( 86
)
From Eq. ( 78 ), we see that depending on the detuning parameter ∆ω = ω p -ω T , the damping rate of the oscillator due to the detector back reaction can be either negative or positive. Specifically, positive damping requires the following condition on the detuning parameter:
∆ω < - |c| 2 ω 2 T K 2 T m πω m = - 2I 2 0 Z p K 2 T m ω T γ pT ω m (γ 2 T + ∆ω 2 ) . ( 87
)
B. Displacement sensitivity
In the absence of a classical force acting on the mechanical oscillator, from Eq. (79) the mechanical oscillator thermal noise displacement signal spectral density takes the familiar Lorentzian form:
S x (ω)| signal = 2R γ γ bm (ω -ω p ∓ R ω ω m ) 2 + (R γ γ bm ) 2 2mR ω ω m [2n (R ω ω m ) + 1] . ( 88
)
25 In order to be able to resolve this mechanical signal, the detector noise (81) referred to the mechanical oscillator input must be smaller than (88). The detector noise spectral density at the input is
S x (ω = ω p ± R ω ω m )| noise = 2 R γ γ bm ∓1 + |c| 2 K 2 T m ω 2 T γ T πγ bm γ 2 T + (∆ω ∓ ω m ) 2 + 2πR γ γ 2 T + (∆ω ± ω m ) 2 |c| 2 K 2 T m ω 2 T γ pT 2mR ω ω m , ( 89
)
where the first term on the right hand side is the back reaction noise acting on the mechanical oscillator and the second term is the output, probe line zero-point noise referred to the input.
Note that the noise has been evaluated at ω = ω p ±R ω ω m , the maximum of the back reaction Lorentzian.
If the detector output is to depend linearly on the mechanical oscillator signal input (i.e., function as a linear amplifier), then back reaction effects must be small. In particular, we require that γ back ≪ γ bm , i.e., R γ ≈ 1. With |c| being proportional I 0 , we see from Eq. (89) that increasing the drive current amplitude I 0 increases the back reaction noise, but decreases the probe line noise referred to the mechanical oscillator input. Thus, there is an optimum I 0 such that the sum S x | noise is a minimum. Making the approximation R γ = 1 and R ω = 1 in Eq. ( 89 ) and optimizing with respect to |c|, we find
S x (ω = ω p ± R ω ω m )| noise-optimum = mω m γ bm ∓1 + 2 γ T γ pT γ 2 T + (∆ω ± ω m ) 2 γ 2 T + (∆ω ∓ ω m ) 2 . ( 90
)
From Eq. ( 90 ), we see that the noise is further reduced if (i) the dominant source of transmission line mode dissipation is due to energy loss through the coupled probe (information gathering) line: 12 γ T ≈ γ pT ; (ii) the detuning frequency is chosen to be ∆ω = ∓ γ 2 T + ω 2 m , where the minus (plus) sign corresponds to phase preserving (conjugating) detection. With this detuning choice, the condition R γ ≈ 1 requires (ω m /γ T ) 2 ≪ 1 and so the minimum detector noise is
S x (ω = ω p ± R ω ω m )| noise-optimum = mω m γ bm 2 ∓ 1 + O (ω m /γ T ) 2 , ( 91
)
where in order to determine the O ((ω m /γ T ) 2 ) term, the full form of R γ given in Eq. (78) must be used in Eq. (89) when optimizing. Comparing with Eq. (88) for the signal noise, we see that to leading order the detector noise effectively doubles the zero-point signal in 26 1 1.5 2 2.5 3 3.5 4 I 0 10 8 A 0.5 1 1.5 2 2.5 3 3.5 4 S x mΩ m Γ bm FIG. 2: Displacement detector noise spectral density (solid line) and lower bound (dashed line) versus drive current amplitude. The noise densities are evaluated at ω = ω p +R ω ω m , corresponding to phase preserving detection.
the phase preserving case. This exceeds the lower bound on the detector noise derived from Eq. (75), which is zero to leading order in the phase preserving case.
We now numerically evaluate Eq. (89) for the detector noise. The feasible example parameter values we use are: 14 B ext = 0.005 Tesla, Z p = 50 Ohms, ω T /2π = 3×10 9 s -1 , Q T = ω T /(2γ T ) = 100, γ T = 9.4 × 10 7 s -1 , l osc = 5 µm, λ = 1 (geometrical correction factor), m = 10 -16 kg, ω m = 2.5 × 10 7 s -1 , and Q bm = ω m /(2γ bm ) = 10 3 . These values give a mechanical oscillator zero-point uncertainty ∆x zp = 1.45 × 10 -13 m, a zero-point displacement noise /(mω m γ bm ) = 3.4 × 10 -30 m 2 /Hz, and a dimensionless coupling strength K T m = -1.1 × 10 -5 , where we assume that in the expression (22) for K T m , Φ ext can be chosen such that the dimensionless factor Φ 0 4πL T lIc tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) ≈ 1 (matching condition). We also suppose that γ T ≈ γ pT , i.e., the transmission line mode damping is largely due to the probe line coupling.
Fig. 2
shows S x (ω = ω p + R ω ω m )| noise × mω m γ bm /
and also the lower bound on the detector noise that follows from Eq. (75) for phase preserving detection. Note that the minimum detector noise is approximately 0.8 /(mω m γ bm ). Thus, for this example, the next-to-leading O ((ω m /γ T ) 2 ) term in Eq. (91) is approximately -0.2. Note also that the detector noise coincides with the lower bound in the small drive limit.
Consider a monochromatic classical driving force with frequency ω 0 ∼ R ω ω m acting on the oscillator: F ext (ω) = F 0 δ(ω -ω 0 ). The force signal spectral density is then S F (ω)| signal = F 2 0 δ(ω -ω 0 ). For force detection operation, the mechanical oscillator is included as part of the detector degrees of freedom. From Eqs. (79-82), the force noise spectral density evaluated at ω = ω p ± ω 0 is
S F (ω = ω p ± ω 0 )| noise = 2m ω m γ bm 2n (ω 0 ) + 1 ∓ 1 + |c| 2 K 2 T m ω 2 T γ T πγ bm γ 2 T + (∆ω ∓ ω m ) 2 + π (ω 0 -R ω ω m ) 2 + (R γ γ bm ) 2 γ 2 T + (∆ω ± ω m ) 2 γ bm |c| 2 K 2 T m ω 2 T γ pT . ( 92
)
Comparing the displacement noise (89) with the force noise (92), we see that the latter includes the additional 2m ω m γ bm [2n (ω 0 ) + 1] mechanical quantum thermal displacement noise term. Since the mechanical oscillator forms part of the force detector, it need not necessarily be weakly driven and or weakly coupled to the transmission line; as explained in Sec. III A, the present analysis employs a linear response approximation for force detection, not displacement detection. Thus, in determining the optimum I 0 (and or K T m ) and ∆ω such that S F | noise is a minimum, we should not assume a priori the restrictions R γ , R ω ≈ 1.
2 4 6 8 10 12 14 I 0 10 8 A 2 3 4 5 6 S F 2m Ω m Γ bm
FIG. 3: Force detector noise spectral density versus drive current amplitude for detuning ∆ω = 0 (solid line), ∆ω = -5ω m (dashed line), and ∆ω = -10ω m (dotted line) . The noise densities are evaluated at ω = ω p + R ω ω m , corresponding to phase preserving detection.
Fig. 3 shows the results of numerically evaluating the force noise spectral density given by Eq. (92) for phase preserving detection (ω = ω p + ω 0 ) and a range of detuning values.
The same example parameters are used as in the above displacement sensitivity analysis, with n(ω 0 ) = 0 and ω 0 = R ω ω m . The force noise is expressed in units 2m ω m γ bm = 6.6 × 10 -39 N 2 /Hz. Note that the minimum force noise is exactly 2 in these units, independently of the detuning, with the minimum occuring at larger I 0 values as the detuning is made progressively more negative.
From Eq. ( 86 ), we see that the net, thermal average occupation number n net of the mechanical oscillator's fundamental mode decreases as R γ increases. Thus, by increasing the drive and or coupling strength such that γ back ≫ γ bm , the mechanical oscillator can be effectively cooled at the expense of increasing its damping rate. 3, 8, 9, 21, 26, 40, 41, 42, 43, 44, 45, 46, 47, 48 Consider sufficiently negative detuning such that -∆ω ≫ |c| 2 ω 2 T K 2 T m /(πω m ) [see Eq. (87)]. Substituting definition (78) for R γ and definition (82) for N + into Eq. ( 86 ) and supposing R γ is large enough that we can neglect the external damping term γ bm , we obtain approximately for the phase preserving case:
n + net ≈ n (R ω ω m ) R γ + n + back , ( 93
) where n + back ≈ - γ 2 T + (∆ω + ω m ) 2 4∆ωω m - 1 2 . ( 94
)
This expression agrees with that derived in Ref. 26, apart from the 1/2 which is simply due to a small difference in the way we define n ± back in Eq. ( 83 ). Choosing optimum detuning ∆ω = -γ 2 T + ω 2 m to minimize n + back in Eq. ( 94 ), we therefore have
n + net ≈ n (R ω ω m ) R γ + 1 2 1 + (γ T /ω m ) 2 -1. ( 95
)
How much cooling can be achieved depends on (i) how large R γ can be, subject to the above inequality on -∆ω; (ii) making the ratio γ T /ω m as small as possible. 26
Using the same example parameter values as above, but taking instead a larger but still realistic quality factor Q bm = 10 4 for the mechanical oscillator, 6 the resulting numerically evaluated effective occupation number n + net [Eq. ( 86 )] is given in Fig. 4 for a range of external 29 5 10 15 20 25 30 35 40 I 0 10 8 A 1 1.5 2 2.5 3 3.5 4 n net FIG. 4: Net effective average occupation number of the mechanical oscillator versus drive current.
The solid curve is for external bath temperature T = 100 mK [n(R ω ω) = 523], the dashed curve is for T = 10 mK [n(R ω ω) = 52], and the dotted curve is for T = 1 mK [n(R ω ω) = 4.8].
bath occupation numbers n(ω m ). Thus, even for small coupling strengths K T m and drive current amplitudes I 0 , significant cooling of the mechanical oscillator can be achieved. This is in part a consequence of the fact that the quality factor Q bm of the mechanical oscillator when decoupled from the detector is very large.
In the present paper, we have attempted to give a reasonably comprehensive analysis of the quantum-limited detection sensitivity of a DC SQUID for drive currents well below the Josephson junction critical current I c . In this regime, the SQUID functions effectively as a mechanical position-dependent inductance element to a good approximation and the resulting closed system Hamiltonian (24) takes the same form as that for several other types of coupled mechanical resonator-detector resonator systems. Thus, the key derived expressions (69) and (70) for the detector response and detector noise are of more general application.
The main approximation made in analyzing the position and force detection sensitivity, as well as back reaction cooling, was to limit the drive current and or coupling strength according to Eq. (76). This allowed us to find much simpler, analytical approximations to the key expressions, in particular Eqs. (79) and (81). The regime of larger drive currents and 30 or coupling strengths which exceed the limit (76) remains to be explored. However, with the SQUID in mind, it is more appropriate to consider larger drive currents in the context of including the non-linear I/I c corrections to the SQUID effective inductance. This will be the subject of a forthcoming paper. 19
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Whitcomb, C. Wipf, and N. Mavalvala, quant-ph/0612188 (unpublished). | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In a series of recent experiments 1, 2, 3 and related theoretical work, 4, 5, 6, 7, 8, 9 it was demonstrated that a displacement detector based on either a normal or superconducting single electronic transistor (SSET) can resolve the motion of a micron-scale mechanical resonator close to the quantum limit as set by Heisenberg's Uncertainty Principle. 10, 11, 12 The displacement transduction was achieved by capacitively coupling the gated mechanical resonator to the SSET metallic island. When the resonator is voltage biased, motion of the latter changes the island charging energy and hence the Cooper pair tunnel rates. The resulting modulation in the source-drain tunnel current through the SSET is then read out as a signature of the mechanical motion. Given the success of this capacitive-based transduction method in approaching the quantum limit, it is natural to consider complementary, inductive-based transduction methods in which, for example, a superconducting quantum interference device (SQUID) is similarly used as an intermediate quantum-limited stage between the micron-scale mechanical resonator and secondary amplification stages. 13, 14, 15, 16 Unavoidable, fundamental noise sources and how they affect the SSET and SQUID devices are not necessarily the same. Furthermore, achievable coupling strengths between each type of device and a micron-scale mechanical resonator may be different. Therefore, it would be interesting to address the merits of the SQUID in comparison with the established SSET for approaching the quantum limit of displacement detection.\n\nIn the present paper, we analyze a DC SQUID-based displacement detector. The SQUID is integrated with a mechanical resonator in the form of a doubly-clamped beam, shown schematically in Fig. 1 . Motion of the beam changes the magnetic flux Φ threading the SQUID loop, hence modulating the current circulating the loop. We shall address the operation of the SQUID displacement detector in the regime for which the loop current is smaller than the Josephson junction critical current I c and at temperatures well below the superconducting critical temperature. We thus assume that resistive (normal) current flow through the junctions and accompanying current noise can be neglected. (See for example Ref. 17 for a quantum noise analysis of resistively shunted Josephson junctions and Ref. 18 for a related analysis of the DC SQUID.) Such an assumption cannot be made with the usual mode of operation for the SSET devices, where the tunnel current unavoidably involves the 2 quasiparticle decay of Cooper pairs, resulting in shot noise.\n\nAs noise source, we will consider the quantum electromagnetic fluctuations within the pump/probe feedline and also transmission line resonator that is connected to the SQUID. This noise is a consequence of the necessary dissipative coupling to the outside world and affects the mechanical signal output in two ways. First, the noise is added directly to the output in the probe line and, second, the noise acts back on the mechanical resonator via the SQUID, affecting the resonator's motion.\n\nWith the Josephson junction plasma frequencies assumed to be much larger than the other resonant modes of relevance for the device, the SQUID can be modeled to a good approximation as an effective inductance that depends on the external current I entering and exiting the loop, as well as on the applied flux. In this first of two papers, we shall make the further approximation of neglecting the I-dependence of the SQUID effective inductance, which requires the condition I ≪ I c . In the sequel, 19 we will relax this condition somewhat by including the next to leading O(I 2 ) term in the inductance and address the consequences of this non-linear correction for quantum-limited displacement detection.\n\nModeling the SQUID approximately as a passive inductance element, the transmission line resonator-mechanical resonator effective Hamiltonian is given by Eq. (24). This Hamiltonian describes many other detector-oscillator systems that are modeled as two coupled harmonic oscillators, including the examples of an LC resonator capacitively coupled to a mechanical resonator 20, 21 and an optical cavity coupled to a mechanically compliant mirror via radiation pressure; 22, 23, 24, 25, 26 the various systems are distinguished only by the dependences of the coupling strengths on the parameters particular to each system. Thus, many of the results of this paper are of more general relevance.\n\nThe central results of the paper are Eqs. (69) and (70), giving the detector response to a mechanical resonator undergoing quantum Brownian motion and also subject to a classical driving force. In the derivation of these expressions, we do not approximate the response as a perturbation series in the coupling between the SQUID and mechanical resonator as is conventionally done, but rather find it more natural to base our approximations instead on assumed weak coupling between the mechanical resonator and its external heat bath and weak classical driving force. Thus, in the context of the linear response paradigm, our detector should properly be viewed as including the mechanical resonator degrees of freedom as well, with the weak perturbative signal instead consisting of the heat bath force noise 3 and classical drive force acting on the mechanical resonator. Since the quality factors of actual, micron-scale mechanical resonators can be very large at sub-Kelvin temperatures (E.g., Q ∼ 10 5 in the experiments of Refs. 2,3), quantum electromagnetic noise in the transmission line part of the detector can have strong back reaction effects on the motion of the mechanical resonator, even when the coupling between the resonator and the SQUID is very weak. One consequence that we shall consider is cooling of the mechanical resonator fundamental mode, which requires strong back reaction damping combined with low noise.\n\nNevertheless, as we will also show, one can still analyze the quantum-limited detector linear response to the mechanical resonator's position signal using general expressions (69) and (70), under the appropriate conditions of small pump drive and weak coupling between the SQUID and mechanical resonator such that back reaction effects are small.\n\nThe outline of the paper is as follows. In Sec. II, we write down the SQUID-mechanical resonator equations of motion corresponding to the circuit scheme shown in Fig. 1 and then derive the Heisenberg equations for the various mode raising and lowering operators, subject to the above-mentioned approximations. In Sec. III, we solve the equations within the linear response approximation to derive the detector signal response and noise. In Sec. IV, we analyze both the position and force detection sensitivity, and address also back reaction cooling of the mechanical resonator. Sec. V provides concluding remarks."
},
{
"section_type": "OTHER",
"section_title": "A. Transmission line-SQUID-mechanical oscillator Hamiltonian",
"text": "Fig. 1 shows the displacement detector scheme. The device consists of a stripline resonator (transmission line T ) made of two sections, each of length l/2, connected via a DC SQUID (see Refs. 27,28,29,30 for related, qubit detection schemes). The transmission line inductance and capacitance per unit length are L T and C T respectively. The Josephson junctions in each arm of the SQUID are assumed to have identical critical currents I c and capacitances C J . A length l osc segment of the SQUID loop is free to vibrate as a doubly-clamped bar resonator and the fundamental flexural mode of interest (in the plane of the loop) is treated as a harmonic oscillator with mass m, frequency ω m and displacement coordinate y. The total external magnetic flux applied perpendicular to the SQUID loop is given by Φ ext +λB ext l osc y,\n\nwhere Φ ext is the flux corresponding to the case y = 0, B ext is the normal component of the magnetic field at the location of the vibrating loop segment (oscillator), and the dimensionless parameter λ < 1 is a geometrical correction factor accounting for the nonuniform displacement of the doubly-clamped resonator in the fundamental flexural mode. C J C J I c I c y p T T m,ω m γ pT γ eT γ bm Φ FIG. 1: Scheme for the displacement detector showing the pump/probe line 'p', transmission line resonator 'T ', and DC SQUID with mechanically compliant loop segment having effective mass m and fundamental frequency ω m . Note that the scale of the DC SQUID is exaggerated relative to that of the stripline for clarity.\n\nThe transmission line is weakly coupled to a pump/probe feedline (p), with inductance and capacitance per unit length L p and C p respectively, employed for delivering the input and output RF signals; the coupling can be characterized by a transmission line mode amplitude damping rate γ pT (see section II B below). Other possible damping mechanisms in the transmission line may be taken into account by adding a fictitious semi-infinite stripline environment (e), weakly coupled to the transmission line characterized by mode amplitude damping rate γ eT . 31 While γ eT can be made much smaller than γ pT with suitable transmission line resonator design, we shall nevertheless include both sources of damping in our analysis so as to eventually be able to gauge their relative effects on the detector displacement sensitivity [see Eq. (90)]. The SQUID, on the other hand, is assumed to be dissipationless. The 5 mechanical oscillator is also assumed to be coupled to an external heat bath (b), characterized by mode amplitude damping rate γ bm .\n\nA convenient choice of dynamical coordinates for the SQUID are γ ± = (φ 1 ± φ 2 ) /2, where φ 1 and φ 2 are the gauge invariant phases across each of the two Josephson junctions. 32 For the transmission line, we similarly use its phase field coordinate φ(x, t), 30, 33 where x describes the longitudinal location along the transmission line: -l/2 < x < l/2, with the SQUID located at x = 0. In terms of φ, the transmission line current and voltage are I T (x, t) = -Φ 0 2πL T ∂φ(x, t) ∂x (1) and V T (x, t) = Φ 0 2π ∂φ(x, t) ∂t , (\n\n) 2\n\nwhere Φ 0 = h/(2e) is the flux quantum. Neglecting for now the couplings to the feedline, stripline and mechanical oscillator environments, the equations of motion for the closed system comprising the superconducting transmission line-SQUID-mechanical oscillator are as follows (see, e.g., Ref. 14 for a derivation of related equations of motion for a mechanical rf-SQUID): ∂ 2 φ ∂t 2 = (L T C T ) -1 ∂ 2 φ ∂x 2 , (3) ω -2 J γ-+ cos(γ + ) sin(γ -) + 2β -1 L γ --π n + (Φ ext + λB ext l osc y) Φ 0 = 0, (4) ω -2 J γ+ + sin(γ + ) cos(γ -) -\n\nI T 2I c = 0, ( 5\n\n)\n\nand\n\nmÿ + mω 2 m y - Φ 0 πL λB ext l osc γ -= 0, ( 6\n\n)\n\nwhere ω J = 2πI c /(C J Φ 0 ) is the plasma frequency of the SQUID Josephson junctions, the dimensionless parameter β L = 2πLI c /Φ 0 , L is the self inductance of the SQUID, n is an integer arising from the single-valuedness condition for the phase 2γ -around the loop, and I T is shorthand for I T (x = 0, t). Eq. ( 3 ) is simply the wave equation for the phase field coordinate φ(x, t) of the transmission line. Eq. ( 4 ) describes the current circulating the loop, which depends on the external flux threading the loop. Eq. ( 5 ) describes the average current threading the loop, which from current conservation is equal to one-half the transmission line current at x = 0. With the circulating SQUID current given by Φ 0 γ -/(πL) (up to a 6 Φ ext dependent term), we recognize in Eq. ( 6 ) the Lorentz force acting on the mechanical oscillator.\n\nIn addition to the equations of motion, we have the following current and voltage boundary conditions:\n\nI T (x = ±l/2, t) = 0 ( 7\n\n) and ∂ (L eff [Φ ext (y), I T ]I T ) ∂t = V T (0 -, t) -V T (0 + , t), ( 8\n\n)\n\nwhere the external flux and current-dependent, effective inductance L eff [Φ ext (y), I T ] of the SQUID as 'seen' by the transmission line is\n\nL eff [Φ ext (y), I T ] = Φ 0 γ + 2πI T + L 4 , ( 9\n\n)\n\nwith Φ ext (y) = Φ ext + λB ext l osc y. Note that we have set n = 0, since observable quantities do not depend on n.\n\nWe now make the following assumptions and consequent approximations: (a) ω J ≫ ω T ≫ ω m (where ω T is the relevant resonant mode of the transmission line); neglect the SQUID inertia terms ω -2 J γ± . (b) β L ≪ 1; solve for γ ± as series expansions to first order in β L . (c) |B ext l osc y| /Φ 0 ≪ 1; series expand the equations of motion to first order in y(t).\n\n(d) |I T /I c | = Φ 0 2πL T Ic ∂φ(0,t) ∂x\n\n≪ 1; series expand the equations of motion to second order in I T .\n\nWith ω J 's typically in the tens of GHz, assumption (a) is reasonable. From Eq. ( 4 ), we see that a small β L value prevents the γ -coordinate from getting trapped in its various potential minima, causing unwanted hysteresis. With the γ + expansion in I T consisting of only odd powers, approximations (a) and (d) amount to describing the SQUID simply as a current independent, Φ ext -tunable passive inductance element L eff [Φ ext (y)] that also depends on the mechanical oscillator position coordinate y. Including the next-to-leading, I 3 T term in the γ + expansion gives an I 2 T -dependent, nonlinear correction to the SQUID effective inductance. The consequences of including this nonlinear correction term for the quantum-limited displacement detection sensitivity will be considered in a forthcoming paper. 19 Solving for γ + to order I T and substituting in Eq. ( 9 ), we obtain:\n\nL eff [Φ ext (y)] ≈ Φ 0 4πI c sec πΦ ext (y) Φ 0 , ( 10\n\n)\n\nwhere the self inductance L contribution has been neglected since it is of order β L ≪ 1.\n\nSolving for γ -to order I 2 T and substituting into Eq. ( 6 ), we obtain for the mechanical 7 oscillator equation of motion:\n\nmÿ + mω 2 m y - πλB ext l osc I 2 T 8I c tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) = 0, ( 11\n\n)\n\nwhere from (c), we have set y = 0 in the solution for γ -and have dropped an overall constant term. Since the γ -expansion in I T consists only of even powers, we must go to second order in I T so as to have a non-trivial transmission line-oscillator effective coupling.\n\nThus, the SQUID phase coordinates γ ± have been completely eliminated from the equations of motion, a consequence of approximation (a); the SQUID mediates the interaction between the mechanical oscillator coordinate y and transmission line coordinate φ without retardation effects. From Eq. ( 11 ), it might appear that the force on the mechanical oscillator due to the transmission line can be made arbitrarily large by tuning Φ ext close to Φ 0 /2. Note, however, that the proper conditions for the validity of the I T and β L expansions are:\n\nI T I c sec (πΦ ext /Φ 0 ) ≪ 1 ( 12\n\n) and |β L sec (πΦ ext /Φ 0 )| ≪ 1. ( 13\n\n)\n\nWe now restrict ourselves to a single transmission line mode and derive approximate equations of motion for the mode amplitude. Suppose that the mechanical oscillator position coordinate is held fixed at y = 0. The following phase field satisfies the current boundary conditions (7):\n\nφ(x, t) = -φ(t) cos [k 0 (x + l/2)] ; x < 0 +φ(t) cos [k 0 (x -l/2)] ; x > 0 , ( 14\n\n)\n\nwith the wavenumber k 0 determined by the voltage boundary condition (8):\n\nk 0 l 2 tan k 0 l 2 = - L T l L eff (Φ ext ) . ( 15\n\n)\n\nThe wave equation (3) gives for the transmission mode frequency: ω T = k 0 / √ L T C T . Substituting the phase field (14) into the I T part of the oscillator equation of motion (11) furthermore gives the transmission line force acting on the oscillator with fixed coordinate y = 0. Now release the mechanical oscillator coordinate and suppose that for small [condition (c)] , slow [condition (a)] displacements, the force is the same to a good approximation.Then the 8 oscillator equation of motion becomes mÿ(t) + mω 2 m y(t) +\n\n1 4 C T l Φ 0 2π 2 sin 2 (k 0 l/2) × - λB ext l osc (Φ 0 /2π) • Φ 0 4πL T lI c tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) ω 2 T φ 2 (t) = 0, ( 16\n\n)\n\nFrom Eq. ( 16 ), we can determine the mechanical sector of the Lagrangian, along with the interaction potential involving y and the mode amplitude φ. The remaining transmission line sector follows from the wave equation (3) and we thus have for the total Lagrangian:\n\nL φ, y, φ, ẏ = 1 2 m ẏ2 - 1 2 mω 2 m y 2 + 1 2 C T l Φ 0 2π 2 sin 2 (k 0 l/2) × 1 2 φ2 - 1 2 1 - λB ext l osc y (Φ 0 /2π) • Φ 0 4πL T lI c tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) ω 2 T φ 2 . ( 17\n\n)\n\nFrom Eq. ( 17 ), we see that for motion occuring on the much longer timescale ω -1 m ≫ ω -1 T , the mechanical oscillator has the effect of modulating the frequency of the transmission line mode.\n\nThe associated Hamiltonian is\n\nH (φ, y, p φ , p y ) = 2 C T l Φ 0 2π 2 sin 2 (k 0 l/2) 1 2 p 2 φ + 1 2 C T l Φ 0 2π 2 sin 2 (k 0 l/2) × 1 - λB ext l osc y (Φ 0 /2π) • Φ 0 4πL T lI c tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) 1 2 ω 2 T φ 2 + p 2 y 2m + 1 2 mω 2 m y 2 . ( 18\n\n)\n\nLet us now quantize. For the transmission line mode coordinate, the raising(lowering) operator is defined as:\n\nâ± T = 1 2 ω T 1 2 C T l (Φ 0 /2π) 2 sin 2 (k 0 l/2) 1 2 C T l Φ 0 2π 2 sin 2 (k 0 l/2) ω T φ ∓ ip φ ( 19\n\n)\n\nand for the mechanical oscillator\n\nâ± m = 1 √ 2mω (mω ŷ ∓ ip y ) . ( 20\n\n)\n\nIn terms of these operators, the Hamiltionian (18) becomes (for notational convenience we omit from now on the 'hats' on the operators and also the 'minus' superscript on the lowering operator):\n\nH = ω T a + T a T + ω m a + m a m + 1 2 ω T K T m a T + a + T 2 a m + a + m , ( 21\n\n) 9\n\nwhere the dimensionless coupling parameter between the mechanical oscillator and transmission line mode is\n\nK T m = - λB ext l osc ∆x zp (Φ 0 /2π) Φ 0 4πL T lI c tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) , ( 22\n\n)\n\nwith ∆x zp = /(2mω m ) the zero-point uncertainty of the mechanical oscillator. From expression (10) for the effective inductance, another way to express the coupling parameter is as follows:\n\nK T m = - λB ext l osc ∆x zp (Φ 0 /2π) Φ 0 π dL eff /dΦ ext L T l . ( 23\n\n)\n\nFrom Eq. ( 23 ), we see that in order to increase the coupling between the mechanical oscillator and transmission line, the SQUID effective inductance-to-transmission line inductance ratio must be increased. The advantage of using a SQUID over an ordinary, geometrical mutual inductance between a transmission line and micron-sized mechanical oscillator is that the former can give a much larger effective inductance. As we shall see in Sec. IV, just requiring that the inductances be matched such that Φ 0 π dL eff /dΦext L T l ∼ 1 is sufficient for strong back reaction effects with modest drive powers, even though the other term in K T m describing the flux induced for a zero-point displacement is typically very small.\n\nAssuming then that K T m ≪ 1 and making the rotating wave approximation (RWA) for the 'T ' part of the interaction term in the system Hamiltonian (21), i.e., neglecting the terms (a T ) 2 and (a + T ) 2 , we have (up to an unimportant additive constant):\n\nH = ω T a + T a T + ω m a + m a m + ω T K T m a + T a T a m + a + m . ( 24\n\n)\n\nMany other systems are modeled by this form of Hamiltonian, a notable example being the single mode of an optical cavity interacting via radiation pressure with a mechanically compliant mirror. 22, 23, 24, 25, 26 Thus, much of the subsequent analysis will be relevant to a broad class of coupled resonator devices-not to just the transmission line-SQUID-mechanical resonator system."
},
{
"section_type": "OTHER",
"section_title": "B. Open system Heisenberg equations of motion",
"text": "So far, we have treated the transmission line and mechanical resonator as a closed system with SQUID-induced effective coupling . Of course, a real transmission line mode will experience damping and accompanying fluctuations, not least because it must be coupled to the 10 outside world in order for its state to be measured. Furthermore, the mechanical resonator mode will of course be damped even when decoupled from the SQUID. It is straightforward to incorporate the various baths and pump/probe feedline in terms of raising/lowering operators. Assuming weak system-bath couplings, which again justify the RWA, we have for the full Hamiltonian:\n\nH = ω T a + T a T + ω m a + m a m + ω T K T m a + T a T a m + a + m + dωωa + p (ω)a p (ω) + dωωa + e (ω)a e (ω) + dωωa + b (ω)a b (ω) + dω K * pT a + p (ω)a T + K pT a + T a p (ω) + dω K * eT a + e (ω)a T + K pT a + T a e (ω) + dω K * bm a + b (ω)a m + K bm a + m a b (ω) -2mω m (a m + a + m )F ext (t), ( 25\n\n)\n\nwhere a p denotes the pump/probe (p) feed line operator, a e the transmission line bath ('e' for 'environment') operator, and a b the mechanical resonator bath (b) operator. These operators satisfy the usual canonical commutation relations:\n\na i (ω), a + j (ω ′ ) = δ ij δ(ω -ω ′ ). ( 26\n\n)\n\nThe couplings between these baths and the transmission line and mechanical resonator systems are denoted as K pT , K eT , and K bm . Note we have also included for generality a classical driving force F ext (t) acting on the mechanical resonator. This allows us the opportunity to later on analyze quantum limits on force detection in addition to displacement detection.\n\nWithin the RWA, it is straightforward to solve the Heisenberg equations for the bath operators and substitute these solutions into the Heisenberg equations for the transmission line and mechanical oscillator to give\n\nda m dt = -iω m a m + i 2mω m F ext (t) -iω T K T m a + T a T -dω |K T m | 2 t t 0 dt ′ e -iω(t-t ′ ) a m (t ′ ) -i dωK bm e -iω(t-t 0 ) a b (ω, t 0 ) ( 27\n\n)\n\nand\n\nda T dt = -iω T a T -iω T K T m a T a m + a + m -dω |K pT | 2 t t 0 dt ′ e -iω(t-t ′ ) a T (t ′ ) -i dωK pT e -iω(t-t 0 ) a p (ω, t 0 ) 11 -dω |K eT | 2 t t 0 dt ′ e -iω(t-t ′ ) a T (t ′ ) -i dωK eT e -iω(t-t 0 ) a e (ω, t 0 ). ( 28\n\n)\n\nWe now make the so-called 'first Markov approximation', 34, 35 in which the frequency dependences of the couplings to the baths are neglected:\n\nK pT (ω) = γ pT π e iφ pT K eT (ω) = γ eT π e iφ eT K bm (ω) = γ bm π e iφ bm , ( 29\n\n)\n\nwhere the γ's and φ's are independent of ω as stated. The Heisenberg equations of motion (27) and (28) then simplify to\n\nda m dt = -iω m a m + i 2mω m F ext (t) -iω T K T m a + T a T -γ bm a m (t) -i 2γ bm e iφ bm a in b (t) ( 30\n\n) and da T dt = -iω T a T -iω T K T m a T a m + a + m -γ pT a T (t) -i 2γ pT e iφ pT a in p (t) -γ eT a T (t) -i 2γ eT e iφ eT a in e (t), ( 31\n\n)\n\nwhere the γ i 's are the various mode amplitude damping rates (assumed much smaller than their associated mode frequencies) and the 'in' operators 10, 31, 34, 35 are defined as\n\na in i (t) = 1 √ 2π dωe -iω(t-t 0 ) a i (ω, t 0 ), ( 32\n\n)\n\nwith t > t 0 . The time t 0 can be taken to be an instant in the distant past before the measurement commences and when the initial conditions are specified (see below). We can similarly define 'out' operators:\n\na out i (t) = 1 √ 2π dωe -iω(t-t 1 ) a i (ω, t 1 ), ( 33\n\n)\n\nwith t 1 > t. The time t 1 can be taken to be an instant in the distant future after the measurement has finished. From the Heisenberg equations for the bath operators and the definitions of the 'in' and 'out' operators, we obtain the following identities between them: 34,35\n\na out p (t) -a in p (t) = -i 2γ pT e -iφ pT a T ( t\n\n) 12 a out b (t) -a in b (t) = -i 2γ bm e -iφ bm a m (t) a out e (t) -a in e (t) = -i 2γ eT e -iφ eT a T (t). (34)\n\nIn outline, the method of solution runs in principle as follows: 31, 34, 35, 36 (1) specify the 'in' operators. (2) Solve for the system operators a m (t) and a T (t) in terms of the 'in' operators.\n\n(3) Use the relevant identity (34) to determine the 'out' operator a out p (t), which yields the desired probe signal. It is more convenient to solve the Heisenberg equations in the frequency domain with the Fourier transformed operators O(t) = 1 √ 2π ∞ -∞ dωe -iωt O(ω). The equations for the system operators then become\n\na m (ω) = 1 ω -ω m + iγ bm 2γ bm e iφ bm a in b (ω) - 1 √ 2m ω m F ext (ω) + ω T K T m 2 √ 2π ∞ -∞ dω ′ a T (ω ′ )a + T (ω ′ -ω) + a + T (ω ′ )a T (ω + ω ′ ) ( 35\n\n) and a T (ω) = 1 ω -ω T + i(γ pT + γ eT ) 2γ pT e iφ pT a in p (ω) + 2γ eT e iφ eT a in e (ω) + ω T K T m √ 2π ∞ -∞ dω ′ a T (ω ′ ) a m (ω -ω ′ ) + a + m (ω ′ -ω) , ( 36\n\n)\n\nwhile the relevant 'in/out' operator identity becomes\n\na out p (ω) = -i 2γ pT e -iφ pT a T (ω) + a in p (ω). ( 37\n\n)\n\nC. Observables and 'in' states\n\nBefore proceeding with the solution to Eqs. (35) and (36), let us first devote some time to deriving expressions for observables that we actually measure in terms of a out p (ω). Model the pump/probe feedline as a semi-infinite transmission line -∞ < x < 0. Solving the wave equation for the decoupled transmission line and then using the expressions (1), (2) relating the current/voltage to the phase coordinate, we obtain\n\nI out (x, t) = - ∞ -∞ dω ω πZ p sin (ωx/v p ) e -iωt a out p (ω) + e iωt a out+ p (ω) ( 38\n\n)\n\nand\n\nV out (x, t) = i ∞ -∞ dω Z p ω π cos (ωx/v p ) e -iωt a out p (ω) -e iωt a out+ p (ω) , ( 39\n\n) 13\n\nwhere the sinusoidal x dependence in the current expression follows from the vanishing of the current boundary condition at x = 0, the feedline impedance is Z p = L p /C p and the wave propagation velocity is v p = 1/ L p C p . Suppose the current/volt meter is at x → -∞, so that the actual observables correspond to measuring the left-propagating component of the current/voltage. Then decomposing the x-dependent trig terms into their real and imaginary parts, we can identify the left propagating current/voltage operators as\n\nI out (x, t) = -i 4πZ p ∞ 0 dω √ ω e -iω(x/vp+t) a out p (ω) -a out+ p (-ω) +e iω(x/vp+t) a out p (-ω) -a out+ p (ω) ( 40\n\n)\n\nand\n\nV out (x, t) = i Z p 4π ∞ 0 dω √ ω e -iω(x/vp+t) a out p (ω) -a out+ p (-ω) +e iω(x/vp+t) a out p (-ω) -a out+ p (ω) . ( 41\n\n)\n\nThe output signal of interest due to the mechanical oscillator signal input will lie within some bandwidth δω centered at ω s , the 'signal' frequency, and so we define the filtered output current I out (x, t|ω s , δω) and voltage V out (x, t|ω s , δω) to be the same as the above, left-moving operators, but with the integration range instead restricted to the interval [ω s -δω/2, ω s + δω/2].\n\nSince the motion of the mechanical resonator modulates the transmission line frequency, one way to transduce displacements is to measure the relative phase shift between the 'in' pump current and 'out' probe current using the homodyne detection procedure. 35 Another common way is to measure the 'out' power relative to the 'in' power, or equivalently the mean-squared current/voltage (all three quantities differ by trivial factors of Z p ). We will discuss the latter method of transduction; the former, homodyne method can be straightforwardly addressed using similar techniques to those presented here. Thus, we consider the following expectation value:\n\nδI out (x, t|ω s , δω) 2 = I out (x, t|ω s , δω) 2 -I out (x, t|ω s , δω) 2 , ( 42\n\n)\n\nwhere the angle brackets denote an ensemble average with respect to the 'in' states of the various baths and feedline (see below). If the mechanical oscillator is being driven by a classical external force whose fluctuations are invariant under time translations, i.e., 14 F ext (t)F ext (t ′ ) = C(t -t ′ ), then the above, mean-squared current will be time-independent.\n\nAlternatively, if F ext (t) is, e.g., some deterministic, AC drive, then we must also time-average so as to get a time-independent measure of the detector response:\n\n[δI out (x, t|ω s , δω)] 2 = 1 T M T M /2 -T M /2 dt I out (x, t|ω s , δω) 2 , ( 43\n\n)\n\nwhere T M is duration of the measurement, assumed much larger than all other timescales associated with the detector dynamics. We have also assumed that the time-averaged current vanishes in the signal bandwidth of interest: I out (ω s , δω) = 0. Substituting in the expression (40) for I out (x, t|ω s , δω) in terms of the a out p operators, we obtain after some algebra:\n\n[δI out (ω s , δω)] 2 = 1 Z p ωs+δω/2 ωs-δω/2 dω 1 dω 2 2π ω 1 2 (ω 1 -ω 2 ) T M sin [(ω 1 -ω 2 )T M /2] × 1 2 a out p (ω 1 )a out+ p (ω 2 ) + a out+ p (ω 2 )a out p (ω 1 ) . ( 44\n\n)\n\nAs 'in' states, we suppose k B T ≪ ω T , such that the relevant transmission line 'in' bath modes (ω e ∼ ω T ) are assumed to be approximately in the vacuum state. On the other hand, with the mechanical mode typically at a much lower frequency ω m ≪ ω T , we assume that its relevant 'in' bath modes (ω b ∼ ω m ) are in the proper, non-zero temperature thermal state.\n\nFor the pump/probe feedline, we consider the following coherent state:\n\n30 |{α(ω)} p = exp dωα(ω) a in+ p (ω) -a in p (ω) |0 p , ( 45\n\n)\n\nwhere |0 p is the vacuum state and\n\nα(ω) = -I 0 Z p T 2 M 2 e -(ω-ωp) 2 T 2 M /2 √ ω , ( 46\n\n)\n\nnormalized such that the amplitude of the expectation value of I in [the right propagating version of (40) with a out p replaced by a in p ] with respect to this state is just I 0 . Again, we suppose k B T ≪ ω p , so that thermal fluctuations of the feedline are neglected. The frequency width of this pump drive is assumed to be the inverse lifetime of the measurement.\n\nBelow we shall see that the output mechanical signal will appear as two 'satellite' peaks on either side of the central peak at ω p due to the pump signal, i.e, the mechanical signal can be extracted by centering the filter at either of ω s = ω p ± ω m (up to a renormalization of the mechanical oscillator frequency), corresponding to the anti-Stokes and Stokes bands. 15 Note that we do not have to specify the initial t 0 states of the mechanical resonator and transmission line systems; a T (t 0 ) and a m (t 0 )-dependent initial transients have been dropped in the above equations for a T (ω) and a m (ω), since they give a negligible contribution to the long-time, steady-state behavior of interest."
},
{
"section_type": "OTHER",
"section_title": "A. Linear response approximation",
"text": "We are now ready to solve for [δI out ] 2 . Introduce the following shorthand notation:\n\nS T (ω) = 2γ pT e iφ pT a in p (ω) + 2γ eT e iφ eT a in e (ω) S m (ω) = 2γ bm e iφ bm a in b (ω) - 1 √ 2m ω m F ext (ω) K = ω T K T m √ 2π , ( 47\n\n)\n\nand γ T = γ pT +γ eT , the net transmission line mode amplitude dissipation rate due to loss via the probe line and the transmission line bath. Substituting Eq. (35) for a m (ω) into Eq. (36) for a T (ω) yields the following, single equation in terms of a T (ω) only:\n\na T (ω) = ∞ -∞ dω ′ a T (ω -ω ′ )A(ω, ω ′ ) + ∞ -∞ dω ′ B(ω, ω ′ )a T (ω -ω ′ ) × ∞ -∞ dω ′′ a T (ω ′′ )a + T (ω ′′ -ω ′ ) + a + T (ω ′′ )a T (ω ′′ + ω ′ ) + C(ω), ( 48\n\n)\n\nwhere, for the convenience of subsequent calculations, we have made this equation as concise as possible with the following definitions:\n\nA(ω, ω ′ ) = K ω -ω T + iγ T S m (ω ′ ) ω ′ -ω m + iγ bm + S + m (-ω ′ ) -ω ′ -ω m -iγ bm , B(ω, ω ′ ) = K 2 /2 ω -ω T + iγ T 1 ω ′ -ω m + iγ bm + 1 -ω ′ -ω m -iγ bm , C(ω) = S T (ω) ω -ω T + iγ T . ( 49\n\n)\n\nWe expand Eq. (48) for a T (ω) to first order in the mechanical oscillator bath operator a in b (ω) and external driving force\n\nF ext (ω) [equivalently expand in A(ω, ω ′ )]: a T (ω) ≈ a ( 0\n\n) T (ω)+ a (1) T (ω), where a (0) T (ω) = ∞ -∞ dω ′ B(ω, ω ′ )a (0) T (ω -ω ′ ) 16 × ∞ -∞ dω ′′ a (0) T (ω ′′ )a (0)+ T (ω ′′ -ω ′ ) + a (0)+ T (ω ′′ )a ( 0\n\n) T (ω ′′ + ω ′ ) + C(ω) (50) and a (1) T (ω) = ∞ -∞ dω ′ a (0) T (ω -ω ′ )A(ω, ω ′ ) + ∞ -∞ dω ′ B(ω, ω ′ )a ( 1\n\n) T (ω -ω ′ ) × ∞ -∞ dω ′′ a (0) T (ω ′′ )a (0)+ T (ω ′′ -ω ′ ) + a (0)+ T (ω ′′ )a (0) T (ω ′′ + ω ′ ) + ∞ -∞ dω ′ B(ω, ω ′ )a ( 0\n\n) T (ω -ω ′ ) ∞ -∞ dω ′′ a (0) T (ω ′′ )a (1)+ T (ω ′′ -ω ′ ) +a (1)+ T (ω ′′ )a ( 0\n\n) T (ω ′′ + ω ′ ) + a (1) T (ω ′′ )a (0)+ T (ω ′′ -ω ′ ) +a (0)+ T (ω ′′ )a ( 1\n\n) T (ω ′′ + ω ′ ) . ( 51\n\n)\n\nEq. (50) then yields the detector noise, while (51) yields the detector response to the signal within the linear response approximation. Thus, our approach here is to treat the mechanical oscillator as part of the detector degrees of freedom, with the signal defined as the thermal bath fluctuations and classical external force acting on the oscillator. This is the appropriate viewpoint for force detection. On the other hand, if the focus is on measuring the quantum state of the mechanical oscillator itself, then the oscillator should not be included as part of the detector degrees of freedom. Nevertheless, as we shall later see, the latter viewpoint can be straightforwardly extracted from the former under not too strong coupling K T m and pump drive current amplitude I 0 conditions."
},
{
"section_type": "OTHER",
"section_title": "B. Semiclassical approximation",
"text": "The sequence of solution steps to Eqs. (50) and (51) are in principle as follows: (1) Solve first equation (50) for a (0) T (ω) in terms of B(ω, ω ′ ) and C(ω); (2) Substitute the solution for a (0) T (ω) into Eq. (51) for a (1) T (ω) and invert this Eq. (which is linear in a (1) T (ω)) to obtain the solution for a (1) T (ω) in terms of A(ω, ω ′ ), B(ω, ω ′ ), and C(ω). It is not clear how to carry out these steps in practice, however, since the equations involve products of noncommuting operators. Thus, we must find some way to solve by further approximation.\n\nThe key observation is that the feedline is in a coherent state, which is classical-like for sufficiently large current amplitude I 0 so as to ensure signal amplification. We therefore decompose a (0) T (ω) into a classical, expectation-valued part and quantum, operator-valued fluctuation part, a (0) ω), and subtitute into Eq. (50) for a (0) T (ω), 17 linearizing with respect to the quantum fluctuation δa (0) T (ω). This gives two equations, one for the expectation value a (0)\n\nT (ω) = a (0) T (ω) + δa (0) T (\n\nT (ω) = ∞ -∞ dω ′ B(ω, ω ′ ) a (0) T (ω -ω ′ ) ∞ -∞ dω ′′ a (0) T (ω ′′ ) a (0)+ T (ω ′′ -ω ′ ) + a (0)+ T (ω ′′ ) a (0) T (ω ′′ + ω ′ ) + C(ω) ( 52\n\n)\n\nand the other for the quantum fluctuation:\n\nδa ( 0\n\n) T (ω) = ∞ -∞ dω ′ B(ω, ω ′ )δa ( 0\n\n) T (ω -ω ′ ) ∞ -∞ dω ′′ a ( 0\n\n) T (ω ′′ ) a (0)+ T (ω ′′ -ω ′ ) + a (0)+ T (ω ′′ ) a ( 0\n\n) T (ω ′′ + ω ′ ) + ∞ -∞ dω ′ B(ω, ω ′ ) a ( 0\n\n) T (ω -ω ′ ) × ∞ -∞ dω ′′ δa ( 0\n\n) T (ω ′′ ) a (0)+ T (ω ′′ -ω ′ ) + a ( 0\n\n) T (ω ′′ ) δa (0)+ T (ω ′′ -ω ′ ) +δa (0)+ T (ω ′′ ) a ( 0\n\n) T (ω ′′ + ω ′ ) + a (0)+ T (ω ′′ ) δa ( 0\n\n) T (ω ′′ + ω ′ ) + δC(ω). ( 53\n\n)\n\nEq. (51) for a (1) T (ω) is approximated by replacing a (0) T (ω) with its expectation value a (0) T (ω) , i.e., we drop the quantum fluctuation part δa (0) T (ω). This is because Eq. (51) already depends linearly on the quantum fluctuating signal term A(ω, ω ′ ), which we of course want to keep. Dropping the δa (0) T (ω) contribution to Eq. (51) amounts to neglecting multiplicative detector noise, which is reasonable given that we are concerned with large signal amplification."
},
{
"section_type": "OTHER",
"section_title": "C. Complete solution to detector signal response and noise",
"text": "The sequence of solutions steps are therefore in practice as follows: (1) Solve Eq. (52) first for a (0) T (ω) ; (2) Substitute this solution into Eq. (51) for a (1) T (ω) and invert; (3) Substitute the solution for a (0) T (ω) into the Eq. (53) for δa (0) T (ω) and invert; (4) Use these solutions for a (1) T (ω) and δa (0) T (ω) to determine the detector signal and noise terms, respectively. Beginning with step (1), we have\n\nC(ω) = - i 2γ pT e iφ pT γ T -i∆ω a in p (ω) = i 2γ pT e iφ pT γ T -i∆ω • I 0 Z p T 2 M 2 ω e -(ω-ωp) 2 T 2 M /2 , ( 54\n\n)\n\nwhere ∆ω = ω p -ω T is the detuning frequency (not to be confused with the bandwidth δω) and note a in e (ω) = 0 (recall, we assume the transmission line resonant frequency ω T mode is in the vacuum state). Given that T M is the longest timescale in the system dynamics, C(ω) is sharply peaked about the frequency ω p and we will therefore approximate the 18 exponential with a delta function: C(ω) = cδ(ω -ω p ), where\n\nc = i √ 2πe iφ pT γ T -i∆ω I 2 0 Z p γ pT ω p . ( 55\n\n)\n\nConsidering for the moment an iterative solution to Eq. (52) for a (0) T (ω) , we see that a (0) T (ω) must also have the form of a delta function peaked at ω p : a (0) T (ω) = χδ(ω -ω p ). Substituting this ansatz into Eq. (52), we obtain the following equation for χ:\n\nχ = 2χ |χ| 2 B(ω p , 0) + c. ( 56\n\n)\n\nThis equation has a rather involved analytical solution. For sufficiently large |c| 2 |B(ω p , 0)| the response can become bistable (i.e., two locally stable solutions for χ). This region will not be discussed in the present paper, however. When we consider actual device parameters later in Sec. IV, we will assume sufficiently small drive such that χ ≈ c, allowing much simpler analytical expressions to be written down for the detector response.\n\nProceeding now to step (2), we substitute the expectation value a (0) T (ω) = χδ(ω -ω p ) for the operator a (0) T (ω) into Eq. (51) for a (1) T (ω). Carrying out the integrals, we obtain\n\n1 -2 |χ| 2 [B(ω, 0) + B(ω, ω -ω p )] a (1) T (ω) -2χ 2 B(ω, ω -ω p )a (1)+ T (2ω p -ω) = χA(ω, ω -ω p ). ( 57\n\n)\n\nBefore we can invert to obtain a (1) T (ω), we require a second linearly independent equation also involving a (1)+ T (2ω p -ω) and a (1) T (ω). This equation can be obtained by replacing ω with 2ω p -ω in Eq. ( 57 ) and then taking the adjoint:\n\n1 + 2 |χ| 2 [B(ω -2∆ω, 0) + B(ω -2∆ω, ω -ω p )] a (1)+ T (2ω p -ω) +2χ * 2 B(ω -2∆ω, ω -ω p )a (1) T (ω) = -χ * A(ω -2∆ω, ω -ω p ), ( 58\n\n)\n\nwhere we have used the identities 2∆ω, 0). Inverting, we obtain a (1)\n\nA + (2ω p -ω, ω p -ω) = -A(ω -∆ω, ω -ω p ), B * (2ω p - ω, ω p -ω) = -B(ω -2∆ω, ω -ω p ), and B * (2ω p -ω, 0) = -B(ω -\n\nT (ω) = α 1 (ω)A(ω, ω -ω p ) + α 2 (ω)A(ω -2∆ω, ω -ω p ), ( 59\n\n) where α 1 (ω) = D(ω) -1 1 + 2 |χ| 2 [B(ω -2∆ω, 0) + B(ω -2∆ω, ω -ω p )] χ ( 60\n\n)\n\n19 and α 2 (ω) = -2D(ω) -1 |χ| 2 B(ω, ω -ω p )χ, (61) with determinant\n\nD(ω) = 1 -2 |χ| 2 [B(ω, 0) + B(ω, ω -ω p )] × 1 + 2 |χ| 2 [B(ω -2∆ω, 0) + B(ω -2∆ω, ω -ω p )] +4 |χ| 4 B(ω, ω -ω p )B(ω -2∆ω, ω -ω p ). ( 62\n\n)\n\nMoving on now to step (3), we substitute the expectation value a (0) T (ω) = χδ(ω -ω p ) into Eq. (53) for δa (0) T (ω) and carry out the integrals to obtain:\n\n1 -2 |χ| 2 [B(ω, 0) + B(ω, ω -ω p )] δa (0) T (ω) -2χ 2 B(ω, ω -ω p )δa (0)+ T (2ω p -ω) = δC(ω). ( 63\n\n)\n\nReplacing ω with 2ω p -ω in Eq. ( 63 ) and then taking the adjoint:\n\n1 + 2 |χ| 2 [B(ω -2∆ω, 0) + B(ω -2∆ω, ω -ω p )] δa (0)+ T (2ω p -ω) +2χ * 2 B(ω -2∆ω, ω -ω p )δa ( 0\n\n) T (ω) = δC + (2ω p -ω). ( 64\n\n)\n\nInverting Eqs. (63) and (64), we obtain\n\nδa ( 0\n\n) T (ω) = β 1 (ω)δC(ω) + β 2 (ω)δC + (2ω p -ω), ( 65\n\n) where β 1 (ω) = D(ω) -1 1 + 2 |χ| 2 [B(ω -2∆ω, 0) + B(ω -2∆ω, ω -ω p )] ( 66\n\n) and β 2 (ω) = 2D(ω) -1 χ 2 B(ω, ω -ω p ) ( 67\n\n)\n\nWe are now ready to carry out step (4). To obtain the detector response, we substitute into expression (44) for [δI out ] 2 the linear response approximation to the 'out' probe operator [see Eq. (37)]:\n\na out p (ω) = -i 2γ pT e -iφ pT a ( 1\n\n) T (ω) + -i 2γ pT e -iφ pT δa ( 0\n\n) T (ω) + δa in p (ω) . ( 68\n\n)\n\nThe first square-bracketed term will give the signal contribution to the detector response, while the second bracketed term gives the noise contribution. Note that the average values 20 a (0) T (ω) and a in p (ω) are not required in the noise term since they give negligible contribution in the signal bandwidths of interest centered at ω s = ω p ± ω m . Substituting in the signal part of a out p (ω), we obtain after some algebra:\n\n[δI out (ω s , δω)] 2 signal = I 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 ωs+δω/2 ωs-δω/2 dω 2π ω ω p γ 2 T (ω -ω p + ∆ω) 2 + γ 2 T × α 1 (ω) c + α 2 (ω) c ω -ω p + ∆ω + iγ T ω -ω p -∆ω + iγ T 2 × 2γ bm (ω -ω p -ω m ) 2 + γ 2 bm [2n(ω -ω p ) + 1] + 2γ bm (ω p -ω -ω m ) 2 + γ 2 bm [2n(ω p -ω) + 1] + I 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 1 2m ω m γ bm ωs+δω/2 ωs-δω/2 dωdω ′ 2π ω ω p γ 2 T (ω -ω p + ∆ω) 2 + γ 2 T × α 1 (ω) c + α 2 (ω) c ω -ω p + ∆ω + iγ T ω -ω p -∆ω + iγ T 2 × sin [(ω -ω ′ )T M /2] (ω -ω ′ ) T M /2 2γ bm (ω -ω p -ω m ) 2 + γ 2 bm F ext (ω -ω p )F * ext (ω ′ -ω p ) + 2γ bm (ω p -ω -ω m ) 2 + γ 2 bm F ext (ω p -ω)F * ext (ω p -ω ′ ) , ( 69\n\n)\n\nwhere n(ω) = e ω/k B T -1 -1 is the Bose-Einstein thermal occupation number average for bath mode ω. The signal part of the detector response comprises a thermal component and a classical force component. In the limit of weak coupling K T m → 0 and or small drive current amplitude I 0 → 0, we have α 1 (ω)/c → 1, α 2 (ω)/c → 0 and we note that the frequency resolved detector response has the form of two Lorentzians centered at ω p ± ω m .\n\nThe resulting expression for the detector response coincides with an O(K 2 T m ) perturbative solution to the detector response (44) via the linear response Eqs. (50) and (51) (but no semiclassical approximation). However, as shall be described in Sec. IV, when the current drive is not small and or coupling is not weak, then the α i terms will modify this simple form, at the next level of approximation renormalizing the Lorentzians, i.e., shifting their location and changing their width.\n\nSubstituting in the noise part of a out p (ω), we obtain after some algebra:\n\n[δI out (ω s , δω)] 2 noise = Z -1 p ωs+δω/2 ωs-δω/2 dω 2π ω 2γ T γ pT (ω -ω p + ∆ω) 2 + γ 2 T × |β 1 (ω)| 2 + (ω -ω p + ∆ω) 2 + γ 2 T (ω -ω p -∆ω) 2 + γ 2 T |β 2 (ω)| 2 -Re [β 1 (ω)] + (ω -ω p + ∆ω) γ T Im [β 1 (ω)] +Z -1 p ω s 2 δω 2π . ( 70\n\n)\n\n21 The noise part of the detector response comprises a back reaction component (the integral term) where transmission line noise drives the mechanical oscillator via the SQUID coupling, and a component that is added at the output due to zero-point fluctuations in the probe line. While not as obvious given the form of Eq. (70), one may again verify (see Sec. IV) that the detector back reaction on the mechanical oscillator takes the form of two Lorentzians centered at ω p ± ω m in the weak coupling and or weak current drive limit, coinciding with an O(K 2 T m ) perturbative calculation. Eqs. (69) and (70) are the main results of the paper, their sum giving the net output mean-squared current."
},
{
"section_type": "OTHER",
"section_title": "D. Quantum bound on noise",
"text": "As articulated by Caves, 10 the fact that the 'in' and 'out' operators satisfy canonical commutation relations places a lower, quantum limit on the noise contribution to the detector response, Eq. (70). We now derive this quantum limit. First write the 'out' operator (68) as\n\na out p (ω) = -i 2γ pT e -iφ pT a ( 1\n\n) T (ω) + N(ω), ( 71\n\n)\n\nwhere N(ω) = -i 2γ pT e -iφ pT δa (0) T (ω) + δa in p (ω) is the noise part. Taking commutators, we have the following identity relating the noise and signal operator terms:\n\nN(ω), N + (ω ′ ) = δ(ω -ω ′ ) -2γ pT a (1) T (ω), a (1)+ T (ω ′ ) . ( 72\n\n)\n\nNow, from the Heisenberg Uncertainty Principle, one can derive the following general inequality:\n\nN[f ]N + [f ] + N + [f ]N[f ] ≥ N[f ], N + [f ] , ( 73\n\n)\n\nwhere N[f ] = ∞ 0 dωf (ω)N(ω) and f (ω) is an arbitrary function. Inserting the commutator identity (72), Eq. (73) becomes\n\nN[f ]N + [f ] + N + [f ]N[f ] ≥ ∞ 0 dω |f (ω)| 2 -2γ pT a (1) T [f ], a (1)+ T [f ] . ( 74\n\n)\n\nChoosing the 'filter' function f (ω) = ωΘ(ω -ω s + δω/2)Θ(ω s + δω/2 -ω) and evaluating the commutator, we obtain the following lower bound on the detector noise:\n\n[δI out (ω s , δω)] 2 noise ≥ 22 Z -1 p ω s 2 δω 2π - I 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 ωs+δω/2 ωs-δω/2 dω 2π ω ω p γ 2 T (ω -ω p + ∆ω) 2 + γ 2 T × α 1 (ω) c + α 2 (ω) c ω -ω p + ∆ω + iγ T ω -ω p -∆ω + iγ T 2 × 2γ bm (ω -ω p -ω m ) 2 + γ 2 bm - 2γ bm (ω p -ω -ω m ) 2 + γ 2 bm . ( 75\n\n)\n\nIn the next section we will address the extent to which the detector noise can approach the quantum bound on the right hand side of Eq. (75), depending on the current drive amplitude I 0 and other detector parameters."
},
{
"section_type": "OTHER",
"section_title": "A. Analytical approximations",
"text": "To gain a better understanding of the detector response, we now provide analytical approximations to Eqs. (69) and (70) that are valid under the condition 56 )], i.e., the expectation value a (0) T (ω) for the transmission line depends approximately only on the pump/probe feedline state and not on the mechanical oscillator state. Explicitly, this condition reads:\n\n|c| 2 |B(ω p , 0)| ≪ 1 such that χ ≈ c [see Eq. (\n\n2I 2 0 Z p K 2 T m ω T γ pT ω m (γ 2 T + ∆ω 2 ) 3/2 ≪ 1, ( 76\n\n)\n\nplacing an upper limit on I 0 and K T m for the validity of this approximation. We also assume that the mechanical and transmission line mode frequencies are widely separated:\n\nω m ≪ ω T , and with small damping rates: γ bm ≪ ω m , γ T ≪ ω T .\n\nWe do not restrict the relative magnitudes of ω m and γ T , however. A simple picture emerges in which the detector back reaction 'renormalizes' the mechanical oscillator frequency and damping rate:\n\nω m → R ω ω m and γ bm → R γ γ bm , where R ω ω m = ω m + ∆ω + |c| 2 ω 2 T K 2 T m πω m |c| 2 ω 2 T K 2 T m [γ 2 T + ∆ω 2 -ω 2 m ] π γ 2 T + (∆ω + ω m ) 2 γ 2 T + (∆ω -ω m ) 2 ( 77\n\n)\n\nand\n\nR γ γ bm = γ bm -∆ω + |c| 2 ω 2 T K 2 T m πω m 2|c| 2 ω 2 T K 2 T m ω m γ T π γ 2 T + (∆ω + ω m ) 2 γ 2 T + (∆ω -ω m ) 2 , ( 78\n\n) 23\n\nwhere c is defined in Eq. ( 55 ). With the measurement filter bandwidth centered at either of ω s = ω p ± R ω ω m , the approximation to Eq. (69) for the signal response is (with the classical force term omitted):\n\n[δI out (ω s = ω p ± R ω ω m , δω)] 2 signal = I 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 γ 2 T γ 2 T + (∆ω ± ω m ) 2 × ωs+δω/2 ωs-δω/2 dω 2π 2γ bm (ω -ω p ∓ R ω ω m ) 2 + (R γ γ bm ) 2 [2n(R ω ω m ) + 1] . ( 79\n\n)\n\nWhen there is a classical force acting on the mechanical oscillator, we must add to Eq. (79) the term\n\nI 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 1 2m ω m γ bm ωs+δω/2 ωs-δω/2 dωdω ′ 2π γ 2 T (ω -ω p + ∆ω) 2 + γ 2 T × sin [(ω -ω ′ )T M /2] (ω -ω ′ ) T M /2 2γ bm (ω -ω p -R ω ω m ) 2 + (R γ γ bm ) 2 F ext (ω -ω p )F * ext (ω ′ -ω p ) + 2γ bm (ω p -ω -R ω ω m ) 2 + (R γ γ bm ) 2 F ext (ω p -ω)F * ext (ω p -ω ′ ) . ( 80\n\n)\n\nThe approximation to Eq. (70) for the detector noise is\n\n[δI out (ω s = ω p ± R ω ω m , δω)] 2 noise = I 0 K T m ω T γ T 2 γ 2 pT γ 2 T + ∆ω 2 γ 2 T γ 2 T + (∆ω ± ω m ) 2 × ωs+δω/2 ωs-δω/2 dω 2π 2γ bm (ω -ω p ∓ R ω ω m ) 2 + (R γ γ bm ) 2 N ± + Z -1 p ω s 2 δω 2π , ( 81\n\n)\n\nwhere the back reaction noise parameter is\n\nN ± = |c| 2 K 2 T m ω 2 T γ T πγ bm γ 2 T + (∆ω ∓ ω m ) 2 ∓ 1 = 2I 2 0 Z p K 2 T m ω T γ T γ pT γ bm [γ 2 T + ∆ω 2 ] γ 2 T + (∆ω ∓ ω m ) 2 ∓ 1. ( 82\n\n)\n\nThe ∓1 term in the back reaction noise parameter depends on whether the filter is centered\n\nat ω s = ω p + ω m or ω s = ω p -ω m\n\nand corresponds respectively to 'phase preserving' or 'phase conjugating' detection as discussed in Caves. 10 In the limit I 0 → 0 and or K T m → 0, we see from Eqs. (79), (81), and (82) that the back reaction noise amounts to doubling the oscillator quantum zero-point motion signal in the phase conjugating case, while the back reaction noise exactly cancels the quantum zero-point motion signal in the phase preserving case. In both cases, the noise coincides with the lower quantum bound (75). However, in this small drive/coupling limit, we do not have a detector or amplifier but rather an attenuator, which is of only academic interest to us.\n\nComparing the detector response (79) and back reaction part of Eq. ( 81 ), we see that the mechanical oscillator behaves in the steady state as if in contact with a thermal bath. 8, 9, 12, 26, 37, 38, 39 The back reaction of the detector on the mechanical oscillator is effectively that of a thermal bath with damping rate γ back = γ bm (R γ -1) and effective thermal average occupation number n back defined as follows:\n\nγ back (2n ± back + 1) = γ bm N ± . ( 83\n\n)\n\nThus,\n\nn ± back = (R γ -1) -1 1 2 N ± - 1 2 . ( 84\n\n)\n\nThe failure to approach the lower quantum bound (75) when N ± ≫ 1 then translates into having (2n ± back + 1)γ back /γ bm ≫ 1. Thus, to get close to the bound, we necessarily require γ back ≪ γ bm ; 12 the back reaction occupation number n ± back does not have to be small. With the mechanical oscillator also in thermal contact with its external bath, the net damping rate of the oscillator is γ net = γ bm + γ back = R γ γ bm and the net, effective thermal average occupation number n net of the oscillator is defined as follows:\n\nγ net 2n ± net + 1 = γ bm [2n(R ω ω m ) + 1] + γ back 2n ± back + 1 . ( 85\n\n)\n\nThus,\n\nn ± net = R -1 γ n(R ω ω m ) + 1 2 + 1 2 N ± - 1 2 . ( 86\n\n)\n\nFrom Eq. ( 78 ), we see that depending on the detuning parameter ∆ω = ω p -ω T , the damping rate of the oscillator due to the detector back reaction can be either negative or positive. Specifically, positive damping requires the following condition on the detuning parameter:\n\n∆ω < - |c| 2 ω 2 T K 2 T m πω m = - 2I 2 0 Z p K 2 T m ω T γ pT ω m (γ 2 T + ∆ω 2 ) . ( 87\n\n)\n\nB. Displacement sensitivity\n\nIn the absence of a classical force acting on the mechanical oscillator, from Eq. (79) the mechanical oscillator thermal noise displacement signal spectral density takes the familiar Lorentzian form:\n\nS x (ω)| signal = 2R γ γ bm (ω -ω p ∓ R ω ω m ) 2 + (R γ γ bm ) 2 2mR ω ω m [2n (R ω ω m ) + 1] . ( 88\n\n)\n\n25 In order to be able to resolve this mechanical signal, the detector noise (81) referred to the mechanical oscillator input must be smaller than (88). The detector noise spectral density at the input is\n\nS x (ω = ω p ± R ω ω m )| noise = 2 R γ γ bm ∓1 + |c| 2 K 2 T m ω 2 T γ T πγ bm γ 2 T + (∆ω ∓ ω m ) 2 + 2πR γ γ 2 T + (∆ω ± ω m ) 2 |c| 2 K 2 T m ω 2 T γ pT 2mR ω ω m , ( 89\n\n)\n\nwhere the first term on the right hand side is the back reaction noise acting on the mechanical oscillator and the second term is the output, probe line zero-point noise referred to the input.\n\nNote that the noise has been evaluated at ω = ω p ±R ω ω m , the maximum of the back reaction Lorentzian.\n\nIf the detector output is to depend linearly on the mechanical oscillator signal input (i.e., function as a linear amplifier), then back reaction effects must be small. In particular, we require that γ back ≪ γ bm , i.e., R γ ≈ 1. With |c| being proportional I 0 , we see from Eq. (89) that increasing the drive current amplitude I 0 increases the back reaction noise, but decreases the probe line noise referred to the mechanical oscillator input. Thus, there is an optimum I 0 such that the sum S x | noise is a minimum. Making the approximation R γ = 1 and R ω = 1 in Eq. ( 89 ) and optimizing with respect to |c|, we find\n\nS x (ω = ω p ± R ω ω m )| noise-optimum = mω m γ bm ∓1 + 2 γ T γ pT γ 2 T + (∆ω ± ω m ) 2 γ 2 T + (∆ω ∓ ω m ) 2 . ( 90\n\n)\n\nFrom Eq. ( 90 ), we see that the noise is further reduced if (i) the dominant source of transmission line mode dissipation is due to energy loss through the coupled probe (information gathering) line: 12 γ T ≈ γ pT ; (ii) the detuning frequency is chosen to be ∆ω = ∓ γ 2 T + ω 2 m , where the minus (plus) sign corresponds to phase preserving (conjugating) detection. With this detuning choice, the condition R γ ≈ 1 requires (ω m /γ T ) 2 ≪ 1 and so the minimum detector noise is\n\nS x (ω = ω p ± R ω ω m )| noise-optimum = mω m γ bm 2 ∓ 1 + O (ω m /γ T ) 2 , ( 91\n\n)\n\nwhere in order to determine the O ((ω m /γ T ) 2 ) term, the full form of R γ given in Eq. (78) must be used in Eq. (89) when optimizing. Comparing with Eq. (88) for the signal noise, we see that to leading order the detector noise effectively doubles the zero-point signal in 26 1 1.5 2 2.5 3 3.5 4 I 0 10 8 A 0.5 1 1.5 2 2.5 3 3.5 4 S x mΩ m Γ bm FIG. 2: Displacement detector noise spectral density (solid line) and lower bound (dashed line) versus drive current amplitude. The noise densities are evaluated at ω = ω p +R ω ω m , corresponding to phase preserving detection.\n\nthe phase preserving case. This exceeds the lower bound on the detector noise derived from Eq. (75), which is zero to leading order in the phase preserving case.\n\nWe now numerically evaluate Eq. (89) for the detector noise. The feasible example parameter values we use are: 14 B ext = 0.005 Tesla, Z p = 50 Ohms, ω T /2π = 3×10 9 s -1 , Q T = ω T /(2γ T ) = 100, γ T = 9.4 × 10 7 s -1 , l osc = 5 µm, λ = 1 (geometrical correction factor), m = 10 -16 kg, ω m = 2.5 × 10 7 s -1 , and Q bm = ω m /(2γ bm ) = 10 3 . These values give a mechanical oscillator zero-point uncertainty ∆x zp = 1.45 × 10 -13 m, a zero-point displacement noise /(mω m γ bm ) = 3.4 × 10 -30 m 2 /Hz, and a dimensionless coupling strength K T m = -1.1 × 10 -5 , where we assume that in the expression (22) for K T m , Φ ext can be chosen such that the dimensionless factor Φ 0 4πL T lIc tan (πΦ ext /Φ 0 ) sec (πΦ ext /Φ 0 ) ≈ 1 (matching condition). We also suppose that γ T ≈ γ pT , i.e., the transmission line mode damping is largely due to the probe line coupling.\n\nFig. 2\n\nshows S x (ω = ω p + R ω ω m )| noise × mω m γ bm /\n\nand also the lower bound on the detector noise that follows from Eq. (75) for phase preserving detection. Note that the minimum detector noise is approximately 0.8 /(mω m γ bm ). Thus, for this example, the next-to-leading O ((ω m /γ T ) 2 ) term in Eq. (91) is approximately -0.2. Note also that the detector noise coincides with the lower bound in the small drive limit."
},
{
"section_type": "OTHER",
"section_title": "C. Force sensitivity",
"text": "Consider a monochromatic classical driving force with frequency ω 0 ∼ R ω ω m acting on the oscillator: F ext (ω) = F 0 δ(ω -ω 0 ). The force signal spectral density is then S F (ω)| signal = F 2 0 δ(ω -ω 0 ). For force detection operation, the mechanical oscillator is included as part of the detector degrees of freedom. From Eqs. (79-82), the force noise spectral density evaluated at ω = ω p ± ω 0 is\n\nS F (ω = ω p ± ω 0 )| noise = 2m ω m γ bm 2n (ω 0 ) + 1 ∓ 1 + |c| 2 K 2 T m ω 2 T γ T πγ bm γ 2 T + (∆ω ∓ ω m ) 2 + π (ω 0 -R ω ω m ) 2 + (R γ γ bm ) 2 γ 2 T + (∆ω ± ω m ) 2 γ bm |c| 2 K 2 T m ω 2 T γ pT . ( 92\n\n)\n\nComparing the displacement noise (89) with the force noise (92), we see that the latter includes the additional 2m ω m γ bm [2n (ω 0 ) + 1] mechanical quantum thermal displacement noise term. Since the mechanical oscillator forms part of the force detector, it need not necessarily be weakly driven and or weakly coupled to the transmission line; as explained in Sec. III A, the present analysis employs a linear response approximation for force detection, not displacement detection. Thus, in determining the optimum I 0 (and or K T m ) and ∆ω such that S F | noise is a minimum, we should not assume a priori the restrictions R γ , R ω ≈ 1.\n\n2 4 6 8 10 12 14 I 0 10 8 A 2 3 4 5 6 S F 2m Ω m Γ bm\n\nFIG. 3: Force detector noise spectral density versus drive current amplitude for detuning ∆ω = 0 (solid line), ∆ω = -5ω m (dashed line), and ∆ω = -10ω m (dotted line) . The noise densities are evaluated at ω = ω p + R ω ω m , corresponding to phase preserving detection.\n\nFig. 3 shows the results of numerically evaluating the force noise spectral density given by Eq. (92) for phase preserving detection (ω = ω p + ω 0 ) and a range of detuning values.\n\nThe same example parameters are used as in the above displacement sensitivity analysis, with n(ω 0 ) = 0 and ω 0 = R ω ω m . The force noise is expressed in units 2m ω m γ bm = 6.6 × 10 -39 N 2 /Hz. Note that the minimum force noise is exactly 2 in these units, independently of the detuning, with the minimum occuring at larger I 0 values as the detuning is made progressively more negative."
},
{
"section_type": "OTHER",
"section_title": "D. Back reaction cooling",
"text": "From Eq. ( 86 ), we see that the net, thermal average occupation number n net of the mechanical oscillator's fundamental mode decreases as R γ increases. Thus, by increasing the drive and or coupling strength such that γ back ≫ γ bm , the mechanical oscillator can be effectively cooled at the expense of increasing its damping rate. 3, 8, 9, 21, 26, 40, 41, 42, 43, 44, 45, 46, 47, 48 Consider sufficiently negative detuning such that -∆ω ≫ |c| 2 ω 2 T K 2 T m /(πω m ) [see Eq. (87)]. Substituting definition (78) for R γ and definition (82) for N + into Eq. ( 86 ) and supposing R γ is large enough that we can neglect the external damping term γ bm , we obtain approximately for the phase preserving case:\n\nn + net ≈ n (R ω ω m ) R γ + n + back , ( 93\n\n) where n + back ≈ - γ 2 T + (∆ω + ω m ) 2 4∆ωω m - 1 2 . ( 94\n\n)\n\nThis expression agrees with that derived in Ref. 26, apart from the 1/2 which is simply due to a small difference in the way we define n ± back in Eq. ( 83 ). Choosing optimum detuning ∆ω = -γ 2 T + ω 2 m to minimize n + back in Eq. ( 94 ), we therefore have\n\nn + net ≈ n (R ω ω m ) R γ + 1 2 1 + (γ T /ω m ) 2 -1. ( 95\n\n)\n\nHow much cooling can be achieved depends on (i) how large R γ can be, subject to the above inequality on -∆ω; (ii) making the ratio γ T /ω m as small as possible. 26\n\nUsing the same example parameter values as above, but taking instead a larger but still realistic quality factor Q bm = 10 4 for the mechanical oscillator, 6 the resulting numerically evaluated effective occupation number n + net [Eq. ( 86 )] is given in Fig. 4 for a range of external 29 5 10 15 20 25 30 35 40 I 0 10 8 A 1 1.5 2 2.5 3 3.5 4 n net FIG. 4: Net effective average occupation number of the mechanical oscillator versus drive current.\n\nThe solid curve is for external bath temperature T = 100 mK [n(R ω ω) = 523], the dashed curve is for T = 10 mK [n(R ω ω) = 52], and the dotted curve is for T = 1 mK [n(R ω ω) = 4.8].\n\nbath occupation numbers n(ω m ). Thus, even for small coupling strengths K T m and drive current amplitudes I 0 , significant cooling of the mechanical oscillator can be achieved. This is in part a consequence of the fact that the quality factor Q bm of the mechanical oscillator when decoupled from the detector is very large."
},
{
"section_type": "OTHER",
"section_title": "V. CONCLUDING REMARKS",
"text": "In the present paper, we have attempted to give a reasonably comprehensive analysis of the quantum-limited detection sensitivity of a DC SQUID for drive currents well below the Josephson junction critical current I c . In this regime, the SQUID functions effectively as a mechanical position-dependent inductance element to a good approximation and the resulting closed system Hamiltonian (24) takes the same form as that for several other types of coupled mechanical resonator-detector resonator systems. Thus, the key derived expressions (69) and (70) for the detector response and detector noise are of more general application.\n\nThe main approximation made in analyzing the position and force detection sensitivity, as well as back reaction cooling, was to limit the drive current and or coupling strength according to Eq. (76). This allowed us to find much simpler, analytical approximations to the key expressions, in particular Eqs. (79) and (81). The regime of larger drive currents and 30 or coupling strengths which exceed the limit (76) remains to be explored. However, with the SQUID in mind, it is more appropriate to consider larger drive currents in the context of including the non-linear I/I c corrections to the SQUID effective inductance. This will be the subject of a forthcoming paper. 19"
},
{
"section_type": "OTHER",
"section_title": "Acknowledgements",
"text": "We thank A. Armour, A. Rimberg, and P. Nation for helpful discussions. This work was partly supported by the US-Israel Binational Science Foundation (BSF) and by the National Science Foundation (NSF) under NIRT grant CMS-0404031. One of the authors (M.P.B.) thanks the Aspen Center for Physics for their hospitality and support where part of this work was carried out. 1 R. Knobel and A. N. Cleland, Nature 424, 291 (2003). 2 M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, Science 304, 74 (2004).\n\n3 A. Naik, O. Buu, M. D. Lahaye, A. D. Armour, A. A. Clerk, M. P. Blencowe, and K. C. Schwab, Nature 443, 193 (2006). 4 M. P. Blencowe and M. N. Wybourne, Appl. Phys. Lett. 77, 3845 (2000). 5 Y. Zhang and M. P. Blencowe, J. Appl. Phys. 91, 4249 (2002). 6 M. Blencowe, Phys. Rep. 395, 159 (2004). 7 D. Mozyrsky, I. Martin, and M. B. Hastings, Phys. Rev. Lett. 92, 018303 (2004). 8 A. A. Clerk and S. Bennett, N. J. Phys. 7, 238 (2005). 9 M. P. Blencowe, J. Imbers, and A. D. Armour, N. J. Phys. 7, 236 (2005).\n\n10 C. M. Caves, Phys. Rev. D 26, 1817 (1982). 11 V. B. Braginsky and F. Ya. Khalili, Quantum Measurement, (Cambridge University Press, Cambridge, UK, 1992).\n\n12 A. A. Clerk, Phys. Rev. B 70, 245306 (2004). 13 F. Xue, Y. D. Wang, C. P. Sun, H. Okamoto, H. Yamaguchi, and K. Semba, N. J. Phys. 9, 35 (2007). 14 E. Buks and M. P. Blencowe, Phys. Rev. B 74, 174504 (2006). 31 15 X. Zhou and A. Mizel, Phys. Rev. Lett. 97, 267201 (2006). 16 E. Buks, E. Arbel-Segev, S. Zaitsev, B. Abdo, and M. P. Blencowe, quant-ph/0610158 (unpublished).\n\n17 R. H. Koch, D. J. van Harlingen, and J. Clarke, Phys. Rev. Lett. 45, 2132 (1980).\n\n18 R. H. Koch, D. J. van Harlingen, and J. Clarke, Appl. Phys. Lett. 38, 380 (1981). 19 P. Nation, M. P. Blencowe, and E. Buks, in progress.\n\n20 P. A. Truitt, J. B. Hertzberg, C. C. Huang, K. L. Ekinci, and K. C. Schwab, Nano Lett., 7, 120 (2007). 21 D. J. Wineland, J. Britton, R. J. Epstein, D. Leibfried, R. B. Blakestad, K. Brown, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and J. Wesenberg, quant-ph/0606180 (unpublished).\n\n22 A. F. Pace, M. J. Collett, and D. F. Walls, Phys. Rev. A 47, 3173 (1993). 23 S. Mancini and P. Tombesi, Phys. Rev. A 49, 4055 (1994). 24 G. J. Milburn, K. Jacobs, and D. F. Walls, Phys. Rev. A 50, 5256 (1994). 25 K. Jacobs, I. Tittonen, H. M. Wiseman, S. Schiller, Phys. Rev. A 60, 538 (1999). 26 F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, cond-mat/0701416 (unpublished).\n\n27 A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004).\n\n28 A. Wallraff, D. Schuster, A. Blais, L. Frunzo, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 165 (2004). 29 A. Lupaşcu, E. F. C. Driessen, L. Roschier, C. J. P. M. Harmans, and J. E. Mooij, Phys. Rev. . 96, 127003 (2006).\n\n30 G. Johansson, L. Tornberg, V. S. Shumeiko, and G. Wendin, J. Phys.: Condens. Matter 18, S901 (2006). 31 B. Yurke and J. S. Denker, Phys. Rev. A 29, 1419 (1984). 32 T. P. Orlando and K. A. Delin, Chapter 8 of Foundations of Applied Superconductivity, Addison-Wesley Publishing Company, Reading, Massachusetts, 1991).\n\n(\n\n33 M. H. Devoret, in Quantum Fluctuations (Les Houches Session LXIII), edited by S. Reynaud, E. Giacobino, and J. Zinn-Justin (Elsevier Science, 1997), pp. 351-386.\n\n34 C. W. Gardiner and M. J. Collett, Phys. Rev. A 31, 3761 (1985). 35 C. W. Gardiner and P. Zoller, Quantum Noise: 2nd Edition, (Springer-Verlag, Berlin, 2000). 36 B. Yurke and E. Buks, J. Lightwave Tech. 24, 5054 (2006). 32 37 D. Mozyrsky and I. Martin, Phys. Rev. Lett. 89, 018301 (2002).\n\n38 A. D. Armour, M. P. Blencowe, and Y. Zhang, Phys. Rev. B 69, 125313 (2004). 39 M. P. Blencowe, Contemp. Phys. 46, 249 (2005). 40 V. B. Braginsky and S. P. Vyatchanin, Phys. Lett. A 293, 228 (2002). 41 I. Martin, A. Shnirman, L. Tian, and P. Zoller, Phys. Rev. B 69, 125339 (2004). 42 I. Wilson-Rae, P. Zoller, and A. Imamoglu, Phys. Rev. Lett. 92, 075507 (2004).\n\n43 C. Höhberger-Metzger and K. Karrai, Nature 432, 1002 (2004). 44 J. G. E. Harris, B. M. Zwickl, and A. M. Jayich, Rev. Sci. Instrum. 78, 013107 (2007). 45 S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, Nature 444, 67 (2006). 46 O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature 444, 71 (2006).\n\n47 A. Schliesser, P. DelHaye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, Phys. Rev. Lett. 97, 243905 (2006). 48 T. Corbitt, Y. Chen, E. Innerhofer, H. Muller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S.\n\nWhitcomb, C. Wipf, and N. Mavalvala, quant-ph/0612188 (unpublished)."
}
] | |
arxiv:0704.0461 | 0704.0461 | 1 | 10.1103/PhysRevA.76.022102 | 2030e896fe190178eff537073e190c2174761dc1a1cc7d97d92c6f814b29b5e0 | Entanglement increase from local interactions with
not-completely-positive maps | Simple examples are constructed that show the entanglement of two qubits
being both increased and decreased by interactions on just one of them. One of
the two qubits interacts with a third qubit, a control, that is never entangled
or correlated with either of the two entangled qubits and is never entangled,
but becomes correlated, with the system of those two qubits. The two entangled
qubits do not interact, but their state can change from maximally entangled to
separable or from separable to maximally entangled. Similar changes for the two
qubits are made with a swap operation between one of the qubits and a control;
then there are compensating changes of entanglement that involve the control.
When the entanglement increases, the map that describes the change of the state
of the two entangled qubits is not completely positive. Combination of two
independent interactions that individually give exponential decay of the
entanglement can cause the entanglement to not decay exponentially but,
instead, go to zero at a finite time. | [
"Thomas F. Jordan",
"Anil Shaji and E. C. G. Sudarshan"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | We construct simple examples here that show the entanglement of two qubits being both increased and decreased by interactions on just one of them. In our first and basic step, taken in Sec. II, we have one of the two qubits interact with a third qubit, a control, that is never entangled or correlated with either of the two entangled qubits and is never entangled, but becomes correlated, with the system of those two qubits. In Sec. III, we do this for each of the two entangled qubits, and consider the combination of the two interactions, with separate control qubits that are not correlated and do not interact with each other. The two entangled qubits do not interact, but their state can change from maximally entangled to separable or from separable to maximally entangled. Similar changes for the two qubits are made with a swap operation between one of the qubits and a control; then there are compensating changes of entanglement that involve the control. This is described in Sec. II.A. Whenever the entanglement increases, and in some cases where the entanglement decreases, the map that describes the change of the state of the two entangled qubits is not completely positive and does not apply to all states of two qubits. It all depends on whether there are correlations with the controls at the beginning of the interval for which the dynamics is considered. The maps are described in Sec. IV and discussed in Sec. V. The completely positive maps that decrease the entanglement have already been described [1] .
When the interaction of each qubit with its control by itself gives exponential decay of the entanglement, the combination of the two interactions gives exponential decay at the rate that is the sum of the rates for the individual interactions, when the two interactions are made the same way. Making them differently can cause the entanglement to not decay at that rate or at any single rate. Instead, the entanglement goes to zero at a finite time; the state becomes separable and remains separable at later times. This is described in Sec. III.A. Similar behavior has been observed in more physically interesting and mathematically complicated models [2, 3, 4] .
These examples are built on the same framework, but to a very different design, from those we made for Lorentz transformations that entangle spins [5] . There the momenta that played the roles of controls were purposely correlated. Here the controls are kept independent. The framework makes the operations transparent by describing the qubit states with density matrices written in terms of Pauli matrices, so you can see the Pauli matrices being rotated by the interactions. States are shown to be separable by writing out the density matrices explicitly as sums of products for pure states. For each interaction here, the map that makes the change of the density matrix for the entangled qubits is described by a simple rule that particular Pauli matrices in the density matrix are multiplied by a number; equivalently, the map of the state of the entangled qubits is described by a rule that particular mean values are multiplied by a number.
Our examples show that statements like "entanglement should not increase under local operations and classical communication" [6, 7] are not generally true outside the set of local operations considered in the original proof [6] . In our examples, each control qubit interacts with only one of the two entangled qubits. In this sense, the quantum operations are local. Correlation with a control at the beginning of the interval for which the dynamics is considered can give local operations that increase entanglement.
We consider the entanglement of two qubits, A and B. We use Pauli matrices Σ 1 , Σ 2 , Σ 3 for qubit A, and Pauli matrices Ξ 1 , Ξ 2 , Ξ 3 for qubit B. We let qubit A interact with a third qubit, which we call C. We think of C as a control. By interacting with qubit A, it will control the entanglement of qubits A and B. We work with states represented by orthonormal vectors |α and |β for C. We consider a state of the three qubits represented by a density matrix Π = ρ ⊗ 1 2 1 1 C (2.1) with ρ the density matrix for the state of qubits A and B. We follow common physics practice and write a product of operators for separate systems, for example a product of Pauli matrices Σ and Ξ for qubits A and B, simply as ΣΞ, not Σ⊗Ξ.
Occasionally we insert a ⊗ for emphasis or clarity. We write 1 1 A , 1 1 B , 1 1 C , but we do not put labels A and B on the Σ j and Ξ k . The single statement that the Σ j are for qubit A and the Ξ k are for qubit B eliminates the need for continual use of both A and B lalels and ⊗ signs.
Suppose ρ is one of the density matrices
ρ ± = 1 4 (1 1 ± Σ 1 Ξ 1 ± Σ 2 Ξ 2 -Σ 3 Ξ 3 ). (2.2)
Both ρ + and ρ -represent maximally entangled pure states for the two qubits. They are Bell states. The state of zero total spin is represented by ρ -and the state obtained from that by rotating one of the spins by π around the z axis is represented by ρ + . For a rotation W , let D A (W ) be the 2 × 2 unitary rotation matrix made from the Σ j so that
D A (W ) † ΣD A (W ) = W (Σ) (2.3)
where W (Σ) is simply the vector Σ rotated by W . Let W (φ) be the rotation by φ around the z axis, and let D A (φ) be D A (W (φ)).
We consider an interaction between qubits A and C described by the unitary transformation
U = D A (φ)|α α| + D A (-φ)|β β| (2.4)
or, in Hamiltonian form,
U = e -iφH (2.5) with H = Σ 3 1 2 (|α α| -|β β|). (2.6)
This changes the density matrix ρ for qubits A and B to
ρ ′ = Tr C (U ⊗ 1 1 B )Π(U ⊗ 1 1 B ) † = 1 2 D A (φ)ρD A (φ) † + 1 2 D A (-φ)ρD A (-φ) † . (2.7)
For ρ ± this gives
ρ ′ ± = 1 2 1 4 [1 1 ± (Σ 1 cos φ + Σ 2 sin φ)Ξ 1 ± (-Σ 1 sin φ + Σ 2 cos φ)Ξ 2 -Σ 3 Ξ 3 ] + 1 2 1 4 [1 1 ± (Σ 1 cos φ -Σ 2 sin φ)Ξ 1 ± (Σ 1 sin φ + Σ 2 cos φ)Ξ 2 -Σ 3 Ξ 3 ] = 1 4 [1 1 ± (Σ 1 Ξ 1 + Σ 2 Ξ 2 ) cos φ -Σ 3 Ξ 3 ] = ρ ± cos 2 (φ/2) + ρ ∓ sin 2 (φ/2). (2.8)
We focus first on the case where φ is π/2. Then both ρ + and ρ -are changed to
ρ ′ = 1 4 [1 1 -Σ 3 Ξ 3 ] = 1 2 1 2 (1 1 -Σ 3 ) 1 2 (1 1 + Ξ 3 ) + 1 2 1 2 (1 1 + Σ 3 ) 1 2 (1 1 -Ξ 3 ). (2.9)
The density matrix ρ for a maximally entangled state is changed to the density matrix ρ ′ for a separable state that is a mixture of just two products of pure states. The inverse of the unitary dynamics of qubits A and R takes ρ ′ back to ρ; it changes a separable state to a maximally entangled state.
The dynamics continuing forward also changes this separable state to a maximally entangled state. As φ goes from π/2 to π, the density matrix ρ ′ ± changes from that of Eq. (2.9) to
ρ ′ ± = ρ ∓ . (2.10)
There can be revivals of entanglement between two qubits when there is no interaction between them, as well as when [8] there is. Changes in the state of qubits A and B from maximally entangled to separable and back to maximally entangled can also be made very simply with a swap of states[9] between A and C. This can be done with a unitary operator U ⊗ 1 1 B with U a unitary operator for qubits A and C that acts on a basis of product state vectors simply by interchanging the states of A and C. There is interaction between qubits A and C only; qubit B is not involved.
Applied to an initial state described by Eqs. (2.1) and (2.2), where qubits A and B are maximally entangled, this swap operation gives a separable state for A and B. Applied a second time, it restores the initial state where A and B are maximally entangled. For qubits A and B, this is similar to what happens when φ goes from 0 to π/2 to π. For the three qubits, it is different. The swap operation does not change the complete inventory of entanglements for the three qubits. It just moves the entanglements around. In particular, C becomes maximally entangled with B. We will see, in Secs. II.C and 3 D, that the interaction described by Eqs. (2.4), (2.5) and (2.6) does change the complete inventory of entanglements for the three qubits. When the state of qubits A and B changes from maximally entangled to separable and back to maximally entangled, there are no compensating changes of other two-part entanglements. In particular, qubit C never becomes entangled with anything.
The change of entanglement is smaller when φ does not change by π/2. ¿From Eq. (2.8) we have
ρ ′ ± = 1 4 [1 1±(Σ 1 Ξ 1 +Σ 2 Ξ 2 ) cos φ+(Σ 1 Ξ 1 )(Σ 2 Ξ 2 )], (2.11)
after rewriting the last term. This shows that for both ρ ′ + and ρ ′ -the eigenvalues are
1 2 (1 + cos φ), 1 2 (1 -cos φ), 0, 0 (2.12)
because Σ 1 Ξ 1 and Σ 2 Ξ 2 each have eigenvalues 1 and -1 and together they make a complete set of commuting operators: their four different pairs of eigenvalues label a basis of eigenvectors for the space of states for the two qubits. The Wooters concurrence [10] is a measure of the entanglement in a state of two qubits. It is defined by
C(ρ) ≡ max 0, λ 1 -λ 2 -λ 3 -λ 4 (2.13)
where ρ is the density matrix that represents the state and λ 1 , λ 2 , λ 3 , λ 4 are the eigenvalues, in decreasing order, of
ρ Σ 2 Ξ 2 ρ ⋆ Σ 2 Ξ 2 , with ρ ⋆
the complex conjugate that is obtained by changing Σ 2 and Ξ 2 to -Σ 2 and -Ξ 2 . From Eq. (2.11) we have
ρ ′ ± Σ 2 Ξ 2 (ρ ′ ± ) ⋆ Σ 2 Ξ 2 = ρ ′ ± (ρ ′ ± ) ⋆ (Σ 2 Ξ 2 ) 2 = (ρ ′ ± ) 2 (2.14)
so for ρ ′ ± the √ λ i are the eigenvalues of ρ ′ ± and the concurrence is
C(ρ ′ ± ) = | cos φ|. (2.15)
We can consider the change of entanglement as φ changes through any interval. When | cos φ| decreases, the entanglement decreases. When | cos φ| increases, the entanglement increases.
The only two-part entanglements are when qubit A is in one part and qubit B is in the other. There is entanglement between qubit A and the subsystem of two qubits B and C and between qubit B and the subsystem of two qubits A and C, as well as between qubits A and B. There is never entanglement between the state of qubit C and the state of the subsystem of two qubits A and B. The density matrix
(U ⊗ 1 1 B )Π(U ⊗ 1 1 B ) † = 1 2 D A (φ)ρ ± D A (φ) † |α α| + 1 2 D A (-φ)ρ ± D A (-φ) † |β β| (2.16)
is always a mixture of two products of pure states. The reduced density matrix for the subsystem of qubits A and C, obtained by taking the trace over the states of qubit B, is just 1 1 A ⊗ 1 1 C /4, and the reduced density matrix for qubits B and C, obtained by taking the trace over the states of qubit A, is 1 1 B ⊗ 1 1 C /4. There is never entanglement or correlation between qubits A and C or between qubits B and C. The reduced density matrices for the individual single qubits are just 1 1 A /2, 1 1 B /2, and 1 1 C /2. The only subsystem density matrix that carries any information is the density matrix ρ for the qubits A and B, which is changed by the interaction with qubit C. The entropy of the subsystem of qubits A and B can increase or decrease, but there is no change of entropy for any other subsystem or for the entire system of three qubits.
There is three-part entanglement. The state represented by the density matrix (2.16) is called biseparable because it is separable as the state of a system of two parts, with C one part and the subsystem of two qubits A and B the other part. It is not separable as the state of a system of three parts A, B, and C. The density matrix (2.16) is not a mixture of products of density matrices for pure states of the individual qubits A, B, and C. If it were, its partial trace over the states of C, the reduced density matrix that represents the state of the subsystem of the two qubits A and B, would be a mixture of products for pure states of A and B. That happens only when cos φ is 0.
In that case, we can see that the density matrix (2.16) is not a mixture of products for pure states of the individual qubits A, B, and C because its partial transpose obtained by changing Ξ 2 to -Ξ 2 is not a positive matrix.
In the classification of three-part entanglement for qubits, biseparable states are between separable states and states that involve W or GHZ entanglement [11, 12, 13] .
Let Π 1 , Π 2 , Π 3 be Pauli matrices for the qubit C such that |α α| is (1/2)(1 1 + Π 3 ) and |β β| is (1/2)(1 1 -Π 3 ).
Bounds from Mermin witness operators say that for separable or biseparable states 2.17) for j, k = 1, 2, 3 and j = k; a mean value outside these bounds is a mark of W or GHZ entanglement [14] . In our examples, these mean values are always 0. A mean value |GHZ GHZ| larger than 3/4 for the projection operator onto the GHZ state,
-2 ≤ Σ j Ξ j Π j -Σ j Ξ k Π k -Σ k Ξ j Π k -Σ k Ξ k Π j ≤ 2 (
|GHZ = 1 √ 2 |0 |0 |0 + 1 √ 2 |1 |1 |1 , (2.18)
4 is a mark of GHZ entanglement; it can not be larger than 3/4 for a W state [13] . A mean value |GHZ GHZ| larger than 1/2 is a mark of a W state; it can not be larger than 1/2 for a biseparable state [13] . In our examples, |GHZ GHZ| is always 0. A mean value |W W | larger than 2/3 for the projection operator onto the W state,
|W = 1 √ 3 |1 |0 |0 + 1 √ 3 |0 |1 |0 + 1 √ 3 |0 |0 |1 , (2. 19
) is a mark of W entanglement; it can not be larger than 2/3 for a biseparable state [13] . In our examples, |W W | = 1 6 (1 ± cos φ). (2.20) This mean value does not involve either entanglement or correlation of the qubit C; it would be the same if both |α α| and |β β| in the density matrix (2.16) were replaced by (1/2) C , the completely mixed density matrix for C. For any φ, the density matrices (2.16) for the two cases + andare changed into each other by the local unitary transformation that changes the Pauli matrices for one of the qubits A or B by rotating its spin by π around the z axis. As a function of φ, the mean value |W W | changes in opposite directions for the + andcases. So will any mean value for the states described by the density matrices (2.16), if it changes at all. For the states described by the density matrices (2.16), the only nonzero mean values that involve the qubit C are
Σ 1 Ξ 2 Π 3 = ∓ sin φ Σ 2 Ξ 1 Π 3 = ± sin φ. (2.21)
These would be the same if they were calculated with only the |α α| part or only the |β β| part of the density matrix (2.16).
In fact, they are the same as Σ 1 Ξ 2 Π 3 and Σ 2 Ξ 1 Π 3 calculated for one of those parts. Their values do not require either entanglement or correlation of C.
If a control were coupled similarly to qubit B as well, then cos φ would be replaced by cos φ A cos φ B in the next to last line of Eq. (2.8) and in Eqs.(2.11) and (2.15). If the coupling of qubit B is made with a rotation around the x axis instead of the z axis, then the next to last line of Eq. (2.8) becomes
ρ ′ ± = 1 4 [1 1 ± Σ 1 Ξ 1 cos φ A ± Σ 2 Ξ 2 cos φ A cos φ B -Σ 3 Ξ 3 cos φ B ]. (3.1)
Rewriting the last term and looking at eigenvalues in terms of Σ 1 Ξ 1 and Σ 2 Ξ 2 as before yields the concurrence
C(ρ ′ ± )= 1 2 max[0, | cos φ A |+| cos φ A cos φ B |+| cos φ B |-1].
(3.2) When cos φ A is 1, these Eqs.(3.1) and (3.2) describe the result obtained when there is only the interaction of qubit B made with a rotation around the x axis. If neither cos φ A nor cos φ B is 1, the concurrence becomes zero, the state separable, before cos φ A or cos φ B is zero. The interactions of qubits A and B with their controls change maximally entangled states to separable states. The inverses change separable states to maximally entangled states. In the following subsection, we describe the density matrices that show explicitly that the separable states are mixtures of products of pure states.
To describe exponential decay of entanglement we let cos φ A = e -ΓAt , cos φ B = e -ΓB t (3.3) by letting each interaction be modulated by a time-dependent Hamiltonian H(t) that is related to the Hamiltonian H of Eqs.(2.4) and (2.5) by H(t) = H dφ dt = HΓcotφ, (3.4) where φ and Γ are φ A and Γ A or φ B and Γ B . The same result could be produced in different ways. The interactions could be with large reservoirs instead of qubit controls [2, 3, 4] . Each qubit A or B could interact with a stream of reservoir qubits [15] . Here we are interested in the way the entanglement is changed by the combination of the two interactions. That depends only on the changes in the density matrix ρ for qubits A and B, not on the nature of the controls and the interactions. Maps that make the changes in ρ will be described in the next section. If there is only the interaction of qubit A with qubit C, the concurrence is e -ΓAt . If there is only interaction of qubit B with its control, the concurrence is e -ΓB t . If there are both and both are made with rotations around the z axis, the concurrence is e -ΓAt e -ΓB t . If there are both and the interaction of qubit B with its control is made with a rotation around the x axis, the concurrence is C(ρ ′ ± ) = 1 2 max[0, e -ΓAt +e -ΓAt e -ΓBt +e -ΓBt -1]. (3.5) This concurrence (3.5) is zero when e -ΓAt + e -ΓAt e -ΓB t + e -ΓB t = 1. (3.6) Then the state is separable; it is a mixture of six products of pure states: from Eqs.(3.1) and (3.3)
ρ ′ ± = 1 2 e -ΓAt 1 2 (1 1 + Σ 1 ) 1 2 (1 1 ± Ξ 1 ) + 1 2 e -ΓAt 1 2 (1 1 -Σ 1 ) 1 2 (1 1 ∓ Ξ 1 ) + 1 2 e -ΓAt e -ΓBt 1 2 (1 1 + Σ 2 ) 1 2 (1 1 ± Ξ 2 ) + 1 2 e -ΓAt e -ΓBt 1 2 (1 1 -Σ 2 ) 1 2 (1 1 ∓ Ξ 2 ) + 1 2 e -ΓBt 1 2 (1 1 + Σ 3 ) 1 2 (1 1 -Ξ 3 ) + 1 2 e -ΓBt 1 2 (1 1 -Σ 3 ) 1 2 (1 1 + Ξ 3 ). (3.7) 5
The state remains separable at later times; when the sum of the exponential decay factors is less than 1, the density matrix is a mixture in which just a multiple of the density matrix 1/4 for the completely mixed state is added to the terms of Eq. (3.7). This change of maximally entangled states to separable states can be described without reference to exponential decay by continuing to use cos φ A and cos φ B instead of e -ΓAt and e -ΓB t . Similar behavior involving exponential decay has been observed in more physically interesting and mathematically complicated models [2, 3, 4] .
The maps that make the changes in the density matrix ρ for qubits A and B could be described in different ways using various matrix forms. That is not needed here. Writing ρ in terms of Pauli matrices provides a very simple way to describe the maps. For any density matrix
ρ = 1 4 1 1 + 3 j=1 Σ j Σ j + 3 k=1 Ξ k Ξ k + 3 j,k=1 Σ j Ξ k Σ j Ξ k ( 4
.1) for qubits A and B, the result of the interaction of qubit A with qubit C, described by Eq. (2.7), is that in ρ, in both the
Σ j and Σ j Ξ k terms, Σ 1 -→ Σ 1 cos φ A , Σ 2 -→ Σ 2 cos φ A ;
(4.2) the result of the interaction of qubit B with its control is that in ρ
Ξ 1 -→ Ξ 1 cos φ B , Ξ 2 -→ Ξ 2 cos φ B (4.3)
if the interaction is made with a rotation around the z axis; and the result of the interaction of qubit B with its control is that in ρ
Ξ 2 -→ Ξ 2 cos φ B , Ξ 3 -→ Ξ 3 cos φ B (4.4)
if the interaction is made with a rotation around the x axis. The terms with sin φ cancel out because there is an equal mixture of parts with φ and parts with -φ.
The changes in the state of qubits A and B can be described equivalently by maps of mean values that describe the state: the result of the interaction of qubit A with qubit C, described by Eq. (2.7), is that 4.5) for k = 1, 2, 3; the result of the interaction of qubit B with its control is that 4.6) for j = 1, 2, 3 if the interaction is made with a rotation around the z axis; and the result of the interaction of qubit B with its control is that
Σ 1 -→ Σ 1 cos φ A Σ 2 -→ Σ 2 cos φ A Σ 1 Ξ k -→ Σ 1 Ξ k cos φ A Σ 2 Ξ k -→ Σ 2 Ξ k cos φ A (
Ξ 1 -→ Ξ 1 cos φ B Ξ 2 -→ Ξ 2 cos φ B Σ j Ξ 1 -→ Σ j Ξ 1 cos φ B Σ j Ξ 2 -→ Σ j Ξ 2 cos φ B (
Ξ 2 -→ Ξ 2 cos φ B Ξ 3 -→ Ξ 3 cos φ B Σ j Ξ 2 -→ Σ j Ξ 2 cos φ B Σ j Ξ 3 -→ Σ j Ξ 3 cos φ B (4.7)
for j = 1, 2, 3 if the interaction is made with a rotation around the x axis. When φ A and φ B change over intervals from initial values φ Ai and φ Bi to final values φ Af and φ Bf , the cos φ A and cos φ B factors in the maps are replaced by cos φ Af / cos φ Ai and cos φ Bf / cos φ Bi . If either of these factors is larger than 1, the map is not completely positive and does not apply to all density matrices ρ for qubits A and B. This happens whenever the entanglement increases. It also happens in cases where the concurrence (3.2) decreases, when one of cos φ A and cos φ B increases and the other decreases and there is more decrease than increase. The completely positive maps that decrease the entanglement have already been described [1].
Entanglement being increased by local interactions may seem surprising from perspectives framed by experience in common situations where it is impossible. Entanglement is not increased by a completely positive map of the state of two qubits produced by an interaction on one of them. The interaction will produce a completely positive map if it is with a control whose state is initially not correlated with the state of the two qubits, as in Eq. (2.1). In our examples, that happens only when the initial value of φ is 0 or a multiple of π.
Otherwise, the state of the control is correlated with the state of the two qubits as in Eq. (2.16). When a subsystem is initially correlated with the rest of a larger system that is being changed by unitary Hamiltonian dynamics, the map that describes the change of the state of the subsystem generally is not completely positive and applies to a limited set of subsystem states [16, 17] . We see this in our examples whenever the entanglement increases and in some cases when the entanglement decreases.
The map depends on both the dynamics and the initial correlations. It describes the effect of both in one step. Completely positive maps are what you get in the simplest set-up where you bring a system and control together in independent states and consider the effect of the dynamics that begins then. Dynamics over intervals where the maps are not completely positive can be expected to play roles in more complex settings. We should not let expectations for completely positive maps prevent us from seeing things that can happen.
Our perspective is enlarged when we look beyond the map and include the dynamics. We can see the dynamics and the initial preparation as two related but separate steps. We can consider the effect of the dynamics, whatever the preparation may be.
6 Local interactions that increase entanglement are completely outside a perspective that is limited to pure states. An interaction on one of the qubits can not change the entanglement at all if the state of the two qubits remains pure [18] . The entanglement of a pure state of two qubits depends only on the spectrum of the reduced density matrices that describe the states of the individual qubits, which is the same for the two qubits. If that could be changed by an interaction on one of the qubits, there could be a signal faster than light. In our examples, the state of the two qubits is pure only when it is maximally entangled. In our examples, the spectrum of the density matrices for the individual qubits never changes, and gives no measure of the entanglement.
We are grateful to a referee for very helpful suggestions, including the comparison with a swap operation. Anil Shaji acknowledges the support of the US Office of Naval Research through Contract No. N00014-03-1-0426.
[1] M. Ziman and V. Buzek, Phys. Rev. A 72, 052325 (2005). [2] T. Yu and J. H. Eberly, Phys. Rev. Lett. 97, 140403 (2006). [3] X.-T. Liang, Phys. Lett. A 349, 98 (2006). [4] T. Yu and J. H. Eberly, arXiv:quant-ph/0703083 (2007). [5] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A 75, 022101 (2007). [6] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996). [7] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, arXiv:quant-ph/0702225 (2007). [8] K. Zyczkowski, P. Horodecki, M. Horodecki, and R. Horodecki, Phys. Rev. A 65, 012101 (2001). [9] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Phys.
Rev. Lett. 71, 4287 (1993). [10] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [11] W. Dür, J. I. Cirac, and R. Tarrach, Phys. Rev. Lett. 83, 3562 (1999). [12] W. Dür and J. I. Cirac, Phys. Rev. A 61, 042314 (2000). [13] A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, Phys. Rev.
Lett. 87, 040401 (2001). [14] G. Toth, O. Guhne, M. Seevinck, and J. Uffink, Phys. Revs A 72, 014101 (2005). [15] E. C. G. Sudarshan, Chaos, Solitons and Fractals 16, 369 (2003). [16] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A. 70, 52110 (2004). [17] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A. 73, 12106 (2006). [18] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996). | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "We construct simple examples here that show the entanglement of two qubits being both increased and decreased by interactions on just one of them. In our first and basic step, taken in Sec. II, we have one of the two qubits interact with a third qubit, a control, that is never entangled or correlated with either of the two entangled qubits and is never entangled, but becomes correlated, with the system of those two qubits. In Sec. III, we do this for each of the two entangled qubits, and consider the combination of the two interactions, with separate control qubits that are not correlated and do not interact with each other. The two entangled qubits do not interact, but their state can change from maximally entangled to separable or from separable to maximally entangled. Similar changes for the two qubits are made with a swap operation between one of the qubits and a control; then there are compensating changes of entanglement that involve the control. This is described in Sec. II.A. Whenever the entanglement increases, and in some cases where the entanglement decreases, the map that describes the change of the state of the two entangled qubits is not completely positive and does not apply to all states of two qubits. It all depends on whether there are correlations with the controls at the beginning of the interval for which the dynamics is considered. The maps are described in Sec. IV and discussed in Sec. V. The completely positive maps that decrease the entanglement have already been described [1] .\n\nWhen the interaction of each qubit with its control by itself gives exponential decay of the entanglement, the combination of the two interactions gives exponential decay at the rate that is the sum of the rates for the individual interactions, when the two interactions are made the same way. Making them differently can cause the entanglement to not decay at that rate or at any single rate. Instead, the entanglement goes to zero at a finite time; the state becomes separable and remains separable at later times. This is described in Sec. III.A. Similar behavior has been observed in more physically interesting and mathematically complicated models [2, 3, 4] .\n\nThese examples are built on the same framework, but to a very different design, from those we made for Lorentz transformations that entangle spins [5] . There the momenta that played the roles of controls were purposely correlated. Here the controls are kept independent. The framework makes the operations transparent by describing the qubit states with density matrices written in terms of Pauli matrices, so you can see the Pauli matrices being rotated by the interactions. States are shown to be separable by writing out the density matrices explicitly as sums of products for pure states. For each interaction here, the map that makes the change of the density matrix for the entangled qubits is described by a simple rule that particular Pauli matrices in the density matrix are multiplied by a number; equivalently, the map of the state of the entangled qubits is described by a rule that particular mean values are multiplied by a number.\n\nOur examples show that statements like \"entanglement should not increase under local operations and classical communication\" [6, 7] are not generally true outside the set of local operations considered in the original proof [6] . In our examples, each control qubit interacts with only one of the two entangled qubits. In this sense, the quantum operations are local. Correlation with a control at the beginning of the interval for which the dynamics is considered can give local operations that increase entanglement."
},
{
"section_type": "OTHER",
"section_title": "II. ONE INTERACTION",
"text": "We consider the entanglement of two qubits, A and B. We use Pauli matrices Σ 1 , Σ 2 , Σ 3 for qubit A, and Pauli matrices Ξ 1 , Ξ 2 , Ξ 3 for qubit B. We let qubit A interact with a third qubit, which we call C. We think of C as a control. By interacting with qubit A, it will control the entanglement of qubits A and B. We work with states represented by orthonormal vectors |α and |β for C. We consider a state of the three qubits represented by a density matrix Π = ρ ⊗ 1 2 1 1 C (2.1) with ρ the density matrix for the state of qubits A and B. We follow common physics practice and write a product of operators for separate systems, for example a product of Pauli matrices Σ and Ξ for qubits A and B, simply as ΣΞ, not Σ⊗Ξ.\n\nOccasionally we insert a ⊗ for emphasis or clarity. We write 1 1 A , 1 1 B , 1 1 C , but we do not put labels A and B on the Σ j and Ξ k . The single statement that the Σ j are for qubit A and the Ξ k are for qubit B eliminates the need for continual use of both A and B lalels and ⊗ signs.\n\nSuppose ρ is one of the density matrices\n\nρ ± = 1 4 (1 1 ± Σ 1 Ξ 1 ± Σ 2 Ξ 2 -Σ 3 Ξ 3 ). (2.2)\n\nBoth ρ + and ρ -represent maximally entangled pure states for the two qubits. They are Bell states. The state of zero total spin is represented by ρ -and the state obtained from that by rotating one of the spins by π around the z axis is represented by ρ + . For a rotation W , let D A (W ) be the 2 × 2 unitary rotation matrix made from the Σ j so that\n\nD A (W ) † ΣD A (W ) = W (Σ) (2.3)\n\nwhere W (Σ) is simply the vector Σ rotated by W . Let W (φ) be the rotation by φ around the z axis, and let D A (φ) be D A (W (φ)).\n\nWe consider an interaction between qubits A and C described by the unitary transformation\n\nU = D A (φ)|α α| + D A (-φ)|β β| (2.4)\n\nor, in Hamiltonian form,\n\nU = e -iφH (2.5) with H = Σ 3 1 2 (|α α| -|β β|). (2.6)\n\nThis changes the density matrix ρ for qubits A and B to\n\nρ ′ = Tr C (U ⊗ 1 1 B )Π(U ⊗ 1 1 B ) † = 1 2 D A (φ)ρD A (φ) † + 1 2 D A (-φ)ρD A (-φ) † . (2.7)\n\nFor ρ ± this gives\n\nρ ′ ± = 1 2 1 4 [1 1 ± (Σ 1 cos φ + Σ 2 sin φ)Ξ 1 ± (-Σ 1 sin φ + Σ 2 cos φ)Ξ 2 -Σ 3 Ξ 3 ] + 1 2 1 4 [1 1 ± (Σ 1 cos φ -Σ 2 sin φ)Ξ 1 ± (Σ 1 sin φ + Σ 2 cos φ)Ξ 2 -Σ 3 Ξ 3 ] = 1 4 [1 1 ± (Σ 1 Ξ 1 + Σ 2 Ξ 2 ) cos φ -Σ 3 Ξ 3 ] = ρ ± cos 2 (φ/2) + ρ ∓ sin 2 (φ/2). (2.8)"
},
{
"section_type": "OTHER",
"section_title": "A. From maximally entangled to separable and back",
"text": "We focus first on the case where φ is π/2. Then both ρ + and ρ -are changed to\n\nρ ′ = 1 4 [1 1 -Σ 3 Ξ 3 ] = 1 2 1 2 (1 1 -Σ 3 ) 1 2 (1 1 + Ξ 3 ) + 1 2 1 2 (1 1 + Σ 3 ) 1 2 (1 1 -Ξ 3 ). (2.9)\n\nThe density matrix ρ for a maximally entangled state is changed to the density matrix ρ ′ for a separable state that is a mixture of just two products of pure states. The inverse of the unitary dynamics of qubits A and R takes ρ ′ back to ρ; it changes a separable state to a maximally entangled state.\n\nThe dynamics continuing forward also changes this separable state to a maximally entangled state. As φ goes from π/2 to π, the density matrix ρ ′ ± changes from that of Eq. (2.9) to\n\nρ ′ ± = ρ ∓ . (2.10)\n\nThere can be revivals of entanglement between two qubits when there is no interaction between them, as well as when [8] there is. Changes in the state of qubits A and B from maximally entangled to separable and back to maximally entangled can also be made very simply with a swap of states[9] between A and C. This can be done with a unitary operator U ⊗ 1 1 B with U a unitary operator for qubits A and C that acts on a basis of product state vectors simply by interchanging the states of A and C. There is interaction between qubits A and C only; qubit B is not involved.\n\nApplied to an initial state described by Eqs. (2.1) and (2.2), where qubits A and B are maximally entangled, this swap operation gives a separable state for A and B. Applied a second time, it restores the initial state where A and B are maximally entangled. For qubits A and B, this is similar to what happens when φ goes from 0 to π/2 to π. For the three qubits, it is different. The swap operation does not change the complete inventory of entanglements for the three qubits. It just moves the entanglements around. In particular, C becomes maximally entangled with B. We will see, in Secs. II.C and 3 D, that the interaction described by Eqs. (2.4), (2.5) and (2.6) does change the complete inventory of entanglements for the three qubits. When the state of qubits A and B changes from maximally entangled to separable and back to maximally entangled, there are no compensating changes of other two-part entanglements. In particular, qubit C never becomes entangled with anything."
},
{
"section_type": "OTHER",
"section_title": "B. Concurrence",
"text": "The change of entanglement is smaller when φ does not change by π/2. ¿From Eq. (2.8) we have\n\nρ ′ ± = 1 4 [1 1±(Σ 1 Ξ 1 +Σ 2 Ξ 2 ) cos φ+(Σ 1 Ξ 1 )(Σ 2 Ξ 2 )], (2.11)\n\nafter rewriting the last term. This shows that for both ρ ′ + and ρ ′ -the eigenvalues are\n\n1 2 (1 + cos φ), 1 2 (1 -cos φ), 0, 0 (2.12)\n\nbecause Σ 1 Ξ 1 and Σ 2 Ξ 2 each have eigenvalues 1 and -1 and together they make a complete set of commuting operators: their four different pairs of eigenvalues label a basis of eigenvectors for the space of states for the two qubits. The Wooters concurrence [10] is a measure of the entanglement in a state of two qubits. It is defined by\n\nC(ρ) ≡ max 0, λ 1 -λ 2 -λ 3 -λ 4 (2.13)\n\nwhere ρ is the density matrix that represents the state and λ 1 , λ 2 , λ 3 , λ 4 are the eigenvalues, in decreasing order, of\n\nρ Σ 2 Ξ 2 ρ ⋆ Σ 2 Ξ 2 , with ρ ⋆\n\nthe complex conjugate that is obtained by changing Σ 2 and Ξ 2 to -Σ 2 and -Ξ 2 . From Eq. (2.11) we have\n\nρ ′ ± Σ 2 Ξ 2 (ρ ′ ± ) ⋆ Σ 2 Ξ 2 = ρ ′ ± (ρ ′ ± ) ⋆ (Σ 2 Ξ 2 ) 2 = (ρ ′ ± ) 2 (2.14)\n\nso for ρ ′ ± the √ λ i are the eigenvalues of ρ ′ ± and the concurrence is\n\nC(ρ ′ ± ) = | cos φ|. (2.15)\n\nWe can consider the change of entanglement as φ changes through any interval. When | cos φ| decreases, the entanglement decreases. When | cos φ| increases, the entanglement increases."
},
{
"section_type": "OTHER",
"section_title": "C. Two-part entanglements",
"text": "The only two-part entanglements are when qubit A is in one part and qubit B is in the other. There is entanglement between qubit A and the subsystem of two qubits B and C and between qubit B and the subsystem of two qubits A and C, as well as between qubits A and B. There is never entanglement between the state of qubit C and the state of the subsystem of two qubits A and B. The density matrix\n\n(U ⊗ 1 1 B )Π(U ⊗ 1 1 B ) † = 1 2 D A (φ)ρ ± D A (φ) † |α α| + 1 2 D A (-φ)ρ ± D A (-φ) † |β β| (2.16)\n\nis always a mixture of two products of pure states. The reduced density matrix for the subsystem of qubits A and C, obtained by taking the trace over the states of qubit B, is just 1 1 A ⊗ 1 1 C /4, and the reduced density matrix for qubits B and C, obtained by taking the trace over the states of qubit A, is 1 1 B ⊗ 1 1 C /4. There is never entanglement or correlation between qubits A and C or between qubits B and C. The reduced density matrices for the individual single qubits are just 1 1 A /2, 1 1 B /2, and 1 1 C /2. The only subsystem density matrix that carries any information is the density matrix ρ for the qubits A and B, which is changed by the interaction with qubit C. The entropy of the subsystem of qubits A and B can increase or decrease, but there is no change of entropy for any other subsystem or for the entire system of three qubits."
},
{
"section_type": "OTHER",
"section_title": "D. Three-part entanglement",
"text": "There is three-part entanglement. The state represented by the density matrix (2.16) is called biseparable because it is separable as the state of a system of two parts, with C one part and the subsystem of two qubits A and B the other part. It is not separable as the state of a system of three parts A, B, and C. The density matrix (2.16) is not a mixture of products of density matrices for pure states of the individual qubits A, B, and C. If it were, its partial trace over the states of C, the reduced density matrix that represents the state of the subsystem of the two qubits A and B, would be a mixture of products for pure states of A and B. That happens only when cos φ is 0.\n\nIn that case, we can see that the density matrix (2.16) is not a mixture of products for pure states of the individual qubits A, B, and C because its partial transpose obtained by changing Ξ 2 to -Ξ 2 is not a positive matrix.\n\nIn the classification of three-part entanglement for qubits, biseparable states are between separable states and states that involve W or GHZ entanglement [11, 12, 13] .\n\nLet Π 1 , Π 2 , Π 3 be Pauli matrices for the qubit C such that |α α| is (1/2)(1 1 + Π 3 ) and |β β| is (1/2)(1 1 -Π 3 ).\n\nBounds from Mermin witness operators say that for separable or biseparable states 2.17) for j, k = 1, 2, 3 and j = k; a mean value outside these bounds is a mark of W or GHZ entanglement [14] . In our examples, these mean values are always 0. A mean value |GHZ GHZ| larger than 3/4 for the projection operator onto the GHZ state,\n\n-2 ≤ Σ j Ξ j Π j -Σ j Ξ k Π k -Σ k Ξ j Π k -Σ k Ξ k Π j ≤ 2 (\n\n|GHZ = 1 √ 2 |0 |0 |0 + 1 √ 2 |1 |1 |1 , (2.18)\n\n4 is a mark of GHZ entanglement; it can not be larger than 3/4 for a W state [13] . A mean value |GHZ GHZ| larger than 1/2 is a mark of a W state; it can not be larger than 1/2 for a biseparable state [13] . In our examples, |GHZ GHZ| is always 0. A mean value |W W | larger than 2/3 for the projection operator onto the W state,\n\n|W = 1 √ 3 |1 |0 |0 + 1 √ 3 |0 |1 |0 + 1 √ 3 |0 |0 |1 , (2. 19\n\n) is a mark of W entanglement; it can not be larger than 2/3 for a biseparable state [13] . In our examples, |W W | = 1 6 (1 ± cos φ). (2.20) This mean value does not involve either entanglement or correlation of the qubit C; it would be the same if both |α α| and |β β| in the density matrix (2.16) were replaced by (1/2) C , the completely mixed density matrix for C. For any φ, the density matrices (2.16) for the two cases + andare changed into each other by the local unitary transformation that changes the Pauli matrices for one of the qubits A or B by rotating its spin by π around the z axis. As a function of φ, the mean value |W W | changes in opposite directions for the + andcases. So will any mean value for the states described by the density matrices (2.16), if it changes at all. For the states described by the density matrices (2.16), the only nonzero mean values that involve the qubit C are\n\nΣ 1 Ξ 2 Π 3 = ∓ sin φ Σ 2 Ξ 1 Π 3 = ± sin φ. (2.21)\n\nThese would be the same if they were calculated with only the |α α| part or only the |β β| part of the density matrix (2.16).\n\nIn fact, they are the same as Σ 1 Ξ 2 Π 3 and Σ 2 Ξ 1 Π 3 calculated for one of those parts. Their values do not require either entanglement or correlation of C."
},
{
"section_type": "OTHER",
"section_title": "III. TWO INTERACTIONS",
"text": "If a control were coupled similarly to qubit B as well, then cos φ would be replaced by cos φ A cos φ B in the next to last line of Eq. (2.8) and in Eqs.(2.11) and (2.15). If the coupling of qubit B is made with a rotation around the x axis instead of the z axis, then the next to last line of Eq. (2.8) becomes\n\nρ ′ ± = 1 4 [1 1 ± Σ 1 Ξ 1 cos φ A ± Σ 2 Ξ 2 cos φ A cos φ B -Σ 3 Ξ 3 cos φ B ]. (3.1)\n\nRewriting the last term and looking at eigenvalues in terms of Σ 1 Ξ 1 and Σ 2 Ξ 2 as before yields the concurrence\n\nC(ρ ′ ± )= 1 2 max[0, | cos φ A |+| cos φ A cos φ B |+| cos φ B |-1].\n\n(3.2) When cos φ A is 1, these Eqs.(3.1) and (3.2) describe the result obtained when there is only the interaction of qubit B made with a rotation around the x axis. If neither cos φ A nor cos φ B is 1, the concurrence becomes zero, the state separable, before cos φ A or cos φ B is zero. The interactions of qubits A and B with their controls change maximally entangled states to separable states. The inverses change separable states to maximally entangled states. In the following subsection, we describe the density matrices that show explicitly that the separable states are mixtures of products of pure states."
},
{
"section_type": "OTHER",
"section_title": "A. Exponential decay",
"text": "To describe exponential decay of entanglement we let cos φ A = e -ΓAt , cos φ B = e -ΓB t (3.3) by letting each interaction be modulated by a time-dependent Hamiltonian H(t) that is related to the Hamiltonian H of Eqs.(2.4) and (2.5) by H(t) = H dφ dt = HΓcotφ, (3.4) where φ and Γ are φ A and Γ A or φ B and Γ B . The same result could be produced in different ways. The interactions could be with large reservoirs instead of qubit controls [2, 3, 4] . Each qubit A or B could interact with a stream of reservoir qubits [15] . Here we are interested in the way the entanglement is changed by the combination of the two interactions. That depends only on the changes in the density matrix ρ for qubits A and B, not on the nature of the controls and the interactions. Maps that make the changes in ρ will be described in the next section. If there is only the interaction of qubit A with qubit C, the concurrence is e -ΓAt . If there is only interaction of qubit B with its control, the concurrence is e -ΓB t . If there are both and both are made with rotations around the z axis, the concurrence is e -ΓAt e -ΓB t . If there are both and the interaction of qubit B with its control is made with a rotation around the x axis, the concurrence is C(ρ ′ ± ) = 1 2 max[0, e -ΓAt +e -ΓAt e -ΓBt +e -ΓBt -1]. (3.5) This concurrence (3.5) is zero when e -ΓAt + e -ΓAt e -ΓB t + e -ΓB t = 1. (3.6) Then the state is separable; it is a mixture of six products of pure states: from Eqs.(3.1) and (3.3)\n\nρ ′ ± = 1 2 e -ΓAt 1 2 (1 1 + Σ 1 ) 1 2 (1 1 ± Ξ 1 ) + 1 2 e -ΓAt 1 2 (1 1 -Σ 1 ) 1 2 (1 1 ∓ Ξ 1 ) + 1 2 e -ΓAt e -ΓBt 1 2 (1 1 + Σ 2 ) 1 2 (1 1 ± Ξ 2 ) + 1 2 e -ΓAt e -ΓBt 1 2 (1 1 -Σ 2 ) 1 2 (1 1 ∓ Ξ 2 ) + 1 2 e -ΓBt 1 2 (1 1 + Σ 3 ) 1 2 (1 1 -Ξ 3 ) + 1 2 e -ΓBt 1 2 (1 1 -Σ 3 ) 1 2 (1 1 + Ξ 3 ). (3.7) 5\n\nThe state remains separable at later times; when the sum of the exponential decay factors is less than 1, the density matrix is a mixture in which just a multiple of the density matrix 1/4 for the completely mixed state is added to the terms of Eq. (3.7). This change of maximally entangled states to separable states can be described without reference to exponential decay by continuing to use cos φ A and cos φ B instead of e -ΓAt and e -ΓB t . Similar behavior involving exponential decay has been observed in more physically interesting and mathematically complicated models [2, 3, 4] ."
},
{
"section_type": "OTHER",
"section_title": "IV. MAPS",
"text": "The maps that make the changes in the density matrix ρ for qubits A and B could be described in different ways using various matrix forms. That is not needed here. Writing ρ in terms of Pauli matrices provides a very simple way to describe the maps. For any density matrix\n\nρ = 1 4 1 1 + 3 j=1 Σ j Σ j + 3 k=1 Ξ k Ξ k + 3 j,k=1 Σ j Ξ k Σ j Ξ k ( 4\n\n.1) for qubits A and B, the result of the interaction of qubit A with qubit C, described by Eq. (2.7), is that in ρ, in both the\n\nΣ j and Σ j Ξ k terms, Σ 1 -→ Σ 1 cos φ A , Σ 2 -→ Σ 2 cos φ A ;\n\n(4.2) the result of the interaction of qubit B with its control is that in ρ\n\nΞ 1 -→ Ξ 1 cos φ B , Ξ 2 -→ Ξ 2 cos φ B (4.3)\n\nif the interaction is made with a rotation around the z axis; and the result of the interaction of qubit B with its control is that in ρ\n\nΞ 2 -→ Ξ 2 cos φ B , Ξ 3 -→ Ξ 3 cos φ B (4.4)\n\nif the interaction is made with a rotation around the x axis. The terms with sin φ cancel out because there is an equal mixture of parts with φ and parts with -φ.\n\nThe changes in the state of qubits A and B can be described equivalently by maps of mean values that describe the state: the result of the interaction of qubit A with qubit C, described by Eq. (2.7), is that 4.5) for k = 1, 2, 3; the result of the interaction of qubit B with its control is that 4.6) for j = 1, 2, 3 if the interaction is made with a rotation around the z axis; and the result of the interaction of qubit B with its control is that\n\nΣ 1 -→ Σ 1 cos φ A Σ 2 -→ Σ 2 cos φ A Σ 1 Ξ k -→ Σ 1 Ξ k cos φ A Σ 2 Ξ k -→ Σ 2 Ξ k cos φ A (\n\nΞ 1 -→ Ξ 1 cos φ B Ξ 2 -→ Ξ 2 cos φ B Σ j Ξ 1 -→ Σ j Ξ 1 cos φ B Σ j Ξ 2 -→ Σ j Ξ 2 cos φ B (\n\nΞ 2 -→ Ξ 2 cos φ B Ξ 3 -→ Ξ 3 cos φ B Σ j Ξ 2 -→ Σ j Ξ 2 cos φ B Σ j Ξ 3 -→ Σ j Ξ 3 cos φ B (4.7)\n\nfor j = 1, 2, 3 if the interaction is made with a rotation around the x axis. When φ A and φ B change over intervals from initial values φ Ai and φ Bi to final values φ Af and φ Bf , the cos φ A and cos φ B factors in the maps are replaced by cos φ Af / cos φ Ai and cos φ Bf / cos φ Bi . If either of these factors is larger than 1, the map is not completely positive and does not apply to all density matrices ρ for qubits A and B. This happens whenever the entanglement increases. It also happens in cases where the concurrence (3.2) decreases, when one of cos φ A and cos φ B increases and the other decreases and there is more decrease than increase. The completely positive maps that decrease the entanglement have already been described [1]."
},
{
"section_type": "OTHER",
"section_title": "V. RECONCILIATION",
"text": "Entanglement being increased by local interactions may seem surprising from perspectives framed by experience in common situations where it is impossible. Entanglement is not increased by a completely positive map of the state of two qubits produced by an interaction on one of them. The interaction will produce a completely positive map if it is with a control whose state is initially not correlated with the state of the two qubits, as in Eq. (2.1). In our examples, that happens only when the initial value of φ is 0 or a multiple of π.\n\nOtherwise, the state of the control is correlated with the state of the two qubits as in Eq. (2.16). When a subsystem is initially correlated with the rest of a larger system that is being changed by unitary Hamiltonian dynamics, the map that describes the change of the state of the subsystem generally is not completely positive and applies to a limited set of subsystem states [16, 17] . We see this in our examples whenever the entanglement increases and in some cases when the entanglement decreases.\n\nThe map depends on both the dynamics and the initial correlations. It describes the effect of both in one step. Completely positive maps are what you get in the simplest set-up where you bring a system and control together in independent states and consider the effect of the dynamics that begins then. Dynamics over intervals where the maps are not completely positive can be expected to play roles in more complex settings. We should not let expectations for completely positive maps prevent us from seeing things that can happen.\n\nOur perspective is enlarged when we look beyond the map and include the dynamics. We can see the dynamics and the initial preparation as two related but separate steps. We can consider the effect of the dynamics, whatever the preparation may be.\n\n6 Local interactions that increase entanglement are completely outside a perspective that is limited to pure states. An interaction on one of the qubits can not change the entanglement at all if the state of the two qubits remains pure [18] . The entanglement of a pure state of two qubits depends only on the spectrum of the reduced density matrices that describe the states of the individual qubits, which is the same for the two qubits. If that could be changed by an interaction on one of the qubits, there could be a signal faster than light. In our examples, the state of the two qubits is pure only when it is maximally entangled. In our examples, the spectrum of the density matrices for the individual qubits never changes, and gives no measure of the entanglement."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "We are grateful to a referee for very helpful suggestions, including the comparison with a swap operation. Anil Shaji acknowledges the support of the US Office of Naval Research through Contract No. N00014-03-1-0426.\n\n[1] M. Ziman and V. Buzek, Phys. Rev. A 72, 052325 (2005). [2] T. Yu and J. H. Eberly, Phys. Rev. Lett. 97, 140403 (2006). [3] X.-T. Liang, Phys. Lett. A 349, 98 (2006). [4] T. Yu and J. H. Eberly, arXiv:quant-ph/0703083 (2007). [5] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A 75, 022101 (2007). [6] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996). [7] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, arXiv:quant-ph/0702225 (2007). [8] K. Zyczkowski, P. Horodecki, M. Horodecki, and R. Horodecki, Phys. Rev. A 65, 012101 (2001). [9] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Phys.\n\nRev. Lett. 71, 4287 (1993). [10] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [11] W. Dür, J. I. Cirac, and R. Tarrach, Phys. Rev. Lett. 83, 3562 (1999). [12] W. Dür and J. I. Cirac, Phys. Rev. A 61, 042314 (2000). [13] A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, Phys. Rev.\n\nLett. 87, 040401 (2001). [14] G. Toth, O. Guhne, M. Seevinck, and J. Uffink, Phys. Revs A 72, 014101 (2005). [15] E. C. G. Sudarshan, Chaos, Solitons and Fractals 16, 369 (2003). [16] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A. 70, 52110 (2004). [17] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A. 73, 12106 (2006). [18] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996)."
}
] | |
arxiv:0704.0476 | 0704.0476 | 1 | 10.1103/PhysRevA.76.033615 | 491c6baa121b1c5f43f439a50b386e310d14dba84727c26df92a3b0724362aaf | Geometric phase of an atom inside an adiabatic radio frequency potential | We investigate the geometric phase of an atom inside an adiabatic radio
frequency (rf) potential created from a static magnetic field (B-field) and a
time dependent rf field. The spatial motion of the atomic center of mass is
shown to give rise to a geometric phase, or Berry's phase, to the adiabatically
evolving atomic hyperfine spin along the local B-field. This phase is found to
depend on both the static B-field along the semi-classical trajectory of the
atomic center of mass and an ``effective magnetic field'' of the total B-field,
including the oscillating rf field. Specific calculations are provided for
several recent atom interferometry experiments and proposals utilizing
adiabatic rf potentials. | [
"P. Zhang and L. You"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | Magnetic trapping is an important enabling technology for the active research field of neutral atomic quantum gases. A variety of trap potentials can be developed using magnetic (B-) fields with different spatial distributions and time variations. For instance, the widely used quadrupole trap and the Ioffe-Pritchard trap [1] are usually created with static B-fields, while the time averaged orbiting potential (TOP) [2] and time orbiting ring trap (TORT) [3, 4] are created using oscillating B-fields with frequencies larger than the effective trap frequencies. Atom chips [5] have brought further developments to magnetic trap technology, as they can provide larger B-fields and gradients at reduced power-consumptions or electric currents using micro-fabricated coils. Today, magnetic trapping is a versatile tool used in many laboratories around the world for controlling atomic spatial motion in regions of different scales and geometric shapes, e.g., 3D or 2D traps, double well traps, and storage ring traps [1, 2, 3, 4, 5, 6, 7] .
Recently, magnetic traps based on adiabatic microwave [8, 9] and adiabatic radio frequency (rf) potentials (ARFP) [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] have attracted considerable attention. An ARFP is typically created with the combination of a static B-field and an rf field. The idea for an ARFP has been around for some time [10, 11, 12] , and experimental demonstrations recently have been carried out for confining both thermal [13] and Bose condensed atoms [14, 15, 16, 17] . Further development of improved ARFP with atom chip technology likely will assist in practical applications of atom interferometry. For instance, a double well potential was constructed recently using low order multi-poles capable of atomic beam splitting while maintaining tight spatial confinement [21] . Several interesting recent proposals outline the construction of small storage rings with radii of the order 1µm [18, 19, 20, 21, 22] , which could become useful if implemented for atom Sagnac interferometry [27] setups on atom chips.
When a neutral atom is confined in a magnetic potential, its hyperfine spin is assumed to follow adiabatically the spatial variation of the B-field direction during its spatial translational motion. As a result of this adiabatic approximation, the center of mass motion for the atom experiences an induced gauge field [28] , giving rise to a geometric phase (or Berry's phase) to the atomic internal spin state [29, 30] . The effect of this geometric phase is widely known, and is first addressed carefully in a meaningful way for atomic quantum gases by an explicit calculation of the resulting geometric phase in a static or a time averaged magnetic trap in Ref. [31] . Several important consequences are predicted to occur for a magnetically trapped atomic condensate in a quadrupole trap, a Ioffe-Pritchard trap [31, 32] , or a TORT based storage ring [33] . To our knowledge, this geometric phase effect has not been investigated in any detail for an atom inside an ARFP.
In a recent paper, we show that this geometric phase causes an effective Aharonov-Bohm-type [34] phase shift in a magnetic storage ring based atom interferometer [33] . In addition, our studies imply that the spatial fluctuation of the geometric phase can lead to a reduction of the visibility of the interference contrast. In view of this, we decided to carry out this study as reported here for the atomic geometric phase in an ARFP in order to shed light on the proposed high precision atom Sagnac interference experiment [27] . Analytical derivations for this study at some places become rather tedious and complicated. We therefore first will summarize our major results here for readers who may not be interested in the intricate details. We find that the geometric phase in an ARFP generally takes a more complicated form in comparison to the case of a static trap or a time averaged trap. In an ARFP, this phase factor is found to be determined by the trajectory of the time independent component of the trap field as well as an "effective B-field" that depends on the total B-field. In contrast to the earlier result found for a static trap or a time averaged trap [31, 33] , the final result turns out to be not expressible as a functional of the trajectory 2 for the direction of the total B-field in the parameter space. This paper is organized as follows. In sec. II, we generalize the semi-classical approach as outlined in Ref. [21] for the operating principle of an ARFP to a form more convenient for discussing the geometric phase. Section III parallels that of sec. II by reformulating a full quantum theory for discussing the geometric phase inside an ARFP [22] . The explicit expression for the geometric phase inside an ARFP is given in a readily adaptable form for specifical calculations. In sec. IV, we discuss the effect of the geometric phase in several types of ARFP recently proposed for atomic splitters and storage rings [18, 19, 20, 21, 22] and beam splitters [14, 21] . Finally, concluding remarks are given in sec. V.
In this section, we provide a semi-classical formulation for calculating the atomic geometric phase inside an ARFP. The semi-classical working principle for an ARFP is described in Ref. [21] , although only for the special case when the atomic center of mass is assumed at a fixed location. In order to calculate the geometric phase, our formulation allows for the explicit consideration of atomic center of mass motion classically. In our approach, the geometric phase is obtained naturally, and the validity conditions for both the adiabatic and the rotating wave approximations are clearly shown for an ARFP.
Inside an ARFP [21] , the total B-field B( r, t) is the sum of a static field component B s ( r) and an oscillatory rf field B o ( r, t), which conveniently is expressed as B o ( r, t) = B (a) rf ( r, t) cos(ωt) + B (b) rf ( r, t) cos(ωt + η). (1)
where r is the spatial position vector of the atom, ω is frequency of the rf field, and η is a relative phase factor.
In this section, we will assume that the atomic spatial motion is pre-determined, i.e., r(t) is given (as a slowly varying function of time t). For weak B-fields, the system Hamiltonian is simply the linear Zemman interaction
H(t) = g F µ B F • B[ r(t), t], ( 2
)
where g F is the corresponding Lande g-factor and µ B denotes the Bohr magneton. = 1 is assumed. For a static or a time averaged magnetic trap, the Hamiltonian (2) varies slowly over time scales of the Larmor precession of the atomic spin in the total B-field.
During the effectively slow trapped motion, the atomic hyperfine spin is assumed to be fixed at the instantaneous eigenstate of the Hamiltonian (2). The geometric phase then can be calculated straightforwardly from the variation of the B-field direction in the parameter space [31, 33] .
In an ARFP, the situation is more complicated. Although the variation of B s [ r(t)] remains much slower than the Larmor precession, the rf frequency ω usually is assumed to be nearly resonant with the precession frequency. Thus, the Hamiltonian (2) contains both fast and slow time varying components, making the direct calculation of the geometric phase a more involved task.
In the following, we will proceed step by step, clarifying the various approximations adopted along the way.
To understand the working principle for an ARFP, we first decompose the Hamiltonian H(t) (2) into the following form
H(t) = H s [ r(t)] + H + [ r(t)]e -iωt + H -[ r(t)]e iωt , ( 3
)
where H s and H ± are all slow varying functions of time and are given by
H s [ r(t)] = g F µ B F • B s [ r(t)], H + [ r(t)] = 1 2 g F µ B F • B (a) rf [ r(t)] + e -iη B (b) rf [ r(t)] , H -[ r(t)] = H † + [ r(t)]. ( 4
)
H s is diagonal in the spin angular momentum basis defined along the local direction of the static Bfield B s [ r(t)]. The eigenstate takes the familiar form |m F [ r(t)] s , quantized along the direction of B s [ r(t)], with the eigenvalue
m F |B s [ r(t)]| for B s [ r(t)] • F and m F ∈ [-F, F ], in analogy with the usual case of the z- quantized representation result of F z |m F z = m F |m F z .
Next we introduce a unitary transformation
U (t) = F mF =-F |m F z s m F [ r(t)]|e imF κωt , ( 5
)
with κ = sign(g F ) for the rotating wave approximation. The quantum state in the interaction picture |Ψ(t) I = U (t)|Ψ(t) defined by U (t) is governed by the Schroedinger equation
i∂ t |Ψ(t) I = H I (t)|Ψ(t) I , with
the Hamiltonian in the interaction picture given by
H I (t) = U HU † + i(∂ t U )U † = F m=-F mκ∆[ r(t)]|m z z m| -i F m,n=-F |m z s m[ r(t)]| d dt |n[ r(t)] s z n|e i(m-n)κωt
3 + F m=-F +1 h (+) m [ r(t)]|m z z m -1| + h (-) m [ r(t)]|m z z m -1|e 2iκωt + h.c.
+ F m=-F h m [ r(t)]|m z z m|e iκωt + h.c. , ( 6
)
where the time dependent parameters are defined as
∆[ r(t)] = µ B |g F B[ r(t)]| -ω, h (±) m [ r(t)] = s m[ r(t)]|H ± [ r(t)]|(m -1)[ r(t)] s , and (7) h m [ r(t)] = s m[ r(t)]|H ± [ r(t)]|m[ r(t)] s .
The above result is obtained easily if we note that the matrix element s m[ r(t)|H ± (t)|m ′ [ r(t) s is non-zero only when m -m ′ = 0, ±1. So far, we have always assumed that |m[ r(t)] s is a single valued function of the atomic position r. A careful examination shows that the eigenstate |m[ r(t)] s cannot be determined uniquely because of the presence of the U (1) gauge freedom for selecting a local phase factor exp{iφ[ r(t)]}, which consequently affects the resulting expressions for h
(±) m (t) and s m[ r(t)]|d/dt|m ′ [ r(t)] s .
The rotating wave approximation neglects of the oscillating terms proportional to e imωt (m = 0) in the Hamiltonian H I (6) . The error for this approximation is estimated easily from a time dependent perturbation calculation. The sufficient condition for its validity requires that all factors such as
t 0 dt ′ h m (t ′ ) exp[iκωt ′ ], t 0 dt ′ h (-) m (t ′ )ξ m,m-1 (t ′ ) exp[iκ(2ω + ∆)t ′ ], and t 0 dt ′ m[ r(t ′ )]|d/dt ′ |n[ r(t ′ )] ξ mn (t ′ ) exp[i(m -n)κ(ω + ∆)t ′ ] are negligible, where ξ mn (t) = exp t 0 dt ′ s m[ r(t ′ )]| d dt ′ |m[ r(t ′ )] s × exp - t 0 dt ′ s n[ r(t ′ )]| d dt ′ |n[ r(t ′ )] s . ( 8
)
Thus, the gauge independent factors h (-)
m ξ m,m-1 , m s |d/dt|n s ξ mn , h m ,
and ∆ should all vary slowly with time and with the modulus of their amplitudes much less than ω.
The effective Hamiltonian in the interaction picture under the rotating wave approximation then becomes
H (I) eff (t) = µ B g F F • B eff [ r(t)] -i F m=-F |m z s m[ r(t)]| d dt |m[ r(t)] s z m|, ( 9
)
where the first term resembles a coupling between the atomic spin and an "effective B-field" B eff ( r), whose components in real space are given by
B eff x ( r) = Re 2 s m( r)|H ± ( r)|(m -1)( r) s µ B g F (F + m)(F -m -1) , B eff y ( r) = -Im 2 s m( r)|H ± ( r)|(m -1)( r) s µ B g F (F + m)(F -m -1) , and
B eff z ( r) = B s ( r) - ω µ B |g F | . ( 10
)
Clearly, the x-and y-components of the effective field B eff ( r) depend on the explicit form of the eigenstate |m( r) s . In fact, it easily can be seen that different choices of the local phase factor for the |m( r) s actually lead to different values of B eff ( r) related to each other through r-dependent rotations in the x-y plane. In practice, the eigenstate |n( r) s and the effective field B eff can sometimes be constructed more simply, as in Ref. [21] . For any spatial position r, we first choose a rotation R[ m( r), χ( r)] along the axis m( r) with an angle χ( r
) that satisfies R[ n( r), χ( r)] B s ( r) = | B s ( r)|ê z . It is then easy to show that the eigenstate |n[ r(t)] s can be chosen as |n( r) s = exp i F • m( r)χ( r) |n z . ( 11
)
Unfortunately, the choice for R is not unique in a given static field B s ( r), an analogous result to the U (1) gauge freedom for the the egienstate |n( r) s . Corresponding to the choice (11) given above for |n( r) s , the unitary transformation U defined in (5) would become
U (t) = exp(-iF z ωt) • exp -i F • m( r)χ( r) , ( 12
)
and the transverse components of the "effective B-field" given by
B eff x,y ( r) = B x,y ( r)/2 [21] with B( r) = R[ m( r), χ( r)] B (a) rf ( r) +R[ê z , -κη]R[ m( r), χ( r)] B (b) rf ( r). (13)
In earlier discussions of an ARFP [21, 22] , the atomic internal state is assumed uniformly to remain adiabatically in a certain eigenstate of the first term of H (I) eff (t). To fully appreciate this adiabatic approximation and to calculate the geometric phase, we expand |Ψ(t) I into the instantaneous eigenstate basis |n[ r(t)] eff quantized along the direction of the effective B-field
B eff accord- ing to |Ψ(t) I = n C n (t)|n[ r(t)] eff .
The first term of H (I) eff (t) is simply the effective Zemman interaction between the atomic hyperfine spin and the effective B-field. The corresponding Schroedinger equation for the Hamiltonian H (I) eff (t) of (9) then becomes
i d dt C n (t) = [ǫ (n) I (t) + ν nn (t)]C n (t) + m =n ν nm (t)C m (t), 4 with ǫ (n) I (t) = nµ B g F | B eff (t)|, ν pq (t) = -i l eff p[ r(t)]|l z s l[ r(t)]| d dt |l[ r(t)] s z l|q[ r(t)] eff -i eff p[ r(t)]| d dt |q[ r(t)] eff . ( 14
)
Under the adiabatic approximation, the atomic internal state remains in a given eigenstate |n[ r(t)] eff with transitions to states |m[ r(t)] eff (m = n) being negligibly small. Thus, the transition probability, as estimated from the first order perturbation theory,
t 0 dt ′ ν nm (t ′ )e i R t ′ 0 dt ′′ [ǫ (n) I (t ′′ )+νnn(t ′′ )-ǫ m I (t ′′ )-νmm(t ′′ )]
should be much less than one. As before, we find the sufficient condition for the validity of the adiabatic approximation is given by
|ν mn (t)| |ǫ m I (t ′ ) -ǫ (n) I (t ′ )| ≪ 1, ( 15
) provided that ν mn (t) exp[i t ′ 0 [ν nn (t ′′ ) -ν mm (t ′′
)]dt ′ ], which is independent of the local phase factor for |n[ r(t)] eff and |n[ r(t)] s , remains a slowly varying function of time.
A straight forward calculation from the effective Hamiltonian (9) then gives the general expression for the geometric phase in an ARFP
γ n (t) = t 0 ν nn (t ′ )dt ′ = -i t 0 l | eff n[ r(t ′ )|l z | 2 × s l[ r(t ′ )]| d dt ′ |l[ r(t ′ )] s dt ′ +γ (I) n (t), ( 16
) and γ (I) n (t) = -i t 0 dt ′ eff n[ r(t ′ )|d/dt ′ |n[ r(t ′ ) eff .
During the adiabatic motion in a given internal state, the time evolution of the coefficient C n (t) takes the form (16) is the central result of this work. The geometric phase of an atom inside an ARFP is shown to contain two parts. The second part, γ 9) , with its value determined by the trajectory of the direction for the "effective B-field" B eff . The first part arises from the second term of H (I) eff (9). It is determined by the trajectories of both the static field B s and the effective B-field B eff . The expression for γ n in an ARFP is complicated because the internal quantum state in an ARFP is assumed to be adiabatically kept in an eigenstate of µ B g F F • B eff , rather than an eigenstate of the total interaction Hamiltonian H (I) eff (t). In section IV, we will perform explicit calculations for several examples of ARFP proposed for various applications: e.g., as atomic storage rings or atomic beam splitters. Most often we find that only the first part of Eq. (16) contributes a non-zero value to the geometric phase.
C n (t) = C n (0)e -i R t 0 ǫ (n) I (t ′ )dt ′ e -iγn(t) . ( 17
) Equation
(I) n (t), is clearly due to the interaction term µ B g F F • B eff in H (I) eff (
Before proceeding to the next section for a quantal treatment of the geometric phase, we find the time evolution of the atomic spin state in the Schroedinger picture
|Ψ(t) = ml C l (0) z m|l[ r(t)] eff × e -i R t 0 ǫ l I (t ′ )dt ′ e -iγ l (t) e -imωt |m[ r(t)] s , ( 18
) obtained directly from |Ψ(t) = U † (t)|Ψ(t) I after the
applications of the rotating wave and adiabatic approximations. When the atom is prepared initially in a specific adiabatic state |n[ r(t)] eff of the interaction picture, we arrive at the simple case of C l (0) = δ ln .
In the previous section, we provided the result for the geometric phase γ n (t) in an ARFP based on a semiclassical approach, where the atomic center of mass motion is described classically. A clear physical picture exists in this case for the appearance of the geometric phase in a certain parameter space. The validity conditions for the rotating wave and the adiabatic approximations as obtained above are all formulated in terms of gauge independent forms. However, if the influence of the geometric phase on the atomic spatial motion is to be included, e.g., as in the Aharonov-Bohm-type, phase shift, interference arrangement in an atomic Sagnac interferometer discussed earlier [33] , we would need an improved description where both the atomic spin and its center of mass motion are treated quantum mechanically.
In a full quantum treatment of the atomic motion, the quantum state of an atom can be expressed as |Φ(t) = φ l ( r, t)|l z , where φ l ( r, t) is the atomic spatial wave function for the internal state |l z of F z . The state |Φ(t) then satisfies the Schroedinger equation governed by the Hamiltonian
H = P 2 2M + g F µ B F • B( r, t), ( 19
)
with P being the kinetic momentum and M the atomic mass. The rotating wave and adiabatic approximations can be introduced now by defining the interaction picture 5 with the unitary transformation
U(t) = F m=-F |m z eff m( r)| × F n=-F |n z s n( r)|e inκωt . ( 20
)
The state in the interaction picture |Φ(t) I = U(t)|Φ(t) now is governed by the Schroedinger equation with the Hamiltonian H eff = UHU † . Under the rotating wave and adiabatic approximations, we neglect transitions between states |m z and |n z (m = n) as well as the rapidly oscillating terms. We then obtain
H eff ≈ n |n z z n|H eff |n z z n| ≈ n H (n) ad |n z z n|, ( 21
)
where the adiabatic Hamiltonian H (n) ad for the n-th adiabatic branch is defined as
H (n) ad = P -A n 2 2M + ǫ (n) I ( r), ( 22
)
with the effective gauge potential
A n ( r) = -i l | eff n( r)|l z | 2 s l( r)|∇|l( r) s -i eff n( r)|∇|n( r) eff . ( 23
)
In this form, it is well known that the geometric phase γ n can be expressed as the integral of the gauge potential A n along the spatial trajectory for the atomic center of mass in an ARFP, i.e., one would expect generally that γ n = A n • d r. Similar to the result of the semiclassical approach, the gauge potential A n ( r) can be expressed as the sum of two parts. The first part in Eq. (23) is the weighted sum of the atomic gauge potential -i s l( r)|∇|l( r) s from the static field B s , while the second term is the atomic gauge potential from the "effective B-field" B eff .
A full quantum treatment for atomic motion in an ARFP has been attempted earlier [22] . In fact, many of our formulations are identical to the results of Ref. [22] . For instance, it is easy to show that the unitary transformations U S , U R , and U F in [22]
are related directly to ours as U † F U † R U † S = U.
The only difference concerns the gauge potential A n that was neglected in Ref. [22] . Thus, they did not give the expression for the gauge potential, and the result for the geometric phase was not obtained either [22] . Our study shows that the neglect of the adiabatic gauge potential potentially can give rise to a final result, dependent on the choice of the local phase factors for the internal eigenstate.
In the above two sections, we obtain the expression for the atomic geometric phase in an ARFP. This section is devoted to the calculations of the geometric phases for several proposed applications of ARFP, such as storage rings or beam splitters for neutral atoms [18, 19, 20, 21, 22] .
Before presenting our results for the more specific cases, we provide some general discussions of the geometric phases in several ARFP based storage rings. As was pointed out earlier, the geometric phase γ n is given by the line integral of the gauge potential A n along the trajectory for the atomic center of mass motion. For a closed path in the storage ring at a fixed ρ = ρ c and z = z c , this can be further reduced to
γ n = q 2π 0 A (φ) n (ρ, φ, z)ρdφ, ( 24
)
where the integer q is the winding number of the path and A (φ) n is the component of A n along the azimuthal direction êφ of the familiar cylindrical coordinate system (ρ, φ, z). Without loss of generality, we take q = 1 in this paper. For the storage rings proposed in Refs. [18, 19, 20, 21, 22] , the gauge potentials A (φ) n (ρ, φ, z) are actually independent of the angle φ. Therefore, the geometric phase is simply given by
γ (c) n = 2πρ c A (φ) n (ρ c , z c ), ( 25
)
given out in explicit forms for different storage ring schemes [18, 19, 20, 21, 22] .
In reality, because of thermal motion or when the atomic transverse motional state is considered, the center of mass for an atom can deviate from (ρ c , z c ) even for a closed trajectory. This uncertainty in the exact shape of the closed trajectory gives rise to a fluctuating geometric phase and is usually difficult to study. Assuming a simple closed path at fixed ρ and z, we have found previously that the subsequently fluctuations could decrease the visibility of the interference pattern [33] . Quantum mechanically, such destructive interference can be explained as resulting from entanglement between the freedoms for φ and (ρ, z) because of the dependence of the gauge potential A φ n on ρ and z. Therefore, it is important to investigate this dependence near the trap center.
For simplicity, our discussions below will focus on the closed loops where ρ and z are φ-independent constants. In this case, the geometric phase can be expressed as ρ, z) . We will show numerically the distributions for γ n (ρ, z) obtained this way near the central region of (ρ c , z c ). If needed, a more rigorous approach can be developed to investigate the fluctuations of the resulting geometric phase from the gauge potential A (φ) n (ρ, z).
γ n (ρ, z) = 2πρA (φ) n (
A. The storage ring proposals of Refs. [18, 19, 20] This subsection is devoted to a detailed calculation of the geometric phases for the ARFP storage ring proposals of Refs. [18, 19, 20] . We will derive the analytical expressions for the azimuthal component A (φ) n of the gauge potential that arises in both cases from cylindrically symmetric static B-field and rf fields. Because of the cylindrical symmetry, the angle β s (ρ, z) between the local static B-field and the z-axis is required to be analytical in the region near the storage ring. Therefore, the eigenstate |n( r) s of F • B s can be chosen as
|n( r) s = exp{-i[ F • êφ β s (ρ, z) + nφ]}|n z . ( 26
)
Consequently, B eff ( r) is also cylindrically symmetric, which leads to the eigenstate |n( r
) eff of F • B eff as |n( r) eff = exp{-i[ F • n eff ⊥ ( r)β eff (ρ, z) + nφ]}|n z , ( 27
)
with the unit vector n eff ⊥ ( r) in the x-y plane orthogonal to B eff ( r) and β eff (ρ, z) denoting the angle between B eff ( r) and the z-axis. We note that the unit vector field n eff ⊥ ( r) also possesses cylindrical symmetry, i.e., remains invariant under rotation around the z-axis. The expressions of (26) and (27) allow us to obtain the simple expression of the gauge potential
A (φ) n (ρ, z) = - n ρ cos β eff (ρ, z) cos β s (ρ, z), ( 28
)
after straightforward calculations. In the scheme of Ref. [18] , the static B-field is a "ringshaped quadrupole field" that vanishes along a circle of a radius ρ 0 in the x-y plane. Near ρ = ρ 0 , the B-field is given approximately by
B s ( r) = B ′ (ρ -ρ 0 )ê ρ -B ′ zê z , ( 29
) like a quadrupole field, while the rf-field takes a complicated form
B o ( r, t) = a √ 2 cos(ωt) + b √ 2 cos(ωt + ϕ) êρ + - a √ 2 sin(ωt) + b √ 2 sin(ωt + ϕ) êz , ( 30
)
with constants a and b independent of r. From the expression of (26) for the eigenstate |n( r) s , the "effective B-field" B eff becomes
B eff ( r) = B ′ [ (ρ -ρ 0 ) 2 + z 2 -r 0 ]ê z - b √ 2 cos(θ + ϕ) + a √ 2 cos θ êρ + b √ 2 sin(θ + ϕ) - a √ 2 sin θ êφ , ( 31
)
FIG. 1: (Color online) A cross-sectional view for the storage ring of Ref. [18] . The static field is zero in the ring at the fixed radius ρ0. The addition of rf-fields creates an ARFP centered at a ring through (ρc, zc). The distance from the trap center to the ring with radius ρ0 in the plane z = 0 is r0.
where r 0 and θ are given by
r 0 = ω |µ B g F B ′ | , cos θ(ρ, z) = ρ -ρ 0 (ρ -ρ 0 ) 2 + z 2 , sin θ(ρ, z) = z (ρ -ρ 0 ) 2 + z 2 . ( 32
)
In an ARFP, as discussed here, the trap center at (ρ c , z c ) is determined by minimizing both the zcomponent and the transverse component of B eff . Without loss of generality, we will assume a, b > 0. Then, (ρ c , z c ) is found to satisfy
θ(ρ c , z c ) = -ϕ/2, (ρ c -ρ 0 ) 2 + z 2 c = r 0 , ( 33
)
i.e., the trap center lies on the surface of the "resonance toroid" at ρ = ρ 0 with a radius r 0 as shown in Fig. 1 . The relative angle of the trap center with respect to the center of the toroid cross-section is given by -ϕ/2. On this "resonance toroid," the rf-field is resonant with the static field, i.e., B eff z vanishes. As a result, the "effective B-field" lies again in the x-y plane on the "resonance toroid," which gives cos β eff (ρ c , z c ) = 0 and leads to the result A (φ) n (ρ c , z c ) = γ n = 0 as shown in the trap center for the storage ring considered before in Ref. [18] .
From the expression (29) of the static field and the definition of the angle θ(ρ, z), we find a simple relationship β s (ρ, z) = π/2 + θ(ρ, z), with which the gauge potential A (φ) n (ρ, φ) in (28) can be further simplified as
A (φ) n (ρ, z) = n ρ cos β eff (ρ, z) sin θ(ρ, z) 7 ≈ n ρ cos β eff (ρ, z) sin θ(ρ c , z c ), ( 34
)
near the trap center. Thus, the spatial fluctuation of the gauge potential A (φ) n (ρ, z) in the region around the trap center is closely related to the angle θ(ρ c , z c ) of the trap center, or the parameter ϕ of the oscillating field B o . When ϕ = 0, the atom is trapped in the region with θ ≈ 0 or π, where the fluctuation of A (φ) n (ρ, z) is suppressed significantly due to the small value of sin θ. On the other hand, if the angle ϕ is set to π with the trap center located in the region with θ ≈ ±π/2, the fluctuation of the gauge potential becomes amplified.
In Fig. 2 , we illustrate numerical results for the distribution of the geometric phase γ 1 (ρ, z) = 2πρA φ 1 (ρ, z) in the region near the trap center at ϕ = 0, π/2, π. We see clearly decreased fluctuations of γ 1 when the absolute value of sin θ(ρ c , z c ) = -sin(ϕ/2) is decreased.
Next we turn to the storage ring of Ref. [19] constructed from a quadrupole static B-field B s ( r) = B ′ (x, y, -2z) and an r-independent rf field B o = B rf cos(ωt)ê z along the z direction. The resulting ARFP provides a 2D ring shaped trap in the x-y plane. In addition, a 1D optical potential along the z direction is employed to confine atoms in the transverse plane at z = 0 [19]. The "effective B-field" takes the form
B eff ( r) = B ′ (ρ -ρ 0 )ê z - 1 2 B rf êρ , ( 35
)
in the plane at z = 0, with
ρ 0 = ω/|µ B g F B ′ |.
Because the strength of B eff is near minimum at the ring ρ = ρ 0 , the trap center for this storage ring is located at ρ c = ρ 0 and z c = 0. At the trap center, the "effective Bfield" is along the direction of êρ . Thus, according to Eq. ( 28 ), the geometric phase γ (c) n at the trap center again vanishes.
In Fig. 3 , we show the distribution of the geometric phase γ 1 in the region near the trap center for B rf = 0.05|B ′ |ρ 0 and B rf = 0.15|B ′ |ρ 0 . We see that the fluctuation is relatively small when the strength of the rf-field is large. This can be explained by Eq. (28) , which shows that A (φ) n is proportional to cos β eff and can be approximated as 2B eff z /B rf near the trap center. When B rf is large, the gauge potential becomes a relatively slow varying function of ρ and z. In this case, the presence of a 1D optical potential allows for the possibility of tuning the trap center position to a nonzero value of z, with the storage ring remaining in the x-y plane. Then cos β s is assumed to a nonzero value, leading to increased fluctuations for the geometric phase.
Finally, we discuss the geometric phase in the "time averaged" ARFP storage ring proposed in Ref. [20] . Unlike previously considered ARFP based storage rings, the time dependence now exists in both the "static B-field" and the frequency of the rf field given by Color online) The distribution of the geometric phase γ1 near the trap center (ρc, zc) of the storage ring proposed in Ref. [18] at (a) ϕ = 0, (b) ϕ = π/2, and (c) ϕ = π, clearly displaying the sin(ϕ/2) dependence.
B s ( r, t) = B ′ ρê ρ -2B ′ zê z + B m sin(ω m t)ê z , -0.1 0 0.1 -0.1 0 0.1 -1 0 1 (z-z c )/ρ 0 γ 1 (a) (ρ-ρ c )/ρ 0 -0.1 0 0.1 -0.1 0 0.1 -5 0 5 (z-z c )/ρ 0 (ρ-ρ c )/ρ 0 γ 1 (b) -0.1 0 0.1 -0.1 0 0.1 -10 0 10 (z-z c )/ρ 0 γ 1 (c) (ρ-ρ c )/ρ 0 FIG. 2: (
B o (t) = B rf sin[ω(t)t]ê z , ω(t) = ω 0 1 + (B m /B ′ ρ 0 ) 2 sin 2 (ω m t) . (36)
The frequency ω m is assumed to be much smaller than ω 0 but much larger than the trap frequency. The radius ρ 0 is now defined as
ρ 0 = ω 0 /|µ B g F B ′ |,
and the "effective 8 -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 (a) -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 (b) z/ρ 0 (ρ-ρ 0 )/ρ 0 γ 1 FIG. 3: (Color online) The geometric phase γ1 for the storage ring of Ref. [19] with (a) B rf = 0.15B ′ ρ0 and (b) B rf = 0.05B ′ ρ0.
B-field" takes the form
B eff ( r, t) = ∆( r, t)ê z - B ′ ρ |2 B s ( r, t)| B rf êφ . ( 37
)
The operating principle for the time averaged storage ring of Ref. [20] is similar to the well-known TOP [2] and TORT traps [3, 4] . The effective trap potential experienced by the atom is proportional to the time averaged value of the "effective B-field"
2π/ωm 0 | B eff ( r, t)|dt.
When B rf and B m are much smaller than B ′ ρ 0 , the center of the storage ring is located approximately at ρ c = ρ 0 , z c = 0. Using the earlier result [33] , we find that in the time averaged storage ring, the effective gauge potential Ã(φ) n (ρ, z) is reduced simply to the time averaged instantaneous gauge potential 28) . The geometric phase then is given approximately by γ n (ρ, z) = 2π Ã(φ) n (ρ, z). In this case, we find that the geometric phase always vanishes at -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 z/ρ 0 (ρ-ρ 0 )/ρ 0 (a) γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 Color online) The geometric phase γ1 for the storage ring of Ref. [20] at (a) B rf = 0.3B ′ ρ0 and (b) B rf = 0.1B ′ ρ0. Bm = 0.05B ′ ρ0.
Ã(φ) n (ρ, z) = ω m 2π 2π/ωm 0 A (φ) n (ρ, z, t)dt, ( 38
) with A (φ) n (ρ, z, t) given in (
(b) (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 FIG. 4: (
the trap center (ρ c , z c ). Figure 4 illustrates the distribution of the geometric phase in the region near the trap center for two different values of the rf-field amplitude B rf . Similar to the storage ring of Ref. [19] , the fluctuation of the geometric phase is suppressed in this case for large B rf .
B. The storage ring proposals of Refs. [21, 22] Next we consider the ARFP based storage ring proposed in Refs. [21, 22] . In this case, the static B-field is that of a Ioffe-Pritchard trap on an atom chip. In the Cartesian coordinate (x, y, z), it takes the form
B s = B ′ xê x -B ′ yê y + B ′ Lê z , ( 39
)
where B ′ is the B-field gradient and the bias field along the z-direction is denoted as B ′ L. The amplitudes B
(a) rf and B (b) rf (z) of the rf field are B
(a) rf = [B rf (z)/ √ 2]ê x and B (b) rf = [B rf (z)/ √ 2]ê y with B rf (z) = B ( 0
) rf + B ′′ z 2 . ( 40
)
9 In the schemes of Ref. [21, 22] considered earlier, the phase η of the rf field is assumed to be κπ/2. The xand y-components of the "effective B-field" B eff ( r) then become
B eff x ( r) = B rf (z) 2 √ 2 (1 + cos β s (ρ, z)), B eff y ( r) = 0, ( 41
)
according to Eq. (10). Then the strength of the "effective B-field" B eff has its minimum along a circle with a nonzero radius ρ c , provided a positive detuning ∆ exists at the origin (0, 0, 0) [21, 22] . The "effective B-field" B eff is easily shown to lie in the x-z plane along the trap bottom mapped out by the atomic center of mass motion. This gives rise to a vanishing γ (I) F . With a proper choice for the local phase of |n( r) eff , the gauge potential A (φ) n takes the form
A (φ) n (ρ, z) = n ρ cos β eff (ρ, z) (1 -cos β s (ρ, z)) . ( 42
)
Figure 5 displays the geometric phase along a closed path for a spin-1 atom as a function of ρ c for the ARFP storage ring proposed in Refs. [21, 22] . The parameter λ is defined as
λ = √ 2 ∆[ r = 0] |g F µ B B (0) rf | . ( 43
)
To assure the validity of the rotating wave approximation, we find that the maximal values of ∆[ r = 0]/(|g F |µ B ) and B (0) rf / √ 2 must be restricted to the region of λ ∈ [0, 0.15].
As shown in Fig. 1 , the geometric phase remains much smaller than 2π in this situation. This fact can be appreciated easily if we look at the distribution of the "effective B-field" B eff . According to Eq. (41), the component B eff x has a nonzero minimal value B rf /(2 √ 2), while |B eff z | can become arbitrarily small, although not necessarily zero in general. Therefore, at the trap center where | B eff | is a minimum, the value of cos β eff = B eff z /| B eff | can become very small, leading to small geometric phases. Yet, despite the relatively small geometric phase found here, our result remains important because it could represent a systematic error if not properly included in a Sagnac interference experiment.
In Fig. 6 , we show the spatial distribution of the geometric phase γ 1 around the trap center with λ = 1/3 and λ = 3. The fluctuation is found to be relatively small when λ is small or when the rf-field amplitude B rf is large.
Although not discussed in Refs. [21, 22] , a ring shaped trap also can be realized if we take η = -κπ/2. The "effective B-field" B eff still lies in the x-y plane
B eff x ( r) = - B rf (z) 2 √ 2 cos(2φ)(1 -cos β s (ρ, z)), B eff y ( r) = B rf (z) 2 √ 2 sin(2φ)(1 -cos β s (ρ, z)), ( 44
) 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 -0.08 -0.06 -0.04 -0.02 0 λ=1/3 λ=3 λ=1 ρ c /L γ (c) 1
FIG. 5: (Color online) The geometric phase γ1 is plotted against the radius ρc for the ARFP storage ring of Ref. [21, 22] with η = κπ/2 at λ = 3, λ = 1, and λ = 1/3. To assure the validity of the rotating wave approximation, in the solid lines, the maximal value of ∆[ r = 0]/|gF µBB ′ ρ0| or B rf /( √ 2B ′ ρ0) are restricted to be smaller than 0.15. The extending dashed line is beyond the rotating wave approximation for λ = 1/3 and B rf /( √ 2B ′ ρ0) ∈ [0.15, 0.3].
clearly giving rise to a non-zero solid angle with respect to a closed path along the storage ring. Therefore, the term γ (eff) F is non-zero in this case. We choose the eigenstates |n( r
) s and |n( r) eff as |n( r) s = exp[-i F • ns ( r)β s (ρ, z)]|n z , |n( r) eff = exp[-i F • n eff ⊥ ( r)β eff (ρ, z)]|n z . ( 45
)
with the unit vector ns ⊥ ( r) in the x-y plane orthogonal to B s ( r). In this case, the gauge potential A (φ) n can be expressed as
A (φ) n (ρ, z) = n ρ cos β eff (ρ, z)[1 + cos β s (ρ, z)]. ( 46
)
In Figure 7 , we show the fluctuation of the geometric phase γ 1 for a closed path with a new parameter
λ ′ = 6 √ 2 ∆[ r = 0] |g F µ B B (0) rf | , ( 47
)
equal to 3 and 1/3. The fluctuation for γ 1 is found to be much larger than the case of η = κπ/2, which can be explained by the transverse components B eff x,y of the "effective B-field." Because cos β s is always close to unity.
In the case of η = -κπ/2, B eff x,y can take only small positive values. Therefore, at the minimum of the ARFP
trap ρ = ρ 0 of | B eff |, both B eff z
and B eff x,y have to be close to zero. In this case the value for cos β eff becomes a rapidly changing function of ρ in the region near ρ c .
Our above calculations have obtained analytical expressions of the geometric phases in an ARFP based storage ring for η = ±κπ/2. We have further investigated the 10 -0.05 0 0.05 -0.05 0 0.05 -0.06 -0.02 0.02 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.6 0 0.6 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 6: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ = 1/3 and (b) λ = 3. η = κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.
fluctuations of the geometric phase for the two cases of η = ±κπ/2. It seems one benefits from implementing a Sagnac interferometer in the discussed ARFP storage ring with η = κπ/2 and operating at a relatively large λ.
Before proceeding onto the concluding section, we will discuss the geometric phase in an ARFP based beam splitter created via a double potential [14, 21] . In such an implementation, the static field B s is created from a Ioffe-Pritchard trap, while the oscillating rf field components are B (a) rf = B rf [z]ê x and B (b) rf = 0. By spatially tuning the amplitude of B rf from zero to a significant value, in the x-y plane, an ARFP can be tuned from a single well centered near the origin to a double well with two minimal points at the point with nonzero radius ρ 0 and φ = 0, π. Therefore, a Y-shaped atom beam splitter can be accomplished when the B rf [z] initially is increased along the z-axis to a large value, and then decreased to zero. In such an arrangement, the atom beam moving along the z direction can be separated into two beams that move along the z-axis at φ = 0, π for a while, and then can be recombined again into a single beam.
-0.05 0 0.05 -0.05 0 0.05 -10 0 10 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -15 0 15 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 7: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ ′ = 3 and (b)λ ′ = 1/3. η = -κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.
In the atom interferometer considered above, both the static field B s and the "effective B-field" B eff are limited to the x-z plane. Therefore, for motion along the closed path of the trap bottom, the solid angle enclosed by the trajectory of B eff is zero. Thus, the geometric phase in (16) can be expressed as
γ n (t) = -i t 0 l | eff n( r)|l z | 2 s l( r)|∇|l( r) s • vdt ′ . ( 48
)
We can show that the product r) s is a function of ρ c and is independent of z. Thus, the geometric phase can be expressed as an integral of this function with respect to ρ c , from zero to a large value and then back to zero. Therefore, the value of the geometric phase would be zero in the end.
eff n( r)|l z | 2 s l( r)|∇|l(
11 V. CONCLUSION
In this study, we develop theoretical formalisms for the calculation of the atomic geometric phase inside an ARFP. We show that, due to the complexity of the ARFP, the geometric phase depends on the spatial variation of both the static field and an "effective B-field" B eff . We provide general expressions for the geometric phase and the corresponding adiabatic gauge potential in Eqs. (16) and (23), respectively.
To shed light on actual applications of the atomic geometric phase, we investigate the distribution of atomic geometric phases for several proposed or ongoing experiments with ARFP based storage rings and atom beam splitters. We prove rigorously that the geometric phase in the center of the storage rings proposed in Refs. [18, 19] is always zero. In addition, we find that in the storage ring of Ref. [18] , the spatial fluctuation of the geometric phase sensitively depends on the position of the trap center on the "resonance toroid." In the proposals of Refs. [19, 20, 21, 22] , the fluctuation for the geometric phase becomes significantly suppressed when the amplitude B rf of the rf-field is large. In the proposals of [21, 22] , the fluctuations of the geometric phase also is suppressed if the angle η is set to be κ2π. In the beam splitter realized with the double well potential ARFP [14, 21], the geometric phase is shown to be zero.
Our work helps to clarify the working principle of trapping neutral atoms in an ARFP and the validity conditions for the various approximations involved. We hope our results will shine new light on the proposed inertial sensing experiments based on trapped atoms in ARFP.
We thank Dr. T. Uzer and Dr. B. Sun for helpful discussions. This work is supported by NASA, NSF, CNSF, and the 863 and 973 programs of the MOST of China.
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{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Magnetic trapping is an important enabling technology for the active research field of neutral atomic quantum gases. A variety of trap potentials can be developed using magnetic (B-) fields with different spatial distributions and time variations. For instance, the widely used quadrupole trap and the Ioffe-Pritchard trap [1] are usually created with static B-fields, while the time averaged orbiting potential (TOP) [2] and time orbiting ring trap (TORT) [3, 4] are created using oscillating B-fields with frequencies larger than the effective trap frequencies. Atom chips [5] have brought further developments to magnetic trap technology, as they can provide larger B-fields and gradients at reduced power-consumptions or electric currents using micro-fabricated coils. Today, magnetic trapping is a versatile tool used in many laboratories around the world for controlling atomic spatial motion in regions of different scales and geometric shapes, e.g., 3D or 2D traps, double well traps, and storage ring traps [1, 2, 3, 4, 5, 6, 7] .\n\nRecently, magnetic traps based on adiabatic microwave [8, 9] and adiabatic radio frequency (rf) potentials (ARFP) [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] have attracted considerable attention. An ARFP is typically created with the combination of a static B-field and an rf field. The idea for an ARFP has been around for some time [10, 11, 12] , and experimental demonstrations recently have been carried out for confining both thermal [13] and Bose condensed atoms [14, 15, 16, 17] . Further development of improved ARFP with atom chip technology likely will assist in practical applications of atom interferometry. For instance, a double well potential was constructed recently using low order multi-poles capable of atomic beam splitting while maintaining tight spatial confinement [21] . Several interesting recent proposals outline the construction of small storage rings with radii of the order 1µm [18, 19, 20, 21, 22] , which could become useful if implemented for atom Sagnac interferometry [27] setups on atom chips.\n\nWhen a neutral atom is confined in a magnetic potential, its hyperfine spin is assumed to follow adiabatically the spatial variation of the B-field direction during its spatial translational motion. As a result of this adiabatic approximation, the center of mass motion for the atom experiences an induced gauge field [28] , giving rise to a geometric phase (or Berry's phase) to the atomic internal spin state [29, 30] . The effect of this geometric phase is widely known, and is first addressed carefully in a meaningful way for atomic quantum gases by an explicit calculation of the resulting geometric phase in a static or a time averaged magnetic trap in Ref. [31] . Several important consequences are predicted to occur for a magnetically trapped atomic condensate in a quadrupole trap, a Ioffe-Pritchard trap [31, 32] , or a TORT based storage ring [33] . To our knowledge, this geometric phase effect has not been investigated in any detail for an atom inside an ARFP.\n\nIn a recent paper, we show that this geometric phase causes an effective Aharonov-Bohm-type [34] phase shift in a magnetic storage ring based atom interferometer [33] . In addition, our studies imply that the spatial fluctuation of the geometric phase can lead to a reduction of the visibility of the interference contrast. In view of this, we decided to carry out this study as reported here for the atomic geometric phase in an ARFP in order to shed light on the proposed high precision atom Sagnac interference experiment [27] . Analytical derivations for this study at some places become rather tedious and complicated. We therefore first will summarize our major results here for readers who may not be interested in the intricate details. We find that the geometric phase in an ARFP generally takes a more complicated form in comparison to the case of a static trap or a time averaged trap. In an ARFP, this phase factor is found to be determined by the trajectory of the time independent component of the trap field as well as an \"effective B-field\" that depends on the total B-field. In contrast to the earlier result found for a static trap or a time averaged trap [31, 33] , the final result turns out to be not expressible as a functional of the trajectory 2 for the direction of the total B-field in the parameter space. This paper is organized as follows. In sec. II, we generalize the semi-classical approach as outlined in Ref. [21] for the operating principle of an ARFP to a form more convenient for discussing the geometric phase. Section III parallels that of sec. II by reformulating a full quantum theory for discussing the geometric phase inside an ARFP [22] . The explicit expression for the geometric phase inside an ARFP is given in a readily adaptable form for specifical calculations. In sec. IV, we discuss the effect of the geometric phase in several types of ARFP recently proposed for atomic splitters and storage rings [18, 19, 20, 21, 22] and beam splitters [14, 21] . Finally, concluding remarks are given in sec. V."
},
{
"section_type": "METHOD",
"section_title": "II. A SEMI-CLASSICAL APPROACH",
"text": "In this section, we provide a semi-classical formulation for calculating the atomic geometric phase inside an ARFP. The semi-classical working principle for an ARFP is described in Ref. [21] , although only for the special case when the atomic center of mass is assumed at a fixed location. In order to calculate the geometric phase, our formulation allows for the explicit consideration of atomic center of mass motion classically. In our approach, the geometric phase is obtained naturally, and the validity conditions for both the adiabatic and the rotating wave approximations are clearly shown for an ARFP.\n\nInside an ARFP [21] , the total B-field B( r, t) is the sum of a static field component B s ( r) and an oscillatory rf field B o ( r, t), which conveniently is expressed as B o ( r, t) = B (a) rf ( r, t) cos(ωt) + B (b) rf ( r, t) cos(ωt + η). (1)\n\nwhere r is the spatial position vector of the atom, ω is frequency of the rf field, and η is a relative phase factor.\n\nIn this section, we will assume that the atomic spatial motion is pre-determined, i.e., r(t) is given (as a slowly varying function of time t). For weak B-fields, the system Hamiltonian is simply the linear Zemman interaction\n\nH(t) = g F µ B F • B[ r(t), t], ( 2\n\n)\n\nwhere g F is the corresponding Lande g-factor and µ B denotes the Bohr magneton. = 1 is assumed. For a static or a time averaged magnetic trap, the Hamiltonian (2) varies slowly over time scales of the Larmor precession of the atomic spin in the total B-field.\n\nDuring the effectively slow trapped motion, the atomic hyperfine spin is assumed to be fixed at the instantaneous eigenstate of the Hamiltonian (2). The geometric phase then can be calculated straightforwardly from the variation of the B-field direction in the parameter space [31, 33] .\n\nIn an ARFP, the situation is more complicated. Although the variation of B s [ r(t)] remains much slower than the Larmor precession, the rf frequency ω usually is assumed to be nearly resonant with the precession frequency. Thus, the Hamiltonian (2) contains both fast and slow time varying components, making the direct calculation of the geometric phase a more involved task.\n\nIn the following, we will proceed step by step, clarifying the various approximations adopted along the way.\n\nTo understand the working principle for an ARFP, we first decompose the Hamiltonian H(t) (2) into the following form\n\nH(t) = H s [ r(t)] + H + [ r(t)]e -iωt + H -[ r(t)]e iωt , ( 3\n\n)\n\nwhere H s and H ± are all slow varying functions of time and are given by\n\nH s [ r(t)] = g F µ B F • B s [ r(t)], H + [ r(t)] = 1 2 g F µ B F • B (a) rf [ r(t)] + e -iη B (b) rf [ r(t)] , H -[ r(t)] = H † + [ r(t)]. ( 4\n\n)\n\nH s is diagonal in the spin angular momentum basis defined along the local direction of the static Bfield B s [ r(t)]. The eigenstate takes the familiar form |m F [ r(t)] s , quantized along the direction of B s [ r(t)], with the eigenvalue\n\nm F |B s [ r(t)]| for B s [ r(t)] • F and m F ∈ [-F, F ], in analogy with the usual case of the z- quantized representation result of F z |m F z = m F |m F z .\n\nNext we introduce a unitary transformation\n\nU (t) = F mF =-F |m F z s m F [ r(t)]|e imF κωt , ( 5\n\n)\n\nwith κ = sign(g F ) for the rotating wave approximation. The quantum state in the interaction picture |Ψ(t) I = U (t)|Ψ(t) defined by U (t) is governed by the Schroedinger equation\n\ni∂ t |Ψ(t) I = H I (t)|Ψ(t) I , with\n\nthe Hamiltonian in the interaction picture given by\n\nH I (t) = U HU † + i(∂ t U )U † = F m=-F mκ∆[ r(t)]|m z z m| -i F m,n=-F |m z s m[ r(t)]| d dt |n[ r(t)] s z n|e i(m-n)κωt\n\n3 + F m=-F +1 h (+) m [ r(t)]|m z z m -1| + h (-) m [ r(t)]|m z z m -1|e 2iκωt + h.c.\n\n+ F m=-F h m [ r(t)]|m z z m|e iκωt + h.c. , ( 6\n\n)\n\nwhere the time dependent parameters are defined as\n\n∆[ r(t)] = µ B |g F B[ r(t)]| -ω, h (±) m [ r(t)] = s m[ r(t)]|H ± [ r(t)]|(m -1)[ r(t)] s , and (7) h m [ r(t)] = s m[ r(t)]|H ± [ r(t)]|m[ r(t)] s .\n\nThe above result is obtained easily if we note that the matrix element s m[ r(t)|H ± (t)|m ′ [ r(t) s is non-zero only when m -m ′ = 0, ±1. So far, we have always assumed that |m[ r(t)] s is a single valued function of the atomic position r. A careful examination shows that the eigenstate |m[ r(t)] s cannot be determined uniquely because of the presence of the U (1) gauge freedom for selecting a local phase factor exp{iφ[ r(t)]}, which consequently affects the resulting expressions for h\n\n(±) m (t) and s m[ r(t)]|d/dt|m ′ [ r(t)] s .\n\nThe rotating wave approximation neglects of the oscillating terms proportional to e imωt (m = 0) in the Hamiltonian H I (6) . The error for this approximation is estimated easily from a time dependent perturbation calculation. The sufficient condition for its validity requires that all factors such as\n\nt 0 dt ′ h m (t ′ ) exp[iκωt ′ ], t 0 dt ′ h (-) m (t ′ )ξ m,m-1 (t ′ ) exp[iκ(2ω + ∆)t ′ ], and t 0 dt ′ m[ r(t ′ )]|d/dt ′ |n[ r(t ′ )] ξ mn (t ′ ) exp[i(m -n)κ(ω + ∆)t ′ ] are negligible, where ξ mn (t) = exp t 0 dt ′ s m[ r(t ′ )]| d dt ′ |m[ r(t ′ )] s × exp - t 0 dt ′ s n[ r(t ′ )]| d dt ′ |n[ r(t ′ )] s . ( 8\n\n)\n\nThus, the gauge independent factors h (-)\n\nm ξ m,m-1 , m s |d/dt|n s ξ mn , h m ,\n\nand ∆ should all vary slowly with time and with the modulus of their amplitudes much less than ω.\n\nThe effective Hamiltonian in the interaction picture under the rotating wave approximation then becomes\n\nH (I) eff (t) = µ B g F F • B eff [ r(t)] -i F m=-F |m z s m[ r(t)]| d dt |m[ r(t)] s z m|, ( 9\n\n)\n\nwhere the first term resembles a coupling between the atomic spin and an \"effective B-field\" B eff ( r), whose components in real space are given by\n\nB eff x ( r) = Re 2 s m( r)|H ± ( r)|(m -1)( r) s µ B g F (F + m)(F -m -1) , B eff y ( r) = -Im 2 s m( r)|H ± ( r)|(m -1)( r) s µ B g F (F + m)(F -m -1) , and\n\nB eff z ( r) = B s ( r) - ω µ B |g F | . ( 10\n\n)\n\nClearly, the x-and y-components of the effective field B eff ( r) depend on the explicit form of the eigenstate |m( r) s . In fact, it easily can be seen that different choices of the local phase factor for the |m( r) s actually lead to different values of B eff ( r) related to each other through r-dependent rotations in the x-y plane. In practice, the eigenstate |n( r) s and the effective field B eff can sometimes be constructed more simply, as in Ref. [21] . For any spatial position r, we first choose a rotation R[ m( r), χ( r)] along the axis m( r) with an angle χ( r\n\n) that satisfies R[ n( r), χ( r)] B s ( r) = | B s ( r)|ê z . It is then easy to show that the eigenstate |n[ r(t)] s can be chosen as |n( r) s = exp i F • m( r)χ( r) |n z . ( 11\n\n)\n\nUnfortunately, the choice for R is not unique in a given static field B s ( r), an analogous result to the U (1) gauge freedom for the the egienstate |n( r) s . Corresponding to the choice (11) given above for |n( r) s , the unitary transformation U defined in (5) would become\n\nU (t) = exp(-iF z ωt) • exp -i F • m( r)χ( r) , ( 12\n\n)\n\nand the transverse components of the \"effective B-field\" given by\n\nB eff x,y ( r) = B x,y ( r)/2 [21] with B( r) = R[ m( r), χ( r)] B (a) rf ( r) +R[ê z , -κη]R[ m( r), χ( r)] B (b) rf ( r). (13)\n\nIn earlier discussions of an ARFP [21, 22] , the atomic internal state is assumed uniformly to remain adiabatically in a certain eigenstate of the first term of H (I) eff (t). To fully appreciate this adiabatic approximation and to calculate the geometric phase, we expand |Ψ(t) I into the instantaneous eigenstate basis |n[ r(t)] eff quantized along the direction of the effective B-field\n\nB eff accord- ing to |Ψ(t) I = n C n (t)|n[ r(t)] eff .\n\nThe first term of H (I) eff (t) is simply the effective Zemman interaction between the atomic hyperfine spin and the effective B-field. The corresponding Schroedinger equation for the Hamiltonian H (I) eff (t) of (9) then becomes\n\ni d dt C n (t) = [ǫ (n) I (t) + ν nn (t)]C n (t) + m =n ν nm (t)C m (t), 4 with ǫ (n) I (t) = nµ B g F | B eff (t)|, ν pq (t) = -i l eff p[ r(t)]|l z s l[ r(t)]| d dt |l[ r(t)] s z l|q[ r(t)] eff -i eff p[ r(t)]| d dt |q[ r(t)] eff . ( 14\n\n)\n\nUnder the adiabatic approximation, the atomic internal state remains in a given eigenstate |n[ r(t)] eff with transitions to states |m[ r(t)] eff (m = n) being negligibly small. Thus, the transition probability, as estimated from the first order perturbation theory,\n\nt 0 dt ′ ν nm (t ′ )e i R t ′ 0 dt ′′ [ǫ (n) I (t ′′ )+νnn(t ′′ )-ǫ m I (t ′′ )-νmm(t ′′ )]\n\nshould be much less than one. As before, we find the sufficient condition for the validity of the adiabatic approximation is given by\n\n|ν mn (t)| |ǫ m I (t ′ ) -ǫ (n) I (t ′ )| ≪ 1, ( 15\n\n) provided that ν mn (t) exp[i t ′ 0 [ν nn (t ′′ ) -ν mm (t ′′\n\n)]dt ′ ], which is independent of the local phase factor for |n[ r(t)] eff and |n[ r(t)] s , remains a slowly varying function of time.\n\nA straight forward calculation from the effective Hamiltonian (9) then gives the general expression for the geometric phase in an ARFP\n\nγ n (t) = t 0 ν nn (t ′ )dt ′ = -i t 0 l | eff n[ r(t ′ )|l z | 2 × s l[ r(t ′ )]| d dt ′ |l[ r(t ′ )] s dt ′ +γ (I) n (t), ( 16\n\n) and γ (I) n (t) = -i t 0 dt ′ eff n[ r(t ′ )|d/dt ′ |n[ r(t ′ ) eff .\n\nDuring the adiabatic motion in a given internal state, the time evolution of the coefficient C n (t) takes the form (16) is the central result of this work. The geometric phase of an atom inside an ARFP is shown to contain two parts. The second part, γ 9) , with its value determined by the trajectory of the direction for the \"effective B-field\" B eff . The first part arises from the second term of H (I) eff (9). It is determined by the trajectories of both the static field B s and the effective B-field B eff . The expression for γ n in an ARFP is complicated because the internal quantum state in an ARFP is assumed to be adiabatically kept in an eigenstate of µ B g F F • B eff , rather than an eigenstate of the total interaction Hamiltonian H (I) eff (t). In section IV, we will perform explicit calculations for several examples of ARFP proposed for various applications: e.g., as atomic storage rings or atomic beam splitters. Most often we find that only the first part of Eq. (16) contributes a non-zero value to the geometric phase.\n\nC n (t) = C n (0)e -i R t 0 ǫ (n) I (t ′ )dt ′ e -iγn(t) . ( 17\n\n) Equation\n\n(I) n (t), is clearly due to the interaction term µ B g F F • B eff in H (I) eff (\n\nBefore proceeding to the next section for a quantal treatment of the geometric phase, we find the time evolution of the atomic spin state in the Schroedinger picture\n\n|Ψ(t) = ml C l (0) z m|l[ r(t)] eff × e -i R t 0 ǫ l I (t ′ )dt ′ e -iγ l (t) e -imωt |m[ r(t)] s , ( 18\n\n) obtained directly from |Ψ(t) = U † (t)|Ψ(t) I after the\n\napplications of the rotating wave and adiabatic approximations. When the atom is prepared initially in a specific adiabatic state |n[ r(t)] eff of the interaction picture, we arrive at the simple case of C l (0) = δ ln ."
},
{
"section_type": "OTHER",
"section_title": "III. A QUANTUM MECHANICAL TREATMENT",
"text": "In the previous section, we provided the result for the geometric phase γ n (t) in an ARFP based on a semiclassical approach, where the atomic center of mass motion is described classically. A clear physical picture exists in this case for the appearance of the geometric phase in a certain parameter space. The validity conditions for the rotating wave and the adiabatic approximations as obtained above are all formulated in terms of gauge independent forms. However, if the influence of the geometric phase on the atomic spatial motion is to be included, e.g., as in the Aharonov-Bohm-type, phase shift, interference arrangement in an atomic Sagnac interferometer discussed earlier [33] , we would need an improved description where both the atomic spin and its center of mass motion are treated quantum mechanically.\n\nIn a full quantum treatment of the atomic motion, the quantum state of an atom can be expressed as |Φ(t) = φ l ( r, t)|l z , where φ l ( r, t) is the atomic spatial wave function for the internal state |l z of F z . The state |Φ(t) then satisfies the Schroedinger equation governed by the Hamiltonian\n\nH = P 2 2M + g F µ B F • B( r, t), ( 19\n\n)\n\nwith P being the kinetic momentum and M the atomic mass. The rotating wave and adiabatic approximations can be introduced now by defining the interaction picture 5 with the unitary transformation\n\nU(t) = F m=-F |m z eff m( r)| × F n=-F |n z s n( r)|e inκωt . ( 20\n\n)\n\nThe state in the interaction picture |Φ(t) I = U(t)|Φ(t) now is governed by the Schroedinger equation with the Hamiltonian H eff = UHU † . Under the rotating wave and adiabatic approximations, we neglect transitions between states |m z and |n z (m = n) as well as the rapidly oscillating terms. We then obtain\n\nH eff ≈ n |n z z n|H eff |n z z n| ≈ n H (n) ad |n z z n|, ( 21\n\n)\n\nwhere the adiabatic Hamiltonian H (n) ad for the n-th adiabatic branch is defined as\n\nH (n) ad = P -A n 2 2M + ǫ (n) I ( r), ( 22\n\n)\n\nwith the effective gauge potential\n\nA n ( r) = -i l | eff n( r)|l z | 2 s l( r)|∇|l( r) s -i eff n( r)|∇|n( r) eff . ( 23\n\n)\n\nIn this form, it is well known that the geometric phase γ n can be expressed as the integral of the gauge potential A n along the spatial trajectory for the atomic center of mass in an ARFP, i.e., one would expect generally that γ n = A n • d r. Similar to the result of the semiclassical approach, the gauge potential A n ( r) can be expressed as the sum of two parts. The first part in Eq. (23) is the weighted sum of the atomic gauge potential -i s l( r)|∇|l( r) s from the static field B s , while the second term is the atomic gauge potential from the \"effective B-field\" B eff .\n\nA full quantum treatment for atomic motion in an ARFP has been attempted earlier [22] . In fact, many of our formulations are identical to the results of Ref. [22] . For instance, it is easy to show that the unitary transformations U S , U R , and U F in [22]\n\nare related directly to ours as U † F U † R U † S = U.\n\nThe only difference concerns the gauge potential A n that was neglected in Ref. [22] . Thus, they did not give the expression for the gauge potential, and the result for the geometric phase was not obtained either [22] . Our study shows that the neglect of the adiabatic gauge potential potentially can give rise to a final result, dependent on the choice of the local phase factors for the internal eigenstate."
},
{
"section_type": "OTHER",
"section_title": "IV. GEOMETRIC PHASES IN ARFP BASED APPLICATIONS",
"text": "In the above two sections, we obtain the expression for the atomic geometric phase in an ARFP. This section is devoted to the calculations of the geometric phases for several proposed applications of ARFP, such as storage rings or beam splitters for neutral atoms [18, 19, 20, 21, 22] .\n\nBefore presenting our results for the more specific cases, we provide some general discussions of the geometric phases in several ARFP based storage rings. As was pointed out earlier, the geometric phase γ n is given by the line integral of the gauge potential A n along the trajectory for the atomic center of mass motion. For a closed path in the storage ring at a fixed ρ = ρ c and z = z c , this can be further reduced to\n\nγ n = q 2π 0 A (φ) n (ρ, φ, z)ρdφ, ( 24\n\n)\n\nwhere the integer q is the winding number of the path and A (φ) n is the component of A n along the azimuthal direction êφ of the familiar cylindrical coordinate system (ρ, φ, z). Without loss of generality, we take q = 1 in this paper. For the storage rings proposed in Refs. [18, 19, 20, 21, 22] , the gauge potentials A (φ) n (ρ, φ, z) are actually independent of the angle φ. Therefore, the geometric phase is simply given by\n\nγ (c) n = 2πρ c A (φ) n (ρ c , z c ), ( 25\n\n)\n\ngiven out in explicit forms for different storage ring schemes [18, 19, 20, 21, 22] .\n\nIn reality, because of thermal motion or when the atomic transverse motional state is considered, the center of mass for an atom can deviate from (ρ c , z c ) even for a closed trajectory. This uncertainty in the exact shape of the closed trajectory gives rise to a fluctuating geometric phase and is usually difficult to study. Assuming a simple closed path at fixed ρ and z, we have found previously that the subsequently fluctuations could decrease the visibility of the interference pattern [33] . Quantum mechanically, such destructive interference can be explained as resulting from entanglement between the freedoms for φ and (ρ, z) because of the dependence of the gauge potential A φ n on ρ and z. Therefore, it is important to investigate this dependence near the trap center.\n\nFor simplicity, our discussions below will focus on the closed loops where ρ and z are φ-independent constants. In this case, the geometric phase can be expressed as ρ, z) . We will show numerically the distributions for γ n (ρ, z) obtained this way near the central region of (ρ c , z c ). If needed, a more rigorous approach can be developed to investigate the fluctuations of the resulting geometric phase from the gauge potential A (φ) n (ρ, z).\n\nγ n (ρ, z) = 2πρA (φ) n (\n\nA. The storage ring proposals of Refs. [18, 19, 20] This subsection is devoted to a detailed calculation of the geometric phases for the ARFP storage ring proposals of Refs. [18, 19, 20] . We will derive the analytical expressions for the azimuthal component A (φ) n of the gauge potential that arises in both cases from cylindrically symmetric static B-field and rf fields. Because of the cylindrical symmetry, the angle β s (ρ, z) between the local static B-field and the z-axis is required to be analytical in the region near the storage ring. Therefore, the eigenstate |n( r) s of F • B s can be chosen as\n\n|n( r) s = exp{-i[ F • êφ β s (ρ, z) + nφ]}|n z . ( 26\n\n)\n\nConsequently, B eff ( r) is also cylindrically symmetric, which leads to the eigenstate |n( r\n\n) eff of F • B eff as |n( r) eff = exp{-i[ F • n eff ⊥ ( r)β eff (ρ, z) + nφ]}|n z , ( 27\n\n)\n\nwith the unit vector n eff ⊥ ( r) in the x-y plane orthogonal to B eff ( r) and β eff (ρ, z) denoting the angle between B eff ( r) and the z-axis. We note that the unit vector field n eff ⊥ ( r) also possesses cylindrical symmetry, i.e., remains invariant under rotation around the z-axis. The expressions of (26) and (27) allow us to obtain the simple expression of the gauge potential\n\nA (φ) n (ρ, z) = - n ρ cos β eff (ρ, z) cos β s (ρ, z), ( 28\n\n)\n\nafter straightforward calculations. In the scheme of Ref. [18] , the static B-field is a \"ringshaped quadrupole field\" that vanishes along a circle of a radius ρ 0 in the x-y plane. Near ρ = ρ 0 , the B-field is given approximately by\n\nB s ( r) = B ′ (ρ -ρ 0 )ê ρ -B ′ zê z , ( 29\n\n) like a quadrupole field, while the rf-field takes a complicated form\n\nB o ( r, t) = a √ 2 cos(ωt) + b √ 2 cos(ωt + ϕ) êρ + - a √ 2 sin(ωt) + b √ 2 sin(ωt + ϕ) êz , ( 30\n\n)\n\nwith constants a and b independent of r. From the expression of (26) for the eigenstate |n( r) s , the \"effective B-field\" B eff becomes\n\nB eff ( r) = B ′ [ (ρ -ρ 0 ) 2 + z 2 -r 0 ]ê z - b √ 2 cos(θ + ϕ) + a √ 2 cos θ êρ + b √ 2 sin(θ + ϕ) - a √ 2 sin θ êφ , ( 31\n\n)\n\nFIG. 1: (Color online) A cross-sectional view for the storage ring of Ref. [18] . The static field is zero in the ring at the fixed radius ρ0. The addition of rf-fields creates an ARFP centered at a ring through (ρc, zc). The distance from the trap center to the ring with radius ρ0 in the plane z = 0 is r0.\n\nwhere r 0 and θ are given by\n\nr 0 = ω |µ B g F B ′ | , cos θ(ρ, z) = ρ -ρ 0 (ρ -ρ 0 ) 2 + z 2 , sin θ(ρ, z) = z (ρ -ρ 0 ) 2 + z 2 . ( 32\n\n)\n\nIn an ARFP, as discussed here, the trap center at (ρ c , z c ) is determined by minimizing both the zcomponent and the transverse component of B eff . Without loss of generality, we will assume a, b > 0. Then, (ρ c , z c ) is found to satisfy\n\nθ(ρ c , z c ) = -ϕ/2, (ρ c -ρ 0 ) 2 + z 2 c = r 0 , ( 33\n\n)\n\ni.e., the trap center lies on the surface of the \"resonance toroid\" at ρ = ρ 0 with a radius r 0 as shown in Fig. 1 . The relative angle of the trap center with respect to the center of the toroid cross-section is given by -ϕ/2. On this \"resonance toroid,\" the rf-field is resonant with the static field, i.e., B eff z vanishes. As a result, the \"effective B-field\" lies again in the x-y plane on the \"resonance toroid,\" which gives cos β eff (ρ c , z c ) = 0 and leads to the result A (φ) n (ρ c , z c ) = γ n = 0 as shown in the trap center for the storage ring considered before in Ref. [18] .\n\nFrom the expression (29) of the static field and the definition of the angle θ(ρ, z), we find a simple relationship β s (ρ, z) = π/2 + θ(ρ, z), with which the gauge potential A (φ) n (ρ, φ) in (28) can be further simplified as\n\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z) sin θ(ρ, z) 7 ≈ n ρ cos β eff (ρ, z) sin θ(ρ c , z c ), ( 34\n\n)\n\nnear the trap center. Thus, the spatial fluctuation of the gauge potential A (φ) n (ρ, z) in the region around the trap center is closely related to the angle θ(ρ c , z c ) of the trap center, or the parameter ϕ of the oscillating field B o . When ϕ = 0, the atom is trapped in the region with θ ≈ 0 or π, where the fluctuation of A (φ) n (ρ, z) is suppressed significantly due to the small value of sin θ. On the other hand, if the angle ϕ is set to π with the trap center located in the region with θ ≈ ±π/2, the fluctuation of the gauge potential becomes amplified.\n\nIn Fig. 2 , we illustrate numerical results for the distribution of the geometric phase γ 1 (ρ, z) = 2πρA φ 1 (ρ, z) in the region near the trap center at ϕ = 0, π/2, π. We see clearly decreased fluctuations of γ 1 when the absolute value of sin θ(ρ c , z c ) = -sin(ϕ/2) is decreased.\n\nNext we turn to the storage ring of Ref. [19] constructed from a quadrupole static B-field B s ( r) = B ′ (x, y, -2z) and an r-independent rf field B o = B rf cos(ωt)ê z along the z direction. The resulting ARFP provides a 2D ring shaped trap in the x-y plane. In addition, a 1D optical potential along the z direction is employed to confine atoms in the transverse plane at z = 0 [19]. The \"effective B-field\" takes the form\n\nB eff ( r) = B ′ (ρ -ρ 0 )ê z - 1 2 B rf êρ , ( 35\n\n)\n\nin the plane at z = 0, with\n\nρ 0 = ω/|µ B g F B ′ |.\n\nBecause the strength of B eff is near minimum at the ring ρ = ρ 0 , the trap center for this storage ring is located at ρ c = ρ 0 and z c = 0. At the trap center, the \"effective Bfield\" is along the direction of êρ . Thus, according to Eq. ( 28 ), the geometric phase γ (c) n at the trap center again vanishes.\n\nIn Fig. 3 , we show the distribution of the geometric phase γ 1 in the region near the trap center for B rf = 0.05|B ′ |ρ 0 and B rf = 0.15|B ′ |ρ 0 . We see that the fluctuation is relatively small when the strength of the rf-field is large. This can be explained by Eq. (28) , which shows that A (φ) n is proportional to cos β eff and can be approximated as 2B eff z /B rf near the trap center. When B rf is large, the gauge potential becomes a relatively slow varying function of ρ and z. In this case, the presence of a 1D optical potential allows for the possibility of tuning the trap center position to a nonzero value of z, with the storage ring remaining in the x-y plane. Then cos β s is assumed to a nonzero value, leading to increased fluctuations for the geometric phase.\n\nFinally, we discuss the geometric phase in the \"time averaged\" ARFP storage ring proposed in Ref. [20] . Unlike previously considered ARFP based storage rings, the time dependence now exists in both the \"static B-field\" and the frequency of the rf field given by Color online) The distribution of the geometric phase γ1 near the trap center (ρc, zc) of the storage ring proposed in Ref. [18] at (a) ϕ = 0, (b) ϕ = π/2, and (c) ϕ = π, clearly displaying the sin(ϕ/2) dependence.\n\nB s ( r, t) = B ′ ρê ρ -2B ′ zê z + B m sin(ω m t)ê z , -0.1 0 0.1 -0.1 0 0.1 -1 0 1 (z-z c )/ρ 0 γ 1 (a) (ρ-ρ c )/ρ 0 -0.1 0 0.1 -0.1 0 0.1 -5 0 5 (z-z c )/ρ 0 (ρ-ρ c )/ρ 0 γ 1 (b) -0.1 0 0.1 -0.1 0 0.1 -10 0 10 (z-z c )/ρ 0 γ 1 (c) (ρ-ρ c )/ρ 0 FIG. 2: (\n\nB o (t) = B rf sin[ω(t)t]ê z , ω(t) = ω 0 1 + (B m /B ′ ρ 0 ) 2 sin 2 (ω m t) . (36)\n\nThe frequency ω m is assumed to be much smaller than ω 0 but much larger than the trap frequency. The radius ρ 0 is now defined as\n\nρ 0 = ω 0 /|µ B g F B ′ |,\n\nand the \"effective 8 -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 (a) -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 (b) z/ρ 0 (ρ-ρ 0 )/ρ 0 γ 1 FIG. 3: (Color online) The geometric phase γ1 for the storage ring of Ref. [19] with (a) B rf = 0.15B ′ ρ0 and (b) B rf = 0.05B ′ ρ0.\n\nB-field\" takes the form\n\nB eff ( r, t) = ∆( r, t)ê z - B ′ ρ |2 B s ( r, t)| B rf êφ . ( 37\n\n)\n\nThe operating principle for the time averaged storage ring of Ref. [20] is similar to the well-known TOP [2] and TORT traps [3, 4] . The effective trap potential experienced by the atom is proportional to the time averaged value of the \"effective B-field\"\n\n2π/ωm 0 | B eff ( r, t)|dt.\n\nWhen B rf and B m are much smaller than B ′ ρ 0 , the center of the storage ring is located approximately at ρ c = ρ 0 , z c = 0. Using the earlier result [33] , we find that in the time averaged storage ring, the effective gauge potential Ã(φ) n (ρ, z) is reduced simply to the time averaged instantaneous gauge potential 28) . The geometric phase then is given approximately by γ n (ρ, z) = 2π Ã(φ) n (ρ, z). In this case, we find that the geometric phase always vanishes at -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 z/ρ 0 (ρ-ρ 0 )/ρ 0 (a) γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 Color online) The geometric phase γ1 for the storage ring of Ref. [20] at (a) B rf = 0.3B ′ ρ0 and (b) B rf = 0.1B ′ ρ0. Bm = 0.05B ′ ρ0.\n\nÃ(φ) n (ρ, z) = ω m 2π 2π/ωm 0 A (φ) n (ρ, z, t)dt, ( 38\n\n) with A (φ) n (ρ, z, t) given in (\n\n(b) (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 FIG. 4: (\n\nthe trap center (ρ c , z c ). Figure 4 illustrates the distribution of the geometric phase in the region near the trap center for two different values of the rf-field amplitude B rf . Similar to the storage ring of Ref. [19] , the fluctuation of the geometric phase is suppressed in this case for large B rf .\n\nB. The storage ring proposals of Refs. [21, 22] Next we consider the ARFP based storage ring proposed in Refs. [21, 22] . In this case, the static B-field is that of a Ioffe-Pritchard trap on an atom chip. In the Cartesian coordinate (x, y, z), it takes the form\n\nB s = B ′ xê x -B ′ yê y + B ′ Lê z , ( 39\n\n)\n\nwhere B ′ is the B-field gradient and the bias field along the z-direction is denoted as B ′ L. The amplitudes B\n\n(a) rf and B (b) rf (z) of the rf field are B\n\n(a) rf = [B rf (z)/ √ 2]ê x and B (b) rf = [B rf (z)/ √ 2]ê y with B rf (z) = B ( 0\n\n) rf + B ′′ z 2 . ( 40\n\n)\n\n9 In the schemes of Ref. [21, 22] considered earlier, the phase η of the rf field is assumed to be κπ/2. The xand y-components of the \"effective B-field\" B eff ( r) then become\n\nB eff x ( r) = B rf (z) 2 √ 2 (1 + cos β s (ρ, z)), B eff y ( r) = 0, ( 41\n\n)\n\naccording to Eq. (10). Then the strength of the \"effective B-field\" B eff has its minimum along a circle with a nonzero radius ρ c , provided a positive detuning ∆ exists at the origin (0, 0, 0) [21, 22] . The \"effective B-field\" B eff is easily shown to lie in the x-z plane along the trap bottom mapped out by the atomic center of mass motion. This gives rise to a vanishing γ (I) F . With a proper choice for the local phase of |n( r) eff , the gauge potential A (φ) n takes the form\n\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z) (1 -cos β s (ρ, z)) . ( 42\n\n)\n\nFigure 5 displays the geometric phase along a closed path for a spin-1 atom as a function of ρ c for the ARFP storage ring proposed in Refs. [21, 22] . The parameter λ is defined as\n\nλ = √ 2 ∆[ r = 0] |g F µ B B (0) rf | . ( 43\n\n)\n\nTo assure the validity of the rotating wave approximation, we find that the maximal values of ∆[ r = 0]/(|g F |µ B ) and B (0) rf / √ 2 must be restricted to the region of λ ∈ [0, 0.15].\n\nAs shown in Fig. 1 , the geometric phase remains much smaller than 2π in this situation. This fact can be appreciated easily if we look at the distribution of the \"effective B-field\" B eff . According to Eq. (41), the component B eff x has a nonzero minimal value B rf /(2 √ 2), while |B eff z | can become arbitrarily small, although not necessarily zero in general. Therefore, at the trap center where | B eff | is a minimum, the value of cos β eff = B eff z /| B eff | can become very small, leading to small geometric phases. Yet, despite the relatively small geometric phase found here, our result remains important because it could represent a systematic error if not properly included in a Sagnac interference experiment.\n\nIn Fig. 6 , we show the spatial distribution of the geometric phase γ 1 around the trap center with λ = 1/3 and λ = 3. The fluctuation is found to be relatively small when λ is small or when the rf-field amplitude B rf is large.\n\nAlthough not discussed in Refs. [21, 22] , a ring shaped trap also can be realized if we take η = -κπ/2. The \"effective B-field\" B eff still lies in the x-y plane\n\nB eff x ( r) = - B rf (z) 2 √ 2 cos(2φ)(1 -cos β s (ρ, z)), B eff y ( r) = B rf (z) 2 √ 2 sin(2φ)(1 -cos β s (ρ, z)), ( 44\n\n) 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 -0.08 -0.06 -0.04 -0.02 0 λ=1/3 λ=3 λ=1 ρ c /L γ (c) 1\n\nFIG. 5: (Color online) The geometric phase γ1 is plotted against the radius ρc for the ARFP storage ring of Ref. [21, 22] with η = κπ/2 at λ = 3, λ = 1, and λ = 1/3. To assure the validity of the rotating wave approximation, in the solid lines, the maximal value of ∆[ r = 0]/|gF µBB ′ ρ0| or B rf /( √ 2B ′ ρ0) are restricted to be smaller than 0.15. The extending dashed line is beyond the rotating wave approximation for λ = 1/3 and B rf /( √ 2B ′ ρ0) ∈ [0.15, 0.3].\n\nclearly giving rise to a non-zero solid angle with respect to a closed path along the storage ring. Therefore, the term γ (eff) F is non-zero in this case. We choose the eigenstates |n( r\n\n) s and |n( r) eff as |n( r) s = exp[-i F • ns ( r)β s (ρ, z)]|n z , |n( r) eff = exp[-i F • n eff ⊥ ( r)β eff (ρ, z)]|n z . ( 45\n\n)\n\nwith the unit vector ns ⊥ ( r) in the x-y plane orthogonal to B s ( r). In this case, the gauge potential A (φ) n can be expressed as\n\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z)[1 + cos β s (ρ, z)]. ( 46\n\n)\n\nIn Figure 7 , we show the fluctuation of the geometric phase γ 1 for a closed path with a new parameter\n\nλ ′ = 6 √ 2 ∆[ r = 0] |g F µ B B (0) rf | , ( 47\n\n)\n\nequal to 3 and 1/3. The fluctuation for γ 1 is found to be much larger than the case of η = κπ/2, which can be explained by the transverse components B eff x,y of the \"effective B-field.\" Because cos β s is always close to unity.\n\nIn the case of η = -κπ/2, B eff x,y can take only small positive values. Therefore, at the minimum of the ARFP\n\ntrap ρ = ρ 0 of | B eff |, both B eff z\n\nand B eff x,y have to be close to zero. In this case the value for cos β eff becomes a rapidly changing function of ρ in the region near ρ c .\n\nOur above calculations have obtained analytical expressions of the geometric phases in an ARFP based storage ring for η = ±κπ/2. We have further investigated the 10 -0.05 0 0.05 -0.05 0 0.05 -0.06 -0.02 0.02 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.6 0 0.6 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 6: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ = 1/3 and (b) λ = 3. η = κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.\n\nfluctuations of the geometric phase for the two cases of η = ±κπ/2. It seems one benefits from implementing a Sagnac interferometer in the discussed ARFP storage ring with η = κπ/2 and operating at a relatively large λ.\n\nBefore proceeding onto the concluding section, we will discuss the geometric phase in an ARFP based beam splitter created via a double potential [14, 21] . In such an implementation, the static field B s is created from a Ioffe-Pritchard trap, while the oscillating rf field components are B (a) rf = B rf [z]ê x and B (b) rf = 0. By spatially tuning the amplitude of B rf from zero to a significant value, in the x-y plane, an ARFP can be tuned from a single well centered near the origin to a double well with two minimal points at the point with nonzero radius ρ 0 and φ = 0, π. Therefore, a Y-shaped atom beam splitter can be accomplished when the B rf [z] initially is increased along the z-axis to a large value, and then decreased to zero. In such an arrangement, the atom beam moving along the z direction can be separated into two beams that move along the z-axis at φ = 0, π for a while, and then can be recombined again into a single beam.\n\n-0.05 0 0.05 -0.05 0 0.05 -10 0 10 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -15 0 15 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 7: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ ′ = 3 and (b)λ ′ = 1/3. η = -κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.\n\nIn the atom interferometer considered above, both the static field B s and the \"effective B-field\" B eff are limited to the x-z plane. Therefore, for motion along the closed path of the trap bottom, the solid angle enclosed by the trajectory of B eff is zero. Thus, the geometric phase in (16) can be expressed as\n\nγ n (t) = -i t 0 l | eff n( r)|l z | 2 s l( r)|∇|l( r) s • vdt ′ . ( 48\n\n)\n\nWe can show that the product r) s is a function of ρ c and is independent of z. Thus, the geometric phase can be expressed as an integral of this function with respect to ρ c , from zero to a large value and then back to zero. Therefore, the value of the geometric phase would be zero in the end.\n\neff n( r)|l z | 2 s l( r)|∇|l(\n\n11 V. CONCLUSION\n\nIn this study, we develop theoretical formalisms for the calculation of the atomic geometric phase inside an ARFP. We show that, due to the complexity of the ARFP, the geometric phase depends on the spatial variation of both the static field and an \"effective B-field\" B eff . We provide general expressions for the geometric phase and the corresponding adiabatic gauge potential in Eqs. (16) and (23), respectively.\n\nTo shed light on actual applications of the atomic geometric phase, we investigate the distribution of atomic geometric phases for several proposed or ongoing experiments with ARFP based storage rings and atom beam splitters. We prove rigorously that the geometric phase in the center of the storage rings proposed in Refs. [18, 19] is always zero. In addition, we find that in the storage ring of Ref. [18] , the spatial fluctuation of the geometric phase sensitively depends on the position of the trap center on the \"resonance toroid.\" In the proposals of Refs. [19, 20, 21, 22] , the fluctuation for the geometric phase becomes significantly suppressed when the amplitude B rf of the rf-field is large. In the proposals of [21, 22] , the fluctuations of the geometric phase also is suppressed if the angle η is set to be κ2π. In the beam splitter realized with the double well potential ARFP [14, 21], the geometric phase is shown to be zero.\n\nOur work helps to clarify the working principle of trapping neutral atoms in an ARFP and the validity conditions for the various approximations involved. We hope our results will shine new light on the proposed inertial sensing experiments based on trapped atoms in ARFP."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "We thank Dr. T. Uzer and Dr. B. Sun for helpful discussions. This work is supported by NASA, NSF, CNSF, and the 863 and 973 programs of the MOST of China.\n\n[1] D. E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983). [2] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A.\n\nCornell, Phys. Rev. Lett. 74, 3352 (1995). [3] A.S. Arnold and E. Riis, J. Mod. Opt. 49, 959 (2002); C. S. Garvie, E. Riis, and A. S. Arnold, Laser Spectroscopy XVI, edited by P. Hannaford et al. (World Scientific, Singapore, 2004), p. 178, see also www.photonics.phys.strath.ac.uk . [4] S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, Phys. Rev. Lett. 95, 143201 (2005); K. W. Murch, K. L. Moore, S. Gupta, and D.M. Stamper-Kurn, Phys. Rev. Lett. 96, 013202 (2005). [5] R. Folman, P. Krüger, J. Schmiedmayer, J. Denschlag, and C. Henkel, Advances in Atomic, Molecular, and Optical Physics, vol. 48, 263 (2002). [6] J. A. Sauer, M. D. Barrett, and M. S. Chapman, Phys.\n\nRev. Lett. 87, 270401 (2001). [7] A.S. Arnold, C.S. Garvie, and E. Riis, Phys. Rev. A 73, 041606(R) (2006). [8] C. C. Agosta, I. F. Silvera, H. T. C. Stoof, and B. J. Verhaar, Phys. Rev. Lett. 62, 2361 (1989). [9] Z. Zhao, I. F. Silvera, and M. Reynolds, Jour. Low. Temp. Phys. 89, 703 (1992). [10] A. J. Moerdijk, B. J. Verhaar, and T. M. Nagtegaal, Phys. Rev. A 53, 4343 (1996). [11] H. Zhang, P. Zhang, X. Xu, J. Han, and Y. Wang, Chin. Phys. Lett. 22, 83 (2001). [12] O. Zobay and B. M. Garraway, Phys. Rev. Lett. 86, 1195 (2001); Phys. Rev. A 69, 023605 (2004). [13] Y. Colombe, E. Knyazchyan, O. Morizot, B. Mercier, V.\n\nLorent, and H. Perrin, Europhys. Lett. 67, 593 (2004). [14] S. Hofferberth, I. Lesanovsky, B. Fischer, J. Verdu, and J. Schmiedmayer, Nature Physics 2, 710 (2006). [15] T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, Nature Physics 1, 57 (2005). [16] G.-B. Jo, Y. Shin, S. Will, T. A. Pasquini, M. Saba, W. Ketterle, D. E. Pritchard, M. Vengalattore, and M. Prentiss, Phys. Rev. Lett. 98, 030407 (2007). [17] M. White, H. Gao, M. Pasienski, and B. DeMarco, Phys.\n\nRev. A 74, 023616 (2006). [18] T. Fernholz, C. R. Gerritsma, P. Krüger, and R. J. C. Spreeuw, arXiv:physics/0512017. [19] O. Morizot, Y. Colombe, V. Lorent, and H. Perrin, arXiv: physics/0512015. [20] I. Lesanovsky and W. von Klitzing, arXiv: cond-mat/0612213. [21] I. Lesanovsky, T. Schumm, S. Hofferberth, L. M. Andersson, P. Krüger, and J. Schmiedmayer, Phys. Rev. A 73, 033619 (2006). [22] I. Lesanovsky, S. Hofferberth, J. Schmiedmayer, and P. Schmelcher, Phys. Rev. A 74, 033619 (2006). [23] C. L. G. Alzar, H. Perrin, H. B. M. Garraway, and V. Lorent, arXiv: physics/0608088. [24] X. Li, H. Zhang, M. Ke, B. Yan, and Y. Wang, arXiv: physics/0607034. [25] S. Hofferberth, B. Fishcher, T. Schumm, J. Schmiedmayer, and I. Lesanovsky, arXiv: quan-ph/0611240. [26] Ph.W. Courteille, B. Deh, J. Fortágh, A. Günther, S.\n\nKraft, C. Marzok, S. Slama, and C. Zimmermann, J. Phys. B 39, 1055 (2006). [27] M. G. Sagnac, C. R. Hebd. Seances Acad. Sci. 157, 708 (1913). [28] C. A. Mead and D. G. Truhlar, J. Chem. Phys. 70, 2284 (1979); C. A. Mead, Phys. Rev. Lett. 59, 161 (1987); C. P. Sun and M. L. Ge, Phys. Rev. D 41, 1349 (1990). [29] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984). [30] J. Schmiedmayer, M. S. Chapman, C. R. Ekstrom, T. D. Hammond, D. K. Kokorowski, A. Lenef, R. A. Rubenstein, E. T. Smith, and D. E. Pritchard, p. 72, Atom interferometry, edited by P. Berman, (Academic Press, N.Y. 1997). [31] T. Ho and V. B. Shenoy, Phys. Rev. Lett. 77, 2595 (1996).\n\n[32] P. Zhang, H. H. Jen, C. P. Sun, and L. You, Phys. Rev. Lett. 98, 030403 (2007). [33] P. Zhang and L. You, Phys. Rev. A 74, 062110 (2006). [34] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959)."
}
] | |
arxiv:0704.0482 | 0704.0482 | 1 | 10.1103/PhysRevA.76.062311 | 4caec5185f281f351422ba4711e16fc0b298b4789732d3ff316689474088ac07 | Implementation of holonomic quantum computation through engineering and
manipulating environment | We consider an atom-field coupled system, in which two pairs of four-level
atoms are respectively driven by laser fields and trapped in two distant
cavities that are connected by an optical fiber. First, we show that an
effective squeezing reservoir can be engineered under appropriate conditions.
Then, we show that a two-qubit geometric CPHASE gate between the atoms in the
two cavities can be implemented through adiabatically manipulating the
engineered reservoir along a closed loop. This scheme that combines engineering
environment with decoherence-free space and geometric phase quantum computation
together has the remarkable feature: a CPHASE gate with arbitrary phase shift
is implemented by simply changing the strength and relative phase of the
driving fields. | [
"Zhang-qi Yin",
"Fu-li Li",
"Peng Peng"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | Quantum computation, attracting much current interest since Shor's algorithm [1] was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevitably coupled to their environment so that entanglement is too fragile to be retained. This makes the realization of quantum computation extremely difficult in the real world. In order to overcome this difficulty, one proposed the decoherence-free space concept [2, 3] . It is found that when qubits involved in quantum computation collectively interact with a same environment there exists a "protected" subspace in the entire Hilbert space, in which the qubits are immune to the decoherence effects induced by the environment. This subspace is called decoherence-free space (DFS). To perform quantum computation in a DFS, one has to design the specific Hamiltonian containing controlling parameters, which eigenspace is spanned by DFS states and the state-unitary manipulation related to quantum computation goal is implemented by changing the controlling parameters [4] .
As well known, instantaneous eigenstates of a quantum system with the time-dependent Hamiltonian may acquire a geometric phase when the time-dependent parameters adiabatically undergo a closed loop in the parameter space [5] . The phase depends only on the swept solid angle by the parameter vector in the parameter space. This feature can be utilized to implement geometric quantum computation (GQC) which is resilient to stochastic control errors [6, 7, 8] . On combining the DFS approach with the GQC scheme, one may build quantum gates which may be immune to both the environment-induced decoherence effects and the control-led errors [9] . In the scheme, quantum logical bits are represented by degenerate eigenstates of the parameterized Hamiltonian. These states have the features: they belong to DFS, and unitarily evolve in time and acquire a geometric phase when the controlling parameters adiabatically vary and undergo a closed loop.
In the recent paper [10], Carollo and coworkers showed that a cascade three-level atom interacting with a broadband squeezed vacuum bosonic bath can be prepared in a state which is decoupled to the environment. This state depends on the reservoir parameters such as squeezing degree and phase angle. As the squeezing parameters smoothly vary, the atomic state can unitarily evolve in time and always be in the manifold of the DFS. Moreover, after a cyclic evolution of the squeezing parameters, the state acquires a geometric phase. This investigation has been generalized to cases where both quantum systems and manipulated reservoir under consideration are not restricted to cascade three-level atoms and squeezed vacuum [11] . These results strongly inspire us that instead of engineering Hamiltonian one may implement the decoherence-free GQC by engineering and manipulating reservoir.
In this paper, we propose a scheme in which the quantum-reservoir engineering [12, 13, 14] is combined with DFS and Berry phase together to realize a twoqubit CPHASE gate [15] . We show that atomic states can unitarily evolve in time in a DFS if the change rate of reservoir parameters is much smaller than the characteristic relaxation time of an atom-reservoir coupled system. Moreover, we find that as the reservoir parameters adiabatically change in time along an appropriate closed loop, the atomic state in the DFS acquires a Berry phase and a CPHASE gate with arbitrary phase shift can be realized. To our knowledge, it is the first proposal for the realization of quantum gates by engineering and steering the environment.
This paper is organized as follows. In Sec. II, we introduce a cavity-atom coupling model in which two pairs of four-level atoms are respectively trapped in two distant cavities that are connected by an optical fiber. In the model, each of pairs of the atoms are simultaneously driven by laser fields and coupled to the local cavity 2 modes through the double Raman transition configuration. Under large detuning and bad cavity limits, we investigate to engineer an effective broadband squeezing reservoir for the atoms. In Sec. III, we analyze how to realize controlling gates between the atoms trapped in the two cavities by steering the squeezing reservoir. Section IV contains conclusions of our investigations.
FIG. 1. Atom-field coupling scheme.
|g jn |r jn |s jn ∆ r jn g s jn ∆ s jn δ jn |e jn g r jn Ω r jn Ω s jn
FIG. 2. Atomic level configuration for atom j in cavity n.
Our scheme is shown in Fig. 1 . A pair of four-level atoms are trapped in each of two distant cavities, respectively, which are connected through an optical fiber. In the short fiber limit [16, 17, 18] , only one fiber mode b is excited and coupled to cavity modes a 1 and a 2 with strength ν [19] . We assume that the cavity modes and the fiber mode have the same frequency ω. The level scheme of atoms is shown in Fig. 2 . Atom j in cavity n is labeled by the index jn with j, n = 1, 2. The distance between the atoms in the same cavity is assumed to be large enough that there is no direct interaction between the atoms. The levels |g jn and |e jn of atom j in cavity n, with j, n = 1, 2 are stable with a long life time. The energy of the level |g jn is taken to be zero as the energy reference point. The lower lying level |e jn , and upper levels |r jn and |s jn have the energy δ jn , and ω r jn and ω s jn , respectively, in the unit with = 1. Transitions |g jn ↔ |s jn and |e jn ↔ |r jn are driven by laser fields of frequencies ω Ls jn and ω Lr jn with Rabi frequencies Ω s jn and Ω r jn and relative phase ϕ, respectively. Transitions |g jn ↔ |r jn and |e jn ↔ |s jn are coupled to the cavity mode a n with the strengths g r jn and g s jn , respectively. Here, we set ∆ r jn = ω r jn -ω = ω r jn -ω Lr jn -δ jn , and ∆ s jn = ω s jn -ω -δ jn = ω s jn -ω Ls jn . Under the Markovian approximation, the master equation of the density matrix for the whole system under consideration can be written as [14]
ρT = -i[H, ρ T ] + L cav1 ρ T + L cav2 ρ T + L f iber ρ T , (1) where H = H 0 + H d + H ac + H cf with H 0 = 2 j,n=1 ω r jn |r jn r jn | + ω s jn |s jn s jn | + δ jn |e jn e jn |) + ω( 2 n=1 a † n a n + b † b), H d = 2 j,n=1 Ω s jn 2 e -iω Ls jn t |s jn g jn | + Ω r jn 2 e -i(ω Lr jn t+ϕ) |r jn e jn | + H.c. , H ac = 2 j,n=1 (g r jn |r jn g jn |a n + g s jn |s jn e jn |a n + H.c.), H cf =ν b(a † 1 + a † 2 ) + H.c. . (2)
Here, H 0 is the free energy of atoms and cavity fields, H d is the interaction energy between the atoms and laser fields, H ac is the interaction energy between the atoms and the cavity fields, and H cf describes the interaction between the cavity modes and the fiber mode. The last three terms in (1) describe the relaxation processes of the cavity and fibre modes in the usual vacuum reservoir, taking the forms
L cavn ρ T =κ n (2a n ρ T a † n -a † n a n ρ T -ρ T a † n a n ), L f iber ρ T =κ f (2bρ T b † -b † bρ T -ρ T b † b), ( 3
)
where κ n is the leakage rate of photons from cavity n, and κ f is the decay rate of the fiber mode.
Let's introduce collective basis:
|a n = (|g 1n |e 2n - |e 1n |g 2n )/ √ 2, | -1 n = |g 1n |g 2n , |0 n = (|g 1n |e 2n + |e 1n |g 2n )/ √ 2, |1 n = |e 1n |e 2n .
The states |a n and | -1 n are taken as a qubit n for quantum computation. In the large detuning limit, adiabatically eliminating the excited states and setting 2 ), we obtain the effective interaction Hamiltonian
Ω 1n g r 1n 2∆ r 1n = Ω r 2n g r 2n 2∆ r 2n = β r n and Ω s 1n g s 1n 2∆ s 1n = Ω s 2n g s 2n 2∆ s 2n = β s n , from (
H ef f = n √ 2 a n (β r n e iϕ S + n + β s n S n ) + H.c. + H cf , ( 4
) where S + n = |0 nn -1|+|1 nn 0|.
In the derivation of (4), we have assumed the resonant condition
g s jn 2 ∆ s jn a † n a n + 3 Ω r jn 2 4∆ r jn = Ω s jn 2 4∆ s jn + g r jn 2 ∆ r jn a † n a n + δ ′ jn .
In order to satisfy the condition with the flexible choice of Ω r jn , Ω s jn , ∆ r jn and ∆ s jn , we have introduced additional ac-Stark shifts δ ′ jn to states |g jn , which can be generated by using a laser field to couple the level |g jn to an ancillary level.
We now introduce three normal modes c and c ± with frequencies ω and ω ± √ 2ν by use of the unitary transfor-
mation a 1 = 1 2 (c + + c -+ √ 2c), a 2 = 1 2 (c + + c -- √ 2c), b = 1 √ 2 (c + -c -) [17, 18]. In the limit ν ≫ |β r j |, |β s j |,
neglecting the far off-resonant modes c ± and setting β p 1 = -β p 2 = β p with p = r, s, we can approximately write the effective Hamiltonian (4) as
H ef f = (β r e iϕ S + + β s S)c + H.c., ( 5
)
where
S + = S + 1 + S + 2 .
Since the modes c ± are nearly not excited and decoupled with the resonant mode c, the fiber mode b is mostly in the vacuum state, therefore, L f iber ρ T can be neglected, and L cav1 ρ T + L cav2 ρ T can be approximated as
L cav ρ T = κ(2cρ T c † -c † cρ T -ρ T c † c), ( 6
)
where κ = (κ 1 + κ 2 )/2. In the bad cavity limit, κ ≫ β, adiabatically eliminating the mode c [12, 14] , from Eq. ( 1 ) with the replacement of the Hamiltonian (2) and the relaxation terms (3) by the effective Hamiltonian (5) and the relaxation term (6), respectively, we can obtain the master equation for the density matrix of the atoms
ρ = - Γ 2 (R + Rρ + ρR + R -2RρR + ), ( 7
)
where ρ = Tr f (ρ T ), R = S cosh r + e iϕ S † sinh r, r = cosh -1 (β r / β r 2 -β s 2
) and Γ = 2(β r 2 -β s2 )/κ. Eq. ( 7 ) describes the collective interaction of two cascade threelevel atoms with the effective squeezed vacuum reservoir [10] . The parameters β r , β s and ϕ are easily changed and controlled at will by varying the strength and phase of the driving lasers [8] . We will show that a geometric phase gate can be realized through changing these parameters.
The DFS of the atomic system is spanned by the states which satisfy the equation R(r, ϕ
)|ψ DF (r, ϕ) = 0 [10]. In terms of basis states |e 1 = |a 1 |a 2 , |e 2 = |a 1 | -1 2 , |e 3 = | -1 1 |a 2 , |e 4 = |a 1 |0 2 , |e 5 = |0 1 |a 2 , |e 6 = |a 1 |1 2 , |e 7 = |1 1 |a 2 , |e 8 = |1 1 |1 2 , |e 9 = 1 √ 2 (|1 1 |0 2 + |0 1 |1 2 ), |e 10 = | - 1 1 | -1 2 , |e 11 = 1 √ 2 (|0 1 | -1 2 + | -1 1 |0 2 ), |e 12 = 1 √ 6 (|1 1 | -1 2 + | -1 1 |1 2 ) + 2 √ 6 |0 1 |0 2 )
, the DFS states can be written as
|ψ DF (r, ϕ) 1 =|e 1 , |ψ DF (r, ϕ) j = cosh r √ cosh 2r |e j -e iϕ sinh r √ cosh 2r |e j+4 , j = 2, 3, |ψ DF (r, ϕ) 4 = e 2iϕ (tanh r) 2 |e 8 -2 3 e iϕ tanh r|e 12 + |e 10 (tanh r) 4 + 2 3 (tanh r) 2 + 1 . ( 8
)
Let's introduce a unitary transformation O(r, ϕ) 1, 2, 3, 4. For the transformed density matrix ρ = O † ρO, we have
by |φ i = 12 i=1 O ij (r, ϕ)|e j , where |φ i = |ψ DF i for i =
dρ dt = i[G, ρ] + O † dρ dt O, ( 9
)
where G(r, ϕ) = iO † dO dt = iO † [ ṙ dO dr + φ dO dϕ ]. To solve Eq. (9) in the DFS, let's define the time-independent projector Π(0 9) , we obtain the equation of motion for ρDF = Π(0)ρΠ(0)
) = O † Π(r, ϕ)O = 4 i=1 O † |φ i φ i |O = 3 j=1 |e j e j | + |e 10 e 10 | onto the DFS. From (
dρ DF dt =i[G DF , ρDF ] + iΠ(0)GΠ ⊥ (0)ρΠ( 0
)
-iΠ(0)ρΠ ⊥ (0)GΠ(0) + Π(0)O † dρ dt OΠ(0), (10) where Π ⊥ (0) = 1 -Π(0) and G DF = Π(0)GΠ(0). In the limit of ṙ, φ ≪ Γ, the last three terms in Eq. (10) can be neglected [11] . In this way, Eq. (10) is reduced to
dρ DF dt = i[G DF , ρDF ]. ( 11
)
Therefore, in the frame dragged adiabatically by the reservoir, the state of the atoms in the DFS unitarily evolves in time.
In this section, we investigate how to realize a CPHASE gate through manipulating the engineered reservoir. Suppose that at the initial time the laser field driving the transition |g ↔ |s is switched off but the laser field driving the transition |r ↔ |e is switched on and the atoms are in the DFS state |Ψ(0
) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ) = 4 j=1 |ψ DF (0, 0) j /2.
To generate a geometric phase for the atomic state, we smoothly change the parameters of the engineered reservoir along a closed loop, which is divided into the following three steps: (1) From time 0 to T 1 , hold on ϕ = 0, and adiabatically increase the parameter r from 0 to r 0 ; (2) From time T 1 to T 2 , hold on r = r 0 , and adiabatically change the phase ϕ from 0 to ϕ 0 ; (3) From time T 2 to T 3 , hold on ϕ = ϕ 0 , and adiabatically decrease r from r 0 to 0. When the cyclic evolution ends, the atomic state becomes
|Ψ(T 3 ) a = 1 2 (|e 1 + e iχ1 |e 2 + e iχ1 |e 3 + e iχ12 |e 10 ), (12) where geometric phases χ 1 = -ν 1 ϕ 0 , χ 12 = -ν 12 ϕ 0 with ν 1 = sinh 2 r0 sinh 2 r0+cosh 2 r0 , ν 12 = 2 tanh 4 r0+ 2 3 tanh 2 r0 tanh 4 r0+ 2 3 tanh 2 r0+1 . By per- forming local transformations U 1 = e -iχ1 |-1 11 -1| and 4 U 2 = e -iχ1 | -1 22 -1|,
the state (12) can be written as
|Ψ ′ (T 3 ) a = U 1 U 2 |Ψ(T 3 ) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + e i∆ | -1 1 | -1 2 ), where ∆ = χ 12 -2χ 1 = (2ν 1 -ν 12 )ϕ 0 .
Thus, the CPHASE gate with the phase shift ∆ is realized. If both the atoms in cavity 1 and the atoms in cavity 2 "see" different environments, |χ 12 | must be equal to |2χ 1 | and ∆ = 0. Therefore, the phase shift ∆ results from the collective coupling of the atoms in both cavities with the same engineered environment.
If r 0 = atanh( 4/3 -1) ≃ 0.4157, |ν 12 | = |ν 1 |. Under this condition with ϕ 0 = π/ν 1 ,
the state of the atoms at the time T
3 is |Ψ ′′ (T 3 ) a = -1 2 (-|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ).
In this case, the Controlled-Z gate between the two qubits is realized without local transformations. 0 0.2 0.4 0.6 0.8 0.9965 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 r 0 F r T=100/Γ T=200/Γ T=400/Γ (a)
FIG. 3. Fidelity F r of the atomic state. 0 0.2 0.4 0.6 0.8 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 r 0 F p T = 200/Γ T = 400/Γ T = 1000/Γ (b) FIG. 4. Fidelity F p of the atomic state.
The above results depend on the adiabatical approximation. To check the adiabatical condition, we numerically simulate the following two examples. In the first example, we suppose that at the initial time the atoms are in the state |Ψ 1 a = (|a 1 |a 2 + |ψ DF (0, 0) 2 )/ √ 2 and the laser field driving the transition |e ↔ |r are turned on. Then, by slowly switching the laser field driving the transition |g ↔ |s , we increase the parameter r from 0 to r 0 according to the linear function r(t) = r 0 t/T . In the adiabatical limit (T ≫ Γ -1 ), the atomic state becomes
|Ψ ′ 1 a = (|a 1 |a 2 + |ψ DF (r 0 , 0) 2 )/ √ 2
at the time T . On the other hand, in the Hilbert space spanned by the basis states {|e i } for i = 1, 2, • • • , 12 , we can numerically solve Eq. ( 7 ) and obtain the density matrix ρ 1 (T ) of the atoms. Let's define
F r = a Ψ ′ 1 |ρ 1 (T )|Ψ ′ 1 a as
the fidelity for this process. As shown in Fig. 3 , if T > 100/Γ, F r is always bigger than 0.997 if r ∈ (0, 0.8), corresponding to the almost perfect evolution.
In the second example, we suppose that the atoms are initially in the state
|Ψ 2 a = (|a 1 |a 2 + |ψ DF (r, 0) 4 )/
√ 2 and all the driving fields are turned on to hold the parameters r = r 0 and ϕ = 0. By adiabatically changing the phase ϕ from 0 to 2π at the rate φ = 2π/T , the atomic state at the time
T becomes |Ψ ′ 2 a = (|a 1 |a 2 + e iχ12 |ψ DF (r, 2π) 4 )/ √ 2.
Let's define the fidelity for this example as
F p = a Ψ ′ 2 |ρ(T )|Ψ ′ a ,
where ρ(T ) is the numerical solution of Eq. ( 7 ). As shown in Fig. 4 , F p increases as T increases but decreases as the parameter r 0 increases. If T > 1000/Γ, F p is larger than 0.992 for 0 < r 0 < 0.8. From these two examples, we find that to fulfill the adiabatical condition the time used in the step 2 should be much longer than in the steps 1 and 3.
A controlled-Z gate has been numerically simulated by directly solving Eq. ( 7 ) with r 0 = 0.5, and ϕ
0 = π/|2ν 1 -ν 12 |.
In the simulation, we set ṙ = r 0 /T 1 in the steps 1 and 3, and φ = ϕ 0 /(T 2 -T 1 ) in the step 2 with T 1 = 0.05T 3 and T 2 -T 1 = 0.90T 3 . If T 3 > 1100/Γ, we find that the fidelity F = a Ψ(T 3 )|ρ(T 3 )|Ψ(T 3 ) a is larger than 0.95. For an almost perfect controlled-Z gate with F > 0.99, we find that T 3 must be longer than 6000/Γ. Now let's briefly discuss the effects of the atomic spontaneous emission, the fiber mode decay and cavity photon leakage. For simplicity but without the loss of generality, we suppose that atomic spontaneous emission rates of the excited levels are equal to γ. In the large detunig limit, the characteristic spontaneous emission rate of the atoms is γ ef f = γ(Ω 2 /2∆ 2 ) [14, 16] and the effective decay rate of the fiber mode is
κ ef f = κ f Ω 2 g 2 /(4∆ 2 ν 2 ). If κ f ≤ γ and g 2 ≪ ν 2 , κ
ef f be much smaller than γ ef f . Under this condition, the present scheme is feasible if Γ ≫ γ ef f . In the current cavity quantum dynamic (CQED) experiment, the parameters (g, κ, γ) = (2000, 10, 10) MHz could be available [20] . If setting Ω/(2∆) = 1 √ 2 × 10 -3 , we have Γ ≃ 4 × 10 4 γ ef f . The condition is held. In the present scheme, the large cavity decay rate is required to ensure that the cavity modes are in a broadband squeezed vacuum reservoir and then the atoms always "see" the broadband squeezed vacuum reservoir during the dynamic evolution. For an arbitrary small but nonzero value of the squeezing degree of the reservoir, a CPHASE gate with arbitrary high fidelity can always be realized in the represent scheme. The cavity decay does 5 not directly affect the fidelity of the realized CPHASE gates. However, the larger the decay rate is, the longer the operation time of the CPHASE gates is. Thus, we have the condition κ >> β, γ ef f for realizing the reliable CPHASE gates. Based on the parameters quoted above, this condition can be well satisfied. With the parameters of the current CQED experiment, we find that the operation time of the controlled-Z gate, with fidelity larger than 0.95, is about 2.8 ms. It is much shorter than both 1/γ ef f and the single-atom trapping time in cavity [21] . On the other hand, the present scheme needs a strong coupling between the cavity and the fiber. This could be realized at the current experiment [22] . Therefore, the requirement for the realization of the present scheme can be satisfied with the current technology.
We propose a cavity-atom coupled scheme for the realization of quantum controlling gates, in which each of two pairs of four-level atoms in two distant cavities connected by a short optical fibre are simultaneously driven by laser fields and coupled to the local cavity modes through the double Raman transition configuration. We show that an effective squeezing reservoir coupled to the multilevel atoms can be engineered under appropriate driving condition and bad cavity limit. We find that in the scheme a CPHASE gate with arbitrary phase shift can be implemented through adiabatically changing the strength and phase of driving fields along a closed loop. It is also noticed that the larger the effective coupling strength between the environment and the atoms is, the more reliable the realized CPHASE gate is.
We thank Yun-feng Xiao and Wen-ping He for valuable discussions and suggestions. This work was supported by the Natural Science Foundation of China (Grant Nos. 10674106, 60778021 and 05-06-01).
[1] P. W. Shor, in Proceeings, 35 th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Press, Los Almitos, CA, 1994), p. 124. [2] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997). [3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). [4] D. A. Lidar and et al., Phys. Rev. Lett. 81, 2594 (1998). [5] M. V. Berry, Proc. R. Soc. A 392, 45 (1984). [6] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [7] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301 (2005). [8] L.-M. Duan and et al., Science 292, 1695 (2001). [9] L.-A. Wu and et al., Phys. Rev. Lett. 95, 130501 (2005). [10] Angelo Carollo et al., Phys. Rev. Lett. 96, 150403 (2006). [11] Angelo Carollo et al., Phys. Rev. Lett. 96, 020403 (2006). [12] J. I. Cirac, Phys. Rev. A 46, 4354 (1992). [13] C. J. Myatt and et al., Nature (London) 403, 269 (2000). [14] S. G. Clark and A. S. Parkins, Phys. Rev. Lett. 90, 047905 (2003). [15] S. Lloyd, Phys. Rev. Lett. 75, 346 (1995). [16] T. Pellizzari, Phys. Rev. Lett. 79, 5242 (1997). [17] A. Serafini and et al., Phys. Rev. Lett. 96, 010503 (2006) . [18] Zhang-qi Yin and Fu-li Li, Phys. Rev. A 75, 012324 (2007). [19] S. J. van Enk and et al., Phys. Rev. A 59, 2659 (1999). [20] S. M. Spillane et al., Phys. Rev. A 71, 013817 (2005). [21] Stefan Nuβmann and et al., Nature Phys. 1, 122 (2005). [22] S. M. Spillane et al., Phys. Rev. Lett. 91, 043902 (2003). | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Quantum computation, attracting much current interest since Shor's algorithm [1] was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevitably coupled to their environment so that entanglement is too fragile to be retained. This makes the realization of quantum computation extremely difficult in the real world. In order to overcome this difficulty, one proposed the decoherence-free space concept [2, 3] . It is found that when qubits involved in quantum computation collectively interact with a same environment there exists a \"protected\" subspace in the entire Hilbert space, in which the qubits are immune to the decoherence effects induced by the environment. This subspace is called decoherence-free space (DFS). To perform quantum computation in a DFS, one has to design the specific Hamiltonian containing controlling parameters, which eigenspace is spanned by DFS states and the state-unitary manipulation related to quantum computation goal is implemented by changing the controlling parameters [4] .\n\nAs well known, instantaneous eigenstates of a quantum system with the time-dependent Hamiltonian may acquire a geometric phase when the time-dependent parameters adiabatically undergo a closed loop in the parameter space [5] . The phase depends only on the swept solid angle by the parameter vector in the parameter space. This feature can be utilized to implement geometric quantum computation (GQC) which is resilient to stochastic control errors [6, 7, 8] . On combining the DFS approach with the GQC scheme, one may build quantum gates which may be immune to both the environment-induced decoherence effects and the control-led errors [9] . In the scheme, quantum logical bits are represented by degenerate eigenstates of the parameterized Hamiltonian. These states have the features: they belong to DFS, and unitarily evolve in time and acquire a geometric phase when the controlling parameters adiabatically vary and undergo a closed loop.\n\nIn the recent paper [10], Carollo and coworkers showed that a cascade three-level atom interacting with a broadband squeezed vacuum bosonic bath can be prepared in a state which is decoupled to the environment. This state depends on the reservoir parameters such as squeezing degree and phase angle. As the squeezing parameters smoothly vary, the atomic state can unitarily evolve in time and always be in the manifold of the DFS. Moreover, after a cyclic evolution of the squeezing parameters, the state acquires a geometric phase. This investigation has been generalized to cases where both quantum systems and manipulated reservoir under consideration are not restricted to cascade three-level atoms and squeezed vacuum [11] . These results strongly inspire us that instead of engineering Hamiltonian one may implement the decoherence-free GQC by engineering and manipulating reservoir.\n\nIn this paper, we propose a scheme in which the quantum-reservoir engineering [12, 13, 14] is combined with DFS and Berry phase together to realize a twoqubit CPHASE gate [15] . We show that atomic states can unitarily evolve in time in a DFS if the change rate of reservoir parameters is much smaller than the characteristic relaxation time of an atom-reservoir coupled system. Moreover, we find that as the reservoir parameters adiabatically change in time along an appropriate closed loop, the atomic state in the DFS acquires a Berry phase and a CPHASE gate with arbitrary phase shift can be realized. To our knowledge, it is the first proposal for the realization of quantum gates by engineering and steering the environment.\n\nThis paper is organized as follows. In Sec. II, we introduce a cavity-atom coupling model in which two pairs of four-level atoms are respectively trapped in two distant cavities that are connected by an optical fiber. In the model, each of pairs of the atoms are simultaneously driven by laser fields and coupled to the local cavity 2 modes through the double Raman transition configuration. Under large detuning and bad cavity limits, we investigate to engineer an effective broadband squeezing reservoir for the atoms. In Sec. III, we analyze how to realize controlling gates between the atoms trapped in the two cavities by steering the squeezing reservoir. Section IV contains conclusions of our investigations."
},
{
"section_type": "OTHER",
"section_title": "laser laser fibre",
"text": "FIG. 1. Atom-field coupling scheme.\n\n|g jn |r jn |s jn ∆ r jn g s jn ∆ s jn δ jn |e jn g r jn Ω r jn Ω s jn\n\nFIG. 2. Atomic level configuration for atom j in cavity n."
},
{
"section_type": "OTHER",
"section_title": "II. ENGINEERING A SQUEEZING ENVIRONMENT AND GENERATING A DECOHERENCE-FREE SUBSPACE",
"text": "Our scheme is shown in Fig. 1 . A pair of four-level atoms are trapped in each of two distant cavities, respectively, which are connected through an optical fiber. In the short fiber limit [16, 17, 18] , only one fiber mode b is excited and coupled to cavity modes a 1 and a 2 with strength ν [19] . We assume that the cavity modes and the fiber mode have the same frequency ω. The level scheme of atoms is shown in Fig. 2 . Atom j in cavity n is labeled by the index jn with j, n = 1, 2. The distance between the atoms in the same cavity is assumed to be large enough that there is no direct interaction between the atoms. The levels |g jn and |e jn of atom j in cavity n, with j, n = 1, 2 are stable with a long life time. The energy of the level |g jn is taken to be zero as the energy reference point. The lower lying level |e jn , and upper levels |r jn and |s jn have the energy δ jn , and ω r jn and ω s jn , respectively, in the unit with = 1. Transitions |g jn ↔ |s jn and |e jn ↔ |r jn are driven by laser fields of frequencies ω Ls jn and ω Lr jn with Rabi frequencies Ω s jn and Ω r jn and relative phase ϕ, respectively. Transitions |g jn ↔ |r jn and |e jn ↔ |s jn are coupled to the cavity mode a n with the strengths g r jn and g s jn , respectively. Here, we set ∆ r jn = ω r jn -ω = ω r jn -ω Lr jn -δ jn , and ∆ s jn = ω s jn -ω -δ jn = ω s jn -ω Ls jn . Under the Markovian approximation, the master equation of the density matrix for the whole system under consideration can be written as [14]\n\nρT = -i[H, ρ T ] + L cav1 ρ T + L cav2 ρ T + L f iber ρ T , (1) where H = H 0 + H d + H ac + H cf with H 0 = 2 j,n=1 ω r jn |r jn r jn | + ω s jn |s jn s jn | + δ jn |e jn e jn |) + ω( 2 n=1 a † n a n + b † b), H d = 2 j,n=1 Ω s jn 2 e -iω Ls jn t |s jn g jn | + Ω r jn 2 e -i(ω Lr jn t+ϕ) |r jn e jn | + H.c. , H ac = 2 j,n=1 (g r jn |r jn g jn |a n + g s jn |s jn e jn |a n + H.c.), H cf =ν b(a † 1 + a † 2 ) + H.c. . (2)\n\nHere, H 0 is the free energy of atoms and cavity fields, H d is the interaction energy between the atoms and laser fields, H ac is the interaction energy between the atoms and the cavity fields, and H cf describes the interaction between the cavity modes and the fiber mode. The last three terms in (1) describe the relaxation processes of the cavity and fibre modes in the usual vacuum reservoir, taking the forms\n\nL cavn ρ T =κ n (2a n ρ T a † n -a † n a n ρ T -ρ T a † n a n ), L f iber ρ T =κ f (2bρ T b † -b † bρ T -ρ T b † b), ( 3\n\n)\n\nwhere κ n is the leakage rate of photons from cavity n, and κ f is the decay rate of the fiber mode.\n\nLet's introduce collective basis:\n\n|a n = (|g 1n |e 2n - |e 1n |g 2n )/ √ 2, | -1 n = |g 1n |g 2n , |0 n = (|g 1n |e 2n + |e 1n |g 2n )/ √ 2, |1 n = |e 1n |e 2n .\n\nThe states |a n and | -1 n are taken as a qubit n for quantum computation. In the large detuning limit, adiabatically eliminating the excited states and setting 2 ), we obtain the effective interaction Hamiltonian\n\nΩ 1n g r 1n 2∆ r 1n = Ω r 2n g r 2n 2∆ r 2n = β r n and Ω s 1n g s 1n 2∆ s 1n = Ω s 2n g s 2n 2∆ s 2n = β s n , from (\n\nH ef f = n √ 2 a n (β r n e iϕ S + n + β s n S n ) + H.c. + H cf , ( 4\n\n) where S + n = |0 nn -1|+|1 nn 0|.\n\nIn the derivation of (4), we have assumed the resonant condition\n\ng s jn 2 ∆ s jn a † n a n + 3 Ω r jn 2 4∆ r jn = Ω s jn 2 4∆ s jn + g r jn 2 ∆ r jn a † n a n + δ ′ jn .\n\nIn order to satisfy the condition with the flexible choice of Ω r jn , Ω s jn , ∆ r jn and ∆ s jn , we have introduced additional ac-Stark shifts δ ′ jn to states |g jn , which can be generated by using a laser field to couple the level |g jn to an ancillary level.\n\nWe now introduce three normal modes c and c ± with frequencies ω and ω ± √ 2ν by use of the unitary transfor-\n\nmation a 1 = 1 2 (c + + c -+ √ 2c), a 2 = 1 2 (c + + c -- √ 2c), b = 1 √ 2 (c + -c -) [17, 18]. In the limit ν ≫ |β r j |, |β s j |,\n\nneglecting the far off-resonant modes c ± and setting β p 1 = -β p 2 = β p with p = r, s, we can approximately write the effective Hamiltonian (4) as\n\nH ef f = (β r e iϕ S + + β s S)c + H.c., ( 5\n\n)\n\nwhere\n\nS + = S + 1 + S + 2 .\n\nSince the modes c ± are nearly not excited and decoupled with the resonant mode c, the fiber mode b is mostly in the vacuum state, therefore, L f iber ρ T can be neglected, and L cav1 ρ T + L cav2 ρ T can be approximated as\n\nL cav ρ T = κ(2cρ T c † -c † cρ T -ρ T c † c), ( 6\n\n)\n\nwhere κ = (κ 1 + κ 2 )/2. In the bad cavity limit, κ ≫ β, adiabatically eliminating the mode c [12, 14] , from Eq. ( 1 ) with the replacement of the Hamiltonian (2) and the relaxation terms (3) by the effective Hamiltonian (5) and the relaxation term (6), respectively, we can obtain the master equation for the density matrix of the atoms\n\nρ = - Γ 2 (R + Rρ + ρR + R -2RρR + ), ( 7\n\n)\n\nwhere ρ = Tr f (ρ T ), R = S cosh r + e iϕ S † sinh r, r = cosh -1 (β r / β r 2 -β s 2\n\n) and Γ = 2(β r 2 -β s2 )/κ. Eq. ( 7 ) describes the collective interaction of two cascade threelevel atoms with the effective squeezed vacuum reservoir [10] . The parameters β r , β s and ϕ are easily changed and controlled at will by varying the strength and phase of the driving lasers [8] . We will show that a geometric phase gate can be realized through changing these parameters.\n\nThe DFS of the atomic system is spanned by the states which satisfy the equation R(r, ϕ\n\n)|ψ DF (r, ϕ) = 0 [10]. In terms of basis states |e 1 = |a 1 |a 2 , |e 2 = |a 1 | -1 2 , |e 3 = | -1 1 |a 2 , |e 4 = |a 1 |0 2 , |e 5 = |0 1 |a 2 , |e 6 = |a 1 |1 2 , |e 7 = |1 1 |a 2 , |e 8 = |1 1 |1 2 , |e 9 = 1 √ 2 (|1 1 |0 2 + |0 1 |1 2 ), |e 10 = | - 1 1 | -1 2 , |e 11 = 1 √ 2 (|0 1 | -1 2 + | -1 1 |0 2 ), |e 12 = 1 √ 6 (|1 1 | -1 2 + | -1 1 |1 2 ) + 2 √ 6 |0 1 |0 2 )\n\n, the DFS states can be written as\n\n|ψ DF (r, ϕ) 1 =|e 1 , |ψ DF (r, ϕ) j = cosh r √ cosh 2r |e j -e iϕ sinh r √ cosh 2r |e j+4 , j = 2, 3, |ψ DF (r, ϕ) 4 = e 2iϕ (tanh r) 2 |e 8 -2 3 e iϕ tanh r|e 12 + |e 10 (tanh r) 4 + 2 3 (tanh r) 2 + 1 . ( 8\n\n)\n\nLet's introduce a unitary transformation O(r, ϕ) 1, 2, 3, 4. For the transformed density matrix ρ = O † ρO, we have\n\nby |φ i = 12 i=1 O ij (r, ϕ)|e j , where |φ i = |ψ DF i for i =\n\ndρ dt = i[G, ρ] + O † dρ dt O, ( 9\n\n)\n\nwhere G(r, ϕ) = iO † dO dt = iO † [ ṙ dO dr + φ dO dϕ ]. To solve Eq. (9) in the DFS, let's define the time-independent projector Π(0 9) , we obtain the equation of motion for ρDF = Π(0)ρΠ(0)\n\n) = O † Π(r, ϕ)O = 4 i=1 O † |φ i φ i |O = 3 j=1 |e j e j | + |e 10 e 10 | onto the DFS. From (\n\ndρ DF dt =i[G DF , ρDF ] + iΠ(0)GΠ ⊥ (0)ρΠ( 0\n\n)\n\n-iΠ(0)ρΠ ⊥ (0)GΠ(0) + Π(0)O † dρ dt OΠ(0), (10) where Π ⊥ (0) = 1 -Π(0) and G DF = Π(0)GΠ(0). In the limit of ṙ, φ ≪ Γ, the last three terms in Eq. (10) can be neglected [11] . In this way, Eq. (10) is reduced to\n\ndρ DF dt = i[G DF , ρDF ]. ( 11\n\n)\n\nTherefore, in the frame dragged adiabatically by the reservoir, the state of the atoms in the DFS unitarily evolves in time."
},
{
"section_type": "OTHER",
"section_title": "III. REALIZING CONTROLLING PHASE GATES THROUGH MANIPULATING THE SQUEEZING ENVIRONMENT",
"text": "In this section, we investigate how to realize a CPHASE gate through manipulating the engineered reservoir. Suppose that at the initial time the laser field driving the transition |g ↔ |s is switched off but the laser field driving the transition |r ↔ |e is switched on and the atoms are in the DFS state |Ψ(0\n\n) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ) = 4 j=1 |ψ DF (0, 0) j /2.\n\nTo generate a geometric phase for the atomic state, we smoothly change the parameters of the engineered reservoir along a closed loop, which is divided into the following three steps: (1) From time 0 to T 1 , hold on ϕ = 0, and adiabatically increase the parameter r from 0 to r 0 ; (2) From time T 1 to T 2 , hold on r = r 0 , and adiabatically change the phase ϕ from 0 to ϕ 0 ; (3) From time T 2 to T 3 , hold on ϕ = ϕ 0 , and adiabatically decrease r from r 0 to 0. When the cyclic evolution ends, the atomic state becomes\n\n|Ψ(T 3 ) a = 1 2 (|e 1 + e iχ1 |e 2 + e iχ1 |e 3 + e iχ12 |e 10 ), (12) where geometric phases χ 1 = -ν 1 ϕ 0 , χ 12 = -ν 12 ϕ 0 with ν 1 = sinh 2 r0 sinh 2 r0+cosh 2 r0 , ν 12 = 2 tanh 4 r0+ 2 3 tanh 2 r0 tanh 4 r0+ 2 3 tanh 2 r0+1 . By per- forming local transformations U 1 = e -iχ1 |-1 11 -1| and 4 U 2 = e -iχ1 | -1 22 -1|,\n\nthe state (12) can be written as\n\n|Ψ ′ (T 3 ) a = U 1 U 2 |Ψ(T 3 ) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + e i∆ | -1 1 | -1 2 ), where ∆ = χ 12 -2χ 1 = (2ν 1 -ν 12 )ϕ 0 .\n\nThus, the CPHASE gate with the phase shift ∆ is realized. If both the atoms in cavity 1 and the atoms in cavity 2 \"see\" different environments, |χ 12 | must be equal to |2χ 1 | and ∆ = 0. Therefore, the phase shift ∆ results from the collective coupling of the atoms in both cavities with the same engineered environment.\n\nIf r 0 = atanh( 4/3 -1) ≃ 0.4157, |ν 12 | = |ν 1 |. Under this condition with ϕ 0 = π/ν 1 ,\n\nthe state of the atoms at the time T\n\n3 is |Ψ ′′ (T 3 ) a = -1 2 (-|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ).\n\nIn this case, the Controlled-Z gate between the two qubits is realized without local transformations. 0 0.2 0.4 0.6 0.8 0.9965 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 r 0 F r T=100/Γ T=200/Γ T=400/Γ (a)\n\nFIG. 3. Fidelity F r of the atomic state. 0 0.2 0.4 0.6 0.8 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 r 0 F p T = 200/Γ T = 400/Γ T = 1000/Γ (b) FIG. 4. Fidelity F p of the atomic state.\n\nThe above results depend on the adiabatical approximation. To check the adiabatical condition, we numerically simulate the following two examples. In the first example, we suppose that at the initial time the atoms are in the state |Ψ 1 a = (|a 1 |a 2 + |ψ DF (0, 0) 2 )/ √ 2 and the laser field driving the transition |e ↔ |r are turned on. Then, by slowly switching the laser field driving the transition |g ↔ |s , we increase the parameter r from 0 to r 0 according to the linear function r(t) = r 0 t/T . In the adiabatical limit (T ≫ Γ -1 ), the atomic state becomes\n\n|Ψ ′ 1 a = (|a 1 |a 2 + |ψ DF (r 0 , 0) 2 )/ √ 2\n\nat the time T . On the other hand, in the Hilbert space spanned by the basis states {|e i } for i = 1, 2, • • • , 12 , we can numerically solve Eq. ( 7 ) and obtain the density matrix ρ 1 (T ) of the atoms. Let's define\n\nF r = a Ψ ′ 1 |ρ 1 (T )|Ψ ′ 1 a as\n\nthe fidelity for this process. As shown in Fig. 3 , if T > 100/Γ, F r is always bigger than 0.997 if r ∈ (0, 0.8), corresponding to the almost perfect evolution.\n\nIn the second example, we suppose that the atoms are initially in the state\n\n|Ψ 2 a = (|a 1 |a 2 + |ψ DF (r, 0) 4 )/\n\n√ 2 and all the driving fields are turned on to hold the parameters r = r 0 and ϕ = 0. By adiabatically changing the phase ϕ from 0 to 2π at the rate φ = 2π/T , the atomic state at the time\n\nT becomes |Ψ ′ 2 a = (|a 1 |a 2 + e iχ12 |ψ DF (r, 2π) 4 )/ √ 2.\n\nLet's define the fidelity for this example as\n\nF p = a Ψ ′ 2 |ρ(T )|Ψ ′ a ,\n\nwhere ρ(T ) is the numerical solution of Eq. ( 7 ). As shown in Fig. 4 , F p increases as T increases but decreases as the parameter r 0 increases. If T > 1000/Γ, F p is larger than 0.992 for 0 < r 0 < 0.8. From these two examples, we find that to fulfill the adiabatical condition the time used in the step 2 should be much longer than in the steps 1 and 3.\n\nA controlled-Z gate has been numerically simulated by directly solving Eq. ( 7 ) with r 0 = 0.5, and ϕ\n\n0 = π/|2ν 1 -ν 12 |.\n\nIn the simulation, we set ṙ = r 0 /T 1 in the steps 1 and 3, and φ = ϕ 0 /(T 2 -T 1 ) in the step 2 with T 1 = 0.05T 3 and T 2 -T 1 = 0.90T 3 . If T 3 > 1100/Γ, we find that the fidelity F = a Ψ(T 3 )|ρ(T 3 )|Ψ(T 3 ) a is larger than 0.95. For an almost perfect controlled-Z gate with F > 0.99, we find that T 3 must be longer than 6000/Γ. Now let's briefly discuss the effects of the atomic spontaneous emission, the fiber mode decay and cavity photon leakage. For simplicity but without the loss of generality, we suppose that atomic spontaneous emission rates of the excited levels are equal to γ. In the large detunig limit, the characteristic spontaneous emission rate of the atoms is γ ef f = γ(Ω 2 /2∆ 2 ) [14, 16] and the effective decay rate of the fiber mode is\n\nκ ef f = κ f Ω 2 g 2 /(4∆ 2 ν 2 ). If κ f ≤ γ and g 2 ≪ ν 2 , κ\n\nef f be much smaller than γ ef f . Under this condition, the present scheme is feasible if Γ ≫ γ ef f . In the current cavity quantum dynamic (CQED) experiment, the parameters (g, κ, γ) = (2000, 10, 10) MHz could be available [20] . If setting Ω/(2∆) = 1 √ 2 × 10 -3 , we have Γ ≃ 4 × 10 4 γ ef f . The condition is held. In the present scheme, the large cavity decay rate is required to ensure that the cavity modes are in a broadband squeezed vacuum reservoir and then the atoms always \"see\" the broadband squeezed vacuum reservoir during the dynamic evolution. For an arbitrary small but nonzero value of the squeezing degree of the reservoir, a CPHASE gate with arbitrary high fidelity can always be realized in the represent scheme. The cavity decay does 5 not directly affect the fidelity of the realized CPHASE gates. However, the larger the decay rate is, the longer the operation time of the CPHASE gates is. Thus, we have the condition κ >> β, γ ef f for realizing the reliable CPHASE gates. Based on the parameters quoted above, this condition can be well satisfied. With the parameters of the current CQED experiment, we find that the operation time of the controlled-Z gate, with fidelity larger than 0.95, is about 2.8 ms. It is much shorter than both 1/γ ef f and the single-atom trapping time in cavity [21] . On the other hand, the present scheme needs a strong coupling between the cavity and the fiber. This could be realized at the current experiment [22] . Therefore, the requirement for the realization of the present scheme can be satisfied with the current technology."
},
{
"section_type": "CONCLUSION",
"section_title": "IV. CONCLUSIONS",
"text": "We propose a cavity-atom coupled scheme for the realization of quantum controlling gates, in which each of two pairs of four-level atoms in two distant cavities connected by a short optical fibre are simultaneously driven by laser fields and coupled to the local cavity modes through the double Raman transition configuration. We show that an effective squeezing reservoir coupled to the multilevel atoms can be engineered under appropriate driving condition and bad cavity limit. We find that in the scheme a CPHASE gate with arbitrary phase shift can be implemented through adiabatically changing the strength and phase of driving fields along a closed loop. It is also noticed that the larger the effective coupling strength between the environment and the atoms is, the more reliable the realized CPHASE gate is."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "We thank Yun-feng Xiao and Wen-ping He for valuable discussions and suggestions. This work was supported by the Natural Science Foundation of China (Grant Nos. 10674106, 60778021 and 05-06-01).\n\n[1] P. W. Shor, in Proceeings, 35 th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Press, Los Almitos, CA, 1994), p. 124. [2] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997). [3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). [4] D. A. Lidar and et al., Phys. Rev. Lett. 81, 2594 (1998). [5] M. V. Berry, Proc. R. Soc. A 392, 45 (1984). [6] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [7] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301 (2005). [8] L.-M. Duan and et al., Science 292, 1695 (2001). [9] L.-A. Wu and et al., Phys. Rev. Lett. 95, 130501 (2005). [10] Angelo Carollo et al., Phys. Rev. Lett. 96, 150403 (2006). [11] Angelo Carollo et al., Phys. Rev. Lett. 96, 020403 (2006). [12] J. I. Cirac, Phys. Rev. A 46, 4354 (1992). [13] C. J. Myatt and et al., Nature (London) 403, 269 (2000). [14] S. G. Clark and A. S. Parkins, Phys. Rev. Lett. 90, 047905 (2003). [15] S. Lloyd, Phys. Rev. Lett. 75, 346 (1995). [16] T. Pellizzari, Phys. Rev. Lett. 79, 5242 (1997). [17] A. Serafini and et al., Phys. Rev. Lett. 96, 010503 (2006) . [18] Zhang-qi Yin and Fu-li Li, Phys. Rev. A 75, 012324 (2007). [19] S. J. van Enk and et al., Phys. Rev. A 59, 2659 (1999). [20] S. M. Spillane et al., Phys. Rev. A 71, 013817 (2005). [21] Stefan Nuβmann and et al., Nature Phys. 1, 122 (2005). [22] S. M. Spillane et al., Phys. Rev. Lett. 91, 043902 (2003)."
}
] | |
arxiv:0704.0488 | 0704.0488 | 1 | 10.1007/s10714-007-0431-5 | 5164ad34ca4de631f8a4c37f1aba38127c537f07d91eec4dfc4322512c2ad2cd | Teleparallel Version of the Stationary Axisymmetric Solutions and their
Energy Contents | This work contains the teleparallel version of the stationary axisymmetric
solutions. We obtain the tetrad and the torsion fields representing these
solutions. The tensor, vector and axial-vector parts of the torsion tensor are
evaluated. It is found that the axial-vector has component only along $\rho$
and $z$ directions. The three possibilities of the axial vector depending on
the metric function $B$ are discussed. The vector related with spin has also
been evaluated and the corresponding extra Hamiltonian is furnished. Further,
we use the teleparallel version of M$\ddot{o}$ller prescription to find the
energy-momentum distribution of the solutions. It is interesting to note that
(for $\lambda=1$) energy and momentum densities in teleparallel theory are
equal to the corresponding quantities in GR plus an additional quantity in
each, which may become equal under certain conditions. Finally, we discuss the
two special cases of the stationary axisymmetric solutions. | [
"M. Sharif and M. Jamil Amir"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-04 | 2026-02-26 | This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along ρ and z directions. The three possibilities of the axial vector depending on the metric function B are discussed. The vector related with spin has also been evaluated and the corresponding extra Hamiltonian is furnished. Further, we use the teleparallel version of Möller prescription to find the energy-momentum distribution of the solutions. It is interesting to note that (for λ = 1) energy and momentum densities in teleparallel theory are equal to the corresponding quantities in GR plus an additional quantity in each, which may become equal under certain conditions. Finally, we discuss the two special cases of the stationary axisymmetric solutions.
The attempts made by Einstein and his followers to unify gravitation with other interactions led to the investigation of structures of gravitation other than the metric tensor. These structures yield the metric tensor as a by product. Tetrad field is one of these structures which leads to the theory of teleparallel gravity (TPG) [1, 2] . TPG is an alternative theory of gravity which corresponds to a gauge theory of translation group [3, 4] based on Weitzenböck geometry [5] . This theory is characterized by the vanishing of curvature identically while the torsion is taken to be non-zero. In TPG, gravitation is attributed to torsion [4] which plays a role of force [6] . In General Relativity (GR), gravitation geometrizes the underlying spacetime. The translational gauge potentials appear as a non-trivial part of the tetrad field and induce a teleparallel (TP) structure on spacetime which is directly related to the presence of a gravitational field. In some other theories [3] [4] [5] [6] [7] [8] , torsion is only relevant when spins are important [9] . This point of view indicates that torsion might represent additional degrees of freedom as compared to curvature. As a result, some new physics may be associated with it. Teleparallelism is naturally formulated by gauging external (spacetime) translations which are closely related to the group of general coordinate transformations underlying GR. Thus the energy-momentum tensor represents the matter source in the field equations of tetradic theories of gravity like in GR.
There is a large literature available [10] about the study of TP versions of the exact solutions of GR. Recently, Pereira, et al. [11] obtained the TP versions of the Schwarzschild and the stationary axisymmetric Kerr solutions of GR. They proved that the axial-vector torsion plays the role of the gravitomagnetic component of the gravitational field in the case of slow rotation and weak field approximations. In a previous paper [12] , we have found the TP versions of the Friedmann models and of the Lewis-Papapetrou spacetimes, and also discussed their axial-vectors.
There has been a longstanding, controversial and still unresolved problem of the localization of energy (i.e., to express it as a unique tensor quantity) in GR [13] . Einstein [14] introduced the energy-momentum pseudo-tensor and then Landau-Lifshitz [15] , Papapetrou [16] , Bergmann [17] , Tolman [18] and Weinberg [19] proposed their own prescriptions to resolve this issue. All these prescriptions can provide meaningful results only in Cartesian coordinates. But Möller [20] introduced a coordinate-independent prescription. The idea of coordinate-independent quasi-local mass was introduced [21] by associ-ating a Hamiltonian term to each gravitational energy-momentum pseudotensor. Later, a Hamiltonian approach in the frame of Schwinger condition [22] was developed, followed by the construction of a Lagrangian density of TP equivalent to GR [4, 6, 23, 24] . Many authors explored several examples in the framework of GR and found that different energy-momentum complexes can give either the same [25] or different [26] results for a given spacetime.
Mikhail et al. [27] defined the superpotential in the Moller's tetrad theory which has been used to find the energy in TPG. Vargas [28] defined the TP version of Bergman, Einstein and Landau-Lifshitz prescriptions and found that the total energy of the closed Friedman-Robinson-Walker universe is zero by using the last two prescriptions. This agrees with the results of GR available in literature [29] . Later, many authors [30] used TP version of these prescriptions and showed that energy may be localized in TPG. Keeping these points in mind, this paper is addressed to the following two problems: We obtain TP version of the stationary axisymmetric solutions and then calculate the axial-vector part of the torsion tensor. The energymomentum distribution of the solutions is explored by using the TP version of Möller prescription.
The scheme adopted in this paper is as follows. In section 2, we shall review the basic concepts of TP theory. Section 3 contains the TP version of the stationary axisymmetric solutions and the tensor, vector and axial-vector parts of the torsion tensor. Section 4 is devoted to evaluate the energymomentum distribution for this family of solutions using the TP version of Möller prescription. In section 5, we present two special solutions for this class of metrics and investigate the corresponding quantities. The last section contains a summary and a discussion of the results obtained.
In teleparallel theory, the connection is a Weitzenböck connection given as [31] Γ
θ µν = h a θ ∂ ν h a µ , (1)
where h a ν is a non-trivial tetrad. Its inverse field is denoted by h a µ and satisfy the relations
h a µ h a ν = δ µ ν ; h a µ h b µ = δ a b . (2)
In this paper, the Latin alphabet (a, b, c, ... = 0, 1, 2, 3) will be used to denote tangent space indices and the Greek alphabet (µ, ν, ρ, ... = 0, 1, 2, 3) to denote spacetime indices. The Riemannian metric in TP theory arises as a by product [4] of the tetrad field given by
g µν = η ab h a µ h b ν , (3)
where η ab is the Minkowski metric η ab = diag(+1, -1, -1, -1). For the Weitzenböck spacetime, the torsion is defined as [2]
T θ µν = Γ θ νµ -Γ θ µν (4)
which is antisymmetric w.r.t. its last two indices. Due to the requirement of absolute parallelism, the curvature of the Weitzenböck connection vanishes identically. The Weitzenböck connection also satisfies the relation
Γ 0 θ µν = Γ θ µν -K θ µν , (5)
where
K θ µν = 1 2 [T µ θ ν + T ν θ µ -T θ µν ] (6)
is the contortion tensor and Γ 0 θ µν are the Christoffel symbols in GR. The torsion tensor of the Weitzenböck connection can be decomposed into three irreducible parts under the group of global Lorentz transformations [4] : the tensor part
t λµν = 1 2 (T λµν + T µλν ) + 1 6 (g νλ V µ + g νµ V λ ) - 1 3 g λµ V ν , (7)
the vector part
V µ = T ν νµ ( 8
)
and the axial-vector part
A µ = 1 6 ǫ µνρσ T νρσ . (9)
The torsion tensor can now be expressed in terms of these irreducible components as follows
T λµν = 1 2 (t λµν -t λνµ ) + 1 3 (g λµ V ν -g λν V µ ) + ǫ λµνρ A ρ , (10)
where
ǫ λµνρ = 1 √ -g δ λµνρ . (11)
Here δ = {δ λµνρ } and δ * = {δ λµνρ } are completely skew symmetric tensor densities of weight -1 and +1 respectively [4] . TP theory provides an alternate description of the Einstein's field equations which is given by the teleparallel equivalent of GR [24, 31] . Mikhail et al. [27] defined the super-potential (which is antisymmetric in its last two indices) of the Möller tetrad theory as
U µ νβ = √ -g 2κ P τ νβ χρσ [V ρ g σχ g µτ -λg τ µ K χρσ -(1 -2λ)g τ µ K σρχ ], (12)
where
P τ νβ χρσ = δ χ τ g νβ ρσ + δ ρ τ g νβ σχ -δ σ τ g νβ χρ ( 13
)
and g νβ ρσ is a tensor quantity defined by
g νβ ρσ = δ ρ ν δ σ β -δ σ ν δ ρ β . ( 14
)
K σρχ is the contortion tensor given by Eq.( 6 ), g is the determinant of the metric tensor g µν , λ is the free dimensionless coupling constant of TPG, κ is the Einstein constant and V µ is the basic vector field given by Eq.( 8 ). The energy-momentum density is defined as
Ξ ν µ = U νρ µ , ρ , (15)
where comma means ordinary differentiation. The momentum 4-vector of Möller prescription can be expressed as
P µ = Σ Ξ 0 µ dxdydz, (16)
where P 0 gives the energy and P 1 , P 2 and P 3 are the momentum components while the integration is taken over the hypersurface element Σ described by x 0 = t = constant. The energy may be given in the form of surface integral [20] as
E = lim r→∞ r=constant U 0 0ρ u ρ dS, (17)
where u ρ is the unit three-vector normal to the surface element dS.
Tupper [32] found five classes of non-null electromagnetic field plus perfect fluid solutions in which the electromagnetic field does not inherit one of the symmetries of the spacetime. The metric representing the stationary axisymmetric solutions is given by [32 ]
ds 2 = dt 2 -e 2K dρ 2 -(F 2 -B 2 )dφ 2 -e 2K dz 2 + 2Bdtdφ, (18)
where B = B(ρ, z), K = K(ρ, z) and F = F (ρ) are such functions which satisfy the following relations . The metric given by Eq.( 18 ) represents five classes of non-null electromagnetic field and perfect fluid solutions which possesses a metric symmetry not inherited by the electromagnetic field and admits a homothetic vector field. Two of these classes contain electrovac solutions as special cases, while the other three necessarily contain some fluid. The generalization of this metric is given in [34] .
Ḃ = F W ′ , B ′ = - 1 4 aF ( Ẇ 2 -W ′2 ), K ′ = - 1 2 aF Ẇ W ′ , Ẅ + Ḟ F -1 Ẇ + W ′′ = 0, (19)
Using the procedure adopted in the papers [11, 12] , the tetrad components of the above metric can be written as
h a µ = 1 0 B 0 0 e K cos φ -F sin φ 0 0 e K sin φ F cos φ 0 0 0 0 e K (20)
with its inverse
h a µ = 1 0 0 0 B F sin φ e -K cos φ -1 F sin φ 0 -B F cos φ e -K sin φ 1 F cos φ 0 0 0 0 e -K . ( 21
)
The non-vanishing components of the torsion tensor are
T 0 12 = Ḃ + B F (e K -Ḟ ), T 0 32 = B ′ , T 1 13 = -K ′ , T 2 12 = 1 F ( Ḟ -e K ), T 3 31 = -K. ( 22
)
Using these expressions in Eqs.( 7 )-( 9 ), we obtain the following non-zero components of the tensor part
t 001 = 1 3 [ K + 1 F ( Ḟ -e K )], t 003 = 1 3 K ′ , t 010 = 1 6 { 1 F (e K -Ḟ ) -K} = t 100 , t 030 = - 1 6 K ′ = t 300 , t 012 = 1 2 Ḃ + B 6 { 1 F (e K -Ḟ ) -K} = t 102 , t 021 = - 1 2 Ḃ - B 3 { 1 F (e K -Ḟ ) -K} = t 201 , t 023 = - 1 2 B ′ + 1 3 BK ′ = t 203 , t 032 = 1 2 B ′ - 1 6 BK ′ = t 302 , t 122 = 1 2 {F (e K -Ḟ ) + B Ḃ} + 1 6 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K} = t 212 , t 120 = B 6 { 1 F (e K -Ḟ ) -K} = t 210 , t 113 = 2K ′ 3 e 2K , t 131 = - K ′ 3 e 2K = t 311 , t 133 = - e 2K 6 { 1 F (e K -Ḟ ) + 2 K} = t 313 , t 221 = -F (e K -Ḟ ) -B Ḃ - 1 3 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K}, t 223 = -BB ′ + K ′ 3 (B 2 -F 2 ), t 331 = e 2K 3 { 1 F (e K -Ḟ ) + 2 K}, t 322 = 1 2 BB ′ - K ′ 6 (B 2 -F 2 ) = t 232 , t 320 = - 1 6 BK ′ = t 232 , (23)
the vector part
V 1 = - 1 F ( Ḟ -e K ) -K, (24)
V 3 = -K ′ , (25)
and the axial-vector part
A 1 = B ′ 3F e -2K , (26)
A 3 = Ḃ 3F e -2K , (27)
respectively. The axial-vector component along the φ-direction vanishes and hence the spacelike axial-vector can be written as
A = √ -g 11 A 1 êρ + √ -g 33 A 3 êz , (28)
where êρ and êz are unit vectors along the radial and z-directions respectively. Substituting A 1 , A 3 , g 11 and g 33 in Eq.( 28 ), it follows that
A = e -K 3F (B ′ êρ + Ḃê z ). ( 29
)
This shows that the axial-vector lies along radial direction if B = B(z), along z-direction if B = B(ρ) and vanishes identically if B is constant. As the axialvector torsion represents the deviation of axial symmetry from cylindrical symmetry, the symmetry of the underlying spacetime will not be affected even for B constant. Also, the torsion plays the role of the gravitational force in TP theory, hence a spinless particle will obey the force equation [11, 24]
du ρ ds -Γ µρν u µ u ν = T µρν u µ u ν . ( 30
)
The left hand side of this equation is the Weitzenböck covariant derivative of u ρ along the particle world-line. The appearance of the torsion tensor on its right hand side indicates that the torsion plays the role of an external force in TPG. It has been shown, both in GR and TP theories, by many authors [4, 35] that the spin precession of a Dirac particle in torsion gravity is related to the torsion axial-vector by
dS dt = -b × S, ( 31
)
where S is the spin vector of a Dirac particle and b = 3 2 A, with A the spacelike part of the torsion axial-vector. Thus
b = e -K 2F {B ′ êρ + Ḃê z }. (32)
The corresponding extra Hamiltonian [36] is given by
δH = -b.σ, ( 33
)
where σ is the spin of the particle [35] . Using Eq.( 32 ), this takes the form
δH = - e -K 2F (B ′ êρ + Ḃê z ).σ. ( 34
)
4 Teleparallel Energy of the Stationary Axisymmetric Solutions
In this section we evaluate the component of energy-momentum densities by using the teleparallel version of Möller prescription. Multiplying Eqs.( 24 ) and ( 25 ) by g 11 and g 33 respectively, it follows that
V 1 = Ke -2K + e -2K F ( Ḟ -e K ), (35)
V 3 = K ′ e -2K . (36)
In view of Eqs.( 6 ) and ( 22 ), the non-vanishing components of the contorsion tensor are
K 100 = -e -2K { B 2 F 3 (e K -Ḟ ) + B Ḃ F 2 } = -K 010 , K 300 = - BB ′ F 2 e -2K = -K 030 , K 122 = - e -2K F 3 (e K -Ḟ ) = -K 212 , K 133 = Ke -4K = -K 313 , K 311 = K ′ e -4K = -K 131 , K 102 = K 120 = e -2K { B F 3 (e K -Ḟ ) + Ḃ 2F 2 } = -K 012 = -K 210 , K 302 = K 320 = K 023 = B ′ 2F 2 e -2K = -K 032 = -K 230 = -K 203 , K 021 = Ḃ 2F 2 e -2K = -K 201 . ( 37
)
It should be mentioned here that the contorsion tensor is antisymmetric w.r.t. its first two indices. Making use of Eqs.( 35 )- (37) in Eq.( 12 ), we obtain the required independent non-vanishing components of the supperpotential in Möller's tetrad theory as
U 01 0 = 1 κ [e K -Ḟ -F K + 1 2 (1 + λ) B Ḃ F ] = -U 10 0 , U 03 0 = 1 κ [-F K ′ + 1 2 (1 + λ) BB ′ F ] = -U 30 0 , U 21 0 = - 1 2κ (1 + λ) Ḃ F = -U 12 0 , U 23 0 = - 1 2κ (1 + λ) B ′ F = -U 32 0 , U 01 2 = 1 κ [B(e K -Ḟ ) + 1 2 (1 + λ) B 2 Ḃ F + 1 2 (1 -λ) ḂF ] = -U 10 2 , U 03 2 = 1 κ [ 1 2 (1 + λ) B 2 B ′ F + 1 2 (1 -λ)B ′ F ] = -U 30 2 , U 02 1 = 1 2κF (λ -1) Ḃe 2K = -U 20 1 , U 02 3 = 1 2κF (λ -1)B ′ e 2K = -U 30 1 . (38)
It is worth mentioning here that the supperpotential is skew symmetric w.r.t. its last two indices. When we make use of Eqs.( 15 ), ( 37 ), (38) and take λ = 1, the energy density turns out to be
Ξ 0 0 = 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ ) + 1 F 2 {BF ( B + E ′′ ) + ( Ḃ2 + B ′ 2 )F -B Ḃ Ḟ }]. ( 39
)
This implies that
E d T P T = E d GR + 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ )], (40)
where E d stands for energy density. The only non-zero component of momentum density is along φ-direction and (for λ = 1) it takes the form
Ξ 0 2 = 1 κF 2 {F 3 ( B + B ′′ ) + B 2 F ( B + B ′′ ) + 2BF ( Ḃ2 + B ′ 2 ) -Ḃ Ḟ (B 2 + F 2 )} + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (41)
that is,
M d T P T = M d GR + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (42)
where M d stands for momentum density.
In this section, we evaluate the above quantities for some special cases of the non-null Einstein Maxwell solutions.
A special case of the non-null Einstein-Maxwell solutions can be obtained by choosing
B = m n e nρ , F = e nρ , K = 0. ( 43
)
This is known as electromagnetic generalization of the Gödel solution [32] .
When we make use of Eq.( 43 ) in Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42), the corresponding results reduce to
t 001 = 1 3 (n -e -nρ ), t 010 = 1 6 (e -nρ -n) = t 100 , t 012 = m 6n (1 + 2ne nρ ) = t 102 , t 021 = - m 3n (1 + 2ne nρ ) = t 201 , t 122 = e nρ 6n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ } = t 212 , t 120 = m 6n (1 -ne nρ ) = t 210 , t 133 = 1 6 (n -e -nρ ) = t 313 , t 221 = - e nρ 3n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ },
t 331 = - 1 3 (n -e -nρ ), (44)
V 1 = e -nρ -n, V 3 = 0, (45)
A 1 = 0, A 3 = m 3 , (46)
A = m 3 êz , b = m 2 êz , (47)
δH = m 2 êz .σ, (48)
Ξ 0 0 = 1 κ (m 2 -n 2 )e nρ , (49)
E d T P T = E d GR - n 2 κ e nρ (50)
Ξ 0 2 = 1 κ ( 2m 3 n ) + m κ (1 -2ne nρ )e nρ , (51)
M d T P T = M d GR + m κ (1 -2ne nρ )e nρ . (52)
The metric (43) reduces to the usual perfect fluid solution when m = √ 2n [32] , i.e., B = √ 2e nρ . The corresponding energy and momentum densities take the form as
E d T P T = E d GR - n 2 κ e nρ (53)
M d T P T = M d GR + √ 2n κ (1 -2ne nρ )e nρ . (54)
When we choose B = e aρ , F = e aρ √ 2 and K = 0, the metric given by Eq.( 18 ) reduces to the Gödel metric [32] . The results corresponding to Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42) take the following form
t 001 = 1 3 (a - √ 2e -aρ
), t 010 = -1 6 (a -√ 2e -aρ ) = t 100 , t 012 = 1 6 ( √ 2 + 2ae aρ ) = t 102 , t 021 = -1 6 (2 √ 2 + ae aρ ) = t 102 , t 122 = e aρ 6 (2 √ 2 + ae aρ ) = t 212 , t 120 = 1 6 ( √ 2ae aρ ) = t 210 , t 133 = 1 6 (a -√ 2e -aρ ) = t 313 , t 221 = -e aρ 3 (2 √ 2 + ae aρ ), (55)
t 331 = 1 3 (a - √ 2e -aρ ), ( 56
) V 1 = √ 2e -aρ -a, V 3 = 0, (57)
A 1 = 0, A 3 = √ 2a 3 , (58)
A = √ 2a 3 êz , b = a √ 2 êz , (59)
δH = a √ 2 êz .σ, (60)
Ξ 0 0 = √ 2 κ a 2 e aρ - a 2 κ √ 2 e aρ , (61)
E d T P T = E d GR - a 2 κ √ 2 e aρ (62) Ξ 0 2 = a 2 κ √ 2 e 2aρ + a κ (1 - √ 2ae aρ )e aρ , (63)
M d T P T = M d GR + a κ (1 - √ 2ae aρ )e aρ . ( 64
)
The purpose of this paper is twofold: Firstly, we have found the TP version of the non-null Einstein Maxwell solutions. This provides some interesting features about the axial vector and the corresponding quantities. Secondly, we have used the TP version of Möller prescription to evaluate the energymomentum distribution of the solutions. The axial-vector torsion of these solutions has been evaluated. The only non-vanishing components of the vector part are along the radial and the z-directions due to the cross term dx 0 dx 2 involving in the metric. This corresponds to the case of Kerr metric [11] , which involves the cross term dx 0 dx 3 . We also find the vector b which is related to the spin vector [4] as given by Eq.( 32 ). The axial-vector torsion lies in the ρz-plane, as its component along the φ-direction vanishes everywhere.
The non-inertial force on the Dirac particle can be represented as a rotation induced torsion of spacetime. There arise three possibilities for the axial-vector, depending upon the metric function B(ρ, z). When B is a function of z only, the axial-vector lies only along the radial direction. When B is a function of ρ only, the axialvector will lie along z-direction. The axial-vector vanishes identically for B to be constant. As the axial-vector represents the deviation from the symmetry of the underlying spacetime corresponding to an inertial field with respect to the Dirac particle, the symmetry of the spacetime will not be affected in the third possibility. Consequently there exists no inertial field with respect to the Dirac particle and the spin vector of the Dirac particle becomes constant. The corresponding extra Hamiltonian is expressed in terms of the vector b which vanishes when the metric function B is constant, i.e., when the axialvector becomes zero.
The energy-momentum distribution of the non-null Einstein-Maxwell solutions has been explored by using the TP version of Möller prescription. It is found that energy in the TP theory is equal to the energy in GR (as found by Sharif and Fatima [37] ) plus some additional part. If, for a particular case, we have K = 0 and K ′ , Ḟ = constant (or if Ḟ , K = 0 and K
′ = constant), then E d T P G = E d GR . (65)
On the other hand, the only non-vanishing component of the momentum density lies along φ-direction, similar to the case of Kerr metric [11] , due to the cross term appearing in both the metrics. When we choose λ = 1, it becomes equal to be the momentum in GR [37] plus an additional quantity. If F , Ḃ, B ′′ , K all vanish, then
M d T P G = M d GR . (66)
By taking particular values of E, F and K, we obtain the electromagnetic generalization of the Gödel solution and the Gödel metric as two special cases. The corresponding results for both the special cases are obtained. It is shown that, for the electromagnetic generalization of the Gödel solution, Eq.(65) does not hold, while Eq.(66) holds when m = 0. However, for the perfect fluid case, i.e., when m = √ 2n, both Eqs.( 65 ) and (66) hold by taking n = 0. In the case of the Gödel metric, these equations hold if we choose the arbitrary constant a = 0. For the special solutions, the vector part lies along the radial direction while the axial-vector part along z-direction.
We would like to re-iterate here that the tetrad formalism itself has some advantages which comes mainly from its independence from the equivalence principle and consequent suitability to the discussion of quantum issues. In TPG, an energy-momentum gauge current j i µ for the gravitational field can be defined. This is covariant under a spacetime general coordinate transformation and transforms covariantly under a global tangent space Lorentz transformation [38] . It, then, follows that j i µ is a true spacetime tensor but not a tangent space tensor. When we re-write the gauge field equations in a purely spacetime form, they lead to the Einstein field equations and the gauge current j i µ reduces to the canonical energy-momentum pseudo-tensor of the gravitational field. Thus TPG seems to provide a more appropriate environment to deal with the energy problem.
Finally, it is pointed out that we are not claiming that this paper has resolved the problems of GR using the TPG. This is an attempt to touch some issues in TPG with the hope that this alternative may provide more feasible results. Also, it is always an interesting and enriching to look at things from another point of view. This endeavor is in itself commendable. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along ρ and z directions. The three possibilities of the axial vector depending on the metric function B are discussed. The vector related with spin has also been evaluated and the corresponding extra Hamiltonian is furnished. Further, we use the teleparallel version of Möller prescription to find the energy-momentum distribution of the solutions. It is interesting to note that (for λ = 1) energy and momentum densities in teleparallel theory are equal to the corresponding quantities in GR plus an additional quantity in each, which may become equal under certain conditions. Finally, we discuss the two special cases of the stationary axisymmetric solutions."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The attempts made by Einstein and his followers to unify gravitation with other interactions led to the investigation of structures of gravitation other than the metric tensor. These structures yield the metric tensor as a by product. Tetrad field is one of these structures which leads to the theory of teleparallel gravity (TPG) [1, 2] . TPG is an alternative theory of gravity which corresponds to a gauge theory of translation group [3, 4] based on Weitzenböck geometry [5] . This theory is characterized by the vanishing of curvature identically while the torsion is taken to be non-zero. In TPG, gravitation is attributed to torsion [4] which plays a role of force [6] . In General Relativity (GR), gravitation geometrizes the underlying spacetime. The translational gauge potentials appear as a non-trivial part of the tetrad field and induce a teleparallel (TP) structure on spacetime which is directly related to the presence of a gravitational field. In some other theories [3] [4] [5] [6] [7] [8] , torsion is only relevant when spins are important [9] . This point of view indicates that torsion might represent additional degrees of freedom as compared to curvature. As a result, some new physics may be associated with it. Teleparallelism is naturally formulated by gauging external (spacetime) translations which are closely related to the group of general coordinate transformations underlying GR. Thus the energy-momentum tensor represents the matter source in the field equations of tetradic theories of gravity like in GR.\n\nThere is a large literature available [10] about the study of TP versions of the exact solutions of GR. Recently, Pereira, et al. [11] obtained the TP versions of the Schwarzschild and the stationary axisymmetric Kerr solutions of GR. They proved that the axial-vector torsion plays the role of the gravitomagnetic component of the gravitational field in the case of slow rotation and weak field approximations. In a previous paper [12] , we have found the TP versions of the Friedmann models and of the Lewis-Papapetrou spacetimes, and also discussed their axial-vectors.\n\nThere has been a longstanding, controversial and still unresolved problem of the localization of energy (i.e., to express it as a unique tensor quantity) in GR [13] . Einstein [14] introduced the energy-momentum pseudo-tensor and then Landau-Lifshitz [15] , Papapetrou [16] , Bergmann [17] , Tolman [18] and Weinberg [19] proposed their own prescriptions to resolve this issue. All these prescriptions can provide meaningful results only in Cartesian coordinates. But Möller [20] introduced a coordinate-independent prescription. The idea of coordinate-independent quasi-local mass was introduced [21] by associ-ating a Hamiltonian term to each gravitational energy-momentum pseudotensor. Later, a Hamiltonian approach in the frame of Schwinger condition [22] was developed, followed by the construction of a Lagrangian density of TP equivalent to GR [4, 6, 23, 24] . Many authors explored several examples in the framework of GR and found that different energy-momentum complexes can give either the same [25] or different [26] results for a given spacetime.\n\nMikhail et al. [27] defined the superpotential in the Moller's tetrad theory which has been used to find the energy in TPG. Vargas [28] defined the TP version of Bergman, Einstein and Landau-Lifshitz prescriptions and found that the total energy of the closed Friedman-Robinson-Walker universe is zero by using the last two prescriptions. This agrees with the results of GR available in literature [29] . Later, many authors [30] used TP version of these prescriptions and showed that energy may be localized in TPG. Keeping these points in mind, this paper is addressed to the following two problems: We obtain TP version of the stationary axisymmetric solutions and then calculate the axial-vector part of the torsion tensor. The energymomentum distribution of the solutions is explored by using the TP version of Möller prescription.\n\nThe scheme adopted in this paper is as follows. In section 2, we shall review the basic concepts of TP theory. Section 3 contains the TP version of the stationary axisymmetric solutions and the tensor, vector and axial-vector parts of the torsion tensor. Section 4 is devoted to evaluate the energymomentum distribution for this family of solutions using the TP version of Möller prescription. In section 5, we present two special solutions for this class of metrics and investigate the corresponding quantities. The last section contains a summary and a discussion of the results obtained."
},
{
"section_type": "OTHER",
"section_title": "An Overview of the Teleparallel Theory",
"text": "In teleparallel theory, the connection is a Weitzenböck connection given as [31] Γ\n\nθ µν = h a θ ∂ ν h a µ , (1)\n\nwhere h a ν is a non-trivial tetrad. Its inverse field is denoted by h a µ and satisfy the relations\n\nh a µ h a ν = δ µ ν ; h a µ h b µ = δ a b . (2)\n\nIn this paper, the Latin alphabet (a, b, c, ... = 0, 1, 2, 3) will be used to denote tangent space indices and the Greek alphabet (µ, ν, ρ, ... = 0, 1, 2, 3) to denote spacetime indices. The Riemannian metric in TP theory arises as a by product [4] of the tetrad field given by\n\ng µν = η ab h a µ h b ν , (3)\n\nwhere η ab is the Minkowski metric η ab = diag(+1, -1, -1, -1). For the Weitzenböck spacetime, the torsion is defined as [2]\n\nT θ µν = Γ θ νµ -Γ θ µν (4)\n\nwhich is antisymmetric w.r.t. its last two indices. Due to the requirement of absolute parallelism, the curvature of the Weitzenböck connection vanishes identically. The Weitzenböck connection also satisfies the relation\n\nΓ 0 θ µν = Γ θ µν -K θ µν , (5)\n\nwhere\n\nK θ µν = 1 2 [T µ θ ν + T ν θ µ -T θ µν ] (6)\n\nis the contortion tensor and Γ 0 θ µν are the Christoffel symbols in GR. The torsion tensor of the Weitzenböck connection can be decomposed into three irreducible parts under the group of global Lorentz transformations [4] : the tensor part\n\nt λµν = 1 2 (T λµν + T µλν ) + 1 6 (g νλ V µ + g νµ V λ ) - 1 3 g λµ V ν , (7)\n\nthe vector part\n\nV µ = T ν νµ ( 8\n\n)\n\nand the axial-vector part\n\nA µ = 1 6 ǫ µνρσ T νρσ . (9)\n\nThe torsion tensor can now be expressed in terms of these irreducible components as follows\n\nT λµν = 1 2 (t λµν -t λνµ ) + 1 3 (g λµ V ν -g λν V µ ) + ǫ λµνρ A ρ , (10)\n\nwhere\n\nǫ λµνρ = 1 √ -g δ λµνρ . (11)\n\nHere δ = {δ λµνρ } and δ * = {δ λµνρ } are completely skew symmetric tensor densities of weight -1 and +1 respectively [4] . TP theory provides an alternate description of the Einstein's field equations which is given by the teleparallel equivalent of GR [24, 31] . Mikhail et al. [27] defined the super-potential (which is antisymmetric in its last two indices) of the Möller tetrad theory as\n\nU µ νβ = √ -g 2κ P τ νβ χρσ [V ρ g σχ g µτ -λg τ µ K χρσ -(1 -2λ)g τ µ K σρχ ], (12)\n\nwhere\n\nP τ νβ χρσ = δ χ τ g νβ ρσ + δ ρ τ g νβ σχ -δ σ τ g νβ χρ ( 13\n\n)\n\nand g νβ ρσ is a tensor quantity defined by\n\ng νβ ρσ = δ ρ ν δ σ β -δ σ ν δ ρ β . ( 14\n\n)\n\nK σρχ is the contortion tensor given by Eq.( 6 ), g is the determinant of the metric tensor g µν , λ is the free dimensionless coupling constant of TPG, κ is the Einstein constant and V µ is the basic vector field given by Eq.( 8 ). The energy-momentum density is defined as\n\nΞ ν µ = U νρ µ , ρ , (15)\n\nwhere comma means ordinary differentiation. The momentum 4-vector of Möller prescription can be expressed as\n\nP µ = Σ Ξ 0 µ dxdydz, (16)\n\nwhere P 0 gives the energy and P 1 , P 2 and P 3 are the momentum components while the integration is taken over the hypersurface element Σ described by x 0 = t = constant. The energy may be given in the form of surface integral [20] as\n\nE = lim r→∞ r=constant U 0 0ρ u ρ dS, (17)\n\nwhere u ρ is the unit three-vector normal to the surface element dS."
},
{
"section_type": "OTHER",
"section_title": "Teleparallel Version of the Stationary Axisymmetric Solutions",
"text": "Tupper [32] found five classes of non-null electromagnetic field plus perfect fluid solutions in which the electromagnetic field does not inherit one of the symmetries of the spacetime. The metric representing the stationary axisymmetric solutions is given by [32 ]\n\nds 2 = dt 2 -e 2K dρ 2 -(F 2 -B 2 )dφ 2 -e 2K dz 2 + 2Bdtdφ, (18)\n\nwhere B = B(ρ, z), K = K(ρ, z) and F = F (ρ) are such functions which satisfy the following relations . The metric given by Eq.( 18 ) represents five classes of non-null electromagnetic field and perfect fluid solutions which possesses a metric symmetry not inherited by the electromagnetic field and admits a homothetic vector field. Two of these classes contain electrovac solutions as special cases, while the other three necessarily contain some fluid. The generalization of this metric is given in [34] .\n\nḂ = F W ′ , B ′ = - 1 4 aF ( Ẇ 2 -W ′2 ), K ′ = - 1 2 aF Ẇ W ′ , Ẅ + Ḟ F -1 Ẇ + W ′′ = 0, (19)\n\nUsing the procedure adopted in the papers [11, 12] , the tetrad components of the above metric can be written as\n\nh a µ = 1 0 B 0 0 e K cos φ -F sin φ 0 0 e K sin φ F cos φ 0 0 0 0 e K (20)\n\nwith its inverse\n\nh a µ = 1 0 0 0 B F sin φ e -K cos φ -1 F sin φ 0 -B F cos φ e -K sin φ 1 F cos φ 0 0 0 0 e -K . ( 21\n\n)\n\nThe non-vanishing components of the torsion tensor are\n\nT 0 12 = Ḃ + B F (e K -Ḟ ), T 0 32 = B ′ , T 1 13 = -K ′ , T 2 12 = 1 F ( Ḟ -e K ), T 3 31 = -K. ( 22\n\n)\n\nUsing these expressions in Eqs.( 7 )-( 9 ), we obtain the following non-zero components of the tensor part\n\nt 001 = 1 3 [ K + 1 F ( Ḟ -e K )], t 003 = 1 3 K ′ , t 010 = 1 6 { 1 F (e K -Ḟ ) -K} = t 100 , t 030 = - 1 6 K ′ = t 300 , t 012 = 1 2 Ḃ + B 6 { 1 F (e K -Ḟ ) -K} = t 102 , t 021 = - 1 2 Ḃ - B 3 { 1 F (e K -Ḟ ) -K} = t 201 , t 023 = - 1 2 B ′ + 1 3 BK ′ = t 203 , t 032 = 1 2 B ′ - 1 6 BK ′ = t 302 , t 122 = 1 2 {F (e K -Ḟ ) + B Ḃ} + 1 6 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K} = t 212 , t 120 = B 6 { 1 F (e K -Ḟ ) -K} = t 210 , t 113 = 2K ′ 3 e 2K , t 131 = - K ′ 3 e 2K = t 311 , t 133 = - e 2K 6 { 1 F (e K -Ḟ ) + 2 K} = t 313 , t 221 = -F (e K -Ḟ ) -B Ḃ - 1 3 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K}, t 223 = -BB ′ + K ′ 3 (B 2 -F 2 ), t 331 = e 2K 3 { 1 F (e K -Ḟ ) + 2 K}, t 322 = 1 2 BB ′ - K ′ 6 (B 2 -F 2 ) = t 232 , t 320 = - 1 6 BK ′ = t 232 , (23)\n\nthe vector part\n\nV 1 = - 1 F ( Ḟ -e K ) -K, (24)\n\nV 3 = -K ′ , (25)\n\nand the axial-vector part\n\nA 1 = B ′ 3F e -2K , (26)\n\nA 3 = Ḃ 3F e -2K , (27)\n\nrespectively. The axial-vector component along the φ-direction vanishes and hence the spacelike axial-vector can be written as\n\nA = √ -g 11 A 1 êρ + √ -g 33 A 3 êz , (28)\n\nwhere êρ and êz are unit vectors along the radial and z-directions respectively. Substituting A 1 , A 3 , g 11 and g 33 in Eq.( 28 ), it follows that\n\nA = e -K 3F (B ′ êρ + Ḃê z ). ( 29\n\n)\n\nThis shows that the axial-vector lies along radial direction if B = B(z), along z-direction if B = B(ρ) and vanishes identically if B is constant. As the axialvector torsion represents the deviation of axial symmetry from cylindrical symmetry, the symmetry of the underlying spacetime will not be affected even for B constant. Also, the torsion plays the role of the gravitational force in TP theory, hence a spinless particle will obey the force equation [11, 24]\n\ndu ρ ds -Γ µρν u µ u ν = T µρν u µ u ν . ( 30\n\n)\n\nThe left hand side of this equation is the Weitzenböck covariant derivative of u ρ along the particle world-line. The appearance of the torsion tensor on its right hand side indicates that the torsion plays the role of an external force in TPG. It has been shown, both in GR and TP theories, by many authors [4, 35] that the spin precession of a Dirac particle in torsion gravity is related to the torsion axial-vector by\n\ndS dt = -b × S, ( 31\n\n)\n\nwhere S is the spin vector of a Dirac particle and b = 3 2 A, with A the spacelike part of the torsion axial-vector. Thus\n\nb = e -K 2F {B ′ êρ + Ḃê z }. (32)\n\nThe corresponding extra Hamiltonian [36] is given by\n\nδH = -b.σ, ( 33\n\n)\n\nwhere σ is the spin of the particle [35] . Using Eq.( 32 ), this takes the form\n\nδH = - e -K 2F (B ′ êρ + Ḃê z ).σ. ( 34\n\n)\n\n4 Teleparallel Energy of the Stationary Axisymmetric Solutions\n\nIn this section we evaluate the component of energy-momentum densities by using the teleparallel version of Möller prescription. Multiplying Eqs.( 24 ) and ( 25 ) by g 11 and g 33 respectively, it follows that\n\nV 1 = Ke -2K + e -2K F ( Ḟ -e K ), (35)\n\nV 3 = K ′ e -2K . (36)\n\nIn view of Eqs.( 6 ) and ( 22 ), the non-vanishing components of the contorsion tensor are\n\nK 100 = -e -2K { B 2 F 3 (e K -Ḟ ) + B Ḃ F 2 } = -K 010 , K 300 = - BB ′ F 2 e -2K = -K 030 , K 122 = - e -2K F 3 (e K -Ḟ ) = -K 212 , K 133 = Ke -4K = -K 313 , K 311 = K ′ e -4K = -K 131 , K 102 = K 120 = e -2K { B F 3 (e K -Ḟ ) + Ḃ 2F 2 } = -K 012 = -K 210 , K 302 = K 320 = K 023 = B ′ 2F 2 e -2K = -K 032 = -K 230 = -K 203 , K 021 = Ḃ 2F 2 e -2K = -K 201 . ( 37\n\n)\n\nIt should be mentioned here that the contorsion tensor is antisymmetric w.r.t. its first two indices. Making use of Eqs.( 35 )- (37) in Eq.( 12 ), we obtain the required independent non-vanishing components of the supperpotential in Möller's tetrad theory as\n\nU 01 0 = 1 κ [e K -Ḟ -F K + 1 2 (1 + λ) B Ḃ F ] = -U 10 0 , U 03 0 = 1 κ [-F K ′ + 1 2 (1 + λ) BB ′ F ] = -U 30 0 , U 21 0 = - 1 2κ (1 + λ) Ḃ F = -U 12 0 , U 23 0 = - 1 2κ (1 + λ) B ′ F = -U 32 0 , U 01 2 = 1 κ [B(e K -Ḟ ) + 1 2 (1 + λ) B 2 Ḃ F + 1 2 (1 -λ) ḂF ] = -U 10 2 , U 03 2 = 1 κ [ 1 2 (1 + λ) B 2 B ′ F + 1 2 (1 -λ)B ′ F ] = -U 30 2 , U 02 1 = 1 2κF (λ -1) Ḃe 2K = -U 20 1 , U 02 3 = 1 2κF (λ -1)B ′ e 2K = -U 30 1 . (38)\n\nIt is worth mentioning here that the supperpotential is skew symmetric w.r.t. its last two indices. When we make use of Eqs.( 15 ), ( 37 ), (38) and take λ = 1, the energy density turns out to be\n\nΞ 0 0 = 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ ) + 1 F 2 {BF ( B + E ′′ ) + ( Ḃ2 + B ′ 2 )F -B Ḃ Ḟ }]. ( 39\n\n)\n\nThis implies that\n\nE d T P T = E d GR + 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ )], (40)\n\nwhere E d stands for energy density. The only non-zero component of momentum density is along φ-direction and (for λ = 1) it takes the form\n\nΞ 0 2 = 1 κF 2 {F 3 ( B + B ′′ ) + B 2 F ( B + B ′′ ) + 2BF ( Ḃ2 + B ′ 2 ) -Ḃ Ḟ (B 2 + F 2 )} + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (41)\n\nthat is,\n\nM d T P T = M d GR + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (42)\n\nwhere M d stands for momentum density."
},
{
"section_type": "OTHER",
"section_title": "Special Solutions of the Non-Null Einstein Maxwell Solutions",
"text": "In this section, we evaluate the above quantities for some special cases of the non-null Einstein Maxwell solutions."
},
{
"section_type": "OTHER",
"section_title": "Electromagnetic Generalization of the Gödel Solution",
"text": "A special case of the non-null Einstein-Maxwell solutions can be obtained by choosing\n\nB = m n e nρ , F = e nρ , K = 0. ( 43\n\n)\n\nThis is known as electromagnetic generalization of the Gödel solution [32] .\n\nWhen we make use of Eq.( 43 ) in Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42), the corresponding results reduce to\n\nt 001 = 1 3 (n -e -nρ ), t 010 = 1 6 (e -nρ -n) = t 100 , t 012 = m 6n (1 + 2ne nρ ) = t 102 , t 021 = - m 3n (1 + 2ne nρ ) = t 201 , t 122 = e nρ 6n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ } = t 212 , t 120 = m 6n (1 -ne nρ ) = t 210 , t 133 = 1 6 (n -e -nρ ) = t 313 , t 221 = - e nρ 3n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ },\n\nt 331 = - 1 3 (n -e -nρ ), (44)\n\nV 1 = e -nρ -n, V 3 = 0, (45)\n\nA 1 = 0, A 3 = m 3 , (46)\n\nA = m 3 êz , b = m 2 êz , (47)\n\nδH = m 2 êz .σ, (48)\n\nΞ 0 0 = 1 κ (m 2 -n 2 )e nρ , (49)\n\nE d T P T = E d GR - n 2 κ e nρ (50)\n\nΞ 0 2 = 1 κ ( 2m 3 n ) + m κ (1 -2ne nρ )e nρ , (51)\n\nM d T P T = M d GR + m κ (1 -2ne nρ )e nρ . (52)\n\nThe metric (43) reduces to the usual perfect fluid solution when m = √ 2n [32] , i.e., B = √ 2e nρ . The corresponding energy and momentum densities take the form as\n\nE d T P T = E d GR - n 2 κ e nρ (53)\n\nM d T P T = M d GR + √ 2n κ (1 -2ne nρ )e nρ . (54)"
},
{
"section_type": "OTHER",
"section_title": "The Gödel Metric",
"text": "When we choose B = e aρ , F = e aρ √ 2 and K = 0, the metric given by Eq.( 18 ) reduces to the Gödel metric [32] . The results corresponding to Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42) take the following form\n\nt 001 = 1 3 (a - √ 2e -aρ\n\n), t 010 = -1 6 (a -√ 2e -aρ ) = t 100 , t 012 = 1 6 ( √ 2 + 2ae aρ ) = t 102 , t 021 = -1 6 (2 √ 2 + ae aρ ) = t 102 , t 122 = e aρ 6 (2 √ 2 + ae aρ ) = t 212 , t 120 = 1 6 ( √ 2ae aρ ) = t 210 , t 133 = 1 6 (a -√ 2e -aρ ) = t 313 , t 221 = -e aρ 3 (2 √ 2 + ae aρ ), (55)\n\nt 331 = 1 3 (a - √ 2e -aρ ), ( 56\n\n) V 1 = √ 2e -aρ -a, V 3 = 0, (57)\n\nA 1 = 0, A 3 = √ 2a 3 , (58)\n\nA = √ 2a 3 êz , b = a √ 2 êz , (59)\n\nδH = a √ 2 êz .σ, (60)\n\nΞ 0 0 = √ 2 κ a 2 e aρ - a 2 κ √ 2 e aρ , (61)\n\nE d T P T = E d GR - a 2 κ √ 2 e aρ (62) Ξ 0 2 = a 2 κ √ 2 e 2aρ + a κ (1 - √ 2ae aρ )e aρ , (63)\n\nM d T P T = M d GR + a κ (1 - √ 2ae aρ )e aρ . ( 64\n\n)"
},
{
"section_type": "DISCUSSION",
"section_title": "Summary and Discussion",
"text": "The purpose of this paper is twofold: Firstly, we have found the TP version of the non-null Einstein Maxwell solutions. This provides some interesting features about the axial vector and the corresponding quantities. Secondly, we have used the TP version of Möller prescription to evaluate the energymomentum distribution of the solutions. The axial-vector torsion of these solutions has been evaluated. The only non-vanishing components of the vector part are along the radial and the z-directions due to the cross term dx 0 dx 2 involving in the metric. This corresponds to the case of Kerr metric [11] , which involves the cross term dx 0 dx 3 . We also find the vector b which is related to the spin vector [4] as given by Eq.( 32 ). The axial-vector torsion lies in the ρz-plane, as its component along the φ-direction vanishes everywhere.\n\nThe non-inertial force on the Dirac particle can be represented as a rotation induced torsion of spacetime. There arise three possibilities for the axial-vector, depending upon the metric function B(ρ, z). When B is a function of z only, the axial-vector lies only along the radial direction. When B is a function of ρ only, the axialvector will lie along z-direction. The axial-vector vanishes identically for B to be constant. As the axial-vector represents the deviation from the symmetry of the underlying spacetime corresponding to an inertial field with respect to the Dirac particle, the symmetry of the spacetime will not be affected in the third possibility. Consequently there exists no inertial field with respect to the Dirac particle and the spin vector of the Dirac particle becomes constant. The corresponding extra Hamiltonian is expressed in terms of the vector b which vanishes when the metric function B is constant, i.e., when the axialvector becomes zero.\n\nThe energy-momentum distribution of the non-null Einstein-Maxwell solutions has been explored by using the TP version of Möller prescription. It is found that energy in the TP theory is equal to the energy in GR (as found by Sharif and Fatima [37] ) plus some additional part. If, for a particular case, we have K = 0 and K ′ , Ḟ = constant (or if Ḟ , K = 0 and K\n\n′ = constant), then E d T P G = E d GR . (65)\n\nOn the other hand, the only non-vanishing component of the momentum density lies along φ-direction, similar to the case of Kerr metric [11] , due to the cross term appearing in both the metrics. When we choose λ = 1, it becomes equal to be the momentum in GR [37] plus an additional quantity. If F , Ḃ, B ′′ , K all vanish, then\n\nM d T P G = M d GR . (66)\n\nBy taking particular values of E, F and K, we obtain the electromagnetic generalization of the Gödel solution and the Gödel metric as two special cases. The corresponding results for both the special cases are obtained. It is shown that, for the electromagnetic generalization of the Gödel solution, Eq.(65) does not hold, while Eq.(66) holds when m = 0. However, for the perfect fluid case, i.e., when m = √ 2n, both Eqs.( 65 ) and (66) hold by taking n = 0. In the case of the Gödel metric, these equations hold if we choose the arbitrary constant a = 0. For the special solutions, the vector part lies along the radial direction while the axial-vector part along z-direction.\n\nWe would like to re-iterate here that the tetrad formalism itself has some advantages which comes mainly from its independence from the equivalence principle and consequent suitability to the discussion of quantum issues. In TPG, an energy-momentum gauge current j i µ for the gravitational field can be defined. This is covariant under a spacetime general coordinate transformation and transforms covariantly under a global tangent space Lorentz transformation [38] . It, then, follows that j i µ is a true spacetime tensor but not a tangent space tensor. When we re-write the gauge field equations in a purely spacetime form, they lead to the Einstein field equations and the gauge current j i µ reduces to the canonical energy-momentum pseudo-tensor of the gravitational field. Thus TPG seems to provide a more appropriate environment to deal with the energy problem.\n\nFinally, it is pointed out that we are not claiming that this paper has resolved the problems of GR using the TPG. This is an attempt to touch some issues in TPG with the hope that this alternative may provide more feasible results. Also, it is always an interesting and enriching to look at things from another point of view. This endeavor is in itself commendable."
}
] | |
arxiv:0704.0489 | 0704.0489 | 1 | 10.2478/s11534-008-0018-0 | 05a07c71e5ae114221137d0f5aa9063589e159ec0dabb4c42451345ca0534803 | Relativistic treatment in D-dimensions to a spin-zero particle with
noncentral equal scalar and vector ring-shaped Kratzer potential | The Klein-Gordon equation in D-dimensions for a recently proposed Kratzer
potential plus ring-shaped potential is solved analytically by means of the
conventional Nikiforov-Uvarov method. The exact energy bound-states and the
corresponding wave functions of the Klein-Gordon are obtained in the presence
of the noncentral equal scalar and vector potentials. The results obtained in
this work are more general and can be reduced to the standard forms in
three-dimensions given by other works. | [
"Sameer M. Ikhdair",
"Ramazan Sever"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics. The Klein-Gordon equation has also been used to understand the motion of a spin-0 particle in large class of potentials. In recent years, much efforts have been paid to solve these relativistic wave equations for various potentials by using different methods. These relativistic equations contain two objects: the four-vector linear momentum operator and the scalar rest mass. They allow us to introduce two types of potential coupling, which are the four-vector potential (V) and the space-time scalar potential (S).
Recently, many authors have worked on solving these equations with physical potentials including Morse potential [3], Hulthen potential [4] , Woods-Saxon potential [5] , Pösch-Teller potential [6] , reflectionless-type potential [7] , pseudoharmonic oscillator [8] , ring-shaped harmonic oscillator [9], V 0 tanh 2 (r/r 0 ) potential [10] , five-parameter exponential potential [11] , Rosen-Morse potential [12] , and generalized symmetrical double-well potential [13] , etc. It is remarkable that in most works in this area, the scalar and vector potentials are almost taken to be equal (i.e., S = V ) [2, 14] . However, in some few other cases, it is considered the case where the scalar potential is greater than the vector potential (in order to guarantee the existence of Klein-Gordon bound states) (i.e., S > V ) [15] [16] [17] [18] [19] . Nonetheless, such physical potentials are very few. The bound-state solutions for the last case is obtained for the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is greater than the vector potential [15] .
The study of exact solutions of the nonrelativistic equation for a class of non-central potentials with a vector potential and a non-central scalar potential is of considerable interest in quantum chemistry [20] [21] [22] . In recent years, numerous studies [23] have been made in 2 analyzing the bound states of an electron in a Coulomb field with simultaneous presence of Aharanov-Bohm (AB) [24] field, and/or a magnetic Dirac monopole [25] , and Aharanov-Bohm plus oscillator (ABO) systems. In most of these works, the eigenvalues and eigenfunctions are obtained by means of seperation of variables in spherical or other orthogonal curvilinear coordinate systems. The path integral for particles moving in non-central potentials is evaluated to derive the energy spectrum of this system analytically [26] . In addition, the idea of SUSY and shape invariance is also used to obtain exact solutions of such noncentral but seperable potentials [27, 28] . Very recently, the conventional Nikiforov-Uvarov (NU) method [29] has been used to give a clear recipe of how to obtain an explicit exact bound-states solutions for the energy eigenvalues and their corresponding wave functions in terms of orthogonal polynomials for a class of non-central potentials [30] .
Another type of noncentral potentials is the ring-shaped Kratzer potential, which is a combination of a Coulomb potential plus an inverse square potential plus a noncentral angular part [31, 32] . The Kratzer potential has been used to describe the vibrational-rotational motion of isolated diatomic molecules [33] and has a mixed-energy spectrum containing both bound and scattering states with bound-states have been widely used in molecular spectroscopy [34] . The ring-shaped Kratzer potential consists of radial and angular-dependent potentials and is useful in studying ring-shaped molecules [22] . In taking the relativistic effects into account for spin-0 particle in the presence of a class of noncentral potentials, Yasuk et al [35] applied the NU method to solve the Klein-Gordon equation for the noncentral Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [36] also used the same method to solve the Klein-Gordon equation for the Kratzer-type potential.
Recently, Chen and Dong [37] proposed a new ring-shaped potential and obtained the exact solution of the Schrödinger equation for the Coulomb potential plus this new ringshaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. This type of potential used by Chen and Dong [37] appears to be very similar to the potential used by Yasuk et al [35] . Moreover, Cheng and Dai [38] , proposed a new potential consisting from the modified Kratzer's potential [33] plus the 3 new proposed ring-shaped potential in [37] . They have presented the energy eigenvalues for this proposed exactly-solvable non-central potential in three dimensional (i.e., D = 3)-Schrödinger equation by means of the NU method. The two quantum systems solved by Refs [37, 38] are closely relevant to each other as they deal with a Coulombic field interaction except for a slight change in the angular momentum barrier acts as a repulsive core which is for any arbitrary angular momentum ℓ prevents collapse of the system in any dimensional space due to the slight perturbation to the original angular momentum barrier. Very recently, we have also applied the NU method to solve the Schrödinger equation in any arbitrary Ddimension to this new modified Kratzer-type potential [39, 40] .
The aim of the present paper is to consider the relativistic effects for the spin-0 particle in our recent works [39, 40] . So we want to present a systematic recipe to solving the D-dimensional Klein-Gordon equation for the Kratzer plus the new ring-shaped potential proposed in [38] using the simple NU method. This method is based on solving the Klein-Gordon equation by reducing it to a generalized hypergeometric equation.
This work is organized as follows: in section II, we shall present the Klein-Gordon equation in spherical coordinates for spin-0 particle in the presence of equal scalar and vector noncentral Kratzer plus the new ring-shaped potential and we also separate it into radial and angular parts. Section III is devoted to a brief description of the NU method.
In section IV, we present the exact solutions to the radial and angular parts of the Klein-Gordon equation in D-dimensions. Finally, the relevant conclusions are given in section V.
In relativistic quantum mechanics, we usually use the Klein-Gordon equation for describing a scalar particle, i.e., the spin-0 particle dynamics. The discussion of the relativistic behavior of spin-zero particles requires understanding the single particle spectrum and the 4 exact solutions to the Klein Gordon equation which are constructed by using the four-vector potential A λ (λ = 0, 1, 2, 3) and the scalar potential (S). In order to simplify the solution of the Klein-Gordon equation, the four-vector potential can be written as A λ = (A 0 , 0, 0, 0).
The first component of the four-vector potential is represented by a vector potential (V ), i.e., A 0 = V. In this case, the motion of a relativistic spin-0 particle in a potential is described by the Klein-Gordon equation with the potentials V and S [1]. For the case S ≥ V, there exist bound-state (real) solutions for a relativistic spin-zero particle [15] [16] [17] [18] [19] . On the other hand, for S = V, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby the bound-state solutions are easily obtained by using the well-known methods developed in nonrelativistic quantum mechanics [2] .
The Klein-Gordon equation describing a scalar particle (spin-0 particle) with scalar S(r, θ, ϕ) and vector V (r, θ, ϕ) potentials is given by [2,14]
P 2 -[E R -V (r, θ, ϕ)/2] 2 + [µ + S(r, θ, ϕ)/2] 2 ψ(r, θ, ϕ) = 0, ( 1
)
where E R , P and µ are the relativistic energy, momentum operator and rest mass of the particle, respectively. The potential terms are scaled in (1) by Alhaidari et al [14] so that in the nonrelativistic limit the interaction potential becomes V.
In this work, we consider the equal scalar and vector potentials case, that is, S(r, θ, ϕ) = V (r, θ, ϕ) with the recently proposed general non-central potential taken in the form of the Kratzer plus ring-shaped potential [38-40]:
V (r, θ, ϕ) = V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ , ( 2
) V 1 (r) = - A r + B r 2 , V 2 (θ) = Cctg 2 θ, V 3 (ϕ) = 0, ( 3
)
where A = 2a 0 r 0 , B = a 0 r 2 0 and C is positive real constant with a 0 is the dissociation energy and r 0 is the equilibrium internuclear distance [33] . The potentials in Eq. ( 3 ) introduced by Cheng-Dai [38] reduce to the Kratzer potential in the limiting case of C = 0 [33]. In fact the energy spectrum for this potential can be obtained directly by considering it as special case of the general non-central seperable potentials [30] .
In the relativistic atomic units (h = c = 1), the D-dimensional Klein-Gordon equation in (1) becomes [41]
1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 -(E R + µ) V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ + E 2 R -µ 2 ψ(r, θ, ϕ) = 0. ( 4
)
with ψ(r, θ, ϕ) being the spherical total wave function separated as follows
ψ njm (r, θ, ϕ) = R(r)Y m j (θ, ϕ), R(r) = r -(D-1)/2 g(r), Y m j (θ, ϕ) = H(θ)Φ(ϕ). ( 5
)
Inserting Eqs (3) and (5) into Eq. ( 4 ) and using the method of separation of variables, the following differential equations are obtained:
1 r D-1 d dr r D-1 dR(r) dr - j(j + D -2) r 2 + α 2 2 α 2 1 - A r + B r 2 R(r) = 0, ( 6
) 1 sin θ d dθ sin θ d dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ + j(j + D -2) H(θ) = 0, ( 7
) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 8
)
where α 2 1 = µ -E R , α 2 2 = µ + E R , m
and j are constants and with m 2 and λ j = j(j + D -2) are the separation constants.
For a nonrelativistic treatment with the same potential, the Schrödinger equation in spherical coordinates is
1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 + 2µ E N R -V 1 (r) - V 2 (θ) r 2 - V 3 (ϕ) r 2 sin 2 θ ψ(r, θ, ϕ) = 0. ( 9
)
where µ and E N R are the reduced mass and the nonrelativistic energy, respectively. Besides, the spherical total wave function appearing in Eq. (9) has the same representation as in Eq. 6 (5) but with the transformation j → ℓ. Inserting Eq. (5) into Eq. (9) leads to the following differential equations [39,40]:
1 r D-1 d dr r D-1 dR(r) dr - λ D r 2 -2µ E N R + A r - B r 2 R(r) = 0, ( 10
) 1 sin θ d dθ sin θ d dθ - m 2 + 2µC cos 2 θ sin 2 θ + ℓ(ℓ + D -2) H(θ) = 0, ( 11
) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 12
)
where m 2 and λ ℓ = ℓ(ℓ + D -2) are the separation constants. Equations (6)-(8) have the same functional form as Eqs (10)-(12). Therefore, the solution of the Klein-Gordon equation can be reduced to the solution of the Schrödinger equation with the appropriate choice of parameters: j → ℓ, α 2 1 → -E N R and α 2 2 → 2µ.
The solution of Eq. ( 8 ) is well-known periodic and must satisfy the period boundary condition Φ(ϕ + 2π) = Φ(ϕ) which is the azimuthal angle solution:
Φ m (ϕ) = 1 √ 2π exp(±imϕ), m = 0, 1, 2, ..... ( 13
)
Additionally, Eqs (6) and (7) are radial and polar angle equations and they will be solved by using the Nikiforov-Uvarov (NU) method [29] which is given briefly in the following section.
The NU method is based on reducing the second-order differential equation to a generalized equation of hypergeometric type [29] . In this sense, the Schrödinger equation, after employing an appropriate coordinate transformation s = s(r), transforms to the following form:
ψ ′′ n (s) + τ (s) σ(s) ψ ′ n (s) + σ(s) σ 2 (s) ψ n (s) = 0, ( 14
) 7
where σ(s) and σ(s) are polynomials, at most of second-degree, and τ (s) is a first-degree polynomial. Using a wave function, ψ n (s), of the simple ansatz:
ψ n (s) = φ n (s)y n (s), ( 15
)
reduces (14) into an equation of a hypergeometric type
σ(s)y ′′ n (s) + τ (s)y ′ n (s) + λy n (s) = 0, ( 16
)
where
σ(s) = π(s) φ(s) φ ′ (s) , ( 17
)
τ (s) = τ (s) + 2π(s), τ ′ (s) < 0, ( 18
)
and λ is a parameter defined as
λ = λ n = -nτ ′ (s) - n (n -1) 2 σ ′′ (s), n = 0, 1, 2, .... ( 19
)
The polynomial τ (s) with the parameter s and prime factors show the differentials at first degree be negative. It is worthwhile to note that λ or λ n are obtained from a particular solution of the form y(s) = y n (s) which is a polynomial of degree n. Further, the other part y n (s) of the wave function (14) is the hypergeometric-type function whose polynomial solutions are given by Rodrigues relation
y n (s) = B n ρ(s) d n ds n [σ n (s)ρ(s)] , ( 20
)
where B n is the normalization constant and the weight function ρ(s) must satisfy the condition [29]
d ds w(s) = τ (s) σ(s) w(s), w(s) = σ(s)ρ(s). ( 21
)
The function π and the parameter λ are defined as
π(s) = σ ′ (s) -τ (s) 2 ± σ ′ (s) -τ (s) 2 2 -σ(s) + kσ(s), ( 22
) 8 λ = k + π ′ (s). ( 23
)
In principle, since π(s) has to be a polynomial of degree at most one, the expression under the square root sign in (22) can be arranged to be the square of a polynomial of first degree [29] . This is possible only if its discriminant is zero. In this case, an equation for k is obtained. After solving this equation, the obtained values of k are substituted in (22) . In addition, by comparing equations (19) and (23), we obtain the energy eigenvalues.
We seek to solving the radial and angular parts of the Klein-Gordon equation given by Eqs (6) and (7). Equation (6) involving the radial part can be written simply in the following form [39-41]:
d 2 g(r) dr 2 - (M -1)(M -3) 4r 2 -α 2 2 A r - B r 2 + α 2 1 α 2 2 g(r) = 0, ( 24
)
where
M = D + 2j. ( 25
)
On the other hand, Eq. ( 7 ) involving the angular part of Klein-Gordon equation retakes the simple form
d 2 H(θ) dθ 2 + ctg(θ) dH(θ) dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ -j(j + D -2) H(θ) = 0. ( 26
)
Thus, Eqs (24) and (26) have to be solved latter through the NU method in the following subsections. 9 B. Eigenvalues and eigenfunctions of the angle-dependent equation
In order to apply NU method [29, 30, 33, 35, 36, [38] [39] [40] [42] [43] [44] , we use a suitable transformation variable s = cos θ. The polar angle part of the Klein Gordon equation in (26) can be written in the following universal associated-Legendre differential equation form [38-40]
d 2 H(s) ds 2 - 2s 1 -s 2 dH(s) ds + 1 (1 -s 2 ) 2 j(j + D -2)(1 -s 2 ) -m 2 -Cα 2 2 s 2 H(θ) = 0. ( 27
)
Equation (27) has already been solved for the three-dimensional Schrödinger equation through the NU method in [38] . However, the aim in this subsection is to solve with different parameters resulting from the D-space-dimensions of Klein-Gordon equation. Further, Eq.
(27) is compared with (14) and the following identifications are obtained
τ (s) = -2s, σ(s) = 1 -s 2 , σ(s) = -m ′2 + (1 -s 2 )ν ′ , ( 28
)
where
ν ′ = j ′ (j ′ + D -2) = j(j + D -2) + Cα 2 2 , m ′2 = m 2 + Cα 2 2 . ( 29
)
Inserting the above expressions into equation (22), one obtains the following function:
π(s) = ± (ν ′ -k)s 2 + k -ν ′ + m ′2 , ( 30
)
Following the method, the polynomial π(s) is found in the following possible values
π(s) = m ′ s for k 1 = ν ′ -m ′2 , -m ′ s for k 1 = ν ′ -m ′2 , m ′ for k 2 = ν ′ , -m ′ for k 2 = ν ′ . ( 31
)
Imposing the condition τ ′ (s) < 0, for equation (18), one selects
k 1 = ν ′ -m ′2 and π(s) = -m ′ s, ( 32
) which yields 10 τ (s) = -2(1 + m ′ )s. ( 33
)
Using equations (19) and (23), the following expressions for λ are obtained, respectively,
λ = λ n = 2 n(1 + m ′ ) + n( n -1), ( 34
) λ = ν ′ -m ′ (1 + m ′ ). ( 35
)
We compare equations (34) and (35), the new angular momentum j values are obtained as
j = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -4Cα 2 2 -1, ( 36
) or j ′ = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -1. ( 37
)
Using Eqs (15)-(17) and (20)-(21), the polynomial solution of y n is expressed in terms of Jacobi polynomials [39,40] which are one of the orthogonal polynomials:
H n (θ) = N n sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ), ( 38
) where N n = 1 2 m ′ ( n+m ′ )! (2 n+2m ′ +1)( n+2m ′ )! n! 2
is the normalization constant determined by +1 -1 [H n (s)] 2 ds = 1 and using the orthogonality relation of Jacobi polynomials [35, 45, 46] .
Besides n = - (1 + 2m ′ ) 2 + 1 2 (2j + 1) 2 + 4j(D -3) + 4Cα 2 2 , ( 39
)
where m ′ is defined by equation (29).
The solution of the radial part of Klein-Gordon equation, Eq. ( 24 ), for the Kratzer's potential has already been solved by means of NU-method in [39] . Very recently, using the same method, the problem for the non-central potential in (2) has been solved in three 11 dimensions (3D) by Cheng and Dai [36] . However, the aim of this subsection is to solve the problem with a different radial separation function g(r) in any arbitrary dimensions. In what follows, we present the exact bound-states (real) solution of Eq. (24). Letting
ε 2 = α 2 1 α 2 2 , 4γ 2 = (M -1)(M -3) + 4Bα 2 2 , β 2 = Aα 2 2 , ( 40
)
and substituting these expressions in equation (24), one gets
d 2 g(r) dr 2 + -ε 2 r 2 + β 2 r -γ 2 r 2 g(r) = 0. ( 41
)
To apply the conventional NU-method, equation (41) is compared with (14), resulting in the following expressions:
τ (r) = 0, σ(r) = r, σ(r) = -ε 2 r 2 + β 2 r -γ 2 . ( 42
)
Substituting the above expressions into equation (22) gives
π(r) = 1 2 ± 1 2 4ε 2 r 2 + 4(k -β 2 )r + 4γ 2 + 1. ( 43
)
Therefore, we can determine the constant k by using the condition that the discriminant of the square root is zero, that is
k = β 2 ± ε 4γ 2 + 1, 4γ 2 + 1 = (D + 2j -2) 2 + 4Bα 2 2 . ( 44
)
In view of that, we arrive at the following four possible functions of π(r) :
π(r) = 1 2 + εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 -εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 + εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1, 1 2 -εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1. ( 45
)
The correct value of π(r) is chosen such that the function τ (r) given by Eq. (18) will have negative derivative [29] . So we can select the physical values to be
k = β 2 -ε 4γ 2 + 1 and π(r) = 1 2 -εr - 1 2 4γ 2 + 1 , ( 46
)
12 which yield τ (r) = -2εr + (1 + 4γ 2 + 1), τ ′ (r) = -2ε < 0. (47) Using Eqs (19) and (23), the following expressions for λ are obtained, respectively,
λ = λ n = 2nε, n = 0, 1, 2, ..., ( 48
) λ = δ 2 -ε(1 + 4γ 2 + 1). ( 49
)
So we can obtain the Klein Gordon energy eigenvalues from the following relation:
1 + 2n + (D + 2j -2) 2 + 4(µ + E R )B µ -E R = A µ + E R , ( 50
)
and hence for the Kratzer plus the new ring-shaped potential, it becomes
1 + 2n + (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 51
)
with j defined in (36) . Although Eq. (51) is exactly solvable for E R but it looks to be little complicated. Further, it is interesting to investigate the solution for the Coulomb potential.
Therefore, applying the following transformations: A = Ze 2 , B = 0, and j = ℓ, the central part of the potential in (3) turns into the Coulomb potential with Klein Gordon solution for the energy spectra given by
E R = µ 1 - 2q 2 e 2 q 2 e 2 + (2n + 2ℓ + D -1) 2 , n, ℓ = 0, 1, 2, ..., ( 52
)
where q = Ze is the charge of the nucleus. Further, Eq. ( 52 ) can be expanded as a series in the nucleus charge as
E R = µ - 2µq 2 e 2 (2n + 2ℓ + D -1) 2 + 2µq 4 e 4 (2n + 2ℓ + D -1) 4 -O(qe) 6 , ( 53
)
The physical meaning of each term in the last equation was given in detail by Ref. [36] .
Besides, the difference from the conventional nonrelativistic form is because of the choice of the vector V (r, θ, ϕ) and scalar S(r, θ, ϕ) parts of the potential in Eq. ( 1 ). 13 Overmore, if the value of j obtained by Eq.(36) is inserted into the eigenvalues of the radial part of the Klein Gordon equation with the noncentral potential given by Eq. (51), we finally find the energy eigenvalues for a bound electron in the presence of a noncentral potential by Eq. (2) as
1 + 2n + (2j ′ + D -2) 2 + 4(a 0 r 2 0 -C)(µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 54
)
where m ′ = m 2 + C(µ + E R ) and n is given by Eq. (39). On the other hand, the solution of the Schrödinger equation, Eq. ( 9 ), for this potential has already been obtained by using the same method in Ref. [39] and it is in the Coulombic-like form:
E N R = - 8µa 2 0 r 2 0 2n + 1 + (2ℓ ′ + D -2) 2 + 8µ(a 0 r 2 0 -C) 2 , n = 0, 1, 2, ... ( 55
) 2ℓ ′ + D -2 = (D -2) 2 + (2 n + 2m ′ + 1) 2 -1, ( 56
)
where m ′ = √ m 2 + 2µC. Also, applying the following appropriate transformation: 54 ) provides exactly the nonrelativistic limit given by Eq. (55).
µ+E R → 2µ, µ -E R → -E N R , j → ℓ to Eq. (
In what follows, let us now turn attention to find the radial wavefunctions for this potential. Substituting the values of σ(r), π(r) and τ (r) in Eqs (42), (45) and (47) into Eqs.
(17) and (21), we find φ(r) = r (ζ+1)
/2 e -εr , ( 57
) ρ(r) = r ζ e -2εr , ( 58
)
where ζ = √ 4γ 2 + 1. Then from equation (20), we obtain
y nj (r) = B nj r -ζ e 2εr d n dr n r n+ζ e -2εr , ( 59
)
and the wave function g(r) can be written in the form of the generalized Laguerre polynomials as 14 g(ρ) = C nj ρ 2ε
(1+ζ)/2
e -ρ/2 L ζ n (ρ), ( 60
)
where for Kratzer's potential we have
ζ = (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ), ρ = 2εr. ( 61
)
Finally, the radial wave functions of the Klein-Gordon equation are obtained R(ρ) = C nj ρ 2ε
(ζ+2-D)/2
e -ρ/2 L ζ n (ρ), ( 62
)
where C nj is the normalization constant to be determined below. Using the normalization condition, ∞ 0 R 2 (r)r D-1 dr = 1, and the orthogonality relation of the generalized Laguerre polynomials, (2n+η+1)(n+η)! n! , we have
∞ 0 z η+1 e -z [L η n (z)] 2 dz =
C nj = 2 µ 2 -E 2 R 1+ ζ 2 n! (2n + ζ + 1) (n + ζ)! . ( 63
)
Finally, we may express the normalized total wave functions as
ψ(r, θ, ϕ) = 2 µ 2 -E 2 R 1+ ζ 2 2 m ′ ( n + m ′ )! (2 n + 2m ′ + 1)( n + 2m ′ )! n!n! 2π (2n + ζ + 1) (n + ζ)! ×r (ζ+2-D) 2 exp(-µ 2 -E 2 R r)L ζ n (2 µ 2 -E 2 R r) sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ) exp(±imϕ). ( 64
)
where ζ is defined in Eq. ( 61 ) and m ′ is given after the Eq. (54).
The relativistic spin-0 particle D-dimensional Klein-Gordon equation has been solved easily for its exact bound-states with equal scalar and vector ring-shaped Kratzer potential through the conventional NU method. The analytical expressions for the total energy levels and eigenfunctions of this system can be reduced to their conventional three-dimensional space form upon setting D = 3. Further, the noncentral potentials treated in [30] can be introduced as perturbation to the Kratzer's potential by adjusting the strength of the 15 coupling constant C in terms of a 0 , which is the coupling constant of the Kratzer's potential.
Additionally, the radial and polar angle wave functions of Klein-Gordon equation are found in terms of Laguerre and Jacobi polynomials, respectively. The method presented in this paper is general and worth extending to the solution of other interaction problems. This method is very simple and useful in solving other complicated systems analytically without given a restiction conditions on the solution of some quantum systems as the case in the other models. We have seen that for the nonrelativistic model, the exact energy spectra can be obtained either by solving the Schrödinger equation in (9) (cf. Ref. [39] or Eq. ( 55 )) or by applying appropriate transformation to the relativistic solution given by Eq. ( 54 ). Finally, we point out that these exact results obtained for this new proposed form of the potential (2) may have some interesting applications in the study of different quantum mechanical systems, atomic and molecular physics.
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[20] M. Kibler and T. Negadi, Int. J. Quantum Chem. 26 (1984) 405; İ. Sökmen, Phys. Lett. 118A (1986) 249; L.V. Lutsenko et al., Teor. Mat. Fiz. 83 (1990) 419; H. Hartmann et al., Theor. Chim. Acta 24 (1972) 201; M.V. Carpido-Bernido and C. C. Bernido, Phys. Lett. 134A (1989) 315; M.V. Carpido-Bernido, J. Phys. A 24 (1991) 3013; O.
F. Gal'bert, Y. L. Granovskii and A. S. Zhedabov, Phys. Lett. A 153 (1991) 177; C. Quesne, J. Phys. A 21 (1988) 3093.
[21] M. Kibler and P. Winternitz, J. Phys. A 20 (1987) 4097.
[22] H. Hartmann and D. Schuch, Int. J. Quantum Chem. 18 (1980) 125. [23] M. Kibler and T. Negadi, Phys. Lett. A 124 (1987) 42; A. Guha and S. Mukherjee, J.
Math. Phys. 28 (1989) 840; G. E. Draganescu, C. Campiogotto and M. Kibler, Phys.
Lett. A 170 (1992) 339; M. Kibler and C. Campiogotto, Phys. Lett. A 181 (1993) 1; V. M. Villalba, Phys. Lett. A 193 (1994) 218. [24] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [25] P. A. M. Dirac, Proc. R. Soc. London Ser. A 133 (1931) 60. [26] B.P. Mandal, Int. J. Mod. Phys. A 15 (2000) 1225.
[27] R. Dutt, A. Gangopadhyaya and U.P. Sukhatme, Am. J. Phys. 65 (5) (1997) 400. [28] B. Gönül and İ. Zorba, Phys. Lett. A 269 (2000) 83. [29] A.F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics 18 (Birkhauser, Bassel, 1988). [30] S.M. Ikhdair and R. Sever, to appear in the Int. J. Theor. Phys. (preprint quantph//0702186). [31] H.S. Valk, Am. J. Phys. 54 (1986) 921. [32] Q.W. Chao, Chin. Phys. 13 (5) (2004) 575.
[33] C. Berkdemir, A. Berkdemir and J.G. Han, Chem. Phys. Lett. 417 (2006) 326. [34] A. Bastida et al, J. Chem. Phys. 93 (1990) 3408. [35] F. Yasuk, A. Durmuş and I. Boztosun, J. Math. Phys. 47 (2006) 082302. [36] C. Berkdemir, Am. J. Phys. 75 (2007) 81. [37] C.Y. Chen and S.H. Dong, Phys. Lett. A 335 (2005) 374. [38] Y.F. Cheng and T.Q. Dai, Phys. Scr. 75 (2007) 274. [39] S.M. Ikhdair and R. Sever, preprint quant-ph/0703008; quant-ph/0703042. [40] S.M. Ikhdair and R. Sever, preprint quant-ph/0703131. [41] S.M. Ikhdair and R. Sever, Z. Phys. C 56 (1992) 155; C 58 (1993) 153; D 28 (1993) 1; Hadronic J. 15 (1992) 389; Int. J. Mod. Phys. A 18 (2003) 4215; A 19 (2004) 1771; A 20 (2005) 4035; A 20 (2005) 6509; A 21 (2006) 2191; A 21 (2006) 3989; A 21 (2006) 6699; Int. J. Mod. Phys. E (in press) (preprint hep-ph/0504176); S. Ikhdair et al, Tr. J. Phys. 16 (1992) 510; 17 (1993) 474. [42] S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. (DOI 10.1007/s10773-006-9317-7; J.
Math. Chem. (DOI 10.1007/s10910-006-9115-8). [43] S.M. Ikhdair and R. Sever, Ann. Phys. (Leipzig) 16 (3) (2007) 218. [44] S.M. Ikhdair and R. Sever, to appear in the Int. J. Mod. Phys. E (preprint quant-19 ph/0611065). [45] G. Sezgo, Orthogonal Polynomials (American Mathematical Society, New York, 1939). [46] N.N. Lebedev, Special Functions and Their Applications (Prentice-Hall, Englewood Cliffs, NJ, 1965). 20 | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics. The Klein-Gordon equation has also been used to understand the motion of a spin-0 particle in large class of potentials. In recent years, much efforts have been paid to solve these relativistic wave equations for various potentials by using different methods. These relativistic equations contain two objects: the four-vector linear momentum operator and the scalar rest mass. They allow us to introduce two types of potential coupling, which are the four-vector potential (V) and the space-time scalar potential (S).\n\nRecently, many authors have worked on solving these equations with physical potentials including Morse potential [3], Hulthen potential [4] , Woods-Saxon potential [5] , Pösch-Teller potential [6] , reflectionless-type potential [7] , pseudoharmonic oscillator [8] , ring-shaped harmonic oscillator [9], V 0 tanh 2 (r/r 0 ) potential [10] , five-parameter exponential potential [11] , Rosen-Morse potential [12] , and generalized symmetrical double-well potential [13] , etc. It is remarkable that in most works in this area, the scalar and vector potentials are almost taken to be equal (i.e., S = V ) [2, 14] . However, in some few other cases, it is considered the case where the scalar potential is greater than the vector potential (in order to guarantee the existence of Klein-Gordon bound states) (i.e., S > V ) [15] [16] [17] [18] [19] . Nonetheless, such physical potentials are very few. The bound-state solutions for the last case is obtained for the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is greater than the vector potential [15] .\n\nThe study of exact solutions of the nonrelativistic equation for a class of non-central potentials with a vector potential and a non-central scalar potential is of considerable interest in quantum chemistry [20] [21] [22] . In recent years, numerous studies [23] have been made in 2 analyzing the bound states of an electron in a Coulomb field with simultaneous presence of Aharanov-Bohm (AB) [24] field, and/or a magnetic Dirac monopole [25] , and Aharanov-Bohm plus oscillator (ABO) systems. In most of these works, the eigenvalues and eigenfunctions are obtained by means of seperation of variables in spherical or other orthogonal curvilinear coordinate systems. The path integral for particles moving in non-central potentials is evaluated to derive the energy spectrum of this system analytically [26] . In addition, the idea of SUSY and shape invariance is also used to obtain exact solutions of such noncentral but seperable potentials [27, 28] . Very recently, the conventional Nikiforov-Uvarov (NU) method [29] has been used to give a clear recipe of how to obtain an explicit exact bound-states solutions for the energy eigenvalues and their corresponding wave functions in terms of orthogonal polynomials for a class of non-central potentials [30] .\n\nAnother type of noncentral potentials is the ring-shaped Kratzer potential, which is a combination of a Coulomb potential plus an inverse square potential plus a noncentral angular part [31, 32] . The Kratzer potential has been used to describe the vibrational-rotational motion of isolated diatomic molecules [33] and has a mixed-energy spectrum containing both bound and scattering states with bound-states have been widely used in molecular spectroscopy [34] . The ring-shaped Kratzer potential consists of radial and angular-dependent potentials and is useful in studying ring-shaped molecules [22] . In taking the relativistic effects into account for spin-0 particle in the presence of a class of noncentral potentials, Yasuk et al [35] applied the NU method to solve the Klein-Gordon equation for the noncentral Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [36] also used the same method to solve the Klein-Gordon equation for the Kratzer-type potential.\n\nRecently, Chen and Dong [37] proposed a new ring-shaped potential and obtained the exact solution of the Schrödinger equation for the Coulomb potential plus this new ringshaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. This type of potential used by Chen and Dong [37] appears to be very similar to the potential used by Yasuk et al [35] . Moreover, Cheng and Dai [38] , proposed a new potential consisting from the modified Kratzer's potential [33] plus the 3 new proposed ring-shaped potential in [37] . They have presented the energy eigenvalues for this proposed exactly-solvable non-central potential in three dimensional (i.e., D = 3)-Schrödinger equation by means of the NU method. The two quantum systems solved by Refs [37, 38] are closely relevant to each other as they deal with a Coulombic field interaction except for a slight change in the angular momentum barrier acts as a repulsive core which is for any arbitrary angular momentum ℓ prevents collapse of the system in any dimensional space due to the slight perturbation to the original angular momentum barrier. Very recently, we have also applied the NU method to solve the Schrödinger equation in any arbitrary Ddimension to this new modified Kratzer-type potential [39, 40] .\n\nThe aim of the present paper is to consider the relativistic effects for the spin-0 particle in our recent works [39, 40] . So we want to present a systematic recipe to solving the D-dimensional Klein-Gordon equation for the Kratzer plus the new ring-shaped potential proposed in [38] using the simple NU method. This method is based on solving the Klein-Gordon equation by reducing it to a generalized hypergeometric equation.\n\nThis work is organized as follows: in section II, we shall present the Klein-Gordon equation in spherical coordinates for spin-0 particle in the presence of equal scalar and vector noncentral Kratzer plus the new ring-shaped potential and we also separate it into radial and angular parts. Section III is devoted to a brief description of the NU method.\n\nIn section IV, we present the exact solutions to the radial and angular parts of the Klein-Gordon equation in D-dimensions. Finally, the relevant conclusions are given in section V."
},
{
"section_type": "OTHER",
"section_title": "II. THE KLEIN-GORDON EQUATION WITH EQUAL SCALAR AND VECTOR POTENTIALS",
"text": "In relativistic quantum mechanics, we usually use the Klein-Gordon equation for describing a scalar particle, i.e., the spin-0 particle dynamics. The discussion of the relativistic behavior of spin-zero particles requires understanding the single particle spectrum and the 4 exact solutions to the Klein Gordon equation which are constructed by using the four-vector potential A λ (λ = 0, 1, 2, 3) and the scalar potential (S). In order to simplify the solution of the Klein-Gordon equation, the four-vector potential can be written as A λ = (A 0 , 0, 0, 0).\n\nThe first component of the four-vector potential is represented by a vector potential (V ), i.e., A 0 = V. In this case, the motion of a relativistic spin-0 particle in a potential is described by the Klein-Gordon equation with the potentials V and S [1]. For the case S ≥ V, there exist bound-state (real) solutions for a relativistic spin-zero particle [15] [16] [17] [18] [19] . On the other hand, for S = V, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby the bound-state solutions are easily obtained by using the well-known methods developed in nonrelativistic quantum mechanics [2] .\n\nThe Klein-Gordon equation describing a scalar particle (spin-0 particle) with scalar S(r, θ, ϕ) and vector V (r, θ, ϕ) potentials is given by [2,14]\n\nP 2 -[E R -V (r, θ, ϕ)/2] 2 + [µ + S(r, θ, ϕ)/2] 2 ψ(r, θ, ϕ) = 0, ( 1\n\n)\n\nwhere E R , P and µ are the relativistic energy, momentum operator and rest mass of the particle, respectively. The potential terms are scaled in (1) by Alhaidari et al [14] so that in the nonrelativistic limit the interaction potential becomes V.\n\nIn this work, we consider the equal scalar and vector potentials case, that is, S(r, θ, ϕ) = V (r, θ, ϕ) with the recently proposed general non-central potential taken in the form of the Kratzer plus ring-shaped potential [38-40]:\n\nV (r, θ, ϕ) = V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ , ( 2\n\n) V 1 (r) = - A r + B r 2 , V 2 (θ) = Cctg 2 θ, V 3 (ϕ) = 0, ( 3\n\n)\n\nwhere A = 2a 0 r 0 , B = a 0 r 2 0 and C is positive real constant with a 0 is the dissociation energy and r 0 is the equilibrium internuclear distance [33] . The potentials in Eq. ( 3 ) introduced by Cheng-Dai [38] reduce to the Kratzer potential in the limiting case of C = 0 [33]. In fact the energy spectrum for this potential can be obtained directly by considering it as special case of the general non-central seperable potentials [30] .\n\nIn the relativistic atomic units (h = c = 1), the D-dimensional Klein-Gordon equation in (1) becomes [41]\n\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 -(E R + µ) V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ + E 2 R -µ 2 ψ(r, θ, ϕ) = 0. ( 4\n\n)\n\nwith ψ(r, θ, ϕ) being the spherical total wave function separated as follows\n\nψ njm (r, θ, ϕ) = R(r)Y m j (θ, ϕ), R(r) = r -(D-1)/2 g(r), Y m j (θ, ϕ) = H(θ)Φ(ϕ). ( 5\n\n)\n\nInserting Eqs (3) and (5) into Eq. ( 4 ) and using the method of separation of variables, the following differential equations are obtained:\n\n1 r D-1 d dr r D-1 dR(r) dr - j(j + D -2) r 2 + α 2 2 α 2 1 - A r + B r 2 R(r) = 0, ( 6\n\n) 1 sin θ d dθ sin θ d dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ + j(j + D -2) H(θ) = 0, ( 7\n\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 8\n\n)\n\nwhere α 2 1 = µ -E R , α 2 2 = µ + E R , m\n\nand j are constants and with m 2 and λ j = j(j + D -2) are the separation constants.\n\nFor a nonrelativistic treatment with the same potential, the Schrödinger equation in spherical coordinates is\n\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 + 2µ E N R -V 1 (r) - V 2 (θ) r 2 - V 3 (ϕ) r 2 sin 2 θ ψ(r, θ, ϕ) = 0. ( 9\n\n)\n\nwhere µ and E N R are the reduced mass and the nonrelativistic energy, respectively. Besides, the spherical total wave function appearing in Eq. (9) has the same representation as in Eq. 6 (5) but with the transformation j → ℓ. Inserting Eq. (5) into Eq. (9) leads to the following differential equations [39,40]:\n\n1 r D-1 d dr r D-1 dR(r) dr - λ D r 2 -2µ E N R + A r - B r 2 R(r) = 0, ( 10\n\n) 1 sin θ d dθ sin θ d dθ - m 2 + 2µC cos 2 θ sin 2 θ + ℓ(ℓ + D -2) H(θ) = 0, ( 11\n\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 12\n\n)\n\nwhere m 2 and λ ℓ = ℓ(ℓ + D -2) are the separation constants. Equations (6)-(8) have the same functional form as Eqs (10)-(12). Therefore, the solution of the Klein-Gordon equation can be reduced to the solution of the Schrödinger equation with the appropriate choice of parameters: j → ℓ, α 2 1 → -E N R and α 2 2 → 2µ.\n\nThe solution of Eq. ( 8 ) is well-known periodic and must satisfy the period boundary condition Φ(ϕ + 2π) = Φ(ϕ) which is the azimuthal angle solution:\n\nΦ m (ϕ) = 1 √ 2π exp(±imϕ), m = 0, 1, 2, ..... ( 13\n\n)\n\nAdditionally, Eqs (6) and (7) are radial and polar angle equations and they will be solved by using the Nikiforov-Uvarov (NU) method [29] which is given briefly in the following section."
},
{
"section_type": "METHOD",
"section_title": "III. NIKIFOROV-UVAROV METHOD",
"text": "The NU method is based on reducing the second-order differential equation to a generalized equation of hypergeometric type [29] . In this sense, the Schrödinger equation, after employing an appropriate coordinate transformation s = s(r), transforms to the following form:\n\nψ ′′ n (s) + τ (s) σ(s) ψ ′ n (s) + σ(s) σ 2 (s) ψ n (s) = 0, ( 14\n\n) 7\n\nwhere σ(s) and σ(s) are polynomials, at most of second-degree, and τ (s) is a first-degree polynomial. Using a wave function, ψ n (s), of the simple ansatz:\n\nψ n (s) = φ n (s)y n (s), ( 15\n\n)\n\nreduces (14) into an equation of a hypergeometric type\n\nσ(s)y ′′ n (s) + τ (s)y ′ n (s) + λy n (s) = 0, ( 16\n\n)\n\nwhere\n\nσ(s) = π(s) φ(s) φ ′ (s) , ( 17\n\n)\n\nτ (s) = τ (s) + 2π(s), τ ′ (s) < 0, ( 18\n\n)\n\nand λ is a parameter defined as\n\nλ = λ n = -nτ ′ (s) - n (n -1) 2 σ ′′ (s), n = 0, 1, 2, .... ( 19\n\n)\n\nThe polynomial τ (s) with the parameter s and prime factors show the differentials at first degree be negative. It is worthwhile to note that λ or λ n are obtained from a particular solution of the form y(s) = y n (s) which is a polynomial of degree n. Further, the other part y n (s) of the wave function (14) is the hypergeometric-type function whose polynomial solutions are given by Rodrigues relation\n\ny n (s) = B n ρ(s) d n ds n [σ n (s)ρ(s)] , ( 20\n\n)\n\nwhere B n is the normalization constant and the weight function ρ(s) must satisfy the condition [29]\n\nd ds w(s) = τ (s) σ(s) w(s), w(s) = σ(s)ρ(s). ( 21\n\n)\n\nThe function π and the parameter λ are defined as\n\nπ(s) = σ ′ (s) -τ (s) 2 ± σ ′ (s) -τ (s) 2 2 -σ(s) + kσ(s), ( 22\n\n) 8 λ = k + π ′ (s). ( 23\n\n)\n\nIn principle, since π(s) has to be a polynomial of degree at most one, the expression under the square root sign in (22) can be arranged to be the square of a polynomial of first degree [29] . This is possible only if its discriminant is zero. In this case, an equation for k is obtained. After solving this equation, the obtained values of k are substituted in (22) . In addition, by comparing equations (19) and (23), we obtain the energy eigenvalues."
},
{
"section_type": "OTHER",
"section_title": "IV. EXACT SOLUTIONS OF THE RADIAL AND ANGLE-DEPENDENT EQUATIONS A. Separating variables of the Klein-Gordon equation",
"text": "We seek to solving the radial and angular parts of the Klein-Gordon equation given by Eqs (6) and (7). Equation (6) involving the radial part can be written simply in the following form [39-41]:\n\nd 2 g(r) dr 2 - (M -1)(M -3) 4r 2 -α 2 2 A r - B r 2 + α 2 1 α 2 2 g(r) = 0, ( 24\n\n)\n\nwhere\n\nM = D + 2j. ( 25\n\n)\n\nOn the other hand, Eq. ( 7 ) involving the angular part of Klein-Gordon equation retakes the simple form\n\nd 2 H(θ) dθ 2 + ctg(θ) dH(θ) dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ -j(j + D -2) H(θ) = 0. ( 26\n\n)\n\nThus, Eqs (24) and (26) have to be solved latter through the NU method in the following subsections. 9 B. Eigenvalues and eigenfunctions of the angle-dependent equation\n\nIn order to apply NU method [29, 30, 33, 35, 36, [38] [39] [40] [42] [43] [44] , we use a suitable transformation variable s = cos θ. The polar angle part of the Klein Gordon equation in (26) can be written in the following universal associated-Legendre differential equation form [38-40]\n\nd 2 H(s) ds 2 - 2s 1 -s 2 dH(s) ds + 1 (1 -s 2 ) 2 j(j + D -2)(1 -s 2 ) -m 2 -Cα 2 2 s 2 H(θ) = 0. ( 27\n\n)\n\nEquation (27) has already been solved for the three-dimensional Schrödinger equation through the NU method in [38] . However, the aim in this subsection is to solve with different parameters resulting from the D-space-dimensions of Klein-Gordon equation. Further, Eq.\n\n(27) is compared with (14) and the following identifications are obtained\n\nτ (s) = -2s, σ(s) = 1 -s 2 , σ(s) = -m ′2 + (1 -s 2 )ν ′ , ( 28\n\n)\n\nwhere\n\nν ′ = j ′ (j ′ + D -2) = j(j + D -2) + Cα 2 2 , m ′2 = m 2 + Cα 2 2 . ( 29\n\n)\n\nInserting the above expressions into equation (22), one obtains the following function:\n\nπ(s) = ± (ν ′ -k)s 2 + k -ν ′ + m ′2 , ( 30\n\n)\n\nFollowing the method, the polynomial π(s) is found in the following possible values\n\nπ(s) = m ′ s for k 1 = ν ′ -m ′2 , -m ′ s for k 1 = ν ′ -m ′2 , m ′ for k 2 = ν ′ , -m ′ for k 2 = ν ′ . ( 31\n\n)\n\nImposing the condition τ ′ (s) < 0, for equation (18), one selects\n\nk 1 = ν ′ -m ′2 and π(s) = -m ′ s, ( 32\n\n) which yields 10 τ (s) = -2(1 + m ′ )s. ( 33\n\n)\n\nUsing equations (19) and (23), the following expressions for λ are obtained, respectively,\n\nλ = λ n = 2 n(1 + m ′ ) + n( n -1), ( 34\n\n) λ = ν ′ -m ′ (1 + m ′ ). ( 35\n\n)\n\nWe compare equations (34) and (35), the new angular momentum j values are obtained as\n\nj = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -4Cα 2 2 -1, ( 36\n\n) or j ′ = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -1. ( 37\n\n)\n\nUsing Eqs (15)-(17) and (20)-(21), the polynomial solution of y n is expressed in terms of Jacobi polynomials [39,40] which are one of the orthogonal polynomials:\n\nH n (θ) = N n sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ), ( 38\n\n) where N n = 1 2 m ′ ( n+m ′ )! (2 n+2m ′ +1)( n+2m ′ )! n! 2\n\nis the normalization constant determined by +1 -1 [H n (s)] 2 ds = 1 and using the orthogonality relation of Jacobi polynomials [35, 45, 46] .\n\nBesides n = - (1 + 2m ′ ) 2 + 1 2 (2j + 1) 2 + 4j(D -3) + 4Cα 2 2 , ( 39\n\n)\n\nwhere m ′ is defined by equation (29)."
},
{
"section_type": "OTHER",
"section_title": "C. Eigensolutions of the radial equation",
"text": "The solution of the radial part of Klein-Gordon equation, Eq. ( 24 ), for the Kratzer's potential has already been solved by means of NU-method in [39] . Very recently, using the same method, the problem for the non-central potential in (2) has been solved in three 11 dimensions (3D) by Cheng and Dai [36] . However, the aim of this subsection is to solve the problem with a different radial separation function g(r) in any arbitrary dimensions. In what follows, we present the exact bound-states (real) solution of Eq. (24). Letting\n\nε 2 = α 2 1 α 2 2 , 4γ 2 = (M -1)(M -3) + 4Bα 2 2 , β 2 = Aα 2 2 , ( 40\n\n)\n\nand substituting these expressions in equation (24), one gets\n\nd 2 g(r) dr 2 + -ε 2 r 2 + β 2 r -γ 2 r 2 g(r) = 0. ( 41\n\n)\n\nTo apply the conventional NU-method, equation (41) is compared with (14), resulting in the following expressions:\n\nτ (r) = 0, σ(r) = r, σ(r) = -ε 2 r 2 + β 2 r -γ 2 . ( 42\n\n)\n\nSubstituting the above expressions into equation (22) gives\n\nπ(r) = 1 2 ± 1 2 4ε 2 r 2 + 4(k -β 2 )r + 4γ 2 + 1. ( 43\n\n)\n\nTherefore, we can determine the constant k by using the condition that the discriminant of the square root is zero, that is\n\nk = β 2 ± ε 4γ 2 + 1, 4γ 2 + 1 = (D + 2j -2) 2 + 4Bα 2 2 . ( 44\n\n)\n\nIn view of that, we arrive at the following four possible functions of π(r) :\n\nπ(r) = 1 2 + εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 -εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 + εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1, 1 2 -εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1. ( 45\n\n)\n\nThe correct value of π(r) is chosen such that the function τ (r) given by Eq. (18) will have negative derivative [29] . So we can select the physical values to be\n\nk = β 2 -ε 4γ 2 + 1 and π(r) = 1 2 -εr - 1 2 4γ 2 + 1 , ( 46\n\n)\n\n12 which yield τ (r) = -2εr + (1 + 4γ 2 + 1), τ ′ (r) = -2ε < 0. (47) Using Eqs (19) and (23), the following expressions for λ are obtained, respectively,\n\nλ = λ n = 2nε, n = 0, 1, 2, ..., ( 48\n\n) λ = δ 2 -ε(1 + 4γ 2 + 1). ( 49\n\n)\n\nSo we can obtain the Klein Gordon energy eigenvalues from the following relation:\n\n1 + 2n + (D + 2j -2) 2 + 4(µ + E R )B µ -E R = A µ + E R , ( 50\n\n)\n\nand hence for the Kratzer plus the new ring-shaped potential, it becomes\n\n1 + 2n + (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 51\n\n)\n\nwith j defined in (36) . Although Eq. (51) is exactly solvable for E R but it looks to be little complicated. Further, it is interesting to investigate the solution for the Coulomb potential.\n\nTherefore, applying the following transformations: A = Ze 2 , B = 0, and j = ℓ, the central part of the potential in (3) turns into the Coulomb potential with Klein Gordon solution for the energy spectra given by\n\nE R = µ 1 - 2q 2 e 2 q 2 e 2 + (2n + 2ℓ + D -1) 2 , n, ℓ = 0, 1, 2, ..., ( 52\n\n)\n\nwhere q = Ze is the charge of the nucleus. Further, Eq. ( 52 ) can be expanded as a series in the nucleus charge as\n\nE R = µ - 2µq 2 e 2 (2n + 2ℓ + D -1) 2 + 2µq 4 e 4 (2n + 2ℓ + D -1) 4 -O(qe) 6 , ( 53\n\n)\n\nThe physical meaning of each term in the last equation was given in detail by Ref. [36] .\n\nBesides, the difference from the conventional nonrelativistic form is because of the choice of the vector V (r, θ, ϕ) and scalar S(r, θ, ϕ) parts of the potential in Eq. ( 1 ). 13 Overmore, if the value of j obtained by Eq.(36) is inserted into the eigenvalues of the radial part of the Klein Gordon equation with the noncentral potential given by Eq. (51), we finally find the energy eigenvalues for a bound electron in the presence of a noncentral potential by Eq. (2) as\n\n1 + 2n + (2j ′ + D -2) 2 + 4(a 0 r 2 0 -C)(µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 54\n\n)\n\nwhere m ′ = m 2 + C(µ + E R ) and n is given by Eq. (39). On the other hand, the solution of the Schrödinger equation, Eq. ( 9 ), for this potential has already been obtained by using the same method in Ref. [39] and it is in the Coulombic-like form:\n\nE N R = - 8µa 2 0 r 2 0 2n + 1 + (2ℓ ′ + D -2) 2 + 8µ(a 0 r 2 0 -C) 2 , n = 0, 1, 2, ... ( 55\n\n) 2ℓ ′ + D -2 = (D -2) 2 + (2 n + 2m ′ + 1) 2 -1, ( 56\n\n)\n\nwhere m ′ = √ m 2 + 2µC. Also, applying the following appropriate transformation: 54 ) provides exactly the nonrelativistic limit given by Eq. (55).\n\nµ+E R → 2µ, µ -E R → -E N R , j → ℓ to Eq. (\n\nIn what follows, let us now turn attention to find the radial wavefunctions for this potential. Substituting the values of σ(r), π(r) and τ (r) in Eqs (42), (45) and (47) into Eqs.\n\n(17) and (21), we find φ(r) = r (ζ+1)\n\n/2 e -εr , ( 57\n\n) ρ(r) = r ζ e -2εr , ( 58\n\n)\n\nwhere ζ = √ 4γ 2 + 1. Then from equation (20), we obtain\n\ny nj (r) = B nj r -ζ e 2εr d n dr n r n+ζ e -2εr , ( 59\n\n)\n\nand the wave function g(r) can be written in the form of the generalized Laguerre polynomials as 14 g(ρ) = C nj ρ 2ε\n\n(1+ζ)/2\n\ne -ρ/2 L ζ n (ρ), ( 60\n\n)\n\nwhere for Kratzer's potential we have\n\nζ = (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ), ρ = 2εr. ( 61\n\n)\n\nFinally, the radial wave functions of the Klein-Gordon equation are obtained R(ρ) = C nj ρ 2ε\n\n(ζ+2-D)/2\n\ne -ρ/2 L ζ n (ρ), ( 62\n\n)\n\nwhere C nj is the normalization constant to be determined below. Using the normalization condition, ∞ 0 R 2 (r)r D-1 dr = 1, and the orthogonality relation of the generalized Laguerre polynomials, (2n+η+1)(n+η)! n! , we have\n\n∞ 0 z η+1 e -z [L η n (z)] 2 dz =\n\nC nj = 2 µ 2 -E 2 R 1+ ζ 2 n! (2n + ζ + 1) (n + ζ)! . ( 63\n\n)\n\nFinally, we may express the normalized total wave functions as\n\nψ(r, θ, ϕ) = 2 µ 2 -E 2 R 1+ ζ 2 2 m ′ ( n + m ′ )! (2 n + 2m ′ + 1)( n + 2m ′ )! n!n! 2π (2n + ζ + 1) (n + ζ)! ×r (ζ+2-D) 2 exp(-µ 2 -E 2 R r)L ζ n (2 µ 2 -E 2 R r) sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ) exp(±imϕ). ( 64\n\n)\n\nwhere ζ is defined in Eq. ( 61 ) and m ′ is given after the Eq. (54)."
},
{
"section_type": "CONCLUSION",
"section_title": "V. CONCLUSIONS",
"text": "The relativistic spin-0 particle D-dimensional Klein-Gordon equation has been solved easily for its exact bound-states with equal scalar and vector ring-shaped Kratzer potential through the conventional NU method. The analytical expressions for the total energy levels and eigenfunctions of this system can be reduced to their conventional three-dimensional space form upon setting D = 3. Further, the noncentral potentials treated in [30] can be introduced as perturbation to the Kratzer's potential by adjusting the strength of the 15 coupling constant C in terms of a 0 , which is the coupling constant of the Kratzer's potential.\n\nAdditionally, the radial and polar angle wave functions of Klein-Gordon equation are found in terms of Laguerre and Jacobi polynomials, respectively. The method presented in this paper is general and worth extending to the solution of other interaction problems. This method is very simple and useful in solving other complicated systems analytically without given a restiction conditions on the solution of some quantum systems as the case in the other models. We have seen that for the nonrelativistic model, the exact energy spectra can be obtained either by solving the Schrödinger equation in (9) (cf. Ref. [39] or Eq. ( 55 )) or by applying appropriate transformation to the relativistic solution given by Eq. ( 54 ). Finally, we point out that these exact results obtained for this new proposed form of the potential (2) may have some interesting applications in the study of different quantum mechanical systems, atomic and molecular physics."
},
{
"section_type": "OTHER",
"section_title": "ACKNOWLEDGMENTS",
"text": "This research was partially supported by the Scientific and Technological Research Council of Turkey. S.M. Ikhdair wishes to dedicate this work to his family for their love and assistance. 16 REFERENCES [1] T.Y. Wu and W.Y. Pauchy Hwang, Relativistic Quantum Mechanics and Quantum Fields (World Scientific, Singapore, 1991). [2] W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd edn (springer, Berlin, 2000). [3] A.D. Alhaidari, Phys. Rev. Lett. 87 (2001) 210405; 88 (2002) 189901.\n\n[4] G. Chen, Mod. Phys. Lett. A 19 (2004) 2009; J.-Y. Guo, J. Meng and F.-X. Xu, Chin.\n\nPhys. Lett. 20 (2003) 602; A.D. Alhaidari, J. Phys. A: Math. Gen. 34 (2001) 9827; 35 (2002) 6207; M. S ¸imşek and H. Egrifes, J. Phys. A: Math. Gen. 37 (2004) 4379.\n\n[5] J.-Y. Guo, X.-Z. Fang and F.-X. Xu, Phys. Rev. A 66 (2002) 062105; C. Berkdemir, A.\n\nBerkdemir and R. Sever, J. Phys. A: Math. Gen. 39 (2006) 13455. [6] G. Chen, Acta Phys. Sinica 50 (2001) 1651; Ö. Yeşiltaş, Phys. Scr. 75 (2007) 41.\n\n[7] G. Chen and Z.M. Lou, Acta Phys. Sinica 52 (2003) 1071. [8] S.M. Ikhdair and R. Sever, J. Mol. Structure:THEOCHEM 806 (2007) 155; G. Chen, Z.D. Chen and Z.M. Lou, Chin. Phys. 13 (2004) 279. [9] W.C. Qiang, Chin. Phys. 12 (2003) 136.\n\n[10] W.C. Qiang, Chin. Phys. 13 (2004) 571. [11] G. Chen, Phys. Lett. A 328 (2004) 116; Y.F. Diao, L.Z. Yi and C.S. Jia, Phys. Lett. A 332 (2004) 157. [12] L.Z. Yi et al, Phys. Lett. A 333 (2004) 212.\n\n[13] X.Q. Zhao, C.S. Jia and Q.B.Yang, Phys. Lett. A 337 (2005) 189. [14] A.D. Alhaidari, H. Bahlouli and A. Al-Hasan, Phys. Lett. A 349 (2006) 87.\n\n[15] G. Chen, Phys. Lett. A 339 (2005) 300. 17 [16] A. de Souza Dutra and G. Chen, Phys. Lett. A 349 (2006) 297. [17] F. Dominguez-Adame, Phys. Lett. A 136 (1989) 175. [18] A.S. de Castro, Phys. Lett. A 338 (2005) 81. [19] G. Chen, Acta Phys. Sinica 53 (2004) 680; G. Chen and D.F. Zhao, Acta Phys. Sinica 52 (2003) 2954.\n\n[20] M. Kibler and T. Negadi, Int. J. Quantum Chem. 26 (1984) 405; İ. Sökmen, Phys. Lett. 118A (1986) 249; L.V. Lutsenko et al., Teor. Mat. Fiz. 83 (1990) 419; H. Hartmann et al., Theor. Chim. Acta 24 (1972) 201; M.V. Carpido-Bernido and C. C. Bernido, Phys. Lett. 134A (1989) 315; M.V. Carpido-Bernido, J. Phys. A 24 (1991) 3013; O.\n\nF. Gal'bert, Y. L. Granovskii and A. S. Zhedabov, Phys. Lett. A 153 (1991) 177; C. Quesne, J. Phys. A 21 (1988) 3093.\n\n[21] M. Kibler and P. Winternitz, J. Phys. A 20 (1987) 4097.\n\n[22] H. Hartmann and D. Schuch, Int. J. Quantum Chem. 18 (1980) 125. [23] M. Kibler and T. Negadi, Phys. Lett. A 124 (1987) 42; A. Guha and S. Mukherjee, J.\n\nMath. Phys. 28 (1989) 840; G. E. Draganescu, C. Campiogotto and M. Kibler, Phys.\n\nLett. A 170 (1992) 339; M. Kibler and C. Campiogotto, Phys. Lett. A 181 (1993) 1; V. M. Villalba, Phys. Lett. A 193 (1994) 218. [24] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [25] P. A. M. Dirac, Proc. R. Soc. London Ser. A 133 (1931) 60. [26] B.P. Mandal, Int. J. Mod. Phys. A 15 (2000) 1225.\n\n[27] R. Dutt, A. Gangopadhyaya and U.P. Sukhatme, Am. J. Phys. 65 (5) (1997) 400. [28] B. Gönül and İ. Zorba, Phys. Lett. A 269 (2000) 83. [29] A.F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics 18 (Birkhauser, Bassel, 1988). [30] S.M. Ikhdair and R. Sever, to appear in the Int. J. Theor. Phys. (preprint quantph//0702186). [31] H.S. Valk, Am. J. Phys. 54 (1986) 921. [32] Q.W. Chao, Chin. Phys. 13 (5) (2004) 575.\n\n[33] C. Berkdemir, A. Berkdemir and J.G. Han, Chem. Phys. Lett. 417 (2006) 326. [34] A. Bastida et al, J. Chem. Phys. 93 (1990) 3408. [35] F. Yasuk, A. Durmuş and I. Boztosun, J. Math. Phys. 47 (2006) 082302. [36] C. Berkdemir, Am. J. Phys. 75 (2007) 81. [37] C.Y. Chen and S.H. Dong, Phys. Lett. A 335 (2005) 374. [38] Y.F. Cheng and T.Q. Dai, Phys. Scr. 75 (2007) 274. [39] S.M. Ikhdair and R. Sever, preprint quant-ph/0703008; quant-ph/0703042. [40] S.M. Ikhdair and R. Sever, preprint quant-ph/0703131. [41] S.M. Ikhdair and R. Sever, Z. Phys. C 56 (1992) 155; C 58 (1993) 153; D 28 (1993) 1; Hadronic J. 15 (1992) 389; Int. J. Mod. Phys. A 18 (2003) 4215; A 19 (2004) 1771; A 20 (2005) 4035; A 20 (2005) 6509; A 21 (2006) 2191; A 21 (2006) 3989; A 21 (2006) 6699; Int. J. Mod. Phys. E (in press) (preprint hep-ph/0504176); S. Ikhdair et al, Tr. J. Phys. 16 (1992) 510; 17 (1993) 474. [42] S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. (DOI 10.1007/s10773-006-9317-7; J.\n\nMath. Chem. (DOI 10.1007/s10910-006-9115-8). [43] S.M. Ikhdair and R. Sever, Ann. Phys. (Leipzig) 16 (3) (2007) 218. [44] S.M. Ikhdair and R. Sever, to appear in the Int. J. Mod. Phys. E (preprint quant-19 ph/0611065). [45] G. Sezgo, Orthogonal Polynomials (American Mathematical Society, New York, 1939). [46] N.N. Lebedev, Special Functions and Their Applications (Prentice-Hall, Englewood Cliffs, NJ, 1965). 20"
}
] | |
arxiv:0704.0490 | 0704.0490 | 1 | 10.1103/PhysRevD.76.111701 | 33a1d1b81936bb4d490d360820f2123e47c0f5994f09751e7382ad683a799b76 | Long Distance Signaling Using Axion-like Particles | The possible existence of axion-like particles could lead to a new type of
long distance communication. In this work, basic antenna concepts are defined
and a Friis-like equation is derived to facilitate long-distance link
calculations. An example calculation is presented showing that communication
over distances of 1000 km or more may be possible for $m_{a}< 3.5$ meV and
$g_{a\gamma \gamma} > 5 \times 10^{- 8} {\text{GeV}}^{- 1}$. | [
"Daniel D. Stancil"
] | [
"hep-ph",
"astro-ph",
"hep-th"
] | hep-ph | [] | 2007-04-04 | 2026-02-26 | The possible existence of axion-like particles could lead to a new type of long distance communication. In this work, basic antenna concepts are defined and a Friis-like equation is derived to facilitate long-distance link calculations. An example calculation is presented showing that communication over distances of 1000 km or more may be possible for 3.5
suggested coupling between the axion and electromagnetic fields that was much larger than thought possible based on solar axion observations by the CAST collaboration [5] .
Although mechanisms have been proposed to reconcile the reports [6] , the PVLAS collaboration recently retracted the results [7] and an independent group has reported a negative result from a photon regeneration experiment that excludes the PVLAS result [8] .
There does not now appear to be any experimental evidence of a coupling strength inconsistent with CAST observations. However, since recent work has suggested mechanisms whereby such strong coupling may be possible, I believe it remains interesting to consider the implications of stronger-than-expected axion-photon coupling.
In particular, I would like to call attention to the observation that a new type of longdistance signaling and communication may be possible. It may be possible to construct a communication system that cannot be blocked-even communicating directly through the diameter of the earth. This would make reliable worldwide signaling possible without the use of either satellites or the ionosphere, and would enable communication to locations previously inaccessible, such as submarines at the bottom of the sea and mines deep beneath the earth. The signal would also be very difficult to intercept since the axion beam would be essentially as narrow as a laser beam used to create it, and most of the path would be underground. With advances in power and sensitivity, it may also be possible to use axion signaling in space communications. For example, using such a system, communication with points on the far side of the moon may be possible without the use of lunar satellites.
Communication systems using neutrinos have also been proposed [9], and would have many of the same characteristics as the proposed axion system. However, the generation and detection of neutrinos requires massive particle accelerators and scintillation detectors [10] . Also, full deflection over 4π steradians would not be practical, though limited beam steering could be achieved using a magnetic field to deflect the precursor pion beam. Finally, it would be difficult to consider modulation techniques more sophisticated than simple amplitude modulation. In contrast, using axion mass and coupling values not yet experimentally explored, it appears that worldwide communication would be possible with a fully steerable axion system about the size of a medium-size telescope. Further, since the signals at the input and output would be electromagnetic waves, any existing modulation technology could be used. However, as with neutrinos, the lack of strong interactions with matter presents challenges with respect to the generation and detection of axions. Sikivie proposed an experimental approach for detecting axions via their coupling to the electromagnetic field [11] . The coupling was obtained by considering the Lagrangian density
2 2 1 1 1 1 4 4 2 2 a a L F F g aF F a a m a μν μν μ μν γγ μν μ = - - + ∂ ∂ - , (1)
where
F A A μν μ ν ν μ = ∂ -∂ is the electromagnetic field tensor, ( ) , A V A α =
where V is the electric scalar potential and A is the magnetic vector potential, a is the axion field, a g γγ is the coupling constant between the electromagnetic and axion fields, and the electromagnetic dual tensor is given by 1 2 F F γδ αβ αβγδ ε = . In these equations we have
taken 1 h c = = .
As an example, consider the coupling between plane waves propagating along the z direction caused by a strong static magnetic field parallel to the polarization of the incident electromagnetic wave. If the time dependence is exp(
) i t ω -
, where ω is the frequency of the incident linearly polarized electromagnetic wave, the equations of motion obtained from (1) reduce to the coupled equations ( )
2 2 2 2 2 0 0 2 2 , . a a a a A m a i g B A A i g B a z z γγ γγ ω ω ω ω ∂ ∂ + - =- + = ∂ ∂ (2)
Thus the static magnetic field B 0 couples the photon and axion fields. An apparatus for the generation and detection of axions based on this coupling is shown in Figure 1(a) . This is sometimes referred to as an "invisible light through walls" experiment [4, 12, 13 ].
An electromagnetic wave with amplitude (0) A enters a region of magnetic field of strength B OT extending over a distance L T . If the coupling is sufficiently weak that the change in A over the transmit conversion region and the change in a over the receive conversion region are negligible, then the conversion loss through the system will be [12] 0 / in R T
P P p p = , (3)
where in P is the optical input power, 0 P is the optical output power, T p , R p are the probabilities of photon-axion (and axion-photon) conversion in the transmitter and receiver, respectively, and the conversion probability is [11, 12, 14] 2
0 1 sin /2 2 /2 a a qL p g BL k q L γγ ω ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ . (4)
Here a q k k γ =indicates the phase mismatch between the photon and axion fields.
The efficiency of generating axions and regenerating photons can be greatly increased by adding electromagnetic resonators, as shown in Figure 1 (b) [4, 15] . In this light to pass through the magnetic field multiple times, increasing the conversion probability by the factor 2 / T F π , where F T is the finesse of the resonator in the axion generator (transmitter). This factor can be interpreted as the effective number of photon passes in the resonator. Axions are also emitted in the backward direction owing to the counter-propagating light in the resonator, resulting in half the particles traveling in an unwanted direction. Consequently, the probability of conversion in a given direction is increased by the factor / T F π . As also shown in Figure 1(b), a resonator on the photon regenerator (receiver) likewise increases the axion-photon conversion probability [15]. Since regenerated photons will be emitted in both directions, detectors are placed on both ends of the receiving optical resonator, enhancing the photon regeneration probability by the factor 2 / R F π , where F R is the finesse of the receiving resonator. (If the power from a single end is collected, the factor would be / R F π , as with the transmitter.) Finally, to turn this into a communication system, we add appropriate electromagnetic wave modulators and detectors as shown in Figure 1(b).
The conversion loss equation ( 3 ) is valid when the transmitter and receiver are sufficiently close together that beam diffraction can be neglected, and when the transmitter and receiver have equal cross-sectional areas. For signaling over long distances, neither assumption will be valid in general. To treat the long-distance case, we first calculate the radiated axion field, then calculate the regenerated photons resulting when the radiated field reaches the receiver.
The general solution to Eq. ( 2 ) is given by [12] 3 0
( ) ( ) ( ) 4 a ik r r a V e a r i g d r A r B r r r γγ ω π ′ - ′ ′ ′ = ′ - ∫ i . ( 5
)
If the observation point r is very far away from all points in the source volume V, then we obtain the far-field approximation for the axion field
T A x F A ik z γ π =
, where 0 A is the amplitude of the incident electromagnetic wave, and the cylinder is contained in a resonant cavity with finesse T F .
Using Eq. ( 6 ), the far-field potential is found to be
( ) ( ) ( ) 1 0 0 sin / 2 ( ) 2 / 2 a ik r az T a T T a T T T a T az T k k L J k R F e a r i g A B L s r k R k k L γ ρ γγ ρ γ ω π π ⎡ ⎤⎡ ⎤ - ⎣ ⎦ ≈ ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ , (7)
where sin , cos
1 0 lim / 1/ 2 a a T a T k J k R k R ρ ρ ρ → ⎡ ⎤ = ⎣ ⎦ .
Consequently, the axion field on axis is
[ ] 0 0 sin / 2 ( ) 4 / 2 a ik r T T a T T T T qL F e a r i g A B L s r q L γγ ω π π ≈ . ( 8
)
The time averaged transmitted power density is ( )
2 2 2 1 ( ) ( ) 2 2 a T a a T Ti n k F S r k a r s p P r ω π π = = , (9)
where (0)
in T P S s γ = , and
( ) 2 1 (0) 0 2 S k A γ γ ω = .
If this axion power density is incident upon a photon regenerator at distance r that is perfectly aligned with the transmitter, then the received power is
0 (2 / ) ( ) R R a R P F p S r s π = . ( 10
)
Substituting Eq. (9) for the power flux ( )
a S r gives 2 0 2 2 4 in a R T R R T T P k F F P ps ps r π π π π = . ( 11
)
This expression can be understood in terms of antenna theory for electromagnetic waves.
In this context, we refer to the apparatus consisting of the resonator and the structure creating the magnetic field as an axion antenna. In analogy with conventional antenna theory, we define the directivity as 2 ( ) /( 4 )
a rad S r D P r π = , ( 12
)
where P rad is the total power radiated by the transmitting antenna. To obtain the total power radiated, we could integrate the power flux (9) over a sphere enclosing the antenna. However, it is easier to do the calculation in the near field using the axion field at the aperture of the transmitting antenna. Using ( ) ( / ) (0)
a T T T S L F p S γ π = , we have 2 ( ) (2 / ) rad a T T T T in P S L s F p P π = = . (13)
Substituting ( 13 ) for the total radiated power and (9) for the power flux, the directivity simplifies to
2 2 (2 / ) (4 / )( /2) T a T a T D s s π λ π λ = = . ( 14
)
The relation between the directivity and physical area ( 14 ) is ½ that found in conventional antenna theory, or equivalently, the area appears to be half the physical area. This results from the bi-directional radiation properties of the resonator. Defining an efficiency as /
rad in P P η = , we also define the antenna gain as 2 (2 / ) (2 / ) T T T T T a T G D F p s η π π λ = = , (15) where (2 / ) T T T F p η π = .
Next suppose that at some distant location this transmitted field is incident upon a receive antenna with length L R , radius R R , and finesse F R . From (10) and assuming the photons emitted from both ends of the receive antenna are collected, the total power collected will be 0 As with electromagnetic antennas, the ratio of effective area to gain is found to be independent of the details of the antenna, other than whether or not photons are collected from both ends of the antenna when used to receive:
2 , , / / / (4 ) . e R R e T T c a s G s G n λ π = = ( 17
)
If photons were collected only from one end of the receive antenna ( 1 c n = ), then Eq. ( 17 ) would be identical to conventional antenna theory.
With these definitions, the expression for the received power ( 11 ) can be interpreted as a Friis-like equation:
( ) ( )( )
2 2 2 0 , , , / / 4 / ( ) / in c T R a e T c T a e R P P n G G r s n s r λ π λ = = . (18) Here , c T n is the value used to compute , e T s according to Eq. (16). The ratio , , / e T c T s n is independent of the choice of , c T
n , as it should be since the number of detectors that might be used on receive is independent of the transmit properties of the antenna.
It is also useful to note that if the magnetic field is uniform (i.e., wigglers, or quasi-phase matching, are not used [12] ), then there is an optimum length for the conversion region of an antenna. This occurs when ( )
sin / 2 1 qL = , or qL π = . The optimum length is found to be 2 ( / ) opt a L m γ λ ω = . ( 19
)
In obtaining this expression, we have used 2 /(2 ) a q m ω ≈ , which is valid for a m ω .
The diffraction-limited power pattern of the radiated axion field is determined by the aperture size in wavelengths through the Bessel function term in ( 7 ):
( ) ( )
2 1 4 sin /( sin ) d a a P J k R k R θ θ θ = ⎡ ⎤ ⎣ ⎦ . ( 20
)
The diffraction beam width between first nulls is determined by the first zero of the Airy disc, and for small angles is given by the well-known expression
1.22 / d FWFN T R γ θ λ ≈ . (21)
Similarly, the diffraction beam width at half maximum is determined by the roots of ( )
) / 0.514 / d FWHM a T T R R γ θ πλ λ = ≈ . 1/ 2 d P θ = , or (1.616 /
In contrast, the conversion-limited power pattern is given by ( ) ( )
2 sin cos / 2 ( ) 2 cos / 2 a c a k k L P k k L γ γ θ π θ θ ⎡ ⎤ ⎡ ⎤ - ⎣ ⎦ ⎢ ⎥ = ⎢ ⎥ - ⎣ ⎦ , (23)
and depends on both the length in wavelengths and the velocity mismatch. For the optimum length L given by ( 19 ), the conversion beam widths are approximately given by 2( / ), 1.06( / )
c c FWFN a FWHM a m m θ ω θ ω ≈ ≈ . ( 24
)
For quantum-limited detection, the channel capacity is [16] log [17], and c is the velocity of light. Equation (25) can be combined with (11) or (18) to find the axion parameters that would permit a particular channel capacity at a given distance for a particular experimental apparatus.
(1 / ) d R C N ν η ν = Δ + Δ , (25)
As an example, consider transmitters and receivers with 1064 nm (1.17 eV)
γ λ = , 10 W in P = , 2 0.01 m s = , 0 10 T B = , 3 L = m, 5 3.1 10 F = × , 0.5 d η =
, and a minimum information capacity of 1 bps. Figure 3 shows the inverse coupling strength curves
( ) 1/ a a M m g γγ =
for distances of 1000 km and the diameter of the earth. Also shown is the curve for communication between the earth and the far side of the moon using the "4+4" experimental apparatus proposed in [15] . The shaded regions are excluded by the results from the BFRT collaboration [13] and Robilliard et al. [8] . The dots represent extensions of the curves if quasi-phase matching (QPM) is used by periodically reversing the magnetic field [12] . The minimum period of the reversal is taken to be twice the beam diameter for the 3 m system example, and 28.6 m for the 4+4 system. From the figure, communication over distances in excess of 1000 km should be possible for 3.5
a m < meV and 7 2 10 GeV M < × ( 8 1 5 10 GeV a g γγ -- Note that the receive resonator must be tuned to the same frequency as the transmit resonator to within a fraction of the resonator line width which is 161 Hz in this example. This will present a significant challenge, especially since the resonators are remote from one another. A possible approach would be to use atomic clocks to stabilize both the transmit frequency and a local reference frequency at the receiver. The receive resonator would then be locked to the reference to get as close as possible to the correct frequency, then slowly tuned until the signal is located. A feedback loop could then be closed to lock the receive resonator to the signal.
In summary, for 3.5 > × ), it may be possible to realize a new type of wireless signaling that cannot be blocked or shielded. An example calculation shows that communication between points located diametrically opposite on the earth should be possible. This could enable world-wide communication without the use of satellites or the ionosphere. However, with present knowledge, the signaling will be limited to low data rates, perhaps on the order of a few bits per second for terrestrial links. This estimate assumes 3 m long generation/regeneration regions to allow fully steerable instruments. Though not easily steerable, the apparatus in the "4+4" experiment proposed by Sikivie et al., [15] may enable communication to the far side of the moon for 0.3 a m < meV and 6 6 10 GeV M < × . I would like to acknowledge helpful discussions with Jim Lesh, Rich Holman, Jeff Peterson, Pierre Sikivie, and David Tanner during the development of these ideas. Electronic address: stancil@cmu.edu 17. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6 th Edition (Oxford Univ. Press, New York, 2007), pp. 171. 18. C-C. Chen, op cit., p. 106. B 0R L T L R Mod Laser Demod Data In Data Out B 0T Optical Detectors B 0T B 0R Photon input Photon output Axions penetrate barrier opaque to photons L T L R (a) Photons Axions Cavity mirrors (b) Figure 1, Stancil, Phys. Rev. D. x y z r θ -L T /2 L T /2 | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The possible existence of axion-like particles could lead to a new type of long distance communication. In this work, basic antenna concepts are defined and a Friis-like equation is derived to facilitate long-distance link calculations. An example calculation is presented showing that communication over distances of 1000 km or more may be possible for 3.5"
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "suggested coupling between the axion and electromagnetic fields that was much larger than thought possible based on solar axion observations by the CAST collaboration [5] .\n\nAlthough mechanisms have been proposed to reconcile the reports [6] , the PVLAS collaboration recently retracted the results [7] and an independent group has reported a negative result from a photon regeneration experiment that excludes the PVLAS result [8] .\n\nThere does not now appear to be any experimental evidence of a coupling strength inconsistent with CAST observations. However, since recent work has suggested mechanisms whereby such strong coupling may be possible, I believe it remains interesting to consider the implications of stronger-than-expected axion-photon coupling.\n\nIn particular, I would like to call attention to the observation that a new type of longdistance signaling and communication may be possible. It may be possible to construct a communication system that cannot be blocked-even communicating directly through the diameter of the earth. This would make reliable worldwide signaling possible without the use of either satellites or the ionosphere, and would enable communication to locations previously inaccessible, such as submarines at the bottom of the sea and mines deep beneath the earth. The signal would also be very difficult to intercept since the axion beam would be essentially as narrow as a laser beam used to create it, and most of the path would be underground. With advances in power and sensitivity, it may also be possible to use axion signaling in space communications. For example, using such a system, communication with points on the far side of the moon may be possible without the use of lunar satellites.\n\nCommunication systems using neutrinos have also been proposed [9], and would have many of the same characteristics as the proposed axion system. However, the generation and detection of neutrinos requires massive particle accelerators and scintillation detectors [10] . Also, full deflection over 4π steradians would not be practical, though limited beam steering could be achieved using a magnetic field to deflect the precursor pion beam. Finally, it would be difficult to consider modulation techniques more sophisticated than simple amplitude modulation. In contrast, using axion mass and coupling values not yet experimentally explored, it appears that worldwide communication would be possible with a fully steerable axion system about the size of a medium-size telescope. Further, since the signals at the input and output would be electromagnetic waves, any existing modulation technology could be used. However, as with neutrinos, the lack of strong interactions with matter presents challenges with respect to the generation and detection of axions. Sikivie proposed an experimental approach for detecting axions via their coupling to the electromagnetic field [11] . The coupling was obtained by considering the Lagrangian density\n\n2 2 1 1 1 1 4 4 2 2 a a L F F g aF F a a m a μν μν μ μν γγ μν μ = - - + ∂ ∂ - , (1)\n\nwhere\n\nF A A μν μ ν ν μ = ∂ -∂ is the electromagnetic field tensor, ( ) , A V A α =\n\nwhere V is the electric scalar potential and A is the magnetic vector potential, a is the axion field, a g γγ is the coupling constant between the electromagnetic and axion fields, and the electromagnetic dual tensor is given by 1 2 F F γδ αβ αβγδ ε = . In these equations we have\n\ntaken 1 h c = = .\n\nAs an example, consider the coupling between plane waves propagating along the z direction caused by a strong static magnetic field parallel to the polarization of the incident electromagnetic wave. If the time dependence is exp(\n\n) i t ω -\n\n, where ω is the frequency of the incident linearly polarized electromagnetic wave, the equations of motion obtained from (1) reduce to the coupled equations ( )\n\n2 2 2 2 2 0 0 2 2 , . a a a a A m a i g B A A i g B a z z γγ γγ ω ω ω ω ∂ ∂ + - =- + = ∂ ∂ (2)\n\nThus the static magnetic field B 0 couples the photon and axion fields. An apparatus for the generation and detection of axions based on this coupling is shown in Figure 1(a) . This is sometimes referred to as an \"invisible light through walls\" experiment [4, 12, 13 ].\n\nAn electromagnetic wave with amplitude (0) A enters a region of magnetic field of strength B OT extending over a distance L T . If the coupling is sufficiently weak that the change in A over the transmit conversion region and the change in a over the receive conversion region are negligible, then the conversion loss through the system will be [12] 0 / in R T\n\nP P p p = , (3)\n\nwhere in P is the optical input power, 0 P is the optical output power, T p , R p are the probabilities of photon-axion (and axion-photon) conversion in the transmitter and receiver, respectively, and the conversion probability is [11, 12, 14] 2\n\n0 1 sin /2 2 /2 a a qL p g BL k q L γγ ω ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ . (4)\n\nHere a q k k γ =indicates the phase mismatch between the photon and axion fields.\n\nThe efficiency of generating axions and regenerating photons can be greatly increased by adding electromagnetic resonators, as shown in Figure 1 (b) [4, 15] . In this light to pass through the magnetic field multiple times, increasing the conversion probability by the factor 2 / T F π , where F T is the finesse of the resonator in the axion generator (transmitter). This factor can be interpreted as the effective number of photon passes in the resonator. Axions are also emitted in the backward direction owing to the counter-propagating light in the resonator, resulting in half the particles traveling in an unwanted direction. Consequently, the probability of conversion in a given direction is increased by the factor / T F π . As also shown in Figure 1(b), a resonator on the photon regenerator (receiver) likewise increases the axion-photon conversion probability [15]. Since regenerated photons will be emitted in both directions, detectors are placed on both ends of the receiving optical resonator, enhancing the photon regeneration probability by the factor 2 / R F π , where F R is the finesse of the receiving resonator. (If the power from a single end is collected, the factor would be / R F π , as with the transmitter.) Finally, to turn this into a communication system, we add appropriate electromagnetic wave modulators and detectors as shown in Figure 1(b).\n\nThe conversion loss equation ( 3 ) is valid when the transmitter and receiver are sufficiently close together that beam diffraction can be neglected, and when the transmitter and receiver have equal cross-sectional areas. For signaling over long distances, neither assumption will be valid in general. To treat the long-distance case, we first calculate the radiated axion field, then calculate the regenerated photons resulting when the radiated field reaches the receiver.\n\nThe general solution to Eq. ( 2 ) is given by [12] 3 0\n\n( ) ( ) ( ) 4 a ik r r a V e a r i g d r A r B r r r γγ ω π ′ - ′ ′ ′ = ′ - ∫ i . ( 5\n\n)\n\nIf the observation point r is very far away from all points in the source volume V, then we obtain the far-field approximation for the axion field\n\nT A x F A ik z γ π =\n\n, where 0 A is the amplitude of the incident electromagnetic wave, and the cylinder is contained in a resonant cavity with finesse T F .\n\nUsing Eq. ( 6 ), the far-field potential is found to be\n\n( ) ( ) ( ) 1 0 0 sin / 2 ( ) 2 / 2 a ik r az T a T T a T T T a T az T k k L J k R F e a r i g A B L s r k R k k L γ ρ γγ ρ γ ω π π ⎡ ⎤⎡ ⎤ - ⎣ ⎦ ≈ ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ , (7)\n\nwhere sin , cos\n\n1 0 lim / 1/ 2 a a T a T k J k R k R ρ ρ ρ → ⎡ ⎤ = ⎣ ⎦ .\n\nConsequently, the axion field on axis is\n\n[ ] 0 0 sin / 2 ( ) 4 / 2 a ik r T T a T T T T qL F e a r i g A B L s r q L γγ ω π π ≈ . ( 8\n\n)\n\nThe time averaged transmitted power density is ( )\n\n2 2 2 1 ( ) ( ) 2 2 a T a a T Ti n k F S r k a r s p P r ω π π = = , (9)\n\nwhere (0)\n\nin T P S s γ = , and\n\n( ) 2 1 (0) 0 2 S k A γ γ ω = .\n\nIf this axion power density is incident upon a photon regenerator at distance r that is perfectly aligned with the transmitter, then the received power is\n\n0 (2 / ) ( ) R R a R P F p S r s π = . ( 10\n\n)\n\nSubstituting Eq. (9) for the power flux ( )\n\na S r gives 2 0 2 2 4 in a R T R R T T P k F F P ps ps r π π π π = . ( 11\n\n)\n\nThis expression can be understood in terms of antenna theory for electromagnetic waves.\n\nIn this context, we refer to the apparatus consisting of the resonator and the structure creating the magnetic field as an axion antenna. In analogy with conventional antenna theory, we define the directivity as 2 ( ) /( 4 )\n\na rad S r D P r π = , ( 12\n\n)\n\nwhere P rad is the total power radiated by the transmitting antenna. To obtain the total power radiated, we could integrate the power flux (9) over a sphere enclosing the antenna. However, it is easier to do the calculation in the near field using the axion field at the aperture of the transmitting antenna. Using ( ) ( / ) (0)\n\na T T T S L F p S γ π = , we have 2 ( ) (2 / ) rad a T T T T in P S L s F p P π = = . (13)\n\nSubstituting ( 13 ) for the total radiated power and (9) for the power flux, the directivity simplifies to\n\n2 2 (2 / ) (4 / )( /2) T a T a T D s s π λ π λ = = . ( 14\n\n)\n\nThe relation between the directivity and physical area ( 14 ) is ½ that found in conventional antenna theory, or equivalently, the area appears to be half the physical area. This results from the bi-directional radiation properties of the resonator. Defining an efficiency as /\n\nrad in P P η = , we also define the antenna gain as 2 (2 / ) (2 / ) T T T T T a T G D F p s η π π λ = = , (15) where (2 / ) T T T F p η π = .\n\nNext suppose that at some distant location this transmitted field is incident upon a receive antenna with length L R , radius R R , and finesse F R . From (10) and assuming the photons emitted from both ends of the receive antenna are collected, the total power collected will be 0 As with electromagnetic antennas, the ratio of effective area to gain is found to be independent of the details of the antenna, other than whether or not photons are collected from both ends of the antenna when used to receive:\n\n2 , , / / / (4 ) . e R R e T T c a s G s G n λ π = = ( 17\n\n)\n\nIf photons were collected only from one end of the receive antenna ( 1 c n = ), then Eq. ( 17 ) would be identical to conventional antenna theory.\n\nWith these definitions, the expression for the received power ( 11 ) can be interpreted as a Friis-like equation:\n\n( ) ( )( )\n\n2 2 2 0 , , , / / 4 / ( ) / in c T R a e T c T a e R P P n G G r s n s r λ π λ = = . (18) Here , c T n is the value used to compute , e T s according to Eq. (16). The ratio , , / e T c T s n is independent of the choice of , c T\n\nn , as it should be since the number of detectors that might be used on receive is independent of the transmit properties of the antenna.\n\nIt is also useful to note that if the magnetic field is uniform (i.e., wigglers, or quasi-phase matching, are not used [12] ), then there is an optimum length for the conversion region of an antenna. This occurs when ( )\n\nsin / 2 1 qL = , or qL π = . The optimum length is found to be 2 ( / ) opt a L m γ λ ω = . ( 19\n\n)\n\nIn obtaining this expression, we have used 2 /(2 ) a q m ω ≈ , which is valid for a m ω .\n\nThe diffraction-limited power pattern of the radiated axion field is determined by the aperture size in wavelengths through the Bessel function term in ( 7 ):\n\n( ) ( )\n\n2 1 4 sin /( sin ) d a a P J k R k R θ θ θ = ⎡ ⎤ ⎣ ⎦ . ( 20\n\n)\n\nThe diffraction beam width between first nulls is determined by the first zero of the Airy disc, and for small angles is given by the well-known expression\n\n1.22 / d FWFN T R γ θ λ ≈ . (21)\n\nSimilarly, the diffraction beam width at half maximum is determined by the roots of ( )\n\n) / 0.514 / d FWHM a T T R R γ θ πλ λ = ≈ . 1/ 2 d P θ = , or (1.616 /\n\nIn contrast, the conversion-limited power pattern is given by ( ) ( )\n\n2 sin cos / 2 ( ) 2 cos / 2 a c a k k L P k k L γ γ θ π θ θ ⎡ ⎤ ⎡ ⎤ - ⎣ ⎦ ⎢ ⎥ = ⎢ ⎥ - ⎣ ⎦ , (23)\n\nand depends on both the length in wavelengths and the velocity mismatch. For the optimum length L given by ( 19 ), the conversion beam widths are approximately given by 2( / ), 1.06( / )\n\nc c FWFN a FWHM a m m θ ω θ ω ≈ ≈ . ( 24\n\n)\n\nFor quantum-limited detection, the channel capacity is [16] log [17], and c is the velocity of light. Equation (25) can be combined with (11) or (18) to find the axion parameters that would permit a particular channel capacity at a given distance for a particular experimental apparatus.\n\n(1 / ) d R C N ν η ν = Δ + Δ , (25)\n\nAs an example, consider transmitters and receivers with 1064 nm (1.17 eV)\n\nγ λ = , 10 W in P = , 2 0.01 m s = , 0 10 T B = , 3 L = m, 5 3.1 10 F = × , 0.5 d η =\n\n, and a minimum information capacity of 1 bps. Figure 3 shows the inverse coupling strength curves\n\n( ) 1/ a a M m g γγ =\n\nfor distances of 1000 km and the diameter of the earth. Also shown is the curve for communication between the earth and the far side of the moon using the \"4+4\" experimental apparatus proposed in [15] . The shaded regions are excluded by the results from the BFRT collaboration [13] and Robilliard et al. [8] . The dots represent extensions of the curves if quasi-phase matching (QPM) is used by periodically reversing the magnetic field [12] . The minimum period of the reversal is taken to be twice the beam diameter for the 3 m system example, and 28.6 m for the 4+4 system. From the figure, communication over distances in excess of 1000 km should be possible for 3.5\n\na m < meV and 7 2 10 GeV M < × ( 8 1 5 10 GeV a g γγ -- Note that the receive resonator must be tuned to the same frequency as the transmit resonator to within a fraction of the resonator line width which is 161 Hz in this example. This will present a significant challenge, especially since the resonators are remote from one another. A possible approach would be to use atomic clocks to stabilize both the transmit frequency and a local reference frequency at the receiver. The receive resonator would then be locked to the reference to get as close as possible to the correct frequency, then slowly tuned until the signal is located. A feedback loop could then be closed to lock the receive resonator to the signal.\n\nIn summary, for 3.5 > × ), it may be possible to realize a new type of wireless signaling that cannot be blocked or shielded. An example calculation shows that communication between points located diametrically opposite on the earth should be possible. This could enable world-wide communication without the use of satellites or the ionosphere. However, with present knowledge, the signaling will be limited to low data rates, perhaps on the order of a few bits per second for terrestrial links. This estimate assumes 3 m long generation/regeneration regions to allow fully steerable instruments. Though not easily steerable, the apparatus in the \"4+4\" experiment proposed by Sikivie et al., [15] may enable communication to the far side of the moon for 0.3 a m < meV and 6 6 10 GeV M < × . I would like to acknowledge helpful discussions with Jim Lesh, Rich Holman, Jeff Peterson, Pierre Sikivie, and David Tanner during the development of these ideas. Electronic address: stancil@cmu.edu 17. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6 th Edition (Oxford Univ. Press, New York, 2007), pp. 171. 18. C-C. Chen, op cit., p. 106. B 0R L T L R Mod Laser Demod Data In Data Out B 0T Optical Detectors B 0T B 0R Photon input Photon output Axions penetrate barrier opaque to photons L T L R (a) Photons Axions Cavity mirrors (b) Figure 1, Stancil, Phys. Rev. D. x y z r θ -L T /2 L T /2"
}
] | |
arxiv:0704.0495 | 0704.0495 | 1 | 10.3842/SIGMA.2007.075 | 1901032a0ef23b90804048b2aef0d82a6e8e98bc6f56ec72e49cd08936aac7b0 | The Veldkamp Space of Two-Qubits | Given a remarkable representation of the generalized Pauli operators of
two-qubits in terms of the points of the generalized quadrangle of order two,
W(2), it is shown that specific subsets of these operators can also be
associated with the points and lines of the four-dimensional projective space
over the Galois field with two elements - the so-called Veldkamp space of W(2).
An intriguing novelty is the recognition of (uni- and tri-centric) triads and
specific pentads of the Pauli operators in addition to the "classical" subsets
answering to geometric hyperplanes of W(2). | [
"Metod Saniga (ASTRINSTSAV)",
"Michel Planat (FEMTO-ST)",
"Petr Pracna\n (JH-Inst)",
"Hans Havlicek (TUW)"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2 N -dimensional Hilbert spaces is embodied in the geometry of the symplectic polar space of rank N and order two [1, 2, 3] . The case of two-qubit operator space, N = 2, was scrutinized in very detail [1, 3] by explicitly demonstrating, in different ways, the correspondence between various subsets of the generalized Pauli operators/matrices and the fundamental subgeometries of the associated rank-two polar space -the (unique) generalized quadrangle of order two. In this paper we will reveal another interesting geometry hidden behind the Pauli operators of two-qubits, namely that of the Veldkamp space defined on this generalized quadrangle.
In this section we will briefly highlight the basics of the theory of finite generalized quadrangles [4] and introduce the concept of the Veldkamp space of a point-line incidence geometry [5] to be employed in what follows. 2 M. Saniga, M. Planat, P. Pracna and H. Havlicek A finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), is an incidence structure S = (P, B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [4]: (i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique pair (y, M ) ∈ P × B for which xIM IyIL; from these axioms it readily follows that |P | = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). It is obvious that there exists a point-line duality with respect to which each of the axioms is self-dual. Interchanging points and lines in S thus yields a generalized quadrangle S D of order (t, s), called the dual of S. If s = t, S is said to have order s. The generalized quadrangle of order (s, 1) is called a grid and that of order (1, t) a dual grid. A generalized quadrangle with both s > 1 and t > 1 is called thick.
Given two points x and y of S one writes x ∼ y and says that x and y are collinear if there exists a line L of S incident with both. For any x ∈ P denote x ⊥ = {y ∈ P |y ∼ x} and note that x ∈ x ⊥ ; obviously, x ⊥ = 1 + s + st. Given an arbitrary subset A of P , the perp(-set) of A, A ⊥ , is defined as A ⊥ = {x ⊥ |x ∈ A} and A ⊥⊥ := (A ⊥ ) ⊥ . A triple of pairwise non-collinear points of S is called a triad; given any triad T , a point of T ⊥ is called its center and we say that T is acentric, centric or unicentric according as |T ⊥ | is, respectively, zero, non-zero or one. An ovoid of a generalized quadrangle S is a set of points of S such that each line of S is incident with exactly one point of the set; hence, each ovoid contains st + 1 points.
The concept of crucial importance is a geometric hyperplane H of a point-line geometry Γ(P, B), which is a proper subset of P such that each line of Γ meets H in one or all points [6] . For Γ = GQ(s, t), it is well known that H is one of the following three kinds: (i) the perp-set of a point x, x ⊥ ; (ii) a (full) subquadrangle of order (s, t ′ ), t ′ < t; and (iii) an ovoid.
Finally, we need to introduce the notion of the Veldkamp space of a point-line incidence geometry Γ(P, B), V(Γ) [5] . V(Γ) is the space in which (i) a point is a geometric hyperplane of Γ and (ii) a line is the collection i = 1, 2) , where H 1 and H 2 are distinct points of V(Γ). 1 If Γ = S, from the preceding paragraph we learn that the points of V(S) are, in general, of three different types.
H 1 H 2 of all geometric hyperplanes H of Γ such that H 1 H 2 = H 1 H = H 2 H or H = H i (
The smallest thick GQ is obviously the one with s = t = 2, dubbed the "doily." This quadrangle has a number of interesting representations of which we mention the most important two [4]. One, frequently denoted as W 3 (2) or simply W (2), is in terms of the points of P G(3, 2) (i.e., the three-dimensional projective space over the Galois field with two elements) together with the totally isotropic lines with respect to a symplectic polarity. The other, usually denoted as Q(4, 2), is in terms of points and lines of a parabolic quadric in P G (4, 2) . By abuse of notation, any GQ isomorphic to W (2) will also be denoted by this symbol. From the preceding section we readily get that W (2) is endowed with 15 points/lines, each line contains three points and, dually, each point is on three lines; moreover, it is a self-dual object, i.e., isomorphic to its dual. W (2) features all the three kinds of hyperplanes, of the following cardinalities [5]: 15 perp-sets, x ⊥ , seven points each; 10 grids (of order (2, 1)), nine points each; and six ovoids, five points each -as depicted in Fig. 1 . The quadrangle exhibits two distinct kinds of triads, viz. unicentric and tricentric. A point of W (2) is the center of four distinct unicentric triads (Fig. 2 , left ); hence, 1 It is important to mention here that the definition of Veldkamp space given by Shult in [7] is more restrictive than that of Buekenhout and Cohen [5] adopted in this paper.
The Veldkamp Space of Two-Qubits 3 Figure 1. The three kinds of geometric hyperplanes of W (2). The points of the quadrangle are represented by small circles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the quadrangle. The upper panel shows the points' perp-sets (yellow bullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouring will become clear later. Each pictureexcept that in the bottom right-hand corner -stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon.
Figure 2. Left: -The four distinct unicentric triads (grey bullets) and their common center (black bullet); note that the triads intersect pairwise in a single point and their union covers fully the center's perp-set. Right: -A grid (red bullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); the two triads are also seen to comprise a dual grid (of order (1, 2)). 4 M. Saniga, M. Planat, P. Pracna and H. Havlicek Figure 3. The five different kinds of the lines of V(W (2)), each being uniquely determined by the properties of its core-set (black bullets). Note that the "yellow" hyperplanes (i.e., perp-sets) occur in each type, and yellow is also the colour of two homogeneous (i.e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also worth mentioning that the cardinality of core-sets is an odd number not exceeding five. The three hyperplanes of any line are always in such relation to each other that their union comprises all the points of W (2).
The Veldkamp Space of Two-Qubits 5 the number of such triads is 4 × 15 = 60. Tricentric triads always come in "complementary" pairs, one representing the centers of the other, and each such pair is the complement of a grid of W (2) (Fig. 2, right); hence, the number of such triads is 2 × 10 = 20. A unicentric triad is always a subset of an ovoid, which is never the case for a tricentric triad; the latter, in graphcombinatorial terms, representing a complete bipartite graph on six vertices. Now, we have enough background information at hand to reveal the structure of the Veldkamp space of our "doily", V(W (2)). 2
From the definition given in Section 2, we easily see that V(W (2)) consists of 31 points of which 15 are represented/generated by single-point perp-sets, 10 by grids and six by ovoids. The lines of V(W (2)) feature three points each and are of five distinct types, as illustrated in Fig. 3. These types differ from each other in the cardinality and structure of "core-sets", i.e., the sets of points of W (2) shared by all the three hyperplanes forming a given line. As it is obvious from Fig. 3, the lines of the first three types (the first three rows of the figure) have the core-sets of the same cardinality, three, differing from each other only in the structure of these sets as being unicentric triads, tricentric triads and triples of collinear points, respectively. The lines of the fourth type have as core-sets pentads of points, each being a quadruple of points collinear with a given point of W (2), whereas core-sets of the last type's lines feature just a single point. A much more interesting issue is the composition of the lines. Just a brief look at Fig. 3 reveals that geometric hyperplanes of only one kind, namely perp-sets, are present on each line of V(W (2)); grids and ovoids occur only on two kinds of the lines. We also see that the purely homogeneous types are those whose core-sets feature collinear triples and tricentric triads, the most heterogeneous type -the one exhibiting all the three kinds of hyperplanes -being that characterized by unicentric triads. We also notice that there are no lines comprising solely grids and/or solely ovoids, nor the lines featuring only grids and ovoids, which seems to be connected with the fact that the cardinality of a core-set is an odd number. From the properties of W (2) and its triads as discussed above it readily follows that the number of the lines of type one to five is 60, 20, 15, 45 and 15, respectively, totalling 155. All these observations and facts are gathered in Table 1 . We conclude this section with the observation that V(W (2)) has the same number of points (31) and lines (155) as P G(4, 2), the four-dimensional projective space over the Galois field of two elements [8] ; this is not a coincidence, as the two spaces are, in fact, isomorphic to each other [5].
4 Pauli operators of two-qubits in light of V(W (2))
As discovered in [1] (see also [3]), the fifteen generalized Pauli operators/matrices associated with the Hilbert space of two-qubits (see, e.g., [9]) can be put into a one-to-one correspondence with the fifteen points of the generalized quadrangle W (2) in such a way that their commutation algebra is completely and uniquely reproduced by the geometry of W (2) in which the concept commuting/non-commuting translates into that of collinear/non-collinear. Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [1, 3] for more details). Yet, V(W (2)) puts this bijection in a different light, in which other three subsets of the Pauli operators come into play, namely those represented by the two types of a triad and by the specific pentads occurring as the core-sets of the lines of V(W (2)) (Table 1 ). As already mentioned, the role of tricentric triads of the operators has 2 As this paper is primarily aimed at physicists rather than mathematicians, in what follows we opt for an elementary and self-contained exposition of the Veldkamp space of W (2); this explanation is based only upon some very simple properties of W (2) readily to be grasped from its depiction as "the doily", and does not presuppose/require any further background from the reader. 6 M. Saniga, M. Planat, P. Pracna and H. Havlicek Table 1. A succinct summary of the properties of the five different types of the lines of V(W (2)) in terms of the core-sets and the types of geometric hyperplanes featured by a generic line of a given type. The last column gives the total number of lines per the corresponding type.
Type of Core-Set Perp-Sets Grids Ovoids # Single Point 1 0 2 15 Collinear Triple 3 0 0 15 Unicentric Triad 1 1 1 60 Tricentric Triad 3 0 0 20 Pentad 1 2 0 45 Table 2. Three kinds of the distinguished subsets of the generalized Pauli operators of two-qubits (PO) viewed as the geometric hyperplanes in the generalized quadrangle of order two (GQ) [1, 3].
PO set of five mutually set of six operators nine operators of non-commuting operators commuting with a given one a Mermin's square GQ ovoid perp-set\{reference point} grid been recognized in disguise of complete bipartite graphs on six vertices [3]. A true novelty here is obviously unicentric triads and pentads of the generalized Pauli operators as these are all intimately connected with single-point perp-sets; given a point of W (2) (i.e., a generalized Pauli operator of two-qubits), its perp-set fully encompasses four unicentric triads (Fig. 2 , left) and three pentads (Fig. 3 , 4th row) of points/operators. This feature has also a very interesting aspect in connection with the conjecture relating the existence of mutually unbiased bases and finite projective planes raised in [10] , because with each point x of W (2) there is associated a projective plane of order two (the Fano plane) whose points are the elements of x ⊥ and whose lines are the spans {u,
v} ⊥⊥ , where u, v ∈ x ⊥ with u = v [4].
Identifying the Pauli operators of a two-qubit system with the points of the generalized quadrangle of order two led to the discovery of three distinguished subsets of the operators in terms of geometric hyperplanes of the quadrangle. Here we go one level higher, and identifying these subsets with the points of the associated Veldkamp space leads to recognition of nother remarkable subsets of the Pauli operators, viz. unicentric triads and pentads. It is really intriguing to see that these are the core-sets of the two kinds of lines that both feature grids alias Mermin squares. As it is well known, Mermin squares, which reveal certain important aspects of the entanglement of the system, play a crucial role in the proof of the Kochen-Specker theorem in dimension four and our approach gives a novel geometrical meaning to this [3, 11] . At the Veldkamp space level it turns out of particular importance to study relations between eigenvectors of the above-mentioned unicentric triads and pentad of operators in order to reveal finer, hitherto unnoticed traits of the structure of Mermin squares. These seem to be intimately connected with the existence of outer automorphisms of the symmetric group on six letters, which is the full group of automorphisms of our quadrangle; as this group is the only symmetric group possessing (non-trivial) outer automorphisms, this implies that two-qubits have a rather special footing among multiple qubit systems. All these aspects deserve special attention and will therefore be dealt with in a separate paper.
Concerning three-qubits, our preliminary study indicates that the corresponding finite geometry differs fundamentally from that of W (2) in the sense that it contains multi-lines, i.e., two or more lines passing through two distinct points [12] . As we do not have a full picture at hand yet, we cannot see if it admits hyperplanes and so lends itself to constructing the corresponding Veldkamp space. If the latter does exist, it is likely to differ substantially from that of twoqubits, which would imply the expected difference between entanglement properties of the two
The Veldkamp Space of Two-Qubits 7 kinds of systems; if not, this will only further strengthen the above-mentioned uniqueness of two-qubits.
By employing the concept of the Veldkamp space of the generalized quadrangle of order two, we were able to recognize other, on top of those examined in [1, 2, 3] , distinguished subsets of generalized Pauli operators of two-level quantum systems, namely unicentric triads and pentads of them. It may well be that these two kinds of subsets of the two-qubit Pauli operators hold an important key for getting deeper insights into the nature of finite geometries underlying multiple higher-level quantum systems [12, 13] , in particular when the dimension of Hilbert space is not a power of a prime [14].
This work was partially supported by the Science and Technology Assistance Agency under the contract # APVT-51-012704, the VEGA grant agency projects # 2/6070/26 and # 7012 (all from Slovak Republic), the trans-national ECO-NET project # 12651NJ "Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases" (France), the CNRS-SAV Project # 20246 "Projective and Related Geometries for Quantum Information" (France/Slovakia) and by the Action Austria-Slovakia project # 58s2 "Finite Geometries Behind Hilbert Spaces". | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2 N -dimensional Hilbert spaces is embodied in the geometry of the symplectic polar space of rank N and order two [1, 2, 3] . The case of two-qubit operator space, N = 2, was scrutinized in very detail [1, 3] by explicitly demonstrating, in different ways, the correspondence between various subsets of the generalized Pauli operators/matrices and the fundamental subgeometries of the associated rank-two polar space -the (unique) generalized quadrangle of order two. In this paper we will reveal another interesting geometry hidden behind the Pauli operators of two-qubits, namely that of the Veldkamp space defined on this generalized quadrangle."
},
{
"section_type": "OTHER",
"section_title": "Finite generalized quadrangles and Veldkamp spaces",
"text": "In this section we will briefly highlight the basics of the theory of finite generalized quadrangles [4] and introduce the concept of the Veldkamp space of a point-line incidence geometry [5] to be employed in what follows. 2 M. Saniga, M. Planat, P. Pracna and H. Havlicek A finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), is an incidence structure S = (P, B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [4]: (i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique pair (y, M ) ∈ P × B for which xIM IyIL; from these axioms it readily follows that |P | = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). It is obvious that there exists a point-line duality with respect to which each of the axioms is self-dual. Interchanging points and lines in S thus yields a generalized quadrangle S D of order (t, s), called the dual of S. If s = t, S is said to have order s. The generalized quadrangle of order (s, 1) is called a grid and that of order (1, t) a dual grid. A generalized quadrangle with both s > 1 and t > 1 is called thick.\n\nGiven two points x and y of S one writes x ∼ y and says that x and y are collinear if there exists a line L of S incident with both. For any x ∈ P denote x ⊥ = {y ∈ P |y ∼ x} and note that x ∈ x ⊥ ; obviously, x ⊥ = 1 + s + st. Given an arbitrary subset A of P , the perp(-set) of A, A ⊥ , is defined as A ⊥ = {x ⊥ |x ∈ A} and A ⊥⊥ := (A ⊥ ) ⊥ . A triple of pairwise non-collinear points of S is called a triad; given any triad T , a point of T ⊥ is called its center and we say that T is acentric, centric or unicentric according as |T ⊥ | is, respectively, zero, non-zero or one. An ovoid of a generalized quadrangle S is a set of points of S such that each line of S is incident with exactly one point of the set; hence, each ovoid contains st + 1 points.\n\nThe concept of crucial importance is a geometric hyperplane H of a point-line geometry Γ(P, B), which is a proper subset of P such that each line of Γ meets H in one or all points [6] . For Γ = GQ(s, t), it is well known that H is one of the following three kinds: (i) the perp-set of a point x, x ⊥ ; (ii) a (full) subquadrangle of order (s, t ′ ), t ′ < t; and (iii) an ovoid.\n\nFinally, we need to introduce the notion of the Veldkamp space of a point-line incidence geometry Γ(P, B), V(Γ) [5] . V(Γ) is the space in which (i) a point is a geometric hyperplane of Γ and (ii) a line is the collection i = 1, 2) , where H 1 and H 2 are distinct points of V(Γ). 1 If Γ = S, from the preceding paragraph we learn that the points of V(S) are, in general, of three different types.\n\nH 1 H 2 of all geometric hyperplanes H of Γ such that H 1 H 2 = H 1 H = H 2 H or H = H i ("
},
{
"section_type": "OTHER",
"section_title": "The smallest thick GQ and its Veldkamp space",
"text": "The smallest thick GQ is obviously the one with s = t = 2, dubbed the \"doily.\" This quadrangle has a number of interesting representations of which we mention the most important two [4]. One, frequently denoted as W 3 (2) or simply W (2), is in terms of the points of P G(3, 2) (i.e., the three-dimensional projective space over the Galois field with two elements) together with the totally isotropic lines with respect to a symplectic polarity. The other, usually denoted as Q(4, 2), is in terms of points and lines of a parabolic quadric in P G (4, 2) . By abuse of notation, any GQ isomorphic to W (2) will also be denoted by this symbol. From the preceding section we readily get that W (2) is endowed with 15 points/lines, each line contains three points and, dually, each point is on three lines; moreover, it is a self-dual object, i.e., isomorphic to its dual. W (2) features all the three kinds of hyperplanes, of the following cardinalities [5]: 15 perp-sets, x ⊥ , seven points each; 10 grids (of order (2, 1)), nine points each; and six ovoids, five points each -as depicted in Fig. 1 . The quadrangle exhibits two distinct kinds of triads, viz. unicentric and tricentric. A point of W (2) is the center of four distinct unicentric triads (Fig. 2 , left ); hence, 1 It is important to mention here that the definition of Veldkamp space given by Shult in [7] is more restrictive than that of Buekenhout and Cohen [5] adopted in this paper.\n\nThe Veldkamp Space of Two-Qubits 3 Figure 1. The three kinds of geometric hyperplanes of W (2). The points of the quadrangle are represented by small circles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the quadrangle. The upper panel shows the points' perp-sets (yellow bullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouring will become clear later. Each pictureexcept that in the bottom right-hand corner -stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon.\n\nFigure 2. Left: -The four distinct unicentric triads (grey bullets) and their common center (black bullet); note that the triads intersect pairwise in a single point and their union covers fully the center's perp-set. Right: -A grid (red bullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); the two triads are also seen to comprise a dual grid (of order (1, 2)). 4 M. Saniga, M. Planat, P. Pracna and H. Havlicek Figure 3. The five different kinds of the lines of V(W (2)), each being uniquely determined by the properties of its core-set (black bullets). Note that the \"yellow\" hyperplanes (i.e., perp-sets) occur in each type, and yellow is also the colour of two homogeneous (i.e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also worth mentioning that the cardinality of core-sets is an odd number not exceeding five. The three hyperplanes of any line are always in such relation to each other that their union comprises all the points of W (2).\n\nThe Veldkamp Space of Two-Qubits 5 the number of such triads is 4 × 15 = 60. Tricentric triads always come in \"complementary\" pairs, one representing the centers of the other, and each such pair is the complement of a grid of W (2) (Fig. 2, right); hence, the number of such triads is 2 × 10 = 20. A unicentric triad is always a subset of an ovoid, which is never the case for a tricentric triad; the latter, in graphcombinatorial terms, representing a complete bipartite graph on six vertices. Now, we have enough background information at hand to reveal the structure of the Veldkamp space of our \"doily\", V(W (2)). 2\n\nFrom the definition given in Section 2, we easily see that V(W (2)) consists of 31 points of which 15 are represented/generated by single-point perp-sets, 10 by grids and six by ovoids. The lines of V(W (2)) feature three points each and are of five distinct types, as illustrated in Fig. 3. These types differ from each other in the cardinality and structure of \"core-sets\", i.e., the sets of points of W (2) shared by all the three hyperplanes forming a given line. As it is obvious from Fig. 3, the lines of the first three types (the first three rows of the figure) have the core-sets of the same cardinality, three, differing from each other only in the structure of these sets as being unicentric triads, tricentric triads and triples of collinear points, respectively. The lines of the fourth type have as core-sets pentads of points, each being a quadruple of points collinear with a given point of W (2), whereas core-sets of the last type's lines feature just a single point. A much more interesting issue is the composition of the lines. Just a brief look at Fig. 3 reveals that geometric hyperplanes of only one kind, namely perp-sets, are present on each line of V(W (2)); grids and ovoids occur only on two kinds of the lines. We also see that the purely homogeneous types are those whose core-sets feature collinear triples and tricentric triads, the most heterogeneous type -the one exhibiting all the three kinds of hyperplanes -being that characterized by unicentric triads. We also notice that there are no lines comprising solely grids and/or solely ovoids, nor the lines featuring only grids and ovoids, which seems to be connected with the fact that the cardinality of a core-set is an odd number. From the properties of W (2) and its triads as discussed above it readily follows that the number of the lines of type one to five is 60, 20, 15, 45 and 15, respectively, totalling 155. All these observations and facts are gathered in Table 1 . We conclude this section with the observation that V(W (2)) has the same number of points (31) and lines (155) as P G(4, 2), the four-dimensional projective space over the Galois field of two elements [8] ; this is not a coincidence, as the two spaces are, in fact, isomorphic to each other [5].\n\n4 Pauli operators of two-qubits in light of V(W (2))\n\nAs discovered in [1] (see also [3]), the fifteen generalized Pauli operators/matrices associated with the Hilbert space of two-qubits (see, e.g., [9]) can be put into a one-to-one correspondence with the fifteen points of the generalized quadrangle W (2) in such a way that their commutation algebra is completely and uniquely reproduced by the geometry of W (2) in which the concept commuting/non-commuting translates into that of collinear/non-collinear. Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [1, 3] for more details). Yet, V(W (2)) puts this bijection in a different light, in which other three subsets of the Pauli operators come into play, namely those represented by the two types of a triad and by the specific pentads occurring as the core-sets of the lines of V(W (2)) (Table 1 ). As already mentioned, the role of tricentric triads of the operators has 2 As this paper is primarily aimed at physicists rather than mathematicians, in what follows we opt for an elementary and self-contained exposition of the Veldkamp space of W (2); this explanation is based only upon some very simple properties of W (2) readily to be grasped from its depiction as \"the doily\", and does not presuppose/require any further background from the reader. 6 M. Saniga, M. Planat, P. Pracna and H. Havlicek Table 1. A succinct summary of the properties of the five different types of the lines of V(W (2)) in terms of the core-sets and the types of geometric hyperplanes featured by a generic line of a given type. The last column gives the total number of lines per the corresponding type.\n\nType of Core-Set Perp-Sets Grids Ovoids # Single Point 1 0 2 15 Collinear Triple 3 0 0 15 Unicentric Triad 1 1 1 60 Tricentric Triad 3 0 0 20 Pentad 1 2 0 45 Table 2. Three kinds of the distinguished subsets of the generalized Pauli operators of two-qubits (PO) viewed as the geometric hyperplanes in the generalized quadrangle of order two (GQ) [1, 3].\n\nPO set of five mutually set of six operators nine operators of non-commuting operators commuting with a given one a Mermin's square GQ ovoid perp-set\\{reference point} grid been recognized in disguise of complete bipartite graphs on six vertices [3]. A true novelty here is obviously unicentric triads and pentads of the generalized Pauli operators as these are all intimately connected with single-point perp-sets; given a point of W (2) (i.e., a generalized Pauli operator of two-qubits), its perp-set fully encompasses four unicentric triads (Fig. 2 , left) and three pentads (Fig. 3 , 4th row) of points/operators. This feature has also a very interesting aspect in connection with the conjecture relating the existence of mutually unbiased bases and finite projective planes raised in [10] , because with each point x of W (2) there is associated a projective plane of order two (the Fano plane) whose points are the elements of x ⊥ and whose lines are the spans {u,\n\nv} ⊥⊥ , where u, v ∈ x ⊥ with u = v [4].\n\nIdentifying the Pauli operators of a two-qubit system with the points of the generalized quadrangle of order two led to the discovery of three distinguished subsets of the operators in terms of geometric hyperplanes of the quadrangle. Here we go one level higher, and identifying these subsets with the points of the associated Veldkamp space leads to recognition of nother remarkable subsets of the Pauli operators, viz. unicentric triads and pentads. It is really intriguing to see that these are the core-sets of the two kinds of lines that both feature grids alias Mermin squares. As it is well known, Mermin squares, which reveal certain important aspects of the entanglement of the system, play a crucial role in the proof of the Kochen-Specker theorem in dimension four and our approach gives a novel geometrical meaning to this [3, 11] . At the Veldkamp space level it turns out of particular importance to study relations between eigenvectors of the above-mentioned unicentric triads and pentad of operators in order to reveal finer, hitherto unnoticed traits of the structure of Mermin squares. These seem to be intimately connected with the existence of outer automorphisms of the symmetric group on six letters, which is the full group of automorphisms of our quadrangle; as this group is the only symmetric group possessing (non-trivial) outer automorphisms, this implies that two-qubits have a rather special footing among multiple qubit systems. All these aspects deserve special attention and will therefore be dealt with in a separate paper.\n\nConcerning three-qubits, our preliminary study indicates that the corresponding finite geometry differs fundamentally from that of W (2) in the sense that it contains multi-lines, i.e., two or more lines passing through two distinct points [12] . As we do not have a full picture at hand yet, we cannot see if it admits hyperplanes and so lends itself to constructing the corresponding Veldkamp space. If the latter does exist, it is likely to differ substantially from that of twoqubits, which would imply the expected difference between entanglement properties of the two\n\nThe Veldkamp Space of Two-Qubits 7 kinds of systems; if not, this will only further strengthen the above-mentioned uniqueness of two-qubits."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusion",
"text": "By employing the concept of the Veldkamp space of the generalized quadrangle of order two, we were able to recognize other, on top of those examined in [1, 2, 3] , distinguished subsets of generalized Pauli operators of two-level quantum systems, namely unicentric triads and pentads of them. It may well be that these two kinds of subsets of the two-qubit Pauli operators hold an important key for getting deeper insights into the nature of finite geometries underlying multiple higher-level quantum systems [12, 13] , in particular when the dimension of Hilbert space is not a power of a prime [14]."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgements",
"text": "This work was partially supported by the Science and Technology Assistance Agency under the contract # APVT-51-012704, the VEGA grant agency projects # 2/6070/26 and # 7012 (all from Slovak Republic), the trans-national ECO-NET project # 12651NJ \"Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases\" (France), the CNRS-SAV Project # 20246 \"Projective and Related Geometries for Quantum Information\" (France/Slovakia) and by the Action Austria-Slovakia project # 58s2 \"Finite Geometries Behind Hilbert Spaces\"."
}
] | |
arxiv:0704.0505 | 0704.0505 | 1 | 10.1103/PhysRevD.76.085001 | aeda426d1071e75e895afdc97caa1fa9e76af03cbfa583d2180a72529e4ba2a9 | Exact Solutions of Einstein-Yang-Mills Theory with Higher-Derivative
Coupling | We construct a classical solution of an Einstein-Yang-Mills system with a
fourth order term with respect to the field strength of the Yang-Mills field.
The solution provides a spontaneous compactification proposed by Cremmer and
Scherk; ten-dimensional space-time with a cosmological constant is compactified
to the four-dimensional Minkowski space with a six-dimensional sphere S^6 on
which an instanton solution exists. The radius of the sphere is not a modulus
but is determined by the gauge coupling and the four-derivative coupling
constants and the Newton's constant. We also construct a solution of
ten-dimensional theory without a cosmological constant compactified to AdS_4 x
S^6. | [
"Hironobu Kihara",
"Muneto Nitta"
] | [
"hep-th",
"gr-qc"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term with respect to the field strength of the Yang-Mills field. The solution provides a compactification proposed by Cremmer and Scherk; ten-dimensional space-time with a cosmological constant is compactified to the four-dimensional Minkowski space with a six-dimensional sphere S 6 on which an instanton solution exists. The radius of the sphere is not a modulus but is determined by the gauge coupling and the four-derivative coupling constants and the Newton's constant. We also construct a solution of ten-dimensional theory without a cosmological constant compactified to AdS 4 × S 6 .
Unification of fundamental forces with space-time and matter often requires higherdimensional space-time rather than our four-dimensional Universe. The early Kaluza-Klein theory unifies gravity and the electro-magnetic interaction by considering five-dimensional space-time with one direction compactified into a circle S 1 [1] . This old idea has been revisited several times. After supergravity was discovered many people tried to unify all forces and matter in higher-dimensional space-time compactified on various internal manifolds [2] .
String theory was proposed as the most attractive candidate of unification, but it is defined only in ten-dimensional space-time. In order to realize four-dimensional Universe one has to find a suitable six-dimensional internal space. So many candidates of such spaces were proposed; Calabi-Yau manifolds and orbifold models. Internal manifolds can be deformed with satisfying the Einstein equation, and these degrees of freedom are called the moduli.
The moduli introduce unwanted massless particles in four-dimensional world. Recently a new mechanism has been suggested to fix these moduli by turning on the Ramond-Ramond flux on the internal space [3, 4] . This flux compactification has been extensively studied in these years.
We would like to revise the compactification scenario with fixed moduli proposed by Cremmer and Scherk long time ago [5] (see also [6, 7] ) in a theory with a cosmological constant. By placing solitons on a compact internal space they showed decompactifying limit with large radius of the internal space is disfavored and the radius is fixed to a certain value determined by coupling constants. They considered the 't Hooft-Polyakov monopole [8] on S 2 and the Yang-Mills instanton [9] on S 4 , both of which can satisfy, with proper coupling constants, the first order (self-dual) equations rather than the second order equations of motion, but their solutions on higher dimensional sphere are not the case. Since string theory is defined in ten dimensions, it is natural to consider this scenario with stable BPS solitons on a six-dimensional internal space like S 6 .
Higher dimensional generalization of self-dual equations was suggested by Tchrakian some years ago [10] . Eight dimensional case is known as octonionic instantons [11] . Though several works have been done for generalized self-dual equations [12, 13] , a six-dimensional case was not discussed because of the lack of conformal property. Recently we have found a new solution to the generalized self-dual equations in an SO (6) pure Yang-Mills theory with a fourth order term with respect to the field strength of the Yang-Mills field (a four-derivative term) on a six-dimensional sphere S 6 [14] .
In this letter we propose to use this solution in the context of a compactification of the Cremmer-Scherk type. In our model ten-dimensional space-time with (without) a cosmological constant is compactified to a four-dimensional Minkowski space M 4 (anti de Sitter space AdS 4 ) with a six-dimensional sphere S 6 , where dimensionality of the internal space, six, is required by the four derivative term. Unlike the case of the absence of gravity [14] the four-derivative coupling constant α can differ from the constant β in the generalized self-dual equations. When the relation α = β holds the generalized self-dual equations become the Bogomol'nyi equations and solutions are BPS. We find for both M 4 × S 6 and AdS 4 × S 6 that certain relations exist between the radius of S 6 , the gauge coupling, the four-derivative coupling α and the gravitational coupling constants. When the four-derivative coupling constant α vanishes in the case of M 4 × S 6 , these relations reduce to those of the original work by Cremmer and Scherk. The advantage of our model to the Cremmer-Scherk model is that the Yang-Mills soliton in our model satisfies the self-dual equations (the Bogomol'nyi equations for α = β) rather than usual equations of motion in the case of the Cremmer-Scherk model. This ensures the stability of configuration at least for the sector of Yang-Mills fields.
Let us consider that space-time is a ten-dimensional manifold. We consider an Einstein-Yang-Mills theory. Our action contains as dynamical variables the Yang-Mills (gauge) fields
A [ab] μ
and a graviton field or the metric ĝμν . Indices with a hat "ˆ" will refer to a tendimensional space-time (X 0 , X 1 , • • • , X 9 ). Latin indices (a, b, • • • ) run from 1 to 6 and refer to an internal space. The Clifford algebra associated with the orthogonal group SO( 6 ) is useful and we represent generators of the Lie algebra so(6) as their elements. The Clifford algebra is defined by gamma matrices {Γ a } which satisfy the following anti-commutation relations, {Γ a , Γ b } = 2δ ab . These matrices can be realized as 8 × 8 matrices with complex coefficients. The generators of so( 6 ) are represented by
Γ ab = 1 2 [Γ a , Γ b ].
We often abbreviate the Yang-Mills fields as
A μ = 1 2 A [ab]
μ Γ ab and we also use notations with differential forms. Thus the gauge fields are expressed as A = A μdX μ. In this notation, the corresponding field strength F is written as F = dA + eA ∧ A, where e is a gauge coupling. Covariant derivative
D μ on an adjoint representation Y = 1 2 Y [ab] Γ ab is defined as D μY = ∂ μY + e(A μY -Y A μ),
where Y is a scalar multiplet. The action S total consists of two parts. One is the Einstein-Hilbert action S E and the other S Y M T is a Yang-Mills action with a term which is the fourth power of the field strength F . Such a quartic term has been studied by Tchrakian [10] and so we call it the Tchrakian term. The total action is:
S total = S E + S Y M T , S E = 1 16πG dvR , S Y M T = 1 16 Tr -F ∧ * F + α 2 (F ∧ F ) ∧ * (F ∧ F ) -V 0 dv . (1)
Here the 10-form dv is an invariant volume form with respect to the metric ĝ and is written as dv = √ -ĝd 10 X in a local patch. The scalar curvature is denoted by R. The asterisk " * " denotes the Hodge dual operator. This operator defines an inner product over differential forms, and for a given form ω, ω ∧ * ω is proportional to the invariant volume form dv. 1 The parameters of this action are the Newton's gravitational constant G, the gauge coupling e, the four-derivative coupling α and the cosmological constant V 0 .
We show the explicit form of the Yang-Mills part with components of A and F ,
S Y M T = -dv 1 8 F [ab] μν F μν,[ab] + α 2 8 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 16 S μν ρσ S μν ρσ + 1 2 V 0 , (2)
F = 1 4 F [ab] μν dX μ ∧ dX ν Γ ab , S μν ρσ = F [ab] μν F [ab] ρσ + F [ab] μρ F [ab] σ ν + F [ab] μσ F [ab] ν ρ , (3)
T [ab][cd] μν ρσ = F [ab] μν F [cd] ρσ + F [ab] μρ F [cd] σ ν + F [ab] μσ F [cd] ν ρ , T [abcd] μν ρσ = 1 6 T [ab][cd]
μν ρσ + T
[ac][db] μν ρσ + T [ad][bc] μν ρσ + T [cd][ab] μν ρσ + T [db][ac] μν ρσ + T [bc][ad] μν ρσ .
The Euler-Lagrange equations from these actions read the usual Einstein equation and the equations for the Yang-Mills fields:
R μν - 1 2 ĝμν R = 8πGT μν , D μ √ -gF μν -2α 2 √ -gF [μν F ρσ] F ρσ = 0 . (5)
Here the energy-momentum tensor T μν is obtained by the variation of the Yang-Mills part with respect to the metric:
T μν = 1 2 F μ ρ,[ab] F [ab] ν ρ + α 2 T [abcd] μρσ τ T ν ρστ ,[ab][cd] + α 2 3 • 2 S μρσ τ S ν ρσ τ - 1 2 g μν χ χ = 1 4 F [ab] μν F μν,[ab] + α 2 4 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 8 S μν ρσ S μν ρσ + V 0 . (6)
To solve these equations, we make an ansatz which is the same as that of Cremmer-Scherk.
Our ansatz for the metric is the following:
ds 2 = η µν dx µ dx ν + δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J = ĝμν dX μdX ν , y 2 = 6 a=1 (y I ) 2 , ( 7
)
1 The Hodge dual operator acting on a differential form on a space with Minkowski signature satisfies the following relation:
(F µν dx µν ) ∧ * (F ρσ dx ρσ ) = -F µν F µν dv.
where the coordinates X are the total space-time coordinates. The metric η µν = diag(-+ ++) is the Lorentz metric on the four-dimensional Minkowski space. Greek indices without a hat "ˆ", for instance µ will refer to the first four variables. Capital indices (I, J, • • • ) run from one to six and refer to the compact space. The six-dimensional space is taken as a sphere with a radius R 0 . The Riemann tensor, Ricci tensor and scalar curvature are
R I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ , R = 30 R 2 0 . (8)
The rest components of the curvature tensor vanish. In this space, the Einstein equations in (5) reduce to simple equations,
- 1 2 η µν 30 R 2 0 = 8πGT µν , 0 = T µI , - 1 2 20 R 2 0 g IJ = 8πGT IJ . (9)
We now make ansatzes for the gauge fields. We assume that the fields A do not depend on the four-dimensional directions, ∂ µ A = 0, and they have no four-dimensional components A µ = 0. This implies that the field strengths are two forms on the six-dimensional sphere:
A = A I (y)dy I , F = 1 2
F IJ dy I ∧ dy J . With these ansatzes, the four-dimensional part of the energy-momentum tensor becomes -1 2 η µν χ, and the equation reduces to 30/R 2 0 = 8πGχ. This equation requires that the χ is a constant. Suppose that the field strength fulfils the generalized self-dual condition
F = iβγ 7 * 6 (F ∧ F ), ( 10
)
where β is a real parameter. Here " * 6 " means the Hodge dual on the six-dimensional sphere.
Then the second part of the equations of motion ( 5 ) is fulfilled automatically by the relation DF = 0, where the exterior covariant derivative is defined as DF = dF +e (A ∧ F -F ∧ A).
In fact we have an explicit solution to the self-dual equation:
A = 1 4eR 2 0 y a e b Γ ab , F = 1 4eR 2 0 e a ∧ e b Γ ab , β = eR 2 0 3 . ( 11
)
Here we identify the internal space index and the sphere index. The energy-momentum tensor of this configuration becomes
ζ ≡ α 2 β 2 , χ = (1 + ζ) 15 4e 2 R 4 0 + V 0 , T IJ = - 1 2 (1 -ζ) 5 4e 2 R 4 0 + V 0 g IJ . (12)
With these ansatzes we obtain algebraic equations from the Einstein equations:
30 R 2 0 = 8πG 1 2 (1 + ζ) 15 2e 2 R 4 0 + V 0 , 10 R 2 0 = 8πG (1 -ζ) 5 8e 2 R 4 0 + V 0 2 , (13)
From these we finally obtain
1 πG = 1 e 2 R 2 0 (2 + 4ζ) , V 0 = 15 4e 2 R 4 0 (1 + 3ζ) . (14)
When the four-derivative coupling vanishes, α = 0 and therefore ζ = 0, these relations reduce to those of the Cremmer and Scherk [5] . 2 When the relation α = β holds (ζ = 1) our solution saturates the Bogomol'nyi bound and becomes a BPS state. The energy density is given by an integral over S 6 as follows:
E S 6 Y M T = 1 16 S 6 Tr -F ∧ * 6 F + α 2 (F ∧ F ) ∧ * 6 (F ∧ F ) = 1 16 S 6 Tr (iF ∓ αγ 7 * 6 (F ∧ F )) ∧ * 6 (iF ∓ αγ 7 * 6 (F ∧ F )) ± i 8 α S 6 Trγ 7 F ∧ F ∧ F ≥ ± i 8 α S 6 Trγ 7 F ∧ F ∧ F = ∓ 1 2 3 S 6 ǫ abcdef F [ab] ∧ F [cd] ∧ F [ef ] ≡ ±Q , (15)
where the field strength F has only components along S 6 . If the coupling α is equal to β, the solution of eq. ( 10 ) satisfies the Bogomol'nyi equation and the energy attains the local minimum. We can also consider a system coupled with scalar fields. Suppose that scalar fields Q m transform as a representation of SO (6) . The index m labels the representation space. Let us add an action S Q of the scalar fields Q with a Higgs potential
S Q = 1 2 dvD μQ m D μQ m + V (Q 2 ) , D μQ m = ∂ μQ m - 1 2 ieA [ab] μ R(Γ ab ) mm ′ Q m ′ (16)
to S total . The equations of motion are modified. In general, our solution mentioned above does not satisfy the modified equations any more. However, for the scalars which fulfil the covariantly constant condition D μQ m = 0 and attain the absolute minimum V (Q) = 0, the configurations of A and g in equations ( 7 ), (11) are still solutions for the modified equations. Here the constant value of the minimum is shifted to 0. Thus we can argue the Higgs mechanism around our solutions.
Next we suppose that the four-dimensional part is an anti-de Sitter space AdS 4 of radius R A . Our ansatz for the metric is the following:
ds 2 = η µν (x)dx µ dx ν + g IJ (y)dy I dy J = ĝμν dX μdX ν , (17)
g IJ (y)dy
I dy J = δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J y 2 = 6 a=1 (y I ) 2 , η µν (x)dx µ dx ν = R 2 A cos 2 θ -dτ 2 + dθ 2 + sin 2 θdΩ 2 , dΩ 2 = |dz| 2 (1 + |z| 2 /4) 2 , (18)
where z parametrizes a whole complex plane. The metric η µν (x) is a maximally symmetric metric on the four-dimensional anti-de Sitter space. The Riemann tensor and the Ricci tensor are
R µ νρσ = - 1 R 2 A δ µ ρ η νσ -δ µ σ η νρ , R µν = - 3 R 2 A η µν , R I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ . (19)
The total scalar curvature is obtained by a summation of those of two parts: R = -
12 R 2 A + 30 R 2 0 .
In this space, the Einstein equations are
R µν - 1 2 η µν R = 8πGT µν , R IJ - 1 2 g IJ R = 8πGT IJ . (20)
The ansatz for the gauge fields is the same as previous one and the energy momentum tensor does not change. With these ansatzes, we obtain algebraic equations from the Einstein
equations as 3 R 2 A - 15 R 2 0 = -4πG (1 + ζ) 15 4e 2 R 4 0 + V 0 , 6 R 2 A - 10 R 2 0 = -4πG (1 -ζ) 5 4e 2 R 4 0 + V 0 . (21)
We are interested in a possible relation to string theory and therefore we consider the case with the vanishing cosmological constant, V 0 = 0. In this case, the radii (R A , R 0 ) are written by the couplings,
R 2 0 = (5 + 7ζ) πG 4e 2 , R 2 A = 5 + 7ζ 5 + 15ζ R 2 0 . (22)
Thus the additional higher derivative coupling term of the Tchrakian type does not affect critically to the equations of motion. When ζ = 1 our solution becomes a solution of the Bogomol'nyi equation again.
Our solutions introduced in this letter are new solutions of the system with a Tchrakian term. The origin of this term has not been clear so far but it seems rather universal in order to construct solitons with codimensions higher than four: for instance it has played a crucial role to construct a finite energy monopole (with codimension five) in a six-dimensional spacetime [13] . Though the parameter ζ(= α 2 /β 2 ) is a free parameter, we expect that the system goes to ζ = 1 because it becomes BPS. There are several discussions on the (in)stability of higher-dimensional Yang-Mills theories [15] . To compute the mass spectra of the fluctuations around our solutions is a future work. When the scalar fields Q m are non-trivially coupled, the system may allow BPS composite solitons which are made of solitons with different codimensions, as in the case of usual self-dual Yang-Mills equations coupled to Higgs fields [16] .
Finally our solution of AdS 4 × S 6 may have a relation with D2-branes, and we hope that there exists some impact on AdS/CFT duality [17] . | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term with respect to the field strength of the Yang-Mills field. The solution provides a compactification proposed by Cremmer and Scherk; ten-dimensional space-time with a cosmological constant is compactified to the four-dimensional Minkowski space with a six-dimensional sphere S 6 on which an instanton solution exists. The radius of the sphere is not a modulus but is determined by the gauge coupling and the four-derivative coupling constants and the Newton's constant. We also construct a solution of ten-dimensional theory without a cosmological constant compactified to AdS 4 × S 6 ."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Unification of fundamental forces with space-time and matter often requires higherdimensional space-time rather than our four-dimensional Universe. The early Kaluza-Klein theory unifies gravity and the electro-magnetic interaction by considering five-dimensional space-time with one direction compactified into a circle S 1 [1] . This old idea has been revisited several times. After supergravity was discovered many people tried to unify all forces and matter in higher-dimensional space-time compactified on various internal manifolds [2] .\n\nString theory was proposed as the most attractive candidate of unification, but it is defined only in ten-dimensional space-time. In order to realize four-dimensional Universe one has to find a suitable six-dimensional internal space. So many candidates of such spaces were proposed; Calabi-Yau manifolds and orbifold models. Internal manifolds can be deformed with satisfying the Einstein equation, and these degrees of freedom are called the moduli.\n\nThe moduli introduce unwanted massless particles in four-dimensional world. Recently a new mechanism has been suggested to fix these moduli by turning on the Ramond-Ramond flux on the internal space [3, 4] . This flux compactification has been extensively studied in these years.\n\nWe would like to revise the compactification scenario with fixed moduli proposed by Cremmer and Scherk long time ago [5] (see also [6, 7] ) in a theory with a cosmological constant. By placing solitons on a compact internal space they showed decompactifying limit with large radius of the internal space is disfavored and the radius is fixed to a certain value determined by coupling constants. They considered the 't Hooft-Polyakov monopole [8] on S 2 and the Yang-Mills instanton [9] on S 4 , both of which can satisfy, with proper coupling constants, the first order (self-dual) equations rather than the second order equations of motion, but their solutions on higher dimensional sphere are not the case. Since string theory is defined in ten dimensions, it is natural to consider this scenario with stable BPS solitons on a six-dimensional internal space like S 6 .\n\nHigher dimensional generalization of self-dual equations was suggested by Tchrakian some years ago [10] . Eight dimensional case is known as octonionic instantons [11] . Though several works have been done for generalized self-dual equations [12, 13] , a six-dimensional case was not discussed because of the lack of conformal property. Recently we have found a new solution to the generalized self-dual equations in an SO (6) pure Yang-Mills theory with a fourth order term with respect to the field strength of the Yang-Mills field (a four-derivative term) on a six-dimensional sphere S 6 [14] .\n\nIn this letter we propose to use this solution in the context of a compactification of the Cremmer-Scherk type. In our model ten-dimensional space-time with (without) a cosmological constant is compactified to a four-dimensional Minkowski space M 4 (anti de Sitter space AdS 4 ) with a six-dimensional sphere S 6 , where dimensionality of the internal space, six, is required by the four derivative term. Unlike the case of the absence of gravity [14] the four-derivative coupling constant α can differ from the constant β in the generalized self-dual equations. When the relation α = β holds the generalized self-dual equations become the Bogomol'nyi equations and solutions are BPS. We find for both M 4 × S 6 and AdS 4 × S 6 that certain relations exist between the radius of S 6 , the gauge coupling, the four-derivative coupling α and the gravitational coupling constants. When the four-derivative coupling constant α vanishes in the case of M 4 × S 6 , these relations reduce to those of the original work by Cremmer and Scherk. The advantage of our model to the Cremmer-Scherk model is that the Yang-Mills soliton in our model satisfies the self-dual equations (the Bogomol'nyi equations for α = β) rather than usual equations of motion in the case of the Cremmer-Scherk model. This ensures the stability of configuration at least for the sector of Yang-Mills fields.\n\nLet us consider that space-time is a ten-dimensional manifold. We consider an Einstein-Yang-Mills theory. Our action contains as dynamical variables the Yang-Mills (gauge) fields\n\nA [ab] μ\n\nand a graviton field or the metric ĝμν . Indices with a hat \"ˆ\" will refer to a tendimensional space-time (X 0 , X 1 , • • • , X 9 ). Latin indices (a, b, • • • ) run from 1 to 6 and refer to an internal space. The Clifford algebra associated with the orthogonal group SO( 6 ) is useful and we represent generators of the Lie algebra so(6) as their elements. The Clifford algebra is defined by gamma matrices {Γ a } which satisfy the following anti-commutation relations, {Γ a , Γ b } = 2δ ab . These matrices can be realized as 8 × 8 matrices with complex coefficients. The generators of so( 6 ) are represented by\n\nΓ ab = 1 2 [Γ a , Γ b ].\n\nWe often abbreviate the Yang-Mills fields as\n\nA μ = 1 2 A [ab]\n\nμ Γ ab and we also use notations with differential forms. Thus the gauge fields are expressed as A = A μdX μ. In this notation, the corresponding field strength F is written as F = dA + eA ∧ A, where e is a gauge coupling. Covariant derivative\n\nD μ on an adjoint representation Y = 1 2 Y [ab] Γ ab is defined as D μY = ∂ μY + e(A μY -Y A μ),\n\nwhere Y is a scalar multiplet. The action S total consists of two parts. One is the Einstein-Hilbert action S E and the other S Y M T is a Yang-Mills action with a term which is the fourth power of the field strength F . Such a quartic term has been studied by Tchrakian [10] and so we call it the Tchrakian term. The total action is:\n\nS total = S E + S Y M T , S E = 1 16πG dvR , S Y M T = 1 16 Tr -F ∧ * F + α 2 (F ∧ F ) ∧ * (F ∧ F ) -V 0 dv . (1)\n\nHere the 10-form dv is an invariant volume form with respect to the metric ĝ and is written as dv = √ -ĝd 10 X in a local patch. The scalar curvature is denoted by R. The asterisk \" * \" denotes the Hodge dual operator. This operator defines an inner product over differential forms, and for a given form ω, ω ∧ * ω is proportional to the invariant volume form dv. 1 The parameters of this action are the Newton's gravitational constant G, the gauge coupling e, the four-derivative coupling α and the cosmological constant V 0 .\n\nWe show the explicit form of the Yang-Mills part with components of A and F ,\n\nS Y M T = -dv 1 8 F [ab] μν F μν,[ab] + α 2 8 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 16 S μν ρσ S μν ρσ + 1 2 V 0 , (2)\n\nF = 1 4 F [ab] μν dX μ ∧ dX ν Γ ab , S μν ρσ = F [ab] μν F [ab] ρσ + F [ab] μρ F [ab] σ ν + F [ab] μσ F [ab] ν ρ , (3)\n\nT [ab][cd] μν ρσ = F [ab] μν F [cd] ρσ + F [ab] μρ F [cd] σ ν + F [ab] μσ F [cd] ν ρ , T [abcd] μν ρσ = 1 6 T [ab][cd]\n\nμν ρσ + T\n\n[ac][db] μν ρσ + T [ad][bc] μν ρσ + T [cd][ab] μν ρσ + T [db][ac] μν ρσ + T [bc][ad] μν ρσ .\n\nThe Euler-Lagrange equations from these actions read the usual Einstein equation and the equations for the Yang-Mills fields:\n\nR μν - 1 2 ĝμν R = 8πGT μν , D μ √ -gF μν -2α 2 √ -gF [μν F ρσ] F ρσ = 0 . (5)\n\nHere the energy-momentum tensor T μν is obtained by the variation of the Yang-Mills part with respect to the metric:\n\nT μν = 1 2 F μ ρ,[ab] F [ab] ν ρ + α 2 T [abcd] μρσ τ T ν ρστ ,[ab][cd] + α 2 3 • 2 S μρσ τ S ν ρσ τ - 1 2 g μν χ χ = 1 4 F [ab] μν F μν,[ab] + α 2 4 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 8 S μν ρσ S μν ρσ + V 0 . (6)\n\nTo solve these equations, we make an ansatz which is the same as that of Cremmer-Scherk.\n\nOur ansatz for the metric is the following:\n\nds 2 = η µν dx µ dx ν + δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J = ĝμν dX μdX ν , y 2 = 6 a=1 (y I ) 2 , ( 7\n\n)\n\n1 The Hodge dual operator acting on a differential form on a space with Minkowski signature satisfies the following relation:\n\n(F µν dx µν ) ∧ * (F ρσ dx ρσ ) = -F µν F µν dv.\n\nwhere the coordinates X are the total space-time coordinates. The metric η µν = diag(-+ ++) is the Lorentz metric on the four-dimensional Minkowski space. Greek indices without a hat \"ˆ\", for instance µ will refer to the first four variables. Capital indices (I, J, • • • ) run from one to six and refer to the compact space. The six-dimensional space is taken as a sphere with a radius R 0 . The Riemann tensor, Ricci tensor and scalar curvature are\n\nR I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ , R = 30 R 2 0 . (8)\n\nThe rest components of the curvature tensor vanish. In this space, the Einstein equations in (5) reduce to simple equations,\n\n- 1 2 η µν 30 R 2 0 = 8πGT µν , 0 = T µI , - 1 2 20 R 2 0 g IJ = 8πGT IJ . (9)\n\nWe now make ansatzes for the gauge fields. We assume that the fields A do not depend on the four-dimensional directions, ∂ µ A = 0, and they have no four-dimensional components A µ = 0. This implies that the field strengths are two forms on the six-dimensional sphere:\n\nA = A I (y)dy I , F = 1 2\n\nF IJ dy I ∧ dy J . With these ansatzes, the four-dimensional part of the energy-momentum tensor becomes -1 2 η µν χ, and the equation reduces to 30/R 2 0 = 8πGχ. This equation requires that the χ is a constant. Suppose that the field strength fulfils the generalized self-dual condition\n\nF = iβγ 7 * 6 (F ∧ F ), ( 10\n\n)\n\nwhere β is a real parameter. Here \" * 6 \" means the Hodge dual on the six-dimensional sphere.\n\nThen the second part of the equations of motion ( 5 ) is fulfilled automatically by the relation DF = 0, where the exterior covariant derivative is defined as DF = dF +e (A ∧ F -F ∧ A).\n\nIn fact we have an explicit solution to the self-dual equation:\n\nA = 1 4eR 2 0 y a e b Γ ab , F = 1 4eR 2 0 e a ∧ e b Γ ab , β = eR 2 0 3 . ( 11\n\n)\n\nHere we identify the internal space index and the sphere index. The energy-momentum tensor of this configuration becomes\n\nζ ≡ α 2 β 2 , χ = (1 + ζ) 15 4e 2 R 4 0 + V 0 , T IJ = - 1 2 (1 -ζ) 5 4e 2 R 4 0 + V 0 g IJ . (12)\n\nWith these ansatzes we obtain algebraic equations from the Einstein equations:\n\n30 R 2 0 = 8πG 1 2 (1 + ζ) 15 2e 2 R 4 0 + V 0 , 10 R 2 0 = 8πG (1 -ζ) 5 8e 2 R 4 0 + V 0 2 , (13)\n\nFrom these we finally obtain\n\n1 πG = 1 e 2 R 2 0 (2 + 4ζ) , V 0 = 15 4e 2 R 4 0 (1 + 3ζ) . (14)\n\nWhen the four-derivative coupling vanishes, α = 0 and therefore ζ = 0, these relations reduce to those of the Cremmer and Scherk [5] . 2 When the relation α = β holds (ζ = 1) our solution saturates the Bogomol'nyi bound and becomes a BPS state. The energy density is given by an integral over S 6 as follows:\n\nE S 6 Y M T = 1 16 S 6 Tr -F ∧ * 6 F + α 2 (F ∧ F ) ∧ * 6 (F ∧ F ) = 1 16 S 6 Tr (iF ∓ αγ 7 * 6 (F ∧ F )) ∧ * 6 (iF ∓ αγ 7 * 6 (F ∧ F )) ± i 8 α S 6 Trγ 7 F ∧ F ∧ F ≥ ± i 8 α S 6 Trγ 7 F ∧ F ∧ F = ∓ 1 2 3 S 6 ǫ abcdef F [ab] ∧ F [cd] ∧ F [ef ] ≡ ±Q , (15)\n\nwhere the field strength F has only components along S 6 . If the coupling α is equal to β, the solution of eq. ( 10 ) satisfies the Bogomol'nyi equation and the energy attains the local minimum. We can also consider a system coupled with scalar fields. Suppose that scalar fields Q m transform as a representation of SO (6) . The index m labels the representation space. Let us add an action S Q of the scalar fields Q with a Higgs potential\n\nS Q = 1 2 dvD μQ m D μQ m + V (Q 2 ) , D μQ m = ∂ μQ m - 1 2 ieA [ab] μ R(Γ ab ) mm ′ Q m ′ (16)\n\nto S total . The equations of motion are modified. In general, our solution mentioned above does not satisfy the modified equations any more. However, for the scalars which fulfil the covariantly constant condition D μQ m = 0 and attain the absolute minimum V (Q) = 0, the configurations of A and g in equations ( 7 ), (11) are still solutions for the modified equations. Here the constant value of the minimum is shifted to 0. Thus we can argue the Higgs mechanism around our solutions.\n\nNext we suppose that the four-dimensional part is an anti-de Sitter space AdS 4 of radius R A . Our ansatz for the metric is the following:\n\nds 2 = η µν (x)dx µ dx ν + g IJ (y)dy I dy J = ĝμν dX μdX ν , (17)\n\ng IJ (y)dy\n\nI dy J = δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J y 2 = 6 a=1 (y I ) 2 , η µν (x)dx µ dx ν = R 2 A cos 2 θ -dτ 2 + dθ 2 + sin 2 θdΩ 2 , dΩ 2 = |dz| 2 (1 + |z| 2 /4) 2 , (18)\n\nwhere z parametrizes a whole complex plane. The metric η µν (x) is a maximally symmetric metric on the four-dimensional anti-de Sitter space. The Riemann tensor and the Ricci tensor are\n\nR µ νρσ = - 1 R 2 A δ µ ρ η νσ -δ µ σ η νρ , R µν = - 3 R 2 A η µν , R I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ . (19)\n\nThe total scalar curvature is obtained by a summation of those of two parts: R = -\n\n12 R 2 A + 30 R 2 0 .\n\nIn this space, the Einstein equations are\n\nR µν - 1 2 η µν R = 8πGT µν , R IJ - 1 2 g IJ R = 8πGT IJ . (20)\n\nThe ansatz for the gauge fields is the same as previous one and the energy momentum tensor does not change. With these ansatzes, we obtain algebraic equations from the Einstein\n\nequations as 3 R 2 A - 15 R 2 0 = -4πG (1 + ζ) 15 4e 2 R 4 0 + V 0 , 6 R 2 A - 10 R 2 0 = -4πG (1 -ζ) 5 4e 2 R 4 0 + V 0 . (21)\n\nWe are interested in a possible relation to string theory and therefore we consider the case with the vanishing cosmological constant, V 0 = 0. In this case, the radii (R A , R 0 ) are written by the couplings,\n\nR 2 0 = (5 + 7ζ) πG 4e 2 , R 2 A = 5 + 7ζ 5 + 15ζ R 2 0 . (22)\n\nThus the additional higher derivative coupling term of the Tchrakian type does not affect critically to the equations of motion. When ζ = 1 our solution becomes a solution of the Bogomol'nyi equation again.\n\nOur solutions introduced in this letter are new solutions of the system with a Tchrakian term. The origin of this term has not been clear so far but it seems rather universal in order to construct solitons with codimensions higher than four: for instance it has played a crucial role to construct a finite energy monopole (with codimension five) in a six-dimensional spacetime [13] . Though the parameter ζ(= α 2 /β 2 ) is a free parameter, we expect that the system goes to ζ = 1 because it becomes BPS. There are several discussions on the (in)stability of higher-dimensional Yang-Mills theories [15] . To compute the mass spectra of the fluctuations around our solutions is a future work. When the scalar fields Q m are non-trivially coupled, the system may allow BPS composite solitons which are made of solitons with different codimensions, as in the case of usual self-dual Yang-Mills equations coupled to Higgs fields [16] .\n\nFinally our solution of AdS 4 × S 6 may have a relation with D2-branes, and we hope that there exists some impact on AdS/CFT duality [17] ."
}
] | |
arxiv:0704.0507 | 0704.0507 | 1 | 10.1103/PhysRevD.76.124023 | 22f44bb72e223a65c9213c7d3684c85f57dc3b9bd321cac32466088966d0a47e | E_6 and the bipartite entanglement of three qutrits | Recent investigations have established an analogy between the entropy of
four-dimensional supersymmetric black holes in string theory and entanglement
in quantum information theory. Examples include: (1) N=2 STU black holes and
the tripartite entanglement of three qubits (2-state systems), where the common
symmetry is [SL(2)]^3 and (2) N=8 black holes and the tripartite entanglement
of seven qubits where the common symmetry is E_7 which contains [SL(2)]^7. Here
we present another example: N=8 black holes (or black strings) in five
dimensions and the bipartite entanglement of three qutrits (3-state systems),
where the common symmetry is E_6 which contains [SL(3)]^3. Both the black hole
(or black string) entropy and the entanglement measure are provided by the
Cartan cubic E_6 invariant. Similar analogies exist for ``magic'' N=2
supergravity black holes in both four and five dimensions. | [
"M. J. Duff and S. Ferrara"
] | [
"hep-th",
"quant-ph"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | 1 m.duff@imperial.ac.uk 2 Sergio.Ferrara@cern.ch Contents 1 D = 4 black holes and qubits 1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 1.2 N = 2 black holes and the bipartite entanglement of two qubits . . . . . . . 1.3 N = 8 black holes and the tripartite entanglement of seven qubits . . . . . . 1.4 Magic supergravities in D = 4 . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Five-dimensional supergravity 3 D = 5 black holes and qutrits 3.1 N = 2 black holes and the bipartite entanglement of two qutrits . . . . . . . 3.2 N = 8 black holes and the bipartite entanglement of three qutrits . . . . . . 3.3 Magic supergravities in D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Conclusions 5 Acknowledgements 2 1 D = 4 black holes and qubits
It sometimes happens that two very different areas of theoretical physics share the same mathematics. This may eventually lead to the realisation that they are, in fact, dual descriptions of the same physical phenomena, or it may not. Either way, it frequently leads to new insights in both areas. Recent papers [1, 2, 3, 4, 5, 6] have established an analogy between the entropy of certain four-dimensional supersymmetric black holes in string theory and entanglement measures in quantum information theory. In this paper we extend the analogy from four dimensions to five which also involves going from two-state systems (qubits) to three-state systems (qutrits). We begin by recalling the four-dimensional examples:
1.1 N = 2 black holes and the tripartite entanglement of three qubits
The three qubit system (Alice, Bob, Charlie) is described by the state
|Ψ = a ABC |ABC (1.1)
where A = 0, 1, so the Hilbert space has dimension 2 3 = 8. The complex numbers a ABC transforms as a (2, 2, 2) under SL(2, C) A × SL(2, C) B × SL(2, C) C . The tripartite entanglement is measured by the 3-tangle [7, 8]
τ 3 (ABC) = 4|Det a ABC |. (1.2)
where Det a ABC is Cayley's hyperdeterminant [9] .
Det a = - 1 2 ǫ A 1 A 2 ǫ B 1 B 2 ǫ A 3 A 4 ǫ B 3 B 4 ǫ C 1 C 4 ǫ C 2 C 3 a A 1 B 1 C 1 a A 2 B 2 C 2 a A 3 B 3 C 3 a A 4 B 4 C 4 (1.
3) The hyperdeterminant is invariant under SL(2) A × SL(2) B × SL(2) C and under a triality that interchanges A, B and C. In the context of stringy black holes the 8 a ABC are the 4 electric and 4 magnetic charges of the N = 2 STU black hole [10] and hence take on real (integer) values. The ST U model corresponds to N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . The Bekenstein-Hawking entropy of the black hole, S, was first calculated in [11] . The connection to quantum information theory arises by noting [1] that it can also be expressed in terms of Cayley's hyperdeterminant
S = π |Det a ABC |.
(1.4) One can then establish a dictionary between the classification of various entangled states (separable A-B-C; bipartite entangled A-BC, B-CA, C-AB; tripartite entangled W; tripartite entangled GHZ) and the classfication of various "small" and "large" BPS and non-BPS black holes [1, 2, 3, 4, 5, 6] . For example, the GHZ state [12] |Ψ ∼ |111 + |000 (1.5) 3 with Det a ABC ≥ 0 corresponds to a large non-BPS 2-charge black hole; the W-state |Ψ ∼ |100 + |010 + |001 (1.6) with Det a ABC = 0 corresponds to a small-BPS 3-charge black hole; the GHZ-state
|Ψ = -|000 + |011 + |101 + |110 (1.7)
corresponds to a large BPS 4-charge black hole.
1.2 N = 2 black holes and the bipartite entanglement of two qubits
An even simpler example [2] is provided by the two qubit system (Alice and Bob) described by the state |Ψ = a AB |AB (1.8) where A = 0, 1, and the Hilbert space has dimension 2 2 = 4. The a AB transforms as a (2, 2) under SL(2, C) A × SL(2, C) B . The entanglement is measured by the 2-tangle τ 2 (AB) = C 2 (AB) (1.9) where C(AB) = 2 |det a AB | (1.10) is the concurrence. The determinant is invariant under SL(2, C) A × SL(2, C) B and under a duality that interchanges A and B. Here it is sufficient to look at N = 2 supergravity coupled to just one vector multiplet and the 4 a AB are the 2 electric and 2 magnetic charges of the axion-dilaton black hole with entropy S = π|det a AB | (1.11) For example, the Bell state |Ψ ∼ |11 + |00 (1.12) with det a AB ≥ 0 corresponds to a large non-BPS 2-charge black hole.
We recall that in the case of D = 4, N = 8 supergravity, the the 28 electric and 28 magnetic charges belong to the 56 of E 7 (7) . The black hole entropy is [15, 18] (1.13) where J 4 is Cartan's quartic E 7 invariant [13, 14] . It may be written
S = π |J 4 |
J 4 = P ij Q jk P kl Q li -1 4 P ij Q ij P kl Q kl + 1 96 ǫ ijklmnop Q ij Q kl Q mn Q op + ǫ ijklmnop P ij P kl P mn P op . (1.14) 4
where P ij and Q jk are 8 × 8 antisymmetric matrices. The qubit interpretation [4] relies on the decomposition E 7 (C) ⊃ [SL(2, C)] 7 (1.15) under which 56 → 2, 2, 1, 2, 1, 1, 1) + (1, 2, 2, 1, 2, 1, 1 1, 1, 2, 2, 1, 2, 1) +(1, 1, 1, 2, 2, 1, 2) +(2, 1, 1, 1, 2, 2, 1) + (1, 2, 1, 1, 1, 2, 2 2, 1, 2, 1, 1, 1, 2) (1.16) suggesting the tripartite entanglement of seven qubits (Alice, Bob, Charlie, Daisy, Emma, Fred and George) described by the state.
(
) +(
) +(
|Ψ = a ABD |ABD +b BCE |BCE +c CDF |CDF +d DEG |DEG +e EF A |EF A +f F GB |F GB +g GAC |GAC (1.17)
where A = 0, 1, so the Hilbert space has dimension 7.2 3 = 56. The a, b, c, d, e, f, g transform as a 56 of E 7 (C). The entanglement may be represented by a heptagon where the vertices A,B,C,D,E,F,G represent the seven qubits and the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC represent the tripartite entanglement. See Figure 1 . Alternatively, we can use the Fano plane. See Figure 2 . The Fano plane also corresponds to the multiplication table of the octonions 3 The measure of the tripartite entanglement of the seven qubits is provided by the 3-tangle
τ 3 (ABCDEF G) = 4|J 4 | (1.18) with J 4 ∼ a 4 + b 4 + c 4 + d 4 + e 4 + f 4 + g 4 +
3 Not the "split" octonions as was incorrectly stated in the published version of [4].
A B C D E F G Figure 1: The E 7 entanglement diagram. Each of the seven vertices A,B,C,D,E,F,G represents a qubit and each of the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC describes a tripartite entanglement.
2[a 2 b 2 + b 2 c 2 + c 2 d 2 + d 2 e 2 + e 2 f 2 + f 2 g 2 + g 2 a 2 + a 2 c 2 + b 2 d 2 + c 2 e 2 + d 2 f 2 + e 2 g 2 + f 2 a 2 + g 2 b 2 + a 2 d 2 + b 2 e 2 + c 2 f 2 + d 2 g 2 + e 2 a 2 + f 2 b 2 + g 2 c 2 ] +8[bcdf + cdeg + def a + ef gb + f gac + gabd + abce] (1.19)
where products like a 4 = (ABD)(ABD)(ABD)(ABD)
= ǫ A 1 A 2 ǫ B 1 B 2 ǫ D 1 D 4 ǫ A 3 A 4 ǫ B 3 B 4 ǫ D 2 D 3 a A 1 B 1 D 1 a A 2 B 2 D 2 a A 3 B 3 D 3 a A 4 B 4 D 4 (1.20)
exclude four individuals (here Charlie, Emma, Fred and George), products like
a 2 b 2 = (ABD)(ABD)(F GB)(F GB) = ǫ A 1 A 2 ǫ B 1 B 3 ǫ D 1 D 2 ǫ F 3 F 4 ǫ G 3 G 4 ǫ B 2 B 4 a A 1 B 1 D 1 a A 2 B 2 D 2 b F 3 G 3 B 3 b F 4 G 4 B 4 (1.21)
exclude two individuals (here Charlie and Emma), and products like
abce = (ABD)(BCE)(CDF )(EF A) = ǫ A 1 A 4 ǫ B 1 B 2 ǫ C 2 C 3 ǫ D 1 D 3 ǫ E 2 E 4 ǫ F 3 F 4 a A 1 B 1 D 1 b B 2 C 2 E 2 c C 3 D 3 F 3 e E 4 F 4 A 4 (1.22)
exclude one individual (here George) 4 . Once again large non-BPS, small BPS and large BPS black holes correspond to states with J 4 > 0, J 4 = 0 and J 4 < 0, respectively.
4 This corrects the corresponding equation in the published version of [4] which had the wrong index contraction.
F E B G C A D
Figure 2: The Fano plane has seven points, representing the seven qubits, and seven lines (the circle counts as a line) with three points on every line, representing the tripartite entanglement, and three lines through every point.
The black holes described by Cayley's hyperdeterminant are those of N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . In [4] the following four-dimensional generalizations were considered: 1) N = 2 supergravity coupled to l vector multiplets where the symmetry is SL(2, Z) × SO(l -1, 2, Z) and the black holes carry charges belonging to the (2, l + 1) representation (l + 1 electric plus l + 1 magnetic).
2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SL(2, Z) × SO(6, m, Z) where the black holes carry charges belonging to the (2, 6 + m) representation (m + 6 electric plus m + 6 magnetic).
3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 7(7) (Z) and the black holes carry charges belonging to the fundamental 56-dimensional representation (28 electric plus 28 magnetic).
In all three case there exist quartic invariants akin to Cayley's hyperdeterminant whose square root yields the corresponding black hole entropy. In [4] we succeeded in giving a quantum theoretic interpretation in the N = 8 case together with its truncations to N = 4 (with m = 6) and N = 2 (with l = 3, the case we already knew [1]).
However, as suggested by Levay [5], one might also consider the "magic" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 4 supergravity coupled to 6, 9, 15 and 27 vector multiplets with symmetries Sp(6, Z), SU(3, 3), SO * (12) and E 7(-25) , respectively. Once again, as has been shown just recently [20] , in all cases there are quartic invariants whose square root yields the corresponding black hole entropy.
Here we demonstrate that the black-hole/qubit correspondence does indeed continue to 7 hold for magic supergravities. The crucial observation is that, although the black hole charges a ABC are real (integer) numbers and the entropy (1.13) is invariant under E 7 (7)(Z), the coefficients a ABC that appear in the qutrit state (1.17) are complex. So the three tangle (1.18) is invariant under E 7 (C) which contains both E 7(7) (Z) and E 7(-25) (Z) as subgroups.
To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that
E 7(7) (Z) ⊃ [SL(2)(Z)] 7 (1.23) but E 7(-25) (Z) ⊃ [SL(2)(Z)] 7 (1.24)
is irrelevant. All that matters is that
E 7 (C) ⊃ [SL(2)(C)] 7 (1.25)
The same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic) . Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) above. When we looked at the seven qubit subsector E 7 (C) ⊃ SL(2, C) × SO(12, C), we gave an N = 4 supergravity interpretationwith symmetry SL(2, R)×SO(6, 6) [4], but we could equally have given an interpretation in terms of N = 2 supergravity coupled to 11 vector multiplets with symmetry SL(2, R) × SO(10, 2).
Moreover, SO(l -1, 2) is contained in SO(l + 1, C) and SO(6, m) is contained in SO(12 + m, C) so we can give a qubit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(4n, C) which contains [SL(2, C)] 2n .
In five dimensions we might consider: 1) N = 2 supergravity coupled to l+1 vector multiplets where the symmetry is SO(1, 1, Z)× SO(l, 1, Z) and the black holes carry charges belonging to the (l + 1) representation (all electric) .
2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SO(1, 1, Z)× SO(m, 5, Z) where the black holes carry charges belonging to the (m + 5) representation (all electric).
3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 6(6) (Z) and the black holes carry charges belonging to the fundamental 27-dimensional representation (all electric).
The electrically charged objects are point-like and the magnetic duals are one-dimensional, or string-like, transforming according to the contrgredient representation. In all three cases above there exist cubic invariants akin to the determinant which yield the corresponding black hole or black string entropy.
In this section we briefly describe the salient properties of maximal N = 8 case, following [16] . We have 27 abelian gauge fields which transform in the fundamental representation of 8 6(6) . The first invariant of E 6(6) is the cubic invariant [13, 17, 16, 18, 19]
E
J 3 = q ij Ω jl q lm Ω mn q np Ω pi (2.1)
where q ij is the charge vector transforming as a 27 which can be represented as traceless Sp(8) matrix. The entropy of a black hole with charges q ij is then given by
S = π |J 3 | (2.2)
We will see that a configuration with J 3 = 0 preserves 1/8 of the supersymmetries. If J 3 = 0 and ∂J 3 ∂q i = 0 then it preserves 1/4 of the supersymmetries, and finally if ∂J 3 ∂q i = 0 (and the charge vector q i is non-zero), the configuration preserves 1/2 of the supersymmetries. We will show this by choosing a particular basis for the charges, the general result following by U-duality.
In five dimensions the compact group H is USp(8). We choose our conventions so that USp(2) = SU(2). In the commutator of the supersymmetry generators we have a central charge matrix Z ab which can be brought to a normal form by a USp(8) transformation. In the normal form the central charge matrix can be written as
e ab = s 1 + s 2 -s 3 0 0 0 0 s 1 + s 3 -s 2 0 0 0 0 s 2 + s 3 -s 1 0 0 0 0 -(s 1 + s 2 + s 3 ) × 0 1 -1 0 (2.3) we can order s i so that s 1 ≥ s 2 ≥ |s 3 |.
The cubic invariant, in this basis, becomes
J 3 = s 1 s 2 s 3 (2.4)
Even though the eigenvalues s i might depend on the moduli, the invariant (2.4) only depends on the quantized values of the charges. We can write a generic charge configuration as UeU t , where e is the normal frame as above, and the invariant will then be (2.4). There are three distinct possibilities
J 3 = 0 s 1 , s 2 , s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 , s 2 = 0, s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 = 0, s 2 , s 3 = 0 (2.5)
Taking the case of type II on T 5 we can choose the rotation in such a way that, for example, s 1 corresponds to solitonic five-brane charge, s 2 to fundamental string winding charge along some direction and s 3 to Kaluza-Klein momentum along the same direction. We can see that in this specific example the three possibilities in (2.5) break 1/8, 1/4 and 1/2 supersymmetries. The respective orbits are
E 6 (6) F 4(4) E 6 (6) SO(5, 4) ×T 16 9 E 6 (6) SO(5, 5) ×T 16 (2.6)
This also shows that one can generically choose a basis for the charges so that all others are related by U-duality. The basis chosen here is the S-dual of the D-brane basis usually chosen for describing black holes in type II B on T 5 . All others are related by U-duality to this particular choice. Note that, in contrast to the four-dimensional case where flipping the sign of J 4 (1.14) interchanges BPS and non-BPS black holes, the sign of the J 3 (2.4) is not important since it changes under a CPT transformation. There is no non-BPS orbit in five dimensions.
In five dimensions there are also string-like configurations which are the magnetic duals of the configurations considered here. They transform in the contragredient 27 ′ representation and the solutions preserving 1/2, 1/4, 1/8 supersymmetries are characterized in an analogous way. We could also have configurations where we have both point-like and string-like ch the point-like charge is uniformly distributed along the string, it is more natural to consider this configuration as a point-like object in D = 4 by dimensional reduction.
It is useful to decompose the U-duality group into the T-duality group and the S-duality group. The decomposition reads E 6 → SO(5, 5) × SO(1, 1), leading to
27 → 16 1 + 10 -2 + 1 4 (2.7)
The last term in (2.7) corresponds to the NS five-brane charge. The 16 correspond to the D-brane charges and the 10 correspond to the 5 directions of KK momentum and the 5 directions of fundamental string winding, which are the charges that explicitly appear in string perturbation theory. The cubic invariant has the decomposition (27) 3 → 10 -2 10 -2 1 4 + 16 1 16 1 10 -2 (2.8) This is saying that in order to have a non-zero area black hole we must have three NS charges (more precisely some "perturbative" charges and a solitonic five-brane); or we can have two D-brane charges and one NS charge. In particular, it is not possible to have a black hole with a non-zero horizon area with purely D-brane charges. Notice that the non-compact nature of the groups is crucial in this classification.
So far, all the quantum information analogies involve four-dimensional black holes and qubits. In order to find an analogy with five-dimensional black holes we invoke three state systems called qutrits.
The two qutrit system (Alice and Bob) is described by the state
|Ψ = a AB |AB 10
where A = 0, 1, 2, so the Hilbert space has dimension 3 2 = 9. The a AB transforms as a (3, 3) under SL(3) A × SL(3) B . The bipartite entanglement is measured by the concurrence [21]
C(AB) = 3 3/2 |det a AB |. (3.1)
The determinant is invariant under SL(3, C) A × SL(3, C) B and under a duality that interchanges A and B. The black hole interpretation is provided by N = 2 supergravity coupled to 8 vector multiplets with symmetry SL(3, C) where the black hole charges transform as a 9. The entropy is given by
S = π|det a AB | (3.2)
3.2 N = 8 black holes and the bipartite entanglement of three qutrits
As we have seen in section (2) in the case of D = 5, N = 8 supergravity, the black hole charges belong to the 27 of E 6(6) and the entropy is given by (2.2).
The qutrit interpretation now relies on the decomposition
E 6 (C) ⊃ SL(3, C) A × SL(3, C) B × SL(3, C) C (3.3) under which 27 → (3, 3, 1) + (3 ′ , 1, 3) + (1, 3 ′ , 3 ′ ) (3.4)
suggesting the bipartite entanglement of three qutrits (Alice, Bob, Charlie). However, the larger symmetry requires that they undergo at most bipartite entanglement of a very specific kind, where each person has bipartite entanglement with the other two:
|Ψ = a AB |AB + b B C |BC + c CA |CA (3.5)
where A = 0, 1, 2, so the Hilbert space has dimension 3.3 2 = 27. The three states transforms as a pair of triplets under two of the SL(3)'s and singlets under the remaining one. Individually, therefore, the bipartite entanglement of each of the three states is given by the determinant (3.1). Taken together however, we see from (3.4) that they transform as a complex 27 of E 6 (C). The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements. See Fig. 3. The N=2 truncation of section 3.1 is represented by just the line AB with endpoints A and B. Note that: 1) Any pair of states has an individual in common 2) Each individual is excluded from one out of the three states The entanglement measure will be given by the concurrence
C(ABC) = 3 3/2 |J 3 | (3.6)
J 3 being the singlet in 27 × 27 × 27:
J 3 ∼ a 3 + b 3 + c 3 + 6abc (3.7) 11 A B C
Figure 3: The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements.
where the products
a 3 = ǫ A 1 A 2 A 3 ǫ B 1 B 2 B 3 a A 1 B 1 a A 2 B 2 a A 3 B 3 (3.8) b 3 = ǫ B 1 B 2 B 3 ǫ C 1 C 2 C 3 b B 1 C 1 b B 2 C 2 b B 3 C 3 (3.9) c 3 = ǫ C 1 C 2 C 3 ǫ A 1 A 2 A 3 c C 1 A 1 c C 2 A 2 c C 3 A 3 (3.10)
exclude one individual (Charlie, Alice, and Bob respectively), and the product abc = a AB b B C c CA (3.11) excludes none.
Just as in four dimensions, one might also consider the "magic" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 5 supergravity coupled to 5, 8, 14 and 26 vector multiplets with symmetries SL(3, R), SL(3, C), SU * (6) and E 6(-26) respectively. Once again, in all cases there are cubic invariants whose square root yields the corresponding black hole entropy [20] . Here we demonstrate that the black-hole/qubit correspondence continue to hold for these D = 5 magic supergravities, as well as D = 4 . Once again, the crucial observation is that, although the black hole charges a AB are real (integer) numbers and the entropy (2.2) is invariant under E 6(6) (Z), the coefficients a AB that appear in the wave function (3.5) are complex. So the 2-tangle (3.6) is invariant under E 6 (C) which contains both E 6(6) (Z) and E 6(-26) (Z) as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that
E 6(6) (Z) ⊃ [SL(3)(Z)] 3 (3.12) 12 but E 6(-26) (Z) ⊃ [SL(3)(Z)] 3 ( 3
.13) is irrelevant. All that matters is that
E 6 (C) ⊃ [SL(3)(C)] 3 (3.14)
The same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic). In fact, the example of section 3.1 corresponds to the complex case. Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) of section (2). SO(l, 1) is contained in SO(l + 1, C) and SO(m, 5) is contained in SO(5 + m, C), so we can give a qutrit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(6n, C) which contains [SL(3, C)] n .
We note that the 27-dimensional Hilbert space given in (3.4) and (3.5) is not a subspace of the 3 3 -dimensional three qutrit Hilbert space given by (3, 3, 3) , but rather a direct sum of three 3 2 -dimensional Hilbert spaces. It is, however, a subspace of the 7 3 -dimensional three 7-dit Hilbert space given by (7, 7, 7) . Consider the decomposition 3, 3, 1) + (3, 1, 3) + (1, 3, 3) 1, 1, 1) This contains the subspace that describes the bipartite entanglement of three qutrits, namely
SL(7) A × SL(7) B × SL(7) C → SL(3) A × SL(3) B × SL(3) C under which (7, 7, 7) → (3 ′ , 3 ′ , 3 ′ ) + (3 ′ , 3 ′ , 3) + (3 ′ , 3, 3 ′ ) + (3, 3 ′ , 3 ′ ) + (3 ′ , 3, 3) + (3, 3 ′ , 3) + (3, 3, 3 ′ ) + (3, 3, 3) +(3 ′ , 3 ′ , 1) + (3 ′ , 1, 3 ′ ) + (1, 3 ′ , 3 ′ ) + (3 ′ , 1, 3) + (3 ′ , 3, 1) + (1, 3, 3 ′ ) +(
+ (3, 1, 3 ′ ) + (3, 3 ′ , 1) + (1, 3 ′ , 3) +(3 ′ , 1, 1) + (1, 3 ′ , 1) + (1, 1, 3 ′ ) + (3, 1, 1) + (1, 3, 1) + (1, 1, 3) +(
(3 ′ , 3, 1) + (3, 1, 3) + (1, 3 ′ , 3 ′ )
So the triangle entanglement we have described fits within conventional quantum information theory. Our analogy between black holes and quantum information remains, for the moment, just that. We know of no physics connecting them.
Nevertheless, just as the exceptional group E 7 describes the tripartite entanglement of seven qubits [4, 5] , we have seen is this paper that the exceptional group E 6 describes the bipartite entanglement of three qutrits. In the E 7 case, the quartic Cartan invariant provides both the measure of entanglement and the entropy of the four-dimensional N = 8 black hole, 13 whereas in the E 6 case, the cubic Cartan invariant provides both the measure of entanglement and the entropy of the five-dimensional N = 8 black hole.
Moreover, we have seen that similar analogies exist not only for the N = 4 and N = 2 truncations, but also for the magic N = 2 supergravities in both four and five dimensions (In the four-dimensional case, this had previously been conjectured by Levay[4, 5]). Murat Gunaydin has suggested (private communication) that the appearance of octonions implies a connection to quaternionic and/or octonionic quantum mechanics. This was not apparent (at least to us) in the four-dimensional N = 8 case [4], but the appearance in the five dimensional magic N = 2 case of SL(3, R), SL(3, C), SL(3, H) and SL(3, O) is more suggestive.
MJD has enjoyed useful conversations with Leron Borsten, Hajar Ebrahim, Chris Hull, Martin Plenio and Tony Sudbery. This work was supported in part by the National Science Foundation under grant number PHY-0245337 and PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The work of S.F. has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 Constituents, fundamental forces and symmetries of the universe, in association with INFN Frascati National Laboratories and by the D.O.E grant DE-FG03-91ER40662, Task C. The work of MJD is supported in part by PPARC under rolling grant PPA/G/O/2002/00474, PP/D50744X/1. | [
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"section_title": "Untitled Section",
"text": "1 m.duff@imperial.ac.uk 2 Sergio.Ferrara@cern.ch Contents 1 D = 4 black holes and qubits 1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 1.2 N = 2 black holes and the bipartite entanglement of two qubits . . . . . . . 1.3 N = 8 black holes and the tripartite entanglement of seven qubits . . . . . . 1.4 Magic supergravities in D = 4 . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n2 Five-dimensional supergravity 3 D = 5 black holes and qutrits 3.1 N = 2 black holes and the bipartite entanglement of two qutrits . . . . . . . 3.2 N = 8 black holes and the bipartite entanglement of three qutrits . . . . . . 3.3 Magic supergravities in D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n4 Conclusions 5 Acknowledgements 2 1 D = 4 black holes and qubits\n\nIt sometimes happens that two very different areas of theoretical physics share the same mathematics. This may eventually lead to the realisation that they are, in fact, dual descriptions of the same physical phenomena, or it may not. Either way, it frequently leads to new insights in both areas. Recent papers [1, 2, 3, 4, 5, 6] have established an analogy between the entropy of certain four-dimensional supersymmetric black holes in string theory and entanglement measures in quantum information theory. In this paper we extend the analogy from four dimensions to five which also involves going from two-state systems (qubits) to three-state systems (qutrits). We begin by recalling the four-dimensional examples:\n\n1.1 N = 2 black holes and the tripartite entanglement of three qubits\n\nThe three qubit system (Alice, Bob, Charlie) is described by the state\n\n|Ψ = a ABC |ABC (1.1)\n\nwhere A = 0, 1, so the Hilbert space has dimension 2 3 = 8. The complex numbers a ABC transforms as a (2, 2, 2) under SL(2, C) A × SL(2, C) B × SL(2, C) C . The tripartite entanglement is measured by the 3-tangle [7, 8]\n\nτ 3 (ABC) = 4|Det a ABC |. (1.2)\n\nwhere Det a ABC is Cayley's hyperdeterminant [9] .\n\nDet a = - 1 2 ǫ A 1 A 2 ǫ B 1 B 2 ǫ A 3 A 4 ǫ B 3 B 4 ǫ C 1 C 4 ǫ C 2 C 3 a A 1 B 1 C 1 a A 2 B 2 C 2 a A 3 B 3 C 3 a A 4 B 4 C 4 (1.\n\n3) The hyperdeterminant is invariant under SL(2) A × SL(2) B × SL(2) C and under a triality that interchanges A, B and C. In the context of stringy black holes the 8 a ABC are the 4 electric and 4 magnetic charges of the N = 2 STU black hole [10] and hence take on real (integer) values. The ST U model corresponds to N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . The Bekenstein-Hawking entropy of the black hole, S, was first calculated in [11] . The connection to quantum information theory arises by noting [1] that it can also be expressed in terms of Cayley's hyperdeterminant\n\nS = π |Det a ABC |.\n\n(1.4) One can then establish a dictionary between the classification of various entangled states (separable A-B-C; bipartite entangled A-BC, B-CA, C-AB; tripartite entangled W; tripartite entangled GHZ) and the classfication of various \"small\" and \"large\" BPS and non-BPS black holes [1, 2, 3, 4, 5, 6] . For example, the GHZ state [12] |Ψ ∼ |111 + |000 (1.5) 3 with Det a ABC ≥ 0 corresponds to a large non-BPS 2-charge black hole; the W-state |Ψ ∼ |100 + |010 + |001 (1.6) with Det a ABC = 0 corresponds to a small-BPS 3-charge black hole; the GHZ-state\n\n|Ψ = -|000 + |011 + |101 + |110 (1.7)\n\ncorresponds to a large BPS 4-charge black hole.\n\n1.2 N = 2 black holes and the bipartite entanglement of two qubits\n\nAn even simpler example [2] is provided by the two qubit system (Alice and Bob) described by the state |Ψ = a AB |AB (1.8) where A = 0, 1, and the Hilbert space has dimension 2 2 = 4. The a AB transforms as a (2, 2) under SL(2, C) A × SL(2, C) B . The entanglement is measured by the 2-tangle τ 2 (AB) = C 2 (AB) (1.9) where C(AB) = 2 |det a AB | (1.10) is the concurrence. The determinant is invariant under SL(2, C) A × SL(2, C) B and under a duality that interchanges A and B. Here it is sufficient to look at N = 2 supergravity coupled to just one vector multiplet and the 4 a AB are the 2 electric and 2 magnetic charges of the axion-dilaton black hole with entropy S = π|det a AB | (1.11) For example, the Bell state |Ψ ∼ |11 + |00 (1.12) with det a AB ≥ 0 corresponds to a large non-BPS 2-charge black hole."
},
{
"section_type": "OTHER",
"section_title": "N = 8 black holes and the tripartite entanglement of seven qubits",
"text": "We recall that in the case of D = 4, N = 8 supergravity, the the 28 electric and 28 magnetic charges belong to the 56 of E 7 (7) . The black hole entropy is [15, 18] (1.13) where J 4 is Cartan's quartic E 7 invariant [13, 14] . It may be written\n\nS = π |J 4 |\n\nJ 4 = P ij Q jk P kl Q li -1 4 P ij Q ij P kl Q kl + 1 96 ǫ ijklmnop Q ij Q kl Q mn Q op + ǫ ijklmnop P ij P kl P mn P op . (1.14) 4\n\nwhere P ij and Q jk are 8 × 8 antisymmetric matrices. The qubit interpretation [4] relies on the decomposition E 7 (C) ⊃ [SL(2, C)] 7 (1.15) under which 56 → 2, 2, 1, 2, 1, 1, 1) + (1, 2, 2, 1, 2, 1, 1 1, 1, 2, 2, 1, 2, 1) +(1, 1, 1, 2, 2, 1, 2) +(2, 1, 1, 1, 2, 2, 1) + (1, 2, 1, 1, 1, 2, 2 2, 1, 2, 1, 1, 1, 2) (1.16) suggesting the tripartite entanglement of seven qubits (Alice, Bob, Charlie, Daisy, Emma, Fred and George) described by the state.\n\n(\n\n) +(\n\n) +(\n\n|Ψ = a ABD |ABD +b BCE |BCE +c CDF |CDF +d DEG |DEG +e EF A |EF A +f F GB |F GB +g GAC |GAC (1.17)\n\nwhere A = 0, 1, so the Hilbert space has dimension 7.2 3 = 56. The a, b, c, d, e, f, g transform as a 56 of E 7 (C). The entanglement may be represented by a heptagon where the vertices A,B,C,D,E,F,G represent the seven qubits and the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC represent the tripartite entanglement. See Figure 1 . Alternatively, we can use the Fano plane. See Figure 2 . The Fano plane also corresponds to the multiplication table of the octonions 3 The measure of the tripartite entanglement of the seven qubits is provided by the 3-tangle\n\nτ 3 (ABCDEF G) = 4|J 4 | (1.18) with J 4 ∼ a 4 + b 4 + c 4 + d 4 + e 4 + f 4 + g 4 +\n\n3 Not the \"split\" octonions as was incorrectly stated in the published version of [4].\n\nA B C D E F G Figure 1: The E 7 entanglement diagram. Each of the seven vertices A,B,C,D,E,F,G represents a qubit and each of the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC describes a tripartite entanglement.\n\n2[a 2 b 2 + b 2 c 2 + c 2 d 2 + d 2 e 2 + e 2 f 2 + f 2 g 2 + g 2 a 2 + a 2 c 2 + b 2 d 2 + c 2 e 2 + d 2 f 2 + e 2 g 2 + f 2 a 2 + g 2 b 2 + a 2 d 2 + b 2 e 2 + c 2 f 2 + d 2 g 2 + e 2 a 2 + f 2 b 2 + g 2 c 2 ] +8[bcdf + cdeg + def a + ef gb + f gac + gabd + abce] (1.19)\n\nwhere products like a 4 = (ABD)(ABD)(ABD)(ABD)\n\n= ǫ A 1 A 2 ǫ B 1 B 2 ǫ D 1 D 4 ǫ A 3 A 4 ǫ B 3 B 4 ǫ D 2 D 3 a A 1 B 1 D 1 a A 2 B 2 D 2 a A 3 B 3 D 3 a A 4 B 4 D 4 (1.20)\n\nexclude four individuals (here Charlie, Emma, Fred and George), products like\n\na 2 b 2 = (ABD)(ABD)(F GB)(F GB) = ǫ A 1 A 2 ǫ B 1 B 3 ǫ D 1 D 2 ǫ F 3 F 4 ǫ G 3 G 4 ǫ B 2 B 4 a A 1 B 1 D 1 a A 2 B 2 D 2 b F 3 G 3 B 3 b F 4 G 4 B 4 (1.21)\n\nexclude two individuals (here Charlie and Emma), and products like\n\nabce = (ABD)(BCE)(CDF )(EF A) = ǫ A 1 A 4 ǫ B 1 B 2 ǫ C 2 C 3 ǫ D 1 D 3 ǫ E 2 E 4 ǫ F 3 F 4 a A 1 B 1 D 1 b B 2 C 2 E 2 c C 3 D 3 F 3 e E 4 F 4 A 4 (1.22)\n\nexclude one individual (here George) 4 . Once again large non-BPS, small BPS and large BPS black holes correspond to states with J 4 > 0, J 4 = 0 and J 4 < 0, respectively.\n\n4 This corrects the corresponding equation in the published version of [4] which had the wrong index contraction.\n\nF E B G C A D\n\nFigure 2: The Fano plane has seven points, representing the seven qubits, and seven lines (the circle counts as a line) with three points on every line, representing the tripartite entanglement, and three lines through every point."
},
{
"section_type": "OTHER",
"section_title": "Magic supergravities in D = 4",
"text": "The black holes described by Cayley's hyperdeterminant are those of N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . In [4] the following four-dimensional generalizations were considered: 1) N = 2 supergravity coupled to l vector multiplets where the symmetry is SL(2, Z) × SO(l -1, 2, Z) and the black holes carry charges belonging to the (2, l + 1) representation (l + 1 electric plus l + 1 magnetic).\n\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SL(2, Z) × SO(6, m, Z) where the black holes carry charges belonging to the (2, 6 + m) representation (m + 6 electric plus m + 6 magnetic).\n\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 7(7) (Z) and the black holes carry charges belonging to the fundamental 56-dimensional representation (28 electric plus 28 magnetic).\n\nIn all three case there exist quartic invariants akin to Cayley's hyperdeterminant whose square root yields the corresponding black hole entropy. In [4] we succeeded in giving a quantum theoretic interpretation in the N = 8 case together with its truncations to N = 4 (with m = 6) and N = 2 (with l = 3, the case we already knew [1]).\n\nHowever, as suggested by Levay [5], one might also consider the \"magic\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 4 supergravity coupled to 6, 9, 15 and 27 vector multiplets with symmetries Sp(6, Z), SU(3, 3), SO * (12) and E 7(-25) , respectively. Once again, as has been shown just recently [20] , in all cases there are quartic invariants whose square root yields the corresponding black hole entropy.\n\nHere we demonstrate that the black-hole/qubit correspondence does indeed continue to 7 hold for magic supergravities. The crucial observation is that, although the black hole charges a ABC are real (integer) numbers and the entropy (1.13) is invariant under E 7 (7)(Z), the coefficients a ABC that appear in the qutrit state (1.17) are complex. So the three tangle (1.18) is invariant under E 7 (C) which contains both E 7(7) (Z) and E 7(-25) (Z) as subgroups.\n\nTo find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\n\nE 7(7) (Z) ⊃ [SL(2)(Z)] 7 (1.23) but E 7(-25) (Z) ⊃ [SL(2)(Z)] 7 (1.24)\n\nis irrelevant. All that matters is that\n\nE 7 (C) ⊃ [SL(2)(C)] 7 (1.25)\n\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic) . Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) above. When we looked at the seven qubit subsector E 7 (C) ⊃ SL(2, C) × SO(12, C), we gave an N = 4 supergravity interpretationwith symmetry SL(2, R)×SO(6, 6) [4], but we could equally have given an interpretation in terms of N = 2 supergravity coupled to 11 vector multiplets with symmetry SL(2, R) × SO(10, 2).\n\nMoreover, SO(l -1, 2) is contained in SO(l + 1, C) and SO(6, m) is contained in SO(12 + m, C) so we can give a qubit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(4n, C) which contains [SL(2, C)] 2n ."
},
{
"section_type": "OTHER",
"section_title": "Five-dimensional supergravity",
"text": "In five dimensions we might consider: 1) N = 2 supergravity coupled to l+1 vector multiplets where the symmetry is SO(1, 1, Z)× SO(l, 1, Z) and the black holes carry charges belonging to the (l + 1) representation (all electric) .\n\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SO(1, 1, Z)× SO(m, 5, Z) where the black holes carry charges belonging to the (m + 5) representation (all electric).\n\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 6(6) (Z) and the black holes carry charges belonging to the fundamental 27-dimensional representation (all electric).\n\nThe electrically charged objects are point-like and the magnetic duals are one-dimensional, or string-like, transforming according to the contrgredient representation. In all three cases above there exist cubic invariants akin to the determinant which yield the corresponding black hole or black string entropy.\n\nIn this section we briefly describe the salient properties of maximal N = 8 case, following [16] . We have 27 abelian gauge fields which transform in the fundamental representation of 8 6(6) . The first invariant of E 6(6) is the cubic invariant [13, 17, 16, 18, 19]\n\nE\n\nJ 3 = q ij Ω jl q lm Ω mn q np Ω pi (2.1)\n\nwhere q ij is the charge vector transforming as a 27 which can be represented as traceless Sp(8) matrix. The entropy of a black hole with charges q ij is then given by\n\nS = π |J 3 | (2.2)\n\nWe will see that a configuration with J 3 = 0 preserves 1/8 of the supersymmetries. If J 3 = 0 and ∂J 3 ∂q i = 0 then it preserves 1/4 of the supersymmetries, and finally if ∂J 3 ∂q i = 0 (and the charge vector q i is non-zero), the configuration preserves 1/2 of the supersymmetries. We will show this by choosing a particular basis for the charges, the general result following by U-duality.\n\nIn five dimensions the compact group H is USp(8). We choose our conventions so that USp(2) = SU(2). In the commutator of the supersymmetry generators we have a central charge matrix Z ab which can be brought to a normal form by a USp(8) transformation. In the normal form the central charge matrix can be written as\n\ne ab = s 1 + s 2 -s 3 0 0 0 0 s 1 + s 3 -s 2 0 0 0 0 s 2 + s 3 -s 1 0 0 0 0 -(s 1 + s 2 + s 3 ) × 0 1 -1 0 (2.3) we can order s i so that s 1 ≥ s 2 ≥ |s 3 |.\n\nThe cubic invariant, in this basis, becomes\n\nJ 3 = s 1 s 2 s 3 (2.4)\n\nEven though the eigenvalues s i might depend on the moduli, the invariant (2.4) only depends on the quantized values of the charges. We can write a generic charge configuration as UeU t , where e is the normal frame as above, and the invariant will then be (2.4). There are three distinct possibilities\n\nJ 3 = 0 s 1 , s 2 , s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 , s 2 = 0, s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 = 0, s 2 , s 3 = 0 (2.5)\n\nTaking the case of type II on T 5 we can choose the rotation in such a way that, for example, s 1 corresponds to solitonic five-brane charge, s 2 to fundamental string winding charge along some direction and s 3 to Kaluza-Klein momentum along the same direction. We can see that in this specific example the three possibilities in (2.5) break 1/8, 1/4 and 1/2 supersymmetries. The respective orbits are\n\nE 6 (6) F 4(4) E 6 (6) SO(5, 4) ×T 16 9 E 6 (6) SO(5, 5) ×T 16 (2.6)\n\nThis also shows that one can generically choose a basis for the charges so that all others are related by U-duality. The basis chosen here is the S-dual of the D-brane basis usually chosen for describing black holes in type II B on T 5 . All others are related by U-duality to this particular choice. Note that, in contrast to the four-dimensional case where flipping the sign of J 4 (1.14) interchanges BPS and non-BPS black holes, the sign of the J 3 (2.4) is not important since it changes under a CPT transformation. There is no non-BPS orbit in five dimensions.\n\nIn five dimensions there are also string-like configurations which are the magnetic duals of the configurations considered here. They transform in the contragredient 27 ′ representation and the solutions preserving 1/2, 1/4, 1/8 supersymmetries are characterized in an analogous way. We could also have configurations where we have both point-like and string-like ch the point-like charge is uniformly distributed along the string, it is more natural to consider this configuration as a point-like object in D = 4 by dimensional reduction.\n\nIt is useful to decompose the U-duality group into the T-duality group and the S-duality group. The decomposition reads E 6 → SO(5, 5) × SO(1, 1), leading to\n\n27 → 16 1 + 10 -2 + 1 4 (2.7)\n\nThe last term in (2.7) corresponds to the NS five-brane charge. The 16 correspond to the D-brane charges and the 10 correspond to the 5 directions of KK momentum and the 5 directions of fundamental string winding, which are the charges that explicitly appear in string perturbation theory. The cubic invariant has the decomposition (27) 3 → 10 -2 10 -2 1 4 + 16 1 16 1 10 -2 (2.8) This is saying that in order to have a non-zero area black hole we must have three NS charges (more precisely some \"perturbative\" charges and a solitonic five-brane); or we can have two D-brane charges and one NS charge. In particular, it is not possible to have a black hole with a non-zero horizon area with purely D-brane charges. Notice that the non-compact nature of the groups is crucial in this classification."
},
{
"section_type": "OTHER",
"section_title": "D = 5 black holes and qutrits",
"text": "So far, all the quantum information analogies involve four-dimensional black holes and qubits. In order to find an analogy with five-dimensional black holes we invoke three state systems called qutrits."
},
{
"section_type": "OTHER",
"section_title": "N = 2 black holes and the bipartite entanglement of two qutrits",
"text": "The two qutrit system (Alice and Bob) is described by the state\n\n|Ψ = a AB |AB 10\n\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3 2 = 9. The a AB transforms as a (3, 3) under SL(3) A × SL(3) B . The bipartite entanglement is measured by the concurrence [21]\n\nC(AB) = 3 3/2 |det a AB |. (3.1)\n\nThe determinant is invariant under SL(3, C) A × SL(3, C) B and under a duality that interchanges A and B. The black hole interpretation is provided by N = 2 supergravity coupled to 8 vector multiplets with symmetry SL(3, C) where the black hole charges transform as a 9. The entropy is given by\n\nS = π|det a AB | (3.2)\n\n3.2 N = 8 black holes and the bipartite entanglement of three qutrits\n\nAs we have seen in section (2) in the case of D = 5, N = 8 supergravity, the black hole charges belong to the 27 of E 6(6) and the entropy is given by (2.2).\n\nThe qutrit interpretation now relies on the decomposition\n\nE 6 (C) ⊃ SL(3, C) A × SL(3, C) B × SL(3, C) C (3.3) under which 27 → (3, 3, 1) + (3 ′ , 1, 3) + (1, 3 ′ , 3 ′ ) (3.4)\n\nsuggesting the bipartite entanglement of three qutrits (Alice, Bob, Charlie). However, the larger symmetry requires that they undergo at most bipartite entanglement of a very specific kind, where each person has bipartite entanglement with the other two:\n\n|Ψ = a AB |AB + b B C |BC + c CA |CA (3.5)\n\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3.3 2 = 27. The three states transforms as a pair of triplets under two of the SL(3)'s and singlets under the remaining one. Individually, therefore, the bipartite entanglement of each of the three states is given by the determinant (3.1). Taken together however, we see from (3.4) that they transform as a complex 27 of E 6 (C). The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements. See Fig. 3. The N=2 truncation of section 3.1 is represented by just the line AB with endpoints A and B. Note that: 1) Any pair of states has an individual in common 2) Each individual is excluded from one out of the three states The entanglement measure will be given by the concurrence\n\nC(ABC) = 3 3/2 |J 3 | (3.6)\n\nJ 3 being the singlet in 27 × 27 × 27:\n\nJ 3 ∼ a 3 + b 3 + c 3 + 6abc (3.7) 11 A B C\n\nFigure 3: The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements.\n\nwhere the products\n\na 3 = ǫ A 1 A 2 A 3 ǫ B 1 B 2 B 3 a A 1 B 1 a A 2 B 2 a A 3 B 3 (3.8) b 3 = ǫ B 1 B 2 B 3 ǫ C 1 C 2 C 3 b B 1 C 1 b B 2 C 2 b B 3 C 3 (3.9) c 3 = ǫ C 1 C 2 C 3 ǫ A 1 A 2 A 3 c C 1 A 1 c C 2 A 2 c C 3 A 3 (3.10)\n\nexclude one individual (Charlie, Alice, and Bob respectively), and the product abc = a AB b B C c CA (3.11) excludes none."
},
{
"section_type": "OTHER",
"section_title": "Magic supergravities in D = 5",
"text": "Just as in four dimensions, one might also consider the \"magic\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 5 supergravity coupled to 5, 8, 14 and 26 vector multiplets with symmetries SL(3, R), SL(3, C), SU * (6) and E 6(-26) respectively. Once again, in all cases there are cubic invariants whose square root yields the corresponding black hole entropy [20] . Here we demonstrate that the black-hole/qubit correspondence continue to hold for these D = 5 magic supergravities, as well as D = 4 . Once again, the crucial observation is that, although the black hole charges a AB are real (integer) numbers and the entropy (2.2) is invariant under E 6(6) (Z), the coefficients a AB that appear in the wave function (3.5) are complex. So the 2-tangle (3.6) is invariant under E 6 (C) which contains both E 6(6) (Z) and E 6(-26) (Z) as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\n\nE 6(6) (Z) ⊃ [SL(3)(Z)] 3 (3.12) 12 but E 6(-26) (Z) ⊃ [SL(3)(Z)] 3 ( 3\n\n.13) is irrelevant. All that matters is that\n\nE 6 (C) ⊃ [SL(3)(C)] 3 (3.14)\n\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic). In fact, the example of section 3.1 corresponds to the complex case. Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) of section (2). SO(l, 1) is contained in SO(l + 1, C) and SO(m, 5) is contained in SO(5 + m, C), so we can give a qutrit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(6n, C) which contains [SL(3, C)] n ."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusions",
"text": "We note that the 27-dimensional Hilbert space given in (3.4) and (3.5) is not a subspace of the 3 3 -dimensional three qutrit Hilbert space given by (3, 3, 3) , but rather a direct sum of three 3 2 -dimensional Hilbert spaces. It is, however, a subspace of the 7 3 -dimensional three 7-dit Hilbert space given by (7, 7, 7) . Consider the decomposition 3, 3, 1) + (3, 1, 3) + (1, 3, 3) 1, 1, 1) This contains the subspace that describes the bipartite entanglement of three qutrits, namely\n\nSL(7) A × SL(7) B × SL(7) C → SL(3) A × SL(3) B × SL(3) C under which (7, 7, 7) → (3 ′ , 3 ′ , 3 ′ ) + (3 ′ , 3 ′ , 3) + (3 ′ , 3, 3 ′ ) + (3, 3 ′ , 3 ′ ) + (3 ′ , 3, 3) + (3, 3 ′ , 3) + (3, 3, 3 ′ ) + (3, 3, 3) +(3 ′ , 3 ′ , 1) + (3 ′ , 1, 3 ′ ) + (1, 3 ′ , 3 ′ ) + (3 ′ , 1, 3) + (3 ′ , 3, 1) + (1, 3, 3 ′ ) +(\n\n+ (3, 1, 3 ′ ) + (3, 3 ′ , 1) + (1, 3 ′ , 3) +(3 ′ , 1, 1) + (1, 3 ′ , 1) + (1, 1, 3 ′ ) + (3, 1, 1) + (1, 3, 1) + (1, 1, 3) +(\n\n(3 ′ , 3, 1) + (3, 1, 3) + (1, 3 ′ , 3 ′ )\n\nSo the triangle entanglement we have described fits within conventional quantum information theory. Our analogy between black holes and quantum information remains, for the moment, just that. We know of no physics connecting them.\n\nNevertheless, just as the exceptional group E 7 describes the tripartite entanglement of seven qubits [4, 5] , we have seen is this paper that the exceptional group E 6 describes the bipartite entanglement of three qutrits. In the E 7 case, the quartic Cartan invariant provides both the measure of entanglement and the entropy of the four-dimensional N = 8 black hole, 13 whereas in the E 6 case, the cubic Cartan invariant provides both the measure of entanglement and the entropy of the five-dimensional N = 8 black hole.\n\nMoreover, we have seen that similar analogies exist not only for the N = 4 and N = 2 truncations, but also for the magic N = 2 supergravities in both four and five dimensions (In the four-dimensional case, this had previously been conjectured by Levay[4, 5]). Murat Gunaydin has suggested (private communication) that the appearance of octonions implies a connection to quaternionic and/or octonionic quantum mechanics. This was not apparent (at least to us) in the four-dimensional N = 8 case [4], but the appearance in the five dimensional magic N = 2 case of SL(3, R), SL(3, C), SL(3, H) and SL(3, O) is more suggestive."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgements",
"text": "MJD has enjoyed useful conversations with Leron Borsten, Hajar Ebrahim, Chris Hull, Martin Plenio and Tony Sudbery. This work was supported in part by the National Science Foundation under grant number PHY-0245337 and PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The work of S.F. has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 Constituents, fundamental forces and symmetries of the universe, in association with INFN Frascati National Laboratories and by the D.O.E grant DE-FG03-91ER40662, Task C. The work of MJD is supported in part by PPARC under rolling grant PPA/G/O/2002/00474, PP/D50744X/1."
}
] | |
arxiv:0704.0511 | 0704.0511 | 1 | 10.1007/s10946-007-0032-5 | be892cb8194712d11f25f536c9a91df2f945e518862e27506bb42e0662579c98 | A unified approach to SIC-POVMs and MUBs | A unified approach to (symmetric informationally complete) positive operator
valued measures and mutually unbiased bases is developed in this article. The
approach is based on the use of operator equivalents expanded in the enveloping
algebra of SU(2). Emphasis is put on similarities and differences between
SIC-POVMs and MUBs. | [
"O. Albouy (IPNL)",
"M. R. Kibler (IPNL)"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | The importance of finite-dimensional spaces for quantum mechanics is well recognized (see for instance [1]-[3]). In particular, such spaces play a major role in quantum information theory, especially for quantum cryptography and quantum state tomography [4]- [27] .
Along this vein, a symmetric informationally complete (SIC) positive operator valued measure (POVM) is a set of operators acting on a finite Hilbert space [4]-[14] (see also [3] for an infinite Hilbert space) and mutually unbiased bases (MUBs) are specific bases for such a space [15]- [27] .
The introduction of POVMs goes back to the seventies [4]- [7] . The most general quantum measurement is represented by a POVM. In the present work, we will be interested 1 in SIC-POVMs, for which the statistics of the measurement allows the reconstruction of the quantum state. Moreover, those POVMs are endowed with an extra symmetry condition (see definition in Sec. 2). The notion of MUBs (see definition in Sec. 3), implicit or explicit in the seminal works of [15] - [18] , has been the object of numerous mathematical and physical investigations during the last two decades in connection with the so-called complementary observables. Unfortunately, the question to know, for a given Hilbert space of finite dimension d, whether there exist SIC-POVMs and how many MUBs there exist has remained an open one.
The aim of this note is to develop a unified approach to SIC-POVMs and MUBs based on a complex vector space of higher dimension, viz. d 2 instead of d. We then give a specific example of this approach grounded on the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) recently used for a study of entanglement of rotationally invariant spin systems [28] and for an angular momentum study of MUBs [26, 27] .
Most of the notations in this work are standard. Let us simply mention that I is the identity operator, the bar indicates complex conjugation, A † denotes the adjoint of the operator A, δ a,b stands for the Kronecker symbol for a and b, and ∆(a, b, c) is 1 or 0 according as a, b and c satisfy or not the triangular inequality.
Let C d be the standard Hilbert space of dimension d endowed with its usual inner product denoted by | . As is usual, we will identify a POVM with a nonorthogonal decomposition of the identity. Thus, a discrete SIC-POVM is a set {P x : x = 1, 2, • • • , d 2 } of d 2 nonnegative operators P x acting on C d , such that:
• they satisfy the trace or symmetry condition
Tr (P x P y ) = 1 d + 1 , x = y; ( 1
)
moreover, we will assume the operators P x are normalized, thus completing this 2 condition with
Tr P 2 x = 1; ( 2
)
• they form a decomposition of the identity
1 d d 2 x=1 P x = I; ( 3
)
• they satisfy a completeness condition: the knowledge of the probabilities p x defined by p x = Tr(P x ρ) is sufficient to reconstruct the density matrix ρ.
Now, let us develop each of the operators P x on an orthonormal (with respect to the Hilbert-Schmidt product) basis {u i : i = 1, 2, • • • , d 2 } of the space of linear operators on
C d P x = d 2 i=1 v i (x)u i , ( 4
)
where the operators u i satisfy Tr(u † i u j ) = δ i,j . The operators P x are thus considered as vectors
v(x) = (v 1 (x), v 2 (x), • • • , v d 2 (x)) ( 5
)
in the Hilbert space C d 2 of dimension d 2 and the determination of the operators P x is equivalent to the determination of the components v i (x) of v(x). In this language, the trace property (1) together with the normalization condition (2) give y) is the usual Hermitian product in C d 2 . In order to compare Eq. ( 6 ) with what usually happens in the search for SIC-POVMs, we suppose from now on that the operators P x are rank-one operators. Therefore, by putting
v(x) • v(y) = 1 d + 1 (dδ x,y + 1) , ( 6
) where v(x) • v(y) = d 2 i=1 v i (x)v i (
P x = |Φ x Φ x | ( 7
) 3 with |φ x ∈ C d , the trace property (1, 2) reads | Φ x |Φ y | 2 = 1 d + 1 (dδ x,y + 1) . ( 8
)
From this point of view, to find d 2 operators P x is equivalent to finding d 2 vectors |φ x in C d satisfying Eq. ( 8 ). At the price of an increase in the number of components from d 3 (for d 2 vectors in C d ) to d 4 (for d 2 vectors in C d 2 ), we have got rid of the square modulus to result in a single scalar product (compare Eqs. (6) and (8)), what may prove to be suitable for another way to search for SIC-POVMs. Moreover, our relation (6) is independent of any hypothesis on the rank of the operators P x . In fact, there exists a lot of relations among these d 4 coefficients that decrease the effective number of coefficients to be found and give structural constraints on them. Those relations are highly sensitive to the choice of the basis {u i : i = 1, 2, • • • , d 2 } and we are going to exhibit an example of such a set of relations by choosing the basis to consist of Racah unit tensors.
The cornerstone of this approach is to identify C d with a subspace ε(j) of constant angular momentum j = (d -1)/2. Such a subspace is spanned by the set {|j, m : m = -j, -j + 1, • • • , j}, where |j, m is an eigenvector of the square and the z-component of a generalized angular momentum operator. Let u (k) be the Racah unit tensor [29]
of order k (with k = 0, 1, • • • , 2j) defined by its 2k + 1 components u (k) q (where q = -k, -k + 1, • • • , k) through u (k) q = j m=-j j m ′ =-j (-1) j-m j k j -m q m ′ |j, m j, m ′ |, ( 9
)
where (• • •) denotes a 3-jm Wigner symbol. For fixed j, the (2j + 1) 2 operators u
(k) q (with k = 0, 1, • • • , 2j and q = -k, -k + 1, • • • , k) act on ε(j) ∼ C d and form a basis
of the Hilbert space C N of dimension N = (2j + 1) 2 , the inner product in C N being the Hilbert-Schmidt product. The formulas (involving unit tensors, 3-jm and 6-j symbols) relevant for this work are given in Appendix (see also [29] to [31] ). We must remember that those Racah operators are not normalized to unity (see relation (46)). So this will generate an extra factor when defining v i (x).
4 Each operator P x can be developed as a linear combination of the operators u (k) q .
Hence, we have
P x = 2j k=0 k q=-k c kq (x)u (k) q , ( 10
)
where the unknown expansion coefficients c kq (x) are a priori complex numbers. The determination of the operators P x is thus equivalent to the determination of the coefficients c kq (x), which are formally given by
c kq (x) = (2k + 1) Φ x |u (k) q |Φ x , ( 11
)
as can be seen by multiplying each member of Eq. (10) by the adjoint of u (ℓ) p and then using Eq. (46) of Appendix.
By defining the vector
v(x) = (v 1 (x), v 2 (x), • • • , v N (x)), N = (2j + 1) 2 ( 12
) via v i (x) = 1 √ 2k + 1 c kq (x), i = k 2 + k + q + 1, ( 13
)
the following properties and relations are obtained.
• The first component v 1 (x) of v(x) does not depend on x since c 00 (x) = 1 √ 2j + 1 ( 14
) for all x ∈ {1, 2, • • • , (2j + 1) 2 }.
Proof: Take the trace of Eq. (10) and use Eq. (48) of Appendix.
• The components v i (x) of v(x) satisfy the complex conjugation property described by c kq (x) = (-1) q c k-q (x) ( 15
) for all x ∈ {1, 2, • • • , (2j + 1) 2 }, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.
Proof: Use the Hermitian property of P x and Eq. (43) of Appendix.
• In terms of c kq , Eq. (6) reads
2j k=0 1 2k + 1 k q=-k c kq (x)c kq (y) = 1 2(j + 1) [(2j + 1)δ x,y + 1] ( 16
) for all x, y ∈ {1, 2, • • • , (2j + 1) 2 },
where the sum over q is SO(3) rotationally invariant.
Proof: The proof is trivial.
• The coefficients c kq (x) are solutions of the nonlinear system given by
1 2K + 1 c KQ (x) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ k ℓ K -q -p Q × k ℓ K j j j c kq (x)c ℓp (x) ( 17
) for all x ∈ {1, 2, • • • , (2j+1) 2 }, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K+1, • • • , K}.
Proof: Consider P 2 x = P x and use the coupling relation (51) of Appendix involving a 3-jm and a 6-j Wigner symbols.
As a corollary of the latter property, by taking K = 0 and using Eqs. (47) and (50) of Appendix, we get again the normalization relation v(x
) 2 = v(x) • v(x) = 1.
• All coefficients c kq (x) are connected through the sum rule
(2j+1) 2 x=1 2j k=0 k q=-k c kq (x) j k j -m q m ′ = (-1) j-m (2j + 1)δ m,m ′ , ( 18
)
which turns out to be useful for global checking purposes.
Proof: Take the jm-jm ′ matrix element of the resolution of the identity in terms of the operators P x /(2j + 1).
A complete set of MUBs in the Hilbert space
C d is a set of d(d + 1) vectors |aα ∈ C d such that | aα|bβ | 2 = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 19
) 6 where a = 0, 1, • • • , d and α = 0, 1, • • • , d -1.
The indices of type a refer to the bases and, for fixed a, the index α refers to one of the d vectors of the basis corresponding to a.
We know that such a complete set exists if d is a prime or the power of a prime (e.g., see [16]-[24]).
The approach developed in Sec. 2 for SIC-POVMs can be applied to MUBs too. Let us suppose that it is possible to
find d + 1 sets S a (with a = 0, 1, • • • , d) of vectors in C d , each set S a = {|aα : α = 0, 1, • • • , d -1} containing d vectors |aα such that Eq. ( 19
)
be satisfied. This amounts to finding d(d + 1) projection operators
Π aα = |aα aα| ( 20
)
satisfying the trace condition
Tr (Π aα Π bβ ) = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 21
)
where the trace is taken on C d . Therefore, they also form a nonorthogonal decomposition of the identity
1 d + 1 d a=0 d-1 α=0 Π aα = I. ( 22
)
As in Sec. 2, we develop each operator Π aα on an orthonormal basis with expansion coefficients w i (aα). Thus we get vectors w(aα) in
C d 2 w(aα) = (w 1 (aα), w 2 (aα), • • • , w d 2 (aα)) ( 23
) such that w(aα) • w(bβ) = δ α,β δ a,b + 1 d (1 -δ a,b ) ( 24
) for all a, b ∈ {0, 1, • • • , d} and α, β ∈ {0, 1, • • • , d -1}.
Now we draw the same relations as for POVMs by choosing the Racah operators to be our basis in C d 2 . We assume once again that the Hilbert space C d is realized by ε(j) with j = (d -1)/2. Then, each operator Π aα can be developed on the basis of the (2j + 1) 2 7 operators u (k) q as
Π aα = 2j k=0 k q=-k d kq (aα)u (k) q , ( 25
)
to be compared with Eq. (10). The expansion coefficients are d kq (aα) = (2k + 1) aα|u
(k) q |aα ( 26
) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1,
• • • , 2j} and q ∈ {-k, -k + 1, • • • , k}. For a and α fixed, the complex coefficients d kq (aα) define a vector
w(aα) = (w 1 (aα), w 2 (aα), • • • , w N (aα)) , N = (2j + 1) 2 ( 27
)
in the Hilbert space C N , the components of which are given by
w i (aα) = 1 √ 2k + 1 d kq (aα), i = k 2 + k + q + 1. ( 28
)
We are thus led to the following properties and relations. The proofs are similar to those in Sec. 2.
• First component w 1 (aα) of w(aα):
d 00 (aα) = 1 √ 2j + 1 ( 29
) for all a ∈ {0, 1, • • • , 2j + 1} and α ∈ {0, 1, • • • , 2j}.
• Complex conjugation property:
d kq (aα) = (-1) q d k-q (aα) ( 30
) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.
• Rotational invariance:
2j k=0 1 2k + 1 k q=-k d kq (aα)d kq (bβ) = δ α,β δ a,b + 1 2j + 1 (1 -δ a,b ) ( 31
) for all a, b ∈ {0, 1, • • • , 2j + 1} and α, β ∈ {0, 1, • • • , 2j}.
• Tensor product formula: 1 2K + 1 d KQ (aα) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ
k ℓ K -q -p Q × k ℓ K j j j d kq (aα)d ℓp (aα) ( 32
) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K + 1, • • • , K}.
• Sum rule:
2j+1 a=0 2j α=0 2j k=0 k q=-k d kq (aα) j k j -m q m ′ = (-1) j-m 2(2j + 1)δ m,m ′ ( 33
)
which involves all coefficients d kq (aα).
Although the structure of the relations in Sec. 1 on the one hand and Sec. 2 on the other hand is very similar, there are deep differences between the two sets of results. The similarities are reminiscent of the fact that both MUBs and SIC-POVMs can be linked to finite affine planes [12, 13, 22, 23, 25] and to complex projective 2-designs [8, 10, 19, 24] .
On the other side, there are two arguments in favor of the differences between relations (6) and (24). First, the problem of constructing SIC-POVMs in dimension d is not equivalent to the existence of an affine plane of order d [12, 13] . Second, there is a consensus around the conjecture according to which there exists a complete set of MUBs in dimension d if and only if there exists an affine plane of order d [22].
In dimension d, to find d 2 operators P x of a SIC-POVM acting on the Hilbert space
C d amounts to find d 2 vectors v(x) in the Hilbert space C N with N = d 2 satisfying v x = 1, v(x) • v(y) = 1 d + 1 for x = y ( 34
)
(the norm v(x) of each vector v(x) is 1 and the angle ω xy of any pair of vectors v(x) and v(y) is ω xy = cos -1 [1/(d + 1)] for x = y).
In a similar way, to find d + 1 MUBs of C d is equivalent to find d + 1 sets S a (with a = 0, 1, • • • , d) of d vectors, i.e., d(d + 1) vectors in all, w(aα) in C N with N = d 2 satisfying w(aα) • w(aβ) = δ α,β , w(aα) • w(bβ) = 1 d for a = b (35) (each set S a consists of d orthonormalized vectors and the angle ω aαbβ of any vector w(aα) of a set S a with any vector w(bβ) of a set S b is ω aαbβ = cos -1 (1/d) for a = b).
According to a well accepted conjecture [8, 10], SIC-POVMs should exist in any
dimension. The present study shows that in order to prove this conjecture it is sufficient to prove that Eq. (34) admits solutions for any value of d.
The situation is different for MUBs. In dimension d, it is known that there exist d + 1 sets of d vectors of type |aα in C d satisfying Eq. (19) when d is a prime or the power of a prime. This shows that Eq. ( 35 ) can be solved for d prime or power of a prime. For d prime, it is possible to find an explicit solution of Eq. ( 19 ). In fact, we have [26, 27] |aα = 1 √ 2j + 1 j m=-j ω (j+m)(j-m+1)a/2+(j+m)
α |j, m , ( 36
) ω = exp i 2π 2j + 1 , j = 1 2 (d -1) ( 37
) for a, α ∈ {0, 1, • • • , 2j} while |aα = |j, m ( 38
) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j. Then, Eq. (26) yields d kq (aα) = 2k + 1 2j + 1 j m=-j j m ′ =-j ω θ(m,m ′ ) (-1) j-m j k j -m q m ′ , ( 39
) θ(m, m ′ ) = (m -m ′ ) 1 2 (1 -m -m ′ )a + α ( 40
) for a, α ∈ {0, 1, • • • , 2j} while d kq (aα) = δ q,0 (2k + 1)(-1) j-m j k j -m 0 m ( 41
) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j.
It can be shown that Eqs. (40) and (41) are in agreement with the results of Sec. 3. We thus have a solution of the equations for 10 the results of Sec. 3 when d is prime. As an open problem, it would be worthwhile to find an explicit solution for the coefficients d kq (aα) when d = 2j + 1 is any positive power of a prime. Finally, note that to prove (or disprove) the conjecture according to which a complete set of MUBs in dimension d exists only if d is a prime or the power of a prime is equivalent to prove (or disprove) that Eq. ( 35 ) has a solution only if d is a prime or the power of a prime.
APPENDIX: WIGNER-RACAH ALGEBRA OF SU(2) ⊃ U(1)
We limit ourselves to those basic formulas for the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) which are necessary to derive the results of this paper. The summations in this appendix have to be extended to the allowed values for the involved magnetic and angular momentum quantum numbers.
The definition (9) of the components u (k) q of the Racah unit tensor u (k) yields
j, m|u (k) q |j, m ′ = (-1) j-m j k j -m q m ′ , ( 42
)
from which we easily obtain the Hermitian conjugation property
u (k) q † = (-1) q u (k) -q . ( 43
)
The 3-jm Wigner symbol in Eq. (42) satisfies the orthogonality relations
mm ′ j j ′ k m m ′ q j j ′ ℓ m m ′ p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j ′ , k) ( 44
) and kq (2k + 1) j j ′ k m m ′ q j j ′ k M M ′ q = δ m,M δ m ′ ,M ′ . ( 45
)
The trace relation on the space ε(j)
Tr u (k) q † u (ℓ) p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j, k) ( 46
)
11 easily follows by combining Eqs. (42) and (44). Furthermore, by introducing
j j ′ 0 m -m ′ 0 = δ j,j ′ δ m,m ′ (-1) j-m 1 √ 2j + 1 ( 47
)
in Eq. ( 44 ), we obtain the sum rule
m (-1) j-m j k j -m q m = 2j + 1δ k,0 δ q,0 ∆(j, k, j), ( 48
)
known in spectroscopy as the barycenter theorem.
There are several relations involving 3-jm and 6-j symbols. In particular, we have
mm ′ M (-1) j-M j j -m q M j ℓ j -M p m ′ j K j -m Q m ′ = (-1) 2j-Q k ℓ K -q -p Q k ℓ K j j j , ( 49
)
where {• • •} denotes a 6-j Wigner symbol (or W Racah coefficient). Note that the introduction of k ℓ 0 j j J = δ k,ℓ (-1) j+k+J 1 (2k + 1)(2j + 1) (50) in Eq. (49) gives back Eq. ( 44 ). Equation (49) is central in the derivation of the coupling relation
u (k) q u (ℓ) p = KQ (-1) 2j-Q (2K + 1) k ℓ K -q -p Q k ℓ K j j j u (K) Q . ( 51
)
Equation (51) makes it possible to calculate the commutator [u
(k) q , u (ℓ)
p ] which shows that the set {u (k) q
: k = 0, 1, • • • , 2j; q = -k, -k + 1, • • •
, k} can be used to span the Lie algebra of the unitary group U(2j + 1). The latter result is at the root of the expansions (17) and (32).
After the submission of the present paper for publication in Journal of Russian Laser Research, a pre-print dealing with the existence of SIC-POVMs was posted on arXiv [32] .
The main result in [32] is that SIC-POVMs exist in all dimensions. As a corollary of this result, Eq. (34) admits solutions in any dimension.
12
This work was presented at the International Conference on Squeezed States and Uncertainty Relations, University of Bradford, England (ICSSUR'07). The authors wish to thank the organizer A. Vourdas and are grateful to D. M. Appleby, V. I. Man'ko and M.
Planat for interesting comments. 13 References [1] A. Peres, "Quantum Theory: Concepts and Methods", Dordrecht: Kluwer (1995) [2] A. Vourdas, J. Phys. A: Math. Gen. 38, 8453 (2005) [3] W. M. de Muynck, "Foundations of Quantum Mechanics, an Empiricist Approach", Dordrecht: Kluwer (2002) [4] J. M. Jauch and C. Piron, Helv. Phys. Acta 40, 559 (1967) [5] E. B. Davies and J. T. Levis, Comm. Math. Phys. 17, 239 (1970) [6] E. B. Davies, IEEE Trans. Inform. Theory IT-24, 596 (1978) [7] K. Kraus, "States, Effects, and Operations", Lect. Notes Phys. 190 (1983) [8] G. Zauner, Diploma Thesis, University of Wien (1999) [9] C. M. Caves, C. A. Fuchs and R. Schack, J. Math. Phys. 43, 4537 (2002) [10] J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, J. Math. Phys. 45, 2171 (2004) [11] D. M. Appleby, J. Math. Phys. 46, 052107 (2005) [12] M. Grassl, Proc. ERATO Conf. Quant. Inf. Science (EQIS 2004) ed. J. Gruska, Tokyo (2005) [13] M. Grassl, Elec. Notes Discrete Math. 20, 151 (2005) [14] S. Weigert, Int. J. Mod. Phys. B 20, 1942 (2006) [15] J. Schwinger, Proc. Nat. Acad. Sci. USA 46, 570 (1960) [16] P. Delsarte, J. M. Goethals and J. J. Seidel, Philips Res. Repts. 30, 91 (1975) 14 [17] I. D. Ivanović, J. Phys. A: Math. Gen. 14, 3241 (1981) [18] W. K. Wootters, Ann. Phys. (N.Y.) 176, 1 (1987) [19] H. Barnum, Preprint quant-ph/0205155 (2002) [20] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, Algorithmica 34, 512 (2002) [21] A. O. Pittenger and M. H. Rubin, Linear Alg. Appl. 390, 255 (2004) [22] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum Semiclassical Opt. 6, L19 (2004) [23] I. Bengtsson and Å. Ericsson, Open Syst. Inf. Dyn. 12, 107 (2005) [24] A. Klappenecker and M. Rötteler, Preprint quant-ph/0502031 (2005) [25] W. K. Wootters, Found. Phys. 36, 112 (2006) [26] M. R. Kibler and M. Planat, Int. J. Mod. Phys. B 20, 1802 (2006) [27] O. Albouy and M. R. Kibler, SIGMA 3, article 076 (2007) [28] H.-P. Breuer, J. Phys. A: Math. Gen. 38, 9019 (2005) [29] G. Racah, Phys. Rev. 62, 438 (1942) [30] U. Fano and G. Racah, "Irreducible Tensorial Sets", New York: Academic (1959) [31] M. Kibler and G. Grenet, J. Math. Phys. 21, 422 (1980) [32] J.L. Hall and A. Rao, Preprint quant-ph/0707.3002v1 (20 July 2007) 15 | [
{
"section_type": "BACKGROUND",
"section_title": "INTRODUCTION",
"text": "The importance of finite-dimensional spaces for quantum mechanics is well recognized (see for instance [1]-[3]). In particular, such spaces play a major role in quantum information theory, especially for quantum cryptography and quantum state tomography [4]- [27] .\n\nAlong this vein, a symmetric informationally complete (SIC) positive operator valued measure (POVM) is a set of operators acting on a finite Hilbert space [4]-[14] (see also [3] for an infinite Hilbert space) and mutually unbiased bases (MUBs) are specific bases for such a space [15]- [27] .\n\nThe introduction of POVMs goes back to the seventies [4]- [7] . The most general quantum measurement is represented by a POVM. In the present work, we will be interested 1 in SIC-POVMs, for which the statistics of the measurement allows the reconstruction of the quantum state. Moreover, those POVMs are endowed with an extra symmetry condition (see definition in Sec. 2). The notion of MUBs (see definition in Sec. 3), implicit or explicit in the seminal works of [15] - [18] , has been the object of numerous mathematical and physical investigations during the last two decades in connection with the so-called complementary observables. Unfortunately, the question to know, for a given Hilbert space of finite dimension d, whether there exist SIC-POVMs and how many MUBs there exist has remained an open one.\n\nThe aim of this note is to develop a unified approach to SIC-POVMs and MUBs based on a complex vector space of higher dimension, viz. d 2 instead of d. We then give a specific example of this approach grounded on the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) recently used for a study of entanglement of rotationally invariant spin systems [28] and for an angular momentum study of MUBs [26, 27] .\n\nMost of the notations in this work are standard. Let us simply mention that I is the identity operator, the bar indicates complex conjugation, A † denotes the adjoint of the operator A, δ a,b stands for the Kronecker symbol for a and b, and ∆(a, b, c) is 1 or 0 according as a, b and c satisfy or not the triangular inequality."
},
{
"section_type": "OTHER",
"section_title": "SIC-POVMs",
"text": "Let C d be the standard Hilbert space of dimension d endowed with its usual inner product denoted by | . As is usual, we will identify a POVM with a nonorthogonal decomposition of the identity. Thus, a discrete SIC-POVM is a set {P x : x = 1, 2, • • • , d 2 } of d 2 nonnegative operators P x acting on C d , such that:\n\n• they satisfy the trace or symmetry condition\n\nTr (P x P y ) = 1 d + 1 , x = y; ( 1\n\n)\n\nmoreover, we will assume the operators P x are normalized, thus completing this 2 condition with\n\nTr P 2 x = 1; ( 2\n\n)\n\n• they form a decomposition of the identity\n\n1 d d 2 x=1 P x = I; ( 3\n\n)\n\n• they satisfy a completeness condition: the knowledge of the probabilities p x defined by p x = Tr(P x ρ) is sufficient to reconstruct the density matrix ρ.\n\nNow, let us develop each of the operators P x on an orthonormal (with respect to the Hilbert-Schmidt product) basis {u i : i = 1, 2, • • • , d 2 } of the space of linear operators on\n\nC d P x = d 2 i=1 v i (x)u i , ( 4\n\n)\n\nwhere the operators u i satisfy Tr(u † i u j ) = δ i,j . The operators P x are thus considered as vectors\n\nv(x) = (v 1 (x), v 2 (x), • • • , v d 2 (x)) ( 5\n\n)\n\nin the Hilbert space C d 2 of dimension d 2 and the determination of the operators P x is equivalent to the determination of the components v i (x) of v(x). In this language, the trace property (1) together with the normalization condition (2) give y) is the usual Hermitian product in C d 2 . In order to compare Eq. ( 6 ) with what usually happens in the search for SIC-POVMs, we suppose from now on that the operators P x are rank-one operators. Therefore, by putting\n\nv(x) • v(y) = 1 d + 1 (dδ x,y + 1) , ( 6\n\n) where v(x) • v(y) = d 2 i=1 v i (x)v i (\n\nP x = |Φ x Φ x | ( 7\n\n) 3 with |φ x ∈ C d , the trace property (1, 2) reads | Φ x |Φ y | 2 = 1 d + 1 (dδ x,y + 1) . ( 8\n\n)\n\nFrom this point of view, to find d 2 operators P x is equivalent to finding d 2 vectors |φ x in C d satisfying Eq. ( 8 ). At the price of an increase in the number of components from d 3 (for d 2 vectors in C d ) to d 4 (for d 2 vectors in C d 2 ), we have got rid of the square modulus to result in a single scalar product (compare Eqs. (6) and (8)), what may prove to be suitable for another way to search for SIC-POVMs. Moreover, our relation (6) is independent of any hypothesis on the rank of the operators P x . In fact, there exists a lot of relations among these d 4 coefficients that decrease the effective number of coefficients to be found and give structural constraints on them. Those relations are highly sensitive to the choice of the basis {u i : i = 1, 2, • • • , d 2 } and we are going to exhibit an example of such a set of relations by choosing the basis to consist of Racah unit tensors.\n\nThe cornerstone of this approach is to identify C d with a subspace ε(j) of constant angular momentum j = (d -1)/2. Such a subspace is spanned by the set {|j, m : m = -j, -j + 1, • • • , j}, where |j, m is an eigenvector of the square and the z-component of a generalized angular momentum operator. Let u (k) be the Racah unit tensor [29]\n\nof order k (with k = 0, 1, • • • , 2j) defined by its 2k + 1 components u (k) q (where q = -k, -k + 1, • • • , k) through u (k) q = j m=-j j m ′ =-j (-1) j-m j k j -m q m ′ |j, m j, m ′ |, ( 9\n\n)\n\nwhere (• • •) denotes a 3-jm Wigner symbol. For fixed j, the (2j + 1) 2 operators u\n\n(k) q (with k = 0, 1, • • • , 2j and q = -k, -k + 1, • • • , k) act on ε(j) ∼ C d and form a basis\n\nof the Hilbert space C N of dimension N = (2j + 1) 2 , the inner product in C N being the Hilbert-Schmidt product. The formulas (involving unit tensors, 3-jm and 6-j symbols) relevant for this work are given in Appendix (see also [29] to [31] ). We must remember that those Racah operators are not normalized to unity (see relation (46)). So this will generate an extra factor when defining v i (x).\n\n4 Each operator P x can be developed as a linear combination of the operators u (k) q .\n\nHence, we have\n\nP x = 2j k=0 k q=-k c kq (x)u (k) q , ( 10\n\n)\n\nwhere the unknown expansion coefficients c kq (x) are a priori complex numbers. The determination of the operators P x is thus equivalent to the determination of the coefficients c kq (x), which are formally given by\n\nc kq (x) = (2k + 1) Φ x |u (k) q |Φ x , ( 11\n\n)\n\nas can be seen by multiplying each member of Eq. (10) by the adjoint of u (ℓ) p and then using Eq. (46) of Appendix.\n\nBy defining the vector\n\nv(x) = (v 1 (x), v 2 (x), • • • , v N (x)), N = (2j + 1) 2 ( 12\n\n) via v i (x) = 1 √ 2k + 1 c kq (x), i = k 2 + k + q + 1, ( 13\n\n)\n\nthe following properties and relations are obtained.\n\n• The first component v 1 (x) of v(x) does not depend on x since c 00 (x) = 1 √ 2j + 1 ( 14\n\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }.\n\nProof: Take the trace of Eq. (10) and use Eq. (48) of Appendix.\n\n• The components v i (x) of v(x) satisfy the complex conjugation property described by c kq (x) = (-1) q c k-q (x) ( 15\n\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\n\nProof: Use the Hermitian property of P x and Eq. (43) of Appendix.\n\n• In terms of c kq , Eq. (6) reads\n\n2j k=0 1 2k + 1 k q=-k c kq (x)c kq (y) = 1 2(j + 1) [(2j + 1)δ x,y + 1] ( 16\n\n) for all x, y ∈ {1, 2, • • • , (2j + 1) 2 },\n\nwhere the sum over q is SO(3) rotationally invariant.\n\nProof: The proof is trivial.\n\n• The coefficients c kq (x) are solutions of the nonlinear system given by\n\n1 2K + 1 c KQ (x) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ k ℓ K -q -p Q × k ℓ K j j j c kq (x)c ℓp (x) ( 17\n\n) for all x ∈ {1, 2, • • • , (2j+1) 2 }, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K+1, • • • , K}.\n\nProof: Consider P 2 x = P x and use the coupling relation (51) of Appendix involving a 3-jm and a 6-j Wigner symbols.\n\nAs a corollary of the latter property, by taking K = 0 and using Eqs. (47) and (50) of Appendix, we get again the normalization relation v(x\n\n) 2 = v(x) • v(x) = 1.\n\n• All coefficients c kq (x) are connected through the sum rule\n\n(2j+1) 2 x=1 2j k=0 k q=-k c kq (x) j k j -m q m ′ = (-1) j-m (2j + 1)δ m,m ′ , ( 18\n\n)\n\nwhich turns out to be useful for global checking purposes.\n\nProof: Take the jm-jm ′ matrix element of the resolution of the identity in terms of the operators P x /(2j + 1).\n\nA complete set of MUBs in the Hilbert space\n\nC d is a set of d(d + 1) vectors |aα ∈ C d such that | aα|bβ | 2 = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 19\n\n) 6 where a = 0, 1, • • • , d and α = 0, 1, • • • , d -1.\n\nThe indices of type a refer to the bases and, for fixed a, the index α refers to one of the d vectors of the basis corresponding to a.\n\nWe know that such a complete set exists if d is a prime or the power of a prime (e.g., see [16]-[24]).\n\nThe approach developed in Sec. 2 for SIC-POVMs can be applied to MUBs too. Let us suppose that it is possible to\n\nfind d + 1 sets S a (with a = 0, 1, • • • , d) of vectors in C d , each set S a = {|aα : α = 0, 1, • • • , d -1} containing d vectors |aα such that Eq. ( 19\n\n)\n\nbe satisfied. This amounts to finding d(d + 1) projection operators\n\nΠ aα = |aα aα| ( 20\n\n)\n\nsatisfying the trace condition\n\nTr (Π aα Π bβ ) = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 21\n\n)\n\nwhere the trace is taken on C d . Therefore, they also form a nonorthogonal decomposition of the identity\n\n1 d + 1 d a=0 d-1 α=0 Π aα = I. ( 22\n\n)\n\nAs in Sec. 2, we develop each operator Π aα on an orthonormal basis with expansion coefficients w i (aα). Thus we get vectors w(aα) in\n\nC d 2 w(aα) = (w 1 (aα), w 2 (aα), • • • , w d 2 (aα)) ( 23\n\n) such that w(aα) • w(bβ) = δ α,β δ a,b + 1 d (1 -δ a,b ) ( 24\n\n) for all a, b ∈ {0, 1, • • • , d} and α, β ∈ {0, 1, • • • , d -1}.\n\nNow we draw the same relations as for POVMs by choosing the Racah operators to be our basis in C d 2 . We assume once again that the Hilbert space C d is realized by ε(j) with j = (d -1)/2. Then, each operator Π aα can be developed on the basis of the (2j + 1) 2 7 operators u (k) q as\n\nΠ aα = 2j k=0 k q=-k d kq (aα)u (k) q , ( 25\n\n)\n\nto be compared with Eq. (10). The expansion coefficients are d kq (aα) = (2k + 1) aα|u\n\n(k) q |aα ( 26\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1,\n\n• • • , 2j} and q ∈ {-k, -k + 1, • • • , k}. For a and α fixed, the complex coefficients d kq (aα) define a vector\n\nw(aα) = (w 1 (aα), w 2 (aα), • • • , w N (aα)) , N = (2j + 1) 2 ( 27\n\n)\n\nin the Hilbert space C N , the components of which are given by\n\nw i (aα) = 1 √ 2k + 1 d kq (aα), i = k 2 + k + q + 1. ( 28\n\n)\n\nWe are thus led to the following properties and relations. The proofs are similar to those in Sec. 2.\n\n• First component w 1 (aα) of w(aα):\n\nd 00 (aα) = 1 √ 2j + 1 ( 29\n\n) for all a ∈ {0, 1, • • • , 2j + 1} and α ∈ {0, 1, • • • , 2j}.\n\n• Complex conjugation property:\n\nd kq (aα) = (-1) q d k-q (aα) ( 30\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\n\n• Rotational invariance:\n\n2j k=0 1 2k + 1 k q=-k d kq (aα)d kq (bβ) = δ α,β δ a,b + 1 2j + 1 (1 -δ a,b ) ( 31\n\n) for all a, b ∈ {0, 1, • • • , 2j + 1} and α, β ∈ {0, 1, • • • , 2j}.\n\n• Tensor product formula: 1 2K + 1 d KQ (aα) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ\n\nk ℓ K -q -p Q × k ℓ K j j j d kq (aα)d ℓp (aα) ( 32\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K + 1, • • • , K}.\n\n• Sum rule:\n\n2j+1 a=0 2j α=0 2j k=0 k q=-k d kq (aα) j k j -m q m ′ = (-1) j-m 2(2j + 1)δ m,m ′ ( 33\n\n)\n\nwhich involves all coefficients d kq (aα)."
},
{
"section_type": "CONCLUSION",
"section_title": "CONCLUSIONS",
"text": "Although the structure of the relations in Sec. 1 on the one hand and Sec. 2 on the other hand is very similar, there are deep differences between the two sets of results. The similarities are reminiscent of the fact that both MUBs and SIC-POVMs can be linked to finite affine planes [12, 13, 22, 23, 25] and to complex projective 2-designs [8, 10, 19, 24] .\n\nOn the other side, there are two arguments in favor of the differences between relations (6) and (24). First, the problem of constructing SIC-POVMs in dimension d is not equivalent to the existence of an affine plane of order d [12, 13] . Second, there is a consensus around the conjecture according to which there exists a complete set of MUBs in dimension d if and only if there exists an affine plane of order d [22].\n\nIn dimension d, to find d 2 operators P x of a SIC-POVM acting on the Hilbert space\n\nC d amounts to find d 2 vectors v(x) in the Hilbert space C N with N = d 2 satisfying v x = 1, v(x) • v(y) = 1 d + 1 for x = y ( 34\n\n)\n\n(the norm v(x) of each vector v(x) is 1 and the angle ω xy of any pair of vectors v(x) and v(y) is ω xy = cos -1 [1/(d + 1)] for x = y).\n\nIn a similar way, to find d + 1 MUBs of C d is equivalent to find d + 1 sets S a (with a = 0, 1, • • • , d) of d vectors, i.e., d(d + 1) vectors in all, w(aα) in C N with N = d 2 satisfying w(aα) • w(aβ) = δ α,β , w(aα) • w(bβ) = 1 d for a = b (35) (each set S a consists of d orthonormalized vectors and the angle ω aαbβ of any vector w(aα) of a set S a with any vector w(bβ) of a set S b is ω aαbβ = cos -1 (1/d) for a = b).\n\nAccording to a well accepted conjecture [8, 10], SIC-POVMs should exist in any\n\ndimension. The present study shows that in order to prove this conjecture it is sufficient to prove that Eq. (34) admits solutions for any value of d.\n\nThe situation is different for MUBs. In dimension d, it is known that there exist d + 1 sets of d vectors of type |aα in C d satisfying Eq. (19) when d is a prime or the power of a prime. This shows that Eq. ( 35 ) can be solved for d prime or power of a prime. For d prime, it is possible to find an explicit solution of Eq. ( 19 ). In fact, we have [26, 27] |aα = 1 √ 2j + 1 j m=-j ω (j+m)(j-m+1)a/2+(j+m)\n\nα |j, m , ( 36\n\n) ω = exp i 2π 2j + 1 , j = 1 2 (d -1) ( 37\n\n) for a, α ∈ {0, 1, • • • , 2j} while |aα = |j, m ( 38\n\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j. Then, Eq. (26) yields d kq (aα) = 2k + 1 2j + 1 j m=-j j m ′ =-j ω θ(m,m ′ ) (-1) j-m j k j -m q m ′ , ( 39\n\n) θ(m, m ′ ) = (m -m ′ ) 1 2 (1 -m -m ′ )a + α ( 40\n\n) for a, α ∈ {0, 1, • • • , 2j} while d kq (aα) = δ q,0 (2k + 1)(-1) j-m j k j -m 0 m ( 41\n\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j.\n\nIt can be shown that Eqs. (40) and (41) are in agreement with the results of Sec. 3. We thus have a solution of the equations for 10 the results of Sec. 3 when d is prime. As an open problem, it would be worthwhile to find an explicit solution for the coefficients d kq (aα) when d = 2j + 1 is any positive power of a prime. Finally, note that to prove (or disprove) the conjecture according to which a complete set of MUBs in dimension d exists only if d is a prime or the power of a prime is equivalent to prove (or disprove) that Eq. ( 35 ) has a solution only if d is a prime or the power of a prime.\n\nAPPENDIX: WIGNER-RACAH ALGEBRA OF SU(2) ⊃ U(1)\n\nWe limit ourselves to those basic formulas for the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) which are necessary to derive the results of this paper. The summations in this appendix have to be extended to the allowed values for the involved magnetic and angular momentum quantum numbers.\n\nThe definition (9) of the components u (k) q of the Racah unit tensor u (k) yields\n\nj, m|u (k) q |j, m ′ = (-1) j-m j k j -m q m ′ , ( 42\n\n)\n\nfrom which we easily obtain the Hermitian conjugation property\n\nu (k) q † = (-1) q u (k) -q . ( 43\n\n)\n\nThe 3-jm Wigner symbol in Eq. (42) satisfies the orthogonality relations\n\nmm ′ j j ′ k m m ′ q j j ′ ℓ m m ′ p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j ′ , k) ( 44\n\n) and kq (2k + 1) j j ′ k m m ′ q j j ′ k M M ′ q = δ m,M δ m ′ ,M ′ . ( 45\n\n)\n\nThe trace relation on the space ε(j)\n\nTr u (k) q † u (ℓ) p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j, k) ( 46\n\n)\n\n11 easily follows by combining Eqs. (42) and (44). Furthermore, by introducing\n\nj j ′ 0 m -m ′ 0 = δ j,j ′ δ m,m ′ (-1) j-m 1 √ 2j + 1 ( 47\n\n)\n\nin Eq. ( 44 ), we obtain the sum rule\n\nm (-1) j-m j k j -m q m = 2j + 1δ k,0 δ q,0 ∆(j, k, j), ( 48\n\n)\n\nknown in spectroscopy as the barycenter theorem.\n\nThere are several relations involving 3-jm and 6-j symbols. In particular, we have\n\nmm ′ M (-1) j-M j j -m q M j ℓ j -M p m ′ j K j -m Q m ′ = (-1) 2j-Q k ℓ K -q -p Q k ℓ K j j j , ( 49\n\n)\n\nwhere {• • •} denotes a 6-j Wigner symbol (or W Racah coefficient). Note that the introduction of k ℓ 0 j j J = δ k,ℓ (-1) j+k+J 1 (2k + 1)(2j + 1) (50) in Eq. (49) gives back Eq. ( 44 ). Equation (49) is central in the derivation of the coupling relation\n\nu (k) q u (ℓ) p = KQ (-1) 2j-Q (2K + 1) k ℓ K -q -p Q k ℓ K j j j u (K) Q . ( 51\n\n)\n\nEquation (51) makes it possible to calculate the commutator [u\n\n(k) q , u (ℓ)\n\np ] which shows that the set {u (k) q\n\n: k = 0, 1, • • • , 2j; q = -k, -k + 1, • • •\n\n, k} can be used to span the Lie algebra of the unitary group U(2j + 1). The latter result is at the root of the expansions (17) and (32)."
},
{
"section_type": "OTHER",
"section_title": "Note added in version 3",
"text": "After the submission of the present paper for publication in Journal of Russian Laser Research, a pre-print dealing with the existence of SIC-POVMs was posted on arXiv [32] .\n\nThe main result in [32] is that SIC-POVMs exist in all dimensions. As a corollary of this result, Eq. (34) admits solutions in any dimension.\n\n12"
},
{
"section_type": "OTHER",
"section_title": "Acknowledgements",
"text": "This work was presented at the International Conference on Squeezed States and Uncertainty Relations, University of Bradford, England (ICSSUR'07). The authors wish to thank the organizer A. Vourdas and are grateful to D. M. Appleby, V. I. Man'ko and M.\n\nPlanat for interesting comments. 13 References [1] A. Peres, \"Quantum Theory: Concepts and Methods\", Dordrecht: Kluwer (1995) [2] A. Vourdas, J. Phys. A: Math. Gen. 38, 8453 (2005) [3] W. M. de Muynck, \"Foundations of Quantum Mechanics, an Empiricist Approach\", Dordrecht: Kluwer (2002) [4] J. M. Jauch and C. Piron, Helv. Phys. Acta 40, 559 (1967) [5] E. B. Davies and J. T. Levis, Comm. Math. Phys. 17, 239 (1970) [6] E. B. Davies, IEEE Trans. Inform. Theory IT-24, 596 (1978) [7] K. Kraus, \"States, Effects, and Operations\", Lect. Notes Phys. 190 (1983) [8] G. Zauner, Diploma Thesis, University of Wien (1999) [9] C. M. Caves, C. A. Fuchs and R. Schack, J. Math. Phys. 43, 4537 (2002) [10] J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, J. Math. Phys. 45, 2171 (2004) [11] D. M. Appleby, J. Math. Phys. 46, 052107 (2005) [12] M. Grassl, Proc. ERATO Conf. Quant. Inf. Science (EQIS 2004) ed. J. Gruska, Tokyo (2005) [13] M. Grassl, Elec. Notes Discrete Math. 20, 151 (2005) [14] S. Weigert, Int. J. Mod. Phys. B 20, 1942 (2006) [15] J. Schwinger, Proc. Nat. Acad. Sci. USA 46, 570 (1960) [16] P. Delsarte, J. M. Goethals and J. J. Seidel, Philips Res. Repts. 30, 91 (1975) 14 [17] I. D. Ivanović, J. Phys. A: Math. Gen. 14, 3241 (1981) [18] W. K. Wootters, Ann. Phys. (N.Y.) 176, 1 (1987) [19] H. Barnum, Preprint quant-ph/0205155 (2002) [20] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, Algorithmica 34, 512 (2002) [21] A. O. Pittenger and M. H. Rubin, Linear Alg. Appl. 390, 255 (2004) [22] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum Semiclassical Opt. 6, L19 (2004) [23] I. Bengtsson and Å. Ericsson, Open Syst. Inf. Dyn. 12, 107 (2005) [24] A. Klappenecker and M. Rötteler, Preprint quant-ph/0502031 (2005) [25] W. K. Wootters, Found. Phys. 36, 112 (2006) [26] M. R. Kibler and M. Planat, Int. J. Mod. Phys. B 20, 1802 (2006) [27] O. Albouy and M. R. Kibler, SIGMA 3, article 076 (2007) [28] H.-P. Breuer, J. Phys. A: Math. Gen. 38, 9019 (2005) [29] G. Racah, Phys. Rev. 62, 438 (1942) [30] U. Fano and G. Racah, \"Irreducible Tensorial Sets\", New York: Academic (1959) [31] M. Kibler and G. Grenet, J. Math. Phys. 21, 422 (1980) [32] J.L. Hall and A. Rao, Preprint quant-ph/0707.3002v1 (20 July 2007) 15"
}
] | |
arxiv:0704.0516 | 0704.0516 | 1 | 4a119e65b80170869ee3687296117b16f01ab82d1c97a18278fec0312c5f9da7 | Effects of Imperfect Gate Operations in Shor's Prime Factorization
Algorithm | The effects of imperfect gate operations in implementation of Shor's prime
factorization algorithm are investigated. The gate imperfections may be
classified into three categories: the systematic error, the random error, and
the one with combined errors. It is found that Shor's algorithm is robust
against the systematic errors but is vulnerable to the random errors. Error
threshold is given to the algorithm for a given number $N$ to be factorized. | [
"Hao Guo",
"Gui Lu Long and Yang Sun"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | Shor's factorization algorithm [1] is a very important quantum algorithm, through which one has demonstrated the power of quantum computers. It has greatly promoted the worldwide research in quantum computing over the past few years. In practice, however, quantum systems are subject to influence of environment, and in addition, quantum gate operations are often imperfect [2, 3] . Environment influence on the system can cause decoherence of quantum states, and gate imperfection leads to errors in quantum computing. Thanks to Shor's another important work, in which he showed that quantum error correlation can be corrected [4] . With quantum error correction scheme, errors arising from both decoherence and imperfection can be corrected.
There have been several works on the effects of decoherence on Shor's algorithm. Sun et al. discussed the effect of decoherence on the algorithm by modeling the environment [5] . Palma studied the effects of both decoherence and gate imperfection in ion trap quantum computers [6] . There have also been many other studies on the quantum algorithm [7, 8, 9, 10] .
The error correction scheme uses available resources. Thus it is important to study the robustness of the algorithm itself so that one can strike a balance between the amount of quantum error correction and the amount of qubits available. In this paper, we investigate the effects of gate imperfection on the efficiency of Shor's factorization algorithm. The results may guide us in practice to suppress deliberately those errors that influence the algorithm most sensitively. For those errors that do not affect the algorithm very much, we may ignore them as a good approximation. In addition, study of the robustness of algorithm to errors is important where one can not apply the quantum error correction at all, for instance, in cases that there are not enough qubits available.
The paper is organized as follows. Section II is devoted to an outline of Shor's algorithm and different error's modes. In Section III, we present the results. Finally, a short summary is given in Section IV.
Shor's algorithm consists of the following steps: 1) preparing a superposition of evenly distributed states |ψ = 1 √ q q-1 a=0 |a |0 ,
where q = 2 L and N 2 ≤ q ≤ 2N 2 with N being the number to be factorized; 2) implementing y a modN and putting the results into the 2nd register
|ψ 1 = 1 √ q q-1 a=0 |a |y a modN ;
3) making a measument on the 2nd register; The state of the register is then
|φ 2 = 1 √ A + 1 A j=0 |jr + l |z = y l = y jr+l modN
where j ≤ q-l r = A.
4) performing discrete Fourier transformation (DFT) on the first
register |φ 3 = c f (c) |c |z , where f (c) = √ r q q r -1 j=0 exp 2πi(jr + l) q = √ r q e 2πilc q q r -1 j=0 exp 2πijrc q .
2 This term is nonzero only when c = k q r , with k = 0, 1, 2...r -1, which correspond to the peaks of the distribution in the measured results, and thus this term becomes f (c) = 1 √ r e 2πilc q . The Fourier transformation is important because it makes the state in the first register the same for all possible values in the 2nd register. The DFT is constructed by two basic gate operations: the single bit gate operation A j = 1 √ 2 1 1 1 -1 , which is also called the Walsh-Hadmard transformation, and the 2-bits controlled rotation
B jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e iθ jk with θ jk = π 2 k-j .
The gate sequence for implementing DFT is
(A q-1 )(B q-2q-1 A q-2 ) . . . (B 0q-1 B 0q-2 . . . B 01 A 0 ). Errors can occur in both A j and B jk . A j is actually a rotation about y-axis through π 2 A j (θ) = e i h Syθ = I cos( θ 2 )-i sin( θ 2 )σ y = cos( θ 2 ) -sin( θ 2 ) sin( θ 2 ) cos( θ 2 )
.
If the gate operation is not perfect, the rotation is not exactly π 2 . In this case, A j is a rotation of π 2 + 2δ A j (δ) = 1 √ 2 cos(δ) -sin(δ) -(sin(δ) + cos(δ)) sin(δ) + cos(δ) cos(δ) -sin(δ) .
If δ is very small, we have:
A j (θ) = 1 √ 2 1 -δ -(1 + δ) 1 + δ+ 1 -δ .
Similarly, errors in B jk can be written as
B jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e i(θ jk +δ) .
With these errors, the DFT becomes
|a → 1 √ q q-1 c=0 e i( 2π q/c +δc)a (1 + δ ′ c )|c = 1 √ q q-1 c=0 e i( 2πc q +δc)a (1 + δ ′ c )|c ,( 1
)
where δ c and δ ′ c denote the error of A j and B jk , respectively.
Let us assume the following error modes: 1) systematic errors, where δ c or δ ′ c in (1) can only have systematic errors (EM 1 ); 2) random errors (EM 2 ), for which we assume that δ c or δ ′ c can only be random errors of the Gaussian or the uniform type; 3) coexistence of both systematic and random errors (EM 3 ). In the next section, we shall present the results of numerical simulations and discuss the effects of imperfect gate operation on the DFT algorithm, and thus on the Shor's algorithm.
We first discuss the influence of imperfect gate operations in the initial preparation
A l-1 A l-2 ...A 0 |0...0 = 1 √ 2 (|0 + |1 + δ 1 (|0 -|1 )) ⊗ (|0 + |1 + δ 2 . = 1 √ 2 l 1 i1i2...in=0 |i 1 i 2 ...i n + 1 √ 2 l n R=1 δ n 1 i
If the errors are systematic, for instance, caused by the inaccurate calibration of the rotations, then δ 1 = δ 2 = . . . = δ n = δ. In this case, we can write the 2nd term as
|ψ = 1 √ 2 l δ 1 i1i2...in=0 (2s -n)|i 1 i 2 ...i n ,
where s stands for the number of 1's, and 2sn = s -(ns) is the difference in the number of 1's and 0's. Thus the results after the first procedure is
1 √ 2 l 2 L -1 a=0 (|a + δ(2s -n)|a ) = 1 √ 2 l a=0 (1 + δ a )|a . (2)
This implies that after the procedure, the amplitude of each state is no longer equal, but have slight difference. Combining the effect in the initialization and in the DFT, we have
(1 + δ a )(1 + δ c )e i( 2πc q +δ ′ c )a . = (1 + δ ′′ )e i( 2πc q +δ ′ c )a , where δ ′′ c = δ c + δ a . In the DFT, we have |ψ ⇒ √ r q q-1 c=0 q r -1 j=0 (1 + δ j )e i( 2πc q +δ ′ j )(jr+l) |c ,
where we have rewrite δ ′′ as δ j here. Let P c denote the probability of getting the state |c after we perform a measurement, we have 3 ), we find that after the last measurement, each state can be extracted with a probability which is nonzero, and the offset l can't be eliminated. Eq. ( 3 ) is very complicated, so we will make some predigestions to discuss different error modes for convenience. Generally speaking, the influence of exponential error δ j is more remarkable than δ j , so we can omit the error δ j , thus
P c = r q 2 q r -1 m=0 q r -1 k=0 (1 + δ m )(1 + δ k )e i( 2πc q +δ ′ m )(mr+l) ×e -i( 2πc q +δ ′ k )(kr+l) = r q 2 m k (1 + δ m )(1 + δ k ) cos[ 2πc q r(m -k) + (mr + l)δ ′ m -( From Eq. (
DFT q |φ = √ r q q-1 c=0 q r j=0 e i( 2πc q +δ ′ j )(jr+l) |c . 3
A. Case 1 If only systematic errors (EM 1 ) are considered, namely, all the δ j 's are equal, then f (c) can be given analytically f (c) = √ r q q r -1 j=0 e i( 2πc q +δ)(jr+l) = √ r q e il( 2πc q +δ) 1e i( 2πc q +δ)q
1 -e i( 2πc q +δ)r ( 4
)
The relative probability of finding c is
P c = f (c) 2 = r q 2 sin 2 ( δq 2 ) sin 2 ( πcr q + δr 2 ) , and if c = k q r , then P c = r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) .
It can be easily seen that lim δ→0 P c = 1 r , which is just the case that no error is considered.
When δ takes certain values, say, δ = 2 r (k -r q )π where k is an integer, then the summation in Eq. (4) is on longer valid. In our simulation, δ does not take these values. Here we consider the case where q = 2 7 = 128 and r = 4. For comparisons, we have drawn the relative probability for obtaining state c in Fig.1. for this given example. We have found the following results: (i) When δ is small, the errors do hardly influence the final result, for instance when c = k q r , then
lim δ→0 P c = lim δ→0 r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) = 1 r .
The probability distribution is almost identical to those without errors.
(ii) Let us increase δ gradually, from Fig. 2 , we see that a gradual change in the probability distribution takes place. (Here, we again consider the relative probabilities) When δ is increased to certain values, the positions of peaks change greatly. For instance at δ = 0.05, there appears a peak at c=127, whereas it is P c = 0 when no systematic errors are present. In general, the influence of systematic errors on the algorithm is a shift of the peak positions. This influences the final results directly.
When both random errors and systematic errors are present, we add random errors to the simulation. To see the effect of different mode of random errors, we use two random number generators. One is the Gaussian mode and the other is the uniform mode. In this case, the error has the form δ = δ 0 + s, where δ 0 is the systematic error. s has a probability distribution with respect to c, depending on the uniform or the Gaussian distribution. When δ 0 = 0, we have only random errors which is our error mode 2. When δ 0 = 0, we have error mode 3. For the uniform distribution, s ∼ ±s max × u(0, 1) where u(0, 1) is evenly distributed in [0, 1] . s max indicates the maximum deviation from δ 0 . For Gaussian distribution, s ∼ N (0, σ 0 ). Through the figure, we see the following: (1) When only random errors are present (δ 0 = 0), the peak positions are not affected by these random errors. However, different random error modes cause similar results. The results for uniform random error mode are shown in Fig. 3 . For the uniform distribution error mode, with increasing δ max , the final probability distribution of the final results become irregular. In particular, when δ max is very large, all the patterns are destroyed and is hardly recognizable. Many unexpected small peaks appear. For the Gaussian distribution error mode, as shown in Fig. 4 , the influence of the error is more serious. This is because in Gaussian distribution, there is no cut-off of errors. Large errors can occur although their probability is small. The influence of σ 0 on the final results is also sensitive, because it determines the shape of the distribution. When σ 0 increases, the final probability distribution becomes very messy. A small change in σ 0 can cause a big change in the final results. (2) When δ 0 = 0, which corresponds to error mode 3, the effect is seen as to shift the positions of the peaks in addition to the influences of the random errors.
To summarize, we have analyzed the errors in Shor's factorization algorithm. It has been seen that the effect of the systematic errors is to shift the positions of the peaks, whereas the random errors change the shape of the probability distribution. For systematic errors, the shape of the distribution of the final results is hardly destroyed, though displaced. We can still use the result with several trial guesses to obtain the right results because the peak positions are shifted only slightly. However, the random errors are detrimental to the algorithm and should be reduced as much as possible. It is different from the case with Grover's algorithm where systematic errors are disastrous while random errors are less harmful [10].
[1] P.W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S.
Goldwasser (IEEE Computer Society Press, Los Alamitos, CA, 1994) p.124.
-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 p c FIG. 1: Relative probability for finding state c in the absence of errors.
[2] A. Ekert and R. Jozsa, Rev. Mod. Phys. 68 (1996) 733. [3] W.G. Unruh, Phys. Rev A51 (1995) 992. [4] I. Chuang and R. laflamme, "Quantum error correction by codding" (1995) quant-ph/9511003. [5] C.P. Sun, H. Zhan and X.F. Liu, Phys. Rew. A58 (1998) 1810. [6] G.M. Palma, K.A. Suominen and A.K. Ekert, Proc. R. Soc. London, A 452 (1996) 567. [7] R.P. Feynman, Int. J. Theo. Phys., 21 (1982) 467. [8] D. Deutsch, Proc. R. Soc. Land. A 400 (1985) 97. [9] L.K. Grover, Phys. Rev, Lett, 79 (1997) 325. [10] G.L. Long, Y.S. Li, W.L. Zhang, C.C. Tu, Phys. Rev. A 61 (2000) 042305. [11] L.K. Grover, Phys. Rev. Lett, 80 (1998) 4329.
-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (4) (3) c P (2) -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 c P -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c FIG. 2: The same as Fig.1. with systematic errors. In subfigures (1), (2), (3), (4), δ are 0.02, 0.03, 0.05 respectively. In sub-figure (4), the curve with solid circles(with higher peaks) is the result with δ = 0.1, and the one without solid circles(with lower peaks) denotes the result with δ = 0.33.
-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (3) c P (1) -20 0 20 40 60 80 100 120 140 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (4) c P (2)
FIG. 3: The same as Fig.1. with uniform random errors. In sub-figures (1), (2), (3), (4), smax are set to 0.01, 0.03, 0.05, 0.1 respectively.
-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 (2) P c -20 0 20 40 60 80 100 120 140 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 c P (3) -20 0 20 40 60 80 100 120 140 0.00 0.02 0.04 0.06 0.08 c P (4)
FIG. 4: The same as Fig.1. with Gaussian random errors and systematic errors. In sub-figures (1), (2), and (3) τ are set to 0.01, 0.03 and 0.05 respectively, and δ0 = 0(without systematic errors). In sub-figure (4), both systematic and random Gaussian errors exist, where δ0 = 0.33, τ = 0.02. | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Shor's factorization algorithm [1] is a very important quantum algorithm, through which one has demonstrated the power of quantum computers. It has greatly promoted the worldwide research in quantum computing over the past few years. In practice, however, quantum systems are subject to influence of environment, and in addition, quantum gate operations are often imperfect [2, 3] . Environment influence on the system can cause decoherence of quantum states, and gate imperfection leads to errors in quantum computing. Thanks to Shor's another important work, in which he showed that quantum error correlation can be corrected [4] . With quantum error correction scheme, errors arising from both decoherence and imperfection can be corrected.\n\nThere have been several works on the effects of decoherence on Shor's algorithm. Sun et al. discussed the effect of decoherence on the algorithm by modeling the environment [5] . Palma studied the effects of both decoherence and gate imperfection in ion trap quantum computers [6] . There have also been many other studies on the quantum algorithm [7, 8, 9, 10] .\n\nThe error correction scheme uses available resources. Thus it is important to study the robustness of the algorithm itself so that one can strike a balance between the amount of quantum error correction and the amount of qubits available. In this paper, we investigate the effects of gate imperfection on the efficiency of Shor's factorization algorithm. The results may guide us in practice to suppress deliberately those errors that influence the algorithm most sensitively. For those errors that do not affect the algorithm very much, we may ignore them as a good approximation. In addition, study of the robustness of algorithm to errors is important where one can not apply the quantum error correction at all, for instance, in cases that there are not enough qubits available.\n\nThe paper is organized as follows. Section II is devoted to an outline of Shor's algorithm and different error's modes. In Section III, we present the results. Finally, a short summary is given in Section IV."
},
{
"section_type": "OTHER",
"section_title": "II. SHOR'S ALGORITHM AND ERROR'S MODES",
"text": "Shor's algorithm consists of the following steps: 1) preparing a superposition of evenly distributed states |ψ = 1 √ q q-1 a=0 |a |0 ,\n\nwhere q = 2 L and N 2 ≤ q ≤ 2N 2 with N being the number to be factorized; 2) implementing y a modN and putting the results into the 2nd register\n\n|ψ 1 = 1 √ q q-1 a=0 |a |y a modN ;\n\n3) making a measument on the 2nd register; The state of the register is then\n\n|φ 2 = 1 √ A + 1 A j=0 |jr + l |z = y l = y jr+l modN\n\nwhere j ≤ q-l r = A.\n\n4) performing discrete Fourier transformation (DFT) on the first\n\nregister |φ 3 = c f (c) |c |z , where f (c) = √ r q q r -1 j=0 exp 2πi(jr + l) q = √ r q e 2πilc q q r -1 j=0 exp 2πijrc q .\n\n2 This term is nonzero only when c = k q r , with k = 0, 1, 2...r -1, which correspond to the peaks of the distribution in the measured results, and thus this term becomes f (c) = 1 √ r e 2πilc q . The Fourier transformation is important because it makes the state in the first register the same for all possible values in the 2nd register. The DFT is constructed by two basic gate operations: the single bit gate operation A j = 1 √ 2 1 1 1 -1 , which is also called the Walsh-Hadmard transformation, and the 2-bits controlled rotation\n\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e iθ jk with θ jk = π 2 k-j .\n\nThe gate sequence for implementing DFT is\n\n(A q-1 )(B q-2q-1 A q-2 ) . . . (B 0q-1 B 0q-2 . . . B 01 A 0 ). Errors can occur in both A j and B jk . A j is actually a rotation about y-axis through π 2 A j (θ) = e i h Syθ = I cos( θ 2 )-i sin( θ 2 )σ y = cos( θ 2 ) -sin( θ 2 ) sin( θ 2 ) cos( θ 2 )\n\n.\n\nIf the gate operation is not perfect, the rotation is not exactly π 2 . In this case, A j is a rotation of π 2 + 2δ A j (δ) = 1 √ 2 cos(δ) -sin(δ) -(sin(δ) + cos(δ)) sin(δ) + cos(δ) cos(δ) -sin(δ) .\n\nIf δ is very small, we have:\n\nA j (θ) = 1 √ 2 1 -δ -(1 + δ) 1 + δ+ 1 -δ .\n\nSimilarly, errors in B jk can be written as\n\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e i(θ jk +δ) .\n\nWith these errors, the DFT becomes\n\n|a → 1 √ q q-1 c=0 e i( 2π q/c +δc)a (1 + δ ′ c )|c = 1 √ q q-1 c=0 e i( 2πc q +δc)a (1 + δ ′ c )|c ,( 1\n\n)\n\nwhere δ c and δ ′ c denote the error of A j and B jk , respectively.\n\nLet us assume the following error modes: 1) systematic errors, where δ c or δ ′ c in (1) can only have systematic errors (EM 1 ); 2) random errors (EM 2 ), for which we assume that δ c or δ ′ c can only be random errors of the Gaussian or the uniform type; 3) coexistence of both systematic and random errors (EM 3 ). In the next section, we shall present the results of numerical simulations and discuss the effects of imperfect gate operation on the DFT algorithm, and thus on the Shor's algorithm."
},
{
"section_type": "OTHER",
"section_title": "III. INFLUENCE OF IMPERFECT GATE OPERATIONS",
"text": "We first discuss the influence of imperfect gate operations in the initial preparation\n\nA l-1 A l-2 ...A 0 |0...0 = 1 √ 2 (|0 + |1 + δ 1 (|0 -|1 )) ⊗ (|0 + |1 + δ 2 . = 1 √ 2 l 1 i1i2...in=0 |i 1 i 2 ...i n + 1 √ 2 l n R=1 δ n 1 i\n\nIf the errors are systematic, for instance, caused by the inaccurate calibration of the rotations, then δ 1 = δ 2 = . . . = δ n = δ. In this case, we can write the 2nd term as\n\n|ψ = 1 √ 2 l δ 1 i1i2...in=0 (2s -n)|i 1 i 2 ...i n ,\n\nwhere s stands for the number of 1's, and 2sn = s -(ns) is the difference in the number of 1's and 0's. Thus the results after the first procedure is\n\n1 √ 2 l 2 L -1 a=0 (|a + δ(2s -n)|a ) = 1 √ 2 l a=0 (1 + δ a )|a . (2)\n\nThis implies that after the procedure, the amplitude of each state is no longer equal, but have slight difference. Combining the effect in the initialization and in the DFT, we have\n\n(1 + δ a )(1 + δ c )e i( 2πc q +δ ′ c )a . = (1 + δ ′′ )e i( 2πc q +δ ′ c )a , where δ ′′ c = δ c + δ a . In the DFT, we have |ψ ⇒ √ r q q-1 c=0 q r -1 j=0 (1 + δ j )e i( 2πc q +δ ′ j )(jr+l) |c ,\n\nwhere we have rewrite δ ′′ as δ j here. Let P c denote the probability of getting the state |c after we perform a measurement, we have 3 ), we find that after the last measurement, each state can be extracted with a probability which is nonzero, and the offset l can't be eliminated. Eq. ( 3 ) is very complicated, so we will make some predigestions to discuss different error modes for convenience. Generally speaking, the influence of exponential error δ j is more remarkable than δ j , so we can omit the error δ j , thus\n\nP c = r q 2 q r -1 m=0 q r -1 k=0 (1 + δ m )(1 + δ k )e i( 2πc q +δ ′ m )(mr+l) ×e -i( 2πc q +δ ′ k )(kr+l) = r q 2 m k (1 + δ m )(1 + δ k ) cos[ 2πc q r(m -k) + (mr + l)δ ′ m -( From Eq. (\n\nDFT q |φ = √ r q q-1 c=0 q r j=0 e i( 2πc q +δ ′ j )(jr+l) |c . 3\n\nA. Case 1 If only systematic errors (EM 1 ) are considered, namely, all the δ j 's are equal, then f (c) can be given analytically f (c) = √ r q q r -1 j=0 e i( 2πc q +δ)(jr+l) = √ r q e il( 2πc q +δ) 1e i( 2πc q +δ)q\n\n1 -e i( 2πc q +δ)r ( 4\n\n)\n\nThe relative probability of finding c is\n\nP c = f (c) 2 = r q 2 sin 2 ( δq 2 ) sin 2 ( πcr q + δr 2 ) , and if c = k q r , then P c = r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) .\n\nIt can be easily seen that lim δ→0 P c = 1 r , which is just the case that no error is considered.\n\nWhen δ takes certain values, say, δ = 2 r (k -r q )π where k is an integer, then the summation in Eq. (4) is on longer valid. In our simulation, δ does not take these values. Here we consider the case where q = 2 7 = 128 and r = 4. For comparisons, we have drawn the relative probability for obtaining state c in Fig.1. for this given example. We have found the following results: (i) When δ is small, the errors do hardly influence the final result, for instance when c = k q r , then\n\nlim δ→0 P c = lim δ→0 r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) = 1 r .\n\nThe probability distribution is almost identical to those without errors.\n\n(ii) Let us increase δ gradually, from Fig. 2 , we see that a gradual change in the probability distribution takes place. (Here, we again consider the relative probabilities) When δ is increased to certain values, the positions of peaks change greatly. For instance at δ = 0.05, there appears a peak at c=127, whereas it is P c = 0 when no systematic errors are present. In general, the influence of systematic errors on the algorithm is a shift of the peak positions. This influences the final results directly."
},
{
"section_type": "OTHER",
"section_title": "B. Case 2",
"text": "When both random errors and systematic errors are present, we add random errors to the simulation. To see the effect of different mode of random errors, we use two random number generators. One is the Gaussian mode and the other is the uniform mode. In this case, the error has the form δ = δ 0 + s, where δ 0 is the systematic error. s has a probability distribution with respect to c, depending on the uniform or the Gaussian distribution. When δ 0 = 0, we have only random errors which is our error mode 2. When δ 0 = 0, we have error mode 3. For the uniform distribution, s ∼ ±s max × u(0, 1) where u(0, 1) is evenly distributed in [0, 1] . s max indicates the maximum deviation from δ 0 . For Gaussian distribution, s ∼ N (0, σ 0 ). Through the figure, we see the following: (1) When only random errors are present (δ 0 = 0), the peak positions are not affected by these random errors. However, different random error modes cause similar results. The results for uniform random error mode are shown in Fig. 3 . For the uniform distribution error mode, with increasing δ max , the final probability distribution of the final results become irregular. In particular, when δ max is very large, all the patterns are destroyed and is hardly recognizable. Many unexpected small peaks appear. For the Gaussian distribution error mode, as shown in Fig. 4 , the influence of the error is more serious. This is because in Gaussian distribution, there is no cut-off of errors. Large errors can occur although their probability is small. The influence of σ 0 on the final results is also sensitive, because it determines the shape of the distribution. When σ 0 increases, the final probability distribution becomes very messy. A small change in σ 0 can cause a big change in the final results. (2) When δ 0 = 0, which corresponds to error mode 3, the effect is seen as to shift the positions of the peaks in addition to the influences of the random errors."
},
{
"section_type": "OTHER",
"section_title": "IV. SUMMARY",
"text": "To summarize, we have analyzed the errors in Shor's factorization algorithm. It has been seen that the effect of the systematic errors is to shift the positions of the peaks, whereas the random errors change the shape of the probability distribution. For systematic errors, the shape of the distribution of the final results is hardly destroyed, though displaced. We can still use the result with several trial guesses to obtain the right results because the peak positions are shifted only slightly. However, the random errors are detrimental to the algorithm and should be reduced as much as possible. It is different from the case with Grover's algorithm where systematic errors are disastrous while random errors are less harmful [10].\n\n[1] P.W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S.\n\nGoldwasser (IEEE Computer Society Press, Los Alamitos, CA, 1994) p.124.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 p c FIG. 1: Relative probability for finding state c in the absence of errors.\n\n[2] A. Ekert and R. Jozsa, Rev. Mod. Phys. 68 (1996) 733. [3] W.G. Unruh, Phys. Rev A51 (1995) 992. [4] I. Chuang and R. laflamme, \"Quantum error correction by codding\" (1995) quant-ph/9511003. [5] C.P. Sun, H. Zhan and X.F. Liu, Phys. Rew. A58 (1998) 1810. [6] G.M. Palma, K.A. Suominen and A.K. Ekert, Proc. R. Soc. London, A 452 (1996) 567. [7] R.P. Feynman, Int. J. Theo. Phys., 21 (1982) 467. [8] D. Deutsch, Proc. R. Soc. Land. A 400 (1985) 97. [9] L.K. Grover, Phys. Rev, Lett, 79 (1997) 325. [10] G.L. Long, Y.S. Li, W.L. Zhang, C.C. Tu, Phys. Rev. A 61 (2000) 042305. [11] L.K. Grover, Phys. Rev. Lett, 80 (1998) 4329.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (4) (3) c P (2) -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 c P -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c FIG. 2: The same as Fig.1. with systematic errors. In subfigures (1), (2), (3), (4), δ are 0.02, 0.03, 0.05 respectively. In sub-figure (4), the curve with solid circles(with higher peaks) is the result with δ = 0.1, and the one without solid circles(with lower peaks) denotes the result with δ = 0.33.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (3) c P (1) -20 0 20 40 60 80 100 120 140 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (4) c P (2)\n\nFIG. 3: The same as Fig.1. with uniform random errors. In sub-figures (1), (2), (3), (4), smax are set to 0.01, 0.03, 0.05, 0.1 respectively.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 (2) P c -20 0 20 40 60 80 100 120 140 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 c P (3) -20 0 20 40 60 80 100 120 140 0.00 0.02 0.04 0.06 0.08 c P (4)\n\nFIG. 4: The same as Fig.1. with Gaussian random errors and systematic errors. In sub-figures (1), (2), and (3) τ are set to 0.01, 0.03 and 0.05 respectively, and δ0 = 0(without systematic errors). In sub-figure (4), both systematic and random Gaussian errors exist, where δ0 = 0.33, τ = 0.02."
}
] | ||
arxiv:0704.0520 | 0704.0520 | 1 | e2ed075f4ade80918e070091647f48ec5d73e5a1179cb737dc87b8224a6ec3ab | A critical theory of quantum entanglement for the Hydrogen molecule | In this paper we investigate some entanglement properties for the Hydrogen
molecule considered as a two interacting spin 1/2 (qubit) model. The
entanglement related to the $H_{2}$ molecule is evaluated both using the von
Neumann entropy and the Concurrence and it is compared with the corresponding
quantities for the two interacting spin system. Many aspects of these functions
are examinated employing in part analytical and, essentially, numerical
techniques. We have compared analogous results obtained by Huang and Kais a few
years ago. In this respect, some possible controversial situations are
presented and discussed. | [
"Tina A.C. Maiolo",
"Luigi Martina",
"Giulio Soliani"
] | [
"quant-ph",
"cond-mat.other",
"physics.atom-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | Entanglement is a physical observable measured by the von Neumann entropy or, alternatively, by the Concurrence of the system under consideration.
The concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The electron correlation is not directly observable, since it is defined as the difference between the exact ground state energy of the many electrons Schrödinger equation and the Hartree-Fock energy.
In this paper we discuss the Hamiltonian which describes the Hydrogen molecule regarded as a two interacting spin 1 2 (qubit) model. In [1] it was argued that the entanglement (a quantum observable) can be used in analyzing the so-called correlation energy which is not directly observable. From our point of view, the Hydrogen molecule is dealt with a bipartite system governed by the Hamiltonian
H H2 = - J 2 (1 + g)σ 1 ⊗ σ 1 - J 2 (1 -g)σ 2 ⊗ σ 2 -B(σ 3 ⊗ σ 3 + σ 0 ⊗ σ 3 ), ( 1
) 1
where σ i stand for the Pauli matrices (σ 0 = I). Actually, this model was considered in [1] in order to illustrate their method. However, here we will make some interpretative changes. Indeed, from our point of view, the states of an isolated atom are strongly reduced to a system with two energy levels related to the intensity of the magnetic field B. Relatively to this scale, the exchange interaction constant J is usually smaller than B, in order to represent the residual interatomic interactions. From the point of view of quantum chemistry, one may interpret the discrete spectrum as provided by the Hartree-Fock calculations, while the interaction coupling J models the residual multielectronic effects, not taken into account by the mean field approximation. For simplicity we limit ourselves to the ferromagnetic phase with J > 0. The parameter g, such that 0 ≤ g ≤ 1, describes the degree of anisotropy corresponding for g = 0 to the completely isotropic XY spin model. Conversely, g = 1 provides the anisotropic XY spin model, the so-called Ising model.
We notice that when the atoms are far apart, their interaction is quite weak. This corresponds to a vanishing value of J. In this situation the state of the system is completely factorized in the product state of the ground states of the indipendent spins. The corresponding total energy, in unit of B, is just the sum of the two fundamental levels, E 0 = -2, which we may consider as the Hartree-Fock approximated fundamental level in molecular structure calculations.
When J = 0, the fundamental energy eigenvalue is E= -4 + g 2 λ 2 in Region I defined by 0 < λ ≤ 2 √ 1-g 2 , otherwise E = -λ (λ means the coupling constant) in Region II, which is the complement of I which respect to positive real axis. The corresponding (non normalized) eigenstates are |Ψ I = √ g 2 λ 2 +4+2 gλ , 0, 0, 1 and |Ψ II = 0, 1, 1, 0 , respectively. In both cases the state is entangled.
Since we are dealing with pure states, the von Neumann entropy [2]
S vN = -Tr ρ 1 log 2 ρ 1 ( 2
)
is chosen to be a measurement of the entanglement, where ρ 1 is the 1-particle reduced density matrix. However, for general mixed states other entanglement estimators (for instance, the Concurrence [4]) have to be used. In the considered case, one has S vN,I = -g 2 g 2 λ 2 + 4 log 1 2 -1 g 2 λ 2 + 4 λ 2 -g 2 g 2 λ 2 + 4 + 4 λ 2 + 8 g 2 λ 2 + 4 + 2 log 1 2 + 1 g 2 λ 2 + 4 1 g 2 λ 2 + 4 g 2 λ 2 + 2 g 2 λ 2 + 4 + 4 log(4) (3) S vN,II = 1. (4) 2 Scrutinizing Eq. (3) and Eq. ( 4 ) it emerges that the entropy is an increasing function of the coupling constant λ in Region I, but the state is maximally entangled in Region II independently from the anisotropy parameter g. One sees that, as it arises graphycally, for g = 1 the entanglement is a monotonic increasing function of the interaction coupling λ. Moreover for weak (< 1) coupling values it is always less than the 30%. Of course, for large coupling constants the entropy approaches 1, meaning that all levels are equiprobably visited by the considered spin.
Limiting all further considerations to the case of weak interaction, we observe that at the boundary point λ b = 2 √ 1-g 2 a discontinuity occurs, signaling a crossing of the lowest eigenvalues and, in a more general context, a quantum phase transition [5] .
As it was pointed out in [6] , for quantifying the entanglement we can resort to the reduced density matrix. Furthermore, in [7] , Wootters has shown that for a pair of binary qubits one can use the concept of Concurrence C to measure the entanglement.
The Concurrence reads
C(ρ) = max(0, ν 1 -ν 2 -ν 3 -ν 4 ), ( 5
)
where the ν i 's are the eigenvalues of the Hermitian matrix
R = √ ρρ √ ρ 1 2 ,
where ρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ), ρ * being the complex conjugate of ρ taken in the standard basis [7] . Some interesting results on the simple model (1) of the Hydrogen molecule can be achieved by realizing a comparative study of the von Neumann entropy and the Concurrence.
To this aim, we compute the Concurrence C I and C II , i. e.
C I = gλ 1 g 2 λ 2 + 4 , C II = 1. ( 6
)
where I and II refer to Regions I and II, where 0 ≤ λ ≤ 2 1-g 2 , and E = -λ, respectively.
In Figure 1 a comparison between the Concurrence and the von Neumann entropy for two spins system as a function of the coupling λ for g = 1 is presented. Sec. 2 contains a comparison between the entanglement and the correlation energy. In Sec. 3 the Configuration Interaction method is introduced to compare entanglement and correlation energy. In Sec. 4 some differences between the Configuration Interaction approach and the two spin Ising model are presented. Finally, our main results are summarized in Sec. 5. 3 1 2 3 4 5 Λ 0.2 0.4 0.6 0.8 1 S vN Conc.
Figure 1: Comparison between the Concurrence and the von Neumann entropy for the two spins system as a function of the coupling constant λ for g = 1.
2 A comparison between the entanglement and the correlation energy
Now we look for a comparison between the entanglement with the energy correlation, which as we have already recalled, it is understood as the difference of the fundamental energy level compared with respect to the corresponding value at vanishing coupling constant λ. For g = 1 and in unities of B it is given by
E corr = |E 0 | -2 = 4 + λ 2 -2. ( 7
)
We observe that the entanglement measure is always bounded, while E corr is a divergent function of λ. So it does not make much sense to look for simple relations valid on the entire λ-axes. Consequently, limiting ourselves to weak couplings, for 0 ≤ λ ≤ 1, we minimize the mean squared deviation
I α = 1 0 ∆S 2 α dλ, with ∆S α = E corr -α S vN . ( 8
)
Thus the minimizing parameter α min will be given by
α min = 1 0 E corr S vN dλ 1 0 S 2 vN dλ ≈ -0.691217. ( 9
)
A formula analogous to (9) can be obtained by using the Concurrence as a measure of entanglement. In this case, by minimizing the mean squared deviation we have
I C α ′ = 1 0 ∆C 2 α ′ dλ, with ∆C α ′ = E corr -α ′ C. ( 10
)
Now, in order to estimate the relative deviation of
S vN with respect to E corr , let us report |∆S αmin |/S vN and |∆S αmin /E corr | as
functions of λ at the optimal value α min . The graphs of these functions are shown in Figure 2. 4 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min S vN 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min E corr Figure 2: The relative quadratic deviation between the von Neumann entropy and the correlation energy with respect to the former and the latter, respectively, at the optimal value α min as a function of the coupling constant λ for g = 1.
In Figure 3 , the relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, at the optimal values α ′ min , is represented.
0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min C 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min E corr Figure 3: The relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, respectively, at the optimal value α ′ min as a function of the coupling constant λ for g = 1.
From these graphs, one can argue that the agreement between the two functions S vN and E corr is only qualitatively good, in fact, for very small λ, it is not good at all. However, in an intermediate range of values, i. e., 0.6 ≤ λ ≤ 1 the two functions are almost proportional within the 10%. Analogously, the same is true between energy and Concurrence. Even, the agreement becomes worst comparing the relative deviation of the Concurrence with respect to the correlation energy, since the range in which the relative deviations become smaller than 10% are narrower. Then, the question is whether the above results are i) sufficient to justify the conjecture advanced in [1] , i.e., entanglement can be considered as an estimation of correlation energy; ii) if such a relation has a 5 more concrete physical meaning, in particular whether the minimizing parameter α min and the vanishing point of ∆S αmin does possess any physical meaning (or α ′ min and the vanishing point of ∆C α ′ min ). Notice that in the case of the comparison for the Concurrence simpler analytical expressions appear. For instance one finds ∆C α ′ min = 0.383249 λ
√ λ 2 +4 - √ λ 2 + 4 + 2 2 . Remark 2
We note that in an interval of values around α min , the deviation function (8) possesses a minimum in the interval of interest 0 ≤ λ ≤ 1, otherwise the minimum is achieved at larger value of λ, or the function is monotonically increasing (see Figure 4 ).
0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 S Α 0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 d S Α dΛ
Figure 4: The deviation ∆S α and its derivative with respect to λ are computed for values of -1.29(red) ≤ α ≤ -0.091(violet), for steps of 0.06. The curve drawn thicker corresponds to α min This behavior suggests to consider the function ∆S αmin as a sort of "free energy" , where α min mimics the "temperature" specific of the system. If, for some reason, we allow λ to change, then we expect that spontaneously the interaction coupling adjusts itself to the minimum of ∆S αmin . Similar considerations can be made looking at the graphs drawn for the function ∆C α ′ min and its derivative with respect to λ (see Figure 5 ). The function ∆S αmin or, alternatively, the minimum of ∆C α ′ min can be obtained algebraically. Such a minimum is at the value of the coupling constant λ SvN min ≈ 0.485 and λ C min ≈ 0.371, respectively. The authors in [1] studied numerically the von Neumann entropy and the correlation function for a Hydrogen molecule, using an old result by Herring and Flicker [8], going back to an oldest idea by Heitler and London [9], which consists in substituting the molecular binding with a position dependent exchange coupling:
J(r) ≈ 1.641 r 5 2 e -2 r Ry, ( 11
)
where r is given in Bohr radius, see Figure 6 . The maximum value taken by this function is at the point r max = 1.25. Assuming B = 0.5 Ry, i.e. 1 2 of the fundamental level of the Hydrogen atom, the maximum value λ ′ max = J(r max )/B ≈ 6 0.2 0.4 0.6 0.8 1 Λ -0.2 -0.1 0.1 0.2 0.3 C Α' 0.2 0.4 0.6 0.8 1 Λ -0.4 -0.2 0.2 0.4 d C Α' dΛ 5: The deviation ∆C α ′ and its derivative with respect to λ are computed for values of -0.98(red) ≤ α ′ ≤ 0.22(violet), for steps of 0.06. The curve drawn thicker corresponds to α ′ min 0.5 1 1.5 2 2.5 3 r 0.05 0.1 0.15 0.2 J r Ry
Figure 6: The effective interaction Hydrogen-Hydrogen atom 0.470628 < λ SvN min , i.e. the value of the effective interaction value is less than the minimum for the deviation function ∆S αmin . Then, the equilibrium balance between entanglement (as von Neumann entropy) and correlation energy predicts a length of the molecule equal to r max (see the first panel of Figure 7 ). On the other hand, if we consider the energy gap 2B = 3/4 Ry, i.e. the energy step to the first excited state, one obtains the new value λ ′′ max ≈ 0.628, which goes beyond λ min , even if it is always less than 1. Now, the deviation function ∆S αmin has two minima as seen in the second panel of Figure 7 , one of which is at r ′′ -≈ 0.76 , the other one being at r ′′ + ≈ 1.91. These results should be compared with the experimental equilibrium length of the Hydrogen molecule, which is r exp ≈ 2.0. We point out that although the spin-model described by the Hamiltonian (1) is characterized by features which are essentially rough, however we are induced to answer positively to the quest for a physical meaning of the deviation function ∆S αmin . Indeed, the results elucidated in Figure 7 encourage, on one part, improvement of the computation of r in order to make more accurate the comparison with the experimental value r exp . 7 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .5 Ry 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .375 Ry r'' r''
Figure 7: The von Neumann entropy for the 2-spin model for B = .5 Ry (left panel) and for B = .375 Ry (right panel) and the position depending interaction given by ( 11 ).
The first question to answer is whether this draft works also for the Concurrence. A statement about it is not obvious, since the von Neumann entropy is a nonlinear function of the Concurrence in the 2-qubits case. However, from Figure 8 one can see that the minimized deviation of the Concurrence takes one minimum for relatively large intensity of the magnetic field ( say B ≥ 0.6 Ry), while for weak fields two minima appear, corresponding to the situation depicted nearby. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 r Bohr B Ry 0.2 0.4 0.6 0.8 1 0 1 2 3 4 B(Ry) r(Bohr)
Figure 8: Two contour plots of the minimized deviation of the Concurrence as a function of the magnetic field B (Ry) and of the internuclear distance r, as given by (11) . The range of values divided by the contour lines is [-0.038, 0, 04] for the left panel and [-0.03705, -0, 03000] for the right one that approximatively corresponding to the black area in the left panel.
In correspondence of the same values considered above, for B = 0.5 Ry the function ∆C α ′ min (r) has two minima at r = 0.79 and r = 1.88, while for B = 0.375 Ry they are located at r = 0.60 and r = 2.25. So one sees that the resulting equilibrium configurations are not much very close to the experimental one. The equilibrium configuration more closest to the experimental one is the minimum occurring at r = 1.88 (B = 1 2 Ry) for the function ∆C α ′ min (r).
8 One sees that one of the resulting equilibrium configurations is only roughly close to the experimental one.
In other words, to conclude monitoring numerically B the equilibrium configuration more closest to experimental one in the minimum occurring at r = 1.88 for B = 1 2 Ry and at r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r).
In this Section we represent the results produced in [1], where the electron entanglement in the Hydrogen molecule, calculated by the von Neumann entropy of the reduced density matrix ρ 1 , is obtained starting by the excitation coefficients of the wave function expanded by a configuration interaction method:
S ρ CISD 1 = -T r ρ CISD 1 log 2 ρ CISD 1 = = - m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 log 2 m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 + -|c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 log 2 |c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 , ( 12
)
where c 1 is the coefficient for a single excitation, and c 1,2 is the double excitation (in Appendix A of [10] more details are shown).
In this framework, entanglement (S) and correlation energy (E corr ), as functions of nucleus -nucleus separation are those in Figure 9 0 0.01 0.02 0.03 0.04 0.05 E c [ a. u. ] E c 0 1 2 3 4 5 R ( Å ) 0 0.2 0.4 0.6 0.8 1 Entanglement (S) S ( ρ 1 CISD ) H 2 Figure 9: Comparison between the entanglement, calculated by the von Neumann entropy of the reduced density matrix, and the electron correlation energy in the Hydrogen molecule.
By the results given by this model, we want to discuss and to suggest some answers to the questions i) and ii) presented in Remark 1. Even if, in order 9 to represent correlation energy and entanglement, we use two different scales, in Figure 9 we can see that entanglement has a small value in the united atom limit after it is growing for small distances till it arrives at a maximum value then it decrease till it assumes zero value at the separated atom limit and it is exactly the progress of the correlation curve.
In order to compare the entropy S with the electron correlation energy E corr , we rescale S with the parameter α min calculated with some procedure illustrated in Eq. ( 8 ) and Eq. (9) replacing the integration variable λ with R; in this way we extract
α = E corr S vN dR S vN dR ≈ 0.009. ( 13
)
The corresponding ∆S αmin = E corr -αS allows us to answer to the question ii); in fact, as it is shown in Figure 10 , the vanishing point of ∆S αmin is, according to the two -spin Ising model, nearby R ≈ 2 Å that corresponds to the equilibrium configuration of the Hydrogen molecule.
0 1 2 3 4 5 R(Å) -0.01 0 0.01 0.02 0.03 0.04 Ec-αS
Figure 10: ∆S αmin for the H 2 molecule as a function of nucleus-nucleus distance.
The model proposed in Sec. 1 provides us with a measurement of entanglement: indeed, Eq. (3) describes the von Neumann entropy as a function of coupling constant λ, for small λ. By using Eq. ( 7 ), we can express λ in terms of correlation energy and substituting it in Eq. ( 3 ) we can obtain the variation of S vN in terms of E corr .
S vN = - E corr Log Ecorr 2(Ecorr +2) + (E corr + 4)Log Ecorr +4 2(Ecorr +2) (E corr + 2)Log4 . ( 14
)
10 In order to calculated the coefficient of proportionality among S vN and E corr we make an expansion of S vN for E corr → 0 (or equivalently for λ → 0) at the first order, obtaining a straight line characterized by an angular coefficient given by m SvN (Ecorr) = ( 1 4 )(1 + 1 Log2 ). Since this behavior is uncorrect to represent the logatithmic singularity of S vN in the origin, we make an expansion of Eq. ( 14 ), preserving the logarithmic deviation, and we obtain an expression of the form
S vN = AE corr + BE corr Log(E corr ), ( 15
)
where A = 1/2 and B = -1/(4Log2).
0.02 0.04 0.06 0.08 0.1 0.025 0.05 0.075 0.1 0.125 0.15 Ecorr S Linear AE+BELogE Figure 11: A comparison among the behavior of Eq. ( 14 ) and its linear approximation and the logarithmic one, for the Ising model.
In order to compare the behavior of S vN in Eq. ( 14 ), we have organized the numerical data, calculated with the method proposed in [1] , by making a correspondence between each value of E corr and its respective value of S vN , obtaining the plot in Figure 12 0 0.01 0.02 0.03 0.04 0.05 E c 0 0.2 0.4 0.6 0.8 1 S Figure 12: A correspondence of E corr and S vN by the numerical procedure suggested by [1] Of particular significance is the fact that, in the range where S is monotonically increasing, the correlation energy has its maximum, consequently S seems to be not a function. Moreover, it is important to note that E corr begins to decrease for R > 1 Å, region where the states become mixed, i. e. ,T rρ = T rρ 2 ; as depicted in Figure 13. 11 0 1 2 3 4 R(Å ) 0.4 0.6 0.8 1 1.2 1.4 Trρ 2
Figure 13: The increasing of the degree of mixing in the two electron state: in black we depict the trace of ρ, in red the trace of ρ 2 .
Probably, for this reason, the procedure adopted in [1] seems to be not correct: the density matrix, in fact, is calculated starting by the excitation coefficient of a wave function obtained developping with the Configuration Interaction Single Double method a pure two electrons state.
However, even if we consider only the first branch of the plot in Figure 12 , i.e. , the numerical values of S vN corresponding with increasing values of E corr , and we fit the values around E corr → 0 with a F = AE corr + BE corr Log(E corr ) we draw out numerical values of the coefficient different from the ones used in Eq. (15) . This result is shown in Figure 14. 0.01 0.02 0.03 0.04 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 S Ecorr A=17.1 B=3.3 Figure 14: A fit of S vN as a function of E corr , around the origin, with a function of the form F = AE corr + BE corr Log(E corr ) whose coefficients A and B assume the numerical values in Figure.
In particular the arithmetic sign of the coefficient B in the two models are opposite and this implies the opposite concavity of the curve.
This fact, clearly demonstrates a not satisfactory agreement between the Ising model and the one proposed in [1] .
We have explored the role of entanglement in the model of two qubits describing the Hydrogen molecule (1), considered as a bipartite system. In our discussion we have limited to the ferromagnetic case governed by the interaction coupling parameter J > 0.
The concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The entanglement can be measured by using the von Neumann entropy or, alternatively, the notion of Concurrence [7] . To compute the entanglement it is convenient to consider two Regions, say I and II, which provide two different reduced density matrices. The entropy turns out to be an increasing function of the coupling constant λ in Region I, but the state under consideration is maximally entangled in Region II indipendently from the anisotropy parameter g.
An interesting result is that for large coupling constants the entropy approach 1, meaning that all levels are equiprobably visited by the considered spin.
For weak interactions, at the boundary point λ b = 2 √ 1-g 2 the von Neumann entropy admits a discontinuity, indicating a crossing of the lowest eigenvalues and, in a more general constext, a quantum phase transition [5] .
In Sec. 2 a comparison between the entanglement and the correlation energy is performed.
To quantifying the entanglement we resort to the reduced density matrix. The entanglement can also be measured by exploiting the concept of Concurrence.
The entanglement measure is always bounded, while the energy correlation,
E corr = |E 0 | -2 =
√ 4 + λ 2 -2, is a divergent function of λ. This fact tells us that to look for simple relations valid on the whole λ-axes has no sense.
Thus, by limiting ourselves to weak couplings, we have minimized the mean square deviation given by Eq. ( 8 ). This procedure leads to the value α min ≈ -0.691217 for the minimizing parameter (see Eq. (9)). Sec. 1 contains a comparison between the von Neumann entropy and the Concurrence. Such a comparison is illustrated in Figure 1 , for two spin system as a function of the coupling λ for g = 1. Some important points are commented in Remark 1 and Remark 2 .
In Figure 4 the deviation ∆S α and its derivatives with respect to λ are computed and α min is evaluated for α ranging in the interval -1.29 ≤ α ≤ -0.091.
In Figure 5 the minimized Concurrence deviation ∆C α (i) α ′
for the four eigenstates of the 2-spin model is shown.
We point out the existence of a perfect symmetry among the Concurrence deviations for pairs of eigenstates of opposite eigenvalues.
Formula (11), due to Heitler-London [9], is reported, where the position dependent exchange coupling J(r) is expressed in term of the length r of the 13 nucleus-nucleus separation in the Hydrogen molecule.
To conclude, the magnetic field B has been monitored such that the equilibrium configuration more closest to the experimental one, r ≈ 2.00, is the minimum occurring at r = 1.88 for B = 1 2 Ry and r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r). We observe also that in the intermediate range of values, i. e., for 0.6 ≤ λ ≤ 1, the two functions SvN and the correlation energy are almost proportional within the 10%.
However, when we organized the pairs of points (E corr , S vN ) calculated by following the procedure described by [1], it is clear that the von Neumann entropy cannot be considered a function of correlation energy. The principle cause is that the function E corr presents a maximum in the region where S vN is monotonically increasing.
The reversing behavior of correlation energy occurs in correspondence with an increase of the mixing degree of the two electrons state. The function E corr in terms of the nucleus -nucleus distance R, increases till the state is pure, on the contrary, when T r(ρ 2 ) becomes discordant from T r(ρ), the function E corr decreases.
This fact suggests us that the numerical model based on the calculation of S vN starting by the excitation coefficients c i , isn't completley correct because the density matrix is obtained as a product of two electron pure states. However, even if we consider only a branch of the plot in Figure 12 , the function obtained by the two spin Ising model, i. e., Eq. ( 14 ), is unsuitable for fitting these numerical data.
On the basis of our results, essentially grounded on numerical considerations, in the near feature we would explore more complicated systems of molecules, such as for example the ethylene or other hydrocarbons, and compare these studies with the goals obtained for the Hydrogen molecule.
The authors acknowledge the Italian Ministry of Scientific Researches (MIUR) for partial support of the present work under the project SINTESI 2004/06 and the INFN for partial support under the project Iniziativa Specifica LE41. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction and the model",
"text": "Entanglement is a physical observable measured by the von Neumann entropy or, alternatively, by the Concurrence of the system under consideration.\n\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The electron correlation is not directly observable, since it is defined as the difference between the exact ground state energy of the many electrons Schrödinger equation and the Hartree-Fock energy.\n\nIn this paper we discuss the Hamiltonian which describes the Hydrogen molecule regarded as a two interacting spin 1 2 (qubit) model. In [1] it was argued that the entanglement (a quantum observable) can be used in analyzing the so-called correlation energy which is not directly observable. From our point of view, the Hydrogen molecule is dealt with a bipartite system governed by the Hamiltonian\n\nH H2 = - J 2 (1 + g)σ 1 ⊗ σ 1 - J 2 (1 -g)σ 2 ⊗ σ 2 -B(σ 3 ⊗ σ 3 + σ 0 ⊗ σ 3 ), ( 1\n\n) 1\n\nwhere σ i stand for the Pauli matrices (σ 0 = I). Actually, this model was considered in [1] in order to illustrate their method. However, here we will make some interpretative changes. Indeed, from our point of view, the states of an isolated atom are strongly reduced to a system with two energy levels related to the intensity of the magnetic field B. Relatively to this scale, the exchange interaction constant J is usually smaller than B, in order to represent the residual interatomic interactions. From the point of view of quantum chemistry, one may interpret the discrete spectrum as provided by the Hartree-Fock calculations, while the interaction coupling J models the residual multielectronic effects, not taken into account by the mean field approximation. For simplicity we limit ourselves to the ferromagnetic phase with J > 0. The parameter g, such that 0 ≤ g ≤ 1, describes the degree of anisotropy corresponding for g = 0 to the completely isotropic XY spin model. Conversely, g = 1 provides the anisotropic XY spin model, the so-called Ising model.\n\nWe notice that when the atoms are far apart, their interaction is quite weak. This corresponds to a vanishing value of J. In this situation the state of the system is completely factorized in the product state of the ground states of the indipendent spins. The corresponding total energy, in unit of B, is just the sum of the two fundamental levels, E 0 = -2, which we may consider as the Hartree-Fock approximated fundamental level in molecular structure calculations.\n\nWhen J = 0, the fundamental energy eigenvalue is E= -4 + g 2 λ 2 in Region I defined by 0 < λ ≤ 2 √ 1-g 2 , otherwise E = -λ (λ means the coupling constant) in Region II, which is the complement of I which respect to positive real axis. The corresponding (non normalized) eigenstates are |Ψ I = √ g 2 λ 2 +4+2 gλ , 0, 0, 1 and |Ψ II = 0, 1, 1, 0 , respectively. In both cases the state is entangled.\n\nSince we are dealing with pure states, the von Neumann entropy [2]\n\nS vN = -Tr ρ 1 log 2 ρ 1 ( 2\n\n)\n\nis chosen to be a measurement of the entanglement, where ρ 1 is the 1-particle reduced density matrix. However, for general mixed states other entanglement estimators (for instance, the Concurrence [4]) have to be used. In the considered case, one has S vN,I = -g 2 g 2 λ 2 + 4 log 1 2 -1 g 2 λ 2 + 4 λ 2 -g 2 g 2 λ 2 + 4 + 4 λ 2 + 8 g 2 λ 2 + 4 + 2 log 1 2 + 1 g 2 λ 2 + 4 1 g 2 λ 2 + 4 g 2 λ 2 + 2 g 2 λ 2 + 4 + 4 log(4) (3) S vN,II = 1. (4) 2 Scrutinizing Eq. (3) and Eq. ( 4 ) it emerges that the entropy is an increasing function of the coupling constant λ in Region I, but the state is maximally entangled in Region II independently from the anisotropy parameter g. One sees that, as it arises graphycally, for g = 1 the entanglement is a monotonic increasing function of the interaction coupling λ. Moreover for weak (< 1) coupling values it is always less than the 30%. Of course, for large coupling constants the entropy approaches 1, meaning that all levels are equiprobably visited by the considered spin.\n\nLimiting all further considerations to the case of weak interaction, we observe that at the boundary point λ b = 2 √ 1-g 2 a discontinuity occurs, signaling a crossing of the lowest eigenvalues and, in a more general context, a quantum phase transition [5] .\n\nAs it was pointed out in [6] , for quantifying the entanglement we can resort to the reduced density matrix. Furthermore, in [7] , Wootters has shown that for a pair of binary qubits one can use the concept of Concurrence C to measure the entanglement.\n\nThe Concurrence reads\n\nC(ρ) = max(0, ν 1 -ν 2 -ν 3 -ν 4 ), ( 5\n\n)\n\nwhere the ν i 's are the eigenvalues of the Hermitian matrix\n\nR = √ ρρ √ ρ 1 2 ,\n\nwhere ρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ), ρ * being the complex conjugate of ρ taken in the standard basis [7] . Some interesting results on the simple model (1) of the Hydrogen molecule can be achieved by realizing a comparative study of the von Neumann entropy and the Concurrence.\n\nTo this aim, we compute the Concurrence C I and C II , i. e.\n\nC I = gλ 1 g 2 λ 2 + 4 , C II = 1. ( 6\n\n)\n\nwhere I and II refer to Regions I and II, where 0 ≤ λ ≤ 2 1-g 2 , and E = -λ, respectively.\n\nIn Figure 1 a comparison between the Concurrence and the von Neumann entropy for two spins system as a function of the coupling λ for g = 1 is presented. Sec. 2 contains a comparison between the entanglement and the correlation energy. In Sec. 3 the Configuration Interaction method is introduced to compare entanglement and correlation energy. In Sec. 4 some differences between the Configuration Interaction approach and the two spin Ising model are presented. Finally, our main results are summarized in Sec. 5. 3 1 2 3 4 5 Λ 0.2 0.4 0.6 0.8 1 S vN Conc.\n\nFigure 1: Comparison between the Concurrence and the von Neumann entropy for the two spins system as a function of the coupling constant λ for g = 1.\n\n2 A comparison between the entanglement and the correlation energy\n\nNow we look for a comparison between the entanglement with the energy correlation, which as we have already recalled, it is understood as the difference of the fundamental energy level compared with respect to the corresponding value at vanishing coupling constant λ. For g = 1 and in unities of B it is given by\n\nE corr = |E 0 | -2 = 4 + λ 2 -2. ( 7\n\n)\n\nWe observe that the entanglement measure is always bounded, while E corr is a divergent function of λ. So it does not make much sense to look for simple relations valid on the entire λ-axes. Consequently, limiting ourselves to weak couplings, for 0 ≤ λ ≤ 1, we minimize the mean squared deviation\n\nI α = 1 0 ∆S 2 α dλ, with ∆S α = E corr -α S vN . ( 8\n\n)\n\nThus the minimizing parameter α min will be given by\n\nα min = 1 0 E corr S vN dλ 1 0 S 2 vN dλ ≈ -0.691217. ( 9\n\n)\n\nA formula analogous to (9) can be obtained by using the Concurrence as a measure of entanglement. In this case, by minimizing the mean squared deviation we have\n\nI C α ′ = 1 0 ∆C 2 α ′ dλ, with ∆C α ′ = E corr -α ′ C. ( 10\n\n)\n\nNow, in order to estimate the relative deviation of\n\nS vN with respect to E corr , let us report |∆S αmin |/S vN and |∆S αmin /E corr | as\n\nfunctions of λ at the optimal value α min . The graphs of these functions are shown in Figure 2. 4 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min S vN 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min E corr Figure 2: The relative quadratic deviation between the von Neumann entropy and the correlation energy with respect to the former and the latter, respectively, at the optimal value α min as a function of the coupling constant λ for g = 1.\n\nIn Figure 3 , the relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, at the optimal values α ′ min , is represented.\n\n0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min C 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min E corr Figure 3: The relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, respectively, at the optimal value α ′ min as a function of the coupling constant λ for g = 1."
},
{
"section_type": "OTHER",
"section_title": "Remark 1",
"text": "From these graphs, one can argue that the agreement between the two functions S vN and E corr is only qualitatively good, in fact, for very small λ, it is not good at all. However, in an intermediate range of values, i. e., 0.6 ≤ λ ≤ 1 the two functions are almost proportional within the 10%. Analogously, the same is true between energy and Concurrence. Even, the agreement becomes worst comparing the relative deviation of the Concurrence with respect to the correlation energy, since the range in which the relative deviations become smaller than 10% are narrower. Then, the question is whether the above results are i) sufficient to justify the conjecture advanced in [1] , i.e., entanglement can be considered as an estimation of correlation energy; ii) if such a relation has a 5 more concrete physical meaning, in particular whether the minimizing parameter α min and the vanishing point of ∆S αmin does possess any physical meaning (or α ′ min and the vanishing point of ∆C α ′ min ). Notice that in the case of the comparison for the Concurrence simpler analytical expressions appear. For instance one finds ∆C α ′ min = 0.383249 λ\n\n√ λ 2 +4 - √ λ 2 + 4 + 2 2 . Remark 2\n\nWe note that in an interval of values around α min , the deviation function (8) possesses a minimum in the interval of interest 0 ≤ λ ≤ 1, otherwise the minimum is achieved at larger value of λ, or the function is monotonically increasing (see Figure 4 ).\n\n0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 S Α 0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 d S Α dΛ\n\nFigure 4: The deviation ∆S α and its derivative with respect to λ are computed for values of -1.29(red) ≤ α ≤ -0.091(violet), for steps of 0.06. The curve drawn thicker corresponds to α min This behavior suggests to consider the function ∆S αmin as a sort of \"free energy\" , where α min mimics the \"temperature\" specific of the system. If, for some reason, we allow λ to change, then we expect that spontaneously the interaction coupling adjusts itself to the minimum of ∆S αmin . Similar considerations can be made looking at the graphs drawn for the function ∆C α ′ min and its derivative with respect to λ (see Figure 5 ). The function ∆S αmin or, alternatively, the minimum of ∆C α ′ min can be obtained algebraically. Such a minimum is at the value of the coupling constant λ SvN min ≈ 0.485 and λ C min ≈ 0.371, respectively. The authors in [1] studied numerically the von Neumann entropy and the correlation function for a Hydrogen molecule, using an old result by Herring and Flicker [8], going back to an oldest idea by Heitler and London [9], which consists in substituting the molecular binding with a position dependent exchange coupling:\n\nJ(r) ≈ 1.641 r 5 2 e -2 r Ry, ( 11\n\n)\n\nwhere r is given in Bohr radius, see Figure 6 . The maximum value taken by this function is at the point r max = 1.25. Assuming B = 0.5 Ry, i.e. 1 2 of the fundamental level of the Hydrogen atom, the maximum value λ ′ max = J(r max )/B ≈ 6 0.2 0.4 0.6 0.8 1 Λ -0.2 -0.1 0.1 0.2 0.3 C Α' 0.2 0.4 0.6 0.8 1 Λ -0.4 -0.2 0.2 0.4 d C Α' dΛ 5: The deviation ∆C α ′ and its derivative with respect to λ are computed for values of -0.98(red) ≤ α ′ ≤ 0.22(violet), for steps of 0.06. The curve drawn thicker corresponds to α ′ min 0.5 1 1.5 2 2.5 3 r 0.05 0.1 0.15 0.2 J r Ry\n\nFigure 6: The effective interaction Hydrogen-Hydrogen atom 0.470628 < λ SvN min , i.e. the value of the effective interaction value is less than the minimum for the deviation function ∆S αmin . Then, the equilibrium balance between entanglement (as von Neumann entropy) and correlation energy predicts a length of the molecule equal to r max (see the first panel of Figure 7 ). On the other hand, if we consider the energy gap 2B = 3/4 Ry, i.e. the energy step to the first excited state, one obtains the new value λ ′′ max ≈ 0.628, which goes beyond λ min , even if it is always less than 1. Now, the deviation function ∆S αmin has two minima as seen in the second panel of Figure 7 , one of which is at r ′′ -≈ 0.76 , the other one being at r ′′ + ≈ 1.91. These results should be compared with the experimental equilibrium length of the Hydrogen molecule, which is r exp ≈ 2.0. We point out that although the spin-model described by the Hamiltonian (1) is characterized by features which are essentially rough, however we are induced to answer positively to the quest for a physical meaning of the deviation function ∆S αmin . Indeed, the results elucidated in Figure 7 encourage, on one part, improvement of the computation of r in order to make more accurate the comparison with the experimental value r exp . 7 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .5 Ry 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .375 Ry r'' r''\n\nFigure 7: The von Neumann entropy for the 2-spin model for B = .5 Ry (left panel) and for B = .375 Ry (right panel) and the position depending interaction given by ( 11 ).\n\nThe first question to answer is whether this draft works also for the Concurrence. A statement about it is not obvious, since the von Neumann entropy is a nonlinear function of the Concurrence in the 2-qubits case. However, from Figure 8 one can see that the minimized deviation of the Concurrence takes one minimum for relatively large intensity of the magnetic field ( say B ≥ 0.6 Ry), while for weak fields two minima appear, corresponding to the situation depicted nearby. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 r Bohr B Ry 0.2 0.4 0.6 0.8 1 0 1 2 3 4 B(Ry) r(Bohr)\n\nFigure 8: Two contour plots of the minimized deviation of the Concurrence as a function of the magnetic field B (Ry) and of the internuclear distance r, as given by (11) . The range of values divided by the contour lines is [-0.038, 0, 04] for the left panel and [-0.03705, -0, 03000] for the right one that approximatively corresponding to the black area in the left panel.\n\nIn correspondence of the same values considered above, for B = 0.5 Ry the function ∆C α ′ min (r) has two minima at r = 0.79 and r = 1.88, while for B = 0.375 Ry they are located at r = 0.60 and r = 2.25. So one sees that the resulting equilibrium configurations are not much very close to the experimental one. The equilibrium configuration more closest to the experimental one is the minimum occurring at r = 1.88 (B = 1 2 Ry) for the function ∆C α ′ min (r).\n\n8 One sees that one of the resulting equilibrium configurations is only roughly close to the experimental one.\n\nIn other words, to conclude monitoring numerically B the equilibrium configuration more closest to experimental one in the minimum occurring at r = 1.88 for B = 1 2 Ry and at r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r)."
},
{
"section_type": "OTHER",
"section_title": "A quantum chemical framework to compare entanglement and correlation energy",
"text": "In this Section we represent the results produced in [1], where the electron entanglement in the Hydrogen molecule, calculated by the von Neumann entropy of the reduced density matrix ρ 1 , is obtained starting by the excitation coefficients of the wave function expanded by a configuration interaction method:\n\nS ρ CISD 1 = -T r ρ CISD 1 log 2 ρ CISD 1 = = - m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 log 2 m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 + -|c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 log 2 |c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 , ( 12\n\n)\n\nwhere c 1 is the coefficient for a single excitation, and c 1,2 is the double excitation (in Appendix A of [10] more details are shown).\n\nIn this framework, entanglement (S) and correlation energy (E corr ), as functions of nucleus -nucleus separation are those in Figure 9 0 0.01 0.02 0.03 0.04 0.05 E c [ a. u. ] E c 0 1 2 3 4 5 R ( Å ) 0 0.2 0.4 0.6 0.8 1 Entanglement (S) S ( ρ 1 CISD ) H 2 Figure 9: Comparison between the entanglement, calculated by the von Neumann entropy of the reduced density matrix, and the electron correlation energy in the Hydrogen molecule.\n\nBy the results given by this model, we want to discuss and to suggest some answers to the questions i) and ii) presented in Remark 1. Even if, in order 9 to represent correlation energy and entanglement, we use two different scales, in Figure 9 we can see that entanglement has a small value in the united atom limit after it is growing for small distances till it arrives at a maximum value then it decrease till it assumes zero value at the separated atom limit and it is exactly the progress of the correlation curve.\n\nIn order to compare the entropy S with the electron correlation energy E corr , we rescale S with the parameter α min calculated with some procedure illustrated in Eq. ( 8 ) and Eq. (9) replacing the integration variable λ with R; in this way we extract\n\nα = E corr S vN dR S vN dR ≈ 0.009. ( 13\n\n)\n\nThe corresponding ∆S αmin = E corr -αS allows us to answer to the question ii); in fact, as it is shown in Figure 10 , the vanishing point of ∆S αmin is, according to the two -spin Ising model, nearby R ≈ 2 Å that corresponds to the equilibrium configuration of the Hydrogen molecule.\n\n0 1 2 3 4 5 R(Å) -0.01 0 0.01 0.02 0.03 0.04 Ec-αS\n\nFigure 10: ∆S αmin for the H 2 molecule as a function of nucleus-nucleus distance."
},
{
"section_type": "METHOD",
"section_title": "Differences between the Configuration Interaction approach and the two-spin Ising model",
"text": "The model proposed in Sec. 1 provides us with a measurement of entanglement: indeed, Eq. (3) describes the von Neumann entropy as a function of coupling constant λ, for small λ. By using Eq. ( 7 ), we can express λ in terms of correlation energy and substituting it in Eq. ( 3 ) we can obtain the variation of S vN in terms of E corr .\n\nS vN = - E corr Log Ecorr 2(Ecorr +2) + (E corr + 4)Log Ecorr +4 2(Ecorr +2) (E corr + 2)Log4 . ( 14\n\n)\n\n10 In order to calculated the coefficient of proportionality among S vN and E corr we make an expansion of S vN for E corr → 0 (or equivalently for λ → 0) at the first order, obtaining a straight line characterized by an angular coefficient given by m SvN (Ecorr) = ( 1 4 )(1 + 1 Log2 ). Since this behavior is uncorrect to represent the logatithmic singularity of S vN in the origin, we make an expansion of Eq. ( 14 ), preserving the logarithmic deviation, and we obtain an expression of the form\n\nS vN = AE corr + BE corr Log(E corr ), ( 15\n\n)\n\nwhere A = 1/2 and B = -1/(4Log2).\n\n0.02 0.04 0.06 0.08 0.1 0.025 0.05 0.075 0.1 0.125 0.15 Ecorr S Linear AE+BELogE Figure 11: A comparison among the behavior of Eq. ( 14 ) and its linear approximation and the logarithmic one, for the Ising model.\n\nIn order to compare the behavior of S vN in Eq. ( 14 ), we have organized the numerical data, calculated with the method proposed in [1] , by making a correspondence between each value of E corr and its respective value of S vN , obtaining the plot in Figure 12 0 0.01 0.02 0.03 0.04 0.05 E c 0 0.2 0.4 0.6 0.8 1 S Figure 12: A correspondence of E corr and S vN by the numerical procedure suggested by [1] Of particular significance is the fact that, in the range where S is monotonically increasing, the correlation energy has its maximum, consequently S seems to be not a function. Moreover, it is important to note that E corr begins to decrease for R > 1 Å, region where the states become mixed, i. e. ,T rρ = T rρ 2 ; as depicted in Figure 13. 11 0 1 2 3 4 R(Å ) 0.4 0.6 0.8 1 1.2 1.4 Trρ 2\n\nFigure 13: The increasing of the degree of mixing in the two electron state: in black we depict the trace of ρ, in red the trace of ρ 2 .\n\nProbably, for this reason, the procedure adopted in [1] seems to be not correct: the density matrix, in fact, is calculated starting by the excitation coefficient of a wave function obtained developping with the Configuration Interaction Single Double method a pure two electrons state.\n\nHowever, even if we consider only the first branch of the plot in Figure 12 , i.e. , the numerical values of S vN corresponding with increasing values of E corr , and we fit the values around E corr → 0 with a F = AE corr + BE corr Log(E corr ) we draw out numerical values of the coefficient different from the ones used in Eq. (15) . This result is shown in Figure 14. 0.01 0.02 0.03 0.04 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 S Ecorr A=17.1 B=3.3 Figure 14: A fit of S vN as a function of E corr , around the origin, with a function of the form F = AE corr + BE corr Log(E corr ) whose coefficients A and B assume the numerical values in Figure.\n\nIn particular the arithmetic sign of the coefficient B in the two models are opposite and this implies the opposite concavity of the curve.\n\nThis fact, clearly demonstrates a not satisfactory agreement between the Ising model and the one proposed in [1] ."
},
{
"section_type": "OTHER",
"section_title": "5 Concluding remarks",
"text": "We have explored the role of entanglement in the model of two qubits describing the Hydrogen molecule (1), considered as a bipartite system. In our discussion we have limited to the ferromagnetic case governed by the interaction coupling parameter J > 0.\n\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The entanglement can be measured by using the von Neumann entropy or, alternatively, the notion of Concurrence [7] . To compute the entanglement it is convenient to consider two Regions, say I and II, which provide two different reduced density matrices. The entropy turns out to be an increasing function of the coupling constant λ in Region I, but the state under consideration is maximally entangled in Region II indipendently from the anisotropy parameter g.\n\nAn interesting result is that for large coupling constants the entropy approach 1, meaning that all levels are equiprobably visited by the considered spin.\n\nFor weak interactions, at the boundary point λ b = 2 √ 1-g 2 the von Neumann entropy admits a discontinuity, indicating a crossing of the lowest eigenvalues and, in a more general constext, a quantum phase transition [5] .\n\nIn Sec. 2 a comparison between the entanglement and the correlation energy is performed.\n\nTo quantifying the entanglement we resort to the reduced density matrix. The entanglement can also be measured by exploiting the concept of Concurrence.\n\nThe entanglement measure is always bounded, while the energy correlation,\n\nE corr = |E 0 | -2 =\n\n√ 4 + λ 2 -2, is a divergent function of λ. This fact tells us that to look for simple relations valid on the whole λ-axes has no sense.\n\nThus, by limiting ourselves to weak couplings, we have minimized the mean square deviation given by Eq. ( 8 ). This procedure leads to the value α min ≈ -0.691217 for the minimizing parameter (see Eq. (9)). Sec. 1 contains a comparison between the von Neumann entropy and the Concurrence. Such a comparison is illustrated in Figure 1 , for two spin system as a function of the coupling λ for g = 1. Some important points are commented in Remark 1 and Remark 2 .\n\nIn Figure 4 the deviation ∆S α and its derivatives with respect to λ are computed and α min is evaluated for α ranging in the interval -1.29 ≤ α ≤ -0.091.\n\nIn Figure 5 the minimized Concurrence deviation ∆C α (i) α ′\n\nfor the four eigenstates of the 2-spin model is shown.\n\nWe point out the existence of a perfect symmetry among the Concurrence deviations for pairs of eigenstates of opposite eigenvalues.\n\nFormula (11), due to Heitler-London [9], is reported, where the position dependent exchange coupling J(r) is expressed in term of the length r of the 13 nucleus-nucleus separation in the Hydrogen molecule.\n\nTo conclude, the magnetic field B has been monitored such that the equilibrium configuration more closest to the experimental one, r ≈ 2.00, is the minimum occurring at r = 1.88 for B = 1 2 Ry and r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r). We observe also that in the intermediate range of values, i. e., for 0.6 ≤ λ ≤ 1, the two functions SvN and the correlation energy are almost proportional within the 10%.\n\nHowever, when we organized the pairs of points (E corr , S vN ) calculated by following the procedure described by [1], it is clear that the von Neumann entropy cannot be considered a function of correlation energy. The principle cause is that the function E corr presents a maximum in the region where S vN is monotonically increasing.\n\nThe reversing behavior of correlation energy occurs in correspondence with an increase of the mixing degree of the two electrons state. The function E corr in terms of the nucleus -nucleus distance R, increases till the state is pure, on the contrary, when T r(ρ 2 ) becomes discordant from T r(ρ), the function E corr decreases.\n\nThis fact suggests us that the numerical model based on the calculation of S vN starting by the excitation coefficients c i , isn't completley correct because the density matrix is obtained as a product of two electron pure states. However, even if we consider only a branch of the plot in Figure 12 , the function obtained by the two spin Ising model, i. e., Eq. ( 14 ), is unsuitable for fitting these numerical data.\n\nOn the basis of our results, essentially grounded on numerical considerations, in the near feature we would explore more complicated systems of molecules, such as for example the ethylene or other hydrocarbons, and compare these studies with the goals obtained for the Hydrogen molecule."
},
{
"section_type": "OTHER",
"section_title": "Acknowledgments",
"text": "The authors acknowledge the Italian Ministry of Scientific Researches (MIUR) for partial support of the present work under the project SINTESI 2004/06 and the INFN for partial support under the project Iniziativa Specifica LE41."
}
] | ||
arxiv:0704.0523 | 0704.0523 | 1 | 10.1103/PhysRevA.76.042103 | 1ca28034a1181cbdbddecf9d042ad8b097ea11a943736a666791eb33df4d9ea3 | Quantum superpositions and entanglement of thermal states at high
temperatures and their applications to quantum information processing | We study characteristics of superpositions and entanglement of thermal states
at high temperatures and discuss their applications to quantum information
processing. We introduce thermal-state qubits and thermal-Bell states, which
are a generalization of pure-state qubits and Bell states to thermal mixtures.
A scheme is then presented to discriminate between the four thermal-Bell states
without photon number resolving detection but with Kerr nonlinear interactions
and two single-photon detectors. This enables one to perform quantum
teleportation and gate operations for quantum computation with thermal-state
qubits. | [
"Hyunseok Jeong and Timothy C. Ralph"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states without photon number resolving detection but with Kerr nonlinear interactions and two singlephoton detectors. This enables one to perform quantum teleportation and gate operations for quantum computation with thermal-state qubits.
In many problems considered within the framework of quantum physics, physical systems are treated as pure states that can be represented by state vectors, or equivalently, by wave functions. Even though such an approach is simple and useful to address certain problems, it could often be quite different from real conditions of physical systems. This may be particularly true when one deals with macroscopic physical systems in terms of quantum physics. A macroscopic object is a complex open system which cannot avoid continuous interactions with the environment. Such a physical system is generally in a significantly mixed state and cannot be represented by a state vector. In general, mixed states are subtle objects whose properties are significantly more difficult to characterize than pure states.
Schrödinger's famous cat paradox is a typical example where a massive classical object was assumed to be a pure state. It describes a counter-intuitive feature of quantum physics which dramatically appears when the principle of quantum superposition is applied to macroscopic objects. In the original paradox and its various explanations, the initial cat isolated in the steel chamber is considered a pure state that can be represented by a state vector such as |alive (or a wave function such as ψ alive ). The cat isolated from the environment is then assumed to interact with a microscopic superposition state, (|g +|e )/ √ 2, where |g and |e are the ground and excited states of a two-level atom. The cat will be dead if the atom is found in the excited state, |e , while it will remain alive if otherwise. Thus in Schrödinger's gedanken experiment the cat is entangled with the atom as (|g |alive + |e |dead ) √ 2, where the alive and dead statuses of the cat are described by the state vectors |alive and |dead . If one measures out the atomic system on the superposed basis, (|g ± |e )/ √ 2, the cat will be in a superposition of alive and dead states such as (|alive ± |dead )/ √ 2. It is often argued that such superposed states and entangled states can theoretically exist but are virtually impossible to observe because one cannot perfectly isolate a macroscopic object such as the cat from its environment [4] .
However, this explanation is not fully satisfactory because the cat, a macroscopic object, is a complex open system which cannot be represented by a state vector. One may argue that the cat could be assumed to be in an unknown pure state such that the cat was certainly alive but the exact state of the cat was unknown. However, the interactions between the cat and its environment can cause the cat to become entangled with the environment [5] . In such a case, even though one can perfectly isolate the cat in the steel chamber from the enviroment, the cat will remain entangled with the environment due to its pre-interactions with the environment. Therefore, strictly speaking, even to assume a cat as an unknown pure state in the steel chamber is not legitimate. Thus a key point here is that it is unsatisfactory to describe the cat by a pure state such as |alive and |dead . We may need a more realistic assumption that the "cat" in Schrödinger's paradox was in a significantly mixed classical state. An intriguing question is then whether the quantum properties of the resulting state would still remain or diminish under such an assumption.
Recently, such an analogy of Schrödinger's cat paradox, where the state corresponding to the virtual cat is a significantly mixed thermal state, was investigated [6] . A thermal state with a high temperature is considered a classical state in quantum optics. As the temperature of the thermal state increases, the degree of mixedness, which can be quantified by linear entropy, rapidly approaches the maximum value. When the temperature approaches infinity, the thermal state does not show any quantum properties. As a comparison, coherent states with large amplitudes are known as the most classical pure states [7] , and their superposition is often regarded as a superposition of classical states [8] . However, coherent states are still pure states which may not well represent truly classical systems, and they display some nonclassical features [9] . In Ref. [6] , it was shown that prominent quantum properties can actually be transferred from a microscopic superposition to a significantly mixed thermal state (i.e. a thermal state of which the degree of mixedness is close to the maximum value) at a high temperature through an experimentally feasible process. This result clarifies that unavoidable initial mixedness of the cat does not preclude strong quantum phenomena.
One of the results in Ref. [6] is that quantum entanglement can be produced between thermal states with nearly the maximum Bell-inequality violation when the temperatures of both modes goes to infinity. In previous related results, Bose et al. showed that entanglement can arise when two systems interact if one of the system are pure even when the other system is extremely mixed [10] . There is an interesting previous example shown by Filip et al. for the maximum violation of Bell's inequality when one of the modes is an extremely mixed thermal state [11] . Very recently, Ferreira et al. showed that entanglement can be generated at any finite temperature between high Q cavity mode field and a movable mirror thermal state [12] . However, in these example [10, 11, 12] only one of the modes is considered a large thermal state [10, 11, 12] and entanglement vanishes in the infinite temperature limit [10, 12] , which is obviously in contrast to the result presented in Ref. [6] . Entanglement for both of the modes at the thermal limit of the infinitely high temperature has not been found before. Remarkably, the violation of Bell's inequality in our examples reaches up to Cirel'son's bound [13] even in this infinite-temperature limit for both modes. As Vedral [14] and Ferreira et al. [12] pointed out it is believed that high temperatures reduce entanglement and all entanglement vanishes if the temperature is high enough, which is obviously not the case in Ref. [6] .
The purpose of this paper is twofold. Firstly, we review and further investigate various properties of superpositions and entanglement of thermal states at high temperatures [6] . In particular, we investigate two classes of highly mixed symmetric states in the phase space. Both the classes of these states do not show typical interference patterns in the phase space while they manifest strong singular behaviors. Interestingly, the first class of states has neither squeezing properties nor negative values in their Wigner functions, however, they are found to be highly nonclassical states. The second class of states has the maximum negativity in the Wigner function. Further, we discuss the possibility of quantum information processing with thermal-state qubits. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure Bell states. We show that four thermal-Bell states can be well discriminated by nonlinear interactions without photon number resolving measurements. Quantum teleportation and gate operations for thermal-state qubits can be realized using the Bell measurement scheme.
This paper is organized as follows. In Sec. II, we review the generation process of superpositions of thermal states and study their characteristics. In Sec. III, we study en-tanglement of thermal states, i.e., Bell inequality violations. In Sec. IV, we discuss the possibility of quantum information processing using thermal states. We first define the thermal-state qubit and the Bell-basis states using thermal-state entanglement. We then show that the four Bell states can be well discriminated by homodyne detection and two Kerr nonlinearities. It follows that quantum teleportation and quantum gate operations can be realized with thermal-state qubits. We conclude with final remarks in Sec. V.
Let us first consider a two-mode harmonic oscillator system. A displaced thermal state can be defined as
ρ th (V, d) = d 2 αP th (V, d)|α α| ( 1
)
where |α is a coherent state of amplitude α and
P th α (V, d) = 2 π(V -1) exp[- 2|α -d| 2 V -1 ] (2)
with variance V and displacement d in the phase space.
The thermal temperature τ increases as V increases as e ν/τ = (V + 1)/(V -1), where is Planck's constant and ν is the frequency [15] . Suppose that a microscopic superposition state
|ψ a = 1 √ 2 (|0 a + |1 a ), (3)
where |0 and |1 are the ground and first excited states of the harmonic oscillator, interacts with a thermal state ρ th b (V, d) and the interaction Hamiltonian is
H K = λâ † âb † b (4)
which corresponds to the cross Kerr nonlinear interaction. The resulting state is then
ρ ent ab = 1 2 d 2 αP th (V, d) |0 0| ⊗ |α α| + |1 0| ⊗ |αe iϕ α| + |0 1| ⊗ |α αe iϕ | + |1 1| ⊗ |αe iϕ αe iϕ | ( 5
)
and ϕ is determined by the strength of the nonlinearity λ and the interaction time. The Wigner representation of ρ ent ab is
W ent ab (α, β) = 1 π e -2|α| 2 W th (β; d) + 2αV c (β; d) + 2[αV c (β; d)] * + (4|α| 2 -1)W th (β; de iϕ ) (6)
where α and β are complex numbers parametrizing the phase spaces of the microscopic and macroscopic systems respectively and
W th (α; d) = 2 πV exp[- 2|α -d| 2 V ], (7)
V c (α; d) = 2 πJK exp[- 2 K (1 -e iϕ )d 2 - 1 J (α - 2e iϕ d K )(α * - 2d K )], (8)
K = 2 + (V -1)(1 -e iϕ
), J = (sin ϕ/2 + iV cos ϕ/2)/(2V sin ϕ/2 + 2i cos ϕ/2), and d has been assumed real without loss of generality. If one traces ρ ent ab over mode a, the remaining state will be simply in a classical mixture of two thermal states and its Wigner function will be positive everywhere. However, if one measures out the "microscopic part" on the superposed basis, i.e., (|0 a ± |1 a )/ √ 2, the "macroscopic part" for mode b may not lose its nonclassical characteristics. Such a measurement on the the superposed basis will reduce the remaining state to
ρ sup(±) = N ± s d 2 αP th (V, d) |α α| ± |αe iϕ α| ± |α αe iϕ | + |αe iϕ αe iϕ | , (9)
where N ± s are the normalization factors, and its Wigner function is
W sup(±) (α) = N ± s {W th (α; d) ± V c (α; d) ± {V c (α; d)} * + W th (α; de iϕ )}. ( 10
)
The ± signs in Eqs. ( 8 ) and (9) correspond to the two possible results from the measurement of the microscopic system. The state in Eq. ( 10 ) is a superposition of two thermal states. A feasible experimental setup to generate superpositions of thermal states is atom-field interactions in cavities, where a π/2 pulse can be used to prepare the atom in a superposed state. This type of experiment has already been performed to produce a superposition of coherent states [16] . In our cases, simply thermal states can be used instead of coherent states. Another possible setup is an all-optical scheme with free-traveling fields and a cross-Kerr medium, where a standard singlephoton qubit could be used as the microscopic superposition. Recently, there have been theoretical and experimental efforts to produce and observe giant Kerr nonlinearities using electromagnetically induced transparency [17] . Furthermore, it was shown that a weak Kerr nonlinearity can still be useful if a initially strong field is employed in this type of experiment [18] . We shall further explain this with examples in Sec. III.
The negativity of the Wigner function is known as an indicator of non-classicality of quantum states. In order to observe negativity of the Wigner function in a real experiment, its absolute minimum negativity should be large enough. The minimum negativity of the Wigner function in Eq. ( 6 ) for V = 1 is -0.144 for d = 0 and -0.246 for d → ∞. Now suppose the initial state can be considered a classical thermal state by letting V ≫ 1. One might expect that the negativity would be washed out as the initial state becomes mixed, but this is not the case. The minimum negativity actually increases as V gets larger. If V → ∞, the minimum negativity of the Wigner function ( 6 ) is -0.246 regardless of d: no matter how mixed the initial thermal state was, the minimum negativity of Wigner function is found to be a large value. The point in the phase space which gives the minimum negativity when V ≫ 1 or d ≫ 0 is (-1 2 , 0) and has negativity
W neg ≡ W ent ab (- 1 2 , 0) = 2(-2 + 1 V exp[-2d 2 V ]) π 2 √ e . (11)
It can be shown that
W neg approaches -4/(π 2 √ e) ≈ -0.246 when either d → ∞ or V → ∞.
This effect is obviously due to the interaction between the microscopic superposition and the macroscopic thermal state. If the initial microscopic state is not superposed, e.g., |ψ a = |1 a , the resulting state will be a simple direct product, (|1 1|) a ⊗ ρ th b (V, -d). Whilst for V = 1 this state will exhibit negativity, this is washed out and tends to zero as V → ∞. Needless to say, if it was |0 a instead of |1 a , the resulting Wigner function will be a direct product of two Gaussian states whose Wigner fucntion can never be negative. The superpositon state (3) plays the crucial role in making the minimum negativity of the resulting Wigner function always saturate to a certain negative value no matter how mixed and classical the initial state of the other mode becomes. The Wigner functions of the single-mode states, W sup(±) (α), in Eq. (10) show large negative values. The minimum negativity of the Wigner function W sup(-) (α) is W sup(-) (0) = 2/π regardless of the values of V and d. On the other hand, the minimum negativity of the Wigner function W sup(+) (α) approaches 2/π for d → ∞ and disappears when d = 0.
When ϕ = π, the state (9) becomes (12) where
ρ ± = N (ρ th (V, d)±σ(V, d)±σ(V, -d)+ρ th (V, -d)),
σ(V, d) = d 2 αP th (V, d)| -α α| and N = 2 1 ± exp[-2d 2 V ] V 2 . ( 13
)
If the initial state for mode b is a pure coherent state, i.e., V = 1, the measurement on the superposed basis for mode a will produce a superposition of two pure coherent states as
| Ψ ± = 1 √ 1 ± e -2|α| 2 (|α ± | -α ), (14)
where α = d. The probability P ± to obtain the state ρ ± is obtained as [19 ] where
P ± = ψ ± |Tr b [ρ ent ab ]|ψ ± = 1 2 (1 ± exp[-2d 2 V ] V ), ( 15
)
|ψ ± = (|0 ±|1 )/ √ 2.
The probability approaches P ± = 1/2 when either d or V becomes large.
As an analogy of Schrödinger's cat paradox, the variance V corresponds to the size the initial "cat", and the distance d between the two thermal component states corresponds to distinguishability between the "alive cat" and the "dead cat". Suppose that both V and d are very large for the initial thermal state. The two thermal states ρ th (V, ±d) become macroscopically distinguishable when d ≫ √ V , and our example may become a more realistic analogy of the cat paradox in this limit. Both the states ρ ± in this case show probability distributions with two Gaussian peaks and interference fringes [6] . Figure 1 presents the probability distributions of x (≡ Re[α]) and p (≡ Im[α]) for ρ -(a) when V = 100 and d = 100 and (b) when V = 1000 and d = 300. The probability distribution of x (p) for ρ ± can be obtained by integrating the Wigner function of ρ ± over p (x). The two Gaussian peaks along the x axis and interference fringes along the p axis shown in Fig. 1 are a typical signature of a quantum superposition between macroscopically distinguishable states. The visibility v of the interference fringes is defined as [15]
v = I max -I min I max + I min , (16)
where I = dxW sup(-) (α) and the maximum should be taken over p. It can be simply shown that the visibility v is always 1 regardless of the value of V . Note that d should increase proportionally to √ V to maintain the condition of classical distingushability between the two component thermal states ρ th (V, ±d). The interference fringes with high visibility are incompatible with classical physics and evidence of quantum coherence. The fringe spacing (the distance between the fringes) does not depend on V but only on d, i.e., a pure superposition of coherent states shows the same fringe spacing for a given d. We emphasize that the states shown in Fig. 1 are "superpositions" of severely mixed thermal states.
An experimental realization of a nonlinear effect corresponding to ϕ = π is very demanding particularly in the presence of decoherence. Here we point out that the method using a weak nonlinear effect (ϕ ≪ π) combined with a strong field (d ≫ 1) [18] can be useful to generate a thermal-state superposition with prominent interference patterns. In Fig. 2 , we have used experimentally accessible values, V = 5, d = 2000 and ϕ = π/1000, but the fringe visibility is still 1. In this case, decoherence during the nonlinear interaction would be significantly reduced because of the decrease of the interaction time [18] . Note also that, if required, the state in Fig. 2 can be moved to the center of the phase space, for example, using a biased beam splitter (BS) and a strong coherent field [18] .
Let us assume that d = 0, i.e., the initial state is the thermal state, ρ th (V, 0), at the origin of the phase space. In this case, the thermal-state superpositions, ρ ± , are produced with probabilities, P ± = (1/2){1 ± (1/V )}, respectively. Figure 3 shows the Wigner functions of ρ + dependent on the interaction time between the macro-scopic thermal state and the microscopic superposition in a cross Kerr medium. The state is always symmetric in the phase space regardless of the interaction time as shown in Fig. 3 . In this figure, the initial state is a thermal state of V = 100 (Fig. 3(a) ). In a relatively short time (θ = π/32 and θ = π/16), the state shows some interference patterns. When θ = π, the evolved state looks very localized around the origin as shown in Fig 3 . The generated state at θ = π does not show negativity of the Wigner function nor squeezing properties. On the other hand, a well defined P function does not exist for this state.
In the case of ρ -, with the same assumption d = 0, the Wigner function at ϕ = π has the minimum negativity (-2/π) at the origin regardless of V . As a result of the interaction with the microscopic superposition, a deep hole to the negative direction below zero has been formed around the origin for ρ -as shown in Fig. 4 .
Entanglement between macroscopic objects and its Bell-type inequality tests are an important issue. In this section, we shall show that entanglement can be generated between high-temperature thermal states even when the temperature of each mode goes to infinity. 2 200 1 400 600 800 1000 V 2.1 2.2 2.3 B (a) 50 100 150 d 2.2 2 0 2.4 2.6 2.8 B (b) FIG. 5: (a) The optimized violation, B ≡ |B + |max, of Bell-
CHSH inequality for the "thermal-state entenglement", ρ+, of V = 1000 (solid curve) and V = 100 (dashed curve). The Bell-violation of a pure entangled coherent state, i.e., V = 1, has been plotted for comparison (dotted curve). The Bell-violation B approaches its maximum bound, 2 √ 2, when d ≫ √ V regardless of the level of the mixedness V . (b) The optimized Bell-violation B against d for the different type of thermal-state entanglement generated using a 50:50 beam splitter from ρ + . V = 1000 (solid curve), V = 100 (dashed curve) and V = 1 (dotted curve).
If the microscopic superposition interacts with two thermal states, ρ th b (V, d) and ρ th c (V, d), and the micro-scopic particle is measured out on the superposed basis, the resulting state will be
ρ tm(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (17)
where
N t = 2 1 ± exp[-4d 2 V ] V 2 . ( 18
)
Such two-mode thermal-state entanglement can be generated using two cavities and an atomic state detector [20] .
Extending the two cavities to N cavities, entanglement of N -mode thermal states can also be generated. Such a state is an analogy of the N -mode pure GHZ state [21] but each mode is extremely mixed. Here we shall consider the Bell-CHSH inequality [22, 23] with photon number parity measurements [20, 24] . The parity measurements can be performed in a high-Q cavity using a far-off-resonant interaction between a two-level atom and the field [25] . The Bell-CHSH inequality can be represented in terms of the Winger function as [24 ]
|B (±) | = π 2 4 |W tm(±) (α, β) + W tm(±) (α, β ′ ) + W tm(±) (α ′ , β) -W tm(±) (α ′ , β ′ )| ≤ 2, (19)
where W tm(±) (α, β) is the Wigner function of ρ tm(±) in Eq. ( 17 ). As shown in Fig. 5 , the Bell-violation approaches the maximum bound for a bipartite measurement, 2 √ 2 [13] , when d ≫ √
V regardless of the level of the mixedness V , i.e., the temperatures of the thermal states. Note that it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(a) . This implies that entanglement of nearly 1 ebit has been produced between the two significantly mixed thermal states for d ≫ √ V , and such "thermal-state entanglement" cannot be described by a local theory.
A different type of macroscopic entanglement can be generated by applying the beam splitter operation
exp[θ/2(e iφ â † s âd -e -iφ â † d âs )], (20)
on the "thermal-state superpositions" in Eq. ( 9 ). The state after passing through a 50:50 beam splitter can be represented as
N d 2 αP th α (V, d) | α √ 2 , - α √ 2 ± | - α √ 2 , α √ 2 α √ 2 , - α √ 2 | ± - α √ 2 , α √ 2 | , (21
) 200 400 600 800 1000 V 2.1 2 0 2.2 2.3 B FIG. 6: The optimized Bell-violation B against V for the slightly different type of thermal-state entanglement generated using a 50:50 beam splitter using ρ + when d = 0. 3.13 3.14 3.15 3.16 ϕ 1.6 2.2 2 2.4 2.6 2.8 B (a) 3.13 3.14 3.15 3.16 ϕ 1.6 2 2.2 2.4 2.6 2.8 B (b) where N is defined in Eq. ( 13 ). When d is large, this state violates the Bell-CHSH inequality to the maximum bound 2 √ 2 regardless of the level of mixedness V as shown in Fig. 5(b ). Again, it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(b ). Furthermore, these states severely violate Bell's inequality even when d = 0 as V increases as shown in Fig. 6 . We have found that the optimized Bell violation of these states approaches 2.32449 for V → ∞. Interestingly, this value is exactly the same as the optimized Bell-CHSH violation for a pure two-mode squeezed state in the infinite squeezing limit [26] . Note that multilmode entangled states can be generated using multiple beam splitters.
It should be noted that the Bell violations are more sensitive to the interaction time when either V or d is larger. Figure 7 clearly shows this tendency. Therefore, in order to observe the Bell violations using the mixed state of V (and d) large, the interaction time in the Kerr medium should be more accurate.
In this section, we discuss the possibility of quantum information processing with thermal-state qubits and thermal-state entanglement.
We introduce a thermal-state qubit
ρ ψ = |a| 2 ρ th (V, d) ± ab * σ(V, d) ± a * bσ(V, -d) + |b| 2 ρ th (V, -d), (22)
where a and b are arbitrary complex numbers. The basis states, ρ th (V, d) and ρ th (V, -d), can be well discriminated by a homodyne measurement when d is larger than V . The thermal state qubit (22) can be re-written as
ρ ψ = d 2 αP th α (V, d) a|α + b| -α a * α| + b * -α| , (23)
which can be understood as a generalization of the coherent state qubit, a|d + b|d , where |d is a coherent state of amplitude d. The thermal-state qubit (23) becomes identical to the coherent-state qubit when V = 1.
We also define four thermal-Bell states as
ρ Φ(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (24) ρ Ψ(±) = N t ρ th (V, d) ⊗ ρ th (V, -d) ± σ(V, d) ⊗ σ(V, -d) ± σ(V, -d) ⊗ σ(V, d) + ρ th (V, -d) ⊗ ρ th (V, d) (25)
where N t was defined in Eq. ( 18 ). The thermal-Bell states can be written as For quantum information processing applications, it is an important task to discriminate between the four Bell states. Here we discuss two possible ways to discriminate between the thermal-Bell states (25) . We shall only briefly describe the first scheme using photon number resolving measurements and focus on the second scheme using nonlinear interactions. The first method is to simply use a 50-50 beam splitter and two photon number resolving detectors as shown in Fig. 8(a) . This scheme is basically the same as the Bellstate measurement scheme with pure entangled coherent states [27, 28] . Let us suppose that the amplitude, d, is large enough, i.e., d ≫ √ V . If the incident state was ρ Φ(+) or ρ Φ(-) , most of the photons are detected on detector A in in Fig. 8(a) . Meanwhile, most of the photons are detected on detector B when the incident state was ρ Ψ(+) or ρ Ψ(-) . The average photon numbers between the "many-photon case" and the "few-photon case" are compared in Fig. 9 . Furthermore, the states ρ Ψ(+) and ρ Φ(+) contain only even numbers of photons while ρ Ψ(-) and ρ Φ(-) contain only odd numbers of photons. Therefore, all the four Bell states can be well discriminated by analyzing numbers of photons detected at detectors A and B. For example, if detector A detects many photons while detector B detects few and the total photon number detected by the two detectors are even, this means that state ρ Φ(+) was measured by the thermal-Bell measurement. The nonzero failure probability can be made arbitrarily small by increasing d.
ρ Φ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, β ± | -α, -β α, β| ± -α, -β| , ( 26
)
ρ Ψ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, -β ± | -α, β α, -β| ± -α, β| . (27)
However, the average photon numbers of the thermal-Bell states are high when V ≫ 1 and d ≫ 1. In this case, it would be unrealistic to use photon number resolving detectors. It would be an interesting question whether
0.5 1 1.5 2 2.5 3 d 10 0 0 20 30 N (a) 0.5 1 1.5 2 2.5 3 d 0 0 10 20 30 N (b)
FIG. 9: The average photon number N for the "many-photon case" (solid line) and the "few-photon case" (dashed line) for V = 10 against d (a) when the input state is either ρ Φ(+) or ρ Ψ(+) and (b) when the input state is either ρ Φ(-) or ρ Ψ(-) .
these four thermal-Bell states can be distinguished by classical measurements, such as homodyne detection, instead of photon number resolving detection. Our alternative scheme employs cross-Kerr nonlinearities and single photon detectors as shown in Fig. 8(b ). Let us first suppose that the input field was ρ Φ(+) . The incident twomode state passes through a 50-50 beam splitter, BS1. The state after passing through the 50:50 beam splitter, BS1, is
ρ B = N t d 2 αd 2 βP th α (V, d)P th β (V, d) |η, -ξ η, -ξ| + |η, -ξ -η, ξ| + | -η, ξ η, -ξ| + | -η, ξ -η, ξ| (28)
where η = (α+β)/ √ 2 and ξ = (α-β)/ √ 2. Two dual-rail single photon qubits, |ψ + ee ′ and |ψ + f f ′ , where
|ψ + = 1 √ 2 (|0 |1 + |1 |0 ), (29)
are prepared using two single photons and 50:50 beam splitters, BS2 and BS3, as shown in Fig. 8(b ). Then, traveling fields at modes c and d interacts with those of modes e and f , respectively, in cross-Kerr nonlinear media. We suppose that the interaction time is t = π/λ, and the resulting state is then
ρ B ′ = U ce U df ρ B cd ρ q ee ′ ρ q f f ′ U † ce U † df (30)
where U ce = exp[iπH K ce /λ ] and ρ q = |ψ q ψ q |. An explicit form of Eq. ( 30 ) can then be simply obtained using the identity
U ce |α c |0 e = |α c |0 e , U ce |α c |1 e = | -α c |1 e ( 31
)
where |α is a coherent state. However, we omit such an explicit expression in this paper for it is too lengthy. After the nonlinear interactions, the qubit parts, modes e, e ′ , f and f ′ , should be measured with the measurement basis
{| + + , | + -, | -+ , | --} (32)
where
| + + = |ψ + ee ′ |ψ + f f ′ , | + -= |ψ + ee ′ |ψ -f f ′ , | -+ = |ψ -ee ′ |ψ + f f ′ , | --= |ψ -ee ′ |ψ -f f ′ , and |ψ -= (|0 |1 -|1 |0 )/ √ 2.
This measurement can be performed using two 50:50 beam splitters, BS4 and BS5, and four detectors, A1, A2, B1 and B2, as shown in Fig. 8(b ). If detector A1 and B1 click, i.e., the measurement result is | + + , the resulting state at modes c and d is
ρ ++ = N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η + | -η )( η| + -η|) c ⊗ (|ξ + | -ξ )( ξ| + -ξ|) d (33)
Note that state ρ ++ is not normalized, which implies that the probability of obtaining the corresponding measurement result is not unity. The probability of obtaining this result is
P ++ = (V + 1)(V + e -4d 2 V ) 2(V 2 + e -4d 2 V ) . (34)
When the result is either | +or | -+ , the result is
ψ 2 |ρ B ′ |ψ 2 = ψ 3 |ρ B ′ |ψ 3 = 0, (35)
which obviously means that the probability of the obtaining this result is zero. When the result is | --, i.e., detector A2 and B2 click,
ρ --= N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η -| -η )( η| --η|) c ⊗ (|ξ -| -ξ )( ξ| --ξ|) d , (36)
which is not normalized. The probability of obtaining this result is
P --= (V -1)(V -e -4d 2 V ) 2(V 2 + e -4d 2 V ) , (37)
and it can be simply verified that P ++ +P --= 1. Therefore, only the measurement results | + + and | -can be obtained in the case of the input state ρ Φ(+) . This is exactly the same for the case of ρ Ψ(+) . In the same way, it can be shown that if either the input state was ρ Φ(-) or ρ Ψ(-) , only the measurement results | +and | -+ can be obtained. In other words, the parity of the toincoming state is perfectly well discriminated by the measurements on single-photon qubits.
Subsequently, a homodyne measurement is performed for mode c by homodyne detector C as shown in Fig. 8(b) . We assume that ideal homodyne measurements are performed, i.e., when a homodyne measurement is performed the state is projected onto eigenstate |x of operator X with eigenvalue x, where
X = 1 √ 2 (a + a † ). ( 38
)
Let us first consider the case when the measurement result for the single photon qubits is | + + . In this case, the remaining state is ρ ++ in Eq. ( 33 ). The probability distribution P ++ Φ (+) for the homodyne measurement at detector C is
P ++ Φ (+) = x|Tr d [ρ ++ ]|x = V 1 2 (e -V x 2 + e -x 2 V ) π 1 2 (V + 1)
. (39) Note that the superscript, ++, denotes that the qubit measurement result was | + + , and the subscript, Φ (+) , denotes that the input state was ρ Φ (+) . These notations will be used also for the other cases in this section. The same analysis can be performed for the other possible measurement outcome | --:
P -- Φ (+) = x|Tr d [ρ --]|x = V 1 2 (e -V x 2 -e -x 2 V ) π 1 2 (V -1)
. (40)
In the same way, for another input state, ρ Φ(-) , it is straightforward to show:
P +- Φ (-) = P ++ Φ (+) , P -+ Φ (-) = P -- Φ (+) , (41)
and P ++ Φ (-) = P -- Φ (-) = 0. On the other hand, if the input state was ρ Ψ(+) , the probability distributions P ++ Ψ (+) and P -- Ψ (+) at detector C are It is straightforward to show for the other input state ρ Ψ(-) :
P ++ Ψ (+) = x|Tr c [ρ ++ ]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V + 2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) + 1 , (42)
P -- Ψ (+) = x|Tr c [ρ --]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V -2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) -1 . ( 43
P +- Ψ (-) = P -- Ψ (+) , P -+ Ψ (-) = P ++ Ψ (+) . (44)
The probability distributions P ++ Φ (±) and P ++ Ψ (±) are plotted in Fig. 10 . Figure 10 shows that when the input state was ρ Φ(+) or ρ Φ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Φ (+) and P -- Φ (+) , is located around the origin. However, when the input state was ρ Ψ(+) or ρ Ψ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Ψ (+) and P -- Ψ (+) , is located far from the origin. Therefore, two of the Bell states, ρ Φ(+) or ρ Φ(-) , can be well distinguished from the other two by the homodyne detector C for the case of the measurement outcome | + + . Finally, by combining the homodyne measurement result and the qubit measurement result, all four Bell states can be effectively distinguished. For example, let us assume that the measurement outcome of the single photon detectors was | + + and the homodyne detection outcome was around the origin, i.e., x ≈ 0. Then, one can say that state ρ Ψ(-) has been measured for the result of the thermal-Bell measurement.
As implied in Fig. 10 , the overlaps between the probability distributions around the origin, P ++ Φ (+) and P -- Φ (+) , and the other distributions, P ++ Ψ (+) and P -- Ψ (+) , are extremely small for a sufficiently large d. In other words, the distinguishability by the homodyne detection rapidly approaches 1 as d increases. As an example, we can calculate the distinguishability between the states ρ Ψ(+) and ρ Φ(+) by the homodyne measurement by detector C. The distinguishability by homodyne detection is Note also that the second scheme using homodyne detection is robust to detection inefficiency compared with the first scheme using photon number resolving measurements. In the first scheme, even if a detector misses only one photon, it will result in a completely wrong measurement outcome. In the second scheme, however, the measurement outcome will not be affected in that way. If a single photon detector misses a photon, it will be immediately recognized. Such a case can simply be discarded so that it will only degrade the success probability of the Bell measurement. The homodyne detection inefficiency will not significantly affect the result when the distributions around the origin and the distributions far from the origin are well separated, i.e., when d ≫ √ V , as shown in Fig. 10 . On the other hand, loss in the Kerr medium will have a detrimental affect.
P s = 1 2 |x|<d dxP ++ c (x) + |x|≥d dxP ++ d (x) (45
Quantum teleportation of a thermal-state qubit can be performed using one of the Bell states as the quantum channel. Let us assume that Alice needs to teleport a thermal-state qubit, ρ ψ , to Bob using a thermal-state entanglement, ρ Ψ(-) , shared by the two parties. The total state can be represented as
Alice first needs to perform the thermal-Bell measurement described in the previous subsection. To complete the teleportation process, Bob should perform an appropriate unitary transformation on his part of the quantum channel according to the measurement result sent from Alice via a classical channel. It is straightforward to show that the required transformations are exactly the same to those for the coherent-state qubit [27] . When the measurement outcome is ρ Ψ(-) , Bob obtains a perfect replica of the original unknown qubit without any operation.
When the measurement outcome is ρ Φ(-) , Bob should perform |α ↔ |α on his qubit in Eq. ( 23 ). Such a phase shift by π can be done using a phase shifter whose action is described by P (ϕ) = e iϕa † a , where a and a † are the annihilation and creation operators. When the outcome is ρ Ψ(+) , the transformation should be performed as |α → |α and |α → -|α . It is known that the displacement operator is a good approximation of this transformation for d ≫ 1 [29] . This transformation can also be achieved by teleporting the state again locally and repeating until the required phase shift is obtained [30] . When the outcome is ρ Φ(+) , σ x and σ z should be successively applied.
In this paper, we have studied characteristics of superpositions and entanglement of thermal states at high temperatures and discussed their applications to quantum information processing. The superpositions and entanglement of thermal states show various nonclassical properties such as interference patterns, negativity of the Wigner functions, and violations of the Bell-CHSH inequality. The Bell violations are more sensitive to the interaction time during the generation process when the thermal temperature (i.e. mixedness) of the thermalstate entanglement is larger. Therefore, in order to observe the Bell violations using the mixed state at a high temperature, the interaction time in the Kerr medium should be accurate. We have pointed out that certain superpositions of high-temperature thermal states, symmetric in the phase space, can also be generated. Some of these states have neither squeezing properties nor negative values in their Wigner functions but they are found to be highly nonclassical.
We have introduced the thermal-state qubit and thermal-Bell states for applications to quantum information processing. We have presented two possible methods for the Bell-state measurement. The Bell-state measurement enables one to perform quantum teleportation and gate operations for quantum computation with thermalstate qubits. The first scheme uses two photon number resolving detectors and a 50-50 beam splitter to discriminate the thermal-Bell states. Using the second scheme, it is possible to effectively discriminate the thermal-Bell states without photon number resolving detection. The required resources for the second scheme are two Kerr nonlinear interactions, two single photon detectors, two 50:50 beam splitters and one homodyne detector. The second scheme is more robust to inefficiency of the detectors: the inefficiency of the single photon detectors only degrades the success probability of the Bell measurement. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states without photon number resolving detection but with Kerr nonlinear interactions and two singlephoton detectors. This enables one to perform quantum teleportation and gate operations for quantum computation with thermal-state qubits."
},
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In many problems considered within the framework of quantum physics, physical systems are treated as pure states that can be represented by state vectors, or equivalently, by wave functions. Even though such an approach is simple and useful to address certain problems, it could often be quite different from real conditions of physical systems. This may be particularly true when one deals with macroscopic physical systems in terms of quantum physics. A macroscopic object is a complex open system which cannot avoid continuous interactions with the environment. Such a physical system is generally in a significantly mixed state and cannot be represented by a state vector. In general, mixed states are subtle objects whose properties are significantly more difficult to characterize than pure states.\n\nSchrödinger's famous cat paradox is a typical example where a massive classical object was assumed to be a pure state. It describes a counter-intuitive feature of quantum physics which dramatically appears when the principle of quantum superposition is applied to macroscopic objects. In the original paradox and its various explanations, the initial cat isolated in the steel chamber is considered a pure state that can be represented by a state vector such as |alive (or a wave function such as ψ alive ). The cat isolated from the environment is then assumed to interact with a microscopic superposition state, (|g +|e )/ √ 2, where |g and |e are the ground and excited states of a two-level atom. The cat will be dead if the atom is found in the excited state, |e , while it will remain alive if otherwise. Thus in Schrödinger's gedanken experiment the cat is entangled with the atom as (|g |alive + |e |dead ) √ 2, where the alive and dead statuses of the cat are described by the state vectors |alive and |dead . If one measures out the atomic system on the superposed basis, (|g ± |e )/ √ 2, the cat will be in a superposition of alive and dead states such as (|alive ± |dead )/ √ 2. It is often argued that such superposed states and entangled states can theoretically exist but are virtually impossible to observe because one cannot perfectly isolate a macroscopic object such as the cat from its environment [4] .\n\nHowever, this explanation is not fully satisfactory because the cat, a macroscopic object, is a complex open system which cannot be represented by a state vector. One may argue that the cat could be assumed to be in an unknown pure state such that the cat was certainly alive but the exact state of the cat was unknown. However, the interactions between the cat and its environment can cause the cat to become entangled with the environment [5] . In such a case, even though one can perfectly isolate the cat in the steel chamber from the enviroment, the cat will remain entangled with the environment due to its pre-interactions with the environment. Therefore, strictly speaking, even to assume a cat as an unknown pure state in the steel chamber is not legitimate. Thus a key point here is that it is unsatisfactory to describe the cat by a pure state such as |alive and |dead . We may need a more realistic assumption that the \"cat\" in Schrödinger's paradox was in a significantly mixed classical state. An intriguing question is then whether the quantum properties of the resulting state would still remain or diminish under such an assumption.\n\nRecently, such an analogy of Schrödinger's cat paradox, where the state corresponding to the virtual cat is a significantly mixed thermal state, was investigated [6] . A thermal state with a high temperature is considered a classical state in quantum optics. As the temperature of the thermal state increases, the degree of mixedness, which can be quantified by linear entropy, rapidly approaches the maximum value. When the temperature approaches infinity, the thermal state does not show any quantum properties. As a comparison, coherent states with large amplitudes are known as the most classical pure states [7] , and their superposition is often regarded as a superposition of classical states [8] . However, coherent states are still pure states which may not well represent truly classical systems, and they display some nonclassical features [9] . In Ref. [6] , it was shown that prominent quantum properties can actually be transferred from a microscopic superposition to a significantly mixed thermal state (i.e. a thermal state of which the degree of mixedness is close to the maximum value) at a high temperature through an experimentally feasible process. This result clarifies that unavoidable initial mixedness of the cat does not preclude strong quantum phenomena.\n\nOne of the results in Ref. [6] is that quantum entanglement can be produced between thermal states with nearly the maximum Bell-inequality violation when the temperatures of both modes goes to infinity. In previous related results, Bose et al. showed that entanglement can arise when two systems interact if one of the system are pure even when the other system is extremely mixed [10] . There is an interesting previous example shown by Filip et al. for the maximum violation of Bell's inequality when one of the modes is an extremely mixed thermal state [11] . Very recently, Ferreira et al. showed that entanglement can be generated at any finite temperature between high Q cavity mode field and a movable mirror thermal state [12] . However, in these example [10, 11, 12] only one of the modes is considered a large thermal state [10, 11, 12] and entanglement vanishes in the infinite temperature limit [10, 12] , which is obviously in contrast to the result presented in Ref. [6] . Entanglement for both of the modes at the thermal limit of the infinitely high temperature has not been found before. Remarkably, the violation of Bell's inequality in our examples reaches up to Cirel'son's bound [13] even in this infinite-temperature limit for both modes. As Vedral [14] and Ferreira et al. [12] pointed out it is believed that high temperatures reduce entanglement and all entanglement vanishes if the temperature is high enough, which is obviously not the case in Ref. [6] .\n\nThe purpose of this paper is twofold. Firstly, we review and further investigate various properties of superpositions and entanglement of thermal states at high temperatures [6] . In particular, we investigate two classes of highly mixed symmetric states in the phase space. Both the classes of these states do not show typical interference patterns in the phase space while they manifest strong singular behaviors. Interestingly, the first class of states has neither squeezing properties nor negative values in their Wigner functions, however, they are found to be highly nonclassical states. The second class of states has the maximum negativity in the Wigner function. Further, we discuss the possibility of quantum information processing with thermal-state qubits. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure Bell states. We show that four thermal-Bell states can be well discriminated by nonlinear interactions without photon number resolving measurements. Quantum teleportation and gate operations for thermal-state qubits can be realized using the Bell measurement scheme.\n\nThis paper is organized as follows. In Sec. II, we review the generation process of superpositions of thermal states and study their characteristics. In Sec. III, we study en-tanglement of thermal states, i.e., Bell inequality violations. In Sec. IV, we discuss the possibility of quantum information processing using thermal states. We first define the thermal-state qubit and the Bell-basis states using thermal-state entanglement. We then show that the four Bell states can be well discriminated by homodyne detection and two Kerr nonlinearities. It follows that quantum teleportation and quantum gate operations can be realized with thermal-state qubits. We conclude with final remarks in Sec. V."
},
{
"section_type": "OTHER",
"section_title": "II. SUPERPOSITIONS OF THERMAL STATES A. Generation of thermal-state superpositions",
"text": "Let us first consider a two-mode harmonic oscillator system. A displaced thermal state can be defined as\n\nρ th (V, d) = d 2 αP th (V, d)|α α| ( 1\n\n)\n\nwhere |α is a coherent state of amplitude α and\n\nP th α (V, d) = 2 π(V -1) exp[- 2|α -d| 2 V -1 ] (2)\n\nwith variance V and displacement d in the phase space.\n\nThe thermal temperature τ increases as V increases as e ν/τ = (V + 1)/(V -1), where is Planck's constant and ν is the frequency [15] . Suppose that a microscopic superposition state\n\n|ψ a = 1 √ 2 (|0 a + |1 a ), (3)\n\nwhere |0 and |1 are the ground and first excited states of the harmonic oscillator, interacts with a thermal state ρ th b (V, d) and the interaction Hamiltonian is\n\nH K = λâ † âb † b (4)\n\nwhich corresponds to the cross Kerr nonlinear interaction. The resulting state is then\n\nρ ent ab = 1 2 d 2 αP th (V, d) |0 0| ⊗ |α α| + |1 0| ⊗ |αe iϕ α| + |0 1| ⊗ |α αe iϕ | + |1 1| ⊗ |αe iϕ αe iϕ | ( 5\n\n)\n\nand ϕ is determined by the strength of the nonlinearity λ and the interaction time. The Wigner representation of ρ ent ab is\n\nW ent ab (α, β) = 1 π e -2|α| 2 W th (β; d) + 2αV c (β; d) + 2[αV c (β; d)] * + (4|α| 2 -1)W th (β; de iϕ ) (6)\n\nwhere α and β are complex numbers parametrizing the phase spaces of the microscopic and macroscopic systems respectively and\n\nW th (α; d) = 2 πV exp[- 2|α -d| 2 V ], (7)\n\nV c (α; d) = 2 πJK exp[- 2 K (1 -e iϕ )d 2 - 1 J (α - 2e iϕ d K )(α * - 2d K )], (8)\n\nK = 2 + (V -1)(1 -e iϕ\n\n), J = (sin ϕ/2 + iV cos ϕ/2)/(2V sin ϕ/2 + 2i cos ϕ/2), and d has been assumed real without loss of generality. If one traces ρ ent ab over mode a, the remaining state will be simply in a classical mixture of two thermal states and its Wigner function will be positive everywhere. However, if one measures out the \"microscopic part\" on the superposed basis, i.e., (|0 a ± |1 a )/ √ 2, the \"macroscopic part\" for mode b may not lose its nonclassical characteristics. Such a measurement on the the superposed basis will reduce the remaining state to\n\nρ sup(±) = N ± s d 2 αP th (V, d) |α α| ± |αe iϕ α| ± |α αe iϕ | + |αe iϕ αe iϕ | , (9)\n\nwhere N ± s are the normalization factors, and its Wigner function is\n\nW sup(±) (α) = N ± s {W th (α; d) ± V c (α; d) ± {V c (α; d)} * + W th (α; de iϕ )}. ( 10\n\n)\n\nThe ± signs in Eqs. ( 8 ) and (9) correspond to the two possible results from the measurement of the microscopic system. The state in Eq. ( 10 ) is a superposition of two thermal states. A feasible experimental setup to generate superpositions of thermal states is atom-field interactions in cavities, where a π/2 pulse can be used to prepare the atom in a superposed state. This type of experiment has already been performed to produce a superposition of coherent states [16] . In our cases, simply thermal states can be used instead of coherent states. Another possible setup is an all-optical scheme with free-traveling fields and a cross-Kerr medium, where a standard singlephoton qubit could be used as the microscopic superposition. Recently, there have been theoretical and experimental efforts to produce and observe giant Kerr nonlinearities using electromagnetically induced transparency [17] . Furthermore, it was shown that a weak Kerr nonlinearity can still be useful if a initially strong field is employed in this type of experiment [18] . We shall further explain this with examples in Sec. III."
},
{
"section_type": "OTHER",
"section_title": "B. Negativity of the Wigner function",
"text": "The negativity of the Wigner function is known as an indicator of non-classicality of quantum states. In order to observe negativity of the Wigner function in a real experiment, its absolute minimum negativity should be large enough. The minimum negativity of the Wigner function in Eq. ( 6 ) for V = 1 is -0.144 for d = 0 and -0.246 for d → ∞. Now suppose the initial state can be considered a classical thermal state by letting V ≫ 1. One might expect that the negativity would be washed out as the initial state becomes mixed, but this is not the case. The minimum negativity actually increases as V gets larger. If V → ∞, the minimum negativity of the Wigner function ( 6 ) is -0.246 regardless of d: no matter how mixed the initial thermal state was, the minimum negativity of Wigner function is found to be a large value. The point in the phase space which gives the minimum negativity when V ≫ 1 or d ≫ 0 is (-1 2 , 0) and has negativity\n\nW neg ≡ W ent ab (- 1 2 , 0) = 2(-2 + 1 V exp[-2d 2 V ]) π 2 √ e . (11)\n\nIt can be shown that\n\nW neg approaches -4/(π 2 √ e) ≈ -0.246 when either d → ∞ or V → ∞.\n\nThis effect is obviously due to the interaction between the microscopic superposition and the macroscopic thermal state. If the initial microscopic state is not superposed, e.g., |ψ a = |1 a , the resulting state will be a simple direct product, (|1 1|) a ⊗ ρ th b (V, -d). Whilst for V = 1 this state will exhibit negativity, this is washed out and tends to zero as V → ∞. Needless to say, if it was |0 a instead of |1 a , the resulting Wigner function will be a direct product of two Gaussian states whose Wigner fucntion can never be negative. The superpositon state (3) plays the crucial role in making the minimum negativity of the resulting Wigner function always saturate to a certain negative value no matter how mixed and classical the initial state of the other mode becomes. The Wigner functions of the single-mode states, W sup(±) (α), in Eq. (10) show large negative values. The minimum negativity of the Wigner function W sup(-) (α) is W sup(-) (0) = 2/π regardless of the values of V and d. On the other hand, the minimum negativity of the Wigner function W sup(+) (α) approaches 2/π for d → ∞ and disappears when d = 0."
},
{
"section_type": "OTHER",
"section_title": "C. Quantum interference in the phase space",
"text": "When ϕ = π, the state (9) becomes (12) where\n\nρ ± = N (ρ th (V, d)±σ(V, d)±σ(V, -d)+ρ th (V, -d)),\n\nσ(V, d) = d 2 αP th (V, d)| -α α| and N = 2 1 ± exp[-2d 2 V ] V 2 . ( 13\n\n)\n\nIf the initial state for mode b is a pure coherent state, i.e., V = 1, the measurement on the superposed basis for mode a will produce a superposition of two pure coherent states as\n\n| Ψ ± = 1 √ 1 ± e -2|α| 2 (|α ± | -α ), (14)\n\nwhere α = d. The probability P ± to obtain the state ρ ± is obtained as [19 ] where\n\nP ± = ψ ± |Tr b [ρ ent ab ]|ψ ± = 1 2 (1 ± exp[-2d 2 V ] V ), ( 15\n\n)\n\n|ψ ± = (|0 ±|1 )/ √ 2.\n\nThe probability approaches P ± = 1/2 when either d or V becomes large.\n\nAs an analogy of Schrödinger's cat paradox, the variance V corresponds to the size the initial \"cat\", and the distance d between the two thermal component states corresponds to distinguishability between the \"alive cat\" and the \"dead cat\". Suppose that both V and d are very large for the initial thermal state. The two thermal states ρ th (V, ±d) become macroscopically distinguishable when d ≫ √ V , and our example may become a more realistic analogy of the cat paradox in this limit. Both the states ρ ± in this case show probability distributions with two Gaussian peaks and interference fringes [6] . Figure 1 presents the probability distributions of x (≡ Re[α]) and p (≡ Im[α]) for ρ -(a) when V = 100 and d = 100 and (b) when V = 1000 and d = 300. The probability distribution of x (p) for ρ ± can be obtained by integrating the Wigner function of ρ ± over p (x). The two Gaussian peaks along the x axis and interference fringes along the p axis shown in Fig. 1 are a typical signature of a quantum superposition between macroscopically distinguishable states. The visibility v of the interference fringes is defined as [15]\n\nv = I max -I min I max + I min , (16)\n\nwhere I = dxW sup(-) (α) and the maximum should be taken over p. It can be simply shown that the visibility v is always 1 regardless of the value of V . Note that d should increase proportionally to √ V to maintain the condition of classical distingushability between the two component thermal states ρ th (V, ±d). The interference fringes with high visibility are incompatible with classical physics and evidence of quantum coherence. The fringe spacing (the distance between the fringes) does not depend on V but only on d, i.e., a pure superposition of coherent states shows the same fringe spacing for a given d. We emphasize that the states shown in Fig. 1 are \"superpositions\" of severely mixed thermal states.\n\nAn experimental realization of a nonlinear effect corresponding to ϕ = π is very demanding particularly in the presence of decoherence. Here we point out that the method using a weak nonlinear effect (ϕ ≪ π) combined with a strong field (d ≫ 1) [18] can be useful to generate a thermal-state superposition with prominent interference patterns. In Fig. 2 , we have used experimentally accessible values, V = 5, d = 2000 and ϕ = π/1000, but the fringe visibility is still 1. In this case, decoherence during the nonlinear interaction would be significantly reduced because of the decrease of the interaction time [18] . Note also that, if required, the state in Fig. 2 can be moved to the center of the phase space, for example, using a biased beam splitter (BS) and a strong coherent field [18] ."
},
{
"section_type": "OTHER",
"section_title": "D. Symmetric macroscopic quantum states",
"text": "Let us assume that d = 0, i.e., the initial state is the thermal state, ρ th (V, 0), at the origin of the phase space. In this case, the thermal-state superpositions, ρ ± , are produced with probabilities, P ± = (1/2){1 ± (1/V )}, respectively. Figure 3 shows the Wigner functions of ρ + dependent on the interaction time between the macro-scopic thermal state and the microscopic superposition in a cross Kerr medium. The state is always symmetric in the phase space regardless of the interaction time as shown in Fig. 3 . In this figure, the initial state is a thermal state of V = 100 (Fig. 3(a) ). In a relatively short time (θ = π/32 and θ = π/16), the state shows some interference patterns. When θ = π, the evolved state looks very localized around the origin as shown in Fig 3 . The generated state at θ = π does not show negativity of the Wigner function nor squeezing properties. On the other hand, a well defined P function does not exist for this state.\n\nIn the case of ρ -, with the same assumption d = 0, the Wigner function at ϕ = π has the minimum negativity (-2/π) at the origin regardless of V . As a result of the interaction with the microscopic superposition, a deep hole to the negative direction below zero has been formed around the origin for ρ -as shown in Fig. 4 ."
},
{
"section_type": "OTHER",
"section_title": "III. ENTANGLEMENT BETWEEN THERMAL STATES",
"text": "Entanglement between macroscopic objects and its Bell-type inequality tests are an important issue. In this section, we shall show that entanglement can be generated between high-temperature thermal states even when the temperature of each mode goes to infinity. 2 200 1 400 600 800 1000 V 2.1 2.2 2.3 B (a) 50 100 150 d 2.2 2 0 2.4 2.6 2.8 B (b) FIG. 5: (a) The optimized violation, B ≡ |B + |max, of Bell-\n\nCHSH inequality for the \"thermal-state entenglement\", ρ+, of V = 1000 (solid curve) and V = 100 (dashed curve). The Bell-violation of a pure entangled coherent state, i.e., V = 1, has been plotted for comparison (dotted curve). The Bell-violation B approaches its maximum bound, 2 √ 2, when d ≫ √ V regardless of the level of the mixedness V . (b) The optimized Bell-violation B against d for the different type of thermal-state entanglement generated using a 50:50 beam splitter from ρ + . V = 1000 (solid curve), V = 100 (dashed curve) and V = 1 (dotted curve)."
},
{
"section_type": "OTHER",
"section_title": "A. Entanglement using two initial thermal states",
"text": "If the microscopic superposition interacts with two thermal states, ρ th b (V, d) and ρ th c (V, d), and the micro-scopic particle is measured out on the superposed basis, the resulting state will be\n\nρ tm(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (17)\n\nwhere\n\nN t = 2 1 ± exp[-4d 2 V ] V 2 . ( 18\n\n)\n\nSuch two-mode thermal-state entanglement can be generated using two cavities and an atomic state detector [20] .\n\nExtending the two cavities to N cavities, entanglement of N -mode thermal states can also be generated. Such a state is an analogy of the N -mode pure GHZ state [21] but each mode is extremely mixed. Here we shall consider the Bell-CHSH inequality [22, 23] with photon number parity measurements [20, 24] . The parity measurements can be performed in a high-Q cavity using a far-off-resonant interaction between a two-level atom and the field [25] . The Bell-CHSH inequality can be represented in terms of the Winger function as [24 ]\n\n|B (±) | = π 2 4 |W tm(±) (α, β) + W tm(±) (α, β ′ ) + W tm(±) (α ′ , β) -W tm(±) (α ′ , β ′ )| ≤ 2, (19)\n\nwhere W tm(±) (α, β) is the Wigner function of ρ tm(±) in Eq. ( 17 ). As shown in Fig. 5 , the Bell-violation approaches the maximum bound for a bipartite measurement, 2 √ 2 [13] , when d ≫ √\n\nV regardless of the level of the mixedness V , i.e., the temperatures of the thermal states. Note that it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(a) . This implies that entanglement of nearly 1 ebit has been produced between the two significantly mixed thermal states for d ≫ √ V , and such \"thermal-state entanglement\" cannot be described by a local theory."
},
{
"section_type": "OTHER",
"section_title": "B. Entanglement using a beam splitter",
"text": "A different type of macroscopic entanglement can be generated by applying the beam splitter operation\n\nexp[θ/2(e iφ â † s âd -e -iφ â † d âs )], (20)\n\non the \"thermal-state superpositions\" in Eq. ( 9 ). The state after passing through a 50:50 beam splitter can be represented as\n\nN d 2 αP th α (V, d) | α √ 2 , - α √ 2 ± | - α √ 2 , α √ 2 α √ 2 , - α √ 2 | ± - α √ 2 , α √ 2 | , (21\n\n) 200 400 600 800 1000 V 2.1 2 0 2.2 2.3 B FIG. 6: The optimized Bell-violation B against V for the slightly different type of thermal-state entanglement generated using a 50:50 beam splitter using ρ + when d = 0. 3.13 3.14 3.15 3.16 ϕ 1.6 2.2 2 2.4 2.6 2.8 B (a) 3.13 3.14 3.15 3.16 ϕ 1.6 2 2.2 2.4 2.6 2.8 B (b) where N is defined in Eq. ( 13 ). When d is large, this state violates the Bell-CHSH inequality to the maximum bound 2 √ 2 regardless of the level of mixedness V as shown in Fig. 5(b ). Again, it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(b ). Furthermore, these states severely violate Bell's inequality even when d = 0 as V increases as shown in Fig. 6 . We have found that the optimized Bell violation of these states approaches 2.32449 for V → ∞. Interestingly, this value is exactly the same as the optimized Bell-CHSH violation for a pure two-mode squeezed state in the infinite squeezing limit [26] . Note that multilmode entangled states can be generated using multiple beam splitters.\n\nIt should be noted that the Bell violations are more sensitive to the interaction time when either V or d is larger. Figure 7 clearly shows this tendency. Therefore, in order to observe the Bell violations using the mixed state of V (and d) large, the interaction time in the Kerr medium should be more accurate."
},
{
"section_type": "OTHER",
"section_title": "IV. QUANTUM INFORMATION PROCESSING WITH THERMAL-STATE QUBITS",
"text": "In this section, we discuss the possibility of quantum information processing with thermal-state qubits and thermal-state entanglement."
},
{
"section_type": "OTHER",
"section_title": "A. Qubits and Bell-state measurements",
"text": "We introduce a thermal-state qubit\n\nρ ψ = |a| 2 ρ th (V, d) ± ab * σ(V, d) ± a * bσ(V, -d) + |b| 2 ρ th (V, -d), (22)\n\nwhere a and b are arbitrary complex numbers. The basis states, ρ th (V, d) and ρ th (V, -d), can be well discriminated by a homodyne measurement when d is larger than V . The thermal state qubit (22) can be re-written as\n\nρ ψ = d 2 αP th α (V, d) a|α + b| -α a * α| + b * -α| , (23)\n\nwhich can be understood as a generalization of the coherent state qubit, a|d + b|d , where |d is a coherent state of amplitude d. The thermal-state qubit (23) becomes identical to the coherent-state qubit when V = 1.\n\nWe also define four thermal-Bell states as\n\nρ Φ(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (24) ρ Ψ(±) = N t ρ th (V, d) ⊗ ρ th (V, -d) ± σ(V, d) ⊗ σ(V, -d) ± σ(V, -d) ⊗ σ(V, d) + ρ th (V, -d) ⊗ ρ th (V, d) (25)\n\nwhere N t was defined in Eq. ( 18 ). The thermal-Bell states can be written as For quantum information processing applications, it is an important task to discriminate between the four Bell states. Here we discuss two possible ways to discriminate between the thermal-Bell states (25) . We shall only briefly describe the first scheme using photon number resolving measurements and focus on the second scheme using nonlinear interactions. The first method is to simply use a 50-50 beam splitter and two photon number resolving detectors as shown in Fig. 8(a) . This scheme is basically the same as the Bellstate measurement scheme with pure entangled coherent states [27, 28] . Let us suppose that the amplitude, d, is large enough, i.e., d ≫ √ V . If the incident state was ρ Φ(+) or ρ Φ(-) , most of the photons are detected on detector A in in Fig. 8(a) . Meanwhile, most of the photons are detected on detector B when the incident state was ρ Ψ(+) or ρ Ψ(-) . The average photon numbers between the \"many-photon case\" and the \"few-photon case\" are compared in Fig. 9 . Furthermore, the states ρ Ψ(+) and ρ Φ(+) contain only even numbers of photons while ρ Ψ(-) and ρ Φ(-) contain only odd numbers of photons. Therefore, all the four Bell states can be well discriminated by analyzing numbers of photons detected at detectors A and B. For example, if detector A detects many photons while detector B detects few and the total photon number detected by the two detectors are even, this means that state ρ Φ(+) was measured by the thermal-Bell measurement. The nonzero failure probability can be made arbitrarily small by increasing d.\n\nρ Φ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, β ± | -α, -β α, β| ± -α, -β| , ( 26\n\n)\n\nρ Ψ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, -β ± | -α, β α, -β| ± -α, β| . (27)\n\nHowever, the average photon numbers of the thermal-Bell states are high when V ≫ 1 and d ≫ 1. In this case, it would be unrealistic to use photon number resolving detectors. It would be an interesting question whether\n\n0.5 1 1.5 2 2.5 3 d 10 0 0 20 30 N (a) 0.5 1 1.5 2 2.5 3 d 0 0 10 20 30 N (b)\n\nFIG. 9: The average photon number N for the \"many-photon case\" (solid line) and the \"few-photon case\" (dashed line) for V = 10 against d (a) when the input state is either ρ Φ(+) or ρ Ψ(+) and (b) when the input state is either ρ Φ(-) or ρ Ψ(-) .\n\nthese four thermal-Bell states can be distinguished by classical measurements, such as homodyne detection, instead of photon number resolving detection. Our alternative scheme employs cross-Kerr nonlinearities and single photon detectors as shown in Fig. 8(b ). Let us first suppose that the input field was ρ Φ(+) . The incident twomode state passes through a 50-50 beam splitter, BS1. The state after passing through the 50:50 beam splitter, BS1, is\n\nρ B = N t d 2 αd 2 βP th α (V, d)P th β (V, d) |η, -ξ η, -ξ| + |η, -ξ -η, ξ| + | -η, ξ η, -ξ| + | -η, ξ -η, ξ| (28)\n\nwhere η = (α+β)/ √ 2 and ξ = (α-β)/ √ 2. Two dual-rail single photon qubits, |ψ + ee ′ and |ψ + f f ′ , where\n\n|ψ + = 1 √ 2 (|0 |1 + |1 |0 ), (29)\n\nare prepared using two single photons and 50:50 beam splitters, BS2 and BS3, as shown in Fig. 8(b ). Then, traveling fields at modes c and d interacts with those of modes e and f , respectively, in cross-Kerr nonlinear media. We suppose that the interaction time is t = π/λ, and the resulting state is then\n\nρ B ′ = U ce U df ρ B cd ρ q ee ′ ρ q f f ′ U † ce U † df (30)\n\nwhere U ce = exp[iπH K ce /λ ] and ρ q = |ψ q ψ q |. An explicit form of Eq. ( 30 ) can then be simply obtained using the identity\n\nU ce |α c |0 e = |α c |0 e , U ce |α c |1 e = | -α c |1 e ( 31\n\n)\n\nwhere |α is a coherent state. However, we omit such an explicit expression in this paper for it is too lengthy. After the nonlinear interactions, the qubit parts, modes e, e ′ , f and f ′ , should be measured with the measurement basis\n\n{| + + , | + -, | -+ , | --} (32)\n\nwhere\n\n| + + = |ψ + ee ′ |ψ + f f ′ , | + -= |ψ + ee ′ |ψ -f f ′ , | -+ = |ψ -ee ′ |ψ + f f ′ , | --= |ψ -ee ′ |ψ -f f ′ , and |ψ -= (|0 |1 -|1 |0 )/ √ 2.\n\nThis measurement can be performed using two 50:50 beam splitters, BS4 and BS5, and four detectors, A1, A2, B1 and B2, as shown in Fig. 8(b ). If detector A1 and B1 click, i.e., the measurement result is | + + , the resulting state at modes c and d is\n\nρ ++ = N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η + | -η )( η| + -η|) c ⊗ (|ξ + | -ξ )( ξ| + -ξ|) d (33)\n\nNote that state ρ ++ is not normalized, which implies that the probability of obtaining the corresponding measurement result is not unity. The probability of obtaining this result is\n\nP ++ = (V + 1)(V + e -4d 2 V ) 2(V 2 + e -4d 2 V ) . (34)\n\nWhen the result is either | +or | -+ , the result is\n\nψ 2 |ρ B ′ |ψ 2 = ψ 3 |ρ B ′ |ψ 3 = 0, (35)\n\nwhich obviously means that the probability of the obtaining this result is zero. When the result is | --, i.e., detector A2 and B2 click,\n\nρ --= N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η -| -η )( η| --η|) c ⊗ (|ξ -| -ξ )( ξ| --ξ|) d , (36)\n\nwhich is not normalized. The probability of obtaining this result is\n\nP --= (V -1)(V -e -4d 2 V ) 2(V 2 + e -4d 2 V ) , (37)\n\nand it can be simply verified that P ++ +P --= 1. Therefore, only the measurement results | + + and | -can be obtained in the case of the input state ρ Φ(+) . This is exactly the same for the case of ρ Ψ(+) . In the same way, it can be shown that if either the input state was ρ Φ(-) or ρ Ψ(-) , only the measurement results | +and | -+ can be obtained. In other words, the parity of the toincoming state is perfectly well discriminated by the measurements on single-photon qubits.\n\nSubsequently, a homodyne measurement is performed for mode c by homodyne detector C as shown in Fig. 8(b) . We assume that ideal homodyne measurements are performed, i.e., when a homodyne measurement is performed the state is projected onto eigenstate |x of operator X with eigenvalue x, where\n\nX = 1 √ 2 (a + a † ). ( 38\n\n)\n\nLet us first consider the case when the measurement result for the single photon qubits is | + + . In this case, the remaining state is ρ ++ in Eq. ( 33 ). The probability distribution P ++ Φ (+) for the homodyne measurement at detector C is\n\nP ++ Φ (+) = x|Tr d [ρ ++ ]|x = V 1 2 (e -V x 2 + e -x 2 V ) π 1 2 (V + 1)\n\n. (39) Note that the superscript, ++, denotes that the qubit measurement result was | + + , and the subscript, Φ (+) , denotes that the input state was ρ Φ (+) . These notations will be used also for the other cases in this section. The same analysis can be performed for the other possible measurement outcome | --:\n\nP -- Φ (+) = x|Tr d [ρ --]|x = V 1 2 (e -V x 2 -e -x 2 V ) π 1 2 (V -1)\n\n. (40)\n\nIn the same way, for another input state, ρ Φ(-) , it is straightforward to show:\n\nP +- Φ (-) = P ++ Φ (+) , P -+ Φ (-) = P -- Φ (+) , (41)\n\nand P ++ Φ (-) = P -- Φ (-) = 0. On the other hand, if the input state was ρ Ψ(+) , the probability distributions P ++ Ψ (+) and P -- Ψ (+) at detector C are It is straightforward to show for the other input state ρ Ψ(-) :\n\nP ++ Ψ (+) = x|Tr c [ρ ++ ]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V + 2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) + 1 , (42)\n\nP -- Ψ (+) = x|Tr c [ρ --]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V -2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) -1 . ( 43\n\nP +- Ψ (-) = P -- Ψ (+) , P -+ Ψ (-) = P ++ Ψ (+) . (44)\n\nThe probability distributions P ++ Φ (±) and P ++ Ψ (±) are plotted in Fig. 10 . Figure 10 shows that when the input state was ρ Φ(+) or ρ Φ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Φ (+) and P -- Φ (+) , is located around the origin. However, when the input state was ρ Ψ(+) or ρ Ψ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Ψ (+) and P -- Ψ (+) , is located far from the origin. Therefore, two of the Bell states, ρ Φ(+) or ρ Φ(-) , can be well distinguished from the other two by the homodyne detector C for the case of the measurement outcome | + + . Finally, by combining the homodyne measurement result and the qubit measurement result, all four Bell states can be effectively distinguished. For example, let us assume that the measurement outcome of the single photon detectors was | + + and the homodyne detection outcome was around the origin, i.e., x ≈ 0. Then, one can say that state ρ Ψ(-) has been measured for the result of the thermal-Bell measurement.\n\nAs implied in Fig. 10 , the overlaps between the probability distributions around the origin, P ++ Φ (+) and P -- Φ (+) , and the other distributions, P ++ Ψ (+) and P -- Ψ (+) , are extremely small for a sufficiently large d. In other words, the distinguishability by the homodyne detection rapidly approaches 1 as d increases. As an example, we can calculate the distinguishability between the states ρ Ψ(+) and ρ Φ(+) by the homodyne measurement by detector C. The distinguishability by homodyne detection is Note also that the second scheme using homodyne detection is robust to detection inefficiency compared with the first scheme using photon number resolving measurements. In the first scheme, even if a detector misses only one photon, it will result in a completely wrong measurement outcome. In the second scheme, however, the measurement outcome will not be affected in that way. If a single photon detector misses a photon, it will be immediately recognized. Such a case can simply be discarded so that it will only degrade the success probability of the Bell measurement. The homodyne detection inefficiency will not significantly affect the result when the distributions around the origin and the distributions far from the origin are well separated, i.e., when d ≫ √ V , as shown in Fig. 10 . On the other hand, loss in the Kerr medium will have a detrimental affect.\n\nP s = 1 2 |x|<d dxP ++ c (x) + |x|≥d dxP ++ d (x) (45"
},
{
"section_type": "OTHER",
"section_title": "B. Quantum teleportation and computation",
"text": "Quantum teleportation of a thermal-state qubit can be performed using one of the Bell states as the quantum channel. Let us assume that Alice needs to teleport a thermal-state qubit, ρ ψ , to Bob using a thermal-state entanglement, ρ Ψ(-) , shared by the two parties. The total state can be represented as\n\nAlice first needs to perform the thermal-Bell measurement described in the previous subsection. To complete the teleportation process, Bob should perform an appropriate unitary transformation on his part of the quantum channel according to the measurement result sent from Alice via a classical channel. It is straightforward to show that the required transformations are exactly the same to those for the coherent-state qubit [27] . When the measurement outcome is ρ Ψ(-) , Bob obtains a perfect replica of the original unknown qubit without any operation.\n\nWhen the measurement outcome is ρ Φ(-) , Bob should perform |α ↔ |α on his qubit in Eq. ( 23 ). Such a phase shift by π can be done using a phase shifter whose action is described by P (ϕ) = e iϕa † a , where a and a † are the annihilation and creation operators. When the outcome is ρ Ψ(+) , the transformation should be performed as |α → |α and |α → -|α . It is known that the displacement operator is a good approximation of this transformation for d ≫ 1 [29] . This transformation can also be achieved by teleporting the state again locally and repeating until the required phase shift is obtained [30] . When the outcome is ρ Φ(+) , σ x and σ z should be successively applied."
},
{
"section_type": "CONCLUSION",
"section_title": "V. CONCLUSION",
"text": "In this paper, we have studied characteristics of superpositions and entanglement of thermal states at high temperatures and discussed their applications to quantum information processing. The superpositions and entanglement of thermal states show various nonclassical properties such as interference patterns, negativity of the Wigner functions, and violations of the Bell-CHSH inequality. The Bell violations are more sensitive to the interaction time during the generation process when the thermal temperature (i.e. mixedness) of the thermalstate entanglement is larger. Therefore, in order to observe the Bell violations using the mixed state at a high temperature, the interaction time in the Kerr medium should be accurate. We have pointed out that certain superpositions of high-temperature thermal states, symmetric in the phase space, can also be generated. Some of these states have neither squeezing properties nor negative values in their Wigner functions but they are found to be highly nonclassical.\n\nWe have introduced the thermal-state qubit and thermal-Bell states for applications to quantum information processing. We have presented two possible methods for the Bell-state measurement. The Bell-state measurement enables one to perform quantum teleportation and gate operations for quantum computation with thermalstate qubits. The first scheme uses two photon number resolving detectors and a 50-50 beam splitter to discriminate the thermal-Bell states. Using the second scheme, it is possible to effectively discriminate the thermal-Bell states without photon number resolving detection. The required resources for the second scheme are two Kerr nonlinear interactions, two single photon detectors, two 50:50 beam splitters and one homodyne detector. The second scheme is more robust to inefficiency of the detectors: the inefficiency of the single photon detectors only degrades the success probability of the Bell measurement."
}
] | |
arxiv:0704.0526 | 0704.0526 | 1 | a1a26ce33595a638cf1c21c15b33b4f6450ff8e123099f4f85a016d59c12c65c | Fractional WKB Approximation | Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is
investigated in this paper using the fractional calculus. In the fractional
case the wave function is constructed such that the phase factor is the same as
the Hamilton's principle function "S". To demonstrate our proposed approach two
examples are investigated in details. | [
"Eqab M. Rabei",
"Ibrahim M. A. Altarazi",
"Sami I. Muslih",
"Dumitru Baleanu"
] | [
"math-ph",
"hep-th",
"math.MP"
] | math-ph | [] | 2007-04-04 | 2026-02-26 | Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function "S". To demonstrate our proposed approach two examples are investigated in details.
Fractional calculus is a branch of mathematics that deal with a generalization of well-known operations of differentiations and integrations to arbitrary non-integer order, which can be real non-integer or even imaginary number.
Nowadays physicists have used this powerful tool to deal with some problems which were not solvable in the classical sense. Therefore, the fractional calculus became one of the most powerful and widely useful tools in describing and explaining some physical complex systems.
Recently, the Euler-Lagrange equations has been presented for unconstrained and constrained fractional variational problems [1 and other references] . This technique enable us to solve some problems including describing the behavior of non-conservative systems developed by Riewe [2] , where he used the fractional derivative to construct the Lagrangian and Hamiltonian for non-conservative systems.
From these reasons in [3] was developed a general formula for the potential of any arbitrary force conservative or not conservative, which leads directly to the consideration of dissipative effect in Lagrangian and Hamiltonian formulation. Also, the canonical quantization of non-conservative systems has been carried out in [4] .
Starting from a Lagrangian containing a fractional derivative, the fractional Hamiltonian is achieved in [5] . In addition, the passage from Hamiltonian containing fractional derivatives to the fractional Hamilton-Jaccobi is achieved by Rabei et.al [6] . The equations of motion are obtained in a similar manner to the usual mechanics.
All these outstanding results using the fractional derivative make us concentrate on another branch of quntam physics. WKB approximation [7, 8, 9, 10, 14] . In this paper we are mainly interested to construct the solution of Schroödinger equation in an exponential form (Griffith 1995) starting from fractional Hamilton-Jaccobi equation and how it leads naturally to this semi-classical approximation namely fractional WKB.
The purpose of this paper is to find the solution of Schrödinger equation for some systems that have a fractional behavior in their Lagrangians and obey the WKB approximation assumptions.
The plan of this paper is as follows: In section II the derivation of generalized Hamilton-Jaccobi partial differential equation which given in [6] is briefly reviewed. In section III the fractional WKB approximation is derived. In Section IV some examples with the fractional WKB technique is reported. Section V is dedicated to conclusions.
The left and right Reimann-Loville fractional derivative are defined as follows [3] The left Riemann-Liouville fractional derivative is given by
( ) τ τ τ α α α d f x dx d n x f D x a n n x a ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - Γ = ) ( ) ( 1 ) ( 1 (1)
The right Riemann-Liouville fractional derivative has the form
( ) τ τ τ α β β d f x dx d n x f D b x n n b x ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - Γ = ) ( ) ( 1 ) ( 1 (2)
Here α, β are the order of derivation such that n-1≤α <n, n-1≤β<n, and they are not zero.
If α is an integer, these derivatives are defined in usual sense as
) ( ) ( x f dx d x f D x a α α ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = (3-a) ) ( ) ( x f dx d x f D b x β β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = (3-b)
Hamilton formalism with fractional derivative was proposed in [5]
namely ) , , , ( ) , , , ( t q D q D q L q D p q D p t p p q H b t t a b t t a β α β β α α β α - + = , ( 4
) where L represents the fractional Lagrangian obtained by replacing the classical derivatives with the corresponding fractional ones [5] .
Hamilton's equations of motion are obtained as follows [5]
t L t H ∂ ∂ - = ∂ ∂ ; ; q D p H t a α α = ∂ ∂ ; q D p H b t β β = ∂ ∂ α α β β p D p D q H b t t a + = ∂ ∂ (5)
In [6] based on the sequential derivatives the fractional Hamilton-Jacobi partial differential equation is obtained. The Hamilton-Jacobi function in configuration space is written in a similar manner to the usual mechanics by using the Reimann-Loville fractional derivative. In [6] the following generating function is used, where α and β are bigger or equal to 1. Thus, the new Hamiltonian is expressed as
S t P P q D q D F F b t t a = = - - ) , , , , ( 1 1 2 β α β α ) , , , ( ' ) , , , ( t Q D Q D Q L Q D P Q D P t P P Q K b t t a b t t a β α β β α α β α - + = (7)
It is concluded that, the following relation relates the two Hamiltonians
dt dF K Q D P Q D P H q D p q D p b t t a b t t a + - + = - + β β α α β β α α (8)
According to reference [6] the function F is proposed as ) , , , , (
1 1 t P P q D q D S F b t t a β α β α - - = Q D P Q D P B t t a 1 1 - - - - β β α α , (9)
The function S is called Hamilton's principle function.
Therefore, requiring that the transformed Hamiltonian K shall be zero the Hamilton-Jacobi equation is satisfied. In other words Q, P α , P β are constants.
0 = ∂ ∂ + t S H (10)
Since Q, P α , P β are constants, The Hamilton's principle function is written as (11) where
) , , , , ( 2 1 1 1 t E E q D q D S S b t t a - - = β α
1 E P = α 2 E P = β
If the Hamiltonian is explicitly independent of time, then S can be written as follows (12) ) , , ( ) , ( ) , (
2 1 2 1 2 1 1 1 t E E f E q D W E q D W S b t t a + + = - - β α
W represents the Hamilton's characteristic function; therefore, the following equations of motion are obtained in [6] as:
q D W P t a 1 1 - ∂ ∂ = α α q D W P b t 1 1 - ∂ ∂ = β β (13) 1 1 1 1 λ α = ∂ ∂ = - E W Q D t a 2 2 2 1 λ β = ∂ ∂ = - E W Q D b t (14)
Here λ 1 ,λ 2 are constants.
The outstanding result regarding the meaning of the state function ψ and its relationship to Hamilton's principle function S enables us to write the exponential solution of Schrödinger equation [13] .
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = h ) , ( exp ) , ( t q S i t q ψ (15)
The phase of state function obeys the same mathematical equation, as does Hamilton's principle function S. The physical significance of S in classical mechanics is that it represents the generator of trajectories [12] for fractional systems; the fractional Hamilton's principle function is become the phase of the state function ψ. One can write the solution of Schrödinger equation under the postulated constrains by the WKB approximation and using the fractional Hamilton's principle function eq (12) . Thus we propose the fractional state function as:
( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t q D q D S i t q D q D b t t a b t t a , , exp ) , , ( 1 1 1 1 β α β α ψ h (16)
From the quantization using WKB approximation [7, 8, 9, 10, 14] a general solution of Schrödinger equation is obtained using the expansion for S and then using the transformation to the N-dimensional system as:
( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ∏ = h t q S i q i N i i o i , exp ) ( 1 ψ ψ (17)
where ( )
) ( 1 i i io q p q = ψ (18)
In our case, S behaves like a 2-dimensional problem with two distinct momenta. Thus,
(19) q D q t a 1 1 - ≡ α α P P 1 ≡ (20) q D q b t 1 2 - ≡ β β P P 2 ≡
And the momenta are defined as operators. Therefore, we can propose the wave function ψ of the fractional system in the following form
( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t E E q D q D S i P P t q D q D b t t a b t t a , , , , exp 1 ) , , ( 2 1 1 1 1 1 β α β α β α ψ h (21)
and the momenta operators in the form
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 , ˆβ β α α h h (22)
We conclude that (21) is the solution of Schrödinger equation for any given fractional systems. If α and β both are equal to unity, then we will return to the usual classical solution of Schrödinger equation, also we can notice how the probability is inversely proportional to the momentum . ) (
1 2 q p ≅ ψ IV. Examples
As a first model let us consider the following fractional Lagrangian,
( ) ( ) 2 1 2 0 2 1 2 1 q D q D L t t β α + = (23)
The fractional Hamilton-Jacobi equation for this fractional Lagrangian can be calculated as:
( ) ( ) . 0 2 1 2 1 2 2 = ∂ ∂ + + t S P P β α (24) where q D L P t α α 0 ∂ ∂ = ; q D L P t β β 1 ∂ ∂ =
Making use of equation ( 13 ), the fractional Hamilton-Jacobi equation (24) becomes:
0 2 1 2 1 2 1 1 2 2 1 0 1 = ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - t S q D W q D W t t β α (25) Taking into account t S H ∂ ∂ - = (26)
If we apply (26) on a wave function it gives:
) ( 2 1 E E E t S + - ≡ - = ∂ ∂ (27)
By using the fact that E is the total energy of the system and taking into account (27) we obtain 0 2
1 2 1 2 2 1 1 2 1 2 1 0 1 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - E q D W E q D W t t β α (28)
Thus, both sides of (28) should be zero, and we obtain
q D E W q D E W t t 1 1 2 2 1 0 1 1 2 , 2 - - = = β α
(29) By using ( 12 ) and ( 21 ) we obtain It's the same as the classical solution. Also, when applying the energy operator it gives the energy eigenvalues:
( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + = - - - - t E q D E q D E i P P t q D q D t t t t 1 1 2 1 0 1 1 1 1 0 2 2 exp 1 ) , , (
( ) ( )ψ ψ ψ β α 2 2 2 1 2 1 P P H + = ( ) ψ 2 1 E E + = (34)
as in the classical case.
As a second example let us consider the following fractional Lagrangian ( ) ( )
2 1 0 2 1 2 0 2 1 2 1 2 1 q q D q D q D q D L t t t t + + + + = β α β α (35)
The corresponding fractional Hamilton is calculated as follows
( ) ( ) 2 2 2 2 1 1 2 1 1 2 1 q P P H - - + - = β α (36)
Thus, the fractional Hamilton-Jacobi equation becomes ( ) ( )
0 2 1 1 2 1 1 2 1 2 2 2 = ∂ ∂ + - - + - t S q P P β α (37)
The fractional Hamilton's principle function is calculated as,
) ( )
t E E q D E q D E q S t t ) ( 1 2 1 2 2 1 1 1 2 1 0 1 2 + - + + + + = - - β α (38)
As a result the wave function can be written in the form ( )
( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + + + + = - - - - t E q D E q D E q i P P t q D q D t t t t 1 1 2 1 0 1 2 1 1 1 0 1 2 1 2 exp 1 ) , , ( β α β α β α ψ h (39)
To identify the influence of the operators let us test the effect of the momenta
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 ˆβ β α α h h (40)
Using the characteristic equations, it can be shown that
1 2 , 1 2 ˆ2 1 2 + = + + = E P E q P β α (41) The result shown in (41 ) is the same classical solution. When applying the energy operator it will give the energy eigenvalues ( ) ( ) ψ ψ ψ ψ β α 2 2 2 2 1 1 2 1 1 2 1 q P P H --+ -= (42) ( ) ( ) ψ ψ ψ ψ β α β α 2 2 2 2 1 2 1 q P P P P -+ + -+ = ( ) ( ) ψ } 2 1 1 2 2 2 2 2 2 2 2 2 2 1 { 2 2 1 2 2 1 2 2 q E E q E E q E q -+ + + + -+ + + + + = Then we get
ψ ψ E H = (43)
which is exactly the total energy as the case for the classical systems.
We use the generating function "S" of the Hamilton-Jaccobi equation in its fractional form to be the phase factor of the wave function describing some potentials valid for the assumptions suggested by the WKB approximation The proof of our results arises from the new proposed concepts of the momentum and energy operators, that they give the same eigenvalues producing the ordinary results achieved by the classical approach.
Giving the same eigenvalues that means this form of fractional operator also eigen, valid, and useful in effecting on a state functions. | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function \"S\". To demonstrate our proposed approach two examples are investigated in details."
},
{
"section_type": "BACKGROUND",
"section_title": "I. Introduction",
"text": "Fractional calculus is a branch of mathematics that deal with a generalization of well-known operations of differentiations and integrations to arbitrary non-integer order, which can be real non-integer or even imaginary number.\n\nNowadays physicists have used this powerful tool to deal with some problems which were not solvable in the classical sense. Therefore, the fractional calculus became one of the most powerful and widely useful tools in describing and explaining some physical complex systems.\n\nRecently, the Euler-Lagrange equations has been presented for unconstrained and constrained fractional variational problems [1 and other references] . This technique enable us to solve some problems including describing the behavior of non-conservative systems developed by Riewe [2] , where he used the fractional derivative to construct the Lagrangian and Hamiltonian for non-conservative systems.\n\nFrom these reasons in [3] was developed a general formula for the potential of any arbitrary force conservative or not conservative, which leads directly to the consideration of dissipative effect in Lagrangian and Hamiltonian formulation. Also, the canonical quantization of non-conservative systems has been carried out in [4] .\n\nStarting from a Lagrangian containing a fractional derivative, the fractional Hamiltonian is achieved in [5] . In addition, the passage from Hamiltonian containing fractional derivatives to the fractional Hamilton-Jaccobi is achieved by Rabei et.al [6] . The equations of motion are obtained in a similar manner to the usual mechanics.\n\nAll these outstanding results using the fractional derivative make us concentrate on another branch of quntam physics. WKB approximation [7, 8, 9, 10, 14] . In this paper we are mainly interested to construct the solution of Schroödinger equation in an exponential form (Griffith 1995) starting from fractional Hamilton-Jaccobi equation and how it leads naturally to this semi-classical approximation namely fractional WKB.\n\nThe purpose of this paper is to find the solution of Schrödinger equation for some systems that have a fractional behavior in their Lagrangians and obey the WKB approximation assumptions.\n\nThe plan of this paper is as follows: In section II the derivation of generalized Hamilton-Jaccobi partial differential equation which given in [6] is briefly reviewed. In section III the fractional WKB approximation is derived. In Section IV some examples with the fractional WKB technique is reported. Section V is dedicated to conclusions."
},
{
"section_type": "OTHER",
"section_title": "II. Basic Tools",
"text": "The left and right Reimann-Loville fractional derivative are defined as follows [3] The left Riemann-Liouville fractional derivative is given by\n\n( ) τ τ τ α α α d f x dx d n x f D x a n n x a ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - Γ = ) ( ) ( 1 ) ( 1 (1)\n\nThe right Riemann-Liouville fractional derivative has the form\n\n( ) τ τ τ α β β d f x dx d n x f D b x n n b x ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - Γ = ) ( ) ( 1 ) ( 1 (2)\n\nHere α, β are the order of derivation such that n-1≤α <n, n-1≤β<n, and they are not zero.\n\nIf α is an integer, these derivatives are defined in usual sense as\n\n) ( ) ( x f dx d x f D x a α α ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = (3-a) ) ( ) ( x f dx d x f D b x β β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = (3-b)\n\nHamilton formalism with fractional derivative was proposed in [5]\n\nnamely ) , , , ( ) , , , ( t q D q D q L q D p q D p t p p q H b t t a b t t a β α β β α α β α - + = , ( 4\n\n) where L represents the fractional Lagrangian obtained by replacing the classical derivatives with the corresponding fractional ones [5] .\n\nHamilton's equations of motion are obtained as follows [5]\n\nt L t H ∂ ∂ - = ∂ ∂ ; ; q D p H t a α α = ∂ ∂ ; q D p H b t β β = ∂ ∂ α α β β p D p D q H b t t a + = ∂ ∂ (5)\n\nIn [6] based on the sequential derivatives the fractional Hamilton-Jacobi partial differential equation is obtained. The Hamilton-Jacobi function in configuration space is written in a similar manner to the usual mechanics by using the Reimann-Loville fractional derivative. In [6] the following generating function is used, where α and β are bigger or equal to 1. Thus, the new Hamiltonian is expressed as\n\nS t P P q D q D F F b t t a = = - - ) , , , , ( 1 1 2 β α β α ) , , , ( ' ) , , , ( t Q D Q D Q L Q D P Q D P t P P Q K b t t a b t t a β α β β α α β α - + = (7)\n\nIt is concluded that, the following relation relates the two Hamiltonians\n\ndt dF K Q D P Q D P H q D p q D p b t t a b t t a + - + = - + β β α α β β α α (8)\n\nAccording to reference [6] the function F is proposed as ) , , , , (\n\n1 1 t P P q D q D S F b t t a β α β α - - = Q D P Q D P B t t a 1 1 - - - - β β α α , (9)\n\nThe function S is called Hamilton's principle function.\n\nTherefore, requiring that the transformed Hamiltonian K shall be zero the Hamilton-Jacobi equation is satisfied. In other words Q, P α , P β are constants.\n\n0 = ∂ ∂ + t S H (10)\n\nSince Q, P α , P β are constants, The Hamilton's principle function is written as (11) where\n\n) , , , , ( 2 1 1 1 t E E q D q D S S b t t a - - = β α\n\n1 E P = α 2 E P = β\n\nIf the Hamiltonian is explicitly independent of time, then S can be written as follows (12) ) , , ( ) , ( ) , (\n\n2 1 2 1 2 1 1 1 t E E f E q D W E q D W S b t t a + + = - - β α\n\nW represents the Hamilton's characteristic function; therefore, the following equations of motion are obtained in [6] as:\n\nq D W P t a 1 1 - ∂ ∂ = α α q D W P b t 1 1 - ∂ ∂ = β β (13) 1 1 1 1 λ α = ∂ ∂ = - E W Q D t a 2 2 2 1 λ β = ∂ ∂ = - E W Q D b t (14)\n\nHere λ 1 ,λ 2 are constants."
},
{
"section_type": "OTHER",
"section_title": "III. Fractional WKB approximation",
"text": "The outstanding result regarding the meaning of the state function ψ and its relationship to Hamilton's principle function S enables us to write the exponential solution of Schrödinger equation [13] .\n\n⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = h ) , ( exp ) , ( t q S i t q ψ (15)\n\nThe phase of state function obeys the same mathematical equation, as does Hamilton's principle function S. The physical significance of S in classical mechanics is that it represents the generator of trajectories [12] for fractional systems; the fractional Hamilton's principle function is become the phase of the state function ψ. One can write the solution of Schrödinger equation under the postulated constrains by the WKB approximation and using the fractional Hamilton's principle function eq (12) . Thus we propose the fractional state function as:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t q D q D S i t q D q D b t t a b t t a , , exp ) , , ( 1 1 1 1 β α β α ψ h (16)\n\nFrom the quantization using WKB approximation [7, 8, 9, 10, 14] a general solution of Schrödinger equation is obtained using the expansion for S and then using the transformation to the N-dimensional system as:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ∏ = h t q S i q i N i i o i , exp ) ( 1 ψ ψ (17)\n\nwhere ( )\n\n) ( 1 i i io q p q = ψ (18)\n\nIn our case, S behaves like a 2-dimensional problem with two distinct momenta. Thus,\n\n(19) q D q t a 1 1 - ≡ α α P P 1 ≡ (20) q D q b t 1 2 - ≡ β β P P 2 ≡\n\nAnd the momenta are defined as operators. Therefore, we can propose the wave function ψ of the fractional system in the following form\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t E E q D q D S i P P t q D q D b t t a b t t a , , , , exp 1 ) , , ( 2 1 1 1 1 1 β α β α β α ψ h (21)\n\nand the momenta operators in the form\n\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 , ˆβ β α α h h (22)\n\nWe conclude that (21) is the solution of Schrödinger equation for any given fractional systems. If α and β both are equal to unity, then we will return to the usual classical solution of Schrödinger equation, also we can notice how the probability is inversely proportional to the momentum . ) (\n\n1 2 q p ≅ ψ IV. Examples"
},
{
"section_type": "OTHER",
"section_title": "IV. a) Example 1:",
"text": "As a first model let us consider the following fractional Lagrangian,\n\n( ) ( ) 2 1 2 0 2 1 2 1 q D q D L t t β α + = (23)\n\nThe fractional Hamilton-Jacobi equation for this fractional Lagrangian can be calculated as:\n\n( ) ( ) . 0 2 1 2 1 2 2 = ∂ ∂ + + t S P P β α (24) where q D L P t α α 0 ∂ ∂ = ; q D L P t β β 1 ∂ ∂ =\n\nMaking use of equation ( 13 ), the fractional Hamilton-Jacobi equation (24) becomes:\n\n0 2 1 2 1 2 1 1 2 2 1 0 1 = ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - t S q D W q D W t t β α (25) Taking into account t S H ∂ ∂ - = (26)\n\nIf we apply (26) on a wave function it gives:\n\n) ( 2 1 E E E t S + - ≡ - = ∂ ∂ (27)\n\nBy using the fact that E is the total energy of the system and taking into account (27) we obtain 0 2\n\n1 2 1 2 2 1 1 2 1 2 1 0 1 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - E q D W E q D W t t β α (28)\n\nThus, both sides of (28) should be zero, and we obtain\n\nq D E W q D E W t t 1 1 2 2 1 0 1 1 2 , 2 - - = = β α\n\n(29) By using ( 12 ) and ( 21 ) we obtain It's the same as the classical solution. Also, when applying the energy operator it gives the energy eigenvalues:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + = - - - - t E q D E q D E i P P t q D q D t t t t 1 1 2 1 0 1 1 1 1 0 2 2 exp 1 ) , , (\n\n( ) ( )ψ ψ ψ β α 2 2 2 1 2 1 P P H + = ( ) ψ 2 1 E E + = (34)\n\nas in the classical case."
},
{
"section_type": "OTHER",
"section_title": "IVb.)Example 2:",
"text": "As a second example let us consider the following fractional Lagrangian ( ) ( )\n\n2 1 0 2 1 2 0 2 1 2 1 2 1 q q D q D q D q D L t t t t + + + + = β α β α (35)\n\nThe corresponding fractional Hamilton is calculated as follows\n\n( ) ( ) 2 2 2 2 1 1 2 1 1 2 1 q P P H - - + - = β α (36)\n\nThus, the fractional Hamilton-Jacobi equation becomes ( ) ( )\n\n0 2 1 1 2 1 1 2 1 2 2 2 = ∂ ∂ + - - + - t S q P P β α (37)\n\nThe fractional Hamilton's principle function is calculated as,\n\n) ( )\n\nt E E q D E q D E q S t t ) ( 1 2 1 2 2 1 1 1 2 1 0 1 2 + - + + + + = - - β α (38)\n\nAs a result the wave function can be written in the form ( )\n\n( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + + + + = - - - - t E q D E q D E q i P P t q D q D t t t t 1 1 2 1 0 1 2 1 1 1 0 1 2 1 2 exp 1 ) , , ( β α β α β α ψ h (39)\n\nTo identify the influence of the operators let us test the effect of the momenta\n\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 ˆβ β α α h h (40)\n\nUsing the characteristic equations, it can be shown that\n\n1 2 , 1 2 ˆ2 1 2 + = + + = E P E q P β α (41) The result shown in (41 ) is the same classical solution. When applying the energy operator it will give the energy eigenvalues ( ) ( ) ψ ψ ψ ψ β α 2 2 2 2 1 1 2 1 1 2 1 q P P H --+ -= (42) ( ) ( ) ψ ψ ψ ψ β α β α 2 2 2 2 1 2 1 q P P P P -+ + -+ = ( ) ( ) ψ } 2 1 1 2 2 2 2 2 2 2 2 2 2 1 { 2 2 1 2 2 1 2 2 q E E q E E q E q -+ + + + -+ + + + + = Then we get\n\nψ ψ E H = (43)\n\nwhich is exactly the total energy as the case for the classical systems."
},
{
"section_type": "CONCLUSION",
"section_title": "V. Conclusions",
"text": "We use the generating function \"S\" of the Hamilton-Jaccobi equation in its fractional form to be the phase factor of the wave function describing some potentials valid for the assumptions suggested by the WKB approximation The proof of our results arises from the new proposed concepts of the momentum and energy operators, that they give the same eigenvalues producing the ordinary results achieved by the classical approach.\n\nGiving the same eigenvalues that means this form of fractional operator also eigen, valid, and useful in effecting on a state functions."
}
] | ||
arxiv:0704.0527 | 0704.0527 | 1 | 10.1103/PhysRevD.75.125023 | edd179bddd702d53493d85c4e4f40b62a69e7e5d63dd63196e931a133fab844c | Towards Skyrmion Stars: Large Baryon Configurations in the
Einstein-Skyrme Model | We investigate the large baryon number sector of the Einstein-Skyrme model as
a possible model for baryon stars. Gravitating hedgehog skyrmions have been
investigated previously and the existence of stable solitonic stars excluded
due to energy considerations. However, in this paper we demonstrate that by
generating gravitating skyrmions using rational maps, we can achieve
multi-baryon bound states whilst recovering spherical symmetry in the limit
where B becomes large. | [
"B.M.A.G. Piette and G.I. Probert"
] | [
"hep-th"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in a series of papers as a non-linear field theory of pions [2] , [3] . Skyrme's initial idea was to think of baryons (in particular the nucleons) as secondary structures arising from a more fundamental mesonic fluid. The key property of the model was that the baryons arose as solitons in a topological manner and thus possessed a conserved topological charge identified with the baryon number.
The lowest energy stable solutions of the model are termed Skyrmions and can be thought of as baryonic solitons. The Skyrme model has been very successful in modelling the structures of various nuclei and has been shown by Witten et al. [4] to possess the general features of a low energy effective field theory for QCD. Some studies of the Skyrme model coupled to gravity have previously been undertaken [1], [5] , [6] , mainly with the motivation of a comparison of its features with those of other nonlinear field theories coupled to gravity. Of particular note is the Einstein-Yang-Mills theory, in which gravitationally bound configurations of non-abelian gauge fields are produced.
Other reasons for studying the Einstein-Skyrme model are cosmological and astrophysical ones. Various authors have studied black hole formation in the model, with the conclusion that the so-called no-hair conjecture may not hold [7] , [8] .
The purpose of this paper is to study large baryon number Skyrmions or configurations of Skyrmions in the Einstein-Skyrme model. In particular, we wish to investigate if stable solitonic stars could exist within the model and to compare their properties to those of neutron stars. Preliminary studies of Skyrmion stars have predicted instability to single particle decay [1] . However this was done using the hedgehog ansatz for baryon number larger than 1 which is known to lead to unstable solutions even for the usual Skyrme model. Since then, it has been shown that the Skyrme model has stable shell-like solutions [9] which can be well approximated by the so called rational map ansatz [10] .
In this paper we use the rational map ansatz and its extension to multiple shells to construct configurations in the Gravitating Skyrme model that have a very large number of baryon. We show that those configurations, contrary to the hedgehog ansatz are bound even for very large baryon numbers.
To construct configurations that have a baryon number comparable to that of neutron star, we have to introduce a further approximation, which we call the ramp ansatz. We show 2 that this anstaz introduces further errors of only a few percent and we use it to compute very large Skyrmion configurations.
The paper is organised as follow: first we outline the Einstein-Skyrme model and discuss the main features of the results on static gravitating SU (2) hedgehogs obtained by Bizon and Chmaj [1] . We then use the rational map ansatz to construct shell like gravitating multibaryon configurations and show that for a fixed value of the coupling constant, the configurations exist only when the baryon number is below a certain critical value. Finally we introduce a ramp profile approximation to construct solutions with extremely high baryon numbers. We show how accurate it is and use it to construct Skyrmion stars configuration.
The action for gravitating Skyrmions is formed from the standard Skyrme action for the matter field and the Einstein-Hilbert action for the gravitational field.
S = Z M √ -g " L Sk - R 16πG « d 4 x. ( 1
)
Here L Sk is the Lagrangian density for the Skyrme model defined on the manifold M :
L Sk = F 2 π 16
T r(∇µU ∇ µ U -1 ) + 1 32e 2 T r[(∇µU )U -1 , (∇νU )U -1 ] 2 , (
) 2
where U belongs to SU (2). As we eventually wish to study baryon stars, we take a spherically symmetric metric, such as associated with the line element ds 2 = -A 2 (r) " 1 -2m(r) r « dt 2 + " 1 -2m(r) r
« -1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), ( 3
)
where A(r) and m(r) are two profile functions that must be determined by solving the Einstein equations for the model. Our choice of ansatz is motivated by the fact that although in some cases we will be studying non-spherical Skyrmion configurations, the regime we are primarily interested in (i.e. Skyrmions of extremely high baryon number) will be shown to admit quasi spherical solutions. Also, for realistic values of the couplings, the gravitational interaction is small compared to the Skyrme interaction and thus the use of a spherical metric even with non-spherical configurations, is not a great problem.
From (3), it can be shown that the Ricci scalar is
R = -2 Ar 2 `-A ′′ r 2 -2A ′ r + 2A ′′ rm + A ′ m + 3A ′ rm ′ + Arm ′′ + 2Am ′ ´( 4
)
which, after integrating various terms by parts and noting that asymptotic flatness requires both A(r) and m(r) to take a constant value at spatial infinity, reduces the gravitational part of the action to Sgr = Z A(r) " -m ′ (r) G « dr + m(∞) G . (5) For what follows, it will be convenient to scale to dimensionless variables by defining x = eFπr and µ(x) = eFπm(r)/2, resulting in one dimensionless coupling parameter for the model, α = πF 2 π G. We note that taking Fπ = 186M ev and G = 6.72 × 10 -45 M ev -2 , then the physical value of the coupling is α = 7.3 × 10 -40 .
As the Skyrme field is an SU (2) valued scalar field, at any given time one can think of it as a map from R 3 to the SU (2) manifold. Finite energy considerations impose that the field at spatial infinity should map to the same point on SU (2), say the identity. Thus, one can simply think of the Skyrme field as a map between three-spheres. All such maps fall into disjoint homotopy classes characterised by their winding number. This winding number is a conserved topological charge because no continuous deformation of the field and thus no time evolution, can allow transitions between homotopy classes. It is this topological charge that is interpreted as the baryon number.
Gravitating Skyrmions were first studied by Bizon and Chmaj[1] who analysed the properties of static spherically symmetric gravitating SU (2) skyrmions. Taking the Hedgehog Ansatz for the Skyrme field
U = exp(i -→ σ .rF (r)) ( 6
)
subject to the boundary conditions
F (r = 0) = Bπ ( 7
) F (r = ∞) = 0 ( 8
)
where B is the Baryon number associated with the Skyrmion configuration, they derived the Euler-Lagrange equation for the profiles F (r), (A(r) and m(r) and found that the model admit two branches of global solitonic solutions at each given baryon number, which annihilate at a critical value of the coupling parameter. Above αcrit no further solutions were found. In particular the value of the critical coupling decreased quite considerably with increasing baryon number as αcrit ≈ 0.040378/B 2 . It appears that the existence of a 4 critical coupling does not signal the collapse of a Skyrmion to form a black hole. In fact the metric factor S(x) = (1 -2µ(x) x ) is non-zero at αcrit; there simply ceases to be any stationary points of the action above the critical coupling.
The major problem with the ansatz (7) is that it leads to unstable solutions, i.e. for any given value of α, MADM (B = N ) > N MADM (B = 1). This is actually the case for the pure Skyrme model as well where the hedgehog anstaz (7) with B > 1 does not correspond to the lowest energy solution for the model. The solutions of the pure Skyrme model when B > 1 are known not to be spherically symmetric [11] but are stable i.e. E(B = N ) < N * E(B = 1).
It was actually shown by Houghton et al [10] , [12] that the multi-baryon solutions of the pure Skyrme model can be well approximated by the so called rational maps ansatz which is a generalisation of the hedgehog ansatz. While not radially symmetric, the ansatz separates its radial and angular dependence through a profile function and a rational map respectively.
In the following sections we will generalise the construction of Houghton et al to approximate the solution of the Einstein-Skyrme model.
The rational map ansatz introduced by Houghton et al.[10] works by decomposing the field into angular and radial parts. Using the polar coordinates in R 3 and defining the stereographic coordinates z = tan(θ/2) exp iφ the ansatz reads [10]
U = exp (i σ • nRF (r, t)) ( 9
) where nR = 1 1 + |R| 2 `2ℜ(R), 2ℑ(R), 1 -|R| 2 ´( 10
)
is a unit vector where R is a rational function of z.
It can be shown that the baryon number for Skyrmions constructed in this way, is equal to the degree of the rational map providing we take the boundary conditions
F (r = 0) = π F (r = ∞) = 0. ( 11
)
Substituting the ansatz (9) into the action for the model and scaling to dimensionless
5
variables as earlier, we obtain the following reduced Hamiltonian
H = 16πFπ e »Z ∞ 0 A(x) " 1 2 S(x)F (x) ′2 x 2 + Bsin 2 F (x)(1 + S(x)F (x) ′2 ) + Isin 4 F (x) 2x 2 - µ ′ (x) α « dx + µ(∞) α - ( 12
)
where
S(x) = 1 - 2µ(x) x ( 13
)
From which one obtains the following field equations
µ(x) ′ = α " 1 2 S(x)x 2 F (x) ′2 + B sin 2 F (x) + S(x)BF (x) ′2 sin 2 F (x) + I sin 4 F (x) 2x 2 « ( 14
) F (x) ′′ = 1 S(x)V (x) » sin 2F (x) " B + S(x)BF (x) ′2 + I sin 2 F (x) x 2 « - αS(x)F (x) ′3 V (x) 2 x -S(x) ′ F (x) ′ V (x) -S(x)F (x) ′ V (x) ′ ˜( 15
) and A(x) ′ = αA(x)F (x) ′2 " x + 2B sin 2 F (x) x « ( 16
)
where, for convenience, we have defined V (x) as
V (x) = x 2 + 2B sin 2 F (x). ( 17
)
B is the baryon number and
I = 1 4π Z " 1 + |z| 2 1 + |R| 2 ˛dR dz ˛«4 2idzdz (1 + |z| 2 ) 2 ( 18
)
Its value depends on the chosen rational map R. To compute low energy configurations for a given baryon charge B one must find the rational map R or degree B that minimize I.
This has been done in [10] and [11] for several values of B. Moreover when b is large, one can use the approximation[11] I ≈ 1.28B 2 . The value of I so obtained is then used as a parameter and one can solve equations (14) -(16) for the radial profiles F (x), A(x) and µ(x).
We should point out here that for the pure Skyrme model the rational map ansatz produce very good approximation to the multi skyrmion solutions [10]: the energies are only 3 or 4 percent higher and the energy densities exhibit the same symmetries and differ by very little. All the solutions computed by Battye and Sutcliffe [11] , when B is not too small, have somehow the shape of a hollow shell. The baryon density is very small everywhere outside the shell, while on the shell itself, it forms a lattice of hexagons and pentagons.
Using the rational map ansatz, we will now solves the field equations (14) -(16) to compute some low action configurations. These solutions will correspond, initially, to a hollow shell of Skyrmions similar to the configuration obtained with the rational map anstaz for the pure Skyrme model. In the following sections we will show how our ansatz can be generalised to allow for more realistic configuration made out of embedded shells.
The first thing to note about our solutions is that we again obtain two branches of solutions at each baryon number (Fig. 1 ). Obtaining this same qualitative behaviour is not surprising when one considers that the B = 1 rational map Skyrmion reproduces the usual B = 1 hedgehog. However, the behaviour of the critical coupling itself is drastically altered for the rational map generated configurations. Namely, we observe that it decreases as approximately 0.040378/B 1 2 (Fig. 2 ). In particular this means that for a given value of the coupling, the rational map generated skyrmions can possess a much higher topological charge than their hedgehog counterparts, before there ceases to be any solutions. Quantitatively if B hedgehog is the maximum baryon number for which hedgehog solutions can be found at a given value of the coupling, then the highest baryon number rational map solution found at the same value of α will be approximately B 4 hedgehog . Again we observe that the metric function S(x) is non-zero at the critical coupling for all the solutions we have found and as such a horizon has not formed.
In Table 1 we present the radius, ADM mass per baryon and minimum value of the metric function, S(x), for configurations up to the maximum baryon number allowed at α = 1 × 10 -6 . These values were obtained by direct numerical solution of equations (14) -(16), where we have used the boundary data as specified in (11) .
We didn't didn't use the physical value of α (7.3×10 -40 ) because for this value, the ratio between the width of the shell and its radius is so small when we reach the maximum value of B that it becomes very difficult to solve the equation reliably. The value α = 1 × 10 -6 is small enough to allow for a shell with a large baryon number to exist but large enough to make it possible to compute these solution nears the critical value of B for a single shell configuration. 7 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 M ADM α Figure 1: Plot of the two branches of solutions found for B=2 configurations generated with the rational map ansatz.
The major difference between these configuration and the solutions of Bizon and Chmadj is that the rational map ansatz configurations become more bound when the baryon number increases This suggests the possibility that giant gravitating Skyrmions can be bound and consequently, that the Skyrme model can be used to study baryon stars.
Another interesting feature of the data is the observed change in the radius of the solutions with increasing baryon number. We note that the radius grows as approximately B 1 2 .
However there are two main deviations from this. Firstly, the constant of proportionality relating the radius to the square root of the baryon number decreases slightly but persistently as we increase the baryon number, indicating the gravitational interaction becoming more important as the number of baryons increases.
As we approach the maximum baryon charge that can exist at α = 1 × 10 -6 , we also notice that the radius of the skyrmion actually decreases as we add more baryons. This shows that the gravitation pull plays a crucial role near the critical value of the skyrmion.
This is a tantalising property when one considers that generally a neutron star's radius must decrease for an increase in mass in order to achieve sufficient degenerate neutron pressure 8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 2 4 6 8 10 12 14 16 18 20 α crit B Figure 2: Plot of the decrease in α crit with increasing baryon number, for configurations generated with the rational map ansatz. +: α c r for the minimum value of I; curve: α c r = 0.0404B -1/2 .
to support the star.
To motivate the further approximation that we will introduce in the next section, we now look at the profiles of the configuration that we have computed. First of all, we observe that the profile function F (x) stays approximately at its boundary value, π, for a finite radial distance before decreasing monotonically over some small region and finally attaining its second boundary value, 0. A similar behaviour is seen for both the mass field µ(x) and the metric field A(x) (see Fig. 3 ). Furthermore, as we increase the baryon number the structure becomes more pronounced, with the distance before the fields change (shell radius) increasing significantly, whilst the distance over which the fields change (shell width) settles to a constant size. We conclude that at large baryon numbers, those configurations correspond to hollow shells where the baryons are distributed on a tight lattice over the shell. As such the, structures are nearly spherical, validating our choice of radial metric.
Such structures immediately pose an interesting question. Can the gravitating Skyrmions exist as shells with more than one layer? To investigate this we note that it is possible to 9 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 1 0.8763 1.2315 1.0000 4 1.7728 1.1365 1.0000 8 2.5065 1.1180 1.0000 100 8.6829 1.0845 0.9999 500 19.3994 1.0827 0.9998 1 × 10 3 27.4314 1.0825 0.9997 1 × 10 4 86.7192 1.0821 0.9989 1 × 10 5 274.0397 1.0814 0.9963 1 × 10 6 864.6968 1.0792 0.9883 1 × 10 7 2715.0729 1.0722 0.9628 1 × 10 8 8377.4601 1.0500 0.88192 1 × 10 9 23585.5315 0.9743 0.6107 1.5 × 10 9 26860.2040 0.9463 0.5020 1.8 × 10 9 27470.2449 0.9302 0.4256 1.81 × 10 9 27456.5804 0.9296 0.4225 1.85 × 10 9 27357.9201 0.9274 0.4090 1.9 × 10 9 27078.6014 0.9246 0.3886 1.95 × 10 9 26126.5508 0.9217 0.3517 1.951 × 10 9 26050.7695 0.9217 0.3495 1.952 × 10 9 25937.4210 0.9216 0.3463 Table 1: Properties of the one shell low energy configuration for α = 1 × 10-6 10 0 0.5 1 1.5 2 2.5 3 3.5 4 1210 1215 1220 1225 1230 1235 F(x) x 0 2 4 6 8 10 1210 1215 1220 1225 1230 1235 µ(x) x 0.99 0.995 1 1.005 1.01 1210 1215 1220 1225 1230 1235 A(x) x Figure 3: Numerical solutions for the profiles F (x), µ(x) and A(x) when B = 2 × 10 6 and α = 1 × 10 -6 .
modify the boundary condition (11) to read
F (r = 0) = N π ( 19
) F (r = ∞) = 0 ( 20
)
whilst still ensuring that the Skyrme field is well defined at the origin. This idea was first used in [12] to construct two shell configurations for the pure Skyrme model.
The baryon charge is now N times the degree of the rational map. Fig. 4 shows the structure of the solutions we find in this case when N = 2. They suggest that the Skyrmion now exists as a N -layered structure. This is exhibited in the form of the profile, mass and metric functions which interpolate between the boundary values in N distinct steps of equal 11 0 1 2 3 4 5 6 7 1200 1205 1210 1215 1220 1225 1230 F(x) x 0 5 10 15 20 1200 1205 1210 1215 1220 1225 1230 µ(x) x 0.995 1 1.005 1.01 1.015 1.02 1200 1205 1210 1215 1220 1225 1230 A(x) x Figure 4: Numerical solutions for for the profiles F (x), µ(x) and A(x) for 2 layers configurations (F (0) = 2π) when B = 2 × 10 6 and α = 1 × 10 -6 .
size stacked next to each other.
We can therefore think of this as a naive way of constructing a gravitating Skyrmion.
Instead of using the boundary conditions as in (11) and a rational map of degree B we consider constructing the B-Skyrmion using a rational map of degree B/N (with the associated value of I) and the boundary condition (20). This is a crude construction as we are effectively considering N adjacent shells of baryons, all with the same baryon number. We might realistically expect that the baryon number per shell and distribution of shells may vary significantly for the minimum energy configuration. Nevertheless we shall study the properties of such structures. In fact, in the case where the baryon number is large and the number of shells is small, we expect this crude construction to be quite valid. That is, we do not expect the baryon number to change significantly over the few shells at large radius.
12 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 4 1.2898 1.6179 1.0000 8 1.7858 1.4072 1.0000 100 6.1754 1.1363 0.9999 1 × 10 3 19.4157 1.0913 0.9996 1 × 10 4 61.3207 1.0833 0.9985 1 × 10 5 193.7006 1.0812 0.9949 1 × 10 6 610.6271 1.0779 0.9835 1 × 10 7 1911.3704 1.0680 0.9475 1 × 10 8 5825.2626 1.0362 0.8325 9.0 × 10 8 13736.9982 0.9302 0.4258 9.7 × 10 8 13263.0853 0.9224 0.3644 9.76 × 10 8 12998.2817 0.9217 0.3480 9.764 × 10 8 12931.5189 0.9216 0.3444 9.7647 × 10 8 12895.6984 0.9216 0.3425 9.76472 × 10 8 12891.4247 0.9216 0.3423 9.764724 × 10 8 12889.8645 0.9216 0.3422 Table 2: Table of properties of double layer solutions obtained numerically at α = 1 × 10-6
For the remainder of this section we will restrict ourselves to the case where N = 2.
Table 2 summarises the properties of double layered gravitating skyrmions up to the maximum baryon charge allowed at α = 1 × 10 -6 . Briefly, we note the main features.
Firstly, for all baryon numbers, the radius of the double layered solutions is significantly less than their single layered counterparts. This is not surprising as the baryon charge exists over a thicker region and so the mean radius can decrease with the baryon density remaining the same. Secondly, when B is large enough, i.e. when the double layer starts to make sense, the double layer solutions are energetically favourable when one compares the ADM mass with the single layer solutions. Finally we note that the maximum baryon number allowed (at the given coupling) is almost twice as much in the case of the single layer skyrmions.
Of course the results of this section are not really the main regime of interest. We clearly 13 need to study configurations of extremely high baryon number (of order 10 58 ) relevant for baryon stars.
We will now discuss this high baryon number regime.
Unfortunately, at very high baryon numbers, eqns. ((14) -(16)) become difficult to handle numerically. This is largely because the radius of the solutions becomes much larger than the distance over which the fields change. That is, we need to integrate over a region which is much less than 10 -16 radius, and so even double precision data types have insufficient precision.
Moreover, single shell configurations are not physically relevant and multiple shells will only yield configuration that looks like a star if the number of layers is very large, typically well over 10 17 . With such a large number of layers we won't be able to solve the equation numerically as we will need at least 10 times as many sampling points for the profile functions.
We must thus resort to another level of approximation: approximate the profile functions by profiles that are piecewise linear. This is inspired by the work of Kopeliovich [13] [14] except that our ansatz has to be piecewise linear to be able to generate configurations with a huge number of layers. After defining the ansatz for an arbitrary number of layers, we will show that for a single layer configuration the ansatz produces configurations that are in good agreements with the rational map ansatz configuration. Then we will use the new ansatz to construct configurations that are made out of a very large no of layers.
We have shown, in the previous section, that one can construct shell like structures with very large Baryon numbers. At large baryon numbers, the Skyrmions resemble shell like structures. That is, the fields are constant nearly everywhere except in a small region corresponding to the shell. In that region, the profile look like linear functions smoothly linked to the constant parts at the edges (cfr. Fig 3 ). Motivated by this we approximate the fields by the ramp-functions
F (x) = N π 2 -(x -x0) π W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 21
) µ(x) = M 2 + (x -x0) M N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 22
) A(x) = (1 + A0) 2 + (x -x0) (1 -A0) N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 23
)
In the above there are four free parameters, namely the central radius x0 of the shell
over which the fields change, the width of the shell W , the mass field at spatial infinity M and the value of the metric field at the origin A0 such that limx→∞ = 0. N is the number of layers we wish to study and, as such, is treated as an input parameter.
The picture is of a gravitating skyrmion with very high baryon number existing as N thin layers or shell of small thickness.
The above ansatz, allow us to find an approximation to the integrated energy. To do this we use the fact that the shell width is much smaller than the radius at large baryon numbers. In particular to evaluate the action integral we can approximate expressions of the type R G(x) sin p F (x) for any function G(x) that varies very little over the width of the shell by R G(x0) sin p F (x). We then use the fact that
Z x 0 +NW/2 x 0 -NW/2 sin p F (x) = N π W Z π 0 sin p y dy. ( 24
)
This leads to the following expression for the energy:
E = - 16πFπ e » M α " 1 + A0 2 « - π 2 W » " 1 + A0 2 « " x 2 0 -M x0 + W 2 12 - M W 6 « + " 1 -A0 W « " W 2 x0 6 - M W 2 12 - M W x0 6 « - + B " 1 + A0 2 « 1 2 " M π 2 W x0 -W - π 2 W « - 3IW 16x 2 0 " 1 + A0 2 « - M α - ( 25
)
To find the configurations which minimize this energy we first minimised it with respect to A0 and M algebraically in order to find an expression for the energy as a function of the width and radius only. Then we minimised this numerically using Mathematica. We will now discuss the features of these configurations.
First of all, we must compare the results obtained with the ramp-profile when N = 1 and compare them to the result obtained with the full profile. Tables. 3 and 4 show the properties of solutions we obtained using the ramp-profile approximation, again at α = 1 × 10 -6 . All the general features of the full numerical solutions are reproduced. In particular, the approximate B 1 2 scaling and then decrease of the radius, the decreasing ADM mass and the differences between the double and single layer solutions are all exhibited by the data obtained using the ramp-profile approximation.
Quantitatively though, there are some differences. The approximation allows a significant increase in the maximum allowed baryon charge. Also, the radius of configurations obtained using the approximation, tend to be smaller than those obtained numerically. If we concentrate on the baryon numbers greater than 10 5 so as to ensure our approximation, 15 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 8.3063 3.1286 1.1023 0.9999 500 18.6031 3.1386 1.1160 0.9997 1 × 10 3 26.313 3.1397 1.1195 0.9996 1 × 10 4 83.206 3.1396 1.1254 0.9987 1 × 10 5 262.94 3.1357 1.1266 0.9960 1 × 10 6 829.60 3.1230 1.1251 0.9872 1 × 10 7 2604.2 3.0825 1.1186 0.9595 1 × 10 8 8032.8 2.9512 1.0972 0.8713 1 × 10 9 22899 2.4837 1.0272 0.5772 2 × 10 9 29121 2.1092 0.9818 0.3645 2.8 × 10 9 29098 1.6623 0.9505 0.1380 2.83 × 10 9 28514 1.6066 0.9495 0.1119 2.839 × 10 9 28024 1.5671 0.94922 0.09373 2.8397 × 10 9 27869.3 1.5556 0.94924 0.08845 2.83975 × 10 9 27869.8 1.5524 0.94925 0.08699 2.839752 × 10 9 27822 1.5521 0.94925 0.08687 Table 3: Table of properties of the single layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6 16 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 5.7924 3.0428 1.0692 0.9999 1 × 10 3 18.5788 3.1305 1.1047 0.9995 1 × 10 4 58.8202 3.1380 1.1201 0.9983 1 × 10 5 185.8420 3.1332 1.1246 0.9944 1 × 10 6 585.7950 3.1153 1.1233 0.9820 1 × 10 7 1833.0500 3.0578 1.1143 0.9428 1 × 10 8 5587.3600 2.8688 1.0840 0.8172 9 × 10 8 14147.1782 2.1859 0.9900 0.4065 9.764724 × 10 8 14472.3851 2.1276 0.9837 0.3746 1 × 10 9 14560.5000 2.1092 0.9818 0.3646 1.4 × 10 9 14549.0000 1.6623 0.9505 0.1381 1.41963 × 10 9 13994.0523 1.5644 0.9492 0.0926 1.419635134 × 10 9 13993.2000 1.5643 0.9492 0.0925 Table 4: Table of properties of the double layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6
that the width is much smaller than the radius, is valid, then at worst we find a discrepancy in the ADM mass of 11% and in the radius of 7%.
In general then, the data seems to confirm the reliability the ramp-profile approximation. In fact the approach will be even more reliable at the extremely high values of the baryon number that we are interested in. This is because the radius of solutions is of orders of magnitudes greater than the width in such a regime, consistent with the approximations we have made.
Moreover, whilst searching for minima of the energy does not allow us to probe both branches of solutions, it does allow us to locate the value of αcrit. We again obtain the approximate trend αcrit ∝ B -1 2 , for large B. Now in order to say anything about the possibility of baryon stars in the Skyrme model we need to be able to verify that the decrease in the ADM mass per baryon we observed at low and moderate baryon numbers, extends to baryon numbers of order 10 58 for α = 7.3 × 10 -40 .
Table. 5 summarizes our solutions in such a regime. Firstly we consider constructing a single layer self-gravitating Skyrmion with these values. We do indeed see that the configuration is bound. This is verified by checking that the ADM mass is lower (even at this significantly lower value of α) than for the B = 1 hedgehog. So the possibility of baryon stars in the Einstein-Skyrme model cannot be ruled out on the grounds of energy.
The Skyrmion exists as a giant thin shell, and the large baryon charge is distributed as a tight lattice over this. However a hollow shell is clearly not a realistic construction for a neutron star. This fact manifests itself in the extremely high radius of the configuration.
Transferring to standard units, the single layer B = 10 58 gravitating Skyrmion has a radius of 2.42 × 10 10 km ! To address this issue, we can use a large number of layered Skyrmions as discussed earlier. This has several benefits. Firstly, as we are distributing the baryon number through a larger volume, then at a given baryon density the necessary radius can decrease. Similar to what we see in the double layer results. On top of this, we expect the radius to decrease further due to extra gravitational compression, as the outer layers of the Skyrmions feel the attraction of inner layers. Finally, the many layer approach is also a more realistic construction of a solid baryon star.
The results for using more and more layers in the construction (for fixed B and α), are also presented in Fig. 5 . We note that not only does the radius decrease significantly, but
the added gravitational binding further improves the energies of the configurations, reflected in the low ADM masses obtained. There appears to be a critical number of layers that can be used before there ceases to be any solutions and although the value of Smin is close zero at this point, the star still has not collapsed to form a black hole. Finally, we note that the radius of the Skyrmion at the critical number of layers is approximately 20.91km. This is comparable to a real neutron star, with a typical radius of 10km.
We reemphasise here that our approach to embedding shells of baryons is quite crude.
For few shells and large baryon number, we might reasonably believe that baryon number does not chance significantly from one shell to the next. However, when we embed many shells we should really consider that the baryon number of the inner most shells would likely be significantly less than the that of the outer shells. Nevertheless, our naive embedding has produced some interesting properties. In a future work we hope to improve our multi-layer construction to obtain a more realistic description of a baryon star. 19 N Shell R( 2 Fπe ) W M ADM /(6π 2 B) S min 1 × 10 2 8.3236 × 10 27 3.1416 1.1285 1.0000 1 × 10 3 2.6321 × 10 27 3.1416 1.1285 1.0000 1 × 10 4 × 10 26 3.1416 1.1285 1.0000 1 × 10 5 2.6321 × 10 26 3.1416 1.1285 1.0000 1 × 10 6 8.3236 × 10 25 3.1416 1.1285 1.0000 1 × 10 7 2.6321 × 10 25 3.1416 1.1285 1.0000 1 × 10 8 8.3236 × 10 24 3.1416 1.1285 1.0000 1 × 10 9 2.6321 × 10 24 3.1415 1.1285 1.0000 1 × 10 10 8.3234 × 10 23 3.1415 1.1285 0.9999 1 × 10 11 2.6319 × 10 23 3.1412 1.1285 0.9997 1 × 10 12 8.3216 × 10 22 3.1402 1.1283 0.9991 1 × 10 13 2.6301 × 10 22 3.1373 1.1278 0.9971 1 × 10 14 8.3034 × 10 21 3.1280 1.1263 0.9907 1 × 10 15 2.6118 × 10 21 3.0986 1.1213 0.9705 1 × 10 16 8.1147 × 10 20 3.0037 1.1057 0.9063 1 × 10 17 2.4001 × 10 20 2.6810 1.0552 0.6977 5 × 10 17 8.2066 × 10 19 1.7888 0.9552 0.2036 5.3 × 10 17 7.4172 × 10 19 1.6227 0.9491 0.1247 5.33 × 10 17 7.1871 × 10 19 1.5625 0.94866 0.0971 5.3306 × 10 17 7.1597 × 10 19 1.5549 0.94868 0.0936 5.33065 × 10 17 7.1525 × 10 19 1.5528 0.948694 0.0927 5.330657 × 10 17 7.1506 × 10 19 1.5523 0.948692 0.0924 Table 5: Table of properties of the step ansatz configurations for varying the number of embedded shells at fixed B = 10 58 and α = 7.3 × 10 -40 .
Previous work on the Einstein-Skyrme model highlighted a considerable problem with using the Skyrmions as a model for baryon stars. Namely, multibaryon hedgehog Skyrmions were simply not energetically favourable states. We have shown that this is simply a consequence of a poor ansatz for the true Skyrmion and, having used the more appropriate rational map ansatz, we have generated energetically favourable configurations of multibaryons.
We also observe the interesting property that near the critical coupling, the Skyrmions can decrease in radius as we add more baryons. This hints towards the similar behaviour exhibited by real neutron stars.
Although the rational map ansatz does not have an exact radial symmetry, at large scale it does. The anisotropy only appears at the nucleon scale.
Finally, since we started with the motivation of studying baryon stars within the Skyrme model, it is interesting to compare the features of our configurations with those of neutron stars. For realistic values, B = 10 58 and α = 7.3 × 10 -40 we find a minimal energy single layer configuration with radius=2.42 × 10 10 km. This is clearly too large for a neutron star (which is of order 10km. in radius). This is to be expected however due to the shell model we have taken. Firstly, as we are distributing the baryons over the surface area rather than throughout the volume of the star we naturally must require a much larger star for a given baryon number. This effect is two-fold in that if we were distributing the baryons throughout the volume, outer layers would feel the attraction of inner layers and enhanced radial compression would occur. The loss of such an effect is pronounced when we are considering realistically small values of the coupling.
It seems therefore that the way to construct baryon stars in the Skyrme model is to consider embedding shells of baryons within shells. This gives rise to more appropriate specifications for the star and is also more realistic. We do indeed observe such improvements for a many layered configuration. In fact the radius of B = 10 58 gravitating Skyrmion (at realistic α), can be decreased in this manner to approximately 20.91km.
We note however that this approach to shell embedding has only be done naively thus far.
We have only considered the case where the baryon number is equal for each shell. We really should allow the baryon number(and hence the rational map quantities) to vary over the shells. One approach towards this would be to assume that the baryon density is a constant over the shells. An even better approach would be to allow this to be a smoothly varying function that must be determined by minimising the energy. This will give a more realistic 21 description of baryon stars within the Einstein-Skyrme model, as traditional descriptions of neutron stars also involve many strata, of differing neutron density. We are currently investigating such configurations.
[13] V. B. Kopeliovich "The Bubbles of Matter from MultiSkyrmions" JETP Lett. 73 (2001), 587-591; Pisma Zh.Eksp.Teor.Fiz. 73 (2001), 667-671 [14] V. B. Kopeliovich "MultiSkyrmions and Baryonic Bags" J.Phys. G28 (2002), 103-120 23
GIP is supported by a PPARC studentship. | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in a series of papers as a non-linear field theory of pions [2] , [3] . Skyrme's initial idea was to think of baryons (in particular the nucleons) as secondary structures arising from a more fundamental mesonic fluid. The key property of the model was that the baryons arose as solitons in a topological manner and thus possessed a conserved topological charge identified with the baryon number.\n\nThe lowest energy stable solutions of the model are termed Skyrmions and can be thought of as baryonic solitons. The Skyrme model has been very successful in modelling the structures of various nuclei and has been shown by Witten et al. [4] to possess the general features of a low energy effective field theory for QCD. Some studies of the Skyrme model coupled to gravity have previously been undertaken [1], [5] , [6] , mainly with the motivation of a comparison of its features with those of other nonlinear field theories coupled to gravity. Of particular note is the Einstein-Yang-Mills theory, in which gravitationally bound configurations of non-abelian gauge fields are produced.\n\nOther reasons for studying the Einstein-Skyrme model are cosmological and astrophysical ones. Various authors have studied black hole formation in the model, with the conclusion that the so-called no-hair conjecture may not hold [7] , [8] .\n\nThe purpose of this paper is to study large baryon number Skyrmions or configurations of Skyrmions in the Einstein-Skyrme model. In particular, we wish to investigate if stable solitonic stars could exist within the model and to compare their properties to those of neutron stars. Preliminary studies of Skyrmion stars have predicted instability to single particle decay [1] . However this was done using the hedgehog ansatz for baryon number larger than 1 which is known to lead to unstable solutions even for the usual Skyrme model. Since then, it has been shown that the Skyrme model has stable shell-like solutions [9] which can be well approximated by the so called rational map ansatz [10] .\n\nIn this paper we use the rational map ansatz and its extension to multiple shells to construct configurations in the Gravitating Skyrme model that have a very large number of baryon. We show that those configurations, contrary to the hedgehog ansatz are bound even for very large baryon numbers.\n\nTo construct configurations that have a baryon number comparable to that of neutron star, we have to introduce a further approximation, which we call the ramp ansatz. We show 2 that this anstaz introduces further errors of only a few percent and we use it to compute very large Skyrmion configurations.\n\nThe paper is organised as follow: first we outline the Einstein-Skyrme model and discuss the main features of the results on static gravitating SU (2) hedgehogs obtained by Bizon and Chmaj [1] . We then use the rational map ansatz to construct shell like gravitating multibaryon configurations and show that for a fixed value of the coupling constant, the configurations exist only when the baryon number is below a certain critical value. Finally we introduce a ramp profile approximation to construct solutions with extremely high baryon numbers. We show how accurate it is and use it to construct Skyrmion stars configuration."
},
{
"section_type": "OTHER",
"section_title": "The Einstein-Skyrme Model",
"text": "The action for gravitating Skyrmions is formed from the standard Skyrme action for the matter field and the Einstein-Hilbert action for the gravitational field.\n\nS = Z M √ -g \" L Sk - R 16πG « d 4 x. ( 1\n\n)\n\nHere L Sk is the Lagrangian density for the Skyrme model defined on the manifold M :\n\nL Sk = F 2 π 16\n\nT r(∇µU ∇ µ U -1 ) + 1 32e 2 T r[(∇µU )U -1 , (∇νU )U -1 ] 2 , (\n\n) 2\n\nwhere U belongs to SU (2). As we eventually wish to study baryon stars, we take a spherically symmetric metric, such as associated with the line element ds 2 = -A 2 (r) \" 1 -2m(r) r « dt 2 + \" 1 -2m(r) r\n\n« -1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), ( 3\n\n)\n\nwhere A(r) and m(r) are two profile functions that must be determined by solving the Einstein equations for the model. Our choice of ansatz is motivated by the fact that although in some cases we will be studying non-spherical Skyrmion configurations, the regime we are primarily interested in (i.e. Skyrmions of extremely high baryon number) will be shown to admit quasi spherical solutions. Also, for realistic values of the couplings, the gravitational interaction is small compared to the Skyrme interaction and thus the use of a spherical metric even with non-spherical configurations, is not a great problem.\n\nFrom (3), it can be shown that the Ricci scalar is\n\nR = -2 Ar 2 `-A ′′ r 2 -2A ′ r + 2A ′′ rm + A ′ m + 3A ′ rm ′ + Arm ′′ + 2Am ′ ´( 4\n\n)\n\nwhich, after integrating various terms by parts and noting that asymptotic flatness requires both A(r) and m(r) to take a constant value at spatial infinity, reduces the gravitational part of the action to Sgr = Z A(r) \" -m ′ (r) G « dr + m(∞) G . (5) For what follows, it will be convenient to scale to dimensionless variables by defining x = eFπr and µ(x) = eFπm(r)/2, resulting in one dimensionless coupling parameter for the model, α = πF 2 π G. We note that taking Fπ = 186M ev and G = 6.72 × 10 -45 M ev -2 , then the physical value of the coupling is α = 7.3 × 10 -40 .\n\nAs the Skyrme field is an SU (2) valued scalar field, at any given time one can think of it as a map from R 3 to the SU (2) manifold. Finite energy considerations impose that the field at spatial infinity should map to the same point on SU (2), say the identity. Thus, one can simply think of the Skyrme field as a map between three-spheres. All such maps fall into disjoint homotopy classes characterised by their winding number. This winding number is a conserved topological charge because no continuous deformation of the field and thus no time evolution, can allow transitions between homotopy classes. It is this topological charge that is interpreted as the baryon number."
},
{
"section_type": "OTHER",
"section_title": "Gravitating Hedgehog Skyrmions",
"text": "Gravitating Skyrmions were first studied by Bizon and Chmaj[1] who analysed the properties of static spherically symmetric gravitating SU (2) skyrmions. Taking the Hedgehog Ansatz for the Skyrme field\n\nU = exp(i -→ σ .rF (r)) ( 6\n\n)\n\nsubject to the boundary conditions\n\nF (r = 0) = Bπ ( 7\n\n) F (r = ∞) = 0 ( 8\n\n)\n\nwhere B is the Baryon number associated with the Skyrmion configuration, they derived the Euler-Lagrange equation for the profiles F (r), (A(r) and m(r) and found that the model admit two branches of global solitonic solutions at each given baryon number, which annihilate at a critical value of the coupling parameter. Above αcrit no further solutions were found. In particular the value of the critical coupling decreased quite considerably with increasing baryon number as αcrit ≈ 0.040378/B 2 . It appears that the existence of a 4 critical coupling does not signal the collapse of a Skyrmion to form a black hole. In fact the metric factor S(x) = (1 -2µ(x) x ) is non-zero at αcrit; there simply ceases to be any stationary points of the action above the critical coupling.\n\nThe major problem with the ansatz (7) is that it leads to unstable solutions, i.e. for any given value of α, MADM (B = N ) > N MADM (B = 1). This is actually the case for the pure Skyrme model as well where the hedgehog anstaz (7) with B > 1 does not correspond to the lowest energy solution for the model. The solutions of the pure Skyrme model when B > 1 are known not to be spherically symmetric [11] but are stable i.e. E(B = N ) < N * E(B = 1).\n\nIt was actually shown by Houghton et al [10] , [12] that the multi-baryon solutions of the pure Skyrme model can be well approximated by the so called rational maps ansatz which is a generalisation of the hedgehog ansatz. While not radially symmetric, the ansatz separates its radial and angular dependence through a profile function and a rational map respectively.\n\nIn the following sections we will generalise the construction of Houghton et al to approximate the solution of the Einstein-Skyrme model."
},
{
"section_type": "OTHER",
"section_title": "The Rational Map Ansatz",
"text": "The rational map ansatz introduced by Houghton et al.[10] works by decomposing the field into angular and radial parts. Using the polar coordinates in R 3 and defining the stereographic coordinates z = tan(θ/2) exp iφ the ansatz reads [10]\n\nU = exp (i σ • nRF (r, t)) ( 9\n\n) where nR = 1 1 + |R| 2 `2ℜ(R), 2ℑ(R), 1 -|R| 2 ´( 10\n\n)\n\nis a unit vector where R is a rational function of z.\n\nIt can be shown that the baryon number for Skyrmions constructed in this way, is equal to the degree of the rational map providing we take the boundary conditions\n\nF (r = 0) = π F (r = ∞) = 0. ( 11\n\n)\n\nSubstituting the ansatz (9) into the action for the model and scaling to dimensionless\n\n5\n\nvariables as earlier, we obtain the following reduced Hamiltonian\n\nH = 16πFπ e »Z ∞ 0 A(x) \" 1 2 S(x)F (x) ′2 x 2 + Bsin 2 F (x)(1 + S(x)F (x) ′2 ) + Isin 4 F (x) 2x 2 - µ ′ (x) α « dx + µ(∞) α - ( 12\n\n)\n\nwhere\n\nS(x) = 1 - 2µ(x) x ( 13\n\n)\n\nFrom which one obtains the following field equations\n\nµ(x) ′ = α \" 1 2 S(x)x 2 F (x) ′2 + B sin 2 F (x) + S(x)BF (x) ′2 sin 2 F (x) + I sin 4 F (x) 2x 2 « ( 14\n\n) F (x) ′′ = 1 S(x)V (x) » sin 2F (x) \" B + S(x)BF (x) ′2 + I sin 2 F (x) x 2 « - αS(x)F (x) ′3 V (x) 2 x -S(x) ′ F (x) ′ V (x) -S(x)F (x) ′ V (x) ′ ˜( 15\n\n) and A(x) ′ = αA(x)F (x) ′2 \" x + 2B sin 2 F (x) x « ( 16\n\n)\n\nwhere, for convenience, we have defined V (x) as\n\nV (x) = x 2 + 2B sin 2 F (x). ( 17\n\n)\n\nB is the baryon number and\n\nI = 1 4π Z \" 1 + |z| 2 1 + |R| 2 ˛dR dz ˛«4 2idzdz (1 + |z| 2 ) 2 ( 18\n\n)\n\nIts value depends on the chosen rational map R. To compute low energy configurations for a given baryon charge B one must find the rational map R or degree B that minimize I.\n\nThis has been done in [10] and [11] for several values of B. Moreover when b is large, one can use the approximation[11] I ≈ 1.28B 2 . The value of I so obtained is then used as a parameter and one can solve equations (14) -(16) for the radial profiles F (x), A(x) and µ(x).\n\nWe should point out here that for the pure Skyrme model the rational map ansatz produce very good approximation to the multi skyrmion solutions [10]: the energies are only 3 or 4 percent higher and the energy densities exhibit the same symmetries and differ by very little. All the solutions computed by Battye and Sutcliffe [11] , when B is not too small, have somehow the shape of a hollow shell. The baryon density is very small everywhere outside the shell, while on the shell itself, it forms a lattice of hexagons and pentagons."
},
{
"section_type": "OTHER",
"section_title": "5 Hollow Skyrmion Shells",
"text": "Using the rational map ansatz, we will now solves the field equations (14) -(16) to compute some low action configurations. These solutions will correspond, initially, to a hollow shell of Skyrmions similar to the configuration obtained with the rational map anstaz for the pure Skyrme model. In the following sections we will show how our ansatz can be generalised to allow for more realistic configuration made out of embedded shells.\n\nThe first thing to note about our solutions is that we again obtain two branches of solutions at each baryon number (Fig. 1 ). Obtaining this same qualitative behaviour is not surprising when one considers that the B = 1 rational map Skyrmion reproduces the usual B = 1 hedgehog. However, the behaviour of the critical coupling itself is drastically altered for the rational map generated configurations. Namely, we observe that it decreases as approximately 0.040378/B 1 2 (Fig. 2 ). In particular this means that for a given value of the coupling, the rational map generated skyrmions can possess a much higher topological charge than their hedgehog counterparts, before there ceases to be any solutions. Quantitatively if B hedgehog is the maximum baryon number for which hedgehog solutions can be found at a given value of the coupling, then the highest baryon number rational map solution found at the same value of α will be approximately B 4 hedgehog . Again we observe that the metric function S(x) is non-zero at the critical coupling for all the solutions we have found and as such a horizon has not formed.\n\nIn Table 1 we present the radius, ADM mass per baryon and minimum value of the metric function, S(x), for configurations up to the maximum baryon number allowed at α = 1 × 10 -6 . These values were obtained by direct numerical solution of equations (14) -(16), where we have used the boundary data as specified in (11) .\n\nWe didn't didn't use the physical value of α (7.3×10 -40 ) because for this value, the ratio between the width of the shell and its radius is so small when we reach the maximum value of B that it becomes very difficult to solve the equation reliably. The value α = 1 × 10 -6 is small enough to allow for a shell with a large baryon number to exist but large enough to make it possible to compute these solution nears the critical value of B for a single shell configuration. 7 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 M ADM α Figure 1: Plot of the two branches of solutions found for B=2 configurations generated with the rational map ansatz.\n\nThe major difference between these configuration and the solutions of Bizon and Chmadj is that the rational map ansatz configurations become more bound when the baryon number increases This suggests the possibility that giant gravitating Skyrmions can be bound and consequently, that the Skyrme model can be used to study baryon stars.\n\nAnother interesting feature of the data is the observed change in the radius of the solutions with increasing baryon number. We note that the radius grows as approximately B 1 2 .\n\nHowever there are two main deviations from this. Firstly, the constant of proportionality relating the radius to the square root of the baryon number decreases slightly but persistently as we increase the baryon number, indicating the gravitational interaction becoming more important as the number of baryons increases.\n\nAs we approach the maximum baryon charge that can exist at α = 1 × 10 -6 , we also notice that the radius of the skyrmion actually decreases as we add more baryons. This shows that the gravitation pull plays a crucial role near the critical value of the skyrmion.\n\nThis is a tantalising property when one considers that generally a neutron star's radius must decrease for an increase in mass in order to achieve sufficient degenerate neutron pressure 8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 2 4 6 8 10 12 14 16 18 20 α crit B Figure 2: Plot of the decrease in α crit with increasing baryon number, for configurations generated with the rational map ansatz. +: α c r for the minimum value of I; curve: α c r = 0.0404B -1/2 .\n\nto support the star.\n\nTo motivate the further approximation that we will introduce in the next section, we now look at the profiles of the configuration that we have computed. First of all, we observe that the profile function F (x) stays approximately at its boundary value, π, for a finite radial distance before decreasing monotonically over some small region and finally attaining its second boundary value, 0. A similar behaviour is seen for both the mass field µ(x) and the metric field A(x) (see Fig. 3 ). Furthermore, as we increase the baryon number the structure becomes more pronounced, with the distance before the fields change (shell radius) increasing significantly, whilst the distance over which the fields change (shell width) settles to a constant size. We conclude that at large baryon numbers, those configurations correspond to hollow shells where the baryons are distributed on a tight lattice over the shell. As such the, structures are nearly spherical, validating our choice of radial metric.\n\nSuch structures immediately pose an interesting question. Can the gravitating Skyrmions exist as shells with more than one layer? To investigate this we note that it is possible to 9 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 1 0.8763 1.2315 1.0000 4 1.7728 1.1365 1.0000 8 2.5065 1.1180 1.0000 100 8.6829 1.0845 0.9999 500 19.3994 1.0827 0.9998 1 × 10 3 27.4314 1.0825 0.9997 1 × 10 4 86.7192 1.0821 0.9989 1 × 10 5 274.0397 1.0814 0.9963 1 × 10 6 864.6968 1.0792 0.9883 1 × 10 7 2715.0729 1.0722 0.9628 1 × 10 8 8377.4601 1.0500 0.88192 1 × 10 9 23585.5315 0.9743 0.6107 1.5 × 10 9 26860.2040 0.9463 0.5020 1.8 × 10 9 27470.2449 0.9302 0.4256 1.81 × 10 9 27456.5804 0.9296 0.4225 1.85 × 10 9 27357.9201 0.9274 0.4090 1.9 × 10 9 27078.6014 0.9246 0.3886 1.95 × 10 9 26126.5508 0.9217 0.3517 1.951 × 10 9 26050.7695 0.9217 0.3495 1.952 × 10 9 25937.4210 0.9216 0.3463 Table 1: Properties of the one shell low energy configuration for α = 1 × 10-6 10 0 0.5 1 1.5 2 2.5 3 3.5 4 1210 1215 1220 1225 1230 1235 F(x) x 0 2 4 6 8 10 1210 1215 1220 1225 1230 1235 µ(x) x 0.99 0.995 1 1.005 1.01 1210 1215 1220 1225 1230 1235 A(x) x Figure 3: Numerical solutions for the profiles F (x), µ(x) and A(x) when B = 2 × 10 6 and α = 1 × 10 -6 .\n\nmodify the boundary condition (11) to read\n\nF (r = 0) = N π ( 19\n\n) F (r = ∞) = 0 ( 20\n\n)\n\nwhilst still ensuring that the Skyrme field is well defined at the origin. This idea was first used in [12] to construct two shell configurations for the pure Skyrme model.\n\nThe baryon charge is now N times the degree of the rational map. Fig. 4 shows the structure of the solutions we find in this case when N = 2. They suggest that the Skyrmion now exists as a N -layered structure. This is exhibited in the form of the profile, mass and metric functions which interpolate between the boundary values in N distinct steps of equal 11 0 1 2 3 4 5 6 7 1200 1205 1210 1215 1220 1225 1230 F(x) x 0 5 10 15 20 1200 1205 1210 1215 1220 1225 1230 µ(x) x 0.995 1 1.005 1.01 1.015 1.02 1200 1205 1210 1215 1220 1225 1230 A(x) x Figure 4: Numerical solutions for for the profiles F (x), µ(x) and A(x) for 2 layers configurations (F (0) = 2π) when B = 2 × 10 6 and α = 1 × 10 -6 .\n\nsize stacked next to each other.\n\nWe can therefore think of this as a naive way of constructing a gravitating Skyrmion.\n\nInstead of using the boundary conditions as in (11) and a rational map of degree B we consider constructing the B-Skyrmion using a rational map of degree B/N (with the associated value of I) and the boundary condition (20). This is a crude construction as we are effectively considering N adjacent shells of baryons, all with the same baryon number. We might realistically expect that the baryon number per shell and distribution of shells may vary significantly for the minimum energy configuration. Nevertheless we shall study the properties of such structures. In fact, in the case where the baryon number is large and the number of shells is small, we expect this crude construction to be quite valid. That is, we do not expect the baryon number to change significantly over the few shells at large radius.\n\n12 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 4 1.2898 1.6179 1.0000 8 1.7858 1.4072 1.0000 100 6.1754 1.1363 0.9999 1 × 10 3 19.4157 1.0913 0.9996 1 × 10 4 61.3207 1.0833 0.9985 1 × 10 5 193.7006 1.0812 0.9949 1 × 10 6 610.6271 1.0779 0.9835 1 × 10 7 1911.3704 1.0680 0.9475 1 × 10 8 5825.2626 1.0362 0.8325 9.0 × 10 8 13736.9982 0.9302 0.4258 9.7 × 10 8 13263.0853 0.9224 0.3644 9.76 × 10 8 12998.2817 0.9217 0.3480 9.764 × 10 8 12931.5189 0.9216 0.3444 9.7647 × 10 8 12895.6984 0.9216 0.3425 9.76472 × 10 8 12891.4247 0.9216 0.3423 9.764724 × 10 8 12889.8645 0.9216 0.3422 Table 2: Table of properties of double layer solutions obtained numerically at α = 1 × 10-6\n\nFor the remainder of this section we will restrict ourselves to the case where N = 2.\n\nTable 2 summarises the properties of double layered gravitating skyrmions up to the maximum baryon charge allowed at α = 1 × 10 -6 . Briefly, we note the main features.\n\nFirstly, for all baryon numbers, the radius of the double layered solutions is significantly less than their single layered counterparts. This is not surprising as the baryon charge exists over a thicker region and so the mean radius can decrease with the baryon density remaining the same. Secondly, when B is large enough, i.e. when the double layer starts to make sense, the double layer solutions are energetically favourable when one compares the ADM mass with the single layer solutions. Finally we note that the maximum baryon number allowed (at the given coupling) is almost twice as much in the case of the single layer skyrmions.\n\nOf course the results of this section are not really the main regime of interest. We clearly 13 need to study configurations of extremely high baryon number (of order 10 58 ) relevant for baryon stars.\n\nWe will now discuss this high baryon number regime."
},
{
"section_type": "OTHER",
"section_title": "The Ramp-profile Approximation",
"text": "Unfortunately, at very high baryon numbers, eqns. ((14) -(16)) become difficult to handle numerically. This is largely because the radius of the solutions becomes much larger than the distance over which the fields change. That is, we need to integrate over a region which is much less than 10 -16 radius, and so even double precision data types have insufficient precision.\n\nMoreover, single shell configurations are not physically relevant and multiple shells will only yield configuration that looks like a star if the number of layers is very large, typically well over 10 17 . With such a large number of layers we won't be able to solve the equation numerically as we will need at least 10 times as many sampling points for the profile functions.\n\nWe must thus resort to another level of approximation: approximate the profile functions by profiles that are piecewise linear. This is inspired by the work of Kopeliovich [13] [14] except that our ansatz has to be piecewise linear to be able to generate configurations with a huge number of layers. After defining the ansatz for an arbitrary number of layers, we will show that for a single layer configuration the ansatz produces configurations that are in good agreements with the rational map ansatz configuration. Then we will use the new ansatz to construct configurations that are made out of a very large no of layers.\n\nWe have shown, in the previous section, that one can construct shell like structures with very large Baryon numbers. At large baryon numbers, the Skyrmions resemble shell like structures. That is, the fields are constant nearly everywhere except in a small region corresponding to the shell. In that region, the profile look like linear functions smoothly linked to the constant parts at the edges (cfr. Fig 3 ). Motivated by this we approximate the fields by the ramp-functions\n\nF (x) = N π 2 -(x -x0) π W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 21\n\n) µ(x) = M 2 + (x -x0) M N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 22\n\n) A(x) = (1 + A0) 2 + (x -x0) (1 -A0) N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 23\n\n)\n\nIn the above there are four free parameters, namely the central radius x0 of the shell\n\nover which the fields change, the width of the shell W , the mass field at spatial infinity M and the value of the metric field at the origin A0 such that limx→∞ = 0. N is the number of layers we wish to study and, as such, is treated as an input parameter.\n\nThe picture is of a gravitating skyrmion with very high baryon number existing as N thin layers or shell of small thickness.\n\nThe above ansatz, allow us to find an approximation to the integrated energy. To do this we use the fact that the shell width is much smaller than the radius at large baryon numbers. In particular to evaluate the action integral we can approximate expressions of the type R G(x) sin p F (x) for any function G(x) that varies very little over the width of the shell by R G(x0) sin p F (x). We then use the fact that\n\nZ x 0 +NW/2 x 0 -NW/2 sin p F (x) = N π W Z π 0 sin p y dy. ( 24\n\n)\n\nThis leads to the following expression for the energy:\n\nE = - 16πFπ e » M α \" 1 + A0 2 « - π 2 W » \" 1 + A0 2 « \" x 2 0 -M x0 + W 2 12 - M W 6 « + \" 1 -A0 W « \" W 2 x0 6 - M W 2 12 - M W x0 6 « - + B \" 1 + A0 2 « 1 2 \" M π 2 W x0 -W - π 2 W « - 3IW 16x 2 0 \" 1 + A0 2 « - M α - ( 25\n\n)\n\nTo find the configurations which minimize this energy we first minimised it with respect to A0 and M algebraically in order to find an expression for the energy as a function of the width and radius only. Then we minimised this numerically using Mathematica. We will now discuss the features of these configurations.\n\nFirst of all, we must compare the results obtained with the ramp-profile when N = 1 and compare them to the result obtained with the full profile. Tables. 3 and 4 show the properties of solutions we obtained using the ramp-profile approximation, again at α = 1 × 10 -6 . All the general features of the full numerical solutions are reproduced. In particular, the approximate B 1 2 scaling and then decrease of the radius, the decreasing ADM mass and the differences between the double and single layer solutions are all exhibited by the data obtained using the ramp-profile approximation.\n\nQuantitatively though, there are some differences. The approximation allows a significant increase in the maximum allowed baryon charge. Also, the radius of configurations obtained using the approximation, tend to be smaller than those obtained numerically. If we concentrate on the baryon numbers greater than 10 5 so as to ensure our approximation, 15 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 8.3063 3.1286 1.1023 0.9999 500 18.6031 3.1386 1.1160 0.9997 1 × 10 3 26.313 3.1397 1.1195 0.9996 1 × 10 4 83.206 3.1396 1.1254 0.9987 1 × 10 5 262.94 3.1357 1.1266 0.9960 1 × 10 6 829.60 3.1230 1.1251 0.9872 1 × 10 7 2604.2 3.0825 1.1186 0.9595 1 × 10 8 8032.8 2.9512 1.0972 0.8713 1 × 10 9 22899 2.4837 1.0272 0.5772 2 × 10 9 29121 2.1092 0.9818 0.3645 2.8 × 10 9 29098 1.6623 0.9505 0.1380 2.83 × 10 9 28514 1.6066 0.9495 0.1119 2.839 × 10 9 28024 1.5671 0.94922 0.09373 2.8397 × 10 9 27869.3 1.5556 0.94924 0.08845 2.83975 × 10 9 27869.8 1.5524 0.94925 0.08699 2.839752 × 10 9 27822 1.5521 0.94925 0.08687 Table 3: Table of properties of the single layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6 16 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 5.7924 3.0428 1.0692 0.9999 1 × 10 3 18.5788 3.1305 1.1047 0.9995 1 × 10 4 58.8202 3.1380 1.1201 0.9983 1 × 10 5 185.8420 3.1332 1.1246 0.9944 1 × 10 6 585.7950 3.1153 1.1233 0.9820 1 × 10 7 1833.0500 3.0578 1.1143 0.9428 1 × 10 8 5587.3600 2.8688 1.0840 0.8172 9 × 10 8 14147.1782 2.1859 0.9900 0.4065 9.764724 × 10 8 14472.3851 2.1276 0.9837 0.3746 1 × 10 9 14560.5000 2.1092 0.9818 0.3646 1.4 × 10 9 14549.0000 1.6623 0.9505 0.1381 1.41963 × 10 9 13994.0523 1.5644 0.9492 0.0926 1.419635134 × 10 9 13993.2000 1.5643 0.9492 0.0925 Table 4: Table of properties of the double layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6\n\nthat the width is much smaller than the radius, is valid, then at worst we find a discrepancy in the ADM mass of 11% and in the radius of 7%.\n\nIn general then, the data seems to confirm the reliability the ramp-profile approximation. In fact the approach will be even more reliable at the extremely high values of the baryon number that we are interested in. This is because the radius of solutions is of orders of magnitudes greater than the width in such a regime, consistent with the approximations we have made.\n\nMoreover, whilst searching for minima of the energy does not allow us to probe both branches of solutions, it does allow us to locate the value of αcrit. We again obtain the approximate trend αcrit ∝ B -1 2 , for large B. Now in order to say anything about the possibility of baryon stars in the Skyrme model we need to be able to verify that the decrease in the ADM mass per baryon we observed at low and moderate baryon numbers, extends to baryon numbers of order 10 58 for α = 7.3 × 10 -40 .\n\nTable. 5 summarizes our solutions in such a regime. Firstly we consider constructing a single layer self-gravitating Skyrmion with these values. We do indeed see that the configuration is bound. This is verified by checking that the ADM mass is lower (even at this significantly lower value of α) than for the B = 1 hedgehog. So the possibility of baryon stars in the Einstein-Skyrme model cannot be ruled out on the grounds of energy.\n\nThe Skyrmion exists as a giant thin shell, and the large baryon charge is distributed as a tight lattice over this. However a hollow shell is clearly not a realistic construction for a neutron star. This fact manifests itself in the extremely high radius of the configuration.\n\nTransferring to standard units, the single layer B = 10 58 gravitating Skyrmion has a radius of 2.42 × 10 10 km ! To address this issue, we can use a large number of layered Skyrmions as discussed earlier. This has several benefits. Firstly, as we are distributing the baryon number through a larger volume, then at a given baryon density the necessary radius can decrease. Similar to what we see in the double layer results. On top of this, we expect the radius to decrease further due to extra gravitational compression, as the outer layers of the Skyrmions feel the attraction of inner layers. Finally, the many layer approach is also a more realistic construction of a solid baryon star.\n\nThe results for using more and more layers in the construction (for fixed B and α), are also presented in Fig. 5 . We note that not only does the radius decrease significantly, but\n\nthe added gravitational binding further improves the energies of the configurations, reflected in the low ADM masses obtained. There appears to be a critical number of layers that can be used before there ceases to be any solutions and although the value of Smin is close zero at this point, the star still has not collapsed to form a black hole. Finally, we note that the radius of the Skyrmion at the critical number of layers is approximately 20.91km. This is comparable to a real neutron star, with a typical radius of 10km.\n\nWe reemphasise here that our approach to embedding shells of baryons is quite crude.\n\nFor few shells and large baryon number, we might reasonably believe that baryon number does not chance significantly from one shell to the next. However, when we embed many shells we should really consider that the baryon number of the inner most shells would likely be significantly less than the that of the outer shells. Nevertheless, our naive embedding has produced some interesting properties. In a future work we hope to improve our multi-layer construction to obtain a more realistic description of a baryon star. 19 N Shell R( 2 Fπe ) W M ADM /(6π 2 B) S min 1 × 10 2 8.3236 × 10 27 3.1416 1.1285 1.0000 1 × 10 3 2.6321 × 10 27 3.1416 1.1285 1.0000 1 × 10 4 × 10 26 3.1416 1.1285 1.0000 1 × 10 5 2.6321 × 10 26 3.1416 1.1285 1.0000 1 × 10 6 8.3236 × 10 25 3.1416 1.1285 1.0000 1 × 10 7 2.6321 × 10 25 3.1416 1.1285 1.0000 1 × 10 8 8.3236 × 10 24 3.1416 1.1285 1.0000 1 × 10 9 2.6321 × 10 24 3.1415 1.1285 1.0000 1 × 10 10 8.3234 × 10 23 3.1415 1.1285 0.9999 1 × 10 11 2.6319 × 10 23 3.1412 1.1285 0.9997 1 × 10 12 8.3216 × 10 22 3.1402 1.1283 0.9991 1 × 10 13 2.6301 × 10 22 3.1373 1.1278 0.9971 1 × 10 14 8.3034 × 10 21 3.1280 1.1263 0.9907 1 × 10 15 2.6118 × 10 21 3.0986 1.1213 0.9705 1 × 10 16 8.1147 × 10 20 3.0037 1.1057 0.9063 1 × 10 17 2.4001 × 10 20 2.6810 1.0552 0.6977 5 × 10 17 8.2066 × 10 19 1.7888 0.9552 0.2036 5.3 × 10 17 7.4172 × 10 19 1.6227 0.9491 0.1247 5.33 × 10 17 7.1871 × 10 19 1.5625 0.94866 0.0971 5.3306 × 10 17 7.1597 × 10 19 1.5549 0.94868 0.0936 5.33065 × 10 17 7.1525 × 10 19 1.5528 0.948694 0.0927 5.330657 × 10 17 7.1506 × 10 19 1.5523 0.948692 0.0924 Table 5: Table of properties of the step ansatz configurations for varying the number of embedded shells at fixed B = 10 58 and α = 7.3 × 10 -40 ."
},
{
"section_type": "CONCLUSION",
"section_title": "7 Conclusions - Acknowledgements",
"text": "Previous work on the Einstein-Skyrme model highlighted a considerable problem with using the Skyrmions as a model for baryon stars. Namely, multibaryon hedgehog Skyrmions were simply not energetically favourable states. We have shown that this is simply a consequence of a poor ansatz for the true Skyrmion and, having used the more appropriate rational map ansatz, we have generated energetically favourable configurations of multibaryons.\n\nWe also observe the interesting property that near the critical coupling, the Skyrmions can decrease in radius as we add more baryons. This hints towards the similar behaviour exhibited by real neutron stars.\n\nAlthough the rational map ansatz does not have an exact radial symmetry, at large scale it does. The anisotropy only appears at the nucleon scale.\n\nFinally, since we started with the motivation of studying baryon stars within the Skyrme model, it is interesting to compare the features of our configurations with those of neutron stars. For realistic values, B = 10 58 and α = 7.3 × 10 -40 we find a minimal energy single layer configuration with radius=2.42 × 10 10 km. This is clearly too large for a neutron star (which is of order 10km. in radius). This is to be expected however due to the shell model we have taken. Firstly, as we are distributing the baryons over the surface area rather than throughout the volume of the star we naturally must require a much larger star for a given baryon number. This effect is two-fold in that if we were distributing the baryons throughout the volume, outer layers would feel the attraction of inner layers and enhanced radial compression would occur. The loss of such an effect is pronounced when we are considering realistically small values of the coupling.\n\nIt seems therefore that the way to construct baryon stars in the Skyrme model is to consider embedding shells of baryons within shells. This gives rise to more appropriate specifications for the star and is also more realistic. We do indeed observe such improvements for a many layered configuration. In fact the radius of B = 10 58 gravitating Skyrmion (at realistic α), can be decreased in this manner to approximately 20.91km.\n\nWe note however that this approach to shell embedding has only be done naively thus far.\n\nWe have only considered the case where the baryon number is equal for each shell. We really should allow the baryon number(and hence the rational map quantities) to vary over the shells. One approach towards this would be to assume that the baryon density is a constant over the shells. An even better approach would be to allow this to be a smoothly varying function that must be determined by minimising the energy. This will give a more realistic 21 description of baryon stars within the Einstein-Skyrme model, as traditional descriptions of neutron stars also involve many strata, of differing neutron density. We are currently investigating such configurations.\n\n[13] V. B. Kopeliovich \"The Bubbles of Matter from MultiSkyrmions\" JETP Lett. 73 (2001), 587-591; Pisma Zh.Eksp.Teor.Fiz. 73 (2001), 667-671 [14] V. B. Kopeliovich \"MultiSkyrmions and Baryonic Bags\" J.Phys. G28 (2002), 103-120 23\n\nGIP is supported by a PPARC studentship."
}
] | |
arxiv:0704.0530 | 0704.0530 | 1 | 10.1088/1126-6708/2007/06/065 | e09094677ad52079a512d808ae2511bbf4a65c00c4c0f9afba29ba8ee99b8a1e | Noncommutative Solitons in a Supersymmetric Chiral Model in 2+1
Dimensions | We consider a supersymmetric Bogomolny-type model in 2+1 dimensions
originating from twistor string theory. By a gauge fixing this model is reduced
to a modified U(n) chiral model with N<=8 supersymmetries in 2+1 dimensions.
After a Moyal-type deformation of the model, we employ the dressing method to
explicitly construct multi-soliton configurations on noncommutative R^{2,1} and
analyze some of their properties. | [
"Olaf Lechtenfeld",
"Alexander D. Popov"
] | [
"hep-th"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model, we employ the dressing method to explicitly construct multi-soliton configurations on noncommutative R 2,1 and analyze some of their properties.
In the low-energy limit string theory with D-branes gives rise to noncommutative field theory on the branes when the string propagates in a nontrivial NS-NS two-form (B-field) background [1, 2, 3, 4] . In particular, if the open string has N =2 worldsheet supersymmetry, the tree-level target space dynamics is described by a noncommutative self-dual Yang-Mills (SDYM) theory in 2+2 dimensions [5] . Furthermore, open N =2 strings in a B-field background induce on the worldvolume of n coincident D2-branes a noncommutative Yang-Mills-Higgs Bogomolny-type system in 2+1 dimensions which is equivalent to a noncommutative generalization [6] of the modified U(n) chiral model known as the Ward model [7] . The topological nature of N =2 strings and the integrability of their tree-level dynamics [8] render this noncommutative sigma model integrable. 1 Being integrable, the commutative U(n≥2) Ward model features a plethora of exact scattering and no-scattering multi-soliton and wave solutions, i.e. time-dependent stable configurations on R foot_1 . These are not only a rich testing ground for physical properties such as adiabatic dynamics or quantization, but also descend to more standard multi-solitons of various integrable systems in 2+0 and 1+1 dimensions, such as sine-Gordon, upon dimensional and algebraic reduction. There is a price to pay however: Nonlinear sigma models in 2+1 dimensions may be Lorentz-invariant or integrable but not both [7, 11] . In fact, Derrick's theorem prohibits the existence of stable solitons in Lorentz-invariant scalar field theories above 1+1 dimensions. A Moyal deformation, however, overcomes this hurdle, but of course replaces Lorentz invariance by a Drinfeld-twisted version. There is another gain: The deformed Ward model possesses not only deformed versions of the just-mentioned multi-solitons, but in addition allows for a whole new class of genuinely noncommutative (multi-)solitons, in particular for the U(1) group [12, 13] ! Moreover, this class is related to the generic but perturbatively constructed noncommutative scalar-field solitons [14, 15] by an infinite-stiffness limit of the potential [16] .
In [12, 13] and [17] - [20] families of multi-solitons as well as their reduction to solitons of the noncommutative sine-Gordon equations were described and studied. In the nonabelian case both scattering and nonscattering configurations were obtained. For static configurations the issue of their stability was analyzed [21] . The full moduli space metric for the abelian model was computed and its adiabatic two-soliton dynamics was discussed [16] .
Recall that the critical N =2 string theory has a four-dimensional target space, and its open string effective field theory is self-dual Yang-Mills [8] , which gets deformed noncommutatively in the presence of a B-field [5] . Conversely, the noncommutative SDYM equations are contained [19] in the equations of motion of N =2 string field theory (SFT) [22] in a B-field background. This SFT formulation is based on the N =4 topological string description [23] . It is well known that the SDYM model can be described in terms of holomorphic bundles over (an open subset of) the twistor space 2 [26] CP 3 and the topological N =4 string theory contains twistors from the outset. The Lax pair, integrability and the solutions to the equations of motion by twistor and dressing methods were incorporated into the N =2 open SFT in [27, 28] . However, this theory reproduces only bosonic SDYM theory, its symmetries (see e.g. [29, 30, 31] ) and integrability properties. It is natural to ask: What string theory can describe supersymmetric SDYM theory [32, 33] in four dimensions?
There are some proposals [33, 34, 35, 36] for extending N =2 open string theory (and its SFT) to be space-time supersymmetric. Moreover, it was shown by Witten [37] that N =4 supersymmetric SDYM theory appears in twistor string theory, which is a B-type open topological string with the supertwistor space CP 3|4 as a target space. 3 Note that N <4 SDYM theory forms a BPS subsector of N -extended super Yang-Mills theory, and N =4 SDYM can be considered as a truncation of the full N =4 super Yang-Mills theory [37] . It is believed [43, 39] that twistor string theory is related with the previous proposals [33, 34, 35, 36] for a Lorentz-invariant supersymmetric extension of N =2 (and topological N =4) string theory which also leads to the N =4 SDYM model.
A dimensional reduction of the above relations between twistor strings and N =4 super Yang-Mills and SDYM models was considered in [44, 45, 46, 47] . The corresponding twistor string theory after this reduction is the topological B-model on the mini-supertwistor space P 2|4 . In [47] it was shown that the 2N =8 supersymmetric extension of the Bogomolny-type model in 2+1 dimensions is equivalent to an 2N =8 supersymmetric modified U(n) chiral model on R 2,1 . The subject of the current paper is an 2N ≤8 version of the above supersymmetric Bogomolny-type Yang-Mills-Higgs model in signature (-+ +), its relation with an N -extended supersymmetric modified integrable U(n) chiral model (to be defined) in 2+1 dimensions and the Moyal-type noncommutative deformation of this chiral model. We go on to explicitly construct multi-soliton configurations on noncommutative R 2,1 for the corresponding supersymmetric sigma model field equations. By studying the scattering properties of the constructed configurations, we prove their asymptotic factorization without scattering for large times. We also briefly discuss a D-brane interpretation of these soliton configurations from the viewpoint of twistor string theory.
Space R 2,2 . Let us consider the four-dimensional space R 2,2 = (R 4 , g) with the metric
ds 2 = g µν dx µ dx ν = det(dx α α) = dx 1 1dx 2 2 -dx 2 1dx 1 2 (2.1)
with (g µν ) = diag(-1, +1, +1, -1), where µ, ν, . . . = 1, . . . , 4 are space-time indices and α = 1, 2, α = 1, 2 are spinor indices. We choose the coordinates 4
(x µ ) = (x a , t) = (t, x, y, t) with a, b, . . . = 1, 2, 3 , (2.2)
and the signature (-+ + -) allows us to introduce real isotropic coordinates (cf. [19, 6] )
x 1 1 = 1 2 (t -y) , x 1 2 = 1 2 (x + t) , x 2 1 = 1 2 (x -t) , x 2 2 = 1 2 (t + y) . (2.3)
SDYM. Recall that the SDYM equations for a field strength tensor
F µν on R 2,2 read 1 2 ε µνρσ F ρσ = F µν , (2.4)
where ε µνρσ is a completely antisymmetric tensor on R 2,2 and ε 1234 = 1. In the coordinates (2.3) we have the decomposition
F α α,β β = ∂ α αA β β -∂ β β A α α + [A α α, A β β ] = ε αβ F α β + ε α β F αβ (2.5) with F α β := -1 2 ε αβ F α α,β β and F αβ := -1 2 ε α β F α α,β β , (2.6)
where ε αβ is antisymmetric, ε αβ ε βγ = δ γ α , and similar for ε α β , with ε 12 = ε 1 2 = 1. The gauge potential (A α α) will appear in the covariant derivative
D α β = ∂ α β + [A α β , • ] . (2.7)
In spinor notation, (2.4) is equivalently written as
F α β = 0 ⇔ F α α,β β = ε α β F αβ . (2.8)
Solutions {A α α} to these equations form a subset (a BPS sector) of the solution space of Yang-Mills theory on R 2,2 .
N -extended SDYM in component fields. The field content of N -extended super SDYM is foot_4
N = 0 A α α (2.9a) N = 1 A α α, χ i α with i = 1 (2.9b) N = 2 A α α, χ i α , φ [ij] with i, j = 1, 2 (2.9c)
N = 3 A α α, χ i α , φ [ij] , χ[ijk] α with i, j, k = 1, 2, 3 (2.9d) N = 4 A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl] α β
with i, j, k, l = 1, 2, 3, 4 .
(2.9e)
Here (A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl]
α β ) are fields of helicities (+1, + 1 2 , 0, -1 2 , -1). These fields obey the field equations of the N = 4 SDYM model, namely [33, 37]
F α β = 0 , (2.10a)
D α αχ iα = 0 , (2.10b)
D α αD α αφ ij + 2{χ iα , χ j α } = 0 , (2.10c)
D α α χ α[ijk] -6[χ [i α , φ jk] ] = 0 , (2.10d)
D γ α G [ijkl] γ β + 12{χ [i α , χjkl] β } -18[φ [ij , D α β φ kl] ] = 0 . (2.10e)
Note that the N < 4 SDYM field equations are governed by the first N +1 equations of (2.10), where F α β = 0 is counted as one equation and so on.
N -extended SDYM Superspace R 4|4N .
Recall that in the space R 2,2 = (R 4 , g) with the metric g given in (2.1) one may introduce purely real Majorana-Weyl spinors foot_5 θ α and η α of helicities + 1 2 and -1 2 as anticommuting (Grassmann-algebra) objects. Using 2N such spinors with components θ iα and η α i for i = 1, . . . , N , one can define the N -extended superspace R 4|4N and the N -extended supersymmetry algebra generated by the supertranslation operators
P α α = ∂ α α , Q iα = ∂ iα -η α i ∂ α α and Q i α = ∂ i α -θ iα ∂ α α , (2.11)
where
∂ α α := ∂ ∂x α α , ∂ iα := ∂ ∂θ iα and ∂ i α := ∂ ∂η α i .
(2.12)
The commutation relations for the generators (2.11) read
{Q iα , Q j α} = -2δ j i P α α , [P α α, Q iβ ] = 0 and [P α α, Q i β ] = 0 . (2.13)
To rewrite equations of motion in terms of R 4|4N superfields one uses the additional operators
D iα = ∂ iα + η α i ∂ α α and D i α = ∂ i α + θ iα ∂ α α , (2.14)
which (anti)commute with the operators (2.11) and satisfy
{D iα , D j β } = 2δ j i P α β , [P α α, D iβ ] = 0 and [P α α, D j β ] = 0 . (2.15)
Antichiral superspace R 4|2N . On the superspace R 4|4N one may introduce tensor fields depending on bosonic and fermionic coordinates (superfields), differential forms, Lie derivatives L X etc.. Furthermore, on any such superfield A one can impose the constraint equations L D iα A = 0, which for a scalar superfield f reduce to the so-called antichirality conditions
D iα f = 0 . (2.16)
These are easily solved by using a coordinate transformation on R 4|4N ,
(x α α, η α i , θ iα ) → (x α α = x α α-θ iα η α i , η α i , θ iα ) , (2.17)
under which ∂ α α, D iα and D i α transform to the operators
∂α α = ∂ α α , Diα = ∂ iα and Di α = ∂ i α + 2θ iα ∂ α α . ( 2
.18) Then (2.16) simply means that f is defined on a sub-superspace R 4|2N ⊂ R 4|4N with coordinates xα α and η α i . (2.19)
This space is called antichiral superspace. In the following we will usually omit the tildes when working on the antichiral superspace.
N -extended SDYM in superfields. The N -extended SDYM equations can be rewritten in terms of superfields on the antichiral superspace R 4|2N [33, 48] . Namely, for any given 0 ≤ N ≤ 4, fields of a proper multiplet from (2.9) can be combined into superfields A α α and A i α depending on x α α, η α i ∈ R 4|2N and giving rise to covariant derivatives
∇ α α := ∂ α α + A α α and ∇ i α := ∂ i α + A i α .
(2.20)
In such terms the N -extended SDYM equations (2.10) read
[∇ α α, ∇ β β ] + [∇ α β , ∇ β α] = 0 , [∇ i α, ∇ β β ] + [∇ i β , ∇ β α] = 0 , {∇ i α, ∇ j β } + {∇ i β , ∇ j α} = 0 , (2.21)
which is equivalent to
[∇ α α, ∇ β β ] = ε α β F αβ , [∇ i α, ∇ β β ] = ε α β F i β and {∇ i α, ∇ j β } = ε α β F ij , (2.22)
where F ij is antisymmetric and F αβ is symmetric in their indices.
The above gauge potential superfields (A α α, A i α) as well as the gauge strength superfields (F αβ , F i α , F ij ) contain all physical component fields of the N -extended SDYM model. For instance, the lowest component of the triple (F αβ , F i α , F ij ) in an η-expansion is (F αβ , χ i α , φ ij ), with zeros in case N is too small. By employing Bianchi identities for the gauge strength superfields, one successively obtains [48] the superfield expansions and the field equations (2.10) for all component fields.
It is instructive to extend the antichiral combination in (2.18) to potentials and covariant derivatives, Di
α = ∂ i α + 2 θ iα ∂ α α + + + Ãi α := A i α + 2 θ iα A α α ∇i α := ∇ i α + 2 θ iα ∇ α α ( 2
.23) where ∇ α α, ∇ i α and Di α are given by (2.20) and (2.18), while A i α and A α α depend on x α α and η α i only. With the antichiral covariant derivatives, one may condense (2.21) or (2.22) into the single set { ∇i α, ∇j β
} + { ∇i β , ∇j α} = 0 ⇔ { ∇i α, ∇j β } = ε α β Fij , (2.24)
with Fij =
F ij + 4 θ [iα F j] α + 4 θ iα θ jβ F αβ .
The concise form (2.24) of the N -extended SDYM equations is quite convenient, and we will use it interchangeable with (2.21).
Linear system for N -extended SDYM. It is well known that the superfield SDYM equations (2.21) can be seen as the compatibility conditions for the linear system of differential equations
ζ α(∂ α α + A α α) ψ = 0 and ζ α(∂ i α + A i α) ψ = 0 , (2.25)
where
(ζ β ) = 1 ζ and ζ α = ε α β ζ β . The extra (spectral) parameter 7 ζ lies in the extended complex plane C ∪ ∞ = CP 1 .
Here ψ is a matrix-valued function depending not only on x α α and η α i but also (meromorphically) on ζ ∈ CP 1 . We subject the n×n matrix ψ to the following reality condition:
ψ(x α α, η α i , ζ) ψ(x α α, η α i , ζ) † = 1l , (2.26)
where " †" denotes hermitian conjugation and ζ is complex conjugate to ζ. This condition guarantees that all physical fields of the N -extended SDYM model will take values in the adjoint representation of the algebra u(n). In the concise form the linear system (2.25) is written as
ζ α(∇ i α + 2θ iα ∇ α α) ψ = 0 ⇔ ζ α( Di α + Ãi α) ψ = 0 ⇔ ζ α ∇i α ψ = 0 . (2.27) 2.3 Reduction of N -extended SDYM to 2+1 dimensions
The supersymmetric Bogomolny-type Yang-Mills-Higgs equations in 2+1 dimensions are obtained from the described N -extended super SDYM equations by a dimensional reduction R 2,2 → R 2,1 .
In particular, for the N =0 sector we demand the components A µ of a gauge potential to be independent of x 4 and put A 4 =: ϕ. Here, ϕ is a Lie-algebra valued scalar field in three dimensions (the Higgs field) which enters into the Bogomolny-type equations. Similarly, for N ≥ 1 one can reduce the N -extended SDYM equations on R 2,2 by imposing the ∂ 4 -invariance condition on all the fields (A α α,
χ i α , φ [ij] , χ[ijk] α , G [ijkl]
α β ) from the N =4 supermultiplet or its truncation to N <4 and obtain supersymmetric Bogomolny-type equations on R 2,1 .
Spinors in R 2,1 .
Recall that on R 2,2 both N =4 SDYM theory and full N =4 super Yang-Mills theory have an SL(4, R) ∼ = Spin(3,3) R-symmetry group [33] . A dimensional reduction to R 2,1 enlarges the supersymmetry and R-symmetry to 2N =8 and Spin(4,4), respectively, for both theories (cf. [49] for Minkowski signature). More generally, any number N of supersymmetries gets doubled to 2N in the reduction. Since dimensional reduction collapses the rotation group Spin(2,2)
∼ = Spin(2,1) L ×Spin(2,1) R of R 2,2
to its diagonal subgroup Spin(2,1) D as the local rotation group of R 2,1 , the distinction between undotted and dotted indices disappears. We shall use undotted indices henceforth.
Coordinates and derivatives in R 2,1 . The above discussion implies that one can relabel the bosonic coordinates x α β from (2.3) by x αβ and split them as
x αβ = 1 2 (x αβ + x βα ) + 1 2 (x αβ -x βα ) = x (αβ) + x [αβ]
(2.28) into antisymmetric and symmetric parts,
x [αβ] = 1 2 ε αβ x 4 = 1 2 ε αβ t and x (αβ) =: y αβ , (2.29)
respectively, with
y 11 = x 11 = 1 2 (t -y) , y 12 = 1 2 (x 12 + x 21 ) = 1 2 x , y 22 = x 22 = 1 2 (t + y) . (2.30)
We also have θ iα → θ iα and η α i → η α i for the fermionic coordinates on R 4|4N reduced to R 3|4N . Bosonic coordinate derivatives reduce in 2+1 dimensions to the operators
∂ (αβ) = 1 2 (∂ αβ + ∂ βα ) (2.31)
which read explicitly as
∂ (11) = ∂ ∂y 11 = ∂ t -∂ y , ∂ (12) = ∂ (21) = 1 2 ∂ ∂y 12 = ∂ x , ∂ (22) = ∂ ∂y 22 = ∂ t + ∂ y . (2.32)
We thus have
∂ ∂x αβ = ∂ (αβ) -ε αβ ∂ 4 = ∂ (αβ) -ε αβ ∂ t , (2.33)
where
ε 12 = -ε 21 = -1, ∂ 4 = ∂/∂x 4 and ∂ t = ∂/∂ t.
The operators D iα and D i α acting on t-independent superfields reduce to
D iα = ∂ iα + η β i ∂ (αβ) and D i α = ∂ i α + θ iβ ∂ (αβ) , (2.34)
where
∂ iα = ∂/∂θ iα and ∂ i α = ∂/∂η α i .
Similarly, the antichiral operators Diα and Di α in (2.18) become Diα = ∂ iα and Di α = ∂ i α + 2θ iβ ∂ (αβ) . (2.35) Supersymmetric Bogomolny-type equations in component fields. According to (2.33), the components A α β of a gauge potential in four dimensions split into the components A (αβ) of a gauge potential in three dimensions and a Higgs field
A [αβ] = -ε αβ ϕ, i.e. A αβ = A (αβ) + A [αβ] = A (αβ) -ε αβ ϕ . (2.36)
Then the covariant derivatives D α β reduced to three dimensions become the differential operators
D αβ -ε αβ ϕ = ∂ (αβ) + [A (αβ) , • ] -ε αβ [ϕ, • ] , (2.37)
and the Yang-Mills field strength on R 2,1 decomposes as
F αβ, γδ = [D αβ , D γδ ] = ε αγ f βδ + ε βδ f αγ with f αβ = f βα . (2.38)
Substituting (2.36) and (2.37) into (2.10), i.e. demanding that all fields in (2.10) are independent of x 4 = t, we obtain the following supersymmetric Bogomolny-type equations on R 2,1 :
f αβ + D αβ ϕ = 0 , (2.39a)
D αβ χ iβ + ε αβ [ϕ, χ iβ ] = 0 , (2.39b)
D αβ D αβ φ ij + 2[ϕ, [ϕ, φ ij ]] + 2{χ iα , χ j α } = 0 , (2.39c)
D αβ χβ[ijk] -ε αβ [ϕ, χβ[ijk] ] -6[χ [i α , φ jk] ] = 0 , (2.39d)
D γ α G [ijkl] γβ + [ϕ, G [ijkl] αβ ] + 12{χ [i α , χjkl] β } -18[φ [ij , D αβ φ kl] ] -18ε αβ [φ [ij , [φ kl] , ϕ]] = 0 .(2.
Supersymmetric Bogomolny-type equations in terms of superfields. Translations generated by the vector field ∂ 4 = ∂ t are isometries of superspaces R 4|4N and R 4|2N . By taking the quotient with respect to the action of the abelian group G generated by ∂ 4 , we obtain the reduced full superspace R 3|4N ∼ = R 4|4N /G and the reduced antichiral superspace R 3|2N ∼ = R 4|2N /G. In the following, we shall work on R 3|2N and R 3|2N × CP 1 , since the reduced ψ-function from (2.25) and (2.27) is defined on the latter space.
The linear system stays in the center of the superfield approach to the N -extended SDYM equations. After imposing t-independence on all fields in the linear system (2.27), we arrive at the linear equations
ζ α ∇i α ψ ≡ ζ α ( Di α + Âi α ) ψ = 0 (2.40)
of the same form but with
Di α = ∂ i α + 2θ iβ ∂ (αβ) and Âi α = A i α + 2θ iβ (A (αβ) -ε αβ Ξ) , (2.41)
where A i α , A (αβ) and Ξ are superfields depending on y αβ and η α i only. These linear equations expand again to the pair (cf. (2.25))
ζ β (∂ (αβ) + A (αβ) -ε αβ Ξ) ψ = 0 and ζ α (∂ i α + A i α ) ψ = 0 . (2.42)
The compatibility conditions for the linear system (2.40) read
{ ∇i α , ∇j β } + { ∇i β , ∇j α } = 0 ⇔ { ∇i α , ∇j β } = ε αβ Fij (2.43)
and present a condensed form of (2.39) rewritten in terms of R 3|2N superfields. Similarly, these equations can also be written in more expanded forms analogously to (2.21) or using the superfield analog of (2.37). However, we will not do this since all these sets of equations are equivalent.
As has been known for some time, nonlinear sigma models in 2 + 1 dimensions may be Lorentzinvariant or integrable but not both [7, 11] . We will show that the super Bogomolny-type model discussed in Section 2 after a gauge fixing is equivalent to a super extension of the modified U(n) chiral model (so as to be integrable) first formulated by Ward [7] . Since integrability is compatible with noncommutative deformation (if introduced properly, see e.g. [9] - [20] ) we choose from the beginning to formulate our super extension of this chiral model on Moyal-deformed R 2,1 with noncommutativity parameter θ ≥ 0. Ordinary space-time R 2,1 can always be restored by taking the commutative limit θ → 0.
Star-product formulation. Classical field theory on noncommutative spaces may be realized in a star-product formulation or in an operator formalism 8 . The first approach is closer to the commutative field theory: it is obtained by simply deforming the ordinary product of classical fields (or their components) to the noncommutative star product
(f ⋆ g)(x) = f (x) exp{ i 2 ← - ∂ a θ ab -→ ∂ b } g(x) ⇒ x a ⋆ x b -x b ⋆ x a = iθ ab (3.1)
with a constant antisymmetric tensor θ ab . Specializing to R 2,1 , we use real coordinates (x a ) = (t, x, y) in which the Minkowski metric g on R 3 reads (g ab ) = diag(-1, +1, +1) with a, b, . . . = 1, 2, 3 (cf. Section 2). It is straightforward to generalize the Moyal deformation (3.1) to the superspaces introduced in the previous section, allowing in particular for non-anticommuting Grassmann-odd coordinates. Deferring general superspace deformations and their consequences to future work, we here content ourselves with the simple embedding of the "bosonic" Moyal deformation into superspace, meaning that (3.1) is also valid for superfields f and g depending on Grassmann variables θ iα and η α i . For later use we consider not only isotropic coordinates and vector fields
u := 1 2 (t+y) = y 22 , v := 1 2 (t-y) = y 11 , ∂ u = ∂ t + ∂ y = ∂ (22) , ∂ v = ∂ t -∂ y = ∂ (11) (3.2)
introduced in Section 2, but also the complex combinations
z := x + iy , z := x -iy , ∂ z = 1 2 (∂ x -i∂ y ) , ∂ z = 1 2 (∂ x + i∂ y ) . (3.3)
Since the time coordinate t remains commutative, the only nonvanishing component of the noncommutativity tensor θ ab is Operator formalism. The nonlocality of the star products renders explicit computation cumbersome. We therefore pass to the operator formalism, which trades the star product for operatorvalued spatial coordinates (x, ŷ) or their complex combinations (ẑ, ẑ), subject to
θ xy = -θ yx =: θ > 0 ⇒ θ z z = -θ zz = -2i θ . ( 3
[t, x] = [t, ŷ] = 0 but [x, ŷ] = iθ ⇒ [ẑ, ẑ] = 2 θ . (3.6)
The latter equation suggests the introduction of annihilation and creation operators,
a = 1 √ 2θ ẑ and a † = 1 √ 2θ ẑ with [a , a † ] = 1 , (3.7)
which act on a harmonic-oscillator Fock space H with an orthonormal basis { |ℓ , ℓ = 0, 1, 2, . . .} such that
a |ℓ = √ ℓ |ℓ-1 and a † |ℓ = √ ℓ+1 |ℓ+1 . (3.8)
Any superfield f (t, z, z, η α i ) on R 3|2N can be related to an operator-valued superfield f (t, η α i ) ≡ F (t, a, a † , η α i ) on R 1|2N acting in H, with the help of the Moyal-Weyl map
f (t, z, z, η α i ) → f (t, η α i ) = Weyl-ordered f t, √ 2θa, √ 2θa † , η α i . (3.9)
The inverse transformation recovers the ordinary superfield,
f (t, η α i ) ≡ F (t, a, a † , η α i ) → f (t, z, z, η α i ) = F ⋆ t, z √ 2θ , z √ 2θ , η α i , (3.10)
where F ⋆ is obtained from F by replacing ordinary with star products. Under the Moyal-Weyl map, we have
f ⋆ g → f ĝ and dx dy f = 2π θ Tr f = 2π θ ℓ≥0 ℓ| f |ℓ , (3.11)
and the spatial derivatives are mapped into commutators,
∂ z f → ∂z f = -1 √ 2θ [a † , f ] and ∂ z f → ∂z f = 1 √ 2θ [a , f ] . (3.12)
For notational simplicity we will from now on omit the hats over the operators except when confusion may arise.
Gauge fixing for ψ. Note that the linear system (2.40) and the compatibility conditions (2.43) are invariant under a gauge transformation
ψ → ψ ′ = g -1 ψ , (3.13a)
A → A ′ = g -1 A g + g -1 ∂ g (with appropriate indices) , (3.13b)
Ξ → Ξ ′ = g -1 Ξ g , (3.13c)
where g = g(x a , η α i ) is a U(n)-valued superfield globally defined on the deformed superspace R 3|2N θ × CP 1 . Using a gauge transformation of the form (3.13), we can choose ψ such that it will satisfy the standard asymptotic conditions (see e.g. [51] )
ψ = Φ -1 + O(ζ) for ζ → 0 , (3.14a
)
ψ = 1l + ζ -1 Υ + O(ζ -2 ) for ζ → ∞ , (3.14b)
where the U(n)-valued function Φ and u(n)-valued function Υ depend on x a and η α i . This "unitary" gauge is compatible with the reality condition for ψ,
ψ(x a , η α i , ζ) ψ(x a , η α i , ζ) † = 1l , (3.15)
obtained by reduction from (2.26).
A i 1 = 0 and A i 2 = Φ -1 ∂ i 2 Φ = ∂ i 1 Υ , (3.17)
A (11) = 0 and
A (12) + Ξ = Φ -1 ∂ (12) Φ = ∂ (11) Υ , (3.18)
A (21) -Ξ = 0 and
A (22) = Φ -1 ∂ (22) Φ = ∂ (12) Υ . (3.19)
Using (2.32), we can rewrite the nonzero components as
A := Φ -1 ∂ u Φ = ∂ x Υ , B := Φ -1 ∂ x Φ = ∂ v Υ , C i := Φ -1 ∂ i 2 Φ = ∂ i 1 Υ . (3.20)
Recall that the superfields Φ and Υ depend on x a and η α i .
Linear system. In the above-introduced unitary gauge the linear system (2.42) reads
(ζ∂ x -∂ u -A) ψ = 0 , (ζ∂ v -∂ x -B) ψ = 0 , (ζ∂ i 1 -∂ i 2 -C i ) ψ = 0 , (3.21)
which adds the last equation to the linear system of the Ward model [7] and generalizes it to superfields A(x a , η α j ), B(x a , η α j ) and C i (x a , η α j ). The concise form of (3.21) reads
ζ Di 1 -Di 2 -Âi 2 ψ = 0 (3.22)
or, in more explicit form,
ζ ∂ i 1 + 2θ i1 ∂ v + 2θ i2 ∂ x -∂ i 2 + C i + 2θ i1 (∂ x + B) + 2θ i2 (∂ u + A) ψ = 0 . (3.23)
N -extended sigma model.
The compatibility conditions of this linear system are the Nextended noncommutative sigma model equations Di
1 (Φ -1 Dj 2 Φ) + Dj 1 (Φ -1 Di 2 Φ) = 0 (3.24)
which in expanded form reads
(g ab + v c ε cab ) ∂ a (Φ -1 ∂ b Φ) = 0 ⇔ ∂ x (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ u Φ) = 0 , (3.25a) ∂ i 1 (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ i 2 Φ) = 0 , ∂ i 1 (Φ -1 ∂ u Φ) -∂ x (Φ -1 ∂ i 2 Φ) = 0 , (3.25b)
∂ i 1 (Φ -1 ∂ j 2 Φ) + ∂ j 1 (Φ -1 ∂ i 2 Φ) = 0 . (3.25c)
Here, the first line contains the Wess-Zumino-Witten term with a constant vector (v c ) = (0, 1, 0) which spoils the standard Lorentz invariance but yields an integrable chiral model in 2+1 dimensions. Recall that Φ is a U(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . As discussed in Section 2, the compatibility conditions of the linear equations (3.22) (or (3.21)) are equivalent to the N -extended Bogomolnytype equations (2.39) for the component (physical) fields. Thus, chiral model field equations (3.25) are equivalent to a gauge fixed form of equations (2.39). Υ-formulation. Instead of Φ-parametrization of (A, B, C i ) given in (3.17)-(3.20) we may use the equivalent Υ-parametrization also given there. In this case, the compatibility conditions for the linear system (3.21) reduce to
(∂ 2 x -∂ u ∂ v )Υ + [∂ v Υ , ∂ x Υ] = 0 , (3.26a)
(∂ i 2 ∂ v -∂ i 1 ∂ x )Υ + [∂ i 1 Υ , ∂ v Υ] = 0 , (∂ i 2 ∂ x -∂ i 1 ∂ u )Υ + [∂ i 1 Υ , ∂ x Υ] = 0 , (3.26b) (∂ i 2 ∂ j 1 + ∂ j 2 ∂ i 1 )Υ + {∂ i 1 Υ , ∂ j 1 Υ} = 0 , (3.26c)
which in concise form read
( Di 2 Dj 1 + Dj 2 Di 1 ) Υ + { Di 1 Υ , Dj 1 Υ} = 0 . (3.27)
Recall that Υ is a u(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . For N =4, the commutative limit of (3.27) can be considered as Siegel's equation [33] reduced to 2+1 dimensions. According to Siegel, one can extract the multiplet of physical fields appearing in (2.39) from the prepotential Υ via
∂ i 1 Υ = A i 2 , ∂ i 1 ∂ j 1 Υ = φ ij , ∂ i 1 ∂ j 1 ∂ k 1 Υ = χ[ijk] 2 , ∂ i 1 ∂ j 1 ∂ k 1 ∂ l 1 Υ = G [ijkl] 22 , (3.28a)
∂ (α1) Υ = A (α2) -ε α2 ϕ , ∂ (α1) ∂ i 1 Υ = χ i α , ∂ (α1) ∂ (β1) Υ = f αβ , (3.28b)
where one takes Υ and its derivatives at η 2 i = 0. The other components of the physical fields, i.e. χ[ijk]
1 , G [ijkl] 11 , G [ijkl]
21 , A (11) and A (21) -ϕ, vanish in this light-cone gauge.
Supersymmetry transformations. The 4N supercharges given in (2.11) reduce in 2+1 dimensions to the form
Q iα = ∂ iα -η β i ∂ (αβ) and Q i α = ∂ i α -θ iβ ∂ (αβ) . (3.29)
Their antichiral version, matching to Diα and Dj β of (2.35), reads
Qiα = ∂ iα -2η β i ∂ (αβ) and Qj β = ∂ j β , (3.30)
so that { Qiα , Qj β } = -2 δ j i ∂ (αβ) . (3.31)
On a (scalar) R 3|2N superfield Σ these supersymmetry transformations act as
δ Σ := ε iα Qiα Σ + ε α i Qi α Σ (3.32)
and are induced by the coordinate shifts
δ y αβ = -2ε i(α η β) i and δ η α i = ε α i , (3.33)
where ε iα and ε α i are 4N real Grassmann parameters. It is easy to see that our equations (3.24) and (3.27) are invariant under the supersymmetry transformations (3.32) (applied to Φ or Υ). This is simply because the operators Diα and Dj β anticommute with the supersymmetry generators Qiα and Qj β . Therefore, the equations of motion (3.25) of the modified N -extended chiral model in 2+1 dimensions as well as their reductions to 2+0 and 1+1 dimensions carry 2N supersymmetries and are genuine supersymmetric extensions of the corresponding bosonic equations. Note that this type of extension is not the standard one since the R-symmetry groups are Spin(N , N ) in 2+1 and Spin(N , N )× Spin(N , N ) in 1+1 dimensions, which differ from the compact unitary R-symmetry groups of standard sigma models. Contrary to the standard case of two-dimensional sigma models the above "noncompact" 2N supersymmetries do not impose any constraints on the geometry of the target space, e.g. they do not demand it to be Kähler [52] or hyper-Kähler [53] . This may be of interest and deserves further study.
Action functionals. In either formulation of the N -extended supersymmetric SDYM model on R 2,2 there are difficulties with finding a proper action functional generalizing the one [54, 55] for the purely bosonic case. These difficulties persist after the reduction to 2+1 dimensions, i.e. for the equations (3.25) and (3.26) describing our supersymmetric modified U(n) chiral model. It is the price to be paid for overcoming the no-go barrier N ≤ 4 and the absence of geometric target-space constraints. On a more formal level, the problem is related to the chiral character of (3.24) as well as (3.27) , where only the operators Di α but not Diα appear. Note however, that for N = 4 one can write an action functional in component fields producing the equations (2.39), which are equivalent to the superspace equations (3.24) when i, j = 1, . . . , 4 (see e.g. [47] ).
One proposal for an action functional stems from Siegel's idea [33] for the Υ-formulation of the N -extended SDYM equations. Namely, one sees that ∂ i 2 Υ enters only linearly into the last two lines in (3.26) . Therefore, if we introduce
Υ (1) := Υ| η 2 i =0 (3.34)
then it must satisfy the first equation from (3.26) , and the remaining equations iteratively define the dependence of Υ on η 2 i starting from Υ (1) . Hence, all information is contained in Υ (1) , as can also be seen from (3.28) . In other words, the dependence of Υ on η 2 i is not 'dynamical'. For an action one can then take (cf. [33] )
S = d 3 x d N η 1 Υ (1) ∂ (αβ) ∂ (αβ) Υ (1) + 2 3 Υ (1) ε αβ ∂ (α1) Υ (1) ∂ (β1) Υ (1) . (3.35)
Extremizing this functional yields the first line of (3.26) at η 2 i = 0. Except for the Grassmann integration, this action has the same form as the purely bosonic one [55] . One may apply the same logic to the Φ-formulation where the action for the purely bosonic case is also known [54, 56] .
The existence of the linear system (3.22) (equivalent to (3.21)) encoding solutions of the N -extended U(n) chiral model in an auxiliary matrix ψ allows for powerful methods to systematically construct explicit solutions for ψ and hence for Φ † = ψ| ζ=0 and Υ = lim ζ→∞ ζ (ψ-1l). For our purposes the so-called dressing method [57, 51] proves to be the most practical [12] - [20] , and so we shall use it here for our linear system, i.e. already in the N -extended noncommutative case.
Multi-pole ansatz for ψ. The dressing method is a recursive procedure for generating a new solution from an old one. More concretely, we rewrite the linear system (3.21) in the form
ψ(∂ u -ζ∂ x )ψ † = A , ψ(∂ x -ζ∂ v )ψ † = B , ψ(∂ i 2 -ζ∂ i 1 )ψ † = C i . (4.1)
Recall that ψ † := (ψ(x a , η α i , ζ)) † and (A, B, C i ) depend only on x a and η α i . The central idea is to demand analyticity in the spectral parameter ζ, which strongly restricts the possible form of ψ. One way to exploit this constraint starts from the observation that the left hand sides of (4.1) as well as of the reality condition (3.15) do not depend on ζ while ψ is expected to be a nontrivial function of ζ globally defined on CP 1 . Therefore, it must be a meromorphic function on CP 1 possessing some poles which we choose to lie at finite points with constant coordinates µ k ∈ CP 1 .
Here we will build a (multi-soliton) solution ψ m featuring m simple poles at positions µ 1 , . . . , µ m with foot_8 Im µ k < 0 by left-multiplying an (m-1)-pole solution ψ m-1 with a single-pole factor of the form 1l
+ µ m -μm ζ -µ m P m (x a , η α i ) , (4.2)
where the n×n matrix function P m is yet to be determined. Starting from the trivial (vacuum) solution ψ 0 = 1l, the iteration ψ 0 → ψ 1 → . . . → ψ m yields a multiplicative ansatz for ψ m ,
ψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ , (4.3)
which, via partial fraction decomposition, may be rewritten in the additive form
ψ m = 1l + m k=1 Λ mk S † k ζ -µ k , (4.4)
where Λ mk and S k are some n×r k matrices depending on x a and η α i , with r k ≤ n.
Equations for S k . Let us first consider the additive parametrization (4.4) of ψ m . This ansatz must satisfy the reality condition (3.15) as well as our linear equations in the form (4.1). In particular, the poles at ζ = μk on the left hand sides of these equations have to be removable since the right hand sides are independent of ζ. Inserting the ansatz (4.4) and putting to zero the corresponding residues, we learn from (3.15) that
1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ S k = 0 , (4.5)
while from (4.1) we obtain the differential equations
1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ LA,B,i k S k = 0 , (4.6)
where LA,B,i
k stands for either LA k = ∂ u -μk ∂ x , LB k = µ k (∂ x -μk ∂ v ) or Li k = ∂ i 2 -μk ∂ i 1 . (4.7)
Note that we consider a recursive procedure starting from m=1, and operators (4.7) will appear with k = 1, . . . , m if we consider poles at ζ = μk .
Because the LA,B,i k for k = 1, . . . , m are linear differential operators, it is easy to write down the general solution for (4.6) at any given k, by passing from the coordinates (u, v, x; η 1 i , η 2 i ) to "co-moving coordinates" (w k , wk , s k ; η i k , ηi k ). The precise relation for k = 1, . . . , m is [12, 58]
w k := x + μk u + μ-1 k v = x + 1 2 (μ k -μ -1 k )y + 1 2 (μ k +μ -1 k )t and η i k := η 1 i + μk η 2 i , (4.8)
with wk and ηi k obtained by complex conjugation and the co-moving time s k being inessential because by definition nothing will depend on it. The kth moving frame travels with a constant velocity
(v x , v y ) k = - µ k + μk µ k μk + 1 , µ k μk -1 µ k μk + 1 , (4.9)
so that the static case w k =z is recovered for µ k = -i. On functions of (w k , η i k , wk , ηi k ) alone the operators (4.7) act as
LA k = LB k = (µ k -μ k ) ∂ ∂ wk =: Lk and Li k = (µ k -μ k ) ∂ ∂ ηi k . (4.10)
By induction in k = 1, . . . , m we learn that, due to (4.5), a necessary and sufficient condition for a solution of (4.6) is Lk
S k = S k Zk and Li k S k = S k Zi k (4.11)
with some r k ×r k matrices Zk and Zi k depending on (w k , wk , η j k , ηj k ). Passing to the noncommutative bosonic coordinates we obtain ŵk , ŵk = 2θ ν k νk with
ν k νk = 4i µ k -μ k -µ -1 k +μ -1 k . (4.12)
Thus, we can introduce annihilation and creation operators
c k = 1 √ 2θ ŵk ν k and c † k = 1 √ 2θ ŵk νk so that [c k , c † k ] = 1 (4.13)
for k = 1, . . . , m. Naturally, this Heisenberg algebra is realized on a "co-moving" Fock space H k , with basis states |ℓ k and a "co-moving" vacuum |0 k subject to c k |0 k = 0. Each co-moving vacuum |0 k (annihilated by c k ) is related to the static vacuum |0 (annihilated by a) through an ISU(1,1) squeezing transformation (cf. [12] ) which is time-dependent. The fermionic coordinates η i k and ηi k remain spectators in the deformation. Coordinate derivatives are represented in the standard fashion as
ν k √ 2θ ∂ ∂w k → -[c † k , • ] and νk √ 2θ ∂ ∂ wk → [c k , • ] . (4.14)
After the Moyal deformation, the n×r k matrices S k have become operator-valued, but are still functions of the Grassmann coordinates η i k and ηi k . The noncommutative version of the BPS conditions (4.11) naturally reads
c k S k = S k Z k and ∂ ∂ ηi k S k = S k Z i k (4.15)
where Z k and Z i k are some operator-valued r k ×r k matrix functions of η j k and ηj k .
Nonabelian solutions for S k . For general data Z k and Z i k it is difficult to solve (4.15), but it is also unnecessary because the final expression ψ m turns out not to depend on them. Therefore, we conveniently choose
Z k = c k ⊗ 1l r k ×r k and Z i k = 0 ⇒ S k = R k (c k , η i k ) , (4.16)
where R k is an arbitrary n×r k matrix function independent of c † k and ηi k . 10 It is known that nonabelian (multi-) solitons arise for algebraic functions R k (cf. e.g. [7] for the commutative and [12] for the noncommutative N =0 case). Their common feature is a smooth commutative limit. The only novelty of the supersymmetric extension is the η i k dependence, i.e.
R k = R k,0 + η i k R k,i + η i k η j k R k,ij + η i k η j k η p k R k,ijp + η i k η j k η p k η q k R k,ijpq . (4.17)
Abelian solutions for S k . It is useful to view S k as a map from C r k ⊗H k to C n ⊗H k (momentarily suppressing the η dependence). The noncommutative setup now allows us to generalize the domain of this map to any subspace of C n ⊗ H k . In particular, we may choose it to be finite-dimensional, say C q k , and represent the map by an n×q k array |S k of kets in H. In this situation, Z k and Z i k in (4.15) are just number -valued q k ×q k matrix functions of η j k and ηj k . In case they do not depend on ηj k , we can write down the most general solution as
|S k = R k (c k , η j k ) |Z k exp i Z i k (η j k ) ηi k with |Z k := exp Z k (η j k ) c † k |0 k . (4.18)
As before, we may put Z i k = 0 without loss of generality, but now the choice of Z k does matter. For any given k generically there exists a q k -dimensional basis change which diagonalizes the ket-valued matrix
|Z k → diag e α 1 k c † , e α 2 k c † , . . . , e α q k k c † |0 k = diag |α 1 k , |α 2 k , . . . , |α q k k , (4.19)
where we defined coherent states
|α l k := e α l k c † |0 k so that c k |α l k = α l k |α l k for l = 1, . . . , q k and α l k ∈ C . (4.20)
Note that not only the entries of R k but also the α l k are holomorphic functions of the co-moving Grassmann parameters η j k and thus can be expanded like in (4.17). In the U(1) model, we must use ket-valued 1×q k matrices |S k for all k, yielding rows
|S k = R 1 k |α 1 k , R 2 k |α 2 k , . . . , R q k k |α q k k for k = 1, . . . , m , (4.21)
with functions α l k (η j k ). Here, the R l k only affect the states' normalization and can be collected in a diagonal matrix to the right, hence will drop out later and thus may all be put to one. Formally, we have recovered the known abelian (multi-) soliton solutions, but the supersymmetric extension has generalized |S k → |S k (η j k ) . Explicit form of P k . Let us now consider the multiplicative parametrization (4.3) of ψ m which also allows us to solve (4.5) . First of all, note that the reality condition (3.15) is satisfied if
P k = P † k = P 2 k ⇔ P k = T k (T † k T k ) -1 T † k for k = 1, . . . , m , (4.22)
meaning that P k is an operator-valued hermitian projector (of group-space rank r k ≤ n) built from an n×r k matrix function T k (the abelian case of n=1 is included). The reality condition follows just because The r k columns of T k span the image of P k and obey
1l + µ k -μk ζ -µ k P k 1l + μk -µ k ζ -μk P k =
P k T k = T k ⇔ (1l-P k ) T k = 0 . (4.24)
Furthermore, the equation (4.5) with m = k (induction) rewritten in the form
(1l-P k ) k-1 ℓ=1 1l + µ k-ℓ -μk-ℓ μk -µ k-ℓ P k-ℓ S k = 0 (4.25)
reveals that (cf. (4.24))
T 1 = S 1 and T k = k-1 ℓ=1 1l - µ k-ℓ -μk-ℓ µ k-ℓ -μk P k-ℓ S k for k ≥ 2 , (4.26)
where the explicit form of S k for k = 1, . . . , m is given in (4.16) or (4.18) . The final result reads
ψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ = 1l + m k=1 Λ mk S † k ζ -µ k (4.27)
with hermitian projectors P k given by (4.22), T k given by (4.26) and S k given by (4.16) or (4.18) . The explicit form of Λ mk (which we do not need) can be found in [12] . The corresponding superfields Φ and Υ are
Φ m = ψ † m | ζ=0 = m k=1 (1l -ρ k P k ) with ρ k = 1 - µ k μk , (4.28a)
Υ m = lim ζ→∞ ζ (ψ m -1l) = m k=1 (µ k -μ k ) P k . (4.28b)
From (4.22) it is obvious that P k is invariant under a similarity transformation
T k → T k Λ k ⇔ S k → S k Λ k (4.29)
for an invertible operator-valued r k ×r k matrix Λ k . This justifies putting Z i k = 0 from the beginning and also the restriction to Z k = c k ⊗ 1l r k ×r k in the nonabelian case, both without loss of generality. Hence, the nonabelian solution space constructed here is parametrized by the set {R k } m 1 of matrixvalued functions of c k and η i k and the pole positions µ k . The abelian moduli space, however, is larger by the set {Z k } m 1 of matrix-values functions of η i k which generically contain the coherentstate parameter functions {α l k (η i k )}. Restricting to η i k =0 reproduces the soliton configurations of the bosonic model [12] .
Static solutions. Let us consider the reduction to 2+0 dimensions, i.e. the static case. Recall that static solutions correspond to the choice m = 1 and µ 1 ≡ µ = -i implying w 1 = z, so we drop the index k. Specializing (4.27), we have
ψ = 1l - 2 i ζ + i P so that Φ = Φ † = 1l -2P , (4.30)
where a hermitian projector P of group-space rank r satisfies the BPS equations (1l-P ) a P = 0 ⇒ (1l-P ) a T = 0 ,
(1l-P ) ∂ ∂ ηi P = 0 ⇒ (1l-P ) ∂ ∂ ηi T = 0 , (4.31a)
with P = T (T † T ) -1 T † and η i = η 1 i + iη 2 i . In this case T = S, and for a nonabelian r=1 projector P we get T = T (a, η i ) as an n×1 column. For the simplest case of N =1 we just have (cf. [59] )
T = T e (a) + η T o (a) with η = η 1 + iη 2 , (4.32)
where T e (a) and T o (a) are rational functions of a (e.g. polynomials) taking values in the even and odd parts of the Grassmann algebra. Similarly, an abelian N =1 projector (for n=1) is built from
|T = |α 1 , |α 2 , . . . , |α q . (4.33)
At θ=0, the static solution (4.32) of our supersymmetric U(n) sigma model is also a solution of the standard N =1 supersymmetric CP n-1 sigma model in two dimensions (see e.g. [59] ). 11 For this reason, one can overcome the previously mentioned difficulty with constructing an action (or energy from the viewpoint of 2+1 dimensions) for static configurations. Moreover, on solutions obeying the BPS conditions (4.31) the topological charge
Q = 2πθ dη 1 dη 2 Tr tr Φ D + Φ , D -Φ (4.34)
is proportional to the action (BPS bound)
S = 2πθ dη 1 dη 2 Tr tr D + Φ , D -Φ (4.35)
and is finite for algebraic functions T e and T o . Here, the standard superderivatives D ± are defined as
D + = ∂ ∂η + iη ∂ z and D -= ∂ ∂ η + iη ∂ z . ( 4
One-soliton configuration. For one moving soliton, from (4.27) and (4.28) we obtain
ψ 1 = 1l + µ - μ ζ -µ P with P = T (T † T ) -1 T † (4.37) and Φ = 1l -ρ P with ρ = 1 - µ μ . (4.38)
Now our n×r matrix T must satisfy (putting
Z i = 0 and Z = c ⊗ 1l r×r ) [c , T ] = 0 and ∂ ∂ ηi T = 0 with η i = η 1 i + μ η 2 i , (4.39)
where c is the moving-frame annihilation operator given by (4.13) for k=1.
Recall that the operators c and c † and therefore the matrix T and the projector P can be expressed in terms of the corresponding static objects by a unitary squeezing transformation (see e.g. (4.8) and (4.13)). For simplicity we again consider the case N =1 and a nonabelian projector with r=1. Then (4.39) tells us that T is a holomorphic function of c and η, i.e.
T = T e (c) + η T o (c) = T 1 e (c) + η T 1 o (c)
. . .
T n e (c) + η T n o (c) (4.40)
with polynomials T a e and T a o of order q, say, analogously to the static case (4.32). Note that, for T a o to be Grassmann-odd and nonzero, some extraneous Grassmann parameter must appear. Similarly, abelian projectors for a moving one-soliton obtain by subjecting (4.33) to a squeezing transformation.
For N =1 the moving frame was defined in (4.8) (dropping the index k) via
w = x + 1 2 (μ-μ -1 )y + 1 2 (μ+μ -1 )t and η = η 1 + μη 2 hence ∂ t η = 0 . (4.41)
Consider the moving frame with the coordinates (w, w, s; η, η) with the choice s = t and the related change of the derivatives (see [12, 58] )
∂ x = ∂ w + ∂ w , (4.42a)
∂ y = 1 2 (μ-μ -1 ) ∂ w + 1 2 (µ-µ -1 ) ∂ w , (4.42b)
∂ t = 1 2 (μ+μ -1 ) ∂ w + 1 2 (µ+µ -1 ) ∂ w + ∂ s , (4.42c)
∂ η 1 = ∂ η + ∂ η , (4.42d)
∂ η 2 = μ ∂ η + µ ∂ η . (4.42e)
In the moving frame our solution (4.38) is static, i.e. ∂ s Φ = 0, and the projector P has the same form as in the static case. The only difference is the coefficient ρ instead of 2 in (4.38). Therefore, by computing the action (4.35) in (w, w; η 1 , η 2 ) coordinates, we obtain for algebraic functions T in (4.40) a finite answer, which differs from the static one by a kinematical prefactor depending on µ (cf. [12] for the bosonic case).
Large-time asymptotics. Note that in the distinguished (z, z, t) coordinate frame (4.41) implies that at large times w → κ t with κ = 1 2 (μ+μ -1 ). As a consequence, the t q term in each polynomial in (4.40) will dominate, i.e.
T → t q a 1 + η b 1 . . .
an + η bn =: t q Γ , (4.43)
where Γ is a fixed vector in C n . It is easy to see that in the distinguished frame the large-time limit of Φ given by (4. being the projector on the constant vector Γ.
Consider now the m-soliton configuration (4.28). By induction of the above argument one easily arrives at the m-soliton generalization of (4.44). Namely, in the frame moving with the ℓth lump we have lim t→±∞ Φ m = (1lρ 1 Π 1 ) . . . (1lρ ℓ-1 Π ℓ-1 )(1lρ ℓ P ℓ )(1lρ ℓ+1 Π ℓ+1 ) . . . (1lρ m Π m ) , (4.45) where the Π m are constant projectors. This large-time factorization of multi-soliton solutions provides a proof of the no-scattering property because the asymptotic configurations are identical for large negative and large positive times.
In this paper we introduced a generalization of the modified integrable U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions and considered a Moyal deformation of this model. It was shown that this N -extended chiral model is equivalent to a gauge-fixed BPS subsector of an N -extended super Yang-Mills model in 2+1 dimensions originating from twistor string theory. The dressing method was applied to generate a wide class of multi-soliton configurations, which are time-dependent finite-energy solutions to the equations of motion. Compared to the N =0 model, the supersymmetric extension was seen to promote the configurations' building blocks to holomorphic functions of suitable Grassmann coordinates. By considering the large-time asymptotic factorization into a product of single soliton solutions we have shown that no scattering occurs within the dressing ansatz chosen here.
The considered model does not stand alone but is motivated by twistor string theory [37] with a target space reduced to the mini-supertwistor space [44, 45, 47] . In this context, the obtained multi-soliton solutions are to be regarded as D(0|2N )-branes moving inside D(2|2N )-branes [60] . Here 2N appears due to fermionic worldvolume directions of our branes in the superspace description [60] . Switching on a constant B-field simply deforms the sigma model and D-brane worldvolumes noncommutatively, thereby admitting also regular supersymmetric noncommutative abelian solutions.
Restricting to static configurations, the models can be specialized to Grassmannian supersymmetric sigma models, where the superfield Φ takes values in Gr(r, n), and the field equations are invariant under 2N supersymmetry transformations with 0 ≤ N ≤ 4. This differs from the results for standard 2D sigma models [52, 53] where the target spaces have to be Kähler or hyper-Kähler for admitting two or four supersymmetries, respectively. This difference will be discussed in more details elsewhere.
We derived the supersymmetric chiral model in 2+1 dimensions through dimensional reduction and gauge fixing of the N -extended supersymmetric SDYM equations in 2+2 dimensions. Recall that for the purely bosonic case most (if not all) integrable equations in three and fewer dimensions can be obtained from the SDYM equations (or their hierarchy [25] ) by suitable dimensional reductions (see e.g. [61] - [65] and references therein). Moreover, this Ward conjecture [61] was extended to the noncommutative case (see e.g. [66, 67] ). It will be interesting to consider similar reductions of the N -extended supersymmetric SDYM equations (and their hierarchy [68] ) to supersymmetric integrable equations in three and two dimensions generalizing earlier results [69] . | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model, we employ the dressing method to explicitly construct multi-soliton configurations on noncommutative R 2,1 and analyze some of their properties."
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "In the low-energy limit string theory with D-branes gives rise to noncommutative field theory on the branes when the string propagates in a nontrivial NS-NS two-form (B-field) background [1, 2, 3, 4] . In particular, if the open string has N =2 worldsheet supersymmetry, the tree-level target space dynamics is described by a noncommutative self-dual Yang-Mills (SDYM) theory in 2+2 dimensions [5] . Furthermore, open N =2 strings in a B-field background induce on the worldvolume of n coincident D2-branes a noncommutative Yang-Mills-Higgs Bogomolny-type system in 2+1 dimensions which is equivalent to a noncommutative generalization [6] of the modified U(n) chiral model known as the Ward model [7] . The topological nature of N =2 strings and the integrability of their tree-level dynamics [8] render this noncommutative sigma model integrable. 1 Being integrable, the commutative U(n≥2) Ward model features a plethora of exact scattering and no-scattering multi-soliton and wave solutions, i.e. time-dependent stable configurations on R foot_1 . These are not only a rich testing ground for physical properties such as adiabatic dynamics or quantization, but also descend to more standard multi-solitons of various integrable systems in 2+0 and 1+1 dimensions, such as sine-Gordon, upon dimensional and algebraic reduction. There is a price to pay however: Nonlinear sigma models in 2+1 dimensions may be Lorentz-invariant or integrable but not both [7, 11] . In fact, Derrick's theorem prohibits the existence of stable solitons in Lorentz-invariant scalar field theories above 1+1 dimensions. A Moyal deformation, however, overcomes this hurdle, but of course replaces Lorentz invariance by a Drinfeld-twisted version. There is another gain: The deformed Ward model possesses not only deformed versions of the just-mentioned multi-solitons, but in addition allows for a whole new class of genuinely noncommutative (multi-)solitons, in particular for the U(1) group [12, 13] ! Moreover, this class is related to the generic but perturbatively constructed noncommutative scalar-field solitons [14, 15] by an infinite-stiffness limit of the potential [16] .\n\nIn [12, 13] and [17] - [20] families of multi-solitons as well as their reduction to solitons of the noncommutative sine-Gordon equations were described and studied. In the nonabelian case both scattering and nonscattering configurations were obtained. For static configurations the issue of their stability was analyzed [21] . The full moduli space metric for the abelian model was computed and its adiabatic two-soliton dynamics was discussed [16] .\n\nRecall that the critical N =2 string theory has a four-dimensional target space, and its open string effective field theory is self-dual Yang-Mills [8] , which gets deformed noncommutatively in the presence of a B-field [5] . Conversely, the noncommutative SDYM equations are contained [19] in the equations of motion of N =2 string field theory (SFT) [22] in a B-field background. This SFT formulation is based on the N =4 topological string description [23] . It is well known that the SDYM model can be described in terms of holomorphic bundles over (an open subset of) the twistor space 2 [26] CP 3 and the topological N =4 string theory contains twistors from the outset. The Lax pair, integrability and the solutions to the equations of motion by twistor and dressing methods were incorporated into the N =2 open SFT in [27, 28] . However, this theory reproduces only bosonic SDYM theory, its symmetries (see e.g. [29, 30, 31] ) and integrability properties. It is natural to ask: What string theory can describe supersymmetric SDYM theory [32, 33] in four dimensions?\n\nThere are some proposals [33, 34, 35, 36] for extending N =2 open string theory (and its SFT) to be space-time supersymmetric. Moreover, it was shown by Witten [37] that N =4 supersymmetric SDYM theory appears in twistor string theory, which is a B-type open topological string with the supertwistor space CP 3|4 as a target space. 3 Note that N <4 SDYM theory forms a BPS subsector of N -extended super Yang-Mills theory, and N =4 SDYM can be considered as a truncation of the full N =4 super Yang-Mills theory [37] . It is believed [43, 39] that twistor string theory is related with the previous proposals [33, 34, 35, 36] for a Lorentz-invariant supersymmetric extension of N =2 (and topological N =4) string theory which also leads to the N =4 SDYM model.\n\nA dimensional reduction of the above relations between twistor strings and N =4 super Yang-Mills and SDYM models was considered in [44, 45, 46, 47] . The corresponding twistor string theory after this reduction is the topological B-model on the mini-supertwistor space P 2|4 . In [47] it was shown that the 2N =8 supersymmetric extension of the Bogomolny-type model in 2+1 dimensions is equivalent to an 2N =8 supersymmetric modified U(n) chiral model on R 2,1 . The subject of the current paper is an 2N ≤8 version of the above supersymmetric Bogomolny-type Yang-Mills-Higgs model in signature (-+ +), its relation with an N -extended supersymmetric modified integrable U(n) chiral model (to be defined) in 2+1 dimensions and the Moyal-type noncommutative deformation of this chiral model. We go on to explicitly construct multi-soliton configurations on noncommutative R 2,1 for the corresponding supersymmetric sigma model field equations. By studying the scattering properties of the constructed configurations, we prove their asymptotic factorization without scattering for large times. We also briefly discuss a D-brane interpretation of these soliton configurations from the viewpoint of twistor string theory."
},
{
"section_type": "OTHER",
"section_title": "N -extended SDYM equations in 2+2 dimensions",
"text": "Space R 2,2 . Let us consider the four-dimensional space R 2,2 = (R 4 , g) with the metric\n\nds 2 = g µν dx µ dx ν = det(dx α α) = dx 1 1dx 2 2 -dx 2 1dx 1 2 (2.1)\n\nwith (g µν ) = diag(-1, +1, +1, -1), where µ, ν, . . . = 1, . . . , 4 are space-time indices and α = 1, 2, α = 1, 2 are spinor indices. We choose the coordinates 4\n\n(x µ ) = (x a , t) = (t, x, y, t) with a, b, . . . = 1, 2, 3 , (2.2)\n\nand the signature (-+ + -) allows us to introduce real isotropic coordinates (cf. [19, 6] )\n\nx 1 1 = 1 2 (t -y) , x 1 2 = 1 2 (x + t) , x 2 1 = 1 2 (x -t) , x 2 2 = 1 2 (t + y) . (2.3)\n\nSDYM. Recall that the SDYM equations for a field strength tensor\n\nF µν on R 2,2 read 1 2 ε µνρσ F ρσ = F µν , (2.4)\n\nwhere ε µνρσ is a completely antisymmetric tensor on R 2,2 and ε 1234 = 1. In the coordinates (2.3) we have the decomposition\n\nF α α,β β = ∂ α αA β β -∂ β β A α α + [A α α, A β β ] = ε αβ F α β + ε α β F αβ (2.5) with F α β := -1 2 ε αβ F α α,β β and F αβ := -1 2 ε α β F α α,β β , (2.6)\n\nwhere ε αβ is antisymmetric, ε αβ ε βγ = δ γ α , and similar for ε α β , with ε 12 = ε 1 2 = 1. The gauge potential (A α α) will appear in the covariant derivative\n\nD α β = ∂ α β + [A α β , • ] . (2.7)\n\nIn spinor notation, (2.4) is equivalently written as\n\nF α β = 0 ⇔ F α α,β β = ε α β F αβ . (2.8)\n\nSolutions {A α α} to these equations form a subset (a BPS sector) of the solution space of Yang-Mills theory on R 2,2 .\n\nN -extended SDYM in component fields. The field content of N -extended super SDYM is foot_4\n\nN = 0 A α α (2.9a) N = 1 A α α, χ i α with i = 1 (2.9b) N = 2 A α α, χ i α , φ [ij] with i, j = 1, 2 (2.9c)\n\nN = 3 A α α, χ i α , φ [ij] , χ[ijk] α with i, j, k = 1, 2, 3 (2.9d) N = 4 A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl] α β\n\nwith i, j, k, l = 1, 2, 3, 4 .\n\n(2.9e)\n\nHere (A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl]\n\nα β ) are fields of helicities (+1, + 1 2 , 0, -1 2 , -1). These fields obey the field equations of the N = 4 SDYM model, namely [33, 37]\n\nF α β = 0 , (2.10a)\n\nD α αχ iα = 0 , (2.10b)\n\nD α αD α αφ ij + 2{χ iα , χ j α } = 0 , (2.10c)\n\nD α α χ α[ijk] -6[χ [i α , φ jk] ] = 0 , (2.10d)\n\nD γ α G [ijkl] γ β + 12{χ [i α , χjkl] β } -18[φ [ij , D α β φ kl] ] = 0 . (2.10e)\n\nNote that the N < 4 SDYM field equations are governed by the first N +1 equations of (2.10), where F α β = 0 is counted as one equation and so on."
},
{
"section_type": "OTHER",
"section_title": "Superfield formulation of",
"text": "N -extended SDYM Superspace R 4|4N .\n\nRecall that in the space R 2,2 = (R 4 , g) with the metric g given in (2.1) one may introduce purely real Majorana-Weyl spinors foot_5 θ α and η α of helicities + 1 2 and -1 2 as anticommuting (Grassmann-algebra) objects. Using 2N such spinors with components θ iα and η α i for i = 1, . . . , N , one can define the N -extended superspace R 4|4N and the N -extended supersymmetry algebra generated by the supertranslation operators\n\nP α α = ∂ α α , Q iα = ∂ iα -η α i ∂ α α and Q i α = ∂ i α -θ iα ∂ α α , (2.11)\n\nwhere\n\n∂ α α := ∂ ∂x α α , ∂ iα := ∂ ∂θ iα and ∂ i α := ∂ ∂η α i .\n\n(2.12)\n\nThe commutation relations for the generators (2.11) read\n\n{Q iα , Q j α} = -2δ j i P α α , [P α α, Q iβ ] = 0 and [P α α, Q i β ] = 0 . (2.13)\n\nTo rewrite equations of motion in terms of R 4|4N superfields one uses the additional operators\n\nD iα = ∂ iα + η α i ∂ α α and D i α = ∂ i α + θ iα ∂ α α , (2.14)\n\nwhich (anti)commute with the operators (2.11) and satisfy\n\n{D iα , D j β } = 2δ j i P α β , [P α α, D iβ ] = 0 and [P α α, D j β ] = 0 . (2.15)\n\nAntichiral superspace R 4|2N . On the superspace R 4|4N one may introduce tensor fields depending on bosonic and fermionic coordinates (superfields), differential forms, Lie derivatives L X etc.. Furthermore, on any such superfield A one can impose the constraint equations L D iα A = 0, which for a scalar superfield f reduce to the so-called antichirality conditions\n\nD iα f = 0 . (2.16)\n\nThese are easily solved by using a coordinate transformation on R 4|4N ,\n\n(x α α, η α i , θ iα ) → (x α α = x α α-θ iα η α i , η α i , θ iα ) , (2.17)\n\nunder which ∂ α α, D iα and D i α transform to the operators\n\n∂α α = ∂ α α , Diα = ∂ iα and Di α = ∂ i α + 2θ iα ∂ α α . ( 2\n\n.18) Then (2.16) simply means that f is defined on a sub-superspace R 4|2N ⊂ R 4|4N with coordinates xα α and η α i . (2.19)\n\nThis space is called antichiral superspace. In the following we will usually omit the tildes when working on the antichiral superspace.\n\nN -extended SDYM in superfields. The N -extended SDYM equations can be rewritten in terms of superfields on the antichiral superspace R 4|2N [33, 48] . Namely, for any given 0 ≤ N ≤ 4, fields of a proper multiplet from (2.9) can be combined into superfields A α α and A i α depending on x α α, η α i ∈ R 4|2N and giving rise to covariant derivatives\n\n∇ α α := ∂ α α + A α α and ∇ i α := ∂ i α + A i α .\n\n(2.20)\n\nIn such terms the N -extended SDYM equations (2.10) read\n\n[∇ α α, ∇ β β ] + [∇ α β , ∇ β α] = 0 , [∇ i α, ∇ β β ] + [∇ i β , ∇ β α] = 0 , {∇ i α, ∇ j β } + {∇ i β , ∇ j α} = 0 , (2.21)\n\nwhich is equivalent to\n\n[∇ α α, ∇ β β ] = ε α β F αβ , [∇ i α, ∇ β β ] = ε α β F i β and {∇ i α, ∇ j β } = ε α β F ij , (2.22)\n\nwhere F ij is antisymmetric and F αβ is symmetric in their indices.\n\nThe above gauge potential superfields (A α α, A i α) as well as the gauge strength superfields (F αβ , F i α , F ij ) contain all physical component fields of the N -extended SDYM model. For instance, the lowest component of the triple (F αβ , F i α , F ij ) in an η-expansion is (F αβ , χ i α , φ ij ), with zeros in case N is too small. By employing Bianchi identities for the gauge strength superfields, one successively obtains [48] the superfield expansions and the field equations (2.10) for all component fields.\n\nIt is instructive to extend the antichiral combination in (2.18) to potentials and covariant derivatives, Di\n\nα = ∂ i α + 2 θ iα ∂ α α + + + Ãi α := A i α + 2 θ iα A α α ∇i α := ∇ i α + 2 θ iα ∇ α α ( 2\n\n.23) where ∇ α α, ∇ i α and Di α are given by (2.20) and (2.18), while A i α and A α α depend on x α α and η α i only. With the antichiral covariant derivatives, one may condense (2.21) or (2.22) into the single set { ∇i α, ∇j β\n\n} + { ∇i β , ∇j α} = 0 ⇔ { ∇i α, ∇j β } = ε α β Fij , (2.24)\n\nwith Fij =\n\nF ij + 4 θ [iα F j] α + 4 θ iα θ jβ F αβ .\n\nThe concise form (2.24) of the N -extended SDYM equations is quite convenient, and we will use it interchangeable with (2.21).\n\nLinear system for N -extended SDYM. It is well known that the superfield SDYM equations (2.21) can be seen as the compatibility conditions for the linear system of differential equations\n\nζ α(∂ α α + A α α) ψ = 0 and ζ α(∂ i α + A i α) ψ = 0 , (2.25)\n\nwhere\n\n(ζ β ) = 1 ζ and ζ α = ε α β ζ β . The extra (spectral) parameter 7 ζ lies in the extended complex plane C ∪ ∞ = CP 1 .\n\nHere ψ is a matrix-valued function depending not only on x α α and η α i but also (meromorphically) on ζ ∈ CP 1 . We subject the n×n matrix ψ to the following reality condition:\n\nψ(x α α, η α i , ζ) ψ(x α α, η α i , ζ) † = 1l , (2.26)\n\nwhere \" †\" denotes hermitian conjugation and ζ is complex conjugate to ζ. This condition guarantees that all physical fields of the N -extended SDYM model will take values in the adjoint representation of the algebra u(n). In the concise form the linear system (2.25) is written as\n\nζ α(∇ i α + 2θ iα ∇ α α) ψ = 0 ⇔ ζ α( Di α + Ãi α) ψ = 0 ⇔ ζ α ∇i α ψ = 0 . (2.27) 2.3 Reduction of N -extended SDYM to 2+1 dimensions\n\nThe supersymmetric Bogomolny-type Yang-Mills-Higgs equations in 2+1 dimensions are obtained from the described N -extended super SDYM equations by a dimensional reduction R 2,2 → R 2,1 .\n\nIn particular, for the N =0 sector we demand the components A µ of a gauge potential to be independent of x 4 and put A 4 =: ϕ. Here, ϕ is a Lie-algebra valued scalar field in three dimensions (the Higgs field) which enters into the Bogomolny-type equations. Similarly, for N ≥ 1 one can reduce the N -extended SDYM equations on R 2,2 by imposing the ∂ 4 -invariance condition on all the fields (A α α,\n\nχ i α , φ [ij] , χ[ijk] α , G [ijkl]\n\nα β ) from the N =4 supermultiplet or its truncation to N <4 and obtain supersymmetric Bogomolny-type equations on R 2,1 .\n\nSpinors in R 2,1 .\n\nRecall that on R 2,2 both N =4 SDYM theory and full N =4 super Yang-Mills theory have an SL(4, R) ∼ = Spin(3,3) R-symmetry group [33] . A dimensional reduction to R 2,1 enlarges the supersymmetry and R-symmetry to 2N =8 and Spin(4,4), respectively, for both theories (cf. [49] for Minkowski signature). More generally, any number N of supersymmetries gets doubled to 2N in the reduction. Since dimensional reduction collapses the rotation group Spin(2,2)\n\n∼ = Spin(2,1) L ×Spin(2,1) R of R 2,2\n\nto its diagonal subgroup Spin(2,1) D as the local rotation group of R 2,1 , the distinction between undotted and dotted indices disappears. We shall use undotted indices henceforth.\n\nCoordinates and derivatives in R 2,1 . The above discussion implies that one can relabel the bosonic coordinates x α β from (2.3) by x αβ and split them as\n\nx αβ = 1 2 (x αβ + x βα ) + 1 2 (x αβ -x βα ) = x (αβ) + x [αβ]\n\n(2.28) into antisymmetric and symmetric parts,\n\nx [αβ] = 1 2 ε αβ x 4 = 1 2 ε αβ t and x (αβ) =: y αβ , (2.29)\n\nrespectively, with\n\ny 11 = x 11 = 1 2 (t -y) , y 12 = 1 2 (x 12 + x 21 ) = 1 2 x , y 22 = x 22 = 1 2 (t + y) . (2.30)\n\nWe also have θ iα → θ iα and η α i → η α i for the fermionic coordinates on R 4|4N reduced to R 3|4N . Bosonic coordinate derivatives reduce in 2+1 dimensions to the operators\n\n∂ (αβ) = 1 2 (∂ αβ + ∂ βα ) (2.31)\n\nwhich read explicitly as\n\n∂ (11) = ∂ ∂y 11 = ∂ t -∂ y , ∂ (12) = ∂ (21) = 1 2 ∂ ∂y 12 = ∂ x , ∂ (22) = ∂ ∂y 22 = ∂ t + ∂ y . (2.32)\n\nWe thus have\n\n∂ ∂x αβ = ∂ (αβ) -ε αβ ∂ 4 = ∂ (αβ) -ε αβ ∂ t , (2.33)\n\nwhere\n\nε 12 = -ε 21 = -1, ∂ 4 = ∂/∂x 4 and ∂ t = ∂/∂ t.\n\nThe operators D iα and D i α acting on t-independent superfields reduce to\n\nD iα = ∂ iα + η β i ∂ (αβ) and D i α = ∂ i α + θ iβ ∂ (αβ) , (2.34)\n\nwhere\n\n∂ iα = ∂/∂θ iα and ∂ i α = ∂/∂η α i .\n\nSimilarly, the antichiral operators Diα and Di α in (2.18) become Diα = ∂ iα and Di α = ∂ i α + 2θ iβ ∂ (αβ) . (2.35) Supersymmetric Bogomolny-type equations in component fields. According to (2.33), the components A α β of a gauge potential in four dimensions split into the components A (αβ) of a gauge potential in three dimensions and a Higgs field\n\nA [αβ] = -ε αβ ϕ, i.e. A αβ = A (αβ) + A [αβ] = A (αβ) -ε αβ ϕ . (2.36)\n\nThen the covariant derivatives D α β reduced to three dimensions become the differential operators\n\nD αβ -ε αβ ϕ = ∂ (αβ) + [A (αβ) , • ] -ε αβ [ϕ, • ] , (2.37)\n\nand the Yang-Mills field strength on R 2,1 decomposes as\n\nF αβ, γδ = [D αβ , D γδ ] = ε αγ f βδ + ε βδ f αγ with f αβ = f βα . (2.38)\n\nSubstituting (2.36) and (2.37) into (2.10), i.e. demanding that all fields in (2.10) are independent of x 4 = t, we obtain the following supersymmetric Bogomolny-type equations on R 2,1 :\n\nf αβ + D αβ ϕ = 0 , (2.39a)\n\nD αβ χ iβ + ε αβ [ϕ, χ iβ ] = 0 , (2.39b)\n\nD αβ D αβ φ ij + 2[ϕ, [ϕ, φ ij ]] + 2{χ iα , χ j α } = 0 , (2.39c)\n\nD αβ χβ[ijk] -ε αβ [ϕ, χβ[ijk] ] -6[χ [i α , φ jk] ] = 0 , (2.39d)\n\nD γ α G [ijkl] γβ + [ϕ, G [ijkl] αβ ] + 12{χ [i α , χjkl] β } -18[φ [ij , D αβ φ kl] ] -18ε αβ [φ [ij , [φ kl] , ϕ]] = 0 .(2.\n\nSupersymmetric Bogomolny-type equations in terms of superfields. Translations generated by the vector field ∂ 4 = ∂ t are isometries of superspaces R 4|4N and R 4|2N . By taking the quotient with respect to the action of the abelian group G generated by ∂ 4 , we obtain the reduced full superspace R 3|4N ∼ = R 4|4N /G and the reduced antichiral superspace R 3|2N ∼ = R 4|2N /G. In the following, we shall work on R 3|2N and R 3|2N × CP 1 , since the reduced ψ-function from (2.25) and (2.27) is defined on the latter space.\n\nThe linear system stays in the center of the superfield approach to the N -extended SDYM equations. After imposing t-independence on all fields in the linear system (2.27), we arrive at the linear equations\n\nζ α ∇i α ψ ≡ ζ α ( Di α + Âi α ) ψ = 0 (2.40)\n\nof the same form but with\n\nDi α = ∂ i α + 2θ iβ ∂ (αβ) and Âi α = A i α + 2θ iβ (A (αβ) -ε αβ Ξ) , (2.41)\n\nwhere A i α , A (αβ) and Ξ are superfields depending on y αβ and η α i only. These linear equations expand again to the pair (cf. (2.25))\n\nζ β (∂ (αβ) + A (αβ) -ε αβ Ξ) ψ = 0 and ζ α (∂ i α + A i α ) ψ = 0 . (2.42)\n\nThe compatibility conditions for the linear system (2.40) read\n\n{ ∇i α , ∇j β } + { ∇i β , ∇j α } = 0 ⇔ { ∇i α , ∇j β } = ε αβ Fij (2.43)\n\nand present a condensed form of (2.39) rewritten in terms of R 3|2N superfields. Similarly, these equations can also be written in more expanded forms analogously to (2.21) or using the superfield analog of (2.37). However, we will not do this since all these sets of equations are equivalent."
},
{
"section_type": "OTHER",
"section_title": "Noncommutative N -extended U(n) chiral model in 2+1 dimensions",
"text": "As has been known for some time, nonlinear sigma models in 2 + 1 dimensions may be Lorentzinvariant or integrable but not both [7, 11] . We will show that the super Bogomolny-type model discussed in Section 2 after a gauge fixing is equivalent to a super extension of the modified U(n) chiral model (so as to be integrable) first formulated by Ward [7] . Since integrability is compatible with noncommutative deformation (if introduced properly, see e.g. [9] - [20] ) we choose from the beginning to formulate our super extension of this chiral model on Moyal-deformed R 2,1 with noncommutativity parameter θ ≥ 0. Ordinary space-time R 2,1 can always be restored by taking the commutative limit θ → 0.\n\nStar-product formulation. Classical field theory on noncommutative spaces may be realized in a star-product formulation or in an operator formalism 8 . The first approach is closer to the commutative field theory: it is obtained by simply deforming the ordinary product of classical fields (or their components) to the noncommutative star product\n\n(f ⋆ g)(x) = f (x) exp{ i 2 ← - ∂ a θ ab -→ ∂ b } g(x) ⇒ x a ⋆ x b -x b ⋆ x a = iθ ab (3.1)\n\nwith a constant antisymmetric tensor θ ab . Specializing to R 2,1 , we use real coordinates (x a ) = (t, x, y) in which the Minkowski metric g on R 3 reads (g ab ) = diag(-1, +1, +1) with a, b, . . . = 1, 2, 3 (cf. Section 2). It is straightforward to generalize the Moyal deformation (3.1) to the superspaces introduced in the previous section, allowing in particular for non-anticommuting Grassmann-odd coordinates. Deferring general superspace deformations and their consequences to future work, we here content ourselves with the simple embedding of the \"bosonic\" Moyal deformation into superspace, meaning that (3.1) is also valid for superfields f and g depending on Grassmann variables θ iα and η α i . For later use we consider not only isotropic coordinates and vector fields\n\nu := 1 2 (t+y) = y 22 , v := 1 2 (t-y) = y 11 , ∂ u = ∂ t + ∂ y = ∂ (22) , ∂ v = ∂ t -∂ y = ∂ (11) (3.2)\n\nintroduced in Section 2, but also the complex combinations\n\nz := x + iy , z := x -iy , ∂ z = 1 2 (∂ x -i∂ y ) , ∂ z = 1 2 (∂ x + i∂ y ) . (3.3)\n\nSince the time coordinate t remains commutative, the only nonvanishing component of the noncommutativity tensor θ ab is Operator formalism. The nonlocality of the star products renders explicit computation cumbersome. We therefore pass to the operator formalism, which trades the star product for operatorvalued spatial coordinates (x, ŷ) or their complex combinations (ẑ, ẑ), subject to\n\nθ xy = -θ yx =: θ > 0 ⇒ θ z z = -θ zz = -2i θ . ( 3\n\n[t, x] = [t, ŷ] = 0 but [x, ŷ] = iθ ⇒ [ẑ, ẑ] = 2 θ . (3.6)\n\nThe latter equation suggests the introduction of annihilation and creation operators,\n\na = 1 √ 2θ ẑ and a † = 1 √ 2θ ẑ with [a , a † ] = 1 , (3.7)\n\nwhich act on a harmonic-oscillator Fock space H with an orthonormal basis { |ℓ , ℓ = 0, 1, 2, . . .} such that\n\na |ℓ = √ ℓ |ℓ-1 and a † |ℓ = √ ℓ+1 |ℓ+1 . (3.8)\n\nAny superfield f (t, z, z, η α i ) on R 3|2N can be related to an operator-valued superfield f (t, η α i ) ≡ F (t, a, a † , η α i ) on R 1|2N acting in H, with the help of the Moyal-Weyl map\n\nf (t, z, z, η α i ) → f (t, η α i ) = Weyl-ordered f t, √ 2θa, √ 2θa † , η α i . (3.9)\n\nThe inverse transformation recovers the ordinary superfield,\n\nf (t, η α i ) ≡ F (t, a, a † , η α i ) → f (t, z, z, η α i ) = F ⋆ t, z √ 2θ , z √ 2θ , η α i , (3.10)\n\nwhere F ⋆ is obtained from F by replacing ordinary with star products. Under the Moyal-Weyl map, we have\n\nf ⋆ g → f ĝ and dx dy f = 2π θ Tr f = 2π θ ℓ≥0 ℓ| f |ℓ , (3.11)\n\nand the spatial derivatives are mapped into commutators,\n\n∂ z f → ∂z f = -1 √ 2θ [a † , f ] and ∂ z f → ∂z f = 1 √ 2θ [a , f ] . (3.12)\n\nFor notational simplicity we will from now on omit the hats over the operators except when confusion may arise.\n\nGauge fixing for ψ. Note that the linear system (2.40) and the compatibility conditions (2.43) are invariant under a gauge transformation\n\nψ → ψ ′ = g -1 ψ , (3.13a)\n\nA → A ′ = g -1 A g + g -1 ∂ g (with appropriate indices) , (3.13b)\n\nΞ → Ξ ′ = g -1 Ξ g , (3.13c)\n\nwhere g = g(x a , η α i ) is a U(n)-valued superfield globally defined on the deformed superspace R 3|2N θ × CP 1 . Using a gauge transformation of the form (3.13), we can choose ψ such that it will satisfy the standard asymptotic conditions (see e.g. [51] )\n\nψ = Φ -1 + O(ζ) for ζ → 0 , (3.14a\n\n)\n\nψ = 1l + ζ -1 Υ + O(ζ -2 ) for ζ → ∞ , (3.14b)\n\nwhere the U(n)-valued function Φ and u(n)-valued function Υ depend on x a and η α i . This \"unitary\" gauge is compatible with the reality condition for ψ,\n\nψ(x a , η α i , ζ) ψ(x a , η α i , ζ) † = 1l , (3.15)\n\nobtained by reduction from (2.26)."
},
{
"section_type": "OTHER",
"section_title": "Gauge fixing for",
"text": "A i 1 = 0 and A i 2 = Φ -1 ∂ i 2 Φ = ∂ i 1 Υ , (3.17)\n\nA (11) = 0 and\n\nA (12) + Ξ = Φ -1 ∂ (12) Φ = ∂ (11) Υ , (3.18)\n\nA (21) -Ξ = 0 and\n\nA (22) = Φ -1 ∂ (22) Φ = ∂ (12) Υ . (3.19)\n\nUsing (2.32), we can rewrite the nonzero components as\n\nA := Φ -1 ∂ u Φ = ∂ x Υ , B := Φ -1 ∂ x Φ = ∂ v Υ , C i := Φ -1 ∂ i 2 Φ = ∂ i 1 Υ . (3.20)\n\nRecall that the superfields Φ and Υ depend on x a and η α i .\n\nLinear system. In the above-introduced unitary gauge the linear system (2.42) reads\n\n(ζ∂ x -∂ u -A) ψ = 0 , (ζ∂ v -∂ x -B) ψ = 0 , (ζ∂ i 1 -∂ i 2 -C i ) ψ = 0 , (3.21)\n\nwhich adds the last equation to the linear system of the Ward model [7] and generalizes it to superfields A(x a , η α j ), B(x a , η α j ) and C i (x a , η α j ). The concise form of (3.21) reads\n\nζ Di 1 -Di 2 -Âi 2 ψ = 0 (3.22)\n\nor, in more explicit form,\n\nζ ∂ i 1 + 2θ i1 ∂ v + 2θ i2 ∂ x -∂ i 2 + C i + 2θ i1 (∂ x + B) + 2θ i2 (∂ u + A) ψ = 0 . (3.23)\n\nN -extended sigma model.\n\nThe compatibility conditions of this linear system are the Nextended noncommutative sigma model equations Di\n\n1 (Φ -1 Dj 2 Φ) + Dj 1 (Φ -1 Di 2 Φ) = 0 (3.24)\n\nwhich in expanded form reads\n\n(g ab + v c ε cab ) ∂ a (Φ -1 ∂ b Φ) = 0 ⇔ ∂ x (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ u Φ) = 0 , (3.25a) ∂ i 1 (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ i 2 Φ) = 0 , ∂ i 1 (Φ -1 ∂ u Φ) -∂ x (Φ -1 ∂ i 2 Φ) = 0 , (3.25b)\n\n∂ i 1 (Φ -1 ∂ j 2 Φ) + ∂ j 1 (Φ -1 ∂ i 2 Φ) = 0 . (3.25c)\n\nHere, the first line contains the Wess-Zumino-Witten term with a constant vector (v c ) = (0, 1, 0) which spoils the standard Lorentz invariance but yields an integrable chiral model in 2+1 dimensions. Recall that Φ is a U(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . As discussed in Section 2, the compatibility conditions of the linear equations (3.22) (or (3.21)) are equivalent to the N -extended Bogomolnytype equations (2.39) for the component (physical) fields. Thus, chiral model field equations (3.25) are equivalent to a gauge fixed form of equations (2.39). Υ-formulation. Instead of Φ-parametrization of (A, B, C i ) given in (3.17)-(3.20) we may use the equivalent Υ-parametrization also given there. In this case, the compatibility conditions for the linear system (3.21) reduce to\n\n(∂ 2 x -∂ u ∂ v )Υ + [∂ v Υ , ∂ x Υ] = 0 , (3.26a)\n\n(∂ i 2 ∂ v -∂ i 1 ∂ x )Υ + [∂ i 1 Υ , ∂ v Υ] = 0 , (∂ i 2 ∂ x -∂ i 1 ∂ u )Υ + [∂ i 1 Υ , ∂ x Υ] = 0 , (3.26b) (∂ i 2 ∂ j 1 + ∂ j 2 ∂ i 1 )Υ + {∂ i 1 Υ , ∂ j 1 Υ} = 0 , (3.26c)\n\nwhich in concise form read\n\n( Di 2 Dj 1 + Dj 2 Di 1 ) Υ + { Di 1 Υ , Dj 1 Υ} = 0 . (3.27)\n\nRecall that Υ is a u(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . For N =4, the commutative limit of (3.27) can be considered as Siegel's equation [33] reduced to 2+1 dimensions. According to Siegel, one can extract the multiplet of physical fields appearing in (2.39) from the prepotential Υ via\n\n∂ i 1 Υ = A i 2 , ∂ i 1 ∂ j 1 Υ = φ ij , ∂ i 1 ∂ j 1 ∂ k 1 Υ = χ[ijk] 2 , ∂ i 1 ∂ j 1 ∂ k 1 ∂ l 1 Υ = G [ijkl] 22 , (3.28a)\n\n∂ (α1) Υ = A (α2) -ε α2 ϕ , ∂ (α1) ∂ i 1 Υ = χ i α , ∂ (α1) ∂ (β1) Υ = f αβ , (3.28b)\n\nwhere one takes Υ and its derivatives at η 2 i = 0. The other components of the physical fields, i.e. χ[ijk]\n\n1 , G [ijkl] 11 , G [ijkl]\n\n21 , A (11) and A (21) -ϕ, vanish in this light-cone gauge.\n\nSupersymmetry transformations. The 4N supercharges given in (2.11) reduce in 2+1 dimensions to the form\n\nQ iα = ∂ iα -η β i ∂ (αβ) and Q i α = ∂ i α -θ iβ ∂ (αβ) . (3.29)\n\nTheir antichiral version, matching to Diα and Dj β of (2.35), reads\n\nQiα = ∂ iα -2η β i ∂ (αβ) and Qj β = ∂ j β , (3.30)\n\nso that { Qiα , Qj β } = -2 δ j i ∂ (αβ) . (3.31)\n\nOn a (scalar) R 3|2N superfield Σ these supersymmetry transformations act as\n\nδ Σ := ε iα Qiα Σ + ε α i Qi α Σ (3.32)\n\nand are induced by the coordinate shifts\n\nδ y αβ = -2ε i(α η β) i and δ η α i = ε α i , (3.33)\n\nwhere ε iα and ε α i are 4N real Grassmann parameters. It is easy to see that our equations (3.24) and (3.27) are invariant under the supersymmetry transformations (3.32) (applied to Φ or Υ). This is simply because the operators Diα and Dj β anticommute with the supersymmetry generators Qiα and Qj β . Therefore, the equations of motion (3.25) of the modified N -extended chiral model in 2+1 dimensions as well as their reductions to 2+0 and 1+1 dimensions carry 2N supersymmetries and are genuine supersymmetric extensions of the corresponding bosonic equations. Note that this type of extension is not the standard one since the R-symmetry groups are Spin(N , N ) in 2+1 and Spin(N , N )× Spin(N , N ) in 1+1 dimensions, which differ from the compact unitary R-symmetry groups of standard sigma models. Contrary to the standard case of two-dimensional sigma models the above \"noncompact\" 2N supersymmetries do not impose any constraints on the geometry of the target space, e.g. they do not demand it to be Kähler [52] or hyper-Kähler [53] . This may be of interest and deserves further study.\n\nAction functionals. In either formulation of the N -extended supersymmetric SDYM model on R 2,2 there are difficulties with finding a proper action functional generalizing the one [54, 55] for the purely bosonic case. These difficulties persist after the reduction to 2+1 dimensions, i.e. for the equations (3.25) and (3.26) describing our supersymmetric modified U(n) chiral model. It is the price to be paid for overcoming the no-go barrier N ≤ 4 and the absence of geometric target-space constraints. On a more formal level, the problem is related to the chiral character of (3.24) as well as (3.27) , where only the operators Di α but not Diα appear. Note however, that for N = 4 one can write an action functional in component fields producing the equations (2.39), which are equivalent to the superspace equations (3.24) when i, j = 1, . . . , 4 (see e.g. [47] ).\n\nOne proposal for an action functional stems from Siegel's idea [33] for the Υ-formulation of the N -extended SDYM equations. Namely, one sees that ∂ i 2 Υ enters only linearly into the last two lines in (3.26) . Therefore, if we introduce\n\nΥ (1) := Υ| η 2 i =0 (3.34)\n\nthen it must satisfy the first equation from (3.26) , and the remaining equations iteratively define the dependence of Υ on η 2 i starting from Υ (1) . Hence, all information is contained in Υ (1) , as can also be seen from (3.28) . In other words, the dependence of Υ on η 2 i is not 'dynamical'. For an action one can then take (cf. [33] )\n\nS = d 3 x d N η 1 Υ (1) ∂ (αβ) ∂ (αβ) Υ (1) + 2 3 Υ (1) ε αβ ∂ (α1) Υ (1) ∂ (β1) Υ (1) . (3.35)\n\nExtremizing this functional yields the first line of (3.26) at η 2 i = 0. Except for the Grassmann integration, this action has the same form as the purely bosonic one [55] . One may apply the same logic to the Φ-formulation where the action for the purely bosonic case is also known [54, 56] ."
},
{
"section_type": "OTHER",
"section_title": "N -extended multi-soliton configurations via dressing",
"text": "The existence of the linear system (3.22) (equivalent to (3.21)) encoding solutions of the N -extended U(n) chiral model in an auxiliary matrix ψ allows for powerful methods to systematically construct explicit solutions for ψ and hence for Φ † = ψ| ζ=0 and Υ = lim ζ→∞ ζ (ψ-1l). For our purposes the so-called dressing method [57, 51] proves to be the most practical [12] - [20] , and so we shall use it here for our linear system, i.e. already in the N -extended noncommutative case.\n\nMulti-pole ansatz for ψ. The dressing method is a recursive procedure for generating a new solution from an old one. More concretely, we rewrite the linear system (3.21) in the form\n\nψ(∂ u -ζ∂ x )ψ † = A , ψ(∂ x -ζ∂ v )ψ † = B , ψ(∂ i 2 -ζ∂ i 1 )ψ † = C i . (4.1)\n\nRecall that ψ † := (ψ(x a , η α i , ζ)) † and (A, B, C i ) depend only on x a and η α i . The central idea is to demand analyticity in the spectral parameter ζ, which strongly restricts the possible form of ψ. One way to exploit this constraint starts from the observation that the left hand sides of (4.1) as well as of the reality condition (3.15) do not depend on ζ while ψ is expected to be a nontrivial function of ζ globally defined on CP 1 . Therefore, it must be a meromorphic function on CP 1 possessing some poles which we choose to lie at finite points with constant coordinates µ k ∈ CP 1 .\n\nHere we will build a (multi-soliton) solution ψ m featuring m simple poles at positions µ 1 , . . . , µ m with foot_8 Im µ k < 0 by left-multiplying an (m-1)-pole solution ψ m-1 with a single-pole factor of the form 1l\n\n+ µ m -μm ζ -µ m P m (x a , η α i ) , (4.2)\n\nwhere the n×n matrix function P m is yet to be determined. Starting from the trivial (vacuum) solution ψ 0 = 1l, the iteration ψ 0 → ψ 1 → . . . → ψ m yields a multiplicative ansatz for ψ m ,\n\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ , (4.3)\n\nwhich, via partial fraction decomposition, may be rewritten in the additive form\n\nψ m = 1l + m k=1 Λ mk S † k ζ -µ k , (4.4)\n\nwhere Λ mk and S k are some n×r k matrices depending on x a and η α i , with r k ≤ n.\n\nEquations for S k . Let us first consider the additive parametrization (4.4) of ψ m . This ansatz must satisfy the reality condition (3.15) as well as our linear equations in the form (4.1). In particular, the poles at ζ = μk on the left hand sides of these equations have to be removable since the right hand sides are independent of ζ. Inserting the ansatz (4.4) and putting to zero the corresponding residues, we learn from (3.15) that\n\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ S k = 0 , (4.5)\n\nwhile from (4.1) we obtain the differential equations\n\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ LA,B,i k S k = 0 , (4.6)\n\nwhere LA,B,i\n\nk stands for either LA k = ∂ u -μk ∂ x , LB k = µ k (∂ x -μk ∂ v ) or Li k = ∂ i 2 -μk ∂ i 1 . (4.7)\n\nNote that we consider a recursive procedure starting from m=1, and operators (4.7) will appear with k = 1, . . . , m if we consider poles at ζ = μk .\n\nBecause the LA,B,i k for k = 1, . . . , m are linear differential operators, it is easy to write down the general solution for (4.6) at any given k, by passing from the coordinates (u, v, x; η 1 i , η 2 i ) to \"co-moving coordinates\" (w k , wk , s k ; η i k , ηi k ). The precise relation for k = 1, . . . , m is [12, 58]\n\nw k := x + μk u + μ-1 k v = x + 1 2 (μ k -μ -1 k )y + 1 2 (μ k +μ -1 k )t and η i k := η 1 i + μk η 2 i , (4.8)\n\nwith wk and ηi k obtained by complex conjugation and the co-moving time s k being inessential because by definition nothing will depend on it. The kth moving frame travels with a constant velocity\n\n(v x , v y ) k = - µ k + μk µ k μk + 1 , µ k μk -1 µ k μk + 1 , (4.9)\n\nso that the static case w k =z is recovered for µ k = -i. On functions of (w k , η i k , wk , ηi k ) alone the operators (4.7) act as\n\nLA k = LB k = (µ k -μ k ) ∂ ∂ wk =: Lk and Li k = (µ k -μ k ) ∂ ∂ ηi k . (4.10)\n\nBy induction in k = 1, . . . , m we learn that, due to (4.5), a necessary and sufficient condition for a solution of (4.6) is Lk\n\nS k = S k Zk and Li k S k = S k Zi k (4.11)\n\nwith some r k ×r k matrices Zk and Zi k depending on (w k , wk , η j k , ηj k ). Passing to the noncommutative bosonic coordinates we obtain ŵk , ŵk = 2θ ν k νk with\n\nν k νk = 4i µ k -μ k -µ -1 k +μ -1 k . (4.12)\n\nThus, we can introduce annihilation and creation operators\n\nc k = 1 √ 2θ ŵk ν k and c † k = 1 √ 2θ ŵk νk so that [c k , c † k ] = 1 (4.13)\n\nfor k = 1, . . . , m. Naturally, this Heisenberg algebra is realized on a \"co-moving\" Fock space H k , with basis states |ℓ k and a \"co-moving\" vacuum |0 k subject to c k |0 k = 0. Each co-moving vacuum |0 k (annihilated by c k ) is related to the static vacuum |0 (annihilated by a) through an ISU(1,1) squeezing transformation (cf. [12] ) which is time-dependent. The fermionic coordinates η i k and ηi k remain spectators in the deformation. Coordinate derivatives are represented in the standard fashion as\n\nν k √ 2θ ∂ ∂w k → -[c † k , • ] and νk √ 2θ ∂ ∂ wk → [c k , • ] . (4.14)\n\nAfter the Moyal deformation, the n×r k matrices S k have become operator-valued, but are still functions of the Grassmann coordinates η i k and ηi k . The noncommutative version of the BPS conditions (4.11) naturally reads\n\nc k S k = S k Z k and ∂ ∂ ηi k S k = S k Z i k (4.15)\n\nwhere Z k and Z i k are some operator-valued r k ×r k matrix functions of η j k and ηj k .\n\nNonabelian solutions for S k . For general data Z k and Z i k it is difficult to solve (4.15), but it is also unnecessary because the final expression ψ m turns out not to depend on them. Therefore, we conveniently choose\n\nZ k = c k ⊗ 1l r k ×r k and Z i k = 0 ⇒ S k = R k (c k , η i k ) , (4.16)\n\nwhere R k is an arbitrary n×r k matrix function independent of c † k and ηi k . 10 It is known that nonabelian (multi-) solitons arise for algebraic functions R k (cf. e.g. [7] for the commutative and [12] for the noncommutative N =0 case). Their common feature is a smooth commutative limit. The only novelty of the supersymmetric extension is the η i k dependence, i.e.\n\nR k = R k,0 + η i k R k,i + η i k η j k R k,ij + η i k η j k η p k R k,ijp + η i k η j k η p k η q k R k,ijpq . (4.17)\n\nAbelian solutions for S k . It is useful to view S k as a map from C r k ⊗H k to C n ⊗H k (momentarily suppressing the η dependence). The noncommutative setup now allows us to generalize the domain of this map to any subspace of C n ⊗ H k . In particular, we may choose it to be finite-dimensional, say C q k , and represent the map by an n×q k array |S k of kets in H. In this situation, Z k and Z i k in (4.15) are just number -valued q k ×q k matrix functions of η j k and ηj k . In case they do not depend on ηj k , we can write down the most general solution as\n\n|S k = R k (c k , η j k ) |Z k exp i Z i k (η j k ) ηi k with |Z k := exp Z k (η j k ) c † k |0 k . (4.18)\n\nAs before, we may put Z i k = 0 without loss of generality, but now the choice of Z k does matter. For any given k generically there exists a q k -dimensional basis change which diagonalizes the ket-valued matrix\n\n|Z k → diag e α 1 k c † , e α 2 k c † , . . . , e α q k k c † |0 k = diag |α 1 k , |α 2 k , . . . , |α q k k , (4.19)\n\nwhere we defined coherent states\n\n|α l k := e α l k c † |0 k so that c k |α l k = α l k |α l k for l = 1, . . . , q k and α l k ∈ C . (4.20)\n\nNote that not only the entries of R k but also the α l k are holomorphic functions of the co-moving Grassmann parameters η j k and thus can be expanded like in (4.17). In the U(1) model, we must use ket-valued 1×q k matrices |S k for all k, yielding rows\n\n|S k = R 1 k |α 1 k , R 2 k |α 2 k , . . . , R q k k |α q k k for k = 1, . . . , m , (4.21)\n\nwith functions α l k (η j k ). Here, the R l k only affect the states' normalization and can be collected in a diagonal matrix to the right, hence will drop out later and thus may all be put to one. Formally, we have recovered the known abelian (multi-) soliton solutions, but the supersymmetric extension has generalized |S k → |S k (η j k ) . Explicit form of P k . Let us now consider the multiplicative parametrization (4.3) of ψ m which also allows us to solve (4.5) . First of all, note that the reality condition (3.15) is satisfied if\n\nP k = P † k = P 2 k ⇔ P k = T k (T † k T k ) -1 T † k for k = 1, . . . , m , (4.22)\n\nmeaning that P k is an operator-valued hermitian projector (of group-space rank r k ≤ n) built from an n×r k matrix function T k (the abelian case of n=1 is included). The reality condition follows just because The r k columns of T k span the image of P k and obey\n\n1l + µ k -μk ζ -µ k P k 1l + μk -µ k ζ -μk P k =\n\nP k T k = T k ⇔ (1l-P k ) T k = 0 . (4.24)\n\nFurthermore, the equation (4.5) with m = k (induction) rewritten in the form\n\n(1l-P k ) k-1 ℓ=1 1l + µ k-ℓ -μk-ℓ μk -µ k-ℓ P k-ℓ S k = 0 (4.25)\n\nreveals that (cf. (4.24))\n\nT 1 = S 1 and T k = k-1 ℓ=1 1l - µ k-ℓ -μk-ℓ µ k-ℓ -μk P k-ℓ S k for k ≥ 2 , (4.26)\n\nwhere the explicit form of S k for k = 1, . . . , m is given in (4.16) or (4.18) . The final result reads\n\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ = 1l + m k=1 Λ mk S † k ζ -µ k (4.27)\n\nwith hermitian projectors P k given by (4.22), T k given by (4.26) and S k given by (4.16) or (4.18) . The explicit form of Λ mk (which we do not need) can be found in [12] . The corresponding superfields Φ and Υ are\n\nΦ m = ψ † m | ζ=0 = m k=1 (1l -ρ k P k ) with ρ k = 1 - µ k μk , (4.28a)\n\nΥ m = lim ζ→∞ ζ (ψ m -1l) = m k=1 (µ k -μ k ) P k . (4.28b)\n\nFrom (4.22) it is obvious that P k is invariant under a similarity transformation\n\nT k → T k Λ k ⇔ S k → S k Λ k (4.29)\n\nfor an invertible operator-valued r k ×r k matrix Λ k . This justifies putting Z i k = 0 from the beginning and also the restriction to Z k = c k ⊗ 1l r k ×r k in the nonabelian case, both without loss of generality. Hence, the nonabelian solution space constructed here is parametrized by the set {R k } m 1 of matrixvalued functions of c k and η i k and the pole positions µ k . The abelian moduli space, however, is larger by the set {Z k } m 1 of matrix-values functions of η i k which generically contain the coherentstate parameter functions {α l k (η i k )}. Restricting to η i k =0 reproduces the soliton configurations of the bosonic model [12] .\n\nStatic solutions. Let us consider the reduction to 2+0 dimensions, i.e. the static case. Recall that static solutions correspond to the choice m = 1 and µ 1 ≡ µ = -i implying w 1 = z, so we drop the index k. Specializing (4.27), we have\n\nψ = 1l - 2 i ζ + i P so that Φ = Φ † = 1l -2P , (4.30)\n\nwhere a hermitian projector P of group-space rank r satisfies the BPS equations (1l-P ) a P = 0 ⇒ (1l-P ) a T = 0 ,\n\n(1l-P ) ∂ ∂ ηi P = 0 ⇒ (1l-P ) ∂ ∂ ηi T = 0 , (4.31a)\n\nwith P = T (T † T ) -1 T † and η i = η 1 i + iη 2 i . In this case T = S, and for a nonabelian r=1 projector P we get T = T (a, η i ) as an n×1 column. For the simplest case of N =1 we just have (cf. [59] )\n\nT = T e (a) + η T o (a) with η = η 1 + iη 2 , (4.32)\n\nwhere T e (a) and T o (a) are rational functions of a (e.g. polynomials) taking values in the even and odd parts of the Grassmann algebra. Similarly, an abelian N =1 projector (for n=1) is built from\n\n|T = |α 1 , |α 2 , . . . , |α q . (4.33)\n\nAt θ=0, the static solution (4.32) of our supersymmetric U(n) sigma model is also a solution of the standard N =1 supersymmetric CP n-1 sigma model in two dimensions (see e.g. [59] ). 11 For this reason, one can overcome the previously mentioned difficulty with constructing an action (or energy from the viewpoint of 2+1 dimensions) for static configurations. Moreover, on solutions obeying the BPS conditions (4.31) the topological charge\n\nQ = 2πθ dη 1 dη 2 Tr tr Φ D + Φ , D -Φ (4.34)\n\nis proportional to the action (BPS bound)\n\nS = 2πθ dη 1 dη 2 Tr tr D + Φ , D -Φ (4.35)\n\nand is finite for algebraic functions T e and T o . Here, the standard superderivatives D ± are defined as\n\nD + = ∂ ∂η + iη ∂ z and D -= ∂ ∂ η + iη ∂ z . ( 4\n\nOne-soliton configuration. For one moving soliton, from (4.27) and (4.28) we obtain\n\nψ 1 = 1l + µ - μ ζ -µ P with P = T (T † T ) -1 T † (4.37) and Φ = 1l -ρ P with ρ = 1 - µ μ . (4.38)\n\nNow our n×r matrix T must satisfy (putting\n\nZ i = 0 and Z = c ⊗ 1l r×r ) [c , T ] = 0 and ∂ ∂ ηi T = 0 with η i = η 1 i + μ η 2 i , (4.39)\n\nwhere c is the moving-frame annihilation operator given by (4.13) for k=1.\n\nRecall that the operators c and c † and therefore the matrix T and the projector P can be expressed in terms of the corresponding static objects by a unitary squeezing transformation (see e.g. (4.8) and (4.13)). For simplicity we again consider the case N =1 and a nonabelian projector with r=1. Then (4.39) tells us that T is a holomorphic function of c and η, i.e.\n\nT = T e (c) + η T o (c) = T 1 e (c) + η T 1 o (c)\n\n. . .\n\nT n e (c) + η T n o (c) (4.40)\n\nwith polynomials T a e and T a o of order q, say, analogously to the static case (4.32). Note that, for T a o to be Grassmann-odd and nonzero, some extraneous Grassmann parameter must appear. Similarly, abelian projectors for a moving one-soliton obtain by subjecting (4.33) to a squeezing transformation.\n\nFor N =1 the moving frame was defined in (4.8) (dropping the index k) via\n\nw = x + 1 2 (μ-μ -1 )y + 1 2 (μ+μ -1 )t and η = η 1 + μη 2 hence ∂ t η = 0 . (4.41)\n\nConsider the moving frame with the coordinates (w, w, s; η, η) with the choice s = t and the related change of the derivatives (see [12, 58] )\n\n∂ x = ∂ w + ∂ w , (4.42a)\n\n∂ y = 1 2 (μ-μ -1 ) ∂ w + 1 2 (µ-µ -1 ) ∂ w , (4.42b)\n\n∂ t = 1 2 (μ+μ -1 ) ∂ w + 1 2 (µ+µ -1 ) ∂ w + ∂ s , (4.42c)\n\n∂ η 1 = ∂ η + ∂ η , (4.42d)\n\n∂ η 2 = μ ∂ η + µ ∂ η . (4.42e)\n\nIn the moving frame our solution (4.38) is static, i.e. ∂ s Φ = 0, and the projector P has the same form as in the static case. The only difference is the coefficient ρ instead of 2 in (4.38). Therefore, by computing the action (4.35) in (w, w; η 1 , η 2 ) coordinates, we obtain for algebraic functions T in (4.40) a finite answer, which differs from the static one by a kinematical prefactor depending on µ (cf. [12] for the bosonic case).\n\nLarge-time asymptotics. Note that in the distinguished (z, z, t) coordinate frame (4.41) implies that at large times w → κ t with κ = 1 2 (μ+μ -1 ). As a consequence, the t q term in each polynomial in (4.40) will dominate, i.e.\n\nT → t q a 1 + η b 1 . . .\n\nan + η bn =: t q Γ , (4.43)\n\nwhere Γ is a fixed vector in C n . It is easy to see that in the distinguished frame the large-time limit of Φ given by (4. being the projector on the constant vector Γ.\n\nConsider now the m-soliton configuration (4.28). By induction of the above argument one easily arrives at the m-soliton generalization of (4.44). Namely, in the frame moving with the ℓth lump we have lim t→±∞ Φ m = (1lρ 1 Π 1 ) . . . (1lρ ℓ-1 Π ℓ-1 )(1lρ ℓ P ℓ )(1lρ ℓ+1 Π ℓ+1 ) . . . (1lρ m Π m ) , (4.45) where the Π m are constant projectors. This large-time factorization of multi-soliton solutions provides a proof of the no-scattering property because the asymptotic configurations are identical for large negative and large positive times."
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusions",
"text": "In this paper we introduced a generalization of the modified integrable U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions and considered a Moyal deformation of this model. It was shown that this N -extended chiral model is equivalent to a gauge-fixed BPS subsector of an N -extended super Yang-Mills model in 2+1 dimensions originating from twistor string theory. The dressing method was applied to generate a wide class of multi-soliton configurations, which are time-dependent finite-energy solutions to the equations of motion. Compared to the N =0 model, the supersymmetric extension was seen to promote the configurations' building blocks to holomorphic functions of suitable Grassmann coordinates. By considering the large-time asymptotic factorization into a product of single soliton solutions we have shown that no scattering occurs within the dressing ansatz chosen here.\n\nThe considered model does not stand alone but is motivated by twistor string theory [37] with a target space reduced to the mini-supertwistor space [44, 45, 47] . In this context, the obtained multi-soliton solutions are to be regarded as D(0|2N )-branes moving inside D(2|2N )-branes [60] . Here 2N appears due to fermionic worldvolume directions of our branes in the superspace description [60] . Switching on a constant B-field simply deforms the sigma model and D-brane worldvolumes noncommutatively, thereby admitting also regular supersymmetric noncommutative abelian solutions.\n\nRestricting to static configurations, the models can be specialized to Grassmannian supersymmetric sigma models, where the superfield Φ takes values in Gr(r, n), and the field equations are invariant under 2N supersymmetry transformations with 0 ≤ N ≤ 4. This differs from the results for standard 2D sigma models [52, 53] where the target spaces have to be Kähler or hyper-Kähler for admitting two or four supersymmetries, respectively. This difference will be discussed in more details elsewhere.\n\nWe derived the supersymmetric chiral model in 2+1 dimensions through dimensional reduction and gauge fixing of the N -extended supersymmetric SDYM equations in 2+2 dimensions. Recall that for the purely bosonic case most (if not all) integrable equations in three and fewer dimensions can be obtained from the SDYM equations (or their hierarchy [25] ) by suitable dimensional reductions (see e.g. [61] - [65] and references therein). Moreover, this Ward conjecture [61] was extended to the noncommutative case (see e.g. [66, 67] ). It will be interesting to consider similar reductions of the N -extended supersymmetric SDYM equations (and their hierarchy [68] ) to supersymmetric integrable equations in three and two dimensions generalizing earlier results [69] ."
}
] | |
arxiv:0704.0531 | 0704.0531 | 1 | 10.1088/1126-6708/2007/11/079 | b0e695f3b0dd96c805ccd0cc547d6cced9bf6b91d8fab221fb029b978f81b498 | Gravitational Duality Transformations on (A)dS4 | We discuss the implementation of electric-magnetic duality transformations in
four-dimensional gravity linearized around Minkowski or (A)dS4 backgrounds. In
the presence of a cosmological constant duality generically modifies the
Hamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular
attention to the boundary terms generated by the duality transformations and
discuss their implications for holography. | [
"Robert G. Leigh",
"Anastasios C. Petkou"
] | [
"hep-th"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | We discuss the implementation of electric-magnetic duality transformations in fourdimensional gravity linearized around Minkowski or (A)dS 4 backgrounds. In the presence of a cosmological constant duality generically modifies the Hamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular attention to the boundary terms generated by the duality transformations and discuss their implications for holography. 1 For reviews of higher-spin theories see e.g. [1]. 2 See [4] and [5] for more recent works.
Contents 1. Introduction and Summary 2. Action and Hamiltonian 2.1 The 3 + 1 split 2.2 Shifted Variables 2.3 Linearization 3. Linearized Gravitational Duality and Holography 3.1 Duality and Holography 3.2 Linearized gravitational duality 3.3 Linearized Constraints and Bianchi Identities 3.4 Connection with other known dualities 4. The Effect on the Boundary Theory 5. Conclusions and Outlook 6. Appendix: other duality mappings
Duality has played an important role in our understanding of Yang-Mill theories and it is believed that it will play an important role also in gravity and in higher-spin gauge theories. Indeed, although it is less clear what could be the implications of duality for theories whose quantum versions are still unknown, gravity and higher-spin gauge theories 1 are intimately connected to a quantum string theory where certainly duality plays a crucial role. The recent advent of holography raises some intriguing questions for duality. For example one may wonder what is the holographic image of a duality invariant spectrum, a duality transformation or a possible quantization condition that usually duality implies for charges. Some of these issues were raised by Witten in [2] where it was argued that the standard electric-magnetic duality of a U(1) gauge theory on AdS 4 is responsible for a "natural" SL(2, Z) action on current two-point functions in three-dimensional CFTs. 2 Shortly afterwards it was shown in [3] that such an SL(2, Z) action is intimately related to certain "double-trace" deformations in the boundary, assuming suitable large-N limits and existence of non-trivial fixed points. The latter assumptions are strengthened by the fact that there exist models (e.g., see [6] and references therein) which exhibit the required behavior. In particular, it was shown in [3] that certain "double-trace" deformations induce an SL(2, Z) action on two-point functions of higher-spin (i.e. spin s ≥ 2) currents. This has led to the Duality Conjecture of [3] : linearized higher-spin theories on AdS 4 spaces possess a generalization of electric-magnetic duality whose holographic image is the natural SL(2, Z) action on boundary two-point functions.
Surprisingly, even the duality for linearized spin-2 gauge fields (linearized gravity) was not widely known by the time of this conjecture. 3 Second order linearized gravitational duality was discussed among other in [8, 9, 10, 11, 12] . More recently, the duality properties of linearized gravity around flat space were studied in [13] and were further discussed in [14] . The duality of linearized gravity around dS 4 was later studied in [15] .
In this note we present our calculations regarding the duality properties of gravity in the presence of a cosmological constant. Having in mind applications to higher-spin gauge theories we use forms and work in the first order formalism where duality is also manifested at the level of the action [16] . Moreover, the first order formalism is relevant for applications of duality to holography, since the correlation functions of the boundary theory are essentially determined by the bulk canonical momenta (see e.g. [17] ).
Our aim in this work is to formulate linearized first order gravity using suitable "electric" and "magnetic" variables, in close analogy with electromagnetism. We find that this is possible only when the background geometry is Minkowski or (A)dS 4 . Then we implement the standard electric-magnetic duality rotations. We find that, up to "boundary" terms, the linearized Hamiltonian changes by terms that do not alter the bulk dynamics i.e. do not alter the second order bulk equations of motion. Moreover, the duality rotation interchanges the (linearized) constraints with the (linearized) Bianchi identitites. The "boundary" terms have important holographic consequences since they correspond to marginal "double-trace" deformations [3] that induce the boundary SL(2, Z) action. In the Appendix we exhibit a modified duality rotation that leaves the bulk Hamiltonian invariant and induces "boundary" terms that correspond to relevant deformations as in [3] .
Having in mind the extension of our results to higher-spin gauge theorieswe start from the MacDowell-Mansouri form [18] of the gravitational action foot_1
I M M = 1 2Λ M ǫ abcd R ab ∧ R cd + 2Λe a ∧ e b ∧ R cd + Λ 2 e a ∧ e b ∧ e c ∧ e d , (2.1)
where a, b, ... are Lorentz indices. In this formalism, the vierbein e a and the spin connection ω a b are initially thought of as independent variables. The curvature 2-form is
R a b = dω a b + ω a c ∧ ω c b = 1 2 R a bcd e c ∧ e d .
Varying the action with respect to e a and ω a b , we find
R ab + Λe a ∧ e b = 0 , (2.2)
T a = de a + ω a b ∧ e b = 0 . (2.
3)
The relation to gravity is established via the vanishing torsion equation (2.3), which relates e and ω in the familiar way. The above equations are equivalent to the Einstein equation in metric variables R µν -
1 2 Rg µν = +3Λg µν . (2.4)
and the scalar curvature is R = -d(d -1)Λ = -12Λ. Note that our Λ is related to the cosmological constant in its usual definition via Λ cosm = -6Λ. Λ > 0 corresponds to AdS. Note that this is actually SO(3, 2) covariant, as we can combine ω, e into a super-connection. Note that Λ has units (Length) -2 . In the SO(3, 2)-invariant formalism, I M M arises from
I M M = 1 2Λ ǫ ABCDE V E R AB ∧ R CD , (2.5)
where V E is a non-dynamical 0-form field (that we take to have value V -1 = 1 to gauge back to the SO(3, 1) formalism) and R A B is the curvature of Ω A B ≡ {e a , ω a b }. There are also quasitopological terms of the form
I top = θ 2Λ R A B ∧ R B A + α Λ R A B ∧ R AC V B V C (2.6)
that we could add to the action. In the stated gauge, this reduces to
I top = θ 2Λ P 2 + (θ + α)C N Y + α R ab ∧ e a ∧ e b (2.7)
where
P 2 = R a b ∧ R b a is the Pontryagin class, C N Y = (T a ∧ T a -R ab ∧ e a ∧ e b
) is the Nieh-Yan class and we also note the Euler class E 2 = ǫ abcd R ab ∧ R cd . Note that in the presence of torsion, the action (2.7) contains the non-topological term R ab ∧ e a ∧ e b with "Immirzi parameter" γ = -2/α. In the absence of torsion, this term is a total derivative.
The Hilbert-Palatini action is
I HP = I M M - 1 2Λ E 2 .
(2.8)
It differs from I M M by a boundary term, is smooth as Λ → 0 but is not manifestly SO(3, 2)invariant.
Next, we carefully consider the 3 + 1 split. Although much of the discussion here is familiar from the ADM formalism, we feel it is important to set notation carefully, as we will introduce some new ingredients. To accommodate both AdS and dS signatures simultaneously, we will introduce a 'time' function t and a foliation of space-time Σ t ֒→ M. In dS, t is time-like, and this corresponds to the usual Hamiltonian foliation; in AdS on the other hand, we will take t to be the (space-like) radial coordinate. We will keep track of the resulting signs by a parameter σ ⊥ , equal to ±1 in dS(AdS).
Proceeding as usual then, we get a vector field t that satisfies ∇ t t = 1 ≡ t(t) (so t = ∂ ∂t ) and a 1-form dt. Given a 4-metric, we can introduce the normal 1-form n as
n = σ ⊥ Ndt , (2.9)
which is normalized as (n, n) = σ ⊥ . The dual vector field n can be expanded as
n = 1 N t - 1 N N , (2.10)
where the shift N satisfies (N, n) = 0, and thus (t, n) = σ ⊥ N.
Next, we will locally choose a basis of 1-forms
e 0 = σ ⊥ n = Ndt , (2.11)
e α = ẽα + N α dt .
(2.12)
The ẽα span T * Σ t , and correspond to a 3-metric h ij = ẽα i ẽβ j η αβ . The quantities N α are the components of N: N α = e α i N i . These basis 1-forms are dual to {e 0 = n, e α = ẽα }, with e a (e b ) = δ b a . We expand the spin connection in the same basis 5
ω a b = q a b dt + ωa b , (2.13)
which leads to
R a b = Ra b + dt ∧ r a b , (2.14)
where R is formed from ω and d only, and
r a b = ωa b -dq a b -ωa c q c b + q a c ωc b . (2.15)
Note that these quantities are merely decompositions along T * Σ t in the 4-geometry; we will introduce the intrinsically defined objects shortly.
We then find
I HP = 2ǫ αβγ dt ∧ N( Rαβ + Λẽ α ∧ ẽβ ) ∧ ẽγ -2N α ( R0β ) ∧ ẽγ + r 0α ∧ ẽβ ∧ ẽγ .
(2.16) 5 We have
q a b = N ω 0 a b + N α ω α a b and ωa b ≡ ω α a b ẽα .
As is familiar, the lapse and shift appear as Lagrange multipliers. The constraints that they multiply are of course zero in any background (i.e. vacuum solution), such as (A)dS 4 . The final term in the action contains the real dynamics -r 0α depends on the components R 0α 0β of the Riemann tensor.
Note though that the tensors used here are 4-dimensional. Let us define the "electric field"
K α = σ ⊥ ω0 α = K βα ẽβ .
(2.17)
In the case that ω is the torsion-free Levi-Civita connection, this agrees with the standard definition for extrinsic curvature, regarded as a vector-valued one-form. We then find
Rα β = (3) R α β -σ ⊥ K α ∧ K β , (2.18)
and
R0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ ( DK) α . (2.19)
These equations amount to the Gauss-Codazzi relations. Furthermore, r 0α contains time derivatives of ω0α as well as terms linear in components of q. We find
2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = 2ǫ αβγ σ ⊥ Kα -( Dq) 0α ∧ ẽβ ∧ ẽγ , (2.20)
= 2σ ⊥ ǫ αβγ Kα + q αδ K δ ∧ ẽβ ∧ ẽγ + 4q 0α ǫ αβγ T β ∧ ẽγ up to a total 3-derivative. We have defined the intrinsic 3-torsion T α = dẽ α + ωα β ∧ ẽβ . Since we wish to regard the ẽ as coordinate variables, 6 we integrate the first term by parts to obtain (up to the total time-derivative
∂ ∂t 2σ ⊥ K α ∧ ẽβ ∧ ẽγ ǫ αβγ ) 2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = Π α ∧ ėα + 4q 0α ǫ αβγ T β ∧ ẽγ + 2σ ⊥ q αδ ǫ αβγ K δ ∧ ẽβ ∧ ẽγ . (2.21)
where we have defined the momentum 2-form
Π α = -4σ ⊥ ǫ αβγ K β ∧ ẽγ . (2.22)
The q ab appear as Lagrange multipliers. In particular, the q αβ constraint precisely sets the antisymmetric (torsional) part of the extrinsic curvature tensor K [αβ] to zero. Next, we define the "magnetic field"
B α = 1 2 σ ⊥ ǫ αβγ ωβγ , ω αβ = -ǫ αβγ B γ . (2.23)
and we find that the q 0α constraint
ǫ αβγ T β ∧ ẽγ = ǫ αβγ dẽ β ∧ ẽγ -σ ⊥ B β ∧ ẽβ ∧ ẽα = 0 , (2.24)
involves only the antisymmetric part B [α,β] of the magnetic field B α = B αβ ẽβ . The antisymmetric part of B α spoils the gauge covariance of the constraint (2.24) under an SO(3) rotation of the dreibein ẽα , hence it represents degrees of freedom that can be gauged fixed to zero by an SO(3) rotation. On the other hand, an algebraic equation of motion connects the symmetric part of B αβ to derivatives of ẽα as dẽ α + ǫ αβγ B β ∧ ẽγ = 0 (2.25)
At the end, one is left with the canonically conjugate variables ẽα and Π α . These results are familiar from the metric formalism.
Dropping the torsional terms, we then arrive at the action
I HP = dt ∧ ėα ∧ Π α + 2Nǫ αβγ ( (3) R αβ -σ ⊥ K α ∧ K β + Λẽ α ∧ ẽβ ) ∧ ẽγ -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ . (2.26) Furthermore, using * 3 ẽα ∧ ẽβ = 1 2 η γδ ǫ αβγ ẽδ , we have Πα = * 3 Π α = -2(K αβ -η αβ trK)ẽ β , (2.27)
where trK = η αβ K αβ . We can solve the above equation to get
K α = - 1 2 ( Παβ - 1 2 η αβ tr Π)ẽ β .
(2.28)
As stated above, K αβ (and Παβ ) is symmetric when the torsion vanishes. Finally, with the definition (2.23) we find foot_3
ǫ αβγ (3) R αβ ∧ ẽγ = ǫ αβγ dω αβ + ωα δ ∧ ωδβ ∧ ẽγ = σ ⊥ 2 dB γ + ǫ αβγ B α ∧ B β ∧ ẽγ . (2.29)
Introducing B α is an unusual thing to do but it will play a role in duality: in this form, the Hamiltonian contains terms which are reminiscent of those of the Maxwell theory. The full HP action is of the form
I HP = dt ∧ ėα ∧ Π α -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ +2σ ⊥ N(2 dB γ + ǫ αβγ B α ∧ B β -ǫ αβγ K α ∧ K β + σ ⊥ Λǫ αβγ ẽα ∧ ẽβ ) ∧ ẽγ . (2.30)
Note that the entire contribution of the cosmological constant appears in the last term of the Hamiltonian constraint.
It is possible to make a transformation of the canonical variables in order to absorb the cosmological constant term in (2.30 ). This can be achieved by introducing the new variables
Kα = K α -ρẽ α , (2.31)
and requiring that
ρ 2 = σ ⊥ Λ . (2.32)
This is positive only when σ ⊥ and Λ are simultaneously positive or negative, as it is the case for both AdS 4 (Λ > 0) and dS 4 (Λ < 0). We will often write Λ = σ ⊥ /L 2 where L is a length scale. Under (2.31) the momentum 2-form becomes
Π α → P α -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ . (2.33)
The last term in (2.33) contributes a total time derivative to the action (of the form of a boundary cosmological term). We have introduced a new momentum variable
P α = -4σ ⊥ ǫ αβγ Kβ ∧ ẽγ .
Then, we get the action
I HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K + ρ T ) β ∧ ẽγ - 4 3 σ ⊥ ρǫ αβγ ∂ ∂t (ẽ α ∧ ẽβ ∧ ẽγ ) +2σ ⊥ N 2 d(B α ∧ ẽα ) + 2B γ ∧ T γ -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ .(2.34)
Note that the shift constraint is still written in terms of the ordinary covariant derivative, and thus involves a non-linear term coupling B to K. Consistent with our previous discussion, we drop the terms involving the torsion T , and disregard the boundary term to obtain
I HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K) β ∧ ẽγ +2σ ⊥ N 2 d(B α ∧ ẽα ) -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ . (2.35)
We note that the parameter ρ can be of either sign (although, this sign does not appear in the second order equations of motion).
Next, we linearize the above action around an appropriate fixed background. We expand as
ẽα = ẽα + E α , N = 1 + n, N α = n α , B α = B α + b α , Kα = Kα + k α . (2.36)
The background values should satisfy the constraints. The simplest choice is the background where Kα = 0 = B α .
(2.37)
In fact, reaching this simple form was a motivation for the shift (2.31). Then, to quadratic order in the fluctuating fields the Hamiltonian gives
I HP = dt ∧ Ėα ∧ p α -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + 4σ ⊥ n d(b α ∧ ẽα ) -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β + 2ρk α ∧ E β ∧ ẽγ , (2.38)
where
p α = -4σ ⊥ ǫ αβγ k β ∧ ẽγ (2.39)
are the linearized momentum variables conjugate to E α . In order to reach the form (2.38) the linear terms in the fluctuations must vanish. For this to happen we find the relationships ėα + ρẽ α = 0 .
(2.40)
Notice that we can also write the linearized action in the form
I HP = dt ∧ ( Ėα + ρE α ) ∧ p α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + n 4σ ⊥ db γ + ρp γ ∧ ẽγ . (2.41)
The form of the first term, involving the momentum, makes clear that longitudinal fluctuations are non-dynamical. The natural time dependence of E α is of the form e -ρt (correspondingly, the natural time dependence of p α is e +ρt ). Other than that, we see that in comparing to the flat space action, in these variables, the only change is that the Hamiltonian constraint is modified. The solutions of (2.40) and (2.37) are components of (A)dS 4 spacetimes. We can solve (2.40) to obtain e 0 = dt, e α = e -ρt dx α .
(2.42)
With these we construct the usual Poincaré metric on (A)dS which, however, covers only half of the space even though the parameter t runs from -∞ to +∞. The conformal boundary in these coordinates is at t = +∞. Then we derive
ω α 0 = -ρe -ρt dx α = -ρe α , (2.43)
and so
R α β = -σ ⊥ L 2 e α ∧ e β R α 0 = -1 L 2 e α ∧ e 0 R a b = - σ ⊥ L 2 e a ∧ e b . ( 2
Hence Ric ab = -3σ ⊥ L 2 η ab and R = -12σ ⊥ /L 2 = -12Λ. We also evaluate Π α = -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ , Πα = 4ρẽ α , tr Π = 12ρ (2.45)
B α = 0, K α = ρẽ α ⇒ Kα = 0 (2.46)
Note that in this gauge, ( DK) α = 1 L T α = 0, which solves the shift constraint, while the Hamiltonian constraint is satisfied through a cancellation between the K 2 term and the cosmological term.
Let us summarize what we have obtained so far. In the presence of a cosmological constant we have defined variables such that the action resembles most closely the action without the cosmological constant. This was done in order to look for a suitable background around which linear fluctuations are as simple as possible. Requiring that K (the "electric field") and B (the "magnetic field") vanish in such a background -as they do around flat space -we found that the background should be (A)dS 4 . Quite satisfactorily, both sign choices for ρ in the change of variables (2.31) lead to (A)dS 4 spacetimes.
This is the appropriate point to recall some salient features of duality rotations. In simple Hamiltonian systems the effect of the canonical transformation p → q and q → -p to the action is (see e.g. [19] )
I = t 2 t 1 dt[p q -H(p, q)] → I D = t 2 t 1 dt[-q ṗ -H(q, -p)] . (3.1)
Notice that I D involves the dual variables, for which we have however kept the same notation for simplicity. The transformed Hamiltonian H(q, -p) is in general not related to H(p, q). However, if H(q, -p) = H(p, q) we call the above transformation a duality. It then holds
I D = I -qp t 2 t 1 . (3.2)
The dual action describes exactly the same dynamics as the initial one, up to a modification of the boundary conditions. For example, if I is stationary on the e.o.m for fixed q in the boundary, I D is stationary on the same e.o.m. for fixed p in the boundary. This simple example illustrates the role of duality in holography; a bulk duality transformation corresponds to a particular modification of the boundary conditions. This property of duality transformations is behind the remarkable holographic properties of electormagnetism in (A)dS 4 [2, 3] .
Clearly, the crucial properties of a duality transformation are to be canonical and to leave the Hamiltonian unchanged. However, consider a slight generalization
S = t 2 t 1 dt[p q - 1 2 (p 2 + q 2 + 2λpq)] (3.3)
where λ is an arbitrary parameter. The Hamiltonian now is not invariant under the canonical transformation p → q and q → -p -the pq term changes sign. Consequently, the first order form of the equations of motion are also not duality invariant. Nevertheless, the second order equation of motion is invariant. We will find that gravity in the presence of a cosmological constant follows precisely this model. Of course, gravity is a much more complicated constrained system, but as we will show, the constraints and Bianchi identities transform appropriately.
We also note that the canonical transformation (implemented by a generating functional of the first kind) p → q + 2λp , q → -p . (3.4) is of interest here. The above does not change the Hamiltonian and the transformed action differs from the initial one by total time derivative terms foot_4 S → S D = S -pq
t 2 t 1 -λp 2 t 2 t 1 . (3.5)
As a preamble to gravity we recall the duality properties of Maxwell theory
I M ax = 1 2g 2 dt ∧ Ȧ ∧ * 3 E - 1 2 (E ∧ * 3 E + B ∧ * 3 B) -A 0 d * 3 E , (3.6)
Under the duality E → - * 3 B, B → * 3 E, Ã → ÃD , we find
I M ax → I M ax,D = 1 2g 2 dt ∧ -ȦD ∧ B - 1 2 (E ∧ * 3 E + B ∧ * 3 B) + A 0 dB . (3.7)
E and B in (3.7) should be expressed through ÃD . We observe that the kinetic term has changed sign, while the Hamiltonian remains invariant. In addition, the (Gauss) constraint is dualized to the trivial 'Bianchi' identity dB = 0 for the dual magnetic field.
Next we try to apply a Maxwell-type duality map in gravity. We consider the following transformation around the fixed background (2.40)
k α → -b α , b α → k α . (3.8)
To implement the map (3.8) we need to specify the mapping of E α to a 'dual 3-bein' E α . We do that using the linearized form of (2.25) as
ǫ αβγ b β ∧ ẽγ + dE α = 0 → ǫ αβγ k β ∧ ẽγ + dE α = 0 = dE α - 1 4σ ⊥ p α (3.9) Since p α = 4σ ⊥ dE α , it is natural to define p D,α = 4σ ⊥ dE α = -4σ ⊥ ǫ αβγ b β ∧ ẽγ , (3.10)
and thus the mapping (3.8) is supplemented by
E → E , E → -E , p → -p D , p D → p (3.11)
Now, let us see the effects of the above duality mapping. The action transforms to
I HP → I HP,D = dt ∧ -Ėα ∧ p D,α -ρE α ∧ p D,α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ (3.12) +4σ ⊥ n α ǫ αβγ db β ∧ ẽγ + n 4σ ⊥ dk γ + ρp D,α ∧ ẽγ
where now k α and b α should be expressed in terms of the dual variables E α and p D,α via (3.9) and (3.10). We notice that the 'kinetic' part Ė ∧ p of the action changes sign under the duality map, in direct analogy with the Maxwell case. However, the Hamiltonian is not invariant due to the change of sign of the second term in the first line of (3.12). We will discuss this further in a later section. For now, we note that this sign change would not show up in the equations of motion, written in second order form. It is important to also note that the constraints are transformed into quantities which in the next subsection we will recognize as the linearized Bianchi identities. This is to be expected since the duality transformations are canonical. We also note that it may be possible to choose an alternative canonical transformation, designed to leave the Hamiltonian invariant. The latter is presumably related to the work of Julia et. al. [15] and is considered in the Appendix.
By virtue of the discussion above we may now demonstrate that under the duality mapping (3.8) the linearized constraints transform to the linearized Bianchi identities as
C α ≡ ǫ αβγ dk β ∧ ẽγ → -ǫ αβγ db β ∧ ẽγ (3.13) C 0 ≡ -σ ⊥ db γ -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ → -σ ⊥ dk γ + ρǫ αβγ b α ∧ ẽβ ∧ ẽγ (3.14)
To identify the right hand sides, we first note that the Bianchi identities are
B R a b = dR a b -R a c ∧ ω c b + ω a c ∧ R c b = 0 (3.15) B a T = dT a -R a b ∧ e b + ω a b ∧ T b = 0 (3.16)
which are obtained from the definitions of R a b and T a by exterior differentiation. The first equation is satisfied identically. Since the torsion vanishes, the second equation tells us only that R a b ∧ e b = 0. If we do the 3+1 split, we find two equations. The first is
B T α = -( (3) R α β -σ ⊥ K α ∧ K β ) ∧ ẽβ = 0 (3.17)
which upon using the symmetry of K α linearizes to
B T α = -ǫ αβγ db β ∧ ẽγ + . . . (3.18)
Note that this is the image under duality of the shift constraint as in (3.13) .
The second identity is
B T 0 = -R0 α ∧ ẽα = -σ ⊥ ( DK) α = -σ ⊥ dk α + ρǫ αβγ b β ∧ ẽγ ∧ ẽα = 0 (3.19)
where to arrive in the second line we used (2.46) . This is the image of the Hamiltonian constraint as in (3.14) . Summarizing, the duality transformations between linearized constraints and Bianchi identities are
C α → B T,α C 0 → B 0 T (3.20) B T,α → -C α B 0 T → -C 0 (3.21)
The Maxwell-type duality operation (3. We begin with the spatial 2-forms when we have
Rαβ = -ǫ αβγ dB γ + σ ⊥ (B α ∧ B β -K α ∧ K β ) (3.23) R0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ (
DK) α (3.24) and S0 γ = 1 2 σ ⊥ ǫ αβγ Rαβ (3.25) Sαβ = ǫ αβγ R0γ (3.26) If we linearize these expressions, we find under the duality transformation (3.8) Rab → -σ ⊥ Sab (3.27) Because the expressions (3.24) involve derivatives of B and K, the duality (3.8) is an 'integrated form' of the usual Riemann tensor duality, but implies it. Similarly, if we investigate the spatial 1-forms, we find r ab → -σ ⊥ s ab (3.28)
To arrive at this result we have set to zero the Lagrange multiplier field q.
It is well known that AdS is holographic. We may well ask, in the context of AdS/CFT, how the duality transformation that we have defined here acts in the boundary. We are instructed to consider the on-shell bulk action as a function of bulk fields. So, we evaluate the action on a solution to the equation of motion, resulting in a pure boundary term which is of the form
S bdy = ∂M p α ∧ E α (4.1)
Applying the duality transformation to the bulk theory, although the bulk action is not invariant as we have discussed above, nevertheless it may be easily shown that it induces a simple transformation on the (linearized) boundary term: it simply changes its sign.
S dual bdy = - ∂M p D,α ∧ E α (4.2)
This transformation is exactly analogous to what happens in the Maxwell case: it amounts to the result [21] .
10 G 2 G dual 2 = -1 . (4.3)
Motivated by possible application in holography and in higher-spin gauge theory we have studied the duality properties of gravity in the Hamiltonian formulation. We have presented the gravity action in terms of suitable variables that closely resemble the electric and magnetic fields in Maxwell theory. We have found suitable "electric" and "magnetic" field variables, such that at the linearized level first order gravity most closely resembles electromagnetism. This can be done only around Minkowksi and (A)dS 4 backgrounds. We have implemented duality transformations in the linearized gravity fluctuations around these backgrounds. In the presence of a cosmological constant, the Hamiltonian changes, nevertheless the bulk dynamics remains unaltered, while the linearized lapse and shift constraints are mapped into the linearized Bianchi identities. Moreover, the duality transformations induce boundary terms whose relevance in holography we have briefly discussed. Finally, we have exhibited a modified duality rotation that leaves the bulk Hamiltonian invariant, while it induces boundary terms corresponding to relevant deformations.
The main implication of our results is that certain properties of correlations functions in three-dimensional CFTs mimic the duality of gravity. It would be interesting to extend our results to black-hole backgrounds and also when topological terms are present in the bulk. We also expect that one can analyze the duality of higher-spin gauge theories based on our first-order approach.
The work of A. C. P. was partially supported by the research program "PYTHAGORAS II" of the Greek Ministry of Education. RGL was supported in part by the U.S. Department of Energy under contract DE-FG02-91ER40709.
It is possible to find a transformation that leaves the Hamiltonian unchanged. Consider the following transformation in the fixed background (2.40)
k α → -b α -2ρE α , b α → k α . (6.1)
The mapping to the dual dreibein is still specified by (3.9). A straightforward calculation reveals that the action transforms as The transformations (6.1) leaves unchanged the Hamiltonian and changes the action by the total "time" derivative terms shown in the first line of (6.2). Moreover, the linearized constraints transform into the linearized Bianchi identities. Let us see that in some detail. The second term in the shift constraint is zero since k α is a symmetric one form k α = k αβ ẽβ with k αβ = k βα ; see (2.21) . The term proportional to Λ in the lapse constraint is also zero. This is slightly more involved to see and it is based on the possibility of solving (3.9) for E α after gauge fixing. 11 One way to see this is in components. Write E α = E α β ẽβ and (3.9) becomes
I
∂ α E β γ -∂ γ E β α = ǫ β δα k δ γ -ǫ β δγ k δ α (6.3)
In the "Lorentz gauge" where ∂ α E β α = 0 = ∂ α k β α the above can be inverted as
E α β = 1 ∂ 2 ǫ α δγ ∂ γ k δ β (6.4)
Using (6.4) one verifies that the last term in the lapse constraint vanishes. This modified duality transformation is probably related to the one considered by Julia et. al. in [15] . | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We discuss the implementation of electric-magnetic duality transformations in fourdimensional gravity linearized around Minkowski or (A)dS 4 backgrounds. In the presence of a cosmological constant duality generically modifies the Hamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular attention to the boundary terms generated by the duality transformations and discuss their implications for holography. 1 For reviews of higher-spin theories see e.g. [1]. 2 See [4] and [5] for more recent works."
},
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Contents 1. Introduction and Summary 2. Action and Hamiltonian 2.1 The 3 + 1 split 2.2 Shifted Variables 2.3 Linearization 3. Linearized Gravitational Duality and Holography 3.1 Duality and Holography 3.2 Linearized gravitational duality 3.3 Linearized Constraints and Bianchi Identities 3.4 Connection with other known dualities 4. The Effect on the Boundary Theory 5. Conclusions and Outlook 6. Appendix: other duality mappings"
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction and Summary",
"text": "Duality has played an important role in our understanding of Yang-Mill theories and it is believed that it will play an important role also in gravity and in higher-spin gauge theories. Indeed, although it is less clear what could be the implications of duality for theories whose quantum versions are still unknown, gravity and higher-spin gauge theories 1 are intimately connected to a quantum string theory where certainly duality plays a crucial role. The recent advent of holography raises some intriguing questions for duality. For example one may wonder what is the holographic image of a duality invariant spectrum, a duality transformation or a possible quantization condition that usually duality implies for charges. Some of these issues were raised by Witten in [2] where it was argued that the standard electric-magnetic duality of a U(1) gauge theory on AdS 4 is responsible for a \"natural\" SL(2, Z) action on current two-point functions in three-dimensional CFTs. 2 Shortly afterwards it was shown in [3] that such an SL(2, Z) action is intimately related to certain \"double-trace\" deformations in the boundary, assuming suitable large-N limits and existence of non-trivial fixed points. The latter assumptions are strengthened by the fact that there exist models (e.g., see [6] and references therein) which exhibit the required behavior. In particular, it was shown in [3] that certain \"double-trace\" deformations induce an SL(2, Z) action on two-point functions of higher-spin (i.e. spin s ≥ 2) currents. This has led to the Duality Conjecture of [3] : linearized higher-spin theories on AdS 4 spaces possess a generalization of electric-magnetic duality whose holographic image is the natural SL(2, Z) action on boundary two-point functions.\n\nSurprisingly, even the duality for linearized spin-2 gauge fields (linearized gravity) was not widely known by the time of this conjecture. 3 Second order linearized gravitational duality was discussed among other in [8, 9, 10, 11, 12] . More recently, the duality properties of linearized gravity around flat space were studied in [13] and were further discussed in [14] . The duality of linearized gravity around dS 4 was later studied in [15] .\n\nIn this note we present our calculations regarding the duality properties of gravity in the presence of a cosmological constant. Having in mind applications to higher-spin gauge theories we use forms and work in the first order formalism where duality is also manifested at the level of the action [16] . Moreover, the first order formalism is relevant for applications of duality to holography, since the correlation functions of the boundary theory are essentially determined by the bulk canonical momenta (see e.g. [17] ).\n\nOur aim in this work is to formulate linearized first order gravity using suitable \"electric\" and \"magnetic\" variables, in close analogy with electromagnetism. We find that this is possible only when the background geometry is Minkowski or (A)dS 4 . Then we implement the standard electric-magnetic duality rotations. We find that, up to \"boundary\" terms, the linearized Hamiltonian changes by terms that do not alter the bulk dynamics i.e. do not alter the second order bulk equations of motion. Moreover, the duality rotation interchanges the (linearized) constraints with the (linearized) Bianchi identitites. The \"boundary\" terms have important holographic consequences since they correspond to marginal \"double-trace\" deformations [3] that induce the boundary SL(2, Z) action. In the Appendix we exhibit a modified duality rotation that leaves the bulk Hamiltonian invariant and induces \"boundary\" terms that correspond to relevant deformations as in [3] ."
},
{
"section_type": "OTHER",
"section_title": "Action and Hamiltonian",
"text": "Having in mind the extension of our results to higher-spin gauge theorieswe start from the MacDowell-Mansouri form [18] of the gravitational action foot_1\n\nI M M = 1 2Λ M ǫ abcd R ab ∧ R cd + 2Λe a ∧ e b ∧ R cd + Λ 2 e a ∧ e b ∧ e c ∧ e d , (2.1)\n\nwhere a, b, ... are Lorentz indices. In this formalism, the vierbein e a and the spin connection ω a b are initially thought of as independent variables. The curvature 2-form is\n\nR a b = dω a b + ω a c ∧ ω c b = 1 2 R a bcd e c ∧ e d .\n\nVarying the action with respect to e a and ω a b , we find\n\nR ab + Λe a ∧ e b = 0 , (2.2)\n\nT a = de a + ω a b ∧ e b = 0 . (2.\n\n3)\n\nThe relation to gravity is established via the vanishing torsion equation (2.3), which relates e and ω in the familiar way. The above equations are equivalent to the Einstein equation in metric variables R µν -\n\n1 2 Rg µν = +3Λg µν . (2.4)\n\nand the scalar curvature is R = -d(d -1)Λ = -12Λ. Note that our Λ is related to the cosmological constant in its usual definition via Λ cosm = -6Λ. Λ > 0 corresponds to AdS. Note that this is actually SO(3, 2) covariant, as we can combine ω, e into a super-connection. Note that Λ has units (Length) -2 . In the SO(3, 2)-invariant formalism, I M M arises from\n\nI M M = 1 2Λ ǫ ABCDE V E R AB ∧ R CD , (2.5)\n\nwhere V E is a non-dynamical 0-form field (that we take to have value V -1 = 1 to gauge back to the SO(3, 1) formalism) and R A B is the curvature of Ω A B ≡ {e a , ω a b }. There are also quasitopological terms of the form\n\nI top = θ 2Λ R A B ∧ R B A + α Λ R A B ∧ R AC V B V C (2.6)\n\nthat we could add to the action. In the stated gauge, this reduces to\n\nI top = θ 2Λ P 2 + (θ + α)C N Y + α R ab ∧ e a ∧ e b (2.7)\n\nwhere\n\nP 2 = R a b ∧ R b a is the Pontryagin class, C N Y = (T a ∧ T a -R ab ∧ e a ∧ e b\n\n) is the Nieh-Yan class and we also note the Euler class E 2 = ǫ abcd R ab ∧ R cd . Note that in the presence of torsion, the action (2.7) contains the non-topological term R ab ∧ e a ∧ e b with \"Immirzi parameter\" γ = -2/α. In the absence of torsion, this term is a total derivative.\n\nThe Hilbert-Palatini action is\n\nI HP = I M M - 1 2Λ E 2 .\n\n(2.8)\n\nIt differs from I M M by a boundary term, is smooth as Λ → 0 but is not manifestly SO(3, 2)invariant."
},
{
"section_type": "OTHER",
"section_title": "The 3 + 1 split",
"text": "Next, we carefully consider the 3 + 1 split. Although much of the discussion here is familiar from the ADM formalism, we feel it is important to set notation carefully, as we will introduce some new ingredients. To accommodate both AdS and dS signatures simultaneously, we will introduce a 'time' function t and a foliation of space-time Σ t ֒→ M. In dS, t is time-like, and this corresponds to the usual Hamiltonian foliation; in AdS on the other hand, we will take t to be the (space-like) radial coordinate. We will keep track of the resulting signs by a parameter σ ⊥ , equal to ±1 in dS(AdS).\n\nProceeding as usual then, we get a vector field t that satisfies ∇ t t = 1 ≡ t(t) (so t = ∂ ∂t ) and a 1-form dt. Given a 4-metric, we can introduce the normal 1-form n as\n\nn = σ ⊥ Ndt , (2.9)\n\nwhich is normalized as (n, n) = σ ⊥ . The dual vector field n can be expanded as\n\nn = 1 N t - 1 N N , (2.10)\n\nwhere the shift N satisfies (N, n) = 0, and thus (t, n) = σ ⊥ N.\n\nNext, we will locally choose a basis of 1-forms\n\ne 0 = σ ⊥ n = Ndt , (2.11)\n\ne α = ẽα + N α dt .\n\n(2.12)\n\nThe ẽα span T * Σ t , and correspond to a 3-metric h ij = ẽα i ẽβ j η αβ . The quantities N α are the components of N: N α = e α i N i . These basis 1-forms are dual to {e 0 = n, e α = ẽα }, with e a (e b ) = δ b a . We expand the spin connection in the same basis 5\n\nω a b = q a b dt + ωa b , (2.13)\n\nwhich leads to\n\nR a b = Ra b + dt ∧ r a b , (2.14)\n\nwhere R is formed from ω and d only, and\n\nr a b = ωa b -dq a b -ωa c q c b + q a c ωc b . (2.15)\n\nNote that these quantities are merely decompositions along T * Σ t in the 4-geometry; we will introduce the intrinsically defined objects shortly.\n\nWe then find\n\nI HP = 2ǫ αβγ dt ∧ N( Rαβ + Λẽ α ∧ ẽβ ) ∧ ẽγ -2N α ( R0β ) ∧ ẽγ + r 0α ∧ ẽβ ∧ ẽγ .\n\n(2.16) 5 We have\n\nq a b = N ω 0 a b + N α ω α a b and ωa b ≡ ω α a b ẽα .\n\nAs is familiar, the lapse and shift appear as Lagrange multipliers. The constraints that they multiply are of course zero in any background (i.e. vacuum solution), such as (A)dS 4 . The final term in the action contains the real dynamics -r 0α depends on the components R 0α 0β of the Riemann tensor.\n\nNote though that the tensors used here are 4-dimensional. Let us define the \"electric field\"\n\nK α = σ ⊥ ω0 α = K βα ẽβ .\n\n(2.17)\n\nIn the case that ω is the torsion-free Levi-Civita connection, this agrees with the standard definition for extrinsic curvature, regarded as a vector-valued one-form. We then find\n\nRα β = (3) R α β -σ ⊥ K α ∧ K β , (2.18)\n\nand\n\nR0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ ( DK) α . (2.19)\n\nThese equations amount to the Gauss-Codazzi relations. Furthermore, r 0α contains time derivatives of ω0α as well as terms linear in components of q. We find\n\n2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = 2ǫ αβγ σ ⊥ Kα -( Dq) 0α ∧ ẽβ ∧ ẽγ , (2.20)\n\n= 2σ ⊥ ǫ αβγ Kα + q αδ K δ ∧ ẽβ ∧ ẽγ + 4q 0α ǫ αβγ T β ∧ ẽγ up to a total 3-derivative. We have defined the intrinsic 3-torsion T α = dẽ α + ωα β ∧ ẽβ . Since we wish to regard the ẽ as coordinate variables, 6 we integrate the first term by parts to obtain (up to the total time-derivative\n\n∂ ∂t 2σ ⊥ K α ∧ ẽβ ∧ ẽγ ǫ αβγ ) 2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = Π α ∧ ėα + 4q 0α ǫ αβγ T β ∧ ẽγ + 2σ ⊥ q αδ ǫ αβγ K δ ∧ ẽβ ∧ ẽγ . (2.21)\n\nwhere we have defined the momentum 2-form\n\nΠ α = -4σ ⊥ ǫ αβγ K β ∧ ẽγ . (2.22)\n\nThe q ab appear as Lagrange multipliers. In particular, the q αβ constraint precisely sets the antisymmetric (torsional) part of the extrinsic curvature tensor K [αβ] to zero. Next, we define the \"magnetic field\"\n\nB α = 1 2 σ ⊥ ǫ αβγ ωβγ , ω αβ = -ǫ αβγ B γ . (2.23)\n\nand we find that the q 0α constraint\n\nǫ αβγ T β ∧ ẽγ = ǫ αβγ dẽ β ∧ ẽγ -σ ⊥ B β ∧ ẽβ ∧ ẽα = 0 , (2.24)\n\ninvolves only the antisymmetric part B [α,β] of the magnetic field B α = B αβ ẽβ . The antisymmetric part of B α spoils the gauge covariance of the constraint (2.24) under an SO(3) rotation of the dreibein ẽα , hence it represents degrees of freedom that can be gauged fixed to zero by an SO(3) rotation. On the other hand, an algebraic equation of motion connects the symmetric part of B αβ to derivatives of ẽα as dẽ α + ǫ αβγ B β ∧ ẽγ = 0 (2.25)\n\nAt the end, one is left with the canonically conjugate variables ẽα and Π α . These results are familiar from the metric formalism.\n\nDropping the torsional terms, we then arrive at the action\n\nI HP = dt ∧ ėα ∧ Π α + 2Nǫ αβγ ( (3) R αβ -σ ⊥ K α ∧ K β + Λẽ α ∧ ẽβ ) ∧ ẽγ -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ . (2.26) Furthermore, using * 3 ẽα ∧ ẽβ = 1 2 η γδ ǫ αβγ ẽδ , we have Πα = * 3 Π α = -2(K αβ -η αβ trK)ẽ β , (2.27)\n\nwhere trK = η αβ K αβ . We can solve the above equation to get\n\nK α = - 1 2 ( Παβ - 1 2 η αβ tr Π)ẽ β .\n\n(2.28)\n\nAs stated above, K αβ (and Παβ ) is symmetric when the torsion vanishes. Finally, with the definition (2.23) we find foot_3\n\nǫ αβγ (3) R αβ ∧ ẽγ = ǫ αβγ dω αβ + ωα δ ∧ ωδβ ∧ ẽγ = σ ⊥ 2 dB γ + ǫ αβγ B α ∧ B β ∧ ẽγ . (2.29)\n\nIntroducing B α is an unusual thing to do but it will play a role in duality: in this form, the Hamiltonian contains terms which are reminiscent of those of the Maxwell theory. The full HP action is of the form\n\nI HP = dt ∧ ėα ∧ Π α -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ +2σ ⊥ N(2 dB γ + ǫ αβγ B α ∧ B β -ǫ αβγ K α ∧ K β + σ ⊥ Λǫ αβγ ẽα ∧ ẽβ ) ∧ ẽγ . (2.30)\n\nNote that the entire contribution of the cosmological constant appears in the last term of the Hamiltonian constraint."
},
{
"section_type": "OTHER",
"section_title": "Shifted Variables",
"text": "It is possible to make a transformation of the canonical variables in order to absorb the cosmological constant term in (2.30 ). This can be achieved by introducing the new variables\n\nKα = K α -ρẽ α , (2.31)\n\nand requiring that\n\nρ 2 = σ ⊥ Λ . (2.32)\n\nThis is positive only when σ ⊥ and Λ are simultaneously positive or negative, as it is the case for both AdS 4 (Λ > 0) and dS 4 (Λ < 0). We will often write Λ = σ ⊥ /L 2 where L is a length scale. Under (2.31) the momentum 2-form becomes\n\nΠ α → P α -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ . (2.33)\n\nThe last term in (2.33) contributes a total time derivative to the action (of the form of a boundary cosmological term). We have introduced a new momentum variable\n\nP α = -4σ ⊥ ǫ αβγ Kβ ∧ ẽγ .\n\nThen, we get the action\n\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K + ρ T ) β ∧ ẽγ - 4 3 σ ⊥ ρǫ αβγ ∂ ∂t (ẽ α ∧ ẽβ ∧ ẽγ ) +2σ ⊥ N 2 d(B α ∧ ẽα ) + 2B γ ∧ T γ -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ .(2.34)\n\nNote that the shift constraint is still written in terms of the ordinary covariant derivative, and thus involves a non-linear term coupling B to K. Consistent with our previous discussion, we drop the terms involving the torsion T , and disregard the boundary term to obtain\n\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K) β ∧ ẽγ +2σ ⊥ N 2 d(B α ∧ ẽα ) -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ . (2.35)\n\nWe note that the parameter ρ can be of either sign (although, this sign does not appear in the second order equations of motion)."
},
{
"section_type": "OTHER",
"section_title": "Linearization",
"text": "Next, we linearize the above action around an appropriate fixed background. We expand as\n\nẽα = ẽα + E α , N = 1 + n, N α = n α , B α = B α + b α , Kα = Kα + k α . (2.36)\n\nThe background values should satisfy the constraints. The simplest choice is the background where Kα = 0 = B α .\n\n(2.37)\n\nIn fact, reaching this simple form was a motivation for the shift (2.31). Then, to quadratic order in the fluctuating fields the Hamiltonian gives\n\nI HP = dt ∧ Ėα ∧ p α -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + 4σ ⊥ n d(b α ∧ ẽα ) -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β + 2ρk α ∧ E β ∧ ẽγ , (2.38)\n\nwhere\n\np α = -4σ ⊥ ǫ αβγ k β ∧ ẽγ (2.39)\n\nare the linearized momentum variables conjugate to E α . In order to reach the form (2.38) the linear terms in the fluctuations must vanish. For this to happen we find the relationships ėα + ρẽ α = 0 .\n\n(2.40)\n\nNotice that we can also write the linearized action in the form\n\nI HP = dt ∧ ( Ėα + ρE α ) ∧ p α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + n 4σ ⊥ db γ + ρp γ ∧ ẽγ . (2.41)\n\nThe form of the first term, involving the momentum, makes clear that longitudinal fluctuations are non-dynamical. The natural time dependence of E α is of the form e -ρt (correspondingly, the natural time dependence of p α is e +ρt ). Other than that, we see that in comparing to the flat space action, in these variables, the only change is that the Hamiltonian constraint is modified. The solutions of (2.40) and (2.37) are components of (A)dS 4 spacetimes. We can solve (2.40) to obtain e 0 = dt, e α = e -ρt dx α .\n\n(2.42)\n\nWith these we construct the usual Poincaré metric on (A)dS which, however, covers only half of the space even though the parameter t runs from -∞ to +∞. The conformal boundary in these coordinates is at t = +∞. Then we derive\n\nω α 0 = -ρe -ρt dx α = -ρe α , (2.43)\n\nand so\n\nR α β = -σ ⊥ L 2 e α ∧ e β R α 0 = -1 L 2 e α ∧ e 0 R a b = - σ ⊥ L 2 e a ∧ e b . ( 2\n\nHence Ric ab = -3σ ⊥ L 2 η ab and R = -12σ ⊥ /L 2 = -12Λ. We also evaluate Π α = -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ , Πα = 4ρẽ α , tr Π = 12ρ (2.45)\n\nB α = 0, K α = ρẽ α ⇒ Kα = 0 (2.46)\n\nNote that in this gauge, ( DK) α = 1 L T α = 0, which solves the shift constraint, while the Hamiltonian constraint is satisfied through a cancellation between the K 2 term and the cosmological term."
},
{
"section_type": "OTHER",
"section_title": "Linearized Gravitational Duality and Holography",
"text": "Let us summarize what we have obtained so far. In the presence of a cosmological constant we have defined variables such that the action resembles most closely the action without the cosmological constant. This was done in order to look for a suitable background around which linear fluctuations are as simple as possible. Requiring that K (the \"electric field\") and B (the \"magnetic field\") vanish in such a background -as they do around flat space -we found that the background should be (A)dS 4 . Quite satisfactorily, both sign choices for ρ in the change of variables (2.31) lead to (A)dS 4 spacetimes."
},
{
"section_type": "OTHER",
"section_title": "Duality and Holography",
"text": "This is the appropriate point to recall some salient features of duality rotations. In simple Hamiltonian systems the effect of the canonical transformation p → q and q → -p to the action is (see e.g. [19] )\n\nI = t 2 t 1 dt[p q -H(p, q)] → I D = t 2 t 1 dt[-q ṗ -H(q, -p)] . (3.1)\n\nNotice that I D involves the dual variables, for which we have however kept the same notation for simplicity. The transformed Hamiltonian H(q, -p) is in general not related to H(p, q). However, if H(q, -p) = H(p, q) we call the above transformation a duality. It then holds\n\nI D = I -qp t 2 t 1 . (3.2)\n\nThe dual action describes exactly the same dynamics as the initial one, up to a modification of the boundary conditions. For example, if I is stationary on the e.o.m for fixed q in the boundary, I D is stationary on the same e.o.m. for fixed p in the boundary. This simple example illustrates the role of duality in holography; a bulk duality transformation corresponds to a particular modification of the boundary conditions. This property of duality transformations is behind the remarkable holographic properties of electormagnetism in (A)dS 4 [2, 3] .\n\nClearly, the crucial properties of a duality transformation are to be canonical and to leave the Hamiltonian unchanged. However, consider a slight generalization\n\nS = t 2 t 1 dt[p q - 1 2 (p 2 + q 2 + 2λpq)] (3.3)\n\nwhere λ is an arbitrary parameter. The Hamiltonian now is not invariant under the canonical transformation p → q and q → -p -the pq term changes sign. Consequently, the first order form of the equations of motion are also not duality invariant. Nevertheless, the second order equation of motion is invariant. We will find that gravity in the presence of a cosmological constant follows precisely this model. Of course, gravity is a much more complicated constrained system, but as we will show, the constraints and Bianchi identities transform appropriately.\n\nWe also note that the canonical transformation (implemented by a generating functional of the first kind) p → q + 2λp , q → -p . (3.4) is of interest here. The above does not change the Hamiltonian and the transformed action differs from the initial one by total time derivative terms foot_4 S → S D = S -pq\n\nt 2 t 1 -λp 2 t 2 t 1 . (3.5)"
},
{
"section_type": "OTHER",
"section_title": "Linearized gravitational duality",
"text": "As a preamble to gravity we recall the duality properties of Maxwell theory\n\nI M ax = 1 2g 2 dt ∧ Ȧ ∧ * 3 E - 1 2 (E ∧ * 3 E + B ∧ * 3 B) -A 0 d * 3 E , (3.6)\n\nUnder the duality E → - * 3 B, B → * 3 E, Ã → ÃD , we find\n\nI M ax → I M ax,D = 1 2g 2 dt ∧ -ȦD ∧ B - 1 2 (E ∧ * 3 E + B ∧ * 3 B) + A 0 dB . (3.7)\n\nE and B in (3.7) should be expressed through ÃD . We observe that the kinetic term has changed sign, while the Hamiltonian remains invariant. In addition, the (Gauss) constraint is dualized to the trivial 'Bianchi' identity dB = 0 for the dual magnetic field.\n\nNext we try to apply a Maxwell-type duality map in gravity. We consider the following transformation around the fixed background (2.40)\n\nk α → -b α , b α → k α . (3.8)\n\nTo implement the map (3.8) we need to specify the mapping of E α to a 'dual 3-bein' E α . We do that using the linearized form of (2.25) as\n\nǫ αβγ b β ∧ ẽγ + dE α = 0 → ǫ αβγ k β ∧ ẽγ + dE α = 0 = dE α - 1 4σ ⊥ p α (3.9) Since p α = 4σ ⊥ dE α , it is natural to define p D,α = 4σ ⊥ dE α = -4σ ⊥ ǫ αβγ b β ∧ ẽγ , (3.10)\n\nand thus the mapping (3.8) is supplemented by\n\nE → E , E → -E , p → -p D , p D → p (3.11)\n\nNow, let us see the effects of the above duality mapping. The action transforms to\n\nI HP → I HP,D = dt ∧ -Ėα ∧ p D,α -ρE α ∧ p D,α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ (3.12) +4σ ⊥ n α ǫ αβγ db β ∧ ẽγ + n 4σ ⊥ dk γ + ρp D,α ∧ ẽγ\n\nwhere now k α and b α should be expressed in terms of the dual variables E α and p D,α via (3.9) and (3.10). We notice that the 'kinetic' part Ė ∧ p of the action changes sign under the duality map, in direct analogy with the Maxwell case. However, the Hamiltonian is not invariant due to the change of sign of the second term in the first line of (3.12). We will discuss this further in a later section. For now, we note that this sign change would not show up in the equations of motion, written in second order form. It is important to also note that the constraints are transformed into quantities which in the next subsection we will recognize as the linearized Bianchi identities. This is to be expected since the duality transformations are canonical. We also note that it may be possible to choose an alternative canonical transformation, designed to leave the Hamiltonian invariant. The latter is presumably related to the work of Julia et. al. [15] and is considered in the Appendix."
},
{
"section_type": "OTHER",
"section_title": "Linearized Constraints and Bianchi Identities",
"text": "By virtue of the discussion above we may now demonstrate that under the duality mapping (3.8) the linearized constraints transform to the linearized Bianchi identities as\n\nC α ≡ ǫ αβγ dk β ∧ ẽγ → -ǫ αβγ db β ∧ ẽγ (3.13) C 0 ≡ -σ ⊥ db γ -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ → -σ ⊥ dk γ + ρǫ αβγ b α ∧ ẽβ ∧ ẽγ (3.14)\n\nTo identify the right hand sides, we first note that the Bianchi identities are\n\nB R a b = dR a b -R a c ∧ ω c b + ω a c ∧ R c b = 0 (3.15) B a T = dT a -R a b ∧ e b + ω a b ∧ T b = 0 (3.16)\n\nwhich are obtained from the definitions of R a b and T a by exterior differentiation. The first equation is satisfied identically. Since the torsion vanishes, the second equation tells us only that R a b ∧ e b = 0. If we do the 3+1 split, we find two equations. The first is\n\nB T α = -( (3) R α β -σ ⊥ K α ∧ K β ) ∧ ẽβ = 0 (3.17)\n\nwhich upon using the symmetry of K α linearizes to\n\nB T α = -ǫ αβγ db β ∧ ẽγ + . . . (3.18)\n\nNote that this is the image under duality of the shift constraint as in (3.13) .\n\nThe second identity is\n\nB T 0 = -R0 α ∧ ẽα = -σ ⊥ ( DK) α = -σ ⊥ dk α + ρǫ αβγ b β ∧ ẽγ ∧ ẽα = 0 (3.19)\n\nwhere to arrive in the second line we used (2.46) . This is the image of the Hamiltonian constraint as in (3.14) . Summarizing, the duality transformations between linearized constraints and Bianchi identities are\n\nC α → B T,α C 0 → B 0 T (3.20) B T,α → -C α B 0 T → -C 0 (3.21)"
},
{
"section_type": "OTHER",
"section_title": "Connection with other known dualities",
"text": "The Maxwell-type duality operation (3. We begin with the spatial 2-forms when we have\n\nRαβ = -ǫ αβγ dB γ + σ ⊥ (B α ∧ B β -K α ∧ K β ) (3.23) R0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ (\n\nDK) α (3.24) and S0 γ = 1 2 σ ⊥ ǫ αβγ Rαβ (3.25) Sαβ = ǫ αβγ R0γ (3.26) If we linearize these expressions, we find under the duality transformation (3.8) Rab → -σ ⊥ Sab (3.27) Because the expressions (3.24) involve derivatives of B and K, the duality (3.8) is an 'integrated form' of the usual Riemann tensor duality, but implies it. Similarly, if we investigate the spatial 1-forms, we find r ab → -σ ⊥ s ab (3.28)\n\nTo arrive at this result we have set to zero the Lagrange multiplier field q."
},
{
"section_type": "OTHER",
"section_title": "The Effect on the Boundary Theory",
"text": "It is well known that AdS is holographic. We may well ask, in the context of AdS/CFT, how the duality transformation that we have defined here acts in the boundary. We are instructed to consider the on-shell bulk action as a function of bulk fields. So, we evaluate the action on a solution to the equation of motion, resulting in a pure boundary term which is of the form\n\nS bdy = ∂M p α ∧ E α (4.1)\n\nApplying the duality transformation to the bulk theory, although the bulk action is not invariant as we have discussed above, nevertheless it may be easily shown that it induces a simple transformation on the (linearized) boundary term: it simply changes its sign.\n\nS dual bdy = - ∂M p D,α ∧ E α (4.2)\n\nThis transformation is exactly analogous to what happens in the Maxwell case: it amounts to the result [21] .\n\n10 G 2 G dual 2 = -1 . (4.3)"
},
{
"section_type": "CONCLUSION",
"section_title": "Conclusions and Outlook",
"text": "Motivated by possible application in holography and in higher-spin gauge theory we have studied the duality properties of gravity in the Hamiltonian formulation. We have presented the gravity action in terms of suitable variables that closely resemble the electric and magnetic fields in Maxwell theory. We have found suitable \"electric\" and \"magnetic\" field variables, such that at the linearized level first order gravity most closely resembles electromagnetism. This can be done only around Minkowksi and (A)dS 4 backgrounds. We have implemented duality transformations in the linearized gravity fluctuations around these backgrounds. In the presence of a cosmological constant, the Hamiltonian changes, nevertheless the bulk dynamics remains unaltered, while the linearized lapse and shift constraints are mapped into the linearized Bianchi identities. Moreover, the duality transformations induce boundary terms whose relevance in holography we have briefly discussed. Finally, we have exhibited a modified duality rotation that leaves the bulk Hamiltonian invariant, while it induces boundary terms corresponding to relevant deformations.\n\nThe main implication of our results is that certain properties of correlations functions in three-dimensional CFTs mimic the duality of gravity. It would be interesting to extend our results to black-hole backgrounds and also when topological terms are present in the bulk. We also expect that one can analyze the duality of higher-spin gauge theories based on our first-order approach.\n\nThe work of A. C. P. was partially supported by the research program \"PYTHAGORAS II\" of the Greek Ministry of Education. RGL was supported in part by the U.S. Department of Energy under contract DE-FG02-91ER40709."
},
{
"section_type": "OTHER",
"section_title": "Appendix: other duality mappings",
"text": "It is possible to find a transformation that leaves the Hamiltonian unchanged. Consider the following transformation in the fixed background (2.40)\n\nk α → -b α -2ρE α , b α → k α . (6.1)\n\nThe mapping to the dual dreibein is still specified by (3.9). A straightforward calculation reveals that the action transforms as The transformations (6.1) leaves unchanged the Hamiltonian and changes the action by the total \"time\" derivative terms shown in the first line of (6.2). Moreover, the linearized constraints transform into the linearized Bianchi identities. Let us see that in some detail. The second term in the shift constraint is zero since k α is a symmetric one form k α = k αβ ẽβ with k αβ = k βα ; see (2.21) . The term proportional to Λ in the lapse constraint is also zero. This is slightly more involved to see and it is based on the possibility of solving (3.9) for E α after gauge fixing. 11 One way to see this is in components. Write E α = E α β ẽβ and (3.9) becomes\n\nI\n\n∂ α E β γ -∂ γ E β α = ǫ β δα k δ γ -ǫ β δγ k δ α (6.3)\n\nIn the \"Lorentz gauge\" where ∂ α E β α = 0 = ∂ α k β α the above can be inverted as\n\nE α β = 1 ∂ 2 ǫ α δγ ∂ γ k δ β (6.4)\n\nUsing (6.4) one verifies that the last term in the lapse constraint vanishes. This modified duality transformation is probably related to the one considered by Julia et. al. in [15] ."
}
] |